Anisotropy Correction of Two Dimensional Finite Difference Schemes for Computational Aeroacoustics

Abstract

Because the spatial differencing schemes are analyzed and optimized for one-dimensional test cases, in multidimensional problems they may not have isotropic behavior. In this work, a new class of finite difference schemes for two-dimensional Computational Aeroacoustics are derived which are designed to have improved isotropy compared to existing schemes. The derivation is performed based on both Taylor series expansion and Fourier analysis. Thus far, various explicit centered finite difference schemes and the associated boundary stencils have been derived and analyzed. The isotropy corrector factor, a parameter of the schemes, can be determined in two ways: by minimizing the integrated error between the phase or group velocities, or by simply equalizing them. The order of accuracy of the optimized schemes is the same as that of the classical schemes. The design of the new schemes at the boundaries may provoke problems; special care must be taken at the corners. The schemes are restricted to equally-spaced Cartesian grids, so the generalized curvilinear transformation method and Cartesian grid methods are good candidates.