1. MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS
Summary of Incompressible Flow Over Airfoils
Summary of Thin Airfoil Theory
Example Airfoil Calculation
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk

2. KEY EQUATIONS FOR cl, aL=0, cm,c/4, and xcp
Within these expression we need to evaluate A0, A1, A2, and dz/dx

3. A0, A1, and A2 COEFFICIENTS

4. CENTER OF PRESSURE AND AERODYNAMIC CENTER
Center of Pressure: It is that point on an airfoil (or body) about which the aerodynamic moment is zero
Thin Airfoil Theory:
Symmetric Airfoil:
Cambered Airfoil:
Aerodynamic Center: It is that point on an airfoil (or body) about which the aerodynamically generated moment is independent of angle of attack

5. ACTUAL LOCATION OF AERODYNAMIC CENTER
x/c=0.25
NACA 23012
xA.C. < 0.25c
x/c=0.25
NACA 64212
xA.C. > 0.25 c

6. EXAMPLE OF LEADING EDGE STALL
NACA 4412 Airfoil (12% thickness)
Linear increase in cl until stall
At a just below 15º streamlines are highly curved (large lift) and still attached to upper surface of airfoil
At a just above 15º massive flow-field separation occurs over top surface of airfoil → significant loss of lift
Called Leading Edge Stall
Characteristic of relatively thin airfoils with thickness between about 10 and 16 percent chord

7. EXAMPLE OF TRAILING EDGE STALL
NACA 4421 (21% thickness)
Progressive and gradual movement of separation from trailing edge toward leading edge as a is increased
Called Trailing Edge Stall

8. THIN AIRFOIL STALL
Example: Flat Plate with 2% thickness (like a NACA 0002)
Flow separates off leading edge even at low a (a ~ 3º)
Initially small regions of separated flow called separation bubble
As a increased reattachment point moves further downstream until total separation

9. NACA 4412 VERSUS NACA 4421
Both NACA 4412 and NACA 4421 have same shape of mean camber line
Thin airfoil theory predict that linear lift slope and aL=0 should be the same for both
Leading edge stall shows rapid drop of lift curve near maximum lift
Trailing edge stall shows gradual bending-over of lift curve at maximum lift, “soft stall”
High cl,max for airfoils with leading edge stall
Flat plate stall exhibits poorest behavior, early stalling
Thickness has major effect on cl,max

10. OPTIMUM AIRFOIL THICKNESS
Some thickness vital to achieving high maximum lift coefficient
Amount of thickness will influence type of stalling behavior
Expect an optimum
Example: NACA 63-2XX, NACA looks about optimum
NACA
cl,max

11. AIRFOIL THICKNESS

12. AIRFOIL THICKNESS: WWI AIRPLANES
English Sopwith Camel
Thin wing, lower maximum CL
Bracing wires required – high drag
German Fokker Dr-1
Higher maximum CL
Internal wing structure
Higher rates of climb
Improved maneuverability

13. MODERN LOW-SPEED AIRFOILS
NACA 2412 (1933)
Leading edge radius = 0.02c
NASA LS(1)-0417 (1970)
Whitcomb [GA(w)-1] (Supercritical Airfoil)
Leading edge radius = 0.08c
Larger leading edge radius to flatted cp
Bottom surface is cusped near trailing edge
Discourages flow separation over top
Higher maximum lift coefficient
At cl~1 L/D > 50% than NACA 2412

14. MODERN AIRFOIL SHAPES Boeing 737 Root Mid-Span Tip

15. OTHER CONSIDERATIONS
Note that all airfoils we have seen, even flat plate, will produce lift at some a
Production of lift itself is not difficult
L/D ratio
Production of lift with minimum drag
Measure of aerodynamic efficiency of wing or airplane
Important impact on performance range, endurance
Maximum lift coefficient, CL,max
Effective airfoil shape produces high value of cl,max
Stalling speed of aircraft (take-off, landing)
Improved maneuverability (turn radius, turn rate)

16. HIGH LIFT DEVICES: SLATS AND FLAPS

17. HIGH LIFT DEVICES: FLAPS
Flaps shift lift curve
Act as effective increase in camber of airfoil

18. AIRFOIL DATA: NACA 1408 WING SECTION
Flap extended
Flap retracted

19. HIGH LIFT DEVICES: SLATS
Allows for a secondary flow between gap between slat and airfoil leading edge
Secondary flow modifies pressure distribution on top surface delaying separation
Slats increase stalling angle of attack, but do not shift the lift curve (same aL=0)

20. RECALL BOEING 727 EXAMPLE
cl ~ 4.5

21. EXAMPLE CALCULATION NACA 2412 Root Airfoil: NACA 2412
GOAL: Find values of cl, aL=0, and cm,c/4 for a NACA 2412 Airfoil
Maximum thickness 12 % of chord
Maximum chamber of 2% of chord located 40% downstream of the leading edge of the chord line
Check Out:
NACA 2412
Root Airfoil: NACA 2412
Tip Airfoil: NACA 0012

22. EQUATIONS DESCRIBING MEAN CAMBER LINE: z = z(x)
Equation describes the shape of the mean camber line forward of the maximum camber position (applies for 0 ≤ z/c ≤ 0.4)
Equation describes the shape of the mean camber line aft of the maximum camber position (applies for 0.4 ≤ z/c ≤ 1)

23. EXPRESSIONS FOR MEAN CAMBER LINE SLOPE: dz/dx

24. COORDINATE TRANSFORMATION: x → q, x0 → q0
Equation describes the shape of the mean camber line slope forward of the maximum camber position
Equation describes the shape of the mean camber line slope aft of the maximum camber position

25. EXAMINE LIMITS OF INTEGRATION
Coefficients A0, A1, and A2 are evaluated across the entire airfoil
Evaluated from the leading edge to the trailing edge
Evaluated from leading edge (q=0) to the trailing edge (q=p)
2 equations the describe the fore and aft portions of the mean camber line
Fore equation integrated from leading edge to location of maximum camber
Aft equation integrated from location of maximum camber to trailing edge
The location of maximum camber is (x/c)=0.4
What is the location of maximum camber in terms of q?

26. EXAMPLE: NACA 2412 CAMBERED AIRFOIL
dcl/da = 2p
Thin airfoil theory lift slope:
dcl/da = 2p rad-1 = 0.11 deg-1
What is aL=0?
From data aL=0 ~ -2º
From theory aL=0 = -2.07º
What is cm,c/4?
From data cm,c/4 ~
From theory cm,c/4 =

27. AIRFOIL WEB RESOURCES