1. MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS
Thin Airfoil Theory
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk

2. OVERVIEW: THIN AIRFOIL THEORY
In words: Camber line is a streamline
Written at a given point x on the chord line
dz/dx is evaluated at that point x
Variable x is a dummy variable of integration which varies from 0 to c along the chord line
Vortex strength g=g (x) is a variable along the chord line and is in units of
In transformed coordinates, equation is written at a point, q0. q is the dummy variable of integration
At leading edge, x = 0, q = 0
At trailed edge, x = c, q =p
The central problem of thin airfoil theory is to solve the fundamental equation for g (x) subject to the Kutta condition, g(c)=0
The central problem of thin airfoil theory is to solve the fundamental equation for g (q) subject to the Kutta condition, g(p)=0

3. SUMMARY: SYMMETRIC AIRFOILS

4. SUMMARY: SYMMETRIC AIRFOILS
Fundamental equation of thin airfoil theory for a symmetric airfoil (dz/dx=0) written in transformed coordinates
Solution
“A rigorous solution for g(q) can be obtained from the mathematical theory of integral equations, which is beyond the scope of this book.” (page 324, Anderson)
Solution must satisfy Kutta condition g(p)=0 at trailing edge to be consistent with experimental results
Direct evaluation gives an indeterminant form, but can use L’Hospital’s rule to show that Kutta condition does hold.

5. SUMMARY: SYMMETRIC AIRFOILS
Total circulation, G, around the airfoil (around the vortex sheet described by g(x))
Transform coordinates and integrate
Simple expression for total circulation
Apply Kutta-Joukowski theorem (see §3.16), “although the result [L’=r∞V ∞2G] was derived for a circular cylinder, it applies in general to cylindrical bodies of arbitrary cross section.”
Lift coefficient is linearly proportional to angle of attack
Lift slope is 2p/rad or 0.11/deg

6. 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

7. SUMMARY: SYMMETRIC AIRFOILS
Total moment about the leading edge (per unit span) due to entire vortex sheet
Total moment equation is then transformed to new coordinate system based on q
After performing integration (see hand out, or Problem 4.4), resulting moment coefficient about leading edge is –pa/2
Can be re-written in terms of the lift coefficient
Moment coefficient about the leading edge can be related to the moment coefficient about the quarter-chord point
Center of pressure is at the quarter-chord point for a symmetric airfoil

8. 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

9. CENTER OF PRESSURE AND AERODYNAMIC CENTER
Center of Pressure: Point on an airfoil (or body) about which aerodynamic moment is zero
Thin Airfoil Theory:
Symmetric Airfoil:
Aerodynamic Center: Point on an airfoil (or body) about which aerodynamic moment is independent of angle of attack

10. CAMBERED AIRFOILS: THEORY
In words: Camber line is a streamline
Written at a given point x on the chord line
dz/dx is evaluated at that point x
Variable x is a dummy variable of integration which varies from 0 to c along the chord line
Vortex strength g=g (x) is a variable along the chord line and is in units of
In transformed coordinates, equation is written at a point, q0. q is the dummy variable of integration
At leading edge, x = 0, q = 0
At trailed edge, x = c, q =p
The central problem of thin airfoil theory is to solve the fundamental equation for g (x) subject to the Kutta condition, g(c)=0
The central problem of thin airfoil theory is to solve the fundamental equation for g (q) subject to the Kutta condition, g(p)=0

11. CAMBERED AIRFOILS Fundamental Equation of Thin Airfoil Theory
Camber line is a streamline
Solution
“a rigorous solution for g(q) is beyond the scope of this book.”
Leading term is very similar to the solution result for the symmetric airfoil
Second term is a Fourier sine series with coefficients An. The values of An depend on the shape of the camber line (dz/dx) and a

12. EVALUATION PROCEDURE

13. PRINCIPLES OF IDEAL FLUID AERODYNAMICS BY K
PRINCIPLES OF IDEAL FLUID AERODYNAMICS BY K. KARAMCHETI, JOHN WILEY & SONS, INC., NEW YORK, APPENDIX E

14. PRINCIPLES OF IDEAL FLUID AERODYNAMICS BY K
PRINCIPLES OF IDEAL FLUID AERODYNAMICS BY K. KARAMCHETI, JOHN WILEY & SONS, INC., NEW YORK, APPENDIX E

15. CAMBERED AIRFOILS
After making substitutions of standard forms available in advanced math textbooks
We can solve this expression for dz/dx which is a Fourier cosine series expansion for the function dz/dx, which describes the camber of the airfoil
Examine a general Fourier cosine series representation of a function f(q) over an interval 0 ≤ q ≤ p
The Fourier coefficients are given by B0 and Bn

16. ADVANCED CALCULUS FOR APPLICATIONS, 2nd EDITION BY F. B
ADVANCED CALCULUS FOR APPLICATIONS, 2nd EDITION BY F. B. HILDEBRAND, PRENTICE-HALL, INC., ENGLEWOOD CLIFFS, N.J., 1976

17. ADVANCED CALCULUS FOR APPLICATIONS, 2nd EDITION BY F. B
ADVANCED CALCULUS FOR APPLICATIONS, 2nd EDITION BY F. B. HILDEBRAND, PRENTICE-HALL, INC., ENGLEWOOD CLIFFS, N.J., 1976

18. ADVANCED CALCULUS FOR APPLICATIONS, 2nd EDITION BY F. B
ADVANCED CALCULUS FOR APPLICATIONS, 2nd EDITION BY F. B. HILDEBRAND, PRENTICE-HALL, INC., ENGLEWOOD CLIFFS, N.J., 1976

19. CAMBERED AIRFOILS
Compare Fourier expansion of dz/dx with general Fourier cosine series expansion
Analogous to the B0 term in the general expansion
Analogous to the Bn term in the general expansion

20. CAMBERED AIRFOILS
We can now calculate the overall circulation around the cambered airfoil
Integration can be done quickly with symbolic math package, or by making use of standard table of integrals (certain terms are identically zero)
End result after careful integration only involves coefficients A0 and A1

21. CAMBERED AIRFOILS Calculation of lift per unit span
Lift per unit span only involves coefficients A0 and A1
Lift coefficient only involves coefficients A0 and A1
The theoretical lift slope for a cambered airfoil is 2p, which is a general result from thin airfoil theory
However, note that the expression for cl differs from a symmetric airfoil

22. CAMBERED AIRFOILS From any cl vs. a data plot for a cambered airfoil
Substitution of lift slope = 2p
Compare with expression for lift coefficient for a cambered airfoil
Let aL=0 denote the zero lift angle of attack
Value will be negative for an airfoil with positive (dz/dx > 0) camber
Thin airfoil theory provides a means to predict the angle of zero lift
If airfoil is symmetric dz/dx = 0 and aL=0=0

23. SAMPLE DATA: SYMMETRIC AIRFOIL
Lift Coefficient
Angle of Attack, a
A symmetric airfoil generates zero lift at zero a

24. SAMPLE DATA: CAMBERED AIRFOIL
Lift Coefficient
Angle of Attack, a
A cambered airfoil generates positive lift at zero a

25. SAMPLE DATA Lift (for now) Cambered airfoil has lift at a=0
Lift coefficient (or lift) linear variation with angle of attack, a
Cambered airfoils have positive lift when a = 0
Symmetric airfoils have zero lift when a = 0
At high enough angle of attack, the performance of the airfoil rapidly degrades → stall
Lift (for now)
Cambered airfoil has
lift at a=0
At negative a airfoil
will have zero lift

26. AERODYNAMIC MOMENT ANALYSIS
Total moment about the leading edge (per unit span) due to entire vortex sheet
Total moment equation is then transformed to new coordinate system based on q
Expression for moment coefficient about the leading edge
Perform integration, “The details are left for Problem 4.9”, see hand out
Result of integration gives moment coefficient about the leading edge, cm,le, in terms of A0, A1, and A2

27. AERODYNAMIC MOMENT SUMMARY
Aerodynamic moment coefficient about leading edge of cambered airfoil
Can re-writte in terms of the lift coefficient, cl
For symmetric airfoil
dz/dx=0
A1=A2=0
cm,le=-cl/4
Moment coefficient about quarter-chord point
Finite for a cambered airfoil
For symmetric cm,c/4=0
Quarter chord point is not center of pressure for a cambered airfoil
A1 and A2 do not depend on a
cm,c/4 is independent of a
Quarter-chord point is theoretical location of aerodynamic center for cambered airfoils

28. CENTER OF PRESSURE AND AERODYNAMIC CENTER
Center of Pressure: Point on an airfoil (or body) about which aerodynamic moment is zero
Thin Airfoil Theory:
Symmetric Airfoil:
Cambered Airfoil:
Aerodynamic Center: Point on an airfoil (or body) about which aerodynamic moment is independent of angle of attack

29. 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

30. IMPLICATIONS FOR STALL
Flat Plate Stall
Leading Edge Stall
Trailing Edge Stall
Increasing airfoil
thickness

31. LEADING EDGE STALL NACA 4412 (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

32. 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

33. 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

34. NACA 4412 vs. NACA 4421
NACA 4412 and NACA 4421 have same shape of mean camber line
Theory predicts that linear lift slope and aL=0 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

35. AIRFOIL THICKNESS

36. 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

37. OPTIMUM AIRFOIL THICKNESS
Some thickness vital to achieving high maximum lift coefficient
Amount of thickness influences type of stall
Expect an optimum
Example: NACA 63-2XX, NACA looks about optimum
NACA
cl,max

38. 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 flatten 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

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