ACTA MECHANICA SINICA, VoL10, No.l, February 1994 Science Press, Beijing, China Allerton Press, iNC., New York, U.S.A.

ISSN 0567-7718

SKEWNESS FACTOR OF TURBULENT

VELOCITY DERIVAT IVE*

Qian Jian (J. Qian $~/t~) (Department of Physics, Graduate School of Academia Sinica, Beijing 100039, China)

ABSTRACT: Using nonequilibrium statistical mechanics closure method, it is shown that the skewness factor of the velocity derivative of isotropic turbulence ap- proaches a constant -0.515 when the Reynolds number is very high, which is in agree- ment with the DNS (direct numerical simulation) result of Vincent and Meneguzzi (1991).

KEY WORDS: skewness factor, isotropic turbulence, intermittency, statistical theory of turbulence

I. INTRODUCTION

The skewness factor of turbulent velocity derivative is

( 0ul 3 0Ul 2 3/2 (1)

here Ul is the velocity component along the Xl direction,

Vol.lO, No. l Qian Jian: Skewness Factor of Turbulent Velocity Derivative 13

tube model, fractal model, and Kraichnan's model equation of PDF did not given numerical value of S, which can be compared with the DNS results.

In this paper, we will study the asymptotic behavior of the skewness S from a totally different point of view: it is unnecessary to use a particular model or make a special physical assumption about the small-scale structure of turbulence. Similar to the other statistical properties, the skewness S could be calculated from the basic equation of motion by the method of statistical mechanics. In fact, by applying the nonequilibrium statistical mechan- ics closure method and some basic formulae of spectrum dynamics, we cannot only predict whether ISI approaches a finite constant as R~ -~ oo, but also calculate the numerical value of S and to make a comparison between the present theoretical result and the DNS results.

I I. UPPER L IM IT OF ABSOLUTE VALUE OF SKEWNESS

Suppose r is a random variable, then {((r3)r - (r2}r2) 2) _> O, which implies that Oul

(r3) 2 _< (r2}(r4}. Let r -- Oxl' we obtain

ISI ~ F 1/2 F = ((0~11)4)/((~)2) 2 (3)

Here F is the flatness factor of the turbulent velocity derivative. From the Navier-Stokes equation, by the nonequilibrium statistical mechanics closure method, it has been shown [71 that F approaches a constant (about 15) as the Reynolds number R~ -+ c~. Hence, according to (3), ISI has a upper limit V~5 ~ 3.9 and cannot approach infinity as R~ --+ oo.

Actually we can make a better estimate of the upper limit of ISI than (3) which is valid for any random variable and has not taken advantage of the characteristics of isotropic

Oul turbulence. By using the particular properties of ~xl of isotropic turbulence, the following

relationship can be obtained [s]

ISl ~ (21v~)F 1/2

14 ACTA MECHANICA SINICA 1994

Here E(k) is the energy spectrum, e is the energy dissipation rate, v is the kinematic viscosity, and T(k) is the energy transfer spectrum function. For a stationary turbulence, T(k) = 2vk2E(k), by using (1) (5) and (6), we obtain

S -(12v~-g/7)e-2/ak21~ E(k)k4dk (7)

here kd = (e/u3) 1/4 is the Kolmogorov wavenumber. The Reynolds number

here

RA :< u 2 >1/2 ,,~//,, (8)

I 0 .5

0.6

0.4 i0 l0 2 10 3

R~

Fig.1 Skewness S vs. Reynolds number Ra

In the high-wavenumber range, k >> k0, (13) .becomes (123). When k = (2/3) E(k)dk (10) From (5) (8) (9) and (10), after some manipulation, we obtain

Ra = (2/3)v~(ka/e) 2/3 E(k)dk (11)

By using the nonequilibrium statistical mechanics closure method [9], we have derived the following approximate formula for the energy spectrum in the high-wavenumber range

E(k) = e2/3k-5/3F(k/kd) (12a)

Fix) -- 1.19(1 + 5.3x2/3)exp(-5.4x 4/3) (125)

Let k0 be the characteristic wavenumber of the energy range, and a simple approximate formula of E(k), which is valid in both high-wavenumber and low-wavenumber ranges, is [7'10]

E(k) - e2/3k-5/3F(k/kd)/[1 4- (ko/k) s/3] (13)

Vol.10, No.l Qian Jian: Skewness Factor of Turbulent Velocity Derivative 15

IV . D ISCUSSION

Fig.1 shows that the skewness S approaches a constant S~ = -01515 as R~ ~ co. This theoretical result agrees with (4), and is compatible with the DNS result S = -0 .5 of Vincent and Meneguzzi [6]. Since the R~ of current DNS is of the order 102, it is doubtful whether the skewness S obtained by the DNS can represent the asymptotic value S~. Fig.1 shows that S almost attains the value S~ when R~ ~ 102~ and in fact S differs from S~ by less than 1% when R~ > 60. Therefore, the theoretical result of this paper confirms the following conjecture made by Vincent and Meneguzzi[61: the skewness S obtained by the DNS with R~ ~ 102 is nearly the same as the asymptotic value S~.

The issue, whether the skewness S and the flatness F approach a finite constant as R~ --~ c~, is essentially related to the problem of universality of small-scale structure for high-R~ turbulent flows. If the small-scale structure is universal, the statistical properties of small-scale motion such as the skewness S and the flatness F will be independent of the large-scale motion and the Reynolds number R~ so long as R~ is high enough. By the LN model and the fractal model, IS I and F approach infinity as R~ --* c~, so they are dependent of R~ no matter how high R~ is. According to the theoretical results of this paper and Ref.[7], S and F become universal constants independent of R~ when R~ is high enough, probably implying the universality of small-scale structure of high-R~ turbulent flOWS.

As mentioned above, when k