Laminar-Turbulent Transition Control Using Passivity Analysis of the Orr-Sommerfeld Equation

The control problem for linearized two-dimensional wall-bounded parallel shear flows is considered as a means of preventing the laminar-to-turbulent transition. The linearized Navier-Stokes equations are reduced to the Orr-Sommerfeld equation with wall-normal velocity actuation entering through the boundary conditions on the wall. An analysis of the work-energy balance is used to identify appropriate sensor outputs that could lead to a passive closed-loop system using simple feedback laws. These sensor outputs correspond to the second spatial derivative (normal direction) of the streamwise velocity at the wall, and it is demonstrated that they can be realized by pressure measurements made at appropriate locations along the wall. Spatial discretization of the Orr-Sommerfeld equation in the wall-normal direction is accomplished using Hermite cubic finite elements, and the resulting spectrum is shown to agree with literature values for both plane Poiseuille flow and the Blasius boundary layer. Both cases are shown to be stabilized by simple constant-gain output feedback using the special choice of sensor measurement. In each case, this holds for a single gain and a wide range of Reynolds numbers and wave numbers. However, the closed-loop system produced in the Poiseuille case is shown to be passive (i.e., it has a positive-real transfer function), whereas in the Blasius case, it is nonminimum-phase and hence is not passive.