Daniel ReMine, University of Virginia


Fluid Mechanics Stability Control: Control Stabilization Enhancement of a 3D Linearized Navier-Stokes Fluid Flow in Parallelopiped with a Wall-Normal Boundary Controller

Abstract: Consider a 3-D, linearized Navier Stokes channel flow with periodic boundary conditions in the streamwise direction and subject to a wall-normal control on the top wall. The current research focuses on how to extend a 2-dimensional result (published by Professor Triggiani) into 3 dimensions. The approach is to find an infinite-dimensional subspace E₀ such that the normal component v of the velocity vector, as well as the vorticity ω, are not influenced by the control. The corresponding control-free dynamics for v and ω on E₀ will be inherently exponentially stable, though with limited decay rate. The stability margin of the component v on the complementary space 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, we will try to show its support to 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 ω over Z will also be obtained, to complete the optimal analysis. Alternatively, if this extension of results from 2-D to 3-D cannot be done, the research will focus on the underlying reasons why and what other implications this would have.

Mentor: Irena Lasiecka (University of Virginia)