# Positive Surge Propagation in Sloping Channels

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## Abstract

**:**

## 1. Introduction

## 2. The Model

#### An Approximate Analytical Solution

## 3. Numerical and Laboratory Experiments

#### 3.1. Laboratory Setup and Experimental Procedures

#### 3.2. The Numerical Model

## 4. Discussion

#### 4.1. Comparison between the 0D Model Prediction and the Laboratory Experiments

#### 4.2. Comparison between Model Prediction and the Numerical Experiments

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Numerical Solution of Model Equations

## References

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**Figure 1.**Schematic of the surge propagating upstream against a uniform flow of depth ${Y}_{0}$ and velocity ${U}_{0}$, with notation.

**Figure 2.**Plot of $\Phi \left({F}_{0}\right)$ as a function of ${F}_{0}$. White circles denote the results of the 0D model, the black line is the approximation (10). The inset shows the same plot, with $\Phi \left({F}_{0}\right)$ on the log-scale.

**Figure 3.**(

**a**) coefficients ${k}_{1}$ and ${k}_{2}$ as they vary with ${F}_{0}$ (symbols denote the numerical solution of the 0D model, full lines are the approximation given by Equation (12)); (

**b**) wave front velocity, a, and height, $\Delta Y$, and the position of the front, L, as a function of time for ${F}_{0}=0.65$ (full lines denote the numerical solution of the 0D model, dashed lines denote the analytical solution); (

**c**) same as (

**b**) for ${F}_{0}=0.60$.

**Figure 4.**Free surface elevation, h, measured by the five ultrasonic probes. (

**a**) run 20 (s = 0.001, ${Y}_{0}$ = 0.073 m, ${U}_{0}$ = 0.50 m/s, ${F}_{0}$ = 0.59); (

**b**) run 33 (s = 0.0052, ${Y}_{0}$ = 0.011 m, ${U}_{0}$ = 0.43 m/s, ${F}_{0}$ = 1.31).

**Figure 5.**Dimensionless time variation of the flow depth during the surge propagation: red and black lines denote the 0D model and experimental results, respectively; each curve is offset vertically by 0.2. (

**a**) run 3 (s = 0.0002, ${Y}_{0}$ = 0.084 m, ${U}_{0}$ = 0.34 m/s, ${F}_{0}$ = 0.37); (

**b**) run 12 (s = 0.0005, ${Y}_{0}$ = 0.102 m, ${U}_{0}$ = 0.47 m/s, ${F}_{0}$ = 0.47); (

**c**) run 22 (s = 0.010, ${Y}_{0}$ = 0.058 m, ${U}_{0}$ = 0.46 m/s, ${F}_{0}$ = 0.61); (

**d**) run 32 (s = 0.0022, ${Y}_{0}$ = 0.085 m, ${U}_{0}$ = 0.81 m/s, ${F}_{0}$ = 0.89); (

**e**) run 33 (s = 0.0052, ${Y}_{0}$ = 0.011 m, ${U}_{0}$ = 0.43 m/s, ${F}_{0}$ = 1.31); (

**f**) run 37 (s = 0.0052, ${Y}_{0}$ = 0.032 m, ${U}_{0}$ = 0.81 m/s, ${F}_{0}$ = 1.45).

**Figure 6.**Relative maximum depth at the first wave crest, ${Y}_{max}/{Y}_{0}$, as a function of surge Froude number, ${F}_{S}$. The black curve denotes the conjugate depths ratio as given by Equation (14); the dashed curve is from Equation (15), the dot-dashed line denotes the breaking criterion for a solitary wave $({Y}_{max}-{Y}_{0})/{Y}_{0}$ = 0.78.

**Figure 7.**Comparison between the 0D model (black solid lines) and the numerical solution of the Shallow Water Equations using the Finite Volumes model (red dashed lines), in terms of dimensionless position of the front to dimensionless time (

**a**–

**e**), and dimensionless front height (

**f**–

**h**) and surge celerity (

**i**–

**m**) to dimensionless position of the front; run 3 (s = 0.0002, ${Y}_{0}$ = 0.084 m, ${U}_{0}$ = 0.34 m/s, ${F}_{0}$ = 0.37), run 12 (s = 0.0005, ${Y}_{0}$ = 0.102 m, ${U}_{0}$ = 0.47 m/s, ${F}_{0}$ = 0.47), run 22 (s = 0.001, ${Y}_{0}$ = 0.058 m, ${U}_{0}$ = 0.46 m/s, ${F}_{0}$ = 0.61), Run 30 (s = 0.0022, ${Y}_{0}$ = 0.037 m, ${U}_{0}$ = 0.52 m/s, ${F}_{0}$ = 0.87), and run 33 (s = 0.0052, ${Y}_{0}$ = 0.011 m, ${U}_{0}$ = 0.43 m/s, ${F}_{0}$ = 1.31). Horizontal dashed lines denote the theoretical limit for a stationary jump (panel

**h**, Equation (14) with ${F}_{S}={F}_{0}$) and for an infinitesimal perturbation (panels

**i**–

**l**, $a={c}_{0}-{U}_{0}$, i.e., $a/{c}_{0}=1-{F}_{0}$).

**Figure 8.**Free surface profiles at different times, t/$\tau $, after sluice gate closure; (

**a**) t/$\tau $ = 0.16, 0.32, 0.48, 0.64, 0.80 (run 3); (

**b**) t/$\tau $ = 0.61, 1.21, 1.82, 2.43, 3.03 (run 24). $h/{Y}_{0}$ is the non-dimensional free surface elevation and $xs/{Y}_{0}$ is the non-dimensional coordinate in the flow direction, positive upstream.

**Table 1.**Summary of experimental conditions. Runs 1 to 41 are performed in the long flume, runs 42 to 73 are performed in the short flume; ${F}_{S}$ is the surge Froude number defined by Equation (13).

Run | s | ${{U}}_{0}$ | ${{Y}}_{0}$ | ${{F}}_{0}$ | ${{F}}_{{S}}$ |
---|---|---|---|---|---|

(-) | (m/s) | (m) | (-) | (-) | |

1–7 | 0.0002 | 0.23–0.41 | 0.045–0.165 | 0.31–0.35 | 1.19–1.28 |

8–14 | 0.0005 | 0.27–0.51 | 0.038–0.131 | 0.45–0.46 | 1.31–1.36 |

15–23 | 0.0010 | 0.24–0.60 | 0.022–0.112 | 0.53–0.61 | 1.28–1.49 |

24–32 | 0.0022 | 0.31–0.81 | 0.015–0.085 | 0.70–0.86 | 1.46–1.69 |

33–41 | 0.0052 | 0.43–1.13 | 0.011–0.062 | 1.08–1.41 | 1.75–2.17 |

42–49 | 0.0005 | 0.18–0.43 | 0.047–0.154 | 0.26–0.35 | 1.18–1.25 |

50–57 | 0.00075 | 0.23–0.47 | 0.036–0.142 | 0.38–0.41 | 1.25–1.34 |

58–65 | 0.0010 | 0.25–0.53 | 0.034–0.126 | 0.43–0.48 | 1.28–1.38 |

66–73 | 0.0025 | 0.32–0.72 | 0.026–0.093 | 0.64–0.81 | 1.39–1.67 |

**Table 2.**Summary of numerical conditions. ${k}_{St}$ is the roughness coefficient in the Strickler formula for uniform flow.

s | ${{k}}_{{St}}$ | ${{F}}_{0}$ |
---|---|---|

(-) | (m${}^{1/3}$s${}^{-1}$) | (-) |

0.0002 | 30–90 | 0.31–0.35 |

0.0005 | 25–95 | 0.45–0.46 |

0.0010 | 15–90 | 0.53–0.61 |

0.0020 | 20–90 | 0.53–0.61 |

0.0050 | 80–90 | 1.30–1.75 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Viero, D.P.; Peruzzo, P.; Defina, A.
Positive Surge Propagation in Sloping Channels. *Water* **2017**, *9*, 518.
https://doi.org/10.3390/w9070518

**AMA Style**

Viero DP, Peruzzo P, Defina A.
Positive Surge Propagation in Sloping Channels. *Water*. 2017; 9(7):518.
https://doi.org/10.3390/w9070518

**Chicago/Turabian Style**

Viero, Daniele Pietro, Paolo Peruzzo, and Andrea Defina.
2017. "Positive Surge Propagation in Sloping Channels" *Water* 9, no. 7: 518.
https://doi.org/10.3390/w9070518