A stochastic SPOD-Koopman model for broadband turbulent flows

A data-driven low-order model for broadband turbulent flows that uses spectral proper orthogonal decomposition (SPOD) modes as the modal basis is presented. A discrete Koopman operator obtained via extended dynamic mode decomposition (EDMD) is used to propagate the solution. The proposed stochastic two-level model governs a compound state consisting of modal expansion and forcing coefficients. This approach follows the modeling paradigm that complex nonlinear fluid dynamics can be approximated as stochastically forced linear systems. Under the linear time-invariant assumption, the modal expansion coefficients are advanced by the modified Koopman operator in the first level. The second level governs the forcing coefficients, which compensate for the offset between the linear approximation and the true state. At this level, least squares regression is used to model nonlinear interactions between modes. Closure is achieved by a dewhitening filter that imprints the second-order statistics of the remaining residue onto the forcing coefficients. 

Driven by white noise, the model can be used for Monte Carlo simulation and generates surrogate data that accurately reproduces the second-order statistics and dynamics of the flow. The uncertainty, predictability, and stability of the stochastic model are quantified analytically and through simulations. The model is demonstrated on high-fidelity simulation data of a turbulent jet and the particle image velocity data of an open cavity flow by Zhang et al. (2020).