An asymptotic linear stability theory for channel flow for quantitative comparison with numerical solutions
We consider the linear stability of flows with two and three dimensional disturbances.
Our main focus is the application of matched asymptotic expansions. The model used
throughout is plane Poiseuille flow. The stability properties of this flow have been
analysed before, both analytically and numerically.
Previous applications of matched asymptotics have been developed from two distinct
theories, one relating to the upper branch of the neutral curve and the other
to the lower branch. Our new analytical approach, still based on matched asymptotic
expansions, allows us to develop a single rational asymptotic theory to cover all
waves. This theory enables the construction of a complete neutral curve including both
branches and the kink in the upper branch. Also we investigate the maximum spatial
and temporal growth rates. All asymptotic results are compared against numerical
results obtained using Chebyshev polynomials.
We are able, using the numerical results, to define regions where our asymptotic
theory works well and where it fails, for both two and three dimensional disturbances.
As expected for a theory based around negative powers of the Reynolds number, our
theory produces accurate quantitative predictions for large Reynolds numbers, but not
for those of more relevance to experiment, e.g. for R < 100,000. Throughout, the
asymptotic temporal calculations are considerably less accurate than the spatial calculations.
We find incorporating a small empirical adjustment improves the quantitative
predictions for two dimensional spatial calculations. It reduces the error in the estimate
of the critical point from 18.5% to 0.69%. The adjustment is not as effective for three
Whilst our results are poor for plane Poiseuille flow, our theory should enable the
addition of other effects, such as non-linearity and nonparallel terms to be included,
which are important in boundary layer flows.