Wings and Airfoils C- 3: Panel Methodsmercury.pr.erau.edu/~hayasd87/AE301/AE301_Notes_C-1.pdf · 4...
-
Upload
duongnguyet -
Category
Documents
-
view
222 -
download
1
Transcript of Wings and Airfoils C- 3: Panel Methodsmercury.pr.erau.edu/~hayasd87/AE301/AE301_Notes_C-1.pdf · 4...
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?
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
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
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
(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”
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
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
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________