1. The concept of the airfoil (wing section)
Prandtl’s approach to the analysis of airplane wings:
(1) the study of the section of the wing (the airfoil)
(2) the modification of airfoil properties to account for the complete wing z
What is an airfoil?
an “infinite” wing in 2D flow
the local section of a true wing x y
Airfoil section
Motivation for looking at airfoils:
the wing properties follow from the local airfoil properties
a good model for slender wings (i.e. with large aspect ratio) V
Aerodynamics-B, AE2-115 I, Chapter 4
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2. Aerodynamics-B, AE2-115 I, Chapter 4
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3. Airfoil Nomenclature

4. Airfoil Characteristics
Attached flow:
cl ~ a
(inviscid) airfoil theory
Aerodynamics-B, AE2-115 I, Chapter 4
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5. Limitations of the (inviscid) airfoil theory
Assumptions: - inviscid, irrotational flow
- incompressible
What is correctly predicted: the pressure distribution over the airfoil
lift and pitching moment
What is absent: viscous effects: - boundary layer development friction forces flow separation
no prediction of drag (D = 0!) or maximum lift
Conclusion: airfoil theory can reasonably predict lift and pitching moment as long as the flow does not separate
Aerodynamics-B, AE2-115 I, Chapter 4
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6. Aerodynamics-B, AE2-115 I, Chapter 4
Gerritsma & Van Oudheusden
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7. Aerodynamics-B, AE2-115 I, Chapter 4
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8. Aerodynamics-B, AE2-115 I, Chapter 4
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9. The flow-tangency condition (2)
normal component of the freestream
slope of the camber line
velocity induced by the vortex sheet
(x is fixed;  is running variable)
Aerodynamics-B, AE2-115 I, Chapter 4
Gerritsma & Van Oudheusden
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10. Aerodynamics-B, AE2-115 I, Chapter 4
Gerritsma & Van Oudheusden
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11. Resume: the basic equations of the thin airfoil theory
1. The fundamental equation of the thin airfoil theory:
the flow-tangency condition
(making the camber line z(x) a streamline)
2. The relation that determines the circulation of the airfoil:
the Kutta condition
(making the flow smooth at the trailing edge)
Aerodynamics-B, AE2-115 I, Chapter 4
Gerritsma & Van Oudheusden
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12. Aerodynamics-B, AE2-115 I, Chapter 4
Gerritsma & Van Oudheusden
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13. Aerodynamics-B, AE2-115 I, Chapter 4
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14. The symmetrical airfoil
Coordinate
transformation:
Solution is given by:
Aerodynamics-B, AE2-115 I, Chapter 4
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15. The symmetrical airfoil: lift
Calculation of the lift:
Lift coefficient:
Lift slope:
Aerodynamics-B, AE2-115 I, Chapter 4
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16. The symmetrical airfoil: pitching moment
Calculation of the pitching moment about the leading edge:
Moment coefficient
about leading edge:
Aerodynamics-B, AE2-115 I, Chapter 4
Gerritsma & Van Oudheusden
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17. The symmetrical airfoil: summary
Vorticity distribution (=lift distribution)
Lift coefficient:
Lift slope:
Moment coefficient
about quarter-chord point:
quarter-chord point is both the center of pressure:
and the aerodynamic center: is independent of 
Aerodynamics-B, AE2-115 I, Chapter 4
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18. 4.8 The cambered airfoil TUD
Condition to make the camber line z(x) a streamline of the flow
The solution for this more general problem can be written as a Fourier series:
the coefficients An (n=0,1,2,...) depend on the shape of the camber line z(x)
the coefficients A0 depends also on 
“Basic solution”
for the symmetrical airfoil: A0 = 
Additional terms
Note: () = 0, so the Kutta condition is satisfied
Substitution of the proposed solution in the upper equation gives:
(use again standard integrals)
Aerodynamics-B, AE2-115 I, Chapter 4
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19. The cambered airfoil: finding the coefficients An
The solution can be interpreted as a Fourier expansion of the function dz/dx
This Fourier series can be inverted to find the explicit relations for the individual coefficients An
We can use these expressions in two ways:
1. Analysis: determine the coefficients An for a given camber line z(x)
2. Design: determine camber line z(x) for given coefficients An
Aerodynamics-B, AE2-115 I, Chapter 4
Gerritsma & Van Oudheusden
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20. The cambered airfoil: the aerodynamic coefficients (1)
The lift coefficient:
Note: for the lift coefficient only A0 and A1 required!
Independent of 
Lift slope:
for every (thin) airfoil!
Zero-lift angle:
Aerodynamics-B, AE2-115 I, Chapter 4
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21. The cambered airfoil: the aerodynamic coefficients (2)
The moment coefficient about the LE:
Note: for the moment coefficient only A0, A1 and A2 required!
moment about the quarter-chord point:
Independent of !
For every (thin) airfoil the aerodynamic center is located at the quarter-chord point
The quarter-chord point is (in general) not the center of pressure:
Aerodynamics-B, AE2-115 I, Chapter 4
Gerritsma & Van Oudheusden
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22. The cambered airfoil: summary
Vorticity distribution
(=lift distribution)
Relation with the camber line shape z(x)
Aerodynamic coefficients:
Aerodynamics-B, AE2-115 I, Chapter 4
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23. Aerodynamics-B, AE2-115 I, Chapter 4
Gerritsma & Van Oudheusden
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26. Aerodynamics-B, AE2-115 I, Chapter 4
Gerritsma & Van Oudheusden
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27. Aerodynamics-B, AE2-115 I, Chapter 4
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28. EXAMPLE: NACA 65-006 SYMMETRIC AIRFOIL
dcl/da = 2p
Bell X-1 used NACA (6% thickness) as horizontal tail
Thin airfoil theory lift slope:
dcl/da = 2p rad-1 = 0.11 deg-1
Compare with data
At a = -4º: cl ~ -0.45
At a = 6º: cl ~ 0.65
dcl/da = 0.11 deg-1

29. EXAMPLE: NACA 65-006 SYMMETRIC AIRFOIL
cm,c/4 = 0
Bell X-1 used NACA (6% thickness) as horizontal tail
Thin airfoil theory lift slope:
dcl/da = 2p rad-1 = 0.11 deg-1
Compare with data
At a = -4º: cl ~ -0.45
At a = 6º: cl ~ 0.65
dcl/da = 0.11 deg-1
Thin airfoil theory:
cm,c/4 = 0

30. 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

31. 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 ≤ x/c ≤ 0.4)
Equation describes the shape of the mean camber line aft of the maximum camber position (applies for 0.4 ≤ x/c ≤ 1)
/c ≤ 1)

32. 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

33. 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?

35. 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 =

38. Aerodynamics-B, AE2-115 I, Chapter 4
Gerritsma & Van Oudheusden
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39. Aerodynamics-B, AE2-115 I, Chapter 4
Gerritsma & Van Oudheusden
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40. Aerodynamics-B, AE2-115 I, Chapter 4
Gerritsma & Van Oudheusden
TUD

41. Aerodynamics-B, AE2-115 I, Chapter 4
Gerritsma & Van Oudheusden
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42. Aerodynamics-B, AE2-115 I, Chapter 4
Gerritsma & Van Oudheusden
TUD

43. Aerodynamics-B, AE2-115 I, Chapter 4
Gerritsma & Van Oudheusden
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44. Aerodynamics-B, AE2-115 I, Chapter 4
Gerritsma & Van Oudheusden
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45. TUD

47. Aerodynamics-B, AE2-115 I, Chapter 4
Gerritsma & Van Oudheusden
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48. Aerodynamics-B, AE2-115 I, Chapter 4
Gerritsma & Van Oudheusden
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49. Aerodynamics-B, AE2-115 I, Chapter 4
Gerritsma & Van Oudheusden
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50. 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