A Gas-Kinetic Scheme for Six-Equation Two-Phase Flow Model

Affiliation(s)

Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan.

Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan.

ABSTRACT

A kinetic
flux-vector splitting (KFVS) scheme is applied for solving a reduced six-equation
two-phase flow model of Saurel *et al*.
[1]. The model incorporates single velocity, two pressures and relaxation terms. An additional seventh
equation, describing the total mixture energy, is added to the model to
guarantee the correct treatment
of shocks in the single phase limit. Some salient features of the model are that it is hyperbolic with only three wave
propagation speeds and the volume fraction remains positive. The proposed
numerical scheme is based on the direct splitting of macroscopic flux functions of the system of
equations. The second order accuracy of the scheme is achieved by using MUSCL-type initial reconstruction
and Runge-Kutta time stepping method. Moreover, a pressure relaxation procedure is used to fulfill the
interface conditions. For validation, the results of suggested scheme are compared with those from the high resolution
central upwind and HLLC schemes. The central upwind scheme is also applied for the
first time to this model. The accuracy, efficiency and simplicity of the KFVS scheme
demonstrate its potential for
modeling two-phase flows.

Cite this paper

S. Zia, M. Ahmed and S. Qamar, "A Gas-Kinetic Scheme for Six-Equation Two-Phase Flow Model,"*Applied Mathematics*, Vol. 5 No. 3, 2014, pp. 453-465. doi: 10.4236/am.2014.53045.

S. Zia, M. Ahmed and S. Qamar, "A Gas-Kinetic Scheme for Six-Equation Two-Phase Flow Model,"

References

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[2] M. R. Baer and J. W. Nunziato, “A Two-Phase Mixture Theory for the Deflagration-to-Detonation Transition (DDT) in Reactive Granular Materials,” International Journal of Multiphase Flow, Vol. 12, No. 6, 1986, pp. 861-889.

http://dx.doi.org/10.1016/0301-9322(86)90033-9

[3] R. Abgrall and R. Saurel, “Discrete Equations for Physical and Numerical Compressible Multiphase Mixtures,” Journal of Computational Physics, Vol. 186, No. 2, 2003, pp. 361-396.

http://dx.doi.org/10.1016/S0021-9991(03)00011-1

[4] R. Saurel and R. Abgrall, “A Multiphase Godunov Method for Compressible Multifluids and Multiphase Flows,” Journal of Computational Physics, Vol. 150, No. 2, 1999, pp. 425-467.

http://dx.doi.org/10.1006/jcph.1999.6187

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http://dx.doi.org/10.1016/j.jcp.2010.04.025

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http://dx.doi.org/10.1006/jcph.1999.6280

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http://dx.doi.org/10.1006/jcph.2002.7040

[17] S. Qamar and M. Ahmed, “A High Order Kinetic-Flux-Vector Splitting Method for Reduced Five-Equation Model of Compressible Two-Phase Flows,” Journal of Computational Physics, Vol. 228, No. 24, 2009, pp. 9059-9078.

http://dx.doi.org/10.1016/j.jcp.2009.09.010

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[22] A. Zein, “Numerical Methods for Multiphase Mixture Conservation Laws with Phase Transition,” Ph.D. Thesis, Fakultat fur Mathematik Otto-von-Guericke Universitat Magdeburg, 2010.

[1] R. Saurel, F. Petitpas and A. Berry, “Simple and Efficient Relaxation Methods for Interfaces Separating Compressible Fluids, Cavitating Flows and Shocks in Multiphase Mixtures,” Journal of Computational Physics, Vol. 228, No. 5, 2009, pp. 16781712. http://dx.doi.org/10.1016/j.jcp.2008.11.002

[2] M. R. Baer and J. W. Nunziato, “A Two-Phase Mixture Theory for the Deflagration-to-Detonation Transition (DDT) in Reactive Granular Materials,” International Journal of Multiphase Flow, Vol. 12, No. 6, 1986, pp. 861-889.

http://dx.doi.org/10.1016/0301-9322(86)90033-9

[3] R. Abgrall and R. Saurel, “Discrete Equations for Physical and Numerical Compressible Multiphase Mixtures,” Journal of Computational Physics, Vol. 186, No. 2, 2003, pp. 361-396.

http://dx.doi.org/10.1016/S0021-9991(03)00011-1

[4] R. Saurel and R. Abgrall, “A Multiphase Godunov Method for Compressible Multifluids and Multiphase Flows,” Journal of Computational Physics, Vol. 150, No. 2, 1999, pp. 425-467.

http://dx.doi.org/10.1006/jcph.1999.6187

[5] A. K. Kapila, R. Menikoff, J. B. Bdzil, S. F. Son and D. S. Stewart, “Two-Phase Modeling of Deflagration-to-Detonation Transition in Granular Materials,” Physics of Fluids, Vol. 13, No. 10, 2001, pp. 3002-3024. http://dx.doi.org/10.1063/1.1398042

[6] J. J. Kreeft and B. Koren, “A New Formulation of Kapila’s Five-Equation Model for Compressible Two-Fluid Flow, and Its Numerical Treatment,” Journal of Computational Physics, Vol. 229, No. 18, 2010, pp. 6220-6242.

http://dx.doi.org/10.1016/j.jcp.2010.04.025

[7] J. Wackers and B. Koren, “Five-Equation Model for Compressible Two-Fluid Flow,” Centrum Wiskunde & Informatica, 2004, CWI-Report: MAS-E0414.

[8] A. B. Wood, “A Textbook of Sound,” G. Bell Sons Ltd., London, 1930.

[9] J. C. Mandal and S. M. Deshpande, “Kinetic Flux-Vector Splitting for Euler Equations,” Computer and Fluids, Vol. 23, No. 2, 1994, pp. 447-478.

[10] B. Van Leer, “Flux Vector Splitting for the Euler Equations,” ICASE, 1982, Report No. 82-30.

[11] 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, 1983, pp. 35-62.

http://dx.doi.org/10.1137/1025002

[12] N. P. Weatherill, J. S. Mathur and M. J. Marchant, “An Upwind Kinetic Flux Vector Splitting Method on General Mesh Topologies,” International Journal for Numerical Methods in Engineering, Vol. 37, No. 2, 1994, pp. 623-643.

[13] K. Xu, “Gas-Kinetic Theory Based Flux Slitting Method for Ideal Magneto-Hydrodynamics,” Journal of Computational Physics, Vol. 153, No. 2, 1999, pp. 334-352.

http://dx.doi.org/10.1006/jcph.1999.6280

[14] T. Tang and K. Xu, “A High-Order Gas-Kinetic Method for Multidimensional Ideal Magnetohydrodynamics,” Journal of Computational Physics, Vol. 165, No. 1, 2000, pp. 69-88. http://dx.doi.org/10.1006/jcph.2000.6597

[15] H. Tang, T. Tang and K. Xu, “A Gas-Kinetic Scheme for Shallow-Water Equations with Source Terms,” Zeitschrift für angewandte Mathematik und Physik, Vol. 55, 2004, pp. 365-382.

[16] K. Xu, “A Well-Balanced Gas-Kinetic Scheme for the Shallow-Water Equations with Source Terms,” Journal of Computational Physics, Vol. 178, No. 2, 2002, pp. 533-562.

http://dx.doi.org/10.1006/jcph.2002.7040

[17] S. Qamar and M. Ahmed, “A High Order Kinetic-Flux-Vector Splitting Method for Reduced Five-Equation Model of Compressible Two-Phase Flows,” Journal of Computational Physics, Vol. 228, No. 24, 2009, pp. 9059-9078.

http://dx.doi.org/10.1016/j.jcp.2009.09.010

[18] E. F. Toro, “Riemann Solvers and Numerical Methods for Fluid Dynamics—A Practical Introduction,” 2nd Edition, Springer-Verlag, Berlin Heidelberg, 1999. http://dx.doi.org/10.1007/978-3-662-03915-1

[19] A. Kurganov and E. Tadmor, “New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection Diffusion Equations,” Journal of Computational Physics, Vol. 160, No. 1, 2000, pp. 241-282.

http://dx.doi.org/10.1006/jcph.2000.6459

[20] G. Strang, “On the Construction and Comparison of Difference Schemes,” SIAM Journal on Numerical Analysis, Vol. 5, No. 3, 1968, pp. 506-517. http://dx.doi.org/10.1137/0705041

[21] N. Andrianov, “Analytical and Numerical Investigation of Two-Phase Flows,” Ph.D. Thesis, Otto-von-Guericke University, Magdeburg, 2003.

[22] A. Zein, “Numerical Methods for Multiphase Mixture Conservation Laws with Phase Transition,” Ph.D. Thesis, Fakultat fur Mathematik Otto-von-Guericke Universitat Magdeburg, 2010.