Application Of The Dual Reciprocity MethodTo Acoustic Radiation In A SubsonicNonuniform Flow

Abstract

In this paper, the dual reciprocity method is applied to acoustic radiation in a subsonic nonuniform flow. The BEM formulation is based on a direct boundary integral equation developed very recently by the authors for acoustic radiation in a subsonic uniform flow. All the terms due to the nonuniform-flow effect are treated as source terms. The dual reciprocity method is then used to transform the resulting domain integral into a boundary integral. Numerical results show reasonably good agreement with an available analytical solution for a pulsating sphere submerged in a potential-flow field. INTRODUCTION Simulation of sound radiation and scattering in moving flows has many important applications in aeroacoustics. Examples include prediction of noise radiated from engines of commercial aircraft in steady flight and performance evaluation of silencers containing axial flows. To date, the most frequently used numerical method for sound radiation and scattering in a nonuniform flow field is the finite element method (FEM). The major advantage of the FEM is that it can handle complicated geometries and nonuniform flows without difficulty. However, since the fluid domain involved in an exterior radiation problem is unbounded by nature, the finite element discretization has to be truncated somewhere. In addition, special care has to be taken to insure the Sommerfeld radiation condition at infinitŷ . The BEM has also been applied to aeroacoustics. The main feature of the BEM is that it can handle the Sommerfeld radiation condition automatically. However, previous application of the BEM has been limited to the use of the well-known Helmholtz integral equation, which is valid only under the no-flow condition. In the presence of flows, the governing differential equation has to be transformed into the Helmholtz equation first^. Transactions on Modelling and Simulation vol 8, © 1994 WIT Press, www.witpress.com, ISSN 1743-355X 246 Boundary Element Technology In a very recent work , we have developed a direct BEM formulation for acoustic radiation in a subsonic uniform flow. This direct BEM formulation uses the Green's function derived from the adjoint operator of the governing differential equation for radiation in a uniform flow. Therefore, the convective flow effect is automatically incorporated in the Green's function and no coordinate transformation is needed. In this paper, we extend our BEM formulation to acoustics in a nonuniform flow. The governing differential equation for acoustics in a nonuniform flow is first rearranged in such a way that the left-hand side looks exactly like the acoustic equation in a uniform flow, and all the terms due to the nonuniform flow are taken to the right-hand side and treated as source terms. The differential equation is then recast into an integral equation. In the recasting process, the lefthand side becomes a boundary integral, and the right-hand side (that contains the source terms) becomes a domain integral, which is confined to a finite region only. Then, the so-called Dual Reciprocity Method^ (DRM) is used to convert the domain integral into a boundary integral. The basic idea of the DRM is to approximate the right-hand side by a linear combination of many global interpolating functions, each of which has an analytical particular solution. The domain integral can then be converted into a boundary integral. ACOUSTIC RADIATION IN A NONUNIFORM FLOW The governing differential equation for steady-state linear acoustics in a nonuniform potential flow field is (RJ. Astley2) 2ikM-V M-V(M-V(J)) (y-l)ik(|)V*M