Page 213 - Computational Fluid Dynamics for Engineers

6.6 Applications of the Panel Method 201

6.6.3 Multielement Airfoils

As was discussed in Section 1.3, the maximum lift coefficient, (c/) max , has an
important impact on the performance of an airplane. For a complete flight
vehicle, the maximum lift coefficient determines the stalling speed of the aircraft.
To show this, consider a steady equilibrium flight where the lift L equals the
aircraft weight, w. Thus,
L = w = l -QVlSc t (6.6.4)

This means that the lowest possible speed of the aircraft, the stalling speed,
Vstalh occurs when the lift coefficient is maximum, that is,

-
^ - Vslb (665)

Thus, it is very important to increase the maximum lift coefficient of an airfoil
in order to obtain either a lower stalling speed or a higher payload weight at
the same speed.
Since for an airfoil (c/) max is a function of its shape at a specified Reynolds
number, it is necessary to resort to some special measures in order to increase
its (q) m a x beyond its value fixed by the airfoil shape and Reynolds number.
Such special measures include the use of flaps and/or leading-edge slats; these
are referred to as high-lift devices or multielement airfoils. Their flowfields and
section lift and moment coefficients can be calculated with modifications to the
panel program, as discussed in some detail below.
For a multielement airfoil configuration at an angle of attack a, each element
is represented by a finite number of panels separately. Assuming that the total
number of elements is TV, and the number of panels for the Z-th element is n/,
the source strength qji is taken to be constant over the j-th panel of the Z-th
element and the vorticity strength 77 to be constant on all panels of the Z-th
element. The flow tangency condition, similar to Eq. (6.4.14), can be written
2
2
for i = 1, , . . . , rifc, k — 1, , . . . , N as
N ni N ni
E E AikjiQji + E T i E B ?kjl + ^ o sin(a - 9 ik) = 0 (6.6.6)
1=1 3 = 1 1=1 3 = 1
The Kutta conditions for each element are similar to that given by Eq. (6.5.3)
and can be written for k = 1, , . . . , N with m = n^ as
2
N ni N ni
\Bikjl + B mkji) , .
1=1 j=i l=i j=l [v.v.t)
= -Voo cos(a - 9 lk) - VQO cos(a - # m/c )

Again, the Gaussian elimination method is used to solve for the M (=
n
J2i=i l + N) unknowns qji, r\ from the M equations given above.