Comparisons of exact and approximate convection of plane waves in a simple shear flow

Abstract

In aero- and environmental-acoustic problems, propagation of sound waves through a medium with a nonuniform mean flow has commonly been treated by several different approximate techniques typically involving integral transforms, geometrical acoustics, or other approximate methods. An effectively exact solution technique that properly incorporates the vector character of mean-flow convection has been found [L. Nijs and C. P. A. Wapenaar, J. Acoust. Soc. Am. 87, 1987–1998 (1990)]. However, the formal and numerical complexity of these techniques, typically coupled with nontrivial impedance boundary conditions, has prevented detailed comparisons of acoustic field variables with and without exact treatment of convection. This paper presents results and comparisons between exact and approximate treatments of acoustic plane waves propagating in a simple shear flow. The solution of the linearized time- and space-dependent equations of inviscid fluid motion, as nondimensionalized herein, depends on two-dimensionless parameters, m/f and ft, where m is the convecting flow’s shear rate, f is the acoustic frequency, and t is the length of time that the plane waves interact with the shear flow. The exact treatment involves numerical solution of the governing equations. The approximate treatment is handled with a WKB solution valid for m/f≪1. A comparison of results shows that the approximate treatment of convection produces only single digit percentage errors when (m/f )(ft)=mt≲0.5. However, pressure amplitude differences increase rapidly beyond mt≈0.5 and are found to grow approximately as exp{+0.4(mt)2}. In addition, the computed results display phenomena that are completely absent from the WKB solution such as: (i) dynamic misalignment of the acoustic-velocity vector and the wave front normal; and (ii) the existence of significant acoustic perturbation vorticity.