Numerical analysis of the influence of water/air medium on the muzzle flow field characteristics of machine gun Open Access

The flight of the projectile is deeply influenced by the muzzle flow field, which will affect or even lower the shooting accuracy when the gun is shot. Therefore, it is of great importance to research the flow field around gun muzzle. With the development of the navy, the research on underwater weapons has gradually become a hot spot.

In the past, a large number of experiments, theoretical and numerical analysis were carried out on the flow field of gun launched in air. Based on ALE equations and AUSMDV scheme, calculations with chimera grids and dynamic grid technology were carried out to explore the interaction between the projectile and the muzzle flow field of 7.62mm rifle.^1,2 The dynamic overlapped grids approach and a high-resolution upwind scheme (AUSMPW+) were applied to investigate the complicated transient phenomena in the early launch stages after the projectile moves from the muzzle of the cannon to the free-flight stage.^3,4 Both experimental and numerical studies were conducted to research the interaction between precursor flow field of air and muzzle flow field of the powder gas.^5,6 Through a pressure test of muzzle shock wave of 7.62mm rifle, the propagation of muzzle shock wave in low pressure environment was obtained.⁷ With the Osher scheme, the muzzle flow field of 155mm gun was numerically calculated and analyzed.⁸ However, all the research work above are about the muzzle flow field for gun launched in air, while the research on the muzzle flow field for gun launched underwater has not been published. With the development of underwater weapon, it is extremely urgent to pay attention to the muzzle flow field for gun launched underwater. The interior ballistic model for fully submerged underwater gun was established to guide the experiment, which adopted the way of shortening the tube and reducing the loading density to lower the chamber pressure to ensure the launch safety.^9,10 To obtain the same interior ballistic performance as gun launched in air, a new launch device for underwater gun, named underwater sealed launch, was proposed and designed.¹¹ In order to keep the ambient water from entering into the tube, a baffle is placed at the muzzle. During the interior launching process, the baffle keeps closed until the pressure before the projectile rise to a certain number. Then the air pressure in the tube is higher than the ambient water pressure, so water still cannot enter into the tube to cause high chamber pressure which may lead to chamber premature. By numerically calculation of the underwater sealed launch, it was found that the muzzle velocity reduces and the chamber pressure goes up when compared with the same situation of gun launched in air.¹² Also, the cylindrical shock wave of compressed air inside the muzzle was uncovered. The precursor flow field of underwater sealed launch is formed by compressed air which is driven by the moving projectile. The compressed air pushed away water around the muzzle, and the powder gas flow is the equal of the high-temperature high-pressure gas jet. A lot of experimental and numerical studies about the interaction of gas jet with water have been done by many researchers. Based on the VOF (volume of fluid) multiphase flow model, the underwater gas jet of rocket nozzle was numerically simulated.¹³ With user-defined function (UDF), the standard k − ε turbulent model and the PISO pressure-velocity coupling algorithm were adopted to describe the interaction between gas and liquid. Through the numerically simulation of the horizontal jet of underwater solid rocket motor, it was found that the VOF model can commendably describe the jet structure of the experimental results.¹⁴ A set of experiments were carried out to investigate the behavior of horizontal round noncondensing gas jets that discharge in a stagnant water ambient, considering subsonic and sonic jet exit conditions.¹⁵ It was found that the nozzle diameter and the Froude number had a great influence on the jet shedding and the gas-liquid interface instability. The experimental study on the expansion characteristics of multiple gas jets in a cylindrical filling chamber was carried out and it was found that increasing the injection pressure will accelerate the convergence of multi-jets.¹⁶ The same numerical method as Xu was employed to research the expansion characteristics of combustion-gas jets in cylindrical stepped-wall observation chambers, and the computational results were in good agreement with the experimental results.^17,18 For the purpose of improving the jet stability, a cylindrical four-stage stepped-wall chamber was designed, and both experimentally and numerically verification were obtained.¹⁹ 

Based on a 12.7 mm slide gun, this paper studies the muzzle flow field characteristics for gun sealed launched underwater. By comparatively analysis of muzzle flow field characteristics for gun launched in air and underwater, the influence of different mediums on muzzle flow field characteristics are discussed.

According to the characteristics of the gun sealed launched underwater, the following assumptions are developed to simplify the launching process:

1. The burning of gunpowder follows the law of geometric combustion. The gunpowder particles are burned under mean pressure and follow the law of exponential burning rate. The gunpowder gas follows the Nobel-Abel equation.

2. The heat and temperature generated by the combustion of unit mass gunpowder are constant. During the expansion process of powder gas, the gas composition does not change. And the impetus f, residual capacity α and specific heat ratio k are considered to be constant; Take φ as the coefficient of other minor work.

3. The expansion of the combustion gas jet is considered as a transient process and a two-dimensional axisymmetric problem, approximately. Then, the k − ε model is employed to describe the turbulent effect.

4. The combustion gas jet is also treated as compressible ideal gas jet so that the influence of the volume force is ignorable.

5. The phase change and cavitation of the water near the muzzle are not considered.

According to the physical model established above, the mathematical model of combustion gas jet includes:

1. Continuity equation

Where, two phases, gas and liquid, are considered in this model. α₁ and α₂ are the phase volume fraction and α₁+α₂=1. ρ₁ and ρ₂ are corresponding densities. S_αq is source item. Since the chemical reaction is ignored, S_αq =0.

1. Momentum equation

Where, ρ = α₂ρ₂ + (1 − α₂)ρ₁; p is pressure, and μ₁ is the coefficient of viscosity.

1. Energy equation

Where, E and T are the average energy and temperature, respectively. k_eff is the effective thermal conductivity.

1. Gas State Equation

1. Turbulent flow equation (the turbulence kinetic energy, k, and its rate of dissipation, ε, are obtained from the following transport equations)

Where, σ_k and σ_ε are the turbulent Prandtl numbers, respectively. $ρui'uj'¯$ is the Reynolds stress that expresses the influence of pulsation on the time averaged flow. And the turbulent (or eddy) viscosity, μ_t, is computed from k and ε as μ_t = C_μk²$$ε.

The model constants take the following default values

The interior ballistics equations in numerically simulation are as follows

1. Form function

Where, ψ is mass fraction of the burnt, Z is relative thickness of the burnt. χ, λ and μ are form characteristic quantities.

1. Burning law

Where, u₁ is the burning coefficient, n is the burning exponent, e₁ is half of the web-thickness of the gunpowder, p is the mean pressure in the barrel, and t is time.

1. Momentum equation

Where, p_d is the pressure at the projectile base, p_f is the pressure at the projectile head, A is the sectional area of the projectile, φ is the minor work coefficient, m is the mass of the projectile, v is the velocity of the projectile.

1. Energy equation of interior ballistics

Where, θ = k₀−1 and k₀ is the adiabatic exponent,ω is the charge weight, x is displacement of the projectile, l_ψ is the diameter shrunk length of the free volume and is given by

Where, l₀ is the diameter shrunk length of the chamber. ρ_p is the density of gunpowder, Δ is density of loading, α is the co-volume, V₀ is the volume of the chamber.

1. Motion equation

1. Nobel - Abel equation

Where, v₁ is the specific volume of the gas (or the volume of per mass gas).

The interior ballistics equations are solved utilizing a fourth-order Runge-Kutta method, providing velocity for the projectile when it accelerates in the barrel. Correspondingly, by solving the governing equations and by integrating the pressure over the head and base of projectile, the air resistance and combustion-gas thrust are acquired. In short, the results of the interior ballistics model provide velocity for the projectile, and the results of governing equations provide the value of p_d and p_f for Eq. (10). This coupling between the interior ballistics model and the flow model is implemented by the UDF.

The simulation is carried out by using a CFD solver, and the VOF multiphase flow model and the standard k − ε turbulent flow model are used in the numerically calculation. The phase-coupled-PISO algorithm is adopted for velocity-pressure coupling. Pressure equation adopts the PRESTO! discrete method. Velocity and energy are coupled with PISO solver. The time step is set as 1 × 10⁻⁷s to achieve numerical stability.

The computational domain and the mesh setup are shown in Fig. 1. As can be seen from Fig. 1(a), the computational domain consists of three regions, which are the combustion chamber I, the tube II and the flow field area III around the muzzle. The length of the domain outside of the muzzle is 0.5 m and the radius of this domain is 0.18 m. Here, as a reference point, point O (0, 0) is the muzzle center. Point P (50mm, 19.05mm) is taken as a monitor point for grid independence verification. The grid of the computational domain is shown in Fig. 1(b). All the mesh used here is structured mesh and the size of the minimum mesh cell around the muzzle is 0.2 mm × 0.25 mm.

In Fig. 1(b), the pressure inlet is adopted in combustion chamber. The projectile, which is set as rigid body, starts to move when pressure in combustion chamber reaches the starting pressure. Here, the dynamic layering method is employed to dealing with the moving boundary, and the projectile velocity is computed from interior ballistics equations when the projectile accelerates in the barrel. Before launched, the pressure and temperature in region II and III are the same as the pressure outlet, which are 101,325 Pa and 300 K.

To ensure the optimal configuration of calculation accuracy and computational efficiency, Fig. 2(a) and (b) show grid and time step independence verification, respectively. The grid independence is verified by the number of meshes of 210,000, 170,000 and 130,000, respectively. The pressure at point P in Fig. 1(a) is used as a reference. The time step of 5×10^-8s, 1×10^-7s and 2×10^-7s are adopted to do the time step independence verification. And the muzzle pressure of the combustion-gas is taken as a reference.

It can be seen from Fig. 2(a) that, compared with the pressure at point P of 210 000 meshes, the mean relative error for 170 000 meshes is 4.25% while the mean relative error for 130,000 meshes is up to 21.58%. Thus, this paper takes the 170,000 meshes for numerically calculation. Fig. 2(b) shows that, the mean relative error is about 2.84% for time step 1×10^-7s while the mean relative error is about 7% for time step 2×10^-7s when compared with the calculation result at time step 5×10^-8s. To ensure the accuracy and computational efficiency, the time step is 1×10^-7s.

A numerically calculation of the expansion process of high-pressure round gas jet is carried out for numerically verification in the present paper.

Numerically simulation of the expansion of round gas jet in a cylindrical liquid-filled chamber is performed to compare with the experimental results in article 19. The computational domain and the grids are shown in Fig. 3. All the parameters selected in the simulation are consistent with the experiment parameters. In details, the nozzle diameter is 2 mm. And the length and the diameter of observation chamber are respectively 98 mm and 24 mm. The pressure inlet is set 20 MPa and the temperature is 2200 K. The pressure outlet adopts the parameters which are consistent with the environment, that are 101,325 Pa and 300 K. The shape of the Taylor cavity at 2.0ms is shown in Fig. 4, and the axial extension displacement of the jet head for numerical and experimental results are given in Fig. 5. Fig. 4 and Fig. 5 show that, the shape of the Taylor cavity calculated is in good agreement with the experimental data, and the mean error of jet head extension velocity is less than 2.6%. Therefore, the numerical method employed in this study is appropriate.

The maximum chamber pressure is 320 MPa while the muzzle velocity is up to 810 m⋅s^-1 when the 12.7 mm gun is launched in air. In this paper, the distribution of the muzzle flow field for the 12.7 mm gun sealed launched underwater is numerically analyzed and compared with that for in air. The loading parameters, which are the same as the gun launched in air, are shown in Table I. The maximum chamber pressure and muzzle velocity for underwater sealed launch are 330 MPa and 758 m⋅s^-1, respectively.

In order to understand the precursor flow field characteristics for gun sealed launched underwater, the muzzle pressure field and the Mach number distribution are compared with that for in air at the time of 80 μs before the projectile shoots out in Fig. 6. From Fig. 6(a), pressure before the projectile increases with motion of the projectile when the slide machine gun is sealed launched underwater. However, the compressed air strikes the gas-water interface and bounces due to the high density of water, which is about 800 times heavy of air. Initial air is reflected by the gas-liquid interface, and ultimately forms a series of columnar compressed wave in the chamber, converging into high pressure area. The movement of the air is deflected at the muzzle because of the large resistance for the air to extend in the axial direction, and compressed air escapes along the side of the muzzle. From Fig. 6(b), it can be seen that when the gun is launched in air, because of the low density air outside the muzzle, the high pressure air before the projectile is rapidly expanding to form a bottle-shaped shock wave, which is consistent with previous research.²⁰ By comparing the Mach number distribution in Fig. 6(a) and (b), it can be found that, the velocity of the gas is lower and the expansion direction is deflected at the muzzle, due to the large density of water. Obviously, the precursor flow fields are significantly different when the 12.7 mm machine gun is launched in different mediums.

To understand the characteristics of the jet field formed by the gunpowder gas, the phase distribution for underwater sealed gun is shown in Fig. 7. Correspondingly, the Mach number distribution for underwater launch and air launch are compared in Fig. 8 and Fig. 9.

It can be seen from Fig. 7 that the initial air is located on both sides of the projectile at 0 μs. With the projectile moving forward and the gas injecting, the gas cavity at muzzle is gradually bulging by 30μs, and the bubbles at the jet tail begin to fall off both sides of the muzzle. At 60μs, the small bubbles on both sides of the jet tail are completely separated from the main gas cavity, and the surface of the jet head is wrinkled by the instability of Kelvin-Helmholtz. At 120μs, due to the increased gas-liquid entrainment, cavitation fall off the jet tail one after another and converge with the bubbles fall off previously, and jet head also appears depression. These phenomena become more obvious at 160μs. In Fig. 8, at 0 μs, due to the obstruction of water, the initial air is affected by the projectile movement, resulting in the Mach number on both sides of the muzzle is higher. Then, with the gunpowder gas ejecting and expanding rapidly, the gas jet forms a spindle shock structure while the shock core area was cap-shaped, and Mach disk structure has not formed until to 30 μs. The gunpowder gas jet initially forms the Mach disk until to 60μs, meanwhile, the shock wave structure is still spindle and shock wave area increases gradually. With the projectile moving forward, from 80 μs to 160 μs, the gas jet at the muzzle is fully developed and the shock wave area increases, while the Mach disk gradually changes from arc to perpendicular to the axis. Meanwhile, the shock wave structure gradually changed from the spindle shape into ellipsoid, and there is a strong turbulence at the jet tail, which is consistent with the bubble separation at jet tail in Fig. 7. In Fig. 9, at 30 μs, the gunpowder gas expands rapidly and forms a spherical shock structure after the muzzle. With the gas supplement and the movement of the projectile, from 60 μs to 160 μs, the shock wave head is depressed on account of the influence of the projectile base. Also, the shock wave structure is gradually developed into bowl-shaped, yet the formation of Mach disk has not appeared.

In contrast, under underwater sealed launch condition, there is a violent gas-liquid entrainment near the muzzle, while low-density air does not affect the jet tail for gun launched in air. The shock wave structure is obviously different of the two launching scenarios. The gunpowder gas jet forms the Mach disk faster for gun launched underwater due to the reflection of gas-liquid interface. However, the gas expansion is less obstructed and the shock wave head is affected by the projectile base too long to form the Mach disk for gun launched in air. It is apparent that the shock wave area for gun sealed launched underwater is obviously smaller than that for in air, and the coronal wave is not present at projectile head for the underwater sealed launch.

The temporal and spatial distribution of muzzle pressure fields are presented in Fig. 10 and Fig. 11, respectively. It can be seen from Fig. 10 that pressure at the projectile head is much higher than the combustion gas pressure at the muzzle during the underwater movement owing to the high-density water. However, the pressure at the projectile head caused by the compressed low-density air is much lower than the gunpowder gas pressure at muzzle when the projectile moves in air. It is clear that the large difference in the medium density leads to significant differences in the temporal and spatial distribution of pressure.

From the above temporal and spatial distribution of Mach number and pressure field, it is clear that the launch environment has a great influence on the distribution of the muzzle flow field. In order to further understand the distribution characteristics of the muzzle flow field, the variation curve of the jetting parameters for gun launched in different mediums is presented in Fig. 12. The variation of muzzle pressure and muzzle Mach number under the two launch conditions are given in Fig. 12(a) and 12(b), respectively.

It can be seen that, when the projectile just shoots out the muzzle for underwater launch, gunpowder gas gathered in the muzzle and the pressure is significantly higher than that for in air while the Mach number is lower than that for in air. With the gas jet ejecting out of the muzzle, the pressure drops faster of underwater launch conditions. However, because of the larger water-resistance, the overall expansion of the gas is not as fast as in air, and the muzzle gas pressure is always higher. Referring to Fig. 8, the gunpowder gas is more likely to form a Mach disk at the muzzle under underwater launch conditions, so the gas Mach number rises faster than in air. Therefore, gas Mach number for underwater launch gradually exceeds the muzzle Mach number for air launch 30 μs after the projectile shot out.

Fig. 13 shows the axial distribution curve of pressure and Mach number from the muzzle to the projectile base. The expansion of gunpowder gas is blocked of underwater launch, resulting in a higher muzzle pressure than the air launch. With the gunpowder gas ejecting, the difference of the muzzle gas jet pressure between the two launch conditions is shrunk. As can be seen from Fig. 13(a), at 60 μs, from the muzzle to the projectile base, the variation of pressure distribution along the axial direction are the same of the two launch conditions. That is, the pressure first decreases rapidly along the axial direction, and then rises slightly since the gas is obstructed by the projectile base in axial direction. At the same moment, the projectile position of underwater launch is located behind the location of the projectile in air, so the gas pressure rises in the anterior position. From Fig. 13(b), at 160 μs, the pressure of underwater launch drops rapidly along the axial direction firstly, and then rises with fluctuation. However, the pressure of air launch decreases rapidly and then remains substantially unchanged. The Mach number rises in the axial direction and then plummets of both launch conditions, but there is a certain degree of volatility after the decline under the underwater condition.

Combining Fig. 8 and Fig. 9, at 160μs, a subsonic area locates behind the Mach disk under underwater launch. Unlike the underwater launch, the Mach disk of the air launch has not formed at the same time yet. Influenced by the projectile base and shock wave zone, here the compressed wave oscillates. Fig. 13 further illustrates that, the axial distribution area of muzzle shock wave for underwater launch is significantly smaller when compared with that for in air.

Fig. 14 shows the radial distribution curve of the pressure and Mach number of above two moments. The cross section of x = 928mm for underwater launch and the cross section of x = 939mm for air launch at 60 μs are both the cross section of Mach number axial maximum position. As well, the cross section of x = 944mm for underwater launch and the cross section of x = 1023mm for air launch at 160 μs are both the cross section of Mach number axial maximum position.

It can be seen from Fig. 14 that, the radial distribution law of pressure is roughly the same for underwater launch. Generally, the pressure firstly rises rapidly along the radial direction, where after there is a slight difference in the vicinity of the peak and then slowly descends. Referring to Fig. 8, the radial section is tangent to the head of the Mach disk and passes through the low velocity shock zone on both sides. As the gunpowder gas jet expands, the pressure decreases at the boundary of the shock wave system, and then the pressure decays in the low velocity water. Overall, the pressure is lower than the previous time. Meanwhile, Mach number decreases rapidly along the radial direction. Combined with Fig. 11, the pressure is higher at the axis and the projectile radius at 60 μs for gun launched in air. At 160 μs, the gas jet is less resistant to the projectile base so that the pressure is lower near the axis, then it increases sharply and decreases rapidly within the gas shock wave region. According to Fig. 9, at the earlier time, the Mach number is generally higher near the projectile base, and it decreases rapidly before the radial position crosses the projectile radius position. At 160 μs, Mach number plummets and soars in the vicinity of the axis, and drops rapidly after overing the projectile radius. At this time, the shock is still affected by the projectile base. Obviously, it can be determined that the radial distribution area of the shock wave for underwater launch is much smaller than that for in air.

Through the comparison of the gun muzzle pressure and Mach number distribution in different launch mediums, the following conclusions can be drawn.

1. Through the analysis of the precursor flow field for underwater sealed launch, it is found that the axial expansion of the initial air is greatly blocked due to the obstruction of water. The compressed air impinges on the gas-liquid interface with forming a series of columnar compression waves and local high pressure area inside the barrel. And it is obvious that the air extends outward from the muzzle sidewall. All the characteristics of precursor flow field mentioned above are in wide difference with the muzzle bottle shock for gun launched in air.

2. There is a violent gas-liquid entrainment at the jet tail for gun sealed launched underwater. The Mach disk is initially formed 60 μs after the projectile shot out, and then it gradually changes from arc to vertical to axis. At 30 μs, the structure of the shock wave is a fusiform shape, and then gradually changes into a peach heart shape. Under the air launch condition, there does not have violent suction between the low-density air and gunpowder gas at the gun muzzle. The gas jet fails to develop into the Mach disk at even 160 μs due to the interaction between the projectile base and the shock wave. At 30 μs, the structure of shock wave is spherical and then gradually transforms into a bowl. Compared with the muzzle flow field of gun launched in air, the muzzle shock wave area of underwater sealed launch gun is obviously smaller.

3. After the projectile shot into water, the pressure at the projectile head is much higher than that when the projectile moves in the low density air, and also much larger than the gunpowder gas pressure at muzzle. The gunpowder gas pressure at muzzle is much higher than the compressed air at the projectile head while the projectile flights in air. Under the underwater sealed launch condition, the pressure distribution along the axis from muzzle center to the projectile base is slightly higher than the pressure of gun launched in air. In the radial section of the Mach number at the maximum axial distribution position, the radial distribution gradient of pressure for underwater launch is significantly larger than that for in air.

This work was supported by National Natural Science Foundation of China (No. 11372139).