Determine whether the two-dimensional driven-cavity flow is beyond its first Hopf bifurcation at Re=10,000

Determine whether the steady two-dimensional driven-cavity Navier–Stokes flow at Reynolds number Re=10,000 lies beyond the first Hopf bifurcation, thereby clarifying whether the corresponding flow has transitioned from steady to time-dependent behavior.

Background

The paper uses the two-dimensional driven-cavity problem at Reynolds numbers 5,000 and 10,000 to test viscosity recovery and the CDA-Picard+CDA-Newton solver. The authors note that steady solutions exist at Re=10,000 for their computations, even though the behavior of time-dependent simulations at this parameter remains unsettled.

The unresolved issue concerns whether the Re=10,000 driven-cavity flow has passed the first Hopf bifurcation. Resolving this question would distinguish the existence of a computed steady solution from the dynamical stability of that solution and would provide important context for interpreting the numerical benchmark.

References

This is a harder problem, and although the literature is not settled on whether this is past the first Hopf birfurcation , steady solutions still exist at this $Re$ even if time dependent ones do not converge to those steady solutions .

— Continuous data assimilation in steady Navier-Stokes equations with unknown viscosity: robust and efficient solvers and fast parameter recovery  (2609.02862 - Rebholz et al., 2 Sep 2026) in Section 6.1, “2D driven cavity”