The stability of rotating-disc boundary-layer flow over a compliant wall. Part 1. Type I and II instabilities

Abstract

A theoretical study into the effects of wall compliance on the stability of the rotating-disc boundary layer is described. A single-layer viscoelastic wall model is coupled to a sixth-order system of fluid stability equations which take into account the effects of viscosity, Coriolis acceleration, and streamline curvature. The coupled system of equations is integrated numerically by a spectral Chebyshev-tau technique.Travelling and stationary modes are studied and wall compliance is found to greatly increase the complexity of the eigenmode spectrum. It is effective in stabilizing the inviscid Type I (or cross-flow) instability. The effect on the viscous (Type II) eigenmode is more complex and can be strongly destabilizing. An analysis of the energy flux indicates that this destabilization arises as a result of a large degree of energy production by viscous stresses at the wall/flow interface.The Type I and II instabilities are shown to be negative and positive energy waves respectively. The co-existence of eigenmodes of opposite energy type indicates the possibility of modal interaction and coalescence. It is found that, compared with the rigid disc, wall compliance promotes the interaction and coalescence of the Type I and II eigenmodes. There is an associated strong instability which appears to be characterized by marked horizontal motion of the compliant surface. Modal coalescence is interpreted physically as producing local algebraic growth which could advance the onset of nonlinear effects.