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* - -.1

Pro ceed in g s o f t h e U3SI-IUGG -

I n t e r n a t i o n a l C o l i o q u i u m

on 'Atmospheric Turbulence

a n d Xadio Wave FropaEation'



H. K . M O F F A T T

C a m b r i d g e U n i v e r s i t y ,

C a m b r i d g e , E n g la n d

. v

M O S C O W , J'une 1 9 6 5 .

E d . Yaglorn A . M . & Ta ta r s k y V . I .

Pu bl i s h i ng House 'Kauka ' ,

Floscow 1 9 6 7 .



T h e linear inviscid response of an init ial ly w eak random velocity pertur-ba tio n t o a uniform shearing motion U = (ay, 0, 0) is analysed, first int e r m s of th e individual Fourier components of th e perturbation field, then inte rm s of th e development of i ts spectrum tensor. T his analysis reveals tha tthe dominant contr ibut ion , both to the d is turbance energy and t o the Rey-nolds stress generated, ccme ult imately from eddies having a cylindricalstruc ture, th e axes of th e cylinders being parallel t o the- shear,force. The re-su l ts a re re levant to two aspects of turbulent shear flow: (a ) the equilibriumstruc ture of a small turbulent 'parcel ' of fluid subjected t o persistent almostuniform shear, and (b ) th e st ru ct ur e of the ' lar ge eddies' Ivhich clerhe their#energy directly €rom th e shearing of the mean flow.

Certain aspects of t h e distortion of turbulence b y shear th at is iveaklynonuniform a re also considered, and i t is found tha t the gradient in the

Reynolds stress generated tends t o m ake the mea n velocity profile propagated a s t i c a l l y a s a shear wav e, suggesting that in certa in respects turbu lentfluid 'behaves l ike a visco-elastic fluid in its response t o shear. F jnally , therelevance'of these results t o the phenomenon of th e propagation of sharp turbu -len t-non t ur bulen t interfaces is discussec!.

1. I N T R O D U C T I O N

When homogeneous turbulence is subjected to a uniform irrotationalmean s t ra in ing motion , i t develops an equilibrium structure vi-hich reflectsa balance between the effect of the uniform straining field an d the nonlinearada pta tio n of the turbulence. The experimental evidence for th e struc tural

equilibrium and the dependence of the parameters that describe i t on theparameters describing the strain, have been sumrnzrized by Ton.nsend ( (201,Chap. 4). The init ial s tages of the strain (and the 'cornplete history i i s trainis rapid enough) can be described b y a linear theory (Batchelor an d Proudm an[21; Townsend [191), th e resu l ts of which are in reasonable agreement withexper iment .

The work on irrotational distortion was originally pursued with the aimof gain ing some understanding of the mechanics of shear flow turbulence. I na two-dimensional turbulent flow, such as a , turbulent wake, jet or boundary,layer, any volume element of turbulent fluid, whose scale is smalI comparedwith the lateral scale of the mean flow, is subjected t o the persistent, locally

unifo rm, shear of th e mea n flow. This shear is composed of a rigid body rota-t ion together with an irrotational plane strain with principle a x e a t 45" tothe direction of the mea n flow (Fig. 1). Townsend ((201, $ 6.3) based an impor-ta nt part of his 'large eddy' theory of th e mechanics of shear flon, turbulence onfh e assumption tha t the s t ruc tura l equil ibrium a t ta ined b y the turbulence

i39*Also d i s t r i b u t e d as SUDAZR No. 242 August 1 9 6 5

S t a n f o r d U n i v e r s i t y , S t a n f o r d , C a l i f .

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und er pu re shear is approxim ately the sa m e as that produced by the irrotationafcomponent alone, and he applied the results of th e irrotational str ai n ana-lysis and ex perimen ts t o shear flows, although h e recognized th at this w a sa t best an approximat ion. I t is t rue that r igid body rota t ion would have no

effect i f the principle axes of s t ra in shared this rota t ion; bu t they remain f ixedin space, and therefore ro ta t e re la t ive t o the f lu id , so that the process of stret-ching of vorte x lines is ve ry much less efficient it?hen the rigid bod y ro ta ti onis present than \{,hen is not . In fact , there seems li t t le a pr i or i justif icationfor Townsend’s assumption, and i t seems worthwhile calculating directly

Fig. 1. Decomposi t ion of pure shear i n t o i t s ro ta t iona land i r ro ta t iona l components

the s t ructure imposed by rapid uniform shear , a t any ra te dur ing the ini t ia lstages, when a l inear tre atm ent is val id . This is the prim ary purpose ofthis paper.

The difference between the effects of pure shear an d irrotational stra in ha vebeen emphasized by Pearson [151,who carried out a l inear calculation simi-lar to that of Batchelor and Prou dm an 121, but including the effect of vis-cous forces. On e result which i l lustrates the difference in str ikin g manner isthat th e energy density of th e turbu lenc e ult im ate ly decreases to zero underpureshea r, whereas i t ultim ately increases exp onen tially und er irrotational planestra in (in sp it e of th e acce lerat ing influence of viscosity ). I t seems likely th atthis marked difference in the ’linear response’ of tu rbu len cew ill also carry overt o the nonlinear response when th e am plitud e of the turbulent ,f luct uatio nsbecomes too large to be described adequately by a linear theory, and when(presumably) the nonlinear self-modulation of the velocity field (rather th andirect viscous d issipation) lim its t he orienting effect of th e strain field. Pea r-so n gav e no information about t he orienting effect induced by pure shear, a n dthis aspect of the problem will be emphasized in this paper.

In $$ 2 and 3, th e effect of pu re shear on a random velocity perturbationis analysed , first by examin ing th e effect of shear on a sing le F ourier compo-nent of the field, then by exa mi nin g th e dev elopme nt of t h e spec trum tensor.An imp ortan t result of th e ana lysis is t ha t both t h e energy of the distu rban ceand the Reynolds s t ress tha t develops are ul t imately dominated by contr ibu-tions from a family of eddies of almost cyl indr ical s t ructure , having veryli t t le variation in ’t he direction of the mean flow.

These results seem t o h ave some bearing on th e problem of the generation3f large eddies as described byTownsend ([201, $ 6.1). These large eddies are a nx d e r of magn itude larger than th e energy-containing eddies’ of the turbulence,contain a fraction (about 1/5) of the total turbu lent energy, and ar e believedt o control both the ra te of transfe r of energy from mean flonr to turbulence,an d the rat e of sp read of aturbulenf region in to surrounding laminar f luid.The large eddies derive their energy directly from the m ean flo w, and passi t on to the energy-containing eddies, and the assumption that these tv.0 pro-



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cesses ar e in equi l ibr ium leads t o an im porta nt re la t ion bet iveen the Reynoldsstresses and the mean ra te of s t ra in . I n order to get qua nti ta t ive resul ts , as tr uct ure has t o be assumed for the large eddies , the only constra int beingtha t a random supe rpos ition of these eddie s should lead to correlation func-t ions compatible (a t large separat ions) with the experim ental ly measuredcurves . On the l imited experimental evidence avai l able to him , Townsendchose a cylin drical st ru ct ur e for the large eddies, the axes of the cylinders being

parallel t o the mean flow, and the ana lysis of thi s paper confirms that this isthe typ e of eddy that is ccselected)) by un ifo rm she ar. iVore e xte ns ive m easure-ments , however, b y Gr ant [91 in wakes and boundary layers show that Town-sen d's cylindrical eddies cann ot account for all the experime ntal correlations,

+-? :, I. * a n d m a k e i t clear that in these flows more complicated mechanisms than a' fnere dis tor tion by appr oxim ately uniform shear ar e responsible for theproduc-t ion of lar ge eddies. Nevertheless, this si mp le process (described by Townsend)is an impor tan t one , an d i t is possible that in other shear flows, such asmix ing layers and je t s , i t is the dominant means of production of large ed-d ies .

The relevance of th e an aly sis give n in $9 2 and 3 to the la rge eddy prob-

lem is discussed further in 9 4.The most impor tan t s ing le quan t i ty invo lv ing the tu rbu len t f luctua tions

is the Reynolds stress, since, i f the dis tri bu tio n of th is is known, the meanflow can be determined. The Reynolds stress that develops during the linearst ag e of t he shearing is calculated in 9 3 a n d is found t o be proport ional tothe to ta l s t ra in experienced (for small t imes) suggest ing that the turbulentfluid responds in a n elastic ra ther than in a viscous man ner. Th e consequencesof this conclusion, when the rate of shear is nonuniform, are invest igated in9 5 and a tendency for th e mea n veloci ty prof ile to propagate as a shear wav eis revealed. The resul t has some bearing on the dynam ics of the interfacesepa rat in g rota t ional f rom irrota t ional , f low near the edge of a turb ulent wake

or je t . The exis tence an d pers is teace o fs ha rp wa vy interfaces of th is k ind, cau-sing intermittency in signals received at a fixed point, has been recognizedfor many years , and a f irst a t t em pt t o unders tand t he local dynamics waspublished by Corrs in and Yist ler [41. However , there ar e s t i l l aspects of theproblem tha t have defied explan atio n (see, for e xam ple, Liepm ann [ 13]), andi t is hoped that the approach adopted in 9 6 m a y represent a s tep in the rightdirect on.

Som e of t he results obt ained in 9 3 of this paper have been obtained previo-usly by Deissler [51, a n d further developments hav e been reported bj. Fox I71and Deissler [Sl .

Th e problem has a ls o been t reated b y K hazen [ l O , 1 1 , 121. I t is felt , ho-wever , that the approach adopted in 9 2, in which the history of a singleFourier component is fol lowed, is p h ys ica l ly i l lu min a tin g a n d n a y h elptowards a better. understanding of the com pute d solu tion s of the spectralequations described by Deissler. .

I n atmospheric turbulence, eddy s truc ture is controlled both by n.ind shea rand by thermal s t ra t i f ica t ion . However, even under condit ions of neutrals tra t i f icat ion, a disti nct eddy str uct ure can b e inferred (see, for example ,Lumley and Panofsky [141, Chap. 5), the eddies being pr imari ly of 'corks-screw' type, with axes paral le l t o the wind. Evide nce of sharp tu rbu len tinterfaces in the a tmosphere is less well established, although i t seems atleast a strong possibili ty th at som e of t he ' layer ' rada r echoes described a t thismee t ing a r i se f rom re f rac t iv i ty d iscon tinu it ies a t tu rbu len t -laminar in te r fa -c'es s imi la r t o those a t a tu rbu len t wa ke boundary . The ana lysis of this paperhas therefore a part icular re levance for the topics discussed a t th is meet ing.

.. 5 '


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2 . T H E D I S T O R T I O N O F A P L A N E W A V E B Y U N I F O R I M S H E AR

Consider the action of a uniform shear flow

U = (ay, 0, ) ,


( x ,0)= doeikon.


(2 .2 )

on a weak s inusoidal d is tu rbance U ( x , ) given a t so me in i t i a l i n s t ant t = 0 b y

The linearized equations for the development of the disturbance (assumingincompressibi l i ty) are

ivhere p ( x , t ) i s the associated pressure per turbat ion, and p, a n d v a r e t h edensi ty and kinematic viscosi ty of the f luid . Th e omit te d nonlinear termrt.'i'u is , in fact , ident ical ly zero for a plane wave, but th is is no longertr ue when superposi t ions of waves of diff eren t wa ve vectors ar e considered (as.

i n 9 3). We shall be parti cula rly intereste d in large-scale dist urb ance s for e viscous term in (2 .3) m ay be ignored, an d in $9 2 and 3 we shal l , for s im -pl ic i ty , fol low a n inviscid analysis with v = O:# . Both v iscous and nonl ineareffects become im por tan t afte r a sufficient t i m e has elapsed, a n d in mostcases of in terest i t is the nonlinear forces Lvhich become important first (see

§ 4) .The equations (2.3) (with v = 0) admit the so lu t ion

(2 .4 )

k + (Rax)A + a A , ( l , O ,0)+ a y k l A = k ~ , (295)

k . A = 0 . ( 2 . 6 )

a n d

The coefficients of .t, y and z in equat ion (2.5) must vanish; hence

so t h a t

In these equation s, an d in wh at follows, the suffix 0 re fe rs to condi t ionsa t t = 0, and the suffices 1 , 2 an d 3 refer to comp onents in the x , y an d z

direct ions . The constancy of k , and k , allows us to o mi t the suf fix 0 f o r

these components unless special emphasis of in i t ia l condit ions is required. I fk , = 0, then the wave vector (0, h,, k 3 ) remains cons tan t . I i k , += 0, thenth e effect of the shear is asym ptot ical ly to a l ign the ivave vector in th e (0, 1,O)di rect ion and t o increase its magn i tude l inear ly with time. The effect of t h eshear on wave fronts in these two cases is represented graphically in Fig. 2 aan d 6 . Fig. 2 c an d d show respectiv ely the orienting effect on a sin gle wa ve ve ct orand the orienting effect on those in i t ia l wave vectors on the sphere

T h e n o ta t io n in these figures wil l be referred t o i n what follows.

kl= 0, k, = - a k l , k3= 0 , (2.7).

kl = ko l , kz = k o z- f k o l , k3 = ken. (2.8)

k i l + kiz + k& = k: . (2 9)

* T he o n l y mo dif ica tio n required if viscous forces a re retaine d is the inclusion of a fac-tor

exp[--v 1 2 d t 1 = e x p { - - [ [ k 2 t - k l k o z a t 2 f -k1a 2 t 3 1)0 3

in each component in equation (2.11) below. This causes a viscous da mpi ng of all wave ampli-

tudes and of the disturban ce energy after a sufricient t ime ha s elapsed.



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R - B m - r n2

Fig. 2 . Effect of shear on t h ewave f ront s and wave vec torsof a superposed disturbances

- I

l 2

- Now from equation (2.6), li .A + k - A = 0, so that from equation (2.5),,wr i t ing k2 = k -k ,

from equation ( 2 . 7 ) . The part of equation (2.5) not involving x , y a n d z

is then satisfied provided

- k2 z = i t A + a A& = 2 a A 2 k l

A + a A p ( l , O , O , = - i k n = 2 a A 2 k l k / k 2 . (2.10)

Inte grat ion of thesec ond compon ent of thi s equ atio n, then of the first and thirdcomponents, is ' straightforw ard a nd leads to th e following dependence of th eam pl i tu de components on t ime: .

here, ' 1 2 = k t + k : , t an E) = l / k , (see Fig. 2 4 , and the notat ion. [$ I is used't o i nd i ca t e \c1 ( t ) -9 (0). In th e par ticular case k, = 0, these expressions.reduce to the very s imple form (of course, derivable directly from equation:


A 1 ( t ) = A o 1 - - t t A ? 2 , , '



, . . .

. ,

. . .

I' A2 ( t )= A02,

j . A3 ( t ) = A03.

W e shall be particularly interested i n the behaviour of Ai ( t ) for small and.for large t imes. For utk,< O 2 ,

and . .

k2- : - ayl O 2 , [ e ] - it fkl jk :,.

. .k2 /k2]- (2 ki4 k l k&- i 2k l ) ,


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'so that

A3 = A 0 3

+ 2 a A 0 3 k-2 kl k& k3

+ 0 a ), .

If k, < O 2 , this linear behaviour con tinu es for a long time (until a/,-= k, , /kl ) and I A l J can become very large during this period.

- - .

Fig. . 3 Typica l behaviour of ampli tude kornponents as given by equat lon (2 .1 1 )

For atk, >>n = ( k i 2 + k:)*/*, e have k2- at)? :, 8 -+n, and k2/k2+ ,. -o that

A1 - 01- 02 {hi P 3kl l ki (n- 6,)) + - 2 k , k O 2 ) ,

A2- c2 ki k;' (CC) - 2 ,

A 3- r3 + A o 2 k i Z-3 k3 (n- ,) - -' kW23 ) .


Thus the energy density of the disturbance approaches the constant15 [ I A , (m )

(provided A , , .+ 0, and 8, J= x) this energy density is approximately


+ A , (m ) 2 1 . I f k, is small (so that I = 3) . then in general

1(n- I ) 2 I A I 1 2 kt kL2 k i2 ,

and this is large 'when kl is sm all ; thi s is because wave.vectors.of this kindtake a long t ime to become al igned, and ' I A I I increases linearly throughoutmo st of the alig nm ent time. Typical beha viour of the three am pli tud e compo-



8 nents is sketched in Fig. 3. -

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I f th e ini t ial dis turbance is such that i t can be represented as a superposi-tion of pla ne waves in th e form*

U i ( x ,0)=2 or J d3 ko Aoi (ko)eiko x * ,'

( 3 4, o .

then, dur ing the l inear development of t h e f ield, th e perturba tion a t t ime tis giv en from equation (2.11) b y

u ~ ( x ,) =2 or 1 d3 k Ai ( k , ) e i k x ,(3.2)

Ai ( k , ) = Lij ( k , ) ( k ) ) i (3.3)


where . .


. .

' ,

(3.4) .

. . .. .

. .. .. . . , . . . . . .It o being re la ted to k through equation (2.8). Th e volum e element d3 k in wav evector space is invariant , s ince from equation (2.9) . '


. .The spectrum tensor of t he disturbanc e a t t im e t is

(3.6)il ( k , ) = lim At ( k , ) A; ( k , ) d3 k , . .d 3 k - c o

th e s tar indicat ing a complex conjugate, and th e overbar .an ensemble< . . v e -. . .

.rage. Hence, using equations'- (3.4) and '(3.5), . . . .

' . . ..atj ( k , ) = Lil Lirn QYm (ha) . . , . . . . . (3.7) '

. .here

. ..~O!rn (ko) = im Aol ( kd) A i m ( k o ) ' d 3 k . '

d 3 ko3 0

I , is the spect rum tensor of the ini t ial. disturbance, which may be consideredknown. I f ' k o is replaced by k '+atk, (0, 1, 0), equation (3.7) gives3iii(k, ' t )e,xplici tly in term s of i t s ini t ial va lue . , In order t o go, fur ther i t is expedient t omak e some s imple assumpt ion about th e form of, th e ini t ial .spect rurn tensor ,an d t he argum ent in what follows wi ll be conf ined. to.. two extrem e possibilities. '~ . 'Case,A :. Suppose that the ini t ial dis turbance is two-dimensional with no tlnriationi,n the x-direction; 'i t may be visualised as a superposit ion of 'cyl indrical ed-

. d i e s w i t h .axes paral lel . t o th e m ean shearing flow., (The velocity component u1. need not , however, be' .zero.) Th e Fourier com pone nts of ,.such a f ield 'al l .

h a v e k, = 0, and we have seen in 9 2 th at these a r e the only Fourier components. .

of a.cornpletely random field which can receive.un limited energy through inv is-c id shearing. T he spectrum tensor .of such a distu rban ce . , .is ,of .. .h e. . :orm . . A . ':..

. . . . . . . '(3.8).a , . . I , 'CDjA ( k )=,,Y:mk z , k3) 6 ( k l ) .

* For a stationary rando m function, ' th e Fourier-'Stieit jes ' repr&e ntation 'iC.(.k; o)'=:

=Jeikaax dZo (ko) ould be appro priate. The final ,result .(3.6) s unaffected b y t h i s r s i i nem en t ,

". ,

< '



. . ,

. I: .

. .., .

. ..

. ,* .. .

. . .. .

1 .. .

10 3 a ~ a 3M 2338

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The tensor Lij degenerates in this case to the simple form (from (2.12))

1 - a t 0

il=j 0" 1 . ( 3 9)

The quantities containing the most interesting physical information a re Oiiand cDlz which integrate to giv e t he energy densi ty an d the Reynolds s t resrespectively: -- 1

T ( t ) 5 Ui = \ 0 i i d 3 k , 7 ( t )5- 1 ~2 = S 1 2 d3k. . (3.10)F ,

From equations (3.7) a n d (3.9), i t is evident that, for at 1, cDii is domin a-ted (for almost all values of k ) by those terms on the rig ht for which i = I,I = M = 2 , and i n fact, using equation (3.8),


(3 .12) , _

This energy is concentrated almost entirely in th e component ul. If the tu rbu -

lence is in i t ia l ly isotropic in the y-z plane so t h a t q 2 = = T o , s a y ,then fo r at>

T - (a)' To. (3.13)1

Similarly, from equations (3.7) and (3.9),

0 1 2 ( k , )= - a t ( i t ) , (3.14)

so tha tz t ) = c t To. (3. IS)

Hence th e stress established is proportional to t h e total strain at experienced, elastic rather than a viscous type of response. This idea willbe pursued further in $ 5.

e Case B :Now suppose that th e initial disturban ce is isotropic, so that (Batchelor [ 1 )



.Eo (k , ) is the energy spectrum function satisfying = 1 E, (k , )dko . T he !0

easiest way t o describe the qualitative development of the spectrum funct ionis by reference to Fig. 2 , d . Suppo se fi.rst that .

.E0 ( ko)= To (ko- c ) , (3.17)

where, in this case, T o represents the init ial energy density, so t h a t i n i t i a l l ythe wave vectors of all the Fourier components end on the sphere ko = k ,.Fig. 2 ,d shows how thesewavevectors changewith t ime; a t t ime t , they end onth e sur fac e of th e spheroid


of which a section in a p lan e k, = const is shown in the Figure. T he distanceof this surface from the origin varies con tinu ous ly between a maxim um an da minimum value, which for l a rge at behave l ike and i t is evident that t h e

k m a x- t kc, k m i n ( a )"kc, (3.19)

. I \

2ki + k t

+a t k1)'

+k: = kc,



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at >> 1 ,t<<


Fig. 4. Development of E (k, ) under , shear when E (k, ) = To R--kJ

spectru m becomes severely anisotropic, being dominated asym ptotically b ythose Fourier components for ivhich k, /k , = 0 (at)-l, and th e energy increaseagain being concentrated in theu, component of velocity.

More rigorously, for general E , ( k , ) , we h av e


* the largest contr ibut ion to the in tegral (for at> 1 ) comes from the term forwhich 1 = rn = 2 (du e t o th e presence of th e factor k;' in LIJ , Hence, repla-cing d3k b y d3koa n d p u t t i n g k, , = &t,

T -E f$E0 [ ( 1 2 + ki2)l/a][ an-, - an-l + I 2 '$d k,, dkos


8n 02- W

with 1 = I(&t)2 + k&I1/z. As ut --f 00 , 1+- k O 3 ; this l imiting process can

legitim ately be carried out under t he integral sign in equation (3.21), s ince .the in tegrand is uniform ly convergent for all as at+ CO. (The presence of

the facto rE -2

ensures uniform convergence i n th e range of large th a t would. 1'\ i therwise be troublesome.) Hence, letti ng at+ 00,


This integral (a l inear functional of E , ( k ) ) is certainly convergent for any rea-sonab le E , ( k ) , and independent of t , so the resu l t T - t is true for a ge-neral in i t ia l isotropic spectrum,

Sim ilar ly , the expression for th e R eynolds stress .is

* T =- Llr L,, &$ (&) d3k . . (3.23)

He re we shal l require the behaviour for sma ll as well as for large t imes. Forat < , t h e forms (2.13) lead simply to


(3 .24)

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For at> , the dominant terms in equation (3.23) are again those for which1 = m = 2, and i f we keep, only the first part of L,, which gives the dominantcontr ibut ion t o the integral , we are led t o the form

i: 2::

( 3 . 2 5 )

As at -+ 00, this is dominated by values of qo near 'po = n/2 and 3 4 2 (corres-ponding again to init ial wave vectors which ar e slow t o align). Putting p =

=at (cpo - /2), (and noting that the contributions from the two neighbour-hoods are equal), we have for at> ,



The.increase of T and . ,with at, ar e again due to th e l inear increase i nampl i tude of Fourier components with k , ==: 0. However, T an d T increaseless rapidly in case B th an in. case A , because in cas eB th e Fourier componentswhich experience the l inear increase correspond to an init ia l ' wave-vectorband which decreases in width as (zt)", whereas in case A , the field consistedonly of those comp onents which, , .xperience th e linear increase indefinitely.


The'exis tence of a distinct family'of eddies in turbulent shear f lows, con-taini ng an appreciable fraction of the to ta l turbule nt energy and a scale com-parable with the,scale,o f , th e me an flow,. is .generally accepted (see the discus-sion ' 'in 3 I], bpt there is, as ' ye f , l i t t le theoretical i 'ndication a s t o \(-hat thestatis t ical struc.ture of these eddies m ay b e (and th e experimental evidence isby no means conclusive), and there is no theoretical estimate of the largeedd y ,intensity. Townsend's 'large. eddy theory' is based on an assumption thatthese eddies ar e cri t ically s ta bl e (as far as their energy budget ,is concerned),bu t their actual in tensi ty seems to play no part i n the theory. Can n.e ob ta inan y informat ion on these questions of st ructu re and intensity from the an al y-

" Iin the absence of an y initia l anis trop y, a rnean,shear preferentially amplifiescylindrical eddies :of th e typ e 'proposed by Townsend [201, (see Fig. 5 ) . Howe-yer, th e energy supply to such eddies 'from t h e mean shear does not ceasewhen th e plane of circulation is perpendicular t o Ox as suggested b y Townsend([201, $,6:1); rather , ,,itsenergy increases as .P ,(from 9 3, case A ) indefinitelyfor a cy lindrical . eddy, an d. for a : t ime of .order fo r an e d d y ,of i n i t i a llength: width ratio E .

This energy increa sewill ult im ate ly b e checkedivhen non linear forces becomeimportan t . This happens when th e ' . ma gn i tu de of the-eddy 'vorticity 1o be-

come's comparable with a, ince then th,e rotat iona l'par t of u:.Yq,. i . .e., - .

U A o, ecomes comparable wi th U V U . For t he plane w av e considered i n $ 2 ,

so. that , from equations (2.8) and (2.14), in general, , '

a . . .


sis of th e preced ing sections?. . ' . .

Firs t ly , on the question of str uct ure , i t is clear from $9 2 a n d ' 3 tl-ia i--

. . '

3 ' ' (4: l ). .


. . .. .

. . . .. ... . . . . ... .. .

0= i x ;4 e i k . ~.- .

, ..' i

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where A, is the ul t im ate s teady value for A . [As k , -+ 0, the product k,A,,tends to a f ini te nonzero value.] ,Thus, asymptot ical ly the dis turbance vor tex! i n s t end t o l i e in t he x-z plane. Note t h a t for an initially isotropic field,the vor t ic i ty is not dominated in the l imit by the nearly cylindrical eddieswhich c ontri bute most t o the energy, bu t receives significant contributionsfrom all Fourier components. From equation (4.2), the disturbance vorticity

becomes comparable with a after a time of order (&,/d,)-$ for a plane wa vewith k , + 0, from equation (2.14), this t ime is


When a continuous spectrum of waves is sheared, this means that nonlinearforces become important after a time inversely proportional t o the init ialdis turbance ampli tude.

At this stage, two possible effects can be distinguished, in addi t ion to thedirect interact on of th e ini tia lly independent waves. F ir st , n.hen 10 , I becomes

. -

Fig. 5 . Shearing of a Townsend E d d y

comparable with a, he vortex lines of th e tot al flow und ulat e consid erablyabout th e z-direction. W e have seen th at when the basic shearing motion isuniform, continuous energy transfer is possible only to those eddies having

l i t t l e o r no variation in the x-direction. When the shear is considerablyperturbed, however, th e direction of th e prin cipal axes of st ra in , and thelocal rotat ion axis, will lik ewi seva ry through angles of order z/ 4, and a lar-ger ran ge of.wave vectors n-il l become av ai la bl e for the receipt of energy fromth e perturbed mean flow. I n thi s respect, the large eddies play the rol e ofa n essential intermediary in th e process of energy transfer from th e mean flowt o the turbulence; without them, continuous energy transfer to components ofturbulence with wave vectors in random directions is not possible.

Secondly, as ]a3 increases, points of inflection appear in the profile of thexLcomponent of th e pert urbe d vel ocity field, a n d i t seems likely thzt, nvhenthe vortic ity grad ient becomes large enough, th e perturbed profile will become

- u n s t a b l e to smaller scale disturbances in the manner described by Gill [Sl.I . or this mechanism to operate, the init ial am plitu de must be large enough for.

nonlinea r effects t o become im port ant before viscous forces damp out the distur-bance.

The prediction that the large eddy vorticity must be of the same order asth e mean flow vorticity receives some support from Ton m en d' s experimentales t imates in the case of the turbu lent w ake. T he large ed dy energy density isapproximately 1/8 that of the me an velocity defect; their sca le is som enh at lessthan half the scale of th e mean velocity. Hence, th ev or ti ci ty (= energy)'Vscale)of th e lar ge eddies an d of th e mean flow are of the s am e order of ma gn itu de.This evidence, hode ver, must be viewed w ith cau tion, since, as alrea dy

mentioned in 9 1, the observations of Grant [91, show that other processes inaddi t ion t o distortion by shear, are responsible for large eddy generation; inparticular, Grant suggests that instabilities of the von Karman vortex streetoriginating near the cylinder, an d persistent instab il i ty of the wake associatedwith the continual buil d-up of Reynolds stresses, may gi ve rise t o eddy struc-tures not inconsistent with correlation measurements.





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If homogeneous turbulence is distorted by un iform shear, the n a uniformReynolds stress is established and this has n o effect on the mean flow. I fthe shear is nonuniform, however, i . e., if a = a (y), or i f the turbulence isnonhomogeneous initially, then a nonuniform stress field is established and

this tends to change th e mean flow. For a str ic t ly paral lel flow U = (U (y, t ) ,0, 0), the mean flow equation is


au a tat a y *- = -

Hence U and so a = d U / d y become functions of t , and the to tal s t r a inat that appeared throughout $9 2 and 3, must be replaced by the quant i ty


p = ( a d / .J0

The derivation of the stress ‘G in 9 3 is otherivise unchanged, and gives asbefore

in case A , z = TO P (5.3)

The relat ion ‘1; = c’p with c3 = T o or (2/3) T o (assumed uniform) i n case Aor B respectively, leads to .


We have al ready remarked th at a l inear relat ion between s t ress and s t r ai nimplies an elast ic response, so tha t th e appearance of th e wave equation a s aconsequence is not surpris ing. The derivation requires implici tly t hat . th e scal eof the mean st ra in inhomogeneity should be large compared it-ith th e sca le ofthe strained turb ulence , so th at the assumption of uniform local s train m a y bereasonable. However, i f this is not the case (as, for example, for the largeeddies in wake turbulence), one might-st i l l expect the equation (5.5) to h a v esome qual i tat ive, i f not quanti tat ive, s ignif icance.

Two further qualif icat ions must b e ma de. First , th e stress-strain rela-t ions (5.3) and (5.4) a re valid only du ring the l inear response period t< ,.In a sense, the time f =5: t , marks the ’yield point’ of the turbulent medium .For t 3 (in)he stress presumably approaches an asymptotic level deterrni-ned by the rate of s t r a in . The t ime t , m ay also be interpreted a s the ’ relaxa-t ion t ime’ of the turbulence when the mean rate of strain changes. I f t , i s atim e characteristic of this change, then i f t,,< , the response (for f, < n)will be purely elastic, while i f f,, < , the response will be purely viscous.The analogy between turbulent flow and the flow of a non-Newtonian fluid

was f irs t pointed out by Rivlin 1171, a n d this point of view has been further‘recommended by Liepmann [ 131; the approach adopted here suggests thatturbulent fluid is visco-elastic in its response t o a changing rate of shear.

a point where a’ (y) = 0, then this wil l be a natural source of instabil i t ies



Secondly, i f there is a poin t of inflection in t h e mean velocity profile, i . e., -

I5 0

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which grow exponentia l l y ra ther than l inea r ly wit h t ime. However , in th e l i -mit of uniform shear ( i . e.; a' (y ) = 0 everyw here),- the growth r at e of suchdis turbances is zero, and i t is reasonable t o expect th at for suff ic ient ly sm allc u r v a t u r e in the mean prof i le , the dominan t con t r ibu t ion to the Reynoldss t ress is still given by equa t ion (5.3) or (5.4).

To be more specif ic , consider the fol lowing ini t i a l val ue problem: a t t im e

t = 0, suppose that the mean prof i le is of s im pl e je t- typ e,



U y, 0) = U 0 ech2y / b , (5.6)

an d suppose that the superposed weak turbule nce is of the two-dimensionalty pe considered in case A of 9 3. Th e devel opm ent of t he mean flow is thendescribed by equation (5.5) with c = To , and the so lu t ion is

' represent ing two shear waves t ravel l i ng with velocit ies =k c. If the eddies areof l a te ra l sca le6 (and th is is a ma xim um es t imate ) , the ir vo r t ic i ty a t t ime t isof order (7"%) /3 ==: (c/b)Etwhere Z ==: U,/6 is a m ea nv al ue of th e shear in theje t ; th is vor t ic i ty becomes comp arable Lvith Z a f t e r a t ime t l = b/c , and th isis j u s t t h e t ime th a t t h e in i t i al p ro fi le t ak e s t o sp l i t i n t o its two components .Hence, the l inear development descr ibed by equat ion (5.7) can be va l id on lynea rthe out er edges of the je t where the to ta l s t ra in experienced is s t i l l suff ic ient lysmall ( i . e . , fi ( ( U d c ) .

I f the superposed, tur bul enc e is in i t ia l l y isotropic , then the sam e descr ipt ionholds for fi (( 1, but for U o / c > fi> , t h e a sy m p t o t ic l a w 7- 1/2)T0 log fiis ap p ro pr i a t e. W r i t i n g c2 = (1/2) T o , t h e mean flow equat ion is then most

compactly expressed in terms of fl in the form



p a.- c' - og p.a t 2 , a p

This equa t ion is still hyperbolic , an d presumably exhibi t s solut ions of propaga-t in g type, though the nonlinear i ty suggests tha t th e development of the profi lewill depend s t rong ly on i t s ampl i tude . For fi 3 o/c , the linear theory forthe Reynolds s t ress development breaks down.

6. T H E S T E E P E N I N G A N D P R OP A GA T IO N' I . OF T U R B U L E N T I N T E R F A C E S

, .


I .'



Turbulen t wakes, je t s , m ix ing layers , and boundary layers a re charac te r ized ,i n gener al, by the presence of sha rp un du la tin g interfaces n*hich sep ara teth e rota tion al t ur bu le nt region from a n i rrot atio nal region in \Lrhich turbu len tf luctuat ions decay ra pid ly with dis tanc e fro m the in te rface. The f luctuat ion ener-g y decreases asym pto tica lly a s the inv erse fourt h po\ver of this distance, accor-d ing t o Phi l l ips [ IS] . ) In the i r ro ta t iona l reg ion , the assumpt ion tha t the

- field of fluctuations is homogeneous in dire ctio ns paralle l t o the interface, leads

t o the resul t that the Reynolds s t ress component T =- is zero (Corrsinand Kistler 141). Weak inhomogeneity i n wakes and je t s in the x-direction

does produce a nonzero T, whose only effect, however, is t o mo d ify th e mean, pressure dis tr ibut io n (Stewart [ 181). Th e mean f low in th e i r r ota t ion al region is

the uniform veloci ty a t inf ini ty , together with a component associated withth e f in ite am pli t ude f luctuat ions of the in terface , jvhich, as demonstra ted byStew ar t , can con t r ibu te t o the mean ve loc i ty defec t ( fo r a wake or boundary.layer), or increment (for a jet) . H owever, an y genuine rotatio nal shear that


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is preseiit is necessarily confined to th e rotatio nal region where th e Reynoldsstress is nonzero (and rotational). Fluid is continually entrained across theinterface from the irrotational to the rotational side; equivalently one m aysay t ha t the interface propagates towards t he irrotational region.

Two outstanding questions regarding these interfaces are (a) why do theyremain sharp, and ( 8 ) Fvhat determines their velocity of propagation? Corrsin

an d I(ist1er ha ve proposed the follonring answers t o these questions:( a ) The ra te of pro duction of mean -squ are vo rti ci ty 2 (= U ' say) in hornoge-

neous turbulenc e is proportional t o the mean-square vortici t y already present.I n any local gradient of U' (in a nonhomogeneous turbulence) the higher r at e ofproduction in the region of higher o' -d en si ty wvill tend to steepen th e grad ient.I n pa rtic ula r, near the edge of a free tur bu len t flow wvhere U' decreases fromits va lue well inside the tur bu len t region to zero outside, the gradient of CO'

will increase un ti l direct viscous dissipation intervenes, n-hen the thickness ofthe transit ion layer is of order (Y/u')'~?, (th e local inn er I(o1rnogorov scale). Th isargument predicts steepening of the gradient of o' whether a me an shear ispresent or not. I t i s open to the cri t ic ism that the ra t e of destruction of o'

in homogeneous turbulence by viscous dissipation is also proportional t o &?

and th at th e rates of produ ction and d estruction a r e approxim ately in balance,,.-a t a n y r at e so long as th e turb ulen ce is no t too far from statis tical equilibrium.'

(b ) The propagation of the interface is essentially the transmission of v or - .ticity by viscous stresses through the interface. On dimensional grounds, thepropagation velocity V* must be of order the inner I(olmogorov velocity scale:



th e presence of mea n shear, and therefore of an additional (kinematic) Reynoldsstress, TL (just inside the layer) would tend to augment this velocity: giving

(6.2)' = 0 (TZ+ (vo)2)11,.

An objection t o the form (6.1) raised by Coles [31 is that the mean propzgationv e l o c i t y p is known to be independent of viscosity (at any rate in regions oflow where Reynolds num ber sim ilari ty has an y meaning). The l imiting fo rmof equ atioh (6.2) at high Reynold s num ber, V' = 0 (TJIz, seems reasonable,but i t suffers from the weakne ss (common to a l l results of dimensional analy -

An approach which av oids t he abo ve criticisms (though doubtless i t invitesothers) can be based on the shear wave eq uation derived on th e basis of equ ation s

(5.1) and (5.4a). Let us again formulate an init ial value problem, inter,ded torepresent the situation near the instantaneous boundary of a turbulenl n-akeI(Fig. 6): a t t = 0, let tc (9) = dU/dy be a monotonic function decrea-\sing from a positive valuecc, a t y = - 0 t o zero a t y = + CO, and let T o (y),the (unstrained ) kinetic energy of superposed wea k fluctuations, b e a sim ilarmonotonic decreasing function. The equation for U y, t ) is then

' sis), that the precise propagation mechanism remains obscure.

wherec2ccTo,he constant of proportionali ty being of order u ni ty (an d dependingon th e precise degree of aniso trop y of the turb ulen ce). For a uniq ue solut ion, w e

should also specify d U l d t a t t = 0; this is equivalent t o prescribing the degreet o which the turbulence has been strained prior t o the ini t ia l ins tant . Let uss imply s t ip u la te that dUl d t has. th e form that give s rise t o a singl e wave pro-pagating in the posit ive y-direction. I f To ere uniform, this wave wouldprop agate yitho ut change of shape (the necessary qualif ications about the ' l inear ,

pha se'. of Rey nolds stress developm ent discussed i n 5 being understood):

1 2,


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proceeds. If so me othe r effect doesnot inter ven e first, this develop- y

m en tw ill be checked only when

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I am grateful to the members of the De part me nt of Aeronautics and Astro-nautics, Stanford University, for their generous hospitali ty during the period '

of this research. I a m a ls o i n de b te d t o b r . J o h n E id er f & some encouragingcomments that helped towards the formulation of 5 6 of this paper.



4 .

















G . K. B a t c h e I o r. Homogeneous Turbulence. Cambridge University Press, 1953.G . K. B a t c h e 1 o r and I . P r o U d m a n. The Effect of R ap id Distortio n of a Fluidi n Turbulent Motion. QuarC. J . h fech . and A p p l . M a th . , 1954, 7, 83-103.D. C o 1 e s. Interface and Intermit tency in Turbulent Shear Flow. i j i icanique de la tur-bulence (Col l . In tern . du CNR S a M arsei l le) , Paris , Ed CiYRS, 1962, 229-248.S. C o r r s i n and A . L . K i s t 1 e r . The Free-st ream Boundaries of TurbulentFlows. Na t . Ad v . Co m. Aero n a u f . R e p . , No. 1244, 1955.R . G. D e i s s 1 e r. Eff ect s of Inhom ogeneity and of Shear Flow in Weak Turbulent F ie lds ,Ph ys . F lu id s , 1961, 4 , 1187-1198.R . G. D e i s s 1 e r. Problem of Steady-state Shear-f low Turbulence. Ph ys . F lu id s ,

J . F o x. Veloci ty Correlat ions in Weak Turbulent Shear Flow. Phys. Flu ids , 1964, 7,

A . E . G i 1 1. A Mechanism for Instabili ty of Plane Couet te Flow and of Poiseui l le Flowin a Pipe . J . Fluid Mech ., 1965, 21 . 503-511.

1965, 8 , 391-398.


H . L. G r a n t . The Large Edd ies of Turbulent Not ion . J . Fluid Mechan., 1958, 4 , 149-190.3,M. x 3 e H.

1962, 147, 60-63.3. i. x 3 e H .

1963, 153, 1284-1287.3.M. x 3 e H.



HWIfHefiHaff TeOpHR B03HHKHOBeHMR TYP6Y.leHTHOCTW B I 7 0 T O K a X C r pa -ArreHToar cpeAHeii CKopocTn. AOKA.H CCC P, cepux .+ramex., @u3., 1965, 161, 795-798.

H . W . L i e p rn a n n. FreeTurb ulent F lows . EI icanique de l a turbulence (COIL. In t ern .du CiVRS a M arsei l le) , Paris , E d . CIYRS, 1962, 211-226.I . L U RI 1 e y an d H . P a n o f s k y . The St ruc ture of Atmospheric Turbulence. NewYork -London - ydney, Inferscience, 1964.J . R . A. P e a r s o n. Th e Effect of Uniform Distortion on Weak Homogeneous Turbulence.J . Fluid Mech. , 1959, 5 , Part 2, 274-288.0; M. h i 1 1 i p s. The Irrotat ional Motion Outside a Free Turbulent Boundary. Proc.Cambr. Phi l l . Soc. , 1955, 51, 220-229.R . S. R i v 1 i n. The Rela t ion Between the Flow of No nnewtonian Fluids and T urbule ntNewtonian Flu ids . Qu a rt . Ap p l . Mafh., 957, 1 5 , 212-215.R . W . S t e w a r t . I r rotat ional Motion Associated w ith Free Turbulen t Flows. J . Flu idMech. , 1956, 1, 593-606.A. A . T o w n s e n d . The Uni form Dis tor t ion of Homogerieous Tuibulence. Quart . J .Mech. and A pp l . Math . , 1954, 7 , 104-127.

A . A . T o w n s e n d . The Structu re of T urbulen t Shear Flow. Cambridge U nive rsi ty Press,1956.


Discuss ion

E . A. N o v i k o v-Will th e elas tic response of the turb ulen t flow rem aini f we ta ke i n to account the nonlinear effects?. H. K. M f f a t t - he respon se of t ur bu le nt flow t o sudden disturbancesof th e mean velocity profile has an elastic character but i t has a cer ta in re laxa-t ion t ime and only a t the ini t ia l s tage does i t ha ve elas t ic features. The non-l inear effects certainly modify the simp le elastic response and probably oneshould regard the fluid as having a finite memory.

, W. . J o n e s -You indicate that var ia t ions in the mean flow can bepropagated ((elastically)) by larg e scale turbulence; one c an also thin k, of thisas a propa gation of th e fluid vor ticity . Does this represent th e true transm issionof vort icity from one fluid element t o another, a s is the case in waves, or per-hap s more sim ply the tran spo rt of fluid possessing cert ain vor ticity ?H. . M o f f a t t - he fluid is not transported by this mechanism, but

the mean velocity profile tends to propagate relative to the fluid.





* .