论文标题
使用二次约束对过渡流的非线性稳定性分析
Nonlinear Stability Analysis of Transitional Flows using Quadratic Constraints
论文作者
论文摘要
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与二次非线性之间的相互作用控制。在这项工作中，我们提出了一个非线性稳定性分析框架，该框架利用了以下事实：非线性流相互作用受非线性中编码的物理学的约束。特别是，我们表明，非线性稳定性分析问题可以作为基于Lyapunov矩阵不等式的凸的可行性和优化问题，以及代表非线性流动物理学的一组二次约束。所提出的框架可用于进行全球稳定性，局部稳定性和瞬态能量增长分析。在低维的Waleffe-Kim-Hamilton过渡和持续湍流模型上证明了该方法。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，我们表明该框架可以估计吸引区域的大小以及最大允许的扰动的幅度，从而使所有轨迹都融合到平衡点。此外，我们表明该框架可以预测最大瞬态能量生长的界限。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，可用于提取驱动动力学和相关不稳定性的主要非线性流相互作用。
The dynamics of transitional flows are governed by an interplay between the non-normal 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.
