AE301 Aerodynamics I

UNIT C: 2-D Airfoils

ROAD MAP . . .

C-1: Aerodynamics of Airfoils 1

C-2: Aerodynamics of Airfoils 2

C-3: Panel Methods

C-4: Thin Airfoil Theory

AE301 Aerodynamics I

Unit C-1: List of Subjects

Wings and Airfoils

Lift Curves

Airfoils

NACA Conventional Airfoils

NACA Airfoil Data

NOMENCLATURE FOR WINGS (3-D)

3-D Wing Geometry Nomenclature:

• Leading edge (LE)

• Trailing Edge (TE)

• Airfoil (a cross section of wing)

Recall that the lift, drag, and moment coefficient for 3-D wing can be defined as:

L

LC

q S

D

DC

q S

M

MC

q Sc

NOMENCLATURE FOR AIRFOILS (2-D)

2-D Airfoil Geometry Nomenclature:

• Chord Line

• Mean Camber Line

• Chord (c)

• Thickness (t)

• Camber: (difference between chord line and mean camber line)

For 2-D airfoil, the aerodynamic coefficients are “per unit span” basis:

'

1l

L b Lc

q c q c

'

1d

D b Dc

q c q c

2

'

1m

M b Mc

q cq c c

Note: b = wing span

Unit C-1Page 1 of 10

Wings and Airfoils

LIFT OF AIRFOILS

Lift on an airfoil depends on the following properties:

• V (freestream velocity)

• (freestream density)

• S (wing area)

Hence, lift coefficients are normalized by these properties: L

LC

q S

and '

l

Lc

q c .

LIFT CURVE OF AIRFOILS

The behavior of lift (“lift curve” characteristics) depends on the following properties:

• (angle of attack)

Lift-curve (or often called, “cl - ” curve) provides important relationship between angle of attack and

lift coefficient, under a certain condition of Reynolds number. Interestingly, lift-curve is fairly close to a

linear line, as long as it is not under the “stall” condition (hence, we often assume it is a simple “linear

function” in our aerodynamic analysis for simplification).

• (viscosity)

Lift curve depends on the Reynolds number.

• a (freestream speed of sound, or “compressibility”)

Lift curve will also depend on the compressibility of the flow field (Mach number).

Unit C-1Page 2 of 10

Lift Curves

AIRFOILS

• The “shape” of airfoil: the design of 2-D airfoil will have a significant impact on aircraft

performance.

• Airfoils represent performance of a given cross-section of a wing. The shape of an airfoil has

tremendous effects on the overall performance of wing (thus, airplane).

• Airfoils can be considered as a model for a “unit span” of an infinite wing of constant cross-

section. The performance of an airfoil can be determined by a “quasi-2-D” wind tunnel tests.

• A Quasi-2-D is actually a 3-D, but constant cross section. Thus, the cross-sectional properties (i.e.,

lift, drag, and moment “per unit length”) can be determined.

GEOMETRIC AND AERODYNAMIC TWISTS OF WINGS

• Geometric twist of wing is varying angle of attack along the span, but retains the same airfoil.

• Aerodynamic twist of wing is varying airfoil (cross section of the wing) along the span, but retains

the angle of attack.

Unit C-1Page 3 of 10

Airfoils (1)

http://www.ae.uiuc.edu/m-selig/ads.html

CAMBERED V.S. SYMMETRICAL AIRFOIL

The camber in airfoil is the asymmetry between the top and the bottom curves of an airfoil. Cambered

airfoils generate lift at positive, zero, or even small negative angle of attack, whereas a symmetric airfoil

only has lift at positive angles of attack.

0ldc

ad

: lift curve slope – the slope of the cl – curve (straight line)

The lift curve slope of a “thin” airfoil (either symmetric or cambered) is:

0a 2 (1/rad) = 0.10966 (1/deg)

0L : zero lift angle of attack (“alpha zero-lift”) – the angle of attack (negative value), where the

cambered airfoil generates no lift.

LIFT EQUATIONS

Assuming the “linear” relationship between angle of attack () and lift coefficient (cl), one can

“estimate” the lift coefficient at a given angle of attack:

• For symmetrical airfoil ( 0 0L ):

0lc a

• For cambered airfoil ( 0 0L ):

0 0l Lc a

• NOTE: these lift equations are based on assumptions that angle of attack () and lift coefficient (cl)

are perfectly in linear relationship (these are simple linear algebraic equations, such as: y ax b ).

Is it always true???

Unit C-1Page 4 of 10

Airfoils (2)

(a) Assuming the linear relationship between cl – (valid only in the certain range of ):

0 0( ) 0.11[10 ( 3)]l Lc a 1.43

(b) Upside-down means that the airfoil is now “negatively” cambered. The zero lift AOA is now + 3

degrees. Thus, 10 degrees AOA is essentially equivalent to only 7 degrees AOA, so:

0 0( ) 0.11[10 ( 3)]l Lc a 0.77

(c) In order to maintain the same lift coefficient (1.43 at 10 degrees AOA), the upside-down airfoil must

be pitched to a higher AOA.

0 0( )l Lc a

=> 0

0

1.43(3)

0.11

lL

c

a 16 (degrees)

Unit C-1Page 5 of 10

Class Example Problem C-1-1

Related Subjects . . . “Airfoils”

Can airplane fly upside-down?

To answer this question, make the following simple calculation. Consider a positively

cambered airfoil with a zero-lift angle of attack of 3 degrees. The lift slope of this

airfoil is 0.11 per degree.

(a) Calculate the lift coefficient at an angle of attack of 10 degrees.

(b) Now imagine the same airfoil turned upside-down, but at the same 10 degrees angle

of attack as part (a). Calculate its lift coefficient.

(c) At what angle of attack must the upside-down airfoil be set to generate the same lift

as that when it is right-side-up at a 10 degrees angle of attack?

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NACA AIRFOIL DATA

NACA airfoils are airfoil shapes developed by the National Advisory Committee for Aeronautics

(NACA). The lift, drag, and moment coefficients for these airfoils were obtained through wind tunnel

tests conducted in 1950s (NACA Report 824).

NACA FOUR/FIVE DIGIT SERIES AIRFOILS

The normalized coordinates (x/c, y/c) of NACA 4 digit or 5 digit series conventional airfoils can be

computer-generated (NASA-96-TM4741).

• NACA 4/5 digit series airfoils are usually called as: NACA Conventional Airfoils.

NACA conventional airfoils have maximum thickness (usually) at quarter chord. For a conventional

airfoil, the maximum thickness location is designed to be at the location of aerodynamic center. If the

maximum thickness location is moved (usually “aft” not “forward”), the airfoil design is usually

considered “non-conventional” (i.e., NACA 6-series Natural Laminar Flow or “NLF” airfoils).

Unit C-1Page 6 of 10

NACA Conventional AirfoilsNACA 4-Digit Series: NACA X X XX

One digit describing maximum camber (in % of

chord).

One digit describing the distance to the maximum

camber location measured from the leading edge (in 10% of chord).

Two digits describing maximum thickness of the

airfoil (in % of chord).

NACA 5-Digit Series: NACA X XX XX

One digit, when multiplied by 1.5, gives the lift

coefficient in 1/10.

Two digits, when divided by 2, describe the

distance to the maximum camber location measured from the leading edge in 1/10 of chord.

Two digits describing the maximum thickness of

the airfoil in % of chord.

OTHER NACA AIRFOILS

NACA 1-Series:

Mathematically derived airfoil shape from the desired lift characteristics. Prior to this, airfoil shapes

were only determined using a wind tunnel.

NACA 6-series:

An improvement over 1-series airfoils with emphasis on maximizing natural laminar flow (NLF).

Maximum thickness is moved (close) to the half chord location.

NACA 7-series:

Further advancement in maximizing laminar flow achieved by separately identifying the low pressure

zones on upper and lower surfaces.

NACA 8-series (NASA-SC):

Supercritical (SC) airfoils designed to optimize transonic flow characteristics.

Unit C-1Page 7 of 10

NACA Airfoil Data

THIS FIGURE IS FOR EXPLANATION PURPOSES ONLY = NOT VERY ACCURATE !!!

(a) At standard sea-level condition with 44 m/s airspeed:

= 1.225 kg/m3

= 17.89106 kg/ms

c = 1.0 m

The test section Reynolds number: 6

(1.225)(44)(1)Re

17.89 10

Vc

= 3106

Let us look at the NACA 4412 airfoil data (Reynolds number 3106):

At 2 degrees of AOA:

cl = 0.6 cd = 0.0067 cm,c/4 = 0.08

Unit C-1Page 8 of 10

Class Example Problem C-1-2

Related Subjects . . . “NACA Airfoil Data”

A model wing of constant chord length is placed in a low

speed subsonic wind tunnel, spanning across the test

section (this is called, a quasi-2-D test). The wing has a

NACA 4412 airfoil and a chord length of 1.0 m. The test

section airspeed is 44 m/s at standard sea-level condition.

If the wing is at a 2 degrees angle of attack, determine:

(a) cl, cd, and cm,c/4

(b) the lift, the drag, and the moment about the quarter

chord (per unit span)

NACA 4412

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(b) The dynamic pressure is: 2 21 1

(1.225)(44)2 2

q V = 1,185.8 N/m2

Therefore,

' (1,185.8)(1)(0.6)lL qcc = 711.48 N (per unit span)

' (1,185.8)(1)(0.0067)dD qcc = 7.945 N (per unit span)

2 2

/4 , /4' (1,185.8)(1) ( 0.08)c m cM qc c = 94.86 N (per unit span)

Unit C-1Page 9 of 10

Class Example Problem C-1-2 (cont.)

Related Subjects . . . “NACA Airfoil Data”

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Upside-down flight is equivalent to the NACA data in the range of negative angle of attack. The only

difference is that the negative angle of attack produces “downforce” (means negative lift coefficient),

while upside-down flight produces “lift” (positive lift coefficient). Thus, one can read-out the lift

coefficient from the NACA data (in the range of negative angle of attack).

(a) From NACA 4412 data (assuming the Reynolds number 3106):

At 10 degrees => 1.34lc

(b) If the data is read-off from “negative” angle of attack:

At 10 degrees => 0.64lc

(c) Impossible.

The airfoil stalls out at about “negative” angle of attack of 12 degrees (maximum 0.8lc )

Unit C-1Page 10 of 10

Class Example Problem C-1-3

Related Subjects . . . “NACA Airfoil Data”

Once again, can airplane fly upside-down?

To answer this question, we will use the same wind tunnel test as in Class Example

Problem C-1-2. Consider NACA 4412 airfoil.

(a) Determine the lift coefficient at an angle of attack of 10 degrees.

(b) Now imagine the same airfoil turned upside-down, but at the same 10 degrees angle

of attack as part (a). Determine its lift coefficient.

(c) At what angle of attack must the upside-down airfoil be set to generate the same lift

as that when it is right-side-up at a 10 degrees angle of attack?

NACA 4412

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