A simple and effective axisymmetric convected Helmholtz integral equation

Abstract

In this paper, we develop an axisymmetric boundary integral equation that derives from a reformulation of the 3D Helmholtz integral formula for the acoustic radiation problems in a subsonic uniform flow. Through the use of a new non-standard derivative operator, the axisymmetric convected Helmholtz integral equation substantially reduces the effects of flow incorporated in the classical convected boundary integral formulations, and involved in the normal derivative and the derivative in the flow direction of the axisymmetric convected Green's function. As for the free term derived from the singular integrals, it is given by a new expression independent of complete elliptic integrals and evaluated analytically as a convected angle in the meridian plane. The numerical treatment of singular integrals requires only the use of standard Gauss quadrature rules. Different test cases are presented.

An improved axisymmetric convected boundary element method formulation with uniform flow

“…In addition, the CPU times of the Numerical boundary element method, the finite element method and that the classical boundary element method in Ref. [14] with mean flow are obtained by a machine 2.93 GHz using MATLAB-ACOUSTIC codes, which are given by the following Compared to the classical axisymmetric boundary integral formulae, the proposed method reduces a CPU time equal to 60% of the classical axisymmetric BEM. Also, the relative error for the classical axisymmetric boundary integral equation is less than 2%.…”

“…Using the normal and the flow direction derivatives of the three-dimensional Green's function in Refs. [10,11,14], one obtains the following derivative function…”

“…Following the exclusion procedure [18,19] to isolate the singularity problems in the boundary formula Eq. (14), which can be rewritten as…”

“…Also, the authors in Ref. [14] have been used a reformulation of the threedimensional convected boundary element to solve the axisymmetric acoustic problems. The reformulation based on the non-standard normal derivative similar to the non-flow normal derivative of the Green's kernel.…”

“…The acoustic variable of the classical and reformulation axisymmetric boundary integral representations in Refs. [10,11,14] in terms of the acoustic pressure as well as its normal and tangential derivatives.…”