Error analysis of 3D-PTV through unsteady interfaces

The feasibility of stereoscopic flow measurements through an unsteady optical interface is investigated. Position errors produced by a wavy optical surface are determined analytically, as are the optimal viewing angles of the cameras to minimize such errors. Two methods of measuring the resulting velocity errors are proposed. These methods are applied to 3D particle tracking velocimetry (3D-PTV) data obtained through the free surface of a water flow within a cavity adjacent to a shallow channel. The experiments were performed using two sets of conditions, one having no strong surface perturbations, and the other exhibiting surface gravity waves. In the latter case, the amplitude of the gravity waves was 6% of the water depth, resulting in water surface inclinations of about ${0.2}^{\circ }$. (The water depth is used herein as a relevant length scale, because the measurements are performed in the entire water column. In a more general case, the relevant scale is the maximum distance from the interface to the measurement plane, H, which here is the same as the water depth.) It was found that the contribution of the waves to the overall measurement error is low. The absolute position errors of the system were moderate (1.2% of H). However, given that the velocity is calculated from the relative displacement of a particle between two frames, the errors in the measured water velocities were reasonably small, because the error in the velocity is the relative position error over the average displacement distance. The relative position error was measured to be 0.04% of H, resulting in small velocity errors of 0.3% of the free-stream velocity (equivalent to 1.1% of the average velocity in the domain). It is concluded that even though the absolute positions to which the velocity vectors are assigned is distorted by the unsteady interface, the magnitude of the velocity vectors themselves remains accurate as long as the waves are slowly varying (have low curvature). The stronger the disturbances on the interface are (high amplitude, short wave length), the smaller is the distance from the interface at which the measurements can be performed.