camberline equ

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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

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

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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