Reduced Dimension Models for Hydrodynamical Systems: Experiment, Computation and Theory

本计画旨在发展流体动力系统的降维模型，并促进大学生参与数学研究。虽然描述流体运动的基本方程，即Navier-Stokes方程，是众所周知的，但精确求解完整的方程通常在计算上是昂贵的，并且在分析上是不可行的。为了更好地理解这些系统，有必要转向简化或简化的模型，这些模型是从完整方程的系统近似中得出的。在这个项目中使用的半离散近似技术，主要集中在减少全流体方程的尺寸，利用对称性或广泛不同的长度尺度在特定的系统。从简化模型得到的预测结果将与实验数据进行定性和定量的比较。流体动力学在许多生物，环境和工业应用中出现。该项目旨在开发描述基本流体动力学现象的模型，例如射流冲击冷却系统和具有柔性边界的流体动力学系统中可能出现的水力跃变。后者包括磁盘驱动器等旋转磁性介质，以及昆虫飞行，动脉中的血液流动和鸣禽中的鸣管动力学等生物学问题。本科生的参与是该计划的实验和数学方面的一个组成部分。数学模型将在新的流体动力学实验室进行测试，该实验室将提供一个动手环境，让学生学习将数学与物理现象相结合。这种独特的方法鼓励学生在当代研究的框架内运用他们的数学技能与他们的物理直觉。

This project is designed to develop reduced dimension models for hydrodynamical systems and to foster undergraduate participation in mathematical research. While the fundamental equations describing fluid motion, the Navier-Stokes equations, are well-known, it is generally computationally expensive and analytically infeasible to solve the full equations exactly. To better understand these systems, it is necessary to turn to simplified or reduced models resulting from systematic approximations of the full equations. The semi-discrete approximation techniques which are utilized in this project focus primarily on reducing the dimension of the full fluid equations by exploiting symmetries or widely disparate lengthscales in specific systems. The resulting predictions from the reduced models will be compared qualitatively and quantitatively with experimental data.Fluid dynamics arises in a multitude of biological, environmental and industrial applications. This project is designed to develop models that describe basic hydrodynamical phenomena such as hydraulic jumps that can arise in jet impingement cooling systems and hydrodynamical systems with flexible boundaries. The latter include spinning magnetic media such as disk drives, and biological problems such as insect flight, blood flow in arteries and dynamics of the syrinx in song birds. Undergraduate participation is an integral part of both the experimental and mathematical aspects of the program. Mathematical models will be tested in the new Fluid Dynamics Laboratory which will provide a hands-on environment in which students learn to integrate mathematics with physical phenomena. This unique approach encourages students to apply their mathematical skills in consort with their physical intuition within the framework of contemporary research.