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The sharp-interface approach for fluids with phase change:Riemann problems and ghost fluid techniques

Published online by Cambridge University Press:  15 December 2007

Christian Merkle  and
Christian Rohde
Christian Merkle
Affiliation:
Abteilung für Angewandte Mathematik, lbert-Ludwigs-Universität Freiburg, Hermann-Herder Str. 10, 79104 Freiburg, Germany. christian@mathematik.uni-freiburg.de
Christian Rohde
Affiliation:
Institut für Angewandte Mathematik und Numerische Simulation, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany.
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Abstract


Systems of mixed hyperbolic-elliptic conservation laws can serve as models for the evolution of a liquid-vapor fluid with possible sharp dynamical phase changes. We focus on the equations of ideal hydrodynamics in the isothermal case and introduce a thermodynamically consistent solution of the Riemann problem in one space dimension. This result is the basis for an algorithm of ghost fluid type to solve the sharp-interface model numerically. In particular the approach allows to resolve phase transitions sharply, i.e., without artificial smearing in the physically irrelevant elliptic region. Numerical experiments demonstrate the reliability of the method.

Keywords

Dynamical phase transitions in compressible mediavan-der-Waals pressurekinetic relationsRiemann solverghost fluid approach.

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 6 , November 2007 , pp. 1089 - 1123
Copyright
© EDP Sciences, SMAI, 2007

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References

Abeyaratne, R. and Knowles, J., Kinetic relations and the propagation of phase boundaries in solids. Arch. Ration. Mech. Anal. 114 (1991) 119–154. CrossRef
Abgrall, R. and Karni, S., Compressible multifluid flows. J. Comput. Phys. 169 (2001) 594–623. CrossRef
Abgrall, R. and Saurel, R., Discrete equations for physical and numerical compressible multiphase mixtures. J. Comput. Phys. 186 (2003) 361–396. CrossRef
Aslam, T.D., A level set algorithm for tracking discontinuities in hyperbolic conservation laws II: Systems of equations. J. Sci. Comput. 19 (2003) 37–62. CrossRef
Bedjaoui, N. and LeFloch, P.G., Diffusive-dispersive travelling waves and kinetic relations. II. A hyperbolic-elliptic model of phase-transition dynamics. Proc. Roy. Soc. Edinburgh Sect. A 132A (2002) 1–21.
S. Benzoni-Gavage, Stability of multi-dimensional phase transitions in a van der Waals fluid. Nonlinear Anal., Theory Methods Appl. 31 (1998) 243–263. CrossRef
C. Chalons, Transport-Equilibrium Schemes for Computing Nonclassical Shocks. I. Scalar Conservation Laws. Preprint, Laboratoire Jacques-Louis Lions (2005).
Chalons, C. and LeFloch, P.G., Computing undercompressive waves with the random choice scheme. Interfaces Free Bound. 5 (2003) 129–158. CrossRef
Colombo, R.M. and Corli, A., Continuous dependence in conservation laws with phase transitions. SIAM J. Math. Anal. 31 (1999) 34–62. CrossRef
Colombo, R.M. and Corli, A., Stability of the Riemann semigroup with respect to the kinetic condition. Quart. Appl. Math. 62 (2004) 541–551. CrossRef
C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics. Grundlehren der mathematischen Wisenschaften 325. Springer (2000).
H. Fan and M. Slemrod, Dynamic flows with liquid/vapor phase transitions, in Handbook of mathematical fluid dynamics, Vol. I, North-Holland, Amsterdam (2002) 373–420.
Fedkiw, R.P., Aslam, T., Merriman, B. and Osher, S., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys. 152 (1999) 457–492. CrossRef
Fedkiw, R.P., Aslam, T. and The, S. Xu ghost fluid method for deflagration and detonation discontinuities. J. Comput. Phys. 154 (1999) 393–427. CrossRef
E. Godlewski and P.-A. Raviart, Numerical approximation of hyperbolic systems of conservation laws. Appl. Math. Sci. 118 Springer (1996).
Godlewski, E. and Seguin, N., The Riemann problem for a simple model of phase transition. Commun. Math. Sci. 4 (2006) 227–247. CrossRef
Hayes, B. and LeFloch, P.G., Nonclassical shocks and kinetic relations: strictly hyperbolic systems. SIAM J. Math. Anal. 31 (2000) 941–991. CrossRef
Hou, T.Y., Rosakis, P. and LeFloch, P.G., A level-set approach to the computation of twinning and phase-transition dynamics. J. Comput. Phys. 150 (1999) 302–331. CrossRef
Kac, M., Uhlenbeck, G.E. and Hemmer, P.C., On the van der Waals theory of the vapor-liquid equilibrium. I. Discussion of a one-dimensional model. J. Math. Phys. 4 (1963) 216–228. CrossRef
S. Karni and R. Abgrall, Ghost-Fluids for the Poor: A Single Fluid Algorithm for Multifluids, in Lecture Notes in Mathematics, Proceedings of the 10th International Conference on Hyperbolic problems, theory and numerics, Springer (2001) 293–302.
Le Métayer, O., Massoni, J. and Saurel, R., Modelling evaporation fronts with reactive Riemann solvers. J. Comput. Phys. 205 (2005) 567–610. CrossRef
LeFloch, P.G., Propagating phase boundaries: Formulation of the problem and existence via the Glimm method. Arch. Ration. Mech. Anal. 123 (1993) 153–197. CrossRef
P.G. LeFloch, Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves. Lectures in Mathematics. ETH Zürich, Birkhäuser (2002).
LeFloch, P.G. and Thanh, M.D., Nonclassical Riemann solvers and kinetic relations. II. An hyperbolic-elliptic model of phase transitions. Proc. Royal Soc. Edinburgh A 131A (2001) 1–39.
LeFloch, P.G., Mercier, J.M. and Rohde, C., Fully discrete, entropy conservative schemes of arbitrary order. SIAM J. Numer. Anal. 40 (2002) 1968–1992. CrossRef
Liu, T.G., Khoo, B.C. and Yeo, K.S., Ghost fluid method for strong impacting on material interfaces. J. Comput. Phys. 190 (2003) 651–681. CrossRef
C. Merkle, Dynamical Phase Transitions in Compressible Media. Doctoral dissertation, Albert-Ludwigs-Universität Freiburg (2006) http://www.freidok.uni-freiburg.de/volltexte/2674/.
Merkle, C. and Rohde, C., Computation of dynamical phase transitions in solids. Appl. Numer. Math. 56 (2006) 1450–1463. CrossRef
Mulder, W., Osher, S. and Sethian, J., Computing interface motion in compressible gas dynamics. J. Comput. Phys. 100 (1992) 209–228. CrossRef
Müller, S. and Voß, A., The Riemann problem for the Euler equations with nonconvex and nonsmooth equation of state: construction of wave curves. SIAM J. Sci. Comput. 28 (1992) 651–681. CrossRef
S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces. Appl. Math. Sci. 153. Springer (2003).
Osher, S. and Sethian, J., Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79 (1988) 12–49. CrossRef
Osher, S. and Shu, C.-W., High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal. 28 (1991) 907–922. CrossRef
Peng, D., Merriman, B., Osher, S., Zhao, H.K. and Kang, M., PDE-based, A fast local level set method. J. Comput. Phys. 155 (1999) 410–438. CrossRef
Russo, G. and Smereka, P., A remark on computing distance functions. J. Comput. Phys. 163 (2000) 51–67. CrossRef
D. Serre, Systems of Conservation Laws 1. Cambridge University Press (1999).
Sussman, M., Smereka, P. and Osher, S., A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114 (1994) 146–154. CrossRef
E.F. Toro, Multi-Stage Predictor-Corrector Fluxes for Hyperbolic Equations. Technical Report NI03037-NPA Isaac Newton Institute for Mathematical Sciences (2003).
L. Truskinovsky, Kinks versus Shocks, in Shock induced transitions and phase structures in general media, Springer, New York (1993) 185–229.
L. Truskinovsky and A. Vainchtein, Explicit kinetic relation from “first principles”, in Mechanics of material forces 11, Advances in Mechanics and Mathematics, P. Steinmann and G.A. Maugin (Eds.), Springer (2005) 43–50.
Zhong, X., Hou, T.Y. and LeFloch, P.G., Computational method for propagating phase boundaries. J. Comput. Phys. 124 (1996) 192–216. CrossRef