气动光学效应波前模态的泽尼克-本征正交分解

Zernike and Proper Orthogonal Decomposition of Wavefront Modes of Aero-Optical Effects

本征正交分解(POD)方法已经广泛应用于时间变化随机场的分析,但其直接法或快照法存在自身固有缺陷,前者使关联矩阵求解特征值和特征向量较为困难,后者的有限抽样帧数会影响对随机场的统计特性分析。鉴于此,本文建立了基于泽尼克(Zernike)多项式加权系数的Zernike-POD方法,用于气动光学效应的波前模态统计分析。对于圆域问题,Zernike多项式阶数给定后,各阶加权系数与波前分布是一一对应的,并且通常用几百阶多项式就足以复原各种复杂波面形状。Zernike-POD方法由对波前本身改为对Zernike多项式加权系数的分解和模态计算,由于多项式阶数远少于波前的空间离散点数,因此,关联矩阵的维数降低,计算量减少,计算效率显著提高。Zernike-POD方法不损失空间分辨率,同时不需要限制最高采样帧数,因此,时间统计特性不受影响,预估的波前模态具有较高的时空分辨率。为验证该方法的有效性,用大涡模拟方法开展圆柱绕流数值仿真,并计算圆柱尾迹卡门涡街结构产生的时间系列气动光学效应数据,将该数据用于波前模态分析,波前空间分辨率为100×100,采样帧数为2万帧,Zernike多项式阶数取为217阶。一阶模态与稳态波前分布相似,二阶与三阶、四阶与五阶模态近似互为配对关系;前10阶模态能够基本复原波面形状,前49阶模态含能在97%以上,用完整模态重构的波前与原始波前没有本质差异;模态加权系数及其功率谱随阶数增加而呈下降趋势,前五阶模态加权系数的功率谱尖峰频率与光学窗口中心点脉动速度情况较为一致,对应卡门涡街的主频,斯特劳哈尔数St约为0.22。Zernike-POD方法适用于圆域波前模态分析,也适用于环形域和正方形域,并能够推广到流场结构、图像与信号等处理领域。

Objective Proper orthogonal decomposition(POD)method has been widely applied to time-dependent field analysis,but its direct method and snapshot method both have their inherent problems.The former makes it difficult to solve eigenvalues and eigenvectors of correlation matrices,and the limited sampling number of the latter will affect statistical random field analysis.The direct method needs to solve eigenvalues and eigenvectors of spatial correlation matrices,and the correlation matrix dimensions are the spatial discrete points of the field.When there are more discrete points in the space,the matrix dimensions are high,which results in a large amount of computation,consumed time,occupied memory,and even difficult solutions.The snapshot method is to solve temporal correlation matrices.Generally,by sampling about 200 frames,the correlation matrix dimensions and computation amount are significantly reduced,which makes the POD method practical and operable.However,the few sampling frames will affect the statistical analysis of random field modes,and the calculated modes will vary with the frame number and interval time between frames.Thus,the Zernike and proper orthogonal decomposition(Z-POD)method based on the Zernike polynomial weighted coefficient is established for statistical wavefront mode analysis of aero-optical effects.Methods The Z-POD method which introduces the wavefront reconstruction method based on Zernike polynomials is changed from the decomposition of the wavefront itself to that of the weighted coefficients of Zernike polynomials.For the circle domain,given the Zernike polynomial order,weighted coefficients correspond to wavefront distribution one by one,and polynomials of several hundred orders are usually enough to recover various complex wavefront shapes.Since the polynomial order is far less than the discrete point number in the wavefront space,the correlation matrix dimensions are reduced,with reduced computation amount and significantly improved computation calculation efficiency.The Z-POD method does not lose spatial resolution and does not need to limit the maximum samples.Therefore,the temporal statistical characteristics are not affected and predicted wavefront modes have high spatio-temporal resolution.Results and Discussions To verify the effectiveness of the Z-POD method,we employ the large eddy simulation(LES)method to simulate flow around a cylinder and calculate the time series aero-optical effect wavefront generated by the Karman vortex street structure in the cylinder wake for wavefront modal analysis.The spatial resolution of the wavefront is 100×100,the sampled frame number is 20000,and the order of Zernike polynomials is 217.First-order mode and steady-state wavefront distribution are similar(Fig.7),second-order and third-order modes,and fourth-order and fifth-order modes are approximately paired with each other(Fig.8).The first ten modes can restore the wavefront shape,the first 49 modes contain more than 97%energy,and the wavefront reconstructed with the complete modes has no essential differences from the original wavefront(Figs.9 and 10).The modal weighted coefficients and their power spectrum decrease with increasing order.The peak frequencies of the power spectrum of weighted coefficients of the first five modes are consistent with those of fluctuation velocity at the center point of the optical window,corresponding to the main frequency of Karman vortex street,with the Strauhal number of about 0.22(Figs.3 and 11).Conclusions As it is difficult for us to employ the POD method for statistical analysis of random fields with high spatial resolution and high sampling frames,the Z-POD method is proposed for wavefront modal analysis of time-dependent series aero-optical effects.Based on the original POD method,the Z-POD method introduces wavefront reconstruction based on Zernike polynomials and carries out POD of the weighted coefficients of Zernike polynomials instead of the wavefront itself.Since wavefront reconstruction based on Zernike polynomials has the function of dimensionality reduction for wavefront,the complex wavefront shape can be usually restored with polynomials of several hundred orders,and there is no strict restriction on the number of discrete points and sampling frames of wavefront.Therefore,the correlation matrix dimensions for the Z-POD method are significantly reduced,the computational efficiency is significantly improved,and the wavefront modal analysis can be guaranteed to have a sufficiently high spatio-temporal resolution. In the time series data analysis of wavefront by Karman vortex generated in the wake flow around a cylinder, the Z-POD method also has the advantage of restoring the original wavefront shape with a few modes, and the energy ratios of the first order, 10th order,and 49th order modes are above 44%, 88%, and 97% respectively. Additionally, the wavefront reconstructed with the whole 217 modes is not substantially different from the original wavefront. The Z-POD method has been authorized by a China National invention patent. Since the wavefront reconstruction method based on Zernike polynomials is also applicable to the ring domain and square domain, it is also suitable for statistical analysis of wavefront modes on such domains, and can also be extended to analysis and processing of images, flow fields, and signals on two-dimensional fields.