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

在过去的二十年里，从学术界到工业界，不确定性量化已经引起了广泛的兴趣，人们已经开发出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，以及数学模型和实验数据的相互作用。例如，通过考虑物理参数、初始/边界条件等的不确定性，经典的确定性微分方程被松弛为随机方程。这种策略正越来越多地被应用于工业设计中，以提高稳健性和效率。贝叶斯推理以一种更自然的方式应用于反问题，以处理有噪声的观测，实际上从概率的角度将不适定的确定性反问题变成了适定的反问题。在这个项目中，首席研究人员考虑了动力系统中一个经典问题的随机形式--壁面边界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，无论噪声的幅度多么小，确定性数学模型不可能实现的状态都可以通过随机模型来探索。特别令人感兴趣的问题是向很少发生但有重大影响的反常状态的转变，如系统故障、稳定性丧失等。本项目中的主要数学工具是动力系统小随机扰动的大偏差理论。在许多壁面有界流动中，非线性不稳定性的一个典型情形是Navier-Stokes方程关于雷诺数的亚临界分叉，其中对于某个雷诺数，该方程至少有两个稳定解。首席调查员将在随机环境中重演非线性不稳定，并将其视为由小噪声触发的罕见事件。从数学上讲，考虑的一个问题是当噪声的幅度趋于零时，非线性不稳定性是如何发展的。大偏差原理断言，这样的转变将主要沿着作用泛函的极小化所给出的路径发生。该项目的主要目标有两个：1)开发有效的数值算法来寻找对非线性不稳定性的发展至关重要的最可能的过渡路径；2)广泛的数值研究壁面边界平行剪切流的亚临界分叉。第一个任务将通过时间方向的hp自适应有限元离散和空间离散的谱方法与并行计算相结合来实现。特别是，压力将从使用无散度空间的公式中移除，这样就可以减少自由度数，只关注不稳定性。在第二个任务中，重点研究了基于动作的稳定性理论与经典稳定性理论的关系，如线性稳定性理论、非线性稳定性理论、非模态稳定性理论、边缘状态和最小能量摄动研究。更具体地说，文献中的一些经典确定性模型会加入小噪声，以寻求确定性稳定性理论无法给出的额外信息。

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.