Utilizing the Boussinesq approximation, a double-population incompressible thermal lattice Boltzmann method (LBM) for forced and natural convection in two and three space dimensions is developed and validated. A block-structured dynamic adaptive mesh refinement (AMR) procedure tailored for the LBM is applied to enable computationally efficient simulations of moderate to high Rayleigh number flows which are characterized by a large scale disparity in boundary layers and free stream flow. As test cases, the analytically accessible problem of a two-dimensional (2D) forced convection flow through two porous plates and the non-Cartesian configuration of a heated rotating cylinder are considered. The objective of the latter is to advance the boundary conditions for an accurate treatment of curved boundaries and to demonstrate the effect on the solution. The effectiveness of the overall approach is demonstrated for the natural convection benchmark of a 2D cavity with differentially heated walls at Rayleigh numbers from 103 up to 108. To demonstrate the benefit of the employed AMR procedure for three-dimensional (3D) problems, results from the natural convection in a cubic cavity at Rayleigh numbers from 103 up to 105 are compared with benchmark results.
Abdelhadi, B., Hamza, G., Razik, B. and Raouache, E. (2006). Natural convection and turbulent instability in cavity, WSEAS Transactions on Heat and Mass Transfer 1(2): 179-184.
Alexander, F.J., Chen, S. and Sterling, J.D. (1993). Lattice Boltzmann thermohydrodynamics, Physical Review E 47: 2249-2252.
Azwadi Che Sidik, N. and Irwan, M. (2010). Simplified mesoscale lattice Boltzmann numerical model for prediction of natural convection in a square enclosure filled with homogeneous porous media, WSEAS Transactions on Fluid Mechanics 3(5): 186-195.
Azwadi Che Sidik, N. and Syahrullail, S. (2009). A three-dimension double-population thermal lattice BGK model for simulation of natural convection heat transfer in a cubic cavity, WSEAS Transactions on Mathematics 8(9): 561-571.
Berger, M. and Colella, P. (1988). Local adaptive mesh refinement for shock hydrodynamics, Journal of Computational Physics 82(1): 64-84.
Bhatnagar, P., Gross, E. and Krook, M. (1954). A model for collisional processes in gases, I: Small amplitude processes in charged and in neutral one-component systems, Physical Review 94(3): 511-525.
Bouzidi, M., Firdaouss, M. and Lallemand, P. (2001). Momentum transfer of a Boltzmann-lattice fluid with boundaries, Physics of Fluids 13(11): 3452-3459.
Chen, H., Filippova, O., Hoch, J., Molvig, K., Shock, R., Teixeira, C. and Zhang, R. (2006). Grid refinement in lattice Boltzmann methods based on volumetric formulation, Physica A 362(1): 158-167.
Chen, H. and Teixeira, C. (2000). H-theorem and origins of instability in thermal lattice Boltzmann models, Computer Physics Communications 129(1): 21-31.
Chen, S. and Doolen, G. (1998). Lattice Boltzmann method for fluid flows, Annual Review of Fluid Mechanics 30: 329-364.
Coutanceau, M. and Menard, C. (1985). Influence of rotation on the near-wake development behind an impulsively started circular cylinder, Journal of Fluid Mechanics 158: 399-446.
De Vahl Davis, G. (1983). Natural convection of air in a square cavity a benchmark numerical solution, International Journal for Numerical Methods in Fluids 3(3): 249-264.
Deiterding, R. (2011). Block-structured adaptive mesh refinement-theory, implementation and application, ESAIM Proceedings 34: 97-150.
Deller, P. (2002). Lattice kinetic schemes for magnetohydrodynamics, Journal of Computational Physics 179(1): 95-126.
Dupuis, A. and Chopard, B. (2003). Theory and applications of an alternative lattice Boltzmann grid refinement algorithm, Physica E 67: 066707.
Fusegi, T., Hyun, J., Kuwahara, K. and Farouk, B. (1991). A numerical study of three-dimensional natural convection in a differentially heated cubical enclosure, International Journal of Heat and Mass Transfer 34(6): 1543-1557.
Guo, Z., Shi, B. and Zheng, C. (2002). A coupled lattice BGK model for the Boussinesq equations, International Journal for Numerical Methods in Fluids 39(4): 325-342.
Hähnel, D. (2004). Molekulare Gasdynamik, Springer, Heidelberg.
He, N.-Z., Wang, N.-C., Shi, B.-C. and Guo, Z.-L. (2004). A unified incompressible lattice BGK model and its application to three-dimensional lid-driven cavity flow, Chinese Physics 13(1): 40-46.
He, X., Chen, S. and Doolen, G. (1998). A novel thermal model for the lattice Boltzmann method in incompressible limit, Journal of Computational Physics 146(1): 282-300.
He, X. and Luo, L.-S. (1997). Lattice Boltzmann model for the incompressible Navier-Stokes equation, Journal of Statistical Physics 88(3): 927-944.
Jonas, L., Chopard, B., Succi, S. and Toschi, F. (2006). Numerical analysis of the average flow field in a turbulent lattice Boltzmann simulation, Physica A 32(1): 6-10.
Kuznik, F., Vareilles, J., Rusaouen, G. and Kraiss, G. (2007). A double-population lattice Boltzmann method with non-uniform mesh for the simulation of natural convection in a square cavity, International Journal of Heat and Fluid Flow 28(5): 862-870.
Lai, H. and Yan, Y. (2001). The effect of choosing dependent variables and cell-face velocities on convergence of the SIMPLE algorithm using non-orthohonal grids, International Journal of Numerical Methods for Heat & Fluid Flow 11(6): 524-546.
Lee, T. and Lin, C. (2005). A stable discretization of the lattice Boltzmann equation for simulation of incompressible two-phase flows at high density ratio, Journal of Computational Physics 206(1): 16-47.
Li, L., Mei, R. and Klausner, J. (2013). Boundary conditions for thermal lattice Boltzmann equation method, Journal of Computational Physics 237(1): 366-395.
McNamara, G. and Alder, B. (1993). Analysis of lattice Boltzmann treatment of hydrodynamics, Physica A 194(1): 218-228.
Mohamad, A. (2011). Lattice Boltzmann Method- Fundamentals and Engineering Applications with Computer Codes, Springer, London.
Mohamad, A. and Kuzmin, A. (2010). A critical evaluation of force term in lattice Boltzmann method, natural convection problem, International Journal of Heat and Mass Transfer 53: 990-996.
Peng, Y., Shu, C. and Che, Y. (2003). A 3D incompressible thermal lattice Boltzman model and its application to simulation natural convection in a cubic cavity, Journal of Computational Physics 193(1): 260-274.
Qian, Y. (1993). Simulating thermohydrodynamics with lattice BGK models, Journal of Scientific Computing 8(3): 231-241.
Qian, Y., D’Humires, D. and Lallemand, P. (1992). Lattice BGK models for Navier-Stokes equation, Europhysics Letters 17(6): 479-484.
Succi, S. (2001). The Lattice Boltzmann Equation: For Fluid Dynamics and Beyond, Numerical Mathematics and Scientific Computation, Clarendon Press, Oxford.
Yan, Y. and Zu, Y. (2008). Numerical simulation of heat transfer and fluid flow past a rotating isothermal cylinder-A LBM approach, International Journal of Heat and Mass Transfer 51(9-10): 2519-2536.
Yu, Z. and Fan, L.-S. (2009). An interaction potential based lattice Boltzmann method with adaptive mesh refinement (AMR) for two-phase flow simulation, Journal of Computational Physics 230(17): 6456-6478.