On the Dissipation Rate Coefficient in Homogeneous Isotropic Decaying and Forced Turbulence

TL;DR

This paper derives exact expressions for the dissipation rate coefficient in homogeneous isotropic turbulence, showing its dependence on initial conditions and forcing mechanisms, highlighting non-universality in both decaying and forced cases.

Contribution

The paper provides new exact formulas for the dissipation rate coefficient in isotropic turbulence, revealing its dependence on decay exponent and forcing mechanisms, indicating non-universality.

Findings

Dependence of $C_{\epsilon}$ on decay exponent $n$ in decaying turbulence.

Non-universality of $C_{\epsilon}$ in forced turbulence due to forcing mechanisms.

Lower and less scattered $C_{\epsilon}$ values expected with similar forcing algorithms.

Abstract

The normalized non-dimensional von K\'arm\'an-Howarth equation for isotropic homogeneous decaying and forced steady turbulence is integrated to obtain expressions for the dissipation rate coefficient $C_{\epsilon}=(L \epsilon)/< u^2 >^{3/2}$, where $L$ denotes the longitudinal integral length scale, $\epsilon$ the mean dissipation rate and $< u^2 >$ the mean variance of the longitudinal velocity fluctuations. For decaying turbulence the final exact expressions for $C_{\epsilon}$ for the low and high Reynolds number limit depend on the decay exponent $n$, which is known to depend on the initial velocity structure at the turbulence production. The dependence on $n$ leads to a non-universal coefficient. The expressions for the steady forced case depend on the forcing mechanism and thus are not universal either. Nonetheless, a lower value and considerably less scatter as compared to the decaying turbulence case should be expected when similiar forcing algorithms are employed.