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Point Vortices

Source lecture(s): PC316 Ch. 8

Intuition

Shrink a Rankine vortex core to zero and you get a point vortex: all the circulation on a single point, inducing \(v_\theta = \Gamma/2\pi r\) around itself. The crucial property is what it does not do — a point vortex has no self-induced motion. It cannot push itself. So a collection of them obeys a rule of almost embarrassing simplicity:

Each vortex is advected by the velocity field of all the others.

That single sentence generates orbiting pairs, translating pairs, leapfrogging rings, chaos, and a respectable model of two-dimensional turbulence. Drive it in the point vortex sandbox.

The equations

\[\frac{dx_i}{dt} = -\frac{1}{2\pi}\sum_{j\neq i}\Gamma_j\frac{y_i - y_j}{r_{ij}^2}, \qquad \frac{dy_i}{dt} = +\frac{1}{2\pi}\sum_{j\neq i}\Gamma_j\frac{x_i - x_j}{r_{ij}^2}\]

This is a Hamiltonian system with \(H = -\frac{1}{4\pi}\sum_{i<j}\Gamma_i\Gamma_j\ln r_{ij}\), where \(\Gamma_i x_i\) and \(y_i\) are conjugate variables — an unusual and elegant structure in which position coordinates are canonically conjugate to each other. Conserved: \(H\), the linear impulse \(\sum\Gamma_i\mathbf{r}_i\), and the angular impulse \(\sum\Gamma_i|\mathbf{r}_i|^2\).

Three exact results

Co-rotating pair (equal \(\Gamma\), separation \(d\)). Each drives the other at \(\Gamma/2\pi d\) perpendicular to their line, so the pair orbits its midpoint:

\[T = \frac{2\pi^2 d^2}{\Gamma}\]

Merger of co-rotating vortices is the elementary coarsening step of 2-D turbulence.

Counter-rotating pair (\(+\Gamma\) and \(-\Gamma\), separation \(d\)). Each advects the other in the same direction, so the pair translates in a straight line at

\[V_{\rm pair} = \frac{\Gamma}{2\pi d}\]

This is a wingtip vortex pair seen head-on, and the cross-section of a vortex ring. It explains at once why smoke rings move: the ring propels itself by its own induction.

Vortex near a wall. The no-penetration boundary condition is satisfied by an opposite-sign image vortex mirrored in the wall (the same method of images used in electrostatics). The real vortex is then advected by its image, gliding parallel to the wall at

\[V = \frac{\Gamma}{4\pi h}\]

The dynamics gets rich fast

  • Two vortices: always integrable. They orbit or translate, forever.
  • Three vortices: integrable in general (enough conserved quantities), but capable of intricate motion including collapse in finite time for special initial conditions.
  • Four or more: chaotic. This was one of the first recognised examples of Hamiltonian chaos.
  • Leapfrogging: two counter-rotating pairs in line alternately pass through one another — the 2-D cross-section of two vortex rings chasing each other, and a genuinely beautiful demonstration that these simple rules produce complex behaviour.

Common mistakes

  • Including the self-term. The \(j \neq i\) is essential; a point vortex does not move itself. Including it gives a division by zero, or worse, a plausible-looking wrong answer if you regularise it.
  • Forgetting that images are slaved. The image is not an independent degree of freedom; it must be recomputed from the real vortex at every stage of the integrator, not advanced separately.
  • Expecting energy to be the Hamiltonian. \(H\) is not the kinetic energy (which diverges for point vortices); it is the interaction energy, and it is the generator of the dynamics.

Knowledge graph position

Prerequisites: vorticity, potential flow, Helmholtz theorems. Leads to: vortex-ring dynamics, 2-D turbulence, vortex methods in CFD.

Quiz

Q1 (computational). Two vortices of \(\Gamma = 4\) m²/s sit 2 m apart. What is their orbital period?

Answer

\(T = 2\pi^2d^2/\Gamma = 2\pi^2(4)/4 = 2\pi^2 \approx 19.7\) s.

Q2 (conceptual). Why does a counter-rotating pair translate while a co-rotating pair orbits?

Answer

Each vortex is advected by the other's field, which is perpendicular to the line joining them. For equal signs the two induced velocities are anti-parallel, producing rotation about the midpoint. For opposite signs they are parallel, so both move the same way and the pair translates with fixed separation.

Q3 (MCQ). A vortex at height \(h\) above a wall moves because:

  • (a) the wall exerts a viscous drag
  • (b) its image vortex, of opposite sign at depth \(h\), advects it parallel to the wall
  • (c) of self-induction
  • (d) it does not move
Answer

(b). The image enforces no penetration at the wall, and the real vortex sits in the image's field — a counter-rotating pair of separation \(2h\), giving glide speed \(\Gamma/4\pi h\).