Finite wings Vortices and wings

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  • 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 i for a given section of a finite wing depends on the

    distribution of downwash along the span of the wing

    ii sinLD =

    iisin

  • 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 =

    LC

    Sb

    AR2

    =

    Elliptical lift distribution

  • 4

    Lift curve slope A finite wings 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 =

    ( ) ( )inAReC180

    radinARe

    C

    12

    L

    1

    Li

    =

    =

    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

    mM

    mM

    dD

    lL

    CC

    CC

    CC

    CC

    ====

    00

    00

    00

    1

    lL

    l

    CC

    CeAR

    =+

    Note: do not forget 57.3 deg/rad conversion factor

  • 5

    0

    01

    aa

    aeAR

    =+

    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)

    Prandtls lifting line theory

    Compressibility correction (subsonic flowfield)

    0, 0

    20, 011

    compcomp

    comp

    a aa

    a aM

    eAReAR = =

    ++

    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 = =

    ++

    2

    4

    1compa

    M=

  • 6

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

    Low-aspect-ratio straight wing (incompressible)Helmbolds Equation

    0

    20 01

    aa

    a aAR AR

    = + +

    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 AR

    = + +

    Swept wings

    cosu V=

    0w =

    0w

    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/ M =

    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 AR

    =

    + +

    Swept wing (incompressible)Kuchemann approach

    0

    22 2 0 0

    cos

    cos cos1 cos

    comp

    aa

    a aM

    AR AR

    =

    + +

    Swept wing (subsonic compressible)

    2 2 2,1 1 cosnM M = =

    0 0 /a a , cosnM M =

    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 ==

    LL SCW2

    SCL2

    V

    =

    =

    maxLstall SC

    W2V

    =

    V

    High lift devices

    Other high lift devices include

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