Vortex Sheets, Convective Flows and Singularities

9306720 Caflisch涡旋和对流流体流动对于各种现象都很重要，例如流体混合、围绕翼型和障碍物的流动以及湍流。本项目主要研究三个相关课题：(1)通过分岔分析和数值计算，研究轴对称涡旋片的动力学、不稳定性和奇点形成，并应用于涡旋击穿。(2)研究了具有随机性和时效性的对流流动的有效扩散速率。(3)结合偏微分方程理论、代数几何（突变理论）、符号计算和数值模拟研究了偏微分方程系统的奇异性。本课题的研究是针对复杂流体流动的分析和数值计算，如发生在许多重要的科学技术问题中。有效地理解和应用这些流动需要描述它们的微观和宏观特征。本研究首先关注奇点作为理想流动中的精细尺度现象。流动的宏观描述也通过微观变化的均质化而发展起来。这项研究的结果将对评估复杂流动的定性特征以及开发有效的计算方法具有重要意义。＊＊＊

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. ***