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Building Blocks for a Strictly Conservative Generalized Finite Volume Projection Method for Zero Mach Number Two-Phase Flows

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Transport Processes at Fluidic Interfaces

Part of the book series: Advances in Mathematical Fluid Mechanics ((AMFM))

Abstract

Building blocks for a generalized fully conservative finite volume projection method for numerical simulation of immiscible zero Mach number two-phase flows on Cartesian grids are presented, focusing on the crucial issues of interface propagation, fluid phase conservation and discretization of the singular contribution due to surface tension, each in a discretely conservative fashion. Additionally, a solution approach for solving Poisson-type equations for two-phase flows at arbitrary ratio of coefficients is sketched. Further, (intermediate) results applying these building blocks are presented and open issues and future developments are proposed.

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Notes

  1. 1.

    Small length scales compared to the atmospheric pressure scale height.

  2. 2.

    Changes of fluid phase over time, however, are accounted for.

  3. 3.

    The VoF based interface representation itself is (discretely) conservative by default. However, discretization errors can produce VoF values which can not be assigned a physically reasonable interpretation anymore. Truncation of such over- and undershoots without suitable flux based redistribution leads to a lack of conservation on the discrete level.

References

  1. Almgren, A.S., Bell, J.B., Colella, P., Howell, L.H., Welcome, M.L.: A conservative adaptive projection method for the variable density incompressible Navier-Stokes equations. J. Comput. Phys. 142, 1–46 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bänsch, E.: Finite element discretization of the Navier-Stokes equations with free capillary surface. Numer. Math. 88, 203–235 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bell, J.B., Marcus, D.: A second-order projection method for variable-density flows. J. Comput. Phys. 101, 334–348 (1992)

    Article  MATH  Google Scholar 

  4. Botta, N., Klein, R., Langenberg, S., Lützenkirchen, S.: Well-balanced finite volume methods for nearly hydrostatic flows. J. Comput. Phys. 196, 539–565 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bourlioux, A.: A coupled level-set volume-of-fluid algorithm for tracking material interfaces. In: 3rd Annual Conference of the CFD Society, Banff (1995)

    Google Scholar 

  6. Brackbill, J.U., Kothe, D.B., Zemach, C.: A continuum method for modelling surface tension. J. Comput. Phys. 100, 335–354 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brown, D.L., Cortez, R., Minion, M.L.: Accurate projection methods for the incompressible Navier-Stokes equations. J. Comput. Phys. 168, 464–499 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chorin, A.J.: A numerical method for solving incompressible viscous flow problems. J. Comput. Phys. 2, 12–26 (1967)

    Article  MATH  Google Scholar 

  9. Francois, M.M., Cummins, S.J., Deny, E.D., Kothe, D.B., Sicilian, J.M., Williams, M.W.: A balanced-force algorithm for continuous and sharp interfacial surface tension models with a volume tracking framework. J. Comput. Phys. 213, 141–173 (2006)

    Article  MATH  Google Scholar 

  10. Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43(1), 89–112 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hansbo, A., Hansbo, P.: An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput. Methods Appl. Mech. Eng. 191, 5537–5552 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Harlow, F.H., Welch, J.E.: Numerical calculation of time-dependent viscous incompressible flow of fluid with a free interface. Phys. Fluids 8, 2182–2189 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hartmann, D.: A level-set based method for premixed combustion in compressible flow. Ph.D. thesis, Fakultät für Machinenwesen, Rheinisch-Westfälische Technische Hochschule Aachen, Aachen (2010)

    Google Scholar 

  14. Hartmann, D., Meinke, M., Schröder, W.: Differential equation based constrained reinitialization for level set methods. J. Comput. Phys. 227, 6821–6845 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Klein, R.: Asymptotics, structure, and integration of sound-proof atmospheric flow equations. Theor. Comput. Fluid Dyn. 23(3), 161–195 (2009)

    Article  MATH  Google Scholar 

  16. Kornek, U., Müller, F., Harth, K., Hahn, A., Ganesan, S., Tobiska, L., Stannarius, R.: Oscillations of soap bubbles. New J. Phys. 12, 073031 (2010)

    Article  Google Scholar 

  17. Lakin, S.: Eine implizite Druckrandbedingung für die nichtstationäre Stokes-Gleichung. Master’s thesis, Freie Universität Berlin (2012) [in German]

    Google Scholar 

  18. Lamb, H.: Hydrodynamics. Cambridge University Press, Cambridge (1932)

    MATH  Google Scholar 

  19. Liu, J.-G., Liu, J., Pego, R.L.: Stable and accurate pressure approximation for unsteady incompressible viscous flow. J. Comput. Phys. 229, 3428–3453 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. McCaslin, J.O., Desjardins, O.: A local re-initialization equation for the conservative level set method. J. Comput. Phys. 262, 408–426 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Oevermann, M., Klein, R.: A Cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces. J. Comput. Phys. 219, 749–769 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Oevermann, M., Klein, R.: An asymptotic solution approach for elliptic equations with discontinuous coefficients. J. Comput. Phys. 261, 230–243 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Oevermann, M., Klein, R., Berger, M., Goodman, J.: A projection method for incompressible two-phase flow with surface tension. Technical Report 00-17, Konrad-Zuse-Zentrum, Berlin (2000)

    Google Scholar 

  24. Oevermann, M., Scharfenberg, C., Klein, R.: A sharp interface finite volume method for elliptic equations on Cartesian grids. J. Comput. Phys. 228(14), 5184–5206 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Peskin, C.S.: Numerical analysis of blood flow in the heart. J. Comput. Phys. 25, 220–252 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  26. Popinet, S.: An accurate adaptive solver for surface-tension-driven interfacial flows. J. Comput. Phys. 228, 5838–5866 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Popinet, S., Zaleski, S.: A front-tracking algorithm for accurate representation of surface tension. Int. J. Numer. Methods Fluids 30(6), 775–793 (1999)

    Article  MATH  Google Scholar 

  28. Prosperetti, A., Tryggvason, G. (eds.): Computational Methods for Multiphase Flow. Cambridge University Press, New York (2007)

    Google Scholar 

  29. Schneider, T.: Verfolgung von Flammenfronten und Phasengrenzen in schwachkompressiblen Strömungen. PhD thesis, Fakultät für Machinenwesen, Rheinisch-Westfälische Technische Hochschule Aachen, Germany (2000) [in German]

    Google Scholar 

  30. Schneider, T., Klein, R.: Overcoming mass losses in level-set-based interface tracking schemes. In: 2nd International Conference on Finite Volume for Complex Application (1999)

    Google Scholar 

  31. Schneider, T., Botta, N., Geratz, K.J., Klein, R.: Extension of finite volume compressible flow solvers to multi-dimensional, variable density zero Mach number flows. J. Comput. Phys. 155, 248–286 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  32. Smiljanovski, V., Moser, V., Klein, R.: A capturing-tracking hybrid scheme for deflagration discontinuities. Combust. Theor. Model. 1(2), 183–215 (1997)

    Article  MATH  Google Scholar 

  33. Sussman, M., Smith, K.M., Hussaini, M.Y., Ohta, M., Zhi-Wei, R.: A sharp interface method for incompressible two-phase flow. J. Comput. Phys. 221, 469–505 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. Temam, R.: Suitable initial conditions. J. Comput.Phys. 218, 443–450 (2006)

    Google Scholar 

  35. Terhoeven, P.: Ein numerisches Verfahren zur Berechnung von Flammenfronten bei kleiner Mach-Zahl. PhD thesis, Rheinisch-Westfälische Technische Hochschule Aachen, Aachen (1998) [in German]

    Google Scholar 

  36. Tryggvason, G., Scardovelli, R., Zaleski, S.: Direct Numerical Simulations of Gas-Liquid Multiphase Flows. Cambridge University Press, Cambridge (2011)

    Book  MATH  Google Scholar 

  37. Vater, S., Klein, R.: Stability of a Cartesian grid projection method for zero Froude number shallow water flows. Numer. Math. 113(1), 123–161 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. Waidmann, M.: Towards a strictly conservative hybrid level-set volume-of-fluid finite volume method for zero mach number two-phase flow. PhD thesis, Institut für Mathematik, Freie Universität Berlin, Berlin (2017)

    Google Scholar 

  39. Waidmann, M., Gerber, S., Oevermann, M., Klein, R.: A conservative coupling of level-set, volume-of-fluid and other conserved quantities. In: Fuhrmann, J., Ohlberger, M., Rohde, C. (eds.) Finite Volumes for Complex Applications VII - Methods and Theoretical Aspects. Springer Proceedings in Mathematics and Statistics, vol. 77, pp. 457–465. Springer, Berlin (2014)

    Google Scholar 

  40. Wang, Y., Simakhina, S., Sussman, M.: A hybrid levelset-volume constraint method for incompressible two-phase flow. J. Comput. Phys. 231, 6438–6471 (2012)

    Article  MathSciNet  Google Scholar 

  41. Zhou, Y.C., Liu, J., Harry, D.L.: A matched interface and boundary method for solving multi-flow Navier-Stokes equations with applications to geodynamics. J. Comput. Phys. 231, 223–242 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This project was funded by German Research Foundation (DFG) Priority Program (SPP) 1506 on Transport Processes at Fluidic Interfaces from 2011 to 2016.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Freie Universität Berlin, Arnimallee 6, 14195, Berlin, Germany

    Matthias Waidmann, Stephan Gerber & Rupert Klein

  2. Department for Applied Mechanics, Chalmers University of Technology, Gothenburg, Sweden

    Michael Oevermann

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  1. Matthias Waidmann
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  2. Stephan Gerber
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  3. Michael Oevermann
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  4. Rupert Klein
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Correspondence to Matthias Waidmann .

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Editors and Affiliations

  1. Fachbereich Mathematik, Technische Universität Darmstadt , Darmstadt, Germany

    Dieter Bothe

  2. Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Aachen, Nordrhein-Westfalen, Germany

    Arnold Reusken

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Waidmann, M., Gerber, S., Oevermann, M., Klein, R. (2017). Building Blocks for a Strictly Conservative Generalized Finite Volume Projection Method for Zero Mach Number Two-Phase Flows. In: Bothe, D., Reusken, A. (eds) Transport Processes at Fluidic Interfaces. Advances in Mathematical Fluid Mechanics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-56602-3_4

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