Radiation And Refraction Of Sound Waves Through a Two-Dimensional Shear Layer

This report presents solutions to Problem 1 of Category 4 in the 4th CAA Workshop on Benchmark Problems. The problem is numerically solved on a clustered orthogonal grid using a high-order optimized upwind Dispersion- Relation-Preserving scheme. The Perfectly-Matched-Layer (PML) boundary condition in unsplit physical variables is applied to ensure minimum numerical reflections. The numerical results by solving the linearized Euler equations are the total solution, being the superposition of the acoustic wave solution and the instability wave solution. In order to get the instability wave solution, a source term of the same excitation frequency is effectively placed upstream of the region of interest. Such a source term produces a stronger instability wave of the same excitation frequency but a weaker acoustic wave in the region of interest and thus the total solution in this region can well approximate a pure instability wave. The acoustic wave solution, which is desired in the problem, is then successfully achieved by subtracting the instability wave solution from the total solution. The benchmark problem is also analytically solved to an accuracy sufficient to evaluate the numerical solutions.