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.

Figures (10)

• Figure 1: Graphical representation of the invariant manifold in physical space for a specific point ${\boldsymbol{x}}^*$ (left), and its latent space counterpart (right). The stable () and unstable () steady solutions, and some trajectories () are reported.
• Figure 2: Geometry of the Turek-Schäfer benchmark for the flow around a cylinder in a 2D channel. On the boundary, no-slip (), non-homogeneous Dirichlet (), and homogeneous Neumann () boundary conditions are applied.
• Figure 3: Real (top) and imaginary (bottom) parts of the eigenvalue of the bifurcating mode over a range of $Re$. The values computed by the FOM ($\circ$), requiring a linear eigenproblem per point, are well aligned with those predicted a priori by three ROMs at $Re_0 = 20$ (), $Re_0 = Re_{c}$ () (corresponding to the choice made in Carini2015), and $Re_0 = 70$ (). The real and imaginary parts of the corresponding eigenmode at bifurcation are also shown. Being the eigenmode defined up to an arbitrary phase, their allocation to the top and bottom plots is only for illustrative purposes.
• Figure 4: Bifurcation diagram of the average turbulent kinetic energy with respect to the Reynolds number: FOM ($\circ$); ROMs at $Re_0 = 20$ (), $Re_0 = Re_{c}$ () Carini2015, $Re_0 = 70$ (), and $Re_0 = 80$ (). The expansion point has a large effect on the accuracy of the prediction, both in the vicinity of the bifurcation and at high $Re$.
• Figure 5: Bifurcation diagram of the average turbulent kinetic energy with respect to the Reynolds number: FOM ($\circ$); ROM at $Re_0 = Re_{c}$, using the graph style parametrisation \ref{['eq:constraints_graph_style']}, at orders $3$ (), $5$ (), $7$ (), and $9$ (). The ROMs agree up to $Re \approx 51$, but separate for higher $Re$ where the asymptotic expansion has not reached convergence.