MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS
Thin Airfoil Theory
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk
OVERVIEW: THIN AIRFOIL THEORY
Fundamenta l Equation of
Thin Airfoil Theory :
1 g x dx
dz 

 V    

2p 0 x  x
dx 

c
•
•
•
•
•
Coordinate Transforma tion
c
x  1  cos q 
2
dx  sin qdq
x
c
1  cos q 0 
2
Transforme d Equation
1
2p
g q sin qdq
dz 


V


0 cos q  cos q 0   dx 
p
•
•
•
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
SUMMARY: SYMMETRIC AIRFOILS
Fundamenta l Equation of
Thin Airfoil Theory :
1 g x dx
dz 


V





2p 0 x  x
dx


c
Symmetric airfoils :
dz
0
dx
Coordinate Transforma tion
c
1  cos q 
2
dx  sin qdq
x
x
c
1  cos q 0 
2
Transforme d Equation
1
2p
g q sin qdq
0 cos q  cos q 0  V
p
SUMMARY: SYMMETRIC AIRFOILS
1
2p
g q sin qdq
0 cos q  cos q 0  V
2p

1  cos q 
g q   2V
sin q
0
g p   2V
0
 sin p
g p   2V
0
cos p
• 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.
SUMMARY: SYMMETRIC AIRFOILS
c
G   g x dx
• Total circulation, G, around the airfoil (around the
vortex sheet described by g(x))
0
• Transform coordinates and integrate
p
c
G   g q sin qdq
20
G  pcV
• Simple expression for total circulation
L  r V G  pcr V2
• 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
cl  2p
dcl
 2p
d
EXAMPLE: NACA 65-006 SYMMETRIC AIRFOIL
dcl/d = 2p
•
Bell X-1 used NACA 65-006
(6% thickness) as horizontal tail
•
Thin airfoil theory lift slope:
dcl/d = 2p rad-1 = 0.11 deg-1
•
Compare with data
– At  = -4º: cl ~ -0.45
– At  = 6º: cl ~ 0.65
– dcl/d = 0.11 deg-1
SUMMARY: SYMMETRIC AIRFOILS
c
c
0
0
    xdL   r V 
M LE
1
p
   r V2 c 2
M LE
2
2

M LE
p
cm ,le 

1
2
r V2 Sc
2
c
cm ,le   l
4
cm , c 4
cl
 cm ,le 
4
cm , c 4  0
• Total moment about the leading edge (per
unit span) due to entire vortex sheet
xg x dx
• 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 –p/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
EXAMPLE: NACA 65-006 SYMMETRIC AIRFOIL
•
Bell X-1 used NACA 65-006
(6% thickness) as horizontal tail
•
Thin airfoil theory lift slope:
dcl/d = 2p rad-1 = 0.11 deg-1
•
Compare with data
– At  = -4º: cl ~ -0.45
– At  = 6º: cl ~ 0.65
– dcl/d = 0.11 deg-1
•
Thin airfoil theory:
cm,c/4 = 0
•
Compare with data
cm,c/4 = 0
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:
cp
c
x 
4
• Aerodynamic Center: Point on an airfoil (or body) about which aerodynamic
moment is independent of angle of attack
– Thin Airfoil Theory:
c
• Symmetric Airfoil: x

A.C .
4
CAMBERED AIRFOILS: THEORY
Fundamenta l Equation of
Thin Airfoil Theory :
1 g x dx
dz 

 V    

2p 0 x  x
dx 

c
•
•
•
•
•
Coordinate Transforma tion
c
x  1  cos q 
2
dx  sin qdq
x
c
1  cos q 0 
2
Transforme d Equation
1
2p
g q sin qdq
dz 


V


0 cos q  cos q 0   dx 
p
•
•
•
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
CAMBERED AIRFOILS
1
2p
g q sin qdq
dz 

0 cos q  cos q 0  V   dx 
p
Solution :
 1  cos q 

g q   2V  A0
  An sin nq 
sin q
n 1


Compare :

1  cos q 
g q   2V
sin q
• 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 
EVALUATION PROCEDURE
1
2p
g q sin qdq
dz 

0 cos q  cos q 0  V   dx 
p
 1  cos q 

g q   2V  A0
  An sin nq 
sin q
n 1


1
p

p
0
A0 1  cos q dq 1
An sin nq sin qdq
dz
 
 
cos q  cos q 0
p n 1 0 cos q  cos q 0
dx
 p
PRINCIPLES OF IDEAL FLUID AERODYNAMICS
BY K. KARAMCHETI, JOHN WILEY & SONS, INC., NEW YORK, 1966. APPENDIX E
PRINCIPLES OF IDEAL FLUID AERODYNAMICS
BY K. KARAMCHETI, JOHN WILEY & SONS, INC., NEW YORK, 1966. APPENDIX E
CAMBERED AIRFOILS

dz
A0   An cos nq 0   
dx
n 1

dz
   A0    An cos nq 0
dx
n 1

f q   B0   Bn cos nq
n 1
B0 
0
Bn 
p
f q  cos nqdq

p
2
• 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
p
f q dq

p
1
• After making substitutions of standard
forms available in advanced math
textbooks
0
• The Fourier coefficients are given by
B0 and Bn
ADVANCED CALCULUS FOR APPLICATIONS, 2nd EDITION
BY F. B. HILDEBRAND, PRENTICE-HALL, INC., ENGLEWOOD CLIFFS, N.J., 1976
ADVANCED CALCULUS FOR APPLICATIONS, 2nd EDITION
BY F. B. HILDEBRAND, PRENTICE-HALL, INC., ENGLEWOOD CLIFFS, N.J., 1976
ADVANCED CALCULUS FOR APPLICATIONS, 2nd EDITION
BY F. B. HILDEBRAND, PRENTICE-HALL, INC., ENGLEWOOD CLIFFS, N.J., 1976
CAMBERED AIRFOILS
p
1 dz
  A0   dq 0
p 0 dx
• Compare Fourier expansion of dz/dx
with general Fourier cosine series
expansion
p
1 dz
A0     dq 0
p 0 dx
p
2 dz
An   cos nq 0 dq 0
p 0 dx
• Analogous to the B0 term in the
general expansion
• Analogous to the Bn term in the
general expansion
CAMBERED AIRFOILS
c
G   g x dx
0
p
c
G   g q sin qdq
20
• We can now calculate
the overall circulation
around the cambered
airfoil
Recall general solution for g q  :
 1  cos q 

g q   2V  A0
  An sin nq 
sin q
n 1


p

 p

G  cV  A0  1  cos q dq   An  sin nq sin qdq 
n 1
0
 0

p 

G  cV  pA0  A1 
2 

• 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
CAMBERED AIRFOILS
L  r V G
• Calculation of lift per unit span
p 

G  cV  pA0  A1 
2 

p 

L  r V c pA0  A1 
2 

2
 
cl 
L
 p 2 A0  A1 
1
r V2 S
2
p


1 dz
cl  2p    cos q 0  1dq 0 
p 0 dx


dcl
 2p
d
• 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
CAMBERED AIRFOILS
dcl
   L 0 
cl 
d
cl  2p    L 0 
p


1 dz
cl  2p    cos q 0  1dq 0 
p 0 dx


 L 0
• From any cl vs.  data plot for
a cambered airfoil
• Substitution of lift slope = 2p
• Compare with expression for
lift coefficient for a cambered
airfoil
• Let L=0 denote the zero lift
angle of attack
– Value will be negative for
an airfoil with positive
(dz/dx > 0) camber
p
1 dz
   cos q 0  1dq 0
p 0 dx
• Thin airfoil theory provides a
means to predict the angle of
zero lift
– If airfoil is symmetric
dz/dx = 0 and L=0=0
Lift Coefficient
SAMPLE DATA: SYMMETRIC AIRFOIL
Angle of Attack, 
A symmetric airfoil generates zero lift at zero 
Lift Coefficient
SAMPLE DATA: CAMBERED AIRFOIL
Angle of Attack, 
A cambered airfoil generates positive lift at zero 
SAMPLE DATA
Lift (for now)
• Lift coefficient (or lift) linear
variation with angle of attack, a
– Cambered airfoils have
positive lift when  = 0
– Symmetric airfoils have
zero lift when  = 0
• At high enough angle of attack,
the performance of the airfoil
rapidly degrades → stall
Cambered airfoil has
lift at =0
At negative  airfoil
will have zero lift
AERODYNAMIC MOMENT ANALYSIS
c
c
0
0
    xdL   r V  xg x dx
M LE

g q   2V  A0

cm ,le 
cm ,le
cm ,le

M LE
1  cos q

  An sin nq 
sin q
n 1




M LE
1
1
r V2 Sc
r V2 c 2
2
2
c
2

xg x dx
2 
V c 0
1

2V
cm ,le  
p
• 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
p
c




1

cos
q
g
q
sin qdq

0
A2 
A

A

 0

1
2
2 
2
• Result of integration gives moment
coefficient about the leading edge, cm,le, in
terms of A0, A1, and A2
AERODYNAMIC MOMENT SUMMARY
cm ,le
cm ,le
p
A2 
   A0  A1  
2
2 
 cl p

     A1  A2 
4 4

cm , c 4 
p
4
 A2  A1 

c p
xcp  1   A1  A2 
4  cl

• 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 
• cm,c/4 is independent of 
– Quarter-chord point is theoretical location
of aerodynamic center for cambered airfoils
CENTER OF PRESSURE AND AERODYNAMIC CENTER
• Center of Pressure: Point on an airfoil (or body) about which aerodynamic
moment is zero
– Thin Airfoil Theory:
c
xcp 
• Symmetric Airfoil:
4
• Cambered Airfoil:

c p
xcp  1   A1  A2 
4  cl

• Aerodynamic Center: Point on an airfoil (or body) about which aerodynamic
moment is independent of angle of attack
– Thin Airfoil Theory:
c
x A.C . 
• Symmetric Airfoil:
4
• Cambered Airfoil:
c
x A.C . 
4
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
IMPLICATIONS FOR STALL
• Flat Plate Stall
• Leading Edge Stall
• Trailing Edge Stall
Increasing airfoil
thickness
LEADING EDGE STALL
• NACA 4412 (12% thickness)
• Linear increase in cl until stall
• At  just below 15º streamlines
are highly curved (large lift) and
still attached to upper surface of
airfoil
• At  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
TRAILING EDGE STALL
• NACA 4421 (21% thickness)
• Progressive and gradual movement of separation from trailing edge toward leading
edge as  is increased
• Called Trailing Edge Stall
THIN AIRFOIL STALL
•
•
•
•
Example: Flat Plate with 2% thickness (like a NACA 0002)
Flow separates off leading edge even at low  ( ~ 3º)
Initially small regions of separated flow called separation bubble
As a increased reattachment point moves further downstream until total separation
NACA 4412 vs. NACA 4421
• NACA 4412 and NACA 4421 have
same shape of mean camber line
• Theory predicts that linear lift slope
and L=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
AIRFOIL THICKNESS
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
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 63-212 looks about optimum
NACA 63-212
cl,max
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
MODERN AIRFOIL SHAPES
Boeing 737
Root
Mid-Span
Tip
http://www.nasg.com/afdb/list-airfoil-e.phtml