Response maxima in time-modulated turbulence: Direct numerical simulations

The response of turbulent flow to time-modulated forcing is studied by direct numerical simulations of the Navier-Stokes equations. The large-scale forcing is modulated via periodic energy input variations at frequency $\omega$. The response is maximal for frequencies in the range of the inverse of the large eddy turnover time, confirming the mean-field predictions of von der Heydt, Grossmann and Lohse (Phys. Rev. E, 67 (2003) 046308). In accordance with the theory the response maximum shows only a small dependence on the Reynolds number. At sufficiently high frequencies the amplitude of the kinetic energy response decreases as $1/\omega$. For frequencies beyond the range of maximal response, a significant change in the phase-shift relative to the time-modulated forcing is observed. For large $\omega$ the phase shift approaches roughly $90^{\circ}$ for the total energy and $180^{\circ}$ for the energy dissipation rate.