Interpolation schemes for three-dimensional velocity fields from scattered data using Taylor expansions (Q1902661)

The Lagrangian picture of particle motion in fluid flows using particle tracking velocimetry yields sets of particle positions in a flow volume, and therefore it is required to approximate the distributed velocity field data onto a Cartesian grid since the flow visualization needs graphic packages with input at regular spacing points. The paper introduces an interpolation scheme from data at randomly distributed points within the volume to an arbitrary set of points and shows that the approximation using trivariate polynomial interpolants is equivalent to a Taylor expansion of the field up to the second order terms. The interpolation scheme is carried over locally defined volumes and is shown to be highly accurate over certain ranges of some parameters. The discontinuity in the flow field across neighbouring subregions is tided over by using the excess data in a local volume.