Page 380 - Computational Fluid Dynamics for Engineers

Appendix 12B 371

^-1/2,1+1/2 - ^-1/2,1-1/2
(%)i-l/2,l =
Ay
( i y + 1 + ^-1,1+1 + U iti + ^ - l , l ) / 4 - ( ^ ! - i / 2 + ^i-1,1-1/2)/ 2
Ay
(^,1+1 + ^-1,1+1 + i y + z-i,i)/4 - I^Q + Ui-i$)/2
^
(
Ay
U u
%A - i,l-l/2 __ Ui fi - Uifl
[U> y)i,l-l/2
Ay/2 ~ Ay
t
s
On the op ubdomains i? z ,j, close to he line of ymmetry, j = J
t
s
E
dQi,J _ E i+1/2,J ~ i-1/2,J F i,J+l/2 - F i,J-l/2
dt Ax Ay
1 (Eyh+l/^J- ( v)i-1/2,J (^v)i,J+l/2 _ (^)i,J-l/2
E
+ +
Re Ax Zly
The line of symmetry is at J + 1/2. Assume the ghost points are at J -h 1* and
J + 2*. Then, from symmetry conditions, we can write

Qi,J+l* = Qi,J, Qi,J+2* = £i,j-i
^2,J+1* = ^Z,Jj ^Z,J + 2* = ^2,J-1
^,J+1* = ~ ^ , J , ^i,J+2* = —^i,J-l

Pz,J+l* = Pi,J 5 Pi,J+2* = Pi,J-l

Note that J + 1/2 =^ J + 1

^i.J-i+Fi.j
E
OQ iJ i+l,J ~~ i-l,J E i,J+l
at 2Z\x ^y
1 (^)z+l/2,J _ (Ev)i-l/2,J (Fv)i,J+l - {Fv)i,J-\/2
+ +
R^ Ax Ay

U i+1/2,J+1* - U i+1/2,J-1 _ (^z+l,J + ^i,j) ~ (^i+1,J-l + V>i,J-l)
U
( y)i+1/2,J =
2Ay 44y
( \ ^z,J+l* - Uij
K)i,j+i/2 = ^ = 0

^z+l/2,J+l* - ^z+l/2,J-l _ ~(^+l, J + ^z,j) - (i/i+l,J-l + Ui,J-l)
v
( 2/)i+l/2,J
2^y 4^y