Stochastic Parametric Forcing in Hydrodynamics

直接研究Navier-Stokes方程中的噪声是一项艰巨的任务，因此经常寻求低维的简化模型。研究人员为高维非线性随机偏微分方程的研究引入了一种新的计算框架，并将其应用于有良好控制的精确实验的水动力问题。该方法的主要特点是：(1)以物理动机的方式引入噪声；(2)将随机问题简化为均值流问题和方差问题；(3)利用最先进的谱离散化和随机积分数值技术开发了一种有效的实现方法。控制流体力学的方程，即纳维-斯托克斯方程，已经被人们熟知了一个多世纪，并且成功地描述了许多观察到的物理流动。然而，向复杂流动和湍流的过渡仍然是一个突出的挑战；目前还不清楚噪音在这里扮演什么角色。研究人员开发了一个新的框架来研究噪声对动力学系统模拟流体流动的特性的影响，特别是稳定性和向湍流的过渡。随机过程的数学建模、数值算法的设计、物理问题的计算机模拟以及与实验观察的直接比较相结合，为直接参与该项目的学生提供了丰富的经验。这些想法也被纳入现有的和新的课程。

To study noise in the Navier-Stokes equations directly is aformidable task, and so lower-dimensional reduced models are oftensought. The investigators introduce a new computational frameworkfor the study of high-dimensional nonlinear stochastic partialdifferential equations, and apply this to hydrodynamic problemsfor which there are well-controlled precision experiments withwhich to compare. The main characteristics of the approach are:(1) the noise is introduced in a physically motivated manner; (2)the stochastic problem is reduced to a problem for the mean flowand another for its variance; and (3) an efficient implementationis developed involving state-of-the-art spectral discretizationand stochastic integration numerical techniques. The equations governing hydrodynamics, the Navier-Stokesequations, have been known for well over a century and have beensuccessful in describing many observed physical flows. Nevertheless, the transition to complex flow and turbulenceremains an outstanding challenge; it is not clear what role noiseplays here. The investigators develop a novel framework forstudying the effect of noise on the properties, particularly thestability and transition to turbulence, of dynamical systemsmodeling fluid flows. The combined use of mathematical modelingof stochastic processes, design of numerical algorithms, computersimulations of physical problems and direct comparison withexperimental observations provides an enriching experience for thestudents directly involved in the project. The ideas are alsoincorporated into existing and new courses.