# Hydraulic Characteristics of Lateral Deflectors with Different Geometries in Gentle-Slope Free-Surface Tunnels

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## Abstract

**:**

## 1. Introduction

## 2. Experiment Setup

_{in}= 11.25 cm and volume flow rate Q

_{in}= 57.65 L/s, featuring a Reynolds number $Re=\frac{{Q}_{in}}{{h}_{in}R\upsilon}=$ 1.2 × 10

^{5}(R represents the hydraulic diameter calculated as $\frac{b{h}_{in}}{b+2{h}_{in}}$ and $\upsilon $ = 1 × 10

^{−6}m

^{2}/s is the kinematic viscosity of water). This $Re$ value being larger than 1 × 10

^{5}indicates the scale effect arising from viscous stress can be neglected according to [38]. The corresponding depth-width ratio of 0.6 in the experiment is a representative value for the real-world high-discharge flood tunnels [15,22,23,33,34,35,36] in China. In this study, the flow depth was measured using a fluviograph (accuracy ± 0.18 mm). The water discharge was monitored using an electromagnetic flowmeter (IFM4080K, Jiangsu Runyi Instrument Co., Ltd., Huaian, China), featuring accuracy of 0.1 L/s. The flow velocity was measured with a propeller-type flow meter (LS300-A, Fuzhou Lesida Information Technology Co., Ltd., Fuzhou, China) featuring accuracy of 0.01 m/s.

## 3. Numerical Models and Simulation Setup

#### 3.1. Governing Equations

**u**is velocity, p is the pressure, ρ and υ are the fluid density and kinematic viscosity, respectively.

**f**stands for the body force, k and μ

_{t}stand for the turbulent kinetic energy, and turbulent viscosity. δ

_{ij}is the Kronecker delta, and S

_{ij}is the mean rate of strain tensor calculated as $\frac{1}{2}(\frac{\partial {u}_{i}}{\partial {x}_{j}}+\frac{\partial {u}_{j}}{\partial {x}_{i}})$.

#### 3.2. Simulation Setup

^{®}technique [44,45], the computational domain was rotated by the flume slope of 3% to be aligned with the x-coordinate and the gravity vector

**g**was adjusted accordingly. Moreover, the domain was locally refined to 0.188 cm × 0.094 cm × 0.188 cm in the region of −7.5 < x < 30 cm, −10 < y < −6.25 cm and 6.25 < y < −10 cm, and 0 < z < 12.5 cm.

_{s}= 0.015 mm according to the plexiglass surface properties. The top boundary is specified with fixed relative pressure p = 0 (i.e., atmosphere pressure). As for the inlet boundary, a pre-simulation of the entire physical model (i.e., from the reservoir to the downstream cushion pool) was conducted first, and then all the flow parameters at x = −33.75 cm were mapped onto the inlet of the short domain using the grid overlay BC. The illustrative diagram of the simulation setup is shown in Figure 2.

## 4. Results and Discussion

#### 4.1. Model Validation

#### 4.2. Flow Pattern

#### 4.3. Velocity Distribution

#### 4.4. Energy Dissipation Characteristics

## 5. Conclusions

- The cavity formed behind lateral deflectors usually features a right-angled trapezoid shape with a larger streamwise length at higher elevations because of non-uniform velocity distributions. This makes the deflected flow rise up along the impacting region inside the cavity and potentially induced shock waves depending on the interaction of the rising up water-wings and the jet surfaces.
- The traditional triangular deflector forms an adequately wide cavity that allows for the free rising up of the water-wings inside the cavity, which further contributes to the development of the buddle-type shock wave, whereas the two-arc deflector yields a jet with fluctuating surface, resulting from the non-uniform planar velocity distribution caused by the continuously varying curvature of the arcs. Water-wings also develop inside the cavity and eventually produce a diamond-type shock wave downstream. In contrast, the jet behind the two-arc deflector with a straight guiding line at the tail is stabler and travels a shorter distance before impacting the side wall. The jet could thus restrict the development of the rising flow, and thereby eliminate the formation of water-wings and shock waves. Based on these observations, it is concluded that a continuous variation of the lateral deflector surface at the tail with an additional flow guiding extension is the key to the elimination of the water-wings and shock waves.
- Compared to the triangular deflector and the two-arc deflector, the two-arc deflector with a straight line exhibits more effective energy dissipation, as reflected in the local energy loss coefficient. The underlying reason for its effective energy dissipation is the more intensive turbulence introduced by the stronger interaction between the deflected flow and the jet surface, which also leads to more intensive aeration.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

_{max}in the middle longitudinal section at x = 112.5 cm. The GCI for each grid k is computed as $GC{I}_{k}={F}_{s}\left|{E}_{k}\right|$, where a value of 1.25 is chosen for the factor of safety ${F}_{s}$, following the recommendations of Roache [52].

Mesh | $\mathit{h}$/cm | r | p(u_{max}) | f(u_{max})/(m·s^{−1}) | ε(u_{max})/% | GCI(u_{max})/% |
---|---|---|---|---|---|---|

coarse | 0.496 | - | 1.84 | 2.607 | - | - |

medium | 0.297 | 1.67 | 2.633 | 0.997 | 2.040 | |

fine | 0.149 | 1.99 | 2.621 | 0.456 | 0.224 |

_{x}and turbulent kinetic energy k in the middle longitudinal section at x = 112.5 cm are comparatively shown in Figure A1. From Figure A1, it can be found that the main deviations of the u

_{x}profiles lie in the region close to the water surface and the bottom wall, whereas in the middle, all three meshes return almost identical velocity values. Moreover, the difference between the medium and fine meshes is less obvious compared to that between the coarse and the medium meshes. As for the k profile, the result from the coarse mesh is strikingly different from those from the medium and fine meshes, the difference between which is relatively negligible. Therefore, the fine grid was selected for the simulation.

**Figure A1.**Vertical distributions of streamwise velocity u

_{x}and turbulent kinetic energy k in the middle longitudinal section at x = 112.5 cm.

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**Figure 1.**(

**a**) Schematic diagram of the experimental spillway tunnel and details of (

**b**) the simulated region and (

**c**) the three lateral deflectors (unit is cm).

**Figure 3.**Comparison of the experimental and numerical results: (

**a**) water surface profile; (

**b**) vertical; and (

**c**) horizontal distribution of the streamwise velocity u

_{x}at x = 112.5 cm. The unit of the coordinates is cm.

**Figure 7.**Horizontal velocity distribution around the deflectors at z = 1.0 cm. The vectors are generated using the same density and scale for the three types of deflectors.

**Figure 8.**Velocity distribution near the sidewall (y = −9.3 cm). The vectors are generated using the same density and scale for the three types of deflectors.

**Figure 9.**Streamwise development of the cross-sectional velocity along the flume with three different deflectors and without deflectors. The brown color indicates regions where the cavity range of C (pink) overlaps those of A and B (green).

**Figure 10.**Streamwise development of the flux-averaged hydraulic head (FAHH) along the flume with three different and without deflectors. The brown color indicates regions where the cavity range of C (pink) overlaps those of A and B (green).

**Figure 11.**Horizontal distribution of (

**a**) turbulent kinetic energy k; (

**b**) turbulence dissipation rate ε in deflector and cavity region at z = 1 cm.

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**MDPI and ACS Style**

Da, J.; Wang, J.; Dong, Z.; Du, S. Hydraulic Characteristics of Lateral Deflectors with Different Geometries in Gentle-Slope Free-Surface Tunnels. *Water* **2022**, *14*, 2689.
https://doi.org/10.3390/w14172689

**AMA Style**

Da J, Wang J, Dong Z, Du S. Hydraulic Characteristics of Lateral Deflectors with Different Geometries in Gentle-Slope Free-Surface Tunnels. *Water*. 2022; 14(17):2689.
https://doi.org/10.3390/w14172689

**Chicago/Turabian Style**

Da, Jinrong, Junxing Wang, Zongshi Dong, and Shuaiqun Du. 2022. "Hydraulic Characteristics of Lateral Deflectors with Different Geometries in Gentle-Slope Free-Surface Tunnels" *Water* 14, no. 17: 2689.
https://doi.org/10.3390/w14172689