Instability and Energy Growth in Stratified Shear Flows

Abstract

The dynamics of stratified shear flows are extremely complex due to the interplay between shear and buoyancy effects. In addition, the breakdown from a laminar state into a turbulent one is a process that is far from completely understood, despite aspects of turbulence being studied for over a century, going back to at least Reynolds (1883). A key question in understanding the breakdown is how sufficient energy growth is achieved, with both linear modal instability and transient growth as possible candidates. The introduction of stratification can affect the strength of the instabilities and transient growth by suppressing vertical motions, but it can also introduce new instabilities as a consequence of the supported gravity waves. This thesis explores aspects of modal instability, transient energy growth and nonlinear evolution of stratified shear flows, starting with some background material and key concepts in chapter 1. The linear stability of stratified, rotating plane Couette-Poiseuille flow is the focus of chapter 2, which constitutes the first investigation to combine all of these physical effects with the mixed shear flow. Two well-known instability types are found for the unstratified flow (Tollmein-Schlicting and centrifugal instabilities), which for the most part do not interact as they favour different forms of two-dimensionality. A new three-dimensional instability is found which violates Rayleigh's criterion beyond the expected shear instability. Stable stratification brings new instabilities from resonances between the supported waves. In addition, a stratification modified centrifugal instability is identified, with the three-dimensional variety violating the stratification modified Rayleigh's criterion. Although the resonance instabilities have much weaker growth rates for the most part, the centrifugal instabilities do not appear for cyclonic rotation so the resonance instabilities can dominate. Using the spectral solver Dedalus, chapter 3 presents direct numerical simulations of the saturation, nonlinear evolution and final state of some of the instabilities that are identified in chapter 2, for varying Reynolds numbers. The computational resources only permit this for the two-dimensional stratification modified centrifugal instabilities. The flow remains two-dimensional at late times for small to moderate Reynolds numbers. The dominant wavenumber of the final states match that of the initial condition when Re is small, but increases in a staircase pattern with Re, with the temporal behaviour of the final states fairly unpredictable, switching between steady and oscillating states when Re becomes larger. Even for larger Reynolds numbers, the flow remains largely two-dimensional with only a 4% deviation in the late time kinetic energy between 2D and 3D simulations at Re=100,000. Chapter 4 switches focus to transient growth mechanisms, and first recaps the famous unbounded constant shear model used by Kelvin (1883). Asymptotic results at large Reynolds number reveal that the Orr mechanism reaches its optimal value over the 'slow' viscous timescale when wavenumbers are allowed to be small, rather than the 'fast' timescale often associated with the Orr mechanism when moderate wavenumbers are considered. A streak field is then built in which supplements the well-known lift-up and Orr processes with the recently identified push-over mechanism, and a simplified version of the model reveals that three-dimensional perturbations can grow through a mechanism relying on both push-over and Orr, producing a gain that scales exponentially with the Reynolds number at early times. Restricting to more physically appropriate wavenumbers still allows a large growth, scaling with the exponential of the square root of Reynolds number at 'intermediate' O(√Re) times. The key features of the growth are reproduced by the full numerical data supporting the suggestion that this is in fact the mechanism behind the observations in the literature: both the Orr and push-over mechanisms are crucial for large early time transient growth. Stratification is introduced to the streaky unbounded shear flow model in chapter 5 to investigate whether it simply has a damping effect or introduces more complex behaviour. Making use of another simplified model, it is fairly straightforward to observe that stratification that varies in the spanwise direction will eventually act to completely remove the growth, and exactly when this happens is governed by the critical stratification strength Nc; this appears to be an accurate predictor of the numerical results. When stratification is aligned with the basic unbounded shear, it is not observed to damp the growth at any strength. In fact, the introduction of stratification can actually enhance the optimal growth. The underlying growth mechanism identified from the unstratified flow was found to be possible without the cross-stream velocity perturbation, and that velocity only acted to hinder growth. As the stratification becomes stronger and damps these motions, the transient growth is amplified. A discussion of the key results will conclude the thesis in chapter 6, and some outstanding questions and avenues for further study will be highlighted.