Rankine vortex

{{Short description|Mathematical formula for viscous fluid}}

{{one source |date=April 2024}}

File:RankineVortex.svg

File:Rankine vortex animation.gif

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 $(v\_r,v\_\backslash theta,v\_z)$ of the Rankine vortex, expressed in terms of the cylindrical-coordinate system $(r,\backslash theta,z)$ are given by{{cite book

| title = Elementary Fluid Dynamics

| author = D. J. Acheson

| publisher = Oxford University Press

| year = 1990

| isbn = 0-19-859679-0

}}

:

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

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

:

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.