# Characteristics of Positive Surges in a Rectangular Channel

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{0}= 1.26 to 1.28, whereas it tends to suggest a power law reduction for larger surge Froude numbers. Moreover, the dispersion of undular surges is consistent with the linear wave theory only for surge Froude numbers close to unity. Overall, the present study demonstrates the unique features of positive surges induced by a progressive increase of discharge.

## 1. Introduction

## 2. Experimental Setup and Experiment

_{0}, the drop height H, and the amount of time needed to fully open the gate t

_{v}. In the present study, the initial water depth in the channel was set to 8 cm by referring to the studies of Treske [6] and Koch and Chanson [10]. A summary of the experimental measurements is shown in Table 1. Overall, a large number of experimental runs were carried out in four series, denoted as A, B, C, and D, corresponding to four different times taken to fully open the gate of 20, 25, 30, and 40 s with four different drop heights. The experimental conditions were selected to generate both undular and breaking surges, taking into account the influence of undulation shape deviations on the characteristics of surges with close Froude numbers, which is defined in the system of coordinates in translation with the surge and is given by

## 3. Results and Discussion

#### 3.1. Basic Flow Patterns

- In Stage 1, the front of the positive wave is rather smooth.
- Stage 2 corresponds to the generating process of the undular surges, and can be divided into two sub-stages: Stage2a and Stage2b. In Stage 2a, the front head of the positive wave gradually becomes “rough” and develops into a series of cascading steps at the end of this stage; in Stage 2b, the cascading steps develop into a train of well-formed undulations (i.e., non-breaking undular surges) (Figure 2). Additionally, some sidewall shock waves can be observed in this stage that develop upstream of the first wave crest and intersect next to the first crest. This stage ends when some small waves break at the first wave crest (i.e., breaking undular surges) (Figure 3).
- In Stage 3, the wave amplitude decreases and the free-surface undulations become flatter. This stage ends with the disappearance of fluctuating characteristics of water depth at the surge front.
- During Stage 4, the surge front behaves as a nearly vertical water wall (i.e., breaking surges) (Figure 4).

#### 3.2. Free-Surface Properties

#### 3.2.1. Ratio of Conjugate Depths

_{conj}and h

_{0}are the conjugate and initial depths, respectively. Equation (2) is the famous Bélanger equation. In the present study, the conjugate depth h

_{conj}was defined as the average of the first wave crest and the trough depths for the undular surges, while it was the water depth immediately behind the surge front for breaking surges. The ratio of conjugate depths, h

_{conj}/h

_{0}, is plotted against the surge Froude number, Fr

_{0}, in Figure 5 for both undular and breaking surges. Although the experimental trend is comparable to that produced by Equation (2), the entire dataset shows generally higher values of h

_{conj}/h

_{0}for 1.07 < Fr

_{0}< 1.57. This may be partially attributed to the uncertainty of estimating the conjugate depth; that is, in the undular surges, the conjugate depth was calculated based on a symmetrical undulation profile which is inconsistent with the experimental data, while the measurement of the conjugate depth was affected adversely by the large free-surface fluctuations behind the surge front in the breaking surges. It is worth noting that the data in flow Stage 2b are more scattered than in flow Stages 3 and 4 for a given surge Froude number, which might imply larger deviations of wave shapes for non-breaking surges than for breaking undular and breaking surges.

#### 3.2.2. Free-Surface Undulation Characteristics

_{1c}− h

_{0}) is plotted in a dimensionless form against Fr

_{0}in Figure 6. For non-breaking undular surges in Stage 2b, the values of (h

_{1c}− h

_{0})/h

_{0}show a trend of linear increase with Fr

_{0}in general. For breaking undular surges in Stage 3, a sharp decrease in wave height is observed shortly after the appearance of some wave breaking at the first wave crest, and subsequently, (h

_{1c}− h

_{0})/h

_{0}exhibits the same increasing trend as the data in flow Stage 2b, but at a much smaller rate. It is worth noting that in flow Stage 2b, (h

_{1c}− h

_{0})/h

_{0}is close to the solitary wave theory for surge Froude numbers Fr

_{0}less than 1.1, while the data show consistently higher values of (h

_{1c}− h

_{0})/h

_{0}for larger surge Froude numbers.

_{w}/h

_{0}, L

_{w}/h

_{0}, and a

_{w}/L

_{w}). Both wave amplitude and wave length were calculated according to the definitions by Koch and Chanson [10]; that is, the wave amplitude, a

_{w}, was half of the difference between the water depth at the first wave crest and at the first wave trough, while the wave length, L

_{w}, was defined between the first and second crests. The data are compared with the linear wave theory solution of Lemoine, the Boussinesq equation solution of Anderson, earlier experimental studies and field observations [21,22,23].

_{w}/h

_{0}and Fr

_{0}for undular surges is well described by the following equations

_{w}/L

_{w}shows an increasing and decreasing trend for non-breaking and breaking undular surges, respectively. Additionally exhibited in this figure are the data sets obtained in undular hydraulic jumps [24,25]. For surge Froude numbers less than 1.29, the present data set is in excellent agreement with undular jump data.

_{w}, wave period T and water depth h:

_{0}for undular surges are presented in Figure 9. It is shown that a definite correlation exists between the two dimensionless variables, with essentially no influence of undular surge types. In the experimental range 1.07 < Fr

_{0}< 1.50, the following empirical equation can be proposed by fitting both data sets (i.e., for non-breaking and breaking undular surges)

_{0}less than 1.1, the dispersion parameter D is close to 1, and hence the dispersion relationship based on the linear wave theory is assumed to hold.

#### 3.2.3. Energy and Momentum Fluxes Properties

_{c}is the critical depth in the system of coordinates in translation with the undular surge front, and is defined as

_{c}as a parameter. The function E* − M* has two branches (the red dash line in Figure 10), intersecting at (1.5, 1.5). The lower branch of the curve E* − M* corresponds to a supercritical flow while the upper branch represents a subcritical flow. Figure 10 shows a comparison of M* = f(E*) and the entire dataset in this study. It is expected that the initial flow data are located on the supercritical branch while the corresponding conjugate flow data are on the subcritical branch. It can also be concluded from Figure 10, that at the subcritical branch of the curve E* − M*, some overlaps exist between regions corresponding to different flow stages, indicating marked differences between travelling positive surges and stationary hydraulic jumps [27].

_{0}*) and the corresponding conjugate flow (E

_{conj}*). It can be found that the values of E

_{conj}* are basically larger than that of E

_{0}* for non-breaking undular surges in Stage 2b. This is probably due to the pressure distribution beneath an undular surge that is less than hydrostatic beneath the wave crest and greater than hydrostatic beneath the wave trough. With the ensuing development of undular surges, E

_{conj}* approaches E

_{0}* for breaking undular surges in Stage 3, and is essentially less than E

_{0}* for breaking surges in Stage 4. This may be primarily attributed to energy dissipation due to wave breaking, and hence a smaller specific energy in the conjugated flow than in the initial flow is obtained in breaking surges.

_{0}*) and the corresponding conjugate flow (M

_{conj}*) is presented. The values of M

_{conj}* are consistently larger than that of M

_{0}* for the entire present test range. The main reason for this may be the deviation from the hydrostatic and the complicated velocity distribution in undular and breaking surges.

## 4. Conclusions

_{0}ranging from 1.07 to 1.57. The occurrence of non-breaking undular surges was observed for 1.07–1.10 < Fr

_{0}< 1.26–1.28, breaking undular surges for 1.26–1.28 < Fr

_{0}< 1.45–1.50, and breaking surges for Fr

_{0}> 1.45–1.50. The range of Froude numbers corresponding to each type of surge is consistent with previous findings [11,13,15], although previous studies were concerned with positive surges induced by the rapid closure of a downstream sluice gate.

_{0}< 1.26–1.28), and the wave length data are consistently lower than the values predicted from the above theories. Based on the experimental results, two empirical equations in terms of the surge Froude number were proposed to estimate wave amplitude. Fourth, the dispersion of undular surges is consistent with the linear wave theory only for surge Froude numbers close to unity (Fr

_{0}< 1.1), demonstrating marked differences with undular surges investigated in previous studies [11,12]. Therefore, a novel dimensionless parameter defined as L

_{w}/[(gT

^{2}/2π) × tanh(2πh

_{conj}/L

_{w})] was introduced to characterize the dispersion of undular surges induced by a progressive increase of discharge and it was found to solely depend on the surge Froude number.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Henderson, F.M. Open-Channel Flow; McMillan Publishing Co.: New York, NY, USA, 1966. [Google Scholar]
- Maeck, A.; Lorke, A. Ship-lock induced surges in an im-pounded river and their impact on subdaily flow velocity variation. River Res. Appl.
**2013**, 30, 494–507. [Google Scholar] [CrossRef] - Chanson, H. The Hydraulics of Open Channel Flows: An Introduction; Butterworth-Heinemann: Oxford, UK, 2004; p. 630. [Google Scholar]
- Chen, C.L. Unified Theory on Power Laws for Flow Resistance. J. Hydraul. Eng. ASCE
**1990**, 117, 371–389. [Google Scholar] [CrossRef] - Kjerfve, B.; Ferreira, H.O. Tidal Bores: First Ever Measurements. Ciência e Cultura (J. Braz. Assoc. Adv. Sci.)
**1993**, 45, 135–138. [Google Scholar] - Treske, A. Undular bores (Favre-waves) in open channels—Experimental studies. J. Hydraul. Res.
**1994**, 32, 355–370. [Google Scholar] [CrossRef] - Favre, H. Etude Théoretique et Expérimentale des Ondes de Translation Dans les Canaux Découverts (Theoretical and Experimental Study of Travelling Surges in Open Channels); Dunod: Paris, France, 1935. [Google Scholar]
- Benet, F.; Cunge, J.A. Analysis of experiments on secondary undulations caused by surge waves in trapezoidal channels. J. Hydraul. Res.
**1971**, 9, 11–33. [Google Scholar] [CrossRef] - Soares-Frazão, S.; Zech, Y. Undular bores and secondary waves—Experiments and hybrid finite-volume modeling. J. Hydraul. Res.
**2002**, 40, 33–43. [Google Scholar] [CrossRef] - Koch, C.; Chanson, H. Turbulence measurements in positive surges and bores. J. Hydraul. Res.
**2009**, 47, 29–40. [Google Scholar] [CrossRef] [Green Version] - Chanson, H. Unsteady turbulence in tidal bores: Effects of bed roughness. J. Waterw. Port Coast. Ocean Eng.
**2010**, 136, 247–256. [Google Scholar] [CrossRef] - Gualtieri, C.; Chanson, H. Experimental study of a positive surge. Part 1: Basic flow patterns and wave attenuation. Environ. Fluid Mech.
**2012**, 12, 145–159. [Google Scholar] [CrossRef] [Green Version] - Chanson, H. Undular Tidal Bores: Basic Theory and Free-surface Characteristics. J. Hydraul. Eng. ASCE
**2010b**, 136, 940–944. [Google Scholar] [CrossRef] - Gualtieri, C.; Chanson, H. Experimental study of a positive surge. Part 2: Comparison with literature theories and unsteady flow field analysis. Environ. Fluid Mech.
**2011**, 11, 641–651. [Google Scholar] [CrossRef] - Leng, X.; Chanson, H. Upstream Propagation of Surges and Bores: Free-Surface Observations. Coast. Eng. J.
**2017**, 59, 1750003. [Google Scholar] [CrossRef] - Lemoine, R. Sur les ondes positives de translation dans les canaux et sur le ressaut ondulé de faible amplitude (On the Positive Surges in Channels and on the Undular Jumps of Low Wave Height). Jl La Houille Blanche
**1948**, 2, 183–185. (In French) [Google Scholar] - Andersen, V.M. Undular Hydraulic Jump. J. Hydraul. Div. ASCE
**1978**, 104, 1185–1188. [Google Scholar] - Viero, D.P.; Peruzzo, P.; Defina, A. Positive Surge Propagation in Sloping Channels. Water
**2017**, 9, 518–530. [Google Scholar] [CrossRef] - Peregrine, D.H. Calculations of the development of an undular bore. J. Fluid Mech.
**1966**, 25, 321–330. [Google Scholar] [CrossRef] - McCowan, J. On the solitary wave. Lond. Edinb. Dublin Philos. Mag. J. Sci.
**1891**, 32, 45–58. [Google Scholar] [CrossRef] - Lewis, A.W. Field Studies of a Tidal Bore in the River Dee. Master’s Thesis, University College of North Wales, Bangor, UK, 1972. [Google Scholar]
- Navarre, P. Aspects Physiques du Caracteres Ondulatoire duMacaret en Dordogne (Physical Features of the Undulations of the Dordogne River Tidal Bore). Ph.D. Thesis, University of Bordeaux, Bordeaux, France, 1995. [Google Scholar]
- Wolanski, E.; Williams, D.; Spagnol, S.; Chanson, H. Undular tidal bore dynamics in the Daly Estuary, Northern Australia. Estuar. Coast. Shelf Sci.
**2004**, 60, 629–636. [Google Scholar] [CrossRef] - Montes, J.S.; Chanson, H. Characteristics of undular hydraulic jumps. Results and calculations. J. Hydraul. Eng. ASCE
**1998**, 124, 192–205. [Google Scholar] [CrossRef] - Chanson, H. Physical modelling of the flow field in an undular tidal bore. J. Hydraul. Res.
**2005**, 43, 234–244. [Google Scholar] [CrossRef] [Green Version] - Montes, J.S. Hydraulics of Open Channel Flow; ASCE Press: New York, NY, USA, 1998. [Google Scholar]
- Benjamin, T.B.; Lighthill, M.J. On cnoidal waves and bores. Proc. R. Soc. Lond. Ser. A
**1954**, 224, 448–460. [Google Scholar] [CrossRef]

**Figure 2.**Non-breaking undular surges in Stage 2b: (

**a**) Lateral view; (

**b**) looking upstream at the incoming wave.

**Figure 3.**Breaking undular surges in Stage 3: (

**a**) Lateral view; (

**b**) looking upstream at the incoming wave.

**Figure 8.**Characteristics of undular surges as functions of Fr

_{0}: (

**a**) Dimensionless wave amplitude a

_{w}/h

_{0}; (

**b**) dimensionless wave length L

_{w}/h

_{0}; (

**c**) wave steepness a

_{w}/L

_{w}.

**Figure 10.**Dimensionless relationship between the momentum and energy fluxes for both undular and breaking surges.

Series | h_{0} (m) | t_{v} (s) | H (m) |
---|---|---|---|

A | 0.08 | 20 | 0.1, 0.2, 0.3, 0.4 |

B | 0.08 | 25 | 0.1, 0.2, 0.3, 0.4 |

C | 0.08 | 30 | 0.1, 0.2, 0.3, 0.4 |

D | 0.08 | 40 | 0.1, 0.2, 0.3, 0.4 |

Flow Stage Transition | Surge Froude Number Fr_{0} |
---|---|

1–2a | ≈1.03 |

2a–2b | 1.07–1.10 |

2b–3 | 1.26–1.28 |

3–4 | 1.45–1.50 |

© 2018 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/).

## Share and Cite

**MDPI and ACS Style**

Zheng, F.; Li, Y.; Xuan, G.; Li, Z.; Zhu, L.
Characteristics of Positive Surges in a Rectangular Channel. *Water* **2018**, *10*, 1473.
https://doi.org/10.3390/w10101473

**AMA Style**

Zheng F, Li Y, Xuan G, Li Z, Zhu L.
Characteristics of Positive Surges in a Rectangular Channel. *Water*. 2018; 10(10):1473.
https://doi.org/10.3390/w10101473

**Chicago/Turabian Style**

Zheng, Feidong, Yun Li, Guoxiang Xuan, Zhonghua Li, and Long Zhu.
2018. "Characteristics of Positive Surges in a Rectangular Channel" *Water* 10, no. 10: 1473.
https://doi.org/10.3390/w10101473