Wall imperfections as a triggering mechanism for Stokes-layer transition

The boundary layer generated by the harmonic oscillations of a wavy wall in a fluid otherwise at rest is studied. First the wall waviness is assumed to be of small amplitude and large values of the Reynolds number are considered. The results obtained by means of a linear analysis, where the time variable appears only as a parameter, show that resonance may occur. Indeed it is found that when the Reynolds number is larger than a critical value, an instant within the decelerating part of the cycle exists such that a waviness of infinitesimal amplitude induces unbounded perturbations of the flow in the Stokes layer. The passage through resonance is then studied by means of a multiple-timescale approach, taking into account the damping effect of local acceleration within a small time range around resonance. The asymptotic approach fails beyond a threshold value of the Reynolds number, because the damping effect of the local acceleration terms spreads over the whole cycle. The problem is then tackled by means of an approach that takes into account the above damping effect throughout the whole cycle. Finally, a numerical procedure is used that also allows the inclusion of nonlinear terms and the study of the interactions among forced and free modes. The numerical approach reveals that, even for relatively large values of the amplitude of the wall waviness, nonlinear effects are negligible and the damping of resonance is mainly due to local acceleration effects. The relevance of the results to the understanding of transition to turbulence in Stokes layers is discussed.