Global instability in strongly nonhomogeneous systems
A number of fluid-dynamical systems are considered with regard to their global stability. First, a simple linearised Ginzburg-Landau-type system with periodic coefficients is shown to be absolutely unstable under certain conditions. Some two-fluid channel-flow problems with application to stability of flow through pipes and ducts and to cavity tones are considered next. Firstly, flow through a channel with periodically-deformed walls is shown, using Floquet's theorem, to be absolutely unstable for any case other than that of homogeneous coefficients. This is due to a three-wave interaction. A finite-range variation in the channel shape is then considered. It is found that, for a contraction in the channel, global instability appears for an obstacle of any length. For an expanded channel section, a large length is required for global instability to be observed. As a simplification, a system with two fluids separated by a solid wall containing a finite-length aperture is considered. A finite number of unstable global modes are found and some analysis is done for the infinite-range limit. Next, two systems are investigated of which, if infinite, all harmonic wave like perturbations would be considered to be stable, but in which transient growth is observed in the temporal evolution of disturbances. It is found that the equivalent finite-range formulation shows global instability due to feedback between the ends. The first system exhibits such behaviour due to transient growth of a spatial nature, while the second exhibits purely temporal transients. Finally, a system governed by a Benjamin-Ono equation is considered. It is shown that, for a homogeneous problem, the greatest contribution to the instability of the system can come from a branch point at the origin in the wavenumber plane. Some problems with numerical computation of stability for problems with a Cauchy-type integral are then discussed.