Reconstructing velocity and pressure from noisy sparse particle tracks using constrained cost minimization

Emerging time-resolved volumetric PIV techniques have made simultaneous measurements of velocity and pressure fields possible. Yet, in many experimental setups, satisfying the spatial and temporal resolution requirements is a challenge. To improve the quality of sparse and noisy data, this paper introduces a constrained cost minimization (CCM) technique, which interpolates unstructured particle tracks to obtain the velocity, velocity gradients, material acceleration, hence the pressure, on a Eulerian grid. This technique incorporates physical constraints, such as a divergence-free velocity field and curl-free pressure gradients. The performance is evaluated using synthetic particle tracks for an unsteady double gyre and direct numerical simulations data for a turbulent channel flow, with varying particle concentrations and added errors. The errors in pressure, calculated using omni-directional integration, and correlations with the original data are compared to those obtained using the singular value decomposition (SVD) interpolation technique. The CCM errors are mostly lower, and the correlation is higher and less sensitive to particle sparsity and added errors compared to those of SVD. The synthetic particle traces are also projected onto four planar images to evaluate the performance of the new procedure together with shake-the-box (STB) particle tracking. A comparison of pressure spectra and correlation with the original data show very good agreement for the CCM method. Hence, CCM appears to be an effective method for improving the interpolation of sparse data. Sample experimental data obtained in the shear layer behind a backward-facing step demonstrate the application of STB and CCM to resolve the pressure field in coherent vortex structures. Graphic abstract