# 基于有限体积法Godunov格式的管道泄漏检测模型研究Research on Pipeline Leakage Detection Model Based on Finite Volume Method Godunov Scheme

DOI: 10.12677/MOS.2018.73016, PDF, HTML, XML, 下载: 606  浏览: 1,645  国家自然科学基金支持

Abstract: Finite volume method with first order and second order Godunov schemes is presented to simulate and analyze the pipeline leakage problem. The water hammer equation is discrete by Riemann, and the MUSCL-Hancock method is used to reconstruct the two-order precision Godunov format. In order to avoid the false oscillation, the MINMOD slope limiter is introduced, and the leakage detection model is established by combining the leakage boundary conditions. The two-order Godunov scheme is consistent with the result of the characteristic line method, thus verifying the correctness of the model. The calculation analysis shows that the pressure curve of the leakage pipe is attenuated and deformed, while the pressure curve of the whole pipe does not attenuate and deform; the leakage flux affects the attenuation amplitude of the curve, and the greater the leakage flow, the greater the attenuation amplitude of the curve; the number of leakage points determines the number of inflection points at the peak curve, and the location of the leakage point determines the curve. According to the pressure curve of the leakage pipe, the leakage, the number of leakage points and the location of the leakage can be judged by the time of the inflection point of the shape and the first wave peak. This has important theoretical significance and practical guidance value for pipeline leakage monitoring.

1. 引言

2. 数学模型

2.1. 基本控制方程

$\frac{\partial u}{\partial t}+A\frac{\partial u}{\partial x}=s$ (1)

$\frac{\partial }{\partial t}{\int }_{i-1/2}^{i+1/2}udx+{f}_{i+1/2}-{f}_{i-1/2}={\int }_{i-1/2}^{i+1/2}sdx$ (2)

${f}_{i+1/2}={\stackrel{¯}{A}}_{i+1/2}{u}_{i+1/2}\left(t\right)=\frac{1}{2}{\stackrel{¯}{A}}_{i+1/2}\left\{\left(\begin{array}{cc}1& a/g\\ g/a& 1\end{array}\right){U}_{L}^{n}+\left(\begin{array}{cc}1& -a/g\\ -g/a& 1\end{array}\right){U}_{R}^{n}\right\}$ (3)

2.2. 二阶Godunov格式

${U}_{i}^{L}={U}_{i}^{n}-0.5\Delta xMINMOD\left({\sigma }_{j}^{n},{\sigma }_{j-1}^{n}\right)$ (4)

${U}_{i}^{R}={U}_{i}^{n}+0.5\Delta xMINMOD\left({\sigma }_{j}^{n},{\sigma }_{j-1}^{n}\right)$ (5)

${U}_{i}^{L*}={U}_{i}^{L}-\frac{1}{2}\frac{\Delta t}{\Delta x}\left[f\left({U}_{i}^{R}\right)-f\left({U}_{i}^{L}\right)\right]$ (6)

${U}_{i}^{R*}={U}_{i}^{R}-\frac{1}{2}\frac{\Delta t}{\Delta x}\left[f\left({U}_{i}^{R}\right)-f\left({\text{U}}_{i}^{L}\right)\right]$ (7)

${U}_{L}^{n}={U}_{i}^{R*},$ ${U}_{R}^{n}={U}_{i+1}^{L*}$ (8)

2.3. 泄漏边界条件

(9)

${H}_{leak}={H}_{i-1/2}={H}_{i+1/2}$ (10)

${H}_{i-1/2}+\frac{a}{g}{V}_{i-1/2}={H}_{i}^{n}+\frac{a}{g}{V}_{i}^{n}-\frac{f|{V}_{i}|{V}_{i}}{2gD}\Delta x$ (11)

${H}_{i-1/2}+\frac{a}{g}{V}_{i-1/2}={H}_{i}^{n}+\frac{a}{g}{V}_{i}^{n}-\frac{f|{V}_{i}|{V}_{i}}{2gD}\Delta x$ (12)

${V}_{leak}={C}_{d}{A}_{g}\sqrt{2g\left({H}_{leak}-{H}_{out}\right)}/{A}_{1}$ (13)

3. 算例分析

Figure 1. The schematic diagram of leakage unit

3.1. 完整管道与泄漏管道对比

3.2. 泄漏参数对曲线的影响

(a) 方案1 (b) 方案2 (c) 方案3 (d) 方案4

Figure 2. Pressure head traces at valve

4. 结论

1) 二阶Godunov格式与MOC方法对于无摩阻的泄漏管道数值模拟结果完全一致，从而验证二阶Godunov格式模拟管道泄漏是准确可行的。

2) 管道是否发生泄漏对压力曲线的影响差别较大，管道发生泄漏会令曲线产生衰减，其中泄漏流量影响曲线的衰减幅度，即泄漏流量越大曲线衰减幅度越大。

3) 管道发生泄漏会令曲线产生变形，其中可从峰值处产生的拐点个数判断管道存在的泄漏点数量，然后从曲线向内呈现“凹”形或者向外呈现“凸”形来判断泄漏的大概位置，最后根据第一波峰发生拐点的时间确定管道发生泄漏的具体位置。

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