Lattice Boltzmann Scheme Based on Orthogonal Curvilinear Coordinate Transformations for Simulation of Axisymmetric Thermal Convective Flows

Thermal convective flows with axial symmetry can be represented using the dimensionally reduced axisymmetric Navier-Stokes (NS) and the energy equations. When such flows involve multiple scales or boundary layers, the use of variable or clustered grids enables their simulations more effectively. However, the usual forms of lattice Boltzmann (LB) schemes are limited to using uniform Cartesian grids. We address this issue by constructing a novel LB scheme that solves the NS and energy equations in orthogonal curvilinear coordinates (OCC) that allows the use of the standard collide-and-stream steps while accommodating variable orthogonal grids. This involves designing the collision step, viz., its equilibria and the geometric source terms, based on the metric coefficients and their spatial derivatives the parameterize the use of nonuniform grids achieved via a Chapman-Enskog analysis. A double distribution function approach that is based on numerically more stable collision model involving central moments is used to compute the fluid motion and the temperature field in the OCC-LB framework. We demonstrate the advantages of using our proposed LB scheme based on OCC over the standard LB method using uniform grids for simulating a variety of axisymmetric thermal convective flows.