Path-conservative well-balanced high-order finite-volume solver for the volume-averaged Navier–Stokes equations with discontinuous porosity

Abstract

This study develops a novel numerical scheme to the volume-averaged Navier–Stokes equations, specifically addressing challenges posed by discontinuous porosity fields. Utilizing the path-conservative approach, we propose tailored paths that ensure the conservation of mass and energy across discontinuities. Introducing these paths into the generalized Rankine–Hugoniot relation, we define a stationary wave that incorporates the effects of discontinuous porosity onto the flow. This stationary wave serves as a fundamental component of the exact solution to the Riemann problem. Analyzing the solution space via the L–M/R–M curve approach, we prove that the existence and uniqueness of the solution are guaranteed if the artificial parameter is sufficiently large. Building on the identified structure of the exact solution, a path-conservative well-balanced high-order finite-volume solver is designed. The WENO reconstruction is implemented to achieve high-order accuracy. Mimicking the exact solution structure, we formulate the stationary wave reconstruction to capture the effect of discontinuous porosity on the flow. Lastly, the source term is handled by a well-balanced high-order approximation. Numerical tests were conducted to verify the well-balanced property, high-order accuracy, and shock-capturing capability of the proposed method, demonstrating excellent agreement with reference solutions.