SOME ISSUES OF TURBULENCE

ACTA MECHANICA SINICA, Vol.ll, No.2, May 1995 Science Press, Beijing, China Allerton Press, INC., New York, U.S.A. ISSN 0567-7718 SOME ISSUES OF TURBULENCE STATISTICS* Qian Jian (J. Qian r (Dept. of Phys., Graduate School of Academia Sinica, P.O. Box 3908, Beijing 100039, China) ABSTRACT: The issue of dropping the random force f~ and the arbitrariness of choosing the basic variable in the variational approach to turbulence closure problem, pointed out recently by the Russian scientists Bazdenkov and Kukharkin, are discussed. According to the mean-square estimation method~ the random force fi should be dropped in the error expression of the LFP (Langevin-Fokker-Planck) model. However, f~ is not neglected, its effect has been taken into account by the variational approach. In order to optimize the perturbation solution of the Liouville equation, the LFP model requires that the basic variable is as near to Gaussian as possible. Hence, the velocity, instead of the vorticity, should be chosen as'the basic variable in the three-dimensional turbulence. Although the LFP model and the zero-order Gaussian term of PDF (probability density function) imply whiteness assumption (zero correlation time of f~), the higher-order non-gaussian terms of PDF correspond to the nonwhiteness of turbulence dynamics~ the variational approach does calculate the nonwhiteness effect properly. KEY WORDS: statistical theory of turbulence, variational method, closure problem, optimal control parameter, probability density function I. INTRODUCTION Recently Bazdenkov and Kukharkin [1] made an interesting study of the variational approach to the closure problem of turbulence, with particular attention to the perturbationvariation method of Qian [2]. They pointed out (p2249 of Ref.[1]) that the method of Qian "looks very attractive because of its clear physical grounds" and "succeeded in getting the Kolmogorov scaling and a very good agreement with the Kolmogorov constant value from experiments", however "this method is also not free from arbitrariness". They concluded that "an arbitrariness in some form is intrinsic in all closure methods, since all of them contain a certain approximation", and call it "the principal inevitability of arbitrariness in closure methods" or being "not self-consistent". We have no quarrel with the opinion of Bazdenkov and Kukharkin that all closure methods contain certain approximations. Actually we have remarked [3], "Any approach to turbulence closure problem (so-called closure method) inevitably contains some approximations; of course the approximations should be reasonable and workable". In this paper, the author discusses mainly two interesting problems put forward by Bazdenkov and Kukharkin. The first problem is how to justify dropping the random force fi in the mean-square error Received 18 September 1994 * The work is supported by the National Basic Research Program "Nomlinear Sciences" and the National Natural Science Foundation of China

Vol.ll, No.2 Qian Jian: Some Issues of Turbulence Statistics 123 expression of the LFP (Langevin-Fokker-Planck) model. The second problem is whether the choice of the basic variable is arbitrary. Bazdenkov and Kukharkin have studied both the three-dimensional and two-dimensional cases. In this paper we only discuss the threedimensional turbulence. The two-dimensional case has some peculiar features and will be studied in another paper. We adopt the notations of Refs.[1,2]. The Navier-Stokes equation is d = + F_, A,j xjx (1) Let P be the PDF (probability density function) of turbulence, which satisfies the Liouville 0 equation (~ + L)P ---- 0, L is the Liouville operator, and its structure is determined by the Navier-Stokes equation (1). For stationary turbulence, the Liouville equation becomes LP : 0 (2) The Liouville operator is quite complicated, we do not know how to obtain the exact solution of Eq.(2). Some approximate method has to be used. In the perturbation method used in classical and quantum mechanics, a complicated Hamiltonian H is expressed as the sum of a simpler major part H0 and a perturbation part AH, i.e. H ---- H0 + AH. Similarly we let and Substituting (3) and (4) Z, = + (3) p = p(o) + p0) + p(2) +... into (2), we have ~(o)p(o) = 0.~(o)p(1) A~p(O) ~(O)p(2) = _A~p(1) In order that the perturbation procedure (3)-(5) is reasonable and workable, the' major operator ~(0) is required to satisfy the following two conditions. 1) ~(0) is simpler than the total operator L so that the Eqs.(5) are analytically solvable. 2) The perturbation operator AL -- L - ~(0) is as small as possible. However, the two conditions are contradictory, a compromise has to be made. Some concepts of the estimation and optimization methods can help us to make a compromise between these contradictory requirements. For example, the adjectives "better" and "worse", instead of "correct" and "wrong", are used to describe the result of a compromise. Moreover we hope that there are many candidates for the major operator ~(0), so we can compare them and select the best (or optimal) one. The larger the set of candidates is, the better the result of a compromise is. In Qian's 1983 paper, the candidates for major operator are the Fokker-Planck operator (4) (5a) (5b) (5c) L(f) - ~ Yi ( 0 02 : + (6) i Here r is the average energy of mode i and is related to the energy spectrum of turbulence[2]. The z/i are index parameters of the candidates for major operator, different ~h represent different candidates, so there is an infinite number of candidates. The Fokker-Planck operator

124 ACTA MECHANICA SINICA 1995 (6) satisfies the first condition of major operator, since Eqs.(5) with ~(0) _ ~(I) are analytically solvable. According to the second condition of major operator, the optimal candidate should minimize, = z(71,,~, 7~,.-.) = IIAZ, II = IIZ, - Z~(S)ll (7) which is an appropriate measure of the "smallness" of the perturbation operator, and is a functional of the index parameters 7i of candidates. The explicit expression of I(71,72, 73,'" ") will be discussed in the next section. Hence the index parameters 71 of the optimal candidate satisfy 01/07, = 0 (8) By the terminology of control theory, the index parameters 7i are optimal control parameters. An objection to Eq.(8) has been put forward by McComb [41 who thought 7i should satisfy the energy equation and the variations &/i are not free. In fact, there is no restriction on the admissible values of the index parameters 7i so long as the Fokker-Planck operator (6) satisfies the first condition of major operator. Only the optimal value of ~/i, which satisfies (8), corresponds to the "real physical quantity" which satisfies the energy equation. The optimal value of 7i and the energy spectrum satisfy the energy equation as well as Eq.(8)[ 1,2]. Hence McComb's objection to Eq.(8) is not valid. In order to avoid misunderstanding, we can use two different symbols to represent the admissible and optimal values of 7i respectively. That is not necessary, since the dual role of parameters in an optimization calculation is self-evident. From a different viewpoint, Bazdenkov and Kukharkin [1] have also pointed out that McComb's objection to the correctness of Qian's method is irrelevant and is not valid. II. MEAN-SQUARE ESTIMATION AND CLASSIFICATION STATEMENT The success of the perturbation-variation procedure (3)-(8) depends upon whether we can propose an explicit expression of the error functional (7). The dynamic equation corresponding to the Liouville operator L is the Navier-Stokes equation (1), while the dynamic equation corresponding to the Fokker-Planck operator (6) is the Langevin equation d x~ = -n~x~ + Y~ (9) here fi is a random force of white-noise type. Therefore the smallness of the perturbation operator AL -- L - ~(I) corresponds to the smallness of the error of the approximation j~m AijmXjXm ~ -7iXi + fi (10) here we neglect ui (in the inertial range) for simplicity. Hence we have [2] i However, Bazdenkov and Kukharkin [1] thought that the correct error expression should be i, _- <[ -(-,,x, + :,>]"> i

Vo1.11, No.2 Qian Jian: Some Issues of Turbulence Statistics 125 and they said, "This omission (of fi in (11)), which permits the calculations to proceed, has not been justified, or even discussed by Qian". This issue will be discussed here, and it will be argued that actually (11), instead of (12), should be used. First of all, we forget turbulence for a short while and review a simple example of mean-square estimation. Let X and Z denote the input and output of a noisy transmission channel, and Z =/3X + f (13) here f is a white noise. If there is a serious attenuation,/3 << i, the noise part f might be greater than the signal part fix. We want to make an optimal estimate of the transmission factor /3 by using the statistics of X and Z. According to the mean-square estimation method [5], the optimal estimation is obtained by minimizing ((Z -/3X) 2) instead of ((Z - /3X - f)2). If/3 takes values other than its optimal value, then (Z -/3X) deviates from a white noise. The greater the difference between/3 and its optimal value, the greater the deviation of (Z -/3X) from a white noise. Hence the above estimation problem can also be formulated as follows: the random variable Z is partitioned into a signal part fix and a noise part and it is required to adjust the parameter/3 to make the noise part (Z --/3X) as close to a white noise as possible, then the optimal/3 is obtained by minimizing ((Z -/3X) 2) instead of ((Z -/3X - f)2), which is equivalent to the orthogonality principle. Now back to the approximation (10). The random variable ~ AijmXjX,~ is partitioned into a "signal" part -~ixi and a "noise" part Ni - ~_~ AijmXjX,~ -- (-~RXi) (14) According to (10), r R are adjusted to make the "noise" part Ni as near to a white noise as possible. Similarly, by the mean-square estimation method, the optimal r/i is obtained by minimizing (11) instead of (12). The validity of this process is explained in many books on control theory such as Ref.[5]. Although the random force fi or the "noise" part Ni is implicit in the error expression (11), it is not neglected, and its effect has been taken into account by the perturbationvariation method of Qian in a deliberate way. This point will be discussed further in Section IV. We should notice that the approximation (10) is not affected when the "noise" part Ni is dominant, so long as Ni is as close to a white noise as possible. There are large white noises as well as small white noises. A "white noise" is a class of random variables, similar to a "periodic function" being a class of functions. Hence the approximation (10), or the statement that Ni of (14) is as near to a white noise as possible, is just a qualitative classification statement, in contrast to the quantitative statement such as z = -2x + 3. The mean-square estimation method translates a qualitative classification statement into minimization of an error functional in which the qualitative classification term itself is invisible, and has achieved great success in various applications [5]. Hence, the correct error functional of the LFP model should be (11) instead of (12). Someone might argue that, strictly speaking, the mean-square estimation method with an error functional ((Z -/3X) 2} or (11) can be justified when the noise part f or N/ is white, otherwise the nonwhiteness effect of the noise part is ignored. In numerous successful applications of the mean-square estimation method, the exact property of the noise part is unknown and the noise part might not be white or even is deterministic instead

126 ACTA MECHANICA SINICA 1995 of stochastic [s], and we still obtain satisfactory results. In the case of turbulence closure problem which deals with low-order statistics, the deviation of the noise part Ni from a white noise is not large, and the relevant approximation is reasonable and workable. The error of the approximation has been made as small as possible by our perturbation-variation procedure and by choosing the velocity as the basic variable. Moreover, as shown in Section IV, the nonwhiteness effect of the noise part Ni has been calculated properly by our perturbation-variation method in a deliberate way. III. OPTIMAL BASIC VARIABLE By using (8) and the error expression (11), we can derive an equation for the optimal value of 7/i, which is called the ~/equation. The energy equation and the ~7 equation constitute a closed set of equations of spectral dynamics, solving the closure problem. Bazdenkov and Kukharkin{11 pointed out that the ~/equation depends on the choice of the basic variable. By a change of variable, they obtain the revised error expression, i - F~ = i s (15) Here Fi is a weighting factor and depends upon the choice of the basic variable. When F~ -- 1 or (~ = 0 (14) becomes (11) which corresponds to the velocity being chosen as the basic variable, c~ -- 1 corresponds to the velocity derivative or the vorticity or the rate of strain, c~ -- 2 corresponds to the second derivative of velocity or the energy dissipation rate, and so on[ 1]. Bazdenkov and Kukharkin [1] said, "this choice (of the basic variable) is not restricted by any reason, thus resulting in the presence of arbitrariness in the method". In fact, the essence of the perturbation-variation method places restrictions on the choice of the basic variable. In studying a physical problem such as fluid turbulence, it is natural to use a physical quantity, which has clear physical meaning, as the basic variable. At present it is not clear what physical quantity corresponds to the case c~ = 0.71 or 1.3, hence we are restricted to the case of c~ being integers, corresponding to the velocity, vorticity, energy dissipation, etc. The major restriction on the choice of basic variable is that the PDF of the basic variable should be as near to a Gaussian density function as possible. In the method of Qian, the major operator ~(0) is the Fokker-Planck operator (6), the solution p(0) of (5a) is a Gaussian density function. Hence, in order to minimize the error of the perturbation solution of the Liouville equation, the PDF of the basic variable should be as near to a Gaussian density function as possible. The velocity is nearly Gaussian, while various velocity derivatives, the vorticity, and the energy dissipation are far from Gaussian. Therefore the velocity, instead of the vorticity or the energy dissipation, is the optimal basic variable. In fact, the velocity being nearly Gaussian is the physical fact which leads us to choose the Fokker- Planck operator (6) as the major operator. If the major operator ~(0) is such that the solution p(0) of (5a) is an exponential function, then the optimal basic variable should have a near-exponential PDF. The major operator ~(0) and the basic variable are intimately related, the major operator determines the optimal basic variable. The modal parameters Xi are derived from the basic variables by means of a given linear transform defined in Qian's 1983 paper. The theory of stochastic processes proves that any stochastic process, derived by means of a linear transform from a Gaussian process, is itself a

Vol.ll, No.2 Qian Jian: Some Issues of Turbulence Statistics 127 Oaussian process. It is reasonable to expect that the deviation of the modal parameters from Gaussian increases with the deviation of the corresponding basic variable from Gaussian. IV. NON-WHITENESS AND NON-GAUSSIAN EFFECT Bazdenkov and Kukharkin [1] said, "Another serious arbitrariness of the LFP model is the assumption that the dynamic force fi in the model equation has zero correlation time. In this case one has to drop the forcing term in the expression for the LFP model error ---". This issue has been partially discussed in Section III where we explain why the random force fi should be droppedfin the error expression (11), Another interesting aspect of this issue is how the effect of nonzero correlation time is considered in the method of Qian, which is discussed in this section. The Fokker-Planck operator (6) implies the random force fi being a white noise with zero correlation time, and the corresponding solution p(0) of (5a) is a Gaussian density function. In other words, the whiteness or zero correlation time of the random force corresponds to the Gaussianity of the 'PDF. In fact, the "noise" part Ni of (14) deviates from a white noise and have nonzero correlation time, and the PDF of turbulence deviates from Gaussian. One of possible ways to take into account the effect of "nonwhiteness" or nonzero correlation time, is to study the two-time correlation. The perturbation-variation method of Qian adopts a completely different way to take into account the "nonwhiteness" effect. From the viewpoint of PDF, the whiteness (or zero correlation time) corresponds to a Gaussian density function, and the effect of "nonwhiteness" (or nonzero correlation time) corresponds to the deviation from Gaussian. When the Fokker-Planck operator (6) is the major operator, the perturbation operator AL ---- L - L(f) contains all the "nonwhiteness" effect of the "noise" part N~, hence the solutions p(1) and p(2) of (5b)-(5c) represent the "nonwhiteness" effect or the non-gaussian effect. Although the LFP model and the zeroorder approximation p(0) of the PDF neglect the "nonwhiteness" effect, the p(1) and p(2) does take into account the "nonwhiteness" effect. In fact, the energy equation of Qian's 1983 paper has the same form as Kraichnan's DIA energy equation, the nonwhiteness effect has been calculated in Qian's method as well as in Kraichnan's DIA [2,6]. Of course, their ways of calculating the nonwhiteness effect are completely different. Kraichnan [8] developed a ' theory of two-time correlation of turbulence to study the nonwhiteness effect. Qian calculates the deviation of PDF from the Gaussianity to take into account the nonwhiteness effect, without appealing to an explicit calculation of two-time correlations. For the study of spectral energy transfer [2], it is enough to use p(1) to calculate the nonwhiteness or non-gaussian effect. For the study of intermittency [7], it is necessary to use p(1)q_p(2) to calculate the nonwhiteness or non-gaussian effect. Although the LFP model and the Gaussian term p(0) imply whiteness assumption (zero correlation time of fi), the higher-order non-gaussian terms p(1) and p(2) correspond to the nonwhiteness of turbulence dynamics, the perturbation-variation method of Qian does calculate the nonwhiteness or non-gaussian effect properly. A basic fact of turbulence statistics is that the vorticity (or velocity derivatives) is far from Gaussian while the velocity is close to Gaussian. It is not proper to apply near-gaussian approximation to the vprticity or velocity derivatives. It is also not proper to deny the validity of applying near~gaussian approximation to the velocity. Bazdenkov and Kukharkin has studied how the Kolmogorov constant depends upon the choice of the basic variable.

128 ACTA MECHANICA SINICA 1995 Their result (Fig.1 of Ref.[1]) clearly demonstrates that an unreasonable result is obtained when the basic variable is the vorticity which is far from Gaussian, and a satisfactory result is obtained when the basic variable is the velocity which is nearly Gaussian. This interesting result of Bazdenkov and Kukharkin confirm further that the choice of the velocity as the basic variable and the Fokker-Planck operator (6) as the major operator is reasonable and workable. REFERENCES [1] Bazdenkov SV, Kukharkin NN. On the variational method of closure in the theory of turbulence. Phys Fluids A, 1993, 5(9): 2248 [2] Qian J. Variational approach to the closure problem of turbulence theory. Phys Fluids, 1983, 26(8): 2096 [3] Qian J. Closure problem of turbulence and non-equilibrium statistical mechanics. Mechanics and Practice, 1988, 10 (3): 1 (in Chinese) [4] McComb WD. The Physics of Fluid Turbulence. Oxford: Clarendon Press, 1990 [5] Mendel JM. Discrete Technique of Parameter Estimation. New York: Marcel Dekker, 1973 [6] Leslie DC. Development in the Theory of Turbulence. Oxford: Clarendon Press, 1973 [7] Qian J. A closure theory of intermittency of turbulence. Phys Fluids, 1986, 29 (7): 2166