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An Eulerian interface sharpening algorithm for compressible two-phase flow: the algebraic THINC approach. (English) Zbl 1349.76388
Summary: We describe a novel interface-sharpening approach for efficient numerical resolution of a compressible homogeneous two-phase flow governed by a quasi-conservative five-equation model of G. Allaire et al. [ibid. 181, No. 2, 577–616 (2002; Zbl 1169.76407)]. The algorithm uses a semi-discrete wave propagation method to find approximate solution of this model numerically. In the algorithm, in regions near the interfaces where two different fluid components are present within a cell, the THINC (Tangent of Hyperbola for INterface Capturing) scheme is used as a basis for the reconstruction of a sub-grid discontinuity of volume fractions at each cell edge, and it is complemented by a homogeneous-equilibrium-consistent technique that is derived to ensure a consistent modeling of the other interpolated physical variables in the model. In regions away from the interfaces where the flow is single phase, standard reconstruction scheme such as MUSCL or WENO can be used for obtaining high-order interpolated states. These reconstructions are then used as the initial data for Riemann problems, and the resulting fluctuations form the basis for the spatial discretization. Time integration of the algorithm is done by employing a strong stability-preserving Runge-Kutta method. Numerical results are shown for sample problems with the Mie-Grüneisen equation of state for characterizing the materials of interests in both one and two space dimensions that demonstrate the feasibility of the proposed method for interface-sharpening of compressible two-phase flow. To demonstrate the competitiveness of our approach, we have also included results obtained using the anti-diffusion interface sharpening method.
Reviewer: Reviewer (Berlin)

76M12 Finite volume methods applied to problems in fluid mechanics
76Txx Multiphase and multicomponent flows
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI
[1] Allaire, G.; Clerc, S.; Kokh, S., A five-equation model for the simulation of interface between compressible fluids, J. Comput. Phys., 181, 577-616, (2002) · Zbl 1169.76407
[2] Cassidy, D. A.; Edwards, J. R.; Tian, M., An investigation of interface-sharpening schemes for multi-phase mixture flows, J. Comput. Phys., 228, 5628-5649, (2009) · Zbl 1280.76033
[3] Cheng, J.; Shu, C.-W., A high order ENO conservative Lagrangian type scheme for the compressible Euler equations, J. Comput. Phys., 227, 1567-1596, (2007) · Zbl 1126.76035
[4] Després, B.; Lagoutière, F., Contact discontinuity capturing schemes for linear advection and compressible gas dynamics, J. Sci. Comput., 16, 4, 479-524, (2001) · Zbl 0999.76091
[5] Glimm, J.; McBryan, O. A.; Menikoff, R.; Sharp, D. H., Front tracking applied to Rayleigh-Taylor instability, SIAM J. Sci. Stat. Comput., 7, 1, 230-251, (1986) · Zbl 0582.76107
[6] Gottlieb, S.; Ketcheson, D.; Shu, C.-W., Strong stability preserving Runge-Kutta and multistep time discretizations, (2011), World Scientific · Zbl 1241.65064
[7] Gottlieb, S.; Shu, C.-W.; Tadmor, E., Strong stability preserving high-order time discretization methods, SIAM Rev., 43, 89-112, (2001) · Zbl 0967.65098
[8] Haas, J.-F.; Sturtevant, B., Interaction of weak shock waves with cylindrical and spherical gas inhomogeneities, J. Fluid Mech., 181, 41-76, (1987)
[9] Harten, A.; Lax, P. D.; van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25, 35-61, (1983) · Zbl 0565.65051
[10] Holmes, R. L., A numerical investigation of the Richtmyer-Meshkov instability using front tracking, (August 1994), SUNY at Stony Brook, (unpublished)
[11] Hu, X. Y.; Khoo, B. C.; Adams, N. A.; Huang, F. L., A conservative interface method for compressible flows, J. Comput. Phys., 219, 553-578, (2006) · Zbl 1102.76038
[12] Ii, S.; Sugiyama, K.; Takeuchi, S.; Takagi, S.; Matsumoto, Y.; Xiao, F., An interface capturing method with a continuous function: the THINC method with multi-dimensional reconstruction, J. Comput. Phys., 231, 2328-2358, (2012) · Zbl 1427.76205
[13] Johnsen, E.; Colonius, T., Implementation of WENO schemes in compressible multicomponent flow problems, J. Comput. Phys., 219, 715-732, (2006) · Zbl 1189.76351
[14] Kapila, A. K.; Menikoff, R.; Bdzil, J. B.; Son, S. F.; Stewart, D. S., Two-phase modeling of deflagration-to-denonation transition in granular materials: reduced equations, Phys. Fluids, 13, 10, 3002-3024, (2001) · Zbl 1184.76268
[15] Ketcheson, D. I.; LeVeque, R. J., WENOCLAW: a higher order wave propagation method, (Hyperbolic Problems: Theory, Numerics, Applications, (2008), Springer-Verlag), 609-616 · Zbl 1138.65084
[16] Ketcheson, D. I.; Mandli, K. T.; Ahmadia, A. J.; Alghamdi, A.; De Luna, M. Q.; Parsani, M.; Knepley, M. G.; Emmett, M., PYCLAW: accessible, extensible, scalable tools for wave propagation problems, SIAM J. Sci. Comput., 34, 4, C210-C231, (2012) · Zbl 1253.65220
[17] Ketcheson, D. I.; Parsani, M.; LeVeque, R. J., High-order wave propagation algorithms for hyperbolic systems, SIAM J. Sci. Comput., 35, 1, A351-A377, (2013) · Zbl 1264.65151
[18] Kokh, S.; Lagoutière, F., An anti-diffusive numerical scheme for the simulation of interfaces between compressible fluids by means of a five-equation model, J. Comput. Phys., 229, 2773-2809, (2010) · Zbl 1302.76129
[19] Kucharik, M.; Garimella, R. V.; Schofield, S. P.; Shashkov, M., A comarative study of interface reconstruction methods for multi-material ALE simulations, J. Comput. Phys., 229, 2432-2452, (2010) · Zbl 1423.76343
[20] LeVeque, R. J., Finite volume methods for hyperbolic problems, (2002), Cambridge University Press · Zbl 1010.65040
[21] LeVeque, R. J., Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems, (2007), SIAM Philadelphia · Zbl 1127.65080
[22] LeVeque, R. J.; Berger, M. J., \scclawpack software version 4.5, (2011)
[23] LeVeque, R. J.; Shyue, K.-M., Two-dimensional front tracking based on high resolution wave propagation methods, J. Comput. Phys., 123, 354-368, (1996) · Zbl 0849.65063
[24] Loubére, R.; Maire, P.-H.; Shashkov, M.; Breil, J.; Galera, S., Reale: a reconnection-based arbitrary-Lagrangian-Eulerian method, J. Comput. Phys., 229, 4724-4761, (2010) · Zbl 1305.76067
[25] Luo, H.; Baum, J. D.; Löhner, R., On the computation of multi-material flows using ALE formulation, J. Comput. Phys., 194, 304-328, (2004) · Zbl 1136.76401
[26] Mader, C. L., Numerical modeling of detonations, (1979), University of California Press Berkeley · Zbl 0646.73018
[27] Maire, P.-H.; Abgrall, R.; Breil, J.; Ovadia, J., A cell-centered Lagrangian scheme for two-dimensional compressible flow problems, SIAM J. Sci. Comput., 29, 1781-1824, (2007) · Zbl 1251.76028
[28] Marquina, A.; Mulet, P., A flux-split algorithm applied to conservative models for multicomponent compressible flows, J. Comput. Phys., 185, 120-138, (2003) · Zbl 1064.76078
[29] Marsh, S. P., LASL shock hugoniot data, (1980), University of California Press Berkeley
[30] McQueen, R. G.; Marsh, S. P.; Taylor, J. W.; Fritz, J. N.; Carter, W. J., The equation of state of solids from shock wave studies, (Kinslow, R., High Velocity Impact Phenomena, (1970), Academic Press San Diego), 293-417
[31] Miller, G. H.; Puckett, E. G., A high order Godunov method for multiple condensed phases, J. Comput. Phys., 128, 134-164, (1996) · Zbl 0861.65117
[32] Noh, W. F.; Woodward, P., SLIC (simple line interface calculation), (van de Vooren, A. I.; Zandbergen, P. J., Proc. 5th Intl. Conf. on Numer. Meth. in Fluid Dynamics, (1976), Springer-Verlag) · Zbl 0382.76084
[33] Olsson, E.; Kreiss, G., A conservative level set method for two phase flow, J. Comput. Phys., 210, 225-246, (2005) · Zbl 1154.76368
[34] Osher, S., Convergence of generalized MUSCL schemes, SIAM J. Numer. Anal., 22, 5, 947-961, (1985) · Zbl 0627.35061
[35] Pilliod, J. E.; Puckett, E. G., Second-order accurate volume-of-fluid algorithms for tracking material interfaces, J. Comput. Phys., 199, 465-502, (2004) · Zbl 1126.76347
[36] Pritchett, J. W., An evaluation of various theoretical models for underwater explosion bubble pulsation, (1971), Technical report, IRA-TR-2-71
[37] Quirk, J. J.; Karni, S., On the dynamics of a shock-bubble interaction, J. Fluid Mech., 318, 129-163, (1996) · Zbl 0877.76046
[38] Rider, W. J.; Kothe, D. B., Reconstructing volume tracking, J. Comput. Phys., 141, 112-152, (1998) · Zbl 0933.76069
[39] Saurel, R.; Abgrall, R., A multiphase Godunov method for compressible multifluid and multiphase flows, J. Comput. Phys., 150, 425-467, (1999) · Zbl 0937.76053
[40] Saurel, R.; Petitpas, F.; Abgrall, R., Modelling phase transition in metastable liquids: application to cavitating and flashing flows, J. Fluid Mech., 607, 313-350, (2008) · Zbl 1147.76060
[41] Shu, C.-W., High order weighted essentially nonoscillatory schemes for convection dominated problems, SIAM Rev., 51, 82-126, (2009) · Zbl 1160.65330
[42] Shukla, R. K.; Pantano, C.; Freund, J. B., An interface capturing method for the simulation of multi-phase compressible flows, J. Comput. Phys., 229, 7411-7439, (2010) · Zbl 1425.76289
[43] Shyue, K.-M., A fluid-mixture type algorithm for compressible multicomponent flow with mie-Grüneisen equation of state, J. Comput. Phys., 171, 678-707, (2001) · Zbl 1047.76573
[44] Shyue, K.-M., A volume-fraction based algorithm for hybrid barotropic and non-barotropic two-fluid flow problems, Shock Waves, 15, 6, 407-423, (2006) · Zbl 1195.76269
[45] Shyue, K.-M., A wave-propagation based volume tracking method for compressible multicomponent flow in two space dimensions, J. Comput. Phys., 215, 219-244, (2006) · Zbl 1140.76401
[46] Shyue, K.-M., A simple unified coordinates method for compressible homogeneous two-phase flows, (Tadmor, E.; Liu, J.-G.; Tzavaras, A., Proc. Symp. Appl. Math., vol. 67, (2009), American Mathematical Society), 949-958 · Zbl 1407.76104
[47] Shyue, K.-M., An anti-diffusion based Eulerian interface-sharpening algorithm for compressible two-phase flow with cavitation, (Ohl, C.-D.; Klaseboer, E.; Ohl, S. W.; Gong, S. W.; Khoo, B. C., Proceedings of the 8th International Symposium on Cavitation, (2012), Research Publishing Services), 198
[48] K.-M. Shyue, An Eulerian interface-sharpening algorithm for compressible gas dynamics, 2012, submitted for publication.
[49] Smith, R. W., AUSM (ALE): a geometrically conservative arbitrary Lagrangian-Eulerian flux splitting scheme, J. Comput. Phys., 150, 268-286, (1999) · Zbl 0936.76046
[50] So, K. K.; Hu, X. Y.; Adams, N. A., Anti-diffusion method for interface steepening in two-phase incompressible flow, J. Comput. Phys., 230, 5155-5177, (2011) · Zbl 1416.76334
[51] So, K. K.; Hu, X. Y.; Adams, N. A., Anti-diffusion interface sharpening technique for two-phase compressible flow simulations, J. Comput. Phys., 231, 4304-4323, (2012) · Zbl 1426.76428
[52] Swift, E.; Decius, J. C., Measurement of bubble pulse phenomena, (1946), Technical report, Navord report 97-46
[53] Tang, T., Moving mesh methods for computational fluid dynamics, (Shi, Z.-C.; Chen, Z.; Tang, T.; Yu, D., Contemp. Math., vol. 383, (2005), American Mathematical Society Rhode Island), 620-625
[54] Tiwari, A.; Freund, J. B.; Pantano, C., A diffuse interface model with immiscibility preservation, J. Comput. Phys., 252, 290-309, (2013) · Zbl 1349.76395
[55] Toro, E. F., Riemann solvers and numerical methods for fluid dynamics: A practical introduction, (1999), Springer-Verlag · Zbl 0923.76004
[56] Ullah, M. A.; Gao, W.; Mao, D.-K., Towards front-tracking based on conservation in two space dimensions III: tracking interfaces, J. Comput. Phys., 242, 268-303, (2013) · Zbl 1314.65113
[57] van Leer, B., Towards the ultimate conservative difference scheme V. A second order sequel to Godunov’s method, J. Comput. Phys., 32, 101-136, (1979) · Zbl 1364.65223
[58] Wardlaw, A. B.; Mair, H. U., Spherical solutions of an underwater explosion bubble, Shock Vib., 5, 89-102, (1998)
[59] Wilkins, M. L., Computer simulation of dynamic phenomena, (1999), Springer New York · Zbl 0926.76001
[60] Xiao, F.; Honma, Y.; Kono, T., A simple algebraic interface capturing scheme using hyperbolic tangent function, Int. J. Numer. Mech. Fluids, 48, 1023-1040, (2005) · Zbl 1072.76046
[61] Xiao, F.; Ii, S.; Chen, C., Revisit to the THINC scheme: a simple algebraic VOF algorithm, J. Comput. Phys., 230, 7086-7092, (2011) · Zbl 1408.76547
[62] Yokoi, K., Efficient implementation of THINC scheme: a simple and practical smoothed VOF algorithm, J. Comput. Phys., 226, 1985-2002, (2007) · Zbl 1388.76281
[63] Zein, A.; Hantke, M.; Warnecke, G., Modeling phase transition for compressible two-phase flows applied to metastable liquids, J. Comput. Phys., 229, 2964-2998, (2010) · Zbl 1307.76079
[64] Zhang, R.; Zhang, M.; Shu, C.-W., On the order of accuracy and numerical performance of two classes of finite volume WENO scheme, Commun. Comput. Phys., 9, 807-827, (2011) · Zbl 1364.65176
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