Two-Layer Flow Instability

Abstract

Hydrodynamic stability of a boundary layer flow of two parallel, stratified, but immiscible fluid layers over a flat plate at zero incidence is investigated. The problem is of interest in the context of de-icing of aircraft wings. The describing parameters of this setup of two newtonian fluids are the ratios of viscosity, density and length with respect to the upper layer, as well as Froude number and a surface tension parameter. The linear stability problem is governed two 4th order complex Orr-Sommerfeld equations coupled through four interface conditions. For given values of Reynolds- R and wavenumber alpha, the governing equations are solved numerically for the complex eigenvalue c. It is found that always two stability modes are present, which are called hard and soft due to their different order of magnitude in c. Whereas the former is related to the classical single-layer modes, the latter arises from the interface and is of a different nature. Intensive studies of possible combinations of the relevant parameters show that both modes have to be taken into account for the prediction of flow instability. The results are compared and contrasted against results of asymptotic analyses in the literature. Incidentally the much cited maximum growth rate for the Blasius boundary layer is updated here