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Title: On the weakly non-linear evolution of Tollmien-Schlichting waves in shear flow
Author: Jennings, M. J.
Awarding Body: University of Cambridge
Current Institution: University of Cambridge
Date of Award: 1998
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The thesis essentially consists of two parts. In the first part we consider the evolution of weakly non-linear, modulated disturbances in marginally unstable systems, and address the question of when the Ginzburg-Landau equation is relevant in describing such systems. In the second, we consider secondary instabilities of Tollmien-Schlichting waves in High Reynolds-Number boundary layer and channel flows. In §1.1 we give a brief background and review previous work in the study of modulated disturbances in marginally unstable parallel flows. In §1.2 we briefly outline the approach of Stewartson and Stewart in proposing that the Ginzburg-Landau equation describes the evolution of line disturbances and of Davey et al in proposing that the Davey-Stewartson equations describe the evolution of point disturbances, and propose that a small logarithmic change to the scaling is needed to account for the first effects of weak non-linearity. In §1.3 we give the modifications necessary for a line disturbance and find the resulting amplitude equations for Plane Poiseuille Flow. We show that in the subcritical case the solution terminates in a finite time singularity. In §1.4 and §1.5 we extend the analysis to point disturbances, and again show that a finite-time singularity is encountered for a subcritical system. In §1.6 we extend this analysis to obtain amplitude equations for three dimensional Poiseuille Couette flow, which we rescale to reduce to (essentially) the equations for two dimensional PPF. The implications of this revised analysis are described in §1.7. In §2.1, we give a brief summary and review previous work studying secondary instability of Tollmien-Schlichting waves in high Reynolds Number boundary layer and channel flows, particularly in order to attempt to explain the formation of subharmonic TS modes observed in many experiments. §2.2 summarises the well-known theory of the High Reynolds-Number lower branch of the neutral curve regime for a boundary layer, and describes the evolution of linear TS modes in the high frequency lower branch regime. §2.3 considers weakly non-linear phase-locked resonant triad interactions and demonstrates that a central mode can interact with two oblique subharmonics to cause superexponential growth of the subharmonics.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available