Derivation and implementation of the boundary integral formula for the convective acoustic wave equation in time domain

Abstract

Kirchhoff's formula for the convective wave equation is derived using the generalized function theory. The generalized convective wave equation for a stationary surface is obtained, and the integral formulation, the convective Kirchhoff's formula, is derived. The formula has a similar form to the classical Kirchhoff's formula, but an additional term appears due to a moving medium effect. For convenience, the additional term is manipulated to a final form as the classical Kirchhoff's formula. The frequency domain boundary integral can be obtained from the current time domain boundary integral form. The derived formula is verified by comparison with the analytic solution of source in the uniform flow. The formula is also utilized as a boundary integral equation. Time domain boundary element method (BEM) analysis using the boundary integral equation is conducted, and the results show good agreement with the analytical solution. The formula derived here can be useful for sound radiation and scattering by arbitrary bodies in a moving medium in the time domain.

A new formulation of time domain boundary integral equation for acoustic wave scattering in the presence of a uniform mean flow

“…For sound scattering problems, φ(r s , t ) on the scattering surface S is to be determined by (19) when the boundary condition for φ on S is given. A customary boundary condition on rigid surfaces is that the normal component of the acoustic velocity is zero, i.e., u · n = 0, which, considering (1), leads to…”

“…Indeed, in all the previous literature on wave scattering with a uniform mean flow (e.g., [5,10,13,15,19,28,30]), in both the frequency domain and the time domain, boundary conditions of type (21) have been assumed at solid wall boundaries. To implement such a boundary condition, the combined normal derivative appearing in (19) would then be separated into the normal and tangential components as…”

“…In the present paper, however, we propose an alternative boundary condition to be used at solid surfaces when solving TDBIE (19) in the presence of a uniform flow. The new boundary condition is based on a consideration of the acoustic energy.…”

“…1 The time domain boundary integral equation approach for scattering by moving surfaces was first formulated and studied in [28]. More recent studies of the time domain approach in the presence of a mean flow can be found in [13,16,19].…”

Vector Aeroacoustics for Uniform Mean Flow: Acoustic Velocity and Vortical Velocity

“…Indeed, the convective wave equation of the velocity potential is widely employed to analyze the effect of moving medium on sound propagation [38][39][40][41], and the acoustic pressure and acoustic velocity can be easily calculated from the velocity potential. However, it is difficult to derive a convective wave equation of aeroacoustics, in which the velocity potential is used as the acoustic variable on the left hand side (LHS) meanwhile aerodynamic sources are described on the right hand side (RHS).…”