On quasiperiodic and subharmonic Floquet wake instabilities

The physical characteristics of bifurcated states in systems with inherent symmetry are constrained in ways that those in systems with broken symmetry are not. Here we examine the issue of quasiperiodic versus subharmonic instability modes of time-periodic laminar wakes, and how the relationship between them is influenced by weak symmetry breaking. The examples used are the vortex street wake of a circular cylinder, where symmetry is broken by distorting the cylinder into a ring, and the wake of a square cylinder, where symmetry is broken by a small fixed rotation of the cylinder about its axis. In both cases the symmetric wakes exhibit a quasiperiodic instability mode, with a pair of complex-conjugate Floquet multipliers and which manifests as a traveling wave. As symmetry is broken these multipliers migrate continuously to the real axis, coalesce, and split into a pair of subharmonic multipliers that move apart along the negative real axis. This behavior resolves an apparent dichotomy between the previously established theoretical results and numerical predictions for the symmetric wake systems, and the predictions and experimental observations for systems with weakly broken symmetry.