worksheet8 - Fluid Dynamics 3 2011/12 Sheet 8 Homework to...

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Unformatted text preview: Fluid Dynamics 3 2011/12 Sheet 8 Homework to be handed in 9th December: questions 1,2,3. 1. Consider the two-dimensional, unbounded flow outside a cylinder of radius R centred at the origin. (i) Let the complex potential of the flow in the absence of the cylinder be w ( z ) = f ( z ). Show that w ( z ) = f ( z ) + f ( R 2 / z ) is the solution in the presence of the cylinder, i.e. ={ w ( z ) } = const for | z | = R . (ii) Now consider a vortex at position z outside of the cylinder: f ( z ) =- i 2 ln( z- z ) . Compute the complex potential and thus show that u- iv = i R 2 2 h 2 z h 2- R 2 , where h = | z | is the distance of the vortex from the centre. (iii) Show that the vortex moves around the cylinder in a circle, and compute the angular speed of the motion. 2. Vortices shed from a plane moving along a runway at y = 0 are modelled by line vortices of circulation at ( s,h ) and- at (- s,h ) in the flow domain y > 0. The flow is assumed to be two-dimensional, incompressible and irrotational apart from the vortices (...
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This note was uploaded on 01/20/2012 for the course MATH 33200 taught by Professor Eggers during the Fall '11 term at University of Bristol.

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worksheet8 - Fluid Dynamics 3 2011/12 Sheet 8 Homework to...

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