Computing Circulation in Pipe Flow using Velocity & Angular Direction

[womfalcs3]

I need to find the circulation for a flow in a pipe. I'm using the velocity in the angular direction (cylindrical coordinate system).

Am I right in thinking that an ideal vortex is one where there is no vorticity? That would mean there is no circulation. It's strange, because the velocity is written in terms of circulation, and its circulation is constant. So that can't be correct.

I'm trying to write the circulation in the forms of a surface integral and a line integral. The surface integral contains vorticity, which shows up as zero when I compute it. Say if the velocity is gamma/(2*pi*r)

gamma = circulation

The line integral of the circulation is: gamma = integral (velocity) ds

If I write ds = 2*pi*r, then I get gamma = v*2*pi*r, which just gives me gamma = gamma.

Any help in my thought process please?

 

[mikeph]

If your velocity field in polar coordinates is: vθ = Γ/2πr, vr = 0, then you have an interesting case!

You are correct that the circulation around any circle centred on the origin is Γ (this is a parameter, so don't be confused when you get this for your circulation, that's the way it's meant to come out), but you are also correct in saying the vorticity at any of these points you are integrating over is zero:

vorticity = curl of velocity field = (1/r)[d(r.vθ)/dr - dr/dθ]

d(r.vθ)/dr = 0 because the r cancels, so vorticity is zero.

This may seem to be a problem, because we have a vector theorem which states that the circulation of a conservative vector field around a closed loop (your circulation) is equal to the integral of the curl of that field over the area bounded by that loop, so if there is NO vorticity inside the loop, how can there no non-zero circulation? Very valid question.

I'll put the answer in spoiler because it's quite an interesting resolution... clue: when you are integrating the vorticity over the circular area around which you find the circulation to be Γ, are you actually summing all the points?

Spoiler

The answer is NO... you have not summed the vorticity at the origin! The vorticity is zero for all positive r, but at the origin we have an undefined flow. The non-rigorous explanation is that the entirety of the vorticity that we require to be inside our closed circulation loop is concentrated at the origin, and no-where else!

This is why the circulation is the same for all radii- if there were a bit of vorticity at, say, (r=1,theta=0), then the circulation around a circle radius 0.5 would be smaller than that around one of radius 2, because the circulation must be the sum of all vorticities inside the loop. However, since we can take a loop around any positive radius, arbitrarily small, and still find that the circulation is Γ, we must have all the vorticity at the origin.