MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS

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MAE 3241 AERODYNAMICS AND FLIGHT MECHANICS

• General Lift Distribution
• Mechanical and Aerospace Engineering Department
• Florida Institute of Technology
• D. R. Kirk
• Summer 2006

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SUMMARY PRANDTLS LIFTING LINE THEORY (1/2)
Fundamental Equation of Prandtls Lifting Line
Theory Geometric angle of attack, a, is equal to
sum of effective angle of attack, aeff, plus
induced angle of attack, ai Equation gives
value of Downwash, w, at y0 Equation for
induced angle of attack, ai, along finite wing
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SUMMARY PRANDTLS LIFTING LINE THEORY (2/2)
Lift distribution per unit span given by
Kutta-Joukowski theorem Total lift, L Lift
coefficient, CL Induced drag, Di Induced
drag coefficient, CD,i
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SUMMARY ELLIPTICAL LIFT DISTRIBUTION (1/2)
G/G0
y/b

• Points to Note
• At origin (y 0) G G0
• Circulation and Lift Distribution vary
  elliptically with distance, y, along span, b
• At wing tips G(-b/2) G(b/2) 0
• Circulation and Lift ? 0 at wing tips

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SUMMARY ELLIPTICAL LIFT DISTRIBUTION (2/2)
Downwash is constant over span for an elliptical
lift distribution Induced angle of attack is
constant along span for an elliptical lift
distribution Total lift Alternate expression
for induced angle of attack, expressed in terms
of lift coefficient Induced drag
coefficient For an elliptic lift distribution,
the chord must vary elliptically along the span ?
the wing planform is elliptical in shape
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GENERAL LIFT DISTRIBUTION (1/4)
Fourier sine series expansion for circulation,
G(q) For more details see Principles of
Ideal-Fluid Aerodynamics (Karamcheti) The
unknowns are coefficients An, however they must
satisfy fundamental equation of Prandtls lifting
line theory Next step is to insert expression
for G(q) into fundamental equation of Prandtls
lifting line theory Solve for dG/dy
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GENERAL LIFT DISTRIBUTION (2/4)
Substitute expression for G(q) and dG/dy into
fundamental equation of Prandtls lifting line
theory
Last term on the right (integral term) is a
standard form and may be simplified as
Equation is evaluated at a given spanwise
location (q0), just as fundamental equation of
Prandtls lifting line theory is evaluated at a
given spanwise location (y0) Only unknowns in
equation are Ans Written at q0 equation is 1
algebraic equation with N unknowns Write equation
at N spanwise locations to obtain a system of N
independent algebraic equations with N unknowns
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GENERAL LIFT DISTRIBUTION (3/4)
Lift coefficient can be found once G(q) is
known CL depends only on leading coefficient
A1 Induced drag coefficient, which requires and
expression for the induced angle of attack,
ai Expression for induced angle of attack Last
term on right side is a standard form (see
handout)
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GENERAL LIFT DISTRIBUTION (4/4)
Substitute results into expression for induced
drag coefficient Expression may be simplified
and expressed in terms of AR Further
simplification Note that d 1 and if d 1,
recover expression for an elliptical wing e
span efficiency fact Note that e 1 and if e
1, recover expression for an elliptical wing