Analysis of Lundgren's matched asymptotic expansion approach to the K\'arm\'an-Howarth equation using the EDQNM turbulence closure

TL;DR

This study compares EDQNM turbulence closure results with Lundgren's asymptotic approach, revealing that true inertial range equilibrium is never achieved, but approaches equilibrium near the Taylor length as Reynolds number increases.

Contribution

It demonstrates that EDQNM and Lundgren's asymptotic expansion agree in predicting the non-attainment of inertial range equilibrium in turbulence.

Findings

Inertial range equilibrium is not reached at any Reynolds number.

Equilibrium approaches near the Taylor length as Reynolds number increases.

Departure from equilibrium occurs at scales larger than the Taylor length.

Abstract

In this paper we investigate whether the features of the non-equilibrium cascade, which have been identified in recent studies using high-fidelity tools, can be captured in the case of the classical dissipation scaling by turbulence closures based on the statistical description of freely decaying isotropic turbulence. Numerical results obtained using the EDQNM model over a very large range of Reynolds numbers (from $Re_{\lambda}=50$ up to $Re_{\lambda}=10^6$) are analyzed to perform an extensive investigation of the scaling region identified as inertial range in Kolmogorov's theory. It is observed that EDQNM results are in agreement with the results of Lundgren's matched asymptotic expansion approach to the Karman-Howarth equation. Both predict that the Kolmogorov inertial range equilibrium is never obtained irrespective of Reynolds number. Equilibrium is reached in the vicinity of the Taylor length $\lambda$ (which depends on viscosity) as Reynolds number tends to infinity and there is a gradual departure from equilibrium as the length scale moves away from $\lambda$, in particular towards scales larger than $\lambda$ all the way to the integral length-scale.