Page 239 - Computational Fluid Dynamics for Engineers

228 7. Boundary-Layer Equations

(Wl)j = (n)j - (7ll)j(u>l)j-l - (7l2)j(t«2)j-l - (713)J(W 3 )J-1
(W2)j = (r2)j - (72l)j(^l)j-l - (722)j(«'2)j-l - (723)J(U>3).;-1 (7.4.15b)
(w 3 )j = (r 3 )j

In the backward sweep, fy is computed from the formulas given by Eq. (4.4.34).
With the definitions of 6j, Aj and WJ, it follows from Eq. (4.4.33a) that
Suj = {w 3)j (7.4.16a)

6vj = ^(fliih-exia^j {7AA6h)

Sfj = 6 1 - / t t » ) ^ (7.4.16c)
( a i i ) j
where
ei = (wi)j - (a 12)j8uj
e2 = (w 2 )j - (0^22)JSUJ

The components of 6, for j = J - 1, J - 2 , . . . , 0, follow from Eq. (4.4.33b)
Sv. = (an)j[{w 2)j + e 3 (a22)j] - (^2i)j(^i)j - e 3(a2i)j(<xi2)j ( 7 4 1 7 a )
3
A 2 \ - • )

h l
j± .
SUJ = — ^ f i v j - e 3 (7.4.17b)
2
^ . = ( ^ I ) J ~ (<*i2)jSuj ~ (Qi3)jg^- (7.4.17c)

where
^3 = (w 3 )j - <^j+i + -^Svj+i

= ( a 2 1 ) > 1 2 ) , ^ ± i - (a 2 i),(ai3)j (7.4.17d)
A 2
7 ^
- - ^ ( « 2 2 ) j ( a i l ) j + («23)j(ail)j
To summarize, one iteration of Newton's method is carried out as follows.
The vectors fj defined in Eq. (4.4.30) are computed from Eq. (7.3.24) by using
the latest iterate. The matrix elements of Aj, Bj and Cj defined in Eq. (7.3.28)
are next determined by Eq. (7.3.25a) to (7.3.25f). Using the relations in Eqs.
(4.4.32) and (4.4.33), the matrices Fj and Aj and vectors Wj are calculated.
The matrix elements for Tj denned in Eq. (4.4.32a) are determined from Eq.
(7.4.12). The components of the vector Wj defined in Eq. (7.4.14) are determined
from Eq. (7.4.15). In the backward sweep, the components of 6j are computed
from Eqs. (7.4.16) and (7.4.17). A subroutine which makes use of these formulas
and called SOLV3 is given in Appendix B.