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基于POD-Galerkin降维方法的热毛细对流分岔分析

郭子漪 赵建福 李凯 胡文瑞

郭子漪, 赵建福, 李凯, 胡文瑞. 基于POD-Galerkin降维方法的热毛细对流分岔分析. 力学学报, 2022, 54(5): 1186-1198 doi: 10.6052/0459-1879-21-642
引用本文: 郭子漪, 赵建福, 李凯, 胡文瑞. 基于POD-Galerkin降维方法的热毛细对流分岔分析. 力学学报, 2022, 54(5): 1186-1198 doi: 10.6052/0459-1879-21-642
Guo Ziyi, Zhao Jianfu, Li Kai, Hu Wenrui. Bifurcation analysis of thermocapillary convection based on POD-Galerkin reduced-order method. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(5): 1186-1198 doi: 10.6052/0459-1879-21-642
Citation: Guo Ziyi, Zhao Jianfu, Li Kai, Hu Wenrui. Bifurcation analysis of thermocapillary convection based on POD-Galerkin reduced-order method. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(5): 1186-1198 doi: 10.6052/0459-1879-21-642

基于POD-Galerkin降维方法的热毛细对流分岔分析

doi: 10.6052/0459-1879-21-642
基金项目: 国家自然科学基金(11972353, 12032020)和中国科学院前沿科学重点研究(QYZDY-SSWJSC040)资助项目
详细信息
    作者简介:

    李凯, 研究员, 主要研究方向: 微重力流体物理. E-mail: likai@imech.ac.cn

  • 中图分类号: O357

BIFURCATION ANALYSIS OF THERMOCAPILLARY CONVECTION BASED ON POD-GALERKIN REDUCED-ORDER METHOD

  • 摘要: 作为流动与传热相互耦合的非线性过程, 热毛细对流有着复杂的转捩过程, 探究流场和温度场随参数变化而发生的分岔现象, 是热毛细对流研究的一个重要课题. 基于本征正交分解的POD-Galerkin降维方法可以通过提取特征模态, 构建低维模型, 实现流场的快速计算. 数值分岔方法可以通过求解含参数动力系统的分岔方程, 直接计算稳定解和分岔点. 探究了将直接数值模拟方法、POD-Galerkin降维方法、数值分岔方法的优势结合, 以提高热毛细对流转捩过程分析效率的可行性. 利用直接数值模拟得到的流场和温度场数据, 构建了不同体积比下, 二维有限长液层热毛细对流的POD-Galerkin低维模型, 在低维模型上采用数值积分及数值分岔方法计算了分岔点, 得到了低维方程的分岔图. 在一定参数范围内, 在低维模型上模拟热毛细对流, 对雷诺数和体积比进行参数外推, 通过与直接数值模拟的结果对比, 验证了低维模型的准确性与鲁棒性. 说明了低维方程可以定性反映原高维系统的流动特性, 而定量方面, 由低维模型和直接数值模拟计算得到的周期解频率的相对误差大约为5%. 验证了利用POD-Galerkin降维方法研究热毛细对流的可行性.

     

  • 图  1  有限长二维液层模型

    Figure  1.  Limited liquid layer model

    图  2  DNS与低维模型得到的第一模态系数的时间序列对比

    Figure  2.  The comparison of time evolution of first eigenmode coefficient obtained by DNS and low-order model

    图  3  液层中心点温度振荡时间序列对比

    Figure  3.  The comparison of time evolution of the monitor point at the center of the liquid layer

    图  4  不同Re数下由DNS和低维模型计算出的液层中心点温度振荡时间序列、频谱的对比 (Vr = 1.00)

    Figure  4.  The comparison of time evolution and frequency spectrum of temperature oscillation at the center of the liquid layer obtained by DNS and low-order model for various Re (Vr = 1.00)

    图  5  各体积比下低维方程的分岔图

    Figure  5.  Bifurcation diagram of reduced-order model under different volume ratios

    图  6  体积比为0.95时不同Re下由DNS和低维模型计算出的液层中心点温度振荡时间序列、频谱的对比

    Figure  6.  The comparison of time evolution and frequency spectrum of temperature oscillation at the center of the liquid layer obtained by DNS and low-order model for various Re numbers under Vr = 0.95

    图  7  体积比为1.05时不同Re下由DNS和低维模型计算出的液层中心点温度振荡时间序列、频谱的对比

    Figure  7.  The comparison of time evolution and frequency spectrum of temperature oscillation at the center of the liquid layer obtained by DNS and low-order model for various Re numbers under Vr = 1.05

    表  1  直接数值模拟算法验证

    Table  1.   Verification of DNS method

    Vr = 1.00Recri_1errrel_1/%Recri_2errrel_2/%
    45×15 (uniform) 1930 1.03 4350 13.35
    60×20 (non-uniform q = 1) 1975 1.28 4600 8.37
    60×20 (non-uniform q = 2) 1925 1.28 5050 0.60
    benchmark 1950 5020
    下载: 导出CSV

    表  2  各体积比下的临界雷诺数

    Table  2.   Critical Reynolds number under different volume ratios

    VrRec1Rec2
    0.95 2450 7250
    1.00 1925 5050
    1.05 1650 3550
    下载: 导出CSV

    表  3  不同模态数(n)下的能量占比(Re = 3500, Vr = 1.00)

    Table  3.   The accumulative energy contribution of the first n POD eigenmodes (Re = 3500, Vr = 1.00)

    nEnergy/%
    4 99.17666
    6 99.89763
    8 99.98758
    10 99.99704
    12 99.99893
    14 99.99956
    下载: 导出CSV

    表  4  低维方程得到的温度振荡频率与DNS结果对比(Vr = 1.00)

    Table  4.   The comparison of temperature oscillation frequency obtained by DNS and low-order model

    RefT_PODFT_DNSError/%
    25003.00962.87284.7619
    30006.41516.22643.0306
    40007.29707.39981.3892
    45003.90534.14205.7146
    下载: 导出CSV

    表  5  利用低维方程与DNS计算的临界Re对比

    Table  5.   The comparison of the critical Reynolds number obtained by DNS and reduced-order model

    MethodVr = 0.95Vr = 1.00Vr = 1.05
    Rec1Rec2Rec1Rec2Rec1Rec2
    POD 2020 6520 1620 5020 1420 3420
    DNS 2450 7250 1925 5050 1650 3550
    error/% 17.55 10.07 15.84 0.59 13.94 3.66
    下载: 导出CSV

    表  6  低维方程周期解频率与DNS结果对比(Vr = 1.00)

    Table  6.   The comparison of the frequency of the periodic solution obtained by DNS and reduced-order model (Vr = 1.00)

    ReDNSfDNSRePODfPOD
    2000 2.5937 2004 2.6546
    2500 2.8728 2504 3.0102
    3000 3.0233 3004 3.1878
    3500 3.3259 3504 3.3795
    4000 3.7516 4004 3.6311
    4500 4.1611 4504 3.8926
    5000 4.4706 5004 4.1408
    下载: 导出CSV
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出版历程
  • 收稿日期:  2021-12-03
  • 录用日期:  2022-03-07
  • 网络出版日期:  2022-03-08
  • 刊出日期:  2022-05-01

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