# Investigation of the Fluid Flow in a Large Ball Valve Designed for Natural Gas Pipelines
## 2 Methods
SolidWorks Flow Simulation solves the mass, momentum, and energy conservation laws for fluid flows, fluid state equations, and the empirical dependencies of fluid density, viscosity, and thermal conductivity on temperature. The conservation laws for mass, angular momentum, and energy, along with the additional equations necessary to describe the fluid motion, are presented below as Equations (1)-(16), which were extracted from the manual of the SolidWorks software [26]. \[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho u_{i})=S _{M}^{p} \tag{1}\]  
\[\frac{\partial\rho u_{i}}{\partial t}+\frac{\partial}{\partial x_{i}}\big{(} \rho u_{i}u_{j}\big{)}+\frac{\partial p}{\partial x_{i}}=\frac{\partial}{ \partial x_{j}}\Big{(}\tau_{ij}+\tau_{ij}^{R}\Big{)}+S_{i}+S_{1i}^{p},\ i=1,2,3 ;j=1,2,3 \tag{2}\]  
\[\frac{\partial\rho H}{\partial t}+\frac{\partial\rho u_{i}H}{\partial x_{i}}= \frac{\partial}{\partial x_{i}}\Big{[}u_{j}\Big{(}\tau_{ij}+\tau_{ij}^{R} \Big{)}+q_{i}\Big{]}+\frac{\partial\rho}{\partial t}+S_{i}u_{i}+S_{H}^{p}+Q_ {H} \tag{3}\]  
\[H=h(P,T,y)+\frac{u^{2}}{2}+k \tag{4}\]  
where \(u\) is the fluid velocity, \(\rho\) is the fluid density, and \(S_{i}\) is a mass-distributed external force per unit mass due to the porous media's resistance \(\Big{(}S_{i}^{porous}\Big{)}\), gravity \(\big{(}S_{i}^{gravity}=\rho g_{i}\), where \(g_{i}\) is the gravitational acceleration component along the \(i\)-th coordinate direction), and the coordinate system's rotation \(\big{(}S_{i}^{rotation}\big{)}\). \[S_{i}=S_{i}^{porous}+S_{i}^{gravity}+S_{i}^{rotation} \tag{5}\]  
\(H\) is the total enthalpy in the local reference frame; \(h(P,T,y)\) is the thermal enthalpy at a given pressure \((P)\), temperature \((T)\), and fluid mixture components \(y\); \(y=(y_{1}\dots y_{n})\) is the concentration vector of the fluid mixture components; \(k\) is the kinetic energy of turbulence; \(S_{M}^{p}\), \(S_{i}^{p}\), and \(S_{H}^{p}\) are additional interfacial exchange terms due to Euler-Lagrange particle interaction; \(Q_{H}\) is the heat source or sink per unit volume; \(\tau_{ij}\) is the viscous shear stress tensor; \(q_{i}\) is the diffusive heat flux. The energy equation for calculation with a high Mach number flow is as follows:  
\[\frac{\partial\rho E}{\partial t}+\frac{\partial\rho u_{i}\Big{(}E+\frac{p}{ \rho}\Big{)}}{\partial x_{i}}=\frac{\partial}{\partial x_{i}}\Big{[}u_{j} \Big{(}\tau_{ij}+\tau_{ij}^{R}\Big{)}+q_{i}\Big{]}-\tau_{ij}^{R}\frac{\partial u _{i}}{\partial x_{j}}+\rho\varepsilon+Q_{H} \tag{6}\]  
\[E=e\big{(}\rho,\ T,\ y\big{)}+\frac{u^{2}}{2} \tag{7}\]  
where \(e(\rho,\ T,\ y)\) is the integral energy at a given fluid density \((\rho)\), temperature \((\ T)\), and fluid mixture components \(y\). The viscous shear stress tensor for Newtonian fluids is as follows:  
\[\tau_{ij}=\mu\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{ \partial x_{i}}-\frac{2}{3}\delta_{ij}\frac{\partial u_{k}}{\partial x_{k}}\right) \tag{8}\]  
The Reynolds stress tensor according to the Boussinesq assumption is as follows:  
\[\tau_{ij}^{R}=\mu_{t}\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{ \partial u_{j}}{\partial x_{i}}-\frac{2}{3}\delta_{ij}\frac{\partial u_{k}}{ \partial x_{k}}\right)-\frac{2}{3}\rho k\delta_{ij} \tag{9}\]  
where \(\delta_{ij}\) is the Kronecker delta function, \(\mu\) is the dynamic viscosity, \(\mu_{t}\) is the turbulent eddy viscosity coefficient, and \(k\) is the turbulent kinetic energy.