Crossover from high to low Reynolds number turbulence

The Taylor-Reynolds and Reynolds number ($R{e}_{\lambda }$ and $Re$) dependence of the dimensionless energy dissipation rate $\text{ceps}=\text{eps}L\mathrm{/}\breve{1}rm{s}^3$ is derived for statistically stationary isotropic turbulence, employing the results of a variable range mean field theory. Here, $\text{eps}$ is the energy dissipation rate, $L$ is the (fixed) outer length scale, and $\breve{1}rms$ a rms velocity component. Our fit-parameter free results for $\text{ceps}(R{e}_{\lambda })$ and also for $R{e}_{\lambda }(Re)$ are in good agreement with experimental data. Using the $Re$-dependence of $\text{ceps}$ we account for the time dependence of the mean vorticity $\omega (t)$ for decaying isotropic turbulence, which was recently experimentally examined [M.\ Smith, R.\ J.\ Donelly, N.\ Goldenfeld, and W.
F.\ Vinen, Phys.\ Rev.\ Lett.\ 71, 2583 (1993)]. The lifetime of decaying turbulence, depending on the initial $R{e}_{\lambda ,0}$, is predicted and found to saturate at $0.18L^2\mathrm{/}\nu \propto R{e}_{\lambda ,0}^2$ ($\nu$ is the viscosity) for large $R{e}_{\lambda ,0}$.