Simulations of the macroscopic and mesoscopic natural convection flows within rectangular cavities

Natural convection within closed cavities is of practical and theoretical interest in many nonlinear sciences and industrial applications. Using a simple lattice Boltzmann (LB) thermal model with the Boussinesq approximation, this study investigates 2D natural convection flows with nonlinear phenomena within enclosed rectangular cavities. The simulations are performed at a constant Prandtl number of Pr = 0.71 and the reference Rayleigh numbers of Ra∗ ⩽ 2 × 104 at the macroscopic scale (Kn = 10−4) and the mesoscopic scale (Kn = 10−2), respectively. In every case, an appropriate value of the characteristic velocity, i.e. image, is chosen using a simple model based on the kinetic theory. The simulations commence to identify the convective-dominated stationary, time-independent steady flow (i.e. the primary instability condition). The spectral information of secondary instability with an oscillatory flow is then investigated using a spectrum analysis based on the fast-Fourier transform (FFT) technique. The relationship between the Nusselt number (Nu) and the reference Rayleigh number (Ra∗) is also systematically examined. In general, the simulation results show that unstable flow is generated at particular values of the Rayleigh number, Knudsen number, and cavity aspect ratio. Meanwhile, the Knudsen number and the aspect ratio play key roles in determining the evolution of oscillatory flows beyond the threshold of secondary instability.