1
Finite wings
� Infinite wing (2d) versus finite wing (3d)
– Definition of aspect ratio:
– Symbol changes:
2b bAR AR
S c≡ � =
l L d D m MC C C C C C→ → →
For rectangular platform
Vortices and wings
� What the third dimension does– Difference between upper and lower pressure results in
circulatory motion about the wingtips
– Vortices develop
– Causing downwash
– Drag is increased by this induced downwash
LOWER PRESSURE
HIGHER PRESSURE
VORTEXVORTEX
∞V
∞Vw
∞V WINGTIP VORTEXCAUSES DOWNWASH, w
LOCAL FLOW
2
Origin of induced drag
� Wingtip vortices alter flow field – Resulting pressure distribution increases drag – Rotational kinetic energy is added to the 2-D flow– Lift vector is tilted back
� AOA is effectively reduced� Component of force in drag direction is generated
Induced drag
– The sketch shows– For small angles of attack– The value of αi for a given section of a finite wing depends on the
distribution of downwash along the span of the wing
ii sinLD α=
iisin α≈α
3
� Lift per unit span varies– Chord may vary in length along the wing span– Twist may be added so that each airfoil section is
at a different geometric angle of attack– The shape of the airfoil section may change along
the wing span Lift per unit span as a function of distancealong the span -- also called the lift distribution
The downwash distribution, w, which results from the lift distribution
b
Lift per unit span
� An elliptical lift distribution
– Produces a uniform downwash distribution– For a uniform downwash distribution, incompressible theory
predicts that
� Where is the finite wing (3d) lift coefficient
Aspect Ratio
b
ARCL
i π=α
LC
Sb
AR2
=
Elliptical lift distribution
4
Lift curve slope� A finite wing’s lift curve slope is different from its 2D
lift curve slope
– For an elliptical spanwise lift distribution– Extending this definition to a general platform
ARCL
i π=α
( ) ( )�inAReC180
radinARe
C
12
L
1
Li π
=π
=α
Finite Wing Corrections
� All reference coefficients are not corrected
� Moment coefficients are not corrected� Lift coefficient due to angle of attack is
corrected– AR is the aspect ratio of the wing– e is the Oswald Efficiency Factor
αα mM
mM
dD
lL
CC
CC
CC
CC
====
00
00
00
1
lL
l
CC
CeAR
αα
απ
=+
Note: do not forget 57.3 deg/rad conversion factor
5
0
01
aa
aeARπ
=+
Finite Wing Corrections – High Aspect Ratio Wings (lifting line theory)
High-aspect-ratio straight wing (incompressible)
00, 21
comp
aa
M ∞
=−
Prandtl-Glauert rule (thin airfoil 2D)
Prandtl’s lifting line theory
Compressibility correction (subsonic flowfield)
0, 0
20, 011
compcomp
comp
a aa
a aM
eAReAR ππ ∞
= =− ++
Incompressible lift curve slope
High-aspect-ratio straight wing (subsonic compressible)
2
4
1compa
M∞
=−
High-aspect-ratio straight wing (supersonic compressible)(obtained from supersonic linear theory)
Effect of Mach Number on the Lift Slope
0, 0
20, 011
compcomp
comp
a aa
a aM
eAReAR ππ ∞
= =− ++
2
4
1compa
M∞
=−
6
Finite Wing Corrections – Low Aspect Ratio Wings (lifting surface theory)
Low-aspect-ratio straight wing (incompressible)Helmbold’s Equation
0
20 01
aa
a aAR ARπ π
=� �+ +� �� �
Low-aspect-ratio straight wing (subsonic compressible)
2 2
4 11
1 2 1compa
M AR M∞ ∞
� �� �= −� �− −� �
Low-aspect-ratio straight wing (supersonic compressible)(Hoerner and Borst)
0
22 0 01
comp
aa
a aM
AR ARπ π∞
=� �− + +� �� �
Swept wings
cosu V∞= Λ
0w =
0w ≠Λ
� Subsonically,The purpose of swept wings is to
delay the drag rise associated with wave drag
For a straight wing
Now, sweep theWing by 30°
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Swept and Delta wings
� Supersonically,The goal is to keep wing surfaces inside the mach
cone to reduce wave drag
LEADS TO LOW AR WINGSLEADS TO LOW AR WINGSAND TO HIGH LANDING SPEEDSAND TO HIGH LANDING SPEEDS
( )arcsin 1/ Mµ ∞=
Flow separation from forebodychine and wing leading-edge and roll up to form free vortices
Computational Fluid Dynamics
AGARD WING 445.6
8
Finite Wing Corrections
0
20 0
cos
cos cos1
aa
a aAR ARπ π
Λ=
Λ Λ� �+ +� �� �
Swept wing (incompressible)Kuchemann approach
0
22 2 0 0
cos
cos cos1 cos
comp
aa
a aM
AR ARπ π∞
Λ=
Λ Λ� �− Λ + +� �� �
Swept wing (subsonic compressible)
2 2 2,1 1 cosnM Mβ ∞ ∞= − = − Λ
0 0 /a a β� , cosnM M∞ ∞= Λ
Lift curve slope for an infinite swept wing
0a Lift curve slope airfoil section perpendicular to the leading edge
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Effect of flaps on lift
Medium angle of attack
High angle of attack
Slat opened at high angle of attack
With slats 24 degree angle of
attackWithout slats
15 degree angle of attack
Angle of attack
High lift devices
� Flaps are the most common high lift device
�A PLAIN FLAP DOESNOT CHANGE SLOPE APPRECIABLY
�STALL ANGLE OFATTACK DECREASESWITH FLAP ANGLE
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High lift devices
� The lift equation
� Solving for gives the true airspeed in unaccelerated level flight for a particular CL
� In level unaccelerated flight stall speed occurs when maximum CL occurs
� Flaps are not the only high lift device
L2
L SCV21
SCqL ∞∞∞ ρ==
LL SCW2
SCL2
V∞∞
∞ ρ=
ρ=
maxLstall SC
W2V
∞ρ=
V∞
High lift devices
� Other high lift devices include