Robust wall gradient estimation using radial basis functions and proper orthogonal decomposition (POD) for particle image velocimetry (PIV) measured fields

A robust method for improving the estimation of near-wall velocity gradients from noisy flow data using Gaussian (GA) and generalized multiquadratic (GMQ) radial basis functions (RBFs) that optimizes fitting parameters to minimize the biharmonic equation is introduced. Error analysis of the wall gradient estimation was performed for RBFs, standard finite difference schemes, and polynomial and spline interpolations at various spatial resolutions, interpolation grid sizes and noise levels in synthetically generated Poiseuille and Womersley flow fields. Also, the effectiveness of the methods on digital particle image velocimetry (DPIV) data is tested by processing images generated using velocity fields obtained from direct numerical simulation (DNS) of an open turbulent channel, and the estimated gradients were compared against gradients obtained from DNS data. In the absence of noise, all methods perform well for Poiseuille and Womersley flows yielding a total error under 10% at all resolutions. In the presence of noise, the GMQ performed robustly with a total error under 10–20% even with 10% noise. With DPIV processed data for the turbulent channel flow, the error is on the order of 25–40% using thin plate spline and GMQ interpolations. Optimization of the RBF fitting parameters that minimize the energy functional associated with the analytical surface results in robust velocity gradient estimators but is computationally expensive. This computational expense is reduced and the accuracy of the proposed techniques is further improved by introducing a novel approach that combines the gradient estimators with proper orthogonal decomposition (POD). The implementation of the interpolation schemes on the POD modes results in improving accuracy by 10–15% and reducing the computational cost by approximately 75%.