solution2

solution2 - 650:460 Aerodynamics Homework Assignment No 2...

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650:460 Aerodynamics Homework Assignment No. 2 Prof. Doyle Knight Tel: 732 445 4464, Email: [email protected] OFce hours: Tues and Thur, 4:30 pm - 6:00 pm and by appointment ±all 2009 1
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Problem 3.28 A two-dimensional free vortex is located near an inFnite plane at a distance h above the plane. The pressure at inFnity is p and the velocity at inFnity is U parallel to the plane. ±ind the total force (per unit depth) on the plane if the pressure on the underside of the plane is p . The strength of the vortex is Γ. The ²uid is incompressible and perfect. To what expression does the force simplify if h becomes very large ? Solution The velocity potential for uniform ²ow plus a free vortex is φ ( x , y ) = U x Γ 2 π tan 1 p y h x P where θ = tan 1 ( y h ) / x . Note that θ is measured from the center of the vortex, not the origin of the coordinate system. 2
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Problem 3.28 This velocity potential satisfes Laplace’s equation 2 φ = 0 by the Theorem oF Superposition. However, it does not satisFy the boundary condition that the ±ow is parallel to the x axis. In order to satisFy this boundary condition, an image vortex oF the opposite circulation must be added at ( x , y ) = (0 , h ) φ ( x , y ) = U x Γ 2 π tan 1 p y h x P + Γ 2 π
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This note was uploaded on 12/15/2009 for the course 650 460 taught by Professor Knight during the Spring '09 term at Rutgers.

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solution2 - 650:460 Aerodynamics Homework Assignment No 2...

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