Skin-friction drag reduction in laminar and turbulent boundary layers

The skin-friction drag in a constant, mass-flux plane channel flow is sustained below that corresponding to laminar flow when wall-normal blowing and suction at the upper and lower walls are applied as upstream traveling waves. The control is distributed such that the net mass-flux of the channel is not modified. Direct numerical simulations demonstrate that these upstream traveling waves can induce sublaminar viscous drag in fully developed laminar and turbulent flows. Furthermore, it is found that the observed phenomena can be characterized by the linearized governing equations.; The Navier-Stokes equations can be linearized by describing the flow velocities as perturbations about a mean, resulting in equations that describe the dynamics of the perturbations. A spectral decomposition, involving a two-dimensional Fourier expansion in the streamwise and spanwise directions and a Galerkin projection of Chebyshev polynomials in the wall-normal direction, is used to spatially discretize these equations, leading to a state space model. The traveling wave control is then introduced as a temporally, phase-shifting, wall-normal velocity wall boundary condition at a particular Fourier wavenumber.; This linear model was used to develop reduced-order linear-quadratic-Gaussian (LQG) controllers. While these controls could achieve significant drag reduction, they appeared to be a performance limit. However, they could induce transient sublaminar drag under certain conditions. A nonlinear minimization using full-order nonlinear simulations was attempted to capture this transient behavior on a periodic basis. The upstream traveling wave was discovered in the course of that study.; The same linear models can represent the sustained sublaminar drag. Using a recent formulation which expresses the viscous drag of a fully developed channel flow as a sum of laminar drag and the Reynolds shear stress, it is shown that the Reynolds shear stress calculated from the controlled linear model state provides good predictions of how the nonlinear flow will respond. This provides a computationally efficient environment in which to find the optimal amplitude and speed to set a traveling wave to gain the most predicted drag reduction for a specific level of input power.