The dynamics of transitional flows are governed by an interplay between the nonnormal linear dynamics and quadratic nonlinearity in the incompressible Navier-Stokes equations. In this work, we propose a framework for nonlinear stability analysis that exploits the fact that nonlinear flow interactions are constrained by the physics encoded in the nonlinearity. In particular, we show that nonlinear stability analysis problems can be posed as convex feasibility and optimization problems based on Lyapunov matrix inequalities, and a set of quadratic constraints that represent the nonlinear flow physics. The proposed framework can be used to conduct global stability, local stability, and transient energy growth analysis. The approach is demonstrated on the low-dimensional Waleffe-Kim-Hamilton model of transition and sustained turbulence. Our analysis correctly determines the critical Reynolds number for global instability. For local stability analysis, we show that the framework can estimate the size of the region of attraction as well as the amplitude of the largest permissible perturbation such that all trajectories converge back to the equilibrium point. Additionally, we show that the framework can predict bounds on the maximum transient energy growth. Finally, we show that careful analysis of the multipliers used to enforce the quadratic constraints can be used to extract dominant nonlinear flow interactions that drive the dynamics and associated instabilities.

中文翻译：

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基于二次约束的过渡流非线性稳定性分析

过渡流的动力学由不可压缩的Navier-Stokes方程中的非正态线性动力学和二次非线性之间的相互作用决定。在这项工作中，我们提出了一个非线性稳定性分析的框架，该框架利用了非线性流动相互作用受非线性编码物理约束的事实。特别是，我们表明，基于Lyapunov矩阵不等式以及代表非线性流动物理的一组二次约束，可以将非线性稳定性分析问题提出为凸可行性和优化问题。所提出的框架可用于进行全局稳定性，局部稳定性和瞬态能量增长分析。该方法在过渡和持续湍流的低维Waleffe-Kim-Hamilton模型中得到了证明。我们的分析正确地确定了全球不稳定的关键雷诺数。对于局部稳定性分析，我们表明框架可以估计吸引区域的大小以及最大允许扰动的幅度，从而使所有轨迹都收敛回到平衡点。此外，我们表明该框架可以预测最大瞬态能量增长的界限。最后，我们表明，对用于强制执行二次约束的乘数进行仔细分析，可以得出驱动动力学和相关不稳定性的主要非线性流相互作用。我们表明，该框架可以估计吸引区域的大小以及最大允许扰动的幅度，从而使所有轨迹都收敛回到平衡点。此外，我们表明该框架可以预测最大瞬态能量增长的界限。最后，我们表明，对用于强制执行二次约束的乘数进行仔细分析，可以得出驱动动力学和相关不稳定性的主要非线性流相互作用。我们表明，该框架可以估计吸引区域的大小以及最大允许扰动的幅度，从而使所有轨迹都收敛回到平衡点。此外，我们表明该框架可以预测最大瞬态能量增长的界限。最后，我们表明，对用于强制执行二次约束的乘数进行仔细分析，可以得出驱动动力学和相关不稳定性的主要非线性流相互作用。