solnsheet9 - Fluid Dynamics 3 Solutions to Sheet 9 1(i Use...

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Unformatted text preview: Fluid Dynamics 3 - Solutions to Sheet 9 1. (i) Use the transformation ΞΆ = e z , so the vortex is at ΞΆ = e ib in the upper half plane. Then from the method of images w ( ΞΆ ) = βˆ’ i Ξ“ 2 Ο€ ( ln( ΞΆ βˆ’ ΞΆ ) βˆ’ ln( ΞΆ βˆ’ ΞΆ ) ) in the upper half plane and w ( z ) = βˆ’ i Ξ“ 2 Ο€ ln bracketleftbigg e z βˆ’ e ib e z βˆ’ e βˆ’ ib bracketrightbigg in the channel. (ii) To find the velocity field convecting the vortex, the piece coming from the vortex at z = ib (which is w ( z ) = βˆ’ i Ξ“ 2 Ο€ ln( z βˆ’ ib )), has to be subtracted, giving w s ( z ) = w ( z ) βˆ’ w ( z ) = βˆ’ i Ξ“ 2 Ο€ braceleftbigg ln bracketleftbigg e z βˆ’ e ib z βˆ’ ib bracketrightbigg βˆ’ ln(e z βˆ’ e βˆ’ ib ) bracerightbigg Since we want to know the velocity field at z = ib , we expand about this point: e z = e ib + e ib ( z βˆ’ ib ) + e ib 2 ( z βˆ’ ib ) 2 + O ( z βˆ’ ib ) 3 , giving w s ( z ) = βˆ’ i Ξ“ 2 Ο€ braceleftbigg ln bracketleftbigg e ib + e ib 2 ( z βˆ’ ib ) bracketrightbigg βˆ’ ln(e z βˆ’ e ib ) bracerightbigg . Then the velocity becomes u βˆ’ iv = dw s dz vextendsingle vextendsingle vextendsingle vextendsingle z = ib = βˆ’ i Ξ“ 2 Ο€ bracketleftbigg 1 2 βˆ’ e ib e ib βˆ’ e βˆ’ ib bracketrightbigg . Now e ib e ib βˆ’ e βˆ’ ib = e ib 2 i sin b = βˆ’ i cos b 2 sin b + sin b 2 sin b , which gives the desired result u + iv = Ξ“ 4 Ο€ cot b. 2. (i) The original vortex has images at z 1 = i and z βˆ’ 1 = βˆ’ i , and z 1 has images at z = 0 (upper wall) and z βˆ’ 2 = βˆ’ 2 i (lower wall). In general, the images of z n are at z βˆ’ n βˆ’ 1 and z βˆ’ n +1 . Evidently, this generates an infinity of images at all integer values. (ii) Let the contribution of each vortex to the complex poten- tial be ia n ln( z βˆ’ z n ). To satisfy the boundary conditions at each wall, the amplitude of each images must be of op- posite sign and equal strength: a βˆ’ n βˆ’ 1 = a βˆ’ n +1 = βˆ’ a n . Thus the vortices are alternating in sign and a n = ( βˆ’ 1) n +1 i Ξ“ 2 Ο€ ....
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solnsheet9 - Fluid Dynamics 3 Solutions to Sheet 9 1(i Use...

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