American Institute of Mathematical Sciences

June  2015, 35(6): 2325-2347. doi: 10.3934/dcds.2015.35.2325

Numerical simulation of two-phase flows with heat and mass transfer

 1 AM III, Department Math, Cauerstr. 11 91058 Erlangen, Germany, Germany 2 Reinbeckstrasse 7, 12459 Berlin, Germany

Received  January 2014 Revised  May 2014 Published  December 2014

We present a finite element method for simulating complex free surface flow. The mathematical model and the numerical method take into account two-phase non-isothermal flow of an incompressible liquid and a gas phase, capillary forces at the interface of both fluids, Marangoni effects due to temperature variation of the interface and mass transport across the interface by evaporation/condensation. The method is applied to two examples from microgravity research, for which experimental data are available.
Citation: Eberhard Bänsch, Steffen Basting, Rolf Krahl. Numerical simulation of two-phase flows with heat and mass transfer. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2325-2347. doi: 10.3934/dcds.2015.35.2325
References:
 [1] E. Bänsch, Simulation of instationary, incompressible flows,, Acta Math. Univ. Com., 67 (1998), 101. Google Scholar [2] E. Bänsch, Finite element discretization of the Navier-Stokes equations with a free capillary surface,, Numer. Math., 88 (2001), 203. doi: 10.1007/PL00005443. Google Scholar [3] J. Brackbill, D. Kothe and C. Zemach, A continuum method for modeling surface tension,, Journal of Computational Physics, 100 (1992), 335. doi: 10.1016/0021-9991(92)90240-Y. Google Scholar [4] M.-O. Bristeau, R. Glowinski and J. Pariaux, Numerical methods for the Navier-Stokes equations. applications to the simulation of compressible and incompressible viscous flow,, Computer Physics Report, 6 (1987), 73. doi: 10.1007/978-3-322-87873-1. Google Scholar [5] A. N. Brooks and T. J. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations,, Computer Methods in Applied Mechanics and Engineering, 32 (1982), 199. doi: 10.1016/0045-7825(82)90071-8. Google Scholar [6] S. Das and E. Hopfinger, Mass transfer enhancement by gravity waves at a liquid-vapour interface,, International Journal of Heat and Mass Transfer, 52 (2009), 1400. doi: 10.1016/j.ijheatmasstransfer.2008.08.016. Google Scholar [7] J. Donea, S. Giuliani and J. P. Halleux, An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions,, Comp. Meth. App. MechEng., 33 (1982), 689. doi: 10.1016/0045-7825(82)90128-1. Google Scholar [8] M. E. Dreyer, Free Surface Flows under Compensated Gravity Conditions,, no. 221 in Springer Tracts in Modern Physics, (2007). Google Scholar [9] G. Dziuk, An algorithm for evolutionary surfaces,, Numerische Mathematik, 58 (1991), 603. doi: 10.1007/BF01385643. Google Scholar [10] C. Eck, M. Fontelos, G. Grün, F. Klingbeil and O. Vantzos, On a phase-field model for electrowetting,, Interfaces Free Bound., 11 (2009), 259. doi: 10.4171/IFB/211. Google Scholar [11] E. Fuhrmann and M. Dreyer, Description of the Sounding Rocket Experiment SOURCE,, Microgravity Science and Technology, 20 (2008), 205. doi: 10.1007/s12217-008-9017-4. Google Scholar [12] E. Fuhrmann and M. Dreyer, Heat transfer by thermo-capillary convection,, Microgravity Science and Technology, 21 (2009), 87. doi: 10.1007/s12217-009-9125-9. Google Scholar [13] E. Fuhrmann, M. Dreyer, S. Basting and E. Bänsch, Free surface deformation and heat transfer by thermocapillary convection, 2013,, Submitted for publication., (). Google Scholar [14] J. Gerstmann, Numerische Untersuchung zur Schwingung freier Flüssigkeitsoberflächen,, no. 464 in Fortschritt-Berichte VDI, (2004). Google Scholar [15] J. Gerstmann, M. Michaelis and M. E. Dreyer, Capillary driven oscillations of a free liquid interface under non-isothermal conditions,, PAMM, 4 (2004), 436. doi: 10.1002/pamm.200410199. Google Scholar [16] F. Gibou, L. Chen, D. Nguyen and S. Banerjee, A level set based sharp interface method for the multiphase incompressible Navier-Stokes equations with phase change,, J. Comp. Phys., 222 (2007), 536. doi: 10.1016/j.jcp.2006.07.035. Google Scholar [17] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations,, Springer, (1986). doi: 10.1007/978-3-642-61623-5. Google Scholar [18] S. Gross and A. Reusken, Numerical Methods for Two-phase Incompressible Flows, vol. 40 of Springer Series in Computational Mathematics,, Springer-Verlag, (2011). doi: 10.1007/978-3-642-19686-7. Google Scholar [19] M. E. Gurtin, An Introduction to Continuum Mechanics,, Academic Press, (1981). Google Scholar [20] C. W. Hirt and B. D. Nichols, Volume of fluid (VOF) method for the dynamics of free boundaries,, J. Comp. Phys., 39 (1981), 201. doi: 10.1016/0021-9991(81)90145-5. Google Scholar [21] C. Hirt, A. Amsden and J. Cook, An arbitrary Lagrangian-Eulerian computing method for all flow speeds,, Journal of Computational Physics, 135 (1997), 203. doi: 10.1006/jcph.1997.5702. Google Scholar [22] B. Höhn, Numerik für die Marangoni-Konvektion beim Floating-Zone Verfahren,, Dissertation, (1999). Google Scholar [23] T. J. R. Hughes, W. Liu and T. K. Zimmermann, Lagrangian-Eulerian finite element formulation for incompressible viscous flows,, Computer Methods in Applied Mechanics and Engineering, 29 (1981), 329. doi: 10.1016/0045-7825(81)90049-9. Google Scholar [24] D. Jacqmin, Calculation of two-phase Navier-Stokes flows using phase-field modeling,, J. Comp. Phys., 155 (1999), 96. doi: 10.1006/jcph.1999.6332. Google Scholar [25] D. Jamet, O. Lebaigue, N. Coutris and J. M. Delhaye, The second gradient method for the direct numerical simulation of liquid-vapor flows with phase change,, J. Comp. Phys, 169 (2001), 624. doi: 10.1006/jcph.2000.6692. Google Scholar [26] E. Kennard, Kinetic theory of gases: with an introduction to statistical mechanics,, International series in pure and applied physics, (1938). Google Scholar [27] R. Klein, N. Botta, T. Schneider, C. D. Munz, S. Roller, A. Meister, L. Hoffmann and T. Sonar, Asymptotic adaptive methods for multi-scale problems in fluid mechanics,, Eng. Math., 39 (2001), 261. doi: 10.1023/A:1004844002437. Google Scholar [28] R. Krahl, M. Adamov, M. Lozano Avilés and E. Bänsch, A model for two phase flow with evaporation,, in Two-Phase Flow Modelling and Experimentation 2004 (eds. G. P. Celata, (2004), 2381. Google Scholar [29] R. Krahl, J. Gerstmann, P. Behruzi, E. Bänsch and M. E. Dreyer, Dependency of the apparent contact angle on nonisothermal conditions,, Physics of Fluids, 20 (2008). doi: 10.1063/1.2899641. Google Scholar [30] R. Krahl and E. Bänsch, Impact of marangoni effects on the apparent contact angle - a numerical investigation,, Microgravity Science and Technology, 17 (2005), 39. doi: 10.1007/BF02872086. Google Scholar [31] R. Krahl and E. Bänsch, On the stability of an evaporating liquid surface,, Fluid Dynamics Research, 44 (2012). doi: 10.1088/0169-5983/44/3/031409. Google Scholar [32] R. Krahl and J. Gerstmann, Non-isothermal reorientation of a liquid surface in an annular gap,, in $4^{th}$ International Berlin Workshop - IBW 4 on Transport Phenomena with Moving Boundaries (ed. F.-P. Schindler), (2007), 227. Google Scholar [33] N. Kulev, S. Basting, E. Bänsch and M. Dreyer, Interface reorientation of cryogenic liquids under non-isothermal boundary conditions,, Cryogenics, 62 (2014), 48. doi: 10.1016/j.cryogenics.2014.04.006. Google Scholar [34] N. Kulev and M. Dreyer, Drop tower experiments on non-isothermal reorientation of cryogenic liquids,, Microgravity Science and Technology, 22 (2010), 463. doi: 10.1007/s12217-010-9237-2. Google Scholar [35] D. Meschede (ed.), Gerthsen Physik,, 22nd edition, (2004). Google Scholar [36] M. Michaelis, Kapillarinduzierte Schwingungen Freier Flüssigkeitsoberflächen,, no. 454 in Fortschritt-Berichte VDI, (2003). Google Scholar [37] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations,, J. Comp. Phys., 79 (1988), 12. doi: 10.1016/0021-9991(88)90002-2. Google Scholar [38] S. Ostrach, Low-gravity fluid flows,, Ann. Rev. Fluid. Mech., 14 (1982), 313. doi: 10.1146/annurev.fl.14.010182.001525. Google Scholar [39] L. M. Pismen and Y. Pomeau, Disjoining potential and spreading of thin liquid layers in the diffuse-interface model coupled to hydrodynamics,, Phys. Rev. E, 62 (2000), 2480. doi: 10.1103/PhysRevE.62.2480. Google Scholar [40] M. Rumpf, A variational approach to optimal meshes,, Numerische Mathematik, 72 (1996), 523. doi: 10.1007/s002110050180. Google Scholar [41] Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,, SIAM Sci. Comp., 7 (1986), 856. doi: 10.1137/0907058. Google Scholar [42] R. Scardovelli and S. Zaleski, Direct numerical simulation of free-surface and interfacial flow,, Ann. Rev. Fluid Mech., 31 (1999), 567. doi: 10.1146/annurev.fluid.31.1.567. Google Scholar [43] J. Schlottke and B. Weigand, Direct numerical simulation of evaporating droplets,, J. Comp. Phys., 227 (2008), 5215. doi: 10.1016/j.jcp.2008.01.042. Google Scholar [44] L. E. Scriven, Dynamics of a fluid interface equation of motion for Newtonian surface fluids,, Chem. Eng. Sci., 12 (1960), 98. doi: 10.1016/0009-2509(60)87003-0. Google Scholar [45] J. A. Sethian and P. Smereka, Level set methods for fluid interfaces,, Ann. Rev. Fluid Mech., 35 (2003), 341. doi: 10.1146/annurev.fluid.35.101101.161105. Google Scholar [46] G. Son and V. K. Dhir, Numerical simulation of film boiling near critical pressures with a level set method,, J. Heat Transfer, 120 (1998), 183. doi: 10.1115/1.2830042. Google Scholar [47] M. Sussman, P. Smereka and S. Osher, A level set approach for computing solutions to incompressible two-phase flow,, Journal of Computational Physics, 114 (1994), 146. doi: 10.1006/jcph.1994.1155. Google Scholar [48] S. Tanguy, T. Ménard and A. Berlemont, A level set method for vaporizing two-phase flows,, J. Comp. Phys., 221 (2007), 837. doi: 10.1016/j.jcp.2006.07.003. Google Scholar [49] M. Tenhaeff, Computation of Incompressible, Axisymmetric Flows in Electrically Conducting Fluids Under Influence of Rotating Magnetic Fields (in German),, Diploma thesis, (1997). Google Scholar [50] T. Tezduyar and R. Benney, Mesh moving techniques for fluid-structure interactions with large displacements,, J. Applied Mechanics, 70 (2003), 58. Google Scholar [51] S. W. J. Welch and J. Wilson, A volume of fluid based method for fluid flows with phase change,, J. Comp. Phys, 160 (2000), 662. doi: 10.1006/jcph.2000.6481. Google Scholar [52] T. Wick, Fluid-structure interactions using different mesh motion techniques,, Computers & Structures, 89 (2011), 1456. doi: 10.1016/j.compstruc.2011.02.019. Google Scholar [53] Y. F. Yap, J. C. Chai, K. C. Toh, T. N. Wong and Y. C. Lam, Numerical modeling of unidirectional stratified flow with and without phase change,, J. Int. Heat Mass Transfer, 48 (2005), 477. doi: 10.1016/j.ijheatmasstransfer.2004.09.017. Google Scholar

show all references

References:
 [1] E. Bänsch, Simulation of instationary, incompressible flows,, Acta Math. Univ. Com., 67 (1998), 101. Google Scholar [2] E. Bänsch, Finite element discretization of the Navier-Stokes equations with a free capillary surface,, Numer. Math., 88 (2001), 203. doi: 10.1007/PL00005443. Google Scholar [3] J. Brackbill, D. Kothe and C. Zemach, A continuum method for modeling surface tension,, Journal of Computational Physics, 100 (1992), 335. doi: 10.1016/0021-9991(92)90240-Y. Google Scholar [4] M.-O. Bristeau, R. Glowinski and J. Pariaux, Numerical methods for the Navier-Stokes equations. applications to the simulation of compressible and incompressible viscous flow,, Computer Physics Report, 6 (1987), 73. doi: 10.1007/978-3-322-87873-1. Google Scholar [5] A. N. Brooks and T. J. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations,, Computer Methods in Applied Mechanics and Engineering, 32 (1982), 199. doi: 10.1016/0045-7825(82)90071-8. Google Scholar [6] S. Das and E. Hopfinger, Mass transfer enhancement by gravity waves at a liquid-vapour interface,, International Journal of Heat and Mass Transfer, 52 (2009), 1400. doi: 10.1016/j.ijheatmasstransfer.2008.08.016. Google Scholar [7] J. Donea, S. Giuliani and J. P. Halleux, An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions,, Comp. Meth. App. MechEng., 33 (1982), 689. doi: 10.1016/0045-7825(82)90128-1. Google Scholar [8] M. E. Dreyer, Free Surface Flows under Compensated Gravity Conditions,, no. 221 in Springer Tracts in Modern Physics, (2007). Google Scholar [9] G. Dziuk, An algorithm for evolutionary surfaces,, Numerische Mathematik, 58 (1991), 603. doi: 10.1007/BF01385643. Google Scholar [10] C. Eck, M. Fontelos, G. Grün, F. Klingbeil and O. Vantzos, On a phase-field model for electrowetting,, Interfaces Free Bound., 11 (2009), 259. doi: 10.4171/IFB/211. Google Scholar [11] E. Fuhrmann and M. Dreyer, Description of the Sounding Rocket Experiment SOURCE,, Microgravity Science and Technology, 20 (2008), 205. doi: 10.1007/s12217-008-9017-4. Google Scholar [12] E. Fuhrmann and M. Dreyer, Heat transfer by thermo-capillary convection,, Microgravity Science and Technology, 21 (2009), 87. doi: 10.1007/s12217-009-9125-9. Google Scholar [13] E. Fuhrmann, M. Dreyer, S. Basting and E. Bänsch, Free surface deformation and heat transfer by thermocapillary convection, 2013,, Submitted for publication., (). Google Scholar [14] J. Gerstmann, Numerische Untersuchung zur Schwingung freier Flüssigkeitsoberflächen,, no. 464 in Fortschritt-Berichte VDI, (2004). Google Scholar [15] J. Gerstmann, M. Michaelis and M. E. Dreyer, Capillary driven oscillations of a free liquid interface under non-isothermal conditions,, PAMM, 4 (2004), 436. doi: 10.1002/pamm.200410199. Google Scholar [16] F. Gibou, L. Chen, D. Nguyen and S. Banerjee, A level set based sharp interface method for the multiphase incompressible Navier-Stokes equations with phase change,, J. Comp. Phys., 222 (2007), 536. doi: 10.1016/j.jcp.2006.07.035. Google Scholar [17] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations,, Springer, (1986). doi: 10.1007/978-3-642-61623-5. Google Scholar [18] S. Gross and A. Reusken, Numerical Methods for Two-phase Incompressible Flows, vol. 40 of Springer Series in Computational Mathematics,, Springer-Verlag, (2011). doi: 10.1007/978-3-642-19686-7. Google Scholar [19] M. E. Gurtin, An Introduction to Continuum Mechanics,, Academic Press, (1981). Google Scholar [20] C. W. Hirt and B. D. Nichols, Volume of fluid (VOF) method for the dynamics of free boundaries,, J. Comp. Phys., 39 (1981), 201. doi: 10.1016/0021-9991(81)90145-5. Google Scholar [21] C. Hirt, A. Amsden and J. Cook, An arbitrary Lagrangian-Eulerian computing method for all flow speeds,, Journal of Computational Physics, 135 (1997), 203. doi: 10.1006/jcph.1997.5702. Google Scholar [22] B. Höhn, Numerik für die Marangoni-Konvektion beim Floating-Zone Verfahren,, Dissertation, (1999). Google Scholar [23] T. J. R. Hughes, W. Liu and T. K. Zimmermann, Lagrangian-Eulerian finite element formulation for incompressible viscous flows,, Computer Methods in Applied Mechanics and Engineering, 29 (1981), 329. doi: 10.1016/0045-7825(81)90049-9. Google Scholar [24] D. Jacqmin, Calculation of two-phase Navier-Stokes flows using phase-field modeling,, J. Comp. Phys., 155 (1999), 96. doi: 10.1006/jcph.1999.6332. Google Scholar [25] D. Jamet, O. Lebaigue, N. Coutris and J. M. Delhaye, The second gradient method for the direct numerical simulation of liquid-vapor flows with phase change,, J. Comp. Phys, 169 (2001), 624. doi: 10.1006/jcph.2000.6692. Google Scholar [26] E. Kennard, Kinetic theory of gases: with an introduction to statistical mechanics,, International series in pure and applied physics, (1938). Google Scholar [27] R. Klein, N. Botta, T. Schneider, C. D. Munz, S. Roller, A. Meister, L. Hoffmann and T. Sonar, Asymptotic adaptive methods for multi-scale problems in fluid mechanics,, Eng. Math., 39 (2001), 261. doi: 10.1023/A:1004844002437. Google Scholar [28] R. Krahl, M. Adamov, M. Lozano Avilés and E. Bänsch, A model for two phase flow with evaporation,, in Two-Phase Flow Modelling and Experimentation 2004 (eds. G. P. Celata, (2004), 2381. Google Scholar [29] R. Krahl, J. Gerstmann, P. Behruzi, E. Bänsch and M. E. Dreyer, Dependency of the apparent contact angle on nonisothermal conditions,, Physics of Fluids, 20 (2008). doi: 10.1063/1.2899641. Google Scholar [30] R. Krahl and E. Bänsch, Impact of marangoni effects on the apparent contact angle - a numerical investigation,, Microgravity Science and Technology, 17 (2005), 39. doi: 10.1007/BF02872086. Google Scholar [31] R. Krahl and E. Bänsch, On the stability of an evaporating liquid surface,, Fluid Dynamics Research, 44 (2012). doi: 10.1088/0169-5983/44/3/031409. Google Scholar [32] R. Krahl and J. Gerstmann, Non-isothermal reorientation of a liquid surface in an annular gap,, in $4^{th}$ International Berlin Workshop - IBW 4 on Transport Phenomena with Moving Boundaries (ed. F.-P. Schindler), (2007), 227. Google Scholar [33] N. Kulev, S. Basting, E. Bänsch and M. Dreyer, Interface reorientation of cryogenic liquids under non-isothermal boundary conditions,, Cryogenics, 62 (2014), 48. doi: 10.1016/j.cryogenics.2014.04.006. Google Scholar [34] N. Kulev and M. Dreyer, Drop tower experiments on non-isothermal reorientation of cryogenic liquids,, Microgravity Science and Technology, 22 (2010), 463. doi: 10.1007/s12217-010-9237-2. Google Scholar [35] D. Meschede (ed.), Gerthsen Physik,, 22nd edition, (2004). Google Scholar [36] M. Michaelis, Kapillarinduzierte Schwingungen Freier Flüssigkeitsoberflächen,, no. 454 in Fortschritt-Berichte VDI, (2003). Google Scholar [37] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations,, J. Comp. Phys., 79 (1988), 12. doi: 10.1016/0021-9991(88)90002-2. Google Scholar [38] S. Ostrach, Low-gravity fluid flows,, Ann. Rev. Fluid. Mech., 14 (1982), 313. doi: 10.1146/annurev.fl.14.010182.001525. Google Scholar [39] L. M. Pismen and Y. Pomeau, Disjoining potential and spreading of thin liquid layers in the diffuse-interface model coupled to hydrodynamics,, Phys. Rev. E, 62 (2000), 2480. doi: 10.1103/PhysRevE.62.2480. Google Scholar [40] M. Rumpf, A variational approach to optimal meshes,, Numerische Mathematik, 72 (1996), 523. doi: 10.1007/s002110050180. Google Scholar [41] Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,, SIAM Sci. Comp., 7 (1986), 856. doi: 10.1137/0907058. Google Scholar [42] R. Scardovelli and S. Zaleski, Direct numerical simulation of free-surface and interfacial flow,, Ann. Rev. Fluid Mech., 31 (1999), 567. doi: 10.1146/annurev.fluid.31.1.567. Google Scholar [43] J. Schlottke and B. Weigand, Direct numerical simulation of evaporating droplets,, J. Comp. Phys., 227 (2008), 5215. doi: 10.1016/j.jcp.2008.01.042. Google Scholar [44] L. E. Scriven, Dynamics of a fluid interface equation of motion for Newtonian surface fluids,, Chem. Eng. Sci., 12 (1960), 98. doi: 10.1016/0009-2509(60)87003-0. Google Scholar [45] J. A. Sethian and P. Smereka, Level set methods for fluid interfaces,, Ann. Rev. Fluid Mech., 35 (2003), 341. doi: 10.1146/annurev.fluid.35.101101.161105. Google Scholar [46] G. Son and V. K. Dhir, Numerical simulation of film boiling near critical pressures with a level set method,, J. Heat Transfer, 120 (1998), 183. doi: 10.1115/1.2830042. Google Scholar [47] M. Sussman, P. Smereka and S. Osher, A level set approach for computing solutions to incompressible two-phase flow,, Journal of Computational Physics, 114 (1994), 146. doi: 10.1006/jcph.1994.1155. Google Scholar [48] S. Tanguy, T. Ménard and A. Berlemont, A level set method for vaporizing two-phase flows,, J. Comp. Phys., 221 (2007), 837. doi: 10.1016/j.jcp.2006.07.003. Google Scholar [49] M. Tenhaeff, Computation of Incompressible, Axisymmetric Flows in Electrically Conducting Fluids Under Influence of Rotating Magnetic Fields (in German),, Diploma thesis, (1997). Google Scholar [50] T. Tezduyar and R. Benney, Mesh moving techniques for fluid-structure interactions with large displacements,, J. Applied Mechanics, 70 (2003), 58. Google Scholar [51] S. W. J. Welch and J. Wilson, A volume of fluid based method for fluid flows with phase change,, J. Comp. Phys, 160 (2000), 662. doi: 10.1006/jcph.2000.6481. Google Scholar [52] T. Wick, Fluid-structure interactions using different mesh motion techniques,, Computers & Structures, 89 (2011), 1456. doi: 10.1016/j.compstruc.2011.02.019. Google Scholar [53] Y. F. Yap, J. C. Chai, K. C. Toh, T. N. Wong and Y. C. Lam, Numerical modeling of unidirectional stratified flow with and without phase change,, J. Int. Heat Mass Transfer, 48 (2005), 477. doi: 10.1016/j.ijheatmasstransfer.2004.09.017. Google Scholar
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