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The Vorticity Equation

Source lecture(s): PC316 Ch. 8

Intuition

Take the curl of the Navier–Stokes equation at constant density. The pressure gradient drops out entirely — the curl of a gradient is zero — and what remains is a local transport equation with one genuinely new term.

Vorticity equation

\[\frac{D\boldsymbol{\omega}}{Dt} = \underbrace{(\boldsymbol{\omega}\cdot\nabla)\mathbf{v}}_{\text{stretching + tilting}} + \underbrace{\nu\nabla^2\boldsymbol{\omega}}_{\text{diffusion}}\]

Vorticity is transported with the fluid, diffused by viscosity, and — the new effect — amplified by stretching.

Vortex stretching

Decompose \((\boldsymbol{\omega}\cdot\nabla)\mathbf{v}\) relative to the vorticity direction:

  • The component along \(\boldsymbol{\omega}\) is stretching. If the velocity field stretches a fluid element along its own vorticity vector, the tube thins, and Helmholtz theorem 2 (\(\Gamma = \omega A\) fixed) forces \(\omega\) to rise. It is the figure skater pulling in her arms, and it is conservation of angular momentum.
  • The component perpendicular is tilting: it rotates vortex lines into new directions, converting (say) horizontal vorticity into vertical.

The term that makes 3-D turbulence

\((\boldsymbol{\omega}\cdot\nabla)\mathbf{v}\) is quadratic in the flow and can amplify vorticity without bound (until viscosity intervenes). It is the mechanism of the energy cascade: large eddies stretch smaller ones, spinning them up and making them smaller still, until the Kolmogorov scale is reached and viscosity dissipates the energy.

In two dimensions this term vanishes identically. With \(\boldsymbol{\omega} = \omega\hat{z}\) and \(\mathbf{v}\) in the \(xy\)-plane, \((\boldsymbol{\omega}\cdot\nabla)\mathbf{v} = \omega\,\partial_z\mathbf{v} = 0\). So in 2-D, vorticity is simply advected and diffused:

\[\frac{D\omega}{Dt} = \nu\nabla^2\omega\]

This single missing term is why two-dimensional turbulence is a completely different subject — energy cascades to large scales rather than small ones, coherent vortices survive and merge rather than being shredded, and the atmosphere and oceans (quasi-2D at large scale) behave accordingly. The banded jets of Jupiter and the zonal flows of a tokamak are both consequences.

The viscous term

\(\nu\nabla^2\boldsymbol{\omega}\) diffuses vorticity exactly as heat diffuses. A concentrated vortex spreads: the Lamb–Oseen vortex is the exact solution, with core radius growing as \(\sqrt{4\nu t}\). This is also why the Rankine vortex's sharp corner is an idealisation — viscosity smooths it within moments.

Common mistakes

  • Expecting vortex stretching in 2-D. It is identically zero. If your 2-D code shows vorticity amplification, it is a bug.
  • Thinking the pressure term was neglected. It was eliminated, exactly, by the curl. No approximation.
  • Confusing stretching with tilting. Stretching changes \(|\boldsymbol{\omega}|\); tilting changes its direction. Both come from the same term.

Knowledge graph position

Prerequisites: Navier–Stokes, vorticity, material derivative. Leads to: turbulence, energy cascade, vortex dynamics in 3-D.

Quiz

Q1 (conceptual). Why is two-dimensional turbulence qualitatively different from three-dimensional?

Answer

The stretching/tilting term \((\boldsymbol{\omega}\cdot\nabla)\mathbf{v}\) vanishes identically in 2-D, so vorticity can only be advected and diffused — never amplified. Without the stretching engine there is no forward cascade to small scales; instead energy moves to large scales and coherent vortices persist and merge.

Q2 (computational). A vortex tube of radius 1 cm and vorticity 10 s⁻¹ is stretched until its radius is 2 mm. What is the new vorticity?

Answer

\(\Gamma = \omega A\) is fixed, and \(A \propto r^2\). The area falls by \((1/0.2)^2 = 25\), so \(\omega\) rises by 25 to 250 s⁻¹.

Q3 (MCQ). The pressure gradient does not appear in the vorticity equation because:

  • (a) it is negligible
  • (b) the curl of a gradient is identically zero
  • (c) the flow is incompressible
  • (d) it was absorbed into the viscous term
Answer

(b). It is an exact elimination, not an approximation — and it is precisely why vorticity is such a convenient variable, since pressure in incompressible flow is otherwise a non-local elliptic constraint.