Crossover from High to Low Reynolds Number Turbulence

The Taylor-Reynolds and Reynolds number (Re_λ and Re) dependence of the dimensionless energy dissipation rate c_ε=εLu1,rms³ is derived for statistically stationary isotropic turbulence, employing the results of a variable range mean field theory. Here ε is the energy dissipation rate, L the (fixed) outer length scale, and u1,rms a rms velocity component. Results for c_ε(Re_λ) and also for Re_λ(Re) are in good agreement with experiment. Using the Re dependence of c_ε we account for the time dependence of the mean vorticity ω(t) for decaying isotropic turbulence. The lifetime of decaying turbulence, depending on the initial Re_λ,0 is predicted to saturate at 0.18L²ν∝Re_λ,0² (ν the viscosity) for large Re_λ,0.