Transient acoustic processes in a low-Mach-number shear flow

Abstract

A systematic perturbation procedure, based on a small mean flow Mach number and large duct Reynolds number, is employed to formulate and solve an initial-boundary-value problem for acoustic processes in a shear flow contained within a rigid-walled parallel duct. The results describe the general transient evolution of acoustic waves driven by a plane source located at a given duct cross-section. Forced bulk oscillations near the source and oblique wave generation are shown to result from refraction of the basic planar axial disturbance by the shear flow. Refraction also causes the axial waves to exhibit higher-order amplitude variations in the transverse direction. As the source frequency approaches certain critical values, specific refraction-induced oblique waves evolve into amplifying purely transverse waves. As a result, the magnitude of the refraction effect increases with time, and quasi-steady solutions do not exist. The analysis is extended to the thin acoustic boundary layer adjacent to the solid walls to examine the shear-layer structure induced by the variety of acoustic waves in the core flow. Nonlinear effects and acoustic streaming are shown to be negligibly small on the scale of a few axial wavelengths.