On-the-fly Reduced-Order Modeling of Transient Flow Response Subject to High-Dimensional External Forcing

This work proposes a reduced-order modeling approach based on time-dependent bases (TDBs) to identify instantaneous most amplified external excitation for time-varying base flows. We illustrate that a linear response of the flow to high-dimensional external excitation is extracted and exploited in a low-dimensional subspace using forced optimally time-dependent (f-OTD) modes. Moreover, the f-OTD framework enables reduced-order modeling of the flow response to external forcing for arbitrarily time-varying base flow. The low-rank subspace reveals the hidden important information of high-dimensional forcing. For unsteady base flows, we provide a low-rank approximation of spatial-temporal optimal forcing which leads to most amplification in the flow. We also show that when the flow is linearized around the mean flow under an external harmonic excitation, the dominant f-OTD modes converge to the resolvent operator's optimal forcing and corresponding response mode. To illustrate the performance of the proposed algorithm, three cases are considered: (1) the one-dimensional Burgers' equation, (2) the temporally evolving jet, and (3) two-dimensional decaying isotropic turbulence.