zbMATH — the first resource for mathematics

A second-order cut-cell method for the numerical simulation of 2D flows past obstacles. (English) Zbl 1365.76186
Summary: We present a new second-order method, based on the MAC scheme on cartesian grids, for the numerical simulation of two-dimensional incompressible flows past obstacles. In this approach, the solid boundary is embedded in the cartesian computational mesh. Discretizations of the viscous and convective terms are formulated in the context of finite volume methods ensuring local conservation properties of the scheme. Classical second-order centered schemes are applied in mesh cells which are sufficiently far from the obstacle. In the mesh cells cut by the obstacle, first-order approximations are proposed. The resulting linear system is nonsymmetric but the stencil remains local as in the classical MAC scheme on cartesian grids. The linear systems are solved by a fast direct method based on the capacitance matrix method. The time integration is achieved with a second-order projection scheme. While in cut-cells the scheme is locally first-order, a global second-order accuracy is recovered. This property is assessed by computing analytical solutions for a Taylor-Couette problem. The efficiency and robustness of the method is supported by numerical simulations of 2D flows past a circular cylinder at Reynolds number up to 9500. Good agreement with experimental and published numerical results are obtained.

76M20 Finite difference methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76M12 Finite volume methods applied to problems in fluid mechanics
76M15 Boundary element methods applied to problems in fluid mechanics
Full Text: DOI
[1] Peskin, C.S., The fluid dynamics of heart valves: experimental, theoretical, and computational methods, Ann rev fluid mech, 14, 235-259, (1982)
[2] Peskin, C.S., The immersed boundary method, Acta numer, 11, 1-39, (2002)
[3] Mohd-Yusof J. Combined immersed-boundary/B-Spline methods for simulations of flow in complex geometries. NASA Ames Research Center/Stanford University; 1997. p. 317-27.
[4] Saiki, E.M.; Biringen, S., Numerical simulation of a cylinder in uniform flow: application of a virtual boundary method, J comput phys, 123, 450-465, (1996) · Zbl 0848.76052
[5] Zhang, N.; Zheng, Z., An improved direct-forcing immersed boundary method for finite difference applications, J comput phys, 221, 250-268, (2007) · Zbl 1108.76051
[6] Mittal, R.; Iaccarino, G., Immersed boundary methods, Ann rev fluid mech, 37, 239-261, (2005) · Zbl 1117.76049
[7] Fadlun, E.; Verzicco, R.; Orlandi, P.; Mohd-Yusof, J., Combined immersed-boundary finite difference methods for three-dimensional complex flow simulations, J comput phys, 161, 35-60, (2000) · Zbl 0972.76073
[8] de Tullio M, Cristallo A, Balaras E, Pascazio G, Palma PD, Iaccarino G, Napolitano M, Verzicco R. Recent advances in the immersed boundary method. In: Wesseling P, Oñate E, Périaux J, editors. ECCOMAS CFD; 2006.
[9] Kim, J.; Kim, D.; Choi, H., An immersed-boundary finite volume method for simulation of flow in complex geometries, J comput phys, 171, 132-150, (2001) · Zbl 1057.76039
[10] Muldoon, F.; Acharya, S., A divergence-free interpolation scheme for the immersed boundary method, Int J numer method fluid, 56, 1845-1884, (2008) · Zbl 1262.76077
[11] Angot, P.; Bruneau, C.; Fabrie, P., A penalization method to take into account obstacles in incompressible viscous flows, Numer math, 81, 497-520, (1999) · Zbl 0921.76168
[12] Ye, T.; Mittal, R.; Udaykumar, H.; Shyy, W., Numerical simulation of two-dimensional flows over a circular cylinder using the immersed boundary method, J comput phys, 156, 209-240, (1999) · Zbl 0957.76043
[13] Tucker, P.; Pan, Z., A Cartesian cut-cell method for incompressible viscous flow, Appl math model, 24, 591-606, (2000) · Zbl 1056.76059
[14] Chung, M.-H., Cartesian cut cell approach for simulating incompressible flows with rigid bodies of arbitrary shape, Comput fluids, 35, 6, 607-623, (2006) · Zbl 1160.76369
[15] Mittal, R.; Dong, H.; Bozkurttas, M.; Najjar, F.; Vargas, A.; Loebbecke, A., A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries, J comput phys, 227, 4825-4852, (2008) · Zbl 1388.76263
[16] Cheny, Y.; Botella, O., The ls-stag method: a new immersed boundary/level-set method for the computation of incompressible viscous flows in complex moving geometries with good conservation properties, J comput phys, 229, 1043-1076, (2010) · Zbl 1329.76252
[17] Harlow, F.; Welch, J., Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys fluids, 12, 2182-2189, (1965) · Zbl 1180.76043
[18] Osher, S.; Sethian, J.A., Fronts propagating with curvature dependent speed: algorithms based on hamilton – jacobi formulations, J comput phys, 79, 1, 12-49, (1988) · Zbl 0659.65132
[19] Matsunaga, N.; Yamamoto, Y., Superconvergence of the shortley-weller approximation for Dirichlet problems, J comput appl math, 116, 263-273, (2000) · Zbl 0952.65082
[20] Buzbee, B.; Dorr, F., The direct solution of the biharmonic equation on rectangular regions and the Poisson equation on irregular regions, SIAM J numer anal, 11, 753-763, (1974) · Zbl 0294.65059
[21] Buzbee, B.; Dorr, F.; George, J.; Golub, G., The direct solution of the discrete Poisson equation on irregular regions, SIAM J numer anal, 8, 722-736, (1971) · Zbl 0231.65083
[22] LeVeque, R.J.; Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J numer anal, 31, 1019-1044, (1994) · Zbl 0811.65083
[23] Bouchon, F.; Peichl, G.H., The immersed interface technique for parabolic problems with mixed boundary conditions, SIAM J numer anal, 48, 2247-2266, (2010) · Zbl 1223.65066
[24] Hockney, R., A fast direct solution of poisson’s equation using Fourier analysis, J ACM, 12, 95-113, (1965) · Zbl 0139.10902
[25] Swarztrauber, P., The methods of cyclic reduction, Fourier analysis and the facr algorithm for the discrete solution of poisson’s equation on a rectangle, SIAM rev, 19, 490-501, (1977) · Zbl 0358.65088
[26] Guyon, E.; Hulin, J.; Petit, L.; Mitescu, C., Physical hydrodynamics, (2001), Oxford University Press
[27] Chang, C.-C.; Chern, R.-L., A numerical study of flow around an impulsively started circular cylinder by a deterministic vortex method, J fluid mech, 233, 243-263, (1991) · Zbl 0739.76048
[28] Niu, X.D.; Chew, Y.T.; Shu, C., Simulation of flows around an impulsively started circular cylinder by Taylor series expansion-and least squares-based lattice Boltzmann method, J comput phys, 188, 176-193, (2003) · Zbl 1038.76033
[29] Bouard, R.; Coutanceau, M., Experimental determination of the main features of the viscous flow in the wake of a circular cylinder in uniform translation. part 1: steady flow, J fluid mech, 79, 231-256, (1977)
[30] Dennis, S.; Chang, G., Numerical solutions for steady flow past a circular cylinder at Reynolds number up to 100, J fluid mech, 42, 471-489, (1970) · Zbl 0193.26202
[31] Ding, H.; Shu, C.; Cai, Q., Applications of stencil-adaptative finite difference method to incompressible viscous flows with curved boundary, Comput fluids, 36, 786-793, (2007) · Zbl 1177.76255
[32] Fornberg, B., A numerical study of steady viscous flow past a circular cylinder, J fluid mech, 98, 819-855, (1980) · Zbl 0428.76032
[33] Linnick, M.; Fasel, H., A high-order immersed boundary interface method for simulating unsteady incompressible flows on irregular domains, J comput phys, 204, 157-192, (2005) · Zbl 1143.76538
[34] Taira, K.; Colonius, T., The immersed boundary method: a projection approach, J comput phys, 225, 2118-2137, (2007) · Zbl 1343.76027
[35] Belov A, Martinelli L, Jameson A, A New Implicit Algorithm with Multigrid for Unsteady Incompressible Flow Calculations, AIAA Paper 95-0049, AIAA 33rd Aerospace Sciences Meeting and Exhibit, Reno; 1995.
[36] Liu, C.; Zheng, X.; Sung, C., Preconditioned multigrid methods for unsteady incompressible flows, J comput phys, 139, 35-57, (1998) · Zbl 0908.76064
[37] Rogers, S.; Kwak, D., Upwind differencing scheme for the time-accurate incompressible navier – stokes equations, Aiaa j, 28, 254-262, (1990) · Zbl 0693.76041
[38] Sengupta, T.K.; Sengupta, R., Flow past an impulsively started circular cylinder at high Reynolds number, Comput mech, 14, 298-310, (1994) · Zbl 0800.76280
[39] Bouard, R.; Coutenceau, M., The early stages of development of the wake behind an impulsively started cylinder for 40<re<104, J fluid mech, 101, 583-607, (1980)
[40] Koumoutsakos, P.; Leonard, A., High-resolutions of the flow around an impulsively started cylinder using vortex methods, J fluid mech, 296, 1-38, (1995) · Zbl 0849.76061
[41] George, A.; Huang, L.; Tang, W.; Wu, Y., Numerical simulation of unsteady incompressible flow (re<9500) on the curvilinear half-staggered mesh, SIAM J sci comput, 21, 6, 2331-2351, (2000) · Zbl 0989.76057
[42] George, A.; Tang, W.; Wu, Y., Multilevel one-way dissection factorization, SIAM J matrix anal, 22, 6, 752-771, (2001) · Zbl 0993.65036
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.