Vortex Sheets, Convective Flows and Singularities

小行星9306720 涡流和对流流体流动对于广泛的现象是重要的，例如流体混合、围绕翼型和障碍物的流动以及湍流。 （1）通过分叉分析和数值计算研究了轴对称旋转涡面的动力学、不稳定性和奇异性的形成，并将其应用于涡破裂。 (2)研究了具有随机性和时间依赖性的对流流动的有效扩散率。(3)奇异性研究系统的偏微分方程的通过相结合的偏微分方程理论，代数几何（突变理论），符号计算和数值模拟。 本项目的研究内容是对复杂流体流动的分析和数值计算，例如在许多科学和技术重要性问题中出现的问题。 要有效地理解和应用这些流动，就需要描述它们的微观和宏观特征。 本研究首先集中在奇异性作为一个细尺度的现象，在理想化的流动。 流动的宏观描述也通过微观变化的均匀化发展。 从这项研究的结果将是重要的，在评估复杂的流动的定性特征，以及在有效的计算方法的发展。 ***

9306720 Caflisch Vortical and convective fluid flows are important for a wide range of phenomena, such as fluid mixing, flow around airfoils and obstacles, and turbulence. In this project the investigator undertakes research in three related topics: (1) Dynamics, instabilities and singularity formation for axisymmetric, swirling vortex sheets are studied through bifurcation analysis and numerical computation, with applications to vortex breakdown. (2) The effective diffusion rate for a convective flow is studied for flows with randomness and time-dependence. (3) Singularities are investigated for systems of PDE's through a combination of PDE theory, algebraic geometry (catastrophe theory), symbolic computation, and numerical simulation. The research in this proposal is for analysis and numerical computation of complex fluid flows, such as occur in many problems of scientific and technological importance. Effective understanding and application of these flows requires a description of both their microscopic and their macroscopic features. This investigation first focuses on singularities as a fine-scale phenomena in idealized flows. A macroscopic description of flows is also developed through homogenization of the microscopic variations. The results from this study will be important in assessing the qualitative features of complex flows,as well as in the development of effective computational methods. ***