# Discharge Calculation of Side Weirs with Several Weir Fields Considering the Undisturbed Normal Flow Depth in the Channel

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Discharge Behaviour at Side Weirs in the Context of Flood Risk Management

#### 1.2. Theoretical Background

_{n,u}upstream of the weir, the discharge reduction at the side weir leads to a reduction in the normal depth y

_{n,d}at the downstream end of the weir. Applying a one-dimensional approach, the flow depth decreases continuously from the upstream normal depth (y

_{n,u}) to the upstream end of the weir (Figure 1).

_{x}is the discharge in the channel at position x along the weir, q is the weir discharge per unit length, I

_{S}is the channel slope, I

_{E}is the energy slope, dy/dx is the change in flow depth y with x, and α is the energy correction coefficient for non-uniform velocity distribution (α ≥ 1). The weir discharge per unit length can be calculated with the equation according to Poleni [18], taking a discharge coefficient for side weirs (C

_{Q})and the weir height (w) into account (Equation (2)):

_{W}) is realized when Equation (2) is integrated along the weir length L (Equation (3)):

_{0}remains constant along the weir despite the discharge reduction, approaches based on approximations of the water surface along the weir have been developed for example by De Marchi [2], Schmidt [19], and Dominguez [20]. The common element in these approaches is the need to empirically determine the respective discharge coefficient by means of numerical or physical experiments. Numerous studies have been conducted in past decades, yielding various formulas for estimating these coefficients [2,3,5,6].

_{0}when the kinetic energy is small. This is especially the case in regions close to the channel banks. One simple option to estimate the water level is by using standard formulas for open channel flow. By applying Poleni’s equation, the discharge through the weir is calculated as follows (Equation (4)):

_{b}is the mean discharge coefficient representing all open weir fields, L

_{o}is the weir length that is calculated by the number of open weir fields n

_{o}and the weir field width b (Equation (5)):

_{0}is defined with respect to the weir crest height w, which is related to the channel bottom and refers to the normal flow conditions upstream of the drawdown at the weir (Equation (6)):

_{b}can be determined with Equation (7), depending on specific weir and channel characteristics.

## 2. Materials and Methods

#### 2.1. Discharge Coefficient

_{b}(Equation (7)), which is to be used in an equation similar to Poleni’s equation (Equation (4)), a parametric study was carried out using 3D numerical simulations. Besides the obvious local weir geometry, the following parameters characterizing the setup need to be considered: slope of the main channel I

_{s}, weir height w, weir field width b, normal flow depth at undisturbed conditions closely upstream of the weir h

_{0}, flow depth underwater of the weir h

_{d}, the number of open weir fields n

_{o}, and the main channel width B (Equation (8)):

#### 2.2. Numerical Model

^{®}[24], is widely used and enables modeling of unsteady three-dimensional flows involving complex geometries. The free water level was represented using the volume-of-fluid method according to Hirt and Nichols [25] and the Reynolds-averaged Navier–Stokes equation was solved using the finite difference method on a Cartesian computational grid. The turbulence model used in the simulations was the k-ε model [26].

_{St}= 40 m

^{1/3}/s. In the numerical model, the equivalent grain roughness was used to describe the channel roughness. Calibration led to a value of k

_{s}= 0.1 m for the channel and k

_{s}= 0.005 m for the weir. For analyzing time series of simulation results, monitoring points and planes were defined in the model setup. In total, 68 monitoring points were defined in 12 cross-sections in the area around the weir and another 12 were defined in the main channel. The discharge was determined both in the channel and in the individual weir fields using measuring baffles.

#### 2.3. Validation of the Numerical Simulation Results with a Physical Scale Model

## 3. Results

#### 3.1. Comparison of the Numerical and Physical Model Results

#### 3.2. Influence of Single Parameters on the Discharge Coefficient

_{0}at normal flow conditions upstream of the weir was considered for the determination of the discharge coefficient C

_{b}(Equation (15)), as it not only reflects the actual losses at the weir, but also the impacts of reduced flow depth. As Figure 2 shows, increasing the channel slope yielded significant reductions to the water depth. These circumstances led to reduced weir discharge, even though the models had the same flow depth. The analysis of the discharge coefficient emphasized this effect; models with a slope of 2‰ had the smallest discharge coefficient C

_{b}in almost all variants investigated. Models with 1.0‰ slope showed the largest coefficient C

_{b}, whereas models with 0.5‰ were just slightly lower. As shown in Figure 11a, an extension of the channel width B resulted in an increasing discharge coefficient C

_{b}, whereby models with a channel slope I

_{S}of 0.5 to 1.5‰ were more strongly affected. With the reduction of the channel width B, the impact of the channel slope I

_{S}decreased, which is why the coefficients gradually converge. An increase in weir height relative to the channel bottom also had a positive effect on the discharge coefficient (Figure 11b). As the flow depth above the weir crest declined accordingly, the results showed the opposite behavior. With increasing flow depth h

_{0}, the averaged discharge coefficient C

_{b}decreased (Figure 11c). According to the results in Figure 11d, the coefficient decreases with increasing weir field width b. The arrangement of weir piers and the associated width reduction thus have a positive effect on the discharge capacity. An increasing number of open weir fields leads to a gradual decrease in the discharge coefficient. However, when more weir fields are opened this effect decreases (Figure 11e). The impact of the underwater level shows the typical characteristic as described for example in Aigner and Bollrich [27], for broad-crested weirs. If the flow depth h

_{d}in the underwater rises above 70% in relation to the flow depth h

_{0}, the discharge coefficient is significantly reduced (Figure 11f).

#### 3.3. Regression Analysis

_{b}, which varied between 0.01 and 0.53. If the variable $\left(\frac{{\mathrm{h}}_{\mathrm{d}}}{{\mathrm{h}}_{0}}\right)$ is solely considered in the regression analysis, the estimated value deviates from the simulated coefficient by about 0.076 on average. With an increasing number of variables, this error decreases to a value of 0.022. If all variables are included in the regression analysis, the Equation (15) can be used to determine the discharge coefficient C

_{b}.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Water surface (represented by flow depth y) and energy line (parameter E) in a one-dimensional approach at a side weir at subcritical flow conditions (modified after De Marchi [2]); flow direction from left to right.

**Figure 2.**Spatially distributed drawdown of the water surface in the channels (m) due to water diversion through the side weir with eight open weir fields (blue arrows); channel slope I

_{S}ranges between 0.5 and 2.0‰; the illustrated contour lines reflect the reduction of flow depth in meters compared to the undisturbed normal flow depth; the dash-dot line represents the main channel axis; the main flow direction is from left to right.

**Figure 3.**Schematic sketch of the side weir with eight weir fields—cross-section through the main channel and side weir (

**a**) and plan view (

**b**).

**Figure 4.**Arrangement and dimensions of the Cartesian mesh blocks (border lines in cyan blue), position of the defined monitoring points.

**Figure 5.**Physical scale model of the side weir with four weir fields in an experimental flume; rectifier at the upstream edge of the model (

**a**); height-adjustable gate at the downstream edge of the model (

**b**); channel collecting side weir discharge with a Poncelet weir (

**c**).

**Figure 6.**Arrangement of the 28 monitoring points at 10 cross-sections along the side weir; flow direction from left to right.

**Figure 8.**Profiles of the flow depths h above the weir crest determined in the numerical and in the physical model along the weir depending on the number of weir fields opened for channel slopes of 0.5–2.0‰ (including position of weir fields 1–4); flow direction from left to right.

**Figure 9.**Mean absolute (

**a**) and relative (

**b**) difference of the flow depths h resulting from the numerical and the physical model.

**Figure 10.**Plots of the Froude number and the flow depths in the numerical model as well as photos of the physical model in the configuration with all weir fields (4) opened for channel slopes of 0.5–2.0‰; flow direction from left to right.

**Figure 11.**Influence of the analyzed weir and channel parameters on the discharge coefficient C

_{b}; channel width B (

**a**); weir height w (

**b**); flow depth h

_{0}(

**c**); weir field width b (

**d**); number of open weir fields n

_{o}(

**e**); ratio of underwater flow depth h

_{d}and flow depth h

_{0}at normal flow conditions closely upstream of the weir (

**f**).

Parameters | Dimension |
---|---|

Channel roughness k_{St} | 40 (m^{1/3}/s) |

Channel slope Is | 0.5, 1.0, 1.5, 2.0 (‰) |

Froude number Fr at norm flow conditions | 0.35, 0.50, 0.65, 0.80 (-) |

Channel width B | 56, 76, 96, 116 (m) |

Flow depth h_{0} | 1.6, 2.6, 3.6 (m) |

Flow depth underwater of the weir h_{d} | 0.0, 0.7*h_{0}, 0.8*h_{0}, 0.9*h_{0} (m) |

Weir field width b | 7.0, 15.5, 32.5 (m) |

Weir height w | 2.2, 4.0, 5.8 (m) |

Number of open weir fields n_{o} | 1–8 (-) |

Parameters | Dimensions Numerical Model/Physical Model (1:50) |
---|---|

Channel slope Is | 0.5, 1.0, 1.5, 2.0 (‰) |

Froude number Fr | 0.35, 0.50, 0.65, 0.80 (-) |

Channel width B | 16.0 (m)/32.0 (cm) |

Flow depth h_{0} | 3.6 (m)/7.2 (cm) |

Flow depth underwater of the weir h_{d} | 0.0 (m)/0.0 (cm) |

Weir field width b | 7.0 (m)/14 (cm) |

Weir height w | 2.5 (m)/5 (cm) |

Number of open weir fields n_{o} | 0–4 (-) |

**Table 3.**Successive regression analysis of the averaged discharge coefficient C

_{b}; variables included in the analysis; multiple correlation coefficient R, multiple determination coefficient R², and the standard error.

Variables Included in the Analysis | R | R² | Standard Error | Equation |
---|---|---|---|---|

${\mathrm{C}}_{\mathrm{b}}=\mathrm{f}\left({\left(1-\frac{{\mathrm{h}}_{\mathrm{d}}}{{\mathrm{h}}_{0}}\right)}^{-0.3}\right)$ | 0.664 | 0.414 | 0.076 | (10) |

${\mathrm{C}}_{\mathrm{b}}=\mathrm{f}\left(\frac{\mathrm{b}\times {n}_{\mathrm{o}}}{\mathrm{B}},{\left(1-\frac{{\mathrm{h}}_{\mathrm{d}}}{{\mathrm{h}}_{0}}\right)}^{-0.3}\right)$ | 0.866 | 0.749 | 0.050 | (11) |

${\mathrm{C}}_{\mathrm{b}}=\mathrm{f}\left({\mathrm{I}}_{\mathrm{s}}{}^{5},\frac{\mathrm{b}\times {n}_{\mathrm{o}}}{\mathrm{B}},{\left(1-\frac{{\mathrm{h}}_{\mathrm{d}}}{{\mathrm{h}}_{0}}\right)}^{-0.3}\right)$ | 0.943 | 0.888 | 0.034 | (12) |

${\mathrm{C}}_{\mathrm{b}}=\mathrm{f}\left({\mathrm{I}}_{\mathrm{s}}{}^{5},\text{}\frac{{\mathrm{h}}_{0}}{\mathrm{b}\times {\mathrm{n}}_{\mathrm{o}}},\frac{\mathrm{b}\times {n}_{\mathrm{o}}}{\mathrm{B}},{\left(1-\frac{{\mathrm{h}}_{\mathrm{d}}}{{\mathrm{h}}_{0}}\right)}^{-0.3}\right)$ | 0.954 | 0.911 | 0.030 | (13) |

${\mathrm{C}}_{\mathrm{b}}=\mathrm{f}\left({\mathrm{I}}_{\mathrm{s}}{}^{5},\text{}\frac{{\mathrm{h}}_{0}}{\mathrm{b}\times {n}_{\mathrm{o}}},\frac{\mathrm{b}\times {n}_{\mathrm{o}}}{\mathrm{B}},\frac{\mathrm{w}}{{\mathrm{h}}_{0}},{\left(1-\frac{{\mathrm{h}}_{\mathrm{d}}}{{\mathrm{h}}_{0}}\right)}^{-0.3}\right)$ | 0.977 | 0.954 | 0.022 | (14) |

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**MDPI and ACS Style**

Lindermuth, A.; Ostrander, T.S.P.; Achleitner, S.; Gems, B.; Aufleger, M.
Discharge Calculation of Side Weirs with Several Weir Fields Considering the Undisturbed Normal Flow Depth in the Channel. *Water* **2021**, *13*, 1717.
https://doi.org/10.3390/w13131717

**AMA Style**

Lindermuth A, Ostrander TSP, Achleitner S, Gems B, Aufleger M.
Discharge Calculation of Side Weirs with Several Weir Fields Considering the Undisturbed Normal Flow Depth in the Channel. *Water*. 2021; 13(13):1717.
https://doi.org/10.3390/w13131717

**Chicago/Turabian Style**

Lindermuth, Adrian, Théo St. Pierre Ostrander, Stefan Achleitner, Bernhard Gems, and Markus Aufleger.
2021. "Discharge Calculation of Side Weirs with Several Weir Fields Considering the Undisturbed Normal Flow Depth in the Channel" *Water* 13, no. 13: 1717.
https://doi.org/10.3390/w13131717