Emergence of skewed non-Gaussian distributions of velocity increments in isotropic turbulence

Skewness and non-Gaussian behavior are essential features of the distribution of short-scale velocity increments in isotropic turbulent flows. Yet, although the skewness has been generally linked to time-reversal symmetry breaking and vortex stretching, the form of the asymmetric heavy tails remain elusive. Here we describe the emergence of both properties through an exactly solvable stochastic model with a scale hierarchy of energy transfer rates. From a statistical superposition of a local equilibrium distribution weighted by a background density, the increments distribution is given by a novel class of skewed heavy-tailed distributions, written as a generalization of the Meijer $G$-functions. Excellent agreement in the multiscale scenario is found with numerical data of systems with different sizes and Reynolds numbers. Remarkably, the single-scale limit provides poor fits to the background density, highlighting the central role of the multiscale mechanism. Our framework can be also applied to describe the challenging emergence of skewed distributions in complex systems.

Figure 1

Schematic mixture of Gaussians with nonzero mean yielding a skewed heavy-tailed distribution with zero mean. The uppermost black curve is the sum of the lower curves, which correspond to Gaussians with variances in the interval [0.0002, 0.05] and means as in Eq. (4) with $\mu =-2$, multiplied by weights arbitrarily chosen for convenience of illustration.

Figure 2

Conditional distribution of velocity increments for DNS turbulence data of a system of size ${1024}^3$ and Reynolds number ${\text{Re}}_{\lambda }\approx 433$ [41]. Distinct distributions obtained for the optimal window size $M=19$ have been rescaled and shifted to have the same mean (zero) and variance (unity). A nice agreement is observed with a Gaussian of zero mean and unity variance (dashed line).

Figure 3

(a) Distribution of velocity increments and (b) background density of local variances for the DNS data of Fig. 2 (circles). Excellent fits to the theoretical results (red lines), Eqs. (8) and (9), respectively, are shown for $N=4$ scales. For comparison, the case with $N=1$ (single scale) is also plotted (green lines), displaying much poorer fits. Inset in (b): nice agreement of the empirical distribution of velocity increments (circles) and the compounding integral (blue line), Eq. (3), of the Gaussian and the density $f\left({\epsilon }_N\right)$ obtained from the DNS data.

Figure 4

Relative error for the background distributions $f({\epsilon }_N$) obtained from fits with different hierarchy levels $N$. The optimal values of $N$ for both DNS datasets, i.e., $N=4$ for ${\text{Re}}_{\lambda }\approx 433$ (solid circles) and $N=5$ for ${\text{Re}}_{\lambda }\approx 600$ (crosses), correspond to the estimates provided by comparing the respective integral length scales to the scale $\ell =Mr$ over which ${\epsilon }_N$ is computed (see text).

Figure 5

(a) Distribution of velocity increments and (b) background density for a system with ${4096}^3$ points and Reynolds number ${\text{Re}}_{\lambda }\approx 600$ [45]. The nice fit of the DNS data (circles) to the theoretical model (red lines) occurs for $N=5$ scales. A poor fit is noticed in green lines for $N=1$. Inset: description as in the inset of Fig. 3.