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Sum-of-Squares Approach to Global Stability and Control of Fluid Flows

流体流动全局稳定性和控制的平方和方法

基本信息

批准号：
EP/J011126/1
负责人：
Sergei I. Chernyshenko
金额：
$ 42.94万
依托单位：
Imperial College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2013
资助国家：
英国
起止时间：
2013 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FJ011126%2F1
关键词：
Sum Squares Approach Global Stability

项目摘要

This project aims at developing new methods of analysis of the stability of fluid flows and flow control. Flow control is among the most promising routes for reducing drag, thus reducing carbon emissions, which is the strongest challenge for aviation today. However, the stability analysis of fluid flows poses significant mathematical and computational challenges. The project is based on a recent major breakthrough in mathematics related to positive-definiteness of polynomials. Positive-definiteness is important in stability and control theory because it is an essential property of a Lyapunov function, which is a powerful tool for establishing stability of a given system. For more than a century since their introduction in 1892 constructing Lyapunov functions was dependent on ingenuity and creativity of the researcher. In 2000 a systematic and numerically tractable way of constructing polynomials that are sums of squares and that satisfy a set of linear constraints was discovered. If a polynomial is a sum of squares of other polynomials then it is positive-definite. Thus, systematic, computer-aided construction of Lyapunov functions became possible for systems described by equations with polynomial non-linearity. In the last decade the Sum-of-Squares approach became widely used with significant impact in several research areas.The Navier-Stokes equations governing motion of incompressible fluid have a polynomial nonlinearity. This project will achieve its goals by applying sum-of-squares approach to stability and control of the fluid flows governed by these equations. This will require development of new advanced analytical techniques combined with extensive numerical calculations. The project has a fundamental nature, with main expected outcomes being applicable to a large variety of fluid flows. The rotating Taylor-Couette flow will be the first object to which the developed methods will be applied. Taylor-Couette flow, encountered in a wide range of industrial application, for a variety of reasons has an iconic status in the stability theory, traditionally serving as a test-bench for new methods.In order to maximise the impact of the research, the project collaborators will conduct targeted dissemination activities for industry and academia in the form of informal and formal workshops, in addition to traditional dissemination routes of journal papers and conferences. Selected representatives from industry will be invited to attend the workshops. Wider audience will be reached via a specially created and continuously maintained web page.

该项目旨在开发分析流体流动稳定性和流量控制的新方法。流量控制是减少阻力最有前途的途径之一，从而减少碳排放，这是当今航空业面临的最严峻的挑战。然而，流体流动的稳定性分析提出了重大的数学和计算挑战。该项目基于最近与多项式正定性相关的数学重大突破。正定性在稳定性和控制理论中很重要，因为它是李雅普诺夫函数的基本属性，而李雅普诺夫函数是建立给定系统稳定性的有力工具。自 1892 年提出以来的一个多世纪以来，李亚普诺夫函数的构造一直依赖于研究人员的聪明才智和创造力。 2000 年，人们发现了一种系统化且易于数值处理的构造多项式的方法，该多项式是平方和且满足一组线性约束。如果一个多项式是其他多项式的平方和，那么它是正定的。因此，对于由多项式非线性方程描述的系统，系统地、计算机辅助构建李亚普诺夫函数成为可能。在过去的十年中，平方和方法得到广泛应用，在多个研究领域产生了重大影响。控制不可压缩流体运动的纳维-斯托克斯方程具有多项式非线性。该项目将通过应用平方和方法来稳定和控制由这些方程控制的流体流动来实现其目标。这将需要开发新的先进分析技术并结合广泛的数值计算。该项目具有基本性质，主要预期成果适用于多种流体流动。旋转泰勒-库埃特流将是应用所开发方法的第一个对象。 Taylor-Couette流在广泛的工业应用中遇到，由于多种原因在稳定性理论中具有标志性的地位，传统上作为新方法的测试平台。为了最大限度地发挥研究的影响，除了期刊论文和会议的传统传播途径外，项目合作者还将以非正式和正式研讨会的形式针对工业界和学术界进行有针对性的传播活动。选定的行业代表将被邀请参加研讨会。通过专门创建和持续维护的网页将吸引更广泛的受众。