Nonlinear Instability of Navier-Stokes equations from a probabilistic point of view: Numerics and Simulations

从概率角度看纳维-斯托克斯方程的非线性不稳定性：数值与模拟

• 批准号: 1620026
• 负责人: Xiaoliang Wan
• 金额: $ 18.3万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1620026&HistoricalAwards=false
• 关键词: Nonlinear; Instability; Navier; Stokes; equations

During the last two decades, there has been a widespread interest in uncertainty quantification from academia to industry, where stochastic models and approaches have been developed to effectively describe the propagation of uncertainty of different sources in complex systems, and the interplay of mathematical modeling and experimental data. For example, classical deterministic differential equations have been relaxed to a random one by taking into account the uncertainty in physical parameters, initial/boundary conditions, etc. This strategy is being employed more and more in industrial design to enhance robustness and efficiency. Bayesian inference has been applied to inverse problems in a more natural way to deal with the noisy observations, which actually turns an ill-posed deterministic inverse problem into a well-posed one from the probabilistic point of view. In this project the Principal Investigator considers a stochastic formulation of a classical problem in dynamical system - nonlinear instability of wall-bounded parallel shear flows. The strategy will be based on a very general observation: under the excitation of noise, no matter how small the amplitude of noise is, states that are impossible for a deterministic mathematical model can be explored by a stochastic model. The issue of particular interest is the transitions to the anomalous states that occur rarely but have major impact, such as system failure, loss of stability, etc. The main mathematical tool in this project is the Freidlin-Wentzell theory of large deviations for small random perturbations of dynamical systems. One typical scenario of nonlinear instability in many wall-bounded flows is the subcritical bifurcation of Navier-Stokes equations with respect to the Reynolds number, where the equation can have at least two stable solutions for a certain Reynolds number. The Principal Investigator will recast nonlinear instability in a stochastic setting and regard it as a rare event triggered by small noise. Mathematically, a question considered is how the nonlinear instability develops as the amplitude of noise goes to zero. The large deviation principle asserts that such a transition will occur mainly following a path given by the minimizer of the action functional. The mail goal of this project is twofold: 1) Develop efficient numerical algorithms to seek the most probable transition path that is critical for the development of nonlinear instability; and 2) Extensive numerical studies of the subcritical bifurcation of wall-bounded parallel shear flows. The first task will be achieved by hp adaptive finite element discretization in time direction and spectral method for spatial discretization incorporating with parallel computing. In particular, the pressure will be removed from the formula using a divergence-free space such that one can reduce the number of degrees of freedom and just focus on the instability. In the second task, the focus is on the relation between the action-based stability theory and classical stability theories such as linear stability theory, nonlinear stability theory, nonmodal theory, edge state and minimal energy perturbation study. More specifically, small noise will be added to some classical deterministic models in literature to seek extra information that cannot be given by deterministic stability theories.

在过去的二十年中，从学术界到工业界，对不确定性量化产生了广泛的兴趣，其中已经开发了随机模型和方法来有效地描述复杂系统中不同来源的不确定性的传播，以及数学建模和实验数据的相互作用。例如，经典的确定性微分方程已放宽到随机一个考虑到物理参数的不确定性，初始/边界条件等，这种策略越来越多地在工业设计中，以提高鲁棒性和效率。贝叶斯推理以一种更自然的方式被应用于反问题中，以处理噪声观测，这实际上从概率的角度将不适定的确定性反问题转化为适定的反问题。在这个计画中，主要研究者考虑动力系统中一个经典问题的随机公式化-壁面平行剪切流的非线性不稳定性。 该策略将基于一个非常普遍的观察：在噪声的激励下，无论噪声的幅度有多小，确定性数学模型不可能的状态都可以通过随机模型来探索。特别感兴趣的问题是过渡到异常状态，很少发生，但有重大影响，如系统故障，失去稳定性，在这个项目中的主要数学工具是小随机扰动的动力系统的大偏差的Freidlin-Wentzell理论。在许多有壁流动中，非线性不稳定性的一个典型情形是Navier-Stokes方程相对于雷诺数的亚临界分叉，其中对于一定的雷诺数，方程至少有两个稳定解。主要研究者将在随机环境中重铸非线性不稳定性，并将其视为由小噪声触发的罕见事件。在数学上，考虑的问题是如何发展的非线性不稳定性的振幅的噪声趋于零。大偏差原理断言，这样的转变将主要发生在由作用泛函的最小化者给出的路径之后。 本项目的主要目标是双重的：1）开发有效的数值算法，以寻求最可能的过渡路径，这是发展的非线性不稳定性的关键;和2）广泛的数值研究的亚临界分叉的壁有界平行剪切流。第一个任务是在时间方向上采用hp自适应有限元离散，在空间方向上采用谱方法离散，并结合并行计算。特别地，将使用无发散空间从公式中移除压力，使得可以减少自由度的数量并且仅关注不稳定性。在第二个任务中，重点是基于行动的稳定性理论与经典的稳定性理论，如线性稳定性理论，非线性稳定性理论，非模态理论，边缘状态和最小能量摄动研究之间的关系。更具体地说，小噪声将被添加到文献中的一些经典的确定性模型，以寻求额外的信息，不能由确定性稳定性理论。