Statistical Theories of Turbulence 9781400886890

Table of contents :
PREFACE
CONTENTS
Chapter 1. Basic Concepts
Chapter t. Mathematical Formulation of the Theory of Homogeneous Turbulence
Chapter 3. Physical Aspects of the Theory of Homogeneous Turbulence
Chapter 4. Turbulent Diffusion and Transfer
Chapter 5. Other Aspects of the Problem of Turbulence

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STATISTICAL THEORIES OF TURBULENCE BY C. C. LIN

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CONTENTS Statistical Theories of Turbulence C. C. Lin, Department of ,Mathematics, Massachusetts In­ stitute of Technology, Cambridge, Massachusetts Chapter 1.

1. 2. 3. 4. 5.

3

Basic Concepts

Introduction The Mean Flow and the Reynolds Stresses Frequency Distributions and Statistical Averages Homogeneous Fields of Turbulence Conventional Approach to the Statistical Theory of Turbu­ lence

3 4 5 7 8

Chapter t. Mathemaiical Formulation of the Theory of Homogeneous Turbulence

6. Kinematics of Homogeneous Isotropic Turbulence. Correla­ tion Theory 7. Dynamics of Isotropic Turbulence 8. The Spectral Theory of Isotropic Turbulence 9. Spectral Analysis in One Dimension 10. Spectral Analysis in Three Dimensions 11. General Theory of Homogeneous Anisotropic Turbulence Chapter 3.

Physical Aspects of the Theory of Homogeneous Turbulence

12. 13. 14. 15. 16. 17.

Large Scale Structure of Turbulence Small Scale Structure of Turbulence. Kolmogoroff's Theory Considerations of Similarity The Process of Decay The Quasi-Gaussian Approximation Hypotheses on Energy Transfer

18. 19. 20. 21.

Diffusion by Continuous Movements Analysis Involving More Than One Particle Temperature Fluctuations in Homogeneous Turbulence Statistical Theory of Shear Flow

Chapter 4,

Chapter 5.

22. 23. 24. 25.

9 15 17 21 23 25

26 28 32 37 43 45

Turbulent Diffusion and Transfer

47 50 51 52

Other Aspects of the Problem of Turbulence

Turbulent Motion in a Compressible Fluid Magneto-Hydrodynamic Turbulence Some Aerodynamic Problems Cited References

54 55 56 58

SECTION G ι— ι m

STATISTICAL THEORIES OF TURBULENCE C. C. LIN

CHAPTER

7. £/W/C

CONCEPTS

C,l. Introduction. The general concepts of turbulent motion have been discussed in the previous section. It is recognized that the details of turbulent flow are so complicated that statistical description must be used. Indeed, only statistical properties of turbulent motion are experi­ mentally reproducible. The purpose of the present section is to give a more comprehensive treatment of the statistical theory. Current literature on the statistical theory of turbulence is mainly limited to the treatment of the case of homogeneous turbulence' without any essential mean motion. Superficially, one might think that there is little to be known about such fluid motions. Actually, the very absence of mean motion allows one to go more deeply into the inherent nature of the turbulent flow itself. Many basic concepts have been developed in the study of homogeneous turbulence, and these concepts now gradually find their way into the study of shear flow. Since there is available an account of the theory of homogeneous tur­ bulence [/] with a complete discussion of the mathematical background, a somewhat different presentation is adopted in the present section. Following the historical order, the isotropic ease is taken up first. It is hoped that this will be helpful to those readers who wish to get an idea of the essentials without going through ail the preliminaries required in a complete mathematical treatment. In the later parts of this section, other aspects of the statistical theory and their applications will be treated. 2 We have, however, omitted several other approaches to the problem of turbulence. Among these, the work of Burgers [3,4] and Hopf [6] should especially be mentioned; nor is any attempt made to include a discussion of related mathematical studies, such as that of Hopf [6] and Kampi de Firiet [7], 1

These concept» are explained more precisely in the following pages. a brief survey of some aspects of the problem of turbulent motion, see [·?[.

1 For

C , 2 • THE

MEAN

FLOW

AND

REYNOLDS

STRESSES

C,2. The Mean Flow and the Reynolds Stresses. It is generally assumed that the motion can be separated into a mean flow whose components are and and a superposed turbulent flow whose components are the mean values of which are zero. In taking average values, the following principles will be adopted. If A and B are dependent variables which are being averaged, and S is any one of the space variables or the time t, then where the bar denotes a mean value. When the mean flow is not varying, that is, when the average value defined by

is independent of the time t, the time average is the natural mean value to use. Difficulties arise when the flow is variable, and other types of averages have to be introduced. For instance, in the problem of turbulent flow near an infinite plate moving with variable velocity, the mean values could be taken over planes parallel to the plate. In more general cases, neither the time nor the space mean values can be conveniently defined to possess all the desired properties. We then consider the statistical average over a large (infinite) number of identical systems (ensemble average). The equation of continuity of an incompressible fluid, when averaged, becomes

The Navier-Stokes equations of motion are

where
'-

Ox

.....'-

a

0

...

.0 0

2

x (

.....>: 'Vi 0 cQ)

x

-0

0

0

>-

...0> Q)

c

6

xo

Q)

o.... ..... u Q)

0.

>s 4 ~

V)

)(

2

)(0

b

o

o

)(

100

x

k

200

x

~

300

x

0

400

FrequencYt hertz Fig. C,8. Experimental verification of the Fourier transform relation between space correlation and time spectrum for turbulent fluctuations behind a grid in a wind tunnel (after Stewart and Townsend [221).

This leads to an equation of the form

aF

at

+

W

= -211K2F

(8-7)

In the above equation, W(K, t) is connected with the triple correlation function her, t) by the following relations: W(K, t)

= 2;2 [K 2 H;'(K) < 20

)

- Kl-I;(K)]

(8-8)

C · STATISTICAL THEORIES OF TURBULENCE where KH1(K) = -

h ( r ) sin K r d r (8-9)

It is clear that the quantity t ) in Eq. 8-7 represents the transfer of energy among various frequencies. The above formula for TF(k, t) also shows that

(8-10) which means that no energy is generated or lost while it is redistributed among various scales. The rate of dissipation is obtained from Eq. 8-7 by integrating it with respect to κ from * = 0 to * = ® : d

i=-j~1rd"-2>L'*'Fd*

± », we cannot put u{x) in place of φ(χ) in the above relation. Instead we

C , 9 • SPECTRAL

ANALYSIS

IN ONE

DIMENSION

first consider

and then try to adopt a suitable limiting process as . In fact, we want to consider first the amplitude not at K but associated with a finite range of values of We integrate Eq. 9-1 between k and obtaining

Here, we may take the limit as

, and obtain

since the integral is now convergent. We now form the expression for the measure of energy and calculate its statistical average. Then

where R is the statistical correlation between u(x) and . The inneT integral can be transformed by replacing Then it becomes

and we obtain

We shall now divide both sides by and replace by a new variable Then we obtain a measure of the "density of energy":

It is easy to see that the right-hand side has a limit as W e therefore have the interesting situation that is of the order of Ak and not of the order of Let the limit be denoted by Then

{ 22 )

C . STATISTICAL THEORIES OF TURBULENCE

and y2RW =

i f-"'""

FI(")cued,,

It is dear that F I (,,) must be even when R(~) equations become the same as Eq. 8-1.

IS

real, and the above

C,IO. Spectral Analysis in Three Dimensions. The one-dimensional spectrum, howeyer, does not give an exact representation of the distribution of energy among the seales. Consider a simple harmonic variation 'with waye number" in a direction making an angle 8 with the x axis

Fig. C,IO. Diagram illustrating the relationship between one-dimensional and three-dimensional Fourier analysis of a field of turbulence.

(Fig. C,IO). Its period in the x direction would be longer and the wave number in a harmonic analysis in the x direction is Kz

=

K

(10-1)

cos 8

Thus a modified picture is obtained of the energy distribution among the various scales. In the ca.'!e of isotropic turbulence, as we shall demonstrate below, it is easy to establish the relation between the one-dimensional spectrum F I(K) and the spectrum function F(K) corresponding to a threedimensional Fourier analysis. The relation is F 1(K) =

~ f~ ~~ (K'2

- K2 )F(K')

(10-2)

or, upon differentiation, F(K)

Note that :u 2 = f~F(K)dK

=

=

i[K2F~' (K)

f~Fl(K)dK.

( 23 )

- KF: (K) 1

(10-a)

C , 1 0 • SPECTRAL

ANALYSIS

IN THREE

DIMENSIONS

The analysis of the y component of the motion in the x direction leads to a spectrum

by combining Eq. 10-2 and 10-4, we obtain, after a little calculation,

This relation is more convenient for obtaining from experimental data. It is numerically more accurate than Eq. 10-3 since only one differentiation is involved. To establish the relations (Eq. 10-2 and 10-4) let us write 8 the threedimensional Fourier analysis of the velocity in the following form:

Then, for a wave in the direction of the vector tinuity gives

the equation of con-

This means that all the motion associated with the vector wave number must be perpendicular to this vector. Consider now the contribution to the spectrum of a Fourier analysis in the direction of a component of turbulent motion with vector wave number In the first place, the motion appears to have a space frequency defined by Eq. 10-1. Secondly, the motion has in general all three components. T h e component is (cf. Eq. 10-2 and Fig. C,10).

where is the angle between K, and the x axis, and is the angle which the velocity vector A, makes with the plane containing and the axis. Thus, averaging over the angle we have

Consider now a distribution of energy in the k space. Let the total kinetic energy per unit mass and per unit volume of the K space be is the energy contained in the range To obtain the energy per unit mass lying between and associated with one componentof the motion, one must multiply this expression with the factor and then integrate for all values oi 8 The reasoning here is essentially that used by Heisenberg [23], ( 24 >

C . STA TIS TICAL THEORIES OF TURBULENCE KS

and

K3

while keeping A:l constant. Thus, the one-dimensional spectrum is

:\ow, the three-dimensional spectrum is isotropic, so that

considering all

K)'S

with the same magnitude /(. We have finally

if ( ~

F1\Kl) = F1(Kl)

+ F 1( -/(1)

=

1-

~r~K;)

d/(2d/(3

This is easily transformed into Eq. 10-2 by carrying out the integration in a polar coordinat.e system in the plane of /(2, /(3. ~ll.

General Theory of Homogeneous Anisotropic Turbulence.

The above development of the theory of homogeneous isotropic turbulence can be generalized to remove the restriction of isotropy. Such a generalization is necessary because ani~otropy of turbulence, particularly in the largest eddies. does occur in practice. "\Ve shall outline here only the main features of the developments and conclusions, pointing out especially the difference between the i~otropic and anisotropic cases. The concept of correlation functions requires very little modification, although it is now obviously impossible to represent the double correlation functions, for example, in terms of a single scalar function. The "pectral function must be replaced by a spectral tensor, which may be defined a.'> the three-dimensional Fourier transform of the double correlation tem:or. Thill', (11-1)

and N.)

=

y2

fff

if>'j(/(mJru"")dr(/(m)

(11-2)

It can be shown that if>}J represents the energy density in the wave number space. In the case of isotropic turbulence, (11-3)

Because of the condition of vanishing divergence of the correlation tensor, we obtain (11-4)

( 25 )

C , 1 2 • LARGE

and

SCALE

STRUCTURE

OF

TURBULENCE

can be expressed in the form

where is a scalar function of the vector is a vector perpendicular to and is its complex conjugate. When the turbulence is isotropic, and is a function of the magnitude only. The form (Eq. 11-5) is due to Kamp6 de Fdriet [24], The dynamical equations for anisotropic turbulence are more complicated than those for isotropic turbulence, among other things, by the presence of the pressure terms in the equations of the change of double correlations. In the correlation form, the equations are

where

In the spectral form, we have

where n,k and 8 » are respectively the Fourier transforms of and Obviously so that Thus the pressure fluctuations have no effect on the total energy density their influence produces a redistribution of energy among the various directions. It is not immediately evident whether the net effect is to make the turbulent field more or less isotropic, but general evidence seems to indicate that the former is the case. The above developments are mostly due to Batchelor [25\. Other detailed studies of anisotropic turbulence have been by Batchelor [26], Chandrasekhar [27], and others. The reader is referred to the original papers.

CHAPTER 3. PHYSICAL ASPECTS OF THE THEORY OF HOMOGENEOUS TURBULENCE C,12. Large Scale Structure of Turbulence. In the following articles, we shall make use of the methods developed above—the correlation and spectral theories—to study the nature of turbulent motion. As pointed out above, the theory by itself allows us to reach only partial

< 26 )

C . STATISTICAL THEORIES OF TURBULENCE

results. Some theoretical speculation and assumptions will therefore be introduced in the following discussions for the purpose of reaching definite conclusions. We shall begin by considering the large scale structure of turbulence, which is associated with small values of " in the spectral representation and large values of r in the correlation representation. Let us now consider the second equation in Eq. 8-1, ( 12-1)

and expand cos (Kr) into a power series. We obtain F I(,,) = 2y2 (Jo _ J 2 ,,2

2!

7

+ J 4 ,,4

_

4!

. .)

(12-2)

where (12-3)

Such a step is justified only when the function fer) vanishes sufficiently rapidly at infinity (e.g. as a negative exponential function) so that the integrals J .. are convergent. In that case, one may derive from Eq. 12-2 a power series expansion for the three-dimensional spectrum F(,,) by using Eq. 10-3. This gives F(IC)

y2

2 (J4 = -;"3 ,,4

-

. . .

)

(12-4)

Similarly, assuming that her) also vanishes sufficiently rapidly at infinity, one can show that the transfer function W(", t) behaves as ,,6 for small values of IC. The spectral equation (Eq. 8-7) then shows that

~ (y2J 4)

=

~ [ yz!o" f(r)r 4dr] = 0

(12-5)

It then follows that y2 ( .. f(rJr 4dr = .I, a constant

10

(12-6)

Thus, the large scale motionfl are permanent in the sense that the principal part of F(,,) for small values of " remains unchanged. The above derivation (including explicit statements of the necessary convergence assumptions) was given by Lin [28] for the spectral interpretation of the parameter .I, which was first obtained by Loitsiansky [2.9) from the Karman-Howarth equation. Indeed, if one multiplies that equation by r· and then integrates it with respect to T from zero to infinity, one obtains d dt

[«2

r'" f(T)T 4dr] = 2«3 lim (r 4h)

}o

r-n,

( 27 )

02-7)

G, 13 · S M A L L S C A L E S T R U C T U R E O F T U R B U L E N C E

provided the integral involved is convergent. If, in addition, h ( r , t ) van­ ishes sufficiently rapidly at infinity so that Iim r A h = 0

(12-8)

Γ—» oo

the relation (Eq. 12-5) is obtained. It must be noted that there is no a priori reason® for the convergence of the integrals (Eq. 12-3) and the validity of Eq. 12-8. As a matter of fact, recent investigations of Batchelor and Proudman [31] show that even if f(r) is exponentially small at infinity at an initial instant, because of the influence of the long range pressure forces, one can only be sure that it will be no larger than O (r~ 6 ) when r is large, although the possi­ bility of an exponentially small behavior is by no means excluded. We are therefore only assured of the leading term in Eq. 12-4 and the existence of t h e Loitsiansky p a r a m e t e r , J =

Jo" f(r)r*dr

(12-9)

However, the constancy of J depends on the relation (Eq. 12-8), which is shown to be not generally true by the analysis of Batchelor and Proudman. On the other hand, for low Reynolds numbers based on the turbu­ lence level u, the term on the right-hand side of Eq. 12-7 becomes negli­ gible, and the Loitsiansky parameter is indeed approximately constant. (Cf. Art. 14 and 15 for the part dealing with the final period of decay.) From a physical point of view, any prediction of the behavior of the largest eddies must be regarded with some reserve, since it is expected to be dependent on the experimental apparatus. If the general scale of turbu­ lence is much smaller than the dimensions of the experimental apparatus, it would appear that this complication may be avoided by a proper interpretation of the above results. The integrals (Eq. 12-3) may, for ex­ ample, be considered as extending over a distance much larger than the scale of turbulence but still much smaller than the scale of the apparatus. Generalization of the above discussions to the anisotropic case has been made by Batchelor [£5]. The earlier conclusions are again modified by the work of Batchelor and Proudman [31], In fact, in the anisotropic case, the correlation tensor R lj is shown to be in general of the order of r 5 , so that even the existence of a Loitsiansky parameter is in doubt. C,13. Small Scale Structure of Turbulence. Kolmogoroffs Theory. We now turn to consider the small scale structure of turbu­ lence. Here the formal relations analogous to Eq. 12-1, 12-2, 12-3, and 12-4 are obtained by expanding cos (κτ) into a power series in the first equation in Eq. 8-1: U2f(r) = jg • Cf. BirkhoEf [SO],

F 1 ( K ) cos (n r )dK

(13-1)

C • STATISTICAL

THEORIES

W e then obtain a power series for

OF

TURBULENCE

in the form

with

In terms of the three-dimensional spectrum, these integrals become

Here it is useful to recall that is proportional to the energy, and that is proportional to the rate of energy dissipation. Consider now the dynamical relations in the correlation theory. We expand both and in power series of

(13-5)

and substitute them into the Karm&n-Howarth equation (Eq. 7-4). As observed before (Art. 7) the terms independent of r give the energy relation. The terms in give the vorticity equation in the form

or

where is the vorticity vector, is the mean square value of one component of the vorticity, and is defined by

The second term on the left side of Eq. 13-7 represents the change of vorticity due to stretching or contraction of the vortex tube without the action of viscosity. It is well known that, in a perfect fluid, the circulation around a vortex tube is permanent and hence the vorticity increases at a rate in proportion to its rate of stretching. The right-hand side represents the dissipation of viscosity by viscous forces. Taylor [S2] suggested that this relation represents one of the basic mechanisms in the process of turbulent motion. The rotation of the fluid is being slowed down by the effect of viscosity. This loss is partly com-

( 29 >

C,13 · S M A L L S C A L E S T R U C T U R E O F T U R B U L E N C E

pensated, or even over-compensated, by the stretching of the vortex tubes, due to the diffusive nature of turbulent motion. (Hence one may expect more stretching of the vortex tubes than compression.) Taylor calculated the relative magnitudes of the various quantities by deter­ mining /0' and f'a% and he found that all the three terms in Eq. 13-6 are of the same order of magnitude for his experiments. Such measurements were more accurately made later by Batchelor and Townsend [33] and by Stewart [11]. As noted before (Art. 8), in many experiments the dissipation of energy is practically all associated with the high frequency components which contain a negligible amount of energy. Combining this fact with the mechanism of vortex-stretching just discussed, one can form a reason­ able picture of the process of turbulent motion. There are the large energycontaining eddies which contribute very little to the viscous dissipation directly. By their own diffusive motion, small eddies are formed, i.e. the kinetic energy of turbulent motion goes down to smaller scales. It is at these small scales that viscous forces become most effective and the pre­ dominant part of the energy dissipation occurs. Thus one forms the pic­ ture of an energy reservoir in the large eddies, and a dissipation process in the small eddies which may be presumed to depend very little on the structure of the large eddies except to the extent of the amount of energy supplied to them. This forms the physical basis of Kolmogoroff's theory of locally isotropic turbulence [34]· Before we go on with the discussion of his theory, it should be empha­ sized that the picture is correct only when the diffusive mechanism is strong; i.e. when the inertial forces are large compared with the viscous forces. In other words, the Reynolds number of the turbulent motion must be relatively large. This is well illustrated by the detailed calcu­ lations made by Taylor and Green [35] on a model of isotropic turbu­ lence.10 Indeed, they found that for very low Reynolds numbers of turbu­ lence, defined by

the stretching mechanism is not strong enough, so that the magnitude of the vorticity decreases steadily. On the other hand, if the motion starts out at a fairly high R\, the mean square vorticity (and hence also the rate of energy dissipation) first increases to several times its original value due to the stretching mechanism. The kinetic energy of the motion, how­ ever, decreases steadily. Eventually, it becomes very low, and the stretch­ ing process is so weakened that the vorticity of the motion also decreases steadily. Kolmogoroff's theory. In line with the above ideas, Kolmogoroff pos­ tulates that, at large Reynolds numbers of turbulent motion, the local 10 See

also Goldstein [36],

C ' STATISTICAL THEORIES OF TURBULENCE property of turbulent motion should have a universal character described by the following concepts. First, it is locally isotropic whether the large scale motions are isotropic or not. 1 1 Second, the motion at the very small scales is chiefly governed by the viscous forces and the amount of energy which is handed down to them from the larger eddies. The large eddies tend to break down into smaller eddies due to inertial forces. These in turn break down into still smaller eddies, and so on. At the same time, viscus forces dissipate these eddies at very small scales into heat. In the long series of processes of reaching the smallest eddies, the turbulent mo­ tion adjusts itself to some definite state. The further down the scale, the less is the motion dependent on the large eddies. Furthermore, in line with Taylor's experimental findings, Kolmogoroff essentially postulates that practically all the dissipation of energy occurs at the smallest scales when the Reynolds number of turbulent motion is sufficiently high. To formulate these concepts mathematically, he introduced the corre­ lation functions of the type (« - ν') 1 = H 5 U - /Ml which is the mean square value of the relative velocity of turbulent mo­ tion. The introduction of the relative velocity stresses the local nature. The moments (u — u')" would then be emphasized instead of the usual correlations at two points. (In fact, the third moment (m — u') 3 is pro­ portional to k(r},) The second step in the formulation of the theory is to introduce the assumption that, for small values of r, these correlation functions depend only on the kinematic viscosity e and the total rate of energy dissipation ¢. This is in accordance with the previously discussed physical concepts. One can then make some dimensional analysis and construct universal characteristic velocity and length for motion at very small scales. Indeed, from t and v, one can only construct the length scale η = (jJ

(13-9)

ν = (ve)i

(13-10)

and the velocity scale We may then write (n' — u) 2 =

1 Vf i i Iidd

(u' — Uj z = (vtf'fiddd ( ^ j

(13-11)

(13-12)

where Sdd and S d t i i are universal functions for small values of r. 11

See Sec. B on shear flows for the experimental confirmation of this fact.

C 1 14 · C O N S I D E R A T I O N S O F S I M I L A R I T Y

For very high Reynolds numbers, Kolmogoroff visualizes that, at the larger end of the universal range, there is a range of r for which the vis­ cosity coefficient does not play an explicit role. This range may be con­ veniently referred to as an incrtial subrange. The above relation then implies that (u' — u) 2 • —' (er)' (13-13) A definite form of the correlation function is thereby obtained. The concept of Ivolmogoroff can also be introduced into the spectral formulation. Thus, at high Reynolds numbers the spectrum F(K) at very high frequencies can be expressed as F(K) =

(13-14)

where the function f ( x ) has a universal form for large values of x . For the inertial subrange, the spectral function can again be deter­ mined completely from dimensional arguments. This gives FM ~

(13-15)

This form was first given by Obukhoff [37). It has received some experi­ mental support at high Reynolds numbers. 12 With a spectrum of this form, it can be explicitly demonstrated that the dissipation of energy lies essentially in the universal range of Kolmogoroff (cf. [35]). The actual form of the spectrum in the universal range is obviously of basic theoretical interest. By following the general ideas discussed in this section, Townsend [35] developed a more concrete model giving a definite form for the spectrum of the small eddies. The results are in general agreement with experimental observations. The scales η and ν defined above also occur in the study of the small scale structure even when the Reynolds number is not high. This cannot be interpreted on the basis of Kolmogoroff's theory, but follows from considerations of self-preservation during the process of decay (see next article). C,14. Considerations of Similarity. As noted above, the general theory of turbulent motion, as developed in Chap. 2, cannot lead to specific predictions without auxiliary considerations. For this reason, von Kdrmiin and IIowarth [ 17) introduced the idea of self-preservation of correlation functions. 1 3 In terms of the spectral language, this states that the spectrum remains similar in the course of time. Since the energy distribution among the various frequencies is changing through the trans­ fer mechanism, this may be reasonably expected provided that there is enough time for the necessary adjustments. In this article, we shall con12 Cf. [9 9 1 and { 1 0 1 } for detailed discussions. A different form of the spectrum has been recently obtained by Kraichnan [100}. 13 This article follows closely the treatment of von Kiirmiin and Lin [35, p. 1],

C • STATISTICAL

THEORIES

OF

TURBULENCE

sider the theoretical aspects. Comparison with experiments will be made in the next article. Let us consider the equation (Eq. 8-7) for the change of spectrum

and try to find a similarity solution. If V is a characteristic velocity, and I is a characteristic length, then, from dimensional arguments,

Thus, the above equation becomes

If the similarity solution is to be valid, one must have

where

, and

are all constants. Eq. 14-2 becomes

Besides Eq. 14-3, 14-4, and 14-5, it is evident that the mean square value and the rate of energy dissipation have to satisfy the relations (cf. Eq. 8-6 and 8-11)

Finally, if the convergence criteria for Loitsiansky's relation (Eq. 12-6) are assumed to be valid, we have

This system of equations presumes that the transfer term in Eq. 14-2 is considered generally of equal importance with the term expressing the viscous dissipation. I t has been shown by Dryden {//)] in the equivalent problem of self-preserving correlation functions that such a solution is connected with the statement that the square of the characteristic length is proportional to the time t and the law of decay is expressed by

( 33)

C,14 · CONSIDERATIONS OF SIMILARITY

Heisenberg [41] indicated an equivalent solution for the spectral problem. It is easily seen that these solutions are at variance with Eq. 14-9. In other words, full similarity is only possible when we reject Loitsiansky's theorem. In addition, experimental evidence clearly indicates that the law of decay and the behavior of the characteristic length during decay exclude the possibility of adopting full similarity as a generally valid assumption for all decay processes. Let us now consider two opposite approaches. In the first approach, we assume that Loitsiansky's invariant exists and that it plays a role in the similarity of the spectrum. In the second approach, we assume that similarity of the spectrum is occurring only in the eddies contributing appreciably to the dissipation process, and that the largest eddies play no role in determining the similarity of the spectrum. Clearly, the first approach will not yield valid results unless Loitsiansky's invariant does exist. This is definitely known only in the decay of isotropic turbulence at very low Reynolds numbers (case (a) below). The second approach is naturally independent of Loitsiansky's invariant. Let us consider now two opposite specific cases in the first approach: (a) the transfer term is negligible for all frequencies, and (b) the influ­ ence of viscous dissipation is restricted to high frequencies whereas for low frequencies the transfer term is the prevailing factor. Case (a), w(£) = 0, leads to a solution of Eq. 8-7 which has full simi­ larity for all frequencies and also satisfies Loitsiansky's relation. One ob­ tains with ξ = κΐ and I = -\fvt F = const F2Z£4e-2£!

(14-10)

F = const VH 6 K^ u '"'

(14-11)

or By using the definition of J in Eq. 14-9, we write F = J^e- 2 ""

(14-12)

The corresponding correlation function can be easily shown to be Kr, t) = e~ r ' /8y '

(14-13)

by using Eq. 8-1 and 10-3. This correlation function was noted by von Κάπηάη and Howarth [17], and discussed b y Millionshchikov [42], Loitsiansky [29], and Batchelor and Townsend [43], Kdrmdn and Howarth also obtained a more general self-preserving solution in terms of the Whittaker function, with a spectral form F = It can be easily shown that the solution must specialize into Eq. 14-13 if the Loitsiansky invariant is to be finite. The law of decay in this case is the five-fourths power law: U * ~ ( t - ίο)-»,

λ 2 = Av(t - to)

(14-14)

This law of decay and the corresponding correlation function have been

C · STATISTICAL THEORIES OF TURBULENCE verified experimentally by Batchelor and Townsend for the final stage of decay (see Art. 15 for further details). C a s e ( b ) has also been treated in the theory of self-preserving corre­ lations by von Kdrmsin and Howarth [/7] and later by Kolmogoroff [44]· The former authors came to the conclusion that any power law for the decay-time relation may prevail in the decay process. Kolmogoroff pointed out that if one assumes the validity of Loitsiansky's theorem the relations U t = const t ~ v and X 2 = 7v t (14-15) must apply. 1 4 Von Κάπηάη [45,46} dealt with the corresponding spectral problem in two communications assuming the specific decay law (Eq. 14-15). It should be reiterated, however, that this first approach, especi­ ally in case (b), can only be regarded as tentative because of the un­ certainty in the constancy of the LoitMansky integral. Consider now the second approach. Clearly, the idea of complete similarity, with the rejection of LoitMansky's relation, belongs to this case. However, there are physical and mathematical reasons for believing that the large eddies do not play a significant role in the determination of the similarity characteristics in the smaller eddies. We therefore con­ sider cases where the similarity requirement is relaxed for an increasing range of frequencies at the end of largest eddies. C a s e ( c i . We first consider the assumption that similarity extends over the whole frequency range, with the exception of the lowest. More specifically, we assume that the deviation from similarity shall occur for such small values of κ that, whereas the contribution of the deviation is negligible for computation of t (Eq. 8-11), it enters in the calculation of energy (Eq. 8-6). It is easy to see the corresponding assumption in the correlation for­ mulation by using Eq. 1 '',-2 and 13-4 in the following form: as

-m = - Σ (2irTTKiir+Ji l2ij< / " η —I

04-16,

The above assumptions imply that all the higher moments of F ( k ) are. not appreciably influenced by the deviation from similarity. Hence, they are all proportional to Similarity is therefore assumed for w 5 [l — /(r)]. This form of the similarity hypothesis was introduced by Lin [48]. Assuming the self-preservation of (u -

U' f

= U 2 11 -/0-)1

and ( u — u ' ) ' = 12 u ' h ( r ) " See Frenkicl {/,7\ for some discussion of the comparison of Eq. 14-15 with some experim ente,

C , 1 4 • CONSIDERATIONS

OF

SIMILARITY

he derived the law of decay

where a and b are constants, with This law can be easily obtained from the general relations (Eq. 14-3, 14-4, 14-5, and 14-8), which are valid for any similarity hypothesis. One obtains the positive and negative half-power laws for the change of the characteristic length and the characteristic velocity V, and the inverse square law for the rate of dissipation To be more specific, one finds that I and V may be identified with Kolmogoroff's characteristic quantities (cf. Eq. 13-9 and 13-10)

It can easily be seen by introducing these relations into Eq. 14-3, 14-4, and 14-5 that the law of decay is of the form of Eq. 14-17. It is convenient to rewrite the results as follows, with definite physical interpretations attached to the constants. The law of decay is given by

where u\, is the additive constant giving the departure of the energy content from that in the case of similarity, and D0 is the initial diffusion coefficient

defined according to a formula of the kind suggested earlier by von Kdrmdn The changes with time of the characteristic velocity and scale, and of the Reynolds number of turbulence are given by

where

is the initial Reynolds number of turbulence

It is evident from Eq. 14-19 and 14-21 that the solutions obtained can only be applied to an early stage of the decay process, in which remains small. Case (d). The above assumption is based on the idea that the low frequency components do not have the time to adjust themselves to an equilibrium state. (An investigation of such a concept was made by Lin and will be briefly presented in Art. 17.) I t is specifically assumed that may be calculated by a similarity spectrum. Goldstein further

( 36 )

C · STATISTICAL THEORIES OF TURBULENCE

relaxed the requirement and assumed that the similarity spectrum might be adequate only for the calculation of higher moments of F(κ). If the similarity spectrum is accurate only for the calculation of $όκΨ(κ)άκ and higher moments, Goldstein shows that the law of decay becomes U2(t - to) =

a

+

b(t -

to)

+

c(t

(14-23)

- to)2

This includes one more constant than Eq. 14-17. Further generalization involving higher powers of t - t is immediate. Comparison of the laws of decay with experiments will be made in the next article. 0

C,I5. The Process of Decay. We shall now examine the whole process of decay and compare the above theoretical laws with experi­ ments, whenever such evidence is available. 0.20

0.16

c

0.12

E υ r* ,
= 0, curl u = 0 (22-2) The rotation of the fluid is given by the first part «[" and the com­ pression is given by the second part u$2>. In general, there is a continuous conversion between the rotation component and the compression component of the velocity fluctuations. This additional degree of freedom in the compressible case naturally makes the theory of turbulence much more difficult. In the case of small disturbances from a homogeneous state these modes are separable from each other. 20 The study of small disturbances superimposed on a shear flow is treated in connection with the instability of the boundary layer at high speeds. Attempts have been made to extend directly to the compressible case the various approaches to the theory of turbulence in the incompressible case: e.g. the study of isotropic turbulence by Chandrasekhar [50], and the consideration of von Kiirmdn's similarity theory by Lin and Shen [81] for shear flow. The method discussed in Art. 21 can also be extended to a compressible gas. Obviously, such approaches cannot go beyond the limitations in the incompressible case. It is therefore natural that the more fruitful theoretical results on turbulent motion in a compressible fluid are obtained in connection with the study of the influence of com20

See [79] for a detailed diecussion of this case.

C . STATISTICAL THEORIES OF TURBULENCE

pressibilit.y on turbulent motion , principally for small Mach numbers of ~urbulen('e. It is then plausible that the chief inHuence of compressibility is th.at acoustic energy is constantly being radiated, causing the turbulent mohon to di ssipate faster than in the incompressible case. Lighthill [82] has shown that, in the absence of solid boundaries, turbulent motion acts as quadrupole sources of sound. He also showed that the amount of energy radiated per unit volume of turbulence is proportional to p 1"8 I a~l, where p is the density, V is a typical velocity of the turbulent motion, a is the acoustic speed, and l is a typical linear scale. Since the rate of energy conversion in turbulent motion is proportional to p P / l, the acoustic efficiency is proportional to the fifth power of the root mean square Mach number. At low Mach numbers, this would be a very small amount indeed, if it were not for a numerical factor of proportionality of the order of 40, as shown by Proud man [83J. In the cases where the theory is applicable, the experimental results bear out the general theoretical conclusions. If solid boundaries are present, such as in the problem of the noise from the boundary layer of a flat plate, Phillips [84] found that dipole sources are present if the plate is semi-infinite. Acoustic sources are again of the quadrupole type if the plate is infinite and the motion is statistically the same along the plate. The scattering of energy due to the interaction of turbulence with sound or shock waves has been considered by Lighthill [85] and others. All of the above results are for low Mach numbers of turbulent motion. At the present time, only speculation can be made for the cases of higher ~1ach numbers where shock waves may appear. C,23. )Iagneto-Hydrodynamic Turbulence. In astrophysics, one important problem is the turbulent motion of an electrically conducting gas in the presence of magnetic field s. One is then dealing with the conversion of energy from the mechanical form to the electro-magnetic form. There is an extensive and rapidly growing literature on this subject, and it is perhaps inappropriate to try to survey it at the present time in a volume on high speed aerodynamics. One of the central problems at iBsue is the partition of energy between the two modes. Batchelor [86J noted that the equation for the magnetic field is exactly the same as that for vorticity, and "lUggested that the energy spectrum of magnetic energy is proportional to K2FCK). This would mean that there is little magnetic energy in the large scales. Other authors, however contended that there should be equipartition of energy of the two modes. R ecently, Chandrasekhar [87J undertook a systematic development of the theory of turbulent motion for magneto-hydrodynamics along the lines of Art. 16 and 17, and found solutions which are in agreement with the latter opinion. However, since his assumption limited him

( 55 )

C,24 · S O M E A E R O D Y N A M I C P R O B L E M S

to moderate and small wavelengths, the former opinion is not yet ruled out. For a more detailed discussion of the arguments for and against the two standpoints, see [55, pp. 93-98]. Work decisively distinguishing be­ tween them is clearly needed. In this connection, it would perhaps be worthwhile to obtain some special solutions in magneto-hydrodynamics analogous to those obtained by Taylor and Green [35] in the ordinary case to get an idea of the validity of the existing arguments. C,24. Some Aerodynamic Problems. There are a number of aero­ dynamical problems associated with turbulent motion in which its dif­ fusive nature takes on a secondary role. The random nature of turbulent motion still makes it necessary to use statistical treatments. In this cate­ gory of problems, we briefly discuss (1) the dynamical effects of turbulent motion, (2) the effect of contraction on wind tunnel turbulence, and (3) the effect of damping screens. Dynamical effects of turbulent motion. Dynamical effects caused by turbulent motion are often treated by statistical methods. For example, in the case of a pendulum suspended in a turbulent wind, the spectrum of the motion of the pendulum can be calculated in terms of that of the turbulent motion and the dynamical characteristics of the pendulum \89). Recently, Liepmann [90] tried to apply these methods to the buffeting of airplanes moving through a turbulent stream. Effect of wind tunnel contraction. The effect of wind tunnel contrac­ tion on the intensity of turbulence has been studied by Prandtl and Taylor [91, p. 201]. Recently, Ribner and Tucker {92} applied Taylor's ideas to the study of the influence of the contraction on the spectrum. The combined effect of damping screens and stream convergence have also been studied by Tucker [93]. The reader is referred to the original papers. An experimental investigation of the detailed behavior of the tur­ bulent fluctuations during contraction has been made by Uberoi [9I t ). Effect of damping screens on homogeneous turbulence. Damping screens have long been used for the reduction of turbulence level in the wind tunnel. While these screens no doubt act also as a grid in producing turbu­ lence, the scale of such turbulent motion is usually so small that it damps out at a comparatively small distance behind the screen. The resistance of the screen to the flow, on the other hand, tends to reduce the large scale turbulent motion already existing in front of it. The characteristics of a damping screen are usually described in terms of two force coefficients K b and F 0 - If the screen is placed with its normal at an angle θ relative to a stream of speed U, there is a drop of pressure across it, given by p2 - P i = K i • ^ p U 2

(24-1)

where pi and p 2 are the static pressure upstream and downstream of the

C . STA T]STICAL THEORIES OF TURBULENCE

screen .. At the same time, there is a side force in the plane of the screen per urut area, given by

S = Fe' jpU2

(24-2)

~xp:riments by Schubauer, Spangenberg, and Klebanoff [95) at the 2\ attonal Bureau of Standards (NBS) show that the coefficients Fe and KI are related for usual wire gauze screens. Dryden and Schubauer (96J proposed the relation (24-3) which agrees with experiments for Ke < 1.4. Taylor and Batchelor 183) fitted the XBS data with the empirical formula

Fe _ ')

e - .. -

2.2

(24-4)

VI + Ke

which appears to be a reasonable approximation for 0.7 < Ke < 4. Schubauer, Spangenberg, and KJebanoff also found that Ke/ cOS2 8 can be uniquely related to R