Bifurcation analysis in the regularized four-sided cavity flow: Equilibrium states in a $D 2$-symmetric fluid system

In this study, we investigate a handful of equilibrium states and their hydrodynamic stability for the incompressible flow in a square cavity driven by the tangential simultaneous motion of all of its lids at the same speed. This problem has been investigated in the recent past, albeit only partially and always disregarding the singular boundary conditions at the corners. The implications of the system symmetries have not been rigorously addressed, either. In our study, we introduce a regularized version of the boundary conditions to avoid the corner singularities, in a way that preserves the main features of the original setup. In addition, the Navier–Stokes equations are discretized by means of highly accurate Chebyshev spectral methods that provide exponential convergence of all computed flows and consistently eliminate any potential source of structural instability of the bifurcation scenario. We employ Newton–Krylov solvers, implemented within continuation algorithms, to accurately compute equilibrium solutions. Linear stability analysis of both the primary symmetric base flow and the secondary asymmetric states uncovers new branches of fully asymmetric steady states. The analysis has allowed identification of six previously undetected bifurcations, all of which are associated with the disruption of either the rotational invariance or the reflection symmetry. Some of these bifurcations have been found to be quite clustered in some regions of the parameter space, which points at the underlying action of higher codimension mechanisms. Notably, all the bifurcations reported occur within the range of low to moderate Reynolds numbers, making the regularized four-sided lid-driven cavity flow a reliable benchmark for assessing different numerical schemes in the context of Navier–Stokes equivariant bifurcation theory.