Experimental and numerical simulation of the attenuation effect of blast shock waves in tunnels at different altitudes

Abstract

Traffic engineering such as tunnels in various altitudinal gradient zone are at risk of accidental explosion, which can damage personnel and equipment. Accurate prediction of the distribution pattern of explosive loads and shock wave propagation process in semi-enclosed structures at various altitude environment is key research focus in the fields of explosion shock and fluid dynamics. The effect of altitude on the propagation of shock waves in tunnels was investigated by conducting explosion test and numerical simulation. Based on the experimental and numerical simulation results, a prediction model for the attenuation of the peak overpressure of tunnel shock waves at different altitudes was established. The ` ? results showed that the peak overpressure decreased at the same measurement points in the tunnel entrance under the high altitude condition. In contrast, an increase in altitude accelerated the propagation speed of the shock wave in the tunnel. The average error between the peak shock wave overpressure obtained using the overpressure prediction formula and the measured test data was less than 15%, the average error between the propagation velocity of shock waves predicted values and the test data is less than 10%. The method can effectively predict the overpressure attenuation of blast wave in tunnel at various altitudes.

Headnote

ABSTRACT

Traffic engineering such as tunnels in various altitudinal gradient zone are at risk of accidental explosion, which can damage personnel and equipment. Accurate prediction of the distribution pattern of explosive loads and shock wave propagation process in semi-enclosed structures at various altitude environment is key research focus in the fields of explosion shock and fluid dynamics. The effect of altitude on the propagation of shock waves in tunnels was investigated by conducting explosion test and numerical simulation. Based on the experimental and numerical simulation results, a prediction model for the attenuation of the peak overpressure of tunnel shock waves at different altitudes was established. The ` ? results showed that the peak overpressure decreased at the same measurement points in the tunnel entrance under the high altitude condition. In contrast, an increase in altitude accelerated the propagation speed of the shock wave in the tunnel. The average error between the peak shock wave overpressure obtained using the overpressure prediction formula and the measured test data was less than 15%, the average error between the propagation velocity of shock waves predicted values and the test data is less than 10%. The method can effectively predict the overpressure attenuation of blast wave in tunnel at various altitudes.

ARTICLE INFO

Article history:

Received 29 April 2024

Received in revised form

1 July 2024

Accepted 12 July 2024

Available online 20 July 2024

Keywords:

Shock wave propagation

Tunnel

Altitude effect

Peak overpressure

Shock waves velocity

Engineering safety

(ProQuest: ... denotes formulae omitted.)

1. Introduction

Economic, scientific and technological developments have facilitated the construction of tunnels, subways, underground caverns and other structures in special zones. Owing to the special characteristics of the geographic environment, underground space structures outside the plains face serious security threats such as terrorist attacks and accidental shock explosions, which can lead to the destruction of personnel and equipment in tunnels under different altitudes. The propagation law of explosive shock waves is the basis for evaluating the destructive effect of weapons and the safety performance of the protective engineering. Therefore, a comprehensive theory is needed to predict the propagation pattern of the blast wave in the semi-enclosed structure under different altitudes, and calculate the load distribution inside the structure, so as to rationally design the engineering structure and ensure the safety performance of personnel and infrastructure.

In the study of the propagation law of shock waves influenced by characteristics of air, based on the law of shock wave propagation in a free field in an atmospheric pressure environment [1-3], many scholars have introduced variables of atmospheric parameters to study the relationship between shock wave propagation characteristics and atmospheric parameters. Erilienke [4] studied the shock waves of an air explosion near the ground. Based on the empirical equation of the spherical shock wave produced by the explosion of a Trinitrotoluene (TNT)-concentrated charge in Sadosky's unbounded space, a theoretical calculation method for the initial parameters of the air medium relative to the characteristic parameters of the explosion shock wave at different elevations was presented. Dewey [5] and Sadovskyi [6] combined similarity theory, proposed an empirical equation for calculating the shock wave overpressure caused by a TNT explosion at high altitudes. Kinney et al. [7] proposed a shock wave overpressure and specific impulse calculation model for the atmospheric conversion factor, in which the atmospheric conversion factor not only considers the influence of environmental pressure but also introduces a correction term for temperature. Izadifard et al. [8] studied the influence of explosion shock wave propagation in a high-altitude environment using the AUTODYN finite element software and fitted the correction factor of the explosion shock wave parameters under low pressure. Veldman et al. [9] measured the change in explosion shock wave pressure and specific impulse by adjusting the pressure in the seal. The results showed that the shockwave pressure and specific impulse increased with an increase in the ambient pressure. Huang et al. [10] carried out experiments on explosives detonation in water to study the load distribution and structural response under special pressure environments. Li et al. [11] studied the influence of altitude conditions on the explosion shock wave parameter law, deduced the low temperature and low pressure environment of the Wave front surface parameters of the calculation equation. Wang et al. [12]proposed a computational model for calculating the blast shockwave load distribution based on elevation and scaled distance factors.

The above studies mainly focused on free-field or semi-freefield environments for the propagation of shock waves at different elevations, Compared with the propagation of shock waves in a free field, the propagation process of shock waves in a tunnel is highly complex. For the problem of shock wave propagation in tunnels, scholars at home and abroad have proposed a series of computational models, and the following are commonly used: Calculations from the U.S. Army Corps of Engineers Waterways Experiment Station (WES) [13,14], calculations from the Ernst-Mach-Institut (EMI) in Germany [15], and calculations from the Chinese Army Corps of Engineers Scientific Research Institute (CACRI) [16]; In addition, scholars from Chinese research institutions based on the quantitative analysis and experimental results put forward a variety of explosives (such as TNT explosives, detonating gas, thermobaric explosives, emulsion explosives, etc.) [17-20] in different tunnel-related structures (such as long straight tunnels [21-23], T-shaped tunnels [24], corrugated steel lined tunnels [25,27], the enclosed blast wall structure (EBWS) [26],etc.) in the propagation of the theory of the calculation of the above calculations, combined with the propagation distance, the size of the tunnel and the loading conditions of the coefficients of a variety of shock wave attenuation calculation formulas, which are used in most of the models used to predict the propagation of the shock Wave when the explosion occurs inside the tunnel, but also a large number of different formulae for calculating the results of the large differences. The confinement of the structure complicates the process of shock wave propagation in the building, especially when environmental changes are involved, and the state of motion of the shock wave can be affected substantially [28]. Smirnov et al. [29] investigated the reflection and focusing process of a shock wave Within a wedge structure in a hydrogen-air environment, and optimized the design of a shock wave motion model with turbulence and viscosity. Yin et al. [30] established a Eulerian compressible fluid model to study the propagation law of shock waves Within a double-layer ship plate structure under the underwater high-pressure environment, and combined the isentropic single-fluid model and Newton's second law to analyze the influence of the special environment on the effect of impact.

Most of the existing calculation methods for load distribution in tunnels are based on experimental analyses in atmospheric pressure environments, with insufficient consideration of air parameters, especially when the explosives are detonated outside the tunnel mouth, changes in atmospheric parameters will affect the energy output of the explosion and the propagation mechanism of the tunnel shockwave, and there is a lack of sound calculation methods for predicting the distribution of loads in tunnels on altitudinal zones. Therefore, new influencing factors need to be introduced based on the traditional theoretical model to reveal the propagation law of shock waves in tunnels under different altitude conditions. Based on the test data and numerical simulation results for a tunnel explosion, the altitudinal effect on the propagation of the shock wave in the tunnel was analyzed using quantitative analysis of the determined atmospheric pressure, tunnel dimensions, loading conditions, the peak overpressure and shock Wave velocity as a function of the peak pressure. The atmospheric pressure correction coefficients obtained were used to calculate the peak overpressure and propagation velocity of the blast shock Wave, and the propagation mechanism of the damage elements inside the tunnel after the explosion outside the tunnel portal was investigated at different altitudes. These results provide a basis for establishing damage effect assessment techniques and ensuring process safety performance in unconventional environments such as tunnels under different zones.

2. Test scheme and result analysis

2.1. Test model tunnel

The damage element propagation test model tunnels were built on 4500 m and 200 m environments, respectively. Tunnel dimensions as shown in Fig. 1. The tunnel processing is shown in Fig. 2. The model tunnel is a reinforced concrete structure with a double layer of reinforcing mesh within the concrete lining. Due to site constraints, the tunnel is assembled in segments, with each segment of the tunnel being 3 m long, making a total of 5 segments.

2.2. Explosion test scheme

2.2.1. Layout scheme of the pressure sensor

8 detection sections were demarcated inside the tunnel, and an overpressure sensor was installed at the top of each detection section. The wall pressure sensor adopted was a 102A piezoelectric pressure sensor (PCB Piezotronics Company, America). The sensor models used were 113B21 (Test range 0-1.38 MPa, sensitivity 25 mV/psi), 113B26 (Test range 0-3.4 MPa, sensitivity 10 mV/psi), and 113B24 (Test range 0-6.9 MPa, sensitivity 5 mV/psi). Sensor test section is 0.281 inches in diameter and is acceleration compensated.

The positions of the detection surface, pressure sensor, and sensor-mounting base are shown in Fig. 3 and Table 1.

2.2.2. Data acquisition

The shock wave overpressure acquisition system used was a Swiss Tranas 3 data acquisition instrument with a sampling frequency of 1 MHz. The resolution of the instrument can reach 16-bit, the flash memory of a single channel can reach 128 MS, the accuracy can reach 0.03% of FSR, all channels can be triggered synchronously.

The data-acquisition instrument was connected to the signalconditioning instrument through an acquisition wire. After the signal was processed by the signal-conditioning instrument, it was converted into a digital signal. The signal was analyzed using a computer processing system, and finally, the time curve of the test parameters of each channel was obtained. The operating principle of the data acquisition system is shown in Fig. 4.

2.2.3. Electric cable

The data acquisition equipment is connected to the overpressure sensor via coaxial cable. The coaxial cable used for the test can be used for transmission of both analog and digital signals and can support high-bandwidth communication over relatively long repeaterless lines, and is therefore widely used in the testing part of explosion tests.

2.2.4. High-speed camera

A high-speed camera adjusts the radiant brightness by controlling the exposure time and transmittance to obtain an evolution reflecting the fireball on the millisecond scale. For the explosion test, we used a high-speed camera to film the explosion (Integrated Design Tools, Inc., as shown in Fig. 5). The high-speed camera was placed approximately 50 m from the tunnel entrance, and the camera sampling frequency was 4000 frames/s.

2.3. Experimental design

The explosion source uses cylindrical TNT explosives. The positions of the explosives are shown in Fig. 6. When the explosives explode, it will produce the shock wave and fireball that expands outward rapidly at a very high speed, and the light and shadow signals produced by the flame are captured by the high-speed camera, and the movement process of the flame can be observed through the processing of the image; in addition to this, the shock wave enters into the tunnel and the sensor for measuring the overpressure to produce a violent collision, and the mechanical signals are converted into electronic signals through the contact, and the electronic signals are transmitted through the coaxial wires and the data acquisition wires to the electronic signal is transmitted to the shockwave signal acquisition device through the coaxial wire and data acquisition wire, and the shockwave signal acquisition device will visualize the collected electrical signal and form the pressure curve. Ultimately, the above test method can be a complete measurement of the power field of the explosion shockwave.

To study the propagation law of explosion shock waves in a tunnel structure in different altitude environments, four groups of tunnel explosion test conditions were established to study the influence of the altitudinal environment on the shock wave propagation under different charge quantities and positions. Table 2 lists the layout conditions of each test and the altitude of the test environment; the installation position of the explosive was on the central axis of the equivalent cross-sectional center of the tunnel.

2.4. Test results

2.4.1. Explosive fireball effect

The fireball effect captured by the high-speed camera when the TNT exploded outside the tunnel entrance is shown in Fig. 7. The explosives detonated outside the tunnel entrance produced a rapidly expanding red-orange fireball. Owing to ground constraints, the outward expansion of the fireball following the reflection of the upward movement resulted in a ring of fire circle, whereas owing to the upward propagation of the high-temperature gas and flame, the fireball had the effect of layering, and a large amount of black smoke appeared at the bottom of the ball of fire, which followed the ball of fire to the surrounding area. With the prolongation of the explosion, the images captured by the high-speed camera showed the airwave array in the shape of a hemisphere to the outward expansion of the fireball. The fireball color changed from bright yellow to dark orange, and an increase in black smoke was observed until the explosion process was completed.

2.4.2. Analysis of overpressure time history curve

After the explosion of a 1 kg TNT explosive at 1 m outside the tunnel entrance, the overpressure results at typical locations in the tunnel under the 4500 m zone and 200 m plain were compared (Figs. 8(a) and 8(b), respectively). Next, the overpressure results at typical locations in the tunnels at the high altitude zone and plain were compared after a 2 kg TNT explosion 2 m outside the tunnel entrance (Figs. 9(a) and 9(b), respectively). Under the conditions of the two types of charges, the peak results of the overpressure at the same measuring point in the tunnel in the high altitude zone and plain environments are listed in Table 2, and the comparison results of the peak overpressure are shown in Figs. 10(a) and 10(b), respectively.

According to the time history curve of the overpressure, the shock wave produced by the explosion expanded outward at a very high speed. When the wavefront enters the tunnel, the shock wave acts on each measuring point in the tunnel, and the overpressure on the test surface increases almost vertically to reach its peak. Unlike the steady attenuation process in a free field, the overpressure curve of the shock wave in the tunnel oscillated with multiple peaks and attenuated in a sawtooth shape. It showed the Characteristics of multiple peaks, and the peak value of the subsequent wave was lower than that of the first peak. When the overpressure attenuated to atmospheric pressure, the waveform continued to attenuate. The pressure was less than the atmospheric pressure, and the peak value of the negative pressure was usually less than that of the positive pressure. When the negative pressure ended, it returned to atmospheric pressure. Under the action of subsequent reflections and the current wavefront, a new barotropic band was formed.

The attenuation of shock waves in a long straight tunnel is primarily caused by viscous consumption and irreversible energy loss owing to the shock adiabatic compression of air [27]. When the explosion shock wave propagates in the tunnel, the front end of the wave array propagates at a velocity D higher than the current ambient sound velocity D, and the back end of the barotropic action zone moves forward with the sound velocity са in the air, corresponding to the current altitude environmental pressure Pa. In the area near the tunnel entrance, after the shock wave expanded into the tunnel at high speed, it was reflected violently and flowed around the tunnel wall, and the shock wave overpressure attenuated rapidly. Simultaneously, the fragments and flying rocks produced near the explosion area also had a certain attenuation effect on the shock wave. With an increase in the distance from the explosion source, the peak attenuation velocity of the shock wave overpressure decreased, the duration increased, a flattening effect was formed, and the effect of the reflected wave on the propagation process of the shock wave weakened. The overpressure curve in the tunnel began to exhibit a more regular shape. After surging to the peak value, the overpressure decreased to a negative value and then gradually returned to zero with the pressure fluctuation. The interval of the overpressure arrival time at the adjacent measuring points increased with an increase in the distance from the tunnel entrance, the shock wave propagation velocity slowed down, and the wavefront motion state was similar to the free-field plane wave propagation process.

According to the trend of overpressure in the time history curve illustrated in Figs. 8 and 9, it can be seen that at different elevations, the attenuation law of shock wave overpressure at the same position in the tunnel was similar, and the main difference was in the arrival time and peak value of overpressure. Based on the comparison results, it can be seen that after the explosion outside the tunnel entrance, the shock wave entered the tunnel; at the same position inside the entrance, the overpressure arrived later, and the peak overpressure in the high altitude zone was larger than in the plain environment. According to the peak overpressure at each measuring point presented in Table 3, the peak overpressure did not monotonically decrease with the increase in the propagation distance of the shock wave in the tunnel; owing to the semienclosed structure of the tunnel, the incident shock wave is constrained such that the shock wave is constantly reflected. When the reflected wave of the early batch of the wave catches up with the advancing wave front surface and is superimposed on it, a small increase is observed in the peak overpressure phenomenon.

We considered the time of the shock wave arriving at measuring point 1 as zero and subsequently compared the arrival times of the shock wave at each measuring point. The arrival time of the shock wave at the same location of the tunnel in the high altitude zone and plain environments were compared under the two charge conditions (Figs. 11(a) and 11(b), respectively). According to the comparison between the two altitudes of the arrival time of the shock wave, overall, in the high-altitude environment, the time required for the shock wave to travel the same distance in the tunnel was shorter, the gap in the overpressure arrival time was small in the front part of the tunnel, and the gap in the overpressure arrival time increased with an increase in the propagation distance of the shock wave in the tunnel. This indicates that the highaltitude environment accelerated the propagation of the shock wave in the tunnel.

3. Numerical simulation verification

3.1. Finite element model and simulation condition

In the study of the shock wave propagation law in a tunnel, a tunnel test to reflect the explosion of the shock wave effect, fragmentation, and other destructive effects, is usually combined with a numerical simulation, which provides a complex wavefront surface motion state visualization, simulation of the explosion, and the impact of the kinetic phenomena, to provide an improved method for the real reduction in the movement of matter and serve as a numerical basis for theoretical research. Multiple finite element models were developed using the LS-DYNA software based on blast tests in the high altitude zone and plain. The air parameters of the different altitude environments were calculated according to theoretical formulas in aerodynamics, and the explosion effect of tunnels in different altitude environments was simulated by setting different air parameters. Numerical simulations can visualize complex phenomena, such as the initiation of explosions, propagation of blast shockwaves, and interaction of shockwaves with structures and can further analyze the influence of elevation effects on the propagation law of shockwaves. A three-dimensional finite element model of the test tunnel is shown in Fig. 12.

The geometric model for the numerical simulation was drawn by SOLIDWORKS (Dassault Systemes, United States) modeling software and then imported into Hyper-Mesh software (Altair, United States) to draw the mesh. After cutting the air, tunnels, and other structures into regular hexahedral structures, the geometric parts of the model such as lines, planes, and nodes are deleted, and the final retained part of the model can be regarded as a complete body consisting of several small hexahedra.

In the numerical simulation, the bottom of the model was set to be rigid to simulate the ground, and an ALE mesh was used for the air part of the model, while two sizes of mesh were used, with the size of the mesh on the top part of the tunnel in contact with the air having a size of 3 cm x 3 cm x 3 cm (lengthxwidthxheight), and the rest of the mesh having a size of 5 cm x 3 cm x 3 ст (lengthxwidthx height); the tunnel part is set as a Lagrangian grid with the grid size of 10 cm x 5 cm x 5 ст (length x width x height), and the propagation process of the shock wave in the tunnel is mainly studied in this model, and the tunnel's deformation can be ignored, so the tunnel is set as rigid. The explosive part adopts the filling method, and equal equivalent explosives are set according to the test conditions, so there is no solid grid in the explosive part.

The numerical simulation adopted an explosion of a solid explosive with a simulation mass of M. The explosive mass corresponded to the charge quantity of each working condition. The explosive was in contact with the rigid ground, and the central initiation mode was adopted. The outer surface of the calculation model was set as the transmission boundary, which allows the free flow and material exchange of the medium; the lower surface is the ground, which was set as the rigid boundary, the left and right boundaries were set as symmetrical boundaries, and the explosives and air were considered as uniform continuous media. In the numerical simulation in the present study, eight groups of working conditions were simulated, and the parameters of the various working conditions were set as listed in Table 4.

In the numerical simulation, only the unidirectional propagation process of the explosion shock wave in the tunnel was investigated; the tail of the air domain was set as a nonreflecting boundary, and the air grid nodes were constrained. The constraint parameters set for the air nodes are listed in Table 5.

3.2. Material model and parameters

3.2.1. Air material

In numerical simulations, the key to simulating environmental altitude is the setting of air parameters, and altitude and atmospheric temperature are the main factors that determine air pressure. The existing aerodynamics theory quantitatively provides a correlation model between atmospheric parameters and altitude variation. According to the standard atmospheric regulations [32], the sea level atmospheric temperature is 15 °C (thermodynamic temperature То = 288.16 К), atmospheric pressure ро = 101.35 kPa, and air density ро=1.225 kg/m". A linear relationship exists between the tropospheric temperature T, and environmental altitude h. When the tropospheric altitude increases, the 1 km temperature decreases to 6.5 °C. According to sea level temperatures То and Tp, the calculation formulas for atmospheric pressure py, air density pp, and sound velocity ch at altitude h can be obtained through Egs. (1)-(4). The ratio of high-altitude atmospheric parameters to sea level data calculated based on the correlation equation is shown in Fig. 13. As can be seen from the figure, the atmospheric pressure, density, temperature, and speed of sound all decreased linearly with altitude [33].

... (1)

... (2)

... (3)

... (4)

Based on this relationship, the atmospheric parameters at the corresponding altitudes were calculated (Table 6).

The air model is described by a multilinear equation of state · EOS-LINEAR-POL YNOMIAL, which is expressed as Eq. (5).

... (5)

In the equation: Со-Св and u are the equation coefficients, E represents the internal energy of the air, which can be used for E = Ph/(Y - 1), where y represents the adiabatic index, which generally takes the value of 1.4, and the equation coefficients are shown in Table 7.

3.2.2. Concrete and corrugated steel

The structure in the tunnel does not have sufficient time to move and deform completely under the impact of the shock wave; Table 7 therefore, in the numerical calculation, the tunnel and internal Coefficient of air equation. structure met the conditions of rigid materials. The concrete tunnels and floors were defined as rigid components using the LSDYNA keyword ·MAT RIGID. The parameters of the concrete materials are listed in Table 8 [31].

3.2.3. Explosive material

The explosion process of TNT explosive is described in combination with the following JWL equation of state (6)

... (6)

In the formula, P is the pressure of the detonation product, E is the specific internal energy of the detonation product, V is the relative specific volume of the detonation product, A, В, R1, Ro, and w are the fitting parameters of TNT, and Vo is the initial specific volume of the explosive before detonation. The parameters are set as shown in Table 9.

The JWL equation of state is used in conjunction with a highenergy combustion model (C] model), which determines the time of detonation at each point on the explosive based on the distance of each point from the point of initiation and the explosive detonation velocity. If there are multiple points of initiation, the time of initiation of each unit is calculated based on the distance to the nearest point of initiation. The k-w SST model is used in conjunction with the simulation during the explosion, as it performs better in handling the near-wall region and free shear flow.

3.3. Feasibility verification of numerical model

Explosion tests with the same charging conditions were carried out at the 4500 m high altitude zone and 200 m plain, and finite element models were established for altitudes at 200 m, 1000 m, 2500 m, and 4500 m based on the overpressure time course data at each measurement point. Taking the 4500 m zone test data as an example, the finite element output results were compared with the test data.

The comparison results of the overpressure time history curves at typical locations inside the tunnel under the conditions of an explosion 1 m outside the tunnel entrance with a 1 kg TNT explosive at an altitude of 4500 m (experimental condition 1 and simulation condition 4) and an explosion 2 m outside the tunnel entrance with a 2 kg TNT explosive (experimental condition 2 and simulation condition 8) are shown in Fig. 14. The comparison results of the peak overpressure values at various measurement points in the tunnel under the two loading conditions for the 4500 m high altitude and 200 m plain tests and the numerical simulation tests are shown in Fig. 15.

Taking the arrival time of the overpressure at measurement point 1 as zero, the wave front arrival time in the high-altitude and plain experiments and the numerical simulation calculations were compared. The comparison results are shown in Fig. 16. The numerical simulation results were in good agreement with the experimental data.

In addition to the peak overpressure and the motion time, there are two other important parameters of the shock wave, which are the positive pressure impulse and the duration. Therefore, it is also necessary to compare the experimental and simulation results of these two parameters, so as to further ensure the accuracy of the numerical simulation. The comparison results are shown in Figs. 17 and 18, respectively. According to the comparison of the results, it can be seen that the numerical simulation results are in good agreement with the experimental results at each measurement point location.

3.4. Numerical simulation results

3.4.1. Shock wave propagation process

During the explosion, we mainly observe the motion of the shock wave inside the tunnel, so the main processes are the arrival of the shock wave in the tunnel and after the shock wave enters the tunnel, so the excess air portion is cropped out of the pressure cloud, as shown in Fig. 19. In a semi-enclosed space, the propagation process of shock waves is highly complex because of the reflection and superposition of shock waves on the wall. Fig. 20 shows a pressure cloud map of several typical moments of shockwave propagation after an explosive detonation. As shown in Fig. 20, after the explosive explodes outside the tunnel entrance, the wave front moves upwards under the promotion of the explosive and reflection from the ground, and another part of the wave front spreads in a circular ring shape to the surrounding areas. A flat low-pressure area is formed inside the wavefront, and after a period of time, the two parts of the wavefront merge to form an approximate spherical wave that rapidly expands outward.

After the shock wave reaches the tunnel, it undergoes intense reflection and diffusion along the outer and inner wall surfaces. As the reflection angle increases, Mach waves are formed. At this time, the propagation of the shock wave is affected by multiple factors such as the quality of the charge and shape of the tunnel section, and the propagation process is highly complex. After the shock wave entered the tunnel, the wavefront at the bottom of the tunnel moved faster and continuously extended inward. Subsequently, the wavefront at the arch met the wavefront at the bottom of the tunnel, forming a narrow high-pressure stagnation section at the arch position and a negative pressure zone behind the wavefront. After the wave front moved approximately 2.5 m inside the tunnel, the high-pressure stagnation section at the arch disappeared, and a small high-pressure stagnation point appeared at the bottom of the tunnel. This is due to the strong reflection caused by the three wave points floating up and down in the tunnel during Mach wave propagation. As the shock wave continued to propagate inside the tunnel, under the rectification and directional action of the tunnel wall, the turbulent flow field approximately 5 m away from the tunnel entrance gradually transformed into approximately planar waves. The motion state of the wavefront was in good agreement with the overpressure curve of the experimental test.

3.4.2. Peak value of shock wave overpressure

Under the conditions of 200 m and 4500 m above sea level, after the explosion of 1 kg TNT 1 m outside the tunnel entrance, the shock wave pressure cloud output from the numerical simulation is shown in Fig. 21. According to the numerical simulation results, the propagation processes of the shock wave entering the tunnel after the explosive was detonated were similar at the different altitudes. Considering the same propagation time in the tunnel, the propagation distance of the shock wave in the 4500 m zone simulation was longer, whereas when the shock wave reached the same measurement point, the pressure of the shock wave was greater in the 200 m plain simulation. According to the comparison of overpressure cloud maps, the altitude effect had a certain effect on the magnitude of overpressure and the propagation speed of shock waves. The numerical simulation results were consistent with the experimental results.

According to the numerical simulation output results, the peak values of the shock wave overpressure at various measurement points after an explosion of 1 kg of TNT at a height of 1 m outside the tunnel entrance (simulation conditions 1-4), and at various measurement points after an explosion of 2 kg of TNT at a height of 2 m outside the tunnel entrance (simulation conditions 5-8) were determined as shown in Table 10.

According to the comparison results of the overpressure at various measurement points in Table 10, as the environmental altitude increased, the peak overpressure at the same measurement point decreased. In the front and middle sections of the tunnel, the attenuation amplitude of the overpressure was larger, and the influence of the altitude factors on the peak overpressure was more evident. As the shock wave continued to propagate in the tunnel, the influence of altitude on the peak overpressure of the shock wave decreased. In the numerical simulation results, whether the peak overpressure at the rear measuring point in the middle section of the tunnel was slightly higher than that at the front measuring point was unclear.

3.4.3. Time of overpressure arrival

According to the pressure cloud comparison between simulation conditions 1 and 4 (Fig. 21), the altitude effect influenced the shock wave velocity in the tunnel. According to the numerical simulation calculation results, the arrival times of the wavefront at different measurement points and altitudes under the two charging conditions were compared, as shown in Fig. 22. According to the numerical comparison results, in the front section of the tunnel, the shock wave entered the tunnel owing to severe reflection and flow effects, at this time, the motion state of the wavefront was relatively complex, wavefront movement speed was fast, and the altitude effect had a relatively small effect on the arrival time of the wavefront. As the wavefront continued to propagate, energy dissipated, a plane wave was formed, and the elevation effect played an influential role in the arrival time of the wavefront. Overall, as the elevation increased, the wavefront arrived at the same location earlier, and the numerical simulation results showed that an increase in the ambient elevation accelerated the propagation speed of the wavefront in the tunnel. This was consistent with the experimental results obtained for the high altitude zone and plain environments.

4. Analysis and calculation of overpressure attenuation

4.1. Shock wave propagation process

When an explosion occurs in a vacuum environment, the detonation product is compressed to a certain pressure, barrier-free expansion occurs according to the law of unsteady flow, and the expansion of the explosion product into the vacuum under the condition of zero gravity is unbounded. When explosives explode in other media, the dispersion process of detonation products is different from that in a vacuum environment, and two situations generally occur. In the first case, the detonation products affect the medium, and the detonation products push the shock wave from the interface to the deep layer of the medium. In the second case, sparse waves are immediately introduced into the detonation products. Therefore, the study of the propagation process of detonation products in the medium is more complicated than the diffusion in the vacuum.

The atmosphere is not static, and air undergoes movement, molecular diffusion, and other variations at all times in different quadrant directions. In the explosion process, the flame and blast products along with the air to the outside, in the process of complex gas-phase reaction will occur, the movement of various air masses will bring further mixing of air composition. In particular, with the increase in ambient altitude, the content of nitrogen, oxygen, argon and other components in the air decreases, which has a significant effect on the propagation of the explosive shock wave. According to the test and numerical simulation results, this attenuation effect needs to be further analyzed.

The theoretical model of the detonation wave propagation is shown in Fig. 23. P, р, v, e, and T and po, po, Vo, eo, and То represent the pressure, density, particle velocity, sound velocity, internal energy, and temperature of the detonation products and undisturbed atmosphere, respectively.

According to the mass, momentum and energy conservation theory can be derived from the following Egs. (7)-(10) [34,35], from the theoretical formula can be known shock wave wavefront pressure and air density, internal energy and the energy of the detonation products, so the change of atmospheric parameters will affect the propagation process of the wavefront, according to the results of numerical simulation can be verified in the process of propagation of shock waves overpressure changes, usually take the y-adiabatic index is 1.4.

... (7)

... (8)

... (9)

The trajectory of the shock wave after the explosion of TNT outside the tunnel mouth is shown in Fig. 24. The explosion of explosives to form an outward motion of the shock wave, and reflection with the ground to form a reflected wave, and collision in the explosion near the region, the two waves have different flow angle of rotation, this difference will cause the accumulation of material near the collision point, forcing the intersection of the reflected wave and the incident wave to move with a specific trajectory, the formation of the Mach wave (as shown in Fig. 24 in the part of the C). When these waves continue to move forward to reach the mouth of the tunnel, part of the incident wave reflects with the outer wall surface of the tunnel, and the other part of the incident wave enters the tunnel and reflects with the inner wall surface of the tunnel, and collides with the ground-reflected wave and the Mach wave in secondary collision, and the complex reflection and bypassing phenomenon occurs (as shown in Fig. 20, with T = 2.6 ms), and then superimposed forward motion.

The reflection process between the explosion shock wave and the tunnel wall can generally be assumed to be the coupling of the shock wave load energy on the structure and response of the structure. Therefore, this reflection process can be approximately regarded as the reflection process between the shock wave and the rigid wall. The reflection of the shockwave on the rigid wall is shown in Fig. 25. Regions 0, 1, and 2 represent the undisturbed atmospheric medium, gas after the incident shock wave, and gas reflected by the rigid wall, respectively. The straight line a-n represents the trajectory of the incident wave, and curve n-b represents the trajectory of the reflected wave. As shown in Fig. 25, the shock front pressure is related to the atmospheric pressure, internal energy of the air, and energy of the detonation products. Therefore, the change in atmospheric parameters will affect the propagation process of the wave front.

... (11)

At reflection point n, the particle velocity can be assumed to be zero, and the pressure at n can be determined using Eq. (11), where Pim is the gas pressure on the incident wavefront and y is the atmospheric adiabatic index [4].

4.2. Dimensional analysis and formula fitting

The charge shape of explosives is generally spherical or cylindrical; however, the research object of explosion mechanics is the impact effect, in which the distance of the detonation center is much larger than the size of the charge. Therefore, the charge shape

was not considered. According to the theory of aerodynamics and previous calculation models, a quantitative relationship exists between atmospheric pressure, air density, and internal energy at different elevations; therefore, only the atmospheric pressure factor was considered as the altitude variable in the formula fitting process.

Dimensional analysis and similarity theory were used to establish the functional relationship between the explosion shock wave overpressure of the TNT warhead in the typical position of the tunnel environment and parameters such as tunnel effective section size, proportional distance, and atmospheric pressure. A calculation method for the shock wave in a tunnel under the condition of an external explosion was obtained, and a power prediction model of the shock wave in a tunnel after a TNT explosive explosion outside the tunnel entrance was established. A schematic diagram of the explosion shock wave outside the tunnel entrance is shown in Fig. 26. For an explosion outside the tunnel entrance, the equation for calculating the peak AP, of the wall overpressure at L inside the tunnel entrance is as Eq. (12), the equation for calculating the velocity C of the wall overpressure at L inside the tunnel entrance is as Eq. (13).

... (12)

... (13)

where AP; is the peak overpressure of shock wave (kPa), t is the propagation time of the blast wave, (s), W is the equivalent charge of TNT (kg), C is velocity of shock wave motion (m/s), and R is the distance between the explosion source and the tunnel entrance (m). For the explosion outside the tunnel entrance in the present study, R is negative, D is the tunnel equivalent diameter (m), L is the distance between the cross-section of the measuring point in the tunnel and the tunnel entrance (т), Pa is the atmospheric pressure of the test environment (kPa), Ро is the sea level atmospheric pressure (kPa), E is the instantaneous energy release of unit mass explosive (kJ), pe is the explosive charge density, ye is the explosive product index, and y is the air adiabatic index. There are three basic dimensions among the 11 physical quantities in this equation, and the dimensional power exponents of the dimensional physical quantities in this equation are shown in Tables 11 and 12. According to the T-theorem, eight dimensionless physical quantities can be given as Eq. (14).

... (14)

Thus it can be written as a dimensionless expression as Eq. (15).

... (15)

Existing theoretical work focuses on the study of gas flow, and in most cases, researchers generally adopt isentropic, steady, and adiabatic flow hypotheses. For industrial TNT explosives, the explosive energy release and charge density per unit mass can be regarded as constants, and when exploding in a specific atmospheric environment, the atmospheric adiabatic index and explosive equation of state index are constant. Therefore, in the M-L-T unit system, according to the dimensional analysis theory and 7 theorem, the dimensionless function expression of overpressure peak can be simplified to Eq. (16).

р. (16)

To address the problem of explosions outside the tunnel entrance, the American Engineering Corps Waterway Experiment Station proposed an empirical formula for calculating the peak overpressure in the tunnel. Based on this calculation method, the coefficient was modified and a calculation factor characterizing the altitude environment was introduced to fit the new formula. Through a model test of the propagation law of the TNT explosion damage element outside the tunnel entrance, the overpressure data of different equivalent TNT explosives at each test point after the explosion at different distances from the tunnel entrance were recorded, and the variation rules of the shock wave parameters with an increase in the propagation distance in the tunnel were obtained. The attenuation formula for the peak overpressure of the shock wave after TNT explosion outside the tunnel entrance at different elevations was shown in Eq. (17).

... (17)

4.3. Calculation of the velocity of wavefront motion

After the explosion outside the tunnel entrance, for the wave front surface motion velocity analysis can be divided into two parts, the first stage is between the source of the explosion to the entrance of the tunnel, can be regarded as a shock wave propagation in the free field or semi-free field, the second part of the shock wave in the tunnel after the entry of the state of motion.

Dewey [5] obtained empirical equations for the trajectory of the explosive shock wave propagation of a 1 kg spherical TNT charge in a standard atmospheric environment as Eq. (18) by fitting the experimental data

... (18)

The above formula has certain limitations and cannot describe the propagation process of the explosive shock wave with different charges under low-pressure environment. For this reason, according to the similarity criterion, the above formula is organized, and the formula as Eq. (19) for the propagation process of the shock wave after the explosion of explosives with a mass of W in a pressure is as follows [10].

... (19)

where: г for the blast radius, т; a1 = -0.72, ay = 0.98, аз = -2.77, and a4 = 6.08 for the fitting coefficients; ао = 340.292 m/s for the speed of sound of the air under standard atmospheric conditions; t for the propagation time of the blast wave, s. The velocity of propagation of the shock wave (C) can be obtained by differentiating the time t on both sides, the formula is shown in Eq. (20).

At the entrance to the tunnel, the speed of the wave front can be approximated as equivalent to the expansion of the wave front in the free field velocity, that is, Co. The reflection through the outer wall surface causes some attenuation of Co. In order to establish the velocity of explosion shock wave propagation in the tunnel of the engineering model, the general function of the velocity of propagation of the shock wave propagated along the equal cross-section straight tunnel analyzed with the quantum theory. From the point of view of engineering applications, in order to simplify the processing, without taking into account the changes in explosive density, the simplified engineering model of the shock wave propagation law as Eq. (21).

... (21)

where Kp represents the coefficient of influence of the crosssectional area of the outer wall of the tunnel and the cavity area. The expression for Kr in this model can be simplified to Eq. (22), where S represents the cross-sectional area of the tunnel and 50 represents the wall area of the tunnel mouth, as shown in Fig. 27. After numerical calculations and accuracy checking to obtain, Kr = 0.605 in the tunnel near mouth explosion model.

... (22)

4.4. Comparison of calculation results

4.4.1. Overpressure peak comparison results

To verify the accuracy of the prediction formula, the calculation results of the prediction formula were compared with the experimental results. The comparisons between the test results and the formula calculation results under the explosion conditions of two types of charges at the 4500 m high altitude zone and a 200 m plain environment are shown in Figs. 28 and 29, respectively. The results of the numerical comparison between the formula calculations and the experimental results are shown in Tables 13 and 14, respectively. The comparison results showed that for both the explosion tests in the high altitude zone and plain environments, the error between 80 % of the test data and the predicted results was less than 20%. The average error between the test data and the predicted results was less than 15%.

4.4.2. Shock wave velocity comparison results

The calculated results of the data obtained from the experimental tests are compared with the predicted results of the velocity calculation formula, and the comparison results are shown in Figs. 30 and 31. The results of the numerical comparison are shown in Tables 15 and 16. According to Figs. 30 and 31, it can be seen that the calculated results are in good agreement with the experimental results, and can be used for the design of engineering practice.

5. Conclusions

The research of weapons power in high-altitude environments is of great military and strategic significance. High-altitude areas often have unique environmental characteristics, such as low pressure, low oxygen, cold weather, strong ultraviolet rays, dry air and possible dust storms, which can have a significant impact on the performance of weapons systems. The study of explosions contributes to the understanding of shock waves generated in situations such as explosions and high-velocity impacts, which is critical to improving the design and effects assessment of weapons systems. The propagation process of explosion shock waves in tunnels in a typical altitude environment was obtained by conducting explosion tests outside the tunnel entrance in a highaltitude zone and plain. The propagation and development laws of explosion shock waves in tunnels were analyzed, and the influence and mechanism of the altitude environment were explored. A prediction method for the overpressure attenuation of shock waves in tunnels suitable for different altitude environments was proposed. The main conclusions are as follows.

(1) Differences in altitude resulted in differences in the peak overpressure and propagation speed of explosion shock waves propagating at the same distance within the tunnel. The peak overpressure of the shock waves in the tunnel was negatively correlated with altitude, whereas the velocity of the shock waves in the tunnel was positively correlated with altitude. Air conditions in high-altitude environments accelerate the propagation of shock waves through tunnels.

(2) Based on the calculation method proposed by the American Engineering Corps Waterway Experiment Station, the present study introduces the influence factor of atmospheric pressure to characterize the effect of altitude on the explosion shock wave overpressure. A formula for calculating the peak overpressure of explosion shock waves in tunnels under different altitude environments was established, and the accuracy of the functional relationship was verified using measured data. The verification results indicated that the average error between the experimental and calculated results was less than 15%. Therefore, this function is highly reliable for predicting the peak overpressures of explosion shock waves in tunnels at different altitudes.

(3) Through the combination of experimental data and numerical simulation results, the influence of altitude effect on the propagation speed of shock wave in the tunnel is analyzed, and based on the theory of shock wave propagation in the free field and the magnitude analysis, the formula for the propagation speed of the shock wave in the tunnel under different altitude conditions is proposed, and the predicted results of the formula are in relatively high consistency with the experimental results, the verification results indicated that the average error between the experimental and calculated results was less than 10%.

CRediT authorship contribution statement

Changjiang Liu: Conceptualization, Methodology, Software, Investigation, Formal analysis, Writing - original draft. Hujun Li: Visualization, Writing - review & editing. Zhen Wang: Conceptualization, Funding acquisition, Resources, Supervision, Writing - review & editing. Yong He: Visualization, Writing - review & editing, Resources. Guokai Zhang: Data curation, Writing - original draft. Mingyang Wang: Resources, Supervision.

Declaration of competing interest

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.

Acknowledgment

This work is financially supported by National Natural Science Foundation of China (Grant Nos. 52378401, 52278504), the Fundamental Research Funds for the Central Universities (Grant No. 30922010918).

Footnote

-· Corresponding author.

E-mail address: [email protected] (Z. Wang).

Peer review under responsibility of China Ordnance Society

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