Thayer Watkins

The Vorticity of a Vortex

A general vortex has the velocity field tangential to the concentric circles about some point of origin and the magnitude of the velocity is is constant on any circle and is a function of the radius of the circle, say V(r).

In natural or flow-following coordinates the formula for the point vorticity ζ is:

ζ = V/Rs - ∂V/∂n

For the case of counterclockwise flow Rs=r and ∂/∂n is -∂/∂r

ζ = V(r)/r + ∂V/∂r

Now consider


This is equal to

V + r∂V/∂r
= r(V/r + ∂V/∂r)

The above equations reduce to

∂(rV)/∂r = rζ

or, equivalently

ζ = (∂(rV)/∂r)/r

If a fluid is turning as a disk so that V(r) = ar for some constant a then the vorticity ζ is equal to 2a. If the velocity field were to have some constant component, say V(r) = ar+b, then the vorticity would be

ζ = + 2a + b/r

and thus ζ would go to infinity as r goes to zero.

It could be presumed therefore that V(0)=0. It might also be presumed that beyond some rmax the velocity would go to zero. If beyond some radius the velocity function were c/r then the vorticity would be zero.

A velocity field such that:

V(r,θ) = Vmax(r/rmax)θ for r≤rmax
= Vmax(rmax/r)θ for r>rmax
where θ is the unit tangential vector

is sometimes called a Rankine vortex. For this vortex

ζ = 2Vmax/rmax for r≤rmax
= 0 for r>rmax

Since the circumference of a circle of radius r is 2πr and the time τ taken to traverse this circle when the wind speed is V is 2πr/V the vorticity of a Rankine vortex is given by:

ζ = 4π/τ.

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