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Why is there circulation around a wing?

  1. Dec 5, 2014 #1
    1. The problem statement, all variables and given/known data
    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?"

    3. 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.
     
  2. jcsd
  3. Dec 5, 2014 #2

    Doug Huffman

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    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.)
     
  4. Dec 6, 2014 #3

    haruspex

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    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.
     
  5. Dec 6, 2014 #4

    rcgldr

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    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.
     
  6. Dec 8, 2014 #5

    boneh3ad

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    So then explore why the flow is not irrotational. Do you know what is required in order for a flow to be irrotational?
     
  7. Dec 8, 2014 #6

    haruspex

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    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.
     
  8. Dec 8, 2014 #7

    pasmith

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    This is the case. A finite body surrounded by fluid permits velocity fields with singularities which would lie within the body. Such as: [tex]
    \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[/tex]for constant [itex]k[/itex].

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