Numerical simulation of radiated noise during combustion of energetic materials in a closed bomb

Shi-hui Xiong ,Tuo Yng ,Yu-jun Wu ,Jing-cheng Wng ,Yun Li ,c,Yu-qun Wen ,*

a State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing,100081, China

b Xi‘an Qinghua Cooperation, North Special Energy Group Co., Ltd., Xi'an, 710025, China

c Institute of Extreme Mechanics and School of Aeronautics, Northwestern Polytechnical University, Xi'an, 710072, China

Keywords:Pyrotechnics Noise Energetic materials Fluid-structure interaction Acoustic boundary element

ABSTRACT In this study, based on a closed bomb test combined with computational fluid dynamics, a structural finite element method, and an acoustic boundary element method, a fluid-solid acoustic one-way coupling calculation model is established for the combustion process of energetic materials in a closed bomb,and the effectiveness of the model is verified by experiments.It is found that the maximum peak sound pressure increases exponentially with an increase in loading doses or gas pressure. However, a change in the combustion coefficient of the energetic materials has little effect on the noise generated during the combustion process in the closed bomb. When the combustion coefficient is reduced by a multiple of 16, the maximum transient sound pressure is reduced by 1.79 dB, and the sound pressure level in the frequency band is reduced by 1.75 dB. With an increase in shell thickness, the combustion noise of the energetic materials in the closed bomb decreases,and the reduction range of the combustion noise increases with the increase in shell thickness.

1. Introduction

In existing torpedo weapon systems, the underwater load separation mechanism usually employs explosive bolts, separation nuts, various types of pyrotechnics locks, and/or a pyrotechnic separation push rod, and uses the energy from the explosion or combustion of the energetic materials to drive the functional mechanism to complete specific functions[1-3].The advantages of pyrotechnic driving devices include their large energy storage,fast action speed, simple and compact structure, and small trigger energy. However, there is a problem concerning the loud actuating noise, making it difficult to meet the needs of an underwater vehicle minelaying system. With the improvement of underwater monitoring means, reducing the underwater separation noise emanated from pyrotechnic driving devices is of great military application value for improving the combat performance of underwater weapon systems. However, there are few reports on the mechanisms and influencing factors of the noise generated by energetic materials in the process of energy conversion for pyrotechnic driving devices driven by energetic materials.

At present,the research on the noise of energetic materials can be divided into two aspects. (1) The first aspect concerns research on the application of noise in the process of the energy conversion of energetic materials, mainly focusing on analyzing underwater sound sources, underwater hydrodynamic sound source bubbles and thermal sound mechanisms, the acoustic radiation mechanisms of energetic materials during underwater combustion, and the main factors affecting acoustic radiation characteristics [4-7].(2) The second aspect of research on noise suppression in the process of the energy conversion of energetic materials mainly focuses on the study of the pulse noise of a gas jet at the muzzle or engine nozzle during weapon launching, the high-pressure flow field at that location,and the aerodynamic noise generated from it[8-10]. However, in a pyrotechnic driving device, the energy conversion process of the energetic materials is more complex,and the combustion or explosion of the energetic materials takes place in a closed chamber during the entire process. The noise generation mechanism in such a process is different from the existing underwater combustion noise generation mechanisms of energetic materials and gas jet pulses. When the strength of the shell carrying the energetic material is high enough,the shock wave generated by combustion or explosion can only transmit the gas pulsation through the shock vibration of the shell surface in the shell body,and then propagate through the medium and become noise.Therefore, the unsteady flow of the combustion products of the energetic material in the closed chamber stimulates the shell surface vibration, and the structural vibration noise generated by the vibration excitation is the main noise from the energetic material burning in the closed chamber; it is considered as a fluid-solid interference noise.

The combustion noise generation process of energetic materials in a closed chamber is a complex multi-physical field problem involving the fluid,structural,and acoustic domains.It is necessary to establish a set of multi-physical field-coupled numerical simulation analysis models to study the noise generation mechanism and its influencing factors. A. Bougamra and H. Lu used “ANSYS Fluent”to construct a numerical model for solid energetic material combustion in a closed bomb,which solved the Euler-Euler reactive two-phase flow problem of solid energetic material combustion,and predicted the pressure distribution in the closed bomb [11].Doo Hee Han et al. conducted numerical simulations on the combustion process of zirconium potassium perchlorate in a cylindrical closed bomb using computational fluid dynamics (CFD), analyzed the development of its fluid and the characteristics of its solid combustion, predicted the pressure fluctuation produced by the zirconium potassium perchlorate during the combustion process,and compared and analyzed the different flow characteristics of a closed bomb test in cylindrical and spherical closed bombs[12,13].However, the vibration responses of structures excited by the unsteady flow of combustion products and the acoustic radiation generated by structural vibrations have not been reported.

In this study, to explore the influencing factors of the radiated noise in the combustion process of energetic materials in a closed bomb, based on a closed bomb test, a fluid solid sound one-way coupling calculation model was established for the combustion process of energetic materials in a closed bomb. The effects of the loading doses,gas pressure,combustion coefficient of the energetic materials, and shell thickness of the closed bomb on the combustion noise of energetic materials were studied.The results provide a theoretical basis for the design of low-noise fire-working devices.

2. Multi-parameter test system for closed bomb

The structure of the closed bomb and igniter used in this study is shown in Fig.1.The volume of the closed bomb is about 2.5 mL,the material was 30CrMnSiA steel,and the sealing was reliable and had sufficient strength. The igniter was mainly composed of a shell,electrode plug (with a semiconductor bridge welded), initial charge, main charge, and sealing element. The initial charge comprised a trace of lead styphnate, and the main components of the main charge were 25% Al and 75%KClO.

To measure the gas pressure, shock response, and combustion noise of the igniter main charge after ignition, a multi-parameter test system was designed for the closed bomb. The test system comprised a response plate, closed bomb, igniter, pressure sensor,acceleration sensor,sound pressure sensor,etc.,as shown in Fig.2.The response plate was a 60 cm × 60 cm × 1 cm aluminum plate(suspended by four elastic ropes), and was used to simulate the installation environment of the pyrotechnic driving device[14].The igniter was connected to the closed bomb through the threaded hole, and the latter was installed in the center of the aluminum plate through the threaded hole. Three piezoelectric acceleration sensors (YD-109) were installed at 5 cm, 10 cm, and 15 cm away from the center of the igniter to record the output shock response of the igniter. A pressure sensor (CY-YD-205) was installed on the side of the closed bomb through the adapter. The position of the sensor diaphragm was flush with the inner wall surface.The signal generated by the sensor passed through the amplifier, and the pressure change curve of the inner chamber was recorded at a sampling rate of 500 K/s through the acquisition system. A sound pressure sensor(INV 9204)was installed 300 mm above the axis of the closed bomb.The sampling frequency was 500 K/s for collecting the combustion noise during the test. In this study, the output pressure,shock response, and combustion noise of an igniter with main charges of 100,150,200,and 250 mg were tested and studied.

Fig.1. Structure of closed bomb.

3. Numerical calculation model

3.1. Calculation process

The combustion noise generation process of energetic materials in a closed bomb is a complex multi-physical field problem involving the fluid, structural, and sound domains. Based on the closed bomb test,a fluid solid sound one-way coupling calculation model was established for the combustion process of the energetic materials in the closed bomb. The calculation flow is shown in Fig. 3. The entire calculation process was divided into three main steps:(1)the vibration excitation source of the shell surface during the combustion process of the energetic materials was obtained based on an unsteady flow field calculation; (2) a fluid structure coupling analysis was conducted based on a finite element model of the shell and response plate of the closed bomb under fluid excitation; and (3) the transient vibration response of the shell and response plate of the closed bomb were the boundary conditions of the sound source.

Fig. 2. Schematic diagram of closed bomb multi-parameter test system.

Fig. 3. Simulation calculation process.

3.2. Mesh model

According to the structure of the closed bomb and multiparameter test system, the integrated finite element model of the closed bomb and the shock response plate platform was established, as shown in Fig. 4. To simplify the model, the igniter, cartridge, and other components were deleted, and the charge was simplified to one layer of the main charge.To reduce the calculation scale,a 1/2 3D model was established according to the symmetry of the closed bomb structure. The pre-processing operations, such as model establishment and mesh division, were conducted using workbench software.The gas flow area was divided by a hexahedral grid,the size of the unit was 0.3 mm,and the total number of units was 44044. The closed bomb was divided by tetrahedral solid elements with sizes of 1 mm-1.5 mm,and the total number of units was 23413. The response plate was divided by a mixed grid with sizes of 1.5 mm-3.5 mm,and the total number of units was 75240.Monitoring points were set at different positions on the response board, with distances of 5 cm-15 cm from the center, and 5-cm apart. The acceleration along the Y direction of the monitoring point was extracted for analysis. Because the free boundary conditions were only applied to the shock response plate in the simulation, an elastic rope model was not established. To prevent an unnecessary low-frequency drift caused by rigid body motion,the acceleration data obtained by the simulation were processed by 8-order Butterworth high-pass filter with a cut-off frequency of 60 Hz[15].Finally,the shock response spectrum(SRS)of the closed bomb was calculated using the improved recursive filtering algorithm, and the output shock response characteristics of the closed bomb were analyzed. The validity of the analysis model was verified via comparison with experimental results.

Fig. 4. Integrated finite element model of closed bomb shock response plate.

3.3. Mathematical model

3.3.1. Fluid model

The combustion of energetic materials is a complex physical and chemical process. To simplify the model, the following basic assumptions are made for the entire combustion process [16].

1) The results show that all of the particles burn simultaneously,and satisfy the conditions of the geometric combustion law.

2) When ignoring the influence of the two-phase flow, the gunpowder gas obeys the Nobel equation. The energy released per unit mass of the propellant and gas temperature are regarded as fixed values, whereas the specific heat and residual capacity of the product are regarded as constants(without considering the changes in the gas product composition).

3) As the amounts of the primary explosive and ignition charge are small, it is assumed that the combustion product gas is the gas generated by the combustion of the main charge.

4) By ignoring the delay time of the igniter, it can be considered that the igniter starts to burn at 0.

5) There is no external work during combustion,and any heat loss is not considered.

6) Ignoring the specific process of combustion, the mass, momentum, and energy of the main charge are injected into the closed bomb based on compiling the user-defined function program.

Based on the above assumptions, the mathematical model is established as follows.

(1) Combustion model

The combustion of the propellant satisfies the geometric combustion law. The relationship between the formation rate of the gunpowder gas and change rate of the propellant shape and grain thickness is as follows:

In the above,Z = e/eis the relative thickness of the ignited powder; e is the arc thickness at any time; eis the initial thickness ; ψ is the percentage of the mass of the powder that has been ignited;χ,λ, and μ are the characteristic quantities of the powder shape; and dZ/dt represents the rate of change of the grain thickness, which depends on the burning rate of the propellant.

The burning speed of the powder satisfies the law of Saint Robert [17], as follows:

In the above, mis the powder quality,ψ is the burned mass fraction of the propellant, and ηis the mass fraction of gas in the powder product.

For the entire closed bomb system, according to the law of conservation of energy,a calculation can be derived,as follows:

Here ηis an energy release correction factor; cand care the specific heat of the gas at a constant pressure and constant volume, respectively; Tis the constant pressure explosion temperature; ˙W is the external work rate of the gas;and ˙Qis the heat dissipation rate. This study does not consider the external work and heat loss in the process of combustion, the composition of combustion products does not change with temperature,and the specific heat of the products is the average specific heat c.The temperature change is calculated as follows:

In the closed bomb, the state of gunpowder gas satisfies the Nobel Abel Equation [18,19], as follows:

Momentum equation:

In the above, ρ, u, p, E, and H are the density, velocity separation, pressure, total energy, and total enthalpy, respectively.S, the mass source term, is added to the mass of the gas generated by the combustion of gunpowder; Sis the momentum plus the mass source term along the i direction of the gas generated by the combustion of the propellant;and Sis the energy source term for the gas generated by the combustion of the gunpowder. The other relevant parameters (and their physical significance)can be found in the references[20,21].The calculation formula for each source term is as follows:

(3) Turbulence model and initial conditions

The turbulence model adopted the delayed separation eddy simulation model proposed by Spalart (and based on Mter's K-ω shear stress transport model), which can solve the turbulence kinetic energy and turbulence frequency of the conservative variables at each step [22-24]. The initial conditions of the fluid simulation were a temperature of 300 K, pressure of 1 atm, and velocity of 0.The material parameters of the propellant are shown in Table 1[17,25].

3.3.2. Fluid structure coupling

Fluid-solid coupling follows the most basic conservation principle;thus,the conservation of the stress(τ),displacement(d),heat flow (q), and temperature (T) of the fluid and solid should be satisfied at the interface of the fluid-solid coupling, that is, four equations should be satisfied, as follows [26]:

In the above, dand dare the displacements of the fluid and solid on the fluid solid coupling surface,respectively;τand τare the shear stresses of the fluid and solid on the fluid solid coupling surface, respectively; qand qare the heat fluxes of the fluid and solid on the fluid solid coupling surface,respectively;and Tand Tare the temperatures of the fluid and solid on the fluid solid coupling surface, respectively. Without considering the temperature change,the displacement balance equation and stress balance equation should be satisfied simultaneously,that is,the transfer of the analytical parameters between fluid and solid can be realized by satisfying the conservation of these two variables.In this study,ANSYS Workbench was used to analyze the fluid structure coupling.As the wall deformation of a closed bomb is caused by turbulence but its deformation degree is insufficient for affecting the flow state,this study adopted unidirectional coupling,without damping[27]. According to the relationships between nodes and interpolation between CFD nodes and finite element model grids, the pressure on the inner wall of the closed bomb was converted into the load for the structural simulation,and a transient analysis of the structure was conducted.

3.3.3. Acoustic calculation

In this study, the boundary element method (BEM) of the ‘LMS Virtual Lab’software was used to study the transient radiated noise generated by the vibration response of the closed bomb and the response plate in the combustion process of the propellant.Because the combustion noise of the propellant in a closed bomb is essentially the radiation noise generated by the vibration of the shell surface excited by the fluid,the radiation noise of the dipole source is the main noise. Therefore, in this study, only the dipole source was considered. First, it was transformed into a displacement boundary condition. Then, the velocity boundary condition was loaded into the acoustic boundary element mesh to solve the problem. The Helmholtz integral equation [28,29] was as follows:

In the above,B and C are the coefficient matrices of the BEM,p is the node sound pressure vector,and vis the node normal velocity vector.

The transient acoustic boundary element model for the vibration response of the closed bomb and response plate is shown in Fig. 5 (a); the symmetry plane of the model is Z = 0 mm. The boundary element mesh and field point grid were generated directly in the LMS virtual software. The boundary element mesh was the surface mesh of the closed bomb and response plate,and a triangular element was used to divide the mesh,as shown in Fig.5(b). The grid of the field points was a square area of 600 mm×600 mm in the symmetry plane,and the size of the grid was 2 mm.

4. Numerical simulation results

4.1. Comparison of numerical simulation results and experiments

4.1.1. Fluid simulation results

Taking the main charge of 150 mg as an example, the pressure nephogram of the fluid in the process of gunpowder combustion in the closed bomb is shown in Fig. 6. It can be seen from the figure that after the calculation, owing to the rapid combustion of the gunpowder,the gas pressure of the closed bomb increases rapidly,forming a shock wave. At 0.010 ms, the shock wave collides with the inner wall of the closed bomb and reflects,resulting in a severe pressure pulsation at the inner wall.With the increase in the mass and energy of the source phase, the pressure of the closed bomb increases, and tends to be stable at 0.427 ms.

A comparison between the CFD simulation results for the gas pressure and the experimental results for the closed bomb is shown in Fig. 7. It can be seen that in the closed bomb experiment, thepressure of the gunpowder gas begins to rise rapidly at t(0.117 ms)after ignition, and reaches a maximum value of 39.24 MPa at t(0.773 ms). Because the delay time of the igniter is ignored in the CFD model, it is considered that the igniter starts to burn at 0 ms,and that the particle surface is simultaneously completely burned;therefore, the peak time of the gas pressure obtained by the CFD simulation is smaller than the measured value.The contribution of the primary explosive and ignition powder to the gas pressure is not considered in the CFD model, so the maximum gas pressure obtained by the simulation is 1.73 MPa less than the experimental value (the deviation is 4.4%). In view of the above, the calculation result of the CFD model is considered as reliable.

Table 1 Material parameters of main charge Al/KClO4.

Fig. 5. Transient acoustic boundary element model.

Fig. 6. Pressure nephogram of fluid in the process of gunpowder combustion.

Fig. 7. Comparison of gas pressure experiment and simulation of closed bomb.

4.1.2. Fluid structure coupling

The pressure fluctuation on the inner surface of the closed bomb as excited by the gunpowder gas flow is used as the excitation source, and is loaded into the transient dynamic analysis of the closed bomb structure. Fig. 8 shows the total displacement nephogram of the closed bomb and response plate. It can be seen from the figure that the vibration response mainly occurs on the response plate, and that the maximum displacement is 0.47 mm.The finite element simulation analysis is compared with the measured impact acceleration and response spectrum at a location 5 cm away from the closed bomb on the response plate,as shown in Fig.9.In the time domain,the orders of magnitude of the responses between the simulation results and test results are the same.Compared with the test results, the vibration energy of the simulation results is relatively larger.The main reason is that the model simplification during the modeling process may change the stiffness and mass distribution of the system. It can be seen from the response spectrum that most of the experimental SRS is surrounded by the ±6 dB offset of the simulated SRS. The simulation results are in good agreement with the test data in the time domain and SRS, showing that the finite element model can predict the vibration response of a closed bomb, and that the calculation results are accurate.

Fig. 8. Calculation results of vibration response of closed bomb.

4.1.3. Acoustic calculation

The vibration displacements of the closed bomb and response plate as calculated by ANSYS Workbench software are imported into virtual lab acoustics.Through the data transfer,the calculation results of the finite element model surface mesh are transferred to the boundary element grid. At the acoustic boundary, a transient acoustic analysis is conducted to visualize the acoustic radiation from the transient impact noise. The propagation process of the transient radiation acoustic wave generated by the gunpowder combustion in the closed bomb is shown in Fig.10. It can be seen from Fig.10 that there is an evident fluctuation process in the radiation noise generated by the closed bomb and response plate.The sound pressure amplitude is large in the axial direction of the closed bomb.With the propagation of the radiated sound wave,the noise radiation energy diffuses, and the radiation sound pressure amplitude gradually decreases. Owing to the large vibration response and acoustic radiation area of the response plate under the excitation of the output shock in the process of gunpowder combustion, the radiation noise of the response plate occupies an important proportion of the total noise.

To verify the theoretical model of the gunpowder combustion noise in the closed bomb, the transient sound pressure curve and sound pressure spectrum curve of monitoring point #1 in the sound field are extracted and compared with the test results, as shown in Fig.11.It can be seen from the experimental test results in Fig. 11 (a) that after the igniter receives the ignition signal for 1.20 ms,the noise generated by the combustion of the gunpowder in the closed bomb spreads to the location of the sound pressure monitoring point, and the sound pressure increases sharply. With the increase in time, the sound pressure at the monitoring point fluctuates greatly; nevertheless, the sound pressure value is concentrated between 90 dB and 112 dB, and the maximum peak sound pressure is 111.9 2 dB.It can be seen from the test results of the sound pressure frequency response in Fig.11(b)that the sound energy of the noise generated during the combustion process of the closed bomb is mainly concentrated in the range of 1000 Hz-20000 Hz,and that the maximum sound pressure level is 97.84 dB at 5981 Hz.

In the simulation of the propellant combustion flow field, it is assumed that all particle surfaces burn simultaneously while ignoring the delay time of the igniter,and it is considered that the igniter starts to burn at 0 ms;moreover,some simplifications in the modeling process may change the stiffness and mass distribution of the system. The results show that the time required for transmitting the noise generated by the gunpowder in the closed bomb combustion process to the sound pressure monitoring point is faster than the experimental value, and that the speed of sound pressure increase is also faster. The overall sound pressure amplitude is correspondingly larger, the sound pressure value is denser between 100 dB and 122 dB, and the maximum peak sound pressure is 121.37 dB.In the time range of 1.50 ms-5 ms and frequency range of 2890-Hz-18100 Hz, the calculated results are in good agreement with the experimental data,and the error between the calculation results and test data is within 10%.This shows that the noise calculation model can predict the noise generated during the combustion process of a closed bomb.

4.2. Analysis of influencing factors of combustion noise

4.2.1. Effect of different sound pressure monitoring points on combustion noise of energetic material

Based on the fluid-solid-sound unidirectional coupling calculation model,this paper calculates the noise generated by gunpowder at sound pressure monitoring points #1, #2 and #3 during the combustion process in the closed explosive.The calculation results are shown in Fig.12.Fig.12(a)is the transient sound pressure curve at different sound pressure monitoring points,and Fig.12(b)is the sound pressure spectrum curve at different sound pressure monitoring points.As can be seen from the sound pressure at the sound pressure monitoring points#1 and#2 in Fig.12,with the increase of the axial distance of the sound pressure detection point, the amplitude of the transient radiation sound pressure and the frequency sound pressure level show a decreasing trend, indicating that the sound pressure presents a certain degree of attenuation with the increase of the propagation distance in the propagation process. As can be seen from the sound pressure at the sound pressure detection points#1 and#3 in Fig.12,the change of radial distance has little influence on the transient radiated sound pressure and frequency sound pressure level, indicating that the radiated noise of the response plate is the main component of the total noise, and the propagation of sound wave is approximately plane propagation.

Fig. 9. Comparison between finite element simulation and measured impact response.

Fig.10. Propagation process of transient radiated sound wave.

Fig.11. Comparison of noise simulation results and test results.

Fig.12. Transient sound pressure and frequency sound pressure level under different sound pressure monitoring points.

4.2.2. Effect of charge quantity on combustion noise of energetic material

Based on the fluid-solid sound one-way coupling calculation model, this study calculates the noise generated during the combustion of the propellant in the closed bomb when the main charge is 25 mg,50 mg,100 mg,150 mg,200 mg,and 250 mg,respectively.The calculation results are shown in Fig.13. Fig.13 (a) shows the time history curve of the gas pressure in the closed bomb with the different charge quantities.Fig.13(b)shows the time-domain curve of the impact acceleration at 5 cm away from the center of the igniter under the different charge quantities. Fig.13 (c) shows the transient sound pressure curve at sound pressure monitoring point#1 under the different charge quantities. Fig. 13 (d) shows the sound pressure spectrum curve at sound pressure monitoring point#1 under the different charge quantities.It can be seen from Fig.13 that the gas pressure, impact acceleration, and radiated sound pressure increase with the increase in charge.

Fig.13. Gas pressure, impact acceleration, transient sound pressure, and frequency sound pressure level under different loading capacities.

Taking the loading doses as the independent variable and the transient maximum peak sound pressure at the sound pressure monitoring point under the different charge quantities as the dependent variable,data fitting is conducted.The results are shown in Fig.14,and the fitting correlation is 0.998.The results show that the relationship between the maximum peak sound pressure and loading doses is in accordance with an exponential relationship,as follows:

P=81.36m(19)

In the above, Pis the maximum transient peak sound pressure(dB), and m is the loading doses (mg).

Fig.14. Relationship between loading doses and transient peak sound pressure.

It can be seen from Fig.14 that the actuating noise of the initiating device can be reduced to a certain extent by reducing the loading doses of the gunpowder gas, and the lower the loading doses, the more obvious the reduction effect is.

Since the gas state of gunpowder satisfies the Nobel-Abel equation in a closed chamber, the relationship between the maximum peak sound pressure(P)and peak gas pressure(p) can be obtained based on Eq. (19) as follows:

Therefore, the relation between the maximum peak sound pressure Pand the peak gas pressure pof the gunpowder with the main component of Al/KClO4 in the combustion process in the closed bomb to the exponential relationship, as shown below:

Fig.15. Relationship between gas pressure and maximum peak sound pressure.

The relationship between gas pressure and maximum peak sound pressure are shown in Fig.15.It can be seen from Fig.15 that the actuating noise of the initiating device can be reduced to a certain extent by reducing the charge weight or driving pressure of the gunpowder gas. However, because the pyrotechnic driving device mainly uses the high-temperature and high-pressure gas generated by the combustion of the gunpowder to work externally to realize unlocking and separation, there is a minimum reliable starting pressure [30-32]. Therefore, the effect of reducing the actuating noise of a pyrotechnic driving device by adjusting the pressure of the energetic material gas is relatively limited.

4.2.3. Effect of energetic material burning speed on energetic material combustion noise

To explore the influence of the energetic material combustion rate on the combustion noise, this study takes the Al/KClO4 combustion coefficient u (0.0075) as a reference and calculates the noise generated by the energetic material in the combustion process of the closed explosive when the combustion coefficient is u,u/2, u/4, u/8, and u/16, respectively, while keeping the other parameters unchanged.The results are shown in Figs.16-18.It can be seen from Fig.16 that with a decrease in the combustion coefficient of the energetic material, the rising rate of the gas pressure of the energetic material slows down,and the time to reach the maximum gas pressure is prolonged. As the combustion rate of gunpowder satisfies Saint-Robert's law and the gas state of gunpowder satisfies Nobel-Abel equation, it can be seen from the gunpowder combustion model in this paper that the burning rate coefficient of gunpowder and the time of arrival of gas pressure peak are derivatives of each other, as shown in Fig.17. When the combustion coefficient of gunpowder is only changed, the peak pressure of gunpowder gas and the total energy generated remain unchanged.For the same kind of gunpowder,the peak pressure of gas and the total energy are only related to the charge amount.When the total energy is constant,the amplitude of the shock response caused by it in resonance remains unchanged[33],and then the amplitude of the radiated noise aroused by it also remains basically unchanged.As can be seen from the transient sound pressure curve in Fig.18(a)and sound pressure spectrum curve in Fig.18(b),the change in the combustion coefficient of the energetic material has little effect on the noise generated during the combustion process of the gunpowder in the closed bomb. When the combustion coefficient is reduced by a multiple of 16, the maximum instantaneous sound pressure is reduced by 1.79 dB, and the frequency band sound pressure level is reduced by 1.75 dB. It can be seen that a small reduction in the burning speed and pressure rise rate of the energetic material has little effect on the combustion noise of the energetic material in the closed bomb, and that the noise reduction effect is limited.

Fig.16. Time history curve of gas pressure under different combustion coefficients.

Fig.17. Peak to time of gas pressure under different combustion coefficients.

4.2.4. Influence of shell thickness on combustion noise

To explore the influence of closed bomb shells on gunpowder combustion noise,this study calculates the noise generated by the combustion process of gunpowder in a closed bomb when the shell thickness h is 3 mm, 5 mm, 7 mm, and 9 mm, respectively, under the conditions that the main charge is 150 mg and the other parameters remain unchanged. Taking the shell thickness as the independent variable and the transient maximum peak sound pressure of the different shell thicknesses as the dependent variable,data fitting is conducted.The results are shown in Fig.19,and the fitting correlation is 0.994. This shows that the relationship between the maximum peak sound pressure and shell thickness of gunpowder in the combustion process in a closed explosive conforms to an exponential relationship, as follows:

Fig.18. Transient sound pressure, frequency sound pressure level of energetic material combustion under different combustion coefficients.

Fig. 19. Relation curve between shell thickness and transient maximum peak sound pressure.

Here,Pis the maximum transient peak sound pressure(dB), and h is the shell thickness (mm).

It can be seen from Fig. 19 that with the increase in shell thickness, the combustion noise of the gunpowder in the closed bomb decreases, and with the gradual increase in shell thickness,the reduction range of the combustion noise increases. Thus, the shell thickness of the initiating explosive device can be appropriately increased to achieve noise reduction. However, with the increase in the shell thickness, the volume and mass of the pyrotechnics driving device used temporarily in the system are also increased. Therefore, the appropriate shell thickness should be selected according to the application scenarios of the pyrotechnic driving device.

5. Conclusion

In this study,based on the closed explosive test with Al/KClOas the main charge, combined with CFD, a structural finite element method, and an acoustic boundary element method, a fluid solidacoustic one-way coupling calculation model for the combustion process of energetic materials in a closed bomb was established.The effects of the loading doses, gas pressure, combustion coefficient of the energetic materials, and shell thickness of the closed bomb on the combustion noise of energetic materials were studied.The following conclusions can be drawn.

(1) The maximum peak sound pressure increases exponentially with an increase in loading doses during the combustion process of energetic material in a closed bomb; the specific relationship is P= 81.36 m.

(2) The maximum peak sound pressure increases exponentially with an increase in gas pressure during the combustion process of energetic material in a closed bomb; the specific relationship is P= 91.78p.

(3) When the combustion coefficient is reduced by a multiple of 16, the maximum instantaneous sound pressure is reduced by 1.79 dB, and the frequency band sound pressure level is reduced by 1.75 dB.

(4) With an increase in shell thickness,the combustion noise of the energetic material in the closed bomb decreases,and the reduction range of combustion noise increases, the specific relationship is P= 128.84- 0.336e.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.