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首页> 外文期刊>Journal of Computational Physics >INTERPOLATION SCHEMES FOR THREE-DIMENSIONAL VELOCITY FIELDS FROM SCATTERED DATA USING TAYLOR EXPANSIONS
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INTERPOLATION SCHEMES FOR THREE-DIMENSIONAL VELOCITY FIELDS FROM SCATTERED DATA USING TAYLOR EXPANSIONS

机译：使用泰勒展开法从散射数据中三维速度场的插值方案

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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 delta; i.e., the errors scale with (delta/lambda(3)), where lambda is the length scale 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 defi ned 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 lambda must be greater than 5 delta 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. (C) 1995 Academic Press, Inc. [References: 21]
机译：我们提出了一种数值方案，该方案将三维体积内随机分布的位置处给出的场数据插值到该体积内的任意点集。近似方案使用局部三元多项式插值，并且被证明等效于（速度）场的泰勒展开，直至二阶偏导数。在数据增量（平均）间隔中，它正式是一种三阶方案。即误差用（delta / lambda（3））进行缩放，其中lambda是流场的长度缩放。该方案产生三维速度场（可以是不均匀的和各向异性的）以及速度场的所有27个一阶和二阶偏（空间）导数。将其与自适应高斯窗口法进行比较，结果显示其准确性更高。从某种意义上说，插值方案是本地的，因为它可以在本地定义的体积内对数据进行插值，这些体积定义为具有相同最近邻居的点集（可以设置为10到15之间）。这使得该方案在跨相邻补丁的流场中正式不连续；但是通过利用局部体积内的过量数据，表明，出于实际目的，该方案的确在整个插值体积内产生了连续的流场。该方案通过迭代方法对数据进行插值，在数据中一定程度的错误界限（以及解决方案）可以接受的情况下，该方法非常快。正弦和随机（湍流）测试流场的结果表明，泰勒展开方案适用于速度和一阶导数，并且精度很高，但是，（速度）场λ的最小比例必须大于5德尔塔。最棒的表演。二阶导数的准确性较低。可以以低得多的数据密度准确获得流量，例如流线的分形维数。诸如流的功率谱之类的统计信息也可以准确获得。在速度数据中存在噪声的情况下，少量噪声对获得的速度影响可忽略不计，而对一阶导数则影响不大。二阶导数受到严重影响，只有湍流中最大比例的那些才能得到充分解析。 （C）1995 Academic Press，Inc. [参考：21]

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