# Numerical Investigation of the Influence of Water Jumping on the Local Scour beneath a Pipeline under Steady Flow

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## Abstract

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## 1. Introduction

## 2. Numerical Method

#### 2.1. Flow Model

_{γ}is the curvature of the interface, σ

_{T}is the surface tension constant, and γ is the volume fraction of fluid.

#### 2.2. Turbulence Closure Model

_{w}is the volume rate of water, ρ

_{w}is the density of water, ρ

_{a}is the density of air, k is the turbulence kinetic energy, ε is the dissipation rate of turbulence, C

_{1ε}= 1.44 , C

_{2ε}= 1.99, σ

_{ε}= 1.3 and μ

_{t}is turbulence viscosity.

#### 2.3. Sediment Transport Equations

#### 2.3.1. Bed Load Transport Equations

_{b}is the bed load transport rate per unit width; θ

_{b}is the shields number; g is gravitational acceleration; d is the median grain diameter; s is the relative density of sediment; and θ

_{cr}is the critical shields number, which can also be calculated by Equation (11):

_{cro}is the critical shields number for a flat bed; φ is the angle between the bed’s steepest slope and the bed’s shear stress direction; β is the angle between the bed and horizontal plane; μ

_{s}is the static friction coefficient; and D

_{*}is the dimensionless grain size.

#### 2.3.2. Suspended Load Transport Equations

_{s}is sediment fall velocity in still water; and ν

_{t}is diffusivity of the sediment, which equals turbulence eddy viscosity in the present work.

#### 2.3.3. Boundary Condition

_{b}is the sediment concentrate at a reference level, ρ

_{s}is the density of sand; Δ

_{b}is the distance from the reference level to the bed, T is the non-dimensional excess shear stress, which can be calculated by Equation (16).

_{sc}is the efficiency factor current, τ

_{sc}is the bed shear stress caused by current, and τ

_{cr}is the critical bed shear stress. Other details about these parameters can be found in the user manual of Delft3D-Flow [26].

#### 2.4. Scour Model

_{b}is the bed load transport rate; and D is the deposition rate; E is the erosion rate.

_{b}is the sediment concentration near the bed at a specific location, η is the vector direction that is perpendicular to the bed surface.

#### 2.5. Dynamic Mesh Model

_{1}is the displacement field of mesh points; γ is the diffusion coefficient, which is used to control the mesh motion; the inverse distance diffusion coefficient is chosen in this paper, which is more suitable for scour simulation.

## 3. Numerical Method

#### 3.1. Numerical Schemes of the Flow Model

#### 3.2. Numerical Schemes for Sediment Transport and Scour Model

## 4. Model Validation

#### 4.1. Bed Shear Stress

_{b}should be computed before the friction velocity u

_{*}is calculated.

^{−6}, and the dynamic viscosity coefficient ν is derived from the computed result of the turbulence model:

_{0}× sin (w × t + Q); U

_{0}is the maximum inlet flow velocity (0.32 m/s), ω is the frequency oscillatory flow that can be calculated by ω = 2π/T, T is the period of oscillation flow, i.e., 2.7 s, and Q is the initial phase of model, which is 0 in this model. The Reynolds number is 37,000, which is in the turbulence regime; therefore, the standard k-e model is used. The efficiency roughness height in the model is 2.5 times the sand diameter, which is around 0.0005 m. The bed friction velocity at x = 5 m is collected for comparing with the measured data, and the comparison result is shown in Figure 3.

#### 4.2. Suspended Load Model

#### 4.2.1. Zero Entrainment Experiment

#### 4.2.2. Net Entrainment Experiment

_{50}, which is approximately 0.005 m. Therefore, the concentration at reference height C

_{b}is 4.6 kg/m

^{3}, which is larger than Liang’s [19] result. In OpenFOAM, the basic cell is the mesh volume cell and variations like velocity, pressure, concentration and so on, are stored at the center of the volume. The mesh size in the vertical direction near the bed is 0.01 m, which means the concentration at the mesh center is C

_{b}. In this simulation, the scour model is not included. As for the flow model, the k-epsilon closure model is used to simulate the turbulence and the VOF method is used to capture the water surface. For the turbulence to be fully developed, the length of the rigid bed and loose bed is 20H and 60H, respectively. The suspended concentration of inlet is zero for the clear water condition. The concentration profile at x = 4H, 10H, 20H and 40H is abstracted to validate the simulated result, and x = 0 is the location of the beginning of the loose bed. Figure 7 shows that the sediment concentration profile is parabolic. It can be seen that the sediment concentration decreases rapidly with the height distance from the bed surface, and it becomes a parabolic profiles. In a word, the modeled result matches well with the experimental data.

#### 4.3. Local Scour under a Pipeline

## 5. Discussion

#### 5.1. The Channel without a Pipeline

#### 5.2. The Channel with a Pipeline

#### 5.2.1. Validation of the Free Surface

**/**v is 1.92 × 10

^{5}, and the Froude number Fr = 0.373. Figure 12 shows the comparison between the modeled and measured free surface profiles. The change in free surface elevation above the half cylinder is well predicted by the numerical model.

#### 5.2.2. The Surface Elevation and Bed Shear Stress

#### 5.2.3. The Scour Hole

## 6. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 5.**The model result of the zero erosion experiment (

**a**) x = 0H; (

**b**) x = 4.65H; (

**c**) x = 9.3H; (

**d**) x = 20.9H; (

**e**) x = 37.44H; (

**f**) x = 74.2H.

**Figure 7.**Sediment concentration profile in specified sections (

**a**) x = 4H; (

**b**) x = 10H; (

**c**) x = 20H; (

**d**) x = 40H.

**Figure 9.**The scour profile at different times, (

**a**) T = 10 min; (

**b**) T = 30 min; (

**c**) T = 200 min; (

**d**) T = 370 min.

**Figure 11.**The correlation of the free surface modeled data and the rigid lid model (the circle is the modeled data, the line is the fitted line); (

**a**) Fr = 0.1; (

**b**) Fr = 0.2; (

**c**) Fr = 0.3.

**Figure 14.**The distribution of dimensionless friction velocity under the pipeline at the initial moment.

**Figure 15.**The scour profile of the free surface model and rigid lid assumption at different times in the case of Fr = 0.187, (

**a**) T = 600 s; (

**b**) T = 1800 s; (

**c**) T = 12,000 s.

**Figure 16.**The scour profile of the free surface model and rigid lid assumption at different times in the case of Fr = 0.214, (

**a**) T = 390 s; (

**b**) T = 1890 s; (

**c**) T = 4860 s.

**Figure 17.**The scour profile of the free surface model and rigid lid assumption at different times in the case of Fr = 0.267, (

**a**) T = 90 s; (

**b**) T = 300 s; (

**c**) T = 1800 s.

Location | X = 0D | X = 2D | |
---|---|---|---|

Cases | |||

Experiment data | 0.038 | 0.011 | |

Case1 (mesh size 1 cm) | 0.0326 | 0.0073 | |

Case2 (mesh size 5 mm) | 0.0349 | 0.0089 | |

Case3 (mesh size 1 mm) | 0.0364 | 0.0100 | |

Case4 (mesh size 0.5 mm) | 0.0372 | 0.0105 | |

Case5 (mesh size 0.1 mm) | 0.0374 | 0.0106 |

Cases | Fr = 0.1 | Fr = 0.2 | Fr = 0.3 | |||
---|---|---|---|---|---|---|

H (m) | U (m/s) | Re (10^{4}) | U (m/s) | Re (10^{4}) | U (m/s) | Re (10^{4}) |

0.1 | 0.1 | 1.0 | 0.2 | 4 | 0.3 | 9 |

0.225 | 0.15 | 1.5 | 0.3 | 6 | 0.45 | 13.5 |

0.4 | 0.2 | 2.0 | 0.4 | 8 | 0.6 | 19 |

0.625 | 0.25 | 2.5 | 0.5 | 10 | 0.75 | 22.5 |

0.9 | 0.3 | 3.0 | 0.6 | 12 | 0.9 | 27 |

1.225 | 0.35 | 3.5 | 0.7 | 14 | 1.05 | 31.5 |

1.6 | 0.4 | 4.0 | 0.8 | 16 | 1.2 | 36 |

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**MDPI and ACS Style**

Fan, F.; Liang, B.; Li, Y.; Bai, Y.; Zhu, Y.; Zhu, Z. Numerical Investigation of the Influence of Water Jumping on the Local Scour beneath a Pipeline under Steady Flow. *Water* **2017**, *9*, 642.
https://doi.org/10.3390/w9090642

**AMA Style**

Fan F, Liang B, Li Y, Bai Y, Zhu Y, Zhu Z. Numerical Investigation of the Influence of Water Jumping on the Local Scour beneath a Pipeline under Steady Flow. *Water*. 2017; 9(9):642.
https://doi.org/10.3390/w9090642

**Chicago/Turabian Style**

Fan, Fei, Bingchen Liang, Yaru Li, Yuchuan Bai, Yanjun Zhu, and Zhixia Zhu. 2017. "Numerical Investigation of the Influence of Water Jumping on the Local Scour beneath a Pipeline under Steady Flow" *Water* 9, no. 9: 642.
https://doi.org/10.3390/w9090642