Abstract: While the dynamics of fluids are most naturally described in the time domain, e.g., velocity fields as a function of space and time, there are advantages to working in the frequency domain when developing simplified models. This presentation will highlight two recent examples from my group. First, we have developed a novel flow estimation and control framework based on resolvent analysis, a popular frequency-domain linear model, with several advantages over standard methods. When equivalent assumptions are made, 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. The framework is demonstrated using a series of aerodynamic problems. Second, we are developing predictive nonlinear reduced-order models in the frequency domain based on spectral proper orthogonal decomposition (SPOD), a popular frequency-domain data-mining method. Most model reduction methods for the Navier-Stokes equations and other PDEs spatially reduce the flow state and evolve it forward in time without explicit temporal reduction. In contrast, our new approach expands the state as a sum of temporal Fourier modes with a data-driven orthogonal basis of SPOD modes at each frequency, and we derive near-optimal equations governing the unknown scalar expansion coefficients that define the contribution of each mode to the solution over a time interval of interest. We show that this approach significantly outperforms standard time-domain methods for several example problems.

Contact: Silas Alben