×

zbMATH — the first resource for mathematics

Sharp interface Cartesian grid method. I: An easily implemented technique for 3D moving boundary computations. (English) Zbl 1154.76359
Summary: A Cartesian grid method is developed for the simulation of incompressible flows around stationary and moving three-dimensional immersed boundaries. The embedded boundaries are represented using level-sets and treated in a sharp manner without the use of source terms to represent boundary effects. The narrow-band distance function field in the level-set boundary representation facilitates implementation of the finite-difference flow solver. The resulting algorithm is implemented in a straightforward manner in three-dimensions and retains global second-order accuracy. The accuracy of the finite-difference scheme is established and shown to be comparable to finite-volume schemes that are considerably more difficult to implement. Moving boundaries are handled naturally. The pressure solver is accelerated using an algebraic multigrid technique adapted to be effective in the presence of moving embedded boundaries. Benchmarking of the method is performed against available numerical as well as experimental results.

MSC:
76M20 Finite difference methods applied to problems in fluid mechanics
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
Software:
Gerris; XFEM
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anderson, D.M.; McFadden, G.B.; Wheeler, A.A., Diffuse-interface methods in fluid mechanics, Annu. rev. fluid mech., 30, 139-165, (1998) · Zbl 1398.76051
[2] H. Liu, et al., Sharp interface Cartesian grid method II: a technique for simulating droplet interactions with surfaces of arbitrary shape, J. Comput. Phys., (accepted) · Zbl 1154.76358
[3] Y. Yang, H.S. Udaykumar, Sharp interface Cartesian grid method III: solidification of pure materials and binary solutions, J. Comput. Phys., (accepted) · Zbl 1154.76360
[4] Peskin, C.S., Numerical analysis of blood flow in the heart, J. comput. phys, 25, 220-243, (1977) · Zbl 0403.76100
[5] Tryggvason, G., Computations of multiphase flows, Adv. appl. mech., 39, 81-120, (2003)
[6] Marella, S.V.; Udaykumar, H.S., Computational analysis of the deformability of leukocytes modeled with viscous and elastic structural components, Phys. fluids, 16, 2, 244-264, (2004) · Zbl 1186.76349
[7] 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
[8] Sussman, M.; Smereka, P.; Osher, S., A level set approach for computing solutions to incompressible 2-phase flow, J. comput. phys., 114, 1, 146-159, (1994) · Zbl 0808.76077
[9] Brackbill, J.U.; Kothe, D.B.; Zemach, C., A continuum method for modeling surface-tension, J. comput. phys., 100, 2, 335-354, (1992) · Zbl 0775.76110
[10] Al-Rawahi, N.; Tryggvason, G., Numerical simulation of dendritic solidification with convection: two-dimensional geometry, J. comput. phys., 180, 2, 471-496, (2002) · Zbl 1143.76529
[11] Leveque, R.J.; Li, Z.L., 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
[12] Cheng, T.; Peskin, C.S., Stability and instability in the computation of flows with moving immersed boundaries - a comparison of 3 methods, SIAM J. sci. stat. comput., 13, 6, 1361-1376, (1992) · Zbl 0760.76067
[13] Stockie, J.M.; Wetton, B.R., Analysis of stiffness in the immersed boundary method and implications for time-stepping schemes, J. comput. phys., 154, 1, 41-64, (1999) · Zbl 0953.76070
[14] 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
[15] Roma, A.M.; Peskin, C.S.; Berger, M.J., An adaptive version of the immersed boundary method, J. comput. phys., 153, 2, 509-534, (1999) · Zbl 0953.76069
[16] Lee, L.; Leveque, R.J., An immersed interface method for incompressible Navier-Stokes equations, SIAM J. sci. comput., 25, 3, 832-856, (2003) · Zbl 1163.65322
[17] Glowinski, R., A distributed Lagrange multiplier/fictitious domain method for flows around moving rigid bodies: application to particulate flow, Int. J. numer. meth. fluid, 30, 8, 1043-1066, (1999) · Zbl 0971.76046
[18] Adalsteinsson, D.; Sethian, J.A., A fast level set method for propagating interfaces, J. comput. phys., 118, 2, 269-277, (1995) · Zbl 0823.65137
[19] Patankar, N.A., A new formulation of the distributed Lagrange multiplier/fictitious domain method for particulate flows, Int. J. multiphase flow, 26, 9, 1509-1524, (2000) · Zbl 1137.76712
[20] Wang, X.D.; Liu, W.K., Extended immersed boundary method using FEM and RKPM, Comp. meth. appl. mech. eng., 193, 12-14, 1305-1321, (2004) · Zbl 1060.74676
[21] Fadlun, E.A., Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations, J. comput. phys., 161, 1, 35-60, (2000) · Zbl 0972.76073
[22] Balaras, E., Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations, Comp. fluid, 33, 3, 375-404, (2004) · Zbl 1088.76018
[23] 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
[24] 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
[25] Lai, M.C.; Li, Z.L., A remark on jump conditions for the three-dimensional Navier-Stokes equations involving an immersed moving membrane, Appl. math. lett., 14, 2, 149-154, (2001) · Zbl 1013.76021
[26] Li, Z.L.; 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
[27] Leveque, R.J.; Li, Z.L., The immersed interface method for elliptic-equations with discontinuous coefficients and singular sources, SIAM J. numer. anal., 32, 5, 1704, (1995), (31(4) (1994) 1019-1044) · Zbl 0834.65099
[28] Udaykumar, H.S.; Marella, S.; Krishnan, S., Sharp-interface simulation of dendritic growth with convection: benchmarks, Int. J. heat mass transf., 46, 14, 2615-2627, (2003) · Zbl 1037.76064
[29] Udaykumar, H.S.; Mittal, R.; Rampunggoon, P., Interface tracking finite volume method for complex solid-fluid interactions on fixed meshes, Commun. numer. meth. eng., 18, 2, 89-97, (2002) · Zbl 1093.76543
[30] Udaykumar, H.S., A sharp interface Cartesian grid method for simulating flows with complex moving boundaries, J. comput. phys., 174, 1, 345-380, (2001) · Zbl 1106.76428
[31] Fedkiw, R.P., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. comput. phys., 152, 2, 457-492, (1999) · Zbl 0957.76052
[32] Liu, X.D.; Fedkiw, R.P.; Kang, M.J., A boundary condition capturing method for poisson’s equation on irregular domains, J. comput. phys., 160, 1, 151-178, (2000) · Zbl 0958.65105
[33] Chessa, J.; Belytschko, T., An extended finite element method for two-phase fluids, J. appl. mech. trans. asme, 70, 1, 10-17, (2003) · Zbl 1110.74391
[34] Chessa, J.; Smolinski, P.; Belytschko, T., The extended finite element method (XFEM) for solidification problems, Int. J. numer. meth. eng., 53, 8, 1959-1977, (2002) · Zbl 1003.80004
[35] Sukumar, N., Modeling holes and inclusions by level sets in the extended finite-element method, Comp. meth. appl. mech. eng., 190, 46-47, 6183-6200, (2001) · Zbl 1029.74049
[36] Dolbow, J.; Moes, N.; Belytschko, T., An extended finite element method for modeling crack growth with frictional contact, Comp. meth. appl. mech. eng., 190, 51-52, 6825-6846, (2001) · Zbl 1033.74042
[37] Chen, S., A simple level set method for solving Stefan problems, J. comput. phys., 135, 1, 8-29, (1997) · Zbl 0889.65133
[38] Gibou, F., A second-order-accurate symmetric discretization of the Poisson equation on irregular domains, J. comput. phys., 176, 1, 205-227, (2002) · Zbl 0996.65108
[39] Gibou, F., A level set approach for the numerical simulation of dendritic growth, J. sci. comput., 19, 1-3, 183-199, (2003) · Zbl 1081.74560
[40] Gibou, F.; Fedkiw, R., A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem, J. comput. phys., 202, 2, 577-601, (2005) · Zbl 1061.65079
[41] Yokoi, K., Numerical method for a moving solid object in flows, Phys. rev. E, 67, 4, (2003)
[42] Yokoi, K., Three-dimensional numerical simulation of flows with complex geometries in a regular Cartesian grid and its application to blood flow in cerebral artery with multiple aneurysms, J. comput. phys., 202, 1, 1-19, (2005) · Zbl 1061.76065
[43] Ye, T., An accurate Cartesian grid method for viscous incompressible flows with complex immersed boundaries, J. comput. phys., 156, 2, 209-240, (1999) · Zbl 0957.76043
[44] Udaykumar, H.S.; Mao, L., Sharp-interface simulation of dendritic solidification of solutions, Int. J. heat mass transf., 45, 24, 4793-4808, (2002) · Zbl 1032.76710
[45] Udaykumar, H.S.; Mao, L.; Mittal, R., A finite-volume sharp interface scheme for dendritic growth simulations: comparison with microscopic solvability theory, Numer. heat transf. part B - fundamentals, 42, 5, 389-409, (2002)
[46] Almgren, A.S., A Cartesian grid projection method for the incompressible Euler equations in complex geometries, SIAM J. sci. comput., 18, 5, 1289-1309, (1997) · Zbl 0910.76040
[47] Calhoun, D.; Leveque, R.J., A Cartesian grid finite-volume method for the advection-diffusion equation in irregular geometries, J. comput. phys., 157, 1, 143-180, (2000) · Zbl 0952.65075
[48] Sukumar, N., Extended finite element method for three-dimensional crack modelling, Int. J. numer. meth. eng., 48, 11, 1549-1570, (2000) · Zbl 0963.74067
[49] Jayaraman, V.; Udaykumar, H.S.; Shyy, W.S., Adaptive unstructured grid for three-dimensional interface representation, Numer. heat transf. part B - fundamentals, 32, 3, 247-265, (1997)
[50] Sethian, J.A., Fast marching methods, SIAM rev., 41, 2, 199-235, (1999) · Zbl 0926.65106
[51] Strain, J., Fast tree-based redistancing for level set computations, J. comput. phys., 152, 2, 664-686, (1999) · Zbl 0944.65020
[52] Lin, Q., Stratified flow past a sphere, J. fluid mech., 240, 315-354, (1992)
[53] Zang, Y.; Street, R.L.; Koseff, J.R., A non-staggered grid, fractional step method for time-dependent incompressible Navier-Stokes equations in curvilinear coordinates, J. comput. phys., 114, 1, 18-33, (1994) · Zbl 0809.76069
[54] Sethian, J.A., Evolution, implementation, application of level set and fast marching methods for advancing fronts, J. comput. phys., 169, 2, 503-555, (2001) · Zbl 0988.65095
[55] 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
[56] Sethian, J.A.; Smereka, P., Level set methods for fluid interfaces, Annu. rev. fluid mech., 35, 341-372, (2003) · Zbl 1041.76057
[57] Adalsteinsson, D.; Sethian, J.A., The fast construction of extension velocities in level set methods, J. comput. phys., 148, 1, 2-22, (1999) · Zbl 0919.65074
[58] Sussman, M.; Fatemi, E., An efficient, interface-preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow, SIAM J. sci. comput., 20, 4, 1165-1191, (1999) · Zbl 0958.76070
[59] Sussman, M., An improved level set method for incompressible two-phase flows, Comp. fluid, 27, 5-6, 663-680, (1998) · Zbl 0967.76078
[60] Chopp, D.L., Some improvements of the fast marching method, SIAM J. sci. comput., 23, 1, 230-244, (2001) · Zbl 0991.65105
[61] Strain, J., A fast semi-Lagrangian contouring method for moving interfaces, J. comput. phys., 170, 1, 373-394, (2001) · Zbl 0983.65111
[62] Udaykumar, H.S.; Shyy, W., Simulation of interfacial instabilities during solidification. 1. conduction and capillarity effects, Int. J. heat mass transf., 38, 11, 2057-2073, (1995) · Zbl 0923.76215
[63] Glowinski, R., A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow, J. comput. phys., 169, 2, 363-426, (2001) · Zbl 1047.76097
[64] Anderson, D.M.; McFadden, G.B.; Wheeler, A.A., A phase-field model of solidification with convection, Physica D, 135, 1-2, 175-194, (2000) · Zbl 0951.35112
[65] Popinet, S., Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries, J. comput. phys., 190, 2, 572-600, (2003) · Zbl 1076.76002
[66] Yiu, K.F.C., Quadtree grid generation: information handling, boundary Fitting and CFD applications, Comp. fluid, 25, 8, 759-769, (1996) · Zbl 0892.76074
[67] Losasso, F.; Gibou, F.; Fedkiw, R., Simulating water and smoke with an octree data structure, ACM trans. graphics, 23, 3, 457-462, (2004)
[68] Webster, R., An algebraic multigrid solver for Navier-Stokes problems, Int. J. numer. meth. fluid, 18, 8, 761-780, (1994) · Zbl 0806.76046
[69] Webster, R., An algebraic multigrid solver for Navier-Stokes problems in the discrete second-order approximation, Int. J. numer. meth. fluid, 22, 11, 1103-1123, (1996) · Zbl 0865.76071
[70] Wieselsberger, C., New data on laws of fluid resistance, Naca tn, 84, (1922)
[71] Koopmann, G.H., The vortex wakes of vibrating cylinders at low Reynolds numbers, J. fluid mech., 28, 501, (1967)
[72] Johnson, T.A.; Patel, V.C., Flow past a sphere up to a Reynolds number of 300, J. fluid mech., 378, 19-70, (1999)
[73] Magarvey, R.H.; Bishop, R.L., Transition ranges for three-dimensional wakes, Can. J. phys., 39, 1418-1422, (1961)
[74] A.G. Tomboulides, S.A. Orszag, G.E. Karniadakis, Direct and large-eddy simulation of axisymmetric wakes, AIAA Paper, 1993
[75] Chong, M.S.; Perry, A.E.; Cantwell, B.J., A general classification of three-dimensional flow fields, Phys. fluids, 2, A, 765, (1990)
[76] Lin, Q.; Boyer, D.L.; Fernando, H.J.S., The vortex shedding of a streamwise-oscillating sphere translating through a linearly stratified fluid, Phys. fluids, 6, 1, 239-252, (1994)
[77] Tritton, D.J., Experiments on the flow past a circular cylinder at low Reynolds number, J. fluid mech., 6, 547, (1959) · Zbl 0092.19502
[78] Fornberg, B., A numerical study of steady viscous-flow past a circular-cylinder, J. fluid mech., 98, JUN, 819-855, (1980) · Zbl 0428.76032
[79] R. Mittal, S. Balachandar, Inclusion of three-dimensional effects in simulations of two-dimensional bluff body wake flows, Proceedings, 1997 ASME Fluids Engineering Division Summer Meeting
[80] Williamson, C.H.K., Vortex dynamics in the cylinder wake, Annu. rev. fluid mech., 28, 477-539, (1996) · Zbl 0899.76129
[81] Mittal, R., A Fourier-Chebyshev spectral collocation method for simulating flow past spheres and spheroids, Int. J. numer. meth. fluid, 30, 7, 921-937, (1999) · Zbl 0957.76060
[82] Clift, R.; Grace, J.R.; Weber, M.E., Bubbles, drops and particles, (1978), Academic Press New York
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.