Crossover from High to Low Reynolds Number Turbulence

The Taylor-Reynolds and Reynolds number (${\mathrm{Re}}_{\lambda }$ and Re) dependence of the dimensionless energy dissipation rate $c_{\epsilon }=\frac{\epsilon L}{u_{1,\mathrm{rms}}^3}$ is derived for statistically stationary isotropic turbulence, employing the results of a variable range mean field theory. Here $\epsilon$ is the energy dissipation rate, $L$ the (fixed) outer length scale, and $u_{1,\mathrm{rms}}$ a rms velocity component. Results for $c_{\epsilon }\left({\mathrm{Re}}_{\lambda }\right)$ and also for ${\mathrm{Re}}_{\lambda }\left(\mathrm{Re}\right)$ are in good agreement with experiment. Using the Re dependence of $c_{\epsilon }$ we account for the time dependence of the mean vorticity $\omega \mathrm{}\left(t\right)\mathrm{}$ for decaying isotropic turbulence. The lifetime of decaying turbulence, depending on the initial ${\mathrm{Re}}_{\lambda ,0\mathrm{}}$ is predicted to saturate at $\frac{0.18L^2}{\nu }\propto {\mathrm{Re}}_{\lambda ,0\mathrm{}}^2$ ($\nu$ the viscosity) for large ${\mathrm{Re}}_{\lambda ,0\mathrm{}}$.