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°

7

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

9

10

Computational Fluid Dynamics forWing - Body combinations

11

Flaps

12

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

13

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