(**a**) Absolute coordinate system (**b**) Relative coordinate system

In fluid mechanics, Mach number *M* is an important parameter to describe the flow state. It is defined as the ratio of the velocity *v* of a point in the flow field to the local sound velocity *c* of the point:

$$M = \frac{v}{c} \tag{5}$$

Similar to the concept of local sound speed, the Mach number describes the local properties in the flow field. If the local sound speed of two points in the flow field is different, even if the two points have the same fluid velocity, the Mach number is also different.

The gas Mach number *M* before and after the wave front after the coordinate transformation:

$$M\_1 = \frac{\upsilon\_1}{c\_1} = \frac{\upsilon\_s}{c\_1} \tag{6}$$

$$M\_{\!\!\!2} = \frac{v\_{\!\!2}}{c\_{\!\!2}} = \frac{v\_{\!\!\!suit} - v\_{\!\!\!suit}}{c\_{\!\!2}} \tag{7}$$

In the formula, the subscript of 1 is the gas parameter on the right side of the shock wave, and the subscript of 2 is the gas parameter on the left side of the shock wave.

In Figure 7b, the gas in the imaginary frame is used as the control body, because the standing shock wave is a parameter-stable structure and obeys the mass conservation equation:

$$\iiint\limits\_{S} \rho \vec{V} \cdot d\mathcal{S} = 0 \tag{8}$$

Only the front and back sides of the control body boundary *S* have fluid in and out. *A*<sup>1</sup> is the area of the inlet surface of the standing shock wave, and *A*<sup>2</sup> is the area of the outlet surface of the standing shock wave. The two values are equivalent and the direction is opposite, so *S* = *A*<sup>1</sup> − *A*2, the above formula can be changed into:

$$A\_1 \rho\_1 \upsilon\_s - A\_2 \rho\_2 [(\upsilon\_s - \upsilon\_b)] = 0\tag{9}$$

The formula can be simplified to:

$$
\rho\_1 v\_1 = \rho\_2 v\_2 \tag{10}
$$

The momentum conservation is obeyed in the control body, and the impulse of the pressure resultant force is equal to the difference between the momentum of the outflow fluid and the momentum of the inflow fluid.

$$\iiint\limits\_{S} (\rho \overrightarrow{V} \cdot d\mathcal{S}) \overrightarrow{V} = \iiint\limits\_{S} p dS \tag{11}$$

The above formula can be changed to:

$$p\_1A\_1 - p\_2A\_2 = mv\_2 - mv\_1\tag{12}$$

In the formula, m denotes the mass flow rate of gas in and out of the control body, *m* = *ρvA*. The above formula can be transformed into:

$$
\rho\_1 + \rho\_1 v\_1^2 = p\_2 + \rho\_2 v\_2^2 \tag{13}
$$

Similarly, the energy equation is expressed as:

$$\iiint\_{s} \rho \left( e + \frac{V^{2}}{2} \right) \overrightarrow{V} \cdot d\mathbf{s} = -\underset{S}{\underset{S}{\text{div}}} \, p\overrightarrow{V} \cdot \overrightarrow{dS} \tag{14}$$

The above formula can be transformed into:

$$h\_1 + \frac{v\_1^2}{2} = h\_2 + \frac{v\_2^2}{2} \tag{15}$$

In the formula, *<sup>v</sup>*<sup>1</sup> 2 <sup>2</sup> represents the kinetic energy of the airflow, which is a macroscopic parameter. Enthalpy *h* represents the sum of gas internal energy and pressure potential energy, which is a microscopic parameter.

The momentum Equation (13) is divided by the continuity Equation (10) to obtain:

$$\left(v\_1^2 - v\_2^2\right) = (p\_2 - p\_1)(\frac{1}{\rho\_1} + \frac{1}{\rho\_2})\tag{16}$$

According to:

$$h = \frac{k}{k-1} \frac{p}{\rho} \tag{17}$$

The energy Equation (15) can be transformed into:

$$\left(v\_1\right)^2 - v\_2^2 = \frac{2k}{k-1}(\frac{p\_2}{\rho\_2} - \frac{p\_1}{\rho\_1})\tag{18}$$

Combining Equation (16) with Equation (18):

$$\frac{\rho\_2}{\rho\_1} = \frac{1 + \frac{k+1}{k-1}\frac{p\_2}{p\_1}}{\frac{k+1}{k-1} + \frac{p\_2}{p\_1}}\tag{19}$$

This function represents the relationship between the density ratio and the pressure ratio before and after the normal shock wave, also known as the shock adiabatic relation. In the experiment, the pressure before and after the shock wave can be directly measured by the pressure sensor, and the density cannot be directly measured, so this formula can be used for density calculation.

The isentropic change of gas is a reversible adiabatic process. The relationship between pressure change and density change is:

$$\frac{\rho\_2}{\rho\_1} = \left(\frac{p\_2}{p\_1}\right)^{1/k} \tag{20}$$

The density changes in the isentropic process and the shock process are compared with *k* = 1.4, as shown in Figure 8. It can be seen from the diagram that when the pressure is relatively small, the density ratio of the isentropic process and the shock wave process is not much different. As the pressure ratio increases, the density ratio in the shock wave mutation process becomes significantly smaller. The difference between the shock wave process [31] and the isentropic process curve shows that the shock wave process is entropy-increasing, that is, an irreversible process. In addition, as the pressure ratio further increases, the density ratio of the shock wave process has a limit, and the limit value is:

$$\frac{\rho\_2}{\rho\_1} = \frac{k+1}{k-1} \tag{21}$$

**Figure 8.** Comparison of density changes in isentropic and shock processes.

The calculated limit density ratio is 6. This shows that no matter how strong the shock wave is, the maximum pressure behind the shock wave can reach 6 times the pressure before the shock wave.

Equations (9) and (13) are combined to obtain:

$$w\_s = \sqrt{\frac{\rho\_2}{\rho\_1} \frac{p\_2 - p\_1}{\rho\_2 - \rho\_1}}\tag{22}$$

Combined with the shock adiabatic relationship (19) and the sound velocity relationship, we can get:

$$v\_s = \sqrt{\frac{\rho\_2}{\rho\_1} \cdot \frac{p\_2 - p\_1}{\rho\_2 - \rho\_1}} = c\_1 \sqrt{1 + \frac{k+1}{2k} (\frac{p\_2}{p\_1} - 1)}\tag{23}$$

In the formula, *c*<sup>1</sup> is the sound speed of the gas before the disturbance (before the shock wave). Since the shock wave pressure *p*<sup>2</sup> is always greater than the front atmospheric pressure *<sup>p</sup>*1, *<sup>p</sup>*<sup>2</sup> *p*1 > 1, according to the above formula, it is easy to see that *vs* is greater than *c*1. That is, the velocity of the shock wave front relative to the front gas is theoretically calculated to be supersonic. The greater the shock wave strength, the greater the shock wave propagation speed. When the shock wave strength is very weak *<sup>p</sup>*<sup>2</sup> *<sup>p</sup>*<sup>1</sup> → 1, the shock wave velocity *vs* is infinitely equal to the unperturbed gas sound velocity.

Defined by the shock wave front Mach number:

$$M\_1 = \frac{v\_s}{c\_1} \tag{24}$$

Derived from Equations (23) and (24):

$$M\_1 = \sqrt{1 + \frac{k+1}{2k}(\frac{p\_2}{p\_1} - 1)}\tag{25}$$

The shock wave front Mach number can be obtained from the pressure difference before and after the shock wave. The shock Mach number can be obtained by Equation (24) and Equation (25). However, in Equation (24), the velocity of the wave front needs to be measured, while in Equation (25), the pressure before and after the wave front needs to be measured. The difficulty and accuracy of pressure measurement are higher than that of velocity measurement, so the shock Mach number is generally calculated directly by Equation (24).

In the calculation of shock wave-related parameters, Mach number *M*<sup>1</sup> can be considered an earlier determined parameter. Taking the Mach number as the known number, other parameters can be calculated and deduced.

The inverse operation of Equation (25) is obtained:

$$\frac{p\_2}{p\_1} = \frac{2k}{k+1} M\_1^{\;> \;} - \frac{k-1}{k+1} \tag{26}$$

Using this formula, the pressure behind the shock wave can be calculated by the shock wave Mach number.

The local thermodynamic parameters of a point in the flowing gas in fluid mechanics are called static parameters. If the isentropic velocity of the fluid at this point is reduced to zero, it is called the stagnation state, and then the parameter value corresponding to the static parameter becomes the stagnation parameter. Static parameters are real state parameters, and stagnation parameters are parameters based on theoretical assumptions. The sign of the stagnation parameter is generally 0. The stagnation parameters include stagnation enthalpy *h*0, stagnation temperature *T*0, stagnation pressure *p*0, stagnation sound speed *c*<sup>0</sup> and stagnation density *ρ*0. These parameters are also called total enthalpy, total temperature, total pressure and total density.

According to the Bernoulli equation, any two points on a streamline satisfy the formula:

$$h\_1 + \frac{v\_1^2}{2} = h\_2 + \frac{v\_2^2}{2} = \text{const} \tag{27}$$

According to the definition of stagnation state, *v*<sup>2</sup> = 0, then:

$$h\_0 = h + \frac{1}{2}v^2\tag{28}$$

It can be seen from the Equation (28) that the total enthalpy is the sum of the enthalpy related to the temperature of each point and the enthalpy related to the dynamic pressure of the point. It essentially represents the total energy per unit mass of gas. The Equation (27) also shows that the total temperature (total enthalpy) of the gas remains unchanged during the adiabatic process.

In addition, from the gas state equation:

$$h = c\_p T; c\_p = \frac{kR}{k-1} \tag{29}$$

The total temperature formula is obtained:

$$T\_0 = T(1 + \frac{k-1}{2}\frac{v^2}{c^2})\tag{30}$$

Combined with the definition of the Mach number, the above formula can be written as:

$$T\_0 = T(1 + \frac{k-1}{2}M^2) \tag{31}$$

In the formula, *T* and *M* can be subscripted at any position.

The parameters *p* and *ρ* in the momentum Equation (13) and the energy Equation (15) are eliminated, and only the parameters temperature *T* and velocity are retained. It can be transformed into:

$$\frac{RT\_1}{v\_1} - \frac{RT\_2}{v\_2} + (v\_1 - v\_2) = 0\tag{32}$$

$$\frac{kR}{k-1}T\_1 + \frac{\upsilon\_1^2}{2} = \frac{kR}{k-1}T\_2 + \frac{\upsilon\_2^2}{2} = \frac{kR}{k-1}T\_0\tag{33}$$

From the Equations (32) and (33), we can obtain:

$$\frac{kRT\_0(v\_2 - v\_1)}{v\_1v\_2} + \frac{k+1}{2}(v\_1 - v\_2) = 0\tag{34}$$

By solving the above equation, the sound velocity relationship before and after the shock wave is obtained:

$$v\_1 v\_2 = \frac{2kRT\_0}{k+1} \tag{35}$$

By the critical speed of sound equation:

$$
\omega\_{cr} = \sqrt{\frac{2kRT\_0}{k+1}}\tag{36}
$$

So, there is:

$$
v\_1 v\_2 = c\_{cr}{}^2\tag{37}$$

$$
\lambda\_1 = \frac{v\_1}{c\_{cr}}, \lambda\_2 = \frac{v\_2}{c\_{cr}}, \lambda\_1 \lambda\_2 = 1\tag{38}
$$

By the definition of Equation (35) and the Mach number, it can be concluded that:

$$M\_1 M\_2 \sqrt{T\_1 T\_2} = \frac{2}{k+1} T\_0 \tag{39}$$

Substituting the total temperature Equation (31) into the above formula, we can get:

$$M\_2 = \sqrt{\frac{M\_1^2 + \frac{2}{k-1}}{\frac{2k}{k-1}M\_1^2 - 1}}\tag{40}$$

The Mach number *M*<sup>2</sup> represents the ratio of the airflow velocity to the local sound velocity in the standing shock wave, that is, the Mach number after the coordinate transformation.

From the above theory of total temperature, the shock wave process is an adiabatic process, so the total temperature before and after the wave front remains unchanged, so:

$$T\_1(1 + \frac{k-1}{2}M\_1^{\,\,2}) = T\_2(1 + \frac{k-1}{2}M\_2^{\,\,2})\tag{41}$$

Combining Equation (40) with Equation (41), we can obtain:

$$\frac{T\_2}{T\_1} = \frac{(1 + \frac{k-1}{2}M\_1^{\,2})(\frac{2k}{k-1}M\_1^{\,2} - 1)}{\frac{(k+1)^2}{2(k-1)}M\_1^{\,2}}\tag{42}$$

From the general gas state equation and Equations (26) and (42), it is concluded that:

$$\frac{\rho\_2}{\rho\_1} = \frac{p\_2}{p\_1} \frac{T\_1}{T\_2} = \frac{k+1}{2} (\frac{M\_1^2}{1 + \frac{k-1}{2}M\_1^2})\tag{43}$$

By the continuity Equation (10):

$$\frac{v\_1}{v\_2} = \frac{\rho\_2}{\rho\_1} = \frac{2 + (k - 1)M^2}{(k + 1)M^2} \tag{44}$$

In the continuity equation, *v*<sup>2</sup> is expressed as *v*<sup>2</sup> = (*v*<sup>s</sup> − *v*b). Combining Equation (43) and the sound velocity formula, the expression of the accompanying velocity can be obtained:

$$v\_b = c\_1 \frac{2}{k+1} \frac{M-1}{M} \tag{45}$$

According to the second law of thermodynamics and the definition of entropy:

$$ds = \frac{\delta q}{T} = \frac{dc + p \cdot d(\frac{1}{\rho})}{T} = c\_V \cdot \frac{dT}{T} + R \frac{d(\frac{1}{\rho})}{(\frac{1}{\rho})} \tag{46}$$

Integral on both sides:

$$s\_2 - s\_1 = c\_V \ln(\frac{T\_2}{T\_1}) + R \ln(\frac{\rho\_1}{\rho\_2}) = c\_V \ln\left[\frac{\left(\frac{T\_2}{T\_1}\right)}{\left(\frac{\rho\_2}{\rho\_1}\right)^{k-1}}\right] \tag{47}$$

Substituting the shock temperature ratio Equation (42) and the density ratio Equation (43), we obtain:

$$s\_2 - s\_1 = R \ln \left[ \frac{2 + (k - 1)M\_1^{\cdot 2}}{(k + 1)M\_1^{\cdot 2}} \right]^{\frac{k}{k - 1}} + R \ln \left[ \frac{2kM\_1^{\cdot 2} - (k - 1)}{(k + 1)} \right]^{\frac{1}{k - 1}} \tag{48}$$

*M*<sup>1</sup> > 1 in the process of shock wave is substituted into the above formula:

$$s\_2 - s\_1 \ge 0\tag{49}$$

Therefore, the entropy value increases after the shock wave of a coal and gas outburst, which is an irreversible process.

#### *3.3. The Attenuation of Outburst Shock Wave Intensity*

According to the ideal shock wave theory, when the shock wave source does not attenuate, the ideal gas shock wave front will not weaken with the increase of propagation distance in the process of propagation in the roadway. Even if the shock wave source can only maintain a short time, the propagation of the shock wave in the tunnel may not be affected in a short time, and the specific principle is no longer described. However, the non-attenuation of shock wave is not consistent with the traditional understanding and does not conform to the fact. In the process of real shock wave propagation, due to the influence of viscosity in the moving gas and heat conduction between the roadway, the intensity (or propagation speed) of the shock wave gradually decays, but the attenuation speed is slow [32,33]. In summary, the attenuation of shock waves in an underground ventilation system mainly depends on the bending of facilities such as anti-burst doors or the roadway itself. Some other studies say that there is a linear relationship between the attenuation of the prominent shock wave and the propagation distance of the shock wave, which is not consistent with the facts.

#### **4. Relationship between Gas Pressure and Outburst Shock Wave Intensity**

The above part is the analysis of the shock wave propagation process after the occurrence of a coal and gas outburst and does not involve the analysis of the influence of the initial gas pressure of the coal seam on the shock wave intensity. The following will be combined with the above theory for further analysis to discuss the relationship between gas pressure and shock wave formation strength.

As shown in Figure 9, the region division and pressure magnitude diagram of the shock tube at a certain time are shown. There is a left-lateral sparse wave between zone 3 and zone 4. The high-pressure coal-gas flow in zone 4 accelerates to zone 3 through expansion, and its flow is an isentropic flow. The gas flow in zone 3 and zone 4 on the same streamline conforms to the formula:

$$
\mu\_3 + \frac{2}{k\_3 - 1} c\_3 = \mu\_4 + \frac{2}{k\_4 - 1} c\_4 \tag{50}
$$

**Figure 9.** Pressure distribution in shock tube.

In the formula, *u*<sup>3</sup> and *u*<sup>4</sup> represent the fluid flow velocity in zone 3 and zone 4, respectively, and zone 4 is the static region, so *u*<sup>4</sup> = 0. *k*<sup>3</sup> and *k*<sup>4</sup> represent the specific heat ratio of fluid in zone 3 and zone 4, respectively. Zone 3 and zone 4 are essentially the same substance, so *k*<sup>3</sup> = *k*4. Therefore, the above formula can be rewritten as:

$$
\mu\_3 + \frac{2}{k\_4 - 1} c\_3 = \frac{2}{k\_4 - 1} c\_4 \tag{51}
$$

The isentropic flow satisfies the following equation:

$$\frac{p\_4}{p\_3} = (\frac{c\_4}{c\_3})^{\frac{2k\_4}{k\_4 - 1}}\tag{52}$$

The contact surface of zone 2 and zone 3 meets the compatibility relationship:

$$
\mu\_2 = \mu\_3, \,\mu\_2 = \,\mu\_3 \tag{53}
$$

The pressure relationship between zone 4 and zone 1 satisfies:

$$\frac{p\_4}{p\_1} = \frac{p\_4}{p\_3} \frac{p\_3}{p\_1} = \frac{p\_4}{p\_3} \frac{p\_2}{p\_1} \tag{54}$$

Using the above formula can be obtained:

$$\frac{p\_4}{p\_1} = \frac{2k\_1M^2 - (k\_1 - 1)}{k\_1 + 1} \left\{ 1 - \left(\frac{k\_4 - 1}{k\_1 + 1}\right) \left(\frac{c\_1}{c\_4}\right) \left(M - \frac{1}{M}\right) \right\}^{-\left(\frac{2k\_4}{k\_4 - 1}\right)} \tag{55}$$

This formula represents the relationship between the pressure ratio *p*4/*p*<sup>1</sup> and the shock Mach number *M* (shock wave intensity). The parameters *k*<sup>1</sup> and *c*<sup>1</sup> in the formula are roadway air parameters, which are easy to obtain. *k*<sup>4</sup> and *c*<sup>4</sup> represent the equivalent specific heat ratio and equivalent sound velocity of coal-gas flow. They have no known fixed values and can only be obtained through experiments. Because there are many distribution positions of parameter *k*<sup>4</sup> in Equation (55), it is difficult to unify it. Therefore, this paper sets the parameter as 1.4. The relationship between the theoretical pressure ratio *p*4/*p*<sup>1</sup> and the experimental results is calculated by changing the equivalent sound velocity. Figure 10 shows the ratio of initial gas pressure to atmospheric pressure *p*4/*p*<sup>1</sup> when the ratio of standard atmospheric sound velocity to the equivalent sound velocity of coal-gas flow *c*1/*c*<sup>4</sup> takes different values at Mach number *M* = 1.5. The inverse calculation of Equation (55) can directly calculate the outburst shock wave *M* from the initial gas pressure of the coal seam. If the initial gas pressure *p*<sup>4</sup> of the coal seam and the intensity of the outburst shock wave are known, the equivalent sound velocity of the coal-gas flow under this condition can be calculated.

**Figure 10.** The relationship between equivalent sound ratio and pressure ratio.

It can be seen from Figure 10 that the pressure ratio is very sensitive to the equivalent sound velocity of the coal-gas flow. The pressure ratio is 7 when *c*1/*c*<sup>4</sup> is the same sound velocity in zone 4 and zone 1. When *c*1/*c*4= 1.5, the pressure ratio is 12.6. When *c*1/*c*4= 2, the pressure ratio is 24.

#### **5. Conclusions**

(1) When outburst occurs, a large amount of broken coal and gas migrates from the coal seam to the roadway under the action of ground stress and gas pressure. It can be divided into stable coal seam area, coal-gas flow area, air compression area and roadway unaffected area. The location of these zones will change with the development of the outburst, and the internal of each region is uneven. The outburst energy accumulates continuously at the interface between the air compression zone and the unaffected zone of the roadway, forming an outburst wave front, and its formation and propagation satisfies the aerodynamic theory;


**Author Contributions:** Investigation, D.S., J.C., L.D., R.L. and Y.L.; Methodology, D.S., J.C. and L.D.; Supervision, R.L. and Y.L.; Writing—original draft, D.S., J.C. and L.D.; Writing—review and editing, D.S., J.C., L.D. and R.L; funding acquisition, D.S., J.C., and L.D. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China (No. 51974358, No. 52104239, No. 51774319), Natural Science Foundation of Chongqing (No. CSTB2022NSCQ-MSX1080, No. CSTB2022NSCQ-MSX0379, No. cstc2021jcyj-msxmX0564).

**Data Availability Statement:** The data are available from the corresponding author on reasonable request.

**Acknowledgments:** We also would like to thank the anonymous reviewers for their valuable comments and suggestions that lead to a substantially improved manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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