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256 8. Stability and Transition

8.4 C o m p u t e r P r o g r a m S T P

We now describe a stability-transition program (STP) for calculating the neu-
tral stability curves and transition of two-dimensional flows based on spatial
amplification theory and the numerical method discussed in Sections 8.2 and
8.3. The program (Appendix B) requires that the boundary-layer velocity pro-
files be calculated with the boundary-layer program of Chapter 7 so that u
and u" can be used at each ^-station as a function of r\. For convenience, the
velocity uo and length L scales in the Orr-Sommerfeld equation are chosen to
correspond to
/ i/x x
= u e, L = J— = - = (8.4.1)
u 0
u
V e VRx
with Reynolds number R now given by

R = ^ = y/1£ (8.4.2)

With this choice, the boundary-layer grid of the velocity profiles can be used
in the solution of the Orr-Sommerfeld equation. This means, that u and u" in
the stability-transition program are related to the output of the boundary-layer
program by
u = f, u" = f" (8.4.3)

and that the output of the boundary-layer program can be arranged to calculate
u" once the solutions of the boundary-layer equations are known. The parameter
f" can be obtained either by differentiating f" with respect to r\ or from the
finite-differenced momentum equation, Eq. (7.3.18), which for laminar flow can
be written as

2 n l 1
v'j = U"T = -*i(fv)] + a 2(u j) - a n(v?- f? - f^v?) + R]' (8.4.4)

where i?™ _1 is given by Eq. (7.3.20a).
With boundary-layer profiles known at each x-station, the stability calcula-
tions can be started at any x-station where the critical Reynolds number i? cr
is less than the local Reynolds number used in the boundary-layer calculations.
For external flows, an estimate of R$* can be obtained from Fig. 8.2 with R$*
known from the boundary-layer calculations and included in the output sub-
routine.
The calculations for transition are first performed for a neutral stability
curve at the specified x-location where R (~ yfR^) is known. The calculation
of uo and a requires initial estimates. A convenient procedure to achieve this is
the continuation method discussed in [3] but to retain a comparatively simple
program, this is not part of the present description. The initial estimates of UJ
and a for a given R can be obtained for Blasius flow.