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

Louisiana State University, Baton Rouge LA

Investigators

Abstract

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.