ADVISORY COMMI~~~ FOR ~ERON~~TICS/67531/metadc62897/m... · 2 NACA 94 1377 The first chapter deals...

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NATIONAL ADVISORY C O M M I ~ ~ ~ FOR ~ E R O N ~ ~ T I C S

TECHNICAL MEMORANDUM 1377

Translation of %es Th6ories de la Turbulence. Publications Scientifiques et Techniques du M i n j s t b

de L'Air, No. 237, 1950,

9

Washington October I955

I

1B

PREFACE

The theory of turbulence has made so much progress during these last years that it is of i n t e re s t t o s t a t e exactly the obtained resu l t s , the hypotheses on which these resu l t s are based, and the directions i n which new research is being conducted.

Messrs. h s s and Agostini have undertaken t h i s work and have sliiIimBr- ized our actual knowledge of turbulence i n a series of conferences which took place a t the Sorbonne, within the Paris Ins t i t u t e of Mechanics. The reader w i l l f ind i n the following pages the t ex t of these conferences, perfected and revised by M r . Bass.

I n v i e w of the magnitude of the subject and i t s simultaneously phy- s i c a l and theoret ical aspects it had seemed advisable t o entrust t h i s work t o a team formed by a mathematicxan and a physicist . Mr. Bass had taken the responsibil i ty f o r the theoretical par t , Mr. Agostini f o r the physical par t .

I n i t i a l l y , the report was intended t o contain three theore t ica l and two physical chapters followed by a chapter on the technique of the measurements and on the appratus used: anemometers and s t a t i s t i c a l - measurement apparatus.

The unexpected death of M r . Agostini i n August 1949 unfortunately made modifications of the or iginal project necessary. This premature death deprived us of a highly valuable physicist , a t the peak of i n t e l - l ec tua l maturity, whose current work on these problems showed particu- l a r l y remarkable promise.

M r . Agostini had only j u s t begun drawing up the two last chapters; M r . Bass had t o take up the ed i tor ia l work and t o complete it according t o M r . Agostini 's notes. The chapter on the technique of the measurements has been omitted and w i l l form the object of a l a t e r publication.

I n order to'make up f o r the gap i n the experimental part, the t e x t w a s supplemented by some curves furnished by M r . Favre which w i l l allow u t i l i z a t i o n f o r numerical calculations and w i l l enable the reader t o judge the agreement between theory and t e s t s .

A. Fort ier , Professor a t the Sorbonne

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NACA 'IM 1377

TABLE OF CONTENTS I. 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . .

C " I T R I

GENERAL ASPECTS OF TURBULENCE. THE STATISTICAL METHOD

1. Definition of turbulence . . . . . . . . . . . . . . . . . . . 6 2. Average values - s t a t i s t i c s . . . . . . . . . . . . . . . . . . 8 3 . Random variables and the l a w s of probabili ty . . . . . . . . . 10 4. The concept of random point - velocity

f i e l d - turbulent diffusion . . . . . . . . . . . . . . . . . 13 5 . Equations of development of the laws of probabili ty . . . . . . 15 6. Random velocity. Hydrodynamic equations . . . . . . . . . . . 25 7. Systems of molecules . . . . . . . . . . . . . . . . . . . . . 29

CHAPTER I1

CORRELATIONS AND SPECTRAL FUNCTIONS

8. Introduction - correlations i n

9. Properties of the functions f , g, a, b, c. space - homogeneity, isotropy . . . . . . . . . . . . . . . . 33

Incompressibility . . . . . . . . . . . . . . . . . . . . . . 37 10. Spectral decomposition of the velocity . . . . . . . . . . . . 43 11. Spectral tensor and correlation tensor . . . . . . . . . . . . 45 12. Spectral tensor of isotropic, incompressible turbulence . . . . 50 13. Energy interpretat ion of the spec t ra l function F(k) . . . . . . 55 14. Relations between spectral function F(k)

and correlation functions f ( r ) and g ( r ) . . . . . . . . . . . 58 15. Lateral and longitudinal spectrum . . . . . . . . . . . . . . . 62

CHAPTER I11

DYNAMICS OF TURBUL;ENCE

16. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 64 17. Fundamental equation of turbulent dynamics . . . . . . . . . . 65 18. Case of isotropic turbulence . . . . . . . . . . . . . . . . . 68 19. Local form of the fundamental equation . . . . . . . . . . . . 73 20. Solution of the fundamental equation, when the

21. Solutions involving a s imi la r i ty hypothesis . . . . . . . . . . 80 t r i p l e correlations are disregarded . . . . . . . . . . . . . 76

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NACA '"I 1377

22 . Transformation of the fundamental equation in spectral terms . . . . . . . . . . . . . . . . . 86

23 . First theory of Heisenberg . . . . . . . . . . . . . . . . . . 89 Space-time correlations . . . . . . . . . . . . . . . . . . . 93

25 . Second theory of Heisenberg . . . . . . . . . . . . . . . . . 99 24 . First theory of Heisenberg (continued) .

0

CHAPTER IV

THEORY OF LOCAL ISOTROPY AND STATISTICAL EQUILIBRIUM

26 . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 106 27 . Definition of local homogeneity and local isotropy 108 28 . Similarity hypotheses . Statistical equilibrium . . . . . . . 111 29 . Case of high Reynolds numbers . . . . . . . . . . . . . . . . 114 30 . Validity of the similarity laws . . . . . . . . . . . . . . . . 118

and Heisenberg's theories . . . . . . . . . . . . . . . . . 121

. . . . . .

31 . Interpretation of the laws of statistical equilibrium in spectral terms . Weizsacker's

CHAPTER v DECAY OF TIIE TURBULENCE BEHIND A GRID

32.History . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 33 . Initial and final phase of turbulence . . . . . . . . . . . . 126

final phase of turbulence . . . . . . . . . . . . . . . . . 132 35 . The concept of "dynamic statistical equilibrium" . . . . . . . 135

structure of the spectrum of turbulence . . . . . . . . . . 141

34 . Concepts regarding the structure of the

36 . Synthesis of the results relating to the

INDEX OF PRINCIPAL NOTATIONS. FUNDAMENTAL FORMULAS. AND DIMENSIONAL EQUATIONS . . . . . . . . . . . . . . . . . . . 143

APF%XDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Some experimental results . . . . . . . . . . . . . . . . . . . 147

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

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NATIONAL ADVISORY COMMImE FOR mRONAUTICS

TECHNICAL MEMORANDUM 1377

By L. Agostini and J. Bass

INTRODUCTION

The theory of turbulence reached i t s f u l l growth a t the end of the 19th century as a result of the work by Boussinesq (1877) and Reynolds (1893). It then underwent a long period of stagnation which ended under the impulse given t o it by the development of wind tunnels caused by the needs of aviation. Numerous researchers, mathematicians, aerodynamicists, and meteorologists atteinpted t o put Repalds ' elementary s t a t i s t i c a l theory i n a more precise form, t o define the fundamental quantit ies, t o set up the equations which connect them, and t o explain the pecul ia r i t i es of turbulent flows. This second period of the science of turbulence ended before the w a r and had i t s apotheosis a t the 1938 Congress of Applied Mechanics.

During the war , some isolated sc ien t i s t s - von Weizsgcker and Heisenberg i n Germany, Kolmogoroff i n Russia, Onsager i n the U.S.A. - started a program of research which forms the th i rd period. By a system of assumptions which make it possible t o approach the s t ructure of turbulence i n well-defined l i m i t i n g conditions quantitatively, they obtained a certa'in number of l a w s on the-correlations and the spectrum. These results, once they became known, caused a spate of new researches, the most outstanding of which are those by the team Batchelor-Tomsend a t Cambridge.

The analysis of these works became the subject of a series of lec- tures at the Sorbonne i n February-March 1949, which subsequently were edited and completed. The mathematical theory of turbulence had already been published i n 1946 (ref. 3) but practicaUy ignored a l l publications l a t e r than 1940. language of turbulence, it was deemed advisable t o start with a detailed account of the mathematical methods applicable t o turbulence, inspired at f i r s t by the work of the French school, above aY f o r the basic prin- ciples, then the work of foreigners, above a l l f o r the theory of the spectrum.

Since the late reports have improved the mathematical

*%es Thgories de la Turbulence." Publications Scientif iques e t Techniques du MinistGre de L'Air, No. 237, 1950.

2 NACA 9 4 1377

The f i r s t chapter deals with the precise and elementary def ini t ion of turbulence (sections 1 and 2) and describes the too ls of mathematical statistics on which the ultimate developments are based. Star t ing from paragraph 3, chapter I, the reader should be famil iar with the def ini- t ions of the calculus of probabili t ies, the theorem of t o t a l probabili- t i e s , and the theorem of compound probabi l i t ies . theoretical, and i t s aim i s t o review the methods suggested by the theory of random functions and which seem likely t o be applied t o turbulence. Only the use of Navier's equations has, so f a r , produced posit ive results, and chapters 111, N, and V are largely devoted t o it. However, it should be pointed out that there are theories l e s s familiar t o hydrodynamicists which have been proved i n other branches of physics (kinet ic theory of gases, quantum mechanics). The purely random method i s described i n paragraph 5, and i ts adaptation t o molecular systems (according t o Born and Green) i n paragraph 7. The s t a t i s t i c a l method has the advantage of furnishing a remarkable demonstration of the general equations of hydrodynamics (paragraph 6) and of providing- an exact c lass i f ica t ion of the s t a t i s t i c a l parameters of turbulence (paragraph 4) , which is interest ing t o keep in mind when studying the foreign reports, t o o exclusively devoted t o spat ia l correlations. I n any case, the reader who wants t o read chapters I V and V can pass up most of chapter I, except perhaps paragraphs 1 and 2, without major trouble.

T h i s chapter i s en t i r e ly

I

Chapter I1 deals with the kinematics of s t a t i s t i c a l mediums and, , particularly, isotropic mediums. It seemed prac t ica l t o include at the same time the velocity correlations, the theory of which has been given i n almost f i n a l form by K%dn , a t the end of the second period, and of the spectrum, the theory of which, due t o Taylor's i n i t i a t ive , has only been achieved very recently. t i c a l functions, and t h e i r detailed knowledge i s not indispensable f o r reading the r e s t of the chapter. The resu l t s and the formulas of chapter I1 are constantly applied i n the subsequent chapters, but it i s not necessary t o know the proofs which are, i n most cases, a simple matter of calculation. The most important of these formulas are, moreover, com- piled i n a special section following chapter V.

O n l y paragraphs 10 and 11 refer t o statis-

Chapter I11 i s a mathematical study of the application of Navier's equations t o turbulent motion. mental. Their main purpose i s t o r e c a l l firmiin's results of 1938 with some improvements and some supplements of more recent date. The paragraphs 20 and 21 review a cer ta in number of physically reasonable solu- t ions of the K&&n-Howarth fundamental equation. Some of these assume particular importance i n chapter V but, first, it seems advisable t o give an impartial view of the whole and t o proceed progressively in to the domain of the concrete. by Heisenberg which involves time correlations and from which probably not a l l possible r e su l t s have been extracted. It i s not indispensable t o have knowledge of t h i s i n order t o continue. Paragraph 25 contains a

Paragraphs 17, 18, 19, and 22 are funda-

The paragraphs 23 and 24 deal with an equation

.

.

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NACA !Uvl 1377 3

I

mathematical account of Heisenberg's numerical theory of the spectrum, which is taken up again in chapter V from a more physical point of view and whose examination, contrary to paragraphs 23 and 24, proved useful before attacking chapter V.

Cha@ters IV and V deal with n e w physical theories involving similarity hypotheses and producing numerical laws. works of Kolmogoroff and Weizszcker, chapter V th6se of Heisenberg, Batchelor, and Townsend on the decay of turbulence created by grids.

Chapter IV reviews the

Finally, in an appendix, the theoretical discussions of chapters 111, IV, and V are illustrated by some correlation curves and spectrum curves measured directly in the wind tunnel by A. Favre, in the laboratory of the mechanics of the atmosphere at Marseille, or derived from experimental curves by elementary transformations .

An exhaustive study of modern theories of turbulence calls for some knowledge of the calculation of tensors, probabilities, and statistical analysis, besides the classical conceptions of differential and integral calculus.

As regards the tensors, knowledge of the definitions and fundamental operations with rectangular Cartesian coordinates is sufficient. There are a number of articles on this subject, but they generally lean toward the tensor analysis with curvilinear coordinates for which there is no need. an important part in the theory of the boundary layer around an airf'oil.) Incidentally, there is available a little book recently published, by Lichdrowicz, entitled: Armand Colin). O n mathematical statistics, the book by Darmois, published by Doin (1928), can be consulted. random functions, consult the first part of Bass' report (ref. 3) and the appendix to d'llngot's llcomplements of mathematics" (editions of the Revue d'wtique), edited by Blanc-Lapierre. on statistical functions can be found in Ievy's book: Processes and Brownian Motion" (1948).

(It should be noted that the tensor analysis plays, in contrast,

Elements of tensor calculus (collection

For the elementary theory of

More detailed information "Stochastic

The present report deals only with general theories which are valid whatever the physical or geometric causes of turbulence may be, as is shown in chapter IV. the study involves a problem of decay of turbulence that is compatible with the turbulence in wind tunnels. cable to mediums of extremely diverse scales, from the microturbulence to the terrestrial atmosphere (Dedebant and Wehrle'), to stellar atmos- pheres and to interstellar matter (Weizsacker). However, among the hypotheses there is always that of the incompressibility, which precludes the application to sonic or supersonic flows. This is not an indispensi- ble hypothesis, but it simplifies the calculations considerably, and the

Only in chapter V the assumptions are limited and

These theories are probably appli-

4 NACA '1M 1377

i consequences a re easy t o check. are practically the only ones studied up t o now.

So the incompressible turbulent motions

Among the other hypotheses worthy of discussion, those referr ing t o processes of energy transport (paragraph 24 and chapter V) assume t h a t a l l turbulent energy comes from the motion of the whole, dissociated and broken up by the obstacles of periodic structure. But nature furnishes examples of different turbulences such as the so-called thermal turbulence, f o r example, where the source of energy i s not the fan of a wind tunnel but the s o l a r radiat ion suitably transformed i n t o kinet ic energy. Kolmogoroff's theory of l oca l isotropy applies probably t o thermal turbulence, but the forms of energy transport i n the spectrum must be different from those encountered i n wind tunnels. astronomic turbulent mediums alluded t o previously. So the foregoing remarks l i m i t the scope of the recent theories, in a cer ta in measure.

. The same applies t o tbe

Omitted en t i re ly was the problem of turbulent boundary layer which, experimentally, depends on the same technique, but has been approached by different mathematical methods. Furthermore, it is a complicated problem where the turbulence is neither homogeneous nor isotropic.

c In w h a t measure are the results, suitably demarcated, defined? The s t a t i s t i c a l character of the turbulent velocity seems a c lear and w e l l - established notion and consequently the mechanics of turbulence w i l l be a s t a t i s t i c a l mechanics. cerned, we are therefore on sol id ground. (chapter 111) i t s e l f is likewise w e l l established, by means of the hypothesis of the va l id i ty of the Navier equations. generally adopted because it is convenient and, one m i g h t say, necessary, has however a t times raised considerable doubts. mental verifications do not contradict it, there i s yet no occasion t o r e j ec t it.

As far as the kinematics (chapter 11) a r e con- The dynamics of turbulence

This hypothesis,

However, w h i l e experi-

But, w h a t should be expected from experimental ver i f icat ions? F i r s t of al l , it i s found that the theories are s t i l l rather imperfect. In f ac t , the theories are usually limited, acceptable i n the l imiting con- d i t ions which a re d i f f i c u l t t o a t t a i n actual ly (very high Reynolds numbers, f o r example). the t rue conditions are too far removed from the theore t ica l conditions.

No rigorous ver i f ica t ion should be expected since

O n the other hand, tbe accuracy of measurement i s low. The or ig ina l reason for it l i e s i n the very nature of the turbulent phenomenon, and i ts irregular and badly defined character. Furthermore, the anemometers are coarse instruments, t h e i r operation not suf f ic ien t ly known i n the presence of turbulence, and t h e i r interference with the f l u i d sometimes a l i t t l e mysterious. A l l t h i s helps t o lower the experimental precision. To measure the "length of dissipation," f o r example, two ways are open: one i s t o measure the correlations of the velocity a t two infinitely close

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NACA ?M 1377 5

L points which has no prac t ica l sense since two anemometers cannot be brought together indef in i te ly j the other i s t o use one anemometer f o r measuring the mean square value of the derivative of velocity. operation of taking a derivative of a function as complicated as the velocity is by itself inaccurate and introduces serious scat ter ing of the points of the derivative curve. the results i s extremely limited.

But the

So i n both cases, t h e accuracy of

I n consequence, the experimental ver i f icat ions can be applied only To i l l u s t r a t e : t o the orders of magnitude.

according t o theory, i s constant, one constructs the representative curve of the parameter and, if t h i s c w e has a suf f ic ien t ly extended maximum, one estimates that the experiment closely confirms the theory, conceding that, as the experimental conditions more nearly approach those st ipu- la ted by theory, the maximum f l a t t ens out more and more, and one does not appear too severe i n the examination of the sca t t e r of the tes t points .

t o check whether a parameter,

On t h i s assumption, the verifications of the experimental l a w s of turbulence, due i n par t icular t o Townsend, are encouraging. Therefore, it i s well t o re ta in the hypotheses of chapters 111, IV, and V, although some of them obviously have their limitations, and we ought not hope f o r r e su l t s greater than actual ly can be given. Take an example drawn from the theory of the spectrum, f o r instance. A t the beginning of decay of turbulence i n a tunnel and a t high Reynolds numbers, a ce r t a in spec t ra l function F(k) i s of the form Ck4 f o r the small values of wave nun- ber k It then passes through a maximum, then becomes proportional t o k-’I3, and f i n a l l y approaches zero as k the regions of the axis of k enumerated remain acceptable are badly defined and connected by zones of which the structure i s not known. So the future task of the theo r i s t s w i l l be t o combine these p a r t i a l resul ts into a single acceptable l a w at least t o the extent that it does not become fundamentally incompatible with the nature of turbulence, since k = 0 up t o k = 00. Only then w i l l there be a t rue theory of turbulence.

m, maybe as k’7. But in which the fragments of the l a w s t o be

6

CHAPTER I

GENERAL ASPECTS OF TUFBULENCE

THE STATISTICAL METHOD

4

1. Definition of turbulence:

Our purpose is t o discuss the operation which consists of measuring, a t a given point, the velocity of a flow which, t o avoid every d i f f icu l ty , i s assumed t o be steady and uniform at the usual macroscopic scale. The measuring instrument i s an "anemometer" with approximately determined dimensions and t i m e constant, and of suf f ic ien t ly idea l nature so as not t o disturb the flow by i t s presence. If this anemometer i s large enough, it measures the "velocity of the main flow" of the f lu id . If it is very small, it can be imagined that it operates i n discontinuous manner, never undergoing the influence of more than one molecule a t a time. An anemom- e t e r sensitive t o the individual action of molecules is, of course, unattainable, but the idea of such an instrument is convenient f o r representing the extreme l i m i t of fineness of kinematic measurements i n a f lu id . Between these two extremes, the indications of the anemometer depend upon the structure of the f lu id . It may happen, by exception, that, when i t s dimensions a re progressively reduced, the velocity which it indicates remains unchanged up t o the moment where the individual influence of the molecules starts t o make i t s e l f f e l t and where the indications lose a l l s t a t i s t i c a l significance.

8

The flow i s then said t o be laminar.

But, i n general, the matters are otherwise. We s t a r t with a f i r s t

This anemometer measures the mean speed of the molecules i n a anemometer which, through i t s dimensions, f i xes a cer ta in scale of measurement. cer ta in volume V. i n such a way that volume V decreases progressively. from a certain value V1 of V, the numerical indication supplied by the anemometer changes. If V i s decreased continuously, the new indication remains stable up t o a cer ta in value he intervals ( v ~ , v3), . . . characterize the various scales of turbulence, and the motion of the f l u i d i s said t o be turbulent. More exactly, they are the conditions necessary f o r a f l u i d t o be turbulent, and which must be defined and perfected t o make them suff ic ient and pract ical .

Then it i s replaced successively by smaller anemometers It happens that,

V2, then changes again and so on. v2), (v2,

The last value Vn of the ser ies V1, V2, . . . i s that from which onward, the notion of average loses i t s significance, the number of molecules contained i n the volume

t

Vn not being large enough any longer f o r s t a t i s t i c a l purposes. The ser ies V,, V2, . . . can be discrete or b.

c continuous. I f it i s discrete, it s t i l l does not imply that the c r i t i c a l values V1, V2, . . . are mathematically defined. I n the v ic in i ty of V1, a rapid variation of the anemometer indications occurs, which subsequently become stabi l ized i n the region (Vl, If t h i s stabili- zation is not very clear , the turbulent scales are said t o succeed one another i n continuous fashion.

4

V2), and so for th .

The previous discussion ends with the notion of the turbulent f l u id . But the def in i t ion of the laminar motion given above is a l i t t l e too r e s t r i c t ive and the dis t inct ion between laminar and turbulent s t i l l not precise enough, a s proved by the following example:

Consider the motion of a i r produced by s ta t ionary waves i n a sound tube. The motion of the whole reduces, a t r e s t , t o large scale. But, a t each point there ex i s t s a speed other than zero, a periodic time func- t i on which, a t a given ins tan t , va r i e s periodically from one point t o another. This motion, l y i n g between the system a t rest and the molecular agi ta t ion, has not a turbulent character.

Turbulence, a s shown, implies f i r s t the notion of scale. But it should be added that, a t a given scale, each component of the velocity at a point i s a function of t i m e presenting a character of per iodici ty without fundamental period. This i s not a periodic function but a sum of harmonics, the frequencies of which are not multiples of an ident ica l fundamental frequency. This i r regular i ty of the turbulent ag i ta t ion is essent ia l and distinguishes it from sound agitatiqn, o r preturbulent vor t ica l motions, l i k e the ce l lu la r vortices of Benard. The mathematical symbol f o r the turbulent velocity i s not the ordinary Fourier series, but Fourier 's integral . It w i l l be discussed l a t e r .

*'

c

This concept of i r regular agitation a t a point as function of the time is not itself suff ic ient . It makes it possible t o d i f fe ren t ia te the turbulent ag i ta t ion from the periodic sound ag i ta t ion (musical sound), but not from the noise, which i s an agi ta t ion without def in i te period. What distinguishes the noise from turbulence i s the f a c t that it is propagated by waves t h a t ex i s t on surfaces of equal phase, and consequently have a regular spa t i a l distribution, notwithstanding the i r r egu la r i ty i n

I the time of the loca l velocity.'

'A descriptive and purely kinematic d i s t inc t ion is involved here. Its cause (compressibility) i s not discussed.

The difference between turbulence and sound ag i ta t ion should become plain from the following example: I n a turbulent wind tunnel, we se lec t a t a point the longitudinal component of the velocity with a hot w i r e and send the e l ec t r i c current of the hot Wire t o a loud speaker on the outside of the tunnel. The atmosphere becomes the source of an i r regular ag i - ta t ion, which propagates by waves and i s not turbulence, although, a t a point, the in te rna l turbulent motion and the external sound motion have some important kinematic elements i n common.

J I

, k -J

a NACA 'IM 1377

For a given scale, each component of the turbulent velocity i s an irregularly periodic function of both space and t i m e .

It seems that turbulence i s w e l l defined by these kinematic condi- t ions and hence is distinguished from a l l other more organized f l u i d mot ions.

2. Average values - s t a t i s t i c s :

Figure 1 represents the record of a component u of the turbulent (Record of the velocity of turbulent

airspeed, 20 m/sec; in tens i ty of velocity as function of t h e time. agi ta t ion i n a 20-cm by 30-cm tunnel: turbulence, time of recording i s 0.03 second. )

The most natural method of measuring the mean velocity on the graph consists i n forming the ordinary in tegra l

= 1 f u ( t ) d t T

a

extended over the t o t a l duration T of recording. This method is , i n general, not very sat isfactory because the operat-ion lacks precision when the curve u ( t ) i s complicated.

A more precise method consists i n dividing the graph by para l le l s t o the axis of t, suitably close together i n the ordinates u1, u2, u3, . . ., i n measuring the number ni of points where the l i n e of the

ordinate L(ui + U i + l ) meets the curve u ( t ) and then i n computing the 2

quantity

i i = > u i - ni n i

n = ni is the t o t a l number of points m e t by a l l para l le l s . i For t h i s calculation, a prof i table f i r s t stage consis ts i n first

constructing the graph giving the corresponding s t a t i s t i c a l frequency n.

w f i = -L f o r each velocity u i . Crossing the l i m i t obviously makes it ... I1

possible t o plot a curve of frequency proportion of the values of the velocity comprised between u and u + du

f = f ( u ) ( f ig . 2) such t h a t the

w

NACA TM 1377 9 Y

c i s equal t o f(u)du, the in tegra l s," f (u)du, which replaces

being equal t o unity. The f i n a l expression of Ti is then

It replaces (2-2) and should be compared w i t h (2-1).

The prac t ica l operations enabling the replacement of % by TI correspond t o well-known mathematical operations. The mode of computing ii i s that of a Lebesgue integral , and Um is an in tegra l of the c l a s s i ca l type of Riemann. If % is computable, both methods yield the same result. But it may happen that Riemann's in tegra l does not ex i s t because the function u ( t ) i s too complicated mathematically. However, i n general, Iebesgue's integral ex i s t s ( if and natural ly bounded). p rac t ica l case w h e r e the curye u ( t ) continuation of the integration. he function f (u) , continuous and different iable i n the current cases, is the medium which, determined once f o r a l l , replaces the calcuLation of Lebesgue's in tegra l by that of an ordinary in tegra l by means of the plott ing of curve

u ( t ) i s measurable,

i s too complicated f o r an accurate This mathematical case corresponds t o the

f ( u ) .

From the function f (u), other averages can be computed. For example, the amount of differences of the speed with respect t o i t s mean value can be figured by computing the mean value of (u - E)*. Rather than defining this average by Riemann's in tegra l

T 1 b(t) - %I2dt T O

(2-4)

it is simpler and more.precise to ,use the formula

2 2 ( U - TI) = Jl (u - E) f(u)du (2-5)

which, once the curve f ( u ) i s plotted, c a l l s only f o r operations of a simple character.

*

Thus, it is seen how much the construction of the curve f ( u ) simplifies the numerical calculations of turbulence. mental techniques make f o r d i rec t attainment of t h i s curve without passing through the numerical analysis of a velocity record.

averages such as equipment without f i r s t plot t ing the curve

Various experi-

For measuring the - 2

(u - u) , it i s often advisable a l so t o use specialized f (u) .

3. Random variables and the l a w s of probability:

The theoret ical significance of the function f ( u ) i s analyzed. It groups the s t a t i s t i c a l data contained i n the i n i t i a l curve u ( t ) , with t h i s exception t h a t the chronological order i n which the ve loc i t ies actually follow one another does no longer appear. This l imitat ion i s quite natural, though, and it w i l l be seen later that t h i s order reappears, i n a cer ta in measure, by the introduction of space and time correlations.

Obviously, only s t a t i s t i c a l data can supply stable information on turbulence. times while taking every reasonable precaution so that the conditions are identical , it obviously resu l t s i n curves which absolutely are not superposable. The function u ( t ) has not, therefore, the charac t e r of .permanence that i s suitable f o r representing the l a w s of a physical phenomenon.

or t o averages such as E, (u - E)* whose values a re character is t ic numbers of the investigated flow, and which are derived by simple mathematical operations f r o m the function Hence, we d i r ec t our a t ten- t i o n t o th i s function which can be regarded as representing the f i r s t l a w of turbulence.

When the same record of the velocity i s begun again several

u ( t ) a

But t h i s character is relevant t o the function f ( u )

f ( u ) .

To say t h a t the,veloci ty is characterized by a curve of s t a t i s t i c a l

f ( u ) frequencies i s t o say, by comparing the frequencies w i t h probabi l i t ies and t i c a l quantity; f(u)du the velocity is contained between u and u + du.

with a density of probability, that t h i s velocity i s a statis- i s the probabili ty that the chosen component of

A prior i , such a l a w of probabili ty could be dependent on the time. That would correspond t o a turbulent flow f o r which the l a w s varied w i t h respect t o time. There i s no contradiction t o the i n i t i a l assllmptions of permanence here. It i s a question of scale. I n order f o r the experimental operation by which f ( u ) i s defined t o have any meaning, it is necessary that two conditions be real ized simultaneously:

(1) The number of osci l la t ions i n the time in te rva l must be great enough t o furnish sat isfactory s t a t i s t i c s .

T involved I)

(2) The l a w s of turbulence i n t h i s time in te rva l T must be practic- a l l y permanent. -d

NACA '1M 1377 11

If it i s not so, it might be d i f f i cu l t t o reconcile the s t a t i s t i c a l theory w i t h the experiment. Fortunately, those "ergodic" conditions are pract ical ly always realized i n the usual cases and are therefore taken f o r granted i n the following:

The velocity has three components u p u2, and u3 which are t reated as three random variables, components of a random vector.

It should be noted here that the velocity is perhaps not suff ic ient f o r characterizing the turbulence. It m i g h t appear useful t o introduce other quantit ies, such as the pressure, which should be t reated as a s t a t i s t i c a l quantity. But, owing t o the equations of motion, this then w i l l be a function defined by the velocity and i ts derivatives. present, it is assumed that the turbulent motion i s suf f ic ien t ly w e l l defined by i ts velocity so that the problem narrows down t o the l a w s of probabili ty applied t o the velocity.

For the

The simplest of these laws, that which immediately generalizes the f ( u ) , is the l a w of probabili ty of the system of experimental function

three components ul, u2, u3 of the velocity. This l a w may vary as function of the t i m e t of the space (variation of turbulence i n terns of the distance from the w a l l s ) . It i s therefore a function of t and the ordinates x1, x2, x3 of the point of measurement.' Its density i s denoted by

(problem of spontaneous decay of turbulence) aad

or, abbreviated, f (u; x, t ) . The quantity f du1 due du3 or, abbreviated, f du, represents the

probabili ty tha t , 'at the point XI, x2, x3 (or x ) and a t the instant t, the three velocity components are comprised between ul and ul + dul, ~2 and ~2 + du2, ~3 and ~3 + du3.

But a single law of probability defined i n terms of four parameters XI, x p X3' and t i s not adequate f o r characterizing turbulence. It i s necessary t o introduce the more profound concept of random function and t o consider the turbulent velocity as a random function of space and time. This is the random velocity f ie ld .

This point i s now t o be defined. An isolated random quantity i s defined by i t s l a w of probability. But, t o define a system of coexistant random quantit ies requires more than jus t t h e i r l a w s of individual prob- a b i l i t i e s . The stochastic dependencies or correlations between these

quantities must a l so be known. not give the right t o speak of the system of corresponding random quan- t i t ies without completing the data by those of the correlations. posing that the family i n question depends on a continuous parameter We know then an isolated random quantity U ( t ) f o r each value of t. This immediately suggests grouping the U ( t ) corresponding t o the various values of t i n a well-defined system of s t a t i s t i c a l quantit ies. This ca l l s f o r the introduction of the correlations between the U ( t l ) , U ( t 2 ) , . . . corresponding t o an a rb i t ra ry system E - of the values tl, t2, . . . of the parameter. To proceed thus, means t o define a random function. Naturally, this a l so holds f o r l a w s of probabi l i t ies dependent on several parameters. Thus, when a turbulent medium i s represented as a velocity f i e ld , the system of veloci t ies a t each point and a t each instant precisely const i tutes a system of coexistant s t a t i s t i c a l quantit ies, of which the physical interactions characterizing the structure of turbulence have the correlations f o r mathematical description.

Taking a family of probabili ty l a w s does

Sup- t .

Among the systems E, the simplest are the denumerable systems and even the f i n i t e systems and, among the l a t t e r , the simplest one which i s not t r i v i a l i s that of two elements, that is, of two points of space and t i m e .

The concept of random function thus suggests the comparison of the velocity vectors a t two different points ol? t h e i r f i e l d of definit ion, that is, f o r two different positions x m d x ' (x represents the point of the coordinates xl, x2, x ) and f o r two instants t and t ' It concerns a s t a t i s t i c a l comparison which makes it possible t o define the l a w of probabili ty of the two systems of the velocity components a t points x, x ' and instants t and t ' . This l a w has a very c lear physical meaning and i s easy t o define experimentally or, a t the leas t , t o construct the surface of ( s t a t i s t i c a l ) frequencies corresponding t o one velocity component a t point x and a second component at point x ' , the measurements being spaced at a chosen time in te rva l T .

3

c

I n practice, the question i s frequently handled from a less general point of view. selves as such but only with t h e i r most simple moments, those of the second order which are associated, as w i l l be shown, with cer ta in "physical" aspects of turbulence and, possibly, w i t h ce r ta in moments of the t h i r d order that play a par t i n modern theories. The moments of the second order consti tute the correlation tensor of the l a w of probabili ty of the velocity f i e l d a t two points and a t two instants . They are measured direct, without resorting t o frequency curves or frequency surfaces or t o velocity recordings. A detailed study follows later.

One is not concerned with the l a w s of probabili ty them-

. P

3B

c

C'

.*

4. The concept of random point - velocity f i e l d - turbulent diffusion:

The density of probabili ty f(u; x, t ) of the veloci ty f i e l d contains the velocity of the whole f lu id , but not i t s density. or, abbreviated, p(x, t ) denoting the quotient of the density a t point x by the f l u i d mass, p i s a normalized function, as a density of probabil- i ty , which means that

With p(x1, x2, x3; t )

where dx represents the element of volume dxl dx2 dx3 and the in tegra l i s extended t o the volume V occupied by the f lu id .

I n a l l modern studies on turbulence, the f l u i d i s natural ly assumed

could be simply replaced by a constant; but incompressible, so that p is a constant, equal t o 1/V i n the volume V, and zero a t the outgide. the more general conclusions t o be arrived a t ultimately a re more complete if t h i s simplification i s not made. On the other band, it is interest ing t o foresee, a t a cer ta in stage of the theory, tbe day when it w i l l be possible t o study the turbulent motions i n conditions where the compressi- b i l i t y is no longer negligible. For these reasons, p i s t reated here as a function of x (and even of t, i f necessary).

p

The product R(x, u; t ) = pf i s now formed. It obviously i s normalized with respect t o the system of the s i x variables x, u

R dx du = 1 s (4-2)

R presents thus the characters of a density of probabili ty with respect t o these six variables. The quantities x1, x2, x3 are regarded a s the coordinates of a moving point, u1, u2, u3 t h i s point. These are s i x random quantities of which the l a w of probabili ty z t instant t i s known. Thus a random point can be associated with the turbulent f l u id i n correspondence with a given scale.

as the velocity components of

This point i s now t o be discussed as w a s the velocity f i e l d i n the preceding paragraph.

The posit ion and the velocity of t h i s point are random functions of the time. The theory of random functions suggests the study of the l a w of probabili ty of the system of positions and veloci t ies of this point f o r an arb i t ra ry combination of instants. The i n i t i a l analysis w a s on

1 4 NACA '1M 1377

the l a w of probability a t one instant. The generalization from one t o two instants seems t o us adequate f o r forming a physical theory of turbulence and, i n particular, f o r considering the s t a t i s t i c a l organization i n time of the velocity, and w h a t may be called the "interactions between turbulent par t ic les . "

The values of the positions and of the velocity must therefore be associated t o two instants t and t ' . This association is a stochastic relationship defined by the density of probabili ty of tk following system of 12 s t a t i s t i c a l variables :

a t instant t X I , X2' x3 posi t ion

I a t instant t ' xl', x2', x3' posit ion

velocity J

or, abbreviated,

G(x, x ' , u, u'; t, t ' )

and it i s assumed that it defines the turbulent motion. By w h a t stages can it be measured?

According t o the theorem of compound probabili t ies, G dx dx' du du' i s the product. of two factors:

(1) The probability that at the instants t and t ' the s t a t i s t i c a l image point of the f l u i d m i g h t have positions contained within the in te r - vals (x, x + dx) and (x ' , x' + dx').

(2) The probability (conditional) that, these positions being fixed, the velocit ies are contained in the intervals (u, u + du) and (u ' , u' + du ' ) .

T h i s last probability i s designated by R(u, u' ; x, x', t, t ' ) d u du'. According t o the theorem of compound probabili t ies, the probabili ty (1) i s the product of the probability found, at instant t, i n the interval probability that, the position a t instant t being chosen, i t s posit ion at instant t ' i s found i n the interval x ' , x' + dx' .

p(x, t ) d x that the random point i s x, x + dx, through the (conditional)

This last probabili ty i s designated by p(x'; x, t, t ' ) dx ' . We can write

G(x, x ' , u, u ' , t, t ' ) = H(u, u'; X, x', t, t ' ) p ( x ' ; X, t, t ' ) p ( x ; t )

(4-3)

What I s the significance of the three fac tors of which G i s the product ?

We already know the density of the f lu id .

p, which represents except f o r a numerical f ac to r

The function H and a t two instants . f i e l d already discussed a t the end of paragraph 3. gives no complete picture of turbulence. t i o n p .

i s the l a w of a random velocity f i e l d a t two points It i s natural t o identify it with the law of the

It i s seen t h a t it It does not explain the func-

The function p represents the turbulent diffusion at a chosen scale. It i s the re la t ive density at point x ' and a t instant t ' of f l u i d elements which have passed neighboring point x a t instant t. It should be pointed out t h a t the diffused portion of the f l u i d i s essen- t i a l l y compressible since the density p(x ' ; x, t, t ' ) proportion as the point x' i s removed from the i n i t i a l point x where it i s maximum. The spread can be materialized and p can be measured by introducing with the necessary precautions a t point x a dye that spreads i n the f lu id . But it i s a rather t i ck l i sh matter t o separate the e f f ec t s of the various turbulent scales, especially of the molecular diffusion. It i s accomplished by adapting the pa r t i c l e s of the "dye" t o the chosen scale. fusion of the tunnel flow v i s ib l e i n h i s experiments a t the Ins t i t u t e of Fluid Mechanics, a t Li l le , by injecting soap bubbles at a point i n place of dyed par t ic les . These soap bubbles, because of t h e i r s ize , were sensi t ive t o the turbulent fluctuations and were used successfully f o r measuring a density of turbulent diffusion.

decreases i n

This way bung16 de F h e t rendered the turbulent d i f -

5 . Equations of development of the laws of probability:

The analysis of the turbulent velocity f i e l d made it possible t o represent a turbulent f l u i d by two random functions X ( t ) and U ( t ) playing the par t of the posit ion and of the velocity f o r a random material point. It w a s shown how this concept of random point gives a very com- p l e t e picture of the turbulence. But this picture is qual i ta t ive and must be made more quantitative.

16 NACA ?M 1377

A t the beginning, no dist inct ion i s made between the velocity vector X ( t ) and the posit ion vector U ( t ) . The s i x components of these two vectors are considered as those of a unique vector dimensional space, by putting the density of probabili ty of t h i s vector by the abbreviation of

g(t) i n a s ix- u1 = x4, 9' x5, u3 = X6, and designating

R(x; t ) , this notation being R(xl, x2, x3, x4, x5, x6; t ) , that is, of

R ( q , x2, X3' U l , u2, u3; t ) .

We already had applied (4-3), the theorem of compound probabi l i t ies , t o the law of probabili ty instants t, t ' :

G of the posit ion and the velocity a t two

G(x, x ' , u, u ' ; t, t ' ) = [~(x ; t ) p ( x ' ; x, t, t ' ) H(u, u ' ; x, x ' , t, t ' ) 1 The question involved essent ia l ly the separation of the velocity,

which figures i n the fac tor H, from the posit ion. But there i s another way of applying t h i s theorem. instants t and t ' by writing

It consists i n separating the two

G(x, x ' , U, u'; t, t ' ) = R ( x ' , u ' ; t ' ) K ( X , U; x ' , u ' , t, t ' )

hence it resul ts , according t o the theorm of the t o t a l probabi l i t ies , that

R(x, u; t ) = b ( x ' , u ' ; t ' )K(x, u; x ' , u t , t, t ' ) d x ' du'

(5-2)

K passage'' of the s t a t e of the random point a t instant instant t > t' .

i s a conditional density of probability, that of the "probability of t ' t o i t s state a t

Abbreviated, we get

R(x; t ) = !(XI; t ' )K(x; x ' , t, t ' ) d x ' (5-3)

To exploit t h i s equation, recourse i s bad t o a method patterned after the concept of J. Moyal (ref. 33). l i nea r integral transformation fo r passing from the density of proba- b i l i t y R a t instant t ' t o the same density a t instant t. The density

This idea was t o consider (5-3) as a

NACA '1M 1377

of probabili ty of the passage I

17

i s the kernel of the transformation. With K t t t function

indicating the l inear operation which has as kernel the K(x; x ' , t, t ' ) , one may write symbolically:

R(x; t ) = K t t t R(x ' ; t ' ) (5-4)

This transformation has special properties ra ther d i f f i c u l t t o define. Suffice it t o state that it reduces t o the identicaltransforma- t i o n f o r t = t ' . It i s assumed that it has an inverse = without, however, prejudicing the relations between Kt t t and K t t t .

The infinitesimal transformation applied t o K t t t i s now examined. To t h i s end, R is assumed differentiable with respect t o t and bR/& calculated

lim

If , as assumed, the function R i s different iable with respect t o independent of t ' . It can be computed by giving parameter t ' (independent of variable T) any value not exceeding t, such as t ' = t, f o r example. Hence

t, the l i m i t of the second member exists, and it i s a function of x,t

The operator

(5 -5 )

18 NACA IM 1377

defines the infinitesimal transformation of K, and R confirms the fundamental functional equation

which, theoretically, enables R(x; t) to be computed at instant t when R(x; t') is known at an initial instant t'.

The foregoing calculation cannot be explained in a simple manner K(x, x', t, t') from formula (5-3) because the limit of the function

does nit exist when t' -+ t; this is a symbolical "function of Dirac" which expresses the ideqtity Qt = 1 in the functional formalism. It is preferable to pass, as Moyal did, from the densities of probability to characteristic functions. Moyal's calculation follows:

With the function K is associated the characteristic function of the increment g(t) - X(t'), the value of X_(t') once fixed, or by definition

@(a; x', t, t') = ia(x-x') K(x; x', t, t')dx (5-7)

One assumes likewise :

cp(a, x, t) = seiQxR(x; t)dx (5-8)

the characteristic function of X_(t).

According to the theorem of total probabilities, one has:

cp(a, x, t) = leiax'@(a; x', t, t')R(x'; t')dx' (5 -9 )

This relation replaces (5 -3 ) .

The two members of (5-9) are differentiated with respect to t

R(x'; t')dx' im' @(a; x', t 4- 7, t') - @(a; X, t, t') s T

a - g ! = l i m e at 7 3 0

~ ~~

c

NACA IM 1377

Whereas th- conditi /

ns of differentia, 1

(permutation of signs lim and of 1) cannot be verif ied on (5-3), they

can be here, i n a l l current cases. Since the l i m i t must not depend on t', l e t t ' = t, so that

because @(a,; x, t, t ) = 1, f o r

The l i m i t that f igures under the

B(u; X I , t ' ) , whence follows the basic equation

t ' = t.

sign is a cer ta in function s

equivalent t o (5-6). by taking the Fourier transforms of the two members. Lastly, &/at is expressed i n form of an in tegra l transformation of R, equivalent t o the transformation L, which, after introducing a kernel function L(x, x', t), gives

From that, two probabili ty dens i t ies can be recovered

aR(x, t, = b(X', t)L(x, x', t)dx' a t (5-11)

The second member of (5-11) i s represented i n in tegra l form but, i n many cases, it can be expressed in form of a d i f f e ren t i a l operator of f i n i t e or i n f in i t e order.

Examples. - Supposing the probability of passage obeys the Laplace- Gauss l a w and, t o avoid any confusion of the notations, the previous instant; the differences of X(t0) and z(t) are denoted by So = S (to) and S = S ( t ) , the correlation coefficient between X(t0) and &(t) by r = r to, t . It i s knm that the l a w of probabili ty re la ted t o X ( t ) , - when the value ~0 of X(t0) is given, has f o r density

t o now denotes

c

0

20 I

NACA TM 1377

Using Moyal's method, it is shown that the probability density which assumes K as kernel of the probability of passage satisfies the equation of partial derivatives

R ( x , t)

where r' represents the value of for tl = t, and S '

denotes the derivative - S(t). d dt

The equation (5-13) is, moreover, demonstrated very simply by a direct method. By definition of K

this equation being, in particular, satisfied when

(5-14) .

1 R(x, t) = e

When the two members of (5-14) are differentiated with respect to t and x, equation (5-13) is verified. Reciprocally, the integral of (5-13) which is reduced to R to) for t = to, can be put in the form (5-14).

The construction of the random functions compatible with these laws of probability is an easy matter.

To illustrate:

Let h(s) be a variable random function of the parameter s, obeying the reduced Laplace-Gams l a w (E = 0 , ? = , and so that the increments

tB

c

NACA TM 1377 21

h(s1) - h (s2) and h(S3) - h (“4), corresponding to two separate intervals (sl, “2) and (9, s4 , are independent. The random function 1

n t

is now considered.

It is easily shown2 that &(t)

t3/2 and that, if 7 = t - to for typical difference S(t) =

obeys a Laplace-Gauss l a w having

D-

Without going into details, I t is simply recalled that the demonstration utilizes as intermediary the characteristic function of which the logarithm, owing to the properties of independence o’i; the dh(s), is expressed by an elementary integral.

X ( t ) , of

From these formulas, it follows that

and that R ( x , t) verifies the first-order partial differential equation

*See, for instance, Bass, reference 4, or “The random functions and their mechanical interpretation.” Revue Scientifique, No. 3240, 1943.

22 NACA TM 1377

X ( t ) is a differentiable random function. Its stochastic derivative 7 s deduced from the expression (5-15) by operations of classical form and written as Ltdh(s). It is easily proved that - - is the

2 t related mean of this derivative, if the value x of X is fixed. The equation (5-17) has many solutions which are densitiesof probability. It does not determine the function K.

Next consider the elementary random function

t x(t) = s,

stochastic derivative of the function which has been studied as the first illustrative example. It is shown that X(t) - obeys a Laplace-Gauss law

having S(t) = for typical difference and that, if to < t:

(5-19)

Therefore, R(x, t) verifies the second-order partial derivative a

e quat i on

aR 1 a2R - = - - at 2 ax2

This is the equation of heat. The solution which reduces to a given function %(x), for t = t is given by the classical formula 0'

which can also be shown by using the Fourier transform of its charactistic function.

R, that is,

NACA TM 1377 23

. Thus equation (5-20) defines here the form of the probability of

passage K(x; w, t, to) contrary to what occurred in equation (5-17).

that the function This example presents an unusual peculiarity. It is easily verified

satisfies the functional equation

v called the Chapman-Kolmogoroff equation, which characterizes the Markoff processes (or more generally, the "pseudo-markovian" processes), a functional generalization of simple Markoff chains. This equation can be written in operational notation as

v

.

It expresses that the operations K form a group. The operator

%+Tt' %,'t - I- is then written simply as Kt+Tt - ent of t'. In the general case it iS its limit L Only when 7-0 which must be independent of

. It is independ- 7 T

t, but it itself is not.

Returning to equations (5-12) and (9-13) it now is assumed that R is not dependent on t; S is then a constant and S' = 0. If r ' f 0, equation (5-13) is written as

24 NACA 'I'M 1377

The only solution of this differential equation which defines a law of probability is the density of the Laplace-Gauss l a w

1 2s2 R(x) = - e s 6

(5-25)

This example contains, as a special case, stationary random functions for which the function r(t, to) depends solely on the difference . t - to; r' is then a constant, independent of t.

The final example deals with a vectorial random function of a type to be utilized later. Consider simultaneously the random function

and its derivative

which can play the part of the speed.

It is easy to form the'functions R and K for the vector X having X and XI as components. To find the partial differentiE1 equation verified by the function it is not necessary to first form the kernel K. It is simpler to begin with the expressions of X and X', which, passing through the intermediary of the characteristic function of X, X', gives

R(x, x', t)

(5-28)

.

NACA '1M 1377 25

The operator L is thus a differential operator, of the second order with respect to x':

Returning now to the old notations and adding the letter u to the velocity vector, we get

considering X' (t) as the velocity of point X(t) . The operator L, which is the subject of this example, appertains to the particular class of operators which make it possible to define statistical kinematics and consequently hydrodynamics, as will be proved.

6. Random velocity. Hydrodynamic equations (refs. 3, 4):

The problem involves the separation of what is position and what is velocity in the vectorial random function with six dimensions In classical mechanics the velocity U(t) position X(t) . velocity is the stochastic derivative of the mean square of the position. By theory of random functions, it follows that, if the random function X(t) is completely o r at least known locally, the function U(t) can be defined by operations comparable to those used in classical mechanics to deduce the velocity from the position.

X_(t). is the derivative of the

In the present case, the assumption is made that the

The preceding paragraph contained an example (equations (26) and (27)) of two random functions X(t) and X'(t) = U(t) linked by a relation of this kind, proving in a general way that there is an operator that satisfies the necessary relation:

ARdU = 0 s

26 NACA TM 1377

and i s such tha t the basic equation (5-6) (with or iginal notations) takes the form

If A is a d i f f e ren t i a l operator, it can be expressed by an expan-

- of which the coefficients are functions

sion i n ser ies ( in f in i t e or f i n i t e ) i n terms of the symbolical parers

of the pa r t i a l derivatives

of x and u. The vector ia l operator having fo r components - occurs

only i n the first member, by i ts scalar product with the velocity.

a

a &k

The product G = RK can then a l so be decomposed i n G = ppH (compare formulas (4-3) and (5-1)). The function H defines the correlations i n the velocity f i e l d and function p of turbulent diffusion.

i s the mathematical representation

. Equation (6-2) m u s t therefore play the fundamental par t i n the theory

of turbulence. i s shown that it contains the general equations of hydrodynamics.

No attempt is made here a t par t icular izat ion; it simply

F i rs t , the two members of (6-2) are integrated with respect t o ukj with due regard t o (6-1). We introduce the density of the f l u i d

p(x, t ) = b ( x , u; t )du (6-3)

(or more accurately, a quantity p normalized and numerically propor- t i ona l t o the density) and the re la t ive mean of the velocity for a given position, the components of which a re

We end w i t h an equation of continuity I 6-51

.

NACA TM 1377 27

which simply expresses the f ac t that the velocity i s the (stochastic) derivative of the position. Thus it is seen that the velocity of the whole of the turbulent motion, defined qual i ta t ively i n paragraph 4, has the first of the qual i ta t ive properties of the hydrodynamic velocity i n general. is conserved.

A t the chosen turbulent scale (which is arb i t ra ry) the mass

Now it w i l l be seen that the velocity also satisfies the equations of motion. Multiplying equation (6-2) by the velocity component ui and integrating with respect t o u, gives f o r the f i r s t member

Introducing the speed of fluctuation

and considering the equation of continuity, w e put

The first member becomes

- + E at k

the second member i s wri t ten

r

being a cer ta in function of x which i s deduced from the operator A

and the probabili ty density R . Hence the equations of motion

28 NACA TM 1377

The 7i play the par t of the components of the density of the f i e l d

They are , except fo r -p, the components of the correlation tensors of the velocity

of external forces (gravity, f o r example). The Tik are the components of the s t ress tensor ( w i t h f ixed turbulent scale)3. the factor components a t a point. methods, and the equations of hydrodynamics fo r a given turbulent scale can thus be verified.

They can be measured d i rec t ly by s t a t i s t i c a l

It should be noted tha t i n this case the equations of motion of the ensemble (average) are involved. For the present, nothing about the behavior of the r a t e of f luctuation has been assumed. Later on the usual assumption w i l l be made tha t the velocity satisfies the hydrodynamic equations and, more precisely, the Navier equations. It should be remembered that, based upon t h i s hypothesis, Reynolds w a s able t o es tab l i sh equations similar t o equations (6-7) f o r the mean turbulent motion. It is apparent t h a t , without it being necessary t o repeat Reynolds' calcu- la t ions, the s t a t i s t i c a l theory i n question here i s en t i re ly d i f fe ren t from that of Reynolds. It i s probaljly more complete, but it s t i l l has not been pushed f a r enough to be ver i f ied by experiment, due to a lack of suitable hypothesis.

*

As simple example of the fundamental equation (6-2), the case i s chosen in which the function U ( t ) is i t s e l f different iable ( i n mean squares, r e f s . 15, 16) . I n t h i s case, the s t a t i s t i c a l image point of the f lu id has an instantaneous acceleration. If Tk(x, u, t ) represents

of the acceleration, that is , of the velocity derivative, and the velocity are fixed, it proves tha t the density

X ( t ) and U ( t ) s a t i s f i e s the equation

the eelevant mean when the posit ion of probability of

The operator

and of the first order. The f i e l d of the external forces t o which the f l u i d i s subjected has, necessarily, f o r components, the quantit ies

%his expression of s t resses w a s or iginal ly given by Reynolds (On a

the dynamical of incompressible viscous f lu ids and the determination of the cri terion. proceeding from the Navier equations. Reynolds s t resses i s different from that of

Phil. t rans . Roy. SOC. CLXXXVI, par t I, 123, 1-89?), L

However, the exact meaning of the Tik.

NACA TM 1377 29

c

averages formed of the acceleration components when the posit ion i s fixed, provided only that uiriR to in f in i ty . If , i n par t icular , the ri are not dependent on u, ri i s

ident ical with y i , and R s a t i s f i e s the simple equation

tends toward zero when the velocity increases -

(6 -10)

Once the Tk are given, t h i s equation defines R(x, u, t ) from R(x, u, t o \ . t o form of the or iginal probabili ty say f o r the equation of heat (type (3-19)); this i l l u s t r a t i o n does not appear t o rest on hypotheses suff ic ient ly inspired by r e a l i t y t o serve as basis of a turbulence theory. must be modified, as w i l l be done i n the following paragraph:

But the form of the l i n e a r transformation from R(x, u, to) R(x, u, t) , tha t i s , the probability of t ransi t ion, depends upon the

Rlx, u, t o ), contrary t o w h a t happens,

F i r s t of al l , the s t a r t i ng point

7. Systems of molecules ( re fs . 13 and 14) :

The s t a t i s t i c a l quantit ies t o which the analysis of velocity records leads, represent only cer ta in scales of turbulence, those which correspond t o the ensemble of "vortices" whose dimensions are superior t o a l i m i t approximately f ixed by the employed anemometer. Can the theory be changed so t h a t all the possible scales can be represented simultaneously? It seems t h a t it suff ices f o r t h i s purpose t o start from the f i n e s t scale , tha t is, the molecular scale. of N molecules, and, t o explain the method with as much simplicity as possible, the assumption, which i s not ver i f ied f o r a i r , is made, t h a t the molecules are ident ical , monatomic, comparable to material points subjected t o central interactions. V ( r ) denotes the potent ia l of the force of interact ion of two molecules separated by the distance r. If r is great ( w i t h respect t o the diameter of the molecules, which w i l l not be introduced exp l i c i t l y ) , V ( r ) i s negligible. For the small values of r, V ( r ) expresses the repulsion of the molecules, generalized form of shocks. N o other information about V ( r ) i s needed beforehand, a t least i n a general theory.

The gas i s therefore considered as a system

The motion of this system of molecules i s controlled by the equations of dynamics. But the extreme complication of the t ra jec tor ies of t he

30 NACA TM 1377

molecules prompted the replacement mechanics. We prefer t o introduce i s a random point. By means of an of a molecule can be considered as

T

of ra t iona l mechanics by s t a t i s t i c a l random mechanics where each molecule ergodic hypothesis the random motion being s t a t i s t i c a l l y equivalent t o the

- motion of the ensemble of the f luid, i n quasi-steady conditions, the ensemble of successive states of the visualized molecule replacing the ensemble of the simultaneous s t a t e s of a l l the molecules. The molecule would therefore be the concrete image of the abstract random point serving up t o now f o r representing the f lu id . The molecular scale i s a t rue u l t i - mate scale of turbulence, separated, however, from the actual turbulent scales by a poorly defined but f i n i t e interval . While fo r experimental turbulence the concept of a random image point is a mathematical abstrac- t ion, i t i s a natural idea and a s t a r t i ng point f o r the molecules.

This idea i s now explored but by a method s l igh t ly different from t h a t discussed i n the preceding paragraphs. f x , . . ., xN, u1, . . ., %; t ) of the positions and the veloci t ies of N molecules simultaneously, ra ther than singly, i s introduced. The gas appears then as a random point with three N dimensions, i n a space of configuration, and no longer as a random point of ordinary and physi- c a l space. For the t i m e being, the notations xl, 9, . . ., xN s h a l l have a vectorial character and represent the system of the three coordi-

The probabili ty density

N(

c

nates of the molecules4 numbered 1, 2, . . ., N. The potent ia l of interact ion of the molecules of rank i and j

is indicated by V i j , and the external force t o which the molecule of rank i is subjected, by 7i. If m i s the mass of a molecule, the equations of motion of the molecules read

(7-1) - = m - - = 7 i + ) - dui avi j

d t d t % ui

the summation applying t o all values of j from 1 t o N, when Vii is

assumedto be zero. Naturally Vi j = Vji.

When these equations a re writ ten, the hypothesis is made that the velocity of each molecule i s different iable . It follows, that the random

41n the preceding paragraphs the notation f ( x , t ) represented a l readythe probabili ty density of a vector x, of components x1, x2, x3. W h a t i s used only temporarily i s the meaning of the subscripts, par t icu lar t o t h i s paragraph. cations i n writing which would be more harmful than useful.

6 A change i n notation could be avoided only by compli- -

point of 3N dimensions having for coordinates the ensemble of the coordinates of the molecules is represented by a random vector function of the configuration space of 3 N dimensions doubly different iable . e r ty which, as s ta ted before, i s not necessarily t rue fo r the three- dimensional random point used previously, is the reason, according t o the theory of s t a t i s t i c a l functions, why f N ver i f ies the p a r t i a l derivative equation

This prop-

which expresses that fN is an integral of the equations of motion. It has the same form as Liouville 's equation of s t a t i s t i c a l mechanics. But it should be remembered t h a t i t s original meaning is a l i t t l e different . This equation replaces and defines, i n the space of 3 N dimensions, the equation (6-2), and must now be exploited.

The turbulence involves "particles" or "eddies" formed, at a given scale, by groups of s molecules, s being a very large number, but at the same time very small with respect t o N. functional equation is s a t i s f i e d by the probabili ty density fs r e l a t ive t o the molecules of rank 1, 2, . . ., s . fs is defined by

L e t us investigate what

n

* % I d % (7-3)

The fundamental equation (7-2) m u s t be integrated with respect t o %+1, us+1, * Y XN, w. The operation i s obvious, except f o r the

av, : terms i n 2. , Vi j being function solely of 9 and x the integra- ax : j'

J

t i o n gives a zero r e su l t if i and j are both superior t o s .

avij afs If i and j are both infer ior t o 6 , then - - -.

ax; &4 J J.

If j 5 s, i > s, the integration with respect t o % shows that the result s t i l l is zero.

If, f ina l ly , i , < s, j 7 s, we f i r s t can integrate with respect t o Xk, uk, f o r k # i.

32

fs+l represents the probabili ty hij %+l The result is - - -, where m axj hi -

density re la t ive t o s + 1 molecules of rank 1, 2, . . ., s, j . The last integration

and fs+l a re functions avi j cannot be extended fa r ther , because both - ax3

O f X j .

N a r it w i l l be noted tha t i n the sum

the terms are ident ica l because the molecule j plays an anonymous par t .

The f i n a l result i s

This equation seem capable of serving as basis f o r a theory of 1 S

turbulence. The scale there appears exp l i c i t l y f o r the numbers - and

and the problem (not taken up here) consists i n formulating a reasonable N hypothesis which enables fs+l t o be expressed with the a i d of f,, SO

that (7-4) becomes a functional equation i n f,. -

NACA TM 1377 33

For s = 1, (7-4) is, i n a certain measure, comparable t o (6-2) and reads

(7-5) af, 1 afl - a f l + uI-+ -71- = a t axl m au1

Equation (7-5) differs from (6-2) by the presence of the terms i n The presence of the second member relates (7-5) t o (6-2), but i t s f2.

form makes it different from it. equation i n fl only i f f 2 is t i ed t o f l by a suitable assumption. Such an assumption has been made by J. Yvon who demonstrated with i t s a i d Boltwnann's fundamental equation on which the k ine t ic theory of gases is based. More recently it was taken up again by Born and Green, who a lso studied the case of s = 2, and applied it t o the case of l iqu ids on

Equation (7-5) reduces t o a functional

the basis of an assumption by Kirkwood binding f3 t o f*.

It seems tha t the case of large values of s has never been studied.

Equation (7-5) l i k e (6-2) and despite the not necessarily l i nea r character of the second member with respect t o of hydrodynamics which are easy t o write, i f the method indicated I n paragraph 6 i s applied. i n concrete applications t o turbulence, only these summary indications are given here. d i f f i c u l t i e s which are far from being solved, or even s ta ted but it i s important t o know t ha t this equation exists.

fs , involves equations

Since these equations have so far not been used

The application of equation (7-4) t o turbulence raises

CORRELATIONS AND SPECTRAL FUNCTIONS

8. Introduction - correlations i n space - homogeneity, isotropy:

In t h i s chapter we deal no longer with the general l a w s of proba- We limit ourselves t o the study of the correla-

Following are some b i l i t y of turbulence. t ions of the l a w of the f i e l d preliminary remarks on the subject:

H(u,u'; x , x ' , t , t ' ) .

Experiments furnish the correlations I n form of temporal averages. Chapter I shows how it w a s possible t o change t h e i r interpretat ion f o r converting them into stochastic averages. The assumption is made t h a t

34 NACA TM 1377

th is operation i s always possible. But f o r s ta r t ing , there i s no neces- s i t y f o r knowing whether it has been effected, since only properties of symmetry, of tensional character, are involved here. On the other hand, arr iving a t the dynamics of turbulence we shall see tha t the use of temporal averages leads t o serious d i f f i cu l t i e s , which disappear when they are transformed in to stochastic averages, and our calculations w i l l deal only w i t h stochastic averages. when the calculations have reached the l a w s which must be compared with physical r ea l i t y .

Temporal averages are resorted t o only

The components of the speed fluctuations5 a t point x1,x2,x3 are

designated by u l , 9, and u3. They are s t a t i s t i c a l quantit ies whose mean value i s zero. Time plays no part a t present and the parameter t i n the formulas is disregarded. The correlations of the velocity between the two points x and x ' = x + 5. a re defined by a tensor having f o r components

The scalar is introduced also:

The turbulence is sa id t o be homogeneous (within a cer ta in f i e l d of space) when the RaS are not separately dependent on the two points x ,x ' , but only on t h e i r re la t ive position, that is, on the vector 5 = x ' - x. Then R@( e ) replaces €Q(x,x' ) .

The turbulence i s sa id t o be isotropic a t point x i f it i s homogeneous i n the v ic in i ty of x and the tensor RaP i s invariant t o any rotat ion of the axes and any ref lect ion. on the components of t h i s tensor i s discussed l a t e r .

The form which isotropy imposes

These two definit ions concern only the second-rank tensor Rap. They are "second-rank properties" of the s t a t i s t i c a l vector k. They m u s t be extended t o cer ta in third-rank tensors. can be given a more complete and a l so more r e s t r i c t ive def ini t ion by extension t o a l l possible moments of velocity components taken a t any number of points.

Homogeneity and isotropy

It i s then more convenient t o define the homogeneity a

%hey were cal led u ' ~ , u ' ~ , u ' ~ i n chapter I. But it i s advantageous . t o use thereafter the notation u ' f o r other purposes.

NACA TM 1377 35

as the invariance t o t ranslat ions of the l a w of probabili ty of the ensemble of velocity vectors having for or igin a cer ta in number of a r b i t r a r i l y dis t r ibuted points, and the isotropy, once homogeneity i s achieved, as invariance t o rotations and ref lect ions.

Without r e s t r i c t ive hypotheses the tensor RaS has nine d i s t inc t cmponents, functions of the seven variables If the turbulence i s homogene+ous, the nine functions remain, but of four variables only 5 ,k ,E ,t. T'ne isotropy is now writ ten.

x~,x~,x~,x'~,x'~,x'~,~.

1 2 3

To the tensor R@(E) w e associate the scalar b i l inear form

where Xu and Ya are two arbi t rary vectors. T h i s form is the mean value of the product of two scalar products

b a t i o n , separately l i nea r w i t h respect t o X, and Ya, of invariants of vectors Xa,Ya,5, with respect t o rotations. These invariants are

>ua(x)Xa and %(x')Y,. If it is invariant t o rotations, it is an algebraic corn-

There are therefore two scalars A(r), B ( r ) , functions of r, so that

The ident i f ica t ion shows that, i f BaS represents the c l a s s i ca l symbol of Kronecker, zero i f p # a, equal t o unity when p = a:

NACA TM 1377 36

The tensor Rap, i n the isotropy hypothesis, depends therefore solely on two dis t inct functions of two variables r and t. The notations are changed as usual and one puts w i t h Karman and Hararth (reference 3 0 ) :

f and g being functions of r, t becoming unity f o r r = 0, and ~2 a simple function of t.

If r = 0, Rap i s reduced t o

It is seen that

Also, owing t o the isotropy, the tensor is symmetrical.

(8-7 1

Lastly, it should be noted that the quadratic form > R X X is a$ a B

the f i r s t m e m b e r of the equation of an e l l ipso id of revolution. shows that, i f one takes, no matter how, a symmetrical table of numbers R,,, they are not, i n general, components of a correlation tensor.

nmbers m u s t ver i fy the inequal i t ies which s t a t e that they are coefficients of the first member of the equation of an e l l ipso id . roots of the equation of the th i rd degree S classical) m u s t be posit ive.

T h i s

These

In other words, the (equation i n lRaP - S6apI = 0

A similar method permits the reducing of the components of the tensor

6B

.

.

NACA TM 1377 37

which w i l l be needed l a t e r . This tensor i s symmetrical w i t h respect t o a and p, and the t r i l i n e a r form

i s invariant t o rotations. and Howarth, it is found, a f t e r a few calculations, tha t

By applying the c lass ica l notations of K&&

a, b, and c being three functions of r and t.

9. Properties of the functions f , g, a , b, c. Incompressibility6:

Other forms of symmetry l e s s particular than isotropy can be v i s - ualized, as f o r instance, ax ia l symmetry, o r invariance t o rotations about a given axis, instead of about a point. which the Rap

The case of isotropy i s t h a t i n depend on the smallest number of separate functions.

The functions f and g are correlation coefficients. For example, f

(direct ion of velocity.of the ensemble) a t two points s i tua ted on a p a r a l l e l t o this axis (f ig . 3)

can be defined by taking two velocity components along the axis of x1

g is defined by taking two components s t i l l pa ra l l e l t o the axis of xl, but at two points located on a l i ne perpendicular t o this axis. the

%he authors (reference 30) use q, 2 , k where we use a, b, c. But the l e t t e r k is a l sa used i n a j u s t as c lass ica l although more recent fashion t o designate the spectral frequency and w i l l be used with t h i s meaning i n the present report.

38

For example

NACA ?M 1377

Experience indicates that near r = 0, the functions f and g a re continuous and twice different iable and allow a tangent t o the "horizontal" or ig in (fig. 4 ) .

Theory confirms experience, which unfortunately lacks precision when the distance r becomes small. Much greater accuracy i s obtained by correlation measurements with difference i n t i m e (experiments by Favre, Report to the VIIth Intern. Cong. of Appl. Mech., London, 1948), because only one anemometer i s used and the timing can be reduced as much as desired.

Hence, one may write developments of the form

g ( r ) = 1 + --$'(O) 2 + . . . (9-3)

f"(0) and g " ( 0 ) are negative quantit ies, possibly t i m e functions, having the dimensions of the inverse of the square of a length. The length

i s called length of dissipation.

(9-4)

The t r iple-correlat ion functions a,b,c have i n t e r p r e t a t 2 n s s-mi- 1a.r t o those of f and g. For example:

. (9-5)

NACA TM 1377 39

It follows from formula (8-9) t h a t , i f

l imi t when 5-30, th i s l imi t can only be zero. are zero. In a general way, is an odd function of k, ,5p,5 , . Consequently, considered as function of r, Its development has no term i n $. term i n r, e i ther .

TaPr has a w e l l defined Hence, a(O),b(O),c(O)

c ( r ) i s an odd function. Lastly, it is shown that it has no

The coefficient of t h i s term i s the mean of

or

But on account of the homogeneity extended t o the averages of the

th i rd order, u13(x1 + r,x2,x3) i s equal t o u13(x1,+,x3). Hence the l i m i t i s zero. Lastly, c ( r ) i s an infinitesimal of the t h i r d order w i t h respect t o r. The same holds f o r b ( r ) and a(.).

If the flow is incompressible, the various functions that charac- t e r i ze the correlations of isotropic turbulence are not independent. From

the re la t ion xa”” = 0 it is, i n fac t , immediately deduced t h a t axa

Y - = O aR, (9-6)

T h i s equation is genem?. In the par t icular case of isotropic turbulence it leads t o Idman ‘ s equation

r af g = f + - - 2 ar (9-7)

40 NACA TM 1377

In the same manner, one then f inds that:

So, of the f ive functions f,g,a,b,c t r i p l e correlations of isotropic turbulence, only two remain d i s t inc t i f the flow i s incompressible.

which define the double and

The incompressibility resu l t s , i n par t icular , i n the relat ion g " ( 0 ) = 2f"(0). Therefore, i n terms of dissipation length A:

f ( r ) = 1 - - + . . .

Together w i t h the length A, two other numerical parameters ( t i m e functions) are frequently used t o give an idea of the turbulence.

F i r s t , the correlation length

i s introduced; it depends upon a l l values of f o r 0 < r < m, while h depends only on the form of the function f ( r ) a t the origin. It may be pointed out that L i s expressed by means of the scalar

€3 = >ua(x)ua(x + 6) = ~0 (f + 2g).

f + 2g = 3f + r- = 2f + - (rf ) .

f ( r )

2 O w i n g t o the incompressibility,

If rf (r ) approaches zero when r+: af a ar ar

Jo (f + 2g) dr = 2L (9-11)

NACA 'I'M 1377 41

The in tens i ty of turbulence i n one direction, that of the axis of xl, fo r example, i s the dimensionless quantity

U (9-12)

where U is the m e a n velocity. The t o t a l in tens i ty of turbulence i s the r a t i o

If the turbulence is isotropic, i t s in tens i ty i s the same i n every

direction. It i s equal t o the t o t a l intensi ty and t o -. UO U

I n the case of isotropic turbulence the mean square values of the rotat ional components a re d i rec t ly associated w i t h the quant i t ies and A.

I+,

Introducing the components

au3 au2 aul au3 3% au1 ?I -=- - - ax2 ax3 % ? = - - - ax3 3x1 ? = a x , - - 3x2

- - - of the rotat ional , the averages q 2 , U Q ~ , 32 for homogeneous and

and isotropic turbulence are computed:

42 NACA IM 1377

We note that, f o r .example :

2 u x x x u x 3( 1’ 2’ 3) 3 ( 1jX2 + S , J ~ ~ ) ]

The f i r s t two bracketed terms are equal t o uo2. The th i rd is equal

t o R33(0 5 0) = uo 2 g, or, except fo r the t h i r d order, - ’ 2’ A2,’

Lastly

Likewise

NACA ‘1M 1377 43

Lastly:

Final ly

- - - Naturally, ~ I Q ~ and .)J’ are equal t o y 2 .

10. Spectral decomposition of the velocity:

W e know that the representative curve of a turbulent velocity component u ( t ) phenomenon. The analyt ical representation of u ( t ) is not a periodic Fourier ser ies , but ra ther an almost periodic series o r a Fourier integral .

i n terms of time suggests the idea of an irregular periodic

I n the f irst case, u ( t ) is a sum of harmonics without common base period; by adopting the complex notation

iqt +m

u ( t ) = h e (10-1)

L e pulsations % forming a succession of real numbers increasing w i t h n. It can always be supposed that wmn = -031. u ( t ) is a r e a l quantity i f A_, = &*, the notation &* designating the conjugate imaginary of 4. The ser ies A, m u s t be convergent. l l

44

In the second case:

NACA TM 1377

u ( t ) = rmA(m)eiutdu (10-2)

the function A ( u ) being absolutely summable and such that A(*) = A*(m).

The numbers & or the function A ( u ) depend upon the posit ion of the anemometer.

This representation is prac t ica l when the turbulence i s steady i n time, which enables a vigorous application of the ergodic principle and the calculation of the averages connected with i n a time in te rva l as great as desired. But, among the simplest problems involved, there is , f irst of a l l , t ha t of a turbulence homogeneous i n space ( a t l e a s t i n a suff ic ient ly res t r ic ted range) and developing i n time. periodic character of the speed i s then manifested i n space rather than in time.

u ( t )

The pseudo-

(A detailed study follows.)

The component uu of the velocity of f luctuat ion at point xlY+,x3 . a t a given ins tan t can be developed i n series or by Fourier integral , depending upon whether the spectrum is discrete or continuous

nl, 9, and n3 are three integers that vary independently from

Y

pi7 "2 9 3) --OD t o +m i n formula (10-3), and on which the quant i t ies Z,

-

are dependent. In formula (10-4), I& i s a hl t r i p l e integral extended over the en t i r e space A of wave numbers (The wave number, inverse Of a length, i s the, equivalent i n space t o the f r e - quency, inverse of time. It i s a vector). uu i s a r e a l quantity if -

.

3

.

c

.

*

L

NACA RYI 1377 45

These notations are of c lass ica l form. But, since the ua are

steady, random functions of space, it is preferable t o represent them as stochastic Fourier integrals

I n this formula, the h,(A,t) are random functions of hlrh2,A3 w i t h orthogonal increments (or noncorrelated). h and A ' are two different points i n the space A, the mean'of the product of the two increments d h , ( h , t ) and dh (h ' , t ) is zero. On account of the complex notations it naturally is a question of a product of hemi t i an symmetry. Hence

O r , i n other words, i f

P

dha*(h,t) dhP(h' , t) = 0 i f A ' # h (10-6)

If h' = h at the same point, this average is, i n general, no longer zero. It is in f in i t e ly small of the order of dh. We put

Now the spectrum of turbulence i n the different cases ( l O - 3 ) , (10-4), and (10-5) is defined.

11. Spectral tensor and correlation tensor:

Assuming spa t i a l homogeneity it is now attempted t o form, from the spec t ra l decomposition of the velocity, the expression of the components Rap (k1,k2,k3,t) o r abbreviated, R@(k) of the correlat ion tensor of the velocity at a given instant .

If the formulas (10-3) o r (10-4) are ut i l ized , the spa t i a l averages i n a very great volume, physically limited but prac t ica l ly infinite w i t h respect t o microscopic turbulent lengths, must be used. It involves a .

cube w i t h edges pa ra l l e l t o the axes, of a r b i t r a r i l y large length 28, of which the center, which can be placed anywhere because of the hmogeneity, is placed i n the or igin of the axes of the coordinates.

46 NACA ‘1M 1377

(11-1) .,

independent of xa. consequently

They are therefore equal t o t h e i r mean value and

Firs t , take the case of the Fourier s e r i e s (10-3). TO form Rap, involves, first, the product ua(x)up(x + k ) , by associating an a rb i t r a ry Y

term of the series which represents ua*(x), t o any one term of the ser ies that represents To form the average i n the volume V = 8a3, involves division by V and integration of xa i n space. The exponentials a re res t r ic ted , and likewise t h e i r in tegra ls i n V, so t h a t the quotients by V approach zero when a 4 m . There is an exception f o r the terms of the se r i e s %(x) and u,(x + 6 ) which correspond t o the same wave numbers, and which give the products

ua(x), or ra ther

up(x + 5 ) .

7

This i s none other than the Parseval formula f o r the almost periodic Fourier se r ies of three variables. uniform convergence which need not be defined here.

It naturally assumes conditions of

NACA TM 1377 47

In the case of the Fourier integral , s ( x ) u p ( x + k ) is a sextuple in tegra l

extended over a l l values of A1, +, h3, Atl, At2, and A t 3 .

Integrating under the sign in the f ini te volume V and dividing s by V, we f ind that, mathematically speaking, the mean value which is being sought is the l i m i t fo r an in f in i t e of

s i n ( ~ 1 ~ - %) a s i n ( ~ 1 3 - ~ 3 ) a

A I 2 - b ht3 - A3 (11-4)

Suppose Zu(A) satisfies “Dirlchlet ‘ 8 conditions” or, more gener- a l ly , i s a l imited variation; then the in tegra l which figures i n (11-4) has a f i n i t e l imi t which, when %(A) is continuous, has f o r value

t

48 NACA TM 1377

This is Dirichlet's theorem.

The mean value of %(x)up(x + e ) , the quotient of this integral by L

V = 8a3, thus approaches zero when a 4 w , and the formalism of the Fourier integral does not furnish the spectrum. This conclusion is paradoxical.

But every difficulty disappears when it is assumed that the turbulence might have only a line spectrum, incompatible with the so-called Fourier integral. But such'an assumption does not seem reasonable.

It seems more satisfactory to concede that the functions representing the turbulent velocity are too complicated for applying Dirichlet's theorem and, more precisely, that their oscillations are too crowded to be represented by functions with bounded variation. avoiding this difficulty is to follow.

A mathematical process

But an approximate argument can also be made. In reality, a is very great (with respect to turbulent wave lengths), but finite.

transition to the limit. can be made and assumed that, with suitable accuracy

Hence the mean value of %(x)u,(x + e ) is given by formula (11-4), without I

However, a transition to the approximate limit

SO, if a spectral function cpap(h) is defined by

we get

(11-8)

NACA TM 1377 49 c

. I n a more accurate way, the cpap(h)

tensor, the spec t ra l tensor, transformed from Fourier 's correlation tensor. But the approximate formula (11-8) has the drawback of yielding a spec t ra l tensor of a too r e s t r i c t ive form.

s c r i p t p, that is, the general product of the vector F Z a ( h ) by

i t s e l f . on the s p a t i a l average (11-8). and have recourse t o the representation of the velocity by Fourier 's s to- chastic in tegra l (10-5). The product ua(x)u (x + 6) is a double s to-

chastic in tegra l

const i tute the components of a

In f a c t , there i s no reason t h a t the be the product of a function of subscript a by a function of sub- (pas

v T h i s new defect can be corrected by saperposing a texqoral average

But it is much preferred t o turn elsewhere

P

On averaging, it is seen that the quantity dha+(A)dhB(h') is involved which is, as explained ear l ie r , zero when form qap(h)dh i f A' = h. Hence simply

h' # A, and of the

The correlation tensor Rup(t) i s the Fourier transform of the spec t ra l tensor c ~ , p ( h ) . The (pup (A) here have the generalty desirable. They a r e subject t o the two following conditions:

(a) rp (A) has hermitian syrmnetry ap

50 NACA TM 1377

the hermitian form xaJ (b) Whatever the complex numbers

3

a, p = 1 xaxp*c'pap

which is r ea l according t o the f i r s t condition, cannot be negati re.

Any system of numbers cp consti tuting a tensor and sa t i s fy ing c9 these two conditions can, a pr ior i , be used as spectral tensor.

12. Spectral tensor of isotropic, incompressible turbulence:

I. Suppose, with Heisenberg (reference 2 3 ) , tha t the turbulence i s isotropic. Then, as supplementary condition, the flow i s incompressible.

The method used t o express the isotropy of the correlation tensor The spa t i a l tensor i s is applied t o a l l tensors, whatever i t s meaning.

isotropic when two functions A ( k ) , B ( k ) ex i s t depending, moreover, on . t 6 U C h that

(12-1)

k2 denoting the quantity hl 2 + h2 2 + h3 2 .

If the flow is incompressible, the functions A and B are formed by a relat ion equivalent t o K 6 d n ' s re la t ion (9-7) between the functions

of the correlation f and g . We write, i n f ac t , that - sua = 0.

ax, Since

t p=l (12-2)

NACA TM 1377 51

w e must have ident ical ly

2 ha dha = 0 02-31

hence, w i t h p being fixed

When th i s condition is a p p l i e d t o the components (12-1) of the isotropic spec t ra l tensor, it i s seen that

2 Ak + B = O

W e put

hence

(12-4)

F(k) interpreted l a t e r .

is the spectral function introduced by Heisenberg. It w i l l be

11. With Ka,m$ de Fgriet (reference 26) w e first express incompressi- b i l i t y without disturbing the isotropy.

The cp are subject t o the condition that the hermitian form aB

NACA TM 1377 52

is positive or zero. It is posit ive, except when the vector Xu i s proportional t o vector &, since

(12-6)

on account of the incompressibility condition (12-3).

The hermitian form f i s now reduced. According t o c lass ica l theory and the equation i n S, three real numbers Sl,S2,S3 and three l i nea r forms

can be found w i t h which $ can be expressed i n the form

the three vectors of components If f 2 0, the numbers S1,S2,S3 are posit ive or zero.

au,bu,cu being two by two orthogonal.

But, there ex is t s a par t icular vector 7,, transformed from vector A, by (W-T), so that which $ is the sum, are zero a t the same time. Hence two separate

$ = 0. This can happen only when the three terms of

hypotheses

(a) s2 = o s3 = 0 auAu = o

(12-8)

.

.

NACA TM 1377 53

a being the function of A. T h i s form of cp similar t o (11-8), is

too res t r ic t ive ; U UP '

(b) s3 = 0 E a u & = o x b u h u = o

We w r i t e ra ther

2

and ident i fy w i t h the or iginal form (12-5) of q. It is seen that

T h i s is the most general form of a spec t ra l tensor i n an incompress- ib le medium. In more simple form

p1au = a t u ,/F2bu = b ' U

There are two vectors atu, b', forming w i t h vector h, a t r i r e c - tangular trihedral and such that

'pap = &',*a'p + b',*b' B (12-11)

54

By Pythagorean theorem

NACA !U4 1377

( 12 -12 )

with

By th i s formula, the b t a can be eliminated frm the expression

Of 'pap

Finally we put

This i s the canonical form which Kmpg de Fgriet has given fo r the spectral tensor of an incompressible fluid

(12 -14 )

x i t h

NACA TM 1377 55

The isotropy s t ipu la tes tha t c, = 0, so that formula (12-4) is obtained again.

13. Energy interpretat ion of the spectral function F(k):

Except when s t a t ed otherwise, the isotropic turbulence is assumed t o be i n the incompressible s t a t e .

Hence

The t o t a l energy of the turbulent f luctuations per unit mass is

But

Hence

I n passing t o the polar coordinates i n the space of the wave numbers one f inds tha t

Which, a f t e r minor changes, gives the amount of the energy of the turbulent f luctuations per uni t mass i n the "sphere of the wave numbers"

Ek = L l 0 F(k')dk' (13-5)

k ' being an integration variable.

The "small values of k" correspond, of course, t o the large scales ( large vortices).

The next problem i s t o f ind the expression of the energy E dissipated as heat per uni t mass. nature s t r i c t l y kinematic, the dissipative function here must be so introduced that it yields equations of motion, tha t is, Navier equations; hence, an assumption associated with the dynamics of turbulence, which is discussed i n chapter 111.

While the preceding calculations were by

With v dissipated i n

c

denoting the coefficient of kinematic viscosity, the energy heat by the molecular motion i s -

. . +(3+ ax2 227 ax + . . j (13-6) 4

This i s t o be expressed i n spectral terms and averaged. If

q p d

we ge t

57 NACA TM 1377

and

which, squared, reads

T o obtain the averages, involves operation under the sign. The s r e su l t i s zero, except when the points h and h' coincide i n the space A of the wave numbers. It leaves

Consequently, i f E is t h e mean dissipation of energy per un i t mass

T h i s formula supposes neither incompressibility nor isotropy. If,

now, the turbulence i s incompressible,

isotropy, cpaa = -.

1 h,hP'pup = 0 . UP

Owing t o the

F 2nk

Hence, by integrating i n polar coordinates i n space A

E = 2 v p ( k ) d k

E being a physical quantity, hence f i n i t e , it is seen that the preceding in tegra l is f i n i t e , which gives a first l imitat ion of th6 possible forms of the functions F (k ) .

14 . Relations between spectral function F(k) and correlation func- t i o n s f ( r ) and g ( r ) :

The spectral tensor cp w a s defined by the formulas UP

which assume homogeneity only. added.

Incompressibility and isotropy are t o be

According t o section 8

r 1

and according t o section 12

NACA TM 1377

By Fourier's reciprocity formulas

59

(14-1)

the in tegra l being extended over the en t i r e space. f o r defining F i n terms of f o r g, and f , @I i n terms of F must be particularized .

The general formulas

F 27tk2

It is pointed out that - =yep, i s the scalar invariant of the

spectral tensor rp ("trace of tensor" or contracted tensor). Hence it

seems logica l t o compare with F the analogous invariant of the correla- t ion vector R defined by

ap

aP'

Thus the following problems m u s t be solved:

(a) express F(k) by means of R(r), and conversely

(b) express f ( r ) and g ( r ) by means of R ( r )

( c ) express f ( r ) and g( r ) by means of F(k)

Also t o be defined i n precise manner i s w h a t i s cal led the t rans- versal ( l a t e r a l ) and the longitudinal spectrum of turbulence.

(a) Relation between F(k). ' and R ( r ) :

.

60 NACA TM 1377

To reduce the t r i p l e integral t o a simple integral , simply se lec t

a new axis of the A perpendicular t o the plane

results i n the integral 3

or

04-31

Which, after a short calculation, yields

R ( r ) = 2 [ si:c%(k)dk

C onvers ely

(b) Relations between f ( r ) , g ( r ) , and R ( r ) :

The integration of the d i f fe ren t ia l equation yields

R rf ' + 3f = - uo2

(14-4)

(14-6)

.

B NACA TM 1377

with the condition f(O) = 0. It i s

Uo2f (r) = 1 S , r ' . 2 R ( r 1 )dr' r3

61

(14-7 1

R ( r ) having fo r limit .3u02, vhen r tends toward zero, It con-

tends firms t h a t the in tegra l i s equivalent t o

rather toward unity.

w2r3, and that f (r)

From that, it is deduced, by formula (9-7) that

( c ) Relations between f ( r ) , g ( r ) , and F(k):

A simple calculation gives

1. uo2f(r) = 2 E ( k ) t i n rk - 'Os r3k3 r2k2 1

m s i n rk + s i n r3k3 rk

(14-8)

(14-9)

From t h a t the f i r s t terms of the l imited developments of R ( r ) , f(r), and g ( r ) f o r small values of r are derived i n terms of the dissipated energy E

62

(14-10)

Comparison w i t h (9-9) indicates t h a t the dissipation length h i s re la ted to the dissipated energy E, the viscosity v and the kinet ic

energy of the turbulent f luctuations E = through the formula

(14-11)

15. Lateral and longitudinal spectrum:

The spectral measurements do not provide the spectral function F directly, but merely the spectral terms corresponding t o cer ta in simple associations of velocity components, f o r the observers placed i n the particular re la t ive positions. It concerns longitudinal components of the velocity (para l le l t o the velocity of the main flow), a t points placed either on a para l le l t o the axis o r on a perpendicular t o the axis. The corresponding correlations a re uo2f(r) and w2g(r).

NACA TM 1377

According t o the Qr ig ina l notations of G. I. Taylor

where A(u) functions.

Since

~

63

and B(w) are termed t h e longitudinal and l a t e r a l spectral They now m u s t be expressed by means of F(k) . g = f + -f', elementary calculation shows t i t r

2

B(w) = L[A(cu) - d ' ( u ~ 2

and a t the same time that

05-21

R ( r ) = 2 sin %(k)dk = cos m[A(w) + 2B(o)1 du, (15-3) rk

Elementary Fourier transformations yield

k A(w) + 2B(cu) = 2

A and B are determined by the two equations

1 2B = A - d'

A + 2B = 2

(15-4

05-51

64

The resu l t i s

NACA TM 1377

c m m I11

DYNAMICS OF TURBULENCE

16. Introduction:

To construct a dynamics of turbulence, t ha t is, t o set up the dif- fe ren t ia l or f i n i t e l a w s which govern the development of the s t a t i s t i c a l quantities characterizing the turbulence i n time and space, it i s necessary t o start from elementary l a w s and apply the s t a t i s t i c a l methods t o them. The most natural idea, the only one which actual ly produces concrete results, consists i n u t i l i z ing the Navier equations. To w h a t extent are they applicable t o turbulence? a macroscopic motion with respect t o a f ine r scale motion, and, a t the l i m i t , with respect t o the molecular disturbance. Therefore it is reasonable to believe that it s a t i s f i e s the equations of the mechanics of f lu ids . The next s tep i s t o f ind the solution of the Navier equations which, for cer ta in l i m i t i n g conditions, have the turbulent aspect, and calculate the par t icular averages from these solutions. it i s rather d i f f i c u l t t o define these l imiting conditions. So, the remaining resource is t o examine whether, among all the possible solutions of the Navier equations, there exist any suf f ic ien t ly complicated f o r representing the turbulence, without attempting to determine them logica l ly by the limiting conditions. for the Navier equations are known only f o r simple conditions which are f a r from resembling turbulence. In other words, while conceding t h e i r validity, we pract ical ly do not know how t o solve them.

- The turbulent motion is always

Unfortunately,

But then a new d i f f i cu l ty a r i ses . Solutions

Since it i s not acceptable t o take the averages on the solutions of the Navler equations, it is attempted t o take the averages on the d i f f e r - e n t i a l equations themselves and t o w r i t e the d i f fe ren t ia l equations which verify the s t a t i s t i c a l quant i t ies . Since this method has produced resu l t s , - it is set fo r th i n the simplest case, that of the Kd&n correlation tens’or. -

. c ,

NACA TM 1377 65

17. Fundamental equation of turbulent dynamics:

It is assumed that the total velocity U is uniform and along the axis of xl. The Navier equations read then

in the absence of external forces. constant, and p the pressure. These equations must be supplemented by the equation of continuity

p is the density, usually assumed

The notations are abbreviated by representing the velocity7 at point x', = xu + 5 , by u', = u,(x') = u,(x + E). Multiplication of

(17-1) by u', and summation with respect to a, gives

- - _ 7,1n chapter I, section 6, ~1,9,~3 represented the velocity compo-

nents of the whole and u'1,u'2,u'3, those of the velocity of fluctyations. The over-all velocity having then U,o,o, as components, the notations are changed; ui represents the velocity fluctuations at point x, and U'i the velocity fluctuations at point x'.

Permutation of the points x and XI yields a similar equation, which i s added t o the preceding one

Taking the averages of the two members it i s then assumed that the turbulence i s homogeneous, nothing more. i s a function of sa, and is not individually dependent on, x, and x ' ~ . Consequently

The scalar product uauta

The term containing the over-all velocity U disappears. The following term can be wri t ten

67 t

NACA TM 1777

The last two

L "U* is

Likewise, as

terms disappear as a r e su l t of the equation of continuity.

wri t ten :8

V a re su l t of the homogeneity, 2 becomes P

Since Tup7(-S) = -T ( E ) , the system of these two terms is equal t o UP7

The viscosity term reads

2~ 7 a2 > u ~ u ' , = 2~ A Rm - 2~ AR p & E p a U

07-61

where A is the Laplacian symbol i n three-dimensional space.

Lastly, it i s shown tha t the pressure terms disappear. This is done by extending the def ini t ion of the homogeneity t o the averages containing the pressure, with due regard t o the incompressibility.

as pu', are not functions separately of x and x ' but merely of the

difference 5 = x' - x, so that, on these functions,

The avera,ges such - = - a.

ax ' ax

See the def ini t ion and the properties of the tensors i n chapter 11, sections 8 and 9.

Rap and 8

T a P Y

68 NACA 'IM 1377

Therefore, uta being function of x '

07-71

Likewise fo r the other pressure term. The f ina l result is the fundamental equation (reference 30)

It w a s writ ten with the aid of the components of the tensors RM --+ and Tapr. Introducing the scalar R = Rm, and the vector T which

has f o r components

The equation (17-8) assumes the form (reference 2 )

+ - aR = 2v AR + div T a t

07-91

18. Case of isotropic turbulence :

The equation (17-10) assumes incompressibility and homogeneity but no isotropy. If the turbulence i s isotropic, it is known (14-6) that

t f being the f irst of the two K & m a n correlat ion functions.

OB

*

NACA TM 1377

(section 9 ) : MY According t o the properties of T

The divergence of this vector i s equal t o

Tie fundamental equation (17-10) is now transcribed:

69

(18-1)

(16-2)

I n fact , it is remembered tha t the Laplacian of a function q ( r ) in n-dimensional space is

Multiplication by r2 is followed by integration w i t h respect t o r. It is easily ver i f ied that all the terms cancel out fo r i n mind tha t

r = 0. Bearing

one arr ives a t the equation

(18-4)

70 NACA '1M 1377 .

This is the K&&-Howarth equation ( r e f . 30) i n i t s c lass ica l form.

The general equation (17-10) has the form of the equation of heat propagation i n ordinary three-dimensional space, the quantity being propagated i s the scalar

tensor. The ''second member' ' div T is t i e d t o the t r i p l e correlations as R i s t o the double correlations. The r e su l t i s an equation of

p a r t i a l derivatives between two unknown functions R and div T. For reasons which appear l a t e r on, R i s usually considered as the pr incipal unknown. The problem of solving (17-10) consists i n defining R by means of suitable physical hypotheses on the t r i p l e correlations. The equation w i t h two unknowns f ina l ly arrived a t i s the r e su l t of the nonlinear charac te r of the Navier equations. later.

R, the invariant of the double correlation + 4

Its mode of operation w i l l be explained

The "isotropic" equation (18-5) leaves no t race of the or iginal properties of tensorial symmetry. Its purpose i s t o connect the func- t ions f and c, which can be measured direct ly . If it i s borne i n mind t h a t

4 r

ff l(r) + -fl(r) = %f(r)

then one may write:

It is an "equation five-dimensional space. f i c i a l and without r e a l

(18-6)

of heat," isotropic naturally, but i n a f i c t i t i o u s

physical significance. T h i s remark is interest ing, but it seems arti-

Nevertheless, an unusual property of the functions ug(t) and f ( r , t ) can be derived from it. Multiplying by r4dr followed by integration from 0 t o m yields *

NACA ‘Dl 1377 71

provided that r4f’ and reasonable, and d i f f i c u l t

rn 2 1 Y 4 f dr = 0 (18-8)

4 r c tend toward zero when r + m , which seems t o check. Consequently

i s a numerical constant, independent of t, during the development of the turbulence.

of f luctuat ion E = h2 t o the correlations

It i s the Loitsiansky invariant, which connects the energy

f. 2

It also should be noted that, since the second term of (17-10) i s 4

the divergence of 2v grad R + T, i ts integral , on a sphere of center 0 and i n f i n i t e l y large radius, is zero. Consequently, the in tegra l of

i s zero, and aR a t -

R d x l d”2 dx3 = Cte s (18-10)

the t r i p l e in tegra l lence i s isotropic,

being extended over the en t i r e space. it leaves

If the turbu-

(18-11)

But, according beginning of this paragraph:

t o the expression of R ( r ) , remembered a t the

72 NACA TM 1377

If r 4 f ' 4 0 , r3f tends toward zero also, and this l imi t i s zero

(18-12)

This proves that the function R, which is equal t o

If formula (18-12) i s approximated t o formula (14-5)

3%2 f o r

r = 0, and t o zero f o r r = m, takes negative values f o r suf f ic ien t ly high values of r . which expresses F(h) by means of R ( r ) , it w i l l be found that, providing R ( r ) tends suf f ic ien t ly fast toward zero a t in f in i ty , the development of F(k) i n parers of k starts with a term i n k4; near k = 0 (large s i z e eddies)

F(k) = Ck4 (18-13)

w i t h

The constant C can be expressed by f , because, according t o (14-7)

L+-,~ l r 4 f ( r ) d r = 5 dr L r f 2 R ( r 1 ) d r 1

In order t o change the order of integration, the integrals of the second member are writ ten i n the form ( h g . 5 )

NACA TM 1377

NOW:

73

r t 2 R ( r t ) d r t = ~+,~r?f(r) r2 do-

So, if f ( r ) tends toward zero at i n f i n i t y fast enough so t h a t not only r3f %ut even r 5 f * a2Froaches zero, then

(18-14)

To a factor 1 C is therefore ident ical w i t h Loitsiansky's 3'

invariant (18-9).

These results could be extended t o include the nonisotropic turbulence.

Lastly it should be noted (ref. 2) that, i n the case of isotropic 4

turbulence, the vector T that the TB are proportional t o the 5 In effect :

defined by (18-1) is radial , that is t o say,

8'

Hence one may put:

+ T = T g r a d ' r

T(r) = I+, 3 (c + 28) being a scalar.

19. Local form of the fundamental equation:

(18-16)

The problem i s t o ascertain w h a t the equation (17-10) becomes when r 4 0 .

74 NACA TM 1377

r 2 4 It is known that f(r)--+l. As f = 1 - - + . . ., f " + -f' 2A2 r

5 approaches - - A2 -

4c Lastly, as c is of the th i rd order w i t h respect t o r, c ' .+ - r

tends toward zero.

The f i n a l equation, due t o G. I. Taylor, reads

Recalling that, according t o paragraph 14:

The formula (19-1) becomes thereforeg

It establishes an elementary relationship between the dissipation of energy energy of fluctuation E; the turbulent energy of f luctuat ion i s t o t a l l y dissipated as heat by viscosity, t h i s dissipation following a process t o be analyzed i n chapter V.

E by viscosity and the decay as a function of time of the

The foregoing demonstration s t ipu la tes isotropic turbulence. But the resul t (19-2) i s val id i n more general cases. I n fac t , (19-2) can be derived f r m (17-10). F i r s t it is known that R ( 0 ) = 2E. The connection between A R and E i s established by the intermediary of the spectral functions cp (A) . If the flow i s incompressible, formula (13-9) reads

UP

a a t 9E being functlon of the sole variable t, - can be replaced by

d d t ' a notation which represents an ordinary derivative with respect t o

time (not a derivative w i t h respect t o motion which would be without sense here ) .

NACA TM 1377 75

but on the other hand,

consequently :

and i n par t icular , when approaches zero, vAR is reduced t o P

-v cpa(h)dh = -E.

+ To’prove tha t div T = 0, a t the l i m i t , the simplest way is t o rever t t o the Navier equations.

It is plain that equation (19-21, l i m i t of (17-10) when -0 be obtained without going by way of the f i n i t e values of r. Simply m u l - t i p l y the Navier equations by the disappearance of the pressure terms t o have been proved, it is

can

Assuming u ~ , I + , u ~ , add them up and average.

The term 1 d uu2 i s wri t ten e. The second member,

according t o the foregoing proof, is equal t o -E. Hence it m u s t be

shown that = 0. T h i s quantity r e a d

2 d t d t

By virtue of the homogeneity, ua2uP is not dependent on x, and the f i r s t term i s zero.

Owing t o the incompressibility, the second term is zero. The first member i s therefore zero, which provides the proof. "

20. Solution of the fundamental equation, when the t r i p l e correlations are disregarded:

When the t r i p l e correlations are discounted, equation (17-10) becomes the equation of heat

- aR = 2v m a t (20-1)

This hypothesis, which mathemtically -3 convenient, is physically quite d i f f i cu l t t o jus t i fy . e f fec ts become greater (prevalence of term 2v AR over the term div T ), or as the Reynolds numbers decrease (whatever t he i r def ini t ion) . would be ver i f ied i f the velocity components followed the Laplace- Gaussian law. corresponds t o a s t a t e of turbulence i n which the forces of i n e r t i a are negligible against the forces of viscosity.

I ts accuracy increases as the viscous -3

It a lso

And, w h a t i s more interesting, it may be added that it

Equation (20-1) must thus be resolved knowing R f o r t = 0, and 2 being aware that, i f r + O , R has for l i m i t a f i n i t e quantity 3% (t)

temporarily considered as known. I

NACA TM 1377

I n isotropic turbulence, R depends only on r, t, and [20-1

77

reads

a t (20-2)

To sa t i s fy the l imiting conditions, elementary solutions of the form

are necessary. Hence

The two terms of this equation have a constant value -9. There- 4

fore, R2 s a t i s f i e s the l i nea r d i f fe ren t ia l equation

R t 2 + QsR2 = 0 (20-4)

or, reduced t o c lass ica l form, by the transformation,

Quantity H confirms i n f ac t the Hermitian d i f fe ren t ia l equation

H" - 2s" + (Q - 4 ) H = 0 (20-6)

Since R ( r , t ) must cancel out for r in f in i t e , hence a l so H ( s ) f o r s i n f in i t e , Q - 4 i s necessarily an even integer 2n. In tha t case H is the nth Hermitian polynomial:

and

(20-8)

But R2 must have a f i n i t e l i m i t when s-90, which requires tha t % ( s ) be an odd polynomial; hence that n be an odd integer: n = 2p - 1,

Q = 4p + 2. The function Rl(t) i s then equal t o -. Hence the e le - 1 1

t p + P -

mentary solution:

The general solution i s a superposition of elementary solutions:

(20-10)

the constants

i s equal t o 3u02.

$ being s o chosen tha t the ser ies converges and R(0, t )

Ex<mples: 1. Limited t o the term i n p = 2 it i s found that , by interpreting the Hermitian polynomial, terminate constant by A:

H 3 ( s ) , and denoting an inde-

NACA TM 1377 79

(20-11)

hence n 2 F

T h i s example is rat ional . It is compatible with the hypotheses made a t that t i m e . I n par t icular , rmf approaches zero when r-+, whatever the non-negative number m may be. However, f = 0 f o r t = 0, except when r = 0. A t the i n i t i a l instant the tilrbdence i s concentrated a t one point from which it ultimately spreads throughout the en t i r e f lu id . Such a turbulent structure seems rather d i f f i c u l t t o conceive when assuming it t o start f ron t = 0. An interpretat ion w i l l be given l a t e r (section 34) i n connection w i t h the final phase of decay of turbulence behind a gr id .

For p = 1, one would have

A 8 v t R ( r , t ) = -.;e 3

t p

2 r -- r

R ( r , t ) = & 8 v t r'2e 8 V t b I

3 r3 tp

f = - 3 I (20-12)

f would not converge toward zero when example cannot be sui table f o r a real motion.

r+, which proves tha t t h i s

2. (K&m&-Howarth ( re f . 3 0 ) . Suppose the correlation function f does not depend separately on r and t but only on the vari-

r able s = -.

Since

R = uo2(jr + rg)

NACA 'IT4 1377 80

it then yields

R = u o 2 ( t ) b f ( s ) + s f ' ( s j

so tha t R is the product of a function of t by a function of s. Thus example, corresponding t o a chosen Hermitian function

R i s an elementary solution of the problem treated i n the first

€&l.

The function f ( s ) s a t i s f i e s a d i f f e ren t i a l equation similar t o (20-4) written by K&m& and H a w a r t h . But a great advantage accrues from the use of function R and reduction t o Hermitian polynomials.

Remark. According t o (20-lo), R ( r , t ) i s a sum of "elementary solutions" corresponding t o various values of the integer p, s t a r t i ng from p = 2. For each one of these solutions, except for p = 2, it i s verified that the invariant of Loitsiansky (18-10) i s zero. p = 2

For it has a f i n i t e value.

21. Solutions involving a s imilar i ty hypothesis (see r e f . 7) :

If the t r i p l e correlations are no longer negligible, the fundamental It is assumed tha t the correlat ion equation cannot be solved completely.

functions f ( r , t ) and c ( r , t ) are not separately dependent on the two

variables r,t , but solely on JI = 2 where 2 ( t ) i s a length i n

terms of time. That i s t o say, that a t each ins tan t t the correlat ion curves are superposable, by means of a uni t change on the axis of However, it is recognized tha t t h i s hypothesis, cal led " to ta l s imilar i ty" i s often too specific, and it is therefore replaced by a "par t ia l similarity" which i s ver i f ied only i n a f i n i t e in te rva l

2 (t)

r.

21 < < 22.

For t o t a l s imilar i ty , the Loitsiansky invariant reads

Consequently, w2(t)Z5(t) i s a constant independent of t i n the . course of the motion. similari ty.

This i s no longer t rue i n the case of p a r t i a l

NACA 'iM 1377 81

To transform the fundamental equation, it is possible t o put

(21-2)

a(*) is a function tha t takes the value unity f o r $ = 0; p($) i s a function that caixels out w i t h $. €ll is a Reynolds number i n terms of the t i m e , and T is the scalar which, according t o (18-14), defines the t r i p l e correlations.

Considering the re la t ion (19-1) which affords

and of the length A, the fundamental equation

i n terms of . d t

uo2 a R + - - 2 ~ A R + d i v T a t

i s eas i ly transformed t o

or, schematically arranged:

Equation (21-3) is a sum of four terms each of which is the product of a function of t by a function of J r . How can an equation such as (21-4) be resolved?

82 NACA 'Dl 1377

The q( t ) are considered as the coordinates Of a point A i n four-dimensional space, th i s point describing a curve (A) i n terms of the parameter t. The same applies t o the pi($). It results i n two curves ( f ig . 6 ) such that the s t ra ight l i nes joining the or igin t o any point A of the first curve and t o any point B of the second are always perpendicular. These curves must f i t two orthogonal complemen- ta ry subspaces i n four-dimensional space. Hence there are, a p r io r i , three possible cases :

1. (A) is a l i n e passing through the origin. (B) i s i n the com- plementary three-dimensional space orthogonal t o t h i s l i ne .

2. (A) and (B) are i n two completely orthogonal planes passing through the origin.

3 . (A) is i n a three-dimensional l inear space. (B) i s the orthogonal l i n e .

I n the f i r s t case, the ai(t) are proportional t o constant numbers m i .

I n returning t o the notations of equation (2l-3), the quant i t ies

z2 are proportional t o constant numbers. In other words, Z2, h2' and Ez

are constant. Following th i s , the functions a($) and p ( $ ) are connected by a unique d i f fe ren t ia l equation. Obviously one may assume 2 = h. Then h2 i s a l inear function of t i m e and i s constant.. Considering equation (lg-l), which we r e c a l l here

we obtain of necessity:

NACA TM 1377 83

The Loitsiansky re la t ion i s not verified, so tha t no t o t a l simi- l a r i t y can prevail ; a($) and p($ ) are joined by the re la t ion

(21-6)

which can be subjected t o experimental check.

A complete discussion of the second and th i rd poss ib i l i ty i s fore- gone, i n favor of the case where 2 = h i s prescribed. Then (21-3) becomes

and is represented schematically by

The discussion then deals w i t h three-dimensional rather than four- dimensional space.

Two cases are possible:

(1) The point ai(t) describes a s t r a igh t l i n e (issuing from 0 ) ,

and point pi($) (passing through 0) .

remains i n the plane perpendicular t o t h i s l i ne

(2) The roles of the two points are permutated.

The first case i s not d i s t inc t from tha t which w e have studied. In the second case, there ex is t th ree constants ml,m2,m3, so tha t

84 NACA TM 1377

a3 (21-8) I 1

They are the conditions proposed by Sedov. They can be checked direct ly by experiment; ml cannot be zero, because EA would be = Cte, and a t the same time a($) = Cte, which is not possible; 9 and m 3 cannot be zero simultaneously.

If m 3 # 0, the s imi la r i ty cannot be to t a l , because one would have

%2A5 = Cte and by (19-1)

which i s incompatible with the second equation (21-8).

Following t h i s , L+, and A are solutions e n t i a l equations:

of the system of d i f f e r -

m 3 ml(h2)' + 9 + --u v o A = o

2 0 U

(L+,2)1 = -1ov- A2

a($) ver i f ies the second-order d i f fe ren t ia l equation:

(21-9)

(21-10)

2B

c

.

NACA !I'M 1377

a f t e r which -5 determine by the f i rs t -order d

p ' + + = -$u' m3 0 2vml

f fe ren t ia l equation

(21-11)

The system (21-9) can be reduced t o quadratures. I f the solution cannot be wri t ten i n f i n i t e terms, it s t i l l is desirable t o integrate as far as possible.

When the first equation (21-9) is set i n the form

- = - ( A ) 1 - k 2 1 + k % 1ov

(21-12)

where % and k are constants replacing ml, 9, and m3, it i s found tha t

A = C t e R, 1-k @ - q ) ) k

Whence it is deduced tha t

(21-14 ) at R - - (1 - k ) k \

The problem is thus reduced t o a quadrature. By (21-13), A R h

follows from R, and I+, = F. -

The equation (21-10) generalizes equation (20-4) t o a cer ta in extent, but it is l e s s simple and w i l l not be discussed.

Other types of solutions satisfying a s imi la r i ty hypothesis can be examined, such as the case where A i s very small, that i s , where the dissipation due t o viscosity is low. I n tha t case a length other than

rm f (r)dr . The discussion = $ h must be chosen f o r length 2 , such as

86 NACA TM 1377

proceeds on the assumption t h a t the viscosity terms of the fundamental equation are negligible against the other two. It leaves then

c

The calculations are not developed (for fur ther de t a i l s , consult Batchelor's report (ref. 7) ) . give an idea of the various methods that can be applied t o solve the fundaqental equation on the basis of the s imilar i ty hypothesis. The physical study of the problem i s deferred until la ter .

The sole purpose i n t h i s paragraph w a s t o

22. Transformation of the fundamental equation i n spectral terms:

The formulas of reciprocity between the correlations and the spec- t r a l functions made it seem interest ing t o transcribe the fundamental equation (17-10) i n spectral terms. section 19.)

(Compare spectral intermedium,

By (11-10) :

(22 -1)

with

Likewise

(22-2) -

NACA TM 1377

Hence

3 aR T;- hpkp - - 2v AR = /e P=l (2 + 2vk2cp)dA a t (22-3)

4 + This quantity is equal to div T, where T i s the vector having

f o r components T ~ ~ . We put: U

The fundamental equation becomes

h a t - + 2vk2cp = q (22-4)

In t h i s equation cp and JI are unknown functions of A, t, and

3 k2 = Ap2.

p= l

The t o t a l energy of the fluctuation i s

n

(it was shown i n section 9 tha t $dA = 0 ) . ,irn For isotropic turbulences, the spectral function F(k) = 2rrk2cp,

which does not depend separately on Al,b,h3 fo r cp, but solely on k, i s introduced.

as i n the general case

88 NACA TM 1377

The factor $ can also be expressed by means of the scalar

7

It is actually known tha t

->

which, a f te r elementary calculations, gives

Equation (22-4) reads

- aF + 2vk2F = 25k2$ a t

(22-5)

(22-6)

(22-7)

f i t t ing: 2fik2JI = $ ( k , t ) , where i s a function connected with t r i p l e correlations and such that

yields

aF - + 2vk2F = a t (22-8)

By the use of the formula (22-5) and the l imited development of s i n rk , it is readily proved tha t the function Q(k) i s in f in i t e ly

r

small as kb, provided tha t the correlation function c approaches

NACA TM 1377 89

zero f a s t e r a t i n f i n i t y than - But, with the re la t ion

nothing more i s known a p r i o r i about its form.

r 4'

Equation (22-8) is not essentially different from (17-10). It ra i ses the same d i f f i cu l t i e s and is always a single equation between two unknowns F and $. However, it is much eas ie r t o make mathemati- ca l ly or physically reasonable assumptions about the form of about the form of the t r i p l e correlations, as w i l l be shown i n the f o l - lowing two methods, both due t o Heisenberg, which take advantage of (22-8); the first involves the introduction of the correlations i n time and w i l l go beyond the purpose originally assigned t o it, the second leads effect ively t o cer ta in possible forms of the spectral function F.

$ than

23. F i r s t theory of Heisenberg ( re f . 23) :

The hypotheses t o be examined primarily are those which form the subject of the last par t of Heisenberg's report. We sha l l u t i l i z e the ideas and follow the calculations as closely as possible, but change the notations and subst i tute stochastic averageslo f o r the spa t i a l and temporal averages.

The expressions giving the s t a t i s t i c a l velocity and pressure of homogeneous turbulence read:

The equations of motion are f i r s t wri t ten i n spectral terms. can be assumed tha t h,(h)

the time, and i t s derivative, which corresponds t o - sua, i s designated

It i s derivable on each test w i t h respect t o

a t

loBass, reference 5 . The calculations are rather d i f f i cu l t . The reader who wants t o avoid reading them w i l l f ind a summary of the r e su l t s a t the end of paragraph 24.

The expressions (23-1) must be introduced i n the equations (23-2)

I where U is the constant over-all velocity.

The on ly term of which the equation m e r i t s a more detai led discussion i s

(23-3 1 Putting pp - A t p = and considering the equation of continuity

I yields

which had already been used i n the form

uB7 3% = ihP i , e i I- ' 5 %*(P - h)dhp(P) (23-4) ux!3

the subscript (p) specifying t h a t the integral i s extended over the values of p. This quantity i s the Fourier transform with respect t o A of

-

~~

By means of some elexentary transformations two equations are obtained :

- Fa dq(A) - vk2dha(A)

where designates the derivative of %(A) wit-- respec -3 h.

The demonstration of paragraph 17 from equations (23-5) is then resumed. with respect to a, i is then changed to -i and the two equations obtained are added member by member.

The first equation (23-5) is multiplied by &*(A), added up

Patting

while considering the equation of continuity, gives

R(Z) I denotes the real part of Z..

After averaging, it is known that when the turbulence is isotropic:

Idhu(A)12 = - F dh 2xk2

92 NACA 'I'M 1377

Cons e quentl y

- "(" + 2vk2F) = R_(Z) 2flk2 at

W h a t was called $I i n paragraph 22 i s given by the equation

(23-8)

In order t o f ind a useful expression of Z several hypotheses originating with Heisenberg are made.

F i r s t of all, it is known tha t the s t a t i s t i c a l functions dhu(h) are of orthogonal increments, t ha t is , tha t the averages of the pro- d w t s dhu(A)dhu*(A') are zero, i f the two points h,h' of the wave number space are d i s t inc t . .. variables dhu( A), dhu* (A ) are not only orthogonal but independent ( i n probabili ty). -

The f i r s t hypothesis i s tha t the s t a t i s t i c a l

Recalling that the average values dhu(h) are zero and that dhu*(A) = dhu(-A), it i s seen tha t the average value of

i s zero, except when a t the same time

h = h - p p - h = p h = p

tha t is , jumps of hu(A) are zero, then Z = 0.

h = p = 0 . If it i s assumed that the probabi l i t ies of sudden

Paragraph 20 dealt w i t h the case where Z = 0. It i s cer ta in tha t , i n general, Z is not rigorously zero, but it i s l i ke ly that Z i s s m a l l

compared t o the viscosity term, or the term of development -. In other

words, the hypothesis of orthogonality i s too general, and tha t of the

aF *

a t ~ -

IB

*

t

.

NACA TM 1377 93

independence too r e s t r i c t ive , yet it is d i f f i c u l t t o formulate an in te r - mediary hypothesis. But, s t a r t i ng from the independence it i s possible t o obtain, by means of cer ta in interesting transformations, a s ignif icant expression fo r Z. This expression is defective since there i s contradiction. But it seems l ike ly tha t it represents EL good approximation of Z. We sha l l therefore study it.

24. F i r s t theory of Heisenberg (continued). Space-time correlations :

If the average of R_(Z) disappears, it i s because Z contains an odd number of factors Replace dhP(p) by the t i m e in tegral of i t s derivative &,(p) and

express d$(p) by means of the equations of motion. Certain terms l inea r i n to r s dh appear i n Z, and is not zero. A kind of technique of solving the equations of motion is involved here, which reduces them t o in tegra l equations (bearing on simple averages).

dha(A); the equations of motion are not l inear .

dh are replaced by quadratic terms; products of four fac-

If T i s a very great positive. number

(24-1)

If T i s suf f ic ien t ly large it may be conceded tha t hp(p,-T)

tends toward zero, that is, that the f l u i d has "s tar ted from rest . ' ' Hence

To l ighten the notations, h ' (p) is t o represent the func- P t i on hp(p, t - T ) . Then

z = -2i 1 hp dha*(h) a u * ( P - A)dfq+) (24-3) UP

94 NACA 'IN 1377

and dl;' ( p )

which we removed previously the term of the pressure i n

i s eliminated by means of the equations of motion from a dq:

(24-4)

Now a new assumption i s made according t o which the or iginal assump- t ion of independence (for great values of T , remains valid f o r the whole time in te rva l T , leaving only the averages with four factors

T = 0) , which i s cer ta inly ver i f ied for the

4

The dh being independent i n d i s t inc t points of the frequency space, it i s necessary t o associate the four points which correspond t o the four dh by pa i rs , not forgett ing that d-ha*(A) = &,(-A). Considering

the transformation of the f i r s t integral , the only admissible combinations a re

hence p = 0. No contribution t o the integral .

A = p - p ' p - A = p '

NACA TM 1377 95

conditions which are reduced t o p ' = p - A, and leave an integral other than zero

P7d-hu*(h)ah'p(h) dhU*(P - h)dht7(P - A) (24-6)

A = p ' p - A = p - p '

conditions which are reduced t o p ' = A, and leave behind the integral

After some changes i n the denomination of the subscripts, the first term of assumes the form

Putting p - A = A', while considering the incompressibility condi- t ion, equation (24-8) becomes

A similar calculation which need not be given i n de t a i l gives the - expression of the second term of Z. Finally

(24-10)

96 NACA TM 1377

Now the components of the correlation tensor i n space and time are introduced:

(24- 11)

The uu are expressed i n spectral terms. Putting

it is seen t h a t

a formula which t i e s the correlations Sup t o the generalized spec t ra l -.

tensor 0 UP - Now, i s expressed by means of 0up :

(24-14)

represents s For easier reading of t h i s formula, the term under the sign

the scalar product of the two vectors having fo r components

- The r ea l par t of Z i s 0btained.by transforming the €lap in to

correlations exponentials . SUP by (24-13) and taking the r e a l par ts of the

NACA TM 1377 97

All these calculations do not necessarily assume isotropic turbulence. But i n t h i s par t icular case they a re a l i t t l e simplified and w i l l be termi- nated. If there i s isotropy and incompressibility:

A = 1 XOrn being a real quantity t o be interpreted r igh t away. 2

- - A being rea l , Z i s real and equal t o R_(Z). Therefore

- R(Z) = 2 dh

(24-16 )

To terminate the calculation, the last factors are developed, and w i t h the vector A taken as support of the t h i r d axis one passes t o polar coordinates. The angle 0 of vectors h and A ' i s introduced, defined by

2 h$fj = kk' cos e

Finally, one puts cos 8 = 5 and introduces the elemeqtary function

k + k' log. 31 - (k2 + kf2) (k2 - k12)3 2 kk ' l k - k' 1

NACA TM 1377 98

sent Summary and conclusion.- By the successive assumptions of the pre- paragraph, the fundamental equation (22-8) reads

The function J(k,k ' ) given by (24-17) changes sign when k and k ' respect t o k, from 0 t o a, is therefore zero and (24-18) contains, as it should be, the known equation (19-2):

axe permuted. The integral of the second member of (24-18) with

as it follows from formulas (13-4) and (13-10).

The function A(k,t,-r) i s t i e d t o the space-time correlations:

by the relation

I f the turbulence is isotropic it gives, more simply:

and conversely

sarr being naturally a function of r alone,

(24-20) -.

. If T = 0, the general tensor S a p ( k , t , ~ ) i s reduced to the corre-,

l a t i on tensor i n space Rup(E ,t) and 1 S, becomes R, = R .

F (k , t ) = kk2A(k , t , 0 ) (24-21)

i s ver i f ied whereby the formulas (24-19) and (24-20) contain the formu- l a s (14-4) and (14-5). Therefore, (24-18) i s a nonlinear in tegra l equa- t ion, i n which appear a t the same time A ( k , t , T ) and A(k,t,O) = F(k , t ) and which makes it possible i n principle t o determine the general correlations, i n time and space, of isotropic turbulence by means of t h e i r spectral function A ( k , t , T ) .

25. Second theory of Heisenberg:

T h i s "second theory" came first i n Heisenberg's report ( re f . 2 3 ) . By means of postulates of physical character, it effect ively enables the

But the formula set up s t i l l contains an indeterminate constant. It seems tha t Heisenberg's idea w a s t o carry the expression obtained f o r F i n the

made it possible t o calculate the constant. The theory of sections 23 and 24 does not exactly come up t o these expectations. It results i n a nonlinear in tegra l equation i n which not the spectral function F(k, t ) i t s space-time extension A ( k , t , T ) is involved. I f A ( k , t , O ) = F(k, t ) are known, by the theory w h i c h is t o be discussed, it seems theoret ical ly possible t o deduce from it A(k,t,T) f o r every t i m e in te rva l T. But the mathematical problem thus posed appears d i f f i c u l t and does not f i t i n to any class of known problems, and no attempt i s made t o solve it.

c form of the spectral function F(k) t o be determined i n cer ta in conditions.

.. nonlinear equations deduced from the equations of motion, which would have

but

The dissipation of energy per unit mass i n the ensemble of the spectrum is, as is known

E = 2v k2F(k)dk l (25-1)

100 NACA TM 1377

The dissipated energy per u n i t mass i n the spectral range

1 It 1 h,' < k2, t ha t i s , by the eddies of "dimensions greater than - k and of re la t ively large scale are designated by E k . If k+m,

E k tends toward E , and if k ,O, E k tends toward zero. Heisenberg reduced Ek t o a form similar t o E by introducing an apparent coef- f i c i en t of viscosity yk so tha t

vk must tend toward zero when k 4 a .

Now t h i s f irst hypothesis must be perfected i n such a way that % assumes a usable form.

Lm* Since l i m % = 0, vk can be represented by an in tegra l

What about the function under the sign [P It i s assumed tha t this

function depends on the values k ' ranging between k and m d i rec t ly and by means of it cam be determined by dimensional consideration. having the dimensions of a viscosity

C /?, where C is a purely numerical constant.

k+a

F(k ' ) , but t ha t it does not depend on k. In that case, Its product by dk'

v, it i s necessarily of the form

Hence

Ek is the sum of two terms. The f i rs t corresponds t o the par t O f

the energy of the fluctuations of the spectral range (0,k) whtch i s transformed d i rec t ly in to heat. which, before being transformed in to heat, serves, first, t o maintain

The second i s the par t of that energy

L

4B

c

NACA TM 1377 101

the energy of the fluctuations of the spectral range (k,m) of the "small eddies ."

Now, the fundamental equation (22-8) can be decomposed precisely i n this fashion. Integration member by member from 0 t o k yields

k k k k / F(k')dk' + 2v r k'2F(k1)dk1 = g(k')dk' (25-4) a t 0 ClO

The first member manifests, f i r s t , the energy l o s s of the fluctua- t ions of the range (O,k), then the portion of this energy dissipated d i rec t ly by viscosity. i s transformed in to energy of the fluctuations of the range (k,m), and the formula (25-3) gives ab expression of the function I. Ultimately

The second member is therefore the portion which

This is Heisenberg's fundamental equationll. For the present, it i s not discussed i n i t s most general form, but only i n a special case.

Heisenberg's hypotheses are incompatible with the concept of abso- l u t e steadiness, because (25-5) has no solution independent of chapter V, it w i l l be shown tha t Heisenberg's equation can be applied t o the problem of spontaneous decay of turbulence behind a grid, an essent ia l ly unsteady problem. But, an assumption of p a r t i a l steadiness can be made as w i l l be shown i n chapter I V . For the values of k greater than a fixed approximate value energy dissipated by the large eddies (wave numbers lower than constant. tuations of the small eddies (wave numbers higher than

t. In

b, it can be assumed that the

Only the manner of distribution between the turbulent f luc- k) is a

k) and the

llIt appears i n t h i s form i n the second of Heisenberg's reports (reference f i r s t report (reference

24 ), but the par t icular case i n question i s t reated i n his 23 ) .

..

102 NACA TM 1377

dissipation by viscosity (transformation in to heat, molecular f luctua- t ions) depend cn k, the proportion dissipated as heat increasing with k according t o formula (25-3).

It w i l l be sham ( i n chapter I V ) t ha t this assumption of p a r t i a l steadiness i s equivalent t o an assumption of s t a t i s t i c a l equilibrium or similari ty, which is the more accurate the higher the Reynolds number of the turbulence. O n these premises the hypothesis is discussed from the mathematical point of view. Equation (25-5) must be replaced by

2 + C fs d k ) [kt2F(kt)dk' = Cte = EO i f k > (25-6)

To solve it, simply put

8(k) = [kt2F(kt)dkt F(k) = &'(k) k2

(25-7)

- The function 8(k) cancels out for k = 0 and ver i f ies the equation

which by different ia t ion becomes the elementary d i f f e ren t i a l equation

and introducing an integration constant kl it 4 €0 Putting €13 =

i s seen that

103 NACA Dl 1377

and finally

This formula defines the spectrum of isotropic turbulence for k > k g , in terms of an indeterminate parameter assumptions, themselves suggested by Weizsacker's hypothesis of similarity (reference 42), which will be examined later.

kl, and by means of Heisenberg's

Discussion.- If k is small against kl, F(k) is reasonably pro- 5 --

portional to k 3 . If k is large, F(k) is proportional to k-7 (figure 7). of special interest and used for deriving the corresponding form of the correlation functions.

These two laws have been indicated by Heisenberg. The first is

The formula (25-11) has been established only when k > ko and, actually, has no physical significance if k+O, because F(k) cannot become infinite. It is supposed here, in first approximation, that, if k < ko, the true form of F(k) is F(k) = 0, and, moreover, that k decidedly is smaller than kl- To interpret thi-s hypothesis mathematically is the same as replacing kl by m in (23-11); hence, one uses the physical limiting conditions in which nolds numbers).

kl is infinite (very high Rey-

Putting

F(k) = 0

the correlation function R(r) can be computed.

104 NACA TM 1377

According t o (14-4):

5 5 R ( r ) = 2F(ko)Q7/i sin rk rkk-%k (25-13)

This in tegra l i s convergent when Q > 0; it is not derivable twice under the integration sign. An attempt is made t o f ind the form of func- t i on R ( r ) f o r small values of r . It is shown that there ex is t s a number a such that

A being a constant.

When r+O, it i s eas i ly ver i f ied that R ( r ) has a f i n i t e l i m i t , equal t o

Forming

and making the change of the variable rk = x i n the integral , leaves

NACA TM 1377

Posting

The in tegra l d i f fe rs from I by infinitesimally small

quantit ies with r. Hence, i n f i r s t approximation when r i s not too great,

R ( r ) = 3w2(1 - ~4.1-5)

with

2 2

(25-17)

2 3

The exponent looked for , a, i s equal t o -. T h i s formula i s impor-

t a n t , and we sha l l give l a t e r on demonstrations fo r it, of wholly d i f fe r - ent appearances. The formula tha t gives the correlation function f ( r )

is similar t o (25-17). 2

r3." -

It again i s a "law i n

It is a l imiting l a w f o r in f in i te kl. If kl is f i n i t e , formula (25-11) yields for R ( r ) a law of the ordinary type

R ( r ) = 3% 1 - - + . . 2 ( :2 .)

which proceeds, for of curvature equal t o A. When kl-m, this radius of curvature tends toward zero and the l imiting curve has the form (25-17) (figure 8).

r = 0, w i t h a horizontal tangent and a f i n i t e radius

.

106 NACA 'IM 1377

Note.- Comparison of Heisenberg's equations (25-5) and (24-18). Equation (25-5) i s an equation i n F and can be used t o reduce it t o a (complicated) d i f f e ren t i a l equa- t ion. The calculation w a s made i n the "steady" s t a t e . It leaves, i n the expression of F, the constant not determine. If (24-18) had been an equation i n F too, it could have been used t o determine C. But the procedure that r e su l t s i n this equa- t ion and whose or iginal purpose i s t o eliminate the t r i p l e correlations, meets the expectations only par t ia l ly , because it replaces i n some way the t r i p l e correlations by the correlations of space and of time. Equa- t ion (24-18) serves therefore, as f a r as the mathematical problem is accessible, t o compute the form of the space-time relat ions, once the space correlations are determined by (25-5). the numerical solution of (24-18) has ever been attempted up t o now.

F, which, theoretically, i den t i f i e s c

C, which equation (25-5) can obviously

But it does not appear t ha t

THEORY OF LOCAL ISOTROPY AND STATISTICAL EQUILIBRIUM

I

26. Introduction:

The importance attached t o the study of isotropic turbulence w a s , up t o the l a s t few years, j u s t i f i ed by considerations of, seemingly, essen- t i a l l y pract ical value.

The development of turbulence toward isotropy had i n i t s favor a ra ther feeble theoret ical argument, namely, that, far from the sources Of turbulence, the Reynolds s t resses could be assumed zero (see section 6) , which i s a condition necessary, but far from being suff ic ient , f o r isotropy.

Batchelor (reference 12) proved i n 1948 tha t t h i s argument i s of no value; the elements of the anisotropy of turbulence during i t s formation could be recovered i n the f i n a l phase of i t s evolution.

But, from the experimental point of view, the isotropy of turbulence behind a gr id seems well established, by numerous experimental works, as w i l l be shown l a t e r . seemed that a par t icular case w a s studied here solely by reason of i t s simplicity.

But none of these evolved a general concept; it

The theories t o be examined here form, by way of contrast , an unusual ensemble, not only because of the successful experimental ver i f icat ions but par t icular ly because of the scope i n which they contribute t o fur ther studies of turbulence. They bring t o l i g h t the characters of turbulent motion which are of general significance.

107 NACA TM 1377

The most outstanding r su l t i s perhaps t h be t te r understanding of the ro le played by the viscosity i n turbulence. involved a t the b i r t h of turbulence and a t i ts death; a t i t s b i r t h it is the condition necessary for the creation of the rotat ion and the diffusion of large turbulent masses i n the f lu id ; it intervenes a t i t s death (decay) since the turbulen t energy is ultimately transformed in to heat; but the complexity of the turbulent motion stems precisely from the f a c t t ha t the viscosity plays a negligible par t during the major pa r t of the transformations of turbulent energy.

The viscosi ty is

The fundamental outline i s as follows: re la t ive ly large turbulent masses with re la t ive ly low velocity gradients are formed i n the creative regions of turbulence. The sense of the word re la t ive can be defined by

introducing the Reynolds number -, %Lg where .02 is the mean kinet ic v 2

energy and t o have turbulence i n the proper sense of the word, the r a t i o

the mean quantity or' tnese turbulent masses .12 I n order

must be high. has ca l led the f i n a l period of turbulence, i n which the rotat ion decays, so t o speak, on the spot. The forces of i n e r t i a play a secondary par t . The flow i s quasi-laminar. passes. and Townsend (references 10, 12), and we sha l l come back t o it l a t e r on.

If it is small, we find ourselves a t once i n what Batchelor

This i s the phase through which a l l turbulence It has been experimentally and theoret ical ly studied by Batchelor

If the Reynolds number is high, the vortices undergo a double evolu- t ion, according t o Heisenberg's representation (reference 23) ; they ge t bigger on the one hand and smaller on the other, t h a t is, t h e i r spa t i a l dimensions increase on the one hand, and break up in to smaller vortices on the other. Leaving aside the f i r s t aspect of this evolution, fo r the time being, the theory of s imilar i ty studies the development of the energy passed on t o the small vortices; the l a t t e r undergo a development similar t o that of the generating vortices, they pass on t h e i r energy t o smaller vortices and so fo r th up t o the moment where the s ize of the vortices i s such that, t h e i r Reynolds number is small. energy i s then changed into heat. This somewhat vague representation is defined a l i t t l e more accurately further on, i n par t icular , by the analysis of the rotat ional development.

A bigger and bigger par t of the

12This ra ther vague expression is specified t o some extent by the study of dynamic equilibrium (chapter V) .

108 NACA TM 1377

It is reasonable t o think - and this assumption w i l l be backed up by i t s consequences - t ha t the development of vortices i s the more rapid the smaller t h e i r s ize . On the other hand, the statistical character of the velocity of the fluctuations, which i s prescribed by experience, i s assumed.

Hence it may be imagined that, when the energy has been transmitted t o vortices suff ic ient ly smaller than the i n i t i a l vortices, the s t ructure of these vortices is , on the average, independent of t h a t of the generating vortices, and that the re la t ive ly slow development of the la t ter has no e f fec t on the development of the smaller s i z e vortices. The l a t t e r , therefore, find themselves i n a quasi-balanced s t a t e , one determining fac tor of which i s the energy per uni t mass and time which i s supplied by the bigger t o the smaller vortices.

27. Definition of loca l homogeneity and loca l isotropy:

A t f i r s t w e sha l l follow Kolmogoroff's method (reference 31. more detai ls see Batchelor, reference 6) which seems the most d i rec t as well as the easiest , a t l ea s t , t o begin w i t h . Kolmogoroff introduces quantities which can be completely defined i n a limited spa t i a l domain G and a limited temporal f i e l d T. Assume that M and M ' a re two points of G, tha t t and t ' are two instants of t; we c a l l r the distance MM' , and put T = t - t ' . .

For

The difference w of the veloci t ies a t M and M ' a t ins tan ts t and t ' i s introduced

w i = % ( M , t ) - % ( M ' , t ' ) (27-1)

and defined by i t s l a w of probabili ty i n the space-time domain (G,T). One notices immediately tha t w is l i t t l e dependent on the spectral components of the velocity which develop slowly i n the spa t i a l domain and i n the time interval T.

G

By i ts definit ion w depends a p r i o r i on M and t, as w e l l as on M ' and t ' .

Kolmogoroff defines w h a t he c a l l s the loca l homogeneity and loca l isotropy. A condensed version of these definit ions reads as follows:

The turbulence i s said t o be loca l ly homogeneous when the laws of probability fo r the s t a t i s t i c a l function w depend, i n the domain (G,T),

5B

.

. only on T and the vector (actually, instead of t h i s vector, the vector 2 defined l a t e r on i s involved).

The turbulence i s termed locally isotropic when it i s loca l ly homogeneous and when these probabili ty l a w s depend neither on the or ientat ion nor on the sense of the coordinate axes.

According t o the foregoing, it may be assumed that i n every turbulence having a ra ther high Rey imlds number, there ex i s t s a s p a t i a l domain G which is s m a l l compared to the s ize of the big vortices, and a t i m e in te rva l t o change appreciably, and such that the turbulence is loca l ly homogeneous and isotropic i n the domain (G,T) .

T, short compared to the t i m e necessary f o r their energy

But these definit ions can have no physical significance unless the reference system used by the observer is i t s e l f physically coordinated w i t h the turbulence. The domain G must, i n some fashion, follow the turbulence w i t h the mean flow velocity. Kolmogoroff, i n h i s or ig ina l report (ref. 31) defines the velocity w about a reference point (k,to) i n terms of the components Si of the vector M# and of T = t - by the formulas

_j

then he introduces a vector e having as component

This is a random vector whose physical significance is evident i n f igure 9. Since the essent ia l point of the vector T"(,~o) arises from the "mean velocity" i n the domain G, the use of the coordinates

i s tantamount t o following the turbulence i n i t s en t i r e motion. ei

The correlation measurements in time re la t ive t o the l a w s of proba- b i l i t y of w are f e w and far between. Some measurements have been made i n water and i n air , but none i n the wind tunnel, as far as w e know. Obviously, i n such measurements, time-space intervals must be combined and it must be ascertained whether correlations close t o 1 can be

obtained for a f i n i t e value of the r a t i o

the mean velocity a t the scale involved.

4

E which, i f it exis t s , define. I-

110 NACA TM 1377

. But, a t the present s t a t e of research the study i s l imited t o s i m u l -

taneous quantit ies f o r defining the s t a t i s t i c a l values isotropic turbulence, it i s usually l imited t o the moments of the second and t h i r d order.

w. As i n ordinary, . The tensor of the moments of the second order

simultaneously introduces three points, but can be expressed, on account of the isotropy, homogeneity and incompressibility, as a sum of moments re la t ive t o the couples (MI%), (MIy%), (M,M'):

- 1

I n this case, the tensor

(27-4)

plays a role para l l e l t o t h a t of the tensor i n isotropic turbulence, and by the same argument it i s apparent t h a t the isotropy leads t o w r i t i n g

Rij

where

(27-5)

c

Ud and representing the velocity components along MM' or normal

t o MM'.

A t o t a l moment can be defined too:

B ( r ) = 1 Bii = L (u t i - ui)2 i

The incompressibility involves the re la t ion

equivalent t o K & m & ' s r e la t ion i n isotropic turbulence (9-7). re la t ions can be established, i n particular, the expression of the dissipation of energy by viscosi ty

Other

Likewise, the moments of the t h i r d order can be expressed by the single function

28. Similari ty hypotheses. S t a t i s t i ca l equilibrium:

The def ini t ion of these loca l quantit ies enables the notion of sta- t i s t i c a l equilibrium of small vortices to be defined. According t o our hypotheses, t h i s equilibrium i s not dependent on the par t icu lar charac- t e r i s t i c s of the flow. The only quantities which can be of influence axe :

(1) The energy supplied per unit mass and t i m e t o the s m a l l vortices

(2 ) The factor governing the kinematic viscosity.

by the larger ones;

NACA TM 1377 112

If the fluid is assumed incompressible, the specific mass can be discounted.

Hence Kolmogoroff's postulate of similarity: in a domain where the turbulence is locally homogeneous and isotropic, the laws of probability depend only on r, E , and v.

This postulate justifies the dimensional analysis which results in setting

(28-1)

In this formula, pdd is a universal function, and 2 =

represents a length, called the local scale of turbulence by Kolmogoroff. This length is much smaller than the length of dissipation isotropic turbulence.

h of the It is, in effect (compare (14-11))

whence

where the Reynolds number

conditions of similarity are to be realized. Ilh = - must be at least some tens if the V

The nondimensional statistical quantities must be universal func- r 1

tions of -. Consequently .

c

NACA TM 1377

L J /

U ' d - Ud)3 (coefficient of dissymmetry) S (f\ = (

3 Xi 1-

( U ' d - ")4 (coefficient of f la tness ) Ad(;) =

r(uld - ud)212

(28-2)

and = (28-3)

must be universal constants, which affords a first confirmation of the s imi la r i ty hypothesis.

Townsend ( re f . 37, 38), at Cambridge, developed a method of measuring S(0) and A ( 0 ) . The measurements were f i r s t made i n isotropic turbulence produced by a grid. Reynolds numbers ( ra t ios varying from 1 t o 5 ) and a t varying distances from the gr id (likewise i n a r a t i o near t o 5 ) , tha t is, a t different moments of the development of the turbulence.

These coefficients are constant over a wide range of

Townsend quotes the value 3.49 -f 0.04 f o r A and -0.38 f o r S,' with a l i t t l e lower accuracy. butions the values of A and S are, respectively, 3 and 0.

It should be recalled tha t f o r Gaussian distri-

114 NACA TM 1377

. To check the theory of local ly isotropic turbulence, by which the

structure of the small vortices i s independent of the nature of the large vortices and the formation of turbulence, Townsend made also measurements i n the wake of a cylinder ( re fs . 39, 40, and 41), t ha t i s , i n turbulence neither homogeneous nor isotropic ( i n the or iginal sense). He determined thus Sd(0), Sn(0), b ( 0 ) and An(0) over the en t i r e extent of the wake, a t distances of more than 80 diameters from the cylinder. The values obtained are the same, w i t h suff ic ient accuracy, as those obtained fo r isotropic turbulence behind a grid. Furthermore, the measurement of the quantit ies

"

u1 being pa ra l l e l t o the overall velocity (axis of

f o r isotropic turbulence and throughout the en t i re w a k e , these quant i t ies s a t i s fy the equations of isotropy

xl), shows that, as

By the same argument, the expression of the energy dissipation i s

2 Qvk) , as f o r isotropic turbulence, and as the equations (27-5) and 2

(27-7) may lead t o presume.

These formulas are very important because they show the poss ib i l i ty of defining the dissipation of energy i n any turbulent flow by measurements bearing on a single velocity component.

29. Case of high Reynolds numbers (reference 6):

The study of the s t a t i s t i c a l equilibrium of vortices can be extended fur ther when the Reynolds number i s high enough so that the viscous dissipation i n the la rges t vortices of the s t a t i s t i c a l equilibrium is r e l a - t ive ly low; it can be assumed then tha t the s t a t i s t i c a l equilibrium of these large vortices i s not dependent on

. v .

c

NACA TM 1377 115

c

This i s Kolmogoroff's second hypothesis:

If 2 << r<< L, the l a w s of probability regarding w a re then dependent on E and r only.

T h i s implies that a domain G can be found whose s ize L is large 2 , t ha t i s , t h a t the Reynolds number is very high. compared t o

Dimensional analysis permits then the def ini t ion of the form of the moments of the s t a t i s t i c a l variable w. I n f a c t it i s readi ly apparent

must be of the form tha t , f o r Bdd(r ) not t o be dependent on v,

whence

C being a universal constant (figure 10).

The moment B, m u s t have a similar form. The equation of continuity

4 3

indicates then that Bm( r ) = -Bdd( r) . r 2

On the other hand, when - is s m a l l ,

B,(r) = -B",(O)r2 1 = 2Bdd(r) 2

Likewise, one must have

Bddd(r) = C t e X ET

116

But, i f r i s small

NACA TM 1377

because i n that case, Bddd(r) f Sd(0) bdd(r)]', i n first approximation. Since 2 depends on v, the nondimensional quantit ies such as Sd('\

2 / must be constant, f o r values of r much greater than 1 . On the other hand, is a l so a universal constant, and the experiment seems t o

indicate that these two constants are ident ical . Sd(0)

To t e l l the truth, there i s only one ser ies of measurements of

made by Townsend i n a region where the turbulence w a s isotropic, by measuring the t r i p l e correlation function

Sd(r)

- 2 u u' c ( r ) = -

3 uO

The relat ions

(u - u y = -6u()3c(r) (., - = uo2 E - f(r] (29-4)

a re ut i l ized.

The experimental points are placed near sd = -0.38 fo r values of

of the order of 402. The absolute value of S decreases slowly 2 afterward .

Instead of arguing about the correlation functions, one may j u s t as well apply the dimensional analysis t o the spec t ra l functions F(k) f o r values of k which are suf f ic ien t ly large without being too large, however. Since F(k) must be dependent only on k and E, it i s found that

2 2 - F(k) must be proportional t o E& 3 i n the domain under consideration.

c

6B NACA 'I'M 1377 117

.

- 2 T h i s i s the spectral l a w i n k 3 of Heisenberg's theory (section 25).

spoken of earlier during the discussion

Using the detailed hypotheses of Heisenberg, we have then sham that

r3 and the spec- 2

there i s equivalence between the law of correlation i n

t r d l a w i n k 3. and t h a t it i s not the result of the par t icu lar form of the l a w of energy t ransfer i n the spectrum, but s o l e l y due t o the existence of a general s imi la r i ty hy-pothesis among the hypotheses the u t i l i z a t i o n o f which w a s unimportant.

5 - - It i s now seen t b a t this equivalence w a s inevitable,

BY using the equations df motion the value of Bddd(r) can be defined

and a re la t ion established between the constants C and S, which is sus- ceptible t o experimental verifications.

The equation corresponding t o that of K & d n (18-5) reads here

(29-5 1

The t i m e factor

homogeneity contains

does not appear, because the hy-pothesis of l oca l

t ha t of steadiness with t i m e , so that h2 must dt

be taken equal t o - -E. 2 3

The equation (29-5) multiplied by r 4 , is integrated i n the form

If r i s small, Bddd can be neglected, so that one f inds again

the re la t ion

NACA !I'M 1377

For the values of r i n the in te rva l 2 << r << L, Bddd prevai ls

1 - 7 over €Isdd, because Bddd i s proportional t o r and B'dd t o r

Hence

But i n (29-3) it was seen that, i n first approximation:

Consequently

Therefore, C can be calculated, i f the value of Sd(0) i s known and, particularly, i f the experimental value I n that case

Sd(0) = -0.38 is assumed.

Conversely, the experimental study of the function Bdd(r ) enables

C computed from Sd(0).

t o be measured and the obtained value t o be compared w i t h the value

30. Validity of the s imi la r i ty l a w s :

All second-order moment measurements have been made i n isotropic turbulence. l oca l isotropy if applied t o isotropic turbulence proper.

Hence it i s useful t o examine what becomes of the theory of

.

The f irst point is the determination of the space-time range of

It is b r i e f ly summarized here: val idi ty . Batchelor (ref. 6).

The discussion, necessarily ra ther vague, has been made by

From the spa t ia l point of view, isotropy appears t o be assured for

dimensions of the order of magnitude of the correlation length L =

The d i f f i cu l ty a r i ses from the estimation of the time interval . ,An evaluation of the character is t ic period of the vortices indicates t h a t the r a t i o between the period of the largest and tha t of the smallest

where A i s the length of dissipation. vortices i s of the order of

This r a t i o must, therefore, be great. Experience indicates an order of

magnitude of i n what Batchelor ca l l s the " i n i t i a l phase of decay of

turbulence." i s simply s ta ted tha t it is produced by the turbulence behind a gr id of

mesh s i ze M, when the Reynolds number & = - of the gr id (U = mean

velocity) i s suff ic ient ly high (1,000 t o 3OO,OOO) and the measurements are made a t a distance from the grid of the order of 1,000 M.

2 L h (It i s t o be defined i n chapter V.) For the present, it

UM V

One f inds Uoh then that L i s of the order of !b where 3 = - may be cal led the

h 10' V

Reynolds number of turbulence. During the phase i n question, EA i s

constant and equal t o /?, where A is a number dependent on the

, const i tut ion of the grid.

For values of Rh - not quite up t o 200, which s t i l l gives only values

of r a t i o L h

function g ( r ) . Since

of the order of 20, Townsend has measured the correlat ion

- g(r) must be constant. 2 -

r3

120 NACA TM 1377

The curves plotted for various values of % and have similar *

M shapes. They present a rather f l a t maximum between r = 0.1L and r = O.3L, from which the value of the constant C deduced. The average of the obtained values i s 1.53, with discrepancies of l e s s than 3 percent. Considering the accuracy of the measurements of Sd(0), the agreement with the value 1.64 deduced from

suff ic ient .

(section 29) can be

Sd(0) i s

Other measurements, made at a lower Reynolds number, yield C = 1.33, w h i l e those by Dryden produced C = 1-50. In conclusion, although the range of a va l id i ty of the law i s too r e s t r i c t ed fo r the aspect of the curves t o be convincing, the agreement of the values of C i s quite satisfactory.

The lower l i m i t of va l id i ty of the l a w of loca l isotropy i n these experiments i s of the order of magnitude of h, that is, well superior t o

h 2

2 , since - = 4&€3, i s here of the order of 20 t o 25.

It seems that the Reynolds numbers used are s t i l l too low f o r the - f i e l d of application of Kolmogoroff's second hypothesis t o be extended very far. The first hypothesis, less r e s t r i c t ive , i s much be t te r ver i f ied. Townsend's correlation measurements show, i n f ac t , that the curves repre- *

senting g(r) are when r << h functions of - only, at different dis-

tances from the gr id . The s imi la r i ty i s extended t o values of higher

than unity for the highest values of possesses a domain of val idi ty . €3~ chapter V . ) obviously, not be deduced from the theory of loca l isotropy which assumes E constant.

r h

h RJ, where the second hypothesis

T h i s va l id i ty i s t i e d t o the f a c t that i s constant a t the beginning of the turbulent development. (Compare

T h i s information on the development of turbulence can,

On the other hand, the absence of s imi la r i ty i n the correlation r h

curves at high values of - suggests that the range of application of

the s imilar i ty l a w i s very res t r ic ted , and a l so belief i n the f a c t that the r a t i o of correlation length L t o dissipation length h varies i n reasonable fashion during the evolution of turbulence, even when I31

I .

R remains reasonably constant, because the re la t ion

first approximation.

L = -h h 10

i s only a

In rea l i ty , the domain of s t a t i s t i c a l equilibrium i s extremely extended, but this does not appear i n the s t ructure of the spa t i a l correla t ion tensor. For a c lear def ini t ion of the individual behavior of each

-

c

scale of vortex dimensions the introduction of the spectral tensor i s necessary. Heisenberg's theory is therefore found capable of exceeding the stage reached by the theory of local isotropy, and it likewise permits interpretat ion of the different stages of development of turbulence discussed i n paragraph 31, and i n chapter V.

31. Interpretation of the l a w s of s t a t i s t i c a l equilibrium i n spectral terms - Weizsacker's and Heisenberg's theories (refs. 42 and 23, respectively) :

Kolmogoroff's theory assumes a local ly steady equilibrium and i t s s p a t i a l domain of application i s necessarily res t r ic ted . However, ex&- nation of the correlation curves and of the laws of variation of the correlation length L during the decay of isotropic turbulence behind a gr id leads one t o believe that the scope of s imilar i ty w i l l be extended fa r ther than anticipated.

But t h i s notion of s imi la r i ty can not appear c lear ly unless the influence of the different vortices is separated. This means changing from the correlation functions t o spectral functions.

It w i l l be remembered (compare sections 11 and 12) tha t t o the s p a t i a l correlation tensor Rap(S) = uautP there corresponds the spectral tensor ~ I ~ ~ ( A ) defined by

If there is isotropy, the spectral tensor depends on a single spectral function function R ( r ) f o r example.

F(k), j u s t as the spa t i a l correlation tensor depends on a single

To the properties of R(r) for small values of r correspond those of F(k) f o r great values of k, as results from formula (14-4) which l inks F t o R, and which is recalled here:

R ( r ) = 2 =(k)ak rk

122 NACA TM 1377 c

It has been shown (section 29) t ha t t o the law of correlation i n 2 c -

r3 for R ( r ) , applicable i n the domain 2 << r << L, and resul t ing from Kolmogoroff's second hypothesis, there corresponds a spectral law of the form

where A i s a universal constant. This l a w i s val id i n the domain Q < k < ks and the numbers 25, are of the order of unity.

In the most general case of Kolmogoroff's f irst hypothesis, one can only put: (.

F(k) = Fo@($)

f o r k > ko, FO and being functions of E and v only and Q a universal function.

*

Putting jr(k')dk' = T ( t ) - i n (22-8), gives

h[F(k')dk' a t = -z(k) - v (31-1)

The function T(k) represents the t ransfer of energy by turbulence of the large eddies-to smaller eddies.

In the case i n which the dissipation by viscosity i s negligible, the conditions of s t a t i s t i c a l equilibrium are wri t ten simply T(k) - = cte.

3 5

-- A dimensional analysis affords the spectral l a w i n k . In fac t , it can d

be sa id that x(k) has the dimensions of averages so that u 1 u 2 ~ au1 is

NACA TM 1-57" 123

..

.

vk3k, where k for wave number. Hence one must have vk3k = cte, i n the domain

vk is a cer ta in character is t ic velocity of vortices having

ks<< k<< ko, and

For velocity

1 -- vk is proportional t o k 3 ,

VkJ the quantity I 1 ( uti - ui)2 which, according

t o Kolmogoroff's theory, is proportional t o can be assumed.

r3 i n the chosen conditions, Weizsacker's l a w i s equivalent t o Kolmogoroff's l a w

5 -- vk2 k

because F(k), having the dimensions of -, is proportional t o k 3 .

(Compare section 25.)

The variation with k of other quantit ies associated w i t h d i f fe ren t vortices can also be determined. For example, the character is t ic period

of the vortices is proportional t o -, a quantity which has the dimen- 1 kvk

2

k '. -- sions of a time interval, that is , t o increases, which is i n agreement w i t h the hypothesis according t o which the character is t ic periods of development of the small vortices are shorter than those of large s i ze vortices.

It decreases when. k

It can likewise be ver i f ied that the viscous dissipation, which i s 1

2 3 proportional t o k F, varies l i k e k . Hence, it cannot be neglected f o r great values of k.

-

If now the viscosity i s no longer neglected, the form of the func- t i o n E(k) must be specified. This is what Heisenberg did who, as shown i n section 23, has assumed that could be put i n the same form as the losses of energy by viscosity, the form of the "turbulent viscosity" coefficient result ing from s imi la r i ty considerations.

T(k)

2 The comparison w i t h experiment of the spectral l a w

Nevertheless, i t s range of

k3 has not as yet been carr ied out on flows i n which the Reynolds number is high enough so t h a t the conclusions are w e l l defined.

2 va l id i ty seems more extended than that of the law t ions, because the influence of the various vortex s izes i s d i f fe ren t due t o the use of the spectral analysis.

r7 for the correla-

I24 NACA TM 1377

*

Tarnsend attempted t o check the va l id i ty of the expression of the function z(k) given by Heisenberg, and according t o which F should be

m

i s proportional t o f k%(k)dk. If the l a w i n k-7 were ver i f ied, this J O

integral should be infinite, which s ign i f i e s t h a t the in tegra l ~

k°F(k)dk should increase considerably together with i ts upper

However, the experiment seems t o indicate that this in t eg ra l t limit 5. has a limiting value independent of

turbulence. Therefore, the law i n k-7 cannot be t rue f o r the highest values of

5, and variable during the decay of

k, which limits the range of va l id i ty of Heisenberg's theory.

CHAPTE3 v

DECAY OF TIE TURB-CE BEHIND A GRID

32. History:

The theory of loca l isotropy shows tha t the l a w s of turbulent flows are the same, regardless of the or ig in of turbulence, provided that the conditions are t t local" and the Reynolds numbers high enough. deals w i t h the par t icular l a w s of a turbulent flow of great importance, that which i s produced i n the test chamber of wind tunnels, and i n which turbulence is largely due t o a grid, that is, an obstacle of periodic structure. Turbulence a r i ses from the mixing of w a k e s of g r id elements, and i s dissipated progressively as it moves fa r ther away from the gr id . For an observer carried along with the velocity of the ensemble, the turbulence is therefore subjected t o a "decay" as function of the t i m e . The laws of t h i s decay are, naturally, compatible w i t h the loca l l a w s of Kolmogoroff, but today they are known w i t h a high degree of accuracy i n a much more extended domain, and it can be stated that it is i n the study of this phenomenon that the theory of turbulence, s teadi ly checked by experiment, has made the fastest and most constructive advance.

T h i s chapter

17B

. The basis of this theory was established by G. I. Taylor, who, by

considerations of s imilar i ty , had indicated that, i f the Reynolds number

is high, the turbulent energy must decrease as t'l as a function

of the t i m e , 2

The measurements made since by numerous researchers have sham that this l a w is close t o experimental results; nevertheless the experimental curves were rather of the type

n ranging between 1 and 2, as soon as the turbulence in tens i ty becomes low, the in te rva l t o which the comparisons re fer , is res t r ic ted .

As the precision of the measurement is low

The variation of the length of dissipation h i n terms of time is t i e d d i rec t ly t o tha t of the intensity of turbulence by the re la t ion (19-1)

~

If L+,* varies as t-n, one deduces from it:

2 1ov h =-t n

But, up t o the l a s t few years, the measurements of h offered no poss ib i l i ty of defining the value of n. The value of h is, i n e f f ec t , deduced from the curvature a t the origin of the curve representing the correlat ion function f (r) . e ters , of which the distance r cannot be reduced indef in i te ly fo r manifold reasons (geometrical and physical). i s not well known f o r small values of r.

The measurements are made w i t h two anemom-

As a resu l t , the curve f ( r )

However, an examination of the correlation curves seems t o indicate I that t h e i r shape remains constant i n terms of the distance from the g r i l l ,

during the decay of turbulence.

All these questions have taken a decisive forward step, as a r e s u l t of A. A. Tamsend's new methods which enable d i rec t measurement, by e l e c t r i c a l

~"

*

126

aU a2u ax ax2

methods, of the momentary o r m e a n values of the quant i t ies u, -, - and those of the products of two or three quantit ies, as w e l l as the d i r ec t determination of the s t a t i s t i c a l frequency curve of a quantity i n terms of the velocity.

c

The use of these methods enabled G. K. Batchelor and A. A. Townsend t o determine the law of decay of turbulence produced by a g r i l l with a good degree of accuracy over a wide range of values of the Reynolds numbers of the g r i l l , defined by

& = u M V

(32-1)

where M is the width of the mesh, U the overal l velocity.

When the turbulence i s isotropic

(32-2)

affords the evaluation of h

much more quickly and more accurately than by the procedure employed previously, and consequently defines A i n terms of t i m e , that is, of the distance from the turbulence-producing grid.

"l\'

(K, The d i rec t measurement of

33. I n i t i a l and f i n a l phase of turbulence (Batchelor and Tamsend, re fer - ences 8 and 11):

Townsend's measurements made it possible t o distinguish i n the decay of turbulence a n i n i t i a l phase and a f i n a l phase, separated by a less w e l l defined intermediary phase.

The i n i t i a l phase, c lear ly proved by these experiments, corresponds t o g r i l l distances of l e s s than 100 or 170 mesh widths (variable according t o the Reynolds number). turbulent energy can be represented by an expression proportional t o t'n where the exponent n d i f fe rs from unity by l e s s than 10 percent. It is

sa id that uo2

It is characterized by the f a c t that the decay of

. decreases as t-'.

c

In the f i n a l phase, which corresponds t o the lFmiting s t a t e of decay 5 -- 2 of turbulence, experiments prove that uo2 decreases as t .

The i n i t i a l phase is also characterized by the f a c t that the Reynolds number of turbulence, defined by

R = - "oh -A v (33-1)

remains constant throughout the period of decay t o which it corresponds.

If , i n f ac t , .02 = At'n, it is seen that A2 = at, and that n

Thus - RA remains constant i f n = 1, and i n this case only.

To define the structure of the two essent ia l phases of turbulence, we now introduce the rotat ion (Batchelor and Tmsend , reference 10) of which the relat ions w i t h the correlation functions and the diss ipat ion length h are simple, a t l e a s t when the turbulence is isotropic. The general equations relevant t o the rotation are simplified by the K & d , n - Howarth equation.

The rotat ion of the velocity, defined by the formulas

(33-2) aul au3 a%? au1

m 3 = - - - ?l=--- ax, ax3 9=--- ax3 axl ax1 ax2 au3 8%

s a t i s f i e s t w o well known equations which are obtained by simple combina- t ions of the equations of motion

(33-3)

128 NACA '1M 1377

I f ( 3 3 - 3 ) i s derived w i t h respect t o xi, since one interchanges i and j and subtracts the two equations obtained, the r e su l t is

The two members of (33-4) are multiplied by 2q, or added with

respect t o i, w i t h due allowance f o r incompressibility, the averages

taken and homogeneous turbulence assumed. Putting 3u2 = y2 leaves

(33-5)

I f , i n addition, the turbulence is isotropic, then (section 9 ) :

I n this equation, the first term of the second member represents the increment of the vor t ic i ty due t o the elongation of the vortex filaments, and the second the decrement due t o the influence of the viscosity. It is ver i f ied that this last term rea l ly i s negative because

and - is zero i f the turbulance i s homogeneous. n c

axk

Batchelor and Tamsend masured the three terms of (33-5) di rec t ly . *

But it i s convenient t o give equation (33-5) a s l i g h t l y d i f fe ren t and easier-to-interpret aspect. The Kdrmh-Howarth equation (18-5) can be

.

.

used to evaluate the first member. of f, c, up to the terms in (cf. section 9 )

Introducing the limited developments

i... 3 c "'( 0) c(r) = r -

6

and identifying the terms in r2, gives

Now, the derivatives of the functions f and c can be expressed in terms of the average values bearing on the velocity derivatives. calculation, as done before for the rotation, gives

The

u&"'(o) =

(33-7 1

If therefore the nondimensional quantities

G = - (33-8)

are introduced of which the f irst has already played a prominent p a r t i n section 28 (coefficient of dissymmetry, fo r

t

r = 0), it i s seen t h a t c

A being, as w i l l be remembered, a "length of dissipation" defined by 1 f"(0) = --.

A2

- Since u? = $, equation (33-6) can be writ ten

A2

where F& = - i s the Reynolds n d b e r of turbulence already defined by

equation (33-2).

K&&-Howarth equation of turbulence, equivalent t o (33-5), but superior f o r comparisons w i t h experiments.

V

So i n turbulence language, equation (33-10) i s a consequence of the

The measurements of the various terms of (33-10) during the i n i t i a l period of decay of turbulence confirm, first, that

R+ = cte A2 = l O v t

t being counted from any convenient or igin.

Consequently one may write :

(33-11)

NACA '1M 1377

and (33-10) takes the form

G = & R S + - 30 - 2-1 7

131

(33-13 1

It has been shown (section 28) tha t the neasuremellt,s of the coeff i - c i en t of asymmetry S prove that the l a t t e r i s independent of R,, - and

close t o 0.38. The d i rec t measurements of G are i n good agreement with the above formula, i n which S = 0.38, which consti tutes a check on the qua l i t i es of the measurements.

These results mk& it possible t o eva lmte the ra t io of the two terms of the second member of the equations (33-5) or (33-10). The quo- t i e n t of the term representing the dissipation through viscosi ty t o tha t which expresses the i n e r t i a e f f ec t s is equal t o

. or , replacing S, by i ts experimental value

22 1 + - %

(33-15 1

This r a t i o i s constant throughout the i n i t i a l period of decay of

turbulence. If 22 is considerably less than unity, the contributions EA -

of the viscosity and of the i n e r t i a terms are of comparable a(02 t o - importance and the velocity result ing from the variation of the vo r t i c i ty is of an order of magnitude lower than each one of these two terms. This is the range of high Reynolds numbers

equilibrium (Kolmogoroff, Weizskker) applies.

the wave number k, the spectral law k '3 can be observed there. The

a t

53, where the theory of s t a t i s t i c a l

For large enough values of

- 5

*

measurements show, i n fac t , tha t , i n proportion as

correlation curves approach the l imit ing shape with tangent ve r t i ca l t o the origin as implied by the formula

f4h increases, the

-

The or iginal parabolical

2 - f = 1 - cr3

region becomes consistently smaller as I l l becomes higher (Figure 11; compare a l so the end of section 25).

If 22 approaches unity (Reynolds numbers neither too high nor too El

l o w ) , the results deter iorate and new p a r t i a l o r t o t a l s imi la r i ty hypotheses must be made, resul t ing i n solutions of the type of those studied i n section 21 from the purely repeated here.

22 If - is great ( l o w

correlations is negligible of the decay of turbulence t ions of section 20 can be

EA

mathematical point of v i e w , and which are not

Reynolds numbers), the e f f ec t of the t r i p l e - against the viscous dissipation. The mechanism - is mainly due t o the viscosity, and the s o h - invoked. T h a t i s w h a t happens i n par t icu lar i n

the f i n a l phase of decay which is discussed below.

34. Concepts regarding the s t ructure of the f i n a l phase of turbulence (Batchelor and Tamsend, reference 12; Batchelor, reference 8) :

It is necessary

decay of turbulence, consequently

t o predict by theory the law of the f i n a l phase of G / --

according t o which uo2 decreases as t *, and,

A2 = 4 v t (34-1)

When the turbulence i s suf f ic ien t ly attenuated, the f luctuat ion i s s l igh t , and the i n e r t i a e f f ec t s negligible against the e f f ec t s of v i s - cosity. be discounted, and one may simply write

As a resu l t , the nonlinear terms i n the equations of motion can

18B

.

NACA TM 1377 133

In these conditions the equations of motion can be integrated, the velocity components computed i n terms of t i m e and the i n i t i a l conditions, and the correlations deduced. and it cannot be accomplished except i n very special cases. time correlations can be calculated j u s t as w e l l as the c l a s s i ca l space correlations.

T h i s is the program set fo r th i n sect ion 16, Thus the

Only the case of isotropic turbulence i s analyzed. To neglect the i n e r t i a terms is t o neglect the t r i p l e correlations i n the "fundamental equations." Thus one reverts t o the conditions of sect ion 20, or turns t o the poss ib i l i t y of expressing the spectral equation i n the simplified form

3F - + 2vk2F = 0 at

The general solution of this equation reads

F(k,t) = F(k,tl>e -2vk2 (t-$

(34-2)

(34-3)

tl

t o the spectral equation (34-2), which, with the t r i p l e correlations disregarded, reads

being the ins tan t that makes the start of the f i n a l phase.

One may a l so use equation (20-1), equivalent i n correlat ion terms

- = 2v AR a t of which the general solution (20-9) is a l i t t l e harder t o w r i t e .

But these solutions must have an asymptotic character. It is of no in t e re s t here, except when t - tl is suf f ic ien t ly grea t . F(k , t ) approaches zero when the turbulence decays progressively. So, when t - tl is great, the

only regions of the spectrum continuing t o e x i s t are those for which is small, k2(t - tl) remaining pract ical ly f i n i t e . It has been seen

( i n sect ion 18) that, fo r small values of F(k) w a s necessarily of the form C k , C being a constant independent of the t i m e , and t ied t o the Loitsiansky invariant.

t - tl approaches inf in i ty , as is natural , since

k

k, 4

134 NACA TM 1377

Consequently, for the large values of t, the spec t ra l function takes the form

2 4 -2vk (t-tl) F(k,t) = Ck e (34-4)

val id for small values of k, that is, fo r large-size vortices.

Example 1 i n section 20 gave the corresponding form of the correla- t i on functions R(r , t ) and f(r , t) . I n par t icular it is recal led that

- f(r , t) = e 8 v ‘(t-ti)

and that

Ll()2(t) = 2 - 3 Jo

This is the l a w of decay that had t o be explained. The proof rests on the hypotheses of sect ion 20 and sect ion 18, par t icu lar ly on the fact that the Loitsiansky invariant i s f i n i t e and other than zero. ment of theory and experiment confirms them.

The agree-

I n th i s calculation, it w a s supposed that the turbulence w a s i so t ro- pic. On the other hand, only the large s i ze vortices continued t o e x i s t because the energy contained i n the region of the spectrum corresponding t o substantial values of k becomes negligible. As the constant C i s independent of time, the shape of the spectrum f o r small values of independent of the state of decay of turbulence. contain very l i t t l e energy, but, i n the f ina l phase, the energy of the small vortices i s dissipated and only the large vortices remain visible whose energy, as feeble as it is, has become very great proportionally.

k i s The large-size vort ices

Now, while the turbulence is always loca l ly isotropic and exhibi ts a tendency t o isotropy f o r the small vortices (large values of no longer the same f o r the large-size vortices, i n which the geometrical dissymmetries of the wind tunnel a re exhibited.

k), it i s

.

.

.

A resumption of the calculations, but without introducing the hypothesis of isotropy, produces similar conclusions; the portion of the spectrum (that is , of the spectral tensor ) re la t ive t o small values of remains constant during the decay, while the r e s t of the spectrum vanishes progressively. from the geometrical and mechanical configuration of the f l a w , and which i s prac t ica l ly concealed a t the beginning of the decay of turbulence, must appear i n the f inal phase (Batchelor, r e f . 9). d i f f i c u l t t o check it, since the turbulence is then very weak.

k

It follows that the anisotropy of turbulence that results

Unfortunately, it is rather

35. The concept of "dynamic s t a t i s t i c a l equilibrium" (Heisenberg, ref. 23, Batchelor, r e f . 8):

I n the problem of ' the decay of turbulence behind a g r i l l , the study of the shapes of spectral curves and correlation curves l e d Heisenberg t o believe tha t the notion of similari ty could be extended t o vortices containing the major par t of the turbulent energy. with exception of the region of small wave numbers or large vortices (spectrum i n k4), thus would be i n a dynamic s t a t i s t i c a l equilibrium or quasi-equilibrium, i n which the energy dis t r ibut ion i n the spectrum is modified during the decay of turbulence, the maximum sh i f t ing toward the s m a l l wave numbers, b u t t h e general shape of the curve remaining the same.

The overal l spectrum,

According t o the s t a t i c theory of l oca l isotropy, the spectral func- t ion F(k) can be writ ten i n the fo rm

F(k) = F @(-) O ks

(35-1)

where

region of predominant i ne r t i a forces and that where the viscosity forces become comparable t o them. CP is a nondimensional function. Fo and k are dependent on E and v only, and dimensional analysis shows t h a t

- is a length that f ixes the t rans i t ion between the spec t ra l kS

5 1 3 1 F o = V € iiii k S = v -6 E 6 (35-2)

It w a s shown ( in section 29) that, when the viscosity is negligible, 5 - -

.F becomes proportional t o k 3, which, i n t h i s l imiting case, fixes

. the form of the function 9. In the general case, the form of the func- t i on 9 depends upon that of the function $(k) or T(k) which appears i n the second term of the fundamental equation (22-8) or (3 l - l ) , and which represents the t ransfer of energy through turbulence from the large t o the

small size vortices. 0 is i n a l l cases a universal function of -, and

the s imilar i ty law which it defines i s val id i n a domain where i s an approximate lower l i m i t . T h i s domain is the more extended

the higher the Reynolds number $ = - u’ . T h i s number characterizes the

importance of the portion of the spectrum i n absolute equilibrium. This portion is maintained during the decay of turbulence, so that remains

constant, as well as the nondimensional r a t i o .o’ this number intervenes when the region of absolute equilibrium is l e f t . The quasi-equilibrium therefore depends on the parameters E and

*

k kS

% < k < ks,

ko”

It is found that 6’

3 2 2

v , but, i n addition, on a t h i rd parameter, the t o t a l energy E = -+, , Of . which the diss ipat ion E is the derivative w i t h respect t o time. By

2 U (19-1) and (19-2), 0 is proportional t o the Reynolds number -

Rh = \lEv V

of turbulence. For a quasi-equilibrium t o be possible, €ih must there-

fore be constant during the development of turbulence. T h i s is w h a t the experiment proves during the i n i t i a l phase of decay of turbulence, and this result supplies an argument i n favor of the quasi-equilibrium hypoth- e s i s . I f varies, dynamic s imi l a r i t y ceases, but that does not pre-

vent the possible existence of a region of l oca l isotropy, variable during the development.

By virtue of the known formula (14-11)

2 U h 2 = 15v- 0 E

2 The parameters v, E and uo2 can be replaced by v , h and uo . If

R - - is constant i n a defined regime of decay, the spec t ra l func- Uoh -h - y

t i on F(k,t) depends only on a variable parameter, h fo r example, equi-

valent t o the time, and i n addition, proportional t o 6. .

137 .

NACA TM 1377

We w r i t e

where

number k l . The nondimensiond function 0 is completely defined by the value of EA, t h a t is, by the i n i t i a l conditions a t the gr id . The length of diss ipat ion A can serve as reference length 1. The dimensional

analysis proves then that F(k1,t) is the product resul t ing from %*A,

q = - is a nondimensional parameter dependent on the fixed wave k l

k l

- 2 or k, a quantity proportional t o 2 mil t ip l ied by a quantity con-

A E s t an t during the decay. Introducing a velocity vo and a reference

length -, one may put: 1

kg Z

(35-4)

The form of the function Q, is now defined.by means of the fundamental spec t ra l equation (22-8), which w e recall here:

2 + 2vk2F = iJ a t (35-5)

W e examine first what happens i n case of small values of parameter

q, and that the posit ion of the tangent a t

7. We s h a l l show t h a t f i rs t order with respect t o the or ig in t o the spectral curves F(k,t) is independent of t. For the sake of simplification

@(q) is, according t o (35-4), i n f in i t e ly s m a l l of the

.

Assuming that @l allows a development of the form

. . . n+l @l('ll) = + UlSl + a2v1n+2 +

-2 the first m e m b e r of (35-6) becomes, except f o r the fac tor t 2 ,

. 6 It was seen i n section 22, that $(k) w a s i n f i n i t e l y small as k , *

6 that i s , as q . It follows that, either

n = l a1 = 0 (n + 1 ) q + 4 v q = 0 , . . .

where the first term of (35-7) must be the term i n q6; o r else:

I n the hypothesis n = 6, there would resu l t , f o r F(k, t ) a l a w k 6 which i s incompatible w i t h the exact l a w k4 and w i t h experiment.

For n = 1, there i s for F(k1t) a l a w i n k which is not compatible w i t h experiment when k+O but is ver i f ied f o r small enough values of k, except a t the l i m i t . In that case

*

.

NACA TM 1377

and

c

F(k, t ) = %k + q t k ’ + . . .

%, % being constants independent of t. It follows tha t

(35-8)

= % + %tk2 + . . . ak

approaches a l i m i t independent of t when k-0. A l l spec t ra l curves compatible w i t h the quasi-equilibrium have the same tangent a t the origin.

Therefore, i f the pa r t of the spectrum i n k4 (represented by dotted curves i n f igure 12) i s neglected, the spectral curves are approximately homothetic w i t h respect t o the origin, the maximum decreasing and shif t ing toward the small values of k when the t i m e increases, that is, w i t h progressing decay of turbulence.

The s imi la r i ty is obtained again on the correlation curves f and g. In fact, (14-4)

r s i n --VI R(r) = 2 sin %(k,t)dk = $[ r \Tt (q(-rll)drll (35-9)

rk -71 6

R

0

The functions R t , and hence - f , g, depend therefore on the 2’ U

variable and not separately on r and t. I n other words, the fi

shape of the correlation curves i s maintained, provided that 2 or - r h 6

is taken as variable.

140 NACA 'Dl 1377

Incidentally, can be taken as variable also, because 5

Since i n the region where $ = Cte, uo2 i s inversely proportional L t o t, L i s proportional t o fi as A, and - is constant. h

B u t t h e s imilar i ty with L i s more quickly destroyed during the decay of turbulence than the s imi la r i ty w i t h L, the values of Q1 corresponding t o the small values of v1 are of greater

weight than the others.

longer than the l a w uo2L2 = cte. approached with respect t o A. It can be seen tha t the s imi la r i ty discrepancies.of the correlat ion curves appear, f i r s t , f o r the great values of r, and t h a t they afterward reach the central zone. The reason f o r this zone t o start changing is t h a t the m a x i m u m energy of the spectral curve reaches the spec t ra l region i n k4.

The l a w of decrease of energy can no longer be of the form w2t = Cte.

The turbulence reaches the f i n a l period of decay, i n which the energy

h, because i n the expression of

The l a w uo2h2 = cte prevails therefore much . The explanation is that s imi la r i ty i s

L, while it s t i l l seems t o be ver i f ied f o r

w2t 5 - -

decreases as t 2.

I n this whole analysis it i s not necessary t o assume that 3 is The main point is tha t the i n i t i a l conditions are such that ,the high.

region, i n which the spectral function small. If gh is very high, there exists a region of the spec t ra l curve

i n which F varies as k 3. T h i s region does not e x i s t when Fll i s me d i m .

F varies as k4, be r e l a t ive ly

- 2 *

If it i s desired t o go fa r ther and define the form of the function F, the form of the energy t ransfer $(k) (Heisenberg, reference 24) must be L

9B NACA TM 1377 141

chosen. If %(k) has the form given t o it by Heisenberg (section 251, for instance

the reduced variables (35-4) are employed, and it is proved that (35-5) leads t o the reduced equation

where A = v- kO is the inverse of a Reynolds number. vO

Equation (35-12) is eas i ly reduced t o a d i f f e ren t i a l equation which, contrary t o w h a t occurs i n the case of the steadiness hypothesis (25-6), seems unsolvable. Heisenberg (reference 24) has pointed out approximate solutions, val id f o r great or small values of A. t o Heisenberg's report and a l so t o Batchelor, reference 8).

(The reader i s referred

36. Synthesis of the r e su l t s relating t o the structure of the spectrum of turbulence:

Obviously only isotropic turbulence i s involved. The r e su l t s col- l ec ted regarding Heisenberg's spectral function F(k) which, incidentally, cannot be measured direct ly , but i s deduced, by simple transformations, from the d i rec t ly measurable spectral functions (spectral f'unctions of Taylor, insofar as they .define a spectral d i s t r ibu t ion i n space, and not i n t i m e ) or from the correlation functions.

Whatever the s t ructure of turbulence may be, the spectral l a w s tend, when the Reynolds number becomes high enough, toward an absolute l imi t ing

l a w , where F(k) is proportional t o k 3 (absolute equilibrium). The -2

2 corresponding correlation functions a re of the form f(r) = 1 - C t e r 7.

. / .

142 NACA TM 1377 .

I n the problem of the decay of turbulence behind a grid, an i n i t i a l , an intermediate and a f inal period a re distinguished.

4 k, F(k) is always = Ck , C being a constant independent of time (bound t o the invariant of Lditsiansky).

For small values of c

For the average values of k, and i n the i n i t i a l period, a s t a t e of "dynamic equilibrium" exfsts (figure 13) .

The corresponding portions of the spectral curves are homothetic w i t h respect to the or igin and, when the time of decay Increases, t h e i r maximum approaches the origin. Their shape is known up t o the origin, but the s t r a igh t p a r t that corresponds t o the small values of k and which is . fixed during the decay of turbulence, has no physical r e a l i t y and must be

replaced by the par t of the spectrum i n k . 4

If the Reynolds number i s high enough, the portion of the spectrum corresponding t o great values of k i s defined by the laws of absolute

equilibrium {spectrum i n k '). When k- - - fm the spectrum terminates i n

a curve k-7 (experimentally doubtful). The region k 3 disappeaxs if the Reynolds number is not very high.

5 --

If the Reynolds number is small enough, the region i n dynamical

-2k2 v t equilibrium i s numerically known: F(k) i s proportional t o ke 7

provided that k 2 v t i s not too great.

I n this i n i t i a l period the energy of the fluctuations varies as t-'.

I n the f i n a l period the zone of dynamic s imi la r i ty disappears, and (f ig- the m a x i m u m of the spectral curve intrudes on the region

ure 14 ) . Star t ing from an ins tan t tl marking the beginning of th i s phase w e can write

k4

F(k) = Ck 4 e -2vk2(t-t1)

5 and the energy of the fluctuations decreases as (t - ty.

NACA TM 1377 143 . *,

c

Index of Principal Notations, Fundamental

Formulas, and Dimensional Equations

Certain letters have been employed successively, i n various paragraphs, t o represent different quantit ies without having t o be afraid of confusion. Others, on the contrary, always represent the same quantity; a t least from chapter I1 on. i n to which they enter and t o which one has frequently t o refer:

We shall enumerate them and r e c a l l the pr incipal formulas

X l ~ x 2 ~ X 3 represent the coordinates of a point M

designate the differences of the coordinates of two points M, M', of which the distance is r. (In the first chapter, through a f e w l ines , r represents, by exception, a correlat ion coeff ic ient . )

designate K&~&n's double correlat ion functions. ( In chapter I, f i s used t o represent cer ta in s t a t i s t i c a l frequencies and probabi l i ty dens i t ies . )

These functions are derived from the correlat ion ten-

sor Rup(2;,,E,,E3,t) i n space, of which R = U

i s the scalar invariant. ( In the first chapter, R denotes the density of probabi l i ty of posi t ion and velocity of a s t a t i s t i c a l point.) R should not be confused with R - which denotes a Reynolds number par t icu lar izedby a subscript: EA, %, . . .

4.) , b ( d , c ( r ) denote the t r ip l e correlation functions of K&m.h (section 9 ) which are deduced from the t r i p l e - correlat ion tensor %37

are the vector components of the wave number ( spa t ia l frequency) i n the space A of the wave numbers

k is the length of the vector

E

is Heisenberg 's spectral function, associated w i t h the spec t ra l tensor cp (Al,A2,A3)

UP

denotes the to t a l energy of the f luctuat ion of t u r - bulence per unit mass

144 NACA TM 1377

E *

denotes the energy dissipated as heat per unit mass

V denotes the coefficient of kinematic viscosity *

The letter u always desiwtes a velocity. From chapter I1 on ~ 1 , 3 , ~ 3 represent the components of the velocity of the fluctuations at the point M, and, if the turbulence is isotropic, uo is the mean square value of ui.

h is the dissipation length, L the correlation length

h always represents a statistical function with orthogonal increments

The quantities enumerated above have the following dimensional equations:

F(k) = L%'2 - -

E = L T ' 2 2 - -

2 1

2 -2

v=LT' - -

R = L T - -

= L+'3 - -

By way of comparison, the dimensional equations of classical mechanical quantities read:

Velocity = E-'

Acceleration = - LT'2

Force = MLT'2

Work = ML T'

- 2 2 - -

There follows a list of the principal formulas, referred t o con- s t an t ly i n chapters IV and V. appear f o r the first time are shown i n parentheses.

The sections i n which these formulas

- - (Section 8)

= u$ (yuip + g6@[ ( isotropic turbulence) - -

3

u=l E R a a

i p = f + z F af (incompressibility)

= 2 v k ( k ) d k

R ( r ) = 2Jm =(k)dk rk

m

F(k) = ' rk s i n rk R ( r ) d r J o

(Section 14

146 NACA TM 1377

$ is a function of k and of t, i n f i n i t e l y s m a l l as k6

and such tha t I (Section 22)

The following symbols and abbreviations are used:

X* , is the conjugate complex of X. - U i s the mean value of u. Beginning with chapter I1 the mean

values r e fe r t o a point, considered i n the first chapter as conditional averages. ..

The bold-faced l e t t e r s i n the f irst paragraph, such as K, represent operators.

For simplifying the writing, notations such as f(u; x ) are used i n place of what should be writ ten unabbreviatedly f (u1,u2,u3; xl,x2,x3);

r P P P f(u; x) i s the abbreviation for Jjij’( u1 9 u2 2 u3;

x1’x2Jx3) du 1 du 2 du 3’

The formulas a re numbered i n every section; for example, formula (15-4) i s the fourth formula i n section 15.

Translated by J. Vanier National Advisory Committee for Aeronautics

NACA TM 1377 147

APPEXDIX

Some experimental results.

I n the foregoing analyt ical study of the various recent theories on turbulence, reference was made frequently t o experimental r e su l t s , mostly of English origin, but numerical values w e r e given only rarely. The data contained i n t h i s appendix are intended t o remedy this t o some extent, by way of typical examples of several correlation curves and spectrum curves obtained by d i rec t observations i n the wind tunnel.

These curves have been determined by A . Favr6 and h is collaborators a t the Laboratory of Mechanics of the Atmosphere of the Ins t i t u t e of Fluid Mechanics, a t Marseille, f o r the o f f i c i a l French Aeronautical Research Establishment (0. N. E. R . A . ) . A. Favre,l3 except the spec t ra l curve .F(k) la ted from other data.

They are copied from reports by ( f ig . 20) which w a s calcu-

The turbulent velocity w a s measured with considerably modified hot- wire instruments of the Datwyler type which had undergone important modi- f ica t ions at the Marseille laboratory.

The time-correlation curve w a s obtained by means of a recording device with Tolana magnetic tape, remodeled i n the laboratory so that it could be used f o r aerodynamic purposes. The e l e c t r i c current from the hot w i r e is recorded a t a point on the magnetic tape driven a t uniform forward speed V. Then the record i s read a t t w o points separated by a distance D, before which the tape unrolls. The currents obtained correspond t o two turbulence records separated by a time displacement 2 and

V then conjugated, as customary i n space-correlation measurement, f o r the two currents coming simultaneously from two d i s t inc t hot wires. The method a l so makes it possible t o measure the correlations with time and space displacement. So, from the experimental point of view, the problem of t i m e correlations is solved. I n th i s respect experiment i s ahead of theory which up t o now provides no clear prediction about the t i m e - correlat ion curves. It is t o be noted that the correlations a t a point with t i m e displacement are more precise t o measure than the space correla t ions , because only one hot w i r e i s used, and the time displacement can be diminished as desired, w i t h the employed instrumentation. By way of contrast , fo r the space-correlations, it is d i f f i c u l t t o get two per- f ec t ly ident ica l hot wires, and it i s impossible t o bring them closer together beyond a cer ta in l i m i t .

13A. Favre : S t a t i s t i c a l time-correlation measurements; V I I . International Congress of Applied Mechanics, London, 1948.

c New time-space correlation measurements downstream from a turbulence

Report No. 8/522-~, Dec. 31, 1949. grid, 0. N . E. R . A.

148 NACA TI4 1377

A l l the curves reproduced here correspond t o an overal l speed of 12.2 m/sec. turbulence w a s 1.92 x

For the time-correlation curve ( f ig . 18) the in tens i ty of

The gr id producing the turbulence i n figure 16 w a s of 3.25 inch (8.3cm) mesh width; t h i s value w a s chosen i n order t o f a c i l i t a t e the comparison w i t h H. L. Dryden's measurements. g r id of 1 inch w a s involved. w i r e s from the gr id w a s 40 meshes.

In the other graphs a I n a l l the cases the spacing of the hot

Figure 15 represents a space-correlation curve. The distance r of the hot wires, which were placed on a $?le horizontal perpendicular t o the tunnel axis, serves as abscissa, Karmans's dimensionless func- t i on g ( r ) as ordinate.

Figure 16 represents the same function g ( r ) f o r a different grid. The crosses correspond t o a d i rec t measurement; the round points correspond t o the same measurement, but a f t e r recording on the time-correlation instrument, and r ec t i f i ca t ion with zero time displacement.

This produces a cer ta in check on the accuracy of the time-correlation L measurements. The curve w a s extended up t o near 200 mm.

reasonably constant and below the axis of the It remained

r.

Figure 17 shows the representative curve of the longitudinal corre- l a t ion function f ( r ) , computed from g ( r ) by K&&n's formula, and the

curve representative of the function $(f + 2g), proportional t o the func-

t i on R ( r ) , and obtained by calculation from the curve f and the curve g of f igure 16. The values of R and f are very small and negative s t a r t i ng from r = 120 mm and r = 200 mm, respectively. (corn- pare formula (18-12).)

-

Figure 18 represents a time-correlation curve. It i s seen t o be strongly negative f o r set t ings above 5 milliseconds.

Figure 1.9 represents the spectral function F(n) - of turbulence obtained by transformation of the time-correlation curves. fore the Fourier transform i n cosines of the time-correlation curve.

It i s there- 14

l h o r e precisely, according t o Taylor's formulas, - F(n) i s defined

SO that , if T i s the t i m e displacement, [E(n) cos 25(n~ dn is the

time-correlation function, and, i n par t icu lar , so that l E ( n ) c i n = 1.

In th i s case, i n f in i ty corresponds prac t ica l ly t o 1,000 periods per second.

1

-.

OB

. -i

"I

It was l imited t o the low frequencies of around 16 periods per second. The theory, as far as it can be applied t o the spectral energy dis t r ibu- t ion curves i n time a t a point, suggests that the spectral curve passes through the origin. This corresponds t o the f a c t that the areas of the posit ive and negative par ts of the time correlat ion curve a re equal. It is ver i f ied approximately on figure 18. This ver i f ica t ion is not a l to - gether rigorous, but it does seem that f o r values of more than 30 m i l l i - seconds i n time displacement the curve becomes posit ive again, although remaining very close t o the time a x i s .

Figure 20 gives, computed fromthe curves of figure 17, the in tegra l

equal t o the quotient of the spectral function F(k) by the constant 31+,~. Therefore, w i t h sui table units as ordinates, it is the representative curve of F(k); k is the inverse of a length, while i n figure 19, n is the inverse of a t i m e in te rva l . which represents the spectral energy dis t r ibut ion i n time, and Heisenberg ' s spec t ra l function F(k) which represents the spectral energy d is t r ibu t ion i n space, &re different .

A pr ior i , Taylor's spectral function F(n),

150 NACA TM 1377 U

REFERENCES .

1. Agostini, L.: La Fonction Spectrale de la Turbulence Homog6ne (C. R. Ac. Sc., 228, 1949, p. 736).

2. Agostini, L.: S u r Quelques Rroprie'te's de la Fonction de Corrglation Totale (C. R. Ac. Sc ., 228, 1949, p. 810).

Application au Probl&me de la Turbulence (Rapport Technique No. 28, G. R. A., 1946).

4. Bass, J.: Applications de la Mgcanique Algatoire l'hydrodynamique et & la M4canique Quantique (Pub. Sc. Tech. Min. Air, No. 227,

3. Bass, J.: Les M6thodes Modernes du Calcul des Probabilite's et Leur

1949)

5. Bass, J.: Sur les Bases Ma.th&atiques de la Thdorie de la Turbulence d'Heisenberg (C. R. Ac. Sc., 228, 1949, p. 228).

6. Batchelor, G. K.: Kolmogoroff's theory of Locally Isotropic Turbu- lence (Proc. Cambridge Phil. SOC., 34, 1948, p. 533) . 8..

7. Batchelor, G. K.: Energy Decq and Self Preserving Correlation Func- tions in Isotropic Turbulence (Quarterly of Appl. Math., vol. VI, No. 2, 1948, p. 97) .

8. Batchelor, G. K.: Recent Developments in Turbulence Research (Proc. of the Seventh International Congress of Applied Mechanics, London, 1948) .

9. Batchelor, G. K.: The Role of Big Eddies in Homogeneous Turbulence (Proc. Roy. SOC., A. 193, 1948, p. 539).

10. Batchelor, G. K., and Townsend, A. A.: Decay of Vorticity in Iso- tropic Turbulence (Proc. Roy. SOC., A. 190, 1947, Po 534).

ll. Batchelor, G. K., and Townsend, A. A.: Decay of Isotropic Turbulence in the Initial Period (Proc. Roy. SOC., A. 193, 1948, p. 539).

12. Batchelor, G. K., and Townsend, A. A.: Decay of Turbulence in the Final Period (Proc. Roy. SOC., A. 194, 1948, p. 527).

13. Born, M. and Green, H. S.: A General Kinetic Theory of Liquids I. The Molecular Distribution Functions (Proc. Roy. SOC., A. 188, 1946, p. lo). 1947, P. 435)

111. Dynamical Properties (Proc. Roy. SOC., A. 190,

.'

-.

a.1

c

14 .

15

16

17

18.

19

20 . 21.

22 .

23

24 . 25.

26.

Casd, p.: Etude Des Champs de Vecteurs Algatoires, AppUqude & la Cine'matique des Fluides Turbulents ( B u l l . SOC. Math., UMVII, p. 141, 1949) .

Dedebant, C., Moyal, J., and Wehrlg, Ph. 3 Sur les Equations aux Dkriv6es Pmtielles que V6riPient les Fonctions de Distribution d'un Champ Al6atoire (C. R. Ac. Sc., 210, 1940, p. 243).

Dedebant, G., Moyal, J., and Wehrle', Ph.: Sur L'gquivalent Hydro- -+que d'un Corpuscule Algatoire. Application ; L'e'tablissement des Equations awc Valeurs Probables d'un Fluide Turbulent (C. R. Ac. Sc., 210, 1940, p. 332).

Dedebant, G,, and Wehrle', Ph.: Mdcanique Algatoire, 1. Partie: Le Calcul Algatoire (Portugaliae Physica, 1, Fasc. 3, 1944, p. 95) . 2. Partie: Applications Physiques (Portugaliae Physica, 1, Fasc. 4, 1945, P. 179).

Dryden, H. L.: A Review of the Statistical Theory of Turbulence

Frenkiel, F. No: itude Statistique de la Turbulence. Fonctions Spectrales et Coefficients de Comklation (Rapport Technique, No. 34, G. R. A., 1948).

Vol. 15, No. 1, lgrC8, p. 57).

Frenkiel, F. N.: 1948, p. 311).

(Quarterly of APP~. Math., 1, 1943, P. 7).

Frenkiel, F. N.: On the Kinematics of Turbulence (J. Aer. Sc.,

The Decay of Isotropic Turbulence (J. Appl. Mech.,

Frenkiel, F. N.: On Third Order Correlation and Vorticity in Iso- tropic Turbulence (Quarterly of Appl. Math., Vol. VI, No. 1, 1948, P. 86).

Heisenberg, W. : Zur Statistischen Theorie der Turbulenz (Zeitschrift f%r Physik, 124, 1948, p. 628).

Heisenberg, W.: On the Theory of Statistical and Isotropic Turbu- lence (Proc.. Roy. SOC., A. 195, 1948, p. 402).

Kamp6 de Fdriet, J.: Les Fonctions Algatoires Stationnaires et la Thgorie de la Turbulence Hmogkne (hales SOC. Sc. de Bruxelles, LM, s&ie 1, 1939, p. 145).

K a q 6 de Fkriet, J.: Le Tenseur Spectral de la Turbulence Homogsne (Proc. of the Seventh International Congress for Applied Mechanics, London, 1948).

Y

27. Kampk de F&iet, J.: Confdrences L'Instituto Nacional de Tecnica Aeronautica, 1949. L

(Proc. of the Fifth International Congress for Applied Mechanics, Cambridge (U. S. A.), 1938).

Ac. Sc., 226, 1948, p. 2108) .

28. yon K&mkn, Th.: Some Remazks on the Statistical Theory of Turbulence

29. von K&&n, Th.: Sur la Th6orie Statistique de la Turbulence (C. R.

30. von Kibdn, Th., and Howarth, L.: On the Statistical Theory of Isotropic !Turbulence (Proc. Roy. SOC., A. 164, 1938, p. 192).

31. Kolmogoroff, A. N.: The Local Structure of Turbulence in Incompres- sible Fluid for Very Large Reynolds Nunher (C. R. Ac. Sc. U. R. S. S., 30, 1941, p. 301) - Dissipation of Energy i n the Locally Isotropic Turbulence (C. R. Ac. Sc. U. R. S. S., 32, 1941, p. 16) - On Degeneration of Isotropic Turbulence in an Incompressible Viscous L i w d (C. R. Ac. Sc. U. R. S. S., 31, 1941, P. 538).

32. Martinot-Lagarde, A . : Introduction au Spectre de la Turbulence (Note . Technique, No. 55, G. R. A., 1946). .-.

33. Moyal, J. E.: Quantum Mechanics as a Statistical Theory (Proc. Cambridge Phil. SOC., 45, 1947, p. 99) .

34. Robertson, H. P.: The Invariant Theory of Isotropic Turbulence (Proc . Cambridge Phil. SOC., 36, 1940, p. 209) .

35. Taylor, G. I.: Statistical Theory of Turbulence (Proc. Roy. SOC., A. 151, 1935, P. 421).

36. Taylor, G. I.: The Spectrum of Turbulence (Proc. Roy. SOC., A. 164, 1938, P. 476)

37. Tawnsend, A. A.: The Measurement of Double and Triple Correlation Derivatives in Isotropic Turbulence (Proc . Cambridge P h i l . SOC . , 43, 1947, P. 560).

38. Townsend, A. A.: Experimental Evidence for the Theory of Local Iso- tropy (Proc. Canibrddge Phil. SOC., 44, 1948, p. 560) .

39. Townsend, A. A.: Measurements in the Turbulent Wake of a Cylinder (Proc. Roy. Soc., A. 190, 1947, p. 551).

40. Townsend, A. A,: Momentum and Energy Diffusion in the Turbulent Wake of a Cylinder (Proc. Ray. SOC., A. 197, 1949, p. 124).

c

0

. 41. Townsend, A, A.: Local Isotropy in the 'J!urbulent Wake of a Cylinder

(Australian Journ. of Sc. Research, A., vol. 1, 1948, p. 161) . 42. yon Weizsacker, C. F.: Das Spektrum der 'Jhrbulenz bei Grossen

Reynoldsschen Zahlen (Zeitschrift .€'iir Physik, 124, 1948, p, 614).

43. Yvon, J.: La Thkorie Statistique des Fluides et l1~qUStion d*&t (Act. Sc, hd., Ilermann, 1933).

Figure 1.- Recording of 20 c m x 30 cm. bulence: 5-10'3. HeuGs.)

of the turbulent fluctuation velocity in a wind tunnel Main-flow velocity: 20 m/sec. htensity of t u - The recording duration is 0.03 sec. (Photograph

'I

Figure 2

NACA TM 1377 155 c

c,

Figure 3 LI

.

f 9

r 0

Figure 4

136 . NACA TM 1377

Figure 5

0

Figure 6

\ \ I ' j j 3

? ' I I I

I i 0' ko k

Figure 7

.

Y

i

.

1B

e

L.

.

\

I

01 r

Figure 8

Figure 9

% < Cor relation

function I 6 *4 < 4 3

I I i 01 r

Figure 10

NACA 'D4 1377

1

01 r

Figure 11

Region of quasi -equilibrium I I

F

Direction of increasing t

0 k

Figure 12

NACA '1M 1377

IF

01 k

Figure 13

Figure 14

.

i 160

Figure 15. - Transverse correlation function g(r) in space.

& A+-

NACA TM 1377 e

I .o

0.9

0.8

Figure 16.- Transverse correlation function g(r) in space.

161

162 NACA TM 1377

r mm

Figure 17. - Longitudinal correlation function f(r). Total correlation f(r) + Za(r) function

3

5 IO 15 20 25 30 35 40 - - U,C

Milliseconds

'. Figure 18. - Time correlation function.

NACA TM 1377 163

mm

100 2 0 0 300 400 500 600 700 800 900 lO"o0

Figure 19. - Taylor's spectral function E(n).

4

C '

Figure 20.- Spectral function F(k). Value of the integral

iF Srn kr shkr-dr). 3

( 0

ADVISORY COMMI~~~ FOR ~ERON~~TICS/67531/metadc62897/m... · 2 NACA 94 1377 The first chapter deals...

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