# A Numerical Simulation of the Underwater Supersonic Gas Jet Evolution and Its Induced Noise
## 3 Results and Discussion
### Noise Radiation Characteristics  
Figure 12 shows the locations of the noise measurement points. The measuring points (al-a11) are set on an arc radius of 2.4 m (\(100D_{t}\)), with the nozzle outlet center as the origin to observe the far field noise, and a measuring point is set every 15\({}^{\circ}\) within the range of 0-150\({}^{\circ}\).  
Figure 11: Temperature distribution at different times.  
Figure 13 illustrates the sound pressure curve at measuring point a3 under the under-expanded and fully expanded conditions. The overall trends are similar, particularly after \(t\) = 50 ms, with the sound pressure variations almost coinciding. The differences occur primarily at \(t\) = 20 ms and \(t\) = 40 ms. During \(t\) = 0\(\sim\)10 ms, the pressure decreases steadily from its high value, corresponding to the semicircular high pressure formed when the jet starts. During the period \(t\) = 10\(\sim\)20 ms, the sound pressure curve exhibits a high-frequency and high-amplitude pulsation, originating from the shockwave structure formed by the supersonic jet, related to the nozzle operating conditions [30]. Moreover, the amplitude of the high-frequency noise in the fully expanded condition is smaller than in the under-expanded condition. At \(t\) = 30 ms, based on the pressure contours shown in Figure 3, the jet is fully developed, with the length of the jet core reaching its peak and the high-frequency pulsation decreasing. At \(t\) = 40 ms, the sound pressure curve under the fully expanded condition exhibits a peak, whereas the under-expanded jet produces only a small amplitude of high-frequency pulsation. This corresponds to the earlier description of the jet necking at time \(t\) = 40 ms in Figure 5.  
Figure 12: A schematic diagram of the noise measuring points.  
Figure 13: The time-varying curve of sound pressure at measuring point a3.  
This section describes the separation of bubbles from the nozzle outlet. Jet necking leads to an instantaneous high-pressure zone between the bubbles and the jet. The sound pressure tends to stabilize over time as the high-pressure zone moves away from the nozzle outlet.  
Figure 14 shows the sound pressure level (SPL) spectra at measuring points a3, a7, and a11. The maximum sound pressure level is obtained near 20 Hz, after which the sound pressure level under the fully expanded condition continues to decrease with frequency. Conversely, the sound pressure level under the under-expanded condition increases near 400 Hz, corresponding to the high-frequency and high-amplitude fluctuation of the sound pressure caused by the shockwave structure during \(t\) = 10-20 ms [31]. Moreover, the noise of the under-expanded and fully expanded jets is consistent in the low-frequency band below 400 Hz, whereas the under-expanded jet has a higher sound pressure level in the high-frequency band above 400 Hz. Finally, as the angle with the jet axis increases from measurement points a3-a11, the sound pressure level in the low-frequency band below 400 Hz decreases slightly.  
Figure 14: Sound pressure level spectrum of the measuring points a3, a7, and a11.  
Figure 15 shows the curves of the OASPL versus the angle (\(\theta\)), and the OASPL reaches its maximum value at a jet direction of approximately 15\({}^{\circ}\). Owing to the low-frequency dominant jet noise, the OASPL in both cases gradually decreases with an increase in angle. However, the OASPL difference between the under-expanded and fully expanded jet increases with the angle, corresponding to a higher level of high-frequency noise generated by the fully expanded jet.