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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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MODERN AIRFOIL SHAPES
http://www.nasg.com/afdb/list-airfoil-e.phtml
Root Mid-Span Tip
Boeing 737
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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
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HIGH LIFT DEVICES: SLATS AND FLAPS
max,
2
2
2
2
1
Lstall
L
LL
SC
WV
SC
LV
SCVSCqL
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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)
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EXAMPLE: BOEING 727
cl ~ 4.5
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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
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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
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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
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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
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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