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## Finite line method and its application in coupled heat transfer between fluid-solid domains

Gao Xiao-Wei, Ding Jin-Xing, Liu Hua-Yu
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• #### 摘要

本文创建一种全新的数值计算方法—有限线法, 并将其用于求解流体-固体一体化耦合传热分析. 常用的有限元法是基于体单元的离散方法, 有限容积法是在控制体面上运算的方法, 边界元法是在边界面上离散的方法, 无网格法等是由计算点周围的散点构建计算格式的方法. 本文提出的有限线法是一种基于有限条线段离散的方法, 在每个点只需要两条或三条直线或曲线构成的线系, 则可建立任意高阶的算法格式. 创新性思想是: 通过采用沿线段求方向导数的技术, 由一维拉格朗日插值公式, 建立二维和三维问题的任意高阶线系的空间导数, 并以此直接由问题的控制偏微分方程与边界条件建立离散的系统方程组. 有限线法理论简单、通用性强, 能以统一的格式求解固体与流体力学等偏微分方程边值问题. 在流体方程中, 扩散项采用中心线系保证高精度计算, 而对流项则采用迎风线系体现其方向性特征. 本文将给出有限线法求解流固耦合传热问题的几个算例分析, 验证其正确性与有效性.

#### Abstract

In this paper, a completely new numerical method, called finite line method, is proposed and is used to solve fluid-solid coupled heat transfer problems. The extensively used finite element method is a method based on volume discretization; the finite volume method is a method operated on the surface of the control volume; the boundary element method is the one based on boundary surface discretization; the meshless method is the one constructing the computational algorithm using surrounding scatter points at a collocation point. The method proposed in the work is based on the use of finite number of lines, in which an arbitrarily high-order computational scheme can be established by using only two or three straight or curved lines at each point. The creative idea of the method is that by using a directional derivative technique along a line, high-order two- and three-dimensional spatial partial derivatives with respective to the global coordinates can be derived from the Lagrange polynomial interpolation formulation, based on which the discretized system of equations can be directly formed by the problem-governing partial differential equation and relevant boundary conditions. The proposed finite line method is very simple in theory and robust in universality, by using which the boundary value problems of partial differential equations in solid and fluid mechanics problems can be solved in a unified way. In solving fluid mechanics problems, the diffusion term is simulated by using the central line set to maintain a high efficiency, and the convection term is computed by using an upwind line set to embody its directional characteristic. A few of numerical examples will be given in this paper for fluid-solid coupled heat transfer problems for verifying the correctness and efficiency of the proposed method.

#### 作者及机构信息

###### 通信作者: 高效伟, xwgao@dlut.edu.cn
• 基金项目: 国家自然科学基金(批准号: 12072064)资助的课题.

#### Authors and contacts

###### Corresponding author: Gao Xiao-Wei, xwgao@dlut.edu.cn
• Funds: Project supported by the National Natural Science Foundation of China (Grant No. 12072064).

#### 施引文献

• 图 1  将计算域划分成一系列配置点

Fig. 1.  Discretizing computational domain into a series of collocation points.

图 2  两条线组成的二维(d = 2)线系

Fig. 2.  Line set of 2D (d = 2) consisting of two lines.

图 3  三条线组成的三维(d = 3)线系

Fig. 3.  Line set of 3D (d = 3) consisting of three lines.

图 4  一阶与二阶导数相关的节点

Fig. 4.  Related nodes of the 1st and 2nd order partial derivatives.

图 5  同一配置点的迎风与中心线系

Fig. 5.  Upwind and central line sets at a same collocation point.

图 6  单根管道结构尺寸与边界条件

Fig. 6.  Dimensions and B.C. of a single channel structure.

图 7  不同网格尺寸下管道外表面温度分布

Fig. 7.  Temperature distribution on the channel outer surface under different mesh sizes.

图 8  不同网格尺寸下流固界面温度分布

Fig. 8.  Temperature distribution on the fluid-solid interface under different mesh sizes.

图 9  管道结构左端与中间局部网格放大图

Fig. 9.  Enhanced meshes of the left and middle parts of the channel structure.

图 10  不同流速下管道外表面温度分布

Fig. 10.  Temperature distribution on channel outer surface under different velocities.

图 11  不同流速下流固界面温度分布

Fig. 11.  Temperature distribution on fluid-solid interface under different velocities.

图 12  三根管道结构尺寸与边界条件

Fig. 12.  Dimensions and B.C. of a three-channel structure.

图 13  三管道结构FLM分析线系连成的网格图

Fig. 13.  Mesh connected by line sets of all points for FLM analysis of the three channel structure.

图 14  不同流速下的温度云图　(a) v = 0.002 m/s; (b) v = 0.005 m/s; (c) v = 0.01 m/s; (d) v = 0.5 m/s

Fig. 14.  Contours under different velocities: (a) v = 0.002 m/s; (b) v = 0.005 m/s; (c) v = 0.01 m/s; (d) v = 0.5 m/s.

图 15  不同流速下结构上部外表面温度变化曲线

Fig. 15.  Temperature variation curve on the outer surface of upper structure under different velocities.

图 16  不同流速下斜管流体域下表面温度变化曲线

Fig. 16.  Temperature variation curve on the lower surface of fluid domain of the oblique channel under different velocities.

图 17  计算得到的管道冷却系统温度云图

Fig. 17.  Contours of computed temperature over the channel cooling system.

图 18  计算得到的冷却系统上下表面和各区域界面的温度变化曲线

Fig. 18.  Variation curve of the computed temperature on the upper and lower surfaces as well as on the interfaces of the cooling system.

图 19  含三根管道的三维冷却结构尺寸与边界条件

Fig. 19.  Dimensions and B.C. of a 3D cooling structure with three channels.

图 20  所有配置点的线系连成的网格

Fig. 20.  Mesh connected by line sets of all collocation points

图 21  横截面上的配置点线系连成的网格与局部放大图

Fig. 21.  Mesh connected by line sets on the transverse section and a locally refined part around a corner.

图 22  总体结构上的温度云图

Fig. 22.  Contour plot of computed temperature over the entire structure.

图 23  管道走向垂直中面上的温度云图

Fig. 23.  Contour plot of computed temperature over the vertical middle plan.

图 24  沿图22中所示的线段${L_1}$上的温度分布

Fig. 24.  Computed temperature along line ${L_1}$ marked in Fig.22.

图 25  沿图22中所示的线段${L_2}$上的温度分布

Fig. 25.  Computed temperature along line ${L_2}$ marked in Fig.22.