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

This preview shows pages 1–6. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 ++ =
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 ++=++

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 .
EAS 4101 - Aerodynamics – Spring 2008 5/15 4. Given: Water flows through a water tunnel with an unknown velocity V .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## 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.

### Page1 / 15

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

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online