﻿ 管道中滞留气团排放过程的二维数值模拟研究

# 管道中滞留气团排放过程的二维数值模拟研究A 2-D Numerical Simulation Study on the Process of Air Mass Emission in Pipeline

Abstract: Under severe rainstorm conditions, transient flow of trapped air masses often occurs in urban storm water drainage pipelines. In this paper, VOF (Volume of Fluid Model) model and k-ε turbu-lence model are used to simulate and calculate the dynamic process of the movement and emission of the stranded gas mass under the action of pressure flow. The simulation results are compared with the experimental results. The results show that the motion process and transient pressure variation of the trapped air masses simulated by the VOF model agree well with the experimental results. At the same time, the two simplified methods, such as “equal section ratio” and “equal diameter ratio”, are compared and analyzed in two-dimensional simulation. The comparison results show that the pressure result calculated by the “equal section ratio” method is closer to the experimental value.

1. 引言

2. 数学模型

2.1. 控制方程

$\frac{\partial \rho }{\partial t}+\nabla \cdot \left(\rho v\right)=0$ (1)

$\frac{\partial \rho {u}_{i}}{\partial t}+\frac{\partial \rho {u}_{i}{u}_{j}}{\partial {x}_{j}}=\frac{\partial p}{\partial {x}_{i}}+\frac{\partial {\tau }_{ij}}{\partial {x}_{j}}+\rho {g}_{i}$ (2)

$\frac{\partial {\alpha }_{w}}{\partial t}+v\nabla \cdot {\alpha }_{w}=0$ (3)

$\frac{\partial \rho E}{\partial t}+\frac{\partial }{\partial {x}_{j}}\left[{u}_{i}\left(\rho E+p\right)\right]=\frac{\partial }{\partial {x}_{i}}\left({k}_{eff}\frac{\partial T}{\partial {x}_{i}}\right)$ (4)

$\frac{\partial }{\partial t}\left({\alpha }_{w}{\rho }_{w}\right)+\nabla \cdot \left({\alpha }_{w}{\rho }_{w}U\right)=0$ (5)

${\alpha }_{w}+{\alpha }_{a}=1$ (6)

$\rho =\left({\alpha }_{w}{\rho }_{w}\right)+\left(1-{\alpha }_{w}\right){\rho }_{a}$ (7)

$\left\{\begin{array}{l}\frac{\partial \rho k}{\partial t}+\frac{\partial \rho k{u}_{i}}{\partial {x}_{i}}=\frac{\partial }{\partial {x}_{j}}\left[\left(\mu +\frac{{\mu }_{t}}{{\sigma }_{k}}\right)\frac{\partial k}{\partial {x}_{j}}\right]+{G}_{k}+{G}_{b}+\rho \epsilon -{Y}_{M}\\ \frac{\partial \rho \epsilon }{\partial t}+\frac{\partial \rho \epsilon {u}_{i}}{\partial {x}_{i}}=\frac{\partial }{\partial {x}_{j}}\left[\left(\mu +\frac{{\mu }_{t}}{{\sigma }_{\epsilon }}\right)\frac{\partial \epsilon }{\partial {x}_{j}}\right]+{C}_{1\epsilon }\frac{\epsilon }{k}\left({G}_{k}+{C}_{3\epsilon }{G}_{b}\right)-{C}_{2\epsilon }\frac{\rho {\epsilon }^{2}}{k}\end{array}$ (8)

${\mu }_{t}=\rho {C}_{\mu }\frac{{k}^{2}}{\epsilon }$ (9)

2.2. 求解方法

2.3. 边界条件

1) 进口边界条件设置。进水口断面为压力进口，在实验过程中为系统提供稳定的水压。

2) 出口边界条件设置。出口边界为压力出口，相对压力为零。

3) 管道壁条件设置。壁面为光滑壁面，粗糙度为0。

3. 二维建模

3.1. 算例介绍

3.2. 二维建模的简化方法

$\frac{a}{b}=\frac{d}{D}$ (10)

$\frac{ax}{bx}=\frac{\text{π}{d}^{2}/4}{\text{π}{D}^{2}/4}⇒\frac{a}{b}=\frac{{d}^{2}}{{D}^{2}}$ (11)

3.3. 二维模型

Figure 1. Physical experimental apparatus

Figure 2. Model grid

Table 1. Four calculation model parameters

4. 计算结果及数据分析

4.1. 滞留气团排放过程的动态分析

Figure 3. Working condition (Bd) simplified gas-liquid phase cloud diagram with constant diameter ratio

4.2. 瞬态压力分析

4.3. 两种简化方法对比及分析

$\eta ={d}^{2}/{D}^{2}$ (12)

$\mu =\frac{d\ast h}{D\ast h}=d/D$ (13)

Figure 4. Test data of pressure sensor #1 in working condition A

Figure 5. Test data of pressure sensor #2 in working condition A

Figure 6. Test data of pressure sensor #1 in working condition B

Figure 7. Test data of pressure sensor #2 in working condition A

5. 结论

1) 在二维建模时，“等直径比例”与“等截面比例”两种简化方法的对比表明：可以按实际管道的截面比例建立模型，该模型可以有效的控制模型中横竖管的流通能力，令模拟计算结果更接近实际值。按现实管道的直径比例建立的模型，会放大细管截面积，增大细管流通能力，从而影响计算结果。

2) VOF模型可用于模拟滞留气团的运动过程、瞬态压力变化结果等气水动态行为和瞬变压力，模拟结果完全满足研究需要，并且具有一定的参考价值。

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