Page 382 - Computational Fluid Dynamics for Engineers

Appendix 12B 373

^)l+l/2,j ~ A x / 2 ~ * A x

_ u I+l/2j+l/2 ~ u I-l/2,j+l/2
(U x) Ij+1/2 ~ A x
_ (U 1+ij+i + U!+ij)/2 - Q/J+l + U!j + ii/-lj+l + Ui-ij)/4
~~ Ax

_ U I+l/2,j-l/2 ~ u I-l/2,j-l/2
{u x)
U-l/2 ~ A x
u
( l+i,j + ^J+i,j-i)/2 - (^/j + uij-i + ^J-i,j + ixj-i,j-i)/4
=
~ Ax
On he four corners
t
;
b
Left-bottom corner 4?i,i, close t o oth walls, i = 1, j = 1
<9Ql,l _ ^1 + 1/2,1-^1-1/2,1 ^1,1+1/2 ~ ^1,1-1/2
<9t Zix ziy
1 ( ^ ) l + l/2,l ~ (^)l-l/2, l (^)l,1 + 1/2 ~ ( ^ ) l , 1-1/2
+
Re" Ax Ay
Note that 1 - 1/2 => 0 and 1 - 1/2 =» 0

r
^ 1 + 1,1+^1,11
o ^ / •^1 + 1,1+^1, r?? N \ / ^ i , i + i+-Fi,i 77» N
R
1
oQl,l / — • ~2 • ^ 0 , 1 x / — l 2 ^ ^°
" -
at zlx 4tf
1+1/2,1 _ (^)o,l , (^)l,1 + 1/2 ~ (^)l,0
+ Re Ax + Ay
Right-bottom corner !?/,!, close t o wall and outflow, i — I, j — 1

dQi,i _ E I+1/2,1 - El-1/2,1 F I,l + l/2 ~ ^7,1-1/2
dt Ax Ay
F
(By) 1+1/2,1 - (Ey)i-i/2,i ( v) 1,1+1/2 ~ (^v) 1,1-1/2
+ Re Ax Ay

Note hat/ + 1/2 => / + 1 and 1 - 1/2 => 0
t
oQi,i = _ I /+i,i ~ 2 \ _ / 2 ~ ^/,o
^
^ I Ax J \ Ay

(Ev)i+i,i - (Ev)i-i/2,i (Fv) 1,1+1/2 ~ (Fv)l,0
+ Re Ax + Ay