Page 379 - Computational Fluid Dynamics for Engineers

370 12. Compressible Navier-Stokes Equations

dW? d ON
8Qy
0 0 0
1 0 0
Vy
-[M] + [N y] = 4
> 0 0
-(! - &V - (i - £W - &f a - &)« (! - &)« £ J
(12A.5)
dW 2
[N} =
~dQ~y
0 0 0
(12A.6)
1 0 0
™ = - e 0 4 0
2
2
-(I - £) v - (l - £)u - ^ f (i - £)u (! - £)* £

Appendix 12B
Treatment of he Region Close o he Boundaries
t
t
t
for Eq. (12.5.4)
For the region which has at least one boundary side, Eq. (12.5.4) is approximated
differently from Eq. (12.5.5),

On the bottom subdomains i ? ^ , close to the wall j = 1
9Qi,l _ ^i+1/2,1 _ E i-l/2,l F i,l+l/2 _ -^2,1-1/2
dt Ax Ay
E
1 ( v)i+\/2,\ ~ ( ^ ) i - l / 2 , l ( ^ ) z , l + l/2 ~ ( ^ ) z , l - l / 2
+
Re~ Ax Ay
Note that 1 - 1/2 => 0

£, 2 + 1,1 — ^ 7 - 1 1 2 ^ °
'
^ 2 - 1 , 1
at 2Ax Ay
1 C^v)i+1/2,1 _ (Ev)i-1/2,1 (^v)z,l + l/2 _ (^v)i,0
+ +
Re Ax zAy

^2+1/2,1+1/2 ~~ ^2+1/2,1-1/2
(%)z+l/2,l
Ay
(^2 + 1,1 + 1 + ^2,1 + 1 + ^2+1,1 + ^2,l)/4 ~ ( ^ + 1,1-1/2 + ^ 2 , l - l / 2 ) / 2
Ay
K + 1 , 1 + 1 + ^2,1 + 1 + ^2+1,1 + ^2,l)/4 - (^2+1,0 + U ii0)/2
Ay