# Experimental and Numerical Analysis of a Dam-Break Flow through Different Contraction Geometries of the Channel

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## Abstract

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## 1. Introduction

## 2. Experimental Facility and Measuring Technique

_{0}= 0.25 m in the initial condition; the downstream part of the channel, 4.25 m long, was initially dry and left open in order for the flow to fall freely with no reflection. The water in the reservoir was colored with food dye with the aim of an easier identification of the free surface profiles and evaluation of the behavior of the dam-break flow from the recorded video frames. The obstacles installed to form the local contraction in the channel were made of Plexiglas and were located at a specific distance downstream of the dam on both sidewalls symmetrically. Three different geometries of the contraction were built, namely Triangular, Trapezoidal-A and Trapezoidal-B, dimensions and shapes of which can be seen in Figure 2. The different shapes were selected in order to represent transition from smooth to sudden contraction. The lengths of the obstacles (0.95 m), the maximum contraction width (0.10 m) and the distance from the gate (1.52 m) were chosen equally in order to compare contraction effects.

_{0}/g)

^{1/2}= 0.2 s, where h

_{0}is the initial water level in the reservoir and g is gravity [43]).

## 3. Numerical Simulations

#### 3.1. RANS Equations with k-ε Turbulent Model

_{F}is the fractional volume open to flow, p is the pressure, ρ is the fluid density, u

_{i}is the mean velocity, A

_{i}is fractional area open to flow, g

_{i}is the body acceleration, and f

_{i}is the viscous acceleration in subscript direction, the latter expressed as follows:

_{ij}is given by:

#### 3.2. SWE Equations

_{F}is the volume fraction, p is the pressure, ρ is the fluid density, d is the water depth, u and v are the depth-averaged velocities, ${g}_{x}$ and ${g}_{y}$ the body accelerations along the x and y direction, respectively, and ${\tau}_{b,x}$ and ${\tau}_{b,y}$ represent the x and y components of the bottom shear stress, respectively.

_{F}and the water fraction F variables are used to define a variable bottom contour and fluid depth, respectively [44]. Equations (5)–(7) are expressed in terms of volume, area, and water fractions for flow in a single layer of control volumes used for the application of VOF and FAVOR methods.

_{0}is the atmospheric pressure on the water free surface and H is the height of the free surface above the grid bottom, i.e., the sum of obstacle and water heights:

_{D}represents the drag coefficient and can be calculated as:

_{0}= k

_{s}/30, with k

_{s}as the surface roughness.

#### 3.3. Solution Domain, Boundary and Initial Conditions

#### 3.4. Grid Sensitivity Analysis

## 4. Results and Discussion

#### 4.1. Comparison of Experimental data for Different Contraction Geometries

_{0}was used as denominator of horizontal distance (X = x/h

_{0}) and flow depth (h/h

_{0}), whereas time t was multiplied by (g/h

_{0})

^{1/2}, with g gravity acceleration, to get the non-dimensional form of time T = t (g/h

_{0})

^{1/2}.

_{0}is 0.85 for Trapezoidal-B, 0.84 for Trapezoidal-A and 0.81 for Triangular contraction, i.e., it is higher, since it is reached with more abrupt contraction, in the Trapezoidal-B case. Similar results are obtained observing the graphs of P1, P2 and P3 points. With the Trapezoidal-B contraction, the reflected wave reaches these points earlier and the measured maximum height is higher compared to the other cases; the water level in the upstream sections of the contraction increases significantly with the formation of a negative surge (reflected) wave in all cases. Due to finite reservoir length, the flow rate of the incoming flow decreases after a while and the water accumulated upstream of the contraction also gradually decreases. The reflected wave moving upwards is again reflected from the vertical wall at the upstream end of the channel, and a wave train which moves again downstream is formed. In the plots, while water levels are decreasing, a sudden rise and fluctuation of water levels are observed. The comparison of the reflected waves for all cases shows that the wave reflected in the Trapezoidal-B case is faster than in the other cases. In addition, during the passage of the wave reflected from the upstream channel boundary through the contracted section, the water level increases considerably, especially for the Trapezoidal-B case (before T = 100). Then, the wave reflected from the upstream end of the channel is reflected again from the narrowed sections and starts to move again in the upstream direction. This situation can be seen in the Trapezoidal-B curve at T = 105 for points P2 and P3. In general, when the dam-break wave encounters a cross-sectional change during its propagation, while a part of the flow passes through the existing opening, the rest of it is reflected in the contracted section and forms a reflected wave moving upward between the contraction and the upstream end of the channel until the water completely discharges. As a result, when the dam-break flood wave encounters quite abrupt transitions along its path, stronger reflections, higher water levels and mixed flow conditions occur upstream of the narrowing section. The small oscillations observed in the experimental reconstruction of the water level time histories (Figure 5) with the specific image analysis measuring technique described above are not only the result of the strong reflections of propagating waves upward and downward and their interferences, but even more a result of the mixed flow conditions which occur in such strongly unsteady flows.

#### 4.2. Comparison between Experimental and Numerical Results for Trapezoidal-A Case

- Root Mean Square Error (RMSE)$$\mathrm{RMSE}=\sqrt{\frac{1}{\mathrm{n}}{\displaystyle \sum}_{i=1}^{\mathrm{n}}{\left({y}_{\mathrm{o}}-{y}_{p}\right)}^{2}}$$
- Mean Absolute Percent Error (MAPE)$$\mathrm{MAPE}\%=\frac{100}{\mathrm{n}}\times {\displaystyle \sum}_{i=1}^{\mathrm{n}}\left|\frac{{y}_{\mathrm{o}}-{y}_{p}}{{y}_{0}}\right|$$
- Mean Absolute Error (MAE)$$\mathrm{MAE}=\frac{1}{\mathrm{n}}{\displaystyle \sum}_{\mathrm{i}=1}^{\mathrm{n}}\left|{y}_{\mathrm{o}}-{y}_{p}\right|$$

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Laboratory set-up: (

**a**) Longitudinal view A-A, (

**b**) Top view, (lengths in (cm), Trapezoidal-A contraction).

**Figure 2.**Geometries and dimensions (lengths in (cm)) of the different contractions chosen for the experiments.

**Figure 3.**Comparison of test images for 3 different contraction conditions (dashed lines highlights the contraction position) in terms of water profiles at different times: (

**a**) 1.8 s, (

**b**) 2.4 s, (

**c**) 3.0 s, (

**d**) 4.5 s, respectively, after the gate opening.

**Figure 5.**Comparison of the time variation of water level for the three different contraction cases.

**Figure 6.**Water profiles with formation and propagation of the negative bore for Trapezoidal-A case: (

**a**) numerical solution (RANS), (

**b**) experiment (lengths in m).

**Figure 7.**Mixed flow condition (Froude number): (

**a**) 3D view, (

**b**) plan view, (

**c**) front view, (

**d**) experiment.

**Figure 8.**Comparison of flow depths at initial stages of the dam-break process, calculated, respectively, for Trapezoidal-A case by: (

**a**) SWEs and (

**b**) RANS simulations every 0.5 s from 1.0 to 3.0 s in plan-view (

**c**) 3D-view of SWEs and RANS results at 3.0 s after the gate removal.

**Figure 9.**Comparison between numerically computed and experimentally measured free surface profiles over time for Trapezoidal-A contraction case.

**Figure 10.**Measurement points for Trapezoidal-A case for further experimental evaluations of water level time histories.

**Figure 11.**Comparison of water level changes for experimental and numerical (RANS and SWEs) results at measurement points P1–P8 for Trapezoidal-A case.

Parameter | RANS | SWEs | ||
---|---|---|---|---|

P3 | P6 | P3 | P6 | |

${\varphi}_{1}$ (cm) | 19.8072 | 21.2182 | 20.9844 | 16.4848 |

${\varphi}_{2}$ (cm) | 19.7380 | 21.5719 | 21.0498 | 15.8341 |

${\varphi}_{3}$ (cm) | 19.9245 | 20.1423 | 21.4927 | 17.4143 |

$p$ | 1.43 | 2.02 | 2.76 | 1.28 |

${\varphi}_{\mathrm{ext}}^{21}$ (cm) | 19.8481 | 21.1019 | 20.9730 | 16.5977 |

${e}_{\mathrm{ext}}^{21}\%$ | 0.21 | 0.55 | 0.05 | 0.68 |

${e}_{a}^{21}\%$ | 0.35 | 1.67 | 0.31 | 3.95 |

${e}_{a}^{32}\%$ | 0.94 | 6.63 | 2.10 | 9.98 |

${\mathrm{GCI}}_{\mathrm{fine}}^{21}\%$ | 0.26 | 0.69 | 0.07 | 3.45 |

${\mathrm{GCI}}_{\mathrm{fine}}^{32}\%$ | 0.70 | 2.72 | 0.46 | 8.73 |

${\mathrm{GCI}}_{32}/{r}^{p}{\mathrm{GCI}}_{21}$ | 1.00 | 0.98 | 1.00 | 1.04 |

**Table 2.**Error analysis referred to the comparison between numerical (RANS and SWEs, respectively) and experimental results for water-level time histories at measurement points P1–P8.

Errors | RANS | |||||||

P1 | P2 | P3 | P4 | P5 | P6 | P7 | P8 | |

MAPE % | 3.50 | 4.10 | 4.38 | 5.57 | 5.34 | 7.16 | 7.67 | 7.50 |

RMSE (cm) | 0.82 | 0.79 | 0.80 | 0.80 | 0.96 | 1.45 | 1.20 | 0.65 |

MAE (cm) | 0.42 | 0.45 | 0.53 | 0.63 | 0.56 | 0.73 | 0.53 | 0.46 |

Errors | SWEs | |||||||

P1 | P2 | P3 | P4 | P5 | P6 | P7 | P8 | |

MAPE % | 4.22 | 4.37 | 4.82 | 7.90 | 7.26 | 18.14 | 5.88 | 13.80 |

RMSE (cm) | 0.98 | 0.84 | 0.88 | 1.71 | 1.50 | 2.39 | 0.85 | 1.42 |

MAE (cm) | 0.54 | 0.51 | 0.56 | 0.86 | 0.78 | 2.02 | 0.51 | 1.09 |

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**MDPI and ACS Style**

Kocaman, S.; Güzel, H.; Evangelista, S.; Ozmen-Cagatay, H.; Viccione, G. Experimental and Numerical Analysis of a Dam-Break Flow through Different Contraction Geometries of the Channel. *Water* **2020**, *12*, 1124.
https://doi.org/10.3390/w12041124

**AMA Style**

Kocaman S, Güzel H, Evangelista S, Ozmen-Cagatay H, Viccione G. Experimental and Numerical Analysis of a Dam-Break Flow through Different Contraction Geometries of the Channel. *Water*. 2020; 12(4):1124.
https://doi.org/10.3390/w12041124

**Chicago/Turabian Style**

Kocaman, Selahattin, Hasan Güzel, Stefania Evangelista, Hatice Ozmen-Cagatay, and Giacomo Viccione. 2020. "Experimental and Numerical Analysis of a Dam-Break Flow through Different Contraction Geometries of the Channel" *Water* 12, no. 4: 1124.
https://doi.org/10.3390/w12041124