Modelling detonation waves in condensed energetic materials: multiphase CJ conditions and multidimensional computations

A hyperbolic multiphase flow model with a single pressure and a single velocity but several temperatures is proposed to deal with the detonation dynamics of condensed energetic materials. Temperature non-equilibrium effects are mandatory in order to deal with wave propagation (shocks, detonations) in heterogeneous mixtures. The model is obtained as the asymptotic limit of a total non-equilibrium multiphase flow model in the limit of stiff mechanical relaxation only (Kapila et al. in Phys Fluids 13:3002–3024, 2001). Special attention is given to mass transfer modelling, that is obtained on the basis of entropy production analysis in each phase and in the system (Saurel et al. in J Fluid Mech 607:313–350, 2008). With the help of the shock relations given in Saurel et al. (Shock Waves 16:209–232, 2007) the model is closed and provides a generalized ZND formulation for condensed energetic materials. In particular, generalized CJ conditions are obtained. They are based on a balance between the chemical reaction energy release and internal heat exchanges among phases. Moreover, the sound speed that appears at sonic surface corresponds to the one of Wood (A textbook of sound, G. Bell and Sons LTD, London, 1930) that presents a non-monotonic behaviour versus volume fraction. Therefore, non-conventional reaction zone structure is observed. When heat exchanges are absent, the conventional ZND model with conventional CJ conditions is recovered. When heat exchanges are involved interesting features are observed. The flow behaviour presents similarities with non ideal detonations (Wood and Kirkwood in J Chem Phys 22:1920–1924, 1950) and pathological detonations (Von Neuman in Theory of detonation waves, 1942; Guenoche et al. in AIAA Prog Astron Aeronaut 75: 387–407, 1981). It also present non-conventional behaviour with detonation velocity eventually greater than the CJ one. Multidimensional resolution of the corresponding model is then addressed. This poses serious difficulties related to the presence of material interfaces and shock propagation in multiphase mixtures. The first issue is solved by an extension of the method derived in Saurel et al. (J Comput Phys 228(5):1678–1712, 2009) in the presence of heat and mass transfers. The second issue poses the difficult mathematical question of numerical approximation of non-conservative systems in the presence of shocks associated to the physical question of energy partition among phases for a multiphase shock. A novel approach is used, based on extra evolution equations used to retain the information of the material initial state. This method insures convergence in the post-shock state. Thanks to these various theoretical and numerical ingredients, one-dimensional and multidimensional unsteady detonation waves computations are done, eventually in the presence of material interfaces. Convergence of the numerical hyperbolic solver against ZND multiphase solution is reached. Material interfaces, shocks, detonations are solved with a unified formulation where the same equations are solved everywhere with the same numerical scheme.