Accuracy of Supergrid Boundary Conditions for Incident Disturbances with a Range of Incidence Angles in Uniform and Sheared Flows

We examine the accuracy of supergrid boundary conditions (BC) for problems involving vortical, entropic, and acoustic waves propagating at a range of incidence angles in uniform and sheared flows (Benchmark Problem Category 1, Problem 3). For uniform flow, we compare the accuracy with high-order-accurate Fourier/Laplace BC. Such BC can be extended to arbitrarily order of accuracy, but lead to large errors and instabilities in the presence of shear. The supergrid boundary condition is a buffer layer technique wherein the equations of motion are modified in a region near the boundary in order to attenuate disturbances. The modified equations are developed by analogy with subgrid-scale turbulence models, except that the role of small and large scales is interchanged. For uniform flow, both methods studied can give arbitrarily small errors, but the Fourier/Laplace BC are much more efficient than the supergrid BC. However, the supergrid BC produces similar results in the presence of shear as well as in cases involving sonic points. For the benchmark problem, relatively large buffers were needed to achieve acceptably small errors. The cause of the inefficiency is the incidence of faint, repeated reflections from distant image sources from the initial condition. In cases where such waves are not an important feature (many relevant CAA problems), the supergrid BC produces good results with relatively small buffers.