Fabien Petitpas

Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows and shocks in multiphase mixtures

Numerical approximation of the five-equation two-phase flow of Kapila et al. [A.K. Kapila, R. Menikoff, J.B. Bdzil, S.F. Son, D.S. Stewart, Two-phase modeling of deflagration-to-detonation transition in granular materials: reduced equations, Physics of Fluids 13(10) (2001) 3002-3024] is examined. This model has shown excellent capabilities for the numerical resolution of interfaces separating compressible fluids as well as wave propagation in compressible mixtures [A. Murrone, H. Guillard, A five equation reduced model for compressible two phase flow problems, Journal of Computational Physics 202(2) (2005) 664-698; R. Abgrall, V. Perrier, Asymptotic expansion of a multiscale numerical scheme for compressible multiphase flows, SIAM Journal of Multiscale and Modeling and Simulation (5) (2006) 84-115; F. Petitpas, E. Franquet, R. Saurel, O. Le Metayer, A relaxation-projection method for compressible flows. Part II. The artificial heat exchange for multiphase shocks, Journal of Computational Physics 225(2) (2007) 2214-2248]. However, its numerical approximation poses some serious difficulties. Among them, the non-monotonic behavior of the sound speed causes inaccuracies in waves transmission across interfaces. Moreover, volume fraction variation across acoustic waves results in difficulties for the Riemann problem resolution, and in particular for the derivation of approximate solvers. Volume fraction positivity in the presence of shocks or strong expansion waves is another issue resulting in lack of robustness. To circumvent these difficulties, the pressure equilibrium assumption is relaxed and a pressure non-equilibrium model is developed. It results in a single velocity, non-conservative hyperbolic model with two energy equations involving relaxation terms. It fulfills the equation of state and energy conservation on both sides of interfaces and guarantees correct transmission of shocks across them. This formulation considerably simplifies numerical resolution. Following a strategy developed previously for another flow model [R. Saurel, R. Abgrall, A multiphase Godunov method for multifluid and multiphase flows, Journal of Computational Physics 150 (1999) 425-467], the hyperbolic part is first solved without relaxation terms with a simple, fast and robust algorithm, valid for unstructured meshes. Second, stiff relaxation terms are solved with a Newton method that also guarantees positivity and robustness. The algorithm and model are compared to exact solutions of the Euler equations as well as solutions of the five-equation model under extreme flow conditions, for interface computation and cavitating flows involving dynamics appearance of interfaces. In order to deal with correct dynamic of shock waves propagating through multiphase mixtures, the artificial heat exchange method of Petitpas et al. [F. Petitpas, E. Franquet, R. Saurel, O. Le Metayer, A relaxation-projection method for compressible flows. Part II. The artificial heat exchange for multiphase shocks, Journal of Computational Physics 225(2) (2007) 2214-2248] is adapted to the present formulation.

Modelling phase transition in metastable liquids: application to cavitating and flashing flows

A hyperbolic two-phase flow model involving five partial differential equations is constructed for liquid-gas interface modelling. The model is able to deal with interfaces of simple contact where normal velocity and pressure are continuous as well as transition fronts where heat and mass transfer occur, involving pressure and velocity jumps. These fronts correspond to extra waves in the system. The model involves two temperatures and entropies but a single pressure and a single velocity. The closure is achieved by two equations of state that reproduce the phase diagram when equilibrium is reached. Relaxation toward equilibrium is achieved by temperature and chemical potential relaxation terms whose kinetics is considered infinitely fast at specific locations only, typically at evaporation fronts. Thus, metastable states are involved for locations far from these fronts. Computational results are compared to the experimental ones. Computed and measured front speeds are of the same order of magnitude and the same tendency of increasing front speed with initial temperature is reported. Moreover, the limit case of evaporation fronts propagating in highly metastable liquids with the Chapman-Jouguet speed is recovered as an expansion wave of the present model in the limit of stiff thermal and chemical relaxation.

Modelling dynamic and irreversible powder compaction

A multiphase hyperbolic model for dynamic and irreversible powder compactionis built. Four important points have to be addressed in this case. The first one isrelated to the irreversible character of powder compaction. When a granular media issubjected to a loading–unloading cycle, the final volume is lower than the initial one.To deal with this hysteresis phenomenon, a multiphase model with relaxation is built.During loading, mechanical equilibrium is assumed corresponding to stiff mechanicalrelaxation, while during unloading non-equilibrium mechanical transformation isassumed. Consequently, the sound speed of the limit models are very different duringloading and unloading. These differences in acoustic properties are responsible forirreversibility in the compaction process. The second point is related to dynamiceffects, where pressure and shock waves play an important role. Wave dynamics isguaranteed by the hyperbolic character of the equations. Phase compressibility aswell as configuration energy are taken into account. The third point is related tomulti-dimensional situations that involve material interfaces. Indeed, most processeswith powder compaction entail free surfaces. Consequently, the model should be ableto solve interfaces separating pure fluids and granular mixtures. Finally, the fourthpoint is related to gas permeation that may play an important role in some specificpowder compaction situations. This poses the difficult question of multiple-velocitydescription. These four points are considered in a unique model fitting the frameof multiphase theory of diffuse interfaces (Saurel & Abgrall, J. Comput. Phys.,vol. 150, 1999, p. 425; Kapila et al., Phys. Fluids, vol. 13, 2001, p. 3002; Saurel et al.,J. Comput. Phys., vol. 228, 2009, p. 1678). The ability of the model to deal with thesevarious effects is validated on basic situations, where each phenomenon is consideredseparately. Except for the material EOS (hydrodynamic and granular pressures andenergies), which are determined on the basis of separate experiments found in theliterature, the model is free of adjustable parameter.Key words: granular media, particle/fluid flows, shock waves

A model and numerical method for compressible flows with capillary effects

A new model for interface problems with capillary effects in compressible fluids is presented together with a specific numerical method to treat capillary flows and pressure waves propagation. This new multiphase model is in agreement with physical principles of conservation and respects the second law of thermodynamics. A new numerical method is also proposed where the global system of equations is split into several submodels. Each submodel is hyperbolic or weakly hyperbolic and can be solved with an adequate numerical method. This method is tested and validated thanks to comparisons with analytical solutions (Laplace law) and with experimental results on droplet breakup induced by a shock wave.

Toward a Thermal Disequilibrium Multiphase Model for High Explosives Containing Metallic Particles

To investigate the effects of explosive composition on Al combustion, in particular regarding its oxygen balance, several liquid mixtures are experimentally studied with varying oxygen balance. They are then loaded with Al particles and the velocity of detonation (VOD) is recorded. Computational results with the help of conventional Chapman Jouguet (CJ) codes are compared but fail to reproduce experimental observations. A new multiphase flow model out of thermal equilibrium is then considered. Two options are considered as limiting cases: stiff thermal relaxation and vanishing heat exchange between Al and detonation products. With this last option, predictions are in excellent agreement with the experiments. This suggests that temperature disequilibrium plays a major role in heterogeneous explosives detonation dynamics.

Progress in the Development of Compressible, Multiphase Flow Modeling Capability for Nuclear Reactor Flow Applications

In nuclear reactor safety and optimization there are key issues that rely on in-depth understanding of basic two-phase flow phenomena with heat and mass transfer. Within the context of multiphase flows, two bubble-dynamic phenomena – boiling (heterogeneous) and flashing or cavitation (homogeneous boiling), with bubble collapse, are technologically very important to nuclear reactor systems. The main difference between boiling and flashing is that bubble growth (and collapse) in boiling is inhibited by limitations on the heat transfer at the interface, whereas bubble growth (and collapse) in flashing is limited primarily by inertial effects in the surrounding liquid. The flashing process tends to be far more explosive (and implosive), and is more violent and damaging (at least in the near term) than the bubble dynamics of boiling. However, other problematic phenomena, such as crud deposition, appear to be intimately connecting with the boiling process. In reality, these two processes share many details.

A hyperbolic non equilibrium model for cavitating flows

A hyperbolic two-phase flow model involving five partial differential equations is built for liquid-gas interface modelling. The model is able to deal with interfaces of simple contact where normal velocity and pressure are continuous as well as transition fronts where heat and mass transfer occur, involving pressure and velocity jumps. These fronts correspond to extra waves into the system. The model involves two temperatures and entropies but a single pressure and a single velocity. The closure is achieved by two equations of state that reproduce the phase diagram when equilibrium is reached. Relaxation toward equilibrium is achieved by temperature and chemical potential relaxation terms whose kinetics is considered infinitely fast only at specific locations, typically at evaporation fronts. Doing so, metastable states are involved for locations far from these fronts. Specific numerical hyperbolic and relaxation solver are built to solve the non-conservative system. Computational tests are done in 1D and 2D and are compared to experimental observations.