Bifurcation sequence in the wakes of a sphere and a cube

A literature review suggests that the flows past simply connected bodies with aspect ratio close to unity and symmetries aligned with the flow follow a consistent sequence of regimes (steady, periodic, quasiperiodic) as the Reynolds number increases. However, evidence is fragmented, and studies are rarely conducted using comparable numerical or experimental set-ups. This paper investigates the wake dynamics of two canonical bluff bodies with distinct symmetries: a cube (discrete) and a sphere (continuous). Employing three-dimensional (3-D) global linear stability analysis and nonlinear simulations within a unified numerical framework, we identify the bifurcation sequence driving these regime transitions. The sequence: a pitchfork bifurcation breaks spatial symmetry; a Hopf bifurcation introduces temporal periodicity ( $St_1$ ); a Neimark–Sacker bifurcation destabilises the periodic orbit, leading to quasiperiodic dynamics with two incommensurate frequencies ( $St_1, St_2$ ). A Newton–Krylov method computes the unstable steady and periodic base flows without imposing symmetry constraints. Linear stability reveals similarities between the cube and sphere in the spatial structure of the leading eigenvectors and in the eigenvalue trajectories approaching instability. This study provides the first confirmation of a Neimark–Sacker bifurcation to quasiperiodicity in these 3-D wakes, using Floquet stability analysis of computed unstable periodic orbits and their Floquet modes. The quasiperiodic regime is described in space and time by the Floquet modes’ effects on the base flow and a spectrum dominated by the two incommensurate frequencies and tones arising from nonlinear interactions. Although demonstrated for a cube and a sphere, this bifurcation sequence, leading from steady state to quasiperiodic dynamics, suggests broader applicability beyond these geometries.