Revisiting the momentary stability analysis of the Stokes boundary layer

Abstract

The stability of the boundary layer generated by the harmonic oscillations of a plate in its own plane in a fluid otherwise at rest (Stokes boundary layer) is investigated by considering the time development of perturbations of small amplitude and introducing a momentary criterion of instability. The temporal scale of the perturbations is assumed to be much smaller than the period of the plate oscillations because transition takes place at values of the Reynolds number much larger than one. The results confirm that the Stokes boundary layer is linearly unstable when the Reynolds number is larger than a first critical value equal to which is almost coincident with that determined by Von Kerczek & Davis (J. Fluid Mech. vol. 62 issue 4 1974 753-773) for a Stokes boundary layer in a fluid domain which is bounded by a second fixed plate at a distance from the oscillating one. For values of the Reynolds number close to the instability is restricted to phases close to the inversion of the plate velocity. When the Reynolds number becomes larger than a second threshold value close to the instability rapidly pervades a large part of the cycle. However only when the Reynolds number becomes larger than a third critical value equal to is the instability present during the whole cycle. Heuristically these three critical values of the Reynolds number can be associated with the transition from the laminar regime to the disturbed laminar the intermittently turbulent and fully turbulent regimes.