Section: New Results

Mathematical Modeling

Geometric evolution of the Reynolds stress tensor in three-dimensional turbulence

Participants : Sergey Gavrilyuk, Henri Gouin [ M2P2, Aix Marseille University, France ] .

The dynamics of the Reynolds stress tensor is described with an evolution equation coupling both geometric effects and turbulent source terms. The effects of the mean flow geometry are shown up when the source terms are neglected : the Reynolds stress tensor is then expressed as the sum of three tensor products of vector fields which are governed by a distorted gyroscopic equation. Along the mean flow trajectories and in the directions of the vector fields, the fluctuations of velocity are described by differential equations whose coefficients only depend on the mean flow deformation. If the mean flow vorticity is small enough, an approximate turbulence model is derived, and its application to shear shallow water flows is proposed. In particular, it is proven that the approximate turbulence model admits a variational formulation [10] .

A numerical scheme for the Green-Naghdi model

Participants : Olivier Le Métayer, Sergey Gavrilyuk, Sarah Hank.

For this work, a hybrid numerical method using a Godunov type scheme is proposed to solve the Green-Naghdi model describing dispersive shallow water waves. The corresponding equations are rewritten in terms of new variables adapted for numerical studies. In particular, the numerical scheme preserves the dynamics of solitary waves. Some numerical results are shown and compared to exact and/or experimental ones in different and significant configurations. A dam break problem and an impact problem where a liquid cylinder is falling to a rigid wall are solved numerically. This last configuration is also compared with experiments leading to a good qualitative agreement [11] .

Shallow water model for lakes with friction and penetration

Participants : Nikolay V. Chemetov, Fernanda Cipriano, Sergey Gavrilyuk.

We consider the flow of an ideal fluid in a 2D-bounded domain, admitting flows through the boundary of this domain. The flow is described by the Euler equations with non-homogeneous Navier slip boundary conditions. We establish the solvability of this problem in the class of solutions with L_p -bounded vorticity, p(2, ] . To prove the solvability we realize the passage to the limit in Navier-Stokes equations with vanishing viscosity [8] .

Diffuse solid-fluid interface model in cases of extreme deformations

Participants : Nicolas Favrie, Sergey Gavrilyuk, Richard Saurel.

A diffuse interface model for elastic solid – fluid coupling in Eulerian formulation is built. This formulation generalizes the diffuse interface models for compressible multi-fluid computations [40] , [46] , [25] , [48] , [16] . Elastic effects are included following the Eulerian conservative formulation proposed by Godunov in 1978 [34] ), by Miller and Colella (2001) [45] , Godunov and Romenskii (2003) [35] , Plohr and Plohr (2005) [49] , Gavrilyuk et al . (2008) [31] . The aim is to derive an extended system of hyperbolic partial differential equations, valid at each mesh point (pure fluid, pure elastic solid, and mixture cells) to be solved by a unique hyperbolic solver. The model is derived with the help of Hamilton's principle of stationary action. In the limit of vanishing volume fractions the Euler equations of compressible fluids and a conservative hyperelastic model are recovered. The model is hyperbolic and compatible with the entropy inequality. Special attention is paid to the approximation of geometrical equations, as well as the fulfilment of solid-fluid interface conditions. Capabilities of the model and methods are illustrated on hypervelocity impacts of solids [9] .

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

Participants : Fabien Petitpas, Richard Saurel, Erwin Franquet [ University of Pau, France ] , Ashwin Chinnayya [ University of Rouen, France ] .

A hyperbolic multiphase flow model with a single pressure and a single velocity but several temperatures is presented to deal with the detonation dynamics of condensed high energetic materials. Temperature non-equilibrium is mandatory in order to deal with realistic wave propagation (shocks, detonations) in heterogenous mixtures. The model is obtained as the asymptotic limit of a total non-equilibrium multiphase flow model in the limit of stiff mechanical relaxation only [40] . Special attention is given to mass transfer modeling, that is obtained with the help of entropy production analysis in each phase and in the system [51] . With the help of the shock relations given in [52] the model is closed and provides a generalized ZND formulation for condensed energetic materials. In particular, generalized CJ conditions are obtained. They are based on a balance between the chemical reaction energy release and internal heat exchanges between phases. Moreover, the sound speed that appear at sonic surface corresponds to the non-monotonic one of Wood (1930) [57] . Therefore, non-conventional reaction zone structure is observed. When heat exchanges are absent, the conventional ZND model with conventional CJ conditions is recovered. When heat exchanges are involved, a behaviour similar to non-ideal explosives is observed, even in absence of front curvature effects (Wood and Kirkwood, 1954, [56] ).

Multidimensional resolution of the corresponding model is then addressed. This poses serious difficulties related to the presence of material interfaces and shock propagation in multiphase mixtures. The first issue is solved by an extension of the method derived in [16] in the presence of heat and mass transfers. The second issue poses the difficult mathematical question of numerical approximation of non-conservative systems in the presence of shocks associated to the physical question of energy partition between phases for a multiphase shock. A novel approach is used, based on extra evolution equations used to retain the information of the material initial state. This method insure convergence of the method in the post-shock state.

Thanks to these various theoretical and numerical ingredients, one-dimensional and multidimensional unsteady detonation waves computations are done, eventually in the presence of material interfaces. Convergence of the numerical hyperbolic solver against ZND multiphase solution is reached. Material interfaces, shocks, detonations are solved with a unified formulation where the same equations are solved everywhere with the same numerical scheme. Method convergence is reached at material interfaces even in the presence of very high density and pressure ratios, as well as convergence in the multiphase detonation wave reaction zone [13] .

Reduced models for compaction

Participants : Marie-Hélène Lallemand, Richard Saurel.

The aim of this study is to find a model that accounts for hysteresis effects due to compaction in multiphase flows where some of the phases are constituted by small solid grains (powder). Here, we are concerned with dynamic compaction, which means compaction is due to the pressure and velocity of the gas phase acting on the solid grains. We define an additional bulk pressure, assumed to represent the compaction pressure of the solid phase. That compaction pressure is supposed to represent the resulting forces due to material resistance of the solid phase under compression. This resistance is also first assumed to be in the elastic limit of the material. As a matter of fact, we first restrict our study to that limit, plasticity and rupture will be part of future work. For that purpose, an additional potential energy, the compaction energy, from which the compaction pressure is derived, is introduced. That energy is supposed to depend only on the volume fraction of the solid phase and will only act in a certain range of values (starting with a lower-limit value for which compaction begins to be effective, and ending with a upper-limit value depending on the elastic limit of the material). The parent model, written for two phases and in one dimension, is first introduced, and we derive several reduced models, resulting from asymptotic analysis around different equilibrium states (velocity equilibrium and pressure equilibrium), since we are interested in flows for which those mechanical equilibria are done in a very small time scale (dealing with very high gas pressure and velocity). Discussion about relaxation coefficients are also done together with links with the study done by M. Labois in [44] .

We are reporting the present study in [24] .

Modeling dynamic and irreversible powder compaction

Participants : Richard Saurel, Nicolas Favrie, Fabien Petitpas, Marie-Hélène Lallemand, Sergey Gavrilyuk.

A multiphase hyperbolic model for dynamic and irreversible powder compaction is built. Three major issues have to be addressed in this aim. The first one is related to the irreversible character of powder compaction. When a granular media is subjected to a loading-unloading cycle the final volume is lower than the initial one. To deal with this hysteresis phenomenon a multiphase model with relaxation is built. During loading, mechanical equilibrium is assumed corresponding to stiff mechanical relaxation, while during unloading non-equilibrium mechanical transformation is assumed. Consequently, the sound speeds of the limit models are very different during loading and unloading. These differences in acoustic properties are responsible for irreversibility in the compaction process. The second issue is related to dynamic effects where pressure and shock waves play important role. Wave dynamics is guaranteed by the hyperbolic character of the equations. Phases  compressibility is considered, as well as configuration pressure and energy. The third issue is related to multidimensional situations that involve material interfaces. Indeed, most processes with powder compaction entail  free surfaces . Consequently the model has to be able to solve interfaces separating pure fluids and granular mixtures. These various issues are solved by a unique model fitting the frame of multiphase theory of diffuse interfaces (Saurel and Abgrall, 1999, Kapila et al., 2001, Saurel et al., 2009). Model s ability to deal with these various effects is validated on basic situations, where each phenomenon is considered separately. Special attention is paid to the validation of the hysteresis phenomenon that occurs during powder compaction. Basic experiments on energetic material (granular HMX) and granular NaCl compaction are considered and are perfectly reproduced by the model. Excepting the materials equations of state (hydrodynamic and granular pressures and energies) that are determined on the basis of separate experiments found in the literature, the model is free of adjustable parameter. Its ability to reproduce the hysteresis phenomenon is due to a relaxation parameter that tends either to infinity in the loading regime, or to zero in the unloading stage. Discontinuous evolution of this relaxation parameter is explained [15] .

Shock-bubbles interaction : a test configuration for two-fluid modeling

Participants : François Renaud [ CEA / DAM, Bruyères le Châtel, France ] , Richard Saurel, Georges Jourdan, Lazhar Houas [ IUSTI, Aix Marseille University ] .

Direct numerical simulation of two nonmiscible fluids mixing under shock waves is extremely expensive in computational resources. This is incompatible with design process. It is thus necessary to develop models of mixing in order to reduce this cost. This must be made on simple mixing flows representative of studied configurations. Doing so we can combine efficiently modeling and experimental validation. We illustrate this matter on the development of a one-dimensional two-fluid model [14] .

Diffuse interface model for high speed cavitating underwater systems

Participants : Fabien Petitpas, Jacques Massoni, Richard Saurel, Emmanuel Lapébie [ DGA, Centre d'Études de Gramat, France ] , Laurent Munier [ DGA, Centre d'Études de Gramat, France ] .

High speed underwater systems involve many modeling and simulation difficulties related to shock, expansion waves and evaporation fronts. Modern propulsion systems like underwater solid rocket motors 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 Wood (1930) 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 multidimensions against experiments of high velocity projectile impact onto a liquid tank. Simulation 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 [12] .