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# Wake states and frequency selection of a streamwise oscillating cylinder

## Abstract

This paper presents the results of an in-depth study of the flow past a streamwise oscillating cylinder, examining the impact of varying the amplitude and frequency of the oscillation, and the Reynolds number of the incoming flow. These findings are presented in a framework that shows that the relationship between the frequency of vortex shedding ${f}_{s}$  and the amplitude of oscillation ${A}^{\ast }$  is governed by two primary factors: the first is a reduction of ${f}_{s}$  proportional to a series in ${A}^{\ast 2}$  over a wide range of driving frequencies and Reynolds numbers; the second is nonlinear synchronization when this adjusted ${f}_{s}$  is in the vicinity of $N= {(1- {f}_{s} / {f}_{d} )}^{- 1}$ , where $N$ is an integer. Typically, the influence of higher-order terms is small, and truncation to the first term of the series ( ${A}^{\ast 2}$ ) well represents the overall trend of vortex shedding frequency as a function of amplitude. However, discontinuous steps are overlaid on this trend due to the nonlinear synchronization. When ${f}_{s}$  is normalized by the Strouhal frequency ${f}_{St}$  (the frequency of vortex shedding from an unperturbed cylinder), the rate at which ${f}_{s} / {f}_{St}$ decreases with amplitude, at least for ${f}_{d} / {f}_{St} = 1$ , shows a linear dependence on the Reynolds number. For a fixed $\mathit{Re}= 175$ , the truncated series shows that the rate of decrease of ${f}_{s} / {f}_{St}$ with amplitude varies as ${(2- {f}_{d} / {f}_{St} )}^{- 1/ 2}$ for $1\leqslant {f}_{d} / {f}_{St} \leqslant 2$ , but is essentially independent of ${f}_{d} / {f}_{St}$  for ${f}_{d} / {f}_{St} \lt 1$ . These trends of the rate of decrease of ${f}_{s}$  with respect to amplitude are also used to predict the amplitudes of oscillation around which synchronization occurs. These predicted amplitudes are shown to fall in regions of the parameter space where synchronized modes occur. Further, for the case of varying ${f}_{d} / {f}_{St}$ , a very reasonable prediction of the amplitude of oscillation required for the onset of synchronization to the mode where ${f}_{s} = 0. 5{f}_{d}$ is given. In a similar manner, amplitudes at which ${f}_{s} = 0$ are calculated, predicting where the natural vortex shedding is completely supplanted by the forcing. These amplitudes are found to coincide approximately with those at which the onset of a symmetric vortex shedding mode is observed. This result is interpreted as meaning that the symmetric shedding mode occurs when the dynamics crosses over from being dominated by the vortex shedding to being dominated by the forcing.

## References

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