1D Exact Elastic-Perfectly Plastic Solid Riemann Solver and Its Multi-Material Application

The equation of state (EOS) plays a crucial role in hyperbolic conservation laws for the compressible fluid. Whereas, the solid constitutive model with elastic-plastic phase transition makes the analysis of the solid Riemann problem more difficult. In this paper, one-dimensional elastic-perfectly plastic solid Riemann problem is investigated and its exact Riemann solver is proposed. Different from previous works treating the elastic and plastic phases integrally, we resolve the elastic wave and plastic wave separately to understand the complicate nonlinear waves within the solid and then assemble them together to construct the exact Riemann solver for the elastic-perfectly plastic solid. After that, the exact solid Riemann solver is associated with the fluid Riemann solver to decouple the fluid-solid multi-material interaction. Numerical tests, including gas-solid, water-solid high-speed impact problems are simulated by utilizing the modified ghost fluid method (MGFM).

Extension and Comparative Study of AUSM-Family Schemes for Compressible Multiphase Flow Simulations

Several recently developed AUSM-family numerical flux functions (SLAU, SLAU2, AUSMM+-up2, and AUSMPW+) have been successfully extended to compute compressible multiphase flows, based on the stratified flow model concept, by following two previous works: one by M.-S. Liou, C.-H. Chang, L. Nguyen, and T.G. Theofanous [AIAA J. 46:2345-2356, 2008], in which AUSM+-up was used entirely, and the other by C.-H. Chang, and M.-S. Liou [J. Comput. Phys. 225:840-873, 2007], in which the exact Riemann solver was combined into AUSM+-up at the phase interface. Through an extensive survey by comparing flux functions, the following are found: (1) AUSM+-up with dissipation parameters of Kp and Ku equal to 0.5 or greater, AUSMPW+, SLAU2, AUSM+-up2, and SLAU can be used to solve benchmark problems, including a shock/water-droplet interaction; (2) SLAU shows oscillatory behaviors [though not as catastrophic as those of AUSM+ (a special case of AUSM+-up with Kp = Ku = 0)] due to insufficient dissipation arising from its ideal-gas-based dissipation term; and (3) when combined with the exact Riemann solver, AUSM+-up (Kp = Ku = 1), SLAU2, and AUSMPW+ are applicable to more challenging problems with high pressure ratios.

An exact Riemann-solver-based solution for regular shock refraction

We study the classical problem of planar shock refraction at an oblique density discontinuity, separating two gases at rest. When the shock impinges on the density discontinuity, it refracts, and in the hydrodynamical case three signals arise. Regular refraction means that these signals meet at a single point, called the triple point. After reflection from the top wall, the contact discontinuity becomes unstable due to local Kelvin–Helmholtz instability, causing the contact surface to roll up and develop the Richtmyer–Meshkov instability (RMI). We present an exact Riemann-solver-based solution strategy to describe the initial self-similar refraction phase, by which we can quantify the vorticity deposited on the contact interface. We investigate the effect of a perpendicular magnetic field and quantify how its addition increases the deposition of vorticity on the contact interface slightly under constant Atwood number. We predict wave-pattern transitions, in agreement with experiments, von Neumann shock refraction theory and numerical simulations performed with the grid-adaptive code AMRVAC. These simulations also describe the later phase of the RMI.