# A Numerical Simulation of the Underwater Supersonic Gas Jet Evolution and Its Induced Noise
## 3 Results and Discussion
### The Development of Underwater Supersonic Jets
The peak value of the pressure formed under the fully expanded condition with a total pressure \(p_{0}\) = 10 MP is nearly twice as much as under the under-expanded condition with a total pressure of \(p_{0}\) = 15 MP. This means that as jet necking occurs, the disturbances of the jet become more severe as the jet approaches full expansion. Figure 6 shows that the stable flow at the nozzle outlet is disrupted instantaneously owing to the high pressure caused by jet necking, resulting in the transition of the shock structure from an expansion wave to an oblique shockwave over a short period. The high pressure formed under the under-expanded condition is considerably lower than that formed under the fully expanded condition, indicating that a higher NPR can help mitigate the effects of jet necking. Figure 5: Instantaneous pressure distribution of the jet axis. Figures 7 and 8 illustrate the distributions of the mean pressure and root-mean-square (RMS) values along the jet axis during \(t\) = 90-210 ms. Influenced by the interaction of gas and water and the instability of the jet itself, it is more challenging to ensure a stable underwater jet than a jet in the air. The pressure distribution along the jet axis exhibits three peaks, with two peaks within the core area of the jet corresponding to the double-expansion compression process after ejection from the nozzle outlet. The contour of the instantaneous density gradient, represented by \(\log_{10}(|\
abla\rho|)\) in Figure 9, confirms this point. Specifically, it indicates that the pressure fluctuations experience a sharp increase at the intersection of barrel shocks. This phenomenon is attributed to the instability of supersonic jets, which is amplified by the two-phase interaction. The under-expanded jet has a longer jet core at the same water depth as the fully expanded jet. The axial position of the RMS pressure fluctuation peak downstream from the jet core is close to the position at which the local Mach number approaches unity, as indicated by the shaded region in Figures 8 and 10. This indicates that the jet becomes highly unstable when the local Mach number approaches the speed of sound, which is consistent with the results of previous studies [5]. Figure 6: Pressure distributions when bubbles leave the nozzle. Figure 7: The average pressure distribution curve of the jet axis. Figure 8: Root-mean-square distribution curve of the jet axial pressure. Figure 10: The average Mach number distribution curve of the jet axis. Figure 9: Instantaneous density gradient (\(\log_{10}(|\
abla\rho|)\) distributions at time \(t=20\) ms. Figure 11 shows the temperature distributions of the flow field at various times. The jet temperature fields are very close when the total pressures are \(p_{0}=15\) and 10 MPa. During the initial stage of jet development, the jet collides with the water medium and stagnates to form a high-temperature gas bubble. Due to the low pressure in the bubble, the kinetic and pressure energies of the jet are converted into internal energy. Simultaneously, the high-temperature area of the jet is concentrated on the isosurface of the gas phase volume fraction of 0.1. The temperature is attenuated to the ambient temperature at the isosurface of the gas-phase volume fraction of 0.9, indicating that the influence of the high-temperature jet in directly heating the water medium can be ignored and the phase transition of the water during the jet process can be neglected. As the bubble moves away from the nozzle, its temperature quickly decreases to the ambient temperature.