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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• 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 is a dummy variable of integration which varies from 0 to c along the chord line
• Vortex strength = () is a variable along the chord line and is in units of
• In transformed coordinates, equation is written at a point, 0. is the dummy variable of integration
– At leading edge, x = 0, = 0
– At trailed edge, x = c, =• The central problem of thin airfoil theory is
to solve the fundamental equation for () subject to the Kutta condition, (c)=0
• The central problem of thin airfoil theory is to solve the fundamental equation for () subject to the Kutta condition, ()=0
dx
dzV
d
cx
dd
c
dx
dzV
x
dc
0 0
0
0
coscos
sin
2
1
Equation dTransforme
cos12
sin
cos12
tionTransforma Coordinate
2
1
:Theory AirfoilThin
ofEquation lFundamenta
SUMMARY: SYMMETRIC AIRFOILS
Vd
cx
dd
c
dx
dz
dx
dzV
x
dc
0 0
0
0
coscos
sin
2
1
Equation dTransforme
cos12
sin
cos12
tionTransforma Coordinate
0
:airfoils Symmetric
2
1
:Theory AirfoilThin
ofEquation lFundamenta
SUMMARY: SYMMETRIC AIRFOILS
0cos
sin2
0
02
sin
cos12
coscos
sin
2
1 2
0 0
V
V
V
Vd
• Fundamental equation of thin airfoil theory for a symmetric airfoil (dz/dx=0) written in transformed coordinates
• Solution
– “A rigorous solution for () 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 ()=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• Total circulation, , around the airfoil (around the
vortex sheet described by ())
• Transform coordinates and integrate
• Simple expression for total circulation
• Apply Kutta-Joukowski theorem (see §3.16), “although the result [L’=∞V ∞
2] 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 2/rad or 0.11/deg
2
2
sin2
2
0
0
d
dc
c
VcVL
cV
dc
d
l
l
c
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 = 2 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
dcl/d = 2
SUMMARY: SYMMETRIC AIRFOILS
0
4
4
221
22
1
4,
,4,
,
2,
22
00
cm
llemcm
llem
LElem
LE
cc
LE
c
ccc
cc
ScV
Mc
cVM
dVdLM
• 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
• After performing integration (see hand out, or Problem 4.4), resulting moment coefficient about leading edge is –/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 = 2 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:
• Aerodynamic Center: Point on an airfoil (or body) about which aerodynamic moment is independent of angle of attack
– Thin Airfoil Theory:
• Symmetric Airfoil:
4
cxcp
4..
cx CA
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 is a dummy variable of integration which varies from 0 to c along the chord line
• Vortex strength = () is a variable along the chord line and is in units of
• In transformed coordinates, equation is written at a point, 0. is the dummy variable of integration
– At leading edge, x = 0, = 0
– At trailed edge, x = c, =• The central problem of thin airfoil theory is
to solve the fundamental equation for () subject to the Kutta condition, (c)=0
• The central problem of thin airfoil theory is to solve the fundamental equation for () subject to the Kutta condition, ()=0
dx
dzV
d
cx
dd
c
dx
dzV
x
dc
0 0
0
0
coscos
sin
2
1
Equation dTransforme
cos12
sin
cos12
tionTransforma Coordinate
2
1
:Theory AirfoilThin
ofEquation lFundamenta
CAMBERED AIRFOILS• Fundamental Equation of
Thin Airfoil Theory• Camber line is a streamline
• Solution– “a rigorous solution for
() 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
sin
cos12
:Compare
sinsin
cos12
:Solution
coscos
sin
2
1
10
0 0
V
nAAV
dx
dzV
d
nn
EVALUATION PROCEDURE
dx
dzdnAdA
nAAV
dx
dzV
d
n
n
nn
1 0 00 0
0
10
0 0
coscos
sinsin1
coscos
cos11
sinsin
cos12
coscos
sin
2
1
PRINCIPLES OF IDEAL FLUID AERODYNAMICSBY K. KARAMCHETI, JOHN WILEY & SONS, INC., NEW YORK, 1966. APPENDIX E
PRINCIPLES OF IDEAL FLUID AERODYNAMICSBY K. KARAMCHETI, JOHN WILEY & SONS, INC., NEW YORK, 1966. APPENDIX E
CAMBERED AIRFOILS
0
0
0
10
100
100
cos2
1
cos
cos
cos
dnfB
dfB
nBBf
nAAdx
dz
dx
dznAA
n
nn
nn
nn
• 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() over an interval 0 ≤ ≤
• The Fourier coefficients are given by B0 and Bn
ADVANCED CALCULUS FOR APPLICATIONS, 2nd EDITIONBY F. B. HILDEBRAND, PRENTICE-HALL, INC., ENGLEWOOD CLIFFS, N.J., 1976
ADVANCED CALCULUS FOR APPLICATIONS, 2nd EDITIONBY F. B. HILDEBR AND, PRENTICE-HALL, INC., ENGLEWOOD CLIFFS, N.J., 1976
ADVANCED CALCULUS FOR APPLICATIONS, 2nd EDITIONBY F. B. HILDEBRAND, PRENTICE-HALL, INC., ENGLEWOOD CLIFFS, N.J., 1976
CAMBERED AIRFOILS
0
00
0
00
0
00
cos2
1
1
dndx
dzA
ddx
dzA
ddx
dzA
n
• 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
CAMBERED AIRFOILS
10
0 1 0
0
10
0
0
2
sinsincos1
sinsin
cos12
:for solution general Recall
sin2
AAcV
dnAdAcV
nAAV
dc
d
nn
nn
c
• 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
CAMBERED AIRFOILS
2
1cos1
2
2
21
2
2
0
00
102
102
10
d
dc
ddx
dzc
AASV
Lc
AAcVL
AAcV
VL
l
l
l
• 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 2, which is a general result from thin airfoil theory
• However, note that the expression for cl differs from a symmetric airfoil
CAMBERED AIRFOILS
0
000
0
00
0
0
1cos1
1cos1
2
2
ddx
dz
ddx
dzc
c
d
dcc
L
l
Ll
Ll
l
• From any cl vs. data plot for a cambered airfoil
• Substitution of lift slope = 2
• 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
• Thin airfoil theory provides a means to predict the angle of zero lift– If airfoil is symmetric
dz/dx = 0 and L=0=0
SAMPLE DATA: SYMMETRIC AIRFOIL
Lif
t Coe
ffic
ient
Angle of Attack,
A symmetric airfoil generates zero lift at zero
SAMPLE DATA: CAMBERED AIRFOIL
Lif
t Coe
ffic
ient
Angle of Attack,
A cambered airfoil generates positive lift at zero
SAMPLE DATA• 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
Lif
t (fo
r no
w)
Cambered airfoil haslift at =0At negative airfoilwill have zero lift
AERODYNAMIC MOMENT ANALYSIS
22
sin2
cos12
1
2
21
21
sinsin
cos12
210,
0
,
02,
222,
10
00
AAAc
dc
Vc
dcV
c
cV
M
ScV
Mc
nAAV
dVdLM
lem
lem
c
lem
LELElem
nn
cc
LE
• 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
• 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
AERODYNAMIC MOMENT SUMMARY
21
124,
21,
210,
14
4
44
22
AAc
cx
AAc
AAc
c
AAAc
lcp
cm
llem
lem
• 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:
• Symmetric Airfoil:
• Cambered Airfoil:
• Aerodynamic Center: Point on an airfoil (or body) about which aerodynamic moment is independent of angle of attack
– Thin Airfoil Theory:
• Symmetric Airfoil:
• Cambered Airfoil:
2114
4
AAc
cx
cx
lcp
cp
4
4
..
..
cx
cx
CA
CA
ACTUAL LOCATION OF AERODYNAMIC CENTER
NACA 23012xA.C. < 0.25c
NACA 64212xA.C. > 0.25 c
x/c=0.25
x/c=0.25
IMPLICATIONS FOR STALL
• Flat Plate Stall
• Leading Edge Stall
• Trailing Edge Stall
Increasing airfoilthickness
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
German Fokker Dr-1
Higher maximum CL
Internal wing structureHigher rates of climbImproved maneuverability
Thin wing, lower maximum CL
Bracing wires required – high drag
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
cl,max
NACA 63-212
MODERN LOW-SPEED AIRFOILSNACA 2412 (1933)Leading edge radius = 0.02c
NASA LS(1)-0417 (1970)Whitcomb [GA(w)-1] (Supercritical Airfoil)Leading edge radius = 0.08cLarger leading edge radius to flatten cp
Bottom surface is cusped near trailing edgeDiscourages flow separation over topHigher maximum lift coefficientAt cl~1 L/D > 50% than NACA 2412
MODERN AIRFOIL SHAPES
http://www.nasg.com/afdb/list-airfoil-e.phtml
Root Mid-Span Tip
Boeing 737