• Source: 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.
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, one can also use the parameter



Ω
=
Γ

/

(
2
π

a

2


)


{\displaystyle \Omega =\Gamma /(2\pi a^{2})}

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


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




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.

Kata Kunci Pencarian:

• Pusaran
• Aerodinamika
• Rankine vortex
• Vortex
• William Rankine
• Vorticity
• Quantum turbulence
• Lamb–Oseen vortex
• Rankine
• Lofoten Vortex
• Kaufmann vortex
• Glossary of tornado terms