Multidimensional Detonation Solutions from Reactive Navier-Stokes Equations

This study will describe multi-dimensional detonation wave solutions of the compressible reactive Navier-Stokes equations. As discussed in detail by Fickett and Davis [1], a steady onedimensional detonation with a spatially resolved reaction zone structure is known as ZND wave, named after Zeldovich, von Neumann, and Döring. In experiments [2] and calculations with simplified models [3], [4], [5], [6], it has been observed and predicted that these ZND waves are unstable. In the experiments, detonation in a tube with walls coated with a thin layer of soot etches detailed regular patterns on the tube walls, indicating the existence of cellular detonation wave structure. Linear analysis [3] demonstrates the fundamental instability of the one-dimensional ZND structure. This is extended by analysis of the full one-dimensional unsteady Euler equations to describe galloping detonations [4]. In two-dimensional calculations [5] it is found that complex cellular structures and transverse waves evolve from the original one-dimensional, steady detonation, for cases in which the steady one-dimensional structure is unstable. Grismer and Powers [6] have shown numerically that detonations which are guaranteed stable in one dimension can be unstable when the geometry is relaxed to include two-dimensional effects. Most calculations are done with compressible reactive Euler equations, and two-dimensional cell size is often predicted to be dependent on grid resolution, which indicates numerical viscosity is playing a determining role in predicting the physics. To remedy this, we reintroduce in this study the usually-neglected physical mechanisms of mass, momentum, and energy diffusion to the conservation equations. In this abstract, we give results of our initial calculations which are very similar to those of Lindström [7]. The full paper will extend these results to consider the effects of diffusion on one-dimensional structure, wall boundary layer effects, and the corrections for diffusion coefficients with dependency on thermodynamic properties. If feasible, the full paper will also present some three-dimensional results. The model equations are as follows, employing essentially the same non-dimensionalization as that of Ref. [5], with extensions made to account for the diffusion coefficients: