# Design of a New Supersonic Shock Wave Generator and Application in Power Generation
## 3 Supersonic Shock Wave Monitoring
### Numerical Simulation of the Shock Wave Monitoring  
#### 3.2.1 Governing Equations and Preconditioning System  
For the numerical simulation of supersonic, high-temperature, and high-pressure flow fields in a long cylindrical shock tube [23], this study uses a multi-block grid approach, using a CFD (computational fluid dynamics) code based on the control volume method and the preconditioning method for solving Navier-Stokes equations to model compressible and incompressible coupling problems. We define the inviscid flux vector \(\overset{\rightarrow}{F}\) in the standard conservation form as follows:  
\[\overset{\rightarrow}{F}= \begin{bmatrix}\rho\overset{\rightarrow}{V}\\ \rho\overset{\rightarrow}{V}v_{x}+p\overset{\rightarrow}{i}\\ \rho\overset{\rightarrow}{V}v_{y}+p\overset{\rightarrow}{j}\\ \rho\overset{\rightarrow}{V}v_{z}+p\overset{\rightarrow}{k}\\ \rho\overset{\rightarrow}{V}E+p\overset{\rightarrow}{V}\end{bmatrix}  \tag{1}\]  
where \(\rho\) is the density, \(\overset{\rightarrow}{V}\) is the velocity vector, \(v_{x}\), \(v_{y}\), and \(v_{z}\) are the velocity components in the \(x\), \(y\), and \(z\) directions, respectively, \(p\) is the pressure, \(E\) is the total energy per unit mass, and the relationship between \(E\) and the total enthalpy \(H\) is \(H=E-p/\rho\), where \(H=C_{p}T+V^{2}/2\) (\(C_{p}\) is the constant pressure specific heat and \(T\) is the temperature). This system is closed by the equation of state, which usually has the form \(\rho=\rho(p,T)\). The low diffusion flux splitting method of AUSM+ is used here [24], then the inviscid interface flux \(F_{i+1/2}\) in the \(x\) direction can be decomposed into the sum of the convection contribution \(F_{1/2}^{r}\) and the pressure contribution \(F_{1/2}^{p}\), where the subscript \(1/2\) represents the center of the grid, and the convection term \(F_{1/2}^{c}\) and the pressure term \(F_{1/2}^{p}\) can be defined as follows:  
Figure 5: The architecture of the shock wave monitoring system. \[F_{1/2}^{c}=(\rho v_{x})_{1/2} \begin{bmatrix}1\\ v_{x}\\ v_{y}\\ v_{z}\\ H\end{bmatrix} _{i/i+1}\quad\quad,\;F_{1/2}^{p}=(\rho v_{x})_{1/2} \begin{bmatrix}0 \\ p_{1/2}\\ 0\\ 0\\ 0\end{bmatrix}  \tag{2}\]  
In the convection term \(F_{1/2}^{c}\), if interface mass flux \((\rho v_{x})_{1/2}\) is non-negative, then state \(i\) is selected for column vector (1, \(v_{x}\), \(v_{y}\), \(v_{z}\), \(H\))\({}^{T}\); if \((\rho v_{x})_{1/2}\) is negative, then state \(i+1\) is selected. The interface quantities \((\rho v_{x})_{1/2}\) and \(p_{1/2}\) are defined as follows:  
\[(\rho v_{x})_{1/2}=a_{1/2}\Big{(}\rho_{i}m_{1/2}^{+}+\rho_{i+1}m_{1/2}^{-}\Big{)} \tag{3}\]  
\[p_{1/2}=P_{(5)}^{+}(M_{i})p_{i}+P_{(5)}^{-}(M_{i+1})p_{i+1} \tag{4}\]  
Among them, \(a_{1/2}\) is the interface sound speed, \(M_{i}\) is the MACH number, \(M_{i}=v_{xi}/a_{1/2}\), and \(m_{1/2}^{+}\) and \(m_{1/2}^{-}\) are defined as follows:  
\[m_{1/2}^{+}=M_{(4)}^{+}(M_{i})+M_{(4)}^{-}(M_{i+1}) \tag{5}\]  
\[m_{1/2}^{-}=\frac{1}{2}(M_{1/2}+|M_{1/2}|) \tag{6}\]  
\(P_{(5)}^{+}\) and \(P_{(5)}^{-}\) in Equation (4) are both fifth-degree polynomials of \(M\), and their definitions are as follows:  
\[P_{(5)}^{\pm} \begin{cases}\frac{1}{4}(M\pm 1)^{2}(2\mp M)\pm\frac{3}{16}M\big{(}M ^{2}-1\big{)}^{2},\;|M|<1\\ \frac{1}{2M}(M+|M|),\;\;\;\text{otherwise}\end{cases}  \tag{7}\]  
\(M_{(4)}^{+}\) and \(M_{(4)}^{-}\) in Equation (5) are both fourth-degree polynomials of \(M\), and their definitions are as follows:  
\[M_{(4)}^{\pm} \begin{cases}\pm\frac{1}{4}(M\pm 1)^{2},\;|M|<1\\ \frac{1}{2}(M\pm|M|),\;\;\text{otherwise}\end{cases}  \tag{8}\]  
Next, we use the following Navier-Stokes equations as governing equations:  
\[\frac{\partial}{\partial t}\iiint Wd\Omega+\iint\left[\overset{\to}{F}- \overset{\to}{G}\right]\cdot\overset{\to}{A}=0 \tag{9}\]  
where \(\Omega\) is an arbitrary control volume, \(\overset{\to}{A}\) is the surface area, \(W\) is the dependent vector of conservative variables, and \(\overset{\to}{G}\) is the viscous flux vector in the standard conservation form.