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 Unit C-1: List of Subjects Wings and Airfoils Lift Curves Airfoils NACA Conventional Airfoils NACA Airfoil Data
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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)
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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?