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Unformatted text preview: vector Fluids – Lecture 16 Notes 1. Vortex 2. Lifting ﬂow about circular cylinder Reading: Anderson 3.14 – 3.16 Vortex Flowfield Definition A vortex ﬂow has the following radial and tangential velocity components C V r = 0 , V θ = r where C is a scaling constant. The circulation around any closed circuit is computed as θ 2 C Γ ≡ − V · dvectors = − V θ r dθ = − r dθ = − C ( θ 2 − θ 1 ) θ 1 r y y x V ds θ d θ r d The integration range θ 2 − θ 1 = 2 π if the circuit encircles the origin, but is zero otherwise. − 2 πC , (circuit encircles origin) Γ = 0 , (circuit doesn’t encircle origin) y y θ θ 2 1 x θ θ 2 1 x In lieu of C , it is convenient to redefine the vortex velocity field directly in terms of the circulation of any circuit which encloses the vortex origin. Γ V θ = − 2 π r 1 x A positive Γ corresponds to clockwise ﬂow, while a negative Γ corresponds to counterclock- wise ﬂow. Cartesian representation The cartesian velocity components of the vortex are u ( x, y ) = Γ 2 π y x 2 + y 2 v ( x, y ) = − Γ 2 π x x 2 + y 2 and the corresponding potential and stream functions are as follows....
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- Fall '05
- Fluid Dynamics, Lift, Circular Cylinder, Vθ, Vortex Flow, vortex velocity ﬁeld