# A Gas-Kinetic BGK Scheme for Natural Convection in a Rotating Annulus
## 1 Introduction  
Natural convection in rotating annuli has attracted a great deal of attention due to its widespread presence in both scientific and industrial applications [1]. A common example is the sealed rotating cavity formed between two compressor or turbine disks and inner and outer cylinders, where natural convection could occur if the temperature difference exists between the surfaces. Depending on the direction of the heat flux inside the cavity, there are two major situations [2; 3]. One is axial heat flux configuration, which refers to a cavity with insulated cylindrical walls and differentially heated disks, and the other is radial heat flux configuration, which contains insulated disks and differentially heated cylindrical walls. In both situations, the convective flows are induced by the centrifugal buoyancy force and the Coriolis force, leading to diverse flow structures and heat transfer properties. This work mainly focuses on the radial heat flow, and investigations can be made in an annulus provided that the flow is invariant in the axial direction (also referred to as the Taylor-Proudman theorem). In this case, like in the flow between two horizontal plates with the lower one heated, the well-known Rayleigh-Benard convection can also appear in the rotating annulus with the outer surface heated [4]. During the past decades, a lot of numerical studies have been carried out on natural convection in rotating cavities or annuli. For example, [PERSON] et al. [5] studied the basic behavior of the buoyancy-induced flow in a simplified cavity with axial heat transfer. In their work, the complete steady-state N-S equations with some assumptions are solved by using the Semi-Implicit Method for Pressure Linked Equations (SIMPLE) pressure correction method. It is found that fluid motion and heat transfer can be attenuated by the Coriolis forces. [PERSON] et al. [6] expressed the incompressible N-S equations with the Boussinesq approximation in streamfunction-vorticity and velocity-vorticity formulations for the two-dimensional (2D) and 3D computations, respectively. The governing equations are solved using second-order finite difference approximations in space and are advanced in time using the Du Fort-Frankel method. Through the computations for a rotating annulus with radial heat transfer, they obtained the streamlines and isothermals at different Rayleigh numbers. [PERSON] et al. [7] also solved the N-S equations subject to the Boussinesq approximation and studied the 2D dynamics of gravity and centrifugally-driven convection in a horizontal rotating cylinder. Later, they discussed the role of Coriolis force in the evolution of a two-dimensional thermally driven flow in a rotating enclosure of arbitrary geometry [8]. [PERSON] et al. [9] used two different solvers to perform the 3D simulations for thermal convection in a closed rotating cavity with inclusion of side walls: one is a commercial code \"Fluent\" with the SIMPLE algorithm, and the other one is a compressible code \"Hydra\" with a low-speed preconditioner. The computed Nusselt numbers showed good agreements with the correlations obtained from [PERSON] et al. [3]. For more numerical studies on the buoyancy-induced flows in rotating cavities with either an axial throughflow or a radial inflow, one may refer to [10] and references therein. All the numerical studies mentioned above are based on solving macroscopic N-S equations, from which a lot of theoretical and experimental investigations are verified. In recent years, the use of Boltzmann-type methods such as the lattice Boltzmann method (LBM) and the gas-kinetic scheme (GKS) for convective flows have become increasingly popular. Particularly, the Bhatnagar-Gross-Krook (BGK) model based GKS (BGK scheme for short) has been proven to give accurate N-S solutions for low speed flows [11; 12; 13], despite it was originally developed for compressible flows [14; 15]. As an example, [PERSON] et al.