Numerical methods for the simulation of reactive and non-reactive compressible flow for materials with general equations of state

Abstract

In the second part of this thesis a model for multi-material flow is described, again assuming general equations of state of Mie-Gr\"uneisen form for the constituents. A corresponding high-resolution quasi-conservative Godunov method is developed. The numerical approach employs a Riemann solver based on local approximations to the equations of state for the constituent materials. An energy correction term is included in the numerical method to capture uniform-pressure-velocity flow across material interfaces. A series of two-material flow tests is presented to verify the accuracy and robustness of the numerical method in a variety of physical situations. These include flow subsequent to the head-on collision of a planar shock with a planar interface, the collision of a planar shock with an inclined interface, and the collision of a planar shock with a spherical~inclusion.

In this thesis numerical methods for multi-phase and multi-material compressible flows are developed, verified and applied to physical problems of current interest. Here, the term `multi-phase' refers to the coexistence of the constituent materials of the model at each point, while the term `multi-material' refers to the coexistence of the constituent materials only at a diffuse interface. The thesis consists of two parts.

In the first part, a high-resolution Godunov method is developed for a two-phase model of reactive flow. The model considers general equations of state of Mie-Gr\"uneisen form for the constituents and makes different choices for the reaction rate, both relevant to simulations of the dynamical behavior of detonations in PBX-type granular explosives. The numerical approach employs a Riemann solver, and various options ranging from an exact solver to an approximate solver of HLLC type are explored. A principal aim of the study is an assessment of the merits of the approximate Riemann solvers in terms of accuracy and efficiency. Specifically, this study considers evolution to detonation subsequent to the impact of an impulsive piston. The transient phase of evolution, consisting of compaction and transition to detonation, is examined, as well as the approach to a steady compaction-led or reaction-led detonation. For these stages, the convergence properties of the numerical approach are examined for the different choices of the Riemann solver, as is the error in the solutions at lower grid resolution. Finally, the computational performance of the time-stepping scheme is assessed for the different solvers.