Laboratoire de Mathématiques, UMR CNRS 5127Université de Savoie, 73376 Le Bourget du Lac, France

AMIS 2012

Applied Mathematics In Savoie

19-22 June 2012

“Multiphase phase flow in industrial and environmental engineering”A meeting between Engineers and “Applied Mathematicians”

With the financial support of :

Laboratoire de MathématiquesThomas Lachand-Robert Conference Room

Conference Program

Tuesday 19 June9:00-10:00 - Welcome10:00-10:45 - Philippe Helluy : Computing bubble oscillations on GPU10:45-11:30 - Olivier Le Métayer : Dynamic relaxation processes in compressible multiphaseflows12:00-1400: - Lunch at the Restaurant “Le Prieuré”, Le Bourget du Lac14:00-14:45 - Enrique Nieto : Two-phase models for debris flows with dilatancy effects14:45-15:30 - Raphaèle Herbin : Staggered schemes for compressible flows15:30-16:00 - Coffee break16:00-16:45 - Marica Pelanti : A mixture-energy-consistent 6-equation two-phase numericalmodel for fluids with interfaces and cavitation phenomena

Wednesday 20 June

9:00-9:45 -Richard Saurel : Diffuse interfaces and transformation fronts modelling in compress-ible materials9:45-10:30 - Debora Amadori : Solutions for a hyperbolic model of multi-phase flow and relatednumerical issues10:30-11:00 - Coffee break11:00-11:45 - Jean-Marc Hérard : A few results on the modelling of multiphase flows12:15-14:00 - Lunch at the Restaurant “Le Prieuré”, Le Bourget du Lac14:00-17:00 - Social Event: guided visit of Chambery city center (to be confirmed)19:30 - Conference Dinner at the Restaurant “Les gourmands disent”, Chambéry

Thursday 21 June

9:00-9:45 - Alfredo Soldati : Physics of Turbulent Dispersed Flows and Simulation Approaches9:45-10:30 - Daniel Chauveheid : Some recent developments in the computation of complexmultimaterial flows10:30-11:00 - Coffee break11:00-11:45 - Benjamin Dewals : Multiphase flow from a civil engineering perspective12:15-14:00 - Lunch at the Restaurant “Le Prieuré”, Le Bourget du Lac14:00-14:45 - Yujie Liu : Computing water-hammer flows with two-fluid models14:45-15:30 - Frank Bierbrauer : Drop Pinch-Off for Discrete Flows from a Capillary15:30 - 16:00 - Coffee break16:00-16:45 - Aymen Laadhari: An Eulerian Finite Element Method for the Simulation of Fluid-Structure Interaction

Friday 22 June

9:00-9:45 - Gijsbert Ooms : Numerical study of eccentric core-annular flow9:45-10:30 - Nicolas Seguin : Compressible two-phase flow models: asymptotics, coupling andadaptation10:30-11:00 - Coffee break11:00-11:45 - Tran Quang Huy : Exact relaxation and accuracy enhancement of numericalschemes for some hyperbolic conservation laws11:45-12:00 - Closing12:15-14:00 -Lunch at the Restaurant “Le Prieuré”, Le Bourget du Lac

Solutions for a hyperbolic model of multi-phase flowand related numerical issues

Debora Amadori

Department of Pure and Applied Mathematics

University of L’Aquila, Italy

e-mail: amadori@univaq.it

In the first part of the talk we present a model for the flow of an inviscid fluid admitting liquid and vapor

phases, as well as a mixture of them. The model is one-dimensional and the state variables are the specific

volume, the velocity and the mass density fraction of vapor in the fluid. A reaction term for the equation of the

mass density fraction drives the fluid towards certain regimes, leading to stable and metastable configurations.

For this system we prove the existence of solutions, possibly containing large shocks, that are defined for all

positive times, and prove the relaxation limit to a reduced system. (joint work with Andrea Corli, Univ. Ferrara).

The second part of the talk will concern the numerical study of conservation laws with source term. In the

approximation of non-resonant balance laws, the ability of Well-Balanced (WB) schemes to capture very accu-

rately steady-state regimes has been thoroughly illustrated since its introduction by Greenberg and LeRoux. We

aim at showing, by means of rigorous C0t (L1

x) estimates, that these schemes deliver an increased accuracy in time-

dependent regimes too. Namely, after explaining that the C0t (L1

x) error of conventional fractional-step numerical

approximations may grow exponentially in time like exp(max(g′)t)√

∆x (where g represents the nonlinear term

in the source), it is shown that WB schemes involving an exact Riemann solver suffer from a much smaller error

amplification in time: their error can grow at most linearly. (joint work with Laurent Gosse, IAC-CNR, Rome).

Some recent developments in the computation of complex

multimaterial flows

Daniel Chauveheid∗

CEA, DAM, DIF, F-91297 Arpajon, France

CMLA, ENS Cachan and CNRS UMR 8536, 61 avenue du President Wilson F-94235 CACHAN CEDEX

The FVCF-ENIP method (Finite Volume with Characteristic Fluxes - Enhanced Natural InterfacePositioning), [1, 2], is a hybrid Eulerian/Lagrangian finite volume method that has been designed forsimulating non-miscible multimaterial flows. Pure parts of the computational domain, that is partswhere only one fluid is present, are treated with a classical upwind finite volume scheme with co-locatedvariables, namely the FVCF scheme [3]. On the contrary, near the material interfaces, intermediatestructures are constructed that condensate mixed cells together. These structures, called condensates,allow the writing of a conservative lagrangian scheme over mono-material controle volumes evolving intime. Time evolution in these controle volumes is obtained by adding ingoing and outgoing fluxes andtherefore allow to take precisely into account interface velocities and pressures which is necessary forseveral physical models.

A full description of the numerical method will be given. Afterwards, the talk will focus on the majorrecent developments it has known until then. Actually a wide range of new perspectives will be discussed,according to the three following categories : technical achievements (hybrid MPI-OpenMP paralleliza-tion, multi-material simulations involving three or more different materials), numerical improvements(extension of the original scheme for almost incompressible fluid flows, that is in the low Mach numberlimit and 3D flows with cylindrical symmetry) and finally physical extensions of the method to variousmodels (radiation hydrodynamics models, numerical modeling of surface tension).

References

[1] J.-P. Braeunig, B. Desjardins and J.-M. Ghidaglia, A totally Eulerian Finite Volume solver for multi-material fluid flows, Eur. J. Mech. B/Fluids, 28, 2009, pp.475-485

[2] R. Loubere, J.-P. Braeunig and J.-M. Ghidaglia, A totally Eulerian finite volume solver for multi-material fluid flows: Enhanced Natural Interface Positioning (ENIP), Eur. J. Mech. B/Fluids, 31,2012, pp.1-11

[3] J.-M. Ghidaglia, A. Kumbaro and G. Le Coq, On the numerical solution to two fluid models via acell centered finite volume method, Eur. J. Mech. B/Fluids, 20, 2001, pp.841-867

[4] D. Chauveheid, Ecoulements multi-materiaux et multi-physiques : solveur volumes finis co-localiseavec capture d’interfaces, analyse et simulations, Thesis, ENS-Cachan, July 2012

∗daniel.chauveheid@cmla.ens-cachan.fr

1

MULTIPHASE FLOW FROM A CIVIL ENGINEERING PERSPECTIVE

Benjamin Dewals, Sebastien Erpicum, Pierre Archambeau & Michel Pirotton

Hydraulics in Environmental and Civil Engineering (HECE), University of Liège, Belgium

E-mail: b.dewals@ulg.ac.be, hece@ulg.ac.be

For over 15 years, our research group at the University of Liege has been conducting basic and

applied research on flow and transport phenomena involved in environmental and civil engineering

applications. The research topics cover a wide range of multiphase flow, including aerated flow and

sediment-laden flow. The research group has developed the modelling system WOLF, which

performs 1D, 2D-horizontal, 2D-vertical and 3D simulations of free surface and pressurized flow,

coupled with air- or sediment-transport as well as morphodynamic simulations [9]. An original

depth-averaged k- model is used to evaluate the eddy viscosity and diffusivity [5]. In a composite

modelling approach [e.g., 2, 8], the numerical simulations have systematically been carried out in

parallel with experimental research in flumes and scale models of hydraulic structures [3, 4, 7].

A particularly challenging issue in multiphase flow modelling is the need to handle accurately and

efficiently a wide range of time scales involved in the relevant phenomena (e.g., sediment transport).

Therefore, the modelling system provides a series of complementary numerical schemes designed to

be combined for covering the whole range of relevant time scales.

This presentation will provide examples of analysis of aerated flow, such as on stepped spillways

and in penstocks [6], and of reservoir sedimentation. In particular, recent experimental findings have

revealed that the flow pattern in rectangular shallow reservoirs is considerably modified at a macro-

scale when suspended load is added to the flow [1]. This sudden change in flow pattern has been

observed to take place quickly after the beginning of the experiment, so that it is very unlikely to

result from morphodynamic changes of the reservoir bottom. Therefore, the numerical model WOLF

has been used to investigate the effect of two other possible feedback mechanisms of sediment

deposits and suspended load on the overall flow pattern, namely (i) increased bottom roughness and

(ii) turbulence damping. From the simulations results, it can be argued that turbulence intensity is

probably the main cause for the flow pattern to change at a macro-scale, as suspended load exerts a

turbulence damping effect at a micro-scale in the two-phase flow (water-sediment mixture).

Finally, the analysis of flow and sediment transport in shallow rectangular reservoirs also highlights

the need, when structures are designed based on numerical simulations, to carefully check the

stability of the computed flow fields needs by conducting sensitivity analyses, not only with respect

to the modelling parameters but also with respect to the initial conditions.

References

[1] Camnasio, E., S. Erpicum, E. Orsi, M. Pirotton, A. Schleiss, & B. Dewals. Feedback from

reservoir sedimentation on the flow pattern in rectangular basins. in 6th International

Conference on Scour and Erosion (ICSE). 2012. Paris, France.

[2] Dewals, B.J., S. Kantoush, S. Erpicum, M. Pirotton, & A. Schleiss, Experimental and numerical

analysis of flow instabilities in rectangular shallow basins. Environmental Fluid Mechanics,

2008. 8: 31-54.

[3] Dufresne, M., B.J. Dewals, S. Erpicum, P. Archambeau, & M. Pirotton, Experimental

investigation of flow pattern and sediment deposition in rectangular shallow reservoirs.

International Journal of Sediment Research, 2010. 25: 258-270.

[4] Dufresne, M., B. Dewals, S. Erpicum, P. Archambeau, & M. Pirotton, Flow patterns and

sediment deposition in rectangular shallow reservoirs. Water and Environment Journal, 2012.

In press.

[5] Erpicum, S., T. Meile, B.J. Dewals, M. Pirotton, & A.J. Schleiss, 2D numerical flow modeling in

a macro-rough channel. International Journal for Numerical Methods in Fluids, 2009. 61: 1227-

1246.

[6] Kerger, F., S. Erpicum, B.J. Dewals, P. Archambeau, & M. Pirotton, 1D unified mathematical

model for environmental flow applied to steady aerated mixed flows. Advances in Engineering

Software, 2011. 42: 660-670.

[7] Machiels, O., S. Erpicum, P. Archambeau, B.J. Dewals, & M. Pirotton, Experimental

observation of flow characteristics over a Piano Key Weir. Journal of Hydraulic Research, 2011.

49: 359-366.

[8] Roger, S., B.J. Dewals, S. Erpicum, D. Schwanenberg, H. Schüttrumpf, J. Köngeter, & M.

Pirotton, Experimental und numerical investigations of dike-break induced flows. Journal of

Hydraulic Research, 2009. 47: 349-359.

[9] Rulot, F., B. Dewals, S. Erpicum, P. Archambeau, & M. Pirotton, Modelling sediment transport

over partially non-erodible bottoms. International Journal for Numerical Methods in Fluids,

2012. In press.

Two-phase models for debris flows with dilatancy effects

E.D. Fernandez-Nieto ∗

Abstract: In the first part of the talk we present the mathematical modelling of somegeophysical mass flows containing a mixture of solid material and interstitial fluid in orderto simulate avalanches evolution.

Savage and Hutter presented in 1991 a pioneering work on the study of aerial granularavalanches, obtaining a model of shallow water type in local coordinates on an inclinedplane. This model includes a Coulomb friction terms that depends on the internal frictionangle.

Iverson and Denlinger in 2001 proposed a model for the study of shallow partiallyfluidized avalanches, a mixture of a granular material and a fluid.

In [1] Anderson and Jackson propose two-phase models to study partially fluidizedflows by including buoyancy effects. Pitman and Le propose in [6] a model based on themodel proposed in [1]. They proposed an averaged model considering mass and momentumequations for both the liquid phase and the solid phase. Pailha and Pouliquen in [4] alsouse the two phase approach but they introduce an additional closure equation relatedto dilatancy effects. Alternatively, George and Iverson in [3] deal with the mass andmomentum equation of the mixture. They also include dilatancy effects that involves thepore fluid pressure.

In this talk we present a two-phase model for the study of partially fluidized aerialavalanches. It considers mass and momentum equations for both liquid and solid phases,buoyancy and dilatancy effects. The model has an associated energy.

In the second part of the talk we present a numerical method to aproximate the proposedmodel. Finally, some numerical tests will be presented.

References

[1] T.B. Anderson, R. Jackson. A fluid mechanical description of fluidized beds. Ind. Eng.Chem. Fundam. 6, 527–539, 1967.

[2] R.M Iverson, R.P. Denlinger.: Flow of variably fluidized ranular masses across three-dimensional terrain 1: Coulomb mixture theory. J. Geophys. Res. 106, 537–552, 2001.

[3] D.L. George, R.M. Iverson: A two-phase debris-flow model that includes coupled evolu-tion of volume fractions, granular dilatancy, and pore-fluid pressure. Italian Journal ofEngineering Geology and Environment, In press, (2011).

[4] M. Pailha, O. Pouliquen: A two-phase flow description of the initiation of underwatergranular avalanches. J. Fluid Mech. vol. 633, 115-135 (2009).

[5] M. Pelanti, F. Bouchut, A. Mangeney. A Roe-type scheme for two-phase shallow gran-ular flows over variable topography. M2AN 42, 851–885, 2008.

[6] E.B. Pitman, L. Le. A two-fluid model for avalanche and debris flows. Phil. Trans. R.Soc. A 363, 1573–1601, 2005.

[7] S. B. Savage, K. Hutter. The dynamics of avalanches of granular materials from initi-ation to runout. Acta Mech. 86, 201–223, 1991.

∗ Dpto. Matemtica Aplicada I,Universidad de Sevilla (Spain),(edofer@us.es).

This is a joint work withA. Mangeney (IPG Paris),F. Bouchut (U. Marne-la-Vallee) andG. Narbona-Reina (U. Sevilla).

Computing bubble oscillations on GPU

Philippe Helluy

IRMA, Université de Strasbourg7 rue René Descartes, 67084 Strasbourg Cedex, France

helluy@math.unistra.fr

We compute the oscillations of a spherical gas bubble inside a compressible liquid. Despite its one-dimensional nature, this problem is computationally intensive. We use a recently developed Glimmprojection technique in order to capture the gas-liquid interface. We also implement the algorithmon GPU in order to get fast and highly resolved numerical results. 2D results will also be presentedon several test cases : bubble collapses and wave breaking.

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A few results on the modelling of multiphase flows

Jean-Marc Herard ∗

The numerical modelling of multiphase flows is a rather wide topic that has been inves-tigated from long by engineers, physicians and also in the mathematical community. Therange of industrial applications is actually quite wide, and, among many others, compa-nies and research institutes that are involved in the simulation of these flows for nuclearsafety applications are looking forward to getting an accurate representation of difficultflow configurations such as the boiling crisis, and the loss of coolant accidents, among oth-ers. Multiphase codes that have been built since the 1980’s rely either on the single fluidapproach (THYC, FLICA, GENEPI in France), or alternatively on the two-fluid approach(CATHARE or NEPTUNE CFD in France). Both of these have well-known advantagesand drawbacks,7,8, 17,26,30 and these codes are chosen for their ability to account for therelevant patterns depending on situations. The present talk will focus on the multi-fluidapproach, and more precisely, we will try to present in a synthetic way some work that hasbeen achieved during the last ten years, focusing on both the mathematical and physicalmodelling. Numerical algorithms will not be discussed in detail herein, and we will onlygive the main numerical ingredients that have been developed with co-workers and PhDstudents, refering to the existing literature on that topic.

Modelling of multi-phase flows.We will explain the main guidelines that led to models and closure laws. These are ba-sically grounded on two physical requirements : (i) an entropy inequality should governevolutions of the state variable ; (ii) unique jump conditions should hold when neglectingviscous contributions. Based on these, and focusing on two-fluid models, a class of modelsmay be exhibited, that complies with the former two constraints (see6,8–10,19), and includesthe Baer-Nunziato model.3 Evenmore, these results may be extended to the porous frame-work,11,15 and to three-phase flow models.14 A more recent -and preliminary- result16

tends to confirm the strategy, allowing -at least formally- the simulation of the transitionfrom water with dilute gas bubbles to steam with liquid droplets. A brief comparison ofthese models with those arising from2,4, 10,12,20–22 will be done.

Numerical simulation of multi-phase flow models.Many strategies may be introduced in order to compute approximate solutions of the abovementionned systems. Roughly speaking, ours consists in the development of fractional stepmethods, which enable us to develop and use explicit solvers for the convective part, andnon linear implicit solvers in order to account for interfacial transfer terms, while comply-ing with the global entropy inequality. Techniques will be briefly discussed: however, sincemost of existing papers provide several schemes to cope with the hyperbolic part of PDEs(see1,5, 9, 11,13,19,23,24,27–29,31), special focus will be given on the treatment of source terms(see references8,9, 18,19). Drawbacks and advantages of current methods, and also possiblenew directions will be eventually discussed.

Acknowledgments: this work has benefited from several contributions by F. Coquel, T.Gallouet, L. Girault, V. Guillemaud, P. Helluy, O. Hurisse, Y. Liu, K. Saleh, N. Seguin.Part of the work has been achieved within the framework of NEPTUNE project (withfinancial support by EDF/CEA/AREVA/IRSN) and SITAR project (EDF).

∗EDF, R&D, MFEE, 6 quai Watier, 78400, Chatou, France. (jean-marc.herard@edf.fr)

1

References

1 Ambroso A., Chalons C., Coquel F., Galie T., ”Relaxation and numerical approximation of a two-fluid two-pressurediphasic model ” Math. Model. and Numer. Anal., vol. 43(6), pp. 1063–1098, 2009.

2 Bo W., Jin H., Kim D., Liu X., Lee H. Pestiau N., Yu Y., Glimm J., Grove J.W., ”Comparison and validation ofmultiphase closure models”, Computers and Mathematics with Applications, vol. 56, pp. 1291-1302, 2008.

3 Baer M.R., Nunziato J.W., ”A two-phase mixture theory for the deflagration to detonation transition (DDT) inreactive granular materials”, Int. J. Multiphase Flow, vol. 12(6), pp. 861–889, 1986.

4 Bdzil J.B., Menikoff R., Son S.F., Kapila A.K., Stewart D.S., ”Two-phase modeling of a DDT in granular materials:a critical examination of modeling issues”, Phys. of Fluids, vol. 11, pp. 378-402, 1999.

5 Chalons C., Coquel F., Kokh S., Spillane N. ”A relaxation method to compute approximations of the Baer-Nunziatomodel” Proceedings of Finite Volumes Complex Applications VI, Fort, Furst, Halama, Herbin, Hubert editors, Springer, 2011.

6 Coquel F., Gallouet T., Herard J.M., Seguin N., ”Closure laws for a two-fluid two-pressure model”, C. R. Acad.Sci. Paris, vol. I-332, pp. 927–932, 2002.

7 Drew D.A., Passman S.L., ”Theory of multi-component fluids”, Applied Mathematical Sciences, vol. 135, Springer,1999.

8 Gallouet T., Helluy P., Herard J.-M., Nussbaum J., ”Hyperbolic relaxation models for granular flows”, Math.Model. and Numer. Anal., vol.44(2), pp.371-400, 2010.

9 Gallouet T., Herard J.-M., Seguin N., ”Numerical modelling of two phase flows using the two-fluid two-pressureapproach”, Math. Mod. Meth. in Appl. Sci., vol. 14(5), pp. 663-700, 2004.

10 Gavrilyuk S., Saurel R., ”Mathematical and numerical modelling of two phase compressible flows with inertia”, J.Comp. Phys., vol. 175, pp. 326-360, 2002.

11 Girault L., Herard J.-M., ”A two-fluid hyperbolic model in a porous medium”, Math. Model. and Numer. Anal., vol.44(6), pp. 1319-1348, 2010.

12 Glimm J., Saltz D., Sharp D.H., ”Two-phase flow modelling of a fluid mixing layer”, J. Fluid Mechanics, vol. 378,pp. 119-143, 1999.

13 Guillemaud V., ”Modelisation et simulation numerique d’ecoulements diphasiques par une approche bifluide a deuxpressions”, PhD thesis, Universite Aix-Marseille I, Marseille, France, 2007.

14 Herard J.-M., ”A three-phase flow model”, Mathematical Computer Modelling, vol. 45, pp. 432-455, 2007.15 Herard J.M., ”An hyperbolic two-fluid model in a porous medium”, C. R. Mecanique, vol. 336, pp. 650-655, 2008.16 Herard J.-M., ”Une classe de modeles diphasiques bi-fluides avec changement de regime”, EDF report H-I81-2010-

0486-FR, 2010.17 Herard J.-M., Hurisse O., ”A simple method to compute standard two-fluid models”, Int. J. Comput. Fluid Dynamics,

vol. 19(7), pp. 475-482, 2005.18 Herard J.-M., Hurisse O., ”Schemas d’integration du terme source de relaxation des pressions phasiques pour un

modele bifluide hyperbolique”, EDF report, H-I81-2009-1514-FR, in French, 2009.19 Herard J.-M., Hurisse O., ” A fractional step method to compute a class of compressible gas-liquid flows”, Computers

and Fluids, vol.55, pp. 57-69, 2012.20 Jin H., Glimm J., Sharp D.H., ”Compressible two-pressure two-phase models”, Physics Letters A, vol. 353, pp. 469-474,

2006.21 Kapila A.K., R. Menikoff R., Bdzil J.B., Son S.F., Stewart D.S., ”Two-phase modeling of a DDT in granular

materials: reduced equations”, Phys. of Fluids, vol. 13, pp. 3002-3024, 2001.22 Kapila A.K., Son S.F., Bdzil J.B., Menikoff R., Stewart D.S., ”Two-phase modeling of a DDT: structure of the

velocity relaxation zone”, Phys. of Fluids, vol. 9(12), pp. 3885–3897, 1997.23 Karni S., Hernandez-Duenas G., ”A hybrid algorithm for the Baer Nunziato model using the Riemann invariants”,

SIAM J. of Sci. Comput., vol.45, pp.382-403, 2010.24 Liu Y., PhD thesis, Universite Aix Marseille I, Marseille, France, in preparation, 2013.25 Papin M., Abgrall R., ”Fermetures entropiques pour les modeles bifluides a sept equations”, C. R. Mecanique, vol. 333,

pp. 838–842, 2005.26 Ransom V., Hicks D.L., ”Hyperbolic two-pressure models for two-phase flow”, J. Comp. Phys., vol.53, pp.124-151,

1984.27 Saleh K., PhD thesis, Universite Pierre et Marie Curie, Paris, France, in preparation, 2012.28 Saurel R., Abgrall R., ” A multiphase Godunov method for compressible multifluid and multiphase Flows ”, J.

Comp. Physics., vol.150, pp.425-467, 1999.29 Schwendeman D.W., Wahle C.W., Kapila A.K., ” The Riemann problem and a high-resolution Godunov method for

a model of compressible two-phase flow ”, J. Comp. Phys., vol.212, pp.490-526, 2006.30 Stewart H.B., Wendroff B., ” Two-phase flow: models and methods ”, J. Comp. Phys., vol.56, pp.363-409, 1984.31 Tokareva S.A., Toro E.F., ” HLLC type Riemann solver for the Baer-Nunziato equations of compressible two-phase

flow”, J. Comp. Phys., vol.229, pp.3573-3604, 2010.

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Staggered schemes for compressible flows

Raphaèle Herbin

Centre de Mathématiques et InformatiqueUniversité d’Aix-Marseille

39 rue Joliot Curie13453 Marseille 13, France

raphaele.herbin@latp.univ-mrs.fr

We propose staggered implicit and pressure correction schemes for the compressible Euler equa-tions, which include the discretization of the internal energy balance equation. This offers two mainadvantages : first, we avoid the space discretization of the total energy, which involves cell-centeredand face-centered variables in the staggered framework ; second, we obtain algorithms which boildown to usual schemes in the incompressible limit. To obtain correct weak solutions (in particular,with shocks satisfying the Rankine-Hugoniot conditions), we need to introduce a corrective termin the internal energy balance, which we build as follows. We first derive a discrete kinetic energybalance. This relation involves source terms, which are then, in some way, compensated in the in-ternal energy balance. Since the kinetic and internal energy equation are associated to the primaland dual mesh respectively, they cannot be summed to obtain a total energy balance. However, wetheoretically prove, in the 1D case, that, if the scheme converges, the limit indeed satisfies a weakform of this latter equation. Finally, we present numerical results which confort this theory.

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Dynamic relaxation processes in compressible multiphase flows

Olivier Le Métayer

Aix-Marseille University, IUSTI - SMASH Group, 5 rue E. Fermi, 13453 Marseille Cedex 13, France

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 occur. 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. In this model initially introduced by (Baer & Nunziato, 1986), each phase obeys its own equation of state and has its own set of equations and variables (pressure, temperature, velocity, energy, entropy,…). In this context the multiphase mixture instantaneously tends towards a mechanical or thermodynamic equilibrium at each point of the flow. For example, when considering pressure and velocity relaxation effects, a single pressure and single velocity model is solved intrinsically (Kapila & al., 2001; Saurel & al., 2008). In addition 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. First the numerical treatment aimed to solve the non-equilibrium compressible multiphase flow model will be presented. This numerical method named Discrete Equations Method has shown its ability to solve multiphase compressible flows in very different configurations (Abgrall & Saurel, 2003; Saurel & al., 2003; Chinnayya & al, 2004; Le Métayer & al., 2005; Saurel & al., 2007; Le Métayer & al., 2011) such as mixtures with several velocities or interface problems. Then a description of the various relaxation processes will be given as well as a presentation of physical examples.

Key words: compressible multiphase flows, relaxation procedures Abgrall R. & Saurel R. (2003) Discrete equations for physical and numerical compressible multiphase mixtures. Journal of Computational Physics

186, 361-396 Baer M.R. & Nunziato J.W. (1986) A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials.

International Journal of Multiphase Flows 12, 861-889 Chinnayya A., Daniel E. & Saurel R. (2004) Modelling detonation waves in heterogeneous energetic materials. Journal of Computational Physics

196, 490-538 Kapila A., Menikoff R., Bdzil J., Son S., Stewart D. (2001) Two-phase modeling of DDT in granular materials: reduced equations, Physics of

Fluids 13, 3002-3024 Le Métayer O., Massoni J. & Saurel R. (2005) Modelling evaporation fronts with reactive Riemann solvers. Journal of Computational Physics 205,

567-610 Le Métayer O., Massol A., Favrie N. & Hank S. (2011) A discrete model for compressible flows in heterogeneous media. Journal of

Computational Physics 230, 2470-2495 Saurel R., Gavrilyuk S. & Renaud F. (2003) A multiphase model with internal degrees of freedom : application to shock-bubble interaction.

Journal of Fluid Mechanics 495, 283-321 Saurel R., Massoni J. & Renaud F. (2007) A numerical method for one-dimensional compressible multiphase flows on moving meshes.

International Journal for Numerical Methods in Fluids 54, 1425-1450 Saurel R., Petitpas F., Abgrall R. (2008), Modelling phase transition in metastable liquids. Application to cavitating and flashing flows, Journal of

Fluid Mechanics 607, 313-350

Numerical study of eccentric core-annular owG. Ooms1, M.J.B.M. Pourquie1 and P. Poesio21 J.M. Burgerscentrum, Delft University of Technology,Laboratory for Aero- and Hydrodynamics,Leeghwaterstraat 21, 2628 CA Delft, The Netherlands2 Universita degli Studi di Brescia,Dipartimento di Ingegneria e Industriale,Via Branze 38, 25123 Brescia, ItalyAbstractA numerical study (taking into account inertial -, viscous - and pressure forces) has been madeof eccentric core-annular ow through a horizontal pipe, special attention being paid to the verticalforce on the core. The viscosity of the core is assumed to be so large that it behaves as a rigid solid. Awave is present at its surface. The shape of the wave is based on experimental results published earlierin the open literature. Due to the eccentricity the centre line of the core is shifted in the upwardvertical direction with respect to the centre line of the tube. The vertical force on the core was foundto be dependent on the Reynolds number: at small values of the Reynolds number the force is in theupward vertical direction, at large values the force is downward. This means that at large values ofthe Reynolds number an upward buoyancy force on the core due to a density di�erence between coreand annulus can be counterbalanced. So a stationary core-annular ow is then possible.

1

Diffuse interfaces and transformation fronts modelling in compressible materials

Richard Saurel

Aix-Marseille University, IUSTI - SMASH Group, 5 rue E. Fermi, 13453 Marseille Cedex 13, France

Diffuse interfaces are a consequence of contact discontinuities numerical diffusion. They appear with any Eulerian hyperbolic solver and result in computational mixture cells. This has serious consequences on the thermodynamic state computation as the equations of state of the fluids in contact are discontinuous. To circumvent this difficulty artificial mixture cells are considered as true multiphase mixtures with stiff mechanical relaxation effects (Saurel and Abgrall, 1999). This method has been simplified by Kapila et al. (2001) with the help of asymptotic analysis, resulting in a single velocity, single pressure but multi-temperature flow model. This model presents serious difficulties for its numerical resolution, as one of the equations is non-conservative, but is an excellent candidate to solve mixture cells as well as pure fluids. In the presence of shocks, jump conditions have to be provided. They have been determined in Saurel et al. (2007) in the weak shock limit. When compared against experiments for both weak and strong shocks, excellent agreement is observed. These relations are accepted as closure relations for the Kapila et al. (2001) model in the presence of shocks. Mass transfer modeling in this model has been addressed in Saurel et al. (2008) in the context of evaporation and flashing fronts. With the help of corresponding heat and mass transfer terms, it became possible to deal with high speed cavitating flows (Petitpas et al., 2009). Oppositely to the previous example of endothermic phase transition, when exothermic effects are considered as for example with high energetic materials, detonation waves appear. With the help of the shock relations and governing equations inside the reaction zone, generalized Chapman-Jouguet conditions have been obtained as well as detonation wave structure of heterogeneous explosives (Petitpas et al., 2009). Within this multiphase flow approach, the same equations are solved at each mesh point with the same numerical scheme (Saurel et al., 2009). Extra multiphysics extensions have been addressed in:

- Perigaud and Saurel (2005), to model capillary effects. - Favrie et al. (2009) to model hyperelastic materials and interactions with fluids. - Saurel et al. (2010) to deal with dynamic and irreversible powder compaction.

Unstable turbulent interfaces where mixing layer appear between various gases are under consideration. An introduction to multiphase flow modeling for diffuse interfaces will be given as well as an overview of possible multiphysic extensions.

Key words: multiphase flow modelling, hyperbolic systems with relaxation Favrie N., Gavrilyuk S. and Saurel R. (2009) Solid-fluid diffuse interface model in cases of extreme deformations. Journal of Computational

Physics 228(16), 6037-6077 Kapila A., Menikoff R., Bdzil J., Son S., Stewart D. (2001) Two-phase modeling of DDT in granular materials: reduced equations, Physics of

Fluids 13, 3002-3024 Perigaud G., Saurel R. (2005) A compressible flow model with capillary effects, Journal of Computational Physics 209, 139-178 Petitpas F., Saurel R., Franquet E. and Chinnayya A. (2009) Modelling detonation waves in condensed energetic materials: Multiphase CJ

conditions and multidimensional computations. Shock Waves 19(5), 377-401 Petitpas, F., Massoni, J., Saurel, R., Lapebie, E. and Munier, L. (2009) Diffuse interface model for high speed cavitating underwater systems. Int. J.

Multiphase Flow 35, 747-759 Saurel R. and Abgrall R. (1999) A multiphase Godunov method for compressible multifluid and multiphase flows. Journal of Computational

Physics 150, 425-467 Saurel R., Petitpas F., Abgrall R. (2008), Modelling phase transition in metastable liquids. Application to cavitating and flashing flows, Journal of

Fluid Mechanics 607, 313-350 Saurel R., Petitpas F. and Berry R.A. (2009) Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows

and shocks in multiphase mixtures. Journal of Computational Physics 228, 1678-1712 Saurel R., Favrie N., Petitpas F., Lallemand M.H. and Gavrilyuk S. (2010) Modelling irreversible dynamic compaction of powders. Journal of Fluid

Mechanics 664, 348-396

Compressible two-phase flow models:asymptotics, coupling and adaptationNicolas Seguin

In the context of simulations of water flows in pressurised water reactors, sev-eral models of two-phase flows are used. They may vary according to the locationin the water cooling system and to the local properties of the expected flow. Allthese models can be linked by asymptotic limits, through relaxation processes.

First, this presentation aims at providing the different models we are inter-ested in together with their asymptotic compatibilities. In general, the couplingtechniques introduce spurious perturbations which increase if the coupling inter-faces are ill located. To overcome this problem, we propose several methods toestimate the optimal position of the coupling interface — the optimality has tobe understood in the sense of minimizing the use of the more complex modelswhile minimizing the global error due to the coupling. This enables us to developdynamical model adaptation. Several numerical examples illustrate the good be-havior of this strategy.

Physics of Turbulent Dispersed Flows and Simulation Approaches

Alfredo Soldati

Centro Interdipartimentale di Fluidodinamica e Idraulica, Universita di Udine, 33100, Udine, Italy& Department of Fluid Mechanics, International Center for Mechanical Sciences, 33100 Udine, Italy

soldati@uniud.it

One particle with inertia larger than that of the surrounding fluid will undergo a centrifugal motion when entrained ina vortex of suitable scale. One bubble will undergo an opposite centripetal motion. Turbulent flows are populated bymany vortical structures: In fact an entire turbulence spectrum ranging from tiny to extremely large scales. Particles,droplets and bubbles are inertial and their behavior in turbulence is characterized by a lag time with respect to theforcing action of the surrounding fluid: This phenomenon, which scales with the inertia of the entrained object, leadsto preferential distribution and accumulation into specific flow regions, with a pattern fully determined, and yet nontrivially explainable, by the corresponding complex pattern of coherent fluid motions [1, 4].

Preferential distribution controls the rate at which sedimentation and re-entrainment occur, reaction rates in burners orreactors and can also determine raindrop formation and, through plankton, bubble and droplet dynamics, the rate ofoxygen-carbon dioxide exchange at the ocean-atmosphere interface.

In this talk, we will review a number of physical phenomena in which particle segregation in turbulence is a crucialeffect describing the physics by means of Direct Numerical Simulation of turbulence. We will elucidate conceptsand modeling ideas derived from a systematic numerical study of the turbulent flow field coupled with Lagrangiantracking of particles under different modeling assumptions [5, 6]. We will underline the presence of the strong shearwhich flavors wall turbulence with a unique multiscale aspect and adds intricacy to the role of inertia and gravity ininfluencing particle motion [2]. Through a number of physical examples of practical interest such as boundary layers,free-surface and stratified flows [3], we will show that a sound rendering of turbulence mechanisms is required toproduce a physical understanding of particle trapping, segregation and ultimately macroscopic flows such as settlingand re-entrainment.

Remarking that fluid motions responsible for particle segregation which control the transport of the dispersed speciesmay be ignored or filtered by turbulence solving strategies such as Large Eddy Simulation, we will pinpoint modellingissues also providing prospects and guidelines [7, 8].

References

[1] C. Marchioli and A. Soldati, Mechanisms for Particle Transfer and Segregation in Turbulent Boundary Layer, J.Fluid Mech., 468, 283-315, 2002.

[2] V. Lavezzo, A. Soldati, S. Geraschenko, Z. Warhaft, and L. R. Collins, On the role of gravity and shear on inertialparticle accelerations in near-wall turbulence, J. Fluid Mech., 658, 229-246, 2010.

[3] F. Zonta, M. Onorato and A. Soldati, Turbulence and internal waves in stably-stratified channel flow withtemperatre-dependent fluid properties, J. Fluid Mech., 2012.

[4] A. Soldati, and C. Marchioli, Physics and modelling of turbulent particle deposition and entrainment: Review ofa systematic study, Int. J. Multiphase Flow, 35, 827-839, 2009.

[5] D. Molin, C. Marchioli and A. Soldati, Turbulence modulation and microbubble dynamics in vertical channelflow, Int. J. Multiphase Flow, 2012, 42, 80-95.

[6] C. Marchioli, M. Fantoni, and A. Soldati, Orientation, distribution and deposition of elongated, inertial fibers inturbulent channel flow, Phys. Fluids, 22, 033301, 2010.

[7] C. Marchioli, M.V. Salvetti, and A. Soldati, Some issues concerning Large-Eddy Simulation of inertial particledispersion in turbulent bounded flows, Phys. Fluids, 20, 040603, 2008.

[8] F. Bianco, S. Chibbaro, C. Marchioli, M.V. Salvetti, A. Soldati, Intrinsic filtering errors of Lagrangian particletracking in LES flow fields, Phys. Fluids, 24, 2012.

Exact relaxation and accuracy enhancement of numerical

schemes for some hyperbolic conservation laws

Quang Huy Tran

IFP Energies nouvelles, Departement Mathematiques Appliquees

1 et 4 avenue de Bois Preau, 92852 Rueil-Malmaison Cedex, France

quang-huy.tran@ifpen.fr

We propose a novel approach to enhance the space-time accuracy of relaxation schemes forscalar conservation laws and some hyperbolic systems, such as the p-system. Our constructionrelies on a Strang splitting of an exact augmented relaxation model, “exact” in the sense that ithas the same solution as the original model. By means of some additional technicalities similar tolimited reconstructions, we are able to ensure that the computed solution is second-order accuratewith respect to the original model in smooth regions, while degenerating to a first-order scheme inthe neighborhood of shocks. In any cases, maximum and/or positivity principles are guaranteed.

1

Drop Pinch-Off for Discrete Flows from a Capillary

F. Bierbrauer,1 N. Kapur,

2 M. Wilson

2

1School of Computing, Mathematics and Digital Technology, Manchester Metropolitan

University, Manchester, M1 5GD, UK

2School of Mechanical Engineering, University of Leeds, Leeds, LS2 9JT, UK

The problem of drop formation and pinch-off from a capillary tube under the influence of

gravity has been extensively studied when the internal capillary pressure is constant. This

ensures a continuous time independent flow field inside the capillary tube typically of the

Poiseuille flow type. Characteristic drop ejection behaviour includes: periodic drop ejection,

drop ejection with associated satellite production, chaotic behaviour and jetting. It is well

known that this characteristic behaviour is governed by the Weber (We) and Ohnesorge (Oh)

(for a given Bond number) numbers and may be delineated in a We verses Oh operability

diagram.

An in-depth physical understanding of drop ejection is also of great importance to industry

where the tight control of drop size and ejection velocity are of critical importance in

industrial processes such as water-spray cooling, spray painting and for agricultural

insecticide sprays. However, the use of such a continuous flow approach for drop ejection in

industry is often impractical since such continuous flows cannot be operator controlled. For

this reason it is important to investigate so-called discrete pipe flows where the flow can be

turned on and off at will. This means the flow inside the pipe is now time-dependent being

controlled in a step-wise fashion.

As a first stage in the investigation of drop pinch-off behaviour in discrete pipe flows this

paper will study the critical pinch-off time required for drop ejection. This is the discrete

amount of time the pipe flow is turned on for in order for a drop to be ejected from the

capillary. A Newtonian incompressible free-surface CFD flow code developed at the

University of Leeds is used to investigate the critical pinch-off time for a range of internal

pipe velocities (the central flow maximum in Poiseuille flow). It is found that the time

required for drop ejection to occur decreases exponentially with internal pipe velocity. These

characteristic times are also far smaller than typical static drop release times expected from

Harkins-Brown analyses. The phenomenology of the process is due to the creation of a

capillary wave at the pipe exit upon the sudden turning on of the flow inside the pipe. The

capillary wave acts to transport fluid from the upper part of the forming pendant drop at the

end of the capillary to the lower part of the drop both lowering the pendant drop centre-of-

mass and thinning the neck region connecting the drop to the pipe. This allows the drop to be

pinched off at an earlier than expected time as compared to static drop release times.

Computing water-hammer flows with two-fluid models

Fabien Crouzet Frederic Daude ∗ Pascal Galon † Philippe Helluy ‡

Jean-Marc Herard Olivier Hurisse § Yujie Liu ¶

Abstract submitted for contribution to AMIS2012 - Chambery, France, june 19-22, 2012.

We focus in this paper on the computation of water-hammer flows with two-fluid modelsof compressible water-vapour flows (see3,4, 6, 7). The two-fluid model contains stiff sourceterms associated with pressure-velocity-temperature-Gibbs potential relaxation terms, andthe non-linear convective part of the model is not under conservative form. Thus the com-putation of the whole model is rather difficult, and it requires the development of specificalgorithms. Once the two-fluid model and its properties have been recalled, we detail theFinite volume algorithms that are used, giving special emphasis on the pressure relaxationschemes that have been developed for practical simulations, when a stiffened gas EOS isretained for the liquid phase. These schemes, which rely on early propositions detailedin5,8 , require a strong coupling of source terms in total energy and statistical void fractionequations (see9). The accurate computation of gas-liquid flows without any mass transferis not obvious at all, but simulations involving mass transfer are even more questionable.In order to illustrate these difficulties, some simulations of water-hammer situations willbe presented and discussed. The paper will also provide some numerical convergence ratesthat have been obtained while computing several Riemann problems (1,2, 10).

This work has been achieved in the framework of the SITAR project.

References

1 Ambroso A., Chalons C., Coquel F., Galie T., ”Relaxation and numerical approximation of a two-fluid two-pressurediphasic model ” Math. Model. and Numer. Anal., vol. 43(6), pp. 1063–1098, 2009.

2 Ambroso A., Chalons C., Raviart P.A., ”A Godunov-type method for the seven-equation model of compressibletwo-phase flow ” Computers and Fluids, vol. 54, pp67-91, 2012.

3 Baer M.R., Nunziato J.W., ”A two-phase mixture theory for the deflagration to detonation transition (DDT) inreactive granular materials”, Int. J. Multiphase Flow, vol. 12(6), pp. 861–889, 1986.

4 Bdzil J.B., Menikoff R., Son S.F., Kapila A.K., Stewart D.S., ”Two-phase modeling of a DDT in granular materials:a critical examination of modeling issues”, Phys. of Fluids, vol. 11, pp. 378-402, 1999.

5 Gallouet T., Helluy P., Herard J.-M., Nussbaum J., ”Hyperbolic relaxation models for granular flows”, Math.Model. and Numer. Anal., vol.44(2), pp.371-400, 2010.

6 Gallouet T., Herard J.-M., Seguin N., ”Numerical modelling of two phase flows using the two-fluid two-pressureapproach”, Math. Mod. Meth. in Appl. Sci., vol. 14(5), pp. 663-700, 2004.

7 Gavrilyuk S., Saurel R., ”Mathematical and numerical modelling of two phase compressible flows with inertia”, J.Comp. Physics., vol. 175, pp. 326-360, 2002.

8 Herard J.-M., Hurisse O., ”Schemas d’integration du terme source de relaxation des pressions phasiques pour unmodele bifluide hyperbolique”, EDF report, H-I81-2009-1514-FR, in French, 2009.

9 Herard J.-M., Hurisse O., ” A fractional step method to compute a class of compressible gas-liquid flows”, Computersand Fluids, vol.55, pp. 57-69, 2012.

10 Schwendeman D.W., Wahle C.W., Kapila A.K., ” The Riemann problem and a high-resolution Godunov method fora model of compressible two-phase flow ”, J. Comp. Physics., vol. 212, pp. 490-526, 2006.

∗EDF, R&D, AMA, and LaMSID, UMR EDF/CNRS/CEA 2832, 1 avenue du General de Gaulle, 92141, Clamart, France.†CEA, Saclay, France, and LaMSID, UMR EDF/CNRS/CEA 2832, 1 avenue du General de Gaulle, 92141, Clamart, France.‡IRMA, 7 rue Descartes, Universite de Strasbourg, 67084, Strasbourg, France.§EDF, R&D, MFEE, 6 quai Watier, 78400, Chatou, France.¶Corresponding author, EDF, R&D, AMA, and LaMSID, UMR EDF/CNRS/CEA 2832, 1 avenue du General de Gaulle,

92141, Clamart, France. PhD student in LATP-Universite Aix-Marseille, 39 rue Joliot Curie, 13453, Marseille , France.

1

Aymen Laadhari

AutorsAymen Laadhari , Ricardo Ruiz-Baier and Alfio Quarteroni.

AffiliationChair of Modelling and Scientific Computing. École Polytechnique Fédérale de Lausanne,Switzerland.Abstract

Title : An Eulerian Finite Element Method for the Simulation of Fluid-Structure InteractionBiophysics and biomechanics are two fields where Fluid-Structure Interactions play an important role, bothfrom the modeling and computing points of view. In many applications, flow and solid models coexist withbiochemical systems. For such problems, it is desirable to have an accurate and efficient tool coupling modelsof different nature, typically Eulerian for fluids and Lagrangian for solids. In this talk, we aim at providingsuitable tools based on a Level Set approach for the numerical simulation of a multiphasic flow modelling anisolated cardiomyocyte rhytmic contraction driven by intracellular calcium waves.We propose an Eulerian Finite Element formulation for the numerical simulation of incompressible hyperelasticmaterials possibly immersed in a Newtonian fluid, and their interaction with chemical processes related tothe release of cytosolic and sarcoplasmic calcium governed by reaction-diffusion equations. The mechanicaldescription of the phenomenon relies in a virtual multiplicative decomposition of the deformation gradient intoan elastic passive response, and an anisotropic active component depending on the calcium release. The solid-fluid interface is captured using a level set approach. We apply a consistent Newton-Raphson linearization ofthe Cauchy stress tensor for the solid, and we describe the associated finite element framework. Finally, wereport several numerical experiments.

Updated: May 31, 2012.

MATHICSE-CMCS, École Polytechnique Fédérale de Lausanne, CH-1015Switzerland – T

• B aymen.laadhari@epfl.ch • http://sma.epfl.ch/˜ laadhari/

A mixture-energy-consistent 6-equation two-phase numerical model forfluids with interfaces and cavitation phenomena

Marica Pelanti1,∗ and Keh-Ming Shyue2

1ENSTA ParisTech, Unite de Mecanique, Chemin de la Huniere, 91761 Palaiseau Cedex, France.2National Taiwan University, Department of Mathematics, Taipei 10617, Taiwan.

∗Presenting author. marica.pelanti@ensta-paristech.fr

The modelling of cavitating flows is relevant in numerous areas of engineering, from naval industry to rocket

propulsion technology. In this work we are interested in developing new methods for the simulation of cavitating

flow processes in the framework of diffuse-interface compressible multi-phase flow models. We begin by considering

the hyperbolic 6-equation single-velocity two-phase model with instantaneous pressure relaxation of Saurel–

Petitpas–Berry [1]. We adopt a variant of this model by using phasic total energy equations instead of phasic

internal energy equations, in contrast with the more classical approach [1, 2]. We numerically solve the proposed

two-phase flow model in two dimensions by a fully-discretized high-resolution wave-propagation scheme based

on a hybrid HLLC/Roe Riemann solver. The alternative formulation with phasic total energies allows us to

write discrete non-conservative phasic energy equations whose sum exactly recovers the conservative discrete

form of the total energy equation for the mixture. A first advantage of this method is that there is no need to

augment the 6-equation model with an extra equation for the mixture total energy as done in [1, 2] to correct

the thermodynamic state resulting from the non-conservative energy equations. Moreover, the consistence of the

computed phasic energies with conservation of the mixture total energy enables us to ensure agreement of the

relaxed pressure with the mixture equation of state, in combination with a simple pressure relaxation procedure.

Temperature and Gibbs free energy relaxation terms can be also included (see e.g. [2]) in a way that preserves

consistency with the mixture energy at the discrete level. Several two-dimensional numerical experiments of

cavitation appearance and dynamic creation of interfaces are presented to show the efficiency of the numerical

model.

References

[1] R. Saurel, F. Petitpas, and R. A. Berry, Simple and efficient relaxation methods for interfaces separating compressiblefluids, cavitating flows and shocks in multiphase mixture, J. Comput. Physics, Vol. 228 (2009), pp. 1678–1712.

[2] A. Zein, M. Hantke, and G. Warnecke, Modeling phase transition for compressible two-phase flows applied tometastable liquids, J. Comput. Physics, Vol. 229 (2010), pp. 2964–2998.