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

TL;DR

This paper presents a parametric reduced order model for incompressible flows experiencing Hopf bifurcations, constructed via invariant manifolds, enabling efficient simulations across parameter ranges without full-order computations.

Contribution

It introduces a novel, simulation-free reduced order modeling approach based on invariant manifolds that works directly on governing equations for bifurcation analysis.

Findings

Accurately captures steady states, bifurcation points, and limit cycles.

Provides significant computational speed-up over full simulations.

Demonstrates effectiveness across a range of bifurcation parameters.

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.