Velocity from Flow Visualizations

This chapter discusses extraction of high-resolution velocity (or displacement) fields from flow visualization images, focusing on the physical foundations of the optical flow method (OFM) for global flow diagnostics. The projected motion equations are derived for typical flow visualizations based on projection of the transport equations and continuity equation in the 3D object space onto the image plane. Here, flow visualization images include laser sheet-induced fluorescence images, images of density-varying flow (Schlieren images, shadowgraph images, and transmittance images), transmittance images through scattering particulate flow, scattering images toward incident direction from particulate flow, laser sheet-illuminated particle images, and neutron radiography images. The projected motion equations provide the relation between the radiance projected to a digital camera and the light-path-averaged velocity field weighted in a relevant field quantity. These equations for different flow visualizations have the same mathematical form, which can be recast to the optical flow equation in the image plane where the optical flow is proportional to the light-path-averaged velocity. To determine the optical flow as an inverse problem, a variational formulation is proposed, and the Euler-Lagrange equation with the Neumann condition is given. As examples, OFM is applied to planetary cloud tracking, neutron radiography (NR) of two-phase flow, particle image velocimetry (PIV), and background-oriented Schlieren (BOS) visualization.