An adaptive coupled level-set/volume-of-fluid interface capturing method for unstructured triangular grids. (English) Zbl 1160.76377
Summary: We present an adaptive coupled level-set/volume-of-fluid (ACLSVOF) method for interfacial flow simulations on unstructured triangular grids. At each time step, we evolve both the level set function and the volume fraction. The level set function is evolved by solving the level set advection equation using a discontinuous Galerkin finite element method. The volume fraction advection is performed using a Lagrangian-Eulerian method. The interface is reconstructed based on both the level set and the volume fraction information. In particular, the interface normal vector is calculated from the level set function while the line constant is determined by enforcing mass conservation based on the volume fraction. Different from previous works, we have developed an analytic method for finding the line constant on triangular grids, which makes interface reconstruction efficient and conserves volume of fluid exactly. The level set function is finally reinitialized to the signed distance to the reconstructed interface. Since the level set function is continuous, the normal vector calculation is easy and accurate compared to a classic volume-of-fluid method, while tracking the volume fraction is essential for enforcing mass conservation. The method is also coupled to a finite element based Stokes flow solver. The code validation shows that our method is second order and mass is conserved very accurately. In addition, owing to the adaptive grid algorithm we can resolve complex interface changes and interfaces of high curvature efficiently and accurately.
MSC:
| 76M12 | Finite volume methods applied to problems in fluid mechanics |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
Software:
PROSTReferences:
| [1] | Harlow, F. H.; Welch, J. E., Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids, 8, 12, 2182-2189 (1965) · Zbl 1180.76043 |
| [2] | Osher, S. J.; Sethian, J. A., Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79, 12-49 (1988) · Zbl 0659.65132 |
| [3] | Sethian, J. A.; Smereka, P., Level set methods for fluid interfaces, Annu. Rev. Fluid Mech., 35, 341-372 (2003) · Zbl 1041.76057 |
| [4] | Osher, S.; Fedkiw, R. P., Level set methods: an overview and some recent results, J. Comput. Phys., 169, 463-502 (2001) · Zbl 0988.65093 |
| [5] | Hirt, C. W.; Nichols, B. D., Volume of fluid (VOF) method for the dynamics of free boundaries, J. Comput. Phys., 39, 201-225 (1981) · Zbl 0462.76020 |
| [6] | Rider, W. J.; Kothe, D. B., Reconstructing volume tracking, J. Comput. Phys., 141, 112-152 (1998) · Zbl 0933.76069 |
| [7] | Scardovelli, R.; Zaleski, S., Direct numerical simulation of free-surface and interfacial flow, Annu. Rev. Fluid Mech., 31, 567-603 (1999) |
| [8] | Unverdi, S. O.; Tryggvason, G., A front-tracking method for viscous, incompressible, multi-fluid flows, J. Comput. Phys., 100, 25-37 (1992) · Zbl 0758.76047 |
| [9] | Tryggvason, G.; Bunner, B.; Esmaeeli, A.; Juric, D.; Al-Rawahi, N.; Tauber, W.; Han, J.; Nas, S.; Jan, Y.-J., A front-tracking method for the computations of multiphase flow, J. Comput. Phys., 169, 708-759 (2001) · Zbl 1047.76574 |
| [10] | Badalassi, V. E.; Ceniceros, H. D.; Banerjee, S., Computation of multiphase systems with phase field models, J. Comput. Phys., 190, 371-397 (2003) · Zbl 1076.76517 |
| [11] | Kim, J. S.; Kang, K.; Lowengrub, J., Conservative multigrid methods for Cahn-Hilliard fluids, J. Comput. Phys., 193, 511-543 (2004) · Zbl 1109.76348 |
| [12] | Jacqmin, D., Calculation of two-phase Navier-Stokes flows using phase-field modeling, J. Comput. Phys., 155, 96-127 (1999) · Zbl 0966.76060 |
| [13] | Anderson, D. M.; McFadden, G. B.; Wheeler, A. A., Diffuse-interface methods in fluid mechanics, Annu. Rev. Fluid Mech., 30, 139-165 (1998) · Zbl 1398.76051 |
| [14] | Keestra, B. J.; Puyvelde, P. C.J. V.; Anderson, P. D.; Meijer, H. E.H., Diffuse interface modeling of the morphology and rheology of immiscible polymer blends, Phys. Fluids, 15, 9, 2567-2575 (2003) · Zbl 1186.76273 |
| [15] | W.J. Rider, D.B. Kothe, Stretching and tearing interface tracking methods, Tech. Rep. AIAA 95-1717, AIAA, the 12th AIAA CFD Conference, San Diego, 1995.; W.J. Rider, D.B. Kothe, Stretching and tearing interface tracking methods, Tech. Rep. AIAA 95-1717, AIAA, the 12th AIAA CFD Conference, San Diego, 1995. |
| [16] | Sussman, M.; Smereka, P.; Osher, S., A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., 114, 146-159 (1994) · Zbl 0808.76077 |
| [17] | Sussman, M.; Fatemi, E.; Smereka, P.; Osher, S., An improved level set method for incompressible two-phase flows, Comput. Fluids, 27, 663-680 (1998) · Zbl 0967.76078 |
| [18] | Sussman, M.; Fatemi, E., An efficient, interface-preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow, SIAM J. Sci. Comput., 20, 4, 1165-1191 (1999) · Zbl 0958.76070 |
| [19] | S.J. Mosso, B.K. Swartz, D.B. Kothe, R.C. Ferrell, A parallel, volume-tracking algorithm for unstructured meshes, in: Parallel Computational Fluid Dynamics’96, Italy, 1996.; S.J. Mosso, B.K. Swartz, D.B. Kothe, R.C. Ferrell, A parallel, volume-tracking algorithm for unstructured meshes, in: Parallel Computational Fluid Dynamics’96, Italy, 1996. |
| [20] | G.R. Price, G.T. Reader, R.D. Rowe, J.D. Bugg, A piecewise parabolic interface calculation for volume tracking, in: Proceedings of the Sixth Annual Conference of the Computational Fluid Dynamics Society of Canada, vol. VIII, 1998, pp. 71-77.; G.R. Price, G.T. Reader, R.D. Rowe, J.D. Bugg, A piecewise parabolic interface calculation for volume tracking, in: Proceedings of the Sixth Annual Conference of the Computational Fluid Dynamics Society of Canada, vol. VIII, 1998, pp. 71-77. |
| [21] | Renardy, Y.; Renardy, M., PROST: a parabolic reconstruction of surface tension for the volume-of-fluid method, J. Comput. Phys., 183, 400-421 (2002) · Zbl 1057.76569 |
| [22] | Ginzburg, I.; Wittum, G., Two-phase flows on interface refined grids modeled with VOF, staggered finite volumes, and spline interpolants, J. Comput. Phys., 166, 302-335 (2001) · Zbl 1030.76035 |
| [23] | Scardovelli, R.; Zaleski, S., Interface reconstruction with least-square fit and split Eulerian-Lagrangian advection, Int. J. Numer. Methods Fluids, 41, 251-274 (2003) · Zbl 1047.76080 |
| [24] | Shahbazi, K.; Paraschivoiu, M.; Mostaghimi, J., Second order accurate volume tracking based on remapping for triangular meshes, J. Comput. Phys., 188, 100-122 (2003) · Zbl 1127.76337 |
| [25] | Ashgriz, N.; Barbat, T.; Wang, G., A computational Lagrangian-Eulerian advection remap for free surface flows, Int. J. Numer. Methods Fluids, 44, 1-32 (2004) · Zbl 1062.76041 |
| [26] | D.B. Kothe, W.J. Rider, S.J. Mosso, J.S. Brock, Volume tracking of interfaces having surface tension in two and three dimensions, Tech. Rep. AIAA 96-0859, AIAA, the 34th Aerospace sciences meeting and exhibit, 1996.; D.B. Kothe, W.J. Rider, S.J. Mosso, J.S. Brock, Volume tracking of interfaces having surface tension in two and three dimensions, Tech. Rep. AIAA 96-0859, AIAA, the 34th Aerospace sciences meeting and exhibit, 1996. |
| [27] | M.W. Williams, D.B. Kothe, E.G. Puckett, Convergence and accuracy of kernel-based continuum surface tension models, in: Proceedings of the Thirteenth US National Congress of Applied Mechanics, Gainesville, FL, 1998.; M.W. Williams, D.B. Kothe, E.G. Puckett, Convergence and accuracy of kernel-based continuum surface tension models, in: Proceedings of the Thirteenth US National Congress of Applied Mechanics, Gainesville, FL, 1998. · Zbl 0979.76014 |
| [28] | Sussman, M.; Puckett, E. G., A coupled level set and volume-of-fluid method for computing 3d and axisymmetric incompressible two-phase flows, J. Comput. Phys., 162, 301-337 (2000) · Zbl 0977.76071 |
| [29] | Sussman, M., A second order coupled level set and volume-of-fluid method for computing growth and collapse of vapor bubbles, J. Comput. Phys., 187, 110-136 (2003) · Zbl 1047.76085 |
| [30] | Enright, D.; Fedkiw, R.; Ferziger, J.; Mitchell, I., A hybrid particle level set methods for improved interface capturing, J. Comput. Phys., 183, 83-116 (2002) · Zbl 1021.76044 |
| [31] | Aulisa, E.; Manservisi, S.; Scardovelli, R., A mixed markers and volume-of-fluid method for the reconstruction and advection of interfaces in two-phase and free-boundary flows, J. Comput. Phys., 188, 611-639 (2003) · Zbl 1127.76346 |
| [32] | Greaves, D., A quadtree adaptive method for simulating fluid flows with moving interfaces, J. Comput. Phys., 194, 35-56 (2004) · Zbl 1136.76408 |
| [33] | Sussman, M., A parallelized, adaptive algorithm for multiphase flows in general geometries, J. Comput. Struct., 83, 435-444 (2005) |
| [34] | M. Sussman, M.Y. Hussaini, K. Smith, R.-Z. Wei, A second order adaptive sharp interface method for incompressible multiphase flow, in: Proceedings of the Third International Conference on Computational Fluid Dynamics, Toronto, Canada, 2004.; M. Sussman, M.Y. Hussaini, K. Smith, R.-Z. Wei, A second order adaptive sharp interface method for incompressible multiphase flow, in: Proceedings of the Third International Conference on Computational Fluid Dynamics, Toronto, Canada, 2004. |
| [35] | Anderson, A.; Zheng, X.; Cristini, V., Adaptive unstructured volume remeshing-I: the method, J. Comput. Phys., 208, 616-625 (2005) · Zbl 1075.65119 |
| [36] | Zheng, X.; Lowengrub, J.; Anderson, A.; Cristini, V., Adaptive unstructured volume remeshing-II: application to two- and three-dimensional level-set simulations of multiphase flow, J. Comput. Phys., 208, 626-650 (2005) · Zbl 1075.65120 |
| [37] | Cristini, V.; Blawzdziewicz, J.; Loewenberg, M., An adaptive mesh algorithm for evolving surfaces: simulations of drop breakup and coalescence of interfaces in two-phase and free-boundary flows, J. Comput. Phys., 168, 445-463 (2001) · Zbl 1153.76382 |
| [38] | X. Yang, A.J. James, Analytic relations for reconstructing piecewise linear interfaces in triangular and tetrahedral grids, J. Comput. Phys., in press, doi:10.1016/j.jcp.2005.09.002; X. Yang, A.J. James, Analytic relations for reconstructing piecewise linear interfaces in triangular and tetrahedral grids, J. Comput. Phys., in press, doi:10.1016/j.jcp.2005.09.002 · Zbl 1137.76456 |
| [39] | Cockburn, B.; Hou, S.; Shu, C.-W., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Math. Comput., 54, 545-581 (1990) · Zbl 0695.65066 |
| [40] | Cockburn, B.; Shu, C.-W., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16, 173-261 (2001) · Zbl 1065.76135 |
| [41] | Gelder, A. V., Efficient computation of polygon area and polyhedron volume, (Paeth, A. W., Graphics Gems V (1995), Academic Press), 35-41 |
| [42] | Sutherland, I. E.; Hodgeman, G. W., Reentrant polygon clipping, Communications of the ACM, 17, 1, 32-42 (1974) · Zbl 0271.68065 |
| [43] | K. Weiler, P. Atherton, Hidden surface removal using polygon area sorting, in: SIGGRAPH, 1977, pp. 214-222.; K. Weiler, P. Atherton, Hidden surface removal using polygon area sorting, in: SIGGRAPH, 1977, pp. 214-222. |
| [44] | Arnold, D. N.; Brezzi, F.; Fortin, M., A stable finite element for the Stokes equations, Calcolo, 21, 337-344 (1984) · Zbl 0593.76039 |
| [45] | Elman, H. C.; Golub, G. H., Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal., 31, 6, 1645-1661 (1994) · Zbl 0815.65041 |
| [46] | Pilliod, J. E.; Puckett, E. G., Second-order accurate volume-of-fluid algorithms for tracking material interfaces, J. Comput. Phys., 199, 2, 465-502 (2004) · Zbl 1126.76347 |
| [47] | Zalesak, S. T., Fully multidimensional flux-corrected transport algorithms for fluids, J. Comput. Phys., 31, 335-362 (1979) · Zbl 0416.76002 |
| [48] | López, J.; Hernández, J.; Gómez, P.; Faura, F., A volume of fluid method based on multidimensional advection and spline interface reconstruction, J. Comput. Phys., 195, 718-742 (2004) · Zbl 1115.76358 |
| [49] | Rudman, M., Volume-tracking methods for interfacial flow calculations, Int. J. Numer. Methods Fluids, 24, 671-691 (1997) · Zbl 0889.76069 |
| [50] | Bell, J. B.; Colella, P.; Glaz, H. M., A second order projection method of the incompressible Navier-Stokes equations, J. Comput. Phys., 85, 257-283 (1989) · Zbl 0681.76030 |
| [51] | Leveque, R. J., High-resolution conservative algorithms for advection in incompressible flow, SIAM J. Numer. Anal., 33, 627-665 (1996) · Zbl 0852.76057 |
| [52] | Aulisa, E.; Manservisi, S.; Scardovelli, R., A surface marker algorithm coupled to an area-preserving marker redistribution method for three-dimensional interface tracking, J. Comput. Phys., 197, 555-584 (2004) · Zbl 1079.76605 |
| [53] | Taylor, G. I., The formation of emulsions in definable fields of flow, Proc. R. Soc. Lond. Ser. A, 146, 501-523 (1934) |
| [54] | Bentley, B. J.; Leal, L. G., An experimental investigation of drop deformation and breakup in steady, two-dimensional linear flow, J. Fluid Mech., 167, 241-283 (1986) · Zbl 0611.76117 |
| [55] | Hu, Y. T.; Pine, D. J.; Leal, L. G., Drop deformation, breakup, and coalescence with compatibilizer, Phys. Fluids, 12, 3, 484-489 (2000) · Zbl 1149.76413 |
| [56] | Siegel, M., Cusp formation for time-evolving bubbles in two-dimensional Stokes flow, J. Fluid Mech., 412, 227-257 (2000) · Zbl 0967.76028 |
| [57] | Stone, H. A., Dynamics of drop deformation and breakup in viscous fluids, Annu. Rev. Fluid Mech., 26, 65-102 (1994) · Zbl 0802.76020 |
| [58] | James, A. J.; Lowengrub, J., A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant, J. Comput. Phys., 201, 685-722 (2004) · Zbl 1061.76062 |
| [59] | X. Yang, A.J. James, An arbitrary Lagrangian-Eulerian method for interfacial flows with insoluble surfactants, in preparation.; X. Yang, A.J. James, An arbitrary Lagrangian-Eulerian method for interfacial flows with insoluble surfactants, in preparation. · Zbl 1153.76401 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
