## 1 Introduction
[13] both researched slug flow patterns in curved pipes with different radii of the curvature  
Figure 1: Subsea M-shaped jumper [2]. and used dynamic slug flow models to simulate completions and throttling under site conditions. [PERSON] et al. [14] simulated a slug flow in a 90-degree bend by fluid-structure interaction and obtained the flow pattern distribution and structural response in the pipe. A higher superficial gas velocity of the slug flows had a more severe periodic effect on the pipeline. [PERSON] et al. [15] carried out an experimental study using a 90-degree bend with an inner diameter of 52.5 mm to investigate the pulsation forces induced by internal two-phase flow over the bend and showed that the dominant frequency of the pulsation forces peaked in the slug flow region and increased monotonically with the superficial gas velocity. [PERSON] et al. [16] analyzed the vibrations caused by bubble flow, slug flow, wave flow, and annular flow in U-shaped tubes and proposed a mathematical model to predict the impact force of the U-tubes. The geometry of the jumper is more complex than that of the elbow, which consists of multiple bends and a long overhanging section. Slug flow exhibits a more complex behavior as it flows through the jumper, where successive elbows cause the transition of the slug characteristics. To the authors' knowledge, the investigations on the two-phase flow in the subsea jumper have mostly been conducted by numerical simulations. [PERSON] et al. [17] analyzed the vibration induced by the slug flow using a combination of CFD and computational structural dynamics (CSD) simulations in a subsea rigid jumper and obtained the frequency of the slug flow by tracking the slug in the pipe using the VOF model. [PERSON] et al. [18] used a three-beam model to analyze the vibration displacement response of the jumper at different slug velocities and found that the fatigue damage degree of the jumper correlated with the flow rate of the slug. [PERSON] et al. [19] analyzed the flow-induced vibrations in a subsea jumper using bi-directional fluid-structure interaction (FSI) simulations, evaluated the impact on the fatigue life of the jumper, and finally concluded that the FSI can be captured using a one-way fluid-structure coupling. [PERSON] et al. [20] set the length and frequency of the slug by numerical simulation to study the interaction characteristics of the two gas-liquid phases in the jumper interior. [PERSON] et al. [21] carried out numerical simulations on the slug in a multi-plane jumper and used a finite element model (FEM) to analyze the fatigue damage of the jumper. [PERSON] et al. [22] conducted a three-dimensional numerical simulation of the vibration process caused by the slug flow in the jumper, estimated the structural response of the jumper, including the fatigue life, and found that the main frequency of the pipe vibration was related to modes 1-4, and the stress is highest when the gas volume fraction was 55%. Later, [PERSON] et al. [23] evaluated the vibration fatigue caused by a flow of water injection across a pipe using a thermoplastic composite pipe (TCP) and used a three-layer screening method to evaluate the degree of vibration fatigue damage expected in this TCP jumper application. [PERSON] et al. [24] evaluated the dynamic response of a jumper through a one-way fluid-structure interaction method and discussed the potential resonance through modal analysis. [PERSON] et al. [25] analyzed the flow-induced vibrations caused by the slug flow in a jumper through numerical simulations and compared the results with current examples in subsea pipeline design to ensure the accuracy of the simulations. In summary, the current research on slug flow in the literature has mainly focused on simple pipelines, such as horizontal straight pipes and 90-degree bend pipes, and the slug flow in a subsea jumper has rarely been involved.