Rankine vortex
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The Rankine vortex is a simple mathematical model of a
vortex In fluid dynamics, a vortex ( : vortices or vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in th ...
in a
viscous The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inter ...
fluid. It is named after its discoverer,
William John Macquorn Rankine William John Macquorn Rankine (; 5 July 1820 – 24 December 1872) was a Scottish mechanical engineer who also contributed to civil engineering, physics and mathematics. He was a founding contributor, with Rudolf Clausius and William Thomson ( ...
. The vortices observed in nature are usually modelled with an
irrotational In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not c ...
(potential or free) vortex. However, in 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,\quad v_\theta(r) = \frac\begin r/a^2 & r \le a, \\ 1/ r & r > a \end, \quad 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 In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
, one can also use the parameter \Omega =\Gamma/(2\pi a^2) to characterize the vortex. The
vorticity In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along wi ...
field (\omega_r,\omega_\theta,\omega_z) associated with the Rankine vortex is :\omega_r=0,\quad \omega_\theta=0, \quad \omega_z = \begin 2\Omega & r \le a, \\ 0 & r > a \end. Inside the core of the Rankine vortex, the vorticity is constant and twice the angular velocity, whereas outside the core, the flow is irrotational. In reality, vortex cores are not always exactly circular; and vorticity is not uniform within the vortex core.


See also

*
Kaufmann (Scully) vortex The Kaufmann vortex, also known as the Scully model, is a mathematical model for a vortex taking account of viscosity.Mahendra J. Bhagwat and J. Gordon LeishmanGeneralized Viscous Vortex Model for Application to Free-Vortex Wake and Aeroacoustic Ca ...
– an alternative mathematical simplification for a vortex, with a smoother transition. *
Lamb–Oseen vortex In fluid dynamics, the Lamb–Oseen vortex models a line vortex that decays due to viscosity. This vortex is named after Horace Lamb and Carl Wilhelm Oseen. Mathematical description Oseen looked for a solution for the Navier–Stokes equati ...
– the exact solution for a free vortex decaying due to viscosity. *
Burgers vortex In fluid dynamics, the Burgers vortex or Burgers–Rott vortex is an exact solution to the Navier–Stokes equations governing viscous flow, named after Jan Burgers and Nicholas Rott. The Burgers vortex describes a stationary, self-similarity, self- ...


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

{{reflist Equations of fluid dynamics Vortices