以流体中向turbulence过渡为例分析非线性稳定性的优化方法
An optimisation approach for analysing nonlinear stability with transition to turbulence in fluids as an exemplar
摘要：
本文介绍，并回顾了最近的工作，使用一个简单的优化技术，分析在一个动态系统的状态的非线性稳定性。该技术可用于识别干扰系统的最有效方式，使其从一个稳定状态过渡到另一个稳定状态。的关键思想是在一个有限维的一组常微分方程（ODE）的框架内，然后说明了一个非常简单的系统2 ODE具有双稳态。然后，在流体力学中的湍流问题的过渡，以显示该技术可以制定一个空间扩展的系统所描述的偏微分方程（著名的纳维-斯托克斯方程）。在这种情况下，优化技术的桥梁之间的差距（线性）最优扰动理论和（非线性）动态系统的方法来流体流动。事实上，该技术最近已被证明在这种非常高维的设置工作，预示着它在其他物理系统的实用性。
英文摘要：
This article introduces, and reviews recent work using, a simple optimisation technique for analysing the nonlinear stability of a state in a dynamical system. The technique can be used to identify the most efficient way to disturb a system such that it transits from one stable state to another. The key idea is introduced within the framework of a finite-dimensional set of ordinary differential equations (ODEs) and then illustrated for a very simple system of 2 ODEs which possesses bistability. Then the transition to turbulence problem in fluid mechanics is used to show how the technique can be formulated for a spatially-extended system described by a partial differential equation (the well-known Navier-Stokes equation). Within that context, the optimisation technique bridges the gap between (linear) optimal perturbation theory and the (nonlinear) dynamical systems approach to fluid flows. The fact that the technique has now been recently shown to work in this very high dimensional setting augurs well for its utility in other physical systems.
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