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Two studies in hydrodynamic stability : interfacial instabilities and applications of bounding theory Mandre, Shreyas

Abstract

Interfacial phenomena, viz. roll waves and elastic tremor, are considered in the first part, whereas in the second bounding theory is applied to double-diffusive convection and shear flows. Roll waves are investigated using shallow-water equations with bottom drag and diffusivity. We explore the effect of bottom topography on linear stability of turbulent flow, followed by an investigation of the nonlinear dynamics. Low-amplitude topography and hydraulic jumps are found to destabilize turbulent roll waves, while higher amplitude topography stabilizes them. The nonlinear dynamics of these waves is explored with numerical and asymptotic solutions of the shallow-water equations. We find that trains of roll waves undergo coarsening dynamics, however coarsening does not continue indefinitely but becomes interrupted at intermediate scales, creating patterns with preferred wavelengths. We quantify the coarsening dynamics in terms of linear stability of steady roll-wave trains. For elastic tremor, e.g. observed in musical reed instruments and vocal cords, a first principles explanation is given for the onset of these oscillations using linear stability theory. An analytical solution is built on the assumptions of thin-film flow and stiff elastic material and the criterion for the destabilization of natural elastic oscillations is derived. Acoustic excitation (e.g. organ pipes) is treated as an analogue, with compressibility playing the role of elasticity, with similar mechanism possibly at work. In double diffusive convection, the flux of the unstably stratified species is bounded using the background method in the presence of opposite stratification of the other species. In order to incorporate a dependence of the bound on the stably stratified component, Joseph's (Stability of fluid motion, 1976, Springer-Verlag) energy stability 112 analysis is extended. At large Rayleigh number, the bound is found to behave like R[sub T][sup ½] for fixed ratio R[sub S]/R[sub T], where R[sub T] and R[sub S] are the Rayleigh numbers of the unstably and stably stratified components, respectively. The energy stability of plane Couette flow is improved for two dimensional perturbations. The energy is chosen from a family of norms so as to maximize the critical Reynolds number. An explicit relation for the critical Reynolds number is found in terms of the perturbation direction.

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