From 2D to 3D wings

Finite wings 

Infinite wing (2d) versus finite wing (3d)

Definition of aspect ratio:

 – 

2

b ≡R  A

b =R A c

For rectangular platform

Symbol changes:

 – 

Cl → CL Cd → CD Cm → CM  

 

Vortices and wings 

What the third dimension does  – 

Difference between upper and lower pressure results in circulatory motion about the wingtips LOWER PRESSURE 

VORTEX 

VORTEX  HIGHER PRESSURE 

V ∞

 – 

Vortices develop

 – 

Causing downwash

V ∞

LOCAL FLOW 

 – 

V ∞

WINGTIP VORTEX  CAUSES DOWNWASH, w 

w 

Drag is increased by this induced downwash

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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 D i  = L sin α i  For small angles of attack sin α i  ≈ α i  The value of α i for a given section of a finite wing depends on the distribution of downwash along the span of the wing

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Lift per unit span 

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 distance  along the span -- also called the lift distribution 

b  The downwash distribution, w, which results from the lift distribution 

Elliptical lift distribution 

An elliptical lift distribution

b   –   – 

Produces a uniform downwash distribution For a uniform downwash distribution, incompressible theory predicts that C  α i  = L πAR  

Where C L is the finite wing (3d) lift coefficient b 2  Aspect Ratio AR =

S 

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Lift curve slope 

A finite wing’s lift curve slope is different from its 2D lift curve slope

 –   – 

C 

L α i  = For an elliptical spanwise lift distribution πAR  Extending this definition to a general platform

α i  =

C L 180 C  (in rad ) = 2  L in  πe 1 AR  π e 1 AR 

( ) 

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

C  L 0 = C l 0 C  D 0 = C d 0 C  M 0 = C m 0 C  M α  = C mα  C  Lα  = 1+

C lα  C lα 

π eAR

Note: do not forget 57.3 deg/rad conversion factor

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Finite Wing Corrections – High Aspect Ratio Wings (lifting line theory) a0 a0

a= 1+

High-aspect-ratio straight wing (incompressible) Prandtl’s lifting line theory

π eAR

Incompressible lift curve slope

2

1 − M ∞

a0, comp

acomp =

1+

acomp =

Compressibility correction (subsonic flowfield) Prandtl-Glauert rule (thin airfoil 2D)

a0

a0, comp =

a0, comp

a0

=

2

1 − M ∞ +

π eAR 4

a0

π eAR

High-aspect-ratio straight wing (subsonic compressible)

High-aspect-ratio straight wing (supersonic compressible) (obtained from supersonic linear theory)

2

 M ∞ − 1

Effect of Mach Number on the Lift Slope acomp =

a0, comp 1+

a0, comp

π eAR

=

a0 2

1 − M ∞ +

a0

π eAR

acomp =

4

 M ∞2 − 1

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Finite Wing Corrections – Low Aspect Ratio Wings (lifting surface theory) a0

a=

 a0 1+  π

Helmbold’s Equation

a0

acomp = 2

 a0 

1 − M ∞ + 

π

acomp =

Low-aspect-ratio straight wing (incompressible)

2

a  + 0  AR  π  AR

2

 + π 

AR 

Low-aspect-ratio straight wing (subsonic compressible)

a0 AR

  1 1 −  2  2 AR M2 −  1  M − 1 ∞ ∞   4

Low-aspect-ratio straight wing (supersonic compressible) (Hoerner and Borst)

Swept wings 

Subsonically, The purpose of swept wings is to delay the drag rise associated with wave drag w=0

For a straight wing

Now, sweep the Wing by 30°

Λ

w≠0

u = V ∞ cos Λ

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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 LOW AR WINGS WINGS AND TO HIGH LANDING SPEEDS

 µ  = arcsin (1 /  M ∞ )

Computational Fluid Dynamics Flow separation from forebody chine and wing leading-edge and roll up to form free vortices AGARD WING 445.6

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Finite Wing Corrections a0 Lift curve slope airfoil section perpendicular to the leading edge Lift curve slope for an infinite swept wing a0 cos Λ

a=

Swept wing (incompressible)

2

a cos Λ  a0 cos Λ  Kuchemann approach + 0  π  AR AR  π 

1+ 

a0  a0 /  β 

 M ∞, n = M ∞ cos Λ

 β  = 1 − M ∞2 , n = 1 − M ∞2 cos2 Λ acomp =

a0 cos Λ 2

Swept wing (subsonic

a cos Λ  a cos Λ  compressible) 1 − M ∞ cos Λ +  0 + 0  π  AR AR  π  2

2

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Computational Fluid Dynamics for Wing - Body combinations

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Flaps

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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 attack Without slats 15 degree angle of attack

Angle of attack

High lift devices 

Flaps are the most common high lift device

A PLAIN FLAP DOES 

NOT CHANGE SLOPE  APPRECIABLY  STALL ANGLE OF  ATTACK DECREASES  WITH FLAP ANGLE 

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High lift devices 

L = q ∞SC L =

The lift equation 

Solving for V ∞ gives the true airspeed in unaccelerated level flight for a particular CL

V ∞ = 



1 ρ∞V ∞2 SC L 2 

2 L = ρ∞SC L

2 W  ρ∞SC L

In level unaccelerated flight stall speed occurs when maximum CL occurs 2 W  V stall  = ρ∞SC Lmax 

Flaps are not the only high lift device

High lift devices 

Other high lift devices include

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