turbulent heat transfer

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Turbulent Heat Transfer

Scott Stolpa

April 30, 2004


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Preface

This paper is an independent research project that combines many different sources in the

field of heat transfer. Its goal is to collect and organize many of the turbulent heat transfer princi-

ples that are fundamental to heat transfer textbooks and combine those with state of the art find-

ings by some of the top researchers in the field. The ultimate aim is to connect everything to the

atmospheric boundary layer, a subject that is central to my own research. The audience is engi-

neering graduate students who have had some heat transfer and some turbulence. It is also neces-

sary to have some understanding of basic fluid mechanics.


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CONTENTS

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

CHAPTER 1: INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Principles of Heat Transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..1

1.2 Fundamental Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..2

1.3 Modes of Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..4

1.4 Laminar Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..5

1.5 Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..6

1.6 External Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..61.6.1 Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.6.2 Free Turbulent Flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.7 Internal Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..9

1.7.1 Entrance Region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.7.2 Fully-Developed Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.8 Principles of Scaling and Non-Dimensionalization . . . . . . . . . . . . . . . . . . . . .11

CHAPTER 2: FORCED CONVECTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..15

2.2 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..15

2.3 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .162.4 The Turbulent Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..17

2.4.1 Turbulence Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 Boundary Layer Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..22

2.5.1 Mixing Length Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5.2 Solving the Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5.3 Wall Heat Flux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.6 Other External Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..32

2.6.1 Cylinder in Cross Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.6.2 Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.6.3 General Spheroid Body Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.6.4 Array of Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.7 Internal flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..36

2.7.1 Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

CHAPTER 3: NATURAL AND MIXED CONVECTION . . . . . . . . . . . . . . . . . . . . . . 42

3.1 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42

3.2 Boussinesq Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44

3.3 Natural Convection with a Vertical Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . ..46


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3.4 Thermal Stratification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..49

3.5 Natural Convection Over a Flat Horizontal Plate. . . . . . . . . . . . . . . . . . . . . ..49

3.6 Vertical Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..50

3.7 Mixed Convection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..52

3.7.1 Vertical Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.7.4 Horizontal Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

CHAPTER 4: ATMOSPHERIC BOUNDARY LAYER . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..55

4.2 Important Notes on Variable Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..56

4.3 Fluid Mechanics and Governing Equations for the ABL . . . . . . . . . . . . . . . ..56

4.4 Flat, Infinite, Uniform Terrain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..58

4.4.1 ABL Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.4.2 Stability Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.4.3 The Three States of the ABL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.4.4 Properties of the Surface Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.4.5 Monin-Obukhov Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.4.6 Spectra and Cospectra for the Surface Layer. . . . . . . . . . . . . . . . . . 67

4.4.7 Beyond the Surface Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.5 Plant Canopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..72

4.6 Changing Terrain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..72

4.7 Hills. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..73

CHAPTER 5: MEASUREMENTS TECHNIQUES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.1 Velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..74

5.2 Temperature and Heat Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..77

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78


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CHAPTER 1

INTRODUCTION

1.1 Principles of Heat Transfer

Heat transfer is essentially made up of two different mechanisms. Radiation and conduc-

tion. A third mechanism, convection, is the conduction of heat between two objects with a rela-

tive velocity between them. There are many different scenarios under which convection occurs.

It can occur in horizontal flow or vertical flow. It can take place in a duct or in the open air. It can

be laminar or turbulent. Natural convection requires no external flow at all. Buoyancy effects

will create air flow and heat transfer will occur. Convection laws rely on the fundamental princi-

ples of both heat transfer and fluid flow. This includes obeying the laws of mass conservation,

momentum conservation, and energy conservation. Convection is also governed by the first and

second laws of thermodynamics. The simple equation for the convective heat transfer between a

flat plate and a fluid is Newtons Law of Cooling:

. (1.1)

The symbols q'', h, To, and T represent the heat wall heat flux, the heat transfer coeffi-

cient, the temperature of the flat plate and the temperature of the fluid respectively. Near the wall,

the fluid must obey the no-slip condition, and the velocity is essentially zero at the wall. Without

a relative velocity, the heat transfer is governed by pure conduction and the equation:

q'' h To T( )=


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. (1.2)

These two equations represent the instantaneous heat transfer between two points for conduction

and convection. In order to get the total heat transfer between a fluid and a flat plate, one must

solve the integral:

(1.3)

where q is the total heat transfer and W is the dimension of the plate perpendicular to the stream-

wise direction. The rest of this paper will deal with ways to adapt these equations to find the total

heat transfer between a surface and a fluid or between two fluids.

1.2 Fundamental Equations

The equations for all convective heat transfer are derived from the fundamental principles

that were stated in Section 1.1. This amounts to solving four nonlinear partial differential equa-

tions for the conservation of mass, momentum and energy. In their most general, incompressible,

Boussinesq (which will be discussed in subsequent chapters) forms, these equations read:

. (1.4)

(1.5)

(1.6)

One important thing that was neglected in the energy equation is a viscous dissipation term. The

viscous dissipation term is negligible at low Mach numbers. Since we are assuming incompress-

ible flow, we must also assume low Mach number since at M > 0.3 the fluid can no longer be

q'' kTy------

y 0=

=

q qW xd

0

L

=

ivi 0=

Dv i

Dt-------- fi ip ijvi+=

cp

DT

Dt-------- k

ijT=


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assumed to be incompressible. The viscous dissipation does not disappear because of incom-

pressible effects, but because of low Mach number limits. In most cases considered however, the

two effects will be equivalent. Eqns. 1.4-1.6 are the building blocks for all subsequent heat trans-

fer analysis. They are general and valid for steady and unsteady incompressible flow with con-

stant properties. The material derivative, D/Dt, is defined as

. (1.7)

Assuming steady state flow would allow further simplification of the equations. In the case of

steady state flow, the partial time derivative (/t) vanishes. These are also subject to boundary

conditions that are unique to a given flow configuration. For example, a two dimensional bound-

ary layer would have the following boundary conditions:

, (1.8)

, (1.9)

Eqn. 1.8 is a consequence of the no-slip condition and Eqn. 1.9 is a condition of having a solid

wall. The other condition that is required at the wall is information on the temperature distribu-

tion or heat transfer. The most typical conditions are a uniform wall temperature or a uniform

wall heat flux. In the freestream, the following conditions typically apply:

, (1.10)

, (1.11)

. (1.12)

These three equations are the properties of the uniform flow in the freestream, that is the area infi-

nitely away from the wall in which there are no friction or wall effects. Each area of convective

D

Dt------ o ujj+=

u 0=

v 0=

u U=

v 0=

T T=
http://-/?-http://-/?-


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heat transfer will have some variation of these equations that will be cultivated to the specific cir-

cumstances.

1.3 Modes of Heat Transfer

Conduction: Conduction is the most basic method of heat transfer. It involves direct heat trans-

fer from one object to another object. The objects can be gases, fluids, or solids.

Convection: The key to understanding convection is remembering that its is conduction between

two objects (or gases, fluids, or a mixture) in which the two substances have a relative velocity

between themselves. For example, with a gas moving over a hot plate, the plate transfers heat to

the gas and the heated molecules move on. Cooler molecules move into the vacated area, and

they are heated as well. Convection typically transfers more heat because the heated particles

move on and are replaced by cooler ones, allowing for a greater temperature difference for a

longer time.

Forced Convection: Forced convection is the transfer of heat by convection in which the

relative velocity is created by some outside source. For instance, a fan blowing air over a

Figure 1.1: Flow Over a Flat Plate


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plate is forced convection. Any time there is a freestream velocity, there is going to be

forced convection. Chapter 2 deals with forced convection.

Natural Convection: Natural Convection has no freestream velocity, but there is move-

ment of air that is created by buoyancy effects. When air is heated it expands and

becomes less dense. Hot air will rise and cooler air will replace it. The cycle will con-

tinue as long as there is a temperature difference between the top of a room and the floor.

The movement that is created by this buoyant air displacement is the basis of natural con-

vection. Chapter 3 focuses on natural convection.

Mixed Convection: In a situation where there is both forced convection and natural con-

vection, there exists mixed convection. Usually one of the methods is the dominant mode

of heat transfer, and there are ways to determine how to handle these situations. Natural

convection usually serves to affect the up and downward components of velocity in these

cases.

Radiation: Radiation is a type of heat transfer between two objects that are not in contact with

each other. The transfer can occur through gas or fluid or a vacuum. The heat transfer is indepen-

dent of the medium through which it travels. The air can be heated by radiation if there are water

molecules in the air. The water is heated by the sun and then the air appears to be heated. The sun

heats the earth through the process of radiation. Radiation is not covered in this paper in detail,

but it is referred to on a couple of occasions.

1.4 Laminar Flow

The simplest type of convection is that which takes place in laminar flow. Laminar flow is

easy to predict and has very little fluctuation in it. There are several approaches to solving the

flow in a laminar boundary layer. Most situations begin as laminar flows and then later transition


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to turbulence. Laminar flows are more structured than turbulent flows, but are still not fully

understood. In order to get solutions for laminar flow, assumptions must be made because the

equations cannot be analytically solved as they are. One must be careful to note when a flow tran-

sitions to turbulent, because it becomes more complicated and unpredictable. Although most real

flows are turbulent in nature, it is often helpful to understand the nature of the laminar flows

before attempting to tackle turbulent flows. Laminar flow principles are simpler and help one get

an idea of how the flow behaves before adding the complexities of turbulent effects.

1.5 Turbulent Flow

In contrast with laminar flow, turbulent flow is less structured and predictable. Structures

called eddies dominate the flow. The eddies can be any size, ranging from the entire width of the

boundary layer to microscopic structures. These eddies rotate and mix the flow in such a way that

there it is almost impossible to solve analytically. Turbulent flow problems involve systems of

equations in which there are more unknowns than equations. Assumptions and models reduce the

number of unknowns, but these assumptions are not perfect. Turbulence modelling software

relies on the accuracy of the models themselves. Researchers are always working to improve

these models in order to get a better understanding of turbulent flow. Typically all high Reynolds

number flows are turbulent. Low Reynolds number flow can be turbulent or laminar. It is easy to

make a flow transition to turbulence through methods such as tripping the flow, where roughness

is added to a surface in order to facilitate the transition. Turbulence is a phenomenon whose char-

acteristics are still being investigated by researchers. Until the turbulence problem can be solved,

researchers must rely on idealized laminar solutions and turbulence models.

1.6 External Flows


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The most common type of external flow is flow over an object. The object is usually a flat

plate as the flat plate can simulate a number of different real applications such as the wing of an

airplane. Another common object is a cylinder. Boundary layers are important concepts in exter-

nal flows and they will be addressed in Section 1.6.1. Section 1.6.2 covers other types of external

flow that are called free turbulent flows. The effects of friction from walls are unimportant in

these situations. Although heat transfer is present in these situations, quantities are difficult to

measure because the transfer is from fluid to fluid (or gas to gas). They are also less important in

application and are only briefly covered.

1.6.1 Boundary Layers

Boundary layers are central to the idea of forced convection. Any flow that is bounded by

a surface will develop a region that is adjacent to the surface, in which the flow properties are dif-

ferent from the freestream. Friction is the primary reason for the existence of the boundary layer.

There are an infinite number of thin regions that can be considered a boundary layer. Each one is

based upon a property that varies from a value at the wall to a freestream value away from the

wall. The two most common types of boundary layers are velocity and thermal. In a velocity

boundary layer, the flow velocity is zero at the wall because of the no-slip condition. The velocity

increases as you move away from the wall until it eventually becomes a constant and is equal (or

close enough) to the freestream value. In a thermal boundary layer, the temperature varies from a

temperature To at the wall to the freestream value T at the edge of the thermal boundary layer.

The lengthscale of the velocity boundary layer is typically given the symbol while the thermal

boundary layer has the scale T. In general these scales are not equal. The relative size of these

scales is a condition of a parameter known as the Prandtl number. It will be introduced later in

this chapter, and its specific function will be discussed in Chapter 3.


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The boundary layer is an important concept because it is the region in which heat transfer

between a gas or a fluid and a surface take place. It is distinct from the rest of the flow, which is

usually considered to have constant freestream properties. The nature of the boundary layer is

determined by the surface fluid interaction. This includes the roughness and the shear stress at the

wall. The velocity distribution within a boundary layer is important because velocity plays an

important part in the mass momentum and energy equations. The temperature is the main compo-

nent of the energy equation and as such the thermal boundary layer is important to fluid mechan-

ics. Understanding the boundary layer is the first step in analyzing external fluid flow and

external convection.

1.6.2 Free Turbulent Flows

In a free turbulent flow, the flow is different from a typical external flow in that it is

located far enough away from all surfaces that there are no effects from the wall. Free turbulent

flows come in several different forms, but they all share the common theme of being unaffected

by walls. The flow is built on the interaction of two different fluids or gases mixing with each

other. The first fluid or gas has some velocity and it is released into a stagnant fluid or gas. Free

turbulent flows are built on the concept of shear layers. A shear layer is the area of interaction

that develops between the two fluids. The area of interaction usually grows at an angle from the

point where the first fluid leaves the pipe or duct in which it had been travelling. Laminar and tur-

bulent shear layers are both possible, with the laminar almost always becoming turbulent after

sufficient distance. A shear layer usually develops on all sides of the duct from which the fluid is

released. When these shear layers combined, the flow becomes what is known as a jet. The most

typical examples of a jet occur when the fluid is released from a slit rather than a large duct. The


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shear layers combine almost instantaneously and the entire flow is a jet essentially. Often similar-

ity is used to find the solution to jet flow.

The final two examples of shear layers are the wake and the plume. When a fluid flows

over an object such as a cylinder, it creates a wake downstream from that object. Much like the

shear layer, the affected region expands at a certain angle from the object. The governing equa-

tions differ slightly from those of a jet. If the object is heated, there will be temperature effects

that complicate the situation more. A plume is a region of heated air that rises from a heated

source into cooler air. The source may be a cylinder or a heated plate. This is a prime example of

natural convection. The heated air near the source becomes less dense than the air surrounding it.

It rises into the air and in turn creates an angled area of interaction with the stagnant air around it.

These plumes are important in the atmospheric boundary layer and will be discussed further in

Chapter 4.

1.7 Internal Flows

In contrast to external flows, internal flows are completely encompassed by a pipe, a duct

or some other fluid carrier. Probably the most common type of internal flow is flow of water

through round pipes. In a situation involving round pipes, the easiest thing to do is to change the

coordinates to cylindrical. This is because the flow is virtually identical in all radial directions if

the pipe is full and the effects of gravity can be ignored. In internal flows, the boundary layers

grow to the point where they meet in the middle of the pipe, presenting a new type of flow called

fully developed.


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1.7.1 Entrance Region

When a fluid leaves a reservoir and first enters a pipe, the flow cannot truly be considered

internal even though it is flowing inside of a pipe or duct. In this region, all of the principles of

external boundary layer flow apply. In a circular pipe, the boundary layers have not grown large

enough to interact, so there is no difference between flow that is bounded on all sides and flow

that is bounded on one side. It does help to change the coordinate system, but this is only to facil-

itate the analysis. There is also no rule for the entrance region being laminar or turbulent. If the

fluid only travels at a low speed, the flow may stay laminar. In most cases the fluid will reach a

certain transition Reynolds number and the flow will become turbulent. In either case, external

flow guidelines apply until the flow becomes fully developed.

1.7.2 Fully-Developed Region

The fully developed region of pipe flow refers to the area where the boundary layers com-

bine in order to form one flow in the entire pipe or duct. There is no longer any distinguishable

Figure 1.2: Internal Pipe Flow


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freestream. Although internal, fully-developed flow retains some of the principles of boundary

layer and external flow, it is governed by additional theories. Most of the equations for fully

developed flow are empirical in nature. This is especially the case when the fully developed flow

is turbulent. The equations will be examined in depth in Chapter 2.

1.8 Principles of Scaling and Non-Dimensionalization

In order to get a better understanding of the relative sizes of each term in the fundamental

equations, one can identify scaling arguments. Not only does it give the researcher a better idea of

what to expect in terms of size, but certain terms can be eliminated in the equation. If L is the

streamwise length, then one can make the assumption that

(1.13)

indicating that the boundary layer is slender. With this assumption one can examine the terms in

Eqns. 1.4 to 1.6 in order to discover their relative sizes. Scaling uses one constant term that is rep-

resentative of the flow variable. It is often the freestream or sometimes the maximum value for

the variable. For instance, the y direction is scaled by and the distance x is scaled by L, the total

streamwise length. If there were a third dimension, z would be scaled by W the width of the plate

or the duct. The temperature T and the velocity u are scaled by their freestream properties T (or

some function of it) and U. Anything that is a constant does not need to be scaled. Scaling

allows simplifications in equation because one term may dominate another term, making the sec-

ond term inconsequential. If we assume steady state flow (/t = 0), the mass continuity equation

(1.5) requires that

. (1.14)

L

U

L-------

v

------
http://-/?-http://-/?-


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in order for the magnitudes of the two terms to be equal. This necessitates the conclusion that

both terms in the mass continuity equation are on the order of U /L. This is important because it

allows some easier substitutions in the momentum equations. There are three different types of

forces in the momentum equation: inertia, pressure, and friction. The inertia forces are the terms

that have a squared velocity term in the numerator and scale as

. (1.15)

The pressure forces are the terms with a pressure term in the numerator and they scale as

. (1.16)

The final force term is a friction force. This case has a second derivative of velocity and they can

be found to scale as

. (1.17)

If we assume steady state flow, the x-momentum equation contains two inertia terms (once the

material derivative is expanded), a body force term, a pressure term, and two friction terms. Let

us consider a common case in which there is no external body force. The two inertia terms are of

the same order as a consequence of the continuity equation. They both scale as U2/L. Neither of

these terms are dominant. Looking at the friction terms on the right side of the equation, one term

scales as U/L2 and the other scales as U/

2. If we continue with the assumption of Eqn. 1.13,

the second term is much larger than the first term. As such we can essentially ignore the second

derivative of x. This is an example of how scaling can be used in order to simplify equations. It

will be further used in subsequent chapters, and it is important to understand the process.

U

U

L------- v

U

-------,

P

L------

U

L2

------- U

2

-------,


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Often it is useful to work independent of units. This is helpful because solutions can apply

to any system of measurement one chooses to use in an analysis. The principle behind it is to turn

each and every variable into a new dimensionless variable. In most cases this means dividing by

a constant of the same units. In some cases one must divide by multiple constants in order to get

the units to cancel. All velocities are usually scaled by the same freestream velocity U. Dis-

tances often have a choice in scaling. Normal distances are usually divided by a channel half

width or radius. Sometimes they are divided by the streamwise length. The streamwise distance

usually uses the streamwise length as its scaling variable, as long as the distance is not considered

to be an infinite pipe. In a case such as that, there is usually no variation in x anyway so those

terms are ignored. Temperature can often be trickier, as often it is scaled as

(1.18)

in order to have the term vary from 0 to 1. Once all variables have been taken care of, the equa-

tion needs to be cleaned up in order to get rid of all dimensions. This gives rise to a number of

dimensionless constants that are a combination of other constants. Some of the most common

dimensionless constants are

Reynolds Number = (1.19)

Nusselt Number = (1.20)

Peclet Number = (1.21)

Prandtl Number = (1.22)

T To

T To------------------=

eLu

----------

Lu

------= =

NuhL

k------=

PeUL

-----------=

Pr---=


18/83

14

Many of these terms are important not only in nondimensionalization, but also in classifying

flows in all studies of heat transfer and fluid flow. The Reynolds number plays an important part

in wind tunnel tests when it comes to scaling models in order to apply work to real air and seac-

raft. Non-dimensionalization will be used in order to find some solutions later in this paper. The

principles outlined above remain the same no matter the situation.


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15

CHAPTER 2

FORCED CONVECTION

2.1 Introduction

The most common type of convection is referred to as forced convection. As defined in

Chapter 1, it is the transfer of heat from one substance to another in which a relative velocity

exists between the two mediums. Most commonly this exchange is between a solid and a gas or a

liquid. Section 2.2 highlights the general equations that are used for forced convection and gives

a brief overview of the force convection in laminar flow. Section 2.3 introduces the reader to the

turbulent flow equations that become important for the study of turbulent heat transfer. Section

2.4 adapts the turbulent flow equations to boundary layer flow. This amounts to making some

assumptions in order to simplify the equations.

2.2 References

The information in this chapter is primarily taken from Bejan [1]. There are many other

good sources that one can investigate that will give them information about turbulence and about

heat transfer. Schlicting [17] and White [19] have written some of the pioneering texts on fluid

flow in a boundary layer. Schlicting was one of the first to develop boundary layer theory, while

White examined the effect of friction on flows. Some of the leading text on turbulence has been

written by Hinze [8], Pope [16], and Bernard and Wallace [2]. These authors focus on the concept

of turbulence primarily, though they do touch on heat transfer. Some of the better convective heat


20/83

16

transfer texts include Bejan, Kays and Crawford [11], and Burmeister [4]. A couple of introduc-

tory heat transfer textbooks include Janna [9] and Oosthuizen and Naylor [14]. For specific areas

of heat transfer, the reader is directed to Kutateladze and Leonitev [12] (boundary layer heat trans-

fer) and Petukhov and Polyakov [15] (turbulent mixed convection). The reader is directed to

Adrian et al. [20] for a journal article on a relative experiment involving heat transfer in turbulent

boundary layers.

2.3 Equations

It is often necessary to make certain assumptions about physical flows in order to facilitate

the analysis. The general equations 1.4 through 1.6 are written in cartesian notation, remember-

ing that incompressible (low Mach number) and constant property flow have already been

assumed. Another assumption that is commonly made about a flow of this nature is the Bouss-

inesq approximation. The Boussinesq approximation states that in a flow where the density

changes (not by compressibility effects, but by temperature differences) are small but not zero, the

density change is important only in relation to the body force. This body force is usually gravity,

and is typically important only in the vertical momentum equation. The Boussinesq approxima-

tion will be discussed further in the next chapter when talking about natural convection. All other

density gradients and fluctuations may be neglected. Recalling the equations in cartesian nota-

tion:

(2.1)

. (2.2)

ui

xi------- 0=

gc-----

Dui

Dt--------- Fi

Pxi-------

2uixjxj---------------+=


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17

. (2.3)

There is some further simplification that can be done if one wants to consider only steady flow. In

this case all partial derivatives with respect to time can be eliminated. However turbulent flows

are often unsteady and as such often contain time dependent terms. For completeness, we will not

ignore the unsteady terms at this time.

2.4 The Turbulent Equations

By nature turbulent flows are irregular. In the past turbulent flows have been considered

to be completely random and unpredictable. Some recent research suggests that there is an under-

lying structure to turbulent flow. Flow visualization has shown ordered structures composed of

eddies that provide the large scale order to the turbulent flow. It is still impossible to predict

instantaneous results in a turbulent flow, and as such time averaged methods are used to examine

turbulent flows. Transforming the equations from general to turbulent requires the following def-

initions for the flow:

, (2.4)

, (2.5)

, (2.6)

, (2.7)

, (2.8)

where the ( ) indicates a time averaged quantity and () denotes an instantaneous fluctuation. The

general equations are valid for both turbulent and laminar flow. In order to make them specifi-

cally applicable to turbulent flow, Eqns. 2.4 to 2.8 are inserted into the general equations. The

cDT

Dt-------- k

2T

xixi---------------=

u u u'+=

v v v'+=

w w w'+=

T T T'+=

P P P '+=


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18

equations are still valid for laminar flow by considering the fluctuations to be negligible. Further

manipulation of the equations can be done to compress some of the terms. The transformation is

done by substituting the instantaneous quantities and then integrating each term over time. We

recall the definitions:

(2.9)

(2.10)

Eqn. 2.9 is the rule for time averaging a quantity and Eqn. 2.10 states that all fluctuating quantities

time averaged over a period are 0 by definition. Some of the other algebraic properties that will

help us arrive at the turbulent equations are:

(2.11)

(2.12)

(2.13)

(2.14)

(2.15)

(2.16)

(2.17)

u1

--- u td

0

=

u' td0

0=

u v+ u v+=

uu' 0=

uv uv u'v'+=

u2

u2

u'2

+=

ux------

ux------=

ut------ 0=

ut------ 0=


23/83


24/83

20

Although this form is helpful, we can use Eqn. 2.13 in order to show the effects of the instanta-

neous fluctuations. This transformation reads:

(2.24)

The term is called the Reynolds stress and is an important quantity in measuring turbulence.

Applying the mass continuity equation to the momentum equation yields the final form of the

momentum equation:

. (2.25)

The final form shows how the Reynolds stress affects the flow in a turbulent situation. The final

equation is the energy equation. In much the same way that the mass and momentum equations

were converted to turbulent form, the energy equation can be transformed into the following.

(2.26)

Analogous to the Reynolds stress is the turbulent heat flux which appears in Eqn. 2.26 and is writ-

ten . It is as important to turbulent heat transfer as the Reynolds stress is to turbulent flow.

Eqns. 2.19, 2.25, and 2.26 are the turbulent flow equations. They contain 17 unknowns, but only

represent 5 equations (since the momentum equation can be expressed in three different dimen-

sions). This is referred to as the closure problem that makes turbulent flow so difficult to analyze

and predict. Depending on the situations, flow-specific assumptions are often used to reduce the

number of unknowns. Although it is impossible to realistically reduce the number of unknowns

to be equal to the number of equations without introducing some inaccuracy, we can get close by

uiujxj------------

u'iu'jxj--------------+

Fi

P

xi-------

2ui

xjxj---------------+=

u'iu'j

ui

ui

xj-------

u'iu'j

xj--------------+ FiP

xi-------

2ui

xjxj---------------+=

uiTxi-------

u'iT'

xi-------------+

2T

xjxj---------------=

u'iT'


25/83

21

making reasonable assumptions and estimations concerning the flow properties. Part of the prob-

lem is handled with turbulence models.

2.4.1 Turbulence Models

The use of models is situation dependent. There is no one model that is universally appli-

cable. Typically boundary layer flow (which will be discussed in the next section) relies on the

mixing length model for analytical solutions. Computer turbulence modeling often relies on com-

plicated models, but they are difficult to solve analytically. One of these models is known as the

k- model. It is the most popular model used in computer turbulence solutions. The definition of

k, the turbulence energy, is:

, (2.27)

which is complemented by a model for the momentum eddy diffusivity that reads:

(2.28)

C is an experimentally determined constant and L is a yet to be determined length scale similar

to the mixing length described above. The term is the dissipation rate or the rate of k destruc-

tion. Details of this derivation can be found in Bejan [1] and other sources. Only the fundamental

equations will be presented here. The dissipation rate is defined as

. (2.29)

The length scale can be eliminated by using these two equations and setting CD to be 1.

(2.30)

Using this equation and the following two equations for k and

k1

2--- u'( )

2v'( )

2w'( )

2+ +[ ]=

M Ck1 2

L=

CDk

3 2

L----------=

M Ck

2

-----=


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22

, (2.31)

(2.32)

one can solve for the three unknowns (k, , M). The final values in these equations are empirical

constants that have typical values of

, , , , . (2.33)

which can be found in Bejan [1]. Similar constants are valid for most forms of flow. The number

of equations now matches the number of unknowns. The equation that completes the solution of

the energy equation involves the turbulent Prandtl number, which is defined as:

. (2.34)

This is the last equation needed to determine a full solution. The turbulent Prandtl number is typ-

ically assigned a value of 0.9, based on empirical data. One can see why this model is used only

in computer modelling. The mixing length model will be introduced in the next section and will

be used for boundary layer flow when computer modelling is not involved. Bejan [1] has pro-

vided much of the information for the modelling in this section, but the reader is also referred to

Launder [13] for more information on turbulence modelling. The next section will address the

specifics of the boundary layer flow configuration.

2.5 Boundary Layer Flow

The boundary layer was first introduced in Section 1.6.1. Its properties and many of its

assumptions were addressed in that section, but they will be quickly reviewed in this section. The

turbulent flow equations 2.19, 2.25, and 2.26 can be adapted to a boundary layer flow in order to

Dk

Dt-------

y-----

Mk------

ky-----

Muy------

2

+=

D

Dt

-------

y-----

M

------

y-----

C1Mu

y------

2

k

-- C22

k

-----+=

C 0.09= C1 1.44= C2 1.92= k 1= 1.3=

PrtMH------=


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23

reduce the number of unknowns. Consider a freestream flow with freestream properties U, T,

andP. A boundary layer flow is usually symmetric. If the axes are chosen so that the x direction

is in the direction of the mean flow, and the y direction is the normal direction away from the wall

as shown in Fig. 2.1, the symmetric properties allow the elimination of all /z terms. Scaling

assumptions allow the neglect of several other quantities. The velocity fluctuations u and v are

both of the same magnitude because they are both products of the same mechanism, namely a

rotating eddy. While the eddy rotates it produces fluctuations of equal magnitude in the x and y

direction, while w fluctuations reduce to zero because of symmetric properties. Although eddys

are three dimensional in nature, they extend primarily in only the x and y directions. The relative

size of the w term is small enough comparatively to ignore.

The equal magnitudes of the u and v fluctuations mean that scaling laws allow partial

derivatives of the Reynolds stresses in the x-direction to be ignored. This goes back to the equa-

tion that assumed that


28/83

24

Typically boundary layer flow also allows the assumption that pressure is a function of the x-

direction only, that is:

. (2.35)

Boundary layer flow considers only the x momentum equation along with the mass and energy

equations. Taking all of the preceding assumptions and assuming that there is no body force in

the x-direction and applying them to the turbulent flow equations, we get the following three

boundary layer equations:

, (2.36)

, (2.37)

. (2.38)

These are the important equations in turbulent boundary layer flow. They are similar to the equa-

tions that are used in laminar boundary layer flow. The laminar flow equations can be recovered

by assuming that all fluctuations are zero.

Another common way that the momentum and energy equations can be rewritten is:

, (2.39)

(2.40)

which can easily be seen using simple algebra and calculus rules. The importance of these

expressions lies in the final term of each category: and . The first term one should recog-

dP

dx

-------P

x------=

u

x------v

y-----+ 0=

uux------ v

uy------+

1

---

dP

dx-------

2u

y2

--------u'v'y

-----------+=

uTx------ v

Ty------+

2T

y2---------

v'T'y

-----------=

uux------ v

uy------+

1

---

dP

dx-------

1

---

y-----

u

y------ u'v'+

+=

uTx------ v

Ty------+

1

cP---------

y----- k

Ty------ cPv'T'

=

u'v' v'T'


29/83

25

nize as the Reynolds stress, and the second term is the turbulent heat flux. If these terms vanish,

then the equation will be reduced to the boundary layer equations for laminar flow. For this rea-

son the importance of these two terms to the analysis of turbulent flow is evident. Typically both

of these terms have a negative value at all points in a boundary layer profile. As a step towards

reducing the number of unknowns and to give the reader a physical meaning to these terms, let us

introduce the definitions:

, eddy shear stress (2.41)

. eddy heat flux (2.42)

The terms M and H are known as the momentum eddy diffusivity and the thermal eddy diffusiv-

ity. They are empirical correlations and must be determined experimentally. They are not flow

properties independent of the situation, for each flow will have a different value although some

may have similar results. Substitution of the new terms into the previous turbulent flow equations

yields:

(2.43)

(2.44)

Despite giving physical meaning to our two new terms, this method has not reduced the number

of unknowns that we have been given. The closure problem still remains with 5 unknowns and

only 3 equations in the turbulent boundary layer. However we can now use another model to esti-

mate the diffusivities. Before we move on let us define an apparent shear stress, app and an

apparent heat flux qapp. Both quantities will be important in later analysis.

u'v' Muy------=

cPv'T' cPHTy------=

ut------ u+

ux------ v

uy------+

1

---

dP

dx-------

y----- M+( )

uy------

+=

Tt------ u+

Tx------ v

Ty------+

y----- H+( )

Ty------

=


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26

(2.45)

. (2.46)

These two terms can be found explicitly in the momentum equation (2.43) and the energy equa-

tion (2.44).

2.5.1 Mixing Length Model

This boundary layer specific model is known as the mixing length model. Using scaling

laws and boundary layer assumptions, an approximation for M can be found in terms of the other

unknowns and the known constants. Details can be found in Bejan [1]. The expression for the

mixing length model is:

. (2.47)

is an empirical constant that must be found by experiment. This is a minor complication in that

solving these equations still relies on some experimentation, but this is the best way to deal with

the closure problem. Other uses of scaling in the problem can advance the theoretical equation

further, but the mass and momentum equations can now be integrated numerically as a two equa-

tion, two unknown system. The reader is referred to books such as Bejan for further details.

Further analysis of the boundary layer makes use of the assumption that the left hand side

of the momentum can be neglected in the region near the wall due to small velocities and .

Let us assume that in an area near the wall, the velocities are small enough as to make the left side

of Eqn. 2.43 negligible, a reasonable assumption based on the no slip condition and wall impene-

app M+( )uy------=

q''app cp H+( )Ty------=

M 2y

2 uy------=

u v


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27

trability. The apparent shear stress,app, is then constant in this small region near the wall. Let us

redefine this constant as:

, (2.48)

with o being the actual shear stress at the wall. The friction velocity u is defined as:

. (2.49)

which is dependent only on x-position in the flow, and can often be taken as a constant every-

where in the flow. It becomes necessary at this point to introduce wall coordinate notation. Wall

coordinates are non-dimensional variables that are often used in boundary layers in order to uni-

versalize equations. They are as follows:

(2.50)

(2.51)

(2.52)

, (2.53)

(2.54)

Inserting the wall coordinates into Eqn. 2.48, the following expression is derived:

, (2.55)

M

+( )u

y------

o

-----=

uo-----

1

2---

=

u+ u

u-----=

v+ v

u

-----=

x+ xu

--------=

y+ yu

--------=

1M

------+ du

+

dy+

--------- 1=
http://-/?-http://-/?-


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28

One can envision two different situations that factor into the result of this equation. The first is

the region immediately bordering the wall in which >>M. This region is known as the viscous

sublayer where Eqn. 2.55 reduces to

(2.56)

Upon integration with the initial condition u+(0)=0, the velocity distribution for the viscous sub-

layer reads:

. (2.57)

The other situation that arises is one in which the momentum diffusivity dominates the viscosity,

that is M>>. In this case Eqn. 2.55 reduces to

. (2.58)

The mixing length model transforms this equation to

(2.59)

Integration of this equation from the outer point of the viscous sublayer region where y+= y+VSL,

and from Eqn.2.57 that says u+= y+VSL, the velocity distribution in this region is

, (2.60)

where experimentally , the von Karmann constant has been determined to be approximately 0.41

and B has been found to be approximately 5.5. This is known the law of the wall and extends

from the law of the wall out through the inner region of the boundary layer. The place where the

flow begins to deviate from the law of the wall is flow dependent. Other researchers have found

du+

dy+

--------- 1=

u+

y+

=

M

------du

+

dy+

--------- 1=

2 y+( )2 du+

dy+

---------

2

1=

u+ 1

--- y

+B+ln=


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29

different relations for the velocity distributions in the viscous sublayer and the inner region, but

this is the most common representation.

2.5.2 Solving the Energy Equation

Since the mass and momentum equations have been solved, we are left with one equation

and two variables: T and H. The solution of this equation is an extension of the mixing length

model used for the momentum diffusivity. In the same way that we neglected the left hand side

of the momentum equation, let us neglect the left hand side of the energy equation as a result of

the small velocities near the wall. That is equivalent to saying that the apparent heat flux is not

dependent on the distance from the wall as long as the region is sufficiently close to the surface.

To put it another way,

(2.61)

Insertion of the wall coordinates into Eqn. 2.61 leads to the following expression:

. (2.62)

From this expression we can get the proper non-dimensionalization of the Temperature T in +

units:

. (2.63)

Placing T+ into Eqn. 2.62 then gives the integral expression:

. (2.64)

H+( )Ty------

q''o

cP----------=

cPuq''o

-------------- y+-------- T To( ) 1 H +

------------------------------=

T+

x+y

+( , ) To T( )

cPuq''o

--------------=

T+ y

+d

1 Pr 1 Prt( ) M ( )+----------------------------------------------------------

0

y+

=


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30

Unfortunately we still must deal with the M and Prt terms in order to make this equation solvable

analytically. The assumptions that we used to find two different situations for Eqn. 2.55 were that

in a region close to the wall viscous effects would dominate, and as one moved away from the

wall the momentum diffusivity would grow proportionately. Using these same assumptions, and

dividing the thermal boundary layer into two regions, one can see that the two terms in the

denominator of Eqn. 2.64 will dominate in different regions. When viscous effects are large, the

first term will dominate, making the second term negligible, and when momentum effects domi-

nate further away from the wall, the first term will be negligible. Instead of the viscous sublayer,

let us refer to this sublayer as a conductive sublayer, with its outermost point denoted by y+CSL.

Introducing these terms and assumptions we then can transform the integral to

(2.65)

Integrating this equations gives the following piece-wise solution

and (2.66)

This solution relies on knowing three experimentally determined constants. We have already

given a typical value for = 0.41. The outer point of the conduction sublayer is typically valued

around y+ = 13.2. The final value is the turbulent Prandtl number which had been previously

defined as the ratio of the molecular diffusivity to the heat diffusivity. Studies have indicated that

the turbulent Prandtl number starts high near the wall and levels out around 0.85 as it progresses

away from the wall. One theory, the details of which will not be presented here is called the Rey-

nolds analogy. It states that the rate of thermal diffusivity is equal to that of the heat diffusivity.

T+ y

+d

1 Pr-------------

y+

d

1

Prt-------

M

------

---------------

yCS L+

y+

+0

yCS L+

=

T+

Pr y+

= y+

yCSL+< T+ Pr y CSL

+ Prt

-------

y+

yCSL+

-----------ln+= y+

yCSL+>


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31

Although this is a somewhat reasonable assumption, it is not entirely accurate throughout the

flow. For this study we will use a value of Prt = 0.9. The solution of the equation in the inner

region of the boundary layer (y+ > y+CSL) using these values is then

(2.67)

These solutions are valid as long as our previous assumptions of qapp and app remain constant

or reasonably so.

2.5.3 Wall Heat Flux

As long as this solution holds throughout the boundary layer, an expression for the wall

heat flux can be developed. Using the boundary condition that is equal to T at y = and

expanding some of the wall coordinates in Eqn. 2.66 gives the following expression:

. (2.68)

The thickness of the boundary layer, , is still unknown, but let us use the law of the wall to obtain

the expression for the velocity at the edge of the boundary layer:

(2.69)

Also let us define the following quantity:

(2.70)

where Cf,x is the coefficient of friction and is a function of x in the same way that the friction

velocity u is. Combing the three previous equations we obtain the following expression:

T+

2.195 y+

13.2Pr 5.66+ln=

T

cPuTo T

qo------------------ Pr yCSL

+ Prt

-------

u

yCSL+

---------------

ln+=

U

u-------

1

---

u

-------- B+ln=

Cf x,

2u

U-------

2

=


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32

(2.71)

The left side of the equation is an important quantity known as the local Stanton number. In rela-

tion to other dimensionless constants it is defined as:

. (2.72)

Eqn. 2.71 has a denominator of O(1) and is Prandtl number dependent, when experimentally

determined constants are substituted in the equation. Colburn simplified the situation further by

suggesting the following formula

. (2.73)

Experimental data has been found to confirm this relation for Prandtl numbers between 0.6 and

60. This formula can be rearranged using the previous relations to find that

(2.74)

. (2.75)

Other models for the velocity distribution can produce estimates and models for C f,x in these

equations. One example is the 1/7 power law developed by Prandtl. Further details can be found

in sources such as Bejan.

2.6 Other External Flows

Not all flows are best modeled by a flow over a flat plate. Other simple geometries have

their own heat transfer correlations that are better suited for that individual situation. These rela-

h

cPU-----------------

1

2---Cf x,

Prt1

2---Cf x,

1 2

Pr yCSL+

BPrtPrt

------- yCSL

+ln

+

-----------------------------------------------------------------------------------------------------------------=

Stxh

cPU-----------------

Nux

Pex---------

Nux

RexPr---------------= = =

StxPr2 3 1

2---Cf x,=

Nux1

2---Cf x, RexPr1 3

= Pr 0.5( )

NuL1

2---Cf 0-x, ReLPr

1 3= Pr 0.5( )


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33

tions will not be presented in detail here, but the important relations will be given. They have

been developed in a method similar to the flow over a flat plate, though portions of each equation

rely on empirical data.

2.6.1 Cylinder in Cross Flow

. (2.76)

This formula is valid as long as PeD is greater than 0.2 and was developed by Churchill and Bern-

stein and listed in Bejan. If PeD is less than 0.2 and the substance is air then the following rela-

tion, credited to Nakai and Okazaki and listed in Bejan, is more accurate

(2.77)

2.6.2 Sphere

Flow over a sphere has the following Nusselt number correlation:

. (2.78)

This relationship has been confirmed to be accurate for fluids with 0.71 < Pr < 380 and flows with

3.5 < ReD < 7.6 x 104. The equation was developed by Whitaker and listed in Bejan.

2.6.3 General Spheroid Body Shapes

Yovanovich, referenced in Bejan, is credited with developing a formula that is applicable

to a general spheroid geometry. The semiaxis that is aligned with the freestream is C and the

other semiaxis is B. Representing the area of the objects surface as A, the length scale is written

as

NuD 0.30.62ReD

1 2Pr

1 3

1 0.4 Pr( )2 3+[ ]-------------------------------------------- 1

ReD

282000------------------

5 8

+4 5

+=

NuD1

0.8237 0.5 PeDln----------------------------------------------=

NuD 2 0.4ReD1 2

0.06ReD2 3

+( )Pr0.4w------

1 4

+=


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34

. (2.79)

The Reynolds number and Nusselt Number are then represented as

and (2.80)

The Nusselt number equation, accurate for the ranges 0 < Re 0.7, and C/B < 5, is

then represented by

. (2.81)

where p is the maximum perimeter of the object and is found from Table 2.1.

2.6.4 Array of Cylinders

A group of cylinders is usually positioned in an array of staggered cylinders or in an array

of aligned cylinders. Each arrangements has its own set of Reynolds number dependent Nusselt

Object

Sphere 3.545

Bi-Sphere 3.475

Cube 3.388

Cylinder 3.444

Prolate Spheroid(C/B = 1.93

3.566

Oblate Spheroid

(C/B = 0.5)

3.529

Oblate Spheroid

(C/B = 0.1)

3.342

Table 2.1: Constants for General Shape Heat Correlation

A1 2

=

Re

U

------------= Nuhk-------=

Nu Nu0 0.15

p

----

1 2

Re1 2

0.35Re0.566

+ Pr1 3

+=

u0

u0


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36

2.7 Internal flow

Convection within a duct or pipe is very similar to boundary layer flow, mostly because it

begins as a boundary layer flow. The most practical application for turbulent duct flow is flow

within a circular pipe (see Fig. 1.2). Although ducts sometimes shapes other than circles for their

cross sectional areas, most real world fluid flows occur in circular pipes. This means changing

coordinates to cylindrical and making the same boundary layer assumptions that we made in Sec-

tion 2.5. The mass, momentum, and energy equations read as follows:

(2.84)

(2.85)

. (2.86)

Until the boundary layers merge and the flow becomes fully developed, internal flow is no differ-

ent than internal boundary layer flow. The Eqns. 2.84-2.86 are no different then Eqns.2.36-2.38

Figure 2.2: Cn for an Array of Cylinders

ux------

1

r---

r----- rv( )+ 0=

uux------ v

ur------+

1

---

dP

dx-------

1

r---

r----- r M+( )

ur------+=

uTx------ vT

r------+ 1

r---

r----- H+( )

Tr------=


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37

except that they are in different coordinate systems. Fully developed flow occurs at approxi-

mately

. (2.87)

The entrance length for turbulent flow is shorter than the entrance length in laminar flow. In fully

developed flow, all of the inertia terms are neglected. This means that all terms on the left hand

side of the momentum equation (2.85) vanish. There is also no room in the center for a wake

region. Let us now consider something called the control volume force balance that reads:

. (2.88)

Considering this equation and integrating Eqn. 2.85, the following relation is found:

(2.89)

where the y coordinate is defined as the distance away from the wall, y = r o- r. The apparent shear

stress is redefined slightly as

. (2.90)

The law of the wall is the same as it is in external flow, but it breaks down near the centerline in

fully developed flow. Unfortunately these equations produce a velocity profile that has a finite

slope at the channel centerline, when it should be zero for a continuity. An empirical relation that

does produce a zero slope was developed by Reichardt

. (2.91)

2.7.1 Heat Transfer

X

D

---- 10

dP

dx------- 2

oro-----=

appo

---------- 1y

ro----=

app M+( )uy------=

u+

2.53 1 r ro+( )

2 1 2 r ro( )+[ ]------------------------------------y

+5.5+

ln=


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38

The momentum equation is rewritten for fully developed flow as

(2.92)

Integrate this equation two different ways to get:

(2.93)

(2.94)

Dividing these two equations and defining the term M to be

(2.95)

gives the following relation for the apparent heat flux

. (2.96)

In most practical cases, M is equal to 1, so let us assume that quantity for now. The following

analysis was developed by Prandtl and recounted in Bejan. Let us divide Eqn. 2.89 by Eqn. 2.96,

assume that M = 1, and substitute the values definitions for the apparent shear stress and the

apparent heat flux to get:

(2.97)

When we examined boundary layer flow, we noted that there was a region in which the viscosity

and conduction dominate in the area near the wall. Outside of that area, the momentum and heat

eddy diffusivities are dominant. In the same way let us imagine that region in the circular pipe as

cPuTx------

1

r---

r----- rqapp( )=

cPuTx------r rd

0

r

rqapp=

cPuTx------r rd

0

r0

rq0=

M

2

r2

---- uTx------r rd

0

r

2

r02

---- uTx------r rd

0

r0

------------------------------=

qapp

q0------------- M 1

y

r0----

=

M+0----------------du cP H+( )q0

--------------------------- dT=


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40

Other authors have improved this formula by being more specialized. Dittus and Boelter pro-

posed

(2.103)

where n is 0.3 if the fluid is cooled by the wall and 0.4 if the wall is heating the fluid. If the fluid

properties are heavily influenced by the temperature, Sieder and Tate proposed using:

(2.104)

Gnielinskis developed a correlation that is accurate within 10%:

(2.105)

Eqns. 2.102 - 2.105 are valid for constant wall heat flux and constant wall temperature. Eqn.

2.105 can be simplified to the following two equations:

(2.106)

(2.107)

All of the above equations are valid only for liquids and gases. One must find different correla-

tions for liquid metals. They will not be covered here.

The total heat transfer rate in internal flow is defined as:

(2.108)

NuD 0.023ReD4 5 Prn=

0.7 Pr 120

2500 ReD 1.24 105( )L D 60>

NuD 0.027ReD4 5

Pr1 3

0-----

0.14=0.7 Pr 16700

NuD

f 2( ) ReD 103

( )Pr

1 12.7 f 2( )1 2 Pr2 3 1( )+---------------------------------------------------------------------=

0.5 Pr 106


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41

where Aw is the total wall area, perimeter times length (Aw = pL). The term must be found

to make this analysis complete. Let us define the terms:

. (2.109)

Details of this analysis will be skipped, but the result for an isothermal wall is:

. (2.110)

For a wall that provides a uniform heat flux, the solution is:

(2.111)

Tlm

Tin T0 Tin= and Tout T0 Tin=

TlmTin Tout

TinTout-------------

ln

-------------------------------=

Tlm

Tin

Tout

= =


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42

CHAPTER 3

NATURAL AND MIXED CONVECTION

Natural Convection is governed by the same general principles as forced convection does.

However it must be treated slightly differently than forced convection because there is no external

flow to carry heat away from a surface. Instead, natural convection induces flow in a fluid by

heating a portion of that fluid. The heated fluid expands, becoming less dense than the cooler air

around it. The heated air rises due to buoyancy effects, and the cooler air descends to replace it.

Now if the surface that is heating the fluid is horizontal for example, the cold air will then heat up

and the hot air cool, and the flow will continue. Another area of question is whether this flow is

laminar or turbulent. Typically low speed flow is considered laminar. In this case however, there

is little predictable about the flow. Turbulent flow mist be considered as well, especially when it

combines with forced convection to form mixed convection. This chapter will primarily focus on

laminar natural convection since it is the most common due to the low induced speed. This chap-

ter serves mostly to give the reader an idea of natural convection so that the subsequent chapters

can be properly understood. As such it is not of primary importance to learn about turbulent nat-

ural convection.

3.1 Equations

The objective of natural convection heat transfer problems is to solve the equation:


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43

, (3.1)

that is to solve for the total heat transfer when the wall and fluid temperatures are known. HW is

the wall area. The conservation equations that are used for heat transfer begin with the equations

that are found in Chapter 1 (1.4-1.6). The equations are expanded and slightly modified as fol-

lows:

(3.2)

(3.3)

(3.4)

(3.5)

The only major difference is that the equations have been simplified to two dimensions only, and

the body force has been modified. The x-momentum equation neglects the body force while the

y-momentum equation retains the term and defines it as g, the gravity force. As was mentioned

above in the description of natural convection, the gravity is the driving force behind the motion.

Much like forced convection, boundary layer assumptions can be made for natural convection to

simplify the problem. In previous boundary layer assumptions, the motion was horizontal. The

motion in natural convection is vertical. Instead of neglecting the y-momentum equation, the x-

momentum equation becomes negligible because of the minimal movement in the x-direction.

The boundary layer assumptions we used for forced convection are then applied to natural con-

vection, except now the flow is in the vertical direction. Consider that the scale of the boundary

layer, or T are much smaller than the vertical scale H. This allows us to neglect the

Q HW( )h0 h T0 T( )=

ux------

vy-----+ 0=

uu

x------ v

uy------+

Px------ 2u+=

uvx----- v

vy-----+

Py------

2u g+=

uTx------ v

Ty------+

2T=

2

y2


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44

term in the momentum and energy equation due to scaling laws that were developed previously.

The velocity boundary layer in this case does not vary from 0 at the wall to some freestream

velocity. It starts at 0 at the wall and then rises to reach a maximum before falling to zero again at

the outer edge of the boundary layer. The final general equations for natural convection are then

reduced to:

(3.6)

(3.7)

(3.8)

Note that the variables still remain general and have not been converted to their turbulent forms.

3.2 Boussinesq Approximation

One common approach to natural convection is to simplify the density term. This simpli-

fication is called the Boussinesq approximation. The y-momentum equation (3.4) can be simpli-

fied by removing the variable , the density. In a boundary layer, the pressure is effectively a a

function of the primary direction of motion only. We can write

. (3.9)

Also a fluid of density has a hydrostatic pressure gradient

. (3.10)

This eliminates all pressure terms from the equation and allows us to work with only density.

This may not seem like it has simplified the problem much, but the Boussinesq approximation

ux------

vy-----+ 0=

uv

x----- v

vy-----+

Py------

2v

x2

-------- g+=

uTx------ v

Ty------+

2T

x2---------=

Py------

dP

dy-------

dP

dy----------= =

dP

dy---------- g=


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45

allows a simplification of the equations further. It allows us to rewrite the momentum equation

as:

. (3.11)

Remembering the ideal gas law,

, (3.12)

we can rewrite the density terms in Eqn. 3.11 as:

, (3.13)

which can be restated as

. (3.14)

In the limit that (T-T)


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46

3.3 Natural Convection with a Vertical Wall

The flow induced by natural convection is considered to be laminar as long as the wall

height is not great enough to make the Rayleigh number surpass a critical value. The Rayleigh

number is defined as:

. (3.18)

The critical Ray is dependent on the Prandtl number of the fluid. Defining the Grashof number as

being Ray/Pr, the Grashof number (Gry) must not exceed 109 as long as the Prandtl number is

between 10-3 and 103. The Grashof number is defined as:

. (3.19)

Beyond this point the flow will become turbulent.

Scaling analysis (details can be found in Bejan [1]) reveals that natural convection is influ-

enced by three terms: inertia, friction and buoyancy. In high Prandtl number flow, the boundary

layer is governed by a friction-buoyancy balance. In low Prandtl number fluids, the boundary

layer will feature buoyancy balanced by inertia.

High Pr fluids have a different set of relations that characterize their flow. For a fluid with

a high Pr,

(3.20)

(3.21)

(3.22)

RaygTy3

-------------------=

GrygTy

3

2-------------------

RaH

Pr----------= =

T HRaH1 4

v

H----RaH

1 2

HRaH1 4

Pr1 2


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47

(3.23)

The velocity boundary layer is larger than the temperature boundary layer because the moving

fluid will drag some fluid with it outside of the region with a nonzero temperature gradient. So

although the temperature gradient is the driving force behind natural convection, friction effects

will cause fluid flow beyond the area where there is a temperature difference. The maximum

velocity is indicated by the term v, which is also the velocity scale. The boundary layer in high Pr

fluids is dominated by friction and buoyancy in the area up until the velocity peak, also indicated

by the edge of the thermal boundary layer. Outside of this region, until the end of the boundary

layer, the flow is driven by a friction and inertia balance.

For low Pr fluids, the resulting balances of the conservation equations lead to the follow-

ing observations

(3.24)

(3.25)

(3.26)

(3.27)

The term is the viscous sublayer where friction is important, and it is the area that exists in low

Prandtl number fluids where the velocity has not yet reached its maximum. It is smaller than both

the thermal and velocity boundary layers. Low Pr numbers also have a boundary layer divided

into two subsections. Near the wall, in the boundary layer v the flow is governed by a friction

buoyancy balance. Outside of this layer, until the edge of the thermal boundary layer, the flow is

NuhH

k------- RaH

1 4=

T H RaHPr( )1 4

H

---- RaHPr( )1 2

HGrH1 4

NuhH

k------- RaHPr( )

1 4=


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48

dominated by an inertia-buoyancy force balance. Since the quantity RaHPr seems to be prevalent

in this analysis, let us define the Boussinesq number:

. (3.28)

The dimensionless quantities are very useful in giving us an idea of how thick the boundary layer

is. The following three relations show an interesting effect of these numbers:

(3.29)

These numbers show just how slender the boundary layer is in relation to the total wall height.

Churchill and Chu developed an empirical relation for the Nusselt number if the wall is

isothermal. It holds for all Prandtl numbers and 10-1 < Ray < 1012.

. (3.30)

All fluid properties are evaluated at the film temperature, defined as (Tw + T)/2.

The case of a uniform heat flux led Vliet and Liu to produce the following relations for

turbulent flow:

. (3.31)

This equation is valid for 1013 < Ra*y < 1016. Ra*y is a Rayleigh number based on the heat flux:

BoH RaHPr

gTH3

2---------------------= =

wall height

thermal boundary layer thickness------------------------------------------------------------------------------- Ra

1 4 Pr 1>

wall height

thermal boundary layer thickness------------------------------------------------------------------------------- BoH

1 4 Pr 1

turbulent heat transfer

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