Current Particle Image Velocimetry (PIV) and Lagrangian Particle Tracking (LPT) provide flow velocity measurements at high spatial resolution. The velocimetry results can be used to reconstruct other fields of fluid flow such as vorticity, pressure, and even sound field. However, PIV or LPT results are noisy and gappy. A key question we wat to answer is how to reconstruct the pressure field, load on surfaces, and sound field from the imperfect velocimetry results with confidence? This is not an easy problem, as it requires a comprehensive understanding of the experimental, analytical, and numerical aspects of the problem (see the tangram below).
In our group, we attack this problem from a unique angle: understand how the error and uncertainty propagate from the velocity field to the pressure field first, and then control and minimize the error propagation to achieve the optimal full field reconstruction. Data assimilation, control theory, uncertainty quantification, and analysis of PDEs are employed in this investigation. Our series works can be found in special issues in Meas. Sci. Technol. (2016, 2020, 2021) and pre-prints.
Error propagation of the PIV/LPT-based pressure field reconstruction can be thought of as three Tangrams. A Tangram is a geometric Chinese puzzle that can be rearranged to make various shapes. Here, the left puzzle represents the inherent uncertainty quantification of PIV and LPT. The puzzle in the middle represents mathematical analysis of the error propagation problem, focusing on the details of the mathematical construction of the pressure field. The puzzle to the right represents the numerical issues that arise.