A Framework for Input-Output Analysis of Wall-Bounded Shear Flows

Mohamadreza Ahmadi (1) , Giórgio Valmórbida (2, 3) , Dennice Gayme (4) , Antonis Papachristodoulou (5)

1 University of Texas at Austin [Austin]
2 DISCO - Dynamical Interconnected Systems in COmplex Environments
3 UNICAMP - Universidade Estadual de Campinas = University of Campinas
4 JHU - Johns Hopkins University
5 Department of Engineering Science

We propose a new framework to evaluate input-output amplification properties of nonlinear models of wall-bounded shear flows, subject to both square integrable and persistent disturbances. We focus on flows that are spatially invariant in one direction and whose base flow can be described by a polynomial, e.g. streamwise constant channel, Couette and pipe flows. Our methodology is based on the notion of dissipation inequalities in control theory and provides a single unified approach to examining flow properties such as energy growth, worst case disturbance amplification, and stability to persistent excitation (i.e., input-to-state stability). It also enables direct analysis of the nonlinear partial differential equation (PDE) rather than of a discretized form of the equations, thereby removing the possibility of truncation errors. We demonstrate how to numerically compute the input-output properties of the flow as the solution of a (convex) optimization problem. We apply our theoretical and computational tools to plane Couette, channel and pipe flows. Our results demonstrate that the proposed framework leads to results that are consistent with theoretical and experimental amplification scalings obtained in the literature.

Mots clés: Control theory; Navier-Stokes equations; Transition to turbulence; Nonlinear instability; Channel flow

Domaines: Automatique / Robotique

Mohamadreza Ahmadi, Giórgio Valmórbida, Dennice Gayme, Antonis Papachristodoulou. A Framework for Input-Output Analysis of Wall-Bounded Shear Flows. Journal of Fluid Mechanics, 2019, 873, pp.742-785. ⟨10.1017/jfm.2019.418⟩. ⟨hal-02430328⟩