Page 260 - Computational Fluid Dynamics for Engineers

250 8. Stability and Transition

Jr [ Ji (8.2.14b)
^ 0 da r) \0a r
where
4 dUX (dfiY (BUY/dlS" (a2 i4c)
° - m) m - igj m -

To evaluate the derivatives of f r and fi with respect to a r and CJ, we need only
to differentiate Eq. (4.4.29) and, since the vector r*is independent of a r and a;,
we get:

We shall refer to the above equations as the variational equations of Eq. (^.J^.29)
with respect to a r and u, respectively. Thus, to obtain the required derivatives,
we need to solve only two linear systems with the same coefficient matrix A
already computed and factored for Eq. (4.4.29). The vectors on the right-hand
side of Eqs. (8.2.15a) are determined from Eqs. (8.2.3), (8.2.5b) and (8.2.7). For
J
Eq. (8.2.15a) (ri)j = (r 2)j = 0 for 0 < j < , but (r 3)j and (r 4 )j for 1 < j < J
are given by
(r 2 {S2 m
*-' = (£:) *•*-* -

2
'
("»'-= (S)^» + 2 (£)*'-» (8216b)

{r )j <82iM)
' =-(£)/-' -

For Eq. (8.2.15b), again {r\)j = {r2)j = 0 for 0 < j < J, but with c\, C2 and
;
C3 being independent of to for j < J — 1, the coefficients {rz)j and {r±)j for
1 < j < J are given by
(r 3 )j-i = 0 (8.2.17a)

(r4) 1 2 8 8 2 17b
^ = (tl) M ( - - )

^' — (fO/' (8217c)
--

( r 4 ) j = ( 8 2 1 7 d )
- ( ^ ) / - ' - -