HW2soln - EAS 4101 - Aerodynamics Spring 2008 1/24/08, 2nd...

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EAS 4101 - Aerodynamics – Spring 2008 1/15 1/24/08, 2 nd Homework 1. Given: - Cartesian form of the N-S equation Find: - Derive Bernoulli’s equation for incompressible, inviscid, and irrotational flow. - Please state all assumptions and show all steps. Solution: Basic Equations: Vector form of the Navier-Stokes equations: 2 b DV p fV Dt ρρ μ = −∇ + + ∇ G G G Assumptions: 1) Incompressible flow / constant thermodynamic properties, Stokes hypothesis 2) Steady 3) Inviscid flow 4) Irrotational flow Solution: Applying assumptions to the N-S equation we obtain 2 b DV p Dt = −∇ + + ∇ G G G V t ρ G () (2) b VV pf ρμ ⎛⎞ ⎜⎟ +⋅ = + + ⎝⎠ G GG (3) 2 V G Then ( ) b ⋅∇ =−∇ + G G G Recall the vector identity 2 1 2 V V V ∇= ∇− × × G G Since flow is irrotational this equation reduces to 2 1 2 V = ∇ G G Plugging back into the reduced N-S equation we obtain 2 1 2 b Vp f + G or 2 1 2 b f + G Where b f G is the gravity force acting on the fluid: b f gz =−∇ G Then we obtain 2 1 2 g z =−∇ − or
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EAS 4101 - Aerodynamics – Spring 2008 2/15 2 11 0 2 Vp g z ρ ⎛⎞ ++= ⎜⎟ ⎝⎠ Then the Bernoulli’s equation is 2 2 V p gz const ++ =
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EAS 4101 - Aerodynamics – Spring 2008 3/15 2. Given: The integral form of the energy equation, Find: Derive Bernoulli’s equation for a conservative fluid system. Please state all assumptions and show all steps. Solution: Basic Equations: Integral form of the energy equation () s shear other QW W W ed e p v Vd A t ρρ −− = + + ∫∫∫ ∫∫ G G ± ±± ± 2 2 V eu g z = ++ Assumptions: 1) 0 s W = ± , 0 other W = ± 2) 0 shear W = ± (wall velocities = 0) 3) Steady flow 4) Incompressible flow 5) Internal energy, pressure and velocity uniform across inlet and outlet control surfaces (reduce control surfaces to point 1 and point 2) 6) No heat transfer 7) No change in internal energy from 1 to 2 Solution: Applying the assumptions to the integral form of the energy equation found in Fox & McDonald, s shear other W A t = + + G G ± ± reduces to 22 21 0 p pV V mu u m mgz z m ⎛⎞ =− + + + ⎜⎟ ⎝⎠ ± ± and then to 11 2 2 12 p Vp V gz gz ++=++
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EAS 4101 - Aerodynamics – Spring 2008 4/15 3. Given: The definition of a streamline (2-D, incompressible) Find : Prove that the difference in value between two streamlines in a planar flow is equal to the volumetric flow rate per unit depth. Solution: x 1 ψ 2 3 dx dy Recall the definition of a streamline ,v u yx ∂∂ == and the definition of a total differential, dd x d y xy =+ . This can be rewritten using the velocity-streamfunction relationships, v x u d y =− + . If we integrate along the x-axis (i.e., holding y-constant, we obtain, 22 11 v x x x ∫∫ . Note, that from the geometry above, dx is negative, therefore ( ) 2 1 21 v depth depth x x Vnd A Q dx −= = = G G . Similarly, integrating along the y-axis and holding x- constant yields ( ) 2 1 depth depth y y A Q udy = = G G .
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EAS 4101 - Aerodynamics – Spring 2008 5/15 4. Given: Water flows through a water tunnel with an unknown velocity V .
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This note was uploaded on 03/18/2008 for the course EAS 4101 taught by Professor Sheplak during the Spring '08 term at University of Florida.

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HW2soln - EAS 4101 - Aerodynamics Spring 2008 1/24/08, 2nd...

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