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A local directional ghost cell approach for incompressible viscous flow problems with irregular boundaries. (English) Zbl 1388.76199
Summary: An immersed boundary method for the incompressible Navier-Stokes equations in irregular domains is developed using a local ghost cell approach. This method extends the solution smoothly across the boundary in the same direction as the discretization it will be used for. The ghost cell value is determined locally for each irregular grid cell, making it possible to treat both sharp corners and thin plates accurately. The time stepping is done explicitly using a second order Runge-Kutta method. The spatial derivatives are approximated by finite difference methods on a staggered, Cartesian grid with local grid refinements near the immersed boundary. The WENO scheme is used to treat the convective terms, while all other terms are discretized with central schemes. It is demonstrated that the spatial accuracy of the present numerical method is second order. Further, the method is tested and validated for a number of problems including uniform flow past a circular cylinder, impulsively started flow past a circular cylinder and a flat plate, and planar oscillatory flow past a circular cylinder and objects with sharp corners, such as a facing square and a chamfered plate.

##### MSC:
 76M20 Finite difference methods applied to problems in fluid mechanics 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs 76D05 Navier-Stokes equations for incompressible viscous fluids
ITSOL; SPARSKIT
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##### References:
 [1] Balaras, E., Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations, Comput. fluids, 33, 3, 375-404, (2004) · Zbl 1088.76018 [2] Bearman, P.W.; Downie, M.J.; Graham, J.M.R.; Obasaju, E.D., Forces on cylinders in viscous oscillatory flow at low keulegan – carpenter numbers, J. fluid mech., 154, 337-356, (1985) · Zbl 0586.76051 [3] Berger, M.; Rigoutsos, I., An algorithm for point clustering and grid generation, IEEE trans. syst. man cybernet., 21, 5, 1278-1286, (1991) [4] Berthelsen, P.A., A decomposed immersed interface method for variable coefficient elliptic equations with non-smooth and discontinuous solutions, J. comput. phys., 197, 1, 364-386, (2004) · Zbl 1052.65100 [5] Bouard, R.; Coutanceau, M., The early stage of development of the wake behind an impulsively started cylinder for $$40 < \mathit{Re} < 10^4$$, J. fluid mech., 101, 3, 583-607, (1980) [6] M. Bozkurttas, H. Dong, V. Seshadri, R. Mittal, F. Najjar, Towards numerical simulation of flapping foils on fixed Cartesian grids, in: AIAA 43rd Aerospace Sciences Meeting and Exhibit, Reno, Nevada, USA, 10-13th January, 2005, AIAA 2005-0079. [7] M. Bozkurttas, H. Dong, R. Mittal, P. Madden, G.V. Lauder, Hydrodynamic performance of deformable fish fins and flapping foils, in: AIAA 44th Aerospace Sciences Meeting and Exhibit, Reno, Nevada, USA, 9-12th January, 2006, AIAA 2006-1392. [8] Braza, M.; Chassaing, P.; Ha Minh, H., Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder, J. fluid mech., 165, 79-130, (1986) · Zbl 0596.76047 [9] Calhoun, D., A Cartesian grid method for solving the two-dimensional streamfunction-vorticity equations in irregular regions, J. comput. phys., 176, 2, 231-275, (2002) · Zbl 1130.76371 [10] Chorin, A.J., Numerical solution of the navier – stokes equations, Math. comput., 22, 742-762, (1968) · Zbl 0198.50103 [11] Coutanceau, M.; Bouard, R., 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, 2, 231-256, (1977) [12] Dennis, S.C.R.; Chang, G.-Z., Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100, J. fluid mech., 42, 3, 471-489, (1970) · Zbl 0193.26202 [13] DeZeeuw, D.; Powell, K.G., An adaptively refined Cartesian mesh solver for the Euler equations, J. comput. phys., 104, 1, 56-68, (1993) · Zbl 0766.76066 [14] Fadlun, E.A.; Verzicco, R.; Orlandi, P.; Mohd-Yusof, J., Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulation, J. comput. phys., 161, 1, 35-60, (2000) · Zbl 0972.76073 [15] Faltinsen, O.M., Sea loads on ships and offshore structures, (1990), Cambridge University Press Cambridge, England [16] Ferziger, J.H.; Perić, M., Computational methods for fluid dynamics, (2002), Springer-Verlag Berlin, Germany · Zbl 0869.76003 [17] Forrer, H.; Jeltsch, R., A higher-order boundary treatment for Cartesian-grid methods, J. comput. phys., 140, 2, 259-277, (1998) · Zbl 0936.76041 [18] R. Ghias, R. Mittal, T.S. Lund, A non-body conformal grid method for simulation of compressible flows with complex immersed boundaries, in: 42nd AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, USA, 5-8 January, 2004. [19] Gibou, F.; Fedkiw, R.P.; Cheng, L.-T.; Kang, M., A second-order-accurate symmetric discretization of the Poisson equation on irregular domains, J. comput. phys., 176, 1, 205-227, (2002) · Zbl 0996.65108 [20] Gilmanov, A.; Sotiropoulos, F.; Balaras, E., A general reconstruction algorithm for simulating flows with complex 3D immersed boundaries on Cartesian grids, J. comput. phys., 191, 2, 660-669, (2003) · Zbl 1134.76406 [21] Gilmanov, A.; Sotiropoulos, F., A hybrid Cartesian/immersed boundary method for simulating flows with 3D, geometrically complex, moving bodies, J. comput. phys., 207, 2, 457-492, (2005) · Zbl 1213.76135 [22] Goldstein, D.; Handler, R.; Sirovich, L., Modeling a no-slip flow boundary with an external force field, J. comput. phys., 105, 2, 354-366, (1993) · Zbl 0768.76049 [23] Graham, J.M.R., The forces on sharp-edged cylinders in oscillatory flow at low keulegan – carpenter numbers, J. fluid mech., 97, 331-346, (1980) [24] Grove, A.S.; Shair, F.H.; Petersen, E.E.; Acrivos, A., An experimental investigation of the steady separated flow past a circular cylinder, J. fluid mech., 19, 60-80, (1964) · Zbl 0117.42506 [25] He, X.; Doolen, G., Lattice Boltzmann method on curvilinear coordinate system: flow around a circular cylinder, J. comput. phys., 134, 2, 306-315, (1997) · Zbl 0886.76072 [26] Henderson, R.D., Details of the drag curve near the onset of vortex shedding, Phys. fluids, 7, 9, 2102-2104, (1995) [27] Henderson, R.D., Nonlinear dynamics and pattern formation in turbulent wake transition, J. fluid mech., 352, 65-112, (1997) · Zbl 0903.76070 [28] K. Herfjord, A study of two-dimensional separated flow by a combination of the finite element method and Navier-Stokes equations, Dr. Ing.-Thesis, Norwegian Institute of Technology, Department of Marine Hydrodynamics, Trondheim, Norway, 1996. [29] Honji, H., Streaked flow around an oscillating circular cylinder, J. fluid mech., 107, 509-520, (1981) [30] Jiang, G.-S.; Peng, D., Weighted ENO schemes for hamilton – jacobi equations, SIAM J. sci. comput., 21, 6, 2126-2143, (2000) · Zbl 0957.35014 [31] Jiang, G.-S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. comput. phys., 126, 1, 202-228, (1996) · Zbl 0877.65065 [32] Kang, M.; Fedkiw, R.P.; Liu, X.-D., A boundary condition capturing method for multiphase incompressible flow, J. sci. comput., 15, 3, 323-360, (2000) · Zbl 1049.76046 [33] Keulegan, G.H.; Carpenter, L.H., Forces on cylinders and plates in an oscillating fluid, J. res. natl. bur. standards, 60, 5, 423-440, (1958) [34] Kim, J.; Kim, D.; Choi, H., An immersed-boundary finite-volume method for simulations of flow in complex geometries, J. comput. phys., 171, 1, 132-150, (2001) · Zbl 1057.76039 [35] Kirkpatric, M.P.; Armfield, S.W.; Kent, J.H., A representation of curved boundaries for the solution of the navier – stokes equations on a staggered three-dimensional Cartesian grid, J. comput. phys., 184, 1, 1-36, (2003) · Zbl 1118.76350 [36] Koumoutsakos, P.; Leonard, A., High-resolution simulations of the flow around an impulsively started cylinder using vortex methods, J. fluid mech., 296, 1-38, (1995) · Zbl 0849.76061 [37] Koumoutsakos, P.; Shiels, D., Simulations of the viscous flow normal to an impulsively started and uniformly accelerated flat plate, J. fluid mech., 328, 177-227, (1996) · Zbl 0890.76061 [38] Lai, M.-C.; Peskin, C.S., An immersed boundary method with formal second-order accuracy and reduced numerical viscosity, J. comput. phys., 160, 2, 705-719, (2000) · Zbl 0954.76066 [39] Le, D.V.; Khoo, B.C.; Peraire, J., An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries, J. comput. phys., 220, 1, 109-138, (2006) · Zbl 1158.74349 [40] Lecointe, Y.; Piquet, J., On the use of several compact methods for the study of unsteady incompressible viscous flow round a circular cylinder, Comput. fluids, 12, 4, 255-280, (1984) · Zbl 0619.76023 [41] Leveque, R.J.; Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. numer. anal., 31, 4, 1019-1044, (1994) · Zbl 0811.65083 [42] Leveque, R.J.; Li, Z., Immersed interface method for Stokes flow with elastic boundaries or surface tension, SIAM J. sci. comput., 18, 3, 709-735, (1997) · Zbl 0879.76061 [43] Li, Z.; Lai, M.-C., The immersed interface method for the navier – stokes equations with singular forces, J. comput. phys., 171, 2, 822-842, (2001) · Zbl 1065.76568 [44] Li, Z.; Wang, C., A fast finite difference method for solving navier – stokes equations on irregular domains, Commun. math. sci., 1, 1, 180-196, (2003) · Zbl 1082.76077 [45] N Linnick, M.; Fasel, H.F., A high-order immersed interface method for simulating unsteady compressible flows on irregular domains, J. comput. phys., 204, 1, 157-192, (2005) · Zbl 1143.76538 [46] Liu, H.; Krishnan, S.; Marella, S.; Udaykumar, H.S., Sharp interface Cartesian grid method II: a technique for simulating droplet interactions with surfaces of arbitrary shape, J. comput. phys., 210, 1, 32-54, (2005) · Zbl 1154.76358 [47] Liu, X.-D.; Fedkiw, R.P.; Kang, M., A boundary condition capturing method for poisson’s equation on irregular domains, J. comput. phys., 160, 1, 151-178, (2000) · Zbl 0958.65105 [48] Majumdar, S.; Iaccarino, G.; Durbin, P., RANS solvers with adaptive structured boundary non-conforming grids, Annu. res. briefs, cent. turbul. res., 353-366, (2001) [49] Marella, S.; Krishnan, S.; Liu, H.; Udaykumar, H.S., Sharp interface Cartesian grid method I: an easily implemented technique for 3D moving boundary computations, J. comput. phys., 210, 1, 1-31, (2005) · Zbl 1154.76359 [50] D.F. Martin, K. Cartwright, Solving Poisson’s equation using adaptive mesh refinement, Technical Report UCB/ERI M96/66, U.C. Berkeley, California, USA, 1996. . [51] Mittal, R.; Iaccarino, G., Immersed boundary methods, Annu. rev. fluid mech., 37, 239-261, (2005) · Zbl 1117.76049 [52] Niu, X.D.; Chew, Y.T.; Shu, C., Simulation of flows around an impulsively started circular cylinder by Taylor series expansion- and least square-based lattice Boltzmann method, J. comput. phys., 188, 1, 176-193, (2003) · Zbl 1038.76033 [53] Pan, D., An immersed boundary method for incompressible flows using volume of body function, Int. J. numer. methods fluids, 50, 6, 733-750, (2006) · Zbl 1086.76055 [54] Park, J.; Kwon, K.; Choi, H., Numerical solutions of flow past a circular cylinder at Reynolds numbers up to 160, KSME int. J., 12, 1200-1205, (1998) [55] Peskin, C.S., Flow patterns around heart valves: a numerical method, J. comput. phys., 10, 2, 252-271, (1972) · Zbl 0244.92002 [56] Peskin, C.S., Numerical analysis of blood flow in the heart, J. comput. phys., 25, 3, 220-252, (1977) · Zbl 0403.76100 [57] Quirk, J.J., An alternative to unstructured grids for computing gas dynamic flows around arbitrarily complex two-dimensional bodies, Comput. fluids, 23, 1, 125-142, (1994) · Zbl 0788.76067 [58] Russel, D.; Wang, Z.J., A Cartesian grid method for modeling multiple moving objects in 2D incompressible viscous flow, J. comput. phys., 191, 1, 177-205, (2003) · Zbl 1160.76389 [59] Y. Saad, SPARSKIT: a basic tool kit for sparse matrix computations, ver. 2, Technical Report, Research Institute for Advanced Computer Science, NASA Ames Research Center, Moffet Field, California, USA, 1994. . [60] Saad, Y., Iterative methods for sparse linear systems, (2003), Society for Industrial and Applied Mathematics Philadelphia, Pennsylvania, USA · Zbl 1002.65042 [61] Saiki, E.M.; Biringen, S., Numerical simulation of a cylinder in uniform flow: application of a virtual boundary method, J. comput. phys., 123, 2, 450-465, (1996) · Zbl 0848.76052 [62] Sarpkaya, T., Force on a circular cylinder in viscous oscillatory flow at low keulegan – carpenter numbers, J. fluid mech., 165, 61-71, (1986) [63] Su, S.-W.; Lai, M.-C.; Lin, C.-A., An immersed boundary technique for simulating complex flows with rigid boundary, Comput. fluids, 36, 2, 313-324, (2007) · Zbl 1177.76299 [64] Sumer, B.M.; Fredsøe, J., () [65] Loc, Ta Phuoc, Numerical analysis of unsteady secondary vortices generated by an impulsively started circular cylinder, J. fluid mech., 100, 1, 111-128, (1980) · Zbl 0441.76034 [66] Tanji, S.; Honji, H., Unsteady flow past a flat plate normal to the direction of motion, J. phys. soc. jpn., 30, 1, 262-272, (1971) [67] F. Tremblay, Direct and large-eddy simulation of flow around a circular cylinder at subcritical Reynolds numbers, Dr.Ing. Dissertation, Technischen Universität München, Fachgebiet Strömungsmechanik, Germany, 2001. . [68] F. Tremblay, R. Friedrich, An algorithm to treat flows bounded by arbitrarily shaped surfaces with Cartesian meshes, in: Proceedings of AGSTAB Conference, University of Stuttgart, Germany, 15-17th November, 2000. · Zbl 1112.76412 [69] Tritton, D.J., Experiments on the flow past a circular cylinder at low Reynolds numbers, J. fluid mech., 6, 547-567, (1959) · Zbl 0092.19502 [70] Tseng, Y.-H.; Ferziger, J.H., A ghost-cell immersed boundary method for flow in complex geometry, J. comput. phys., 192, 2, 593-623, (2003) · Zbl 1047.76575 [71] Tseng, Y.-H.; Ferziger, J.H., Large-eddy simulation of turbulent wavy boundary flow – illustration of vortex dynamics, J. turbulence, 5, (2004) · Zbl 1083.76544 [72] R. Tønnessen, A finite element method applied to unsteady viscous flow around 2D blunt bodies with sharp corners, Dr.Ing.-Thesis, Norwegian University of Science and Technology, Department of Marine Hydrodynamics, Trondheim, Norway, 1999. [73] Udaykumar, H.S.; Shyy, W.; Rao, M.M., ELAFINT: a mixed eulerian – lagrangian method for fluid flows with complex and moving boundaries, Int. J. numer. methods fluids, 22, 8, 691-712, (1996) · Zbl 0887.76059 [74] Udaykumar, H.S.; Mittal, R.; Shyy, W., Computation of solid – liquid phase fronts in the sharp interface limit on fixed grids, J. comput. phys., 153, 2, 535-574, (1999) · Zbl 0953.76071 [75] A. Varges, R. Mittal, Aerodynamic performance of biological airfoils, in: 2nd Flow Control Conference, Portland, Oregon, USA, 28th June-1st July, 2004, AIAA 2004-2319. [76] Wang, C.-Y., On high-frequency oscillatory viscous flows, J. fluid mech., 32, 55-68, (1968) · Zbl 0155.55201 [77] Wiegmann, A.; Bube, K.P., The explicit-jump immersed interface method: finite difference methods for PDEs with piecewise smooth solutions, SIAM J. numer. anal., 37, 3, 827-862, (2000) · Zbl 0948.65107 [78] Williamson, C.H.K., Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds numbers, J. fluid mech., 206, 579-627, (1989) [79] Williamson, C.H.K., Three-dimensional wake transition, J. fluid mech., 328, 345-407, (1996) · Zbl 0899.76129 [80] White, F.M., Viscous fluid flow, (1991), McGraw-Hill Singapore [81] Xu, S.; Wang, Z.J., An immersed interface method for simulating the interaction of a fluid with moving boundaries, J. comput. phys., 216, 2, 454-493, (2006) · Zbl 1220.76058 [82] Yang, J.; Balares, E., An embedded-boundary formulation for large-eddy simulation of turbulent flows interacting with moving boundaries, J. comput. phys., 215, 1, 12-40, (2006) · Zbl 1140.76355 [83] Yang, G.; Causon, D.M.; Ingram, D.M., Cartesian cut-cell method for axisymmetric separating body flows, Aiaa j., 37, 8, 905-911, (1999) [84] Yang, Y.; Udaykumar, H.S., Sharp interface Cartesian grid method III: solidification of pure materials and binary solutions, J. comput. phys., 210, 1, 54-74, (2005) · Zbl 1154.76360 [85] 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, 2, 209-240, (1999) · Zbl 0957.76043 [86] Zou, J.-F.; Ren, A.-L.; Deng, J., Study on flow past two spheres in tandem arrangement using a local mesh refinement virtual boundary method, Int. J. numer. meth. fluids, 49, 5, 465-488, (2005) · Zbl 1079.76061 [87] Zhou, Y.C.; Zhao, S.; Feig, M.; Wei, G.W., High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources, J. comput. phys., 213, 1, 1-30, (2006) · Zbl 1089.65117
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