The hydrodynamic stability of channel flow with compliant boundaries

Abstract

An asymptotic theory is developed for the hydrodynamic stability of an incompressible fluid flowing in a channel in which one wall is rigid and the other is compliant. We exploit the multideck structure of the flow to investigate theoretically the development of disturbances to the flow in the limit of large Reynolds numbers. A simple spring-plate model is used to describe the motion of the compliant wall, and this study considers the effect of the various wall parameters, such as tension, inertia, and damping, on the stability properties. An amplitude equation for a modulated wavetrain is derived and the properties of this equation are studied for a number of cases including linear and nonlinear theory. It is shown that in general the effect of viscoelastic damping is destabilizing. In particular, for large damping, the analysis points to a fast travelling wave, short-scale instability, which may be related to a flutter instability observed in some experiments. This work also demonstrates that the conclusions obtained by previous investigators in which the effect of tension, inertia, and other parameters is neglected, may be misleading. Finally it is shown that a set of compliant-wall parameters exists for which the Haberman type of critical layer analysis leads to stable equilibrium amplitudes, in contrast to many other stability problems where such equilibrium amplitudes are unstable.