OJFD  Vol.4 No.1 , March 2014
Sharpening Diffuse Interfaces with Compressible Flow Solvers
ABSTRACT
Diffuse interfaces appear with any Eulerian discontinuity capturing compressible flow solver. When dealing with multifluid and multimaterial computations, interfaces smearing results in serious difficulties to fulfil contact conditions, as spurious oscillations appear. To circumvent these difficulties, several approaches have been proposed. One of them relies on multiphase flow modelling of the numerically diffused zone and is based on extended hyperbolic systems with stiff mechanical relaxation (Saurel and Abgrall, 1999 [4], Saurel et al., 2009 [6]). This approach is very robust, accurate and flexible in the sense that many physical effects can be included: surface tension, phase transition, elastic-plastic materials, detonations, granular effects etc. It is also able to deal with dynamic appearance of interfaces. However it suffers from an important drawback when long time evolution is under interest as the interface becomes more and more diffused. The present paper addresses this issue and provides an efficient way to sharpen interfaces. A sharpening flow model is used to correct the solution after each time step. The sharpening process is based on a hyperbolic equation that produces a steady shock in finite time at the interface location. This equation is embedded in a “sharpening multiphase model” redistributing volume fractions, masses, momentum and energy in a consistent way. The method is conservative with respect to the masses, mixture momentum and mixture energy. It results in diffused interfaces sharpened in one or two mesh points. The method is validated on test problems having exact solutions.



Cite this paper
Favrie, N. , Gavrilyuk, S. , Nkonga, B. and Saurel, R. (2014) Sharpening Diffuse Interfaces with Compressible Flow Solvers. Open Journal of Fluid Dynamics, 4, 44-68. doi: 10.4236/ojfd.2014.41004.
References
[1]   Hirt, C.W. and Nichols, B.D. (1981) Volume of Fluid (VOF) METHOD for the Dynamics of Free Boundaries. Journal of Computational Physics, 39, 201-255. http://dx.doi.org/10.1016/0021-9991(81)90145-5

[2]   Youngs, D.L. (1989) Modelling Turbulent Mixing by Rayleigh-Taylor Instability. Physica D: Nonlinear Phenomena, 37, 270-287. http://dx.doi.org/10.1016/0167-2789(89)90135-8

[3]   Fedkiw, R.P., Aslam, T., Merriman, B. and Osher, S. (1999) A Non-Oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method). Journal of Computational Physics, 152, 457-492. http://dx.doi.org/10.1006/jcph.1999.6236

[4]   Saurel, R. and Abgrall, R. (1999) A Multiphase Godunov Method for Compressible Multifluid and Multiphase Flows. Journal of Computational Physics, 150, 425-467. http://dx.doi.org/10.1006/jcph.1999.6187

[5]   Kapila, A.K., Menikoff, Bdzil, J.B., Son, S.F. and Stewart, D.S. (2001) Two-Phase Modeling of Deflagration to Detonation Transition in Granular Materials: Reduced Equations. Physics of Fluids, 13, 3002-3024. http://dx.doi.org/10.1063/1.1398042

[6]   Saurel, R., Petitpas, F. and Berry, R.A. (2009) Simple and Efficient Relaxation for Interfaces Separating Compressible Fluids, Cavitating Flows and Shocks in Multiphase Mixtures. Journal of Computational Physics, 228, 1678-1712. http://dx.doi.org/10.1016/j.jcp.2008.11.002

[7]   Perigaud, G. and Saurel, R. (2005) A Compressible Flow Model with Capillary Effects. Journal of Computational Physics, 209, 139-178. http://dx.doi.org/10.1016/j.jcp.2005.03.018

[8]   Braconnier, B. and Nkonga, B. (2009) An All-Speed Relaxation Scheme for Interface Flows with Surface Tension. Journal of Computational Physics, 228, 5722-5739. http://dx.doi.org/10.1016/j.jcp.2009.04.046

[9]   Saurel, R., Petitpas, F. and Abgrall, R. (2008) Modeling Phase Transition in Metastable Liquids. Application to Flashing and Cavitating Flows. Journal of Fluid Mechanics, 607, 313-350. http://dx.doi.org/10.1017/S0022112008002061

[10]   Favrie, N., Gavrilyuk, S. and Saurel, R. (2009) Solid-Fluid Diffuse Interface Model in Cases of Extreme Deformations. Journal of Computational Physics, 228, 6037-6077. http://dx.doi.org/10.1016/j.jcp.2009.05.015

[11]   Favrie, N. and Gavrilyuk, S. (2012) Diffuse Interface Model for Compressible Fluid-Compressible Elastic-Plastic Solid Interaction. Journal of Computational Physics, 231, 2695-2723. http://dx.doi.org/10.1016/j.jcp.2011.11.027

[12]   Petitpas, F., Saurel, R., Franquet, E. and Chinnayya, A. (2009) Modelling Detonation Waves in Condensed Materials: Multiphase CJ Conditions and Multidimensional Computations. Shock Waves, 19, 377-401. http://dx.doi.org/10.1007/s00193-009-0217-7

[13]   Saurel, R., Favrie, N., Petitpas, F., Lallemand, M.H. and Gavrilyuk, S. (2010) Modelling Dynamic and Irreversible Powder Compaction. Journal of Fluid Mechanics, 664, 348-396. http://dx.doi.org/10.1017/S0022112010003794

[14]   Kokh, S. and Lagoutière, F. (2010) An Anti-Diffusive Numerical Scheme for the Simulation of Interfaces between Compressible Fluids by Means of a Five-Equation Model. Journal of Computational Physics, 229, 2773-2809. http://dx.doi.org/10.1016/j.jcp.2009.12.003

[15]   So, K.K., Hu, X.Y. and Adams, N.A. (2012) Anti-Diffusion Interface Sharpening Technique for Two-Phase Compressible Flow Simulations. Journal of Computational Physics, 231, 4304-4323. http://dx.doi.org/10.1016/j.jcp.2012.02.013

[16]   Harten, A. (1977) The Artificial Compression Method for Computation of Shocks and Contact Discontinuities. I. Single Conservation Laws. Communications on Pure and Applied Mathematics, 30, 611-638. http://dx.doi.org/10.1002/cpa.3160300506

[17]   Harten, A. (1978) The Artificial Compression Method for Computation of Shocks and Contact Discontinuities: III. Self-Adjusting Hybrid Schemes. Mathematics of Computation, 32, 363-389.

[18]   Shukla, R.K., Pantano, C. and Freund, J.B. (2010) An Interface Capturing Method for the Simulation of Multi-Phase Compressible Flows. Journal of Computational Physics, 229, 7411-7439. http://dx.doi.org/10.1016/j.jcp.2010.06.025

[19]   Allaire, G., Clerc, S. and Kokh, S. (2002) A Five-Equation Model for the Simulation of Interfaces between Compressible Fluids. Journal of Computational Physics, 181, 577-616. http://dx.doi.org/10.1006/jcph.2002.7143

[20]   Massoni, J., Saurel, R., Nkonga, B. and Abgrall, R. (2002) Proposition de Méthodes et Modèles Eulériens pour les Problèmes à Interfaces Entre Fluides Compressibles en Présence de Transfert de Chaleur: Some Models and Eulerian Methods for Interface Problems between Compressible Fluids with Heat Transfer. International Journal of Heat and Mass Transfer, 45, 1287-1307. http://dx.doi.org/10.1016/S0017-9310(01)00238-1

[21]   Wood, A.B. (1930) A Textbook of Sound. G. Bell and Sons LTD, London.

[22]   Saurel, R., Le Metayer, O., Massoni, J. and Gavrilyuk, S. (2007) Shock Jump Relations for Multiphase Mixtures with Stiff Mechanical Relaxation. Shock Waves, 16, 209-232. http://dx.doi.org/10.1007/s00193-006-0065-7

[23]   Saurel, R., Gavrilyuk, S. and Renaud, F. (2003) A Multiphase Model with Internal Degree of Freedom: Application to Shock-Bubble Interaction. Journal of Fluid Mechanics, 495, 283-321. http://dx.doi.org/10.1017/S002211200300630X

[24]   Toro, E.F., Spruce, M. and Speares, W. (1994) Restoration of the Contact Surface in the HLL Riemann Solver. Shock Waves, 4, 25-34. http://dx.doi.org/10.1007/BF01414629

 
 
Top