How turbulence is generated from zero—and the interscale dynamics in stationary state

A large database of direct numerical simulations of isotropic turbulence, with Taylor-Reynolds numbers ranging from $R_\lambda \approx 1$ to $300$ and grid sizes of up to $2048^3$, is analyzed to understand how broadband turbulence develops from the forcing at a few low wavenumbers. Different forcing schemes at large scales are used. The energy growth in various wavenumbers follows power laws to very good accuracy. The energy in the first few wavenumbers close to the forcing band grows simultaneously with forcing; but that in higher
wavenumbers occurs after a time lag of the order of a large-eddy time scale, qualitatively consistent with the cascade picture. However, it appears that part of the energy leaks to dissipative wavenumbers as an instantaneous reaction to forcing. Once in a steady sate, while Gaussian statistics are observed for temporal fluctuations in the inertial range consistent with the fluctuations in forcing, increasingly skewed probability density functions emerge at higher wavenumbers. In particular, fluctuations from the mean in the far-dissipation range, defined here as wavenumbers larger than twice the mean Kolmogorov wavenumber, are very large.
Different transfer models to predict the observed behavior are discussed.