Dynamic mode decomposition of a flexible flag behind a semi-circular cylinder

Abstract

The dynamics of a single flexible flag behind a semi-circular cylinder are investigated using vortex interaction study and dynamic mode decomposition (DMD). The problem is numerically solved using the immersed boundary method. For Reynolds number 300, by the variation of the streamwise gap between flag and bluff body (Dx), five regions, each exhibiting its unique pattern of flapping, from symmetric and periodic to chaotic, are identified by vortex interaction study. DMD and kernel-DMD are utilized for modal analysis and reconstruction of a viscous flow behind a semi-circular cylinder with a flexible flag located at a streamwise distance of Dx = 1.8 for moderate Reynolds number (Re = 30–500) with emphasis on representing the dynamics of the system using as few DMD modes as is practically possible. Sampling rate sensitivity study shows that low sampling rate data induce an additional frequency in the decomposition, which actually make up for the high frequency content in the vicinity of the flag and single frequency system is decomposed as the quasiperiodic system. According to the modal analysis, the fundamental frequency mode conjugate pair has the same frequency as the flag's vertical flapping and lift. It accounts for vortex production and advection from the bluff and flag and lift phenomenon. The first harmonic mode contains information regarding vortex shedding from the bluff body edges and flag tip and drag as they share same frequencies. DMD reconstruction demonstrates that 97.65% of the Re 100 post-transient system can be reconstructed using 28 DMD modes, whereas 86.71% of the Re 300 post-transient system requires 25 DMD modes. For fully transient cases, poor performance is achieved when using the DMD. The kernel-DMD application to full transient cases yields a non-oscillatory “mean mode,” a “shift mode,” and stable harmonic modes that are also present in post-transient analysis. The shift mode is famous in the modal analysis community, and it accounts for the correction to the mean mode for the transient region. In addition, these modes, there are additional modes that represent the transient dynamics of the system. 50 DMD modes reconstruct 86.98% and 77.65% of the Re 100 and Re 300 full-transient system, respectively.