ME521_Lecture_22

ME521_Lecture_22 - ME 521 Fall 2007 Professor John M...

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ME 521 Fall 2007 Professor John M. Cimbala Lecture 22 10/17/2007 Today, we will : Continue to discuss 2-D, irrotational, incompressible flow – the complex potential, analytic functions, the complex velocity, building block potential flows Last time, we defined the complex potential function w ( z ) = φ ( x , y ) + i ψ ( x , y ) , where z = x + iy .
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Elementary Planar Irrotational Flows in Complex Variables Author: John M. Cimbala, Penn State University Latest revision: 17 October 2007 Note : Consider steady, incompressible, irrotational, Newtonian fluid flow in which gravity is neglected. The flow is assumed to be two-dimensional in the x-y or r- θ plane. Summary of the Equations Complex potential: ( ) ( ) () , , wz xy i φψ =+ or ( ) ( ) , , r i r φ θψθ , where i zxi yr e =+ = , u x y ψ == , v yx ∂∂ , 1 r u rr , and 1 u . Complex velocity: i (, ) (,) (, i r dw uxy i vxy q e u r iu r e dz α θθ =− = = where 22 qu v , 1 tan v u ⎛⎞ = ⎜⎟ ⎝⎠ . Elementary Planar Irrotational Flows a. Uniform stream in the x -direction : 0 uU v = = , w z Uz Ux iUy ==+ , dw U dz = , Ux = , Uy = b. Uniform stream in an
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ME521_Lecture_22 - ME 521 Fall 2007 Professor John M...

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