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The Space-Time CE/SE Method for Solving Reduced Two-Fluid Flow Model

Published online by Cambridge University Press:  20 August 2015

Shamsul Qamar ,
Munshoor Ahmed  and
Ishtiaq Ali
Shamsul Qamar*
Affiliation:
Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Chak Shahzad Islamabad, Pakistan Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg, Germany
Munshoor Ahmed*
Affiliation:
Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Chak Shahzad Islamabad, Pakistan
Ishtiaq Ali*
Affiliation:
Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Chak Shahzad Islamabad, Pakistan
*
Corresponding author.Email:[email protected]
Email:[email protected]
Email:[email protected]
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Abstract

The space-time conservation element and solution element (CE/SE) method is proposed for solving a conservative interface-capturing reduced model of compressible two-fluid flows. The flow equations are the bulk equations, combined with mass and energy equations for one of the two fluids. The latter equation contains a source term for accounting the energy exchange. The one and two-dimensional flow models are numerically investigated in this manuscript. The CE/SE method is capable to accurately capture the sharp propagating wavefronts of the fluids without excessive numerical diffusion or spurious oscillations. In contrast to the existing upwind finite volume schemes, the Riemann solver and reconstruction procedure are not the building block of the suggested method. The method differs from the previous techniques because of global and local flux conservation in a space-time domain without resorting to interpolation or extrapolation. In order to reveal the efficiency and performance of the approach, several numerical test cases are presented. For validation, the results of the current method are compared with other finite volume schemes.

Keywords

76T9965Y9965M9935L4535L6535L67Reduced modelspace-time CE/SE methodcentral schemesconservation lawshyperbolic systemsshock solutions
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 4 , October 2012 , pp. 1070 - 1095
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Abgrall, R. and Saurel, R., Discrete equations for physical and numerical compressible multiphase mixtures, J. Comput. Phys., 186 (2003), 361–396.CrossRefGoogle Scholar
[2]Allaire, G., Clerc, S. and Kokh, S., A five-equation model for the simulation of interfaces between compressible fluids, J. Comput. Phys., 181 (2002), 577–616.Google Scholar
[3]Baer, M. R. and Nunziato, J. W., A two-phase mixture theory for the deflagration-to-detonation transition in reactive granular materials, Int. J. Multiphase Flows, 12 (1986), 861–889.Google Scholar
[4]H. van Brummelen, E. and Koren, B., A pressure-invariant conservative Godunov-type method for barotropic two-fluid flows, J. Comput. Phys., 185 (2003), 289–308.CrossRefGoogle Scholar
[5]Chang, S. C., The method of space time conservation element and solution element-a new approach for solving the Navier-Stokes and Euler equations, J. Comput. Phys., 119 (1995), 295–324.Google Scholar
[6]Chang, S. C., Wang, X. Y. and Chow, C. Y., New Developments in the Method of SpaceTime Conservation Element and Solution Element-Applications to Two-Dimensional Time-Marching Problems, NASA TM 106758, 1994.Google Scholar
[7]Chang, S. C., Wang, X. Y. and Chow, C. Y., The space-time conservation element and solution element method: a new high resolution and genuinely multidimensional paradigm for solving conservation laws, J. Comput. Phys., 156 (1999), 89–136.Google Scholar
[8]Chang, S. C., Wang, X. Y. and To, W. M., Application of the space-time conservation element and solution element method to one-dimensional convection-diffusion problems, J.Comput. Phys., 165 (2000), 189–215.Google Scholar
[9]Clerc, S., Numerical simulation of the homogeneous equilibrium model for two-phase flows, J. Comput. Phys., 161 (2000), 354–375.Google Scholar
[10]Guillard, H. and Labois, M., Numerical Modeling of Compressible Two-Phase Flows, in: Wesseling, P., Onñate, E., Périaux (Eds.), J., ECCOMAS CFD, 2006.Google Scholar
[11]‐M. Ghidaglia, J., Kumbaro, A. and Coq, G. L., On the numerical solution to two fluid models via a cell centered finite volume method, Euro. J. Mech. B Fluids, 20 (2001), 841–867.CrossRefGoogle Scholar
[12]‐F. Haas, J. and Sturtevant, B., Interaction of weak shock waves with cylinderical and spherical gas inhomogeneities, J. Fluid Mech., 181 (1987), 41–76.Google Scholar
[13]Hirt, C. W. and Nichols, B. D., Volume of fluid (VOF) method for the dynamics of free boundaries, J. Comput. Phys., 39 (1981), 201–225.Google Scholar
[14]‐S. Jaing, G. and Tadmor, E., Non-oscillatory central schemes for multidimensional hyperbolic conservation laws, SIAM J. Sci. Comput., 19 (1998), 1892–1917.Google Scholar
[15]Kapila, A. K., Menikoff, R., Bdzil, J. B., Son, S. F. and Stewart, D. S., Two-phase modeling of deflagration-to-detonation transition in granular materials: reduced equations, Phys. Fluids, 13 (2001), 3002–3024.CrossRefGoogle Scholar
[16]Karni, S.1, Kirr, E., Kurganov, A. and Petrova, G., Compressible two-phase flows by central and upwind schemes, ESAIM: M2AN, 38 (2004), 477–493.Google Scholar
[17]Kreeft, J. J. and Koren, B., A new formulation of Kapila’s five-equation model for compressible two-fluid flow and its numerical treatment, J. Comput. Phys., 229 (2010), 6220–6242.Google Scholar
[18]Loh, C. Y., Hultgren, L. S., Chang, S. C. and Jorgenson, P. C. E., Noise Computation of a Shock-Containing Supersonic Axisymmetric Jet by the CE/SE Method, AIAA Paper 2000–0475, presented at the 38th AIAA Aerospace Sciences Meeting, January 10–13, Reno, NV, 2000.Google Scholar
[19]Loh, C. Y., Hultgren, L. S. and Chang, S. C., Wave computation in compressible flow using the space-time conservation element and solution element method, AIAA J., 39 (2001), 794–801.Google Scholar
[20]Liu, M., Wang, J. B. and ‐Q. Wu, K., The direct aero-acoustics simulation of flow around a square cylinder using the CE/SE scheme, J. Alg. Comput. Tech., 1 (2007), 525–537.Google Scholar
[21]Loh, C. Y. and Zaman, K. B. M. Q., Numerical investigation of transonic resonance with a convergent-divergent nozzle, AIAA J., 40 (2002), 2393–2401.CrossRefGoogle Scholar
[22]Marquina, A. and Mulet, P., A flux-split algorithm applied to conservative model for multi-component compressible flows, J. Comput. Phys., 185 (2003), 120–138.Google Scholar
[23]Menikoff, R. and Plohr, B. J., The Riemann problem for fluid flow of real materials, Rev. Mod. Phys., 61 (1989), 75–130.Google Scholar
[24]Mulder, W., Osher, S. and Sethian, J. A., Computing interface motion in compressible gas dynamics, J. Comput. Phys., 100 (1992), 209–228.Google Scholar
[25]Murrone, A. and Guillard, H., A five equation reduced model for compressible two phase flow problems, J. Comput. Phys., 202 (2005), 664–698.Google Scholar
[26]Nessyahu, H. and Tadmor, E., Non-oscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys., 87 (1990), 408–463.CrossRefGoogle Scholar
[27]Qamar, S. and Ahmed, M., A high order kinetic flux-vector splitting method for the reduced five-equation model of compressible two-fluid flows, J. Comput. Phys., 228 (2009), 9059–9078.Google Scholar
[28]Qamar, S. and Mudasser, S., On the application of a variant CE/SE method for solving two-dimensional ideal MHD equations, Appl. Numer. Math., 60 (2010), 587–606.Google Scholar
[29]Saurel, R. and Abgrall, R., A multiphase Godunov method for compressible multifluids and multiphase flows, J. Comput. Phys., 150 (1999), 425–467.CrossRefGoogle Scholar
[30]Saurel, R., Franquet, E., Daniel, E. and Metayer, O. L., A relaxation-projection method for compressible flows-part I: the numerical equation of state for the Euler equations, J. Comput. Phys., 223 (2007), 822–845.Google Scholar
[31]Saurel, R., Petitpas, F. and Berry, R. A., Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows and shocks in multiphase mixtures, J. Comput. Phys., 228 (2009), 1678–1712.Google Scholar
[32]Sussman, M., Smereka, P. and Osher, S., A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., 114 (1994), 146–159.Google Scholar
[33]Romenski, E., Resnyansky, A. D. and Toro, E. F., Conservative hyperbolic formulation for compressible two-phase flow with different phase pressures and temperatures, Quart. Appl. Math., 65 (2007), 259–279.Google Scholar
[34]Wang, X. Y., Chen, C. L. and Liu, Y., The space-time CE/SE method for solving Maxwell’s equations in time domain, Antennas and Propagation Society International Symposium, IEEE 02/2002, 1 (2002), 164–167.Google Scholar
[35]Zhang, Z. C., Yu, S. T. and Chang, S. C., A space-tome conservation element and solution element method for solving the two- and three-dimensional unsteady Euler equations using quadrilateral and hexhedral meshes, J. Comput. Phys., 175 (2002), 168–199.CrossRefGoogle Scholar
[36]Zhou, J., Cai, L., ‐H. Feng, J. and ‐X. Xie, W., Numerical simulation for two-phase flows using hybrid scheme, Appl. Math. comput., 186 (2007), 980–991.Google Scholar