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:

b2 b
AR ≡ ⇒ AR = for rectangular planforms
S c

AR ⇒ ∞ for the airfoil … wingtip effects vanish

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

AR ~ 4

AR ~ 10

AR ~ 20

AR ~ 3

AR ~ 8

AR ~ 20
 AIRFOILS

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
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, 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 N
R
L

α D
V∞ A

Lift = L and Drag = D Forces r r r

L: Force perpendicular to V∞ R= L+D
D: Force parallel to V∞

Normal = N and Axial = A Forces r r r

N: Force perpendicular to chord R=N+A
A: Force parallel to chord

L = cos α N − sin α A  L  c α − sα   N   N   cα sα   L 
 =    or   =   
D = sin α N + cos α A  D   sα cα   A   A   − sα c α   D 
 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

z L M X = M LE + xL
MLE MX
x
There is no aerodynamic moment at the center of pressure, thus
M CP = 0 ⇒ 0 = M LE + xL ⇒ xCP = − M LE / L
 x – y – z = “roll – pitch – yaw”
( positive is roll right, pitch up, yaw right )

z
The Aerodynamic Moment
… moments may be transferred.

a moment may be about the leading edge

M X = M L.E . + xL

or, from the quarter chord

z L
Mc/4 MX M X = M c /4 + x ' L

x’
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:

L = f1 (V∞ , ρ∞ , µ ∞ ,S , a∞ )
D = f2 (V∞ , ρ∞ , µ ∞ ,S , a∞ )
M = f3 (V∞ , ρ∞ , µ ∞ ,S , a∞ )

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:

L = Z (V∞a , ρ ∞b , S d , a∞e , µ∞f )

Z , a, b, d , e, f are
DIMENSIONLESS CONSTANTS
 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:
ml
L∝ 2 L = f (V∞ , ρ ∞ , S , a∞ , µ ∞ )
t

EQUATING THE DIMENSIONS ON LEFT AND RIGHT OF THE LIFT FORCE
EQUATION a b e f
ml  l   m  2 d  l   m 
2
=    3  (l )    
t t  l   t   lt 

 EQUATING MASS EXPONENTS 1= b+f

 EQUATING LENGTH EXPONENTS 1 = a − 3 b + 2d + e − f

 EQUATING TIME EXPONENTS − 2 = −a − e − f

DIMENSIONAL ANALYSIS

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

1− f
V∞ 2 − e − f ρ1∞− f
L = Z( ) ,S 2 ,ae ,µf
∞ ∞
e f
a   µ∞ 
REARRANGING : L = ZV∞2 S  ∞   
ρ V S 
 V∞   ∞ ∞ 

a∞ 1

NOTING THAT = AND S HAS UNITS OF LENGTH,
V∞ M ∞
WE CHOOSE c AS OUR CHARACTERISTIC LENGTH

µ∞ µ∞

THEN WE CAN REPLACE WITH ρ V c
ρ∞V∞ S ∞ ∞

NOW, THE LIFT EQUATION IS OF THE FORM

e f
 1   1 
L = ZV∞2 S    
 M∞   Rec 
FORCE / MOMENT COEFFICIENTS
 NOW WE DEFINE THE AIRFOIL’S SECTION LIFT COEFFICIENT
e f
cl  1   1  1
≡ Z     ⇒ L= ρ∞V∞2 Scl
2  M∞   Rec  2

OR WE COULD HAVE SIMPLY DEFINED LIFT COEFFICIENT AS

L from ch. 4 Df
cl ≡ cf ≡
q∞S q∞S
 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

cl = f (α, M∞ , Re)
FORCE / MOMENT COEFFICIENTS
 A SIMILAR ANALYSIS LEADS TO

DRAG COEFFICIENT D = q∞Scd

MOMENT COEFFICIENT M = q∞Sccm

 Moment has length, c , due to FORCE x LENGTH

 Moment must be referenced to the point where the moment is taken …
L D Mx
 THUS, cl ≡ cd ≡ cmx ≡
q∞S q∞S q∞Sc
WHERE cl = f1 (α, M ∞ , Re) cd = f2 (α, M ∞ , Re) cm = f3 (α, M ∞ , Re)

 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!
AIRFOIL DATA … refer to Appendix D
AIRFOIL CLASSIFICATIONS

NACA 2412

NACA 23012

NACA 63-210
 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

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
AIRFOIL DATA

 EXPERIMENTAL DATA
ARE ESSENTIAL TO
AIRCRAFT DESIGN

NACA / NASA DATA

APPENDIX D

 cl VARIES LINEARLY
WITH α

CAMBER CHANGES αL= 0

 THIS LINEAR RELATIONSHIP

BREAKS DOWN WHEN STALL
OCCURS
AIRFOIL DATA

AT HIGH α, THE
BOUNDARY LAYER
WILL SEPARATE

LIFT DECREASES

DRAG INCREASES

AERO MOMENT
BECOMES
NOSE DOWN ( - )

… ahhh …vicious viscous

 Cl ,max
Cl

∆Cl
Clα =0 = ao
∆α
α l =0
α
α stall
AIRFOIL DATA
 EXPERIMENTAL DATA ( Appendix D )

NACA data

Incompressible flow

Re specified

Cl Cd

α Cl

Cm c/4 Cm ac
AIRFOIL DATA
 EXPERIMENTAL DATA ( Appendix D )

NACA data

Incompressible flow

Re specified

Cl Cd

α Cl

Cm c/4 Cm ac
 α

Cm c/4

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

M x = M c / 4 + xL ⇒ 0 = M c /4 + xc. p. L ⇒ 0 = qScCmc/4 + xc. p.qSCl

− qScCmc/4 −cCmc/4 xc. p. −Cmc/4

xc. p. = = ⇒ = =
qSCl Cl c Cl
AIRFOIL DATA – The Effect of a Flap

Cl

‘straight & level’

L=W
L = qSCL = W
q = W/SCL
α
ଶௐ
V=
ఘௌ஼ಽ
(CL/CD) max and α (CL/CD) max (leads to ‘best’ A/C performance)

Cl Cd

α Cl

α 0 2 4 8 8 10

Cl Note: this is not at

CL max or CD min or αmax
Cd

Cl/Cd
Cd
tangent
⇒ (Cl / Cd ) MAX

Cl
Cl ( C /C
l d ) MAX

Cl ( C /C ⇒ α ( Cl / Cd )MAX
l d ) MAX
Cd = Cd , friction + Cd , pressure = Cd , profile

Cd

Cd, pressure
≈ Cd , friction
Cl
AIRFOILS
 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.
 p − p∞ p − p∞
PRESSURE COEFFICIENT Cp ≡
q∞
=
1
ρ∞V∞2

Cp < 0 … “suction” 2
… a “pull”

Cp > 0 … “positive pressure”

… a “push”

Let’s examine a special case …

incompressible via Bernoulli

V∞2 V2
P∞ + ρ =P+ρ
2 2  V∞2 − V 2 
ρ 
 V∞2 − V 2  P − P∞  2  = C =1− V 2
P − P∞ = ρ   = P
 2  ρ V∞2 / 2 ρ V∞2 / 2 V∞2
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)
 p − p∞ p − p∞
Cp ≡ =
q∞ 1
ρ∞V∞2
2
AIRFOILS
 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 α, THE LOCATION OF THE c.p. VARIES WITH α

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)

OBTAINING CL FROM Cp
 LIFT PER UNIT SPAN

dscos θ = dx
θ
LE (leading edge) dx

TE (trailing edge)

LIFT … NET UPWARD FORCE DUE TO THE PRESSURE DIFFERENCE

BETWEEN THE LOWER SURFACE AND THE UPPER SURFACE

TE TE
L=∫ pl cos θds − ∫ pu cos θds
LE LE

Note, ds cos θ = dx c c
L = ∫ pl dx − ∫ pu dx
0 0
CL FROM Cp
 LIFT PER UNIT SPAN
c c

Add & subtract p∞ L = ∫ (pl − p∞ )dx − ∫ (pu − p∞ )dx
0 0

Use the def’n of the lift coefficient

L L L
cl ≡ = =
q ∞ S q ∞ c (1 ) q ∞ c

combine
1 c pl − p∞ 1 c pu − p∞
cl = ∫ dx − ∫ dx CL is the area of
c 0 q∞ c 0 q∞
the Cp profile

p − p∞ p − p∞
where C p ,l ≡ l AND C p ,u ≡ u
q∞ q∞

Thus:
1 c
(
cl = ∫ C p ,l − C p ,u dx
c 0
)
CL FROM Cp p − p∞ p − p∞
Cp ≡ =
q∞ 1
ρ∞V∞2
 SUMMARY 2

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)

1 c 1 x
cl = ∫ ( C p ,l − C p ,u ) dx or cl = ∫ ( C p ,l − C p ,u ) d
c 0 0 c
 The centroid of the area is the center of pressure
 p − p∞ p − p∞
Cp ≡ =
PRESSURE COEFFICIENT q∞ 1
ρ∞V∞2
 PRESSURE COEFFICIENT VERSUS MACH 2

Cp
C p ,0

C p ,0
 PRANDTL-GLAUERT RULE Cp =
1 − M∞2
 Assumed Valid to M ∞ ≤ 0.8 , given a Cpo , determines the Cp at the higher M∞
or, given Cp at the higher M∞, finds corresponding Cpo
CORRECTION FOR COMPRESSIBILITY
C p ,0
THE PRANDTL-GLAUERT RULE Cp =
2
1 − M∞

SUBSTITUTING Cp FROM THE PRANDTL-GLAUERT EQUATION
INTO THE Cl DEFINITION
( )
1 c C p ,l − C p ,u 0 1 1 c
cl = ∫
c 0 2
dx =
2 c ∫0
( 0
)
C p,l − C p ,u dx
1 − M∞ 1 − M∞

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

THUS, THE SECTION LIFT COEFFICIENT IS: Cl =
2
1 − M∞

FOR SUBSONIC SPEEDS

( LESS THAN M∞ = 1 ),
as the case for Cp ,
Cl VARIES INVERSELY WITH M∞
 p − p∞ p − p∞
What happens if we combine Cp ≡ =
1
at M∞ < 0.3 (“Cp,0”)
q∞ ρ∞V∞2
Cp ,0 2
with Cp = for 0.3 < M∞ < 0.8 ?
1 − M∞2
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

M∞ = 0.3

M = 0.772
M∞ = 0.5 Mpeakpeak
= 0.435

M∞ = 0.61 Mpeak = 1.0

Mcrit ≡ CRITICAL MACH NUMBER

M∞ >> 0.61

SHOCK-INDUCED
FLOW SEPARATION
THE ADVERSE CONSEQUENCE OF Exceeding Mcrit :
GREATLY INCREASED DRAG,
(THE SHOCK WAVE PRODUCES A LARGER SEPARATED WAKE)

… leads to “Drag Divergence” = Mdd

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

b
a b

c
DRAG-DIVERGENCE MACH NO.

 DIFFERENT DEFINITIONS USED BY DIFFERENT COMPANIES

DOUGLAS DEFINITION BOEING DEFINITION

Critical Pressure Coefficient “Cpcr” pcr − p∞
Cp cr ≡
q∞
 AT THE AIRFOIL’S MINIMUM PRESSURE POINT

Cp

Thick airfoil Critical

Pressure Thick airfoil
Coefficients
Medium airfoil
Medium airfoil
Cp,crit = f (M∞ )
Thin airfoil
Thin airfoil

Mcrit 1.0
Mcrit (thick) Mcrit Freestream Mach
(medium) (thin)
Critical Pressure Coefficient “Cpcr”
ANALYTICAL EXPRESSION

THE PRESSURE COEFFICIENT

p − p∞ p∞  p 
Cp ≡ = 
 − 1 
q∞ q∞  p∞ 

FROM THE DEF’N OF q
1 1 ρ∞ q∞ γ V∞2 γ V∞2
q∞ ≡ ρ ∞V∞2 = γ p∞V∞2 ⇒ = =
2 2 γ p∞ p∞ 2γ ( p∞ / ρ∞ ) 2γ RT∞

And with the Def’n of the Speed of Sound
q∞ γ M ∞2
2
a = γ RT∞
∞ ⇒ =
p∞ 2

FOR ISENTROPIC FLOW
γ
p0  γ −1 2  γ −1
= 1 + M 
p  2 
γ
p0  γ − 1 2  γ −1
= 1 + M∞ 
p∞  2 
Critical Pressure Coefficient “Cpcr”

DIVIDING THESE TWO PRESSURE RATIOS

γ
 γ− 1 2  γ −1
p  1 + M∞ 
= 2 
p∞  1 + γ −1 M 2 
 
 2 

SUBSTITUTING INTO THE Cp DEFINITION
γ
 γ− 1 2  γ −1
2  1+ M∞ 
Cp =  2  −1
2  γ −1 2 
γM ∞ 1 + M 
 2 

SPECIALIZE THIS EXPRESSION TO THE POINT WHERE M = 1

Critical Pressure Coefficient “Cpcr”

 FINALLY …

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

γ
2  2 + (γ − 1)M ∞
2  γ −1
C p ,crit =   −1
2  γ +1 
γM ∞  
Cp
Thick airfoil

THE DASHED CURVE = CP,crit

Medium airfoil

THE SOLID CURVE IS A

PLOT OF THE Thin airfoil
PRANDTL GLAUERT Cp,crit
EXPRESSION

C p ,0 Mcrit 1.0
Cp = Mcrit
(thick) Mcrit Freestream Mach

1 − M∞2 (medium) (thin)

SUPERCRITICAL AIRFOILS

 TAILORED CAMBER LINE

LITTLE HIGHLY
CAMBER CAMBERED
V∞

 DELAYED DRAG DIVERGENCE

cd
CONVENTIONAL
AIRFOILS

SUPERCRITICAL
AIRFOILS

MACH NUMBER
PRESSURE WAVES
& MACH WAVES Vwave = a∞

source

no sound

Sound & Aerodynamic disturbances are pressure waves

that travel at the speed of sound
V∞ < a∞ M ∞ < 1
subsonic

V∞ = a∞ M ∞ = 1
sonic

V∞ > a∞ M ∞ > 1
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, α
We may approximate the
lift and drag coefficients

4α
cl ≈
M ∞2 − 1
4α 2
cd , w ≈ ≈ clα
M ∞2 − 1
but only for M∞ > 1 !!
Airfoil drag

4α
cl ≈
M ∞2 − 1
cd only for M∞ > 1
4α 2
cd ≈ c l α ≈
M∞2 − 1

cd , profile = cd , frict + cd , pres

ρ∞V∞2 γ p∞ M ∞2 M∞
D = q∞ Scd = Scd = Scd
2 2