Interpolation Schemes for Three-Dimensional Velocity Fields from Scattered Data Using Taylor Expansions

Abstract

We present a numerical scheme that interpolates field data given at randomly distributed locations within a three-dimensional volume to any arbitrary set of points within that volume. The approximation scheme uses local trivariate polynomial interpolants and it is shown to be equivalent to a Taylor expansion of the (velocity) field up to second-order partial derivatives. It is formally a third-order scheme in the (mean) spacing of the data δ; i.e., the errors scale with (δ/λ,)3, where λ is the length scare of the flow field. The scheme yields the three-dimensional velocity field (which can be inhomogeneous and anisotropic) and all the 27 first- and second-order partial (spatial) derivatives of the velocity field. It is compared with the adaptive Gaussian window method and shown to be considerably more accurate. The interpolation scheme is local in the sense that it interpolates the data within locally defined volumes defined as the set of points with the same nearest neighbours (which may be set at between 10 and 15 in number). This makes the scheme formally discontinuous in the flow field across neighbouring patches; but by making use of the excess data within a local volume, it is shown that for practical purposes the scheme does yield a continuous flow field throughout the entire interpolation volume, The scheme interpolates the data by an iterative method which is extremely fast in situations where a certain level of error bounds in the data (and, hence, also the solution) is acceptable. Results from sinusoidal and stochastic (turbulent) test flow fields show that the Taylor expansion scheme is widely applicable and highly accurate for the velocity and first derivatives. However, the smallest scale of the (velocity) field λ must be greater than 5δ for the best performance. Second-order derivatives are less accurate. Flow quantities such as the fractal dimension of streamlines can be obtained accurately with much lower data density. Statistics like the power spectrum of the flow can also be obtained accurately. In the presence of noise in the velocity data, small levels of noise have negligible effect on the obtained velocities and a modest effect on the first derivatives. The second derivatives are seriously affected, and only those of the largest scales in a turbulent flow can be adequately resolved.