The linear-time-invariance notion of the Koopman analysis. Part 2. Dynamic Koopman modes, physics interpretations and phenomenological analysis of the prism wake

• Published in: Journal of fluid mechanics Vol. 959
• Main Authors: Li, Cruz Y., Chen, Zengshun, Tse, Tim K.T., Weerasuriya, Asiri Umenga, Zhang, Xuelin, Fu, Yunfei, Lin, Xisheng
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 25.03.2023
• Subjects: Algorithms; Analysis; Analytical methods; Crosswinds; Data science; Dynamics; Energy exchange; Entrainment; Excitation; Fluid flow; Fluid mechanics; Fluid-structure interaction; Fluids; Harmonics; Invariance; JFM Papers; Modes; Multimedia; Nonlinear systems; Physics; Shear layers; Stochastic systems; Stochasticity; Turbulence; Velocity; Vortices; wakes; flow-structure interactions; low-dimensional models
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/jfm.2023.36

This serial work presents a linear-time-invariance (LTI) notion to the Koopman analysis, finding consistent and physically meaningful Koopman modes and addressing a long-standing problem of fluid mechanics: deterministically relating the fluid excitations and corresponding structure reactions. Part 1 (Li et al., Phys. Fluids, vol. 34, no. 12, p. 125136) developed the Koopman-LTI architecture and applied it to a pedagogical prism wake. By a systematic analytical procedure, the Koopman-LTI generated sampling-independent linear models that captured all the recurring dynamics embedded in the input data, finding six corresponding, orthogonal, and in-synch fluid–structure mechanisms. This Part 2 analyses the six modal duplets to underpin their physical implications, providing a phenomenological analysis of the subcritical prism wake. Visualizing the newly proposed dynamic Koopman modes, results show that two mechanisms at St1 = 0.1242 and St5 = 0.0497 describe shear layer dynamics, the associated Bérnard–Kármán shedding and turbulence production, which together overwhelm the upstream and crosswind walls by instigating a reattachment-type of reaction. The on-wind walls’ dynamical similarity renders them a spectrally unified fluid–structure interface. Another four harmonic counterparts, namely the subharmonic at St7 = 0.0683, the second harmonic at St3 = 0.2422, and two ultra-harmonics at St7 = 0.1739 and St13 = 0.1935, govern the downstream wall. Finally, this work discovered the vortex breathing phenomenon, describing the constant energy exchange in the wake's circulation-entrainment-deposition processes. With the Koopman-LTI, one may pinpoint the exact excitations responsible for a specific structure reaction, benefiting future investigations into fluid–structure interactions and nonlinear, stochastic systems.