MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS

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MAE 3241 AERODYNAMICS ANDFLIGHT MECHANICS

• Finite Wings General Lift Distribution
• April 2, 2007
• Mechanical and Aerospace Engineering Department
• Florida Institute of Technology
• D. R. Kirk

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PRANDTLS LIFTING LINE EQUATION

• Fundamental Equation of Prandtls Lifting Line
  Theory
• In Words Geometric angle of attack is equal to
  sum of effective angle of attack plus induced
  angle of attack
• Mathematically a aeff ai
• Only unknown is G(y)
• V8, c, a, aL0 are known for a finite wing of
  given design at a given a
• Solution gives G(y0), where b/2 y0 b/2 along
  span

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WHAT DO WE GET OUT OF THIS EQUATION?

• Lift distribution
• Total Lift and Lift Coefficient
• Induced Drag

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ELLIPTICAL LIFT DISTRIBUTION

• For a wing with same airfoil shape across span
  and no twist, an elliptical lift distribution is
  characteristic of an elliptical wing planform

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SPECIAL SOLUTIONELLIPTICAL LIFT DISTRIBUTION

• Points to Note
• At origin (y0) GG0
• Circulation varies elliptically with distance y
  along span
• At wing tips G(-b/2)G(b/2)0
• Circulation and Lift ? 0 at wing tips

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SPECIAL SOLUTIONELLIPTICAL LIFT DISTRIBUTION

• Elliptic distribution
• Equation for downwash
• Coordinate transformation ? q
• See reference for integral

Downwash is constant over span for an elliptical
lift distribution Induced angle of attack is
constant along span Note w and ai ? 0 as b ? 8
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SPECIAL SOLUTIONELLIPTICAL LIFT DISTRIBUTION
We can develop a more useful expression for
ai Combine L definition for elliptic profile
with previous result for ai Define AR because
it occurs frequently Useful expression for
ai Calculate CD,i
CD,i is directly proportional to square of
CL Also called Drag due to Lift
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GENERAL LIFT DISTRIBUTION (5.3.2)

• Circulation distribution
• Transformation
• At q0, y-b/2
• At qp, yb/2
• Circulation distribution in terms of q suggests a
  Fourier sine series for general circulation
  distribution
• N terms
• now as many as we want for accuracy
• Ans are unkowns, however must satisfy
  fundamental equation of Prandtls lifting-line
  theory

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GENERAL LIFT DISTRIBUTION (5.3.2)

• General circulation distribution
• Lifting line equation
• Finding dG/dy
• Transform to q
• Last integral has precise form for simplification

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GENERAL LIFT DISTRIBUTION (5.3.2)

• Evaluated at a given spanwise location, q0 is
  specified
• Givens
• b wingspan
• c(q0) chord at the given location for evaluation
• The zero lift angle of attack, aL0(q0), for the
  airfoil at this specified location
• Note that the airfoil may vary from location to
  location, and hence the zero lift angle of attack
  may vary from location to location
• Can put twist into the wing
• Geometric twist
• Aerodynamic twist
• This is one algebraic equation with N unknowns
  written at q0
• Must choose N different spanwise locations to
  write the equation to give N independent equations

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WING TWIST
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GENERAL LIFT DISTRIBUTION (5.3.2)

• General expression for lift coefficient of a
  finite wing
• Substitution of expression for G(q) and
  transformation to q
• Integral may be simplified
• CL depends only on leading coefficient of the
  Fourier series expansion (however must solve for
  all Ans to find leading coefficient A1)

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GENERAL LIFT DISTRIBUTION (5.3.2)

• General expression for induced drag coefficient
• Substitution of G(q) and transformation to q
• Expression contains induced angle of attack,
  ai(q)
• Expression for induced angle of attack
• Can be mathematically simplified
• Since q0 is a dummy variable which ranges from 0
  to p across the span of wing, it can simply be
  replaced with q

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GENERAL LIFT DISTRIBUTION (5.3.2)

• Expression for induced drag coefficient
• Expression for induced angle of attack
• Substitution of ai(q) in CD,i
• Mathematical simplification of integrals
• More simplifications leads to expression for
  induced drag coefficient

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GENERAL LIFT DISTRIBUTION (5.3.2)

• Repeat of expression for induced drag coefficient
• Repeat of expression for lift coefficient
• Substituting expression for lift coefficient into
  expression for induced drag coefficient
• Define a span efficiency factor, e, and note that
  e 1
• e1 for an elliptical lift distribution

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VARIOUS PLANFORMS FOR STRAIGH WINGS
Elliptic Wing
Rectangular Wing
cr
ct
Tapered Wing
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INDUCED DRAG FACTOR, d (e1/(1d))