Time-dependent operator reveals transient flow dynamics

To capture the dynamics of nonlinear transient flows, we propose a mode decomposition technique based on time-dependent operators. Our method is based on dynamic mode decomposition (DMD) but allows the computation of instantaneous linear operators, along with their eigenvalues and eigenvectors, at each time instant during a transient process. To obtain a low-dimensional subspace that accurately represents the instantaneous eigenvectors of the linear operator, we introduce a data acquisition strategy based on a time-stepping approach. The proposed method is applied to a transient process in which the Kármán vortex street in the wake of a cylinder is disturbed by an external gust vortex. The resulting eigenvalues of the operators capture energy amplification and frequency modulation of the vortex street induced by the gust. Moreover, the corresponding eigenvectors capture the influence of the gust disturbance on each frequency component of the Kármán vortex street, including the fundamental shedding frequency and its higher harmonics. These eigenvalues and eigenvectors thus provide critical insight into the transient dynamics of the flow.