波茲曼晶格方法應用於滑動與熱邊界條件對 Rayleigh-Benard 熱對流影響之研究

Abstract

This work studies the effects of hydrodynamic and thermal slip boundaries on Rayleigh-B′enard convection (RB convection) using lattice Boltzmann method (LBM). Firstly, the theoretical relations of the critical Rayleigh number (Rc) and corresponding wavenumber (ac) of RB convection with partially slippery boundary conditions on infinite horizontal plates are derived using the linear stability analysis. The results make previous studies by others under both slip and nonslip boundary conditions as special cases of the general relations. Secondly, before numerical study of the effects of various slip side walls on RB convection in a 2D box, a new implementation of partial slip boundary conditions in LBM is developed by using the parameter of tangential momentum accommodation coefficient (TMAC,σ). This new implementation utilizes the native expressions of velocity gradients in LBM and then eliminates the need for information of neighbor nodes to estimate the velocity and temperature gradients, which is inevitable for conventional numerical technique like finite-difference method. Thirdly, the numerical simulations show that the vertical slip side walls have similar impacts as the horizontal slip plates on the determination of Rc and the relations for infinite horizontal plates can be used as guidelines for the cases with side walls. The observations of pattern selection in the box with aspect ratio equal to 2 show that when σh (σ of horizontal plates) is less than 0.02, the preferred pattern is the one-roll mode. When σh ≥ 0.02, the fluid prefers the two-roll mode in which the fluid moves upwards in the center of the box if σv ≤ 0.1 (σ of vertical side walls), while the fluid switches the rotation directions if σv ≥ 0.2. The investigation of initial disturbance indicates the existence of the threshold of the initial amplitude for the desired mode. This result reveals the importance of initial conditions to RB convection.