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worksheet9 - Fluid Dynamics 3 2011/12 Sheet 9 Homework to...

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Fluid Dynamics 3 2011/12 Sheet 9 Homework to be handed in 16th December: questions 1,3,5. 1. Consider a two-dimensional channel with walls located at y = 0 and y = π . A point vortex with circulation Γ > 0 is placed at z 0 = ib , with 0 < b < π . (i) Write down the complex potential for the flow in the channel, generated by the vortex. (ii) Show that the vortex moves parallel to the channel walls, at speed Γ 4 π cot b . Hint: the contribution coming from the vortex inside the channel has to be subtracted. 2. Once more consider a vortex with circulation Γ > 0 in a channel, but find the complex potential for the flow using the method of images. For simplicity, let the vortex be at z 0 = 0 in the centre of the channel, whose walls are at at y = i/ 2 and y = i/ 2. (i) Show that to satisfy the boundary conditions at both walls, image vortices are needed at z n = in, n Z . (ii) Show that the complex potential is w ( z ) = i Γ 2 π summationdisplay n = -∞ ( 1) n ln( z ni ).
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  • Fall '11
  • Eggers
  • Lift, Trailing edge, complex potential, horizontal potential flow, Joukowski mapping, symmetric Joukowski wing

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