BIFURCATION ANALYSIS OF THERMOCAPILLARY CONVECTION BASED ON POD-GALERKIN REDUCED-ORDER METHOD
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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.Abstract: Thermocapillary convection is driven by surface tension gradient caused by temperature gradient. The flow is subject to nonlinear interactions between convection and heat transfer, so it has complex transition behaviors. It is significant to investigate the flow bifurcation phenomenon as parameters in the governing equations change. The POD-Galerkin reduced-order method is a fast fluid computational method, based on proper orthogonal decomposition and Galerkin projection. The numerical bifurcation method finds the parameter values at which bifurcation exists by computing the asymptotic flow states and bifurcation points directly. In order to tackle flow transition problems in a more efficient way, a combination of direct numerical simulation, POD-Galerkin reduced-order method and numerical bifurcation method is applied to investigate the transition behavior of thermocapillary convection in a liquid layer. The POD reduced-order model of thermocapillary convection in a 2D cavity under different volume ratios is established and its bifurcation diagram is obtained by numerical bifurcation method. The validity of such a model for Reynolds numbers and volume ratios that are different from those for which the model is derived is studied and the possibility of modelling thermocapillary flow in a simple geometry over a range of flow parameters is assessed. Compared with the results obtained by direct numerical simulation, the accuracy and robustness of the low-order model are verified. The results show that the reduced-order model reflects qualitatively similar flow characteristics to the original high-order system, and quantitively, the relative error of frequency of periodic solution of the reduced-order model to that obtained by the direct numerical simulation is around 5%. Hence, the feasibility of the POD-Galerkin reduced-order method on thermocapillary convection is confirmed.
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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 = 1.00 Recri_1 errrel_1/% Recri_2 errrel_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 — 
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表 2 各体积比下的临界雷诺数
Table 2. Critical Reynolds number under different volume ratios
Vr Rec1 Rec2 0.95 2450 7250 1.00 1925 5050 1.05 1650 3550
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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)
n Energy/% 4 99.17666 6 99.89763 8 99.98758 10 99.99704 12 99.99893 14 99.99956
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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
Re fT_POD FT_DNS Error/% 2500 3.0096 2.8728 4.7619 3000 6.4151 6.2264 3.0306 4000 7.2970 7.3998 1.3892 4500 3.9053 4.1420 5.7146
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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
Method Vr = 0.95 Vr = 1.00 Vr = 1.05 Rec1 Rec2 Rec1 Rec2 Rec1 Rec2 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
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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)
ReDNS fDNS RePOD fPOD 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
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