Mathematical Sciences: Formation Process and 3-D Dynamics of Vortex Rings

9408697 Nitsche本项目的目标是进行数值研究，以更好地了解涡环动力学。考虑的情况是通过从开口喷出一定量的流体来形成涡环。在项目的第一部分，重点将放在轴对称环的初始形成过程上。在初始时间间隔内，粘性效应主导流动，并影响旋涡轨迹和总的脱落环量。采用一个成功的二维N-S方程求解高雷诺数流动，研究了粘性对轴对称流动的影响。还将研究使用涡片模式的流动的无粘性效应，以确定是否可以调整现有的相似理论预测以更好地预测初始流动。该项目的第二部分涉及在开口处形成的涡环的三维动力学。发展一种计算边缘三维涡片分离的数值方法将有助于研究这些流动的稳定性，以及非轴对称开口和非轴对称强迫的影响。为此，将开发一种三维涡丝方法，该方法在尖锐的边缘结合涡旋分离，并实施快速求和算法以实现高分辨率计算。了解涡环的动力学对于了解更复杂的流动是至关重要的，例如在燃烧过程中发生的流动，或对飞机构成危险的空气中的涡旋结构。建立了圆形开口轴对称涡环的无粘性数值模型。通过与实验的对比，验证了该模型的有效性，恢复了真实流动的详细信息。在本文中，该模型将扩展到三维流动，并将用于研究流动的稳定性，非轴对称开口和非轴对称强迫的影响，以及本理论结果对流动预测的潜在适用性。这些流动的几个方面很难通过实验或分析来理解，计算有望使我们对流动动力学有更深入的了解。为了完成这项工作，目前可用于二维流动的数值工具将扩展到三维。

9408697 Nitsche The goal of the present project is to perform numerical investigations to develop a better understanding of vortex ring dynamics. The case is considered in which a vortex ring is formed by ejecting an amount of fluid from an opening. In the first part of the project, the focus will be on the initial formation process of axisymmetric rings. Viscous effects dominate the flow in an initial time interval and affect the vortex trajectory and total shed circulation. The effects of viscosity by adapting a successful 2-dimensional Navier Stokes solver for high Reynolds number flow to the axisymmetric case will be investigated. Inviscid effects of the flow using a vortex sheet model will also be studied in order to determine whether present similarity theory predictions can be adjusted to better predict the initial flow. The second part of the project concerns 3-dimensional dynamics of vortex rings formed at an opening. The development of a numerical method to compute 3-dimensional vortex sheet separation at an edge will enable the study of the stability of these flows, as well as the effects of nonaxisymmetric openings and nonaxisymmetric forcing. For this purpose, a 3-dimensional vortex filament method will be developed which incorporates vortex separation at a sharp edge and implements a fast summation algorithm to enable high resolution calculations. Understanding the dynamics of vortex rings is essential to understand more complicated flows such as those that occur in combustion processes, or in the airborne vortex structures presenting a hazard to aircraft. An inviscid numerical model has been developed for axisymmetric vortex rings generated at a circular opening. This model was proven by comparison with experiment to recover detailed information about the real flow. In the present work, this model will be extended to 3-dimensional flows, and will be used to study the stability of the flows, the effects of non-axisymmetric openings a nd nonaxisymmetric forcing, as well as the potential applicability of present theoretical results to predict the flow. Several of these aspects of the flow are difficult to understand experimentally or analytically, and the computations promise to give a deeper insight into the flow dynamics. In order to perform this work, current numerical tools available for 2-dimensional flows will be expanded to 3-dimensions.