Page 368 - Computational Fluid Dynamics for Engineers

12.4 Beam-Warming Method 359

where / is the unity matrix given by

1 0 0 0
0 1 0 0
(12.4.14)
0 0 1 0
0 0 0 1

and the Jacobian matrices A, P, R, R x, B, M, iV, N y are given in Appendix
12. We note that in Eq. (12.4.13)
M-Ny
- I B -
dx V Re Redx J Re
(12.4.15a)
are equivalent to
d_ P-R* J_d_ 7] 5 M-Ny 1 <9 \ 1
A R)AQ [(-
dx Re Redx" Re
(12.4.15b)
The left-hand-side of Eq. (12.4.13) can be factored and expressed in the form

OAt
[I} + 2 [i?]"
l + £ Redx
9At n ^ 2
[/] + I ^ 7 ^ W n ZiQ"
+ £ Re<9j/ (12.4.16)
d
At
-{ 1 + C r - ^ M ( - ^ "
&
1
OAt 1 I^-'I + IW- ! £ 7 1 - 1 '
+ l + £Re + :(zAQ
The solution of Eq. (12.4.16) is obtained in two steps
Step 1:

9 At d ( u] _ [P] - [R x 1 d 2 n-1/2
[I] [Al 2 [R] n zAQ
die \ Re Re dx
At
(12.4.17)
1 + C dx \ Re J dy Re
6At 1 d_ 1 | ^r') n - l i
+ l + £Re dx (Avr ) + ( + (4Q
Step 2:

5 [M] n n 1 2
M + ARI - W r>„ a„,2 ^ J Z\Q = AQ ~ /
2
Redy
(12.4.18)