Rankine vortex

The Rankine vortex is a simple mathematical model of a vortex in a viscous fluid. It is named after its discoverer, William John Macquorn Rankine.

Velocity distribution in a Rankine vortex.

Animation of a Rankine vortex. Free-floating test particles reveal the velocity and vorticity pattern.

The vortices observed in nature are usually modelled with an irrotational (potential or free) vortex. However, in a potential vortex, the velocity becomes infinite at the vortex center. In reality, very close to the origin, the motion resembles a solid body rotation. The Rankine vortex model assumes a solid-body rotation inside a cylinder of radius $a$ and a potential vortex outside the cylinder. The radius $a$ is referred to as the vortex-core radius. The velocity components $(v_{r},v_{\theta },v_{z})$ of the Rankine vortex, expressed in terms of the cylindrical-coordinate system $(r,\theta ,z)$ are given by

$v_{r}=0,\quad v_{\theta }(r)={\frac {\Gamma }{2\pi }}{\begin{cases}r/a^{2}&r\leq a,\\1/r&r>a\end{cases}},\quad v_{z}=0$

where $\Gamma$ is the circulation strength of the Rankine vortex. Since solid-body rotation is characterized by an azimuthal velocity $\Omega r$, where $\Omega$ is the constant angular velocity, one can also use the parameter $\Omega =\Gamma /(2\pi a^{2})$ to characterize the vortex.

The vorticity field $(\omega _{r},\omega _{\theta },\omega _{z})$ associated with the Rankine vortex is

$\omega _{r}=0,\quad \omega _{\theta }=0,\quad \omega _{z}={\begin{cases}2\Omega &r\leq a,\\0&r>a\end{cases}}.$

At all points inside the core of the Rankine vortex, the vorticity is uniform at twice the angular velocity of the core; whereas vorticity is zero at all points outside the core because the flow there is irrotational.

In reality, vortex cores are not always circular; and vorticity is not exactly uniform throughout the vortex core.

See also

• Kaufmann (Scully) vortex – an alternative mathematical simplification for a vortex, with a smoother transition.
• Lamb–Oseen vortex – the exact solution for a free vortex decaying due to viscosity.
• Burgers vortex

External links

• Streamlines vs. Trajectories in a Translating Rankine Vortex: an example of a Rankine vortex imposed on a constant velocity field, with animation.