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The Application of HLLC Numerical Solver to the Reduced Multiphase Model
The performance of high-resolution schemes is investigated for unsteady, inviscid and compressible multiphase flows. An Eulerian diffuse interface approach has been chosen for the simulation of multicomponent flow problems. The reduced fiveequation and seven equation models are used with HLL and HLLC approximation. The authors demonstrated the advantages and disadvantages of both seven equations and five equations models studying their performance with HLL and HLLC algorithms on simple test case. The seven equation model is based on two pressure, two velocity concept of Baer–Nunziato [10], while five equation model is based on the mixture velocity and pressure. The numerical evaluations of two variants of Riemann solvers have been conducted for the classical one-dimensional air-water shock tube and compared with analytical solution for error analysis.
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[1] G. Allaire, S. Clerc, and S. Kokh, "A five-equation model for the numerical simulation of interfaces in two-phase flows," Comptes Rendus De L Academie Des Sciences Serie I-Mathematique, vol. 331, pp. 1017-1022, 2000.
[2] N. Andrianov, R. Saurel, and G. Warnecke, "A simple method for compressible multiphase mixtures and interfaces," International journal for numerical methods in fluids, vol. 41, pp. 109-131, 2003.
[3] C. E. Brennen, Fundamentals of multiphase flow, 1st ed., United States of America, Cambridge University Press, 2005.
[4] J. Cocchi, and R. Saurel, "A Riemann Problem Based Method for the Resolution of Compressible Multimaterial Flows," Journal of Computational Physics, vol. 137, pp. 265-298, 1997.
[5] F. Coquel, K. El Amine, E. Godlewski, B. Perthame, and P. Rascle, "A Numerical Method Using Upwind Schemes for the Resolution of Two- Phase Flows," Journal of Computational Physics, vol. 137, pp. 272-288, 1997.
[6] D. A. Drew, and S. L. Passman, Theory of Multicomponent Fluids. Springer: New York, 1999.
[7] D. A. Drew, "Mathematical Modeling of two-phase flow," Annual review of fluid mechanics, vol. 15, pp. 261-291, 1983.
[8] R. J. Leveque, Finite volume methods for hyperbolic problem-, 1st ed., Cambridge, University Printing House, 2002.
[9] M. Ishii, Thermo-Fluid Dynamic Theory of Two-Phase Flow. Paris: Eyrolles, 1975.
[10] M.R. Bear and J.W. Nunziato, "A two phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials," Journal of Multiphase Flow, vol. 2, pp. 861-889, 1986.
[11] A. Murrone and H. Guillard, "A five equation reduced model for compressible two phase flow problems," Journal of Computational Physics, vol. 202, pp. 664-698, 2005.
[12] A. F. Nowakowski "Numerical simulation of microbubbles in a liquidfilled flexible tube," Bangkuk paper submitted for WACBE world congress of bioengineering, 2007.
[13] G. Perigaudand and R. Saurel, "A compressible flow model with capillary effects," Journal of Computational Physics, vol. 209, pp. 139- 178, 2005.
[14] R. Saurel, and R. Abgrall, "A Multiphase Godunov Method for Compressible Multifluid and Multiphase Flows," Journal of Computational Physics, vol. 150, pp. 425-467, 1999.
[15] R. Saurel and O. Lemetayer, "A multiphase model for compressible flows with interfaces, shocks, detonation waves and cavitation," Joirnal of fluid mechanics, vol. 431, pp. 239-271, 2001.
[16] E. F. Toro, Riemann solvers and numerical methods for fluid dynamics, 1st ed., Berlin Heidelberg, Springer-Verlag, 1997.
[17] A. Harten, P.D. Lax, and B. van Leer, "On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws," SIAM Review, vol. 25, no. 1, pp. 35-61, 1983.
[18] E.F. Toro and P.L. Roe, "A Hybrid Scheme for the Euler Equations Using the Random Choice and Roe-s Methods," In Numerical Methods for Fluid Dynamics III. The Institute of Mathematics and its Applications Conference Series, New series No. 17, Morton and Baines (Editors), Oxford University Press, New York, vol. 17, pp. 391-402. 1988.
[19] Li Qiang, Feng Jian-Hu, Cai Ti-min and Hu Chun-bo, "Difference scheme for two-phase flow," Applied Mathematical and Mechanics (English edition), Shanghai University, China. vol. 25, no. 5, May 2004.
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