# Large-Eddy Simulation of Compound Channels with Staged Floodplains: Flow Interactions and Turbulent Structures

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## Abstract

**:**

## 1. Introduction

- The study’s experimental arrangements and flow setup are first described for the three test cases. The mathematical and numerical solution method is then discussed with a computational model setup.
- The validation of the experimental time-averaged primary mean velocity and secondary current are presented for three different depth ratios of multiple staged compound channels.
- Then, the streamwise primary velocity distribution, turbulence statistics and vortical structures to define the mass and momentum exchange are presented.
- The numerical model results and conclusions are made for the new flow configuration with staged floodplains.

**Figure 1.**(

**a**) Cross-sectional view of the staged floodplains and main channel of the experimental test case and (

**b**) pictorial view from the downstream end of the compound channel with synthetic grass turf used for the floodplain. Note: ${H}_{c}$ denotes the flow depth in the main channel; $B$ is the top width; $c$ is the main channel width; ${h}_{i}$ and ${b}_{fi}$ are floodplains’ depth and bed width, respectively.

## 2. Experimental Arrangements

^{2}and 8.3 mm, respectively (see Figure 1b). The bankfull height (${h}_{i}$) of the first stage floodplain was 4 cm and the second stage floodplain had 8 cm, where $i=1$ or 2. With the dual existence of the interfacial section and bankfull height, the relative depth ratio $Dr$ was distinctively variable at each stage. The flow regime definition undertaken in this analysis is defined based on ${h}_{2}=8$ cm. However, it should be noted that $Dr$ for the first stage, the floodplain (${h}_{1}=4$ cm), was always a high-flow regime when $Dr\ge 0.5$. The measurement cross-section was located 11 m away from the flume inlet. An electromagnetic flowmeter was used to measure the upstream discharge. A honeycomb structure was installed in the stilling tank before the flume entrance to regulate the development and uniformity of the flow. The uniform section was maintained at all times by keeping the bed slope $\left({S}_{o}=0.003\right)$ and the water surface slope $\left({S}_{w}\right)$equal through the downstream tailgate settings. The free water surface level was measured using a point gauge to obtain a free surface parallel to the channel bed.

## 3. Numerical Model

#### Governing Equations

## 4. Results and Discussion

#### 4.1. Distribution of the Mean Streamwise Velocity and Secondary Current

#### 4.2. Distribution of Instantaneous Streamwise Velocity

#### 4.3. Distribution of Turbulent Kinetic Energy

#### 4.4. Power Density Spectra (PSD) Using Time Series at Interfaces of Multistage Floodplains

#### 4.5. Vortical Structures

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Abril, J.; Knight, D. Stage-discharge prediction for rivers in flood applying a depth-averaged model. J. Hydraul. Res.
**2004**, 42, 616–629. [Google Scholar] [CrossRef] - Knight, D. Hydraulic problems in flooding: From data to theory and from theory to practice. In Experimental and Computational Solutions of Hydraulic Problems; Springer: Berlin/Heidelberg, Germany, 2013; pp. 19–52. [Google Scholar]
- Cappato, A.; Baker, E.A.; Reali, A.; Todeschini, S.; Manenti, S. The role of modeling scheme and model input factors uncertainty in the analysis and mitigation of backwater induced urban flood-risk. J. Hydrol.
**2022**, 614, 128545. [Google Scholar] [CrossRef] - Knight, D.W.; Shiono, K. Turbulence measurements in a shear layer region of a compound channel. J. Hydraul. Res.
**1990**, 28, 175–196. [Google Scholar] [CrossRef] - Peltier, Y. Physical Modelling of Overbank Flows with a Groyne Set on the Floodplain. Ph.D. Thesis, Université Claude Bernard-Lyon I, Lyon, France, 2011. [Google Scholar]
- Atabay, S. Stage-Discharge, Resistance, and Sediment Transport Relationships for Flow in Straight Compound Channels. Doctoral Dissertation, University of Birmingham, Birmingham, UK, 2001. Available online: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.427466 (accessed on 14 February 2023).
- Knight, D.W.; Demetriou, J.D. Flood Plain and Main Channel Flow Interaction. J. Hydraul. Eng.
**1983**, 109, 1073–1092. [Google Scholar] [CrossRef] - Singh, P.; Tang, X. Interfacial compound section transverse flow variation in symmetric and asymmetric compound open channel flow. In Proceedings of the 9th International Symposium on Environmental Hydraulics, Seoul, Republic of Korea, 18–22 July 2021; pp. 174–175. [Google Scholar]
- Singh, P.; Tang, X.; Rahimi, H.R. Modelling of the apparent shear stress for predicting zonal discharge in rough and smooth asymmetric compound open channels. In River Flow 2020: Proceedings of the 10th Conference on Fluvial Hydraulics, Delft, The Netherlands, 7–10 July 2020; CRC Press/Balkema (Taylor & Francis): Leiden, The Netherlands, 2020; pp. 84–94. [Google Scholar]
- Nicollet, G.; Uan, M. Ecoulements permanents à surface libre en lits composes. La Houille Blanche
**1979**, 21–30. [Google Scholar] [CrossRef][Green Version] - Knight, D.W.; Hamed, M.E. Boundary Shear in Symmetrical Compound Channels. J. Hydraul. Eng.
**1984**, 110, 1412–1430. [Google Scholar] [CrossRef] - Tominaga, A.; Nezu, I. Turbulent Structure in Compound Open-Channel Flows. J. Hydraul. Eng.
**1991**, 117, 21–41. [Google Scholar] [CrossRef] - Fernandes, J. Compound channel uniform and non-uniform flows with and without vegetation in the floodplain. Ph.D. Thesis, Departamento de Engenharia Civil, Instituto Superior Técnico da Universidade Técnica de Lisboa, Lisboa, Portugal, 2013. [Google Scholar]
- Tang, X.; Knight, D.W. Sediment Transport in River Models with Overbank Flows. J. Hydraul. Eng.
**2006**, 132, 77–86. [Google Scholar] [CrossRef] - Pasche, E.; Rouvé, G. Overbank flow with vegetatively roughened flood plains. J. Hydraul. Eng.
**1985**, 111, 1262–1278. [Google Scholar] [CrossRef] - Kozioł, A.P. Three-Dimensional Turbulence Intensity in a Compound Channel. J. Hydraul. Eng.
**2013**, 139, 852–864. [Google Scholar] [CrossRef] - Dupuis, V.; Proust, S.; Berni, C.; Paquier, A. Mixing layer development in compound channel flows with submerged and emergent rigid vegetation over the floodplains. Exp. Fluids
**2017**, 58, 1–18. [Google Scholar] [CrossRef][Green Version] - Tang, X.; Rahimi, H.R.; Wang, Y.U.; Zhao, Y.U.; Lu, Q.I.; Wei, Z.I.; Singh, P.R. Flow characteristics of open-channel flow with partial two-layered vegetation. In Proceedings of the 38th IAHR World Congress, Panama City, Panama, 1–6 September 2019; pp. 1–6. [Google Scholar]
- Tang, X.; Guan, Y.; Rahimi, H.; Singh, P.; Zhang, Y. Discharge and velocity variation of flows in open channels partially covered with different layered vegetation. E3S Web Conf.
**2021**, 269, 03001. [Google Scholar] [CrossRef] - Proust, S.; Fernandes, J.N.; Leal, J.B.; Rivière, N.; Peltier, Y. Mixing layer and coherent structures in compound channel flows: Effects of transverse flow, velocity ratio, and vertical confinement. Water Resour. Res.
**2017**, 53, 3387–3406. [Google Scholar] [CrossRef] - Proust, S.; Fernandes, J.N.; Peltier, Y.; Leal, J.B.; Riviere, N.; Cardoso, A.H. Turbulent non-uniform flows in straight compound open-channels. J. Hydraul. Res.
**2013**, 51, 656–667. [Google Scholar] [CrossRef][Green Version] - Naik, B.; Khatua, K.K.; Padhi, E.; Singh, P. Loss of Energy in the Converging Compound Open Channels. Arab. J. Sci. Eng.
**2017**, 43, 5119–5127. [Google Scholar] [CrossRef] - Rajaratnam, N.; Ahmadi, R. Hydraulics of channels with floodplains. J. Hydraul. Res.
**1981**, 19, 43–60. [Google Scholar] [CrossRef] - Kara, S.; Stoesser, T.; Sturm, T.W. Turbulence statistics in compound channels with deep and shallow overbank flows. J. Hydraul. Res.
**2012**, 50, 482–493. [Google Scholar] [CrossRef] - Naot, D.; Nezu, I.; Nakagawa, H. Hydrodynamic Behavior of Compound Rectangular Open Channels. J. Hydraul. Eng.
**1993**, 119, 390–408. [Google Scholar] [CrossRef] - Lin, B.; Shiono, K. Numerical modelling of solute transport in compound channel flows. J. Hydraul. Res.
**1995**, 33, 773–788. [Google Scholar] [CrossRef] - Sofialidis, D.; Prinos, P. Numerical Study of Momentum Exchange in Compound Open Channel Flow. J. Hydraul. Eng.
**1999**, 125, 152–165. [Google Scholar] [CrossRef] - Cokljat, D.; Younis, B.A. Second-Order Closure Study of Open-Channel Flows. J. Hydraul. Eng.
**1995**, 121, 94–107. [Google Scholar] [CrossRef] - Kang, H.; Choi, S.-U. Turbulence modeling of compound open-channel flows with and without vegetation on the floodplain using the Reynolds stress model. Adv. Water Resour.
**2006**, 29, 1650–1664. [Google Scholar] [CrossRef] - Jing, H.; Li, C.; Guo, Y.; Xu, W. Numerical simulation of turbulent flows in trapezoidal meandering compound open channels. Int. J. Numer. Methods Fluids
**2011**, 65, 1071–1083. [Google Scholar] [CrossRef] - Singh, P.K.; Tang, X.; Rahimi, H. A Computational Study of Interaction of Main Channel and Floodplain: Open Channel Flows. J. Appl. Math. Phys.
**2020**, 08, 2526–2539. [Google Scholar] [CrossRef] - Wang, W.; Huai, W.-X.; Gao, M. Numerical investigation of flow through vegetated multi-stage compound channel. J. Hydrodyn.
**2014**, 26, 467–473. [Google Scholar] [CrossRef] - Chen, G.; Zhao, S.; Huai, W.; Gu, S. General model for stage–discharge prediction in multi-stage compound channels. J. Hydraul. Res.
**2018**, 57, 517–533. [Google Scholar] [CrossRef] - Menter, F.; Schutze, J.; Kurbatskii, K.; Lechner, R.; Gritskevich, M.; Garbaruk, A. Scale-Resolving simulation techniques in industrial CFD. In Proceedings of the 6th AIAA Theoretical Fluid Mechanics Conference, Honolulu, HI, USA, 27–30 June 2011; p. 3474. [Google Scholar]
- Thomas, T.; Williams, J. Large eddy simulation of turbulent flow in an asymmetric compound open channel. J. Hydraul. Res.
**1995**, 33, 27–41. [Google Scholar] [CrossRef] - Cater, J.E.; Williams, J.J.R. Large eddy simulation of a long asymmetric compound open channel. J. Hydraul. Res.
**2008**, 46, 445–453. [Google Scholar] [CrossRef] - Xie, Z.; Lin, B.; Falconer, R.A. Large-eddy simulation of the turbulent structure in compound open-channel flows. Adv. Water Resour.
**2012**, 53, 66–75. [Google Scholar] [CrossRef] - Goring, D.G.; Nikora, V.I. Despiking acoustic Doppler velocimeter data. J. Hydraul. Eng.
**2002**, 128, 117–126. [Google Scholar] [CrossRef][Green Version] - Nikitin, N.V.; Nicoud, F.; Wasistho, B.; Squires, K.; Spalart, P.R. An approach to wall modeling in large-eddy simulations. Phys. Fluids
**2000**, 12, 1629–1632. [Google Scholar] [CrossRef] - Shur, M.L.; Spalart, P.R.; Strelets, M.K.; Travin, A.K. A hybrid RANS-LES approach with delayed-DES and wall-modelled LES capabilities. Int. J. Heat Fluid Flow
**2008**, 29, 1638–1649. [Google Scholar] [CrossRef] - Zeng, C.; Bai, Y.; Zhou, J.; Qiu, F.; Ding, S.; Hu, Y.; Wang, L. Large Eddy Simulation of Compound Open Channel Flows with Floodplain Vegetation. Water
**2022**, 14, 3951. [Google Scholar] [CrossRef] - Ding, S.-W.; Zeng, C.; Zhou, J.; Wang, L.-L.; Chen, C. Impact of depth ratio on flow structure and turbulence characteristics of compound open channel flows. Water Sci. Eng.
**2021**, 15, 265–272. [Google Scholar] [CrossRef] - Hussaini, M.Y.; Zang, T.A. Spectral methods in fluid dynamics. Annu. Rev. Fluid
**1987**, 19, 339–367. [Google Scholar] [CrossRef] - Hirsch, C. Numerical Computation of Internal and External Flows: The Fundamentals of Computational Fluid Dynamics; Elsevier: Amsterdam, The Netherlands, 2007. [Google Scholar]
- Uijttewaal, W.S.J.; Booij, R. Effects of shallowness on the development of free-surface mixing layers. Phys. Fluids
**2000**, 12, 392–402. [Google Scholar] [CrossRef] - Batchelor, G.K. Computation of the Energy Spectrum in Homogeneous Two-Dimensional Turbulence. Phys. Fluids
**1969**, 12, II-233–II-239. [Google Scholar] [CrossRef] - Hunt, J.C.; Wray, A.A.; Moin, P. Eddies, streams, and convergence zones in turbulent flows. In Studying Turbulence Using Numerical Simulation Databases, 2. Proceedings of the 1988 Summer Program; NASA: Washington, DC, USA, 1988; p. 19890015184142023. Available online: https://ntrs.nasa.gov/citations/19890015184 (accessed on 14 February 2023).

**Figure 2.**Contour mapping of the streamwise mean velocity for experimental and numerical results for three depth ratios as (

**a**) Dr = 0.1, (

**b**) Dr = 0.3 and (

**c**) Dr = 0.5.

**Figure 3.**The distribution of the instantaneous streamwise velocity $u$ and the velocity vectors ($u$, $w$) for (

**a**) $Dr=0.1$, (

**b**) $Dr=0.3$ and (

**c**) $Dr=0.5$ to illustrate the shear layer at a horizontal plane at second stage $Z=0.08\mathrm{m}$, for a constant time interval of $0.1$ times of flow cycle. The values of $w$ are increased by a factor of 10 for clarity.

**Figure 4.**The distribution of the instantaneous streamwise velocity u and the velocity vectors (u, w) for (

**a**) Dr = 0.1, (

**b**) Dr = 0.3 and (

**c**) Dr = 0.5 to illustrate the shear layer at a horizontal plane at the first stage Z = 0.04 m, for a constant time interval of 0.1 times of flow cycle. A factor of 10 for clarity increases the values of w.

**Figure 5.**Contour mapping of the turbulent kinetic energy in m

^{2}/s

^{2}for (

**a**) Dr = 0.1, (

**b**) Dr = 0.3 and (

**c**) Dr = 0.5.

**Figure 6.**Time series of streamwise (${u}^{\prime}$) and lateral velocities (${v}^{\prime}$) for depth ratio (Dr) being (

**a**) 0.1, (

**b**) 0.3 and (

**c**) 0.5 at the interface of ${y}_{int=2}=20$ cm with h = 0.85 cm and ${y}_{int=1}=30$ cm with h = 0.45 cm. The black line is the raw signal, the blue line is the low-pass filtered signal and the solid red line is the mean value of the raw signal.

**Figure 7.**Power spectral density (PSD) of transverse velocity fluctuations v′, as a function of frequency for the three depth ratios.

**Figure 8.**Instantaneous vortical structure plotted as iso-surfaces of ${\lambda}_{2}=-20$ for Dr at (

**a**) 0.1, (

**b**) 0.3 and (

**c**) 0.5, respectively, and coloured by the x-velocity component in the multistage compound open-channel flows.

**Table 1.**Flow conditions of test cases. Note that ${Q}_{t}$ is the total discharge and ${R}_{e}$ denotes the Reynolds number.

$\mathit{D}\mathit{r}$ | ${\mathit{H}}_{\mathit{c}}\left(\mathbf{m}\right)$ | ${\mathit{Q}}_{\mathit{t}}({\mathbf{m}}^{3}/\mathbf{s})$ | ${\mathit{U}}_{*}$$(\times {10}^{-3})(\mathbf{m}/\mathbf{s})$ | $\mathbf{Friction}\mathbf{Factor}(\mathit{f})$$(\times {10}^{-5})$ | ${\mathit{R}}_{\mathit{e}}$ $\left(\times {10}^{5}\right)$ |
---|---|---|---|---|---|

0.1 | 0.0910 | 0.02536 | 1.27 | 9.44 | 0.86 |

0.3 | 0.1108 | 0.03542 | 1.46 | 7.40 | 1.36 |

0.5 | 0.1617 | 0.06075 | 1.76 | 7.62 | 2.39 |

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**MDPI and ACS Style**

Singh, P.K.; Tang, X.; Rahimi, H. Large-Eddy Simulation of Compound Channels with Staged Floodplains: Flow Interactions and Turbulent Structures. *Water* **2023**, *15*, 983.
https://doi.org/10.3390/w15050983

**AMA Style**

Singh PK, Tang X, Rahimi H. Large-Eddy Simulation of Compound Channels with Staged Floodplains: Flow Interactions and Turbulent Structures. *Water*. 2023; 15(5):983.
https://doi.org/10.3390/w15050983

**Chicago/Turabian Style**

Singh, Prateek Kumar, Xiaonan Tang, and Hamidreza Rahimi. 2023. "Large-Eddy Simulation of Compound Channels with Staged Floodplains: Flow Interactions and Turbulent Structures" *Water* 15, no. 5: 983.
https://doi.org/10.3390/w15050983