A roe-average algorithm for a granular-gas model with non-conservative terms

A Roe-average algorithm has been derived for a granular-gas model, proposed by Goldshtein and Shapiro [Goldshtein, Shapiro, Mechanics of collisional motion of granular materials: Part 1. General hydrodynamic equations, J. Fluid Mech. 282 (1995) 75–114], which contains non-conservative terms in the Euler-like hyperbolic governing equations apart from sink terms, which arise from inelastic collision of granules and are present only in the energy equation. The non-conservative terms introduce non-isentropic effects in acoustic-wave propagation within granular media and they also contribute to the Rankine–Hugoniot relations across a discontinuity. A Roe-average algorithm, based on the same granular-gas model, was derived in the literature [V. Kamenetsky, A. Goldshtein, M. Shapiro, D. Degani, Evolution of a shock wave in a granular gas, Phys. Fluids, 12 (2000) 3036–3049] which then required the implementation of a shock-fitting technique at a discontinuity. In the present work, Roe-averaged variables have been obtained from the Rankine–Hugoniot jump relations and the non-conservative terms have been incorporated in the numerical flux formula consistent with upwind principles associated with the granular speed of sound. Results for unsteady one-dimensional granular flows, colliding with a wall, demonstrate the capability of the proposed algorithm to capture strong shocks in addition to flow features not found in molecular gases, such as a fluidized region downstream of the shock and a compacted solid-block region adjacent to the wall.