Modeling flows with curved detonation waves

Abstract

The nonzero width of a curved detonation front has a significant effect on its propagation. Physically, the curvature effect is a small but important correction. In numerical simulations, the curvature effect tends to be greatly exaggerated due to an artificially large width of an underresolved wave. In the context of reactive fluid flow, a detonation wave consists of a lead shock followed by a thin reaction zone. The curvature effect is determined by the dynamics within the reaction zone; in particular, the competition between a source term for the rate of chemical energy release and a geometric source term due to front curvature. When the width of the reaction zone is small compared with the radius of curvature of the front, the reaction zone can be approximated as quasi-steady and be modeled locally by a system of ordinary differential equations (ODEs) in which the front curvature enters as a parameter. Front curvature breaks the Galilean invariance and the quasi-steady approximation is only valid in a distinguished frame determined by both the front curvature and the tangential velocity divergence ahead of the front. The quasi-steady ODEs determine the reaction zone profile to leading order and can be viewed as an extension of the ZND model. However, the source terms in the ODEs lead to modified Hugoniot jump conditions that take into account front curvature and reaction zone width. When the reaction zone is underresolved, a calculation in effect reduces to a “capturing algorithm” in which the burn model plays an analogous role for detonation waves as artificial viscosity does for shock waves. In particular, the reaction zone has an unphysically large width that is proportional to the cell size. As a consequence of the modified jump conditions, the numerical reaction zone width gives rise to an artificial curvature effect. This causes numerical solutions to depend on the cell size and orientation of a detonation front relative to the grid. Two algorithms that eliminate numerical curvature effects are discussed, detonation shock dynamics and front tracking.