The linear-time-invariance notion to the Koopman analysis: The architecture, pedagogical rendering, and fluid–structure association

Abstract

This work augments a Linear-Time-Invariance (LTI) notion to the Koopman analysis, finding an invariant subspace on which consistent Koopman modes are expanded with fluid mechanics implications. The work also develops the Koopman-LTI architecture—a systematic procedure to associate fluid excitation and structure surface pressure by matching Koopman eigen tuples, establishing fluid–structure correspondences that examine fluid–structure interactions (FSIs) at new angles. The data-driven, modular architecture also exhibits the potential to evolve with advances in Koopman algorithms. A pedagogical prism wake example demonstrated that the Koopman-LTI generated a near-perfect linearization of nonlinear FSI dynamics involving inhomogeneous anisotropic turbulence, with mean and root-mean-squared errors of O−12 and O−9, respectively; the infinite-dimensional Koopman modes were also approximated with O−8 error. The subcritical wake during shear layer transition II was also reduced into only six dominant excitation-response Koopman duplets. The upstream and crosswind walls constitute a dynamically unified interface dominated by only two mechanisms. The downstream wall remains a distinct interface and is dominated by four other mechanisms. The complete revelation of the prism wake comes down to understanding the six mechanisms, which Part II [Li et al., “A parametric and feasibility study for data sampling of the dynamic mode decomposition: Range, resolution, and universal convergence states,” Nonlinear Dyn. 107(4), 3683–3707 (2022)] will address by investigating the physics implications of the duplets' in-synch phenomenological features. Finally, the analysis revealed z-velocity's marginal role in the convection-dominated free-shear flow, Reynolds stresses' spectral description of cascading eddies, wake vortices' sensitivity to dilation and indifference to distortion, and structure responses' origin in vortex activities.