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An improved immersed boundary-lattice Boltzmann method for simulating three-dimensional incompressible flows. (English) Zbl 1346.76164
Summary: The recently proposed boundary condition-enforced immersed boundary-lattice Boltzmann method (IB-LBM) [the authors, ibid. 228, No. 6, 1963–1979 (2009; Zbl 1243.76081)] is improved in this work to simulate three-dimensional incompressible viscous flows. In the conventional IB-LBM, the restoring force is pre-calculated, and the non-slip boundary condition is not enforced as compared to body-fitted solvers. As a result, there is a flow penetration to the solid boundary. This drawback was removed by the new version of IB-LBM [loc. cit.], in which the restoring force is considered as unknown and is determined in such a way that the non-slip boundary condition is enforced. Since Eulerian points are also defined inside the solid boundary, the computational domain is usually regular and the Cartesian mesh is used. On the other hand, to well capture the boundary layer and in the meantime, to save the computational effort, we often use non-uniform mesh in IB-LBM applications. In our previous two-dimensional simulations [loc. cit.], the Taylor series expansion and least squares-based lattice Boltzmann method (TLLBM) was used on the non-uniform Cartesian mesh to get the flow field. The final expression of TLLBM is an algebraic formulation with some weighting coefficients. These coefficients could be computed in advance and stored for the following computations. However, this way may become impractical for 3D cases as the memory requirement often exceeds the machine capacity. The other way is to calculate the coefficients at every time step. As a result, extra time is consumed significantly. To overcome this drawback, in this study, we propose a more efficient approach to solve lattice Boltzmann equation on the non-uniform Cartesian mesh. As compared to TLLBM, the proposed approach needs much less computational time and virtual storage. Its good accuracy and efficiency are well demonstrated by its application to simulate the 3D lid-driven cubic cavity flow. To valid the combination of proposed approach with the new version of IBM [loc. cit.] for 3D flows with curved boundaries, the flows over a sphere and torus are simulated. The obtained numerical results compare very well with available data in the literature.

76M28 Particle methods and lattice-gas methods
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[1] Johansen, H.; Colella, P., A Cartesian grid embedded boundary method for poisson’s equation on irregular domains, J. comput. phys., 147, 60-85, (1998) · Zbl 0923.65079
[2] 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
[3] 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
[4] Gilmanov, A.; Sotiropoulos, F., A hybrid Cartesian/immersed boundary method for simulating flows with 3D, geometrically complex, moving bodies, J. comput. phys., 207, 457-492, (2005) · Zbl 1213.76135
[5] Ge, L.; Sotiropoulos, F., A numerical method for solving the 3D unsteady incompressible navier – stokes equations in curvilinear domains with complex immersed boundaries, J. comput. phys., 225, 1782-1809, (2007) · Zbl 1213.76134
[6] Peskin, C.S., Numerical analysis of blood flow in the heart, J. comput. phys., 25, 220-252, (1977) · Zbl 0403.76100
[7] Mittal, R.; Iaccarino, G., Immersed boundary methods, Annu. rev. fluid mech., 37, 239-261, (2005) · Zbl 1117.76049
[8] Chen, S.; Doolen, G.D., Lattice Boltzmann method for fluid flows, Annu. rev. fluid mech., 30, 329-364, (1998) · Zbl 1398.76180
[9] Succi, S., The lattice Boltzmann equation, for fluid dynamics and beyond, (2001), Oxford University Press · Zbl 0990.76001
[10] Feng, Z.; Michaelides, E., The immersed boundary-lattice Boltzmann method for solving fluid – particles interaction problems, J. comput. phys., 195, 602-628, (2004) · Zbl 1115.76395
[11] Feng, Z.; Michaelides, E., Proteus: a direct forcing method in the simulations of particulate flows, J. comput. phys., 202, 20-51, (2005) · Zbl 1076.76568
[12] Niu, X.D.; Shu, C.; Chew, Y.T.; Peng, Y., A momentum exchange-based immersed boundary-lattice Boltzmann method for simulating incompressible viscous flows, Phys. lett. A, 354, 173-182, (2006) · Zbl 1181.76111
[13] Peng, Y.; Shu, C.; Chew, Y.T.; Niu, X.D.; Lu, X.Y., Application of multi-block approach in the immersed boundary-lattice Boltzmann method for viscous fluid flows, J. comput. phys., 218, 460-478, (2006) · Zbl 1161.76552
[14] Wu, J.; Shu, C., Implicit velocity correction-based immersed boundary-lattice Boltzmann method and its applications, J. comput. phys., 228, 1963-1979, (2009) · Zbl 1243.76081
[15] He, X.; Doolen, G.D., Lattice Boltzmann method on a curvilinear coordinate system: vortex shedding behind a circular cylinder, Phys. rev. E, 56, 434-440, (1997)
[16] Mei, R.; Shyy, W., On the finite difference-based lattice Boltzmann method in curvilinear coordinates, J. comput. phys., 143, 426-448, (1998) · Zbl 0934.76074
[17] Peng, G.; Xi, H.; Duncan, C.; Chou, S.H., Finite volume scheme for the lattice Boltzmann method on unstructured meshes, Phys. rev. E, 59, 4675-4682, (1999)
[18] Yu, D.; Mei, R.; Shyy, W., A multi-block lattice Boltzmann method for viscous fluid flows, Int. J. numer. meth. fluid, 39, 99-120, (2002) · Zbl 1036.76051
[19] Shu, C.; Niu, X.D.; Chew, Y.T., Taylor-series expansion and least-squares-based lattice Boltzmann method: two-dimensional formulation and its applications, Phys. rev. E, 65, 036708, (2002)
[20] 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
[21] Qian, Y.H.; d’Humieres, D.; Lallemand, P., Lattice BGK models for navier – stokes equation, Europhys. lett., 17, 479-484, (1992) · Zbl 1116.76419
[22] Peskin, C.S., The immersed boundary method, Acta numer., 11, 479-517, (2002) · Zbl 1123.74309
[23] Babu, V.; Korpela, S.A., Numerical solution of the incompressible three-dimensional navier – stokes equations, Comput. fluid, 23, 675-691, (1994) · Zbl 0811.76041
[24] Shu, C.; Niu, X.D.; Chew, Y.T., Taylor series expansion and least squares-based lattice Boltzmann method: three-dimensional formulation and its applications, Int. J. mod. phys. C, 14, 925-944, (2003) · Zbl 1083.76574
[25] Johnson, T.A.; Patel, V.C., Flow past a sphere up to a Reynolds number of 300, J. fluid mech., 378, 19-70, (1999)
[26] 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
[27] White, F.M., Viscous fluid flow, (1974), McGraw-Hill New York · Zbl 0356.76003
[28] Jeong, J.; Hussain, J., On the identification of a vortex, J. fluid mech., 285, 69-94, (1995) · Zbl 0847.76007
[29] Sheard, G.J.; Thompson, M.C.; hourigan, K., From spheres to circular cylinders: the stability and flow structures of bluff ring wakes, J. fluid mech., 492, 147-180, (2003) · Zbl 1063.76539
[30] Sheard, G.J.; Thompson, M.C.; Hourigan, K., From spheres to circular cylinders: non-axisymmetric transitions in the flow past rings, J. fluid mech., 506, 45-78, (2004) · Zbl 1073.76041
[31] Sheard, G.J.; Hourigan, K.; Thompson, M.C., Computations of the drag coefficients for low-Reynolds-number flow past rings, J. fluid mech., 526, 257-275, (2005) · Zbl 1065.76059
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