Title: MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS
1MAE 3241 AERODYNAMICS ANDFLIGHT MECHANICS
- Finite Wings General Lift Distribution
- April 2, 2007
- Mechanical and Aerospace Engineering Department
- Florida Institute of Technology
- D. R. Kirk
2PRANDTLS 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
3WHAT DO WE GET OUT OF THIS EQUATION?
- Lift distribution
- Total Lift and Lift Coefficient
- Induced Drag
4ELLIPTICAL 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
5SPECIAL 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
6SPECIAL 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
7SPECIAL 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
8GENERAL 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
9GENERAL LIFT DISTRIBUTION (5.3.2)
- General circulation distribution
- Lifting line equation
- Finding dG/dy
- Transform to q
- Last integral has precise form for simplification
10GENERAL 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
11WING TWIST
12GENERAL 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)
13GENERAL 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
14GENERAL 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
15GENERAL 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
16VARIOUS PLANFORMS FOR STRAIGH WINGS
Elliptic Wing
Rectangular Wing
cr
ct
Tapered Wing
17INDUCED DRAG FACTOR, d (e1/(1d))