# Interfacial waves in shear flows

Chen, Michael James 2010 , 'Interfacial waves in shear flows', PhD thesis, University of Tasmania.

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## Abstract

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.

Item Type: Thesis - PhD Chen, Michael James The Author Copyright 2010 the Author – The University is continuing to endeavour to trace the copyright owner(s) and in the meantime this item has been reproduced here in good faith. We would be pleased to hear from the copyright owner(s). Available for use in the Library and copying in accordance with the Copyright Act 1968, as amended. Thesis (PhD)--University of Tasmania, 2010. Includes bibliographical references View statistics for this item