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A kinetic equation for strongly turbulent homogeneous systems
Volume 46A, number 5
PHYSICS LETTERS
14 January 1974
A KINETIC EQUATION FOR STRONGLY TURBULENT HOMOGENEOUS SYSTEMS W.P.M. MALFLIET Departement Natuurkunde, UniversiteitAntwerpen (U.I. A.) Universiteitsplein 1, B-2610 Wilri/k, Belgium Received 5 December 1973 A consistent dynamical equation for the energy-spectrum function has been derived which describes a strong turbulent state of a model system. The main difficulty is treating problems of turbulence with the moments approach is the closure problem. Different theories in this field are proposed by Kraichnan (direct-interaction scheme for instance) [1], Edwards [2], and others, in particular for hydrodynamical turbulence. Along the lines of our earlier work concerning weak turbulence [3], we now develop a formalism for strong turbulence. For simplicity's sake we restrict ourselves to the following one-dimensional equation
aA (k, t) _ at -i
/" , J dk' V ( k ; k , k - k ' ) A ( k ' , t ) A ( k - k , t ) ,
(1)
in which we only take into account the non-linear interaction between the different fluctuations. Hence no linear behaviour is present in this system. The amplitude A(k) represents an individual k-fluctuation ( k ~ L -1) of the form exp (ikx). For convenience the interaction strength V is supposed to be a real quantity. If we introduce the wave correlation functions E(k) (second order), H (third order) and U (fourth order), the first two moment equations of the hierarchy arising from eq. (1) can then be written as:
aE(k)_ at
_ i
_~
aH(k', - k , k - k ' ) at
dk' V(k;k', k - k ' ) H ( k ' , - k , k - k ' )
= f(U) -2iV(k-k
,
+ (k-* - k ) ,
, ,-k, k)E(k')E(k)
(2)
(3)
-2i V(k';k,k'-k)E(k-k')E(k)-2iV(-k;k'-k,-k')E(k-k')E(k'). On the other hand, it can readily be observed that eq. (1) also can be written in the following formal way (see [4] ): +oo
aA (k, t) - -~l(k)A(k, t) - i at
f
dk o V(k;k o, k - k o ) A l ( k o, t ) A l ( k - k o, t),
(4)
_¢¢
where the integral on the right-hand side contains no terms proportional to A(k, t); r/(k) = r/diss ( k ) - r/g r ( k ) which means that there exist a non-linear damping and growth. Again a hierarchy of moment equations can be set up, but now with the aid of eq. (4). In particular the second equation of this hierarchy reads:
a g ( k ' , - k , k - k ' ) _ _ [~?(k') + r?(k) + "q( k - k ' ) ] g(k', - k , k - k ' ) - 2i V ( k - k ' ; - k ' , k ) E 1 (k') El (k) at
(5)
- 2iV(k';k, k ' - k ) El ( k - k ' ) El ( k ) - 2 i V ( - k ; k ' - k , - k ' ) E l ( k - k ' ) E l (k')+ fl (U ). By a careful analysis, with the aid of eq. (2), one can show that E 1(k) ~ E(k) (for all k). So if we replace eq. (3) by eq. (5) and neglect fl (U), the hierarchy is closed; indeed, we are able to determine r/(k) afterwards (see eq. (7)). The common closure, known as the quasi-normal approximation (i.e. the neglect o f f ( U ) in eq. (3)) implied that one did not take into account the non-linear damping (growth) of the third-order correlation function H. 363
Volume 46A, number 5
PHYSICS LETTERS
14 January 1974
If one should solve eq. (5) quite generally and substitute the result into eq. (2), one should find a dynamical equation for E(k) which is similar to the direct-interaction approximation of Kraichman (see eq. (4.1) in ref. [1] ). However, we shall take a particular solution of eq. (5), according to the Bogoliubov-expansion method which we introduced for (weak) turbulence problems [3], and substitute the result in eq. (2). Moreover we assume that aA(k)/at ~ - rl(k)A(k), which justifies the above mentioned closure for strong turbulence. This leads to the following equation for the energy-spectrum function :
~E(k)~t
+=adk' { IV(k;k'k-k')12E(k')E(K-k') _= r~(k)-r~(k')-r~(k- k')
- 4 (
-~ 2rl (k) E(k) = -
2(r/diss (k) - ~ g r (k))
-
2V(k;k',k k')V(k-k';k,-k')E(k')E(k)} ~(k- k')-rl(k)-~(k')
E(k).
(6)
(7)
If the original eq. (1) conserves energy, also this eq. (6) does. We remark further that the whole of the second integrand in eq. (6) contributes to r/(k), while the first integrand only contributes for k' "~ k. In principle one can find back the r7 (k), calculated by Edwards [2], if one assumes that E l (k) is stationary and takes again that particular solution of eq. (5). Obviously only in particular cases we are able to determine rTdiss(k) and r/gr (k) and to prove eq. (7) (which is certainly true for large k-values). This theory will be applied to fluid turbulence for instance which will allow us to discuss in more detail the consequences of the made approximations. The author wishes to thank Prof. Dr. Dirk K. Callebaut for his interest and careful reading of the manuscript. He also is indebted to Dr. M.P.H. Weenink (FOM Jutphaas) for his critical comments in the early stage of this work.
References
[1] [2] [3] [4]
364
S.A. Orszag, J. Fluid Mech. 41 (1970) 363. S.F. Edwards, J. Fluid Mech. 18 (1964) 239. W.P.M. Malfliet, Physica 59 (1972) 321. B.B. Kadomtsev, Plasma Turbulance, Academic Press (1965).
PHYSICS LETTERS
14 January 1974
A KINETIC EQUATION FOR STRONGLY TURBULENT HOMOGENEOUS SYSTEMS W.P.M. MALFLIET Departement Natuurkunde, UniversiteitAntwerpen (U.I. A.) Universiteitsplein 1, B-2610 Wilri/k, Belgium Received 5 December 1973 A consistent dynamical equation for the energy-spectrum function has been derived which describes a strong turbulent state of a model system. The main difficulty is treating problems of turbulence with the moments approach is the closure problem. Different theories in this field are proposed by Kraichnan (direct-interaction scheme for instance) [1], Edwards [2], and others, in particular for hydrodynamical turbulence. Along the lines of our earlier work concerning weak turbulence [3], we now develop a formalism for strong turbulence. For simplicity's sake we restrict ourselves to the following one-dimensional equation
aA (k, t) _ at -i
/" , J dk' V ( k ; k , k - k ' ) A ( k ' , t ) A ( k - k , t ) ,
(1)
in which we only take into account the non-linear interaction between the different fluctuations. Hence no linear behaviour is present in this system. The amplitude A(k) represents an individual k-fluctuation ( k ~ L -1) of the form exp (ikx). For convenience the interaction strength V is supposed to be a real quantity. If we introduce the wave correlation functions E(k) (second order), H (third order) and U (fourth order), the first two moment equations of the hierarchy arising from eq. (1) can then be written as:
aE(k)_ at
_ i
_~
aH(k', - k , k - k ' ) at
dk' V(k;k', k - k ' ) H ( k ' , - k , k - k ' )
= f(U) -2iV(k-k
,
+ (k-* - k ) ,
, ,-k, k)E(k')E(k)
(2)
(3)
-2i V(k';k,k'-k)E(k-k')E(k)-2iV(-k;k'-k,-k')E(k-k')E(k'). On the other hand, it can readily be observed that eq. (1) also can be written in the following formal way (see [4] ): +oo
aA (k, t) - -~l(k)A(k, t) - i at
f
dk o V(k;k o, k - k o ) A l ( k o, t ) A l ( k - k o, t),
(4)
_¢¢
where the integral on the right-hand side contains no terms proportional to A(k, t); r/(k) = r/diss ( k ) - r/g r ( k ) which means that there exist a non-linear damping and growth. Again a hierarchy of moment equations can be set up, but now with the aid of eq. (4). In particular the second equation of this hierarchy reads:
a g ( k ' , - k , k - k ' ) _ _ [~?(k') + r?(k) + "q( k - k ' ) ] g(k', - k , k - k ' ) - 2i V ( k - k ' ; - k ' , k ) E 1 (k') El (k) at
(5)
- 2iV(k';k, k ' - k ) El ( k - k ' ) El ( k ) - 2 i V ( - k ; k ' - k , - k ' ) E l ( k - k ' ) E l (k')+ fl (U ). By a careful analysis, with the aid of eq. (2), one can show that E 1(k) ~ E(k) (for all k). So if we replace eq. (3) by eq. (5) and neglect fl (U), the hierarchy is closed; indeed, we are able to determine r/(k) afterwards (see eq. (7)). The common closure, known as the quasi-normal approximation (i.e. the neglect o f f ( U ) in eq. (3)) implied that one did not take into account the non-linear damping (growth) of the third-order correlation function H. 363
Volume 46A, number 5
PHYSICS LETTERS
14 January 1974
If one should solve eq. (5) quite generally and substitute the result into eq. (2), one should find a dynamical equation for E(k) which is similar to the direct-interaction approximation of Kraichman (see eq. (4.1) in ref. [1] ). However, we shall take a particular solution of eq. (5), according to the Bogoliubov-expansion method which we introduced for (weak) turbulence problems [3], and substitute the result in eq. (2). Moreover we assume that aA(k)/at ~ - rl(k)A(k), which justifies the above mentioned closure for strong turbulence. This leads to the following equation for the energy-spectrum function :
~E(k)~t
+=adk' { IV(k;k'k-k')12E(k')E(K-k') _= r~(k)-r~(k')-r~(k- k')
- 4 (
-~ 2rl (k) E(k) = -
2(r/diss (k) - ~ g r (k))
-
2V(k;k',k k')V(k-k';k,-k')E(k')E(k)} ~(k- k')-rl(k)-~(k')
E(k).
(6)
(7)
If the original eq. (1) conserves energy, also this eq. (6) does. We remark further that the whole of the second integrand in eq. (6) contributes to r/(k), while the first integrand only contributes for k' "~ k. In principle one can find back the r7 (k), calculated by Edwards [2], if one assumes that E l (k) is stationary and takes again that particular solution of eq. (5). Obviously only in particular cases we are able to determine rTdiss(k) and r/gr (k) and to prove eq. (7) (which is certainly true for large k-values). This theory will be applied to fluid turbulence for instance which will allow us to discuss in more detail the consequences of the made approximations. The author wishes to thank Prof. Dr. Dirk K. Callebaut for his interest and careful reading of the manuscript. He also is indebted to Dr. M.P.H. Weenink (FOM Jutphaas) for his critical comments in the early stage of this work.
References
[1] [2] [3] [4]
364
S.A. Orszag, J. Fluid Mech. 41 (1970) 363. S.F. Edwards, J. Fluid Mech. 18 (1964) 239. W.P.M. Malfliet, Physica 59 (1972) 321. B.B. Kadomtsev, Plasma Turbulance, Academic Press (1965).