# Experimental and Numerical Simulation of Water Hammer in Gravitational Pipe Flow with Continuous Air Entrainment

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Modeling and Meshing

#### 2.1. Governing Equations

#### 2.2. Discretization Model

#### 2.3. Meshing and Advancement

#### 2.4. Boundary Conditions

#### 2.4.1. Upstream Boundary Model

#### 2.4.2. Downstream Boundary Model

## 3. Experimental Measurement

#### 3.1. Steady Hydraulic Friction

#### 3.2. Valve Performance

#### 3.3. Water Hammer Measurements

## 4. Simulation and Analysis

#### 4.1. Simulation and Comparison

#### 4.2. Improvement to Approximate the Attenuation

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$g$ | = | acceleration of gravity (m/s^{2}) |

$h$ | = | pressure head (m) |

$x$ | = | distance along pipe from inlet (m) |

$t$ | = | time, as subscript to denote time (s) |

$v$ | = | flow velocity (m/s) |

$f$ | = | Darcy friction factor |

$D$ | = | main pipe diameter (m) |

$\alpha $ | = | the angle between pipe and the horizontal plane. |

$a$ | = | speed of pressure wave (m/s) |

$i$ | = | serial number of nodes (s) |

$\Delta t$ | = | time step (s) |

$n$ | = | number of sections |

$\Delta x$ | = | length of segment (m) |

${h}_{0}$ | = | head of upstream reservoir (m) |

${C}_{d}$ | = | discharge coefficient |

$Z$ | = | the height of reference plane |

${h}_{f}$ | = | head drop due to friction along the pipe (m) |

$\tau $ | = | valve opening ratio |

$K$ | = | volume modulus |

$\rho $ | = | density (kg/m^{3}) |

$E$ | = | elasticity modulus of steel pipe |

$e$ | = | thickness of steel pipe (m) |

$m$ | = | air content occupied in mixed fluid |

$R$ | = | thermodynamics constant (J/(mol∙K)) |

$T$ | = | temperature (K) |

$P$ | = | absolute pressure (pa) |

${f}^{\prime}$ | = | additional friction factor in two-phase flow |

$L$ | = | length of the pipe (m) |

$\varphi $ | = | additional friction factor after valve closed completely |

${t}_{\mathrm{c}}$ | = | time of valve closing operation |

$C$ | = | coefficient in additional friction function |

$\nu $ | = | kinematic viscosity of water (m^{2}/s) |

${R}_{\mathrm{p}}$ | = | hydraulic radius (m) |

${v}_{\mathrm{c}}$ | = | the critical flow velocity between laminar and turbulent flow (m/s) |

$r$ | = | residuals between experimental and simulated peaks (m) |

$S$ | = | the sum of squared residuals |

${h}_{\mathrm{pe}}$ | = | experimental positive peak (m) |

${h}_{\mathrm{ve}}$ | = | experimental negative peak (m) |

${h}_{\mathrm{p}}$ | = | experimental positive peak (m) |

${h}_{\mathrm{v}}$ | = | experimental negative peak (m) |

## Acronyms

MOC | = | Method of Characteristics |

LWM | = | Lax-Wendroff Method |

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Periods | Experimental | Original | Improved | |||||||
---|---|---|---|---|---|---|---|---|---|---|

LM | TM | CM | ||||||||

${\mathit{h}}_{\mathbf{p}\mathbf{e}}$ | ${\mathit{h}}_{\mathbf{v}\mathbf{e}}$ | ${\mathit{h}}_{\mathbf{p}}$ | ${\mathit{h}}_{\mathbf{v}}$ | ${\mathit{h}}_{\mathbf{p}}$ | ${\mathit{h}}_{\mathbf{v}}$ | ${\mathit{h}}_{\mathbf{p}}$ | ${\mathit{h}}_{\mathbf{v}}$ | ${\mathit{h}}_{\mathbf{p}}$ | ${\mathit{h}}_{\mathbf{v}}$ | |

1st | 4.77 | 0.83 | 4.96 | −0.38 | 4.93 | 0.66 | 4.94 | 1.01 | 4.93 | 0.66 |

2nd | 3.50 | 1.53 | 4.84 | −0.27 | 3.22 | 1.68 | 3.08 | 1.65 | 3.23 | 1.68 |

3rd | 2.75 | 1.95 | 4.74 | −0.17 | 2.61 | 2.04 | 2.74 | 1.86 | 2.62 | 2.04 |

4th | 2.41 | 2.16 | 4.65 | −0.09 | 2.39 | 2.18 | 2.60 | 1.96 | 2.39 | 2.18 |

5th | 2.30 | 2.23 | 4.56 | 0.00 | 2.31 | 2.23 | 2.52 | 2.02 | 2.31 | 2.23 |

Case | Wave Speed (m∙s^{−1}) | m | Air Content (%) | Closing Time (s) |
---|---|---|---|---|

#1 | 183 | 0.139 | 0.308 | 2.8 |

#2 | 175 | 0.152 | 0.338 | 2.6 |

#3 | 132 | 0.269 | 0.598 | 3.0 |

#4 | 110 | 0.388 | 0.864 | 3.2 |

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**MDPI and ACS Style**

Zhang, B.; Wan, W.; Shi, M.
Experimental and Numerical Simulation of Water Hammer in Gravitational Pipe Flow with Continuous Air Entrainment. *Water* **2018**, *10*, 928.
https://doi.org/10.3390/w10070928

**AMA Style**

Zhang B, Wan W, Shi M.
Experimental and Numerical Simulation of Water Hammer in Gravitational Pipe Flow with Continuous Air Entrainment. *Water*. 2018; 10(7):928.
https://doi.org/10.3390/w10070928

**Chicago/Turabian Style**

Zhang, Boran, Wuyi Wan, and Mengshan Shi.
2018. "Experimental and Numerical Simulation of Water Hammer in Gravitational Pipe Flow with Continuous Air Entrainment" *Water* 10, no. 7: 928.
https://doi.org/10.3390/w10070928