An input-output inspired method for permissible perturbation amplitude of transitional wall-bounded shear flows

Abstract

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 Lur\'e system that represents the nonlinearity as static feedback). We then construct quadratic constraints of the nonlinear term that is restricted by system physics to be energy-conserving (lossless) and to have bounded input--output energy. Computing the region of attraction of the laminar state (set of safe perturbations) and permissible perturbation amplitude are then reformulated as Linear Matrix Inequalities (LMI), which provides a more computationally efficient solution than prevailing nonlinear approaches based on the sum of squares programming. The proposed framework can also be used for energy method computations and linear stability analysis. We apply our approach to low dimensional nonlinear shear flow models for a range of Reynolds numbers. The results from our analytically derived bounds are consistent with the bounds identified through exhaustive simulations. However, they have the added benefit of being achieved at a much lower computational cost and providing a provable guarantee that a certain level of perturbation is permissible.