Sustaining bursts for linearized Navier-Stokes flows in minimal channels

We investigate the forcing conditions for linearized Navier-Stokes flows to sustain turbulence-like bursts of wall-normal velocities in minimal channels. The investigation is divided into two parts. First, we discuss the forces that are effective in sustaining periodic bursts. The time dependence of a force is modeled as a Dirichlet kernel modulated by a sinusoid. This function preserves the properties of waves or wave packets as in the resolvent analysis, while additionally exhibits long-term periodicity of wave envelopes. For each bursting period prescribed, the most effective wall-normal variation of a force is determined by maximizing the time-averaged or peak value of the bursting energy under a unit-norm force. Second, the effective forces corresponding to various bursting periods are superimposed to mimic the stochastic behavior of bursts in turbulence. The results show that some important properties of bursts in turbulence, including the invariant probability distribution and the short-term mean displacements in the intensity-inclination phase space, can be qualitatively reproduced by superimposing the forces with periods approximately in the range from one to four eddy turnover times. This study can be regarded as a linearized surrogate model of bursts in turbulence that are sustained by essentially minimal forcing and has the potential to shed light on the self-sustaining mechanism of real turbulence.