Na. Malik et T. Dracos, INTERPOLATION SCHEMES FOR 3-DIMENSIONAL VELOCITY-FIELDS FROM SCATTERED DATA USING TAYLOR EXPANSIONS, Journal of computational physics, 119(2), 1995, pp. 231-243
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.