Crossover from high to low Reynolds number turbulence.

Abstract

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