Page 375 - Computational Fluid Dynamics for Engineers

366 12. Compressible Navier-Stokes Equations

and approximated by
= T^i - (Ti£ - Ti ti)/2 (12.6.8)
T w
with the known interior values. The density g w and (Et) w can be calculated
from Eq. (12.6.4).
-WwML Pw
lv L
J
W - oo
Qw = , (Et t w (12.6.9)
T w ' — 7 - 1
The wall temperature T w can also be computed from
V?
CpT w = CpTi + —±- = const. (12.6.10)
2T™
where
_jR_
p = QRT (12.6.11a)
!
7 - l
2
V 2 = u 2 + v , M 2 = - ^ , a 2 = jRT (12.6.11b)
2
a
Since Eq. (12.6.9) is in dimensional form, we express it in dimensionless form,

•Lw _ rp* li. _.*£. V?
J +
rp U>
-* oo J- oo ^-LooCp oo \ ^J-lCp
= II 1 + (7 i + { ^=AM 2
T a i /

or
( 7 - 1 )
r w = Ti 1 + Mi (12.6.12)

Our studies for this particular problem show that both choices (Eqs. (12.6.8)
and (12.6.12)) of computing T w are the same.
On the line of symmetry, y = H, we set

9(gu) dE t
= v = 0, = 0, (12.6.13)
dy dy dy
The outflow boundary conditions at i = I, 0 < j < J are again obtained by
using second-order accurate extrapolation,

2
Qij = Qi-i,j ~ QI-2J
(12.6.14)
(QV)I,J = 2(gv) I- 1j - (gv)i-2j

(Et)ij = 2(E t)i-ij - {Et) 1-2,3