Koopman reduced-order modeling and analysis of flag flapping in the wake of a cylinder

We develop a Koopman reduced-order model (ROM) to analyze the instability mechanism and predict the hydrodynamic behavior for the flag flapping in the wake of a cylinder. The Koopman ROM is constructed using a kernel dynamic mode decomposition method and enhanced through a residual dynamical mode decomposition algorithm, which improves accuracy by identifying and eliminating spurious modes. Our analysis reveals a flow transition from the "2S" mode in the periodic phase to the "2P" mode in the quasiperiodic phase, with the main Koopman mode M₁ providing insights into the instability mechanism. In the case of chaotic flapping at a Reynolds number of Re=1200, the Koopman ROM demonstrates high accuracy in predicting the chaotic fluid-structure interaction flow when comparing the fractal dimension d₂ and the maximum Lyapunov exponent λ₌₀ₗ of the true and reconstructed flow. Additionally, we observe similar flag flapping and vortex shedding characteristics in the near-structure region throughout the investigated Reynolds number range Re∈500, 1200, leading to similar vorticity patterns for M₁. The flag flapping has a local minimum displacement at position x₀≈3. 0 in the flag's displacement envelope. Notably, this position closely matches the critical position x₂ obtained from global linear instability analysis at low Reynolds numbers. This conclusion aligns with the performance of the Koopman ROM using sparse measurement probes attached to the flag. The position of the probes influences the accuracy of the ROM, where higher accuracy corresponds to a larger displacement of the eigenfunction.