Nonlinear stability analysis of transitional flows using quadratic constraints

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.

Figure 1

Lur'e representation of the WKH system $F_u(L,N).$

Figure 2

Illustration of a scalar quadratic function $x=z^2$ that lies inside the sector formed by lines of slope $-R$ and $R$ (red). The blue dashed lines mark the maximum value of the pair $(x,z)$ for a given slope such that $\left|x\right|\le R$.

Figure 3

As the Re is increased, the local stability region $R$ decreases. The red dashed line shows the Re for global stability limit. In panels (a) and (b) as we approach the global stability limit, the size of $R\to \infty$. As $\text{Re}\to \infty$, the size of region $R\to 0$, which corresponds to the linear analysis of infinitesimal perturbations.

Figure 4

The maximum allowable perturbation size $R_0$ as a function of Re for W and B&T parameters compared against the SOS framework and DAL method.

Figure 5

5000 Monte Carlo simulations from various initial conditions such that $\parallel {\mathbf{x}}_0{\parallel }^2\le \frac{R}{\sqrt{q}}$, with $R={10}^{-2}$ and $R={10}^{-4}$ for W parameter and B&T parameters, respectively. The solid blue line shows the MTEG upper bound for nonlinear system predicted by the quadratic constraint framework proposed here. The solid gray curves show the Monte Carlo simulations for various initial conditions and the red curve shows the worst case MTEG of the nonlinear system.

Figure 6

Results obtained from solving $R_0$ in Eq. (20) with W parameters. The dominating nonlinear terms for stability can be identified by analyzing Lagrange multipliers ${\xi }_{p_i}$ in panel (a) and its associated singular values ${\sigma }_{\text{max}}\left({\xi }_{p_i}{M}_i\right)$ in panel (b).

Figure 7

Results obtained from solving for $q^{*}$ in Eq. (24). The dominant nonlinear terms for TEG can be identified by analyzing the dominant Lagrange multipliers ${\xi }_{p_i}$ in panel (a) and its associated singular values ${\sigma }_{\text{max}}\left({\xi }_{p_i}{M}_i\right)$ in panel (b).

Figure 8

Local MTEG analysis for W parameters with global lossless constraint and the two most dominant local constraints compared against the MTEG of system with global and all local constraints for $\text{Re}=100$ and $R=0.01$.

Figure 9

Illustration of a generic sector bounded nonlinearity.