Fig. 4: Kurtosis.
The kurtoses, (a) \({{{{{{{{\mathcal{K}}}}}}}}}_{2}\) and (b) \({{{{{{{\mathcal{K}}}}}}}}\) as a function of the scale r for De = 1, 3 and 9. The red dashed line is at ordinate equal to 3. We also show in (a) a line of slope − 1.6. The scaling exponent of kurtosis, obtained from fitting the data in the gray-shaded region, are: − 1.6 ± 0.3, − 1.6 ± 0.1, and − 1.6 ± 0.1, for De = 1, 3, and 9, respectively. This demonstrates both the non-Gaussian nature of the PDFs and the universality of the exponents with respect to De. The kurtosis of \(\delta u,{{{{{{{\mathcal{K}}}}}}}}\), grows slower as r → 0 and may not be universal. To compare, we also plot, in (b), the corresponding result for Newtonian HIT. Both the kurtoses \({{{{{{{\mathcal{K}}}}}}}},{{{{{{{{\mathcal{K}}}}}}}}}_{2}\to 3\) (shown in dotted-red line) as r → L. This indicates that at large separations the statistics of velocity difference are close to a Gaussian.
