Page 207 - Computational Fluid Dynamics for Engineers

6.5 A Panel Program for Airfoils 195

so that,
l
a N+1J = A{j + A Nj, j = 1,2,..., N (6.5.4a)
N
B
6 5 4 b
B
aN+^N+1 = Y,( h + Nj) ( ' ' )
l
l
where now A\- and A N- are computed from Eq. (6.4.10) and B\- and B N- from
Eq. (6.4.11).
The components of b again follow from Eqs. (6.4.14) and (6.5.2). From Eq.
(6.4.14),
bi = —VQQ sin(a — 0i), i = 1,..., N (6.5.5a)
and from Eq. (6.5.3),

fr/v+i = —Kx) cos(a — 0\) — Foo cos(a — ON) (6.5.5b)
With all the elements of aij determined from Eqs. (6.5.2) and (6.5.4) and
the elements of b from Eq. (6.5.5), the solution of Eq. (4.5.23) can be obtained
with subroutine GAUSS in Table 4.2. The elements of x are given by
N+l
6 ? - D _ E a ? - D . N + ! , . . . , ! (6.5.6)
( i - i ) ij
j=i+l
where

k = 1,...,7V
(k) _ (fc-1) Oife (fc-1) j = fc+l,...,iV+l ^ c ; 7 \
a a
« ~ v (/c-i)%- ' z = fc + ,...,iV + l (,t).D.^aj
l
a
A:fc (0)
x —
zj "U
(fc-1) fc = l , . . . , i V
1
l
6 (*) = 6 ( f c - i ) _ ? L 6 ( * - ) ) t = fc + ,...,JV + l (6.5.7b)
U
(/c-l) fc
>?>=>.
6.5.1 M A I N Program
MAIN contains the input information which comprises (1) the number of panels,
TV, along the surface of the airfoil, NODTOT, (2) airfoil coordinates normal-
ized with respect to its chord c, x/c, y/c, [X(I) and Y(I)], and (3) angle of
attack a (ALPHA) in degrees. The panel slopes are calculated from Eq. (6.4.2).
The subroutine COEF is called to compute A and b in Eq. (4.5.23), subroutine
GAUSS to compute x, subroutine VPDIS to compute the velocity and pres-
sure distributions, and subroutine CLCM to compute the airfoil characteristics
corresponding to lift (CL) and pitching moment (CM) coefficients. (The drag
coefficient CD for irrotational, incompressible flows is always zero, as stated by
D'Alembert Paradox).