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258 8. Stability and Transition
(J) where the profiles are computed. With the specification of U(J) (= ') and
/
V(J) (= ") as a function of ETA(J) (= nj), UUDP(J) (= u'<) are calculated,
/
by differentiating V(J) with respect to 77.
8.4.3 Subroutine CSAVE
This subroutine is used to obtain the solutions of the Orr-Sommerfeld equation
for a given set of a and u when the neutral stability curve is required or a r and
a,i for the determination of the location of the onset of transition. The standard
problem refers to the solution of Eq. (4.4.29) subject to the boundary condition
,f
that (f) (0) (= s 0) is equal to 1. The definitions of {c\)j to (04)j in the Orr-
Sommerfeld equation, as well as the edge definitions, are given by Eqs. (8.2.4)
and (8.2.6), respectively, and the fj terms by Eqs. (8.2.3) and (8.2.7). All values
of fj are zero except for (r2)o which is equal to 1 because of the requirement
that s 0 = 1.
This subroutine also contains the coefficients of the variational equations of
the standard problem, Eq. (4.4.29), with respect to a, u and i?, together with
the right-hand sides of these equations as given by Eq. (8.2.16) for those with
respect to a, Eq. (8.2.17) for CJ, and Eq. (8.2.21) for R.
The variational equations are written with respect to a, a; and i?, but the
coefficient matrix A in Eq. (4.4.29) is the same. For this reason, it is only
necessary to define those (fj) terms that are not zero.
This subroutine also contains the block-elimination algorithm to solve Eq.
(4.4.29) with the procedure described in subsection 4.4.3. Note that AA(1,1,1)
(
and AA(2,2,1) denote a n ) 0 and (0:22)0 and correspond to the "wall" boundary
conditions.
8.4.4 Subroutine N E W T O N
For initial estimates of ALFA(= a r) and OMEGA(= CJ), this subroutine com-
putes the eigenvalues a and u according to the procedure described in Sec-
tion 8.2. The perturbation quantities DALFA(EE 8a r) and DOMEGA(= Sou) are
computed according to Eq. (8.2.14). Upon convergence of the iterations, the di-
mensional frequencies WSO(IX)(= a;*) are calculated and printed for that NX-
station, and for each corresponding frequency, IX, together with the values of a
and UJ which serve as initial estimates for the next NX-station (or i?), are stored.
Also stored are the values of UM(= / 0 ) , UMA(= df 0/da), UMO(= df 0/du),
UMR(= dfo/dR), UE and REY.
8.4.5 Subroutine N E W T O N I
In the calculation of transition, the amplification factor n is computed for a
dimensional frequency ou* according to Eq. (8.3.3) which in terms of the dimen-
sionless quantities used in the solution of the Orr-Sommerfeld equation with