The complexities of transitional and turbulent flows make reduced-complexity models valuable for understanding, predicting, and controlling flow phenomena. One important class of models employs linear stability analysis of the governing Navier-Stokes equations, which describes the behavior of perturbations to the laminar or turbulent mean flow. Linear stability methods are typically classified into two categories: modal theory studies the long-time asymptotic response of the unforced linear system, while non-modal theory investigates the forced response due to either an initial condition (transient growth analysis) or an exogenous forcing (resolvent analysis).
In this seminar, I will discuss three efforts within my group to extend non-modal stability methods beyond their current limitations. First, we developed a new computational algorithm to enable resolvent analysis of large-scale systems of practical engineering interest. Our algorithm combines concepts from randomized linear algebra with an efficient time-stepping method to overcome the primary computational bottlenecks of previous methods, yielding an algorithm that scales linearly with problem size and drastically reduces CPU and memory costs for large systems. Second, we developed a novel resolvent-based flow estimation and control framework with several advantages over standard methods. Under equivalent assumptions, the resolvent-based estimator and controller reproduce the Kalman filter and LQG controller, respectively, but at substantially lower computational cost. Unlike these methods, the resolvent-based approach can naturally accommodate forcing terms (nonlinear terms from Navier-Stokes) with colored-in-time statistics, which significantly improves the accuracy of the estimates. Third, we developed a novel statistical transient growth framework. In addition to the optimal growth curves provided by the standard theory, given statistics of the initial disturbances, our framework predicts the expected (mean) growth, a hierarchy of coherent structures responsible for the expected growth, and the probability of observing any particular level of amplification.