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Multiblock hybrid grid finite volume method to solve flow in irregular geometries. (English) Zbl 1120.76337
Summary: In this work, a finite-volume-based finite-element method is suitably developed for solving incompressible flow and heat transfer on collocated hybrid grid topologies. The method is generally applicable to arbitrarily shaped elements and orientations and, thus, challenges the potential to unify many of the different grid topologies into a single formulation. The key point in this formulation is the correct estimation of the convective and diffusive fluxes at the cell faces using a novel physical influence scheme. This scheme remarkably enhances the achieved solution accuracy. It is shown that the extended formulation is robust enough to treat any combination of multiblock meshes with dual element shape employments. In this regard, the solution domain is broken up into a number of different multiblock arrangements in which each block is filled with only one type of finite element shape. The combined grid can decrease the computational time and memory requirements and increase the numerical accuracy. The results of the extended formulation are validated against different benchmark and other solutions. The current results are in excellent agreement with the other solutions without exhibiting any disturbances around the block boundaries.

MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
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[1] Ainsworth, M.; Coyle, J., Hierarchic hp-edge element families for maxwell’s equations on hybrid quadrilateral/triangular meshes, Comput. methods appl. mech. engrg., 190, 6709-6733, (2001) · Zbl 0991.78031
[2] Alishahi, M.M.; Darbandi, M., Multiple-zone potential solution around wing-body configurations, J. aerospace engrg., 6, 329-346, (1993)
[3] Barrett, J.W.; Morton, K.W., Approximate symmetrization and petrov – galerkin methods for diffusion – convection problems, Comput. methods appl. mech. engrg., 45, 97-122, (1984) · Zbl 0562.76086
[4] Beskok, A.; Warburton, T.C., An unstructured hp finite-element scheme for fluid flow and heat transfer in moving domains, J. thermophys., 174, 492-509, (2001) · Zbl 0995.76043
[5] Brooks, A.; Hughes, T.J.R., Streamline upwind petrov – galerkin formulations of convection dominated flows with particular emphasis on the incompressible navier – stokes equations, Comput. methods appl. mech. engrg., 32, 199-259, (1982) · Zbl 0497.76041
[6] Celia, M.A.; Herrera, I.; Bouloutas, E.; Kindred, J.S., A new numerical approach for the advective – diffusive transport equation, Numer. methods partial differential equations, 5, 203-226, (1989) · Zbl 0678.65083
[7] Celia, M.A.; Russell, T.F.; Herrera, I.; Ewing, R.E., An eulerian – lagrangian localized adjoint method for the advection – diffusion equation, Adv. water resour., 13, 187-206, (1990)
[8] Darbandi, M.; Bostandoost, S.M., A new formulation toward unifying the velocity role in collocated variable arrangement, Numer. heat transfer B, 46, 361-382, (2004)
[9] Darbandi, M.; Mazaheri-Body, K.; Vakilipour, S., A pressure-weighted upwind scheme in unstructured finite-element grids, (), 250-259 · Zbl 1198.76067
[10] Darbandi, M.; Mokarizadeh, V., A modified pressure-based algorithm to solve the flow fields with shock and expansion waves, Numer. heat transfer B, 46, 497-504, (2004)
[11] Darbandi, M.; Schneider, G.E., Momentum variable procedure for solving compressible and incompressible flows, Aiaa j., 35, 1801-1805, (1997) · Zbl 0908.76050
[12] Darbandi, M.; Schneider, G.E., Thermobuoyancy treatment for electronic packaging using an improved advection scheme, ASME J. elect. packag., 125, 244-250, (2003)
[13] M. Darbandi, G.E. Schneider, K. Javadi, N. Solhpour, The performance of a physical influence scheme in structured triangular grids, AIAA Paper 2003-0436, The 41st AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 2003.
[14] M. Darbandi, G.E. Schneider, A.R Naderi, A finite element volume method to simulate flow on mixed element shapes, AIAA Paper 2003-3638, The 36th AIAA Thermophysics Conference, Orlando, FL, June 2003.
[15] M. Darbandi, G.E. Schneider, A.R. Naderi, The mesh orientation impact in performance of physical-based upwinding in structured triangular grids, in: Proc. the 11th Annual CFDSC Conference 2003, CFDSC, Ottawa, Canada, 2003, pp. 688-695.
[16] M. Darbandi, G.E. Schneider, S. Vakilipour, A modified upwind-biased strategy to calculate flow on structured-unstructured grid topologies, AIAA Paper 2004-0435, The 42nd AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 2004.
[17] Davidson, L., A pressure correction method for unstructured meshes with arbitrary control volumes, Int. J. numer. meth. fluids, 22, 265-281, (1996) · Zbl 0863.76050
[18] de Nicola, C.; Pinto, G.; Tognaccini, R., On the numerical stability of block structured algorithms with applications to 1-D advection – diffusion problems, Comput. fluids, 24, 41-54, (1995) · Zbl 0826.76059
[19] de Nicola, C.; Pinto, G.; Tognaccini, R., Stability of two-dimensional model problems for multiblock structured fluid-dynamics calculations, Comput. fluids, 26, 43-58, (1997) · Zbl 0888.76051
[20] De Vahl Davis, G., Natural convection of air in a square cavity: a bench mark numerical solution, Int. J. numer. meth. fluids, 3, 249-264, (1983) · Zbl 0538.76075
[21] Demirdzic, I.; Lilek, Z.; Peric, M., Fluid flow and heat transfer test problems for non-orthogonal grids: bench-mark solutions, Int. J. numer. meth. fluids, 15, 329-354, (1992) · Zbl 0825.76631
[22] Ewing, R.E.; Li, Z.; Lin, T.; Lin, Y., The immersed finite volume element methods for the elliptic interface problems, Math. comput. simul., 50, 63-76, (1999) · Zbl 1027.65155
[23] Feistauer, M.; Felcman, J.; Lukacova-Medvid’ova, M., Combined finite element – finite volume solution of compressible flow, J. comput. appl. math., 63, 179-199, (1995) · Zbl 0852.76040
[24] Fezoui, J.; Stoufflet, B., A class of implicit schemes for Euler simulations with unstructured meshes, J. comput. phys., 84, 174-206, (1989) · Zbl 0677.76062
[25] Fornberg, B., A numerical study of steady viscous flow past a circular cylinder, J. fluid mech., 98, 819-855, (1980) · Zbl 0428.76032
[26] Ghia, U.; Ghia, K.N.; Shin, C.T., High-re solutions for incompressible flow using the navier – stokes equations and a multigrid method, J. comput. phys., 48, 387-411, (1982) · Zbl 0511.76031
[27] Guermond, J.L.; Quartapelle, L., Calculation of incompressible viscous flows by an unconditionally stable projection FEM, J. comput. phys., 132, 12-33, (1997) · Zbl 0879.76050
[28] Hansbo, P.; Szepessy, A., A velocity – pressure streamline diffusion finite element method for the incompressible navier – stokes equations, Comput. methods appl. mech. engrg., 84, 107-129, (1990)
[29] Hayase, T.; Humphrey, J.A.C.; Greif, R., A consistently formulated QUICK scheme for fast and stable convergence using finite-volume iterative calculation procedures, J. comput. phys., 98, 108-118, (1992) · Zbl 0743.76054
[30] Healy, R.W.; Russel, T.F., A finite-volume eulerian – lagrangian localized adjoint method for solution of the advection – dispersion equation, Water resour. res., 29, 2399-2413, (1993)
[31] Hwang, C.J.; Wu, S.J., Adaptive finite volume upwind approach on mixed quadrilateral – triangular meshes, Aiaa j., 31, 61-67, (1993) · Zbl 0779.76070
[32] Johnson, C., Numerical solution of partial differential equations by the finite element method, (1987), Cambridge University Press Cambridge
[33] A. Kassies, R. Tognaccini, Boundary conditions for Euler equations at internal block faces of multi-block domains using local grid refinement, AIAA paper no. 1990-1590, 1990.
[34] Khawaja, A.; Minyard, T.; Kallinderis, Y., Adaptive hybrid grid methods, Comput. methods appl. mech. engrg., 189, 1231-1245, (2000) · Zbl 1003.76055
[35] Kim, D.; Choi, H., A second-order time-accurate finite volume method for unsteady incompressible flow on hybrid unstructured grids, J. comput. phys., 162, 411-428, (2000) · Zbl 0985.76060
[36] Kirby, R.M.; Warburton, T.C.; Lomtev, I.; Karniadakis, G.E., A discontinuous Galerkin spectral/hp method on hybrid grids, Appl. numer. math., 33, 393-405, (2000) · Zbl 0992.76056
[37] Kuan, K.B.; Lin, C.A., Adaptive QUICK-based scheme to approximate convective transport, Aiaa j., 38, 2233-2237, (2000)
[38] Lammer, L.; Burghardt, M., Parallel generation of triangular and quadrilateral meshes, Adv. engrg. softwares, 31, 929-936, (2000) · Zbl 1003.68519
[39] Lange, C.F.; Schafer, M.; Durst, F., Local block refinement with a multigrid flow solver, Int. J. numer. methods fluids, 38, 21-41, (2002) · Zbl 0996.76061
[40] Leonard, B.P., A stable and accurate convective modelling procedure based on quadratic upstream interpolation, Comput. methods appl. mech. engrg., 19, 59-98, (1979) · Zbl 0423.76070
[41] Li, X.; Wu, W.; Zienkiewicz, O.C., Implicit characteristic Galerkin method for convection – diffusion equations, Int. J. numer. methods engrg., 47, 1689-1708, (2000) · Zbl 0977.76047
[42] M.S. Liou, K.H. Kao, Progress in grid generation: from Chimera to DRAGON grids, NASA TM 106709, NASA, August 1994.
[43] Liuo, M.S.; Zheng, Y., A novel approach of three-dimensional hybrid grid methodology: part 2. flow solution, Comput. methods appl. mech. engrg., 192, 4173-4193, (2003) · Zbl 1178.76250
[44] Moinier, P.; Muller, J.D.; Giles, M.B., Edge-based multigrid and preconditioning for hybrid grids, Aiaa j., 40, 1954-1960, (2002)
[45] Patankar, S.V., Numerical heat transfer and fluid flow, (1996), Hemisphere Publishing Co. New York · Zbl 0595.76001
[46] Pironneau, O., Finite element methods for fluids, (1989), John Wiley and Sons Ltd. Chichester · Zbl 0665.73059
[47] Pollard, A.; Siu, A., The calculation of some laminar flows using various discretization schemes, Comput. methods appl. mech. engrg., 35, 293-313, (1982) · Zbl 0493.76007
[48] Prakash, C., Examination of the upwind (donor-cell) formulation in control volume finite-element methods for fluid flow and heat transfer, Numer. heat transfer, 11, 401-416, (1987)
[49] Rai, M.M., An implicit, conservative, zonal-boundary scheme for Euler equation calculations, Comput. fluids, 14, 295-319, (1986) · Zbl 0625.76073
[50] Raithby, G.D.; Torrance, K.E., Upstream-weighted differencing schemes and their application to elliptic problems involving fluid flow, Comput. fluids, 2, 191-206, (1974) · Zbl 0335.76008
[51] Raithby, G.D., Skew upstream differencing schemes for problems involving fluid flow, Comput. methods appl. mech. engrg., 9, 153-164, (1976) · Zbl 0347.76066
[52] 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
[53] Ribbens, C.J.; Watson, L.T.; Wang, C.Y., Steady viscous flow in a triangular cavity, J. comput. phys., 112, 173-181, (1994) · Zbl 0798.76056
[54] Russell, T.F.; Celia, M.A., An overview of research on eulerian – lagrangian localized adjoint methods (ELLAM), Adv. water resour., 25, 1215-1231, (2002)
[55] R. Struijs, P. Vankeirsbilck, H. Deconinck, An adaptive grid polygonal finite volume method for the compressible flow equations, AIAA Paper 1989-1957, 1989.
[56] Toreja, A.J.; Rizwan-uddin, Hybrid numerical methods for convection – diffusion problems in arbitrary geometries, Comput. fluids, 32, 835-872, (2003) · Zbl 1175.76097
[57] Tseng, Y.H.; Ferziger, J.H., A ghost-cell immersed boundary method for flow in complex geometry, J. comput. phys., 192, 593-623, (2003) · Zbl 1047.76575
[58] Wheeler, M.F.; Dawson, C.N., An operator-splitting method for advection – diffusion-reaction problems, The mathematics of finite elements and applications (uxbridge, 1987), vol. VI, (1988), Academic Press London-New York, pp. 463-482 · Zbl 0678.76095
[59] Wood, W.A.; Kleb, W.L., Diffusion characteristic of finite volume and fluctuation splitting schemes, J. comput. phys., 153, 353-377, (1999) · Zbl 0937.65106
[60] Ye, T.; Mittal, R.; Udaykumar, H.S.; Shyy, W., An accurate Cartesian grid method for viscous incompressible flows with complex immersed boundaries, J. comput. phys., 156, 209-240, (1999) · Zbl 0957.76043
[61] Zheng, Y.; Liuo, M.S., A novel approach of three-dimensional hybrid grid methodology: part 1. grid generation, Comput. methods appl. mech. engrg., 192, 4147-4171, (2003) · Zbl 1178.76301
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