Collaborative Research: Beyond Point Vortices: Moving Singularities and Wave Fields in Fluid Mechanics

Kirchhoff的点涡运动方程是将无限维动力系统即不可压缩欧拉方程简化为有限维系统的一个范例。然而，原始的不可压缩欧拉方程本身忽略了在某些应用中可能很重要的物理现象，例如可压缩性。此外，人们还可以研究点涡奇点的其他推广，如偶极子和边界的影响。在本提案中，我们的目标是根据物理原理推导出适合这种情况的方程，并检查由此产生的非线性系统。利用动量守恒理论将点涡向更高奇点的自然扩展遇到了困难：所得到的方程不是唯一指定的，并且存在潜在的不定义正则化。我们使用积分论证的补充工具来解决这个问题，即利用尺度上的差异来获得控制系统大尺度运动的方程，同时考虑到适当的涡旋的小尺度行为。这是一个典型的匹配渐近展开（MAE）问题。这些方法将提供的方程是新的工具，为低阶模型的流体流动超越了众所周知的点涡方程。自亥姆霍兹提出涡度概念以来，它一直是流体力学中的一个基本概念。包括开尔文勋爵在内的苏格兰学派寻求一种“漩涡原子”理论来解释物质的结构。理解涡度动力学对于理解湍流是至关重要的，本质上大规模的流动，无论环境，都是湍流。简化模型是物理、数学和工程中实现这种理解的关键工具，而点涡，二维奇异结构，就是这样一个重要的模型。它们被称为“经典应用数学游乐场”，并被用来理解流体流动控制、生物运动和地球物理学等问题。该项目旨在利用物理原理推导点涡的扩展到更一般的系统。这种模型在考虑可压缩性的大型现代飞机尾迹模型中也有应用。本文中介绍的降阶模型是建模过程中重要的第一步，同时也为比较复杂的CFD（计算流体动力学）计算提供了潜在的基准。该项目将包括数学家和流体动力学家之间的国际合作。

Kirchhoff's equations of motion for point vortices are a paradigm of reduction of an infinite-dimensional dynamical system, namely the incompressible Euler equation, to a finite-dimensional system. Yet the original incompressible Euler equation itself neglects physical phenomena such as compressibility that may be important in certain applications. In addition, one can also examine other generalizations of the point vortex singularity, such as dipoles, and the effect of boundaries. In this proposal, we aim to derive appropriate equations for such situations based on physical principles and examine the resulting nonlinear systems. The natural extension of point vortices to higher singularities using momentum conservation argument encounters difficulties: the resulting equations are not uniquely specified and there is an underlying ill-defined regularization. We tackle this problem using a complementary tool to integral arguments, namely by exploiting the very differences in scales to obtain equations governing the large-scale motion of the system while taking into account the small-scale behavior of the vortices where appropriate. This is an archetypal Matched Asymptotic Expansion (MAE) problem. These approaches will provide equations that are new tools for low-order models of fluid flows beyond the well-known point vortex equations.Vorticity has been a fundamental concept in fluid mechanics since its introduction by Helmholtz. The Scottish school, including Lord Kelvin, sought a theory of `vortex atoms' to explain the structure of matter. Understanding vorticity dynamics is critical to an understanding of turbulence, and essentially large-scale flows, whether environmental, are turbulent. Simplified models a critical tool in physics, mathematics and engineering to achieve this understanding and point vortices, two-dimensional singular structures, are an important such model. They have been called a ``classical applied mathematical playground'' and have been used to understanding problems in the control of fluid flows, biological locomotion and geophysics. This project aims to derive extensions of point vortices to more general systems using physical principles. Such models have applications in models of aircraft wakes for large modern aircraft by taking into account compressibility. Reduced-order models such as the ones presented here are an important first step in the modeling process as well as providing potential benchmarks to compare to complex CFD (computational fluid dynamics) calculations. The project will include an international collaboration between mathematicians and fluid dynamicists.