# Mechanical Interaction in Pressurized Pipe Systems: Experiments and Numerical Models

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Study

#### Laboratory Facility

_{E}= 0.4 s. A sudden change in pressure occurs in PT2, leading to an overpressure after 0.93 s (Figure 2).

_{E}, the pipe reaches the maximum displacement in the x-axis of 3.14 m/s

^{2}(i.e., in longitudinal direction of the pipe) and 2.2 m/s

^{2}in the y-axis. In the z-axis (vertical), the structure vibrates but practically maintains in plan XY, due to the roller supports that allow movements in this plan. Apart from the upstream (i.e., Section 1) and downstream (i.e., ball valve in Section 3) sections of the system, that are fixed to concrete supports, all the remaining supports (i.e., roller) allow movements only in the pipe direction (i.e., x or y, depending on the pipe orientation). Thus, the restrictions imposed by the roller supports change according to the direction of the pipes. After T

_{E}, the structure moves significantly in y-axis, reaching its maximum displacement (2.72 m/s

^{2}) at t = 0.67 s. Compared to the pressure variation, the accelerometer is affected immediately by the wave pressure coming from the ball valve towards the downstream end of the PVC pipe. At t = T

_{E}, a symmetry can be noticed, for about 0.1 s, in x-axis, coinciding with the instant the pressure wave reaches the downstream end and travels downstream towards the ball valve (elastic reflection time). During the relief wave, the pipe represents a harmonic behavior in y-axis until the end of time.

## 3. Methods

#### 3.1. Modified Method of Characteristic Model with Non-Elastic Effects

^{+}and C

^{−}compatibility relations, respectively.

Description | Parameter/Coefficient | Value |
---|---|---|

Upstream head (m) | H | 14.4 |

Discharge (m^{3}/s) | Q | 0.009 |

Wave speed (m/s) | $c$ | 350 |

Time closure of the ball valve (s) | t_{f} | 0.20 |

Total simulation time (s) | tt | 3 |

Head coefficient induced by a discharge variation by non-elastic fluid and pipe deformation (--) | KH | 0.32 |

Discharge coefficient induced by a head variation, due to a non-elastic response in the recuperation phase of the deformation (--) | KQ | 2.9 |

Support Conditions | Non-Dimensional Parameter ($\psi $) |
---|---|

Pipe with frequent expansion joints | $1$ |

Pipe against longitudinal movement throughout its length | $1-{\text{\mu}}^{2}$ |

Pipe against longitudinal movement at the upper end | $1-\frac{\text{\mu}}{2}$ |

Material | PVC | ||
---|---|---|---|

Di (m) | 0.107 | ||

e (m) | 0.0035 | ||

E (GPa) | 2.98 | ||

K (N/m^{2}) | 2.19 × 10^{9} | ||

$\text{\rho}$ (kg/m^{3}) | 1000 | ||

$\text{\upsilon}$ (--) | 0.46 | ||

$\text{\psi}$ | 1 | 0.79 | 0.77 |

c (m/s) | 300 | 350 | 350 |

wave speed adopted (m/s) | 350 |

#### 3.2. FSI Modeling and Solution

#### 3.2.1. Basic Concepts

#### 3.2.2. Geometry and Mesh Adaptation

#### 3.2.3. Boundary Conditions

_{0}and z

_{1}(see detail in Figure 11).

## 4. Results

#### 4.1. Velocity Profiles

_{E}(elastic reflection time), the inside flow is almost stopped and the unsteady friction effect can be neglected. This can also be noticed in Figure 19, where the connection between the closure of the valve and the velocity streamlines are shown. This figure illustrates the velocity vectors in the pipe system from the ball valve to upstream (Section 2 of Figure 14).

#### 4.2. Wave Propagation and Pipe Displacements

**Figure 21.**Fluid velocity (streamlines) and pipe displacement (contours) fields during the first wave period: (

**a**) t = 0.50 s; (

**b**) t = 0.55 s; (

**c**) t = 0.60 s; and (

**d**) t = 0.96 s.

#### 4.3. Deformation Gradient

**Figure 24.**Maximum principal stretches (undeformed, on the left) and strains (deformed, on the right): (

**a**) vertical direction and (

**b**) horizontal direction.

#### 4.4. Stress/Strain Response

**Figure 25.**Different strain responses to a constant load: (

**a**) Hook’s law (instantaneous elastic deformation); (

**b**) viscous behavior (deformation is delayed); and (

**c**) real elastic recovery (time-dependent strain).

## 5. Discussion

**Figure 27.**Pressure variation between experimental tests and simulations: (

**a**) for PT2 and (

**b**) for PT1.

Parameter | Experimental Data | MOC Model | CFD Model |
---|---|---|---|

${P}_{int}^{max}\text{}\left(PT1\right)$ (m) | 18.2 | 16.0 | 17.9 |

${P}_{int}^{max}\text{}\left(PT2\right)$ (m) | 17.5 | 17.0 | 19.0 |

${\delta}_{t}^{max}$ (m) | 0.063 | -- | 0.087 |

${\delta}_{ay}^{max}$ (m) | 0.037 | -- | 0.051 |

${\delta}_{ax}^{max}$ (m) | 0.053 | -- | 0.071 |

Relative error (of pressure) | 12% | 8% |

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Simão, M.; Mora-Rodriguez, J.; Ramos, H.M.
Mechanical Interaction in Pressurized Pipe Systems: Experiments and Numerical Models. *Water* **2015**, *7*, 6321-6350.
https://doi.org/10.3390/w7116321

**AMA Style**

Simão M, Mora-Rodriguez J, Ramos HM.
Mechanical Interaction in Pressurized Pipe Systems: Experiments and Numerical Models. *Water*. 2015; 7(11):6321-6350.
https://doi.org/10.3390/w7116321

**Chicago/Turabian Style**

Simão, Mariana, Jesus Mora-Rodriguez, and Helena M. Ramos.
2015. "Mechanical Interaction in Pressurized Pipe Systems: Experiments and Numerical Models" *Water* 7, no. 11: 6321-6350.
https://doi.org/10.3390/w7116321