Absorbing boundary condition for nonlinear Euler equations in primitive variables based on the Perfectly Matched Layer technique

For aeroacoustics problems, the nonlinear Euler equations are often written in primitive variables in which the pressure is treated as a solution variable. In this paper, absorbing boundary conditions based on the Perfectly Matched Layer (PML) technique are presented for nonlinear Euler equations in primitive variables. A pseudo mean flow is introduced in the derivation of the PML equations for increased efficiency. Absorbing equations are presented in unsplit physical primitive variables in both the Cartesian and cylindrical coordinates. Numerical examples show the effectiveness of the proposed equations although they are not theoretically perfectly matched to the nonlinear Euler equations. The derived equations are tested in numerical examples and compared with the PML absorbing boundary condition in conservation form that was formulated in an earlier work. The performance of the PML in primitive variables is found to be close to that of the conservation formulation. A comparison with the linear PML in nonlinear problems is also considered. It is found that using nonlinear absorbing equations presented in this paper significantly improves the performance of the absorbing boundary condition for strong nonlinear cases.