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 parametrisation 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 pre-critical steady states, the bifurcation point, and post-critical limit cycle oscillations is demonstrated by a strong agreement between the reduced order model and full order simulations, while achieving significant computational speed-up.