Page 180 - Computational Fluid Dynamics for Engineers

5.7 Convergence and Stability 167

e aAt = l + ^^[cos(k mAx)-l] (5.7.9)

Recalling that
. o fk rnAx\ l 1 — cos{k mAx)
sin ' —

Eq. (5.7.9) becomes
4aAt fcrn^SC
eaAt = 1 sin 2 / ^m (5.7.10)
2
{Ax) "" V 2
From Eq. (5.7.4)

aZ\£
= e (5.7.11)
pObL pi-Kiy-fi X
Combining Eqs. (5.7.4), (5.7.10) and (5.7.11), we obtain

_n+l
^m
a At\ 4aAt 2 / r^m.^X
1 - sin < 1 (5.7.12)
jAx)'
the requirement to have a stable solution. The factor
4azl£ . 2fk mAx\
1 — -——TTT in (5.7.13)
s
2
(Ar) V 2 ;
is called the amplification factor and will be denoted by G. In evaluating the
inequality in Eq. (5.7.12), two possible cases must be considered.
(l)If
r 4aAt . 2 (k mAx\
1 ~~ I A NO S 1 1 1 ^ < 1
then
4aAt . 2 / k mAx
sin > 0 (5.7.14)
(Ax) ~"~ V 2
Since 4aAt/(Ax) 2 is always positive, the condition expressed by Eq. (5.7.14)
is always satisfied.
(2) If
4mAtt
r 4aZ\ . o / J L A r M
o
(kmAx
s
1 — -r-:—TTT in '
2
(Ax) > - 1
then
4aAt 2 / rCm^%
sin' 1 < 1 (5.7.15)
For the above condition to hold,
aAt 1
(5.7.16)
— ^ 2
(AxY