1 Validation of a dynamically adaptive lattice Boltzmann method for 2D thermal convection simulations Kai Feldhusen 1,2 , Ralf Deiterding 1 , and Claus Wagner 1,2 Abstract—Utilizing the Boussinesq approxima- tion, a double-population thermal lattice Boltzmann method (LBM) for forced and natural convection in two space dimensions is developed and validated. A block-structured dynamic adaptive mesh refinement procedure tailored for LBM is applied to enable computationally efficient simulations of high Rayleigh number configurations 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 con- sidered. The effectiveness of the overall approach is demonstrated for the 2D natural convection bench- mark of a cavity with differentially heated walls at Rayleigh numbers from 10 3 up to 10 8 . Keywords—Lattice Boltzmann method, thermal con- vection, adaptive mesh refinement I. I NTRODUCTION In recent years, the lattice Boltzmann method has emerged as a powerful alternative to tradi- tional Navier-Stokes (NS) solvers [1]. Instead of discretizing the NS equations directly, the LBM is based on solving a simplified version of the Boltzmann equation in a specially chosen discrete phase space. Using a Chapman-Enskog expansion, it can be shown that the approach recovers the NS equations in the limit of a vanishing Knudsen number [2]. Originally proposed for the isother- mal weakly compressible case, several method enhancements for incompressibility [3], [4] as well as incorporation of a buoyancy-driven temperature 1 German Aerospace Center (DLR), Institute of Aerodynam- ics and Flow Technology, Bunsenstr. 10, 37073 G¨ ottingen, Germany email: kai.feldhusen@dlr.de. 2 Technische Universit¨ at Ilmenau, Institute of Thermo- and Fluiddynamics, 98693 Ilmenau field for thermal convection flows are available [5], [6]. Here, we have chosen to pursue the strictly incompressible double distribution function (DDF) approach proposed by Guo et al. [7]. While the original LBM is formulated on a uni- form Cartesian grid, an increase of local resolution is particularly desirable in the thermal boundary layers close to heated objects and walls. So far, the majority of DDF LBM methods with on-the-fly mesh adaptation has been proposed for isothermal two-phase flows, cf. [8]. Kuznik et al. [9] demon- strated the computational benefit of a non-uniform grid for a thermal DDF LBM method; yet, their approach is restricted to purely Cartesian domains. Our objective is to close this gap by incorporating a DDF LBM method into a block-based dynamic adaptive mesh refinement (AMR) method [10]. The outline of this paper is as follows: In Section II we discuss the details of the numer- ical method, including the adopted thermal lat- tice Boltzmann approach, the block-based AMR method and the treatment of geometrically complex boundaries in the originally Cartesian scheme. Sec- tion III presents the computational results, where the analytic validation example of the 2D flow between two moving porous plates, the 2D flow around a rotating heated cylinder and the well- known benchmark case of a two-dimensional cav- ity with differentially heated walls are considered. The conclusions including a short outlook are given in Section IV. II. NUMERICAL METHOD A. Thermal lattice Boltzmann scheme The incompressible two-dimensional LBM con- structed under Boussinesq approximation used in the present work has been proposed by Guo et al. [7]. Note that the extensions to three dimensions is