## 4 Conclusions  
In this paper, a high-efficiency VLES method is adopted to simulate the VIV phenomenon of the oscillating cylinder at a subcritical Reynolds number \(Re=4000\) combined with the dynamic grid technique. A series of simulations of \(A/D=0.5\) to \(1.5\) and \(f_{c}/\!\!f_{s}=0.5\) to \(1.35\) are conducted, and the temporal evolutions of the lift and drag coefficients, vortex shedding frequency, and wake structures are discussed in detail. The main conclusions are as follows:  
1. The VLES method combined with the dynamic mesh technology is adopted to simulate the flow past an oscillating cylinder in this paper, which is also verified to be able to effectively handle the dynamic boundary problems, such as the oscillating cylinder. 2. The components of vortex shedding frequency of an oscillating cylinder are first determined by a \"lock-in\" bound, which is labelled as \"lock-in\" line at \(f_{c}/\!\!f_{s}=0.75\) in this paper. The vortex shedding is dominated by the excitation frequency and the Strouhl frequency when \(f_{c}/\!\!f_{s}<0.75\), but the vortex shedding is only determined by a dominated vortex shedding frequency when \(f_{c}/\!\!f_{s}\geq 0.75\). The latter is within the range of the \"lock-in\" phenomenon. The temporal evolutions of the lift and drag coefficients show an obvious \"beating\" characteristic in the condition of the \"lock-in\" phenomenon. Even at higher excitation amplitude, multiple nondominant frequencies are also involved. 3. The vortices located in the \"lock-in\" line maintain the same shedding mode and level of strength in the wake per cycle, and the temporal evolutions of lift and drag coefficients also perform periodic fluctuation. For a lower excitation frequency, the vortex shedding mode shows a 2S mode. With an increase in the excitation amplitude, the vortex shedding mode turns into a 2P\({}_{0}\) or 2P mode, corresponding to lower oscillation frequency ratios and higher ones, respectively. As the excitation amplitude continues to increase, the vortex shedding mode exhibits a P + S mode in high oscillation frequency ratios. 4. As the oscillating frequency ratio increases, the vortex shedding mode also changes. The 2P mode characterizes a combined mode including a strong vortex pair and a weak one in a higher oscillation frequency ratio and a medium excitation amplitude. The vortex shedding mode varies from a 2P\({}_{0}\) or 2P mode to a P + S mode under the condition of a high frequency ratio and a  
Figure 13: The vortex formation modes in the computed range at \(Re=4000\). The symbol \(\times\) denotes a case that is not within the “lock-in’ range and the symbol denotes a case that is within the “lock-in” range. P\({}_{s}\), strong vortex pair; P\({}_{w}\), weak vortex pair; S\({}_{w}\), weak single vortex. The unstable status means that the vortex shedding is a vortex pair and a single vortex without regularity. The red arrow represents that the “lock-in” range is located in the right of the “lock-in” line. medium or high excitation amplitude. The vortex shedding mode is an unstable status of the P + S mode for a medium frequency ratio and a high excitation amplitude. However, the case of a higher excitation amplitude and oscillation frequency ratio has a P + S mode featuring a strong vortex pair and a single weak vortex. * The vortex shedding is a lasting process under the condition of a low excitation amplitude, leading the lift and drag coefficients to fluctuate irregularly. For a vortex shedding mode exhibiting a strong vortex pair and a weak vortex pair or a weak single vortex, the temporal evolution of the lift coefficient of the oscillating cylinder shows an irregular \"jumping\" at the specific time per cycle corresponding to the shedding of the strong vortex pair. Moreover, the vortex shedding usually occurs when the direction of oscillation switches. The vortex shedding mode, and the frequency and time of the vortex shedding co-determine the temporal evolutions of the lift and drag coefficients. Investigation, Z.X. ; methodology, Z.X.