Pattern prediction by linear analysis of turbulent flow with drag reduction by wall oscillation

A turbulent flow past a transverse-oscillating wall is considered. The oscillation parameters correspond to the regime where drag reduction is observed. Streak spacing and streak angle obtained from the generalized optimal perturbation approach are compared with results from direct numerical simulations. Other flow features of the generalized optimal perturbation are compared with conditionally-averaged data extracted from numerical simulations. The generalized optimal perturbation at a given instant in time is found to consist of an infinitely long structure at a certain angle to the main flow direction. This angle varies slowly with time for half a period, and then suddenly jumps to a different value, changing both sign and magnitude. The angle variation is shown to be slow, because there is a short time interval in the oscillation period when a small perturbation of a certain angle grows strongly and then remains dominant for almost the entire half-period. The transient growth mechanism of the generalized optimal perturbation is found to be a combination of the On mechanism due to the cross-flow shear, acting at the initial stage, followed by the lift-up mechanism of the velocity component directed along the structure by the wall-normal motion also oriented in the same direction.