# Simulation and Experimental Research on the Energy Loss of Confluence Pipelines
## 2 Numerical Simulation of the Flow Field of a Confluence Pipeline
### Turbulent Flow Modeling of Bus-Tube  
According to the knowledge of fluid mechanics, the fluid in the tube needs to satisfy the law of conservation of mass, that is, satisfy the continuity equation in mathematical form. According to the definition of conservation of mass, the net mass flowing into a tiny cell in unit time is equal to the amount of mass added in the tiny cell in unit time. It can be concluded that the general continuity equation in the circular tube is as follows:  
\[\frac{\partial\rho}{\partial t}+div(\rho\mathbf{u})=0 \tag{1}\]  
If the fluid medium in the pipeline is an incompressible fluid, \(\rho\), it is equal to the constant, so the incompressible fluid equation can be written as Equation (2):  
\[\rho[\frac{\partial u_{x}}{\partial x}+\frac{\partial u_{y}}{\partial y}+\frac{ \partial u_{z}}{\partial z}]=0 \tag{2}\]  
where \(\rho\) is the density, \(t\) is the time, and \(\mathbf{u}\) is the velocity vector of the fluid particle. The turbulent flow in the bus pipe is one of many fluid flow states, and since the medium of the bus pipe is water, it also needs to satisfy the incompressible viscous fluid equation. Therefore, the basic equation of incompressible viscous turbulent flow in a single medium is Equations (3)-(5):  
\[X-\frac{1}{\rho}\frac{\partial P}{\partial x}+\frac{\mu}{\rho}(\frac{\partial^ {2}u_{x}}{\partial x^{2}}+\frac{\partial^{2}u_{x}}{\partial y^{2}}+\frac{ \partial^{2}u_{x}}{\partial z^{2}})=\frac{du_{x}}{dt} \tag{3}\]  
\[Y-\frac{1}{\rho}\frac{\partial P}{\partial y}+\frac{\mu}{\rho}(\frac{\partial^ {2}u_{y}}{\partial x^{2}}+\frac{\partial^{2}u_{y}}{\partial y^{2}}+\frac{ \partial^{2}u_{y}}{\partial z^{2}})=\frac{du_{y}}{dt} \tag{4}\]  
\[Z-\frac{1}{\rho}\frac{\partial P}{\partial z}+\frac{\mu}{\rho}(\frac{\partial^ {2}u_{z}}{\partial x^{2}}+\frac{\partial^{2}u_{z}}{\partial y^{2}}+\frac{ \partial^{2}u_{z}}{\partial z^{2}})=\frac{du_{z}}{dt} \tag{5}\]  
where \(P\) is the pressure on the fluid element body, \(\mu\) is the dynamic viscosity, and \(X\), \(Y\), \(Z\) is the generalized source term of the momentum conservation equation, which is shown in Equations (6)-(8):  
\[\frac{du_{x}}{dt}=\frac{\partial u_{x}}{\partial t}+\frac{\partial({u_{x}}^{2 })}{\partial x}+\frac{\partial(u_{x}u_{y})}{\partial y}+\frac{\partial(u_{x} u_{z})}{\partial z} \tag{6}\]  
\[\frac{du_{y}}{dt}=\frac{\partial u_{y}}{\partial t}+\frac{\partial({u_{y}}^{ 2})}{\partial x}+\frac{\partial(u_{x}u_{y})}{\partial y}+\frac{\partial(u_{y} u_{z})}{\partial z} \tag{7}\]  
\[\frac{du_{z}}{dt}=\frac{\partial u_{z}}{\partial t}+\frac{\partial({u_{z}}^{ 2})}{\partial x}+\frac{\partial(u_{x}u_{z})}{\partial y}+\frac{\partial(u_{y }u_{z})}{\partial z} \tag{8}\]  
This is the momentum conservation equation for viscous incompressible fluids, also known as the Navier-Stokes (N-S) equation. The component form of the N-S equation in cartesian coordinates is shown in Equation (9):  
\[\rho\frac{\partial u_{i}}{\partial t}+\rho u_{j}\frac{\partial u_{i}}{\partial x _{j}}=\rho f_{i}-\frac{\partial p}{\partial x_{i}}+\mu\frac{\partial^{2}u_{ i}}{\partial x_{j}\partial x_{j}} \tag{9}\]  
where \(i,j\) equals, \(X\), \(Y\), \(Z\) and \(f\) is mass force. The basic equations of viscous fluid turbulence are continuity equation and motion equation, namely the N-S equation. Assuming that the turbulence is traversed by each state, the instantaneous value is substituted for the statistical average, and Equation (9) is substituted by Equation (10). \[u=\overline{u}+u\prime \tag{10}\]  
where \(\overline{u}\) is the time-average component of the velocity and \(u^{\prime}\) is the pulsating component of the velocity; taking the time average of the continuous equation, the continuous Equation (11) of the time-average flow is obtained:  
\[\rho[\frac{\partial u_{x}}{\partial x}+\frac{\partial u_{y}}{\partial y}+\frac{ \partial u_{z}}{\partial z}]=0 \tag{2}\]0Equation (12) is subtracted from Equation (2) to obtain the continuous equation of pulsating flow.