a refinement of previous hypotheses concerning the local structure of turbulence in a viscous...

8/12/2019 A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid

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A refinement of previous hypotheses concerningthe local structure of turbulence in a viscousincompressible fluid at high Reynolds number

By A. N. K O L M O G O R O VSteklov Mathematical Institute, Academy of Sciences of the U.S.S.R., Moscow

(Received 10 December 1961)

The hypotheses concerning the local structure of turbulence a t high Reynoldsnumber, developed in the years 1939-41 by myself and Oboukhov (Kolmogorov1941 a, , c ; Oboukhov 1941 a , b )were based physically on Richardson s idea of th eexistence in the turbulent flow of vortices on all possible scales I < r < L betweenthe external scale L and the internal scale and of a certain uniform mechan-ism of energy transfer from the coarser-scaled vortices to the finer.

These hypotheses were arrived a t independently by a number of other authorsand have achieved very wide acceptance. Bu t quite soon after they originated,Landau noticed th at they did not take into account a circumstance which arisesdirectly from th e assumption of the essentially accidental and random characterof the mechanism of transfer of energy from the coarser vortices to the finer:with increase of th e ratio L : the variation of th e dissipation of energy

should increase without limit. More accurately, it is natural to suppose th a t whenLIZ 1 the dispersion of the logarithm of E has the asymptotic behaviour

..fog A + k' log LIZ, (1)

where k' is some universal constant.However, Oboukhov has now discovered how to refine our previous results in a

way which takes Landau s comments into consideration. The method consistsin examining the dissipation

Editor s footnote. This paper is a translation (for which t he editors tak e responsibility)of the Russian text of a lecture given by the author at the Colloque International duC.N.R.S. d e MBcanique d e l a Turbulence, held a t Marseilles from 28 August t o 2 September1961. The work described in it is related closely to a lecture given by A. M . Oboukhovat another symposium in the following week (see the preceding paper in this journal).The two lectures are being included in the published records of the proceedings of therespective symposia, a nd are published here also in view of their interest t o a large numberof E nglish-speaking readers.


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T h e local structure of turbulence at high Reynolds number 83

averaged for a sphere of radius r and in assuming th a t for large L / r the logarithmof c, x, t ) has a normal distribution. It is natura l to suppose that the varianceof log e,.(x, ) s given by

where k is a universal constant, the factor 9 is inserted for later convenience,and A x , )depends on the macrostructure of the flow.

I have formulated appropria te modifications to th e two similarity hypothesestha t I pu t forward in 1941, and I therefore wish to give an interpretation ofOboukhov s new idea in the light of this earlier work. The assumed normality ofth e distribution of log e,, together with formula (2) for its variance, constitutesa third hypothesis.

cr:(x, t ) = A x, )+ QkIogL/r, ( 2 )

I shall show tha t the equation

B d d ( r ) = C r f 8

from Kolmogorov (1941 a ) now takes th e modified form

Bdd(r ) C X, ) W ( L / r ) - k ,

fw = % d d ( r ) / B d ( r )

3 )

where k is the constant in equation (2) and the factor C x,t) epends on themacrostructure of th e flow; an d that the theorem of constancy of skewness

when 1 < r < L, derived in Kolmogorov (1941 b ) , s now replaced by the equation

X ( r )= X O ( L / r ) @ 4)where the coefficient 8, also depends on th e macrostructure of the flow.

velocity

and an internal length scale,

Obviously th e Reynolds number formed from V ,and r is determined by the ratior : , through the equation

Let us associate with the length scale r and th e point (x, ) scales of time an d

T, = r k ; Q , V = r@

1, = v P q 4 .

( 5 )Re, = , r = k) .

V

Let x;,t' be th e co-ordinates of any point (x', ) in the neighbourhood ofof (x, . Then we define dimensionless co-ordinates t x nd time T by equations

X = x a + t X r , t' = t+7T,.,

and dimensionless relative velocities v,( 7 by equations

The rst simi larity hypothesisIf r < L , then for assigned numbers [2 ,7@ a = 1, 2, 3; k = 1, 2, ... n ) theprobability distribution of the 3n variablesv,(Pk), 7 9 or the fixedvalue Re,. = Re,depends only on R e and is the same in all turbulent flows.

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84 A . N . Kolmogorov

T h e second s im ilar ity hypothesis

When Re 1, the distribution given in the k s t hypothesis does not depend onRe.We shall indicate an absolute mathematical expectation (mean) by an overbar.

Since 8 is almost constant in regions which are small in comparison with theexternal scale L when r < L it may be supposed tha t

z = 8 (6)

Let us examine the difference between the linear components of velocities a ttwo points distan t r apart,

From th e definitions of va and V

( 7 )If r 1,where is the upper limit (excluding cases of negligibly small probability)of th e internal scale Z then from equations (6) and 7 ) and the above two hypo-theses, it follows that

where C is an absolute constant.

having a logarithmically normal law of distribution, viz.

A d d ( ? ) = U l ( X 1 - t r , x z x 3 t ) l ( X 1 , xz 3, ) .

A d d ( r )= vl(1 , 0 , 0 , 0 ) k .

lAdd(r)13= Cr8 (8)

We now use the formula pointed out by Oboukhov for moments of quantit ies

= exp ( p m t- i p z a z ) , (9)

where m and a re the mean a nd variance of log g From equations 2) , (8) and(9) , i t follows tha t

I n particular, when p = 2,

IAdd(r)lP= C P ( x , ) r E ) P I 3 L / r ) 4 k P p - 3 J . (10)

A i d ( r )= C2 x, (ri?) L/r)-k,

and this is the same as equation 3).

Bddd(r) -+8r

given in Kolmogorov (1941 b ) remains valid, equation 4 ) now follows from equa-

tion 3).Our presentation can be freed from the special selection of the values of

e,(x, t ) forming the basis of Oboukhovs examination. The two similarityhypotheses are then formulated as follows:

Since the formula

First similarity hypothesisI f Ix Q x [ < L , ( k = 0 , 1 , 2 , . n) hen the probability distribution of the valuesof

U J X ( k ) ) U a ( X )

U,(X(O)) -U, X)___ ___ a = , 2 , 3 ; k = 1 , 2 ... n )

depends only on the Reynolds number


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T h e local structure of turbulence at high Reynolds number 85

Second similarity hypothesis

When R e

as the

1,th e distribution given in th e first hypothesis does not depend on Re.The essential content of Oboukhovs additional assumptions can be formulated

Third hypothesisTwo subsets of values in th e set 1 1) are stochastically independent, if in the

first set Ix ~)-xINaturally t he formulation of this hypothesis must be refined if mathematical

rigour is desired and if it is to be used to derive the logarithmic normality of thedistribution of velocity differences and the formula for the dispersion of thelogarithms of these differences analogous to equations 1 ) and 3).

rl, in th e second Ix@)-xl < r2 and rl r 2 .

R E F E R E N C E S

KOLMOGOROV, . N . 1941a C.R. Acad. Sci. U.R.S.S. 30, 301.KOLMOGOROV, . N. 1941b C.R. Acad. Scd. U.R.S.S. 31, 538.KOLMOGOROV, . N. 1941c C.R. Acad. Sci. U.R.S.S. 32, 16.OBOUKHOV, . M. 1941a C.R. Acad. Sci. U.R.S.S. 32, 19.OBOUKHOV, . M. 1941 b Izv. Nauk. S.S.S.R., Ser. Geogr. i. G w J z . 5 , 453.

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