AdvancedInviscidProblemsSolutions

AdvancedInviscidProblemsSolutions - MECH 660 Advanced Fluid...

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MECH 660 Advanced Fluid Mechanics March 17, 2004 American University of Beirut, Spring 2004 Handout # PS2 Solution Problem Set 2 Solution: Inviscid Flow “Advanced Fluid Mechanics Problems” by Shapiro and Sonin Problems 4.4, 4.7, 4.8, 4.9, 4.10, 4.13, 4.18, 4.19, 4.21, 4.23, 4.24, 4.28. Problem 4.4 Refer to Figure 1 for the schematic. The flow is 1D, inviscid and incompressible. Gravi- A 1 A(x) x A 2 x x+dx A(x) A(x) + dA p n x dS Figure 1: Schematic of Problem 4.4 tational effects are negligible. (a) Applying Bernoulli’s equation along a streamline from location x to location 2, p 2 ρ + u 2 2 2 = p ( x ) ρ + u 2 ( x ) 2 p ( x ) - p ( a ) = ρ 2 u 2 2 - u 2 ( x ) · p ( x ) - p ( a ) = ρQ 2 2 ˆ 1 A 2 2 - 1 A 2 ( x ) ! (b) We consider a differential volume between x and x + dx . The cross sectional areas are respectively A ( x ) and A ( x ) + ( dA/dx ) dx . The nozzle wall area onto which the pressure acts is dS so that the corresponding force x - component is dF x = p ( x ) dS ˆ n · ˆ x = p ( x ) dA , where ˆ n is the unit vector normal to dS and pointing into the wall. The total force component in the x direction is F x = Z 2 1 dF x = Z A 2 A 1 pdA = Z A 2 A 1 ρQ 2 2 ˆ 1 A 2 2 - 1 A 2 ( x ) ! dA F x = ρQ 2 ( A 1 - A 2 ) 2 2 A 1 A 2 2 1
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(c) The product p dA has always the same sign. If A % , dA > 0 u &⇒ p % If A & , dA < 0 u %⇒ p & 2
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Problem 4.7 Refer to Figure 2 for the schematic. D d p a , ρ a p a ρ f u a u f 1 Figure 2: Schematic of Problem 4.7 The fuel-air mass flow rate ratio is α = ˙ m f ˙ m a = ρ f u f ( πd 2 / 4) ρ a u a ( πD 2 / 4) d D = ˆ α ρ a ρ f u a u f ! 1 / 2 In order to determine the ration u a /u f , we apply Bernoulli’s equation along two stream lines, one for air from the ambient to location 1 in the vicinity of the fuel jet and one for fuel from the reservoir to location 1.
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