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

小行星9408697 本项目的目标是进行数值研究，以更好地了解涡环动力学。 考虑的情况下，涡环是通过从开口喷射一定量的流体形成的。 在该项目的第一部分，重点将是轴对称环的初始形成过程。 粘性效应在初始时间间隔内占主导地位，并影响涡轨迹和总脱落环量。 将研究粘性对轴对称情况下高雷诺数流的影响，方法是采用一个成功的二维纳维尔-斯托克斯解算器。 还将研究使用涡面模型的流动的无粘效应，以确定是否可以调整目前的相似理论预测，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面分离的数值方法将能够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，该方法在锐边处结合涡分离，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（例如燃烧过程中发生的流动或对飞机构成危险的空中涡流结构中发生的流动）至关重要。 本文建立了一个无粘流场数值模型，模拟了圆形开口处产生的轴对称涡环。 通过与实验结果的对比，证明了该模型能够恢复出真实的流场的详细信息。 在本工作中，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，以及目前的理论结果预测流动的潜在适用性。 这些方面的流动是很难理解的实验或分析，和计算承诺给一个更深入的了解流动动力学。为了进行这项工作，目前的数值工具可用于二维流动将扩展到三维。

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.