Volume 23, Number 5
Volume 23, Number 5, 2018
160 Years of Vortex Dynamics
Llewellyn Smith S. G., Chang C., Chu T., Blyth M., Hattori Y., Salman H.
Generalized Contour Dynamics: A Review
Abstract
Contour dynamics is a computational technique to solve for the motion of vortices
in incompressible inviscid flow. It is a Lagrangian technique in which the motion of contours
is followed, and the velocity field moving the contours can be computed as integrals along
the contours. Its bestknown examples are in two dimensions, for which the vorticity between
contours is taken to be constant and the vortices are vortex patches, and in axisymmetric
flow for which the vorticity varies linearly with distance from the axis of symmetry. This review
discusses generalizations that incorporate additional physics, in particular, buoyancy effects and
magnetic fields, that take specific forms inside the vortices and preserve the contour dynamics
structure. The extra physics can lead to timedependent vortex sheets on the boundaries, whose
evolution must be computed as part of the problem. The nonBoussinesq case, in which density
differences can be important, leads to a coupled system for the evolution of both mean interfacial
velocity and vortex sheet strength. Helical geometry is also discussed, in which two quantities
are materially conserved and whose evolution governs the flow.

O'Neil K. A.
Dipole and Multipole Flows with Point Vortices and Vortex Sheets
Abstract
An exact method is presented for obtaining uniformly translating distributions of
vorticity in a twodimensional ideal fluid, or equivalently, stationary distributions in the presence
of a uniform background flow. These distributions are generalizations of the wellknown vortex
dipole and consist of a collection of point vortices and an equal number of bounded vortex sheets.
Both the vorticity density of the vortex sheets and the velocity field of the fluid are expressed in
terms of a simple rational function in which the point vortex positions and strengths appear as
parameters. The vortex sheets lie on heteroclinic streamlines of the flow. Dipoles and multipoles
that move parallel to a straight fluid boundary are also obtained. By setting the translation
velocity to zero, equilibrium configurations of point vortices and vortex sheets are found.

Krishnamurthy V. S., Stremler M. A.
Finitetime Collapse of Three Point Vortices in the Plane
Abstract
We investigate the finitetime collapse of three point vortices in the plane utilizing
the geometric formulation of threevortexmotion from Krishnamurthy, Aref and Stremler (2018)
Phys. Rev. Fluids 3, 024702. In this approach, the vortex system is described in terms of the
interior angles of the triangle joining the vortices, the circle that circumscribes that triangle, and
the orientation of the triangle. Symmetries in the governing geometric equations of motion for
the general threevortex problem allow us to consider a reduced parameter space in the relative
vortex strengths. The wellknown conditions for threevortex collapse are reproduced in this
formulation, and we show that these conditions are necessary and sufficient for the vortex
motion to consist of collapsing or expanding selfsimilar motion. The geometric formulation
enables a new perspective on the details of this motion. Relationships are determined between
the interior angles of the triangle, the vortex strength ratios, the (finite) system energy, the time
of collapse, and the distance traveled by the configuration prior to collapse. Several illustrative
examples of both collapsing and expanding motion are given.

García Garrido V. J., Curbelo J., Mancho A. M., Wiggins S., Mechoso C. R.
The Application of Lagrangian Descriptors to 3D Vector Fields
Abstract
Since the 1980s, the application of concepts and ideas from dynamical systems
theory to analyze phase space structures has provided a fundamental framework to understand
longterm evolution of trajectories in many physical systems. In this context, for the study
of fluid transport and mixing the development of Lagrangian techniques that can capture
the complex and rich dynamics of timedependent flows has been crucial. Many of these
applications have been to atmospheric and oceanic flows in twodimensional (2D) relevant
scenarios. However, the geometrical structures that constitute the phase space structures in
timedependent threedimensional (3D) flows require further exploration. In this paper we
explore the capability of Lagrangian descriptors (LDs), a tool that has been successfully
applied to timedependent 2D vector fields, to reveal phase space geometrical structures in 3D
vector fields. In particular, we show how LDs can be used to reveal phase space structures
that govern and mediate phase space transport. We especially highlight the identification
of normally hyperbolic invariant manifolds (NHIMs) and tori. We do this by applying this
methodology to three specific dynamical systems: a 3D extension of the classical linear saddle
system, a 3D extension of the classical Duffing system, and a geophysical fluid dynamics fplane
approximation model which is described by analytical wave solutions of the 3D Euler equations.
We show that LDs successfully identify and recover the template of invariant manifolds that
define the dynamics in phase space for these examples.

Safonova D. V., Demina M. V., Kudryashov N. A.
Stationary Configurations of Point Vortices on a Cylinder
Abstract
In this paper we study the problem of constructing and classifying stationary
equilibria of point vortices on a cylindrical surface. Introducing polynomials with roots at vortex
positions, we derive an ordinary differential equation satisfied by the polynomials. We prove
that this equation can be used to find any stationary configuration. The multivortex systems
containing point vortices with circulation $\Gamma_1$ and $\Gamma_2$ $(\Gamma_2 = \mu\Gamma_1)$ are considered in detail.
All stationary configurations with the number of point vortices less than five are constructed.
Several theorems on existence of polynomial solutions of the ordinary differential equation under
consideration are proved. The values of the parameters of the mathematical model for which
there exists an infinite number of nonequivalent vortex configurations on a cylindrical surface
are found. New point vortex configurations are obtained.

O'Neil K. A.
Relations Satisfied by Point Vortex Equilibria with Strength Ratio $2$
Abstract
Relations satisfied by the roots of the Loutsenko sequence of polynomials are
derived. These roots are known to correspond to families of stationary and uniformly translating
point vortices with two vortex strengths in ratio $2$. The relations are analogous to those
satisfied by the roots of the Adler–Moser polynomials, corresponding to equilibria with ratio
$1$. The proof uses an analysis of the differential equation that these polynomial pairs satisfy.

Wang W., Prants S. V., Zhang J., Wang L.
A Lagrangian Analysis of Vortex Formation in the Wake behind a Transversely Oscillating Cylinder
Abstract
A vortex pair+single vortex (P+S) wake behind a transversely oscillating cylinder
is investigated from the Lagrangian point of view. The Lagrangian coherent structures (LCSs) of
the flow are computed to analyze formation of vortices in the wake. An asymmetric vortex street
is obtained by using a dynamic mesh method. The corresponding vorticity field is found to agree
well with real experiments. The LCSs are approximated by ridges of the finitetime Lyapunov
exponents computed from transient velocity fields. The formation process is investigated using
the vorticity field and the LCSs. It is found that details of the wake pattern are sensitive to
initial oscillation conditions, and that the cylinder motion causes an early rollup of boundary
layers to form new vortex structures in the wake. Lagrangian description of the flow with the
help of the LCSs provides further details about formation of vortices in the cylinder flow and
helps to get a new insight into the flow structure in the wake region.

Calleja R., Doedel E., GarcíaAzpeitia C.
Choreographies in the $n$vortex Problem
Abstract
We consider the equations of motion of $n$ vortices of equal circulation in the plane,
in a disk and on a sphere. The vortices form a polygonal equilibrium in a rotating frame
of reference. We use numerical continuation in a boundary value setting to determine the
Lyapunov families of periodic orbits that arise from the polygonal relative equilibrium. When
the frequency of a Lyapunov orbit and the frequency of the rotating frame have a rational
relationship, the orbit is also periodic in the inertial frame. A dense set of Lyapunov orbits,
with frequencies satisfying a Diophantine equation, corresponds to choreographies of $n$ vortices.
We include numerical results for all cases, for various values of $n$, and we provide key details
on the computational approach.

Borisov A. V., Mamaev I. S., Bizyaev I. A.
Three Vortices in Spaces of Constant Curvature: Reduction, Poisson Geometry, and Stability
Abstract
This paper is concerned with the problem of three vortices on a sphere $S^2$ and the
Lobachevsky plane $L^2$. After reduction, the problem reduces in both cases to investigating a
Hamiltonian system with a degenerate quadratic Poisson bracket, which makes it possible to
study it using the methods of Poisson geometry. This paper presents a topological classification
of types of symplectic leaves depending on the values of Casimir functions and system
parameters.
