Smooth full field reconstruction of velocity and its gradients from noisy scattered velocimetry data in a cross-slot flow

Abstract

We present a method combining generalized Tikhonov regularization with a finite element approximation for reconstructing smooth velocity and velocity gradient fields from spatially scattered and noisy velocity data in a two-dimensional complex flow domain. Synthetic velocity data for a cross-slot geometry are generated using the Oldroyd-B solution, subsequently perturbed by random noise. Performances of diverse finite element continuity-regularization criterion combinations are tested against noise-free data, while the optimum regularization parameter is determined using generalized cross-validation. The best performance is achieved for the velocity field and its gradients simultaneously by C2 continuous Hermite finite elements and minimization of a norm of the velocity’s third derivative. The standard regularization criterion based on the second derivative is shown to lead to systematic distortions in boundary regions, allowing therefore a lower reduction in the statistical error. Furthermore, optical fields are calculated by applying a differential constitutive equation directly to the reconstructed flow kinematics; high quality velocity gradient fields are shown to be an essential prerequisite for their reliable prediction. Overall, the method is expedient to implement and does not require boundary conditions.