Weak wall boundary conditions for compressible flows

Weak imposition of essential boundary conditions (i.e., weak BCs) for the Navier-Stokes equations of incompressible flows allows a certain amount of controlled numerical flow slip on the solid surface. Numerical flow slip mimics the presence of a thin boundary layer that would otherwise need to be captured using a fine mesh resolution. As a result, weak BCs enable the use of coarser meshes near solid walls without sacrificing numerical solution accuracy, which significantly reduces the computational costs, especially for 3D, wall-bounded turbulent flows. However, weak BCs for compressible flows are not as well understood as those for the incompressible-flow case. In particular, numerical instabilities were observed in some cases where the weak BCs were simultaneously imposed for the velocity and temperature fields. In the present effort, to address these stability issues, we develop a methodology for the design of compressible-flow weak BC operators and demonstrate the improved performance of the resulting weak BC formulations using challenging 2D and 3D test cases.