# Investigation of the Interaction Mechanism of Solid Particles under Shock Waves

Authors: Xu Peng; Shunyao Wang; Guoning Rao; Bin Li; Wanghua Chen
Journal: Shock and Vibration
 (2022)

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## Abstract

To deeply understand the interaction mechanism between the two phases, both shock wave experiments and computational fluid dynamics (CFD) simulations are carried out to study the propagation of shock waves and the dispersion process of solid quartz sand particles. The results show that transmission and reflection occur during the interaction between the shock wave and particle. When the Mach number is 1.53, the incident pressure is 154.6 kPa and the reflected pressure is 342.4 kPa. The transmitted pressure rapidly decays due to the exchange of momentum and energy, the particle front gradually changes from horizontal to mushroom shape, and a rising three-dimensional annular vortex is formed obliquely above the particles due to the entrainment effect. The simulated and experimental results are in good agreement, thereby indicating the validity of the numerical model. In addition, based on the response surface method and a series of experimental data, a mathematical model of the dispersion distance of solid particles under the action of shock waves is proposed. It is beneficial for prediction and scientific research for the development of industrial systems.


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## Body

## 1. Introduction

The interaction between shock waves and particles is an essential physical phenomenon in gas-solid two-phase flow [1] and can be widely applied in industry. For example, a fire-extinguishing bomb is loaded with a solid fire-extinguishing agent, which can be launched to the fire area from a long distance using a propellant to effectively spread the fire-extinguishing agent for fire extinguishing and disaster reduction [2]. Carrying out the experimental study of dust dispersion is of great reference value for the prevention of dust explosion accidents and the improvement of the production process [3–7]. High-velocity oxy-fuel is a technology that can accelerate spray particles to supersonic speed through a high-speed jet flame. In this way, the particles will produce an obvious shot peening effect in the deposition process, which can effectively improve the strength of the coating [8]. In the above applications, research on the dispersion of solid particles driven by shock waves is particularly important, especially the dispersion distance, dispersion range, distribution uniformity, and dispersion energy control of particles.The interaction between shock waves and solid particles has been studied by different researchers, and some valuable results have been obtained. For example， Rogue et al. [9] studied the motion state of particles driven by shock waves with different Mach numbers by shadowgraphy. They simulated the experimental process with the finite difference method, and the results showed that the drag force is the main factor for the dynamics of dense particle clouds, and interphase heat transfer with long time scales and interparticle collisions at the early phase of cloud motion have minor influences. Zhang [10] and Frost et al. [11] studied the rapid dispersion process of solid particles in mixed explosives. The interaction between the shock wave and particles was recorded by flash X-ray radiography and high-speed photography. They found that the particle size and the particle density have important influences on the interaction process. Boiko [12] and Kiselev et al. [13] investigated the propagation and attenuation of shock waves in a solid particle layer through a multiframe shadow visualization technique coupled with a laser stroboscopic source of light. They analyzed the formation mechanism of the reflection wave after the shock wave. Theofanous et al. [14–16] studied the interaction between shock waves with different Mach numbers and 1 mm glass particles using shock tube experiments. They obtained the displacement of particles and the change in shock pressure. Zoltak [17] and Jiang et al. [18] simulated the interaction between a shock wave and a single particle and obtained the diffraction process of a shock wave. Osnes et al. [19, 20] used the large eddy simulation (LES) method to study the process of shock waves passing through random particle arrays at different Mach numbers. The results show that the attenuation of the shock wave is closely related to the opacity of the particle cloud. Sugiyama et al. [21, 22] investigated the interaction between shock waves and particles by the CFD-DEM and found that, after the interaction between the gas flow and the PCC, the drag force and heat transfer were activated by the gradients in pressure, velocity, and temperature between them and the gas flow lost momentum and energy, which weakened the transmitted shock wave.However, in previous studies, the interaction between shock waves and solid particles lacks a suitable regular prediction model, which makes it difficult in practical industrial applications and requires a high cost to evaluate the required application scenarios. In addition, the interaction between the shock wave and solid particles is accompanied by very complex physical phenomena, and the interaction time is transient (only a few milliseconds), which makes it difficult to capture some details of the flow field, so the underlying physics is still a source of discussion.Therefore, to compensate for some deficiencies of the previous study, in this paper, a self-designed shock tube experimental apparatus was first employed to record the interaction between the shock wave and solid particles through high-speed shadowgraphy technology. Then, the experimental process was simulated through the computational fluid dynamics (CFD) method to analyze the detailed three-dimensional flow field and the nonequilibrium interaction mechanism between the two phases. Finally, based on the statistical analysis of a large amount of experimental data, a mathematical model of the particle dispersion distance under the action of shock waves was proposed. It is of great significance for scientific research in the field of gas-solid two-phase flow and its application in actual industrial production processes.


## 2. Experimental Setup

The novel shock tube experimental apparatus is mainly composed of three modules: a shock tube, a data acquisition system, and a high-speed shadowgraphy system. The structural diagram is shown in Figure1.Figure 1
Structural diagram of the novel shock tube experimental apparatus.The shock tube consists of three parts: the high-pressure driving section at the bottom, the low-pressure section in the middle, and the metal mesh on top. The high-pressure driving section is connected to the gas cylinder filled with high-pressure air. The low-pressure section is connected to the atmosphere. The high-pressure section and the low-pressure section are separated by a diaphragm. In this paper, hard paper sheets with a thickness of 0.125 mm are used as the diaphragm. When it is broken, the edge of the paper sheet is smooth, and there are no paper scraps. By adjusting the number of paper sheets, shock waves with different Mach numbers can be obtained. In addition, the metal mesh is clamped in the middle of the flange plate and fixed by bolts. The solid quartz sand particles are uniformly placed on the metal mesh. A picture of the metal mesh and the shock tube is shown in Figure2.Figure 2
The picture of the metal mesh and the shock tube.The data acquisition system consists of pressure sensors, an amplifier, a data acquisition instrument, and a controller. The pressure sensor is a piezoelectric sensor produced by the PCB company, with a maximum range of 2 MPa. As shown in Figure1, the shock tube experimental apparatus is equipped with three pressure sensors (P1, P2, and P3). P1 is located under the metal mesh inside the shock tube to measure the pressure before the interaction between the shock wave and particles. P2 and P3 are 100 mm and 200 mm away from the particle surface, respectively, which are used to measure the pressure at different positions after the interaction.The high-speed shadowgraphy system consists of a high-speed camera, timing controller, trigger, multi flashlight source, concave mirror, and multilens camera. The high-speed camera is a YA-16 multiflash high-speed camera. The acquisition rate is 5000 fps, and the resolution is 768 × 480, which is an optical framing high-speed camera with high-spatial resolution.During the experiment, the valve of the gas cylinder was opened by the controller, and the high-pressure driving section of the shock tube was inflated and pressurized. When the ultimate pressure of the paper sheets is reached, the gas breaks through the paper sheets and generates a shock wave with a certain Mach number, which acts on the solid particles on the metal mesh and causes the particles to be scattered. Then, the pressure data in different positions were collected and recorded through the data acquisition system. At the same time, the high-speed shadowgraphy system was triggered to capture the propagation of shock waves at different times and the interaction process between shock waves and solid particles.


## 3. Numerical Model

### 3.1. Geometric Model and Mesh

To simulate the movement process of solid particles outside the shock tube after the interaction by the shock wave, the computational domain includes not only the internal area but also the external flow field of the shock tube, which are both cylindrical. Since the computational domain is axisymmetric, the 2D calculation domain was selected as the research object for simulation to shorten the calculation time. The dimension of the flow field external to the shock tube is 0.5 m wide and 1 m high. In addition, to capture the propagation process of the shock wave more accurately in the simulation, the computational domain was divided into very fine hexahedral meshes by ICEM software. In the simulation, four grid numbers were used in the grid independence verification: 37852, 94266, 154860, and 229878. Through the comparison of the results, we found that the calculation results of the latter two grid numbers were basically the same. Therefore, we considered that the calculation with 154860 grid numbers meets mesh independence concerns. The hexahedral meshes and local details of the computational domain are shown in Figure3, and the minimum mesh size is 0.05 mm.Figure 3
The hexahedral meshes and local details of the computational domain.


### 3.2. Governing Equations

Because the interaction between shock waves and particles is a compressible gas-solid two-phase flow [23–25], based on the conservation of mass, momentum, and energy, a numerical model of shock wave interactions with solid particles was established. The k-ω turbulent model was adopted to describe the turbulence phenomena throughout the process. The model is described by the following equations and solved by the finite volume method in SC Tetra software.Mass conservation equation:(1)∂ρ∂t+∂ρui∂xi=0.Momentum conservation equation:(2)∂ρui∂t+∂ujρui∂xj=−∂σij∂xj+ρgi.Energy conservation equation:(3)∂ρH∂t+∂ujρH∂xj=∂ρ∂t+∂ujρ∂xj+σij∂ui∂xj+∂∂xjK∂T∂xj+Q.Gas equation of state:(4)P=ρRT.Particle force equilibrium equation:(5)dupdt=FDu−up+gxρp−ρρp+Fx,Fv=Cvρρpup∇u−dupdt,Ff=ρρpup∇u,FM=πdp3ρωu−up2,Fs=2kv1/2ρdijρpdpdlkdkl1/4u−up,Ft=−9πapμ¯2ρ¯2Tkf2kf+kp∇T.Reynolds number equation:(6)Re=ρdpup−uμ,where ρ is the density of air, ρp is the density of particles, t is the time, ui is the velocity of the xi direction, σij is the stress tensor, μ is the viscosity, H is the specific enthalpy, gx is gravity, T is the temperature, K is the thermal conductivity, Q is the heat source, FD is the drag force, Fx is the other force in the xi direction, Fv is the virtual mass force, Ff is the pressure gradient force, FM is the Magnus force, FS is the Saffman force, Ft is the thermophoresis force, Re is the Reynolds number, and R is the gas constant.


## 3.1. Geometric Model and Mesh

To simulate the movement process of solid particles outside the shock tube after the interaction by the shock wave, the computational domain includes not only the internal area but also the external flow field of the shock tube, which are both cylindrical. Since the computational domain is axisymmetric, the 2D calculation domain was selected as the research object for simulation to shorten the calculation time. The dimension of the flow field external to the shock tube is 0.5 m wide and 1 m high. In addition, to capture the propagation process of the shock wave more accurately in the simulation, the computational domain was divided into very fine hexahedral meshes by ICEM software. In the simulation, four grid numbers were used in the grid independence verification: 37852, 94266, 154860, and 229878. Through the comparison of the results, we found that the calculation results of the latter two grid numbers were basically the same. Therefore, we considered that the calculation with 154860 grid numbers meets mesh independence concerns. The hexahedral meshes and local details of the computational domain are shown in Figure3, and the minimum mesh size is 0.05 mm.Figure 3
The hexahedral meshes and local details of the computational domain.


## 3.2. Governing Equations

Because the interaction between shock waves and particles is a compressible gas-solid two-phase flow [23–25], based on the conservation of mass, momentum, and energy, a numerical model of shock wave interactions with solid particles was established. The k-ω turbulent model was adopted to describe the turbulence phenomena throughout the process. The model is described by the following equations and solved by the finite volume method in SC Tetra software.Mass conservation equation:(1)∂ρ∂t+∂ρui∂xi=0.Momentum conservation equation:(2)∂ρui∂t+∂ujρui∂xj=−∂σij∂xj+ρgi.Energy conservation equation:(3)∂ρH∂t+∂ujρH∂xj=∂ρ∂t+∂ujρ∂xj+σij∂ui∂xj+∂∂xjK∂T∂xj+Q.Gas equation of state:(4)P=ρRT.Particle force equilibrium equation:(5)dupdt=FDu−up+gxρp−ρρp+Fx,Fv=Cvρρpup∇u−dupdt,Ff=ρρpup∇u,FM=πdp3ρωu−up2,Fs=2kv1/2ρdijρpdpdlkdkl1/4u−up,Ft=−9πapμ¯2ρ¯2Tkf2kf+kp∇T.Reynolds number equation:(6)Re=ρdpup−uμ,where ρ is the density of air, ρp is the density of particles, t is the time, ui is the velocity of the xi direction, σij is the stress tensor, μ is the viscosity, H is the specific enthalpy, gx is gravity, T is the temperature, K is the thermal conductivity, Q is the heat source, FD is the drag force, Fx is the other force in the xi direction, Fv is the virtual mass force, Ff is the pressure gradient force, FM is the Magnus force, FS is the Saffman force, Ft is the thermophoresis force, Re is the Reynolds number, and R is the gas constant.


## 4. Results and Discussion

### 4.1. Pressure Attenuation Data in the Experiments

To verify the reliability of the experimental data, three groups of repeated shock wave experiments were performed (Nos. 1, 2, and 3) and compared with the blank test (No. 4) to analyze the pressure attenuation. During the experiments, a single-layer paper sheet was used, and the Mach number was 1.53. The same batch of paper sheets and quartz sand particles were used to ensure the repeatability of the experiment. The diameter of the quartz sand particles is 0.42 mm, and the filling thickness is 2 mm. The peak pressure data at different positions (P1, P2, and P3, as shown in Figure 1) during the interaction of the shock wave and solid particles are summarized in Table 1. A comparison of the experimental pressure curves with particles (No. 1) and without particles (No. 4) is shown in Figure 4.Table 1
The peak pressure data at different positions during the interaction process.
No.P1 (kPa)P2 (kPa)P3 (kPa)Describe1342.468.324.2With particle2335.562.321.8With particle3337.17122.3With particle4154.6106.232.1Without particleFigure 4
Comparison of experimental pressure curves with particles and without particles.As shown in Table1 and Figure 4, the pressure data of the three groups of experiments (Nos. 1, 2, and 3) are relatively similar, indicating that the experiment has good repeatability. Comparing the experimental results of Nos. 1–3 with the blank test (No. 4), P1 of Nos. 1–3 is significantly greater than that of No. 4 because P1 is the reflected pressure in the presence of particles, while P1 is the incident pressure in the absence of particles. On the contrary, it is found that P2 is smaller in the presence of particles, which indicates that, during the process of interaction between the shock wave and particles, the part of the energy is converted into the kinetic energy of the solid particles, while the other part is used to overcome the solid particle resistance and the air resistance. It can be concluded that there are many reflection and transmission phenomena of shock waves during the interaction between shock waves and solid particles. Therefore, high-speed shadowgraphy technology was used to capture the phenomena related to the interaction between the shock wave and solid particles to analyze the wave structures and particle motion state in the interaction process.


### 4.2. Analysis of High-Speed Shadowgraphy Results

The time when the particles start to be scattered is defined as 0, and the pictures captured by the high-speed shadowgraphy technology at different times during the experiment are shown in Figure5.Figure 5
High-speed shadowgraphy results of the interaction between the shock wave and particles at different times. (a) 30µs, (b) 60 µs, (c) 90 µs, and (d) 120 µs.
(a)(b)(c)(d)As shown in Figure5, during the interaction between the shock wave and solid particles, the movement of solid particles is not synchronous with the shock wave but lags behind the shock wave, and the velocity of particles is obviously smaller than that of the shock wave. The reason is that the interaction process is very short, the propagation speed of the shock wave is very fast, while the solid particles are in a nonequilibrium state after the interaction, and it takes a period of time to reach a new equilibrium state, which is called the relaxation time of particles [26]. In a previous study, the time of shock wave passing through solid particles (tp) was approximately one thousandth of the relaxation time of particles (ts) [27, 28]. Therefore, when the shock wave passes through the particles, the initial velocity of the particles does not change much. The relationship between tp and ts is as follows:(7)tpts≈DpUts≈Dpagts=18μDpρpγRT,where Dp is the diameter of the particle, ag is the sound speed, and γ is the specific heat ratio.According to Figure4, during the process of the particles being scattered upward, the particle front is initially a horizontal plane and then gradually develops into a mushroom shape with a central bulge. The formation mechanism of this phenomenon can be analyzed in detail by numerical simulation. In addition, there is an obvious turbulent vortex area above the particles, which is caused by the complex wave structures that are caused by a large number of transmission, reflection, and diffraction phenomena when the shock wave passes through the metal mesh and particles. Figure 6 indicates the wave structures of the interaction process between the shock wave and solid particles.Figure 6
The wave structures of the interaction process between the shock wave and solid particles.As shown in Figure6, when the shock wave passes through the quartz sand particles, it is accompanied by shock energy loss and momentum and energy exchange between the shock wave and the solid particles. After the collision between the incident wave and the solid particles, the reflected wave and the transmitted wave are generated, and the backpropagated sparse wave is also produced after the incident wavefront. The interaction process is related to parameters such as shock intensity, bulk density, and diameter of solid particles. When the bulk density of solid particles increases, the resistance of the shock wave passing through the solid particles increases, so the intensity of the reflected shock wave increases, and the intensity of the transmitted shock wave decreases.


### 4.3. Simulation of the Interaction between the Shock Wave and Solid Particles

To obtain the three-dimensional detailed flow field in the interaction process between the shock wave and quartz sand particles and to supplement the experimental results and further analyze the nonequilibrium interaction mechanism between the two phases, the interaction process was simulated under the conditions consistent with the above experiment (Ma = 1.53 andd = 0.42 mm). In the simulation, the air was defined as a continuous phase compressible ideal gas, which was described by the Euler method, the quartz sand particles were defined as a discrete phase, which was described by the Lagrangian method, and the coupling effect between the two phases was considered. Similarly, the time when the particles start to scatter is defined as 0. The variation in the particle velocity and air density gradient at different times during the interaction process is shown in Figure 7.Figure 7
Variation in particle velocity and air density gradient at different times during the interaction process. (a) 30µs, (b) 60 µs, (c) 90 µs, and (d) 120 µs.
(a)(b)(c)(d)Due to the abrupt change in the density at the wavefront, the shape of the shock wave can be accurately displayed through the air density gradient according to the principle of shadowgraphy. According to Figures7 and 5, the simulated shock wave shape is consistent with that captured by the experiment, and there is an upward expanding turbulent vortex area above the solid particles in the simulation, which is also similar to the experimental results. Comparing the peak pressure of the wavefront at the P2 position in the experiments and numerical simulation, the simulated peak pressure is 72.4 kPa, which is close to the average value of the peak pressure at the P2 position (67.2 kPa) in the three experiments, thus proving the validity of the numerical model. Figure 7 clearly shows the position of the wavefront and the movement of the particles. The motion of the wavefront is obviously ahead of that of solid particles, and the particle front gradually changes from the initial horizontal surface to the mushroom shape with a central bulge. In a short period of time after the interaction, the velocity of the particles increases continuously. It presents a trend of accelerating upward movement, indicating that the drag force generated by the airflow to the particles is greater than the gravity of the particles.To more clearly reveal the dispersion process of solid particles and the details of the flow field during the interaction, the symmetrical axial plane is selected to draw the motion vector diagram of the gas-solid two phase, as shown in Figure8.Figure 8
Motion vector diagram of gas-solid two-phase flow in the symmetrical axial plane at different times during the interaction process. (a) 30µs, (b) 60 µs, (c) 90 µs, and (d) 120 µs.
(a)(b)(c)(d)Figure8 shows that, after the interaction between the shock wave and solid particles, the particles accelerate continuously in a short time. The velocity of the center particles is approximately twice that of the surrounding particles. During 120 µs, the maximum velocity of the center particles gradually increases from 67 m/s to 97 m/s, while the velocity of the surrounding particles increases from 34 m/s to 52 m/s. During the particle dispersion process, the velocity direction of the central particles is mainly axial, while the surrounding particles have both axial and radial velocities. Thus, the dispersion of the surrounding particles is more prominent. This also explains the gradual evolution of the particle front from horizontal to mushroom-shaped. Moreover, according to the motion vector diagram of air, a rising vortex is formed obliquely above the particles after the interaction, which is caused by the entrainment effect of the fluid. The specific structure of the vortex at 120 µs is shown in Figure 9.Figure 9
The specific structure of the annular vortex at 120µs.As depicted in Figure9, the vortex structure is a three-dimensional torus. The reason is that the vicinity of the central axis is a high-speed airflow region. According to Bernoulli's law, the area with an enormous velocity has low pressure, so a low-pressure region is formed near the central axis. Figure 9 shows that this low-pressure region constitutes the inner torus of the annular vortex. Due to the entrainment effect on the surrounding fluid in the low-pressure region, a complete three-dimensional toroidal vortex structure is formed.


### 4.4. Mathematical Model of the Solid Particle Dispersion Distance

To obtain the mathematical model of the particle dispersion distance under the action of shock waves, which can be better applied in practical engineering, a series of shock wave experiments were carried out by changing different parameters, such as the Mach number (Ma), the average diameter (d) of quartz sand particles, and the filling thickness (δ) of solid particles. The partial experimental data are summarized in Table 2.Table 2
Experimental data of partial shock wave experiments.
MaD (mm)Δ (mm)T (ms)Dh (mm)Dv (mm)1.530.4220.51.15.01.530.4241.55.325.61.530.4260.51.14.91.530.84428.739.21.530.8460.51.14.91.730.4243.517.282.41.730.421012.913.71.730.8462.511.754.51.730.8481.55.425.51.730.8410314.163.11.80.42213.315.61.80.4221.56.229.11.80.421013.013.91.80.84613.415.91.80.841029.442.0The Mach number (Ma), average diameter (d) of quartz sand particles, filling thickness (δ), and time (t) were taken as independent variables, and the horizontal and vertical dispersion distances of solid particles were taken as dependent variables. The experimental data were statistically analyzed and fitted through the linear regression model in SPSS software to establish the mathematical model, and the expression is as follows:(8)Dv=−26.207+12.23Ma−2.248d−0.576δ+25.912t,Dh=−4.527+1.546Ma−0.307d−0.088δ+5.611t,where Dh is the horizontal dispersion distance and Dv is the vertical dispersion distance.Both of the correlation coefficients (R2) of the above two formulas are 0.995, indicating that the mathematical model is in good agreement with the experimental results. Furthermore, the significance character (Sig.) is less than 0.001, indicating that the independent variable has a significant effect on the dependent variable.According to the mathematical model, when the average diameter (d) of quartz sand particles is 0.42 mm and the filling thickness (δ) is 2 mm, the relationship between the horizontal and vertical dispersion distances of solid particles with the Mach number (Ma) and time (t) is shown in Figures 10(a) and 10(b), respectively.Figure 10
The relationship between the (a) horizontal dispersion distance and (b) vertical dispersion distance of solid particles with the Mach number (Ma) and time (t).
(a)(b)Similarly, when the Mach number (Ma) is 1.8 and the time (t) is 2 ms, the relationship between the horizontal and vertical dispersion distances of solid particles with the average diameter (d) and the filling thickness (δ) is shown in Figures 11(a) and 11(b), respectively.Figure 11
The relationship between the (a) horizontal dispersion distance and (b) vertical dispersion distance of solid particles with the average diameter (d) and the filling thickness (δ).
(a)(b)As shown in Figures10 and 11, the Mach number (Ma) and the time (t) are positively correlated with the dispersion distance, while the average diameter (d) and the filling thickness (δ) are negatively correlated with the dispersion distance. Among them, the average diameter (d), filling thickness (δ), and time (t) have a more significant effect on the dispersion distance, while the Mach number (Ma) has no significant effect on the dispersion distance.


## 4.1. Pressure Attenuation Data in the Experiments

To verify the reliability of the experimental data, three groups of repeated shock wave experiments were performed (Nos. 1, 2, and 3) and compared with the blank test (No. 4) to analyze the pressure attenuation. During the experiments, a single-layer paper sheet was used, and the Mach number was 1.53. The same batch of paper sheets and quartz sand particles were used to ensure the repeatability of the experiment. The diameter of the quartz sand particles is 0.42 mm, and the filling thickness is 2 mm. The peak pressure data at different positions (P1, P2, and P3, as shown in Figure 1) during the interaction of the shock wave and solid particles are summarized in Table 1. A comparison of the experimental pressure curves with particles (No. 1) and without particles (No. 4) is shown in Figure 4.Table 1
The peak pressure data at different positions during the interaction process.
No.P1 (kPa)P2 (kPa)P3 (kPa)Describe1342.468.324.2With particle2335.562.321.8With particle3337.17122.3With particle4154.6106.232.1Without particleFigure 4
Comparison of experimental pressure curves with particles and without particles.As shown in Table1 and Figure 4, the pressure data of the three groups of experiments (Nos. 1, 2, and 3) are relatively similar, indicating that the experiment has good repeatability. Comparing the experimental results of Nos. 1–3 with the blank test (No. 4), P1 of Nos. 1–3 is significantly greater than that of No. 4 because P1 is the reflected pressure in the presence of particles, while P1 is the incident pressure in the absence of particles. On the contrary, it is found that P2 is smaller in the presence of particles, which indicates that, during the process of interaction between the shock wave and particles, the part of the energy is converted into the kinetic energy of the solid particles, while the other part is used to overcome the solid particle resistance and the air resistance. It can be concluded that there are many reflection and transmission phenomena of shock waves during the interaction between shock waves and solid particles. Therefore, high-speed shadowgraphy technology was used to capture the phenomena related to the interaction between the shock wave and solid particles to analyze the wave structures and particle motion state in the interaction process.


## 4.2. Analysis of High-Speed Shadowgraphy Results

The time when the particles start to be scattered is defined as 0, and the pictures captured by the high-speed shadowgraphy technology at different times during the experiment are shown in Figure5.Figure 5
High-speed shadowgraphy results of the interaction between the shock wave and particles at different times. (a) 30µs, (b) 60 µs, (c) 90 µs, and (d) 120 µs.
(a)(b)(c)(d)As shown in Figure5, during the interaction between the shock wave and solid particles, the movement of solid particles is not synchronous with the shock wave but lags behind the shock wave, and the velocity of particles is obviously smaller than that of the shock wave. The reason is that the interaction process is very short, the propagation speed of the shock wave is very fast, while the solid particles are in a nonequilibrium state after the interaction, and it takes a period of time to reach a new equilibrium state, which is called the relaxation time of particles [26]. In a previous study, the time of shock wave passing through solid particles (tp) was approximately one thousandth of the relaxation time of particles (ts) [27, 28]. Therefore, when the shock wave passes through the particles, the initial velocity of the particles does not change much. The relationship between tp and ts is as follows:(7)tpts≈DpUts≈Dpagts=18μDpρpγRT,where Dp is the diameter of the particle, ag is the sound speed, and γ is the specific heat ratio.According to Figure4, during the process of the particles being scattered upward, the particle front is initially a horizontal plane and then gradually develops into a mushroom shape with a central bulge. The formation mechanism of this phenomenon can be analyzed in detail by numerical simulation. In addition, there is an obvious turbulent vortex area above the particles, which is caused by the complex wave structures that are caused by a large number of transmission, reflection, and diffraction phenomena when the shock wave passes through the metal mesh and particles. Figure 6 indicates the wave structures of the interaction process between the shock wave and solid particles.Figure 6
The wave structures of the interaction process between the shock wave and solid particles.As shown in Figure6, when the shock wave passes through the quartz sand particles, it is accompanied by shock energy loss and momentum and energy exchange between the shock wave and the solid particles. After the collision between the incident wave and the solid particles, the reflected wave and the transmitted wave are generated, and the backpropagated sparse wave is also produced after the incident wavefront. The interaction process is related to parameters such as shock intensity, bulk density, and diameter of solid particles. When the bulk density of solid particles increases, the resistance of the shock wave passing through the solid particles increases, so the intensity of the reflected shock wave increases, and the intensity of the transmitted shock wave decreases.


## 4.3. Simulation of the Interaction between the Shock Wave and Solid Particles

To obtain the three-dimensional detailed flow field in the interaction process between the shock wave and quartz sand particles and to supplement the experimental results and further analyze the nonequilibrium interaction mechanism between the two phases, the interaction process was simulated under the conditions consistent with the above experiment (Ma = 1.53 andd = 0.42 mm). In the simulation, the air was defined as a continuous phase compressible ideal gas, which was described by the Euler method, the quartz sand particles were defined as a discrete phase, which was described by the Lagrangian method, and the coupling effect between the two phases was considered. Similarly, the time when the particles start to scatter is defined as 0. The variation in the particle velocity and air density gradient at different times during the interaction process is shown in Figure 7.Figure 7
Variation in particle velocity and air density gradient at different times during the interaction process. (a) 30µs, (b) 60 µs, (c) 90 µs, and (d) 120 µs.
(a)(b)(c)(d)Due to the abrupt change in the density at the wavefront, the shape of the shock wave can be accurately displayed through the air density gradient according to the principle of shadowgraphy. According to Figures7 and 5, the simulated shock wave shape is consistent with that captured by the experiment, and there is an upward expanding turbulent vortex area above the solid particles in the simulation, which is also similar to the experimental results. Comparing the peak pressure of the wavefront at the P2 position in the experiments and numerical simulation, the simulated peak pressure is 72.4 kPa, which is close to the average value of the peak pressure at the P2 position (67.2 kPa) in the three experiments, thus proving the validity of the numerical model. Figure 7 clearly shows the position of the wavefront and the movement of the particles. The motion of the wavefront is obviously ahead of that of solid particles, and the particle front gradually changes from the initial horizontal surface to the mushroom shape with a central bulge. In a short period of time after the interaction, the velocity of the particles increases continuously. It presents a trend of accelerating upward movement, indicating that the drag force generated by the airflow to the particles is greater than the gravity of the particles.To more clearly reveal the dispersion process of solid particles and the details of the flow field during the interaction, the symmetrical axial plane is selected to draw the motion vector diagram of the gas-solid two phase, as shown in Figure8.Figure 8
Motion vector diagram of gas-solid two-phase flow in the symmetrical axial plane at different times during the interaction process. (a) 30µs, (b) 60 µs, (c) 90 µs, and (d) 120 µs.
(a)(b)(c)(d)Figure8 shows that, after the interaction between the shock wave and solid particles, the particles accelerate continuously in a short time. The velocity of the center particles is approximately twice that of the surrounding particles. During 120 µs, the maximum velocity of the center particles gradually increases from 67 m/s to 97 m/s, while the velocity of the surrounding particles increases from 34 m/s to 52 m/s. During the particle dispersion process, the velocity direction of the central particles is mainly axial, while the surrounding particles have both axial and radial velocities. Thus, the dispersion of the surrounding particles is more prominent. This also explains the gradual evolution of the particle front from horizontal to mushroom-shaped. Moreover, according to the motion vector diagram of air, a rising vortex is formed obliquely above the particles after the interaction, which is caused by the entrainment effect of the fluid. The specific structure of the vortex at 120 µs is shown in Figure 9.Figure 9
The specific structure of the annular vortex at 120µs.As depicted in Figure9, the vortex structure is a three-dimensional torus. The reason is that the vicinity of the central axis is a high-speed airflow region. According to Bernoulli's law, the area with an enormous velocity has low pressure, so a low-pressure region is formed near the central axis. Figure 9 shows that this low-pressure region constitutes the inner torus of the annular vortex. Due to the entrainment effect on the surrounding fluid in the low-pressure region, a complete three-dimensional toroidal vortex structure is formed.


## 4.4. Mathematical Model of the Solid Particle Dispersion Distance

To obtain the mathematical model of the particle dispersion distance under the action of shock waves, which can be better applied in practical engineering, a series of shock wave experiments were carried out by changing different parameters, such as the Mach number (Ma), the average diameter (d) of quartz sand particles, and the filling thickness (δ) of solid particles. The partial experimental data are summarized in Table 2.Table 2
Experimental data of partial shock wave experiments.
MaD (mm)Δ (mm)T (ms)Dh (mm)Dv (mm)1.530.4220.51.15.01.530.4241.55.325.61.530.4260.51.14.91.530.84428.739.21.530.8460.51.14.91.730.4243.517.282.41.730.421012.913.71.730.8462.511.754.51.730.8481.55.425.51.730.8410314.163.11.80.42213.315.61.80.4221.56.229.11.80.421013.013.91.80.84613.415.91.80.841029.442.0The Mach number (Ma), average diameter (d) of quartz sand particles, filling thickness (δ), and time (t) were taken as independent variables, and the horizontal and vertical dispersion distances of solid particles were taken as dependent variables. The experimental data were statistically analyzed and fitted through the linear regression model in SPSS software to establish the mathematical model, and the expression is as follows:(8)Dv=−26.207+12.23Ma−2.248d−0.576δ+25.912t,Dh=−4.527+1.546Ma−0.307d−0.088δ+5.611t,where Dh is the horizontal dispersion distance and Dv is the vertical dispersion distance.Both of the correlation coefficients (R2) of the above two formulas are 0.995, indicating that the mathematical model is in good agreement with the experimental results. Furthermore, the significance character (Sig.) is less than 0.001, indicating that the independent variable has a significant effect on the dependent variable.According to the mathematical model, when the average diameter (d) of quartz sand particles is 0.42 mm and the filling thickness (δ) is 2 mm, the relationship between the horizontal and vertical dispersion distances of solid particles with the Mach number (Ma) and time (t) is shown in Figures 10(a) and 10(b), respectively.Figure 10
The relationship between the (a) horizontal dispersion distance and (b) vertical dispersion distance of solid particles with the Mach number (Ma) and time (t).
(a)(b)Similarly, when the Mach number (Ma) is 1.8 and the time (t) is 2 ms, the relationship between the horizontal and vertical dispersion distances of solid particles with the average diameter (d) and the filling thickness (δ) is shown in Figures 11(a) and 11(b), respectively.Figure 11
The relationship between the (a) horizontal dispersion distance and (b) vertical dispersion distance of solid particles with the average diameter (d) and the filling thickness (δ).
(a)(b)As shown in Figures10 and 11, the Mach number (Ma) and the time (t) are positively correlated with the dispersion distance, while the average diameter (d) and the filling thickness (δ) are negatively correlated with the dispersion distance. Among them, the average diameter (d), filling thickness (δ), and time (t) have a more significant effect on the dispersion distance, while the Mach number (Ma) has no significant effect on the dispersion distance.


## 5. Conclusions

To further study the interaction mechanism between shock waves and solid particles, shock wave experiments and CFD simulations were combined to study the interaction process in this paper. The main conclusions are as follows:(1)
When the shock wave passes through the solid particles, the reflected wave and the transmitted wave are generated, and the pressure rapidly decays, caused by the exchange of momentum and energy. The motion of solid particles lags behind that of the shock wave, and a new equilibrium can be achieved after the relaxation time of particles.(2)
During the dispersion process of particles, the velocity of the central particles is more significant than that of the surrounding particles, and the moving direction of the central particles is mainly axial, while the surrounding particles have both axial and radial velocity components. Therefore, the particle front gradually changes from horizontal to mushroom-shaped.(3)
After the interaction between the shock wave and the solid particles, due to the high-speed and low-pressure area near the central axis, the entrainment effect is generated on the surrounding fluid. Thus, a rising three-dimensional annular vortex is formed obliquely above the particles. The evolution of the vortex will affect the development of the external flow field.(4)
The mathematical model of the particle dispersion distance under the action of a shock wave is established. All of the parameters, such as the Mach number, diameter of the solid particles, filling thickness, and time, affect the dispersion distance. Among them, the dispersion distance increases with increasing Mach number and time and decreases with increasing particle diameter and filling thickness.



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*Source: 3791156-2022-03-03.xml*