solnsheet5 - Fluid Dynamics 3 - Solutions to Sheet 5 1. (i)...

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Unformatted text preview: Fluid Dynamics 3 - Solutions to Sheet 5 1. (i) The flow field of a uniform stream is u = U z , and thus u r =- 1 r z = 0 , u z = 1 r r = U. From the first equation, = ( r ), and from the second equation = Ur 2 / 2. (ii) According to the lecture, the stream function of the stream and the source is = U 2 r 2 + m 4 1- z z 2 + r 2 . The stagnation point z =- a is where the velocities of the stream and the source com- pensate along the axis, and thus U = m 4 a 2 or a = r m 4 U . Thus = 2 Ua 2 at the stagnation point. The stream function is constant along a streamline and thus along the surface of the body, where the value is 2 Ua 2 . Thus 2 Ua 2 = U 2 r 2 + Ua 2 1- z z 2 + r 2 , from which the shape of the Rankine body fol- lows. (iii) The velocity field is u r = Ua 2 r ( z 2 + r 2 ) 3 / 2 , u z = U + Ua 2 z ( z 2 + r 2 ) 3 / 2 , and so u 2 = U 2 " 1 + 2 a 2 z ( z 2 + r 2 ) 3 / 2 + a 4 ( z 2 + r 2 ) 2 # . Now using z = cos , r = sin we find u 2 = U 2 " 1 + 2 a 2 cos 2 + a 4 4 # , and the surface of the body is given by 2 = 2 a 2 1 + cos sin 2 = 2 a 2 1- cos . Hence on the surface u 2 = U 2 " 1 + cos (1- cos ) + (1- cos ) 2 4 # . (iv) Very far upstream the pressure is atmospheric, and u 2 = U 2 . Thus Bernoulli gives p atm + U 2 / 2 = p ( ) + u 2 / 2 , and therefore p ( ) = p atm- U 2 8 (1- cos )(3cos + 1) ....
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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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solnsheet5 - Fluid Dynamics 3 - Solutions to Sheet 5 1. (i)...

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