非正交多松弛系数轴对称热格子Boltzmann方法

Non-orthogonal multiple-relaxation-time lattice Boltzmann method for axisymmetric thermal ?ows

提出了一种模拟轴对称热流动的非正交多松弛系数格子Boltzmann(MRT-LB)模型.通过采用非正交转换矩阵,建立了基于D2Q9离散速度的求解流场的MRT-LB模型和基于D2Q5离散速度的求解温度场的MRT-LB模型.Chapman-Enskog分析表明,该模型可以恢复对应的柱坐标下的宏观连续方程、动量方程和能量方程.与现有的轴对称MRT-LB模型相比,本文采用的非正交转换矩阵中含有更多的零元素,因而具有更高的计算效率.采用本文模型对几种典型的轴对称热流动问题,包括热Womersley流动、竖直圆柱体内的Rayleigh-Bénard对流和环形柱体内的自然对流进行了数值模拟,通过等温线图和流线图以及定量数据的对比,验证了本文模型的可行性和可靠性.并且数值模拟结果表明,相对现有模型,本文模型具有更好的数值稳定性和计算效率.

Axisymmetric thermal flows in cylindrical systems are widely encountered in engineering practices. Typically, axisymmetric thermal flows belong in three-dimensional （3D） problems. However, taking advantage of the axisymmetric condition, the 3D axisymmetric flows can be reduced to quasi two-dimensional （2D） problems in the meridian plane, which significantly reduces the computational requirements and avoids treating the curved boundary. In recent years, various 2D lattice Boltzmann （LB） models, including single relaxation time LB （SRT-LB, or LBGK） and multiple relaxation time LB （MRT-LB） models, for axisymmetric thermal flows have been proposed. In the LB community, it is well accepted that the MRT-LB is superior to the LBGK in terms of numerical stability. The existing MRT-LB model for axisymmetric thermal flows are developed based on orthogonal basis vectors obtained from the combination of the lattice velocity components, i.e., the transform matrix in the existing MRT-LB is an orthogonal one. Unlike the existing MRT-LB model, in this paper, a non-orthogonal multiple-relaxation-time lattice Boltzmann （MRT-LB） method of simulating axisymmetric thermal flows is proposed. In the proposed MRT-LB method, the velocity field is solved by a D2Q9 discrete velocity set while the temperature by a D2Q5 discrete velocity set. The main advantage of the present MRT-LB model is that the transform matrix of the model is a non-orthogonal one, which is comprised of some proper non-orthogonal basis vectors obtained from the combination of the lattice velocity components. The non-orthogonal transform matrix of the present MRT-LB model contains more zero elements than the classical orthogonal transform matrix, and thus the present MRT-LB model is expected to be more efficient than the existing orthogonal-based MRT-LB model. The equilibrium velocity and temperature moments of the present MRT-LB model are expressed by mapping the equilibrium distribution functions onto their moment spaces through using the non-orthogonal transformation matrix. Also the vectors in the forcing term are modified according to the matrix mapping. Through the Chapman-Enskog analysis, it is demonstrated that the macroscopic governing equations in the cylindrical coordinate can be recovered from the present MRT-LB model. Then several numerical tests, including thermal Womersley flow, Rayleigh-Bénard convection in a vertical cylinder and natural convection in a vertical annulus, are conducted to validate the present model. It is found that the present numerical results are in good agreement with the analytical solutions and/or other numerical results reported in the literature. Numerical stability is also tested, and the results suggest that the present MRT model shows better numerical stability than its LBGK counterpart. Moreover, the numerical results also indicate that the present MRT-LB model is more computationally efficient than the existing MRT-LB model for axisymmetric thermal flow. These findings indicate that the present MRT-LB model can serve as a powerful method of computing the axisymmetric thermal flows.