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12.4 Beam-Warming Method 357

12.4 Beam—Warming Method

The Beam-Warming (B-W) method for the solution of the one-dimensional
time-dependent Euler equations, Eq. (5.1.2), was discussed in Section 5.4 and
its extension to the two-dimensional compressible viscous flows is discussed now,
following [3].
An obvious extension of Eq. (5.4.4) to a two-dimensional flow is,

5
AQ n = 6 At 6At 1 f ( A F » U ° (AF n
1 + e
At d_ ( En E%\ d_
+ dx\ Re/ dy F + Re~
1 + C
+ AQ n-\ (12.4.1)
1 + C
n
The first term on the right-hand-side of Eq. (12.4.1) AE is given by Eq. (5.4.7b)
and, similarly, the second term can be written as
2
AF n = BAQ n + 0{At ) (12.4.2)
where
*"(£* (12.4.3)

1
Since E™ and F™ are functions of Q, Q x and Q y, they can be written as
= Vi(Q,Q x) + V 2(Q,Q y) (12.4.4a)
E v
= W 1(Q,Q x) + W 2(Q,Q y) (12.4.4b)
F v
where

0 0
\\iu x
Vi V 2 = (12.4.5a)
fiv x flUy
HVV X + \\±UU X + ^i^M^pj-Tx fiVUy — ^fXUVy J

0 0
fiu y
Wi = w = (12.4.5b)
/J>V X 2 t^Vy
[fiUUy + zflVVy + J^fM£p- TTy
jJLUVx — ^)1VU X
Then
d 9
Arm A.rn 9 ATrn (12.4.6a)
-AE" V=-AV? + -AV?
(12.4.6b)
= l yAW? l yAW?
l yA K
As with Eq. (5.4.7a), we can write +