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Abstract. In this work we describe an efficient model for the simulation of a two-phase flow made of a gas and a granular solid. The starting point is the two-velocity two-pressure model of Baer and Nunziato [Int. J. Multiph. Flow 16 (1986) . The model is supplemented by a relaxation source term in order to take into account the pressure equilibrium between the two phases and the granular stress in the solid phase. We show that the relaxation process can be made thermodynamically coherent with an adequate choice of the granular stress. We then propose a numerical scheme based on a splitting approach. Each step of the time marching algorithm is made of two stages. In the first stage, the homogeneous convection equations are solved by a standard finite volume Rusanov scheme. In the second stage, the volume fraction is updated in order to take into account the equilibrium source term. The whole procedure is entropy dissipative. For simplified pressure laws (stiffened gas laws) we are able to prove that the approximated volume fraction stays within its natural bounds.Mathematics Subject Classification. 76M12, 65M12.

Since the emergence of the first armoured vehicles on battlefields, armour shield was mainly centred on conventional metallic materials, widespread solutions nowadays. For a long time, weight reduction in armoured protection, which represents the largest part of the vehicle's overall weight, has been the key parameter for vehicle manufacturers looking forward to optimizing fuel consumption, thus increasing the payload and offering increased manoeuvrability to vehicles. The solution generally developed is a combination between those metallic plates and materials lighter than the current steel armour. In this context, the hybridization of some well-known ballistic alloys with textile composite materials appears to be a high-potential solution for armour-plated protection. Indeed, used as a backing, textile composite materials present some worthwhile properties such as having a very low density compared with steel and good behaviour in terms of ballistic efficiency. The use of a textile composite backing allows a reduction in the thickness of the metal plate by a few millimetres, which has a huge impact on the total protection weight. The difficulty of this hybridization is, of course, to reduce the total mass of the protection solution as cautiously as possible while ensuring the safety of the vehicle. The textile composite backing is also efficient in containing the pieces of shrapnel, which might break loose from the metal plate during impact. However, observations from today's military theatres of operation reveal that the spectrum of armoured

Complex phenomena occur in a combustion chamber during a ballistic cycle. From the ignition of the black powder in the primer to the exit of the projectile through the muzzle, two-phase gas-powder mix undertakes various transfers in different forms. A detailed comprehension of these effects is fundamental to predict the behavior of the whole system, considering performances and safety. Although the ignition of the powder bed is three-dimensional due to the primer’s geometry, simulations generally only deal with one- or two-dimensional problem. In this study, we propose a method to simulate the two-phase flows in 1, 2 or 3 dimensions with the same system of partial differential equations. A one-pressure, conditionally hyperbolic model [1] was used and solved by a nonconservative finite volume scheme associated to a fractional step method, where each step is hyperbolic. We extend our study to a two-pressure, unconditionally hyperbolic model [2] in which a relaxation technique was applied in order to recover the one-pressure model by using the granular stress. The second goal of this study is also to propose an improved ignition model of the powder grains, by taking into account simplified chemical kinetics for decomposition reactions in the two phases. Here we consider a 0th-order solid decomposition and an unimolecular, 2nd-order gas reaction. Validation of the algorithm on several test cases is presented.

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