Mathematical Sciences: Analysis of Fluid Motion

比厄 研究者承担了四个与不可压缩流体流动有关的项目。 第一个继续工作的数值方法的运动无粘流体界面，如水波。 在以前的工作中，边界积分型的收敛方法已经开发，使用一种新的方法来分析流体界面。 在这个项目中，更普遍的适用性的方法设计，仍然保持充分的数值稳定性。 在一个相关的项目中，寻求对完全非线性水波方程精确解的存在性和规律性的更好的分析理解。 精确涡环与旋流的不稳定性进行了研究，比较预测从短波渐近分析与直接模拟使用涡的方法，其中计算元素的涡量移动与流动。 这两种非常不同的方法应该允许详细处理这个测试情况下的时间相关的，无粘的三维流动。 最后，研究了粘性流的Navier-Stokes方程的分数时间步长近似，其中粘性流与线性粘性流交替出现，并生成人工边界层。 流体流动在各种应用中都很重要，包括飞机设计、工业过程以及海洋和大气的运动。 定量预测常常是困难的，因为流动中的小尺度的发展--例如，流动的一般行为中的局部扰动。 数学分析的一个作用是设计改进的数值方法。 对于复杂的方程，如无粘流体界面的方程，数值不稳定性是难以避免的。 精心设计的方法，如这里开发的方法，可以用于更好地预测水波的行为。 对解的基本分析理解通常与数值近似的分析密切相关。 预测流体流动不稳定性的两种方法，即短波分析法和涡方法直接模拟法的比较，应有助于确定每种方法的有效范围。 将两者结合起来进行精确解的定量研究，如涡环与旋流，有利于更好地理解现实三维流动中的非线性动力学。 分数阶近似的Navier-Stokes方程取代了两个更特殊的字符部分的完整的发展。 一些数值方法利用了这种对偶结构。 更好地理解这种近似，特别是边界行为，可以提出更准确的版本，这些数值方案。

Beale The investigator undertakes four projects related to incompressible fluid flow. The first continues work on numerical methods for the motion of inviscid fluid interfaces, such as water waves. In previous work, convergent methods of boundary integral type have been developed, using a new approach for the analysis of fluid interfaces. In this project, methods of more general applicability are designed that still maintain full numerical stability. In a related project, improved analytical understanding is sought for existence and regularity in exact solutions of the fully nonlinear equations of water waves. Instabilities in exact vortex rings with swirl are studied, comparing predictions from a short-wave asymptotic analysis with direct simulation using vortex methods, in which computational elements of vorticity move with the flow. These two very different approaches should allow detailed treatment of this test case for time-dependent, inviscid three-dimensional flow. Finally, the approximation of the Navier-Stokes equations of viscous flow by fractional time steps are investigated, in which inviscid flow alternates with linear viscosity, and artificial boundary layers are generated. Fluid flow is important in diverse applications, including aircraft design, industrial processes, and the motion of the oceans and atmosphere. Often quantitative prediction is difficult because of the development of small scales in the flows -- for instance, local disturbances in the general behavior of the flows. One role for mathematical analysis is the design of improved numerical methods. For complicated equations, such as those of inviscid fluid interfaces, numerical instabilities have been difficult to avoid. Carefully designed methods, such as those developed here, could be used for better predictions of the behavior of water waves. Basic analytical understanding of solutions is often closely related to the analysis of numerical approximations. The com parison of two methods for predicting instabilities in fluid flow, short-wave analysis and direct simulation by vortex methods, should help to establish the realm of validity of each approach. The use of the two together for a quantitative study of exact solutions, such as vortex rings with swirl, is advantageous for better understanding of nonlinear dynamics in realistic three-dimensional flow. The fractional step approximation of the Navier-Stokes equations replaces the full evolution by two parts of more special character. Some numerical methods take advantage of this dual structure. Better understanding of such approximations, especially the boundary behavior, could suggest more accurate versions of these numerical schemes.