# A Numerical Simulation of the Underwater Supersonic Gas Jet Evolution and Its Induced Noise
## 3 Results and Discussion
### The Development of Underwater Supersonic Jets  
Figure 3 illustrates the pressure contours of the jet flow field at different times under total pressures \(p_{0}\) = 15 and 10 MPa, where the black solid and dashed lines represent the isolines of the gas phase volume fraction at 0.1 and 0.9, respectively. At the same water depth, the pressure distribution and bubble shape of the under-expanded and fully expanded jets are very similar. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**OASPL** & **Measuring Point A** & **Measuring Point B** & **Measuring Point C** \\ \hline Experimental results & 190 & 189 & 184 \\ Simulation results of the present work & 195.3 & 195.9 & 193.3 \\ Relative error & +2.8\% & +3.7\% & +5.1\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the calculated OASPL at measurement points with the experimental results [23]. Figure 3: Pressure distributions at different times. The underwater supersonic jet gradually evolves from the bubble jet stage before \(t=70\) ms to the conical jet stage, which is consistent with the development pattern of the gas-liquid interface observed in previous experiments [15]. A continuously expanding gas bubble is formed at the nozzle exit during the initial stage. Moreover, due to the blocking of water at \(t=10\) ms, the front of the bubble forms a fan-shaped high-pressure area, whereas the inside of the bubble is a low-pressure area, and the front edge of the bubble is depressed. As the jet develops, the bubble size increases in the axial direction and the front edge depression is filled, resulting in an overall elliptical shape. At approximately \(t=30\) ms, the jet is fully developed, and the radial size of the bubble reaches its maximum. At this time, the bubble begins to move forward and away from the nozzle. At \(t=50\) ms, the bubble moves far away from the nozzle, and a high-pressure zone forms between it and the jet. At \(t=70\) ms, the bubble separates from the jet and gradually breaks. Simultaneously, the tail of the jet bulges again due to the high-pressure zone. After \(t=90\) ms, the underwater jet gradually maintains stability, the shape of the jet transforming from a bubble to a conical jet. At this time, the gas flow velocity near the nozzle outlet is high, and the gas-liquid interface is affected primarily by Kelvin-Helmholtz (K-H) instability. In the middle part of the jet, K-H instability and Rayleigh-Taylor (R-T) instability co-dominate, while downstream from the jet, the axial velocity of the jet decays to almost zero, and R-T instability dominates the gas-liquid interface [15]. The 3D gas-liquid interface of the supersonic jet at \(t=13\) ms is shown in Figure 4, along with the distribution of the radial derivative of axial velocity (\(\partial V_{x}/\partial r\)) at the interface. The gas-liquid interface at the nozzle exit exhibits a large velocity derivative and apparent features of K-H instability. The velocity derivative is relatively small downstream from the jet, and the R-T instability induced by the density gradient results in a pinch-off feature. When the bubble separates from the nozzle, the gas jet comes into contact with the water. Because the pressure of the water is greater than the low pressure in the gas bubble, the jet flow area at the nozzle outlet decreases, resulting in an instantaneous high pressure. This phenomenon is known as jet necking. Figure 5 shows the instantaneous pressure distributions of the jet axis at different times during the period of \(t=10\)-\(60\) ms. During the period \(t=10\)-\(30\) ms, the pressure peak corresponds to the high-pressure zone at the front of the bubble and moves toward the jet direction over time. At \(t=40\) ms, the nozzle outlet  
Figure 4: The gas–liquid interface at \(t=13\) ms.  
suddenly generates high pressure and is accompanied by severe pressure fluctuations near the nozzle outlet at \(t\) = 40-60 ms.