Improved global stability bounds for two-dimensional plane Poiseuille flow

Abstract

This work provides new lower bounds on the global (nonlinear) stability limit of pressure-driven two-dimensional plane Poiseuille flow, improving on the energy stability limit, $Re_E$, originally computed by Orr in 1907. Using a computer we carefully construct quartic Lyapunov functionals of the velocity perturbations about the laminar profile, which certify the nonlinear stability of the flow to arbitrary perturbations. The formulation combines a decomposition of the velocity into finitely many energy eigenmodes, referred to as a 'mode set', and an infinite-dimensional 'tail', together with explicit bounds that recast the Lyapunov inequality conditions as semidefinite programs, whose feasibility is tested. Over the streamwise lengths considered, the certified stability limit exceeds the classical energy bound. In particular, at the critical energy-stable streamwise length, where $Re_E\approx 87.59$, the flow is found to be globally stable up to $Re \approx 106.8$ (representing a $22\%$ improvement). Various modestly-sized mode sets, capable of capturing sufficient features of the nonlinear dynamics of energy growth and subsequent decay, are proposed and found to be successful in producing improved bounds, with the simplest one involving only five modes.