Vortex shedding patterns, their competition, and chaos in flow past inline oscillating rectangular cylinders

Different vortex shedding patterns arising in the flow past inline oscillating rectangular cylinders, at a Reynolds number of 200 is studied numerically in two-dimensions. The S-II mode of symmetric shedding, discovered in 2006, as well as the Couder-Basdevant mode [J. Fluid Mech. 173, 225-251 (1986)], seen in experiments earlier, are found numerically for the first time. Besides, a new mode of symmetric shedding, named here as S-III, is also reported. Chaotic flow in the wake of a circular cylinder, recently reported by Perdikaris \textit{et al}. [Phys. Fluids 21(10), 101705 (2009)] is also seen in flow past the rectangular geometries here, and we show that this is indeed due to mode competition, between antisymmetric and symmetric modes of vortex shedding, in the sense of Ciliberto \& Gollub [Phys. Rev. Lett. 52, 922 (1984)]. A global and reliable parameter has been constructed to ``quantify'' this chaos. The Lattice Boltzmann Method (LBM) has been used to solve for the flow.