搜索

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

有限线法及其在流固域间耦合传热中的应用

高效伟 丁金兴 刘华雩

引用本文:
Citation:

有限线法及其在流固域间耦合传热中的应用

高效伟, 丁金兴, 刘华雩

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

Gao Xiao-Wei, Ding Jin-Xing, Liu Hua-Yu
PDF
HTML
导出引用
  • 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.
    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.
      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.
      Corresponding author: Gao Xiao-Wei, xwgao@dlut.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 12072064).
    [1]

    Zienkiewicz O C, Taylor R L, Fox D 2014 The Finite Element Method for Solid and Structural Mechanics (7th Ed.) (Butterworth-Heinemann: Elsevier)

    [2]

    Zienkiewicz O C, Taylor R L, Nithiarasu P 2014 The Finite Element Method for Fluid Dynamics (7th Ed.)(Butterworth-Heinemann: Elsevier)

    [3]

    Wen P H, Cao P, Korakianitis T 2014 Eng. Anal. Bound. Elem. 46 116Google Scholar

    [4]

    Li M, Wen P H 2014 Int. J. Numer. Methods Eng. 99 372Google Scholar

    [5]

    Gao X W, Huang S Z, Cui M, Ruan B, Zhu Q H, Yang K, Lv J, Peng H F 2017 Int. J. Heat Mass Transf. 115 882Google Scholar

    [6]

    Zheng Y T, Gao X W, Lv J, Peng H F 2020 Int. J. Numer. Methods Eng. 121 3722Google Scholar

    [7]

    陶文铨 2001 数值传热学 (西安: 西安交通大学出版社)

    Tao W Q 2001 Numerical Heat Transfer (Xi’an: Xi’an JiaoTong University Press) (in Chinese)

    [8]

    Moukalled F, Mangani L, Darwish M 2015 The Finite Volume Method in Computational Fluid Dynamics: an Advanced Introduction with OpenFOAM and MATLAB (Cham: Springer)

    [9]

    姚振汉, 王海涛 2010 边界元法 (北京: 高等教育出版社)

    Yao Z H, Wang H T 2010 Boundary Element Methods (Beijing: Higher Education Press) (in Chinese)

    [10]

    胡金秀, 高效伟 2016 物理学报 65 014701Google Scholar

    Hu J X, Gao X W 2016 Acta Phys. Sin. 65 014701Google Scholar

    [11]

    张见明 2010 计算机辅助工程 19 5Google Scholar

    Zhang J M 2010 Comput. Aided Eng. 19 5Google Scholar

    [12]

    袁驷 1992 数值计算与计算机应用 13 252Google Scholar

    Yuan S 1992 J. Num. Methods Comput. Applicat. 13 252Google Scholar

    [13]

    Gao X W, Liang Y, Xu B B, Yang K, Peng H F 2019 Eng. Anal. Bound. Elem. 108 422Google Scholar

    [14]

    张雄, 宋康祖, 陆明万 2003 计算力学学报 20 725Google Scholar

    Zhang X, Song Z K, Lu M W 2003 Chin. J. Comput. Mech. 20 725Google Scholar

    [15]

    王东东, 张汉杰, 梁庆文 2016 计算力学学报 33 605Google Scholar

    Wang D D, Zhang H J, Liang Q W 2016 Chinese Journal of Computational Mechanics 33 605Google Scholar

    [16]

    程玉民 2015 无网格方法 (北京: 科学出版社)

    Cheng Y M 2015 Meshless Methods (Beijing: Sciense Press) (in Chinese)

    [17]

    Karageorghis A, Lesnic D, Marin L 2021 J. Eng. Math. 126 10Google Scholar

    [18]

    傅卓佳, 习强, 黄河 2019 力学与工程——数值计算和数据分析 2019 学术会议论文集 第77页

    Fu Z J, Xi Q, Huang H 2019 Mechanics and Engineering — Numerical Computation and Data Analysis Beijing, China, April 19–21, 2019 p77 (in Chinese)

    [19]

    Lv J, Sheng G Y, Gao X W, Zhang H W 2015 Int. J. Comput. Methods 12 1550026Google Scholar

    [20]

    Dolejší V, Feistauer M 2015 Discontinuous Galerkin Method: Analysis and Applications to Compressible Flow (Cham: Springer)

    [21]

    高效伟, 彭海峰, 杨恺, 王静 2015 高等边界单元法 (北京: 科学出版社)

    Gao X W, Peng H F, Yang K, Wang J 2015 Advanced Boundary Element Method (Beijing: Sciense Press) (in Chinese)

    [22]

    王勖成 2003 有限单元法 (北京: 清华大学出版社)

    Wang X C 2003 Finite Element Method (Beijing: Tsinghua University Press) (in Chinese)

    [23]

    高效伟, 徐兵兵, 吕军, 彭海峰 2019 力学学报 51 703Google Scholar

    Gao X W, Xu B B, Lv J, Peng H F 2019 Chin. J. Theor. Appl. Mech. 51 703Google Scholar

    [24]

    Liu H Y, Gao X W, Xu B B 2019 Comput. Fluids 192 104276Google Scholar

    [25]

    Xu B B, Gao X W, Jiang W W, Cui M, Lv J 2019 Eng. Fract. Mech. 218 106575Google Scholar

    [26]

    Gao X W, Ding J X, Cui M, Yang K 2019 Eng. Anal. Bound. Elem. 109 117Google Scholar

    [27]

    Gao X W 2021 The 2021 International Conference on Applied Mathematics, Modeling and Computer Simulation (AMMCS 2021) Wuhan, China, November 13–14, 2021

    [28]

    Gao X W, Liu H Y, Ruan B 2021 Comput. Struct. 243 106411Google Scholar

  • 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.

    表 1  各计算域的材料参数与入口边界条件

    Table 1.  Material parameters and boundary conditions of each computational domain.

    $ {\varOmega _1} $$ {\varOmega _2} $$ {\varOmega _3} $$ {\varOmega _4} $$ {\varOmega _5} $$ {\varOmega _6} $$ {\varOmega _7} $
    λ1000.62000.62000.610
    $ {T_{{\text{in}}}} $300500300
    v0.10.51.0
    下载: 导出CSV

    表 2  三维流固结构各计算域的材料参数与边界条件

    Table 2.  Material parameters and boundary conditions of each computational domain.

    $ {\varOmega _1} $$ {\varOmega _2} $$ {\varOmega _3} $$ {\varOmega _4} $$ {\varOmega _5} $$ {\varOmega _6} $
    λ10202000.60.60.6
    $ {T_{{\text{in}}}} $300350300
    v0.20.50.2
    下载: 导出CSV

    表 3  三种计算方法的节点数与计算时间比较

    Table 3.  Comparison of total number of nodes and computational time for three methods.

    FLMFLM_fineFLUENTFLUENT_fineCOMSOL
    总节点数1727895827995995194298589570751
    计算时间/s38135115910585
    下载: 导出CSV
  • [1]

    Zienkiewicz O C, Taylor R L, Fox D 2014 The Finite Element Method for Solid and Structural Mechanics (7th Ed.) (Butterworth-Heinemann: Elsevier)

    [2]

    Zienkiewicz O C, Taylor R L, Nithiarasu P 2014 The Finite Element Method for Fluid Dynamics (7th Ed.)(Butterworth-Heinemann: Elsevier)

    [3]

    Wen P H, Cao P, Korakianitis T 2014 Eng. Anal. Bound. Elem. 46 116Google Scholar

    [4]

    Li M, Wen P H 2014 Int. J. Numer. Methods Eng. 99 372Google Scholar

    [5]

    Gao X W, Huang S Z, Cui M, Ruan B, Zhu Q H, Yang K, Lv J, Peng H F 2017 Int. J. Heat Mass Transf. 115 882Google Scholar

    [6]

    Zheng Y T, Gao X W, Lv J, Peng H F 2020 Int. J. Numer. Methods Eng. 121 3722Google Scholar

    [7]

    陶文铨 2001 数值传热学 (西安: 西安交通大学出版社)

    Tao W Q 2001 Numerical Heat Transfer (Xi’an: Xi’an JiaoTong University Press) (in Chinese)

    [8]

    Moukalled F, Mangani L, Darwish M 2015 The Finite Volume Method in Computational Fluid Dynamics: an Advanced Introduction with OpenFOAM and MATLAB (Cham: Springer)

    [9]

    姚振汉, 王海涛 2010 边界元法 (北京: 高等教育出版社)

    Yao Z H, Wang H T 2010 Boundary Element Methods (Beijing: Higher Education Press) (in Chinese)

    [10]

    胡金秀, 高效伟 2016 物理学报 65 014701Google Scholar

    Hu J X, Gao X W 2016 Acta Phys. Sin. 65 014701Google Scholar

    [11]

    张见明 2010 计算机辅助工程 19 5Google Scholar

    Zhang J M 2010 Comput. Aided Eng. 19 5Google Scholar

    [12]

    袁驷 1992 数值计算与计算机应用 13 252Google Scholar

    Yuan S 1992 J. Num. Methods Comput. Applicat. 13 252Google Scholar

    [13]

    Gao X W, Liang Y, Xu B B, Yang K, Peng H F 2019 Eng. Anal. Bound. Elem. 108 422Google Scholar

    [14]

    张雄, 宋康祖, 陆明万 2003 计算力学学报 20 725Google Scholar

    Zhang X, Song Z K, Lu M W 2003 Chin. J. Comput. Mech. 20 725Google Scholar

    [15]

    王东东, 张汉杰, 梁庆文 2016 计算力学学报 33 605Google Scholar

    Wang D D, Zhang H J, Liang Q W 2016 Chinese Journal of Computational Mechanics 33 605Google Scholar

    [16]

    程玉民 2015 无网格方法 (北京: 科学出版社)

    Cheng Y M 2015 Meshless Methods (Beijing: Sciense Press) (in Chinese)

    [17]

    Karageorghis A, Lesnic D, Marin L 2021 J. Eng. Math. 126 10Google Scholar

    [18]

    傅卓佳, 习强, 黄河 2019 力学与工程——数值计算和数据分析 2019 学术会议论文集 第77页

    Fu Z J, Xi Q, Huang H 2019 Mechanics and Engineering — Numerical Computation and Data Analysis Beijing, China, April 19–21, 2019 p77 (in Chinese)

    [19]

    Lv J, Sheng G Y, Gao X W, Zhang H W 2015 Int. J. Comput. Methods 12 1550026Google Scholar

    [20]

    Dolejší V, Feistauer M 2015 Discontinuous Galerkin Method: Analysis and Applications to Compressible Flow (Cham: Springer)

    [21]

    高效伟, 彭海峰, 杨恺, 王静 2015 高等边界单元法 (北京: 科学出版社)

    Gao X W, Peng H F, Yang K, Wang J 2015 Advanced Boundary Element Method (Beijing: Sciense Press) (in Chinese)

    [22]

    王勖成 2003 有限单元法 (北京: 清华大学出版社)

    Wang X C 2003 Finite Element Method (Beijing: Tsinghua University Press) (in Chinese)

    [23]

    高效伟, 徐兵兵, 吕军, 彭海峰 2019 力学学报 51 703Google Scholar

    Gao X W, Xu B B, Lv J, Peng H F 2019 Chin. J. Theor. Appl. Mech. 51 703Google Scholar

    [24]

    Liu H Y, Gao X W, Xu B B 2019 Comput. Fluids 192 104276Google Scholar

    [25]

    Xu B B, Gao X W, Jiang W W, Cui M, Lv J 2019 Eng. Fract. Mech. 218 106575Google Scholar

    [26]

    Gao X W, Ding J X, Cui M, Yang K 2019 Eng. Anal. Bound. Elem. 109 117Google Scholar

    [27]

    Gao X W 2021 The 2021 International Conference on Applied Mathematics, Modeling and Computer Simulation (AMMCS 2021) Wuhan, China, November 13–14, 2021

    [28]

    Gao X W, Liu H Y, Ruan B 2021 Comput. Struct. 243 106411Google Scholar

  • [1] 何欣波, 魏兵. 基于悬挂变量的显式无条件稳定时域有限差分亚网格算法. 物理学报, 2024, 73(8): 080202. doi: 10.7498/aps.73.20231813
    [2] 孙佳坤, 林传栋, 苏咸利, 谭志城, 陈亚楼, 明平剑. 离散Boltzmann方程的求解: 基于有限体积法. 物理学报, 2024, 73(11): 110504. doi: 10.7498/aps.73.20231984
    [3] 于欣如, 崔继峰, 陈小刚, 慕江勇, 乔煜然. 平行板微通道中一类不可压缩微极性流体在高Zeta势下的时间周期电渗流. 物理学报, 2024, 0(0): . doi: 10.7498/aps.73.20240591
    [4] 张天鸽, 任美蓉, 崔继峰, 陈小刚, 王怡丹. 变截面微管道中高zeta势下幂律流体的旋转电渗滑移流动. 物理学报, 2022, 71(13): 134701. doi: 10.7498/aps.71.20212327
    [5] 何郁波, 唐先华, 林晓艳. 基于格子玻尔兹曼方法的一类FitzHugh-Nagumo系统仿真研究. 物理学报, 2016, 65(15): 154701. doi: 10.7498/aps.65.154701
    [6] 张琪, 张然, 宋海明. 美式回望期权定价问题的有限体积法. 物理学报, 2015, 64(7): 070202. doi: 10.7498/aps.64.070202
    [7] 杜超凡, 章定国. 基于无网格点插值法的旋转悬臂梁的动力学分析. 物理学报, 2015, 64(3): 034501. doi: 10.7498/aps.64.034501
    [8] 刘建晓, 张郡亮, 苏明敏. 基于时域有限差分法的各向异性铁氧体圆柱电磁散射分析. 物理学报, 2014, 63(13): 137501. doi: 10.7498/aps.63.137501
    [9] 王光辉, 王林雪, 王灯山, 刘丛波, 石玉仁. K(m,n,p)方程多-Compacton相互作用的数值研究. 物理学报, 2014, 63(18): 180206. doi: 10.7498/aps.63.180206
    [10] 辛成运, 程晓舫, 张忠政. 基于有限立体角测量的谱色测温法. 物理学报, 2013, 62(3): 030702. doi: 10.7498/aps.62.030702
    [11] 乔海亮, 王玥, 陈再高, 张殿辉. 全矢量有限差分法分析任意截面波导模式. 物理学报, 2013, 62(7): 070204. doi: 10.7498/aps.62.070204
    [12] 彭武, 何怡刚, 方葛丰, 樊晓腾. 二维泊松方程的遗传PSOR改进算法. 物理学报, 2013, 62(2): 020301. doi: 10.7498/aps.62.020301
    [13] 鲁思龙, 吴先良, 任信钢, 梅诣偲, 沈晶, 黄志祥. 色散周期结构的辅助场时域有限差分法分析. 物理学报, 2012, 61(19): 194701. doi: 10.7498/aps.61.194701
    [14] 杨秀丽, 戴保东, 栗振锋. 弹性力学的复变量无网格局部 Petrov-Galerkin 法. 物理学报, 2012, 61(5): 050204. doi: 10.7498/aps.61.050204
    [15] 郑保敬, 戴保东. 位势问题改进的无网格局部Petrov-Galerkin法. 物理学报, 2010, 59(8): 5182-5189. doi: 10.7498/aps.59.5182
    [16] 尹经禅, 肖晓晟, 杨昌喜. 光纤中受激Brillouin散射动态弛豫振荡特性及其抑制方法. 物理学报, 2009, 58(12): 8316-8325. doi: 10.7498/aps.58.8316
    [17] 梁 双, 吕燕伍. 有限元法计算GaN/AlN量子点结构中的电子结构. 物理学报, 2007, 56(3): 1617-1620. doi: 10.7498/aps.56.1617
    [18] 谭新玉, 张端明, 李智华, 关 丽, 李 莉. 纳秒脉冲激光沉积薄膜过程中的烧蚀特性研究. 物理学报, 2005, 54(8): 3915-3921. doi: 10.7498/aps.54.3915
    [19] 赵红东, 宋殿友, 张智峰, 孙 静, 孙 梅, 武 一, 温幸饶. n型DBR中电势对垂直腔面发射激光器阈值的影响. 物理学报, 2004, 53(11): 3744-3747. doi: 10.7498/aps.53.3744
    [20] 马坚伟, 杨慧珠, 朱亚平. 多尺度有限差分法模拟复杂介质波传问题. 物理学报, 2001, 50(8): 1415-1420. doi: 10.7498/aps.50.1415
计量
  • 文章访问数:  4042
  • PDF下载量:  77
  • 被引次数: 0
出版历程
  • 收稿日期:  2022-04-27
  • 修回日期:  2022-05-08
  • 上网日期:  2022-09-30
  • 刊出日期:  2022-10-05

/

返回文章
返回