## 1 Introduction  
In recent years, the safety of submarine oil and gas transportation pipelines has become a key priority due to the gradual expansion of oil and gas exploitation in the deep sea, and pipeline vibration is the primary source of fatigue damage. According to a study by the UK Health and Safety Executive (HSE), 21% of pipeline damage in the UK's offshore industry in the North Sea is due to vibration-induced pipeline fatigue failure [1]. Subsea jumpers play a vital role in oil and gas production systems, connecting elements of subsea structures, such as pipe manifolds, Christmas trees, risers, and pipeline terminals. As shown in Figure 1, compared to ordinary subsea pipelines, the subsea jumper has a unique multi-bending structure and long overhanging sections, which leads to a greater impact on the pipeline in the process of oil and gas transportation due to a slug flow formed inside the pipeline. Slug flow is a transient flow. In the classification method of multiphase flow patterns, based on the flow medium's continuity, the state of the medium is diffuse, intermittent, and continuous. Based on this concept, slug flow can be defined as an intermittent flow state. Since the subsea jumper has several elbows and straight pipes, some of the methods and conclusions obtained by studying the flow inside the elbows and straight pipes and their vibration patterns can still be applied to the subsea jumper. For the characteristics of the internal slug flow of straight pipes and the vibration induced by the slug flow of straight pipes, many researchers have carried out investigations by theoretical analysis, experiments, and numerical simulations. To determine the relationships among two-phase flow parameters, including the slug length, slip ratio, void ratio, and pipe vibration, [PERSON] et al. [3] conducted an experimental investigation of the slug flow-induced vibration characteristics of a horizontal pipe with a throttle aperture. [PERSON] [4] used unidirectional fluid-structure interaction simulations and experimental methods to analyze how horizontal pipes and slug flow interacted. The simulation and experimental results were in good agreement. [PERSON]. [5] studied the nonlinear planar vibration of risers excited by a two-phase flow using a model of a steady-state slug flow and provided guidance for the design of a riser considering the slug frequency and slug length. [PERSON] et al. [6] conducted an experimental investigation of slug flow in a horizontal pipe and found that the vibration displacement of the pipe increased with the increasing liquid velocity at a specific air velocity. [PERSON] et al. [7] conducted an experimental study on a free-hanging flexible riser with three slug flow patterns at specific flow rates. [PERSON] et al. [8] conducted an experimental investigation on a gas-liquid two-phase slug flow inside a horizontal pipe with an inner diameter of 26 mm. The mechanism of slug formation and the pressure fluctuations were investigated by visual observation. It was found that the slug was caused by Kelvin-Helmholtz instability. In addition, the slug were formed at high gas velocities by merged multiple wave-like flows. [PERSON] et al. [9] used numerical simulations to explore the Marangoni boundary condition to explore the MHD nonlinear convective nanofluid and hybrid nanofluid flow and heat transport over a vertical flat plate with the Marangoni boundary condition. [PERSON] et al. [10] used a combination of experimental and numerical simulations to study the slug flow characteristics in horizontal bends and summarized a law governing the fluid distribution, void fraction, pressure, and liquid film velocity in the pipe under slug flow conditions. [PERSON] et al. [11] used numerical simulations to deal with a bioconvective 2D THNF flow containing a microbial suspension toward three different geometries (a flat plate, a wedge, and a cone) with the suction/injection effect. [PERSON] et al. [12] and [PERSON] et al.