Error propagation analysis in the continuous limit of the "Omni-Directional Integral" and a novel pressure reconstruction method based on Helmholtz-Hodge decomposition and Curl-free Radial Basis Functions

Reconstructing pressure fields from image velocimetry measurements commonly involves one of two general strategies: 1) solving the pressure Poisson equation (PPE), and 2) recovering pressure directly from measured pressure gradients (e.g., the omni-directional integral (ODI) methods). ODI methods attempt a finite ensemble reconstruction of the pressure field on a discrete mesh satisfying the path independence property (PIP) of the line integral for a scalar field and perform well when white noise is present in the measured pressure gradient. By invoking the Helmholtz-Hodge Decomposition (HHD), which extracts the curl-free components of any vector field, respecting the PIP exactly, our rigorous error propagation analysis on ODI and HHD demonstrates that the continuous limit of ODI is to apply HHD to a measured pressure gradient. We also propose a novel direct HHD-based pressure field reconstruction strategy that offers the following advantages: 1) effective processing of scattered and structured PTV/PIV data using radial basis functions with curl-free kernels, 2) complete elimination of divergence-free bias in measurements, resulting in superior accuracy compared to PPE and ODI, and 3) avoidance of ensemble practices and more than a 100-fold reduction in computational cost compared to conventional ODI methods.