A second-order method for three-dimensional particle simulation. (English) Zbl 1154.76372
Summary: This paper describes a numerical method for the direct numerical simulation of Navier-Stokes flows with one or more solid spheres. The particles may be fixed or mobile, and they may have different radii. The basic idea of the method stems from the observation that, due to the no-slip condition, in the reference frame of each particle, the velocity near the particle boundary is very small so that the Stokes equations constitute an excellent approximation to the full Navier-Stokes problem. The general analytic solution of the Stokes equations can then be used to “transfer” the no-slip condition from the particle surface to the adjacent grid nodes. In this way the geometric complexity arising from the irregular relation between the particle boundary and the underlying mesh is avoided and fast solvers can be used. The method is validated by a detailed comparison with spectral solutions for the flow past a sphere at Reynolds numbers of 50 and 100. The existence in these situations of a Stokes region near the particle is explicitly demonstrated. Other numerical experiments to show the performance of the code are also described. To illustrate the power and efficiency of the method, a simulation of decaying homogeneous turbulence in a cell containing 100 movable spheres is described. As implemented here, the method can only be applied to simple body shapes such as spheres and cylinders. Extensions to more general situations are mentioned in the last section.
MSC:
| 76M25 | Other numerical methods (fluid mechanics) (MSC2010) |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 76D07 | Stokes and related (Oseen, etc.) flows |
References:
| [1] | Takagi, S.; Og̃uz, H.; Z, Z.; Prosperetti, A., PHYSALIS: A new method particle simulation. Part II: Two-dimensional Navier-Stokes flow around cylinders, J. Comput. Phys., 187, 371-390 (2003) · Zbl 1060.76091 |
| [2] | Zhang, Z.; Prosperetti, A., A method for particle simulation, J. Appl. Mech., 70, 64-74 (2003) · Zbl 1110.74798 |
| [3] | Elghobashi, S.; G. C., T., On the two-way interaction between homogeneous turbulence and dispersed solid particles. I: Turbulence modification, Phys. Fluids A, 5, 1790-1801 (1993) · Zbl 0782.76087 |
| [4] | Crowe, C.; Troutt, T.; Chung, J., Numerical models for two-phase turbulent flows, Ann. Rev. Fluid Mech., 28, 11-41 (1996) |
| [5] | Pan, Y.; Banerjee, S., Numerical simulation of particle interactions with wall turbulence, Phys. Fluids, 8, 2733-2755 (1996) |
| [6] | Sundaram, S.; Collins, L., A numerical study of the modulation of isotropic turbulence by suspended particles, J. Fluid Mech., 379, 105-143 (1999) · Zbl 0938.76045 |
| [7] | Armenio, V.; Piomelli, U.; Fiorotto, V., Effect of the subgrid scales on particle motion, Phys. Fluids, 11, 3030-3042 (1999) · Zbl 1149.76308 |
| [8] | Boivin, M.; Simonin, O.; Squires, K., On the prediction of gas-solid flows with two-way coupling using large eddy, Phys. Fluids, 12, 2080-2090 (2000) · Zbl 1184.76064 |
| [9] | Yamamoto, Y.; Potthoff, M.; Tanaka, T.; Kajishima, T.; Tsuji, Y., Large-eddy simulation of turbulent gas-particle flow in a vertical channel: effect of considering inter-particle collisions, J. Fluid Mech., 442, 303-334 (2001) · Zbl 1004.76513 |
| [10] | Fortes, A.; Joseph, D.; Lundgren, T., Nonlinear mechanics of fluidization of beds of spherical particles, J. Fluid Mech., 177, 467-483 (1987) |
| [11] | Hu, H.; Joseph, D.; Crochet, M., Direct simulation of fluid-particle motions, Theor. Comput. Fluid Dynam., 3, 285-306 (1992) · Zbl 0754.76054 |
| [12] | Glowinski, R.; Pan, T.; 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 |
| [13] | 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 |
| [14] | Singh, P.; Hesla, T.; Joseph, D., Distributed Lagrangian multiplier method for particulate flows with collisions, Int. J. Multiphase Flow, 29, 495-509 (2001) · Zbl 1136.76643 |
| [15] | Dong, S.; Liu, D.; Maxey, M. R.; Karniadakis, G. E., Spectral distributed Lagrange multiplier method: algorithm and benchmark tests, J. Comput. Phys., 195, 695-717 (2004) · Zbl 1115.76349 |
| [16] | Johnson, A.; T., T., Simulation of multiple spheres falling in a liquid-filled tube, Comp. Meth. Appl. Mech. Eng., 134, 351-373 (1996) · Zbl 0895.76046 |
| [17] | Johnson, A.; T., T., 3-D simulation of fluid-particle interactions with the number of particles reaching 100, Comp. Meth. Appl. Mech. Eng., 145, 301-321 (1997) · Zbl 0893.76043 |
| [18] | Chattot, J.; Wang, Y., Improved treatment of intersecting bodies with the Chimera method and validation with a simple and fast flow solver, Comput. Fluids, 27, 721-740 (1998) · Zbl 0976.76053 |
| [19] | Nirschl, H.; Dwyer, H.; Denk, V., Three-dimensional calculations of the simple shear flow around a single particle between two moving walls, J. Fluid Mech., 283, 273-285 (1995) · Zbl 0839.76019 |
| [20] | Peskin, C., Numerical analysis of blood flow in the heart, J. Comput. Phys., 25, 220 (1977) · Zbl 0403.76100 |
| [21] | Peskin, C., The immersed boundary method, Acta Numerica, 11, 479-517 (2002) · Zbl 1123.74309 |
| [22] | Pan, Y.; Banerjee, S., Numerical investigation of the effect of large particles on wall turbulence, Phys. Fluids, 9, 3786-3807 (1997) |
| [23] | Takiguchi, S.; Kajishima, T.; Miyake, Y., Numerical scheme to resolve the interaction between solid particles and fluid turbulence, JSME Int. J. B, 42, 411-418 (1999) |
| [24] | Ye, T.; Mittal, R.; Udaykumar, H.; 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 |
| [25] | Udaykumar, H. S.; Mittal, R.; Rampunggoon, P.; Khanna, A., A sharp interface Cartesian grid method for simulating flows with complex moving boundaries, J. Comput. Phys., 174, 345-380 (2001) · Zbl 1106.76428 |
| [26] | Fadlun, E. A.; 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 |
| [27] | Li, Z.; Lai, M.-C., The immersed interface method fro Navier-Stokes equations with singular forces, J. Comput. Phys., 171, 822-842 (2001) · Zbl 1065.76568 |
| [28] | 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 |
| [29] | 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, 660-669 (2003) · Zbl 1134.76406 |
| [30] | Tseng, Y.-H.; Ferziger, J., A ghost-cell immersed boundary method for flow in complex geometry, J. Comput. Phys., 192, 593-623 (2003) · Zbl 1047.76575 |
| [31] | Balaras, E., Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations, Comput. Fluids, 33, 375-404 (2004) · Zbl 1088.76018 |
| [32] | A. Gilmanov, F. Sotiropoulos, A hybrid Cartesian/immersed boundary method for simulating flows with 3D, geometrically complex, moving bodies, J. Comput. Phys., in press.; A. Gilmanov, F. Sotiropoulos, A hybrid Cartesian/immersed boundary method for simulating flows with 3D, geometrically complex, moving bodies, J. Comput. Phys., in press. · Zbl 1213.76135 |
| [33] | Chen, S.; Doolen, G., Lattice Boltzmann method in fluid flows, Annu. Rev. Fluid Mech., 30, 329-364 (1998) · Zbl 1398.76180 |
| [34] | Aidun, C.; Lu, Y.; Ding, E.-J., Direct analysis of particulate suspensions with inertia using the discrete Boltzmann equation, J. Fluid Mech., 37, 287-311 (1998) · Zbl 0933.76092 |
| [35] | Inamuro, T.; Maeba, K.; Ogino, F., Flow between parallel walls containing the lines of neutrally buoyant circular cylinders, Int. J. Multiphase Flow, 26, 1981-2004 (2000) · Zbl 1137.76620 |
| [36] | Hill, R. J.; Koch, D. L., The transition from steady to weakly turbulent flow in a close-packed ordered array of spheres, J. Fluid Mech., 465, 59-97 (2002) · Zbl 1042.76033 |
| [37] | Feng, Z.-G.; Michaelides, E. E., The immersed boundary-lattice Boltzmann method for solving fluid particles interaction problems, J. Comput. Phys., 195, 602-628 (2004) · Zbl 1115.76395 |
| [38] | Feng, Z.-G.; Michaelides, E. E., Proteus: A direct forcing method in the simulations of particulate flows, J. Comput. Phys., 202, 20-51 (2005) · Zbl 1076.76568 |
| [39] | Huang, H.; Takagi, S., PHYSALIS: A new method for particle flow simulation. Part III: Convergence analysis of two-dimensional flows, J. Comput. Phys., 189, 493-511 (2003) · Zbl 1061.76061 |
| [40] | Kalthoff, W.; Schwarzer, S.; Herrmann, H. J., Algorithm for the simulation of particle suspensions with inertia effects, Phys. Rev. E, 56, 2234-2242 (1997) |
| [41] | Martys, N.; Bentz, D.; Garboczi, E., Computer simulation study of the effective viscosity in Brinkman’s equation, Phys. Fluids, 6, 1434-1439 (1994) · Zbl 0825.76819 |
| [42] | 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 |
| [43] | Keller, J.; Givoli, D., Exact nonreflecting boundary conditions, J. Comput. Phys., 82, 172-192 (1989) · Zbl 0671.65094 |
| [44] | Givoli, D., Numerical methods for problems in infinite domains (1992), Elsevier: Elsevier Amsterdam · Zbl 0788.76001 |
| [45] | Givoli, D.; Rivkin, L.; Keller, J., A finite-element method for domains with corners, Int. J. Numer. Meth. Eng., 35, 1329-1345 (1992) · Zbl 0768.73072 |
| [46] | Givoli, D.; Patlashenko, I.; Keller, J., High-order boundary conditions and finite elements for infinite domains, Comput. Method Appl. M, 143, 13-39 (1997) · Zbl 0883.65085 |
| [47] | Bjorstad, P.; Widlund, O., Iterative methods for the solution of elliptic problems on regions partitioned into substructures, SIAM J. Numer. Anal., 23, 1097-1120 (1986) · Zbl 0615.65113 |
| [48] | Nataf, F.; Nier, F., Convergence rate of some domain decomposition methods for overlapping and nonoverlapping subdomains, Numer. Math., 75, 357-377 (1997) · Zbl 0873.65108 |
| [49] | Quarteroni, A.; Valli, A., Somain decomposition methods for partial differential equations (1999), Oxford Science Publishers: Oxford Science Publishers Oxford · Zbl 0931.65118 |
| [50] | Proskurowski, W.; Widlund, O., Numerical solution of Helmholtz’s equation by capacitance matrix method, Math. Comput., 135, 433-468 (1976) · Zbl 0332.65057 |
| [51] | Proskurowski, W., Numerical solution of eigenvalue problem of Laplace operator by a capacitance matrix method, Computing, 20, 139-151 (1978) · Zbl 0402.65058 |
| [52] | Borgers, C., Domain embedding methods for the Stokes equations, Numer. Math., 57, 435-451 (1990) · Zbl 0701.76042 |
| [53] | Lamb, H., Hydrodynamics (1932), Cambridge U.P.: Cambridge U.P. Cambridge · JFM 26.0868.02 |
| [54] | Kim, S.; Karrila, S., Microhydrodynamics (1991), Butterworth-Heinemann: Butterworth-Heinemann Boston |
| [55] | Brown, D.; Cortez, R.; Minion, M., Accurate projection methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 168, 464-499 (2001) · Zbl 1153.76339 |
| [56] | Prosperetti, A.; Og̃uz, H., PHYSALIS: A new o \((N)\) method for the numerical simulation of disperse flows of spheres. Part I: Potential flow, J. Comput. Phys., 167, 196-216 (2001) · Zbl 0988.76072 |
| [57] | Mo, G.; Sangani, A., A method for computing Stokes flow interactions among spherical objects and its application to suspensions of drops and porous particles, Phys. Fluids, 6, 1637-1652 (1994) · Zbl 0829.76019 |
| [58] | Kress, R., Linear integral equations (1989), Springer: Springer Berlin · Zbl 0671.45001 |
| [59] | Press, W.; Vetterling, W.; Teukolsky, S.; Flannery, B., Numerical recipes in FORTRAN (1992), Cambridge U.P.: Cambridge U.P. Cambridge · Zbl 0778.65002 |
| [60] | Bagchi, P.; Balachandar, S., Direct numerical simulation of flow and heat transfer from a sphere in a uniform cross-flow, J. Fluids. Eng., 123, 347-358 (2001) |
| [61] | P. Bagchi, Particle dynamics in inhomogeneous flows at moderate-to-high Reynolds number, Ph.D. Thesis, University of Illinois, Urbana, IL, 2002.; P. Bagchi, Particle dynamics in inhomogeneous flows at moderate-to-high Reynolds number, Ph.D. Thesis, University of Illinois, Urbana, IL, 2002. |
| [62] | Mansfield, J. R.; Knio, O. M.; Meneveau, C., A dynamic LES scheme for the vorticity transport equation: Formulation and a priori tests, J. Comput. Phys., 145, 693-730 (1998) · Zbl 0926.76088 |
| [63] | M., M. I.; Lohse, D.; Toschi, F., On the relevance of the lift force in bubbly turbulence, J. Fluid Mech., 488, 283-313 (2003) · Zbl 1063.76651 |
| [64] | Balucani, U.; Zoppi, M., Dynamics of the liquid state (1994), Clarendon Press: Clarendon Press Oxford |
| [65] | Chaikin, P.; Lubensky, T., Principles of condensed matter physics (1995), Cambridge U.P.: Cambridge U.P. Cambridge, UK |
| [66] | Percus, J.; Yevick, G., Analysis of classical statistical mechanics by means of collective coordinates, Phys. Rev., 110, 1-13 (1958) · Zbl 0096.23105 |
| [67] | Wertheim, M. S., Exact solution of the Percus-Yevick integral equation for hard spheres, Phys. Rev. Lett., 10, 321-323 (1963) · Zbl 0129.44302 |
| [68] | Thiele, E., Equation of state for hard spheres, J. Chem. Phys., 39, 474-479 (1963) |
| [69] | Pozrikidis, C., Boundary integral and singularity methods for linearized viscous flow (1992), Cambridge U.P.: Cambridge U.P. Cambridge · Zbl 0772.76005 |
| [70] | Zhang, Z.; Prosperetti, A., Sedimentation of 1024 particles, (Proceedings of the fluids engineering division summer meeting (2005), ASME: ASME New York) |
| [71] | M. Uhlmann, An immersed boundary method with direct forcing for the simulation of particulate flow, J. Comput. Phys. (2005), in press.; M. Uhlmann, An immersed boundary method with direct forcing for the simulation of particulate flow, J. Comput. Phys. (2005), in press. · Zbl 1138.76398 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
