A Sequel of Inverse Lax–Wendroff High Order Wall Boundary Treatment for Conservation Laws

• Published in: Archives of computational methods in engineering 2021-06, Vol.28 (4), p.2315-2329
• Main Authors: Borges, Rafael B. de Rezende, da Silva, Nicholas Dicati P., Gomes, Francisco A. A., Shu, Chi-Wang, Tan, Sirui
• Format: Article
• Language: English
• Subjects: Boundary conditions; Building codes; Compressibility; Computational fluid dynamics; Conical flow; Conservation laws; Domains; Engineering; Exact solutions; Finite difference method; Fluid flow; Heat exchange; High resolution; Mathematical and Computational Engineering; Nonlinear phenomena; Nozzle flow; Numerical methods; Original Paper; Ringleb flow; Runge-Kutta method; Solvers; Straight lines; Two dimensional flow
• ISSN: 1134-3060; 1886-1784
• DOI: 10.1007/s11831-020-09454-w

When solving CFD problems, the solver, or the numerical code, plays an important role. Depending on the phenomena and problem domain, designing such numerical codes can be hard work. One strategy is to start with simple problems and construct the code as building blocks. The purpose of this work is to provide a detailed review of the theory to compute analytical and exact solutions, and recent numerical methods to construct a code to solve compressible and inviscid fluid flows with high-resolution, arbitrary domains, non-linear phenomena, and on rectangular meshes. We also propose a modification to the inverse Lax–Wendroff procedure solid wall treatment and two-dimensional WENO-type extrapolation stencil selection and weights to handle more generic situations. To test our modifications, we use the finite difference method, Lax–Friedrichs splitting, WENO-Z+ scheme, and third-order strong stability preserving Runge-Kutta time discretization. Our first problem is a simple one-dimensional transient problem with periodic boundary conditions, which is useful for constructing the core solver. Then, we move to the one-dimensional Rayleigh flow, which can handle flows with heat exchange and requires more detailed boundary treatment. The next problem is the quasi-one-dimensional nozzle flow with and without shock, where the boundary treatment needs a few adjustments. The first two-dimensional problem is the Ringleb flow, and despite being smooth, it has a curved wall as the left boundary. Finally, the last problem is a two-dimensional conical flow, which presents an oblique shock and an inclined straight line wall being the cone surface. We show that the designed accuracy is being reached for smooth problems, that high-resolution is being attained for non-smooth problems, and that our modifications produce similar results while providing a more generic way to treat solid walls.