Numerical Analysis, Analysis and Modeling of Fluid Motion

紊流量的准确、高效、可靠的预测（本文拟研究的重点），迫使紊流的建模、分析、数值分析和计算中的基本问题进行系统、综合的对抗，其中许多问题与其他流动问题和其他数学领域的基本问题相联系。相互关联的研究子项目包括：（一）导出大的连贯涡并探测由模型或数值产生的假涡；（二）开发高精度和有效可解的流体流动方程的正则化并进行数值分析；（三）不适定问题的新算法及其应用；（四）开发可压缩湍流的LES模型并分析其噪声预测；（五）时间平均大涡模拟的数学基础；直接数值模拟、大涡模拟和Reynolds平均湍流模型之间的连续转换；（vi）流固耦合、地下水-地表水模型和大气-海洋耦合驱动的流体-流体问题中的解耦多物理场流体流动问题；(v)时间松弛正则化的精确数值分析。在全球气候变化、国土安全（生物或化学制剂的扩散）、污染扩散、能源效率和生物医学设备设计等许多应用中，准确、高效和可靠地预测湍流中的数量是一个至关重要的问题。大涡模拟是一种研究湍流的方法，在这种方法中，大的特征从流动的细微细节中分离出来进行数值模拟。在大涡模拟中积累了丰富的经验和认识，使该方法能够高效、可靠地成功模拟基准湍流。科学和工业应用是流体流动方程与其他物理效应耦合的系统。这些问题需要比基准湍流问题更高的效率和准确性。为了在这些工业和科学应用中取得进展，大涡模拟现在需要系统的方法来评估灵敏度和不确定性，以及在更快、内存更大的计算机上查询从更细的网格中产生的大量数据的方法。这些问题是本文研究的重点。培养博士生为这个重要的、困难的、跨学科的、迷人的和美丽的领域做出贡献是提议努力的很大一部分。

The accurate, efficient and reliable prediction of quantities in turbulent flows (the focus of the proposed research) forces systematic and integrated confrontation of basic issues in the modeling, analysis, numerical analysis and computation of turbulent flows, many of which are linked to basic issues in other flow problems and other areas of mathematics. Interconnected research sub-projects include: (i) Eduction of large coherent vortices and detection of false vortices created by models or numerics, (ii) Development and numerical analysis of high accuracy and efficiently solvable regularizations of fluid flow equations, (iii) New algorithms for ill-posed problems and applications, (iv) Development of LES models for compressible turbulence and analysis of their acoustic noise predictions, (v) Mathematical foundation for time-averaged large eddy simulation: continuous transitioning between direct numerical simulation, large eddy simulation and Reynolds averaged turbulence models, (vi) Uncoupling multi-physics fluid flow problems in fluid-structure interaction, groundwater-surface water models and fluid-fluid problems motivated by atmosphere-ocean coupling, and (v) Precise numerical analysis of time relaxation regularizations. The accurate, efficient and reliable prediction of quantities in turbulent flows is a problem of fundamental importance in many applications ranging from global climate change, homeland security (dispersion of biological or chemical agents), pollution dispersal, energy efficiency and biomedical device design. Large eddy simulation is an approach to turbulent flows in which the large features are separated from the fine details of a flow for a numerical simulation. Hard-won experience and understanding has grown in large eddy simulation so that the methods can now successfully simulate benchmark turbulent flows efficiency and reliably. Scientific and industrial applications are systems of fluid flow equations coupled to other physical effects. These require even greater efficiency and accuracy than benchmark turbulent flow problems. To make progress into these industrial and scientific applications, large eddy simulation now requires systematic methods for assessing sensitivity and uncertainty as well as methods for interrogating the large amounts of data coming out of the finer meshes on faster computers with larger memories. These issues are the focus of the proposed research. Training PhD students to contribute to this important, difficult, interdisciplinary, fascinating and beautiful area is a large part of the proposed effort.