# Numerical study of bed slope change effect of circular channel with side weir in supercritical flow conditions

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## Abstract

In this numerical study, the pattern and the field of the passing flow through circular channels with the side weir in supercritical flow conditions are simulated using the FLOW-3D model. First, the numerical model results are validated with the experimental data and a good agreement between the results is obtained from the numerical simulation and the experimental data. Then, the effects of the slope change of the circular channel bed on the flow free surface, the discharge of the side weir, the Froude number at the end of the side weir upstream, the discharge coefficient of the side weir, the specific energy and the slope of the S3 profile are investigated. In this study, the flow field turbulence and variations of the free surface computational field are simulated using the RNG \(k - \varepsilon\) turbulence model and the volume of fluid (VOF) method, respectively. In practice, transmission channels used in irrigation networks and urban sewage disposal systems are steep. Therefore, the main purpose of this computational fluid dynamics (CFD) study is investigating the effect of the bed slope change of circular channels with side weir in supercritical flow conditions on the pattern and the structure of these types of flows.

## Keywords

Supercritical flow Circular channel Side weir Bed slope of main channel Numerical simulation## List of symbols

- \(A_{x} ,A_{y} ,A_{z}\)
Fractional areas open to flow (m

^{2})- \(A\)
Cross-sectional area (m

^{2})- \(C_{d}\)
Discharge coefficient (–)

- \(C_{u}\)
Constant coefficient (–)

- \(D\)
Channel diameter (m)

- \(E_{1}\)
Specific energy at section 1 in the main channel (m)

- \(E_{2}\)
Specific energy at section 2 in the main channel (m)

- \(F\)
Fluid volume fraction in a cell (–)

- \(F_{1}\)
Froude number at beginning of side weir on axis of main channel (–)

- \(f_{x} ,f_{y} ,f_{z}\)
Viscous accelerations (m/s

^{2})- \(G_{x} ,G_{y} ,G_{z}\)
Body accelerations (m/s

^{2})- \(g\)
Acceleration gravity (m/s

^{2})- \(k_{t}\)
Turbulence kinetic energy (m/s

^{2})- \(L\)
Side weir length (m)

- \(P\)
Side weir height (m)

- \(p\)
Pressure (N/m

^{2})- \(Q_{1}\)
Discharge at section 1 in the main channel (m

^{3}/s)- \(Q_{w}\)
Discharge over the side weir (m

^{3}/s)- \(q\)
Discharge per unit length of the side weir (m

^{2}/s)- \(R_{\text{SOR}}\)
Mass source (–)

- \(S_{f}\)
Friction losses (–)

- \(S_{0}\)
Bed slope of the channel (–)

*T*Width of the channel at the water surface (m)

- \(T_{\text{len}}\)
Turbulent length scale (m)

- \(t\)
Time (s)

- \(u,v,w\)
Velocity components (m/s)

- \(V_{F}\)
Fractional volume open to flow (–)

- \(X,Y,Z\)
Cartesian coordinate directions (m)

- \(x\)
Distance from upstream of the weir (m)

- \(z\)
Depth of the flow (m)

- \(z_{1}\)
Depth of the flow at section 1 in the main channel (m)

- \(z_{2}\)
Depth of the flow at section 2 in the main channel (m)

- \(\alpha\)
Velocity distribution coefficient

- \(\varepsilon_{t}\)
Turbulence dissipation rate (m

^{2}/s^{3})- \(\theta\)
Slope of the S3 profile (rad)

- \(\nu_{t}\)
Turbulent kinematic viscosity (m

^{2}/s)- \(\rho\)
Fluid density (kg/m

^{3})

## Introduction

A side weir is installed on the side wall of the main channel in order to adjust the flow level. Also, flow within open channel along the side weir is considered as spatially varied flow (SVF) with decreasing discharge. Moreover, many experimental and analytical researches and studies have been conducted on the flow pattern of these types of structures. There are so many studies on the flow within rectangular channels along the side weir carried out by various researchers. For example, Ackers (1957), Nandesamoorthy and Thomson (1972), Yu-Tech (1972), RangaRaju et al. (1979), Hager (1982), Uyumaz and Smith (1991), Jalili and Borghei (1996), Borghei et al. (1999), Khorchani and Blanpain (2004), Vatankhah and Bijankhan (2009), Emiroglu et al. (2010, 2011), Bagheri and Heidarpour (2012), Novak et al. (2013), Mohammed (2013) studied the hydraulic behavior of the side weir flow. Additionally, various studies were carried out related to trapezoidal channels along the side weir. For instance, Cheong (1991) presented an equation in order to compute the discharge coefficient of side weirs located on trapezoidal conduit. Also, Vatankhah (2012a) proposed an analytical solution to predict the variations of flow free surface along the side weir for subcritical and supercritical flow regimes. For U-shaped channels along the side weir, Uyumaz (1997) using the specific energy principles and the finite difference method provided an analytical solution for predicting variations of the flow free surface profile along the side weirs. Uyumaz, by conducting some experiments in both subcritical and supercritical flow conditions, showed an acceptable agreement between the experimental and mentioned method results. Vatankhah (2013) introduced a semi-analytical solution for predicting variations of the longitudinal profile of the flow free surface along the side weirs located on U-shaped channels. The method was based on the energy principles obtained by integration solution. Additionally, for triangular channels along the side weirs, Uyumaz (1992) presented an analytical solution for calculating variations of the flow free surface along the side weirs in subcritical and supercritical flow conditions. In addition, Vatankhah (2012b) using integration and the principles of specific energy solved the governing dynamical equation on the passing flow over side weirs located on triangular channels. Vatankhah’s analytical method predicts the free surface flow profile along the side weir with an acceptable accuracy. Moreover, channels with circular cross section are used in urban sewage disposal networks and drainage systems. In association with circular channels along the side weirs, Allen (1957) conducted an experimental study on the hydraulic behavior of the passing flow over rectangular side weirs. Uyumaz and Muslu (1985) by assuming that the flow in the main channel is two dimensional and the pressure is hydrostatical predicted the longitudinal profile of the flow free surface along the side weir and the passing flow over it using the finite element method. Their analytical method for both supercritical and subcritical flow conditions was validated by experimental results. Ramamurthy et al. (1995) through the two-dimensional flow theory provided an analytical method for predicting the passing flow over a side weir located on a circular channel. They introduced the theoretical discharge coefficient of the side weir as a function of the geometric parameters of the side weir. Oliveto et al. (2001) investigated the characteristics of supercritical flow conditions along the side weirs located on circular channels. Their experiments have been focused on the local measurement of the flow output angle and the velocity along the side weir. In all experiments, the side weir upstream flow is subcritical and the flow along the side weir is supercritical. Vatankhah (2012c) by the incomplete elliptic integrals method and the specific energy principle provided a semi-analytical solution for calculating the longitude profile of the flow free surface in both subcritical and supercritical flow conditions. In practice, it is possible that the flow along the side weir be supercritical. Many experimental and theoretical studies have been conducted on supercritical flows passing through channels with side weir. Hager (1994) provided an analytical method for supercritical flows passing through circular channels with side weir. His method is used in both flow conditions: with hydraulic jumps and without hydraulic jumps. He validated the obtained results from his analytical method by the experimental results obtained by other researches. Ghodsian (2003) investigated the passing supercritical flow over the side weir located on a rectangular channel. He introduced the elementary discharge coefficient of the side weir in supercritical flow conditions as a function of the ratio of the head above the weir to the weir crest height and the Froude number. Mizumura et al. (2003) provided an analytical solution for calculating the main channel discharge in a rectangular channel with side weir in supercritical flow conditions. Their analytical method was based on the energy principles, and the main channel discharge was introduced as a function of the Froude number. Pathirana et al. (2006) conducted an experimental study including investigation of the upstream specific energy, the discharge coefficient and the passing flow over the side weir in supercritical flow conditions for a rectangular channel equipped with a side weir. Using the specific energy principles and the regression analysis, they provided a relationship for calculating the discharge coefficient of side weirs in supercritical flow conditions. Rao and Pillai (2008) investigated the discharge coefficient of the side weir, the velocity in the main channel, the flow falling angle from the side weir and measured the flow depth along the weir by conducting an experimental study on a rectangular channel with side weir in supercritical flow conditions. Coonrod et al. (2009) investigated the effect of guide vanes on increasing the diversion flow rate over side weirs located on rectangular channels in supercritical flow conditions. In recent years, computational fluids dynamics (CFD) has been introduced as a powerful tool for understanding the hydrodynamic behavior of fluids and the interaction between the fluid and the structure. Also, many studies were carried out in order to simulate the flow field of main channels along the side weir. For example, Qu (2005) modeled three-dimensional (3D) changes of the flow in rectangular channels along the side weir using the \(k - \omega\) turbulence model and the volume of fluid (VOF) scheme. Tadayon (2009) developed a numerical model to simulate the flow field within rectangular conduits along the side weir. The numerical model simulated the flow turbulence and variations of the flow free surface with the RSM turbulence model and the VOF method, respectively. Aydin (2012) using the FLUENT software simulated the flow free surface within a rectangular channel along the triangular labyrinth side weir by the VOF scheme. Aydin and Emiroglu (2013) using the ANSYS FLUENT software simulated the discharge capacity of labyrinth side weirs. They modeled the free surface changes by the VOF method and the flow turbulence by different turbulence models. Furthermore, they showed that the discharge coefficient increased by increasing the Froude number. Azimi et al. (2014) simulated the velocity field along the side weir in circular channels using RNG \(k - \varepsilon\) and VOF scheme. Also, Azimi and Shabanlou (2015) modeled the flow pattern in triangular channels along the side weir. The effect of the Froude number was examined on the characteristics of the flow field.

Intense studies have been carried out on the application of numerical simulation in modeling engineering problems, with the number of such studies increasing every day. However, according to the literature, despite the importance of awareness about open channels along the side weirs in designing hydraulic structures, very few studies have focused on variations of slop bed. As a result, the current study is deemed preliminary in this direction.

In this numerical simulation, the flow field and the circular channel turbulence are simulated using the FLOW-3D model. In this numerical model, the free surface variations of the flow field are simulated by the VOF method and the flow turbulence is simulated using the RNG \(k - \varepsilon\) turbulence model. In practice, circular channels used in sewage disposal networks are steep. Therefore, in this numerical study, the effects of increasing the main channel bed slope on the hydraulic behavior of supercritical flows along the side weirs are investigated.

## Governing equations

*x-*,

*y-*and

*z*-directions, respectively. Also

*t*, \(\rho\), \(R_{\text{SOR}}\), \(p\) and \(V_{F}\) are the time, the density, the term of the source, the pressure and the fractional volume open to the flow, respectively. In this numerical study, the RNG \(k - \varepsilon\) turbulence model is used for simulating the flow field. Among turbulence methods, the RNG \(k - \varepsilon\) turbulence model is more applicable. Furthermore, this turbulence model describes the flow behavior near the solid wall with a suitable accuracy. In the VOF model, the following continuity equation is solved to calculate the volume component:

*F*is the volume component of the fluid in a specified computational cell. If \(F = 0\), the cell is full of air, and if \(F = 1\) the computational cell is filled with the fluid, and if \(0 < F < 1\) the cell contains air and water.

## Boundary conditions

All of the solid walls including the side walls and the main channel bed are considered as the “wall” boundary condition. The nonslip condition is imposed to the wall boundary condition. At the wall boundary condition, the typical velocity should be considered zero and there is no need to define turbulence parameters such as the turbulence kinetic energy and the turbulence dissipation rate. In this numerical study, the entire upper surface of the air phase is considered as the symmetry boundary conditions. In symmetry boundary conditions, the velocity change is zero across the boundary condition; thus, the turbulence is not generated.

## Experimental data

Number of computational cells used in numerical simulation

Location | Direction | Direction | Direction |
---|---|---|---|

Main channel | 196 | 60 | 58 |

Tank | 60 | 30 | 58 |

*x-*,

*y-*and

*z*-directions are considered close to 1. Therefore, the flow field is separated from the gridding by an optimized and independent mesh. In order to investigate the prediction accuracy of the numerical model, the average percent error (APE) of the flow free surface and the root mean square error percent (RMSE) are calculated using Eqs. (9) and (10), respectively:

Characteristics of griddings, APE and RMSE of simulated free surface profile

Meshing | Number of cells | APE (%) | RMSE (%) |
---|---|---|---|

1 | 85,500 | 8.507 | 7.351 |

2 | 189,920 | 5.036 | 4.744 |

3 | 400,896 | 3.279 | 2.014 |

4 | 786,480 | 1.193 | 0.182 |

5 | 936,448 | 1.191 | 0.181 |

## Results and discussion

### Validation

^{3}/s, 0.1164 m and 0.0806 m, respectively. In the experimental model used by Uyumaz and Muslu, the main channel slope has been considered 0.005 and the flow within the main channel is in supercritical flow conditions. In Fig. 4, the comparison between the simulated and experimental free surface is shown. The values of APE and RMSE of the simulated free surface profile were calculated 1.193% and 0.182%, respectively. Therefore, the numerical model simulates the flow with an acceptable accuracy, so that according to the supercritical flow pattern, the flow depth along the side weir decreases from the upstream end toward the downstream end.

Comparison of experimental and numerical Froude number and discharge coefficient

As shown, the numerical model predicts the side weir discharge coefficient with high accuracy. To evaluate the accuracy of the numerical model, APE and RMSE values of the side weir discharge coefficient are calculated 4.61% and 0.022%, respectively. Uyumaz and Muslu (1985)have calculated the specific energy level at the side weir upstream section equal to 17.99 cm, while the numerical model has predicted the specific energy value at the side weir upstream section equal to 18.67 cm. The relative error percent value is calculated 3.8% that indicates high accuracy of the numerical model in predicting the specific energy value.

### Slope effects of flow field in main channel

#### Slope effect on flow free surface

Changes of \(Q_{w} ,F_{1} ,C_{d}\) versus increasing of main channel bed \((S_{0} )\)

\(\left( {S_{0} } \right)\) | 0.005 | 0.01 | 0.02 |
---|---|---|---|

\(Q_{w}\) (m | 00,054 | 0.0045 | 0.0032 |

\(F_{1}\) | 1.1 | 1.22 | 1.46 |

\(C_{d}\) | 0.358 | 0.352 | 0.339 |

### Specific energy

*E*) is specific energy, (\(\alpha\)) is the velocity distribution coefficient, (\(Q_{1}\)) is the discharge within the main channel and (

*A*) is the flow cross section.

#### Effect of channel slope on specific energy

Changes in simulated specific energy for main channel with different slopes

\(S_{0}\) | 0.005 | 0.01 | 0.02 |
---|---|---|---|

\(E_{1}\) (m) | 0.1867 | 0.1931 | 0.2081 |

\(E_{2}\) (m) | 0.1836 | 0.1926 | 0.2050 |

#### Effect of main channel bed slope on S3 profile slope

Changes in S3 profile slope versus main channel bed slope

\(S_{0}\) | 0.005 | 0.01 | 0.02 |
---|---|---|---|

\(\theta\) (rad) | 0.0187 | 0.0179 | 0.01 |

## Conclusions

Side weirs are used as flow control structures in flood control projects, drainage networks and irrigation lands. Circular channels are widely used in sewage disposal networks. In practice, flood flows, catchment from dams and flows within transmission lines are supercritical. In this numerical modeling, the turbulence and the flow field passing through the circular channel with side weir in supercritical flow conditions were simulated using the FLOW-3D model. The changes in the free surface of the numerical model and the flow turbulence were simulated using the VOF method and the RNG \(k - \varepsilon\) model, respectively. In this numerical study, the comparison of the flow free surface, the discharge coefficient of the side weir \((C_{d} )\), the Froude number of the side weir upstream end \((F_{1} )\), the passing discharge over the side weir \((Q_{w} )\) and the specific energy at the weir upstream indicated a good agreement between the numerical model and experimental results in predicting the flow characteristics. Then, the effects of circular channel slope change on transverse and longitudinal flow profiles along the side weir in supercritical flow conditions were investigated. By increasing the main channel bed slope, the flow depth decreases although this depth decreasing is negligible. Increasing of circular channel bed slope causes reduction in the passing discharge over the side weir and in opposite, the Froude number of the side weir upstream end increases. By increasing the bed slope, the calculated discharge coefficient by Uyumaz and Muslu (1985) shows a decreasing trend. Analysis of the numerical model results shows that the specific energy value at the upstream and downstream ends of the side weir in supercritical flow conditions is almost constant and the specific energy average difference between the beginning and the end of the weir was calculated equal to 2.89%. Then, the effects of the circular channel bed slope change on the specific energy value were investigated, so that the analysis of the numerical results showed increasing trend of the specific energy by increasing of the main channel bed slope. In supercritical flow conditions and steep channels with side weir, the flow depth at the weir downstream end reaches to its lowest value, and then, this depth is connected to the tailwater depth by the S3 profile. Finally, the effects of the main channel bed slope changes on the S3 profile slope were investigated and it was concluded that the increase in the circular channel bed slope \((S_{0} )\) leads to the decrease in the S3 profile slope \((\theta )\).

## Notes

### Compliance with ethical standards

### Conflict of interest

All authors declare they have no conflict of interest.

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