Airfoil versus Wing.pdf

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  1 Finite wings  Infinite wing (2d) versus finite wing (3d)  –  Definition of aspect ratio:  –  Symbol changes: 2 b b AR ARS c ≡    = l L d D m M  C 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 VORTEX VORTEX  ∞ V  ∞ V  w  ∞ V  WINGTIP VORTEX CAUSES 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 α  i for a given section of a finite wing depends on the distribution of downwash along the span of the wing i i    sin LD   α= i i  sin   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 distance along 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 coefficientAspect Ratio b  AR C  Li  π=α L C  S b AR  2  = 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 spanwiselift distribution  –  Extending this definition to a general platform AR C  Li  π=α ( )  ( )  in AR e C 180 rad in AR e C  12 L1Li  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 α α   m M m M d  Dl L C C C C C C C C  ==== 000000 1 l Ll C C C eAR α α α  π  =+ Note: do not forget 57.3 deg/rad conversion factor

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