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Aerodynamics Help

  1. Sep 19, 2007 #1
    1. The problem statement, all variables and given/known data
    A two dimensional free vortex is located near an infinite plane at a distance h above the plane. The pressure at infinity is p_\infty and the velocity at infinity is U_\infty parallel to the plane. Find the total force (per unit depth normal to the paper) on the plane if the pressure on the underside of the plane is p_\infty. The strength of the vortex is \Gamma. The fluid is incompressible and perfect. To what expression does the force simplify if h becomes very large?


    2. Relevant equations

    I know that I will eventually integrate pressure over an area to get the force. I just dont know where to start. I also thought about imposing a mirror image of the cortex under the plane to cancel out the y components of velocity. (Not sure about this)

    3. The attempt at a solution
    It is intuitive that as h increase the force becomes negligable. I just cant find the expression for the force.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 20, 2007 #2

    Astronuc

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    Staff: Mentor

    The net force on the plate would result from the pressure differential across the plate, i.e on the bottom of the plate the pressure is constant P_infty, while above the plate, the vortex reduces the pressure locally, but the pressure would be P_infty as one moves further (laterally) from the vortex.

    Can one determine the pressure field in and around the vortex?
     
  4. Sep 20, 2007 #3
    I can use Bernoulls's equation to find the pressure above the plain right? Then integrate over the area of the vortex?
     
  5. Sep 20, 2007 #4

    Astronuc

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    Staff: Mentor

    Is the 2-D vortex in a plane parallel to the plate, such that the axis is normal to the plate?

    I'm trying to visualize a 2-D vortex, which I assume is circular in 2-D?

    Ostensibly one would have a formula for the pressure within the vortex as a funtion of its rotational velocity.
     
  6. Sep 20, 2007 #5
    I am not sure what you are asking, see the word document i have atached.
     

    Attached Files:

  7. Sep 23, 2007 #6

    Astronuc

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    Staff: Mentor

    That helps.

    There is a relationship between the velocity field around the vortex which is proportional to the vortex strength, [itex]\Gamma[/itex]. Below the vortex, the flow field of the vortex is in the opposite direction of U-infty.

    Is there a discussion in one's textbook that includes something like [tex]U_\theta = \frac{\Gamma}{2\pi{r}}[/tex], where r is the distance measure from the center of the vortex. I think then it is a matter of expressing the flow field around the vortex by r ~ h - y, where y is the elevation from the plane.

    When h gets very large, the pressure equation should be that of Bernoulli's equation for a flow of U_infty and static pressure P_infty.
     
  8. Sep 23, 2007 #7
    Thank you for your help, I have figured out the answer. I took the stream function for the superposition of flows and added a mirror image of the vortex. Then converted the stream function to rectangular coordinates. The took derivative wrt y then integrated pressurewrt to x this game me the lift = density X strength X Velocity (when h goes to infininty)
     
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