Preventing transition to turbulence using streamwise traveling waves: Theoretical analysis

We assess the efficacy of a zero-net-mass-flux blowing and suction in the form of streamwise traveling waves for controlling the onset of turbulence in a channel flow. For small amplitude actuation along the walls, we utilize perturbation analysis to determine modifications in the base flow and to examine the resulting net power balance. Sensitivity of the velocity fluctuations around this base flow is then employed as a basis for selection of traveling wave parameters. Our simulation-free approach reveals that, relative to the uncontrolled flow, the velocity fluctuations around the upstream traveling waves at best exhibit similar sensitivity to background disturbances. In contrast, the downstream traveling waves with properly designed speed and frequency can significantly reduce sensitivity which makes them well-suited for preventing transition. These theoretical predictions are facilitated by perturbation analysis (in the wave amplitude) of the linearized Navier-Stokes equations, and they are verified using full-scale simulations of the nonlinear flow dynamics in companion paper, [1].