Some New Results for the Study of Acoustic Radiation within a Uniform Subsonic Flow Using Boundary Integral Method

Abstract

In this paper, a reformulation of the Helmholtz integral equation for tridimesional acoustic radiation in a uniform subsonic flow is presented. An extension of the Sommerfeld radiation condition, for a free space in a uniform movement, makes possible the determination of the convected Green function, the elementary solution of the convected Helmholtz equation. The gradients of this convected Green function are, so, analyzed. Using these results, an integral representation for the acoustic pressure is established. This representation has the advantage of expressing itself in terms of new surface operators, which simplify the numerical study. For the case of a regular surface, the evaluation of the free term associated with the singular integrals shows that it is independent of the Mach number of the uniform flow. A physical interpretation of this result is offered. A numerical approximation method of the integral representation is developed. Furthermore, for a given mesh, an acoustic discretization criterion in a uniform flow is proposed. Finally, numerical examples are provided in order to validate the integral formula.

A simple and effective axisymmetric convected Helmholtz integral equation

“…The presence of the flow in Eqs. (1)- (4) shows that the integral formula of the acoustic pressure p(M) becomes significantly more complicated than in the no-flow case. To obtain a formulation that is less complicated and suitable for theoretical and numerical developments, the normal derivative and the derivative in the flow direction of the convected Green's function G k c can be converted into a non-standard normal derivative operator DG k…”

“…We note that the idea of introducing the operator DG k c /Dn Q is due to the formulation of a radiation condition at infinity in [4]. Substituting Eqs.…”

“…[3]. It requires the evaluation of the kernel G k c (M, Q) and its derivatives, the normal derivative ∂ G k c (M, Q)/∂n Q and the derivative in the flow direction M ∞ ·grad Q G k c (M, Q), which are given for M = Q by [4]:…”

“…However, this scheme does not describe a physical test-case in R 3 , but is motivated by the obtaining the boundary integral equation. Using radiation conditions at infinity, the acoustic radiation problem in the exterior domain ext is governed by the classical boundary integral representation of the acoustic pressure p given for all point M ∈ ext by [4]:…”

“…The integral representations can be formulated in the transformed acoustic medium [1,2] or in the original acoustic medium [3,4]. However, the advantages of using boundary integral formulations in an untransformed acoustic medium are that the convection effects are explicit and that complex boundary conditions can be handled easily.…”