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Title: An asymptotic linear stability theory for channel flow for quantitative comparison with numerical solutions
Author: Rogers, Stuart John.
Awarding Body: University of Keele
Current Institution: Keele University
Date of Award: 2005
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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 dimensional disturbances. 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.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available