The critical layer in sheared flow

Abstract

Critical layers arise as a singularity of the linearized Euler equations when the phase speed of the disturbance is equal to the mean flow velocity. They are usually ignored when estimating the sound field, with their contribution assumed to be negligible. It is the aim of this paper to study fully both numerically and analytically a simple yet typical sheared ducted flow in order to distinguish between situations when the critical layer may or may not be ignored. The model is that of a linear-then-constant velocity profile with uniform density in a cylindrical duct, allowing for exact Green’s function solutions in terms of Bessel functions and Frobenius expansions. It is found that the critical layer contribution decays algebraically in the constant flow part, with an additional contribution of constant amplitude when the source is in the boundary layer, an additional contribution of constant amplitude is excited, representing the hydrodynamic trailing vorticity of the source. This immediately triggers, for thin boundary layers, the inherent convective instability in the flow. Extra care is required for high frequencies as the critical layer can be neglected only together with the pole beneath it. For low frequencies this pole is trapped in the critical layer branch cut.