On developed hydrodynamic turbulence spectra

Physica 2D (1981) 203-217 @ North-Hol land Pub1 ishing Company

ON DEVEIQEBD HYDRODYNAMIC !I?URBUI;ENCE SPECTRA E.A.

Kuznetsov, V.S. L1vov

I n s t i t u t e of Automation and Electrometry S i b e r i a n Branch of t h e USSR Ac. of S c i . 630090 Novosibirsk USSR

New r e s u l t s i n t h e theory of t h e developed hydrodynamic turbulence s p e c t r a a r e reviewed. Within t h e l i m i t s of t h e hypothesis of i n t e r a c t i o n l o c a l i t y it i s shown t h a t t h e s e r i e s of equations f o r t h e moments has a s c a l e - i n v a r i a n t s o l u t i o n w i t h t h e Kolmogorov index values. With t h e h e l p of t h e Wyld diagram technique t h e equat i o n s i n t h e D i r e c t I n t e r a c t i o n Approximation a r e formulated which a c c u r a t e l y t a k e i n t o account t h e t r a n s f e r e f f e c t and have t h e p r e c i s e s o l u t i o n i n t h e form of t h e Kolmogorov spectrum. I n t h e framework of t h e s e equations t h e correct i o n s t o t h e Kolmogorov spectrum due t o gyrot r o p y and c o m p r e s s i b i l i t y a r e found. INTRODUCTION The n o t i o n of t h e turbulence spectrum o r t h e energy d i s t r i b u t i o n p l a y s an important r o l e i n t h e turbulence theory. The p r e s e n t concept of a developed hydrodynamic turbulence spectrum i s based, t o a g r e a t e x t e n t , on A.N. Kolmogorov i d e a s [I], according t o which t h e small-scale developed hydrodynamic turbulence i s isoits t r o p i c and homogeneous, and w i t h i n t h e i n e r t i a l i n t e r v a l spectrum i s d e f i n e d by t h e s i n g l e dimensional parameter P , t h e energy f l u x . This spectrum JK i s determined up t o some dimensionl e s s factor c :

c

213 -11/3

K

.

(0.1)

Here

I n a s p h e r i c a l normalization t h i s spectrum corresponds t o t h e wellknown Kolmogorov-Obukhov I * fi v e - t h i r d s '* law

confirmed, t o a good accuracy, by the observations of t h e atmosp h e r i c turbulence, i n e r t i a l i n t e r v a l of which changes by f i v e o r d e r s of magnitude [2].

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E.A. Kuznetsov and V.S. L'vov / Hydrodynamic turbulence spectra

Kolmogorov's hypotheses a l s o allow t o determine t h e c h a r a c t e r i s t i c frequency dK of t h e v o r t i c e s of t h e s c a l e K:

and behaviour of t h e higher moments

where

a ; ~= - 10n//3+3.

L a t e r on L.D. Landau and E.N. L i f s h i t s [3] p a i d a t t e r l t i o n t o the f a c t t h a t t h e energy f l u x P i s t a c t u a l l y a f l u c t u a t i n g v a l u e , so t h a t P i s n o t t h e only dimensional c o n s t a n t on which t h e spectrum may depend. Therefore A.N. Kolmogorov and A.M. Obukhov i n troduced some phenomenological changes t o t h e t h e o r y , being e s s e n t i a l l y , a r e f u s a l from t h e i d e a of complete l o c a l i t y and acknowledgement of t h e f a c t t h a t a new dimensionless parameter (kL) can be introduced i n t o t h e ex r e s s i o n f o r J k , t o some unknown, b u t n o t very l a r g e power ~ 2 5 .Thus, t h e phenomenological approach t o t h e hydrodynamical t u r b u l e n c e d e s c r i p t i o n d i d n o t permit t o determine t h e spectrum and s o f a r t h e r e were many a t t e m p t s t o c o n s t r u c t t h e microscopic t h e o r y based d i r e c t l y on t h e Euler equations :

-

It should be noted t h a t i n a number of works t h e s e equations were i n v e s t i g a t e d by means of v a r i o u s hypotheses of t h e higher moments of t h e v e l o c i t y f i e l d , f o r example, t h e k ~ i l l i o n t s h i k o vhypothesis of r e p r e s e n t a t i o n of t h e f o u r t h moments through second moments [ 2 ] . I n f a c t , t h e n o n l i n e a r i t y i n t h e E u l e r e q u a t i o n s i s extremely s t r o n g and, a p p a r e n t l y , t h e r e a r e no s e v e r e r e a s o n s f o r breaking t h e sequence of equations f o r t h e moments. A r e g u l a r procedure f o r i n v e s t i g a t i o n of t h e E u l e r e q u a t i o n s i s t h e diagram technique suggested by kyld [ L C ] :The random Gaussian f o r c e f K L L ) i s introduced i n t h e right-hand s i d e of e q u a t i o n s (0.4) and t h e s o l u t i o n f o r V K i s p r e s e n t e d a s a power s e r i e s i n f The s e r i e s f o r JKw can%e given i n 2 form which a nonzero r e s u l t i s o b t a i n e d a l s o w i t h i n t h e l i m i t f,, -0. The p e c u l i a r i t y of t h e Wyld t e c h n i ue (and a l s o of any technique f o r s t r o n g l y nonequilibrium systems i s t h e appearance of two diagram ser i e s : f o r Green f u n c t i o n GK and second monent JKW , which a r e connected by t h e u n i v e r s a l r a a t i o n o n l y i n t h e thermodynmic e q u i l i b r i u m . Wyld has i n a r t i c u l a r shown t h a t t h e D i r e d l n t e r formulated by Kraichnan C6 ] corresponds a c t i o n Approximation t o t h e approximation, where t h e v e r t i c e s a r e n o t renormalized. Gen e r a l l y speaking, t h i s r e n o r m a l i z a t i o n i s r a t h e r s i g n i f i c a n t and t h u s , i n t h e s t r o n g hydrodynamic t u r b u l e n c e t h e o r y a s w e l l as i n t h e phase t r a n s i t i o n t h e o r y , one should t a k e i n t o account t h e whole diagram s e r i e s r e n o r m a l i z i n g t h e i n t e r a c t i o n . liith t h e account of t h i s f a c t G.A. Kuzmin and A.Z. P a t a s h i n s k i C 7 1 s t u d i e d t h e ways of agreement of v a r i o u s s c a l i n g hypotheses w i t h t h e diagram s e r i e s of t h e Wyld technique.

.

f.51)

E.A.

K u z n e t s o v and V . S . L ' v o v / Hydrodynamic t u r b u l e n c e s p e c t r a

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I n t h i s paper we p r e s e n t t h e review of t h e new r e s u l t s i n t h i s f i e l d , r e s t r i c t i n g o u r s e l v e s t o t h e Authorst works mainly. I n t h e S e c t i o n I we consider some p r o p e r t i e s of t h e E u l e r equations: t h e c o n s e r v a t i o n l a w s and r e l a t i o n s f o r t h e e q u a t i o n c o e f f i c i e n t s r e s u l t i n g from t h e l a w s Besides, t o p o l o g i c a l a s p e c t s of t h e v o r t e x l i q u i d motion a r e d i s c u s s e d , Hamiltonian d e s c r i p t i o n i s given w i t h t h e h e l p of t h e Klebsch v a r i a b l e s . I n t h e second S e c t i o n t h e equat i o n sequence f o r t h e v e l o c i t y f i e l d moments i s analyzed by means of conformal t r a n s f o r m a t i o n s i m i l a r t o t h e t r a n s f o r m a t i o n of t h e k i n e t i c wave e q u a t i o n suggested by V.E. Zakharov [8]. T h i s a l l o w s t o f i n d a n a d d i t i o n a l , as compared t o t h e phase t r a n s i t i o n t h e o r y , s c a l i n g i n d e x appearing i n t u r b u l e n c e t h e o r y i n t h e absence of t h e ther~nodynamic e q u i l i b r i u m . It is i n t h e l i m i t s Kolmogorov hypot h e s i s of t h e i n t e r a c t i o n l o c a l i t y t h a t t h e sequence of e q u a t i o n s f o r t h e moments i s shown t o have a c t u a l l y t h e s o l u t i o n (0.3) w i t h t h e Kolmogorov i n d i c e s . Thus, t h e c e n t e r of t h e problem i s t r a n s f e r r e d t o t h e s t u d y of i n t e r a c t i o n l o c a l i t y i n t h e developed hydrodynamic t u r b u l e n c e . The t h i r d S e c t i o n based on t h e r e s u l t s by one of t h e a u t h o r s [ 9 ] i s devoted t o t h i s s u b j e c t . I n t h i s S e c t i o n diagram d i v e r g e n c i e s i n t h e Wyld technique a r e analyzed. It i s shown t h a t t h e most d i v e r g i n g sequence of t h e diagrams desc r i b e s t h e t r a n s f e r of small v o r t i c e s a s a whole by random l a r g e s c a l e motions. This sequence i s summed t o t h e form:

>

where