Energy transfer and dissipation in forced isotropic turbulence

A model for the Reynolds number dependence of the dimensionless dissipation rate $C_{\epsilon }$ 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 $C_{\epsilon ,\mathrm{\infty }}$ 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 $C_{\epsilon }$ was actually due to the increase in the Taylor surrogate $U^3\mathrm{/}L$. 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 $C_{\epsilon }$ in the infinite Reynolds number limit of $C_{\epsilon ,\mathrm{\infty }}=0.468\pm 0.006$.