Z. Phys. B 101, 157—159 (1996)
Rapid note
Statistical dependency of eddies of different sizes in turbulence J. Peinke1, R. Friedrich2, F. Chilla` 3, B. Chabaud4, A. Naert5 1Experimentalphysik II, Universita¨t Bayreuth, D-95440 Bayreuth, Germany (Fax: 0921/552621, Phone: 0921/553182, e-mail:
[email protected]) 2Institut fu¨r Theoretische Physik und Synergetik, Universita¨t Stuttgart, D-70550 Stuttgart, Germany (Phone: 0711/6854981, e-mail:
[email protected]) 3Ecole Normale Superieure de Lyon, F-69364 Lyon, France 4CRTBT-CRNS, BP 166, F-38042 Grenoble, France 5RIEC, Tohoku University, 2-1-1 Katahira Aobaku, Sendai 980-77, Japan Received: 3 November 1995 / Revised version: 4 April 1996
Abstract. Experimental data of a fluid turbulence is analysed with respect to the statistical dependency of eddies of different length scales. The joint probability distributions of velocity fluctuations of two different length scales ¸ , 1 ¸ are evaluated. We quantify statistical dependency by 2 calculating the correlation function as well as a suitably defined Kullback information.
features. First, the main axis of the distribution changes as a function of ¸ /¸ . Second, they are definitely non-gaus2 1 sian, as can be illustrated by considering the conditional probability distribution p(v¸ Dv¸ ) (c.f. inset of Fig. 2a, b, 2 1 where v¸ "30). Furthermore, the variance of p(v¸ D v¸ ) 1 2 1 changes with v¸ , as it will be discussed later on. 1 A measure for statistical dependency is the correlation function
PACS: 02.50; 47.25
Sv¸ , v¸ T 1 2 C(¸ , ¸ )" 1 2 JSv2 T Sv2 T ¸ 1
The well-known picture of a fully developed turbulent fluid motion [1] is based on the notion of a cascade providing scaling laws for the moments of velocity fluctuations v (x) L v (x)"u(x#¸)!u(x) (1) L where u(x) denotes the velocity at space point x and ¸ is a choosen length scale. Larger eddies decay into smaller eddies eventually leading to the development of the inertial range. Up to now mainly two-point correlations based on (1) have been investigated. The purpose of the present work is to consider the relationship between eddies of different sizes to gain insights into the decay mechanisms leading to the development of the cascade. Our approach is a first attempt to analyse four point correlations. We have analysed the turbulent velocity obtained by the experiment of B. Chabaud et al. described in [2]. Figure 1 shows the third moment v3 as a function of ¸/g, L where g is the Kolmogorov dissipation length. We see that the inertial range lies in the vicinity of 100 g. Next, data sets, consisting of 107 points, were evaluated with respect to the joint probability distribution p(v¸ , v¸ ), where 1
2
v¸ (x)"u(x#¸ /2)!u(x!¸ /2), i"1, 2 (2) i i i This probability distribution measures the simultaneous occurrence of two eddies of sizes ¸ , ¸ characterized by 1 2 the velocities v¸ , v¸ . The scalings ¸ , ¸ are varied separ1 2 1 2 ately; ¸ '¸ for convention. Typical distributions are 1 2 exhibited in Fig. 2a, b. Let us summarize several main
¸
(3)
2
The brackets denote averages. Since the normalized correlation functions C(¸ , ¸ ) are dimensionless quantities, 1 2 dimensional arguments in relationship with the Kolmogorov theory of 41 (K41) c.f. [1] leads us to the ansatz that in the inertial range these two quantities depend only on the ratio of ¸ /¸ . 2 1 Applying naive dimensional arguments commonly used in the statistical treatment of two-point correlations, where one assumes the velocity increments (v )n to scale L according to (v )n+¸n@3, one would expect this correlaL tion function to be constant. Note that after applying refined scaling arguments (like Kolmogorov 62 c.f. [1]), (v )n+¸n/3#kn, a weak power-law decay of the correlation L function could be expected (in fact k/9, k+0.4 for Kolmogorov’s lognormal ansatz). Figure 3 shows the correlation function for the experimental data as a function of ¸ /¸ in two different repres2 1 entations. Evidently the correlation function decays. However, a powerlaw does not fit the data in a satisfactory way. Let us consider the decay of the C(¸ , ¸ ) in more 1 2 detail. To this end we assume that the probability distribution has the following form p(v¸ , v¸ )"p(v¸ !c(¸ , ¸ ) v¸ ) pJ (v¸ ). 1 2 2 1 2 1 1 Then we can evaluate the correlation
(4)
Sv¸ , v¸ T"c(¸ , ¸ ) Sv2¸ T. 1 2 2 2 1 For the third moment we have
(5)
Sv3¸ T"c3(¸ , ¸ ) Sv3¸ T 2 1 2 1
(6)
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Fig. 1. Third moment Sv3T as a function of ¸/g (g denotes the Kolmogorov dissipation Llength). For orientation the solid line shows the power one scaling
Fig. 3. a Log-lin-plot and b log-log-plot of the normalized correlation function C(¸ , ¸ ) as a function of ¸ /¸ for ¸ "78g, 157g, 2 up to bottom) 2 1 1 630g, 2505g, 7500g1 (from
provided that the skewness of the conditional probability distribution p(v¸ !c(¸ , ¸ ) v¸ ) vanishes. Using Kol2 2 1 1 mogorov’s equation Sv3¸ T"!4 e ¸ , (7) 1 5 i which holds in the inertial range (here e denotes the local energy transfer rate), we obtain
A B
¸ 1@3 (8) c(¸ , ¸ )" 2 2 1 ¸ 1 If now the correlation functions is evaluated using (5) we again obtain C(¸ , ¸ )"const for the nonintermittent 1 2 scaling behavior of Sv2¸ T. Therefore the decay of the 1 correlation function indicates a more complicated behaviour of the probability distribution as discussed above. It seems to be important for the decay mechanism of the cascade to understand and quantify deviations of the above mentioned K41 picture. The question whether two velocities v¸ and v¸ are 1 2 statistically dependent can also be quantified by considering the following Kullback information [3] (also known as mutual information) p(v¸ , v¸ ) 1 2 K(¸ , ¸ )": dv¸ : dv¸ p(v¸ , v¸ ) ln , 1 2 1 2 1 2 pJ (v¸ ) pJ (v¸ ) Fig. 2a, b. Contour plots (logarithmic scale) of the probability distribution p(v(¸ ), v(¸ )) from data of [2] for R "690. a ¸ "834g, 2 j the Kolmogorov 1 ¸ "818g b1 ¸ "834g, ¸ "132g (g denotes 2 1 2 units: 50 corresponds to a local dissipation length). The axis Reynold’s number (at the detector) of 47.75. The deviation from Gaussian behavior is illustrated by the conditional probabilities p(v¸ D v¸ ) for v¸ "30 shown in the insets 2
1
1
1
(9)
2
where the distribution function pJ (v¸ ) denotes the prob1 ability distribution of one velocity component v¸ , i"1, 2. 1 This function approaches zero if the two velocity components become statistically independent since then p(v¸ , v¸ ) P pJ (v¸ ) pJ (v¸ ). 1
2
1
2
(10)
159
Furthermore, there appears a singularity for ¸ P ¸ 2 1 since then the probability distribution p(v¸ , v¸ ) is propor1 2 tional to a delta function d(v¸ !v¸ ). 1 2 For the special case of two-dimensional Gaussian distributions the two quantities C(¸ , ¸ ) and K(¸ , ¸ ) are 1 2 1 2 related according to: K(¸ , ¸ )"!1 ln [1!C(¸ , ¸ )2] (11) 2 1 2 1 2 One could expect that for nongaussian statistics the correlation function is an incomplete measure of statistical dependence or independence as it is the case for our experimental data. However, it turns out that formula (11) holds with minor corrections also for our data analysed here. The Kullback information is shown in Fig. 4 for various values of ¸ as a function of ¸ /¸ . As a main result both 1 2 1 quantities decrease over a wide range of ¸ /¸ . Investigat2 1 ing these decays in a double logrithmic plot no evidence for an algebraic decay is found. For larger ¸ values 1 a clear transition to significantly faster decays of the correlation and the Kullback information was found, as it is expected from uncorrelated data. In order to quantify the conditional probability distribution in a proper way we have evaluated the mean value and the variance as function of v¸ (see Fig. 5a, b). A linear 1 fit, Sv¸ D¸ T"c v¸ , yields a dependency of c on (¸ /¸ ) 2 1 c 1 c 2 1 which is compared with the behaviour of (8) in Fig. 6. The variance Sv2¸ D¸ T!Sv¸ D¸ T2 established in Fig. 5b clearly 2 1 2 1 shows increasing dependency on v¸ with decreasing value 1 of ¸ /¸ . This is obviously a signature of increasing inter2 1 mittency. In conclusion we have focused on an evaluation of four-point velocity correlations, based on correlation function, Kullback information and joint probability distribution. The obtained experimental results should settle as a testing ground for theoretical approaches for the turbulent cascade. It seems to be of great importance to compare these results with the evaluation of the corresponding quantities in numerical simulations, like REWA [4]. Furthermore the corresponding statistical dependence of the energy as well as the energy dissipation will be of interest.
Fig. 5. a mean value and b variance of the conditional probability distributions p(v¸ D v¸ ) of Fig. 2 2
1
Fig. 6. Slope c of a linear fit (from !20 to 20) to the mean value of the conditionalc probability distributions see Fig. 5a. For comparison the curve of (8) is shown
We want to acknowledge helpful discussions with S. Grossmann, A. Reeh and D. Lohse and J.P. acknowledges the cooperation with B. Cartaing and B. Hebral. He also acknowledges the financial support of the Deutsche Forschungsgemeinschaft.
References
Fig. 4. Log-in-plot of the Kullback information K(¸ , ¸ ) as a func1 (from 2 tion of ¸ /¸ for ¸ "78g, 157g, 630g, 2505g, 7500g up to 1 bottom) 2 1
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