## 1 Introduction  
As a necessary channel between the offshore platform and the subsea wellhead, the riser is subjected to environmental loads such as waves and earthquakes and impact loads brought by the movement of the upper floating body. Especially when the frequency of the fluid vortex is close to the natural vibration frequency of the riser due to fluid vortex excitation, the structure will move substantially, which directly threatens the safe operation and service life of the riser [1]. In engineering practice, the offshore platform at the top of the riser will move periodically under the action of many complex factors, driving the top of the riser to reciprocate in the water, thus forming a relative oscillating flow between the riser and the water. Under the action of oscillating flow superposition, the response characteristics of the vortex-induced vibration of the riser are quite different from those of steady flow and are more complex. Therefore, it is of great significance to study the vortex-induced vibration response of risers under the action of oscillating flow superposition. [PERSON] et al. [2] have systematically compared the simulation ability of five aerodynamic damping models for the vortex-induced vibration amplitude of linear structures under different mass damping conditions, which is crucial for the accurate evaluation of the vortex-induced vibration of structures. [PERSON] et al. [3] have studied the vortex-induced vibration characteristics of a streamlined box girder with a water-filled barrier through two-dimensional computational fluid dynamics (CFD) simulation, showing that the mean and standard deviation of the pressure coefficient and the superior frequency of the distributed pressure are closely related to each stage of the vortex-induced vibration development process. Based on the slicing method, the Wan, D.C. team [4; 5] has developed a computational fluid dynamics (CFD) solver, vivFOAMSJTU, on the OpenFOAM platform to simulate and verify the standard vortex-induced vibration of slender flexible risers, including accurately predicting the mode and amplitude of the riser in the forward- and cross-flow direction. [PERSON] [6] has numerically simulated the transverse vortex-induced vibration of a cylinder in an oscillating flow. Furthermore, he also found that oscillating flow could excite multi-mode vibration of the cylinder, and that the larger the KC number, the more obvious the modal characteristics. [PERSON] [7] has numerically simulated the vortex-induced vibration of a two-dimensional rigid cylinder that oscillates periodically in still water and found that the lateral vibration of the cylinder exacerbated the variation of the lift coefficient, causing significant changes in the direction of the discharge vortex and the position of the vortex separation point on the surface of the cylinder. [PERSON] [8] has used the slice method to numerically simulate and compare the vortex-induced vibration response characteristics of two end hinged risers and cantilever risers. Additionally, he found that both showed the same variation trend in amplitude and dominant vibration frequency. [PERSON] et al. [9] have simulated the vortex-induced vibration of a cylinder under the combined action of oscillating flow and uniform flow, and they discussed the response of flow ratio a to the cylinder. It was found that when a= 0.8 and \(\mathrm{Vr}=7\), the vortex experienced 2S, 2P and 2T leakage vortex modes in one oscillating flow period. [PERSON] [10] numerically simulated the forced and self-excited vibration of a single degree of freedom elastic supported cylinder under the combined action of uniform flow and oscillating flow at a low Reynolds number and found that the locking range and vibration amplitude of the structure changed significantly when the oscillating flow was involved. [PERSON] et al.