Numerical study on jet noise suppression with water injection during one-nozzle launch vehicle lift-off

ABSTRACT The launch vehicle will experience extreme acoustic environment at lift-off, which will cause the invalidation and destruction of the aerospace equipment, and reducing noise is of great significance for improving launch safety. In this paper, the numerical simulation of gas–liquid two-phase flow and its acoustic field is carried out, and the flow field is simulated by the discrete phase model (DPM), second-order Roe format, Scale-Adaptive Simulation (SAS) and three-dimensional Navier–Stokes. Based on the analysis of the flow field, the acoustic analogy integration method (FW-H) is used to predict the jet noise at the observation position. The influence of the water different injection angles and mass flow rate ratios are studied, and the results show that water injection not only reduces the temperature, velocity and vorticity at the gas–liquid interface, and the temperature at the bottom of the jet deflector, it also significantly reduces the jet noise. The total sound pressure level can be reduced to 6.92 dB by appropriate selection of the water injection angles and water injection mass flow rates. The numerical method established in this paper provides a useful tool for the design and analysis of water injection noise reduction for high-thrust multi-nozzle launch vehicle.


Introduction
Launch vehicles are space vehicles composed of multistage rockets, which can carry payloads such as artificial Earth satellites, manned spacecrafts, space stations and space probes into predetermined orbits. During the ignition and take-off phase of the launch vehicle, the exhaust gas jet from the liquid propulsion engine can cause severe acoustic loads on the ground launch equipment and the launch vehicle (Ahuja et al., 2014;Gao & Fu, 2012;Yu et al., 2014). At the same time, the gas jet impinges on the launch pad, deflector and ground; and the disturbance generated in the impact area as well as the shear layer of the flow field near the wall induces an increase in the acoustic load (Kandula, 2008;Nonomura et al., 2016). This can cause adverse effects such as interference to the astronauts and their precision instruments and equipment in the cabin, and induces launch safety accidents in serious cases. Therefore, reducing the acoustic load is of great significance to improve and enhance launch safety.
The deflector is an important part of the rocket launcher, and its function is to exhaust the hightemperature, high-speed gas jet ejected by the engine CONTACT Guigao Le leguigao@njust.edu.cn away from the launch pad. Closed deflectors are effective in reducing the acoustic environment during launch vehicle lift-off compared to open deflectors, as demonstrated by scaled-down model experiments conducted on the Ariane 5 launch vehicle (Gely et al., 2000). In addition to reducing noise, closed deflectors help to attenuate ignition overpressure (Tsutsumi et al., 2014). Tsutsumi et al. (2013) numerically studied the noise radiation of a solid rocket M-V during takeoff based on the improvement of the original deflector surface, and found that the launch pad with an inclined deflector surface can effectively reduce the acoustic load and improve the acoustic environment around the rocket, because the inclined deflector surface changes the direction of acoustic wave propagation and make the acoustic wave propagate away from the rocket. With the development of numerical method (Ghalandari et al., 2019;Salih et al., 2019), it provides a good analysis method for optimizing the jet deflector. Numerical simulations (Tatsukawa et al., 2014;Tatsukawa et al., 2015;Tatsukawa et al., 2016) found that local angle optimization of the inclined deflector in the impact region is important to reduce the overall sound pressure level (OASPL) near the rocket fairing. Karthikeyan and Venkatakrishnan (2017) experimentally investigated the effect of the launch pad on the acoustic environment around the rocket and showed that the solid launch pad with holes is less noisy compared to the solid launch pad without holes. However, the deflector optimization is limited by the terrain of the launch site, and thus the range of reducing noise during launch vehicle takeoff may be constrained. Spraying water onto the exhaust gas provides effective improvement and protection of the thermal environment of the launch pad and surrounding equipment (Vu et al., 2013;Jiang et al., 2010), and reduces ignition overpressure and noise (Osipov et al., 2015). The momentum exchange between water flow and gas jet can slow down the gas jet, change the velocity distribution and reduce the intensity of gas turbulence (Jiang et al., 2019). In addition, the vaporization of water absorbs a large amount of heat. This momentum and energy exchange can improve the acoustic environment of the launch vehicle at liftoff. Water injection tests with a 1:100 scaled-down model of the launch vehicle and launch pad showed that water injection at a fixed position on the launch pad can effectively suppress the noise during launch vehicle takeoff (Ignatius et al., 2015). The scaled model of Ares I (Garcia, 2012;Panda & Mosher, 2013) experiments showed that water injection near the cut-out of the launch platform, the top of the deflector and the jet deflector trench can effectively reduce the noise generated by the deflector, while water injection on the top of the launch platform can reduce the noise from the impact area and limit the intensity of the noise.
The launch vehicle has the remarkable characteristics of large size of the vehicle and high total pressure of the engine. It has become a challenge to carry out numerical study on jet noise suppression by water injection for full-scale launch vehicles. Most studies in the published literature on acoustic environment of launch vehicle takeoff are based on scaled-down models and experiments, while there are only few researchers on fullscale rocket tests and numerical calculations of water jet noise reduction. In addition, experimental studies using scale models were difficult to satisfy multiple similarity criteria. Numerical methods can overcome these shortcomings in experimental measurement studies, and better describe the wave structure of the gas plume flow field, facilitate the analysis of the flow mechanism and the evolution of induced noise. Moreover, numerical analysis plays a predictive and guiding role for experimental studies, providing a certain theoretical basis for those.
The literature (Xing et al., 2020) studied the acoustic environment during the launch vehicle takeoff; however, no effective measures were taken to control the noise. In this paper, we investigate the influence of water injection in the gas jet on the flow field and noise, using the discrete phase model (DPM) for gas-water multiphase flow, the second-order Roe format, the scaleadaptive simulation (SAS) model, acoustic analogy integration method (FW-H) and the three-dimensional unsteady compressible Navier-Stokes equation. The numerical simulations for noise reduction of the onenozzle launch vehicle during takeoff are carried out to study the influence of the water injection angle and mass flow rate ratio on the flow field wave system structure, overall sound pressure level (OASPL) and frequency spectrum. The results provide a good theoretical basis for the further development of water injection system for multi-nozzle high-thrust launch vehicles. Figure 1 shows the calculation model and the schematic diagram of the water injection points around the nozzle outlet. To see the location of the injection point more clearly, a local enlarged display is drawn on the right, where the angle between the central axis of the nozzle and water injection direction is defined as the injection angle α. 72 injection points are arranged around the gas jet, and the injection intersects with the nozzle axis at a distance D e from the nozzle exit. In this paper, three mass flow rate ratios of water and gas (Ignatius et al., 2015;Sankaran et al., 2009) (MFR = 3, 3.6 and 4) and injection angles (α = 50°, 60°and 70°) are studied.

Mathematical model
Within the scope of this paper, the following assumptions are considered: (1) the gas follows the perfect gas law and satisfies the perfect gas equation of state; and (2) the chemical reactions among gas components of the gas are neglected. Based on the above assumptions, the three-dimensional unsteady compressible Navier-Stokes (N-S) equation in the dimensional Cartesian coordinate system is where U is the conservation variable, HG v FG, and are the vectors of the convective flux in the x, y, and z directions, respectively, F v , and H v are respectively the dissipative vectors of the convective flux in the x, y, and where ρ is the density, u, v and w are the velocity components in the x, y and z directions, respectively, p is the fluid pressure, Y k is the mass fraction of species k, Y Ns−1 is calculated by Y Ns−1 = 1 − Ns−1 k=1 Y k , E t is the total energy per unit volume.
where 1, 2, 3 are the components in the x, y and z directions, respectively, σ ij is the viscous stress tensor, V k,i is the velocity of the species relative to the mixture gas, that is, the velocity of species diffusion, q i is the heat flux, ω k is the chemical reaction rate of species k. The SAS turbulence model was used to obtain the wave structure for flow field (Menter & Egorov, 2010). The high-speed hot jet discharged from the engine contains very strong shock waves, Mach disks, expansion waves, contact discontinuities, etc. The flow structure is complex, and the flow equation is highly nonlinear, which requires both a numerical difference format with strong shock wave capture capability and good convergence and stability. For this reason, a high-order Roe format with strong performance, the multi-grid accelerated convergence technique, iterative relaxation system and CFL number are employed to effectively control the numerical stability to ensure that the iterative computation can be carried out smoothly.

Discrete phase model
The study of the acoustic environment with water injection during launch vehicle takeoff involves the high-speed compressible flow of gas, the trajectory of liquid droplets in the flow field and the interaction between water and gas. Water vaporization occurs at the gas-liquid interface, and mass, momentum and energy are exchanged between the continuous and discrete phases. For multiphase flow of water injection in gas flow, this paper uses the discrete phase model (Osipov et al., 2015;Sharma et al., 2021) and the acoustic analogy integration method (FW-H) to solve the flow field and acoustic field. Defining water as a discrete phase, the trajectory of the droplet is predicted by integrating the force balance equation on the water droplet, which is established in the Lagrangian coordinate system. The equilibrium equation for the forces acting on the droplet in the discrete phase model can be described as where m p is the droplet mass, u is the gas flow rate, u p is the droplet velocity, ρ is the gas density, ρ p is the water density, g is the gravitational acceleration, F is the additional force, m p u− u p τ r is the drag force on the droplet, τ r is the relaxation time of the droplet, which is defined as where μ is the gas dynamic viscosity, C d is the drag coefficient, and d p is the droplet diameter. The relative Reynolds number Re is defined as follows: In this paper, there is mass, momentum and energy exchange between low temperature, low velocity water droplets and high temperature, high velocity gas. Therefore, the interaction between gas and droplets is considered and a two-way coupling is adopted: where m p is mass flow rate of droplets,ṁ p,0 is initial mass flow rate of the droplet injection, m p,0 is initial mass of the droplet, u p is velocity of the droplet, u is velocity of the fluid,ṁ p is mass flow rate of the droplet, t is the calculated time step, F other is other interaction forces, m p in is mass of the droplet on cell entry, m p out is mass of the droplet on cell exit, c p p is heat capacity of the droplet, T p in and T p out is temperature of the droplet on cell entry and exit, T ref is reference temperature for enthalpy, and H lat ref is the latent heat at the reference condition.

FW-H acoustic analogy method
In 1955, based on the Lighthill equation (Lighthill, 1952), Curle used the Kirchoff method to consider the effect of noise on a stationary solid wall (Curle, 1955). Ffowcs Williams and Hawkings (1969) used the generalized function method to further extend the acoustic analogy by introducing the Heaviside function and the Dirac delta function to obtain the well-known Ffowcs Williams-Hawkings (FW-H) equation: where p is the sound pressure at the far field, u i is the fluid velocity component in the x i direction; u n is the fluid velocity component perpendicular to the integration surface f = 0, v n is the velocity component of the motion of the integration surface perpendicular to the integration surface, and δ(f ) is the Dirac delta function. H(f ) is the Heaviside function, and T ij is the Lighthill stress tensor. Compared with the Kirchhoff method, the FW-H method has several advantages. It is no longer restricted to the linear, inviscid fluctuation equation, and the FW-H integration surface can be placed anywhere in the jet region. This is true even in the nonlinear flow region, where the Kirchhoff method does not work. For this reason, the FW-H integration surface method is used in this study.

Boundary conditions and mesh model
The boundary condition for nozzle inlet is defined as the total pressure and temperature boundary conditions: the total pressure P 0 = 17.5 MPa and the total temperature T 0 = 3800 K. The external boundary of the computational domain is defined as the atmospheric conditions: the pressure 101,325 Pa and temperature 300 K. The nozzle and other solid walls are defined as adiabatic no-slip wall conditions. Figure 2 shows the grid model of the computational domain, which discretizes the computational region into a hexahedral structured grid. Figure 2 (right) shows the mesh of the region around the nozzle exit.
Grid is an important factor affecting the numerical calculation of the flow field. In this section, we investigate the effect of the number of grid cells on the calculation results, comparing the velocity of the nozzle axis under  the three grids. The results are shown in Figure 3, where Z/D e represents the distance from the nozzle outlet. Grid A, Grid B and Grid C represent grid cells of 1.5, 3 and 4.5 million, respectively. By comparing the results, it is found that in the range of Z/D e = 6 ∼ 11, the velocity with grid cells of 1.5 million has a larger deviation compared with the other two cases with denser grid cells, while there is no significant difference between the results of 3.0 and 4.5 million grids cells. In order to improve the computational efficiency, the number of grid cells of the model used in this paper is controlled between 3.0 and 4.5 million.

Validations of models
Key component in our numerical model, including the second-order Roe format, the turbulence model is SAS model, and the acoustic analogy integration method (FW-H), have been validated in our recent publication (Xing et al., 2020).
For the droplet evaporation and motion, we use the discrete phase model to verify the system in Ref. (Hamey, 1982), where water droplets (T d = 289 K) freely fall in static air with relative humidity RH = 70% and temperature T ref = 293 K. The calculated results are given in   (Hamey, 1982). The decrease in droplet diameter with falling distance, which is caused by evaporation, is in good agreement with the experimental data. The maximum relative errors are 2.2% and 3.4% for droplets with initial diameters of 110 and 115 μm, respectively. This comparison demonstrates that our model has correctly considered the mass, momentum and energy inter between water droplets and the ambient air.
The above verifies the validity of the heat transfer equation, water vapor mass transfer equation and equation of motion adopted for the object of study in this paper, and provides the basis for the study of the numerical calculation model of gas-liquid two-phase flow and aerodynamic noise during launch vehicle takeoff. Figure 5 shows the droplet size distributions and variations during the launch process. Water (MFR = 3, d p = 0.8 mm and T d = 300 K) is injected at an injection angle of 60°near the nozzle outlet, and the water droplets are obstructed by the gas flow and move in the direction of the gas flow. Since there is a velocity difference between the water droplets and the gas flow, the water is subjected to two forces in the vertical direction, namely the gravity of the water droplets and the drag force from the gas on the water droplets. The water droplets continue to move downward through the inlet of the deflector and collide in the impact section of the jet deflector, changing their direction and moving along the transition section of the jet deflector. Due to the impact angle of the impact section and the curvature of the jet deflector, after the collision between the droplet and the impact section, the direction of droplet motion is at a certain angle with the transition section of the jet deflector, as shown in Figure 5(b). Figure 5(c) shows that the water droplets have moved to the baffle section of the jet deflector, because the gas jet impacts the jet deflector flows mainly along the bottom of the jet deflector. Between the outlet of the nozzle and the inlet of the jet deflector, the traveling time is short and the droplet diameter is almost constant. The existence of momentum and energy exchange between water and gas is the minimum droplet diameter of 0.637 mm at 0.1 s, 0.602 mm at 0.16 s, 0.546 mm at 0.28 s, and 0.506 mm at 0.4 s. The longer the water and gas interaction time, the smaller the droplet diameter. At t = 0.16 s, the number of droplets increases between the inlet of the jet deflector and the impact section, causing disturbance on the gas flow field and making some water droplets near the inlet of the jet deflector move towards the rocket body, compared with the other three time instances. This condition is most obvious at t = 0.28 s. Figure 6 gives the temperature contours at the bottom of the jet deflector with water injection (d p = 0.8 mm, α = 60°and T d = 300 K) compared to the case without water injection. The peak temperatures are 3186, 2570, 2464 and 2393 K for the four operating conditions, indicating that water injection helps to reduce the temperature at the bottom of the jet deflector. In addition, the case without water injection, the gas impinges on the bottom of the deflector and then moves in the direction to the side of the deflector, creating a high temperature zone at the junction of the side and the bottom of the deflector, as shown in Figure 6(a). When water injection is employed, water absorbs part of the energy of the gas and reduces the temperature at the junction of the side and bottom of the deflector. Next, the system in Figure 8 with MFR = 3, T d = 300 K and α = 60°is taken as an example to discuss the influence of water injection on the gas flow.

Flow field and noise variation without and with water injection
The vorticity and temperature contours in the verticle y = 0 plane with and without water injection cases, as well as those in five horizontal sections with Z/D e = 1.86, 3.31, 6.44, 7.78 and 9.07 are given in Figure 7 and Figure  8. The temperature contours are within the two dashed lines and the others are vorticity contours. Due to the high velocity of the gas exhausted from the nozzle outlet, the vorticity contours are narrow, and as the gas gradually spreads, the vorticity contours become significantly wider from the first wave node. Figure 7(a,b) are located in the cylindrical section at the inlet of the deflector, where the flow fields show concentric circles due to the structural symmetry. Compared with Z/D e = 1.86, the vorticity radius increases and the temperature decreases at Z/D e = 3.31, due to the gas jet expansion. Figure 7(c-e) show the flow situations below the deflector inlet. With the gradual increase of Z/D e , the jet flow spreads outward. Figure 8 shows the vorticity and temperature contours with water injection. Compared with the results in Figure 7, it is evident that water injection can reduce the temperature and vorticity around the gas jet, mainly because of the energy, momentum and mass exchange between water and high temperature gas. At Z/D e = 1.86, the vorticity diffuses to the surroundings due to the water impacting the gas. The injection of water also causes instability in the vorticity, as shown in Figure 8(b). Figure 9 shows the water vapor and velocity distributions for the MFR = 3 at t = 0.05, 0.06 and 0.07 s, where the lines are velocity contours and the background colors indicate the vapor concentration. As water is injected into the flow field at an angle of 60°, the gas flow shrinks after the water droplets impinge on the gas flow at t = 0.05 s, and the first wave node has the narrowest width compared to the other two later wave. The vaporized water vapor at t = 0.06 s and 0.07 s is mainly located at the first wave node and the second Mach disk, when the water vapor has reached the bottom of the deflector. The maximum value of water vapor mass fraction in the three wave node regions are 0.405, 0.424 and 0.344, indicating that the water vapor mass fraction first increases and then decreases. At t = 0.05 s, water vapor mass fraction is the highest at the first Mach disk for two reasons: first, the injection speed is low; second, as water is injected into the flow field at an angle of 60°, water droplets accumulate at the gas-liquid interface. In addition, due to inertia, the water is not easily folded at the gas-liquid interface. At t = 0.06 s, the wave section is concave in shape and the water is blocked, so the vaporized water vapor reaches its maximum value. Because of the velocity difference between water and gas, the gas has a downward force on the water, driving the water to move to the bottom of the deflector, while playing an accelerating role in the flow of water, so the maximum of water vapor mass fraction at t = 0.07 is at the second Mach disk.
In addition, from the velocity contour, the distance from the nozzle outlet to the lowest point of the contour with a velocity value of 2500 m/s are 6.58D e , 6.1D e , and 6.29D e , indicating that when the maximum vapor fraction is located in the first wave node. It can be clearly observed that as time increases, the contour spacing located between the first wave node and the bottom of the jet deflector becomes narrower, mainly because the water gradually flows down around the gas flow, and this suppresses the expansion of the gas flow.   The velocity and temperature distribution with (MFR = 3) and without water injection are compared in Figure 10. Figure 10(a) shows the velocity profiles along X/D e = 0.5 and 1. Along X/D e = 0.5, water and gas are in full contact, which effectively reduces the velocity of the flow field. The maximum reduction is 1110 m/s at 3.3D e . On the one hand, because the water velocity is much lower than the gas velocity, the momentum is exchanged between the two phases. In addition, water vaporizes and absorbs energy during this process. At X/D e = 1, the velocity of the flow field from Z/D e = 2.22 with the water injection is greater than that without water. This opposite to the situation at X/D e = 0.5, because water impinges the gas-liquid interface, and causes flow instability. Figure 10(b) shows the temperature variations along X/D e = 0.5 and 1. Similar to the velocity curves in Figure 10(b), water injection effectively reduces the temperature along X/D e = 0.5, with a maximum reduction value of 744.32 K at Z = 4.08D e . Figure 11 shows a schematic diagram of the receivers, represented by black circles. The first receiver is at 40D e from the nozzle outlet, and the distance between two consecutive receivers is 5D e . Six receivers are installed in the range of 40D e ∼ 65D e along the center line. In addition, nine at a radius of 4D e in the 50D e section from the nozzle exit, with 22.5°apart from each other, considering the jet deflector symmetry about the o-xz (y = 0) plane. Figure 12 shows the overall sound pressure level (OASPL) without and with water injection. After water injection, the OASPL along the height of the nozzle axis decreases, with the distance from the nozzle outlet (Z/D e ) increases, the OASPL decreases up to 6.92 dB, indicating that the noise source with water injection decays faster with the increase of propagation distance, as shown in Figure 12(a). Figure 12(b) shows the OASPL variation at the circumferential receivers located at Z/D e = 50. With the increase of angle, the difference between OASPL without and with water injection first increases and then gradually decreases, and the difference reaches the maximum at 90°with the value of 7.08 dB. Figure 13 shows the 1/3oct spectra of the sound pressure level (SPL) at the receivers at different locations. Figure 13(a) shows the 1/3oct spectra of the receiver at Z/D e = 40, where the largest reduction is at 160 Hz with a value of 7.75 dB. At 250 Hz, water injection has almost no effect on the SPL. In other frequency ranges, the SPL is reduced by about 5 dB. The 1/3oct spectra of the SPL  at the receiver at Z/D e = 60 shows a similar trend to the receiver at Z/D e = 40, but of smaller SPL values because the noise source gradually decreases from the nozzle exit. The SPL at 160 Hz is reduced most by 8.15 dB; and in the other frequency ranges, the SPL is reduced by about 5.3 dB. It shows that the further the receivers from the nozzle outlet, the greater the reduction in SPL, as shown in Figure 13(b). Figure 14 shows the 1/3oct spectra of the SPL at the circumferential receiver located at the Z/D e = 50 section. For the receiver at 180°, the SPL increases significantly at 200 Hz due to the influence of water injection. In addition, in the other frequency ranges, the decrease is about 3.8 dB. At the receiver at 0°, the SPL increases at 80 Hz compared with that without water injection, and the decrease in SPL is about 5.4 dB from 800 to 10,000 Hz, as shown in Figure 14(b).

Effect of water injection angle
In the case of constant water injection rate and velocity, changing of the injection angle can change the penetration ability of water on the interface between the gas and liquid phases. From the velocity point of view, changing the angle of water injection actually changes the size of the injection velocity in the direction perpendicular to the nozzle axis. For constant mass flow rate, the greater the angle of water injection, the greater the momentum perpendicular to the direction of the nozzle axis, whereas the smaller the momentum perpendicular to the direction of the nozzle axis. In order to study the effect of water injection angle on the flow field and acoustic environment during launch vehicle takeoff, numerical analysis of the flow field, OASPL and spectral characteristics of launch vehicle takeoff is carried out in this section. Figure 15 and Figure 16 show the water vapor and velocity contours at t = 0.05, 0.06 and 0.07 s for injection angles 50°and 70°, where the lines are velocity    contours and the background colors indicate the vapor concentration. Combined with results in Figure 9, at the time of 0.05 s, the maximum values of water vapor corresponding to water injection angles of 50°, 60°and 70°are 0.414,0.405 and 0.267, respectively. The increase of water injection velocity perpendicular to the nozzle axis causes the accumulation of water droplets at the gas-liquid interface. At t = 0.06 s, the amount of water vapor increases with the increase of the angle due to the accumulation of water droplets at the outlet of the nozzle. As the flow time increases, due to the velocity difference of gas and water, the water flow at the gas-liquid interface accelerates. At t = 0.07 s. The water and gas are in full contact, the injection of water mass flow rate remains unchanged, and the amount of water vaporization by the injection angle is very small.

Flow field of the launch vehicle
We now look at the velocity contours. At t = 0.05 s, with the increase of the angle of water injection, the momentum of water in the direction perpendicular to the axis increases, and the gas flow is compressed. As a result, the distance from the lowest point of the contour with the velocity of 2500 m/s to the nozzle exit increases gradually. As the time increases, the water continues to flow downward until t = 0.07 s. The velocity of 2500 m/s at the lowest point of the contour is almost unaffected by the angle. Figure 17(a) shows the temperature variations along the nozzle axis for three injection angles, 50°, 60°and 70°. The temperature peak at Z/D e = 1.5 increases with the increase of injection angle, mainly because the injection angle increases, the more the gas is subjected to the force perpendicular to the axis of the nozzle, while the first peak is the largest, when there is friction between the gas and the air, close to the nozzle outlet, the loss of energy is smaller. At Z/D e = 5.8, it can be clearly observed that, with the increase of the angle, the peak moves toward the nozzle outlet, and the position of the peak value changes slightly when the water injection angle is 60°and 70°. Figure 17(b) displays the radial distributions of temperature at two axial locations Z/D e = 1.87 and 3.31. For X/D e > 0.94, the angle of water injection has no effect on the temperature. In the range of X/D e = 0.48 ∼ 0.94, the temperature increases with the increase in the angle of water injection. For X/D e < 0.48, and the temperature increases with the angle of water injection, which shows that the larger the angle of water injection, the greater the ability of water to penetrate the gas. The temperature variation at Z/D e = 3.31 shows a similar trend to that at Z/D e = 1.87. Since Z/D e = 3.31 is located at the gas expansion region, the temperature change range is inconsistent. Figure 18 shows the velocity variation with water injection angle at two positions X/D e = 0.5 and 0.75. As the water injection angle increases, the water jet penetration into gas flow effectively reduces the velocity around the gas flow. It can be clearly observed in Figure 18(b) that in the range of Z/D e = 1.3 ∼ 5.4, the greater the angle of injection, the larger the velocity, because the angle of injection increases, the velocity component perpendicular to the axis of the nozzle increases. At the same time, the force that impacts the gas-liquid interface increases, compressing the space of gas flow, resulting in gas spilling out of the water droplets, and causing the disturbance of the flow field around it to increase. Figure 19 shows the turbulent kinetic energy and vorticity variations with water injection angle at X/D e = 1. In the range of Z/D e = 2.4 ∼ 5.9, the turbulent kinetic energy increases with injection angle, since the increase of the injection angle increases the disturbance in the gas flow field. After Z/D e = 13.1, the turbulent kinetic energy starts to decrease with the injection angle. According to Figure 18, the increase in injection angle reduces the gas flow velocity at the bottom of the jet deflector. In the range of Z/D e = 2.0 ∼ 5.7, the vorticity increases with the injection angle, which is similar to the trend of turbulent kinetic energy. Figure 20 shows the OASPL variations at different locations, where the columns show the OASPL variations and the lines show the OASPL change due to water injection. The lowest OASPL is observed at 60°, which shows that the noise reduction effect is best when the water injection angle is 60°. Taking the receiver at 50D e as an example. As the water injection angle varies, the OASPL decreases by 3.58 dB at 50°, 6.76 dB at 60°, and 3.03 dB at 70°, as shown in Figure 20(a). Figure 20(b) shows the OASPL variations of the receivers arranged circumferentially with a radius of 4D e at the 50D e section from the nozzle outlet. The OASPL of the receiver located at 90°∼ 180°is larger than those of the receiver located at 0°∼ 90°, i.e. the OASPL at the receiver in the direction of the deflector outlet is greater than that at the receiver away from the deflector outlet. Figure 21 gives the 1/3oct spectra of the SPL of receivers at Z/D e = 40 and Z/D e = 60. Located along the     nozzle axis, the only difference is the distance from the nozzle outlet. Similar to the results in Figure 20, the SPL at these receivers increase with the injection angle first decreases and then increases. The SPL also decreases with the receiver distance from the nozzle outlet. At the injection angle of 70°, the maximum effect on SPL is at 100 Hz, and the SPL at 125 Hz can be reduced. Compared with the water injection angle of 50°, at a frequency greater than 1000 Hz, the influence of the water injection angle of 60°a nd 70°on the SPL of the receiver is decreased by 1.7 dB and increased by 0.55 dB, respectively. Figure 22 shows the 1/3oct spectra of the circumferential receivers at the Z/D e = 50 section. Figure 22(a) is the receiver at 180°. It can be clearly observed that the frequency is greater than 630 Hz, and the SPL reaches the maximum at the injection angle 50°, while the SPL at an angle of 60°and 70°are almost the same. The maximum SPLs of the three injection angles all occur at 100 Hz, and the water injection effect is most significant at 70°. The SPL at the receiver at 0°affected by the angle of water injection is consistent with the variation of the OASPL, i.e. it increases first and then decreases with the injection angle, and the maximum value of the SPL is found near 100 Hz, as shown in Figure 22(b).

Effect of the Mass Flow Rate Ratio
With the same injection speed, position and angle, the greater the mass flow rate of water injection, the greater the evaporation, the more energy absorbed by the water, and thus the greater change to the gas flow field. In addition, with the increase of the water flow rate, part of the water does not vaporize, which also causes disturbance of flow field. Therefore, the mass flow rate of water injection has a direct impact on changing the gas flow field and acoustic environment. The discrete phase model and the acoustic analogy integration method (FW-H) are used to carry out the numerical simulation of the acoustic environment during launch vehicle takeoff suppressed by the MFR in three different cases, i.e. MFR = 3, 3.6 and 4, respectively, to solve for the gas-liquid flow field, flow parameter distribution, OASPL variation and spectral characteristics with different MFR. Figure 23 and Figure 24 show the water vapor and velocity contours at t = 0.05, 0.06 and 0.07 s for MFR = 3.6 and 4, where the lines are velocity contours and the background colors indicate the vapor concentration. Combined with Figure 9, at t = 0.05 s, the momentum component perpendicular to the nozzle axis increases with the increase of the injection mass flow ratio, which is the main reason for the gradual narrowing of the flow width at the first wave node. The water vapor mass fraction reaches the maximum at MFR = 3.6, followed by MFR = 4 and the minimum at MFR = 3. With the increase of mass flow rate of water injection, the energy absorbed by water decreases, and the effect of vaporization on the high-temperature gas flow becomes less significant.

Flow field of the launch vehicle
As can be seen from Figure 23, with the increase of time, the distance from the lowest of the contour with the velocity value of 1875 m/s to the nozzle outlet first decreases and then increases, and the distances to the nozzle outlet are 8.44D e , 7.50D e , and 9.95D e , respectively, when the mass fraction of water vapor is located at the first wave node, which has the greatest effect on suppressing the jet length. As the water continues to flow downward, the gas expansion has is limited and the gas flow length increases.
The temperature variations with different mass flow rate ratios are given in Figure 25. Along the axis, the temperature peaks gradually decrease due to the friction between gas and stationary air, which results in gas energy loss, and begin to increase back as approaching the bottom of the jet deflector. In addition, as the injection flow rate increases, the temperature peak moves toward the outlet of the nozzle. and the temperature peak movement is 0.1D e and 0.25D e for MFR = 3 and 4, respectively, relative to the injection mass flow rate ratio MFR = 3. Figure 25(b) shows the temperature variation at Z/D e = 1.87 and 3.31. It can be observed that, for X/D e > 0.97, the three curves overlap, indicating that different water injection flow rate has nearly no effect on the temperature; For X/D e = 0.59 ∼ 0.97, the greater the injection flow rate, the higher the temperature, since the high-temperature gas flow is squeezed by the water, overflowing to the outside of the water droplets; For X/D e < 0.59, water and gas are in full contact. As the injection flow rate increases, the temperature decreases. In Figure  25(b), the temperature variation at Z/D e = 3.31 is similar to Z/D e = 1.87. With the radial distance increases, the effect of the injection flow rate on the temperature can be divided into three stages: first decrease, then increase, and later become approximately constant. The dividing positions are at 0.94D e , and 1.32D e . Please note that Z/D e = 1.87 and 3.31 are located in the middle of the wave section and Mach disk, and this is the reason for the dividing positions and the difference of temperature peaks.
The velocity variations at X/D e = 0.5 and 0.75 for different mass flow rate ratios are given in Figure 26. It can be observed from Figure 26(a) that the velocity decreases with the mass flow rate of water injection. Since X/D e = 0.5 is closed to the core area of the jet, there are larger velocity and temperature differences between gas and water. The momentum and energy exchange between water and gas reduces the gas velocity. Figure 26(b) shows that the velocity increases with the water injection mass flow rate in the range of Z/D e = 0 ∼ 3.65D e . Because water is injected into the gas flow at an angle near the nozzle exit, the flow field is disturbed, and the space of the gas flow is compressed, causing the gas to spill out of the water droplets. As the distance from the nozzle outlet increases, the increase in the injection flow rate helps to reduce the gas flow velocity. At Z = 13D e , the velocity shows a maximum value, which is largest for MFR = 4. Figure 27 shows the turbulent kinetic energy and vorticity variations at X/D e = 1 for different mass flow rate ratios. It can be observed that, in the range of Z/D e = 0 ∼ 2.05, the three curves overlap, and the effect of the injection mass flow rate on the turbulent kinetic energy is negligible. For the range of Z/D e = 2.05 ∼ 4.38, the turbulent kinetic energy increases with the injection flow rate; with the distance from the nozzle outlet, the turbulent kinetic energy decreases with the increase of the injection mass flow rate. The turbulent kinetic energy reaches its maximum at 13.11D e for MFR = 4. In Figure  27(b), at Z/D e = 1.85 ∼ 4.2, the vortex volume increases with the injection mass flow rate; and at Z/D e = 3.41, the vorticity of MFR = 4 is 32.9% larger than that of MFR = 3. In the range of Z/D e = 3.41 ∼ 13.1, the vorticity decreases with the mass flow rate, indicating that increasing the water injection can reduce the vorticity. Comparing the variations of turbulent kinetic energy and vorticity, both curves show similar trends. Combined with Figure 26(b), it can be seen that water injection changes the velocity of the flow field, which in turn affects the turbulent kinetic energy and vorticity in the flow field. Figure 28(a) shows the OASPL variations along the nozzle axis, where the column shows the OASPL variations and the lines show the OASPL differences. The farther away from the nozzle outlet, the lower the OASPL of receivers, indicating that the acoustic waves are decaying along the flow. In addition, the difference between the OASPL of the two receivers adjacent first increases and then decreases, indicating that the propagation process of the acoustic wave is nonlinear. As the injection mass flow rate increases, the OASPL gradually increases. This phenomenon can be explained as follows. On the one hand, the number of water droplets increases with the injection flow rate, causing greater disturbance in the flow field. On the other hand, water injection can increase the flow  parameters in the local flow field, such as temperature, velocity, turbulence energy and vorticity. We can take the receiver at 50D e as an example. As the water injection mass flow rate increases, the OASPL decreases by 6.76 dB at MFR = 3, 5.37 dB at MFR = 3.6, and 4.29 dB at MFR = 4; as shown in Figure 28(a). This shows that water injection can reduce noise, and the best reduction occurs at MFR = 3.

OASPL and Spectra Variation
In addition, the OASPL difference without and with water injection gradually increases with the increase of distance, indicating that the flow field of water injection can cause the sound waves to attenuate faster in     propagation. Figure 28(b) shows the OASPL variation of the receivers arranged circumferentially with a radius of 4D e on the 50D e section from the nozzle outlet. As the angle increases, the OASPL first increases and then decreases. The OASPL differences at 180°and at 0°are the largest, and the value is 0.9 dB, indicating that the OASPL at the receiver near the outlet of the jet deflector is large, caused by the flow direction and sound source at the jet deflector outlet. Figure 29 gives the 1/3oct spectra of the SPL for receivers at Z/D e = 40 and 60. At Z/D e = 40, the frequency is higher than 1000 Hz. The SPL increases approximately linearly with the injection flow rate. The changes in SPL at most locations are less than 1000 Hz. The maximum SPLs at MFR = 3, 3.6 and 4 are 114.66, 112.66 and 112.62 dB, respectively, as shown in Figure 29(a). The receiver located at Z/D e = 60 shows a similar trend as that at Z/D e = 40. Since the receiver at Z/D e = 60 is farther from the nozzle outlet, the SPL is small at the same frequency, as shown in Figure 29(b). Figure 30 shows that the 1/3oct spectra of the SPL at the circumferential receiver is located at Z/D e = 50. Compared with MFR = 3, the higher flow rate MFR = 3.6 has little effect on the SPL for frequencies higher than 500 Hz at 180°, but the SPL at MFR = 4 increases by 2.73 dB. The frequencies corresponding to the maximum SPLs are 100 Hz for MFR = 3, 100 Hz for MFR = 3.6, and 200 Hz for MFR = 4, respectively, as shown in Figure 30(a). The SPL increases gradually with MFR at 0°with a frequency greater than 630 Hz, but the effect of MFR = 4 on the SPL is not significant. The flow rate MFR = 3.6 can reduce the SPL in the range of 125 Hz to 160 Hz, as shown in Figure 30(b).

Conclusion
Based on the discrete phase model (DPM) and acoustic analogy integration method (FW-H), this paper establishes a model of jet noise suppression with water injection during one-nozzle launch vehicle lift-off. This model is then used to study the influence of water injection angle and water injection mass flow rate ratio on the acoustic environment, focusing on the analysis of the gas flow field, the OASPL and spectral characteristics of the receiver without and with water injection. According to the results obtained in this research, the following conclusions can be drawn.
For the droplet evaporation and motionïĳŇWe use discrete phase model to verify the droplet evaporation and motion. The calculated droplet diameter variation is in good agreement with the experimental data, the validity of the discrete phase model is verified.
Due to the exchange of mass, momentum and energy between water and gas, water injection can reduce the temperature at the bottom of the jet deflector, and reduce the vorticity and turbulent kinetic energy at the gas-liquid interface. The overall sound pressure level can be reduced by 6.92 dB.
The overall sound pressure level varies with the injection angle, with the best noise reduction is obtained when the water injection angle is 60°.
The overall sound pressure level increases with increasing the injection mass flow rate ratio. When the mass flow rate ratio is 3, the best noise reduction is obtained. At axis receivers, the frequency is higher than 1000 Hz, and the SPL increases approximately linearly with the injection flow rate.
In this paper, a numerical study is carried out on jet noise suppression by water injection during one-nozzle launch vehicle lift-off. Due to the heavy workload of the numerical calculation model of the launch vehicle acoustic environment and the limitation of computational resources, the effect of the launch vehicle motion on the noise is not taken into account. In future research, it is necessary to use dynamic mesh to further study the impact of launch vehicle motion on the acoustic environment.

Disclosure statement
No potential conflict of interest was reported by the author(s).