N.W. Mureithi*

Department of Mechanical Engineering, École Polytechnique, Box 6079, Station Centre-ville, Montréal, QC, H3C 3A7, Canada

**ABSTRACT**

A fundamental symmetry group theoretic analysis of the periodically forced cylinder wake dynamics problem is presented. The analysis yields a clear and concise explanation of experimentally observed cylinder wake dynamics.

First it is shown that the underlying problem can be posed as a mode interaction problem subject to *O*(2) symmetry constraints. An analysis of the amplitude equations for **S**/**K** mode interactions predicts the experimentally observed period-doubling instabilities when λ_{s}/λ_{k} = 1/1. For 2/1 wavelength ratio, it is shown that traveling wave solutions are to be expected. Furthermore, the apparent lack of interactions for 3/1 ratio is shown to be due to the fact that important nonlinear coupling terms appear only at 7th order.

For asymmetrical **K1**/**K** mode interactions, the symmetry 'compatibility', via common subgroups, explains the strong resonances observed experimentally for 1/1 and 1/3 frequency ratios. For a frequency ratio 1/2, it is shown that **K1** and **K** mode symmetries are incompatible. Consequently, traveling wave solutions most likely underlie the organized wake topology observed in experiments.

*Keywords: *
Karman wake; Mode interactions; Bifurcations; Symmetry group; Normal forms