Page 367 - Computational Fluid Dynamics for Engineers

358 12. Compressible Navier-Stokes Equations

2
yn+1 = yn + <L Vn + Q(At )
At
and, with the chain rule and noting that V™ is a function of Q and Q x

yn+1 = yn + Atf
~~dQ~~dt + dQ x~dt
(12.4.7)

where
(12.4.8)

Analogous to Eq. (5.4.7b), we can write
Ayn = pn AQn + Rn QU + 0(^2)
A
or
AV? = (P n - K)AQ n + (RAQ)x (12.4.9)
Similarly, noting that AW? is a function of Q and Q y, we can write

AW? = (M n - N£)AQ n + (NAQ)% (12.4.10)

(12.4.11)
\8Q> \8QJ
1
Since AV^ and AW™ are functions of Q, Q x and Q y, the terms

-AV^ -AW?

in Eq. (12.4.6) represent mixed derivatives and are approximated by

1 2
^-AV£ = dx + 0(At ) (12.4.12a)
dx
-^AV?-
1 1 2
^-AW? = —AWl ' + 0(At ) (12.4.12b)
dy dy
Equations (5.4.7a), (12.4.2), (12.4.9), (12.4.10), and (12.4.12) are now substi-
tuted into Eq. (12.4.1) to yield
6 At d P-R x
A-
I 1 + £ I dx Re Re Ox
d ( M-JV, 1 d
r
y AQ '
Redy
6 At d_ (AY?- 1 \ ^ d^ (AW?- 1 (12.4.13)
1 + C dx \ Re + dy Re
At d En , V? + V?\ 9 n W? + W 2 W
l + £[dx\ i Re J dy -F + Re
n l
+ 1 + ^ AQ ~