A new method of identifying self-similarity in isotropic turbulence

TL;DR

The paper introduces a novel method for analyzing self-similarity in isotropic turbulence by plotting the ratio of structure functions, revealing that the second-order exponent decreases with Reynolds number, supporting finite-viscosity corrections.

Contribution

It proposes a new technique for analyzing structure functions in turbulence, differing from ESS, and provides new insights into the behavior of the second-order exponent at high Reynolds numbers.

Findings

The new method shows $ ext{zeta}_2$ approaches 0.67 as Reynolds number increases.

The method supports finite-viscosity corrections to Kolmogorov's theory.

Contradicts previous ESS-based results regarding $ ext{zeta}_2$.

Abstract

In order to analyse results for structure functions, $S_n(r)$, we propose plotting the ratio $|S_n(r)/S_3(r)|$ against the separation $r$. This method differs from the extended self-similarity (ESS) technique, which plots $S_n(r)$ against $S_3(r)$, where $S_3(r) \sim r$. Using this method in conjunction with pseudospectral evaluation of structure functions, for the particular case of $S_2(r)$ we obtain the new result that the exponent $\zeta_2$ decreases as the Taylor-Reynolds number increases, with $\zeta_2 \to 0.67 \pm 0.02$ as $R_\lambda \to \infty$. This supports the idea of finite-viscosity corrections to the K41 prediction for $S_2$, and is the opposite of the result obtained by ESS.