The Geometry and Building Blocks of Chaotic Fluid Convection

包含由外部来源驱动的流体的大型系统是当今科学和工程面临的许多重要挑战的核心。例如天气预报、海洋动态和管道中的湍流。这项研究的目标是建立描述、预测和控制这样的系统所需的洞察力。这项研究将通过关注从下面加热的浅层流体的复杂动力学来研究这些问题。动力系统理论将被用来通过利用算法的最新进展和大型超级计算资源的可用性来提供对这些动态的看法。这个项目将提高我们对构成混沌流体动力学的基本构件的理解。该项目将支持本科生和研究生，该项目将为本科工程学学生开发一门新课程，内容涉及动力系统的令人兴奋的进步和涉及流体动力学的重要问题的大规模计算。本研究运用动力系统理论探讨了混沌流体动力学的基本性质。该项目专注于浅层流体在引力场中从下加热的混沌对流。这个项目利用精确相干结构和协变Lyapunov矢量的思想，将流体动力学理解为通过高维状态空间的轨迹。精确相干结构是不稳定和不变的解，它在状态空间中提供了一个刚性骨架，轨迹必须在该骨架上导航。协变Lyapunov向量描述了动力学小扰动的增长或衰减，并量化了动力学的不稳定和稳定流形。这些量将被用来建立对状态空间几何的理解，并用于识别流体运动的构件。一个重要的成果将是根据所获得的洞察力制定动态的统计描述。该项目将为理论思想的发展提供重要的指导，以描述广泛的流体系统，以及其他具有社会意义的重要的大耗散系统。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.