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Mie–Grüneisen type equations of state (EOS) are widely used to describe the thermodynamics of solids, liquids, and gases in a variety of physics problems, including shock wave dynamics in condensed materials and the thermodynamic behavior of dense gases generated by detonation waves. The Jones–Wilkins–Lee (JWL), Cochran–Chan, and original Mie–Grüneisen EOSs are relevant examples. However, these EOSs present several major difficulties. First, their range of validity is limited because these EOSs are fitted to a reference curve and tolerate only small deviations from that curve. The second difficulty lies in the complexity of the mathematical formulations, affecting the efficiency of pressure relaxation solvers. These two difficulties give rise to a third. Under extreme flow conditions, computational failures are quite common. Methods for "extending" the equations of state are often used to continue the computations. For example, below a certain arbitrary density, the JWL EOS is sometimes extended to the ideal-gas EOS. A fundamentally new method is presented in this work. It consists of using the much simpler Noble–Abel-stiffened-gas EOS as a predictor, to close the corresponding flow model. A thermodynamic relaxation step follows the prediction step. The solution is projected onto the target EOS, in this case the Mie–Grüneisen EOS, with the help of additional transport equations. Therefore, the flow model is an extended version, much more efficient for numerical resolution. This method solves the three problems above by extending the validity domain of the thermodynamic formulation, making the relaxation solvers much faster, and dramatically increasing the robustness of computations. [ABSTRACT FROM AUTHOR]

The Noble–Abel (NA) equation of state (EOS) is widely used in the 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 formulation 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, co-volume, and condensed material energy, for both formulations (NA and VO1) are presented. For this, a combination of closed bomb vessel experiments and thermochemical code computations is needed. An extended VO1 EOS with temperature-dependent specific heat is examined. Then, its 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 have emerged. First, a consideration of the specific heat temperature dependance in computations of the interior ballistics of guns appeared insignificant. Second, VO1 appeared to be more accurate than NA, particularly when gas density is beyond the range used for EOS parameter determination. Last, regarding mixtures of condensed reactive materials, producing burnt gas mixtures, NA again appeared to be less accurate than VO1. However, its formulation is explicit, while VO1 requires numerical solving of a non-linear equation, with consequences on computational cost. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Diffuse interface methods with compressible fluids, considered through hyperbolic multiphase flow models, have demonstrated their capability to solve a wide range of complex flow situations in severe conditions (both high and low speeds). These formulations can deal with the presence of shock waves, chemical and physical transformations, such as cavitation and detonation. Compared to existing approaches able to consider compressible materials and interfaces, these methods are conservative with respect to mixture mass, momentum, energy and are entropy preserving. Thanks to these properties they are very robust. However, in many situations, typically in low transient conditions, numerical diffusion at material interfaces is excessive. Several approaches have been developed to lower this weakness. In the present contribution, a specific flux limiter is proposed and inserted into conventional MUSCL type schemes, in the frame of the diffuse interface formulation of Saurel et al. (2009). With this limiter, interfaces are captured with 3 ± 1 mesh points depending on the test problem, showing significant improvement in interface representation compared to conventional limiters, such as for example Superbee. The method works on both structured and unstructured meshes and its implementation in existing codes is simple. Computational examples showing method capabilities and accuracy are presented. [ABSTRACT FROM AUTHOR]

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 are able to cover accurately the entire range, in particular regarding waves propagation. In the present work, an alternative to the Baer and Nunziato (BN) model [Baer, M. R. and Nunziato, J.W., "A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials," Int. J. Multiphase Flow 12(6), 861 (1986)], 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 model [Marble, F. E., "Dynamics of a gas containing small solid particles," Combustion and Propulsion (5th AGARD Colloquium) (Pergamon Press, 1963), Vol. 175] 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 characteristicwave speeds as well, but with different velocities. Last, the two sub-models with 4waves are combined in a unique formulation, valid for the full range of volume fractions. It involves the same 6 wave 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 pressurewaves 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 showing clear improvements. [ABSTRACT FROM AUTHOR]

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 hyperbolic 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," 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Compressible granular materials are involved in many applications, some of them being related to energetic porous media. Gas permeation effects are important during their compaction stage, as well as their eventual chemical decomposition. Also, many situations involve porous media separated from pure fluids through two-phase interfaces. It is thus important to develop theoretical and numerical formulations to deal with granular materials in the presence of both two-phase interfaces and gas permeation effects. Similar topic was addressed for fluid mixtures and interfaces with the Discrete Equations Method (DEM) [R. Abgrall and R. Saurel, "Discrete equations for physical and numerical compressible multiphase mixtures," J. Comput. Phys. 186(2), 361-396 (2003)] but it seemed impossible to extend this approach to granular media as intergranular stress [K. K. Kuo, V. Yang, and B. B. Moore, "Intragranular stress, particle-wall friction and speed of sound in granular propellant beds," J. Ballist. 4(1), 697-730 (1980)] and associated configuration energy [J. B. Bdzil, R. Menikoff, S. F. Son, A. K. Kapila, and D. S. Stewart, "Two-phase modeling of deflagration-to-detonation transition in granular materials: A critical examination of modeling issues," Phys. Fluids 11, 378 (1999)] were present with significant effects. An approach to deal with fluid-porous media interfaces was derived in Saurel et al. ["Modelling dynamic and irreversible powder compaction," J. Fluid Mech. 664, 348-396 (2010)] but its validity was restricted to weak velocity disequilibrium only. Thanks to a deeper analysis, the DEM is successfully extended to granular media modelling in the present paper. It results in an enhanced version of the Baer and Nunziato ["A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials," Int. J. Multiphase Flow 12(6), 861-889 (1986)] model as symmetry of the formulation is now preserved. Several computational examples are shown to validate and illustrate method's capabilities. [ABSTRACT FROM AUTHOR]

The aim of the present work is to account for polydisperse effects in a two-phase flow with a simple and fast method. Polydisperse two-phase flows arise in numerous applications. Fire sprinkler systems are relevant examples as they release clouds of polydisperse droplets. Another relevant example is the polydisperse two-phase flow created by the detonation of an explosive charge surrounded by a liquid layer. In such a situation, material interfaces are initially present and the created two-phase flow consists of a carrier gas phase and a liquid phase involving many droplets of various sizes. Spherical particles or droplets are usually assumed in two-phase flow computations. When dealing with explosion situations involving both dense and dilute flow regimes, multiple particle diameters can be addressed but at the price of introducing as many additional equations that describe mass, momentum and energy balance of the various particle classes. Consequently, the computation time needed to address numerical resolution increases tremendously. Under explosion situations involving many particle diameters, the method becomes intractable and is usually reduced to a single diameter, which is often insufficient. A simplified approach is developed in the present work to account for a substantial number of particles of different sizes with few extra computational cost. The approach is said to be simplified as a single velocity and a single temperature are considered for all the spherical particles, regardless of their diameters. This type of modeling seems apt for the target explosion situations. The focus is placed on the interfacial area, which is the main parameter involved in the coupling of the two phases. In the present work, Gamma-like continuous probability distributions are considered to address the various sizes of particles. The effects of the size distribution are only summarized in the specific interfacial area, yielding consequently few code modifications while taking into account the polydisperse aspect of the two-phase flow. [ABSTRACT FROM AUTHOR]

Abstract: For the simulation of light water nuclear reactor coolant flows, general two-phase models (valid for all volume fractions) have been generally used which, while allowing for velocity disequilibrium, normally force pressure equilibrium between the phases (see, for example, the numerous models of this type described in ()). These equations are not hyperbolic, their physical wave dynamics are incorrect, and their solution algorithms rely on dubious truncation error induced artificial viscosity to render them numerically well-posed over a portion of the computational spectrum. The inherent problems of the traditional approach to multiphase modeling, which begins with an averaged system of (ill-posed) partial differential equations (PDEs) which are then discretized to form a numerical scheme, are avoided by employing a new homogenization method known as the discrete equation method (DEM) (). This method results in well-posed hyperbolic systems, this property being important for transient flows. This also allows a clear treatment of non-conservative terms (terms involving interfacial variables and volume fraction gradients) permitting the solution of interface problems without conservation errors, this feature being important for the direct numerical simulation of two-phase flows. Unlike conventional methods, the averaged system of PDEs for the mixture are not used, and the DEM method directly obtains a well-posed discrete equation system from the single-phase conservation laws, producing a numerical scheme which accurately computes fluxes for arbitrary number of phases. The method effectively uses a sequence of single-phase Riemann problem solutions. Phase interactions are accounted for by Riemann solvers at each interface. Non-conservative derivative product terms are correctly approximated. Some of the closure relations missing from the traditional approach are automatically obtained. Lastly, the continuous equation system resulting from the discrete equations can be identified by taking the continuous limit with weak-wave assumptions. In this work, this approach is tested by constructing a DEM model for the flow of two compressible phases in 1-D ducts of spatially varying cross-section with explicit time integration. An analytical equation of state is included for both water vapor and liquid phases, and a realistic interphase mass transfer model is developed based on interphase heat transfer. A robust compliment of boundary conditions is developed and discussed. Though originally conceived as a first step toward implict time integration of the DEM method (to relieve time step size restrictions due to stiffness and to achieve tighter coupling of equations) in multidimensions, this model offers some unique capabilities for incorporation into next generation light water reactor safety analysis codes. We demonstrate, on a converging-diverging two-phase nozzle, that this well-posed, 2-pressure, 2-velocity DEM model can be integrated to a realistic and meaningful steady state with both phases treated as compressible. [Copyright &y& Elsevier]

Abstract: A new projection method is developed for the Euler equations to determine the thermodynamic state in computational cells. It consists in the resolution of a mechanical relaxation problem between the various sub-volumes present in a computational cell. These sub-volumes correspond to the ones traveled by the various waves that produce states with different pressures, velocities, densities and temperatures. Contrarily to Godunov type schemes the relaxed state corresponds to mechanical equilibrium only and remains out of thermal equilibrium. The pressure computation with this relaxation process replaces the use of the conventional equation of state (EOS). A simplified relaxation method is also derived and provides a specific EOS (named the Numerical EOS). The use of the Numerical EOS gives a cure to spurious pressure oscillations that appear at contact discontinuities for fluids governed by real gas EOS. It is then extended to the computation of interface problems separating fluids with different EOS (liquid–gas interface for example) with the Euler equations. The resulting method is very robust, accurate, oscillation free and conservative. For the sake of simplicity and efficiency the method is developed in a Lagrange-projection context and is validated over exact solutions. In a companion paper [F. Petitpas, E. Franquet, R. Saurel, A relaxation-projection method for compressible flows. Part II: computation of interfaces and multiphase mixtures with stiff mechanical relaxation. J. Comput. Phys. (submitted for publication)], the method is extended to the numerical approximation of a non-conservative hyperbolic multiphase flow model for interface computation and shock propagation into mixtures. [Copyright &y& Elsevier]

Abstract: A quasi-conservative formulation for compressible flows with interfaces including both capillary and viscous effects is developed. The model involves: (i) acoustic and convective transport; (ii) surface tension effects introduced as an extension of the CSF method (Brackbill et al.) to compressible flows; (iii) viscous effects. The interfaces are considered as diffused zones. Every point of the flow is correctly described thanks to a mixture equation of state based on energy conservation and pressure equilibrium. The model is thus valid in each pure fluid as well as at interfaces. A Godunov type method, that enables interface capturing, is developed and used whatever the mesh point. Numerical tests are performed over a wide range of physical situations involving surface tension, compressibility, gravity, viscosity and large density ratios. Tests involving break-up and coalescence are considered to show the ability of the method to deal with dynamic appearance and disappearance of interfaces in an Eulerian framework. [Copyright &y& Elsevier]

We have recently proposed, in , a compressible two-phase unconditionally hyperbolic model able to deal with a wide range of applications: interfaces between compressible materials, shock waves in condensed multiphase mixtures, homogeneous two-phase flows (bubbly and droplet flows) and cavitation in liquids. One of the difficulties of the model, as always in this type of physical problems, was the occurrence of non-conservative products. In , we have proposed a discretisation technique that was without any ambiguity only in the case of material interfaces, not in the case of shock waves. This model was extended to several space dimensions in , In this paper, thanks to a deeper analysis of the model, we propose a class of schemes that are able to converge to the correct solution even when shock waves interact with volume fraction discontinuities. This analysis provides a more accurate estimate of closure terms, but also an accurate resolution method for the conservative fluxes as well as non-conservative terms even for situations involving discontinuous solutions. The accuracy of the model and method is clearly demonstrated on a sequence of difficult test problems. [Copyright &y& Elsevier]

A new model with full coupling between micro- and macroscale motion is developed for compressible multiphase mixtures. The equations of motion and the coupling microstructural equation (an analogue of the Rayleigh–Lamb equation) are obtained by using the Hamilton principle of stationary action. In the particular case of bubbly fluids, the resulting model contains eight partial differential equations (one-dimensional case) and is unconditionally hyperbolic. The equations are solved numerically by an adapted Godunov method. The model and methods are validated for two very different test problems. The first one consists of a wave propagating in a liquid containing a small quantity of gas bubbles. Computed oscillating shock waves fit perfectly the experimental data. Then the one-dimensional multiphase model is used as a reduction tool for the multidimensional interaction of a shock wave with a large bubble. Good agreement is again obtained. [Copyright &y& Elsevier]

A simple second order accurate and fully Eulerian numerical method is presented for the simulation of multifluid compressible flows, governed by the stiffened gas equation of state, in hydrodynamic regime. Our numerical method relies on a second order Godunov-type scheme, with approximate Riemann solver for the resolution of conservation equations, and a set of nonconservative equations. It is valid for all mesh points and allows the resolution of interfaces. This method works for an arbitrary number of interfaces, for breakup and coalescence. It allows very high density ratios (up to 1000). It is able to compute very strong shock waves (pressure ratio up to 105). Contrary to all existing schemes (which consider the interface as a discontinuity) the method considers the interface as a numerical diffusion zone as contact discontinuities are computed in compressible single phase flows, but the variables describing the mixture zone are computed consistently with the density, momentum and energy. Several test problems are presented in one, two, and three dimensions. This method allows, for example, the computation of the interaction of a shock wave propagating in a liquid with a gas cylinder, as well as Richtmeyer­Meshkov instabilities, or hypervelocity impact, with realistic initial conditions. We illustrate our method with the Rusanov flux. However, the same principle can be applied to a more general class of schemes. [ABSTRACT FROM AUTHOR]

• A Riemann solver with internal reconstruction (RSIR) has been built as an extension of Linde [18] solver. • RSIR solver for the Euler equations is developed and compared to the HLLC solver. • The method is extended to a two-phase non-equilibrium model developed by the authors. • RSIR is applied to a problem of fingering instability in explosive granular media. • First computed results of this instability. 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. [ABSTRACT FROM AUTHOR]

• A diffuse interface model with enhanced physics and thermochemistry is built for the computation of the sodium-water reaction. • HLLC-type Riemann solvers with both gravity and surface tension are built. • The thermo-chemical Leidenfrost-type effect responsible for autonomous drop motion is qualitatively reproduced. • Explosion effect with shock wave emission and sodium ejection is reproduced as well. 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, heat and mass diffusion as well as 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 modeling and numerical issues are present and addressed in the present contribution. Explosion phenomenon is addressed in this paper and is reproduced at least qualitatively thanks to additional ingredients such as turbulent mixing of sodium and water vapors in the gas film and delayed ignition modeled through finite rate chemistry. Shock wave emission from the thermo-chemical Leidenfrost-type flow is observed as well as liquid sodium ejection off the water surface, as reported in related experiments. [ABSTRACT FROM AUTHOR]

• Automatic differentiation using operator overloading for implicit schemes. • Exact differentiation gives improved convergence of the unsteady residual. • Single and two phase fully compressible models tested successfully. Implicit time integration schemes are widely used in computational fluid dynamics 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 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 (ADOO). 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 [28] and a symmetric variant [44] of the two-phase flow model of Baer and Nunziato [2] dealing with mixtures in total disequilibrium. [ABSTRACT FROM AUTHOR]

Abstract A simple method is developed to couple accurately translational 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 [9]. 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 [13] 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 cylindrical 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. Highlights • A simple and efficient method is developed for rigid solids/fluid coupling on unstructured meshes. • The method addresses solid rigid body motion on fixed meshes, simpler than those based on moving meshes. • The present approach doesn't seem more accurate than existing ones but seems simpler and easier to implement. [ABSTRACT FROM AUTHOR]

This paper presents under-resolved simulations of compressible flows in complex structures like buildings. The mean pressure field is the primary focus as it is of high importance in many cases, such as the overpressure generated by the detonation of an explosive charge. In such high-risk situations, the mean pressure in the different building rooms has to be determined as quickly as possible, in order to help first responders evaluating the risk level of entering the building right after an explosion for instance. Conventional computational fluid dynamics methods are obviously able to deliver those requested pressure fields. However, they also require a long and tedious meshing process due to the presence of small openings in the computational domain (doors, windows or stairwells). In turn, these small openings need spatial and temporal resolution having significant consequences on computation time. But when dealing with pressing situations, a long pre-processing is not acceptable. The present paper proposes a two-step solution method. First, a very rough mesh is constructed which does not consider small openings. The mesh is constructed in an easy and particular way such that simplifications can be made when computing the fluid flux through these openings. Second, a specific Riemann solver dealing with geometric discontinuities is developed to estimate the fluid flow. The Riemann solver takes advantage of the simplifications provided by the meshing method, and accounts for both unchoked and choked flows through openings. The overall method then consists of both a specific meshing process that requires very little pre-processing, which considerably reduces the time needed to produce a complete simulation, and a Riemann solver. The proposed method is named MUZO in reference to its "MUlti-ZOne" flow solver. The MUZO solver is validated against resolved computations and provides accurate simulated mean pressure fields within a realistic building in a few seconds. • 3D simulations of compressible flow discharge in buildings are addressed through a very fast numerical framework. • Only 2D footprints without geometric details (doors, windows, etc.) are required to generate 3D meshes. • The previously omitted geometric restrictions are considered through a specific Riemann solver. [ABSTRACT FROM AUTHOR]

This paper addresses post detonation modelling in spherical explosions. One of the challenges is thus related to compressible turbulent mixing layers modelling. A one-dimensional flow model is derived consisting in a reduced two-phase compressible flow model with velocity drift. To reduce the number of model parameters, the stiff velocity relaxation limit is considered. A semi-discrete analysis is used resulting in a specific artificial viscosity formulation embedded in the diffuse interface model of Kapila et al. [Phys. Fluids 13(10), 3002-3024 (2001)]. Thanks to the velocity non-equilibrium model and semi discrete formulation, the model fulfils the second law of thermodynamics in the global sense and uses a single parameter. Multidimensional mixing layer effects occurring at gas-gas unstable interfaces are thus summarized as artificial viscosity effects. Model's predictions are compared against experimental measurements of mixing layer growth in shock tubes at moderate initial pressure ratios as well as fireball radius evolutions in air explosions at high initial pressure ratios. Also, pressure signals recorded at various stations are compared, showing excellent agreement for the leading shock wave as well as the secondary one. With the help of various experiments in the low and high initial pressure ratios bounds, estimates for the interpenetration parameter are given. [ABSTRACT FROM AUTHOR]

Compression of monodisperse powder samples in quasistatic conditions is addressed in a pressure range such that particles fragmentation occurs while the solid remains incompressible (typical pressure range of 1-300 MPa for glass powders). For a granular bed made of particles of given size, the existence of three stages is observed during compression and crush up. First, classical compression occurs and the pressure of the granular bed increases along a characteristic curve as the volume decreases. Then, a critical pressure is reached for which fragmentation begins. During the fragmentation process, the granular pressure stays constant in a given volume range. At the end of this second stage, 20%-50% of initial grains are reduced to finer particles, depending on the initial size. Then the compression undergoes the third stage and the pressure increases along another characteristic curve, in the absence of extra fragmentation. The present paper analyses the analogies between the phase transition in liquid-vapour systems and powder compression with crush-up. Fragmentation diagram for a soda lime glass is determined by experimental means. The analogues of the saturation pressure and latent heat of phase change are determined. Two thermodynamic models are then examined to represent the crush-up diagram. The first one uses piecewise functions while the second one is of van derWaals type. Both equations of state relate granular pressure, solid volume fraction, and initial particle diameter. The piecewise functions approach provides reasonable representations of the phase diagram while the van der Waals one fails. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: High speed underwater systems involve many modelling and simulation difficulties related to shocks, expansion waves and evaporation fronts. Modern propulsion systems like underwater missiles also involve extra difficulties related to non-condensable high speed gas flows. Such flows involve many continuous and discontinuous waves or fronts and the difficulty is to model and compute correctly jump conditions across them, particularly in unsteady regime and in multi-dimensions. To this end a new theory has been built that considers the various transformation fronts as ‘diffuse interfaces’. Inside these diffuse interfaces relaxation effects are solved in order to reproduce the correct jump conditions. For example, an interface separating a compressible non-condensable gas and compressible water is solved as a multiphase mixture where stiff mechanical relaxation effects are solved in order to match the jump conditions of equal pressure and equal normal velocities. When an interface separates a metastable liquid and its vapor, the situation becomes more complex as jump conditions involve pressure, velocity, temperature and entropy jumps. However, the same type of multiphase mixture can be considered in the diffuse interface and stiff velocity, pressure, temperature and Gibbs free energy relaxation are used to reproduce the dynamics of such fronts and corresponding jump conditions. A general model, based on multiphase flow theory is thus built. It involves mixture energy and mixture momentum equations together with mass and volume fraction equations for each phase or constituent. For example, in high velocity flows around underwater missiles, three phases (or constituents) have to be considered: liquid, vapor and propulsion gas products. It results in a flow model with 8 partial differential equations. The model is strictly hyperbolic and involves waves speeds that vary under the degree of metastability. When none of the phase is metastable, the non-monotonic sound speed is recovered. When phase transition occurs, the sound speed decreases and phase transition fronts become expansion waves of the equilibrium system. The model is built on the basis of asymptotic analysis of a hyperbolic total non-equilibrium multiphase flow model, in the limit of stiff mechanical relaxation. Closure relations regarding heat and mass transfer are built under the examination of entropy production. The mixture equation of state (EOS) is based on energy conservation and mechanical equilibrium of the mixture. Pure phases EOS are used in the mixture EOS instead of cubic one in order to prevent loss of hyperbolicity in the spinodal zone of the phase diagram. The corresponding model is able to deal with metastable states without using Van der Waals representation. The model’s predictions are validated in multi-dimensions against experiments of high velocity projectile impact onto a liquid tank. Simulations are compared to experiments and reveal excellent quantitative agreement regarding shock and cavitation pocket dynamics as well as projectile deceleration versus time. Then model’s capabilities are illustrated for flow computations around underwater missiles. [Copyright &y& Elsevier]

Abstract: The relaxation-projection method developed in Saurel et al. [R. Saurel, E. Franquet, E. Daniel, O. Le Metayer, A relaxation-projection method for compressible flows. Part I: The numerical equation of state for the Euler equations, J. Comput. Phys. (2007) 822–845] is extended to the non-conservative hyperbolic multiphase flow model of Kapila et al. [A.K. Kapila, 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]. This model has the ability to treat multi-temperatures mixtures evolving with a single pressure and velocity and is particularly interesting for the computation of interface problems with compressible materials as well as wave propagation in heterogeneous mixtures. The non-conservative character of this model poses however computational challenges in the presence of shocks. The first issue is related to the Riemann problem resolution that necessitates shock jump conditions. Thanks to the Rankine–Hugoniot relations proposed and validated in Saurel et al. [R. Saurel, O. Le Metayer, J. Massoni, S. Gavrilyuk, Shock jump conditions for multiphase mixtures with stiff mechanical relaxation, Shock Waves 16 (3) (2007) 209–232] exact and approximate 2-shocks Riemann solvers are derived. However, the Riemann solver is only a part of a numerical scheme and non-conservative variables pose extra difficulties for the projection or cell average of the solution. It is shown that conventional Godunov schemes are unable to converge to the exact solution for strong multiphase shocks. This is due to the incorrect partition of the energies or entropies in the cell averaged mixture. To circumvent this difficulty a specific Lagrangian scheme is developed. The correct partition of the energies is achieved by using an artificial heat exchange in the shock layer. With the help of an asymptotic analysis this heat exchange takes a similar form as the ‘pseudoviscosity’ introduced by Von Neumann and Richtmyer [J. Von Neumann, R.D. Richtmyer, A method for the numerical calculation of hydrodynamic shocks, J. Appl. Phys. 21 (1950) 232–237]. The present Lagrangian numerical scheme thus combines Riemann solvers and artificial heat exchanges. An Eulerian variant is then obtained by using the relaxation-projection method developed earlier by the authors for the Euler equations. The method is validated against exact solutions based on the multiphase shock relations as well as exact solutions of the Euler equations in the context of interface problems. The method is able to solve interfaces separating pure fluids or heterogeneous mixtures with very large density ratio and with very strong shocks. [Copyright &y& Elsevier]

The computation of detonation waves in heterogeneous explosives involves compressible multiphase mixtures due to the chemical decomposition of the energetic material as well as its heterogeneous initial formulation. Also material interfaces are present between the explosive and the surrounding inert or reactive materials. We develop a new method for the modelling of interface problems and multiphase mixtures in the particular limit where phases pressures and velocities relax towards equilibrium very fast. This method is a variant of the discrete equations for multiphase mixtures proposed by Abgrall and Saurel [J. Comput. Phys. 186 (2) (2003) 361]. The new discrete model is adapted to reacting flows with mass, momentum and energy transfer. The model is based on the pure material equation of state only (no mixture equation of state is used) for the computation of the reaction zone of detonation waves. The algorithm is full Eulerian and fulfills interface conditions between mixtures and pure materials automatically. It is validated over a set of difficult test problems with exact solution and its multi-dimensional capabilities are shown over problems involving a large number of materials. [Copyright &y& Elsevier]

Abstract: Ammonium-nitrate-based explosives used by the mining industry exhibit strong non-ideal detonation behaviour. Detonation velocities in rate-sticks with radii close to the failure radius, can be as low as one third of the ideal detonation velocity, which poses a significant challenge for their accurate predictive computational modelling. Given that these emulsions are highly heterogeneous, multi-phase formulations are well suited for their representation in numerical hydrocodes. To this end, a single-pressure, single-velocity multi-phase model is employed for the simulation of an explosive emulsion widely used by the mining industry. The model is modified to rectify a problem related to the calculation of a unique detonation state, and is evaluated using a high-resolution, shock-capturing Riemann problem-based scheme. In order to perform high-resolution numerical simulations at a reduced cost, a shock-following method is implemented and validated against the full-domain solutions. An improved iterative fitting procedure for steady-state detonation kinetics is also presented. Validation against experimental evidence shows that the model can reproduce confined VOD experimental data, solely by adjusting the reaction kinetics to match unconfined experimental VOD data. Furthermore, the model can match experimental front curvature measurement without further adjustments. [Copyright &y& Elsevier]

• Sodium vaporization is key to triggering sodium-water reaction (SWR) runaway. • SWR proceeds first with a surface reaction and then a gas-phase reaction. • The delay time of SWR decreases exponentially as water temperature is increased. • A simple differential model reproduces SWR mechanism and experimental data. Sodium-Water Reaction (SWR) is a notorious and complex interaction between two condensed phase reactants. It involves both physical and chemical processes, is fast, exothermic, and can be explosive under specific conditions. The fine-scale processes of SWR, and in particular the runaway mechanism eventually leading to explosive effects, are not yet fully understood. This paper focuses on how SWR runaway is triggered. Experiments investigating the processes of SWR with a high-speed camera are first described. These experiments suggest that sodium vaporization is responsible for provoking runaway. The main experimental observations are discussed and a scenario for SWR runaway is proposed. Finally, a semi-analytical model based on that mechanism is developed. Physical considerations and simplifying assumptions allow reducing the model to a single differential equation in the configuration of current experiments. The results of this model are consistent with experimental observations, and confirm the critical role of sodium vaporization in the onset of SWR runaway. [ABSTRACT FROM AUTHOR]