Mathematical Sciences: Analysis of Fluid Motion

Beale研究员承担了四个与不可压缩流体流动有关的项目。第一部分继续研究无粘流体界面运动的数值方法，例如水波。在以前的工作中，已经发展了边界积分型收敛方法，使用了一种新的方法来分析流体界面。在这个项目中，设计的方法具有更广泛的适用性，仍然保持完全的数值稳定性。在一个相关的项目中，改进了对完全非线性水波方程精确解的存在性和正则性的解析理解。研究了具有旋涡的精确涡环中的不稳定性，比较了短波渐近分析的预测结果和涡量计算元素随流动移动的涡量直接模拟结果。这两种非常不同的方法应该允许对依赖时间的、无粘的三维流动的测试用例进行详细的处理。最后，对粘性流动的Navier-Stokes方程进行了分数阶时间步长逼近，其中无粘流与线性粘性流交替，并产生了人工边界层。流体流动在各种应用中都很重要，包括飞机设计、工业流程以及海洋和大气的运动。通常，由于流动中小尺度的发展--例如，流动一般行为中的局部扰动--定量预测是困难的。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，例如无粘流体界面的方程，数值不稳定性是很难避免的。精心设计的方法，比如这里开发的方法，可以用来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。比较两种预测流体流动不稳定性的方法--短波分析方法和涡流直接模拟方法，应该有助于确定每种方法的有效范围。将两者结合起来对涡环等精确解进行定量研究，有利于更好地理解现实三维流动中的非线性动力学。Navier-Stokes方程的分步近似用更特殊的两部分代替了完整的演化。一些数值方法利用了这种双重结构。更好地理解这种近似，特别是边界行为，可以建议这些数值格式的更准确的版本。

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.