turbulent heat transfer
TRANSCRIPT
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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=
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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
------
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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---=
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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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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
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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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. (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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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=
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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'
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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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, (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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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
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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'
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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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(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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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=
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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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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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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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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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(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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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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. (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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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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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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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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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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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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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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, (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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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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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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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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(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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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