# A Gas-Kinetic BGK Scheme for Natural Convection in a Rotating Annulus
## 1 Introduction
[16] constructed a gas-kinetic BGK model for the incompressible N-S equations with thermal effects, where the flow field and temperature field are described by two coupled BGK models. By using the developed method, they studied the Rayleigh-Benard thermal convection between horizontal plates. [PERSON] et al. [17] simulated the 2D compressible convection by using a developed multidimensional BGK scheme with inclusion of the gravitational force effects. The predicted critical Rayleigh numbers agree with those from the linear analysis. The importance of including the source effect in the BGK model and the necessity of treating the source term consistently with the flux calculation have been demonstrated in detail. Similar to the method of [PERSON] et al. [16], by describing velocity and temperature fields independently, [PERSON] et al. [18] developed a thermal and coupled discrete unified gas-kinetic (DUGKS) scheme for the Boussinesq flows. The computed results show that the coupled DUGKS can well describe the convection phenomena from laminar to turbulent flows, with Rayleigh number up to \(10^{10}\). Recently, they extended it for 3D natural convection in a differentially heated cubical cavity [19]. The above numerical examples by using the BGK scheme are all about buoyancy-induced flows in stationary enclosures. As for convection in rotating cavities, however, there seems to be a lack of systematic studies. Therefore, several important issues are addressed here for the development and application of the present method. Firstly, unlike most existing evolution models for the incompressible N-S equations, we derive a single BGK model for the compressible full N-S equations, where the density is calculated by the ideal gas law. This is due to the fact that in practical high-speed engines, large temperature differences exist leading to a much higher Rayleigh number (possibly greater than \(10^{12}\)) and the Boussinesq approximation may become invalid [4]. The derived model can be more general than the Boussinesq-based incompressible flow model. Secondly, it is known that the application of compressible N-S solvers to extremely low-speed flows could suffer due to two difficulties: one is the accuracy degradation due to the inappropriate numerical dissipation in upwind schemes, and the other is the low convergence rate due to the system stiffness. However, it is herein shown that the accuracy of the developed BGK scheme would not be greatly affected when modeling low-speed thermal flows. This is actually not surprising since the BGK scheme has a delicate dissipation mechanism [15]. The latter difficulty currently seems difficult to be perfectly resolved and some discussion will be presented. Thirdly, for the rotating fluid applications, a rotating frame of reference is usually used, thus the non-inertial effects (also referred to sources of buoyancy force) should be considered in constructing the BGK model. In the recent work of [PERSON] et al. [20], a moving reference frame based BGK scheme (MRF-BGK) was developed for 2D viscous flows around arbitrarily moving bodies, where the source effects are accounted for by adding specifically-designed particle acceleration terms into the Boltzmann equation. It is proven that with such a gas evolution model, the N-S equations formulated in a moving frame can be correctly recovered. However, the formulations of the part of fluxes resulted from these acceleration terms can be complex and also it is difficult to obtain the explicit expression of the distribution function. Since this is not the only choice for considering the source effect, as pointed out in [21], we overcome these drawbacks by replacing the particle acceleration terms with a simple microscopic source term, which is derived directly from the macroscopic source terms. This paper is organized as follows. In Section 2, we introduce the macroscopic equations for natural convection in a closed rotating annulus.