Hummingbirds generate bilateral vortex loops during hovering: evidence from flow visualization

This article derives a method to estimate and correct the bias error of the shift vector’s absolute length in the presence of curved streamlines. The main idea is to identify the most likely streamline with constant curvature from the second-order shift vector and its gradient. The work establishes a theoretical framework including the systematic errors of the first-order and second-order shift vector’s absolute value and angle. Synthetic images of a stationary vortex are used to validate the proposed method. The curvature-correction is also applied to a synthetic flow field with non-constant curvature to demonstrate its potential for more realistic flow fields. The results reveal that second-order accurate vector fields suffer from a biased shift vector length depending on the streamline’s curvature and on the shift vector length. The bias error is negligible for vector fields with a shift vector length below the streamline curvature radius. For large shift vectors or strong curvatures, the bias error can be significantly reduced with the developed method. The approach is very general and can be applied to any vector field obtained from window-correlation particle image velocimetry (PIV), single-pixel ensemble-correlation PIV, particle tracking velocimetry or optical flow methods. It also works for all 3D extensions of the techniques, such as 3D-PTV or tomographic PIV.