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 {\displaystyle a} and a potential vortex outside the cylinder. The radius a {\displaystyle a} is referred to as the vortex-core radius. The velocity components ( v r , v θ , v z ) {\displaystyle (v_{r},v_{\theta },v_{z})} of the Rankine vortex, expressed in terms of the cylindrical-coordinate system ( r , θ , z ) {\displaystyle (r,\theta ,z)} are given by
v r = 0 , v θ ( r ) = Γ 2 π { r / a 2 r ≤ a , 1 / r r > a , v z = 0 {\displaystyle 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 Γ {\displaystyle \Gamma } is the circulation strength of the Rankine vortex. Since solid-body rotation is characterized by an azimuthal velocity Ω r {\displaystyle \Omega r} , where Ω {\displaystyle \Omega } is the constant angular velocity, the parameter Ω = Γ / ( 2 π a 2 ) {\displaystyle \Omega =\Gamma /(2\pi a^{2})} can also be used to characterize the vortex. The vorticity field ( ω r , ω θ , ω z ) {\displaystyle (\omega _{r},\omega _{\theta },\omega _{z})} associated with the Rankine vortex is
ω r = 0 , ω θ = 0 , ω z = { 2 Ω r ≤ a , 0 r > a . {\displaystyle \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 Burgers vortex 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.
References
External links Streamlines vs. Trajectories in a Translating Rankine Vortex: an example of a Rankine vortex imposed on a constant velocity field, with animation.
