Dynamics and Control of a Nine-Mode Reduced-Order Model of the 2D Navier-Stokes Equations

This paper investigates the dynamics and control of the two-Dimensional (2D) Kolmogorov flow, a canonical model in fluid dynamics governed by the 2D incompressible Navier-Stokes equations with periodic boundary conditions and a sinusoidal external forcing in the x-direction. To study this system, a Fourier-Galerkin spectral method is used to derive a reduced-order model consisting of nine nonlinear Ordinary Differential Equations (ODEs) that capture the essential features of the Kolmogorov flow. Compared to an earlier model, the addition of two extra modes allows the new system to exhibit dynamical features such as hysteresis and more intricate bifurcation patterns, which were not captured in previous formulations. The resulting ninth-order ODE system is analyzed in detail to explore its rich dynamical behavior and underlying symmetries across a range of Reynolds numbers 0 < Re < 30. Numerical simulations demonstrate the model's ability to reproduce diverse dynamical regimes, including steady states, periodic orbits, and chaotic attractors, confirming the fidelity of the reduced-order representation. Moreover, a Lyapunov-based feedback control strategy is formulated to regulate the system and stabilize it toward desired invariant sets, such as equilibria, periodic trajectories, or chaotic attractors. Numerical experiments are conducted to validate the effectiveness of the proposed control scheme. These findings contribute to broader understanding of nonlinear flow dynamics and control in reduced-order settings and may inform future developments in flow regulation and turbulence management.