Applications of the optimized upwind dispersion-relation-preserving schemes for multi-dimensional acoustic problems

Abstract

A high order optimized upwind Disperion-RelationPreserving (DRP) finite difference scheme is developed and implemented for multi-dimensional acoustic problems. A sequence of numerical simulations of acoustic wave propagation problems is carried out to evaluate the robustness and accuracy of the optimized upwind DRP scheme and its potential for solving complex aeroacoustics problems. The results of the upwind DRP scheme are compared with that of the central DRP scheme and the analytic solutions whenever it is possible. It is concluded from the current investigations that the optimized upwind DRP scheme not only can be successfully implemented to multi-dimensional aeroacoustic computations, but can accurately predict acoustics wave propagation without adding an artificial damping term. Introduction In contrast to Computational Fluid Dynamics (CFD), which has advanced to a fairly mature state, Computational Aeroacoustics (CAA) has only recently emerged as a separate area of study. Although Aeroacoustics problems are governed by the same equations as those in aerodynamics, acoustic waves have their own characteristics which makes the computation challenging. Acoustic waves are inherently unsteady, and their amplitudes are several order smaller than the mean flow and their frequencies are generally very high. These require that computational schemes be high order in both space and time with least dispersion and dissipation [1,2]. Many CFD schemes such as MacCormack scheme, upwind schemes and ENO schemes etc., have been extended to high order using more stencil points and applied to the computations of acoustic problems [3-7]. Many compact and noncompact optimized schemes [8-12] including Tam and Webb's Dispersion-Relation-Preserving (DRP) scheme, which were designed for the linear acoustic waves, were recently reviewed by Zigg [13]. It has been shown that for waves with high wavenumbers (short waves) the optimized schemes require less grid points per wavelength (PPW) than traditional high order CFD methods. This property of requiring less PPW is essential in CAA, since large computational domain is usually required. Most optimized schemes, however, are restricted to central difference algorithms. This restriction inevitably leads to stability problems which mu5t be dealt with through the use of filters or explicit dissipation terms. Although the deliberate filter or dissipation terms are proved quite successful in many acoustic problems [3,9], they are problem-dependent and require the prior knowledge of the problems. Upwind schemes have been widely used in CFD and have been shown very efficient and robust. Upwind schemes ensure that waves propagate in * current address: CFD Research Corporation, Huntsville, AL Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc. the correct physical direction. With the build-in dissipation upwind schemes automatically damp out high wavenumbers component of solution. Since the dissipation does not distinguish the spurious waves from real acoustic waves, upwind schemes are necessary to be optimized such that the dissipation has little effect on large range of acoustic waves. Because of the success of the optimized upwind DRP scheme in solving one-dimensional aeroacoustics problems [14], the objectives of the current investigations are aimed at (1) to develop and to implement the optimized upwind DRP scheme to multi-dimensional aeroacoustics problems, (2) to evaluate the accuracy and robustness of the scheme and its potential for solving complex aeroacoustics problems. Problem Formulation 1. The Optimized upwind DRP Scheme Optimized shcemes preserve wave propagation characteristics for relative large range of wave numbers and requires less grid points per wavelength (PPW). They are usually constructed by optimizing the finite difference approximations of the space and time derivatives in the wave and frequency space [8,13]. Consider the approximation of the first spatial derivative du/dxby the finite difference equation, which is given for a uniform grid of spacing Ax. Suppose M values of u to the right and N values of u to the left of the point x are used in the finite difference equation where x is a continuous variable, i.e. du(x) 1 jAx) 0) The Fourier transform and its inverse of a function are related by dx (2)