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

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.