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Circulation and Kelvin's Theorem

Source lecture(s): PC316 Ch. 8

Intuition

Circulation is the total swirl around a closed loop:

\[\Gamma = \oint_C \mathbf{v}\cdot d\boldsymbol{\ell} = \int_S \boldsymbol{\omega}\cdot d\mathbf{S}\]

the second equality being Stokes' theorem — circulation is the flux of vorticity through any surface spanning the loop. Circulation already appeared in this course as the sole ingredient of lift, \(L' = \rho U\Gamma\). Kelvin's theorem tells you it is also nearly indestructible.

Kelvin's circulation theorem

Kelvin's circulation theorem

For inviscid, barotropic flow (\(\rho = \rho(p)\)) with conservative body forces, the circulation around any loop moving with the fluid is constant: $\(\frac{D\Gamma}{Dt} = 0\)$

Proof sketch. Differentiate \(\Gamma = \oint\mathbf{v}\cdot d\boldsymbol{\ell}\) following the loop. Two terms appear. The first, \(\oint \frac{D\mathbf{v}}{Dt}\cdot d\boldsymbol{\ell}\), becomes by Euler's equation \(-\oint\frac{dp}{\rho} - \oint d\Phi_{\rm grav}\) — both closed integrals of exact differentials, hence zero when \(\rho = \rho(p)\). The second comes from the stretching of the loop elements, \(\oint\mathbf{v}\cdot d\mathbf{v} = \oint d(\tfrac12 v^2) = 0\). ∎

Every hypothesis earns its place: inviscid (viscosity diffuses vorticity across the loop), barotropic (otherwise \(\oint dp/\rho \neq 0\) — the baroclinic term), and conservative body forces (otherwise the force integral survives).

The plasma twin

Kelvin's theorem is the exact fluid analogue of the frozen-in flux theorem of MHD: there, magnetic flux through a co-moving loop is conserved when resistivity vanishes; here, vorticity flux is conserved when viscosity vanishes. The two subjects share the structure and, consequently, share their failure mode — both conservation laws break down precisely in thin layers where the neglected diffusivity finally matters, producing reconnection in one case and boundary-layer vorticity generation in the other.

The starting vortex

The most striking consequence. A flow started from rest has \(\Gamma = 0\) around every loop. Kelvin says it can never acquire net circulation by inviscid means. But a wing needs bound circulation \(+\Gamma\) to generate lift.

The resolution: take a large loop enclosing both the wing and its wake. That loop's circulation must remain zero, so the wing's \(+\Gamma\) must be paid for by shedding a starting vortex of circulation \(-\Gamma\), left behind in the fluid at the moment lift begins.

This is not a theoretical device — you can watch it. Dip a canoe paddle and pull: a vortex rolls off the trailing edge with every stroke. Every aircraft that has ever taken off left a starting vortex at the runway.

Common mistakes

  • Applying it to a fixed loop. The loop must be material — made of the same fluid particles for all time. Circulation around a loop fixed in space changes freely.
  • Forgetting barotropicity. Baroclinic vorticity generation (\(\nabla\rho\times\nabla p \neq 0\)) is how sea breezes, thermals and shock-driven instabilities create vorticity from nothing.
  • Thinking viscosity destroys circulation. It diffuses vorticity; the total circulation of an isolated vortex in an unbounded fluid is conserved even with viscosity. What viscosity does is let vorticity cross a material loop.

Knowledge graph position

Prerequisites: vorticity, Euler's equation, Stokes' theorem. Leads to: Helmholtz theorems, wingtip vortices, airfoil theory.

Quiz

Q1 (conceptual). Why must an airfoil shed a starting vortex?

Answer

A flow from rest has zero circulation on every material loop, and Kelvin's theorem preserves that. A loop enclosing both wing and wake must therefore still have \(\Gamma = 0\), so the bound circulation \(+\Gamma\) needed for lift must be balanced by \(-\Gamma\) shed into the fluid.

Q2 (conceptual). Which hypothesis of Kelvin's theorem fails in a sea breeze, and what does it generate?

Answer

Barotropicity. Land heats faster than sea, so density varies horizontally while pressure varies vertically; \(\nabla\rho\times\nabla p \neq 0\). This baroclinic term generates circulation from rest — the sea breeze is vorticity created by the failure of Kelvin's theorem.

Q3 (MCQ). Kelvin's theorem applies to a loop that is:

  • (a) fixed in space
  • (b) material — always composed of the same fluid particles
  • (c) any closed loop
  • (d) perpendicular to the flow
Answer

(b). The theorem is about a loop advected by the flow. For a loop fixed in space, circulation can change simply because different fluid occupies it.