Phase II Final Report - NASA's Institute for Advanced Concepts

Chapter 3.0 Vehicle Design
3.3 Wing Aerodynamics
C()
φ
:=
J1 k()
φ
( ) ⋅ ( J1( k()
φ ) − Y0( k()
φ ))
+ 1i ⋅
Y1 k()
φ
+
( J1( k()
φ ) + Y0( k()
φ )) 2 + Y1 k φ
Y1( k()
φ ) ⋅ Y0( k()
φ ) + J1 k()
φ ⋅
( J1( k()
φ ) + Y0( k()
φ )) 2 + Y1 k()
φ −
( ) ⋅ ( Y1( k()
φ ) − J0( k()
φ ))
( ( ()) − J0( k()
φ )) 2
( ) J0( k()
φ )
( ( ) J0( k()
φ )) 2
...
For determination of finite wing correction factors, the following methodology is adopted:
a. Choose span locations where the correction factors and amplitude of the three dimensional
circulation function is to be determined. This span has been divided into twenty
one segments in units of semispan.
Cos(φ) = 0, 0.05, 0.10, ………, 1.0
b. Calculate values of normalized chord lengths, reduced frequencies, at these stations, and
then calculate the two dimensional circulation function at these span locations.
c. Calculate the coefficients A(n,φ), where n is like the harmonic number. Note that if the
wing flapping is symmetrical about the midspan, only odd values of n occur, while for
the asymmetrical case, even values of n will occur. Our case is symmetric, so odd values
of n have been taken as 1,3,5,7,9.
d. Solve the system of equations with complex coefficients to find Knj. Since there are
twenty span locations, and only five Knj values need to be found, only five span locations
will be used. This procedure has been solved by using Crout’s Method for solution
of equations with complex numbers.
e. Calculate the values of the three dimensional circulation function at all span locations.
f. Calculate the values of correction factors for flapping and pitching.
g. Add the real part of the correction term to F(φ) and imaginary part to G(φ) to get the
solution with finite span and planform correction
h. To get the total forces on the wing, the forces of all segments (span locations) must be
integrated and then divided by span. The solution obtained is asymptotic near the wing
root, so an assumption has been made based on propeller theory that the force coefficients
at the 75% span location are representative of the entire wing.
µ () φ
:=
π ⋅
k()
φ
⋅
J0( k()
φ ) − 1i ⋅ J1( k()
φ )
⎡⎣ ( J0( k()
φ ) − Y1( k()
φ )) − 1i ⋅ ( J1( k()
φ ) + Y0( k()
φ ))
⎤⎦
Ωh2()
φ :=
()
( ( ) Y1( k()
φ ))
4⋅
1i⋅
C φ
k() φ ⋅ J1 k()
φ −
⋅ e
1i ⋅ km ( φ )
⋅ 1i ⋅
k()
φ
⋅
cos () φ
⋅ hamp
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