On the existence and scaling of structure functions in turbulence according to the data

TL;DR

The paper investigates the existence and scaling of turbulence structure functions by constructing a data-informed 1D stochastic model that matches a Kolmogorov-type spectrum and observed skewness. It shows that the third-order scaling $S_3(r) \sim r$ can be maintained even with intermittency, but higher-order moments diverge above a threshold near $p \approx 3.3$ due to heavy-tailed distributions, signaling incomplete similarity and Reynolds-number dependence. The authors provide a computational framework using Meyer wavelets and reversible RNGs to generate velocity fields and analyze $S_p(r)$, offering insights into when Kolmogorov's scaling remains valid and when intermittency alters high-order statistics. These results inform theoretical turbulence models and suggest that intermittency corrections may be sufficient at low orders but lead to fundamental limitations for high-order moments in non-Gaussian flows.

Abstract

We sample a velocity field that has an inertial spectrum and a skewness that matches experimental data. In particular, we compute a self-consistent correction to the Kolmogorov exponent and find that for our model it is zero. We find that the higher order structure functions diverge for orders larger than a certain threshold, as theorized in some recent work. The significance of the results for the statistical theory of homogeneous turbulence is reviewed.

Figures (5)

• Figure 1: Non-Gaussian probability density functions, obtained by the algorithm in the text
• Figure 2: Skewness as a function of separate $r/M$, from Stewart (stewart); $M$ is a reference length in that paper. Reproduced with permission.
• Figure 3: Exponent $\xi_3$ of the structure function of order 3 versus $\alpha$ in the relation $E(k)\sim k^{\frac{-5}{3}+\alpha}$. The intermittency correction should be where $\xi_3$=1.
• Figure 4: Cumulative distribution function of $S_1=|\Delta u|$ for a Gaussian and our non-Gaussian fields at $r/M=1$. M is again a reference length in the tabulated data.
• Figure 5: Cumulative distribution functions of $S_1$ obtained from a Gaussian field (Top), and a non-Gaussian field, at several separations.