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International audience; The Noble-Abel (NA) equation of state (EOS) is widely used in interior ballistics of guns as well as rocket propulsion computations. Its simplicity and accuracy are key points for intensive computations with hyperbolic two-phase flow models considered in interior ballistics codes. An alternative is examined in the present contribution through a first-order virial (VO1) equation of state. Appropriate methods for the determination of related parameters, such as specific gas constant, covolume and condensed material energy for both formulations (NA and VO1) are presented. Combination of closed bomb vessel experiments and thermochemical code computations are needed. An extended VO1 EOS with temperature dependent specific heat is examined. Then extension to multiple reactive materials is addressed. Examples are examined for each formulation (NA and VO1) and comparisons are done with the Becker-Kistiakowsky-Wilson (BKW) EOS as reference. Several conclusions emerged. First, consideration of specific heat temperature dependance in interior ballistics of guns computations appeared insignificant. Second, VO1 appeared more accurate than NA, particularly when gas density comes out of the range used for the EOS parameters determination. Last, regarding mixtures of condensed reactive materials, producing burnt gas mixtures, NA appeared again less accurate than VO1. However, its formulation is explicit, while VO1 requires numerical solving of a non-linear equation, with consequences on computational cost.

International audience; Computation of gas dispersal in urban places or hilly grounds requires a large amount of computer time when addressed with conventional multidimensional models. Those are usually based on two-phase flow or Navier-Stokes equations. Different classes of simplified models exist. Among them, two-layer shallow water models are interesting to address large-scale dispersion. Indeed, compared to conventional multidimensional approaches, 2D simulations are carried out to mimic 3D effects. The computational gain in CPU time is consequently expected to be tremendous. However, such models involve at least three fundamental difficulties. The first one is related to the lack of hyperbolicity of most existing formulations, yielding serious consequences regarding wave propagation. The second is related to the non-conservative terms in the momentum equations. Those terms account for interactions between fluid layers. Recently, these two difficulties have been addressed in Chiapolino and Saurel (2018) and an unconditional hyperbolic model has been proposed along with a Harten-Lax-van Leer (HLL) type Riemann solver dealing with the non-conservative terms. In the same reference, numerical experiments showed robustness and accuracy of the formulation. In the present paper, a third difficulty is addressed. It consists of the determination of appropriate drag effect formulation. Such effects also account for interactions between fluid layers and become of particular importance when dealing with heavy-gas dispersion. With this aim, the model is compared to laboratory experiments in the context of heavy gas dispersal in quiescent air. It is shown that the model accurately reproduces experimental results thanks to an appropriate drag force correlation. This function expresses drag effects between the heavy and light gas layers. It is determined thanks to various experimental configurations of dam-break test problems.

A new method is developed to approximate a first-order Hamilton–Jacobi equation. The constant motion of an interface in the normal direction is of interest. The interface is captured with the help of a “Level-Set” function approximated through a finite-volume Godunov-type scheme. Contrarily to most computational approaches that consider smooth Level-Set functions, the present one considers sharp “Level-Set”, the numerical diffusion being controlled with the help of the Overbee limiter (Chiapolino et al. in J Comput Phys 340:389–417, 2017). The method requires gradient computation that is addressed through the least squares approximation. Multidimensional results on fixed unstructured meshes are provided and checked against analytical solutions. Geometrical properties such as interfacial area and volume computation are addressed as well. Results show excellent agreement with the exact solutions.

Solid and liquid sodium combustion with liquid water occurs through a thin gas layer where exothermic reactions happen with sodium and water vapors. It thus involves multiple interfaces separating liquid and gas in the presence of surface tension, phase transition, and surface reactions. The gas-phase reaction involves compressible effects resulting in possible shock wave appearance in both gas and liquid phases. To understand and predict phenomena occurring during sodium combustion with water, a diffuse interface flow model is built. This formulation enables flow resolution in multidimension in the presence of complex motion, such as, for example, Leidenfrost-type thermochemical flow. More precisely sodium drop autonomous motion on the liquid surface is computed. Various modeling and numerical issues are present and addressed in the present contribution. In the author’s knowledge, the first computed results of such type of combustion phenomenon in multidimensions are presented in this paper thanks to the diffuse interface approach. Explosion phenomenon is addressed as well and is reproduced at least qualitatively thanks to extra ingredients such as turbulent mixing of sodium and water vapors in the gas film and finite rate chemistry. Shock wave emission from the thermochemical Leidenfrost-type flow is observed as well as sodium projection, as reported in related experiments.

A new Riemann solver is built to address numerical resolution of complex flow models. The research direction is closely linked to a variant of the Baer and Nunziato [2] model developed in Saurel et al. [31] . This recent model provides a link between the Marble [19] model for two-phase dilute suspensions and dense mixtures. As the Marble model, the Saurel et al. one is weakly hyperbolic with the same 4 characteristic waves, while the system involves 7 partial differential equations. It implies serious theoretical and practical issues rendering the building of a simple and accurate flow solver difficult. To overcome related difficulties the Riemann solver of Linde [18] is revisited. The method is first examined in the simplified context of the compressible Euler equations. Physical considerations are introduced in the solver improving robustness and accuracy of the Linde method. With these modifications, accuracy is shown to be comparable to the one of the HLLC solver of Toro et al. [35] . Second the two-phase flow model is considered. A locally conservative formulation is built and validated, removing issues related to the non-conservative terms. However, two extra major issues arise from numerical experiments. The solution appears not self-similar and multiple contact waves emerge in the dispersed phase. The design of a Riemann solver is consequently challenging. The modified Linde method is thus examined for the considered flow model. Some basic properties of the equations are used, such as shock relations of the dispersed phase and jump conditions across the contact wave. Thanks to these ingredients the new Riemann solver with internal reconstruction (RSIR), modification of the Linde method, handles stationary volume fraction discontinuities, presents low dissipation for transport waves and handles shocks and expansion waves accurately. It is validated on various test problems showing method's accuracy and versatility for complex flow models. Its capabilities are illustrated on a difficult two-phase flow instability problem, unresolved before.

International audience; Two-layer shallow water models present at least two fundamental difficulties that are addressed in the present contribution. The first one is related to the lack of hyperbolicity of most existing models. By considering weak compressibility of the phases, a strictly hyperbolic formulation with pressure relaxation is obtained. It is shown to tend to the conventional two-layer model in the stiff pressure relaxation limit. The second issue is related to the non-conservative terms in the momentum equations. Analyzing the Riemann problem structure, local constants appear precisely at locations where the non-conservative products need definition. Thanks to these local constants, a locally conservative formulation of the equations is obtained, simplifying the Riemann problem resolution through a HLL-type Riemann solver. The method is compared to literature data, showing accurate and oscillation free solutions. Additional numerical experiments show robustness and accuracy of the method.

International audience; Numerical investigation of two-phase loop thermosyphon (2PLT) in steady and transient states is addressed. A one-dimensional two-phase flow model describing a liquid-gas mixture in both mechanical and thermal equilibrium but out of thermodynamic equilibrium is developed. The model considers subcooled liquid and over heated vapor as well as phase transition (evaporation and condensation). The flow model is solved with a specific hyperbolic solver based on Godunov method and Harten-Lax-van Leer-Contact (HLLC) Riemann solver. A parametric study on the thermal power at the evaporator is performed in steady and transient states, the aim being to determine the effects of thermal power increase at the evaporator on the loop behavior. The comparison between Goodwin and Stiffened Gas (SG) equation of state (EOS) models shows fair agreement for latent heat of vaporization, specific volume and enthalpy for both liquid and vapor phases. Simulation of four test cases, corresponding to different evaporator thermal loads, is also carried out in transient state showing that loop response is correctly reproduced by this numerical approach, novel in the context of thermosyphon loops.

Solid and liquid sodium combustion with liquid water occurs through a thin gas layer where exothermic reactions happen with sodium and water vapors. It thus involves multiple interfaces separating liquid and gas in the presence of surface tension, phase transition and surface reactions. The gas phase reaction involves compressible effects resulting in possible shock wave appearance in both gas and liquid phases. To understand and predict the complexity of sodium combustion with water a diffuse interface flow model is built. This formulation enables flow resolution in multidimension in the presence of complex motion, such as for example Leidenfrost-type thermo-chemical flow. More precisely sodium drop autonomous motion on the liquid surface is computed. Various modelling and numerical issues are present and addressed in the present contribution. In the author's knowledge, the first computed results of such type of combustion phenomenon in multidimensions are presented in this paper thanks to the diffuse interface approach. Explosion phenomenon is addressed as well and is reproduced at least qualitatively thanks to extra ingredients such as turbulent mixing of sodium and water vapors in the gas film and delayed ignition. Shock wave emission from the thermo-chemical Leidenfrost-type flow is observed as reported in related experiments., Journal of Computational Physics, Elsevier, In press

International audience; The aim of the present work is to progress in the identification of the effects responsible for the formation of jets in heterogeneous gas-particle cylindrical and spherical explosions. In this direction three two-phase flow models are considered, namely Baer and Nunziato's (BN) (1986) model, Marble's (1963) model and the dense-dilute model of Saurel et al. (2017). The first and third ones involve both non-conservative terms and viscous drag effects while the second one involves viscous drag only as interaction force. Computed results show that viscous drag alone is unable to reproduce finger-like instabilities. The BN model and the dense-dilute one differ significantly by their acoustic properties. It is shown that the only model able to reproduce qualitatively finger-like jets is the dense-dilute model. Mesh dependence of the results is studied as well as presence or absence of viscous drag. It appears that the non-conservative terms seem responsible for jetting effects.

Implicit time integration schemes are widely used in computational fluid dynamics numerical codes to speed-up computations. Indeed, implicit schemes usually allow for less stringent time-step stability constraints than their explicit counterpart. The derivation of an implicit scheme is however a challenging and time-consuming task, increasing substantially with the model equations complexity since this method usually requires a fairly accurate evaluation of the spatial scheme's matrix Jacobian. This article presents a flexible method to overcome the difficulties associated to the computation of the derivatives, based on the forward mode of automatic differentiation using operator overloading. Flexibility and simplicity of the method are illustrated through implicit resolution of various flow models of increasing complexity such as the compressible Euler equations, a two-phase flow model in full equilibrium (Le Martelot, et al., 2014) and a symmetric variant (Saurel , et al., 2003) of the two-phase flow model of (Baer \& Nunziato, 1986) dealing with mixtures in total disequilibrium., Journal of computational physics, In press

Summary Determining liquid–vapor phase equilibrium is often required in multiphase flow computations. Existing equilibrium solvers are either accurate but computationally expensive or cheap but inaccurate. The present paper aims at building a fast and accurate specific phase equilibrium solver, specifically devoted to unsteady multiphase flow computations. Moreover, the solver is efficient at phase diagram bounds, where non-equilibrium pure liquid and pure gas are present. It is systematically validated against solutions based on an accurate (but expensive) solver. Its capability to deal with cavitating, evaporating, and condensing two-phase flows is highlighted on severe test problems both 1D and 2D. Copyright © 2016 John Wiley & Sons, Ltd.

International audience; Hot spots ignition and shock to detonation transition modeling in pressed explosives is addressed in the frame of multiphase flow theory. Shock propagation results in mechanical disequilibrium effects between the condensed phase and the gas trapped in pores. Resulting subscale motion creates hot spots at pore scales. Pore collapse is modeled as a pressure relaxation process, during which dissipated power by the ‘configuration’ pressure produces local heating. Such an approach reduces 3D micromechanics and subscale contacts effects to a ‘granular’ equation of state. Hot spots criticity then results of the competition between heat deposition and conductive losses. Heat losses between the hot solid-gas interface at pore's scale and the colder solid core grains are determined through a subgrid model using two energy equations for the solid phase. The conventional energy balance equation provides the volume average solid temperature and a non-conventional energy equation provides the solid core temperature that accounts for shock heating. With the help of these two temperatures and subscale reconstruction, the interface temperature is determined as well as interfacial heat loss.The overall flow model thus combines a full disequilibrium two-phase model for the mean solid-gas flow variables with a subgrid model, aimed to compute local solid-gas interface temperature. Its evolution results of both subscale motion dissipation and conductive heat loss. The interface temperature serves as ignition criterion for the solid material deflagration. There is no subscale mesh, no system of partial differential equations solved at grain scale.The resulting model contains less parameter than existing ones and associates physical meaning to each of them. It is validated against experiments in two very different regimes: Shock to detonation transition, that typically happens in pressure ranges of 50 kbar and shock propagation that involves pressure ranges 10 times higher.

International audience; Simulation of compressible flows became a routine activity with the appearance of shock-/contact-capturing methods. These methods can determine all waves, particularly discontinuous ones. However, additional difficulties may appear in two-phase and multimaterial flows due to the abrupt variation of thermodynamic properties across the interfacial region, with discontinuous thermodynamical representations at the interfaces. To overcome this difficulty, researchers have developed augmented systems of governing equations to extend the capturing strategy. These extended systems, reviewed here, are termed diffuse-interface models, because they are designed to compute flow variables correctly in numerically diffused zones surrounding interfaces. In particular, they facilitate coupling the dynamics on both sides of the (diffuse) interfaces and tend to the proper pure fluid–governing equations far from the interfaces. This strategy has become efficient for contact interfaces separating fluids that are governed by different equations of state, in the presence or absence of capillary effects, and with phase change. More sophisticated materials than fluids (e.g., elastic–plastic materials) have been considered as well.

International audience; A simple method is developed to couple accurately the motion of rigid bodies to compressible fluid flows. Solid rigid bodies are tracked through a Level-Set function. Numerical diffusion is controlled thanks to a compressive limiter (Overbee) in the frame of MUSCL type scheme, giving an excellent compromise between accuracy and efficiency on unstructured meshes (Chiapolino et al., 2017). The method requires low resolution to preserve solid bodies' volume. Several coupling methods are then addressed to couple rigid body motion to fluid flow dynamics: a method based on stiff relaxation and two methods based on Ghost cells (Fedkiw et al., 1999) and immersed boundaries. Their accuracy and convergence rates are compared against an immersed piston problem in 1D having exact solution. The second Ghost cell method is shown to be the most efficient. It is then extended to multidimensional computations on unstructured meshes and its accuracy is checked against flow computations around blunt bodies. Reference results are obtained when the flow evolves around a rigid body at rest. The same rigid body is then considered with prescribed velocity moving in a flow at rest. Computed results involving wave dynamics match very well. The method is then extended to two-way coupling and illustrated to several examples involving shock wave interaction with solid particles as well as phase transition induced by projectiles motion in liquid-gas mixtures. e-mails

International audience; Hot spots ignition and shock to detonation transition modeling in pressed explosives is addressed in the frame of multiphase flow theory. Shock propagation results in mechanical disequilibrium effects between the condensed phase and the gas trapped in pores. Resulting subscale motion creates hot spots at pore scales. Pore collapse is modeled as a pressure relaxation process, during which dissipated power by the ‘configuration’ pressure produces local heating. Such an approach reduces 3D micromechanics and subscale contacts effects to a ‘granular’ equation of state. Hot spots criticity then results of the competition between heat deposition and conductive losses. Heat losses between the hot solid-gas interface at pore's scale and the colder solid core grains are determined through a subgrid model using two energy equations for the solid phase. The conventional energy balance equation provides the volume average solid temperature and a non-conventional energy equation provides the solid core temperature that accounts for shock heating. With the help of these two temperatures and subscale reconstruction, the interface temperature is determined as well as interfacial heat loss.The overall flow model thus combines a full disequilibrium two-phase model for the mean solid-gas flow variables with a subgrid model, aimed to compute local solid-gas interface temperature. Its evolution results of both subscale motion dissipation and conductive heat loss. The interface temperature serves as ignition criterion for the solid material deflagration. There is no subscale mesh, no system of partial differential equations solved at grain scale.The resulting model contains less parameter than existing ones and associates physical meaning to each of them. It is validated against experiments in two very different regimes: Shock to detonation transition, that typically happens in pressure ranges of 50 kbar and shock propagation that involves pressure ranges 10 times higher.

Many two-phase flow situations, from engineering science to astrophysics, deal with transition from dense (high concentration of the condensed phase) to dilute concentration (low concentration of the same phase), covering the entire range of volume fractions. Some models are now well accepted at the two limits, but none is able to cover accurately the entire range, in particular regarding waves propagation. In the present work an alternative to the Baer and Nunziato (1986) (BN for short) model, initially designed for dense flows, is built. The corresponding model is hyperbolic and thermodynamically consistent. Contrarily to the BN model that involves 6 wave speeds, the new formulation involves 4 waves only, in agreement with the Marble (1963) model based on pressureless Euler equations for the dispersed phase, a well-accepted model for low particle volume concentrations. In the new model, the presence of pressure in the momentum equation of the particles and consideration of volume fractions in the two phases render the model valid for large particle concentrations. A symmetric version of the new model is derived as well for liquids containing gas bubbles. This model version involves 4 wave speeds as well, but with different wave's speeds. Last, the two sub-models with 4 waves are combined in a unique formulation, valid for the full range of volume fractions. It involves the same 6 wave's speeds as the BN model, but at a given point of space 4 waves only emerge, depending on the local volume fractions. The non-linear pressure waves propagate only in the phase with dominant volume fraction. The new model is tested numerically on various test problems ranging from separated phases in a shock tube to shock – particle cloud interaction. Its predictions are compared to BN and Marble models as well as against experimental data. Emails: Richard.Saurel@univ-amu.fr; Ashwin.Chinnayya@ensma.fr; Quentin.Carmouze@rs2n.eu 2 I. Introduction It is well accepted that hyperbolic models are mandatory to deal with phenomena involving wave propagation. This is the case for multiphase flow mixtures in many situations such as in particular shocks and detonations propagation in granular explosives and in fuel suspensions, as well as liquid-gas mixtures with bubbles, cavitation and flashing, as soon as motion is intense and governed by pressure gradients. This is thus the case of most unsteady two-phase flows situations. Wave propagation is important as it carries pressure, density and velocity disturbances. Sound propagation is also very important as it determines critical (choked) flow conditions and associated mass flow rates. It has also fundamental importance on sonic conditions of detonation waves when the two-phase mixture is exothermically reacting (Petitpas et al., 2009). Hyperbolicity is also related to the causality principle, meaning that initial and boundary conditions are responsible of time evolution of the solution. When dealing with first-order partial differential equations it means that the Riemann problem must have a solution, and the Riemann problem is correctly posed only if the equations are hyperbolic. However, only a few two-phase flow models are hyperbolic in the whole range of parameters. The Baer and Nunziato (1986) model seems to be the only formulation able to deal with such requirement. Its symmetric extension (Saurel et al., 2014) facilitates the Riemann problem resolution as the corresponding model involves 7 wave's speeds (instead of 6 in the original version). See also Ambroso et al. (2012) for similar conclusions. However, in the dilute limit at least, the acoustic properties of this model seem inconsistent (Lhuillier et al., 2013). Indeed with this model, the dispersed phase sound speed corresponds to the one of the pure phase, while this phase is not continuous and unable to propagate sound in reality, at least at a scale larger than particle's one. When the phase is not continuous (dispersed drops in a gas, dispersed bubbles in a liquid), the associated sound speed should vanish, such effect being absent in the formulation. In the low particles concentration limit, the Marble (1963) model is preferred. This model corresponds to the Euler equations with source terms for the gas phase and pressureless gas dynamic equations for the particle phase (see also Zeldovich, 1970). This model is thermodynamically consistent and hyperbolic as well, except that the particle phase equations are hyperbolic degenerate. In this model, contrarily to the BN model, sound doesn't propagate in the particles phase, this behaviour being more physical in this limit. However, the Marble model has a limited range of validity as the volume of the dispersed phase is neglected, this assumption having sense only for low (less than per cent) condensed phase volume fraction. There are thus fundamental differences between these two models:-The volume occupied by the condensed phase is considered in BN while it is neglected in the dilute model, restricting its validity to low dispersed phase volume fractions.-Condensed phase compressibility is considered in BN while incompressible particles are assumed in the dilute formulation.-Acoustic properties of the BN model are well accepted in the dense domain but seem inappropriate in the dilute limit. Even if these two models can be used in the entire space of two-phase flow variables without yielding computational failure (this is characteristic of thermodynamically consistent hyperbolic models) validity of their results is questionable when they are used out of their range of physical validity. This issue has been clearly understood in Lhuillier et al. (2013), McGrath et al. (2016) and Houim and Oran (2016) where various attempts to build new formulations are reported. In Lhuillier et al. (2013) discussion on the volume fraction equation is done, but no explicit flow model is given. In McGrath et al. (2016) a model is given with conditional hyperbolicity. Same issue is present with different cause in Houim and Oran (2016). The aim of the present paper is to build an alternative to the BN model with improved acoustic properties, while remaining unconditionally hyperbolic and thermodynamically consistent. The new model is derived from number density and particle radius (or bubble radius) evolution equations resulting in a volume fraction evolution equation expressed in conservation form with

During the last decade, investigations have been achieved to determine the physical mechanism which governs particle jet formation induced by the dispersion of a granular medium exposed to an impulsive pressure load, i.e., by a shock or a blast wave. This kind of such physical mechanism is observed during explosions or in nature as volcanic eruptions [1]. The formation of particle jets is also observed when a solid projectile impacts a particle layer [2]. Previous experiments have been conducted so far in three-dimensional spherical configurations using explosives surrounded by a granular layer [3–7]. In these experiments, the particle jet formation was obtained and clearly observed. The attempt was to correlate the particle jet distribution to the initial parameters like the particle diameter and density, the particle layer thickness and the strength of the incident shock wave in order to understand the breaking mode of a solid particle cluster. Even if it has been shown that the formation of particle jets depends both on the particle material properties and the ratio between the particle layer and the explosive mass [3], it was quite difficult to accurately relate the jet distribution with the initial conditions. More recently, cylindrical experiments were performed by Frost et al. [8] where it was easier to observe the particle jet formation. In spite of these studies, the selection mechanism is still not understood. The present experimental study was conducted in quasi two-dimensional geometry in order to ensure accurate characterization of the phenomenon. We present the concept of the two-dimensional experiments carried out. Parameters such as material density, strength of the incident shock wave, and the initial geometry of particle rings can be changed in order to find out what parameters can control the breaking mode of the particle clusters [9].

International audience; A flow model is derived for the numerical simulation of interfacial flows with phase transition. The model arises from the classical multi-component Euler equations, but is associated to a non-classical thermodynamic closure: each phase is compressible and evolves in its own subvolume, with phases sharing common pressure, velocity and temperature, leading to non-trivial thermodynamic relations for the mixture. Phase transition is made possible through the introduction of Gibbs free energy relaxation terms in the equations. Capillary effects and heat conduction—essential in boiling flows—are introduced as well. The resulting multi-phase flow model is hyperbolic, valid for arbitrary density jumps at interfaces as well as arbitrary flow speeds. Its capabilities are illustrated successively through examples of nozzle induced cavitation and heated wall induced boiling.

International audience; The present paper aims at building a fast and accurate phase transition solver dedicated to unsteady multiphase flow computations. In a previous contribution (Chiapolino et al. 2017), such a solver was successfully developed to compute thermodynamic equilibrium between a liquid phase and its corresponding vapor phase. The present work extends the solver's range of application by considering a multicomponent gas phase instead of pure vapor, a necessary improvement in most practical applications. The solver proves easy to implement compared to common iterative procedures, and allows systematic CPU savings over 50%, at no cost in terms of accuracy. It is validated against solutions based on an accurate but expensive iterative solver. Its capability to deal with cavitating, evaporating and condensing two-phase flows is highlighted on severe test problems both 1D and 2D.

Diffuse interfaces appear with any Eulerian discontinuity capturing compressible flow solver. When dealing with multifluid and multimaterial computations, interfaces smearing results in serious difficulties to fulfil contact conditions, as spurious oscillations appear. To circumvent these difficulties, several approaches have been proposed. One of them relies on multiphase flow modelling of the numerically diffused zone and is based on extended hyperbolic systems with stiff mechanical relaxation (Saurel and Abgrall, 1999 [4], Saurel et al., 2009 [6]). This approach is very robust, accurate and flexible in the sense that many physical effects can be included: surface tension, phase transition, elastic-plastic materials, detonations, granular effects etc. It is also able to deal with dynamic appearance of interfaces. However it suffers from an important drawback when long time evolution is under interest as the interface becomes more and more diffused. The present paper addresses this issue and provides an efficient way to sharpen interfaces. A sharpening flow model is used to correct the solution after each time step. The sharpening process is based on a hyperbolic equation that produces a steady shock in finite time at the interface location. This equation is embedded in a “sharpening multiphase model” redistributing volume fractions, masses, momentum and energy in a consistent way. The method is conservative with respect to the masses, mixture momentum and mixture energy. It results in diffused interfaces sharpened in one or two mesh points. The method is validated on test problems having exact solutions.

This special issue is dedicated to recent developments concerning modern numerical methods for time dependent partial differential equations that can be used to simulate complex nonlinear flow and transport processes and tries to cover a wide spectrum of different applications and numerical approaches. Most of the models under consideration here are nonlinear hyperbolic conservation laws, which are a very powerful mathematical tool to describe a large set of problems in science and engineering. Hyperbolic PDE are based on first principles, such as mass, momentum and total energy conservation as well as the second law of thermodynamics. These principles are universally valid and have been successfully used for the simulation of a very wide class of problems, ranging from astrophysics over geophysics to complex flows in engineering and computational biology. Most of the seemingly different applications listed above have a common mathematical description under the form of nonlinear hyperbolic systems of partial differential equations, possibly containing also higher-order derivative terms, non-conservative products and nonlinear (potentially stiff) source terms. From the mathematical point of view, the major difficulties in these systems arise from the nonlinearities that allow the formation of non-smooth solutions such as shock waves, material interfaces or other discontinuities, but nonlinear conservation laws can simultaneously also contain smooth features like sound waves and vortex flows. Even nowadays the key challenge for the construction of successful numerical methods for nonlinear hyperbolic PDE is therefore to capture at the same time shock waves and discontinuities robustly without producing spurious oscillations and exhibit little numerical dissipation and dispersion errors in smooth regions of the solution. For this reason, the development of robust and accurate numerical methods for nonlinear flow and transport problems remains a great challenge even after several decades of successful research, which started back in the 1940s and 1950s with the pioneering works of von Neumann [1] and Godunov [2] . The present special issue is published on the occasion of the 70th birthday of Prof. Eleuterio F. Toro and in honor of his outstanding contributions to the fields of applied mathematics, scientific computing and computational physics with his well-known numerical methods for the solution of hyperbolic conservation laws and their application to computational fluid dynamics, computational physics and computational biology.

Phase changes and heat exchanges are examples of physical processes appearing in many industrial applications involving multiphase compressible flows. Their knowledge is of fundamental importance to reproduce correctly the resulting effects in simulation tools. A fine description of the flow topology is thus required to obtain the interfacial area between phases. This one is responsible for the dynamics and the kinetics of heat and mass transfer when evaporation or condensation occurs. Unfortunately this exchange area cannot be obtained easily and accurately especially when complex mixtures (drops, bubbles, pockets of very different sizes) appear inside the transient medium. The natural way to solve this specific trouble consists in using a thin grid to capture interfaces at all spatial scales. But this possibility needs huge computing resources and can be hardly used when considering physical systems of large dimensions. A realistic method is to consider instantaneous exchanges between phases by the way of additional source terms in a full non-equilibrium multiphase flow model [2,15,17]. In this one each phase obeys its own equation of state and has its own set of equations and variables (pressure, temperature, velocity, energy, entropy,...). When enabling the relaxation source terms the multiphase mixture instantaneously tends towards a mechanical or thermodynamic equilibrium state at each point of the flow. This strategy allows to mark the boundaries of the real flow behavior and to magnify the dominant physical effects (heat exchanges, evaporation, drag,...) inside the medium. A description of the various relaxation processes is given in the paper. Les changements de phase et les transferts de chaleur sont des exemples de phénomènes physiques présents dans de nombreuses applications industrielles faisant intervenir des écoulements compressibles multiphasiques. La connaissance des mécanismes associés est primordiale afin de reproduire correctement leurs effets à travers des outils de simulation. Dans ce cadre, une description fine de la topologie d’un écoulement est nécessaire afin de connaître précisément l’aire interfaciale entre toutes les phases. Celle-ci est en effet responsable de la dynamique et de la cinétique des transferts de masse et de chaleur lorsque de l’évaporation et de la condensation se produisent. Malheureusement cette aire interfaciale est difficilement accessible particulièrement lorsque des mélanges complexes se forment (gouttes, bulles, inclusions de différentes tailles) au sein du milieu. La façon la plus naturelle de résoudre ce problème est d’utiliser un maillage suffisamment fin afin de capturer toutes les interfaces présentes à toutes les échelles. Cependant cette possibilité demanderait des ressources informatiques démesurées au vue de certains systèmes pouvant être de très grande taille. Une méthode plus réaliste est de considérer que les échanges entre les phases s’effectuent instantanément. Des termes sources de relaxation liés à ces échanges sont utilisés dans un modèle d’écoulement compressible à phases séparées en déséquilibre [2,15,17]. Dans celui-ci, chaque phase possède son propre jeu d’équations et ses propres variables (pression, vitesse, température, énergie, entropie, ...). Quand les termes de relaxation sont activés, le mélange multiphasique tend instantanément en chaque point de l’écoulement vers un état d’équilibre prédéfini. Cette approche permet également de borner les conditions réelles d’écoulement et de souligner les effets physiques dominants (transfert de chaleur, évaporation, trainée, ...). Une description des différents processus de relaxation est proposée dans ce papier.

Computation of interfaces separating compressible materials is related to mixture cells appearance. These mixture cells are consequences of fluid motion and artificial smearing of discontinuities. The correct computation of the entire flow field requires perfect fulfillment of the interface conditions. In the simplest situation of contact interfaces with perfect fluids, these conditions correspond to equal normal velocities and equal pressures. To compute compressible flows with interfaces two main classes of approaches are available. In the first one, the interface is considered as a sharp discontinuity. Lagrangian, Front Tracking and Level Set methods belong to this class. The second option consists in the building of a flow model valid everywhere, in pure materials and mixture cells, solved routinely with a unique Eulerian algorithm [37]. In this frame, the interface is considered as a numerically diffused zone, captured by the algorithm. There are some advantages with this approach, as the corresponding flow model is not only valid in artificial mixture cells, but it also describes accurately true multiphase mixtures of materials. The [37] approach has been simplified by [22] with the help of asymptotic analysis, resulting in a single velocity, single pressure but multi-temperature flow model. This reduced model presents however difficulties for its numerical resolution as one of the equations is non-conservative. In the presence of shocks, jump conditions have been provided by [42], determined in the weak shock limit. When compared against experiments for both weak and strong shocks, excellent agreement was observed. These relations have been accepted as closure shock relations for the [22] model and allowed the study of detonation waves in heterogeneous energetic materials. Generalized Chapman-Jouguet conditions were obtained as well as heterogenous explosives (non-ideal) detonation wave structures [36]. Oppositely to the previous example of exothermic reactions and high speed flows, endothermic reactions are considered in [43] to deal with cavitating and flashing flows. In conjunction with capillary [33] and diffusive effects, it has been possible to deal with boiling flows [25]. Extra multiphysic extensions such as dynamic powder compaction [38], solid-fluid coupling in extreme deformations [12] have been investigated too. La résolution numérique des problèmes à interfaces entre milieux compressibles est fortement liée à l’apparition de ’mailles de mélange’. Celles-ci sont liées au mouvement des fluides sur maillages fixes, induisant une diffusion artificielle des discontinuités de contact. La résolution numérique de l’ensemble de l’écoulement nécessite le respect des conditions d’interface, qui correspondent pour les fluides parfaits à l’égalité des vitesses normales et à l’égalité des pressions. Pour résoudre les écoulements de fluides compressibles en présence d’interfaces matérielles, essentiellement deux catégories de méthodes sont disponibles. Dans la première catégorie, l’interface est considérée comme une discontinuité. Les méthodes lagrangiennes, de ’suivi de front’ et Level Set appartiennent à cette catégorie. La seconde option consiste en la construction d’un modèle d’écoulement valide partout, dans les milieux purs ainsi que dans les zones de mélange, résolu routinièrement avec un unique solveur eulérien [37]. Dans ce contexte, les interfaces sont traitées en tant que zones de diffusion numérique, capturées par l’algorithme. Cette approche présente certains avantages. Par exemple, le modèle d’écoulement n’est pas seulement valable dans les mailles de mélange artificiel, mais aussi dans les zones d’écoulement diphasiques d’origine physique et ayant des évolutions hors d’équilibre. L’approche de [37] a été simplifiée par [22] sur la base d’une analyse asymptotique dans la limite de forts coefficients de relaxation des vitesses et des pressions. Cette analyse conduit à un modèle à une seule vitesse et une seule pression, mais en déséquilibre de températures. La résolution numérique de ce modèle présente néanmoins certaines difficultés, en raison du caractère non conservatif d’une des équations. En présence d’ondes de choc, des relations de saut ont été proposées [42] et justifiées dans la limite des chocs faibles. Ces relations ont été validées par rapport à toutes les expériences disponibles, à la fois pour les chocs faibles et forts. Elles ont donc été acceptées comme relations de fermeture aux chocs pour le modèle de [22] et ont permis l’étude des ondes de détonation dans les matériaux énergétiques hétérogènes. Des conditions de Chapman-Jouguet généralisées ont été obtenues ainsi que la structure des ondes de détonation correspondantes (non idéales) [36]. A l’opposé des situations précédentes de transformations exothermiques dans des écoulements à grandes vitesses, des réactions d’évaporation endothermiques ont été considérées [43] pour traiter les écoulements cavitants et l’évaporation ’flash’. Lorsque cette modélisation de la transition de phase est combinée aux effets capillaires [33] et diffusifs, il est possible de traiter la simulation numérique directe de l’ébullition nucléée [25]. Enfin, d’autres extensions sont aussi envisageables, telles que la compaction dynamique des poudres [38] ou le couplage solide-fluide en déformations extrêmes [12].

The dispersion of particles by blast or shock waves induces the formation of coherent structures taking the shape of particle jets. In the present study, a blast wave, issued from an open shock tube, is generated at the center of a granular ring initially confined in a Hele-Shaw cell. With the present experimental setup, solid particle jet formation is clearly observed in a quasi-two-dimensional configuration. In all instances, the jets are initially generated inside the particle ring and thereafter expelled outward. Furthermore, thanks to the two-dimensional experimental configuration, a general study of the main parameters involved in these types of flows can be performed. Among them, the particle diameter, the density of the particles, the initial size of the ring, the shape of the overpressure generated and the surface friction of the Hele-Shaw cell are investigated. Empirical relationships are deduced from experimental results.

International audience; Hyperbolic two-phase flow models have shown excellent ability for the resolution of a wide range of applications ranging from interfacial flows to fluid mixtures with several velocities. These models account for waves propagation (acoustic and convective) and consist in hy-perbolic systems of partial differential equations. In this context, each phase is compressible and needs an appropriate convex equation of state (EOS). The EOS must be simple enough for intensive computations as well as boundary conditions treatment. It must also be accurate , this being challenging with respect to simplicity. In the present approach, each fluid is governed by a novel EOS named 'Noble Abel Stiffened Gas' (NASG), this formulation being a significant improvement of the popular 'Stiffened Gas' (SG) EOS. It is a combination of the so-called 'Noble-Abel' and 'Stiffened Gas' equations of state that adds repulsive effects to the SG formulation. The determination of the various thermodynamic functions and associated coefficients is the aim of this article. We first use thermodynamic considerations to determine the different state functions such as the specific internal energy, enthalpy and entropy. Then we propose to determine the associated coefficients for a liquid in the presence of its vapor. The EOS parameters are determined from experimental saturation curves. Some examples of liquid-vapor fluids are examined and associated parameters are computed with the help of the present method. Comparisons between analytical and experimental saturation curves show very good agreement for wide ranges of temperature for both liquid and vapor.

Dispersed liquid droplet flows with evaporation and condensation in multi-component gas mixture made of vapor and other gas phase chemical species such as air occur in many engineering applications dealing with two-phase flows. However, existing models are essentially derived for vaporization occurring in sprays combustion. It means that the energy is transferred from a hot gas to the liquid to produce its phase change. This is thus a non-symmetric approach as in some situations the energy is already stored in the liquid phase and flashing occurs as a consequence of pressure drop. In the present paper a droplet mass transfer model is derived and is valid in any situation: evaporation, flashing and condensation. It accounts for: - coupled heat and mass diffusion in the gas phase, - thermodynamics of the multi-component gas mixture, - heat diffusion inside the liquid droplet, enabling consideration of both droplet heating and cooling. These effects are important in evaporating and flashing situations respectively. The resulting model consists in an algebraic non-linear system of three equations giving the interface temperature, the mass flow rate and vapor species concentration at the interface. These interfacial variables enable computation of the mass species, momentum and energy transfer rates appearing in volume averaged two-phase flow models. Computational examples are shown with this mass transfer model embedded in a compressible two-phase flow model of Baer and Nunziato (1986) type.

The diffuse interface model of Saurel et al. (2008) is used for the computation of compressible cavitating flows around underwater missiles. Such systems use gas injection and natural cavitation to reduce drag effects. Consequently material interfaces appear separating liquid and gas. These interfaces may have a really complex dynamics such that only a few formulations are able to predict their evolution. Contrarily to front tracking or interface reconstruction method the interfaces are computed as diffused numerical zones, that are captured in a routinely manner, as is done usually with gas dynamics solvers for shocks and contact discontinuity. With the present approach, a single set of partial differential equations is solved everywhere, with a single numerical scheme. This leads to very efficient solvers. The algorithm derived in Saurel et al. (2009) is used to compute cavitation pockets around solid bodies. It is first validated against experiments done in cavitation tunnel at CNU. Then it is used to compute flows around high speed underwater systems (Shkval-like missile). Performance data are then computed showing method ability to predict forces acting on the system.

Conventional modeling of two-phase dilute suspensions is achieved with the Euler equations for the gas phase and gas dynamics pressureless equations for the dispersed phase, the two systems being coupled by various relaxation terms. The gas phase equations form a hyperbolic system but the particle phase corresponds to a hyperbolic degenerated one. Numerical difficulties are thus present when dealing with the dilute phase system. In the present work, we consider the addition of turbulent effects in both phases in a thermodynamically consistent way. It results in two strictly hyperbolic systems describing phase’s dynamics. Another important feature is that the new model has improved physical capabilities. It is able, for example, to predict particle dispersion, while the conventional approach fails. These features are highlighted on several test problems involving particles jets dispersion and are compared against experimental data. With the help of a single parameter (a turbulent viscosity), excellent agreement is obtained for various experimental configurations studied by different authors