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Implementation of a high-order compact finite-difference lattice Boltzmann method in generalized curvilinear coordinates. (English) Zbl 1349.76475
Summary: In this work, the implementation of a high-order compact finite-difference lattice Boltzmann method (CFDLBM) is performed in the generalized curvilinear coordinates to improve the computational efficiency of the solution algorithm to handle curved geometries with non-uniform grids. The incompressible form of the discrete Boltzmann equation with the Bhatnagar-Gross-Krook (BGK) approximation with the pressure as the independent dynamic variable is transformed into the generalized curvilinear coordinates. Herein, the spatial derivatives in the resulting lattice Boltzmann (LB) equation in the computational plane are discretized by using the fourth-order compact finite-difference scheme and the temporal term is discretized with the fourth-order Runge-Kutta scheme to provide an accurate and efficient incompressible flow solver. A high-order spectral-type low-pass compact filter is used to regularize the numerical solution and remove spurious waves generated by boundary conditions, flow non-linearities and grid non-uniformity. All boundary conditions are implemented based on the solution of governing equations in the generalized curvilinear coordinates. The accuracy and efficiency of the solution methodology presented are demonstrated by computing different benchmark steady and unsteady incompressible flow problems. A sensitivity study is also conducted to evaluate the effects of grid size and filtering on the accuracy and convergence rate of the solution. Four test cases considered herein for validating the present computations and demonstrating the accuracy and robustness of the solution algorithm are: unsteady Couette flow and steady flow in a 2-D cavity with non-uniform grid and steady and unsteady flows over a circular cylinder and the NACA0012 hydrofoil at different flow conditions. Results obtained for the above test cases are in good agreement with the existing numerical and experimental results. The study shows the present solution methodology based on the implementation of the high-order compact finite-difference Lattice Boltzmann method (CFDLBM) in the generalized curvilinear coordinates is robust, efficient and accurate for solving steady and unsteady incompressible flows over practical geometries.

76M20 Finite difference methods applied to problems in fluid mechanics
76M28 Particle methods and lattice-gas methods
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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[1] Imamura, T.; Suzuki, K.; Nakamura, T.; Yoshida, M., Acceleration of steady-state lattice Boltzmann simulations on non-uniform mesh using local time step method, Comput. Phys., 202, 645, (2005) · Zbl 1076.82032
[2] He, X.; Luo, L.-S., Theory of the lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation, Phys. Rev. E, 53, 6811-6817, (1997)
[3] Wu, J.; Shu, C., A solution-adaptive lattice Boltzmann method for two-dimensional incompressible viscous flows, Comput. Phys., 230, 2246-2269, (2011) · Zbl 1391.76643
[4] Yu, D.; Mei, R.; Shyy, W., A multi-block lattice Boltzmann method for viscous fluid flows, Numer. Methods Fluids, 39, 99-120, (2002) · Zbl 1036.76051
[5] Tolke, J.; Krafczyk, M., Second order interpolation of the flow field in the lattice Boltzmann method, Comput. Math. Appl., 58, 898-902, (2009) · Zbl 1189.76416
[6] Janssen, C.; Krafczyk, M., A lattice Boltzmann approach for free-surface-flow simulations on non-uniform block-structured grids, Comput. Math. Appl., 59, 2215-2235, (2010) · Zbl 1193.76088
[7] Schönherr, M.; Kucher, K.; Geier, M.; Stiebler, M.; Freudiger, S.; Krafczyk, M., Multi-thread implementations of the lattice Boltzmann method on non-uniform grids for CPUs and gpus, Comput. Math. Appl., 61, 3730-3743, (2011)
[8] Wolf-Gladrow, D. A., Lattice gas cellular automata and lattice Boltzmann models, (2000), Springer Verlag Berlin · Zbl 0999.82054
[9] Reider, M. B.; Sterling, J. D., Accuracy of discrete-velocity BGK models for the simulation of the incompressible Navier-Stokes equations, Comput. Fluids, 24, 4, 459-467, (1995) · Zbl 0845.76086
[10] Mei, R.; Shyy, W., On the finite difference-based lattice Boltzmann method in curvilinear coordinates, Comput. Phys., 143, 426-448, (1998) · Zbl 0934.76074
[11] Tolke, J.; Krafczyk, M.; Schulz, M.; Rank, E., Implicit discretization and nonuniform refinement approaches for FD discretizations of LBGK models, Modern Phys. C, 9, 8, 1143-1157, (1998)
[12] Guo, Z.; Zhao, T. S., Explicit finite-difference lattice Boltzmann method for curvilinear coordinates, Phys. Rev. E, 67, 066709, (2003)
[13] Sofonea, V.; Sekerka, R. F., Viscosity of finite difference lattice Boltzmann models, Comput. Phys., 184, 422-434, (2003) · Zbl 1062.76556
[14] Watari, M.; Tsutahara, M., Two-dimensional thermal model of the finite-difference lattice Boltzmann method with high spatial isotropy, Phys. Rev. E, 67, 036306, (2003)
[15] Xu, A., Finite-difference lattice-Boltzmann methods for binary fluids, Phys. Rev. E, 71, 066706, (2005)
[16] Shi, Yong; Zhao, T. S.; Guo, Z. L., Finite difference-based lattice Boltzmann simulation of natural convection heat transfer in a horizontal concentric annulus, Comput. Fluids, 35, 1-15, (2006) · Zbl 1134.76440
[17] Wu, L.; Tsutahara, M.; Tajiri, S., Finite difference lattice Boltzmann method for incompressible Navier-Stokes equation using acceleration modification, Fluids Sci. Technol., 2, 1, (2007)
[18] Fu, S. C.; So, R. M.C.; Leung, W. W.F., Stochastic finite difference lattice Boltzmann method for steady incompressible viscous flows, Comput. Phys., 229, 6084-6103, (2010) · Zbl 1425.76061
[19] Chen, H., Volumetric formulation of the lattice Boltzmann method for fluid dynamics: basic concept, Phys. Rev. E, 58, 3955-3963, (1998)
[20] Xi, H.; Peng, G.; Chou, S.-H., Finite volume lattice Boltzmann method, Phys. Rev. E, 59, 6202, (1999)
[21] Ubertini, S.; Bella, G.; Succi, S., Unstructured lattice Boltzmann method: further development, Phys. Rev. E, 68, 016701, (2003)
[22] Stiebler, M.; Tolke, J.; Krafczyk, M., An upwind discretization scheme for the finite volume lattice Boltzmann method, Comput. Fluids, 35, 814-819, (2006) · Zbl 1177.76329
[23] Dubois, F.; Lallemand, P., On lattice Boltzmann scheme, finite volumes and boundary conditions, Prog. Comput. Fluid Dyn., 8, 11-24, (2008) · Zbl 1169.76049
[24] Patil, V.; Lakshmisha, K. N., Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh, Comput. Phys., 228, 5262-5279, (2009) · Zbl 1280.76054
[25] Lee, T.; Lin, C.-L., A characteristic Galerkin method for discrete Boltzmann equation, Comput. Phys., 171, 336-356, (2001) · Zbl 1017.76043
[26] Lee, T.; Lin, C.-L., An Eulerian description of the streaming process in the lattice Boltzmann equation, Comput. Phys., 185, 445-471, (2003) · Zbl 1047.76106
[27] Li, Y.; LeBoeuf, E. J.; Basu, P. K., Least-squares finite-element scheme for the lattice Boltzmann method on an unstructured mesh, Phys. Rev. E, 72, 046711, (2005)
[28] Shi, X.; Lin, J.; Yu, Z., Discontinuous Galerkin spectral element lattice Boltzmann method on triangular element, Numer. Methods Fluids, 42, 1249-1261, (2003) · Zbl 1033.76046
[29] Duster, A.; Demkowicz, L.; Rank, E., High-order finite elements applied to the discrete Boltzmann equation, Numer. Methods Eng., 67, 1094-1121, (2006) · Zbl 1113.76049
[30] Min, M.; Lee, T., A spectral-element discontinuous Galerkin lattice Boltzmann method for nearly incompressible flows, Comput. Phys., 230, 245-259, (2011) · Zbl 1427.76189
[31] Hejranfar, K.; Ezzatneshan, E., A high-order compact finite-difference lattice Boltzmann method for simulation of steady and unsteady incompressible flows, J. Numer. Methods Fluids, (2014), submitted for publication
[32] Bhatnagar, P.; Gross, E. P.; Krook, M., A model for collision processes in gases I: small amplitude processes in charged and neutral one-component systems, Phys. Rev. E, 94, 511-525, (1954) · Zbl 0055.23609
[33] He, X.; Luo, L., Lattice Boltzmann model for incompressible Navier-Stokes equation, Stat. Phys., 88, 3, (1997)
[34] Luo, L. S., Theory of the lattice Boltzmann method: lattice Boltzmann models for nonideal gases, Phys. Rev. E, 62, 4982-4996, (2000)
[35] Hirsch, S. R., Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique, Comput. Phys., 19, 90-109, (1975) · Zbl 0326.76024
[36] Lele, S. K.; Tatineni, M., Compact finite different schemes with spectral-like resolution, Comput. Phys., 103, 16-42, (1992) · Zbl 0759.65006
[37] Carpenter, M. H.; Gottlieb, D.; Abarbanel, S., The stability of numerical boundary treatments of compact high-order finite-difference schemes, Comput. Phys., 108, 272-295, (1993) · Zbl 0791.76052
[38] Visbal, M. R.; Gaitonde, D. V., On the use of higher-order finite-difference schemes on curvilinear and deforming meshes, Comput. Phys., 181, 155-185, (2002) · Zbl 1008.65062
[39] Ricot, D.; Marié, S.; Sagaut, P.; Bailly, C., Lattice Boltzmann method with selective viscosity filter, Comput. Phys., 228, 4478-4490, (2009) · Zbl 1395.76073
[40] Brownlee, R. A.; Levesley, J.; Packwood, D.; Gorban, A. N., Add-ons for lattice Boltzmann methods: regularization, filtering and limiters, Prog. Comput. Phys., vol. 3, Part II, (2013), Bentham Science Publishers
[41] Visbal, M. R.; Gaitonde, D. V., High-order accurate methods for complex unsteady subsonic flows, AIAA J., 37, 10, 1231-1239, (1999)
[42] Gaitonde, D. V.; Shang, J. S.; Young, J. L., Practical aspects of higher-order numerical schemes for wave propagation phenomena, Numer. Methods Eng., 45, 1849-1869, (1999) · Zbl 0959.65103
[43] D.V. Gaitonde, M.R. Visbal, Further development of a Navier-Stokes solution procedure based on higher-order formulas, AIAA Technical Paper, 99-0557 (1999).
[44] Hoffmann, K. A.; Chiang, S. T., Computational fluid dynamics, vol. 1, (2000), Engineering Education System Wichita, Kansas, USA
[45] Ghia, U.; Ghia, K. N.; Shin, C. T., High-re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, Comput. Phys., 48, 387-411, (1982) · Zbl 0511.76031
[46] Erturk, E.; Corke, T. C.; Gokcol, C., Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers, Numer. Methods Fluids, 48, 747-774, (2005) · Zbl 1071.76038
[47] Hejranfar, K.; Khajeh-Saeed, A., Implementing a high-order accurate implicit operator scheme for solving steady incompressible viscous flows using artificial compressibility method, Numer. Methods Fluids, 66, 939-962, (2010) · Zbl 1285.76026
[48] White, F. M., Viscous fluid flow, (2006), McGraw-Hill
[49] 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, Fluid Mech., 79, 231-256, (1977)
[50] Tritton, D. J., Experiments on the flow past a circular cylinder at low Reynolds numbers, Fluid Mech., 6, 547-567, (1959) · Zbl 0092.19502
[51] Dennise, S. C.R.; Chang, G. Z., Numerical solutions for steady flow past a circular cylinder at Reynolds number up to 100, Fluid Mech., 42, 471-489, (1970) · Zbl 0193.26202
[52] Nieuwstadt, F.; Keller, H. B., Viscous flow past circular cylinders, Comput. Fluids, 59, (1973) · Zbl 0328.76022
[53] He, X.; Luo, L. S.; Dembo, M., Some progress in lattice Boltzmann method. part 1. nonuniform mesh grids, Comput. Phys., 129, 357, (1996) · Zbl 0868.76068
[54] Grove, A. S.; Shair, F. H.; Petersen, E. E.; Acrivos, A., An experimental investigation of the steady separated flow past a circular cylinder, Fluid Mech., 19, 60, (1964) · Zbl 0117.42506
[55] Berger, E.; Wille, R., Periodic flow phenomena, Annu. Rev. Fluid Mech., 4, 313-340, (1972)
[56] Liu, C.; Sheng, X.; Sung, C. H., Preconditioned multigrid methods for unsteady incompressible flows, Comput. Phys., 139, 35-57, (1998) · Zbl 0908.76064
[57] Wright, J. A.; Smith, R. W., An edge-based method for the incompressible Navier-Stokes equations on polygonal meshes, Comput. Phys., 169, 24-43, (2001) · Zbl 0989.76054
[58] Calhoun, D., A Cartesian grid method for solving the two-dimensional streamfunction-vorticity equations in irregular regions, Comput. Phys., 176, 231-275, (2002) · Zbl 1130.76371
[59] Russell, D.; Wang, Z. J., A Cartesian grid method for modeling multiple moving objects in 2D incompressible viscous flow, Comput. Phys., 191, 177-205, (2003) · Zbl 1160.76389
[60] Chiu, P. H.; Lin, R. K.; Sheu, T. W.H., A differentially interpolated direct forcing immersed boundary method for predicting incompressible Navier-Stokes equations in time-varying complex geometries, Comput. Phys., 229, 4476-4500, (2010) · Zbl 1305.76073
[61] Steger, J. L.; Sorenson, R. L., Automatic mesh-point clustering near a boundary in grid generation with elliptic partial differential equations, Comput. Phys., 33, 405-410, (1979)
[62] Hafez, M.; Shatalov, A.; Nakajima, M., Improved numerical simulations of incompressible flows based on viscous/inviscid interaction procedures, Comput. Fluids, 36, 1588-1591, (2007) · Zbl 1194.76249
[63] Imamura, T.; Suzuki, K.; Nakamura, T.; Yoshida, M., Flow simulation around an airfoil by lattice Boltzmann method on generalized coordinates, AIAA J., 43, (2005)
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