# Review and Comparison of Numerical Simulations of Secondary Flow in River Confluences

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Description of the Main Flow and Secondary Circulation

## 3. Boundary Conditions

#### 3.1. Wall Function Boundaries

$${u}_{i}\frac{\partial k}{\partial {x}_{i}}$$
| = |
$$\frac{\partial}{\partial {x}_{i}}(\frac{{\nu}_{t}}{{\sigma}_{k}}\frac{\partial k}{\partial {x}_{i}})$$
| − |
$${\nu}_{t}(\frac{\partial {u}_{i}}{\partial {x}_{j}}+\frac{\partial {u}_{j}}{\partial {x}_{i}})\frac{\partial {u}_{i}}{\partial {x}_{j}}$$
| − | ε | (10) |

convection owing to transport by mean flow | diffusion into mean flow |
$$\mathrm{production}\text{}(=P)$$
| viscous dissipation rate |

#### 3.2. Free Surface

#### 3.3. Inlet

#### 3.4. Outlet

## 4. Numerical Research of River Confluence Flows

#### 4.1. Secondary Flow in a 90° Confluent Channel with Bed Concordance

#### 4.1.1. RANS Models

#### 4.1.2. LES Models

#### 4.1.3. ANN Models

#### 4.2. Secondary Flow in Differently Angled Confluent Channels with Bed Concordance

#### 4.2.1. RANS Models

#### 4.2.2. DES Models

#### 4.3. Secondary Flow in Confluent Channel with Bed Discordance

#### 4.3.1. RANS Models

#### 4.3.2. LES Models

#### 4.3.3. DES Models

## 5. Discussion

## 6. Future Research Needs

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Streamlines in an equal width 90° confluence for ${Q}_{r}=0.50$ in different sections (Shakibaeinia et al., 2010) [3].

**Figure 4.**Main channel cross section showing calculated flow pattern and secondary currents in confluences for ${Q}_{r}=0.5$, ${W}_{r}=100,$ and four different confluence angles. (

**a**) α = 15°, (

**b**) α = 45°, (

**c**) α = 90°, (

**d**) α = 105° (Shakibaeinia et al., 2010) [3].

**Figure 5.**Model setup for the confluent channel (Shaheed et al., 2018) [17].

**Figure 6.**Resultant velocities across dimensionless transverse distance for confluent channel: (

**a**) at x* = −1.0; (

**b**) at x* = −1.667; and (

**c**) at x* = −3.0, (

**d**) at x* = −4.0 (Shaheed et al., 2018) [17].

**Figure 7.**Kaskaskia River—copper slough confluence (Gökçen, 2015) [47].

**Figure 8.**Geometry of the five laboratory simulations: (

**a**) symmetrical confluence, (

**b**) single-meander bend, (

**c**) asymmetrical confluence with width ratio (WR) of combined upstream width of tributaries to width of post-confluence channel of 1.0, (

**d**) asymmetrical confluence with WR = 0.75, (

**e**) asymmetrical confluence with WR = 0.5, where “A” indicates the upstream junction corner and “B” indicates the downstream junction corner (Bradbrook et al., 2000a) [8].

**Figure 10.**Geometry of separation zone: (

**a**) experimental; (

**b**) k-ε; (

**c**) RNG k-ε; (

**d**) k-ω; (

**e**) SST k-ω; (

**f**) EARSM. (Brito et al., 2014 [44]).

**Figure 12.**Location of confluence of Bayonne and Berthier rivers, Québec, Canada (Biron et al., 2004) [13].

**Figure 13.**Y-shaped open junction (Wang and Yan, 2007) [56].

**Figure 14.**Inlet and outflow boundary conditions (Wang and Yan, 2007) [56].

**Figure 16.**Bed elevation discordance in (

**a**) tributary; (

**b**) main channel; (

**c**) locations of backward facing steps in tributary and main channels (Dordevica and Stojnic, 2016) [57].

**Figure 17.**View of the junction of the Bayonne and Berthier Rivers (Boyer et al., 2006) [58].

**Table 1.**Summary of selected studies about secondary flow in confluent channels used in Section 4.1.1.

Shape of Channel | Numerical Model | Key Findings | Reference |
---|---|---|---|

Asymmetrical, shaped | k-ω model | The strength of the secondary flow was underpredicted | (Huang et al., 2002) [12] |

Asymmetrical, shaped | The CFD package, PHOENICS (version 3.5), the turbulence model was not specified in this paper | There was a good agreement at the upstream end of the junction, and the discrepancy increased at other downstream locations | (Sivakumar et al., 2004) [6] |

Asymmetrical, shaped | 3D k-ω model | The results of the numerical model agreed well with the experimental data | (Zhang et al., 2009 [25]) |

Asymmetrical, shaped | RNG form of k-ε model | The secondary flows appeared directly after the junction in approximately three counter-rotating helical cells | (Shakibaeinia et al., 2010) [3] |

Asymmetrical, shaped | A hybrid RANS-LES model was developed | The new modeling approach is more accurate than the RANS approach in addition to its ability of saving computational effort comparing to the LES approach | (Zeng and Li, 2010) [26] |

Asymmetrical, shaped | Standard k-ε, RNG k-ε, and RSM turbulence models | At the section immediately after the junction, the secondary flow predicted by the numerical models was smaller than that of the experiment | (Yang et al., 2011) [27] |

Asymmetrical, shaped | RNG form of k-ε model | The results showed good agreement between the simulation and the measured data | (Mignot et al., 2012) [28] |

Asymmetrical, shaped | Standard k-ε, realizable k-ε, and k-ω | The preferable model for the confluence flow simulation was k-ω | (Yang et al., 2013) [2] |

Asymmetrical, shaped | Reynolds Stress Modeling (RSM) | Flow pattern could be improved significantly by employing some geometrical adjustments | (Mohammadiun et al., 2015) [29] |

Asymmetrical, shaped | Standard k–ε model and the realizable k–ε model | The realizable k–ε model is better | (Shaheed et al., 2018) [17] |

Asymmetrical, shaped | Reynolds averaged Navier–Stokes equations and Reynolds stress turbulence model (RANS and RSM) | The contaminants mixing basically happens in the mixing layer at the interface of two confluent flows | (Tang et al., 2018) [30] |

Asymmetrical, shaped | A modified 1D nonlinear dynamic model and fully 3D non-hydrostatic, Reynolds-averaged Navier–Stokes equations (RANS) model | The strategies of 1D and 3D modeling could be applied to other flow diversion problems like river meander | (Luo et al., 2018) [31] |

**Table 2.**Flow conditions in the experiment and five simulation runs (Yang et al., 2013) [2].

Flow Conditions in the Experiment | ||||||

Runs | ${Q}_{b}({m}^{3}$/s) | ${Q}_{m}({m}^{3}$/s) | ${Q}_{d}({m}^{3}$/s) | ${q}_{b}({m}^{2}$/s) | ${q}_{d}({m}^{2}$/s) | ${R}_{q}({q}_{b}$$/{q}_{d}$) |

1 | 0.127 | 0.042 | 0.169 | 0.139 | 0.185 | 0.750 |

Runs for Three Dimensional Simulation of Confluence Flow | ||||||

Runs | Cases | Turbulence Model | Surface Treating | Meshes Adopted | Computing Time (hours) | |

1 | Standard k-ε | Dynamic meshes | 165,456 | 118 | ||

2 | Realizable k-ε | Dynamic meshes | 165,456 | 124 | ||

3 | ${R}_{q}=0.75$ | k-ω | Dynamic meshes | 165,456 | 130 | |

4 | k-ω | Rigid lid | 165,456 | 50 | ||

5 | k-ω | VOF | 232,200 | 245 |

**Table 3.**Summary of selected studies about secondary flow in confluent channels used in Section 4.1.2.

Shape of Channel | Numerical Model | Key Findings | Reference |
---|---|---|---|

Asymmetrical, shaped | LES | A reasonable agreement with experimental data and analytical solutions was achieved. | (Liu et al., 2009) [36] |

Asymmetrical, shaped | LES | New features of the flow patterns were induced due to the flow of the tributary. | (Schindfessel et al., 2015) [37] |

Asymmetrical, shaped | LES | There was a significant difference in the separation zone dimensions for non-rectangular shapes. | (Schindfessel et al., 2017) [38] |

Asymmetrical, shaped | LES | Oversimplification of free surface numerical processing leads to lower accuracy of secondary flow and turbulent kinetic energy predictions. | (Ramos et al., 2019) [39] |

**Table 4.**Summary of selected studies about secondary flow in confluent channels used in Section 4.2.1.

Shape of Channel | Numerical Model | Key Findings | Reference |
---|---|---|---|

Asymmetrical, 60° angle | Standard k–ε turbulence model | The comparison between the numerical and experimental results was good | (Weerakoon and Tamai, 1989) [21] |

Asymmetrical, 60° angle | Standard k–ε turbulence model | Experimental comparison good, but size of lateral separation zone underpredicted | (Weerakoon et al., 1991) [43] |

Different styles of channels | Turbulence model based on a renormalized group (RNG) | As asymmetry increases, the structure of back-to-back helical cells thought to be less representative of the flow field | (Bradbrook et al., 2000a) [8] |

Asymmetrical, 60° angle | Turbulence model based on a renormalized group (RNG) | As asymmetry increases, the structure of back-to-back helical cells thought to be less representative of the flow field | (Bradbrook et al., 2000a) [8] |

Asymmetrical, 30° angle | RNG k-ε model | The mixing was faster at higher junction angles, especially for concordant beds | (Biron et al., 2004) [13] |

Asymmetrical, 70° angle | k-ε, RNG k-ε, k-ω, SST k-ω, EARSM | The numerical results indicated the importance of simulating the secondary flows due to their great impact on the separation zone and velocity contour lines | (Brito et al., 2014) [44] |

Ten different confluences from 45° to 90° | k-ε turbulence closure model | The increase in the confluence angle caused a wider and longer retardation zone at the corner of upstream junction and the separation zone | (Penna et al., 2018) [45] |

**Table 5.**Summary of selected studies used in Section 4.3.1 about secondary flow in confluent channels.

Shape of Channel | Numerical Model | Key Findings | Reference |
---|---|---|---|

Asymmetrical, 30° angle | RNG k-ε model | The efficiency of secondary flows in the bend was less for mixing at the low flow than for the high flow. | (Biron et al., 2004) [13] |

Asymmetrical, 60° angle | RNG k-ε model | The efficiency of secondary flows in the bend was less for mixing at the low flow than for the high flow. | (Biron et al., 2004) [13] |

Symmetrical, shaped | Linear Renormalization Group (RNG) 𝑘-ε turbulence model | The results appeared to be an accurate prediction by the numerical model. | (Wang and Yan, 2007) [56] |

Asymmetrical, shaped | Standard k–ε turbulence model | The perpendicular velocities were enhanced by the tributary bed steps along the side wall of the intersection and by the main channel bed steps along the opposite wall. | (Dordevica and Stojnic, 2016) [57] |

**Table 6.**Summary of selected studies used in Section 4.3.2 about secondary flow in confluent channels.

Shape of Channel | Numerical Model | Key Findings | Reference |
---|---|---|---|

Asymmetrical, parallel channels | LES | The numerical simulation agreed well with the laboratory experiments. | (Bradbrook et al., 2000b) [9] |

Asymmetrical, 65° angle | LES | The mixing data and the detailed flow provided by the model were helpful in understanding the production and evolution of large-scale turbulence features in confluence channels. | (Bradbrook et al., 2000b) [9] |

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**MDPI and ACS Style**

Shaheed, R.; Yan, X.; Mohammadian, A.
Review and Comparison of Numerical Simulations of Secondary Flow in River Confluences. *Water* **2021**, *13*, 1917.
https://doi.org/10.3390/w13141917

**AMA Style**

Shaheed R, Yan X, Mohammadian A.
Review and Comparison of Numerical Simulations of Secondary Flow in River Confluences. *Water*. 2021; 13(14):1917.
https://doi.org/10.3390/w13141917

**Chicago/Turabian Style**

Shaheed, Rawaa, Xiaohui Yan, and Abdolmajid Mohammadian.
2021. "Review and Comparison of Numerical Simulations of Secondary Flow in River Confluences" *Water* 13, no. 14: 1917.
https://doi.org/10.3390/w13141917