Uncovering a Log-Normal Cascade Process in High Reynolds Number Turbulence from Wavelet Analysis

We present a generalization of the Castaing et al. [1] approach of velocity intermittency using the wavelet transform (WT). This description consists in looking for a multiplicative cascade process directly on the velocity field assuming that the probability density function (pdf) of the modulus maxima of the WT (WTMM) at a given scale a, P a (T), can be expressed as a weighted sum of dilated pdf’s at a larger scale a’ > a: 
 
$$ {{P}_{a}}(T) = \int {{{G}_{{aa'}}}} (u){{P}_{{a'}}}({{e}^{{ - u}}}T){{e}^{{ - u}}}du,\quad fora' > a. $$ 
 
(1)

We present a generalization of the Castaing et al. [1] approach of velocity intermittency using the wavelet transform (WT) . This description consists in looking for a multiplicative cascade process directly on the velocity field assuming that the probability density function (pdf) of the modulus maxima of the WT (WTMM) at a given scale a, Pa(T), can be expressed as a weighted sum of dilated pdf's at a larger scale a' > a: (1) The reader is referred to Ref. [2] for the computation of the WT of the velocity field and the restriction of the analysis to the modulus maxima of the WT (WTMM method). Our numerical method for estimating the Fourier transform Gaa' of the kernel Gaa' is described in Refs. [3,4]. This wavelet-based method is applied to a turbulent velocity signal recorded in the Modane wind tunnel and kindly provided by Y. Gagne and Y. Malecot. The Taylor scale based Reynolds number is R>. c::: 2000 and the sample is 2.5 · 10 8 points long, with a resolution of roughly 3ry (where ry is the Kolmogorov scale), corresponding to 25000 integral scales. We then propose a two-point statistical analysis based on space-scale correlations [5].

Experimental results of the one-point statistical approach
The computation of Gaa' for various pairs of inertial scales a < a' reveals the existence of a function s( a, a') and of a single kernel G such that Gaa' = (js(a,a') (i.e. formally Gaa' = GIZls(a,a')). According to the definitions given in Ref. [1], this means that the underlying cascade process is self-similar  The first three cumulants of Gaa' vs the sample length. Turbulent velocity signal for a = 770ry and a' = 1540ry ( o and dashed line), log-normal numerical process with m= 0.32 and rr 2 = 0.03 ( • and solid line) and log-Poisson numerical process with A = 2, 6 = 0.89 and 1 = -0.082 (triangles and dots) for the two corresponding scales a = 2 8 and a' = 2 9 • The analyzing wavelet is a first order compactly-supported wavelet [4]. and that s( a, a') accounts for the number of cascade steps from scale a' to scale a. Such a cascade is said scale-similar (or scale-invariant) ifs( a, a') = ln( a'/ a). In the present case, s( a, a') turns out to be very well fitted by the functional form s( a, a') = ( a-!3 -a'-f3)j (3, where (3 ~ 0.095 quantifies the departure from scale-similarity (scale-invariance is restored for (3 ~ 0) [6]. Thus, the cascade process is self-similar but not scale-invariant.
To analyze precisely the shape of G, we use the Taylor series expansion: G(p) = exp (2::::~1 q (ift) , where the (real valued) coefficients Ck are the cumulants of G. Figure 1 shows the first three cumulants Ck = s(a, a') Ck (for a given pair of inertial scales a < a') for the turbulence data and for both a log-normal (mean m and variance 0" 2 ) and a log-Poisson (with the >., 8 and 1 parameters consistent with the ones proposed by She and Leveque [7]) synthetic numerical processes. Even though both numerical processes perfectly fit the first two cumulants, the log-Poisson model yields a third order cumulant that is more than one order of magnitude higher than the experimental one, whereas the very small (theoretically zero) third order cumulant of the log-normal numerical process still remains within the error bars of the experimental one. We thus exclude log-Poisson statistics as proposed in Ref. [7] and conclude that the statistics underlying the velocity fluctuations are log-normal, as soon as R).. > 1000.
Testing the convolution formula (1) on the WTMM pdf's using a Gaussian kernel yields results in very good agreement with the log-normal cascade picture [4]. A second test oflog-normality lies in the scaling exponents (q of the velocity structure functions. As shown in Fig. 2(a), the experimental spectrum, obtained using extended self-similarity (ESS) [8] and extended to negative values of q thanks to the WTMM method, remarkably coincides with the quadratic log-normal prediction. The log-Poisson prediction [7] provides a good approximation of (q for q E [ -6, 6). However, plotting the deviation of the (g's from the Kolmogorov (1941) linear q/3 spectrum (Fig. 2(b)), reveals a systematic departure ofthe log-Poisson prediction from the experimental spectrum, whereas the log-normal model still perfectly fits the experimental data.

Space-scale correlations of the WT: a two-point statistical analysis
To get a deeper insight into the nature of the statistics, we study the spacescale correlations of the "magnitude" w( x, a) of the velocity field at point x and scale a [5]. w(x, a) is defined as the logarithm of a local average over a size a of the wavelet coefficients at scale a around the point x.
Here, we use the velocity increments to compute w(x, a). Figure 3 shows various correlation functions C(~x,a,a') = w(x,a)w(x + ~x,a'), where the overline stands for ensemble average and w for the centered process