Interpretation of Omnidirectional Integration using the Green's Function: Inversion from Error-contaminated Pressure Gradient to Pressure FIelds*

Accurate and efficient measurement of the pressure field is crucial in fluid mechanics. This research introduces a novel method for reconstructing the instantaneous pressure field from error-embedded pressure gradient data using the Green's function of the Laplacian operator as the convolution kernel. The method establishes a mathematical connection of the Green's Function Integral (GFI) with the omnidirectional integration (ODI) for pressure reconstruction, but with improved computational simplicity. The GFI method is applied to simple canonical and isotropic turbulence flows in two-dimensional and three-dimensional domains, respectively, while considering uncertainty quantification through eigenanalysis. Results demonstrate that the accuracy of GFI is comparable to ODI, yet with significantly improved computational efficiency. This method presents a promising approach for the reconstruction of not only pressure field, but also other types of scalar potential quantities, such as density, temperature, stream function, velocity potential, etc., from their error-contaminated gradient measurement data in various fluid mechanics applications.