Chapter_1.pdf

Chapter 1

INTRODUCTION

1.1 Generalities

The subject of turbulent shear flow is not simply connected. Some

organization is essential, and I have tried to arrange that material

required in a particular discussion appears earlier in the text. Thus

pipe flow is discussed first because it provides the best evidence for

the existence and value of Karman’s two constants in Prandtl’s law

of the wall. It might seem easier to begin with the simpler topic

of free shear flows, such as the plane jet. However, there is then a

difficulty with the natural progression to wall jets, impinging jets,

and other topics that require experience with Karman’s constants.

The main advantage of pipe flow is that the magnitude of the wall

friction in fully developed flow can be obtained unambiguously from

the pressure gradient, although a preliminary study is needed to

determine what conditions guarantee that a given pipe flow is axially

symmetric and fully developed.

It is an accepted axiom in basic research on the classical turbu-

lent shear flows, elegantly expressed by NARASIMHA (1984), that

there exists in each case a unique equilibrium state that can be re-

alized in different experiments and thus made the basis of a general

synthesis of empirical knowledge. The equilibrium may be station-

ary (as in pipe or channel flow) or dynamic or developing (as in the

CHAPTER 1.

INTRODUCTION

plane jet, the boundary layer, and most other flows). This axiom

will be tested repeatedly in various parts of this monograph, usually

by an emphasis on similarity laws, an emphasis that generates its

own problems. There are no widely accepted similarity laws for sev-

eral important turbulent flows, including boundary-layer flows with

compressibility, mass transfer, heat transfer, or lateral or longitudi-

nal curvature. Other difficult cases are the wall jet and the three-

dimensional boundary layer. Because most of these problems involve

walls, some attention is paid to the issue of the behavior of various

mean quantities in turbulent flow near a wall. For example, there

was at one time some ambiguity in the literature about the lead-

ing term in the expansion for the Reynolds shearing stress

−

′

This leading term has sometimes been identified as a term in

, and

sometimes as a term in

(see HINZE 19XX, p. 621; SPALDING

1981). In fact, a rigorous argument is analytically straightforward,

and generalizations that take into account changes in coordinates and

boundary conditions may eventually shed some light on the nature of

the proper similarity variables for some of the flows just mentioned.

Experimental evidence for the magnitude of the first few coefficients

is unreliable, but these coefficients can sometimes be estimated from

numerical work on solutions of the full Navier-Stokes equations for

flow in channels and boundary layers. There is a clear and present

need for detail here when choosing boundary conditions for large-

eddy simulations.

It is remarkable that one of the ostensibly most difficult prob-

lems in fluid mechanics, the problem of surface roughness, should

be in a relatively comfortable state. Missing are sound methods for

characterizing roughness. It is also remarkable that an apparently

unrelated problem, the flow of a dilute solution of a high-molecular-

weight polymer, exhibits properties that might be more easily un-

derstood if there were such a thing as negative roughness. Although

expectations are not high, the prospect of finding possible connec-

tions in the transport mechanisms for these two problems is certainly

worth some effort.

A topic that involves the mechanisms of turbulence in an es-

sential way is the problem of relaxation, especially from one classical

1.1. GENERALITIES

flow to another. One example is the strong plane jet into a moving

fluid (

∼

), with a final state describable by a linearized analysis

(

∼

). The round jet into a moving fluid has an equivalent

behavior. Another example is the turbulent boundary layer on a fi-

nite flat plate (

∼

) which relaxes downstream from the trailing

edge to a plane wake (

∼

). Several experimental studies exist

of pipe, channel, or boundary-layer flows during a smooth-rough or

rough-smooth transition of the wall boundary condition. Another

instructive case is relaxation of a rectangular wake or jet to a round

one. Such flows often overshoot the final state at least once. Finally,

there are several flows with initially variable density that relax to-

ward constant density as mixing proceeds. These include jets into

a different medium, as well as plumes with finite initial momentum.

A global view of these problems may lead to useful inferences about

characteristic scales in time or space. The list of issues mentioned

here is not intended to be comprehensive, but only to suggest various

approaches that may or may not be productive in the future, given

the fact that insight cannot be programmed.

It is also relevant that papers on models and mechanisms of

turbulence tend to cite a limited standard set of experimental papers

(these papers are sometimes different from one discipline to another).

Endorsement by repetition often fails to present the best evidence.

A positive development is that survey papers on various topics in

turbulent shear flow are an increasingly important component of the

contemporary literature. In a smaller setting, such surveys are also

a common ingredient in Ph.D. theses although most of these latter

surveys are not as critical as they could be. In any event, all of this

material is a valuable resource for this monograph.

Sources of information about turbulent flow exist on several

levels. In decreasing order of authority, I distinguish

1. the laws of mechanics

2. expert measurements (or numerical simulations)

3. insight

4. peripheral vision

5. brute force.

CHAPTER 1.

INTRODUCTION

In what follows I will emphasize the first three levels of information.

By “the laws of mechanics” I mean the Navier-Stokes equations and

their boundary conditions. The question of “expert measurements”

is more subjective. In any survey of experimental data, it is nec-

essary somehow to assign a degree of confidence to each particular

measurement. If the measurement has been made many times by dif-

ferent observers, like the pressure drop in a pipe, this is fairly easy to

do. But if the measurement has been made only a few times, or even

only once, judgment has to be backed by experience. “Insight” tends

to be rare. Instances occur at intervals typically measured in years.

The problem is then to pick signal out of noise in the literature, and

the turbulence community is reasonably effective at doing so. An

example of “peripheral vision” is the use of power laws that are not

intrinsic, such as in Bradshaw’s treatment of equilibrium turbulent

boundary layers. What I hope to establish in this book is a set of

ideas, chosen according to criteria defined by evidence rather than

by faith or tradition. I will try to avoid the last level, “brute force,”

except when I am obliged to live up to the claim that part of my

purpose is to deal with technical problems.

Two constraints dominate the whole subject of turbulent shear

flow at the contemporary stage of development. One is the boundary-

layer approximation and the other is the idea of Reynolds stresses. I

will almost always be considering incompressible fluids in the sense

that variations in density are caused by variations in temperature,

not by high velocity, and are important mainly because of associated

body forces in a gravitational field.

The following two paragraphs appeared at this point in the 1997 draft of this

work:

The notation of this monograph tends toward usage in aeronautical engineering.

I have tried to use mnemonic notation where this is possible, and I have therefore

avoided arbitrary use of Greek symbols except where these are firmly established

in the literature.

The literature in Russian is not well represented, primarily because the text is

usually terse, the figures small, and tables nonexistent. The Russian subliterature

— institute reports and theses — is not accessible at all.

1.2. ANALYTICAL PROLOGUE

1.2 Analytical prologue

1.2.1

Definitions and identities

Effects of compressibility are deliberately not emphasized in this

monograph. However, effects of buoyant body forces and of heat

transfer at walls are considered, so that the density of the fluid can-

not always be taken as constant. In particular, I want to comment

on something called the BOUSSINESQ approximation (1903), which

I believe is not always well presented in the engineering literature. I

will therefore outline briefly the structure of the Navier-Stokes equa-

tions for a compressible fluid and consider the limiting form of these

equations, first as the Mach number approaches zero, and then as

the Froude number approaches zero.

The first part of the discussion, and the notation, are taken

from the classical article by LAGERSTROM (1964, 1996) in Vol-

ume IV of the Princeton handbook series. Many details are omitted

here that can be found in Lagerstrom’s article. The main reason for

this choice of model is that the compact notation of vector calculus,

with an appropriate generalization to operations on tensors, allows

easy manipulations whose results are independent of any particular

coordinate system.

A number of definitions and identities will be used in this and

later sections of this monograph. In what follows an arrow over a

symbol indicates a vector, and an underline indicates a tensor.

Some definitions from vector geometry and vector calculus,

with generalizations, include the divergence of a vector,

∫ ∫ ∫

div

∫ ∫

S ,

(1.1)

where

is a stationary control volume bounded by a surface

, and

is the unit outward normal.

The divergence of a tensor is defined similarly,

∫ ∫ ∫

div

∫ ∫

S ,

(1.2)

CHAPTER 1.

INTRODUCTION

where

is a multiplication defined by (

)

The gradient of a scalar is

grad

= d

(1.3)

and of a vector is

(grad

) d

= d

~a .

(1.4)

The dyadic product of two vectors is

(

◦

)

(

)

arbitrary

(1.5)

The deformation tensor is symmetric;

def

= grad

+ (grad

)

∗

(1.6)

where

∗

means the transpose. The corresponding antisymmetric ten-

sor defines the curl operator;

(curl

)

[

grad

−

(grad

)

∗

]

arbitrary

(1.7)

The substantial derivative of a scalar is

Dα

∂α

∂t

+ grad

(1.8)

and of a vector is

D~a

∂~a

∂t

+ (grad

)

~u .

(1.9)

The identity tensor

is defined by

(1.10)

and the scalar product of two tensors by

∑

i,j

(1.11)

Various identities are also useful. These are written here in a form

independent of the choice of coordinates, and are easily verified in

any convenient orthogonal coordinate system, say rectangular;

div (

α~a

) =

div

grad

;

(1.12)

1.2. ANALYTICAL PROLOGUE

div (

αI

) = grad

;

(1.13)

div curl

= 0 ;

(1.14)

div (

◦

) = (grad

)

+ (div

)

;

(1.15)

div (

) = (div

∗

)

∗

grad

;

(1.16)

div (

) =

curl

−

curl

;

(1.17)

div (grad

)

∗

= grad (div

) ;

(1.18)

curl (grad

) = 0 ;

(1.19)

curl (

α~a

) =

curl

+ (grad

)

;

(1.20)

curl (

) = div

[

(

◦

)

−

(

◦

)

]

;

(1.21)

(

)

= (

)

~a , A

symmetric ;

(1.22)

(

) =

(

)

−

(

) ;

(1.23)

( grad

)

= grad

+ (curl

)

~a , α

~a .

(1.24)

The three relations (1.19), (1.14), and (1.1) might be considered the

basis of a

carpe diem

school of mechanics. See a gradient, take the

curl. See a curl, take the divergence. See a divergence, integrate over

a control volume.

1.2.2

Equations of motion

Mass.

I have grown up with a derivation of the Navier-Stokes equa-

tions of motion using the device of a stationary control volume. Con-

servation of mass is expressed with complete clarity by the relation

∫ ∫ ∫

−

∫ ∫

ρ~u

S ,

(1.25)

where

is density,

is velocity, and d

and d

are elements of

volume and surface, respectively. The negative sign on the right-hand

side is required by the convention that

is the unit outward normal

to the surface of the control volume. No provision is ordinarily made

CHAPTER 1.

INTRODUCTION

for sources or sinks within the control volume. If these are needed,

they can be added at an appropriate later stage. With the aid of

the definition (1.1) for divergence, equation (1.25) can be rewritten

in terms of volume integrals only;

∫ ∫ ∫

(

∂ρ

∂t

+ div

ρ~u

)

= 0

(1.26)

Because the control volume is arbitrary, the integrand must be zero

everywhere. Conservation of mass therefore requires

∂ρ

∂t

+ div

ρ~u

= 0

(1.27)

A different form is obtained by use of the vector identity (1.12);

∂ρ

∂t

grad

div

= 0

(1.28)

The first two terms are now a rate of change of density following an

element of the fluid, already defined by equation (1.8). A final form

for the continuity equation is therefore

Dρ

div

= 0

(1.29)

Momentum.

Conservation of momentum is expressed in the

same simple terms by

∫ ∫ ∫

ρ~u

−

∫ ∫

ρ~u

(

) d

∫ ∫ ∫

∫ ∫

S .

(1.30)

The term on the left is the time rate of change of the vector momen-

tum within the control volume. The first term on the right is flux

of momentum through the boundary, with a negative sign for the

same reason given earlier. In this term the velocity

appears twice

in different roles. The first is as vector momentum per unit mass.

The second is as volume flux per unit area per unit time. I consider

the distinction to be important and will maintain it throughout this

1.2. ANALYTICAL PROLOGUE

monograph. The quantity

is an internal body force per unit mass,

usually due to gravity. The surface stress

is a tensor, or linear vec-

tor operator, with

the vector force per unit area on the boundary

of the control volume.

Two steps are required to obtain a differential equation. The

first step introduces the dyadic product of two vectors, (

◦

), de-

fined by equation (1.5) with

ρ~u

, and

. The second

step introduces the generalized divergence of a tensor by use of equa-

tion (1.2). With these relationships, equation (1.30) can be written

in terms of volume integrals only

;

∫ ∫ ∫

(

∂ρ~u

∂t

+ div

( ̃u

◦

̃u

)

−

div

)

= 0

(1.31)

Since the control volume is arbitrary, it follows that

∂ρ~u

∂t

+ div

( ̃u

◦

̃u

) =

+ div

(1.32)

everywhere.

This form for the transport terms is well suited for the intro-

duction of what are called Reynolds stresses in turbulent flow. It is

also the most useful form when the objective is to derive integral laws

(such as Karman’s momentum integral) from the differential equa-

tions, because the volume integral of a divergence usually begins life

as the surface integral of a flux. It is therefore often convenient to

return to the control volume for this operation.

Another form for the transport terms follows from the identity

(1.15) with

ρ~u

, so that

div

( ̃u

◦

̃u

) =

(grad

)

div

ρ~u .

(1.33)

Now the momentum equation (1.32) takes the form

∂~u

∂t

∂ρ

∂t

div

ρ~u

(grad

)

+ div

(1.34)

The tilde notation is not defined here but section 2.1.2 states

, ”[T]he tilde,

here and elsewhere, is intended as a mnemonic for an integral mean value.”

CHAPTER 1.

INTRODUCTION

The second and third terms drop out, by virtue of the continuity

equation (1.27), leaving

[

∂~u

∂t

+ (grad

)

]

+ div

(1.35)

The quantity in brackets on the left is the substantial derivative

(1.9) (the derivative following a fluid element) of the vector

, so

that finally

D~u

+ div

(1.36)

This equation may also have explicit source or sink terms for mo-

mentum, although these are usually left in implicit form. A notation

sometimes used in equation (1.36) is

D~u

∂~u

∂t

+ (

grad)

~u .

(1.37)

I think that this notation might be misleading in any coordinate

system except a rectangular one.

The body force

will normally be a gravity force,

−

where

is a unit vector directed vertically upward. It might also be a

local force expressed as a Dirac

-function to represent a concentrated

source or sink of momentum; e.g., thrust or drag. For a Newtonian

fluid, the tensor

is symmetric, isotropic, and linear in the spatial

first derivatives of the velocity. The most general form meeting these

conditions is (cite Jeffries, Stefan)

−

div

~uI

def

−

p I

(1.38)

where

is the identity tensor, defined by

, and where

div

~uI

def

(1.39)

The stress tensor

thus involves two viscosities,

and

, of which

the first is immaterial if div

= 0. In this formulation the three

state variables

are introduced in a single operation and are

not conceptually different.

1.2. ANALYTICAL PROLOGUE

With the aid of the identity (1.13) with

, the momentum

equation can be written finally as

D~u

−

grad

+ div

(1.40)

Energy.

A law for conservation of energy can be derived from

first principles by visualizing a molecular structure for the fluid, al-

though the result is often classified as part of continuum mechanics.

Hard spherical molecules in a state of agitation have two kinds of

energy: internal kinetic energy

, associated with random motion

and represented as temperature, and directed or organized motion

~u/

2 associated with bulk velocity. Energy can be added or sub-

tracted in the interior of a control volume by at least two processes.

One is heat release

per unit mass by chemical reactions, including

phase changes such as evaporation and condensation. The other is

work done by body forces

. Energy can also be transferred at

the boundary of a control volume by heat conduction

and by work

done by surface forces

. The latter two processes will appear as di-

vergence terms and are neutral in the interior of the control volume.

Thus write

∫ ∫ ∫

(

)

−

∫ ∫

(

)

∫ ∫ ∫

ρQ

∫ ∫ ∫

−

∫ ∫

(

)

(1.41)

where

. The three surface integrals can be converted to

volume integrals using equations (1.1) and (1.2), with the result,

after use of the identity (1.22) in the last term,

∂

∂t

(

)

+ div

(

)

ρQ

−

div

+ div(

)

(1.42)

CHAPTER 1.

INTRODUCTION

The first two terms can be modified by differentiating the second

term as a product and using the continuity equation (1.27). The last

term can be modified using equation (1.38) for

and the identity

(1.10). Finally, with the definition (1.8) for the derivative of a scalar

following a fluid element, there is obtained

(

)

ρQ

−

div

−

div

p~u

+ div

~u .

(1.43)

For a Fourierian fluid, the heat conduction vector is linear in the

spatial first derivatives of the temperature,

−

grad

T ,

(1.44)

with a scalar heat conductivity

. The negative sign indicates that

energy is transferred down the temperature gradient.

There is also available a mechanical energy equation, derived

independently of the thermodynamic equation (1.43) by taking the

scalar product of the momentum equation (1.40) with the vector

velocity

to obtain

−

grad

div

(1.45)

Equations (1.43) and (1.45) and an obvious formula for

(

p/ρ

)

/Dt

lead, with the aid of the identity (1.16), to the array of energy equa-

tions displayed in TABLE 1.1. The scalar product of two tensors is

defined by the identity

(summed over

and

The middle three equations, the last three, and the first three

(the equation for

is listed twice) form natural groups that describe

the evolution of the quantities

and

2 and their sum in equation

(1.43), and the evolution of the terms appearing in the definitions

for static and stagnation enthalpy,

(1.46)

and

(1.47)

1.2. ANALYTICAL PROLOGUE

CHAPTER 1.

INTRODUCTION

So far the nature of the fluid is left open. In this monograph

the fluid will be either an ordinary liquid or a thermally perfect gas

with an equation of state

ρRT .

(1.48)

The gas will also be assumed to be calorically perfect; that is, the

specific heats

and

will be taken as constant in the definitions

;

(1.49)

;

(1.50)

and in the combinations

−

, γ

(1.51)

Finally, the scalar quantities

, and

are state variables that can

be taken to depend on temperature only.

These full equations of motion for a compressible fluid are so

complex as to be intractable. Analytical progress toward their solu-

tion therefore tends to occur in small increments, in which the equa-

tions are truncated in various ways and solved for special classes of

problems. The simplest method of truncation is brute force. For

example, it can be stipulated that the density of a fluid is constant,

or that the viscosity and heat conductivity are zero, or that the flow

depends on only one space coordinate and time. More systematic

truncations can often be based on dimensional considerations. For

example, suppose that the ostensible data for a class of problems, in-

cluding boundary conditions and characteristic fluid properties, are

sufficient to define a complete set of global scales for length, velocity,

temperature, and so on. Then the equations of motion can imme-

diately be put in non-dimensional form. The relative magnitude of

various terms can be estimated, and certain terms can be discarded

as negligible, along with the physical processes that they model. A

limit process is often involved. If the limit is regular, suitable expan-

sions define themselves. However, the expansion procedure is usually

neither transparent nor trivial, so that it is best illustrated by a few

examples.

1.2. ANALYTICAL PROLOGUE

1.2.3

Incompressible fluids

A large part of fluid mechanics deals either with liquids or with gases

moving at low speeds, so that effects of compressibility are not im-

portant. The limit process that allows a gas to be treated as incom-

pressible was first accurately described by LAGERSTROM (1964) in

his handbook article on laminar flow. This limit process preserves

the mechanical role of the pressure in the momentum equation (1.40)

while suppressing the thermodynamic role of the pressure in the en-

ergy equation (1.43) and the state equation (1.48). The reasoning

here proceeds from the general to the particular, on the premise that

it is logically easier (and safer) to derive the correct equations for a

compressible fluid from first principles, and then to apply the correct

limit, than it is to go in the opposite direction. However, it is impor-

tant to keep in mind that the reasoning is also

ad hoc

, being strictly

valid in each instance only for a particular class of flows specified in

advance.

Consider the class of flows of a viscous perfect gas past a finite

body. Begin by converting the equations of motion to dimensionless

form. Assume that there is a constant reference length

in the prob-

lem, together with a constant reference velocity

and a reference

fluid state in which

have the values

(

for ambient,

usually at infinity), and similarly for

and

. The body force per

unit mass,

, is made dimensionless with

, the acceleration of grav-

ity. Nothing essential is lost if the flow is taken to be steady, with

= 0, and if

and

are both represented by

. With an over-

bar to indicate dimensionless variables like

~u/

and

ρ/ρ

(and, implicitly, dimensionless operators div, grad, and

D/Dt

) in

coordinates

~x/

, the equations of motion can be written

div

= 0 ;

(1.52)

−

γ M

grad

div

;

(1.53)

(

−

)

grad

−

div

−

grad

; (1.54)

CHAPTER 1.

INTRODUCTION

T .

(1.55)

Several dimensionless parameters materialize as coefficients of terms

on the right-hand sides. They are called, respectively, the Mach

number

, the Froude number

, the Reynolds number

, the

P ́eclet number

, and the ratio of specific heats

;

γ p

/ρ

;

(1.56)

;

(1.57)

;

(1.58)

;

(1.59)

(1.60)

The ratio

e/Re

, which compares the relative rates of diffu-

sion for heat and vorticity, is a derived parameter called the Prandtl

number

;

(1.61)

I was not aware, until some of my students pointed it out to me, that

the definition of Froude number in current textbooks and mono-

graphs on fluid mechanics is not uniform. When I made an infor-

mal survey of books within easy reach in my office, I found seven

authors, some of them very distinguished, who use the definition

. Eleven other authors, equally distinguished, use the

definition

. I will adopt the second definition, as in

equation (1.57) above; first, because it is the form originally pro-

posed by FROUDE (ref); second, because it is the form commonly

used by writers on topics that directly involve surface waves or buoy-

ancy forces; and third, and most important, because the symbol

then runs in parallel with the symbol

as the ratio of a fluid velocity

to a characteristic wave velocity.

1.2. ANALYTICAL PROLOGUE

It is not necessary to use or even to know the equations of mo-

tion in order to discover these five dimensionless parameters. The

Buckingham Π theorem (see SECTION X)

is sufficient, given the

presence of nine independent physical quantities in the problem, to-

gether with four independent physical units (mass, length, time, and

temperature). For each of the dimensionless parameters just listed,

and others to come, experience shows that much of fluid mechan-

ics and the associated applied mathematics is concentrated near the

three special values 0, 1,

∞

. Lagerstrom discusses several of these

special values at length, especially the difficult cases

→

0 and

→ ∞

. The case at hand, the case

→

0, is relatively straight-

forward because the perturbation is entirely regular.

For each class of problems, the global reference values used to

make the variables and operators dimensionless should be chosen in

such a way that the essential terms in the equations, weighted by

their dimensionless coefficients, are of order unity in the limit. In

the present instance of flow past a body, the transport terms on the

left in the momentum equation (1.53) and in the energy equation

(1.54) are essential by assumption. So is the pressure gradient term

in equation (1.53). All other terms can be left to follow these leaders.

This format is not universal. For example, the transport terms in

the momentum equation are not important in lubrication theory, and

different arguments are needed. Fortunately, in such cases the equa-

tions are usually capable of indicating the direction the argument

should take as well as the nature of higher approximations.

After these preliminaries, consider the limit

→

0 in equa-

tions (1.52)–(1.55). There is an obvious difficulty in the momentum

equation, where the pressure term blows up. This problem can be

avoided by making the pressure dimensionless with the dynamic pres-

sure rather than the static pressure. Instead of

(1.62)

define

γ M

(1.63)

Sections that discuss Buckingham

include 2.4.1, 9.1.2, and 11.1.1

CHAPTER 1.

INTRODUCTION

The difficulty in the momentum equation clears up, but a new prob-

lem appears in the state equation (1.55), which becomes

γ M

p ,

(1.64)

with a right-hand side that vanishes in the limit. This new prob-

lem is caused by the fact that

is defined only within an additive

constant in grad

, but is defined in an absolute sense in the state

equation. Both cases are accounted for if the dimensionless pres-

sure is expressed in the form usually called a pressure coefficient by

aeronautical and mechanical engineers. Thus define

−

γ M

−

γ M

−

(1.65)

With this change, the dimensionless equations become

div

= 0 ;

(1.66)

−

grad

div

;

(1.67)

= (

−

grad

−

div

+ (

−

grad

;

(1.68)

= 1 +

γ M

p .

(1.69)

Now in the limit

→

0 the flow of a viscous perfect gas about

a finite body is described by the equations

div

= 0 ;

(1.70)

−

grad

div

;

(1.71)

−

div

;

(1.72)

= 1

(1.73)

1.2. ANALYTICAL PROLOGUE

In particular, the pressure-work term and the dissipation term drop

out of the energy equation (1.68), leaving only conduction to balance

transport of heat. Restored to dimensional form, the equations for

→

0 are

Dρ

div

= 0 ;

(1.74)

D~u

−

grad

+ div

;

(1.75)

−

div

;

(1.76)

ρT

(1.77)

The limit

→

0 can also be approached more directly. For a

perfect gas, the definition of stagnation enthalpy,

(1.78)

and the definition of local Mach number,

ρu

γp

(1.79)

imply

= 1 +

−

(1.80)

Now suppose that the flow is steady and isentropic (inviscid, adia-

batic), so that

(

)

−

(1.81)

Then

ρu

(

−

)





(

)

−





(1.82)

Finally, suppose that

p/p

is nearly unity. Put

= 1 +

and

take the first term in an expansion in powers of

of the quantity in

brackets in equation (1.82);

ρu

(

−

) [

−

]

p ,

(1.83)

CHAPTER 1.

INTRODUCTION

M=oo

o /

Figure 1.1:    The Bernoulli integral for an

incompressible fluid and for a perfect gas with

= 1

ρu

−

p .

(1.84)

Thus the low-speed Bernoulli integral is recovered in the limit

→

Comparison of equations (1.79) and (1.83) shows that

γM

so that this is also the limit

→

0. It is equation (1.82) and not

equation (1.78) that should be referred to as the Bernoulli integral for

a compressible perfect gas. To illustrate this limit graphically, the

relationship between

p/p

and

ρu

from equations (1.82) and

(1.84) is shown in FIGURE 1.1 for

= 7

5. The absence of a lower

bound for

in an incompressible fluid is evident, as is the basis of

1.2. ANALYTICAL PROLOGUE

an expansion for

p/p

near unity.

Throughout this discussion of the limit

→

0, the main issue

is the behavior of the variable called the pressure. In its thermody-

namic role, the pressure must be non-negative. In its dynamic role,

there is no lower limit for the pressure. For an incompressible fluid

it can go to negative infinity, according to the Bernoulli equation

(1.84), when the velocity

goes to positive infinity, as for potential

flow at a sharp external corner or for flow at a source or sink. The

issue is resolved formally by measuring the magnitude of the pressure

in both of its roles from a local reference value, here called

, and re-

quiring changes in

to be small when compared with

but of order

unity when compared with

. It should not be surprising, with

the pressure almost constant, that the only thermodynamic process

that leaves any residue in the limit is the process at constant pres-

sure, as represented by the enthalpy

. The specific heat at

constant volume

has dropped out, along with the internal energy

and the state constants

and

−

The argument just presented does not require the density

be constant. It does require changes in density to be associated with

changes in temperature, not with compressibility. If the temperature

is constant, the energy equation (1.76) with

−

grad

is moot,

and the state equation (1.77) requires the density

to be constant

also. The continuity equation (1.74) is then reduced to div

= 0,

and the fluid is effectively incompressible. No distinction need be

made between a gas and a liquid.

1.2.4

Low-speed heat transfer

For many low-speed flows, variations in temperature are forced by the

boundary conditions. If these variations are small, both gases and

liquids can be accommodated as working fluids through linearization

of the state equation. Consider again the low-speed equations (1.74)–

(1.77) in dimensional form. Omit the body-force term temporarily,

and take

. Use Newton’s hypothesis

def

for the

viscous terms, and Fourier’s hypothesis

−

grad

for the heat-

conduction terms. For low-speed thermal problems, replace the state

CHAPTER 1.

INTRODUCTION

equation (1.77) by a tangent approximation,

−

βρ

(

−

)

(1.85)

where the constant

−

(

∂ρ/∂T

)

/ρ

is called the volume coeffi-

cient of expansion and depends on the reference temperature

. If

the inventors of the linearized state equation (1.85) had chosen to

include a factor

in the denominator of the right-hand side, then

the parameter

would represent a dimensionless slope in logarith-

mic coordinates and would have the value unity for a perfect gas.

The reference parameter

would still have to be specified for any

other fluid, as it does now. There are also problems that may require

more complex measures; for example,

→

0 for water near 4

◦

C, or

→∞

for a liquid near its critical point.

The dimensional equations of motion become

Dρ

div

= 0 ;

(1.86)

D~u

−

grad

+ div (

def

) ;

(1.87)

ρ c

= div (

grad

) ;

(1.88)

together with the state equation (1.85). It will be trivial in the sequel

that

and

can be taken as constant.

An important issue involves the limiting form of the continuity

equation (1.86). Because the energy equation (1.88) with constant

is linear and homogeneous in

, it can be written with the aid of

equation (1.85) as an equation for

;

Dρ

ρc

div grad

ρ .

(1.89)

Note that the commonly accepted form of the continuity equation

(1.86) for low-speed flow with heat transfer is

div

= 0

(1.90)

1.2. ANALYTICAL PROLOGUE

and that this seems to imply, according to equation (1.86),

Dρ

= 0

(1.91)

Thus

Dρ/Dt

is both zero and not zero. Some writers ignore the in-

consistency. Others (see, for example, CHANDRASEKHAR (1961,

16–17) explain it by estimates that depend on the smallness of

the parameter

in equation (1.85). However, I find it more useful to

think that it is the temperature difference (

−

) in this equation

that is small in some appropriate sense. The argument proceeds in

the same spirit as that of the preceding section, and again contem-

plates flow around a body. Denote a constant reference temperature

(

for wall) and an ambient temperature by

. Define two

dimensionless parameters (the symbol P should be read as a Greek

capital

rho

)

P =

−

Θ =

−

(1.92)

and consider the limit P

→

0, Θ

→

0. Suitable non-dimensional

variables of order unity suggest themselves as

−

(1.93)

Now write the dimensionless equations of motion (1.70)–(1.72) in

terms of

+ (1 + P

) div

= 0 ;

(1.94)

(1 + P

)

−

grad

div

def

;

(1.95)

(1 + P

)

div

grad

;

(1.96)

−

β T

;

(1.97)

and take the limit P

→

0, Θ

→

0. The incompressible form div

= 0

is evidently the correct limit of the continuity equation. The deriva-

tive

ρ/Dt

is not zero in the limit; only its coefficient P is zero.

CHAPTER 1.

INTRODUCTION

There is no effect on the momentum and energy equations. The

state equation remains in the form (1.85).

In this development, the special case of a perfect gas turns out

to be in no way special. In terms of the new variables (1.93), the

state equation (1.73) for a perfect gas becomes

(1 + P

)(1 + Θ

) = 1

(1.98)

To first order in the small quantities P and Θ, this is

+ Θ

= 0

(1.99)

or, in physical variables,

−

(

−

)

(1.100)

This equation is the same as equation (1.85) if

= 1

The equations (1.86)–(1.88) have now become, in physical vari-

ables,

div

= 0 ;

(1.101)

D~u

−

grad

div grad

;

(1.102)

ρc

div grad

;

(1.103)

with

all constant. These equations are the stuff of low-speed

heat transfer. The form of the viscous terms in the momentum equa-

tion (1.102) now takes account of vector identity (1.18) (which states

that div (grad

)

∗

= 0 if div

= 0). The state equation has been

discarded. The momentum and energy equations are uncoupled in

the sense that the momentum equation does not involve the temper-

ature. The energy equation is linear and homogeneous in

, so that

solutions can be superposed as long as the variable coefficients

and

in the transport terms remain fixed. A large literature testifies to

the importance of this property.

1.2. ANALYTICAL PROLOGUE

1.2.5

The Boussinesq approximation

It remains to consider the body-force term in the momentum equa-

tion. Suppose first that the gravitational force is normal to the gen-

eral direction of flow, as in a free-surface water channel. In rectan-

gular coordinates, the body-force term is then

F =

−

g~ı

(1.104)

where

~ı

is a unit vector directed vertically upward (the notation is

that of the literature of meteorology). This force does not necessarily

play a part in the dynamics of the fluid motion. Take the velocity

to be zero in the vertical component of the momentum equation

(1.75) (more accurately, take

Dv/DT

). In this hydrostatic

limit, denoted by the subscript zero, vertical equilibrium implies

0 =

−

g .

(1.105)

If the density is constant, the integral is

−

gz ,

(1.106)

where

is the pressure at

= 0 (

for sea level, say). If the fluid

is a perfect gas and is compressible but isothermal, with temperture

, equation (1.106) is replaced by

−

gz/RT

(1.107)

(Comment on Chapman’s paper.)

A more accurate model of

the neutral atmosphere would be isentropic (well mixed) rather than

isothermal, in which case

[

−

(

−

)

]

−

(1.108)

Equations (1.107) and (1.108) both reduce to equation (1.106) if

small enough. For an isothermal atmosphere, the pressure and den-

sity decrease exponentially with increasing height, with an

-folding

distance

given by

g z

/RT

γ g z

= 1, where

is the speed

CHAPTER 1.

INTRODUCTION

of sound. For the earth’s atmosphere,

is several thousand me-

ters. For any motion that occupies a sufficiently small fraction of

this distance, the atmosphere can be treated as homogeneous.

Next, subtract equation (1.105) from equation (1.75) to obtain

D~u

−

grad (

−

)

−

(

−

)

g~ı

+ div

(1.109)

In particular, if the density is constant the hydrostatic pressure is

irrelevant, because pressure changes associated with fluid motion are

measured from a variable hydrostatic datum

. An exception may

occur if the vertical acceleration

w∂w/∂z

is not negligibly small com-

pared with the acceleration of gravity. This will certainly be the case

if a liquid has a free surface that is not sensibly flat, so that the hy-

drostatic datum

is itself disturbed.

Now to the main point in connection with buoyancy effects,

and the reason that an argument for the Boussinesq approximation

is needed. The energy equation (11.3) can be written,

with the aid

of equation (11.4), in the form

Dρ

ρh

div

−

(1.110)

whereas the continuity equation (11.1) reads

Dρ

−

div

~u .

(1.111)

The Boussinesq approximation sets the two sides of (1.111) sepa-

rately to zero, but leaves (1.110) intact, in order to retain heat con-

duction as an essential process.

This apparent inconsistency can be resolved with the aid of an-

other limiting process, this time involving the Froude number. Sup-

pose that the motion is driven entirely by buoyancy forces, and that

density changes are small enough to permit linearization of the rela-

tionship

(

). The linearized state equation is usually taken in the

form

−

βρ

(

−

)

(1.112)

The equations cited in this paragraph are discussed in Section 11.1.1.