An immersed boundary method with direct forcing for the simulation of particulate flows. (English) Zbl 1138.76398

Summary: We present an improved method for computing incompressible viscous flow around suspended rigid particles using a fixed and uniform computational grid. The main idea is to incorporate C.S. Peskin’s regularized delta function approach [Acta Numerica 11, 479–517 (2002; Zbl 1123.74309)] into a direct formulation of the fluid-solid interaction force in order to allow for a smooth transfer between Eulerian and Lagrangian representations while at the same time avoiding strong restrictions of the time step. This technique was implemented in a finite-difference and fractional-step context. A variety of two- and three-dimensional simulations are presented, ranging from the flow around a single cylinder to the sedimentation of 1000 spherical particles. The accuracy and efficiency of the current method are clearly demonstrated.


76M20 Finite difference methods applied to problems in fluid mechanics
74S10 Finite volume methods applied to problems in solid mechanics


Zbl 1123.74309
Full Text: DOI arXiv


[1] Sundaresan, S., Modeling the hydrodynamics of multiphase flow reactors: current status and challenges, Aiche j., 46, 6, 1102-1105, (2000)
[2] Moin, P.; Mahesh, K., Direct numerical simulation: a tool in turbulence research, Ann. rev. fluid mech., 30, 539-578, (1998) · Zbl 1398.76073
[3] Hu, H.; Patankar, N.; Zhu, N., Direct numerical simulation of fluid-solid systems using the arbitrary Lagrangian Eulerian technique, J. comput. phys., 169, 427-462, (2001) · Zbl 1047.76571
[4] Höfler, K.; Schwarzer, S., Navier-Stokes simulation with constraint forces: finite-difference method for particle-laden flows and complex geometries, Phys. rev. E, 61, 6, 7146-7160, (2000)
[5] Kim, J.; Kim, D.; Choi, H., An immersed-boundary finite-volume method for simulations of flow in complex geometries, J. comput. phys., 171, 132-150, (2001) · Zbl 1057.76039
[6] Kajishima, T.; Takiguchi, S., Interaction between particle clusters and particle-induced turbulence, Int. J. heat fluid flow, 23, 639-646, (2002)
[7] Glowinski, R.; Pan, T.; Hesla, T.; Joseph, D.; Périaux, J., 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, 363-426, (2001) · Zbl 1047.76097
[8] Zhang, Z.; Prosperetti, A., A method for particle simulation, J. appl. mech., 70, 64-74, (2003) · Zbl 1110.74798
[9] Feng, Z.-G.; Michaelides, E., The immersed boundary-lattice Boltzmann method for solving fluid-particles interaction problems, J. comput. phys., 195, 2, 602-628, (2004) · Zbl 1115.76395
[10] C. Peskin, Flow patterns around heart valves: a digital computer method for solving the equations of motion, Ph.D. thesis, Albert Einstein College of Medicine, 1972
[11] Fogelson, A.; Peskin, C., A fast numerical method for solving the three-dimensional stokes’ equations in the presence of suspended particles, J. comput. phys., 79, 50-69, (1988) · Zbl 0652.76025
[12] Lai, M.-C.; Peskin, C., An immersed boundary method with formal second-order accuracy and reduced numerical viscosity, J. comput. phys., 160, 705-719, (2000) · Zbl 0954.76066
[13] Saiki, E.; 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
[14] Lee, C., Stability characteristics of the virtual boundary method in three-dimensional applications, J. comput. phys., 184, 559-591, (2003) · Zbl 1073.76609
[15] Goldstein, D.; Handler, R.; Sirovich, L., Modeling a no-slip boundary with an external force field, J. comput. phys., 105, 354-366, (1993) · Zbl 0768.76049
[16] 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
[17] M. Uhlmann, First experiments with the simulation of particulate flows, Technical Report No. 1020, CIEMAT, Madrid, Spain, ISSN 1135-9420, 2003
[18] Glowinski, R.; Pan, T.-W.; Hesla, T.; Joseph, D., A distributed Lagrange multiplier/fictitious domain method for particulate flows, Int. J. multiphase flow, 25, 755-794, (1999) · Zbl 1137.76592
[19] Pan, T.; Joseph, D.; Bai, R.; Glowinski, R.; Sarin, V., Fluidization of 1204 spheres: simulation and experiment, J. fluid mech., 451, 169-191, (2002) · Zbl 1037.76037
[20] Patankar, N.; Singh, P.; Joseph, D.; Glowinski, R.; Pan, T.-W., A new formulation of the distributed Lagrange multiplier/fictitious domain method for particulate flows, Int. J. multiphase flow, 26, 1509-1524, (2000) · Zbl 1137.76712
[21] Patankar, N., A formulation for fast computations of rigid particulate flows, CTR res. briefs, 185-196, (2001)
[22] N. Sharma, N. Patankar, A fast computation technique for the direct numerical simulation of rigid particulate flows, J. Comput. Phys., in press, doi:10.1016/j.jcp.2004.11.012 · Zbl 1087.76533
[23] Peskin, C., The immersed boundary method, Acta numerica, 11, 1-39, (2002)
[24] M. Uhlmann, New results on the simulation of particulate flows, Technical Report No. 1038, CIEMAT, Madrid, Spain, ISSN 1135-9420, 2004
[25] Roma, A.; Peskin, C.; Berger, M., An adaptive version of the immersed boundary method, J. comput. phys., 153, 509-534, (1999) · Zbl 0953.76069
[26] Rai, M.; Moin, P., Direct simulation of turbulent flow using finite-difference schemes, J. comput. phys., 96, 15-53, (1991) · Zbl 0726.76072
[27] Verzicco, R.; Orlandi, P., A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates, J. comput. phys., 123, 402-414, (1996) · Zbl 0849.76055
[28] Schumann, U.; Sweet, R., A direct method for the solution of poisson’s equation with Neumann boundary conditions on a staggered grid of arbitrary size, J. comput. phys., 20, 171-182, (1976)
[29] Liu, C.; Zheng, X.; Sung, C., Preconditioned multigrid methods for unsteady incompressible flows, J. comput. phys., 139, 35-57, (1998) · Zbl 0908.76064
[30] Behr, M.; Hastreiter, D.; Mittal, S.; Tezduyar, T., Incompressible flow past a circular cylinder: dependence of the computed flow field on the location of the lateral boundaries, Comput. methods appl. mech. eng., 123, 309-316, (1995)
[31] Linnick, M.N.; Fasel, H.F., A high-order immersed interface method for simulating unsteady incompressible flows on irregular domains, J. comput. phys., 204, 1, 157-192, (2005) · Zbl 1143.76538
[32] Park, J.; Kwon, K.; Choi, H., Numerical solutions of flow past a circular cylinder at Reynolds numbers up to 160, KSME int. J., 12, 6, 1200-1205, (1998)
[33] Lu, X.Y.; Dalton, C., Calculation of the timing of vortex formation from an oscillating cylinder, J. fluids structures, 10, 527-541, (1996)
[34] Mordant, N.; Pinton, J.-F., Velocity measurement of a settling sphere, Eur. phys. J. B, 18, 343-352, (2000)
[35] M. Uhlmann, Simulation of particulate flows on multi-processor machines with distributed memory, CIEMAT Technical Report No. 1039, Madrid, Spain, ISSN 1135-9420, 2003
[36] Saff, E.; Kuijlaars, A., Distributing many points on a sphere, Math. intelligencer, 19, 1, 5-11, (1997) · Zbl 0901.11028
[37] Aris, R., Vectors, tensors, and the basic equations of fluid mechanics, (1962), Dover Science and Maths · Zbl 0123.41502
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