## 5 Conclusions  
Through the multiphase flow and vibration experimental system of Dalian Maritime University, the properties of a gas-liquid slug flow and its flow-induced vibration in an M-shaped jumper were studied through simulations and experiments. Analysis of the slug characteristics and their excitation characteristics of the slug flow in the M-shaped jumper at various gas-liquid superficial velocities was conducted, and the results of the simulation were compared with the experimental data. The evolution of the flow pattern and the corresponding pressure variation of the slug flow in the M-shaped jumper were investigated. The conclusions obtained from the study are as follows:  
As the slug flow went through the jumper, it ascended in the shape of a bullet bubble in section S1. In section S2, the flow pattern of the slug flow underwent a transition between the wave flow and the slug flow.  
When the slug flow flowed through the jumper, there were periodic fluctuations in the pressure as it passed through the pressure-monitoring point. When passing through the elbow, the pressure decreased due to friction loss and the interaction between the gas and the liquid. In the first half of this section, there were obvious spikes in the pressure, and a clear separation of the gas and liquid phases was observed.  
After passing through continuous bends and long overhanging sections, the gas-liquid phase was greatly disturbed, and the pressure experienced secondary fluctuations. However, due to the influence of the flow rate in the first half, the second half exhibited different pressure variation characteristics from the previous part. In addition to the flow pattern and pressure fluctuation structure in the time domain, it can also be seen that at P6, P7, and P8, there were obvious second harmonic frequencies in addition to the main frequency.  
When the gas-liquid mixing superficial velocity was larger than 1.5 m/s, it was discovered that the slug characteristics, such as the slug frequency and slug velocity in the M-shaped jumper, increased with the increase in the gas-liquid mixing velocity. It is worth mentioning that increasing the superficial gas velocity resulted in a faster increase in the slug frequency than increasing the superficial liquid velocity. The vibration characteristics of the jumper were highly correlated with the flow characteristics of the slug flow. The increase in the slug frequency increased the vibration amplitude of the jumper. It is important to note that when the slug frequency was close to the natural frequency of the jumper, resonance occurred, causing the jumper to exhibit large vibrations.  
The employed VOF and one-way fluid-structure coupling model of the present numerical simulations showed good applicability by validation against the experimental data. The results of the slug characteristics obtained from the numerical simulations are in good agreement with the experimental results.  
Conceptualization, W.L., J.L., G.Y. and M.C.O.; methodology, W.L., J.L., G.Y. and M.C.O.; software, W.L. and J.L.; validation, W.L. and J.L.; formal analysis, W.L. and J.L.; investigation, W.L., J.L., G.Y. and M.C.O.; resources, W.L.; data curation, W.L. and J.L.; writing--original draft preparation, W.L. and J.L.; writing--review and editing, W.L., J.L., G.Y. and M.C.O.; visualization, W.L. and J.L.; project administration, W.L.; funding acquisition, W.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (52271258), the Liaoning Revitalization Talents Program (XLYC2007092), the 111 Project (B18009), the Open Fund of the National Center for International Research of Subsea Engineering Technology and Equipment (No. 3132023352), and the Cultivation Program for the Excellent Doctoral Dissertation of Dalian Maritime University (2022 YBPY002).  
Not applicable.  
The authors declare no conflict of interest.