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. HILDEBRAND, 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