Determination of Velocity and Skin Friction Fields from Images by Solving Projected Motion Equations

This paper describes a general method to extract velocity and skin friction fields from images based on the projected motion equations. The generic form of the projected motion equations and the optical flow equation are given for typical flow visualizations, where the optical flow is proportional to the path-averaged velocity of fluid or particles weighted with a specific field quantity like dye concentration, fluid density and particle density. The variational formulation with a smoothness constraint is used to calculate the optical flow in the image plane that is proportional to the path-averaged velocity. This method allows determination of velocity fields from various visualization images of complex flows from large atmospheric storms to supersonic mixing layers. Similarly, the projected motion equation for oil doped with luminescent molecules is derived under an assumption that the luminescent intensity is proportional to the oil thickness when the oil film is optically thin. The equation is solved using the variational method to extract a skin friction field from images of the evolution of the luminescent oil film in flows. Examples are given to illustrate applications of the proposed method.