INTERPOLATION SCHEMES FOR 3-DIMENSIONAL VELOCITY-FIELDS FROM SCATTERED DATA USING TAYLOR EXPANSIONS

We present a numerical scheme that interpolates field data given at ra ndomly distributed locations within a three-dimensional volume to any arbitrary set of points within that volume. The approximation scheme u ses local trivariate polynomial interpolants and it is shown to be equ ivalent to a Taylor expansion of the (velocity) field up to second-ord er partial derivatives. It is formally a third-order scheme in the (me an) spacing of the data delta; i.e., the errors scale with (delta/lamb da(3)), where lambda is the length scale of the flow field. The scheme yields the three-dimensional velocity field (which can be inhomogeneo us and anisotropic) and all the 27 first- and second-order partial (sp atial) derivatives of the velocity field. It is compared with the adap tive 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 defi ned as the set of points with the same nearest neighbours (which may be set at between 10 and 1 5 in number). This makes the scheme formally discontinuous in the flow field across neighbouring patches; but by making use of the excess da ta within a local volume, it is shown that for practical purposes the scheme does yield a continuous flow field throughout the entire interp olation volume. The scheme interpolates the data by an iterative metho d which is extremely fast in situations where a certain level of error bounds in the data (and, hence, also the solution) is acceptable. Res ults from sinusoidal and stochastic (turbulent) test flow fields show that the Taylor expansion scheme is widely applicable and highly accur ate for the velocity and first derivatives, However, the smallest scal e of the (velocity) field lambda must be greater than 5 delta for the best performance. Second-order derivatives are less accurate. Flow qua ntities such as the fractal dimension of streamlines can be obtained a ccurately with much lower data density. Statistics like the power spec trum of the flow can also be obtained accurately. In the presence of n oise in the velocity data, small levels of noise have negligible effec t on the obtained velocities and a modest effect on the first derivati ves. The second derivatives are seriously affected, and only those of the largest scales in a turbulent flow can be adequately resolved. (C) 1995 Academic Press, Inc.