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MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS
Summary of Incompressible Flow Over Airfoils
Summary of Thin Airfoil Theory
Example Airfoil Calculation
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk
2
KEY EQUATIONS FOR cl, L=0, cm,c/4, and xcp
• Within these expression we need to evaluate A0, A1, A2, and dz/dx
21
124,
0
0
00
10
14
4
1cos1
2
AAc
cx
AAc
ddx
dz
AAc
lcp
cm
L
l
3
A0, A1, and A2 COEFFICIENTS
0
00
0
00
cos2
1
dndx
dzA
ddx
dzA
n
0
002
0
001
0
00
2cos2
cos2
1
ddx
dzA
ddx
dzA
ddx
dzA
4
CENTER OF PRESSURE AND AERODYNAMIC CENTER
• Center of Pressure: It is that point on an airfoil (or body) about which the aerodynamic moment is zero
– Thin Airfoil Theory:
• Symmetric Airfoil:
• Cambered Airfoil:
• Aerodynamic Center: It is that point on an airfoil (or body) about which the aerodynamically generated moment is independent of angle of attack
– Thin Airfoil Theory:
• Symmetric Airfoil:
• Cambered Airfoil:
2114
4
AAc
cx
cx
lcp
cp
4
4
..
..
cx
cx
CA
CA
5
ACTUAL LOCATION OF AERODYNAMIC CENTER
NACA 23012xA.C. < 0.25c
NACA 64212xA.C. > 0.25 c
x/c=0.25
x/c=0.25
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EXAMPLE OF LEADING EDGE STALL• NACA 4412 Airfoil
(12% thickness)
• Linear increase in cl until stall
• At just below 15º streamlines are highly curved (large lift) and still attached to upper surface of airfoil
• At just above 15º massive flow-field separation occurs over top surface of airfoil → significant loss of lift
• Called Leading Edge Stall• Characteristic of relatively thin
airfoils with thickness between about 10 and 16 percent chord
7
EXAMPLE OF TRAILING EDGE STALL
• NACA 4421 (21% thickness)• Progressive and gradual movement of separation from trailing edge toward
leading edge as is increased
• Called Trailing Edge Stall
8
THIN AIRFOIL STALL• Example: Flat Plate with 2% thickness (like a NACA 0002)• Flow separates off leading edge even at low ( ~ 3º)
• Initially small regions of separated flow called separation bubble
• As a increased reattachment point moves further downstream until total separation
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NACA 4412 VERSUS NACA 4421• Both NACA 4412 and NACA 4421
have same shape of mean camber line
• Thin airfoil theory predict that linear lift slope and L=0 should be the same for both
• Leading edge stall shows rapid drop of lift curve near maximum lift
• Trailing edge stall shows gradual bending-over of lift curve at maximum lift, “soft stall”
• High cl,max for airfoils with leading edge stall
• Flat plate stall exhibits poorest behavior, early stalling
• Thickness has major effect on cl,max
10
OPTIMUM AIRFOIL THICKNESS• Some thickness vital to achieving high maximum lift coefficient
• Amount of thickness will influence type of stalling behavior
• Expect an optimum
• Example: NACA 63-2XX, NACA 63-212 looks about optimum
cl,max
NACA 63-212
11
AIRFOIL THICKNESS
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AIRFOIL THICKNESS: WWI AIRPLANES
English Sopwith Camel
German Fokker Dr-1
Higher maximum CL
Internal wing structureHigher rates of climbImproved maneuverability
Thin wing, lower maximum CL
Bracing wires required – high drag
13
MODERN LOW-SPEED AIRFOILSNACA 2412 (1933)Leading edge radius = 0.02c
NASA LS(1)-0417 (1970)Whitcomb [GA(w)-1] (Supercritical Airfoil)Leading edge radius = 0.08cLarger leading edge radius to flatted cp
Bottom surface is cusped near trailing edgeDiscourages flow separation over topHigher maximum lift coefficientAt cl~1 L/D > 50% than NACA 2412
14
MODERN AIRFOIL SHAPES
http://www.nasg.com/afdb/list-airfoil-e.phtml
Root Mid-Span Tip
Boeing 737
15
OTHER CONSIDERATIONS• Note that all airfoils we have seen, even flat
plate, will produce lift at some • Production of lift itself is not that difficult
• L/D ratio
– Production of lift with minimum drag
– Measure of aerodynamic efficiency of wing or airplane
– Important impact on performance range, endurance
• Maximum lift coefficient, CL,max
– Effective airfoil shape produces high value of cl,max
– Stalling speed of aircraft (take-off, landing)
– Improved maneuverability (turn radius, turn rate)
final
initial
D
L
W
W
C
C
SFCR ln
2
12
12
123
2 initialfinalD
L WWSC
C
SFCE
V
ng
R
V
dt
d
ng
VR
1
12
2
2
16
HIGH LIFT DEVICES: SLATS AND FLAPS
max,
2
2
2
2
1
Lstall
L
LL
SC
WV
SC
LV
SCVSCqL
17
HIGH LIFT DEVICES: FLAPS
• Flaps shift lift curve
• Act as effective increase in camber of airfoil
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Flap extended
Flap retracted
AIRFOIL DATA: NACA 1408 WING SECTION
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HIGH LIFT DEVICES: SLATS
• Allows for a secondary flow between gap between slat and airfoil leading edge
• Secondary flow modifies pressure distribution on top surface delaying separation
• Slats increase stalling angle of attack, but do not shift the lift curve (same L=0)
20
EXAMPLE: BOEING 727
cl ~ 4.5
21
EXAMPLE CALCULATION• GOAL: Find values of cl, L=0, and cm,c/4 for a NACA 2412 Airfoil
– Maximum thickness 12 % of chord
– Maximum chamber of 2% of chord located 40% downstream of the leading edge of the chord line
• Check Out: http://www.pagendarm.de/trapp/programming/java/profiles/
Root Airfoil: NACA 2412Tip Airfoil: NACA 0012
NACA 2412
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EQUATIONS DESCRIBING MEAN CAMBER LINE: z = z(x)
• Equation describes the shape of the mean camber line forward of the maximum camber position (applies for 0 ≤ z/c ≤ 0.4)
• Equation describes the shape of the mean camber line aft of the maximum camber position (applies for 0.4 ≤ z/c ≤ 1)
2
2
2.00555.0
8.0125.0
c
x
c
x
c
z
c
x
c
x
c
z
aft
fore
23
EXPRESSIONS FOR MEAN CAMBER LINE SLOPE: dz/dx
c
x
dx
dz
c
x
dx
dz
c
x
c
x
c
z
fore
fore
fore
25.01.0
28.0125.0
8.0125.02
c
x
dx
dz
c
x
dx
dz
c
x
c
x
c
z
aft
aft
aft
111.00444.0
28.00555.0
2.00555.02
24
COORDINATE TRANSFORMATION: x → , x0 → 0
025.0cos125.0
cos12
25.01.0
25.01.0
fore
fore
fore
dx
dz
dx
dz
c
x
dx
dz
0111.0cos0555.0
cos12
111.00444.0
111.00444.0
aft
aft
aft
dx
dz
dx
dz
c
x
dx
dz
2
cos1
c
x
• Equation describes the shape of the mean camber line slope forward of the maximum camber position
• Equation describes the shape of the mean camber line slope aft of the maximum camber position
25
EXAMINE LIMITS OF INTEGRATION• Coefficients A0, A1, and A2 are evaluated across the entire airfoil
– Evaluated from the leading edge to the trailing edge
– Evaluated from leading edge (=0) to the trailing edge (=)
• 2 equations the describe the fore and aft portions of the mean camber line
– Fore equation integrated from leading edge to location of maximum camber
– Aft equation integrated from location of maximum camber to trailing edge
– The location of maximum camber is (x/c)=0.4
– What is the location of maximum camber in terms of ?
rad 3694.1
463.78
2.0cos
4.02
cos1
cambermax
cambermax
cambermax
cambermax
c
x
26
EXAMPLE: NACA 2412 CAMBERED AIRFOIL
• Thin airfoil theory lift slope:
dcl/d = 2 rad-1 = 0.11 deg-1
• What is L=0?
– From data L=0 ~ -2º
– From theory L=0 = -2.07º
• What is cm,c/4?
– From data cm,c/4 ~ -0.045
– From theory cm,c/4 = -0.054
dcl/d = 2
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