zbMATH — the first resource for mathematics

Coupling staggered-grid and MPFA finite volume methods for free flow/porous-medium flow problems. (English) Zbl 1453.76109
Summary: A discretization is proposed for models coupling free flow with anisotropic porous-medium flow. Our approach employs a staggered-grid finite volume method for the Navier-Stokes equations in the free-flow subdomain and a MPFA finite volume method to solve Darcy flow in the porous medium. After appropriate spatial refinement in the free-flow domain, the degrees of freedom are conveniently located to allow for a natural coupling of the two discretization schemes. In turn, we automatically obtain a more accurate description of the flow field surrounding the porous medium. Numerical experiments highlight the stability and applicability of the scheme in the presence of anisotropy and second-order grid convergence was found in both domains, verifying our approach.
76M12 Finite volume methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
76N06 Compressible Navier-Stokes equations
Full Text: DOI
[1] Vanderborght, J.; Fetzer, T.; Mosthaf, K.; Smits, K. M.; Helmig, R., Heat and water transport in soils and across the soil-atmosphere interface, 1: theory and different model concepts, Water Resour. Res., 53, 2, 1057-1079 (2017)
[2] Gurau, V.; Mann, J. A., A critical overview of computational fluid dynamics multiphase models for proton exchange membrane fuel cells, SIAM J. Appl. Math., 70, 2, 410-454 (2009) · Zbl 1404.92228
[3] Verboven, P.; Flick, D.; Nicolaï, B.; Alvarez, G., Modelling transport phenomena in refrigerated food bulks, packages and stacks: basics and advances, Int. J. Refrig., 29, 6, 985-997 (2006)
[4] Dahmen, W.; Gotzen, T.; Müller, S.; Rom, M., Numerical simulation of transpiration cooling through porous material, Int. J. Numer. Methods Fluids, 76, 6, 331-365 (2014)
[5] Bear, J., Dynamics of Fluids in Porous Media (1972), Courier Dover Publications · Zbl 1191.76001
[6] Whitaker, S., The Method of Volume Averaging (1999), Kluwer Academic
[7] Brinkman, H. C., A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Appl. Sci. Res., 1, 1, 27-34 (1949) · Zbl 0041.54204
[8] Ochoa-Tapia, J. A.; Whitaker, S., Momentum transfer at the boundary between a porous medium and a homogeneous fluid, I: theoretical development, Int. J. Heat Mass Transf., 38, 14, 2635-2646 (1995) · Zbl 0923.76320
[9] Layton, W. J.; Schieweck, F.; Yotov, I., Coupling fluid flow with porous media flow, SIAM J. Numer. Anal., 40, 6, 2195-2218 (2002) · Zbl 1037.76014
[10] Jamet, D.; Chandesris, M.; Goyeau, B., On the equivalence of the discontinuous one- and two-domain approaches for the modeling of transport phenomena at a fluid/porous interface, Transp. Porous Media, 78, 3, 403-418 (2009)
[11] K. Mosthaf, K. Baber, B. Flemisch, R. Helmig, A. Leijnse, I. Rybak, B. Wohlmuth, A coupling concept for two-phase compositional porous-medium and single-phase compositional free flow, Water Resour. Res 47 (10). · Zbl 1343.76024
[12] Hassanizadeh, S. M.; Gray, W. G., Derivation of conditions describing transport across zones of reduced dynamics within multiphase systems, Water Resour. Res., 25, 3, 529-539 (1989)
[13] Fetzer, T.; Grüninger, C.; Flemisch, B.; Helmig, R., On the conditions for coupling free flow and porous-medium flow in a finite volume framework, (International Conference on Finite Volumes for Complex Applications (2017), Springer), 347-356 · Zbl 1365.76306
[14] Iliev, O.; Laptev, V., On numerical simulation of flow through oil filters, Comput. Vis. Sci., 6, 2-3, 139-146 (2004) · Zbl 1299.76255
[15] 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
[16] Discacciati, M.; Quarteroni, A., Navier-Stokes/Darcy coupling: modeling, analysis, and numerical approximation, Rev. Mat. Complut., 22, 2, 315-426 (2009) · Zbl 1172.76050
[17] Huber, R.; Helmig, R., Node-centered finite volume discretizations for the numerical simulation of multiphase flow in heterogeneous porous media, Comput. Geosci., 4, 2, 141-164 (2000) · Zbl 0973.76059
[18] Rybak, I.; Magiera, J.; Helmig, R.; Rohde, C., Multirate time integration for coupled saturated/unsaturated porous medium and free flow systems, Comput. Geosci., 19, 2, 299-309 (2015) · Zbl 1392.76090
[19] Masson, R.; Trenty, L.; Zhang, Y., Coupling compositional liquid gas Darcy and free gas flows at porous and free-flow domains interface, J. Comput. Phys., 321, 708-728 (2016) · Zbl 1349.65376
[20] Versteeg, H. K.; Malalasekera, W., An Introduction to Computational Fluid Dynamics: The Finite Volume Method (2007), Pearson Education
[21] Aavatsmark, I., An introduction to multipoint flux approximations for quadrilateral grids, Comput. Geosci., 6, 3-4, 405-432 (2002) · Zbl 1094.76550
[22] Iliev, O.; Kirsch, R.; Lakdawala, Z.; Printsypar, G., MPFA algorithm for solving Stokes-Brinkman equations on quadrilateral grids, (Finite Volumes for Complex Applications VII-Elliptic, Parabolic and Hyperbolic Problems (2014), Springer), 647-654 · Zbl 1426.76378
[23] Beavers, G. S.; Joseph, D. D., Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30, 01, 197-207 (1967)
[24] Saffman, P. G., On the boundary condition at the surface of a porous medium, Stud. Appl. Math., 50, 2, 93-101 (1971) · Zbl 0271.76080
[25] Yang, G.; Coltman, E.; Weishaupt, K.; Terzis, A.; Helmig, R.; Weigand, B., On the Beavers-Joseph interface condition for non-parallel coupled channel flow over a porous structure at high Reynolds numbers, Transp. Porous Media, 128, 2, 431-457 (2019)
[26] Edwards, M. G.; Rogers, C. F., Finite volume discretization with imposed flux continuity for the general tensor pressure equation, Comput. Geosci., 2, 4, 259-290 (1998) · Zbl 0945.76049
[27] Aavatsmark, I.; Eigestad, G.; Mallison, B.; Nordbotten, J., A compact multipoint flux approximation method with improved robustness, Numer. Methods Partial Differ. Equ., Int. J., 24, 5, 1329-1360 (2008) · Zbl 1230.65114
[28] Edwards, M.; Zheng, H., Quasi-positive families of continuous Darcy-flux finite volume schemes on structured and unstructured grids, J. Comput. Appl. Math., 234, 2152-2161 (2010) · Zbl 1402.76078
[29] Koch, T.; Gläser, D.; Weishaupt, K.; Ackermann, S.; Beck, M.; Becker, B.; Burbulla, S.; Class, H.; Coltman, E.; Fetzer, T.; Flemisch, B.; Grüninger, C.; Heck, K.; Hommel, J.; Kurz, T.; Lipp, M.; Mohammadi, F.; Schneider, M.; Seitz, G.; Scholz, S.; Weinhardt, F., Dumux 3.0.0 (Dec. 2018)
[30] Blatt, M.; Burchardt, A.; Dedner, A.; Engwer, C.; Fahlke, J.; Flemisch, B.; Gersbacher, C.; Gräser, C.; Gruber, F.; Grüninger, C.; Kempf, D.; Klöfkorn, R.; Malkmus, T.; Müthing, S.; Nolte, M.; Piatkowski, M.; Sander, O., The distributed and unified numerics environment, Arch. Numer. Softw., 4, 100, 13-29 (2016), version 2.4
[31] Davis, T. A., Algorithm 832: UMFPACK V4.3 - an unsymmetric-pattern multifrontal method, ACM Trans. Math. Softw., 30, 2, 196-199 (2004) · Zbl 1072.65037
[32] Eymard, R.; Fuhrmann, J.; Linke, A., On mac schemes on triangular Delaunay meshes, their convergence and application to coupled flow problems, Numer. Methods Partial Differ. Equ., 30, 4, 1397-1424 (2014) · Zbl 1383.76353
[33] Jambhekar, V.; Helmig, R.; Schröder, N.; Shokri, N., Free-flow-porous-media coupling for evaporation-driven transport and precipitation of salt in soil, Transp. Porous Media, 110, 2, 251-280 (2015)
[34] Gläser, D.; Helmig, R.; Flemisch, B.; Class, H., A discrete fracture model for two-phase flow in fractured porous media, Adv. Water Resour., 110, 335-348 (2017)
[35] Schneider, M.; Flemisch, B.; Helmig, R.; Terekhov, K.; Tchelepi, H., Monotone nonlinear finite-volume method for challenging grids, Comput. Geosci., 22, 2, 565-586 (2018) · Zbl 1405.65145
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.