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.
This letter describes a method for estimating regions of attraction and bounds on permissible perturbation amplitudes in nonlinear fluids systems. The proposed approach exploits quadratic constraints between the inputs and outputs of the nonlinearity on elliptical sets. This approach reduces conservatism and improves estimates for regions of attraction and bounds on permissible perturbation amplitudes over related methods that employ quadratic constraints on spherical sets. We present and investigate two algorithms for performing the analysis: an iterative method that refines the analysis by s…
A method to bound the maximum energy perturbation for which regional stability of transitional fluid flow models can be guaranteed is introduced. The proposed method exploits the fact that the fluid model's nonlinearities are both lossless and locally bounded and uses the axes lengths of the ellipsoids for the trajectory set containment as variables in the stability conditions. Compared to existing approaches, the proposed method leads to an average increase in the maximum allowable energy perturbation of 29% for the Waleffe-KimHamilton (WKH) shear flow model and of 38% for the 9-state reduced…
The choice and placement of sensors and actuators is an essential factor determining the performance that can be realized using feedback control. This determination is especially important, but difficult, in the context of controlling transitional flows. The highly non-normal nature of the linearized Navier-Stokes equations makes the flow sensitive to small perturbations, with potentially drastic performance consequences on closed-loop flow control performance. Full-information controllers, such as the linear quadratic regulator (LQR), have demonstrated some success in reducing transient energ…
Although the roll/streak structure is ubiquitous in pre-transitional wall-bounded shear flow, this structure is linearly stable if the idealization of laminar flow is made. Lacking an instability, the large transient growth of the roll/streak structure has been invoked to explain its appearance as resulting from chance occurrence in the free-stream turbulence (FST) of perturbations configured to optimally excite it. However, there is an alternative interpretation which is that FST interacts with the roll/streak structure to destabilize it. Statistical state dynamics (SSD) provides analysis met…
The 2D second-mode is a potent instability in hypersonic boundary layers (HBLs). We study its linear and nonlinear evolution, followed by its role in transition and eventual breakdown of the HBL into a fully turbulent state. Linear stability theory (LST) is utilized to identify the second-mode wave through FS-synchronization, which is then recreated in linearly and nonlinearly forced 2D direct numerical simulations (DNSs). The nonlinear DNS shows saturation of the fundamental frequency, and the resulting superharmonics induce tightly braided ``rope-like'' patterns near the generalized inflecti…
The precise set of parameters governing the transition to turbulence in wall-bounded shear flows remains an open question; many theoretical bounds have been obtained, but there is not yet a consensus between these bounds and experimental/simulation results. In this work, we focus on a method to provide a provable Reynolds number dependent bound on the amplitude of perturbations a flow can sustain while maintaining the laminar state. Our analysis relies on an input--output approach that partitions the dynamics into a feedback interconnection of the linear and nonlinear dynamics (i.e., a Luré sy…
Modeling realistic fluid and plasma flows is computationally intensive, motivating the use of reduced-order models for a variety of scientific and engineering tasks. However, it is challenging to characterize, much less guarantee, the global stability (i.e., long-time boundedness) of these models. The seminal work of Schlegel and Noack (JFM, 2015) provided a theorem outlining necessary and sufficient conditions to ensure global stability in systems with energy-preserving, quadratic nonlinearities, with the goal of evaluating the stability of projection-based models. In this work, we incorporat…
Verifying nonlinear stability of a laminar fluid flow against all perturbations is a central challenge in fluid dynamics. Past results rely on monotonic decrease of a perturbation energy or a similar quadratic generalized energy. None show stability for the many flows that seem to be stable despite these energies growing transiently. Here a broadly applicable method to verify global stability of such flows is presented. It uses polynomial optimization computations to construct non-quadratic Lyapunov functions that decrease monotonically. The method is used to verify global stability of 2D plan…
