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

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Document pages: 39 pages

Abstract: The precise set of parameters governing the transition to turbulence inwall-bounded shear flows remains an open question; many theoretical bounds havebeen obtained, but there is not yet a consensus between these bounds andexperimental simulation results. In this work, we focus on a method to providea provable Reynolds number dependent bound on the amplitude of perturbations aflow can sustain while maintaining the laminar state. Our analysis relies on aninput--output approach that partitions the dynamics into a feedbackinterconnection of the linear and nonlinear dynamics (i.e., a Luré systemthat represents the nonlinearity as static feedback). We then constructquadratic constraints of the nonlinear term that is restricted by systemphysics to be energy-conserving (lossless) and to have bounded input--outputenergy. Computing the region of attraction of the laminar state (set of safeperturbations) and permissible perturbation amplitude are then reformulated asLinear Matrix Inequalities (LMI), which provides a more computationallyefficient solution than prevailing nonlinear approaches based on the sum ofsquares programming. The proposed framework can also be used for energy methodcomputations and linear stability analysis. We apply our approach to lowdimensional nonlinear shear flow models for a range of Reynolds numbers. Theresults from our analytically derived bounds are consistent with the boundsidentified through exhaustive simulations. However, they have the added benefitof being achieved at a much lower computational cost and providing a provableguarantee that a certain level of perturbation is permissible.

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