Stability of wall modes in fluid flow past a flexible surface

Abstract

The stability of wall modes in fluid flow past a flexible surface is analyzed using asymptotic and numerical methods. The fluid is Newtonian, while two different models are used to represent the flexible wall. In the first model, the flexible wall is modeled as a spring-backed, plate-membrane-type wall, while in the second model the flexible wall is considered to be an incompressible viscoelastic solid of finite thickness. In the limit of high Reynolds number (Re), the vorticity of the wall modes is confined to a region of thickness O(Re−1/3) in the fluid near the wall of the channel. An asymptotic analysis is carried out in the limit of high Reynolds number for Couette flow past a flexible surface, and the results show that wall modes are always stable in this limit if the plate-membrane wall executes motion purely normal to the surface. However, the flow is shown to be unstable in the limit of high Reynolds number when the wall can deform in the tangential direction. The asymptotic results for this case are in good agreement with the numerical solution of the complete governing stability equations. It is further shown using a scaling analysis that the high Reynolds number wall mode instability is independent of the details of the base flow velocity profile within the channel, and is dependent only on the velocity gradient of the base flow at the wall. A similar asymptotic analysis for flow past a viscoelastic medium of finite thickness indicates that the wall modes are unstable in the limit of high Reynolds number, thus showing that the wall mode instability is independent of the wall model used to represent the flexible wall. The asymptotic results for this case are in excellent agreement with a previous numerical study of Srivatsan and Kumaran.