## 1 Introduction  
As the exploitation of offshore oil and gas expands into deep water and regions with strong ocean currents, the significance of vortex-induced vibration (VIV) has attracted the attention of engineers. However, some grave engineering problems are due mainly to VIV, similar to fatigue failure on submarine pipelines. Therefore, the prediction of the hydrodynamic forces on an oscillating cylinder is still paramount to the design of the structures. Numerical and experimental studies on the flow around a cylinder oscillating in the fluid have always been the focus of related studies for many years. However, it is still a significant primary task because of the complexity of flow characteristics. At present, one of the common methods to determine the VIV phenomenon is a circular cylinder that is forced to oscillate with a preset motion including an excitation amplitude (\(A\)) and a frequency (\(f_{c}\)) approximating VIV. Given the richness of the research on VIV, only a forced oscillating cylinder problem is included in the following discussion. [PERSON] et al. conducted an experimental study on the hydrodynamic forces acting on a circular cylinder and found that an abrupt jump existed in the phase angle between the cylinder motion and the lift force when the oscillation frequency (\(f_{c}\)) approximately equaled the natural shedding frequency (\(f_{s}\)) of the stationary cylinder [1]. In addition, the average amplitude of the lift and drag increased when the frequency ratio satisfied \(f_{c}f_{s}\approx 1\). This condition is also known as a \"lock-in\" phenomenon, which signifies that the vortex shedding frequency of the oscillating cylinder (\(f_{v}\)) primarily depends on the oscillation frequency \(f_{c}\), i.e., \(f_{v}=f_{c}\). Additionally, it also means that the vortex shedding frequency \(f_{v}\) is no longer determined by the Strouhal number (\(St\)). [PERSON] and [PERSON] studied the flow characteristics of a circular cylinder at various Reynolds numbers (_Re_), _Re_ = 300 to 1000 [2] and obtained three main vortex shedding modes that existed in their experiments when meeting _fe_/_f_s \(\approx\) 1, including 2S, 2P, and P + S modes. The 2S mode signifies two single vortices shed per oscillation cycle, which is also the natural Karman mode of shedding. The 2P mode indicates two count-rotating vortex pairs formed per oscillation cycle, and the P + S asymmetric mode shows a vortex pair on one side and a single vortex on the other side. Moreover, the vortex shedding mode changes from 2P to 2S when _fe_/_f_s increased through unity. [PERSON] and [PERSON] experimentally investigated the amplitude-response prediction of a forced oscillation with much higher resolution. In a high-amplitude regime, they found two vortex shedding modes overlap at _Re_ = 4000 and 12,000 [3]. This vortex formation mode includes two vortex pairs per oscillation cycle. Importantly, the primary vortex in each vortex pair is stronger than the secondary, which also is recognized as a 2P overlap mode (2P\({}_{0}\)) in the condition of _fe_/_f_s \(\approx\) 1. [PERSON] et al. researched the wake state by using controlled oscillation experiments based on amplitude-to-cylinder ratios (_A_/_D_) that increased from 0.25 to 0.6 at _Re_ = 2300, 4410, and 9100 [4], where \(D\) is the cylinder diameter. The high- and low-frequency wake states showed the 2S and 2P modes, respectively, which also impacted the near wake structure and the lift. The high- and low-frequency wake states were transited through an intermediate wake state. [PERSON] et al. conducted a high-fidelity large-eddy simulation (LES) study to investigate the two-degree-of-freedom (2 DOF) VIV phenomenon of a low mass ratio circular cylinder at _Re_ = 3900, 5300, and 11000 [5; 6]. Their simulations presented that a 2T vortex formation mode, namely two triplets of vortices per cycle, was an inherent feature of the flow in the super-upper branch. [PERSON] et al.