"Numerical Methods for Problems in
Computational Aeroacoustics"

ABSTRACT

A goal of computational aeroacoustics is the accurate calculation of noise from a jet in the far field. This work concerns the numerical aspects of accurately calculating acoustic waves over large distances and long time. More specifically, the stability, efficiency, accuracy, dispersion and dissipation in spatial discretizations, time stepping schemes, and absorbing boundaries for the direct solution of wave propagation problems are determined.

Efficient finite difference methods developed by Tam and Webb, which minimize dispersion and dissipation, are commonly used for the spatial and temporal discretization. Alternatively, high order pseudospectral methods can be made more efficient by using the grid transformation introduced by Kosloff and Tal-Ezer. Work in this dissertation confirms that the grid transformation introduced by Kosloff and Tal-Ezer is not spectrally accurate because, in the limit, the grid transformation forces zero derivatives at the boundaries. If a small number of grid points are used, it is shown that approximations with the Chebyshev pseudospectral method with the Kosloff and Tal-Ezer grid transformation are as accurate as with the Chebyshev pseudospectral method. This result is based on the analysis of the phase and amplitude errors of these methods, and their use for the solution of a benchmark problem in computational aeroacoustics. For the grid transformed Chebyshev method with a small number of grid points it is, however, more appropriate to compare its accuracy with that of high-order finite difference methods. This comparison, for an order of accuracy $10^{-3}$ for a benchmark problem in computational aeroacoustics, is performed for the grid transformed Chebyshev method and the fourth order finite difference Solutions with the finite difference method are as accurate, and the finite difference method is more efficient than, the Chebyshev pseudospectral method with the grid transformation.

The efficiency of the Chebyshev pseudospectral method is further improved by developing Runge-Kutta methods for the temporal discretization which maximize imaginary stability intervals. Two new Runge-Kutta methods, which allow time steps almost twice as large as the maximal order schemes, while holding dissipation and dispersion fixed, are developed. In the process of studying dispersion and dissipation, it is determined that maximizing dispersion minimizes dissipation, and vice versa.

In order to determine accurate and efficient absorbing boundary conditions, absorbing layers are studied and compared with one way wave equations. The matched layer technique for Maxwell equations is equivalent to the absorbing layer technique for the acoustic wave equation introduced by Kosloff and Kosloff. The numerical implementation of the perfectly matched layer for the acoustic wave equation with a large damping parameter results in a small portion of the wave transmitting into the absorbing layer. A large damping parameter also results in a large portion of the wave reflecting back into the domain. The perfectly matched layer is implemented on a single domain for the solution of the second order wave equation, and when implemented in this manner shows no advantage over the matched layer. Solutions of the second order wave equation, with the absorbing boundary condition imposed either by the matched layer or by the one way wave equations, are compared. The comparison shows no advantage of the matched layer over the one way wave equation for the absorbing boundary condition. Hence there is no benefit to be gained by using the matched layer, which necessarily increases the size of the computational domain.


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