A High-Resolution Numerical Method for a Two-Phase Model of Deflagration-to-Detonation Transition

A conservative, upwind numerical method is formulated for the solution of a two-phase (reactive solid and inert gas) model of deflagration-to-detonation transition (DDT) in granular energetic solids. The model, which is representative of most two-phase DDT models, accounts for complete nonequilibrium between phases and constitutes a nonstrictly hyperbolic system of equations having parabolic degeneracies. The numerical method is based on Godunovʹs methodology and utilizes a new approximate solution for the two-phase Riemann problem for arbitrary equations of state. The approximate solution is similar to the Roe-type Riemann solution for single-phase systems. The method is able to accurately capture strong shocks associated with each phase without excessive smearing or spurious oscillations and can accurately resolve fine-scale detonation structure resulting from interaction between phases. The utility of the method is demonstrated by comparing numerical predictions with known solutions for three test cases: (1) a two-phase shock tube problem; (2) the evolution of a steady compaction wave in a granular material resulting from weak piston impact (∼100 m/s); and (3) the evolution of a steady two-phase detonation wave in an energetic granular material resulting from weak piston impact. The nominally second-order accurate numerical method is shown to have global convergence rates of 1.001 and 1.670 for inert test cases with (case 1) and without (case 2) discontinuities, respectively. For the reactive test case having a discontinuity (case 3), a convergence rate of 1.834 was predicted for coarse grids that seemed to be approaching the expected value of unity with increasing resolution.