A circulant-matrix-based hybrid optical flow method for PIV measurement with large displacement

The accurate measurement of fluid flows with large velocity gradients and a high velocity range, such as vortex flow and jet flow, is still paramount and challenging in fluid dynamics research. Particle image velocimetry (PIV) is a well-established experimental technique for determining fluid velocities; however, it cannot obtain an accurate result when the displacements of tracer particles are much larger than their sizes (Liu in J Fluids Eng 142(5):054051, 2020). In this study, we propose a novel hybrid optical flow formulation that consists of two parts, namely a circulant-matrix-based correlation and an adaptive weight optical flow, to solve this problem. First, a novel correlation method based on a circulant matrix is proposed, where the interrogation window is cyclically sampled to improve the measurable velocity range and measurement accuracy. Subsequently, we design a specific optical flow whose regularization parameters can be adjusted with changes in the velocity gradient and introduce a non-quadratic penalty function to achieve better stability and convergence. The solution provided by the circulant-matrix-based correlation is used as an initialization to account for large displacements with small particles and corrects the following adaptive weight optical flow in real time to accurately capture small vortex and turbulent structures. Velocity fields are estimated over synthetic and experimental particle images, and the velocity results are compared with advanced PIV methods. The results and comparisons show that the proposed method successfully achieves good performance in capturing small-scale vortices and turbulent structures when measuring fluid flows with large velocity gradients and a high velocity range. Graphic abstract