tu r bu l e nc e - aaa scientists › aliabadi 2018.pdf · theory and applications of turbulence a...

TU R BU L E NC EAMIR A. ALIABADI

Theory and Applications of TurbulenceA Fundamental Approach for Scientists and Engineers

Amir A. Aliabadi

School of EngineeringUniversity of Guelph

i

c©2018 Amir A. Aliabadi Publications

All rights reserved. No part of this book may be reproduced, in any form or by anymeans, without permission in writing from the publisher.

ISBN: 978-0-9809704-9-4

Atmospheric Innovations Research (AIR) Laboratory, Environmental Engineering,School of Engineering, RICH 2515, University of Guelph, Guelph, ON N1G 2W1,Canada

ii

Dedication

Fariborz Aliabadi

iii

Preface

Amir A. Aliabadi

Turbulence is a ubiquitous phenomenon found in many natural and man-madesystems from the core of the earth to the top of the atmosphere. It is also found inouter space, influencing how galaxies form and evolve. The most simple and curiousmind can put an intuitive definition to turbulence. Some examples include presence

of chaos in a system or the lack of accurate prediction for the evolution of a system.Yet, providing an exact definition and developing the mathematical framework todescribe turbulence can become complex and confusing. I have been studying thesubject for more than ten years, yet many of my exposures to the topic have beendisappointing. I have often found myself lost in the equations describing turbulence,neither completely understanding the mathematical rigour, nor fully comprehendingparticular models that attempt to predict a system’s evolution.

In my opinion, the first reason for my unsuccessful attempts was that the topicwas always introduced to me without enough coverage of fundamental concepts.Too often the material was focused on practical aspects of a particular analysis.The second reason was that most tools I have been using to model turbulence werehighly integrated and automated application software, thanks to the developers, en-abling me to perform a complex simulation with selecting a few menu items andthen pushing a few run buttons. While efficient, such application software have al-ways masked the inner workings of the equations, approximations, and solutionmethods throughout a modelling exercise. The third reason was that the subjectmatter has always been difficult by its very nature. As Horace Lamb have said at ameeting of the British Association in London in 1932: “I am an old man now, and

when I die and go to Heaven there are two matters on which I hope for enlight-

enment. One is quantum electrodynamics and the other is the turbulent motion of

fluids. And about the former I am really rather optimistic.”

To overcome the shortcomings in my own understanding of the topic, I de-cided to write this text and develop a few computer laboratories in support of agraduate course in turbulence suitable for majors in engineering and science. The

iv

text has been made as brief as possible, only covering fundamentals, but with adeep level of mathematical rigour. In addition, the computer laboratories solve onlysimple turbulence problems, but they start from scratch and develop fundamentalcomputer code to formulate and solve equations of turbulence. It is hoped that thisfundamental approach will help students appreciate the topic at a more profoundlevel and develop an understanding that can help them tackle more sophisticatedturbulence problems encountered in science and engineering.

The mathematical notation discussed in this book closely follows the graduatetext book Turbulent Flows by professor Stephen B. Pope (Pope, 2000). In closing,I encourage and thank the readers to critique this text and provide me feedback andpossibly corrections to the material studied here.

v

Acknowledgment

I thank my family for their support while I spent countless hours to research, col-lect, and compose the material found in this text and the accompanying computerlaboratory instructions. I am indebted to Reza Aliabadi and Homa Ansari for thecover design and execution of this text. I am grateful to Massachusetts Instituteof Technology (MIT) Press for allowing me to use their LATEX template to type-set this text. Finally, I appreciate the patience of all the readers of the text and theaccompanying computer laboratory instructions for their patience, hard work, andpotentially feedback toward improving the quality of the material.

– AAA

vi

Contents

Preface ivby Amir A. Aliabadi

Part I: Fundamentals 1

Chapter 1: Introduction 2Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Chapter 2: Equations of Fluid Motion 5The Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 5The Momentum Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 5Conserved Passive Scalars . . . . . . . . . . . . . . . . . . . . . . . . . 8The Vorticity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Fluid Element Deformation . . . . . . . . . . . . . . . . . . . . . . . . . 10Similitude and Non-dimensional Transport Equations . . . . . . . . . . . 11Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Chapter 3: Statistical Description of Turbulent Flows 16Mean and Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Standardization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Joint Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Normal and Joint-normal Distributions . . . . . . . . . . . . . . . . . . . 19Random Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Random Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Statistically Stationary, Homogeneous, and Axisymmetric Turbulent Flows 26Isotropic and Anisotropic Turbulence . . . . . . . . . . . . . . . . . . . . 26Two-point Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Wavenumber Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Types of Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

vii

Chapter 4: Mean Flow Equations 32Tensor Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Mean Scalar Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Gradient-diffusion and Turbulent-viscosity Hypotheses . . . . . . . . . . 36Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Chapter 5: Wall Flows 45Transport Equations and the Balance of Mean Forces . . . . . . . . . . . 45The Shear Stress Near Wall . . . . . . . . . . . . . . . . . . . . . . . . . 48Viscous, Buffer, and Log-law Sublayers . . . . . . . . . . . . . . . . . . 50Law of the Wall for Temperature . . . . . . . . . . . . . . . . . . . . . . 50Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Chapter 6: Free Shear Flows 57Round Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Axial Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Axial Variation of Scales . . . . . . . . . . . . . . . . . . . . . . . . . . 60Self-similarity of a Round Jet . . . . . . . . . . . . . . . . . . . . . . . . 60Reynolds Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Mean Continuity and Momentum Equations for a Jet . . . . . . . . . . . 62Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Chapter 7: Scales of Turbulent Motion 67The Energy Cascade and Kolmogorov Hypotheses . . . . . . . . . . . . . 67The Energy Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Two-point Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Structure Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Taylor Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Chapter 8: Time and Frequency Domains 83

viii

Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . 83Nyquist Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Discrete Energy Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 87Discrete Energy Density Spectrum . . . . . . . . . . . . . . . . . . . . . 88Spectra of Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 88Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Part II: Turbulence Measurement Techniques 92

Chapter 9: Sonic and Ultrasonic Techniques 93Ultrasonic Anemometer . . . . . . . . . . . . . . . . . . . . . . . . . . . 93SOnic Detection And Ranging (SODAR) . . . . . . . . . . . . . . . . . 94

Chapter 10: Electro-magnetic Techniques 96Particle Image Velocimetry (PIV) and Particle Tracking Velocimetry (PTV) 96Schlieren Image Velocimetry (SIV) . . . . . . . . . . . . . . . . . . . . . 97Laser Doppler Velocimetry (LDV) . . . . . . . . . . . . . . . . . . . . . 98Radiometry and Pyrometry . . . . . . . . . . . . . . . . . . . . . . . . . 99Light Detection And Ranging (LiDAR) . . . . . . . . . . . . . . . . . . 100

Chapter 11: In-situ Techniques 102Hot Wire Anemometry (HWA) . . . . . . . . . . . . . . . . . . . . . . . 102Pitot Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Balloons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Part III: Turbulence Modelling and Simulation 104

Chapter 12: Introduction to Modelling and Simulation 105Summary of Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 105Model or Simulation Completeness . . . . . . . . . . . . . . . . . . . . . 106Turbulence Model or Simulation Closure Problem . . . . . . . . . . . . . 107Digital Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Chapter 13: Turbulent Viscosity Models 109

ix

Algebraic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Spalart-Allmaras Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 112Turbulent Kinetic Energy Models . . . . . . . . . . . . . . . . . . . . . . 112The k − ε Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115The k − ω Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Chapter 14: Large-eddy Simulation Models 122Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123Filtered Conservation Equations . . . . . . . . . . . . . . . . . . . . . . 123The Smagorinsky Model . . . . . . . . . . . . . . . . . . . . . . . . . . 125One-equation Turbulent Kinetic Energy Model . . . . . . . . . . . . . . . 125The Problem of Inlet Condition . . . . . . . . . . . . . . . . . . . . . . . 126Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Chapter 15: Direct Numerical Simulation 135Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Chapter 16: Wall Models 138Point-wise Standard Wall Function . . . . . . . . . . . . . . . . . . . . . 138Integrated Werner-Wengle Wall Function . . . . . . . . . . . . . . . . . . 141van Driest Near Wall Treatment . . . . . . . . . . . . . . . . . . . . . . . 142Wall Function Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 143Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Chapter 17: Model Evaluation 150Verification and Validation . . . . . . . . . . . . . . . . . . . . . . . . . 150Time and Space Discretization Error Estimation . . . . . . . . . . . . . . 150Observations and Model Error Quantification . . . . . . . . . . . . . . . 155Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Part IV: Applications 158

Chapter 18: Engineering 159

x

Liquid-liquid Extraction Industries . . . . . . . . . . . . . . . . . . . . . 159Coalescer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159Waste Water Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . 160Desalination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160Combustion Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160Indoor Ventilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161Aeronautics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161Renewable Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161River Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Chapter 19: Sciences 163Meteorology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163Oceanography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Part V: Fundamental Analysis Tools and Principles 168

Chapter 20: Statistics 169Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169Event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169Cumulative Distribution Function . . . . . . . . . . . . . . . . . . . . . . 170Probability Density Function . . . . . . . . . . . . . . . . . . . . . . . . 170Mean and Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 171Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

Chapter 21: Mathematics 173Einstein’s Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173Kronecker Delta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173Alternating Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174Position Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

xi

Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177Dot Product of Two Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 177Cross Product of Two Vectors . . . . . . . . . . . . . . . . . . . . . . . . 178Material or Substantial Derivative . . . . . . . . . . . . . . . . . . . . . 178Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

Chapter 22: Numerical Methods 182Taylor Series Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 182The Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . . 182Newton’s Method for Solving Non-linear System of Equations . . . . . . 183Explicit and Implicit Euler Methods . . . . . . . . . . . . . . . . . . . . 184Under Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

References 189

List of Contributors 200

Index 201

xii

Part I

Fundamentals

1

Chapter 1

Introduction

In fluid dynamics, turbulent flow is a flow regime characterized by chaotic property

changes with time. For instance flow velocity, temperature, or concentration of atransported substance may exhibit such chaotic changes. Turbulence is the domi-nating force in the physics and chemistry of the atmosphere and hydrosphere. Itis ubiquitous in everyday life from airflow in cities and buildings to water flow inrivers and oceans. Even voice generation by humans is only possible by the turbu-lent flow of air through the vocal tract. Figure 1.1 shows the turbulent flow of acigarette plume. Yet turbulence is difficult to understand, measure, model, or simu-late, and it has been considered as one of the classical unsolved problems of physics.

At a fundamental level, turbulence is caused by growth of instabilities in fluidmotion, which typically occurs in many practical applications in the atmosphere orhydrosphere. The pioneering work in turbulence was conducted by Osborn Reynolds(1842-1912). He performed a series of experiments involving the injection of a dyeat the centre of water flowing through a pipe. He observed that at some critical watervelocity, the dispersion of the dye in the water transitions from laminar to turbulentregime. As was discovered by Reynolds, the transition from laminar to turbulentregimes was governed by the non-dimensional Reynolds number

Re =ULν, (1.1)

where U and L are characteristic velocity and length scales of the flow, and ν is thekinematic viscosity of the fluid. In physics, a characteristic quantity is a quantitythat defines the scale of a physical system or phenomena. For instance, the charac-teristic length defines the size of a system, and the characteristic velocity definesthe velocity of flow through or around the system. The Reynolds number quantifiesthe ratio of inertial to viscous forces. If Re < 1000, the flow is in the laminar range;if 1000 < Re < 4000, the flow is in the transitional stage; and if Re > 4000,the flow is in the turbulent range. These limits for the Re number are provided for

2

Figure 1.1: A cigarette plume exhibits turbulence, which is evident from thechaotic property changes in velocity, temperature, and smoke concentration.

demonstration. In fact, these limits could be different based on the type of flow: in-ternal flows (pipes, ducts, ...), external flows (atmospheric boundary layer, airfoils,...), and more.

Turbulence can be perceived as the presence of eddies in the fluid (Richard-son, 1920). An eddy is a local structure of a fluid that spins. Further, the turbulent

energy cascade implies that turbulence begins with formation of large eddies dueto instability that subsequently break down to form smaller and smaller eddies dueto subsequent instabilities until they are so small that they are damped by viscous

dissipation. This phenomenon was observed in the early twentieth century and ex-pressed poetically by Lewis Richardson (1881-1953):

“Big whirls have little whirls that feed on their velocity, and little whirls have

lesser whirls and so on to viscosity”.

3

The study of turbulence may fall under one of three major categories: (i) Dis-

covery: experimental or simulation studies providing qualitative or quantitative in-formation about particular flows; (ii) Modelling: theoretical, modelling, or simu-lation studies, aimed at developing rigorous mathematical models that can predictproperties of turbulent flows; and (iii) Control: studies aimed at manipulating orcontrolling turbulent flow in a beneficial way.

Exercises

1) For flow of air in a ventilation duct the characteristic velocity scale is U =

1 m s−1 and the characteristic lengthscale, i.e. the hydraulic diameter of the duct, isL = 0.3 m. Assuming that the kinematic viscosity of air is ν = 1.5× 10−5m2 s−1,calculate the Reynolds number of the flow and state if the flow is in the laminar orturbulent regime.2) Provide an argument in favour of more aggressive instability growth in fluidmotion when flow length and velocity scales are are higher, and when dynamicviscosity of the fluid is lower. Does this demonstrate why level of turbulence in aflow is correlated with the Reynolds number?

4

Chapter 2

Equations of Fluid Motion

The Continuity Equation

For a flow with variable density ρ and velocity vector U the continuity equation isgiven as

∂ρ

∂t︸︷︷︸Storage

+∇.(ρU)︸︷︷︸Advection

= 0, (2.1)

where t is time. If we assume ρ to be independent of space and time, i.e. to be con-stant, the continuity equation simplifies to the kinematic condition that the velocityfield should be solenoidal or divergence-free:

∇.U = 0. (2.2)

The Momentum Equation

The momentum equation in fluid dynamics is based on Newton’s second law, whichrelates the fluid particle acceleration, i.e. the material derivative of fluid velocityDU/Dt, to the surface forces and body-forces experienced by the fluid particle.In general, the surface forces are described by the stress tensor τij(x, t), with theproperty that is symmetric (τij = τji). Here x indicates the position vector. Thestress tensor is shown in Figure 2.1.

The body force of interest is usually gravity. If Ψ is the gravitational potential

(i.e. the potential energy per unit mass associated with gravity), the body force perunit mass is

g = −∇Ψ. (2.3)

5

Figure 2.1: Surface forces expressed as stress tensor components for a Cartesianfluid element. The stress tensor contains six shear components and three normal

components.

For a constant gravitational field the potential is Ψ = gz, where g is the grav-itational acceleration and z is the vertical coordinate. These forces cause the fluidparticle to accelerate according to the momentum equation given by

ρDUjDt︸︷︷︸

Material Derivative

=∂τij∂xi︸︷︷︸

Surface Force

− ρ∂Ψ

∂xj︸︷︷︸Body Force

. (2.4)

We now develop the momentum equation for flows of constant-property New-

tonian fluids, e.g. constant density, the fundamental class of flows we consider inthis book. In this case, the stress tensor is

τij︸︷︷︸Surface Force

= − Pδij︸︷︷︸Surface Normal Force

+µ

(∂Ui∂xj

+∂Uj∂xi

)︸︷︷︸

Surface Shear Force

, (2.5)

where P is the pressure and µ is a constant coefficient of dynamic viscosity. Byinserting this stress tensor in the general momentum equation, and recalling thatboth ρ and µ are assumed constants and that∇.U = 0, we obtain the Navier-Stokesequation for momentum

6

ρDUjDt︸︷︷︸

Material Derivative

= µ∂2Uj∂xi∂xi︸︷︷︸

Surface Shear Force

− ∂P

∂xj︸︷︷︸Surface Normal Force

− ρ∂Ψ

∂xj︸︷︷︸Body Force

. (2.6)

We can further compress this equation in shorter form by defining the modified

pressure p by

p = P + ρΨ, (2.7)

which simplifies the general momentum equation to

DU

Dt︸︷︷︸Material Derivative

= − 1

ρ∇p︸︷︷︸

Surface Normal and Body Forces

+ ν∇2U︸︷︷︸Surface Shear Force

(2.8)

where ν = µ/ρ is the kinematic viscosity. Note that instead of Einstein’s notation,this momentum equation is now shown in vector form for velocity. In fact thislast equation represents the three momentum equations at once and is valid foreach velocity component U1 = U , U2 = V , and U3 = W . The Newtonian fluidswith constant property are specified with this momentum equation together with thesolenoidal condition∇.U = 0, which stipulates mass conservation.

In many practical applications, the viscous term containing µ is negligible com-pared to the pressure term containing p. Such flows are termed inviscid flows, inwhich the stress tensor is given as

τij = −Pδij. (2.9)

In this case the general momentum equation is further simplified to the Euler

equation given by

DU


= − 1

ρ∇p︸︷︷︸

Surface Normal and Body Forces

, (2.10)

which does not contain the second partial derivative of velocity. Named after Leon-hard Euler (1707-1783), the Euler equation has applications in airfoil theory tocalculate the lift force generated from an aerodynamic wing.

7

Conserved Passive Scalars

The Navier-Stokes equation can be extended to express the transport of a con-

served passive scalar denoted by φ(x, t). In constant-property flows, the conser-vation equation for φ is

Dφ


= Γ∇2φ︸︷︷︸Diffusion

, (2.11)

where Γ is the constant and uniform diffusivity. For a scalar quantity φ to be passive,it must be conserved, i.e. there is no source or sink term in the transport equation.It is termed a passive since its value has no effect on material properties such asdensity, viscosity, and diffusivity. In other words, the flow will actually behave thesame way whether or not there is a passive scalar present.

Usually the scalar quantity can represent various physical properties such asconcentration or temperature. However, a passive scalar must only be present insmall amounts, e.g. low concentrations or small excess temperatures so that it doesnot affect the material properties in the flow. High concentrations of another sub-stance in the flow or an excessive temperature difference cause density and viscosityvariations. It must be noted that the Navier-Stokes equations for scalars in generalhave more terms involved that are not discussed in the interest of the simplicity ofthe transport equations for our purposes in this book.

When the passive scalar is temperature, the diffusivity is called the thermaldiffusivity. The thermal diffusivity is non-dimensionalized using the kinematic vis-cosity in the so called Prandtl number in honour of Ludwig Prandtl (1875-1953)by

Pr =ν

Γ. (2.12)

When the passive scalar is concentration of trace species, the diffusivity iscalled the molecular diffusivity. The molecular diffusivity is non-dimensionalizedusing the kinematic viscosity in the so called Schmidt number in honour of ErnstHeinrich Wilhelm Schmidt (1892-1975) by

8

Sc =ν

Γ. (2.13)

For air at standard conditions the Prandtl number is in the range from 0.7 to 0.8.However, the Schmidt number depends on the type of dilute mixture containing thevapor of a substance. For instance the Schmidt number can be as low as 0.22 forhelium and as high as 2.66 for octane.

The Vorticity Equation

Turbulent flows contain eddies, which by nature spin. As a result turbulent flowsare rotational, i.e. they have non-zero vorticity. The vorticity ω(x, t) is defined asthe curl of the velocity

ω = ∇×U, (2.14)

and is equal to twice the rate of rotation of the fluid at any given space and time(x, t). The transport equation for vorticity can be obtained by taking the curl of theNavier-Stokes equations

Dω


= ν∇2ω︸︷︷︸Diffusion

+ ω.∇U︸︷︷︸Vortex Stretching

, (2.15)

where the pressure term (−∇×∇p/ρ) in the Navier-Stokes equation vanishes forconstant-density flows. Vorticity has received much attention in turbulence researchdue to convenient properties. Unlike flow velocity, vorticity cannot be created nordestroyed within the interior of a fluid and is rather transported within the flow byadvective and diffusion processes. When we talk of a turbulent eddy we really meana blob of vorticity and its associated rotational and irrotational motion (Davidson,2009).

9

Fluid Element Deformation

Fluid elements experience deformation due to body and surface forces exerted onthem. The deformation can be expressed as a combined effect of a) translation, b)linear strain, c) shear strain, and d) rotation as shown in Figure 2.2.

Figure 2.2: Deformation of a fluid element in the Cartesian coordinates can beunderstood as a combined effect of a) translation, b) linear strain, c) shear strain,

and d) rotation.

These deformations can be expressed in terms of fluid velocity and velocitygradients. As noted earlier, the velocity gradients ∂Ui/∂xj are the components of asecond-order stress tensor. For constant-density flow, the composition of ∂Ui/∂xjinto symmetric-deviatoric and antisymmetric parts is

∂Ui∂xj︸︷︷︸

Deformation

= Sij︸︷︷︸Symmetric-deviatoric Deformation

+ Ωij︸︷︷︸Antisymmetric Deformation

. (2.16)

Sij is the symmetric-deviatoric rate-of-strain tensor

Sij ≡1

2

(∂Ui∂xj

+∂Uj∂xi

), (2.17)

and Ωij is the antisymmetric rate-of-rotation tensor

Ωij ≡1

2

(∂Ui∂xj− ∂Uj∂xi

). (2.18)

10

The rate-of-strain and rate-of-rotation tensors are very useful because transportequations can be expressed using them in shorter form. For instance, the Newtonianstress law can be re-expressed as

τij = −Pδij + 2µSij, (2.19)

indicating that the viscous stress depends linearly on the rate-of-strain independentof the rate-of-rotation. The vorticity and the rate-of-rotation are related by

ωi = −εijkΩjk, (2.20)

Ωij = −1

2εijkωk, (2.21)

where εijk is the alternating symbol. Therefore, both rate-of-rotation and vorticityhave the same information. However, while rate-of-rotation is a tensor, vorticity isa vector.

Similitude and Non-dimensional Transport Equations

A relevant question in fluid dynamics is whether or not two flow systems behave inthe same fashion regardless of their size or properties. This is the concept of simili-

tude in physics. Two systems may be similar in three ways. 1) Geometric similarity

requires that the ratio of any two linear dimensions of two systems be the same. 2)Kinematic similarity is the similarity of time as well as geometry. It exists betweentwo systems if the paths of moving particles are geometrically similar and if theratios of the velocities of particles are similar. 3) Dynamic similarity exists betweentwo systems when forces at corresponding points are similar. In fluid mechanics,to achieve dynamic similarity all the non-dimensional numbers relevant to the flowmust be preserved between the two systems, such as Reynolds, Grashof, and othernumbers (Ettema, 2000). This is extremely limiting since in practice at most one ortwo non-dimensional numbers of the flow can be preserved.

Two geometrically similar systems may have different sizes, densities, and vis-cosities but the same velocity fields when they are appropriately scaled and referred

11

to an appropriate coordinate system. Similar systems can be studied by the transfor-

mation properties of the Navier-Stokes equations. These properties are also calledinvariance properties or symmetries.

It can be shown that two flow systems are similar if they are geometricallysimilar and if they have the same non-dimensional transport equations. Considera system with characteristic length L and characteristic velocity U . The character-istic length and velocity can be used to define the non-dimensional independent

variables,

x = x/L, (2.22)

t = tU/L, (2.23)

and also the dependent variables given the non-dimensional independent variablesand system properties such as density,

U(x, t) = U(x, t)/U , (2.24)

p(x, t) = p(x, t)/(ρU2). (2.25)

In applying these simple scaling transformations to the continuity and momen-tum equations, we will obtain the non-dimensional continuity and momentum equa-tions

∂Ui∂xi

= 0, (2.26)

DUi

Dt=

1

Re

∂2Uj∂xi∂xi

− ∂p

∂xj, (2.27)

where Re = UL/ν is the familiar Reynolds number. The non-dimensional trans-port equations are now only a function of the Re parameter. In other words, theseequations are identical as long as Re is the same between two geometrically similarsystems regardless of size or property differences. This ensures that the two flowsystems be similar too.

12

Exercises

1) A two-dimensional fluid velocity vector field is given as U = (x+y2)i+(x2+y)j

in units of m s−1. Show that the vorticity of the flow as a function of x and y, ins−1, is given by

ω = (2x− 2y)k. (2.28)

2) For a simple shear flow near a wall, the one-dimensional fluid velocity vectorfield can be given as U = yi in units of m s−1, where unit vector i is the directionof flow along x axis parallel to the wall and unit vector j is the direction normal tothe wall along the y axis. Show that for this flow the rate-of-strain tensor Sij andthe rate-of-rotation tensor Ωij , in s−1, are given by

Sij =1

2, (2.29)

Ωij =1

2, (2.30)

suggesting that even for the simplest of shear flows there could be fluid elementstrain and rotation.3) In a two dimensional flow, a fluid is spinning around the vertical z axis in thedirection of the unit vector k. The flow field is given as U = −Ωyi+ Ωxj, where Ω

is a constant, also known as angular velocity. Here the unit vector i is the directionof flow along x axis and unit vector j is the direction of the flow along the y axis.Show that this flow is a rotational flow, i.e. a flow in which the vorticity is non-zero.In addition, show that for this flow the vorticity is equal to the following constantvalue in the entire flow domain, i.e.

ω = ∇×U = 2Ωk. (2.31)

4) In a two dimensional flow, a fluid is spinning around the vertical z axis in thedirection of the unit vector k. The flow field is given as U = − y

x2+y2i + x

x2+y2j.

Here the unit vector i is the direction of flow along x axis and unit vector j is thedirection of the flow along the y axis. Show that this flow is an irrotational flow, i.e.a flow in which the vorticity is precisely zero in the domain, i.e.

13

ω = ∇×U = 0. (2.32)

5) In the previous problem show that the rate-of-rotation tensor is also identicallyequal to zero, hinting that for two-dimensional flows the rate-of-rotation tensor andvorticity give the same information, i.e.

Ωij = 0. (2.33)

6) Use suffix notation to verify the following relations:

∇.ω = 0, (2.34)

∇×∇φ = 0, (2.35)

∇× (∇×U) = ∇(∇.U)−∇2U, (2.36)

U× ω =1

2∇(U.U)−U.∇U. (2.37)

7) A steady state velocity field in a fluid flow is given by U = −3xi − 3yj + 6zk.Show that the material derivative of this velocity field itself is given by

DU

Dt=∂U

∂t+ U.∇U = 9xi + 9yj + 36zk. (2.38)

8) State if the following statement is true or false, why? the material derivative

of a scalar quantity is a scalar quantity, while the material derivative of a vector

quantity is a vector quantity.

9) The classical Kelvin-Stokes theorem relates the surface integral of the curl of avector field F over a surface S in Euclidean three dimensional space to the line in-tegral of the vector field F over its boundary ∂S, which is a closed circuit enclosingS, i.e. ∫∫

S

(∇× F).dS =

∮∂S

F.dr (2.39)

14

In fluid dynamics, circulation is the line integral around a closed curve of thevelocity field. Circulation is normally denoted by Γ and is shown as

Γ =

∮∂S

U.dl. (2.40)

Using the classical Kelvin-Stokes theorem show that the circulation and vortic-ity are related in such a way that circulation is the surface integral of the dot productof vorticity vector ω with differential area vector dS on a surface S enclosed by ∂S,i.e.

Γ =

∮∂S

U.dl =

∫∫S

ω.dS. (2.41)

15

Chapter 3

Statistical Description of Turbulent Flows

In turbulent flow, the velocity field U(x, t) is random despite the fact that theNavier-Stokes equations are deterministic, i.e. given the right boundary and initialconditions, the equations should predict an exact solution in a prognostic manner.How can this be? The answer lies in the combination of two observations: 1) in anyturbulent flow there are unavoidable perturbations in initial conditions, boundaryconditions, and material properties; 2) turbulent flow fields display an acute sensi-tivity to such perturbations (Pope, 2000). These facts help explain why it is difficultto perform an exact prediction of turbulent flows even though the equations describ-ing them are deterministic. An analogy is an experienced golf player who aims theball on a target at a very far distance. However, the location of landing for the ballwill exhibit errors due to minute initial misalignments in the club swing, imperfec-tions in the club and the ball, and perturbations in air, e.g. winds, that all contributeto deviate the ball off target (Figure 3.1). Another point is that such minute mis-alignments, imperfections, and perturbations in air are never the same for two clubswings, so every realization of the golf ball trajectory will be different under differ-ent conditions from one experiment to another.

Figure 3.1: Golf ball trajectories are severely affected by minute initialmisalignments in the club swing, imperfections in the club and the ball, and

perturbations in air.

16

For laminar flow conditions, the effect of perturbations in boundary conditions,initial conditions, and material properties are insignificant in creating instabilities,therefore a measurement and prediction of flow using the Navier-Stokes equationswill agree with a high degree of confidence. Navier-Stokes equations apply equallyto turbulent flows, but the objective of the theory is different. In turbulent flows,since U is a random variable, its value is unpredictable, so a theory cannot performan exact prediction. However, a theory can either try to determine the probabilityof events that depend on the random variable U or try to determine some statisticaldescription of the random variable U, such as mean or standard deviation of therandom variable.

Mean and Moments

The mean of the random variable U is defined by

〈U〉 ≡∫ ∞−∞

V f(V )dV (3.1)

where the Probability Density Function (PDF) f(V ) is the probability per unit dis-tance in the sample space, hence the term probability density function. The PDFf(V ) has the dimensions of the inverse of U , while the Cumulative Distribution

Function (CDF) and the product f(V )dV are non-dimensional. Note that the meanas defined here does not necessitate any averaging by time or space, but it simplydefines the mean as variations of the random variable itself. 〈U〉 is the probability-weighted average of all possible values of U . More generally, if Q(U) is any func-tion of U , the mean of Q(U) is given by

〈Q(U)〉 ≡∫ ∞−∞

Q(V )f(V )dV. (3.2)

If Q(U) and R(U) are two functions of the random variable U , the rules fortaking means satisfy

〈[aQ(U) + bR(U)]〉 = a〈Q(U)〉+ b〈R(U)〉. (3.3)

17

In other words, 〈〉 behaves as a linear operator. The fluctuation of U is defined by uas

u ≡ U − 〈U〉. (3.4)

and variance is defined to be the mean-square fluctuation by

〈u2〉 =

∫ ∞−∞

(V − 〈U〉)2f(V )dV. (3.5)

The square-root of the variance is the standard deviation and denoted by 〈u2〉1/2.In many text books the fluctuations are also shown as u′. The square-root of vari-ance is also shown by σu. In general the nth central moment is defined as

µn ≡ 〈un〉 =

∫ ∞−∞

(V − 〈U〉)nf(V )dV. (3.6)

Standardization

It is possible to standardize random variable fluctuations. The standardized randomvariable fluctuation has zero mean and unit variance. The standardized random vari-able U corresponding to U is

U ≡ (U − 〈U〉)/σu, (3.7)

and its PDF, i.e. the standardized PDF of U , is

f(V ) = σuf(〈U〉+ σuV ). (3.8)

The moments of U , the standardized moments of U , are

µn =〈un〉σnu

=µnσnu

=

∫ ∞−∞

V nf(V )dV . (3.9)

Evidently we have µ0 = 1, µ1 = 0, and µ2 = 1. In statistics, the third stan-dardized moment, µ3, is called skewness. Skewness is a measure of the asymmetryof the probability distribution of a real-valued random variable about its mean. Theskewness value can be positive or negative. The fourth standardized moment, µ4, is

18

called kurtosis or flatness. Kurtosis is a measure of the tailedness of the probabilitydistribution of a real-valued random variable. In a similar way to the concept ofskewness, kurtosis is a descriptor of the shape of a probability distribution.

Joint Random Variables

The concept of a single random variable U can be extended to two or more ran-dom variables. For instance we can take the three components of a velocity vector(U1, U2, U3) representing three random variables. A Joint Probability Density Func-

tion (JPDF) can represent the density function for a pair of two random variables,such as f12(V1, V2). The covariance of U1 and U2 is the mixed second moment andcan be expressed using the JPDF as

〈u1u2〉 =

∫ ∞−∞

∫ ∞−∞

(V1 − 〈U1〉)(V2 − 〈U2〉)f12(V1, V2)dV1dV2, (3.10)

and the correlation coefficient is

ρ12 ≡〈u1u2〉

[〈u21〉〈u2

2〉]1/2. (3.11)

As shown in Figure 3.2 a positive correlation coefficient occurs when a posi-tive fluctuation of one random variable is most likely accompanied with a positivefluctuation of the other random variable as well (the same can be said for a negativefluctuation). A negative correlation coefficient occurs when a positive fluctuationof one random variable is most likely accompanied with a negative fluctuation ofthe other random variable. In Figure 3.2 each axis represents one of the randomvariables, and the mean for both random variables occurs at the centre of the scatterplot.

Normal and Joint-normal Distributions

The central-limit theorem and Gaussian distribution function play an importantrole in the theory of probability and that of turbulent flows. It is necessary to be-gin the theory by the concept of ensemble average. Assume U is a component of

19

Figure 3.2: Demonstration of correlation coefficient between two randomvariables from a perfectly anti-correlated case (ρ12 = −1) to a perfectly correlated

case (ρ12 = 1).

velocity at a particular space and time in a repeatable turbulent flow experiment,and that U (n) denotes the nth repetition of the experiment, i.e. coming from an-other measurement of U at the same location and time. The set of random variablesU (1), U (2), U (3), ..., U (N) are independent and have the same distribution, i.e. thatof U , and are said to be independent and identically distributed. The ensemble av-

erage is calculated by

20

〈U〉N ≡1

N

N∑n=1

U (n). (3.12)

Note that an ensemble average results from a finite number of measurementsand is different from 〈U〉. The ensemble average itself is a random variable. It isstraightforward to show that its mean and variance are given as

〈〈U〉N〉 = 〈U〉, (3.13)

〈(〈U〉N − 〈U〉)2〉 =σ2u

N. (3.14)

So even though the means of the ensemble average and the random variableare the same, the standard deviation for an ensemble average reduces comparedto the standard deviation for the random variable as the number of measurementsincrease. In the limit of N → ∞, if the ensemble average is calculated over a verylarge sample, then its standard deviation approaches zero.

It is possible to standardize the ensemble average random variable fluctuations.Again, the standardized random variable has zero mean and unit variance. The stan-dardized random variable U corresponding to 〈U〉N is defined by

U = [〈U〉N − 〈U〉]N1/2/σu. (3.15)

The central-limit theorem states that, as N → ∞, the PDF of U , i.e. f(V ),tends to the standardized Gaussian distribution or normal distribution

f(V ) =1√2π

exp(−1

2V 2

). (3.16)

This is an important result, indicating that regardless of the nature of fluctua-tions, i.e. they could be expressed using any other PDF, the standardized fluctuationsof the ensemble averaged random variable will have a Gaussian or normal PDF.

Other ensemble-averaged joint random variables also exhibit joint Gaussian ornormal distribution in homogeneous turbulence. For instance, the velocity compo-nents and a conserved passive scalar are found to be joint-normally distributed.

21

Random Processes

In many instances, particularly when observing natural phenomena such as atmo-spheric or oceanic flows, it is not possible to repeat an experiment in order to obtainensemble averages. In those cases, it is useful to develop theories for the same ex-periment but only as a function of time, for instance velocity as a function of timeU(t). Such a time dependent random variable is called a random process. Manyrandom processes are statistically stationary. A process is statistically stationary ifall multi-time statistics are invariant under a shift in time. For instance, in a statis-tically stationary atmosphere, if one performs a five-minute average of wind speedat one particular time, one would obtain the same average as another five-minutesample taken at another time. A turbulent flow, after an initial transient period, canreach a statistically stationary state, where even though the flow variables vary withtime, the statistics are independent of time.

For a statistically stationary process, the simplest multi-time statistic that canbe considered is the autocovariance given by

R(s) ≡ 〈u(t)u(t+ s)〉 (3.17)

where a flow quantity is measured at the same location with a time shift s over an in-definite time period. Note that autocovariance is not a function of time t but insteada function of time shift s. In the normalized form, this is called the autocorrelation

function defined by

ρ(s) ≡ 〈u(t)u(t+ s)〉〈u(t)2〉

(3.18)

where u(t) ≡ U(t) − 〈U(t)〉 is the velocity fluctuation. The autocorrelation func-tion is the correlation coefficient between a function and itself at time t and t + s.Consequently ρ(0) = 1 and ρ(s) ≤ 1.

Related to the concept of autocorrelation is the concept of integral timescale.Usually the autocorrelation function diminishes quickly as a function of time shift.This means that usually the integral of the autocorrelation function over time shiftconverges to the integral timescale define as

22

τ =

∫ ∞0

ρ(s)ds. (3.19)

In essence, the integral timescale looks at the overall memory of the process andhow strongly it is influenced or correlated by state of the flow in a previous time.A vivid example is a person standing in a gusty wind. The timescale for person’ssensation of each turbulent eddy arriving at the face is determined by the integraltimescale. If the memory of the turbulent flow is short, the person will feel veryquick bursts of eddies at the face where the integral timescale is short. If the memoryof the turbulent flow is longer, the person will feel longer bursts of eddies at the facewhere the integral timescale is longer.

The autocovariance R(s) ≡ 〈u(t)u(t + s)〉 = 〈u(t)2〉ρ(s) and twice the fre-

quency spectrum E(ω) form a Fourier-transform pair:

E(ω) ≡ 1

π

∫ ∞−∞

R(s)e−iωsds

=2

π

∫ ∞0

R(s) cos(ωs)ds, (3.20)

and

R(s) ≡ 1

2

∫ ∞−∞

E(ω)eiωsdω

=

∫ ∞0

E(ω) cos(ωs)dω, (3.21)

where ω = 2πf is frequency in radians per second, calculated from frequency f incycles per seconds in Hz. Both R(s) and E(ω) contain the same information aboutthe flow but in different forms. The Fourier-transform pair can be better understoodusing Euler’s (1707-1783) formula:

eix = cos(x) + i sin(x), (3.22)

which in our case can be re-written as

eiωs = cos(ωs) + i sin(ωs). (3.23)23

Note that since the integration is performed from −∞ to∞ the i sin(ωs) term,which is an odd function, is integrated to zero and only the term cos(ωs) remains,which is an even function. Due to this, both R(s) and E(ω) are even functions.

A fundamental property of the frequency spectrum is that for a range of fre-quencies ωa < ω < ωb the integral∫ ωb

ωa

E(ω)dω (3.24)

is the contribution to the variance 〈u(t)2〉 of all modes in the frequency range ωa <ω < ωb. In other words, this integral gives the amount of variance 〈u(t)2〉 dueto turbulent fluctuations in that particular frequency range. Of course, the rest ofthe total amount of variance comes from turbulent fluctuations with frequenciesoutside this range. Given the frequency spectrum it is easily possible to calculatethe variance by

R(0) = 〈u(t)2〉 =

∫ ∞0

E(ω)dω. (3.25)

Another relationship can be established between the frequency spectrum andthe autocorrelation function. It can be shown that

τ =πE(0)

2〈u(t)2〉. (3.26)

Not all random processes are differentiable, but random processes that arisefrom turbulent flows are differentiable so that by definition we have

dU(t)

dt= lim

∆t→0

(U(t+ ∆t)− U(t)

∆t

). (3.27)

It is also possible to calculate the mean for the differentiation of a randomprocess, realizing that the mean and taking the limit commute, so that

24

〈dU(t)

dt〉 = 〈 lim

∆t→0

(U(t+ ∆t)− U(t)

∆t

)〉

= lim∆t→0

(〈U(t+ ∆t)〉 − 〈U(t)〉

∆t

)=d〈U(t)〉dt

(3.28)

This result is very important and reads as the mean of the derivative of a random

process is equal to the derivative of the mean of the random process. This result isextremely useful for deriving transport equations for turbulent flows.

Random Fields

In the previous section we discussed a random process at a particular location in aturbulent flow. It is possible to extend the concept of random process to a region ofspace or field as well. The velocity U(x, t) is a time-dependent random vector field.The fluctuating velocity field is defined by

u(x, t) ≡ U(x, t)− 〈U(x, t)〉. (3.29)

The one-point and one-time covariance of the velocity is 〈ui(x, t)uj(x, t)〉.This covariance is computed for every spatial location in the flow field. This co-variance is not to be confused with autocovariance discussed in the autocorrelationfunction, where a time shift was considered at the same spatial point. Rather thiscovariance is calculated at the same time for different components of the velocityin the same spatial location, for instance the horizontal and vertical components.These covariances for the velocity field are called Reynolds stresses, and are writ-ten as 〈uiuj〉, with dependencies on x and t being understood and the notationcompressed without writing x and t.

Turbulent velocity fields are also differentiable with respect to time and space,and the mean and the differentiation commute so that

〈∂Ui∂t〉 =

∂〈Ui〉∂t

, (3.30)

25

〈∂Ui∂xj〉 =

∂〈Ui〉∂xj

. (3.31)

Statistically Stationary, Homogeneous, and Axisymmetric Turbulent Flows

The random field U(x, t) is statistically stationary if all statistics are invariant undera shift in time. Similarly, the field is statistically homogeneous if all statistics areinvariant under a shift in position. Finally, the field is statistically axisymmetric if allstatistics are independent of the circumferential coordinate. While the statisticallystationary or axisymmetric conditions are more common to reach, the statisticallyhomogeneous condition occurs less frequently in applied cases, unless consideringa finite region of space. For instance, most turbulent flows are statistically stationaryif boundary conditions are not time varying. For example, flow over a test articlein a wind tunnel exhibits the statistically stationary condition. Most circular jetsare statistically axisymmetric if there is no body force, such as gravity, exertednormal to the axis of the flow. For instance, a round nozzle sprinkler that ejectswater upward is statistically axisymmetric. The statistically homogenous conditionmay be created in laboratory experiments or within a subset of flow region far awayfrom boundaries.

Isotropic and Anisotropic Turbulence

A statistically homogeneous field U(x, t) is, by definition, statistically invariantunder translation. If the field is also invariant under rotations and reflections of thecoordinate system, then it is statistically isotropic. For instance, in a statisticallyisotropic field 〈u1u2〉 = 〈u1u3〉 = 〈u2u3〉 since the same Reynolds stresses must beobtained regardless of the rotational orientation of the coordinate system. Otherwisethe field is statistically anisotropic. Most of the turbulent theory is centred aroundthis concept. In fact, many turbulence models assume a statistically isotropic condi-tion to be valid. However, most turbulent flows deviate from the isotropic condition.For instance, the atmosphere is found to be anisotropic near the earth surface, i.e.in the atmospheric boundary layer. Fortunately, many turbulence models are devel-oped to treat such cases.

26

Two-point Correlation

In a previous section we discussed the one-point and two-timeR(s) autocovariance,where covariance in one location of the flow was calculated as a function of a timeshift s. It is also possible to define a two-point and one-time autocovariance definedby

Rij(r,x, t) ≡ 〈ui(x, t)uj(x + r, t)〉, (3.32)

which is also known as a two-point correlation. With this statistic it is possible todefine an integral lengthscale by

L11(x, t) ≡ 1

R11(0,x, t)

∫ ∞0

R11(e1r,x, t)dr, (3.33)

where e1 is the unit vector in the x1-coordinate direction. The integral lengthscalemeasures the correlation distance of a process in terms of space or time. In essence,it looks at the overall memory of the process and how it is influenced by previouspositions and parameters. An intuitive example would be the case in which you havea very low Reynolds number flow such as a laminar flow, where the flow is fullyreversible and thus fully correlated with previous particle positions. This conceptmay be extended to turbulence, where it may be thought of as the time during whicha particle is influenced by its previous position.

Wavenumber Spectra

We begin this section by defining the concept of wavenumber vector. The wavenum-ber is the spatial frequency of a wave, either in cycles per unit distance or radiansper unit distance. It can be envisaged as the number of waves that exist over a speci-fied distance (analogous to frequency being the number of cycles or radians per unittime). The wave number can be defined by

|κ| = 1

λ, (3.34)

|κ| = 2π

λ, (3.35)

27

where λ is the wavelength in the direction of the wavenumber vector κ, with thefirst definition in units of cycles per unit distance and the second definition in unitsof radians per unit distance. Figure 3.3 shows the visual representation of the wavepropagation and the wavelength. Note that the wavenumber vector will be in thesame direction as the distance vector shown.

Figure 3.3: Visual representation of wave propagation and the wavelength.

In a previous section we established a relationship between the one-point andtwo-time autocovariance R(s) and the frequency spectrum E(ω). This relationshipwas in the form of a Fourier-transform pair. In this section we seek a similar re-lationship but between the two-point and one-time autocovariance Rij(r,x, t) andvelocity spectrum tensor Φij(κ, t), which is a function of wavenumber vector κ,

Φij(κ, t) =1

(2π)3

∫∫∫ ∞−∞

e−iκ.rRij(r, t)dr, (3.36)

and

Rij(r, t) =

∫∫∫ ∞−∞

eiκ.rΦij(κ, t)dκ, (3.37)

where dr = dr1 dr2 dr3 and dκ = dκ1 dκ2 dκ3. Note that in these relationshipsRij(r, t) is considered independent of x, assuming that turbulence is homogeneous.By setting r = 0 the second integral can give us the Reynolds stress

Rij(0, t) = 〈uiuj〉 =

∫∫∫ ∞−∞

Φij(κ, t)dκ, (3.38)

The velocity spectrum tensor Φij(κ, t) represents the contribution to the co-variance of velocity 〈uiuj〉 at wavenumber κ.

28

Types of Averaging

We have defined the velocity vector mean in turbulent flow as a function of PDF.For instance the mean at any time t was given by

〈U(t)〉 ≡∫ ∞−∞

V f(V ; t)dV (3.39)

However, in turbulent flow an exact PDF is not known or is not possible to mea-sure or calculate. Instead, various types of averaging techniques are performed insimulation and experiments in order to determine an estimate for the mean 〈U(t)〉.We already introduced the concept of an ensemble average, where an identical ex-periment is repeated N number of times and a velocity component is measured atsome location and time N times. The ensemble average is given by

〈U(t)〉N ≡1

N

N∑n=1

U (n)(t) (3.40)

where U (n)(t) is the nth measurement or experiment. Alternatively, and for statisti-cally stationary flow it is possible to take a time average starting at some arbitrarytime t for a period of T , given by

〈U(t)〉T ≡1

T

∫ t+T

t

U(t′)dt′. (3.41)

where an integral is used, assuming a continuous measurement of velocity is avail-able. Where a time discrete measurement is available, it is necessary to replace theintegral with a sum to calculate the time average. In simulations of homogeneousturbulence, it is also possible to calculate a spatial average in a cubic domain ofside L of a component of velocity field U(x, t), given by

〈U(t)〉L ≡1

L3

∫∫∫ L0

U(x, t)dx1 dx2 dx3. (3.42)

where an integral is used, assuming a continuous measurement of velocity is avail-able. Where a space discrete measurement is available, it is necessary to replace theintegral with a sum to calculate the spatial average. These averages, i.e. 〈U(t)〉N ,

29

〈U(t)〉T , and 〈U(t)〉L are random variables themselves, but they can be used toestimate 〈U(t)〉, but not to estimate it with certainty.

Exercises

1) In a one-dimensional mechanical wave the wavelength is λ = 5 m. What is thevalue and units of the wave number κ? Calculate the wave number in both units ofcycles per unit distance and radians per unit distance.2) Derive the following formula, also known as the Euler’s identity, which wasknown as early as the 18th century,

eiπ + 1 = 0. (3.43)

3) The autocorrelation function ρ(s) for a flow is given as a function of s in secondsfor the range s = 0 s to s =∞s as,

ρ(s) =1

(1 + s)2. (3.44)

Show that the integral timescale of the flow is equal to

τ = 1 s. (3.45)

4) The two-point correlation in a flow is given by,

R11(e1r,x, t) = f(x, t)e−r/a, (3.46)

where f(x, t) is a well-behaved function of time and space that is continuous, dif-ferentiable, and integrable, e is the Euler number, and a is a positive real number. Ifr is distance in meters, show that for this flow the integral lenghscale in meters canbe given by

L11(x, t) = a. (3.47)

5) A combustion scientist has developed a chamber for combustion experiments,where he premixes air and fuel and then pressurizes the mixture for auto-ignition,i.e. self-ignition without having to use a spark. He is interested in the turbulence

30

properties of combustion at the centre of the chamber in a region of space 2 ×2 × 2cm3. He develops a fast camera system to take two-dimensional images ona 2 × 2cm2 plane slicing the centre of this region to obtain images of temperaturefluctuations. For each combustion experiment he varies the air and fuel proportionsand then takes images at multiple instances of time before and after complete com-bustion. He then averages temperature fluctuations for each image at a specific timeand reports a mean for temperature and a variance for temperature fluctuations as afunction of time. a) Has he assumed the temperature field to be stationary or non-stationary? b) Has he assumed the temperature field to be statistically homogeneousor non-homogeneous? c) What type of averaging has he performed, i.e. ensemble,time, or spatial averaging?

31

Chapter 4

Mean Flow Equations

In the last chapter we discussed various statistical quantities that described variousaspects of turbulent velocity fields. These included PDF, mean, moments, Reynoldsstress, one-point, and two-point correlations. The goal of this chapter is to developthe Navier-Stokes transport equations for various statistical quantities in turbulentflow, as opposed to the equations for the instantaneous quantities. The most basicof these equations are those derived for mean velocity field 〈U(x, t)〉. As discussedearlier, the velocity field can be decomposed into the mean velocity field and thefluctuating velocity field using Reynolds decomposition, i.e.

U(x, t)︸︷︷︸Instantaneous Velocity

= 〈U(x, t)〉︸︷︷︸Mean Velocity

+ u(x, t)︸︷︷︸Fluctuating Velocity

. (4.1)

The continuity equation implies that all U(x, t), 〈U(x, t)〉, and u(x, t) aresolenoidal, given the fact that the mean and differentiation operations commuteso that 〈∇.U〉 = ∇.〈U〉 = 0. This results in

∇.〈U(x, t)〉 = 0, (4.2)

∇.u(x, t) = 0. (4.3)

We next focus our attention on the material or substantial derivative to obtainan expression for the material derivative of mean velocity field. By definition ofmaterial derivative we can arrive at the expressions

DUjDt︸︷︷︸

Material Derivative

=∂Uj∂t︸︷︷︸

Storage

+∂

∂xi(UiUj)︸︷︷︸

Advection

, (4.4)

〈DUjDt〉 =

∂〈Uj〉∂t

+∂

∂xi〈UiUj〉. (4.5)

32

It is then needed to find an expression for 〈UiUj〉. This expression can be ob-tained as follows

〈UiUj〉 = 〈(〈Ui〉+ ui)(〈Uj〉+ uj)〉

= 〈〈Ui〉〈Uj〉+ ui〈Uj〉+ uj〈Ui〉+ uiuj〉

= 〈Ui〉〈Uj〉+ 〈uiuj〉. (4.6)

Note that the mean for each of the terms ui〈Uj〉 and uj〈Ui〉 is equal to zero sinceeither ui or uj can be given by PDFs centred at zero and symmetric with respect tozero. In other words, the fluctuating velocity field has the same likelihood of beingpositive or negative and with equal distributions on either side, so that the terms〈ui〈Uj〉〉 and 〈uj〈Ui〉〉 will be zero. With this development the mean of materialderivative for velocity field can be given as

〈DUjDt〉 =

∂〈Uj〉∂t

+∂

∂xi(〈Ui〉〈Uj〉+ 〈uiuj〉)

=∂〈Uj〉∂t

+ 〈Ui〉∂

∂xi〈Uj〉+

∂

∂xi〈uiuj〉, (4.7)

where the second step follows from ∂〈Ui〉/∂xi = 0, which is another manifestationof the continuity equation. A useful notation to develop is the mean substantial

derivative given by

D

Dt≡ ∂

∂t+ 〈U〉.∇. (4.8)

For any property in the flow, this mean substantial derivative represents its rateof change as the property is following a point moving with the local mean velocity.Using this notation the mean of material or substantial derivative of velocity fieldcan be shown as

〈DUjDt〉︸︷︷︸

Mean of Material Derivative

=D

Dt〈Uj〉︸︷︷︸

Material Derivative of Mean

+∂

∂xi〈uiuj〉︸︷︷︸

Reynolds Stresses

. (4.9)

Certainly the mean of material or substantial derivative of velocity field shallnot be confused with, nor is it equal to, the mean substantial derivative. With this

33

consideration, it is possible to develop the equation for the mean momentum suchthat

D〈Uj〉Dt︸︷︷︸


= ν∇2〈Uj〉︸︷︷︸Surface Forces

− ∂〈uiuj〉∂xi︸︷︷︸

Reynolds Stresses

− 1

ρ

∂〈p〉∂xj︸︷︷︸

Normal and Body Forces

. (4.10)

The momentum equations for instantaneous and mean velocity field are re-markably similar, but with the crucial difference of the Reynolds stress term. Asshall be seen, much of the statistical turbulence modelling theory revolves aroundparameterizing this Reynolds stress term. This term shows an instance of the closure

problem, where in turbulent flow equations one always encounters more unknownsthan the number of equations available to close the system of equations.

Tensor Properties

It is worth remembering that Reynolds stresses 〈uiuj〉 are components of a second-order tensor, with the property that it is symmetric, i.e. 〈uiuj〉 = 〈ujui〉. The diago-nal components of this tensor, i.e. 〈u2

1〉 = 〈u1u1〉, 〈u22〉, and 〈u2

3〉 are called normal

stresses, while the off-diagonal components, e.g. 〈u1u2〉, are called shear stresses.The Reynolds stress tensor can be shown in the matrix as follows 〈u

21〉〈u1u2〉〈u1u3〉

〈u2u1〉〈u22〉〈u2u3〉

〈u3u1〉〈u3u2〉〈u23〉

.Given the symmetric property of the Reynolds stress tensor, the matrix is sym-

metric about the diagonal. The turbulent kinetic energy k(x, t) is defined to be halfthe trace of the Reynolds stress tensor. i.e.

k ≡ 1

2〈u.u〉 =

1

2〈uiui〉. (4.11)

34

Anisotropy

The difference between shear stresses and normal stresses is dependent on thechoice of a coordinate system. For instance, if the coordinate system is rotated,the Reynolds stress tensor components may change. It is useful to express the stresstensor as the sum of isotropic and anisotropic parts. This intrinsic distinction im-plies that the isotropic component of the stresses would never change, regardless ofthe choice of the reference coordinate system, but the anisotropic part may change.This can be done as follows

〈uiuj〉 = 〈uiuj〉 −2

3kδij︸︷︷︸

Anisotropic Part

+2

3kδij︸︷︷︸

Isotropic Part

. (4.12)

The anisotropic part is also known as anisotropy and is notated with the symbolaij ≡ 〈uiuj〉− 2

3kδij . An important concept is that it is only the anisotropic compo-

nent that is effective in turbulent transport of momentum. Therefore, the combina-tion of Reynolds stress term and the pressure term in the momentum equation canbe written as

ρ∂〈uiuj〉∂xi

+∂〈p〉∂xj

= ρ∂aij∂xi

+∂

∂xj(〈p〉+

2

3ρk), (4.13)

demonstrating that the isotropic component 23k can be absorbed in the modified

mean pressure.

Mean Scalar Equation

The Navier-Stokes transport or conservation equation for a passive scalar fieldφ(x, t) can be developed further for the mean passive scalar field 〈φ(x, t)〉 as well.Again, the Reynolds decomposition can be applied to the passive scalar field

φ(x, t)︸︷︷︸Instantaneous Scalar

= 〈φ(x, t)〉︸︷︷︸Mean Scalar

+ φ′(x, t)︸︷︷︸Fluctuating Scalar

. (4.14)

We begin with the conservation equation for the instantaneous passive scalarfield

35

∂φ


+∇.(Uφ)︸︷︷︸Advection

= Γ∇2φ︸︷︷︸Diffusion

. (4.15)

The only nonlinear term involves the convective flux Uφ, for which we canobtain the following mean

〈Uφ〉 = 〈(〈U〉+ u)(〈φ〉+ φ′)〉

= 〈U〉〈φ〉+ 〈uφ′〉. (4.16)

The velocity-scalar covariance 〈uφ′〉 is a vector known as scalar flux. It quanti-fies the flow rate of the scalar, per unit time and unit area, due to fluctuating velocityfield. This mechanism for transporting the scalar is just as important as the mech-anism due to mean velocity field. In fact, in many flows that do not exhibit meanvelocity field in some particular direction, the scalar flux is the only mechanismthat transports the scalar in that direction. Many boundary-layer flows, such as theatmospheric boundary layer, can exhibit this condition under certain circumstances.Taking the mean of the scalar transport equation, we obtain the following results

∂〈φ〉∂t

+∇.(〈U〉〈φ〉+ 〈uφ′〉) = Γ∇2〈φ〉, (4.17)

D〈φ〉Dt

= ∇.(Γ∇〈φ〉 − 〈uφ′〉). (4.18)

This equation shows another instance of the turbulence closure problem wherethe term 〈uφ′〉 has to be modelled further or parameterized in order to be able tosolve the passive scalar equation.

Gradient-diffusion and Turbulent-viscosity Hypotheses

The first attempt to model or parameterize the scalar flux was made by JosephValentin Boussinesq (1842-1929). He hypothesized that the mean scalar flux is inthe direction and proportional to the negative mean scalar gradient, i.e. ∇〈φ〉. Thisis known as the gradient-diffusion hypothesis or turbulent-viscosity hypothesis. The

36

constant of proportionality is the turbulent diffusivity that is itself a function ofspace and time, ΓT (x, t), i.e.

〈uφ′〉 = −ΓT∇〈φ〉. (4.19)

The subscript T is for turbulence, and ΓT is turbulent diffusivity and shouldnot be confused with molecular diffusivity Γ. This hypothesis is similar to Fourier’slaw of heat conduction and Fick’s law of molecular diffusion, where the heat orscalar flux is always in the direction from a hot or high concentration region towarda cold or low concentration region. In other words, the diffusion is in the negativedirection of the gradient. Because of this, another term for this hypothesis is thedown-gradient hypothesis. It is remarkable that, at least conceptually, the scalarturbulent flux is formulated in the similar manner as the molecular flux. Due to thisreason it is possible to combine the molecular and turbulent diffusivities to arrive atan effective diffusivity given by

Γeff (x, t)︸︷︷︸Effective Diffusivity

= Γ︸︷︷︸Molecular Diffusivity

+ ΓT (x, t)︸︷︷︸Turbulent Diffusivity

. (4.20)

The effective diffusivity may be referred to as effective heat diffusivity associ-ated with a temperature or effective mass diffusivity associated with a passive scalarconcentration. With this consideration, it is possible to express the mean scalar con-servation equation as

D〈φ〉Dt︸︷︷︸


= ∇.(Γeff∇〈φ〉)︸︷︷︸Diffusion of Mean

. (4.21)

For the mean momentum transport equation, the gradient-diffusion hypothesisis more delicate to implement. In this case we should relate the anisotropic part ofthe Reynolds stress 〈uiuj〉 − 2

3kδij to the mean rate of strain by

〈uiuj〉 −2

3kδij = −νT

(∂〈Ui〉∂xj

+∂〈Uj〉∂xi

)= −2νTSij, (4.22)

37

where the positive scalar coefficient νT is the momentum diffusivity also known asturbulent viscosity or eddy viscosity. With these considerations, the mean momen-tum equation becomes

D

Dt〈Uj〉︸︷︷︸


=∂

∂xi

[νeff

(∂〈Ui〉∂xj

+∂〈Uj〉∂xi

)]︸︷︷︸

Surface Forces and Reynolds Stress

− 1

ρ

∂

∂xj(〈p〉+

2

3ρk)︸︷︷︸

Modified Pressure

,

(4.23)

νeff (x, t)︸︷︷︸Effective Viscosity

= ν︸︷︷︸Molecular Viscosity

+ νT (x, t)︸︷︷︸Turbulent Viscosity

, (4.24)

where νeff is the effective viscosity and 〈p〉 + 23ρk is the modified pressure. It

must be recalled that the gradient diffusion hypothesis can only be applied to theanisotropic part of the Reynolds stress because the momentum is transported onlyby the anisotropic part of the Reynolds stress. This consideration requires the mod-ified pressure instead of pressure to remain in the transport equation.

Similar to molecular diffusivity, turbulent diffusivity can be non-dimensionalizedas well. The turbulent Prandtl number gives the ratio of momentum diffusivity toheat diffusivity by

PrT =νTΓT. (4.25)

Likewise, the turbulent Schmidt number gives the ratio of momentum diffusiv-ity to mass diffusivity by

ScT =νTΓT. (4.26)

The turbulent Prandtl and Schmidt numbers are not material constants and mayvary greatly according to flow conditions. In addition, they may not even be equalto each other. However, under special cases, for instance for air, where the Lewisnumber Le = Sc/Pr ≈ 1, we can employ the concept of Reynolds analogy andassume Pr = Sc and PrT = ScT (Reynolds, 1975; Flores et al., 2013).

It is worth noting that the gradient-diffusion hypothesis is a simple hypothesisthat enables closing a turbulence equation. However, this hypothesis is not valid

38

for many flows, even the simplest flows, and can lead to significant errors in thecalculation. So it must be used with extreme care. It is only in light of historical de-velopments and its limited use that we introduce this hypothesis as a closure schemefor a turbulence equation. Furthermore, the turbulent diffusivity ΓT itself needs tobe modelled accurately in order to arrive at more accurate closure schemes. Again,much of the theory of turbulence modelling is focused around the parameterizationfor the turbulent diffusivity, of course for gradient-diffusion models.

Exercises

1) The simplest model for the turbulent Prandtl number assumes PrT = 1. How-ever, numerous experimental observations suggest that PrT can be variable evenfor the same fluid under different flow circumstances. For instance, flow near wallsexhibits a different turbulent Prandtl number compared to flow in the internal regionof a fluid domain away from the walls (Reynolds, 1975). Assuming PrT = 0.85,express a relationship between momentum diffusivity νT and heat diffusivity ΓT .2) Using the gradient-diffusion hypothesis, the surface and Reynolds stress forcesin the momentum equation are expressed by the term

∂

∂xi

[νeff

(∂〈Ui〉∂xj

+∂〈Uj〉∂xi

)]. (4.27)

Reason why this term cannot be expressed and is not equal to the followingterm under general conditions

νeff∂

∂xi

[(∂〈Ui〉∂xj

+∂〈Uj〉∂xi

)]. (4.28)

3) Suppose that in a flow the mean passive scalar field 〈φ〉 can be expressed as afunction of x, y, and z coordinates along the Cartesian system with unit vectors i,j, and k, respectively, such that

〈φ〉 = φ0(x+ y2 − z3), (4.29)

39

where φ0 is a constant. Assuming that the gradient-diffusion hypothesis applies tothis flow and passive scalar field, show that the passive scalar flux 〈uφ′〉 is alongthe following unit vector at location (x, y, z) = (1, 1, 1), i.e.

〈uφ′〉|〈uφ′〉|

|(1,1,1) = − 1√14

i− 2√14

j +3√14

k. (4.30)

4) Given a constant effective viscosity νeff that accounts for effects of molecularand turbulent diffusion, we wish to develop a one-dimensional (1D) momentumtransport equation. Suppose we use the Cartesian coordinate system with coordinateaxes of x, y, and z, and velocities corresponding to these axes being U = 〈U〉+ u,V = 〈V 〉+ v, and W = 〈W 〉+w, respectively. Further, we assume that mean flowis only in the x direction parallel to the surface and that the direction z is normal tothe surface, i.e. 〈V 〉 = 〈W 〉 = 0. We assume steady state conditions. In addition,we assume that the mean velocity 〈U〉 in the x and y directions does not change.Also we assume that the modified pressure has a constant gradient in the x direction.Show that the 1D momentum equation then simplifies to

0 = νeffd2〈U〉dz2

− τ. (4.31)

where τ is the constant modified pressure gradient in the x direction divided by den-sity. Demonstrate why the storage and advection terms in the momentum transportequation have vanished.5) Given a non-constant turbulent viscosity νT that accounts for effects of molecularand turbulent diffusion and a turbulent Prandtl number PrT = 1, we wish to developa one-dimensional (1D) heat (temperature) transport equation. Suppose we use theCartesian coordinate system with coordinate axes of x, y, and z, and velocitiescorresponding to these axes being U = 〈U〉+u, V = 〈V 〉+ v, and W = 〈W 〉+w,respectively. Further, we assume that mean flow is only in the x direction parallelto the surface and that the direction z is normal to the surface, i.e. 〈V 〉 = 〈W 〉 = 0.We assume steady state conditions. In addition, we assume that the mean velocity〈U〉 in the x and y directions does not change. We also assume a constant heat sinkor source for temperature by a uniform rate of cooling or heating in the domain.Show that the 1D heat equation then simplifies to

40

0 =∂

∂z

(νT∂〈T 〉∂z

)− γ, (4.32)

where the term γ is a sink or source for temperature by uniform rate of cooling orheating in the domain. Demonstrate why the storage and advection terms in the heattransport equation have vanished.6) Given a non-constant turbulent viscosity νT that accounts for effects of molec-ular and turbulent diffusion and a turbulent Schmidt number ScT = 1, we wish todevelop a transient one-dimensional (1D) passive scalar transport equation. Sup-pose we use the Cartesian coordinate system with coordinate axes of x, y, and z,and velocities corresponding to these axes being U = 〈U〉 + u, V = 〈V 〉 + v, andW = 〈W 〉 + w, respectively. Further, we assume that mean flow is only in the xdirection parallel to the surface and that the direction z is normal to the surface, i.e.〈V 〉 = 〈W 〉 = 0. In addition, we assume that the mean velocity 〈U〉 in the x and ydirections does not change. Show that the 1D passive scalar transport equation thensimplifies to

∂〈φ〉∂t

=∂

∂z

(νT∂〈φ〉∂z

). (4.33)

Demonstrate why the advection terms in the passive scalar transport equation havevanished.7) The turbulence intensity, also often referred to as turbulence level, is defined as

I ≡ u

〈U〉, (4.34)

Where u is the root-mean-square of the turbulent velocity fluctuations and 〈U〉 isthe mean velocity. Assuming that turbulence is isotropic, show that

I ≡

√23k

〈U〉, (4.35)

where k is the turbulent kinetic energy. For atmospheric flows I is typically lessthan one and under gusty conditions it can be equal to or larger than one.8) A meteorologist is working on the following problem. The atmospheric boundary-layer can be approximated by a one-dimensional transport equation for heat, i.e.

41

temperature, in the vertical direction (z) over short distances, where atmosphericpressure can be assumed constant with no temperature lapse rate. The one-dimensionalassumption is based on horizontal homogeneity of the atmosphere and no meanvertical motion of air. Using this approximation, the vertical transport of heat ortemperature can be expressed as

∂〈T 〉∂t

= −∂〈tw〉∂z

, (4.36)

where T = 〈T 〉 + t is temperature in Kelvin [K] expressed as the sum of meanand turbulent fluctuating parts, and W = 〈W 〉 + w is wind speed in the verticaldirection in [m s−1] expressed as the sum of mean and turbulent fluctuating parts.Of course it is assumed that 〈W 〉 = 0 m s−1. Assume that the turbulent heat flux in[K m s−1] in a range of heights from 10 to 30 m for a diurnal time is given as

〈tw〉 = − 1

1 + z, (4.37)

where z is height in m. Help the meteorologist calculate how much the temperatureof the atmosphere changes at a height of z = 15 m over a time duration of 100 s. Isthe atmosphere cooling or heating in this scenario?9) A meteorologist is deriving the energy equation (i.e. temperature equation) nearthe earth surface. The atmospheric boundary-layer can be approximated by a one-dimensional transport equation for heat, i.e. temperature, in the vertical direction(z) over short distances, where atmospheric pressure can be assumed constant withno temperature lapse rate. The one-dimensional assumption is based on horizontalhomogeneity of the atmosphere and no mean vertical motion of air. a) Using thisapproximation, show that the temperature near the surface can be expressed as

0 =d

dz

(Γd〈T 〉dz− 〈tw〉

), (4.38)

where T = 〈T 〉+t is temperature in [K] expressed as the sum of mean and turbulentfluctuating parts, W = 〈W 〉 + w is wind speed in the vertical direction in [m s−1]expressed as the sum of mean and turbulent fluctuating parts, and Γ is thermaldiffusivity in [m2 s−1]. b) Using the turbulent viscosity assumption show that this

42

equation can be written as

0 =d

dz

[(ν

Pr+

νTPrT

)d〈T 〉dz

]. (4.39)

10) Consider a two dimensional flow field in which velocity can be given by timeaveraging along x and z components using Reynolds averaging such that U =

〈U〉 + u and W = 〈W 〉 + w. Consider that there is also a passive scalar that isbeing transported by the velocity field and can be shown using Reynolds averagingby S = 〈S〉+s. Of course the velocity and passive scalar are functions of space andtime. Suppose that one wishes to calculate the total flux of the passive scalar alongthe x and z directions at a given point. Show that the total fluxes can be calculatedusing the following expressions:

〈SU〉︸︷︷︸Total flux

= 〈S〉〈U〉︸︷︷︸Advective flux

+ 〈su〉︸︷︷︸Turbulent flux

, (4.40)

〈SW 〉︸︷︷︸Total flux

= 〈S〉〈W 〉︸︷︷︸Advective flux

+ 〈sw〉︸︷︷︸Turbulent flux

. (4.41)

In another word, the total flux can be calculated as the sum of advective fluxand turbulent flux.11) A meteorologist is developing a model for vertical transport of atmosphericspecies within the atmospheric boundary layer. The boundary layer is a two dimen-sional flow field in which velocity can be given by time averaging along x and zcomponents using Reynolds averaging such that U = 〈U〉+ u and W = 〈W 〉+w.Here x is along wind direction, or parallel to earth surface, and z is normal to earthsurface positive upward. Consider that there is also a passive scalar that is beingreleased at the surface and transported upward due to molecular and turbulent dif-fusion transport mechanisms. Using Reynolds averaging the passive scalar can beshown by S = 〈S〉+ s. The vertical flux of the species along the z direction can begiven by

FS︸︷︷︸Vertical flux of S

= − ν

Sc

d〈S〉dz︸︷︷︸

Molecular diffusion flux

+ 〈sw〉︸︷︷︸Turbulent diffusion flux

, (4.42)

43

where ν = 1.5× 10−5 m2 s−1 is molecular viscosity of air and Sc = 1 is molecularSchmidt number. Here 〈W 〉 = 0, therefore there is no advective flux. Consider thatthe mean passive scalar varies exponentially away from the surface, i.e. 〈S〉 = e−z,with units of [kg m−3]. a) Perform unit analysis to find the appropriate unit for FS .b) Use the turbulent-diffusion hypothesis to express the flux only in terms of ν,Sc, νT , ScT , and 〈S〉, where subscript T denotes turbulent. c) Now consider that atz = 1 m, νT = 5 m2 s−1 and ScT = 1. Calculate FS . d) In the previous calculation,what fraction of the flux is due to molecular diffusion and what fraction of the fluxis due to turbulent diffusion? What do you conclude?

44

Chapter 5

Wall Flows

Most turbulent flows are bounded by at least in part one or more solid surfaces.For instance flows in pipes or buildings are fully surrounded by surfaces and areso called internal flows. Flows over the earth surface, around a ship hull, and overairfoil of an aircraft or vehicle are partially surrounded by surfaces and are so calledexternal flows. Understanding flows near surfaces or walls are of fundamental im-portance in turbulence studies.

Transport Equations and the Balance of Mean Forces

We do not restrict our discussion to internal or external flows but will discuss fully

developed flows near walls, defined as a state where velocity statistics no longer varyin the direction of the flow. For instance, this is the case in short range for atmo-spheric flows near the earth surface on flat terrain (Aliabadi et al., 2016). Supposethe flow near the wall is in the x direction, the direction normal to the wall is y, andthe spanwise direction is z, in the Cartesian coordinate system. The correspondingmean velocities are 〈U〉, 〈V 〉, and 〈W 〉, respectively.

As shown in Figure 5.1, suppose that the influence of the wall on the flow isdiminished at a distance of δ away from the wall in the normal direction, where thefree stream velocity is almost achieved completely, say by 99%, 0.99U0 = 〈U〉y=δ.The distance δ represents the thickness of the boundary layer that is a thin layer ofviscous fluid close to the solid surface of a wall in contact with a moving stream, inwhich the flow velocity varies from zero at the wall up to close to the value of thefree stream velocity at the boundary (usually 99% of the free stream velocity). δ isalso known as boundary-layer thickness. The bulk velocity of the flow within theboundary layer is given by

U ≡ 1

δ

∫ δ

0

〈U〉dy. (5.1)

45

Figure 5.1: Schematic of two-dimensional wall flow and boundary layer for thefully developed flow near the wall.

Since 〈U〉 does not vary as a function of x for fully developed flow and that〈W 〉 = 0, the continuity equation reduces to

d〈V 〉dy

= 0. (5.2)

Note that the continuity equation is written with derivative as opposed to par-tial derivative. This is the case since all mean variables in the fully developed regionare only a function of y. The boundary condition at the wall is the no-slip bound-

ary condition implying that the fluid does not move near a stationary wall, whichimplies that 〈V 〉 = 0 everywhere. This implies that V = v, i.e. the velocity in they direction is equal to the turbulent fluctuation in that direction. With this develop-ment, the mean-momentum equation in the y direction can be written as

0 = − d

dy〈v2〉︸︷︷︸

Reynolds Stress

− 1

ρ

∂〈p〉∂y︸︷︷︸


. (5.3)

Again note that in the Reynolds stress term we have a derivative but in thenormal and body forces term we have a partial derivative. This equation can be in-tegrated from y = 0 to an arbitrary height. At the wall 〈v2〉y=0 = 0 but 〈p(x)〉y=0 =

Pw(x), therefore the integration of the mean momentum equation in the y directioncan be written as

46

〈v2〉+ 〈p〉/ρ = pw(x)/ρ. (5.4)

If we now take the derivative of this equation with respect to x we find thatthe partial derivative of the mean pressure at any height is equal to the derivative ofpressure at the wall in the x direction, i.e.

∂〈p〉∂x

=dpw(x)

dx, (5.5)

which states that the mean axial pressure drop is uniform across the flow, no matterhow close or far a location is from the wall. We can now write the momentumequation in the x direction as

0 = νd2〈U〉dy2︸︷︷︸

Surface Forces

− d

dy〈uv〉︸︷︷︸

Reynolds Stress

− 1

ρ

∂〈p〉∂x︸︷︷︸


. (5.6)

If we group the first two terms on the right hand side, this equation can bewritten as

dτ

dy=dpw(x)

dx(5.7)

where τ is the total shear stress accounting for surface forces and the Reynoldsstress, but not the normal and body forces, which are accounted for in the pressureterm. The total shear stress can be written as

τ = ρνd〈U〉dy− ρ〈uv〉. (5.8)

Since there is no net acceleration in this flow, the axial normal stress gradi-ent dpw(x)/dx is balanced by the cross-stream shear-stress gradient dτ/dy. Thedifferential equation written for this balance states that the shear-stress is only afunction of y, while the normal stress is only a function of x. This implies that bothdpw(x)/dx and dτ/dy must be constants. The value of τ at the wall, i.e. τw is knownas wall shear stress

τw = τ(y = 0). (5.9)

47

The wall shear stress can be normalized by a reference velocity to give theskin-friction coefficient. This type of friction is analogous to the coefficient of dragin bluff body flow. In fact skin-friction imposes a type of drag on the flow. Theskin-friction coefficient can be given as either of

cf ≡τw

12ρU2

0

, (5.10)

Cf ≡τw

12ρU

2 . (5.11)

The Shear Stress Near Wall

The total shear stress at any distance away from the wall, i.e. τ(y), is the sum ofthe viscous stress ρνd〈U〉/dy and the Reynolds stress −ρ〈uv〉. We know that atthe wall U(x, t) = 0, and therefore there cannot be any turbulent fluctuations andconsequently no Reynolds stress. Consequently, the wall shear stress is only causedby surface forces, i.e.

τw ≡ ρν

(d〈U〉dy

). (5.12)

The kinematic viscosity ν, density ρ, and shear stress near the wall τw are im-portant parameters in turbulence studies. After all, they determine the skin frictionand other flow properties. From these quantities we can define viscous scales thatare appropriate velocity scales and lengthscales associated with the near wall re-gion. The friction velocity and viscous lengthscale are defined as

uτ ≡√τwρ, (5.13)

δv ≡ ν

√ρ

τw=

ν

uτ. (5.14)

The friction Reynolds number is defined with friction velocity but the entireheight of the boundary layer, i.e.

Reτ ≡uτδ

ν=

δ

δv. (5.15)

48

An important parameter in the study of turbulence near walls is the distancefrom the wall measured in viscous lengths, or wall units, denoted by

y+ ≡ y

δv=uτy

ν. (5.16)

This quantity is similar to a local Reynolds number, so that its magnitude can beexpected to determine the relative importance of viscous and turbulent processes. Ify+ is small, then the relative importance of viscous processes is higher, while if y+

is large, the relative importance of turbulent processes is higher. In fact, knowledgeof the magnitude of y+ is sufficient to understand which wall regime is present withrespect to viscous and turbulent processes. Figure 5.2 shows the classification ofwall layers and regions (sublayers). In general, we have two layers: the inner layer

and the outer layer that of course overlap significantly.

Figure 5.2: Wall layers (black-filled) and regions (grey-filled) defined in terms ofy+.

Another important parameter in the study of turbulence near walls is the non-dimensional mean velocity, which is mean velocity divided by friction velocity,defined by

u+ ≡ 〈U〉uτ

. (5.17)

A critical task in studies of turbulence near walls is to find a functional rela-tionship between y+ and u+, also known as the law of the wall, i.e.

49

u+ = fw(y+). (5.18)

Of course there is no universal law of the wall since the law of the wall canchange according to the physics of the flow. However, for typical flows, the lawof the wall yields three distinct wall regions or sublayers: viscous sublayer, buffer

sublayer, and log-law sublayer.

Viscous, Buffer, and Log-law Sublayers

The viscous, buffer, and log-law sublayers were first studied by Theodore vonKarman (1881-1963) (von Karman, 1931). In the viscous sublayer y+ < 5, thefunctional relationship between y+ and u+ is linear, so that it can be given as

u+ = y+. (5.19)

In the buffer sublayer 5 < y+ < 30, there is no simple relationship betweeny+ and u+. This is the region where both viscous and turbulent processes can bepresent. In the log-law sublayer y+ > 30 the relationship between y+ and u+ islogarithmic such that

u+ =1

κln y+ +B, (5.20)

κ = 0.41, B = 5.2. (5.21)

where κ is the von Karman constant and B is another constant. These constants canbe slightly different depending on the physics of the flow. The law of the wall forthe viscous, buffer, and log-law sublayers are demonstrated in Figure 5.3.

Law of the Wall for Temperature

It is possible to develop a law of the wall for the energy (temperature) equation.Consider a flow along x axis near a wall with axis y normal to the wall. Assuminghorizontal homogeneity and using the turbulent viscosity assumption the energyequation can be written as

50

Figure 5.3: Law of the wall and classifications of viscous, buffer, and log-lawsublayers.

0 =d

dy

[(µ

Pr+

µTPrT

)d〈T 〉dy

], (5.22)

where 〈T 〉 is ensemble mean temperature in [K], µ is dynamic viscosity in [kg m−1 s−1],Pr is Prandtl number, µT is turbulent dynamic viscosity in [kg m−1 s−1], and PrTis turbulent Prandtl number. This differential equation requires that the term in thesquare bracket be a constant, i.e.

C = − qwCp

=

(µ

Pr+

µTPrT

)d〈T 〉dy

, (5.23)

where qw is the specific heat flux between the wall and fluid in [J m−2 s−1] and Cpis heat capacity of the fluid at constant pressure in [J kg−1 K−1]. This is a separabledifferential equation that can be written as∫ 〈T 〉

Tw

d〈T 〉 = − qwCp

∫ y

0

dyµPr

+ µTPrT

, (5.24)

51

where Tw is the temperature of the wall. We can introduce the normalized valuessuch that

y+ ≡ yuτν, (5.25)

T+ ≡ (Tw − 〈T 〉)ρCpuτqw

, (5.26)

where ν is kinematic viscosity in [m2 s−1], ρ is the fluid density in [kg m−3], anduτ is friction velocity in [m s−1]. Using the relationships above we can derive thefollowing equation for T+

T+ =

∫ y+

0

dy+

1Pr

+ (νT /ν)PrT

, (5.27)

where νT is turbulent viscosity in [m2 s−1]. This integral is not useful practically.However, we can show that if two layers are considered, one laminar where y+ =

0→ y+crit and one turbulent where y+ > y+

crit, T+ can be written as

T+ =

∫ y+crit

0

Prdy+ +

∫ y+

y+crit

dy+

(νT /ν)PrT

. (5.28)

Using the mixing length theory we can relate turbulent viscosity and distanceaway from the wall by νt = uτκy where κ = 0.41 is the von Karman constant. Wecan show that in such a case the T+ equation can be written as (Bredberg, 2000)

T+ = y+critPr +

PrTκ

ln

(y+

y+crit

), (5.29)

which is known as the law of the wall for the temperature field. For instance, for air,if Pr = 0.7, PrT = 0.85, and y+

crit = 13.2, then the following expression can beobtained as the law of the wall if y+ is in the turbulent layer (Kays and Crawford,1993)

T+ = 2.075 ln(y+) + 3.9. (5.30)

52

Exercises

1) Show that the Reynolds number based on the viscous scales near a wall is iden-tically unity, i.e.

uτδvν

= 1. (5.31)

2) The law of the wall states that for the viscous sublayer u+ = y+ and for thelog-law sublayer u+ = 1

κln y+ + B, where κ = 0.41 and B = 5.2. It is desired

to calculate the point of intersection of the two equations, i.e. the value of y+ forwhich the same value of u+ is obtained using either equation. An exact calculationof such y+ requires solving a non-linear equation involving both y+ and ln y+.However, a first approximation for the suitable y+ can be obtained by replacing thelog-law equation with its first order Taylor expansion expressed about y+ = 10.Doing so the non-linear equation can be linearized and solved conveniently. Usingthis approach, show that the value of y+ in question is approximately

y+ ≈ 11.1. (5.32)

3) Show that in general

cf = 2

(uτU0

)2

. (5.33)

4) Various thickness quantities can be defined to characterize wall flows. Thesethickness quantities themselves can be a function of distance x downstream of theflow. In this chapter, the boundary-layer thickness was defined and noted with δ(x).The displacement thickness can be defined as

δ∗(x) ≡∫ ∞

0

(1− 〈U〉(y)

U0

)dy, (5.34)

where y is the axis normal to the wall, 〈U〉(y) is mean velocity at any height, and U0

is freestream mean velocity. Alternatively, the momentum thickness can be definedas

θ(x) ≡∫ ∞

0

〈U〉(y)

U0

(1− 〈U〉(y)

U0

)dy. (5.35)

53

Provide an argument to rank these thicknesses, i.e. δ(x), δ∗(x), and θ(x) atany x, from largest to smallest. Likewise, various Reynolds numbers can be definedassociated with each thickness quantity, such that

Reδ(x) =U0δ(x)

ν,Reδ∗(x) =

U0δ∗(x)

ν,Reθ(x) =

U0θ(x)

ν. (5.36)

Provide an argument to rank these Reynolds numbers from largest to smallest.5) A hypothetical boundary layer is characterized by the following velocity profileat position x. 〈U〉(y) = y m s−1 for y < 10 m

〈U〉(y) = U0 = 10 m s−1 for y ≥ 10 m.

Using the definitions in the previous problem, calculate δ(x), δ∗(x), θ(x),Reδ(x),Reδ∗(x), and Reθ(x). Can you confirm the ranks from largest to smallest hypothe-sized in the previous problem?6) A Large-Eddy Simulation (LES) is used to model airflow inside a wind tunnel.The tunnel has a smooth surface and various vertical profiles of non-dimensionalvelocity U+ versus non-dimensional distance normal to the wall (z − d)+ are ob-tained at various distances downstream of the flow in the x direction. Here d isdisplacement height that is used to normalize normal distance to the wall. Usually dis in the order of roughness characteristic length of a surface. For a smooth surfaced ≈ 0. Furthermore, the LES model is compared with experimental measurementsof Fernholz and Finley (1996).

For each LES profile, provide approximate ranges of (z − d)+ associated withthe inner layer and the outer layer. In addition, provide approximate ranges of (z −d)+ for the viscous sublayer, buffer sublayer, viscous wall region, overlap region,and the log-law region. Is there a unique slope that characterizes the log-law region?Using the experimental profile explain what happens after the log-law region forvery large values of (z − d)+?7) A mechanical engineer is analyzing a hypothetical boundary layer characterizedby the following velocity profile at position x.

54

Figure 5.4: Profiles of non-dimensional horizontal velocity versusnon-dimensional distance normal to the wall for a smooth surface. Profiles 1, 2, 3,and 4 are obtained using a Large-Eddy Simulation (LES) model that resolves flow

near the walls. The experimental profile is obtained from Fernholz and Finley(1996).

〈U〉(y) = y1/2 m s−1 for y < 9 m

〈U〉(y) = U0 = 3 m s−1 for y ≥ 9 m,

where, flow is in the x direction and y is the direction normal to the surface. Usingthe definition for the momentum thickness, θ(x), and the Reynolds number associ-ated with this thickness, Reθ(x), help this engineer calculate both θ(x) and Reθ(x).Assume that the kinematic viscosity of air is ν = 1.5× 10−5m2 s−1.

55

8) Scales of turbulence near walls for most engineering applications are much finerthan elsewhere in a system. Therefore, near walls, say in the log-law region, it canbe assumed that the production rate and dissipation rate of turbulent kinetic energyare in equilibrium, i.e. they are the same. Suppose near a wall the mean flow is inthe x direction with mean velocity 〈U〉 and y is the direction normal to the wall.Also suppose there is horizontal homogeneity. The equivalency of production rateand dissipation rate of turbulent kinetic energy can be stated as

Pk = ε, (5.37)

where the production rate of turbulent kinetic energy can be expressed as Pk =

−〈uv〉d〈U〉dy

. Here u and v are turbulent fluctuations in the x and y directions. As-suming that the region of interest is the log-law region, express the Reynolds stress〈uv〉 as a function of friction velocity uτ . Also, assuming that the region of interestis the log-law region, express the gradient of mean velocity d〈U〉

dyas a function of

friction velocity uτ , von Karman constant κ, and distance away from the wall y.Using these assumptions show that

ε =u3τ

κy. (5.38)

56

Chapter 6

Free Shear Flows

Free shear flows are those that are remote from walls or surfaces. Typical examplesinclude jets, wakes, and mixing layers. Turbulence in shear flows arise because ofthe existence of mean velocity gradients.

Round Jet

A common shear flow is the round jet. This flow is created by a fluid exiting anozzle with diameter d, that produces approximately a flat-topped and uniform ve-locity profile UJ . The flow then enters an ambient background at rest. The flow isstatistically stationary and axisymmetric. Therefore, the statistics of the flow wouldonly depend on axial and radial coordinates, i.e. x and r, but are independent oftime and circumferential coordinate, θ. The velocity components in the x, r, and θcoordinates are denoted by U , V , and W . This is shown in Figure 6.1.

Figure 6.1: Schematic of a round jet showing the polar-cylindrical coordinatesystem.

In the ideal scenario the jet can be specified by UJ , d, and ν. The jet Reynoldsnumber can also be defined using these parameters as

57

Re =UJν

d. (6.1)

However, in practice the nozzle dimension and the surroundings have someeffect on the jet so more specifications can be given to characterize a jet (Husseinet al., 1994).

The mean velocity of a jet is predominantly in the axial direction. Note that ata radial distance of r = 0 the jet axis is defined, about which the profiles of themean axial velocity 〈U〉 are symmetric. Due to symmetry, the mean circumferentialvelocity is zero, i.e. 〈W 〉 = 0, while the mean radial velocity is not zero but anorder of magnitude smaller than the mean axial velocity, i.e. 〈V 〉〈U〉.

Axial Velocity

The centre-line mean axial velocity can be defined in terms of the mean axial ve-locity, 〈U(x, r, θ)〉 such that

U0(x) ≡ 〈U(x, 0, 0)〉. (6.2)

The jet’s half-width r1/2(x) is defined as a radial distance where the mean axialvelocity is half of the centre-line mean axial velocity, i.e.

〈U(x, r1/2(x), 0)〉 ≡ 1

2U0(x). (6.3)

Note that the jet’s half-width is not constant but a function of axial distance. Infact this half-width constant increases further away from the nozzle.

Self-similarity

Self-similarity is an important concept in the study of turbulent flows. In mathemat-ics, a self-similar object is exactly or approximately similar to a part of itself (i.e.the whole has the same shape as one or more of the parts). Many objects in the realworld, such as coastlines, are statistically self-similar: parts of them show the samestatistical properties at many scales. Figure 6.2 shows two instances of self-similarobjects in nature.

58

Figure 6.2: Self-similar objects in nature.

Consider a quantityQ(x, y) that depends on two independent variables x and y.As functions of x, characteristic scalesQ0(x) and δ(x) are defined for the dependentvariable Q and the independent variable y, respectively. Then scaled variables aredefined by

ξ ≡ y

δ(x), (6.4)

Q(ξ, x) ≡ Q(x, y)

Q0(x). (6.5)

If the scaled dependent variable is independent of x, i.e. there is a functionQ(ξ) such that

Q(ξ, x) = Q(ξ), (6.6)

then Q(x, y) is self-similar. In this case, Q(x, y) can be expressed in terms of func-tions of single independent variables, Q0(x), δ(x), and Q(ξ). In a successful for-mulation of self-similar phenomena, it is important to chose scales Q0(x) and δ(x)

appropriately. In some circumstances, more general transformations are required. Itis also important to note that self-similar behaviour may not be observed over theentire range of independent variables, but only for a limited range.

59

Axial Variation of Scales

To characterize a jet we need to determine variations of two scales, i.e. the velocityscale U0(x) and lengthscale r1/2(x). It is experimentally verified that the inverseof U0(x), specifically UJ/U0(x), plotted against x/d, falls on a straight line. Theintercept of this line with the abscissa defines the virtual origin, denoted by x0, sothat the straight line corresponds to

U0(x)

UJ=

B

(x− x0)/d, (6.7)

where B ≈ 5.9 is an empirical constant. It has been experimentally observed thatx0/d ≈ 4 (Hussein et al., 1994). The virtual origin x0 represents, mathematically,the location where the self-similar behaviour of a jet begins.

The spreading of a jet characterizes how the jet grows in the radial directionas a function of axial distance away from the nozzle. The jet spreading rate can bedefined using the differential equation

S ≡dr1/2(x)

dx, (6.8)

which is a constant quantity in the self-similar region of the jet. The spreadingof a jet has been measured experimentally as S ≈ 0.102 (Hussein et al., 1994).Integrating the spreading rate, we can obtain a formula for the the jet’s half-widthin the self-similar region

r1/2(x) = S(x− x0). (6.9)

Self-similarity of a Round Jet

In the well-behaved self-similar region of a round jet (x/d > 30) and a highReynolds number turbulent jet (Re > 104), the centre-line velocity U0(x) and thehalf-width r1/2(x) vary according to equations in the previous section. The empir-ical constants, B and S, are independent of the Reynolds number. A cross-streamsimilarity variable can be taken as either of

60

ξ ≡ r

r1/2

, (6.10)

η ≡ r

x− x0

. (6.11)

These two variables are in fact related by η = Sξ. The self-similar mean ve-locity profile can be defined by

f(η) = f(ξ) =〈U(x, r, 0)〉U0(x)

, (6.12)

where functions can be fitted for η. An approximation is f(η) = (1+aη2)−2, wherea is a constant. The mean lateral velocity in the self-similar region, i.e. 〈V 〉, can bedetermined from 〈U〉 via the continuity equation. Likewise, self-similar profiles of〈V 〉/U0 can be obtained. The mean lateral velocity is usually found to be more thanone order of magnitude smaller than the mean axial velocity, i.e. |〈V 〉| ≈ 0.03|〈U〉|.At the edge of the jet, the mean lateral velocity is negative, i.e. the ambient fluid isbeing pulled toward the centre-line axis. This is known as entrainment.

Reynolds Stresses

For a round jet, the fluctuating velocity components in the x, r, and θ coordinatedirections are denoted by u, v, and w. The Reynolds-stress tensor is〈u

2〉〈uv〉 0

〈uv〉〈v2〉 0

0 0 〈w2〉

,where due to circumferential symmetry, 〈uw〉 and 〈vw〉 are zero. The geometryof the flow dictates that the normal stresses be even functions of r, while the shearstress 〈uv〉 is an odd function. As the r approaches zero, the radial V and circumfer-ential W components of velocity become indistinguishable. Therefore on the axisof the jet 〈v2〉 and 〈w2〉 become equal.

Consider the root-mean-square of axial velocity on the centreline defined as

u′0(x) ≡ 〈u2〉1/2r=0. (6.13)61

We wish to characterize how u′0(x) will vary as a function of x, or equivalentlyin terms of non-dimensional quantities, how the ratio u′0(x)/U0(x) varies with x/d.It has been observed that after the development region of the jet, i.e. after the virtualorigin, both u′0(x) and U0(x) decay as x−1.

In a similar way to mean velocities, it has been found that the Reynolds stressesbecome self-similar. That is, profiles of 〈uiuj〉/U2

0 (x) plotted against r/r1/2 or η ≡r/(x− x0) collapse for all x beyond the development region on the same curve.

The Reynolds stress exhibits significant anisotropy. The relative magnitude ofthe shear stress compared to the normal stresses can be quantified using 〈uv〉/k ≈0.27. The velocity fluctuations of u and v are found to be correlated with a correla-tion coefficient of ρuv ≈ 0.4.

Numerous functions have been fitted to describe round jets, such as those pro-vided by Hussein et al. (1994), where model fitting to experimental data is achievedby method of least squares to all measured profiles with a similarity variable andfitted an even function given below to calculate mean flow and turbulent properties,

p(η) =[C0 + C2η

2 + C4η4 + ...

]exp

(−Aη2

). (6.14)

The multiplication of the polynomial and exponential functions provides anexcellent fit over the range in which data were taken (η ≡ r/(x − x0) < 0.2), andcare must be given not to apply these fits beyond this range. Table below shows thefitting parameters.

Mean Continuity and Momentum Equations for a Jet

We wish to derive the mean continuity and momentum equations for a two-dimensionaljet. The jet axis is in the x direction and the lateral direction is y. Far away fromthe jet in the lateral direction y → ∞ the fluid is quiescent at a pressure p0. Thecontinuity equation is given as

∂〈U〉∂x

+∂〈V 〉∂y

= 0. (6.15)

The momentum equations in the x and y directions can be written as follows.Note that terms in curly brackets can be assumed negligible.

62

Table 6.1: Constants to determine turbulent properties of a self-similar round jet(Hussein et al., 1994)

p(η) C0 C2 C4 C6 A

〈U〉/U0(x) 1.0 –1.925 0.0 0.0 63〈u2〉/U2

0 (x) 7.778e–2 2.79e1 –2.02e3 4.3e5 257〈v2〉/U2

0 (x) 5.457e–2 0.355 –4.298e1 0.0 89〈w2〉/U2

0 (x) 5.78e–2 –1.71 2.73e–1 0.0 42〈uv〉/U2

0 (x) 4.375e–1 –3.931e1 1.55e2 1.342e4 90ε/ [U3

0 (x)/(x− x0)] 0.3549 11.99 –1635 43470 201

〈U〉∂〈U〉∂x

+ 〈V 〉∂〈U〉∂y︸︷︷︸

Advection

= ν ∂2〈U〉∂x2

+ ν∂2〈U〉∂y2︸︷︷︸

Surface Force

− ∂〈uv〉∂y︸︷︷︸

Shear Stress Force

− ∂〈u2〉

∂x︸︷︷︸

Normal Stress Force

− 1

ρ

∂〈p〉∂x︸︷︷︸


, (6.16)

〈U〉∂〈V 〉∂x+ 〈V 〉∂〈V 〉

∂y︸︷︷︸

Advection

= ν ∂2〈V 〉∂x2

+ ν ∂2〈V 〉∂y2

︸︷︷︸Surface Force

− ∂〈uv〉∂x︸︷︷︸

Shear Stress Force

− ∂〈v2〉∂y︸︷︷︸

Normal Stress Force

− 1

ρ

∂〈p〉∂y︸︷︷︸


. (6.17)

The rationale for the neglected terms can be explained conveniently. Allterms containing a partial derivative of 〈V 〉 are neglected since mean velocity in thelateral direction is small. Also all partial derivatives for components of Reynoldsstress, involving both normal or shear stresses, with respect to x are neglected be-cause these derivatives are insignificant with respect to derivatives in the y direction.Finally, second derivatives with respect to x are neglected since variations in theaxial direction exhibit negligible curvature. The mean lateral momentum equationsimplifies to

63

1

ρ

∂〈p〉∂y

+∂〈v2〉∂y

= 0, (6.18)

which can be integrated from an arbitrary y to y →∞ to give an expression relatingmean pressure to far field pressure p0 and v2 such that

〈p〉/ρ = p0/ρ− 〈v2〉. (6.19)

This equation can now be differentiated with respect to x to give the axialpressure gradient as a function of far field pressure p0 and 〈v2〉 such that

1

ρ

∂〈p〉∂x

= 1

ρ

dp0

dx − ∂〈v

2〉∂x ≈ 0, (6.20)

which is valid since in a quiescent background p0 is uniform and that we alreadyassumed partial derivatives of Reynolds stress components in the x direction arenegligible. These developments result in a much shorter version of the mean mo-mentum equation in the axial direction given by

〈U〉∂〈U〉∂x

+ 〈V 〉∂〈U〉∂y︸︷︷︸

Advection

= ν∂2〈U〉∂y2︸︷︷︸

Surface Force

− ∂〈uv〉∂y︸︷︷︸

Shear Stress Force

. (6.21)

It is important to note that this equation does not have a pressure term, suggest-ing that the pressure variation in a jet with quiescent background is small so that alltransport of momentum is performed by advection, surface, and shear stress forces.

Exercises

1) A round steady jet is produced by a nozzle with diameter d = 0.01 m. Thejet exit velocity is UJ = 1m s−1. Making the necessary assumptions, at positionx = 0.5 m, show that the jet’s mean axial velocity on the centreline U0(x) and thejet’s half-width r1/2(x) are given by

U0(x = 0.5 m) = 0.13 m s−1, (6.22)

r1/2(x = 0.5 m) = 0.047 m. (6.23)64

2) In the previous problem, show that the mean velocity in the axial direction atx = 0.5 m and r = 0.02 m is given by

〈U〉(x = 0.5 m, r = 0.02 m) = 0.12 m s−1. (6.24)

3) The Fibonacci Sequence is the series of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...where the next number in the series is found by adding up the two numbers beforeit, i.e.

xn = xn−1 + xn−2. (6.25)

This series exhibits self similarity because each term depends on part of previ-ous terms. A surprising result is that we can calculate any term of this series usingthe Golden Ratio φ = 0.618034... such that

xn =φn − (1− φ)n√

5. (6.26)

For example,

x6 =0.618034...6 − (1− 0.618034...)6

√5

= 8. (6.27)

So it appears that the Golden Ratio is a number in nature that describes self sim-ilarity! In mathematics, two quantities form a golden ratio if their ratio is the sameas the ratio of their sum to the larger of the two quantities. Expressed algebraically,for quantities a and b with a > b > 0, they form a Golden Ratio if

a+ b

a=a

b≡ φ. (6.28)

Use basic algebra to show that

φ =1 +√

5

2. (6.29)

As shown in Fig. 6.3, it is inspiring that the Golden Ratio and the FibonacciSequence express many natural phenomena based on self similarity across scalesfrom ovary of an anglerfish, nautilus shells, hurricanes, to spiral galaxies.

65

Figure 6.3: Fibonacci Sequence expressed in geometric form is found in natureacross scales from ovary of an anglerfish, nautilus shells, hurricanes, to spiralgalaxies.

66

Chapter 7

Scales of Turbulent Motion

The study of scales of turbulent motion revolves around the understanding of vari-ous time, length, and velocity scales at which turbulent motion forms and progressesinto turbulent motion of different scales, for instance different time, length, and ve-locity scales. This hints that turbulent motion is intrinsically transient and movestoward various states governed by known hypotheses.

The energy cascade hypotheses was first introduced by Lewis Fry Richardson(1881-1953). According to this hypothesis, turbulent kinetic energy is producedand enters a turbulent flow at the largest lengthscales of motion. This energy is thentransferred by inviscid processes to smaller and smaller scales until, at the smallestscales, the energy is dissipated by viscous action. In a famous poem Richardsondescribes this hypothesis:

“Big whirls have little whirls that feed on their velocity, and little whirls have

lesser whirls and so on to viscosity”.Andrey Nikolaevich Kolmogorov (1903-1987) later quantified the energy cas-

cade hypothesis using mathematical formulations. Kolmogorov performed this cal-culation by other refined hypotheses known as the Kolmogorov hypotheses.

The Energy Cascade and Kolmogorov Hypotheses

Consider a turbulent flow with characteristic velocity scale U and characteristiclengthscale L and a Reynolds number Re = UL/ν that is very high. Richardsonviewed the turbulent flow to be composed of eddies of different sizes. Eddies ofsize ` have a characteristic velocity u(`) and timescale τ(`) ≡ `/u(`). Each eddyis conceived to be a turbulent motion, localized within a region of size ` that has acoherent structure. For instance, it can be a blob of fluid spinning.

The largest eddies that are formed in turbulent flow are designated with sub-script 0, i.e. `0, and are comparable to the flow lengthscale L. Likewise, the veloci-

67

ties associated with these eddies, i.e. u0 ≡ u(`0), are comparable to the flow veloc-ity scale U . These velocities are in the order of velocity fluctuations and related tothe root mean square turbulent kinetic energy, i.e. u′ ≡ (2

3k)1/2. It is possible to as-

sociate a Reynolds number to these eddiesRe0 = u0`0/ν, which is also comparableto the flow Reynolds number Re. In this scale the effect of viscosity is negligiblesince the flow is dominated by inertial effects.

The larger eddies continue to break-up into smaller and smaller eddies, eachcharacterized by a Reynolds numberRe(`) = u(`)`/ν. When this Reynolds numberis sufficiently small, at some point the viscous effects dominate and the kineticenergy dissipates.

The rate at which turbulent kinetic energy dissipates by viscous effects is calledthe dissipation rate ε. In statistically stationary turbulent flow the energy transfer

rate TEI at each eddy lengthscale is the same, which provides a convenient methodfor the calculation of the dissipation rate. At the beginning of the process, the ed-dies have energy in the order of u2

0 and timescale in the order of τ0 = `0/u0, sothe rate of transfer of energy and ultimately the dissipation rate is in the order ofu2

0/τ0 = u30/`0. An interesting result is that this rate is independent of flow kine-

matic viscosity ν, provided that the Reynolds number of the flow is sufficientlyhigh. It is now time to introduce Kolmogorov hypotheses (Kolmogorov, 1941) todescribe and quantify the energy cascade hypothesis:

Kolmogorov hypothesis of local isotropy states that “At sufficiently high Reynolds

number, the small-scale turbulent motions (` `0) are statistically isotropic”.In other words, at the isotropic scales, the directional information of the large

scale eddies is lost as the energy passes down the cascade. It is useful to intro-duce the lengthscale ÈI , which is usually a fraction of `0 as the scale at which theanisotropic larger eddies break-up into isotropic smaller eddies.

Kolmogorov’s first similarity hypothesis states that “In every turbulent flow at

sufficiently high Reynolds number, the statistics of the small-scale motions (` ≤ÈI) have a universal form that is uniquely determined by kinematic viscosity ν and

dissipation rate ε”.This lengthscale range of ` ≤ ÈI is known as the universal equilibrium range,

while the range ÈI ≤ ` is known as the energy-containing range. In this range

68

small eddies quickly adapt to maintain a dynamic equilibrium with the energy trans-fer rate TEI imposed by the larger eddies. Given ν and ε, there are unique length,velocity, and time scales that can be formed. These are known as the Kolmogorov

scales:

η ≡(ν3

ε

)1/4

, (7.1)

uη ≡ (εν)1/4 , (7.2)

τη ≡(νε

)1/2

. (7.3)

The Kolmogorov scales characterize the very smallest and dissipative eddiesthat do not break-up into smaller eddies. The Reynolds number based on the Kol-mogorov scales is unity, i.e. Re(η) = ηuη/ν = 1. In addition, the dissipation rateis given by

ε = ν

(uηη

)2

=ν

τ 2η

. (7.4)

On the smallest scales in high Reynolds number turbulent flows, turbulent ve-locity fields are statistically similar; that is they are statistically identical when theyare scaled by the Kolmogorov scales. The ratios of the smallest to largest scales aregiven by

η

`0

∼ Re−3/4, (7.5)

uηu0

∼ Re−1/4, (7.6)

τητ0

∼ Re−1/2. (7.7)

These relationships are extremely useful because without having to even mea-sure or simulate turbulent flow, it is possible to infer the Kolmogorov scales havingthe characteristic length, velocity, and time scales of the flow that are in the orderof the largest eddy scales as well.

69

Kolmogorov’s second similarity hypothesis states that “In every turbulent flow

at sufficiently high Reynolds number, the statistics of the motions of scale ` in the

range η ` `0 have a universal form that is uniquely determined by ε indepen-

dent of ν”.This hypothesis builds on the previous hypotheses in a way that if ` is sig-

nificantly larger than the Kolmogorov lengthscale η, then only inertial effects aredominant and the viscous effects should not determine the statistics of the turbulentmotions. For this hypothesis, it is useful to define another lengthscale `DI that is asignificant multiple of η. We can then specify a range for ` such as `DI ≤ ` ≤ `EI

known as the inertial subrange and another range for ` such as η ≤ ` ≤ `DI knownas the dissipation range. Figure 7.1 demonstrates the range of eddy lengthscalesand the energy cascade ranges.

Figure 7.1: Eddy size ` and various ranges of energy cascade and eddylengthscales for very high Reynolds number flows.

For the inertial subrange, having ` and ε, it is possible to estimate characteristicvelocity and time scales for an eddy lengthscale of ` as follows

u(`) = (ε`)1/3, (7.8)

τ(`) = (`2/ε)1/3. (7.9)

70

It is worth reiterating that for statistically stationary flow, and in the inertialsubrange, the rate of energy transfer T (`) from eddies larger than ` to eddies smallerthan ` is approximately the same as the dissipation rate ε and is in the order of

T (`) ∼ ε ∼ u(`)2

τ(`). (7.10)

The Energy Spectrum

The turbulent kinetic energy in a flow is distributed among different eddies withdifferent lengthscales (and the associated time and velocity scales). This distributioncan be quantified by the energy spectrum function E(κ), where κ = 2π/` is thewave number associated with each eddy size `. The turbulent kinetic energy k(κa,κb)

in the wave number range (κa, κb) can be given by

k(κa,κb) =

∫ κb

κa

E(κ)dκ. (7.11)

Likewise, the dissipation rate ε for the same range of wave numbers can begiven by

ε(κa,κb) =

∫ κb

κa

2νκ2E(κ)dκ. (7.12)

For homogeneous turbulence and in the inertial subrange (κEI < κ < κDI) theenergy spectrum can be given by well-defined functions. One example is

E(κ) = Cε2/3κ−5/3, (7.13)

where C is a constant. This is also known as Kolmogorov −5/3 spectrum and cansuccessfully describe the energy spectrum for many turbulent flows.

Two-point Correlation

The autocorrelation function was previously defined for statistically stationary tur-bulence as a one-point and two-time statistic, i.e. it quantified the correlation of

71

velocity fluctuations at one point in the flow but with a variable time shift. Here weredefine autocorrelation function as a two-point and one-time statistic given by

Rij(r,x, t) ≡ 〈ui(x + r, t)uj(x, t)〉. (7.14)

If such a correlation sharply decreases with the increasing distance betweenthe two points, then turbulent flow ought to exhibit small turbulent eddies. On theother hand if this correlation is significant even at larger distances between the twopoints, then the flow exhibits large turbulent eddies. In the limit of no separationdistance between the two points, the correlation represent the Reynolds stresses. Inthe limit of a very large distance between the two points, the correlation ought todrop to zero since a flow cannot exhibit infinitely large turbulent eddies (Davidson,2009).

Some simplifications can be made. For homogeneous and isotropic turbulencethe dependence on x can be removed, i.e. onlyRij(r, t) shall be considered. Further,since the coordinate system can be freely rotated, the vector r can be replaced byscalar r, so we shall only consider Rij(r, t). It is convenient to express componentsof the autocorrelation function after the rotation of the Cartesian coordinate systemsuch that the coordinate axis e1 is aligned with vector r. Note that this is perfectlyallowed for isotropic turbulence. The other two coordinate axes e2 and e3 thenbecome normal or perpendicular to r. This is depicted in Figure 7.2

After the rotation of the Cartesian coordinate system, the components of theautocorrelation function are R11 = RLL and R22 = R33 = RNN that are knownas longitudinal autocorrelation function and transverse autocorrelation function,respectively. Note that the isotropic condition requires R22 = R33, but there is noreason for the longitudinal and transverse autocorrelation functions to be the same.

It was stated earlier that if the separation distance between the two points iszero, then the autocorrelation reduces to Reynolds stresses so that

Rij(0, t) = 〈uiuj〉. (7.15)

Due to isotropic condition this implies that 〈u2〉 ≡ 〈u21〉 = 〈u2

2〉 = 〈u23〉. The

autocorrelation functions can be expressed in the general forms

72

Figure 7.2: Velocity components involved in longitudinal and transverseautocorrelation functions for r = e1r.

R11(r, t) = RLL(r, t) = 〈u2〉f(r, t), (7.16)

R22(r, t) = R33(r, t) = RNN(r, t) = 〈u2〉g(r, t), (7.17)

where f(0, t) = g(0, t) = 1 by definition. Typical autocorrelation functions nor-malized by 〈u2〉 are plotted in Figure 7.3.

Figure 7.3: The shape of the longitudinal and transverse autocorrelation functionsnormalized by 〈u2〉.

73

It is possible to calculate integral lengthscales if the autocorrelation functionsare available. The first of the lengthscales obtained from f(r, t) is the longitudinal

integral lengthscale given by

L11(t) = LLL(t) ≡∫ ∞

0

f(r, t)dr, (7.18)

which is characteristic of larger eddies. In isotropic turbulence, the transverse inte-

gral lengthscale is obtained from g(r, t) given by

L22(t) = L33(t) = LNN(t) ≡∫ ∞

0

g(r, t)dr, (7.19)

which is characteristic of smaller eddies. If desired, an average of the two eddylengthscales can be used to report one eddy lengthscale for the turbulent flow ofinterest.

Structure Functions

The second-order velocity structure functions are useful statistics to describe thenature of turbulent flows, particularly the dissipation rate (Hocking, 1999; Dehghanet al., 2014). The second-order velocity structure function is the covariance of thedifference in velocity between two points x and x + r:

Dij(r,x, t) ≡ 〈[Ui(x + r, t)− Ui(x, t)][Uj(x + r, t)− Uj(x, t)]〉. (7.20)

It is understood that the structure function is computed as an ensemble averagefor a given location x and the separation distance r, and at a particular time t for aturbulent flow. It seems that only eddies of size |r| or smaller can make a significantcontribution to the structure function (Davidson, 2009). In other words, if an eddyis much larger than |r|, then it affects velocities at two locations in a similar way,which does not alter the correlation.

We can make some simplifications. In homogeneous turbulence the velocityfluctuations are independent of location, so the structure function does not dependon location x, i.e. we only have Dij(r, t). Using Kolmogorov hypothesis of localisotropy for r L we can consider Dij(r, t) as an isotropic function of r. It is

74

convenient to express components of the structure function after the rotation of theCartesian coordinate system such that the coordinate axis e1 is aligned with vectorr. Note that this is perfectly allowed for isotropic turbulence. The other two coordi-nate axes e2 and e3 then become normal or perpendicular to r. This is depicted inFigure 7.4

Figure 7.4: Velocity components involved in longitudinal and transverse structurefunctions for r = e1r.

After the rotation of the Cartesian coordinate system, the components of thevelocity structure function are D11 = DLL and D22 = D33 = DNN that are knownas longitudinal structure function and transverse structure function, respectively.Note that the isotropic condition requires D22 = D33, but there is no reason for thelongitudinal and transverse structure functions to be the same.

Another simplification that can be made is that in isotropic turbulenceDLL andDNN are independent of the direction of the vector r, so that the vector quantity r

can be replaced by scalar r. Given the first Kolmogorov hypothesis, when r L,Dij can be uniquely described by ν and ε in the universal equilibrium range. Further,when η r L, Dij can be uniquely described by ε in the inertial subrange. Forthe inertial subrange and statistically homogeneous and isotropic conditions, it hasbeen shown that

DLL(r, t) = C2(εr)2/3, (7.21)

75

DNN(r, t) =4

3C2(εr)2/3, (7.22)

where C2 ≈ 2.0 is a universal constant.

Taylor Hypothesis

Measurement of the one-time and two-point velocity correlation function Rij(r)

experimentally is very difficult since this requires two stationary probes that needto be used in a large number of experiments with variable separation distance r.Alternatively, it is possible to use a moving probe in turbulent flow that travels atvery high speed V along r. Let us assume that r is in the direction of the firstcomponent of the Cartesian coordinate system, i.e. r = re1, where e1 is the unitvector along the first component. Now if the autocorrelation for the moving probewith a time shift s is calculated, we obtain R

(m)ij (s), where the superscript (m)

signifies the moving probe. On the other hand, for a time shift of s, the probe hasmoved equal to

r = V s. (7.23)

It is possible to show that if the probe is moving infinitely fast in the flow, thenthis autocorrelation is equal to the one-time and two-point velocity correlation, i.e.

R(m)ij (s) = Rij(r). (7.24)

This concept is the basis of turbulence measurements using flying hot-wireanemometers, where a hot-wire anemometer that is very sensitive to flow measure-ments with a large sampling rate is flown in a turbulent flow to reveal one-time andtwo-point velocity correlation functions (Hussein et al., 1994). In other atmosphericmeasurement studies using aircraft, the same approximation can be made given thefact that aircraft velocity is usually very fast so that the timescale involved in fly-ing through turbulence structures is significantly shorter compared to the timescaleof atmospheric eddies, at least eddies in the inertial subrange or energy containingrange (Willis and Deardorff, 1976; Stull, 1988; Aliabadi et al., 2016).

76

In many studies, particularly in atmospheric flows, there is a simpler approachto use. In this technique, only a single stationary probe is required for the approxi-mation to be valid so that

Rij(s) = Rij(r), (7.25)

where now V = r/s represents the average wind speed. The approximation of spa-tial correlations by temporal correlations is known as the Taylor hypothesis (Taylor,1938) and is only valid for the frozen turbulence approximation. This approxima-tion states that eddies can be conceived as frozen and moving with the flow as theytravel past a stationary probe. This condition occurs in the atmosphere when turbu-lent fluctuations are much smaller in magnitude than the average flow velocity, forinstance

u1

〈U1〉 1. (7.26)

Exercises

1) Show that at Kolmogorov scales the Reynolds number of the flow is unity, i.e.

Re(η) =ηuην

= 1. (7.27)

2) Show that ratios of the smallest Kolmogorov to the largest scales of turbulencein a flow are given by

η

`0

∼ Re−3/4, (7.28)

uηu0

∼ Re−1/4, (7.29)

τητ0

∼ Re−1/2. (7.30)

3) For the inertial subrange and very high Reynolds number turbulent flows, showthat it is possible to approximate characteristic velocity and time scales for an eddylengthscale of ` as follows

77

u(`) ∼ u0(`/`0)1/3, (7.31)

τ(`) ∼ τ0(`/`0)2/3. (7.32)

4) For homogeneous and statistically stationary turbulence, the energy spectrumfunction for the inertial subrange can be given using a power law such that

E(κ) = Aκ−p, (7.33)

where A and p are constants. Show that the turbulent kinetic energy and the dissi-pation rate over specific ranges of wave numbers below can be given by

k(κ,∞) =A

p− 1κ−(p−1), (7.34)

ε(0,κ) =2νA

3− pκ3−p. (7.35)

5) For homogeneous and isotropic turbulence, sketch an approximate plot of thelogarithm of velocity structure functions DLL(r, t) and DNN(r, t) versus r. Identifythe zero and the horizontal asymptotic limits for the logarithm of the structure func-tions. Identify the portion of the plot for which you know the slope of the curves.Identify the dissipation range, inertial subrange, and the energy-containing rangeon the plot. Within the inertial subrange, identify which curve is above the other.6) Show by definition why for isotropic turbulence f(0, t) = g(0, t) = 1.7) For a flow with isotropic turbulence conditions the autocorrelation functions at aparticular time t in the longitudinal and transverse directions are given as

f(r) =1

1 + r2and g(r) = cos(r)e−r, (7.36)

where r is distance in meters. Show that for this flow and at this particular time, thelongitudinal integral lengthscale and the transverse integral lengthscale are givenby

LLL =π

2m and LNN =

1

2m. (7.37)

78

8) For an experiment, in which flow is homogeneous and statistically stationary, theenergy spectrum function is obtained in a few discrete points. Figure 7.5 shows thebase ten logarithm of energy spectrum function E(κ) versus the base ten logarithmof wave number κ.

Figure 7.5: Energy spectrum function: base ten logarithm of energy spectrumfunction E(κ) versus base ten logarithm of wave number κ.

Provide a range for wave numbers in the energy-containing range, i.e. κ < κEI ;provide a range for wave numbers in the inertial subrange, i.e. κEI < κ < κDI ; andprovide a range for wave numbers in the dissipation range, i.e. κ > κDI . Notethat units for wave number κ is m−1. The base ten logarithm of wave number istaken for a unitless wave number normalized by unit wave number, i.e. 1 m−1. Notethat units for energy spectrum function E(κ) is m3 s−2. The base ten logarithm ofenergy spectrum function is taken for a unitless energy spectrum normalized by unitspectrum 1 m3 s−2.

79

9) Show that the units for energy spectrum function E(κ) is m3 s−2.10) A mathematician is challenged with the following question. For a particularairflow system, the energy spectrum function in a specific range of wave numbersκ [Rad m−1] within the inertial subrange is given as

E(κ) = e−κ, (7.38)

in units of [m3 s−2]. Assuming that the kinematic viscosity of air is ν = 1.5 ×10−5m2 s−1, help her calculate the turbulent kinetic energy k and the turbulentkinetic energy dissipation rate ε contained in the range of wave numbers [1, 10]

Rad m−1 within the inertial subrange. Hint: both of these quantities can be obtainedby integrating a function containing the energy spectrum.11) The model spectrum for energy density provided thus far only describes theinertial subrange. However, other model spectra have been proposed that covermore subranges of the energy cascade for statistically stationary but isotropic oranisotropic turbulence, such as energy-containing, inertial, and dissipation sub-ranges. One such model can be written as

E(κ) = Cε2/3κ−5/3fL(κL)fη(κη), (7.39)

where C and L are constants, ε is the turbulent kinetic energy dissipation rate, κ isthe wave number, and η is the Kolmogorov length scale. Here fL(κL) and fη(κη)

are multiplying functions to correct the model spectrum for the energy-containingand dissipation subranges, respectively (Pope, 2000). fL(κL) is defined as

fL(κL) =

(κL

[(κL)2 + cL]1/2

)5/3+p0

, (7.40)

where cL and p0 are constants. p0 is taken as a different constant for anisotropic andisotropic flows (von Karman, 1948; Kaimal et al., 1972, 1976). fη(κη) is defined as

fη(κη) = exp−β[(κη)4 + c4η]

1/4 − cη, (7.41)

where β and cη are constants. a) Show that in the limit where κ → 0, i.e. whenthe energy-containing range is considered, fη(κη) behaves like a constant, and themodel spectrum behaves like

80

E(κ) ∼ κp0 . (7.42)

b) Assuming cη = 0, show that in the limit where κ → ∞, i.e. when thedissipation range is considered, fL(κL) approaches unity, and the model spectrumbehaves like

E(κ) ∼ κ−5/3 exp(−βκη). (7.43)

Note that since the velocity field is infinitely differentiable, for large κ, thespectrum function decays more rapidly than any power of κ, say−5/3, so the dissi-pation range is described by the model spectrum appropriately. Also note that in theinertial subrange both fL(κL) and fη(κη) approach near unity so the Kolmogorov−5/3 spectrum is produced by the model spectrum. Figure 7.6 shows the energyspectrum function for velocity components U and W from a Large-eddy Simu-lation (LES) of anisotropic atmospheric flows (Aliabadi et al., 2018). Attempt toidentify the slopes for the energy-containing and inertial subranges.

81

Figure 7.6: Energy spectrum function from a Large-eddy Simulation (LES) show-ing spectrum slopes for the energy-containing and inertial subranges suitable foratmospheric flows: base ten logarithm of energy spectrum function E(κ) versusbase ten logarithm of wave number κ. Figure is extracted from Aliabadi et al.(2018).

82

Chapter 8

Time and Frequency Domains

Physical phenomena that are represented as a function of time, i.e. in time domain,can also be represented as a function of frequency, i.e. in frequency domain. Thisis true since most time-evolving phenomena exhibit various repeating patterns thatoccur at various frequencies. The frequency domain is the description of such rep-etitions that occur at different levels of strength.

Discrete Fourier Transform

From Fourier analysis in calculus we know that any well-behaved continuous func-tion can be described by an infinite Fourier series, i.e. the sum of an infinite numberof sine and cosine terms. However, most turbulence signals that are measured andrepresented with time series are discrete functions of time. In the case of discretetime series with a finite number of points, we only need a finite number of sineand cosine terms to fit our time series points exactly, i.e. we can have a perfecttransformation of a signal from the time domain to the frequency domain.

We begin with a time series function A(k), with N discrete timestamps fromk = 0 to k = N − 1 with a discretization of ∆t. This time series fills an entireperiod of time equal to P = N∆t. The idea is to represent this time series functionin frequency domain as a sum of sines and cosines with amplitudes FA(n). FA(n)

is a complex number, where the real part represents the amplitude of the cosinewaves and the imaginary part is the sine wave amplitude. FA(n) is a function of fre-quency n because the waves of different frequencies must be multiplied by differentamplitudes to reconstruct the original time series. n is the frequency in number ofcycles per time period P , and FA(n) is the discrete Fourier transform of functionA(k) and hence is written with subscript A. There are a number of ways to describefrequency:

83

n = number of cycles per entire time period of signal P , (8.1)

f = number of cycles per second =n

P=

n

N∆t, (8.2)

ω = radians per second = 2πf =2πn

N∆t. (8.3)

A frequency of zero (n = 0) denotes a mean value. The fundamental frequency,where n = 1, means that exactly one wave fills the entire time period P . Higherfrequencies correspond to harmonics of the fundamental frequency. For instance,n = 5 means that exactly 5 waves fill the period P .

Using Euler’s (1707-1783) formula, eix = cos(x) + i sin(x), as a short nota-tion for sines and cosines, we can write the Fourier-transform pair to express therelationship between A(k) and FA(n) using the discrete Fourier transform

FA(n) =N−1∑k=0

[A(k)

N

]e−i2πnk/N , (8.4)

and

A(k) =N−1∑n=0

FA(n)ei2πnk/N , (8.5)

which are termed the forward transform and inverse transform, respectively. If theoriginal time series A(k) is known, then the FA(n) coefficients can be found fromthe forward transform, and once these coefficients are known, it is possible to re-construct the time series using the inverse transform. Conversion of a signal fromthe time domain to the frequency domain is also known as the Fourier decompo-

sition. These expressions state that for a discrete time series with no more than Ndata points, we do not need more than N different frequencies to describe it. In factwe need less than N frequencies.

84

Example

Eight data points of specific humidity q have been collected as a function of time fora turbulence measurement in the atmosphere using a fixed probe. The data pointsare presented in Table 8.1.

Table 8.1: Specific humidity measurements in an atmospheric turbulence study.

Index (k) 0 1 2 3 4 5 6 7

Time (UTC) 1200 1215 1230 1245 1300 1315 1330 1345q (g kg−1) 8 9 9 6 10 3 5 6

By performing the forward Fourier transform we can obtain eight coefficientsFq(n). These coefficients are complex, i.e. Fq(n) = Freal(n) + iFimag(n). SinceN = 8 and ∆t = 15 min, the total period is P = N∆t = 2 hr. The eight co-efficients can be calculated using the following formulat and are shown in Table8.2.

Fq(n) =1

N

N−1∑k=0

q(k) cos(2πnk/N)− i

N

N−1∑k=0

q(k) sin(2πnk/N). (8.6)

This calculation can be checked by the inverse transform to ensure that the timeseries signal can be recovered using the following formula

q(k) =N−1∑n=0

Freal(n) cos(2πnk/N)−N−1∑n=0

Fimag(n) sin(2πnk/N). (8.7)

These inverse transform sums are in fact four sums as opposed to two sums.The remaining two sums consist of the real part of F times the imaginary factori sin(...), and the imaginary part of F times the real factor cos(...). Because thelast half of the Fourier transforms are the complex conjugates of the first half (notcounting the mean), these two sums identically cancel, leaving the two listed above.By performing the calculation for the above sum, it can be verified that the signalq(k) can be recovered.

85

Table 8.2: Coefficients for the forward discrete Fourier transform of Specific hu-midity measurements in an atmospheric turbulence study.

n Fq(n)

0 7.0

1 0.28− 1.03i

2 0.5

3 −0.78− 0.03i

4 1.0

5 −0.78 + 0.03i

6 0.5

7 0.28 + 1.03i

Nyquist Frequency

Almost all measured time series are discrete in nature because every measurementprobe has a sampling frequency and cannot measure continuously. The rule ofthumb in discrete data analysis is that at least two data points are needed per periodor wavelength in order to resolve a wave with that period or wavelength. For exam-ple, discrete Fourier analysis involves decomposing a time series signal into wavesof different frequencies. If we have a total ofN data points with a constant samplingfrequency, then the highest frequency that we can resolve in our Fourier transformis nf = N/2, also called the Nyquist frequency, named after Harry Nyquist (1889-1976). In another word, if a wave period as small as 0.1 s must be measured, thenthe time series must be digitized at least every 0.05 s. Similarly, when flying a mov-ing probe into a turbulent flow, if a wavelength as small as 1 m is to be measured,then the signal must be digitized at least once every 0.5 m.

If higher frequencies in a time series or other signals are present but cannot bemeasured due to the limited sampling frequency of a probe, then those frequenciesthat are higher than the Nyquist frequency are folded or aliased into lower frequen-cies. In other words, our analysis will show artificially higher amplitudes for lowfrequency decomposition of the signal. Common strategies to eliminate folding or

86

aliasing is to use an analog electronic low pass filter to remove high frequenciesfrom the signal before it is sampled. Alternatively, the time series can be block-averaged, for instance by averaging ten adjacent data points.

Discrete Energy Spectrum

In turbulence studies, it is often desired to know how much of the variance of afluctuating time series signal is associated with a particular frequency or range offrequencies. The answer to this question is possible using the discrete Fourier trans-form. The square of the norm of the complex Fourier transform for any frequencyn is given by

|FA(n)|2 = [Freal(n)]2 + [Fimag(n)]2. (8.8)

Where |FA(n)|2 is summed over frequencies n = 1 to N − 1, the result equalsthe total biased variance of the original time series, i.e.

σ2A =

1

N

N−1∑k=0

(A(k)− 〈A(k)〉T )2 =N−1∑n=1

|FA(n)|2 (8.9)

where the time average 〈〉T is the only average available to us for calculating thevariance. Note that the square of the norm of the complex Fourier transform issummed starting at n = 1 instead of n = 0. This is trivial since there are noturbulent fluctuations associated with n = 0.

We can interpret |FA(n)|2 as the portion of variance explained by waves offrequency n. Likewise the contribution of frequencies from n = n1 to n = n2 tothe variance can be found out by summing |FA(n)|2 from n = n1 to n = n2.

For frequencies greater than the Nyquist frequency, the |FA(n)|2 values areidentically equal to those at the corresponding folded lower frequencies becausethe Fourier transform of high frequencies are the same as those for the low fre-quencies, except for a sign change in the imaginary part. Frequencies higher thanthe Nyquist frequency cannot be resolved by Fourier transform, therefore, |FA(n)|2

values at high frequencies should be folded back and added to those at the lowerfrequencies. Therefore, the discrete spectral intensity or discrete spectral energy,

87

EA(n), is defined as EA(n) = 2|FA(n)|2, for n = 1 to n = nf , with N beingodd, while for N being even, EA(n) = 2|FA(n)|2 for frequencies from n = 1 ton = nf − 1, but EA(n) = |FA(n)|2 for n = nf .

Discrete Energy Density Spectrum

Related to the concept of energy is energy density. If we divide EA(n) by ∆n weobtain the spectral energy density

SA(n) =EA(n)

∆n(8.10)

which has the units of A squared per unit frequency. The advantage of spectralenergy density if that instead of summing the discrete spectral energy over a rangefor n to yield variance for that range, one can simply integrate SA(n) over the samerange if a well defined function can be fitter to SA(n). For the entire range of n thevariance of A can be given by

σ2A =

∫ nf

1

SA(n)dn. (8.11)

Example

For the previous example, having |Fq(n)|, it is possible to calculate the spectralenergy and spectral density according to table 8.3. Note that in this example N iseven so the spectral energy at n = nf has not been multiplied by two. Also notethat |Fq(n)|2 is symmetric about n = nf showing the nature of high frequenciesthat have been folded on lower frequencies. It can be verified that the sum of the|Fq(n)|2 or spectral energy is indeed equal to the total biased variance of q(k).

Spectra of Two Variables

In turbulence studies it is often very useful to analyze the spectrum for a productof two variables. For instance the product of the fluctuations of velocities in thedirections of e1 and e2 coordinate vectors, i.e. u1u2. Cross-spectrum analysis relatesthe spectra of two variables. The phase refers to the position within one wave, such

88

Table 8.3: Spectral energy and spectral density for specific humidity measure-ments in an atmospheric turbulence study.

n Fq(n) |Fq(n)|2 Eq(n) Sq(n)

0 7.0 (=mean)1 0.28− 1.03i 1.14 2.28 2.282 0.5 0.25 0.5 0.53 −0.78− 0.03i 0.61 1.22 1.224 = nf 1.0 1.0 1.0 1.05 −0.78 + 1.03i 0.616 0.5 0.257 0.28 + 1.03i 1.14

Sum 5.0 5.0 5.0

as the crest or the trough, and is given as an angle. Phase shift refers to the anglebetween one part of a wave, such as a crest, to that of another wave. The equationfor a single sine wave of amplitude C that is shifted by angle Φ to the right is

A(k, n) = C(n) sin

(2πkn

N− Φ(n)

)(8.12)

In this equation the k index signifies time stamp, i.e. represents a particular datapoint of a time series, while n is a measure of the frequency of the wave. The sameequation can be shown as the sum of a sine and a cosine wave by

A(k, n) = Cs(n) sin

(2πkn

N

)+ Cc(n) cos

(2πkn

N

)(8.13)

where Cs = C cos(Φ) and Cc = −C sin(Φ). As was discussed, the Fourier trans-forms give the amplitudes of sine and cosine terms in the spectral decomposition ofthe original time series. Therefore, we can also interpret the spectra in terms of anamplitude and phase shift for waves of each frequency.

We can define GA(n) = |FA(n)|2 or GA = |FA|2 for short as the unfoldedspectral energy for variable A and frequency n. We can write GA = F ∗A.FA, whereF ∗A is the complex conjugate of FA, and where the dependence on n is still im-

89

plied. This can be shown by considering the real and imaginary parts of FA, i.e.FA = FAr + iFAi, where subscripts r and i signify the real and imaginary parts,respectively,

GA = F ∗A.FA

= (FAr − iFAi)(FAr + iFAi)

= F 2Ar + iFAiFAr − iFAiFAr − i2F 2

Ai

= F 2Ar + F 2

Ai

= |FA|2 (8.14)

We can now define the cross spectrum between two variables A and B by

GAB = F ∗A.FB

= (FAr − iFAi)(FBr + iFBi)

= FArFBr + iFArFBi − iFAiFBr − i2FAiFBi (8.15)

If we now collect the real parts and the imaginary parts, the real part is definedas the cospectrum, Co, and the imaginary part is defined as quadrature spectrum,Q:

GAB = Co+ iQ, (8.16)

where

Co = FArFBr + FAiFBi (8.17)

Q = FArFBi − FAiFBr. (8.18)

It must be reminded that both FA and FB are still functions of frequency n,resulting in both cospectrum and quadrature spectrum functions of frequency n, i.e.Co(n) and Q(n).

90

The cospectrum is frequently used in turbulence studies because the sum overall frequencies of cospectrum amplitudes is equal to the covariance of turbulentfluctuations in A and B, i.e. a and b, such that

∑n

Co(n) = 〈ab〉T , (8.19)

where again it is assumed that only time averaging is possible for time series as aclose approximate for covariance. It is important to note that the cospectrum com-puted this way is not equal to the spectrum of the time series of the product ab.

Exercises

1) A time series has a period of P = 15 s. This time series is composed of numerousharmonic frequencies. Calculate the third harmonic frequency, i.e. n = 3, in unitsof cycles per second (f ) and radians per second (ω).2) A time series signal has N = 100 timestamps with a constant sampling intervalof ∆t = 2 s. Show that the highest frequency that a Fourier transform of this signalcan resolve, i.e. the Nyquist frequency, is, nf = 50 cycles per the entire time periodof the signal, or equivalently f = 0.25 Hz cycles per second.3) What is folding or aliasing?4) The spectral energy density for a component of momentum, in [m2 s−2 per num-ber of cycles per the entire time period of the flow], for a turbulent flow in part ofthe dissipation range is given by the following expression

S(n) =100

1 + n2, (8.20)

where n is the number of cycles per entire time period of the flow. Note that thisexpression is only valid from n = 100 to n = 200. Calculate the amount of varianceσ2 for the component of momentum in this flow associated with number of cyclesper entire time period of the flow from n1 = 120 to n2 = 140. What is the unit forthe calculated variance?

91

Part II

Turbulence Measurement Techniques

92

Chapter 9

Sonic and Ultrasonic Techniques

Sonic and ultrasonic techniques are among the most practical and convenient tech-niques to measure turbulent flows. These techniques rely on the measurement ofpressure waves in a fluid system. The main difference between the two approachesis the range of frequencies employed for the measurements. Sonic frequencies arewithin the human audible range from about 20 Hz to 20 kHz, while the ultrasonicfrequencies are above the audible range.

Sonic and ultrasonic measurements for atmospheric and oceanic studies havebeen around since the early 1970s. The periodic cost of maintenance is minimalas the instruments are solid state with no moving parts. The starting threshold andresponse time, for both flow speed and direction, are essentially zero. These twocharacteristics make the sonic or ultrasonic techniques good candidates for meteo-rological, air pollution, and dispersion studies among other applications requiringhigh accuracy at very low flow speeds (Aliabadi et al., 2019).

Ultrasonic Anemometer

A conventional ultrasonic anemometer generates a small amplitude pressure dis-turbance in the fluid, which travels at the speed of a mechanical wave, determinedfrom the physical properties of the fluid. The absolute velocity of pressure distur-bance propagation in a moving fluid is the algebraic sum of the fluid velocity andthe pressure disturbance velocity. Knowing the velocity of the pressure disturbance,the fluid velocity could then be calculated. Such anemometers need pairs of acoustictransducers, usually two pairs for two-dimensional velocity vector measurementsand three pairs for three-dimensional velocity vector measurements (Ghaemi-Nasabet al., 2018). The speed of measurement is determined by the frequency response ofthe transducer operation (Brock, 2001). For atmospheric measurements the speedof measurement, i.e. sampling frequency, is usually between 10 Hz and 40 Hz. Usu-

93

ally, the higher the sampling rate, the finer time and length scales of turbulence canbe detected. Numerous studies report ultrasonic anemometer sampling of the atmo-sphere at 10Hz (Eliasson et al., 2006; Klein and Clark, 2007; Nelson et al., 2007;Ramamurthy et al., 2007; Nelson et al., 2007; Balogun et al., 2010; Zajic et al.,2011; Klein and Galvez, 2015), 20Hz (Ramamurthy et al., 2007; Barlow et al.,2015; Blackman et al., 2015; Giometto et al., 2016), or greater than 20Hz (Loukaet al., 2000; Rotach et al., 2005; Nelson et al., 2007; Inagaki and Kanda, 2008,2010; Giometto et al., 2016) for field campaigns focused on turbulence measure-ments. Ultrasonic anemometers reach accuracies and precisions within centimetresper second.

SOnic Detection And Ranging (SODAR)

SOnic Detection And Ranging (SODAR) has been used successfully during the lastfew decades for measurement of atmospheric wind profiles from a few tens of me-ters up to a few kilometres in altitude. Also known as wind profilers, SODARs areused to measure the scattering of sound waves by atmospheric turbulence. SODARsystems are used to measure wind speed and direction at various heights abovethe ground and the thermodynamic structure of the lower layer of the atmosphere(Beyrich, 1997).

SODARs are equivalent of SOund NAvigation Ranging (SONAR) systemsused in the air rather than in water. SODARs use the Doppler effect with a multi-beam configuration to determine the wind speed and direction. SODARs consist ofantennas that transmit and receive acoustic signals. The horizontal components ofthe wind velocity are calculated from the radially measured Doppler shifts and thespecified tilt angle from the vertical. The tilt angle, or zenith angle, is generally 15to 30 degrees, and the horizontal beams are typically oriented at right angles to oneanother. Since the Doppler shift of the radial components along the tilted beamsincludes the influence of both the horizontal and vertical components of the wind, acorrection for the vertical velocity is needed in systems with zenith angles less than20 degrees. In addition, if the system is located in a region where vertical veloc-

94

ities may be greater than about 0.2 m s−1, corrections for the vertical velocity areneeded, regardless of the beam’s zenith angle.

The vertical range of SODARs is approximately 0.2 to 2 km and is a func-tion of frequency, power output, atmospheric stability, turbulence, and, most impor-tantly, the noise environment in which a SODAR is operated. Operating frequenciesvary in range from less than 1000 Hz to over 4000 Hz, with power levels up to sev-eral hundred watts. SODARs reach accuracies and precisions within centimetresper second. SODARs are most effective in the measurement of mean wind speedand direction. Given the time delay for the acoustic wave to travel back and forth,SODARs are only able to detect large scale fluctuations of turbulence in the atmo-sphere.

95

Chapter 10

Electro-magnetic Techniques

Electro-magnetic techniques in measuring turbulent flows all have one character-istic in common: they all function on principles of electro-magnetic radiation. Theelectromagnetic source and detection technology in each individual technique mayvary.

Particle Image Velocimetry (PIV) and Particle Tracking Velocimetry (PTV)

Particle image velocimetry (PIV) is an optical method of flow visualization whereone can obtain instantaneous velocity measurements and related properties in flu-ids. The fluid is seeded with tracer particles which, for sufficiently small particles,are assumed to faithfully follow the flow dynamics (the degree to which the parti-cles faithfully follow the flow is represented by the Stokes number). The fluid withentrained particles is illuminated so that particles are visible. The motion of theseeding particles is used to calculate speed and direction (the velocity field) of theflow being studied.

Other techniques used to measure flows are Laser Doppler Velocimetry (LDV)and Hot Wire Anemometry (HWA). The main difference between PIV and thosetechniques is that PIV produces two-dimensional or even three-dimensional vectorfields, while the other techniques measure the velocity at a point. During PIV, theparticle concentration is such that it is possible to identify individual particles inan image, but not with certainty to track it between images. When the particle con-centration is so low that it is possible to follow an individual particle, the techniqueis called Particle Tracking Velocimetry (PTV). In its simplest form, PIV acquirestwo consecutive images (with a very small time delay) of flow field seeded by thesetracer particles, and the particle images are then cross-correlated to yield the in-stantaneous fluid velocity field. Also in the simplest form, PTV acquires two con-secutive images (with a very small time delay) of flow field seeded by these tracer

96

particles, and the particle images are then cross-correlated to track movement ofindividual particles and hence determination of the flow field.

A typical PIV or PTV apparatus consists of a camera (normally a digital camerawith a CCD chip in modern systems), a strobe or laser with an optical arrangementto limit the physical region illuminated (normally a cylindrical lens to convert a lightbeam to a line), a synchronizer to act as an external trigger for control of the cameraand laser, the seeding particles, and the fluid under investigation. Normally a PIVor PTV software is used to post-process the optical images to obtain the velocityfield (Aliabadi et al., 2011).

Schlieren Image Velocimetry (SIV)

Unlike PIV and PTV, Schlieren Image Velocimetry (SIV) is truly a nonintrusivetechnique that relies on the fact that the change in refractive index causes light todeviate due to optical inhomogeneities present in the medium. Schlieren methodscan be used for a broad range of high-speed turbulent flows containing refractive in-dex gradients in the form of identifiable and distinguishable flow structures. In SIVtechniques, the eddies in a turbulent flow field serve as PIV particles. Unlike PIV,there are no seeding particles in SIV. As the eddy length scale decreases with theincreasing Reynolds number, the length scales of the turbulent eddies become ex-ceptionally important. These self-seeded successive Schlieren images with a smalltime delay between them can be correlated to find velocity field information. Thus,the analysis of Schlieren images is of great importance in the field of fluid mechan-ics since this system enables the visualization and flow field calculation of unseededflow (Hargather et al., 2011).

The classical implementation of an optical Schlieren system uses light from asingle collimated source shining on, or from behind, a target object. Variations inrefractive index caused by density gradients in the fluid distort the collimated lightbeam. This distortion creates a spatial variation in the intensity of the light, whichcan be visualized directly with a shadowgraph system.

In classical Schlieren photography, the collimated light is focused with a con-verging optical element (usually a lens or curved mirror), and a knife edge is placed

97

at the focal point, positioned to block about half the light. In flow of uniform densitythis will simply make the photograph half as bright. However, in flow with densityvariations the distorted beam focuses imperfectly, and parts that have been focusedin an area covered by the knife edge are blocked. The result is a set of lighter anddarker patches corresponding to positive and negative fluid density gradients in thedirection normal to the knife edge. When a knife edge is used, the system is gener-ally referred to as a Schlieren system, which measures the first derivative of densityin the direction of the knife edge. If a knife edge is not used, the system is gener-ally referred to as a shadowgraph system, which measures the second derivative ofdensity.

If the fluid flow is uniform, the image will be steady, but any turbulence willcause scintillation, the shimmering effect that can be seen on hot surfaces on a sunnyday. To visualize instantaneous density profiles, a short-duration flash (rather thancontinuous illumination) may be used.

Laser Doppler Velocimetry (LDV)

Laser Doppler Velocimetry (LDV), also known as Laser Doppler Anemometry(LDA), is the technique of using the Doppler shift in a laser beam to measure thevelocity in transparent or semi-transparent fluid flows. The measurement with LDVor LDA is absolute, linear with velocity, and requires no pre-calibration.

In its simplest form, LDV crosses two beams of collimated, monochromatic,and coherent laser light in the flow of the fluid being measured. The two beams areusually obtained by splitting a single beam, thus ensuring coherence between thetwo. Lasers with wavelengths in the visible spectrum (390-750 nm) are commonlyused; these are typically He-Ne, Argon ion, or laser diode, allowing the beam pathto be observed. A transmitting optics focuses the beams to intersect at their waists(the focal point of a laser beam), where they interfere and generate a set of straightfringes. As particles (either naturally occurring or induced) entrained in the fluidpass through the fringes, they reflect light that is then collected by a receiving opticsand focused on a photodetector (typically an avalanche photodiode).

98

The reflected light fluctuates in intensity, the frequency of which is equivalentto the Doppler shift between the incident and scattered light, and is thus proportionalto the component of particle velocity which lies in the plane of two laser beams. Ifthe sensor is aligned to the flow such that the fringes are perpendicular to the flowdirection, the electrical signal from the photodetector will then be proportional tothe full particle velocity. By combining three devices (e.g. He-Ne, Argon ion, andlaser diode) with different wavelengths, all three flow velocity components can besimultaneously measured (Drain, 1980).

Radiometry and Pyrometry

Radiometry is a set of techniques for measuring electro-magnetic radiation, includ-ing visible light. Radiometric techniques in optics characterize the distribution ofthe radiation’s power in space, as opposed to photometric techniques, which charac-terize the light’s interaction with the human eye. Radiometry is distinct from quan-tum techniques such as photon counting. The use of radiometers to determine thetemperature of objects and gasses by measuring radiation flux is called Pyrometry.Handheld pyrometer devices are often marketed as infrared thermometers. Radiom-etry is important in astronomy, especially radio astronomy, and plays a significantrole in Earth remote sensing. The measurement techniques categorized as radiom-etry in optics are called photometry in some astronomical applications, contrary tothe optics usage of the term.

Microwave radiometry has been used successfully to measure vertical profilesof atmospheric temperature and humidity, which are important drivers of atmo-spheric turbulence, from near surface level up to an altitude of about 10 km (Can-dlish et al., 2012; Ramamurthy et al., 2017). In addition, Doppler RAdio DetectionAnd Ranging (RADAR) has been used to derive the vertical-wind field in the atmo-sphere at a scale of 1-2 km by performing vertical scans in the plane of the meanwind and tracking turbulent features on a scale of several hundred metres in theradial-velocity field between consecutive scans (Hogan et al., 2008).

99

Light Detection And Ranging (LiDAR)

Light Detection And Ranging (LiDAR) is a surveying method that measures dis-tance to a target by illuminating the target with pulsed laser light and measuringthe reflected pulses with a sensor. Differences in laser return times and wavelengthscan then be used to make digital three-dimensional representations of the target.LiDARS have terrestrial, airborne, and mobile applications.

LiDAR systems are used to perform a range of measurements that include pro-filing clouds, measuring winds, studying aerosols, and quantifying various atmo-spheric components. Atmospheric components can in turn provide useful informa-tion including surface pressure (by measuring the absorption of oxygen or nitro-gen), greenhouse gas emissions (carbon dioxide and methane), photosynthesis (car-bon dioxide), fires (carbon monoxide), and humidity (water vapor). AtmosphericLiDAR remote sensing works in two ways: 1) by measuring backscatter from theatmosphere, and 2) by measuring the scattered reflection off the ground (when theLiDAR is airborne) or other hard surfaces.

Backscatter from the atmosphere directly gives a measure of clouds and aerosols.Other derived measurements from backscatter such as winds or cirrus ice crystalsrequire careful selection of the wavelength and/or polarization detected. Doppler Li-DAR and Rayleigh Doppler LiDAR are used to measure temperature and/or windspeed along the beam by measuring the frequency of the backscattered light. TheDoppler broadening of gases in motion allows the determination of properties viathe resulting frequency shift. Scanning LiDARs have been used to measure atmo-spheric wind velocity.

Doppler LiDAR systems are also now beginning to be successfully appliedin the renewable energy sector to acquire wind speed, turbulence, wind veer, andwind shear data. Both pulsed and continuous wave systems are being used. Pulsedsystems use signal timing to obtain vertical distance resolution, whereas continuouswave systems rely on detector focusing.

For turbulence measurements in the atmosphere, LiDARs actually interact withatmospheric constituents such as particulates or fine water droplets that enable map-ping three-dimensional wind fields at much higher frequency than SODARs, given

100

the fact light travels much faster than sound. Nevertheless, the sampling frequencyof a LiDAR is limited by speed and efficiency of the software algorithms that con-vert the light signal to velocity measurements (Huang et al., 2017; Halios and Bar-low, 2018).

101

Chapter 11

In-situ Techniques

In-situ techniques for measuring turbulence are intrusive, meaning that the mea-suring probe will be placed at the location where the flow is to be measured. Thismay likely influence, or disturb, the flow. However, since the probe is placed at thevery location of the flow to be measured, the technique may be more accurate thannon-intrusive techniques such as sonic, ultrasonic, or electro-magnetic techniques.

Hot Wire Anemometry (HWA)

Hot wire anemometers use a fine wire (on the order of several micro-metres) elec-trically heated to some temperature above the ambient. Air flowing past the wirecools the wire. As the electrical resistance of most metals is dependent upon thetemperature of the metal (tungsten is a popular choice for hot-wires), a relationshipcan be obtained between the resistance of the wire and the flow speed.

Several ways of implementing this exist, and hot-wire devices can be furtherclassified as Constant Current Anemometer (CCA), Constant Voltage Anemometer(CVA), and Constant Temperature Anemometer (CTA). The voltage output fromthese anemometers is thus the result of some sort of circuit within the device tryingto maintain a specific variable (current, voltage, or temperature) constant, followingOhm’s law.

Additionally, Pulse-Width Modulation (PWM) anemometers are used, whereinthe velocity is inferred by the time length of a repeating pulse of current that bringsthe wire up to a specified resistance and then stops until a threshold floor is reached,at which time the pulse is sent again.

Hot-wire anemometers, while extremely delicate, have extremely high frequency-response and fine spatial resolution compared to other measurement methods, andas such are almost universally employed for the detailed study of turbulent flows,or any flow in which rapid velocity fluctuations are of interest. For special appli-

102

cations, frequencies as high as 1 MHz can be achieved (Suminska, 2005), whiletypical frequencies are around 1 kHz (Hussein et al., 1994).

Pitot Tube

A pitot-static tube, which is a pitot tube with two ports: pitot and static, is normallyused in measuring the airspeed of aircraft. The pitot port measures the dynamicpressure of the open mouth of a tube with pointed head facing wind, and the staticport measures the static pressure from small holes along the side on that tube. Thepitot tube is connected to a tail so that it always makes the tube’s head to face thewind. Additionally, the tube is heated to prevent rime ice formation on the tube.There are two lines from the tube down to the devices to measure the difference inpressure of the two lines. The measurement devices can be manometers, pressuretransducers, or analog chart recorders. Pitot tubes have been successfully used inaircraft to measure atmospheric turbulence (Aliabadi et al., 2016).

Balloons

Balloons are among the earliest in-situ techniques that have been used to measureatmospheric turbulence. Balloons use either hot-air or a light-gas, employing he-lium or other light gases, to levitate a suite of sensors in the atmosphere by buoy-ancy force. Balloons can either be launched once without returning, the so calledradio sondes, or they can be tethered for multiple use (Suminska, 2005; Aliabadiet al., 2019).

103

Part III

Turbulence Modelling and Simulation

104

Chapter 12

Introduction to Modelling and Simulation

More than one century of experience has shown that the turbulence problem is in-conveniently difficult. Though conceptually simple, the turbulence problem is theclassical unsolved problem of physics. In other words, there is no simple analyticaltheory that completely describes physics of turbulence. Instead, engineers and sci-entists rely on the ever-increasing power of digital computers to model or simulate

turbulence for a given application to calculate the relevant properties of turbulentflows. Modelling or simulating turbulence exactly is still out of reach for practicalproblems given today’s computational power accessible to engineers and scientists.Instead, a variety of models and simulation tools are accessible that calculate fewor many relevant properties of turbulent flows.

It is worth distinguishing the technical difference between a turbulence model

and a turbulence simulation. In a turbulence model, equations are solved to givesome mean quantities, for instance mean velocity or Reynolds stress. In contrast,in a turbulence simulation, equations are solved for time-dependent properties offlow for a particular realization of the flow. Given enough realizations, it is stillpossible to derive mean quantities by performing statistical analysis on simulationresults in the post processing phase as opposed to the solution calculation phase.Nevertheless, the terms model and simulation are often freely interchanged in theliterature, so it is the responsibility of the careful reader to infer whether an approachactually refers to a model or a simulation.

Summary of Approaches

Turbulent viscosity models are among the most computationally affordable, but notnecessarily conceptually or mathematically simple, models. As a class of Reynolds-Averaged Navier-Stokes (RANS) models, turbulent viscosity models calculate mean

105

properties of the flow, such as velocity or Reynolds stress. Two common examplesof such models are the mixing-length and the k − ε models.

In Large-Eddy Simulation (LES), equations are solved for filtered propertiesof the flow, such as velocity, which are actually resolved and give a realization ofthe flow at the larger scale of turbulent motion. Smaller scales of motion, however,are not resolved but modelled. LES requires significantly higher computational re-sources, but provides more accurate results and is practical for many problems. Therational behind suitability of LES is that for many applications the physics of theflow is dominated by large-scale fluctuations, while smaller-scale fluctuations playa lesser signficant role and do not have to be resolved or realized. With exponentialincrease in computational power, LES has gained popularity in the recent decadesand has been applied to many practical engineering and science problems.

In Direct Numerical Simulation (DNS), equations are solved for properties ofthe flow, such as velocity, which are resolved and give a realization of the flowacross all spatial and temporal scales. DNS is computationally very expensive andnot practical for most problems. However, it can calculate all turbulence propertiesof the flow. The computational requirements for DNS are so prohibitive that it isstill primarily used as a research tool for fundamental studies.

Model or Simulation Completeness

A turbulence model or simulation is termed complete if its constituent equationsare free from flow-dependent specifications. Such specifications include materialproperties (density and viscosity), initial and boundary conditions, and numericaldiscretization. For example, DNS and the k − ε turbulent viscosity model are com-plete since their equations do not depend on flow-dependent specifications. On theother hand LES and the mixing-length model are incomplete. LES results dependon numerical discretization and the mixing-length model needs to be specified by amixing-length, which is flow dependent.

106

Turbulence Model or Simulation Closure Problem

When modelling or simulating turbulence, most often the number of unknowns inthe set of equations for turbulent flow is larger than the number of equations. Whenone tries to include new equations to balance the number of unknowns and equa-tions, one finds out that the number of unknowns further increase so that, again,the number of unknowns is larger than the number of equations considered. This iscalled the closure problem and is attributed to the non-linear characteristics of tur-bulence. One approach is to use only a finite number of equations, i.e. stop writingdown more equations, and then approximate the remaining unknowns in terms ofthe known quantities (Stull, 1988).

The art of modelling and simulating turbulence is to chose or develop an ap-proach that utilizes the least number of equations and unknowns but provides areasonably acceptable solution for a given problem. It is also known that more equa-tions and unknowns do not necessarily guarantee a more acceptable solution for agiven problem. For instance, it has been shown that for planetary boundary layeratmospheric flows, only a few equations suffice to arrive at an acceptable solution(Mellor and Yamada, 1974). The proper choice or development of a suitable modelor simulation tool usually follows from years of practice and experience in the field.

Digital Computation

There has been significant advancement in speed and volume of calculations en-abled by digital computers over the last fifty years. The amount of computation inflops (floating-point operations per second) over a specified time is a measure of acomputer’s ability to perform calculations. For instance a processor technology isspecified by the number of gigaflops, teraflops, or petaflops per second.

High Performance Computing (HPC) hardware configurations have been de-veloped on Central Processing Units (CPU) and Graphical Processing Units (GPU)for the purpose of turbulence modelling and simulation. Both homogeneous andheterogeneous hardware architectures have been considered: in the homogeneousconfiguration only multiple CPUs are considered, while in the heterogeneous con-figuration both CPUs and GPUs are integrated. It is known that a GPU consists of

107

thousands of computational cores that can, theoretically, perform arithmetic com-putations faster than a multi-threaded (parallel) CPU. However, a GPU requirestime for communicating data with the main computer memory. This factor slowsdown the speed of overall computation. On the other hand, a multi-threaded CPUhas direct access to the computer memory and benefits from fast data transfer butperforms arithmetic computations more slowly than a GPU. As a trade-off, mostrecent research efforts to accelerate models and simulations have considered usingvery few GPUs to off-load the solver module of the model or simulation from CPU,i.e. the part of the code that solves a linear system of equations and is most compu-tationally expensive (Codyer et al., 2012). This is also known as GPU-acceleratedcomputing.

Exercises

1) Describe the difference between turbulence modelling and turbulence simulation.2) What is flops?3) Describe the pros and cons of CPUs and GPUs.4) Most recently and alternative to CPUs and GPUs, Field-Programmable Gate Ar-rays (FPGAs) have been used for digital computation. FPGAs contain an array ofprogrammable logic blocks and a hierarchy of reconfigurable interconnects that al-low the blocks to be wired together. Logic blocks can be configured to performsimple or complex logic operations (e.g. AND, OR, etc.). In most FPGAs, logicblocks also include memory elements, which may be simple flip-flops or more com-plete blocks of memory. Although programmable to some degree, an FPGA doesnot function similar to a processor, so it cannot run any general program storedin the memory. Rather, an FPGA is meant to perform specific computational tasksor algorithms for a particular application. However, because of its parallel and re-configurable architecture, an FPGA can run logic operations much faster and moreenergy efficiently than CPUs or a GPUs. Discuss ways in which an FPGA can beused to accelerate a turbulence model or simulation.

108

Chapter 13

Turbulent Viscosity Models

Turbulent viscosity models are based on the turbulent-viscosity hypothesis, in whichthe Reynolds stresses are given as

〈uiuj〉 =2

3kδij − νT

(∂〈Ui〉∂xj

+∂〈Uj〉∂xi

). (13.1)

These models are a class of Reynolds-Averaged Navier-Stokes (RANS) models,in which the Reynolds equations are solved for the mean velocity field. In a simpleshear flow, the shear stress is given by

〈uv〉 = −νT∂〈U〉∂y

, (13.2)

in which u is velocity parallel to a wall in the x direction, and v is velocity normal tothe wall in the y direction. As was described before, νT is not a constant but a fieldthat depends on both position and time, i.e. νT (x, t). If this field can be modelledconveniently, this provides a convenient closure scheme for turbulence modelling.Turbulent viscosity can be written as the product of a velocity scale u∗(x, t) andlengthscale `∗(x, t), both of which are fields themselves

νT = u∗`∗. (13.3)

This is analogous to the kinetic theory of gasses, with the kinematic viscositygiven by

ν ∼ 1

2Cλ, (13.4)

where C is the mean molecular speed and λ is the mean free path (Sonntag, 1966).The main task in turbulent viscosity models is to specify these velocity and lengthscales. In mixing length models, `∗ is specified on the basis of the geometry ofthe flow. In one-equation turbulent kinetic energy models, `∗ is still specified on

109

the basis of the geometry of the flow, while u∗ is specified by the turbulent kineticenergy k, for which a transport equation is solved. In two-equation models, e.g. thek − ε model, both `∗ and u∗ are expressed as functions of k and dissipation rate ε,for which transport equations are solved.

The turbulent-viscosity hypothesis is based on two assumptions. First, the in-trinsic assumption is that at each point and time the Reynolds-stress anisotropy

aij ≡ 〈uiuj〉− 23kδij is determined by the mean velocity gradients ∂〈Ui〉/∂xj . Sec-

ond, there is the specific assumption that the relationship between aij and ∂〈Ui〉/∂xjis

〈uiuj〉 −2

3kδij = −νT

(∂〈Ui〉∂xj

+∂〈Uj〉∂xi

), (13.5)

or equivalently

aij = −2νTSij, (13.6)

where Sij is the mean rate-of-strain tensor. This concept, is directly analogous tothe relation for the viscous stress in a Newtonian fluid:

−(τij + Pδij)/ρ = −2νSij. (13.7)

Related to the concept of turbulent-viscosity hypothesis is the gradient-diffusion

hypothesis introduced earlier as

〈uφ′〉 = −ΓT∇〈φ〉, (13.8)

which states that the scalar flux 〈uφ′〉 is aligned with the mean scalar gradient. Itwas stated earlier that ΓT is turbulent diffusivity and should not be confused withmolecular diffusivity.

The following sections present the turbulent viscosity models developed andordered with increasing level of sophistication and accuracy. From historical pointof view, it must be understood that the turbulence modelling field made significantprogress in the recent decades albeit slowly given the difficulty of the problem.Therefore, the reader is encouraged to appreciate all the developments that con-

110

tinue to have applications today given the the level of sophistication and accuracyrequired for a problem.

Algebraic Models

The algebraic models are classified into uniform turbulent viscosity models andmixing-length models. The mixing-length model is frequently used, particularly insimple shear flows over flat surface in atmospheric and oceanic applications. In thismodel, the mixing length `m(x, y) is specified as a function of position, and thenthe turbulent viscosity is obtained as

νT = `2m|∂〈U〉∂y|, (13.9)

where 〈U〉 is mean velocity parallel to the surface and y is the direction normal tothe surface. As was stated earlier, in the log-law region, mixing length and turbulentviscosity can be specified by

`m = κy, (13.10)

νT = uτκy. (13.11)

where κ is the von Karman constant (von Karman, 1931). Mixing length modelscan be applied for generalized flows with the following formulation

νT = `2m(2SijSij)

1/2 = `2mS, (13.12)

where Sij is the mean rate-of-strain tensor. Other generalized flows formulate themixing length model with

νT = `2m(2ΩijΩij)

1/2 = `2mω, (13.13)

where Ωij is the mean rate-of-rotation tensor. Note that in these models, `m muststill be specified given some information about the geometry of the flow. As a gen-eral rule, mixing length approaches zero at walls or surfaces and reaches somemaximum in the interior of the flow or far away from walls.

111

Spalart-Allmaras Model

The next level of sophistication in turbulence modelling is possible by consideringa transport model for the turbulent viscosity itself. This is motivated by the fact that,after all, turbulent viscosity is not constant but a field that varies spatiotemporally.Therefore, its transport must be governed by an equation. Spalart and Allmaras(1994) proposed a one-equation model, although with similar preceding proposalsin the literature, developed for aerodynamic applications, where a single modeltransport equation is solved for turbulent viscosity νT . The equation is given as

DνT


≡ ∂νT∂t︸︷︷︸

Storage

+ 〈U〉.∇νT︸︷︷︸Advection

= ∇.(νTσν∇νT

)︸︷︷︸

Turbulent Viscosity Flux Divergence

+ Sv︸︷︷︸Source/Sink

, (13.14)

where σν is the Prandtl number for turbulent viscosity and Sv is the source or sinkterm that depends on many variables such as laminar viscosity ν, turbulent viscos-ity νT , mean vorticity or rate of rotation Ω, turbulent viscosity gradient ∇νT , anddistance to the nearest wall `w. This model is successful for aerodynamic flows,but has limitations as a general model. For instance it cannot account for decay ofturbulent viscosity in isotropic turbulence (Pope, 2000).

Turbulent Kinetic Energy Models

It was introduced earlier that turbulent viscosity can be written as νT = u∗`∗, i.e. theproduct of a velocity scale and a lengthscale. In the mixing length model `∗ = `m,and the velocity scale was given as

u∗ = `m|∂〈U〉∂y|. (13.15)

This approximation requires that u∗ be zero wherever |∂〈U〉/∂y| is zero. Thisapproximation is far from reality for many cases where |∂〈U〉/∂y| is zero but the ve-locity scale u∗ is not necessarily zero. It has been suggested by Kolmogorov (1942)and Prandtl (1945), independently, that the velocity scale can be better formulatedby the turbulent kinetic energy, i.e.

112

u∗ = ck1/2, (13.16)

where c is a constant. If the lengthscale is again taken to be the mixing length, thenthe turbulent viscosity can be formulated as

νT = c`mk1/2. (13.17)

The constant c can be fitted for various turbulent regimes. For instance c ≈0.55 yields a correct approximation in the log-law region. This formulation requiresknowledge of k(x, t), i.e. the turbulent kinetic energy field. If we assume we havea knowledge of `m or we can formulate it based on known quantities, we can thendevelop a transport equation for k, which can be solved for k. Kolmogorov (1942)and Prandtl (1945) suggested the following transport equation

Dk


≡ ∂k


+ 〈U〉.∇k︸︷︷︸Advection

= −∇.T′︸︷︷︸Energy Flux Divergence

+ P︸︷︷︸Production

− ε︸︷︷︸Dissipation

.

(13.18)Note that the production term P is referred to as a mechanism that generates

turbulent kinetic energy, and hence the term has a positive sign in the equation.On the other hand, the dissipation term ε is responsible for consuming the kineticenergy, and hence the term has a negative sign. The total derivative Dk/Dt andproduction terms P are in closed form, but the energy flux divergence −∇.T′ anddissipation terms ε are unknowns and should be modelled or closed further.

It was discussed extensively that the dissipation rate ε scales as u30/`0, where u0

and `0 are the velocity scale and the lengthscale of the energy-containing motions.The same scaling can be used to formulate the dissipation rate

ε = CDk3/2

`m, (13.19)

where CD is a model constant. Indeed, an examination of the log-law region yieldsthat CD = c3. This modelling approach in fact eliminates `m since it vanishes in theturbulent viscosity equation.

113

νT = cCDk2

ε. (13.20)

The energy flux T′ can be modelled using the gradient-diffusion hypothesissuch that

T′ = −νTσk∇k, (13.21)

where σk is the turbulent Prandtl number for turbulent kinetic energy, generallyassumed to be one (Pope, 2000). The energy flux divergence term in the turbulentkinetic energy equation accounts for the flux of k down the gradient of k due tovelocity and pressure fluctuations. This term ensures that the resulting transportequation model for k yields smooth solutions, and that a boundary condition canbe imposed on k everywhere in the boundary of the domain. Otherwise the modelmay diverge if other transport mechanisms for k are much smaller than this term.In summary, the one-equation model based on k consists of the following transportequation and the closure schemes

Dk


≡ ∂k


+ 〈U〉.∇k︸︷︷︸Advection

= ∇.(νTσk∇k)

︸︷︷︸Energy Flux Divergence



, (13.22)

νT = ck1/2`m, (13.23)

ε = CDk3/2

`m, (13.24)

`m(x, t) known. (13.25)

This model belongs to a class of models known as one-equation models. Thesemodels are called so because only one extra equation has to be solved to close theturbulence model.

114

The k − ε Model

The k − ε model is among the most popular turbulence models, particularly withinthe engineering community and comprehensively cited in the 1970s (Launder andSpalding, 1974). This model belongs to a class of models known as two-equation

models where two extra equations have to be solved to close the turbulence model.The two unknowns in this case are turbulent kinetic energy k and dissipation rateε. From these two unknowns and equations other quantities can be formed. Forinstance a lengthscale in turbulence can be formulated as L = k3/2/ε, a timescalecan be formulated as τ = k/ε, and a turbulent viscosity can be formulated as νT =

Ck2/ε. Since all quantities can be formed having these variables, the two-equationmodels are complete models. For instance, flow-dependent specifications such as`m(x, t) are not required.

The model transport equation for k is already provided in the previous section.The specification of turbulent viscosity is provided as

νT = Cµk2/ε, (13.26)

where Cµ = 0.09 is one of five model constants. This relationship implies that νTonly depends on k and ε, and not gradients of mean velocity.

Unlike the model equation for k that can be derived, the model equation for ε ismainly empirical, and in some authors’ opinion a pure invention (Davidson, 2009)

Dε


≡ ∂ε


+ 〈U〉.∇ε︸︷︷︸Advection

= ∇.(νTσε∇ε)

︸︷︷︸Dissipation Flux Divergence

+ Cε1Pεk︸︷︷︸

Production

− Cε2ε2

k︸︷︷︸Dissipation

.

(13.27)There are many variations of the k − ε model. The constants for the standard

model are provided as (Launder and Spalding, 1974)

Cµ = 0.09, Cε1 = 1.44, Cε2 = 1.92, σk = 1.0, σε = 1.3. (13.28)

115

The k − ω Model

Two-equation models are numerous and expand beyond the k − ε model. In manyof such models k is taken as one of the variables, however, there are diverse optionsfor the second variable. The k−ω model was first proposed by Kolmogorov (1942),where the second variable was defined as ω ≡ ε/k. It has been suggested that thechoice of this second variable mainly impacts simulations of non-homogeneousturbulence (Pope, 2000) because for homogeneous turbulence the equation for thesecond variable reduces to the ε equation provided earlier. Therefore, the carefulchoice of this second variable may help developing better models for specific non-homogeneous flows.

For non-homogeneous flows, the difference usually lies in the flux divergenceterm. The model equation for ω can be given as

Dω


≡ ∂ω


+ 〈U〉.∇ω︸︷︷︸Advection

= ∇.(νTσω∇ω)

︸︷︷︸Dissipation Flux Divergence

+Cω1Pωk︸︷︷︸

Production

− Cω2ω2︸︷︷︸

Dissipation

.

(13.29)If one tries to derive this equation from the ε equation in the k − ε model, the

difference between the models can be observed. Assuming σk = σε = σω, the ωequation can be derived as

Dω

Dt= ∇.

(νTσω∇ω)

+ (Cε1 − 1)Pωk− (Cε2 − 1)ω2 +

2νTσω∇ω.∇k. (13.30)

For homogeneous turbulence the ω and ε equations are the same, and thechoices of model constants are Cω1 = Cε1 − 1 and Cω2 = Cε2 − 1. However,for non-homogeneous flows, the two equations will be different since there will bean extra term in the ε equation for the k − ε model written as a k − ω model.

Menter (1994) successfully combined functionalities of the k − ε and k − ω

models such that the two are blended. In the free stream, and away from the walls,the blended model behaves as a k − ε model, which is known to perform better inhomogeneous turbulence; however, near the walls, the blended model behaves as ak − ω model with superior performance for non-homogeneous turbulence.

116

Exercises

1) Explain why the algebraic and turbulent kinetic energy models are incompleteturbulence models, while the k − ε model is a complete turbulence model.2) Consider the log-law region of a wall-bounded flow. Suppose that in the log-lawregion the turbulent kinetic energy and the friction velocity can be related usinguτ = C

1/4µ k1/2, where Cµ = 0.09. Also consider that in the log-law region the

friction velocity is dominated by the Reynolds stress, i.e. −〈uv〉 = u2τ . Use the log-

law and the specification `m = κy to show that the appropriate value of the constantc in the relation νT = ck1/2`m is

c ≈ 0.55. (13.31)

3) An atmospheric scientist wishes to develop a transient one-dimensional (1D)momentum transport model. She closes her turbulence model by parameterizinga mixing length that also formulates the turbulent viscosity. Suppose she uses theCartesian coordinate system with coordinate axes of x, y, and z, and velocitiescorresponding to these axes being U = 〈U〉+u, V = 〈V 〉+ v, and W = 〈W 〉+w,respectively. Further, she assumes that mean flow is only in the x direction parallelto the surface and that the direction z is normal to the surface. She assumes meanvelocities in y and z directions being zero, i.e. 〈V 〉 = 〈W 〉 = 0. In addition, sheassumes that the mean velocity 〈U〉 in the x and y directions does not change. Shealso assumes that the modified pressure has a constant gradient in the x direction.Her 1D transport model is written as

∂〈U〉∂t

=∂

∂z

(νT∂〈U〉∂z

)− τ, (13.32)

νT = `2m|

d〈U〉dz|,

`m = κz/(1 + κz`0

).

In the momentum equation identify the following terms: storage, surface forcesand Reynolds stress, and modified pressure forces. In the above equations, iden-tify the parameterizations for the following terms: turbulent viscosity and mixinglength. Demonstrate why the advection terms in the momentum transport equation

117

have vanished. Which classification of turbulence model is this simple model ac-cording to this chapter?4) An atmospheric scientist wishes to develop a transient one-dimensional (1D)momentum and turbulent kinetic energy transport model. She closes her turbulencemodel by assuming turbulent Prandtl number σk = 1 and parameterizing a mixinglength that also formulates the turbulent viscosity and the turbulent kinetic energydissipation rate. Suppose she uses the Cartesian coordinate system with coordinateaxes of x, y, and z, and velocities corresponding to these axes being U = 〈U〉+ u,V = 〈V 〉+v, andW = 〈W 〉+w, respectively. Further, she assumes that mean flowis only in the x direction parallel to the surface and that the direction z is normalto the surface. She assumes mean velocities in y and z directions being zero, i.e.〈V 〉 = 〈W 〉 = 0. In addition, she assumes that the mean velocity 〈U〉 in the x andy directions does not change. She also assumes that the modified pressure has aconstant gradient in the x direction. Her 1D transport model is written as

∂〈U〉∂t

=∂

∂z

(νT∂〈U〉∂z

)− τ, (13.33)

∂k

∂t=

∂

∂z

(νTσk

∂k

∂z

)+ νT

(∂〈U〉∂z

)2

− ε, (13.34)

νT = Ck`mk

1/2,

ε = Cε`−1m k3/2,

`m = κz/(1 + κz`0

).

In the above, identify the momentum and turbulent kinetic energy equations.In the momentum equation identify the following terms: storage, surface forces andReynolds stress, and modified pressure forces. In the turbulent kinetic energy equa-tion identify the following terms: storage, energy flux divergence, shear production,and turbulent kinetic energy dissipation. In the above equations, identify the param-eterizations for the following terms: turbulent viscosity, turbulent kinetic energydissipation rate, and mixing length. Demonstrate why the advection terms in themomentum and turbulent kinetic energy transport equations have vanished. Whichclassification of turbulence model is this simple model according to this chapter?

118

5) An atmospheric scientist wishes to develop a steady one-dimensional (1D) heat,momentum, and turbulent kinetic energy transport model. He assumes a non-constantturbulent viscosity νT that accounts for effects of molecular and turbulent diffusionand a turbulent Prandtl number PrT = σk = 1. He assumes a Cartesian coordinatesystem with coordinate axes of x, y, and z, and velocities corresponding to theseaxes being U = 〈U〉+ u, V = 〈V 〉+ v, and W = 〈W 〉+ w, respectively. Further,he assumes that mean flow is only in the x direction parallel to the surface and thatthe direction z is normal to the surface, i.e. 〈V 〉 = 〈W 〉 = 0. He assumes steadystate conditions. In addition, he assumes that the mean velocity 〈U〉 in the x and ydirections does not change. He assumes a constant heat sink or source for temper-ature by a uniform rate of cooling or heating in the domain. He also assumes thatthe modified pressure has a constant gradient in the x direction. His 1D transportmodel is written as

0 =∂

∂z

(νTPrT

∂〈T 〉∂z

)− γ, (13.35)

0 =∂

∂z

(νT∂〈U〉∂z

)− τ, (13.36)

0 =∂

∂z

(νTσk

∂k

∂z

)+ νT

(∂〈U〉∂z

)2

− g

T0

νTPrT

∂〈T 〉∂z− ε, (13.37)

νT = Ck`mk

1/2,

ε = Cε`−1m k3/2,

`m = κz/(1 + κz`0

).

In the above, identify the heat, momentum, and turbulent kinetic energy equa-tions. In the heat equation, identify the following terms: storage, diffusion of meantemperature, and rate of heat sink or source. In the momentum equation identify thefollowing terms: storage, surface forces and Reynolds stress, and modified pres-sure forces. In the turbulent kinetic energy equation identify the following terms:storage, energy flux divergence, shear production, buoyant production or sink, andturbulent kinetic energy dissipation. In the above equations, identify the parameter-izations for the following terms: turbulent viscosity, turbulent kinetic energy dissi-

119

pation rate, and mixing length. Demonstrate why the advection terms in the heat,momentum, and turbulent kinetic energy transport equations have vanished. Whichclassification of turbulence model is this simple model according to this chapter?6) An atmospheric scientist wishes to develop a transient one-dimensional (1D) pas-sive scalar, momentum, and turbulent kinetic energy transport model. He assumes anon-constant turbulent viscosity νT that accounts for effects of molecular and turbu-lent diffusion, a turbulent Prandtl number σk = 1, and a turbulent Schmidt numberScT = 1. He assumes a Cartesian coordinate system with coordinate axes of x, y,and z, and velocities corresponding to these axes being U = 〈U〉+u, V = 〈V 〉+v,and W = 〈W 〉+ w, respectively. Further, he assumes that mean flow is only in thex direction parallel to the surface and that the direction z is normal to the surface,i.e. 〈V 〉 = 〈W 〉 = 0. He assumes steady state conditions. In addition, he assumesthat the mean velocity 〈U〉 in the x and y directions does not change. He also as-sumes that the modified pressure has a constant gradient in the x direction. His 1Dtransport model is written as

∂〈φ〉∂t

=∂

∂z

(νT∂〈φ〉∂z

), (13.38)

∂〈U〉∂t

=∂

∂z

(νT∂〈U〉∂z

)− τ, (13.39)

∂k

∂t=

∂

∂z

(νTσk

∂k

∂z

)+ νT

(∂〈U〉∂z

)2

− ε, (13.40)

νT = Ck`mk

1/2,

ε = Cε`−1m k3/2,

`m = κz/(1 + κz`0

).

In the above, identify the passive scalar, momentum, and turbulent kinetic en-ergy equations. In the passive scalar equation, identify the following terms: storageand diffusion of mean passive scalar. In the momentum equation identify the follow-ing terms: storage, surface forces and Reynolds stress, and modified pressure forces.In the turbulent kinetic energy equation identify the following terms: storage, en-ergy flux divergence, shear production, and turbulent kinetic energy dissipation. In

120

the above equations, identify the parameterizations for the following terms: turbu-lent viscosity, turbulent kinetic energy dissipation rate, and mixing length. Demon-strate why the advection terms in the passive scalar, momentum, and turbulent ki-netic energy transport equations have vanished. Which classification of turbulencemodel is this simple model according to this chapter?7) Explain what a mixing length physically describes in turbulence modelling.8) In wall flows, the mixing-length algebraic turbulence model provides the turbu-lent viscosity as

νT = `2m|∂〈U〉∂y|, (13.41)

where `m is the mixing length, 〈U〉 is mean flow parallel to the wall surface, and yis the wall-normal direction. In the log-law region, the mixing length can be givenby `m = κy, where κ is the von Karman constant. Show that the turbulent viscosityin the log-law region can be given by

νT = uτκy, (13.42)

where uτ is the friction velocity.

121

Chapter 14

Large-eddy Simulation Models

In Large-Eddy Simulation (LES) , the larger three-dimensional unsteady turbulentmotions are directly represented, while the effects of the smaller scale motions aremodelled. In terms of computational expense, LES lies between Reynolds-stressmodels and Direction Numerical Simulation (DNS). It is motivated by the limita-tions of each of these approaches. In addition, for many applications, most domi-nant transport mechanisms occur at larger scales (e.g. pollution and heat transport)hence justifying simulating large scales accurately while modelling small scales.From theoretical point of view, large scales of turbulent motion are dependent ongeometry and conditions of the flow, while smaller scales can be universal. Thisin turn motivates use of LES for more accurate predictions. LES was pioneeredfor applications in meteorology first in the 1960s and 1970s (Smagorinsky, 1963;Deardorff, 1974).

LES requires four conceptual steps. First, a filtering operation is required todecompose the velocity U(x, t) into the sum of a filtered (or resolved) componentU(x, t) and a residual or Subgrid-scale (SGS) component u′(x, t). The filtered ve-locity field differs from a mean 〈U(x, t)〉 because it is time dependent and repre-sents the motion of the large eddies,

U(x, t) = U(x, t) + u′(x, t). (14.1)

Second, the Navier-Stokes momentum equation can be developed for the evo-lution of the filtered velocity. This equation will contain a residual stress tenor orSGS tensor that arises from the residual motions.

Third, the Navier-Stokes momentum equation must be closed by modellingthe residual stress tensor or SGS tenser, usually using an eddy-viscosity (or down-gradient diffusion).

Fourth, the model equations are solved numerically for U(x, t), which providesan approximation of the large-scale motions in one realization of the turbulent flow.

122

Filtering

Mathematically, the filtering operation is described by integrating a function overits entire domain. For turbulence, the function can be velocity vector, the integrand,and can be integrated over the entire flow domain

U(x, t) =

∫G(r,x)U(x− r, t)dr (14.2)

where the specified filter function G must satisfy the normalization condition∫G(r,x)dr = 1. (14.3)

The filtering operation can be demonstrated for a 1D velocity field U(x) witha homogeneous filter. A commonly used filter is the box filter. With this filter, U(x)

is simply the average of U(x′) in the interval x− 12∆ < x′ < x+ 1

2∆, where ∆ can

be understood as the filter width. The box filter G(r, x) can be given as 1∆

if |x− r| < 12∆

0 otherwise.

Filtered Conservation Equations

By applying the filtering operation to the Navier-Stokes equations we can obtainthe governing equations for filtered quantities. When spatially uniform filters areused, such as the box filter, the filtering and differentiation operations commute.The filtered continuity equation is(

∂Ui∂xi

)=∂Ui∂xi

= 0. (14.4)

This also implies a continuity equation for the SGS velocity. Therefore, boththe filtered field U and the SGS field u′ are solenoidal, i.e.

∂u′i∂xi

=∂

∂xi(Ui − U i) = 0. (14.5)

The filtered momentum conservation equation can be expressed as,123

∂U j


+∂UiUj∂xi︸︷︷︸

Advection

= ν∂2U j

∂xi∂xi︸︷︷︸Surface Forces

− 1

ρ

∂p

∂xj︸︷︷︸Normal and Body Forces

, (14.6)

where p(x, t) is the filtered pressure field. This equation needs further attentionsince the filtered product UiUj is different than the product of the filtered velocitiesU iU j . The difference of the two is the residual-stress tensor defined by

τRij ≡ UiUj − U iU j. (14.7)

This is analogous to the Reynolds-stress tensor 〈uiuj〉 ≡ 〈UiUj〉 − 〈Ui〉〈Uj〉.The residual kinetic energy is defined as

kr ≡ τRii . (14.8)

The anisotropic residual-stress tensor is defined by

τ rij = τRij −2

3krδij. (14.9)

It is possible to absorb the isotropic residual stress in the pressure to obtain themodified filtered pressure such that

pm ≡ p+2

3ρkr. (14.10)

With these considerations it is possible to obtain the momentum equation forthe filtered velocity as a function of anisotropic residual-stress tensor such that

D U j


= ν∂2U j

∂xi∂xi︸︷︷︸Surface Forces

−∂τ rij∂xi︸︷︷︸

Anisotropic Residual Stress Forces

− 1

ρ

∂pm

∂xj︸︷︷︸Modified Pressure Forces

.

(14.11)The material or substantial derivative based on the filtered velocity can be de-

fined in a similar way to the mean velocity, i.e.

D

Dt≡ ∂

∂t+ U.∇. (14.12)

124

The closure for the filtered momentum equation can be obtained by modellingthe anisotropic residual or SGS stress tensor. For brevity, sometimes this stress ten-sor is referred to as residual or SGS stress tensor, with the word anisotropic dropped.

The Smagorinsky Model

The simplest model to close the turbulence model for LES was proposed by Smagorin-sky (1963). First, the model employs the linear eddy-viscosity relationship to givethe anisotropic residual-stress tensor

τ rij = −2νrSij, (14.13)

where Sij is the filtered rate of strain. Second, the model specifies the residual eddyviscosity νr(x, t) by employing a mixing length relationship

νr = `2SS = (CS∆)2S, (14.14)

where S is the characteristic filtered rate-of-strain. `S is Smagorinsky lengthscale,analogous to the mixing length, which through constant CS can be related to thefilter width ∆. In a simplistic model it can be assumed CS = 0.09. In many LESnumerical simulation models ∆ is taken as the geometric average of the grid ele-ment dimensions

∆ = (∆x∆y∆z)1/3. (14.15)

One-equation Turbulent Kinetic Energy Model

Another recent approach to close the turbulence model for LES is the one-equationturbulent kinetic energy model (Li et al., 2010; Aliabadi et al., 2017, 2018). In thismodel, an extra transport equation is solved for the SGS Turbulent Kinetic Energy(TKE) ksgs given as

125

Dksgs


≡ ∂ksgs∂t︸︷︷︸

Storage

+U.∇ksgs︸︷︷︸Advection

= ∇.(

νrσksgs

∇ksgs)




.

(14.16)Here, the SGS Turbulent Kinetic Energy (TKE) ksgs does not represent fluc-

tuations at all scales but only at smallest scales below the filter width ∆. U is thespatially- and temporally-resolved velocity vector and σksgs = 1 is the turbulentPrandtl number for SGS TKE. The production term P is closed as

P = −τ rijSij = 2νr(Sij)2, (14.17)

where the residual viscosity νr is closed using the following relationship employingthe SGS TKE ksgs and more constants

νr = Ckk1/2sgs l. (14.18)

HereCk = 0.094 is a constant, and l = C∆(∆x∆y∆z)1/3 is the subgrid mixinglength. Constant C∆ is usually in the order of one but can be optimized to controlthe SGS dissipation rate of turbulent kinetic energy (Aliabadi et al., 2018). Finallythe dissipation rate of turbulent kinetic energy ε is closed using one more constant

ε = Cεk

3/2sgs

l, (14.19)

with Cε = 1.048. The one-equation turbulent kinetic energy model for closing LESis advantageous over the Smagorinsky model, especially for free shear flows wheregradients of velocity may approach zero in the domain while the residual viscosityis not negligible. This is analogous to the advantage of turbulent kinetic energymodels over mixing length models for turbulent viscosity models.

The Problem of Inlet Condition

As appealing as LES may appear, it has a serious limitation: realistic inlet bound-ary condition. In order to build a robust LES model, introducing realistic turbulentfluctuations at the inlet are required that would evolve in the entire domain. From a

126

theoretical stand point, the fluctuations must meet several criteria: a) they must bestochastically varying, on scales down to the spatial and temporal filter scales; b)they must be compatible with the Navier-Stokes equations; c) they must be com-posed of coherent eddies across a range of spatial scales down to the filter length;d) they must allow easy specification of turbulence properties; and e) they must beeasy to implement (Tabor and Baba-Ahmadi, 2010).

Two of the most common approaches to generate the inlet turbulent fluctu-ations for LES models are the synthetic and precursor methods. In the syntheticmethod, random fields are constructed at the inlet, while in the precursor methodan additional simulation is performed to generate the desired fluctuations. Precur-sor methods are shown to be more accurate but more computationally demandingand more difficult to implement (Tabor and Baba-Ahmadi, 2010). These methodshave been reviewed in the literature (Tabor and Baba-Ahmadi, 2010; Castro andPaz, 2013). The synthetic method is more popular for practical applications and isdescribed further below.

Lund et al. (1998) developed a synthetic model, originally introduced by Spalart(1988), to generate the inlet turbulent fluctuations by rescaling the velocity field ata downstream station, and re-introducing it as a boundary condition at the inlet,and hence developing spatial and temporal turbulent boundary layers economically(Lund et al., 1998; Cao, 2014). Compared to primitive methods of random inclu-sion of perturbations at inlet, it has been shown that this synthetic method reducesadaptation distance upstream of the flow significantly, resulting in smaller domainsand more economical simulations (Lund et al., 1998). Another common syntheticmodel is the vortex method originally developed by Sergent (2002) and later refinedby Benhamadouche et al. (2006), Mathey et al. (2006), and Xie (2016) that insertsrandom two-dimensional vortices at the inlet boundary that evolve into the simula-tion domain. These vortices are parameterized by realistic lengthscales, timescales,and vorticity magnitudes, formulated from mean flow information and grid spac-ing. Aboshosha et al. (2015) developed a method based on synthesizing randomdivergence-free turbulence velocities with consideration of spectra and coherencyfunctions that match the flow statistics. The method is also known as ConsistentDiscrete Random Field Generation (CDRFG) . This scheme maintains both the tur-

127

bulence spectra and coherency function, which are essential for proper simulationof interaction of turbulent flow with flexible structures, such as buildings, proneto flow-induced dynamic excitation. Another notable synthetic method useful foranalysis of flow-induced excitation of structures is known as Modified Discretiz-ing and Synthesizing Random Flow Generation (MDSRFG) that was developedrecently by Castro et al. (2017) and Ricci et al. (2017). This method is based onproviding a true representation of the coherency of the velocity field at the inlet. Analternative technique to the classical velocity perturbations is the Temperature Per-turbation Method (TPM) developed by Buckingham et al. (2017) to generate inletturbulence fluctuations. This method relies on the creation of turbulent structuresthrough a buoyancy triggered mechanism by seeding the flow with random temper-ature perturbations at the inlet. Buckingham et al. (2017) found that the temperatureperturbation method can result in long adaptation distances. Although this methodbenefits from the simplicity of not requiring prior knowledge of second order mo-ments or integral lengthscales at the inlet, additional refinements for developingflows that include physical temperature effects are required.

Exercises

1) Explain why the Large-eddy Simulation (LES) is an incomplete turbulence model.2) A computational model domain in Cartesian coordinates in consideration forLarge-eddy Simulation (LES) has uniform mesh spacing with ∆x = 4 m, ∆y =

2 m, and ∆z = 1 m, show that the filter width is calculated as

∆ = 2 m. (14.20)

3) Some LES models parameterize the residual eddy viscosity νr(x, t) independentof the local filter width ∆ (Mason and Callen, 1986). Suppose a simple LES modelis to be developed with such a formulation of the residual eddy viscosity. For in-stance, the atmospheric boundary layer wind is considered. The axis normal to theearth surface is y and wind flows in the x direction parallel to the earth surface.A formulation for residual eddy viscosity is proposed employing a mixing length

128

relationship, in which the mixing length is only a function of distance away fromthe earth surface y, i.e.

νr = (`m(y))2 S. (14.21)

It is desired to formulate mixing length `m(y) in such a way that it approachesa maximum value `0 far away from the earth surface, while it approaches κy closeto the surface, where κ = 0.41 is the von Karman constant, i.e.y → 0 `m(y)→ κy

y →∞ `m(y)→ `0.

Show that the following formulation for `m(y) has this property, where

1

`m(y)=

1

`0

+1

κy. (14.22)

4) Consider the previous problem. The proposed formulation of the mixing length`m(y) was idealistic because near the earth surface, mixing length, and thereforeresidual eddy viscosity, should approach zero according to this formulation. In otherwords, the earth surface was assumed absolutely smooth for the first numerical cellsuch that turbulent flow would vanish in this cell. In practical modelling, however,the first numerical cell adjacent to the earth surface is large enough so that theearth surface is rough in comparison to the characteristic size of the first numericalcell. In this scale, there will be turbulence and hence non-zero mixing length, andtherefore non-zero residual eddy viscosity. To remedy this, the mixing length canbe parameterized alternatively.

Consider that the roughness of the earth surface can by characterized by length-scale y0 such that y0 <<< `0. It is desired to formulate mixing length `m(y) in sucha way that it approaches a maximum value `0 far away from the earth surface, whileit approaches κ(y + y0) close to the surface, where κ = 0.41 is the von Karmanconstant, i.e. y → 0 `m(y)→ κ(y + y0)

y →∞ `m(y)→ `0.

129

Show that the following formulation for `m(y) has this property, where

1

`m(y)=

1

`0

+1

κ(y + y0). (14.23)

5) An atmospheric scientist has developed an LES code to simulate the airflow overa flat farm land. Flow parallel to the earth surface is in the direction of positive x.The direction normal to the earth surface is positive y. The model height is H . Thescientist has developed a numerical grid that is uniform in the x and z directionsbut varies in the y direction. That is, the grid is more refined near the surface but isless refined far away from the surface. The scientist has generated three numericalgrids: fine, medium, and coarse. The fine mesh has the smallest cell sizes while thecoarse mesh has the largest cell sizes. The local grid size in the domain is given as∆. It is known that the Kolmogorov scales anywhere in a fluid domain are given by

η =

(ν3

ε

)1/4

, (14.24)

where ν is fluid’s molecular kinematic viscosity and ε is the local dissipation rate forturbulent kinetic energy. It has been suggested that the maximum dissipation takesplace corresponding to a lengthscale of about 24η. Since at least two grid points areneeded to resolve a flow feature, a grid spacing of ∆ = 12η is required to resolvefeatures of the flow having a scale of 24η (Frohlich et al., 2005). This justifiescalculating ∆/η and evaluating where in the domain it is less or greater than 12.The scientist has produced this plot shown in Figure 14.1, where non-dimensionalheight y/H is plotted versus ∆/η.

Provide an argument and analyze this plot to determine for each simulationwhere in the domain the flow features are resolved associated with lengthscale of24η, i.e. the lengthscale associated with maximum dissipation.6) An atmospheric scientist has developed an LES code to simulate the airflow overan urban area. Flow parallel to the earth surface is in the direction of positive x.The direction normal to the earth surface is positive y. The model height is H . Thescientist has generated three numerical grids: fine, medium, and coarse. The finemesh has the smallest cell sizes while the coarse mesh has the largest cell sizes.It has been suggested that the ratio of the resolved and modelled features of the

130

Figure 14.1: LES case: non-dimensional altitude y/H plotted versus the ratio ofgrid size to Kolmogorov scales ∆/η.

flow are controlled by the ratio of the residual to molecular viscosities, i.e. νr/ν(Frohlich et al., 2005). This justifies calculating this ratio and evaluating where inthe domain it is low or high. The scientist has produced this plot shown in Figure14.2, where non-dimensional height y/H is plotted versus νr/ν.

Figure 14.2: LES case: non-dimensional altitude y/H versus νr/ν.

Provide an argument and analyze this plot to determine for each simulationwhere in the domain the flow features are more resolved than modelled, or where in

131

the domain the flow features are more modelled than resolved. Why does generallya coarser mesh result in more flow features being modelled as opposed to resolved?7) A scientist is developing a Large Eddy Simulation (LES) code for a two-dimensionalflow. He is considering a wall flow for a region near the wall given by the followingmean velocity profile in units of [m s−1]

〈U〉 = ln y, (14.25)

in which the x direction is along the wall in the streamwise direction, 〈U〉 is themean flow parallel to the wall along the x direction, and y is the direction normalto the wall. Note that the mean velocity normal to the wall is given by 〈V 〉 = 0. Inhis model the residual viscosity is given as νr = 0.2 m2 s−1. Help him calculate theanisotropic residual-stress tensor τ rxy at y = 2 m. What is the units of this quantity?8) A meteorologist is developing a one-equation turbulent kinetic energy model toclose the system of equations for a Large-eddy Simulation (LES). He wishes to addthe temperature and passive scalar equations to the continuity, momentum, and Sub-grid Scale (SGS) turbulent kinetic energy equations. For this model, the momentum,temperature, and passive scalar equations for the filtered velocity, temperature, andpassive scalar are given by

D U i


= ν∂2U i

∂xj∂xj︸︷︷︸Surface Forces

+∂

∂xj

(νr∂U i

∂xj

)︸︷︷︸

Anisotropic Residual Stress Forces

(14.26)

− 1

ρ

∂pm

∂xi︸︷︷︸Modified Pressure Forces

− gδi3︸︷︷︸Buoyancy Body Force

,

D T


= αθ∂2T

∂xj∂xj︸︷︷︸Molecular Heat Diffusion

+∂

∂xj

(αθr

∂T

∂xj

)︸︷︷︸

Turbulent Heat Diffusion

, (14.27)

D S


= αS∂2S

∂xj∂xj︸︷︷︸Molecular Passive Scalar Diffusion

+∂

∂xj

(αSr

∂S

∂xj

)︸︷︷︸

Turbulent Passive Scalar Diffusion

. (14.28)

132

Note that the buoyancy body force due to gravitational acceleration is not ab-sorbed in the modified pressure force term in this formulation. The term δi3 ensuresthat the gravitational acceleration is only considered in the vertical direction, i.e.z direction or the third component of the Cartesian coordinate system. ν and νr

are molecular and residual viscosities. αθ and αθr are molecular and residual ther-mal diffusivities. Likewise, αS and αSr are molecular and residual passive scalardiffusivities. The turbulent kinetic energy model is given by

Dksgs


= ∇.(

νrσksgs

∇ksgs)


+ P︸︷︷︸Shear Production

+ B︸︷︷︸Buoyant Production


,

(14.29)where P and B account for shear and buoyant production of turbulent kinetic en-ergy, respectively. Here, the SGS Turbulent Kinetic Energy (TKE) ksgs does notrepresent fluctuations at all scales but only at smallest scales below the filter width∆. σksgs is the turbulent Prandtl number for SGS TKE. The production terms areclosed as

P = 2νr(Sij)2, (14.30)

B = −gαθr∂T

∂z, (14.31)

where buoyant production is only related to vertical gradient of the resolved temper-ature. The residual viscosity νr is closed using the following relationship employingthe SGS TKE ksgs and more constants

νr = Ckk1/2sgs l. (14.32)

HereCk = 0.094 is a constant, and l = C∆(∆x∆y∆z)1/3 is the subgrid mixinglength. Constant C∆ is usually in the order of one but can be optimized to controlthe SGS dissipation rate of turbulent kinetic energy. Finally the dissipation rate ofturbulent kinetic energy ε is closed using one more constant

ε = Cεk

3/2sgs

l, (14.33)

133

with Cε = 1.048. As noted above the turbulence model is not closed yet, becausefurther parameterizations are necessary for αθ, αθr, αS , and αSr. Provide simpleparameterizations for these terms as functions of either ν or νr.

134

Chapter 15

Direct Numerical Simulation

For the first time Orszag and Patterson (1972) showed that it is possible to per-form computer simulations of a fully developed turbulent flow without the needto model or parameterize turbulence in a closure-like approach. In the so-calledDirect Numerical Simulation (DNS) every eddy from the largest to the smallestis computed. In this approach the Navier-Stokes equations are integrated forwardgiven some initial and boundary conditions. Using this approach the entire velocityfield is available at all spatial and temporal scales of turbulence.

Although attractive, there are serious computational limits in using DNS giventoday’s computer technology. It was discussed earlier that the Kolmogorov lengthscale in turbulent flow scales with η ∼ Re−3/4`0, where `0 is the size of energy-containing eddies andRe is the Reynolds number of the flow. If a DNS shall resolveall eddies, then the numerical grid must be as spatially refined as the Kolmogorovlength scale. In other words, given the Cartesian coordinate, the grid resolutionshould satisfy

∆x ∼ ∆y ∼ ∆z ∼ Re−3/4`0. (15.1)

This requirement can quickly become inhibiting with increasing Reynolds num-ber. The number of grid points required at any instant for a three-dimensional sim-ulation is therefore

Nx ∼(Lbox

∆x

)3

∼(Lbox

`0

)3

Re9/4, (15.2)

where Lbox is a typical dimension of the computational domain. We can immedi-ately spot the problem, that a large Reynolds number requires a large number of gridpoints for a practical DNS. Furthermore, we can assess the number of timesteps re-quired for a DNS given the Kolmogorov lengthscale. The maximum permissibletimestep in a simulation is in the order of ∆t ∼ ∆x/〈U〉 ∼ η/〈U〉 since, in order

135

to maintain numerical stability and accuracy, we cannot allow a fluid parcel travelmore than on grid spacing per timestep. If T is the total duration of the simulation,then the minimum number of timesteps required for a DNS is

Nt ∼T

∆t∼ T

η/〈U〉∼ T

`0/〈U〉Re3/4. (15.3)

The number of computer operations required for DNS is thus obtained by mul-tiplying Nx and Nt, so the computational cost scales with

Computational Cost ∼ NxNt ∼(

T

`0/〈U〉

)(Lbox

`0

)3

Re3. (15.4)

Given today’s computational power, size of the domain, and the Reynolds num-ber, a DNS for a practical problem may take anywhere between few hours to cen-turies to run. At the moment DNS investigators have spent most of the effort onsimple possible geometries such as a periodic cube or box turbulence (Davidson,2009). Someday with the advent of quantum computing, it may be possible to applyDNS to more and more practical problems with higher domains and larger Reynoldsnumbers.

Exercises

1) Assuming that Lbox = 1 m and that Re = 10, 000, estimate the number of spatialgrid points for a three-dimensional DNS. How did you estimate `0?2) Many practical DNS models do not use Kolmogorov length and time scales todetermine the grid resolution or time step. Instead they choose larger length andtime scales that correspond to the finest scales of the inertial subrange. This way,turbulent fluctuations finer than these scales are ignored and models can performsimulations more computationally efficiently. In the inertial subrange, the velocityand time scales for an eddy of size ` are given by

u(`) = (ε`)1/3, (15.5)

τ(`) = (`2/ε)1/3, (15.6)136

where ε is the dissipation rate of the turbulent kinetic energy. If one rearranges theexpressions for the velocity and time scales, one can obtain the eddy length and timescales as functions of velocity scale and dissipation rate. The eddy length and timescales can be used to estimate the computational cost of a DNS model assumingthat the grid resolution is in the order of ` and simulation time step is in the orderof τ(`). Assuming that Lbox is a typical dimension of the computational domainand that T is the total duration of the simulation, show that the computational costscales with

Computational Cost ∼ NxNt ∼ L3boxT

ε4

u(`)11. (15.7)

137

Chapter 16

Wall Models

The near-wall region in a turbulent flow adds complexity and computational ex-pense to the task of performing calculations for accurately predicting turbulentflows. Given the very steep profiles for most solution variables near the wall, suchas k, ε, 〈U〉, etc., resolving the near-wall regions in a turbulent simulation is verycostly. As a result, alternative approaches must be found to model or approximatethe solution behaviour near the walls using some form of algebraic models, withouthaving to resolve the solution profiles in detail near the walls.

The wall function approach, first introduced by Launder and Spalding (1972),applies boundary conditions at some distance away from the wall, so that turbulence-model equations do not have to be solved close to the wall, i.e. between the walland the location at which the boundary conditions are applied. Generally, if thefirst computational cell adjacent to a wall lies entirely inside the viscous sublayer,then a wall function is not required. However, if the first computational cell coversparts of the buffer layer and beyond, then a wall function is necessary for accuratesimulation of turbulence.

Point-wise Standard Wall Function

Figure 16.1 shows the placement of the first computational cell with respect to wallregions according to the law of the wall. The entire height of the first computationalcell is ∆y = 2yp, where yp is the distance from wall to the centre of the com-putational cell. The standard wall function assumes that point yp is located in thelog-law region. The wall-function boundary conditions are applied at this location,i.e. y = yp. The subscript p indicates quantities evaluated at yp, e.g. 〈U〉p, kp, andεp. For a high-Reynolds-number zero-pressure-gradient boundary layer, the log-lawequation is given as

138

Figure 16.1: Placement of the first computational cell with respect to wall regionsaccording to the law of the wall.

u+ ≡ 〈U〉uτ

=1

κln y+ +B. (16.1)

It must be remembered that uτ ≡√τw/ρ is defined as the friction velocity,

where τw is the shear stress at the wall. The balance of the production rate anddissipation rate of turbulent kinetic energy near the wall yields

ε =u3τ

κy. (16.2)

In addition, using the k − ε model, and the expression for the turbulent viscos-ity for simple shear flows, it is possible to relate shear stress near the wall to theturbulent kinetic energy by

−〈uv〉 = u2τ = C1/2

µ k. (16.3)

The standard wall function uses these relations to provide a robust boundaryconditions under all circumstances at location yp. First a nominal friction velocity

is defined using the value of turbulent kinetic energy at distance yp, i.e. kp

u∗τ ≡ C1/4µ k1/2

p . (16.4)

139

In turbulent simulations, an exact value of y+p ≡ uτy/ν is not known since an

accurate estimate of uτ is not available. However, once the nominal friction veloc-ity is calculated, it is possible to estimate the corresponding y+

p by the followingrelationship

y∗p ≡ypu

∗τ

ν. (16.5)

The nominal mean velocity is then obtained from the log-law relationship,which can approximate the true mean velocity at position yp, which lies in the log-law region

〈U〉∗p = u∗τ

(1

κln y∗p +B

). (16.6)

The boundary condition at yp for the mean momentum equation is not appliedby specifying a 〈U〉p but instead by specifying a shear stress as

−〈uv〉p = u∗2τ〈U〉p〈U〉∗p

. (16.7)

The boundary condition for ε can be conveniently defined having the nominalfriction velocity as

εp =u∗3τκyp

, (16.8)

while zero-normal gradient conditions are applied to k and to the normal stresses. Infinite volume simulations of turbulent flow, the location of yp is taken to be the firstgrid node away from the wall. Wall functions in general introduce yp as an artificialparameter. For boundary-layer flows for which the log-law relations are accurate,the overall solution is insensitive to the choice of yp, as long as it is within the log-law region. However, in other flows it is found that the overall solution is sensitiveto this choice. As a result, it may not be possible to obtain numerically accurate andgrid-independent solutions, since refining grids usually means reducing yp.

140

Integrated Werner-Wengle Wall Function

As was seen in the standard wall function, one needs to solve and resolve k near thewall, at least at point yp, to be able to estimate the friction velocity by the nominalfriction velocity. For the wall function to provide the shear stress boundary condi-tion at point yp in this way requires an iterative approach that could impose com-putational cost. Alternatively, a wall function was proposed by Werner and Wengle(1991) that eliminated the need for a solution of k because it enabled a closed formsolution for the shear stress near the wall given other known parameters. In addition,this wall function integrated the entire profile of u+ over the first computational cellin the direction normal to the wall to arrive at a better estimate of u+ compared topoint-wise models. The Werner-Wengle wall function provides the following rela-tionship between u+ and y+

u+ = y+ if y+ ≤ 11.81

u+ = A(y+)B if y+ > 11.81,

where A = 8.3 and B = 1/7. The two near-wall regions intersect at the value ofy+ = 11.81 known as the intersecting y+ or y+

i , which can be given in terms of Aand B, as

y+i = A

11−B . (16.9)

It is possible to find an average value for u+ over the entire computational cellin the wall normal direction, i.e. from y+ = 0 to y+ = ∆y+ = 2y+

p , by performingthe following integral

u+avg =

1

2y+p

∫ 2y+p

0

u+(y+)dy+

=1

2y+p

(∫ y+i

0

y+dy+ +

∫ 2y+p

y+i

A(y+)Bdy+

), (16.10)

where the integral is split into two integrals appropriate for each near-wall region.This definite integral can be evaluated such that

141

2u+avgy

+p =

y+2i

2+

A

1 +B

((2y+

p )1+B − (y+i )1+B

), (16.11)

which by substitution of u+avg, y

+p , and y+

i can be re-expressed as

2〈U〉avguτ

ypuτν

=1

2A

21−B +

A

1 +B

((2ypuτν

)1+B

− A1+B1−B

). (16.12)

Note that 〈U〉avg represents the control volume-averaged mean velocity, notto be confused with 〈U〉p, which represented the mean velocity at point p in thestandard wall function (Efros, 2006). This formula can be rearranged to give thesquare of friction velocity, or alternatively wall shear stress, i.e. −〈uv〉 = u2

τ , as anexplicit function of other known variables such as 〈U〉avg, yp, A, B, and ν, withoutthe need for knowledge of k, such that

−〈uv〉avg =

[1 +B

A

(ν

2yp

)B〈U〉avg +

(ν

2yp

)1+B (A

1+B1−B − 1 +B

2AA

21−B

)] 21+B

.

(16.13)Note that in control volume schemes for computational fluid dynamics anal-

ysis, most solution variables are in fact averaged throughout each control volume.The Werner-Wengle wall function is consistent with control volume schemes suchthat the boundary condition for the wall shear stress in the mean momentum equa-tion for the first computational cell adjacent to the wall can be conveniently applied.

van Driest Near Wall Treatment

In LES based on the Smagorinsky closure scheme (Smagorinsky, 1963) the Smagorin-

sky lengthscale was introduced as `S = CS∆, where CS is a constant and ∆ is filterlength, typically chosen as the geometric average grid size, i.e. ∆ = (∆x∆y∆z)1/3.However, near walls the actual lengthscale `S can reduce more significantly thancalculated using the Smagorinsky lengthscale formula. If this reduction is not takeninto account, then excessive dissipation of turbulent kinetic energy near the walls

142

may be predicted. To prevent this, van Driest (1956) suggested that `S can bedamped near walls according to the van Driest function

`S = CS∆

[1− exp

(−y+

A+

)](16.14)

with a typical value for A+ = 25. Note that for LES the van Driest function canbe used in combination with a wall function such as the Werner-Wengle model toresult in more accurate simulations without having to resolve turbulence near thewalls.

Wall Function Summary

Wall functions are widely used to economize turbulent simulations. However, theycan be much more complex than introduced here. For instance, other wall laws maybe assumed, such as multi-layer laws, considering more than two layers discussedhere (Temmerman et al., 2003). Another complexity that arises is wall modellingfor very rough walls with possibly vertical and horizontal heterogeneous roughnessstructures. For such cases the law of the wall may change and require application-specific wall models (Raupach et al., 1980, 1991; Raupach, 1992; Jimenez, 2004;Blocken et al., 2007; Qi et al., 2018).

Exercises

1) For the Werner-Wengle wall function (Werner and Wengle, 1991) show that

y+i = A

11−B . (16.15)

2) Suppose that the Werner-Wengle wall function (Werner and Wengle, 1991) is tobe used in the point-wise form, i.e. the profile of u+ is not supposed to be integrated,but instead the value of the shear stress 〈uv〉p at yp is desired as a function of 〈U〉p,ν, yp, A, and B. If point yp is located in the power-law region, show that

−〈uv〉p = u2τ =

(νB〈U〉pAyBp

) 21+B

. (16.16)

143

3) In Computational Fluid Dynamics (CFD) models that use wall functions, it isvery critical for the wall law to be valid in the majority of the spatial extent of thefirst computational cell adjacent to the wall. The coarser the first computational celladjacent to the wall, the higher the likelihood that the wall law may not satisfy thisrequirement. If an incompetent CFD modeller uses a very coarse wall-adjacent cell,then the profile of the u+ in the cell according to the wall law may over estimate orunder estimate the actual flow profile of u+. Assuming the flow velocity, 〈U〉p or〈U〉avg, is the same, argue whether the flow friction velocity uτ as a result of usinga wall function will be over or under predicted based on whether the profile of u+ isover or under estimated. Is your answer the same for point-wise and integrated wallfunctions? Hint: you may draw the profile of u+ versus y+ and use the definition ofu+.4) When implementing the standard wall model for numerical simulations, weshould express the Reynolds stress term in the mean momentum equation,

D〈Uj〉Dt︸︷︷︸


= ν∇2〈Uj〉︸︷︷︸Surface Forces

− ∂〈uiuj〉∂xi︸︷︷︸

Reynolds Stresses

− 1

ρ

∂〈p〉∂xj︸︷︷︸


, (16.17)

only in terms of constants and other solution variables, namely average velocity,turbulent kinetic energy, and turbulent kinetic energy dissipation rate. Show thatthe standard wall model can be used to express Reynolds stress in a form that doesnot include the nominal friction velocity, u∗τ , so that

−〈uv〉p =c

1/4µ k

1/2p 〈U〉p

1κ

ln(c1/4µ k

1/2p ypν

)+B

. (16.18)

5) Various specific wall models are used to describe atmospheric flows. Wall modelsfor the atmosphere describe vertical variation of horizontal wind speed within thefirst few tens of meters away from the earth surface. For instance, Seinfeld andPandis (2006) provide the following logarithmic wall model

〈U〉(z) =uτκ

ln

(z

z0

), (16.19)

where 〈U〉(z) is horizontal wind speed as a function of height z, uτ is frictionvelocity, κ is the von Karman constant, and z0 is the earth surface aerodynamic

144

roughness height. For instance, over grassland it may be assumed z0 = 0.1 m, whilein low-rise residential urban areas it may be assumed z0 = 10 m. Assuming uτ =

1 m s−1, calculate horizontal wind speed at z = 100 m over grassland versus low-rise residential areas. What is the difference between the calculated horizontal windspeeds?6) Some atmospheric boundary-layer wall models assume a displacement heightd0, which will shift the wind profile in the z direction with magnitude d0. Therationale for this modification is the observation particularly in the densely builturban environment (Raupach et al., 1991). Raupach et al. (1991) and Graf et al.(2014) provide one such wall model as

〈U〉(z) =uτκ

ln

(z − d0

z0

), (16.20)

where 〈U〉(z) is horizontal wind speed as a function of height z, uτ is frictionvelocity, κ is the von Karman constant, and z0 is the earth surface aerodynamicroughness height. Assuming z0 = 10 m, uτ = 1 m s−1, and d0 = 5 m calculatehorizontal wind speed at z = 100 m. What is the difference between this calculationcompared to the same calculation in the previous problem?7) Alternative to the logarithmic wall model, a power law wall model has beenproposed for the atmospheric boundary layer. Seinfeld and Pandis (2006) providethis wall model as

〈U〉(z) = 〈U〉(zref )

(z

zref

)α, (16.21)

where zref is some reference height, 〈U〉(zref ) is horizontal wind speed at thisreference height, and α is a fitted constant. We wish to express α as a functionof zref and z0 if the logarithmic and power law wall functions are to be matched atreference height zref . To do this, follow these steps. First rearrange the power lawwall model to express α as

α =ln(〈U〉(z)〈U〉(zref )

)ln(

zzref

) . (16.22)

145

Next substitute both 〈U〉(z) and 〈U〉(zref ) by their logarithmic wall model ex-pressions. After doing this α can be expressed as a function of z, zref , and z0. Nexttake the limit of

limz→zref

(α(z, zref , z0)) . (16.23)

This limit can be conveniently taken by invoking the L’Hospital’s Rule. Aftertaking this limit, obtain the following relationship for α

α =1

ln(zrefz0

) . (16.24)

8) An oceanographic scientist has developed a Large-Eddy Simulation (LES) modelto simulate water flow in the ocean on top of a rough ocean bed. Suppose the char-acteristic length at the ocean floor is L. For instance, this could be the depth ofan ocean valley or the height of an ocean hill. Flow is in the positive x directionparallel to ocean floor while the direction normal to the ocean floor is positive y.The scientist creates three numerical grids: fine, medium, and coarse. The fine gridhas the smallest numerical cells while the coarsest grid has the largest numericalcells. The scientist simulates ocean flow at two water velocities that correspond toReynolds numbers 10,000 and 100,000. In total, the scientist has six simulations.For each simulation, the scientist provides a plot of y+ against non-dimensional dis-tance x/L. y+ is calculated using the height of the centre of the first cell adjacentto the ocean bed. This plot is provided in Figure 16.2.

Provide an argument to explain which of the simulations above require a wallmodel to provide accurate results. Why does y+ increase by coarsening the numer-ical grid? Why does y+ increase by increasing the flow Reynolds number?9) In atmospheric science, surface layer is defined as a layer of flow adjacent to theearth surface within the boundary layer where the magnitude of the friction velocityis constant (Stull, 1988). This friction velocity in atmospheric flows is denoted byuτ,ABL, where ABL stands for Atmospheric Boundary Layer (ABL). The heightof the surface layer is usually about 10% of the height of ABL. In fact in manyCFD analyses applied to ABL, where the height of the computational domain isless than 10% of the height of ABL, it is fair to assume a constant atmospheric

146

Figure 16.2: Ocean flow Large-Eddy Simulation (LES): y+ versus x/L.

friction velocity uτ,ABL throughout all elevations for upstream flow. With analogyto the material found in this chapter, provide an argument to support the followingrelationships

〈U〉(y) =uτ,ABLκ

ln

(y

y0

), (16.25)

k(y) =u2τ,ABL√Cµ

, (16.26)

ε(y) =u3τ,ABL

κy, (16.27)

where y is vertical distance normal to the earth surface, y0 is the aerodynamic rough-ness height of the earth surface, 〈U(y)〉 is mean horizontal velocity in the x direc-tion, κ is the von Karman constant, k(y) is turbulent kinetic energy, Cµ is a modelconstant, and ε(y) is turbulent kinetic energy dissipation rate. Comment if the tur-bulent kinetic energy is or is not a function of height y within the surface layer.10) In this chapter the log-law of the wall was introduced as u+ = 1

κln(y+) + B.

This law is valid for rather smooth walls. It has been shown that this model must bemodified for rough walls to arrive at a new law that is the log-law for a rough wall(Blocken et al., 2007). To express this new law first we define the dimensionlessphysical roughness height such that

147

k+S =

uτkSν

, (16.28)

where kS is the roughness characteristic height. It has been reported that the log-lawof the wall for a rough wall does not exhibit a viscous or buffer sublayer. In otherwords, these sublayers are destroyed or eliminated. The new law is simply obtainedby shifting the intercept of the curve for log-law (Blocken et al., 2007) such that

u+ =1

κln(y+) +B −∆B(k+

S ), (16.29)

where the magnitude of the shift in intercept ∆B(k+S ) is a function of k+

S . Thisintercept shift is reported as

∆B(k+S ) =

1

κln(k+

S )− 3.3. (16.30)

This equation is valid for dimensionless physical roughness heights as large ask+S = 10, 000. Show that ifB = 5.2 then the log-law for a rough wall can be written

as

u+ =1

κln

(y+

k+S

)+ 8.5. (16.31)

11) In the previous two problems two wall models have been proposed for the at-mospheric boundary layer. These were

〈U〉(y) =uτκ

ln

(y

y0

), (16.32)

u+ =1

κln

(y+

k+S

)+ 8.5. (16.33)

If these two wall models were to be matched, then there will be a relationshipbetween kS and y0, i.e. kS will be a multiplier of y0. If κ = 0.4, show that thisrelationship can be approximated by

kS ≈ 30y0. (16.34)

In fact both kS and y0 are measures of earth surface roughness characterizinga flat horizontal surface. y0 is known as aerodynamic roughness height while kS

148

is known as sand-grain roughness height (Blocken et al., 2007). From the point ofview of meteorology, earth surface is never smooth but rough due to trees, buildings,etc.12) An engineer is designing two types of wall functions for temperature in aboundary layer flow along a surface with direction z normal to the surface. Us-ing Reynolds averaging, vertical velocity normal to the surface can be expressed asW = 〈W 〉+w, where 〈W 〉 = 0, and temperature can be expressed as T = 〈T 〉+ t.For the wall, the surface heat flux is specified as qw in [W m−2]. This is the amountof heat transferred to the fluid, positive when heat is added to the fluid, normal tothe surface. The engineer wishes to use qw and implement a) wall function 1 forforcing the turbulent kinematic heat flux at point p, i.e. 〈wt〉p, away from the sur-face and in the middle of the first computational cell adjacent to the wall and b) wallfunction 2 for forcing mean temperature gradient at point p, i.e. ∂〈T 〉

∂z|p, away from

the surface and in the middle of the first computational cell adjacent to the wall. a)For wall function 1, perform unit analysis to show that the dynamic and kinematicheat fluxes are related in such a way to result in the following simple wall function,

〈wt〉p =qwρCp

, (16.35)

where ρ is fluid density and Cp is the heat capacity of the fluid. Argue in whichregion within the law of the wall for temperature the viscous heat transfer may beignored so that this wall function is valid. b) For wall function 2, use the gradientdiffusion hypothesis to show that the wall function can be written as

∂〈T 〉∂z|p = − qw

ρCpαt= −qwPrT

ρCpνt, (16.36)

where νT is turbulent viscosity and PrT is turbulent Prandtl number. Again, arguefor which region within the law of the wall for temperature the viscous heat transfermay be ignored so that this wall function is valid.

149

Chapter 17

Model Evaluation

Any turbulence model must be carefully assessed before it could be concluded thatit is suitable for a particular application or analysis. Especially after introducingnumerous models in this book, it is necessary to consider a variety of tools andtechniques in assessing turbulence models.

Verification and Validation

One of the main pillars of numerical simulation is the validation and verificationof the calculations performed using a specific code or technique. Validation is theprocess of solving the right equations for the simulated physics, while verification

is solving those equations in a proper manner. Based on the two definitions, onecannot validate a whole numerical code, but only a specific set of calculations for acase study performed using the code (Roache, 1997).

Time and Space Discretization Error Estimation

Discretization convergence is an analysis that must be performed for most complete

models of turbulence, such as the k−εmodel, that are solved numerically. However,although informative, this analysis is not necessarily conclusive when applied toincomplete models of turbulence, such as the LES or mixing length models.

Spatial and temporal discretization of a model domain is a crucial compo-nent for numerical simulation. The number of timesteps or grid elements, or nodes,which can be created in one domain may vary significantly depending on the timemarch sequence, size of that domain, and the sizes of the grid elements themselves.As the numerical solution of the governing equations is obtained for each timestepand each spatial element or node of the grid, the number of timesteps, elements, or

150

nodes and the way they are arranged in the grid can notably affect the accuracy ofthe numerical results.

The most simple test is applied for spatial discretization, where a procedureis performed for quantifying the degree of independence of the numerical solutionfrom the grid size and configuration changes, which is called Grid Independence

Test (GIT). The grid independence test is usually performed for three different lev-els of grid fineness: coarse, medium and fine. The methodology that is used forgrid independence testing consists of a comparison between the obtained solution(velocity, temperature, concentration, etc.) on a continuous spatial segment, such asline or surface, for each grid level. The grid level that exhibits enough grid indepen-dency of its solution (shows no significant change in the solution with the changein grid size to a finer level) is chosen for further use. Sometimes grid economy isconsidered in the independency test. If a finer grid shows no considerable changein the solution than a coarser grid, the latter is “independent enough” and is chosento accelerate the solution process in later case studies (Elmaghraby et al., 2018).

Order of Convergence

The order of grid convergence involves the behaviour of the solution error definedas the difference between the discrete solution and the exact solution,

E = f(h)− fexact = Chp +H.O.T., (17.1)

where C is a constant, h is some measure of mesh or grid spacing, and p is theorder of convergence. The Higher Order Terms (H.O.T.) are negligible comparedto Chp. A representative cell mesh size h can be defined as

h =

(1

N

N∑i=1

∆Vi

)1/3

, (17.2)

where ∆Vi is the volume of cell i, and N is the total number of cells. Note that thisis defined for Control Volume Schemes (CVS), but for Finite Difference Schemes

(FDS) or Finite Element Schemes (FES), the representative cell mesh size h can bedefined in a similar fashion.

151

A second-order discretization for either space or time means that p is equal,or at least very close, to two. A numerical code uses a numerical algorithm thatwill provide a theoretical order of convergence; however, the boundary conditions,numerical models, and mesh will reduce this order so that the observed order ofconvergence will likely be lower. Neglecting H.O.T and taking the logarithm ofboth sides of the above equation result in

lnE = lnC + p lnh. (17.3)

The order of convergence p can be obtained from the slope of the curve of lnE

versus lnh. If such data points are available, the slope can be read from the graphor the slope can be computed from a least-squares fit to the data.

A more direct evaluation of p can be obtained from three solutions. Suppose,we select three significantly different sets of meshes and run our simulations to de-termine values of key solutions needed for an error estimation study. For example,assume φ is a solution being reported. We assume h1 < h2 < h3. We define themesh refinement ratio to be rmn = hm/hn and further the difference between so-lutions at two difference mesh levels be φmn = φm − φn. If using a constant meshrefinement ratio, i.e. if r = r32 = r21, then

p =ln(φ32/φ21)

ln r. (17.4)

The order of convergence is determined by the order of the leading term ofthe truncation error and is represented with respect to the scale of the discretiza-tion, h. The local order of convergence is the order for the stencil representing thediscretization of the equation at one location in the mesh, for instance interior orboundary locations. The global order of convergence considers the propagation andaccumulation of errors outside the stencil. This propagation causes the global orderof convergence to be less than the local order of convergence in the interior of adomain. The order of convergence for the boundary conditions can be one orderlower than the interior order of convergence without degrading the overall globalorder of convergence significantly. This is due to the fact that in three dimensionalgeometries boundary cells are outnumbered by interior cells.

152

Assessing the order of convergence of a code and calculations requires oneto sufficiently refine the mesh such that the solution is in the asymptotic rangeof convergence. The asymptotic range of convergence is obtained when the meshspacing is such that the various mesh spacings h and errorsE result in the constancyof C, i.e.

C =E

hp(17.5)

Grid Convergence Index

A representative measure for grid refinement studies was proposed by Roache (1994),called Grid Convergence Index (GCI). The GCI is extracted from the generalizedRichardson extrapolation theory, and uses an asymptotic approach for calculatingthe amount of uncertainty in grid convergence (Roache, 1994). Similar to the sim-pler grid independence test, the GCI makes use of the solution on three differentgrid size levels, such grids that can be created through grid coarsening and not nec-essarily by grid refinement (Roache, 1997). The GCI reports a numeric value thatshows how much convergence is achieved in the solution between two successivegrid levels, or between the coarsest grid level, taken as a reference, and each one ofthe two other grids.

A consistent numerical analysis is one which provides a result approachingan asymptotic value as the mesh resolution approaches zero. Thus, the discretizedequations will approach the solution of the original differential equations. One sig-nificant issue in numerical computations is to decide what level of mesh resolutionis appropriate. This is a function of the flow conditions, type of analysis, geometry,and other variables. One is often left to start with a coarse mesh resolution and thenconduct a series of mesh refinements to assess the effect of mesh resizing. This isknown as a mesh refinement study.

One must recognize the distinction between a numerical result that approachesan asymptotic value and one that approaches the true solution. Even when theasymptotic solution to a set of differential equations is found, it may be differ-ent from the true physical solution. The GCI is a measure of the percentage thecomputed solution is away from the asymptotic computed solution. It indicates an

153

error band on how far the solution is from the asymptotic value and how much thesolution would change with a further refinement of the mesh. A small value of GCIindicates that the computation is within the asymptotic range. The GCI is definedas

GCImn =Fs|εmn|rp − 1

, (17.6)

where Fs is a factor of safety. The refinement may be in either space or time. Thefactor of safety is recommended to be 3.0 for comparisons of two meshes and 1.25for comparison over three meshes or more. The relative error εmn is defined by

εmn =φm − φnφn

. (17.7)

It is assumed that the mesh refinement ratio r is applied equally in all coordi-nate directions (i, j, k) for steady state solutions and also time t for time dependentsolutions. If this is not the case, then the grid convergence indices can be computedfor each direction independently and then added to give the overall grid convergenceindex by

GCI = GCIt +GCIx +GCIy +GCIz. (17.8)

It must be noted that the concept of grid convergence does not always ap-ply to models that are not complete (or incomplete), such as the LES or mixinglength models. A model is termed complete if its constituent equations are freefrom flow-dependent specifications. Such specifications include material properties(density and viscosity), initial and boundary conditions, and numerical discretiza-tion. In such a case, a GCI may not necessarily approach zero or even reduce byfurther refining the mesh, which is the case for models that are redefined at a spe-cific lengthscale. For example, the LES model formulates and solves different setsof partial differential equations at above-grid and subgrid scales (Roache, 1997;Poletto et al., 2013; Aliabadi et al., 2017).

154

Observations and Model Error Quantification

Error quantification between the experimentally observed or measured quantitiesand the numerically predicted quantities is performed in various ways. Most com-monly, the simple Bias (B) or Root Mean Square Error (RMSE) are quantified andused to report the amount of shift and spread, respectively, between the observedand predicted values, with the observed quantity usually used as the reference.These errors are defined as

B = 〈φp − φo〉, (17.9)

RMSE =√〈(φp − φo)2〉, (17.10)

where φp and φo are the predicted and observed quantities. Although providingabsolute measures of error, this method may yield some exaggerated, unrepresenta-tive, and undefined error estimates in some cases. Alternatively, Hanna and Chang(2012) proposed two performance measures to express the error between the ob-served and predicted quantities: the Fractional mean Bias (FB), and the Normalized

Mean Square Error (NMSE), defined as

FB =2(〈φp − φo〉)〈φp〉+ 〈φo〉

, (17.11)

NMSE =〈(φp − φo)2〉〈φp〉〈φo〉

. (17.12)

Unlike the previous method, FB and NMSE define some specific attributesfor the calculated error. Averaged over all data points, the FB represents the shiftbetween the observed and predicted values, while the NMSE gives the spread ofone side of the values with respect to the other. For a theoretically perfect model,the FB and NMSE should be equal to 0. The FB and NMSE can be used with anyphysical quantity. The caveat in using FB and NMSE is that if either the predictedor observed quantity are near zero, then this method predicts large and exaggeratedamounts of error.

155

Exercises

1) A model error quantification is to be performed against experimental observa-tions. The following set of observations and the corresponding model predictionsare provided.

φo = 1, 2, 3, 4, 5 (17.13)

φp = 0, 1, 1, 2, 3 (17.14)

Show that the fractional bias and the normalized mean square error for theabove sets can be calculated as

FB = −0.73, (17.15)

NMSE = 0.67. (17.16)

2) Assume that the order of convergence for a discretized model is exactly p = 2.Show that if the resolution of discretization is increased by a factor of two, i.e.h2 = 1

2h1, then the discretization error is reduced by a factor of four, i.e.

E2 =1

4E1. (17.17)

3) Reason why the safety factor Fs = 3.0 is larger when calculating GCI for twolevels of mesh compared to the safety factor Fs = 1.25 used when calculating GCIfor tree levels of mesh.4) A marine engineer is performing convergence analysis of a Reynolds-AveragedNavier Stokes (RANS) turbulence model for an engineering problem involvingocean flow around a massive oil tanker. He has run the model on various repre-sentative cell mesh sizes h and obtained various errors E for a particular solutionof the model. Table below shows numerous pairs of h and E.

Help him write a Python code to calculate the order of convergence p for thisRANS model. This can be achieved by fitting the data. Show your a) Python code,

156

Table 17.1: Various runs of a RANS model given different pairs of cell mesh sizeh and a solution error E.

Run 1 2 3 4 5 6

h [m] 2.72 7.40 20.1 54.6 148 403E [unitless] 8.17 44.7 493 3640 19930 59874

b) its console output, and c) plot of the curve for natural logarithm of error versusnatural logarithm of mesh cell size.5) An engineer is calculating the Grid Convergence Index (GCI) for her turbulencemodel studying the solution φ obtained on two levels of grid, a coarse and a finegrid, with a grid refinement ratio of r = 1.5. The model exhibits an order of con-vergence of p = 1.8. The two successive solutions obtained on each grid result inthe solution φm = 2 on the coarse grid and solution φn = 2.1 on the fine grid.Assuming Fs = 1.25, calculate the GCImn.

157

Part IV

Applications

158

Chapter 18

Engineering

Fluid flow occurs in many fields in the natural and technical environment. Suchflows are either laminar or turbulent. Three characteristics of the fluids that are ofspecial importance are viscosity, density, and compressibility.

Liquid-liquid Extraction Industries

The liquid-liquid extraction is the process of separating a liquid solution contain-ing more than one liquid component. One typical process is solvent partitioning,where compounds having different solubilities can be separated in two differentimmiscible liquids. Solvent extraction is widely used in various industries such asproduction of vegetable oils and biodiesel, processing of perfumes, and reprocess-ing of nuclear fuels.

Coalescer

A coalescer is a device that divides an emulsion into specific components. It is pri-marily used in oil refining to remove water from hydrocarbon liquids and gases toproduce high quality hydrocarbon products. For instance, in natural gas industries,a coalescer is used to recover a lube oil from natural gas at the downstream loca-tion of a compressor. In a coalescer, droplets will stay in the streamlines arounda wire or fibre target, where they are expected to be collected. Usually laminarflow conditions are required in a coalescer since high fluid velocities overcome sur-face tension forces and strip droplets out of the coalescer medium. This results inreentrainment in flow and prevents droplets from being collected. Slower velocitiesresult in greater residence time in the media and therefore more time for droplets tobe collected. In coalescers, turbulent flow conditions must be prevented.

159

Waste Water Treatment

In waste water treatment, the process of mixing is characterized by high flow veloc-ities that create turbulent eddies and disperse components in a fluid. For example,the mixer in the flocculation tank mixes the water and the unwanted particles. Theturbulent flow will then slowly change to the laminar flow as it moves toward thesedimentation tank. An understanding of the flow regime, either laminar or turbu-lent, in a sedimentation tank is also essential to properly predict the settling veloc-ity of components. This leads to proper design of the tank for adequate settling ofcomponents (Gao and Stenstrom, 2018). Such processes require a detailed under-standing of turbulence.

Desalination

Desalination is the process of obtaining water with low mineral concentrations fromwater with high mineral concentrations. A typical application is obtaining freshwater from oceans. Desalinators use a direct contact membrane, into which flowsa low viscosity liquid in the turbulent regime. Turbulent flow is desired in suchmembranes to reduce head losses along the membrane.

Combustion Devices

An essential indicator of combustion performance is how well a combustion devicecan mix the fuel and air so that combustion chemistry can occur at stoichiometricconditions, i.e. the right amount of air and fuel concentrations can be achieved overthe entire combustion domain for complete combustion. Turbulent mixing is usuallydesigned to be maximized in combustion devices to ensure this condition. Thesedevices are ubiquitous in automobile engines, aircraft engines or turbines, powerplant turbines, furnaces, and boilers.

160

Indoor Ventilation

Ventilation is the intentional introduction of fresh or recirculated air into a space.Ventilation is mainly used to control indoor air quality by diluting and displacingindoor pollutants; it can also be used for purposes of thermal comfort or dehumidi-fication when the introduction of air will help achieve desired indoor psychrometricconditions. The intentional introduction of air can be categorized as either mechan-ical ventilation or natural ventilation. Mechanical ventilation uses fans to drive theflow of air into a space. Natural ventilation is the intentional passive flow of air intoa space through planned openings. Natural ventilation does not require mechanicalsystems to move air because it relies entirely on passive physical phenomena, suchas diffusion, wind pressure, or the stack effect. Mixed mode ventilation systems useboth mechanical and natural processes. Airflow in buildings and enclosed spacessuch as transportation devices, typically occurs in the turbulent regime, so a throughunderstanding of turbulent flow is necessary for proper design and maintenance ofventilation systems.

Aeronautics

Aircraft fly in the turbulent atmosphere at high speeds. In addition, aircraft them-selves create wake turbulence behind them as they move through the atmosphere.The presence of turbulent air around an aircraft has implications in lift and dragforces and ultimately in the navigability of the aircraft. The design and operation ofaircraft require a full understanding of turbulence so that they can be built econom-ically and operated safely. Aircraft flying at a considerable fraction of the speed ofsound enter a flow regime called compressible flow. In this regime density varia-tions of air must be considered around the aircraft and the transport equations mustbe altered and solved accordingly to allow for this important feature of the flow.

Renewable Energy

Many renewable energy conversion devices involve fluid flows, particularly in theturbulent regime. Some examples include wind turbines, hydro power turbines, and

161

archimedes screws. Again, an understanding of turbulent flows will help designing,operating, and maintaining such devices efficiently.

River Engineering

River engineering involves human intervention in the course, characteristics, or flowof a river with the intention to produce benefits for humans. Some examples includewater resource management, flood protection, and hydropower. Water flow in riversis usually in the turbulent regime and involves detailed understanding of turbulenttransport processes. Rivers are also studied for their sedimentation and erosion be-haviour, which are processes that are influenced significantly by turbulent transport(Cheng et al., 2018; He and Nguyen, 2019).

Exercises

1) Would you expect the Reynolds number of flows around an airplane be higher orthe Reynolds number of flows around ships? Why?2) Speculate why turbulence inside internal combustion engines, where air and fuelmix before burning, is desired.

162

Chapter 19

Sciences

Turbulence is the dominating force in nature from the Earth to the outer space.Our understanding of turbulent mechanisms in the nature has been improving bycontinuous observations and study of the relevant processes through the scientificefforts.

Meteorology

Atmospheric turbulence are small-scale and irregular air motions characterized bywinds that vary in speed and direction. Turbulence is important because it mixesand churns the atmosphere and causes water vapour, smoke, and other substances,as well as energy, to become distributed both vertically and horizontally.

Atmospheric turbulence near the Earth’s surface differs from that at higher lev-els. At low levels (within a few hundred metres of the surface), turbulence has amarked diurnal variation under partly cloudy and sunny skies, reaching a maximumabout midday. This occurs because, when solar radiation heats the surface, the airabove it becomes warmer and more buoyant, and cooler, denser air descends todisplace it. The resulting vertical movement of air, together with flow disturbancesaround surface obstacles, makes low-level winds extremely irregular. At night thesurface cools rapidly, chilling the air near the ground; when that air becomes coolerthan the air above it, a stable temperature inversion is created, and wind speed andgustiness both decrease sharply. When the sky is overcast, low-level air tempera-tures vary much less between day and night, and turbulence remains nearly con-stant (Stull, 1988). Several studies have investigated atmospheric turbulence in ur-ban (Aliabadi et al., 2017, 2019) and remote areas (Wyngaard, 2004; Flores et al.,2014).

At altitudes of several thousand metres or more, frictional effects of surfacetopography on the wind are greatly reduced, and the small-scale turbulence charac-

163

teristic of the lower atmosphere is absent. Although upper-level winds are usuallyrelatively regular, they sometimes become turbulent enough to affect aviation.

Oceanography

The ocean circulation is turbulent in the sense that motions on a wide range ofscales from a few centimetres to thousand of kilometres continuously interact. Inorder to develop theories of the large-scale circulation, which affects our climate,we need to understand these interactions. Researchers tackle motions on all of thesescales using a combination of observation, turbulence theories, and high resolutionnumerical models. One of the major challenges is to understand how energy is trans-ferred from the thousand of kilometres scales, where oceanic motions are forced bythe large-scale atmospheric winds, heat, and freshwater fluxes, to the centimetresscales, where energy is dissipated as heat. The first major transfer is from the large-scale currents to the mesoscale eddies. The large-scale ocean currents are unstableto baroclinic instability, which generate eddies with scales of ten to one hundredkilometres, the mesoscales. The mesoscale eddies then interact and generate sub-mesoscale turbulent filaments on scales from ten kilometres to one hundred meters.These motions are primarily horizontally constrained by the ocean stratification androtation. Only at scales below approximately one hundred meters, the turbulencebecomes three-dimensional and is described as stratified microscale turbulence.

The ocean circulation is dominated by geostrophic eddies, i.e. cyclones and an-ticyclones with radii of ten to one hundred kilometres. These eddies are the oceanequivalent of the storms we experience in the atmosphere as weather. Eddies play animportant role in the transport of heat, carbon, and other climatically important trac-ers across the oceans. Researchers develop theories for the physics of ocean eddies,their role in climate, and their representation in numerical models used for climatestudies. A major outcome of these studies is that lateral mixing by mesoscale eddiesis suppressed across strong currents, while it is strong on the flanks of the currents.Researchers have also shown that these variations in lateral mixing are crucial toquantify the impact of mesoscale eddies on the large-scale ocean circulation andclimate.

164

Submesoscale flows are a convoluted web of fronts, with horizontal scalesbetween ten kilometres and one hundred meters, that separate waters of differenttemperatures and salinities. The importance of submesoscale fronts have increas-ingly come into focus as an important component of ocean dynamics. The subme-soscale fronts are most energetic close the ocean surface. Just as alveoli facilitaterapid exchange of gases in the lungs, fronts are the ducts through which heat, car-bon, oxygen, and other climatically important tracers enter into the deep ocean.Researchers have shown that the submesoscale fronts are generated through insta-bilities of mesoscale currents confined to the surface mixed layer. The instabilitiesare stronger in winter and thus submesoscale flows are stronger in winter than insummer.

Oceanic motions with horizontal scales larger than one kilometre and verticalscales larger than one hundred meters are constrained to flow along density surfacesby the Earth’s rotation and the density stratification. However at smaller scales, theso-called microscales, these constraints become weak and turbulent motions crossdensity surfaces. In the upper ocean, microscale turbulence is generated by surfacewinds, air-sea cooling or evaporation. In the ocean interior, microscale turbulencedevelops when internal waves develop strong shears and overturn and break, muchlike surface gravity waves. These breaking events play a fundamental role in theocean circulation, because they mix the densest waters at the ocean bottom withthe lighter waters above, thereby allowing the densest waters to come back to thesurface. Researchers have shown that most of these transformations happen alongdeep ocean boundaries and not in the ocean interior, as previously believed.

Space

Space weather is driven by the sun, which has a continually changing magneticfield. When this field accumulates excess energy, it can erupt and send energeticparticles into space, sometimes toward Earth. Scientists believe that such violentemissions of charged particles, known as coronal mass ejections, are in part fu-elled by gusty space winds, or turbulence. Scientists have known of the existenceof space turbulence, but did not really understand its properties, or how it works.

165

Researchers, recently measured space turbulence directly for the first time in thelaboratory and finally were able to see the dynamics behind it.

Space turbulence is not the same as Earth’s turbulence, the sensation people ex-perience when they are flying or sailing. Unlike wind gusts that create air turbulenceon Earth, it turns out that space turbulence—also known as plasma turbulence—results from the interaction of so-called Alfven waves, often described as resem-bling waves that run along a stretched piece of string. It is critically important tounderstand coronal mass ejections because they have the potential to cause harmfuland dramatic effects if they strike the Earth’s own magnetic field. They could crip-ple space and satellite systems, GPS, military satellites, and the power grid, amongother things.

The surface of the sun is about 6,000 degrees Celsius, but if you go into thesolar corona, it is about 1 million degrees. We do not understand why that is—whatleads to this very hot plasma in the solar corona. The theory is that turbulence playsa very important role. And it is those high temperatures that are responsible for thegeneration of solar wind. The entire sun is a turbulent ball of plasma. Activity onthe sun can generate big blobs of plasma launched from the sun that will collidewith the Earth’s magnetic field, resulting in some serious consequences. About 500magnetic storms occur during a typical eleven-year solar cycle, and already havecaused damage. In March 1989, for example, a solar storm caused the entire Quebecpower grid to collapse in less than two minutes, affecting six million people in themiddle of a Canadian winter. A 2003 Halloween storm prompted a massive blackoutin the Northeast, and extensive satellite problems, including the loss of the $450million Midori-2 research satellite.

The worst solar storm in history occurred earlier, in 1859, when a magneticexplosion on the sun, known as the Carrington Event, after the British astronomerwho identified it, prompted a worldwide breakdown in communication systems. Inthose days, however, the major casualties were to telegraphs—primitive comparedto the elaborate electronics currently in use. An event of that magnitude today couldprove disastrous. A doomsday scenario could knock out power for months. Imaginelife without electricity for months. It would change life as we know it. It is veryreal, and it could happen. It has only happened once before in history, and that was

166

before we had this extensive power grid. Now the world is very different today andramifications of such events are different too.

Understanding how space turbulence works may tell us how these coronal ejec-tions occur in the first place, and may give us enough information to predict whenone is coming and protect us against it. To see how Alfven waves produce turbu-lence, the researchers took a plasma cylinder, eighteen meters long and one meterin diameter, and recreated Alfven waves in the laboratory, launching two along thecylinder—one from the top going down, and one from the bottom going up. Whenthose two waves crossed through each other, they generated a third Alfven wave.Scientists were able to measure it and determined that it had all the properties weexpected it to have. It was shown that the interaction of two Alfven waves willcreate a third. This is the fundamental building block of space turbulence, whichconsists of thousands of these building blocks. Only with a combination of experi-mental, theoretical, and numerical efforts can we hope to understand the nature ofturbulence in space, with the potential to solve some of the long-standing mysteriesof the solar system and the universe.

Exercises

1) Provide an argument to justify whether atmospheric flows or oceanic flows ex-hibit higher Reynolds numbers? why?2) Can you provide an example of a turbulent flow within living systems in whichthe fluid of concern is neither air nor water?

167

Part V

Fundamental Analysis Tools and Principles

168

Chapter 20

Statistics

Random Variables

In probability and statistics, a random variable is a variable quantity whose valuedepends on possible outcomes. As a function, a random variable is required to bemeasurable. For example, a velocity component in turbulent flow U is a randomvariable.

Event

In probability theory, an event is a set of outcomes of an experiment (a subset ofthe sample space), to which a probability is assigned. A single outcome may be anelement of many different events, and different events in an experiment are usuallynot equally likely, since they may include very different groups of outcomes. Forexample, a set of velocity components in turbulent flow that are all less than aparticular value constitute an event A ≡ U < 10 m s−1.

Probability

Probability is the measure of the likelihood that an event will occur. Probabilityis quantified as a number between 0 and 1, where, loosely speaking, 0 indicatesimpossibility and 1 indicates certainty. The higher the probability of an event, themore certain that the event will occur. For instance the probability of event A ≡U < 10 m s−1 is

p = P (A) = PU < 10 m s−1 (20.1)

The probability of any event always satisfies 0 ≤ p ≤ 1. In other words, aprobability can never be less than zero or greater than 1.

169

Cumulative Distribution Function

The probability of any event can be determined from the Cumulative Distribution

Function (CDF) defined by

F (V ) ≡ PU < V (20.2)

where U is the random variable and V is one outcome from the sample space. Forexample, for two events B ≡ U < Vb and C ≡ Va < U < Vb the probabilitiesP (B) and P (C) can be calculated as

P (B) = PU < Vb = F (Vb), (20.3)

P (C) = PVa < U < Vb = PU < Vb − PVa < U = F (Vb)− F (Va).

(20.4)The three basic properties of CDF are as follows

F (V → −∞) = 0, (20.5)

F (V →∞) = 1, (20.6)

Vb > Va → F (Vb) > F (Va). (20.7)

Probability Density Function

The Probability Density Function (PDF) is defined as the derivative of CDF,

f(V ) ≡ dF (V )

dV(20.8)

The PDF has the following four properties

f(V ) ≥ 0, (20.9)

170

PV ≤ U ≤ V + dV = F (V + dV )− F (V ) = f(V )dV, (20.10)

PVa ≤ U ≤ Vb = F (Vb)− F (Va) =

∫ Vb

Va

f(V )dV, (20.11)

∫ ∞−∞

f(V )dV = 1. (20.12)

Mean and Moments

If U is a random variable given with PDF f(V ), then the mean or expectation of Uis given by

〈U〉 ≡∫ ∞−∞

V f(V )dV. (20.13)

The moments of random variable U describe the shape of its PDF f(V ). Thenth central moment of U is defined by

µn ≡∫ ∞−∞

(V − 〈U〉)nf(V )dV. (20.14)

Note that the zeroth moment of U is equal to 1 (µ0 = 1), while the first momentof U is equal to zero (µ1 = 0). These are evident from evaluating the momentintegral.

Probability Distributions

If U is uniformly distributed in the interval a ≤ V < b, then the PDF of uniform

distribution for U , f(V ) is 1b−a if a ≤ V < b,

0 if V < a or V ≥ b.

If U is exponentially distributed with parameter λ, then the PDF of exponential

distribution for U , f(V ) is171

1λ

exp(−V/λ) if V ≥ 0,

0 if V < 0.

Of fundamental importance in probability theory and theory of turbulent flowsis the Gaussian distribution or the normal distribution with mean µ and standarddeviation σ. The PDF for this distribution is given by

f(V ) = N (V ;µ, σ2) ≡ 1

σ√

2πexp[−1

2(V − µ)2/σ2]. (20.15)

Exercises

1) Consider that a random variable U is exponentially distributed with parameter λ,for which the PDF for U , f(V ) is 1

λexp(−V/λ) if V ≥ 0,

0 if V < 0.

Derive an expression for the Cumulative Distribution Function (CDF) for U , orF (V ), as a function of V .2) Compare the number of parameters needed to define the PDFs for a uniform, ex-ponential, and Gaussian distributions. Which distribution(s) requires the least num-ber of parameters? Which distribution(s) requires the most number of parameters?

172

Chapter 21

Mathematics

Einstein’s Notation

In mathematics, especially in applications of linear algebra to physics, the Ein-stein’s notation or Einstein summation convention is a notational convention thatimplies summation over a set of indexed terms in a formula, thus achieving nota-tional brevity. According to this convention, when an index variable appears twicein a single term and is not otherwise defined, it implies summation of that term overall the values of the index. In fluid mechanics the indices can range over the setof 1, 2, 3 for three dimensional space. For instance the expression below can beexpanded as the summation

Ui∂

∂xi= U1

∂

∂x1

+ U2∂

∂x2

+ U3∂

∂x3

, (21.1)

where i represents the index and the summation is carried out over a set of three, i.e.for three dimensional space. In the more familiar notation for Cartesian coordinates,x1 = x, x2 = y, and x3 = z are coordinate axes and U1 = U , U2 = V , and U3 = W

are fluid velocities along the coordinate axes.

Kronecker Delta

In mathematics, the Kronecker delta, δij , is a function of two variables, usually justpositive integers. The function is 1 if the variables are equal, and 0 otherwise:δij = 1 if i = j

δij = 0 if i 6= j

This function is named after Leopold Kronecker (1823-1891).

173

Alternating Symbol

In mathematics, the alternating symbol, or the Levi-Civita symbol, is a functionof three variables, usually positive integers 1, 2, and 3. The function value is 1 ifthe ordering of the variables are cyclic and −1 if the ordering of the variables areanti-cyclic,

εijk = 1 if (i, j, k) are cyclic,

εijk = −1 if (i, j, k) are anti-cyclic,

εijk = 0 otherwise.

Cyclic orderings are 123, 231, and 312, while anti-cyclic orderings are 321,132, and 213; otherwise two or more of the suffixes are the same. The alternatingsymbol, or Levi-Civita symbol, can be remembered using the visual aid in Figure21.1.

Figure 21.1: Visual aid for remembering the Levi-Civita symbol.

Position Vector

In Cartesian coordinates, the position vector x gives the position of a general pointP using three coordinates relative to the origin O. It is given by

x = xiei = x1e1 + x2e2 + x3e3. (21.2)

In the more familiar notation for Cartesian coordinates, x1 = x, x2 = y, andx3 = z are coordinate axes and e1 = i, e2 = j, and e3 = k are the unit vectorsalong the coordinate axes.

174

Divergence

In vector calculus, divergence is a vector operator that produces a signed scalar fieldgiving the quantity of a vector field’s source at each point. More technically, thedivergence represents the volume density of the outward flux of a vector field froman infinitesimal volume around a given point. In physical terms, the divergence ofa three-dimensional vector field is the extent to which the vector field flow behaveslike a source at a given point. It is a local measure of its “outgoingness”, i.e. theextent to which there is more of some quantity exiting an infinitesimal region ofspace than entering it. If the divergence is nonzero at some point then there mustbe a source or sink at that position. In Cartesian coordinates, if U is flow’s velocityvector, then divergence is defined as

∇.U =∂Ui∂xi

=∂U1

∂x1

+∂U2

∂x2

+∂U3

∂x3

. (21.3)

In the more familiar notation for Cartesian coordinates, x1 = x, x2 = y, andx3 = z are coordinate axes and U1 = U , U2 = V , and U3 = W are fluid velocitiesalong the coordinate axes.

Gradient

In mathematics, the gradient is a multi-variable generalization of the derivative.While a derivative can be defined on functions of a single variable, for functions ofseveral variables, the gradient takes its place. The gradient is a vector-valued func-tion, as opposed to a derivative, which is scalar-valued. If f is a differentiable andreal-valued function of several variables, its gradient is the vector whose compo-nents are the partial derivatives of f . Like the derivative, the gradient represents theslope of the tangent of the graph of the function. More precisely, the gradient pointsin the direction of the greatest rate of increase of the function, and its magnitude isthe slope of the graph in that direction. The gradient (or gradient vector field) of ascalar function f is given by

∇f =∂f

∂xiei =

∂f

∂x1

e1 +∂f

∂x2

e2 +∂f

∂x3

e3. (21.4)

175

In the more familiar notation for Cartesian coordinates, x1 = x, x2 = y, andx3 = z are coordinate axes and e1 = i, e2 = j, and e3 = k are the unit vectorsalong the coordinate axes.

Curl

In vector calculus, the curl is a vector operator that describes the infinitesimal ro-tation of a three-dimensional vector field. At every point in the field, the curl ofthat point is represented by a vector. The attributes of this vector (length and direc-tion) characterize the rotation at that point. The direction of the curl is the axis ofrotation, as determined by the right-hand rule, and the magnitude of the curl is themagnitude of rotation. If the vector field represents the flow velocity of a movingfluid, then the curl is the circulation density of the fluid. A vector field whose curlis zero is called irrotational. The curl is a form of differentiation for vector fields.The curl of the velocity vector U is given by

∇×U =

∣∣∣∣∣∣∣e1 e2 e3

∂∂x1

∂∂x2

∂∂x3

U1 U2 U3

∣∣∣∣∣∣∣∇×U =

(∂U3

∂x2

− ∂U2

∂x3

)e1 +

(∂U1

∂x3

− ∂U3

∂x1

)e2 +

(∂U2

∂x1

− ∂U1

∂x2

)e1. (21.5)

In the more familiar notation for Cartesian coordinates, where x1 = x, x2 = y,x3 = z, U1 = U , U2 = V , and U3 = W the curl of velocity vector is given by

∇×U =

∣∣∣∣∣∣∣i j k∂∂x

∂∂y

∂∂z

U V W

∣∣∣∣∣∣∣∇×U =

(∂W

∂y− ∂V

∂z

)i +

(∂U

∂z− ∂W

∂x

)j +

(∂V

∂x− ∂U

∂y

)k. (21.6)

176

Laplacian

In mathematics, the Laplace operator or Laplacian is a differential operator given bythe divergence of the gradient of a function. In a Cartesian coordinate system, theLaplacian is given by the sum of second partial derivatives of the function with re-spect to each independent variable. The Laplace operator is named after the Frenchmathematician Pierre-Simon de Laplace (1749-1827), who first applied the operatorto the study of celestial mechanics. The Laplacian occurs in differential equationsthat describe many physical phenomena, such as electric and gravitational poten-tials, the diffusion equation for heat and fluid flow, wave propagation, and quan-tum mechanics. The Laplacian represents the flux density of the gradient flow of afunction. For instance, the net rate, at which a chemical dissolved in a fluid movestoward or away from some point, is proportional to the Laplacian of the chemicalconcentration at that point. The Laplacian is given by

∇2 = ∇.∇ =

(ei

∂

∂xi

).

(ej

∂

∂xj

)=

∂2

∂xi∂xi(21.7)

For example the Laplacian of function f is given by

∇2f =∂2f

∂x21

+∂2f

∂x22

+∂2f

∂x23

. (21.8)

In the more familiar notation for Cartesian coordinates, x1 = x, x2 = y, andx3 = z.

Dot Product of Two Vectors

In mathematics, the dot product or scalar product is an algebraic operation that takestwo equal-length sequences of numbers (usually coordinate vectors) and returns asingle number. Sometimes it is called inner product in the context of Euclideanspace, or rarely projection product for emphasizing the geometric significance. Al-gebraically, the dot product is the sum of the products of the corresponding entriesof the two sequences of numbers. Geometrically, it is the product of the Euclideanmagnitudes of the two vectors and the cosine of the angle between them. If u and v

177

represent two vectors expressed in the Cartesian coordinate system, the dot productof the two is given by

u.v = (eiui).(ejvj) = uivi = u1v1 + u2v2 + u3v3. (21.9)

Cross Product of Two Vectors

In mathematics and vector algebra, the cross product or vector product (occasion-ally directed area product to emphasize the geometric significance) is a binary op-eration on two vectors in three-dimensional space. Given two linearly independentvectors, the cross product is a vector that is perpendicular to both vectors and there-fore normal to the plane containing them. Because the magnitude of the cross prod-uct goes by the sine of the angle between its arguments, the cross product can bethought of as a measure of perpendicularity in the same way that the dot product is ameasure of parallelism. If u and v represent two vectors expressed in the Cartesiancoordinate system, the cross product of the two is given by

u× v =

∣∣∣∣∣∣∣e1 e2 e3

u1 u2 u3

v1 v2 v3

∣∣∣∣∣∣∣ = εijkujvkei

u× v = (u2v3 − u3v2)e1 + (u3v1 − u1v3)e2 + (u1v2 − u2v1)e3 (21.10)

Remarkably the alternating symbol expresses the cross product in a very con-cise manner. Component i of the cross product is given by εijkujvk. For instancethe first component of the cross product can be calculated by

ε123u2v3 + ε132u3v2 = u2v3 − u3v2. (21.11)

Material or Substantial Derivative

Material derivative or substantial derivative describes the time rate of change ofsome physical quantity of a material element that is subjected to a space-and-time-dependent macroscopic velocity field variations of that physical quantity. In fluid

178

dynamics, the velocity field is the flow velocity, and the quantity of interest may bethe density, momentum, or temperature of the fluid. In this case, the total derivativedescribes the density, momentum, or temperature change of a certain fluid parcelwith time, as it flows along its path line or trajectory. The total derivative is definedas

D

Dt≡ ∂

∂t+ Ui

∂

∂xi=

∂

∂t+ U.∇. (21.12)

For instance the material derivative of fluid density ρ is

Dρ

Dt≡ ∂ρ

∂t+ Ui

∂ρ

∂xi=∂ρ

∂t+ U.∇ρ. (21.13)

Tensors

Given a coordinate basis or fixed frame of reference, a tensor can be represented asan organized multidimensional array of numerical values. The order (also degree orrank) of a tensor is the dimensionality of the array needed to represent it, or equiv-alently, the number of indices needed to label a component of that array. For exam-ple, a linear map is represented by a matrix (a two-dimensional array) in a basis,and therefore is a second-order tensor. A vector is represented as a one-dimensionalarray in a basis, and is a first-order tensor. Scalars are single numbers and are thuszeroth-order tensors. Because tensors express a relationship between vectors, ten-sors themselves must be independent of a particular choice of coordinate system.The coordinate independence of a tensor then takes the form of a transformationlaw that relates the array computed in one coordinate system to that computed inanother one.

A zeroth-order tensor is a scalar. It has 30 = 1 component, which has the samevalue in every coordinate system. Examples are physical quantities such as density,temperature, and pressure.

A first-order tensor, e.g. u, has 31 = 3 components. Some examples includeposition, velocity, acceleration, and vorticity. In the Cartesian system, the first-ordertensor can be represented using unit vectors along the coordinate axes

179

u = eiui = e1u1 + e2u2 + e3u3. (21.14)

A second-order tensor, b, has 32 = 9 components. An example is the stresstensor. In the Cartesian system, the second-order tensor can be represented usingcombination of unit vectors along the coordinate axes

b = eiejbij = e1e1b11 + e1e2b12 + e1e3b13

+ e2e1b21 + e2e2b22 + e2e3b23

+ e3e1b31 + e3e2b32 + e3e3b33. (21.15)

Note that for the second-order tensors, the Einsteins notation requires that thesummation over repeated indices be expanded in two dimensions. In the formulaabove since both indices i and j have been repeated, the summation requires nineterms over two dimensions to include all combinations of indices.

Exercises

1) How many components are there for a third-order tensor?2) A velocity vector field is given by the following expression. Calculate the curlof this velocity vector, i.e. ∇ ×U. Also evaluate the curl at the point given by thecoordinates x = 1, y = 2, and z = 3.

U = xyzi + x2y2z2j + x−1y−1z−1k (21.16)

3) An environmental engineer is analyzing a transient velocity field in a fluid flowgiven by

U = −3xti− 3ytj + 6ztk, (21.17)

where x, y, and z are cartesian coordinates and t is time. In addition, the tem-perature field in this flow is given by

T = e−t(x+ y + z). (21.18)

180

Help this engineer find an expression for the material derivative of temperaturein this velocity field, as a function of x, y, z, and t, given by

DT

Dt=∂T

∂t+ U.∇T. (21.19)

181

Chapter 22

Numerical Methods

Taylor Series Expansion

Any continuous differentiable function φ(x) can, in the vicinity of xi, be expressedas a Taylor series

φ(x) = φ(xi) + (x− xi)(∂φ

∂x

)i

+(x− xi)2

2!

(∂2φ

∂x2

)i

+(x− xi)3

3!

(∂3φ

∂x3

)i

+ ...+(x− xi)n

n!

(∂nφ

∂xn

)i

+H, (22.1)

where H means higher-order terms. The Taylor series expansion is at the core ofnumerical methods.

The Finite Difference Method

In finite difference methods, the derivatives in transport equations are replaced byfinite differences. This is supported by the Taylor series expansion, which can berearranged to give a derivative having differences and derivatives of other order.The core idea in finite difference methods is to approximate a derivative by finitedifferences and ignore other order derivatives (Ferziger and Peric, 2002). Supposea problem is discretized with ∆x = xi+1−xi for all i. The first order derivative canthen be approximated using (

∂φ

∂x

)i

≈ φi+1 − φixi+1 − xi

, (22.2)

(∂φ

∂x

)i

≈ φi − φi−1

xi − xi−1

, (22.3)

(∂φ

∂x

)i

≈ φi+1 − φi−1

xi+1 − xi−1

, (22.4)

182

which are known in order as, Forward-Difference Scheme (FDS), Backward-Difference

Scheme (BDS), and Central-Difference Scheme (CDS).The same technique can be used to approximate second derivatives. For in-

stance, for equidistant spacing the second derivative can be approximated usingcentral differencing such that(

∂2φ

∂x2

)i

≈ φi+1 − 2φi + φi−1

(∆x)2. (22.5)

More complex second order derivatives can be approximated in situations whereinner and outer derivatives exists with some variable inside the outer derivative be-side the inner derivative. For example,

[∂

∂x

(Γ∂φ

∂x

)]i

≈

(Γ∂φ∂x

)i+ 1

2

−(Γ∂φ∂x

)i− 1

2

12(xi+1 − xi−1)

≈Γi+ 1

2

φi+1−φixi+1−xi − Γi− 1

2

φi−φi−1

xi−xi−1

12(xi+1 − xi−1)

, (22.6)

where Γ is a variable and has to be evaluated at half integer indices, i.e. i + 12

andi− 1

2. To simplify the approximation the inner derivatives have been approximated

using forward and backward schemes (Ferziger and Peric, 2002).

Newton’s Method for Solving Non-linear System of Equations

The master method in solving non-linear equations or system of equations is theNewton’s method. This method is also called the linearization method, in which anon-linear function at a point can be replaced by the terms of Taylor series that onlycontain constants or first order derivatives.

For a uni-variable non-linear function f(x), the value of the function around x0

can be replaced by

f(x) ≈ f(x0) +

(df

dx

)x0

(x− x0). (22.7)

For instance if f(x) = x2 then f(x) ≈ x20 + 2x0(x−x0) = −x2

0 + 2x0x, whichis a linear function. This method is extremely useful since it can convert any non-

183

linear system of equations to a linear system of equations. The effectiveness of thismethod for solving a non-linear system of equations depends on other properties ofthe non-linear system.

The Newton method can be applied to multi-variable non-linear functions aswell. For instance, if f is a non-linear function of three variables x, y, and z, thenthe linearized form of f around x0, y0, and z0 is

f(x, y, z) ≈ f(x0, y0, z0) +

(∂f

∂x

)x0,y0,z0

(x− x0)

+

(∂f

∂y

)x0,y0,z0

(y − y0)

+

(∂f

∂z

)x0,y0,z0

(z − z0), (22.8)

where the derivatives have now been replaced by partial derivatives. This methodcan also be extremely effective in reducing many highly non-linear system of equa-tions to a linear system of equations.

Explicit and Implicit Euler Methods

Many transport problems are transient in nature. That is a solution of interest issought for not only in the spatial domain but also in the temporal domain. Therefore,numerical methods must be developed to solve a transport equation in both spaceand time. The Euler’s method is the most common approach for this purpose, andthere are two classes of Euler’s method: explicit and implicit. We demonstrate thesemethods using one-dimensional transport equation for a scalar φ, e.g. concentrationof a species in a fluid domain, with constant fluid velocity u. The transport equationis given as

∂φ


= −u∂φ∂x︸︷︷︸

Advection

+Γ

ρ

∂2φ

∂x2︸︷︷︸Diffusion

(22.9)

Consider that a Central-Difference Scheme (CDS) is used to discretize the equa-tion using the finite difference method. Assume φni represents the solution in spatial

184

location i and timestep n and that the spatial and temporal domains are discretizedby ∆x and ∆t, respectively. The finite difference version of the transport equationcan be shown using

φn+1i − φni

∆t= −u

φni+1 − φni−1

2∆x+

Γ

ρ

φni+1 − 2φni + φni−1

(∆x)2(22.10)

where the spatial derivatives are discretized at timestep n. This equation can be sim-plified by grouping solutions at a particular spatial location and timestep to arriveat the following equation.

φn+1i =

(d+

c

2

)φni−1 + (1− 2d)φni +

(d− c

2

)φni+1 (22.11)

where we have introduced the dimensionless parametersd = Γ∆tρ(∆x)2

c = u∆t∆x.

The parameter d is the ratio of timestep ∆t to the characteristic diffusion timeρ(∆x)2/Γ, which is roughly the time required for a disturbance to be transmittedby diffusion over distance ∆x. The second quantity, parameter c, is the ratio ofthe timestep ∆t to the characteristic convection time u/∆x, the time required forthe disturbance to be convected a distance of ∆x. This ratio is called the Courant

number

Co =u∆t

∆x. (22.12)

The equations over all spatial locations can be combined so that the set ofequations can be written in matrix form

φn+1 = Aφn, (22.13)

where the boundary conditions can be absorbed in the system of equations as well.This suggests that to solve the transient transport equation one needs to chose aninitial solution φ0 and then repetitively perform the matrix multiplication over afinite number of timesteps to obtain new solutions φn. This is called the explicit

185

Euler method. As convenient as it appears, this method is severely limited becauseit necessitates the discretization to be extremely refined. So the method is onlyconditionally stable if ∆x and ∆t meet specific criteria. For instance, for flows thatare dominated by convection, where diffusion is negligible, such as atmosphericflows, the Courant number has to be less than one, i.e.

Co =u∆t

∆x< 1. (22.14)

Alternatively, the transport equation can be discretized in such a way that thespatial derivatives are discretized at timestep n+ 1. i.e.

φn+1i − φni

∆t= −u

φn+1i+1 − φn+1

i−1

2∆x+

Γ

ρ

φn+1i+1 − 2φn+1

i + φn+1i−1

(∆x)2. (22.15)

This equation can be simplified by grouping solutions at a particular spatiallocation and timestep to arrive at the following equation

(− c

2− d)φn+1i−1 + (1 + 2d)φn+1

i +( c

2− d)φn+1i+1 = φni . (22.16)

The resulting system of equations can be written in matrix form such that

Aφn+1 = φn. (22.17)

Now it appears that to arrive at the solution for a new timestep, not only onemust have a solution in the previous timestep, but also one needs to solve a linearsystem of equations. This is the implicit Euler method. It happens that this methodresults in better stability and poses less restrictions on resolution of the discretiza-tion, i.e. the magnitude of ∆x and ∆t.

Under Relaxation

When solving a system of equations iteratively, it is sometimes more stable to onlypartially update a solution after each iteration. This is known as under relaxation.Consider that φn−1 is the solution space found in the previous iteration and φnew is

186

the newly found solution. With the under relaxation factor 0 < α < 1, the solutioncan be updated for the next iteration such that

φn = φn−1 + α(φnew − φn−1). (22.18)

Particularly, whenever solving non-linear system of equations, this method im-proves stability of obtaining a numerical solution.

Exercises

1) A continuous differentiable 1D function φ(x) is being considered that only de-pends on x as an independent variable. We like to approximate the first and secondorder derivatives of φ(x), i.e.

(∂φ∂x

)i

and(∂2φ∂x2

)i

at point i, using the finite differ-ence method. Consider a uniform discretization of ∆x = xi+1 − xi = 2 m, givenany i. Suppose the values of φ at points i = 1, 2, and 3 are given as, 2, 4, and 7,respectively. The function φ is unitless. Show that the first derivative of φ at pointi = 2 using the Forward-Difference Scheme (FDS), Backward-Difference Scheme(BDS), and Central-Difference Scheme (CDS) is approximated, respectively, by(

∂φ

∂x

)i=2

≈ 1.5, 1, 1.25m−1. (22.19)

Also show that the second order derivative of φ at point i = 2 using the Central-Differencing Scheme (CDS) is approximated by(

∂2φ

∂x2

)i=2

≈ 0.25m−2. (22.20)

2) A continuous differentiable 1D function φ(x) = ex is being considered that onlydepends on x as an independent variable. This function is to be approximated usingthe first three terms of its Taylor series expansion around point x = 1. Show thatthe approximation as a function of x is given by

φ(x) ≈(x+

(x− 1)2

2

)e. (22.21)

Now use this approximation to evaluate the function at x = 1.5. How close isthis approximation to the true value of the function, i.e. φ(x = 1.5)?

187

3) A mathematician wishes to approximate a continuous differentiable 1D functionφ(x) = sinx, that only depends on x as an independent variable, by the first threeterms of its Taylor series expansion around point x = 1. a) What is the functionalform of this approximation, i.e. as a function of x? b) Now use this approximationto evaluate the function at x = 1.2. How close is this approximation to the truevalue of the function, i.e. φ(x = 1.2)?

188

Bibliography

Aboshosha, H., G. Bitsuamlak, and A. El Damatty (2015). LES of ABL flow in thebuilt-environment using roughness modeled by fractal surfaces. Sustain. Cities

Soc. 19, 40–60.

Aliabadi, A. A., E. S. Krayenhoff, N. Nazarian, L. W. Chew, P. R. Armstrong,A. Afshari, and L. K. Norford (2017). Effects of roof-edge roughness on airtemperature and pollutant concentration in urban canyons. Boundary-Layer Me-

teorol. 164(2), 249–279.

Aliabadi, A. A., K. W. Lim, S. N. Rogak, and S. I. Green (2011). Steady andtransient droplet dispersion in an air-assist internally mixing cone atomizer. At-

omization Spray. 21(12), 1009–1031.

Aliabadi, A. A., M. Moradi, D. Clement, W. D. Lubitz, and B. Gharabaghi (2019).Flow and temperature dynamics in an urban canyon under a comprehensive setof wind directions, wind speeds, and thermal stability conditions. Environ. Fluid

Mech. 19(1), 81–109.

Aliabadi, A. A., R. M. Staebler, J. de Grandpre, A. Zadra, and P. A. Vaillancourt(2016). Comparison of estimated atmospheric boundary layer mixing height inthe Arctic and southern Great Plains under statically stable conditions: experi-mental and numerical aspects. Atmos.-Ocean 54(1), 60–74.

Aliabadi, A. A., R. M. Staebler, M. Liu, and A. Herber (2016). Characterization andparametrization of Reynolds stress and turbulent heat flux in the stably-stratifiedlower Arctic troposphere using aircraft measurements. Boundary-Layer Meteo-

rol. 161(1), 99–126.

Aliabadi, A. A., N. Veriotes, and G. Pedro (2018). A Very Large-Eddy Simulation(VLES) model for the investigation of the neutral atmospheric boundary layer. J.

Wind Eng. Ind. Aerod. 183, 152–171.

189

Balogun, A. A., A. S. Tomlin, C. R. Wood, J. F. Barlow, S. E. Belcher, R. J. Smalley,J. J. N. Lingard, S. J. Arnold, A. Dobre, A. G. Robins, D. Martin, and D. W.Shallcross (2010). In-street wind direction variability in the vicinity of a busyintersection in central London. Boundary-Layer Meteorol. 136(3), 489–513.

Barlow, J. F., C. H. Halios, S. E. Lane, and C. R. Wood (2015). Observations ofurban boundary layer structure during a strong urban heat island event. Environ.

Fluid Mech. 15(2), 373–398.

Benhamadouche, S., N. Jarrin, Y. Addad, and D. Laurence (2006). Synthetic turbu-lent inflow conditions based on a vortex method for large-eddy simulation. Prog.

Comput. Fluid Dy. 6(1-3), 50–57.

Beyrich, F. (1997). Mixing height estimation from sodar data - a critical discussion.Atmos. Environ. 31(23), 3941–3953.

Blackman, K., L. Perret, E. Savory, and T. Piquet (2015). Field and wind tunnelmodeling of an idealized street canyon flow. Atmos. Environ. 106, 139–153.

Blocken, B., T. Stathopoulos, and J. Carmeliet (2007). CFD simulation of the at-mospheric boundary layer: Wall function problems. Atmos. Environ. 41(2), 238–252.

Bredberg, J. (2000). On the wall boundary condition for turbulence models. Internalreport 00/4, Department of Thermo and Fluid Dynamics, Chalmers University ofTechnology, Goteborg, Sweden.

Brock, F. V. (2001). Meteorological measurement systems. Cambridge: OxfordUniversity Press.

Buckingham, S., L. Koloszar, Y. Bartosiewicz, and G. Winckelmans (2017). Op-timized temperature perturbation method to generate turbulent inflow conditionsfor LES/DNS simulations. Comput. Fluids 154, 44–59.

Candlish, L. M., R. L. Raddatz, M. G. Asplin, and D. G. Barber (2012). Atmo-spheric temperature and absolute humidity profiles over the Beaufort Sea and

190

Amundsen Gulf from a microwave radiometer. J. Atmos. Ocean. Tech. 26(9),1182–1201.

Cao, S. (2014). Advanced physical and numerical modeling of atmospheric bound-ary layer. Journal of Civil Engineering Research 4(3A), 14–19.

Castro, H. G. and R. R. Paz (2013). A time and space correlated turbulence synthe-sis method for large eddy simulations. J. Comput. Phys. 235, 742–763.

Castro, H. G., R. R. Paz, J. L. Mroginski, and M. A. Storti (2017). Evaluation ofthe proper coherence representation in random flow generation based methods.J. Wind Eng. Ind. Aerodyn. 168, 211–227.

Cheng, W., H. Fang, H. Lai, L. Huang, and S. Dey (2018). Effects of biofilmon turbulence characteristics and the transport of fine sediment. J. Soil. Sedi-

ment. 18(10), 3055–3069.

Codyer, S. R., M. Raessi, and G. Khanna (2012, 8-12 July). Using graphical pro-cessing units to accelerate numerical simulations of interfacial incompressibleflows. Rio Grande, Puerto Rico. Proceedings of the ASME 2012 Fluids Engi-neering Summer Meeting.

Davidson, P. A. (2009). Turbulence: an introduction for scientists and engineers.Oxford, U.K.: Oxford University Press.

Deardorff, J. W. (1974). Three-dimensional numerical study of the height and meanstructure of a heated planetary boundary layer. Boundary-Layer Meteorol. 7(1),81–106.

Dehghan, A., W. K. Hocking, and R. Srinivasan (2014). Comparisons betweenmultiple in-situ aircraft turbulence measurements and radar in the troposphere. J.

Atmos. Sol-Terr. Phy. 118, 64–77.

Drain, L. E. (1980). The Laser Doppler Technique. Michigan: John Wiley & Sons,Inc.

191

Efros, V. (2006). Large eddy simulation of channel flow using wall functions. The-sis, Department of Thermo and Fluid Dynamics, Chalmers University of Tech-nology, Goteborg, Sweden.

Eliasson, I., B. Offerle, C. S. B. Grimmond, and S. Lindqvist (2006). Wind fieldsand turbulence statistics in an urban street canyon. Atmos. Environ. 40(1), 1–16.

Elmaghraby, H. A., Y. W. Chiang, and A. A. Aliabadi (2018). Ventilation strategiesand air quality management in passenger aircraft cabins: A review of experimen-tal approaches and numerical simulations. Sci. Technol. Built En. 24(2), 160–175.

Ettema, R. (2000). Hydraulic Modelling: Concepts and Practice. Reston, Virginia:American Society of Civil Engineering (ASCE).

Fernholz, H. and P. J. Finley (1996). The incompressible zero-pressure-gradientturbulent boundary layer: An assessment of the data. Prog. Aerospace Sci. 32(4),245–311.

Ferziger, J. H. and M. Peric (2002). Computational Methods for Fluid Dynamics

(3rd ed.). Berlin: Springer.

Flores, F., R. Garreaud, and R. C. M. noz (2014). OpenFOAM applied to the CFDsimulation of turbulent buoyant atmospheric flows and pollutant dispersion insidelarge open pit mines under intense insolation. Comput. Fluids 90(72-87).

Flores, F., R. Gerreaud, and R. C. Munoz (2013). CFD simulations of turbulentbuoyant atmospheric flows over complex geometry: solver development in Open-FOAM. Comput. Fluids 82, 1–13.

Frohlich, J., C. P. Mellen, W. Rodi, L. Temmerman, and M. A. Leschziner (2005).Highly resolved large-eddy simulation of separated flow in a channel withstreamwise periodic constrictions. J. Fluid Mech. 526, 19–66.

Gao, H. and M. Stenstrom (2018). Evaluation of three turbulence models in pre-dicting the steady state hydrodynamics of a secondary sedimentation tank. Water

Res. 143, 445–456.

192

Ghaemi-Nasab, M., S. Franchini, A. R. Davari, and F. Sorribes-Palmer (2018).A procedure for calibrating the spinning ultrasonic wind sensors. Measure-

ment 114, 365–371.

Giometto, M. G., A. Christen, C. Meneveau, J. Fang, M. Krafczyk, and M. B. Par-lange (2016). Spatial characteristics of roughness sublayer mean flow and turbu-lence over a realistic urban surface. Boundary-Layer Meteorol. 160(3), 425–452.

Graf, A., A. van de Boer, A. Moene, and H. Vereecken (2014). Intercomparison ofmethods for the simultaneous estimation of zero-plane displacement and aerody-namic roughness length from single-level eddy-covariance data. Boundary-Layer

Meteorol. 151(2), 373–387.

Halios, C. H. and J. F. Barlow (2018). Observations of the Morning Developmentof the Urban Boundary Layer Over London, UK, Taken During the ACTUALProject. Bound.-Lay. Meteorol. 166(3), 395–422.

Hanna, S. and J. Chang (2012). Acceptance criteria for urban dispersion modelevaluation. Meteorol. Atmos. Phys. 116(3), 133–146.

Hargather, M. J., M. J. Lawson, G. S. Settles, and L. M. Weinstein (2011). Seedlessvelocimetry measurements by Schlieren image velocimetry. AIAA J. 49(3), 611–620.

He, C. and D. Nguyen (2019). Erodibility study of sediment in a fast-flowing river.Int. J. Sediment Res. 34(2), 144–154.

Hocking, W. K. (1999). The dynamical parameters of turbulence theory as theyapply to middle atmosphere studies. Earth Planets Space 51(7-8), 525–541.

Hogan, R. J., A. J. Illingworth, and K. Halladay (2008). Estimating mass and mo-mentum fluxes in a line of cumulonimbus using a single high-resolution Dopplerradar. Q. J. Roy. Meteor. Soc. 134(634), 1127–1141.

Huang, M., Z. Gao, S. Miao, F. Chen, M. A. LeMone, J. Li, F. Hu, and L. Wang(2017). Estimate of Boundary-Layer Depth Over Beijing, China, Using DopplerLidar Data During SURF-2015. Bound.-Lay. Meteorol. 162(3), 503–522.

193

Hussein, H. J., S. P. Capp, and W. K. George (1994). Velocity measurements ina high-Reynolds-number momentum-conserving, axisymmetric, tubulent jet. J.

Fluid Mech. 258, 31–75.

Inagaki, A. and M. Kanda (2008). Turbulent flow similarity over an array of cubesin near-neutrally stratified atmospheric flow. J. Fluid Mech. 615, 101–120.

Inagaki, A. and M. Kanda (2010). Organized structure of active turbulence over anarray of cubes within the logarithmic layer of atmospheric flow. Boundary-Layer

Meteorol. 135(2), 209–228.

Jimenez, J. (2004). Turbulent flows over rough walls. Annu. Rev. Fluid Mech. 36(1),173–196.

Kaimal, J. C., J. C. Wyngaard, D. A. Haugen, O. R. Cote, Y. Izumi, S. J. Caughey,and C. J. Readings (1976). Turbulence structure in the convective boundary layer.J. Atmos. Sci. 33(11), 2152–2169.

Kaimal, J. C., J. C. Wyngaard, Y. Izumi, and O. R. Cote (1972). Spectral character-istics of surface-layer turbulence. Q. J. Roy. Meteor. Soc. 98(417), 563–589.

Kays, W. M. and M. E. Crawford (1993). Convective heat and mass transfer (3rded.). New York: McGraw-Hill Inc.

Klein, P. and J. V. Clark (2007). Flow variability in a North American downtownstreet canyon. J. Appl. Meteorol. Clim. 46, 851–877.

Klein, P. M. and J. M. Galvez (2015). Flow and turbulence characteristics in asuburban street canyon. Environ. Fluid Mech. 15(2), 419–438.

Kolmogorov, A. N. (1941). The local structure of turbulence in incompressibleviscous fluid for very large reynolds numbers. Dokl. Akad. Nauk SSSR 30, 299–303.

Kolmogorov, A. N. (1942). The equations of turbulent motion in an incompressiblefluid. Izvestia Acad. Sci., USSR; Phys. 6, 56–58.

194

Launder, B. E. and D. B. Spalding (1972). Mathematical models of turbulence.London: Academic Press.

Launder, B. E. and D. B. Spalding (1974). The numerical computation of turbulentflows. Comp. Meth. Appl. Mech. Eng. 3(2), 269–289.

Li, X.-X., R. E. Britter, T. Y. Koh, L. K. Norford, C.-H. Liu, D. Entekhabi, andD. Y. C. Leung (2010). Large-eddy simulation of flow and pollutant transportin urban street canyons with ground heating. Boundary-Layer Meteorol. 137(2),187–204.

Louka, P., S. E. Belcher, and R. G. Harrison (2000). Coupling between air flowin streets and the well-developed boundary layer aloft. Atmos. Environ. 34(16),2613–2621.

Lund, T. S., X. Wu, and K. D. Squires (1998). Generation of turbulent inflow datafor spatially-developing boundary layer simulations. J. Comput. Phys. 140(2),233–258.

Mason, P. J. and N. S. Callen (1986). On the magnitude of the subgrid-scaleeddy coefficient in large-eddy simulations of turbulent channel flow. J. Fluid

Mech. 162, 439–462.

Mathey, F., D. Cokljat, J. P. Bertoglio, and E. Sergent (2006). Assessment of thevortex method for large eddy simulation inlet conditions. Prog. Comput. Fluid

Dy. 6(1-3), 58–67.

Mellor, G. L. and T. Yamada (1974). A hierarchy of turbulence closure models forplanetary boundary layers. J. Atmos. Sci. 31, 1791–1806.

Menter, F. (1994). Two-equation eddy-viscosity turbulence models for engineeringapplications. AIAA J. 32(8), 1598–1605.

Nelson, M. A., E. R. Pardyjak, M. J. Brown, and J. C. Klewicki (2007). Propertiesof the wind field within the Oklahoma City Park Avenue street canyon. Part II:spectra, cospectra, and quadrant analyses. J. Appl. Meteorol. Clim. 46(12), 2055–2073.

195

Nelson, M. A., E. R. Pardyjak, J. C. Klewicki, S. U. Pol, and M. J. Brown (2007).Properties of the wind field within the Oklahoma City Park Avenue street canyon.Part I: mean flow and turbulence statistics. J. Appl. Meteorol. Clim. 46(12), 2038–2054.

Orszag, S. A. and G. S. Patterson (1972). Numerical simulation of three-dimensional homogeneous isotropic turbulence. Phys. Rev. Lett. 28, 76–79.

Poletto, R., T. Craft, and A. Revell (2013). A new divergence free synthetic eddymethod for the reproduction of inlet flow conditions for LES. Flow Turb. Com-

bust. 91(3), 519–539.

Pope, S. B. (2000). Turbulent flows. Cambridge, U.K.: Cambridge University Press.

Prandtl, L. (1945). Uber ein neues Formelsystem fur die ausgebildete Turbulenz.Nachr. Akad. Wiss. Gottingen Math-Phys. K1(6-19).

Qi, M., J. Li, Q. Chen, and Q. Zhang (2018). Roughness effects on near-wall tur-bulence modelling for open-channel flows. J. Hydraul. Res. 56(5), 648–661.

Ramamurthy, P., J. Gonzalez, L. Ortiz, M. Arend, and F. Moshary (2017). Impactof heatwave on a megacity: an observational analysis of New York City duringJuly 2016. Environ. Res. Lett. 12(5), 054011.

Ramamurthy, P., E. R. Pardyjak, and J. C. Klewicki (2007). Observations of the ef-fects of atmospheric stability on turbulence statistics deep within an urban streetcanyon. J. Appl. Meteorol. Clim. 46(12), 2074–2085.

Raupach, M. R. (1992). Drag and drag partition on rough surfaces. Boundary-Layer

Meteorol. 60(4), 375–395.

Raupach, M. R., R. A. Antonia, and S. Rajagopalan (1991). Rough-wall turbulentboundary layers. Appl. Mech. Rev. 44(1), 1–25.

Raupach, M. R., A. S. Thom, and I. Edwards (1980). A wind-tunnel study of tur-bulent flow close to regularly arrayed rough surfaces. Boundary-Layer Meteo-

rol. 18(4), 373–397.

196

Reynolds, A. J. (1975). The prediction of turbulent Prandtl and Schmidt numbers.Int. J. Heat Mass Transfer 18(9), 1055–1069.

Ricci, M., L. Patruno, and S. de Miranda (2017). Wind loads and structural re-sponse: Benchmarking LES on a low-rise building. Eng. Struct. 144, 26–42.

Richardson, L. F. (1920). The supply of energy from and to atmospheric eddies.Proc. R. Soc. Lond. A 97(686), 354–373.

Roache, P. J. (1994). A method for uniform reporting of grid refinement studies. J.

Fluid Eng. 116, 405–413.

Roache, P. J. (1997). Quantification of uncertainty in computational fluid dynamics.Annu. Rev. Fluid Mech. 29, 123–160.

Rotach, M. W., R. Vogt, C. Bernhofer, E. Batchvarova, A. Christen, A. Clappier,B. Feddersen, S. E. Gryning, G. Martucci, H. Mayer, V. Mitev, T. R. Oke, E. Par-low, H. Richner, M. Roth, Y. A. Roulet, D. Ruffieux, J. A. Salmond, M. Schatz-mann, and J. A. Voogt (2005). BUBBLE–an urban boundary layer meteorologyproject. Theor. Appl. Climatol. 81(3), 231–261.

Seinfeld, J. H. and S. N. Pandis (2006). Atmospheric Chemistry and Physics From

Air Pollution to Climate Change (2nd ed.). Hoboken, New Jersey, United Statesof America: John Wiley & Sons, Inc.

Sergent, M. E. (2002). Vers une methodologie de couplage entre la simulation desgrandes echelles et les modeles statistique.

Smagorinsky, J. (1963). General circulation experiments with the primitive equa-tions: I. The basic equations. Mon. Weather Rev. 91, 99–164.

Sonntag, R. E. (1966). Fundamentals of Statistical Thermodynamics. Michigan:Wiley.

Spalart, P. R. (1988). Direct simulation of a turbulent boundary layer up to Rθ =

1410. J. Fluid Mech. 187, 61–98.

197

Spalart, P. R. and S. R. Allmaras (1994). A one-equation turbulence model foraerodynamic flows. Recherche Aerospatiale 1, 5–21.

Stull, R. B. (1988). An introduction to boundary layer meteorology. Dordrecht, TheNetherlands: Kluwer Academic Publishers.

Suminska, O. A. (2005). Application of constant temperature anemometer forballoon-borne stratospheric turbulence soundings.

Tabor, G. R. and M. H. Baba-Ahmadi (2010). Inlet conditions for large eddy simu-lation: A review. Comput. Fluids 39(4), 553–567.

Taylor, G. I. (1938). The spectrum of turbulence. Proc. R. Soc. Lond. A 164, 476–490.

Temmerman, L., M. A. Leschziner, C. P. Mellen, and J. Frohlich (2003). Investi-gation of wall-function approximations and subgrid-scale models in large eddysimulation of separated flow in a channel with streamwise periodic constrictions.Int. J. Heat Fluid Fl. 24(2), 157–180.

van Driest, E. R. (1956). On turbulent flow near a wall. J. Aeronaut. Sci. 23(11),1007–1011.

von Karman, T. (1931). Mechanical similitude and turbulence. Technical report,National Advisory Committee for Aeronautics, Washington DC.

von Karman, T. (1948). Progress in the statistical theory of turbulence. Proc. Natl.

Acad. Sci. U. S. A. 34(11), 530–539.

Werner, H. and H. Wengle (1991, September). Large-eddy simulation of turbulentflow over and around a cube in a plate channel. In 8th Symposium on Turbulent

Shear Flows, Technical University of Munich.

Willis, G. E. and J. W. Deardorff (1976). On the use of Taylor’s translation hypoth-esis for diffusion in the mixed layer. Q. J. Roy. Meteor. Soc. 102, 817–822.

Wyngaard, J. C. (2004). Toward Numerical Modeling in the ”Terra Incognita”. J.

Atmos. Sci. 61(14), 1816–1826.

198

Xie, B. (2016). Improved vortex method for LES inflow generation and applicationsto channel and flat-plate flows.

Zajic, D., H. J. S. Fernando, R. Calhoun, M. Princevac, M. J. Brown, and E. R.Pardyjak (2011). Flow and turbulence in an urban canyon. J. Appl. Meteorol.

Clim. 50(1), 203–223.

199

List of Contributors

Amir A. Aliabadi received his bachelor’s and master’s degrees in Mechanical En-gineering, in 2006 and 2008 respectively, from University of Toronto, Toronto,Canada, and his doctoral degree in Mechanical Engineering in 2013 from Univer-sity of British Columbia, Vancouver, Canada. He is an assistant professor of en-gineering in the Environmental Engineering program at the University of Guelph,Canada. He is specialized in applications of thermo-fluids in buildings and the en-vironment. Prior to this position he was a visiting research fellow at Air QualityResearch Division, Environment and Climate Change Canada from 2013 to 2015in Toronto, Canada, and a research associate in Department of Architecture at theMassachusetts Institute of Technology (MIT) from 2015 to 2016 in Cambridge,U.S.A.

Reza Aliabadi graduated from University of Tehran, Tehran, Iran, in 1999 witha master’s degree in Architecture, and founded the “Reza Aliabadi Building Work-shop”. After completing a post-professional master’s of Architecture at McGill Uni-versity, Montreal, Canada, in 2006 and obtaining the OAA license in 2010, theworkshop was reestablished in Toronto as atelier Reza Aliabadi “rzlbd”. He hasestablished a strong reputation in both national and international architectural com-munities. Local and global media have published many of rzlbd’s projects. He hasbeen invited to install in Toronto Harbourfront Centre, sit at peer assessment com-mittee of Canada Council for the Art, speak at CBC Radio, give lectures at art andarchitecture schools and colleges, be a guest reviewer at design studios, and mentora handful of talented interns in the Greater Toronto Area. He also had a teachingposition at the School of Fine Arts at the University of Tehran and was a guest lec-turer in the doctoral program at the same university. Artifice has recently publishedReza’s first monograph “rzlbd hopscotch”. He maintains an ongoing interest in ar-chitectural research in areas such as microarchitecture, housing ideas for the future,and other dimensions of urbanism such as compactness and intensification. Besidehis architectural practice, Reza also publishes a periodical zine called rzlbdPOST.

200

Index

aerodynamic roughness height, 144, 145,147, 148

algebraic models, 111aliased, 86alternating symbol, 11anisotropic residual-stress tensor, 124anisotropy, 35, 110antisymmetric, 10Atmospheric Boundary Layer (ABL),

146autocorrelation function, 22autocovariance, 22

Backward-Difference Scheme (BDS),183

Bias (B), 155body-forces, 5boundary layer, 45boundary-layer thickness, 45, 53box turbulence, 136buffer layer, 50buffer sublayer, 50

central moment, 18Central Processing Units (CPU), 107Central-Difference Scheme (CDS), 183,

184central-limit theorem, 19, 21chaotic property changes, 2characteristic filtered rate-of-strain, 125characteristic quantity, 2

circulation, 15classical Kelvin-Stokes theorem, 14closure problem, 34, 36, 107coalescer, 159complete models, 115, 150compressible flow, 161conditionally stable, 186conserved passive scalar, 8Consistent Discrete Random Field Gen-

eration (CDRFG), 127constant-property Newtonian fluids, 6Control Volume Schemes (CVS), 151correlation coefficient, 19cospectrum, 90Courant number, 185covariance, 19Cross spectrum, 88Cumulative Distribution Function (CDF),

17, 170

dependent variables, 12Desalination, 160diffusion, 37dimensionless physical roughness height,

147direct contact membrane, 160Direct Numerical Simulation (DNS),

106, 135discrete Fourier transform, 83, 84discrete spectral energy, 87

201

discrete spectral intensity, 87displacement height, 145displacement thickness, 53dissipation range, 70dissipation rate, 68divergence-free, 5Doppler effect, 94Doppler shift, 98down-gradient hypothesis, 37Dynamic similarity, 11

eddy, 3eddy viscosity, 38effective diffusivity, 37effective viscosity, 38energy cascade, 67energy spectrum function, 71energy transfer rate, 68energy-containing range, 68ensemble average, 19, 20, 29entrainment, 61Euler equation, 7event, 169expectation, 171explicit Euler method, 185exponential distribution, 171external flows, 45

Field-Programmable Gate Array (FPGA),108

filtered, 106filtering operation, 122Finite Difference Schemes (FDS), 151

Finite Element Schemes (FES), 151first-order tensor, 179flatness, 19flops, 107fluctuation, 18folded, 86forward transform, 84Forward-Difference Scheme (FDS), 183Fourier decomposition, 84Fourier-transform pair, 23, 28, 84Fractional mean Bias (FB), 155frequency domain, 83frequency spectrum, 23friction Reynolds number, 48friction velocity, 48, 139frozen turbulence approximation, 77fully developed flows, 45fundamental frequency, 84

Gaussian distribution, 19, 21, 172Geometric similarity, 11GPU-accelerated computing, 108gradient-diffusion hypothesis, 36, 110Graphical Processing Units (GPU), 107gravitational potential, 5Grid Convergence Index (GCI), 153Grid Independence Test (GIT), 151

harmonics, 84heterogeneous hardware architecture,

107High Performance Computing (HPC),

107

202

higher-order terms, 182homogeneous hardware architecture,

107Hot Wire Anemometry (HWA), 102

implicit Euler method, 186incomplete models, 150independent variables, 12inertial subrange, 70inner layer, 49integral lengthscale, 27integral lengthscales, 74integral timescale, 22internal flows, 45invariance properties, 12inverse transform, 84inviscid flows, 7irrotational flow, 13

Joint Probability Density Function (JPDF),19

Kinematic similarity, 11Kolmogorov hypotheses, 68Kolmogorov hypothesis, 67Kolmogorov scales, 69, 130kurtosis, 19

L’Hospital’s Rule, 146Large-Eddy Simulation (LES), 106, 122laser, 100Laser Doppler Anemometry (LDA), 98Laser Doppler Velocimetry (LDV), 98law of the wall, 49, 50

Light Detection And Ranging (LiDAR),100

linearization method, 183log-law for a rough wall, 147log-law region, 111log-law sublayer, 50logarithmic wall model, 144longitudinal autocorrelation function,

72longitudinal lengthscale, 74longitudinal structure function, 75

material derivative, 178mean, 17, 171mean rate-of-rotation tensor, 111mean rate-of-strain tensor, 110, 111mean substantial derivative, 33mixing length, 111mixing-length, 106mixing-length models, 111model, 105Modified Discretizing and Synthesiz-

ing Random Flow Generation(MDSRFG), 128

modified filtered pressure, 124modified mean pressure, 35modified pressure, 7molecular diffusivity, 37, 110momentum thickness, 53

Newton’s method, 183no-slip boundary condition, 46nominal friction velocity, 139

203

normal distribution, 21, 172normal stresses, 34Normalized Mean Square Error (NMSE),

155Nyquist frequency, 86

one-equation models, 112, 114order of convergence, 151outer layer, 49

Particle image velocimetry (PIV), 96Particle Tracking Velocimetry (PTV),

96periodic cube, 136phase, 88Phase shift, 89Pitot Tube, 103power law wall model, 145Probability, 169Probability Density Function (PDF),

17, 170prognostic, 16Pyrometry, 99

quadrature spectrum, 90

RAdio Detection And Ranging (RADAR),99

Radiometry, 99random process, 22random variable, 169rate-of-strain tensor, 10residual, 122residual kinetic energy, 124

residual stress tensor, 122residual-stress tensor, 124Reynolds analogy, 38Reynolds decomposition, 32, 35Reynolds number, 2Reynolds stress tensor, 124Reynolds stresses, 25Reynolds-Averaged Navier-Stokes (RANS),

105, 109Root Mean Square Error (RMSE), 155rotational flow, 13

sand-grain roughness height, 149scalar flux, 36Schlieren Image Velocimetry (SIV), 97Schlieren system, 98second-order discretization, 152second-order tensor, 180Self-similarity, 58SGS tensor, 122SGS Turbulent Kinetic Energy (TKE),

125shadowgraph system, 97shear stresses, 34similarity variable, 60similitude, 11simulation, 105skewness, 18skin-friction coefficient, 48Smagorinsky lengthscale, 125, 142solenoidal, 5solvent partitioning, 159solver, 108

204

SOnic Detection And Ranging (SO-DAR), 94

SOund NAvigation Ranging (SONAR),94

Spalart-Allmaras Model, 112spatial average, 29spectral energy density, 88spreading, 60spreading rate, 60standard deviation, 18standard model, 115standard wall function, 138, 139statistically anisotropic, 26statistically axisymmetric, 26statistically homogeneous, 26statistically isotropic, 26statistically stationary, 22, 26Stokes number, 96stress tensor, 5Subgrid-scale (SGS), 122substantial derivative, 178surface forces, 5symmetric-deviatoric, 10symmetries, 12

tailedness, 19Taylor hypothesis, 77Taylor series, 182Temperature Perturbation Method (TPM),

128time average, 29time domain, 83total shear stress, 47

transformation properties, 12transverse autocorrelation function, 72transverse integral lengthscale, 74transverse structure function, 75turbulence intensity, 41turbulence level, 41turbulence problem, 105turbulent diffusivity, 37, 110turbulent eddy, 9turbulent energy cascade, 3turbulent flow, 2turbulent kinetic energy, 34turbulent viscosity, 38Turbulent viscosity models, 109turbulent-viscosity hypothesis, 36, 109,

110two-equation models, 115two-point correlation, 27

Ultrasonic Anemometer, 93under relaxation, 186under relaxation factor, 187uniform distribution, 171uniform turbulent viscosity models, 111universal equilibrium range, 68

Validation, 150van Driest function, 143variance, 18vector field, 25velocity field, 25velocity structure function, 74Ventilation, 161

205

verification, 150virtual origin, 60viscous dissipation, 3viscous lengthscale, 48viscous scales, 48viscous sublayer, 50vorticity, 9

wall function, 138wall shear stress, 47wall units, 49waste water treatment, 160wavenumber vector, 27, 28Werner-Wengle wall function, 141

zeroth-order tensor, 179

206

ISBN 978-0-9809704-9-4

THEORY AND APPLICATIONS OF

TURBULENCE

A FUNDAMENTAL APPROACH FOR SCIENTISTS AND ENGINEERS

tu r bu l e nc e - aaa scientists › aliabadi 2018.pdf · theory and applications of turbulence a...

Documents