Efficient evaluation of three-dimensional Helmholtz Green's functions tailored to arbitrary rigid geometries for flow noise simulations

Abstract

The Lighthill's wave equation provides an accurate characterization of the hydrodynamic noise due to the interaction between a turbulent flow and an obstacle, that is needed to get in many industrial applications. In the present study, to solve the Lighthill's equation expressed as a boundary integral equation, we develop an efficient numerical method to determine the three-dimensional Green's function of the Helmholtz equation in presence of an obstacle of arbitrary shape, satisfying a Neumann boundary condition. This so-called tailored Green's function is useful to reduce the computational costs to solve the Lighthill's equation. The first step consists in deriving an integral equation to express the tailored Green's function thanks to the free space Green's function. Then a Boundary Element Method (BEM) is used to compute tailored Green's functions. Furthermore, an efficient method is performed to compute the second derivatives needed for accurate flow noise determinations. The proposed approach is first tested on simple geometries for which analytical solutions can be determined (sphere, cylinder, half plane). In order to consider realistic geometries in a reasonable amount of time, fast BEMs are used: fast multipole accelerated BEM and hierarchical matrix based BEM. A discussion on the numerical efficiency and accuracy of these approaches in an industrial context is finally proposed.