## Mathematics & Statistics Theses & Dissertations

Spring 2008

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics and Statistics

#### Program/Concentration

Computational and Applied Mathematics

Fang Q. Hu

Oktay Baysal

#### Committee Member

D. Glenn Lasseigne

Li-Shi Luo

Ruhai Zhou

#### Abstract

The Perfectly Matched Layer (PML) was originally proposed by Berenger as an absorbing boundary condition for Maxwell's equations in 1994 and is still used extensively in the field of electromagnetics. The idea was extended to Computational Aeroacoustics in 1996, when Hu applied the method to Euler's equations. Since that time much of the work done on PML in the field of acoustics has been specific to the case where mean flow is perpendicular to a boundary, with an emphasis on Cartesian coordinates. The goal of this work is to further extend the PML methodology in a two-fold manner: First, to handle the more general case of an oblique mean flow, where mean velocities strike the boundary at an arbitrary angle, and second, to adapt the equations for use in a cylindrical coordinate system. These extensions to the PML methodology are effectively carried out in this dissertation. Perfectly Matched Layer absorbing boundary conditions are presented for the linearized and nonlinear Euler equations in two dimensions. Such boundary conditions are presented in both Cartesian and cylindrical coordinates for the case of an oblique mean flow. In Cartesian coordinates, the PML equations for the side layers and corner layers of a rectangular domain will be derived independently. The approach used in the formation of side layer equations guarantees that the side layers will be perfectly matched at the interface between the interior and PML regions. Because of the perfect matching of the side layers, the equations are guaranteed to be stable. However, a somewhat different approach is used in the formation of the corner layer equations. Therefore, the stability of linear waves in the corner layer is analyzed. The results of the analysis indicate that the proposed corner equations are indeed stable. For the PML equations in cylindrical coordinates, there is no need for separate derivations of side and corner layers, and in this case, the stability of the equations is achieved through an appropriate space-time transformation. As is shown, such a transformation is needed for correcting the inconsistencies in phase and group velocities which can negatively affect the stability of the equations. After this correction has been made, the cylindrical PML can be implemented without risk of instability. In both Cartesian and cylindrical coordinates, the PML for the linearized Euler equations are presented in primitive variables, while conservation form is used for the nonlinear Euler equations. Numerical examples are also included to support the validity of the proposed equations. Specifically, the equations are tested for a combination of acoustic, vorticity and entropy waves. In each example, high-accuracy solutions are obtained, indicating that the PML conditions are effective in minimizing boundary reflections.

#### DOI

10.25777/6j3k-ky24

9780549614593

COinS