# Three-Dimensional Turbulence Numerical Simulation of Flow in a Stepped Dropshaft

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Simulation

#### 2.1. Volume of Fluid Method

#### 2.2. Turbulence Model

^{®}, Canonsburg, PA, USA), Fluent [43], was utilized to investigate the flow over the stepped dropshaft. The equations of the turbulent kinetic energy, k, and its dissipation rate, ε, are as follows:

#### 2.3. Numerical Algorithm

#### 2.4. Geometric Model

#### 2.5. Boundary Conditions and Cases

- (1)
- Inlet boundary: the velocity inlet was used for the intake, which was set at 0.89–2.69 m/s;
- (2)
- Outlet boundary: the outlet boundary was set as pressure outlet and the normal gradient of all variables was equal to 0;
- (3)
- Free surface: the free surface of water was assumed to be the pressure inlet and the pressure value was P = 0; and
- (4)
- Wall boundary: no-slip velocity boundary condition; the near-wall regions of the flow were analyzed using the method of standard wall function.

#### 2.6. Verification

#### 2.6.1. Grid Testing

^{3}/s. The data obtained from grid 2 were close to the data obtained with the grid 1 and the computational efficiency decreased by 20%. In Figure 2c,e, the maximum uncertainties in pressure and tangential velocity were approximately 3.12% and 4.17%, respectively, implying that the discretization uncertainties were little in most locations. For the consideration of computational efficiency and accuracy, the grid number of 0.46 million (grid 2) was adopted in the present study.

#### 2.6.2. Model Verification

^{3}/s was smaller than that of Q = 12.75 m

^{3}/s; and (2) for Q = 12.75 m

^{3}/s, there was an increase in the water level when flow entered the next step, but for Q = 41.5 m

^{3}/s, the increase in water level occurred after the nappe flow impacted the horizontal step. As can be seen from Figure 3b,d, the simulated flow patterns were in good agreement with experimental results.

^{3}/s and Q = 41.5 m

^{3}/s were 7.5% and 10%, respectively, indicating that the numerical simulations produced reliable and acceptable results.

#### 2.6.3. Fluctuation of Calculation Results

## 3. Results and Analysis

#### 3.1. Region Division in the Flow

^{3}/s was analyzed.

#### 3.2. Regional Scope

#### 3.3. Water Depth

_{max}is the maximum water depth and ${\alpha}_{\mathrm{max}}$ is the relative position of the maximum water depth. The distance between the steps is h

_{s}, and the relative position of the vertical plane of the upper step is ${\alpha}_{s}$. A dimensionless height, h

_{s}/h, for $\theta $ = 120° and $\theta $ = 180° is, respectively, 3 and 2, and the location, ${\alpha}_{s}$, for both $\theta $ = 120° and $\theta $ = 180° is 0. However, for $\theta $ = 150°, h

_{s}/h and ${\alpha}_{s}$ vary significantly with h

_{s}/h ranging from 3 to 2 and ${\alpha}_{s}$ ranging from 0° to 60°.

_{max}/h and ${\alpha}_{\mathrm{max}}$ increased as Q increased.

_{max}/h and α

_{max}showed no obvious trend as $\theta $ increased.

_{max}/h decreased initially and then increased, whereas ${\alpha}_{\mathrm{max}}$ exhibited the opposite phenomenon.

_{max}/h increased and ${\alpha}_{\mathrm{max}}$ presented no evident change.

_{max}/h depended on the discharge and the central angle of step, whereas the range of ${\alpha}_{\mathrm{max}}$ was associated with the central angle of step. For $\theta $ = 180°, when h

_{max}/h = 2 (Q = 48 m

^{3}/s, 80 m

^{3}/s), the flow impacted the bottom of the upper step, which can only occur on some steps when $\theta $ = 150° for Q = 80 m

^{3}/s. For $\theta $ = 120°, h

_{max}/h was considerably lower than the height of the upper step in all discharge.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Rajaratnam, N.; Mainali, A.; Hsung, C. Observations on flow in vertical dropshafts in urban drainage systems. J. Environ. Eng.
**1997**, 123, 486–491. [Google Scholar] [CrossRef] - Chanson, H. Hydraulics of rectangular dropshafts. J. Irrig. Drain. Eng.
**2004**, 130, 523–529. [Google Scholar] [CrossRef] - Chanson, H. Understanding air–water mass transfer in rectangular dropshafts. J. Environ. Eng. Sci.
**2004**, 3, 319–330. [Google Scholar] [CrossRef] [Green Version] - Adriana Camino, G.; Zhu, D.Z.; Rajaratnam, N. Flow observations in tall plunging flow dropshafts. J. Hydraul. Eng.
**2014**, 141, 06014020. [Google Scholar] [CrossRef] - Granata, F.; de Marinis, G.; Gargano, R.; Hager, W.H. Hydraulics of circular drop manholes. J. Irrig. Drain. Eng.
**2010**, 137, 102–111. [Google Scholar] [CrossRef] - Ma, Y.; Zhu, D.Z.; Rajaratnam, N.; van Duin, B. Energy dissipation in circular drop manholes. J. Irrig. Drain. Eng.
**2017**, 143, 04017047. [Google Scholar] [CrossRef] - Jain, S.C. Air transport in vortex-flow drop shafts. J. Hydraul. Eng.
**1988**, 114, 1485–1497. [Google Scholar] [CrossRef] - Vischer, D.; Hager, W. Vortex drops. In Energy Dissipators; Routledge: Abingdon, UK, 2018; pp. 167–181. ISBN 9781351451345. [Google Scholar]
- Odgaard, A.J.; Lyons, T.C.; Craig, A.J. Baffle-drop structure design relationships. J. Hydraul. Eng.
**2013**, 139, 995–1002. [Google Scholar] [CrossRef] - Stirrup, M.; Margevicius, T.; Hrkac, T.; Baca, A. A baffling solution to Sewage Conveyance In York Region, Ontario. Proc. Water Environ. Fed.
**2012**, 2012, 74–90. [Google Scholar] [CrossRef] - Kennedy, J.F.; Jain, S.C.; Quinones, R.R. Helicoidal-ramp dropshaft. J. Hydraul. Eng.
**1988**, 114, 315–325. [Google Scholar] [CrossRef] - Tamura, S.; Matsushima, O.; Yoshikawa, S. Helicoidal-ramp type drop shaft to deal with high head drop works in manholes. Proc. Water Environ. Fed.
**2010**, 2010, 4991–5002. [Google Scholar] [CrossRef] - Jain, S.C.; Kennedy, J.F. Vortex-Flow Drop Structures for the Milwaukee Metropolitan Sewerage District Inline Storage System; Iowa Institute of Hydraulic Research, The University of Iowa: Iowa City, IA, USA, 1983. [Google Scholar]
- Hager, W.H. Wastewater Hydraulics: Theory and Practice; Springer Science & Business Media: Berlin, Germany, 2010. [Google Scholar]
- Zhao, C.H.; Zhu, D.Z.; Sun, S.K.; Liu, Z.P. Experimental study of flow in a vortex drop shaft. J. Hydraul. Eng.
**2006**, 132, 61–68. [Google Scholar] [CrossRef] - Del Giudice, G.; Gisonni, C. Vortex dropshaft retrofitting: Case of Naples city (Italy). J. Hydraul. Res.
**2011**, 49, 804–808. [Google Scholar] [CrossRef] - Natarius, E.M. Aeration performance of vortex flow insert assemblies in sewer drop structures. Proc. Water Environ. Fed.
**2008**, 2008, 842–851. [Google Scholar] [CrossRef] - Yu, D.; Lee, J.H. Hydraulics of tangential vortex intake for urban drainage. J. Hydraul. Eng.
**2009**, 135, 164–174. [Google Scholar] [CrossRef] - Wu, J.H.; Yang, T.; Sheng, J.Y.; Ren, W.C.; Fei, M.A. Hydraulic characteristics of stepped spillway dropshafts with large angle. Chin. J. Hydrodyn.
**2018**, 33, 176–180. (In Chinese) [Google Scholar] [CrossRef] - Christodoulou, G.C. Energy dissipation on stepped spillways. J. Hydraul. Eng.
**1993**, 119, 644–650. [Google Scholar] [CrossRef] - Sorensen, R.M. Stepped spillway hydraulic model investigation. J. Hydraul. Eng.
**1985**, 111, 1461–1472. [Google Scholar] [CrossRef] - Frizell, K.W.; Renna, F.M.; Matos, J. Cavitation potential of flow on stepped spillways. J. Hydraul. Eng.
**2012**, 139, 630–636. [Google Scholar] [CrossRef] - Wu, J.H.; Ren, W.C.; Ma, F. Standing wave at dropshaft inlets. J. Hydrodyn. Ser. B
**2017**, 29, 524–527. [Google Scholar] [CrossRef] - Hirt, C.W.; Nichols, B.D. Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys.
**1981**, 39, 201–225. [Google Scholar] [CrossRef] - Peng, Y.; Zhou, J.G.; Burrows, R. Modelling the free surface flow in rectangular shallow basins by lattice Boltzmann method. J. Hydrau. Eng.
**2011**, 137, 1680–1685. [Google Scholar] [CrossRef] - Peng, Y.; Zhou, J.G.; Burrows, R. Modelling solute transport in shallow water with the lattice Boltzmann method. Comput. Fluids.
**2011**, 50, 181–188. [Google Scholar] [CrossRef] - Peng, Y.; Zhang, J.M.; Zhou, J.G. Lattice Boltzmann Model Using Two-Relaxation-Time for Shallow Water Equations. J. Hydrau. Eng.
**2016**, 142, 06015017. [Google Scholar] [CrossRef] - Peng, Y.; Zhang, J.M.; Meng, J.P. Second order force scheme for lattice Boltzmann model of shallow water flows. J. Hydraul. Res.
**2017**, 55, 592–597. [Google Scholar] [CrossRef] - Peng, Y.; Mao, Y.F.; Wang, B.; Xie, B. Study on C-S and P-R EOS in pseudo-potential lattice Boltzmann model for two-phase flows. Int. J. Mod. Phys. C
**2017**, 28, 1750120. [Google Scholar] [CrossRef] - Peng, Y.; Wang, B.; Mao, Y.F. Study on force schemes in pseudopotential lattice Boltzmann model for two-phase flows. Math. Probl. Eng.
**2018**. [Google Scholar] [CrossRef] - Galván, S.; Reggio, M.; Guibault, F. Assessment study of k-ε turbulence models and near-wall modeling for steady state swirling flow analysis in draft tube using fluent. Eng. Appl. Comput. Fluid
**2011**, 5, 459–478. [Google Scholar] [CrossRef] - Morovati, K.; Eghbalzadeh, A.; Javan, M. Numerical investigation of the configuration of the pools on the flow pattern passing over pooled stepped spillway in skimming flow regime. Acta Mech.
**2015**, 227, 1–14. [Google Scholar] [CrossRef] - Devolder, B.; Troch, P.; Rauwoens, P. Performance of a buoyancy-modified k-ω and k-ω SST turbulence model for simulating wave breaking under regular waves using OpenFOAM
^{®}. Coast. Eng.**2018**, 138, 49–65. [Google Scholar] [CrossRef] - Fuhrman, D.R.; Dixen, M.; Jacobsen, N.G. Physically-consistent wall boundary conditions for the k-ω turbulence model. J. Hydraul. Res.
**2010**, 48, 793–800. [Google Scholar] [CrossRef] - Bai, Z.L.; Zhang, J.M. Comparison of different turbulence models for numerical simulation of pressure distribution in V-shaped stepped spillway. Math. Probl. Eng.
**2017**, 2017, 1–9. [Google Scholar] [CrossRef] - Li, S.; Zhang, J. Numerical investigation on the hydraulic properties of the skimming flow over pooled stepped spillway. Water
**2018**, 10, 1478. [Google Scholar] [CrossRef] - Chan, S.; Qiao, Q.; Lee, J.H. On the three-dimensional flow of a stable tangential vortex intake. J. Hydro-Environ. Res.
**2018**, 21, 29–42. [Google Scholar] [CrossRef] - Gao, X.P.; Zhang, H.; Liu, J.J.; Sun, B.; Tian, Y. Numerical investigation of flow in a vertical pipe inlet/outlet with a horizontal anti-vortex plate: Effect of diversion orifices height and divergence angle. Eng. Appl. Comput. Fluid Mech.
**2018**, 12, 182–194. [Google Scholar] [CrossRef] - Yakhot, V.; Orszag, S.A. Renormalization group analysis of turbulence. I. Basic theory. J. Sci. Comput.
**1986**, 57, 1722. [Google Scholar] [CrossRef] - Liu, Z.P.; Guo, X.L.; Xia, Q.F.; Fu, H.; Wang, T.; Dong, X.L. Experimental and numerical investigation of flow in a newly developed vortex drop shaft spillway. J. Hydraul. Eng.
**2018**, 144, 04018014. [Google Scholar] [CrossRef] - Zhang, W.; Wang, J.; Zhou, C.; Dong, Z.; Zhou, Z. Numerical simulation of hydraulic characteristics in a vortex drop shaft. Water
**2018**, 10, 1393. [Google Scholar] [CrossRef] - Guo, X.L.; Xia, Q.F.; Fu, H.; Yang, K.L.; Li, S.J. Numerical study on flow of newly vortex drop shaft spillway. J. Hydraul. Eng.
**2016**, 47, 733–741. (In Chinese) [Google Scholar] [CrossRef] - ANSYS Fluent Theory Guide; Release 16.0; ANSYS Inc.: Canonsburg, PA, USA, 2015.
- Celik, I.B.; Ghia, U.; Roache, P.J. Procedure for estimation and reporting of uncertainty due to discretization in CFD applications. J. Fluids Eng.
**2008**, 130. [Google Scholar] [CrossRef] - Iaccarino, G.; Mishra, A.A.; Ghili, S. Eigenspace perturbations for uncertainty estimation of single-point turbulence closures. Phys. Rev. Fluids
**2017**, 2, 024605. [Google Scholar] [CrossRef] - Mishra, A.A.; Iaccarino, G. Uncertainty Estimation for Reynolds-Averaged Navier–Stokes Predictions of High-Speed Aircraft Nozzle Jets. AIAA J.
**2017**, 3999–4004. [Google Scholar] [CrossRef] - Del Giudice, G.; Gisonni, C.; Rasulo, G. Design of a scroll vortex inlet for supercritical approach flow. J. Hydraul. Eng.
**2010**, 136, 837–841. [Google Scholar] [CrossRef] - Hager, W.H. Vortex drop inlet for supercritical approaching flow. J. Hydraul. Eng.
**1990**, 116, 1048–1054. [Google Scholar] [CrossRef] - Kawagoshi, N.; Hager, W. Wave type flow at abrupt drops Wave type flow at abrupt drops: I. Flow geometry. J. Hydraul. Res.
**1990**, 28, 235–252. [Google Scholar] [CrossRef]

**Figure 2.**Grid convergence index value for different grid densities: (

**a**) location of radial line; (

**b**) pressure profile in models with different mesh numbers; (

**c**) fine-grid solution, with discretization error bars computed using Equation (11); (

**d**) tangential velocity profile in models with different mesh numbers; (

**e**) fine-grid solution, with discretization error bars computed using Equation (11).

**Figure 3.**Comparison of experimental and calculated flow pattern: (

**a**,

**c**) flow pattern in experiment; (

**b**,

**d**) flow pattern in simulation.

**Figure 4.**Comparison of numerical and experimental data: (

**a**) location of the measuring points (m); (

**b**) pressure distribution along the flow direction; (

**c**) pressure distribution along the radical direction. Notes: L

_{p}represents the arc length of the center line; b is the width of the horizontal step.

**Figure 6.**Flow pattern in the cross-section for different configurations, where ${\alpha}^{\prime}$ is the relative location of the cross-section.

**Figure 8.**Pressure distribution for R = 0.25 m in different $\theta $: (

**a**) 120°; (

**b**) 150°; and (

**c**) 180°.

**Figure 9.**Flow region: (

**a**) range of flow regions and (

**b**) range of flow regions under different configurations.

**Figure 12.**Water depth characteristics: (

**a**) the relationship between h

_{max}/h, ${\alpha}_{\mathrm{max}}$ and Q and (

**b**) the relationship between h

_{max}/h, ${\alpha}_{\mathrm{max}}$ and $\theta $.

h (m) | D | $\mathbf{\theta}$ (°) | i | Q (m^{3}/s) | Fr | Case |
---|---|---|---|---|---|---|

0.131 | 0.6 | 150 | 0.20 | 12.75 | 0.58 | test1 |

41.50 | 1.90 | test2 | ||||

120 | 0.25 | 80.00 | 3.67 | 1 | ||

48.00 | 2.20 | 2 | ||||

26.50 | 1.22 | 3 | ||||

150 | 0.20 | 80.00 | 3.67 | 4 | ||

48.00 | 2.20 | 5 | ||||

26.50 | 1.22 | 6 | ||||

180 | 0.17 | 80.00 | 3.67 | 7 | ||

48.00 | 2.20 | 8 | ||||

26.50 | 1.22 | 9 |

**Table 2.**Summary of the standard deviations of pressure and water surface in different locations at the same time period, where $E{r}_{ave}$ represents the average relative errors of the mass flow rate of inlet and outlet; ${\sigma}_{p/h}$ and ${\sigma}_{{h}_{w}/h}$ are the standard deviation of pressure and water surface.

${\mathit{Er}}_{\mathit{ave}}$ (%) | 10° | 30° | 60° | 90° | 120° | ||
---|---|---|---|---|---|---|---|

case1 | 8.25 | ${\sigma}_{p/h}$ | 0.0707 | 0.0692 | 0.0723 | 0.0818 | 0.0726 |

${\sigma}_{{h}_{w}/h}$ | 0.0852 | 0.0912 | 0.0823 | 0.0797 | 0.0906 | ||

case2 | 7.18 | ${\sigma}_{p/h}$ | 0.0675 | 0.0704 | 0.0681 | 0.0619 | 0.0621 |

${\sigma}_{{h}_{w}/h}$ | 0.0823 | 0.0834 | 0.0885 | 0.0818 | 0.0911 | ||

case3 | 4.58 | ${\sigma}_{p/h}$ | 0.0754 | 0.0652 | 0.0801 | 0.0781 | 0.0702 |

${\sigma}_{{h}_{w}/h}$ | 0.0925 | 0.0922 | 0.0879 | 0.0921 | 0.0942 | ||

case4 | 7.32 | ${\sigma}_{p/h}$ | 0.0583 | 0.0621 | 0.0612 | 0.0565 | 0.0669 |

${\sigma}_{{h}_{w}/h}$ | 0.0725 | 0.0822 | 0.0850 | 0.0818 | 0.0861 | ||

case5 | 7.07 | ${\sigma}_{p/h}$ | 0.0754 | 0.0689 | 0.0692 | 0.0717 | 0.0722 |

${\sigma}_{{h}_{w}/h}$ | 0.0628 | 0.0587 | 0.0603 | 0.0614 | 0.0592 | ||

case6 | 5.68 | ${\sigma}_{p/h}$ | 0.0718 | 0.0782 | 0.0811 | 0.0777 | 0.0798 |

${\sigma}_{{h}_{w}/h}$ | 0.0823 | 0.0898 | 0.0912 | 0.0884 | 0.0879 | ||

case7 | 8.12 | ${\sigma}_{p/h}$ | 0.0905 | 0.0972 | 0.0883 | 0.0878 | 0.0928 |

${\sigma}_{{h}_{w}/h}$ | 0.0923 | 0.0985 | 0.0984 | 0.1008 | 0.0954 | ||

case8 | 7.22 | ${\sigma}_{p/h}$ | 0.0661 | 0.0622 | 0.0704 | 0.0683 | 0.0688 |

${\sigma}_{{h}_{w}/h}$ | 0.0921 | 0.0918 | 0.0856 | 0.0885 | 0.0892 | ||

case9 | 6.64 | ${\sigma}_{p/h}$ | 0.0775 | 0.0721 | 0.0605 | 0.0644 | 0.0786 |

${\sigma}_{{h}_{w}/h}$ | 0.0858 | 0.0805 | 0.0734 | 0.0713 | 0.0728 |

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**MDPI and ACS Style**

Qi, Y.; Wang, Y.; Zhang, J.
Three-Dimensional Turbulence Numerical Simulation of Flow in a Stepped Dropshaft. *Water* **2019**, *11*, 30.
https://doi.org/10.3390/w11010030

**AMA Style**

Qi Y, Wang Y, Zhang J.
Three-Dimensional Turbulence Numerical Simulation of Flow in a Stepped Dropshaft. *Water*. 2019; 11(1):30.
https://doi.org/10.3390/w11010030

**Chicago/Turabian Style**

Qi, Yongfei, Yurong Wang, and Jianmin Zhang.
2019. "Three-Dimensional Turbulence Numerical Simulation of Flow in a Stepped Dropshaft" *Water* 11, no. 1: 30.
https://doi.org/10.3390/w11010030