Charting the State Space of Plane Couette Flow: Equilibria, Relative Equilibria, and Heteroclinic Connections
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The study of turbulence has been dominated historically by a bottom-up approach, with a much stronger emphasis on the physical structure of ﬂows than on that of the dynam- ical state space. Turbulence has traditionally been described in terms of various visually recognizable physical features, such as waves and vortices. Thanks to recent theoretical as well as experimental advancements, it is now possible to take a more top-down approach to turbulence. Recent work has uncovered non-trivial equilibria as well as relative periodic orbits in several turbulent systems. Furthermore, it is now possible to verify theoretical results at a high degree of precision, thanks to an experimental technique known as Particle Image Velocimetry. These results squarely frame moderate Reynolds number Re turbulence in boundary shear ﬂows as a tractable dynamical systems problem. In this thesis, I intend to elucidate the ﬁner structure of the state space of moderate Re wall-bounded turbulent ﬂows in hope of providing a more accurate and precise description of this complex phenomenon. Computation of new undiscovered equilibria, relative equilibria, and their heteroclinic connections provide a skeleton upon which a numerically accurate description of turbulence can be framed. The behavior of the equilibria under variation of Reynolds number and cell aspect ratios is also examined. It is hoped that this description of the state space will provide new avenues for research into nonlinear control systems for shear ﬂows as well as quantitative predictions of transport properties of moderate Re ﬂuid ﬂows.