A Numerical Simulation of the Underwater Supersonic Gas Jet Evolution and Its Induced Noise

Abstract

To explore the complex flow field and noise characteristics of underwater high-speed gas jets, the mixture multiphase model, large eddy simulation method, and Ffowcs Williams–Hawking (FW–H) acoustic model were used for simulations, and the numerical methods were validated by the gas jet noise experimental results. The results revealed that during the initial stages, the jet collided with the water surface and created low-pressure high-temperature gas bubbles, accompanied by much high-frequency noise. When the jet reached its maximum length, its impact weakened, the bubble broke, the jet transformed into a conical shape, and the jet noise changed from high- to low-frequency. The pressure fluctuation peaked near the position at which the Mach number reached 1, indicating that the jet was the most unstable at the sonic point. Additionally, at low frequencies, the sound pressure levels between jets with different nozzle pressure ratios were similar, whereas above 400 Hz, under-expanded jets had higher sound pressure levels. This paper provides theoretical guidance for the study of underwater jet noise.

1. Introduction

Underwater high-speed gas jet technology has broad application prospects in areas such as ocean resource exploitation and the design of underwater weapon systems. The underwater jet propulsion system has a simpler structure and higher propulsion efficiency compared to traditional propeller propulsion methods [1]. However, a profound understanding of the intricate jet flow fields and noise characteristics is indispensable to designing a high-speed, efficient, stable, and low-noise underwater jet propulsion system. The main challenge lies in the complex interaction between the high-temperature, high-pressure gas jet and the significantly denser water.

Early underwater high-speed gas jet research focused on experimental measurements and theoretical analyses. Unlike a low-speed jet, where the bubbles break and rise at the nozzle outlet under the action of gravity or density differences [2], an underwater high-speed gas jet usually starts to break up and spray downstream from the jet. Mori et al. [3] experimentally studied the transition of a high-speed underwater jet from bubbles to a jet; McNallan et al. [4] observed that the jet changed from bubbles to a jet near the sonic point; and Chen et al. [5] attributed this phenomenon to compressibility effects, surmising that the temporal and spatial growth rates of the disturbance increased with an increase in the subsonic gas velocity. The disturbance growth rate also peaked when it reached the speed of sound, after which it decreased rapidly to zero with increasing gas velocity. Shi et al. [6] established a two-dimensional underwater high-speed gas jet experimental system to study the instability of subsonic and supersonic jets. Weida et al. [7] measured the spatial and temporal changes in the jet boundary directly through high-speed imaging. They found that the large density difference between the gas and water led to the enhancement of turbulent mixing, shortened jet kernel, and a relationship between the jet stability and Mach number.

Owing to the challenges of observing the complex underwater high-speed jet processes, traditional experimental methods can be costly, and capturing detailed flow information can be difficult. Consequently, numerical simulations have become increasingly important for their study. However, the numerical simulation of a three-dimensional underwater supersonic jet takes time and has poor stability. Most of the numerical simulations use two-dimensional axisymmetric assumptions, the research content covering the thrust characteristics of the engine [8,9,10,11], the structure of the jet wave system [12], the development of the gas–liquid interface [9,11,13,14,15], and the jet instability [14,15,16], which provides a theoretical basis for the flow process of underwater high-speed jets. Considering that a two-dimensional axisymmetric jet cannot precisely capture the influence of three-dimensional fluid motion on the air–liquid interface, Fronzeo et al. [17] used the delayed detached eddy simulation (DDES) turbulence model to study the effect of different environmental densities on a supersonic jet. They found that the turbulence intensity at the air–liquid interface increased considerably with an increase in fluid density, resulting in poorer jet stability. Regarding the noise characteristics of underwater gas jets, the noise from low-speed jets is primarily caused by bubbles [18,19,20,21,22]. However, based on the research conducted by Liu et al. [23], who used a large eddy simulation (LES) to analyze the flow structure and noise radiation of underwater supersonic jets, the noise generated by high-speed jets originated primarily from the shockwave structure of the jet.

In this study, three-dimensional unsteady numerical simulations of underwater supersonic gas jets were performed using LES. The noise characteristics of the jet were analyzed using the Ffowcs Williams–Hawkings (FW–H) model [24], which was compared with the experimental noise data to validate the accuracy of the numerical method. The development processes of under-expanded and fully expanded underwater supersonic jets were investigated, and their noise characteristics were analyzed and compared. Consequently, the results of this study provide a better understanding of the flow mechanics and noise characteristics of underwater supersonic jets, thereby providing a theoretical foundation for the design of underwater propulsion systems.

2. Numerical Methods

2.1. Mathematical Method

The numerical simulation of an underwater supersonic jet and its noise involves calculating compressible flow, two-phase interaction, turbulence, and jet noise [25]. A mixture multiphase model with a hybrid sharp/dispersed interface modeling method was used to simulate multiphase flow with solid coupling, allowing all phases to move at the same speed [26]. This model is particularly appropriate for small-scale multiphase flow with slip velocities between the phases and locally isotropic assumptions. This paper focuses on two-phase flow (n = 2) in which the primary phase is compressible air, assuming an ideal gas, and the secondary phase is incompressible water, the effects of gravity, and evaporation–condensation being neglected [27]. The continuity, momentum, energy, and volume fraction equations for the mixture can be expressed as follows:

$$\frac{\partial }{\partial t}\left({\rho }_m\right)+\nabla ·\left({\rho }_m{\overrightarrow{v}}_m\right)=0$$

(1)

$$\frac{\partial }{\partial t}\left({\rho }_m{\overrightarrow{v}}_m\right)+\nabla ·\left({\rho }_m{\overrightarrow{v}}_m{\overrightarrow{v}}_m\right)=-\nabla p+\nabla ·\left[{\mu }_m\left(\nabla {\overrightarrow{v}}_m+\nabla {\overrightarrow{v}}_m^T\right)\right],$$

(2)

$$\frac{\partial }{\partial t}{\sum }_k\left({\alpha }_k{\rho }_k{E}_k\right)+\nabla ·{\sum }_k\left({\alpha }_k\overrightarrow{v}\left({\rho }_k{E}_k+p\right)\right)=\nabla ·\left(k_{eff}\nabla T+\left({\stackrel{̳}{\tau }}_{eff}·\overrightarrow{v}\right)\right),$$

(3)

$$\frac{\partial }{\partial t}\left({\alpha }_p{\rho }_p\right)+\nabla ·\left({\alpha }_p{\rho }_p{\overrightarrow{v}}_m\right)=0$$

(4)

where the subscript p denotes the secondary phase, ${\overrightarrow{v}}_m$ denotes the mass-averaged velocity, ${\rho }_m$ denotes the density of the mixture, ${\mu }_m$ denotes the mixture dynamical viscosity coefficient, and $k_{eff}$ denotes the effective thermal conductivity; the corresponding expressions are as follows:

$${\overrightarrow{v}}_m=\frac{{\sum }_{k=1}^n{\alpha }_k{\rho }_k}{{\rho }_m}\overrightarrow{v}, {\rho }_m={\sum }_{k=1}^n{\alpha }_k{\rho }_k, {\mu }_m={\sum }_{k=1}^n{\alpha }_k{\mu }_k, k_{eff}=\sum {\alpha }_k\left(k_k+k_t\right),$$

(5)

where ${\alpha }_k$ denotes the volume fraction of the kth component, $k_t$ denotes the turbulent thermal conductivity coefficient, and ${\stackrel{̳}{\tau }}_{eff}$ denotes the effective viscosity stress tensor.

2.2. Numerical Scheme

The software version used for numerical simulations was ANSYS-fluent 19.0. The transient simulations in this study were based on the finite volume method using the pressure velocity coupling SIMPLE algorithm, a bounded second-order implicit method being adopted for the transient formulation. The SIMPLE algorithm can obtain good results when simulating the transient characteristics of the flow [15,28]. The LES method with the wall-adapting local eddy viscosity (WALE) model [29] was adopted to ensure the accuracy of the sound field calculation. The FW–H model was adopted to compute the far-field noise.

2.3. Initial and Boundary Conditions

The expansion ratio of the nozzle model in the present study was 4, the throat diameter of the nozzle was D_t = 24 mm, and the exit diameter was 2D_t. Assuming a specific heat ratio $\gamma =1.4$, according to Equations (6) and (7), the design Mach number of the nozzle can be computed to be $M_d$ = 2.94, corresponding to a design nozzle pressure ratio (NPR) of 33.56. Two working conditions with the same total temperature and water depth but different total pressures were set, corresponding to the under-expanded and fully expanded conditions. The detailed parameters are listed in

Table 1. Nozzle calculation conditions.

	Total Pressure at Nozzle Inlet p0 (MPa)	Total Temperature at Nozzle Inlet T0 (K)	Water Depth H (m)	NPR p0/pb
Under-expansion condition	15	2300	20	50.45
Full expansion condition	10	2300	20	33.63

.

$$\frac{A_e}{A_t}=\frac{1}{M_d}{\left[\frac{2}{\gamma +1}\left(1+\frac{\gamma -1}{2}{M}_d^2\right)\right]}^{\frac{\gamma +1}{2(\gamma -1)}},$$

(6)

$${\left(\frac{p_0}{p_b}\right)}_d={\left(1+\frac{\gamma -1}{2}{M}_d^2\right)}^{\frac{\gamma }{\gamma -1}}$$

(7)

A schematic of the simulation domain of the jet flow field is shown in Figure 1. The simulation domain is divided into three regions: the nozzle area, the jet area, and the buffer zone. A damping region, also referred to as a buffer zone, is established near the outer boundary to minimize the impact of the finite boundaries on the acoustic field calculation results. The entire simulation domain is cylindrical with an axial length of 120$D_t$, an upper diameter of 50$D_t$, and a lower diameter of 100$D_t$. Between the jet area and buffer zone is a cylindrical sound source integration surface with an axial length of 86$D_t$ and a radial length from 16$D_t$ to 33$D_t$.

The outer boundary of the flow field is set as the pressure outlet and outlet pressure ($p_b$) corresponding to the water depth, and the temperature is 300 K. The total pressure and temperature at the pressure inlet are consistent with the total pressure ($p_0$) and temperature ($T_0$) at the nozzle inlet. The remaining flow field boundaries are set as non-slip walls. The minimum grid size of the jet core area is 8 × 10⁻⁴ m, and the maximum grid size downstream from the jet area is 1 × 10⁻³ m. The grid size of the first layer of the nozzle boundary layer is 3.3 × 10⁻⁶ m, the calculated y+ is less than 5, and the total number of grids is 2.2 million. In the initial state, the gas inside the nozzle is still and the pressure is consistent with the outlet pressure ($p_b$). The unsteady time step is 1 × 10⁻⁵ s, approximately 0.21 s in total.

3. Results and Discussion

3.1. Verification with the Experimental Results

The numerical method was verified using the experimental and simulation results of the underwater supersonic jet used in the present study [23], where the design Mach number of the nozzle was 3.41 and the throat diameter D* was 14 mm. The test facility has a nozzle, a combustion engine, and a pressure water tank. The combustion gas comes out of the engine through a Laval nozzle and is produced by the combustion of air and methane. After leaving the nozzle exit, they inject into the surrounding water environment. The inlet of the nozzle was set to a high-temperature gas with a total pressure of 9.6 MPa and a total temperature of 2200 K, the ambient pressure was 0.1 MPa, and the temperature was 300 K. A schematic diagram of the noise measurement points is shown in Figure 2.

Table 2. Comparison of the calculated OASPL at measurement points with the experimental results [23].

OASPL	Measuring Point A	Measuring Point B	Measuring Point C
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%

compares the overall sound pressure level (OASPL) computed at the measurement points in the present study with the experimental results [23]. Point A is used to monitor the noise in the far field, while points B and C are used to monitor the noise in the near field to observe the jet noise results comprehensively. The OASPL computed by simulation in the current study is slightly larger than the experimental results, with a maximum relative error of 5.1%, indicating the reliability of the calculation method.

3.2. The Development of Underwater Supersonic Jets

Figure 3 illustrates the pressure contours of the jet flow field at different times under total pressures p₀ = 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.

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 (∂V_x/∂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 suddenly generates high pressure and is accompanied by severe pressure fluctuations near the nozzle outlet at t = 40~60 ms. The peak value of the pressure formed under the fully expanded condition with a total pressure p₀ = 10 MP is nearly twice as much as under the under-expanded condition with a total pressure of p₀ = 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 7 and Figure 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 ${\log }_{10}\left(\left|\nabla \mathsf{\rho }\right|\right)$ 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 Figure 8 and Figure 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 11 shows the temperature distributions of the flow field at various times. The jet temperature fields are very close when the total pressures are p₀ = 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.

3.3. Noise Radiation Characteristics

Figure 12 shows the locations of the noise measurement points. The measuring points (a1–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° within the range of 0–150°.

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 = 0~10 ms, the pressure decreases steadily from its high value, corresponding to the semicircular high pressure formed when the jet starts. During the period t = 10~20 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.

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 15 shows the curves of the OASPL versus the angle (θ), and the OASPL reaches its maximum value at a jet direction of approximately 15°. 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.

4. Conclusions

Three-dimensional unsteady numerical simulations of under-expanded and fully expanded underwater supersonic jets were conducted to analyze their complex flow field and noise characteristics. The main conclusions to be drawn are as follows.

• During the initial stage of an underwater jet, the supersonic jet forms a continuously expanding gas bubble enveloped in water, and high-frequency noise is generated by the interaction of the shockwaves formed by the supersonic jet. As the jet reaches its maximum length, the impact force gradually weakens, and the gas bubble ruptures and transforms into a conical jet. Consequently, the high-frequency noise transitions into low-frequency noise.
• When the gas bubble detaches from the nozzle, jet necking occurs at the nozzle outlet, leading to intense jet instability and severe pressure fluctuations. The closer the jet is to full expansion, the more severe the disturbance the jet is subjected to.
• After the jet transforms into a conical jet, the pressure fluctuations along the centerline of the jet exhibit three peaks. The first two peaks within the core area of the jet correspond to shock interaction in a supersonic jet, which is amplified by the two-phase interaction. The third peak corresponds to the streamwise location where the local Mach number approaches unity, indicating that the jet is most unstable at the sonic point.
• In cases with equal water depth, the overall trend of the sound pressure and sound pressure level spectrum of the under-expanded and fully expanded jets remains consistent. In particular, the low-frequency noise in both cases is almost identical. However, the high-frequency noise (above 400 Hz) in the under-expanded jet is greater than in the fully expanded jet. Thus, the OASPL of the under-expanded jet is also larger.