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Vorticity

Source lecture(s): PC316 Ch. 8

Intuition

Vorticity is the curl of the velocity field:

\[\boldsymbol{\omega} = \nabla\times\mathbf{v}\]

It equals twice the local angular velocity of a fluid element: a tiny paddle wheel dropped into the flow spins at \(\boldsymbol{\omega}/2\). The single most important thing to understand is that vorticity measures local spin, not curved motion — and those two are genuinely independent.

Two examples that separate spin from curvature

Plane shear flow, \(\mathbf{v} = (ky,\,0,\,0)\). Perfectly straight, parallel streamlines. No fluid element travels a curved path. Yet

\[\omega_z = \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y} = -k \neq 0\]

The paddle wheel spins because its top edge is in faster fluid than its bottom edge. Straight streamlines, non-zero vorticity.

The irrotational vortex, \(v_\theta = \Gamma/2\pi r\). Circular streamlines; every fluid element orbits the axis. Yet \(\boldsymbol{\omega} = 0\) everywhere except on the axis itself. As an element goes round, the shear from the \(1/r\) profile rotates it backwards at exactly the rate its orbit rotates it forwards. The two cancel. It is a gondola on a Ferris wheel: it circles the centre without ever turning over. Curved streamlines, zero vorticity.

Getting these two straight is most of the conceptual work of the chapter.

Why it is worth changing variables

Taking the curl of Navier–Stokes eliminates the pressure entirely — see the vorticity equation. Pressure in incompressible flow is not a thermodynamic variable but a Lagrange multiplier enforcing \(\nabla\cdot\mathbf{v}=0\), determined globally by an elliptic problem. Vorticity obeys a local transport equation instead.

Vorticity is also compact where velocity is not. A vortex ring occupies a small region; the velocity field it induces fills all space. Tracking the vorticity means tracking a small object and reconstructing the rest via Biot–Savart.

Where it comes from

In an ideal barotropic fluid, Kelvin's theorem says circulation cannot be created at all. So all vorticity in an ordinary flow is generated at boundaries — the no-slip condition at a wall is a vorticity source — or by baroclinicity, when \(\nabla\rho\) and \(\nabla p\) are not parallel (which is why a horizontal density gradient in a gravitational field spins up a flow, and why sea breezes exist).

Common mistakes

  • "Curved streamlines mean vorticity." The irrotational vortex is the counterexample, and it is the single most common error in the chapter.
  • "Straight streamlines mean no vorticity." Plane shear is the counterexample. Any boundary layer is full of vorticity with nearly straight streamlines.
  • Confusing vorticity with circulation. Vorticity is a local vector field; circulation is its integral over a surface. One is a density, the other a total.

Knowledge graph position

Prerequisites: Eulerian vs Lagrangian, material derivative, vector calculus. Leads to: circulation, vorticity equation, turbulence.

Quiz

Q1 (conceptual). A flow has perfectly circular streamlines. Is it rotational?

Answer

Not necessarily. If \(v_\theta = \Gamma/2\pi r\), the vorticity is zero everywhere off the axis: the orbital rotation of a fluid element is exactly cancelled by the shear of the \(1/r\) profile. If instead \(v_\theta \propto r\) (solid-body rotation), the vorticity is uniform and non-zero. Streamline shape alone tells you nothing.

Q2 (computational). Find the vorticity of \(\mathbf{v} = (-\Omega y,\ \Omega x,\ 0)\) and interpret it.

Answer

\(\omega_z = \partial_x v_y - \partial_y v_x = \Omega - (-\Omega) = 2\Omega\). This is solid-body rotation at angular velocity \(\Omega\), and the vorticity is exactly twice it — the general rule.

Q3 (MCQ). Taking the curl of the incompressible Navier–Stokes equation is useful mainly because it:

  • (a) linearises the equation
  • (b) eliminates the pressure, which is otherwise a non-local elliptic constraint
  • (c) removes the viscous term
  • (d) makes the equation one-dimensional
Answer

(b). The pressure gradient is a gradient, and the curl of a gradient vanishes. What remains is a local transport equation for a compactly supported quantity — a much better variable to compute with.