Mathematical Sciences: A Numerical Simulator for the Taylor-Couette Problem

本项目的主要目的是为Taylor-Couette实验装置建立一个能够模拟实验室实验的数值模拟器。该项目将涉及开发一个强大的数值方案，能够模拟泰勒-库埃特流的大范围参数值，包括从库埃特流到湍流前流的所有已知流型。这个项目的重点是物理实验的重复。此外，还将使用Taylor-Couette模拟器对已经得到的一些数学结果进行研究，特别是那些与实验结果不一致的结果。该项目将包括在矢量和/或并行计算机上有效地实施三维方案（使用多重网格加速收敛），以及一个完整的图形输出包，该输出包将模拟可用的实验输出并提供额外的结果（在物理实验中通常无法获得），这些结果将更清楚地显示这些流动中涉及的一些现象和机制。数值计算产生的数据将使用生物驱动的“神经”特征检测器进行分析，该检测器提供高维数据集的低维表示。可视化技术的基础上预处理数据使用神经网络，自组织特征图，以及更标准的Karhunen-Loeve分解将被研究。这种模式分析的重点将通过直接比较在数值模拟器和实验室实验中出现的大中型连贯结构来直接验证模型。在许多涉及开发和制造的领域，数值模拟正成为开发和制造过程中不可或缺的一部分。长期以来，有一种流行的观点认为，人们可以“抛弃风洞”，……并在计算机上完成所有的实验。“然而，并没有很多研究清楚地证明计算机模拟可以在多大程度上取代或增强实验获得的信息。大型计算机代码被用来解释和预测复杂的现象（比如天气预报），由于运行这些代码的成本，这些代码从未被仔细验证过。泰勒-库埃特流所表现出的不稳定性和流动模式多年来一直挑战着实验者和理论家。所获得的结果包括在最终过渡到湍流过程中遇到的各种复杂流动形式。因此，Taylor-Couette问题是评估计算机模型执行物理实验的能力的绝佳选择。泰勒-库埃特问题的结果非常复杂，需要精确的数值模型、高效的求解器和大量的网格点。由于问题的这些方面，结果将说明计算机模型如何在广泛的参数范围内模拟物理问题，将表明在大型问题上可能的数值效率，并将清楚地说明与产生大量数据的长期数值模拟相关的许多问题。

The main objective of this project is to build a numerical simulator for the Taylor-Couette experimental device which is capable of emulating laboratory experiments. The project will involve developing a robust numerical scheme capable of simulating the Taylor-Couette flow over a wide range of parameter values, including all of the known flow regimes from Couette flow to the pre-turbulence flows. The emphasis of this project is on the duplication of physical experiments. In addition, the Taylor-Couette simulator will be used to investigate some of the mathematical results that have been obtained, especially those which do not agree with the experimental results. The project will involve an efficient implementation of a three-dimensional scheme on a vector and/or parallel computer (using multigrid to speed convergence) along with a full graphical output package that will both imitate the available experimental outputs and give additional results (not generally obtainable in physical experiments) that will show more clearly some of the phenomena and mechanisms that are involved in these flows. The data generated by the numerical computations will be analyzed using biologically motivated ``neural'' feature detectors which provide low-dimensional representations of high-dimensional data sets. Visualization techniques based on preprocessing the data using neural networks, self organizing feature maps, in addition to the more standard Karhunen-Loeve decomposition will be investigated. This emphasis on pattern analysis will be directed towards validation of the model by directly comparing large and medium-scale coherent structures which appear both in the numerical simulator and laboratory experiments. In many areas involving development and manufacturing, numerical simulation is becoming an integral part of the development and manufacturing processes. There has been a longstanding popular argument that one might ``throw away the wind tunnels,.. and perform all of the experiments on the comp uter.'' However, there have not been many studies which clearly demonstrate to what extent computer simulations can replace or augment information obtained experimentally. Large computer codes are used to explain and predict complex phenomena (say in weather forecasting) that, due to the cost of running these codes, have never been carefully validated. The instabilities and flow patterns exhibited by the Taylor-Couette flow have challenged experimenters and theorists for many years. The results that have been obtained include a rich assortment of complex flow regimes encountered during the eventual transition to turbulence. Hence, the Taylor-Couette problem is an excellent candidate to assess how well a computer model can perform physical experiments. The Taylor-Couette problem is such that the results are sufficiently complex to require an accurate numerical model, an efficient solver and a large number of grid points. Because of these aspects of the problem, the results will illustrate how well a computer model can simulate a physical problem over a broad range of parameters, will indicate what numerical efficiencies are possible on large problems and will clearly illustrate many of the problems associated with long term numerical simulations that produce large amounts of data.