| In this thesis, linear mechanisms in wall-bounded turbulent shear flows have been investigated, and some linear optimal controls based on a linearized model of flow dynamics are designed to achieve viscous drag reduction.; It is shown that the linear coupling term in the linearized Navier-Stokes equations, which enhances non-normality of the system, plays an important role in turbulent shear flows. In the absence of the linear coupling term, near-wall turbulence is shown to decay. On the other hand, without the nonlinear terms in the Navier-Stokes equations, near-wall coherent structures are not formed in the commonly observed scales in turbulent flows, indicating that the formation of the near-wall turbulence structures in turbulent flows is essentially nonlinear. However, the linear coupling mechanism is observed to be a critical factor in the onset of the formation of coherent structures. Therefore, a proper linear optimal controller that can minimize the effect of the linear coupling mechanism could suppress the formation of these structures, which in turn may lead to a substantial drag reduction in turbulent shear flows.; A state-space representation of the linearized Navier-Stokes equations for wall-bounded flows is formulated for the design of various linear optimal controls. The controls are designed to minimize a cost function, which consists of a flow quantity and the control input. Several cost functions are tested for maximum mean drag reduction in a turbulent channel flow. When the flow information is fully available, control inputs based on the linear quadratic regulator (LQR) algorithm are able to produce a significant mean drag reduction for all the cost functions tested. Various control methods studied here provide a robust performance within a range of Reynolds numbers, but an adaptive approach (‘gain-scheduling’) has to be taken for further drag reduction. An observer-based optimal control scheme is also built using the linear quadratic Gaussian (LQG) algorithm in order to make control more practical. The LQG control requires only wall-measured flow quantities, but it produces less drag reduction than the LQR control does. The choice of the measurement and the cost function affects the performance of the LQG control significantly. It is shown that the measurements near the control input location are sufficient for feedback information. A simple ad-hoc control law, mimicking the relation of the measurement and the control input from LQG, is developed to result in the same amount of drag reduction. Other possibilities and limitations of the linear control designs for nonlinear turbulent flows have been discussed as well. |
