Title:

On the weakly nonlinear evolution of TollmienSchlichting waves in shear flow

The thesis essentially consists of two parts. In the first part we consider the evolution of weakly nonlinear, modulated disturbances in marginally unstable systems, and address the question of when the GinzburgLandau equation is relevant in describing such systems. In the second, we consider secondary instabilities of TollmienSchlichting waves in High ReynoldsNumber 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 GinzburgLandau equation describes the evolution of line disturbances and of Davey et al in proposing that the DaveyStewartson 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 nonlinearity. 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 finitetime 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 TollmienSchlichting 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 wellknown theory of the High ReynoldsNumber 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 nonlinear phaselocked resonant triad interactions and demonstrates that a central mode can interact with two oblique subharmonics to cause superexponential growth of the subharmonics.
