zbMATH — the first resource for mathematics

PROST: a parabolic reconstruction of surface tension for the volume-of-fluid method. (English) Zbl 1057.76569
Summary: Volume-of-fluid (VOF) methods are popular for the direct numerical simulation of time-dependent viscous incompressible flow of multiple liquids. As in any numerical method, however, it has its weaknesses, namely, for flows in which the capillary force is the dominant physical mechanism. The lack of convergence with spatial refinement, or convergence to a solution that is slightly different from the exact solution, has been documented in the literature. A well-known limiting case for this is the existence of spurious currents for the simulation of a spherical drop with zero initial velocity. These currents are present in all previous versions of VOF algorithms. In this paper, we develop an accurate representation of the body force due to surface tension, which effectively eliminates spurious currents. We call this algorithm PROST: parabolic reconstruction of surface tension. There are several components to this procedure, including the new body force algorithm, improvements in the projection method for the Navier-Stokes solver, and a higher order interface advection scheme. The curvature to the interface is calculated from an optimal fit for a quadratic approximation to the interface over groups of cells.

76M25 Other numerical methods (fluid mechanics) (MSC2010)
76D45 Capillarity (surface tension) for incompressible viscous fluids
76T30 Three or more component flows
Full Text: DOI
[1] Brackbill, J.U.; Kothe, D.B.; Zemach, C., A continuum method for modeling surface tension, J. comput. phys., 100, 335, (1992) · Zbl 0775.76110
[2] Coward, A.V.; Renardy, Y.; Renardy, M.; Richards, J.R., Temporal evolution of periodic disturbances in two-layer Couette flow, J. comput. phys., 132, 346, (1997) · Zbl 0880.76055
[3] Rider, W.J.; Kothe, D.B., Reconstructing volume tracking, J. comput. phys., 141, 112, (1998) · Zbl 0933.76069
[4] Lafaurie, B.; Nardone, C.; Scardovelli, R.; Zaleski, S.; Zanetti, G., Modelling merging and fragmentation in multiphase flows with SURFER, J. comput. phys., 113, 134, (1994) · Zbl 0809.76064
[5] Scardovelli, R.; Zaleski, S., Direct numerical simulation of free surface and interfacial flow, Annu. rev. fluid mech., 31, 567, (1999)
[6] D. B. Kothe, M. W. Williams, and, E. G. Puckett, Accuracy and convergence of continuum surface tension models, in, Fluid Dynamics at Interfaces, edited by, W. Shyy and R. Narayanan, Cambridge Univ. Press, Cambridge, UK, 1998, p, 294.
[7] Renardy, Y.; Renardy, M.; Cristini, V., A new volume-of-fluid formulation for surfactants and simulations of drop deformation under shear at a low viscosity ratio, Eur. J. mech. B/fluids, 21, 49, (2002) · Zbl 1063.76077
[8] Renardy, Y.; Cristini, V.; Li, J., Drop fragment distributions under shear with inertia, Int. J. mult. flow, 28, 1125, (2002) · Zbl 1137.76723
[9] M. Meier, G. Yadigaroglu, and, B. L. Smith, A novel technique for including surface tension in PLIC-VOF methods, Preprint. · Zbl 1064.76084
[10] 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, (2001) · Zbl 1030.76035
[11] Popinet, S.; Zaleski, S., A front-tracking algorithm for the accurate representation of surface tension, Int. J. numer. methods fluids, 30, 775, (1999) · Zbl 0940.76047
[12] Torres, D.; Brackbill, J., The point-set method: front tracking without connectivity, J. comput. phys., 165, 620, (2000) · Zbl 0998.76070
[13] Li, J., Calcul d’interface affine par morceaux (piecewise linear interface calculation), C. R. acad. sci. Paris, 320, 391, (1995) · Zbl 0826.76065
[14] 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
[15] V. Cristini, S. Guido, A. Alfani, J. Blawzdziewicz, and, M. Loewenberg, Drop Breakup in Shear Flow, Preprint.
[16] Zaleski, S.; Li, J.; Succi, S., Two-dimensional Navier-Stokes simulation of deformation and break-up of liquid patches, Phys. rev. lett., 75, 244, (1995)
[17] Zaleski, S., simulation of high Reynolds breakup of liquid-gas interface, Lecture series 1996-2, (1996)
[18] Li, J.; Renardy, Y., Direct simulation of unsteady axisymmetric core-annular flow with high viscosity ratio, J. fluid mech., 391, 123, (1999) · Zbl 0973.76067
[19] Renardy, Y.; Li, J., Comment on ‘A numerical study of periodic disturbances on two-layer Couette flow phys fluids 10 (12), pp. 3056-3071’, Phys. fluids, 11, 3189, (1999) · Zbl 1149.76521
[20] Li, J.; Renardy, Y., Shear-induced rupturing of a viscous drop in a Bingham liquid, J. non-newt. fluid mech., 95, 235, (2000) · Zbl 0994.76005
[21] Li, J.; Renardy, 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
[22] Renardy, Y.; Li, J., Numerical simulation of two-fluid flows of viscous immiscible liquids, Proceedings of the IUTAM symposium on nonlinear waves in multiphase flow, 117, (2000)
[23] Y. Renardy, and, J. Li, Parallelized simulations of two-fluid dispersions, SIAM News, December, p, 1, 2000.
[24] Li, J.; Renardy, Y., Numerical study of flows of two immiscible liquids at low Reynolds number, SIAM rev., 42, 417, (2000) · Zbl 0981.76062
[25] Renardy, M.; Renardy, Y.; Li, J., Numerical simulation of moving contact line problems using a volume-of-fluid method, J. comput. phys., 171, 243, (2001) · Zbl 1044.76051
[26] Renardy, Y.; Li, J., Merging of drops to form bamboo waves, Int. J. mult. flow, 27, 753, (2001) · Zbl 1137.76725
[27] Dennis, J.E.; Schnabel, R.B., numerical methods for unconstrained optimization and nonlinear equations, (1983), Prentice Hall New York · Zbl 0579.65058
[28] Chorin, A.J., A numerical method for solving incompressible viscous flow problems, J. comput. phys., 2, 12, (1967) · Zbl 0149.44802
[29] Li, J.; Renardy, Y., Shear-induced rupturing of a viscous drop in a Bingham liquid, J. non-newt. fluid mech., 95, 235, (2000) · Zbl 0994.76005
[30] Cristini, V.; Blawzdziewicz, J.; Loewenberg, M., Drop breakup in three-dimensional viscous flows, Phys. fluids, 10, 1781, (1998)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.