Brinkman Penalization Method for Computation of Acoustic Scattering from Complex Geometry

Abstract

In this study, Brinkman penalization method (BPM) is extended for prediction of acoustic scattering from complex geometries. The main idea of the BPM is to model the solid obstacle as a porous material with zero porosity and permeability. With the aim of increasing the spatial accuracy at the immersed boundaries, computation is carried out on the boundary-fitted Cartesian-like grid with a high-order compact scheme combined with one-side differencing/filtering technique at the boundaries, while a slip boundary condition at the wall is imposed by introducing the ‘anisotropic’ penalization terms to the momentum equations. Several test cases are considered to demonstrate the accuracy, robustness and feasibility of the BPM. Numerical results are in excellent agreement with the analytic solutions for single and two cylinder scattering problems. The present BPM is then used to solve the acoustic scattering from a three-element high-lift wing (30P30N model). I. Introduction UMERICAL simulation of acoustic scattering from complex geometries has received attention in a wide range of aeroacoustic problems, such as slat noise and flap side-edge noise from a multi-element airfoil in high-lift configuration, rotor-stator interaction noise in turbo-machinery, etc. There are two main strategies in direct simulation of acoustic scattering from solid boundaries of complexity, i.e. structured/unstructured body-fitted grid methods [1-3] and immersed boundary methods [4-6]. In the former, implementation of wall boundary condition is straightforward, attaining a desired degree of accuracy at the boundaries. However, when complex geometries are concerned, the structured body-fitted grid method or even the overset grid method [7-9] often meets difficulties associated with the grid generation as well as with the quality of the grids. A discontinuous Galerkin (DG) method [10-12] based on the unstructured grids promises success for real complex geometries but computational cost has always been an issue. In this regard, an immersed boundary technique can be considered as an alternative because of its simple and efficient implementation for arbitrarily shaped surfaces with reasonable computational cost. Following the pioneering work of Peskin [13,14], a number of immersed boundary methods have been proposed to handle the complex geometries [4,15,16]. Among them, a Brinkman penalization method (BPM) [17], which was originally developed to model the fluid flow in porous media, appears attractive because of its easiness to handle the solid obstacle by simply treating as a porous medium of high impedance. In BPM, porosity and permeability in the penalty terms which are added in the compressible Navier-Stokes equations are set to zero in the solid region to impose the immersed boundary effect on the fluids. A no-slip boundary condition is therefore enforced naturally at the solid boundary. There are, however, two inherent limitations with this penalization technique. First, it requires a large number of grid points in solid region to retain the order of accuracy at the wall, thus making the method impractical at highly sophisticated geometries. Another drawback is that only the no-slip boundary condition is satisfied at the solid wall, whereas a slip boundary condition has to be met with the full or linearized Euler equations. In the present study, we address these numerical issues. With aim of increasing the spatial accuracy at the embedded boundary, we conform the immersed boundary grids to the actual shape of the surface following the idea of reshaped cell approach [18,19]. The slip boundary condition at the solid surface is imposed by introducing the ‘anisotropic’ penalization terms in the momentum equations. The validity of the present method is then assessed by considering the acoustic scattering from i) a single cylinder, ii) two circular cylinders, and iii) three element high-lift wing with the deployed slat and flap. We also discuss numerical issues related to the implementation of the reshaped cell approach and to the stiffness due to the penalty terms.