Page 371 - Computational Fluid Dynamics for Engineers

362 12. Compressible Navier-Stokes Equations

dE OF 1 (dE v dF v
/ / dx dy + Re \ dx + dy dxdy
i ,3
Vj + l/2 x i + l/2
dE OF 1 (dE v dF v
• I I dx dy Re \ dx dy dxdy
2 / 7 - 1 / 2 ^ - 1 / 2
Xi+1/2
Vj + l/2
E
j ( i+i/2 ~ Ei_ 1/2)dy - j (Fj+i/2 - F j_ l/2)dx
yj-1/2 c i - l / 2
x
/ yj+1/2 i + l/2 \
1
d
E
+ / (^Wl/2 - vi-l/2)dy + / (^j+1/2 _ F vj-l/2) 2
R^
Vi-1/2 ^i-1/2 /
E
= ~(Ei+i/2,j - i-i/2,j)Ay - (Fij+x/2 - F itj_ 1/2)Ax
E _ F x
+ p ^ [(^m+l/2j - vi-l/2j)Ay + (^zJ+1/2 viJ-l/2)^ \
(12.5.2b)
We then obtain
dQi 7
- ^ ArZh/ = -(E i+1/2J - Ei_ 1/2j)Ay - (F iJ+1/2 - F^_ 1/2)Ax
E
E
( vi+i/2j ~ vi-i/2j)Ay + (F vij +1/ 2 - F vij_ 1/2)Ax\ (12.5.3)
+ Re
Dividing both sides of Eq. (12.5.3) by Ax Ay results in a system of ordinary
differential equations, applicable to the interior region i?^j including the region
close to the boundaries.
E
dQ E i+l/2J ~ i-l/2j F iJ+l/2 ~ F iJ-l/2
1,3
dt Ax Ay
E
E
E
E
( v)i+1/2J ~ ( v)i-l/2,j ( v)i,j+l/2 - ( v)ij-l/2
+ Re Ax + Ay (12.5.4)
We first discuss the solution for the interior region Q^ and approximate Eq.
(12.5.4) by
E
dQ E i+l,j — i-lJ Fjj + i — Fjj-i
1,3
dt 2Ax 2Ay
F
F
E
i+l/2J ~ ( v)i-l/2,j ( v)ij+l/2 ~ ( v)iJ-l/2
(12.5.5)
+ Re Ax Ay
The elements in the dissipation terms can be approximated by the following
central differences.
^i+l,j ^ij
[u : x)i+l/2J {Ux)i-1/2J - ^
~ Ax