Analysis and Computational Fluid Dynamics for the Stabilization and Control of 3-Dimensional Navier-Stokes Fluid Channel Flows by a Wall-Normal Boundary Controller

The field of fluid control seeks to analyze our ability to control fluid-mechanical systems and stabilize their flows towards desired outcomes. This work considers the extension of exponential stability enhancement results from a previously analyzed 2-D linear Navier-Stokes channel flow case to that of 3-D. We consider a 3-D, linear Navier-Stokes channel flow with periodic boundary conditions in the streamwise direction and subject to a wall-normal control on the top wall. There exists an infinite-dimensional subspace E, where the normal component v of the velocity vector, as well as the vorticity omega, are not influenced by the control. The corresponding control-free dynamics for $v$ and omega on E are inherently exponentially stable, though with limited decay rate. In the case of the linear 3-D channel, the stability margin of the component v on the complementary space Z can be enhanced by a prescribed decay rate, by means of an explicit, 3-D wall-normal controller acting on the top wall, whose space component is subject to algebraic rank conditions. Moreover, its support may be arbitrarily small. Corresponding optimal decays, by the same 3-D wall-normal controller, of the tangential component u of the velocity vector; of the pressure p; and of the vorticity omega over Z are also obtained, to complete the optimal analysis. Incompressible computational fluid dynamics methods are used to simulate the analysis of the 2-D channel flow case. Computational fluid dynamics methods and control simulation algorithms are presented.