A General Buffer Zone-type Non-Reflecting Boundary Condition for Computational Aeroacoustics

Abstract

A non-reflecting buffer zone-type boundary condition based on grid stretching and attenuation methods is presented for computational aeroacoustics. Energy of outgoing wave content is aggressively transferred into increasingly higher wavenumber modes via grid stretching and this high wavenumber (frequency) content is annihilated using a high-order, implicit low-pass filter. The linearized Euler equations in strong conservative form are recast using a general curvilinear coordinate transformation. Spatial discretization is achieved using a sixth-order compact difference operator and the solution is timemarched with a fourth-order Runge-Kutta integration technique. Numerical examples with uniform mean flow are presented for a periodic acoustic source, convection of a vortical disturbance, and a wallbounded acoustic disturbance. The numerical solutions match analytic, reference, and published data very well, and demonstrate the effectiveness of this non-reflecting boundary treatment for computational aeroacoustics. INTRODUCTION The equations describing the propagation of acoustic signals, the linearized Euler equations, require numerical methods that are essentially free of dispersion and dissipation. Therefore, high-order spatial discretization schemes and temporal integration techniques that have minimal affect on the resolution characteristics of the spatial scheme are required for accurate simulation. The lack of a viscous (damping) effect in the describing equations means that any spurious errors generated due to numerical boundary treatment may propagate freely, corrupting the acoustic field. Assistant Professor, Member AIAA Technical Area Leader, Associate Fellow AIAA The need for highly accurate non-reflecting boundary conditions to truncate the domain of interest for computational aeroacoustics (CAA) is well established. A variety of techniques have been proposed, and can be grouped as: (a) asymptotic approximations, (b) characteristics-based techniques, (c) Perfectly Matched Layers (PML), (d) absorbing layers, and (e) hybrid approaches. In this paper, a non-reflecting boundary condition utilizing grid stretching and various attenuation methods is considered. The non-reflecting property is obtained by an energy transfer of wave content into increasingly higher order wavenumber (spatial frequency) modes, and then annihilating this high frequency content by filtering or damping. This energy transfer and annihilation (ETA) boundary condition is used for both outflow and radiation boundaries for acoustic and vortical disturbances. Variations of this method have been considered by Colonius, et.al. and Bogey, et.al. as an outflow boundary condition. The present study is a continuation of earlier work by Visbal and Gaitonde for computational acoustics. The focus of this work is to establish the effectiveness of the ETA boundary conditions as a general non-reflecting boundary treatment for problems that arise in computational acoustics. In the next section, the specific form of the equations describing propagation of acoustic signals is presented. The numerical methodology and application specifics of the ETA boundary conditions are discussed next, followed by a series of numerical examples that are used to evaluate the effectiveness of the proposed treatment. These include a periodic source in a uniform mean flow, convection of a vortical disturbance, and the propagation of a wallbounded acoustic disturbance subjected to a uniform mean flow. 9th AIAA/CEAS Aeroacoustics Conference and Exhibit 12-14 May 2003, Hilton Head, South Carolina AIAA 2003-3300 Copyright © 2003 by the American Institute of Aeronautics and Astronautics, Inc. The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental purposes. All other rights are reserved by the copyright owner. 2 American Institute of Aeronautics and Astronautics DESCRIBING EQUATIONS The Euler equations, linearized about a uniform mean flow, are used to describe the propagation of small amplitude acoustic signals and vortical disturbances. A generalized curvilinear coordinate transformation is used to map the physical Cartesian system (x,y) into a uniform computational space (ξ,η). The two-dimensional form of the equations in computational space can be written as: S F E U = η ∂ ∂ + ξ ∂ ∂ + ∂ ∂