Measurements of the unsteady wall shear stress vector using multi-aperture defocusing microscopic particle tracking velocimetry

A volumetric, three component microscopic particle tracking velocimetry (μPTV) system is presented which relies on single window optical access for both high-speed tracer illumination and image recording. Similar to the triple aperture “defocusing” concept originally introduced by Willert and Gharib (1992), the wall distance of individual particles is obtained from the size of projected particle image triplets formed by a triplet of apertures on the entrance pupil of the microscope lens. The measurement principle is validated with particle tracking measurements of a canonical turbulent boundary layer (TBL) within the closed test section of a wind tunnel at free-stream velocities of $5.2\le U_{\infty }\le 20\text{m/s}$ with corresponding shear Reynolds numbers of $560\le {\text{Re}}_{\tau }\le 1630$. Velocity profiles and higher order statistics are obtained by bin averaging of the particle velocity data up to the inner turbulence peak at a wall distance of $y^{+}=\nu /u_{\tau }\approx 15$ with a spatial resolution better than 5 µm. Excellent agreement with DNS data was obtained at similar Reynolds numbers. The unsteady wall shear stress (WSS) is estimated from particle tracking data sampled in the viscous sub-layer ($y^{+}\le 4$). The joint probability density distributions of stream- and spanwise WSS components are reliably obtained down to probability densities of $10^{-3}$ which, to date, has rarely been achieved through measurements. Fluctuations of the WSS components follow the Reynolds number dependency of the correlations reported in the literature but were found to be systematically underestimated with increasing distance of the sampling volume from the wall, affecting the spanwise component to a higher degree. A correction method is suggested.