Page 342 - Computational Fluid Dynamics for Engineers

332 11. Incompressible Navier-Stokes Equations

Substituting Eq. (11.4.4) into Eq. (11.4.3) and noting that the D's are vec-
tors, leads to the following delta form

n
_/_ (dR\ r n
AD -R (11.4.5)
+
AT \dDJ
X
n
n+l
where AD n = D -D n and AT = T + -T n and / is the 3 x 3 identity matrix,
or in compact form
'BAD 71 = -R n (11.4.6)
where
_/_ (dR\ n (11.4.7)
zi7 + \dD)

t
11.4.2 Discretization of he Convective Fluxes
The analysis in Section 11.2 showed that the system is hyperbolic, and the
upwind method is applied as discussed in Section 5.5. The Jacobian matrices of
the convective flux E and F are

o p o 0 0 (3
A 9 E B - 9 F -
A 1 2u 0 B 0 v u (11.4.8)
= 0D = -dD~
0 v u 1 0 2v
and the eigenvalues A and eigenvectors X of A and B can be written as
iW
u 0 0
Ai = X^AXi = ,(1) 0 u + c\ 0
A (i) 0 0 u-c\\
(2)
A v 0 0
L (2)
A 2~-= X^ BX 2 = A 0 v + c 2 0
(2) 0 0
A v-c 2 (11.4.9)
6
o Clp " - Clp •
Xl 0 u(ci + u) + /3 u(ci-u) + P
= Wr
\_2f3 v{c\ + u) v(c\—u)
' 0 C20 -C2/3
X2 = -2/3 + v) - v)
W 2 0 v(c 2 + v) + (3 v(c 2-v) + (3}
u(c 2
u(c 2
where
c\ (11.4.10a)

and
(11.4.10b)