# Simulation and Experimental Research on the Energy Loss of Confluence Pipelines
## 2 Numerical Simulation of the Flow Field of a Confluence Pipeline
### Simulation Setup and Result Analysis  
The inlet-outlet configurations of confluence pipelines with different shapes are shown in Figure 2. For the T-shaped pipeline, the left and right sides are water inlets, the vertical direction is a water outlet, and the cross-sectional areas of the inlet and outlet are equal. For the Y-shaped confluence pipeline, the left and right sides are water inlets, the vertical direction is a water outlet, the cross-sectional areas of the inlet and outlet are equal, and the angle between the two water inlets is 45\({}^{\circ}\). The vertical upper end and the right branch pipe of the bifurcated confluence pipeline are the water inlets, and the vertical lower end is the water outlet, as shown by the arrows. The left and right sides of the arc confluence pipeline are water inlets, and the vertical direction is a water outlet; the inlet and outlet cross-sectional areas are equal. In the arc bending stage, the angle is 90\({}^{\circ}\). For the manifold tubes of different structures studied in this paper, the same boundary conditions are adopted, that is, the boundary conditions of velocity inlet are adopted at inlet 1 and inlet 2, the boundary conditions of pressure outlet are adopted, as shown as red arrows, and the wall surface is non-slip wall, as shown as blue dotted lines. The RNG turbulence model was selected. By correcting the turbulent dynamic viscosity coefficient and adding the term of turbulent energy dissipation rate, the accuracy of the turbulent energy dissipation rate equation has been effectively improved. The SIMPLE algorithm, due to its ability to handle pressure-velocity coupling, iterative solutions, and suitability for complex flow problems, makes it an appropriate choice for solving turbulent flow models in a confluence pipe. The relevant parameters of the numerical simulation were set according to the environmental conditions in the actual measurement process. Under the local atmospheric pressure (about 101.3 kPa), the liquid used in the experiment was water, the temperature was roughly 23 \({}^{\circ}\)C, and the density was roughly 997.54 kg/m\({}^{3}\). In addition, the gravity acceleration was roughly 9.8035 m/s\({}^{2}\). The inner surface roughness of the arc confluence pipeline made of polymethyl methacrylate referenced in this paper was measured using a roughness measurement instrument, and the measurement result is 0.149 \(\upmu\)m.  
Figure 2: Schematic diagram of the structure of confluence pipelines with different shapes. (**a**) T shape; (**b**) Y shape; (**c**) bifurcated; and (**d**) arc shape. The unstructured grid was generated using the grid generation software ICEM 2022 R2. Compared with the division of structured grids, the structure of unstructured grids is complex and the number of grids generated is larger. The results can be more similar to the real flow state of the fluid. The results of unstructured grid division of different structure bus tubes are shown in Figure 3. Based on the conclusions of the study, grid independence in the RNG turbulence model is verified. To determine the flow in the confluence pipeline, the energy loss coefficient of different structures is calculated according to the pressure difference between the inlet and outlet according to [PERSON]'s equation [29], as shown in Equation (15):  
\[\frac{p_{1}}{\rho}+z_{1}+\alpha\frac{v_{1}^{2}}{2g}=\frac{p_{2}}{\rho}+z_{2}+ \alpha\frac{v_{2}^{2}}{2g}+h_{w} \tag{15}\]  
where \(p_{1}\), \(z_{1}\), and \(v_{1}\) are the average pressure, average position head, and average velocity of the section at the inlet of the confluence pipeline; \(p_{2}\), \(z_{2}\), and \(v_{2}\) refer to the average pressure, average position head, and average velocity of the section at the outlet of the confluence pipeline; \(\alpha\) is the correction coefficient ( generally, \(\alpha\) = 1 is selected); and \(h_{w}\) is the head loss between the inlet and outlet sections.