zbMATH — the first resource for mathematics

A balanced-force algorithm for two-phase flows. (English) Zbl 1349.76510
Summary: Numerical methods for imposing body forces in two-phase flow simulations are discussed. Numerical schemes are presented to avoid the inaccurate solutions that result from inconsistent implementation of forces. First, the momentum equations are discretized so that they accurately accommodate the discontinuity in fluid properties at an interface. Consistent numerical estimations for different body forces such as interfacial (including Marangoni), gravity and electromagnetic forces are discussed. Then, it is shown that the standard pressure-velocity coupling scheme for collocated algorithms is not sufficient for multiphase flows, and therefore a new pressure-velocity coupling is devised and tested for both single and two-phase flows. Finally, to advect the level set function, a cost effective fifth-order WENO method is developed. These formulations are accurate and efficient both for uniform and non-uniform meshes. Several test cases are presented and compared with a standard implementation of body forces to demonstrate the efficiency of the proposed algorithm.

76M20 Finite difference methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76T99 Multiphase and multicomponent flows
Full Text: DOI
[1] Armfield, S. W., Finite difference solutions of the Navier-Stokes equations on staggered and non-staggered grids, Comput. Fluids, 20, 1, 1-17, (1991) · Zbl 0731.76044
[2] Ashrafizadeh, A.; Rezvani, M.; Bakhtiari, B., Pressure-velocity coupling on co-located grids using the method of proper closure equations, Numer. Heat Transf., Part B, Fundam., 56, 3, 259-273, (2009)
[3] Chapra, S. C.; Canale, R. P., Numerical methods for engineers, (2010), McGrawHill
[4] Dalal, A.; Eswaran, V.; Biswas, G., A finite-volume method for Navier-Stokes equations on unstructured meshes, Numer. Heat Transf., Part B, Fundam., 54, 238-259, (2008)
[5] Darbandi, M.; Bostandoost, S. M., A new formulation toward unifying the velocity role in collocated variable arrangement, Numer. Heat Transf., Part B, Fundam., 47, 361-382, (2005)
[6] Darbandi, M.; Schneider, G. E., Momentum variable procedure for solving compressible and incompressible flows, AIAA J., 35, 1801-1805, (1997) · Zbl 0908.76050
[7] Darwish, M.; Sraj, I.; Moukalled, F., A coupled finite volume solver for the solution of incompressible flows on unstructured grids, J. Comput. Phys., 228, 180-201, (2009) · Zbl 1277.76051
[8] Das, K. S.; Ward, C. A., Surface thermal capacity and its effects on the boundary conditions at fluid-fluid interfaces, Phys. Rev. E, 75, 065303, (2007), (R)
[9] Date, A. W., Complete pressure correction algorithm for solution of incompressible Navier-Stokes equations on a non-staggered grid, Numer. Heat Transf., Part B, Fundam., 29, 441-458, (1996)
[10] Deng, G. B.; Piquet, J.; Queutey, P.; Visonneau, M., Incompressible flow calculations with a consistent physical interpolation finite volume approach, Comput. Fluids, 23, 1029-1047, (1994) · Zbl 0816.76066
[11] Ferziger, J. H.; Peric, M., Computational methods for fluid dynamics, (1999), Springer-Verlag Berlin, Heildelberg · Zbl 0943.76001
[12] Francois, M. M.; Cummins, S. J.; Dendy, E. D.; Kothe, D. B.; Sicilian, J. M.; Williams, M. W., A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework, J. Comput. Phys., 213, 141-173, (2006) · Zbl 1137.76465
[13] Gerolymos, G. A.; Sénéchal, D.; Vallet, I., Very-high-order WENO schemes, J. Comput. Phys., 228, 23, 8481-8524, (2009) · Zbl 1176.65088
[14] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S., Uniformly high order essentially non-oscillatory schemes, III, J. Comput. Phys., 71, 231-303, (1987) · Zbl 0652.65067
[15] Herrmann, M., A balanced force refined level set grid method for two-phase flows on unstructured flow solver grids, J. Comput. Phys., 227, 2674-2706, (2008) · Zbl 1388.76252
[16] Hong, W.-L.; Walker, D. T., Reynolds-averaged equations for free-surface flows with application to high-Froude-number jet spreading, J. Fluid Mech., 417, 183-209, (2000) · Zbl 0983.76030
[17] Jiang, G.-S.; Peng, D., Weighted ENO schemes for Hamilton-Jacobi equations, SIAM J. Sci. Comput., 21, 6, 2126-2143, (2000) · Zbl 0957.35014
[18] Jiang, G.-S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126, 1, 202-228, (1996) · Zbl 0877.65065
[19] Kang, M.; Fedkiw, R. P.; Liu, X., A boundary condition capturing method for multiphase incompressible flow, J. Sci. Comput., 15, 3, 323-360, (2000) · Zbl 1049.76046
[20] Karimian, S. M.H.; Schneider, G. E., Pressure-based computational method for compressible and incompressible flows, J. Thermophys. Heat Transf., 8, 267-274, (1994)
[21] Kim, J.; Moin, P., Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59, 2, 308-323, (1985) · Zbl 0582.76038
[22] Liu, X.; Fedkiw, R. P.; Kang, M., A boundary condition capturing method for poissonʼs equation on irregular domains, J. Comput. Phys., 160, 151-178, (2000) · Zbl 0958.65105
[23] Liu, X.-D.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115, 1, 200-212, (1994) · Zbl 0811.65076
[24] Mazhar, Z., A procedure for the treatment of the velocity-pressure coupling problem in incompressible fluid flow, Numer. Heat Transf., Part B, Fundam., 39, 91-100, (2001)
[25] Montazeri, H., A consistent numerical method for simulating interfacial turbulent flows, (2010), University of Toronto Toronto, PhD thesis
[26] Montazeri, H.; Bussmann, M.; Mostaghimi, J., Accurate implementation of forcing terms for two-phase flows into SIMPLE algorithm, Int. J. Multiph. Flow, 45, 40-52, (2012)
[27] Osher, S.; Fedkiw, R., Level set methods and dynamic implicit method, (2003), Springer-Verlag New York · Zbl 1026.76001
[28] Patankar, S. V., Numerical heat transfer and fluid flow, (1980), Hemisphere Pub. Corp. · Zbl 0521.76003
[29] Perić, M.; Kessler, R.; Scheuerer, G., Comparison of finite-volume numerical methods with staggered and colocated grids, Comput. Fluids, 16, 4, 389-403, (1988) · Zbl 0672.76018
[30] Rahman, M. M.; Miettinen, A.; Siikonen, T., Modified simple formulation on a collocated grid with an assessment of the simplified quick scheme, Numer. Heat Transf., Part B, Fundam., 30, 3, 291-314, (1996)
[31] Rahman, M. M.; Siikonen, T., A dual-dissipation scheme for pressure-velocity coupling, Numer. Heat Transf., Part B, Fundam., 42, 231-242, (2002)
[32] Rauwoens, P.; Vierendeels, J.; Merci, B., A solution for the odd-even decoupling problem in pressure-correction algorithms for variable density flows, J. Comput. Phys., 227, 79-99, (2007) · Zbl 1126.76039
[33] Ray, S.; Date, A. W., A calculation procedure for solution of incompressible Navier-Stokes equations on curvilinear non-staggered grids, Numer. Heat Transf., Part B, Fundam., 38, 93-131, (2000)
[34] Rhie, C. M.; Chow, W. L., Numerical study of the turbulent flow past an airfoil with trailing edge separation, AIAA J., 21, 1525-1532, (1983) · Zbl 0528.76044
[35] Schneider, G. E.; Raw, M. J., Control volume finite-element method for heat transfer and fluid flow using colocated variables-1, computational procedure, Numer. Heat Transf., Part B, Fundam., 11, 363-390, (1987) · Zbl 0631.76107
[36] Tafti, D., Alternate formulations for the pressure equation Laplacian on a collocated grid for solving the unsteady incompressible Navier-Stokes equations, J. Comput. Phys., 116, 1, 143-153, (1995) · Zbl 0817.76048
[37] Tomar, G.; Gerlach, D.; Biswas, G.; Alleborn, N.; Sharma, A.; Durst, F.; Welch, S. W.J.; Delgado, A., Two-phase electrohydrodynamic simulations using a volume-of-fluid approach, J. Comput. Phys., 227, 1267-1285, (2007) · Zbl 1126.76044
[38] Wang, R.; Feng, H.; Spiteri, R. J., Observations on the fifth-order WENO method with non-uniform meshes, Appl. Math. Comput., 196, 1, 433-447, (2008) · Zbl 1134.65060
[39] Wolf, W. R.; Azevedo, J. L.F., High-order ENO and WENO schemes for unstructured grids, Int. J. Numer. Methods Fluids, 5, 917-943, (2007) · Zbl 1388.76217
[40] Young, N. O.; Goldstein, J. S.; Block, M. J., The motion of bubbles in a vertical temperature gradient, J. Fluid Mech., 6, 350-356, (1959) · Zbl 0087.19902
[41] Zhang, S.; Jiang, S.; Shu, C.-W., Development of nonlinear weighted compact schemes with increasingly higher order accuracy, J. Comput. Phys., 227, 15, 7294-7321, (2008) · Zbl 1152.65094
[42] Arvanitis, C.; Makridakis, C.; Sfakianakis, N. I., Entropy conservative schemes and adaptive mesh selection for hyperbolic conservation laws, J. Hyperbolic Differ. Equ., 7, 3, 383-404, (2010) · Zbl 1204.65118
[43] Sfakianakis, N., Finite difference schemes on non-uniform meshes for hyperbolic conservation laws, (2009), University of Crete Heraklion, PhD thesis
[44] Berger, M.; Oliger, J., Adaptive mesh refinement for hyperbolic partial differential equation, J. Comput. Phys., 53, 484-512, (1984) · Zbl 0536.65071
[45] Srinivasa, N. A., Adaptive mesh refinement for a finite difference scheme using a quadtree decomposition approach, (2010), Texas A&M University, Master thesis
[46] Herrmann, H.; Lopez, J. M.; Brady, P.; Raessi, M., Thermocapillary motion of deformable drops and bubbles, (Center for Turbulence Research Proceedings of the Summer Program, vol. 155, (2008))
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.