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Reinitialization of the Level-Set Function in 3d Simulation of Moving Contact Lines

Part of: Two-phase and multiphase flows

Published online by Cambridge University Press:  02 November 2016

Shixin Xu  and
Weiqing Ren
Shixin Xu*
Affiliation:
Department of Mathematics, National University of Singapore, 119076, Singapore
Weiqing Ren*
Affiliation:
Department of Mathematics, National University of Singapore, 119076, Singapore Institute of High Performance Computing, Agency for Science, Technology and Research, 138632, Singapore
*
*Corresponding author. Email addresses:[email protected] (S. Xu), [email protected] (W. Ren)
*Corresponding author. Email addresses:[email protected] (S. Xu), [email protected] (W. Ren)
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Abstract

The level set method is one of the most successful methods for the simulation of multi-phase flows. To keep the level set function close the signed distance function, the level set function is constantly reinitialized by solving a Hamilton-Jacobi type of equation during the simulation. When the fluid interface intersects with a solid wall, a moving contact line forms and the reinitialization of the level set function requires a boundary condition in certain regions on the wall. In this work, we propose to use the dynamic contact angle, which is extended from the contact line, as the boundary condition for the reinitialization of the level set function. The reinitialization equation and the equation for the normal extension of the dynamic contact angle form a coupled system and are solved simultaneously. The extension equation is solved on the wall and it provides the boundary condition for the reinitialization equation; the level set function provides the directions along which the contact angle is extended from the contact line. The coupled system is solved using the 3rd order TVD Runge-Kutta method and the Godunov scheme. The Godunov scheme automatically identifies the regions where the angle condition needs to be imposed. The numerical method is illustrated by examples in three dimensions.

Keywords

Level set methodtwo-phase flowsmoving contact linescontact anglereinitializationlevel set function

MSC classification

Secondary: 65Z05: Applications to physics 76T10: Liquid-gas two-phase flows, bubbly flows
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 5 , November 2016 , pp. 1163 - 1182
Copyright
Copyright © Global-Science Press 2016 

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References

[1] Dussan, E. B., On the spreading of liquids on solid surfaces: Static and dynamic contact lines, Annu. Rev. Fluid Mech. 11 (1979) 371–400.Google Scholar
[2] de Gennes, P. G., Wetting: statics and dynamics, Rev. Mod. Phys. 57 (1985) 827–863.Google Scholar
[3] Blake, T. D., The physics of moving wetting lines, J. Colloid Interface Sci. 299 (2006) 1–13.Google Scholar
[4] Bonn, D., Eggers, J., Indekeu, J., Meunier, J., Rolley, E., Wetting and spreading, Rev. Mod. Phys. 81 (2009) 739–805.Google Scholar
[5] de Gennes, P.-G., Brochard-Wyart, F., Quéré, D., Capillarity and wetting phenomena: drops, bubbles, pearls, waves, Springer, New York (2003).Google Scholar
[6] Starov, V. M., Velarde, M. G., Radke, C. J., Wetting and spreading dynamics, in: Surfactant Science Series. CRC Press, Taylor & Francis Group, Boca Raton (2007).Google Scholar
[7] Shikhmurzaev, Y. D., Capillary flows with forming interfaces, Chapman & Hall/CRC, Boca Raton (2008).Google Scholar
[8] Afkhami, S., Bussmann, M., Height functions for applying contact angles to 3D VOF simulations, Int. J. Numer. Methods Fluids 61 (2008) 827–847.Google Scholar
[9] Afkhami, S., Zaleski, S., Bussmann, M., A mesh-dependent model for applying dynamic contact angles to VOF simulations, J. Comput. Phys. 228 (2009) 5370–5389.Google Scholar
[10] Renardy, M., Renardy, Y., Li, J., Numerical simulation of moving contact line problems using a volume-of-fluid method, J. Comput. Phys. 171 (2001) 243–263.Google Scholar
[11] Huang, H., Liang, D., Wetton, B., Computation of a moving drop/bubble on a solid surface using a front-tracking method, Commun. Math. Sci. 2 (2004) 535–552.Google Scholar
[12] Lai, M.-C., Tseng, Y.-H., Huang, H., Numerical simulation of moving contact lines with surfactant by immersed boundary method, Commun. Comput. Phys. 8 (2010) 735–757.Google Scholar
[13] Qian, T., Wang, X.-P., Sheng, P., Molecular scale contact line hydrodynamics of immiscible flows, Phys. Rev. E 68 (2003) 016306.Google Scholar
[14] Qian, T., Wang, X.-P., Sheng, P., Molecular hydrodynamics of the moving contact line in two-phase immiscible flows, Commun. Comput. Phys. 1 (2006) 1–52.Google Scholar
[15] Yue, P., Zhou, C., Feng, J. J., Sharp-interface limit of the Cahn-Hilliard model for moving contact lines, J. Fluid Mech. 645 (2010) 279–294.Google Scholar
[16] Gao, M., Wang, X.-P., An efficient scheme for a phase field model for the moving contact line problem with variable density and viscosity, J. Comput. Phys. 272 (2014) 704–718.Google Scholar
[17] Spelt, P. D. M., A level-set approach for simulations of flows with multiple moving contact lines with hysteresis, J. Comput. Phys. 207 (2005) 389–404.Google Scholar
[18] Xu, J.-J., Ren, W., A level-set method for two-phase flows with moving contact line and insoluble surfactant, J. Comput. Phys. 263 (2014) 71–90.Google Scholar
[19] Yokoi, K., Vadillo, D., Hinch, J., Hutchings, I., Numerical studies of the influence of the dynamic contact angle on a droplet impacting on a dry surface, Phys. Fluids 21 (2009) 072102.Google Scholar
[20] Li, Z., Lai, M.-C., He, G., Zhao, H., An augmented method for free boundary problems with moving contact lines, Comput. Fluids 39 (2010) 1033–1040.Google Scholar
[21] Sui, Y., Ding, H., Spelt, P. D. M., Numerical simulations of flows with moving contact lines, Annu. Rev. Fluid Mech. 46 (2014) 97–119.Google Scholar
[22] Osher, S., Fedkiw, R., Level set methods and dynamic implicit surfaces, Applied mathematical science, Springer, New York (2003).Google Scholar
[23] Della Rocca, G., Blanquart, G., Level set reinitialization at a contact line, J. Comput. Phys. 265 (2014) 34–49.Google Scholar
[24] Sussman, M., Uto, S., A computational study of the spreading of oil underneath a sheet of ice, CAM Report, 98–32 (1998).Google Scholar
[25] Sussman, M., An adaptive mesh algorithm for free surface flows in general geometries, In: Vande Wouwer, A., Saucez, Ph., Schiesser, W.E., Adaptive Method of lines. Chapman & Hall/CRC Press (2001) 207–231.Google Scholar
[26] Griebel, M., Klitz, M., Simulation of droplet impact with dynamic contact angle boundary condition, Singular Phenomena and Scaling in Mathematical Models, Springer, Switzerland, 2014.Google Scholar
[27] Xu, J.-J., Li, Z., Lowengrub, J., Zhao, H., A level set method for solving interfacial flows with surfactant, J. Comput. Phys. 212 (2006) 590–616.Google Scholar
[28] Xu, J.-J., Zhao, H., An Eulerian formulation for solving partial differential equations along a moving interface, J. Sci. Comput. 19 (2003) 573–594.Google Scholar
[29] Zhao, H., Osher, S., Fedkiw, R., A variational level set approach to multiphase motion, J. Comput. Phys. 127 (1996) 179–195.Google Scholar
[30] Hartmann, D., Meinke, M., Schröder, W., The constrained reinitialization equation for level set methods, J. Comput. Phys. 229 (2010) 1514–1535.Google Scholar
[31] Jiang, G.-S., Peng, D. P., Weighted ENO schemes for Hamilton-Jacobi equations, SIAM J. Sci. Comput. 21 (2000) 2126–2143.Google Scholar
[32] Adalsteinsson, D., Sethian, J. A., A fast level set method for propagating interfaces, J. Comput. Phys. 118 (1995) 269–277.Google Scholar
[33] Peng, D. P., Merriman, B., Osher, S., Zhao, H., Kang, M., A pde-based fast local level set method, J. Comput. Phys. 155 (1999) 410–438.Google Scholar
[34] Shu, C.-W., Total-variation-diminishing time discretizations, SIAM J. Sci. and Stat. Comput. 9 (1988) 1073–1084.Google Scholar
[35] Ren, W., W. E, , Boundary conditions for the moving contact line problem, Phys. Fluids 19 (2007) 022101.Google Scholar
[36] Ren, W., Hu, D., W. E, , Continuum models for the contact line problem, Phys. Fluids 22 (2010) 102103.Google Scholar
[37] Ren, W., W. E, , Derivation of continuum models for the moving contact line problem based on thermodynamic principles, Commun. Math. Sci. 9 (2011) 597–606.Google Scholar
[38] Ren, W., W. E, , Contact line dynamics on heterogeneous surfaces, Phys. Fluids 23 (2011) 072103.Google Scholar
[39] Guermond, J.-L., Salgado, A., A splitting method for incompressible flows with variable density based on a pressure Poisson equation, J. Comput. Phys. 228 (2009) 2834–2846.Google Scholar
[40] Shen, J., Yang, X., A phase-field model and its numerical approximation for two-phase incompressible flows with different densities and viscosities, SIAM, J. Sci. Comput. 32 (3) (2010) 1159–1179.Google Scholar
[41] Xiang, Y., Cheng, L.-T., Srolovitz, D. J., W. E, , A level set method for dislocation dynamics, Acta Mater. 51 (2003) 5499–5518.Google Scholar
[42] Zhu, A. Y., Jin, C. M., Zhao, D. G., Xiang, Y., Huang, J. F., A numerical scheme for generalized Peierls-Nabarro model of dislocations based on the fast multipole method and iterative grid redistribution, Commun. Comput. Phys. 12 (2012) 226–246.Google Scholar