Linear Problem on Vibrator on a Flat Plate in Subsonic Boundary Layer

Summary

The problem on subsonic flow past a flat plate with a vibrator placed at the distance L ^* from the leading edge is considered. The vibrator comes into operation in an undisturbed boundary layer at the initial moment t ^* = 0. The plate and vibrator are supposed to be insulated, the scales of vibrator and the frequency are such that for Reynolds number Re _L1 the flow is governed by the boundary layer equations with the self-induced pressure included. The amplitude of the vibrator and corresponding disturbances are supposed to be small enough for linearizing governing equations. In the approximation under consideration we do not take into account the boundary layer thickening. The main attention is payed to calculation of the pressure distribution over vibrator as well as upstream and downstream of it. It is shown that the amplitude of the pressure oscillations decreases upstream. As for the region at “moderate” distances downstream the vibrator the pressure decreases only if the frequency ω _o is less than the threshold frequency ω _c* which is fixed by the lower branch of the neutral stability curve. The pressure amplitude is constant if ω _c = ω _c* and grows for ω _o > ω _c* . At the same distances the disturbances take more simple form of the single Tollmien-Schlichting wave in the limit as t → ∞ , the amplitude depending on the vibrator frequency and shape.