Why is there circulation around a wing?

[hyurnat4]

Homework Statement

From my notes: "In an irrotational flow, Γ = 0 for any curve lying wholly within the fluid. But circulation around a wing (airflow) is possible! Why?"

The Attempt at a Solution

The obvious answer is that the air around the wing isn't irrotational. But that seems a bit too simple: they're implying that there's a possible contradiction here. I suspect that the answer is in Bernoulli's or Euler's equations, which I've heard are linked to why planes fly. But that's well ahead in my notes and I don't understand any of that yet.

 

[Doug Huffman]

When I sailed, I took great advantage of Czeslaw A. Marchaj's aerodynamics that used circulation around the entire sail-plan. Sail Performance, Techniques to Maximize Sail Power, Revised edition. (London: Adlard Coles Nautical, 2003.)

 

[haruspex]

"In an irrotational flow, Γ = 0 for any curve lying wholly within the fluid.

I'm no expert on this, but I always thought the condition was that the curve did not go around any point that was not part of the flow. I.e. you could embed the curve in a 2-D simply connected manifold that was entirely contained in the flow.

 

[rcgldr]

In this case "circulation" doesn't mean that the same parcel of air flows all the way around the wing, it's more of a reference to the relative flows at the front, rear, top, and bottom of a wing.

 

[boneh3ad]

The obvious answer is that the air around the wing isn't irrotational. But that seems a bit too simple: they're implying that there's a possible contradiction here.

So then explore why the flow is not irrotational. Do you know what is required in order for a flow to be irrotational?

 

[haruspex]

I found http://en.wikipedia.org/wiki/Vortex#Irrotational_vortices, which supports the explanation I gave at post #3:
"For an irrotational vortex, the circulation is zero along any closed contour that does not enclose the vortex axis and has a fixed value,".
Since the wing may effectively represent a vortex, a contour around the wing can have a nonzero circulation.

 

[pasmith]

I'm no expert on this, but I always thought the condition was that the curve did not go around any point that was not part of the flow.

This is the case. A finite body surrounded by fluid permits velocity fields with singularities which would lie within the body. Such as: $$\mathbf{u}(x,y) = - \frac{k(y - y_0)}{2\pi ((x - x_0)^2 + (y - y_0)^2)}\mathbf{e}_x + \frac{k(x - x_0)}{2\pi ((x - x_0)^2 + (y - y_0)^2)}\mathbf{e}_y$$for constant $k$.

Exercise for the OP: calculate the circulation of this field on a curve consisting of a circle of radius $a > 0$ centered at $(x_0, y_0)$.