Spectral imbalance and the normalized dissipation rate of turbulence

TL;DR

This paper investigates the normalized turbulent dissipation rate in different turbulence regimes, revealing spectral disequilibrium effects and providing analytical models that match numerical simulations.

Contribution

It introduces a spectral imbalance explanation for variations in $C_\epsilon$ and develops simple models that align with simulation results.

Findings

Difference in $C_\epsilon$ between decaying and forced turbulence

Spectral disequilibrium influences $C_\epsilon$ at high Reynolds numbers

Analytical models successfully reproduce numerical observations

Abstract

The normalized turbulent dissipation rate $C_\epsilon$ is studied in decaying and forced turbulence by direct numerical simulations, large-eddy simulations, and closure calculations. A large difference in the values of $C_\epsilon$ is observed for the two types of turbulence. This difference is found at moderate Reynolds number, and it is shown that it persists at high Reynolds number, where the value of $C_\epsilon$ becomes independent of the Reynolds number, but is still not unique. This difference can be explained by the influence of the nonlinear cascade time that introduces a spectral disequilibrium for statistically nonstationary turbulence. Phenomenological analysis yields simple analytical models that satisfactorily reproduce the numerical results. These simple spectral models also reproduce and explain the increase of $C_\epsilon$ at low Reynolds number that is observed in the simulations.