Interfacial waves in shear flows

Interfacial waves in fluid flow have widespread applications in meteorology, oceanography and astrophysics. Several flows are considered in detail in this thesis. A key feature of these flows is that they each possess shear and they are each studied with a view to determining the behaviour of interfacial waves in the flow. The thesis begins with an investigation of a steady three-layer intrusion flow, in which all three layers are in motion with different speeds relative to the observer. Shear is present in the middle layer, and the lowest fluid may even move oppositely to the upper two (so giving an exchange flow). Two thin interfaces are present, above and below the moving middle layer. A linearized analysis is presented for small wave amplitudes. Non-linear periodic solutions are then obtained using a spectral method based on Fourier series, and these reveal a range of non-linear phenomena, including limiting waves, multiple solutions and resonances. The techniques used to study the intrusion flow are then extended to allow for time-dependent behaviour, and applied to the Kelvin-Helmholtz instability for an inviscid fluid. The viscous version of this problem is also considered. Here, two bounded fluid layers flow parallel to each other with the interface between them growing in an unstable fashion when subjected to a small perturbation. In this problem, and the related problem of the vortex sheet, there are several phenomena associated with the evolution of the interface, notably the formation of a finite time curvature singularity for inviscid fluids and the `roll-up' of the interface when viscosity is included. Two contrasting computational schemes will be presented. A spectral method is used to follow the evolution of the interface in the inviscid version of the problem. This allows the interface shape to be computed up to the time that a curvature singularity forms, with several computational difficulties overcome to reach that point. A weakly compressible viscous version of the problem is studied using finite difference techniques and a vorticity-streamfunction formulation. The two versions have comparable, but not identical, initial conditions and so the results exhibit some differences in timing. By including a small amount of viscosity the interface may be followed to the point that it rolls up into a classic `cat's-eye' shape. Particular attention is given to computing a consistent initial condition and solving the continuity equation both accurately and efficiently. The final problem studied in this thesis is a two-layer shear flow for inviscid and viscous fluids. Here, the layers flow between two horizontal walls and are buoyantly stable. Each layer contains a finite amount of shear and the horizontal velocity is specified such that it is continuous when unperturbed. The interface between the two layers is given a small sinusoidal perturbation and the subsequent response of the system is studied. Different solution techniques are employed for the inviscid and viscous flow. These both rely on linearizing the governing equations for each of these flows. As a consequence, several restrictions are placed on how the viscous flow may evolve; namely that as it develops the flow will not differ too greatly from the unperturbed flow. These assumptions are justified since standing wave behaviour is expected in the inviscid case. Solutions are presented for a variety of different values of the shear parameters and the way these parameter choices affect the interaction between vorticity and density in the viscous case is investigated in detail.