Adaptive unstructured volume remeshing. II: Application to two- and three-dimensional level-set simulations of multiphase flow. (English) Zbl 1075.65120
Summary: In Part I [ibid. 208, No. 2, 616–625 (2005; reviewed above)], we presented an adaptive remeshing algorithm that automatically adjusts the size of the elements of meshes of unstructured triangles (2D) and unstructured tetrahedra (3D) with time and position in the computational domain in order to efficiently resolve the relevant physical scales. Here, we illustrate the performance of an implementation of the algorithm in finite-element/level-set simulations of deformable droplet and fluid-fluid interface interactions, breakup and coalescence in multiphase flows. The wide range of length scales characterizing the dynamics are accurately resolved as demonstrated by comparison to experiments and to theoretical and sharp-interface (boundary-integral) numerical results. The computational cost is found to be competitive even with respect to boundary-integral methods. For the first time using an interface-capturing (level-set) method we successfully simulate the inertia driven impact and rebound of a liquid droplet from a liquid interface and find agreement with recent experimental results.
MSC:
| 65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 76T30 | Three or more component flows |
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 76M15 | Boundary element methods applied to problems in fluid mechanics |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
Keywords:
adaptive remeshing algorithm; unstructured triangles; unstructured tetrahedra; finite-element/level-set; fluid-fluid interface interactions; multiphase flows; numerical results; boundary-integral methodsCitations:
Zbl 1075.65113Software:
PROSTReferences:
| [1] | A. Anderson, X. Zheng, V. Cristini, Adaptive unstructured volume remeshing - I: The method, J. Comput. Phys., in press, doi:10.1016/j.jcp.2005.02.023; A. Anderson, X. Zheng, V. Cristini, Adaptive unstructured volume remeshing - I: The method, J. Comput. Phys., in press, doi:10.1016/j.jcp.2005.02.023 · Zbl 1075.65119 |
| [2] | Arnold, D. N.; Brezzi, F.; Fortin, M., A stable finite-element method for the stokes equations, Calcolo, 21, 337 (1984) · Zbl 0593.76039 |
| [3] | Axelsson, O., Iterative solution methods (1994), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0795.65014 |
| [4] | D. Calhoun, P. Smereka, The numerical approximation of a delta function, preprint, 2004; D. Calhoun, P. Smereka, The numerical approximation of a delta function, preprint, 2004 |
| [5] | Carey, G. F.; Oden, J. T., Finite elements: computational aspects, III (1984), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0558.73064 |
| [6] | Chang, Y. C.; Hou, T. Y.; Merriman, B.; Osher, S., A level set formulation of Eulerian interface capturing methods for incompressible fluid flows, J. Comput. Phys., 124, 449 (1996) · Zbl 0847.76048 |
| [7] | Chopp, D., Computing minimal surfaces via level-set curvature flow, J. Comput. Phys., 101, 77 (1993) · Zbl 0786.65015 |
| [8] | Cockburn, B.; Shu, C.-W., The Runge-Kutta local projection discontinous Galerkin finite element method for conservation laws. iv: The multidimensional case, Math. Comput., 54, 545 (1990) · Zbl 0695.65066 |
| [9] | Cockburn, B.; Shu, C.-W., Runge-Kutta discontinous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16, 173 (2001) · Zbl 1065.76135 |
| [10] | Cristini, V.; Blawzdziewicz, J.; Loewenberg, M., Drop breakup in three-dimensional viscous flows, Phys. Fluids, 10, 1781 (1998) |
| [11] | Cristini, V.; Blawzdziewicz, J.; Loewenberg, M., An adaptive mesh algorithm for evolving surfaces: simulations of drop breakup and coalescence, J. Comput. Phys., 168, 445 (2001) · Zbl 1153.76382 |
| [12] | Cristini, V.; Blawzdziewicz, J.; Loewenberg, M.; Collins, L., Breakup in stochastic stokes flows: sub-Kolmogorov drops in isotropic turbulence, J. Fluid Mech., 492, 231 (2003) · Zbl 1063.76567 |
| [13] | Cristini, V.; Guido, S.; Alfani, A.; Blawzdziewicz, J.; Loewenberg, M., Drop breakup and fragment size distribution in shear flow, J. Rheol., 47, 1283 (2003) |
| [14] | Cristini, V.; Macosko, C. W.; Jansseune, T., A note on transient stress calculation via numerical simulations, J. Non-Newtonian Fluid Mech., 105, 177 (2002) · Zbl 1008.76512 |
| [15] | Cristini, V.; Tan, Y.-C., Theory and numerical simulation of droplet dynamics in complex flows - a review, Lab Chip, 4, 257 (2004) |
| [16] | Davis, R. H.; Schonberg, J. A.; Rallison, J. M., The lubrication force between two viscous drops, Phys. Fluids A, 1, 77 (1989) · Zbl 0656.76085 |
| [17] | Cortez, R.; Brown, D. L.; Minion, M. L., Accurate projection methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 168, 464 (2001) · Zbl 1153.76339 |
| [18] | Fortin, M.; Brezzi, F., Mixed and hybrid finite elements (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0788.73002 |
| [19] | Guermond, J.-L.; Quartapelle, L., On the approximation of the unsteady Navier-Stokes equations by finite element projection methods, Numer. Math., 80, 207 (1998) · Zbl 0914.76051 |
| [20] | Gueyffier, D.; Li, J.; Nadim, A.; Scardovelli, R.; Zaleski, S., Volume-of-fluid interface tracking with smoothed surface stress methods for three-dimensional flows, J. Comput. Phys., 152, 423 (1999) · Zbl 0954.76063 |
| [21] | Hackbusch, W., Iterative solution of large sparse systems of equations (1994), Springer-Verlag: Springer-Verlag New York · Zbl 0789.65017 |
| [22] | Anderson, P. D.; Bazhlekov, I. B.; Meijer, H. E.H., Nonsingular boundary integral method for deformable drops in viscous flows, Phys. Fluids, 16, 1064 (2004) · Zbl 1186.76044 |
| [23] | Johnson, C., Numerical solution of partial differential equations by the finite element method (1987), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0628.65098 |
| [24] | Khismatullin, D.; Renardy, Y.; Cristini, V., Inertia-induced breakup of very viscous drops subjected to simple shear, Phys. Fluids, 15, 1351 (2003) · Zbl 1186.76278 |
| [25] | Li, J.; Renardy, Y. Y.; Renardy, M., Numerical simulation of breakup of a viscous drop in simple shear flow through a volume-of-fluid method, Phys. Fluids, 12, 269 (2000) · Zbl 1149.76454 |
| [26] | Smereka, P.; Sussman, M.; Osher, S., A levelset approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., 114, 146 (1994) · Zbl 0808.76077 |
| [27] | Mohamed-Kassim, Z.; Longmire, E. K., Drop impact on a liquid-liquid interface, Phys. Fluids, 15, 3263 (2003) · Zbl 1186.76383 |
| [28] | Osher, S.; Sethian, J., Fronts propagating with curvature-dependent speed: algorithm based on Hamilton-Jacobi formulations, J. Comput. Phys., 79, 12 (1988) · Zbl 0659.65132 |
| [29] | Patel, P. D.; Shaqfeh, E. S.G.; Butler, J. E.; Cristini, V.; Blawzdziewicz, J.; Loewenberg, M., Drop breakup in the flow through fixed fiber beds: an experimental and computational investigation, Phys. Fluids, 15, 1146 (2003) · Zbl 1186.76417 |
| [30] | Peng, D. P.; Merriman, B.; Zhao, H.-K.; Osher, S., A PDE-based fast local level-set method, J. Comput. Phys., 155, 410 (1999) · Zbl 0964.76069 |
| [31] | Renardy, Y.; Cristini, V.; Li, J., Drop fragment distributions under shear with inertia, Int. J. Multiphase Flow, 28, 1125 (2002) · Zbl 1137.76723 |
| [32] | Renardy, Y.; Renardy, M., Prost: a parabolic reconstruction of surface tension for the volume-of-fluid method, J. Comput. Phys, 183, 400 (2002) · Zbl 1057.76569 |
| [33] | Renardy, Y. Y.; Cristini, V., Effect of inertia on drop breakup under shear, Phys. Fluids, 13, 7 (2001) · Zbl 1184.76451 |
| [34] | Renardy, Y. Y.; Cristini, V., Scalings for fragments produced from drop breakup in shear flow with inertia, Phys. Fluids, 13, 2161 (2001) · Zbl 1184.76452 |
| [35] | Shen, J., On error estimates of the projection methods for the Navier-Stokes equations: second-order schemes, Math. Comput., 65, 215, 1039-1065 (1996) · Zbl 0855.76049 |
| [36] | Sussman, M.; Fatemi, E., An efficient, interface preserving level set re-distancing algorithm and its application to interfacial incompressible fluid flow, SIAM J. Sci. Comput., 20, 1165 (1999) · Zbl 0958.76070 |
| [37] | 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., 30, 301 (2000) · Zbl 0977.76071 |
| [38] | Sussman, M.; Smereka, P.; Osher, S., A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., 114, 146 (1994) · Zbl 0808.76077 |
| [39] | Sethian, J. A.; Barth, T. J., Numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains, J. Comput. Phys., 145, 1 (1998) · Zbl 0911.65091 |
| [40] | A.-K. Tornberg, B. Engquist. Numerical approximations of singular source terms in differential equations. J. Comput. Phys., in press (online edition available); A.-K. Tornberg, B. Engquist. Numerical approximations of singular source terms in differential equations. J. Comput. Phys., in press (online edition available) · Zbl 1115.76392 |
| [41] | 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 (2001) · Zbl 1047.76574 |
| [42] | Li, X. F.; Pozrikidis, C., Simple shear flow of suspensions of liquid drops, J. Fluid Mech., 320, 395 (1996) · Zbl 0883.76085 |
| [43] | X. Yang, A.J. James, J. Lowengrub, X. Zheng, V. Cristini. An adaptive coupled level-set/volume-of-fluid interface tracking method for unstructured triangular grids, in review; X. Yang, A.J. James, J. Lowengrub, X. Zheng, V. Cristini. An adaptive coupled level-set/volume-of-fluid interface tracking method for unstructured triangular grids, in review · Zbl 1160.76377 |
| [44] | X. Zheng. Adaptive algorithms for interface capturing methods, Ph.D. Thesis, University of California, Irvine (2005 expected); X. Zheng. Adaptive algorithms for interface capturing methods, Ph.D. Thesis, University of California, Irvine (2005 expected) |
| [45] | Zinchenko, A. Z.; Rother, M. A.; Davis, R. H., A novel boundary integral algorithm for viscous interaction of deformable drops, Phys. Fluids, 9, 1493 (1997) |
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.
