Turbulence suppression in plane Couette flow using reduced-order models

We explore a reduced-order model (ROM) of plane Couette flow with a view to performing turbulence control. The ROM is derived through Galerkin projections of the incompressible Navier-Stokes (NS) system onto a basis composed of controllability modes, truncated to a few streamwise and spanwise wavenumbers. Such ROMs were found to reproduce key aspects of nonlinear turbulence dynamics in Couette flow with only a few hundreds degrees of freedom. Here we use the ROM to devise a control strategy. For that, we consider a ROM with an extra forcing term, consisting in a steady body force. The spatial structure of the forcing is given by a linear combination of Stokes modes, optimised using a gradient-descent algorithm in order to minimise the total fluctuation energy. The optimisation is performed at different Reynolds numbers, with the optimal forcing leading to laminarisation of the flow in all cases. The forcing mechanism acts by strongly reducing the shear in a large central portion of the channel. This disrupts the dynamics of large-scale streaks and rolls and hinders the main energy input to the system. When the forcing is active, the flow reaches a new laminar state which is linearly stable and whose linear transient growth is substantially reduced with respect to that of laminar Couette flow. These features prompt the flow to return to the laminar Couette state when the forcing is switched off. Body forces optimised in the ROM are subsequently applied to the full NS system in direct numerical simulations (DNS) for the same flow configurations. The same control mechanisms are observed in the DNS, where laminarisation is also achieved. The present work opens up interesting possibilities for turbulence control. We show that the ROMs provide an effective framework to design turbulence control strategies, despite the high degree of truncation with respect to the full system.