Reduced-order modeling and control of near-wall turbulent flow

We introduce a new low-dimensional model for the study, simulation and control of near-wall turbulent flow based on the empirical eigenfunctions of the proper orthogonal decomposition (POD). The wall-normal direction is discretized in our model using basis functions derived for the components of velocity as well as the wall-normal vorticity from the POD and the continuity condition, permitting the decoupling of the vertical velocity (roll) and vertical vorticity (streak) modes in our models. The streamwise and spanwise directions are not discretized, permitting some flexibility in the choice of basis functions in these directions. Galerkin projection of the Navier-Stokes equations (in the vertical velocity-vertical vorticity formulation) onto the basis functions yields a set of coupled partial differential equations in the two remaining spatial directions and in time. Modeling of the unresolved, small-scale stresses and the mean velocity profile is required to close the equations. We also discuss our work with the control of near-wall turbulence in simulations of turbulent channel flow in the minimal flow unit. The control focuses on the coherent structures, estimating their strength from the shear stress at the wall and applying control through a time-varying body force. The control is successful in reducing the turbulent skin friction and suppressing the production of turbulence near the wall by the structures.