A Characteristic-Based Volume Penalization Method for Arbitrary Mach Flows Around Solid Obstacles

Volume penalization is a subclass of immersed boundary methods for modeling complex geometry flows, which introduces the effects of obstacles by modifying the governing equations. The method presented in this paper encompasses general boundary conditions as an extension of the Brinkman Penalization Method (BPM) [1], which was originally developed for solid, isothermal obstacles in incompressible flows. A principal strength of Brinkman penalization is that error can be rigorously controlled a priori, with the solution converging to the exact in a predictable fashion [4, 5]. While much work has been done to refine BPM for various numerical techniques and flow regimes, boundary conditions have lacked generality, especially for compressible flows. They have been typically limited to slip and no-slip conditions for the inviscid and viscous flow around isothermal obstacles, though additional boundary conditions have been developed on a problem specific basis. In this way, BPM has been inapplicable and inflexible for many fluid problems, notably those demanding heat-flux and insulating boundary conditions on solid surfaces. The novel Characteristic-Based Volume Penalization method (CBVP), discussed in this paper, employs hyperbolic forcing terms to impose general homogeneous and inhomogeneous Neumann and Robin boundary conditions. The method is flexible and can be applied to parabolic and hyperbolic evolutionary equations. In this paper it is demonstrated for viscous and inviscid flows of arbitrary Mach number. As with BPM, this method maintains rigorous control of the error through a priori chosen parameters for all boundary conditions.