Time-Domain Acoustic Wave Solutions in Sheared Mean Flows

Abstract

The instability waves triggered by an acoustic source of certain frequencies always co-exist with the acoustic waves in the solution of non-homogenous linearized Euler equations (LEE), termed as the total solution, for a given sheared mean flow. The total solution thus contains both the acoustic wave solution and the instability wave solution, the latter referred to as the contained instability wave solution. A time-domain method for obtaining only the acoustic wave solution in sheared mean flows is proposed and verified in this paper. Instead of suppressing or filtering-out the contained instability waves, in the proposed method the acoustic wave solution is achieved by means of subtracting the contained instability wave solution from the total solution of the non-homogenous LEE. The two important aspects of the proposed method are the identification of an appropriate reference point where the instability wave dominates the acoustic wave solution and the development of the approximately pure instability wave solution. The method is feasible to a multi-frequency or broadband sound source and is applicable to mean flows with more relaxed constraints than parallel or slowly varying mean flows. The implementation and effectiveness of the method are demonstrated by solving a test problem of a multi-frequency acoustic source embedded in a sheared mean flow. The acoustic wave solution obtained from the method is verified by the analytical solution.