The Linear-Time-Invariance Notion of the Koopman Analysis-Part 1: The Architecture, Practical Rendering on the Prism Wake, and Fluid-Structure Association

Abstract

This work proposes a Linear-Time-Invariance (LTI) notion to the Koopman analysis, finding an invariant subspace on which Koopman modes are consistent and physically meaningful. It also develops the Koopman-LTI architecture -- a systematic procedure to associate fluid excitation and structure surface pressure by matching Koopman eigen tuples, solving a longstanding problem for fluid-structure interactions. The architecture is data-driven and modular, accommodating all types of data and Koopman algorithms. Through a pedagogical demonstration on a prism wake and the rudimentary Dynamic Mode Decomposition algorithm, results show a near-exact linearization of nonlinear turbulence, with mean and rms errors of O-12 and O-9, respectively. The DMD also approximated the Koopman modes with O-8 error. The LTI reduced the subcritical prism wake during shear layer transition II into only six dominant excitation-response Koopman modal 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 essentially comes down to understanding the six mechanisms, which Part 2 (Li et al., 2022) will address by investigating the physical interpretations of the duplets' in-synch, phenomenological features. Finally, the current analysis also revealed w's trivial role in this convection-dominated free-shear flow, Reynolds stresses' spectral description of cascading eddies, vortices' sensitivity to dilation and indifference to distortion, and structure responses' origin in vortex activities.