Aero II

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Aero II Fundamental Theory of Finite Wings
Lecture 20
J. E. Brandenburg EAS 3010 Spring 2007
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Agenda

• Vorticity
• Rotating Cylinder
• Kutta Jakowski Theorem
• Kelvins Theorem and Starting Vortex
• Wing tip Vortices

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Vorticity and Circulation
Vorticity is important
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Curl
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curl
Important because it gives local vorticity of gas
flow
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Curl Theorem
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Circulation
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Incompressible Inviscid Flow
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Force per unit volume without viscosity
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Bernoulli Equation
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The Pressure Coefficient
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Rankines Ellipse
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Doublet Flow
The limiting case as source and sink separation
goes to zero
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Cylinder in Uniform flow
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Cylinder in Uniform Flow
The addition of a uniform plus doublet flow
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Real Air flow
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Flow Around Cylinder
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Flow Around Rotating Cylinder
Vortex term
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Pressure Field around Non Rotating Cylinder
?
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Pressure Field around Rotating Cylinder
?
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Pressure distribution
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Kutta-Jackowski Theorem
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Rankines Ellipse
Cylinder shape can be modified continuously
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Approximating Flows with Vortices
Using Vortex solutions , airflow can be modeled
continuously
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Drag Polar
Drag is proportional to lift squared
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Trailing Vortex Cont.
Trailing vortex creates tube of rotating air of
approximate circulation ? and approximate Kinetic
energy per unit length Drag Force ½ ?V2???R2 ?
1/8? ? (2?VR)2 1/8? ? ?2 This leads to Drag
relationship D??2?(Lift)2
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Kelvins Theorem
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Kelvins Theorem
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Kelvins Theorem
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Trailing Vortex
At end of wings vorticity does not end but spills
off end of wing
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Aircraft Have Turbulent Wakes
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Starting Vortex
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Homework 4

• 1. (30pts) If an airplane flies at 100 meters a
  second and creates two wing vortices of 10
  meters diameter and 40 meters a second average
  rotational velocity , what is the drag force this
  creates?
• 2. (30pts) Using the graphs following this slide
  , estimate the drag coefficient function as a
  function of lift coefficient for the case of
  Re6x106 ( triangles)
• 3. (30pts) a pumpkin dropped from an airplane
  weighs 20kg and is approximately spherical with
  radius 30cm , A. what is the average suction
  caused by form drag created as it falls at
  terminal velocity, B. what is its approximate
  terminal velocity?

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