Rankine vortex explained

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^[1] v _{r=0,} v _{\theta(r)} = (\Gamma)/(2 \pi) \begin{cases} r/a ^{2} & r \le a, \ 1 / r & r > a \end{cases}, 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,} \omega _{\theta=0,} \omega _{z} = \begin{cases} 2 \Omega & r \le 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.

Notes and References

1. Book: Elementary Fluid Dynamics . D. J. Acheson . . 1990 . 0-19-859679-0 .