## 3 Numerical Simulations of Gas Explosion along Pipelines with Abrupt Cross-Section Area Changes
### Numerical Calculation Methods  
In the numerical simulation, the premixed gas of gas explosion can be regarded as ideal mixed gas, and gas explosion can be characterized by energy conservation, momentum conservation, mass conservation, and composition balance equation [27]. Usually, the Reynolds-averaged N-S equation is used as governing equation [28; 29]. The main equations are as follows:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u)}{\partial x}=0 \tag{6}\]  
\[\frac{\partial\rho u}{\partial t}+\frac{\partial(\rho uu)}{\partial x}=-\frac {\partial p}{\partial x}+\frac{4}{3}\mu_{e}\frac{\partial u}{\partial x} \tag{7}\]  
\[\frac{\partial\rho h}{\partial t}+\frac{\partial}{\partial x}\left[\rho uh- \frac{\mu_{e}}{\partial_{h}}\frac{\partial u}{\partial x}\right]=\frac{Dp}{ Dt}+S_{h} \tag{8}\]  
\[\frac{\partial}{\partial t}(\rho Y_{fu})+\frac{\partial}{\partial x}\left[ \rho uY_{fu}-\frac{\mu_{e}}{\sigma_{fu}}\frac{\partial Y_{fu}}{\partial x} \right]=R_{fu} \tag{9}\]  
where \(P\) is static pressure; \(\rho\) is density; \(u\) is particle velocity; \(\mu_{e}\) is dynamic viscous coefficient; \(h\) is enthalpy; \(Y_{fu}\) is fuel mass fraction; \(R_{fu}\) is volume combustion rate; \(k\) is turbulent kinetic energy; and \(S_{h}\) is the heat source term.  
Turbulence is an important factor characterizing gas explosion or combustion. The turbulence model, turbulent kinetic energy equation, and turbulent flow energy dissipation rate equation are as follows [30]:  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{j}}\left[\rho u _{j}k-\frac{\mu_{e}}{\sigma_{k}}\frac{\partial k}{\partial x_{j}}\right]=G- \rho e \tag{10}\]  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{j}}\left[\rho u _{j}\varepsilon-\frac{\mu_{e}}{\sigma_{k}}\frac{\partial e}{\partial x_{j}} \right]=C_{1}G\frac{\varepsilon}{\bar{k}}-C_{2}\rho\frac{\varepsilon^{2}}{\bar {k}} \tag{11}\]where \(\mu_{e}=\mu_{l}+\mu_{t}\) is the effective viscous coefficient; \(\mu_{l}\) is the laminar viscous coefficient; \(\mu_{t}=C_{\mu}\rho k^{2}/\varepsilon\) is the turbulent viscous coefficient; \(\delta_{ij}\) is the Kronecker operator; \(C_{\mu}\), \(C_{1}\), \(C_{2}\), \(\sigma_{k}\), and \(\sigma_{\varepsilon}\) are the turbulent model constants, respectively; and G is obtained as follows:  
\[G=\frac{\partial u_{i}}{\partial x_{j}}\left[\mu_{e}(\frac{\partial u_{i}}{ \partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}})-\frac{2}{3}\delta_{ij} \big{(}\rho k+\mu_{e}\frac{\partial u_{k}}{\partial x_{k}}\big{)}\right] \tag{12}\]  
Gas explosion is a complex chemical reaction process. Both turbulence effects and chemical reaction kinetics must be considered simultaneously. Therefore, the EBU Arrhenius combustion model is adopted [31]. Usually, turbulent flame velocity is calculated based on flame front model, and turbulent flame velocity \(U_{t}\) is given by the following equation:  
\[U_{t}=A\big{(}u^{\prime}\big{)}^{3/4}U_{l}{}^{1/2}\alpha^{-1/4}{}l_{l}{}^{1/4 }\cdot g=Au^{\prime}\bigg{(}\frac{\tau_{l}}{\tau_{c}}\bigg{)}^{1/4}\cdot g \tag{13}\]  
Turbulence intensity \(u^{\prime}\) and turbulence characteristic scale \(l_{t}\) are calculated as follows:  
\[u^{\prime}=\sqrt{\frac{2}{3}k}\underbrace{l_{t}}_{,}=C_{D}\frac{(u^{\prime}) ^{3}}{\varepsilon} \tag{14}\]  
where \(A\) is model constant, usually 0.52; \(U_{l}\) is laminar flame velocity [m/s]; \(\alpha=k/\rho c_{p}\) is the heat transfer coefficient of temperature, m\({}^{2}\)/s; \(\tau_{l}=l_{t}/u^{\prime}\) is the turbulence time scale; \(\tau_{c}=\alpha/U_{l}{}^{2}\) is the time scale of the chemical reaction; \(g\) is the flame stretching factor (when there is no stretching, \(g=1\)); and \(C_{D}\) is a constant, usually 0.37.  
In the numerical simulations, the SIMPLEC algorithm [32] for pressure-velocity coupling was selected, and the finite volume method [33; 34] was adopted.