The Airfoil and the Wing : chapter 5

The Airfoil represents an infinite wing (2D) with uniform aero data regardless of span

The wing is 3D … which has the affect of wingtips … it has a finite span.

� ASPECT RATIO, AR:

� AR ⇒⇒⇒⇒ ∞ for the airfoil … wingtip effects vanish

� AR is a non-dimensional measure of the slenderness of the wing

planformsrrectangula forc

bAR

S

bAR

2

=⇒≡

AR ~ 4

AR ~ 10

AR ~ 20

AR ~ 3

AR ~ 8

AR ~ 20

AN AIRFOIL IS THE CROSS SECTION OF A WING

(or a vertical fin, or a stabilizer, or a propeller, or a wind turbine blade, … etc.)

The section characteristics may change along the wing (shape, pre-twist, chord)

Cambered

Symmetrical

Laminar Flow

Reflexed

Supercritical

AIRFOILS

AIRFOIL NOMENCLATURE

� Chord Line… the straight line connecting the Leading Edge (LE) and Trailing Edge (TE)

� Mean Camber Line… the locus of points halfway between the upper and lower surface

� Camber… maximum distance between the Mean Camber Line and the Chord Line

� Thickness… the thickness of the airfoil, measured perpendicular to the mean camber line

AIRFOILS

DEFINITIONS

� RELATIVE WIND

DIRECTION OF V ( V∞ ∞ ∞ ∞ )

� ANGLE OF ATTACK (AOA),

α, α, α, α, angle between

relative wind ( V∞ ∞ ∞ ∞ )

& chord

� DRAG, D

AERO FORCE PARALLEL TO V∞∞∞∞

� LIFT, L

AERO FORCE PERPENDICULAR TO V∞∞∞∞

� MOMENT, Mx

PRESSURE DISTRIBUTION PRODUCES A TORQUE

ABOUT POINT x (x may be LE, TE, c/4, … )

AIRFOILS

� Lift = L and Drag = D Forces

L: Force perpendicular to V∞∞∞∞

D: Force parallel to V∞∞∞∞

� Normal = N and Axial = A Forces

N: Force perpendicular to chord

A: Force parallel to chord

L

D

N

A

R

V∞

α

R L D= +r r r

R N A= +rr r

c c

c c

L s N N s Lor

D s A A s D

α α α α

α α α α

− = =

−

cos sin

sin cos

L N A

D N A

α α

α α

= −

= +

The Aerodynamic Moment

Aerodynamic loads may lead to a

aerodynamic moment, Mx, about x,

where x may be LE, TE, c/4, …

By convention, Mx, is defined as positive

if it leads to “positive pitch”

or “leading edge up”

Observe, MLE < 0 and MTE > 0 for the wing … but moments may be transferred.

For example, if we integrate pressures from the leading edge

There is no aerodynamic moment at the center of pressure, thus

z L

MLE MX

x

X LEM M xL= +

0 0 /CP LE CP LEM M xL x M L= ⇒ = + ⇒ = −

x

y

z

x – y – z = “roll – pitch – yaw”

( positive is roll right, pitch up, yaw right )

z L

Mc/4 MX

x’

… moments may be transferred.

a moment may be about the leading edge

or, from the quarter chord

The Aerodynamic Moment

/4 'X c

M M x L= +

. .X L EM M xL= +

DIMENSIONAL ANALYSIS� How do we assure ourselves that data for wings and airplanes

(and wind tunnel tests) are of quality and value ?

� AN AIRPLANE IS DESIGNED FOR DIFFERENT CONDITIONS

(SIZE, SPEED, P, T, etc) ?

� For example, L, D and M are FUNCTIONS OF SEVERAL VARIABLES:

� It is not possible (or wise) to conduct experiments at every possible condition,

we seek to identify key groupings of parameters that assure complete analysis.

� DIMENSIONAL ANALYSIS

� ALLOWS US TO INTELLIGENTLY UNDERSTAND THE VARIABLES

� IS AN APPLICATION OF THE BUCKINGHAM PI THEOREM:

( )

( )

( )∞∞∞∞

∞∞∞∞

∞∞∞∞

µρ=

µρ=

µρ=

a,S,,,VfM

a,S,,,VfD

a,S,,,VfL

3

2

1

( ), , , ,

, , , , ,

a b d e fL Z V S a

Z a b d e f are

DIMENSIONLESS CONSTANTS

ρ µ∞ ∞ ∞ ∞=

DIMENSIONAL ANALYSIS

� PRINCIPLE: DIMENSIONS ON BOTH SIDES OF THE EQUATIONS MUST BE

IDENTICAL

� FUNDAMENTAL UNITS: m, l, t

�ARE RELATED TO PHYSICAL QUANTITIES

�FOR EXAMPLE:

� EQUATING THE DIMENSIONS ON LEFT AND RIGHT OF THE LIFT FORCE

EQUATION

�EQUATING MASS EXPONENTS

�EQUATING LENGTH EXPONENTS

�EQUATING TIME EXPONENTS

2

mlL

t∝

( )2

2 3

a b e fdml l m l m

lt t l t lt

=

fb1 +=

fed2b3a1 −++−=

fea2 −−−=−

),,,,( ∞∞∞∞= µρ aSVfL

DIMENSIONAL ANALYSIS

� SOLVING THE 3 EQUATIONS FOR a, b, AND d (IN TERMS OF e AND f )

� NOTING THAT HAS UNITS OF LENGTH,

WE CHOOSE c AS OUR CHARACTERISTIC LENGTH

� THEN WE CAN REPLACE WITH

� NOW, THE LIFT EQUATION IS OF THE FORM

( )fe

2

fe2f1f1fe2

SVV

aSZVL:GREARRANGIN

,a,S,VZL

ρ

µ

=

µρ=

∞∞

∞

∞

∞∞

∞∞

−−∞

−−∞

SANDM

1

V

a

∞∞

∞ =

SV∞∞

∞

ρ

µ

cV∞∞

∞

ρ

µ

f

c

e2

Re

1

M

1SZVL

=

∞∞

FORCE / MOMENT COEFFICIENTS

� NOW WE DEFINE THE AIRFOIL’S SECTION LIFT COEFFICIENT

� OR WE COULD HAVE SIMPLY DEFINED LIFT COEFFICIENT AS

from ch. 4

�NOTICE THAT cl IS DIMENSIONLESS

�It is a function of M∞

and Re

�DIMENSIONAL ANALYSIS IS FOR A GIVEN AOA & Geometry,depends on these 3 variable

l2

f

c

e

l ScV2

1L

Re

1

M

1Z

2

c∞∞

∞

ρ=⇒

≡

Sq

Lcl

∞

≡

( )Re,M,fcl ∞α=

Sq

Dc f

f

∞

≡

� A SIMILAR ANALYSIS LEADS TO

� DRAG COEFFICIENT

� MOMENT COEFFICIENT

�Moment has length, c , due to FORCE x LENGTH

�Moment must be referenced to the point where the moment is taken …

� THUS,

WHERE

� In summary, we have identified key coefficients Cl, Cd and Cm x ,

in terms of “Similarity Parameters” such as the M and Re.

for the same geometry and AOA ( … leads to same streamlines),

our aero. coefficients for wind tunnel tests are identical to flight conditions!

dScqD ∞=

mSccqM ∞=

Scq

Mc

Sq

Dc

Sq

Lc x

mdl x

∞∞∞

≡≡≡

( ) ( ) ( )Re,M,fcRe,M,fcRe,M,fc 3m2d1l ∞∞∞ α=α=α=

FORCE / MOMENT COEFFICIENTS

AIRFOIL DATA … refer to Appendix D

AIRFOIL CLASSIFICATIONS

NACA 2412

NACA 23012

NACA 63-210

NACA 4 digit airfoil (for example, 2412)

1st digit …maximum camber

in % of chord

2nd digit … location of max camber

in tenths of chord

3rd and 4th digits …maximum thickness

in % of chord

NACA 5 digit airfoil (for example, 23012)

1st digit …multiply by 0.15

to provide design CL

2nd and 3rd digits … divide by 2 to

define location of maximum

camber in percent of chord

4th and 5th digits …maximum

thicknessin % of chord

AIRFOIL DATA

� EXPERIMENTAL DATA

ARE ESSENTIAL TO

AIRCRAFT DESIGN

� NACA / NASA DATA

� APPENDIX D

� cl VARIES LINEARLY

WITH αααα

� CAMBER CHANGES ααααL= 0

� THIS LINEAR RELATIONSHIP

BREAKS DOWN WHEN STALL

OCCURS

AIRFOIL DATA

AT HIGH αααα, THE

BOUNDARY LAYER

WILL SEPARATE

� LIFT DECREASES

� DRAG INCREASES

� AERO MOMENT

BECOMES

NOSE DOWN ( - )

… ahhh …vicious viscous

lC

α

lo

Ca

α

∆=

∆

0lα =

0lCα =

stallα

,maxlC

AIRFOIL DATA� EXPERIMENTAL DATA ( Appendix D )

� NACA data

� Incompressible flow

� Re specified

lC

lC

dC

α

/4cmC

acmC

AIRFOIL DATA� EXPERIMENTAL DATA ( Appendix D )

� NACA data

� Incompressible flow

� Re specified

lC

lC

dC

α

/4cmC

acmC

α

/4cmC

/4

/4 /4 /4

/4 /4 . . . .

. .

. .

0 0c

c c c

x c c c p m c p l

m m mc p

c p

l l l

M M xL M x L qScC x qSC

qScC cC Cxx

qSC C c C

= + ⇒ = + ⇒ = +

− − −= = ⇒ = =

There is no aerodynamic moment at the center of pressure, thus

AIRFOIL DATA – The Effect of a Flap

lC

α

‘straight & level’

L = W

L = qSCL = W

q = W/SCL

V=��

����

(CL/CD) max and α (CL/CD) max (leads to ‘best’ A/C performance)

lC

lC

dC

α

αααα 0 2 4 8 8 10

Cl

Cd

Cl/Cd

Note: this is not at

CL max or CD min or αmax

lC

dC

tangent

( / )l d MAXC C⇒

( / )C Cl d MAXl

C

( / ) ( / )C C l d MAXl d MAXl C C

C α⇒

lC

dC

, , ,d d friction d pressure d profileC C C C= + =

,d frictionC≈

Cd, pressure

� THE AERODYNAMIC CENTER, a.c.

� The a.c. is the location on the airfoil where

M = constant for all changes of AOA

� If L = 0, M is a pure couple = Ma.c.

� Simple airfoil theory shows the a.c. is�At c/4 (“quarter chord”) for low subsonic symmetric airfoils.�At approximately the c/4 for low subsonic non-symmetric airfoils.�At c/2 (“mid-chord”) for supersonic airfoils.

AIRFOILS

PRESSURE COEFFICIENT � Cp < 0 … “suction”

… a “pull”

� Cp > 0 … “positive pressure”

… a “push”

� Let’s examine a special case …

incompressible via Bernoulli

be careful … incompressible only !

but observe … at V = 0 … stagnation … Cp = 1 (maximum)

at V = V∞

… freestream … Cp = 0

at V = 2V∞ … V > V

∞ … Cp = -3 (suction)

2p

V2

1

pp

q

ppC

∞∞

∞

∞

∞

ρ

−=

−≡

2 2

2 2

V VP P

2 2

V VP P

2

ρ ρ

ρ

∞∞

∞∞

+ = +

−− =

2 2

2

P2 2 2

V V

2P P VC 1

V / 2 V / 2 V

ρ

ρ ρ

∞

∞

∞ ∞ ∞

−

− = = = −

2p

V2

1

pp

q

ppC

∞∞

∞

∞

∞

ρ

−=

−≡

� The CENTER OF PRESSURE, c.p.

� THE RESULTANT FORCES (LIFT AND DRAG) ACTING

AT THE c.p. PRODUCE NO MOMENT ( Mcp = 0 )

� SINCE THE PRESSURE DISTRIBUTION OVER THE AIRFOIL

CHANGES WITH αααα, THE LOCATION OF THE c.p. VARIES WITH αααα

� THE MOMENT ABOUT THE c.p. MIGHT NOT BE ZERO AT L = 0

� L, M, D, c.p. depend on shape (camber, chord, thickness, V, AOA)

AIRFOILS

� LIFT PER UNIT SPAN

� LIFT … NET UPWARD FORCE DUE TO THE PRESSURE DIFFERENCE BETWEEN THE LOWER SURFACE AND THE UPPER SURFACE

� Note, ds cos θθθθ = dx

θdscos θ = dx

dxLE (leading edge)

TE (trailing edge)

∫∫ θ−θ=TE

LE uTE

LE l dscospdscospL

∫∫ −=c

0 uc

0 l dxpdxpL

OBTAINING CL FROM Cp

� LIFT PER UNIT SPAN

� Add & subtract p∞∞∞∞

� Use the def’n of the lift coefficient

� combine

where

� Thus:

( ) ( )∫∫ ∞∞ −−−=c

0 uc

0 l dxppdxppL

cq

L

)1(cq

L

Sq

Lcl

∞∞∞

==≡

∫∫∞

∞

∞

∞ −−

−=

c

0uc

0l

l dxq

pp

c

1dx

q

pp

c

1c

∞

∞

∞

∞ −≡

−≡

q

ppCAND

q

ppC u

u,pl

l,p

( )∫ −=c

0 u,pl,pl dxCCc

1c

CL is the area of

the Cp profile

CL FROM Cp

� SUMMARY

� Plots of Cp data provide lift, moment, center of pressure insight

�Cl is the net area between the upper and lower distributions, divided

by chord c (or, an integral from 0 to 1 for x/c)

�The centroid of the area is the center of pressure

( ) ( )1

, , , ,0 0

1 c

l p l p u l p l p u

xc C C dx or c C C d

c c= − = −∫ ∫

CL FROM Cp2

p

V2

1

pp

q

ppC

∞∞

∞

∞

∞

ρ

−=

−≡

� PRESSURE COEFFICIENT VERSUS MACH

� PRANDTL-GLAUERT RULE

� Assumed Valid to , given a Cpo , determines the Cp at the higher M∞

or, given Cp at the higher M∞, finds corresponding Cpo

PRESSURE COEFFICIENT

2

0,pp

M1

CC

∞−=

,0pC

pC

0.8M ∞ ≤

2p

V2

1

pp

q

ppC

∞∞

∞

∞

∞

ρ

−=

−≡

CORRECTION FOR COMPRESSIBILITY

THE PRANDTL-GLAUERT RULE

� SUBSTITUTING Cp FROM THE PRANDTL-GLAUERT EQUATION

INTO THE Cl DEFINITION

� HERE, THE SUBSCRIPT “ 0 ” DENOTES INCOMPRESSIBLE FLOW.

� THUS, THE SECTION LIFT COEFFICIENT IS:

� FOR SUBSONIC SPEEDS

( LESS THAN M∞

= 1 ),

as the case for Cp ,

Cl VARIES INVERSELY WITH M∞

2

0,pp

M1

CC

∞−

=

( )( )∫∫ −

−

=

−

−=

∞∞

c

0 0u,pl,p2

c

0 2

0u,pl,pl dxCC

c

1

M1

1dx

M1

CC

c

1c

2

0,ll

M1

CC

∞−

=

2p

V2

1

pp

q

ppC

∞∞

∞

∞

∞

ρ

−=

−≡

2

0,pp

M1

CC

∞−=

What happens if we combine at M∞

< 0.3 (“Cp,0”)

with for 0.3 < M∞

< 0.8 ?

The Critical Mach No. “Mcr” & the Drag Diverence Mach No. “Mdd”

� CRITICAL MACH NUMBER, Mcr

� Mcr = the LOWEST FREESTREAM Mach No. at which M = 1 FIRST

occurs locally ANYWHERE on the body

THE ADVERSE CONSEQUENCE OF Exceeding Mcrit :

GREATLY INCREASED DRAG,

(THE SHOCK WAVE PRODUCES A LARGER SEPARATED WAKE)

… leads to “Drag Divergence” = Mdd

Mpeak = 0.435Mpeak = 0.772

Mpeak = 1.0

3.0M =∞

5.0M =∞

61.0M =∞

NUMBERMACHCRITICALMcrit ≡

61.0M >>∞

SHOCK-INDUCED

FLOW SEPARATION

DRAG-DIVERGENCE MACH NO. : Mdd

Mdd is the FREESTREAM Mach No. at which cd rises rapidly

� THE PHYSICAL MECHANISM: FLOW SEPARATION INDUCED

BY THE SHOCK WAVE

a

b

c

a b

c

DRAG-DIVERGENCE MACH NO.

� DIFFERENT DEFINITIONS USED BY DIFFERENT COMPANIES

DOUGLAS DEFINITION BOEING DEFINITION

� AT THE AIRFOIL’S MINIMUM PRESSURE POINT

Thick airfoil

Medium airfoil

Thin airfoil

Critical

PressureCoefficients

1.0Freestream Mach

Mcrit

(thick)Mcrit

(medium)

Mcrit

(thin)

Cp

Thick airfoil

Medium airfoil

Thin airfoil

( )∞= MfC crit,p ( )∞= MfC crit,p

∞

∞−≡

q

ppC cr

crp

Critical Pressure Coefficient “Cpcr”

ANALYTICAL EXPRESSION

� THE PRESSURE COEFFICIENT

� FROM THE DEF’N OF q

� And with the Def’n of the Speed of Sound

� FOR ISENTROPIC FLOW

−=

−≡

∞∞

∞

∞

∞ 1p

p

q

p

q

ppCp

( )

2 22 21 1

2 2 2 / 2

q V Vq V p V

p p p RT

ρ γ γρ γ

γ γ ρ γ∞ ∞ ∞ ∞

∞ ∞ ∞ ∞ ∞

∞ ∞ ∞ ∞ ∞

≡ = ⇒ = =

22

2

q Ma RT

p

γγ ∞ ∞

∞ ∞

∞

= ⇒ =

120 M2

11

p

p −γ

γ

−γ+=

120 M2

11

p

p −γ

γ

∞∞

−γ+=

Critical Pressure Coefficient “Cpcr”

� DIVIDING THESE TWO PRESSURE RATIOS

� SUBSTITUTING INTO THE Cp DEFINITION

� SPECIALIZE THIS EXPRESSION TO THE POINT WHERE M = 1

1

2

2

M2

11

M2

11

p

p−γ

γ

∞

∞

−γ+

−γ+

=

1

M2

11

M2

11

M

2C

1

2

2

2p −

−γ+

−γ+

γ=

−γ

γ

∞

∞

1

M2

11

M2

11

M

2C

1

2

2

2p −

−γ+

−γ+

γ=

−γ

γ

∞

∞

Critical Pressure Coefficient “Cpcr”

� FINALLY …

� AT THE CRITICAL PRESSURE COEFFICIENT, THE LOCAL “M” = 1

� THE DASHED CURVE = CP,crit

� THE SOLID CURVE IS A

PLOT OF THE

PRANDTL GLAUERT

EXPRESSION

( )1

1

M12

M

2C

12

2crit,p −

+γ

−γ+

γ=

−γ

γ

∞

∞

Thick airfoil

Medium airfoil

Thin airfoil

1.0Freestream Mach

Mcrit

(thick)Mcrit

(medium)

Mcrit

(thin)

Cp

2

0,pp

M1

CC

∞−=

critpC ,

Critical Pressure Coefficient “Cpcr”

SUPERCRITICAL AIRFOILS

� TAILORED CAMBER LINE

� DELAYED DRAG DIVERGENCE

LITTLECAMBER

HIGHLYCAMBERED

∞V

SUPERCRITICALAIRFOILS

dcCONVENTIONAL

AIRFOILS

MACH NUMBER

waveV a∞

=

Sound & Aerodynamic disturbances are pressure waves

that travel at the speed of sound

PRESSURE WAVES

& MACH WAVES

no sound

source

1V a M∞ ∞ ∞

< <

1V a M∞ ∞ ∞

= =

1V a M∞ ∞ ∞

> >

subsonic

sonic

supersonic

ORIGINS OF WAVE DRAG ( only occurs if M∞

> 1)

WAVE DRAG

� p > p∞∞∞∞ (A SHOCK WAVE

FORMS AT THE L.E.)

� FOR A FLAT PLATE

AT ANGLE OF ATTACK, αααα

We may approximate the

lift and drag coefficients

2

2

,2

4

1

4

1

l

d w l

cM

c cM

α

αα

∞

∞

≈−

≈ ≈−

but only for M∞

> 1 !!

Airfoil drag

, , ,d profile d frict d presc c c= +

2 2

2 2d d d

V p MD q Sc Sc Sc

ρ γ∞ ∞ ∞ ∞∞= = =

M ∞

dc

1

4

2

2

−≈≈

∞Mcc ld

αα

1

4

2 −≈

∞Mc l

α

only for M∞

> 1