Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter July 20, 2019

Eulerian modelling of compressible three-fluid flows with surface tension

  • Chao Zhang EMAIL logo and Igor Menshov


The paper addresses a numerical approach for calculating three-fluid hydrodynamics on Eulerian grids with taking into account surface tension and viscous effects. The medium considered consists of three different compressible fluids separated with interfaces. The fluids are assumed to be immiscible. The three-fluid flow is described by the reduced equilibrium model derived from the non-equilibrium three-phase model by performing an asymptotic analysis in the limit of zero relaxation time. To simulate surface tension effects, we extend the continuum surface force (CSF) model of two-fluid incompressible flow to the case of compressible three-fluid flow. A thermodynamically consistent surface energy of the compressible three-fluid flow is obtained by means of splitting the surface tension between distinct fluids into pairs of specific phase related surface tensions. Some aspects of the numerical method for solving the system of governing equations of the considered three-fluid model are discussed. Numerical results presented demonstrate the accuracy and robustness of the proposed model in simulating dynamics of interfaces and surface tension effects.

MSC 2010: 76N15


[1] R. Abgrall, How to prevent pressure oscillations in multicomponent flow calculations: a quasi conservative approach. J. Comp. Phys. 125 (1996), No. 1, 150–160.10.1006/jcph.1996.0085Search in Google Scholar

[2] G. Allaire, S. Clerc, and S. Kokh, A five-equation model for the simulation of interfaces between compressible fluids. J. Comp. Phys. 181 (2002), No. 2, 577–616.10.1006/jcph.2002.7143Search in Google Scholar

[3] M. Baer and J. Nunziato, A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials. Int. J. Multiphase Flow12 (1986), No. 6, 861–889.10.1016/0301-9322(86)90033-9Search in Google Scholar

[4] P. Batten, N. Clarke, C. Lambert, and D. M. Causon, On the choice of wavespeeds for the HLLC Riemann solver. Comput. Methods Appl. Mech. Engrg. 18 (1997), No. 6, 1553–1570.10.1137/S1064827593260140Search in Google Scholar

[5] J. Brackbill, D. Kothe, and C. Zemach, A continuum method for modeling surface tension. J. Comp. Phys. 100 (1992), No. 2, 335–354.10.1016/0021-9991(92)90240-YSearch in Google Scholar

[6] B. Einfeldt, C. D. Munz, and P. L. Roe, On Godunov-type methods near low densities. J. Comp. Phys. 92 (1991), No. 2, 273–295.10.1016/0021-9991(91)90211-3Search in Google Scholar

[7] M. B. Friess and S. Kokh, Simulation of sharp interface multi-material flows involving an arbitrary number of components through an extended five-equation model. J. Comp. Phys. 273 (2014), No. 273, 488–519.10.1016/ in Google Scholar

[8] J. Glimm, X. Li, Y. Liu, Z. Xu, and N. Zhao, Conservative front tracking with improved accuracy. SIAM J. Numer. Analysis41 (2003), No. 5, 1926–1947.10.1137/S0036142901388627Search in Google Scholar

[9] S. K. Godunov, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Mat. Sb. (N.S.)47(89) (1959), No. 3, 271–306.Search in Google Scholar

[10] D. Gueyffier, J. Li, A. Nadim, R. Scardovelli, and S. Zaleski, Volume-of-fluid interface tracking with smoothed surface stress methods for three-dimensional flows. J. Comp. Phys. 152 (1999), No. 2, 423–456.10.1006/jcph.1998.6168Search in Google Scholar

[11] J.-M. Hérard, A three-phase flow model. Math. Comp. Modelling45 (2007), No. 5, 732–755.10.1016/j.mcm.2006.07.018Search in Google Scholar

[12] C. Hirt, A. Amsden, and J. Cook, An arbitrary lagrangian-eulerian computing method for all flow speeds. J. Comp. Phys. 14 (1974), No. 3, 227–253.10.1016/0021-9991(74)90051-5Search in Google Scholar

[13] 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:reduced equations. Physics of Fluids13 (2001), No. 10, 3002–3024.10.1063/1.1398042Search in Google Scholar

[14] R. LeVeque and Z. Li, Immersed interface methods for stokes flow with elastic boundaries or surface tension. SIAM J. Sci. Comp. 18 (1997), No. 3, 709–735.10.1137/S1064827595282532Search in Google Scholar

[15] I. Menshov and P. Zakharov, On the composite riemann problem for multi-material fluid flows, Int. J. Numer. Methods in Fluids76 (2015), No. 2, 109–127.10.1002/fld.3927Search in Google Scholar

[16] W. Mulder, S. Osher, and J. A. Sethian, Computing interface motion in compressible gas dynamics. J. Comp. Phys. 100 (1992), No. 2, 209–228.10.1016/0021-9991(92)90229-RSearch in Google Scholar

[17] G. Perigaud and R. Saurel, A compressible flow model with capillary effects. J. Comp. Phys. 209 (2005), No. 1, 139–178.10.1016/ in Google Scholar

[18] K.-M. Shyue, A fluid-mixture type algorithm for compressible multicomponent flow with van der waals equation of state. J. Comp. Phys. 156 (1999), No. 1, 43–88.10.1006/jcph.1999.6349Search in Google Scholar

[19] K. M. Shyue and F. Xiao, An eulerian interface sharpening algorithm for compressible two-phase flow: The algebraic THINC approach. J. Comp. Phys. 268 (2014), No. 2, 326–354.10.1016/ in Google Scholar

[20] K. Smith, F. Sons, and D. Chopp, A projection method for motion of triple junctions by level sets. Interfaces and Free Boundaries4 (2002), No. 1, 263–276.10.4171/IFB/61Search in Google Scholar

[21] K. K. So, X. Y. Hu, and N. A. Adams, Anti-diffusion interface sharpening technique for two-phase compressible flow simulations. J. Comp. Phys. 231 (2012), No. 11, 4304–4323.10.1016/ in Google Scholar

[22] M. Sussman, P. Smereka, and S. Osher, A level set approach for computing solutions to incompressible two-phase flow. J. Comp. Phys. 114 (1994), No. 1, 146–159.10.1006/jcph.1994.1155Search in Google Scholar

[23] H. Terashima and G. Tryggvason, A front-tracking/ghost-fluid method for fluid interfaces in compressible flows. J. Comp. Phys. 228 (2009), No. 11, 4012–4037.10.1016/ in Google Scholar

[24] A. Tiwari, J. B. Freund, and C. Pantano, A diffuse interface model with immiscibility preservation. J. Comp. Phys. 252 (2013), No. C, 290–309.10.1016/ in Google Scholar PubMed PubMed Central

[25] E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, Berlin–Heidelberg, 2009.10.1007/b79761Search in Google Scholar

[26] F. Xiao, Y. Honma, and T. Kono, A simple algebraic interface capturing scheme using hyperbolic tangent function. Int. J. Numer. Methods in Fluids48 (2005), No. 9, 1023–1040.10.1002/fld.975Search in Google Scholar

[27] C. Zhang and I. S. Menshov, Continuous method for calculating the transport equations for a multicomponent heterogeneous system on fixed euler grids. Matematicheskoe Modelirovanie31 (2019) No. 4, 111–130 (in Russian).Search in Google Scholar

Received: 2019-02-01
Revised: 2019-05-20
Accepted: 2019-05-21
Published Online: 2019-07-20
Published in Print: 2019-08-27

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 22.3.2023 from
Scroll Up Arrow