# Numerical Simulation of Confluence Flow in a Degraded Bed

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. RANS Simulations

#### 1.2. LES Simulations

#### 1.3. DES Simulations

## 2. Numerical Methodology and Model Verification

#### 2.1. Numerical Framework

^{−5}for the fraction of liquid ($\mathsf{\alpha}$) and 10

^{−8}for pressure and velocity in the Gauss–Seidel method for solving the resulting linear system of equations.

#### 2.2. Hydraulic Conditions

_{Q}= 1.5 was utilized where R

_{Q}= Q

_{t}/Q

_{m}with Q

_{t}and Q

_{m}being the discharges of the upstream tributary channel and main of the open channel, respectively. Consequently, the discharge ratio was identical for all cases and shared the same Froude and Reynolds (Re) numbers, as shown in Table 2.

#### 2.3. Mesh Generation

#### 2.4. Boundary Conditions

#### 2.4.1. Wall Function Boundaries

#### 2.4.2. Free Surface

#### 2.4.3. Bed Morphology

#### 2.5. Flow Visulazation

#### 2.6. Model Verification

#### 2.6.1. Determination of Solution

#### 2.6.2. Governing Equations

#### Large-Eddy Simulation (LES) Models

_{i}= mean velocity in the i-direction (I = x, y, z). The filter width is related to the cell volume. Particularly, the filter width needs to be defined in a way that captures the smallest eddies of interest.

#### Detached Eddy Simulation (DES) Models

#### 2.6.3. Wall-Normal Distance of First Grid Cell

^{+}) from solid boundaries, where

_{τ}is the friction velocity, Y is the absolute distance from the wall, and ν is the kinematic viscosity. However, wall model LES (WMLES), i.e., LES with wall functions, overcomes this requirement by identifying the primary grid point within the log-law region (30 < z+ < 500). Although details of near-bed turbulence are not obtained, the wall-modelled approach has been widely applied to investigate large-scale turbulent phenomena within the outer region of the water column in flume and fluvial applications [29]. A velocity-based wall function was employed to ascertain the near-wall turbulent viscosity and bed shear stress caused by the rough solid boundary. This condition was met for all the simulations, with z+ being usually above 30.

#### 2.6.4. Mesh Sensitivity Analysis and Verification of Turbulent Kinetic Energy (TKE)

_{res}) on the mesh. According to Equation (21), over 80% of the cells in both simulations meet this requirement.

## 3. Results and Discussion

#### 3.1. Model Validation and Comparison

#### 3.1.1. Velocity

#### 3.1.2. Turbulence Characteristics

#### 3.2. LES Deformed Bed Versus Flatbed

#### 3.2.1. Time-Averaged Longitudinal Velocity Field

#### 3.2.2. Streamlines

#### 3.2.3. v-w Vector Characteristics

#### 3.2.4. Vorticity

#### 3.2.5. TKE

## 4. Conclusions

- The simulations captured the flow patterns in the confluence zone, including the formation of recirculation zones and secondary flow, with a strong shear layer forming near the inner bank after the junction. The maximum velocity diverts from the tributary channel to the outer bank of the main channel, with substantial flow curvature through the confluence zone.
- Velocity Prediction: The simulations showed a reasonable level of agreement with the experimental data, particularly for the VoF approach. However, there were areas closest to the bed where improvement is needed. The LES model provided the lowest average velocity error compared to other turbulence models, indicating its reliability in predicting flow velocities in complex flow geometries such as confluent channels. The k-ω SST model was found to be less suitable for simulating cases with complex geometry. In terms of accuracy in both the VoF and rigid-lid approaches, LES best predicted the confluence flow behavior, followed by realizable DES, k-ε, V2F, k-ε, and k-ω SST. Although it had higher cost of CPU time, a good agreement was observed between the LES–VoF results and the experimental data. On the other hand, RANS family models demonstrated relatively identical poor results with minor differences in the cross section immediately after the junction.
- The VoF method was concluded to be a more promising water surface model for complex structures with compound flow behavior than the rigid-lid method.
- The degraded bed scenario exhibited smaller recirculation zones and different secondary flow characteristics compared to the flatbed scenario. The presence of a depositional bar in the degraded bed case affected the flow patterns and secondary circulation.
- In the degraded bed scenario, the recirculation area was discovered to be considerably shorter and narrower or not present at all near the bed as a result of the intricate interplay between the flow and the scour hole and the depositional bar. Secondary circulation in the recirculation zone had different rotation in the degraded bed case than the flatbed case due to the presence of the depositional bar.
- The contraction of the flow in the main channel is weaker in the case of a degraded bed.
- The simulations revealed the formation of vorticity in the confluence zone. The number and behavior of vortices were influenced by the geometry and size of the flow channels. The interactions between separated shear layers played a significant role in the formation and behavior of vortices.
- The generation of vortices in the flow field was mainly ascribed to: (1) variation of the shear layer caused by velocity difference between two channels; and (2) fluid element swirl in the transition zone rooted in the angle between channels.
- The obtained results show the promising applicability of OpenFOAM CFD simulations in resolving the problems associated with the confluence channel design.
- The use of Direct Numerical Simulation (DNS) can improve the accuracy of the study, providing valuable insights into junction areas. However, it is crucial to acknowledge the computational cost associated with DNS, necessitating high-performance computing resources. Future research efforts could focus on optimizing DNS through advancements in computational power or parallelization techniques to make it more feasible for broader applications. Furthermore, to expand our understanding, we propose directing future research towards investigating outfall mixing in the confluence area. This unexplored aspect holds the potential to uncover new dynamics and consequences, contributing to the overall advancement of the field. In balancing the benefits and challenges of DNS, coupled with exploring novel research directions, we aim to provide a more comprehensive overview of our work and stimulate further advancements in the field.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Symbol | Definition |

RANS | Reynolds-Averaged Navier–Stokes |

LES | Large-eddy simulation |

DES | Detached eddy simulation |

VoF | Volume of fluid |

NRMSE | Normalized root-mean-square error |

DNS | Direct Numerical Simulation |

RNG | Renormalization group-based |

FVM | Finite volume method |

LiDAR | Light detection and ranging |

PISO | Pressure implicit with splitting of operator |

SIMPLE | Semi-implicit method for pressure-linked equations |

CFD | Computational fluid dynamic |

NS | Navier–Stokes |

WMLES | Wall model LES |

RMSE | Root-mean-square error |

SGS | Sub Grid-Scale |

α | Confluence junction angle |

R_{Q} | Discharge ratio |

Q_{t} | Tributary discharge |

Q_{m} | Main channel upstream discharge |

Q_{d} | Main channel downstream discharge (Total discharge) |

D | Water depth |

Re | Reynolds number |

Fr | Froude number |

${\mathrm{U}}_{\mathrm{r}}$ | Velocity parallel to the wall |

${\mathrm{U}}_{\mathsf{\tau}}$ | Shear velocity |

$\mathsf{\tau}$ | Bed shear stress |

E | Roughness parameter |

${\mathsf{{\rm Y}}}^{+}$ | Nondimensional wall distance |

$\mathsf{{\rm Y}}$ | Normal distance to the wall |

$\mathsf{\mu}$ | Fluid kinematic viscosity |

$\mathsf{\kappa}$ | von Karman constant |

ρ | Density of the fluid |

P | Pressure |

u_{i} | Mean velocity in the i-direction |

${\mathrm{u}}_{\mathrm{i}}^{\prime}$ | Fluctuating components |

t | Time |

υ | Kinematic viscosity |

$\overline{{\mathrm{u}}_{\mathrm{i}}^{\prime}{\mathrm{u}}_{\mathrm{j}}^{\prime}}$ | Time-averaged turbulent Reynolds shear stresses |

g | Gravitational acceleration |

${\mathrm{u}}_{\mathrm{i}}^{\prime}{\mathrm{u}}_{\mathrm{j}}^{\prime}$ | Turbulent effects |

∆ | Cut-off width in LES models |

${\stackrel{~}{\mathrm{u}}}_{\mathrm{i}}$ | Favre-averaged velocity in tensor notation |

$\mathsf{\rho}$ | Fluid density |

${\mathsf{\vartheta}}_{\mathrm{t}}$ | Eddy viscosity |

${\mathrm{S}}_{\mathrm{i}\mathrm{j}}$ | Local mean strain rate |

Cs | Smagorinsky constant |

${\mathrm{d}}_{\mathrm{wall}}$ | Normal distance to the nearest wall |

h | Mesh interval |

dx, dy, dz | Local mesh dimensions |

Y | Absolute distance from the wall |

U | Velocity streamwise component |

Z | Elevation above the bottom of the flume |

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**Figure 1.**A schematic of the junction presenting different flow zones (Q

_{t}and Q

_{m}being the discharges of the upstream tributary and main channel, respectively).

**Figure 2.**Measured sections, validation profiles for the simulation area, and boundary conditions of the Flume: (

**A**) plan view (

**B**) side view. Q

_{t}is the discharge of tributary channel, while Q

_{m}is the discharge of the main channel. Green circles mark the profiles that are used to compare the modeled and measured data (see more details in Section 3.1).

**Figure 3.**Mesh grids of the domain in OpenFOAM platform, on a scale of 1 to 10 (for RANS turbulence models) in order to visualize the distribution of the mesh grids.

**Figure 4.**Contour map of equilibrium bed morphology in Salome platform. The red solid circle, red dashed circle, red solid rectangle, cyan solid circle, and cyan dashed circle indicate the mid-channel scour hole, downstream scour hole, corner scour hole, upstream bar, and downstream bar, respectively.

**Figure 5.**Measured and modeled streamwise velocities (m/s) (Volume of Fluid) at X = −0.1 and (

**A**) Y = 0.05, (

**B**) Y = 0.21, (

**C**) Y = 0.35. The height of the channel (Z) is identical for all cases.

**Figure 6.**Measured and modeled streamwise velocities (m/s) (Volume of Fluid) at X = 0 and (

**A**) Y = 0.05, (

**B**) Y = 0.21, (

**C**) Y = 0.35. The height of the channel (Z) is identical for all cases.

**Figure 7.**Measured and modeled streamwise velocities (m/s) (Volume of Fluid) at X = 0.3 and (

**A**) Y = 0.05, (

**B**) Y = 0.21, (

**C**) Y = 0. 35. The height of the channel (Z) is identical for all cases.

**Figure 8.**Measured and modeled streamwise velocities (m/s) (Rigid lid) at X = −0.1 and (

**A**) Y = 0.05, (

**B**) Y = 0.21, (

**C**) Y = 0.35. The height of the figures (Z) is identical for all cases.

**Figure 9.**Measured and modeled streamwise velocities (m/s) (Rigid lid) at X = 0 and (

**A**) Y = 0.05, (

**B**) Y = 0.21, (

**C**) Y = 0.35. The height of the figures (Z) is identical for all cases.

**Figure 10.**Measured and modeled streamwise velocities (m/s) (Rigid lid) at X = 0.3 and (

**A**) Y = 0.05, (

**B**) Y = 0.21, (

**C**) Y = 0.35. The height of the channel (Z) is identical for all cases.

**Figure 11.**Comparison of measured and predicted TKE profiles at X = −0.1 and (

**A**) Y = 0.05, (

**B**) Y = 0.21, (

**C**) Y = 0.35.

**Figure 12.**Comparison of measured and predicted TKE (resolved) profiles at X = 0.3 and (

**A**) Y = 0.05, (

**B**) Y = 0.21, (

**C**) Y = 0.35.

**Figure 13.**Distribution of time-averaged longitudinal velocity (m/s) after reaching the steady state (

**top**; degraded bed,

**bottom**; flatbed).

**Figure 14.**Distribution of time-averaged longitudinal velocity (m/s) after the steady state at X = 0.2 (

**Left**; degraded bed,

**Right**; flatbed).

**Figure 15.**Streamlines of simulated time-averaged flow velocity (m/s) in confluence channel with LES model: top, degraded bed; bottom, flatbed. Light blue (upstream main channel) and green (tributary channel) streamlines merge and establish the red streamlines (high velocity) in the downstream channel of the junction. The separation zone is displayed by interwoven blue streamline (

**top**; degraded bed,

**bottom**; flatbed).

**Figure 16.**Cross sections of time-averaged streamwise velocity in the downstream main channel at X = 0.65 (facing downstream), also showing (

**A**,

**B**) in-plane component of time-averaged velocity (v-w vectors), and (

**C**,

**D**) streamlines (

**right**: flatbed case;

**left**: degraded bed case).

**Figure 17.**Cross sections of time-averaged streamwise velocity in the downstream main channel of the flatbed case at X = 0.50 (

**B**) and X = 0.80 (

**D**) and degraded bed at X = 0.50 (

**A**) and X = 0.95 (

**C**).

**Figure 18.**Distribution of the vorticity, in the instantaneous flow (Z = 0.12) (

**top**: degraded bed,

**bottom**: flatbed).

**Figure 19.**Distribution of the vorticity (

**left**: degraded bed,

**right**: flatbed) ((

**A**,

**B**); X = 0.2 and (

**C**,

**D**); X = 0.65).

Research | Model Dimensions | Governing Equations | Model Domain | Specific Investigation |
---|---|---|---|---|

Bradbrook et al. (2000) [14] | 3-D | RANS | Flatbed | Flow structures in symmetrical and asymmetrical confluences |

Huang et al. (2002) [17] | 3-D | RANS | Flatbed | Flow structure with different confluence angle |

Shakibaeinia et al. (2010) [19] | 3-D | RANS | Flatbed | Secondary flow formation with different junction angles |

Constantinescu et al. (2011) [30] | 3-D | DES | Deformed bed | The effect of momentum ratio on the formation of secondary flows |

Song et al. (2012) [20] | 3-D | RANS | Flatbed | The formation of secondary flows |

Sukhodolov et al. (2017) [3] | 3-D | RANS | Deformed bed | The formation of secondary flows with bed discordance |

Schindfessel et al. (2017) [24] | 3-D | LES | Flatbed | The effect of cross-sectional shape on separation zone |

Tang et al. (2018) [22] | 3-D | RANS | Deformed bed | Contaminant transport and pollutant dispersion |

Shaheed et al. (2019) [21] | 3-D | RANS | Flatbed | Secondary flow investigations |

Ramos et al. (2019) [31] | 3-D | LES | Flatbed | Investigation of flow structure by curved rigid lid |

Cheng and Constantinescu (2020) [6] | 3-D | DES | Deformed bed | The impact of stratification on confluence channels |

Horna-Munoz et al. (2020) [13] | 3-D | DES | Deformed bed | Density differences between flows |

Yan et al. (2022) [32] | 2-D | Deformed bed | Modification of anisotropy information | |

This Study | 3-D | LES, DES, RANS | Deformed bed | Investigation of secondary flow with different geometry |

**Table 2.**Summary of the hydraulic and geometric data [33].

Variable | Symbol (Unit) | Value |
---|---|---|

Confluence junction angle | α (-) | 90° |

Discharge ratio | R_{Q} * (-) | 3:2 |

Tributary discharge | Q_{t} (L/s) | 9 |

Main channel upstream discharge | Q_{m} (L/s) | 6 |

Main channel downstream discharge (Total discharge) | Q_{d} (L/s) | 15 |

Water depth | D (m) | 0.16 |

Reynolds number | Re (-) | 0.145 × 10^{5} |

Froude number | Fr (-) | 0.15 |

_{Q}= Q

_{t}/Q

_{m.}

Turbulence Model | Number of Cells | Number of Cells | Number of Cells | Number of Cells | Number of Cells |
---|---|---|---|---|---|

RANS | 801,254 | 1,004,209 | 1,205,052 | - | - |

LES | 801,254 | 1,004,209 | 1,205,052 | 6,376,502 | 7,456,287 |

DES | 801,254 | 1,004,209 | 1,205,052 | 6,376,502 | 7,456,287 |

X (m) | Y (m) | Error | k-ε | Realizable k-ε | k-ω SST | LES | DES | V2F |
---|---|---|---|---|---|---|---|---|

−0.1 | 0.05 | RMSE (m/s) | 0.009 | 0.009 | 0.011 | 0.005 | 0.006 | 0.009 |

NRMSE (-) | 0.10 | 0.10 | 0.12 | 0.06 | 0.07 | 0.10 | ||

−0.1 | 0.21 | RMSE (m/s) | 0.010 | 0.011 | 0.016 | 0.005 | 0.005 | 0.010 |

NRMSE | 0.08 | 0.09 | 0.13 | 0.04 | 0.04 | 0.08 | ||

−0.1 | 0.35 | RMSE (m/s) | 0.010 | 0.011 | 0.012 | 0.002 | 0.005 | 0.011 |

NRMSE | 0.07 | 0.08 | 0.08 | 0.01 | 0.03 | 0.07 | ||

0 | 0.05 | RMSE (m/s) | 0.004 | 0.006 | 0.005 | 0.002 | 0.003 | 0.005 |

NRMSE | 0.03 | 0.05 | 0.04 | 0.02 | 0.02 | 0.04 | ||

0 | 0.21 | RMSE (m/s) | 0.009 | 0.007 | 0.007 | 0.002 | 0.005 | 0.007 |

NRMSE | 0.05 | 0.04 | 0.04 | 0.01 | 0.03 | 0.04 | ||

0 | 0.35 | RMSE (m/s) | 0.012 | 0.009 | 0.011 | 0.003 | 0.003 | 0.008 |

NRMSE | 0.07 | 0.05 | 0.06 | 0.02 | 0.01 | 0.04 | ||

0.3 | 0.05 | RMSE (m/s) | 0.104 | 0.096 | 0.094 | 0.017 | 0.028 | 0.103 |

NRMSE | 0.51 | 0.47 | 0.46 | 0.08 | 0.14 | 0.51 | ||

0.3 | 0.21 | RMSE (m/s) | 0.084 | 0.081 | 0.092 | 0.005 | 0.010 | 0.081 |

NRMSE | 0.28 | 0.27 | 0.31 | 0.02 | 0.03 | 0.27 | ||

0.3 | 0.35 | RMSE | 0.042 | 0.040 | 0.045 | 0.008 | 0.028 | 0.041 |

NRMSE | 0.17 | 0.16 | 0.18 | 0.03 | 0.11 | 0.17 | ||

Average NRMSE | 0.15 | 0.15 | 0.16 | 0.03 | 0.15 | 0.15 |

X | Y | Error | k-ε | Realizable k-ε | k-ω SST | LES | DES | V2F |
---|---|---|---|---|---|---|---|---|

−0.1 | 0.05 | RMSE (m/s) | 0.026 | 0.026 | 0.027 | 0.028 | 0.027 | 0.027 |

NRMSE | 0.28 | 0.28 | 0.30 | 0.31 | 0.30 | 0.30 | ||

−0.1 | 0.21 | RMSE (m/s) | 0.070 | 0.071 | 0.073 | 0.061 | 0.073 | 0.070 |

NRMSE | 0.60 | 0.61 | 0.63 | 0.53 | 0.63 | 0.60 | ||

−0.1 | 0.35 | RMSE (m/s) | 0.053 | 0.053 | 0.051 | 0.049 | 0.051 | 0.053 |

NRMSE | 0.36 | 0.35 | 0.34 | 0.33 | 0.34 | 0.36 | ||

0 | 0.05 | RMSE (m/s) | 0.004 | 0.006 | 0.005 | 0.002 | 0.003 | 0.005 |

NRMSE | 0.03 | 0.05 | 0.04 | 0.02 | 0.02 | 0.04 | ||

0 | 0.21 | RMSE (m/s) | 0.009 | 0.007 | 0.007 | 0.002 | 0.005 | 0.007 |

NRMSE | 0.05 | 0.04 | 0.04 | 0.01 | 0.03 | 0.04 | ||

0 | 0.35 | RMSE (m/s) | 0.012 | 0.009 | 0.011 | 0.003 | 0.003 | 0.008 |

NRMSE | 0.07 | 0.05 | 0.06 | 0.02 | 0.01 | 0.04 | ||

0.3 | 0.05 | RMSE (m/s) | 0.104 | 0.096 | 0.094 | 0.017 | 0.028 | 0.103 |

NRMSE | 0.51 | 0.47 | 0.46 | 0.08 | 0.14 | 0.51 | ||

0.3 | 0.21 | RMSE (m/s) | 0.084 | 0.081 | 0.092 | 0.005 | 0.010 | 0.081 |

NRMSE | 0.28 | 0.27 | 0.31 | 0.02 | 0.03 | 0.27 | ||

0.3 | 0.35 | RMSE (m/s) | 0.042 | 0.040 | 0.045 | 0.008 | 0.028 | 0.041 |

NRMSE | 0.17 | 0.16 | 0.18 | 0.03 | 0.11 | 0.17 | ||

Average NRMSE | 0.26 | 0.25 | 0.26 | 0.15 | 0.18 | 0.26 |

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## Share and Cite

**MDPI and ACS Style**

Behzad, E.; Mohammadian, A.; Rennie, C.D.; Yu, Q.
Numerical Simulation of Confluence Flow in a Degraded Bed. *Water* **2024**, *16*, 85.
https://doi.org/10.3390/w16010085

**AMA Style**

Behzad E, Mohammadian A, Rennie CD, Yu Q.
Numerical Simulation of Confluence Flow in a Degraded Bed. *Water*. 2024; 16(1):85.
https://doi.org/10.3390/w16010085

**Chicago/Turabian Style**

Behzad, Ehsan, Abdolmajid Mohammadian, Colin D. Rennie, and Qingcheng Yu.
2024. "Numerical Simulation of Confluence Flow in a Degraded Bed" *Water* 16, no. 1: 85.
https://doi.org/10.3390/w16010085