Reduced order modelling of Hopf bifurcations for the Navier-Stokes equations  through invariant manifolds.

Abstract

This work introduces a parametric simulation-free reduced-order model for incompressible flows undergoing a Hopf bifurcation, leveraging the parametrization method for invariant manifolds. Unlike data-driven approaches, this method operates directly on the governing equations, eliminating the need for full-order simulations. The proposed model is computed at a single value of the bifurcation parameter yet remains valid over a range of values. The approach systematically constructs an invariant manifold and embedded dynamics, providing an accurate and efficient reduction of the original system. The ability to capture precritical steady states, the bifurcation point, and postcritical limit cycle oscillations is demonstrated by a strong agreement between the reduced-order model and full-order simulations, while achieving significant computational speedup.