## 2 Experimental Work  
As illustrated in Figure 2, an experimental investigation into the slug characteristics of a subsea M-shaped jumper's inner slug flow and the flow-induced vibration was conducted with the support of Dalian Maritime University's multiphase flow experimental platform. The experimental circuit is mainly composed of one three-phase separator, two pumps (oil pump and water pump), a flow-regulating valve, air compressors, an air bottle, a freeze dryer, mixing pipelines, and various experimental circuits (including jumpers). Two air compressors with a combined rated volume flow rate of 1020 m\({}^{3}\)/h and a maximum exhaust pressure of 20 bar are available in the lab to deliver compressed air for the experimental gas phase. The air compressor compresses the air and then enters the air bottle with a volume of 2 m\({}^{3}\) and a 3 MPa maximum operating pressure. The laboratory is equipped with two vertical multistage centrifugal pumps to provide water for the liquid phase in the experiment, for one single pump with a maximum flow rate of 100 m\({}^{3}\)/h. Proportional-integral-derivative (PID) control was utilized during the investigation to precisely control the gas flow-regulating valve and the pump variable frequency. Through the regulation of the ball valve opening after air and water have been combined in the gas-liquid mixing part, the gas-liquid flows into the experimental pipelines for experimental research, flows out from the outlet of the experimental section, and finally, the mixture enters the three-phase separator through the pipeline. After passing through the three-phase separator, the two vertical centrifugal pumps are used to pump the bottom water into the experimental pipe to create a circulating flow, and the separated air enters the atmosphere or is used to maintain a specific back-pressure. Figure 2: Multi-phase flow experimental circuit [2]. The jumper consists of two ascending portions, three horizontal portions, and two descending portions, as depicted in Figure 3. The gas-liquid mixture flow in the jumper passes through six elbows, not including the inlet and outlet connection pipelines. The jumper parts of S1-S7 are flanged connections and are swapped out for transparent windows of the same size to visualize the slug flow inside the jumper. The transparent window is made of Plexiglas material to ensure that it is transparent to meet observational standards and can sustain a fluid pressure of up to 2 MPa. Additionally, reinforced support is used to guarantee the security of the piping connections and transparent windows. Table 1 displays the jumper's physical properties. The experimental jumper has a diameter of 52 mm, an inner diameter of 48 mm, and a wall thickness of 4 mm. The physical properties of the two experiment-related media are listed in Table 2. Given their low flow rates and flow pressures, the compressibility of the fluids was not considered. \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Values** & **Dimension** \\ \hline Total length & 3.6 & m \\ Inner diameter & 0.048 & m \\ Wall thickness & 0.004 & m \\ Curvature radius & 1.5 D & \\ Transparent section length & 0.23 & m \\ Density & 7850 & kg/m\({}^{3}\) \\ Poisson’s ratio & 0.3 & \\ Young’s modulus & \(2.06\times 10^{5}\) & MPa \\ \hline \end{tabular}
\end{table} 
Table 1: The subsea jumper model’s parameters. \begin{table}
\begin{tabular}{c c c c} \hline \(\rho_{g}\) & \(\rho_{1}\) & \({}^{\mu}_{g}\) & \(\mu_{1}\) & **Surface Tension Coefficient** \\ (kg/m\({}^{3}\)) & (kg/m\({}^{3}\)) & (kg/m\(\cdot\)s) & (kg/m\(\cdot\)s) & (N/m) \\ \hline
1.29 & 998.2 & \(1.8\times 10^{-5}\) & \(1.0\times 10^{-3}\) & 0.072 \\ \hline \end{tabular}
\end{table} 
Table 2: The physical properties of the fluids. Figure 3: Model of the M-shaped jumper. A high-speed camera was positioned to capture the slug characteristics of the slug flow in the jumper.