Three-dimensional Floquet instability of the wake of square cylinder

TLDR

The study investigates when a square cylinder’s two‑dimensional periodic wake becomes three‑dimensional. Floquet stability analysis is used to extract and examine the space–time symmetries of the resulting three‑dimensional instability modes. The transition mirrors that of a circular cylinder, with a long‑wavelength mode becoming unstable first at Re≈161, a short‑wavelength mode at Re≈190, and an intermediate subharmonic mode at Re≈200 that breaks reflection symmetry and appears twice.

Abstract

In this study we investigate the onset of three-dimensionality in an otherwise two-dimensional periodic wake of a square cylinder. Floquet stability analysis is employed to extract the different modes of three-dimensional instabilities. It is observed that the three-dimensional transition process for a square cylinder is similar to that of a circular cylinder. Most notably, there is a long-wavelength (mode A) three-dimensional disturbance that becomes unstable first at a Reynolds number of about 161, followed by a short-wavelength (mode B) three-dimensional disturbance that becomes unstable at a Reynolds number of about 190. In addition, a third intermediate-wavelength mode is also observed to become unstable at around Re=200. Unlike modes A and B, the intermediate-wavelength mode is subharmonic with a period of twice the shedding period of the two-dimensional base state. This mode also breaks the reflection translation symmetry observed in the other two modes and as a result appears with multiplicity two. The space–time symmetries of the three modes are explored in detail.