非牛顿自然对流的改进径向基函数生成有限差分分析
Refined radial basis function-generated finite difference analysis of non-Newtonian natural convection
AI总结：
针对封闭差热腔非牛顿自然对流问题，提出改进RBF-FD求解方法，采用无网格离散与节点密度优化，可大幅减少节点数并保持精度，具备几何灵活性。
摘要：
本文提出了一种改进的径向基函数生成有限差分（Radial Basis Function-generated Finite Difference, RBF-FD）求解方法，用于求解封闭差热腔内的非牛顿流体问题。非牛顿行为采用Ostwald-de Waele幂律模型建模，浮力效应采用Boussinesq近似处理。问题域采用离散节点进行离散化，节点之间无需满足拓扑关系。这使得求解过程可轻松推广到复杂的不规则三维（3D）域，本文还通过求解模拟多孔过滤器的二维（2D）和三维几何结构中的该问题，验证了这一推广性。二维结果与两种参考解进行了对比，这两种参考解采用有限体积法结合两种不同的稳定技术，本文方法得到的结果与参考数据吻合良好。改进方案基于一种专用的无网格节点定位算法实现，该算法采用分段线性节点密度函数，在确保域中心节点密度充足的同时，最大化动力学变化最剧烈的边界层处的节点密度。结果表明，与规则离散化相比，采用改进方法以相同精度获得结果所需的节点数减少了5倍以上。本文还讨论了不同场景下改进离散化的收敛性（节点数最高达$2 \cdot 10^5$）、方法参数的影响、边界层内的流动行为、粘度行为以及所提求解过程的几何灵活性。
英文摘要：
In this paper we present a refined Radial Basis Function-generated Finite Difference (RBF-FD) solution for a non-Newtonian fluid in a closed differentially heated cavity. The non-Newtonian behaviour is modelled with the Ostwald-de Waele power law and the buoyancy with the Boussinesq approximation. The problem domain is discretised with scattered nodes without any requirement for a topological relation between them. This allows a trivial generalisation of the solution procedure to complex irregular three dimensional (3D) domains, which is also demonstrated by solving the problem in a two dimensional (2D) and 3D geometry mimicking a porous filter. The results in 2D are compared with two reference solutions that use the Finite volume method in a conjunction with two different stabilisation techniques, where we achieved good agreement with the reference data. The refinement is implemented on top of a dedicated meshless node positioning algorithm using piecewise linear node density function that ensures sufficient node density in the centre of the domain while maximising the node density in a boundary layer where the most intense dynamic is expected. The results show that with a refined approach, more than 5 times fewer nodes are required to obtain the results with the same accuracy compared to the regular discretisation. The paper also discusses the convergence with refined discretisation for different scenarios for up to $2 \cdot 10^5$ nodes, the impact of method parametres, the behaviour of the flow in the boundary layer, the behaviour of the viscosity and the geometric flexibility of the proposed solution procedure.
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