An improved mode time coefficient for dynamic mode decomposition

Dynamic mode decomposition (DMD) is widely used for extracting dominant structures of unsteady flow fields. However, the traditional mode time coefficient of DMD is assumed to change exponentially over the time. Consequently, it cannot deal with the unstable flow fields whose modes present nonexponential evolution regularities. Also, the inaccurate mode time coefficient might cause an unreasonable rank of decomposed modes, leading to the dominant modes to be ignored. To overcome these shortcomings, an improved mode time coefficient based on the Moore-Penrose pseudoinverse is proposed for the DMD, and a new integrated parameter based on the improved mode time coefficient is defined to rank the decomposed modes. The DMD with the improved mode time coefficient (abbreviated as DMD-TC) is expected to accurately describe the temporal evolutions of modes in complex forms for unstable systems and results in a more reasonable rank for the modes. To validate the DMD-TC, two complex analytical functions (a continuous case and an intermittent case) and two typical unstable flows (the flow around a cylinder and the dynamic stall of a pitching airfoil) are investigated. The results indicate that the DMD-TC can accurately describe temporal evolutions of modes with complex nonlinear regularities, including exponential, logarithmic, linear, gradually intermittent, transiently intermittent, and other complex regularities. Also, due to the improved mode time coefficient, the DMD-TC can provide a more reasonable rank for unstable modes. These improvements help to identify instantaneous dominant dynamic modes (even with minor initial amplitudes) of real unstable flow fields and accurately describe their temporal evolutions.