Lecture notes on Turbulent Fluid Motion

LECTURE NOTES

OK TURBUXENT FLUID MOTION

by P r o f . J.M. Burgers Ky. 200 1951 C a l i f o r n i a I n s t i t u t e o f Technology Pasadena^ C a l i f o r n i a

ON TURBULENT FLUID MOTION by J, M. Burgers"''

Chapter I I n t r o d u c t i o n

1, I t i s d i f f i c u l t t o g i v e a c l e a r c u t d e f i n i t i o n of " t u r b u l e n c e . " I n general, vre speak o f tvirbulence when t h e p r e c i s e form o f motion o f a f i e l d does not i n t e r e s t us and we want t o knovf c e r t a i n average or sta= t i s t i c a l aspects of t h i s motion o n l y . Such a s i t u a t i o n presents i t s e l f when t h e motion i s o f a r a t h e r " i r r e g u l a r " character and when i t s

gener-a l p gener-a t t e r n repegener-ats i t s e l f gener-an i n d e f i n i t e number o f times.

A v e r y important case i s t h a t where t h e a c t u a l motion can be considered as f l u c t u a t i n g about a c e r t a i n mean s t a t e so t h a t the mean value o f a q u a n t i t y , taken over a c e r t a i n p e r i o d o f t i m e , tends t o be-come independent o f the l e n g t h o f t h e p e r i o d and of t h e i n s t a n t a t which t h e p e r i o d i s made t o begin. Another case presents i t s e l f when

the character o f the motion appears t o be t h e same over a 3.arge p a r t o f the f i e l d . I n t h e case o f s t a t i o n a r y f l u i d motion through a l o n g tube or canal o f constant s e c t i o n , a combination of t h e tvifo aspects i s foundj a t every p o i n t o f t h e tube t h e f i e l d can be considered as s t a t i o n a r y , w h i l e a t any i n s t a n t t h e general s t a t e o f t h e f i e l d v d . l l be the same

over every s e c t i o n o f t h e tube, p r o v i d e d we a r e f a r enough away from, the entrance and from the e x i t o f t h e tube. I n such cases, mean values can be taken e i t h e r -vïith r e s p e c t t o time a t a given p o i n t , or over a

s t r a i g h t l i n e p a r a l l e l t o t h e a x i s of t h e tube a t any i n s t a n t of t i m e . We may a l s o combine t h e two methods of averaging.

I n other cases we o n l y f i n d a s t a t i o n a r y p a t t e r n v a t h respect t o t i m e , b u t no homogeneity i n spacej f o r instance vfhen v/e consider the s t a t e of motion i n a pump v^orking w i t h constant speed o f r o t a t i o n .

Professor, Laboratorium Voor Aero-en Hydrodynamica der Tech-nische Hoogeschool, D e l f t , Holland; V i s i t i n g Professor, Aero- and Hydro<fynaraics, C a l i f o r n i a I n s t i t u t e of Technology, 19^0-^1,

-2-constant d e l i v e r y , and -2-constant pressiire d i f f e r e n c e .

There can a l s o be cases where t h e s t a t e of motion i s the same over t h e # i o l e f i e l d (so t h a t t h e f i e l d i s s t a t i s t i c a l l y homogeneous), but where the motion g r a d u a l l y dies out so t h a t i t i s not s t a t i o n a r y ' w i t h respect t o t i m e . I n t h a t case space averages must be used .

There a r e , of course, very many cases where t h e r e i s n e i t h e r s t a t i o n a r i t y w i t h respect t o t i m e , nor homogeneity i n space. I t may be, however, t h a t t h e same s i t u a t i o n can be reproduced an i n d e f i n i t e number of times by r e p e a t i n g t h e experiraent which brought i t abou.t. I n such a case vfe can use averages d e f i n e d w i t h r e s p e c t t o the s e r i e s o f repe-t i repe-t i o n s o f repe-t h e experdjnenrepe-t. Such averages are denorepe-ted as "ensemble" averages, since t h e y r e f e r t o an ensemble of cases.

2. The number of p o s s i b l e types of t u r b u l e n t f l u i d motion i s , of course, i n f i n i t e . A t t e n t i o n has been d i r e c t e d mainly t o a few groups of types of such nature t h a t t h e members o f a s i n g l e group appear t o be r e l a t e d amongst each o t h e r .

The t e c h n i c a l l y most important of these groups derives from the s t a t e o f turbiolence which i s found i n a l o n g c y l i n d r i c a l tube m t h constant cross s e c t i o n . The s t a t i s t i c a l p r o p e r t i e s o f t h e f i e l d are independent o f t h e i n s t a n t of time and o f the coordinate x measured along the a x i s of the tube. I n the case of a c i r c u l a r cross s e c t i o n they are moreover symmetrical about t h i s a x i s , but v/ith other forms o f s e c t i o n t h e l a t t e r p r o p e r t y f a l l s away.

I t v a i l not be necessary t o enumerate a l l the types of t u r b u -lence r e l a t e d t o the one mentioned, since these are s t a f f i c i e n t l y Imovm. I n general, one can combine them under the name "boundary l a y e r t u r b u -lence,"

Since t h i s form of turbulence i s s t a t i o n a r y i n t i m e , and since i t i s a l s o knovm t h a t t h e r e i s d i s s i p a t i o n o f energy i n consequence of viscous f r i c t i o n , the motion must be maintained through the i n t r o -d u c t i o n o f energy from the o u t s i -d e . I n the case of t h e motion through a tube, t h i s energy i s d e r i v e d from the pressure drop i n the d i r e c t i o n of the a x i s of the tube. Connected v d t h t h i s circumstance i s the f a c t t h a t the s t a t i s t i c a l p r o p e r t i e s are not homogeneous throughout space: although independent of x, they are dependent on the distance of a p o i n t

from the w a l l , and a l l evidence p o i n t s t o the extreme importance of t h i s dependence i n the energy r e l a t i o n s .

Tijrbi£Lence, homogeneous over a l a r g e space, has never been obt a i n e d i n an exacobt v/ay. I obt i s b e l i e v e d obt h a obt i obt can be obobtained by s u i obt -a b l y s t i r r i n g -a f l u i d i n -a l -a r g e sp-ace -and then l e -a v i n g i t t o i t s e l f under such c o n d i t i o n s t h a t t h e i n f l u e n c e s o f t h e boundary of the f i e l d can be neglected i n t h e p a r t where we d e s i r e t o i n v e s t i g a t e the t u r b u -l e n t motion. T h e o r e t i c a -l -l y t h i s case has been e x t e n s i v e -l y s t u d i e d , be-cause i t has simpler p r o p e r t i e s than non-homogeneous f i e l d s , so t h a t i t i s more amenable t o mathematical a n a l y s i s . The mathematical a n a l y s i s can be extended i n such a vray t h a t c e r t a i n l a r g e scale inhomogeneities can be taken i n t o c o n s i d e r a t i o n . I t i s b e l i e v e d t h a t t h e theory gives r e l a t i o n s which can be a p p l i e d ^uith s u f f i c i e n t approximation t o a ntimber of experimental cases.

These forms of homogeneous turbulence are c o n s t a n t l y d i s s i p a t -i n g energy. The cases s t u d -i e d r e f e r t o f -i e l d s not acted on by e x t e r -i o r f o r c e s ( a f t e r the i n i t i a l s t i r r i n g ) ; hence the turbulence w i l l die o u t . The way i n v/hich i t dies out forms one of the main themes o f the theo-r e t i c a l t theo-r e a t m e n t ,

3. One can ask whether i t virould be p o s s i b l e t o o b t a i n f i e l d s which simultaneously are homogeneous i n space and s t a t i o n a r y i n t i m e . Such f i e l d s n e c e s s a r i l y would need the a c t i o n of e x t e r i o r f o r c e s i n order t o e n t e r t a i n t h e motion. I t v d l l be e v i d e n t , however, t h a t t h e s p a t i a l p a t t e r n of these f o r c e s w i l l i n f l u e n c e the character of the txirbulent motion. Hence homogeneity can be found only when the scale of l e n g t h used i n observation i s so l a r g e t h a t d e t a i l s of t h e f o r c e d i s t r i b u t i o n are elimanated. Since i n general one v d l l have t o apply f o r c e s vfhich are f l u c t u a t i n g or a c t i n t e r m i t t e n t l y , a s i m i l a r p r e c a u t i o n w i l l be necessary i n respect t o time i n order t h a t the f i e l d may be

cons i d e r e d a cons cons t a t i o n a r y . I t conseemcons d i f f i c u l t t o imagine a proper e x p e r i -mental set-up f o r such a case, but one might perhaps t h i n k o f a l a r g e school o f f i s h i n t h e sea, d i s t r i b u t e d v d t h constant average d e n s i t y and p l a y i n g around i n an i r r e g u l a r vray v / i t h s i m i l a r character of motion a t every p o i n t . The most d i r e c t l y i n f l u e n c e d aspect of the turbulence i n such a case v d l l be a type of eddies, compar-able i n e x t e n t v d t h t h e

s i z e and the average distances betv/een the f i s h , „ . and v / i t h periods de-pending on the motion of the f i s h • and on boundary l a y e r phenomena along t h e i r bodies. For an observer of dimensions s m a l l v/ith respect to the f i s h ' ; the f i e l d v r i l l t h e r e f o r e not be homogeneous. Eut when considered over distances l a r g e compared t o the mean distance of t h e f i s h e s , the f i e l d v r i . l l present homogeneous p a t t e r n s . I f we keep i n mind t h a t random d i s t r i b u t i o n of t h e f i s h w i l l present a c e r t a i n amount o f l a r g e scale i r r e g u l a r i t i e s , i t v / i l l be evident t h a t the f i e l d o f t u r b u l e n c e , considered i n the main, v / i l l present f e a t u r e s

(eddies) of dimensions l a r g e compared w i t h the average distance be-tween the f i s h , The study of these l a r g e scale i r r e g u l a r i t i e s may form a s u b j e c t o f great i n t e r e s t . One can, of course, replace " f i s h - ' by any system o f forces i n t h i s example.

U. The type of f i e l d which has been used very e x t e n s i v e l y f o r i n v e s t i g a t i o n s concerning approximately homogeneous, decaying t u r b u -lence, i s the grid-produced vd-nd-tunnel t t i r b u l e n c e . Here we have t o do w i t h a type of turbulence produced by e x t e r i o r f o r c e s , the a c t i o n of which has a s t a t i o n a r y character i n time. The f i e l d obtained i s l i k e -vd.se s t a t i o n a r y i n t i m e , but i t s d i s t r i b u t i o n i n space i s not

homo-geneous. To b r i n g t h i s case i n i t s proper perspective v d t h respect t o the other ones, v/e describe i t as f o l l o w s : v/e consider t h e f i e l d o f motion i n the entrance p a r t of a l a r g e tube; i t i s knovm t h a t by means of proper devices the v e l o c i t y d i s t r i b u t i o n o f the incoming c u r r e n t can be made homogeneous and i t s tvirbiilence can be reduced t o about 0,01^; w h i l e boundary t u r b u l e n c e , which makes i t s appearance a t the w a l l s ,

does not penetrate t o the i n t e r i o r u n t i l one i s f a r dovmstream from the entrance. I n the approximately homogeneous and r e g u l a r entrance c u r r e n t a screen or g r i d i s i n t r o d u c e d , w i t h a mesh s i z e s m a l l i n comparison v/ith t h e cross s e c t i o n of the c u r r e n t . This screen produces turbvilence,

and the eddies formed are c a r r i e d along by the mean motion and gradual-l y die o u t . gradual-lïhen v/e i n t r o d u c e coordinates, x i n the d i r e c t i o n of the main motion, y and z p a r a l l e l t o t h e plane of t h e screen, the average s t a t e o f motion a t any p o i n t v / i l l be s t a t i o n a r y m t h respect t o time; the motion v / i l l be g r e a t l y dependent on y and z near the screen, but i t s average p a t t e r n v d l l become more and more independent of y and z

f u r t h e r downstream; f i n a l l y i t m i l change g r a d u a l l y Yd.th x, but t h e change may be so gradual t h a t over r e s t r i c t e d distances we may consider the motion as being homogeneous w i t h respect t o a l l t h r e e coordinates. I t i s expected t h a t the s t a t i s t i c a l character of the turbulence then w i l l a l s o become i s o t r o p i c .

The type of turbulence obtained i n t h i s case i s v e r y d i f f e r e n t from t h a t obtained i n a c y l i n d r i c a l tube when the boundary l a y e r t u r b u -lence has p e n e t r a t e d so f a r i n t o the i n t e r i o r t h a t a p a t t e r n of motion r e s u l t s which has become independent of x»

I t i s b e l i e v e d , nevertheless, t h a t c e r t a i n c h a r a c t e r i s t i c s o f homogeneous, i s o t r o p i c t u r b u l e n c e , v d l l be present a l s o i n the t u r b u -lence found i n a tube i f a t t e n t i o n i s r e s t r i c t e d t o small scale motions. One of the problems of turbulence theory i s t o f i n d out i n how f a r t h i s i s the case and i n how f a r the d i f f e r e n c e s between the two cases can be understood.

5. I n the g r e a t e r p a r t of the t h e o r e t i c a l i n v e s t i g a t i o n s on t u r b u -lence t h e f l u i d i s considered t o be incompressible. The types of motion p o s s i b l e are,consequently, p o t e n t i a l motion s a t i s f y i n g the o r d i n a r y Laplace equation and incomptcssible v o r t e x flov.-.

Cases where changes of vol-ume or d e n s i t y appear, begin t o a t t r a c t a t t e n t i o n .

One case i s t h a t of a b o i l i n g l i q u i d . I n the f i r s t place, v/e can imagine a f i e l d i n vfhich a great number of bubbles are being formed and are disappearing again w i t h o u t p r e s e n t i n g a mean motion. The appear-ance and disappearappear-ance o f bubbles i s assumed t o be a random e f f e c t ; the c u r r e n t s produced v . d l l be c u r r e n t s produced by sources and s i n k s , and by the l a r g e scale i r r e g u l a r i t i e s i n the d i s t r i b u t i o n of the growing or o f the disappearing bubbles. One might f i n d here a type o f turbulence v/holly o f " p o t e n t i a l " n a t u r e , though v d t h a p o t e n t i a l not s a t i s f y i n g the o r d i n a r y Laplace equation,

A second case v/oxad be t h a t where the bubbles have a c e r t a i n mean motion, and where, moreover, the frequency of appearance or d i s -appearance i s a f u n c t i o n o f the coordinate x, measxured i n t h e d i r e c t i o n

o f t h e mean motion. V/e t h e n come t o t h e s i m p l e s t case of c a v i t a t i o n . I n the general case of f l o w v/ith c a v i t a t i o n the f i e l d i s t h r e e -dimensional. Already i n the one-dimensional case the p h y s i c a l aspect of t h e process f o r c e s us t o i n v e s t i g a t e t h e pressure g r a d i e n t snd t o have r e g a r d t o the e f f e c t of pressure on the appearance and disappearance of bubbles. This becomes much more d i f f i c u l t i n the t h r e e

-dimensional case.

The question a l s o a r i s e s whether t h e r e i s a l o s s o f energj'' i n such cases. I t i s p o s s i b l e t h a t t h e r e v/ould be no d i s s i p a t i o n by v i s c o s i t y so l o n g as the motion of t h e l i q u i d i s p u r e l y p o t e n t i a l and s a t i s f i e s the o r d i n a r y Laplace e q u a t i o n o u t s i d e of the bubbles. The only sources o f d i s s i p a t i o n then v/ould be thermocfynamical through some i r r e v e r s i b i l i t y i n the processes o f evaporation and d i s s i p a t i o n , t i i r o u g h heat conduction, or by means o f sound waves, v/hen the c o m p r e s s i b i l i t y o f t h e l i q u i d i s taken i n t o account.

\Then c o m p r e s s i b i l i t y o f t h e f l u i d becomes o f g r e a t i n f l u e n c e , a l l pressure changes w i l l s e t up a c o u s t i c a l waves so t h a t p a r t o f the energy of t h e f i e l d i s i n the form of edcfy motion, another p a r t i n the form of a c o u s t i c a l v/aves, ViThen t h e f i e l d i s not homogeneous and boundary c o n d i t i o n s have t o be taken i n t o account, t h e r e may be o u t f l o w of b o t h types of energy. Such forms o f t u r b u l e n c e v d . l l be of importance i n h i g h v e l o c i t y boundary f l o w and a l s o i n problems r e f e r r i n g t o s t e l l a r atmospheres or t o i n t e r s t e l l a r gas.

F i n a l l y , t h e r e are forms of t u r b u l e n c e i n which electromagnetic f o r c e s p l a y a p a r t and i n f l u e n c e t h e d i s s i p a t i o n .

Our l i s t i s s t i l l v e r y j.ncomplete. 17e may mention cases v/hich can be found i n the atmosphere, or i n the ocean, v/here t h e r e are s i m u l -taneously present tv/o forms of t t i r b u l e n c e , one l a r g e s c a l e , t h e other s m a l l s c a l e , v d t h a marked gap i n betv/een. The s m a l l scale turbiLLence w i l l then a c t as a k i n d of edc^ v i s c o s i t y f o r the l a r g e scale motions.

The nature of t h i s a c t i o n , and the comparison betv/een t h i s case and cases v/here t h e r e i s a gradual passage from l a r g e scale t o s m a l l scale motions, form problems of g r e a t i n t e r e s t .

_7-6. There i s s t i l l a f i i r t h e r p o i n t t o be considered i n r e l a t i o n t o t u r b u l e n t motion. I n the preceding c o n s i d e r a t i o n s we have r e -peatedly spoken o f turbulence produced by e x t e r i o r f o r c e s . I n c e r t a i n cases we can imagine t h a t these forces a r e g i v e n , i f n o t as exact f t i n c -t i o n s o f -the -time and -the coordina-tes, -then a -t l e a s -t as random f u n c -t i o n s . This was t h e case when we considered the f i s h ; t o a c e r t a i n extent also t h e p r o d u c t i o n o f turbulence i n a vdnd t u n n e l by means o f a g r i d can be brought under t h i s heading. \Te can then assume t h a t the i r r e g u l a r and f l u c t u a t i n g character o f t h e motions p r i m a r i l y i s a consequence o f the randomness of t h e f o r c e system. Since t h e equations o f hydroc^ynamics are n o n - l i n e a r , t h e f l u i d motion cannot be t r e a t e d as a simple super-p o s i t i o n o f d i f f e r e n t tysuper-pes, each c a l l e d f o m m r d by some comsuper-ponent o f the f o r c e system; on t h e c o n t r a r y , t h e v a r i o u s p a t t e r n s o f motion pro-duced by the f o r c e s i n t e r a c t and give r i s e t o nevr p a t t e r n s v/hich may d i f f e r w i d e l y i n character, i n s c a l e , and i n p e r i o d s , from those im-mediately found i n t h e f o r c e s . S t i l l , i n these examples, we d i d not consider a r e a c t i o n from the f i e l d upon the f o r c e system.

I t i s g e n e r a l l y assumed t h a t t h e case i s d i f f e r e n t w i t h t u r b u -lence i n the f l o w through a tube, and vd-th boundary l a y e r t u r b u l e n c e . The e x t e r i o r f o r c e - i n the case of motion tiirough a tube: the press-ure g r a d i e n t - can produce a completely r e g u l a r motion, t h e s o - c a l l e d

laminar f l o w or P o i s e u i l l e f l o w . The f a c t t h a t a c t u a l l y i r r e g u l a r flov/ i s obtained i s supposed t o be a consequence of an i n h e r e n t i n s t a b i l i t y of the laminar flov/, so t h a t s l i g h t d e v i a t i o n s from the mathematically exact p a t t e r n can l e a d t o a complete change of the whole f i e l d . Turbu-lence thus can o r i g i n a t e as i t were "spontaneously,"

I f we l o o k c a r e f u l l y i n t o t h e p i c t u r e , however, v/e may come t o the conclusion t h a t a f t e r a l l the tv/o cases are n o t so very d i f f e r e n t . Let us describe the a c t u a l s t a t e of the f i e l d a t any moment as a super-p o s i t i o n of elementary tysuper-pes of motion, f o r i n s t a n c e , by means of the method of F o u r i e r s e r i e s or of the F o u r i e r i n t e g r a l (althoiigh, i n s t e a d

of simple harmonic components, any other complete system o f normalized s o l u t i o n s o f a l i n e a r d i f f e r e n t i a l equation can be t a k e n ) . I t i s a l -v/ays p o s s i b l e , a t any given i n s t a n t , t o o b t a i n such a r e s o l u t i o n of the

„8-values f o r the amplitudes; we can thus describe t h e h i s t o r y of t h e f i e l d by means o f the time dependence of the v a r i o u s amplitudes. The s t a t i s t i c s of t h e f i e l d then reduce t o the s t a t i s t i c s of t h i s system of a m p l i -tudes .

I n cases v/here t h e behavior o f a f i e l d i s governed by l i n e a r equations, a form of r e s o l u t i o n can be found i n which a l l t h e amplitudes are completely independent o f each o t h e r . Each o f them f o l l o w s i t s ovm course; i t depends o n l y on the way i n which i t may have been e x c i t e d a t an i n i t i a l i n s t a n t , and then e i t h e r grows or decays; or i t may be s t i m u -l a t e d r e p e a t e d -l y but i t i s a-lways independent o f i t s companions. I n the case o f a n o n - l i n e a r system such a r e s o l u t i o n i n t o independent com-ponents i s impossible; every method of r e s o l u t i o n one may apply gives us a s e r i e s o f components whose equations o f motion are i n t e r r e l a t e d and n o n - l i n e a r i n such a way t h a t they cannot be separated. Hence, every component i s coupled v / i t h a l l o t h e r s . I f a t t h e i n i t i a l i n s t a n t only one, or a few components a r e e x c i t e d , other components v r i l l come up soon a f t e r w a r d s and, i n g e n e r a l , v/e can expect t h a t the vihole

spectrum w i l l alv/ays a r i s e . The laws of motion o f those systems which present t u r b u l e n c e now seem t o be such t h a t t h e r e a r e always components v/hd-ch can r i s e t o Ysvy g r e a t amplitudes, even i f t h e y had been very weakly s t i m u l a t e d . This has the consequence t h a t v e r y s m a l l d i s t u r b -ances can produce d i s p r o p o r t i o n a l l y l a r g e e f f e c t s . Since i t i s im-p o s s i b l e t o e l i m i n a t e a l l d i s t u r b i n g e f f e c t s from a r y r e a l case com-p l e t e l y , we can always excom-pect t h a t s m a l l disturbances (e.g., s l i g h t i r r e g u l a r i t i e s o f t h e incoming flov/, s l i g h t disturbances o f the flov/ outside a boimdary l a y e r ) w i l l e x c i t e some p e c u l i a r l y s e n s i t i v e form o f motion and so g i v e r i s e t o t h e appearance of a t u r b u l e n t f i e l d , v r i t h mean amplitudes o f the f l u c t u a t i n g v e l o c i t i e s out o f p r o p o r t i o n t o t h e e f f e c t s which c a l l e d them f o r w a r d .

7, The mathematical problem of turbulence i n a tube (and a l s o o f r e l a t e d forms) i s t o describe the c o u p l i n g between t h e elementary forms of motion i n t o v/hich the f i e l d can be r e s o l v e d , and t o p r e d i c t t h e aver-age d i s t r i b u t i o n o f energy betv/een these forms. I t belongs t o the nature of the equations d e s c r i b i n g the development o f the components, t h a t

^9(;ertain ones o f them can take up energy from the pressure g r a d i e n t a c t -i n g -i n t h e d -i r e c t -i o n o f t h e a x -i s o f the tube. I t w -i l l be ev-ident t h a t those components i / ^ i c h can take up energy most e a s i l y w i l l be the most s e n s i t i v e t o disturbances. I f the t a k i n g up of energy from t h e pres-sure f i e l d i s dependent on terras o f the f i r s t degree i n t h e amplitude, i t w i l l even appear i n a l i n e a r i z e d t h e o r y of the f i e l d , although such a t h e o r y w i l l not b r i n g out t h e coupling between the components. I n c e r t a i n cases i t may be t h a t t h e t a k i n g up of energy w i l l appear only when a t t e n t i o n i s given t o the terms of higher degree than t h e f i r s t . Which o f t h e two cases presents i t s e l f i n t h e t T j r b u l e n c e i n a t u t s ,

forms a problem v/hich has not been f u l l y i n v e s t i g a t e d , but examples of systems can be c o n s t r u c t e d v/here the energy take-up i s given by terms of the f i r s t degree.

I t may be a general character of turbulence t h a t types of

motion p e c u l i a r l y adapted t o t a k i n g up energy are present i n the system^ When t h i s i s t h e case, v/e must expect t h a t t h e average d i s t r i b u t i o n o f

energy between the various components w i l l be determined p r a c t i c a l l y by the r e l a t i o n s between t h e components themselves and w i l l be independent of t h e magnitude and form of t h e s t i m u l a t i n g agency, p r o v i d e d the stimu-l a t i o n xs weakc We w i stimu-l stimu-l then speak of "spontaneous" t u r b u stimu-l e n c e * Yifith s t r o n g s t i m u l a t i o n one must expect t h a t c e r t a i n types of motion d i r e c t -l y r e -l a t e d t o the s t i m u -l a t i n g e f f e c t s may become preponderant. I n such a case one has a type of turbulence which cannot be considered as "spontaneous," but which passes i n t o " s t i m u l a t e d " t u r b u l e n c e . Even i n the case o f turbulence produced by a g r i d or by the school o f f i s h ' ! , v/e have t o do v/ith cases v/hich, s t r i c t l y speaking, come under the f i r s t

category; f o r t h e forces e x c i t e d by the rods o f the g r i d or by the bodies of the f i s h • are due t o boundary l a y e r e f f e c t s , which again produce changes i n the f i e l d of f l o w out o f p r o p o r t i o n t o t h e thickness of t h e boundary l a y e r s . Once s t a r t e d , the disturbances continue t o c a l l f o r w a r d ne^v disturbances, so t h a t the turbulence perpetuates i t s e l f . Nevertheless, f o r an o v e r - a l l i n v e s t i g a t i o n of t h e f i e l d , v/e may some-times neglect t h i s aspect of the problem and assume random system of given f o r c e s t o be the producing agency.

"10-8, One may ask what i s t h e use of t h e preceding c o n s i d e r a t i o n s and how t h e y are r e l a t e d t o p r a c t i c a l problems. The answer i s t h a t t h e i n v e s t i g a t i o n o f argr p r a c t i c a l problem puts us i n t h e p o s i t i o n o f g i v i n g i n f o r m a t i o n about c e r t a i n s t a t i s t i c a l aspects o f the f i e l d o f motion. Cases immediately coming t o the foreground concern the r e l a t i o n between pressure drop and mean v e l o c i t y of f l o w i n a'tUbe, boxmdary l a y e r f r i c t i o n , problems o f d i f f u s i o n and o f heat t r a n s f e r . A l l these depend on c e r t a i n c h a r a c t e r i s t i c s o f t h e f i e l d , and i n order t o be able t o give q u a n t i t a -t i v e i n f o r m a -t i o n an a n a l y s i s of -t h e f i e l d i n -t o som.e sys-tem o f simple components i s an absolute requirement. I t can be surmised t h a t t h e p r a c t i c a l q i e s t i o n s of r e s i s t a n c e , d i f f u s i o n , and heat t r a n s f e r could be t r e a t e d v e r y w e l l i f vre on^y had some Imowledge about t h e components w i t h a r e l a t i v e l y l a r g e scale p a t t e r n . However,, such mathematical a t

-tempts as have been made t o unravel the i n t r i c a c i e s o f t u r b u l e n t motion have g i v e n t h e impression t h a t f u l l i n f o r m a t i o n concerning t h e amplitudes o f t h e coarser components cannot be o b t a i n e d w i t h o u t g i v i n g a t t e n -t i o n -t o -t h e whole spec-trum. Since -t h e r e i s an u n l i m i -t e d supply of ener-gy through the a c t i o n o f t h e pessure f i e l d , t h e mean enerener-gy content o f the tiurbulent motion i s n o t d i r e c t l y determined. Energy i s accumulated and the accumulation goes so f a r t h a t di.ssxpation balances i t . Hence, the mean energy content cannot be found -without g i v i n g r e g a r d t o the d i s s i p a t i o n , and the t o t a l amount o f t h e d i s s i p a t i o n i s v e r y l a r g e l y dependent on the sm.all wavelength end o f t h e spectrum. Thus, the whole s p e c t r m must be a t t a c k e d and we cannot hope t o o b t a i n s a t i s

-f a c t o r y r e s u l t s i -f we should t r y t o evade t h i s .

The problems b e f o r e the i n v e s t i g a t o r d i v i d e themselves i n t o two groups:

(a) To f i n d the way i n which p r a c t i c a l q u a n t i t i e s (determin-i n g , f o r (determin-i n s t a n c e , t r a n s f e r o f moment(determin-im(determin-i o f suspended m a t e r (determin-i a l or o f heat) depend on t h e character o f t h e spectr-um.

(b) To define the spectrum more p r e c i s e l y and t o f i n d the r e l a t i o n s which govern the d i s t r i b u t i o n o f energy over i t .

Serious d i f f i c x i l t i e s present -themselves i n b o t h groups o f problems. Even i f , f o r a given case - say again the f l o w through a

-11-t u b e — w e analyze -11-the f l o w i n -11-t o simple harmonic componen-11-ts and assume t h a t v/e should icnow the mean amplitudes of a l l components, t h e magni-tude of the d i f f u s i o n c o e f f i c i e n t f o r p a r t i c l e s suspended i n the f i e l d cannot be found. This magnitude i s not f u l l y determined by the d i s t r i -b u t i o n of energy over the spectrijm alone. Other r e l a t i o n s , d e f i n i n g the r e l a t i o n betv/een the s t a t e of t h e system a t consecutive i n s t a n t s , are needed. Hence an adequate d e s c r i p t i o n of t h e f i e l d of motion r e -quires more than data about t h e spectrum—^\Te must a l s o have dftta on the p e r s i s t e n c e o f the components.

-12^

Chapter I I

D i f f u s i o n o f P a r t i c l e s i n a Turbulent F i e l d

9. To o b t a i n i n s i g h t i n t o the nature of those c h a r a c t e r i s t i c s o f the t u r b u l e n t f i e l d , which are needed i n the i n v e s t i g a t i o n o f p r a c t i c a l phenomena, we consider t h e d i f f u s i o n o f p a r t i c l e s . We begin by assuming t h a t a l l p a r t i c l e s have t h e same density as t h e f l u i d and t h a t they are s u f f i c i e n t l y s m a l l i n order t o f o l l o w the motion o f the elements o f volume o f the f l u i d w i t h o u t time l a g . We assume t h a t a l a r g e ntimber o f p a r t i c l e s i s f o l l o w e d , s t a r t i n g a l l from the same p o i n t of the f l u i d . I f t h e r e i s a mean motion i n the f i e l d , the p a r t i c l e s w i l l be c a r r i e d along by t h i s mean motion. Let us suppose, a l t h o u g h t h i s w i l l n o t always be the case, t h a t the mean motion i s s t a t i o n a r y , r e c t i l i n e a r and uniform over a c e r t a i n domain so t h a t i t siiaply represents a t r a n s -l a t i o n i n a d e f i n i t e d i r e c t i o n w i t h constant v e -l o c i t y , 1 ^ i n t r o d u c i n g a coordinate system moving w i t h the mean f l o w , Y/e can e l i m i n a t e i t s e f f e c t so t h a t we are only concerned w i t h the t u r b u l e n t motion super-posed on i t .

The t u r b u l e n t motion m i l be i n t l i r e e c o o r d i n a t e s . We r e -s t r i c t t o only one of the-se coordinate-s, -say y.

Observation of the motions o f the i n d i v i d u a l p a r t i c l e s w i l l give data which must be reduced by s t a t i s t i c a l e v a l u a t i o n . We have assumed the f i e l d t o be s t a t i o n a r y and we w i l l , t h e r e f o r e , be i n t e r -e s t -e d i n th-e p o s i t i o n s o f th-e p a r t i c l -e s a f t -e r a c -e r t a i n d u r a t i o n T sinc-e they have s t a r t e d from t h e i r o r i g i n . For any value o f T, the same f o r a l l p a r t i c l e s , we can p i c t u r e the values of t h e coordinates y o f t h e p a r t i c l e s i n a diagram. The d i s t r i b u t i o n obtained may appear t o be simply Gaussian, I f t h i s i s the case, the shape o f the curve can be c h a r a c t e r i g e d by a s i n g l e parameter, f o r which one usvially takes t h e

2 average value of y ,

We w i l l i n v e s t i g a t e how t h i s mean v a l u e , -which i s a f u n c t i o n of T, i s r e l a t e d t o p r o p e r t i e s of the f i e l d ,

10. Vfe consider the v e l o c i t y o f a s i n g l e p a r t i c l e i n the y - d i r e c t i o n as a f u n c t i o n o f two v a r i a b l e s ; the time a t which the p a r t i c l e s s t a r t e d

-13"

and t h e d u r a t i o n T elapsed since t h a t moment. Now: T'

O hence, we havei

and t h e mean value taken over a l a r g e nximber o f p a r t i c l e s becomes:

1'

'T

d T ' c j T " i r ( ^ . / T ' ) ^ ( ^ . ; r ' ) .

O o

We assume t h e t u r b u l e n c e t o be s t a t i o n a r y -vdth respect t o time and homogeneous i n space. I t w i l l be necessary t o assume t h e existence of e x t e r i o r forces i n order t o r e a l i z e these txw c o n d i t i o n s simultane-ously, b u t since a t t h e present we are concerned I'/ith k i n e m a t i c a l r e l a t i o n s o n l y , t t d s w i l l n o t give r i s e t o d i f f i c u l t i e s . The assvimp-t i o n has assvimp-t h e consequence assvimp-t h a assvimp-t assvimp-t h e mean value v ( assvimp-t ^ , T ' J v ( assvimp-t ^ , T'')

w i l l depend on t h e time d i f f e r e n c e T = T' - T " o n l y , and more precise-l y on t h e absoprecise-lute vaprecise-lue (T' -T''| o f t h i s d i f f e r e n c e , since the order i n i t i i c h the two p o s i t i o n s are taken i s i m m a t e r i a l . I f t h e t u r b u l e n c e , although being s t a t i o n a r y i n t i m e , TOvild n o t be homogeneous, the values of T' and T'' w i l l e n t e r . We w i l l keep t o t h e simpler case where T i s the only r e l e v a n t v a r i a b l e ,

-lh-2'

One can now c a l c u l a t e t h e mean value y from t h e f o l l o T f i n g i n t e g r a l :

7 O O which can a l s o be v / r i t t e n : J J 7 J o 0 and can be f u r t h e r transformed i n t o :

T T '

7~ = ^

The i n t e g r a l o c c t i r r i n g i n t h e f i r s t term w i l l approach t o a constant value, when T becomes l a r g e ; we w r i t e :

J

Also t h e second i n t e g r a l w i l l approach t o a constant v a l u e , and we w r i t e

o I n t h i s way we o b t a i n :

-15-Expressions f o r s m a l l values of T can be obtained l i k e w i s e . (They w i l l have a more complicated form.) Tie v / i l l pass over t h e d e t a i l s , however.

The f u n c t i o n , ( f ) i s c a l l e d t h e c o r r e l a t i o n f u n c t i o n f o r the movement o f t h e p a r t i c l e (more p r e c i s e l y , f o r t h e movement i n t h e y - d i r e c t i o n ; other c o r r e l a t i o n f u n c t i o n s may be needed f o r t h e movement i n the x - d i r e c t i o n or i n t h e z - d i r e c t i o n ) . This c o r r e l a t i o n f i m c t i o n , r e f e r r i n g t o t h e h i s t o r y o f a s i n g l e p a r t i c l e , o r , according t o our i n i t i a l s u p p o s i t i o n , t o t h e h i s t o r y o f a s i n g l e element of volume o f t h e f l u i d , i s a Lagrangian c o r r e l a t i o n because i t i s t h e Lagrangian

des-c r i p t i o n o f a f i e l d o f f l u i d motion t h a t a t t e n t i o n i s given t o the h i s t o r y of t h e i n d i v i d u a l elements o f volume.

Most o f our i n f o r m a t i o n about t h e s t a t e o f motion o f a f l u i d , b o t h t h a t r e s x i l t i n g from experimental i n v e s t i g a t i o n and t h a t obtained from t h e o r e t i c a l deduction, i s given i n t h e E u l e r i a n d e s c r i p t i o n , where v e l o c i t i e s are recorded as a f u n c t i o n of coordinates f i x e d i n space, and of time t . This d e s c r i p t i o n does not give a t t e n t i o n t o the h i s t o r y o f a s i n g l e element of volume.

U n f o r t u n a t e l y , t h e Lagrangian c o r r e l a t i o n f u n c t i o n R^ i s not d i r e c t l y r e l a t e d t o t h e data obtained i n t h e E u l e r i a n d e s c r i p t i o n .

E u l e r i a n C o r r e l a t i o n s . E u l e r i a n c o r r e l a t i o n s can r e f e r t o r e -l a t i o n s i n space or t o r e -l a t i o n s i n t i m e , or t o b o t h . L e t us f i r s t take the case o f a homogeneous f i e l d of txirbulence, ¥e consider t h e product v^ Vg, where v^ r e f e r s t o t h e p o i n t y, Vg t o t h e p o i n t y+ "y , both f o r the same i n s t a n t t . Vfe c a l c u l a t e t h e mean value of t h e product by

g i v i n g t o y a l l values i n a c e r t a i n l e n g t h of t h e y r a x i s , keeping a t a f i x e d v a l u e . The mean resvult obtained w i l l be denoted t^y ('^ ) . I n general, t h i s f u n c t i o n v d l l depend on t , t h e i n s t a n t f o r which the c o r r e -l a t i o n was c a -l c u -l a t e d .

I f t h e f i e l d i s s t a t i o n a r y i n t i m e , v/e can c a l c u l a t e mean values of t h e type v ^ ( t ) v ^ ( t + T ) , r e f e r r i n g t o a s i n g l e p o i n t o f the y-axis« This mean value w i l l be denoted by S^(X). I n general, i t w i l l depend on y.

-36-respect t o t i m e , we can d e f i n e a more general type o f c o r r e l a t i o n by mak-i n g r e f e r t o a p a mak-i r o f values y , t , and v^ t o a p a mak-i r of values y + "7 J t + . Tö o b t a i n the meah v a l u e , we proceed as f o l l o w s : i n the y , t - p l a n e a c e r t a i n domain i s chosen v/ith i t s center o f g r a v i t y a t y^, t ^ . The mean value i s c a l c u l a t e d as the average over t h e area o f t h i s domain. I f

the area i s s u f f i c i e n t l y l a r g e , the mean value should become independent b o t h o f the area and o f the p o s i t i o n o f t h e c e n t e r . I t should, moreover, be independent o f t h e form given t o the area p r o v i d e d no e x c e p t i o n a l

choice i s made. Usually i t i s supposed t h a t t h e same mean value should a l s o be o b t a i n e d i n the p a r t i c u l a r cases v/here the domain i s reduced t o a l i n e o f a c e r t a i n l e n g t h , e i t h e r p a r a l l e l t o the t - a x i s (time mean) or p a r a l l e l t o t h e y - a x i s (space mean),

I f , i n s t e a d of time and space averages, "ensemble" averages a r e p r e f e r r e d , we must assume t h a t i n s t e a d o f a s i n g l e f i e l d v ( y , t ) a great many s i m i l a r f i e l d s are given, i n each of v/hich t h e value o f v f o r a

given p a i r o f values y , t j i s determined by a random process. One can then take averages over t h e ensemble f o r every p a i r y , t , and one w i l l thus o b t a i n a s t a t i s t i c a l average f i e l d . This procediu?e can be used a l s o f o r non-homogeneous and n o n - s t a t i o n a r y f i e l d s . I f the f i e l d i s homogeneous and s t a t i o n a r y , the average value o f any q u a n t i t y obtained by ensemble averaging should be the same f o r every p a i r y , t . The en-semble mean value o f t h e product v^ v^ v/ith f i x e d values of y , t , i n p r i n c i p l e can be a f u n c t i o n of a l l f o u r o f these v a r i a b l e s . I n t h e case o f a s t a t i o n a r y f i e l d i t becomes independent of both y and t , and reduces t o a f u n c t i o n of >ƒ and T alone, t o be denoted t y S('>7,T).

12, The i n t r o d u c t i o n of the "ensemble" i s h e l p f u l i n making c l e a r c e r t a i n matters of p r i n c i p l e . I f the random character o f v f o r any p a i r of values y , t , was expressed by g i v i n g a p r o b a b i l i t y f u n c t i o n f o r t h e values o f v f o r t l i a t p a i r , say i n such a way t h a t the p r o b a b i l i t y f o r v t o exceed the value a would be:

(aj y , t )

( w i t h P.^ = 1 £or a = - <»o , and = .0 f o r a = + )', and i f t h i s embodied a l l our knowledge t h e r e would be no basis t o o b t a i n a r e l a t i o n between simultaneous values of v a t d i f f e r e n t p o i n t s y. Our l a c k o f

knowledge would be expressed by the formula; S ,0) = 0,

f o r a l l values of >^ which d i f f e r from zero. I n the same way, we should have

S (0, Z) = 0

f o r a l l values of X d i f f e r e n t from zero, since t h e r e would be no i n f o r -mation about r e l a t i o n s between consecutive values o f v a t a given p o i n t y. I t w i l l be evident t M t more g e n e r a l l y we should w r i t e

s ( r / , -c) = 0

f o r ~>'( or T , or b o t h being d i f f e r e n t from gero.

Hence, virhenever t h e E u l e r i a n c o r r e l a t i o n f u n c t i o n S ( > f , T ) i s not zero f o r some values o f or t. , t h i s means t h a t t h e r e e x i s t r e l a t i o n s between the simultaneous values o f v a t d i f f e r e n t p o i n t s y and between the consecutive values o f v a t a s i n g l e p o i n t . N a t u r a l l y one w i l l expect such r e l a t i o n s t o e x i s t on p h y s i c a l grounds. I t would be absurd t o imagine t h a t i n a moving f l u i d t h e v e l o c i t y c o u l d change from p o i n t t o p o i n t or from i n s t a n t t o i n s t a n t w i t h o u t any r e s t r i c t i o n , however close i n space or i n time the p o i n t s or i n s t a n t s would be chosen. One may ask, t h e r e f o r e , f o r a d e s c r i p t i o n o f the f i e l d , which gives a p i c t u r e o f such r e l a t i o n s , and one may a l s o expect t h a t something about these r e l a t i o n s must f o l l o w from the equations of motion.

I n the d e s c r i p t i o n o f the f i e l d by means of random f u n c t i o n s , as i s the method employed when an "ensemble" of f i e l d s i s considered, these r e l a t i o n s are obscured unless one i n t r o d u c e s s p e c i a l p r o b a b i l i t y f-unetions r e f e r r i n g t o the simultaneous values of v a t d i f f e r e n t p o i n t s and t o consecutive values a t a s i n g l e p o i n t , e t c . The mathematical ex-p r e s s i o n of such ex-p r o b a b i l i t y r e l a t i o n s , hovrever, i s n o t simex-ple and b r i n g s f u r t h e r questions. For most purposes, t h e r e f o r e , we consider the i n -f o r m a t i o n embodied i n the c o r r e l a t i o n -f t m c t i o n S(')-^, T ) as b a s i c , and vre s h a l l make no attempts t o derive i t from p r o b a b i l i t y r e l a t i o n s .

The f o l l o w i n g problems now present themselves:

(a) I s t h e r e a r e l a t i o n between t h e E u l e r i a n c o r r e l a t i o n func-t i o n and func-the Lagrangian c o r r e l a func-t i o n we needed f o r func-t h e problem of d i f f u s i o n ?

(b) Can one i n t e r p r e t the r e l a t i o n s embodied i n t h e f u n c t i o n S(''? ,V) as the r e s u l t o f a c e r t a i n s t r u c t u r e present i n the f i e l d ? This

l a t t e r problem - w i l l be d e f e r r e d t o Chapter I I I .

13, The problem concerning the r e l a t i o n between Lagrangian c o r r e -l a t i o n and E u -l e r i a n c o r r e -l a t i o n has not been so-lved. Not much a t t e n t i o n seems t o have been given t o i t i n p u b l i c a t i o n s . I t has been considered t o some extent i n a paper by F. N. F r e n k i e l , "Comparison Between some T h e o r e t i c a l and Experimental Results on the Decay o f Turbulence," Proc. V l l t h I n t e r n , Congress o f A p p l i e d Mechanics, London, 19li8,

I n order t o pass from the E u l e r i a n d e s c r i p t i o n t o t h e Lagrangian i t i s necessary t o i n t e g r a t e the d i f f e r e n t i a l equation:

I = V ( 7 , t ) .

Let the i n t e g r a l be given by y - (p ( t ; s ) , where s i s the i n t e g r a t i o n constant. Every value of s w i l l c h a r a c t e r i z e an i n t e g r a t i o n curve or p a t h . Having found t h e i n t e g r a l , d i f f e r e n t i a t i o n gives the v e l o c i t y v as a f u n c t i o n o f the time t along a p a r t i c u l a r p a t h . Since i t also may be dependent on the value of the i n t e g r a t i o n constant, we w r i t e :

V = 4^ ( t j s ) .

YiThen the f i e l d i s s t a t i o n a r y w i t h respect t o t i m e , we can now define time mean values along a p a r t i c u l a r path, t h a t i s , f o r a f i x e d value o f s. I n t h i s way vre come back t o Lagrangian mean v a l u e s . I t i s

g e n e r a l l y assumed t h a t t h e m.ean value o f v i t s e l f al ong the p a t h , ob-t a i n e d i n ob-t h i s way, i s equal ob-t o ob-t h e E u l e r i a n mean value; i n p a r ob-t i c i o l a r i f the E u l e r i a n mean value o f v i s zero, i t i s supposed t h a t t h e

Lagrangian mean value of v i s zero too,''^

The Lagrangian c o r r e l a t i o n i s d e f i n e d as t h e time mean value: R^ ( T ) = " ^ . ( t ; s) . ^ . ( t + x; s)

I t i s evident t h a t t h i s f u n c t i o n d i f f e r s w i d e l y from a l l t h r e e E u l e r i a n

l y

-c o r r e l a t i o n f u n -c t i o n s ,

ï/hen the f i e l d i s n o t only s t a t i o n a r y w i t h respect t o t i m e , but a l s o homogeneous i n space, t h e Lagrangian mean value v f i l l be the same f o r every path, so t h a t i t m i l be independent o f s. One might conse-quently d e f i n e a Lagrangian mean value a l s o by means o f an average

taken w i t h respect t o both t and s. For t h i s purpose one c o u l d imagine the f u n c t i o n •4''('^^^) represented i n an a u x i l i a r y plane having s and t as coordinates, so t h a t ever-y v e r t i c a l l i n e corresponds t o a p a r t i c u l a r p a t h . ITe may t h e n define mean values by averages taken over areas i n t h i s plane. I n c e r t a i n cases one may even attempt t o o b t a i n the mean value by averaging over a s e t o f values o f s f o r constant t .

I n t h e l a t t e r concept, however, a d i f f i c u l t y presents i t s e l f , since the parameter s can be chosen i n an i n f i n i t e number o f ways. This means t h a t i n the s, t - p l a n e , we can i n t r o d u c e any t r a n s f o r m a t i o n which s u b s t i t u t e s a new v a r i a b l e s' f o r s, p r o v i d e d t i s l e f t unchanged. Such a t r a n s f o r m a t i o n , however, w i l l i n general a f f e c t the d e f i n i t i o n

of mean v a l u e s . One might attempt t o evade t h i s d i f f i c u l t y t y d e f i n i n g s as the value o f y a t a given i n s t a n t o f t i m e5 b u t i t i s not auto-m a t i c a l l y c e r t a i n t h a t t h i s v / i l l r e a l l y take avray the f u l l d i f f i c u l t y .

We s h a l l r e s t r i c t , t h e r e f o r e , t o time mean values taken along a s i n g l e path, and w i l l assume t h a t these w i l l be independent o f the p a r t i c u l a r p a t h . The i n t r o d u c t i o n , a f t e r w a r d s , o f an average m t h respect t o s v d . l l then n o t b r i n g any u n c e r t a i n t y .

The major problem i s s i t u a t e d i n the i n t e g r a t i o n o f the d i f f e r -e n t i a l -equation. Th-e usual m-ethods o f s -e r i -e s d-ev-elopm-ent do not giv-e s u f f i c i e n t h e l p , since they have a r e s t r i c t e d domain o f convergence, w h i l e the i n t e r e s t i n g aspects o f the c o r r e l a t i o n f u n c t i o n make t h e i r appearance v/hen v/e consider l a r g e time d i f f e r e n c e s . An important ques-t i o n , f o r i n s ques-t a n c e , i s : f o r v/hich ques-time d i f f e r e n c e T v d l l R become zero when we know the behavior o f the f u n c t i o n S ?

Questions o f t h i s nature r e f e r t o p r o p e r t i e s o f t h e i n t e g r a l " i n the l a r g e " according t o modern terminology.

The f o l l o w i n g examples may serve as subjects f o r stucfy: (1) = ^ c o s ( o j t + X y )

w i t h A < «^/X j

ir>\ ^ - A cos '..ot ^ ' dt ~ 1 +. cos >- y T d t h a « 1 ,

-20-The f o l l o w i n g examples can serve as subjects f o r s t u c ^ :

(1) V dt A cos((vOt + A y ) , w i t h A < ^ / \ ; (2) dt A cos U) t _{, w i t h a < C 1 .} V 1 + a cos X y

I n t h e f i r s t example i t w i l l be found t h a t t h e mean value of v along a path i s n o t zero, not w i t h s t a n d i n g t h e f a c t t h a t t h e E u l e r i a n mean values

of V , b o t h w i t h r e s p e c t t o time and w i t h respect t o j, are equal t o zero.

the v e l o c i t y appears t o be zei'o, the same as t h e E u l e r i a n time mean v a l u e . However, the E u l e r i a n mean value w i t h respect t o y ( f o r constant t ) i s not zero.

i n g i n t u r b t i l e n t f i e l d s . W i t h more e l a b o r a t e expressions f o r v , however, i n t e g r a t i o n becomes t o o d i f f i c u l t .

l l j . , Yflien we i n v e s t i g a t e t h e form of a c o r r e l a t i o n curve two charac-t e r i s charac-t i c s are of g r e a charac-t imporcharac-tance: charac-the p o i n charac-t where charac-the c o r r e l a charac-t i o n be-comes zero f o r the f i r s t t i m e , and t h e c u r v a t u r e a t the t o p . The second c h a r a c t e r i s t i c s t i l l belongs t o the domain of "near by" r e l a t i o n s and can be s t u d i e d by means of d i f f e r e n t i a l expressions.

a l l d i f f e r e n t i a t i o n s r e f e r r i n g t o t h e h i s t o r y o f an element of vol-ume, f o l l o w e d d u r i n g i t s motion. For s t a t i o n a r y t u r b u l e n c e , the mean value o f the l e f t hand member w i l l be zero, hence:

I n t h e case o f the second example the Lagrangian mean value o f

Both examples are f a r t o o simple t o represent c o n d i t i o n s e x i s t

-Taking t h e case of the Lagrangian d e s c r i p t i o n , we have

-21

and for tr = 0:

d^R jd\'\ / d v ^ ^ ( I )

dT I d t ^ i , Idt

d i f f e r e n t i a t i o n s and mean values a l l r e f e r r i n g t o the Lagrangian d e s c r i p -t i o n . This r e s u l -t shovfs -t h a -t -the c u r v a -t u r e of -t h e c o r r e l a -t i o n f u n c -t i o n a t i t s t o p ( f o r -T — O) i a uuauected v/ith t h e mean value o f t h e square of t h e d e r i v a t i v e .

I n a s i m i l a r way f o r E u l e r i a n time c o r r e l a t i o n we have:

( I I )

-4-d T \ ? ) t

d e r i v a t i v e s and mean values r e f e r r i n g t o the E u l e r i a n d e s c r i p t i o n . Now we have the w e l l known r e l a t i o n :

d t t>j from which

2 2

\ d t j h > t i

However, t h i s formula cannot be used t o make a comparison between ( l ) and ( I I ) because i n ( I ) we have a Lagrangian mean value and i n ( I I ) an E u l e r i a n one.

-22-Chapter I I I

THE SPBCTBTM ,0F A TURBULENT FIELD

15, We have seen t h a t the E u l e r i a n c o r r e l a t i o n f u n c t i o n gives informa-t i o n abouinforma-t connecinforma-tions i n informa-the f i e l d of f l o w exinforma-tending over space and informa-t i m e . This b r i n g s us t o the problem o f t h e strucb-ure o f t h e f i e l d a t a given i n s t a n t . Various methods are a v a i l a b l e f o r analyzing t h e geometrical p a t t e r n o f t h e motion. We s h a l l consider two o f them.

For s i m p l i c i t y , we consider a t u r b u l e n t f i e l d w i t h o u t mean motion and assume t h e turbulence t o be homogeneous i n space. I t dees not make much d i f f e r e n c e i f we s t a r t v d t h a three-dimensional f i e l d , but i n f u r t h e r work i t i s simpler t o r e s t r i c t t o t h e one-dimensional case and t o postpone

three-dimensional f i e l d s t o a l a t e r occasion.

One method ( a p p l i e d by von Yfeizsyicker and o t h e r s ) s t a r t s from the idea o f t a k i n g average valuesof the v e l o c i t i e s over domains o f decreasing

dimensions. We imagine a scale o f lengths L^, L.^, L 2 , L^, ... i n which LQ i s l a r g e compared w i t h t h e l a r g e s t eddies which observation may

d i s c l o s e i n the f i e l d , v/hile every next term of the s e r i e s i s obtained by t a l d n g 1/2 o f i t s predecessor. ITe nov; associate v d t h evRry p o i n t x, y, z of t h e f i e l d a s e r i e s o f v e l o c i t i e s U^, V^, W^, obtained by averaging the values of t h e a c t u a l v e l o c i t i e s around t h i s p o i n t over a voliMe of

magni-3

tude L^ , having i t s center a t x, y, z. I t can be expected t h a t v&en L„ i s l a r g e enough, the values of U^, V^, Vif^ v d l l be zero f o r every p o i n t X, y, z j whereas more and more d e t a i l s of t h e f i e l d of motion w i l l become apparent when vre go dovm the s c a l e . I t must a l s o be expected t h a t when we have come t o a c e r t a i n very s m a l l l e n g t h Lj^, depending on the nature of the f i e l d but i n a l l normal cases l a r g e compared w i t h molecular d i s -tances, a f u r t h e r s u b d i v i s i o n of L^^ v d l l not r e v e a l new d e t a i l s . This means t h a t t h e r e i s a c e r t a i n minimum scale f o r t h e t u r b u l e n c e , so t h a t the motion shov/s coherence over distances o f t h e order L^^ or smaller, w h i l e i n t h i s process we remain s t i l l f a r from the separate molecular

mo-t i o n s , (This circumsmo-tance was alreacfy broughmo-t forvmrd by Osborne Reynolds i n h i s c l a s s i c a l paper on t u r b u l e n c e ; i t was considered by him t o be a cornerstone o f g r e a t importance i n the a n a l y s i s , )

-23-A l l these c a l c v i l a t i o n s o f averages r e f e r t o one s i n g l e i n s t a n t of t i m e . The a n a l y s i s can be performed f o r a s e r i e s of i n s t a n t s o f t i m e , so t h a t a t every p o i n t o f t h e f i e l d may o b t a i n U^, V^, as f u n c t i o n s o f t h e t i m e . Me now i n t r o d u c e a s e r i e s of ''component f i e l d s " d e f i n e d by t h e formulas from which: \ = \ - \ - l ' U = u + u ,+u „ + + u_ , e t c . n n n - l n-2 0 '

The f i e l d s u^, v^, v/^ (which q u a n t i t i e s o f course are f u n c t i o n s of x, y, z) can be considered as "components" o f the t o t a l f i e l d .

From what has been s a i d concerning t h e behavior o f U , V , W f o r l a r g e n ( i n p a r t i c u l a r f o r n > N), i t f o l l o w s t h a t t h e component f i e l d s u , V , w become zero f o r n "> N,

n^ n

For each o f the component f i e l d s we can c a l c u l a t e t h e mean k i n e t i c energy per u n i t volume (assuming t h a t t h e f i e l d i s s t a t i s t i c a l l y

homo-geneous). This mean I c i n e t i c energy r e f e r s t o a s i n g l e i n s t a n t o f time, v i z , , the i n s t a n t f o r v/hich t h e mean values have been c a l c u l a t e d . By r e -p e a t i n g t h e c a l c u l a t i o n f o r a s e r i e s o f i n s t a n t s , t h e mean k i n e t i c energy o f a f i e l d u , v , w can be obtained as a f u n c t i o n of t h e t i m e ,

n' n n

By means o f t h i s procedure v/e o b t a i n a p i c t u r e o f t h e d i s t r i b u -t i o n o f -t h e energy over -t h e various componen-ts as a f u n c -t i o n o f -the index n, which gives us a k i n d o f "energy spectrum" o f t h e t u r b u l e n t motion.

K i n e t i c energy component of each I

©

4 1 I 9 4 I Q O Q 9 9 » 9 n

-2k~

One would expect t h a t t h e t o t a l energy o f the a c t u a l f i e l d per u n i t volume c o u l d be c a l c u l a t e d by summing t h e energies o f a l l the component f i e l d s . An exact proof of t h i s a s s e r t i o n i s not easy, however, and i t might be t h a t t h e theorem i s not e x a c t l y t r u e .

16, The a n a l y s i s o f a given f i e l d by means o f t h e method of averag-i n g accordaverag-ing t o t h e scheme averag-i n d averag-i c a t e d , s u f f e r s from t h e a r b averag-i t r a r averag-i n e s s i n v o l v e d i n t h e choice o f t h e domains. One can e l i m i n a t e t h e a r b i t r a r i -ness of by i n t r o d u c i n g a s e t of i n c r e a s i n g l e n g t h s L ^, L

d e r i v e d by m u l t i p l y i n g i n steps o f 2. The f i e l d s obtained by averaging over t h e corresponding volumes should be zero, i f LQ had been p r o p e r l y chosenj otherwise a l a r g e r volume must be taken t o s t a r t w i t h . I n s t e a d of d i v i d i n g by 2, one might a l s o d i v i d e by some other number, e i t h e r

l a r g e r or s m a l l e r , perhaps a number close t o u n i t y . I n t h e l a t t e r case t h e number o f component f i e l d s w i l l g r e a t l y increase and each f i e l d w i l l become o f very s m a l l i n t e n s i t y , A p a r t i c u l a r mathematical method must then be i n t r o d u c e d i n order t o r e t a i n d e f i n a b l e components.

One can obviate such d i f f i c u l t i e s by s u b s t i t u t i n g f o r the method of averaging the F o u r i e r a n a l y s i s . R e s t r i c t i n g t o a s i n g l e c o o r d i n a t e , say y, we represent the v e l o c i t y v ( y , t ) - i n the E u l e r i a n d e s c r i p t i o n - by means o f a F o u r i e r i n t e g r a l :

- 2 ^

+ c© .,

(1) V =

r

<p(k) e ' dk

-

CO

where i = " ^ j - 1 . Wow another d i f f i c u l t y presents i t s e l f , since F o u r i e r i n t e g r a l s can be d e f i n e d only f o r f u n c t i o n s which vanish a t i n f i n i t y i n such a v/ay t h a t t h e i r squares admit a f i n i t e i n t e g r a l . To overcome t h i s d i f f i c u l t y one u s u a l l y assumes t h a t we can r e s t r i c t t o the values of v T d t h i n a f i n i t e domain, say v d t h i n t h e p a r t of the y - a x i s from -M t o +M, v/hile o u t s i d e t h i s domain v i s r e p l a c e d by zero. The corresponding F o u r i e r i n t e g r a l v d l l now g i v e the values o f v v d t h i n t h i s domain o n l y , b u t i f M i s l a r g e , t h e i n t e g r a l expression v d l l be s u f f i c i e n t l y r e p r e -s e n t a t i v e . The amplitude f u n c t i o n (p(k) i -s t h e n obtained..from:

(2) 9 ( k ) = -|:p +M

^ -ii<y v ( y ) e dy . 1M

The amplitude f u n c t i o n must s a t i s f y t h e r e l a t i o n (p(-k) = <p-5!-(k) , the a s t e r i s k denoting the complex conjugate, since otherwise v would n o t be a r e a l f u n c t i o n .

The F o u r i e r i n t e g r a l r e f e r s t o t h e course of v a t a s i n g l e i n s t a n t o f t i m e . T/hen the a n a l y s i s i s repeated a t a l a t e r i n s t a n t , t h e amplitude f u n c t i o n (p v d l l , i n general, be d i f f e r e n t . Hence, we must consider <p as a f u n c t i o n both o f t h e frequency k and of t h e time t . P r o v i s i o n a l l y we r e s t r i c t t o t h e s i n g l e i n s t a n t .

17. To o b t a i n the E u l e r i a n c o r r e l a t i o n f u n c t i o n (>?) = S (7^,0) ( w i t h IT equal t o zero, meaning t h a t we consider c o r r e l a t i o n between simultaneous values only)., we form:

V 2

=

i ( k ^ - f k 2 ) y + i k 2 7

^ dk^ ^ dk^ (p(k^) e

-26-I n order t o f i n d t h e mean value v^v^ we i n t e g r a t e w i t h respect t o y from -M^ t o where M. Since the i n t e g r a l s give a value d i f f e r e n t from zero only when y i s s i t u a t e d i n the domain -M... +M, we o b t a i n t h e mean value by d i v i d i n g through 2M. The choice of t h e wider i n t e g r a t i o n l i m i t s has been i n t r o d u c e d t o be able t o make go t o i n f i n i t y a f t e r per-forming t h e i n t e g r a t i o n i n order t o obtain a simp^-ified r e s i o l t . I n t h i s way we f i n d : V 2

=

+ 0 0

+

CO

1

r r s i n ( k + k j

^ dlc^ \ dk^ (p (k^) (pCk^)

e - ^ ^

-

CO

-

CO which f o r -> 00 transforms i n t o :

dk cp (k) (p(«k)

e

- i k

-

ÜO

k having been w r i t t e n f o r k^. Hence

CO On

^1^2

^ ^ ^ 'P(^) ^ * ( k ) COS k ) ^ d ^ . I f vre w r i t e :

(3) r ( k ) = —

<p(k) <p^^(k) ,

M

we can now represent the c o r r e l a t i o n f u n c t i o n by the i n t e g r a l :

CO

(U) S^(7;) = v^Vg = \ p ( k ) cos kT7 di

An i n p o r t a n t p o i n t t o observe i s t h a t the amplitude f u n c t i o n cp depends on M, since M occurs i n the l i m i t s o f the i n t e g r a l (2)» On t h e other hand S^ by i t s nature must be independent of M, and t h e same must

-27-apply t o t h e f u n c t i o n p ( k ) . Hence i t f o l l o w s t h a t the absolute value o f 1/2

the amplitude f u n c t i o n must be p r o p o r t i o n a l t o M ^ . According t o (2) t h e value o f (p i s o b t a i n e d by t h e a d d i t i o n o f a l a r g e number of v e c t o r s

- i k y

v ( y ) e , i n which v ( y ) v a r i e s i n c e s s a n t l y from p o s i t i v e t o negative values and v i c e versa. The f a c t t h a t t h e r e s u l t i n g v e c t o r has a l e n g t h

1/2

p r o p o r t i o n a l t o M ' , i s r e l a t e d t o s i m i l a r r e s u l t s obtained i n t h e problem o f t h e random walk.

A f u r t h e r o b s e r v a t i o n should be made. For convenience, the l i m i t s i n (2) have been w r i t t e n -M, +M. The mean value r e s u l t i n g f o r S^, hovrever, must n o t change i t vre s h i f t t h e i n t e g r a t i o n i n t e r v a l along t h e y - a x i s . Hence the absolute value o f <p (k) must be i n s e n s i t i v e t o such a s h i f t , (The phase angle of (p(k) on the other hand may be s e n s i t i v e t o a s h i f t . This phase angle may a l s o change markedly virhen t h e value of k i s a l t e r e d . )

I f we take ?7 - 0, vre o b t a i n t h e mean square of t h e v e l o c i t y v . Passing t o the mean k i n e t i c energy and o m i t t i n g f o r s i m p l i c i t y the d e n s i t y f a c t o r , we f i n d :

00

(5) E = i

f p

(k) dk .

Hence t h e f x m c t i o n p( k ) gives us t h e d i s t r i b u t i o n of t h e energy over t h e various harmonic components, t h a t i s , what i s commonly c a l l e d the energy

spectrum of t h e t i i r b i i l e n c e ,

I f { 'tj) i s knovm, we can o b t a i n p ( k ) from the i n v e r s i o n o f

(U):

o f

OOS k

(6) r( k ) =

~ j

0

so t h a t the energy spectrum can be c a l c u l a t e d from the c o r r e l a t i o n f i m c t i o n . As mentioned, a l l formulas o f t h i s s e c t i o n r e f e r t o a s i n g l e i n -s t a n t o f t i m e . Hence t h e -spectrum obtained,properly -speaking, i -s an instantaneous spectrum. R e p e t i t i o n o f the c a l c u l a t i o n s f o r consecutive

-28

i n s t a n t s o f time w i l l show whether and how t h e spectrimi changes w i t h time» 18. 1/Vhen t h e tiirbulence i s s t a t i o n a r y one expects t h a t t h e spectrum w i l l be independent of t h e t i m e . This means t h a t we expect JH (k) and

the a b s o l u t e value o f the amplitude f u n c t i o n cp (k) t o have always t h e same v a l u e ; t h e phase o f (p(k) probably w i l l be a r a p i d l y changing f u n c t i o n o f the t i m e .

I n the case of t u r b u l e n c e which i s b o t h homogeneous i n space and s t a t i o n a r y i n t i m e , we can generalize t h e procedure of s e c t i o n 17 and represent the f-unction v ( y , t ) by means o f a double F o u r i e r i n t e g r a l :

+ oo + oo • /I , / > 4-N r f x ( k y + a ; t ) (7) V = I dk \ dv f ( k , Cü ) e i

-

CO

-

OO This r e p r e s e n t a t i o n s h a l l be v a l i d f o r - M<^y<l,+ M ; - T < ^ t <,+ T ; o u t s i d e o f t h i s domain i t w i l l g i v e v = 0.

ITe can t h e n form the complete E u l e r i a n c o r r e l a t i o n f u n c t i o n and o b t a i n :

„ +

CO

+

CO)

.

^1 . / ^ \

^2 p p - i ( k 7 ; + U^T )

S ( ) 7 , -r) = — \ die \ dCO f ( k , UJ) f ( - k,-(^) e

This can a l s o be w r i t t e n : CO CO (8) S(-^,T;) = t dk r dü) ( j ( k , ^ 4 j ) c o s ( k l | ; + T ) + G(k,6c0 Gos(k,^_<iy

J J

where: .2 F ( k , CO ) = f(k,Uj ) £{-k,-Uj) MT G{k,L0) = — f(k,-GU ) f ( -k,<4J ) . MT

-29-I f we take % = O, we come back t o t h e f u n c t i o n S^(>7); hence:

CO

r ( k ) = J dU) |^F(k,Co ) + G(k,aJ f^- , o

I f we take >p = 0, we a r r i v e a t t h e f u n c t i o n ^2^'^ ^' ^ ^ r i t i n g : 00

SgC'F) = r dOj-p *{UJ ) cos U ) T ,

we have:

CO

p = J dk |F(k,6ü ) + G(k, U) )j. . o

The f u n c t i o n p •''''(4>J ) gives the energy spectrtmi w i t h reference t o frequencies i n time, whereas p (k) gives t h e spectrum w i t h reference t o frequencies i n space.

W i t h the a i d o f h o t - m r e anemometers i t i s t h e energy spectrum r e f e r r i n g t o frequencies i n t i m e , which can be measured most e a s i l y and w i t h h i g h r e s o l v i n g power.

I t must be observed t h a t i n d e f i n i n g t h e spectrum sometimes

another p o i n t o f view i s taken. The F o u r i e r a n a l y s i s , ' s a y f o r the •^aluQ o f

V as a time f u n c t i o n a t a given p o i n t o f space, i s made r e p e a t e d l y over a s e t o f consecutive periods o f equal l a r g e d u r a t i o n T, Since we now work w i t h f i n i t e p e r i o d s , we can use a F o u r i e r s e r i e s i n s t e a d of the

i n t e g r a l , so t h a t t h e amplitude f u n c t i o n i s r e p l a c e d by an enumerable s e t of amplitude c o e f f i c i e n t s :

For each p e r i o d T a new s e t o f c o e f f i c i e n t s i s obtained. Consequently, when we consider t h e c o e f f i c i e n t s a^, b of t h e n - t h component, they w i l l show a c o l l e c t i o n o f values v/hich can be described by a p r o b a b i l i t y

"30-t h e n o "30-t i o n o f a p r o b a b i l i "30-t y f u n c "30-t i o n f o r "30-t h e s p e c "30-t r a l ampli"30-tudes.

V/her^, one compares the l a s t mentionèd method w i t h t h a t of t h e F o u r i e r i n t e g r a l , a t t e n t i o n must be given t o the circumstance t h a t we now are r e s t r i c t i n g t o an enumerable s e t o f d i s t i n c t amplitude c o e f f i c i e n t s , each r e f e r r i n g t o a d i s c r e e t frequency; whereas i n the case o f t h e i n t e g r a l we use a continuous frequency s c a l e . The mathematical r e l a t i o n s between

t h e two cases are not simple and r e q u i r e g r e a t care.

periods o f duration T, we can a l s o consider an ensemble o f systems s i m u l -taneously present b e f o r e us and apply t o each t h e a n a l y s i s by means of the Foiurier i n t e g r a l . I t may then be p o s s i b l e t o compare t h e values o f t h e amplitude f u n c t i o n cp ( k ) , f o r a given k, i n the v a r i o u s systems con-s t i t u t i n g the encon-semble, and we can i n v e con-s t i g a t e vfhether a p r o b a b i l i t y f u n c t i o n f o r t h e values o f t h e amplitude f u n c t i o n r e f e r r i n g t o a given k can be found.

t h a t a double F o u r i e r i n t e g r a l l i k e (7) can a l s o be used vrtien t h e t u r b u -lence, although being homogeneous i n space, i s n o t s t a t i o n a r y , b u t i s damped s u f f i c i e n t l y r e p i d l y t o make the i n t e g r a l

convergent ( ^2 s space mean ) . We cannot form S(7^ , t ) i n t h i s case but we can form S (n) as a space mean and o b t a i n :

I n s t e a d o f c a l c u l a t i n g F o u r i e r s e r i e s f o r a consecutive s e t o f

From the mathematical p o i n t of viev/, i t i s of i n t e r e s t t o observe

00

dt

+

CO

+

CO

+

(60