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

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## Abstract

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## 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

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**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