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Unformatted text preview: EGM 6812/ Fluid Mechanics I Homework # 6 Fall 2009 1. ( Bernoulli equation in axisymmetric flows ) For steady axisymmetric flows of an inviscid fluid of constant density , show that + + + F( ) = constant where is the potential for the body force, is the Stokes stream function. Hint : you need to show first, from the vorticity transport equation, that the vorticity (in the direction) is related to as /r = f( ). Solution: Euler equation at steady state: u u = 1 p f  + (1) Note: u u = (u u )  u x ; f =  ; 1 p p = ; (2) => 2 2 u p u + + = (3) For 2D axisymmetric flows, u =( ( , , 0) z r u u , = (0, 0, ) The mass conservation u =0 gives u = 1 e r = 1 1 z r e e r r r z  (4) = r z u u z r  = 2 2 2 2 1 1 r r r r z  + (5) The vorticity transport equation is ( / ) D r Dt = (see problem HW#5.3) Thus, /r = constant along a streamline when the flow is steady. That is = r f( ) (6) Thus, LHS of Eq. (3) becomes u = 1 1 ( ) z r e e rf e r r r z  = ( ) f  (7) (3) & (7) => 2 2 u p + + = ( ) f  (8) Let ( ) ( ) F f d = , then '( ) ( ) F F f = = Hence, 2 2 u p + + = ( ) F  which integrates to + + + F( ) = constant. 2. ( Compressible Bernoulli equation ) From the Bernoulli equation for isentropic compressible flows, show that = [1v 2 /v] /( 1) . For v 2 /v1, show that p = p  [1 v 2 /(4a)+ ...] using Taylor series expansion. If you use the incompressible Bernoulli equation to compute the pressure at v=68 (m/s) when a =340 (m/s), how much relative error in p results in comparison with the dynamic head v 2 /2? Solution: i) Bernoulli equation for isentropic compressible flows : 2 2 V h h + = or 2 2 max 2 2 V V h + = with 2 max 1 2 p V h = = , 1 p h = => 2 2 2 max max 1 2 V h V V =  = / 1 / p p  Since p p = , we have 1 1/ 2 2 max 1 p V p V  =  => /( 1) 2 2 max 1 p V p V  = (1) ii) Since /( 1) 2 2 2 2 2 2 2 max max max 1 1 1 1 ......
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This note was uploaded on 09/09/2010 for the course EGM 6812 taught by Professor Renweimei during the Fall '09 term at University of Florida.
 Fall '09
 RENWEIMEI

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