Sensitivity Analysis of Partial Differential Equations With Applications to Fluid Flow

For over 100 years, researchers have attempted to predict transition to turbulence in fluid flows by analyzing the spectrum of the linearized Navier-Stokes equations. However, for many simple flows, this approach has failed to match experimental results. Recently, new scenarios for transition have been proposed that are based on the non-normality of the linearized operator. These new “mostly linear” theories have increased our understanding of the transition process, but the role of nonlinearity has not been explored. The main goal of this work is to begin to study the role of nonlinearity in transition. We use model problems to illustrate that small unmodeled disturbances can cause transition through movement or bifurcation of equilibria. We also demonstrate that small wall roughness can lead to transition by causing the linearized system to become unstable. Sensitivity methods are used to obtain important information about the disturbed problem and to illustrate that it is possible to have a precursor to predict transition. Finally, we apply linear feedback control to the model problems to illustrate the power of feedback to delay transition and even relaminarize fully developed chaotic flows. This research was supported in part by the Air Force Office of Scientific Research under grant F49620-03-1-0243 and by the DARPA Special Projects Office.