Weighted least-squares finite elements based on particle imaging velocimetry data

The solution of the Navier–Stokes equations requires that data about the solution is available along the boundary. In some situations, such as particle imaging velocimetry, there is additional data available along a single plane within the domain, and there is a desire to also incorporate this data into the approximate solution of the Navier–Stokes equation. The question that we seek to answer in this paper is whether two-dimensional velocity data containing noise can be incorporated into a full three-dimensional solution of the Navier–Stokes equations in an appropriate and meaningful way. For addressing this problem, we examine the potential of least-squares finite element methods (LSFEM) because of their flexibility in the enforcement of various boundary conditions. Further, by weighting the boundary conditions in a manner that properly reflects the accuracy with which the boundary values are known, we develop the weighted LSFEM. The potential of weighted LSFEM is explored for three different test problems: the first uses randomly generated Gaussian noise to create artificial ‘experimental’ data in a controlled manner, and the second and third use particle imaging velocimetry data. In all test problems, weighted LSFEM produces accurate results even for cases where there is significant noise in the experimental data.