# A Numerical Study on the Blast Wave Distribution and Propagation Characteristics of Cylindrical Explosive in Motion

Authors: Haojie Chen; Jianping Yin; Xudong Li; Dong Yang; Wenjie Lu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In order to study the overpressure distribution law of a shock wave under the dynamic explosion of a cylindrical charge and the influence of the charge speed on the overpressure distribution law, a trinitrotoluene (TNT) cylindrical bare charge was selected. Additionally, a numerical simulation of a cylindrical charge explosion under different motion speeds was carried out using the AUTODYN finite element software. The results showed that during the static explosion of a cylindrical charge, the shock wave propagates outward in the form of an ellipsoid, whereas under the dynamic explosion, the shock wave overpressure field is an irregular ellipsoid, and the shock wave overpressure zone shifts to the velocity direction. With an increase in the charge velocity, the peak value of the shock wave overpressure increases gradually in the moving direction. The simulation data were analyzed, revealing that the calculation model of the dynamic explosion shock wave overpressure field of a cylindrical charge with a scaled distance of 1 m·kg−1/3 ≤ R¯ ≤ 1.5 m kg−1/3 can provide a reference basis for the study of dynamic explosion shock waves.


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## Body

## 1. Introduction

Research on the overpressure distribution law of explosive explosion shock waves has always been a hotspot for the relevant scholars worldwide. The calculation formula for the static explosion shock wave overpressure was obtained by fitting and correcting the experimental data based on the explosion similarity law [1]. Commonly used empirical formulas include those of Henrych [2], Sadovsky [3], and Baker [4]. Wisotski and Snyer [5] and Shepherd [6] found that under the static explosion of a cylindrical charge, the axial and radial shock waves reflect each other to form Mach waves and produce secondary shock waves. Knock [7] derived a calculation formula for the overpressure peak value of the axial primary shock wave of a cylindrical charge under a static explosion according to experimental data.In combat, the high-speed movement of the warhead has a significant impact on the characteristics of the explosion shock wave, which changes the distribution of the shockwave overpressure field. Therefore, there is an obvious difference between the dynamic explosion shock wave overpressure field and the static explosion shock wave overpressure field [8]. To calculate the shock wave overpressure field of a moving charge explosion, the equivalent charge quantity method is generally adopted. According to the energy similarity principle, the energy increase caused by the kinetic energy of the moving charge is equivalent to the increase in static charge quantity. Then, the corresponding dynamic explosion shock wave overpressure peak value is obtained by using the static explosion shock wave overpressure calculation formula [9]. This method cannot distinguish the difference in the impact overpressure distribution between the positive and negative motions.Currently, numerical analysis methods are generally used for research on dynamic explosion shock waves. Based on the Baker formula, Nie et al. [10] obtained a calculation model for the dynamic explosion shock wave overpressure field of a spherical charge by establishing the correction factor function of motion speed, azimuth, and scaled charge distance. Jiang et al. [11] obtained the evolution process of a shock wave field and the distribution law of shock wave overpressure under the condition of a dynamic explosion by numerically simulating the shock wave field of a spherical charge explosion at different speeds. They then established a reliable engineering calculation model of the shock wave overpressure field of a spherical charge dynamic explosion based on the simulation results. Chen et al. [12] constructed a correction factor function for the moving speed, azimuth, and scaled distance of a trinitrotoluene (TNT) spherical bare charge through the numerical calculation of the shock wave field of a TNT spherical bare charge at different speeds. They then obtained a dynamic explosion shock wave calculation model based on the Henrych formula. Wang et al. [13] studied the explosion of spherical bare charges at different moving speeds in an infinite air domain and found that the peak overpressure of a dynamic explosion shock wave appears in an area with an angle of 30° with the moving speed of the charge, and the pressure attenuation rate of a dynamic explosion shock wave is faster than that of a static explosion shock wave. At present, research on dynamic explosion shock waves is generally dominated by spherical charges. There is less research on cylindrical charges. Therefore, further research is needed in this field.In this study, the influence of the law of charge motion speed on the dynamic explosion shock wave field of a cylindrical charge was studied. The distribution range and attenuation law of the explosion shock wave of a cylindrical charge under different charge motion speeds were obtained, which will provide a reference for research on dynamic explosion shock wave and shooting range test.


## 2. Numerical Calculation Model

### 2.1. Establishment of a Numerical Calculation Model

The explosive selected was TNT with a mass of 6.23 kg, density of 1.63 g/cm3, and aspect ratio of 4.5 : 1 [14]. Using an axisymmetric calculation model, the air domain was 10 m long and 5m wide, and the Euler algorithm was used for calculation. The flow-out boundary condition was set around the air to eliminate the boundary effect so that the air and explosive gas could flow-out normally when they reached the calculation boundary. The material model was selected directly from the AUTODYN material library. The ideal air state equation was also adopted as follows:(1)p=γ−1ρe,where P is the air pressure (standard atmospheric pressure), γ is the ideal gas adiabatic index (1.4), ρ is the air density (1.225 × 10−3 g/cm3), and e is the initial specific internal energy of air (2.068 × 105 J/m3).TNT adopts the JWL state equation, expressed as(2)P=A1−ωR1Ve−R1V+B1−ωR2Ve−R2V+ωEV.where P is the pressure generated by the explosive; V is the relative volume; E is the internal energy per unit volume; and A, B, R1, R2, and ω are the material constants, as listed in Table 1.Table 1
Material parameters of TNT.
ParameterD/(m/s)a/GPab/GPaR1R2ωE/(J/m3)Value6930373.77374.714.150.90.356.62 × 1010The center point of the TNT is defined as the originO of the coordinate system. The x-axis is the model’s axis of symmetry, the right direction is positive, and the direction of movement of the charge is consistent with the positive direction of the x-axis. To obtain the pressure parameters of the shock wave in the process of propagation, four Gaussian points were set with the explosion center as the circle point and scaled distances R¯ (R¯=R/ω3, where R is the blasting distance, ω is the charge quantity) of 1 m·kg−1/3, 1.5 m·kg−1/3, 2 m·kg−1/3, and 2.5 m·kg−1/3, respectively. The angle between each Gaussian points column was 15°, and 13 columns and 52 Gaussian points were set. The numerical calculation model is shown in Figure 1.Figure 1
Numerical calculation model.


### 2.2. Finite Element Model Verification

To obtain reliable data, a TNT spherical bare charge with a mass of 6.23 kg was selected for the numerical simulation, and the reliability of the finite element model was verified using the Henrych formula. The design grid sizes were 5 mm × 5 mm, 10 mm × 10 mm, and 15 mm × 15 mm, and a reliable finite-element model was obtained by comparing the calculated value of the theoretical formula with the simulation value.(3)Δpm=1.379R¯+0.543R¯2+0.035R¯3+0.0006R¯4,0.05≤R¯≤0.3,0.607R¯−0.032R¯2+0.209R¯3,0.3<R¯≤1.0,0.065R¯+0.397R¯2+0.322R¯3,1.0<R¯≤10,where R¯ is the scaled distance (R¯=R/ω3, where R is the blasting distance (m) and ω is the charge quantity (kg)).The relative errors between the simulation values of different mesh sizes and the Henrych formula are shown in Figure2. According to the relative error comparison diagram, the error of the 10 mm × 10 mm mesh model is relatively small. Therefore, a finite element model with a 10 mm × 10 mm mesh size was selected.Figure 2
Comparison of relative error between numerical calculation value and formula calculation value.


## 2.1. Establishment of a Numerical Calculation Model

The explosive selected was TNT with a mass of 6.23 kg, density of 1.63 g/cm3, and aspect ratio of 4.5 : 1 [14]. Using an axisymmetric calculation model, the air domain was 10 m long and 5m wide, and the Euler algorithm was used for calculation. The flow-out boundary condition was set around the air to eliminate the boundary effect so that the air and explosive gas could flow-out normally when they reached the calculation boundary. The material model was selected directly from the AUTODYN material library. The ideal air state equation was also adopted as follows:(1)p=γ−1ρe,where P is the air pressure (standard atmospheric pressure), γ is the ideal gas adiabatic index (1.4), ρ is the air density (1.225 × 10−3 g/cm3), and e is the initial specific internal energy of air (2.068 × 105 J/m3).TNT adopts the JWL state equation, expressed as(2)P=A1−ωR1Ve−R1V+B1−ωR2Ve−R2V+ωEV.where P is the pressure generated by the explosive; V is the relative volume; E is the internal energy per unit volume; and A, B, R1, R2, and ω are the material constants, as listed in Table 1.Table 1
Material parameters of TNT.
ParameterD/(m/s)a/GPab/GPaR1R2ωE/(J/m3)Value6930373.77374.714.150.90.356.62 × 1010The center point of the TNT is defined as the originO of the coordinate system. The x-axis is the model’s axis of symmetry, the right direction is positive, and the direction of movement of the charge is consistent with the positive direction of the x-axis. To obtain the pressure parameters of the shock wave in the process of propagation, four Gaussian points were set with the explosion center as the circle point and scaled distances R¯ (R¯=R/ω3, where R is the blasting distance, ω is the charge quantity) of 1 m·kg−1/3, 1.5 m·kg−1/3, 2 m·kg−1/3, and 2.5 m·kg−1/3, respectively. The angle between each Gaussian points column was 15°, and 13 columns and 52 Gaussian points were set. The numerical calculation model is shown in Figure 1.Figure 1
Numerical calculation model.


## 2.2. Finite Element Model Verification

To obtain reliable data, a TNT spherical bare charge with a mass of 6.23 kg was selected for the numerical simulation, and the reliability of the finite element model was verified using the Henrych formula. The design grid sizes were 5 mm × 5 mm, 10 mm × 10 mm, and 15 mm × 15 mm, and a reliable finite-element model was obtained by comparing the calculated value of the theoretical formula with the simulation value.(3)Δpm=1.379R¯+0.543R¯2+0.035R¯3+0.0006R¯4,0.05≤R¯≤0.3,0.607R¯−0.032R¯2+0.209R¯3,0.3<R¯≤1.0,0.065R¯+0.397R¯2+0.322R¯3,1.0<R¯≤10,where R¯ is the scaled distance (R¯=R/ω3, where R is the blasting distance (m) and ω is the charge quantity (kg)).The relative errors between the simulation values of different mesh sizes and the Henrych formula are shown in Figure2. According to the relative error comparison diagram, the error of the 10 mm × 10 mm mesh model is relatively small. Therefore, a finite element model with a 10 mm × 10 mm mesh size was selected.Figure 2
Comparison of relative error between numerical calculation value and formula calculation value.


## 3. Numerical Simulation Results

### 3.1. Analysis of Peak Overpressure of Shock Wave

The pressure nephograms of the explosion shock wave with different velocities were obtained by simulation calculations, as shown in Figure3.Figure 3
Shock wave pressure nephogram. (a)v = 0 m/s, (b) v = 340 m/s, (c) v = 680 m/s, and (d) v = 1020 m/s.
(a)(b)(c)(d)Through the shock wave pressure nephogram, it was found that the shock wave formed by charge explosion with different moving speeds propagates in the form of a spherical wave. In the static explosion (v = 0 m/s), the pressure cloud image is ellipsoid, the direction of charge motion is symmetrical, and the shock wave pressure is the largest in the direction of 90° (θ = 90°) with the direction of charge motion. With the increase in the charge velocity, the shock wave pressure cloud is an irregular ellipsoid, and the maximum pressure zone moves in the direction of the velocity with an increase in the charge velocity; that is, the direction angle θ decreases.To study the azimuth of the peak overpressure of the shock wave at different charge velocities, a three-dimensional histogram was drawn based on the data obtained at the same proportional distance, as shown in Figure4.Figure 4
Peak overpressure at different measuring points. (a)R¯ = 1 m·kg−1/3, (b) R¯ = 1.5 m·kg−1/3, (c) R¯ = 2 m·kg−1/3, and (d) R¯ = 2.5 m·kg−1/3.
(a)(b)(c)(d)It can be concluded from Figure4 that when the scaled distances are 1 m·kg−1/3 and 1.5 m·kg−1/3, the peak overpressure of the shock wave increases first and then decreases with the increase of azimuth at the same velocity. When the scaled distance is 2 m·kg−1/3 and the velocity is less than 1133.6 m/s, the peak value of shock wave overpressure increases first and then decreases with the increase of azimuth at the same velocity. When the velocity is greater than 1133.6 m/s, the peak value of shock wave overpressure decreases with the increase of the azimuth angle. When the scaled distance is 2.5 m kg−1/3 and the velocity is less than 906.8 m/s, the peak value of shock wave overpressure increases first and then decreases with the increase of azimuth at the same velocity. When the velocity is greater than 906.8 m/s, the peak value of shock wave overpressure decreases with the increase of azimuth. With an increase in the velocity, the azimuth of the shock wave overpressure peak decreased.


### 3.2. Establishment of the Engineering Calculation Model

To study the azimuth angle of the explosion shock wave overpressure peak at different scaled distances and speeds, the variation curves of the velocity and azimuth at different scaled distances were drawn according to the simulation results, as shown in Figure5.Figure 5
Azimuth angle of peak overpressure at different proportional distances. (a)R¯ = 1m·kg−1/3, (b) R¯ = 1.5 m·kg−1/3, (c) R¯ = 2 m·kg−1/3, and (d) R¯ = 2.5 m·kg−1/3.
(a)(b)(c)(d)The azimuth of the shockwave overpressure peak at different scaled distances and charge velocities is shown in Figure5. By comparing each figure, it can be found that when the scaled distance R¯ < 1.5 m·kg−1/3, with an increase in the motion speed, the azimuth is mainly concentrated at 90°, 75°, 60°, and 45°, and the velocity of charge has little effect on the azimuth of the peak value of shock wave overpressure. When the scaled distance R¯ > 1.5 m·kg−1/3, the azimuth gradient increases, and the velocity of charge has a significant influence on the azimuth of the shock wave overpressure peak.From Figure5, we can obtain the azimuthal comparison concentration of the shock wave overpressure peak when the scaled distance R¯ < 1.5 m·kg−1/3. When the scaled distance R¯ > 2 m·kg−1/3, the azimuth angle of the shock wave overpressure peak becomes divergent, owing to the influence of the charge shape and velocity. Multiple linear fitting was performed on the data at a scaled distance R¯ <1.5 m·kg−1/3 to obtain the numerical calculation model of the peak overpressure of the shock wave, velocity of charge, and scaled distance:(4)P=2002.901−0.047v−1120.505R¯,0≤v≤170,2144.441+0.235v−1273.557R¯,170<v≤680,2568.038+0.413v−1712.858R¯,680<v≤1360,2301.519+0.917v−2057.020R¯,1360<v≤1530,where P is the peak overpressure of the shock wave (kPa), v is the velocity of charge (m/s), and R¯ is the scaled distance (m·kg−1/3).


### 3.3. Model Verification

To verify the reliability of the calculation model, a cylindrical bare charge with charging speeds of 100 m/s, 500 m/s, 1000, and 1500 m/s was numerically simulated, and the overpressure field distribution of the dynamic explosion shock wave was obtained. The simulation and formula calculation values were compared at the scaled distance of 1 m·kg−1/3, 1.2 m·kg−1/3, 1.4 m·kg−1/3, and 1.5 m·kg−1/3, as shown in Figure 6.Figure 6
Comparison between simulation and formula.It can be seen from Figure6 that the calculation results of 8 groups of formulas are consistent with the simulation calculation results. There is little difference between the calculated values of other formulas and the simulation results, and the average error is 8.67%. Therefore, it is considered that the fitting formula has high reliability and can be used to predict the shock wave overpressure when the scaled distance is less than 1.5 m kg−1/3.


## 3.1. Analysis of Peak Overpressure of Shock Wave

The pressure nephograms of the explosion shock wave with different velocities were obtained by simulation calculations, as shown in Figure3.Figure 3
Shock wave pressure nephogram. (a)v = 0 m/s, (b) v = 340 m/s, (c) v = 680 m/s, and (d) v = 1020 m/s.
(a)(b)(c)(d)Through the shock wave pressure nephogram, it was found that the shock wave formed by charge explosion with different moving speeds propagates in the form of a spherical wave. In the static explosion (v = 0 m/s), the pressure cloud image is ellipsoid, the direction of charge motion is symmetrical, and the shock wave pressure is the largest in the direction of 90° (θ = 90°) with the direction of charge motion. With the increase in the charge velocity, the shock wave pressure cloud is an irregular ellipsoid, and the maximum pressure zone moves in the direction of the velocity with an increase in the charge velocity; that is, the direction angle θ decreases.To study the azimuth of the peak overpressure of the shock wave at different charge velocities, a three-dimensional histogram was drawn based on the data obtained at the same proportional distance, as shown in Figure4.Figure 4
Peak overpressure at different measuring points. (a)R¯ = 1 m·kg−1/3, (b) R¯ = 1.5 m·kg−1/3, (c) R¯ = 2 m·kg−1/3, and (d) R¯ = 2.5 m·kg−1/3.
(a)(b)(c)(d)It can be concluded from Figure4 that when the scaled distances are 1 m·kg−1/3 and 1.5 m·kg−1/3, the peak overpressure of the shock wave increases first and then decreases with the increase of azimuth at the same velocity. When the scaled distance is 2 m·kg−1/3 and the velocity is less than 1133.6 m/s, the peak value of shock wave overpressure increases first and then decreases with the increase of azimuth at the same velocity. When the velocity is greater than 1133.6 m/s, the peak value of shock wave overpressure decreases with the increase of the azimuth angle. When the scaled distance is 2.5 m kg−1/3 and the velocity is less than 906.8 m/s, the peak value of shock wave overpressure increases first and then decreases with the increase of azimuth at the same velocity. When the velocity is greater than 906.8 m/s, the peak value of shock wave overpressure decreases with the increase of azimuth. With an increase in the velocity, the azimuth of the shock wave overpressure peak decreased.


## 3.2. Establishment of the Engineering Calculation Model

To study the azimuth angle of the explosion shock wave overpressure peak at different scaled distances and speeds, the variation curves of the velocity and azimuth at different scaled distances were drawn according to the simulation results, as shown in Figure5.Figure 5
Azimuth angle of peak overpressure at different proportional distances. (a)R¯ = 1m·kg−1/3, (b) R¯ = 1.5 m·kg−1/3, (c) R¯ = 2 m·kg−1/3, and (d) R¯ = 2.5 m·kg−1/3.
(a)(b)(c)(d)The azimuth of the shockwave overpressure peak at different scaled distances and charge velocities is shown in Figure5. By comparing each figure, it can be found that when the scaled distance R¯ < 1.5 m·kg−1/3, with an increase in the motion speed, the azimuth is mainly concentrated at 90°, 75°, 60°, and 45°, and the velocity of charge has little effect on the azimuth of the peak value of shock wave overpressure. When the scaled distance R¯ > 1.5 m·kg−1/3, the azimuth gradient increases, and the velocity of charge has a significant influence on the azimuth of the shock wave overpressure peak.From Figure5, we can obtain the azimuthal comparison concentration of the shock wave overpressure peak when the scaled distance R¯ < 1.5 m·kg−1/3. When the scaled distance R¯ > 2 m·kg−1/3, the azimuth angle of the shock wave overpressure peak becomes divergent, owing to the influence of the charge shape and velocity. Multiple linear fitting was performed on the data at a scaled distance R¯ <1.5 m·kg−1/3 to obtain the numerical calculation model of the peak overpressure of the shock wave, velocity of charge, and scaled distance:(4)P=2002.901−0.047v−1120.505R¯,0≤v≤170,2144.441+0.235v−1273.557R¯,170<v≤680,2568.038+0.413v−1712.858R¯,680<v≤1360,2301.519+0.917v−2057.020R¯,1360<v≤1530,where P is the peak overpressure of the shock wave (kPa), v is the velocity of charge (m/s), and R¯ is the scaled distance (m·kg−1/3).


## 3.3. Model Verification

To verify the reliability of the calculation model, a cylindrical bare charge with charging speeds of 100 m/s, 500 m/s, 1000, and 1500 m/s was numerically simulated, and the overpressure field distribution of the dynamic explosion shock wave was obtained. The simulation and formula calculation values were compared at the scaled distance of 1 m·kg−1/3, 1.2 m·kg−1/3, 1.4 m·kg−1/3, and 1.5 m·kg−1/3, as shown in Figure 6.Figure 6
Comparison between simulation and formula.It can be seen from Figure6 that the calculation results of 8 groups of formulas are consistent with the simulation calculation results. There is little difference between the calculated values of other formulas and the simulation results, and the average error is 8.67%. Therefore, it is considered that the fitting formula has high reliability and can be used to predict the shock wave overpressure when the scaled distance is less than 1.5 m kg−1/3.


## 4. Conclusion

In this study, the changes in the dynamic explosion shock wave overpressure field at different speeds were obtained through the numerical simulation of cylindrical charges with different moving speeds. As a result, the following conclusions were drawn:(1)
When the charge velocity was 0 m/s, the shock wave overpressure field was ellipsoid and symmetrical with respect to the charge velocity direction. With an increase in the charge velocity, the shock wave overpressure field became an irregular ellipsoid.(2)
At the same proportional distance, with an increase in the moving speed of the charge, the shock wave pressure increased gradually in the moving direction of the charge and decreased gradually in the negative direction of the moving direction of the charge(3)
When the scaled distance was less than 1.5 m·kg−1/3, the charge velocity had little influence on the distribution of the shock wave overpressure peak. When it is greater than 1.5 m·kg−1/3, it has a significant influence on the distribution of the shock wave overpressure peak.(4)
Based on the numerical results, an engineering calculation model of dynamic explosion shock wave overpressure field of cylindrical charge with a length diameter ratio of 4.5 : 1 is obtained when the scaled distance is less than 1.5 m·kg−1/3



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*Source: 6446164-2022-03-22.xml*