A relaxation-projection method for compressible flows. Part II: Artificial heat exchanges for multiphase shocks

The relaxation-projection method developed in Saurel et al. [R. Saurel, E. Franquet, E. Daniel, O. Le Metayer, A relaxation-projection method for compressible flows. Part I: The numerical equation of state for the Euler equations, J. Comput. Phys. (2007) 822-845] is extended to the non-conservative hyperbolic multiphase flow model of Kapila et al. [A.K. Kapila, Menikoff, J.B. Bdzil, S.F. Son, D.S. Stewart, Two-phase modeling of deflagration to detonation transition in granular materials: reduced equations, Physics of Fluids 13(10) (2001) 3002-3024]. This model has the ability to treat multi-temperatures mixtures evolving with a single pressure and velocity and is particularly interesting for the computation of interface problems with compressible materials as well as wave propagation in heterogeneous mixtures. The non-conservative character of this model poses however computational challenges in the presence of shocks. The first issue is related to the Riemann problem resolution that necessitates shock jump conditions. Thanks to the Rankine-Hugoniot relations proposed and validated in Saurel et al. [R. Saurel, O. Le Metayer, J. Massoni, S. Gavrilyuk, Shock jump conditions for multiphase mixtures with stiff mechanical relaxation, Shock Waves 16 (3) (2007) 209-232] exact and approximate 2-shocks Riemann solvers are derived. However, the Riemann solver is only a part of a numerical scheme and non-conservative variables pose extra difficulties for the projection or cell average of the solution. It is shown that conventional Godunov schemes are unable to converge to the exact solution for strong multiphase shocks. This is due to the incorrect partition of the energies or entropies in the cell averaged mixture. To circumvent this difficulty a specific Lagrangian scheme is developed. The correct partition of the energies is achieved by using an artificial heat exchange in the shock layer. With the help of an asymptotic analysis this heat exchange takes a similar form as the 'pseudoviscosity' introduced by Von Neumann and Richtmyer [J. Von Neumann, R.D. Richtmyer, A method for the numerical calculation of hydrodynamic shocks, J. Appl. Phys. 21 (1950) 232-237]. The present Lagrangian numerical scheme thus combines Riemann solvers and artificial heat exchanges. An Eulerian variant is then obtained by using the relaxation-projection method developed earlier by the authors for the Euler equations. The method is validated against exact solutions based on the multiphase shock relations as well as exact solutions of the Euler equations in the context of interface problems. The method is able to solve interfaces separating pure fluids or heterogeneous mixtures with very large density ratio and with very strong shocks.

[1]  R. Abgrall.  How to Prevent Pressure Oscillations in Multicomponent Flow Calculations , 1996 .

[2]  Barry Koren,et al.  A PRESSURE-INVARIANT AND CONSERVATIVE METHOD FOR TWO-FLUID FLOWS , 2003 .

[3]  R. Trunin,et al.  Shock compression of condensed materials (laboratory studies) , 2001 .

[4]  G. Strang.  On the Construction and Comparison of Difference Schemes , 1968 .

[5]  E. Lee,et al.  Adiabatic Expansion of High Explosive Detonation Products. , 1968 .

[6]  Richard Saurel,et al.  A multiphase model with internal degrees of freedom: application to shock–bubble interaction , 2003, Journal of Fluid Mechanics.

[7]  E. Sinibaldi.  Implicit preconditioned numerical schemes for the simulation of three-dimensional barotropic flows , 2007 .

[8]  Smadar Karni,et al.  Multicomponent Flow Calculations by a Consistent Primitive Algorithm , 1994 .

[9]  Rémi Abgrall,et al.  Modelling phase transition in metastable liquids: application to cavitating and flashing flows , 2008, Journal of Fluid Mechanics.

[10]  Rémi Abgrall,et al.  Discrete equations for physical and numerical compressible multiphase mixtures , 2003 .

[11]  Richard Saurel,et al.  A compressible flow model with capillary effects , 2005 .

[12]  R. D. Richtmyer,et al.  A Method for the Numerical Calculation of Hydrodynamic Shocks , 1950 .

[13]  Richard Courant,et al.  Supersonic Flow And Shock Waves , 1948 .

[14]  Randall J. LeVeque,et al.  One-Dimensional Front Tracking Based on High Resolution Wave Propagation Methods , 1995, SIAM J. Sci. Comput..

[15]  S. Osher,et al.  A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method) , 1999 .

[16]  M. Baer,et al.  A two-phase mixture theory for the deflagration-to-detonation transition (ddt) in reactive granular materials , 1986 .

[17]  S. Marsh.  Lasl Shock Hugoniot Data , 1980 .

[18]  Richard Saurel,et al.  Shock jump relations for multiphase mixtures with stiff mechanical relaxation , 2007 .

[19]  Qiang Zhang,et al.  Three-Dimensional Front Tracking , 1998, SIAM J. Sci. Comput..

[20]  Richard Saurel,et al.  Modelling detonation waves in heterogeneous energetic materials , 2004 .

[21]  Richard Saurel,et al.  Modelling evaporation fronts with reactive Riemann solvers , 2005 .

[22]  Keh-Ming Shyue,et al.  An Efficient Shock-Capturing Algorithm for Compressible Multicomponent Problems , 1998 .

[23]  D. Stewart,et al.  Two-phase modeling of deflagration-to-detonation transition in granular materials: Reduced equations , 2001 .

[24]  Richard Saurel,et al.  A relaxation-projection method for compressible flows. Part I: The numerical equation of state for the Euler equations , 2007, J. Comput. Phys..

[25]  R. Abgrall,et al.  A Multiphase Godunov Method for Compressible Multifluid and Multiphase Flows , 1999 .

[26]  Rémi Abgrall,et al.  Asymptotic Expansion of a Multiscale Numerical Scheme for Compressible Multiphase Flow , 2006, Multiscale Model. Simul..

[27]  Grégoire Allaire,et al.  A five-equation model for the simulation of interfaces between compressible fluids , 2002 .

[28]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[29]  Olivier Coutier-Delgosha,et al.  Stability of preconditioned Navier–Stokes equations associated with a cavitation model , 2005 .

[30]  J. Chan,et al.  Shock initiation and detonation models in one and two dimensions. [PBX-9404, RX-03-BB] , 1979 .

[31]  Brij Lal,et al.  A textbook of sound , 1995 .

[32]  Richard Saurel,et al.  A multiphase model for compressible flows with interfaces, shocks, detonation waves and cavitation , 2001, Journal of Fluid Mechanics.

[33]  Rémi Abgrall,et al.  Proposition de méthodes et modèles eulériens pour les problèmes à interfaces entre fluides compressibles en présence de transfert de chaleur , 2002 .

[34]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[35]  Richard Saurel,et al.  A mechanistic model for shock initiation of solid explosives , 1999 .

[36]  Hervé Guillard,et al.  A five equation reduced model for compressible two phase flow problems , 2005 .