Stabilization

Abstract

The steady-state solutions, or equilibrium velocity profiles, obtained for the prototype flows of Section 2.6, are parabolic in shape in the streamwise direction, and zero in the other directions. Thus, the flows consist of parallel layers of fluid moving in a very regular and deterministic way. These are examples of so-called laminar flows. For wall-bounded laminar flows, wall friction, or drag, is favorably low, and these flows are therefore target flows in drag reduction applications. Unfortunately, they are rarely stable. In fact, stability is assured at small Reynolds number, only. An unstable flow is characterized by the fact that small perturbations from the equilibrium velocity profile will grow, and eventually cause the flow to transition to turbulent flow. Turbulent flow is characterized by small scale, apparently stochastic, velocity components, which lead to substantially higher drag than what is present in laminar flow. Being able to relaminarize a turbulent flow is therefore of great importance, and can be achieved in the prototype flows studied here, by stabilizing the parabolic equilibrium profile using boundary control. Boundary control implies specifying the flow field dynamically on the boundary of the domain, in this case on the channel or pipe walls, possibly based on values of flow variables taken at the boundary (boundary feedback). In this work, we assume that there exist sensors that provide distributed flow information at the wall, and actuators that can set prescribed distributed velocities. Chapter 6 reviews a selection of sensors and actuators that accomplish this task.