Fluid motion is a common phenomenon in observed nature and utilized in industries. Mastering the fluid flow is an important prerequisite for an in-depth study of fluid mechanics. The particle image velocimetry (PIV) is a non-contact global flow-field measurement and display technology that provides accurate data for flow-field measurements without affecting the flow field. The particle image velocimetry is mainly divided into two categories: cross-correlation and optical flow algorithms. The optical flow algorithm is primarily used in small-displacement scenarios. When the particle displacement is significantly larger than the particle size, the optical flow method cannot yield accurate results. The cross-correlation algorithm is mainly used in large displacement scenarios, and the combination of the two algorithms can satisfy more application scenarios. Although the hybrid algorithm has higher accuracy than the traditional algorithm in large- and small-displacement scenarios, the angle information is not well retained in the case of complex fluid. Because the image of the particle conforms to the Airy spot model and the light intensity satisfies the two-dimensional Gaussian distribution, if the Gaussian radial basis function interpolation is used, the velocity field refinement will be transformed into a surface reconstruction problem, and the reconstructed velocity field will have a higher accuracy. Therefore, we propose a cross-correlation optical flow mixing algorithm based on the Gaussian radial basis function interpolation to reduce the angular error.

Based on the traditional hybrid algorithm, in this study, the Gaussian radial basis function interpolation is used to replace bicubic interpolation and design a cross-correlation optical flow hybrid algorithm. First, a pair of particle images is inputted, and a cross-correlation method is used to extract the relatively large particle motion in each query window. A Gaussian radial basis function is used for data interpolation to fill the speed vector in each pixel. For each pixel, the image displacement is processed to remove the speed vector detected in the image. Subsequently, the initial velocity vector is determined using the HS optical flow method, and the residual velocity field is refined using the variable spectral flow method based on the dynamic illumination equation. The Gaussian radial basis function interpolation method is used to interpolate the velocity field at each layer, and the more refined velocity field vectors are obtained. Finally, the velocity field vectors obtained by the cross-correlation and optical flow algorithms are superimposed to obtain an accurate velocity field. The algorithm is quantitatively evaluated through a Rankine vortex simulation experiment. The influence of displacement and particle size on the accuracy of the algorithm is studied. Subsequently, a two-dimensional PIV experimental system is built, and rotation and water injection experiments are performed to simulate the vortex current field and jet field, respectively. The practicability of the proposed algorithm is verified.

In the Rankine vortex simulation experiment, the manifold reconstructed by the proposed method is more in line with the characteristics of the Rankine vortex and closer to the ground truth (Fig. 3). The root mean square error (RMSE) and average angular error of the cross-correlation optical flow hybrid algorithm based on Gaussian radial basis function interpolation are 27.36% and 38.32% lower than those of the Hybrid method 2020, respectively (Table 1). With an increase in the maximum displacement, the root mean square error gradually increases. In most cases, the hybrid algorithm based on Gaussian radial basis function interpolation is superior to the Hybrid method 2020. In the case of a small displacement, the RMSE can be decreased by approximately 45%, whereas in the case of a large displacement, the RMSE can be decreased by approximately 15% (Fig. 5). With an increase in particle size, the angle error first decreases and then the best reconstruction result is obtained when the particle size is 3 pixel. The proposed method can obtain good reconstruction results in the cases of both small and large particle sizes. In the case of particle size of 2-4 pixel, the average angle error of the proposed method is approximately 15% lower than that of the Hybrid method 2020 (Fig. 6). The results of water injection and rotation experiments verify the performance of the proposed algorithm in practical applications.

In this study, based on the traditional hybrid algorithm, the Gaussian radial basis function interpolation is used to replace bicubic interpolation, and a cross-correlation optical flow hybrid algorithm based on Gaussian radial basis function interpolation is proposed. This approach preserves the angle information in the complex flow field, which is not possible using the traditional hybrid algorithm. It changes considerably with velocity, and the method can accurately reconstruct flow fields. The proposed algorithm and the Hybrid method 2020 algorithm are used to reconstruct the velocity field in an experiment. The results show that the two algorithms can maintain high consistency in the entire manifold, and the proposed algorithm can retain more angle information. This verifies that the proposed algorithm can accurately reconstruct the actual complex flow field and has potential for practical applications.