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.
Dynamical systems theory concerns the study of the qualitative behavior of differential equations including those describing classical physics. The motion of fluids is described by the Euler equations when the viscosity is small; solving these nonlinear partial differential equations pose a considerable challenge. For systems that are nearly two-dimensional and characterized by strong swirling motion, Kirchhoff's equations of motion for point vortices (PVE) give a reduction of an infinite-dimensional dynamical system. The goal of CMMI-0970113 was to derive appropriate equations for such situations based on physical principles and to examine the resulting nonlinear systems. A major goal was to understand desingularized vortical structures as representations of point vortices. Progress was accomplished in a number of areas. The physical background and justification for the point vortex equation and its generalizations were examined in Llewellyn Smith (2011). The results can be summarized using the ``highlights'' of the paper: ``The justification for the equation governing point vortex motion is examined. The original argument to Helmholtz is still presented in most references. A method using a generalized momentum flux argument is presented. Issues with regularization remain.'' Vortex dipoles were investigated further by Llewellyn Smith & Nagem (2013). A fundamental theme of CMMI-0970113 was how to move beyond the idea of moving point vortex singularities by considering desingularizations of point vortices. A well-known family of solutions to the Euler equations is that of vortex patches. Less well known are hollow vortices, originally defined as vortices with no flow inside them, but more profitably viewed as equilibrium vortex sheets. Llewellyn Smith (2014) calculates the correction to the propagation velocity of point vortex equilibria using a matched asymptotic expansion. Llewellyn Smith & Crowdy (2012) found solutions for hollow vortices in strain and examined their stability. Crowdy & Green (2011) found new solutions for the von Kármán vortex street. Finding these solutions required using new techniques of conformal geometry combining the Schottky-Klein prime function with logarithmic mappings. Crowdy, Llewellyn Smith & Freilich (2013) obtained Pocklington's hollow vortex pair by a new method using the Schottky-Klein prime function, allowing the stability of the pair to be calculated. Crowdy & Roenby (2014) found two new classes of analytical solutions for hollow vortex equilibria. These equilibrium fluid regions have a mathematical interpretation as an abstract class of planar domains known as double quadrature domains. Crowdy visited UCSD several times during the award, and he and SGLS have built up a strong collaboration resulting in a number of published manuscripts. Daniel Freilich has been carrying out PhD research and receiving training in applied mathematics, computational techniques and fluid mechanics.
