Rankine vortex

Velocity distribution in a Rankine vortex.

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

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.

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

$v_r=0,v_{\theta }(r)=\frac{\mathrm{\Gamma }}{2\pi }\{\begin{array}{cc}r/a^2& r\le a,\\ 1/r& r>a\end{array},v_z=0$

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

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

${\omega }_r=0,{\omega }_{\theta }=0,{\omega }_z=\{\begin{array}{cc}2\mathrm{\Omega }& r\le a,\\ 0& r>a\end{array}.$

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.

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