1. MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS
Further Examples of Infinite Wing Implications
April 9, 2007
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk

2. RECALL U2 VS. F-15 EXAMPLE U2 F-15
Cruise at 70,000 ft
Air density highly reduced
Flies at slow speeds, low q∞ → high angle of attack, high CL
AR ~ 14.3
Flies at high speed (and lower altitudes), so high q∞ → low angle of attack, low CL
AR ~ 3
AR ↑ and Di ↓, but which to control b2 or S?

3. 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

4. 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

5. 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 aL=0 same for infinite or finite wings

6. 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

7. EXAMPLE: FINITE WING COMPOSED OF NACA 23012 AIRFOIL
Consider a wing with AR=10 and NACA 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

8. EXAMPLE: U2 VS. F-15 U2
F-15
Cruise at 70,000 ft
Air density highly reduced
Flies at slow speeds, low q∞ → high angle of attack, high CL
AR ~ 14.3
Flies at high speed (and lower altitudes), so high q∞ → low angle of attack, low CL
AR ~ 3
Which of airplane is more sensitive to atmospheric turbulence?