A high-order implicit radial basis function-based differential quadrature-finite volume (IRBFDQ-FV) method is presented in this work to efficiently simulate incompressible flows with heat transfer on unstructured mesh. The velocity and temperature fields are solved by locally using the lattice Boltzmann flux solver and the high-order finite volume method. Specifically, the proposed highly accurate finite volume method utilizes a high-order Taylor polynomial to approximate the solution within every control cell. Spatial derivatives are the corresponding coefficients in the polynomial, and they are approximated by the meshless radial basis function-based differential quadrature (RBFDQ) method. The diffusive and convective fluxes at each cell interface are simultaneously evaluated through local reconstruction of lattice Boltzmann solution using D2Q9 lattice velocity model. To efficiently calculate the solution with high-order accuracy, an implicit time-marching method incorporating the lower-upper symmetric Gauss-Seidel (LU-SGS) and the explicit first stage, singly-diagonally implicit Runge-Kutta (ESDIRK) approaches is devised. The proposed method is comprehensively validated by a series of numerical experiments containing both steady-state and time-dependent heat transfer problems with/without curved boundaries at a wide variety of Rayleigh numbers and Grashof numbers. The obtained results demonstrate a high degree of accuracy and reliability of the proposed method for complex flows on unstructured mesh. In comparison with the classical second-order method, the proposed high-order method has better computational efficiency when comparable results are achieved.

在这项工作中，提出了一种基于高阶隐式径向基函数的微分正交有限体积（IRBFDQ-FV）方法，以有效地模拟非结构化网格上具有传热的不可压缩流动。采用格子玻尔兹曼通量求解器和高阶有限体积法对速度场和温度场进行局部求解。具体来说，所提出的高精度有限体积方法利用高阶泰勒多项式来近似每个控制单元内的解。空间导数是多项式中的对应系数，它们通过基于无网格径向基函数的微分求积（RBFDQ）方法进行近似。通过使用 D2Q9 晶格速度模型对晶格 Boltzmann 解进行局部重构，同时评估每个单元界面处的扩散通量和对流通量。为了有效地计算具有高阶精度的解，设计了一种结合上下对称高斯-赛德尔 (LU-SGS) 和显式第一阶段的隐式时间推进方法，单对角隐式龙格-库塔 (ESDIRK) 方法. 所提出的方法通过一系列数值实验得到了全面验证，这些实验包含在各种瑞利数和格拉肖夫数下有/没有弯曲边界的稳态和时间相关传热问题。所获得的结果证明了所提出的方法在非结构化网格上的复杂流动具有高度的准确性和可靠性。