Energy transfer and dissipation in forced isotropic turbulence

TL;DR

This paper develops a theoretical model for how the energy dissipation rate in forced isotropic turbulence depends on Reynolds number, supported by DNS data showing a power-law decay and asymptotic behavior.

Contribution

It introduces a new Reynolds number-dependent model for the dissipation rate based on the Kármán-Howarth equation, validated with extensive DNS simulations.

Findings

Dissipation rate decreases as R_L^{-1} with increasing Reynolds number.

The decay of the dimensionless dissipation follows a power law with exponent -1.

Asymptotic dissipation rate at infinite Reynolds number is approximately 0.468.

Abstract

A model for the Reynolds number dependence of the dimensionless dissipation rate $C_{\varepsilon}$ was derived from the dimensionless K\'{a}rm\'{a}n-Howarth equation, resulting in $C_{\varepsilon}=C_{\varepsilon, \infty} + C/R_L + O(1/R_L^2)$, where $R_L$ is the integral scale Reynolds number. The coefficients $C$ and $C_{\varepsilon,\infty}$ arise from asymptotic expansions of the dimensionless second- and third-order structure functions. This theoretical work was supplemented by direct numerical simulations (DNSs) of forced isotropic turbulence for integral scale Reynolds numbers up to $R_L=5875$ ($R_\lambda=435$), which were used to establish that the decay of dimensionless dissipation with increasing Reynolds number took the form of a power law $R_L^n$ with exponent value $n = -1.000\pm 0.009$, and that this decay of $C_{\varepsilon}$ was actually due to the increase in the Taylor surrogate $U^3/L$. The model equation was fitted to data from the DNS which resulted in the value $C=18.9\pm 1.3$ and in an asymptotic value for $C_\varepsilon$ in the infinite Reynolds number limit of $C_{\varepsilon,\infty} = 0.468 \pm 0.006$.