The Geometry and Building Blocks of Chaotic Fluid Convection

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Large systems containing fluids that are driven by an external source are at the core of many important challenges facing science and engineering today. Examples include weather prediction, the dynamics of the oceans, and turbulent flow in a pipe. The goal of this study is to build the insights that will be needed to describe, predict, and control systems such as these. The research will study these questions by focusing on the complex dynamics of a shallow layer of fluid that is heated from below. Dynamical systems theory will be used to provide a view into these dynamics by exploiting recent advances in algorithms and the availability of large supercomputing resources. This project will improve our understanding of the basic building blocks that compose chaotic fluid dynamics. This project will support undergraduate and graduate students, and the project will develop a new course for undergraduate engineering students on the exciting advances of dynamical systems and large-scale computations for important problems involving the dynamics of fluids. This study explores the basic nature of chaotic fluid dynamics using dynamical systems theory. The project focusses on the chaotic convection of a shallow layer of fluid that is heated from below in a gravitational field. This project builds an understanding of the fluid dynamics as a trajectory through a high-dimensional state space using the ideas of exact coherent structures and covariant Lyapunov vectors. Exact coherent structures are unstable and invariant solutions that provide a rigid skeleton in state space upon which the trajectory must navigate. The covariant Lyapunov vectors describe the growth or decay of small perturbations to the dynamics and quantify the unstable and stable manifolds of the dynamics. These quantities will be used to build an understanding of the geometry of state space and to identify the building blocks of the fluid motion. An important outcome will be the development of a statistical description of the dynamics based upon the gained insights. This project will provide important guidance for the development of theoretical ideas to describe a wide range of fluid systems, as well as other important large dissipative systems that are of societal interest.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.