Multiphase Flows: Compressible Multi-Hydrodynamics

  • Daniel Lhuillier
  • Theo G. Theofanous
  • Meng-Sing Liou
Reference work entry


Effective field modeling of two-phase flow has provided a critical part of the foundation upon which light water (power) reactor technology was made to rest some 20-30 years ago. We can envision a similarly significant role in the future as simulation capabilities are poised to meet new kinds of practical demands at the interplay between economics, safety assurance, and regulatory needs. These new demands will require better predictive reliability for larger departures from past practices, and this in turn will require strengthening of the scientific component along with translating past empiricism into more and more fundamental terms. In this perspective, the mathematical formulation of the effective field model, as well as the numerical implementation of this formulation needs to be revisited and reassessed. Helping respond to this need is the purpose of this chapter.

We delineate a conceptual framework for addressing prediction of multiphase flows at the three-dimensional, phase distribution level. This is in terms of a local, disperse system description (bubbles/drops in a continuous liquid/vapor phase). The requirement that follows is a well-posed formulation and a high-fidelity numerical treatment that allows capturing of shocks and contact discontinuities over all (relative) flow speeds, consistently with what is physically allowable according to the density ratios involved — in particular, high relative Mach numbers for droplet/particle flows. The importance of inviscid interactions (between the phases) in this context is highlighted.

The scope is for disperse-phase volume fractions up to about 20% as pertinent to fluid-fluid systems. This provides the basis for addressing phase transitions through coalescence as this process becomes significant at still higher volume fractions. The theoretical framework provides also the basis for extensions (outlined only in general terms here) to the high-volume fractions pertinent to dense solid-particle systems. The computational approach is readily applicable to both these extensions.

A general disperse system formulation is derived by means of a new, “hybrid” method that incorporates features of a statistical approach and reveals more clearly the nature of phase interactions at the individual particle scale. Moreover in this manner the formulation lends itself to elaboration of the constitutive treatment by means of numerical simulations (based on the direct solution of the Navier-Stokes equations) resolved at the particle scale. The formulation is exemplified by successive applications to various increasingly complex situations, starting with non-dissipative systems, where one or the other phase may be incompressible. At each step we examine the hyperbolic character of the system of equations, and we include consideration of high (relative) Mach numbers. The basic constitutive treatment concerns pseudo-turbulent fluctuations of the continuous phase, and the resulting systems of equations are fully closed and hyperbolic even in their non-dissipative form (ready for computation), except for a non-hyperbolic corridor around the transonic region. Results obtained are discussed in relation to formulations that form the basis of current numerical tools (codes) employed in nuclear reactor design and safety analyses (mostly addressing bubbly flows), as well as formulations found in other contexts.

This mathematical formulation is pursued further to its numerical implementation. With an emphasis on flow compressibility, we focus on capturing shocks and contact discontinuities robustly for all flow speeds and at arbitrarily high spatial resolutions. As we learn from relatively recent progress in single-phase flows, the key role is that of “up-winding” applied on the basis of a scheme that emphasizes conservative discretization. This background is briefly reviewed, culminating with a rather detailed exposition of the most recent advance in this line of development: the Advection Upstream Splitting Method (AUSM). The essential and new step here is to extend the basic ideas of the AUSM to the compressible multi-hydrodynamics problems of interest here, and the above-mentioned EFM in particular. We also include calculations illustrative of numerical performance.

The presentation is arranged into two autonomous parts: Part I addresses the formulation of the EFM and Part II deals with the numerical implementation and testing. An overall summarization of where we stand at the completion of this work and what we see as needed future developments is provided in the following.

One basic objective was to address inviscid interactions by means of a coherent theoretical formulation and through to computational testing. In particular, we wanted to access high-fidelity simulations of the type needed to address flow regimes at all flow speeds; especially at the high relative Mach numbers pertinent to disperse particle/droplet flows. This requires that the system of equations be hyperbolic, and we wanted to achieve a solid foundation rather than adopting any of the several ad hoc constitutive models as post facto “remedying” the problem.

On theory, we begin with entropy rather than energy transport equations and we derive, consistently with thermodynamics and the momentum equations, a condition for satisfying conservation of total energy. This condition is of utmost importance showing the tight link between the conservation laws employed, and the transport equations of volume fraction and of pseudo-turbulent kinetic energies of the continuous (included) and disperse (not yet included in the derivation) phases. On this basis we demonstrate a systematic way to deduce closed systems of equations for non-dilute disperse flows, and thusly we arrive at an EFM that is hyperbolic except for a “corridor” around the transonic region. The key is a function of the disperse phase volume fraction E(α d ). It enters as a coefficient of the disperse phase pseudo-turbulent kinetic energy. Awaiting further definition as a function of the Mach number, by means of the type of direct simulations noted above, it is employed here throughout in its zero-Mach form. A much needed extension would also involve the pseudo-turbulent kinetic energy of the disperse phase, along with physics of dense dispersions (collisions etc.).

While terms such as those proposed previously for “interfacial pressure” and “added mass” phenomena can be identified, the complete formulation is not reducible to any of those ad hoc models. Notably, the disperse phase pressure appears nowhere in the momentum equations. Also we find that the claimed as hyperbolic, Baer-Nunziato model involves a volume fraction transport equation, which is not physically tenable for dispersions, or is it an appropriate means to dealing with ill-posedness. On the other hand, we find perfect agreement with the formulations obtained at the incompressible limit by Geurst (1985), employing a complex variational approach, and by Wallis (1989), employing a rather involved development based on potential flow theory.

On computations our objective is to capture shocks and contact discontinuities, for conditions that are within the hyperbolic regions in the Mach number space, and to explore (1) behaviors within the non-hyperbolic corridor, and (2) means of stabilization as necessary. Given the EFM development needs expressed above, it is understood that this testing in the Mach number space is strictly provisional. We begin with an adaptive mesh refinement infrastructure, and the Advection Upstream Splitting Method (AUSM), currently the method of choice for single-phase compressible flows. A key point of adaptation to our EFM is treating the pseudo-turbulent stresses within the pressure flux splitting, and ensuring that the discretization of the nonconservative terms is done in a way that satisfies propagation of contact discontinuities in uniform steady flow without disturbing the pressure field. Our approach is readily extendable to any equation of state and to adding any number of equations (volume fraction transport, multiple equations for the disperse phase for tracking multiple length scales as may be found when the disperse phase is subject to fragmentation). The testing performed for this work was done on 1D problems only. Extending this testing to 2D and 3D problems is underway.

Testing was carried out independently with two computer codes: ARMS (all-regime multiphase simulation) and MuSiC-ARMS (Multi-scale Simulation Code-ARMS). The ARMS was built on an open access platform, the structured adaptive mesh refinement infrastructure (SAMRAI) developed at Lawrence Livermore National Laboratory. The MuSiC-ARMS was built, more recently, on the MuSiC platform, our own specialized code, using irregular grids to “fit” areas of highest refinement (shocks, interfaces, etc.), which are embedded in a multilevel (adaptive) Cartesian mesh. This platform is also used for a DNS code, the MuSiC-SIM, and a pseudo-compressible (incompressible) code, the MuSiC-ISIM. We focus on dispersed being the heavy phase (droplet/particle flows) so as to access realistically high Mach numbers, and significant inviscid interactions.

The test cases were selected to include various kinds of Riemann problems with discontinuities in (a) Mach number only (Fitt’s problems) and (b) pressure, or pressure and disperse phase volume fraction (shock tube problems). In the Fitt’s problem case, we include parametric studies on the value of C that appears in function E(α d ). In addition, we consider shock wave “impact” problems on particle clouds that are either with sharp or smooth (in particle volume fraction) outer boundaries, and as part of this class also the case of dilute clouds for which we have the analytic solution for comparison. Finally, we considered the capturing of contact discontinuities in “mild” situations such as the so-called Faucet problem and the simple convection of a coherent second phase by uniform flow. The Faucet problem is well known to be failed under grid refinement in all published tests to date. The convection problem is important check of the pressure non-disturbing condition, a requirement that is hard to meet due to the non-conservative terms found in all effective field models.

The emphasis being on stability and convergence under grid refinement, all problems were carried out in the inviscid limit (no interfacial drag), and all cases passed the test except for the high pressure ratio shock tube problems where instabilities developed within the expansion wave. However, these cases were stabilized with a minimal amount of dissipation effected by adding a small amount of interfacial drag (roughly one tenth of the normal amount). These numerical results render support to the idea that, notwithstanding the “mild” non-hyperbolic corridor found in the analysis of Part I, the present effective field model is hyperbolic, and along with the numerical treatment employed they provide access to rather extreme two-phase flow conditions in a robust and accurate manner.

In an overall perspective of computational fluid dynamics, the presently offered capability is complementary to that already available through the “standard”, non-hyperbolic two-fluid model as already found in the computational frameworks of the ICE (Harlow and Amsden 1968) and SIMPLE (Patankar and Spalding 1972) methods. The special purposes aimed here are to overcome limitations in grid refinement and to approach flows where the phasic-relative velocities are high enough to introduce significant compressibility effects. Rapid advancement in hardware makes computational analysis of complex multiphase flows, even direct numerical simulations, increasingly more practical and reliable. High-fidelity/resolution techniques such as those employed here can address problems of varying time and length scales and this paves the way for actual simulations of multiphase physics at the effective field level, and even allowing a seamless analysis transitioning across regimes of multiphase flows.


Mach Number Velocity Fluctuation Riemann Problem Contact Discontinuity Carrier Fluid 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Part I

  1. Abgrall R, Saurel R (2003) Discrete equations for physical and numerical compressible multiphase mixtures. J comput Phys 186:361–396MathSciNetMATHCrossRefGoogle Scholar
  2. Baer MR, Nunziato JW (1986) A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials. Int J Multiphase Flow 12:861–889MATHCrossRefGoogle Scholar
  3. Batchelor GK (1970) The stress system in a suspension of force-free particles. J Fluid Mech 41:545–570MathSciNetMATHCrossRefGoogle Scholar
  4. Bdzil J, Menikoff R, Son S, Kapila A, Steward D (1999) Two-phase modeling of a deflagration-to-detonation transition in granular materials: a critical examination of modeling issues. Phys Fluids 11:378–402MATHCrossRefGoogle Scholar
  5. Bestion D (1990) The physical closure laws in the CATHARE code. Nucl Eng Design 124:229–245CrossRefGoogle Scholar
  6. Biesheuvel A, Spoelstra S (1989) The added-mass coefficient of a dispersion of spherical gas bubbles in liquid. Int J Multiphase Flow 15:911–924MATHCrossRefGoogle Scholar
  7. Bronshtein IN, Semendyayev KA, Musiol G, Muehlig H (2007) Handbook of mathematics, 5th edn. Springer, BerlinMATHGoogle Scholar
  8. Bulthuis HF, Prosperetti A, Sangani AS (1995) Particle stress in disperse two-phase potential flow. J Fluid Mech 294:1–16MathSciNetMATHCrossRefGoogle Scholar
  9. Buyevich YuA, Shchelchkova IN (1978) Flow of dense suspensions. Prog Aerospace Sci 18:121–150CrossRefGoogle Scholar
  10. Doi M, Ohta T (1991) Dynamics and rheology of complex interfaces. J Chem Phys 95:1242–1248CrossRefGoogle Scholar
  11. Drew DA, Passman SL (1998) Theory of multicomponents fluids. Springer-Verlag, New-YorkGoogle Scholar
  12. Fletcher DF, Theofanous TG (1997) Heat transfer and fluid-dynamics aspects of explosive melt-water interactions. Adv Heat Transfer 29:129–156 and 157–213Google Scholar
  13. Geurst JA (1985) Virtual mass in two-phase bubbly flow. Physica 129A:233–261Google Scholar
  14. Gidaspow D (1974) Modeling of two-phase flow, round table discussions. In Proceeding of the 5th International Heat Transfer Conference, Tokyo, JapanGoogle Scholar
  15. Hancox WT, Frech RL, Liu WS, Niemann RE (1980) One-dimensional models for transient gas-liquid flows in ducts. Int J Multiphase Flow 6:25–40MATHCrossRefGoogle Scholar
  16. Ishii M, Hibiki T (2006) Thermo-fluid dynamics of two-phase flow. Springer, New YorkMATHGoogle Scholar
  17. Jackson R (1997) Locally averaged equations of motion for a mixture of identical spherical particles and a Newtonian fluid. Chem Engng Sci 52:2457–2469CrossRefGoogle Scholar
  18. Jones AV, Prosperetti A (1985) On the suitability of first-order differential models for two-phase flow predictions. Int J Multiphase Flow 11:133–148MATHCrossRefGoogle Scholar
  19. Kapila AK, Menikoff R, Bdzil JB, Son SF, Stewart DS (2001) Two-phase modeling of deflagration-to-detonation transition in granular materials: reduced equations. Phys Fluids 13:3002–3024CrossRefGoogle Scholar
  20. Keller J, Miksis M (1980) Bubble oscillations of large amplitude. J Acoust Soc Am 68:628–633MATHCrossRefGoogle Scholar
  21. Lhuillier D (1992) Ensemble averaging in slightly non-uniform suspensions. Eur J Mech B/Fluids 11:649–661MathSciNetMATHGoogle Scholar
  22. Lhuillier D (2003a) A mean-field description of two-phase flows with phase changes. Int J Multiphase Flows 29:511–525MATHCrossRefGoogle Scholar
  23. Lhuillier D (2003) Dynamics of interfaces and rheology of immiscible liquid-liquid mixtures. CR Mecanique 331:113–118MATHGoogle Scholar
  24. Marble FE (1969) Some gas dynamic problems in the flow of condensing vapors. Astronaut Acta 14:585–614Google Scholar
  25. MATHEMATICA (2008) Wolfram Research, Inc., Version 7.0, Champaign, ILGoogle Scholar
  26. Nigmatulin RI (1991) Dynamics of multiphase media, vols 1 and 2. Hemisphere Publishing, New YorkGoogle Scholar
  27. Palierne JF (1990) Linear rheology of viscoelastic emulsions with interfacial tension. Rheol Acta 29:204–214CrossRefGoogle Scholar
  28. Park JW, Drew DA, Lahey RT (1998) The analysis of void wave propagation in adiabatic monodispersed bubbly two-phase flows using an ensemble averaged two-fluid model. Int J Multiphase Flow 24:1205–1244MATHCrossRefGoogle Scholar
  29. Prosperetti A, Satrape JV (1990) Stability of two-phase flow models. In: Joseph DD, Schaeffer DG (eds) Two phase flows and waves. Springer, pp 98–117Google Scholar
  30. Prosperetti A, Marchioro M, Tanksley M (1998) Closure of averaged equations for non-uniform disperse flows by direct numerical simulations, presented at ICMF’98, Lyon, FranceGoogle Scholar
  31. Ransom VH, Hicks DL (1984) Hyperbolic two-pressure models for two phase flows. J Comput Phys 53:124–151 and 75:498–504 (1988)Google Scholar
  32. RELAP5/MOD3.3 Code Manual Volume 1: Code structure, system models and solution methods (2001)Google Scholar
  33. Sangani AS, Didwania AK (1993) Dispersed-phase stress tensor in flows of bubbly liquids at large Reynolds numbers. J Fluid Mech 248:27–54MATHCrossRefGoogle Scholar
  34. Stuhmiller JH (1977) The influence of interfacial pressure forces on the character of two-phase flow model equations. Int J Multiphase Flow 3:551–560MATHCrossRefGoogle Scholar
  35. Sushchikh S, Chang C-H (2009) Hyperbolicity analysis and numerical stability of a fully compressible effective field model, Internal Report CRSS/UCSB, October 2009Google Scholar
  36. Theofanous TG, Dinh TN (2003) On the prediction of flow patterns as a principal scientific issue in multifluid flow: contribution to the task group on flow regimes. Multiphase Sci Technol 15(1–4):57–76CrossRefGoogle Scholar
  37. Theofanous TG, Hanraty TJ (2003) Appendix 1: report of study group on flow regimes in multifluid flow. Int J Multiphase Flow 29(7): 1061–1068MATHCrossRefGoogle Scholar
  38. Theofanous TG, Dinh TN, Tu JP, Dinh AT (2002) The boiling crisis phenomenon Part 1: nucleation and nucleate boiling heat transfer, V.26 (6–7), pp 775–792. Part 2: Dryout dynamics and burnout. J Exp Thermal Fluid Sci 26(6–7):793–810Google Scholar
  39. Theofanous TG, Nourgaliev RR, Dinh TN (2004) Compressible multi-hydrodynamics: emergent needs, approaches and status, IUTAM Symposium Computational Approaches to Disperse Multiphase Flow. ANL Oct. 2004. In: Prosperetti A, Balachandar S (eds) Computational approaches to disperse multiphase flow. Springer Verlag, HeidelbergGoogle Scholar
  40. Tucker CL, Moldenaers P (2002) Microstructural evolution in polymer blends. Ann Rev Fluid Mech 34:177–210MathSciNetCrossRefGoogle Scholar
  41. Wallis GB (1989) Inertial coupling in two-phase flow: macroscopic properties of suspensions in an inviscid fluid. Multiphase Sci Technol 5:239–361Google Scholar
  42. Wallis GB (1991) The averaged Bernoulli equation and macroscopic equations of motion for the potential flow of a two-phase dispersion. Int J Multiphase Flow 17:638–695Google Scholar
  43. Young JB (1995) The fundamental equations of gas-droplet multiphase flow. Int J Multiphase Flow 21:175–191MATHCrossRefGoogle Scholar
  44. Zhang DZ, Prosperetti A (1994) Averaged equations for inviscid disperse two-phase flow. J Fluid Mech 267:185–219MathSciNetMATHCrossRefGoogle Scholar

Part II

  1. Amsden AA, Harlow FH (1975) KACHINA: an Eulerian computer program for multifield fluid flows, Tech. rep. la-5680, Scientific Laboratory, Los AlamosGoogle Scholar
  2. Bestion D, Barre F, Faydide B (1999) Methodology, status and plans for development and assessment of CATHARE code, in Proceedings of the International Conference of OECD/CSNI. Annapolis, USAGoogle Scholar
  3. Chakravarthy SR, Osher S (1983) High resolution applications of the Osher upwind scheme for the Euler equations, AIAA Paper 83–1943Google Scholar
  4. Chang C-H, Liou M-S (2003) A new approach to the simulation of compressible multifluid flows with AUSM + scheme, AIAA paper 03–4170Google Scholar
  5. Chang C-H, Liou M-S (2007) A robust and accurate approach to computing compressible multiphase flow: stratified flow model and AUSM + -up scheme. J Comput Phys 225:840–873.MathSciNetMATHCrossRefGoogle Scholar
  6. Chang C-H, Liou M-S, Loc N, Sushchikh S, Wilkin L, Theofanous TG (2009) The all regime multiphase simulation (ARMS) code: formulation, numerics, verification and user’s manual, ver. 3.0, Report CRSS-09–1, Center for Risk Studies and Safety, University of California, Santa BarbaraGoogle Scholar
  7. Chang C-H, Deng X, Sushchikh S, Liou M-S, Theofanous TG (2010a) A multi-scale simulation code for compressible multi-hydrodynamics (MuSiC-ARMS): formulation, numerics, verification and user’s manual, ver. 1.0, Report CRSS-10–2, Center for Risk Studies and Safety, University of California, Santa BarbaraGoogle Scholar
  8. Chang C-H, Deng X, Theofanous TG (2010b) Numerical prediction of interfacial instabilities. Part II: the SIM extended to high speed flows. J Comput Phys (Submitted)Google Scholar
  9. Colella P, Glaz HM (1985) Efficient solution algorithm for the Riemann problem for real gases. J Comput Phys 59:264–289MathSciNetMATHCrossRefGoogle Scholar
  10. Concentration Heat and Momentum Ltd.,
  11. Courant R, Issacson E, Rees M (1952) On the solution of nonlinear hyperbolic differential equations by finite differences. Commun Pure Appl Math 5:243–255MATHCrossRefGoogle Scholar
  12. Drew DA, Passman SL (1999) Theory of multicomponent fluids, Springer-Verlag, BerlinGoogle Scholar
  13. Drew D, Cheng L, Lahey RT (1979) Analysis of virtual mass effects in two-phase flow. Int J Multiphase Flow 5:233–242MATHCrossRefGoogle Scholar
  14. Einfeldt B, Munz CC, Roe PL, Sjogreen B (1991) On Godunov-type methods near low densities. J Comput Phys 92:273–295MathSciNetMATHCrossRefGoogle Scholar
  15. Fitt AD (1989) The numerical and analytical solution of ill-posed systems ofconservation laws. Appl Math Modelling 13:618–631MathSciNetMATHCrossRefGoogle Scholar
  16. Fluent Inc. Joins ANSYS Inc,
  17. Geurst JA (1985) Virtual mass in two-phase bubbly flow. Physica 129A:233–261Google Scholar
  18. Gidaspow D (1974) Modeling of two-phase flow, round table discussions, In 5th International Heat Transfer Conference, Tokyo, JapanGoogle Scholar
  19. Godunov SK (1959) A difference method for the numerical calculation of discontinuous solutions of hydrodynamic equations. Mat Sb 47:271–306MathSciNetGoogle Scholar
  20. Hänel D, Schwane R, Seider G (1987) On the accuracy of upwind schemes for the solution of the Navier-Stokes equations, In 8th AIAA CFD Conference AIAA Paper 87–1105-CPGoogle Scholar
  21. Hancox W, Ferch R, Liu W, Nieman R (1980) One-dimensional models for transient gas–liquid flows inducts. Int J Multiphase Flow 6:25–40MATHCrossRefGoogle Scholar
  22. Harlow FH, Amsden AA (1968) Numerical calculation of almost incompressible flows. J Comput Phys 3(80) (LA-DC-9496)Google Scholar
  23. Harlow FH, Welch JE (1965) Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys Fluids 8:2182–2189MATHCrossRefGoogle Scholar
  24. Harten A (1983) High resolution schemes for hyperbolic conservation laws. J Comput Phys 49:357–393MathSciNetMATHCrossRefGoogle Scholar
  25. Harten A, Hyman JM (1983) Self-adjusting grid methods for one dimensional hyperbolic conservation laws. J Comput Phys 50:235–269MathSciNetMATHCrossRefGoogle Scholar
  26. Jameson A (1983) Solution of the Euler equation for two dimensional transonic flow by a multigrid method. Appl Math Comput 13:327–355MathSciNetMATHCrossRefGoogle Scholar
  27. Kashiwa BA, Rauenzahn RM (1994) A cell-centered ICE method for multiphase flow simulations, Tech. rep. la-ur-93–3922, Los Alamos National LaboratoryGoogle Scholar
  28. Kitamura K, Roe P, Ismail F (2007) An evaluation of Euler fluxes for hypersonic flow computations, 18th AIAA Computational Fluid Dynamics Conference, AIAA 2007–4465Google Scholar
  29. Lax PD (1954) Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun Pure Appl Math 7:159–193MathSciNetMATHCrossRefGoogle Scholar
  30. LeVeque RJ (1990) Numerical Methods for Conservation Laws. Birkhauser Verlag, BaselMATHGoogle Scholar
  31. Liou M-S (1996) A sequel to AUSM: AUSM + . J Comput Phys 129:364–382MathSciNetMATHCrossRefGoogle Scholar
  32. Liou M-S (2006) A sequel to AUSM, Part II: AUSM + -up for all speeds. J Comput Phys 214:137–170MathSciNetMATHCrossRefGoogle Scholar
  33. Liou M-S, Edwards JR (1999) AUSM schemes and extensions for low Mach and multiphase flows, Lecture Series 1999–03, Von Karman Institute, Belgium, March 8–12, 1999Google Scholar
  34. Liou M-S, Steffen CJ (1993) A new flux splitting scheme. J Comput Phys 107:23–39MathSciNetMATHCrossRefGoogle Scholar
  35. Liou M-S, van Leer B, Shuen J-S (1990) Splitting of inviscid fluxes for realgases. J Comput Phys 87:1–24MathSciNetMATHCrossRefGoogle Scholar
  36. Nourgaliev RR, Theofanous TG (2007) High-fidelity interface tracking in compressible flows: unlimited anchored adaptive level set. J Comput Phys 224:836–866MATHCrossRefGoogle Scholar
  37. Nourgaliev RR, Dinh TN, Theofanous TG (2006) Adaptive characteristics-based matching for compressible multifluid dynamics. J Comput Phys 213:500–529MATHCrossRefGoogle Scholar
  38. Nourgaliev RR, Liou M-S, Theofanous TG (2008) Numerical prediction of interfacial instabilities. Part I: sharp interface method (SIM). J Comput Phys 227:3940–3970MathSciNetMATHCrossRefGoogle Scholar
  39. Osher S, Solomon F (1982) Upwind difference schemes for hyperbolic systems of conservation laws. Math Comput 38(158):339–374MathSciNetMATHCrossRefGoogle Scholar
  40. Patankar SV, Spalding DB (1972) A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int J Heat Mass Transf 15:1787–1806MATHCrossRefGoogle Scholar
  41. Piltch MM, Yan H, Theofanous TG (1994) The probability of containment failure by direct containment heating in Zion, NUREG/CR-6075, SAND93–1535; also appeared in Nucl Eng Design 164:1–36 (1996)Google Scholar
  42. Prosperetti A (1999) Some consideration on the modeling of disperse multiphase flows by averaged equation, JSME International Journal. Series B 42:573–585Google Scholar
  43. Prosperetti A, Satrape JV (1990) Stability of two-phase flow models. In: Joseph DD, Schaeffer DG (eds) Two phase flows and waves, Springer-VerlagGoogle Scholar
  44. Ransom VH (1987) Numerical benchmark tests. In: Hewitt GF, Delhay JM, Zuber N (eds) Multiphase science and technology, vol 3. Hemisphere Publishing, Washington, DCGoogle Scholar
  45. RELAP5-3D Manuals, Idaho National Laboratory (2006),
  46. Rivard W, Torrey M (1977) K-FIX: a computer program for transient, two dimensional, two-fluid flow. Tech. rep, Los Alamos Scientific LaboratoryGoogle Scholar
  47. Roe PL (1981) Approximate Riemann solvers, parameter vectors, and difference scheme. J Comput Phys 43:357–372MathSciNetMATHCrossRefGoogle Scholar
  48. Roe PL (1986) Characteristic-based schemes for the Euler equations. Ann Rev Fluid Mech 18:337–365MathSciNetCrossRefGoogle Scholar
  49. SAMRAI: Structured Adaptive Mesh Refinement Application Infrastructure,
  50. Shuen J-S, Liou M-S, van Leer B (1990) Inviscid flux-splitting algorithms for real gases with nonequilibrium chemistry. J Comput Phys 90:371–395MATHCrossRefGoogle Scholar
  51. Smoller J (1983) Shock waves and reaction-diffusion equations. Springer-Verlag, New YorkMATHCrossRefGoogle Scholar
  52. Steger JL, Warming RW (1981) Flux vector splitting of the inviscid gas dynamic equations with applications to finite-difference methods. J Comput Phys 40:263–293MathSciNetMATHCrossRefGoogle Scholar
  53. Stuhmiller JH (1977) The influence of interfacial pressure forces on the character of two-phase flow model equations. Int J Multiphase Flow 3:551–560MATHCrossRefGoogle Scholar
  54. Theofanous TG, Dinh TN (2008) Integration of multiphase science and technology with risk management in nuclear power reactors. Multiphase Sci Technol 20Google Scholar
  55. Theofanous TG, Yuen WW, Angelini S (1998a) Premixing of steam explosions: PM-ALPHA verification studies, DOE/ID-10504; also in Nucl Eng Design 189:59–102 (1999)Google Scholar
  56. Theofanous TG, Yuen WW, Freeman K, Chen X (1998b) Escalation and propagation of steam explosions: ESPROSE.m verification studies, DOE/ID-10503; also in Nucl Eng Design 189:103–138 (1999)Google Scholar
  57. Theofanous TG, Yuen WW, Angelini S, Sienicki JJ, Freeman K, Chen X, Salmassi T (1999) Lower head integrity under steam explosion loads. Nucl Eng Design 189:7–57CrossRefGoogle Scholar
  58. Theofanous TG, Nourgaliev R, Li G, Dinh TN (2006) Compressible multihydrodynamics(CMH): breakup, mixing and dispersal of liquids/solids inhigh speed flows. In: Prosperetti A, Balachandar S (eds) Computational approaches to disperse multiphase flow. Springer Verlag, HeidelbergGoogle Scholar
  59. Toro EF (1997) Riemann solvers and numerical methods for fluid dynamics: a practical introduction. Springer-Verlag, HeidelbergMATHGoogle Scholar
  60. Toumi I, Kumbaro A, Paillere H (1999) Approximate Riemann solvers and flux vector splitting schemes for two-phase flow, VKI Lecture Series 1999–03(von Karman Institute for Fluid Dynamics. Rhode Saint Genese, BelgiumGoogle Scholar
  61. van Albada GD, van Leer B, Roberts WW (1982) A comparative study of computational methods in cosmic gas dynamics. Astron Astrophys 108:76–84MATHGoogle Scholar
  62. van Leer B (1979) Towards the ultimate conservation difference scheme V. A second order sequel to Godunov’s method. J Comput Phys 32 (1979)101–136CrossRefGoogle Scholar
  63. van Leer B (1982) Flux vector splitting for the Euler equations. Lecture Notes Phys 170:507–512CrossRefGoogle Scholar
  64. Wada Y, Liou M-S (1997) An accurate and robust flux splitting scheme for shock and contact discontinuities. SIAM J Sci Stat Comput 18:633–657MathSciNetMATHCrossRefGoogle Scholar
  65. Wallis GB (1991) The averaged Bernoulli equation and macroscopic equations of motion for the potential flow of a two-phase dispersion. Int J Multiphase Flow 17:638–695Google Scholar
  66. Yuen WW, Theofanous TG (1999) On the existence of multiphase thermal detonations. Int J Multiphase Flow 25:1505–1519MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Daniel Lhuillier
    • 1
  • Theo G. Theofanous
    • 2
  • Meng-Sing Liou
    • 3
  1. 1.Institut Jean le Rond d’AlembertCNRS and University ParisParisFrance
  2. 2.Department of Chemical Engineering, Department of Mechanical Engineering, Center for Risk Studies and SafetyUniversity of CaliforniaSanta BarbaraUSA
  3. 3.NASA Glenn Research CenterClevelandUSA

Personalised recommendations