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L.P. Yarin
The Pi-Theorem
Applications to Fluid Mechanicsand Heat and Mass Transfer
L.P. YarinTechnion-Israel Institute of TechnologyDept. of Mechanical EngineeringTechnion City32000 HaifaIsrael
ISBN 978-3-642-19564-8 e-ISBN 978-3-642-19565-5DOI 10.1007/978-3-642-19565-5Springer Heidelberg Dordrecht London New York
Library of Congress Control Number: 2011944650
# Springer-Verlag Berlin Heidelberg 2012This work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publicationor parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer. Violationsare liable to prosecution under the German Copyright Law.The use of general descriptive names, registered names, trademarks, etc. in this publication does notimply, even in the absence of a specific statement, that such names are exempt from the relevant protec-tive laws and regulations and therefore free for general use.
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To the blessed memory of my parents Professor Peter Yarinand Mrs. Leah Aranovich
.
Preface
The book is devoted to the Buckingham Pi-theorem and its applications to various
phenomena in nature and engineering. The accent is made on problems character-
istic of heat and mass transfer in solid bodies, as well as in laminar and turbulent
flows of liquids and gases. Such choice is not accidental. It is dictated by the
requirements of modern technology and encompasses a vast majority of important
problems related with drag and heat transfer experienced by solid bodies moving in
viscous fluids. These problems involve the evaluation of temperature fields in
media with constant and temperature-dependent thermal diffusivity, heat and
mass transfer in boundary layers, pipe and jet flows, as well as thermal processes
occurring in reactive media. In all these cases a uniform approach to the
corresponding complex thermohydrodynamical problems is used. It is based on
the direct application of the Pi-theorem to the analysis of two types of problems:
those which admit a rigorous mathematical formulation, as well as those for which
such formulation is unavailable. For the former problems our attention will be
focused on the establishment of self-similarity which reduces the governing partial
differential equations to the ordinary ones by means of the Pi-theorem, whereas for
the latter problems the Pi-theorem will be used to reveal a set of the governing
dimensionless groups. To a certain degree the choice of the problems is subjective.
However, it allows the evaluation of the range of possible applications of the
Pi-theorem and the peculiarities characteristic of the complex thermohydrodyna-
mical processes in continuous media.
The book consists of nine chapters. They deal with the basics of the dimensional
analysis, the application of the Pi-theorem to find self-similarities and reduce partial
differential equations to the ordinary ones. Then, such interrelated topics as the drag
force, laminar flows in channels, pipes and jets are covered in detail. The discussion
also involves kindred heat and mass transfer in natural, forced and mixed convec-
tion and in situations with phase change and chemical reactions. Some problems of
turbulence theory are also covered in the framework of the Pi-theorem. In addition
to the in-depth exposition of the basic theory and the generic problems, a number of
worked examples of problems related to the application of the Pi-theorem to
different hydrodynamic, heat and mass transfer questions are presented in the end
of each chapter. They can be interest to the engineering and physics students.
vii
The book is intended to scientists and engineers interested in hydrodynamic and
heat and mass transfer problems. It could also be useful to graduate students
studying mechanical, civil and chemical engineering, as well as applied physics.
L.P. Yarin
viii Preface
Acknowledgment
I am especially grateful and deeply indebted to my son Professor Alexander Yarin
for some special consultations related to the applications of the dimensional
analysis to thermohydrodynamics problems, many insightful suggestions and dis-
cussions, as well as multiple comments on the contents of the book.
I am deeply obligated to my daughter Mrs. Elena Yarin and my granddaughter
Miss Inna Yarin. Without their help this book would not have materialized.
ix
.
Contents
1 The Overview and Scope of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Basics of the Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2.1 Dimensional and Dimensionless Parameters . . . . . . . . . . . . . . . . . . . . . 3
2.2.2 The Principle of Dimensional Homogeneity . . . . . . . . . . . . . . . . . . . . . 7
2.3 Non-Dimensionalization of the Governing Equations . . . . . . . . . . . . . . . . 11
2.4 Dimensionless Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.1 Characteristics of Dimensionless Groups . . . . . . . . . . . . . . . . . . . . . . 18
2.4.2 Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 The Pi-Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5.2 Choice of the Governing Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Application of the Pi-Theorem to Establish Self-Similarity
and Reduce Partial Differential Equations to the Ordinary Ones . . . . 39
3.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Flow over a Plane Wall Which Has Instantaneously Started
Moving from Rest (the Stokes Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3 Laminar Boundary Layer over a Flat Plate (the Blasius Problem) . . . 47
3.4 Laminar Submerged Jet Issuing from a Thin Pipe
(the Landau Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.5 Vorticity Diffusion in Viscous Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.6 Laminar Flow near a Rotating Disk (the Von Karman Problem) . . . . . 55
3.7 Capillary Waves after a Weak Impact of a Tiny Object onto
a Thin Liquid Film (the Yarin-Weiss Problem) . . . . . . . . . . . . . . . . . . . . . . . 58
3.8 Propagation of Viscous-Gravity Currents over a Solid Horizontal
Surface (the Huppert Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.9 Thermal Boundary Layer over a Flat Wall
(the Pohlhausen Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
xi
3.10 Diffusion Boundary Layer over a Flat Reactive Plate
(the Levich Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4 Drag Force Acting on a Body Moving in Viscous Fluid . . . . . . . . . . . . . . . 71
4.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Drag Action on a Flat Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2.1 Motion with Constant Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2.2 Oscillatory Motion of a Plate Parallel to Itself . . . . . . . . . . . . . . . . . 75
4.3 Drag Force Acting on Solid Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.1 Drag Experienced by a Spherical Particle at Low,
Moderate and High Reynolds Numbers . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.2 The Effect of Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3.3 The Effect of Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.3.4 The Effect of the Free Stream Turbulence . . . . . . . . . . . . . . . . . . . . . 81
4.3.5 The Influence of the Particle-Fluid Temperature
Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.4 Drag of Irregular Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.5 Drag of Deformable Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.6 Drag of Bodies Partially Submerged in Liquid . . . . . . . . . . . . . . . . . . . . . . . 86
4.7 Terminal Velocity of Small Spherical Particles Settling
in Viscous Liquid (the Stokes Problem for a Sphere) . . . . . . . . . . . . . . . . . 87
4.8 Sedimentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.8.1 Dimensionless Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.8.2 Terminal Velocity of Heavy Grains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.8.3 The Critical State of a Fluidized Bed . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.9 Thin Liquid Film on a Plate Withdrawn Vertically from
a Pool Filled with Viscous Liquid (the Landau-Levich
Problem of Dip Coating) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5 Laminar Flows in Channels and Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2 Flows in Straight Pipes of Circular Cross-Section . . . . . . . . . . . . . . . . . . . 106
5.2.1 The Entrance Flow Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.2.2 Fully Developed Region of Laminar Flows
in Smooth Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.2.3 Fully Developed Laminar and Turbulent
Flows in Rough Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.3 Flows in Irregular Pipes and Ducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.4 Microchannel Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.5 Non-Newtonian Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
xii Contents
5.6 Flows in Curved Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.7 Unsteady Flows in Straight Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6 Jet Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2 The Far Field of Submerged Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.3 The Dimensionless Groups of Jet Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.4 Plane Laminar Submerged Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.5 Laminar Wake of a Blunt Solid Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.6 Wall Jets over Plane and Curved Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.7 Buoyant Jets (Plumes) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7 Heat and Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.2 Conductive Heat and Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.2.1 Temperature Field Induced by Plane Instantaneous
Thermal Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.2.2 Temperature Field Induced by a Pointwise Instantaneous
Thermal Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.2.3 Evolution of Temperature Field in Medium with
Temperature-Dependent Thermal Diffusivity
(The Zel’dovich-Kompaneyets Problem) . . . . . . . . . . . . . . . . . . . . . 162
7.3 Heat and Mass Transfer Under Conditions of Forced Convection . . 165
7.3.1 Heat Transfer from a Hot Body Immersed in Fluid Flow . . . . 165
7.3.2 The Effect of Particle Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
7.3.3 The Effect of the Free Stream Turbulence . . . . . . . . . . . . . . . . . . . . 171
7.3.4 The Effect of Energy Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
7.3.5 The Effect of Velocity Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7.3.6 Mass Transfer to Solid Particles and Drops Immersed
in Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.4 Heat and Mass Transfer in Channel and Pipe Flows . . . . . . . . . . . . . . . . . 178
7.4.1 Couette Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
7.4.2 The Entrance Region of a Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
7.4.3 Fully Developed Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
7.5 Thermal Characteristics of Laminar Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.6 Heat and Mass Transfer in Natural Convection . . . . . . . . . . . . . . . . . . . . . . 186
7.6.1 Heat Transfer from a Spherical Particle Under the
Conditions of Natural Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
7.6.2 Heat Transfer from Spinning Particle Under the
Condition of Mixed Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
Contents xiii
7.6.3 Mass Transfer from a Spherical Particle Under the
Conditions of Natural and Mixed Convection . . . . . . . . . . . . . . . . 189
7.6.4 Heat Transfer From a Vertical Heated Wall . . . . . . . . . . . . . . . . . . 190
7.6.5 Mass Transfer to a Vertical Reactive Plate Under the
Conditions of Natural Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
7.7 Heat Transfer From a Flat Plate in a Uniform Stream of Viscous,
High Speed Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
7.8 Heat Transfer Related to Phase Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
7.8.1 Heat Transfer Due to Condensation of Saturated Vapor
on a Vertical Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
7.8.2 Freezing of a Pure Liquid (The Stefan Problem) . . . . . . . . . . . . . 202
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
8 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
8.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
8.2 Decay of Isotropic Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
8.3 Turbulent Near-Wall Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
8.3.1 Plane-Parallel Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
8.3.2 Pipe Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
8.3.3 Turbulent Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
8.4 Friction in Pipes and Ducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
8.4.1 Friction in Smooth Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
8.4.2 Friction in Rough Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
8.5 Turbulent Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
8.5.1 Eddy Viscosity and Thermal Conductivity . . . . . . . . . . . . . . . . . . . . 224
8.5.2 Plane and Axisymmetric Turbulent Jets . . . . . . . . . . . . . . . . . . . . . . . 229
8.5.3 Inhomogeneous Turbulent Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
8.5.4 Co-flowing Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
8.5.5 Turbulent Jets in Crossflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
8.5.6 Turbulent Wall Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
8.5.7 Impinging Turbulent Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
9 Combustion Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
9.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
9.2 Thermal Explosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
9.3 Combustion Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
9.4 Combustion of Non-premixed Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
9.5 Diffusion Flame in the Mixing Layer of Parallel Streams
of Gaseous Fuel and Oxidizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
xiv Contents
9.6 Gas Torches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
9.7 Immersed Flames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
Contents xv
.
Nomenclature
Chapter 2:
Ar Archimedes number
Bi Biot number
Bo Bond number
Br Brinkman number
C Speed of sound
Cd Drag coefficient
c Concentration
Ca Capillary number
cP Specific heat
Da Damkohler number
Da Darcy number
De Dean number
De Deborah number
D Diffusivity
D� Permeability coefficient of porous medium
Ec Eckert number
Ek Ekman number
Eu Euler number
Fd Drag force
Fr Froude number
Fg Gravity force
f Frequency
g Gravity acceleration
Gr Grashof number
h Heat transfer coefficient, or enthalpy
hm Mass transfer coefficient
Ja Jacob number
k Thermal conductivity
kB Boltzmann’s constant
Kn Knudsen number
Ku Kutateladze number
L Characteristic length scale
Lh Height of liquid layer
xvii
l Length of a pipe
Le Lewis number
m Mass of a particle
M Mach number
Nu Nusselt number
P Pressure
DP Pressure drop
Pe Peclet number
Ped Peclet number (for diffusion)
Pr Prandtl number
Qv Volumetric flow rate
q Heat of reaction
R Gas constant, or radius of curvature
r Cross-sectional radius of a pipe
Ra Rayleigh number
Re Reynolds number
Ri Richardson number
Ro Rossby number
rv Latent heat of vaporization
Sc Schmidt number
Se Semenov number
Sh Sherwood number
St Stanton number
St Strouhal number
Ta Taylor number
T Temperature, or torque
DT Temperature difference
t Time
tr Relaxation time
t0 Observation time
u Particle velocity
v Velocity vector with components u, v and w in projections to the Cartesian axes x, y and z
v Specific volume
Vm Mass flow rate
W Rate of chemical reaction rate, or power (Watt)
We Weber number
x, y, z Cartesian coordinates
Greek Symbols
b Coefficient of bulk expansion
g Ratio of the specific heat at constant pressure to the specific heat at
constant volume (the adiabatic index)
d Boundary layer thickness
dT Thermal boundary layer thickness
y Dimensionless temperature
L Angle between the axis of Earth rotation and the direction of fluid motion
l Mean free path
m Viscosity
n Kinematic viscosity
r Density
s Surface tension
xviii Nomenclature
t Time
tr Relaxation time
t0 Observation time
f Dissipation function
o Angular velocity
oe Angular velocity of Earth’s rotation
Subscripts
f Fluid
v Vapor
w Wall
1 Undisturbed fluid at infinity
Chapter 3:
a Thermal diffusivity
D Diffusivity
g Gravity acceleration
h Thickness of liquid layer
J Total momentum flux in jet
j Diffusion flux
P Pressure
Pr Prandtl number
Q Source strenght
r Radial coordinate
r; y; ’ Spherical coordinates
S Surface or surface area
Sc Schmidt number
t Time
U Plate or flow velocity in x directionu Fluid velocity
v Velocity vector with components vr ; vy; and v’ in spherical coordinate system
Greek Symbols
a Thermal diffusivity, exponent
G Strength of an infinitely thin vortex line
d Thickness of the boundary layer
� Dimensionless variable
# Dimensionless temperature
n Kinematic viscosity
r Density
s Surface tension
t Shear stress
’ Polar angle, dimensionless function
O Vorticity component normal to the flow plane, or angular velocity
Nomenclature xix
Subscript
1 Undisturbed fluid
Chapter 4:
Ac Acceleration parameter
cd Drag coefficient
cl Lift coefficient
d Diameter
fd Drag force
fl Lift force
g Gravity acceleration
l Scale of turbulence, length of plate
P Pressure
Q Volumetric flow rate
R Radius
T Dimensionless turbulence intensity
u0 Root-mean square of turbulent fluctuations
u; v;w Velocity components
v Velocity vector
Fr Froud number
Re Reynolds number
We Weber number
Greak Symbols
a Angle
m Viscosity
n Kinematic viscosity
r Density
s Surface tension
t Shear stress at the wall
g Dimensionless angular velocity
o Angular velocity
Subscripts
d Drag
l Lift
p Particle
1 Ambient
Chapter 5:
d Diameter
FI Inertial force
Fc Centrifugal force
xx Nomenclature
Fn Friction force
Fo Fouier number
k Dean number, or roughness
K Modified Dean number
l The entrance length of pipe
l� Characteristic length of pipe
P Pressure
DP Pressure drop
Po Poiseuille number
Q Volumetric flow rate
R Radius of curvature of a torus
Re Reynolds number
r0 Cross-sectional radius of a pipe
t Time
u; v;w Velocity components
u0 Initial velocity
umax Maximum velocity
v Velocity vector
w0 Mean velocity
x, y, z Cartesian coordinates
r; y; x Cylindrical coordinates
Greek Symbols
a Large semi-axis of an ellipse
b Small semi-axis of an ellipse
g Shear rate
d Ratio of pipe radius to its curvature
l Friction factor
m Viscosity
m0 Viscosity of Binham fluid
n Kinematic viscosityQBingham number
r Density
t Shear stress, geometric torsion
t0 Yield stress
Chapter 6:
h Enthalpy
Ix Kinematic momentum flux
Jx Momentum flux
k Thermal conductivity
Mx Total moment-of-momentum flux
P Pressure
Pr Prandtl number
Red Local Reynolds number
T Temperature
u Longitudinal velocity component
v Transversal velocity component
Nomenclature xxi
Greek Symbols
b Thermal expansion coefficient
d Jet thickness
m Viscosity
n Kinematic viscosity
# Excessive temperature
r Density
Subscripts
1 Undisturbed fluid
m Jet axis
Chapter 7:
c Specific heat capacity, concentration
cP Specific heat at constant pressure
cv Specific heat at constant volume
D Diffusivity
d Diameter
E Pointwise energy release
g Gravity acceleration
H Channel height
h Heat transfer coefficient, rate of heat transfer, enthalpy
j Mechanical equivalent of heat
k Thermal conductivity
kB Boltzmann’s constant
l Turbulence scale
P Pressure
Q Strength of thermal source
q Heat flux
ql Latent heat of freezing
r Radius
T Temperature
Tu Turbulence intensity
v; u Velocityev0 Velocity fluctuation
Ec Eckert number
Gr Grashof number
M Mach number
Nu Nusselt number
Pe Peclet number
Pr Prandtl number
Ra Rayleigh number
Re Reynolds number
Reo Rotational Reynolds number
Sh Sherwood number
St Stephan number
xxii Nomenclature
Greek Symbols
a Thermal diffusivity
b Thermal expansion coefficient
g Ratio of specific heat at constant pressure to specific heat at constant volume
(the adiabatic index)
d Delta function; boundary layer thickness
w Radiant thermal diffusivity
m Viscosity
n Kinematic viscosity
r Density
Subscripts
en Entrance
f Front of thermal wave
P Pressure
T Thermal
W Wall
1 Undisturbed flow
Chapter 8:
A Cross-sectional area of a jet
C Concentration
d0 Nozzle diameter
dc Nozzle width
Fr Froude number
Gx Total mass flux
H Distance between the nozzle exit and the unperturbed liquid surface
hc Cavity depth
I0 The exit kinematic momentum flux
Jx Total momentum flux
l Characteristic length
Pr Prandtl number
P Pressure
Re Reynolds number
T Temperature
u,v Velocity components
um Centerline velocity
We Weber number
Greek Symbols
aT Eddy thermal diffusivity
d Jet half-width
� Dimensionless variable
m Viscosity
Nomenclature xxiii
mT Eddy viscosity
n Kinematic viscosity
nT Eddy kinematic viscosity
r Density
s Surface tension
Subscripts
G Gas
L Liquid
Chapter 9:
c Reactant concentration
cP Specific heat
D Diffusivity
E Activation energy
h Enthalpy
k Chemical reaction constant; thermal conductivity
k0 Pre-exponential
Le Lewis number
lf Flame length
P Pressure
Pe Peclet number
Q1 Heat release
Q2 Heat losses
q Heat of reaction
R The universal gas constant
Re Reynolds number
T Temperature
uf Speed of combustion wave
u0 Speed of reactive mixture at the nozzle exit
W Rate of chemical reaction
Wj Rate of conversion of the j-th species
z Pre-exponential
Greek Symbols
a Thermal diffusivity
d Frank-Kamenetskii parameter
m Viscosity
n Kinematic viscosity
r Density
tk Characteristic kinetic time
tD Characteristic diffusion time
O Stoichiometric oxidizer-to-fuel mass ratio
xxiv Nomenclature
Subscripts
f Fuel
o Oxidizer
m Maximum; axis
0 Initial state
� Gas-liquid interface
Nomenclature xxv
.
Chapter 1
The Overview and Scope of the Book
The present book deals with the concepts and methods of the dimensional analysis
and their applications to various thermohydrodynamic phenomena in continuous
media. A comprehensive exposition of the results of systematic analysis of a
number of important problems in this area in the framework of the Pi-theorem is
given in nine chapters. In Chap. 2 the basics of the dimensional analysis are
discussed. In particular, the principle of dimensional homogeneity and non-
dimensionalization of the mass, momentum, energy and diffusion equations and
the corresponding initial and boundary conditions are described in this chapter. This
is complemented by the introduction of several dimensionless groups and similarity
criteria characteristic of hydrodynamic and heat and mass transfer problems. The
Buckingham Pi-theorem is also formulated in Chap. 2.
In Chap. 3 the Pi-theorem is used to establish self-similarity if it is admitted by a
particular problem and reduce the corresponding partial differential equations to the
ordinary ones. This approach to the search of self-similarity is illustrated with a
number of generic situations corresponding to the Stokes, Blasius, Landau, von
Karman, Yarin-Wess and Huppert hydrodynamic problems and the Pohlhausen and
Levich heat and mass transfer problems.
Chapter 4 deals with the drag force acting on a body moving in viscous fluid. The
attention is focused on drag experienced by spherical particles at low, moderate and
high Reynolds numbers. Such additional effects on the drag force as particle
rotation, free stream turbulence and particle-fluid temperature difference are also
analyzed. Then, some problems related to sedimentation are considered in the
framework of the dimensional analysis. Finally, the Landau-Levich withdrawal
problem on the thickness of thin liquid film on a vertical plate in dip coating process
is tackled. As before, the consideration is based on the Pi-theorem.
Chapter 5 is devoted to laminar channel and pipe flows. In this chapter the Pi-
theorem is applied to study stationary flows of Newtonian and non-Newtonian
fluids in straight smooth and rough pipes, as well as in curved channels and
pipes. In addition, some transient flows of Newtonian fluids are considered.
L.P. Yarin, The Pi-Theorem, Experimental Fluid Mechanics,
DOI 10.1007/978-3-642-19565-5_1, # Springer-Verlag Berlin Heidelberg 2012
1
The application of the Pi-theorem to the laminar submerged viscous jets is
discussed in Chap. 6. These include jets propagating in an infinite space, as well
as wall jets and wakes of solid bodies.
Chapter 7 deals with the heat and mass transfer phenomena. The results
presented in this chapter are related to the application of the Pi-theorem to conduc-
tive heat transfer in media with constant and temperature-dependent thermal con-
ductivity, convective heat and mass transfer in forced, natural and mixed
convection. The Pi-theorem is also used for the analysis of heat and mass transfer
associated with hot particles immersed in fluid flow, channel and pipe flows, high
speed gas flows, as well as flows with phase changes.
Chapter 8 is devoted to turbulence. Here the Pi-theorem is used to study
problems related to a decay of the uniform and isotropic turbulence, turbulent
near-wall flows and submerged and wall turbulent jets. The results are used to
interpret the wide range experimental data.
Chapter 9 is related with the application of the Pi-theorem to combustion
processes. A number of important problems of the combustion theory are consid-
ered in this chapter. These include the thermal explosion, propagation of combus-
tion waves and aerodynamics of gas torches.
2 1 The Overview and Scope of the Book
Chapter 2
Basics of the Dimensional Analysis
2.1 Preliminary Remarks
In this introductory chapter some basic ideas of the dimensional analysis are
outlined using a number of the instructive examples. They illustrate the applications
of the Pi-theorem in the field of hydrodynamics and heat and mass transfer.
The systems of units and dimensional and dimensionless quantities, as well as
the principle of dimensional homogeneity are discussed in Sect. 2.2. Section 2.3
deals with non-dimensionalization of the mass and momentum balance equations,
as well as the energy and diffusion equations. In Sect. 2.4 the dimensionless
groups characteristic of hydrodynamic and heat and mass transfer phenomena are
presented. Here the physical meaning of several dimensionless groups and simi-
larity criteria is discussed, In addition, similitude and modeling characteristic of the
experimental investigations of thermohydrodynamic processes are considered.
The Pi-theorem is formulated in Sect. 2.5.
2.2 Basic Definitions
2.2.1 Dimensional and Dimensionless Parameters
Momentum, heat and mass transfer in continuous media occur in processes
characterized by the interaction and coupling of the effects of hydrodynamic and
thermal nature. The intensity of these interactions and coupling is determined by
the magnitudes of physical quantities involved which characterize the physical
properties of the medium, its state, motion and interactions with the surrounding
boundaries and penetrating fields. The magnitudes of these quantities are deter-
mined experimentally by comparing the readings of the measuring devices
with some chosen scales, which are taken as units of the measured characteristics,
L.P. Yarin, The Pi-Theorem, Experimental Fluid Mechanics,
DOI 10.1007/978-3-642-19565-5_2, # Springer-Verlag Berlin Heidelberg 2012
3
e.g. length, mass, time, etc. For example, an actual pipe diameter, fluid velocity or
temperature are expressed as
d ¼ nL�; v ¼ mV�; T ¼ kT� (2.1)
where n; m and k are some numbers, whereas L�;V� and T� are units of length,
velocity and temperature, respectively.
The quantities which characterize flow and heat and mass transfer of fluids
are related to each other by certain expressions based on the laws of nature. For
example, the volumetric flow rate Qv of viscous fluid through a round pipe of radius
r, and the drag force Fd acting on a small spherical particle slowly moving with
constant velocity in viscous fluid are expressed by the Poiseuille and Stokes laws
Qv ¼ pr4DP8ml
(2.2)
Fd ¼ 6pmur (2.3)
In (2.2) and (2.3) DP is the pressure drop on a length l; m is the fluid viscosity,
and u is the particle velocity. Equations 2.2 and 2.3 show that units of the
volumetric flow rate Qv and drag force Fd can be expressed as some combinations
of the units of length, velocity, viscosity and pressure drop. In particular, the unit of
r coincides with the unit of length L, of u is expressed through the units of length
and time as LT�1, the unit of m½ � ¼ L�1MT�1 in addition involves the unit of mass,
as well as the unit of the pressure drop DP½ � ¼ L�1MT�2 (cf. Table 2.1). Here and
hereinafter symbol A½ � denotes units of a dimensional quantity A.
It is emphasized that the units of numerous physical quantities can be expressed
via a few fundamental units. For example, we have just seen that the units of
volumetric flow rate and drag force are expressed via units of length, mass and time
only, as Qv½ � ¼ L3T�1; and Fd½ � ¼ LMT�2. A detailed information the units of
measurable quantities is available in the book by Ipsen (1960). The possibility to
express units of any physical quantities as a combination of some fundamental units
allows subdividing all physical quantities into two characteristic groups, namely (1)
primary or fundamental quantities, and (2) derivative (secondary or dependent)
ones. The set of the fundamental units of measurements that is sufficient for
expressing the other measurement quantities of a certain class of phenomena is
called the system of units. Historically, different systems of units were applied to
physical phenomena (Table 2.2).
In the present book we will use mainly the International System of Units
(Table 2.3).
In this system of units (hereinafter called SI Units) an amount of a substance is
measured with a special unit- mole (mol). Also, two additional dimensionless units:
one for a plane angle- radian (rad), and another one for a solid angle- steradian (sr),
are used. A detailed description of the SI Units can be found in the books of
Blackman (1969) and Ramaswamy and Rao (1971).
4 2 Basics of the Dimensional Analysis
The numerical values of the physical quantities expressed through fundamental
units depend on the scales of arbitrarily chosen for the latter in any given system of
units. For example, the velocity magnitude of a solid body moving in fluid, which is
1 m/s in SI units is 100 cm/s in the Gaussian CGS (centimeter, gram, second)
System of Units. The physical quantities whose numerical values depend on the
Table 2.1 Physical
quantitiesQuantity Dimensions Derived units
A. (Mechanical quantities)
Acceleration LT�2 m�s�2
Action ML2T�1 kg�m2� s
�1
Angle (plane) 1 rad:
Angle (solid) 1 sterad:
Angular acceleration T�2 rad�s�2
Angular momentum ML2T�1 kg�m2� s
�1
Area L2 m2
Curvature L�1 m�1
Surface tension MT�2 kg�s�2
Density ML�3 kg�m�3
Elastic modulus ML�1T�2 kg�m�1� s�2
Energy (work) ML2T2 J
Force MLT�2 N
Frequency T�1 s�1
Kinematic viscosity L2T�1 m2� s
�1
Mass M kg
Momentum MLT�1 kg�m�s�1
Power ML2T�3 W
Pressure ML�1T�2 N�m�2
Time T s
Velocity LT�1 m�s�1
Volume L3 m3
B. (Thermal quantities)
Enthalpy ML2T2 J
Entropy ML2T2y�1 J�K�1
Gas constant L2T�1y�1 J�kg��1K�1
Heat capacity per unit mass L2T�2y�1 J�kg�1� K�1
Heat capacity per unit volume ML�1T�2y�1 J�m��3K�1
Internal energy ML2T2 J
Latent heat of phase change L2T�2 J�kg�1
Quantity of heat ML2T�2 J
Temperature y K
Temperature gradient L�1y K�m�1
Thermal conductivity MT�3Ly�1 W�m�1� K�1
Thermal diffusivity L2T�1 m2� s
�1
Heat transfer coefficient MT3y�1 W�m2� K
�1
2.2 Basic Definitions 5
fundamental units are called dimensional. For such quantities, units are derivative
and are expressed through the fundamental unites according to the physical
expressions involved. For example, units of the gravity force Fg ¼ mg are
expressed through the fundamental units bearing in mind the previous expression
and the fact that m½ � ¼ M; and g½ � ¼ LT�2 as
Fg
� � ¼ LMT�2 (2.4)
In fact, units of any physical quantity can be expressed through a power law1
A½ � ¼ La1Ma2Ta3 (2.5)
where the exponents ai are found by using the principle of dimensional
homogeneity.
The quantities whose numerical values are independent of the chosen units of
measurements are called dimensionless. For example, the relative length of a pipe
l ¼ ld (where l and d are the length and diameter of the pipe, respectively) is
dimensionless. Formally this means that l� � ¼ 1:
In the general case, physical quantities can be characterized by their magnitude
and direction. Such quantities as, for example, temperature and concentration are
scalar and are characterized only by their magnitudes, whereas such quantities as
velocity and force are vectors and are characterized by their magnitudes and
directions. Vectors can also be characterized by introducing a so-called vector
length L (Williams 1892). Projections of the vector length L on, say, the axes of
Table 2.2 Systems of units
Quantity
Absolute Technical
CGS MKS FPS CGS MKS FPS
Mass Gram Kilogram Pound 9.81 g 9.81 kg Slug
Force Dyne Newton Poundal Gram-force Kilogram-force Pound-force
Length Centimeter Meter Foot Santimeter Meter Foot
Time Second Second Second Second Second Second
Table 2.3 International system of units-SI
Quantity Units Abbreviation
Mass Kilogram kg
Length Meter m
Time Second s
Temperature Kelvin K
Electric current Ampere A
Luminous intensity Candela cd
1A demonstration of this statement can be found in Sedov (1993).
6 2 Basics of the Dimensional Analysis
a Cartesian coordinate system x; y and z are denoted as Lx; Ly and Lz, respectively.A number of instructive examples of application of vector length for studying different
problems of applied mechanics are presented in the monographs by Huntley (1967)
and Douglas (1969). The application of the idea of vector length in studying of
drag and heat transfer at a flat plate subjected to a uniform flow of the incompressible
fluid is discussed by Barenblatt (1996) and Madrid and Alhama (2005).
The expansion of a number of the fundamental units allows a significant
improvement of the results of the dimensional analysis. For this aim it is useful to
consider different properties the mass: (1) mass as the quantity of matter Mm, and
(2) mass as the quantity of the inertia Mi. Similarly, using projections of a vector Lon the Cartesian coordinate axes as the fundamental units it is possible to express
the units of such derivative (secondary) quantities as volume V and velocity vector
v as V½ � ¼ LxLyLz and u½ � ¼ LxT�1; v½ � ¼ LyT
�1; and w½ � ¼ LzT�1 where u,v and
w denote the projections of v on the coordinate axes as is traditionally done in fluid
mechanics. It is emphasized that using two different quantities of mass and
projections of a vector allows one to reveal more clearly the physical meaning
of the corresponding quantities. For example, the dimensions of work W in a
rectilinear motion and torque T in rotation system of units LMT are the same
L2MT�2;whereas in the system of unitsLxLyLzMT they are different, namely
W½ � ¼ L2xMT�2; whereas T½ � ¼ LxLyMT�2:
2.2.2 The Principle of Dimensional Homogeneity
Principle of dimensional homogeneity expresses the key requirements to a structure
of any meaningful algebraic and differential equations describing physical phe-
nomena, namely: all terms of these equations must to have the same dimensions.
To illustrate this principle, we consider first the expression for the drag force acting
on a spherical particle slowly moving in highly viscous fluid. The Stokes formula
describing Fd reads
Fd ¼ 6pmur (2.6)
Here Fd½ � ¼ LMT�2 is the drag force, m½ � ¼ L�1MT�1 is the viscosity of the
fluid, u½ � ¼ LT�1 and r½ � ¼ L are the particle velocity and its radius, respectively.
It is easy to see that (2.6) satisfies the principle dimensional homogeneity. Indeed,
substitution of the corresponding dimensions to the left hand side and the right hand
side of (2.6) results in the following identity
LMT�2 ¼ ðL�1MT�1ÞðLT�1ÞðLÞ ¼ LMT�2 (2.7)
As a second example, we consider the Navier–Stokes and continuity equations.
For flows of incompressible fluids they read
2.2 Basic Definitions 7
@v
@tþ ðv � rÞv ¼ � 1
rrPþ nr2v (2.8)
r � v ¼ 0 (2.9)
where v ¼ LT�1½ � is the velocity vector, r½ � ¼ L�3M, n½ � ¼ L2T�1 and P½ � ¼L�1MT�2 are the density, kinematic viscosity n and pressure, respectively.
It is seen that all the terms in (2.8) have dimensions LT�2 and in (2.9) have
dimensions T�1.
There are a number of important applications of the principle of the dimensional
homogeneity. For example, it can be used for correcting errors in formulas or
equations, which is advisable to students. Take the expression for the volumetric
rate of incompressible fluid through a round pipe of radius r as
Qv ¼ pr2
8mDPl
� �(2.10)
where Qv is the volumetric flow rate, DP is the pressure drop over an arbitrary
section of the pipe length of length l.The dimension of the term on the left hand side in (2.10) is L3T�1, whereas of the
one on the right hand side of this equation is LT�1. Thus, (2.10) does not satisfy
the principle of dimensional homogeneity. In order to find the correct form of the
dependence of the volumetric flow rate on the governing parameters, we present
(2.10) as follows
Qv ¼ p8ra1ma2
DPl
� �a3
(2.11)
where ai are unknown exponents.
Bearing in mind the dimensions of Qv; r; m and DPl
� �, we arrive at the following
system of algebraical equations for the exponents ai
a1 � a2 � 2a3 ¼ 3
a2 þ a3 ¼ 0
�a2 � 2a3 ¼ �1 (2.12)
From (2.12) it follows that the exponents ai are equal a1 ¼ 4; a2 ¼ �1;and a3 ¼ 1. Then, the correct form of (2.10) reads as
Qv ¼ pr4
8mDPl
� �(2.13)
8 2 Basics of the Dimensional Analysis
The third example concerns the application the principle of dimensional homo-
geneity to determine the dimensionless groups from a set of dimensional
parameters. Consider a set of dimensional parameters
a1; a2 � � � ak; akþ1 � � � an (2.14)
Assume that k parameters have independent dimensions. Accordingly, the
dimensions of the other n� k parameters can be expressed as
akþ1½ � ¼ a1½ �a01 � � � � � ak½ �a
0k
� � � � � � � � � � � � � � � � � � � � � � ��an½ � ¼ a1½ �an�k
1 � � � � ak½ �an�kk (2.15)
Therefore, the ratios
akþ1
aa01
1 � � � aa0k
k
¼ P1
� � � � � � � � � � � � � � � � � � ��an
an�k1 � � � an�k
k
¼ Pn�k (2.16)
are dimensionless. Requiring that the dimensions of the numerator and denominator
in the ratios (2.16) will be the same, we arrive at the system of algebraical equations
for the unknown exponents.
In conclusion, we give one more instructive example of the application of the
principle of dimensional homogeneity for the description of the equation of state of
perfect gas. The general form of the equation of state reads (Kestin v.1 (1966) and
v.2 (1968)):
FðP; vs; TÞ ¼ 0 (2.17)
where P; vsand T are the pressure, specific volume and temperature, respectively.
Equation 2.17 can be solved (at least in principle), with respect to any one of the
three variables involved. In particular, it can be written as
P ¼ f ðvs; TÞ (2.18)
The set of the governing parameters involved in (2.18) is incomplete since the
dimension of pressure P½ � ¼ L�1MT�2 cannot be expressed in the form of any
combination of dimensions of specific volume vs½ � ¼ L3M�1 and temperature
T½ � ¼ y. Therefore, the function f on right hand side in (2.18) must include some
dimensional constant c
2.2 Basic Definitions 9
P ¼ f ðc; vs; TÞ (2.19)
It is reasonable to choose as such a constant the gas constant R that account
for the physical nature of the gas, but does not depend on its specific volume,
pressure and temperature. Assuming that c ¼ R g= (g is a dimensionless constant),
we write the dimension of this constant as c½ � ¼ L2T�2y�1: All the parameters
in (2.19) have independent dimensions. Then, according to the Pi-theorem (see
Sect. 2.5), (2.19) takes the form
P ¼ g1ca1va2s T
a3 (2.20)
where g1 is a dimensionless constant.
Using the principle of the dimensional homogeneity, we find the values of
the exponents ai as a1 ¼ 1; a2 ¼ �1; a3 ¼ 1: Assuming g ¼ g1, we arrive at the
Clapeyron equation
P ¼ RrT (2.21)
The equation of state of perfect gas can be also derived directly by applying the
Pi-theorem to solve the problems of the kinetic theory and accounting for the fact
pressure of perfect gas results from atom (molecule) impacts onto a solid wall.2
Considering perfect gas as an ensemble of rigid spherical atoms (or molecules)
moving chaotically in the space, we can assume that pressure of such gas is deter-
mined by atom (or molecule) mass m, their number per unit volume N and the
average velocity squared <v2>
P ¼ f ðm;N; <v2>Þ (2.22)
The dimensions of P and the governing parameters m;N and <v2> are
P½ � ¼ L�1MT�2; m½ � ¼ M; N½ � ¼ L�3; <v2>� � ¼ L2T�2 (2.23)
All the governing parameters have independent dimensions. Therefore, the
difference between the number of the governing parameters n and the number of
the parameters with independent dimensions k equals zero. In this case the pressurecan be expressed as Sedov (1993);
P ¼ gma1Na2<v2>a3 (2.24)
where g is a dimensionless constant.
2 This idea was expressed first by D. Bernoulli in 1727 who wrote that pressure of perfect gas is
related to molecule velocities squared.
10 2 Basics of the Dimensional Analysis
Using the principle of dimensional homogeneity, we find the values of the
exponents in (2.24) as a1 ¼ a2 ¼ a3 ¼ 1: Then, (2.24) takes the form
P ¼ gmN<v2> (2.25)
Bearing in mind that m<v2> is directly proportional kBT (m<v2> ¼g1kBT; where g1 is a dimensionless constant), we arrive at the following equation
P ¼ ekBTN (2.26)
Here e ¼ gg1 is a dimensionless constant, kB½ � ¼ L2MT�2y�1 is Boltzmann’s
constant, T½ � ¼ y is the absolute temperature.
Applying (2.26) to a unit mole of a perfect gas, we can write the known
thermodynamic relations as
N ¼ Nm; kB ¼ mRNm
; mvs ¼ constant (2.27)
Here Nm is the Avogadro number, m is the molecular mass, vs is the specific
volume, and R½ � ¼ L2T�2y�1 is the gas constant. Then, (2.27) takes the form
P ¼ rRT (2.28)
Summarizing, we see that the pressure of perfect gas is directly proportional to
the product of the gas density, gas constant and the absolute temperature and does
not depend on the mass of individual atoms (molecules). Note that (2.28) can be
obtained directly from the functional equation P ¼ f ðm;N; T; kBÞ(Bridgman 1922).
2.3 Non-Dimensionalization of the Governing Equations
It is beneficial in the analysis complex thermohydrodynamic phenomena to trans-
form the system of mass, momentum, energy and species balance equations into a
dimensionless form. The motivation for such transformation comes from two
reasons. The first reason is related with the generalization of the results of theoreti-
cal and experimental investigations of hydrodynamics and heat and mass transfer in
laminar and turbulent flows by presentation the data of numerical calculation and
measurements in the form of dependences between dimensionless parameters.
The second reason is related to the problem of modeling thermohydrodynamic
processes by using similarity criteria that determine the actual conditions of
the problem. The procedure of non-dimensionalization of the continuity (mass
balance), momentum, energy and species balance equations is illustrated below
by transforming the following model equation
2.3 Non-Dimensionalization of the Governing Equations 11
Xnj¼1
AðiÞj ¼ 0 (2.29)
where AðiÞj includes differential operators, some independent variables, as well as
constants; superscript i refers to the momentum ði ¼ 1Þ; energy ði ¼ 2Þ; species
ði ¼ 3Þ and continuity ði ¼ 4Þ equations, n is the total number of terms in a given
equation.
The terms in (2.29) account for different factors that affect the velocity, temper-
ature and species fields: the inertia features of fluid, viscous friction, conductive and
convective heat transfer, etc. These terms are dimensional. The dimension of AðiÞj in
the system of units LMTy is
AðiÞj
h i¼ La
ðiÞj MbðiÞ
j TgðiÞj ye
ðiÞj (2.30)
where the values of the exponents a; b; g and e are determined by the magnitude of
i and j; all the terms that correspond to a given i have the same dimension:
AðiÞ1
h i¼ A
ðiÞ2
h i¼ � � � AðiÞ
j
h i¼ � � � AðiÞ
n
h i(2.31)
The variables and constants included in (2.29) may be rendered dimensionless
by using some characteristic scales of the density r�½ � ¼ L�3M; velocity v�½ � ¼LT�1; length l�½ � ¼ L; time t�½ � ¼ T, etc. Then, the dimensionless variables and
constants of the problem are expressed as
r ¼ rr�
; v ¼ v
v�;T ¼ T
T�; c ¼ c
c�; t ¼ t
t�;P ¼ P
P�; m ¼ m
m�; k ¼ k
k�;D
¼ D
D�; g ¼ g
g�(2.32)
where the asterisks denote the characteristic scales, and the dimensionless
parameters are denoted by bars. In addition, k� ¼ LMT�3y�1� �
;D� ¼ L2T�1½ �;and g� ¼ LT�2½ � are the characteristic scales of thermal conductivity, diffusivity
and gravity acceleration, respectively.
Taking into account (2.32), we can present all terms of (2.29) as follows
AðiÞj ¼ A
ðiÞj� A
ðiÞj (2.33)
where AðiÞj� is the corresponding dimensional multiplier comprised of the character-
istic scales, AðiÞj ¼ A
ðiÞj =A
ðiÞj� is the dimensionless form of the jth term in (2.29).
The exact form of the multipliers AðiÞj� is determined by the actual structure of the
terms AðiÞj . For example, the multiplier of the first term of the momentum balance
equation is found from
12 2 Basics of the Dimensional Analysis
AðiÞ1 ¼ r
@v
@t¼ r�v�
t�
@ðv=v�Þ@ðt=t�Þ ¼ A
ðiÞ1�A
ðiÞ1 (2.34)
where AðiÞ1� ¼ r�v�
t�, A
ðiÞ1 ¼ @v
@t.
The substitution of the expression (2.33) into (2.29) yields
Xnj¼1
AðiÞj� A
ðiÞj ¼ 0 (2.35)
Dividing the left and right hand sides of (2.35) by a multiplier AðiÞk� ð1 � k � nÞ,
we arrive at the dimensionless form of the conservation equations
AðiÞk þ
Xk�1
j¼1
YðiÞj�
AðiÞj þ
Xnj¼kþ1
YðiÞj�
AjðiÞ
( )¼ 0 (2.36)
whereQðiÞ
j� ¼ Aj�=Ak� are the dimensionless groups.
To illustrate the general approach described above, we render dimensionless
the Navier–Stokes equations, the energy and species balance equations, as well
as the continuity equation. For incompressible fluids these equations read
r@v
@tþ r v � rð Þv ¼ �rPþ mr2vþ rg (2.37)
rcp@T
@tþ rcP v � rð ÞT ¼ kr2T þ f (2.38)
r@cx@t
þ r v � rð Þcx ¼ rDr2cx (2.39)
r � v ¼ 0 (2.40)
where r; v T; P and cx are the density, velocity vector, the temperature, pressure
and the concentration of the species x. In particular, let us use the Cartesian
coordinate system where vector v has components u; v and w in projections
to the x; y and z axes. In addition, m; k andD are the viscosity, thermal conductivity
and diffusivity which are assumed to be constant, g the magnitude of the gravity
acceleration g, f is the dissipation function f ¼ 2m ð@u=@xÞ2 þ ð@v=@yÞ2þh
ð@w=@zÞ2� þ mð@u=@yþ @v=@xÞ2 þ mð@v=@zþ @w=@yÞ2þ mð@w=@xþ @u=@zÞ2.The multipliers A
ðiÞj� in (2.37)–(2.40) are listed below
Að1Þ1� ¼ r�v�
t�;A
ð1Þ2� ¼ r�v2�
l�;A
ð1Þ3� ¼ P�
l�;A
ð1Þ4� ¼ r�g� (2.41)
2.3 Non-Dimensionalization of the Governing Equations 13
Að2Þ1� ¼ r�cP�T�
t�; A
ð2Þ2� ¼ r�cP�v�T�
l�;A
ð2Þ3� ¼ k�T�
l�;A
ð2Þ4� ¼ m�v2�
l�
Að3Þ1� ¼ r�c�
t�;A
ð3Þ2� ¼ r�c�
l�;A
ð3Þ3� ¼ r�D�c�
l2�
Að4Þ1� ¼ v�
l�;A
ð4Þ2� ¼ v�
l�
Dividing the multipliers Að1Þj� by A
ð1Þ2� ; A
ð2Þj� by A
ð2Þ2� ; A
ð3Þj� by A
ð3Þ2� and A
ð4Þj� by A
ð4Þ2� ,
we arrive at the following system of dimensionless equations
St@v
@tþ v � rð Þv ¼ �EurPþ 1
Rer2vþ 1
Fr(2.42)
St@T
@tþ v � rð ÞT ¼ 1
Per2T þ Br
Ref (2.43)
St@cx@t
þ v � rð Þcx ¼ 1
Pedr2cx (2.44)
r � v ¼ 0 (2.45)
where St ¼ l�=v�t�; Eu ¼ P�=r�v2�; Re ¼ v�l�=n�; Pe ¼ v�l�=a�; Ped ¼ v�l�=D�,
Fr ¼ v2�=g�l�; Br ¼ m�v2�=k�T� are the Strouhal, Euler and Reynolds numbers,
as well as the thermal and diffusion Peclet numbers, and the Froude and Brinkman
numbers, respectively, n and a are the kinematic viscosity and thermal diffusivity,
and the dimensionless dissipation function f ¼ f= mðv�=l�Þ2h i
; v ¼ v v�= ;P ¼P rv2��
; T ¼ T T�= and cx ¼ c c�= are the dimensionless variables.
The non-dimensionalization of the initial and boundary conditions is similar to
the one described above. In that case each of the independent variables x; y; z and t,as well as the flow characteristics u; v; T and cx are also rendered dimensionless by
using some scales that have the same dimensions as the corresponding parameters.
For example, consider the non-dimensionalization of the initial and boundary
conditions for the following three problems of the theory of viscous fluid flows:
(1) steady flow in laminar boundary layer over a flat plate, (2) laminar flow about
a flat plate which instantaneous started to move in parallel to itself, and (3) sub-
merged laminar jet issued from a round nozzle.
In case (1), let the velocity and temperature of the undisturbed fluid far enough
from the plate be u1, T1, and the wall temperature be Tw ¼ const: Then, theboundary conditions read
14 2 Basics of the Dimensional Analysis
x ¼ 0; 0 � y � 1; u ¼ u1; T ¼ T1 (2.46)
x> 0, y ¼ 0, u ¼ v ¼ 0; T ¼ Tw; y ! 1, u ! u1, T ! T1
Introducing as the scales of length some L, velocity u1 and temperature
Tw � T1, we rearrange (2.46) to the following dimensionless form3
x ¼ 0; 0 � y � 1 u ¼ 1; DT ¼ 1 (2.47)
x> 0, y ¼ 0 u ¼ v ¼ 0; DT ¼ 0; y ! 1 u ! 1; DT ! 1
where x ¼ x=L; y ¼ y=L; u ¼ u=u1; v ¼ v=u1; DT ¼ ðTw � TÞ=ðTw � T1Þ.The equation for the heat flux at the wall is used to introduce the heat transfer
coefficient h:
h Tw � T1ð Þ ¼ �k@T
@y
� �y¼0
(2.48)
Being rendered dimensionless, the heat transfer coefficient is expressed in the
following form
Nu ¼ @DT@y
� �y¼0
(2.49)
where Nu ¼ hL=k is the dimensionless heat transfer coefficient is called the Nusselt
number.
In case (2), the initial and boundary conditions of the problem on a plate starting
to move from rest with velocityU in the x-direction in contact with the viscous fluidread
t ¼ 0; 0 � y � 1 u ¼ 0 (2.50)
t> 0, y ¼ 0 u ¼ U; y ¼ 1, u ¼ 0
Since no time or length scales are given, we use as the characteristic time scale
t� ¼ n=U2 and as the characteristic length scale n=U. Then, (2.50) take the follow-ing dimensionless form
t ¼ 0; 0 � y � 1 u ¼ 0; t> 0; y ¼ 0 u ¼ 1; y ! 1 u ! 0 (2.51)
In case (3), the boundary conditions for a submerged laminar jet are
3 It is emphasized that in the problem on flow in the boundary layer over a semi-infinite plate,
a given characteristic scale L is absent. According to the self-similar Blasius solution of this
problem, the dimensionless coordinate y ¼ y=ðnx=u1Þ1=2 with ðnx=u1Þ1=2 playing the role of the
length scale (Sedov 1993).
2.3 Non-Dimensionalization of the Governing Equations 15
x ¼ 0; 0 � y � r0; u ¼ u0; T ¼ T0; y> r0 u ¼ 0; T ¼ T1 (2.52)
x> 0; y ¼ 0,@u
@y¼ 0,
@T
@y¼ 0; y ! 1, u ! 0, T ! T1
where r0is the nozzle radius.The dimensionless form of the conditions (2.52) is
x ¼ 0; 0 � y � 1; u ¼ 1 DT ¼ 1; y>1; u ! 0; DT ! 0 (2.53)
x> 0, y ¼ 0,@u
@y¼ 0,
@DT@y
¼ 0; y ¼ 1, u ! 0, DT ! 0
where x ¼ x=r0; y ¼ y=r0; u ¼ u=u0; DT ¼ ðT1 � TÞ=ðT1 � T0Þ:At large enough distance from the jet origin at x=r0>> 1, it is possible to use the
integral conditionR10
u2ydy ¼ const; instead of the condition (2.52) at x ¼ 0. Note
that there is another way of rendering the system of fundamental equations of
hydrodynamics and heat and mass transfer theory dimensionless. It consists in
rendering dimensionless each quantity in these equations using for this aim the
scales of the density, velocity, temperature, etc. Requiring that the convective terms
of these equations do not contain any dimensional multipliers, it is not easy to arrive
at the equations identical to (2.42)–(2.45). To illustrate this approach to non-
dimensionalization of the mass, momentum, energy and species conservation
equations, consider, for example, the system of equations describing flows of
reactive gases
@r@t
þr � ðrvÞ ¼ 0 (2.54)
r@v
@tþ rðv � rÞv ¼ �rPþr � ðmrvÞ þ rg (2.55)
r@h
@tþ rðv � rÞh�r � ðkrTÞ ¼ qWk (2.56)
r@ck@t
þ rðv � rÞck �r � ðrDrckÞ ¼ �Wk (2.57)
P ¼ g� 1
grh (2.58)
where v is the velocity vector, r; P; h and T are the density, pressure, enthalpy
and temperature, ck ¼ rk=r is the relative concentration of the kth species,
r ¼ Srk; with rk being density of the kth species, Wkðck; TÞ and W are the chemi-
cal reaction rates, q is the heat of the overall reaction, and g ¼ cp=cvis the ratio of
16 2 Basics of the Dimensional Analysis
specific heat at constant pressure to the one at constant volume (the adiabatic index).
Note that in the energy balance equation (2.56) the dissipation term is neglected.
Introducing dimensionless parameters as follows a ¼ aa�
(the asterisk denotes the
scale of a parameter a), we arrive at the following equations
r�t�
@r@t
þ r�v�L�
r � ðrvÞ ¼ 0 (2.59)
r�v�t�
@v
@tþ r�v�
L�rðv � rÞv ¼ �P�
L�rPþ m�v�
L2�r � ðmrvÞ þ r�g�rg (2.60)
r�h�t�
r@h
@tþ r�v�h�
L�rðv � rÞh� k�T�
L2�r � ðkrTÞ ¼ qWk:�Wk (2.61)
r�t�r@ck@t
þ r�v�L�
rðv � rÞck � r�D�L2�
r � ðrDrckÞ ¼ �Wk:��Wk (2.62)
P ¼ g� 1
gr�h�P�
rh (2.63)
where r�; v�; P�; T�; h� and L� are the scales of density, velocity, pressure,
temperature, enthalpy and length, respectively.
Requiring that the second terms on left hand sides in (2.59)–(2.62) do not contain
any dimensionless multipliers and also accounting for the fact that for perfect gas
r�h�=P� ¼ g=ðg� 1Þ, we obtain
St@r@t
þr � ðrvÞ ¼ 0 (2.64)
St@v
@tþ rðv � rÞv ¼ �EurPþ 1
Rer � ðmrvÞ þ 1
Frrg (2.65)
St@T
@tþ rðv � rÞT � 1
Per � ðkrTÞ ¼ Da3Wk (2.66)
St@ck@t
þ rðv � rÞck � 1
Pedr � ðrDrckÞ ¼ Da1Wk (2.67)
P ¼ rh (2.68)
where in addition to previously introduced Strouhal, Reynolds, Euler, the
thermal and diffusion Peclet numbers, and the Froude number, two Damkohler
numbers Da1 ¼ Wk:�L�=r�v�; and Da3 ¼ qWk:�L�=r�v�h� (defined according to
the Handbook of Chemistry and Physics,1968) appear.
2.3 Non-Dimensionalization of the Governing Equations 17
2.4 Dimensionless Groups
2.4.1 Characteristics of Dimensionless Groups
As was shown in Sect. 2.3, the dimensionless momentum, energy and diffusion
equations contain a number of dimensionless groups, which represent themselves
some combinations of the physical properties of fluid, acting forces, heat fluxes, etc.
The physical meaning and number of these groups is determined by a specific
situation, as well as by a particular model used for description of the physical
phenomena characteristic of that situation (Table 2.4).4
Consider in detail some particular dimensionless groups. The Prandtl, Schmidt
and Lewis numbers belong to a subgroup of dimensional groups that incorporate
only quantities that account for the physical properties of fluid. They are expressed
as the following ratios (cf. Table 2.4)
Pr ¼ na; Sc ¼ n
D; Le ¼ a
D(2.69)
where n; a and D are the kinematic viscosity, thermal diffusivity and diffusivity,
respectively.
Consider, for example the Prandtl number. It represents itself the ratio of
kinematic viscosity to thermal diffusivity, i.e. of the characteristics of fluid respon-
sible for the intensity of momentum and heat transfer. Accordingly, the Prandtl
number can be considered as a parameter that characterizes the ratio of the extent of
propagation of the dynamic and thermal perturbations. Therefore, at very low
Prandtl numbers (for example, in flows of liquid metals), the thickness of the
thermal boundary layer dT is much larger than the thickness of the dynamical
one, d: In contrast, at Pr >> 1 (in flows of oils) the equality d>> dT is valid. The
Schmidt number is the diffusion analog of the Prandtl number. It determines the
ratio of the thicknesses of the dynamical and diffusion boundary layers.
The Reynolds number belongs to the subgroup of the dimensionless groups
which are ratios of the acting forces. It can be considered as the ratio of the inertia
force Fito the friction force Ff
4 Dimensionless groups can be also found directly by transformation of the functional equations of
a specific problem using the Pi-theorem (see Sect. 2.5). A detailed list of dimensionless groups
related to flows of incompressible and compressible fluids in adiabatic and diabatic conditions,
flows of non-Newtonian fluids and reactive mixtures can be found in Handbook of Chemistry and
Physics, 68th Edition, 1987–1988, CBC Inc. Boca Roton, Florida, and in Chart of Dimensionless
Numbers, OMEGA Technology Company. See also Lykov and Mikhailov (1963) and Kutateladze
(1986).
18 2 Basics of the Dimensional Analysis
Table 2.4 Dimensionless groups
Name Symbol Definition Comparison ratio Field of use
Archimedes
number
Ar gL3rm2 ðr� rf Þ Gravity force to viscous
force
Motion of fluid due to
density
differences
(buoyancy)
Biot number Bi hLks
Convection heat transfer to
conduction heat transfer
Heat transfer
Bond
number
Bo rgL2
sGravitaty force to surface
tension
Motion of drops and
bubbles.
Atomization
Brinkman
number
Br mv2
kDTHeat dissipation to heat
transferred
Viscous flows
Capillary
number
Ca mvs Viscous force to surface
tension force
Two-phase flow.
Atomization.
Moving contact
lines
Damkohler
number
Da1Da3
WLVmqWL
rvcPDT
Chemical reaction rate to
bulk mass flow rate.
Heat released to
convected heat
Chemical reactions,
momentum, and
heat transfer
Darcy
number
Da2 vLD�
Inertia force to permeation
force
Flow in porous media
Dean
number
De vRrm
ffiffiffiRr
qCentrifugal force to inertial
force
Flow in curved
channels and
pipes
Deborah
number
De trt0
Relaxation time to the
characteristic
hydrodynamic time
Non-Newtonian
hydrodynamics.
Rheology
Eckert
number
Ec v21cPDT
Kinetic energy to thermal
energy
Compressible flows
Ekman
number
Ek m2roL2
�1=2 (Viscous force to Coriolis
force)1=2Rotating flows
Euler
number
Eu rv2
DPPressure drop to dynamic
pressure
Fluid friction in
conduits
Grashof
number
Gr r2gbL3DTm2
Buoyancy force to viscous
force
Natural convection
Jacob
number
Ja cPrfDTrrV
Heat transfer to heat of
evaporation
Boiling
Knudsen
number
Kn lL
Mean free path to
characteristic dimension
Rarefied gas flows
and flows in
micro- and nano-
capillaries
Kutateladze
number
K rvcPDT
Latent heat of phase change
to convective heat
transfer
Combined heat and
mass transfer in
evaporation
Lewis
number
Le krcPD
Thermal diffusivity to
diffusivity
Combined heat and
mass transfer
Mach
number
M vC Flow speed to local speed of
sound
Compressible flows
Nu hLk
Forced convection
(continued)
2.4 Dimensionless Groups 19
Re ¼ vL
n¼ rv2
m v L=ð Þ ¼rv2 L=
m v L2=ð Þ (2.70)
where r; m and L are the density, viscosity and the characteristic length.
The dimensions of the numerator and denominator in right hand side ratio in
(2.70) are rv2 L=½ � ¼ m v L2�� �� � ¼ L�2MT�2, i.e. the same as the dimensions of the
terms r @v=@tþ v � rð Þv½ � and mr2v accounting for the inertia and viscous forces
in the momentum balance equation. The terms rv2=L and mv=L2 can be treated as
the specific inertia and viscous forces fi ¼ Fi V= and ff ¼ Ff V= , respectively, with
the dimensions Fi½ � ¼ LMT�2, Ff
� � ¼ LMT�2, and V½ � ¼ L3.At small Reynolds numbers when the influence of viscosity is dominant, any
chance perturbations of the flow field decay very quickly. At large Re such
perturbations increase and result in laminar-turbulent transition. Therefore, the
Table 2.4 (continued)
Name Symbol Definition Comparison ratio Field of use
Nusselt
number
Total heat transfer to
conductive heat transfer
Peclet
number
Pe Lrvcpk
Bulk heat transfer to
conductive heat transfer
Forced convection
Prandtl
number
Pr mcPk Momentum diffusivity to
thermal diffusivity
Heat transfer in fluid
flows
Rayleigh
number
Ra gbL3r2cPmk
Thermal expansion to
thermal diffusivity and
viscosity
Natural convection
Richardson
number
Ri � gr
@P@Lh
�@v@Lh
�w
.Gravity force to the inertia
force
Stratified flow of
multilayer
systems
Rossby
number
Ro voL sinL The inertia force to Coriolis
force
Geophysical flows.
Effect of earth’s
rotation on flow in
pipes
Schmidt
number
Sc mrD Kinematic viscosity to
molecular diffusivity
Diffusion in flow
Senenov
number
Se hmK
Intensity of heat transfer to
intensity of chemical
reaction
Reaction kinetics.
Convective heat
transfer.
Sherwood
number
Sh hmLD
Mass diffusivity to
molecular diffusitivy
Mass transfer
Stenton
number
St hrvcP
Heat transferred to thermal
capacity of fluid
Forced convection
Strouhal
number
St fLv
Time scale of flow to
oscillation period
Unsteady flow.
Vortex shedding
Taylor
number
Ta 2oL2rm
�2 (Coriolis force to viscous
force)2Effect of rotation on
natural convection
Weber
number
We v2rLs
The dynamic pressure to
capillary pressure
Bubble formation,
drop impact
20 2 Basics of the Dimensional Analysis
Reynolds number is sensitive indicator of flow regimes. For example, in flows of an
incompressible fluid in a smooth pipe, three kinds of flow regime can be realized
depending on the value of the Reynolds number: (1) laminar (Re � 2300), transi-
tional (2300 � Re � 3500), and developed turbulent (Re > 3500).
The Peclet number is an example of a dimensionless group that is a ratio of heat
fluxes of different nature. It reads
Pe ¼ vL
a¼ rvcPDT
k DTL
� � (2.71)
where k and cP are the thermal conductivity and specific heat at constant pressure,
DT is the characteristic temperature difference.
The Peclet number is the ratio of the heat flux due to convection to the heat flux
due to conduction. It can be considered as a measure of the intensity of molar to
molecular mechanisms of heat transfer.
We mention also the Damkohler number that characterize the conditions of
chemical reaction which proceeds in a reactive mixture, i.e. in the process
accompanied by consumption of the initial reactants, formation of the combustion
products, as well as an intensive heat release. Under these conditions the evolution
of the temperature and concentration fields is determined by two factors: (1)
hydrodynamics of the flow of reacting mixture, and (2) the rate of chemical
reaction. The contribution of each of these factors can be estimated by the ratio
of the characteristic hydrodynamic time th � W�1 to the chemical reaction time
tr � V�1v i.e. by the Damkohler number
Da1 ¼ thtr
(2.72)
If the Damkohler number is much less than unity, the influence of the chemical
reaction on the temperature (concentration) field is negligible. At large values of
Da1 the effect of the chemical reaction and its heat release is dominant.
2.4.2 Similarity
Before closing the brief comments on the dimensionless groups, we outline how
such groups are used in modeling of hydrodynamic and thermal phenomena. For
this aim, we turn back to (2.64)–(2.68) that describe the mass, momentum, heat and
species transfer in flows of incompressible fluids with constant physical properties.
These equations contain eight dimensionless groups, namely, St; Re; Pe; Ped;Eu; Fr; Da1 andDa3: If the initial and boundary conditions of a particular problemdo not contain any additional dimensionless groups (as, for example, the conditions
y ¼ 0 v ¼ 0; T ¼ 0; ck ¼ 0, y ! 1 v ¼ 1; T ¼ 1; ck ¼ 1), the velocity,
2.4 Dimensionless Groups 21
temperature and concentration fields determined by (2.64)–(2.68) can be expressed
as follows
v ¼ fvðx; y; z; St;Re;Eu;FrÞ (2.73)
T ¼ ftðx; y; z; St;Pe;Da1Þ (2.74)
ck ¼ fcðx; y; z; St;Ped;Da3Þ (2.75)
In (2.73) and (2.75)T ¼ ðT�TwÞ=ðT1�TwÞ; and ck ¼ ðck� ck;wÞ=ðck;1� ck;wÞ;subscripts w; and1 correspond to the values at the wall and in undisturbed fluid.
The expressions (2.73)–(2.75) are universal in a sense that the fields of dimen-
sionless velocity, temperature and concentration determined by these expressions
do not depend on the absolute values of the characteristic scales. That means that in
geometrically similar systems (for example, cylindrical pipes of different diameter)
values of dimensionless velocity, temperature and concentration at any similar
point (with x1 ¼ x2 ¼ � � � ¼ xi; y1 ¼ y2 ¼ � � � ¼ yi; z1 ¼ z2 ¼ � � � ¼ zi) are the
same if the values of the corresponding dimensionless groups are the same. Thus,
the necessary conditions of the dynamic and thermal similarity in geometrically
similar systems consist in equality of dimensionless groups (similarity numbers)
relevant for the compared systems, i.e.
St ¼ idem; Re ¼ idem; Eu ¼ idem; Fr ¼ idem; Pe ¼ idem;
Ped ¼ idem; Da1 ¼ idem; Da3 ¼ idem(2.76)
for a considered class of flows. It is emphasized that in geometrically similar
systems the boundary conditions should also be identical in such comparisons.
The conditions (2.76) allow modeling the momentum, heat and mass transfer
processes in nature and technical applications by using the results of the
experiments with miniature geometrically similar models. Note that among the
totality of similarity numbers it is possible to select a family of dimensionless
groups that contain combinations of only scales of the considered flow family and
the physical parameters of a medium involved in a situation under consideration.
Such similarity numbers are called similarity criteria (Loitsyanskii 1966). A num-
ber of similarity criteria can be less than the number of similarity numbers. For
example, hydraulic resistance of cylindrical pipes with fully developed incompress-
ible viscous fluid flow with a given throughput is characterized by two similarly
numbers, namely, the Reynolds and Euler numbers. The first of them Re ¼ v0d=n isthe similarity criterion, since it contains known parameters: the average velocity of
fluid v0, its viscosity n and pipe diameter d. In contrast, the Euler number is not
a similarity criterion, since it contains an unknown pressure drop which has to be
found by solving the problem or measured experimentally (Loitsyanskii 1966).
22 2 Basics of the Dimensional Analysis
2.5 The Pi-Theorem
2.5.1 General Remarks
This whole book is devoted to the Buckingham Pi-theorem (1914), which is widely
used in a number of important problems of modern physics and, in particular,
mechanics. The proof of this theorem, as well as numerous instructive examples
of its applications for the analysis of various scientific and technical problems are
contained in the monographs by Bridgman (1922), Sedov (1993), Spurk (1992)
and Barenblatt (1987). Referring the readers to these works, we restrict our consid-
eration by applications of the Pi-theorem to problems of hydrodynamics and the
heat and mass transfer only.
The study of thermohydrodynamical processes in continuous media consists in
establishing the relations between some characteristic quantities corresponding to
a particular phenomenon and different parameters accounting for the physical
properties of the matter, its motion and interaction with the surrounding medium.
Such relations can be expressed by the following functional equation
a ¼ f ða1; a2 � � � anÞ (2.77)
where a is the unknown quantities (for example, velocity, temperature, heat or mass
fluxes, etc.), a1; a2; � � �an are the governing parameters (the characteristics of an
undisturbed fluid, physical constants, time and coordinates of a considered point).
Equation 2.77 indicates only the existence of some relation between the unknown
quantities and the governing parameters. However, it does not express any particular
form of such relation. There are two approaches to determine an exact form of
a relation of the type of (2.77): one is experimental, and the other one theoretical.
The first approach is based on generalization of the results of measurements of
unknown quantities a while varying the values of the governing parameters
a1; a2; � � �an: The second, theoretical, approach relies on the analytical or numerical
solutions of the mass, momentum, energy and species balance equations. In both
cases the establishment of a particular exact form of (2.77) does not entail significant
difficulties while studying the simplest one-dimensional problemswhen (2.77) takes
the form a ¼ f ða1Þ: On the contrary, a comprehensive experimental and theoretical
analysis of amultiparametric equation a ¼ f ða1; a2 � � � anÞ is extremely complicated
and often represents itself an insoluble problem. The latter can be illustrated by the
problem on a drag force acting on a body moving with a constant velocity in an
infinite bulk of incompressible viscous fluid. In this case the drag force Fd acting
from the fluid to the body depends on four dimensional parameters, namely, the fluid
density r and viscosity m, a characteristic size of the body d, and its velocity v. Then,the functional equation (2.77) takes the form
Fd ¼ f ðr; m; d; vÞ (2.78)
2.5 The Pi-Theorem 23
In order to find experimentally the drag force, it is necessary to put the body into
a wind tunnel and measure the drag force at a given velocity by an aerodynamic
scale. That is the experimental way of solving the problem under consideration but
only for one point on the parametric plane drag force-velocity. To determine the
dependence of the drag force on velocity within a certain range of velocity v, it isnecessary to reiterate the measurement of Fd at N values of v to determine the
dependence Fd ¼ f ðvÞ within a range ½v1; v2� at fixed values of r; m and d. If wewant to find the dependence Fd on all four governing parameters, we have to
perform N4 measurement.5 Therefore, if the number of data points forFd at varying
one governing parameter is N ¼ 102; the total number of measurements that one
needs will be equal to 108! It is evident that such number of measurements is
practically impossible to perform. Moreover, even if we have an experimental
data bank with 108 measurement points, we cannot say anything about the behavior
of the function Fd ¼ f ðr; m; v; dÞ outside the studied range of the governing para-
meters. An analytical or numerical calculation of the dependence of drag force on
density, viscosity, velocity and size of the body is also an extremely complicated
problem in the general case (at the arbitrary values of r; m; v; and d) due to the
difficulties involved in integrating the system of nonlinear partial differential
equations of hydrodynamics.
Essentially both approaches to study the dependence of drag force on density,
viscosity, velocity and size of the body allow a significant simplification of the
problem by using the Pi-theorem. The latter points at the way of transformation
of the function of n dimensional variables into a function of m ðwith m < nÞdimensionless variables. As a matter of fact, the Pi-theorem suggests how many
dimensionless variables are needed for describing a given problem containing
n dimensional parameters.
The Pi-theorem can be stated as follows. Let some dimension physical quantities
a depend on n dimensional parameters a1; a2 � � � an; where k of them have an
independent dimension. Then the functional equation for the quantities a
a ¼ f ða1; a2 � � � ak; akþ1 � � � anÞ (2.79)
can be reorganized to the form of the dimensionless equation
P ¼ ’ðP1;P2 � � �Pn�kÞ (2.80)
that contain n� k dimensionless variables. The latter are expressed as
P1 ¼ a1
aa01
1 aa02
2 � � �aa0k
k
; P2¼ a2
aa001
1 aa002
2 � � �aa00k
k
� � � Pn�k ¼ an
aan�k1
1 aan�k2
2 � � �aan�kk
k
(2.81)
The dimensionless form of the unknown quantities a is
5With an equal number of data points for each one of the four governing parameters.
24 2 Basics of the Dimensional Analysis
P ¼ a
aa11 aa22 � � � aakk
(2.82)
To illustrate the application of the Pi-theorem to hydrodynamic problems, return
to the drag force acting on a body moving in viscous fluid. The unknown quantities
and governing parameters of the corresponding problem have the following
dimensions
Fd½ � ¼ LMT�2; r½ � ¼ L�3M; m½ � ¼ L�1MT�1; d½ � ¼ L; v½ � ¼ LT�1 (2.83)
Three from the four governing parameters of this problem have independent
dimensions. That means that a dimension of any governing parameters in this case
can be expressed as a combination of dimensions of the three others. The dimension
of the unknown quantity is also expressed as a combination of the governing
parameters having independent dimensions Fd½ � ¼ LMT�2 ¼ rv2d2½ � ¼ m2=r½ � ¼mvd½ �:In accordance with the Pi-theorem, (2.78) takes the form
P ¼ ’ðP1Þ (2.84)
where P ¼ Fd
ra1 va2da3 ; and P1 ¼ m
ra01 v
a02d
a03
:
Taking into account the dimension of the drag forceFd and governing parameters
with independent dimension r; v and d and using the principle of the dimensional
homogeneity, we find the values of the exponents ai and a0i
a1 ¼ 1; a2 ¼ 2; a3 ¼ 2; a01 ¼ 1; a
02 ¼ 1; a
03 ¼ 1 (2.85)
Then (2.84) reads
Cd ¼ ’ðReÞ (2.86)
where Cd ¼ Fd=rv2d2 is the drag coefficient, and Re¼rvd=m is the Reynolds
number.
The exact form of the function ’ðReÞ cannot be determined by means of the
dimensional analysis. However, this fact does not diminish the importance of
the obtained result. Indeed, the dependence of the drag coefficient on only one
dimensionless group (the Reynolds number) allows generalization of the experi-
mental data on drag related to motions of bodies of different sizes moving with
different velocities in fluids with different densities and viscosities. All this data can
be presented in a collapsed form of a single curve CdðReÞ. Moreover, in some
limiting cases corresponding to motion with low velocities (the so-called, creeping
flows with Re<< 1) or high speeds when Re>> 1, it is possible to determine the
exact forms of the dependence of the drag coefficient on Re.
2.5 The Pi-Theorem 25
In particular, at Re << 1 the inertia effects become negligible. Returning to
(2.78), we can assume that the drag force depends on fluid viscosity, body size and
its velocity
Fd ¼ f ðm; v; dÞ (2.87)
All the governing parameters in (2.87) have independent dimensions
(n� k ¼ 0). Therefore, in this case (2.87) reduces to
Fd ¼ cma1va2da3 (2.88)
where c is a dimensionless constant and a1 ¼ 1; a2 ¼ 1; a3 ¼ 1.
Substituting the values of the exponents a1; a2 and a3 into (2.88) leads to the
following expression for the drag coefficient
Cd ¼ c
Re(2.89)
It is evident that to determine the dependence CdðReÞ at Re<< 1 it is sufficient to
perform only one measurement in order to establish the value of the constant c.It is emphasized that the efficiency of using the Pi-theorem in studies of physical
phenomena is determined by the value of the difference n� k, i.e. by the number of
the governing dimensionless groups. In all cases (excluding k ¼ 0) the transforma-
tion of the functional equation by the Pi-theorem allows one to decrease number of
variables. The most interesting two cases correspond to the difference n� k being
either 0 or 1. In the first case the functional equation takes the form
a ¼ caa11 aa22 � � � aann (2.90)
In the second one it becomes
P ¼ ’ðP1Þ (2.91)
where P represents itself the dimensionless group corresponding to the unknown
parameter. Decreasing the number of dimensionless variables in (2.91) to only one
is equivalent to the transformation of partial differential equations into the ordinary
ones. A number of examples of transformation of the functional equations similar to
(2.77) to a dimensionless form, as well as transformations of partial differential
equations into the ordinary ones is given in the following sections.
2.5.2 Choice of the Governing Parameters
The theoretical study of hydrodynamic and heat and mass transfer processes is
based on the system of partial differential equations that include the mass,
26 2 Basics of the Dimensional Analysis
momentum, energy and species conservation balances. This system of equations is
supplemented by an equation of state and correlations determining the physical
properties of the medium. The exact and approximate solutions of hydrodynamic
and heat transfer problems in the framework of the continuum approach yield
comprehensive answers to different problems of the theory. In distinction the
dimensional analysis of hydrodynamic and heat and mass transfer problems
meets some difficulties that arise already at the first step of the investigation
when choosing the governing parameter of the problem. They stem from certain
vagueness in choosing the governing parameters beginning from a pure intuitive
evaluation of the features of a phenomenon under consideration. In addition,
such approach to choosing the governing parameters often involves a number of
parameters whose influence will appear to be negligible at the end. The latter makes
it difficult to foresee the results of the dimensional analysis from scratch in
generalizing hydrodynamic and heat and mass transfer. In order to improve the
procedure of choosing the governing parameters and simplify and the following
analysis, it is possible to use the system of the mass, momentum, energy and species
balance equations.
Let us illustrate such an approach by the following examples.6 We begin with the
drag force acting on a spherical particle moving with a constant velocity in an
infinite bulk of viscous incompressible fluid. It is reasonable to assume that the
force that acts on the particle depends on its size d; velocity v and physical
properties of the fluid, namely its density r and viscosity m: In this case the
functional equation for the drag force Fd reads
Fd ¼ f ðr; m; d; vÞ (2.92)
The dimensional analysis of (2.92) leads to the following transformation in the
form of the drag coefficient
Cd ¼ ’�ðReÞ (2.93)
where Cd ¼ Fd=rv2d2 is the drag coefficient, and Re ¼ vd=n is the Reynolds
number.
It is emphasized that the function ’�ðReÞon the right hand side of (2.93) can be
presented as c’ðReÞ, where c is a dimensionless constant with its value being
chosen according to the experimental data. For example, for creeple motion of
a small spherical particle when the drag force is given by the Stokes law
Fd ¼ 3pmvd, constant c ¼ 8 p= : Then the expression for the drag coefficient takes
the form Cd ¼ 24=Re:To determine the exact form of the dependence (2.93), one needs to integrate the
continuity and the Navier–Stokes equations subjected to the no-slip condition at the
particle surface. In some limiting cases corresponding to special conditions of
6A detailed analysis of these problems see in Chaps. 4 and 7
2.5 The Pi-Theorem 27
particle motion (say, very slow or fast), it is possible to find the exact form of the
function ’�ðReÞ in (2.93) using these equations only for determining the set of
the governing parameters. For example, in the case of slow motion (creeping flows)
the inertia terms on the left hand side of the Navier–Stokes equations rðv � rÞv is
much less than the terms in on the right hand side of these equations. That allows
one to omit the inertial term and thereby exclude density from the governing
parameters. As a result, the functional equation for the drag force reduces to the
form of (2.6). Such simplification of the problem formulation is a key element
which allows establishing an exact form of the dependence of the drag force on
viscosity, velocity and diameter of the particle as Fd mvd that coincide (up to a
numerical factor) with the exact result (the Stokes force) derived from the
Navier–Stokes equations.
In the second case corresponding to a rapid body motion (the case of a large
Reynolds number) the dominant role belongs to the turbulent transfer. The average
characteristics of fully developed turbulent flows are governed by the Reynolds
equations (Hinze 1975; Loitsyanskii 1966)
@vi@t
þ vj@vi@xj
¼ � 1
r@P
@xiþ nr2vi þ 1
r@
@xjð�rv0
iv0jÞ (2.94)
(bars over parameters denote the average values).
In high Reynolds number flows the term nr2vi associated with the effect of the
molecular momentum transfer through molecular viscosity mechanism can be
omitted. Then, assuming steady state average turbulent flow, the drag force which
does not depend on molecular viscosity and time is given by
Fd ¼ f ðr; v; dÞ (2.95)
Applying the Pi-theorem to (2.95), we can rearrange it to the following form
Fd r�v2d2 (2.96)
which agrees with the Newton law for drag.
Another example of employing the conservation equations to facilitate the
dimensional analysis of complicated hydrodynamic and heat transfer problems is
related to mass transfer to a vertical reactive plate in contact with a liquid solution
of a reactive species (a reagent) which is initially at rest. When the rate of
a heterogeneous reaction at the plate surface is much larger than the rate of
diffusion transport of the reagent toward the surface, its concentration there equals
zero, whereas far from the surface it is equal c1: The gradient of the reagent
concentration across the thickness of the diffusion boundary layer results in
a non-uniform density field. That, in turn, triggers buoyancy force which results
in liquid motion near the wall. It is reasonable to assume that the velocity
and concentration of the reactive species in the dynamic and diffusion boundary
layers are determined by four parameters r1; c1; n; g and two independent
variables x and y
28 2 Basics of the Dimensional Analysis
u ¼ fuðr1; c1; n; g; x; yÞ (2.97)
c ¼ fcðr1; c1; n; g; x; yÞ (2.98)
where we consider for brevity only one component of the velocity vector u; nand D are the kinematic viscosity and diffusivity, and gis the acceleration due
to gravity.
The functional equations (2.97) and (2.98) contain six governing parameters.
Three of them have independent dimensions. Choosing r1; n and g or r1; D and gas parameters with the independent dimensions, we transform (2.97) and (2.98) to
the following form
Pu ¼ ’uðP1;P2;P3Þ (2.99)
Pc ¼ ’cðP1�;P2�;P3�Þ (2.100)
where Pu ¼ u= gnð Þ1=3; P1 ¼ c1=r1; P2 ¼ x=ðn2=gÞ1=3; P3 ¼ y=ðn2=gÞ1=3;and Pc ¼ c=r1; P1� ¼ c1=r1; P2� ¼ x=ðD2=gÞ1=3; P3� ¼ y=ðD2=gÞ1=3:
Equations (2.99) and (2.100) show that the dimensionless velocity and concen-
tration of the reactive species are the function of three dimensionless groups, which
makes the analysis of the problem under consideration difficult. Therefore, employ
also the conservation equations. The momentum and species balance equations that
describe flow in the boundary layer and mass transfer to the vertical reactive wall
read (Levich 1962) (see Sect. 3.10)
u@u
@xþ v
@u
@y¼ n
@2u
@y2þ g�c� (2.101)
u@c�@x
þ v@c�@y
¼ D@2c�@y2
(2.102)
where g� ¼ gðc1=rÞð@r=@cÞc¼c1 ; c� ¼ ðc1 � cÞ=c1; and r ¼ rðcÞ.The boundary conditions for (2.101) and (2.102) are
u ¼ v ¼ 0 c� ¼ 1 at y ¼ 0; u ¼ v ¼ 0 c� ¼ 0 at y ! 1 (2.103)
Equations (2.101) and (2.102) and the boundary conditions (2.103) contain four
parameters that determine the local velocity and concentration fields
u ¼ fuðx; y; n; g�Þ (2.104)
c� ¼ fcðx; y;D; g�Þ (2.105)
2.5 The Pi-Theorem 29
Applying the Pi-theorem to transform (2.104) and (2.105) to the dimensionless
form, we obtain
Pu ¼ cuðP1Þ (2.106)
Pc ¼ ccðP1�Þ (2.107)
or equivalently,
u ¼ xg�ð Þ1=2cuð�Þ (2.108)
c� ¼ ccð�ffiffiffiffiffiSc
pÞ (2.109)
where Pu ¼ u= xg�ð Þ1=2; P1 ¼ y g�=xn2ð Þ1=4; Pc ¼ c�; P1� ¼ y g�=xD2ð Þ1=4,� ¼ y g�=xn2ð Þ1=4; and Sc ¼ n/D is the Schmidt number.
A number of instructive examples of application of the mass, momentum, energy
and species conservation equations for dimensional analysis of the hydrodynamic
and heat and mass transfer problems can be found in Chap. 7.
Problems
P.2.1. Transform the van der Waals equation (Kestin 1966; Jones and Hawkis 1986)
to the dimensionless form. Show that such form is universal for any van der Waals
gas if one uses the critical values of the pressure, volume and temperature as the
characteristic scales.
In order to transform equation the van der Waals equation ðPþ a=V2ÞðV � bÞ ¼RT(where a and b are constants) to dimensionless form, we present this equation as
ðA1 þ A2ÞðA3 þ A4Þ ¼ A5 (P.2.1)
where A1 ¼ P; A2 ¼ a=V2; A3 ¼ V; A4 ¼ �b; A5 ¼ RT with P;V and Tbeing pressure, molar volume and temperature, respectively.
We can introduce some still undefined scales of pressure P0; volume V0 and
temperature T0 and write the expressions for scales of Aj as
A1� ¼ P�; A2� ¼ a
V2�; A3� ¼ V�; A4� ¼ �b; A5� ¼ RT� (P.2.2)
Then (P.2.1) reduces to the form
ðA1 þ aA2ÞðbA3 þ gAÞ ¼ eA5 (P.2.3)
30 2 Basics of the Dimensional Analysis
where A1 ¼ A1=A1� ¼ P=P; A2 ¼ A2 A2�; ¼ V V�=ð Þ�2;.
A3 ¼ A3=A3� ¼ ðV=V�Þ;A4 ¼ A4 A4� ¼ 1= A5 ¼ A5=A5� ¼ T=T� are the dimensionless variables, and a ¼A2�=A1� ¼ a= V2
�P�� �
; b ¼ A3�=A1� ¼ V�=P�; g ¼ A4�=A1� ¼ �b=P�; and e ¼A5�=A1� ¼ RT�=P� are the dimensionless constants.
Equation (P.2.3) is the dimensionless van der Waals equation. For its further
transformation one should define the characteristic scales of pressure, volume and
temperature. For that purpose, take as the scales P�; V� and T� the critical values
of pressure, volume and temperature Pcr; Vcr and Tcr, respectively. Bearing
in mind that the critical point is the inflection point where ð@P=@VÞT ¼ 0;
and ð@2P=@V2ÞT ¼ 0 , we find
a ¼ 27
64
R2T2cr
Pcr; b ¼ Vcr
3(P.2.4)
Pcr ¼ 1
27
a
b2; Vcr ¼ 3b; Tcr ¼ 8a
27bR(P.2.5)
Using as the characteristic scales the critical values of pressure, temperature and
specific volume, we transform (P.2.3) to the following final form
ðpþ 3
o2Þð3o� 1Þ ¼ 8t (P.2.6)
where p ¼ P=Pcr; o ¼ V=Vcr; and t ¼ T=Tcr.Equation (P.2.6) does not contain any constants accounting for the physical
properties of any particular gas and, thus, is universal. It holds for any van der
Waals gas.
P.2.2. (i) Transform the momentum and continuity equations for laminar flow of
incompressible fluid over a plane plate in the boundary layer approximation to the
dimensionless form using the LMT and LxLyLzMT systems of units. (ii) Show that
the LxLyLzMT system of units cannot be used for transformation of the
Navier–Stokes equations to the dimensionless form.
(i)-A: The LMTsystem of units. The boundary layer and continuity equations
read
u@u
@xþ v
@u
@y¼ n
@2u
@y2(P.2.7)
@u
@xþ @v
@yþ 0 (P.2.8)
where the dimensions of u; v; n; x and yare as follows
u½ � ¼ LT�1; v½ � ¼ LT�1; n½ � ¼ L2T�1; x½ � ¼ L; y½ � ¼ L (P.2.9)
Problems 31
All the terms in (P.2.7) have the dimension LT�2; whereas the dimension of the
terms in (P.2.8) is T�1: That shows that (P.2.7) and (P.2.8) can be transformed to
the dimensionless form by using the multipliers N1½ � ¼ ðLT�2Þ�1and N2½ � ¼
ðT�1Þ�1; respectively. Introducing the scales of length L�; velocity V�and kine-
matic viscosity n� ¼ n, we write the expressions for the coefficients AðiÞj� as follows
Að1Þ1� ¼ A
ð1Þ2� ¼ V2
�L�
;Að1Þ3� ¼ n�
V�L2�
;Að2Þ1� ¼ A
ð2Þ2� ¼ V�
L�(P.2.10)
Bearing in mind the dimensions of AðiÞj� , we express the multipliers N1 and N2 as
N1 ¼ 1
Að1Þ1�
; N2 ¼ 1
Að2Þ1�
(P.2.11)
Then (P.2.7) and (P.2.8) reduce to the following form
u@u
@xþ v
@u
@y¼ 1
Re
@2u
@y2(P.2.12)
@u
@xþ @v
@y¼ 0 (P.2.13)
where u ¼ u=V�; v ¼ v=V�; x ¼ x=L�; y ¼ y=L�; andRe ¼ V�L�=n�.The coefficients A
ðiÞj� for the Navier–Stokes and continuity equations
u@u
@xþ v
@u
@y¼ n
@2u
@x2þ @2u
@y2
� �(P.2.14)
v@u
@xþ v
@v
@y¼ n
@2v
@x2þ @2v
@y2
� �(P.2.15)
@u
@xþ @v
@y¼ 0 (P.2.16)
are defined as follows
Að1Þ1� ¼A
ð1Þ2� ¼
V2�
L�;A
ð1Þ3� ¼n�
V�L�
;Að2Þ1� ¼A
ð2Þ2� ¼
V2�
L�;A
ð2Þ3� ¼n�
V�L�
;Að3Þ1� ¼A
ð3Þ2� ¼
V�L�
(P.2.17)
Using the multipliers N1 ¼ 1=Að1Þ1� ;N2 ¼ 1=A1
ð2Þ� ; and N3 ¼ 1=A
ð3Þ1� , we reduce
(P.2.14)–(P.2.16) to the following dimensionless form
u@u
@xþ v
@u
@y¼ 1
Re
@2u
@x2þ @2u
@y2
� �(P.2.18)
32 2 Basics of the Dimensional Analysis
v@u
@xþ v
@v
@y¼ 1
Re
@2v
@x2þ @2v
@y2
� �(P.2.19)
@u
@xþ @v
@y¼ 0 (P.2.20)
(i)-B: The LxLyLzMT system of units. The dimensions of u; v; x and y are
u½ � ¼ LxT�1; v½ � ¼ LyT
�1; x½ � ¼ Lx; y½ � ¼ Ly (P.2.21)
where Lx and Ly are the scales of length in the x and ydirections.Introducing the characteristic scales of u; v; n; x and y as U�½ � ¼ LxT
�1; V�½ � ¼LyT
�1; and n�½ � ¼ n½ �, we transform first of all the boundary layer and continuity
equations (P.2.7) and (P.2.8). To this aim, we write the expressions for the coeffi-
cients AðiÞj� as
Að1Þ1� ¼ U2
�Lx
; Að1Þ2� ¼ U�V�
Ly; A
ð1Þ3� ¼ n�U�
L2y; A
ð2Þ1� ¼ U�
Lx; A
ð2Þ2� ¼ V�
Ly(P.2.22)
Then (P.2.7) and (P.2.8) are transformed to
u@u
@xþ V�Lx
U�Ly
� �v@u
@y¼ n�Lx
L2yU�
!@2u
@y2(P.2.23)
@u
@xþ V�Lx
U�Ly
� �@v
@y¼ 0 (P.2.24)
where u ¼ u=U�; v ¼ v=V�; x ¼ x=Lx; y ¼ y=Ly; and the multipliers before the
second terms on left hand side of the boundary layer and continuity equations are
dimensionless, i.e. V�Lx=U�Ly� � ¼ 1.
In planar viscous flows in the x-direction with shear in the y-direction an
important role is played by the shear component tyx of the stress tensor. The
shear stress tyx can be presented as the ratio of the force Fyx to the surface area
Szx which have the following dimensions: Fyx
� � ¼ MLxT�2; and Szx½ � ¼ LzLx:
Then the dimension of the shear stress is tyx� � ¼ ML�1
z T�2: For viscous Newtonianfluids tyx ¼ mdu=dy, where m is the viscosity. Then, we find the dimension of the
viscosity in the LxLyLzMT system of units as
m½ � ¼ tyx du dy=ð Þ=� � ¼ L�1
x LyL�1z MT�1 (P.2.25)
Problems 33
Bearing in mind that the dimension of density in the LxLyLzMT system of units is
r½ � ¼ L�1x L�1
y L�1z M, we determine the dimension of the kinematic viscosity n as
n½ � ¼ m r=½ � ¼ L2yT�1 (P.2.26)
Thus, the multiplier n�Lx L2yU�. �
on the right hand side of (P.2.23) is dimen-
sionless. It can be presented as Re�1� , where Re� ¼ U�L2y n�Lx=
�is a modified
Reynolds number. Taking into account that the characteristic scales Lx; Ly;U� andVx are arbitrary, it is possible to assume that the ratio U�Ly V�Lx=
� � ¼ 1. Then,
(P.2.23) and (P.2.24) take the following form
u@u
@xþ v
@u
@y¼ 1
Re�
@2u
@y2(P.2.27)
@u
@xþ @v
@y0 (P.2.28)
The Navier–Stokes and continuity equations (P.2.14) and (P.2.16) can be
presented as
Að1Þ1� A
ð1Þ1 þ A
ð1Þ2� A
ð1Þ2 ¼ A
ð1Þ3� A
ð1Þ3 þ A
ð1Þ4� A
ð1Þ4 (P.2.29)
Að2Þ1� A
ð2Þ1 þ A
ð2Þ2� A
ð2Þ2 ¼ A
ð2Þ3� A
ð2Þ3 þ A
ð2Þ4� A
ð2Þ4 (P.2.30)
Að3Þ1� A
ð3Þ1 þ A
ð3Þ2� A
ð3Þ2 ¼ 0 (P.2.31)
where Að1Þ1� ¼ U2
�=Lx; Að1Þ2� ¼ U�V�=Ly; A
ð1Þ3� ¼ n�U�=L2x; A
ð1Þ4� ¼ n�U�=L2y , A
ð2Þ1� ¼
U�V=Lx; Að2Þ2� ¼ V2
�=Ly; Að2Þ3� ¼ n�V�=L2x ; A
ð2Þ4� ¼ n�V�=L2y , A
ð3Þ1� ¼ U�=Lx;A
ð3Þ2� ¼
V�=Ly, Að1Þ1 ¼ u@u=@x; A
ð1Þ2 ¼ v@u=@y; A
ð1Þ3 ¼ @2u=@x2; A
ð1Þ4 ¼ @2u=@y2, A
ð2Þ1 ¼
v@u=@x, Að2Þ2 ¼ v@v=@y, A
ð2Þ3 ¼ @2v=@x2;A
ð2Þ4 ¼ @2v=@y2, A
ð3Þ1 ¼ @u=@x; and
Að3Þ2 ¼ @[email protected] (P.2.18)–(P.2.20) take the form
u@u
@xþ V�Lx
U�Ly
� �v@u
@y¼ n�Lx
U�L2y
!LyLx
� �2 @2u
@x2þ @2u
@y2
( )(P.2.32)
v@u
@xþ V�Lx
U�Ly
� �v@v
@y¼ n�Lx
U�L2y
!LyLx
� �2 @2v
@x2þ @2v
@y2
( )(P.2.33)
@u
@xþ V�Lx
U�Ly
� �@v
@y¼ 0 (P.2.34)
34 2 Basics of the Dimensional Analysis
The system of Eqs. (P.2.32)–(P.2.34) can be written as
u@u
@xþ v
@u
@y¼ 1
Re
LyLx
� �2 @2u
@x2þ @2u
@y2
( )(P.2.35)
v@u
@xþ v
@v
@y¼ 1
Re�
LyLx
� �2 @2v
@x2þ @2u
@y2
( )(P.2.36)
@u
@xþ @v
@y¼ 0 (P.2.37)
if we account for the fact that the dimension of the kinematic viscosity n�½ � ¼L2yT
�1: However, even in this case (P.2.35) and (P.2.36) are not dimensionless,
since the dimension of the ratio Ly Lx= is not 1. Moreover, (P.2.35) and (P.2.36) do
not satisfy the principle of the dimensional homogeneity under any assumption on
the dimension of the kinematic viscosity. The latter shows that applying the
LxLyLzMT system of units to transformation of the Navier–Stokes is incorrect.
P.2.3. (Reynolds 1886) Determine the resistance force acting on each of two
circular disks of radii R which approach each other along the joint axis of symmetry
with a constant velocity u, while the gap between the disks and the surrounding
space are filled with incompressible viscous fluid. The pressure in the surrounding
fluid far from the disks is equal P�.The liquid flow in the gap is axisymmetric. Therefore, we use cylindrical
coordinates z; r; ’ with the origin at the center of the lower disk which is assumed
to be motionless (z and r correspond to the vertical and radial directions, respec-
tively). Consider the low velocity case when the inertial effects are negligible. The
effect of the gravity force we also will neglected. Then, it is possible to assume that
the pressure gradient DP=r ðDP ¼ P� P�Þ is determined by the speed of the upper
disk u; liquid viscosity m; the instantaneous height of the gap h, and the radial
position r
DPr
¼ f ðu; m; h; rÞ (P.2.38)
For analyzing the problem, we use two different systems of units with a single
(L) and two (Lz; Lr) length scales. In the first case the dimensions of the pressure
gradient and the governing parameters can be expressed as
DPr
� ¼ L�2MT�2; u½ � ¼ LT�1; m½ � ¼ L�1MT�1; h½ � ¼ L; r½ � ¼ L (P.2.39)
Three of the four governing parameters in (P.2.38) have independent
dimensions. Choosing u; m; and r as the parameters with the independent
dimensions, we reduce (P.2.38) according to the Pi-theorem to the following form
Problems 35
P ¼ ’ðP1Þ (P.2.40)
where P ¼ DP=rð Þ=ua1ma2ra3 and P1 ¼ h=ua01ma
02 ra
03 .
Using the principle of the dimensional homogeneity, we find the values of
the exponents ai and a0i: a1 ¼ 1; a2 ¼ 1; a3 ¼ �2; a
01 ¼ 0; a
02 ¼ 0; and a
03 ¼ 1.
Accordingly, we arrive at the following expression
DPr
¼ umr�2’h
r
� �(P.2.41)
The force acting at the disk is found as
Fd ¼ 2pZR
DPrdr (P.2.42)
Substituting the expression (P.2.41) into (P.2.42), we obtain
Fd ¼ 2pumRZ10
’ex
� �dx (P.2.43)
where e ¼ h=R; x ¼ r=R; andR10
’ e=xð Þdx ¼ c eð Þ.Equation P.2.43 shows that the resistance force acting on a disk is directly
proportional to its velocity, the radius of the disk, viscosity of the liquid, as well
as a function of the ratio of the gap to the disk radius.
Additionally we transform (P.2.38) using the system of units with the two length
scales Lz and Lr in the z and r directions, respectively. First, we determine the
dimensions of the governing parameters and pressure gradient. The dimensions of
the velocity u, gap thickness h and r are
u½ � ¼ LzT�1; h½ � ¼ Lz; r½ � ¼ Lr (P.2.44)
To determine the dimensions of viscosity m and pressure gradient DP=r, we takeinto account the fact that in flows of viscous fluids in a narrow gap the dominant role
is played by the radial velocity component, since the axial one is typically much
smaller, vz << vr: In this case the force acting in the r-direction is much larger than
in the z-direction, so that its dimension is Fr½ � ¼ MLrT�2: Accordingly, the dimen-
sion of the shear stress tzr ¼ Fr Srr= ð Srr½ � ¼ L2r Þ is tzr½ � ¼ ML�1r T�2: For Newtonian
viscous fluids tzr ¼ m dvr dz=ð Þ. As a result, we find the dimension of viscosity m½ � ¼ML�2
r LzT�1: The dimensions of pressure and its gradient are
DP½ � ¼ Fr
Srz¼ MLrT
�2
LrLz¼ ML�1
z T�2 (P.2.45)
36 2 Basics of the Dimensional Analysis
DPr
� ¼ ML�1
z T�2
Lr¼ ML�1
z L�1r T�2 (P.2.46)
Thus, the dimensions of all the governing parameters are expressed in the system
of units with two length scales are independent. Then, according to the Pi-theorem,
(P.2.38) takes the form
DPr
¼ cua1ma2ha3ra4 (P.2.47)
where c is a dimensionless constant.
Determining the values of the exponents ai using the principle of the dimen-
sional homogeneity as a1 ¼ 1; a2 ¼ 1; a3 ¼ �3 and a4 ¼ 1, we obtain
DPr
¼ cumr
h3(P.2.48)
Then, the substitution of (P.2.48) into (P.2.42) yields
Fd ¼ c
2pumR
R
h
� �3
(P.2.49)
The exact solution of this problem reads (Landau and Lifshitz 1987)
Fd ¼ 3
2pmuR
R
h
� �3
(P.2.50)
The comparison of (P.2.49) and (P.2.50) shows that the exact solution and the
result of the dimensional analysis agree up to a dimensionless numerical factor. At
the same time, the dimensional analysis of the problem using the system of units
with a single length scale yields a less informative result, since (P.2.43) contains an
unknown function c h R=ð Þ:
References
Barenblatt GI (1987) Dimensional analysis. Gordon and Breach Science Publication, New York
Barenblatt GI (1996) Similarity, self-similarity, and intermediate asymptotics. Cambridge Uni-
versity Press, Cambridge
Blackman DR (1969) SI units in engineering. Macmillan, Melbourne
Bridgman PW (1922) Dimension analysis. Yale University Press, New Haven
Buckingham E (1914) On physically similar system: illustrations of the use of dimensional
equations. Phys Rev 4:345–376
Chart of Dimensionless Numbers (1991) OMEGA Technology Company.
References 37
Douglas JF (1969) An introduction to dimensional analysis for engineers. Isaac Pitman and Sons,
London
Hinze JO (1975) Turbulence, 2nd edn. McGraw-Hill, New York
Huntley HE (1967) Dimensional analysis. Dover Publications, New York
Ipsen DC (1960) Units, dimensions, and dimensionless numbers. McGraw-Hill, New York
Jones JB, Hawkis GA (1986) Engineering thermodynamics. An introductory textbook. John Wiley
& Sons, New York
Kestin J (v.1, 1966; v.2, 1968) A course in thermodynamics. Blaisdell Publishing Company, New
York
Kutateladze SS (1986) Similarity analysis and physical models. Nauka, Novosibirsk (in Russian)
Landau LD, Lifshitz EM (1987) Fluid mechanics, 2nd edn. Pergamon, Oxford
Levich VG (1962) Physicochemical hydrodynamics. Prentice-Hill, Englewood Cliffs
Loitsyanskii LG (1966) Mechanics of liquid and gases. Pergamon, Oxford
Lykov AM, Mikhailov Yu A (1963) Theory of heat and mass transfer. Gosenergoizdat Moscow-
Leningrad (English translation 1965. Published by the Israel Program for Scientific Transla-
tion. Jerusalem)
Madrid CN, Alhama F (2005) Discriminated dimensional analysis of the energy equation:
application to laminar forced convection along a flat plate. Int J Thermal Sci 44:331–341
Ramaswamy GS, Rao VVL (1971) SI units. A source book. McGraw-Hill, Bombay
Reynolds O (1886) On the theory of lubrication. Philos Trans R Soc 177:157–233
Sedov LI (1993) Similarity and dimensional methods in mechanics, 10th edn. CRC Press, Boca
Raton
Spurk JH (1992) Dimensionsanalyse in der Stromungslehre. Springer-Verland in Berlin, New
York
Weast RC Handbook of chemistry and physics, 68th edition. 1987–1988. CRC, Boca Raton, FL.
Williams W (1892) On the relation of the dimensions of physical quantities to directions in space.
Philos Mag 34:234–271
38 2 Basics of the Dimensional Analysis
Chapter 3
Application of the Pi-Theorem to EstablishSelf-Similarity and Reduce Partial DifferentialEquations to the Ordinary Ones
3.1 General Remarks
Chapter 3 deals with the application of the Pi-theorem to reduce partial differential
equations (PDEs) of certain hydrodynamic and heat transfer problems to the
ordinary differential equations (ODEs). In the cases which allow for such transfor-
mation (including the initial and boundary conditions), solution of the problem
reduces to a much simpler problem posed for an ODE, i.e. depends on a single
compound variable. The latter represent itself a combination of variables and
dimensional constants involved in the problem formulation. Such solutions are
called self-similar, since a single fixed value of the compound single variable
corresponds to numerous combinations of, say, coordinates or coordinates and
time, which make them identical in the “space” of the single compound variable.
The general consideration in Chap. 3 is followed by a number of examples
illustrating the usage of the Pi-theorem for establishing self-similar solutions.
They include flows of viscous incompressible fluid (the Stokes, Landau, and von
Karman problems), hydrodynamic and thermal (diffusion) boundary layers (the
Blasius, Pohlhausen and Levich problems), as well as some special hydrodynamical
problems, e.g. propagation of viscous-gravity currents (the Huppert problem)
and capillary waves on a thin liquid after a weak impact of a tiny droplet (the
Yarin-Weiss problem).
Below we consider the applications of the Pi-theorem for solving the differential
equations of hydrodynamics and the theory of heat and mass transfer. In the frame
of continuum approach, fluid flows and heat and mass transfer are described by a set
of PDEs that include the continuity, momentum (the Navier–Stokes), energy and
species balance equations (Kays and Crawford, 1993; Baehr and Stephan 1998;
Schlichting 1979). Solving these equations is extremely difficult because of the
non-linearity of the inertial terms of the Navier–Stokes equations and dependences
of the velocity, temperature and species concentration fields on several variables: in
the general case on three spatial coordinates and time. An essential simplification of
the problem can be achieved by reducing the number of independent variables by
L.P. Yarin, The Pi-Theorem, Experimental Fluid Mechanics,
DOI 10.1007/978-3-642-19565-5_3, # Springer-Verlag Berlin Heidelberg 2012
39
means of a transformation of the problem PDEs to a set of ODEs if the initial and
boundary conditional of a specific problem admit that as well. As a matter of fact,
such a reduction can be considered as a typical problem of the dimensional analysis,
namely, the transition from n dimensional variables to n� k dimensionless ones. In
the case where the number of dimensionless groups n� k ¼ 1, a multi-dimensional
problem reduces to a one-dimensional one.
In order to demonstrate the transformation of PDEs into ODEs by using the
Pi-theorem, we consider a process in which some unknown dimensional
characteristics a (say, velocity, temperature, etc.) is expected to be determined by
dimension parameters a1; a2; � � �an (say, viscosity, density, velocity of the undis-
turbed flow, a characteristic size of a body, etc.)
a ¼ f ða1; a2 � � � ak � � � anÞ (3.1)
where a1; a2; � � �ak are the parameters with independent dimensions.
Assume that two parameters ai and aj among the n governing parameters
are independent variables (say, a coordinate and time) and all the other
n� 2 parameters are constants (say, the undisturbed velocity, viscosity, etc.).
Then the unknown characteristic a is described by PDE and the boundary and
initial conditions which contain all the n� 2 constants of the problem, the indepen-
dent variables ai and aj, as well as the derivatives @a=@ai; @a=@aj;
@2a=@a2i ; @2a=@a2j , etc. In order to reduce the problem to the integration of a
set of ODEs with the appropriate boundary conditions, one should find such a
transformation of the unknown characteristics a and the governing parameters
a1;a2; � � �an that the dimensionless unknown characteristics (denoted as P) will
become a function of a single dimensionless group P1 formed from the governing
parameters of the problem. The latter is possible when the number of the governing
parameters equals k þ 1: Then (3.1) reduces to
P ¼ ’ðP1Þ (3.2)
where P ¼ a=aa11 aa22 � � � aakk , and P1 ¼ an=a
a01
1 aa02
2 � � � aa0k
k , with ai and a0i (i ¼ 1,
2,. . . kÞ being some exponents.
It is emphasized that P is described by an ODE (or a system of ODEs), since
it is a function of a single variableP1: The above consideration shows that
transformation of a PDE into an ODE is determined by the dimensions of the
governing parameters. For example, consider a particular case when the unknown
characteristics a depends on four dimension parameters: two constants a1; a2 andtwo independent variables a3; a4
a ¼ f ða1; a2; a3; a4Þ (3.3)
Let the dimensions of the unknown characteristics a and governing parameters
a1; a2; a3 and a4 be
40 3 Application of the Pi-Theorem to Establish Self-Similarity
a½ � ¼ Le1Me2Te3 ; a1½ � ¼ Le01Me
02Te
03 ; a2½ � ¼ Le
001Me
002 Te
003 ; a3½ �
¼ Le0001 Me
0002 Te
0003 ; a½ � ¼ Le
IV1 MeIV
2 TeIV3 (3.4)
where ei; e0i; e
00i ; e
000i and eIVi (i ¼ 1,2,3) are some known exponents.
When three governing parameters, for example, a1; a2 and a3 have independentdimensions, (3.3) takes the form of (3.2) with P ¼ a=aa11 a
a22 a
a33 and P1 ¼
a4=aa01
1 aa02
2 aa03
3 : Bearing in mind the dimensions of the parameters a; a1; a2; a3 anda4, we arrive at the following sets of the algebraic equations for the exponents
ai and a0i:
e01a1 þ e
001a2 þ e
0001 a3 ¼ e1
e02a1 þ e
002a2 þ e
0002 a3 ¼ e2
e03a1 þ e
003a2 þ e
0003 a3 ¼ e3
(3.5)
and
e01a
01 þ e
001a
02 þ e
0001 a
03 ¼ eIV1
e02a
01 þ e
002a
02 þ e
0002 a
03 ¼ eIV2
e03a
01 þ e
003a
02 þ e
0003 a
03 ¼ eIV3
(3.6)
Solution of the systems of (3.5) and (3.6) exists when
e01 e
001 e
0001
e02 e
002 e
0002
e03 e
003 e
0003
������������ 6¼ 0 (3.7)
The inequality (3.7) is the condition under which transformation of a PDE into
an ODE is possible. This inequality determines only the necessary rather than the
sufficient condition of the existence of self-similar solutions. In addition to
satisfying the ODEs, such solutions should also satisfy the initial and boundary
conditions of the problem.
The transformation of a PDE into an ODE is also possible with n� k ¼ 2; 3; � � �lwhen all the constants in (3.1) have the same dimensions, which are different from
the dimensions of the independent variables ai and aj: Indeed, (3.1) can be recast intothe following form
a ¼ f ða1; a2 � � � ai � � � ak � � � aj � � � anÞ (3.8)
The latter equation is reduced to the dimensionless form when the Pi-theorem is
applied
P ¼ ’ðP1;P2 � � �Pj � � �Pn�kÞ (3.9)
3.1 General Remarks 41
where P ¼ a=aa11 aa22 � � � aaii � � � aakk ; P1 ¼ akþ1=a
a01
1 aa02
2 � � � aa0i
i � � � aa0k
k ; P2 ¼akþ2=a
a001
1 aa002
2 � � � aa00i
i � � � aa00k
k ;Pj ¼ aj=aaj1
1 aaj2
2 � � � aaji
i � � � aajk
k ; and Pn�k ¼ an�k=
aan�k1
1 aan�k2
2 � � � aan�ki
i � � � aan�kk
k .
Due to the fact that the dimensions of all the constants are different from the
dimensions of variable ai; we find that the exponents a0i, a
00i ; � � �an�k
i are equal to
zero. Then (3.9) takes the following form
P ¼ ’ðc1; c2 � � �Pj � � � cn�kÞ ¼ ’ðPjÞ (3.10)
where c1; c2; � � �cn�k are the dimensionless constants: c1 ¼akþ1=a
a01
1 aa02
2 � � � aa0i�1
i�1aa0iþ1
iþ1 � � � aa0k
k , c2 ¼ akþ2=aa001
1 aa002
2 � � � aa00i�1
i�1aa00iþ1
iþ1 � � �aa00k
k ; . . ..cn�k ¼an�k=a
an�k1
1 aan�k2
2 � � � aan�ki�1
i�1 aan�kiþ1
iþ1 � � � aan�kk
k :
It is emphasized that the possibility to transform the problem PDEs into ODEs
can be recognized by a simple analysis of the dimensions of constants involved in
the problem formulation, i.e. in the governing equations, the initial and boundary
conditions, as well as the additional characteristics which are given, such as the
invariant overall momentum flux in a submerged jet, etc. The rule that such a
transformation is possible (and thus, a self-similar solution of the PDE-based
problem exists) can be formulated as follows: the transition of PDEs into ODEs
is possible when scales of the independent variables cannot be constructed using the
scales of the problem constants (Loitsyanskii 1966). In order to illustrate this
statement, we consider the radial flow in a plane wedge. The radial component of
fluid velocity v in this case depends on the strength of a source Q located at the
wedge apex, kinematic viscosity n of fluid, as well as the radial coordinate r and thepolar angle ’ of the cylindrical coordinate system
v ¼ f�ðQ; n; r; ’Þ (3.11)
The dimensions of the given constants Q and n are: Q½ � ¼ L2T�1 (per unit length
of a source or sink normal to the wedge plane) and n½ � ¼ L2T�1: It is easy to see thatit is impossible to express the dimension of r via the dimensions of the constants
Q and n: This fact points out at the possibility of transformation of the PDE
describing the flow under consideration into an ODE. Indeed, accounting for the
fact that two from the four governing parameters have independent dimensions, we
can rewrite (3.11) as follows
v ¼ f1Q
n; ’
� �¼ f ð’Þ (3.12)
where v ¼ v= nr
� �, and Q=n are dimensionless constants.
42 3 Application of the Pi-Theorem to Establish Self-Similarity
Substituting the expression (3.12) into the set of the governing equations
vdw
dr¼ � 1
r@P
@rþ n
@2v
@r2þ 1
r2@2v
@’2þ 1
r
@v
@r� v
r2
� �(3.13)
� 1
rr@P
@’þ 2v
r2@v
@’¼ 0 (3.14)
@ðrvÞ@r
¼ 0 (3.15)
we arrive at the following ODE (Hamel 1917; see also Rosenhead 1963 and
Loitsyanskii 1966)
d2f
d’2þ 4f þ 6f 2 ¼ 2c1 (3.16)
where c1 is a constant.The constant c1, as well as the two additional constants c2 and c3 that appear
when integrating (3.16), are determined by the no-slip boundary conditions at the
wedge walls v �a=2ð Þ ¼ 0 and the condition that a constant mass flow rate is given.
It is emphasized that a similar analysis of a flow in a cone (instead of a wedge)
where Q has the dimension L3T�1 shows that transformation of the PDE describing
this flow into an ODE is impossible since the dimension of L is provided by the ratio
Q=n.Reduction of PDEs to ODEs is possible under a certain idealization of real
phenomena. The absence of a characteristic scale (for example, the length scale is
missing in flows about semi-infinite plates, in submerged jet flows issued from a
point wise source of momentum flux, etc.) is a sign of the existence a self-similar
solution of PDEs and a possibility of their transformation to ODEs (Sedov 1993).
Solving such problems in dimensionless form requires introduction of some arbi-
trary scales instead of the missing ones. Only then the governing equations and the
initial and boundary conditions can be transformed to the dimensionless form. As a
result, the unknown characteristics can be expressed as a dimensionless function of
dimensionless variables only. The requirement that the arbitrary length scale should
not be involved in the dimensional solution of an idealized problem allows reveal-
ing the form of this function. It is emphasized that the absence of a characteristic
length scale in the problem formulation is an essential but not a sufficient condition
for the existence of a self-similar solution of a set of the corresponding PDEs. For
example, the problem on flow in an axisymmetric diffuser with circle cross-section
has no self-similar solution.
Below we consider a number of examples of applications of the Pi-theorem for
reduction of PDEs describing incompressible fluid flows and heat transfer in media
at rest to ODEs, and thus finding self-similar solutions.
3.1 General Remarks 43
3.2 Flow over a Plane Wall Which Has Instantaneously StartedMoving from Rest (the Stokes Problem)
Consider an upper half-space filled with a viscous incompressible fluid in contact
with a flat plate corresponding to y ¼ 0 (Fig. 3.1). (Stokes 1851)
This wall has instantaneously started to move horizontally with a constant
velocity U. The wall motion is transmitted to the fluid due to the action of viscous
forces. As a result, the fluid is entrained into horizontal motion as well. The fluid
flow is subjected to the no-slip boundary condition at the wall surface and is
described as follows
@u
@t¼ n
@2u
@y2(3.17)
t ¼ 0 : 0 � y � 1 u ¼ 0; t>0 : y ¼ 0 u ¼ U; y ! 1 u ¼ 0 (3.18)
where u is the fluid velocity and n is the kinematic viscosity.
Equation (3.17) with the conditions (3.18) show that fluid velocity depends on
four parameters: two independent variables t and yand two constants n and U :
u ¼ f ðU; n; y; tÞ (3.19)
First of all, let us ascertain the possibility of reduction of (3.17) to an ODE. For
this aim we make use of the above-mentioned signs pointing at such a transforma-
tion. The lack of a given characteristic length in (3.17) and the conditions (3.18)
points at the possibility of reduction of (3.17) to an ODE.
To apply the Pi-theorem to transform (3.17) into an ODE, it is necessary to
consider the dimensions of the unknown characteristics u and the governing
parameters n; U; y and t: In principle, it is possible to use different systems of
fundamental units, in particular, the LMT system. Then, we have the dimensions as
u½ � ¼ LT�1; U½ � ¼ LT�1; n½ � ¼ L2T�1; y½ � ¼ L; t½ � ¼ T (3.20)
t > 0
U > 0
y
u0
t = 0
U = 0
y
u0
Fig. 3.1 Scheme of flow over
plane wall has instantaneous
by started moving from rest
44 3 Application of the Pi-Theorem to Establish Self-Similarity
Two from the four governing parameters have independent dimensions, so that
n� k ¼ 2: In accordance with that, we obtain
u ¼ ’ðt; yÞ (3.21)
where u ¼ u=U; t ¼ t= n=U2ð Þ; y ¼ y=ðn=UÞ.Equation (3.21) shows that u depends on two dimensionless groups that seem-
ingly shows that it is impossible of transform (3.17) into an ODE. This result
follows directly from the analysis of the dimensions of the parameters involved.
Indeed, we can construct the length and time scales, L� ¼ n=U, T� ¼ n=U2, that
shows that it is impossible to express u as a function of a single dimensionless
variable. At the first sight this result contradicts to the expectations based on
the absence of the characteristic length scale in the problem formulation as in
the present case. The apparent contradiction can be explained as follows. The
Pi-theorem determines only the number of dimensionless groups which can be
constructed from n governing parameters including k parameters with independent
dimensions. The number n is determined by the physical essence of the problem,
whereas the number k can be changed depending on the system of units used. Thus,
the difference n� k that determines the number of dimensionless variables depends
also on the system of units used.
Let us extend the system of units by introducing three different length scales Lxand Ly for x and y directions (along and normal to the wall in Fig. 3.1), and Lz forthe z direction normal to the xy plane. This means that the LxLyLzMT system of units
is used. Taking into account that the wall and the velocity component u, as well asthe velocity of the unperturbed flow U are directed along the x-axis, we define theirdimensions as
u½ � ¼ LxT�1; U½ � ¼ LxT
�1 (3.22)
where T is the time scale, t½ � ¼ T:The dimension of the kinematic viscosity, is
n½ � ¼ L2yT�1 (3.23)
since in the case under consideration viscosity transmits information about the
wall motion into the liquid bulk in the y direction. Indeed, the dimension of
viscosity m can be found directly from the rheological constitutive equation of the
Newtonian fluid tyx ¼ m du=dyð Þ as the ratio of the shear stress to the velocity
gradient (Huntley 1967; Douglas 1969). Bearing in mind that tyx ¼ Fyx=Sxz, wedetermine the dimension of tyx
tyx� � ¼ L�1
z MT�2 (3.24)
where Fyx and Sxz are the force in the x direction acting at the surface element in the
xz plane, respecticaly; Fyx
� � ¼ LxMT�2; Sxz½ � ¼ LxLz.
3.2 Flow over a Plane Wall Which Has Instantaneously Started Moving from Rest 45
Since the dimension of the velocity gradient du=dy is LxL�1y T�1, the dimension
of viscosity is expressed as
m½ � ¼¼ L�1x LyL
�1z MT�1 (3.25)
Then the dimension of the kinematic viscosity is
n½ � ¼ mr
¼ L2yT
�1 (3.26)
where r½ � ¼ L�1x L�1
y L�1z M is the fluid density.
As a result, we guarantee that the dimensions of all the terms in (3.17) are the
same: @u @t=½ � ¼ LxT�2; and n@2u @y2
�� � ¼ LxT�2.
In the framework of the LxLyLzMT system among the four governing parameters
there are three parameters with independent dimensions
U½ � ¼ Le01x L
e02y T
e03 ; n½ � ¼ L
e001x L
e002y T
e003 ; t½ � ¼ L
e0001x L
e0002y Te
0003 (3.27)
where e01 ¼ 1; e
02 ¼ 0; e
03 ¼ �1; e
001 ¼ �1; e
002 ¼ 2; e
003 ¼ �1; e
0001 ¼ 0; e
0002 ¼ 0; and
e0003 ¼ 1.
At such values of the exponents e0i; e
00i and e
000i determinant (3.7) is not equal to
zero. In this case (3.19) takes the form of (3.2) with P ¼ u=Ua1na2 ta3 and P1 ¼y=Ua
01na
02 ta
03 : Taking into account the dimensions of u and U; n; t; y; we arrive
at the system of the algebraic equations for the exponents ai and a0i(i ¼ 1; 2; 3Þ
SLx1� a1 ¼ 0; a01 ¼ 0
SLy a2 ¼ 0; 1� 2a02 ¼ 0
STa1 � a3 � 1 ¼ 0; a02 � a
03 ¼ 0 (3.28)
where the symbols SLx ;SLy and ST refer to the summation of the exponents of
Lx; Ly and T; respectively.From (3.28) it follows
a1 ¼ 1; a2 ¼ 0; a3 ¼ 0; a01 ¼ 0; a
02 ¼
1
2; a
03 ¼
1
2(3.29)
Then (3.19) reduces to
u
U¼ ’
yffiffiffiffint
p� �
(3.30)
The substitution of the derivatives @u=@t ¼ �U�’0=2t and @2u=@y2 ¼ U’
00=nt
into (3.17) leads to the following ODE determining the function ’
’00 þ �
2’
0 ¼ 0 (3.31)
where ’ ¼ ’ð�Þ; ’0 ¼ d’=d�, and ’00 ¼ d2’=d�2, with � ¼ y=
ffiffiffiffint
p.
46 3 Application of the Pi-Theorem to Establish Self-Similarity
3.3 Laminar Boundary Layer over a Flat Plate(the Blasius Problem)
The previous example was related to flow development in fluid that has initially
been at rest and started moving being entrained by a plate. Below we consider an
application of the Pi-theorem for transformation of the boundary layers equations
into an ODE in the case of fluid flow about a motionless wall.
Consider a flow over a semi-infinite plate. The flow is assumed to be incom-
pressible and fluid velocity is considered to be uniform far away from the plate
surface (Fig. 3.2). (Blasius 1980)
The system of the governing equations in this case reads
u@u
@xþ v
@v
@y¼ n
@2u
@y2(3.32)
@u
@xþ @v
@y¼ 0 (3.33)
Equations (3.32) and (3.33) should be integrated subjected to the no-slip boundary
conditions at the plate surface, as well as and a given constant velocity of the stream
parallel to the plate is prescribed far away from the plate
y ¼ 0; u ¼ v ¼ 0; y ! 1; u ! U (3.34)
We use the Blasius problem to demonstrate the efficiency of using the ordinary
LMT and modified LxLyLzMT systems of units for dimensional analysis of
thermohydrodynamic problems. First of all, we consider the application of the
Pi-theorem to the Blasius problem using the LMTsystem of units. Equations
d (x)
y
x0
u
Fig. 3.2 The flow in the
boundary layer over a flat
plate
3.3 Laminar Boundary Layer over a Flat Plate (the Blasius Problem) 47
(3.32–3.33) and the boundary conditions (3.34) show that flow velocity within the
boundary layer over a flat plate depends on four dimensional parameters: two
independent variables x; y and two constants n and U: Therefore, we can write the
following functional equation for the longitudinal velocity component u
u ¼ f ðx; y; n;UÞ (3.35)
where the dimensions of u; x; y; n and U are expressed as
u½ � ¼ LT�1; x½ � ¼ L; y½ � ¼ L; n½ � ¼ L2T�1; U½ � ¼ LT�1 (3.36)
It is seen that of the four governing parameters, two parameters possess indepensdent
dimensions. That means that the difference n� k ¼ 2, so that the dimensionless
velocity is a function of two dimensionless groups. Choosing n and U as the
parameters with independent dimensions, we tramsform (3.35) to the dimensionless
form using the Pi-theorem. As a result, we arrive at the following equation
u ¼ ’ðx; �Þ (3.37)
where u ¼ u=U; x ¼ xU=n; � ¼ yU=n:Consider (3.37) from the point of view of generalization of the experimental data
for flows over flat plates, as well as the theoretical analysis of the corresponding
problem. Assume that an experimental data bank for the velocity at a number of
points within the boundary layer is available. According to (3.37), these data
determine a surface in the parametric space u� x� �: A section of this surface
by a plane x ¼ const determines the velocity distribution in given cross-section of
the boundaty layer. The totality of the velocity profiles corresponding to different
values of x determines the flow field within the boundary layer. It is obvious that
usefulness of such an approach for the generalization of the experimental data
would be low, since it requires many diagrams corresponding to different cross-
sections of the boundary layer, which makes it extremely laborious.
On the other hand, we apply now (3.37) for the theoretical analysis of the Blasius
problem. For this aim we rewrite (3.32) and (3.33) and the boundary conditions
(3.34) using the variables u; x and �. Taking into account that
u@u
@x¼ U3
n
� �u@u
@x; v
@u
@y¼ U3
n
� �v@u
@�; n
@2u
@y2¼ U3
n
� �@2u
@�2(3.38)
and
@u
@x¼ U2
n
� �@u
@x;@v
@y¼ U2
n
� �@v
@�(3.39)
we arrive at the equations
48 3 Application of the Pi-Theorem to Establish Self-Similarity
u@u
@xþ v
@u
@�¼ @2u
@�2(3.40)
@u
@xþ @v
@�¼ 0 (3.41)
Their solutions are subject to the following boundary conditions
� ¼ 0; u ¼ v ¼ 0; � ! 1; u ! 1 (3.42)
were v ¼ v=U.
Is easy to see that the transformation of (3.32) and (3.33) and the boundary
conditions (3.34) using the LMTsystem of units does not lead to any simplification
of the theoretical analysis of the Blasius problem. The latter still reduces to
integrating the system of the partial differential equations (3.41) and (3.42).
Accordingly, for the analysis of the planar boundary layer problems, it is conve-
nient to use the modified LMT system that includes two different scales of length Lx,Ly and Lz for the x, y and z directions, respectively, where the x axis is parallel to theplate in the flow direction, while the y and z axes are normal to it (cf. Fig. 3.2). It is
easy to show that the introduction of the two additional length scales does not affect
the dimension uniformity of the terms of the boundary layer and continuity
equations. Indeed, assuming that the dimensions of x½ � ¼ Lx and t½ � ¼ T; we find
that the corresponding dimension of the longitudinal velocity component is
u½ � ¼ LxT�1 (3.43)
Requiring that both terms of the continuity equation (3.30) possess the same
dimensions @u @x=½ � ¼ T�1; and @v @y=½ � ¼ T�1; we find the dimensions of v and y as
n½ � ¼ LyT�1; y½ � ¼ Ly (3.44)
Then the dimensions of the terms in the momentum equation (3.32) become
u@u
@x
¼ LxT
�2; v@u
@y
¼ LxT
�2; n@2u
@y2
¼ LxT
�2 (3.45)
where the dimension of kinematic viscosity n is L2yT�1.
First we estimate the thickness of the boundary layers d: It is clear that
d can be a function of a single independent variable x, as well as of the two
constants of the problem: the kinematic viscosity n and the free stream velocity
U½ � ¼ LxT�1
d ¼ fdðU; n; xÞ (3.46)
3.3 Laminar Boundary Layer over a Flat Plate (the Blasius Problem) 49
Since all the governing parameters in (3.46) possess independent dimensions, the
difference n� k ¼ 0 and the thickness of the boundary layer can be expressed as
d ¼ cna1xa2Ua3 (3.47)
where d½ � ¼ Ly; c is a dimensionless constant and the exponents a1; a2 and a3 areequal to 1=2; 1=2 and � 1=2; respectively.
As a result, we obtain
d ¼ c
ffiffiffiffiffinxU
r(3.48)
The velocity at any point in the boundary layer depends on the variables x½ � ¼Lx; y½ � ¼ Ly and constants n and U
u ¼ fuðU; n; x; yÞ (3.49)
Three governing parameters in (3.49) possess independent dimensions. There-
fore, in accordance with the Pi-theorem, (3.49) can be reduced to the form of (3.2)
with P ¼ u=U and P1 ¼ y=ffiffiffiffiffiffiffiffiffiffiffinx=U
p, i.e.
u
U¼ ’
0uð�Þ (3.50)
where ’0u ¼ d’u=d�, � ¼ y=
ffiffiffiffiffiffiffiffiffiffiffinx=U
p.
Equation (3.50) shows that the dimensionless velocity u ¼ u U= is determined by
a single variable �: That allows one to generalize the experimental data for the
velocity distribution in different cross-sections of the boundary layer over a flat
plate in the form of a single curve uð�Þ. Naturally such presentation of the results ofexperimental investigations has a significant advantage compared to the presenta-
tion of the experimental data in the form of a surface in the parametrical space u�x� � discussed before. The theoretical analysis of the Blasius problem is also
significantly simplified by using the LxLyLzMT system of units, since the problem is
reduced in this case to integrating an ordinary differential equation. Indeed, the
substitution of the expression (3.50) into (3.32) and (3.33) results in the following
ODE for the unknown function ’uð�Þ
2’000u þ ’
0u’
00u ¼ 0 (3.51)
with the boundary conditions
� ¼ 0; ’u ¼ 0 ’0u ¼ 0; � ! 1’
0u ¼ 1 (3.52)
The shear stress at the wall tw ¼ mð@u=@yÞ0 ¼ mffiffiffiffiffiffiffiffiffiffiffiffiffiU3=nx
p’
0 ð0Þ;where’
0 ð0Þ ¼ du=d�j0.It is emphasized that there is another way of transforming (3.32) and (3.33) into
the ODE. It is based on the assumption that velocity at any point of the boundary
layer is determined by three governing parameters, namely, the free stream velocity
50 3 Application of the Pi-Theorem to Establish Self-Similarity
U½ � ¼ LxT�1; the thickness of the boundary layer d½ � ¼ Ly and the distance from the
plate to a point under consideration y½ � ¼ Ly
u ¼ fuðU; y; dÞ (3.53)
Since two of the three governing parameters in (3.53) possess independent
dimensions, we obtain
u
U¼ ’
0u
y
d
�(3.54)
where the dependence dðxÞ is given by (3.48).
The instructive examples of the applications of the Pi-theorem for the analysis of
the Stokes and Blasius problems allow one to evaluate the true value of the LMTand LxLyLzMT systems of units. The comparison of the results produced by both
systems of units shows that the expansion of the system of units by introducing
different length scales in the x; y and z directions allows one to reduce the number
of the dimensionless groups and significantly simplifies generalization of the
experimental data and theoretical analysis of these problems. As a matter of fact,
the rationale for choosing a system of units (LMT or LxLyLzMTÞ should be based
on the comparison of the number of parameters with independent dimensions in the
set of the governing parameters determining the problem. Indeed, since the total
number of the governing parameters n does not depend on the system of units, the
number of the dimensionless groups in any given problem, n� k, is fully deter-
mined by the number of parameters with independent dimensions k. Therefore, thechoice of the LxLyLzMT system of units is desirable when
k�� > k� (3.55)
where subscripts � and � � correspond to the LMT and LxLyLzMT systems of units,
respectively.
Thus, the LMT system of units should be used when ðn� kÞ� equals zero or
unity. In the case when ðn� kÞ� > 1, it is preferable to use the LxLyLzMT system of
units. In future we will use both systems of units without an additional discussion
of the reasons for choosing a given system.
3.4 Laminar Submerged Jet Issuing from a Thin Pipe(the Landau Problem)
Let an incompressible fluid be issued from a thin pipe into an infinite space filled
with the same medium (with the same physical properties as those of the jet).
As a result of the laminar jet flow, mixing of the issuing and the ambient fluids
takes place
3.4 Laminar Submerged Jet Issuing from a Thin Pipe (the Landau Problem) 51
The flow is described by the Navier–Stokes and continuity equations
ðr � vÞv ¼ � 1
rrPþ nr2v (3.56)
r � v ¼ 0 (3.57)
where v is the velocity vector with the components vr; vy; and v’, and r; y; and ’are the spherical coordinates with the y axis (the polar axis) in the direction of the
jet and centered at its origin; P is the pressure. The sketch of this flow is shown in
Fig. 3.3 (Landau 1944).
Let us assume that there is no swirl and v’ ¼ 0: In addition, due to the assumed
axial symmetry of the flow about the polar axis (y ¼ 0Þ, the velocity components vrand vy are the function of only two variables: r and y: The velocity components also
depend on viscosity, as well as on the kinematic momentum flux J ¼ I r= (I is thetotal momentum flux in the jet which is determined by the pipe flow and is given).
Thus, we can write the functional equations for the velocity components vr and v’and pressure P in the following form
vr ¼ f1ðr; y; n; JÞ (3.58)
vy ¼ f2ðr; y; n; JÞ (3.59)
P ¼ f3ðr; y; n; JÞ (3.60)
where r; n, y and J have the following dimensions
r½ � ¼ L; n½ � ¼ L2T�1; y½ � ¼ 1; J½ � ¼ L4T�2 (3.61)
It is seen that two of the four dimensional parameters in (3.58–3.60) possess
independent dimensions (n� k ¼ 2Þ: In this case the Pi-theorem yields
Pi ¼ ’iðP1i;P2iÞ (3.62)
thin pipe
Fig. 3.3 Stream lines in flow
is induced laminar jet issuing
from a thin pipe
52 3 Application of the Pi-Theorem to Establish Self-Similarity
where Pi ¼ Ni=ra1ina2i ; P1:i ¼ y=ra
01ina
02i ; P2;i ¼ J=ra
001ina
002i ; N1 ¼ vr; N2 ¼ vy; and
N3 ¼ P=r; with i ¼ 1; 2; 3.Bearing in mind the dimensions of vr, vy; P; J and y; we find the values of the
exponents in (3.62)
a11 ¼ �1; a21 ¼ 1; a12 ¼ �1; a22 ¼ 1; a13 ¼ �2; a23 ¼ 2
a011 ¼ 0; a
021 ¼ 0; a
012 ¼ 0; a
022 ¼ 0; a
013 ¼ 0; a
023 ¼ 0
a0011 ¼ 0; a
0021 ¼ 0; a
0012 ¼ 0; a
0022 ¼ 2; a
0013 ¼ 0; a
0023 ¼ 0 (3.63)
Then, the dimensionless groups in (3.62) become
P1i ¼ y; P21 ¼ J
n2¼ const: (3.64)
for i ¼ 1; 2; 3; and
P1 ¼ vrv�
; P2 ¼ vyv�
; P3 ¼ P
rv2�(3.65)
where v� ¼ n=r:Accordingly, we obtain the following expressions for the velocity components
and pressure
vr ¼ nr’1ðyÞ; vy ¼
nr’2ðyÞ;
P
r¼ n2
r2’3ðyÞ (3.66)
Substituting the expressions (3.66) into (3.56) and (3.57), we arrive at the
following system of ODEs
’001 þ ’
01ðctgy� ’2Þ þ ’2
1 þ ’22 � 2’3 ¼ 0 (3.67)
’2’02 � ’
01 � ’
03 ¼ 0 (3.68)
’1 þ ’02 � ’2ctgy ¼ 0 (3.69)
Excluding ’3 from (3.67–3.69), we obtain the following system of ODEs for the
unknown functions ’1 and ’2
’0001 þ ð’0
1ctgyÞ0 þ ð’2’
01Þ
0 þ 2’1’01 þ 2’
01 ¼ 0 (3.70)
’1 þ ’01 � ’2ctgy ¼ 0 (3.71)
A solution of (3.70) and (3.71) corresponding to the issuing viscous fluid from a
thin pipe (a point wise source of momentum flux) was found by Landau (1944)
3.4 Laminar Submerged Jet Issuing from a Thin Pipe (the Landau Problem) 53
’1 ¼ �2þ 2ðA2 � 1ÞðAþ cos yÞ2 ; ’2
2 sin yAþ cos y
(3.72)
where A is a constant of integration which is related to the total momentum flux of
the jet J by the following expression
J ¼ 16pn2A 1þ A
3ðA2 � 1Þ �A
2lnAþ 1
A� 1
(3.73)
A detailed analysis of the flow in a submerged jet issued from a thin pipe
following the original work of Landau (1944). It can be also found in the
monographs of Landau and Lifshitz (1987), Sedov(1993), Vulis and Kashkarov
(1965).
3.5 Vorticity Diffusion in Viscous Fluid
Consider transformation of the PDE into an ODE in the problem which describes
the evolution of an initially infinitely thin vortex line of strength G. Assume that the
vortex line is normal to the flow plane (Fig. 3.4 a).
The vorticity transport equation reads (Batchelor 1967)
@O@t
¼ nr
@
@rr@O@r
� �(3.74)
where O is the vorticity component (the only one which is non-zero and normal to
the flow plane).
The unknown characteristics O½ � ¼ T�1 depends on two variables-time t½ � ¼ Tand the radial coordinate reckoned from the location of the initial vortex line
r
ϕ
rΩ
a b
t0
Fig. 3.4 Diffusion of vorticity in viscous fluid. (a) Stream lines. (b) The dependence of vorticityon time for different values of radial coordinate
54 3 Application of the Pi-Theorem to Establish Self-Similarity
r½ � ¼ L, as well as on two constants of the problem-the vortex strength G½ � ¼ L2T�1
and kinematic viscosity n½ � ¼ L2T�1. From the initial condition of the problem
GR ¼ G at t ¼ 0 (with GR being the circulation over a circle of radius r ¼ Rwhere Ris arbitrary) and the fact that (3.74) is linear, it follows that O is directly propor-
tional to G (Sedov 1993)
O ¼ Gf1ðn; r; tÞ (3.75)
Two from the three governing parameters in (3.75) have independent
dimensions. Therefore, the difference n� k ¼ 1: Then, in accordance with the
Pi-theorem (3.75) takes the form
P ¼ ’ðP1Þ (3.76)
where P ¼ O=Gna1 ta2 ; and P1 ¼ r=na01 ta
02 .
Bearing in mind the dimensions of O; G; n; r; and t, we find the values of
the exponents ai and a0i as : a1 ¼ �1; a2 ¼ �1; a
01 ¼ 1=2; and a
02 ¼ 1=2: Then,
(3.76) takes the form
O ¼ Gnt’
rffiffiffiffint
p� �
(3.77)
Substituting the expression (3.77) into (3.74), we arrive at the ODE
2ð�’0 Þ0 þ �ð2’þ �’0 Þ ¼ 0 (3.78)
with � ¼ r=ffiffiffiffint
p.
Its solution with the account for the initial condition yields the following well-
known vorticity distribution (cf. Sherman 1990)
O ¼ G4pnt
exp � r2
4nt
� �(3.79)
depicted in Fig. 3.4 b.
3.6 Laminar Flow near a Rotating Disk (the Von KarmanProblem)
The flow sketch is presented in Fig. 3.5 (Karman 1921). The velocity vector of
flow over a rotating disk has three projections u; v and w on the radial, azimuthal
and axial axes of the cylindrical coordinate system associated with the center of
the disk.
3.6 Laminar Flow near a Rotating Disk (the Von Karman Problem) 55
The system of the governing Navier–Stokes and continuity equations
corresponding to this flow takes the following form
u@u
@r� v2
rþ w
@u
@z¼ � 1
r@P
@rþ n
@2u
@r2þ @
@r
u
r
�þ @2u
@z
� �(3.80)
u@v
@rþ uv
rþ w
@v
@z¼ n
@2v
@r2þ @
@r
v
r
�þ @2v
@z2
� �(3.81)
u@w
@rþ w
@w
@z¼ � 1
r@P
@zþ n
@2w
@r2þ 1
r
@w
@rþ @2w
@z2
� �(3.82)
@u
@rþ u
rþ @w
@z¼ 0 (3.83)
The boundary conditions for (3.80–3.83) read
z ¼ 0; u ¼ 0 v ¼ rO w ¼ 0; z ¼ 1; u ¼ v ¼ 0 (3.84)
where it is assumed that the disk rotates with the angular velocity O.Assume that velocity components or pressure at any point of a thin liquid layer
over a rotating disk depend on some characteristic velocity (or pressure), the axial
distance from the disk z and the layer thickness d. Then, the functional equations forthe velocity components and pressure can be written as
u ¼ f1ðu�; z; dÞ (3.85)
v ¼ f2ðv�; z; dÞ (3.86)
z
Ω
w
u
v
0
ϕ
P
r
Fig.3.5 Flow over rotating
disk in liquid at rest
56 3 Application of the Pi-Theorem to Establish Self-Similarity
w ¼ f3ðw�; z; dÞ (3.87)
P ¼ f4ðP�; z; dÞ (3.88)
where the velocity component and pressure scales for a given radial position r are
denoted with the asterisks.
The dimensions of the governing parameters in (3.80–3.83) are
u�½ � ¼ LT�1; v�½ � ¼ LT�1; w�½ � ¼ LT�1; z½ � ¼ L; d½ � ¼ L; P½ � ¼ L�1MT�2 (3.89)
It is seen that two of the three governing parameters on the right hand side in
(3.85–3.88) possess independent dimensions. Accordingly, these equations can be
presented in the form
u
u�¼ ’1
z
d
�(3.90)
v
v�¼ ’2
z
d
�(3.91)
w
w�¼ ’3
z
d
�(3.92)
P
P�¼ ’4
z
d
�(3.93)
It is easy to show that the thickness of the fluid layer carried by the disk d is of
the order offfiffiffiffiffiffiffiffin=O
p. Then, taking as the characteristic scales of u; v; w and P as
u� ¼ rO; v� ¼ rO; w� ¼ffiffiffiffiffiffiffinO;
pP� ¼ rnO (3.94)
we arrive at the following expressions
u ¼ rO’1ð�Þ (3.95)
v ¼ rO’2ð�Þ (3.96)
w ¼ffiffiffiffiffiffinO
p’3ð�Þ (3.97)
P ¼ rnO’4ð�Þ (3.98)
where � ¼ ffiffiffiffiffiffiffiffin=O
p.
Using the expressions (3.95–3.98), we transform (3.80–3.83) into the following
ODEs
3.6 Laminar Flow near a Rotating Disk (the Von Karman Problem) 57
2’1 þ ’3 ¼ 0 (3.99)
’21 þ ’
01’3 � ’2
2 � ’001 ¼ 0 (3.100)
2’1’2 þ ’3’02 � ’
002 ¼ 0 (3.101)
’4 þ ’3’03 � ’
003 ¼ 0 (3.102)
where differentiation by � is denoted by prime.
The boundary conditions for (3.99–3.102) become
� ¼ 0; ’1 ¼ 0 ’2 ¼ 1 ’3 ¼ 0 ’4 ¼ 0; � ! 1; ’1 ¼ 0 ’2 ¼ 0 (3.103)
Note that above approach dealing with the flow over an infinite disk can also be
used for the evaluation of flow characteristics in the case of a finite radius disk if the
latter is much larger than the thickness of the liquid layer adjacent to the disk
surface (Schlichting 1979).
3.7 Capillary Waves after a Weak Impact of a Tiny Object ontoa Thin Liquid Film (the Yarin-Weiss Problem)
The flow in a planar thin liquid film on a solid surface after an impact of a tiny wire
(similarly to the axisymmetric case shown in Fig. 3.6) is governed by the following
system of PDEs (the beam equations; Yarin and Weiss 1995)
@2w@t2
¼ a2@4w@x4
(3.104)
@2v
@t2¼ a2
@4v
@x4(3.105)
where w ¼ Dh=h0 is the small dimensionless perturbation of the liquid layer
thickness, with h0 and h being the unperturbed and perturbed thicknesses,
Dh ¼ h� h0; v is the liquid velocity in the x-direction (along the surface),
a ¼ sh0=rð Þ1=2, where s is the surface tension and r the density, t is time.
Equations (3.104) and (3.105) correspond to the situations where gravity and
viscous effects are negligible and perturbations of the liquid layer thickness and
flow velocity are sufficiently small. All these assumptions are realized after impacts
of tiny wire as in Fig. 3.6. Moreover, these objects should be assumed to be point
wise. Then, a given length scale disappears from the problem, and there should exist
a self-similar solution, which we are searching for below.
58 3 Application of the Pi-Theorem to Establish Self-Similarity
These equations show that w and v depends on two variables x and t and one
constant a. Therefore, the functional equations for w and x read
w ¼ f1ða; x; tÞ (3.106)
v ¼ f2ða; x; tÞ (3.107)
The dimensions of w; v; a; x and t are
w½ � ¼ 1; n½ � ¼ L2T�1; a½ � ¼ L2T�1; x½ � ¼ L; t½ � ¼ T (3.108)
It is seen that (3.106) and (3.107) contain three governing parameters, whereas
two of them have independent dimensions. In accordance with the Pi-theorem,
(3.106) and (3.107) can be presented in the following dimensionless form
Pw ¼ ’wðP1wÞ (3.109)
Pv ¼ ’vðP1vÞ (3.110)
where Pw ¼ w=aa1 ta2 ; P1w ¼ x=aa01 ta
02 ; Pv ¼ v=aa1� ta2� ; and P1v ¼ x=aa
01� ta
02� .
The exponents ai; ai�; a0i and a
0i� found by applying the principle of the dimen-
sional homogeneity are equal to
a1 ¼ 0; a2 ¼ 0; a01 ¼
1
2; a
02 ¼
1
2; a1� ¼ 1
2; a2� ¼ � 1
2; a
01� ¼
1
2; (3.111)
Fig. 3.6 A system of concentric waves propagating over a thin liquid film on a solid surface from
the impact point of a thin stick seen at the center of the image Reprinted from Yarin and Weiss
(1995) with permission
3.7 Capillary Waves after a Weak Impact of a Tiny Object 59
Accordingly, (3.106) and (3.107) take the form
w ¼ ’wð�Þ (3.112)
v ¼ffiffiffia
t
r� ’vð�Þ (3.113)
where � ¼ x=ffiffiffiffiat
p:
Substituting the expressions (3.112) and (3.113) into (3.104) and (3.105) yields
the following ODEs for the functions ’wð�Þ and ’vð�Þ
’IVw þ 1
4�2’
00w þ
3
4�’
0w ¼ 0 (3.114)
’IVv þ 1
4�2’
00v þ
5
4�’
0v þ
3
4’v ¼ 0 (3.115)
Similarly, in the axisymmetric case corresponding to a weak impact of a tiny
droplet or a stick (Fig. 3.6) the equation for the surface perturbation w
@2w@t2
þ a2
r
@
@rr@3w@r3
� �(3.116)
with r being the radial coordinate can be transformed to the following ODE
’IVw þ 1
�’
000w þ �2
4’
00w þ
3
4�’
0w ¼ 0 (3.117)
where � ¼ r=ffiffiffiffiat
p.
It is emphasized that the solutions corresponding to the self-similar capillary
waves generated by impacts of poitwise objects in reality correspond to remote
asymptotics of capillary waves generated by weak impacts of small but finite
objects.
3.8 Propagation of Viscous-Gravity Currents over a SolidHorizontal Surface (the Huppert Problem)
Gravity currents belong to a wide class of flows in which one fluid with density r1 isintruding into another fluid with a different density r2. Such flows are characteristicof many natural phenomena and various engineering processes (Hoult 1972;
Simpson 1982). Below we consider one type of gravity currents, namely viscous-
gravity currents over a rigid surface (Fig. 3.7). (Huppert, 1982)
60 3 Application of the Pi-Theorem to Establish Self-Similarity
Let a layer of a denser fluid of density r invades into a thicker layer of another
fluid of a lower density r� Dr under the action of gravity. Assume that the volume
of the denser fluid increases as ta; where t is time and a½ � ¼ 1 is a constant. The
system of the governing equations that describes such flow in lubrication approxi-
mation reads (Huppert, 1982)
@h
@t� 1
3
g0
n@
@xh3
@h
@x
� �¼ 0 (3.118)
ðxN0
hdx ¼ qta (3.119)
where h is the thickness of the invading fluid layer, q is a constant, xN is the distance
from x ¼ 0 to the leading edge of the invading fluid layer, g0 ¼ Dr=rð Þg, and n is
the kinematic viscosity of the invading denser fluid (cf. Fig. 3.7).
The parameters that are involved in the problem formulation, (3.118) and
(3.119) have the following dimensions
h L½ �; t½ � ¼ T; g0
h i¼ LT�2; n½ � ¼ L2T�1; x½ � ¼ L; xN½ � ¼ L; q½ �
¼ L2T�a; a½ � ¼ 1 (3.120)
It is possible to reduce the number of parameters involved in (3.118) and (3.119)
by introducing new generalized parameters:
h� � ¼ h
q
¼ L�1Ta; A½ � ¼ 1
3
g0q3
n
¼ L5T�ð3aþ1Þ (3.121)
Then, (3.118) and (3.119) take the following form
z
P = P0
h(x, t)
xN(t) x0
r – Δr, na
r, νFig. 3.7 Scheme of viscous
gravity current over a solid
horizontal surface
3.8 Propagation of Viscous-Gravity Currents over a Solid Horizontal Surface 61
@h
@t� A
@
@xh3 @h
@x
� �¼ 0 (3.122)
ðxN0
hdx ¼ ta (3.123)
The position of the leading edge of the gravity-driven current xN depends on one
independent variable t and a generalized parameter A. Therefore, the functional
equation for xN has the form
xN ¼ f x;Að Þ (3.124)
The governing parameters in (3.124) have independent dimensions. In accor-
dance with the Pi-theorem, (3.124) can be reduced to the form
xN ¼ cAa1 ta2 (3.125)
where c½ � ¼ 1 is a constant, and the exponents a1 and a2 are equal to: a1 ¼ 1=5;and a2 ¼ 3aþ 1ð Þ=5, respectively.
Accordingly, the coordinate of the leading edge xN can be expressed as
xN ¼ cA1=5tð3aþ1Þ=5 (3.126)
The thickness h of the gravity-driven current is determined by two independent
variable x and t, as well as by the position of the leading edge of the denser layer
xN [the latter involves the constants A and a, as per (3.126)]. Accordingly, the
functional equation for h reads
h ¼ f ðx; xN; tÞ (3.127)
Two from the three governing parameters involved in (3.127) possess indepen-
dent dimensions. Applying the Pi-theorem to (3.127) we arrive at the following
dimensionless equation
P ¼ ’ðP1Þ (3.128)
where P ¼ h=xb1N tb2 ; and P1 ¼ x=xN; the exponents b1 and b2 are equal: b1 ¼ �1;and b2 ¼ a, respectively.
Bearing in mind the values of the exponents b1 and b2, as well as the expression(3.126), we rewrite (3.128) as
h ¼ x�1N ta’ð�Þ (3.129)
62 3 Application of the Pi-Theorem to Establish Self-Similarity
where � ¼ x=xN ¼ c0xA�1=5t�3ðaþ1Þ=5; and c
0 ¼ 1=c.Substitution of the expression (3.126) into (3.129) yields
h ¼ c0A�1=5tð2a�1Þ=5’ð�Þ (3.130)
Calculating the derivatives @h=@t and @h=@x, we transform PDE (3.122) into the
following ODE
c3c0�
�0
�þ 1
53aþ 1ð Þ�c0
� � 2a� 1ð Þcn o
¼ 0 (3.131)
where c ¼ c0� �5=3
’.Equation (3.23) takes the form
c0
�5=3 ð10
cð�Þ ¼ 1 (3.132)
Equations (3.131) and (3.132) manifest the fact that self-similar solutions of the
nonlinear partial differential equations (3.118) and (3.119) do exist. The solutions
of the ODEs corresponding to the plane and axisymmetric problems were found by
Huppert (1982).
Theoretical predictions were compared with the experimental data for the
axisymmetric spreading of silicon oil puddles into air for the release rates corres-
ponding to a ¼ 0 and a ¼ 1 in (2.219). Comparisons were also done between the
results of the theoretical analysis and data for the axisymmetric spreading of
salt water into sweet water in the experiments of Didden and Maxworthy (1982)
and Britter (1979). A good agreement of the theoretical predictions with the
experimental data was demonstrated.
3.9 Thermal Boundary Layer over a Flat Wall (the PohlhausenProblem)
Consider the thermal field over a hot or cold semi-infinite flat wall subjected to
a parallel uniform flow of an incompressible fluid of a different temperature T1far from the wall (Pohlhausen 1921). We assume that the difference between the
fluid and plate temperatures is sufficiently small, as well as neglect dissipation
kinetic energy. We also neglect dependence of the physical properties of the fluid
(the kinematic viscosity and thermal diffusivity) on temperature. In this case the
system of the governing equations reads
3.9 Thermal Boundary Layer over a Flat Wall (the Pohlhausen Problem) 63
u@u
@xþ v
@u
@y¼ n
@2u
@y2(3.133)
@u
@xþ @v
@y¼ 0 (3.134)
u@DT@x
þ v@DT@y
¼ a@2DT@y2
(3.135)
where DT ¼ T � T1:The boundary conditions for (3.133–3.134) are as follows
y ¼ 0; u ¼ v ¼ 0 DT ¼ DTw; y ! 1; u ! U DT ¼ 0 (3.136)
if the wall temperature Tw is given. Another type of the thermal boundary condition
at the plate might be that of thermal insulation. Then, the thermal boundary
condition at the wall in (3.136) is replaced by @DT=@y ¼ 0 at y ¼ 0:Under the assumptions made, the dynamic and thermal problems are uncoupled.
Then, the flow field is described by the self-similar Blasius solution of (3.133) and
(3.134) (see Sect. 3.3 and Schlichting 1979)
u ¼ U’0; v ¼ 1
2
ffiffiffiffiffiffinUx
rð�’0 � ’Þ (3.137)
where ’ ¼ ’ð�Þ is the function determined by (3.51), � ¼ yffiffiffiffiffiffiffiffiffiffiffiU=nx
p; and prime
denotes differentiation by �:The temperature at any point of the thermal boundary layer depends on the
temperature difference DTw ¼ Tw � T1; flow velocity, kinematic viscosity and
thermal diffusivity of fluid, as well as on the location
DT ¼ FðDTw;U; x; y; n; aÞ (3.138)
The dimensions of the governing parameters in (3.126) are
DT½ � ¼ y; U½ � ¼ LxT�1; x½ � ¼ Lx; y½ � ¼ Ly; n½ � ¼ L2yT
�1; a½ � ¼ L2yT�1 (3.139)
Since four of the six governing parameters in (3.138) have independent
dimensions, it can be reduced to the following dimensionless equation
P ¼ #ðP1;P2Þ (3.140)
where P ¼ DT=DTa1w Ua2xa3na4 ; P1 ¼ y=DT
a01
w Ua02xa
03na
04 ; and P2 ¼
a=DTa001
w Ua002 xa
003 na
004 .
64 3 Application of the Pi-Theorem to Establish Self-Similarity
Bearing in mind the dimensions of DT; DTw; U; x; y; n and a, we find the
exponents ai a0i and a
000 as
a1 ¼ 1; a2 ¼ 0; a3 ¼ 0; a4 ¼ 0
a01 ¼ 0; a
02 ¼ � 1
2; a
03 ¼
1
2; a
04 ¼
1
2
a001 ¼ 0; a
002 ¼ 0; a
003 ¼ 0; a
004 ¼ 1 (3.141)
Accordingly, (3.140) takes the form
DTDTw
¼ #ð�; PrÞ (3.142)
where Pr ¼ n=a is the Prandtl number.
Substituting the expression (3.142) into (3.135), we arrive at the following ODE
#00 þ 1
2Pr’#
0 ¼ 0 (3.143)
The boundary conditions for (3.143) read
# ¼ 1at � ¼ 0; # ¼ 0 at � ! 1 (3.144)
Then, the solution (3.143) is found in the following form
#ð�; PrÞ ¼
Ð1�
’00 ðxÞ� �Pr
dx
Ð10
’00 ðxÞ½ �Prdx(3.145)
where ’ð�Þ is determined Eq (3.51).
At Pr ¼1
#ð�Þ ¼ 1� ’0 ð�Þ ¼ 1� u
u1(3.146)
i.e. the dimensionless excess temperature and velocity fields coincide.
3.10 Diffusion Boundary Layer over a Flat Reactive Plate(the Levich Problem)
Consider distribution of liquid (or gaseous) reactant in the boundary layer over a flat
reactive plate (Levich, 1962). Assume that the rate of an exothermal hetorogenous
reaction at the plate surface exceeds significantly the diffusion flux toward the
3.10 Diffusion Boundary Layer over a Flat Reactive Plate (the Levich Problem) 65
surface, and also neglect the influence of the heat release due on the flow field.
Then, the field of the reactant concentration is described by the following problem
u@c
@xþ v
@c
@y¼ D
@2c
@c2(3.147)
y ¼ 0 c ¼ 0; y ! 1 c ! c0 (3.148)
where the velocity components u and v are determined by the Blasius solution (Sect.
3.3), and c0 is the concentration of the reagent in the undisturbed flow, and D is the
diffusion coefficient.
The governing parameters that determined the concentration field at any point
of the boundary layer are: the concentration c0, the velocity of the undisturbed
flow U, the kinematic viscosity of the liquid or gaseous carrier and diffusity n andD, respectively, as well as the coordinates of the point of interest x; and y: Accord-ingly, the functional equation for the reactant concentration c reads
c ¼ f ðc0;U; x; y; n;DÞ (3.149)
The dimensions of the governing parameters are as follows
c0½ � ¼ L�3M; U½ � ¼ LT�1; x½ � ¼ L; y½ � ¼ L; n½ � ¼ L2T�1; D½ �¼ L2T�1 (3.150)
Since four of the six governing parameters have independent dimensions, (3.49)
takes the following dimensionless form
P ¼ cðP1;P2Þ (3.151)
where P ¼ c=ca10 Ua2xa3na4 ; P1 ¼ y=c
a01
0 Ua02xa
03na
04 ; and P2 ¼ D=c
a001
0 Ua002 xa
003 na
004 :
Taking into account the principle of dimensional homogeneity, we find the
following values of the exponents ai; a0i and a
00i
a1 ¼ 1; a2 ¼ 0; a3 ¼ 0; a4 ¼ 0
a01 ¼ 0; a
02 ¼ � 1
2; a
03 ¼
1
2; a
04 ¼
1
2
a001 ¼ 0; a
002 ¼ 0; a
003 ¼ 0; a
004 ¼ 1
(3.152)
Then, (3.151) takes the following form
c
c0¼ cð�; ScÞ (3.153)
66 3 Application of the Pi-Theorem to Establish Self-Similarity
where � ¼ yffiffiffiffiffiffiffiffiffiffiffiU=nx
p; and Sc ¼ n=D is the Schmidt number.
Substituting the expression (3.153) into (3.47), we arrive at the following ODE
c00 þ 1
2Sc � c0 ¼ 0 (3.154)
The solution of (3.154) with the corresponding boundary conditions following
from (3.148) has the form
cð�; ScÞ ¼
Ð1�
’00ðxÞ½ �ScdxÐ10
’00 ðxÞ½ �Scdx(3.155)
where ’00 ð�Þis determined by Eq. (1.40).
The diffusion flux of the reactant at the wall is found as
j ¼ �D@c
@y
� �0
¼ Dc0 f ðScÞffiffiffiffiffiU
nx
r(3.156)
where the function f ðScÞ equals to: 0.332Sc1/3 for 0.6<Sc<10, 0:564Sc1=2 for
Sc ! 0, and 0:339Sc1=3 for Sc ! 1(Schlichting 1979).
Problems
P.3.1. Show that flow of an incompressible fluid in a cone does not correspond to
aself-similar solution.
The steady-state flow in a cone is described by the Navier–Stokes and continuity
equations
ðr � vÞv ¼ � 1
rrPþ nr2v (P.3.1)
r � v ¼ 0 (P.3.2)
where v is the velocity vector with the radial and two azimuthal components vr; vy;and v’, respectively, in the spherical coordinate system centered at the cone tip.
The flow in the cone is axially symmetric relative to the axis y ¼ 0: Let the swirlis absent, so that v’ ¼ 0. Under these conditions the velocity components vr and vydepend on two given constants of the problem, namely, the kinematic viscosity of
the fluid ½n� ¼ L2T�1 and the volumetric flow rate ½Q� ¼ L3T�1; as well as on two
Problems 67
coordinates r L½ � and y 1½ �: Then, the functional equation for the velocity components
vr and vy are
vr ¼ frðn;Q; r; yÞ (P.3.3)
vy ¼ fyðn;Q; r; yÞ (P.3.4)
From the three dimensional parameters n; Q and r one dimensionless group can
be constructed, namely, R ¼ r=ðQ=nÞ: Accordingly, (P.3.3) and (P.3.4) take the thefollowing form
vr ¼ ’rðR; yÞ (P.3.5)
vy ¼ ’yðR; yÞ (P.3.6)
where vr ¼ vr Q=n2ð Þ; and vy ¼ vy= Q=n2ð Þ.Thus, vr and vy are the functions of two dimensionless variables R and y that
does not allow to reduce PDEs (P.3.1) and (P.3.2) to ODEs.
P.3.2. Find self-similarity for the flow in the boundary layer of an incompress-
ible fluid near a solid wall in the case of a power-law velocity distribution far from
the plate (Flakner and Skan 1931).
In this case the velocity of the undisturbed flow far from the solid wall is given
by the power law U ¼ cxm; where c is a dimensional and m is a dimensionless
constants, respectively. The boundary layer and continuity equations read
u@u
@xþ v
@u
@y¼ n
@2u
@y2� U
dU
dx(P.3.7)
@u
@xþ @v
@y¼ 0 (P.3.8)
The boundary conditions are as following
y ¼ 0; u ¼ v ¼ 0; y ! 1; u ¼ UðxÞ ¼ cxm (P.3.9)
The governing parameters of the problem are: the dimensional constants c½ � ¼L1�mx T�1; and n½ � ¼ L2yT
�1 and the independent variables x½ � ¼ Lx and y½ � ¼ Ly:
Accordingly, the functional equation for u is
u ¼ fuðc; n; x; yÞ (P.3.10)
It is seen that three governing parameters have independent dimensions, so that
n-k ¼ 1. Then the dimensionless form of (P.3.10) is
68 3 Application of the Pi-Theorem to Establish Self-Similarity
P ¼ ’0 ðP1Þ (P.3.11)
where P ¼ u=ca1na2xa3 ; P1 ¼ y=ca01na
02xa
03 ; ’ ¼ ’ðP1Þ and ’
0 ¼ du=dP1.
Taking into account the dimensions of u; c; n; x and y; we find that
a1 ¼ 1; a2 ¼ 0; a3 ¼ m; a01 ¼ � 1
2; a
02 ¼
1
2; a
03 ¼
1� m
2(P.3.12)
Accordingly, (P.3.11) takes the form
u
U¼ ’
0y
ffiffiffiffiffiffiffiffiffiffiffifficxm�1
n
r !(P.3.13)
Substituting the expression (P.3.13) into (P.3.7) and (P.3.8), we obtain the follow-
ing ODE
’000 þ mþ 1
2’’
00 ¼ mð’02 � 1Þ (P.3.14)
where prime denotes differentiation by � ¼ yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficxm�1=n
p:
References
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Batchelor GK (1967) An introduction to fluid dynamics. Cambridge University Press, Cambridge
Britter RE (1979) The spread of a negatively plum in calm environment. Atmos Environ
13:1241–1247
Blasius H (1908) Grenzschichten in Flussigkeiten mit kleiner Reibung. Z Math Phys 56:1–37
Didden N, Maxworthy T (1982) The viscous spreading of plane and axisymmetric gravity
currents. J Fluid Mech 121:27–42
Douglas JF (1969) An introduction to dimensional anlysis for engineers. Pitman, London
Flakner VM, Skan SW (1931) Some approximate solutions of the boundary layer equiation. Phil
Mag 12:865–896
Hamel G (1917) Spiralformige Bewegungen zahen Flussigkeiten. Jahr-Ber Dtsch Math Ver
25:34–60
Hoult DP (1972) Oil spreading on the sea. Ann Rev Fluid Mech 4:341–368
Huntley HE (1967) Dimensional analysis. Dover Publications, New York
Huppert HE (1982) The propagation of two–dimensional and axisymmetric viscous- gravity
currents over a rigid horizontal surface. J Fluid Mech 121:43–58
Karman Th (1921) Uber laminare and turbulente Reibung. ZAMM 1:233–252
Kays WM, Crowford ME (1993) Convective heat and mass transfer, 3rd edn. McGraw-Hill, New
York
Landau LD (1944) New exact solution of the navie-stokes equation. DAN SSSR 44:311–314
Landau LD, Lifshitz EM (1987) Fluid mechnics, 2nd edn. Pergamon, Oxford
Levich VG (1962) Physicochemical hydrodynamics. Prentice Hall, Englewood Cliffs
Loitsyanskii LG (1966) Mechanics of liquid and gases. Pergamon, Oxford
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Pohlhausen E (1921) Der Warmeaustaush zwichen festen Korpern and Flussigkeiten mit kleiner
Reibung and kleiner Warmeleitung. ZAMM 1:115
Rosenhead L (ed) (1963) Laminar boundary layers. Clarendon, Oxford
Sedov LI (1993) Similarity and dimensional methods in mechanics, 10th edn. CRC, Boca Raton
Schlichting H (1979) Boundary layer theory. McGraw-Hill, New York
Sherman FS (1990) Viscous flow. McGraw-Hill, New York
Simpson JE (1982) Gravity currents in the laboratory, atmosphere, and ocean. Ann Rev Fluid
Mech 14:213–234
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Vulis LA, Kashkarov VP (1965) Theory of viscous liquid jets. Nauka, Moscow (in Russian)
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70 3 Application of the Pi-Theorem to Establish Self-Similarity
Chapter 4
Drag Force Acting on a Body Movingin Viscous Fluid
4.1 Introductory Remarks
Drag force is one of the most important factors that determine dynamics of solid
bodies moving in viscous fluids. The knowledge of this force is essential for a
number of applications in engineering, in particular, for the evaluation the engine
power to ensure a desirable velocity of the airplanes, ships, etc., as well as for
analyzing the behavior of solid particles, droplets and bubbles in two-phase flows.
Numerous experimental and theoretical investigations dealing with drag of bodies
of different shapes moving with low and high velocities in viscous fluid were
performed during the last three centuries. The results of these researches are
generalized in monographs by Landau and Lifshitz (1987), Batchelor (1967),
Happel and Brenner (1983), and Soo (1990). The detailed data on drag of bubbles,
droplets and solid particles can be found in the monograph by Clift et al. (1978).
The effect of heating, evaporation and combustion on drag force of small particles
is discussed in the monograph by Yarin and Hetsroni (2004). The application of the
dimensional analysis in studies of drag force of bodies moving in viscous fluid are
discussed in Sedov (1993). The readers are referred to the above-mentioned works,
whereas we discuss briefly some principles that are essential for problems dealing
with drag force acting on bodies moving in viscous fluid and the related
applications of the Pi-theorem.
First we pose the problem on a steady-state linear translation of a solid body of a
given shape with velocity v1 in an infinite uniform incompressible fluid. This
problem is equivalent to the problem on the solid body subjected to a uniform
fluid flow with the undisturbed and constant velocity v1. The velocity and pressure
fields in such flow are determined by the Navier-Stokes and continuity equations
rðv � rÞv ¼ �rPþ mr2v (4.1)
r � v ¼ 0 (4.2)
L.P. Yarin, The Pi-Theorem, Experimental Fluid Mechanics,
DOI 10.1007/978-3-642-19565-5_4, # Springer-Verlag Berlin Heidelberg 2012
71
subjected to the no-slip conditions at the body surface y�ðxÞ:v ¼ 0 at y ¼ y�ðxÞ (4.3)
v ¼ v1 at y ! 1 (4.4)
At low Reynolds number (the creeping flow with the Reynolds number
Re � 1) the convective terms in the momentum (4.1) can be omitted (the Stokes
approximation) and the system of (4.1) and (4.2) takes the following form
DP� mr2 v ¼ 0 (4.5)
r � v ¼ 0 (4.6)
Equations (4.1) and (4.2) and the conditions (4.3) and (4.4) show that the
situation under consideration is determined by the following four dimensional
parameters r; m; v1; and d�, which determine the velocity and pressure fields in
fluid, as well as the drag force.1 Then, the functional equation for the drag force
acting on a solid body moving in an unbounded fluid acquires the following form
Fd ¼ f ðr; m; v1; d�Þ (4.7)
where Fd½ � ¼ LMT�2 is the drag force.
The set of the governing parameters corresponding to solid body motion in this
case is
r½ � ¼ L�3M; m½ � ¼ L�1MT�1; v1½ � ¼ LT�1; d�½ � ¼ L (4.8)
Taking into account that three governing parameters in (4.8) have independent
dimensions, we can transform the functional equation for the drag force (4.7) into
the following dimensionless form
P� ¼ ’�ðP1Þ (4.9)
where P� ¼ Fd=ðrv21d2�Þ and P1 ¼ v1d�=n is the Reynolds number.
On the other hand, when a solid body moves in liquid near a gas-liquid interface,
it excites perturbations of the interface. The perturbed interface is wavy either due
to the effect of gravity or/and surface tension. In this case the set of the governing
parameters also includes gravity acceleration g or/and surface tension s. In this casethe set of the parameters determining the drag force reads
r½ � ¼ L�3M; m½ � ¼ L�1MT�1; v1½ � ¼ LT�1; d�½ � ¼ L; g½ �¼ LT�2; or=and, s½ � ¼ MT�2 (4.10)
1Here d� is characteristic size of the body which is fully determined by its shape.
72 4 Drag Force Acting on a Body Moving in Viscous Fluid
Then, the drag coefficient in the case of a body moving in liquid near a gas-liquid
interface becomes
P� ¼ ’1�ðP1;P2Þ; or=and P� ¼ ’2�ðP1;P2;P3Þ (4.11)
where P2 ¼ v1=ffiffiffiffiffiffiffigd�
pand P3 ¼ m=
ffiffiffiffiffiffiffiffiffiffirsd�
pare the Froude and Ohnesorge
numbers, respectively.
In the general case, the set of the governing parameters can also include a
number of parameters that account for roughness of the solid body surface, turbu-
lence of the free stream, etc. In accordance with that the dimensionless form of
the functional equation for the drag force (the functional equation for the drag
coefficient) takes the following form
P� ¼ ’ðn�1Þ�ðP1;P2; :::PnÞ (4.12)
where ’ðn�1Þ� accounts for the effect of the nth factor.It is emphasized that the unknown function ’i� in the expression for the drag
coefficient can be presented as ’i� ¼ a’i; where a is a dimensionless constant. The
numerical value of this constant is chosen in accordance with an accepted protocol
of processing of experimental data. As a rule, the value of the coefficient a is
assumed to be equal to 1=2. Then, (4.9), (4.11) and (4.12) determine the
dependences of the drag coefficient cd ¼ P� 1 2=ð Þ= ¼ Fd= ð1=2Þrv21d2�� �
on the
dimensionless groups P1;P2 � � �Pn characteristic of the considered problem.
For a spherical body moving with small velocity ( the creeping flow Stokes
approximation) when the drag force is expressed as Fd ¼ 3pmv1d�, it is assumed
that a ¼ p 8= : That leads to the generally accepted expression for the drag coeffi-
cient in the form cd ¼ P� p 8=ð Þ= ¼ 24 Re= where cd is the drag coefficient, Re ¼v1d�=n is the Reynolds number, v1 is the velocity of the undisturbed fluid (or a
particle moving in fluid at rest), and n is the kinematic viscosity.
4.2 Drag Action on a Flat Plate
4.2.1 Motion with Constant Speed
Consider viscous drag of a thin flat plate subjected to a parallel uniform fluid stream
(Fig. 4.1).
At sufficiently high values of the Reynolds number (Re>> 1), the boundary layers
are formed over both sides of the plate. The thicknesses of the boundary layers
increase downstream (cf. Fig. 4.1). The drag force that act on the plate is determined as
Fd ¼ 2b
ðl0
twdx (4.13)
4.2 Drag Action on a Flat Plate 73
where Fd is the drag force, tw is the shear stress at the plate surface, and b and l arethe width and length of the plate.
Bearing in mind the character of flow over the plate, we can assume that drag
force is determined by density and viscosity of the fluid, the undisturbed flow
velocity u1, as well as the length and width of the plate. Then, we can present
(4.13) as follows
Fd ¼ 2bf ðr; m; u1; lÞ (4.14)
where Fd½ � ¼ LMT�2; r½ � ¼ L�3M; m½ � ¼ L�1MT�1; u1½ � ¼ LT�1; l½ � ¼ L.Applying the Pi-theorem to (4.14), we arrive at the following expression for the
drag coefficient
cd ¼ ’ðReÞ (4.15)
where cd ¼ Fd=ru122bl, u1 is the velocity of the undisturbed fluid is the drag
coefficient, and Re ¼ u1l=n is the Reynolds number.
In order to reveal an explicit form of the dependence cdðReÞ, we use the
expression for the shear stress at the plate surface that was found in Chap. 3 by
via the dimensional analysis of laminar flow over a plate
tw ¼ m
ffiffiffiffiffiffiu31nx
r’
00 ð0Þ (4.16)
where ’ð�Þ is determined by solving the Blasius equation (3.51), and
� ¼ y=ffiffiffiffiffiffiffiffiffiffiffiffiffinx=u1
pis the dimensionless variable.
Substitution of the expression (4.16) into (4.13) yields
Fd ¼ 4b’00 ð0Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffimrlu31
q(4.17)
y
0x
δ(x)
Plate
u∞Fig. 4.1 A thin flat plate
subjected to a uniform
parallel flow
74 4 Drag Force Acting on a Body Moving in Viscous Fluid
and
cd ¼ 4’00 ð0ÞffiffiffiffiffiffiRe
p (4.18)
where ’00 ð0Þ is constant.
4.2.2 Oscillatory Motion of a Plate Parallel to Itself
The flow in the vicinity of an oscillating plate is determined by the unsteady boundary
layer equations. Transient and oscillatory flows in the boundary layers are discussed in
the monographs of Schlichting (1979) and Loitsyanskii (1967). In the framework
of the dimensional analysis the problem on a drag Fd experienced by an oscillating
plate involves choosing a set of the governing parameters and subsequent transforma-
tion of the functional equation for Fd to a dimensionless form using the Pi-theorem.
The set of the governing parameters in this case includes the parameters responsible
for the physical properties of the fluid (its density and viscosity r and mÞ; sizes of
the plate (l and bÞ, as well as such flow characteristics as its period t½ � ¼ T(or frequency) of the oscillations and the maximum velocity um that plays the role
of the velocity scale. Then, the functional equation for Fd takes the form
Fd ¼ 2bf1ðr; m; um; l; tÞ (4.19)
It is seen that the present problem contains five governing parameters, three of
which have independent dimensions. Therefore, according to the Pi-theorem (4.19)
can be reduced to the following form
P ¼ ’ðP1;P2Þ (4.20)
where P ¼ Fd=2bra1ua2m la3 ; P1 ¼ m=ra
01u
a02
m la03 ; and P2 ¼ t=ra
001 u
a002
m la003 .
Determining the values of the exponents ai; a0i and a
00i with the help of the
principle of dimensional homogeneity, we find that a1 ¼ 1; a2 ¼ 2; a3 ¼ 1;
a01 ¼ 1; a
02 ¼ 1; a
03 ¼ 1; a
001 ¼ 0; a
002 ¼ �1; and a
003 ¼ 1. Then, we arrive at the
following expression for the drag coefficient
cd ¼ ’1ðRe;KsÞ (4.21)
where cd ¼ Fd 2bru2ml�
is the drag coefficient, Re ¼ uml=n is the Reynolds number,
Ks ¼ St�1; where St is the Strouhal number determined by the maximum velocity
and length of the plate, while Ks ¼ Ks�b; with Ks� ¼ tum=b being the Keulegan-
Carpenter number, and b ¼ b=l. Shih and Buchanan (1971) studied experimentally
the dependence cd ¼ ’1ðRe;KsÞ. It was shown that the drag coefficient of an
4.2 Drag Action on a Flat Plate 75
oscillating plate decreases as the Reynolds number increases. An increase in Ks�also leads to decreasing cd. For the engineering applications the following empirical
correlation is useful
cd ¼ 15ðKsÞ exp 1:88
Re0:547�
� �(4.22)
where Re� ¼ umb=n.The forces acting on cylinders in viscous oscillatory flow are also determined
by Ks� at low values of the Keulegan-Carpenter numbers (Graham 1980; Bearman
et al 1985).
4.3 Drag Force Acting on Solid Particles
4.3.1 Drag Experienced by a Spherical Particle at Low, Moderateand High Reynolds Numbers
In Sect. 4.1 we discussed briefly the application of the Pi-theorem for evaluating
drag force experienced by a solid body moving in viscous fluid. In the present
section we consider this problem in more detail, in particular, dealing with the drag
force acting on a spherical particle at low, moderate and high Reynolds numbers.
The drag of a spherical particle moving in viscous fluid represents the total force
exerted by the surrounding fluid on the particle surface. This force depends on the
physical properties of the fluid, as well as on particle size and its velocity
fd ¼ f ðr; m; d; uÞ (4.23)
where fd is the drag force, d is the particle diameter, and uis the particle velocity
relative to fluid at infinity. The drag force fd and the governing parameters
r; m; d; and u have the following dimensions
fd½ � ¼ LMT�2; r½ � ¼ L�3M; m½ � ¼ L�1MT�1; d½ � ¼ L; u½ � ¼ LT�1
(4.24)
It is seen that three from the four governing parameters have independent
dimensions. Then, in accordance with the Pi-theorem we transform (4.23) into
the following dimensionless form
P� ¼ ’�ðP1Þ (4.25)
where P� ¼ fd=ra1ua2da3 and P1 ¼ m=ra01ua
02da
03 , and the exponents ai and a
0i are
determined from the principle of dimensional homogeneity. They are found as
76 4 Drag Force Acting on a Body Moving in Viscous Fluid
follows: a1 ¼ 1; a2 ¼ 2; a3 ¼ 2; a01 ¼ 1; a
02 ¼ 1; and a
03 ¼ 1. Then, we obtain
thatP� ¼ fd=ru2d2; and P1 ¼ m=rud ¼ Re�1 and (4.25) takes the following form
cd ¼ ’ðReÞ (4.26)
where cd ¼ P� p 8=ð Þ= and Re are the drag coefficient and the Reynolds number,
respectively.
The explicit forms of the dependence (4.26) can be found in the framework of
the dimensional analysis for two limiting cases corresponding to very small and
very large Reynolds number (see Problems P.4.1 and P.4.2).
Equation 4.26 indicates that the drag coefficient of a spherical particle depends
on a single dimensionless group, namely, the Reynolds number. In order to deter-
mine an exact form of the dependence cdðReÞ, it is necessary to either solve the
hydrodynamic problem on flow of viscous fluid about the particle, or to study it
experimentally. The structure of such flow determines the normal and shear stresses
at the particle surface, i.e. the total drag force.
The flow about a spherical particle that moves rectilinearly with a constant
velocity in fluid is described by the system of the Navier-Stokes and continuity
equations
rðv � rÞv ¼ �rPþ mr2v (4.27)
r � v ¼ 0 (4.28)
which are subjected to the following boundary conditions
v ¼ 0; r ¼ R; v1 ¼ �u; r ¼ 1 (4.29)
where r and m are the density and viscosity of the fluid, R is the particle radius, v isfluid velocity relative the spherical coordinate system associated with the center of
the moving particle, u is the absolute particle velocity, r is the radial coordinate, P is
the pressure; the boldface symbols represent vector quantities.
The inertial term rðv � rÞv on the left-hand side of (4.27) is negligible at low
Reynolds numbers. The problem is thus can be simplified significantly and reduced
to the integration of the linear Stokes equations
rP� mr2v ¼ 0 (4.30)
r � v ¼ 0 (4.31)
subjected to the boundary conditions (4.29).
The solution of (4.30) and (4.31) results in the following Stokes expression for
the drag force
fd ¼ ff þ fp ¼ 3pmud (4.32)
4.3 Drag Force Acting on Solid Particles 77
where ff ¼ � Ðp0
tRy sin y � 2pa2 sin ydy ¼ 2pmud, fd ¼ � Ðp0
P cos y � 2pa2 sin ydy ¼pmud are the contributions to the total drag force from the viscous friction (the shear
stresses) and pressure, respectively, tRy; is the shear stress at the surface, P is the
pressure at the surface, a is the sphere radius, R and y are the radial and angular
coordinates in the spherical coordinate system (Stokes 1851)).
The drag coefficient for a spherical particle becomes accordingly
cd ¼ 24
Re(4.33)
The Stokes’ law (4.32) and (4.33) is valid only for low Reynolds numbers Re �0:1 (cf. Fig. 4.2). The deviation of the predicted values of cd from the experimental
data for the drag coefficient does not exceed 2% at Re � 0.24 and 20% at
Re � 0.75. The experimental data show that the dependence of cd ¼ cdðReÞ has arather complicated shape when a wider range of the Reynolds number values is
considered (Fig. 4.2). In the range of 1 < Re < 800 the drag coefficient is accu-
rately expressed by the empirical Schiller and Naumann law (Clift et al. 1978)
cD ¼ ð24=ReÞ 1þ 0:15Re0:687�
(4.34)
Significant deviations from Stokes’ law are related to the growth of the so-called
form drag component of the drag force at higher Reynolds numbers. It is associated
Fig. 4.2 Drag coefficient of a spherical particle: the solid line – the dependence of the drag
coefficient on the Reynolds number, the dotted line – the Stokes’ law
78 4 Drag Force Acting on a Body Moving in Viscous Fluid
with the development of the boundary layer near the particle surface and its
separation at the rear part. The later results in a stagnation zone behind the particle
and a reduced pressure at the rear compared to the full dynamic pressure acting at
the front part of the particle.
In range of the Reynolds number 750 � Re � 3 � 105 the drag coefficient is closeto a constant value of 0.445 (the Newton law). At higher Re, the drag coefficient
reveals a dimple at about Re � 2 � 105. The latter is a result of the change in the flowstructure, when transition to turbulence happens in the boundary layer at the sphere
surface, which leads to the flow reattachment to the surface and diminishes the form
drag.
4.3.2 The Effect of Rotation
Particle rotation is a cause of lift force fl, which is directed normally to the plane
formed by the particle velocity and angular velocity vectors v and o, respectively.The magnitude of this force, which is the cause of the Magnus effect, depends on
the physical properties of the fluid and diameter of a spherical particle, as well as on
its velocity (relative to the fluid) u ¼ vj j and the magnitude of the angular velocity
o. Therefore, the functional equation for the lift force reads
fl ¼ f ðr; m; u; d;oÞ (4.35)
The problem at hand involves five governing parameters, three of them with
independent dimensions. Then, in accordance with the Pi-theorem, we find that the
lift force coefficient cl ¼ 4f= ru2pd2=2ð Þ is given by the following expression
cl ¼ ’ðRe; gÞ (4.36)
where g ¼ od=2u is the dimensionless angular velocity.
A lift force also acts at a spherical particle moving in a simple shear flow
characterized by velocity gradient du=dy (Saffman 1965, 1968). In this case the
functional equation for the Saffman lift force flS is
flS ¼ f r; m; u; d;du
dy
�(4.37)
Applying the Pi� theorem, we arrive at the following dimensionless expression
for the Saffman lift force coefficient clS normalized as cl before
clS ¼ ’ðRe; gÞ (4.38)
where g ¼ ðdu=dyÞd2=n.
4.3 Drag Force Acting on Solid Particles 79
The important results regarding the Saffman lift force were obtained by Dandy
and Dwyer (1990), McLaughlin (1991), Anton (1987) and Mei (1992). In particu-
lar, Dandy and Dwyer (1990) showed that at a fixed shear rate the lift and drag
coefficients for a spherical particle, normalized using the uniform flow velocity are
approximately constant over the range 40 � Re � 100. On the other hand, the drag
and lift coefficients cd and cl increase sharply as the Reynolds number decreases in
the range Re<10.
4.3.3 The Effect of Acceleration
Transient motions of particles result in additional forces imposed on them by the
surrounding fluid. These forces are related to the acceleration of the surrounding
fluid (the added mass force), as well as to the viscous effects due to delay in flow
development as the velocity changes with time (the Basset force). Transient
motions of spherical particles in an incompressible fluid at rest were studied by
Boussinesq (1903), Oseen (1910, 1927) and Basset (1961). Recently a number of
modified model equations describing transient particle motions in steady-state or
weakly fluctuating flows were proposed (Maxey and Riley 1983; Berlemont et al.
1990; Mei et al. 1991; Mei 1994; Chang and Maxey 1994, 1995). These
modifications were mostly dealing with the history term in the drag force. In
particular, they accounted for the effect of the initial velocity difference between
the fluid and a particle. At the same time, some other approaches to determine the
unsteady drag force are based on the empirical correlations constructed in accor-
dance with the Pi-theorem (Odar and Hamilton 1964; Karanfilian and Kotas 1978).
They account for the influence of particle acceleration du/dt on drag coefficient.
Accordingly, assuming that the transient drag force depends not only on particle
velocity but also on its acceleration, we present the functional equation for fd in thefollowing form
fd ¼ f r; m; u; d;du
dt
�(4.39)
It is easy to see that three from the five governing parameters involved have
independent dimensions. Then, in accordance with the Pi-theorem, we obtain the
drag coefficient based on the previous non-dimensionalization in the following
dimensionless form
cd ¼ ’ Re;Acð Þ (4.40)
where Ac ¼ u2=½ðdu=dtÞd� is the acceleration parameter.
An explicit expression for cd Re;Acð Þ reads (Karanfilian and Kotas 1978)
80 4 Drag Force Acting on a Body Moving in Viscous Fluid
cd ¼ cd Reð Þ 1þ Ac�1� �n
(4.41)
where n ¼ 1:2� 0:03. The expression (4.41) gives a fairly good agreement with
experiments in the range 102 � Re � 104 and 0 � Ac<1:05.The forces exerted on an “infinite” cylindrical particle in an oscillatory flow
normal to its axis were studied by Kenlegan and Carpenter (1958), Graham (1980)
and Bearmen et al. (1985). There are three factors that determine the drag force: (1)
the inertia of the accelerating outer flow, (2) the influence of the viscous boundary
layers, and (3) separation of these boundary layers leading to vortex shedding. The
drag coefficient of such particles is determined by two dimensionless groups: the
Reynolds and Kenlegan-Carpenter numbers.
4.3.4 The Effect of the Free Stream Turbulence
There is a number of additional factors which affect particle drag. Turbulence of
free stream impinging on a particle is among them. In order to account for the effect
of turbulence intensity on particle drag, it is necessary to extend the set of the
governing parameters by including it in the set. For example, in the case when a
particle moves in a turbulent flow, the functional equation for the drag force fd reads
fd ¼ f ðr; m; u; u0; lÞ (4.42)
where u0 ¼
ffiffiffiffiffiffiu02
pis the root-mean square of turbulent fluctuations of the carrier
fluid, l is the turbulence length scale (it is implied that u0and l are some characteris-
tic values of the turbulent fluctuations and of the turbulence length scale).
Then, the dimensional analysis yields the following equation for the drag
coefficient
cd ¼ ’ðRe; Tu; lÞ (4.43)
where Tu ¼ u0=u is the dimensionless turbulence intensity, and l ¼ l=d is the
dimensionless turbulence scale.
At a given l , the effect of u0on the drag coefficient depends on the Reynolds
number. At sufficiently high Re, close to transition to turbulence in the particle
boundary layer, an increase in the turbulence intensity of the free stream is
accompanied by a decrease in cd. This is a result of a shift of the boundary layer
separation point towards the rear stagnation point. In the range of relatively low
Reynolds numbers, the drag coefficient slightly increase with u0. This effect is due
to the intensified viscous dissipation. The effect of the turbulence scale on the drag
coefficient depends on l . At l<<1 the effect of the turbulence scale is negligible,
whereas at l>1, the drag coefficient increases with l.
4.3 Drag Force Acting on Solid Particles 81
4.3.5 The Influence of the Particle-Fluid Temperature Difference
A difference between the particle and surrounding temperature can also affect the
drag force. For the flow of viscous incompressible fluid the functional equation for
the drag force fd in this case reads
fd ¼ f r1; m1; d; u1; T1; TPð Þ (4.44)
where subscripts P and 1 refer to the particle and the ambient parameters.
Applying the Pi-theorem to transform (4.44) to dimensionless form, we arrive at
the following equation
cd ¼ ’ðRe;cÞ (4.45)
where cd ¼ 4fd=p 1 2=ð Þr1u21d2 is the drag coefficient, and c ¼ TP=T1 is the
temperature ratio.
It is emphasized that when transforming (4.44) to the dimensionless form, we do
not account for the effect of temperature on the physical properties of the fluid. At
large enough values of the temperature difference TP � T1, a significant variation
of density and viscosity of fluid within the boundary layer can take place. In this
case the situation becomes complicated. In accordance with that, a set of the
governing parameters should includes thermal conductivity and heat capacity.
The effect of variation of the physical properties of the fluid within the boundary
layer on the drag force was studied by Fendell et al. (1966), Kassoy et al. 1966) and
Dwyer (1989). In particular, Kassoy et al. (1966) evaluated the drag coefficient of a
spherical particle moving in a high-velocity flow of a perfect gas. Assuming a linear
dependence of the viscosity and thermal conductivity on temperature, as well as
constant specific heat and Prandtl number, they derived the following relation for
the drag coefficient
cd ¼ 24
Re
16C
3K� K
3
�(4.46)
where K ¼ OðOþ 2Þ and C is tabulated function of the parameter O ¼ ðc� 1Þ.According to (4.46), the particle drag coefficient increases almost linearly with
O. When O is of the order of one, the drag coefficient at low Reynolds numbers
increases up to 70% over the isothermal value corresponding to the Stokes law.
4.4 Drag of Irregular Particles
Consider the factors which determine drag force of an irregular body moving in an
unbounded viscous incompressible fluid with a constant velocity. From the physical
point of view it is clear that the drag force of such a body should depend on the fluid
82 4 Drag Force Acting on a Body Moving in Viscous Fluid
properties, the body center-of-mass velocity and orientation relatively to the veloc-
ity vector, as well as the body geometry. The latter (at fixed configuration of an
irregular body) can be determined by one characteristic length, for example, the
length of the wing chord (Fig. 4.3).
The pressure of the undisturbed fluid P1 does not affect the flow field in the
incompressible fluid. Indeed, in this case instead of the total pressure P it is
always possible to consider the difference P� P1: Thus, the value of P1 is
immaterial. It can be excluded from the set of the governing parameters of the
problem (Sedov 1993). Therefore, we can write the functional equation for the
drag force as follows
fd ¼ f ðr; m; u; d; aÞ (4.47)
where d is the characteristic length, and a is the angle of inclination (cf. Fig. 4.3).
By using the Pi-theorem, (4.47) can be transformed to into the following
dimensionless form
cd ¼ ’ðRe; aÞ (4.48)
where cd ¼ 4fd= 1 2=ð Þru2d2 is the drag coefficient.
The Reynolds number characterizes the ratio of the inertia and viscous forces.
The contribution of the viscous forces to the drag force decreases as the viscosity mdecreases, the Reynolds number grows. At large values of Re, the dominant role is
played by fluid inertia. In such cases it is possible to neglect the effect of viscosity
and to reduce the number of the governing parameters to four. Then we obtain
fd ¼ rd2u2’ðaÞ (4.49)
Equation 4.49 shows that in an ideal (inviscid) fluid the drag force is propor-
tional to the velocity squared.
Note that the effect of particle shape becomes important only at sufficiently high
Reynolds numbers when a vortex zone forms behind the particle. In creeping flow
this effect is less expressed. For example, for a thin disk oriented normally to the
flow, the factor in the Stokes’ law equals 8, whereas for a spherical particle it equals
3 p (Lamb 1959). In the general case, the drag coefficient of irregular particles cd
y
0x
α
d
Fig. 4.3 Orientation of an
irregular body in a uniform
flow directed along the x-axis
4.4 Drag of Irregular Particles 83
depends on two dimensionless groups: the Reynolds number and the shape factor
(Boothroyd 1971)
cd ¼ cdðRe;fÞ (4.50)
where f ¼ Seq=S is the shape factor, Seq is the surface area of a volume-equivalent
sphere, and S is the actual surface area. The surface area and diameter of the
volume-equivalent sphere are defined as Seq ¼ p1=3ð6VÞ2=3 and deq ¼ 6V=pð Þ1=3;respectively, where V is the particle volume. In (4.50) the Reynolds number is
based on the equivalent diameter deq.The dependence cdðRe;fÞ is presented in Fig. 4.4 as a family of curves cdðReÞ
corresponding to different values of f Boothroyd (1971).
It is seen that the drag coefficient of irregular particles is larger than the one for a
volume-equivalent spherical particle. The difference increases significantly with
the Reynolds number.
4.5 Drag of Deformable Particles
In distinction from rigid particles, the drag of drops and bubbles depends not only
on the outside velocity distribution but also on their deformation and the inside
flow. The particle deformation is controlled by the competition between the surface
tension and the hydrodynamic forces arising as a result of the particle-fluid interac-
tion. For sufficiently small Weber numbers (We ¼ ru2d=s; where s is the surface
tension), particles remain spherical for any finite Reynolds number. At Re 1 the
functional equation for the drag force of a droplet (a bubble) with viscosity m2moving in a fluid with viscosity m1 has the following form
fd ¼ f ðm1; m2; d; uÞ (4.51)
Fig. 4.4 The drag coefficient
of an irregular particle
84 4 Drag Force Acting on a Body Moving in Viscous Fluid
Three governing parameters from the four in (4.51) have independent
dimensions. Therefore, (4.51) can be reduced to
P ¼ ’ðP1Þ (4.52)
where P ¼ fd=ma11 d
a2ua3 ; and P1 ¼ m2=ma01
1 da02ua
03 .
Using the principle of the dimensional homogeneity, we find the values of the
exponents in (4.52) as a1 ¼ a2 ¼ a3 ¼ 1; and a01 ¼ 1; a
02 ¼ a
03 ¼ 0. Then, we
arrive at the following equation for the drag force for a small deformable particle
(a drop or a bubble
fd ¼ m1du’ðm21Þ (4.53)
where m21 ¼ m2=m1.Equation 4.53 can be transformed to the following form for the drag coefficient
cd ¼ ’ðm21ÞRe
(4.54)
The function ’ðm21Þ is strongly dependent on the fluid circulation inside a drop,
which is determined by the ratio of fluid viscosities inside and outside it m21. An
increase in the value of m21 is accompanied by flow weakening inside the drop. In
the limiting case of an infinitely large m21, the flow of the inner fluid completely
stops and the drag coefficient approaches to the drag coefficient of a rigid spherical
particle. Therefore, ’ð1Þ ¼ 24. In the second limiting case corresponding to a very
small viscosity of the inner fluid (m21 � 0), the drag coefficient approaches to the
drag coefficient of a bubble and thus ’ð0Þ ! const:An explicit form of (4.54) can be found by integrating the following set of the
momentum balance (in the creeping flow, inertialess approximation) and continuity
equations
�rPþ mðiÞr2vðiÞ ¼ 0 (4.55)
r � vðiÞ ¼ 0; (4.56)
where i ¼ 1 and 2 for the outer and inner fluids, respectively.
The solution of (4.55) and (4.56) is subject to the following boundary conditions:
(1) a given uniform flow at infinity, (2) finite velocity everywhere inside the
particle. Also, at the drop interface with the surrounding fluid: (3) continuity of
tangential stress components, and a jump of the normal stresses due to surface
tension; (4) continuity of the velocity components of the inner and outer liquids. It
is also assumed that the drop keeps its spherical shape during its motion in an
immiscible fluid with different physical properties. This is possible when the drop
diameter is sufficiently small for the surface tension force being dominant com-
pared to the hydrodynamic tractions tending to distort the droplet shape.
The problem (4.55) and (4.56) was solved for a perfectly spherical drop by
Hadamard (1911) and Rybczynski (1911). They derived the following relation for
the drag force acting on a drop moving in uniform fluid
4.5 Drag of Deformable Particles 85
fd ¼ 3pm1u1d2m1 þ 3m23ðm1 þ m2Þ
(4.57)
Accordingly, the drag coefficient becomes
cd ¼ 16
Re
1þ 3m21=21þ m21
(4.58)
Equation 4.58 acquires the simplest forms corresponding to the drag coefficient
for a rigid particle as m2i ! 1 and cd ¼ 24=Re. Another important limit
corresponds to bubbles when m21 ! 0 and cd ¼ 16=Re. Note that the drag on a
spherical particle given by (4.57) and (4.58) does not depend on the density ratio of
the inner and outer fluids, because the creeping flow approximation was employed.
4.6 Drag of Bodies Partially Submerged in Liquid
Consider the drag of a spherical body moving with a constant velocity u along an
air-water interface. Let the density and viscosity of water and air be r1; r2 and
m1; m2, respectively, and the vertical size of the underwater part be l. The latter is
determined by the body weight G ¼ r1gD, where g and D are the gravity accelera-
tion and body displacement, respectively. Thus the set of the governing parameters
in this case includes seven dimensional parameters r1; m1; r2; m2; D; g, and u.The contribution to the drag force from the air-body interaction, as a rule, is
negligible in comparison with the effect of water-body interaction. This makes
the effect of the air density and viscosity r2; and m2 immaterial. Therefore, the
number of the governing parameters can be reduced to only five and, correspond-
ingly, the functional equation for the total drag force acting on a partially
submerged body moving along the air-water interface can be written as
fb ¼ f ðr; m; g;D; uÞ (4.59)
Here and hereinafter subscripts 1 for r and m are omitted. Three from the five
governing parameters in (4.59) have independent dimensions. Then, according to
the Pi-theorem the dimensionless form of (4.59) reads P ¼ ’ðP1;P2Þ, which is
equivalent to
fb ¼ ’ðFr;ReÞ (4.60)
where fb ¼ fb=L3=2mg1=2 is the dimensionless drag force, Fr ¼ u=
ffiffiffiffiffiffigL
pand Re ¼
uL=n are the Froude and Reynolds numbers, and L ¼ D1=3 is the characteristic size
of the body.
86 4 Drag Force Acting on a Body Moving in Viscous Fluid
Equation 4.60 shows that the drag force of a partially submerged body is
determined by the dimensionless groups with different dependences on the charac-
teristic scale L. Namely, the Reynolds is proportional to L, whereas the Froude
number is inverse to L. This circumstance does not allow modeling of drag force
acting on a partially submerged body (for example, a ship) when using the same
liquid in laboratory experiments as in reality (Sedov 1993).
It is emphasized that drag acting on a solid body partially submerged in viscous
fluid is determined by two different phenomena. (1) Namely, by viscous friction
resulting from body-water interaction, and (2) generation of gravitational waves on
the account of the kinetic energy of motion. Accordingly, the total drag force can be
presented approximately as a sum of the friction ff and wave fw components, as
fb ¼ ff þ fw (4.61)
Introducing the corresponding drag coefficients as
cf ¼ ffð1=2Þru2S ; cw ¼ fw
rgL(4.62)
where S is the cross-sectional area of the body, one can present (4.61) as follows
fb ¼ cfru2S2
þ cwrgL (4.63)
where cf ¼ cf ðReÞ; and cw ¼ cwðFrÞ.A detailed consideration of the applications of the dimensional analysis to
motion of partially submerged solid bodies (in particular, ships) can be found in
the monograph by Sedov (1993).
4.7 Terminal Velocity of Small Spherical Particles Settling inViscous Liquid (the Stokes Problem for a Sphere)
The problem on a small heavy spherical particle settling in viscous liquid was
solved by Stokes (1851). Later on, Bridgman (1922) studied this problem in the
framework of the dimensional analysis. He considered the steady-state stage of
settling (which follows the initial transient stage) when the equilibrium between the
weight, buoyancy and viscous drag has already been established. Under such
conditions particles settle steadily with the so-called terminal velocity v, which is
determined by the particle and liquid densities, particle radius, liquid viscosity and
gravity acceleration g. Then, the functional equation for the terminal velocity reads
v ¼ f ðd; r1; r2; m; gÞ (4.64)
4.7 Terminal Velocity of Small Spherical Particles Settling 87
where r is the particle radius, r1 and r2 are the particle and liquid density,
respectively, and m is the liquid viscosity.
The terminal velocity v and the governing parameters d; r1; r2; m and g have
the following dimensions in the LMT System
v½ � ¼ LT�1; d½ � ¼ L; r1½ � ¼ L�3M; r2½ � ¼ L�1M; m½ � ¼ L�1MT�1;
g½ � ¼ LT�2(4.65)
It is seen that three of the five governing parameters in (4.64) have independent
dimensions, so that the difference n� k ¼ 2: Then, in accordance with the
Pi-theorem, (4.64) reduces to the following expression
P ¼ ’ðP1;P2Þ (4.66)
where P ¼ v= m=r1dð Þ; P1 ¼ r1=mð Þ2gd3; and P2 ¼ r2=r1.Regarding (4.66) Bridgman noticed that “we evidently can say nothing about the
effect on the velocity of any of the elements taken themselves, since they all occur
under the arbitrary functional symbol”. In order to further simplify the analysis,
Bridgman uses the system of units, which includes units of the length, mass, time
and force. At small Reynolds numbers when the effect of liquid inertia is negligible,
the force is treated as its own compensating dimensional constant. In this case
the dimensions of viscosity and gravity acceleration are expressed as it follows
from their definitions: m½ � ¼ F=S du dy=ð Þ0 ¼ FL�2T and g½ � ¼ F m=½ � ¼ FM�1,
where, m is the particle mass, S is its surface area, and du=dyð Þ0 is the velocity
gradient at the surface.
Then, the set of the governing parameters includes five parameters of which four
parameters have independent dimensions, i.e. the difference n� k ¼ 1: Accord-ingly, (4.64) takes the form
P ¼ ’ P1ð Þ (4.67)
where P ¼ v= r2r1g=mð Þ; and P1 ¼ r2=r1.Huntley (1967) obtained (4.67) by using the LxLyLzMT System of Units for the
dimensional analysis the present problem. If the z axis is in the direction of gravity,it is possible to say that the viscous drag depends on the particle cross-section
normal to the direction of its motion, i.e. on the scales Lx and Ly: Since, the diameter
with the dimension Lx has an equal status with the diameter with the dimension
Ly(owing to the axial symmetry of the problem), one can define the characteristic
particle size d½ � as L1=2x L1=2y . Bearing in mind that viscosity is defined as the ratio of
the shear stress to the corresponding velocity gradient, it is possible to determine the
expressions for the “directional” viscosities mx and my, the x and y directions,
respectively,
mx½ � ¼ LxL�1y L�1
z MT�1; my� � ¼ L�1
x LyL�1z MT�1 (4.68)
88 4 Drag Force Acting on a Body Moving in Viscous Fluid
Taking into account the symmetry of the problem, we write the dimension of
viscosity as m½ � ¼ m1=2x m1=2y
h i¼ L�1
z MT�1� �
. Thus, the dimensions of the governing
parameters are
r1½ � ¼ L�1x L�1
y L�1z M; d½ � ¼ L
12xL
12y; r2½ � ¼ L�1
x L�1y L�1
z M; m½ � ¼ L�1z MT�1 g½ �
¼ LxT�2 (4.69)
Four of the five governing parameters in (4.69) have independent dimensions, so
that the difference n� k ¼ 1: In this case, in accordance with the Pi-theorem, (4.64)
reduces to the following one
P ¼ ’ðP1Þ (4.70)
where P ¼ v=ra11 da2ma3ga4 and P1 ¼ r2=r
a01
1 da02ma
03ga
04 .
The exponents ai and a0i are equal to: a1¼ 1; a2¼ 2; a3 ¼�1; a4¼ 1;
a01 ¼ 1; a
02 ¼ a
03 ¼ a
04¼ 0: Accordingly, (4.70) takes the form of (4.67)
Equation 4.67 determines the dimensionless settling velocity as a function of
only one dimensionless parameter, namely, the ratio of the liquid density to the
particle density. However, the explicit form of this dependence is unknown, which
is a drawback inherent in both (4.66) and (4.67). It should be also emphasized that
the previous consideration is related to a particular case of very-low-Reynolds
number particles when r1dv1=dt ¼ 0; and r2 @v2i=@tþ v2i@v2i=@xkð Þ ¼ 0: In
fact, in this case the influence of the inertial forces is completely neglected.
Obviously, in this case densities of the settling particle and the surrounding liquid
do not influence the process and have to be excluded from the set of the governing
parameters.
The above-mentioned drawbacks can be removed by a correct choice of the set
of the governing parameters beginning from the analysis of the actual physical
factors determining settling of particles in viscous liquid. The latter means that
particle and liquid densities should be accounted for even when the inertial effects
are expected to be small. In particular, settling of a particle in viscous fluid is due to
the excess of particle weight compared to its buoyancy force. Then, the force
responsible for the particle settling is the net weight-buoyancy force
Fnet ¼ 4
3pðr1 � r2Þgr3 (4.71)
Equation (4.71) shows that the difference of the particle and fluid densities
r1 � r2ð Þ, as a factor in the product r1 � r2ð Þg (but not separately r1 and r2 or
their ratio) is one of the governing parameters that affect the driving force and thus,
particle settling. In the set of the governing parameters one should also include fluid
viscosity and particle radius as the factors which the drag and driving forces.
Therefore, the functional equation for the settling velocity v takes the following
form
4.7 Terminal Velocity of Small Spherical Particles Settling 89
v ¼ f ðr; m; gDrÞ (4.72)
where Dr ¼ r1 � r2, and all the governing parameters of the problem have
independent dimensions in the LMT System of Units: r½ � ¼ L; m½ � ¼L�1MT�1; and gDr½ � ¼ L�2MT�2.
Applying the Pi-theorem to (4.72), we obtain
v ¼ cra1ma2ðgDrÞa3 (4.73)
where c is a dimensionless constant.
Finding the values of the exponents using the principle of dimensional homoge-
neity, as a1 ¼ 2; a2 ¼ �1; and a3 ¼ 1, we arrive at the following expression
v ¼ cr2gðr1 � r2Þ
m(4.74)
It agrees (up to a dimensionless factor c) with the exact analytical solution of theproblem.
In conclusion, we mention an additional simple approach to determine terminal
velocity from the equilibrium of forces acting on a particle settling in fluid (Lamb
1959). The dimensional analysis shows that the drag force acting on a spherical
particle at Re < 1 can be expressed as Fd ¼ cmrv (with c being a dimensionless
constant). The driving force Fnet is determined by (4.71). The equality of Fd and
Fnet results in (4.74).
4.8 Sedimentation
4.8.1 Dimensionless Groups
Sedimentation of granular materials in fluid flows exerts considerable influence on
various physical processes in nature and engineering. In the present Section we
consider in accordance with Yalin (1972) only one aspect of sedimentation,
namely the application of the Pi-theorem in the sedimentation context. We begin
with the parameters that significantly affect flows carrying heavy granules. Namely,
the density and viscosity of the carrier fluid, the density of granules and their sizes,
the average flow depth h and its slope j ( j ¼ sin a; with a being the slope angle), aswell as gravity acceleration. Correspondingly, the set of the governing parameters
that determine characteristics of uniform two-phase stationary flows carrying
cohesionless granular material consists of the following list
r; m; rs; d; h; j; g (4.75)
90 4 Drag Force Acting on a Body Moving in Viscous Fluid
where subscript s corresponds to granular material.
Sedimentation inevitably involves the compound parameter involved in weight
and buoyancy forces (cf. Sect. 4.7), gs ¼ gðrs � rÞ, as well as a characteristic
velocity v� ¼ ðgjhÞ1=2, which also accounts for the flow inclination. Then, the set
of the governing parameters takes the following form
r; m; rs; d; h; v�; gs (4.76)
In this case, the functional equation for a flow characteristics A reads
A ¼ f ðr; m; rs; d; h; v�; gsÞ (4.77)
where A is, for example, the average velocity of the fluid, or the rate of the granule
transport, etc.
The dimensions of the governing parameters in (4.77) are as follows
r½ � ¼ L�3M; m½ � ¼ L�1MT�1; rs½ � ¼ L¼3M; d½ � ¼ L; h½ � ¼ L; gs½ �¼ L�2MT�2 (4.78)
Three of the seven governing parameters have independent dimensions, so that
the difference n� k ¼ 4: In accordance with that, (4.77) can be reduced to the
following dimensionless form
P ¼ ’ðP1;P2;P3;P4Þ (4.79)
where P¼A=ra1va2� da3 ;P1¼dv�=n;P2¼rv2�=gsd;P3¼h=d; and P4¼rs=r, and
the exponents ai are determined for any given A by using the principle of dimen-
sional homogeneity.
The first dimensionless groupP1 represents itself the grain-size-based Reynolds
number, which expresses the ratio of the inertia and viscous forces acting on an
individual grain. The second group P2 characterizes the ratio of the hydrodynamic
and buoyancy forces acting on a grain. This group is referred to as the mobility
number and it plays an important role in studies grain motion. The limiting case
P2 ! 0(gs ! 1Þ corresponds to a rough rigid bed. The dimensionless groups P3
and P4 express the influence of the flow depth and the grain material density on
sedimentation, respectively.
4.8.2 Terminal Velocity of Heavy Grains
In the general case the sedimentation velocity of a heavy grain depends on
the physical properties of the carrier fluid (its density and viscosity r and m,respectively), the grain density rs, gravity acceleration g, as well as on time t:
4.8 Sedimentation 91
As it was established earlier, sedimentation is characterized rather by the product
gs ¼ gðrs � rÞ than by g. In this case the functional equation for the terminal
velocity of sedimentation vt takes the form
vt ¼ f ðr; m; rs; gs; tÞ (4.80)
At sufficiently large values of time t, the grain velocity reaches its stationary
value called the terminal velocity. In such a non-accelerating, steady-state motion,
the grain density rs and time tshould be excluded from (4.80). Then, it takes the
following form
vt ¼ f ðr; m; d; gsÞ (4.81)
Since, three of the four governing parameters in (4.81) have independent
dimensions, it can be reduced to the following dimensionless form
P ¼ ’ðP1Þ (4.82)
where P ¼ vtdr=m; P1 ¼ gsrd3=m2 ¼ ðgs=gÞ d3g=n2ð Þ; and g ¼ rg:
Measurements of terminal sedimentation velocity show that all the experimental
data for grains of different types ð1bgs=g<38Þ collapse onto a single curve P ¼’ P1ð Þ (Yalin 1972). For sedimentation of very small grains, when the effect of the
fluid inertia is negligible (in the Stokes creeping flow regime), (4.81) inevitably
takes the scaling form
vt ¼ cma1da2ga3s (4.83)
where the exponents ai are equal a1 ¼ �1; a2 ¼ 2; and a3 ¼ 1, as it follows from
the principle of the dimensional homogeneity.
As a result, we arrive at (4.74).
4.8.3 The Critical State of a Fluidized Bed
Consider detachment of an individual grain from a fluidized bed assuming that the
height of the sand roughness equals grain size and the lower surface of the fluidized
bed is approximately a horizontal plane. The critical state that corresponds to
the onset of the instability refers to a situation, in which grains are beginning
to settle down from the lower surface of the bed. In order to find a relation between
the critical parameters corresponding to onset of such settling, we use (4.79) written
for the sedimentation transport rate
Pqs ¼ fqsðP1;P2;P3;P4Þ (4.84)
92 4 Drag Force Acting on a Body Moving in Viscous Fluid
where Pqs ¼ qs=rv2�; with qsbeing the total transport rate (the total load per unit
width of the flow).
The positive values of the parameter Pqs ðPqs>0Þ correspond to all possible
states of two-phase flow in the fluidized bed, whereas the casePqs ¼ 0 corresponds
to the critical state. Since the granular material does not settle from the lower side of
the fluidized bed in the critical state, the dimensionless group P4 ¼ rs=r can be
excluded from the set of the governing parameters. In addition, it is also possible to
omit the dimensionless group P3 ¼ h=d; since at the critical state grain motion is
entirely due to the action of the flow in the vicinity of the lower surface of the
fluidized bed. Accordingly, we arrive at the following equation
’ðP1;cr;P2;crÞ ¼ 0 (4.85)
Which is equivalent to
P2;cr ¼ ’ðP1;crÞ (4.86)
For small and large values of P1;cr, the following expressions are valid
P2;cr ¼ FðP1;crÞ ¼ const
P1;cr(4.87)
P2;cr ¼ FðP1;crÞ ¼ const (4.88)
Note, that the first of these equations follows directly from the assumption that
the critical stage at smallP1;cr corresponds to the equality of the hydrodynamic lift
force of Saffman (1965) and the grain weight, i.e to the condition Fl G ¼ 1= ; whereFl andG are the lift force and the grain weight, respectively. At large values ofP1;cr
the fluid viscosity, and accordingly, P1;cr, are no longer the characteristic
parameters. In this case corresponding to very small m, it is possible to assume
that P2;cr is constant.
The predictions of (4.86–4.88) are in a good agreement with the experimental
data (Yalin 1972).
4.9 Thin Liquid Film on a Plate Withdrawn Vertically from aPool Filled with Viscous Liquid (the Landau-LevichProblem of Dip Coating)
In dip coating process a thin film of viscous liquid is withdrawn by a flat plate from
a pool filled with that liquid, as shown in Fig. 4.5. The film is withdrawn by the
moving vertical plate due to the action of viscous forces, which overbear counter-
action of gravity and surface tension. As can be seen in Fig. 4.5, it is possible to
4.9 Thin Liquid Film on a Plate Withdrawn Vertically from a Pool Filled 93
distinguish two characteristic flow domains located at different distances from the
moving plate. In the first of these domains the free surface elevations are small and
the dominant role is played by the capillary force and gravity forces. On the other
hand, in the second domain close to the plate, viscous forces are dominant.
An asymptotic solution of the dip coating problem consists in the calculations
within each one of the above-mentioned domains 1 and 2 and matching the results,
which allows determination of the withdrawn film thickness (Landau and Levich
1942; Levich 1962). Referring the readers to the original work by Landau and
Levich, we restrict our discussion to the analysis of several particular cases of film
withdrawal in dip coating in the framework of the Pi-theorem.
We will consider plate motion with sufficiently small velocities, as specified
below, when the inertia effect is negligible. In this case the thickness of the
withdrawn liquid film h depends on plate velocity v, gravity acceleration g, liquidspecific weight, viscosity and surface tension rg (with r being density), m and s,respectively, as well as on the coordinate along the plate x. Accordingly, thefunctional equation for the thickness of the withdrawn liquid film reads
h ¼ f ðrg; m; v; s; xÞ (4.89)
It is emphasized that the set of the governing parameters does not contain liquid
density alone because the inertia effect is assumed to be negligibly small.
The dimensions of the governing parameters that determine the film thickness
are
ðrgÞ½ � ¼ L�2MT�2; m½ � ¼ L�1MT�1; v½ � ¼ LT�1; s½ � ¼ MT�2; x½ �¼ L (4.90)
Three of the five governing parameters have independent dimensions. Choosing
as these parameters (rgÞ; v and s, we transform (4.89) using the Pi-theorem to the
following dimensionless form
P ¼ ’ P1;P2ð Þ (4.91)
x
h(x)
3
1
y0
2
g
Fig. 4.5 Withdrawn liquid
film. 1-Liquid pool, 2-the free
surface, 3-plate moving
upwards
94 4 Drag Force Acting on a Body Moving in Viscous Fluid
where P ¼ h=ðrgÞa1va2sa3 ; P1 ¼ m=ðrgÞa01va
02sa
03 ; and P2 ¼ x=ðrgÞa
001 va
002 sa
003 .
The corresponding exponents are found using the principle of the dimensional
homogeneity as a1 ¼ �1=2; a2 ¼ 0; a3 ¼ 1=2; a01 ¼ 0; a
02 ¼ �1; a
03 ¼ 1; and
a001 ¼ �1=2; a
002 ¼ 0; a
003 ¼ 1=2:
Then (4.91) takes the following form
h ¼ srg
�1=2
’mvs;
x2rgs
�1=2" #
(4.92)
In the limit x>>1; it is possible to assume that the effect of the dimensionless
group P2 ¼ x2rg=sð Þ1=2 has already been saturated and it does not affect the
withdrawn film thickness anymore. Then, (4.92) reduces to the following
expression
h ¼ srg
�1=2
’mvs
� (4.93)
It is convenient to present (4.93) as a dependence of the dimensionless with-
drawn film h ¼ h=l; where l ¼ s=rgð Þ1=2 is the capillary length, on the capillary
number Ca ¼ mv=s
h ¼ ’ðCaÞ (4.94)
The function ’ðCaÞ can be either found experimentally, or through the original
asymptotic solution of Landau and Levich. It has the following form (Levich 1962)
’ðCaÞ ¼ 0:93Ca1=6 (4.95)
at Ca<<1; and
’ðCaÞ � 1 (4.96)
at Ca � 1.
At a high withdrawal velocity the thickness of the withdrawn liquid film h shouldnot depend on the surface tension, since both the effects of gravity and viscous
forces become dominating. Therefore, the functional equation for the thickness of
the withdrawn liquid layer becomes
h ¼ f ðrg; m; vÞ (4.97)
All the parameters in (4.97) have independent dimensions. Therefore, it takes the
following form
h ¼ cðrgÞa1ma2va3 (4.98)
where c is a dimensionless constant.
4.9 Thin Liquid Film on a Plate Withdrawn Vertically from a Pool Filled 95
The exponents ai are found from the principle of the dimensional homogeneity
as a1 ¼ �1=2; a2 ¼ 1=2; and a3 ¼ 1=2. Accordingly, (4.98) becomes
h ¼ cmvrg
�1=2
(4.99)
The numerical value of the constant c can be found experimentally and approxi-
mately equals to 2/3 (Derjaguin and Levi 1964).
Problems
P.4.1. Show that at low Reynolds numbers the drag coefficient of a spherical
particle moving with a constant velocity in an incompressible viscous fluid is
inversely proportional to the Reynolds number.
The terms rðv � rÞv and mr2v in the Navier-Stokes equation have the order
of rv2x=lx and mvx=l2x , respectively, whereas and their ratio is of the order of
rv2x=lx�
= mvx=l2x� ¼ Re<<1 (vx and lx are the characteristic scales of the velocity
and length in the direction of the particle motion). That allows to omit the term
rðv � rÞv from left hand side of the Navier-Stokes equation and thus eliminate
the fluid density from the set of the governing parameters. Then the functional
equation for the drag force takes the following form
fd ¼ f ðm; u; dÞ (P.4.1)
All the governing parameters in (P.4.1) have independent dimensions, so that the
difference n� k ¼ 0. Therefore, in accordance with the Pi-theorem we obtain
fd ¼ Cma1ua2da3 (P.4.2)
where C is a dimensionless constant.
Taking into account the dimensions of fd; m; u and d, we find the values of the
exponents ai from the principle of dimensional homogeneity as a1 ¼ a2 ¼ a3 ¼ 1.
Accordingly, the drag force and the drag coefficient are expressed as
fd ¼ Cmdu (P.4.3)
cd ¼ C1
Re(P.4.4)
Equations P.4.3 and P.4.4 coincide with the Stokes law up to a dimensionless
factor. The values of the constants C and C1, which can be either determined from
96 4 Drag Force Acting on a Body Moving in Viscous Fluid
the exact solution of the Stokes equations or measured experimentally are 3 and 24,
respectively.
P.4.2. Show that the drag coefficient of a spherical particle does not depend on
Re at the high values of the Reynolds number.
At high values of the Reynolds number the inertia forces play dominant role.
Under these conditions it is possible to omit fluid viscosity from the set of the
governing parameters. That leads to (4.48) for the drag force. For a spherical
particle ’ðaÞ is constant. Then, we obtain from (4.48) the following expression
for the drag coefficient
cd ¼ fd1=2ð Þru2d2 ¼ const: (P.4.5)
P.4.3. Derive a dimensionless equation for the drag force acting on an infinitely
long cylinder in subjected to a uniform laminar flow of incompressible fluid normal
to its axis.
Let the fluid density and viscosity be r and m, respectively, and velocity of the
undisturbed flow and radius of cylinder be u and R, respectively. Then the drag
force fc is determined by the following fourth governing parameters
fc ¼ f ðr; m; u;RÞ (P.4.6)
That have the dimensions as listed: r½ � ¼ L�3M; m½ � ¼ L�1MT�1; u½ � ¼ LT�1
and R L½ �. The dimension of the drag force acting at a unit length of the cylinder is
fc½ � ¼ MT�2. Since three of the four governing parameters have independent
dimensions, (P.4.6) reduces to the following form
P ¼ ’ðP1Þ (P.4.7)
where P ¼ fc=ra1ma2ua3 ; and P1 ¼ R=ra01ma
02ua
03 .
Bearing in mind the dimensions of the drag force and the governing parameters
and applying the principle of the dimensional homogeneity, we find values of the
exponents ai and a0i as: a1 ¼ 0; a2 ¼ 1; a3 ¼ 1; a
01 ¼ �1; a
02 ¼ 1; and a
03 ¼ �1.
Then, the drag force fc can be expressed as
fc ¼ mu’ðReÞ (P.4.8)
where Re ¼ ruR=m is the Reynolds number.
Note, that the exact solution of the problem based on Oseen’s equation reads
(Lamb 1959)
fc ¼ 4pmu1=2� Cþ lnðReÞ ¼
4pmulnð37=ReÞ (P.4.9)
Problems 97
where C ¼ 0:577::::is the Euler constant.Comparing the expressions (P.4.8) and (P.4.9), we find that ’ðReÞis
’ðReÞ ¼ 4plnð37=ReÞ (P.4.10)
P.4.4. Show that in the frame of Stokes’ approximation it is impossible to
determine an steady-state drag force acting on a unit length of an infinitely long
cylinder.
Stokes’ approximation is based on the assumption that the inertial term rðv �rÞvin (P.4.6) can be neglected and, thus, the drug force acting on a unit length of a
cylinder is written as
fc ¼ f ðm; u;RÞ (P.4.11)
Applying the Pi-theorem to (P.4.11) leads to an unrealistic result: the drag force
does not depend on the body size, i.e. its radius (Landau and Lifshitz. 1987).
P.4.5. Find the drag force acting on two round coaxial disks submerged in a fluid
and moving towards each other.
Let the lower disk is motionless, whereas the upper one moves with a constant
velocity u towards the lower one. The approach of the disks inevitable squeezes thefluid from the gap in between. The force acting on the disks due to the difference of
pressure in the gap P and in the surrounding fluid P0 is
fdisk ¼ 2pðR0
ðP� P0Þrdr (P.4.12)
The force fc ¼ LMT�2 is determined by the following four governing
parameters: the fluid viscosity m½ � ¼ L�1MT�1, the velocity of the moving disk
u½ � ¼ LT�1, the disk radii R½ � ¼ L, and the current gap width between the disks
h½ � ¼ L. Therefore, it is given by the following equation
fc ¼ f ðm; u;R; hÞ (P.4.13)
Three of the four governing parameters have independent dimensions, which
allows us to reduce (P.4.13) to the following dimensionless form
P ¼ ’ðP1Þ (P.4.14)
where P ¼ fdisk=ma1ua2Ra3 ; and P1 ¼ h=ma01ua
02Ra
03 .
Taking into account the dimensions of fdisk and h; m; u; and R and applying the
principle of the dimensional homogeneity, we find values of the exponents ai and a0i
as: a1 ¼ a2 ¼ a3 ¼ 1; a01 ¼ a
02 ¼ 0; and a
03 ¼ 1. Then, we obtain the following
expression for the drag force
98 4 Drag Force Acting on a Body Moving in Viscous Fluid
fdisk ¼ muR’h
R
�(P.4.15)
It is seen that the drag acting on a disk is directly proportional to the product of
the viscosity, velocity of the disk, its radius, as well as a function of the ratio h/R.This function cannot be established in the framework of the dimensional analysis.
An exact solution of the problem yields
fdisk ¼ 3p2muR
R
h
�3
(P.4.16)
P.4.6. Determine the dependence of the friction force acting on a sphere rotating
in an infinite viscous fluid.
The friction force ff� � ¼ LMT�2 depends on the following three governing
parameters: the fluid viscosity m½ � ¼ L�1MT�1, sphere radius R½ � ¼ L, and the
angular velocity of rotation o½ � ¼ T�1. These parameters have independent
dimensions. According to the Pi-theorem, we have the following functional equa-
tion for the drag force
ff ¼ Cma1Ra2oa3 (P.4.17)
where C is a constant.
Bearing in mind the dimensions of ff ; m; R and o, we arrive at the following
result
ff ¼ CmR2o (P.4.18)
P.4.7. Determine the dependence of the semi-axes ratio of a droplet moving in
air, which acquires an approximately spheroidal shape, on the physical properties of
liquid and velocity of motion.
The surface of a relatively small droplet moving in air approximately resembles
a spheroid. We adopt the Cartesian axes OX; OY; and OZ with the center O at the
droplet center and the axis OZ directed with the undisturbed flow experienced by an
observer moving with the droplet. The droplet surface is approximated as a spher-
oid with semi-axes a and b (with a>b)
x2 þ y2
a2þ z2
b2¼ 1 (P.4.19)
The deviation of the droplet surface from a sphere is characterized by the
difference a� b that depends on the characteristic size of the droplet, as well as
the density, velocity and surface tension of the liquid
a� b ¼ f ðb; r; u; sÞ (P.4.20)
Problems 99
The three governing parameters in (P.4.20) have independent dimensions.
Therefore, the dimensionless form of (P.4.20) reads
P ¼ ’ðP1Þ (P.4.21)
Where P ¼ ða� bÞ=ra1ua2sa3 ; and P1 ¼ b=ra01ua
02sa
03 .
Determining the values of the exponents ai and a0i, we arrive at the following
equation
w ¼ 1þ cðWeÞ (P.4.22)
where w ¼ a=b, and We ¼ ru2b=s is the Weber number. The latter can equally be
based on the volume-equivalent droplet diameter d.P.4.8.Determine the moment of force required for a slow steady-state rotation of
an infinitely long solid cylinder of radius R1 with the angular velocity o about its
axis. Consider two cases: (1) the cylinder rotates in an infinite viscous fluid; and (2)
the cylinder rotates inside a cylindrical shell of radius R2, which is sufficiently
larger than R1, filled with a viscous fluid.
1. The moment of force required for swirling of a cylinder in viscous fluid is given
by the expression
Mf ¼ð2p0
tSd’ (P.4.23)
where t is the surface traction, S ¼ 2pR1 is the area of the lateral surface of the
cylinder of unit length.
The functional equation for the moment of force in the inertialess case of slow
swirling considered here reads
Mf ¼ f ðm;o;RÞ (P.4.24)
All the governing parameters in (P.4.24) have independent dimensions,
namely m½ � ¼ L�1MT�1; o½ � ¼ T�1; and R½ � ¼ L. Then, in accordance with
the Pi-theorem, (P.4.24) reduces to the following one
Mf ¼ cma1oa2Ra31 (P.4.25)
where c is a constant.
Accounting for the dimension of the moment of force Mf
� � ¼ LMT�2, we find
the values of the exponents as a1 ¼ 1; a2 ¼ 1; and a3 ¼ 2: Then, (P.4.25) takesthe following form
Mf ¼ cmoR2 (P.4.26)
100 4 Drag Force Acting on a Body Moving in Viscous Fluid
2. In the second case the functional equation for the moment of force reads
Mf ¼ f ðm;o;R1; e; eÞ (P.4.27)
where e ¼ R2 � R1 is the gap between the cylinder and the surrounding shell, and eis the eccentricity ðe L½ �; and e L½ �Þ:
Three of the five governing parameters in (P.4.27) possess independent
dimensions, so that the difference n� k ¼ 2: Therefore, the dimensionless
form of (P.4.27) reads
P ¼ ’ðP1;P2Þ (P.4.28)
where P ¼ Mf =moR21; P1 ¼ e=R1; and P2 ¼ e=R1:
In the particular case of concentric cylinder and shell ðe ¼ 0Þ, (P.4.28) takes theform
Mf ¼ moR21’
eR1
�(P.4.29)
In the general case the expression for Mf has the form (Loitsyanskii 1966)
Mf ¼ 4pmoR3
ecðlÞ (P.4.30)
where cðlÞ ¼ 1þ 2l2�
= 2þ l2� ffiffiffiffiffiffiffiffiffiffiffiffiffi
1� l2ph i
; and l ¼ e=e.
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102 4 Drag Force Acting on a Body Moving in Viscous Fluid
Chapter 5
Laminar Flows in Channels and Pipes
5.1 Introductory Remarks
Fluid flow in pipes and ducts was a subject of numerous experimental and theoretical
investigations performed during the last two centuries. Beginning from the seminal
works of Hagen (1839) and Poiseuille (1840), a detailed data on flows of incom-
pressible viscous fluids in pipes and ducts of different geometry was obtained.
These results are presented in many review articles, monographs and textbooks.
A comprehensive analysis of problems related to laminar and turbulent flows in
pipes and ducts (the physical foundations of the theory and its mathematical
formulation) can be found in such widely known books as Schlichting (1979),
Landau and Lifshitz (1987), Loitsyanskii (1966) and Ward-Smith (1980). Refering
the readers to these monographs, we focus on the applications of the Pi-theorem for
the analysis of pipe and duct flows.
First we discuss some specific features of laminar flows of incompressible
viscous fluid outflowing from a large tank through a conical nozzle into a straight
pipe or duct. A sketch of such flow within the entrance and fully developed sections
of a pipe located, correspondingly, near and far from the inlet is shown in Fig. 5.1. It
is seen that the velocity profile which is uniform at the entrance cross-section of the
pipe becomes non-uniform at x>0 as a consequence of fluid-wall interaction at the
flow periphery.
Under the conditions corresponding to relatively large Reynolds numbers deter-
mined by the mean velocity, pipe diameter and fluid viscosity, laminar flows in
straight pipes can be schematically presented as follows. Near the pipe inlet, the
boundary layer forms over the wall. The thickness of the layer increases downstream
(cf. the left part of Fig. 5.1). The longitudinal velocity component u increases withinthe boundary layer from zero at the wall to the velocity of the undisturbed fluid flow
in the potential core. The velocity in the potential core gradually increases from an
initial velocity u0 at x ¼ 0 up to the maximum one, umax, in the cross-section x ¼ lenwhere the boundary layers from different sides merge. Approximately at x ¼ len thevelocity profile approaches to the parabolic Poiseuille profile, the flow becomes fully
L.P. Yarin, The Pi-Theorem, Experimental Fluid Mechanics,
DOI 10.1007/978-3-642-19565-5_5, # Springer-Verlag Berlin Heidelberg 2012
103
developed, and does not vary with x anymore. Thus, the following two flow regions
in the pipe flows are distinguished. Namely, (1) the entrance flow region (x<len), andthe fully developed flow region (x>len). The flow characteristics within the entrance
region are mostly determined by the boundary layer, whereas within the fully
developed region all fluid elements move parallel to the pipe axis x with the
longitudinal velocity u ¼ uðyÞ; and v ¼ w ¼ 0, where x and y are the longitudinaland transversal coordinates, respectively. Schlichting (1979).
Stationary flows of incompressible viscous fluid in straight pipes are described
by the Navier–Stokes and continuity equations
v � rv ¼ � 1
rrPþ nr2v (5.1)
r � v ¼ 0 (5.2)
where r; P and v are the density, pressure and velocity vector, respectively, n is thekinematic viscosity.
Solutions of (5.1) and (5.2) should satisfy the no-slip boundary conditions at the
wall, as well as to correspond to given velocity distributions v0 and v00 at the inlet(x ¼ 0) and outlet (x ¼ L) of the pipe
v ¼ v0 at x ¼ 0; v ¼ v00 at x ¼ L; v ¼ 0 at the pipe wall (5.3)
The approximate analytical solutions of the problem on the flow in the entrance
region of a plane channel was obtained by Schlichting (1934) and Schiller
(1922). In this cases the length of the entrance region len is determined by the
expression
lenl�Re
¼ c (5.4)
Fig. 5.1 Velocity profiles in laminar flows in straight cylindrical pipes
104 5 Laminar Flows in Channels and Pipes
where l� is the characteristic size of a channel or a pipe, a semi-height or a radius,
respectively. Re is the Reynolds number based on the channel height or pipe
diameter, and c is a dimensionless constant.
The numerical solution of the Navier–Stokes equations (Friedmann et al. 1968),
as well as the experimental studies of Emery and Chen (1968) and Fargie and
Martin (1971) show that c in (5.4) depends on the Reynolds number at Re<500:Within this range of the Reynolds number, c decreases monotonously as Re
increases. In the range Re>500 c is practically constant (cf. Fig. 5.2).
Accordingly, in the framework of the two region model described above, the
problem on the laminar flow in a pipe of circular cross-section amounts to solving
the following system of equations (Loitsyanskii 1966)
ru@u
@xþ rv
@u
@y¼ � dP
dxþ m
1
yk@
@yyk@u
@y
� �(5.5)
@u
@xþ 1
yk@vyk
@y¼ 0 (5.6)
for the entrance region (in the boundary layer approximation), and
dP
dx¼ mr2u (5.7)
for the fully developed region. In (5.5) and (5.6) k ¼ 0 or 1 for the plane or
axisymmetric flows, respectively.
Fig. 5.2 The dependence of the length of the entrance section on the Reynolds number in laminar
flow in a straight cylindrical pipe. 1-Calculations by Friedman et al. (1968), 2-Data by Emery and
Chen (1968)
5.1 Introductory Remarks 105
The boundary conditions for (5.5) and (5.6) read
y ¼ 0 u ¼ v ¼ 0; y ¼ yd u ¼ ud (5.8)
(with yd being the boundary layer thickness and u ¼ udðxÞ the velocity of the
external flow velocity in the potential core), and for (5.7)
y ¼ 0@u
@y¼ 0; y ¼ r0 u ¼ 0 (5.9)
In (5.5), (5.6) and (5.8) y is understood as the distance from the wall to a point
within the boundary layer. On the other hand, in (5.7) and (5.9) y is reckoned from
the channel axis and r0 is the cross-sectional radius of the pipe.It is emphasized thatwithin the entrance region of the flow the pressure gradient is a
function on x, whereas within the fully developed flow region it is constant. That
allows us to understand, as usual for the Poiseuille flows, that in (5.7)
dP=dx ¼ �DP=L; where DP is the pressure drop over a pipe section of length l.As usual, two types of questions can be posed regarding flows in pipes (and
similarly, in planar channels). In the first one, a volumetric flow rate Q is given,
while the corresponding pressure gradient required to sustain such flow, DP=lshould be found. In the second one, a pressure drop DP=l is given, while the
corresponding volumetric flow rate Q is to be found.
5.2 Flows in Straight Pipes of Circular Cross-Section
5.2.1 The Entrance Flow Region
The flow of viscous fluid within the entrance region of a pipe is illustrated in
Fig. 5.1. As a result of the fluid friction at the wall, the boundary layer is formed.
The thickness of the boundary layer dðxÞ increases along the pipe, with the
longitudinal coordinate x. At some distance from the inlet x ¼ len, the boundary
layer fills the whole pipe cross-section and thus dðlenÞ ¼ d=2:Having in mind the two-region pattern of pipe flows, we evaluate their
characteristics using the dimensional analysis. First, we evaluate the dependence
of the length of the entrance region len on the flow parameters. It is natural to
assume that the thickness of the boundary layer depends on the fluid density and
viscosity r and m, velocity in the undisturbed core u, as well as the distance from
the inlet to a cross-section under consideration. Then, we have the functional
equation for dðxÞ as follows
dðxÞ ¼ f ðr; m; u; xÞ (5.10)
In the framework of the boundary layer theory expediently it is natural to use the
System of Units LxLyLzMT with three different length scales Lx,Ly and Lz in the x,y
106 5 Laminar Flows in Channels and Pipes
and z directions, respectively. Using this system of units, we express the
dimensions of the governing parameters as r½ � ¼ L�1x L�1
y L�1z M; m½ � ¼
L�1x LyL
�1z MT�1; u½ � ¼ LxT
�1, x½ � ¼ L, and the thickness of the boundary layer as
d½ � ¼ Ly. Taking into account that all the governing parameters have independent
dimensions, we rewrite (5.10) as follows
dðxÞ ¼ cra1ma2ua3xa4 (5.11)
where c is a dimensionless constant and the exponents ai found using the
principle of the dimensional homogeneity, read a1 ¼ �1=2; a2¼ 1=2; a3 ¼ �1=2; and a4 ¼ 1=2.
Then (5.11) takes the form
dðxÞ ¼ cmxru
� �1=2
(5.12)
or
dðxÞd
¼ cnxd2u
� �1=2(5.13)
Bearing in mind that dðlenÞ ¼ d=2, we obtain
lend
¼ c�Red (5.14)
where1 Red ¼ ud=n; and c� ¼ ð2cÞ�21
.
The dependence (5.14) qualitatively agrees with the theoretically predicted
value of len ¼ 0:04dRe (Schlichting 1979). The results of the numerical investiga-
tion of the dependence lenðReÞ for flows in straight cylindrical pipes are presented inFig. 5.2. They show that the value of the ratio len=dRe is approximately constant at
Re > 500 when the boundary layer approximations are valid.
In addition, we address the pressure drop over the entrance region of the pipe
flow. Equation (5.14) shows that the length of the entrance region len depends on
pipe diameter and the Reynolds number. That allows us to use len as a generalizedparameter of the problem. Namely, assume that the ratio dP=dxð Þ=ru2 depends on asingle dimensional parameter len and one variable x. Then, the functional equationfor this ratio becomes
dP
dx
� �ru2� �� ¼ f ðlen; xÞ (5.15)
1 Re½ � ¼ 1 since it is defined by the parameter magnitudes.
5.2 Flows in Straight Pipes of Circular Cross-Section 107
In (5.15) one of the two governing parameters has an independent dimension.
Then, in accordance with the Pi-theorem, (5.15) reduces to the dimensionless
form
P ¼ ’ðP1Þ (5.16)
where P ¼ dP=dxð Þ= ru2ð Þ=laen; and P1 ¼ x=la0
en.
Bearing in mind the dimension of dP=dx½ � ¼ L�1y L�1
Z MT�2, we find the
values of the exponents ai and a0i as a ¼ �1; a
0 ¼ 1. Accordingly, (5.16) takes
the form
dP=dxð Þlenru2
¼ ’x
len
� �(5.17)
Substituting the expression (5.14) for len into (5.17), we arrive at the
dependence
Po ¼ ’ðXÞ (5.18)
where the Poiseuille numberPo¼ c�lRed; l¼ dP=dxð Þd=ru2; and X¼ x= c�dRedð Þ.Equation (5.18) shows that the Poiseuille number is a function of one dimen-
sionless variable X alone (Fig. 5.3).
Fig. 5.3 The dependence of the Poiseuille number on X
108 5 Laminar Flows in Channels and Pipes
5.2.2 Fully Developed Region of Laminar Flows in Smooth Pipes
Consider pressure drop in the fully developed flow region. To choose the governing
parameters for this flow, it is necessary to account for such the specific features as
(1) constancy of the pressure gradient DP=l; and (2) fluid motion parallel to the pipe
axis with a constant mean velocity when density does not affect the flow. Under
these conditions the functional equation for the pressure drop takes the form
DPl
¼ f ðm; u0; dÞ (5.19)
where m½ � ¼ L�1MT�1 is the dimension of the viscosity, u0½ � ¼ LT�1 is dimension
of the mean velocity and d½ � ¼ L (with d ¼ 2r0 being the cross-sectional diameter),
i.e. they have independent dimensions.
Then, using the Pi-theorem, we can transform (5.19) to the following one
DPl
¼ cma1ua20 da3 (5.20)
where c is a dimensionless constant, and the exponents ai are equal to a1 ¼ 1;a2 ¼ 1 and a3 ¼ �2:
As a result, we arrive at the expression
DPl
¼ cmu0d2
(5.21)
Dividing both sides of (5.21) by ru2, we obtain the following expression for the
friction factor l
l ¼ c�Re
(5.22)
where l ¼ 2 DP=lð Þ d=ru20� �
is the friction factor, c� ¼ 2c is a dimensionless
constant.
As follows from the exact analytical solution of the Hagen-Poiseuille problem
on laminar flows in round smooth pipes, the factor c� in (5.22) equals 64.
5.2.3 Fully Developed Laminar and Turbulent Flowsin Rough Pipe
Turbulent pipe flows were studied experimentally in Nikuradse (1933) who used
uniform sand grains of different sizes glued to the inside walls of pipes to vary
roughness in a controlled manner. It is emphasized that in stable laminar pipe flows
5.2 Flows in Straight Pipes of Circular Cross-Section 109
at sufficiently low values of the Reynolds number (Re � 2000) the wall roughness
was assumed to have minor effect. In stable pipe flows perturbations introduced by
roughness elements were expected to fade sufficiently fast to affect flow field
globally (Schlichting 1979). Later on, Kandlicar (2005) revisited the results of
Nikuradse’s experiments in the light of the recent studies of laminar flows in
micro-channels with rough walls and concluded that surface roughness plays an
important role in fluid flow in micro and mini-channels. The numerical calculations
by Hezwig et al. (2008) showed that in laminar flows the resistance of rough-wall
channels with regular roughness significantly depends on the relative roughness
size. A number of experimental studies of laminar flows in micro-channels also
suggest some influence of wall roughness on resistance to flow in such channels
(Yarin et al. 2009). Nevertheless, in spite of the existence of a number of theoretical
and experimental works devoted to laminar flows in rough pipes and channels, the
role of roughness and mechanisms of its influence on the resistance of such
channels is not clear yet.
Below we attempt to reveal some special features characteristic of resistance of
rough straight pipes on fully developed laminar flow by using the dimensional
analysis. Suppose that a rough pipe is characterized by two dimensional parameters,
namely, its effective diameter def and the height of roughness elements k. Inaddition, the set of the governing parameters of such flows should include three
dimensional parameters that characterize the physical properties of fluid-its density
r and viscosity m; as well as the average flow velocity u: (A discussion concerning
the choice of the characteristic size of rough elements see in Hezwig et al. 2008).
Then, the functional equation for pressure drop over pipe section of length l reads
DPl
¼ f ðr; m; u; def ; kÞ (5.23)
The governing parameters in (5.23) have the following dimensions:
r½ � ¼ L�3M; m½ � ¼ L�1MT�1; u½ � ¼ LT�1; def ¼ L; k½ � ¼ L: Three of the five
governing parameters have independent dimensions. Therefore, according to the
Pi-theorem, (5.23) can be transformed to the form
P ¼ ’ P1;P2ð Þ (5.24)
where P ¼ 2DPd=lru2 ¼ l is the friction factor, P1 ¼ m=rudef ¼ Re�1;
with Re ¼ rudef =m, which is the Reynolds number, and P2 ¼ k=def ¼ k is the
relative roughness of pipe wall.
The dimensional analysis does not allow finding an exact form of the depen-
dence of the drag friction factor on the dimensionless groups- the Reynolds number
and relative roughness of the wall. However, it is possible to evaluate some
particular trends of the friction factor when Re and k are changing. Indeed. At
large values of the Reynolds number corresponding to fully developed turbulence,
the inverse Reynolds number P1 is much less than the value of the relative
110 5 Laminar Flows in Channels and Pipes
roughnessP2. In this case the effect of viscosity is negligible, and the friction factor
is determined mostly by the relative roughness l � ’ðkÞ. In the second
corresponding to relatively small Reynolds number (laminar flows), the inverse
Reynolds numberP1 can be much large than the relative roughnessP2. Therefore,
in the latter case the roughness effects are weak. It is emphasized that at any finite
values of Re the influence of wall roughness on the friction factor of a pipe exists.
Only at Re ! 0, (5.24) reduces to the form l ¼ ’ðReÞ, whereas within the range ofmoderate Reynolds numbers the friction factor depends on both dimensionless
groups, namely l ¼ lðRe; kÞ. The latter form practically encompasses the whole
bulk of the available experimental data (Moody 1948; Schlichting 1979).
5.3 Flows in Irregular Pipes and Ducts
At applying the Pi-theorem to flows in irregular (non-circular) pipes or channels, it
is convenient to use an effective size def L½ � expressed through the cross-section
geometry. For example, the so-called hydraulic diameter dh defined as 4S P= , where
S is the cross-sectional area and P the wetted perimeter is used as an effective size
of irregular pipes and channels. In this case the form of the functional equation for
the pressure drop in irregular pipes and ducts remains the same as the regular ones
(with circular cross-section). However, as the parameter d in (5.19) the effective
size of an irregular pipe def is to be used. In particular, the Reynolds number Re is
defined based on def .As an example, we mention that the effective size of pipes with the elliptic cross-
section is defined as follows
def ¼ 2ffiffiffi2
p b
ð1þ e2Þ1=2(5.25)
where e ¼ b=a; and a and b are the large and small semi-axes of the ellipse.
The detailed data on pressure drop in the irregular pipes and ducts is available in
monographs of Shah and London (1978), Ward-Smith (1980), Ma and Peterson
(1997) and White (2008). Some results that display the effect of pipe shape on
resistance to laminar flows in the irregular pipes are presented in Table 5.1.
Table 5.1 Characteristics of irregular pipes
Shape of pipe Caracteristic size Constant in (5.22) Reynolds number
Circular def ¼ d 64 udn
Elliptic def ¼ 2ffiffiffi2
p ab
ða2þb2Þ1=264 udef
n
Equilateral triangular def ¼ h 160 uhn
Square def ¼ H 128 f ðxÞ= u2Hn
5.3 Flows in Irregular Pipes and Ducts 111
In this Table in addition to the previously introduced notation, h is a side of an
equilateral triangle, H is a side of a square, and f ðxÞ is a tabulated function of the
height to width ratio [ f ðxÞ changes from 2.253 to 5.333 when x varies from 1 to1(Loistyanskii, 1966)]. Note that the factor c in (5.22) for the friction factor for
laminar pipe flows is equal to 128, 160, and 64 for pipes with rectangular, equilat-
eral triangular and elliptic cross-sections, respectively (Table 5.1).
The friction factor for laminar flows in concentric and eccentric annuli can be
also presented in the form of (5.22). In these cases the factor c depends on the ratioof the effective diameters of the external and internal pipes, as well as on the value
of their eccentricity (see the Problems at the end of this Chapter).
5.4 Microchannel Flows
A detailed data on flows in microchannels with circular, rectangular, triangular and
trapezoidal cross-sections with hydraulic diameters in the range from 10�5m to
10�3m is available in a number of monographs (Incropera 1999; Celata 2000;
Kakas et al. 2005; Yarin et al. 2009) and review articles (Gad-el-Hak 1999;
Garimella and Sobhan 2003; Hetsroni et al. 2005a, b). In spite of the existence of
the numerous experimental and theoretical studies devoted to hydrodynamics and
heat transfer in laminar flows in microchannels, there is no consensus, in particular,
between the data on the friction factor of rough pipes and ducts, the effect of energy
dissipation, thermal conductivity of the wall and fluid, etc. Such situation
complicates understanding of the physical phenomena occurring in flows in
microchannels and sometimes leads to questionable ‘discoveries’ of specific
‘micro-effects’ (Duncan and Peterson, 1994; Ho and Tai 1988; Plam 2000;
Hezwig 2002; Hezwig andHausner 2003; Gad-el-Hak 2003). The shaky foundations
of a number of such results were revealed by comparison of the experimental data
with predictions of the conventional theory based on the Navier–Stokes equations.
In the framework of this theory, solutions corresponding to developed laminar flow
in straight microcannels involve a number of assumptions on flow conditions. They
are as follows: (1) the flow is driven by a static pressure drop in the fluid, (2) the
flow is stationary and fully developed, i.e. strictly axial, (3) the flow is laminar, (4)
the Knudsen number is small enough so that the fluid can be considered a contin-
uum, (5) there is no slip at the wall, (6) the fluid is incompressible Newtonian with
constant viscosity, (7) there is no heat transfer to (from) the ambient medium, (8)
the energy dissipation is negligible, (9) there is no fluid-wall interaction (except
purely viscous, so the electric, dispersive London and van der Waals forces are
neglected), (10) the micro-channel walls are smooth. Under these conditions
problems on flows in microchannels reduces to integrating (5.7) with no-slip
conditions over the wetted perimeter of the channel. The corresponding solutions
lead to the following results: the Poiseuille number Po is a constant, which is
determined by the micro-channel shape. The discrepancy between this result and
112 5 Laminar Flows in Channels and Pipes
the experimental data on microchannel resistance to flow are interpreted sometimes
as a manifestation of some new effects inherent to microchannel flows. The
puzzling situation was considered in Hetsroni et al. (2005a) where the influence
of different factors that affect hydrodynamics in microchannels (e.g. the change of
the physical properties of fluid, the energy dissipation, etc.) was evaluated. In
addition, the conformity of the actual experimental conditions with those assumed
in the theoretical description of micro-channel flows was considered. It was shown
that only in certain cases the experimental conditions were consistent with the
theoretical assumptions. The experimental results corresponding to these cases
agree fairly well with the theory. In particular, in single-phase fluid flow in smooth
micro-channels of hydraulic diameter in the range from 15 mm to 4010 mm the
Poiseuille number is independent on the Reynolds number and equals 64. In single-
phase gas flows in micro-channels of hydraulic diameter in the range from 10.1 mmto 4010 mm and the Knudsen number 0:001 � Kn � 0:38; the friction factor agreesfairly well with the theoretical predictions for fully developed laminar flows (Yarin
et al. 2009). Therefore, it was concluded that the experiments on flows in smooth
micro-channels show no difference with macro-scale flow and no new physical
effects should be invoked. From the point of view of the dimensional that means
that the approach which is used for study of friction factors in macroscopic channels
can be equally used for micro-scale channels. In both cases the problem consists in
choosing as the governing parameters of fluid viscosity, mean velocity and macro-
or micro-channels diameter followed by transformation of the functional equation
for pressure drop to dimensionless form using the Pi-theorem. It is emphasized that
this statement refers to the application of the Pi-theorem to flows in smooth micro-
channels only. A number of experiments show that the Poiseuille number in rough
micro-tubes exceeds significantly the one in smooth micro-tubes (Qu et al. 2000;
Pfund et al. 2000; Li et al. 2003; Bahrami et al. 2006; Wang and Wang 2007; Li
et al. 2007). The latter shows that the set of the governing parameters responsible
for the friction factor of rough micro-channels has to include roughness of the wall.
5.5 Non-Newtonian Flows
Non-Newtonian fluids are those for which the rheological constitutive equation
differs from the Newton-Stokes one. In particular, it means that such fluids can
possess a yield stress or/and in simple shear flow the dependence of shear stress on
shear rate becomes nonlinear (i.e. the shear viscosity of such fluids is not constant at
given pressure and temperature as Fig. 5.4 shows).
The rheological constitutive equations of non-Newtonian fluids are more complex
than that of Newtonian ones. They can contain a number of parameters that account
for such specific features of non-Newtonian fluids as consistency index, the exponent,
yields stress which should be exceeded before flow starts, and viscoelastic relaxation
time (Wilkinson and Chen 1960; Astarita and Marrucci 1974; Bird et al. 1977).
5.5 Non-Newtonian Flows 113
As an example that demonstrate the application of the Pi-theorem in hydrody-
namics of non-Newtonian fluids, we consider laminar flow in a straight cylindrical
pipe of Bingham fluid possessing yield stress. Our consideration will be restricted to
the analysis of fully developed flows in the two cases corresponding to: (1) a given
volumetric flow rate (i.e. a given mean velocity in the pipe), and (2) a given
pressure drop along the pipe. In the first case the problem consists in finding the
friction factor, whereas in the second one in finding the volumetric flow rate. In
both cases we use the Pi-theorem to eatablish dependences of the unknown
characteristics on flow and geometrical parameters.
The shear behavior of Bingham fluids is given by the following equation
(Wilkinson and Chen 1960)
t� t0 ¼ m0 _g; t>t0 (5.26)
where t is the shear stress, t0 is the yield stress, m0 is the viscosity, and _g is the rateof shear.
We assume that flow of Bingham fluid in a pipe is dominated by shear near the
walls and thus consider (5.26) to be an adequate representative of the overall
(tensorial) rheological constitutive equation of the fluid. It is seen that (5.26)
contains two characteristic parameters: t0 and m0 which have the following
dimensions
t0½ � ¼ L�1MT�2; m½ � ¼ L�1MT�1 (5.27)
The pressure drop in flows of Bingham fluids in straight cylindrical pipes should
depend on the mean velocity u and pipe diameter d in addition to the rheological
parameters. Then, the functional equation for the pressure drop DP=l reads
Fig. 5.4 Flow curves for
time-independent non-
Newtonian fluids. 1-Bingham
plastic, 2-pseudoplastic fluid,
3-Newtonian viscous fluid,
4 dilatant fluid
114 5 Laminar Flows in Channels and Pipes
DPl
¼ f ðm0; t0; u; dÞ (5.28)
Three governing parameters in (5.28) have independent dimensions, so that the
difference n� k ¼ 1: Then, in accordance with the Pi-theorem (5.28) reduces to the
following one
P� ¼ ’ðPÞ (5.29)
where P� ¼ DP=lð Þ=ma10 ua2da3 ; and P ¼ t0=ma01
0 ua02da
03 .
Bearing in mind that DP=l½ � ¼ L�2MT�2 and u½ � ¼ LT�1, we find that
the exponents ai and a0i are as follows: a1 ¼ 1; a2 ¼ 1; a3 ¼ �2; a
01 ¼ 1; a
02 ¼ 1;
and a03 ¼ �1: In accordance with that, (5.29) takes the form
f ¼ c
Re’�ðPÞ (5.30)
where f ¼ 2 DP=lð Þd=ru2 is the friction factor, c�’� ¼ 2’, c is dimensionless
constant equels 64 for flow in round pipe, and P ¼ t0d=m0u is the dimensionless
yield stress parameter called the Bingham number.
The dependences of the friction factor on the Reynolds number in (5.30) shows
that friction factor f is a function of two dimensionless groups, namely, the
Reynolds number Re and the Bingham number P. This dependence is shown
in Fig. 5.5 in log-log coordinates for different values of the Bingham number P.
For the lowest line P ¼ 0, which corresponds to Newtonian fluid. The kink on the
lowest curve corresponds to laminar-turbulent transition.
In the second case when the pressure drop DP=l is given, the volumetric flow rate
depends on the parameters m0; t0 and r0 (r0 is the radius of pipe)
Π
104
5.103
Rough
Smooth
Frict
ion
fact
or
0.01 0.1 1.0 10 100 .104
0.1
1.0
10
100
1000
Fig. 5.5 Friction factor
versus Reynolds number for
Bingham plastic fluids in
laminar regime
5.5 Non-Newtonian Flows 115
Q ¼ f ðDPl; m0; t0; r0Þ (5.31)
Three governing parameters in (5.31) have independent dimensions. Then,
according to the Pi-theorem, (5.31) transforms to the following one
P ¼ ’ðP1Þ (5.32)
where P ¼ Q= DP l=ð Þa1ma2o ra30�
;P1 ¼ t0=fðDP=lÞa01ma
02
0 ra03
0 g::.Bearing inmind the dimensions ofQ; r0; mo; DP=l and t0, we find the exponents
ai and a0i: They are as follows: a1 ¼ 1; a2 ¼ �1; a3 ¼ 4; a
01 ¼ 1; a
02 ¼ 0; a
03 ¼ 1:
Then (5.32) takes the form
Q ¼ DPl
r4om0
’tolroDP
� �(5.33)
The analytical solution of this problem leads to Buckingham’s equation
(Wilkinson and Chen 1960)
Q ¼ p8
DPl
r40m0
1� 4
3
2lt0r0DP
� �þ 1
3
2lt0r0DP
� �4( )
(5.34)
5.6 Flows in Curved Pipes
Flows in curved toroidal pipes demonstrate a number of characteristic features,
which result from the action of the centrifugal forces. The centrifugal-force–
induced pressure gradient generates the secondary flow in the pipe cross-section.
In the general case flows in curved pipes depend on the inertia, centrifugal and
viscous friction forces, which, in its turn, are determined by the pipe geometry, as
well as the physical properties of fluid. At a given pressure gradient the local
velocity in a curved pipe depends on the fluid density and viscosity, radius of the
pipe and its curvature, as well as the coordinates of a given point. The mean flow
characteristics, in particular, the pressure drop DP=l, are determined by the physical
properties of fluid, pipe geometry and mean velocity w0
DPl
¼ f ðr; m; r0;R;w0Þ (5.35)
where r0 and R are the radius of the pipe cross-section and the radius of curvature.
Equation (5.35) contains five governing parameters, whereas (5.15) corresponding
to flows in straight pipes only three of them: m; r0; and u0: The additional parameters
116 5 Laminar Flows in Channels and Pipes
r and R are included in the set of the governing parameters in (5.35) in order to
account for the effect of the centrifugal acceleration of fluid in a curved pipe and
actual pipe geometry. The governing parameters in (5.35) have the following
dimensions
r½ � ¼ L�3M; m½ � ¼ L�1MT�1; r0½ � ¼ L; R½ � ¼ L; ½w0� ¼ LT�1 (5.36)
It is seen that in the present case the difference n� k ¼ 2, and (5.35) reduces to
the following form
P ¼ f ðP1;P2Þ (5.37)
where P ¼ DP=lð Þ=ra1wa20 r
a30 ; P1 ¼ R=ra
01w
a02
0 ra03
0 ; and P2 ¼ m=ra001 w
a002
0 ra003
0 .
In addition, the exponents ai; a0i and a
00i are found from the principle of dimen-
sional homogeneity as a1 ¼ 1; a2 ¼ 2; a3 ¼ �1; a01 ¼ 0; a
02 ¼ 0; a
03 ¼ 1; a
001 ¼ 1;
a002 ¼ 1; a
003 ¼ 1 Then, (5.37) takes the form
l ¼ ’ðd;ReÞ (5.38)
where l ¼ 2 DP=lð Þ 2r0ð Þ=rw20 is the friction factor, Re ¼ w0r0=n is the Reynolds
number, and d ¼ r0=R is the ratio of the cross-sectional radius of pipe to its radius
of curvature.
The above result is obviously very weak. Indeed, in writing the functional
equation (5.35), we implicitly assumed that flows in curved pipes are determined
by the radius of curvature R. Under this assumption, (5.35) is identical to the
functional equation for flows in straight irregular ducts, in particular, to the one
for rectangular channels where the friction factor depends on two dimensionless
groups: (1) the Reynolds number, and (2) the ratio of channel width to its depth. In
contrast with the flow in a straight pipe, when studying flows in curved pipes it is
necessary to account for a number of different factors that affect such flows through
the interaction of the inertial, centrifugal and viscous forces. Accordingly, the
functional equation for the pressure drop in a curved pipe as a dependence of
DP l=ð Þ on the acting forces reads
DPl
� �¼ f ðfi; fc; ff Þ (5.39)
where fi; fc and ff are the specific inertial, centrifugal and viscous forces,
respectively, with fj ¼ L�2MT�2 and subscript j ¼ i; c; f .
All the governing parameters in (5.39) have independent dimensions. Then,
according to the Pi-theorem, this equation takes the form
DPl
� �¼ cf a1i f a2c f a3f (5.40)
5.6 Flows in Curved Pipes 117
or
l ¼ c1fa1�i f a2c f a3f (5.41)
where l ¼ 2 DP=lð Þf�1i is the friction factor, ci ¼ 2c; and a1� ¼ a1 � 1.
Since the dimension of the friction factor l½ � ¼ 1; the dimension of the right
hand side of (5.41) has to be equal 1. There is a number of different combinations
of fi; fc and ff that are dimensionless. However, only one of them, namely,
f1=2i f
1=2c =ff has a clear physical meaning. It can be interpreted as the ratio of the
inertial forces to the viscous ones, i.e. as the natural analog of the Reynolds number
for flows in curved pipes.
Then, we can present (5.41) in the following form
l ¼ c1f1=2i f
1=2c
ff
!n
(5.42)
with n being a dimensionless constant.
Assuming that all the velocity components are proportional to the mean velocity
w0, which is defined as the ratio of the flow rate to the cross-sectional area of a
curved pipe, we can estimate fi; fc and ff as follows
fi � rw20
r0; fc � rw2
0
R; ff � m
w0
r20(5.43)
Substituting (5.43) into (5.42), one finds the following equation for the friction
factor
l ¼ ’ðkÞ (5.44)
where k ¼ d1=2Re is the Dean number, and ’ðkÞ is the function of the Dean number.
Thus, we arrive at a very impotent result: the friction factor of a slightly curved
pipe is determined by a single dimensionless group-the Dean number. Naturally,
this result can be obtained directly by transforming the Navier–Stokes equations to
the dimensionless form that contain two dimensionless groups, d and Re (Berger
et al. 1983). A number of dimensionless groups that characterize flows in curved
pipes can be reduced to a single one in the particular case of a flow in a slightly
curved pipe, i.e. d<<1 (Dean 1927, 1928). When the order of magnitude of the
inertial, centrifugal and viscous forces is the same, the system of dimensionless
equations for the fully developed flows in slightly curved pipes takes the form.
@u
@rþ u
rþ 1
r
@v
@a¼ 0 (5.45)
118 5 Laminar Flows in Channels and Pipes
u@u
@rþ v
r
@u
@a� v2
r� w2 cos a ¼ � @P1
@r� 2
k
1
r
@
@a@v
@rþ v
r� 1
r
@u
@r
� �(5.46)
u@v
@rþ v
rþ uv
rþ w2 sin a ¼ � 1
r
@P1
@aþ 2
k
@
@r
@v
@rþ v
r� @u
@a
� �(5.47)
u@w
@rþ v
r
@w
@a¼ � @P0
@zþ 2
k
@
@rþ 1
r
� �@w
@rþ 1
r2@2w
@a2
� �(5.48)
u ¼ u0w0; v= ¼ v0 w0= ;w ¼ w
0w0= are the velocity components in the toroidal
coordinate system r0; a; y (cf. Fig. 5.6), P ¼ P rw2
0
�is the pressure
P ¼ PðzÞ þ Pðr; z; aÞ½ �, r ¼ r0=r0; s ¼ s
0=r0 ¼ Ry=r0 ¼ d1=2z, r
0denotes the dis-
tance from the center of circular pipe cross-section in its plane, a is the angle
between the radius vector r0and the plane of symmetry, y is the angular distance of
the cross-section from the pipe entrance.
Therefore, according to (5.45–5.48) the velocity components and pressure will
depend on the sole dimensionless group- the Dean number. As a result, all the
integral characteristics of flow, in particular, the axial pressure drop are also
functions of the Dean number. The friction factor of a curved pipe lc can be
expressed at small values of the Dean number k as (White 1929)
l ¼ 1þ 0:00306K
576
� �2
þ 0:0110K
576
� �4
þ ::: (5.49)
where l ¼ lc=ls; with lc and ls being the friction factors of the flow in curved and
the corresponding straight pipe, respectively, and K ¼ 2dRe2.At large values of the Dean number the asymptotic expressions for the ration of
the friction factors l are given by (Adler 1934; Ito 1959; Barua 1963; Mori and
Nakayama 1965)
l � 0:1k1=2 (5.50)
θ
S¢ = Rq
P(r′, a, q)α
v′w′
u′
R
r00
r′
z
Fig. 5.6 Pipe cross-section
and the toroidal coordinate
system
5.6 Flows in Curved Pipes 119
and (Van Dyke 1978)
l ¼ 0:4713k1=2 (5.51)
The correlation (5.49) agrees fairly well with the measurements of White (1929),
Adler (1934) and Ito (1959) at relatively small values of the Dean number ðk<102Þand small values of d ðd<10�3Þ. At a fixed d, an increase in the Dean number (i.e.
an increase in Re) results in a significant disagreement of the theoretical predictions
with experiment. The asymptotic formula of Van Dyke (1978) corresponding to
very large Dean numbers (k ! 1) agrees well with the experimental data at d<4 �10�3 and k<102: In all cases the disagreement of the theoretical and experimental
results stems from the change in the flow structure as the Reynolds number
increases.
In flows in helical pipes (Fig. 5.7) the set of the governing parameters is
supplemented by an additional dimensional parameter, the geometric torsion t½ � ¼L�1: Accordingly, the flow in helical pipes is determined by two dimensionless
groups (Germano 1989)
K ¼ 2dRe2; T ¼ t=r�Re
(5.52)
where r� ¼ R= R2 þ l2ð Þ is the modified curvature, and t ¼ l= R2 þ l2ð Þ is the
modified torsion.
5.7 Unsteady Flows in Straight Pipes
Consider axisymmetric flows of incompressible viscous fluids in straight cylindri-
cal pipes driven by given pressure gradient DP=l ¼ const, which is imposed
instantaneously on fluid at rest at t ¼ 0 (the startup flow). The reduced form of
the Navier–Stokes equations, which corresponds to this flow reads (Loitsyanskii
1966)
@u
@t� n
@2u
@r2þ 1
r
@u
@r
� �¼ 1
rDPl
(5.53)
The boundary and initial conditions corresponding in this case are
u ¼ 0 at r ¼ r0; u ¼ 0 at t ¼ 0 (5.54)
Equation (5.53) subjected to the conditions (5.54) shows that the velocity at any
point of pipe cross-section depends on six dimensional parameters. Four of the are
the given constants, namely, DP=l; r; m and r0, whereas the two others are
120 5 Laminar Flows in Channels and Pipes
variables r and t (the radial coordinate in the cross-section and time). Accordingly,
the mean flow characteristics, in particular, the volumetric flow rate, depend on five
dimensional parameters (without r). Therefore, we can write the following func-
tional equations for the volumetric rate Q½ � ¼ L3T�1 and local velocity u½ � ¼ LT�1
Q ¼ f ðDPl; m; r0; r; tÞ (5.55)
u ¼ f1ðDPl; m; r0; r; r; tÞ (5.56)
The governing parameters in (5.55) and (5.56) have the following dimensions
DPl
� �¼ L�2MT�2; m½ � ¼ L�1MT�1; r0½ � ¼ L; r½ � ¼ L�3M; t½ � ¼ T (5.57)
Fig. 5.7 Helical pipe with the modified curvature R= R2 þ l2ð Þ and torsion t ¼ l= R2 þ l2ð Þ
5.7 Unsteady Flows in Straight Pipes 121
First we consider (5.55). It contains five governing parameters, three of them
have independent dimensions. Therefore, the difference n� k ¼ 2: In this case
(5.55) reduces to the following dimensionless equation
P ¼ ’ðP1;P2Þ (5.58)
with the dimensionless groups being given by the following expressions:
P ¼ Q= DP=lð Þa1ma2ra30a
; P1 ¼ r= DP=lð Þa01ma
02r
a02
o
� �; P2 ¼ t= DP=lð Þa
001 ma
002 r
000
h i:
Bearing in mind the dimensions of the volumetric flow rate and the governing
parameters, we find the values of the exponents ai; a0i and a
00i as
a1 ¼ 1; a2 ¼ �1; a3 ¼ 4; a01 ¼ �1; a
02 ¼ 2; a
03 ¼ �3; a
001 ¼ �1; a
002
¼ 1; a003 ¼ �1 (5.59)
Then, the expressions for the dimensionless groups P; P1 and P2 become
P ¼ QmDP=lð Þr40
; P1 ¼ r DP=lð Þr30m2
¼ c1; P2 ¼ t DP=lð Þr0m
¼ c1Fo (5.60)
For given conditions (the physical properties of fluid, the pipe radius and the
pressure gradient DP=lÞ the dimensionless group P1 ¼ c1 is a constant ( c1½ � ¼ 1)
whereas the dimensionless group P2 ¼ c1Fo where Fo ¼ tn=r20 is kindred to the
Fourier number, with nbeing kinematic viscosity.
Accordingly, (5.58) takes the form
Q ¼ DPL
� �r40m’ðFoÞ (5.61)
In order to compare the expression (5.61) with the known analytical solution
corresponding to this case, recast (5.58) as follows
Q ¼ DPl
� �r40m
1� cðFoÞ½ � (5.62)
where cðFoÞ ! 0as Fo ! 1, which means the asymptotic approach to the
Poiseuille law, as expected on the physical grounds.
The exact analytical solution of the problem is available and yields (Loitsyanskii
1966)
Q ¼ p8
DPl
� �r40m
1� 32X1k¼1
expð�l2kFoÞl4k
" #(5.63)
122 5 Laminar Flows in Channels and Pipes
where lk are the roots of the equation J0ðlkÞ ¼ 0; with J0 being the Bessel functionof the zero order.
It is seen that the structure of (5.62) resembles that of (5.63). In both cases the
factors on the right hand side of these equations correspond to the fully developed
laminar flow in a straight pipe, whereas the factor in the parentheses approaches one
at large Fo as expected when transient effects fade.
Applying the Pi-theorem to (5.56), we arrive at
P ¼ ’ðP1;P2;P3Þ (5.64)
whereP ¼ u= DP=lð Þr40=m
; P1 ¼ r=r0; P2 ¼ c1; with c1½ � ¼ 1 being a constant.
Then, we obtain the following expression for the velocity profile u
u ¼ DPl
� �r20m’ðr;FoÞ (5.65)
where r ¼ r=r0:For comparison, the analytical solution for u is (Loitsyanskii 1966)
u ¼ DPl
� �r204m
1� ðrÞ2� �
� 8X1k¼1
expð�lkFoÞ J0ðlkrÞJ1ðlkÞ
( )(5.66)
where J1 is the Bessel function of the first order.
Problems
P.5.1. Determine the dependence of the friction factor of the concentric and
eccentric annuli with fully developed stationary flows on the governing parameters.
A concentric annulus (Fig. 5.8) is characterized by two geometric parameters,
namely: (1) the internal diameter d1 and (2) the external diameter d2. Then, thefunctional equation for the pressure gradient becomes
DPl
¼ f ðm; u�; d1; d2Þ (P.5.1)
where u� is the mean velocity.
Three of the four governing parameters in (P.5.1) have independent dimensions.
Then, according to the Pi-theorem, (P.5.1) reduces to
P ¼ ’ðP1Þ (P.5.2)
where P ¼ DP=lð Þ=ma1ua2� da32 ; and P1 ¼ d1=ma01u
a02� d
a03
2 :
Problems 123
Taking into account the dimensions of the parameters involved, DP l=½ � ¼L�2MT�2; m½ � ¼ L�1MT�1; u�½ � ¼ LT�1; d1½ � ¼ L; and d2½ � ¼ L, we find the
values of the exponents ai and a0i as a1 ¼ 1; a2 ¼ �1; a3 ¼ �2; a
01 ¼ 0; a
02 ¼ 0;
and a03 ¼ 1: Then, (P.5.2) takes the following form
l ¼ 1
Re’
d1d2
� �(P.5.3)
where l ¼ 2 DP=lð Þd2=ru2�; and Re ¼ u�d2=n:It is emphasized that the equivalent diameter de defined as
de ¼ 4S
P¼ d2 � d1 (P.5.4)
where S is the cross-sectional area, and P the wetted perimeter, can be used as one
of the governing parameters for flows in concentric annuli.
In the latter case the form of the dependence of the friction factor on the two
dimensionless groups involved retains the same form as (P.5.3), albeit the Reynolds
number Re should be replaced by Ree based on the equivalent diameter de:The analytical solution of this problem can be found by integrating the Navier–
Stokes equations, which in the present case reduce to
d2d1
d2d1
e
a
b
Fig. 5.8 Concentric (a) andeccentric (b) annuli
124 5 Laminar Flows in Channels and Pipes
dP
dx¼ m
r
d
drrdu
dr
� �(P.5.5)
subjected to the no-slip boundary conditions
u ¼ 0 at r ¼ r1; u ¼ 0 at r ¼ r2 (P.5.6)
The solution leads to the following expression for the friction factor (Ward-
Smith 1980)
l ¼ 64
Ree’
d1d2
� �(P.5.7)
where
’ d1=d2ð Þ ¼ 1� d1=d2½ �2 ln d1=d2ð Þ= 1� d1=d2ð Þ2 þ 1þ d1=d2ð Þ2h i
ln d1=d2ð Þn o
.
In the case of a fully developed stationary flow in a straight eccentric annuls
(Fig. 5.8 b) the functional equation for the dependence of the pressure gradient on
flow and geometric parameters reads
DPl
¼ f1ðm; u�; d1; de; eÞ (P.5.8)
where e is eccentricity (Fig. 5.8b).
Applying the Pi-theorem to (P.5.8) we arrive at the following expression for the
friction factor
l ¼ 1
Ree’1
r1r2;e
re
� �(P.5.9)
where r1 and r2 are the radii involved, and re is the equivalent radius equal to
r2 � r1:The analytical solution describing flows in eccentric annuli is readily available
for comparison (Ward-Smith 1980)
l ¼ 64
Ree’
r1r2;e
re
� �(P.5.10)
In the limit r1=r2 ! 1 the function ’ on the right hand side of (P.5.10) takes the
form ’1 ¼ 2=3ð Þ 1þ 3=2ð Þ e=reð Þ½ �f g2.P.5.2. Determine the velocity profile in the cross-section of a cylindrical pipe
with fully developed laminar flow of Newtonian fluid and a given pressure gradient.
Also, determine the relation between the volumetric flow rate and fluid viscosity,
pressure drop and pipe radius.
Problems 125
At a given pressure gradient DP=l, the functional equation which determines the
velocity profile in the cross-section of a cylindrical pipe reads
u ¼ f m;DPl; r; r0
� �(P.5.11)
where u is the longitudinal velocity component corresponding to the radial coordi-
nate r, r0 is the pipe radius.Applying the Pi-theorem to (P.5.11), we arrive at the following dimensionless
equation
P ¼ ’ðP1Þ (P.5.12)
where P ¼ u ma1 DP l=ð Þa2ra30� �
; and P1 ¼ r=fma01 DP=l:ð Þa02r
a03
0 g:.Taking into account the dimensions of u½ � ¼ LT�1; m½ � ¼ L�1MT�1; DP l=½ � ¼
L�2MT�2; r½ � ¼ L and r0½ � ¼ L, we arrive at the equations� a1 � 2a2 þ a3 � 1 ¼ 0; � a
01 � 2a
02 þ a
03 � 1 ¼ 0
a1 þ a2 ¼ 0; a01 þ a
02 ¼ 0 (P.5.13)
� a1 � 2a2 þ 1 ¼ 0; � a01 � 2a
02 ¼ 0
From (P.5.13) it follows that
a1 ¼ �1; a2 ¼ 1; a3 ¼ 2; a01 ¼ 0; a
02 ¼ 0; a
03 ¼ 1 (P.5.14)
Thus, (P.5.12) takes the following form
u
DPr20=lm� � ¼ ’
r
r0
� �(P.5.15)
For the axial (maximum) velocity u ¼ um at r ¼ 0, (P.5.11) yields the followingfunctional equation
um ¼ f m;DPl; r0
� �(P.5.16)
Since all the governing parameters in (P.5.16) possess independent dimensions,
(P.5.16) takes the form
um ¼ cDPl
� �r20m
(P.5.17)
where c is a constant.
126 5 Laminar Flows in Channels and Pipes
Comparing (P.5.15) with (P.5.17), we obtain
u
um¼ ’�
r
r0
� �(P.5.18)
where ’� r=r0ð Þ ¼ c’ r=r0ð Þ:Calculating the volumetric flow rate using the expression for u
Q ¼ 2pðr00
urdr (P.5.19)
we arrive at the following formula
Q ¼ 2pcDPl
� �r40m
ð10
’ðrÞrdr (P.5.20)
where r ¼ r=r0:The comparison of (P.5.20) with the exact analytical solution of the present
problem shows that cÐ10
’ðrÞrdr ¼ 1=16:
It is emphasized that the form of the dependence Q on the DP=L; m and r can bedirectly revealed by applying the Pi-theorem to the functional equation
Q ¼ fDPl; m; r0
� �(P.5.21)
Bearing in mind the dimensions of the volumetric flow rate, pressure drop, fluid
viscosity and pipe radius, we arrive at the following expression
Q ¼ cra10DPl
� �a2
ma3 (P.5.22)
where the factor c is a constant and a1 ¼ 4; a2 ¼ 1; and a3 ¼ �1.
Thus, for the volumetric flow rate we have the expression
Q ¼ cr40DPl
� �1
m(P.5.23)
The exact analytical solution of the present problem yields a numerical value for
the constant c: It is equal to p=8:P.5.3. Determine the dependence of the friction factor on the Reynolds number
for fully developed flows of viscous incompressible fluid in rough cylindrical pipes.
Let the characteristic sizes of the pipe and its roughness are d and ks; the
mean velocity of the fluid u; and fluid density and viscosity r and m; respectively.Then the functional equation for the pressure gradient is
Problems 127
DPl
¼ f d; k; r; m; uð Þ (P.5.24)
The set of the governing parameters in (P.5.24) includes three parameters that
have independent dimensions. Then, in accordance with the Pi-theorem, (P.5.24)
can be transformed to the following dimensionless form
l ¼ ’1
1
Re; k
� �(P.5.25)
where l ¼ 2 DP=lð Þd=ru2 is the friction factor, Re ¼ ud=n is the Reynolds number,
and k ¼ k=d is the relative roughness.
In the case of fully developed laminar flows the number of the governing
parameters reduces to four (d; k; m; and u), since density of the fluid does not
affect such flows. Then, (P.5.25) takes the following form
DPl
¼ f d; k; m; uð Þ (P.5.26)
Applying the Pi-theorem to (P.5.26), we obtain
l ¼ 1
Re’2 k� �
(P.5.27)
In the fully developed (in average) turbulent flows the effect of molecular
viscosity is negligible, and m can be excluded from the set of governing parameters.
Accordingly, the functional equation for the pressure gradient reads
DPl
¼ f ðd; k; r; uÞ (P.5.28)
The dimensionless form of (P.5.28) becomes
l ¼ ’3ðkÞ (P.5.29)
In connection with the above results, it is necessary to add the following
remarks. An explicit form of the dependences ’iðkÞ cannot be revealed in the
framework of the dimensional analysis. To determine the dependences ’iðkÞ, it isnecessary to recall some additional physical considerations or the experimental
data. The data of Nikuradse (1930) and Schiller (1923) show that the friction factor
of the conventional macroscopic rough pipes does not depend on k in fully
developed laminar flows and corresponds to the one following from the Poiseuille
law. Thus, the function ’2ðkÞ in (P.5.27) can be assumed to be constant equal to 64.
In the fully developed (in average) turbulent flow (at high values of Re) the friction
factor does not depend on Re and is fully determined by relative roughness.
128 5 Laminar Flows in Channels and Pipes
Therefore, the dependence of l on Re for rough conventional pipes can be selected
separately for three characteristic flow regimes corresponding to laminar
ðRe<Recr1Þ, transitional ðRecr1<Re<Recr2Þ and turbulent ðRe>Recr2Þ flows, for
which (P.5.25), (P.5.27) and (P.5.28) are valid, respectively. For micro-channels
ð10 � d � 103mmÞ the recent measurements show that the friction factor
corresponding to fully developed laminar flows depends significantly on the value
of relative roughness (Yarin et al. 2009).
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130 5 Laminar Flows in Channels and Pipes
Chapter 6
Jet Flows
6.1 Introductory Remarks
The subject of the present chapter is the hydrodynamics of laminar submerged jets
in the light of the dimensional analysis. Submerged jets are discussed in detail in
special monographs devoted to the jet theory (Pai 1954; Abramovich 1963; Vulis
and Kashkarov 1965), boundary layer theory (Schlichting 1979), as well as the
theory of turbulent flows (Townsend 1956; Hinze 1959).
Submerged jets belong to the vast class of laminar and turbulent shear flows.
They originate from issuing of viscous fluid into an infinite or semi-infinite space
filled with the same fluid. Jet flows are encountered in numerous engineering
applications such as steam and gas turbines, ejectors, jet engines, industrial
furnaces, pneumatic and ventilation systems, as well as in the phenomena typical
for the environmental flows. A wide variety of flow configurations is characteristic
of jet flows. There are, for example, mixing layers due to the interaction of two
parallel streams moving with different velocities, or jets of viscous fluid issued
from a nozzle or a slit into fluid which is moving or at rest. Wakes behind solid
bodies moving in fluid or plumes due to the action of buoyancy forces also belong
to the class of jet flows. Three main types of jet flows depending on conditions of
their formation can be distinguished: (1) submerged jets, (2) wakes, (3) plumes and
thermals (Fig. 6.1). The existence of convective motion that is predominantly
directed along the flow axis, as well as an intensive transversal mass and momen-
tum transfer due to strong shear in cross-section is typical for all kinds of jet flows.
Submerged jets can be classified by a number of characteristic features: the flow
regime (laminar, turbulent), geometry (plane, axisymmetric), mutual direction of
the jet spreading and motion of the surrounding fluid (co- and counter-flows, a jet in
crossflow), the interaction with the ambient fluid and solid walls (without any
contact with walls, or the wall and impinging jets). A detailed sketch of a planar
submerged jet of an incompressible fluid is shown in Fig. 6.2. It forms as a result of
mixing of viscous fluid issued from a slit into the same fluid as rest. The horizontal
lines a� a subdivide the domains of jet issued from the nozzle and the ambient
L.P. Yarin, The Pi-Theorem, Experimental Fluid Mechanics,
DOI 10.1007/978-3-642-19565-5_6, # Springer-Verlag Berlin Heidelberg 2012
131
fluid at the initial cross-section x ¼ 0 where the jet originates. These are the lines
of a discontinuity of the tangential velocity component. In viscous fluid such
a discontinuity cannot be sustained. As a result of the interaction of the fluid in
the jet with the ambient fluid, a thin mixing layer with a steep continuous velocity
profile forms in the vicinity of the lines a� a. The viscous stresses developed at thejet periphery lead to the expansion of the shear layers downstream. The inner
boundaries dinðxÞ of the shear layers reach the jet axis at some distance xin from
the nozzle exit encasing the core with a plug velocity profile which originated in the
slit. Outside the core domain the velocity profile in the jet cross-section decreases
from its value in the core and asymptotically approaches to zero at the outer
boundaries of the shear layers. In the cross-section x ¼ xin; the shear layers
merge. At x>xin smooth velocity profiles with characteristic maximum at the axis
are observed.
The size and geometry of the slit or a nozzle, as well as the velocity distribution
at its exit affect the flow field in submerged jets. The influence of the issuing
Fig. 6.1 Jet flows.
(a) Submerged jet: 1-constant
velocity core, 2-the far field
of the jet. (b) Wake behind
a solid body 1. (c) Buoyantplume rising from a hot body 1
132 6 Jet Flows
conditions is significant only in a relatively short section of the jet adjacent to the
slit or the nozzle. The length of this section is about 20–30 times the slit or nozzle
cross-sectional size for plane and axisymmetric jets, whereas in three-dimensional
(non-axisymmetric) jets it is significantly larger (Trentacoste and Sforzat 1967;
Krothapalli et al. 1981; Ho and Gutmark 1987; Hussain and Hussain 1989).
Therefore, at large x the flow field of submerged jets practically does not depend
on the details of the issuing conditions and is fully determined by the integral
characteristics of the jets. The latter express conservation of the momentum and
energy fluxes along the jets. For a submerged jet issuing from a thin tube the total
momentum flux is (Landau and Lifshitz 1987)
Jx ¼þPrr cos yds (6.1)
whereQ
rr ¼ Pþ rvr2 � 2m@vr=@r is the component of the tensor of the momen-
tum flux in the r-direction, dS ¼ r2 sin ydyd’ is the surface element, r and P are the
density and pressure, vr is the r-component of the velocity, m is the viscosity;
r; y and ’ are the spherical coordinate frame with the center located at the jet axis
at the tube exit.
The total energy flux in the jet can be expressed as (Vulis and Kashkarov 1965)
Q ¼þ
rvrðh� h1Þ � k@T
@r
� �ds (6.2)
Fig. 6.2 Sketch of
a submerged jet. The inner
and outer boundaries of the
boundary layer are denoted
as din and dout, respectively
6.1 Introductory Remarks 133
where h and T are the enthalpy and temperature, respectively, k is the thermal
conductivity, subscript1 corresponds to the undisturbed fluid far away from the jet
axis.
The submerged jets represent themselves boundary layers. Therefore, the bound-
ary layer theory can be used to study hydrodynamics of submerged jets. For laminar
planar submerged jets issuing from a slit, or a non-swirling (about the jet axis)
axisymmetric jets issuing from a pipe, the boundary layers equations have the
following form
ru@u
@xþ rv
@u
@y¼ m
yj@
@yyj@u
@y
� �(6.3)
ru@h
@xþ rv
@h
@y¼ k
yj@
@yyj@T
@y
� �(6.4)
Here (6.3) expresses the longitudinal momentum balance, (6.4)-the
corresponding energy balance; u and v are the longitudinal and transversal velocitycomponents, j ¼ 0 or 1 for the plane and axisymmetric jets, respectively; pressure
drop is omitted in (6.3), since pressure in the jets is practically constant over the
entire flow field. In (6.4) viscous dissipation is omitted, since it is negligible at
sufficiently small flow velocities.
The corresponding continuity equation reads
@ruyj
@xþ @rvyj
@y¼ 0 (6.5)
Using (6.5), one can transform the momentum and energy balance Eqs. 6.3 and
6.4 to the following form
@ru2yj
@xþ @ruvyj
@y¼ m
@
@yyj@u
@y
� �(6.6)
@ruDhyj
@xþ @rvDhyj
@y¼ k
@
@yyj@T
@y
� �(6.7)
where Dh ¼ h� h1.
The boundary conditions for planar and axisymmetric submerged jets read
y ¼ 0:@u
@y¼ @T
@y¼ @h
@y¼ 0; v ¼ 0 (6.8)
y ! 1 : u ! 0; T ! T1; Dh ! 0 (6.9)
134 6 Jet Flows
Integration of (6.6) and (6.7) across the jet cross-section yields the following
invariants
Jx ¼ð10
ru2yjdy ¼ const: (6.10)
Qx ¼ð10
ruDhyj ¼ const: (6.11)
It is emphasized that (6.10) expresses the fact that the total momentum flux Jxdoes not vary along the jet. Similarly, (6.11) expresses the fact that the total
excessive enthalpy flux Qx does not vary along the jet. Note that the expressions
for Jx and Qx for submerged turbulent jets are similar to those in (6.10) and (6.11).
However, in that case r; u; T and h imply their average values. In addition, it
should be assumed thatÐ10
u02 þ P� �
yjdy ¼ 0; where u0is the velocity fluctuation.
Swirling motion about the jet axis results in a non-uniform pressure distribution
across it. In this case the system of Eqs. 6.3 and 6.4 is supplemented by the
following two equations describing the balance between the centrifugal force due
to swirling and the corresponding pressure gradient across the jet and the azimuthal
momentum balance allowing for finding the swirling velocity component w
jw2
y¼ 1
r@P
@y(6.12)
ru@w
@xþ rv
@w
@yþ r
vw
y¼ m
1
yj@
@yyj@u
@y
� �� w
y2
� (6.13)
In addition, the longitudinal momentum balance of Eq. 6.6 is now modified by
the inclusion the longitudinal pressure gradient � @P=@x on its right-hand side.
Then, the following set of the integral invariants for the axisymmetric swirling jet is
obtained via integration of the modified momentum balance equation, as well as
(6.7), (6.12) and (6.13)
Jx ¼ð10
Pþ ru2 �
ydy (6.14)
Mx ¼ð10
ruwy2dy (6.15)
6.1 Introductory Remarks 135
Qx ¼ð10
ruDhydy (6.16)
where Mx is the total moment-of-momentum flux.
The integral invariants for different types of submerged jets are presented in
Table 6.1.
The integral invariants (6.1) and (6.10) express conservation of the momentum
flux which is brought by the fluid issued from the jet origin. The relations (6.1) and
(6.10) are not identical. The first one accounts for the full momentum flux around
the momentum source, i.e. represent itself the momentum flux through a closed
surface S (Fig. 6.3a), whereas the second one-only the momentum flux through a
cross-section AA normal to the jet axis (Fig. 6.3b). The invariant (6.1) and the
closed surface are used to solve the problem of the submerged jet in the framework
of the full Navier–Stokes equations, whereas the invariant (6.10) and the cross-
sections of the type AA are used to solve the problem in the framework of the
boundary layer theory. It is emphasized that in the framework of the boundary layer
theory the integral invariant (6.10) does not account for the flow ‘history’
(Konsovinous 1978). (Schneider 1985) Nevertheless, the relation (6.10) is a fairly
good approximation of the relation (6.1) and is widely used in the theory of jets.
6.2 The Far Field of Submerged Jets
The understanding that the far field of a jet flow does not depend on the issuing
conditions and the related simplifications plays a pivotal role in the boundary layer
theory of submerged jets. It allows one to decrease the number of factors which
should be accounted for and to develop an asymptotic model of the flow. Namely,
when a jet is assumed to be issued from a pointwise source, it is natural to assume
that the far field of velocity in the jet is determined by physical properties of fluid
and total momentum flux (but not by flow details at the nozzle/pipe exit). Therefore,
we can write the functional equation for the longitudinal velocity component u as
u ¼ fuðr; m; Jx; x; yÞ (6.17)
The functional equation (6.17) contains the set of parameters which determine
the velocity at any point of the far field of a submerged jet. Any change of the flow
conditions (for example the appearance of a co- or counter flow of the surrounding
fluid) requires an expansion of the set of the determining parameters on the right
hand side of Eq. (6.17). In particular, when r and m are constant, the fluid properties
are characterized by a single parameter, namely, the kinematic viscosity n ¼ m=r,as it follows from the boundary layer equations. In this case (6.17) reduces to
136 6 Jet Flows
Tab
le6.1
Jetinvariants
Jetgeometry
Jet’sschem
e
Integralinvariant
Dynam
icproblem
Thermal
problem
Planejet
J x¼Ð1 0
ru2dy
Qx¼Ð1 0
ruc p
T�T1
ðÞdy
Axisymmetricjet
J x¼
2pÐ1 0
ru2ydy
Qx¼
2pÐ1 0
ruc p
T�T1
ðÞdy
Radialjet
J x¼
2px
Ðþ1 �1ru
2dy
Qx¼
2px
Ðþ1 �1ru
c pT�T1
ðÞdy
Walljet
J x¼Ð1 0
ru2Ðy 0
rudy
�� d
yQ
x Pr¼
1¼ð1 0
ruc p
T�T1
ðÞðy 0
rudy
0 @1 A dy
Qx a
d¼ð1 0
ruc p
T�T1
ðÞdy
6.2 The Far Field of Submerged Jets 137
u ¼ fuðn; Ix; x; yÞ (6.18)
where Ix ¼ Jx=r is the kinematic momentum flux.
Similarly we write the functional equation for the jet thickness as
d ¼ fdðr; m; Jx; xÞ (6.19)
or
d ¼ fdðn; Ix; xÞ (6.20)
Since at y ¼ 0; u ¼ um, with um being the axial maximal velocity in a jet cross-
section, we obtain from (6.17) and (6.18) the following equations for the axial
velocity in the planar and axisymmetric (round) jets
um ¼ fuðr; m; Jx; xÞ (6.21)
or
um ¼ fuðn; Ix; xÞ (6.22)
In a cross-section of an axisymmetric jet it is possible to select two characteristic
parameters: the jet (boundary layer) thickness and the axial velocity. These
parameters can be taken as the governing ones. Then, we can assume that velocity
at any point of the jet depends on umðxÞ; d and the distance from the jet axis to the
point under consideration y: Then, (6.18) is replaced with the following one
u ¼ fumðum; d; yÞ (6.23)
Fig. 6.3 Sketch of a jet flow.
(a) Flow in a jet issued from a
thin tube (the Navier–Stokes
approximation). (b) Jet from a
pointwise momentum source
(the boundary layer
approximation)
138 6 Jet Flows
6.3 The Dimensionless Groups of Jet Flows
The dimensions of the governing parameters can be expressed in the framework of
any system of fundamental units. For the dimensional analysis of jet flows it is
convenient to use the modified LMT system that includes three different length
scales Lx, Ly; and Lz, respectively for the x, y and z directions. In this case the
dimensions of the characteristic parameters are
u½ � ¼ LxT�1; v½ � ¼ LyT
�1; x½ � ¼ Lx; y½ � ¼ Ly; r½ � ¼ L�1x L�1
y L�1z M;
m½ � ¼ L�1x LyL
�1z MT�1; n½ � ¼ L2yT
�1; d½ � ¼ Ly (6.24)
In addition, the dimension of the momentum flux is
Jx½ � ¼ Le1x Le2y L
e3z M
e4Te5 (6.25)
where ei 6¼ 0 (i ¼ 1, 2, 3, 4, 5) are constants which depend on the jet configuration
(Table 6.2).
Applying the Pi-theorem to (6.17)–(6.23), we can transform them to the follow-
ing canonical form
P ¼ ’jðP1;P2:::Pn�kÞ (6.26)
where n� k is the number of dimensionless groupsP1; P2:::; whereas n and k arethe number of the governing parameters and the parameters with independent
dimensions, subscript j ¼ u; d refers to the velocity or thickness of the jet,
respectively.
Among the numerous cases corresponding to different values of n� k, the most
important ones are the following two: (i) n� k ¼ 1; and n� k ¼ 0: The former
corresponds to a self-similar flow in which dimensionless velocity depends on a
single dimensionless variable, whereas the latter – to a constant dimensionless
velocity or jet thickness. In the latter case u and d can be expressed in the form of
an explicit function of the governing parameters. As can be seen from (6.24) and
(6.25), the parameters r; m; Jx and x on the right hand side in Eqs. (6.17), (6.19)
and (6.21) have independent dimensions. Accordingly, we obtain the following
equations for the dimensionless velocity and jet thickness
Table 6.2 Exponents eifor different jetsJet geometry e1 e2 e3 e4 e5Plane 1 0 �1 1 �2
Axisymmetric 2 0 �1 1 �2
Radial 2 0 �1 1 �2
Wall 1 0 �2 2 �3
6.3 The Dimensionless Groups of Jet Flows 139
Pu ¼ ’ðP1Þ (6.27)
Pum ¼ c1 (6.28)
Pd ¼ c2 (6.29)
where
Pu ¼ u
ra1ma2Ja3x xa4; P1 ¼ y
ra01ma
02J
a03
x xa04
(6.30)
Pu;m ¼ umra1ma2Ja3x xa4
(6.31)
Pd ¼ d
ra01ma
02J
a03
x xa04
(6.32)
and c1, c2 are constants.The unknown exponents ai and a
0i (i ¼ 1; 2; 3; 4) are found from the expressions
(6.30)–(6.32) accounting for the dimensions of the parameters on the left and right
hand sides of these there. As a result, we arrive at the following sets of algebraic
equations
XLx
: �a1 � a2 þ e1a3 þ a4 ¼ 1
XLy
: �a1 þ a2 þ e2a3 ¼ 0 (6.33)
XLz
: �a1 � a2 þ e3a3 ¼ 0
XM
: a1 þ a2 þ e4a3 ¼ 0
XT
: �a2 þ e5a3 ¼ �1
and
XLx
: �a01 � a
02 þ e1a
03 þ a
04 ¼ 0
140 6 Jet Flows
XLy
: �a01 þ a
02 þ e2a
03 ¼ 1 (6.34)
XLz
: �a01 � a
02 þ e3a
03 ¼ 0
XM
: a01 þ a
02 þ e4a
03 ¼ 0
XT
: �a02 þ e5a
03 ¼ 0
where the symbolsPLx
;PLY
;PLz
;PM;PT
refer to summation of the exponents of
lengths and mass time scales, respectively.
Bearing in mind (6.30)–(6.32), we present (6.27)–(6.29) as follows
u ¼ ’uð�Þ (6.35)
um ¼ c1ra1ma2Ja3x xa4 (6.36)
d ¼ c2ra01ma
02J
a03
x xa04 (6.37)
where u ¼ u=um; � ¼ y=d; and ai and a0i depend on ei which is different for
different jet geometry.
6.4 Plane Laminar Submerged Jet
Consider the application of the Pi-theorem to study characteristics of plane laminar
submerged jets (Fig. 6.4). We deal with only the far-field region of the jet, as
explained before, and select the following governing parameters: r; m; Jx; x and ywhich have dimensions listed in (6.24) and (6.25). Four of the five governing
parameters have independent dimensions, which yields (6.34)–(6.37) that deter-
mine velocity distribution in jet cross-section, the velocity variation along the jet
axis, and the boundary layer thickness. Taking into account the values of the
exponents ei ði ¼ 1; 2; 3; 4Þ in (6.25) (see Table 6.2), we find from (6.33) and
(6.34) the values of the exponents ai and a0i
a1 ¼ � 1
3; a2 ¼ � 1
3; a3 ¼ 2
3; a4 ¼ � 1
3; a
01 ¼ � 1
3; a
02 ¼
2
3; a
03 ¼ � 1
3; a
04
¼ 2
3(6.38)
6.4 Plane Laminar Submerged Jet 141
In accordance with (6.36) and (6.37), we obtain
um ¼ c1J2xr2n
� �1=3
x�1=3 (6.39)
d ¼ c2rn2
Jx
� �1=3
x2=3 (6.40)
Using (6.35), (6.39) and (6.40), it is possible to estimate several important
characteristics of plane laminar submerged jets, in particular, the local Reynolds
number. The latter is defined by the average velocity, the boundary layer thickness
and the kinematic viscosity
Red ¼ <u>dn
(6.41)
where <u> ¼ 1d
Ðd=2�d=2
udy is the average velocity in jet cross-section.
Taking into account (6.35), (6.39) and (6.40), we arrive at the following expres-
sion for the local Reynolds number
Red ¼ c3Jxx
rn2
� �1=3 ð1=2�1=2
’ð�Þd� (6.42)
where c3 ¼ c1c2:
Fig. 6.4 Plane laminar jet
142 6 Jet Flows
Equation 6.42 shows that the local Reynolds number in plane laminar submerged
jet increases downstream as x1=3: At sufficiently large values of x the local Reynoldsnumber exceeds the critical value corresponding to laminar-turbulent transition
(Fig. 6.5).
A higher total momentum flux or a lower fluid viscosity increase the local
Reynolds number Red and the laminar-turbulent transition cross-section appro-
aches the jet origin. The observations show that the laminar sections of plane
submerged jets exist up to Re0 ’ 30 (Re0 is based on the outflow jet velocity
and the slit width, Andrade 1939).
6.5 Laminar Wake of a Blunt Solid Body
As the second example application of the Pi-theorem to jet flows consider plane
laminar wake behind a blunt solid body in a uniform stream of viscous fluid
(Fig. 6.1b). Similarly to flows in submerged jets, flows in laminar wakes can be
characterized by an integral invariant which accounts for the body characteristics.
In order to find this invariant we use the boundary layer equation
ru@u
@xþ rv
@u
@y¼ m
@2u
@y2(6.43)
Fig. 6.5 Laminar-turbulent transition (in cross-section A-A) in a submerged jet
6.5 Laminar Wake of a Blunt Solid Body 143
Consider the relative longitudinal velocity component
u1 ¼ u1 � u (6.44)
where u1 is the free stream velocity; the body is considered to be at rest, while the
stream impinges on it with velocity u1.
At a large distance from the body u1 becomes sufficiently small. Then (6.43) can
be linearized and takes the following form
ru1@u1@x
¼ m@2u1@y2
(6.45)
Integrating (6.45) from y ¼ �1 to y ¼ þ1 across the wake and accounting for
the fact that @u1=@y ! 0 at y ! �1, we obtain
Jx ¼ð1
�1ru1dy ¼ const: (6.46)
Below it will be shown that um depends on the physical properties of fluid,
velocity of the free stream, as well as on the integral invariant Jx
u1 ¼ f ðr; m; Jx; u1x; yÞ (6.47)
where the dimensions of r; m; Jx; u1; x; and y in the LxLyLzMT system of units are
L�1x L�1
y L�1z M; L�1
x LyL�1z MT�1; L�1
z MT�1; LxT�1; Lx and Ly; respectively.
We also write the functional equations for u1m and dw
u1m ¼ fwðr; m; Jx; u1; xÞ (6.48)
dw ¼ fdwðr; m; Jx; u1; xÞ (6.49)
The functional equation for u1 can be rewritten as follows
u1 ¼ f ðu1m; dw; yÞ (6.50)
Equation 6.50 for u1 expresses the physically realistic assumption that velocity
at any point of a wake cross-section is determined by three parameters: velocity at
the wake axis u1m; the boundary layer (wake) thickness dw and the distance from
the wake axis to the point under consideration y:Applying the Pi-theorem to (6.48), we obtain
Pm ¼ ’mðP1Þ (6.51)
144 6 Jet Flows
where Pm ¼ u1m=ra1ma2ua31xa4 ; P1 ¼ Jx=ra01ma
02u
a031xa
04 ; and a1 ¼ a2 ¼ a4 ¼ 0;
a3 ¼ 1; a1 ¼ 1; a01 ¼ a
02 ¼ a
03 ¼ a
04 ¼ 1=2:
Therefore, (6.51) takes the form
u1:mu1
¼ ’m
Jx
rmu1xð Þ1=2u1
( )(6.52)
The application of the Pi-theorem to (6.50), yields
u1 ¼ ’wð�Þ (6.53)
where u1 ¼ u1=u1m; and � ¼ y=dw:In order to determine the dependence of Jx on the drag force acting on a blunt
solid body, we use the overall momentum and mass balances for the rectangular
control volume ABCD encompassing the body (Fig. 6.1b). They read
ru22h�W �ðþh
�h
ru2dy ¼ 0 (6.54)
ru12h�ðþh
�h
rudy ¼ 0 (6.55)
where W is the drag force acting on the solid body, and 2h is height of the contour
ABCD.
From (6.54) and (6.55) it follows that W equals to
W ¼ðþh
�h
ruðu1 � uÞdy (6.56)
Taking into account that in the limit h ! 1, we can rewrite (6.56) as
W ¼ð1
�1rðu1 � uÞu1dy � u1
ð1�1
ru1dy (6.57)
The comparison of (6.46), (6.56) and (6.57), reveals that
Jx ¼ W
u1(6.58)
6.5 Laminar Wake of a Blunt Solid Body 145
Accordingly, (6.52) takes the form
u1mu1
¼ ’m
W
rmu1xð Þ1=2( )
(6.59)
The wake thickness can be found directly from (6.49) or (6.53) and (6.57). From
(6.53) and (6.47) it follows that
dw ¼ W
ru1u1mÐ1
�1’wð�Þd�
(6.60)
6.6 Wall Jets over Plane and Curved Surfaces
Laminar wall jets represent themselves a “compound” boundary layers, which are
similar to the free boundary layer of submerged jets on the one side, and to the near-
wall (Blasius) boundary layers on the other side (close to the wall). The presence of
the wall friction results in variation of the total momentum flux downstream the
wall jets. Akatnov (1953) and Glauert (1956) showed that there is an integral
invariant Jx ¼Ð10
ru2Ðy0
rudy� �
dy which remains constant along the laminar wall
jets over a plane wall.
As in the previous cases of jet flows considered in the present chapter, we begin
our consideration with formulation of the functional equations for the maximal
longitudinal velocity and the boundary layer thickness in the jet. These equations
are identical to (6.19) and (6.21). Since all the governing parameters in the
functional equation for the maximal longitudinal velocity and the boundary thick-
ness have independent dimensions, we obtain
um ¼ cura1ma2Ja3x xa4 (6.61)
d ¼ cdra01ma
02J
a03
x xa04 (6.62)
where cu and cd are constants, and the exponents ai and a0i are given by
a1 ¼ � 1
2; a2 ¼ � 1
2; a3 ¼ 1
2; a4 ¼ � 1
2; a
01 ¼ � 1
4; a
02 ¼
3
4; a
03 ¼ � 1
4; a
04
¼ 3
4(6.63)
Then the expressions for um and dd read
146 6 Jet Flows
um ¼ cuJxrm
� �1=2
x�1=2 (6.64)
dd ¼ cdm3
rJx
� �1=4
x3=4 (6.65)
It is emphasized that (6.39) and (6.40), as well as (6.64) and (6.65) coincide with
the expressions obtained as the exact analytical solutions of the boundary layer
equations (Table 6.3) (Vulis and Kashkarov 1965).
� ¼ 1
2F1ln
Fþ ffiffiffiffiffiffiffiffiffiffiFF1
p þ F1
ð ffiffiffiffiffiffiffiF1
p � ffiffiffiF
p Þ2þ
ffiffiffi3
p
F1ðarctan 2
ffiffiffiF
p þ ffiffiffiffiffiffiffiF1
pffiffiffiffiffiffiffiffiffi3F1
p � arctan1ffiffiffi3
p Þ,
� ¼ Byxb, F1 ¼ 1:7818, the dimensionless value J1 ¼Ð10
FF0d�:
The approach outlined above can be also used for the analysis of wall jets over
curved surfaces. This type of jets was studied theoretically by Wygnanski and
Champagne (1968). They found the integral invariant of the problem and the self-
similar solutions for both concave and convex surfaces in the case when the local
radius of curvature varies as x3=4(with x being reckoned along the surface). Below
we consider wall jets over curved surfaces in the framework of the dimensional
analysis.
In the case when the ratio of the characteristic jet thickness to the local radius of
curvature R is sufficiently small and curvature variations occur in such a way that
dR=dx � 1; the set of the governing boundary layer equations has the following form
ru@u
@xþ 1þ y
R
� �v@u
@yþ r
uv
R¼ � @P
@Xþ m 1þ y
R
� � @2u@y2
þ mR
@u
@r(6.66)
ru2
R¼ @P
@y(6.67)
Table 6.3 Characteristic constants for laminar and turbulent jetsa
Jet geometry
Laminar jet Turbulent jet
’ð�ÞAi Bi ai bi Ai Bi ai bi
Plane jet 12
ffiffi½p 3� 3J2x4r2n
12
ffiffi½p 3� Jx6rn2
� 13
� 23
ffiffiffiffiffiffiffiffiffi3Jx
8rk1=2
q1
2k1=2� 1
2� 1 ’Pð�Þ
Axisym-metric jet 3Jx8prn
ffiffiffiffiffiffiffiffiffi3Jx
8prn2
q � 1 � 1ffiffiffiffiffiffiffi3Jx8prk
q1
k1=2� 1 � 1 ’Að�Þ
Radial jet 14
ffiffi½p 3� 9J2x2p2r2n
12
ffiffi½p 3� 3J2x4prn2
� 1 � 1ffiffiffiffiffiffiffiffiffiffiffiffiffi
3Jx16prk1=2
q1ffiffiffiffi2k
p � 1 � 1 ’Rð�ÞWall jet
ffiffiffiffiffiffiffiffiffiffiJx
4r2nJ1
q12
ffiffiffi4
p 3J2x J�11
4r2n2� 1
2� 3
4� � � � ’Wð�Þ
aA; B; a and b are the coefficients in the expressions for um and ’ :
um ¼ Axa; ’Pð�Þ ¼ ð1� tanh2�Þ; ’A ¼ ð1þ 18�Þ�2; ’Rð�Þ ¼ ð1� tanh2�Þ,
’Wð�Þ ¼ ðF132F
12 � F2Þ, the function F is determined by the transcendental equation
6.6 Wall Jets over Plane and Curved Surfaces 147
@ru@x
þ @
@y1þ y
R
� �rv
n o¼ 0 (6.68)
where x and y are the coordinates along and normal to the surface, u and v are thevelocity components in the x and y directions. When the local radius of curvature
R>0, the wall is convex outwards, whereas R<0 the wall is concave.
Equations (6.66)–(6.68) are subjected to the following boundary conditions at
the wall (y ¼ 0) and far from it
y ¼ 0; u ¼ v ¼ 0; y ¼ 1; u ¼ 0 (6.69)
Integrating (6.66)–(6.68) yields the invariant
Jx ¼ð10
ru 1þ y
R
� � ð10
ru2dy� 1
R
ð10
ru2ydy
8<:
9=;dy ¼ const (6.70)
where Jx½ � ¼ LxL�2z M2T�3:
At R ! 1, the invariant (6.70) takes the form of the Akatnov’s invariant
mentioned above. It is emphasized that there are two identical forms of the
integral invariant of the wall jet near a straight wall: (i) Akatnov’s (1953) invariant-
Jx ¼Ð10
ru2Ð1y
rudy
!dy; and Glauert’s (1956) invariant-Jx ¼
Ð10
ruÐ1y
ru2dy
!dy:
It is easy to show that they are identical.
From the physical point of view, we note that the velocity and the jet thickness in
the far field of the wall jet over a curved surface should depend on the fluid
properties, the integral invariant of the jet, as well as on the radius of curvature of
the wall
u ¼ fuðr; m; Jx;R; x; yÞ (6.71)
um ¼ fumðr; m; Jx;R; xÞ (6.72)
d ¼ fdðr; m; Jx;R; xÞ (6.73)
where u is the longitudinal velocity, um its local maximal value, and d is the jet
thickness.
Equation 6.71 can be presented in the form
u ¼ fuðum; d;R; yÞ (6.74)
Since two of the four governing parameters in (6.74) have independent
dimensions, it reduces to
148 6 Jet Flows
u ¼ ’uð�; rÞ (6.75)
where u ¼ u=um; � ¼ y=d; and r ¼ R=d:Applying the Pi-theorem to (6.72), we arrive at the following expression
um ¼ Jxrm
� �1=21ffiffiffix
p ’um
R
m3x3=rJxð Þ� 1=2
(6.76)
The self-similar solution of the present problem exists when R ¼ cx3=4; where cis a constant (Wygnanski and Champagne 1968). Substitution this expression in
(6.76) yields
um ¼ Jxrm
� �1=21ffiffiffix
p ’um
c
m3=rJxð Þ1=4( )
(6.77)
6.7 Buoyant Jets (Plumes)
The buoyant jets belong to a special class of jet flows developing due to the
influence of buoyancy forces on fluid motion (Fig. 6.1c). They are typical for
a number of phenomena in nature and engineering: mass, momentum and heat
transfer in the atmosphere and oceans, pollutant dispersal, etc. (Turner 1969; 1986;
Jaluria 1980). The buoyant jets were a subject of a number of theoretical and
experimental investigations of complex flows developing under the influence of
the inertial, viscous and buoyancy forces (Zel’dovich 1937; Batchelor 1954;
Morton et al. 1956).
Below we consider only a few simplest examples of buoyant jets that illustrate
the application of the Pi-theorem to buoyancy-driven jet flows. Following
Zel’dovich (1937), we consider laminar vertical buoyant jets (plumes) over
a pointwise horizontal wire or a pointwise sphere treated as sources of heat. In
the framework of the Boussinesq approximation, the boundary layer equations
describing flows in laminar buoyant jet, which are generated by such sources read
u@u
@xþ v
@u
@y¼ n
yk@
@yyk@u
@y
� �þ bg# (6.78)
@u
@xþ 1
yk@vyk
@y¼ 0 (6.79)
u@#
@xþ v
@#
@y¼ a
yk@
@yyk@#
@y
� �(6.80)
6.7 Buoyant Jets (Plumes) 149
where n and a; are the kinematic viscosity and thermal diffusivity, respectively, b is
the thermal expansion coefficient, u and v are the longitudinal and transverse
components of velocity in the vertical and horizontal directions x and y; respec-
tively (cf. Fig. 6.1c), # ¼ T � T1;with T1 being the ambient temperature, g is the
gravity acceleration, k ¼ 0 or 1 for the plane or axisymmetric problems,
respectively.
The solutions of (6.78)–(6.80) are subjected to the following boundary
conditions
y ¼ 0@u
@y¼ 0; v ¼ 0;
@#
@y¼ 0; y ! 1 u ! 0; v ! 0 (6.81)
Rewriting (6.80) in the divergent form
@u#
@xþ 1
yk@v#yk
@y¼ a
yk@
@yyk@#
@y
� �(6.82)
and integrating it in y from the wake center y ¼ 0 to its edge y¼1, we arrive at the
following integral invariant expressing conservation of the excessive convective
heat flux along the plume
Q ¼ð10
u#ykdy ¼ const: (6.83)
Equations 6.78–6.80 with the boundary conditions (6.81) and the integral invari-
ant (6.83) show that velocity and temperature in buoyant laminar vertical jets
depend on six dimensional parameters, namely,
gb½ � ¼ Ly�1T�2; n½ � ¼ L2T�1; a½ � ¼ L2T�1; Q½ � ¼ L2þkyT�1; x½ � ¼ L; y½ �¼ L (6.84)
Therefore, the functional equations for the velocity and temperature fields can be
presented by the following functions
u ¼ f1ðgb; n; a;Q; x; yÞ (6.85)
# ¼ f2ðgb; n; a;Q; x; yÞ (6.86)
In order to reduce the number of the governing parameters it is convenient to
introduce new variables
½ev � ¼ v
n1=2
h i¼ T�1=2; ½y�� ¼ y
n1=2
h i¼ T1=2 (6.87)
150 6 Jet Flows
Then (6.78)–(6.80) take the form
u@u
@xþ ev @u
@ey ¼ 1ey k
@
@ey ey @u@ey
� �þ bg# (6.88)
@u
@xþ 1ey k
@evyk@ey ¼ 0 (6.89)
u@#
@xþ ev @#
@ey ¼ 1
Pr
1ey k
@
@ey ey k @#
@ey� �
(6.90)
The boundary and integral conditions in new variables read
ey ¼ 0@u
@ey ¼ 0; ev ¼ 0;@#
@ey ¼ 0 (6.91)
ey ! 1 u ! 0; # ! 0 (6.92)
Q�1 ¼
ð10
u#ey kdey (6.93)
where Q�1
� ¼ LyTðk�1Þ 2= .
Then, the velocity and temperature fields are given by the following functional
equations (where it is assumed that the Prandtl number equals one)
u ¼ f1ðbg;Q�1; x; ey Þ (6.94)
# ¼ f2ðbg;Q�1; x; ey Þ (6.95)
The axial velocity in the buoyant jet corresponds to ey ¼ 0, which reduces (6.94)
to the following equation
um ¼ f1�ðbg;Q�1; xÞ (6.96)
All the governing parameters in (6.96) have independent dimensions. Therefore,
in accordance with the Pi-theorem, (6.96) takes the form
um ¼ cðbgÞa1ðQ�1Þa2ðxÞa3 (6.97)
where c is a constant and the exponents ai are determined using the principle of the
dimensional homogeneity as
6.7 Buoyant Jets (Plumes) 151
a1 ¼ a2 ¼ 2
5� k; a3 ¼ 1� k
5� k(6.98)
Accordingly, (6.97) reads
um ¼ c bgQ�1
�2=ð5�kÞxð1�kÞ=ð5�kÞ
n o(6.99)
Since Q�1 ¼ Q=nðkþ1Þ=2; (6.99) takes the form
um ¼ bgQð Þ2=ð5�kÞn�ðkþ1Þ=ð5�kÞxð1�kÞ=ð5�kÞn o
(6.100)
Taking in (6.100) k ¼ 0 or 1, we arrive at the following expressions for the axial
velocity in plane and axisymmetric buoyant laminar jets (plumes)
um;plane ¼ c bgQð Þ2=5n�1=5x1=5n o
(6.101)
um;axis: ¼ c bgQð Þ1=2n�1=2n o
(6.102)
According to the Pi-theorem (6.94) corresponds to the case of n�k ¼ 1. There-
fore, it can be reduced to the following dimensionless form
P ¼ ’ðP1Þ (6.103)
where P ¼ u= bgð Þa1ðQ�1Þa2ðxÞa3 ; P ¼ y
�= bgð Þa
01ðQ�
1Þa02ðxÞa
03 ; and a1 ¼ a2 ¼
2=ð5� kÞ; a3 ¼ ð1� kÞ=ð5� kÞ; a01 ¼ a
02 ¼ �1=ð5� kÞ; a0
3 ¼ 2=ð5� kÞ.For k ¼ 0 or 1, (6.103) yields the longitudinal velocity profiles in plane and
axisymmetrical buoyant jets as
uplane ¼ bgQð Þ2=5n�1=5x1=5’plane y bgQð Þ1=5=n3=5x2=5n o
(6.104)
uaxis: ¼ bgQð Þ1=2n�1=2’axis: y bgQð Þ1=4=n3=4x1=2n o
(6.105)
Similarly the temperature distributions in buoyant laminar jets are given by
yplane ¼ Q4=5ðbgÞ�1=5n�2=5x�3=5’�plane yðbgQÞ1=5=x2=5y3=5
n o(6.106)
yaxis: ¼ Qn�1x�1’�axis: yðbgQÞ1=4=x1=2n3=4n o
(6.107)
The expressions (6.104)–(6.107) determine only the form of the self-similar
solutions for the velocity and temperature distributions in laminar buoyant jets.
152 6 Jet Flows
They can be also used to establish the exact analytical solution of this problem. In
particular, substitution of the expressions (6.104)–(6.107) into (6.78)–(6.80) allows
transformation of the system of the governing partial differential equation into
the system of ordinary differential equation for the unknown functions ’plane;’axisymmetric; ’�
plane; and ’�axisymmetric: It is emphasized that that the analytical
solution of this problem is found without any particular hypothesis about the
mechanism of mixing of hot fluid with the surrounding fluid in the buoyant jets.
It is interesting to note that the Reynolds numbers in plane and axisymmetric
laminar buoyant jets increase with x
Replane ¼ bgQx3=n3 (6.108)
Reaxisymmetric ¼ bgQx2=n3 (6.109)
That means that there is some critical distance xcr from the source where
transition of laminar flow into the turbulent one inevitably occur (Zel’dovich 1937).
The developed approach also allows the analysis of the velocity and temperature
fields in turbulent buoyancy jets. In this case the system of the governing equations
is written for the average flow characteristics and should be closed using a semi-
empirical model of turbulence, in particular, the Prandtl mixing length model.
Applying the Pi-theorem it is not possible to obtain the following expressions for
the longitudinal velocity and temperature distributions in plane and axisymmetric
turbulent jets. In particular, for plane turbulent jets
uT ¼ bgQx
� �1=3
’planeðy x= Þ; yT ¼ Q2=3ðbgÞ�1=3x�5=3’planeðy xÞ= (6.110)
and for the axisymmetric jets
uT ¼ bgQ x=ð Þ1=3’axis:ðy x= Þ; yT ¼ Q2=3 bgð Þ�1=3x�5=2’axis: y x=ð Þ (6.111)
Note that the expressions (6.110) and (6.111), as any expressions derived using
semi-empirical models of turbulence incorporate some constants involved in
these models.
Studies of turbulent buoyant jets widely use models based on the so-called
entrainment hypothesis which assumes a certain relation of the mean inflow
velocity across the edge of a turbulent jet with some characteristic velocity in
a given cross-section (Batchelor 1954; Morton et al. 1956). In the framework
of such models the following system of the “entrainment” equations can be
formulated
d
dzb2uum � ¼ 2abuum (6.112)
6.7 Buoyant Jets (Plumes) 153
d
dz
1
2b2uu
2u
� �¼ l2b2ugym (6.113)
d
dz
l2b2uumgym1þ l2
� �¼ �b2uumN
2ðzÞ (6.114)
where a is the entrainment constant, l ¼ by=bu is the ratio of the widths of the
temperature and velocity profiles (l is assumed to be constant-l � 1:2Þ; NðzÞ ¼�g=r1ð Þ dr0=dzð Þ; with r0 and r1 being the density at a given height z and the
reference density in the environment. The change of the characteristic scales of
the problem with height z can be easily found by considering the dimensions of
the parameters involved. As a result, the velocity at the axis of an axisymmetric jet
um is determined as
um ¼ cM1=2z�1 (6.115)
where rM ¼ 2pÐ10
ru2��z¼0
rdr is the momentum flux at z ¼ 0; and c is a constant.
A number of instructive examples of the application of the entrainment model to
flows in buoyant jets of different types can be found in the surveys of Turner (1969)
and List (1982), as well as in the original works of Morton (1957, 1959), Turner
(1966, 1986), Papanicolaou and List (1982), Baines et al. (1990), and Bloomfield
and Kerr (2000).
Problems
P.6.1. Determine the velocity distribution along the axis of plane laminar
submerged jet.
The governing parameters in the present case are: n½ � ¼ L2yT�1; Ix½ � ¼
L2xLyT�2; and x½ � ¼ Lx: Therefore, the functional equation for the axial velocity
reads
um ¼ fmðn; Ix; xÞ (P.6.1)
Since the parameters n; Ix; and x have independent dimensions, (P.4.1) reduces
to the following one
um ¼ c1na1 Ia2x xa3 (P.6.2)
where c1 is constant.
154 6 Jet Flows
Taking into account the dimensions of um; n, Ix and x, we find the values of the
exponents ai: a1 ¼ �1=3; a2 ¼ 2=3; a3 ¼ �1=3. Substitution of the values of a1;a2 and a3 in (P.6.2) yields
um ¼ c13
ffiffiffiffiffiI2xnx
r(P.6.3)
P.6.2. Show that the local Reynolds number Red ¼ <u>d=n in the axisymmetric
laminar submerged jet is independent of x.The functional equations for u; um and d have the following form
u ¼ fuðr; m; Jx; x; yÞ (P.6.4)
um ¼ fu:mðr; m; Jx; xÞ (P.6.5)
d ¼ fdðr; m; Jx; xÞ (P.6.6)
where the governing parameters have the dimensions
r½ � ¼ L�1x L�1
y L�1z M; m½ � ¼ L�1
x LyL�1z MT�1; Jx½ � ¼ LxLyL
�1z MT�2; x½ �
¼ Lx; y½ � ¼ Ly: (P.6.7)
Equation (P.6.4) can be presented in the form
u ¼ fuðum; d; yÞ (P.6.8)
Applying the Pi-theorem to (P.6.8), we reduce it to the dimensionless form
u ¼ ’uð�Þ (P.6.9)
where u ¼ u=um, and � ¼ y=d:Bearing in mind that r; m; Jx and x have independent dimensions, we present
(P.6.5) and (P.6.6) as follows
um ¼ c1ra1ma2Ja3x xa4 (P.6.10)
d ¼ c2ra01ma
02J
a03
x xa04 (P.6.11)
where c1 and c2 are constants, and the exponents ai and a0i are equal to a1 ¼ 0;
a2 ¼ �1; a3 ¼ 1; a4 ¼ �1; a01 ¼ �1=2; a
02 ¼ 1; a
03 ¼ �1=2, and a
04 ¼ 1: Then,
we obtain
Problems 155
um ¼ c1Jxrnx
(P.6.12)
d ¼ c2rnx
rJxð Þ1=2(P.6.13)
Taking into account that <u> ¼ umÐþ1=2
�1=2
’ð�Þ�d�; we find
Red ¼ AJxrn2
� �1=2
¼ const (P.6.14)
where A ¼ c1c2Ð1=2
�1=2
’ð�Þ�d�:
P.6.3 Describe the velocity variation along the axis of a radial laminar
submerged jet (Table 6.1).
The axial velocity of radial jets depends on four governing parameters: ½r� ¼L�1x L�1
y L�1z M; ½m� ¼ L�1
x LyL�1z MT�1; ½Jx� ¼ L�2
y L�1z MT�2 and ½x� ¼ Lx that have
independent dimensions. According to the Pi-theorem,
um ¼ cra1ma2Ja3x xa4 (P.6.15)
where c is a constant, and the exponents ai have the following values: a1 ¼ �1=3;a2 ¼ �1=3; a3 ¼ 2=3; and a4 ¼ �1:
As a result, we obtain the following expression for the axial velocity in radial
laminar submerged jets
um ¼ c3
ffiffiffiffiffiffiffiJ2xr2n
sx�1 (P.6.16)
References
Abramovich GN (1963) The theory of turbulent jets. MIT Press, Boston
Akatnov NI (1953) Development of two-dimensional laminar incompressible jet near a rigid wall.
Proc Leningrad Polytec Inst 5:24–31
Andrade EN (1939) The velocity distribution in liquid-into-liquid jet. The plane jet. Proc Phys Soc
London 51:748–793
Baines WD, Turner JS, Campbell IH (1990) Turbulent fountains in an open chamber. J Fluid Mech
212:557–592
Batchelor GK (1954) Heat convection and buoyancy effects in fluid. Quart J Roy Meteor Soc
80:339–358
Bloomfield LJ, Kerr RC (2000) A theoretical model of a turbulent fountain. J Fluid Mech
424:197–216
156 6 Jet Flows
Glauert MB (1956) The wall jet. J Fluid Mech 1:625–643
Hinze JO (1959) Turbulence. An introduction to its mechanism and theory. McGraw Hill Book
Company, New York
Ho CM, Gutmark E (1987) Vortex induction and mass entrainment in a small-aspect-ratio elliptic
jet. J Fluid Mech 179:383–405
Hussain F, Hussain HS (1989) Elliptic jets. Part 1. Characteristics of unexcited and excited jets.
J Fluid Mech 208:259–320
Jaluria Y (1980) Natural convective heat and mass transfer. Pergamon, Oxford
Konsovinous NS (1978) A note on the conservation of the axial momentum of turbulent jet. J Fluid
Mech 87:55–63
Korthapalli A, Baganoff D, Karamcheti K (1981) On the mixing of rectangular jet. J Fluid Mech
107:201–220
Landau LD, Lifshitz EM (1987) Fluid mechanics, 2nd edn. Pergamon, London
List EJ (1982) Turbulent jets and plumes. Annu Rev Fluid Mech 14:189–212
Morton BR (1957) Buoyant plumes in moist atmosphere. J Fluid Mech 2:127–144
Morton BR (1959) Forced plumes. J Fluid Mech 5:151–163
Morton BR, Taylor GI, Turner JS (1956) Turbulent gravitational convection from maintained and
instantaneous sources. Proc Roy Soc London A 234:1–23
Pai SI (1954) Fluid dynamics of jets. D. Van Nastrand Company, New York
Papanicolaou PN, List EJ (1982) Investigations of round vertical turbulent buoyant jets. J Fluid
Mech 195:341–391
Schlichting H (1979) Boundary layer theory, 8th edn. Springer, Berlin
Schneider W (1985) Decay of momentum flux in submerged jets. J Fluid Mech 154:91–110
Townsend AA (1956) The Structure of turbulent shear flow. Cambridge University Press,
Cambridge
Trentacoste N, Sforzat P (1967) Further experimental results for three-dimensional free jets.
AIAA J 5:885–891
Turner JS (1966) Jets and plumes with negative or reversing buoyancy. J Fluid Mech 26:729–792
Turner JS (1969) Buoyant plumes and thermals. Annu Rev Fluid Mech 1:29–44
Turner JS (1986) Turbulent entrainment: the development of the entrainment assumption and its
application to geophysical flows. J Fluid Mech 173:431–471
Vulis LA, Kashkarov VP (1965) The theory of viscous fluid jets. Nauka, Moscow (in Russian)
Wygnanski I, Champagne FH (1968) The laminar wall jet over curved surface. J Fluid Mech
31:459–465
Zel’dovich YaB (1937) Limiting laws of free-rising convective flows. J Exp Theoret Phys
7:1463–1465. The English translation in: Ya.B. Zel’dovich: Selected Works of Ya.B.
Zel’dovich, vol. 1. Chemical Physics and Hydrodynamics. Limiting laws of freely rising
convective currents (Princeton Univ. Press, Princeton, 1992).
References 157
.
Chapter 7
Heat and Mass Transfer
7.1 Introductory Remarks
This Chapter deals with processes of heat and mass transfer in solid, liquid and
gaseous media. They have important implications for understanding various natural
phenomena as well as technological processes. A vast literature is devoted to heat
and mass transfer in motionless and moving media. Tens of thousands of journal
publications contain detailed information on modern methods of investigation of
heat and mass transfer problems (e.g. mathematical modeling, measuring thermohy-
drodynamical quantities, etc.). The results of numerous theoretical and experimental
researches in this field are generalized in a number of well-known monographs on
fluid dynamics (Landau and Lifshitz 1987; Schlichting 1979; Levich 1962), heat and
mass transfer (Kutateladze 1963; Spalding 1963; Kays 1975;White 1988), as well as
in reference books (Rohsenow et al. 1998; Kaviany 1994).
Heat and mass transfer in various media occur under the interaction of different
factors, which involve the effects of physical properties of the matter, its equation
of state and regime of motion, contact with the surrounding fluid or solid surfaces,
etc. In the general case the relevant physical processes are described by a system of
coupled non-linear partial differential equations that include the continuity,
momentum, energy and species conservation equations. This system of equations
should be supplemented by the equation of state and correlations determining the
dependences of physical properties of the matter on temperature and species
concentrations. Solving such complicated system of equations entails great
difficulties. Therefore, studing heat and mass transfer processes, as a rule, employs
several simplifying assumptions that lead to an approximate solution of the problem
at hand. A qualitative analysis of such complex phenomena as heat and mass
transfer in continous media is signicantly facilitated by applying the dimensional
analysis, in particular, the approach based on the Pi-theorem. The results discuss in
the present chapter demonstrate the applications of the Pi-theorem to conductive
heat transfer in media with constant and temperature-dependent thermal diffusivity,
convective heat transfer under the conditions of forced, natural and mixed
L.P. Yarin, The Pi-Theorem, Experimental Fluid Mechanics,
DOI 10.1007/978-3-642-19565-5_7, # Springer-Verlag Berlin Heidelberg 2012
159
convection, as well as heat and mass transfer in laminar boundary layers and
laminar pipe and jet flows. Using the Pi-theorem for studing heat and mass transfer
under the conditions of phase change is also discussed in this Chapter.
7.2 Conductive Heat and Mass Transfer
7.2.1 Temperature Field Induced by Plane InstantaneousThermal Source
Consider the evolution of the temperature field due to a plane instantaneous thermal
source of strength Q releasing heat at t ¼ 0 and x ¼ 0 in an infinite medium with
constant (temperature-independent) physical properties (Fig. 7.1).
The source can be presented as
q ¼ QdðxÞdðtÞ (7.1)
where Q is a constant, and dð�Þ is the delta-function.The medium is at rest, heat is transferred by conduction only, and the thermal
balance equation that describe the one-dimensional excessive (relative to the initial
one) temperature field Tðx; tÞ at t>0 reads
@T
@x¼ a
@2T
@x2(7.2)
where a is the thermal diffusivity.
Integrating (7.2) by x from �1 to þ1 and accounting for the boundary
conditions at infinity @T=@xj�1 ¼ @T=@xj1 ¼ 0; we find that Q ¼ Ðþ1
�1Tdx
¼ const, i.e. Q is the problem invariant. At the initial time moment t ¼ 0 the
excessive temperature equals zero at any x>0. That means that the previous integral
is indeed equal to the heat source strength, while at t¼0 the temperature field is
Fig. 7.1 Temperature
distribution corresponding to
a plane instantaneous thermal
source acting at t ¼ 0 at
x ¼ 0 in an infinite medium
with constant properties. The
snapshots shown correspond
to ti>0
160 7 Heat and Mass Transfer
given as Tðx; 0Þ ¼ QdðxÞ: Then, the functional equation for the temperature
field reads
T ¼¼ f ða;Q; t; xÞ (7.3)
Analyzing the dimensions of the governing parameters ( a½ � ¼ L2T�1; Q½ � ¼ Ly;t½ � ¼ T; x½ � ¼ L, with y being the temperature scale of temperature and applying the
Pi-theorem, we arrive at the following equation
T ¼ Qffiffiffiffiffiax
p ’xffiffiffiffiat
p� �
(7.4)
where ’ ¼ ’ð�Þ; and � ¼ x=ffiffiffiffiat
p:
7.2.2 Temperature Field Induced by a Pointwise InstantaneousThermal Source
The approach of the previous sub-section can be also applied to the evolution of the
excessive temperature field triggered by a pointwise thermal source of strength Q,which acted at t¼0 and r¼0 in an infinite medium with constant thermal diffusivity
q ¼ QdðrÞdðtÞ (7.5)
where Q is a constant, and r is the radial coordinate in the spherical coordinate
system centered at the heat source.
The thermal balance equation that describe the evolution of the temperature field
at t>0 reads1
@T
@r¼ a
1
r2@
@rr2@T
@r
� �(7.6)
Integrating (7.6) by r from r ¼ 0 to r ¼ 1 and accounting for the boundary
conditions @T=@rjr¼0 ¼ @T=@rjr¼1 ¼ 0, yields the following invariant
ð10
Tr2dr ¼ Q ¼ const: (7.7)
where ½Q� ¼ L3y:
1 Equation 7.6 accounts for the spherical symmetry of the temperature field.
7.2 Conductive Heat and Mass Transfer 161
Then, the excessive temperature field satisfies the following functional equation
T ¼ f ða;Q; t; rÞ (7.8)
Taking into account the dimensions of the governing parameters in (7.8) and
using the Pi-theorem, we arrive at the dimensionless equation
P ¼ ’ðP1Þ (7.9)
where P ¼ T=aa1Qa2 ta3 ; and P1 ¼ r=aa01Qa
02 ta
03 .
Determining the exponent as a1 ¼ �3=2; a2 ¼ 1; a3 ¼ �3=2; a01 ¼ 1=2; a
02 ¼
0 and a03 ¼ 1=2, we arrive at the following expression for the temperature field
T ¼ Q
atð Þ3=2’
r
ðatÞ1=2 !
(7.10)
7.2.3 Evolution of Temperature Field in Mediumwith Temperature-Dependent Thermal Diffusivity(The Zel’dovich-Kompaneyets Problem)
The present sub-section is devoted to the evolution of temperature field in response
to an instantaneous plane energy source in medium which thermal diffusivity
depending on temperature. Very strong heat release in a substance is accompanied
by temperature rise of the order of tens or even hundred of thousands degrees. In
such cases the energy transport occurs mainly by radiation. Under these conditions
the radiant thermal diffusivity coefficient depends on temperature and can be
expressed as (Zel’dovich and Kompaneyets 1970; Zel’dovich and Raizer 2002)
w ¼ aTn (7.11)
where a and n are given constants, in particular, a is dimensional, a½ � ¼ L2T�1y�n
and n is dimensionless, n½ � ¼ 1:According to (7.11), the radiant thermal diffusivity coefficient w approaches to
zero at T ! 0: At high temperature in a heated zone Th � T1 ðTh and T1 are the
temperatures in the heated zone and the surrounding medium, respectively) it is
possible to assume that ambient temperature equals zero, i.e. T1 ¼ 0: In this case
heat can not be transferred instantaneously to large distances from the thermal
source. It spreads over substance with finite speed, so that there exists some
boundary that separate the heated zone from the cooled undisturbed one. In this
case head spreads in the form of a thermal wave as is shown in Fig. 7.2.
162 7 Heat and Mass Transfer
Let at t¼0 in plane x ¼ 0 thermal energy of E (say, Joule) is released per 1m2 of
surface. The evolution of the temperature field at t>0 is described by the thermal
balance equation
@T
@t¼ @
@xw@T
@x
� �(7.12)
with the boundary conditions
x ! 1; T ! 0; x ¼ 0;@T
@x¼ 0 (7.13)
where the dependence of the radiant thermal diffusivity coefficient on temperature
is given by (7.11).
Integrating (7.12) in x from �1 to 1, we obtain the invariant of the present
problem
Q ¼ð1
�1Tdx (7.14)
where ½Q� ¼ ½E=rcP� ¼ Ly, where r and cP are density and the specific heat at
constant pressure of the matter, respectively.
From (7.11), (7.12) and (7.14) it follows that there are the following governing
parameters of the problem: two constants a and Q and two variables x and t
T ¼ f ða;Q; x; tÞ (7.15)
It is seen that three of the four governing parameters have independent
dimensions. Then, in accordance with the Pi-theorem (7.15) reduces to the follow-
ing dimensionless form
P ¼ ’ðP1Þ (7.16)
where P ¼ T=aa1Qa2 ta3 ; and P1 ¼ x=aa01Qa
02 ta
03 .
T
0 x
t1t2t3
Fig. 7.2 Temperature
distribution in response to a
plane instantaneous thermal
source at t ¼ 0 at x ¼ 0 in
medium with temperature-
dependent thermal diffusivity
7.2 Conductive Heat and Mass Transfer 163
Taking into account the dimensions of the parameters involved, we arrive at the
system of the six algebraic equations for the exponents ai and a0i:
2a1 þ a2 ¼ 0; 2a01 þ a
02 ¼ 1
� a1 þ a3 ¼ 0 ;�a01 þ a
03 ¼ 0 (7.17)
� na1 þ a2 ¼ 1; na01 þ a
03 ¼ 0
From (7.17) it follows that
a1 ¼ � 1
nþ 2; a2 ¼ 2
nþ 2; a3 ¼ � 1
nþ 2; a
01
¼ 1
nþ 2; a
02 ¼
n
nþ 2; a
03 ¼
1
nþ 2
(7.18)
Then (7.16) takes the form
T ¼ Q2
at
� �1=ðnþ2Þ’
x
aQntð Þ1=ðnþ2Þ
( )(7.19)
Substituting the expression (7.19) into (7.12) yields the following ODE for the
unknown function ’
ðnþ 2Þ d
dx’n d’
dx
� �þ x
d’
dxþ ’ ¼ 0 (7.20)
The boundary conditions for (7.20) are
’ðxÞ ¼ 0 at x ! 1;d’ðxÞdx
¼ 0 at x ¼ 0 (7.21)
where x ¼ x
aQntð Þ1=ðnþ2Þ :
The solution of (7.20) and (7.21) is (Zel’dovich and Raizer 2002)
’ðxÞ ¼ x20n
2ðnþ 2Þ� �
1� xx0
� �2" #1=n
(7.22)
at x<x0; and
’ðxÞ ¼ 0 (7.23)
164 7 Heat and Mass Transfer
at x ¼ 0, where x0 is a constant which is found from the energy invariant (7.14)
x0 ¼ðnþ 2Þ1þn
21�n
npn=2Gn 1=2þ 1=nð Þ
Gn 1=nð Þ
( )1=ðnþ2Þ(7.24)
In (7.24) Gð�Þis the gamma function. The position and velocity of the thermal
wave front are given by the following expressions
xf ¼ x0 aQntð Þ1=ðnþ2Þ(7.25)
vf ¼ x01
nþ 2
aQn
tnþ1
� �1=ðnþ2Þ(7.26)
where xf ðtÞ and vf ðtÞ are the current coordinate and velocity of the thermal wave front.
7.3 Heat and Mass Transfer Under Conditions of Forced
Convection
7.3.1 Heat Transfer from a Hot Body Immersed in Fluid Flow
The first attempt of theoretical investigation of this problem by applying the
dimensional analysis dates back to Lord Rayleigh (1915). He employed the Pi-
theorem for studying heat transfer from a hot body moving in an incompressible
fluid. Rayleigh assumed that five dimensional parameters, namely, (1) the charac-
teristic size of a body, say, a spherical particle, d; (2) its velocity relative to the
surrounding medium v; (3) the temperature difference between the body and the
undisturbed fluid far away from it DT; as well as (4) the fluid heat capacity c and (5)thermal conductivity k determine the rate of heat transfer h�
h� ¼ f ðd; v;DT; c; kÞ (7.27)
where DT ¼ Tw � T1; Tw and T1 are the body and undisturbed fluid temperature,
respectively.
The unknown rate of heat transfer h� and the governing parameters of the
problem d; v; DT; c and k have the following dimensions
½h�� ¼ JT�1; d½ � ¼ L; v½ � ¼ LT�1; DT½ � ¼ y; c½ � ¼ JL�3y�1; k½ �¼ JL�1T�1y�1 (7.28)
where J and y are the independent units of heat and temperature.
7.3 Heat and Mass Transfer Under Conditions of Forced Convection 165
The set of the five governing parameters contains four parameters with indepen-
dent dimension, so that n� k ¼ 1: Choosing as the parameters with the indepen-
dent dimensions d; v; DT; and k, we write in accordance with the Pi-theorem the
dimensionless form of (7.27) as
P ¼ ’ðP1Þ (7.29)
where P ¼ h�=da1va2DTa3ka4 ; and P1 ¼ c=da01va
02DTa
03ka
04 :
Using the principle of dimensional homogeneity,we find values of the exponents aiand a
0i : a1 ¼ 1; a2 ¼ 0; a3 ¼ 1; a4 ¼ 1; a
01 ¼ �1; a
02 ¼ �1; a
03 ¼ 0; and a
04 ¼ 1:
Then (7.29) takes the form
h�
dkDT¼ ’
dvc
k
� �(7.30)
Defining heat flux as q ¼ h=S; with S being the surface of the body, we rewrite
(7.30) as
Nu ¼ ’ðPeÞ (7.31)
where Nu ¼ qd=kDT; and Pe ¼ dvc=k are the Nusselt and Peclet numbers,
respectively.
Equation 7.31 shows that the Nusselt number depends on a single dimensionless
group Pe. At a fixed Peclet number the rate of heat transfer h is directly proportionalto the temperature difference DT and the characteristic size of the body d, whereasthe heat flux q is inversely proportional to d:
The results of Rayleigh demonstrated the efficiency of application of the dimen-
sional analysis to problems related to convective heat transfer, which attracted a
significant attention to this approach which was thoroughly discussed in the fol-
lowing references: Riabouchinsky (1915), Brigman (1922) and Sedov (1993).
In particular, Riabouchinsky made a remark about the choice of the system of
units of in description of convective heat transfer phenomena. The system of units
that Rayleigh used includes three mechanical units (length L; mass M and time TÞand two independent thermal units for quantity of heat (understood as thermal
energy) J and temperature y: It is emphasized that it is possible to express the
dimension of temperature, and accordingly the other thermal quantities, by means
of the basic mechanical units LMT: Indeed, temperature can be related with the
average kinetic energy of molecules with the dimension ML2T�2½ � in the LMTsystem of units. According to the First Law of thermodynamics, the dimension of
heat in the LMT system of units ss J ML2T�2½ �: Then the dimensions of the
governing parameters in (7.27) are as follows
d½ � ¼ L; v½ � ¼ LT�1; y½ � ¼ ML2T�2; c½ � ¼ L�3; k½ � ¼ L�1T�1 (7.32)
166 7 Heat and Mass Transfer
Three parameters of the five in (7.32) have independent dimensions, so that n�k ¼ 2: In this case the dimensionless form of (7.27) reads
P ¼ ’ðP1;P2Þ (7.33)
where P ¼ h�=kDTd; P1 ¼ dvc=k; and P2 ¼ cd3:Thus, instead of (7.29) that determined the dimensionless rate of heat transfer as
a function of a single dimensionless group P1; we arrive at (7.33) where the
unknown quantity P is a function of two dimensionless groups (P1 and P2) that
makes its much less valuable. Essentially one sees that under the same conditions
(7.29) and (7.33) determine different dependences of the dimensionless heat trans-
fer rate P on the governing parameters. In particular, according to (7.29) the
dimensionless heat transfer rate P does not depend on the heat capacity c at
vc ¼ const, whereas (7.33) shows the existence of such dependence.
The seeming contradiction related to using two different systems of units in the
above-mentioned problem deserves the following comments:
1. By choicing governing parameters, one, in fact, makes some assumptions on the
structure of substance involved. In the frame of the continuum approach and
thermodynamics thermal phenomena are described in the macroscopic approxima-
tion fully the molecular structure of substance. Accoordingly, in this case
Rayleigh’s approach is correct, while Riabouchinsky’s counter-example is illegal.
2. The expression of dimensions of thermal quantities via mechanical units is based
on the First Law of thermodynamics that postulate the equivalence of all kinds of
energy, in particular, the thermal and mechanical ones. Therefore, in the general
case the set of governing parameters that determine the heat transfer rate should be
supplemented by two constants characterizing the relation of thermal energy to
mechanical one. These are the mechanical equivalent of heat j½ � ¼ ML2T�2J�1
and Boltzmann’s constant kB½ � ¼ ML2T�2y�1. Accordingly, Rayleigh’s set of the
governing parameters read
d; v; DT; c; k; j; kB (7.34)
Then the apppplication of the Pi-theorem leads to the following expression for
the heat transfer rate (Sedov 1993)
h
kDTd¼ ’ðdvc
k;jcd3
kBÞ (7.35)
When the effect of transformation of mechanical energy into heat (dissipation) is
negligible, (7.35) reduces to (7.30).
3. Many additional questions were raised by Rayleigh’s analysis (Brigman 1922). In
particular, concernswere driven by the fact that density and viscosity aremissing in
the set of governing parameters. The lack of these quantities in the set of the
governing parameters, seemingly diminishes the value of the result of the dimen-
sional analysis in this case.
7.3 Heat and Mass Transfer Under Conditions of Forced Convection 167
Consider Rayleigh’s problem with the account for the fluid density and viscosity.
Then the functional equation for the heat flux q reads
q ¼ f ðd; v;DT; k; m; cP; rÞ (7.36)
where ½d� ¼ L; ½v� ¼ LT�1; r½ � ¼ L�3M; m½ � ¼ L�1MT�1; cP½ � ¼ JMy�1; and
q½ � ¼ JT�1L�2; k½ � ¼ LMT�3y�1; ½DT� ¼ yThe set of the governing parameters in (7.36) contains seven parameters, five of
them with independent dimensions, so that n� k ¼ 2: Take as the parameters with
independent dimensions d; DT; k; m and r and using the Pi-theorem transform
(7.36) to the following dimensionless form
P ¼ ’ðP1;P2Þ (7.37)
where P ¼ q=da1DTa2ka3ma4ra5 ; P1 ¼ v=da01DTa
02ka
03ma
04ra
05 ; and P2 ¼ cP=d
a001
DTa002 ka
003 ma
004 ra
005 :
Using the principle of dimensional homogeneity, we find the values of the
exponents ai; a0i and a
00i as
a1 ¼ �1; a2 ¼ 1; a3 ¼ 1; a4 ¼ 0; a5 ¼ 0
a01 ¼ �1; a
02 ¼ 0; a
03 ¼ 0; a
04 ¼ 1; a
05 ¼ �1 (7.38)
a001 ¼ 0; a
002 ¼ 0; a
003 ¼ 1; a
004 ¼ �1; a
005 ¼ 0
Then (7.37) takes the form
Nu ¼ ’ðRe; PrÞ (7.39)
where Nu ¼ qd=kDT; Re ¼ vdr=m; and Pr ¼ cpm=k are the Nusselt, Reynolds
and Prandtl numbers, respectively.
In the particular case corresponding to creeping flows with small Reynolds
numbers (Re<1) the inertial effects are negligible. Then it is possible to omit
density from the set of the governing parameters and write (7.36) as follows
q ¼ f ðd; v;DT; k; m; cPÞ (7.40)
The dimensionless form of (7.40) is
Nu ¼ ’ðPrÞ (7.41)
where Pr is the Prandtl number.
The explicit form of (7.31) and (7.37) is determined either experimentally or by
solving the Navier-Stokes and energy equations. For flows of incompressible fluids
168 7 Heat and Mass Transfer
with constant physical properties the momentum and continuity equations can be
uncoupled and integrated independently of the energy equation. In this case the heat
transfer problem reduces to solving the energy equation with a known velocity
field. For creeping flows the expressions for the Nusselt number are typically
presented in the form of the series of the Peclet number. For example, Acrivos
and Taylor (1962) obtained the following expression for the Nusselt number for a
spherical particle at Re � 1 and Pr � 1 using the method of matched asymptotic
expansions
Nu ¼ 2þ Peþ Pe2 ln Peþ 0:829Pe2 þ 1
2Pe3 lnPe (7.42)
For calculating heat transfer from spherical bodies in a wide range of the
Reynolds and Prandtl numbers a number of empirical correlations have been
proposed (Soo 1990). In particular, the correlation valid in the range
1<Re<7� 104, 0.6<Pr<400 reads
Nu ¼ 2þ 0:459Re0:55 Pr0:33
(7.43)
It is seen that the form of the dependences NuðPeÞ and NuðRr; PrÞobtained in theframework of the dimensional analysis is identical to the form of correlations (7.42)
and (7.43) resulting from the analytical solution of the problem and experimental
data.
7.3.2 The Effect of Particle Rotation
The effect of rotation of a spherical particle on heat transfer is due to the influence
of the centrifugal force on fluid around the particle (Kreith 1968). The particle
rotation about an axis through its two poles promotes secondary currents that are
directed toward the poles and outward from the equatorial region. The superposi-
tion of the secondary flows on the main flow (driven by the inertial, buoyancy and
pressure forces) determines the hydrodynamic structure of the overall flow and its
evolution, as well as the general characteristics of heat and mass transfer.
In the general case the intensity of heat transfer from a spinning spherical
particle depends on nine dimensional parameters accounting for the effect of the
inertial, buoyancy and centrifugal forces: r; m; d; u1; k; cP; DT; ðbgÞ; and o.Accordingly, the functional equation for the heat flux from the particle reads
q ¼ f ðr; m; d; u1; k; cP;DT; bg;oÞ (7.44)
where ½b� ¼ y�1 is the thermal expansion coefficient, ½g� ¼ LT�2 is the gravity
acceleration, and ½o� ¼ T�1 is the angular speed of rotation.
7.3 Heat and Mass Transfer Under Conditions of Forced Convection 169
Five governing parameters of the nine in (7.44) have independent dimensions so
that the difference n� k ¼ 4: Then, according to the Pi-theorem, (7.44) reduces to
the following form
P ¼ ’ðP1;P2;P3;P4Þ (7.45)
where P ¼ q=ra1ma2da3ka4DTa5 ; P1 ¼ u=ra01ma
02da
03ka
04DTa
05 ,P2 ¼ cP=ra
001
ma002 da
003 ka
004DTa
005 ; P3 ¼ bg=ra
0001 ma
0002 da
0003 ka
0004 DTa
0005 ; and P4 ¼ o=ra
1V1 ma
1V2
da1V3 ka
1V4 DTa1V
5 .
Taking into account the dimension of the heat flux q and those of the parameters
governing it, we find the values of the exponents ai; a0i; a
00i ; a
000i and a1Vi : They are
equal to
a1 ¼ 0; a2 ¼ 0; a3 ¼ �1; a4 ¼ 1; a5 ¼ 1
a01 ¼ �1; a
02 ¼ þ1; a
03 ¼ �1; a
04 ¼ 0; a
05 ¼ 0
a001 ¼ 0; a
002 ¼ �1; a
003 ¼ 0; a
004 ¼ 1; a
005 ¼ 0 (7.46)
a0001 ¼ �2; a
0002 ¼ 2; a
0003 ¼ �3; a
0004 ¼ 0; a
0005 ¼ �1
a1V1 ¼ �1; a1V2 ¼ 1; a1V3 ¼ �2; a1V4 ¼ 0; a1V5 ¼ 0
Bearing in mind (7.46), we rewrite (7.45) as follows
Nu ¼ ’ðRe; Pr;Gr;ReoÞ (7.47)
where Nu¼hd=k;Re¼u1d=n;Pr¼n=a;Gr¼bgðTw�T1Þd3=n2;and Reo¼od2=nare the Nusselt, Reynolds, Prandtl, Grashof and particle rotational Reynolds num-
bers, respectively, h¼q=DT is the heat transfer coefficient, and a¼k=rcP is the
thermal diffusivity.
The approaches based on the boundary layer theory are used for theoretical
description of heat transfer of a spinning particle (Dorfman 1967; Banks 1965;
Chao and Greif 1974; Lee et al. 1978). Under the conditions of forced convection
due to particle rotation the heat transfer coefficient depends on the rotational
Reynolds and Prandtl numbers, as is anticipated from (7.47). The dependence on
Reo may be expressed as (Kreith 1968)
Nu ¼ 0:43Re0:5o Pr0:4 (7.48)
for Reo<5 105; and
Nu ¼ 0:066Reo0:67Pr0:4 (7.49)
for Reo>5 105.
170 7 Heat and Mass Transfer
The comparison of a number of theoretical predictions with experimental data
on heat transfer of a spherical particle under the conditions of forced convection due
to its rotation shows that all the dependences NuðReo; PrÞ proposed have the form
Nu ¼ ARe0:5Pr0:4; with only the values of the factor A being different (Hussaini and
Sastry 1976). The empirical correlations for the average Nusselt number of a
spherical particle rotating in air at rest or in air stream were suggested by Eastop
(1973). In the former case the correlation reads
Nu ¼ 0:353Re0:5o (7.50)
whereas in the latter one (at Reo=Re>0:54) it has the following form
Nu ¼ 0:288Re0:6 1þ 0:167Reo
Re� 0:54
� �� �(7.51)
7.3.3 The Effect of the Free Stream Turbulence
The free stream turbulence tends to enhance heat transfer from a particle to the
surrounding fluid due to the eddies penetrating from the external flow into the
particle boundary layer. The disturbances of the near-wall flow facilitate transition
to turbulence and shift the boundary layer separation point downstream over the
particle surface. The intensity of heat transfer from a heated spherical particle
immersed into turbulent flow depends on the physical properties of fluid: its density
r½ � ¼ L�3M; viscosity m½ � ¼ L�1MT�1; thermal conductivity k½ � ¼ JT�1y�1L�1;and specific heat cP½ � ¼ Jy�1M�1. It also depends on particle diameter d½ � ¼ L and
its velocity v1½ � ¼ LT�1; the temperature difference between the particle and
surrounding fluid DT½ � ¼ y; the characteristic velocity of turbulent fluctuationsev0h i
¼ LT�1 (ev0 ¼ffiffiffiffiffiv02
pis the root mean square of the turbulence velocity
fluctuations) and the integral scale of turbulence l½ � ¼ L: Accordingly, the func-
tional equation for the heat flux q½ � ¼ JT�1L�2 reads
q ¼ f ðr; m; k; cP; v1;DT; d; ev0 ; lÞ (7.52)
Equation 7.52 contains nine dimensional parameters, five of them having inde-
pendent dimensions. Then, according with the Pi-theorem, the dimensionless form
of (7.52) appears to be
P ¼ ’ðP1;P2;P3;P4Þ (7.53)
7.3 Heat and Mass Transfer Under Conditions of Forced Convection 171
where P ¼ q=ra1ma2ka3DTa4da5 ; P1 ¼ cP=ra01ma
02ka
03DTa
04da
05 ; P2 ¼ v1=ra
001 ma
002 ka
003
DTa004 da
005 ; P3 ¼ ev0=ra
0001 ma
0002 ka
0003 DTa
0004 da
0005 ; and P4 ¼ l=ra
1V1 ma
1V2 ka
1V3 DTa1V
4 da1V5 and the
exponents ai; a0i; a
00i ; a
000i and a1Vi are equal to
a1 ¼ 0; a2 ¼ 0; a3 ¼ 1; a4 ¼ 1; a5 ¼ �1;
a01 ¼ 0; a
02 ¼ �1; a
03 ¼ 1; a
04 ¼ 0; a
05 ¼ 0;
a001 ¼ �1; a
002 ¼ 1; a
003 ¼ 0; a
004 ¼ 0; a
005 ¼ �1; (7.54)
a0001 ¼ �1; a
0002 ¼ 1; a
0003 ¼ 0; a
0004 ¼ 0; a
0005 ¼ �1;
a1V1 ¼ 1; a1V2 ¼ 0; a1V3 ¼ 0; a1V4 ¼ 0; a1V5 ¼ 1;
Taking into account the values of the exponents ai; a0i; a
00i ; a
000i and a1Vi , it is
possible to present (7.52) in the following form
Nu ¼ ’ðPr;Re;ReT ; lÞ (7.55)
where Nu ¼ qd=kDT; Pr ¼ cPm=k; Re ¼ v1dr=m; and ReT ¼ ev0=v1
v1dr=m
¼ TuRe (Tu ¼ ev0v1
is the turbulence intensity) are the Nusselt, Prandtl, Reynolds
and turbulent Reynolds number, respectively, l ¼ l=d is the dimensionless turbu-
lence scale.
Equation 7.55 shows that the contribution of turbulence to heat transfer is
determined by two dimensionless parameters accounting for the turbulence inten-
sity and the size of turbulent eddies. Depending on values of these parameters,
different conditions of heat transfer may be realized. When l 1; the particle
experiences a time-dependent flow, whereas when l<<1, the flow becomes
quasi-steady. In the latter case the heat transfer coefficient depends on the Reynolds
and Prandtl numbers, as well as on the turbulence intensity Tu (Kestin 1966)
Nu ¼ ’ðPr;Re; TuÞ (7.56)
In particular, for a spherical particle immersed in the turbulent flow with
2� 103<Re<6:5� 104 and 1<Tu<17%, (7.55) takes the form (Lavender and Pei
1967)
Nu ¼ 2þ f ðReTÞRe0:5 (7.57)
where f ðReTÞ ¼ 0:629Re0:035T for ReT<103 and f ðReTÞ ¼ 0:145Re0:25T for Re>103:The difference in the dependences f ðReTÞcorresponding to small and large values
of ReT are related to the peculiarities of flow about a particle at low and high
turbulence intensity. The influence of the free stream turbulence on the particle
boundary layer is weak enough at low Tu: When turbulence disturbances reach
172 7 Heat and Mass Transfer
some critical level, laminar-turbulent transition in the particle boundary layer occurs.
It is accompanied by significant changes in the flow structure, reducing dramatically
the particle drag.
7.3.4 The Effect of Energy Dissipation
The internal friction of viscous fluid near the surface of a solid body moving in it is
accompanied by the mechanical energy dissipation. The latter leads to heating the
body, which can result in a significant temperature increase of the body temperature
Tw above the temperature of the undisturbed fluid T1: In flows of viscous incom-
pressible fluid heating of solid bodies depends on physical properties of fluid,
relative velocity of the body to that of fluid and its characteristic size. These
considerations allow us to present the functional equation for equilibrium tempera-
ture difference DT ¼ Tw � T1 as follows
DT ¼ f ðp; m; k; cP; u; d�Þ (7.58)
where p; m and k are the viscosity, thermal conductivity, cp is the specific heat of
fluid, u is the relative velocity of the body, and d� its characteristic size.Since the set of the five governing parameters of the problem contains three
parameters possessing independent dimensions, we can present (7.58) in the fol-
lowing form
P ¼ ’ðP1;P2Þ (7.59)
where P ¼ DT=pa1ma2ca3P ua4 ; P1 ¼ k=pa
01ma
02c
a03
P ua04 ; and P2 ¼ d�=pa
001 ma
002 c
a003
P ua004 .
Determining the values of the exponents ai; a0i and a
00i using the principle of
dimensional homogeneity, we find a1 ¼ 0; a2 ¼ 0; a3 ¼ �1; a4 ¼ 2;
a01 ¼ 0; a
02 ¼ 1; a
03 ¼ 1; a
04 ¼ 0; a
001 ¼ �1; a
002 ¼ 1; a
003 ¼ 0 and a
004 ¼ �1: Then,
we can transform (7.59) to the following form
DT ¼ u2
cP’ðPr;ReÞ (7.60)
The expression (7.60) shows that the temperature difference due to viscous
dissipation is determined by the relative velocity of the body, as well as values of
the Reynolds and Prandtl numbers. The evaluations show that in the two limiting
cases corresponding to small and large Reynolds numbers heating of a body moving
in fluid can be expressed as (Landau and Lifshitz 1987)
Tw � T1 ¼ c Pru2
cP(7.61)
7.3 Heat and Mass Transfer Under Conditions of Forced Convection 173
Tw � T1 ¼ u2
cP’�ðPrÞ (7.62)
respectively. In (7.61) c is a constant that depend on the body shape; ’�ðPrÞ is afunction of the Prandtl number.
Essentially, that in both cases corresponding to small and large Reynolds
number body heating due to viscous dissipation is directly proportional to the
square of the relative velocity.
7.3.5 The Effect of Velocity Gradient
Consider heat flux from a spherical particle immersed in a uniform flow of hot
incompressible fluid with linear velocity distribution. Assume that the total heat
flux is a result of superposition of two components: one – due to the rectilinear
translational motion relative to the surrounding fluid, and the other one – due to
shear. In this case it is possible to write the equation for the total heat flux in the
form
qt ¼ qc þ qsh (7.63)
where qt; qc and qsh are the total heat flux, heat flux due to rectilinear translational
motion, and heat flux due to shear, respectively, [qt] ¼ [qc] ¼ [qsh] ¼ JT�1L�2.
The functional equations for each component of the total heat flux read
qc ¼ fcðr; m; d; u; k; cP;DTÞ (7.64)
qsh ¼ fshðr; m; d; g; k; cP;DTÞ (7.65)
where g T�1½ � is the shear rate, and u is the velocity of the particle, and d is the
diameter of a spherical particle.
Equations 7.64 and 7.65 contain seven governing parameters, five of which
having independent dimensions. According to the Pi-theorem, (7.64) and (7.65)
can be presented as
qc ¼ ra1ma2da3ka4DTa5’u
ra01ma
02da
03ka
03DTa0
5
;cP
ra01ma
02da
03ka
04DTa0
5
!(7.66)
qsh ¼ ra1�ma2�da3�ka4�DTa5�’sh
g
ra01�ma
02�da
03�ka
04�DTa0
5�;
cP
ra001�ma
002�da
003�ka
004�DTa00
5�
!
(7.67)
174 7 Heat and Mass Transfer
Divide the left and right hand sides of (7.63) by the product ra1ma2da3ka4DTa5 . As
a results, we arrive at the following dimensionless equation (accounting for the
equality ai ¼ ai�Þ
P ¼ Pc þPsh (7.68)
where P ¼ qt=ra1ma2da3ka4DTa5 ; Pc ¼ qc=ra1ma2da3ka4DTa5 ; Psh ¼ qsh=ra1ma2
da3ka4DTa5 ;Pc ¼ ’cðP1;P2Þ, Psh ¼ ’shðP1�;P2�Þ, P1 ¼ u=ra01ma
02da
03ka
04DTa
05 ,
P2 ¼ cP=ra001 ma
002 da
003 ka
004DTa
005 , P1:� ¼ g=ra
01�ma
02�da
03�ka
04�DTa
05� , and P2� ¼ cp=ra
001�
ma002�da
003�ka
004�DTa
005� .
Taking into account the dimensions of qt; qc; qsh; u; g and cP; as well as the
fact that the governing parameters r; m; d; k and DTpossess independent
dimensions, we find the values of the exponents ai; a0i; a
00i ; ai�; a
0i�; and a
00i� as
a1 ¼ 0; a2 ¼ 0; a3 ¼ �1; a4 ¼ 1; a5 ¼ 1
a01 ¼ �1; a
02 ¼ 1; a
03 ¼ �1; a
04 ¼ 0; a
05 ¼ 0
a001 ¼ 0; a
002 ¼ �1; a
003 ¼ 0; a
004 ¼ 1; a
005 ¼ 0 (7.69)
a01� ¼ 1; a
02� ¼ 1; a
03� ¼ �2; a
04� ¼ 0; a
05� ¼ 0
a001� ¼ 0; a
002� ¼ �1; a
003� ¼ 0; a
004� ¼ 1; a
005� ¼ 0
According to (7.69), (7.68) takes the form
Nut ¼ Nuc þ Nush (7.70)
Where Nut ¼ qtd=kDT; Nuc ¼ qcd=kDT; Nush ¼ qsh=kDT; P1 ¼ Re ¼ udr=m;P2 ¼ Pr ¼ cPm=k;P1� ¼ d2grcP=k and P2� ¼ Pr ¼ cPm=k.
At low Reynolds number the Nusselt number approaches the value of 2. In this
case (7.70) reads
Nut ¼ 2þ ’shðPe1�; PrÞ (7.71)
The analytical solution of the problem yields the following expression for the
Nusselt number which is valid for low Reynolds and Peclet numbers (Frankel and
Acrivos 1968)
Nu ¼ 2þ 0:9104
2pð Þ1=2Pe (7.72)
where Pe ¼ r2grcP=k and r is the particle radius.
7.3 Heat and Mass Transfer Under Conditions of Forced Convection 175
7.3.6 Mass Transfer to Solid Particles and Drops Immersed inFluid Flow
From the physical point of view the diffusion flux to a reactive spherical particle
immersed in an infinite reactant flow is determined by the physical properties of
liquid (its density, viscosity and diffusivity), particle size, reactant concentration in
the undisturbed flow, fluid velocity, as well as the kinetics of heterogeneous
chemical reaction at the particle surface. When the rate of the surface reaction
exceeds the rate of reactant diffusion in the carrier fluid, the concentration of the
reactant at the particle surface equals zero. In this case the influence of the kinetic
characteristics of the heterogeneous chemical reaction at the particle surface on the
mass transfer rate in the fluid bulk is negligible and these characteristics should be
excluded from the set of the governing parameters. At small values of the Reynolds
number corresponding to the Stokes creeping flow, the fluid density does not affect
the flow field and can be safely excluded from the governing parameters. Assume
that the carrier phase is liquid. Taking into account the fact that typically in liquids
the admixture (e.g., reactant) diffusivities are much less than viscosity (D<<nÞ, it ispossible to omit n from the set of the governing parameters. Indeed, under this
condition the mass transfer results mostly from diffusion rather than convection.
Thus, the functional equation for the diffusion flux of a reactant in a liquid carrier to
a reactive particle takes the form
qm ¼ ’ðD; r0; u1; c1Þ (7.73)
where r0 is the radius of particle, u1 is the undisturbed flow velocity, and c1 is the
reactant concentration in the liquid carrier far from the particle.
The dimensions of the governing parameters are
D½ � ¼ L2T�1; r0½ � ¼ L; u1½ � ¼ LT�1; c1½ � ¼ L�3M (7.74)
Three governing parameters from the set (7.74) have independent dimensions, so
that the difference n� k ¼ 1: Then, accoring to the Pi-theorem, (7.73) transform as
P ¼ ’ðP1Þ (7.75)
where P ¼ qmr0=Dc1 ¼ Sh is the Sherwood number, and P1 ¼ u1r0=D ¼ Ped isthe Peclet number.
Small droplets immersed in liquid flow stay spherical and effectively
undeformable due to the action of the interfacial tension. However, the mass
transfer to the surface of a small reactive droplet of viscosity m2 immersed into a
reactant liquid solution solution of viscosity m1 can differ significantly from the
mass transfer to a solid reactive particle. This stems from the different hydrody-
namic conditions at the liquid-liquid and liquid-solid interfaces. The existence of a
non-zero interfacial tangential velocity at the liquid-liquid interface determined by
176 7 Heat and Mass Transfer
viscosities of two liquids determines the interfacial velocity as characteristic scale
of the problem. Then the functional equation for the diffusion flux qm takes the form
qm ¼ f ðD; r0; v�; c1Þ (7.76)
where v� is the absolute value of the interfacial velocity at the drop equator.
In this case the dimensionless form of (7.76) reads
P ¼ ’ðeP1Þ (7.77)
where e ¼ c m2=m1ð Þ.The explicit forms of the dependences (7.75) and (7.77) can be found only
experimentally or via theoretical solutions of the corresponding problem (Levich
1962). When the rate of the heterogeneous reaction at the particle surface is much
larger than the diffusion rate, the problem reduces to the integration of the species
balance equation. In the framework of the diffusion boundary layer, and for the
creeping flow velocity distribution it reads
vr@c
@rþ v’
r
@c
@y¼ D
@2c
@r2þ 2
r
@c
@r
� �(7.78)
with the boundary conditions
c ! c1 at y ! 1; c ¼ 0 at r ¼ r0 (7.79)
where r; ’; and y are the spherical coordinates with the origin at the particle
center.
The Levich solution results in the following dependence of the Sherwood
number on the Peclet number
Sh ¼ 7:85Pe1=3 (7.80)
for a solid particle, and in the dependence
Sh ¼ 2ffiffiffiffiffiffi6p
p Pe1=2m1
m1 þ m2
� �1=2
(7.81)
for a liquid droplet, where Sh ¼ <qm>r0=Dc1 is the Shrwood number, <qm> ¼I=r20 is the average diffusion flux, I ¼
Rqmds is the total diffusion flux at the particle
or droplet surface.
The comparision of the results of the dimensional analysis given by (7.75) and
(7.77) with the analytical solution of the problem corresponding to (7.80) and (7.81)
reveals the benefits of applying the dimensional anaysis to study mass transfer to
solid particles and droplets in flows of liquid reactant solutions. It is seen that results
7.3 Heat and Mass Transfer Under Conditions of Forced Convection 177
of the dimensional analysis correctly represent the qualitative character of the
dependence of the Sherwood number on the Peclet number, but do not allow finding
the exact form of this dependence.
The knowledge of an exact form of the dependence ShðPeÞ is important for
understanding of the laws of mass transfer to solid particles and droplets. As can be
seen from (7.80) and (7.81), two different scaling laws in the dependence on the
Peclet number exist, with the exponents being one third and one half for particles
and droplets, respectively. That shows that ratio of the mass transfer intensity to a
droplet to that of a comparable solid particle Shd Shp�
depends weekly on the Peclet
number and strongly on the viscosities of the droplet and surroundind medium.
7.4 Heat and Mass Transfer in Channel and Pipe Flows
7.4.1 Couette Flow
Consider heat transfer in fully laminar flow of incompressible fluid between two
parallel closely located infinite plates having different temperature. Let the lower
plate is motionless, whereas the upper one moves with a constant velocity U.
Assuming that temperatures of the lower and upper plates being equal to T0 and
T1, respectively (with T1>T0). In the case when T0 and T1 are constant, the fluid
temperature changes in y-th direction only (coordinate y is normal to the plates).
The set of the governing parameters that determine heat flux from the upper wall to
the fluid q, as well as the fluid temperature # ¼ T � T0 reads
k½ � ¼ LMT�3y�1; m½ � ¼ L�1MT�1; H½ � ¼ L; U½ � ¼ LT�1; #�½ � ¼ y; y½ � ¼ L(7.82)
where H is the gap between the plates, and #� ¼ T1 � T0.Accordingly, the functional equations for q and # have the following form
q ¼ fqðk; m;H;U; #�Þ (7.83)
# ¼ f#ðk;m;H;U; #�; yÞ (7.84)
Four governing parameters listed in (7.83) and (7.84) possess independent
dimension, so that the difference n� k for (7.83) and (7.84) equals 1 and 2,
respectively. Then the dimensionless form of (7.83) and (7.84) is
Pq ¼ ’qðP1qÞ (7.85)
P# ¼ ’#ðP1#;P2qÞ (7.86)
178 7 Heat and Mass Transfer
wherePq ¼ q=ka1Ha2Ua3#a4� ; P1q ¼ m=ka01Ha
02
Ua03#
a04� ; P# ¼ #=ka1�Ha2�Ua3�#a4�� ; P1# ¼ m=ka
01�Ha
02�Ua
03�#a
04� and
P2# ¼ y=ka001�Ha
002�Ua
003�#
a004�� .
Bearing in mind the dimensions of heat flux, temperature and the governing
parameters, we find the values of the exponents ai; a0i; ai�; a
0i and a
00i�as
a1 ¼ 1; a2 ¼ �1; a3 ¼ 0; a4 ¼ 1
a01 ¼ 1; a
02 ¼ 0; a
03 ¼ �2; a
04 ¼ 1
a1� ¼ 0; a2� ¼ 0; a3� ¼ 0; a4� ¼ 1 (7.87)
a01� ¼ 1; a
02� ¼ 0; a
03� ¼ �2; a
04� ¼ 1
a001� ¼ 0; a
002� ¼ 1; a
003� ¼ 0; a
004� ¼ 0
Using (7.87), (7.85) and (7.86) take the following form
Nu ¼ ’qðPrEcÞ (7.88)
# ¼ ’#ð�; PrEcÞ (7.89)
where Nu ¼ qH=kðT1 � T0Þ; Pr ¼ mcP=k; and Ec ¼ U2=cPðT1 � T0Þ are the
Nusselt, Prandtl and Eckert numbers, respectively, # ¼ #=#�; and � ¼ y=H:The analytical solution of the problem on heat transfer in Couette flow reads
(Bayley et al. 1972)
# ¼ � 1þ 1
2PrEcð1� �Þ
� �(7.90)
The heat flux from the top plate to the fluid is expressed using the analytic
solution as
q ¼ k@T
@y
����y¼0
¼ kTs � T0
H1� 1
2PrEc
� �(7.91)
The comparison of the results of the exact analytical solution of the problem
with its analysis based on the Pi-theorem indicates a certain insufficiency of the
latter approach. In particular, the dimensional analysis allows finding the dimen-
sionless groups only, but cannot reveal a number of the important peculiarities of
the process, for example, the reverse of the heat flux at high values of the product
PrEcwhen q changes sign.
7.4 Heat and Mass Transfer in Channel and Pipe Flows 179
7.4.2 The Entrance Region of a Pipe
The evaluation of the thickness of the thermal boundary layer within the entrance
region of heated pipe is tackled in the present subsection. Considering flows of
incompressible fluids it is possible to assume that the local thickness of the thermal
boundary layer dTðxÞ depends on five dimensional parameters: the density r,thermal conductivity k and specific heat of the fluid cP; its velocity in the undis-
turbed core u and the cross-section coordinate x:2 Accordingly, the functional
equation for the thickness of the thermal boundary layer is written as
dTðxÞ ¼ fTðr; k; cP; u; x; Þ (7.92)
In the system of units LxLyLzMTJy the dimensions of the governing parameters
and the thickness of the thermal boundary layer are expressed as
r L�1x L�1
y Lz�1Mh i
; k L�1x LyL
�1z JT�1y�1
�; cP½ � ¼ JM�1y�1; u½ �
¼ LxT�1; x½ � ¼ L; d½ � ¼ Ly dT Ly
�(7.93)
It is seen that all the governing parameters have independent dimensions.
Therefore, in accordance with the Pi-theorem, (7.92) takes the form
dTðxÞ ¼ cTra1ka2ca3p ua4xa5 (7.94)
where cT is a constant.
Accounting for the fact that dT½ � ¼ Ly, we find the values of the exponents ai.They are: a1 ¼ �1=2; a2 ¼ 1=2; a3 ¼ �1=2; a4 ¼ �1=2 and a5 ¼ 1=2. Then
(7.94) takes the form
dTðxÞ ¼ cTkx
rucP
� �1=2
(7.95)
or
dTðxÞ ¼ cTx Pr�1=2
Re�1=2d (7.96)
where Pr ¼ n=a and Re ¼ ud=n are the Prandtl and Reynolds numbers with n and abeing the kinematic viscosity and thermal diffusivity, respectively.
Taking into account that thickness of the thermal boundary layer in the cross-
section that correspond to the end of entrance region equals d=2, we find the thermal
entrance lenth lenT as
lenTd
¼ c�TRed Pr (7.97)
where c�T ¼ ð2cTÞ�2.
180 7 Heat and Mass Transfer
Equations (5.12) and (7.95) show that the dynamic and thermal entrance lengths
of a pipe are determined by its diameter, the physical properties of fluid and flow
velocity. Therefore, it is possible to use len and lenT as some generalized parameters
in studies of flow resistance and heat transfer in the entrance section of pipes.
Introducing the dimensional parameters as dP dx=ð Þ ru2ð Þ� � ¼ L�1 and
qs kDTð Þ=½ � ¼ L�1 (where qs is the heat flux at the pipe wall), we write the func-
tional equations for these parameters as follows
dP=dxð Þru2
¼ fPðlen; xÞ (7.98)
qskDT
¼ fqðlenT ; xÞ (7.99)
In both expressions the difference between the number of dimensional
parameters n and the number of parameters having independent dimensions k is
equal to 1. Then, according to the Pi-theorem, (7.98) and (7.99) are deduced to
dP=dxð Þlenru2
¼ ’P
x
len
� �(7.100)
qslen:TDT
¼ ’qðx
len:TÞ (7.101)
Substituting (4.14) and (6.65) into (7.100) and (7.101), we arrive at the following
dimensionless expressions
l ¼ 1
Red’PðXÞ (7.102)
Nu ¼ 1
Pr Red’qðXTÞ (7.103)
where l ¼ dP=dxð Þdc�½ �=ru2 and Nu ¼ qslenT=kDT are the friction factor and the
Nusselt number, as well as X ¼ x=dc�Red and XT ¼ X=dc�T Pr Red.Comparing (7.95) with (5.12), we find the relation between the thicknesses of the
thermal and dynamical boundary layers as
dTd
Pr�1=2
(7.104)
7.4.3 Fully Developed Flow
Consider heat transfer in laminar pipe flow of incompressible fluid. Let pipe
diameter be denoted as d. We will use the LMTJy system of units. Assume that
physical properties of fluid are constant and flow is fully developed dynamically
7.4 Heat and Mass Transfer in Channel and Pipe Flows 181
and thermally. Let heat flux at the wall qs ¼ hðtw � tmÞ is constant along the tube
with h½ � ¼ JT�1L�2y�1 being the heat transfer coefficient, and tw and tm being the
wall and mean (bulk) fluid temperature, respectively.
It is plausible to assume that heat flux at the pipe wall depends on the
physical properties of fluid ( m½ � ¼ L�1MT�1; k½ � ¼ JT�1L�1y�1; cP½ � ¼ JM�1y�1;
r½ � ¼ L�3MÞ; the mean velocity of flow u½ � ¼ LT�1; the difference between the
wall and bulk temperature Dt ¼ tw � tm½ � ¼ y and the diameter d½ � ¼ L
qs ¼ fqðr; m; k; cP; u;Dt; dÞ (7.105)
Equation 7.105 contains seven governing parameters, five of them with inde-
pendent dimensions. Applying the Pi-theorem, we transform (7.105) to the follow-
ing dimensionless form
P ¼ ’qðP1;P2Þ (7.106)
where P ¼ qs=ra1ma2ka3da4Dta5 ; P1 ¼ u=ra01ma
02ka
03da
04Dta
05 and P2 ¼ cP=ra
001
ma002 ka
003 da
004Dta
005 :
The exponents ai; a0i and a
00i are equal to
a1 ¼ 0; a2 ¼ 0; a3 ¼ 1; a4 ¼ �1; a5 ¼ 1
a01 ¼ �1; a
02 ¼ 1; a
03 ¼ 0; a
04 ¼ �1; a
05 ¼ 0 (7.107)
a001 ¼ 0; a
02 ¼ �1; a
003 ¼ 1; a
004 ¼ 0; a
005 ¼ 0
Then, we obtain
Nu ¼ ’dðRe; PrÞ (7.108)
where Nu ¼ qsd=kDt; Re ¼ udr=m and Pr ¼ cPm=k are the Nusselt number based,
which represents itself in this case the dimensionless heat flux at the pipe wall, the
Reynolds and Prandtl numbers, respectively.
In the particular case of creeping flows corresponding to very small Reynolds
numbers (Re ! 0Þ it is possible to omit u and m from the set of the governing
parameters and write the functional equation for the heat flux in the form
qs ¼ fqðr; k; cP; d;DtÞ (7.109)
All the governing parameters in (7.109) have independent dimensions. Accord-
ingly, this equation should take the following form
qs ¼ cra1ka2ca3P da4Dta5 (7.110)
where c is a constant.
182 7 Heat and Mass Transfer
Taking into account the dimension qS½ � ¼ JT�1L�2, we determine values of the
exponents ai as a1 ¼ 0; a2 ¼ 1; a3 ¼ 0; a4 ¼ �1 and a5 ¼ 1. Accordingly, the
Nusselt number is expressed as
Nu ¼ qsd
kDt¼ c ¼ const: (7.111)
The numerical value of the constant c in (7.111) that follows from the analytical
solution of this problem is 4.364 (Kays and Crawford 1980).
7.5 Thermal Characteristics of Laminar Jets
Consider thermal characteristics of plane submerged laminar jet when a heated (or
cooled) fluid is issuing from a narrow slit in the same cooler (or warmer) fluid,
which is at rest far from the jet origin. We assume that the difference between the
fluid temperature in the jet and the surrounding medium is sufficiently small, which
allows us to consider the fluid density and viscosity being constant. In this case the
flow and temperature fields of fluid in the jet are determined by two parameters: (1)
the kinematic viscosity n, and (2) thermal diffusivity a.Non-isothermal jets, in fact, form two boundary layers, the dynamic and thermal
one. In the general case their thicknesses d and dT are not equal. The relation between dand dT depends on the relative intensity of the momentum and heat transfer which is
characterized by the Prandtl number Pr ¼ n=a:Only when Pr ¼ 1; d ¼ dT ; whereaswhen Pr<1 and Pr>1; dT>d and dT<d; respectively (Fig. 7.3).
There are two underlying physical mechanisms of heat transfer in submerged
jets: (1) convection, and (2) thermal diffusion. Therefore, it is natural to assume that
the temperature field depends on the flow velocity characterizing the convective
Fig. 7.3 Plane jet. Thicknesses of the dynamic and thermal boundary layers at different values of
the Prandtl number
7.5 Thermal Characteristics of Laminar Jets 183
component of the heat transfer, as well as the thermal diffusivity characterizing the
conductive component. Then, the functional equation for the temperature field
reads
DT ¼ f ðu; a; qx; x; yÞ (7.112)
where DT ¼ T � T1½ � ¼ y; T1 is the temperature of the undisturbed fluid, a½ � ¼L2yT
�1 is the thermal diffusivity, qx ¼ Qx rcPð Þ=½ � ¼ LxLyT�1y where the axial
convective enthalpy flux Qx ¼Ð10
ruDhdy� �
¼ L�1z T�1J with h½ � ¼ JM�1 being
the enthalpy, in the LxLyLzMTJy system of units.
For the axial excess (maximal) temperature DTm, and the thermal boundary
thickness dT the following functional equations are valid
DTm ¼ fTðum; a; qx; xÞ (7.113)
dT ¼ fd:Tðum; a; qx; xÞ (7.114)
Since the governing parameters um(the axial maximal velocity in jet cross-
section), a, qx and x have independent dimensions, (7.113) and (7.114) take the form
DT ¼ c1Tua1m a
a2qa3x xa4 (7.115)
dT ¼ c2Tua�1
m aa�2q
a�3x x
a�4 (7.116)
where c1T and c2T are constants.
Then the exponents ai and a�i are found as
a1 ¼ � 1
2; a2 ¼ � 1
2; a3 ¼ 1; a4 ¼ � 1
2; a�1
¼ � 1
2; a�2 ¼
1
2; a3 ¼ 0; a�4 ¼
1
2
(7.117)
Therefore, (7.115) and (7.116) take the form
DTm ¼ c�1TQx
rcP
1
Jxn=rð Þ1=6Pr1=2
x�1=3 (7.118)
dT ¼ c�2T1
J2x=r2n� �1=3 Pr
�1=2
x2=3 (7.119)
where c�1T ¼ c1T=c1 and c�2T ¼ c2T=c1 with c1 being a constant from (5.28).
184 7 Heat and Mass Transfer
Comparing (6.40) and (7.119), we estimate the ratio of the thicknesses of the
dynamic and thermal boundary layers
ddT
ffiffiffiffiffiPr
p(7.120)
Using the Pi-theorem, we transform (7.112) to the following form
DTDTm
¼ ’T
y
dT
� �(7.121)
Bearing in mind (7.120), we present (7.121) in the form
DTDTm
¼ ’T �Pr1=2
(7.122)
where � ¼ y=d.The temperature distribution in cross-sections of plane laminar jet is shown in
Fig. 7.4. It is seen that profiles DTð�Þ=DTm are wider than profiles uð�Þ=um at
Pr<1; and narrower at Pr>1:
Fig. 7.4 Velocity and temperature distributions in cross-sections of plane laminar submerged jet.
1- Velocity and temperature distributions for Pr ¼ 1; 2-Temperature distribution for Pr < 1; 3-
Temperature distribution for Pr > 1
7.5 Thermal Characteristics of Laminar Jets 185
7.6 Heat and Mass Transfer in Natural Convection
7.6.1 Heat Transfer from a Spherical Particle Under theConditions of Natural Convection
Consider heat transfer from a warm spherical particle immersed into still fluid. The
temperature difference between the particle and surrounding fluid is the reason of
natural convection, which arises due to the existence of a non-uniform density field in
fluid with temperature distribution and thermal expansion. Under the conditions of
natural convection it is plausible to assume that heat flux from a warm particle is
determined by the physical properties of fluid- its density r; thermal conductivity k;viscosity m; specific heat cp; the particle and fluid temperature difference Dt ¼ tp �t1; aswell as the product of gravity acceleration g and thermal expansion coefficientb:
Accordingly, the functional equation for the heat flux reads
qt ¼ f ðr; m; k; cP;Dt; gb; r0Þ (7.123)
In the system of units LMTJy the dimensions of the governing parameters are
r½ � ¼ L�3M; m½ � ¼ L�1MT�1; k½ � ¼ JT�1L�1y�1; cP½ � ¼ JM�1y�1 Dt½ �¼ y; gb½ � ¼ y�1LT�2; r0½ � ¼ L (7.124)
while qt½ � ¼ JL�2T�1.
Five of the seven governing parameters in (7.123) possess independent
dimensions. Then, according to the Pi-theorem the dimensionless form of (7.123) is
P ¼ ’ðP1;P2Þ (7.125)
where P ¼ qt=ra1ma2ka3Dta4ra50 ; P1 ¼ cp=ra
01ma
02ka
03Dta
04r
a05
0 and P2¼gb=ra001 ma
002 ka
003
Dta004 r
a005
0 :
Bearing in mind the dimension of the heat flux qt JT�1L�2½ �, we find values of the
exponents ai; a0i and a
00i
a1 ¼ 0; a2 ¼ 0; a3 ¼ 1; a4 ¼ 1; a5 ¼ �1
a01 ¼ 0; a
02 ¼ �1; a
03 ¼ 1; a
04 ¼ 0; a
05 ¼ 0 (7.126)
a001 ¼ �2; a
002 ¼ 2; a
003 ¼ 0; a
004 ¼ �1; a
005 ¼ �3
Accordingly, (7.125) takes the form
Nu ¼ ’ Pr;Grð Þ (7.127)
where Nu ¼ qtr0=kDt; Pr ¼ cpm=k and Gr ¼ r2gbDtr30=m2 are the Nusselt, Prandtl
and Grashof numbers, respectively.
186 7 Heat and Mass Transfer
The explicit form of the dependence of the Nusselt number on Pr and Gr for awarm spherical particle found experimentally has the form (Raithby and Hollands
1998)
Nu ¼ 0:878ARa1=4 (7.128)
where A is a weak function of the Prandtl number, namely, 0.086<A<0.104 for Pr
varying from 2000 to 0.71 and Ra¼PrGr being the Rayleigh number.
At very small values of the Prandtl number (m ! 0Þ the functional equation for
the heat flux from a warm particle has the following form
qt ¼ f ðr; k; cp;Dt; gb; r0Þ (7.129)
Five governing parameters in (7.129) have independent dimensions. Then the
Pi-theorem allows transformation of (7.129) to the following dimensionless form
P ¼ ’ P1ð Þ (7.130)
where P ¼ qt=ra1ca2p ka3Dta4ra50 and P2 ¼ gb=ra
01c
a02
p ka03Dta
04r
a05
0 .
Using the principle of dimensional homogeneity, we find the values of the
exponents ai and a0i as
a1 ¼ 0; a2 ¼ 0; a3 ¼ 1; a4 ¼ 1; a5 ¼ �1 (7.131)
a01 ¼ �2; a
02 ¼ �2; a
03 ¼ 2; a
04 ¼ �1; a
05 ¼ �3
Then (7.129) takes the form
Nu ¼ ðGrPr2 Þ (7.132)
7.6.2 Heat Transfer from Spinning Particle Under the Conditionof Mixed Convection
Consider the general case of heat transfer from a spherical spinning particle under
conditions ofmixed convectionwhen both natural and forced convection are essential.
We assume fluid to be incompressible and possess invariable physical properties. The
factors that determine the heat transfer intensity in the present case are: (1) the physical
properties of fluid-its density r, thermal conductivity k, viscosity m and specific heat
7.6 Heat and Mass Transfer in Natural Convection 187
cP; (2) particle linear and angular velocities relative to fluid u1 and o, respectively,particle diameter d, and temperature difference between the particle and fluid
Dt ¼ tP � t1. In the case when the effect of buoyancy force is significant, it is
necessary to supplement the system of the governing parameters with the product
bg(where b is the thermal coefficient and g gravity acceleration). Therefore, the
functional equation for the heat flux from the particle surface reads
q ¼ f ðr; k; m; cP; u1;o; d; bg;DtÞ (7.133)
In (7.133) the set of the nine governing parameters contains five parameters with
independent dimensions. Then, according to the Pi-theorem, the number of dimen-
sionless groups that determine the dimensionless heat transfer coefficient (the
Nusselt number) equals four. Choosing as parameters with independent dimensions
r; k; m; d and Dt, we arrive at the following equation
P ¼ ’ðP1;P2;P3;P4Þ (7.134)
where
P ¼ q=ra1ka2ma3da4Dta5 ; P1 ¼ u1=ra01ka
02ma
03da
04Dta
05 ; P2 ¼ o=ra
001 ka
002ma
003 da
004Dta
005 ;
P3 ¼ cP=ra0001 ka
0002 ma
0003 da
0004 Dta
0005 ; and P4 ¼ bg=ra
1V1 ka
1V2 ma
1V3 da
1V4 Dta
1V5 :
The values of the exponents ai; a0i; a
00i ; a
000i and a1Vi are equal to
a ¼ 0; a2 ¼ 1; a3 ¼ 0; a4 ¼ �1; a5 ¼ 1
a01 ¼ �1; a
02 ¼ 0; a
03 ¼ 1; a
04 ¼ �1; a
05 ¼ 0
a001 ¼ �1; a
002 ¼ 0; a
003 ¼ 1; a
004 ¼ 2; a
005 ¼ 0 (7.135)
a0001 ¼ 0; a
0002 ¼ 1; a
0003 ¼ �1; a
0004 ¼ 0; a
0005 ¼ 0
a1V1 ¼ �2; a1V2 ¼ 0; a1V3 ¼ 2; a1V4 ¼ �3; a1V5 ¼ �1
Then (7.134) takes the form
Nu ¼ ’ðRe;Reo; Pr;GrÞ (7.136)
where Nu ¼ qd=kDt; Re ¼ u1dr=m; Reo ¼ od2r=m; Pr ¼ mcP=k; and
Gr ¼ bgDtd3r2=m2 are the Nusselt, Reynolds, rotational Reynolds, Prandtl and
Grashof numbers, respectively.
In the particular case corresponding to heat transfer in motionless fluid at an
invariable Prandtl number, the heat transfer coefficient is determined by two
188 7 Heat and Mass Transfer
dimensionless groups, namely, the rotational Reynolds and Grashof numbers. The
measurements show that buoyancy effects are dominant at Reo<103 when the heat
transfer intensity is determined by natural convection (Tieng and Yan 1993). In the
range 103<Reo<9 103 a sharp change in the flow structure is observed. It
manifests itself in formation of a turbulent zone and jet eruptions which affect the
heat transfer process. At Reo>9 103 the forced convection is dominant. In this
case the contribution of the jet eruptions to the heat transfer is essential. For
calculations of the average Nusselt number for mixed convection, Tieng and Yan
(1993) suggested the following empirical relation which is valid at Pr ¼ 0:71 :
Nu3 ¼ Nu
3
n þ Nu3
f (7.137)
where Nu; Nun and Nuf are the overall Nusselt number and the average Nusselt
numbers for the natural and forced convection, respectively which are given by
Nun ¼ 2þ 0:392Gr0:31 (7.138)
at 1<Gr<105 and
Nuf ¼ 2þ 0:175Re0:583o (7.139)
at Gr ¼ 0:10<Reo<104 and Pr ¼ 0:71:
7.6.3 Mass Transfer from a Spherical Particle Under theConditions of Natural and Mixed Convection
Heterogeneous reactions taking place at the liquid-solid interface are the reason of
formation of non-uniform concentration fields in liquid solutions. That leads to
changing density of liquid solution and arising of buoyancy forces which drive
natural convection. When the dependence of liquid density on concentration of
reactant in solution is weak, the following approximation for solution density is
valid (Levich 1962)
rðcÞ rðc1Þ þ @r@c
� �����c¼c1
ðc� c1Þ (7.140)
where c and c1 are the local concentration and concentration far from the particle.
In this case the buoyancy force can be evaluation as follows
fb ¼ g rðc1Þ � rðcÞf g gðc1 � cÞ @r@c
� �����c¼c1
� agc1@r@c
� �����c¼c1
(7.141)
where g is the gravity acceleration and a is a known dimensionless constant (a<1Þ.
7.6 Heat and Mass Transfer in Natural Convection 189
Under the conditions of natural convection, themass flux to a particle is determined
by the buoyancy force, concentration of the reagent in solution, reagent diffusivity and
particle size. Accordingly, the functional equation for the mass flux reads
qm ¼ f ðfb; c1;D; r0Þ (7.142)
Three governing parameters in (7.142) have independent dimensions. Therefore,
it follows from the Pi-theorem that (7.142) can be transformed to the following
dimensionless form
P ¼ ’ðP1Þ (7.143)
where P ¼ qmr=Dc1 and P1 ¼ fbr30=D
2c1 � agr30 @r=@cð Þjc¼c1=D2 are the
Sherwood and Grashof numbers, respectively.
The functional equation for the mass transfer to a particle under the conditions of
mixed convection reads
qm ¼ f ðfb; c1;D; r0; u1Þ (7.144)
where u1 is the velocity far from particle.
In this case the set of the five governing parameters contains three parameters
with independent dimensions. Therefore, the dimensionless form of (7.144) is
P ¼ ’ P1;P2ð Þ (7.145)
where P; P1 and P2 ¼ u1r0=D are the Sherwood, Grashof and Peclet numbers,
respectively.
7.6.4 Heat Transfer From a Vertical Heated Wall
Consider heat transfer from a vertical heated wall under the conditions of natural
convection. In the framework of the boundary layer approximation, the system of
the governing equations describing the flow and heat transfer near the vertical wall
reads
u@u
@xþ v
@u
@y¼ n
@2u
@y2þ gbðT � T1Þ (7.146)
@u
@xþ @v
@y¼ 0 (7.147)
u@T
@xþ v
@T
@T¼ a
@2T
@y2(7.148)
190 7 Heat and Mass Transfer
where n and a are the viscosity and thermal diffusivity, b ¼ 1=T1 is the thermal
expansion coefficient, g gravity acceleration, x and y are the coordinate axes directedalong the wall and normal to it and subscript1 denotes the ambient conditions.
The boundary conditions for (7.146)–(7.148) are as follows
y ¼ 0; u ¼ v ¼ 0 T ¼ Tw; y ! 1; u ! 0 T ! T1 (7.149)
where Tw is the wall temperature.
It is convenient to introduce the excess of temperature DT ¼ T � T1 and rewrite
(7.146) and (7.148) in the following form
u@u
@xþ v
@u
@y¼ n
@2u
@y2þ g�y (7.150)
u@y@x
þ v@y@y
¼ @2y@y2
(7.151)
where y ¼ T � T1ð Þ= Tw � T1ð Þ and g� ¼ g Tw � T1ð Þ=T1.
The boundary conditions for (7.150) and (7.151) read
y ¼ 0 u ¼ v ¼ 0; y ! 1 u ! 1 y ! 0 (7.152)
Equations 7.147, 7.150 and 7.151 and conditions (7.152) contain three dimen-
sional constants ðn; a; and g�Þand two variables ðx and yÞ that determine the veloc-
ity and temperature fields. Therefore, one can assume that the following functional
equations are valid
u ¼ fuðx�; y; n; g�Þ (7.153)
y ¼ fyðx�; y; a; g�Þ (7.154)
where x� ¼ ex with e being a dimensionless constant (its numerical value is
determined from the consideration of dimensions and the condition of the existence
of a self-similar solution), x is the longitudinal coordinate.The dimensions of velocity u, temperature #, as well as of the governing
parameters x�; y; n; a; and g� (in the system of units LxLyLzMTy) are
u½ � ¼ LxT�1; y½ � ¼ 1; x�½ � ¼ Lx; y½ � ¼ Ly; n½ � ¼ L2T�1; a½ � ¼ L2yT
�1; g�½ �¼ LxT
�2 (7.155)
The set of the governing parameters in 7.153 and 7.154 contains three
parameters with independent dimensions, so that the difference n� k ¼ 1. There-
fore, the dimensionless forms of these equations read
7.6 Heat and Mass Transfer in Natural Convection 191
Pu ¼ ’ðP1uÞ (7.156)
Py ¼ #ðP1yÞ (7.157)
where Pu ¼ u=xa1� na2ga3� ; P1u ¼ y=x
a01� na
02g
a03� , Py ¼ y=xa1�� aa2�ga3�� and
P1y ¼ y=xa01�� aa
02�g
a03�� .
Taking into account the dimensions of u; #; x�; y; n and a, we find the values ofthe exponents ai and a
0i
a1 ¼ 1
2; a2 ¼ 0; a3 ¼ 1
2; a
01 ¼
1
4; a
02 ¼
1
2; a
03 ¼ � 1
4(7.158)
a1� ¼ a2� ¼ a3� ¼ 0; a01� ¼
1
4; a
02� ¼
1
2; a
03� ¼ � 1
4
Then we obtain
Pu ¼ u
x�g�ð Þ1=2; P1u ¼ yg
1=4�
x1=4� n1=2
; Py ¼ #; P1y ¼ yg1=4�
x1=4� a1=2
(7.159)
According to the expressions (7.159), (7.156) and (7.157) take the the following
form
u ¼ ðx�g�Þ1=2’ð�Þ (7.160)
y ¼ #ð�ffiffiffiffiffiPr
pÞ (7.161)
where Pr ¼ n=a is the Prandtl number, and � ¼ yg1=4� =x
1=4� n1=2:
Bearing in mind that the stream function is defined as c ¼ Ð udy and
v ¼ �@c=@x, we find
c ¼ x1=4� g1=4� n1=2’ð�Þ (7.162)
v ¼ � e4x1=4� g1=4� n1=2 3’ð�Þ � �’
0 ð�Þn o
(7.163)
Substitution of the expressions for u; v and y into (7.150) and (7.151) yields the
following system of ODEs for the unknown functions ’ð�Þ and #ð� ffiffiffiffiffiPr
p Þ(Pohlhausen, 1921)
’000� þ 3’’�
00 � 2’02� þ # ¼ 0 (7.164)
#00� þ 3 Pr’#
0 ¼ 0 (7.165)
192 7 Heat and Mass Transfer
where subscript � corresponds to differentiation by �: Note, that imposing the
conditions that the equations for the velocity and temperature do not incorporate
an arbitrary constant e, it should be taken as e ¼ 4.
The boundary conditions for (7.164) and (7.165) are
� ¼ 0; ’ ¼ ’0 ¼ 0 # ¼ 1; � ! 1; ’
0� ¼ 0 # ! 0 (7.166)
The local heat flux qðxÞ is expressed as
qðxÞ ¼ �k@T
@y
� �y¼0
¼ �kcx�1=4 @#
@�
� ��¼0
ðTw � T1Þ (7.167)
where k is thermal conductivity, c ¼ g�=4n2ð Þ1=4: The value of @#=@�ð Þ�¼0depends
on the Prandtl number value. It can be found only by solving (7.164) and (7.165).
For example, @#=@�ð Þ�¼0 ¼ 0:508 for Pr ¼ 0:773(Schlichting 1979).
The total heat flux from a plate of length l to the surrounding fluid is
Q ¼ b
ðl0
qðxÞdx ¼ 0:5084
3bl3=4ckðTw � T1Þ (7.168)
where b is the plate width. The average Nusselt number based on Q is expressed as
Nu ¼ 0:478 Grð Þ1=4 (7.169)
where Gr ¼ gl3ðTw � T1Þ=n2T1 is the Grashof number.
7.6.5 Mass Transfer to a Vertical Reactive Plate Under theConditions of Natural Convection
Consider mass transfer to a vertical reactive plate in contact with a liquid reactant
solution. Assume that at the surface of this plate (but not in the solution bulk) takes
place a chemical reaction the rate of which is much large then the rate of reactant
diffusion. In this case the reactant concentration will be equal zero at the wall. As a
result, a not-uniform reactant concentration field arises near the plate. Since the
density of solution depends on reactant concentration, a non-uniform density field
arises as well. The latter results in natural convection that determines the intensity
of mass transfer to the reactive wall.
In the framework of the boundary layer approximation the momentum and
species balance equations of the problem have the form (Levich 1962)
7.6 Heat and Mass Transfer in Natural Convection 193
u@u
@xþ v
@u
@y¼ n
@2u
@y2þ gc�c� (7.170)
u@c�@x
þ v@c�@y
¼ D@2c�@y2
(7.171)
where gc� ¼ ðgc1=rÞ @r=@cð Þc¼c1 ; c� ¼ ðc1 � cÞ=c1; g is gravity acceleration
due, r ¼ rðcÞ; D is the reactant diffusivity coefficient, and subscript 1corresponds to reactant concentration in the undisturbed solution.
The boundary conditions for (7.170) and (7.171) read
u ¼ v ¼ 0 c� ¼ 1 at y ¼ 0; u ¼ v ¼ 0 c� ! 1 at y ! 1 (7.172)
Equations 7.170 and 7.171 and conditions (7.172) formally coincide with
(7.150) and (7.151) and conditions (7.152). Accordingly, the expressions for the
components of velocity and reactant concentration can be recast as follows
u ¼ x�gc�ð Þ1=2’ð�Þ (7.173)
v ¼ �e1
4x�1=4� gc
1=4� D1=2 3’ð�Þ � ’
0 ð�Þn o
(7.174)
c� ¼ #cð�ffiffiffiffiffiffiSm
pÞ (7.175)
where Sc ¼ n=D is the Schmidt number, � ¼ g�=4n2ð Þ1=4y=x1=4.Substitution of the expressions (7.173) and (7.175) into (7.170) and (7.171) leads
(at e ¼ 4Þ to the system of ODEs for the functions ’ð�Þ and #ð� ffiffiffiffiffiSc
p Þ
’000 þ 3’’
00 � 2’02 þ #c ¼ 0 (7.176)
#00c þ 3Sc#
0c ¼ 0 (7.177)
where primes ðÞ0 correspond to differentiation by the dimensionless variable �.The mass flux of reactant at the wall is found by calculating the derivative
@c�=@yð Þy¼0 ¼ @c�=@�ð Þ�¼0@�=@y and using Fick’s law. The expressions that
determine the local j and total J mass flux at the vertical wall of height h and
width b reads (Levich 1962)
j ¼ 0:7Sc1=4Dgc14n2r
@r@c
� �c¼c1
( )1=4c1x1=4
(7.178)
J ¼ 0:9Sc1=4gc14n2r
@r@c
� �c¼c1
( )1=4
bh3=4c1D (7.179)
194 7 Heat and Mass Transfer
7.7 Heat Transfer From a Flat Plate in a Uniform Stream of
Viscous, High Speed Gas
Consider laminar boundary layer over a wall subjected to parallel uniform stream of
viscous, high velocity, perfect gas. It is easy to see that pressure along the wall is
constant in this particular case. Then, the system of equations that correspond to this
problem reads (Loitsyanskii 1966)
ru@u
@xþ rv
@u
@y¼ @
@ym@u
@y
� �(7.180)
@ru@x
þ @rv@y
¼ 0 (7.181)
ru@h
@xþ rv
@h
@y¼ k
mcP
@
@ym@h
@y
� �þ m
@u
@y
� �2
(7.182)
P ¼ g� 1
grh; m ¼ m0f
h
h1
� �(7.183)
where r; u; v; P and h are the density, longitudinal and lateral velocity
components, pressure and enthalpy, respectively; m; k and cp are the viscosity,
thermal conductivity and specific heat of fluid, respectively; g ¼ cP=cV is the
adiabatic index (the ratio of specific heats at constant pressure or constant vol-
ume).The boundary conditions for (7.180)–(7.182) are
y ¼ 0 : u ¼ 0; v ¼ 0 h ¼ hw; y ! 1 : u ! u1; h ! h1; P ! P1 (7.184)
for the plate with a given constant temperature, and
y ¼ 0 : u ¼ 0; v ¼ 0;@h
@y¼ 0; y ! 1 : u ! u1; h ! 1; P ! 1 (7.185)
for thermally insolated wall (subscripts w and 1 refer to the wall and undisturbed
flow, respectively).
The density of the ambient gas (far away from the wall) determines (at a given of
the enthalpy, h1Þ the value of pressure, which follows from the first (7.183). Taking
into account this circumstance, it is possible to write the following functional
equations for the longitudinal velocity component and enthalpy
u ¼ fuðx; y; r1; u1; h1; hw;P1; m1; k1; cP1Þ (7.186)
h ¼ hhðx; y; r1; u1; h1; hw;P1; m1; k1; cPÞ (7.187)
7.7 Heat Transfer From a Flat Plate in a Uniform Stream 195
It shows that u and h are the function of ten dimensional parameters: x½ � ¼ L;
y½ � ¼ L; r1½ � ¼ L�3M; u1½ � ¼ LT�1; P½ � ¼ L�1MT�2; h1½ � ¼ JM�1;
hw½ � ¼ JM�1; k1½ � ¼ JT�1L�1y�1; cP½ � ¼ JM�1y�1 and m½ � ¼ L�1MT�1.
Taking into account that (7.182) includes the ratio k=mcP½ � ¼ 1 as a single
complex, it is possible to diminish the number of the governing parameters in
(7.186) and (7.187) and present them in the following form
u ¼ fuðx; y; r1; u1; h1; hw;P1; m1;k
mcPÞ (7.188)
h ¼ fhðx; y; r1; u1; h1; hw;P1; m1;k
mcPÞ (7.189)
Equations 7.188 and 7.189 contain nine governing parameters, four of which
have independent dimensions. Then, according with the Pi-theorem, (7.188) and
(7.189) reduce to the two following dimensionless equations
Pu ¼ fuðP1;P2;P3;P4;P5Þ (7.190)
Ph ¼ fhðP1;P2;P3;P4;P5Þ (7.191)
where Pu and Ph are the dimensionless velocity and enthalpy, respectively; the
dimensionless groups Pi (i¼1,. . .5) are given by the following expressions:
P1 ¼ y=xa01ra
021u
a031h
a041; P2 ¼ hw=x
a001 ra
0021u
a0031h
a0041; P3 ¼ P1=xa
0001 ra
00021 u
a00031 h
a00041 ; P4 ¼
m1=xa1V1 r
a1V21 u
a1V31 h
a1V41 and P5 ¼ k1 m1cP1=ð Þ=xaV1 raV21u
aV31h
aV41
Bearing in mind that Pu ¼ u=xa1ra21ua31ha41 and Ph ¼ h=xa1�ra2�1 ua3�1 ha4�1 , we can
rewrite (7.190) and (7.191) in the following form
u
u1¼ ’u
y
x;hwh1
;P
r1u21;
m1xr1u1
;k1
m1cP
� �(7.192)
h
h1¼ ’h
y
x;hwh1
;P1
r1u21;
m1xr1u1
;k1
m1cP
� �(7.193)
It is seen that the dimensionless velocity and enthalpy are determined by the
functions of two dimensionless variables y=x and m1=xr1u1and three constants.
That shows that (7.180)–(7.182) cannot be reduce ODEs. In order to solve this
problem it is necessary to use a special transformation (7.180)–(7.182). First of all,
we transform (7.180)–(7.182), with the boundary conditions (7.183) and the
correlations (7.183) to the dimensionless form normalizing the parameters by
their values in the undisturbed flow (u ¼ u=u1; h ¼ h=h1, etc.) and the variables
x and y by some length scaleL (x ¼ x=L and y ¼ y=LÞ. Omiting for brevity bars
196 7 Heat and Mass Transfer
over the dimensionless parameters and assumed that viscosity is a linear function of
temperature, we rewrite (7.180)–(7.183) as follows
ru@u
@xþ rv
@u
@y¼ @
@ym@u
@y
� �(7.194)
@ru@x
þ @rv@y
¼ 0 (7.195)
ru@h
@xþ rv
@h
@y¼ ðg� 1ÞM2
1m@u
@y
� �2
þ 1
Pr
@
@ym@u
@y
� �(7.196)
r ¼ 1
h; m ¼ hn (7.197)
Planar compressible problems described by (7.194)–(7.197) are further
simplified by employing the Dorodnitsyn (1942), Illingworth (1949), Stewartson
(1949) transformation (Loitsyanskii 1966; Schlichting 1979)
x ¼ x; � ¼ðy0
rdy (7.198)
The transformation reduces the compressible (7.194)–(7.197) to the icompressible
boundary layer equations, which take the following dimensionless form
u@u
@xþ ev @u
@�¼ @2u
@�2(7.199)
@u
@xþ @ev@y
¼ 0 (7.200)
u@h
@xþ ev @h
@�¼ 1
Pr
@2h
@h2þ ðg� 1ÞM2
1@u
@�
� �2
(7.201)
with the boundary conditions
� ¼ 0 : u ¼ 0; h ¼ hw; � ! 1 : u ! 1; h ! 1 (7.202)
for plates with a constant temperature given, and
� ¼ 0; u ¼ 0;@h
@�¼ 0; � ! 1 : u ! 1; h ! 1 (7.203)
7.7 Heat Transfer From a Flat Plate in a Uniform Stream 197
In (7.199)–(7.201)
ev ¼ u@�
@xþ rv (7.204)
Equations (7.199) and (7.201) and the boundary conditions (7.202) and (7.203)
show that the dynamic problem formulated in new variables and coordinates
becomes autonomous and the dimensionless velocity u depends on two dimension-
less variables x and � as
u ¼ f ð�; xÞ (7.205)
Any combination of these variables is also dimensionless. Similarly to flows of
incompressible fluid, one can assume that u is self-similar in a sense that f in (7.205)is a function of one dimensionless variable
w ¼ a�
xa(7.206)
where a and a are some constants.
Introducing function ’wðwÞ instead of f in such a way that u is expressed as
u ¼ a’0wðwÞ (7.207)
we find the expressions for the stream function c and a transformed lateral velocity
component ev asc ¼
ðudy ¼ xa’ðwÞ (7.208)
ev ¼ � @c@x
¼ axa�1ða’0wx� ’Þ (7.209)
where prime denotes derivatives in w.Substitution of the expressions for u and ev into (7.194) yields
a
x2a’
000w þ a
x’
00w’ ¼ 0 (7.210)
Requiring that (7.210) become an ODE in w, while variable x cancels, one finds
that a ¼ a ¼ 1=2. Then (7.210) takes the form
’000w þ ’
00w’ ¼ 0 (7.211)
198 7 Heat and Mass Transfer
The boundary conditions for (7.211) are
w ¼ 0; ’ ¼ 0 ’0w ¼ 0; w ! 1; ’
0 ! 2 (7.212)
Correspondingly, the energy equation 7.203 reduces to the following ODE
h00w þ Pr’h
0w þ
Pr
4ðg� 1ÞM2
1’00w ¼ 0 (7.213)
with the boundary conditions
w ¼ 0 : h ¼ hw; w ! 1 : h ! 1 (7.214)
for a constant temperature wall, and
w ¼ 0 :@h
@w¼ 0; w ! 1 : h ! 1 (7.215)
for the thermally-insulated one.
The problem (7.211) and (7.212) formally coincides with the problem on the
incompressible boundary layer near a plane wall (the Blasius problem). Accord-
ingly, the drag coefficient at m ¼ hn and n ¼ 1 is expressed as
cf ¼ 1
2
1ffiffiffiffiffiffiffiffiRex
p ’00wð0Þ (7.216)
where Rex ¼ u1x=n1 and ’00wð0Þ ¼ 1:328.
The solution of (7.213) leads to the following dependence for the the average
Nusselt number for a plate of length L
Nu ¼ Tw � TsTw � T1
f ðPrÞffiffiffiffiffiffiffiffiffiRe1
p(7.217)
where Ts is the stagnation temperature, Re1 ¼ uL=n.
7.8 Heat Transfer Related to Phase Change
7.8.1 Heat Transfer Due to Condensation of Saturated Vaporon a Vertical Wall
Consider the heat transfer process during condensation of saturated vapor on a cold
vertical wall. As a result of condensation, a thin liquid film forms at the wall
surface. This film flows downward due to gravity. Latent heat that is released due
7.8 Heat Transfer Related to Phase Change 199
to vapor condensation is transferred from film to the wall by conduction and
convection. In a thin liquid film flowing with low velocity (Re 1Þthe convectivecomponent of the heat flux qconv is negligible in comparison with the conductive
one qcond, which is supported by the following estimates
qcond � kL@T
@y
� �y¼0
� kLdðTs � TwÞ (7.218)
qconv � rucPdðTs � TwÞ (7.219)
In Eqs. (7.218) and (7.219) d is the liquid film thickness; rL; kL and cPL are thedensity, thermal conductivity and specific heat at constant pressure of the liquid, Tsand Tw are the vapor saturated corresponding to a given pressure and the wall
temperatures, respectively.
The estimations (7.218) and (7.219) show that
qconvqcond
Re Pr 1 at Re 1 and Pr 1 (7.220)
The assumptions adopted here were first used by Nusselt (1916) in a simplified
theory of heat transfer during film condensation of saturated vapor on a cold
vertical wall. Below we consider this problem using the Pi-theorem. First of all,
consider the parameters which can affect the heat transfer from a saturated vapor to
a cold vertical wall. The set of the governing parameters should inevitably include
the liquid density rL and viscosity mL as well as gravity acceleration g: It is
necessary to account for the latent heat of condensation Dhfv;which is the origin
of heat flux due to vapor condensation q. The set of the governing parameters
should also include the vapor density rV (in fact, the difference rL � rVÞ: How-ever, since rL>>rV ; it is possible to omit rV from the set of the governing
parameters. Accordingly, we can write the functional equation for the liquid film
thickness at the wall as follows
d ¼ f ðr; m;Dhfv; g; q; xÞ (7.221)
where x is the longitudinal coordinate reckoned down the wall.
In order to decrease the number of the governing parameters, it is naturally to
assume that the parameters g and mL group with the liquid density rL as the
specific weight grL and kinematic viscosity mL=rL ¼ nL: Then (7.221) transforms
into
d ¼ f ðgrL; nL;Dhfv; q; xÞ (7.222)
In the framework of the boundary layer approximation one can introduce three
different scales of length Lx, Ly and Lz for the longitudinal and two lateral directions
200 7 Heat and Mass Transfer
x, y and z, and use the system of units LxLyLzMTJ: Then the dimensions of the
governing parameters in (7.222) are expressed as
rLg½ � ¼ L�1y L�1
x MT�2; nL½ � ¼ L2yM; Dhfv � ¼ JM�1; q½ �
¼ JT�1L�1x L�1
z ; x½ � ¼ L (7.223)
It is seen that all the governing parameters have independent dimensions, so that
(7.223) takes the form
d ¼ cðrLgÞa1ðnLÞa2ðDhfvÞa3ðqÞa4ðxÞa5 (7.224)
where c is a constant.Taking into account the dimension of d½ � ¼ Ly, we find the values of the
exponents ai as a1 ¼ �1=3; a2 ¼ 1=3; a3 ¼ �1=3; a4 ¼ 1=3 and a5 ¼ 1=3.Then (7.224) takes the form
d ¼ cnLqx
ðrLgÞDhfv
� �1=3(7.225)
Assuming that q ¼ kL @T=@yð Þy¼0 � kL Ts � Twð Þ=d, we obtain
d ¼ c1kLmLðTs � TwÞ
r2LgDhfvx
� �1=4(7.226)
where c1 ¼ c3/4
Under the assumption that the temperature distribution within the liquid film is
linear, the local heat transfer coefficient is found as
a ¼ kLd¼ c2
r2LgDhfvk3L
mLðTs � TwÞ1
x
� �1=4(7.227)
where c2 ¼ c�11 is a constant.
The comparison of the predictions of Nusselt’s theory with the experimental
data on film condensation shows that there is a significant deviation of the
theoretical results from the experimental results (about 25%). The latter stems
from significant oversimplification of the complex phenomenon in Nusselt’s
theory which does not account for a number of factors that affect the heat transfer
intensity. These factors are: wave formation on the film surface, temperature
dependence of the physical properties of fluid, etc. (Baehr and Stephan 1998).
A detailed analysis of steady laminar film condensation (condensation of stagnant
and flowing vapor, the self-similar solution in the cases of free and
forced convection, etc.) can be found in the monographs by Fujii (1991) and
Stephan (1992).
7.8 Heat Transfer Related to Phase Change 201
7.8.2 Freezing of a Pure Liquid (The Stefan Problem)
Let the half-space 0bxb1 be filled with a pure liquid (say, water) at rest with an
initial temperature #21>0(# ¼ T � Tf ; where Tf is the freezing temperature)
(Fig. 7.5). On the left side of the water-filled domain a refrigerator is put at the initial
timemoment t¼0. Its temperature #10 is kept constant below the freezing temperature
#f : #10<#f ¼ 0: Because of the conductive heat transfer to the refrigerator, liquid
temperature decreases in the negative x direction.As a result of liquid cooling, near the
contact with the refrigerator forms an ice layer and its thickness increases in time. The
interface that separates the ice and liquid domains propagates into liquid (in the
positive x direction). Themain aim is in determining the rate of growth of the ice layer.
From the physical point of view it is plausible to assume that the ice layer
location of the freezing interface x is determined by the physical properties of liquid
and its ice (cP1; cP2; k1 and k2;with subscript 1 being used for ice and subscript
2-for liquid), latent heat of solidification qs; the initial temperature of liquid far
away from the refrigerator #20, the refrigerator temperature #10, densities of the ice
and liquid r1 and r2, as well as time t
x ¼ f ðcP1; cP2; k1; k2; #10; #20; r1; r2; qs; tÞ (7.228)
The dimensions of the governing parameters are as follows
cP½ � ¼ JM�1y�1; k½ � ¼ JT�1L�1y�1; #i½ � ¼ y; qs½ � ¼ JM�1; t½ � ¼ T (7.229)
Equations 7.228 and 7.229 show that the present problem contains ten dimen-
sional parameters, five of them possessing independent dimensions. Choosing as
the parameters with the independent dimensions r2; cP2; k2; #21 and t, we
trabsform (7.228) to the following form
P ¼ ’ðP1;P2;P3;P4Þ (7.230)
where the dimensionless groups are expressed as
P ¼ x
ra2cbP2k
g2#
e20t
o; Pi ¼ wi
rai2 cbiP2k
gi2 #
ei20t
oi
; (7.231)
J1 = J2 = Jf = 0
uf
0
Water J2, +>JfJ1.– <Jf ice
Fig. 7.5 Freezing of pure liquid
202 7 Heat and Mass Transfer
with wi being one of the following five parameters r1; cP1; k1; #10 and qs, andi ¼ 1; 2; 3; 4 and 5, respectively.
Determining the values of the exponents a; b; g; e; o; and ai; bi; gi; ei,oi, we
arrive at the following expression
x ¼ a2tð Þ1=2’ðr12; cP12; k12; #12; StÞ (7.232)
where a2 is the thermal diffusivity of liquid, r12 ¼ r1=r2; cP12 ¼ cP1=cp2;k12 ¼ k1=k2; #12 ¼ #10=#20, and St ¼ qs=cP2#20 being the Stanton number.
The speed of the freezing front is
Vs ¼ 1
2
a2t
1=2’ðr12; cP12; k12; #12; StÞ (7.233)
Equations 7.232 and 7.233 show that the freezing front propagates as t1=2,whereas its speed decreases as t�1=2.
In the framework of the dimensional analysis it is impossible to establish the
exact form of the dependence ’ðr12; cP12; k12; #12; StÞ and calculate the exact
values of x and Vs. In order to find the exact form of the solution, the corresponding
equations should be solved exactly. The system of the governing equations of the
Stefan problem reads
@#1
@t¼ a1
@2#1
@t2(7.234)
for the ice domain �1<x<x, and
@#2
@t¼ a2
@2#2
@x2(7.235)
for the liquid domain x<x<1.
The corresponding boundary conditions are
x ¼ x; # ¼ #1 ¼ #2 ¼ #f ¼ 0 (7.236)
x ¼ �1; #1 ¼ #10; x ¼ 1; #2 ¼ #20 (7.237)
In addition, the system of Eqs (7.234) and (7.235) is subjected to the following
boundary condition expressing the thermal balance at the freezing front
k1@#1
@x
����x¼x�
� k2@#2
@x
����x¼xþ
¼ qsrdxdt
(7.238)
This additional condition allows finding the coordinate of the freezing front x asa function of time. It is emphasized that the problem allows determining x only up
7.8 Heat Transfer Related to Phase Change 203
to an additive constant, which corresponds to the initiation of freezing from any
cross-section x (in particular, from x ¼ �1). Equation 7.234 and 7.235 with the
boundary conditions (7.236) and (7.237) admit solutions in the form of the follow-
ing functional equations for the temperature fields in the ice and liquid domains
#1 ¼ f1ða1; #10; x; tÞ (7.239)
#2 ¼ f2ða2; #20; x; tÞ (7.240)
Since the governing parameters in (7.239) as well as in Eq (7.240) contains three
parameters with independent dimensions, these equations can be transformed to the
following self-similar forms
#1 ¼ ’1ð�1Þ (7.241)
#2 ¼ ’2ð�2Þ (7.242)
where #1 ¼ #1=#10 and #2 ¼ #2=#20 with �1 ¼ x=ffiffiffiffiffiffia1t
pand �2 ¼ x=
ffiffiffiffiffiffia2t
p,
respectively.
The conditions (7.236) and (7.237) take the form
x ¼ x; ’1ð�1xÞ ¼ ’2ð�2xÞ ¼ 0 (7.243)
x ¼ �1; ’1ð�1Þ ¼ 1; x ¼ 1; ’2ð1Þ ¼ 1 (7.244)
where �1x ¼ x=ffiffiffiffiffiffia1t
pand �2x ¼ x=
ffiffiffiffiffiffia2t
p.
Accordingly, (7.238) shows that the only possible law of propagation of the
freezing front (up to an additive constant) is given by
x ¼ bffiffit
p(7.245)
where b is a constant (an eigenvalue of the problem) which is also detrmined by
(7.238).
In particular, substituting (7.241)–(7.245) into (7.234)–(7.238), one finds the
self-similar solutions as
’1ð�1Þ ¼ A1 þ B1erf ð�1=2Þ (7.246)
’2ð�2Þ ¼ A2 þ B2efrð�2=2Þ (7.247)
A1 ¼ 1; B1 ¼ 1
erf b=2ffiffiffiffiffia1
p� � ; A2 ¼erf b=2
ffiffiffiffiffia2
p� �1� erf b=2
ffiffiffiffiffia2
p� � ; B2
¼ 1
1� erf b=2ffiffiffiffiffia2
p� � (7.248)
204 7 Heat and Mass Transfer
and the following equation determining b
k1 exp �b=4a21� �
a1erf b=2a1ð Þ þ k2 exp �b=4a22� �
a2 1� erf b=2a2ð Þ½ � ¼ �qsr2bffiffiffip
p2
(7.249)
Problems
P.7.1. Show that the evolution of the thermal field driven by radiative heat flux in a
uniform medium after an instantaneous pointwise energy release by a powerful
source corresponds to a self-similar solution.
The equation that determines temperature field driven by radiative heat flux
reads
@T
@t¼ 1
r2@
@rwr2
@T
@r
� �(P.7.1)
where w ¼ aTn is the radiant thermal diffusivity coefficient, and it is assumed that
the initial energy release happens at r¼0.Integrating (P.7.1) from r ¼ 0 to r ¼ 1 yields the folloving integral invariant
4pð10
Tr2dr ¼ Q (P.7.2)
Equations P.7.1 and P.7.2 show that the problem under consideration contains four
governing parameters: two dimensional constants, a and Q, and two independent
variables r and t. Accordingly, the functional equation for the temperature field reads
T ¼ f ða;Q; r; tÞ (P.7.3)
The dimensions of the governing parameters are
a L2T�1y�n �
; Q yL3 �
; r L½ �; t T½ � (P.7.4)
Since three of the four governing parameters have independent dimensions,
(P.7.3) can be reduced to the following dimensionless equation
P ¼ ’ðP1Þ (P.7.5)
where P ¼ T=aa1Qa2 ta3 and P1 ¼ r=aa01Qa
02 ta
03 :
Problems 205
Bearing in mind the dimensions of T; a; Q; r and t, we arrive at the followingset of algebraical equations determining the exponents ai and a
0i
2a1 þ 3a2 ¼ 0; 2a01 þ 3a
02 ¼ 1
� a1 þ a3 ¼ 0;�a01 þ a
03 ¼ 0 (P.7.6)
� a1nþ a2 ¼ 1;�a01nþ a
02 ¼ 0
Equations P.7.6 yield the following values
a1 ¼ � 1
nþ 2=3; a2 ¼ 2
3
1
nþ 2=3; a3 ¼ � 1
nþ 2=3;
a01 ¼
n
3nþ 2; a
02 ¼
1
3nþ 2; a
03 ¼
1
3nþ 2
(P.7.7)
Then (P.7.5) takes the following form
T ¼ Q2
ðatÞ3 !1=ð3nþ2Þ
’r
ðaQntÞ1=ð3nþ2Þ
( )(P.7.8)
Substitution of the expression (P.7.8) into (P.7.1) leads to the following ODE
determining the function ’ðxÞ
ð’nþ1Þ00x þ2
xð’nþ1Þ0x þ
1
3nþ 2x’
0x þ
3
3nþ 2’ ¼ 0 (P.7.9)
where x ¼ r=ðaQntÞ1=ð3nþ2Þ, and subscript x denotes differentiation by x.
P.7.2. Determine mass flux to a pipe wall in the case of fully developed laminar
flow of liquid reactant solution. The rate of the heterogeneous reaction at the wall is
assumed to be infinite, and accordingly, the reactant concentration at the wall
equals zero.
A sketch of a fully developed (hydrodynamically) laminar flow is shown in
Fig. 5.1. In this case the flow field does not change with x (along the pipe). On the
contrary, the reactant concentration field, and in particular, the thickness of the
diffusion boundary layer near the pipe wall increases in flow direction. The reactant
concentration changes in a thin boundary layer since the kinematic viscosity is
typically much larger than the diffusion coefficient, n � D. Accordingly, the flowstructure and the reactant concentration distribution can be characterized by the
maximum velocity u0 at the pipe axis and the reactant concentration c0 in the core
of the flow. Thus, as the governing parameters that determine the diffusion flux of
the reactant toward the wall one can choose u0; c0, pipe radius R, the diffusion
coefficient D; as well as the longitudinal coordinate x as the parameter responsible
206 7 Heat and Mass Transfer
for the growth of the diffusion boundary layer thickness in flow direction. Then the
functional equation for the reactant diffusion flux reads
qm ¼ f ðu0; c0;R;D; xÞ (P.7.10)
The dimensions of the diffusion flux and the governing parameters are
qm½ � ¼ ML�2T�1; u0½ � ¼ LT�1; c0½ � ¼ L�3M; R½ � ¼ L; D½ � ¼ L2T�1; x½ �¼ L (P.7.11)
Three of the five governing parameters in (P.7.10) have independent dimensions.
Therefore, according to the Pi-theorem, (P.7.10) can be transformed to the follow-
ing form
P ¼ ’ðP1;P2Þ (P.7.12)
where P ¼ qm=ca10 D
a2Ra3 ; P1 ¼ x=ca01
0 Da02Ra
03 and P2 ¼ u0=c
a001
0 Da002Ra
003 :
Determining the values of the exponents ai; a0i and a
00i , we find a1 ¼ 1; a2 ¼ 1;
a3 ¼ �1; a01 ¼ 0; a
02 ¼ 0; a
03 ¼ 1; a
001 ¼ 0; a
002 ¼ 1 and a
003 ¼ �1. Then, we obtain
Sh ¼ ’ðx;PedÞ (P.7.13)
where Sh ¼ qmR=coD and Ped ¼ u0R=D are the Sherwood and the diffusional
Peclet numbers, and x ¼ x=R:Equation P.7.13 shows that the dimensionless mass transfer coefficient is deter-
mined by two dimensionless groups x and Ped. In the framework of the dimensional
analysis it is impossible to specify this dependence further more.. However, this
problem can be solved in the framework of the dimensional analysis if the diffusion
equation is simplified in the boundary layer approximation, which allows another
set of the governing parameters to be chosen. For this aim consider the rigorous
mathematical formulation the problem (but not its exact solution). The reactant
diffusion equation in the thin diffusion layer near the pipe wall reads (Levich 1962)
2u0R
y@c
@x¼ D
@2c
@y2(P.7.14)
c ! c0 at y ! 1; c ¼ 0 at y ¼ 0 (P.7.15)
where y ¼ R� r; and r is the radial coordinate in the cylindrical system reckoned
from the pipe axis. of coordinates.
Introducing the new variable y ¼ y u0=DRð Þ1=3, transform (P.7.14) to the follow-
ing form
@c
@x¼ 1
2
1
y
@2c
@y2(P.7.16)
Problems 207
Equation P.7.16 with the boundary conditions (P.7.15) show that concentration cderends on three dimensional parameters
c ¼ f ðc0; x; yÞ (P.7.17)
Two governing parameters in (P.7.17) have independent dimensions, so that this
equation can be transformed to the following dimensionless form
P ¼ ’ðP1Þ (P.7.18)
where P ¼ c=ca10 xa2 and P1 ¼ y=c
a01
0 xa02 :
The corresponding values of the exponents ai and a0i are found as:
a1 ¼ 1; a2 ¼ 0; a01 ¼ 0 and a
02 ¼ 1=3. Then, (P.7.18) takes the form
c ¼ c0’ð�Þ (P.7.19)
where � ¼ y u0=DRxð Þ1=3.The reactant flux to the pipe wall is
qm ¼ D@c
@y
� �y¼0
¼ @c
@�
� ��¼0
Du0DR
1=3 1
x1=3c0 (P.7.20)
or
Sh ¼ @c
@�
� ��¼0
Ped1=3x�1=3 (P.7.21)
Using the expression (P.7.20), we find the cumulative mass flux of reactant to the
wall in a pipe section of length x as
I ¼ 2pRðqmdx ¼ Ac0DR
u0x2
DR
� �1=3
(P.7.22)
where A ¼ � 2p=3ð Þ @c=@�ð Þ�¼0
The exact analytical solution of the problem reads (Levich 1962)
I ¼ 2:01pc0DRu0x
2
DR
� �1=3
(P.7.23)
208 7 Heat and Mass Transfer
References
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210 7 Heat and Mass Transfer
Chapter 8
Turbulence
8.1 Introductory Remarks
The turbulence represents itself a very complicated hydrodynamic phenomenon
characterized by irregular unsteady fluid motion. It emerges in liquid and gas flows
at sufficiently high Reynolds numbers when laminar flow regime becomes unstable
and strongly perturbed. This process is accompanied by arising turbulent eddies of
different sizes which are, in their turn, sources of velocity disturbances at each point
of the flow field. The amplitudes and frequencies of such disturbances depend
on the Reynolds number value. The scales of these disturbances decrease as
Re increases, whereas their frequency is proportional to the Reynolds number.
An exceptional complexity of turbulence impedes the theoretical analysis of this
phenomenon. In this situation a number of important results (mostly qualitative)
may be obtained using the methods of the similarity theory and dimensional analysis
(Kolmogorov, 1941a, b; Obukhov 1941). Following these works we address briefly
some problems related to the uniform isotropic turbulence, as the examples
illustrating applications of the dimensional considerations to study turbulent flows.
The actual velocity vector v at any point of developed turbulent flows can be
presented as a sum of the mean velocity v (obtained by averaging the actual velocityover a long time interval) and fluctuation velocity v
0. The existence of turbulent
fluctuation velocity and other fluctuating hydrodynamic and thermal characteristics
significantly affect the flow structure, as well as the intensity of processes of
momentum, heat and mass transfer.
Turbulent flows can be represented schematically as a conglomerate of turbulent
eddies of different sizes that generate fluctuations of velocity, temperature, etc. The
influence of large and small eddies on flow field is essentially different. The large
eddies are the bearers of kinetic energy of turbulent flow, which is transferred
though a cascade of smaller and smaller eddies to the smallest ones. On the other
hand, the smallest eddies are subjected to significant viscous forces which dissipate
the transferred kinetic energy into heat. The energy transfer from large to small
eddies occurs practically without dissipation, so that the energy flux from large
L.P. Yarin, The Pi-Theorem, Experimental Fluid Mechanics,
DOI 10.1007/978-3-642-19565-5_8, # Springer-Verlag Berlin Heidelberg 2012
211
eddies equals energy dissipation in the smallest ones. That allows the assumption
that in developed turbulent flows a continuous energy transfer from large to small
eddies takes place. Such peculiarity of turbulent flows has a principal meaning for
the evaluation of a number of important characteristics of developed turbulent
flows. Indeed, using the concept of the energy transfer from large to the smallest
eddies, it is possible to evaluate the dissipation of kinetic energy by analyzing
solely the large scale motions.
Taking into account the above-mentioned peculiarities of the energy transfer, it
is possible to neglect the influence of viscosity on flow characteristics within the
domain of large eddies. In this case the energy dissipation is determined by only
three parameters, namely, the fluid density r, the characteristic size of large eddies lwhich is of the same order as the external flow scale, and the mean velocity change
Du over the length l. Accordingly, the functional equation for the energy dissipationper unit time E takes the form
E ¼ f ðr;Du; lÞ (8.1)
All the governing parameters in (8.1) have independent dimensions. Therefore,
according to the Pi-theorem, (8.1) can be written as
E ¼ cra1Dua2 la3 (8.2)
where c is a dimensionless constant.
Taking into account the dimension E½ � ¼ L�1MT�3, we find the values of the
exponents ai as a1 ¼ 1; a2 ¼ 3 and a3 ¼ �1, which allows transformation of (8.2)
to the following form
e ¼ cDu3
l(8.3)
where e ¼ E=r is the energy dissipation per unit time and unit mass.
In saddition, the functional equation for the turbulent (eddy) viscosity associated
with the turbulent Reynolds stresses can be also expressed through the same set of
the governing parameters
mT ¼ f ðr;Du; lÞ (8.4)
Using the Pi-theorem, we arrive at the following expression
nT ¼ c1lDu (8.5)
where nT ¼ mT=r is the turbulent kinematic viscosity and c1is a dimensionless
constant.
212 8 Turbulence
Then the relation between the turbulent and physical (molecular) viscosities can
be expressed as
nTn¼ c1Re (8.6)
where Re ¼ lDu=n.Equation (8.6) shows that the ratio nT=n increases with the Reynolds number.
Consider now some local properties of turbulence that characterize its features at
the scales of the order l much smaller than the characteristic scale of large eddies lbut much larger than the scale of the smallest dissipative eddies l0. It is possible toassume that turbulence is isotropic in flow regions far away from solid surfaces.
Using this assumption, we evaluate first the change of velocity of turbulent motion
vlover a distance of the order of l. Choosing as the governing parameters the
energy dissipation e and length l, we present the functional equation for vl as
follows
vl ¼ f ðe; lÞ (8.7)
Bearing in mind the dimensions e½ � ¼ L2T�3; l½ � ¼ L and nl½ � ¼ LT�1, we arrive
at the scaling law
vl � ðelÞ1=3 (8.8)
It expresses the Kolmogorov-Obukhov law: the change of velocity over a small
distance l is scales as l1=3. The local Reynolds number for the smallest dissipative
eddies corresponds to the borderline where molecular viscosity begins to play role.
Therefore, this Reynolds number should be of the order of one. On the other hand, it
is determined by velocity vl corresponding to l ¼ l0, the scale of the smallest
eddies responsible for energy dissipation l0, and fluid viscosity n. Accordingly, wearrive at the following estimate of l0
l0 � l
Re3=4(8.9)
In the framework of the dimensional analysis it is possible to evaluate velocity
change over the smallest dissipative eddy scale l0 [which is given by (8.8) and
(8.9)], the frequency of velocity pulsations, as well as the turbulent pulsations of
temperature at l>> l0. In order to estimate the temperature fluctuations in a non-
uniformly heated fluid over a distance l, consider the dependence for the tempera-
ture difference Tl (temperature fluctuations) following Obukhov (1949). For this
aim, we use the expression for the energy dissipation due to thermal conductivity
8.1 Introductory Remarks 213
ET � krTð Þ2T
� kTl l=ð ÞT
(8.10)
where ET is the energy dissipation due to thermal conductivity of fluid
ð ET½ � ¼ L�1MT�3Þ, k is the termal conductivity ( k½ � ¼ LMT�3y�1), Tis the actual
temperature.
Accordingly, the rate of dissipative changing of temperature may be estimated as
follows
E0T � wt
Tl l=ð Þ2T
(8.11)
where E0T is the rate of dissipative changing temperature ( E
0T
� � ¼ yT�1), and wt is theturbulent thermal diffusivity ( wt½ � ¼ L2T�1).
Assuming that E0T ¼ ’ T= , where ’ is a certain factor that determine the local
features of turbulence in a non-uniformly heated fluid, we can postulate the
following relation
wtTl l=ð Þ2T
¼ ’
T(8.12)
At small intensity of temperature fluctuations (Tl T= <<1) it is possible to
replace the actual temperature T by the mean temperature of fluid and assume that
’ ¼ wtTll
� �2
(8.13)
Thus, the factor ’ depends on three dimensional parameters, namely, (1) turbu-
lent thermal diffusivity wt, (2) temperature fluctuations Tl, and (3) the distance l
’ ¼ f ðwt; Tl; lÞ (8.14)
Since all the governing parameters in the functional (8.14) have independent
dimensions, according with the Pi-theorem, it takes the following form
’ ¼ wa1t Ta2l la3 (8.15)
where c is a dimensionless constant.
Bearing in mind the dimension of ’ ( ’½ �¼y2T�1), we find using the principle
of dimensional homogeneity that the values of the exponents aiare:a1¼1;a2¼2;anda3¼�1. Then (8.15) reads
214 8 Turbulence
’ ¼ cwtTll
� �2
(8.16)
Accounting that wt � nt � lvl, and vl ¼ elð Þ1 3= , we arrive at the following
estimate for the temperature fluctuations
T2l � ’e�1=3l2=3 (8.17)
It is seen that temperature fluctuations like the velocity fluctuations are propor-
tional to l1=3 on the scale l>>l0.Below we consider in detail the applications of the Pi-theorem to a number of
important types of turbulent flows, in particular, turbulent near-wall flows, flows in
smooth and rough pipes and channels, as well as various kinds of turbulent jets.
8.2 Decay of Isotropic Turbulence
The behavior of isotropic turbulence is described by the von Karman and Howarth
(1938) [see also Pope (2000)]
@bdd@t
¼ 2n1
r4@
@r
�r4@bdd@r
�þ 1
r4@
@rðr4bdd�dÞ (8.18)
where bdd and bddd are the components of the second and third correlation tensor,
respectively.
At very the smallest-scale dissipative eddies the corresponding small velocity
fluctuations are dominated by the viscous effects and the second term on the right-
hand side in (8.18) can be omitted. Then the von Karman and Howarth equation
takes the form
@bdd@t
¼ 2n1
r4@
@r
�r4@bdd@r
�(8.19)
To find the solution of (8.19), it is necessary to impose the initial distribution of
the second correlation bddðr; tÞ, i.e. bddðr; 0Þ. Equation (8.19) admits the invariant
(the Loitsyanskii invariant L0) which can be determined through the initial condi-
tion bddðr; 0Þ (Loitsyanskii 1939)
L0 ¼ð10
r4bdddr (8.20)
8.2 Decay of Isotropic Turbulence 215
The value of the invariant does not change in time and is valid for all time
moments t > 0.
Consider the problem (8.19) and (8.20) assuming that the value ofL0 is non-zero
and finite, 0<L0<1, as the most plausible assumption (Landau and Lifshitz
1987). Introducing the new variables bdd ¼ bdd=L0 and t� ¼ 2t, we transform
(8.19) and (8.20) to the following form
@bdd@t�
¼ n1
r4@
@r
�r4@bdd@r
�(8.21)
ð10
r4bdddr ¼ 1 (8.22)
From (8.21) and (8.22) it follows that bdd depends on three governing parameters
bdd ¼ f ðt�; n; rÞ (8.23)
These parameters have the following dimensions t�½ � ¼ T; n½ � ¼ L2T�1 and r½ � ¼ L,whereas the dimension of the unknown quantity is bdd
� �¼ L�5. It is seen that two of
the three governing parameters have independent dimensions. Then, in accordance
with the Pi-theorem, the dimensionless form of (8.23) reads
P ¼ ’ðP1Þ (8.24)
where P ¼ bdd=ta1� n
a2 and P1 ¼ r=ta01� na
02 .
Using the principle of the dimensional homogeneity, we find the value of the
exponents ai and a0i as a1 ¼ a2 ¼ �5=2 and a
01 ¼ a
02 ¼ 1=2. Then, (8.24)
transforms to the following expression for bdd
bdd ¼ 1
nt�ð Þ52’
rffiffiffiffiffiffint�
p� �
(8.25)
To find an exact expression for the function ’, one should express the derivativesin (8.21) using (8.25)
@bdd@t�
¼ � 1
2
1
nt�ð Þ5=2t�5’þ x
d’
dx
� �(8.26)
n1
r4@
@rr4@bdd@r
� �¼ 1
nt�ð Þ5=2t�4
xd’
dxþ d2’
dx2
� �(8.27)
216 8 Turbulence
where ’ ¼ ’ðxÞ and x ¼ r= nt�ð Þ1=2.Substituting (8.26) and (8.27) into (8.21), we arrive at the following ODE for the
function ’ ¼ ’ðxÞ
d2’
dx2þ 4
xþ x2
� �d’
dxþ 2’ ¼ 0 (8.28)
The Loitsyanskii invariant takes the form
ð10
x4’ðxÞdx ¼ 1 (8.29)
The solution of (8.28) and (8.29) and a detailed analysis of the problem on decay
of isotropic turbulence can be found in the monographs of Sedov (1993) and
Barenblatt (1996).
8.3 Turbulent Near-Wall Flows
8.3.1 Plane-Parallel Flows
Turbulent flows over smooth and rough walls were a subject of a large number of
experimental and theoretical investigations. The most important results of these
works are combined in several well-known research monographs by Hinze (1975),
Rotta (1962), Schlichting (1979) and Monin and Yaglom (1965-Part 1, 1967-Part 2),
as well as in numerous text- and reference books on hydrodynamics. In the present
sub-section we focus our attention on the application of the Pi-theorem to reveal
some fundamental features of turbulent near-wall flows, while addressing an
interested reader to find the discussion of the other aspects of such flows in the
above-mentioned monographs.
We begin with the general form of the distribution of mean velocity in plane-
parallel flows over a plate. Let vbe the mean velocity vector in a plane-parallel flow
with the components u ¼ uðyÞ; v ¼ 0 and w ¼ 0 in the x, y and z directions,
respectively. It is seen that flow characteristics, in particular, velocity u, in such
flows depend only on the transversal coordinate y normal to the wall. It is plausible
to assume that local velocity uðyÞ is determined by the following four parameters:
the fluid density r½ � ¼ L�3M and viscosity m½ � ¼ L�1MT�1, the shear stress at the
wall t�½ � ¼ L�1MT�2 and the distance from the wall y½ � ¼ L
u ¼ f ðr; m; t�; yÞ (8.30)
8.3 Turbulent Near-Wall Flows 217
Three of the four governing parameters in (8.30) have independent dimensions,
so that according to the Pi-theorem, this equation transforms to the following
dimensionless form
P ¼ ’ P1ð Þ (8.31)
where P ¼ u=ra1ma2ta3� and P1 ¼ y=ra01ma
02t
a03� .
The values of the exponents ai and a0i are found as a1 ¼ �1=2 , a2 ¼ 0 ,
a3 ¼ 1=2; a01 ¼ �1=2; a
02 ¼ 1 and a
03 ¼ �1=2, which allows us to present (8.31)
as follows
u
u�¼ ’
yu�n
� �(8.32)
or
uþ ¼ ’ðyþÞ (8.33)
In (8.33) uþ ¼ u=u� and yþ ¼ yu�=n are the dimensionless velocity and distance
from the wall, respectively, u� ¼ffiffiffiffiffiffiffiffiffiffit�=r
pis the friction velocity.
Equations (8.32) or (8.33) expresses the Prandtl law-of-the wall (Prandtl,
1925b). The exact form of the function ’ yþð Þcan be determined in two limiting
cases corresponding to small or large values of yþ. At small yþ (in the viscous
sublayer) velocity is so low rgat it should not dependend on density, since the
inertia effect is negligible. Then the functional equation for the velocity in the
viscous sublayer reads
u ¼ f ðm; t�; yÞ (8.34)
Since all the governing parameters in (8.34) have independent dimensions, it
takes the form
u ¼ cma1ta2� ya3 (8.35)
where c is a constant.Determining the values of the exponents ai as a1 ¼ �1; a2 ¼ 1 and a3 ¼ 1, we
obtain
u
u�¼ c
yu�n
(8.36)
Equation (8.36) shows that fluid velocity increases linearly in y within viscous
sublayer. This conclusion agrees fairly well with the experimental data in the
0< yþ < 5.
218 8 Turbulence
Consider now the flow far enough from the wall. At large distances from the
wall, turbulent transfer plays the dominant role. That allows one to omit molecular
viscosity in the set of the governing parameters, and assume that flow
characteristics are determined by three parameters: r; t� and y. It is easy to see
that in this case it is impossible to assume m ¼ 0 in (8.30) and write the functional
equation for velocity in the form u ¼ f ðr; t�; yÞ. That would result in the unrealisticoutcome that velocity does not depend on y. Indeed, since all the governing
parameters in this equation have independent dimensions, it takes the following
form u¼ cra1ta2� ya3 , where c is a dimensionless constant, and a1 ¼�1=2;a2¼ 1=2
and a3 ¼ 0. At the same time, this set of governing parameters determines the
velocity gradient far from the wall, so that functional equation for the problem
can be written in the form
du
dy¼ f ðr; t�; yÞ (8.37)
It is emphasized that (8.37) implies a dependence of velocity gradient on r; t�and y due to the fact that at large Reynolds numbers, when the influence of
molecular viscosity is negligible, the value of the velocity gradient at each point
must be determined by only two parameters r; t� and the distance y (Landau and
Lifshitz 1979).
All governing parameters in (8.37) have independent dimensions. In this case the
Pi-theorem determines the following form of (8.37)
du
dy¼ w�r
a1ya2ta3� (8.38)
where w� ¼ x�1 is a constant. Also, a1 ¼ �1=2; a2 ¼ �1 and a3 ¼ 1=2.Then (8.38) takes the form
du
dy¼ u�
wy(8.39)
Integrating (8.39) yields
u ¼ u�wðln yþ cÞ (8.40)
The constant c in (8.40) is found from the matching condition at the outer
boundary layer of the viscous sublayer y0 � n=u�. There the velocity is close to
the friction velocity: u � u�. The latter results in the logarithmic velocity profile
u ¼ u�w
lnyu�n
(8.41)
8.3 Turbulent Near-Wall Flows 219
The accuracy of the law-of-the wall (8.41) can be improved by including an
empirical constant in addition to the logarithmic term. Then, the comparison with
the experimental data shows that the von Karman constant w ¼ 0.4 and the updated
equation (with the additional constant) becomes u ¼ u� 2:5 ln yu�=nð Þ þ 5:1½ � ¼2:5u� ln yu�=0:13nð Þ as suggested by Coles (1955).
8.3.2 Pipe Flows
Consider fully developed turbulent flows in straight pipes with circular cross-
section of radius R. The local velocity in such flows is determined by the distance
from wall y ¼ R� r (r is the corresponding distance from the pipe axis), fluid
density r and viscosity m and friction velocity u� (Monin and Yaglom, 1971)
u ¼ f ðR; y; r;m; u�Þ (8.42)
The transformation of (8.42) to thee dimensionless form by with the help of the
Pi-theorem yields
P ¼ ’ P1;P2ð Þ (8.43)
where P ¼ u=ua1� Ra2ra3 ;P1 ¼ y=u
a01� Ra
02ra
03 and P2 ¼ m=u
a001� Ra
002 ra
003 .
Allying the principle of the dimensional homogeneity and find the values of the
exponents ai; a0i and a
00i as a1 ¼ 1; a2 ¼ 0; a3 ¼ 0; a
01 ¼ 0; a
02 ¼ 1; a
03 ¼ 0;
a001 ¼ 1; a
002 ¼ 1 and a
003 ¼ 1, we transform (8.43) to the form of a dependence of
the dimensionless velocity u=u� on two dimensionless groups y=R and u�R=n
u
u�¼ ’
y
R;u�Rn
� �(8.44)
There are two important cases when the function ’ y=R; u�R=nð Þ can be reducedto a function of one dimensionless variable. The first of them corresponds to small
values of the ratio y=R � 1, at which the dependence of the local velocity on radius
of the pipe becomes unimportant. In this case the functional equation for local
velocity u reduces to
u ¼ f ðy; r; m; u�Þ (8.45)
Transforming (8.45) to dimensionless form by using the Pi-theorem, we arrive at
u
u�¼ ’
u�yn
� �(8.46)
220 8 Turbulence
The second limiting case corresponds to flow in the so-called ‘turbulent core’,
i.e. the region surrounding the pipe axis. In this region the turbulent shear stress is
larger than the viscous one. Correspondingly, the functional equation for the
velocity gradient du=dy should be written as
du
dy¼ f ðy;R; u�Þ (8.47)
Applying the Pi-theorem to (8.47), we reduces it to the form
du
dy¼ u�
R’
y
R
� �(8.48)
Assuming that ’ y=Rð Þ ¼ �Rdc y=Rð Þ=dy with cð0Þ ¼ 0, and integrating (8.48)
from y to R, we arrive at the following equation for the defect of velocity that was
found by von Karman (1930)
u0 � u
u�¼ c
y
R
� �(8.49)
where u0 is the velocity at the boundary of the turbulent core.
Equation (8.49) expresses the law of the defect of velocity. It is valid within the
turbulent core of pipe flows where �1 <�< 1, with � ¼ y=R and �1 and 1 being thedimensionless coordinates of the boundaries of the turbulent core.
8.3.3 Turbulent Boundary Layer
The flow in the turbulent boundary layer over a flat plate is significantly different
from plan-parallel and pipe flows. The differences stem from the different
conditions at the outer boundary of the boundary layer and a pipe axis, as well as
from the dependence flow characteristics on coordinates. In particular, in plane-
parallel and pipe flows velocity depends only on the normal-to-wall coordinate y,whereas in turbulent boundary layers velocity depends on the normal y and longi-
tudinal x coordinates. Accordingly, in turbulent boundary layers the friction veloc-
ity is not constant but changes with x.Assume that the defect of velocity within the outer part of the turbulent boundary
layer u1 � u depends on local parameters of the flow corresponding to a fixed x,e.g. on the thickness of the boundary layer dðxÞ, the friction velocity u�, as well ason the velocity of the undisturbed fluid u1, and distance from the wall y. Then we
can write the following functional equation
8.3 Turbulent Near-Wall Flows 221
ðu1 � uÞ ¼ f ðu1; d; y; u�Þ (8.50)
Applying the Pi-theorem, we transform (8.50) to the following dimensionless
form
P ¼ ’ P1;P2ð Þ (8.51)
where P ¼ u1 � uð Þ=u�; P1 ¼ y=d and P2 ¼ u1=u�.The experimental data on the velocity distribution in turbulent boundary layers
show that the dependence of the dimensionless group P on P1 is very weak
(Clauser 1956). That allows simplification of (8.51) to the form similar to the one
for pipe flows [(8.49)]
u1 � u
u�¼ ’
y
d
� �(8.52)
On the other hand, the flow in a thin viscous sublayer adjacent to the wall is
described by (8.36).
8.4 Friction in Pipes and Ducts
8.4.1 Friction in Smooth Pipes
The application of the Pi-theorem to study friction in smooth pipes of different
geometry in the case of laminar flow of incompressible fluid was considered in
Chap. 4. In the present subsection we briefly address this problem in relation to
friction in fully developed turbulent flows in smooth pipes. It is quite plausible to
assume that friction factor, and thus pressure gradient dP dx=½ � ¼ L�2MT�2, in such
flows is determined by four dimensional parameters: (1) the mean velocity
u½ � ¼ LT�1, pipe diameter d½ � ¼ L, as well as fluid density r� ¼ L�3M½ and viscos-
ity m½ � ¼ L�1MT�1
dP
dx¼ f ðu; d; r; mÞ (8.53)
It is emphasized that there is a principal distinction between the functional
equations for friction or pressure gradient in pipes with fully developed turbulent
and laminar flows. In the case of turbulent pipe flows, (8.53) contains fluid density
as one of the governing parameters. The latter shows that the inertial forces play an
important role in fully developed turbulent flows in pipes. In contrast, in steady-
state, fully developed laminar flow, the inertial forces are immaterial and the fluid
density is absent from the corresponding (8.53).
222 8 Turbulence
In (8.53) three of the four governing parameters have independent dimensions,
which allows transformation of (8.53) to the following dimensionless form
P ¼ ’ P1ð Þ (8.54)
where P ¼ dP=dxð Þ=da1ra2ma3 and P1 ¼ u=da01ra
02ma
03 .
Taking into account the dimensions of the pressure gradient dP=dx and the
governing parameters with independent dimensions, we find the values of the
exponents ai and a0i as a1¼�3; a2¼�1; a3¼�2; a
01¼�1; a
02¼�1 and a
03 ¼ 1.
Then, (8.54) takes the form
l ¼ ’ðReÞRe2
¼ cðReÞ (8.55)
where l ¼ 2 dP=dxð Þd=ru2 is the friction factor and Re ¼ ud=n:The friction factor l can be described by different empirical and semi-empirical
correlations for the function cðReÞproposed for smooth pipes, for example by the
Blasius correlation
cðReÞ ¼ 0:3164Re�1=4 (8.56)
(valid for Re < 105) or the Prandtl universal friction law for smooth pipes
1ffiffiffil
p ¼ 2:01 lg Reffiffiffil
p� �� 0:8 (8.57)
which is valid up to Re ¼ 3.4� 106 (Schlichting 1979).
8.4.2 Friction in Rough Pipes
Numerous experimental investigations performed during the last century show that
roughness significantly affects friction, and thus the pressure gradient, in turbulent
flows in rough pipes. The functional equation for the pressure gradient dP=dx in thiscase can be written as1
1 Strictly speaking, (8.58) should be written as: dP=dx¼ f ðu;def ;m;r;ks;a1;a2 � � �aiÞ; where a1;a2 �� �ai are the parameters characterizing the shape and distribution of rough elements on the surface.
8.4 Friction in Pipes and Ducts 223
dP
dx¼ f ðu; def ; m; r; kÞ (8.58)
with def and k being the effective diameter and characteristic roughness amplitude
(corresponding to sand grains glued to the inner wall) of such pipes (the choice of
the characteristic scales of rough pipes is discussed in Herwig et al. 2008).
Transforming (8.58) to the dimensionless form with the help of the Pi-theorem,
we arrive at the following equation for the friction factor l
l ¼ ’ðRe; kÞ (8.59)
where l ¼ 2 dP=dxð Þd=ru2 and the relative sand roughness is k ¼ k=def .Equation (8.59) shows that the friction factor of rough pipes depends on two
dimensionless parameters: the Reynolds number based on the effective diameter of
a pipe and the relative sand roughness ks. Typically, three different regimes of
turbulent flows in rough pipes are distinguished: (1) the hydraulically smooth
region corresponds to u�k=n< 5, (2) the transition region-to 5< u�k=n< 70; and
(3) the completely rough region with u�k=n>>70. At small enough values of the
dimensionless ratio u�k=n, rough pipes are practically hydraulically smooth since
their friction factor depends only on the Reynolds number [as for example, in the
Blasius correlation (8.56)]. At large values of ks, the riction factor of rough pipes
depends practically only on the relative roughness. In such flows the quadratic
resistance law in which l depends only on ksrather than on Re (i.e. l is proportionalto ru2) is valid.
8.5 Turbulent Jets
8.5.1 Eddy Viscosity and Thermal Conductivity
In the aero- and hydromechanics of submerged turbulent jets the main physical
feature to be accounted for is the fact that the intensity of molecular momentum,
heat and mass transfer in these flows is negligibly small than the intensity of
turbulent transfer due to fluctuation motion of fluid associated with eddies. The
latter allows one to simplify the system of the turbulent Reynolds equations
describing turbulent jets of incompressible fluid
u@u
@xþ v
@u
@y¼ � 1
rdP
dxþ @
@yyj nþ nTð Þ @u
@y
(8.60)
@uyj
@xþ @vyj
@y¼ 0 (8.61)
224 8 Turbulence
where u, v and P are the longitudinal and lateral velocity components and pressure,
respectively (with bars denoting the averaged parameters), n and nT are the molec-
ular and eddy kinematic viscosities of fluid and subscripts j ¼ 0 and 1 corresponds
to the plane and axisymmetric jets, respectively.
The physically plausible assumption that nT >> n allows us to omit n in (8.60)
and reduce the transfer term in this equation to a simpler form
@
@yyj nþ nTð Þ @u
@y
¼ @
@yyjnT
@u
@y
(8.62)
As usual with the turbulent Reynolds equations, the system of (8.60–8.61)
should be supplemented by a semi-empirical correlation that determine the depen-
dence of the eddy viscosity on the characteristics of the mean velocity field. A
number of such correlations were suggested by Prandtl (1925a, 1942), von Karman
(1930) and Taylor (1932) in the framework of the semi-empirical theories of
turbulence.
There are two different approaches to express of the eddy viscosity: (1) differ-
ential approach, and (2) the integral one. The former implies that turbulent transfer
in shear flow (such as submerged jets) is determined by local features of the mean
velocity field in the vicinity of a considered point. In particular, as the governing
parameters that determine turbulent transfer one can take the local velocity gradient
of the mean flow du=dy, the fluid density r, as well as some characteristic length lthat account for the displacement of fluid elements in the lateral y-direction and is
termed the mixing length. Accordingly, the functional equation for the eddy
viscosity in one-dimensional shear flow with zero pressure gradient
dP=dx ¼ 0(including the submerged jet flows) reads
mT ¼ f1 r; l;du
dy
� �(8.63)
where mT is the turbulent eddy viscosity.
Since all the governing parameters in (8.63) have independent dimensions,
namely r½ � ¼ L�3M, l½ � ¼ L and du dy=½ � ¼ T�1, it reduces to
mT ¼ c1ra1 la2du
dy
� �a3
(8.64)
where c1 is a constant.Taking into account the dimension of the eddy viscosity mT½ � ¼ L�1MT�1, we
find the values of the exponents ai as a1 ¼ 1, a2 ¼ 2 and a3 ¼ 1, and obtain the
following relations for mT and nT
mT ¼ c1rl2du
dy(8.65)
8.5 Turbulent Jets 225
nT ¼ c1l2 du
dy(8.66)
In the case when we take into account both the first and second derivatives, the
functional equation for the eddy viscosity reads
mT ¼ f2 r; l;du
dy;d2u
dy2
� �(8.67)
Equation (8.67) contains four governing parameters, three of them have inde-
pendent dimensions. Then (8.67) reduces to
P ¼ ’ðP1Þ (8.68)
whereP¼ mT= ra1 du=dyð Þa2 d2u=dy2ð Þa3� �andP1 ¼ l= ra
01 du=dyð Þa
02 d2u=dy2ð Þa
03
� �.
Finding the values of the exponents ai and a0i as a1 ¼ 1, a2 ¼ 3, a3 ¼ �2; a
01 ¼ 0
a02 ¼ 1 and a
03 ¼ �1, we arrive at the following correlation
mT ¼ rdu=dyð Þ3d2u=dy2ð Þ2 ’
l
du=dyð Þ= d2u=dy2ð Þ
(8.69)
The expression for the eddy viscosity can be significantly simplified by using the
von Karman similarity hypothesis (von Karman 1930) which implies that the
mixing length l is determined by the local characteristics of the mean velocity
field in the vicinity of a point of consideration, in particular, by the values of the
derivatives du=dy and d2u=dy2as
l ¼ cl
du
dy;d2u
dy2
� �(8.70)
Bearing in mind (8.70), we can exclude l from (8.67) and transform this equation
to the following form
mT ¼ f2 r;du
dy;d2u
dy2
� �(8.71)
All the governing parameters in (8.71) have independent dimensions. Therefore,
in accordance with the Pi-theorem, we obtain
mT ¼ c2ra1du
dy
� �a2 d2u
dy2
� �a3
(8.72)
226 8 Turbulence
where c2 is a constant.Taking into account the dimensions of mT , r, du=dy, and d2u=dy2, we find the
values of the exponents ai as a1 ¼ 1, a2 ¼ 3 and a3 ¼ �2. After that, (8.72) takes
the following form
mT ¼ c2rdu=dyð Þ3d2u=dy2ð Þ2 (8.73)
The integral approach to the eddy viscosity implies that turbulent transfer in
shear flows is determined by the integral flow characteristics (in particluar, for the
far field of a turbulent submerged jet by its total momentum flux Jx), as well as fluiddensity and local coordinates x and y
mT ¼ f r; Jx; x; yð Þ (8.74)
where the governing parameters have the following dimensions
r½ � ¼ L�3M; Jx½ � ¼ LjMT�2; x½ � ¼ L; y½ � ¼ L (8.75)
Since in the framework of the Pi-theorem the difference n� k ¼ 1, (8.74) takes
the form
P ¼ ’1ðP1Þ (8.76)
where P ¼ mT=ra1Ja2x xa3 and P1 ¼ y=ra
01J
a02
x xa03 .
For the axisymmetric jets (j ¼ 1) the exponents ai and a0i are found as a1 ¼ 1=2,
a2 ¼ 1=2, a3 ¼ 0; a01 ¼ 0, a
02 ¼ 0 and a
03 ¼ 1 which yields
mT ¼ rJxð Þ1=2’ y
x
� �(8.77)
Within the far field of turbulent axisymmetric jets, the distribution of the
longitudinal velocity component u can be presented as
u
um¼ c
y
d
� �(8.78)
where um is the axial velocity and d is the half-width of the jet.
Indeed, since the velocity profile in the far field of the jet is determined by
three dimensional parameters, namely the velocity at the jet axis um, the jet thicknessd, and a coordinate y of a point under consideration, the functional equation
u ¼ f ðum; d; yÞ is valid. It is easy to see that two of the three governing parameters
in this equation possess independent dimensions. Therefore, applying the Pi-theorem
results in (8.78).
8.5 Turbulent Jets 227
Beating in mind (8.78), and the integral invariant of the axisymmetric jets
Jx ¼Ðy0
ru2ydy ¼ const, we obtain
ffiffiffiffiJxr
s¼ umd
ð10
cð�Þ�d�8<:
9=;
1=2
(8.79)
where � ¼ y=d.Substituting (8.79) into (8.77), we obtain the following expression for the
kinematic eddy viscosity
nT ¼ ðumdÞð10
cð�Þ�d�8<:
9=;
1=2
’y
x
� �(8.80)
In the particular case of ’ y=dð Þ being a week function ofy x= , it is possible to
assume that ’1 is a constant. Then, (8.80) becomes identical Prandtl’s relation for
the eddy viscosity in submerged turbulent jets (Prandtl 1942)
nT ¼ const� ðumdÞ (8.81)
According to (8.81), the kinematic eddy viscosity is constant in the axisymmet-
ric turbulent jets, whereas in plane turbulent jets nT ¼ nTðxÞ (Vilis and Kashkarov
1965). Corrsin and Uberoi (1950), Antonia et al. (1975), Chevray and Tutu (1978),
Chua and Antonia (1990) showed that in reality the eddy viscosity changes across
the axisymmetric turbulent jets. It appears to be constant in the inner part of the jet
flow and decreases rapidly at toward the external boundary of the mixing layer at
the jet edge.
The thermal analog of nT is the eddy thermal diffusivity aT : The experimental
data show that there is a certain difference between the values of nT and aT , so that
the turbulent Prandtl number PrT ¼ nT=aT is not equal to one (Table 8.1, Mayer and
Divoky 1966).
The experimental data also show that nT and aT , as well as the turbulent Prandtlnumber change over the cross-section of submerged turbulent jets (Fig. 8.1). The
mean values of PrT in the far field of submerged jets are about 0.75–0.8 (Vulis and
Kashkarov, 1965; Chua and Antonia, 1990). Note, that the values of the turbulent
Prandtl and Schmidt numbers practically do not depend on the physical properties of
fluid, i.e. on the molecular values of the Prandtl and Schmidt numbers. [The Schmidt
number represents itself the ration of the kinematic viscosity to diffusivity and in
mass transfer processes plays a similar role to that of the Prandtl number in the heat
transfer processes.] The experimental studies of the velocity, temperature or concen-
tration fields in turbulent jets of different fluids (mercury: Pr � 10�2; oil: Pr � 103)
228 8 Turbulence
and aqueous solt solutions (Sc � 103) reveal that the turbulent Prandtl and Schmidt
numbers are always about 0.75–0.8 (Sakipov 1961; Sakipov and Temirbaev 1962;
Forstall and Gaylord 1955). These data confirm that turbulent transfer of momentum,
heat and species is dominant in submerged turbulent jets. A comprehensive analysis
of a number of semi-empirical correlations for the kinematic viscosity and thermal
diffusivity can be found in the monograph by Hinze (1975)
8.5.2 Plane and Axisymmetric Turbulent Jets
Consider velocity distribution in the far field of a plane or axisymmetric turbulent
jet. The governing parameters in this case are selected as: (1) the fluid density r,(2) the total momentum flux Jxwhich is constant along the jet, (3) longitudinal and
Table 8.1 Turbulent Prandtl number in turbulent jet flows
Flow field Type of flow PrT Authors
Planar Wake of a heated cylinder 0.54 Fage and Falkner
Planar Heated jet 0.54 Reichardt
Planar Heated jet 0.42–0.59 Van der Hegge Zijnen
Axisymmetric Heated jet with tracers 0.74 Van der Hegge Zijnen
Axisymmetric Hydrogen jet 0.72 Keary and Weller
Axisymmetric Heated jet 0.7 Corrsin
Axisymmetric Heated jet 0.71 Forstall
Axisymmetric Submerged water jet 0.72–0.83 Forstall and Gaylo-rd
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 0.5 1 1.5 2 2.5
Pr T
η
Fig. 8.1 Turbulent Prandtl number in the axisymmetric turbulent jet. ■-Corrsin and Uberoi
(1950), ~-Chevray and Tutu (1978), ●-Chua and Antonia (1990) (measurements 90 X- probe),� ¼ y=y1=2 is the dimensionless lateral coordinate, y1=2is the half-velocity coordinate where
u ¼ um=2, u and umare the local longitudinal velocity and the axial velocity, respectively.
Reprinted from Chua and Antonia (1990) with permission
8.5 Turbulent Jets 229
lateral coordinates x and y determining the location of a point of consideration. The
eddy viscosity mT , of course, also affects the flow field, however, it is fully
determined by Jx; x and y and thus, is not an independent parameter. Therefore,
the functional equation for the longitudinal mean velocity u reads
u ¼ f r; Jx; x; yð Þ (8.82)
Here and hereinafter bars above mean flow characteristics are omitted for
brevity.
The dimensions of the governing parameters in (8.82) are
r½ � ¼ L�3M; Jx ¼ MT�2; x½ � ¼ L; y½ � ¼ L (8.83)
It is seen that three governing parameters possess independent dimensions, so
that the difference n� k ¼ 1. Then (8.82) reduces to the following form
P ¼ ’1ðP1Þ (8.84)
where P ¼ u=ra1Ja2x xa3 and P1 ¼ y=ra01J
a02
x xa03 .
Taking into account the dimensions of the parameters involved in the
expressions for P and P1 and applying the principle of the dimensional homoge-
neity, we find the values of the exponents ai and a0i as a1 ¼ �1=2, a2 ¼ 1=2,
a3 ¼ �1=2; a01 ¼ 0, a
02 ¼ 0, and a
03 ¼ 1. Accordingly, we obtain
u ¼ffiffiffiffiJxr
s’1
y
x
� �x�1=2 (8.85)
The functional equation for the axial velocity um and the jet thickness d are
um ¼ fu r; Jx; xð Þ (8.86)
d ¼ fd r; Jx; xð Þ (8.87)
Applying the Pi-theorem, we transform (8.86) and (8.87) to the following form
um ¼ c1rb1Jb2x xb3 (8.88)
d ¼ c2rg1Jg2x xg3 (8.89)
where c1 and c2 are constants, and the exponents bi and gi are equal to: b1 ¼ �1=2,b2 ¼ 1=2, b3 ¼ �1=2, g1 ¼ 0, g2 ¼ 0, and g3 ¼ 1.
As a result, we arrive at the following equations
230 8 Turbulence
um ¼ c1
ffiffiffiffiJxr
sx�1=2 (8.90)
d ¼ c2x (8.91)
Bearing in mind (8.89) and (8.90), we can rewrite (8.85) as follows
u
um¼ ’1ð�Þ (8.92)
where � ¼ y=d.The axial velocity um and the jet thickness di are defined by the profile of the i-th
characteristic being used as the characteristic scale, which allows us to present the
profiles of different flow characteristics in the form similar to that of (8.92). Indeed,
the functional equation for any of the flow characteristic Ni reads
Ni ¼ fi um; y; dið Þ (8.93)
where Ni ¼ ui;ffiffiffiffiffiffiu02
p;
ffiffiffiffiffiv02
p; u0v0 . . . which represent any of the mean velocity
components, velocity pulsations, the shear stress, etc.
Applying the Pi-theorem, we arrive at
Pi ¼ ’iðP�Þ (8.94)
where Pi ¼ Ni=uaim , P� ¼ y=di ¼ �, etc., as well as a1 ¼ a2 ¼ a3 ¼ 1 and a4 ¼ 2
for i ¼ 1,. . .4.Accordingly, we arrive at the following equations
u
um¼ ’1ð�Þ;
ffiffiffiffiffiffiu02
pum
¼ ’2ð�Þ;ffiffiffiffiffiv02
pum
¼ ’3ð�Þ;ffiffiffiffiffiffiffiu0v0
pu2m
¼ ’4ð�Þ (8.95)
Choosing as the characteristic scales of temperature and species concentrations
their values at the jet flow axis, as well as the thicknesses of the thermal and
diffusion layers dT and dc, respectively, we express the functional equations for
DT and DCj as follows
DT ¼ fT DTm; dT ; yð Þ (8.96)
DCj ¼ fC DCj; dc; y �
(8.97)
where DT ¼ T � T1, DTm ¼ Tm � T1, DCj ¼ Cj � Cj1, and T1 and Cj1 are the
temperature and concentration of the j� th species in in the undisturbed fluid
outside the jet.
8.5 Turbulent Jets 231
The application of the Pi-theorem to (8.96) and (8.97) leads to
DTDTm
¼ ’Tð�TÞ (8.98)
DCDCm
¼ ’Cð�CÞ (8.99)
where �T ¼ y=dT and �C ¼ y=dC.Similar functional equations for fluctuations of temperature and species
concentrations read
ffiffiffiffiffiffiT 02
pDTm
¼ ’Tð�TÞ (8.100)
ffiffiffiffiffiffiffiC02
pDCm
¼ ’Cð�CÞ (8.101)
The assumption that the axial velocity (temperature or species concentrations)
and the thickness of the dynamic (thermal or diffusional) layer can serve as the
characteristic scales for submerjed jets is based only on the symmetry of these flows
and does not imply whether they are plane or axisymmetric. Therefore,
(8.95–8.101) are valid as for both plane and axisymmetric jets. These equations
can be used for generalizing of the experimental data for the mean and pulsation
velocity, temperature and concentration distributions in the far field of any axisym-
metric jets. The corresponding generalized experimental results for the velocity,
temperature and concentration in plane and axisymmetric turbulent jets are
presented in Figs. 8.2–8.6. It is seen that experimental data obtained at different
jet cross-sections collapse at single curves corresponding to the self-similar behav-
ior uncovered in the present subsection.
8.5.3 Inhomogeneous Turbulent Jets
The inhomogeneous turbulent jets emerge when turbulent mixing of gas streams of
different densities is encountered. The existence of non-uniform density field
significantly affects the aerodynamics of inhomogeneous jets, in particular, the
decay of the mean velocity, velocity pulsations, and concentration along the jet
axis, as well as distributions of these parameters in jet cross-sections, the jet
ejection features, etc. (Abramovich,1974; Panchapakesan and Lumley 1993). The
flow field in turbulent inhomogeneous jets is described by the continuity, momen-
tum and species balance equations for turbulent motion of the inhomogeneous
mixtures of variable density (Shin et al. 1982). Consider a plane or axisymmetric
232 8 Turbulence
turbulent jet of species 1 injected into space submerged by a mixture of species 1
and 2. After a number of simplifications of the mean momentum balance and
species balance equations, the integration of these equations across the jet yields
ð10
ru2yjdy ¼ Jx (8.102)
0
0.2
0.4
0.6
0.8
1
1.2a
b
0 0.5 1 1.5 2 2.5
12
y
y
u u m
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.5 1 1.5 2 2.5 3
12
yy
u'2
u m
Fig. 8.2 Distribution of the mean (a) and (b) velocity pulsations in the cross- sections of turbulentjet. (a)x=d: ♦�118, ■�108, ~�103, ��76, *�65; (b) x=d: ♦�143, ■�129, ~�118, ��106,
*�95. Reprinted from Gutmark and Wygnanski (1976) with permission
8.5 Turbulent Jets 233
ð10
ruDc1yjdy ¼ Gx (8.103)
where r ¼ r1�1 � r2
�1ð Þc1 þ r2�1½ ��1
is the mean local density of the gas mixture,
c1 is the mean local concentration of the injected gas, r1 and r2 are the mean densiy
of the injected (1) and ambient (2) species, Dc1 ¼ c1 � c11 is the excess concen-
tration of the injected gas, subscript 1 corresponds to the ambient conditions far
away from the jet axis, j ¼ 0 or 1 for plane and axisymmetric jets, respectively.
As the governing parameters determining the velocity and concentration fields in
the far field of submerged inhomogeneous turbulent jets it is natural to choose the
0
0.005
0.01
0.015
0.02
0.025
0.03
0 0.5 1 1.5 2
12
yy
u2 m
u′ v
′
Fig. 8.3 The distribution of the turbulent shear stress in cross-sections of planar turbulent jet. x=d:♦�143, ■�129, ~�118, ��106, +�95. Reprinted from Gutmark and Wygnanski (1976) with
permission
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2 2.5
12
yy
ΔT ΔTm
Fig. 8.4 Radial profiles of the mean temperature DT=DTm ¼ T � T1 �
= Tm � T1 �
in the cross-
sections of an axisymmetric turbulent jet x=d: ◊�7.0, □�10, ~�15, ��20.21, +�25.47.
Reprinted from Lockwood and Moneib (1980) with permission
234 8 Turbulence
local density of the gas mixture r½ � ¼ L�3M, the total momentum flux along the jet
Jx ¼ LjMT�2, the total mass flux of the injected gas Gx½ � ¼ Lj�1MT�1; as well as thecoordinates x½ � ¼ L and y½ � ¼ L. (Note that Jxand Gxdo not change along the jet and,
thus are considered as given invariant constants). Accordingly, we can state the
following functional equations for the thickness of a turbulent inhomogeneous jet d,as well as for the axial velocity umand concentration Dc1min it
d ¼ f1 r; Jx;Gx; xð Þ (8.104)
um ¼ f2 r; Jx;Gx; xð Þ (8.105)
Dcm1 ¼ f3 r; Jx;Gx; xð Þ (8.106)
Applying the Pi-theorem to (8.104–8.106), we arrive at the following dimen-
sionless equations
Pk ¼ ’kðP�Þ (8.107)
where k ¼ 1; 2; 3; P1 ¼ d=ra1Ja2x Ga3x , P2 ¼ um=ra
01J
a02
x Ga03
x , P3 ¼ Dcm=ra001 J
a002
x Ga003
x
and P� ¼ x=ra�1J
a�2
x Ga�3x .
Bearing in mind the dimensions of d, um, and Dcm, and applying the principle ofthe dimensional homogeneity, we find the values of the exponents involved in
(8.107) as: a1 ¼ �1=2; a2 ¼ �1=2, a3 ¼ 1; a01 ¼ 0, a
02 ¼ 1, a
03 ¼ �1; a
001 ¼ �1=2,
a002 ¼ �1=2, a
003 ¼ 1; a�1 ¼ �1=2, a�2 ¼ �1=2, a�3 ¼ 1. Accordingly, we obtain the
following equations for d; um, and Dcm
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.5 1 1.5 2 2.5 3
1
yy2
e T e T.m
Fig. 8.5 Radial distributions of the root-mean-square temperature fluctuations in the cross-
sections of an axisymmetric turbulent jet x=d: ◊�7.0, □�10.0, D� 15:0, ��20.21, +�25.47,~�30.75, ♦�36.66; eT-rms of temperature fluctuations. Reprinted from Lockwood and Moneib
(1980) with permission
8.5 Turbulent Jets 235
d
G2x=rJx
�1=2 ¼ ’1
x
G2x=rJx
�1=2( )
(8.108)
umJx=Gxð Þ ¼ ’2 x=
G2x
rJx
� �1=2( )
(8.109)
Dcm1 ¼ ’3 x=G2
x
rJx
� �1=2( )
(8.110)
Taking into account that the invariants Jx and Gx are constants, we find that
Jx ¼ r1u210d
jþ1, Gx ¼ r1u10Dc10djþ1, and Dc10 ¼ c10 ¼ 1 for issuing of pure
injected gas into space filled by another gas, where u10 and c10 are the initial
velocity and concentration of the injected gas and d is the nozzle diameter. Then,
we obtain
0
0.2
0.4
0.6
0.8
1
1.2
0 0.05 0.1 0.15 0.2 0.25η
k
0
0.05
0.1
0.15
0.2
0.25
0.3b
a
0 0.05 0.1 0.15 0.2 0.25η
cC(c
, h)
k
cC¢ rm
s(c,
h)
Fig. 8.6 Concentration distribution in cross-sections of turbulent axisymmetric jet. (a) mean
concentration, (b)concentration pulsations. x d= : □�20, ◊�40, D�60, ��80; Reprinted from
Doweling and Dimotakis (1990) with permission
236 8 Turbulence
JxGx
¼ u10;G2
x
rJx
� �1=2
¼ffiffiffiffiffir1r
rd jþ1ð Þ=2 (8.111)
The gas mixture density in the far field of the jet is of close to the ambient one,
and it is possible to assume that r r2. Then (8.108–8.110) take the form
d ¼ ’1 xffiffiffiffio
p �(8.112)
um ¼ ’2 xffiffiffiffio
p �(8.113)
Dcm1 ¼ ’3 xffiffiffiffio
p �(8.114)
where d ¼ d=dffiffiffiffio
p, um ¼ um=u10, Dcm1 ¼ Dcm1 Dc10 ¼ Dcm1= , ðDcm1 ¼ cm1 when
c11 ¼ 0Þ, x ¼ x=d and o ¼ r2=r1.Thus, the variation of the longitudinal velocity and concentration of the issued
species along the axis of an inhomogeneous turbulent gas jet is determined by a
single dimensionless group xffiffiffiffio
p. The experimental data on the dependence Dcm1 �
xffiffiffiffio
pð Þ are presented in Fig. 8.7. The figure also includes the experimental data on
the axial decay of the excess of enthalpy in high temperature turbulent jets which
is expected to be indentical to the decay of the excess concentration. It is seen
that all the experimental data corresponding to the different inhomogeneous
ð0:27 � o � 8:2Þand high temperature ðo> 15Þ turbulent jets groip around a
single curve cm xffiffiffiffio
pð Þ. (Abramovich et al. 1974)
Choosing as the characteristic scales of velocity, concentration and the
corresponding fluctuations their values at the jet axis, we present distribution of
u,ffiffiffiffiffiffiu02
p,
ffiffiffiffiffiv02
p, and c1 and
ffiffiffiffiffiffiffic102
pas follows
Fig. 8.7 Turbulent mixing, axsisymmetric gas jets; the dependence cm1ðxffiffiffiffio
p Þ, with xffiffiffiffio
p:
□�0.27, ◊�1.3, D�8.2, ��1.5, ~�21, ♦�26
8.5 Turbulent Jets 237
Ni ¼ f1 um;y; d �
(8.115)
Mk ¼ fk cm1; y; dð Þ (8.116)
where Ni ¼ u,ffiffiffiffiffiffiu02
por
ffiffiffiffiffiv02
pand i ¼ 1,2 and 3; also, Mk ¼ c1 or
ffiffiffiffiffiffiffic102
pand k ¼ 4
and 5.
Applying the Pi-theorem to (8.115) and (8.116), we arrive at the following
equations
u
um¼ f1
y
d
� �;
ffiffiffiffiffiffiu02
pum
¼ f2y
d
� �;
ffiffiffiffiffiv02
pum
¼ f3y
d
� �;c1cm1
¼ ’4
y
d
� �;ffiffiffiffiffiffiffi
c102pcm1
¼ ’5
y
d
� �(8.117)
Using a similar approach, we can find the following dimensionless equations for
the turbulent correlations u0v0, v0c0
, etc. in jet cross-sections
u0v0
u2m¼ f6
y
d
� �;v0c1
0
umcm1¼ f7
y
d
� �(8.118)
Equations (8.117) and (8.118) show that the profiles of the dimensional
characteristics corresponding to different cross-sections of the inhomogeneous jet
can be presented in the parametric planes Ni�, andMi� (where � ¼ y=dÞ in the formof universal dependences Nið�Þ and Mið�Þ (cf. Figs. 8.8–8.10)
8.5.4 Co-flowing Jets
Co-flowing turbulent jet is formed when a turbulent jet is issued into a uniform fluid
flow in an infinite domain. Detailed experimental data for the mean and
characteristics and turbulent pulsations in co-flowing turbulent jets were obtained
byMaczynski, 1962; Bradbury and Riley 1967; Antonia and Bigler 1973; Everitt and
Robins 1978; Abramovich et al. 1984; Nickels and Perry 1996. Some general
considerations of the features of co-flowing turbulent jets can be found in the
monograph by Townsend (1956) who analyzed the conditions corresponding to
self-similar flows in such jets. He also dealt with the effect of viscosity on flow
characteristics, e.g. the independence of the decay of the mean and pulsation
velocities along the jet axis on the Reynolds number, etc. For the additional details,
a reader can consult the above-mentioned works, while we restrict our discussion to
the applications of the Pi-theorem to study the aerodynamics of co-flowing turbulent
jets, and especially to generalize the experimental data for the far field in these flows.
238 8 Turbulence
A sketch of co-flowing turbulent jet with a uniform velocity profile at the orifice
exit x ¼ 0 is depicted in Fig. 8.11. At some distance from the jet orifice, the
initialvelocity profile transforms into a smooth profile with a characteristic maxi-
mum at the jet axis, which gradually decreases toward the outer edge of the jet
where the longitudinal velocity u matches the free stream velocity u1. In the
framework of the boundary layer theory the flow in co-flowing turbulent jets is
described by the momentum balance and continuity equations subjected to the
boundary conditions that account for the existence of the outer flow
y ¼ 0;@u
@y¼ 0; v ¼ 0 y ! 1 u ! u1 (8.119)
0
0.2
0.4
0.6
0.8
1
1.2
0 0.05 0.1 0.15 0.2 0.25 0.3yx
0
0.2
0.4
0.6
0.8
1
1.2b
a
0 0.05 0.1 0.15 0.2 0.25 0.3
u u mc 1 c m
1
yx
Fig. 8.8 The mean velocity (a) and mean helium concentration (b) distributions in the axisym-
metric turbulent jet. x d= : □�90, ◊�100, D�110, ��120. Reprinted from Panchapakesan and
Lumley (1993) with permission
8.5 Turbulent Jets 239
where subscript 1 refers to the undisturbed co-flow.
The integration of the momentum balance equation across the jet accounting for
the boundary conditions (8.119) yields the following integral invariant of co-
flowing jets
Jx ¼ð10
ruðu� u1Þyjdy (8.120)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 0.05 0.1 0.15 0.2 0.25 0.3
12
yy
12
y
y
c′1
c m1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4 a
b
0 0.5 1 1.5 2 2.5
u′ u m
Fig. 8.9 The distribution of the intensities of fluctuations, a – the root – mean-square of velocity
(u0=um) and b – concentration (c10=cm1)in cross-sections of the axisymmetric turbulent jet x=d:
□�50, ■�60, D� 70, ��80, ~�90, ♦�100, +� 110. Reprinted from Panchapakesan and
Lumley (1993) with permission
240 8 Turbulence
which is the momentum flux along the jet and where j ¼ 0 or 1correspond to plane
or axisymmetric jets, respectively.
Far from the orifice where the jet ‘forgets’ the initial conditions, the longitudinal
velocity is determined by fluid density r½ � ¼ L�3M, the total momentum flux
Jx½ � ¼ LjMT�2, the free stream velocity u1½ � ¼ LT�1 and the longitudinal and
lateral coordinates x½ � ¼ L and y½ � ¼ L
u ¼ f1 r; Jx; u1; x; yð Þ (8.121)
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0 0.05 0.1 0.15 0.2 0.25 0.3yx
c m1u
m
c 1′v
′
0
0.05
0.1
0.15
0.2
0.25
0.3a
b
0 0.5 1 1.5 2 2.5
u′v′
u m2
12
yy
Fig. 8.10 Variation of the Reynolds stress (u0v0=u2m)�a and radial scalar flux (c1 0v0=cm1um)
�b
across the axisymmetric turbulent jet. x=d: D� 50, □�60, ◊�70, ��90, ■�100, ~�110.
Reprinted from Panchapakesan and Lumley (1993) with permission
8.5 Turbulent Jets 241
Beating in mind the dimensions of the governing parameters we, find that in the
present case the difference n� k ¼ 2 and, accordingly, (8.121) reduces to the
dimensionless equation
P ¼ ’1 P1;P2ð Þ (8.122)
where P ¼ u=u1, P1 ¼ x= Jx=rxjþ1ð Þ1=2 and P2 ¼ y= Jx=rxjþ1ð Þ1=2.Thus, the dimensionless velocity in co-flowing jets is a function of two dimen-
sionless groups. That means that the present problem has no self-similar solution.
Note that excluding u1 from the boundary conditions (8.119) by introducing the
excess of velocity u� u1 does not allow one to decrease the number of the
governing parameters and thereby obtain a self-similar solution, since such trans-
formation introduces u1 into the momentum balance equation.
Within the far field of co-flowing turbulent jets, velocity change across the jet is
small enough compared to the free stream velocity. This allows one to recover an
approximate self-similarity of co-flowing turbulent jets (Townsend 1956). Indeed,
the functional equations for the mean flow characteristics of plane co-flowing
turbulent jets can be written now as
U ¼ f2 r; Jx; x; yð Þ (8.123)
Um ¼ f3 r; Jxxð Þ (8.124)
d ¼ f4 r; Jx; xð Þ (8.125)
where U ¼ u� u1 is the excess of velocity, subscript m refers to the jet axis and ddenotes the jet thickness.
Applying the Pi-theorem to (8.123–8.125), we arrive at the following equations
u� u1ffiffiffiffiffiffiffiffiffiffiffiffiJx=rx
p ¼ ’2
y
x
� �(8.126)
Fig. 8.11 Sketch of co-
flowing turbulent jet with a
step-wise initial velocity
distribution
242 8 Turbulence
um � u1ffiffiffiffiffiffiffiffiffiffiffiffiJx=rx
p ¼ C1 (8.127)
d ¼ C2x (8.128)
where C1 ¼ ’2ð0Þ ¼ const.Equations (8.127) and (8.128) show that the excess of axial velocity in the far
field of plane co-folwing turbulent jets is inversely proportional to x1=2, whereas thejet thickness is directly proportional to x. These results agree fairly well with the
experimental data of Bradbury and Riley (1967) and Everitt and Robins (1978) for
plane co-flowing turbulent jets (cf. Figs. 8.12 and 8.13).
It is seen that d � x (with x ¼ x=y) and Dum � x�1=2 at x> 40 (d ¼ d=y,
x ¼ x=y, Dum ¼ Dum=u1, Dum ¼ um � u1, y ¼ Ðþ1
�1u=u1 u=u1 � 1ð Þdy. It is
emphasized that all the data corresponding to different values of the co-flow
parameter m(the ratio of the exit to the free stream velocity) collapse near single
curves dðxÞ and DumðxÞ in the parametric planes d� x and Dum � x.Using (8.126–8.128), we express the velocity distribution across the co-flowing
jet as
u� u1um � u1
¼ c3’2
y
d
� �(8.129)
where c3 ¼ c�11 .
0
0.5
1
1.5
2
2.5
0 20 40 60 80 100 120xq
,u ∞
qdΔu
m
Fig. 8.12 Variation of the axial velocity and thickness of co-flowing jet along the flow axis.
Reprinted from Bradbury and Riley (1967) with permission
8.5 Turbulent Jets 243
The experimental data on the velocity distribution in different cross-sections of
co-flowing plane turbulent jet are presented in Fig. 8.14. The data related to the
distributions of turbulent pulsations across co-flowing turbulent ‘strong’ and ‘weak’
0
0.5
1
1.5
2
2.5
3
0 20 40 60 80 100 120 140x
q
d q
Fig. 8.13 Variation of the thickness of a plane turbulent co-flowing jet along its axis. y is the
momentum thickness of the jet . u=u1: D�2.60, ◊�3.03, □�3.24, ��3.29, ~�3.78, ■�6.72,
+�17.08. Reprinted from Everitt and Robins (1978) with permission
0
0.2
0.4
0.6
0.8
1
1.2
-3 -2 -1 0 1 2 3
12
yy
U–U
1
U0
Fig. 8.14 Velocity distribution in cross-sections of plane turbulent jet issuing into a parallel
moving air stream. U0=U1: ◊�6.64, □�1.93, D�0,41, ��20.2, ~� Reprinted from Bradbury
and Riley (1967) with permission
244 8 Turbulence
jets (i.e. the jet flows where the excess of center-line velocity is much larger or
much smaller than the free stream velocity) are shown in Fig. 8.15 It is seen that in
both cases the experimental data can be generalized in the form of universal
dependences in the parametric planes Ni � � where Ni ¼ Du=Dum and u02= Duð Þ2.
8.5.5 Turbulent Jets in Crossflow
The aerodynamics of turbulent jets in crossflow was a subject of a number of
experimental works. They contain detailed data on the jet trajectory, decay of
centerline velocity, distributions of the mean and pulsation characteristics, as well
as the vorticity field in such jets (Chassaing et al. 1974; Moussa et al. 1977;
Andreopoulos and Rodi 1985; Fric and Roshko 1994; Kelso et al. 1996; Smith
and Mungal 1998). In addition, a number of the empirical and semi-empirical
correlations for predicting the trajectory of turbulent jets in crossflow and variation
of the flow characteristics along the curved jet axis was proposed (Keffer and
Baines 1963; Abramovich 1963). A self-consistent analysis of the aerodynamics
of turbulent jets in crossflow based on the similarity theory was recently proposed
by Hasselbrink and Mungal (2001). Following the ideas of the latter work, we
outline below the application of the dimensional analysis to study characteristics of
turbulent jets in crossflow.
Our aim is to describe the flow field in an incompressible turbulent jet in
crossflow with a known trajectory ycðxcÞfor a given blowing ratio
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0 0.5 1 1.5 2 2.5 3
12
yy
u'2
U02
Fig. 8.15 Distribution of the velocity fluctuations in cross-sections of plane turbulent jets.
D-strong jet, □-weak jet, ▪▪▪-wake. Reprinted from Everitt and Robins (1978) with permission
8.5 Turbulent Jets 245
ðrju2j r1v21� Þ1 2=
(Fig. 8.16). We will proceed from the assumption that the local
values of any of the flow characteristics Ni ¼ r; u; v � � � are fully determined by the
axial value of a given characteristic in the same cross-section Nic, the jet thickness
there d, the position along the centerline xc, and coordinates x; y of a point under
consideration. It is emphasized that the velocity vector v in the general case has
three components u; v and w, but following Hasselbrink and Mungal (2001), we
consider the flow to be planar and thus, are dealing with only u and v. Then we canwrite the functional equation for Ni as follows
Ni ¼ f Nic; x; y; xc; dð Þ (8.130)
where Ni ¼ r; u; v.The dimensions of Nic are LaMbTg, whereas the dimensions of the other
governing parameters in (8.130) are all L½ �. Therefore, in the case under consider-
ation the total number of the governing parameters is five, whereas the number of
the governing parameters with independent dimensions equals two, so that
n� k ¼ 3. Then, according to the Pi-theorem, (8.130) can be transformed to the
following dimensionless equation
Ni ¼ Nic’ �; g; xcð Þ (8.131)
where �, g and xc are the coordinates x,y and xc normalized by d.The density r and velocity component distributions can be expressed as follows
r ¼ rcðxcÞf1 �; g; xcð Þ (8.132)
u ¼ ucðxcÞf2 �; g; xcð Þ (8.133)
v1 � vð�; g; xcÞ½ � ¼ v1 � vcðxcÞ½ �f3 �; g; xcð Þ (8.134)
Fig. 8.16 Sketch of a jet in
crossflow. The origin of the
coordinates x andyis at thecenter of the nozzle issuing
the jet. Adenotes a jet cross-section with a center at
x ¼ xc and y ¼ yc. Subscriptsj;1and c correspond to the
nozzle, the undisturbed
crossflow and the centerline
of the jet, respectively.
Reprinted from Hasselbrink
and Mungal (2001) with
permission
246 8 Turbulence
where the functions f1, f2, and f3 are assumed to be universal within each of the
characteristic flow domains of jets in crossflow: (1) potential core, (2) the near-field
region, and (3) the far-field region.
In order to determine the value of the flow characteristics at the jet axis, it is
necessary to employ some additional relations that follow from the mass, momen-
tum and momentum excess (based on Dv ¼ v1 � vc) balance equations. In partic-
ular, they have the form
m I1rcucd2 (8.135)
J I2rcu2cd
2 (8.136)
y I3rcd2 v1 � vcð Þuc (8.137)
for the near-field of the jet, and
m I4rcv1d2 (8.138)
J I5rcv1ucd2 (8.139)
y I6rcv1 v1 � vcð Þd2 (8.140)
for the far-field of the jet where m ¼ mj þ m1 is the mass flux,Jand y are the
momentum and momentum excess fluxes, respectively, through a cross-section at a
given distance from the nozzle. In addition, I1 ¼ÐA
f1f2d�dg, I2 ¼ÐA
f1f22 d�dg
I3 ¼ÐA
f1f2f3d�dg, I4 ¼ÐA
f1d�dg, I5 ¼ÐA
f1f2d�dg and I6 ¼ÐA
f1f3d�dg, where A
denotes jet cross-section.
Neglecting the effect of the molecular viscosity, we assume that the functional
equations for the axial velocity and turbulent jet thickness in the near field region of
the jet are
uc ¼ c1 J; r1; xð Þ (8.141)
d ¼ c2 J; r1; xð Þ (8.142)
Taking into account the dimensions of J, r1 and x, we find
uc ¼ c1J
r1
� �1=2
x�1 (8.143)
d ¼ c2x (8.144)
where c1 and c2 are constants.
8.5 Turbulent Jets 247
The dependence of vmon x found using (8.140) is
ðv1 � vcÞ � y
r1=21 J1=2x�1 (8.145)
For the far-field region of the jet flow we find the following expressions for uc; dand vc
uc � J
r1v1x�2 (8.146)
d � x (8.147)
ðv1 � vcÞ � J
r1v1x2 (8.148)
The relations (8.143–8.145) and (8.146–8.148) allow us to evaluate the change
in the velocity and thickness withing the near and far regions of of the jet. It is seen
that the centerline velocity fades according to the different laws at small and large
distances from the nozzle. In particular, at small distance from the nozzle the
centerline velocity fades as x�1, wereas at large x it fades as x�2.In both regions
the jet thickness is proportional to x.
8.5.6 Turbulent Wall Jets
Consider a turbulent wall jet outflowing from a two-dimensional slot in contact with
a wall into a fluid at rest. The velocity profile in a cross-section of the wall jet is
sketched in Fig. 8.17. It results from the interection of the jet with the solid surface,
Fig. 8.17 Sketch of turbulent
wall jet
248 8 Turbulence
as well as with the surrounding fluid. In the far-field region of turbulent wall jets the
profile of the longitudinal mean velocity component has a maximum located at
some distance y ¼ ym from the wall which depends on x. On both sides from the
maximum at y ¼ ym velocity gradually decreases to zero at y ¼ 0 (at the wall, due
to the no-slip condition) and y ! 1 (in the surrounding fluid at rest).
The measurements show that turbulent wall jets have a very complicated struc-
ture because of the interplay of the molecular viscousity and the inertial effects
(Launder and Rodi 1981; 1983). The influence of various factors on flow
characteristics is different in different domains of turbulent wall jets. Within the
near-wall region the molecular viscous effects are dominant, whereas at large
distance from the wall fluid inertia plays an important role. Such structure of
turbulent wall jets allows one to select two characteristic domains: (1) the inner
one close to the wall (with dominant viscous effects), and (2) the outer one in which
the inertial effects are dominant. In the two domains the velocity distribution is
affected by different parameters responsible for specific flow features in the inner
and outer parts of turbulent wall jets. In the framework of such model the charac-
teristic velocity and length can be chosen as the friction velocity u� ¼ tw=rð Þ1=2andlength d� ¼ n=u� in the inner domain, and the maximum longitudinal velocity umand the jet thickness d (e.g. corresponding to the velocity value u ¼ um=2Þ for theouter one.
At the Reynolds number being infinite, there exists a self-similar solution of the
problem (George et al. 2000; Bergstrom and Tachie 2001), and the functional
equation for the longitudinal velocity in the inner and outer domains can be written as
uin ¼ fin u�; y; vð Þ (8.149)
uout ¼ fout um; d; yð Þ (8.150)
Applying the Pi-theorem to (8.149) and (8.150), we transform them to the
following dimensionless form
Pin ¼ ’inðP1inÞ (8.151)
Pout ¼ ’outðP1outÞ (8.152)
where the dimensionless groups read
Pin ¼ u
u�; P1in ¼ yu�
n; Pout ¼ uout
um; P1out ¼ y
d(8.153)
Equations (8.151) and (8.152) show that the dimensionless velocity in the inner
and outer domains is a function of a single dimensionless group. The latter makes it
possible to collapse the experimental data for turbulent wall jets as a single curve
Uþ ¼ u=u� versus yþ ¼ yu�=n for the inner layer, and another single curve
8.5 Turbulent Jets 249
u=u� versus y=d for the outer one .The experimental data for the mean velocity
profiles in the outer (y y1 2=
�) and inner (yþ) coordinates, and the variation of the
half-width with the streamwise distance in turbulent plane jet are shown in
Figs. 8.18, 8.19 and 8.20.
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2 2.5
u u m
12
yy
Fig. 8.18 The mean velocity profiles in the wall jet near a smooth wall.◊�Re0¼12500; x h= ¼50;
□�Re0¼12500; x h¼80= ;■�Re0¼9100; x h= ¼50; ��Re0¼6100; x h= ¼30; ~�Re0¼10000
(data by Karlsson et al., 1993). Reprinted from Tachie et al. (2004) with permission
1.00E+07
1.00E+08
1.00E+09
1.00E+10
1.00E+08 1.00E+09 1.00E+10 1.00E+11
xI0
n2
n2
y 1/2
I 0
Fig. 8.19 Variation of the half-width of turbulent plane wall jet in the streamwise direction. Re0:
�1240, ◊�11900, ~�10300, □�9100, ��6100, ♦�12500, ■�5900. Reprinted from Tachie
et al. (2004) with permission
250 8 Turbulence
In order to determine the character of variation of um and d along the turbulent
wall jet some addition considerations of the governing parameters that determine
flow in the outer domain should be involved. Narasimha et al. suggested that the set
of parameters should also include the momentum flux at the jet origin divided by
density I0½ � ¼ L3T�2 and molecular kinematic viscosity n½ � ¼ L2T�1 (Narasimha
et al. 1973). Accordingly, one can present the functional equations for um and d as
follows
um ¼ f1 n; I0; xð Þ (8.154)
d ¼ f2 n; I0; xð Þ (8.155)
Then, applying the Pi-theorem, we arrive at
umnI0
¼ ’1
xI0n2
� �(8.156)
dI0n2
¼ ’2
xI0n2
� �(8.157)
1.00E+00
1.00E+01
1.00E+02
1.00E+00 1.00E+01 1.00E+02 1.00E+03y+
U+
Fig. 8.20 The mean velocity profile in the inner coordinates in a turbulent plane wall jet. Re:
◊�12500, D�9100, □�6100. Reprinted from Tachie et al. (2004) with permission
8.5 Turbulent Jets 251
8.5.7 Impinging Turbulent Jet
A sketch of the turbulent jet impinging normally onto an infinite solid wall is shown
in Fig. 8.21.
The impinging jets can be subdivided into three characteristic regions. In the first
one (1 in Fig. 8.21) free turbulent mixing of the jet with the surrounding medium is
dominant. In region 2 direct interaction of the jet with the solid wall is accompanied
by an abrupt deceleration of the jet and its deflection in the direction normal to its
initial path. In region 3 the jet-wall interaction resembles that one for a walljet. The
centerline velocity um gradually changes from its initial value at the nozzle exit u0to zero at the stagnation point at the wall surface. The distribution of the centerline
velocity corresponding to different distances from the nozzle exit down to the solid
surface is shown in Fig. 8.22 where u0 and d0 are the jet velocity at the nozzle axis
and the nozzle diameter, respectively. It is seen that the jet-wall interaction affects
significantly the longitudinal velocity umðxÞ.In order to gain an insight into the peculiarities of the velocity distribution we
apply -theorem. In this case the governing parameters determining the centerline
P the (for a ½Jx� ¼ MT�2 the total momentum flux ½r� ¼ L�3M, velocity are: the
fluid density and the ½h� ¼ L plane jet), the distance between the solid surface and
the nozzle exit. Accordingly, the functional equation for the centerline velocity
reads ½x� ¼ L coordinate
um ¼ f r; Jx; h; xð Þ (8.158)
Fig. 8.21 Sketch of the impinging turbulent jet
252 8 Turbulence
Since three of the four governing parameters in (8.158) have independent
dimensions, this equation reduces to the following dimensionless equation
P ¼ ’ðP1Þ (8.159)
where P ¼ um=ra1Ja2x ha3 and P1 ¼ x=ra01J
a02
x ha03 .
Taking into account the dimensions of um, r, Jx, h and x, we find the values of theexponents ai and a
0i as a1 ¼ �1=2, a2 ¼ 1=2, a3 ¼ �1=2; a
01 ¼ a
02 ¼ 0, and a
03 ¼ 1.
Then (8.159) takes the form
umu0
h
d0
� �1=2
¼ ’x
h
� �(8.160)
In (8.160) it is accounted for the fact that Jx � ru20d0.Equation (8.160) shows that dimensionless centerline velocity eu ¼ um=u0ð Þ
h=d0ð Þ1=2 is a universal function of a single dimensionless variable x� ¼ x=h. The
experimental data by Gutmark et al. (1978) on the centerline velocity distribution in
plane impinging turbulent jet are presented in Fig. 8.23. They correspond to different
valuesof the ratio h=d0 and the Reynolds number. Figure 8.23 shows that all experi-
mental data collapse around a single curve u� ðx�Þ according to the results of the
dimensional analysis. It is worth noting that near the wall there is a narrow layer
where the centerline velocity is proportional to the distance from the wall. This linear
relation corresponds to the inviscid stagnation flow. A significant deflection of the
dimensionless velocity u�from the one corresponding to a free jet takes place in the
region 0< x� � 0:2.
Fig. 8.22 Distribution of the centerline velocity at several cross-sections. Curve 1 corresponds
to a free jet, h ¼ h1 ¼ 1. Curves 2–4 correspond to different distances from the nozzle:
h2 > h3 > h4
8.5 Turbulent Jets 253
Problems
P.8.1. Establish the functional dependence of the dimensionless centerline velocity
in the axisymmetric impinging turbulent jet on the dimensionless distance from the
wall.
The governing parameters which determine the centerline velocity in such a jet
are: r½ � ¼ L�3M, Jx½ � ¼ LMT�2, h½ � ¼ L and x½ � ¼ L. Accordingly, the functional
equation for the centerline velocity reads
u ¼ f r; Jxh; xð Þ (P.8.1)
Applying the Pi-theorem to (P.8.1), we obtain
P ¼ ’ðP1Þ (P.8.2)
where P ¼ um=ra1Ja2x ha3 and P1 ¼ x=ra01J
a02
x ha03 .
Taking into account the dimensions of um; r, Jx, h and x, we find the values of
the exponents ai and a0i as a1 ¼ �1=2, a2 ¼ 1=2, a
01 ¼ 0,a
02 ¼ 0, a
03 ¼ 1. Then we
obtain from (P.8.2)
umu0
� �h
d0
� �¼ ’
x
h
� �(P.8.3)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.2 0.4 0.6 0.8 1 1.2x
h
u mh
1 2
u 0d 0
Fig. 8.23 The distribution of the normalized mean longitudinal velocity component
um=u0 h=d0ð Þ1=2along the jet center line. D�Re¼3 �104; h d¼100= , □�Re¼4:3 �104; h d= ¼40,
◊�Re¼5:6 �103; h d= ¼31,��Re¼5:6 �103; h d= ¼43:6, +�Re¼5:6�103; h d= ¼67:5. Reprintedfrom Gutmark et al. (1978) with permission
254 8 Turbulence
Note that in (P.8.3) accounts for the fact that Jx � ru20d20, where d0 is the
diameter of the nozzle.
P.8.2. Consider cavity formation at the free surface of a liquid pull by an
impinging axisymmetric turbulent gas jet. (1) Determine the dimensionless groups
of the flow, and (2) present the experimental data for the cavity depth and diameter
in an appropriate dimensionless form.
The flow under consideration is depicted in Fig. 8.24. An axisymmetric turbulent
gas jet is issued from a nozzle with the exit diameter dj. The jet is directed normally
to the unperturbed free surface of the pool. The distance between the nozzle exit and
the unperturbed surface in the liquid pool is h. The impinging jet causes a depres-
sion of the liquid surface. A sufficiently strong jet creates a visible cavity at the free
surface. Gas penetration into liquid is accompanied by the jet deceleration and
formation of the annular reverse gas flow. The liquid surface is deformed due to the
action of of the dynamic pressure and friction from the gas side as well as the cavity
shape is affected by liquid surface tension. In some cases the cavity surface is
unstable and gas bubble entrainment can take place there, the phenomenon which is
disregarded here. Thus, the cavity formation depends on several competing factors:
the physical properties of the gas and liquid phases, the initial jet diameter and
velocity of the jet, etc.
u0
h
Fig. 8.24 A cavity formed at
the surface of a liquid pool by
an impinging axisymmetric
gas jet
Problems 255
Assuming that the gas velocity distribution at the nozzle exit is uniform, we list
the governing parameters of the flow
rG L�3M� �
; rL L�3M� �
; mG L�1MT�1� �
; mL L�1MT�1� �
; (P.8.4)
s MT�2½ �, g LT�2½ �, uj LT�1½ �, dj L½ �, h L½ �where r and m denote density and viscosity, respectively, s is the surface tension,
g is the gravity acceleration, ujis the jet velocity at the nozzle exit, and subscripts
G and L refer to the gas and liquid, respectively.
Three of the nine governing parameters in (P.8.4) possess independent
dimensions. According to the Pi-theorem, the number of the dimensions groups
that determine the dimensionless characteristics of the cavity equals to six. These
are the following
rGL; h; ReG; ReL; Fr; We (P.8.5)
where rGL ¼ rG=rL, H ¼ h=dj. Also, ReG ¼ ujdj=nG, ReL ¼ ujdj=nL, Fr ¼ u2j =gdjand We ¼ djrGu
2j =s are the two Reynolds numbers, the Froude and Weber num-
bers, respectively, with n being the kinematic viscosity.
In the particular case where the Reynolds and Weber numbers are sufficiently
large and the viscous and surface tension effects are negligible, the number of the
dimensionless groups can be reduced significantly. At a large distance between
the nozzle exit and the unperturbed liquid surface (h >> djÞ the characteristics ofthe gas jet are mostly determined by its total momentum flux Jx ¼ rGu
2j d
2j p=4,
whereas the effect of the gas pressure at the cavity surface (which is determined by
the weight of the liquid displaced from the cavity) can be related to the specific
weight of the liquid g ¼ rLg L�2MT�2½ �. Then, the number of the governing
parameters reduces to three, namely, Jx, g and h, so that the functional equation
for the cavity depth hc becomes
hc ¼ f1 Jx; g; hð Þ (P.8.6)
In (P.8.6) the number of the governing parameters with independent dimensions
equals two. Accordingly, (P.8.6) reduces to the following dimensionless equation
P ¼ ’ðP1Þ (P.8.7)
where P ¼ hc=Ja1x ga2 and P1 ¼ h=J
a01
x ga02 .
Taking into account the dimensions of hc, h, g and Jx, we find that a1 ¼ a01 ¼ 1=2
and a2 ¼ a02 ¼ �1=3. Then (P.8.7) takes the form
hc
Jx=gð Þ1=3¼ ’
h
Jx=gð Þ1=3( )
(P.8.8)
256 8 Turbulence
or
hch¼ Jx
g
� �1=31
h’1
h
Jx=gð Þ1=3( )
¼ F1
Jxgh3
� �(P.8.9)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.1 0.2 0.3 0.4 0.5 0.6
0
0.5
1
1.5
2
2.5
3
3.5a
b
0 0.01 0.02 0.03 0.04 0.05Ix[lb]
d c[in
]h c
[in]
Jx[lb]
Fig. 8.25 Variation of the cavity diameter (a) and depth (b) with the distance between the nozzleand the unperturbed free surface (h) and the momentum flux of the jet. h in½ �: ~�2.0, +�3.0,
��4.0, D�5.0, □�5.0, ◊�7.0. Reprinted from Cheslak et al. (1969) with permission
Problems 257
Since the cavity diameter dc is related with its depth, it is possible to state the
functional equation for dc as follows
dc ¼ f2 Jx; g; hcð Þ (P.8.10)
Applying the Pi-theorem to (P.8.10), we arrive at
dchc
¼ F2
Jxgh3c
� �(P.8.11)
The experimental data on the cavity diameter and depth are presented in
Figs. 8.25a and 8.25b.
It is seen that an increase in h is accompanied by the growth of the cavity
diameter and a decrease in its depth. An increase in the total momentum flux of the
jet leads to an increase of the cavity depth and diameter. Equations (P.8.9) and
(P.8.11) show that the dimensionless cavity depth hc=h and diameter dc=hc are
functions of the dimensionless group Jx=gh3 and Jx=gh3c , respectively. Accordingly,one can expect that all the data points corresponding to different experimental
conditions should collapse at single curves hc=hð Þ Jx=gh3ð Þ and dc=hcð Þ Jx=gh3c �
in
the parametric planes hc=h versus Jx=gh3 and dc=hc versus Jx=gh3c . This result,
indeed, agrees with the data by Banks and Chandrasekhara (1963), as well as
with several other measurements.
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260 8 Turbulence
Chapter 9
Combustion Processes
9.1 Introductory Remarks
Combustion presents itself complicated physicochemical process which proceeds
due to progressively self-accelerating exothermal chemical oxidation reactions
sustained by an intensive heat release. A strong dependence of the chemical
reaction rate on temperature according to the Arrhenius law determines a very
high sensitivity of combustion processes to small disturbances of the governing
parameters. It also determines an almost abrupt transition of reactive systems from
a low temperature state to a high temperature state which is associated with
ignition. The existence of a critical state corresponding to ignition, as well as the
ability of combustion oxidation reactions to sustain a self-propagating flame front
over reactive media represent themselves main features of combustion process.
Combustion of continuous gaseous media is described by the system of
equations including the Navier–Stokes, continuity, energy and species balance
equations
@r@t
þ rðv � rÞv ¼ �rPþrðmrvÞ (9.1)
@r@t
þr � ðrvÞ ¼ 0 (9.2)
r@h
@tþ rðv � rÞh ¼ rðkrTÞ þ qW (9.3)
r@cj@t
þ rðv � rÞcj ¼ rðrDrcjÞ �Wj (9.4)
where r, v, P, T, h and cj are the density, velocity vector, pressure, temperature,
enthalpy, and species concentrations, respectively, q is the heat release of the
combustion oxidation reaction (it is assumed that the whole complicated chemical
L.P. Yarin, The Pi-Theorem, Experimental Fluid Mechanics,
DOI 10.1007/978-3-642-19565-5_9, # Springer-Verlag Berlin Heidelberg 2012
261
process can be reduced to a single equivalent reaction),W is the rate of the chemical
reaction, Wj is the rate of conversion of j� th species (positive for reagents, which
are consumed and negative for the reaction products which are produced), m; k andD are the viscosity, thermal conductivity and diffusivity, respectively. It is
emphasized that for simplicity all the transport coefficients and physical properties
of the parameters are assumed to be identical for all species involved.
The system of (9.1–9.4) should be supplemented by an equation of state of the
gas, a microkinetic law for the rate of chemical reaction and the correlations
determining the dependence of the physical properties on temperature. Solving
the highly nonlinear (9.1), (9.3) and (9.4) is extremely difficult. Therefore, the
analytical models of the combustion processes, as a rule, involve various
simplifications and approximations. Consider briefly some of them. First, we
discuss simplifications related to the chemical reaction rate W. For this aim, we
assume that: (1) the reactive mixture represents itself perfect gases, (2) the thermal
and species diffusivities are equal, and thermal conductivity and specific heat cP areinvariable, so that the enthalpy is expressed as h ¼ cPT, (3) a simple equivalent
single-step chemical reaction with the rate which can be factorized as a product of
two functions depending solely either on temperature or concentration
Wðc; TÞ ¼ ’ðcÞcðTÞ (9.5)
Moreover, we assume that there is only one limiting species in the system (fuel)
and, as a result, only one species (fuel) balance equation is to be considered. Its
concentration is denoted as c.Then, the energy and fuel balance equations read
@h
@tþ ðv � rÞh ¼ ar2hþ qWðc; TÞ (9.6)
@c
@tþ ðv � rÞc ¼ Dr2c�Wðc; TÞ (9.7)
where a and D are the thermal and mass diffusivity coefficients.
Assume that a ¼ D, i.e. the Lewis number Le ¼ a=D ¼ 1. Then, eliminating the
source terms from (9.6) and (9.7), we obtain the equation for the total (physical and
chemical) enthalpy
@H
@tþ ðv � rÞH ¼ ar2H (9.8)
where H ¼ hþ qc:For an isolated system @H=@nð ÞS ¼ 0; with S being the system envelop and n is
the normal to it. In this case (9.8) is integrated as
H ¼ const (9.9)
262 9 Combustion Processes
The constant in (9.9) is found from the condition that the maximum temperature
Tm corresponds to complete fuel consumption, i.e. to c ¼ 0: Accordingly, that theconstant is equal to cPTm, and thus the current fuel concentration is related to the
current temperature as
c ¼ cpðTm � TÞq
(9.10)
Therefore, the function ’ðcÞin (9.5) can be presented as
’ðcÞ ¼ ’cPðTm � TÞ
q
� �¼ ’�ðTÞ (9.11)
Then the reaction rate becomes
Wðc; TÞ ¼ ’�ðTÞcðTÞ ¼ WðTÞ (9.12)
Assuming the nth order reaction depending on temperature via the Arrhenius
law, i.e.Wðc; TÞ ¼ zcn exp �E=RTð Þ, we arrive at the following explicit expressionfor WðTÞ
WðTÞ ¼ zðTm � TÞn exp � E
RT
� �(9.13)
where z is pre-exponent, E is the activation energy and R is the universal gas
constant.
It is emphasized that (9.12) and (9.13) are valid only when the Lewis number
Le ¼ 1. In the general case when Le 6¼ 1; the distribution of the total enthalpy
within a reactive medium is not given by a constant according to (9.9) but is
more complicated. In combustion of gaseous mixtures, which is a particular case
of homogeneous combustion), H has extrema in the vicinity of the flame front. It
was shown in Zel’dovich et al. (1985) that this is the result of the energy redistri-
bution due to different rates of heat and mass transfer at Le 6¼ 1:An additional simplification that is widely use in the theory of thermal explosion
and ignition is related to the Frank-Kamenetskii transformation of the exponent in
the Arrhenius law. Following Frank-Kamenetskii (1969), we present the ratio
E=RTas
E
RT¼ E
RðT� þ DTÞ ¼E
RT�
1
1þ DT=T�ð Þ �E
RT�� E
RT2�DT (9.14)
where as the temperature T� is close to the temperature at which the chemical
reaction proceeds: to the initial temperature T0 in the problem of self-ignition, or to
the maximum temperature Tmin the problem of combustion wave propagation;
D ¼ T � T�:
9.1 Introductory Remarks 263
According to (9.14), the exponent in the Arrhenius law takes the form
e�ERT � e�
ERT�ey (9.15)
where y ¼ E=RT2�
� �ðT � T�Þ.The expression (9.15) is an accurate approximation of the Arrhenius law at
temperatures near to T�, albeit without linearization. The latter feature is important,
since such phenomena as ignition, extinction or thermal explosions are basically
nonlinear and to be able to address them, nonlinear (even though simplified)
nature of the problem should be preserved (Frank-Kamenetskii 1969; Zel’dovich
et al. 1985).
An additional significant simplification of the problems related to combustion of
non-premixed gases can be achieved by means of the Schvab–Zel’dovich trans-
formation (Schvab 1948; Zel’dovich 1948). It allows removal of the non-linear
term WðTÞ from the all but one species balance equations and transforms them to
the form identical to the form for inert gases. In order to illustrate this approach, we
consider application of the Schvab–Zel’dovich transformation to the species
equations in the case of reaction between a single fuels and a single oxidizer
r@ca@t
þ rðv � rÞca ¼ rDr2ca �Wa (9.16)
r@cb@t
þ rðv � rÞcb ¼ rDr2cb �Wb (9.17)
where subscripts aand b refer to fuel and oxidizer, respectively.
Taking into account that the reaction ratesWa andWb are related by the stoichio-
metric relation dictated by the corresponding chemical reaction
Wa ¼ Wb
O(9.18)
with O being the stoichiometric coefficient, we transform the system of equations
(9.16) and (9.17) in the following way. We divide (9.17) by O and subtract (9.17)
from (9.16). As a result, we arrive at the following equation
r@b@t
þ rðv � rÞb ¼ rDr2b (9.19)
where b ¼ ca � cb=O.Equation (9.19) has the same form as the species balance equation the inert flows
without any chemical reactions. This equation admits solutions which differ up to a
constant, i.e. b0 ¼ bþ const: The latter allows one to reduce the boundary
conditions for axisymmetric reactive flows (submerged torches) to the form
264 9 Combustion Processes
y ¼ 0@b0
@y¼ 0; y ! 1 b0 ! 0 (9.20)
where is the radial coordinate.
9.2 Thermal Explosion
Thermal explosions corresponds to unstable states of reactive system at which any
initial temperature distribution leads to a disruption of the thermal equilibrium
resulting from a runaway rate of the exothermal chemical reaction with heat release
higher than the rate of heat removal to the environment. The latter leads to
progressive temperature growth and an abrupt transition from the initial low
temperature state to the final stable high temperature state.
In order to illustrate the nature of thermal explosions, we consider the thermal
regime of combustion of an ideally stirred reactor (for example, a jet-stirred reactor,
JSR) which represents itself a closed volume filled with a homogeneous reactive
mixture of fuel and oxidizer. The wall temperature of JSR is assumed to be fixed and
equal to a low temperature T0 imposed by the surroundingmedium. The temperature
field inside JSR is assumed to be uniform, which corresponds to an infinite rate of
mixing. Changes in the reactant concentrations in time are assumed to be negligible
during the extremely short period of time preceding thermal explosion. Then, the
specific rate of heat release by chemical reaction QI and the rate of heat removal to
the environment QII (both divided by the reactor volume V) read
QI ¼ qW ¼ const� exp � E
RT
� �(9.21)
QII ¼ hS
VðT � T0Þ (9.22)
where q is heat of reaction,W ¼ zceacsb exp �E=RTð Þ is the rate of chemical reaction
(without simplification by means of the Frank-Kamenetskii transformation), z is thepre-exponent, ca and cb are reactant concentrations (fuel and oxidizer, respec-
tively), e and s are constant (they denote the reaction orders in fuel and oxidizer,
respectively), h is the heat transfer coefficient at the outer wall of the reactor, S andVare the surface area and volume of JSR.
The curves of heat release QI and heat removal QII are plotted versus tempera-
ture T in Fig. 9.1. In the general case there are three intersection points which
correspond to the low (1), intermediate (2) and high temperature (3) states. It is easy
to see that intermediate state (2) is unstable whereas the states (1) and (3) are stable.
Indeed, any perturbation increasing the mixture temperature relative to that
corresponding to point 2 (T > T2Þis accompanied by an excess of the heat release
9.2 Thermal Explosion 265
rate over the intensity of the heat removal rate. As a result, temperature T should
keep increasing and thus, the intermediate point 2 is unstable. Also, if due to
a perturbation the temperature decreases (T < T2), the heat removal rate exceeds
the heat release rate. As a result, temperature T keeps decreasing and once more it is
seen that point 2 is unstable. On the other hand, similar arguments show that points
1 and 3 are stable. Among the possible mutual locations of curves QIðTÞ and
QIIðTÞthere are two particular ones where they are tangent to each other. These
two cases signify the two critical states: (1) the transition from the low- to high-
temperature state, which corresponds to the mixture ignition-point I in Fig. 9.1, and
(2) the transition from the high- to low-temperature state, which corresponds the
mixture extinction-point E in Fig. 9.1. Therefore, the thermal explosion
corresponds to the ignition condition, at which a low-temperature stationary state
of a reactive system becomes impossible. This is an outline of the non-stationary
theory of thermal explosion elaborated by Semenov (1935). On the other hand, the
stationary theory of thermal explosion developed by Frank-Kamenetskii (1969)
treats thermal explosion as the situation at which no stationary solution of the
thermal balance equation can be found. A detailed exposition of both theories of
thermal explosion can be found in the monograph by Zel’dovich et al. (1985);
see also Frank-Kamenetskii (1969) and Vulis (1961). Referring the interested
readers to these monographs, we restrict our consideration to the applications of
the Pi-theorem to the stationary problem of thermal explosion.
Consider stationary temperature distribution in a symmetric reactor with charac-
teristic size r0 and wall temperature T0:Assuming as before that changes in reactant
concentrations are negligible, we determine the governing parameters of the
1
I
QII
2
3
QIE
To T
Q
0
Fig. 9.1 The curves
corresponding to heat release
QI, and heat removal QII . The
lower, intermediate and upper
intersection points 1, 2 and 3,
respectively, correspond to
the stable lower temperature
state, intermediate (unstable)
state, and the stable high
temperature state
266 9 Combustion Processes
problem. From the physical point of view the local temperature inside a reactor
depends on the thermal conductivity of gas mixture k½ � ¼ JL�1T�1y�1, kinetic
factors in the Arrhenius law z exp �E=RTð Þ; namely on z½ � ¼ T�1; E½ � ¼Jmol�1 and R½ � ¼ Jy�1mol�1; the heat of reaction q½ � ¼ JL�3; the reactor size
r0½ � ¼ L; the wall temperature T0½ � ¼ y and the coordinate r L½ � of the point under
consideration
T ¼ f ðr0; r; T0; k; z;E;R; qÞ (9.23)
The temperature distribution defined by (9.23) satisfies the following conditions
T ¼ T0 at r ¼ r0;dT
dr¼ 0 at r ¼ 0 (9.24)
Equation (9.23) and the boundary conditions (9.24) contain eight dimensional
parameters including five parameters with independent dimensions. Then,
according to the Pi-theorem, (9.23) reduces to the following dimensionless equation
b ¼ ’ðx; g; b0Þ (9.25)
where b ¼ RT=E; b0 ¼ RT0=E; g ¼ qzRr20=kE and x ¼ r=r0.Equation (9.25) shows that using the Pi-theorem, it was possible to decrease the
number of the governing parameters from eight to three. However, even in this case
the study of the critical state which corresponds to thermal explosion is highly
complicated. In this situation, similarly to the analytical solutions of the combustion
theory discussed above, it is useful to employ a physically-based simplification,
namely the Frank-Kamenetskii transformation. Since the ignition (or thermal
explosion) process occurs at temperatures close to the wall temperature T0, simi-
larly to (9.15) we have
ze�ERT � ze
� ERT0e
E
RT20
ðT�T0Þ(9.26)
Then, (9.23) can be replaced by the following equation
DT ¼ f ðr; r0; k;ez; T0� ; qÞ (9.27)
where ez ¼ z exp �E=RT0ð Þ; T0� ¼ RT2
0=E and DT ¼ T � T0:Applying the Pi-theorem to the simplified (9.27), we arrive at the following
simpler dimensionless equation
# ¼ cðx; dÞ (9.28)
where # ¼ EðT � T0Þ=RT20 and d ¼ qEzr20=kRT
20
� �exp �E=RTð Þ is the Frank-
Kamenetskii parameter, which is the sole dimensionless constant on the right-
hand side in (9.28).
9.2 Thermal Explosion 267
The critical value of dcorresponding to the ignition (or thermal explosion) can be
found only by solving the thermal balance equation
r2x# ¼ �d expð#Þ (9.29)
subjected to the boundary conditions
# ¼ 0 at x ¼ 1;d#
dx¼ 0 at x ¼ 0 (9.30)
The corresponding critical condition found from (9.29) and (9.30) when the
stationary solution becomes impossible reads
d ¼ dcr ¼ const (9.31)
where dcr ¼ 0:88 for plane reactors and 0.33 for the spherical ones.
9.3 Combustion Waves
The present section is devoted to a simple estimate of the speed of combustion wave
that propagate in homogeneous infinite reactive media. The analysis is based on the
approach of the thermal theory of combustion that imply that combustion wave
propagation is a result of heat transfer from a high temperature reaction zone to
a relatively cold fresh mixture of fuel and oxidizer due to thermal conductivity. This
theory was developed by Zel’dovich (cf. Zel’dovich et al. 1985) and Frank-
Kamenetskii (1969). The thermal theory of combustion accounts for the main
features of the process, namely, the sharp dependence of the chemical reaction
rate on temperature, the intensive heat release within a thin reaction front, as well as
for the heat and mass transfer due to molecular thermal conductivity and molecular
diffusion. According to this theory, the mechanism of combustion wave propaga-
tion in homogeneous mixtures is the following. An instantaneous heating of a thin
layer of a preliminarily cold reactive mixture by an external source triggers
chemical reaction within the heated layer (Fig. 9.2). The heat released by the
exothermal chemical reaction, in its turn, leads to a further heating of the mixture
in this layer and its ignition under certain conditions. Heat transfer from the high
temperature zone to the cold mixture ensures heating and ignition of the neighbor-
ing layers, i.e. propagation of a self-sustained chemical reaction zone (the flame
front, or combustion wave) over reactive medium. At the transient stage of the
combustion wave propagation, the process develops under the conditions of a
continuous variation of the temperature and concentration fields. Also, the speed
of the combustion wave varies until its value will not approach the one
corresponding to the stationary regime of combustion.
268 9 Combustion Processes
In the framework of the thermal theory there are two main factors that determine
the speed of combustion wave in homogeneous reactive mixtures: (1) the exother-
mal chemical reaction accompanied by an intense heat release, and (2) the heat
transfer from the high-temperature reaction zone to the cold fresh mixture by
thermal conductivity. Under these conditions the governing parameters of the
process are as follows: the mixture density r½ � ¼ L�3M; thermal conductivity k½ � ¼LMT�3y�1; specific heat cP½ � ¼ L2T�2y�1, and the characteristic time of chemical
reaction tm½ � ¼ T determined by the maximal temperature. In addition, it is
assumed here that the Lewis number Le ¼ 1. Then, the thermal diffusivity a ¼ k/rcP ¼ D, and thus, the diffusion coefficient D should not be included separately in
the set of the governing parameters. Accordingly, the functional equation for the
speed of combustion wave uf� ¼ LT�1 has the form
uf ¼ f ðr; k; cP; tmÞ (9.32)
All the governing parameters in (9.32) have independent dimensions. Then,
according to the Pi-theorem, (9.32) transforms to
uf ¼ cra1ca2p ka3ta4 (9.33)
where c is a dimensionless constant.
Using the principle of the dimensional homogeneity and accounting for the
dimensions of uf ; r; cP; k and tm, we arrive at the following system of equations
for the exponents ai
�3a1 þ 2a2 þ a3 � 1 ¼ 0
a1 þ a2 ¼ 0
�2a1 � 3a3 þ a4 þ 1 ¼ 0
a2 þ a3 ¼ 0
(9.34)
Fig. 9.2 The structure of a
combustion wave at a certain
moment of time. The wave is
propagating from right to left.I-heating zone, II-reaction
zone, III-high temperature
zone
9.3 Combustion Waves 269
Equations (9.34) yield
a1 ¼ � 1
2; a2 ¼ � 1
2; a3 ¼ 1
2; a4 ¼ � 1
2(9.35)
and thus
uf ¼ c
ffiffiffiffiffiatm
r(9.36)
The value of the dimensional constant c depends on the other dimensionless
groups of the problem. They are E=RT0 and E=RTm (Frank-Kamenetskii 1969). On
the other hand, when the Frank-Kamenetskii transformation (9.15) is employed to
simplify the Arrhenius law, the constant c depends a single dimensionless group
ym ¼ EðTm � T0Þ=RT2m combining the two previously mentioned groups.
Using (9.36) it is possible to estimate the effect of pressure on the speed of
combustion wave propagation. Assuming that c is a weak function of the dimension-
less groups E=RT0 and E=RTm, we see that uf �ffiffiffiffiffiffiffiffiffiffia=tm
p: The thermal diffusivity of
gases is inversely proportional to pressure, a � P�1. Based on the chemical kinetics
data, one can also expect that t�1m � rn exp �E=RTmð Þ � Pn exp �E=RTmð Þ; were
n is the reaction order. As a result, we obtain the flame speed as
uf � Pn�12 exp � E
2RTm
� �(9.37)
It is seen that the flame speed uf does not depend on pressure in the case of a firstorder reaction. On the other hand, in the case of a second order reaction the flame
speed uf is proportional toP1=2:
As was noted before, a detailed form of the dependence of the combustion wave
speed on the physicochemical and kinetic parameters can be found by solving the
energy and diffusion equations. At Le ¼ 1when the profiles of fuel concentration
and the normalized temperature are similar to each other, the problem reduces to the
integration of the energy equation. In the frame of reference associated with the
moving combustion front (flame) the equation reads
ruf cPdT
dx¼ d
dxkdT
dx
� �þ qWðTÞ (9.38)
In an infinite premixed mixture of fuel and oxidizer the boundary conditions for
(9.38) have the form
x ¼ �1; T ¼ T0; x ¼ þ1; T ¼ Tm (9.39)
270 9 Combustion Processes
It is easy to see that if T(x) is a solution of the problem (9.38) and (9.39), then
T(x þ c) (with c being an arbitrary constant) is also a solution. The latter means that
the constant c is undetermined in principle, and one of the boundary conditions
(9.39) becomes redundant. However, (9.38) contains a still unknown flame speed uf,which shows that a seemingly redundant boundary condition should be used to
find uf, i.e. the flame speed uf represents itself an eigenvalue of the problem (9.38)
and (9.39). A comprehensive discussion of the analytical solutions for the speed
of combustion waves in homogeneous mixtures can be found in the following
monographs and surveys: Frank-Kamenetskii (1969), Williams (1985), Zel’dovich
et al. (1985), Merzhanov and Khaikin (1992). Numerical solutions of this problem
were discussed in Spalding (1953), Zel’dovich et al. (1985) and Merzhanov
et al. (1969).
9.4 Combustion of Non-premixed Gases
Consider combustion of non-premixed gases in an adiabatic cylindrical chamber
(Fig. 9.3). The gaseous reactants are supplied through a core tube of cross-sectional
radius r1 (fuel) and an annular gap of thickness r2 � r1 (oxidizer). It is assumed
that the velocity distribution at any cross-section of the combustion chamber
(burner) is uniform, i.e. fuel and oxidizer are issued with the same speed and the
effect of viscous friction at the wall is negligible. The mass flux does not change
downstream in the chamber. In addition, it is assumed that the diffusion transfer in
radial direction is much large than in the longitudinal one. Regarding the rate of
chemical reaction at the flame front, it is assumed that it is infinite, and therefore,
concentrations of fuel and oxidizer at the flame front are zero. The assumptions
made follow those in the seminal work of Burke and Schumann (1928), as well as
the detailed analysis of the corresponding problem is covered in the monographs
by Vulis (1961), Williams (1985), Zel’dovich (1948) and Zel’dovich et al. (1985).
Below we discuss briefly the formulation of this problem and concentrate of the
application of the dimensional analysis to in this particular case.
r
x
r2r1Fuel
Oxidizer
Oxidizer
2
1Fig. 9.3 Sketch of a non-
premixed gas burner
9.4 Combustion of Non-premixed Gases 271
The above assumptions and the Schvab-Zel’dovich transformation allow us to
reduce the species balance equations to the following single equation we write the
governing equation in the form
ru@b@x
¼ rD1
r
@
@rr@b@r
� �(9.40)
[cf. (9.19)] where b ¼ ca � cb=O; ca and cb are the concentration of fuel and
oxidizer, respectively, O is the stoichiometric oxidizer-to-fuel mass ratio, ru is
the mass flow rate and rD the product of density and diffusion coefficient; both ruand rD are constant in the present case.
The boundary conditions for (9.40) read
x ¼ 0 : 0 r r1 b ¼ ca0; r1 < r < r2 b ¼ � cb0O
x > 0 : r ¼ 0@b@r
¼ 0; r ¼ r2@b@r
¼ 0
(9.41)
The conditions at x ¼ 0 determine the uniform distribution of fuel and oxidizer
at the burner inlet; the boundary conditions at x> 0 determine the flow symmetry
and correspond to the absence of the chemical reaction at the wall.
The solution of (9.40) with the boundary conditions (9.41) is (Zel’dovich
et al. 1985)
bðr; xÞ ¼ b�X1i¼1
CIJ0ðr’i=r2Þ expð�SI � xÞ (9.42)
The following notation is used in (9.42): b ¼ ca0 � ca0Oþ cb0ð Þ=O½ � r1=r2ð Þ2;Si ¼ rD=ruð Þð’2
i =r22Þ, Ci ¼ 2 cb0=Oð Þð1þ wÞ r1=r2ð ÞJ1ðr1’i r2Þ= = ’i J0ð’iÞ½ �2
n o;
w ¼ ca0O=cb0ð Þ; J0 ð Þ and J1 ð Þ are the Bessel functions of the first kind of zero
and first orders, and ’i are the roots of the equation J1ð’Þ ¼ 0: Concentrations ca0and cb0 correspond to fuel and gas at the burner entrance at x ¼ 0:
The corresponding approximate expression for the flame length xf ¼ lf is
(Zel’dovich et al. 1985)
lf ¼ ur22D’2
1
ln2ð1þ wÞ r1=r2ð Þ2J1 r1’1 r2=ð Þ
’1 J0ð’1Þ½ �2 w� ð1þ wÞ r1=r2ð Þ2h i (9.43)
where ’1 ¼ 3:83 and J0ð’1Þ ¼ �0:4:The expression (9.43) corresponds to combustion at the excess of oxidizer and
the whole fuel is consumed at a finite length xfwhich corresponds to the tip of curve1 in Fig. 9.3. In the opposite case when combustion proceeds at the lack of oxidizer,
the term J0ð’1Þ½ �2 in the denominator of (9.43) should be replaced by J0ð’1Þ½ �.
272 9 Combustion Processes
Then, (9.43) describes the distance at which all oxidizer will be fully consumed,
which corresponds to the right-hand side end of curve 2 in Fig. 9.3.
Consider the Burke-Schumann problem in the framework of the dimensional
analysis. The problem formulation reveals that at Le ¼ 1 when the thermal and
mass diffusivities are equal to each other, the field of the compound concentration
b in coaxial burner is determined by nine parameters
b ¼ f ðu;D; r; x; r1; r2; ca0; cb0;OÞ (9.44)
These governing parameters have the following dimensions
u½ � ¼ LxT�1; D½ � ¼ L2yT
�1; r½ � ¼ Ly; x½ � ¼ Lx; r1½ � ¼ Ly; r2½ � ¼ Ly;
ca0½ � ¼ 1; cb0½ � ¼ 1; O½ � ¼ 1(9.45)
Among of the six dimensional parameters in (9.45), three parameters have
independent dimensions. Therefore, it is possible to form three dimensionless
groups
r1r2;
r
r2;
xD
ur22(9.46)
Then (9.44) reduces to the following dimensionless form
b ¼ ’r
r2;xD
ur22;r1r2; ca0; cb0;O
� �(9.47)
As it was mentioned above, the concentrations of reactants at the combustion
front at xf ¼ xf ðrf Þ are equal to zero, so that b ¼ 0 there. Then, (9.47) yields
’rfr2
xfD
ur22;r1r2; ca0; cb0;O
� �¼ 0 (9.48)
Solving (9.48) relative to the dimensionless group xfD=ur22, we obtain
xfD
ur22¼ c
rfr2;r1r2; ca0; cb0;O
� �(9.49)
Equation (9.49) determines geometry of the diffusion flame of non-premixed
reagents. It contains four constants r1=r2; ca0; cb0 and O that account for the burner
geometry, as well as the characteristics of reactive system. The dependence
xfD=ur22 ¼ c rf =r2
� �found from the exact solution (9.43) is shown in Fig. 9.4. As
discussed, the shape of the diffusion flame depends on a relation between w ¼ca0O=cb0ð Þ and r1=r2ð Þ= 1� r1=r2ð Þ2
h iwhich determines where the system has
9.4 Combustion of Non-premixed Gases 273
either an excess of oxidizer or fuel. In the first case the flame tip is located at the
flow axis, whereas in the second one at the wall of the burner. Assuming in (9.49)
rf ¼ 0, we obtain the following expression for the diffusion flame length xf ¼ lf
lf ¼ Pe� cr1r2; ca0; cb0;O
� �(9.50)
where Pe ¼ ur2=D is the Peclet number and lf ¼ l=r2.Equation (9.50) shows that with r1 r2= ; ca0; cb0 and O being constant, the flame
length lf ¼ lf r2= � Pe i.e. lf � ur22 D= . The volumetric flow rate of the gaseous
phase is Gv � ur2e for the planar flame, and Gv � ur22 for the axisymmetric flame
(e ¼ 1is the unit of length). Then, we arrive at the conclusion that lf ;planar � Gvr2 D=and lf ;axisymm � Gv D= , i.e. the length of the axisymmetric flame does not depend on
the radius of combustion chamber, whereas the length of a planar flame is directly
proportional to r2 whenGv ¼ const. Note, that in the planar case the difference r2 �h is equal to the semi-height of the channel (Vulis 1961).
9.5 Diffusion Flame in the Mixing Layer of Parallel Streams
of Gaseous Fuel and Oxidizer
The flow and flame structure under consideration are sketched in Fig. 9.5. Two
uniform streams of gaseous non-premixed reactants moving over both sides of a
semi-infinite plate which ends at x ¼ 0 come in contact to each other. The fuel
1
2
0.1
0.50
xf D
ur22
rfr2
Fig. 9.4 Configurations of
the diffusion flame of non-
premixed gases. 1: The case
of the excess of oxidizer. 2:
The case of the excess of fuel
274 9 Combustion Processes
stream is supplied at y < 0, whereas the oxidizer-at y > 0. The ignition takes place
at the line x ¼ 0; y ¼ 0: When the rate of chemical reaction is large enough,
conversion of reactants into combustion products occurs within a thin reaction
zone that can be considered practically infinitesimally thin and viewed as the
flame front (Zel’dovich et al. 1985); cf. Fig. 9.5. Then, domain I in Fig. 9.5 is filled
with the oxidizer and fully converted combustion products, whereas domain II is
filled with fuel and combustion products. Chemical reaction takes place neither in
domain I nor in domain II but solely at the flame front. On the other hand, pure
mixing takes place in domains I and II.
In the framework of this model, and assuming the low Mach number (M << 1)
and h ¼ cpT(cP ¼ const), the velocity, temperature and fuel and oxidizer concen-
tration fields are determined by the following equations
ru@u
@xþ rv
@u
@y¼ @
@ym@u
@y
� �(9.51)
@ru@x
þ @rv@y
¼ 0 (9.52)
rucP@T
@xþ rvcP
@T
@y¼ @
@yk@T
@y
� �(9.53)
Fig. 9.5 Diffusion flame in the mixing layer of parallel streams of gaseous and oxidizer
9.5 Diffusion Flame in the Mixing Layer of Parallel Streams 275
ru@cj@x
þ rv@cj@y
¼ @
@yrDj
@cj@y
� �(9.54)
where subscript j corresponds to the j� th species (j ¼ a for fuel, and j ¼ b for
oxidizer).
The system of (9.51–9.54) is supplemented by the equation of state (9.55)
accounting for the fact that pressure is constant in the mixing layer, as well as the
dependences of the physical parameters on temperature (9.56)
rT ¼ const (9.55)
mðTÞ; kðTÞ; rDjðTÞ (9.56)
The boundary conditions at x > 0 for (9.51–9.54) read
y ! þ1; u ! uþ1; T ! Tþ1; ca ! caþ1 (9.57)
y ! �1; u ! u�1; T ! T�1; cb ! cb�1
At the flame front y ¼ yf ðxÞ the reactant concentrations are zero, since the
reaction rate is practically infinite, whereas the diffusion fluxes of fuel and oxidizer
are in stoichiometric ratio
T ¼ Tf ; ca ¼ cb ¼ 0 (9.58)
� Db @cb=@nð ÞfDa @ca=@nð Þf
¼ O (9.59)
where Tf is the flame temperature which is equal to the adiabatic temperature of
combustion of non-premixed fuel and oxidizer, O is the stoichiometric coefficient
and @=@n is the derivative along the normal to the flame front. It is emphasized that
the boundary condition (9.59) allows one to determine the location of the flame front.
The problem we are dealing with in the present section represents itself a
compressible flow. In such cases the Dorodnitsyn–Illingworth–Stewartson trans-
formation discussed previously in Sect. 7.7 of Chap. 7 allows one to reduce
compressible problems to the corresponding incompressible ones. In particular,
introducing new the variables x ¼ x and � ¼ Ry0
rdy and assuming that the
dependences mðTÞ; kðTÞ and rDðTÞ are linear, it is possible to reduce (9.51–9.54)
to the form identical to the incompressible equations corresponding to the same
flow geometry but with r ¼ const: The dimensional analysis of these system of
equations shows that there exists the self-similar solution of the dynamic, thermal
and species balance equations in the following form
u ¼ F0ð’Þ; DT ¼ yð’Þ; cj ¼ #ð’Þ (9.60)
276 9 Combustion Processes
with u ¼ 2u= uþ1 þ u�1ð Þ; DTa ¼ T � Tþ1� �
= Tf � Tþ1� �
; DTb ¼ T � T�1ð Þ=
Tf � T�1� �
; ’ ¼ �xg; � ¼ RY
0
rdy; � ¼ x; y ¼ y=l�ð Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiReð1þ mÞ½ �=2p
; x ¼x=l�; ra ¼ ra=raþ1; rb ¼ rb=rb�1; m ¼ u�1=uþ1; l� is an arbitrary
length scale, x and � are the Dorodnitsy n variables, and g is a constant.The functions Fj
0 ð’Þ; yjð’Þ and #jð’Þ are determined by the following ODEs
Fj000 þ 1
2Fj
00Fj ¼ 0 (9.61)
yj00 þ Pr
2yj
0Fj ¼ 0 (9.62)
#j00 þ Sc
2#j
0Fj ¼ 0 (9.63)
[subscript j ¼ a or b], with the boundary conditions
Fa0 ¼ 2
1þ m; ya ¼ 0; #a ¼ 1 at ’ ! þ1
Fb0 ¼ m
1þ m; yb ¼ 0; #b ¼ 1 at ’ ! �1
ya;b ¼ 1; #a;b ¼ 0 at ’ ¼ ’f
(9.64)
The integration of (9.61–9.63) with the boundary conditions (9.64) leads to the
following expressions for the velocity, temperature and concentration distributions
in the mixing layer
u
uþ1¼ 1
21þ mð Þ þ 1� mð Þerf ð’Þf g (9.65)
ya ¼ 1� erf ð’ ffiffiffiffiffiPr
p Þ1� erf ð’f
ffiffiffiffiffiPr
p Þ ; #a ¼ 1� 1� erf ð’ ffiffiffiffiffiSc
p Þ1� erf ð’f
ffiffiffiffiffiSc
p Þ (9.66)
for ’f < ’ <þ1; and
yb ¼ 1þ erf ð’ ffiffiffiffiffiPr
p Þ1þ erf ð’f
ffiffiffiffiffiPr
p Þ ; #b ¼ 1� 1þ erf ð’ ffiffiffiffiffiSc
p Þ1þ erf ð’f
ffiffiffiffiffiffiffiScÞp (9.67)
for �1 < ’ < ’f :To find the location of the flame front, we take into account the following
evaluations valid for the boundary layer in which the flame front slope relative to
the x-axis is small enough, so that cosðn; xÞ � 0 and cosðn; yÞ � 1: For example, at
9.5 Diffusion Flame in the Mixing Layer of Parallel Streams 277
O ¼ 15; Pr ¼ 1 and Rex ¼ 100� 1000; cosðn; yÞ � 0:99� 0:998. Under these
conditions @=@n ¼ @=@x � cosðn; xÞ þ @=@u � cosðn; yÞ � d=dy. Calculating the
derivatives @ca=@nð Þf and @cb=@nð Þf by using the expressions (9.66) and (9.67)
and substituting them into (9.59), we arrive at the following expression for the
coordinate of the flame front ’f
erf ð’f
ffiffiffiffiffiSc
pÞ ¼ 1� e
1þ e(9.68)
where e ¼ caþ1=cb�1ð Þ Da=Dbð ÞO�1:The results presented above are related to the aerodynamics of non-premixed
diffusion flames at small speed of fluid and oxidizer. Now we consider some
features of the non-premixed diffusion flames in high velocity flows where the
energy dissipation significantly affects the flame characteristics, such as, for
example, the flame front temperature. We will consider diffusion combustion of
non-premixed gases in the boundary layer formed when a high speed uniform semi-
infinite gaseous fuel flow comes in contact with a semi-space filled with gaseous
oxidizer at rest. Mixing of the gaseous fuel with gaseous oxidizer begins at cross-
section x ¼ 0: The ignition of the reactive mixture occurs by an external source
located at point x ¼ 0; y ¼ 0: As a result of the ignition, the reactive mixture in the
boundary layer forms a thin reaction zone that can be presented as an infinitely thin
flame front. The system of the governing equations describing velocity, enthalpy
and species concentration distribution in diffusion combustion of non-premixed
gases in high speed flows takes the following form (in the boundary layer approxi-
mation after the Dorodnitsyn–Illingworth–Stewartson transformation has been
applied; cf. Sect. 7.7, Chap. 7)
u@u
@xþ ev @v
@�¼ 1
Re
@2u
@�2(9.69)
u@h
@xþ ev @h
@�¼ 1
Pr
@2h
@�2þ g� 1ð ÞM2
þ1@u
@�
� �2
(9.70)
u@cj
@xþ ev @cj
@�¼ 1
Sc
@cf
@�2(9.71)
@u
@xþ @ev@�
¼ 0 (9.72)
where u and v are component of the velocity, cj is the concentration
u ¼ u=uþ1; v ¼ v=uþ1ð Þ ffiffiffiffiffiPr
p; ev ¼ rvþ u@�=@x; h ¼ h=hþ1; h ¼ cPT;
ðcP ¼ constÞ r ¼ r=rþ1; x ¼ x=l�; � ¼ �=l�; x and � are the Dorodnitsy
n variables, Re and Mþ1 are the Reynolds and Mach numbers, respectively.
278 9 Combustion Processes
The boundary conditions corresponding to the case of gaseous fuel issuing into
the oxidizer at rest (u�1 ¼ 0) are posed at the both edges of the mixing layer and
the flame front. They are as following
u ¼ 1; h ¼ 1; ca ¼ 1;@u
@�¼ 0 at � ! þ1
u ¼ 0; h ¼ h�1; cb ¼ 1 at � ¼ �1ca ¼ 0; cb ¼ 0 at � ¼ �f
(9.73)
The conditions (9.73) should be also added to the boundary conditions (9.59) to
determine the location of the flame front. Also, the boundary condition describing
the thermal balance at the flame front is needed. The latter is necessary to determine
the combustion temperature, since in high velocity flow it depends not only on the
heat of reaction but also on the heating due to the energy dissipation. The boundary
condition which expresses the thermal balance at the flame front reads
qrf Daf@ca@n
� �f
þ kf@T
@n
� �f
¼ kf@T
@n
� �f
(9.74)
where q is the heat reaction.
The dimensional analysis shows that there exists a self-similar solution of
(9.69–9.72) subjected to all above-mentioned boundary conditions. The self-similar
solution allows us to reduce the system of the partial differential equations of
the problem to the corresponding system of ODEs for the functions F ’ð Þ), yð’Þand #ð’Þ
Fj000 þ 2FjFj
00 ¼ 0 (9.75)
yj00 þ 2 PrFjyj
0 þ Pr g� 1ð ÞM2þ1 Fj
00� �2 ¼ 0 (9.76)
#j00 þ 2ScFj#j
0 ¼ 0 (9.77)
The boundary conditions for (9.75–9.77) read
F0a ¼ 1; ya ¼ 1; #a ¼ 1;F
00a ¼ 0 at c ! þ1
Fb0 ¼ 0; yb ¼ h�1; #b ¼ 1 at c ! �1y ¼ yf ; #a ¼ #b ¼ 0 at c ¼ cf
(9.78)
where F0j ¼ uj uþ1= ; yj ¼ hj hþ1= ; #j ¼ cj cj�1
;c ¼ � 2
ffiffiffix
p ; x and � are the
Dorodnitsyn variables, and Mþ1 is the Mach number of the undisturbed flow.
The boundary conditions (9.78) should be supplemented by the balance relations
(9.59) and (9.74) that determine the position of the flame front, as well as its
9.5 Diffusion Flame in the Mixing Layer of Parallel Streams 279
temperature. The first of the latter is determined (as in the flow with small velocity)
by (9.68). However, in high velocity flows the temperature of the flame front
depends not only on the physicochemical properties of the reactants but also on
the velocity of the undisturbed flow. The transformation of (9.74) leads to the
following relation for the flame temperature
TfTþ1
¼ 1þ qcaþ1cPTþ1
1
1þ eþ g� 1
2M2
þ1e
ð1þ eÞ2 (9.79)
where g ¼ cp cv= is the ratio of the specific heats at constant pressure and volume,
respectively.
The solution of (9.75–9.77) with the boundary conditions (9.58) that determine
the velocity, enthalpy and reactant and combustion products concentration fields, as
well as the configuration of the flame front and its temperature was found by Vulis
et al. (1968). A similar approach can be also used to study combustion of liquid fuel
in a stream of gaseous oxidizer which blows over its surface, for example, the
combustion of large oil spots (Yarin and Sukhov 1987). Solution of the latter
problem is very similar to the one described above, albeit it involves additional
thermal and mass balance conditions, which are required for calculation of the
temperature and vapor concentration at the free surface.
9.6 Gas Torches
Gas torches represent themselves submerged jets in which the intensive exothermal
chemical oxidation reaction (combustion) proceeds. The conversion of the initial
reactants into combustion products occurs in such jets within a thin high tempera-
ture zone that is identified with the flame front. The thickness of this zone can be
estimated by the dimension consideration. Since the combustion process is deter-
mined by the two general factors, (1) kinetics of chemical reactions and (2)
diffusion, we can assume that the thickness of the reaction zone depends on the
rate constant Z½ � ¼ T�1 of the chemical reaction and diffusivity D½ � ¼ L2T�1
d ¼ f ðZ;DÞ (9.80)
where Z ¼ Z0 exp �E=RTð Þ is the Arrhenius factor, k0 is the pre-exponential, E and
R are the activation energy and the universal gas constant, respectively.
According to the Pi-theorem, because k and D have independent dimensions,
(9.80) takes the form
d ¼ c � Za1Da2 (9.81)
where c is a constant.
280 9 Combustion Processes
Taking into account the dimensions of d; Z and D and applying the principle of
dimensional homogeneity, we find the values of the exponents ai as a1 ¼ �1=2;a2 ¼ 1=2 which transforms (9.81) as follows
d ¼ c
ffiffiffiffiD
Z
r(9.82)
Equation (9.82) shows that the characteristic size of the reaction zone in torches
of non-premixed gases (the diffusion flame) is the order of the flame front thickness
in homogeneous mixtures. Indeed,
d �ffiffiffiffiD
Z
r�
ffiffiffiaZ
r� uf
Z(9.83)
where a is the thermal diffusivity, uf �ffiffiffiffiffiffiaZ
pis the speed of combustion wave in
homogeneous mixtures.
The estimate (9.83) reflects the physical similarity of the processes that occur in
combustion of homogeneous reactive mixtures and in reaction zones of torches of
non-premixed gases. Assuming that the characteristic size of the mixing zone in a
gas torch is l (for submerged torches l is on the order of the boundary layer
thickness), we arrive at the following estimate of the relative thickness of the
reaction zone in diffusion flames
d �ffiffiffiffiffiffiffiD
l2Z
r¼
ffiffiffiffiffitktD
rexp
E
2RT
� �(9.84)
where d ¼ d=l; and tk ¼ Z�10 and tD � l2=D are the characteristic kinetic and
diffusion time, respectively.
It is emphasized that the estimate (9.84) is valid not only in laminar torches but
also in turbulent ones. In the latter case (9.84) implies not the molecular diffusion
and thermal conductivity D and k but rather their turbulent analogs. Equation (9.84)shows that the relative thickness of reaction zone d depends essentially on the
values of the kinetic and diffusion times. If the rate of chemical reaction is large
enough so that tk << tD; the relative thickness of the reaction zone is small
enough, since as tk tD ! 0= ; the value of d ! 0:Combustion temperature also affects significantly the thickness of the reaction
zone. An increase in T (due to combustion of high caloric fuels) is accompanied by
decreasing d: That allows one to assume that the chemical reaction of combustion
proceeds within a very thin (in the limiting case of an extremely small ratio tk tD= ,
in an infinitesimally thin) flame front at a temperature close to the maximum one.
Outside of flame front only the inert transfer of mass, momentum and energy take
plays in submerged laminar and turbulent torches. The latter makes it possible to
apply the method of the theory of the gas jets in studying gas torches (Abramovich
1963; Vulis et al. 1968; Vulis and Yarin 1978).
9.6 Gas Torches 281
One can distinguish two main types of gas torches: (1) torches of premixed
homogeneous mixtures, and (2) torches of non-premixed gaseous fuel and oxidizer.
In the first type of torches, a premixed reactive mixture is supplied directly into the
flame front, whereas in the second one a separate supply of reactants into the
reaction zone occurs. Accordingly, in homogeneous torches combustion process
is determined by the rate of chemical reaction, whereas in torches of non-premixed
gases it is determined predominantly by the mixing rates and to some extent by the
reaction rate. Moreover, in the non-premixed torches, as a rule, the mixing is the
limiting process. Therefore, torches of non-premixed gases are longer and less
intense than the homogeneous ones.
In homogeneous torches combustion is fully completed within the entrance
section of the jet, at a distance of about five nozzle calibers. In this case the
flame front is located near the boundary of the potential core of the jet. Under
such conditions the geometry of a homogeneous torch is determined by the velocity
distribution at the nozzle exit, as well as the speed of combustion wave in homo-
geneous mixture. Thus, the functional equation for the length of homogeneous
torch lf is
lf ¼ f ðu0; uf ; dÞ (9.85)
where u0 ¼ u0ðyÞ is the velocity of the reactive mixture at the nozzle exit, with ybeing transversal coordinate. Equation (9.85) also incorporates the speed of com-
bustion wave uf which represents the physicochemical characteristics of the reac-
tive mixture and can be calculated using the well-known methods of the combustion
theory (Zel’dovich et al. 1985; Williams 1985), and the nozzle diameter d.According to the Pi-theorem, (9.85) reduces to the following dimensionless form
lfd¼ ’
u0uf
� �(9.86)
It is seen that the relative lengths of homogeneous torches depend only on the
ratio of the issue velocity to the speed of combustion wave. At a known uf , thelength and shape of homogeneous torches are found by integrating the following
equation
dx
dy¼ �
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu0uf
� �2
� 1
s(9.87)
Integration of (9.87) in the case of a parabolic velocity profile at the nozzle exit
results in the following expression (Khitrin 1957)
lf � xf ¼ N sin’ cos’D’� 1� k2
k2Fð’;KÞ þ 1þ k
k2Eð’; kÞ
� �(9.88)
282 9 Combustion Processes
where sin2’ ¼ yf ðu� 1Þ; k2 ¼ u� 1ð Þ= uþ 1ð Þ; N ¼ ðu� 1Þ ffiffiffiffiffiffiffiffiffiffiffiuþ 1
p; D’ ¼
1� k2sin2’� �1=2
; u ¼ u0=uf ; y ¼ yf =r0; x ¼ xf =r0; lf ¼ lf =r0; r0 ¼ d=2 is the
nozzle radius, Fð’; kÞ and Eð’; kÞ are the elliptic integral the first and second
kind, respectively; cf. Fig. 9.6.
When the velocity distribution at the nozzle exit is uniform, i.e. u0 ¼ const, andthe outflow velocity of reactive mixture significantly exceeds the velocity of
combustion wave, the length of homogeneous torch is given by
lfd¼ u0
uf(9.89)
This relation shows that the length of laminar homogeneous torches is propor-
tional to the outflow velocity. This result agrees well with the experiments on
combustion in laminar torches of premixed gases, whereas the experimental data
for the turbulent torches show that their length weakly depends on u0, since ufT �un0; with ufTbeing the speed of turbulent flame and n � 0:7� 0:8:
Consider now vertical torches of non-premixed gases. First, we reveal the effect
of flow parameters on the length of such torches. For this aim, we begin with the
simplest example, namely a torch which forms when gaseous fuel is issued from a
cylindrical nozzle into a gaseous medium containing oxidizer, which is at rest far
x
y
u0
uf
q
Fig. 9.6 Sketch of a
homogeneous mixture torch
9.6 Gas Torches 283
away in the transverse direction from the nozzle. Assuming that the velocity of the
fuel jet is large enough, we neglect the any possible effect of buoyancy forces,
which might arise from the disparity of fuel and oxidizer densities. We also neglect
the effect of density change and assume that r is constant. At fixed initial
concentrations of fuel in the gas jet and oxidizer in the ambient medium, we state
the functional equation for the torch length lf as follows
lf ¼ f ðu0;D; dÞ (9.90)
Bearing in mind that gas torches features stem from those of submerged jets, we
define the dimensions of the governing parameters in the system of units LxLyMT
u0½ � ¼ LxT�1; D½ � ¼ L2yT
�1; d½ � ¼ Ly (9.91)
Since all the governing parameters possess independent dimensions, (9.91)
reduces to
lf ¼ cua10 Da2da3 (9.92)
where c is a dimensionless constant that depend on concentrations of the fuel in
the gas jet cg and oxygen in the ambientcox, as well as on stoichiometric fuel-to-
oxygen mass ratio O. The latter follows directly from the functional equation
for the torch length written with the account of the dimensional ( u0½ � ¼ LxT�1;
D½ � ¼ L2yT�1; d½ � ¼ Ly) and dimensionless ( O½ � ¼ 1; cg
� ¼ 1; cox½ � ¼ 1) charac-
teristics. Since the dimension of lf� ¼ Lx can be constructed as a combination
of the dimensions of the parameters u0;D and d, such an equation acquires the
form of lf ¼ ua10 Da2da3 f ðO; cg; coxÞ, i.e. the form of (9.92) where the constant
c ¼ f ðO; cg; coxÞ.Taking into account that the dimension of the torch length lf
� ¼ Lx and
applying the principle of the dimensional homogeneity, we find the values of the
exponents ai as a1 ¼ 1; a2 ¼ �1 and a3 ¼ 2: Accordingly, we obtain that in the
case of n ¼ D
lf ¼ cRe (9.93)
where lf ¼ lf d= and Re ¼ u0d=n is the Reynolds number.
It is seen that the length of laminar axisymmetric torches of non-premixed gases
(with n ¼ const) is proportional to the Reynolds number that is defined by the
outflow velocity and the nozzle diameter, whereas the length of turbulent torches
(with nT � u0d) lf does not depend on Re as follows from (9.93). (Hottel,
Hawthorne, 1949) The experimental data shown in Fig. 9.7 agree with these
predictions.
The length of non-premixed gases torches is large enough to assume lf >> 1:Then, the characteristics of non-premixed diffusion flames practically do not
depend on the outflow conditions at the nozzle exit and are determined by the
284 9 Combustion Processes
integral parameters of the flow: the total momentum flux and mass fluxes of species.
In order to determine these characteristics, we use the continuity, momentum and
species balance equations in the boundary layer form
@ruyk
@xþ @rvyk
@y¼ 0 (9.94)
ru@u
@xþ rv
@u
@y¼ 1
yk@
@yykt� �
(9.95)
ru@b@x
þ rv@b@y
¼ 1
yk@
@yykg� �
(9.96)
where t and g are the shear stress and the compound species flux, respectively, the
compound concentration b ¼ ca � cb þ 1; with c1 and c2 being the fuel and oxygenconcentration, respectively, ca ¼ caO=cb1; cb ¼ cb=cb1: Also, O is the stoichio-
metric fuel-to-oxygen mass ratio, subscript1 refers to the ambient conditions, k ¼0 and 1 for the plane and axisymmetric torches, respectively. It is emphasized that
(9.96) for the compound concentration b was obtained from the species balance
equations by using the Schvab–Zel’dovich method, which allows us to exclude the
source terms associated with the rate of chemical reaction.
The boundary conditions for (9.95) and (9.96) read
IIIIII
Re0
f
Fig. 9.7 The dependence lf ðReÞof the length of non-premixed torches on the Reynolds number.
I-laminar torches, II-transitional torches and III-turbulent torches
9.6 Gas Torches 285
u ! 0 b ! 0 at y ! 1;@u
@y¼ @b
@y¼ 0 at y ¼ 0 (9.97)
for submerged torches, and
u ¼ 0 b ¼ 0 at y ! 1; u ¼ 0@b@y
¼ 0 at y ¼ 0 (9.98)
for the wall torches issued from a slit parallel to the adjacent wall (in the case k ¼ 0
only).
By integrating this system of equations with the boundary conditions (9.97), we
arrive (at r ¼ const) the following integral invariants for the free torches
ð10
u2ykdy ¼ eJx (9.99)
ð10
ubykdy ¼ eGx (9.100)
where eJx and eGx are constant determined by the conditions at the nozzle exit and,
thus, are given. It is emphasized that the invariant (9.100) stems from the fact that
the compound concentration b is described by (9.96) which does not involve any
term related to the rate of chemical reaction.
Assuming that the longitudinal convective species fluxes within the far field of
non-premixed torches are determined by the invariants eJx and eGx, we formulate the
functional equation for (ubÞm as follows
ðubÞm ¼ f ðeJx; eGxÞ (9.101)
where subscript m refers to the torch axis.
Taking into account that the dimensions of eJxh i¼ L2xL
1þky T�2 andeGx
h i¼ LxL
1þky T�1 are independent, we arrive at the simpler form of (9.101)
umbm ¼ ceJxfGx
(9.102)
where c is a constant.The dimensional analysis of the flow in the far field of submerged jets shows
(Chap. 6) that the problem under consideration based on (9.95) and (9.96) has the
following self-similar solutions
u
um¼ ’ð�Þ; b
bm¼ cð�Þ; um ¼ Axa (9.103)
286 9 Combustion Processes
where � ¼ y=d; d is the thickness of the boundary layer of the jet, and A and a are
constants, the functions ’ð�Þ and cð�Þ are determined by a system of the ordinary
differential equations that can be found transforming (9.94–9.96) using the Pi-
theorem.
The substitution of the expression (9.103) for the axial velocity um into (9.102)
yields
x�a ¼ bmAc1fGmfJm (9.104)
where c1 ¼ c�1 is the constant.
Taking in (9.104) x ¼ lf and accordingly bm ¼ 1 which corresponds to the
complete combustion at the flame tip, we obtain the following expression for the
length of non-premixed gas torch
l�af ¼ Ac1
fGxeJx (9.105)
Using the known expressions for the coefficients A and a (see Chap. 6) and
accounting for the fact that fGx ¼ pku0b0lkþ1 (where l0 is the nozzle characteristic
size and b0 is the given compound concentration value at the nozzle exit), we arrive
at the following relations for the dimensionless length of torches of non-premixed
gases of different types (Table 9.1). It is seen from Table 9.1 that the length of
laminar non-premixed torches depends on the issue velocity (through the Reynolds
number Re) and parameter e ¼ 1þ ðca0=cb0ÞO accounting for the reactant
concentrations and the stoichiometric fuel-to-oxygen mass ratio. On the other
hand, the length of turbulent torches depends only on the ratio of reactant
concentrations and the stoichiometric fuel-to-oxygen mass ratio. The constants
Ciin the expressions for lf in Table 9.1 depend on the type of the jet flow (free
or wall flow), its geometry (plane or axisymmetric), as well as on the flow
regime (laminar or turbulent). It is emphasized that the dependence of lf on the
reactant concentrations is different for different types of non-premixed torches.
For example, in plane laminar torches lf � e3, whereas in the axisymmetric ones
lf � e. That results from the different mixing intensity in various types of
submerged torches.
Table 9.1 Length of
diffusion torches of different
geometry
Type of flow lfPlane laminar torch C1Ree3
Plane laminar wall-torch over an adiabatic plate C2Ree4
Axisymmetric laminar torch C3ReePlane turbulent torch C4e2
Axisymmetric turbulent torch C5e
9.6 Gas Torches 287
9.7 Immersed Flames
Consider an immersed diffusion flame formed by a jet of gaseous oxidizer issuing
into liquid reagent (fuel); cf. Fig. 9.8. As a result of the gas–liquid interaction,
namely its breakup into bubbles and atomization of liquid in the form of liquid
droplets, a two-phase jet-like flow forms in the liquid medium.
The general characteristics of combustion process in this situation, such as the
intensity of heat release, completeness of combustion, etc., are determined to a
considerable extent by the volumetric content of the gaseous phase mv: Dependingon the value of mv, the immersed jet flow acquires a form of either bubbly or gas-
droplet jets (Abramovich et al. 1984).
At a high initial temperature of reactants (or due to the presence of an external
igniter, chemical reaction between fuel and oxidizer begins. In consequence of
heating and vaporization of liquid reagent due to the chemical reaction between fuel
vapor and the oxidizer supplied as the gas jet, recognizable domains filled with
vapor–oxidizer mixture containing droplets of reactive liquid (fuel) or bubbles
filled by the oxidizer/fuel vapor mixture are formed in the liquid. A high tempera-
ture zone-a diffusion flame is formed in the two-phase mixture. In the case when
combustion products are gaseous, the diffusion flame is located in an open cavity
(Fig. 9.8a). On the other hand, when combustion products represent an immiscible
liquid, the diffusion flame is located in a closed gas cavity (Fig. 9.8b). Its size
depends not only on the physicochemical properties of reactants and the intensity of
their mixing but also on the boiling temperature of combustion products that
determine the position of the condensation zone. Sukhov and Yarin (1981, 1983)
developed a theory of laminar immersed flames in the case when combustion
products are gaseous. In this case, assuming that the rate of chemical reaction is
infinite, the thermal conductivity and diffusion coefficients are constant and the
effect of buoyancy force is negligible, the problem reduces to the following system
of equations
f1
f1
2
2
x
y
x
y0 0
a b
Fig. 9.8 Sketch of the immersed flame. 1: Gaseous phase (a mixture of vapor of reactive liquid,
gaseous oxidizer and combustion products). 2: Reactive liquid
288 9 Combustion Processes
ui@ui@x
þ vi@ui@y
¼ niyk@
@yyk@ui@y
� �(9.106)
@
@xðuiykÞ þ @
@yðviykÞ ¼ 0 (9.107)
ui@Ti@x
þ vi@Ti
@y¼ aiyk
@
@yyk@Ti@y
� �(9.108)
ui@cj@x
þ vi@cj@y
¼ Djyk @
@yyk@cj@y
� �(9.109)
Xj
cj ¼ 1 (9.110)
with the boundary conditions
y ¼ 0@ui@y
¼ 0; vi ¼ 0;@ca@y
¼ @cb@y
¼ 0; ca ¼ 0;@T1@y
¼ 0
y ¼ yf ðxÞ ca ¼ cb ¼ 0; cc ¼ 1; T ¼ Tf
y ¼ y�ðxÞ u1 ¼ u2 ¼ u�;@u1@y
¼ r21n21@u2@y
; cb ¼ 0; T1 ¼ T2 ¼ T�
y ! 1 u2 ¼ 0; T2 ! T21
(9.111)
where x and y are longitudinal and transverse coordinates, u and v are the longitu-dinal and transverse velocity components, T and cj are the temperature and concen-
tration, n; a and Dj are the kinematic viscosity, thermal and species diffusivities,
respectively, k ¼ 0 or 1 for the plane and axisymmetric flows, subscripts f and �correspond to the combustion front and the liquid–gas interface, subscript 1 and
2 refer to the gaseous and liquid phases, and j ¼ a; b and c correspond to the fuel,
oxidizer and combustion products, respectively, r21 ¼ r2 r1 and n21 ¼ n2 n1:==The conditions (9.111) should be also supplemented with the Clausius–
Clapeyron equation determining the equilibrium concentration of vapor at the
liquid–gas interface, as well as by the mass and thermal balances at the flame and
the interface. The Clausius–Clapeyron equation reads
ca� ¼ w exp � qeRaT�
� �(9.112)
where ca� is the vapor concentration at the interface, w is the pre-exponential factor,
qe is the heat of evaporation.
9.7 Immersed Flames 289
In the framework of the boundary layer theory, the slope of the flame front to the
flow axis is sufficiently small, so that @=@nð Þf � @=@yð Þf ; with n being the normal
to the flame front. Then assuming the Lewis number Le ¼ 1; we obtain the balanceconditions at the flame front and the liquid–gas interface in the following form
@cb@y
� �f
þ O@ca@y
� �f
¼ 0 (9.113)
@T
@y
� �af
� @T
@y
� �bf
þ q
rcP
@ca@y
� �f
¼ 0 (9.114)
v1cc� þ D@cc@y
� ��¼ 0 (9.115)
k1@T1@y
� ��� k2
@T2@y
� ��þ r1D
@ca@y
� ��¼ 0 (9.116)
r2v2� � r1v1�ca� þ r1D@ca@y
� ��¼ 0 (9.117)
where k and cP are the thermal conductivity and specific heat, respectively, and q isthe heat of reaction.
The conditions (9.111) should be supplemented with the integral invariants that
are needed to obtain a non-trivial solution. Using the Schvab–Zel’dovich variable
and integrating (9.106) and (9.109) across the immersed jet, we arrive at the
following invariants
ðy�0
u21ykdyþ r21
ð1y�
u22ykdy ¼ Ix ¼ const (9.118)
ðy�0
u1bykdy ¼ Gx ¼ const (9.119)
where b ¼ 1þ Dc; Dc ¼ cb O= � ca and r21 ¼ r2 r1= .
Introducing the dimensionless variables ui ¼ ui=u0 and y ¼ y=y0, we transform(9.118) and (9.119) as follows
ðy�0
u2i ykdyþ r21
ð1y�
u22ykdy ¼ 1 (9.120)
290 9 Combustion Processes
Zy�0
u1bykdy ¼ 1 (9.121)
where u0 and y0 are the scales of velocity and length defined as u0 ¼ Ix Gx=ð Þ andy0 ¼ G2
x Ix=� �1= kþ1ð Þ
.
The totality of the boundary conditions (9.111), balance equations (9.113–9.117)
and the integral relations (9.120) and (9.121) fully determines the problem on the
immersed diffusion torch. It allows finding the profiles of all characteristic para-
meters, values of temperature at the flame front, the temperature and concentrations
at the interface surface, as well as the lengths and configurations of the flame front
and gaseous cavity (Yarin and Sukhov 1987). However, instead of describing the
theoretical solution of (9.106–9.110), following the approach of the present book, we
focus our attention at the calculation of the length and configuration of the immersed
diffusion flame using the dimensional analysis. For this aim, we use the approach
developed inVulis et al. (1968) andVulis andYarin (1978) for the investigation of the
aerodynamics of diffusion flames. It consists in the calculation of the axial velocity
and concentration and then determining the flow parameters corresponding to the
flame front. In the framework of the model of an infinitely thin flame front
(corresponding to the assumption on an infinitely large rate of chemical reaction),
the length of the immersed flame is found from the following conditions: ca ¼ cb ¼ 0
at x ¼ lf , with subscript f corresponding to the tip of the torch.In order to determine the velocity in the immersed flame, we use the approach
developed in Chap. 6 for the free laminar jets. We write the following functional
equations for the local velocity u; the axial velocity um, and the thickness of
gaseous cavity y�
ui ¼ fiðum; y; y�Þ (9.122)
um ¼ fmðIx; n1; xÞ (9.123)
y� ¼ f�ðIx; n1; xÞ (9.124)
where subscripts i ¼ 1 and 2 correspond to the gaseous and liquid phases,
respectively.
Note that the kinematic viscosity of liquid n2; as well as densities of both phasesr1 and r2 are not included in the sets of the governing parameters in (9.122–9.124)
because they are accounted for in the expression for the kinematic momentum Ix:To transform (9.122–9.124) to the dimensionless form, we use the LxLyLzMT
system of units. The dimensions of the parameters involved in (9.122–9.124) in this
system of units are as follows
ui½ � ¼ LxT�1; um½ � ¼ LxT
�1; Ix½ � ¼ L2xLyT�2; n1½ � ¼ L2yT
�1 y½ � ¼ Ly; x½ � ¼ Lx
(9.125)
9.7 Immersed Flames 291
It is seen that two governing parameters in (9.122) have independent
dimensions, whereas the dimensions of the governing parameters in
(9.123–9.124) are independent. Then, according to the Pi-theorem, (9.122–9.124)
take the form
ui ¼ umci ’ð Þ (9.126)
um ¼ Aixe (9.127)
y� ¼ Bxg (9.128)
where Ai ¼ ciu�10 I2x d
kþ10 =nkþ1
1
� �1= 3�kð Þ; B ¼ c� n2d20=Ix
� �1= 3�kð Þ; e ¼ � k þ 1ð Þ=
3� kð Þ, g ¼ 2= 3� kð Þ; ui ¼ u=u0; um ¼ um=u0; y� ¼ y=y0; ’ ¼ y=y�; ci and c�are constants.
Equations (9.126–9.128) show that the system of PDEs determining the velocity
distribution in the immersed laminar torch can be reduced to a system of ODEs.
Therefore, the velocity can be expressed as a function cð’Þ of a single variable
cið’Þ ¼F
0ið’Þ’k
(9.129)
where the function Fð’Þ should to satisfy the following conditions
’ ¼ 0F
01
’k
� �¼ 0;
F01
’k¼ 1; F1 ¼ 0
’ ¼ 1 F01 ¼ n21F
02;
F01
’k
� �0
¼ r21n21F
02
’k
� �0
’ ! 1 F02
’k! 0;
F02
’k
� �0
! 0
(9.130)
Here primes denote differentiation with respect to ’; and n21 ¼ n2 n1=ð Þ.Determine the concentration distribution. Since the physically realistic self-
similar solution of (9.109) is absent under the condition ca� ¼ const, we use the
integral method of calculation distribution of b: Approximate the actual concentra-
tion profile by the series
b ¼X1n¼0
anðxÞy (9.131)
where the coefficients anare found from the conditions at the flow axis and the
interface surface
292 9 Combustion Processes
y ¼ 0 db dy= ¼ 0; b ¼ bm; y ¼ y�ðxÞ b ¼ b� (9.132)
Taking into account three terms of the series (9.131), we arrive at the expression
b ¼ bmð1� ’2Þ þ b�’2 (9.133)
Using (9.121) and (9.133), we find the dependence bmðxÞ as
bm ¼ x�kþ13�kðA1B
kþ1Þ�1 � b�I1n o
F1ð1Þ � I1½ ��1(9.134)
where I1 ¼Ð10
’2F01ð’Þd’.
Bearing in mind that at the flame front bf ¼ 1, and that at the tip of the diffusion
flame bm ¼ 1, we arrive at the following equations for the shape and length of the
immersed torch
yf ¼ Bx2= 3�kð Þ bm � 1
bm � b�
� �1=2
(9.135)
lf ¼ ðA1Bkþ1Þ�1 � b�I1
n o 1�kð Þ= 1þkð ÞF1ð1Þ � I1ð1� b�Þ½ � k�3ð Þ= kþ1ð Þ
(9.136)
where lf is the length of the immersed flame, and subscript f corresponds to the
flame front.
The correlations (9.135) and (9.136) are qualitative, since they contain factors
A and B that incorporate the unknown constants ci and c�, as well as the vapor
concentration at the interface surface b�: The latter can be determined from the
Clapeyron–Clausius equation for a given temperature at the interface T�: The actualvalues of the factors Aand B are found from (9.106) and (9.107) after substituting
into these equations the expressions (9.127–9.129)
Ai ¼ ðI2 þ r21n221I3Þ�2= 3�kð Þ Re
6� 5k
� � 1þkð Þ= 3�kð Þ;A2 ¼ n21A1
B ¼ ðI2 þ r21n221I3Þ1= 3�kð Þ Re1
6� 5k
� ��2= 3�kð Þ(9.137)
where I2 ¼R10
F01
� �2=’kd’; I3 ¼
R11
F02
� �2=’kd’; and Re1 ¼ u0y0=n1.
Accordingly, the expression (9.136) takes the form
lf ¼ Re1
6� 5kðI2 þ r21n
221I3Þ 1�kð Þ= 1þkð Þ F1ð1Þ � I1ð1� b�Þ½ �ðk�3Þ kþ1Þ=
(9.138)
9.7 Immersed Flames 293
The effect of various parameters on the characteristics of the immersed flames is
clearly visible through the dependence of its length on the reactants temperatures,
the issue velocity, composition of gaseous oxidizer, etc. In particular, an increase
in the reactants temperatures is accompanied by shortening of the immersed
torch length. This results from an increase in vapor concentration the interface
(a decrease in b�Þ and an increase of the term I3ð1� b�Þ in (9.138). An opposite
effect takes place at increasing the latent heat of evaporation: an increase in the
value of qe leads to a significant growth of the torch length lf :As with the other types of diffusion flames, the characteristics of the immersed
flames depend on the flow geometry. For example, the length of the plane (k ¼ 0)
and axisymmetric (k ¼ 1) flame is inversely proportional, respectively, to the third
and first powers of the factor F1ð1Þ � I3ð1� b�Þ½ � which accounts for the effect of
vapor concentration at the interface surface. It is emphasized that in both cases the
length of the immersed flames is directly proportional to Re: A possible presence of
an inert admixture in the gas jet also affects the flame characteristics (Sukhov and
Yarin 1983, 1987). An increase in the content of an inert admixture in the gaseous
oxidizer jet is accompanied by a significant decrease in the length of the axisym-
metric and plane flames (cf. Fig. 9.9).
Problems
P.9.1. Evaluate the burning time of a liquid fuel droplet at its combustion in
a stagnant atmosphere that contains gaseous oxidizer.
The process of droplet burning involves a number of simultaneously happening
physical processes such as liquid vaporization, mixing of gaseous reagents and
combustion of the resulting vapor–oxidizer mixture. These processes are also
accompanied by heat and mass transfer, as well as by the diminishment of the
droplet size and surface and the corresponding displacement of the reaction zone.
Accordingly, a theoretical description of droplet combustion implies solving the
coupled non-steady equations governing the mass, momentum, energy and species
transfer in the liquid and gaseous phases (Yarin and Hetsroni 2004).The non-linear
terms accounting for the heat release and species consumption involved in these
equations make the theoretical analysis of the problem extremely difficult.
Fig. 9.9 The effect of an
inert admixture of the length
of plane (k ¼ 0) and
axisymmetric (k ¼ 1)
immersion flames
294 9 Combustion Processes
Therefore, as a rule, droplet burning is studied using models based on a number of
simplifying assumptions: (1) the chemical reaction rate is infinite, (2) the droplet
temperature is uniform, (3) the effects of buoyancy and radiant heat transfer are
negligible, (4) the Lewis number equals one, (5) the physical properties of the liquid
and gaseous phases are constant.
The assumption (1) allows one to consider the reaction zone as an infinitesimally
thin flame front which separates the flow field into two domains: the inner one (near
the droplet surface) filled with a mixture of the fuel vapor and combustion products,
and the outer one filled with a mixture of the oxidizer and combustion products.
Also, in this case the vapor and oxidizer concentrations at the flame front are equal
to zero. The other assumptions make it possible to use the lumped capacitance heat
transfer model for the droplet, as well as to consider a spherically-symmetric flame
(for the burning in a stagnant atmosphere). The flame temperature equals the
adiabatic combustion temperature in this case. Moreover, the droplet surface tem-
perature changes only slightly during the combustion process and can be taken as
a constant equal to the boiling temperature of liquid fuel.
Based on the above-mentioned simplifications, assume that during the combus-
tion process the droplet diameter d depends on: (1) densities of liquid fuel and
gaseous oxidizer, (2) species diffusivity (assumed being identical for all the
components), (3) total enthalpy of fuel, (4) latent heat of evaporation, (5) droplet
initial diameter, and (6) time
d ¼ f ðr1; r2;D; qt; qe; d0; tÞ (P.9.1)
Here r1 and r2 are the density of gaseous and liquid phases, respectively, D is
the diffusivity, qt ¼ c01q=Oþ cPðT1 � TsÞis the total enthalpy of the fuel with qbeing the heat of reaction, c01 the oxidizer concentration in the surrounding
medium, O the stoichiometric oxidizer-to-fuel mass ratio, qe the latent heat of
evaporation, cP the specific heat of gaseous phase, T1 and Ts being the ambient
and saturated temperature, respectively; d0 is the initial droplet diameter, and t istime.
The dimensions of the governing parameters involved are
½r1� ¼ L�3M; ½r2� ¼ L�3M; ½D� ¼ L2T�1; ½qt� ¼ JM�1;
½qe� ¼ JM�1; ½d0� ¼ L; ½t� ¼ T (P.9.2)
Five of the seven governing parameters possess independent dimensions. Then,
according to the Pi-theorem, the number of dimensionless groups of the present
problem is equal two, and (P.9.1) reduces to the following dimensionless equation
P ¼ ’ðP1;P2Þ (P.9.3)
where P ¼ d=d0; P1 ¼ r1=r2ð Þ Dt=d20� �
and P2 ¼ qt=qe ¼ B (B is the Spalding
transfer number.
Problems 295
Droplet is burnt completely at the moment t ¼ tb when d ¼ 0. At that moment
(P.9.3) yields ’ðP1b;P2Þ ¼ 0; where P1b ¼ r1=r2ð Þ Dtb=d20
� �: Solving the latter
equation for P1b, we obtain the following expression for the droplet burning time
tb ¼ r2r1
d20DcðBÞ (P.9.4)
It is emphasized that the analytical solution of the problem yields the following
expression for tb
tb ¼ r2r1
d208D
lnð1þ BÞ½ ��1(P.9.5)
References
Abramovich GN (1963) The theory of turbulent jets. MTI Press, Cambridge
Abramovich GN, Girshovich TA, Krasheninnikov SY, Sekundov AN, Smirnova IP (1984) Theory
of turbulent jets. Nauka, Moscow (in Russian)
Burke SP, Schumann TE (1928) Diffusion flames. Ind Eng Chem 20:998–1004
Frank-Kamenetskii DA (1969) Diffusion and heat transfer in chemical kinetics, 2nd edn. Plenum
Press, New York
Hottel HC, Hawthorne WR (1949) Diffusion in laminar flame jets. In: Proceeding of third
symposium on combustion and flame and explosion phenomena, Williams & Wilkins,
Baltimore, pp 254–266
Khitrin LN (1957) The physics of combustion. Moscow University, Moscow (in Russian)
Merzhanov AG, Khaikin BI (1992) Theory of combustion waves in homogeneous media. AN
SSSR, Chernogolovka (in Russian)
Merzhanov AG, Khaikin BI, Shkadinskii KG (1969) The establishment of a steady-state regime of
flame propagation after gas ignition by an overheated surface. Prikl Mech Tech Phys 5:42–48
Schvab BA (1948). A relation between temperature and velocity fields in gaseous flame. In: The
investigation of the process of fossil fuel combustion, Gosenergoizdat, Moscow-Leningrad, pp
231–248 (in Russian)
Semenov NN (1935) Chemical kinetics and chain reactions. Oxford University Press, Oxford
Spalding DB (1953) Theoretical aspects of flame stabilization: an approximate graphical method
for the flame speed of mixed gases. Aircraft Eng 25:264–276
Sukhov GS, Yarin LP (1981) Combustion of a jet of immiscible fluids. Combust. Explos. Shock.
Waves 17:146–151
Sukhov GS, Yarin LP (1983) Calculating the characteristics of immersion burning. Combust.
Explos. Shock waves 19:155–158
Vulis LA (1961) Thermal regime of combustion. McGraw-Hill, New York
Vulis LA, Yarin LP (1978) Aerodynamics of a torch. Energia, Leningrad (in Russian)
Vulis LA, Ershin SA, Yarin LP (1968) Foundations of the theory of gas torches. Energia,
Leningrad (in Russian)
Williams FA (1985) Combustion theory, 2nd edn. Benjamin-Cummings, Menlo Park
Yarin LP, Hetsroni G (2004) Combustion of two-phase reactive media. Springer, Berlin
Yarin LP, Sukhov GS (1987) Foundations of combustion theory of two-phase media.
Energoatomizdat, Leningrad (in Russian)
Zel’dovich YB (1948) Toward a theory of non-premixed gas combustion. J Tech Phys 19:199–210
Zel’dovich YB, Barenblatt GI, Librovich VB, Makhviladze GM (1985) Mathematical theory of
combustion and explosion. Plenum Press, New York
296 9 Combustion Processes
Author Index
A
Abrahamsson, H., 259
Abramovich, G.N., 131, 156, 232, 237, 238,
245, 258, 281, 288, 296
Acrivos, A., 169, 175, 209
Adamson, T.C., 102
Adler, M., 119, 120, 129
Adrian, R.J., 102
Akatnov, N.I., 146, 148, 156
Alhama, F., 7, 38
Andrade, E.N., 143, 156
Andreopoulos, J., 245, 258
Antonia, R.A., 228, 229, 238, 258, 259
Anton, T.R., 80. 101
Armstrong, R.C., 129
Astarita, G., 113, 129
B
Baehr, H.D., 39, 69, 201, 209
Bagananoff, D., 157
Bahrami, M., 113, 129
Baines, W.D., 154, 156, 245, 259
Balachander, R., 260
Banks, R.B., 258
Banks, W.H.H., 170, 209
Barenblatt, G.I., 7, 23, 37, 217, 258, 296
Barua, S.N., 119, 129
Basset, A.B., 20, 101
Batchelor, G.K., 54, 69, 71, 101, 149, 153, 156
Bayazitoglu, Y., 139
Bayley, F.J., 179, 209
Bearman, P.W., 76, 81, 101
Berger, S.A., 118, 129
Bergstorm, D.J., 249, 258, 260
Berlemont, A., 80, 101
Bernulli, D., 10
Bigler, R.W., 238, 258
Bird, R.B., 113, 129
Blackman, D.R., 4, 37
Blasius, H., 47, 69
Bloonfield, L.J., 154, 156
Boothroyt, R.G., 84, 101
Boussinesq, J., 80, 101
Bradbury, L.I.S., 238, 243, 244, 258
Brenner, H., 71, 102
Bridgmen, P.W., 11, 23, 37, 87, 101, 166,
167, 209
Britter, R.E., 63, 69
Buchanan, H.J., 75, 102
Buckingham, E., 23, 37
Burke, S.P., 271, 296
C
Campbelle, I.H., 156
Carpenter, L.H., 81, 102
Celata, G.P., 112, 129
Champagne, F.H., 147, 149, 157
Chandrasekhara, D.V., 258
Chang, E.J., 80, 101
Chao, B.T., 170. 209
Chassaing, P., 245, 258
Chen, A.M.L., 113, 114, 116, 130
Chen, C.S., 105, 129
Cheslak, F.R., 257, 259
Chevray, R., 228, 229, 259
Cho, Y.I., 209
Chua, L.P., 226, 229, 259
Claria, A., 258
Clauser, F.H., 222, 259
Clift, R., 71, 78, 101
Coles, D., 220, 259
Corrsin, S., 228, 229, 259
L.P. Yarin, The Pi-Theorem, Experimental Fluid Mechanics,
DOI 10.1007/978-3-642-19565-5, # Springer-Verlag Berlin Heidelberg 2012
297
Crawford, M.E., 39, 69, 183, 209
Culham, J., 129
D
Dandy, D.S., 80, 101
Dean, W.R., 118, 129
Derjagin, B.M., 96, 101
Desjanquers, P., 101
De Witt, K.J., 209
Desjonquers, P., 101
Didden, N., 63.69
Dimotakis, P.E., 236, 259
Divoky, D., 228, 259
Dodson, D.S., 102
Dorfman, L.A., 170, 209
Dorodnizin, A.A., 197, 209
Douglas, J.F., 7, 38, 45, 69
Doweling, D.R., 236, 259
Dowirie, M.J., 101
Du, D.X., 130
Dunkan, A.B., 112, 129
Dwyer, H.A., 80, 82, 101, 102
E
Eastor, T.D., 171, 209
Emery, A.E., 105, 129
Eriksson, J., 259
Ershin, S.A., 296
Eskinazi, S., 259
Everitt, K.M., 238, 243–245, 259
F
Fargie, D., 105, 129
Fendell, F.E., 82, 102
Flakner, V.M., 68, 69
Forstall, W., 229, 259
Frankel, N.A., 175, 209
Frank-Kamenetskii, D.A., 263, 264, 266, 268,
270, 271, 296
Fric, T.F., 245, 259
Friedman, M., 105, 129
Fujii, T., 201, 209
G
Gad-el-Hak, M., 112, 129
Garimella, S., 112, 129
Gaylord, E.M., 229, 259
George, J., 249, 258
George, W.K., 249, 259
Germano, M., 120, 129
Gilis, J., 129
Girshovich, T.A., 258, 296
Glauert, M.B., 146, 148, 157
Gloss, D., 130, 259
Couesbet, G., 101
Grace, J.R., 101
Graham, J.M.R., 76, 81, 101, 102
Greif, R., 170, 209
Gua, Z.Y., 130
Gutmark, E., 133, 157, 233, 234, 253, 254, 259
H
Hadamard, J.S., 85, 102
Hagen, G., 103, 129
Hamel, G., 43, 69
Hamilton, W.S., 80, 102
Happel, J., 71, 102
Hartnett, J.P., 209
Hassager, O., 129
Hasselbrink, E.F., 245, 246, 259
Hausner, O., 112, 130
Hawkis, G.A., 30, 38
Hawthorne, W.R., 284, 296
Herwig, H., 110, 112, 129, 130, 224, 259
Hetsroni, G., 71, 102, 112, 113, 130, 294, 296
He, Y.-L., 130
Hinze, J.O., 28, 38, 131, 157, 217, 229, 259
Ho, C.-M., 112, 130, 133, 157
Hollands, K.G.T., 187, 209
Hottel, H.C., 284, 296
Hoult, D.P., 60, 69
Howarth, L., 215, 259
Huntley, H.E., 7, 38, 45, 69, 88, 102
Huppert, H.E., 60, 61, 63, 69
Hussain, F., 133, 157
Hussain, H.S., 133, 157
Hussaini, M.Y., 171, 209
I
Illigworth, C.R., 197, 209
Incorpera. F.P., 112, 130
Ipsen. D.C., 4, 38
Ito. H., 119, 120, 130
J
Jaluria, Y., 149, 157
Jeng, D.R., 209
Jones, J.B., 30, 38
Jonicka, J., 260
298 Author Index
K
Kakas, S., 112, 130
Kandlicar, S.G., 110, 130
Karamcheti, K., 157
Karanfilian, S.K., 80, 102
Karlsson, R.I., 250, 259
Karman, Th., 55, 69, 215, 221, 225, 226, 259
Karthpalli, A., 133, 157
Kashkarov, V.P., 54, 70, 131, 133, 147, 157,
228, 260
Kassoy, D.R., 82, 102
Kaviany, M., 159, 209
Kays, W.M., 39, 69, 159, 183, 209
Keffer, J.F., 245, 259
Kelso, R.M., 245, 259
Kenlegan, G.H., 81, 102
Kerr, R.C., 154, 156
Kestin, J., 9, 30, 38, 172, 209
Khaikin, B.I., 271, 296
Khitrin, L.N., 282, 296
Kolmogorov, A.N., 211, 259
Kompaneyets, A.S., 162, 210
Konsovinous, N.S., 136, 157
Kotas, T.J., 80, 102
Krashenninikov, S.Yu., 258, 296
Kreith, F., 169, 170, 209
Kuta teladse, S.S., 18, 38, 158, 209
L
Lamb, H., 83, 90, 102
Landau, L.D., 37, 38, 52, 53, 54, 69, 71, 94,
95, 98, 102, 103, 130, 133, 157, 159,
173, 209, 216, 219, 259
Launder, B.F., 249, 259
Lavender, W.J., 172, 209
Lawerence, C.J., 102
Lee, M.H., 170, 209
Levich, V.G., 29, 38, 65, 69. 94, 95, 102,
159, 177, 189, 193, 194, 207–209
Levi, S.M., 96, 101
Librovich, V.B., 296
Li, D., 130
Lifshitz, E.M., 37, 38, 54, 69, 71, 98, 102,
103, 130, 133, 157, 159, 173, 209,
216, 219, 259
Lim, T.T., 259
Liron, N., 129
List, E.J., 154, 157
Li, Z., 113, 130
Li, Z.X., 113, 139
Lockwood, F.C., 234, 235, 259
Lofdahl, L., 259
Loitsyanskii, L.G., 22, 28, 38, 42, 43, 69,
75, 101–103, 105, 112, 120. 122,
123, 130, 195,,197, 209, 269
London, A.L., 111, 130
Lumley, J.L., 232, 239–241, 260
Lykov, A.M., 18, 38
M
Maczynski, J.F.J., 238, 259
Madrid, C.N., 7, 38
Ma, H.B., 111, 130
Makhviladze, G.M., 296
Mala, G.M., 130
Marrucci, G., 113, 129
Martin, B.W., 105, 129
Maxey, H.R., 80, 101, 102
Maxworthy, T., 63, 69
Mayer, E., 228, 259
Mc Laughlin, J.B., 80, 102
Mei, R., 80, 102
Merzhanov, A.G., 271, 296
Messiter, L.F., 102
Mikhailov Yu,A., 18, 38
Maneib, H.A., 234, 235, 259
Monin, A.S., 217, 220, 260
Moody, L.F., 111, 130
Mori, Y., 119, 130
Morton, B.R., 149, 153, 154, 157
Mosyak, A., 130
Moussa, Z.M., 245, 259
Mungal, M.G., 245, 246, 259, 260
N
Nakayama, W., 119, 130
Narasimha, R., 251, 260
Narasyan, K.Y., 260
Nicholles, J.A., 259
Nickels, T.B., 238, 260
Nikuradse, J., 109, 129, 130
Nusselt, W., 200, 209
O
Obasaju, E.D., 101
Obukhov, A.M., 211, 213, 260
Odar, F., 80, 102
Oseen, C.W., 80, 102
Owen, J.N., 209
Author Index 299
P
Pai, S.I., 131, 157
Panchapakesan, N.R., 232, 239–241, 260
Papanicolaou, P.N., 154, 157
Parthasarthy, S.R., 260
Pei, D.C.T., 172, 209
Perry, A.E., 238, 259, 260
Persson, J., 259
Peterson, G.P., 111, 112, 129, 130
Pfund, D., 113, 130
Plam, B., 112, 130
Pogrebnyak, E., 130
Pohlhausen, E., 63, 70, 192
Poiseuille, J., 103, 130
Pope, S.P., 215, 260
Prabhu, A., 258
Prandtl, L., 218, 225, 228. 260
Q
Qu, W., 113, 130
R
Raithby, C.A., 187, 209
Raizer, G.P., 162, 164, 209
Ramaswamy, G.S. 4, 38
Rao, V.V.L., 4, 38
Rayleigh, L., 165, 209
Raynolds, O., 35, 38
Rector, D., 130
Riabouchinsky, D., 166, 209
Riley, J.J., 80, 102, 238, 243, 244, 258
Robins, A.G., 238, 243–245, 259
Rodi, W., 245, 249, 258, 259
Rohsenow, W.M., 159, 269
Rosenhead, L., 43, 70
Roshko, A., 245, 259
Rotta, J.C., 217, 245, 260
Rybczynskii, W., 85, 102
S
Saffman, P.S., 79, 93, 102
Sakipov, Z.B., 229, 260
Sananes, F., 258
Sasty, M.S., 171, 209
Schiller, L., 104, 129, 130
Schlichting, H., 39, 52, 58, 64, 67, 70, 75,
102–104, 107, 110, 111, 131, 157, 159,
193, 197, 209, 217, 223, 260
Schneider, W., 136, 157
Schumann, T.E., 271, 296
Schvab, B.A., 264, 296
Sedov, L.I., 6, 10, 15, 23, 38, 43, 54, 55, 70, 71,
83, 87, 102, 166, 209, 217
Sekundov, A.N., 258, 296
Semenov, N.N., 266, 296
Sforzat, P., 133, 157
Shah, R.K., 111, 130
Sherman, F.S., 55, 70
Shekarriz, A., 130
Shih, C.C., 75, 102, 232
Shin, T.-H., 232, 260
Shkadinskii, K.G., 296
Sichel, M., 259
Simpson, J.E., 60, 70
Skan, S.W., 68, 69
Smirnova, I.P., 258, 296
Smith, S.H., 245, 260
Sobhan, C., 112, 129
Soo, S.L., 71, 102, 169, 209
Spolding, D.B., 159, 209, 271, 296
Sprankle, M.L., 102
Spurk, J.H., 23, 38
Stephan, K., 39, 69, 201, 209
Stephenson, S.E., 258
Stewardson, K., 197, 209
Stokes, G.C., 44, 70, 87, 102
Sukhov, G.S., 280, 288, 291, 294, 296
T
Tabol, L., 129
Tachie, M.F., 249–251, 258, 260
Tai, Y.-C., 112, 130
Tang, G.-H., 130
Tao, W.-Q., 130
Taylor, G.I., 157, 225, 260
Taylor, T.D., 169, 209
Temirbaev, D.Z., 229, 260
Tieng, S.M., 189, 210
Towendsend, A.A., 131, 157, 238, 242, 260
Trentacoste, N., 133, 157
Trischka, J.W., 259
Turner, J.S., 149, 154, 156, 157
Turner, A.B., 209
Tutu, N.K., 228, 229, 259
U
Uberoi, M.S., 228, 229, 259
300 Author Index
V
Vasiliev, L.L., 130
Vulis, L.A., 54, 70, 131, 133, 147, 157,
228, 260, 266, 271, 274, 280, 281,
291, 296
Van Dyke, M., 120, 129
W
Wang, H.L., 113, 130
Wang, Y., 113, 130
Ward-Smith, A.C., 103, 111, 125, 130
Weast, R.C., 38
Weber, M.E., 101
Weiss, D.A., 58, 59, 70
Wenterodt, T., 130, 259
White, C.M., 119, 120, 130
White, F.M., 111, 130, 159, 210
Wilkinson, W.L., 113, 114, 116, 130
Williams, F.A., 271, 282, 296
Williams, W., 6, 38
Wolfshtein, M., 259
Wosnik, M., 259
Wygnanscki, I., 147, 149, 157, 233, 234, 259
Y
Yaglom, A.M., 217, 220, 260
Yalin, M.S., 90, 92, 93, 102
Yan, A.S., 189, 210
Yao, L.-S., 129
Yarin, A.L., 58, 59, 70
Yarin, L.P., 71, 102, 110, 112, 113, 129, 130,
280, 281, 288, 291, 294, 296
Yenter, Y., 130
Yavanovich, M.M., 129
Z
Zel’dovich Ya, B., 149, 153, 157, 162, 164,
209, 263, 264, 266, 268, 271, 272, 275,
282, 296
Author Index 301
.
Subject Index
A
Acceleration effect, 80–81
Activation energy, 263, 280
Applying the II-theorem to transform PDE into
ODE, 63
Archimedes number, 19
Arrhenius law, 261, 263–264, 267, 270
B
Bessel function, 123, 272
Biot number, 19
Bond number, 19
Brinkman number, 14, 19
Buoyant jet, 149–154
C
Capillary number, 19, 95
Capillary waves in liquid lamella after a weak
drop impact onto a thin liquid film,
58–60
Clausius–Clapeyron equation, 289, 293
Co-flowing turbulent jets, 238–239, 242–244
Combustion of non-premixed gases, 264,
271–274, 278
Combustion waves, 268–271
Continuity equation, 7, 13, 31–34, 49, 52, 56,
67, 68, 71, 77, 85, 104, 134, 168, 239
Convective heat and mass transfer, 2, 160, 166
Couette flow, 178–179
Critical conditions, 268
D
Damkohler number, 17, 19, 21
Darcy number, 19
Dean number, 19, 118–120
Deborah number, 19
Delta function, 160
Diffusion boundary layer over a flat reactive
plate, 65–67
Diffusion flame, 273–281, 284, 288, 291,
293, 294
Dimensional and dimensionless parameters,
3–7
Dimensionless groups, 1, 3, 9, 13, 18–22, 26,
29, 40, 45, 48, 51, 53, 73, 81, 84, 87,
90–91, 110, 111, 115, 117, 118, 120,
122, 124, 139–141, 167, 179, 188, 189,
196, 202, 207, 220, 242, 249, 255, 256,
270, 273, 295
Dorodnitsyn–Illingworth–Stewardson
transformation, 197, 276, 278
Drag
of a body partially dipped in liquid, 87
of a deformable particle, 84–86
on a flat plate, 73–76
force, 1, 4, 7, 23–28, 71–101, 145
of an irregular particle, 82–84
on a solid particle, 76–82
of a spherical particle at low, moderate and
high Reynolds number, 1, 76–79
E
Eckert number, 19, 179
Eddy viscosity and thermal conductivity,
224–229
Effect
of energy dissipation, 112
of the free-stream turbulence, 1, 81,
171–173
of particle acceleration, 80
L.P. Yarin, The Pi-Theorem, Experimental Fluid Mechanics,
DOI 10.1007/978-3-642-19565-5, # Springer-Verlag Berlin Heidelberg 2012
303
Effect (cont.)of particle-fluid temperature difference,
1, 82
of particle rotation, 1, 79, 169–171
of velocity gradient, 174–175
Ekman number, 19
Enthalpy, 5, 16, 17, 134, 135, 184, 195, 196,
237, 261–263, 278, 280, 295
Entrance
flow regime, 106–108
region of pipe, 106, 107, 180–181
Euler number, 14, 17, 19, 22
F
Flow
in curved pipes, 116–120
in irregular pipes, 111–112
over a plane wall which has instantaneously
started moving from rest, 44–47
in straight rough pipes, 1
Fourier number, 122
Frank-Kamenetskii
approximation, 263, 264
parameter, 266, 267
Freezing of a pure liquid, 202–205
Froude number, 14, 17, 73, 86, 87, 256
Fully developed flow
in rough pipes, 109–111
in smooth pipes, 109
G
Gas torches, 2, 280–287
Grashof number, 19, 170, 186, 188–190, 193
H
Hadamard–Rybczinskii formula, 85
Heat
of reaction, 265, 267, 279, 290, 295
release, 19, 21, 66, 162, 261, 265, 266, 268,
269, 288, 294
Heat transfer
accompanying condensation of saturated
vapor on a vertical wall, 199–201
in channel and pipe flows, 2, 178–183
under the conditions of phase change, 160
from a flat plate in a uniform stream of
viscous high-speed perfect gas,
195–199
in forced convection, 165–178
from a hot particle immersed in fluid flow,
2, 165–169
in mixed convection, 2, 187–188
in natural convection, 186–194
from a spherical particle, 186–187
from a spinning particle, 170, 187–188
from a vertical hot wall, 190–193
I
Ideally stirred reactor, 265
Immersed flame, 288–294
Impinging turbulent jet, 252–254
Inhomogeneous turbulent jet, 232–238
J
Jacob number, 19
Jet flow, 43, 51, 131–156, 160, 225, 228, 229,
231, 245, 248, 287, 288
K
Knudsen number, 19, 112, 113
Kutateladze number, 19
L
Laminar
boundary layer over a flat plate, 14, 47–51
flow near a rotating disk, 55–58
flows in channels and pipes, 103–129
jets issuing from a thin pipe, 134
submerged jet, 51–54, 131, 141–143,
154–156, 185
wake of a solid body, 143–146
Lewis number, 18, 19, 262, 263, 269, 290, 295
M
Mach number, 19, 275, 278, 279
Mass
diffusivity coefficient, 262
flux, 23, 190, 194, 206, 208, 235, 247, 271,
285
transfer from a spherical particle in natural
and mixed convection, 189–190
transfer in forced convection, 165–178
transfer to a vertical reactive plate in natural
convection, 193–194
transfer to solid particles and drops
immersed in fluid flow, 176–178
304 Subject Index
Micro-channel flows, 112–113
Microkinetic law, 262
Mixing length, 153, 225, 226
N
Navier–Stokes equations, 7, 13, 27, 28, 31, 32,
34, 39, 52, 56, 67, 71, 77, 96, 104, 105,
112, 118, 120, 124, 136, 168, 261
Newton’s law, 28, 79
Nondimensionalization of the governing
equations, 1, 16
Non-Newtonian fluid flows, 1, 18, 113–116
Nusselt number, 15, 20, 166, 168–172, 175,
179, 181–183, 186–189, 193, 199
O
Oscillatory motion, 75–76
P
Peclet number, 14, 17, 20, 21, 166, 168, 169,
175–178, 190, 207, 274
Plane jet, 137, 147, 183, 250, 252
Prandtl number, 18, 20, 65, 82, 151, 168–170,
172–174, 179, 180, 182, 183, 186–188,
192, 193, 225, 228–229
Pre-exponential, 280, 289
Propagation of viscous-gravity currents over a
solid horizontal surface, 60–63
R
Rate of conversion, 262
Rayleigh number, 20, 187
Reynolds number, 1, 2, 14, 17, 18, 20–22, 25,
27, 28, 34, 72–84, 86, 88, 89, 91, 96, 97,
103, 105, 107, 110–111, 113, 115, 117,
118, 120, 124, 128, 142, 143, 153, 155,
168–170, 172–176, 180, 182, 188, 189,
211, 213, 219
Richardson number, 20
Rossby number, 20
S
Schmidt number, 18, 20, 30, 67, 194, 228, 229
Schvab–Zel’dovich transformation, 264, 272
Sedimentation, 1, 90–93
Self-similar solution, 39, 41–43, 58, 63, 67,
149, 152, 191, 201, 204, 205, 242, 249,
276, 279, 286
Semenov number,
Shear stress, 33, 36, 45, 50, 74, 77, 78, 88, 113,
114, 217, 221, 231, 234, 285
Sherwood number, 20, 176–178, 190, 207
Similarity, 1, 3, 11, 21–22, 211, 226, 245, 281
Single-one-step chemical reaction, 262
Spalding transfer number, 296
Stoichiometric coefficient, 264, 276
Stokes equations, 97
Strouhal number, 14, 17, 20, 75
T
Taylor number, 20
Temperature field, 150, 151, 153, 160–165,
183, 184, 191, 204, 205, 265
induced by a plane instantaneous thermal
source, 160–161
induced by a pointwise instantaneous
thermal source, 161–162
Terminal velocity of small heavy spherical
particle in viscous liquid, 87–90
Thermal
boundary layer over a flat plate, 63–65
characteristics of laminar jets, 183–185
diffusivity coefficient, 162, 163, 205, 262
explosion, 2, 263–268
particle in viscous liquid, 87–90
Thin liquid film on a plane withdrawn from a
pool filled with viscous liquid, 93–96
Total enthalpy, 263, 295
Total momentum flux, 52, 54, 133, 135, 136,
143, 146, 227, 229, 235, 241, 252, 256,
258, 285
Transfer in turbulent jet, 224
Turbulent
jet, 2, 135, 147, 153, 215, 224–254
wall jet, 248–251
Two-phase flow, 19, 71, 93
U
Unsteady flows in straight pipes, 120–123
V
Van-der Waals equation, 30–31
W
Wall jet over plane and curved surfaces,
146–149
Weber number, 20, 84, 100, 256
Subject Index 305