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.

A swirling flow in a viscous fluid can be characterized by a central core comprising a forced vortex, surrounded by a free vortex. In an inviscid fluid, on the other hand, a swirling flow consists entirely of a free vortex with a singularity at its center point. The tangential velocity[1] of a Rankine vortex with circulation ${\displaystyle \Gamma }$ and radius ${\displaystyle R}$ is

${\displaystyle u_{\theta }(r)={\begin{cases}\Gamma r/(2\pi R^{2})&r\leq R,\\\Gamma /(2\pi r)&r>R.\end{cases}}}$

The remainder of the velocity components are identically zero, so that the total velocity field is ${\displaystyle \mathbf {u} =u_{\theta }\ \mathbf {e_{\theta }} }$.