Crossover from high to low Reynolds number turbulence

Abstract

The Taylor-Reynolds and Reynolds number ($Re_\lambda$ and $Re$) dependence of the dimensionless energy dissipation rate $\ceps =\eps L / \u1rms^3$ is derived for statistically stationary isotropic turbulence, employing the results of a variable range mean field theory. Here, $\eps$ is the energy dissipation rate, $L$ is the (fixed) outer length scale, and $\u1rms$ a rms velocity component. Our fit-parameter free results for $\ceps (Re_\lambda)$ and also for $Re_\lambda (Re)$ are in good agreement with experimental data. Using the $Re$-dependence of $\ceps$ 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 $Re_{\lambda ,0}$, is predicted and found to saturate at $0.18 L^2/\nu \propto Re_{\lambda ,0}^2$ ($\nu$ is the viscosity) for large $Re_{\lambda ,0}$.