Contents I. Introduction
In recent years, the lattice Boltzmann method has emerged as a powerful alternative to traditional 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 isothermal weakly compressible case, several method enhancements for incompressibility [3], [4] as well as incorporation of a buoyancy-driven temperature 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 uniform 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] demonstrated 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].
