A Class of Unsteady Nonlinear Waves in Parallel Flows

Abstract

Studies of instability waves in parallel shear flows generally begin with linearized governing equations, justified by the small amplitude (ε) of the disturbances considered. The fundamental problem of what we will call the viscous theory is the eigenvalue problem for the fourth order Orr-Sommerfeld equation [1]. At high Reynolds number (R), a case of particular interest, the coefficient on the fourth order term is small and the second order Rayleigh equation (inviscid approximation) is valid except near the boundaries and, most important, at the critical layer, where the phase speed c is equal to the flow velocity. The inviscid solution is singular there; viscous effects must be considered in a layer of thickness R^−1/3 to resolve the singularity.