Mapping function model representing the relationship between explosion shock wave pressure and wavefront temperature

Ammunition explosions generate a range of destructive factors, including shock waves, fireballs, fragments, and thermal effects.^1–3 The research on explosion shock wave pressure testing was carried out earlier, and there is a relatively sufficient understanding of the testing methodologies, data processing methods, as well as shock wave pressure propagation and distribution laws. The results obtained from these tests are highly reliable. However, the temperature distribution test is limited by the dynamic characteristics of sensors, sensor installation methods, and the influence of flow field convection characteristics, resulting in inaccurate temperature test results. Therefore, the test results could not be used to accurately evaluate the damage power of ammunition and guide the design of ammunition.^4,5 To address these issues, it is necessary to study the temperature distribution law of the shock wave front of ammunition explosions and clarify the temperature distribution and attenuation characteristics during ammunition explosions. To explore the calculation method of shock wave front temperature based on explosion shock wave pressure data, improve the calculation accuracy of temperature during ammunition explosions, and provide theoretical and scientific data support for ammunition explosion damage power test.

Relevant scholars have carried out research on the distribution law of shock wave pressure and temperature measurement in the process of ammunition explosion and have made certain research results. Hobbs et al.⁶ utilized infrared radiation thermometry to measure the radiative temperature generated by explosions, investigating the early development and expansion processes of ammunition explosions. They combined the measured results with quasi-static pressure gauges, providing a potentially valuable measurement technique for characterizing the early stages of fireball development and expansion. Bai et al.⁷ designed a small linearity shock wave pressure storage test system for measuring the rapid changes in the temperature of the blast field and the harsh field test environment. A numerical finite element analysis of the aluminum-contained explosive temperature of a near-ground secondary explosion was performed using AUTODYN. The effectiveness of the test system was verified by comparing the simulation data with the measured data. Ji et al.⁸ conducted a study on the characteristics of explosion heat effects and shock wave pressure by combining multiple testing techniques, clarifying the variation and regularity of characteristics parameters such as shock wave waveform evolution, shock wave overpressure peak value, positive action time, and a specific impulse of shock waves. Gou et al.⁹ conducted explosive tests of temperature and pressure explosives and trinitrotoluene (TNT) explosives in complex tunnels, studied the propagation law of explosive shock waves in complex tunnel environments, and compared and analyzed the characteristics of explosive effect parameters between temperature and pressure explosives and TNT explosives. Rao et al.¹⁰ conducted numerical simulation tests using AUTODYN for two thermocouple sensor settings to clarify the effect of the tilt angle of the sensors on the ammunition explosion temperature. Xu et al.¹¹ proposed a blast field temperature test method based on high-speed imaging technology and derived the temperature measurement model of a high-speed imaging system based on the radiation principle. They then established the relationship between the grayscale value of high-speed camera images and blast temperature, meeting the need for explosion temperature field measurement. Zhao and Zhang¹² conducted a theoretical analysis of the governing equations to predict the variation in temperature generated by explosive expansion as a function of distance and time. They utilized shock wave theory to calculate the relationship between the initial pressure peak and the temperature of shock waves. Joy et al.¹³ compared the results of fluid dynamics theory with simulation results and found that the predicted radial distribution of thermodynamic quantities did not match the simulated results. To understand the reason for this difference, they validated different assumptions in the fluid dynamics theory during the simulation process and explained the discrepancy by proving that the local pressure, temperature, and density followed the equation of the state of a hard-sphere gas. Kumar et al.¹⁴ developed a thermocouple dynamic calibration system for C-type thermocouples and established a thermocouple dynamic compensation filter model using the particle swarm algorithm and calibration data. The test results indicated a significant improvement in the sensor output response rate, and the compensation temperature reached 200 °C, indicating a good compensation effect. Cheng et al.¹⁵ proposed a method for explosion temperature measurement using infrared thermography to obtain accurate measurements of warm pressure explosive temperatures. This method was successfully applied to PMX explosives for radiation calibration and temperature measurement, resulting in a curve of explosion fireball change over time. The effectiveness and simplicity of this highly stable temperature measurement method were demonstrated, presenting it as a practical solution for explosion field temperature testing. Tian et al.¹⁶ conducted a theoretical analysis and proposed specific measures to address interference factors that arise when using infrared thermal imaging cameras for temperature testing in field environments. By field calibrating the infrared thermal imaging camera using a field-standard black body at the explosion test site of the thermobaric bomb, relevant temperature parameter data were accurately captured, and the temperature change process of the thermobaric bomb explosion was extracted. Wang et al.¹⁷ employed a blackbody to calibrate the infrared thermometer in the field, corrected for the atmospheric transmittance of the blast field, and improved the accuracy of infrared temperature measurement. They recorded the explosion process of a 3 kg solid cloudburst agent and extracted the fireball surface temperature. Zhao et al.¹⁸ utilized an E12 thermocouple and wireless sensing network technology to evaluate the thermal destruction effect of temperature and pressure ammunition by testing the cloudburst ammunition explosion temperature. The test results indicated that the designed system is safe, easy to operate, and has promising applications. Liu et al.¹⁹ proposed a thermocouple structure design method for measuring the explosion temperature, taking into account the transient high-temperature and high-pressure characteristics generated by warm-pressure ammunition explosions. They analyzed the thermocouple’s characteristics and dynamic characteristics in the blast field subjected to shock wave action. The explosion tank test verified that the thermocouple could effectively acquire measurement data, satisfying the explosion temperature measurement accuracy requirements of the temperature–pressure ammunition. Li et al.²⁰ proposed a contact temperature measurement method that uses a tungsten–rhenium thermocouple. They adopted a two-stage amplification strategy and a low-pass filtering method to achieve the conditioning function of the temperature signal. Calibration, testing, and experiments revealed that the static error of the contact temperature measurement method was higher than 0.8% in the calibration temperature range and the dynamic response time was less than 20 µs. Wang et al.²¹ employed an atomic emission spectroscopy dual spectral line temperature measurement system based on the atomic spectroscopy theory to measure the explosion product temperature of aluminum-containing latex explosives in real time. They obtained the transient temperature-time distribution curve of the explosion product and investigated the reasons for the differences in explosion product temperature corresponding to different ratios of explosives.²² To address the issue of poor dynamic characteristics of thermocouples and large dynamic errors in test results, Wang et al.¹⁷ applied a high-power laser as the heat source to construct a traceable dynamic calibration system for the dynamic calibration of thermocouple sensors. They analyzed the dynamic characteristics and prepared a dynamic compensation filtering algorithm. The dynamic response of the compensated thermocouple was faster, and the explosion temperature compensation results exhibited high confidence.

Most studies on the temperature of the ammunition explosion field have focused on improving testing methods, temperature measurement sensors, and dynamic characteristic compensation methods for temperature measurement systems. Furthermore, improvements in the dynamic response characteristics of sensors have enabled researchers to determine the temperature change law in the ammunition explosion process. However, given that the temperature change during ammunition explosions is highly transient, these improvement measures can only achieve limited accuracy improvements. Therefore, the relationship between the explosion shock wave pressure and the wavefront temperature should be investigated using a mapping function to achieve an inversion calculation of the wavefront temperature, thereby enhancing the calculation accuracy of the explosion field temperature.

This study used Autodyn dynamics simulation software to perform numerical simulation analysis of the shock wave pressure and temperature propagation distribution law of the blast field of trinitrotoluene (TNT) explosive masses of 10, 20, 50, and 100 kg. In addition, the shock wave pressure data and temperature were measured at different points. The measurements were combined with the cloud diagrams of pressure evolution at different explosion moments to analyze the shock wave and temperature propagation distribution laws during ammunition explosions. Based on the blast similarity law and the method of dimensional analysis, a mapping function was established to represent the relationship between the shock wave pressure peak and the temperature peak. The accuracy of the mapping model was also verified.

The explosion products of the ammunition explosion will spread outward at a very high velocity to spread outward, resulting in an outward release of energy, thus compressing the surrounding air and forming a high-pressure area; this eventually results in the formation of an initial shock wave.^23,24 The initial shock wave acts as a strong interrupted surface with a large pressure in front of the wave and a small pressure behind the wave. The head of the wave propagates at a supersonic speed, whereas the tail propagates at the speed of sound corresponding to pressure p₀ (the pressure at the tail of the shock wave front, which is affected by the sparse wave, has a peak value that is smaller than the pressure at the head of the wave front). The shock wave continuously expands in the positive pressure zone during propagation.²⁵ The formation and distribution of shock wave is shown in Fig. 1(a). The propagation of shock wave in the air is shown in Fig. 1(b).

The peak pressure and propagation velocity of a shock wave rapidly decrease as it propagates through the air. This can be attributed to the increase in the size of the wavefront with propagation distance, resulting in a rapid decrease in the energy per unit area even when no other energy losses occur. Additionally, the width of the positive pressure area of the blast shock wave increases continuously with distance, affected by the increasing mass of compressed air, causing the average energy per unit mass of air to decrease. Furthermore, the propagation process of shock waves is not isentropic, and the entropy increases at the wavefront surface. Consequently, an irreversible energy loss occurs because of the adiabatic compression of the air during shock propagation. The wavefront pressure decays rapidly in the initial stage of propagation and gradually in the later stage. After propagating to a certain distance, the shock wave decays into an acoustic wave.^26,27 Figure 2 shows the variation curve of the shock wavefront pressure with time.²⁸ 

A finite element numerical simulation method was used to study the distribution laws of explosive shock wave pressure and temperature propagation. TNT explosives with masses of 10, 20, 50, and 100 kg were used with a length-to-diameter ratio of 1:1 and central detonation. The central point of the explosive was 1.5 m above the ground, and the mesh size was 1 × 1 mm², with the mesh type being Lagrange. The size of the visible air domain in the simulation model was 20 000 × 5000 mm², with a mesh size of 5 × 5 mm² and a mesh type of Euler, to simulate the semi-infinite air domain in an actual explosion environment. The upper and left boundaries of the air model were set to pressure flow-out to avoid pressure reflection, while the right boundary was a symmetry axis with no boundary conditions. The ground was set as a rigid boundary condition to simulate the influence of the ground on the propagation of shock wave pressure in actual test sites, and the shock waves were completely emitted on the rigid ground. To obtain data on the variation law of shock wave pressure and temperature with time at different distances from the detonation center, gauges monitoring point were set on the ground every 500 mm. The finite element numerical simulation model established is shown in Fig. 3.

To verify the reliability of the calculated results of the finite element numerical simulation model constructed earlier, we conducted relevant numerical simulation experiments and practical tests to prove the accuracy of the calculated results of the finite element numerical simulation model constructed in this study. For more details, please refer to Ref. 39.

Using the above-mentioned developed model, surface-reflected pressure and temperature data were obtained for different amounts of TNT at a distance of 2–10 m between the measurement point and blast center. The obtained shock wave pressure vs time curve and temperature peak histogram are presented in Figs. 4 and 5, respectively.

The pressure-time curves of the shock wave indicated that the peak pressure at the same position of the explosion shock wave gradually increased with the increase in TNT explosive mass. The closer the measurement point is to the blast center, the faster the peak pressure growth rate of the shock wave. As the distance between the measurement point and the blast center increases, the peak pressure growth rate gradually reduces. Furthermore, a histogram analysis of the changes in the shock wave front peak temperature revealed that the temperature peak gradually decreased with the distance between the measurement point and the blast center of the TNT, and the decay law was similar to that of the shock wave pressure peak. The temperature peak decay rate was higher at a measurement point closer to the location of the burst core, and it gradually decreased with the distance between the measurement point and the blast center. The peak temperature of the wavefront surface increased gradually with the increase in TNT explosive mass, indicating a positive correlation between the mass of the TNT explosive and the peak temperature. The relationship between the shock wavefront temperature and shock wave pressure was analyzed according to the peak pressure and peak temperature of the shock wave at different locations. The pressure and temperature peak change law curves are shown in Fig. 6.

The pressure–temperature curves in Fig. 6 indicated that the peak values of the pressure and temperature of the shock wavefront gradually decreased with increasing distance between the measuring point and the explosion center. The attenuation rate of the peak values was higher at measurement points closer to the blast center. With the increase in distance between the measuring point and the explosion center, the attenuation rate of the peak values gradually decreased until the values were equal to ambient temperature and pressure. The attenuation law of the shock wave pressure peak was highly consistent with that of the wavefront peak temperature. Consequently, changes in the peak pressure directly affected changes in the wavefront temperature. Therefore, the attenuation law of the pressure distribution in a shock wave can reflect the attenuation law of the temperature distribution in the shock wave front. The cloud maps of shock wave pressure evolution were obtained at different explosion times, and the propagation and distribution of the shock wave in the air are illustrated in Fig. 7 (Multimedia views).

The cloud maps of the shock wave pressure evolution at different blast moments indicated that the rear face of the shock wavefront in the near field occurred as a strong sparse wave. This wave induced a rapid decay of pressure at the wavefront, resulting in a higher decay rate of the shock wave pressure, as depicted in Figs. 7(a) and 7(b). With the propagation of the shock wave front in the air, the pressure of the wavefront gradually decreased and tended to be equal to the ambient pressure. Meanwhile, when the sparse wave pressure at the back end of the wavefront increased, the gap between the shock wavefront pressure and the sparse wave pressure gradually decreased, suppressing the attenuation effect of the sparse wave on the wavefront pressure. Consequently, the pressure decay rate of the shock wave in the far field region also decreased. Furthermore, the wavefront temperature of the explosion products at the moment of explosion initiation was considerably high, resulting in an increase in the temperature of the wavefront. As the wavefront propagated in the near-field region, the pressure rapidly decayed because of the effect of strong sparse waves, leading to a sharp decrease in the peak temperature. However, the pressure decay rate gradually decreased with the increase in propagation time, resulting in a decrease in the temperature decay rate. Moreover, the decay law of the pressure peak and temperature peak remained consistent with the increase in TNT explosive mass. Therefore, the decay laws of the peak pressure and temperature of the explosion shock wave propagating in the air were similar, indicating a correlation between temperature and pressure. Moreover, the change law of shock wave pressure can serve as a means to determine the change law of wavefront surface temperature.

The accurate measurement of temperature at the location of a measurement point during ammunition explosions is challenging because of various factors, such as the dynamic characteristics of current temperature sensors, sensor installation, and the role of flow field convection environments, which lead to inaccurate and unreliable temperature measurements. However, the temperature of the munition blast shock wavefront is a very important guide to the assessment of the destructive power of explosives and the reduction of the wavefront temperature to the thermal shock parasitic effect output of the ground reflective pressure sensor. The aforementioned analyses reveal that the change in wavefront surface temperature and shock wave pressure have a significant correlation. Moreover, the results of the shock wave pressure test method and measurement system are accurate, and the pressure test results are reliable. Therefore, the mapping function between the shock wave pressure and wavefront surface temperature must be established using the obtained shock wave pressure distribution data to invert the temperature distribution data. This will improve the calculation accuracy of the near-field temperature distribution law.

In this study, a mapping function model was established to represent the relationship between explosion shock wave pressure and temperature by introducing physical quantities based on the explosion similarity law and the magnitude analysis method. Furthermore, the relationship between physical quantities was analyzed based on the principle of magnitude harmony to construct a mapping function model with a clear physical interpretation.

The four basic physical dimensions of length (L, m), mass (M, kg), time (T, s), and temperature (Θ, K) were introduced to analyze the physical dimensions of Eq. (4), and the dimensions of the physical quantities are listed in Table III.

Considering the parameters that affect the temperature of the shock wave front during ammunition explosions, the propagation velocity of the wavefront V (the propagation velocity of the wavefront is significantly related to the shock wave pressure and wavefront temperature) and the specific heat capacity in the air C affect the temperature of the wavefront, in addition to the shock wave pressure P, the distance from the measurement point r, and the mass of the explosive ω. V and C can be expressed on a scale, as shown in Table IV.

The function in Eq. (7) accounts for each factor influencing the shock wavefront temperature T_temperature. The weightage of each physical quantity is presented in Table V.

Considering the magnitude and physical quantity of each parameter, the distance from the measurement point r, the charge mass ω, the specific heat capacity of air C, and the propagation velocity V of the shock wavefront are selected as the reference physical quantities. Table V shows these reordered physical quantities, and Table VI lists the results.

The physical quantities in Table VI are transformed into rows to obtain the power exponents of each physical quantity, as shown in Table VII.

To determine the specific values of the parameters in Eq. (11), a nonlinear adaptive fitting of the established model was performed using shock wave pressure and temperature data obtained from measured tests and finite element numerical simulations conducted thus far. The fitting results are illustrated in Fig. 8.

The fitting results illustrated in Fig. 8 indicated that some data points did not lie on the fitted function relationship curve. However, they were uniformly distributed on both sides of the curve, indicating that the established function can effectively reflect the relative changes between the peak pressure of the shock wave and the peak temperature.

These results show that the maximum relative error rate of the established mapping function relationship model for the validation data is 10.25%, and the minimum relative error rate is 0.6%. The relative error rates of the remaining data points were within this range. Therefore, the model’s calculation accuracy was higher than 89.75%. These findings suggest that the mapping function model established to study the relationship between the peak pressure and temperature of the shock wavefront yields high calculation accuracy. Consequently, utilizing this model in calculating the peak temperature of the wavefront during an ammunition explosion can considerably improve the accuracy of temperature measurement. Moreover, the model can provide theoretical support for explosion field temperature testing, data acquisition, and data validity testing.

This study addressed the low accuracy and reliability of data obtained from surface temperature testing of explosion shock waves. To this end, a finite element numerical simulation analysis of explosion shock wave pressure and temperature propagation distribution was conducted for TNT masses weighing 10, 20, 50, and 100 kg using Autodyn explicit dynamics simulation software. The time-course curves and peak data of shock wave pressure and wavefront temperature obtained from different measurement locations were analyzed using cloud maps of the pressure propagation evolution of the shock wave at different explosion moments. The analysis results revealed the following:

1. For the same TNT explosive mass, the peaks of the explosion shock wave pressure and wavefront temperature in the air exhibited a high degree of similarity in their decay law. The peak decay rate near the blast center was high and gradually decreased with the distance between the measurement point and the blast center. Finally, the shock wave pressure and temperature decayed to the ambient temperature and pressure; the main origin of this phenomenon is the wavefront surface of the rear face of the strong sparse waves caused by.

2. Based on the explosion similarity law and dimensional analysis method, a mapping function model was established to represent the relationship between the peak shock wave pressure and peak temperature. The model parameters included the explosive mass ω, measurement point radius r, specific heat capacity in the air C, and peak pressure of an explosion shock wave P. The accuracy test of the mapping function model indicated an accuracy higher than 89.75%. Therefore, the proposed model has considerably high accuracy in calculating the temperature of the ammunition explosion shock wavefront. Therefore, this study provides theoretical support for the temperature test, data acquisition, and data validity test of explosion fields. It has important engineering application value for the test and evaluation of the ammunition’s explosive damage power.

The cloud diagrams of the pressure and temperature evolution of the shock wave of 10, 30, 60, and 100 kg of TNT explosives can be found in the supplementary material.

This work was funded by the National Equipment Program of China under Project No. 14021001050206.

The authors have no conflicts to disclose.

Liangquan Wang (王良全): Conceptualization (equal); Data curation (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Deren Kong (孔德仁): Conceptualization (equal); Data curation (equal); Funding acquisition (equal); Project administration (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article and its supplementary material.