Dynamic Mode Decomposition: Theory and Applications

Abstract

Used to analyze the time-evolution of fluid flows, dynamic mode decomposition (DMD) has quickly gained traction in the fluids community. However, the existing DMD literature focuses primarily on applications, rather than theory. In this thesis, we present new results of both types.

First, we propose a new definition in which we interpret DMD as an approximate eigendecomposition of the best-fit (in a least-squares/minimum-norm sense) operator relating two data matrices. This definition preserves the link between DMD and Koopman operator theory; it also highlights the relationship between DMD and linear inverse modeling. Using our definition, we are able to generalize the DMD algorithm to arbitrary datasets, not just sequential time-series (as are typically considered). Then, turning to applications, we use DMD to estimate the slow eigenvectors that dominate the long-time behavior of impulse responses. We use these in developing a variant of balanced proper orthogonal decomposition that is both more accurate and more computationally efficient.

We also apply DMD to analyze oscillatory fluid flows, which is its most common use. In one example, we apply both DMD and proper orthogonal decomposition (POD) to study the effects of zero-net-mass-flux actuation on separated flows. We find a correlation between the separation bubble height and the distribution of energy among the POD modes. We also find that the most effective control strategy is characterized by frequency lock-on between the wake and the shear layer. In another example, we use DMD to investigate the source of low-frequency oscillations in shock-turbulent boundary layer interactions. Using data from direct numerical simulations, we find modes whose characteristics match those suggested by linear stability analysis.

The last part of this thesis deals with issues of time-resolution. DMD requires data that are collected at least twice as fast as any frequency of interest. We propose two approaches for identifying oscillatory flow structures when such sampling rates are not possible. First, we demonstrate a procedure for dynamically estimating a time-resolved trajectory from non-time-resolved data; DMD can computed from the estimated trajectory. Second, we develop a method in which oscillatory modes are computed using compressed sensing techniques.