基于POD-Galerkin降维方法的热毛细对流分岔分析

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. 

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

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.