MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS

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MAE 3241 AERODYNAMICS ANDFLIGHT MECHANICS

• Further Examples of Infinite Wing Implications
• April 9, 2007
• Mechanical and Aerospace Engineering Department
• Florida Institute of Technology
• D. R. Kirk

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RECALL U2 VS. F-15 EXAMPLE
U2
F-15

• Cruise at 70,000 ft
• Air density highly reduced
• Flies at slow speeds, low q8 ? high angle of
  attack, high CL
• AR 14.3

• Flies at high speed (and lower altitudes), so
  high q8 ? low angle of attack, low CL
• AR 3

AR ? and Di ?, but which to control b2 or S?
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WING LOADING (W/S), SPAN LOADING (W/b) AND ASPECT
RATIO (b2/S)
Span loading (W/b), wing loading (W/S) and AR
(b2/S) are related Zero-lift drag, D0 is
proportional to wing area Induced drag, Di, is
proportional to square of span loading Take
ratio of these drags, Di/D0 Re-write W2/(b2S)
in terms of AR and substitute into drag ratio
Di/D0 1 For specified W/S (set by take-off or
landing requirements) and CD,0 (airfoil choice),
increasing AR will decrease drag due to lift
relative to zero-lift drag 2 AR predominately
controls ratio of induced drag to zero lift drag,
whereas span loading controls actual value of
induced drag
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EXAMPLE AIRBUS A380 / BOEING 747 COMPARISON

• Wingspan 79.8 m
• AR 7.53
• GTOW 560 T
• Loading GTOW/b2 87.94

• Wingspan 68.5 m
• AR 7.98
• GTOW 440 T
• Loading GTOW/b2 93.77

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FINITE WING CHANGE IN LIFT SLOPE (? 2p)

• Lift curve for a finite wing has a smaller slope
  than corresponding curve for an infinite wing
  with same airfoil cross-section
• Figure (a) shows infinite wing, ai 0, so plot
  is CL vs. ageom or aeff and slope is a0
• Figure (b) shows finite wing, ai ? 0
• Plot CL vs. what we see, ageom, (or what would be
  easy to measure in a wind tunnel), not what wing
  sees, aeff
• Effect of finite wing is to reduce lift curve
  slope
• Finite wing lift slope a dCL/da ? 2p
• At CL 0, ai 0, so aL0 same for infinite or
  finite wings

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CALCULATING CHANGE IN LIFT SLOPE

• If we know a0 (infinite wing lift slope, say from
  data) how can we find finite wing lift slope, a,
  for wing with given AR?

Lift slope definition for infinite
wing Integrate Substitute definition of
ai Solve for CL Differentiate CL with respect
to a to find lift slope for finite wing Note
Equation is in radians
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EXAMPLE FINITE WING COMPOSED OF NACA 23012
AIRFOIL

• Consider a wing with AR10 and NACA 23012 airfoil
  section, Re 5 million, and span efficiency
  factor, e 0.9. The wing is at an angle of
  attack, a 4º
• Find CL and CD for finite wing

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EXAMPLE U2 VS. F-15
U2
F-15

• Cruise at 70,000 ft
• Air density highly reduced
• Flies at slow speeds, low q8 ? high angle of
  attack, high CL
• AR 14.3

• Flies at high speed (and lower altitudes), so
  high q8 ? low angle of attack, low CL
• AR 3

Which of airplane is more sensitive to
atmospheric turbulence?