Sum-of-squares of polynomials approach to nonlinear stability of fluid flows: an example of application.
Sum-of-squares of polynomials approach to nonlinear stability of fluid flows: an example of application.
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DOI：
10.1098/rspa.2015.0622
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发表时间：
2015-11-08
期刊：
影响因子：
--
通讯作者：
Fuentes F
中科院分区：
文献类型：
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作者：
Huang D;Chernyshenko S;Goulart P;Lasagna D;Tutty O;Fuentes F
与最近提出的方法的应用程序提供的第一个例子的目标，从而证明其能力，原则上给出的结果，旋转库埃特流的一个版本的全局稳定性进行检查。流量取决于雷诺数和表征科里奥利力大小的参数。通过部分Galerkin展开将原始Navier-Stokes方程转化为有限维不确定动力学系统，利用多项式平方和优化得到高次多项式的李雅普诺夫泛函.它表明，所提出的方法允许获得精确的全局稳定性限制，该流量的参数表征的科里奥利力的值的范围内。在此范围之外，获得了全局稳定性极限的下限，该下限仍然优于能量稳定性极限。在研究过程中，还获得了在所使用的方法背景下有意义的几个结果。总体而言，所获得的结果表明，最近提出的方法的流体流动的全局稳定性的适用性。据我们所知，这是第一种情况下，全球稳定性的流体流动已被证明是由一个通用的方法的雷诺数的值大于可以实现的能量稳定性的方法。
With the goal of providing the first example of application of a recently proposed method, thus demonstrating its ability to give results in principle, global stability of a version of the rotating Couette flow is examined. The flow depends on the Reynolds number and a parameter characterizing the magnitude of the Coriolis force. By converting the original Navier–Stokes equations to a finite-dimensional uncertain dynamical system using a partial Galerkin expansion, high-degree polynomial Lyapunov functionals were found by sum-of-squares of polynomials optimization. It is demonstrated that the proposed method allows obtaining the exact global stability limit for this flow in a range of values of the parameter characterizing the Coriolis force. Outside this range a lower bound for the global stability limit was obtained, which is still better than the energy stability limit. In the course of the study, several results meaningful in the context of the method used were also obtained. Overall, the results obtained demonstrate the applicability of the recently proposed approach to global stability of the fluid flows. To the best of our knowledge, it is the first case in which global stability of a fluid flow has been proved by a generic method for the value of a Reynolds number greater than that which could be achieved with the energy stability approach.