# Enhancing the Flow Characteristics in a Branching Channel Based on a Two-Dimensional Depth-Averaged Flow Model

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Data Acquisition

^{3}/s, (b) of Q2 with discharge 127 m

^{3}/s, and (c) of Q3 with discharge 121 m

^{3}/s.

_{50}) was approximately 0.055 mm.

#### 2.2. Critical Velocity of the Sediment Inception Motion

_{50}of 0.055 mm.

_{50}) is known, and the other variables are kept constant for the selected branching of the Tigris River.

#### 2.3. Solver Background, Structure, Characteristics, and Application Procedure

#### 2.3.1. Flow Field Model

- A Galerkin finite element submodel (a type of weighted residual method) is used to discretize continuity and momentum equations.
- Open boundary conditions (upstream and downstream boundaries, etc.) enable the setting up of various conditions, such as time series of flow discharge, time series of water level, and water level discharge.
- The friction of the river bed can be set by using the Manning roughness coefficient. This coefficient can be represented in the model as a polygon for the entire area or for each element (cell), thus providing spatial distribution of roughness.
- Three submodels are available for computing the turbulence field (flows with large and small eddies). These are the zero equation submodel, simple k-ε, and direct input of kinematic eddy viscosity. In this study, the zero equation submodel was selected because of its stability in calculating large and small eddies. The kinematic eddy viscosity, υ, is expressed by using the von Kàrmàn coefficient with $k$(0.40).$$v=\frac{k}{6}{u}_{*}y$$This formula is called the zero equation submodel for turbulence statistics without a transport equation. In flow fields where the water depth and roughness change gradually in the cross-sections of the branching channel, the kinematic eddy viscosity in horizontal and vertical directions is assumed to be in the same order.
- Other effects, such as the effect of wind on the water surface and of vegetation on flow, are available in this solver but were disregarded in this study.

#### 2.3.2. Riverbed Variation Model

- The riverbed variation associated with the flow field model is calculated. This model can calculate the flow field only or together with riverbed variation.
- The riverbed material can be selected from uniform and mixed grain diameters. If a mixed grain diameter is selected, then a variation in grain distribution can be assumed for deep directions and multiple layers.
- Three methods for calculating the total bed load ${q}_{b}$ of depth-averaged flow velocity are available in the Mflow_02 solver, and these are the Meyer-Peter and Müller formula [43], Ashida and Michiue formula [44] and Engrlund–Hansen formula [45]. In this study, the Meyer-Peter and Müller formula was adopted for computing the total bed load.$${q}_{b}=8\sqrt{\left(\frac{\sigma}{\rho}-1\right)g{d}_{50}{}^{3}}{\left({\tau}_{*}^{\prime}-{\tau}_{*c}\right)}^{1.5}$$$\sigma $: Gravel density${\tau}_{*c}$: Critical tractive force (calculated by the Iwagaki formula, [46])${\tau}_{*}^{\prime}$: Calculated by Kishi and Kuroki in the formula below [47].$$\frac{\overline{U}}{{u}_{*}}=\begin{array}{}\mathrm{(6)}& 7.66{\left(\frac{y}{2{d}_{50}}\right)}^{1/6}{\left(\frac{{\tau}_{*}^{\prime}}{{\tau}_{*}}\right)}^{2/3}& \frac{y}{2{d}_{50}}500\mathrm{(7)}& 11.59{\left(\frac{y}{2{d}_{50}}\right)}^{1/10}{\left(\frac{{\tau}_{*}^{\prime}}{{\tau}_{*}}\right)}^{3/5}& \frac{y}{2{d}_{50}}\ge 500\end{array}$$$\overline{U}$: Vertical average flow velocity in the flow direction.The total sediment discharge set by the Meyer-Peter and Müller formula is converted to the normal direction (n) and tangential direction (s) of the streamline, in consideration of the effect of secondary flow and riverbed slope, which is caused by streamline curvature of depth-averaged flow velocity [48].$${q}_{s}={q}_{b}\left(\frac{{\nu}_{b}}{{V}_{b}}-\sqrt{\frac{{\tau}_{*c}}{{\mu}_{s}{\mu}_{k}{\tau}_{*}}}\frac{\partial z}{\partial s}\right)$$$${q}_{n}={q}_{b}\left(\frac{{u}_{b}}{{V}_{b}}-\sqrt{\frac{{\tau}_{*c}}{{\mu}_{s}{\mu}_{k}{\tau}_{*}}}\frac{\partial z}{\partial n}\right)$$${q}_{s}$: (s) direction component of sediment discharge near the riverbed${q}_{n}$: (n) direction component of sediment discharge near the riverbed${V}_{b}$: Absolute value of velocity near the riverbed${\nu}_{b}$: (s) direction component of flow velocity near the riverbed${u}_{b}$: (n) direction component of flow velocity near the riverbed${\mu}_{s}$: Static friction factor${\mu}_{k}$: Kinetic friction factor$z$: Height of the riverbed.
- The scour limit of the riverbed, secondary flow coefficient, and morphological factor can be set accordingly.

#### 2.4. Model Implementation and Boundary Conditions

^{3}/s was set in the inlet boundary condition upstream of the Tigris branching channel. Water levels of 7.205 and 7.2 m above mean sea level were set in outlet boundary conditions 1 and 2, respectively, located downstream of the main and branching channels. Among the turbulence models, a zero-equation model was adopted, and a movable bed computation with a starting time of 800 s was set up to provide stability to the flow field mode operation first and then to the sediment transport and bed deformation. For sediment transport computation, Mflow_02 uses the Ashida and Michiue [44], Meyer-Peter and Müller [43], and Engelund–Hansen [45] equations to compute the total bed load transport. In this study, the Meyer-Peter and Müller equation was selected to compute sediment transport because the equation provides reasonable results for simulated bed elevations. The accuracy of the simulated bed elevations was demonstrated in the calibration and validation results of both models. Other settings related to riverbed morphology and sediment transport were as follows: the scour limit of the riverbed in which the selected branching channel has no limit for scouring was set in the model, the secondary flow coefficient was set to 7 (based on the Engelund model [45]), and the morphological factor (the ratio of bed deformation to flow) was set to 1. The above-mentioned setting was considered for a real simulation, in which an increase in one of the values leads to an increase in the bed deformation of the branching channel.

#### 2.5. Unsubmerged Vanes as Control Structures

- Two vanes 30 m long and an inclination angle of 15° from the axis perpendicular to the flow of the main river
- Two vanes 30 m long perpendicular to the direction of the flow of the main river
- A single vane 50 m long with an inclination angle of 30° from the axis perpendicular to the flow of the main river
- A single vane 50 m long perpendicular to the direction of the flow of the main river.

#### 2.6. Statistical Indices

## 3. Results and Discussion

#### 3.1. Model Calibration

^{−5}m, 0.003 m, and 0.035%, respectively.

#### 3.2. Model Validation

^{3}/s) were compared with measured data for water level, average flow depth, average velocity, and bed elevation, and the results were shown in Table 5 and Table 6, and Figure 11, respectively. The model output showed that the simulation of water levels was highly accurate according to the values of the four statistical indices (Table 5). The values of MAD, MSE, RMSE, and MAPE were 0.002 m, 8 × 10

^{−6}m, 0.003 m, and 0.038%, respectively; the simulated average flow depth was within the observed range, and the values of the statistical indices were 0.407 m, 0.181 m, 0.426 m, and 18.33% respectively. On top of that, a graphical comparison in Figure 12 shows the accuracy of the model validation and the errors of the simulated values from the measured values.

#### 3.3. Model Simulation of Tigris Branching Junction with Vanes

#### 3.3.1. First Configuration: Two Vanes of Length 30 m Each

^{3}/s (dry season) and high discharge of 247 m

^{3}/s (wet season); the greatest change in the hydro-morphodynamic features was associated with the high-flow season. Model outputs of this configuration were compared with the outputs of other configurations (Figure 13 and Figure 14). The results of the simulation with and without obstacles show that the velocity at the upstream location of the junction is always higher than the velocity downstream. The low velocity zones that were recognised at the branching channel of the Tigris River were also observed in other experimental studies [8,16] and denoted as flow separation zones. The first low-velocity zone appears in the main river opposite to the branching channel entrance downstream of the junction. This zone is formed owing to the movement of the streamlines from the main river towards the branching channel. The streamlines attempt to align and transfer to a uniform flow state in the main river after passing the junction. The second low-velocity zone is formed upstream near the right bank of the branching channel. This zone is formed owing to flow contraction at the beginning of the branching channel. For bed topography, two sediment deposition areas are formed with low-velocity zones; the larger of the two areas is located on the right side of the upstream branching channel and considered as one of the most important features that affect the inflow discharge pass through the branching junction. The erosion zone (scour hole) was found on the left side of the flow downstream of the branching channel [23]. This zone was associated with a region of high velocity and vortexes. Vanes were installed to manage the flow at the diversion channel through redirecting the inflow discharge toward the flow separation zone, which consequently led to increased velocity in this area and the deposition zone diminishing over time. The process of deposition zone removal occurs when the velocity of the flow is greater than the critical velocity of the deposition particles. The critical mean velocity was determined based on the Simons and Şentürk equation, and the values for the studied branching channel ranged from 0.26 to 0.32 m/s. The results of this configuration show that there are noticeable changes in the flow depth, velocity distribution with the vectors, and shape of the attached bar at the deposition zone. The flow depths increase at the locations of the vanes, wherein the velocities increased marginally, especially in front of the vane, while the minimum velocity was observed behind the vane (Figure 14b). Furthermore, the simulation results show the existence of a small point bar, which indicates that the velocity and sediment transport distributions in the branching channel section are not uniform. The existing deposition zone area was found to be 3261 m

^{2}, and when the vanes were used, this area reduced to 1422 m

^{2}; as a percentage, this reduction is approximately 56% compared with the case where no vanes are used (Figure 13a).

#### 3.3.2. Second Configuration: Two Vanes of Length 30 m Each and Perpendicular to the Direction of Flow in the Main River

^{2}, which is approximately 72% compared with the case of no vane. Flow depths increased in the deposition zone by 1.2 m, and the volume of sediment removed was approximately 2817 m

^{3}. This increment occurs owing to the removal of sedimentation when the mean velocity increases and exceeds the critical mean velocity of the sediment (Figure 14c). Increasing velocities lead to increases in bottom friction velocity (${u}_{*}$), and this value has a direct effect on the calculation of riverbed shear stress (${\tau}_{*}$). Increasing the riverbed shear stress leads to strengthening of the amount of sediment eroded from the flow separation zone, which in turn increases the flow depth. The velocity distribution and its vectors after the vanes appear to be equally distributed along the transverse direction of the branching channel after the obstacles, with their values ranging between 0.347 and 0.431 m/s. The results of this configuration were compared with the first configuration, and it is shown that the location and angle of the vane are important factors that affect the morphological changes in the branching channel. However, the results of this configuration provide significant improvements to the hydro-morphodynamics of the branching channel.

#### 3.3.3. Third Configuration: A Single Vane of Length 50 m at an Inclination of 30° from the Axis Perpendicular to the Flow in the Main River

#### 3.3.4. Fourth Configuration: A Single Vane of Length 50 m and Perpendicular to the Direction of Flow in the Main River

^{2}, and the volume of sediment removed was around 3200 m

^{3}.

#### 3.4. Discussion on Recommended Configurations

^{2}, respectively, for the first, second, third, and fourth configurations.

_{50}of 0.055 mm, and discharge ratio of 52% (at wet season). Furthermore, simulation results during the dry season showed a good production in managing and reduction of the deposition zone. The average velocity in the deposition zone increased from 0.14 to 0.3 m/s and led to enhancing the sediment transport. As a result, the flow depth increased by approximately 0.5 m at many locations along the deposition zone. Figure 16 shows the simulation results during the dry season with and without vanes (fourth configuration) on the flow depth, and velocity distribution with its vectors.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Characteristics of flow dynamics at a branching channel (Ramamurthy et al. [8]).

**Figure 3.**Observed bathymetry survey, with velocity distribution at each reach of the Tigris branching junction, (

**a**) with discharge 247 m

^{3}/s, (

**b**) with discharge 127 m

^{3}/s, and (

**c**) with discharge 121 m

^{3}/s

**Figure 7.**Simulation results for velocity magnitude at the Tigris branching junction and profiles of average flow velocity in m/s within 24 h at cross-sections CS 1, CS 2, CS 3, CS 4, CS 5, and CS 6.

**Figure 9.**Depth simulation of the Tigris branching junction and profiles of simulated bed elevation (m) within 24 h at cross-sections CS 1, CS 2, CS 3, CS 4, CS 5, and CS 6.

**Figure 10.**Comparison between measured and simulated data used in the model calibration (

**a**) averaged velocity; (

**b**) water level; (

**c**) bed elevation.

**Figure 12.**Comparison between measured and simulated data used in the model validation (

**a**) averaged velocity; (

**b**) water level; (

**c**) bed elevation.

**Figure 13.**Simulated flow depth for the study site (

**a**) without vanes structure; (

**b**) First configuration; (

**c**) Second configuration; (

**d**) Third configuration; (

**e**) Fourth configuration.

**Figure 14.**Simulated velocity distribution with its vectors for the study site (

**a**) without vanes structure; (

**b**) First configuration; (

**c**) Second configuration; (

**d**) Third configuration; (

**e**) Fourth configuration.

**Figure 15.**Impact of using vanes on velocity and reduction area in the deposition zone for the studied configurations.

**Figure 16.**Simulation of flow depth and velocity distribution with its vectors during low flow conditions. (

**a**) Simulated flow depth without vane; (

**b**) Simulated flow depth with vane; (

**c**) Simulated velocity distribution with its vectors without vane; (

**d**) Simulated velocity distribution with its vectors with vane.

**Table 1.**Summary of the studies that used submerged vanes for controlling sediment dynamics at branching channels.

Authors | Main Channel Width (m) | Branching Channel Width (m) | Branching Angle | Submerged Vanes Arrangement and Number | Vane Angle | Nature of Study |
---|---|---|---|---|---|---|

Wang et al. [30] | 90° | Parallel | 20° | Field application | ||

Study 1 | 230 | 25 | 40 | |||

Study 2 | 25 | 10 | 6 | |||

Nakata and Ogden [32] | Missouri River | 90° | Zigzag | Physical model | ||

Study 1 | 22 | 13 | 19.5° | |||

Study 2 | 27.6 | 5 | 20–45° | |||

Study 3 | 8.1 | 17 | 19.5° | |||

Study 4 | 48 | 11 | 22° | |||

Study 5 | 27 | 17 | 22° | |||

Barkdoll et al. [13] | 1.5 | 0.61 | 90° | Parallel with four cases, | 20° | Laboratory |

3 rows/54 | ||||||

3 rows/21 | ||||||

2 rows/20 | ||||||

3 rows/18 | ||||||

Michell et al. [25] | 100 | 26.9 | 90° | Zigzag with two rows/13 | 22° | Field application |

Allahyonesi et al. [33] | 1.5 | 0.6 | 60° | Regular/24 and zigzag/24 | 20° | Laboratory |

AbdelHaleem et al. [34] | 0.6 | 0.2 | 90° | Single row/4 | 10°, 20°, 30°, 40°, 50° | Laboratory |

Moghadam and Keshavarzi [35] | 0.6 | 0.3 | 55° | Zigzag with two rows/10Parallel with three rows/15 | 10°, 20°, 30°, 40° | Laboratory |

Mirzaei et al. [36] | 1 | 0.4 | 90° | Parallel with two rows/10 | 22° | Numerical simulation |

Date | Location | Cross-Section Area (m^{2}) | Discharge (m^{3}/s) | Mean Flow Velocity (m/s) | Mean Flow Depth (m) | Top Width (m) | Water Temp. (°C) | Status |
---|---|---|---|---|---|---|---|---|

11/5/2017 | Q1 | 704 | 247 | 0.351 | 4.19 | 168 | 27.2 | Calibration |

Q2 | 340 | 127 | 0.375 | 3.05 | 111 | 26 | ||

Q3 | 577 | 121 | 0.210 | 3.24 | 186 | 26.2 | ||

20/7/2017 | Q1 | 645 | 119 | 0.185 | 4.11 | 157 | 29.1 | Validation |

Q2 | 220 | 79 | 0.359 | 2.06 | 107 | 28.6 | ||

Q3 | 367 | 41 | 0.112 | 2.25 | 163 | 29.3 |

Cross-section | Average Velocity (m/s) | Error M-S | Absolute Value of Error | Square of Error | Absolute Values of Errors Divided by Measured | Results of Different Statistical Indices | ||
---|---|---|---|---|---|---|---|---|

Measured (M) | Simulated (S) | |||||||

CS 1 | 0.391 | 0.383 | 0.008 | 0.008 | 6 × 10^{−5} | 0.02 | MAD | 0.026 |

CS 2 | 0.375 | 0.383 | −0.008 | 0.008 | 6 × 10^{−5} | 0.021 | MSE | 0.001 |

CS 3 | 0.4 | 0.369 | 0.031 | 0.031 | 1 × 10^{−3} | 0.078 | RMSE | 0.034 |

CS 4 | 0.328 | 0.335 | −0.007 | 0.007 | 5 × 10^{−5} | 0.021 | MAPE | 9.466 |

CS 5 | 0.351 | 0.318 | 0.033 | 0.033 | 0.001 | 0.094 | ||

CS 6 | 0.21 | 0.14 | 0.07 | 0.07 | 0.005 | 0.333 | ||

Sum. | 0.157 | 0.007 | 0.568 |

No | Time (s) | Discharge (m^{3}/s) at the Inlet | Water Level (m) at the Outlet 1 | Water Level (m) at the Outlet 2 |
---|---|---|---|---|

1 | 0 | 119 | 5.935 | 5.93 |

2 | 86,400 | 120 | 5.935 | 5.93 |

3 | 172,800 | 119.3 | 5.935 | 5.93 |

4 | 259,200 | 120 | 5.935 | 5.93 |

5 | 345,600 | 118 | 5.935 | 5.93 |

6 | 432,000 | 117 | 5.93 | 5.925 |

7 | 518,400 | 117 | 5.93 | 5.925 |

Transect Name | Water Level | Results of Different Methods of Errors | Average Flow Depth (m) | Results of Different Methods of Errors | ||||
---|---|---|---|---|---|---|---|---|

Measured | Simulated | Measured | Simulated | |||||

CS 1 | 5.930 | 5.931 | MAD | 0.002 | 1.899 | 2.289 | MAD | 0.407 |

CS 2 | 5.934 | 5.933 | MSE | 8 × 10^{−6} | 1.9 | 2.333 | MSE | 0.181 |

CS 5 | 5.938 | 5.940 | RMSE | 0.003 | 3.927 | 3.7 | RMSE | 0.426 |

CS 6 | 5.935 | 5.940 | MAPE | 0.038 | 2.388 | 2.966 | MAPE | 18.33 |

Transect Name | Average Velocity | Results of Different Methods of Errors | ||
---|---|---|---|---|

Measured | Simulated | |||

CS 1 | 0.375 | 0.312 | MAD | 0.042 |

CS 2 | 0.359 | 0.29 | MSE | 0.002 |

CS 5 | 0.185 | 0.175 | RMSE | 0.049 |

CS 6 | 0.112 | 0.085 | MAPE | 16.38 |

© 2019 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**

Ali, H.L.; Yusuf, B.; Mohammed, T.A.; Shimizu, Y.; Ab Razak, M.S.; Rehan, B.M. Enhancing the Flow Characteristics in a Branching Channel Based on a Two-Dimensional Depth-Averaged Flow Model. *Water* **2019**, *11*, 1863.
https://doi.org/10.3390/w11091863

**AMA Style**

Ali HL, Yusuf B, Mohammed TA, Shimizu Y, Ab Razak MS, Rehan BM. Enhancing the Flow Characteristics in a Branching Channel Based on a Two-Dimensional Depth-Averaged Flow Model. *Water*. 2019; 11(9):1863.
https://doi.org/10.3390/w11091863

**Chicago/Turabian Style**

Ali, Hydar Lafta, Badronnisa Yusuf, Thamer Ahamed Mohammed, Yasuyuki Shimizu, Mohd Shahrizal Ab Razak, and Balqis Mohamed Rehan. 2019. "Enhancing the Flow Characteristics in a Branching Channel Based on a Two-Dimensional Depth-Averaged Flow Model" *Water* 11, no. 9: 1863.
https://doi.org/10.3390/w11091863