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.

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.

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).

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.