# A Design for Vortex Suppression Downstream of a Submerged Gate

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## Abstract

**:**

## 1. Introduction

_{1}$={U}_{1}/\sqrt{g{y}_{1}}$, where $g$ is the gravitational acceleration. The corresponding Reynolds number of the flow is defined based on the hydraulic radius (R) below the gate as R = $4{U}_{1}R/v$, where $v$ is the kinematic viscosity. Here for submerged flow conditions, the tail water depth ${y}_{t}$ is greater than the water depth ${y}_{2}$, which is the subcritical conjugate depth of the ${y}_{1}$ as obtained from the Belanger equation. The corresponding submergence factor can be calculated as $S=\left({y}_{t}-{y}_{2}\right)/{y}_{2}$.

## 2. Flume Experiments

_{1}= 5 are depicted for different spanwise locations in Figure 3a. Dimensionless jet velocity below the gate is about 1.2 and almost independent of the backward effects since the flow with high momentum is strongly in the downstream direction in this region. However, recirculation effects tend to increase near the free surface due to the fact that the baroclinic torque caused by the high gradient of fluid density along the air–water interface generates vortex motion [18]. As seen in Figure 3b, the maximum spanwise velocity is observed at z/b = 0.4, at which station the vorticity effects are expected to be considerable. While normalized Reynolds stresses in horizontal $\left(\sqrt{\overline{{{u}^{\prime}}^{2}}}/{U}_{1}\right)$ and vertical $\left(\sqrt{\overline{{{v}^{\prime}}^{2}}}/{U}_{1}\right)$ directions seem to be identical, the vertical distributions of Reynolds stress in the spanwise direction $\left(\sqrt{\overline{{{w}^{\prime}}^{2}}}/{U}_{1}\right)$ are variable, and the maximum value is observed at z/b = 0.4, which is consistent with the previous observations in the literature [1]. While the normal components of Reynolds stresses in horizontal and vertical directions are identical at each measurement station, the spanwise component of Reynolds stress exhibits a significant variation due to the energetic vortex structures observed on the free surface.

## 3. Computational Model

#### 3.1. Numerical Framework

^{−6}. The computational time step size was calculated according to the Courant stability condition during numerical simulations. The Courant number was set to Cr = 0.5 for the momentum and advection equations in order to reduce truncation errors arising from the discretization of unsteady terms, even though the present solver may yield accurate results for the Courant number up to 6. As discussed in the subsequent part of the study, the Courant number was reduced to 0.3 in order to overcome stability problems arising from the simulation of flows around the vortex breakers, since such local changes applied on the gate required smaller time-step sizes during simulations. Time averaging was started from 120 s of the simulation in order to exclude unphysical unsteady effects arising from the initial conditions, and it was performed for 280 s, which is significantly larger than the time scales in the computational domain.

#### 3.2. Mesh and Boundary Conditions

## 4. Results and Discussion

#### 4.1. Mesh Independent Study and Validation

_{1}

^{+}= 5 and z/b = 0.45 were compared with the experimental measurements using different grids in order to assess the resolution of the computational mesh. Properties of the grids used in the mesh independent study are given in Table 2 along with the dimensionless wall distance as minimum, average, and maximum values over the bottom of the channel. The calculated wall distances over the remaining walls are not shown here. The average value of the ${y}^{+}$ stabilized for Mesh 3 since the variation of ${y}^{+}$ was negligible in Mesh 4.

#### 4.2. Mean Flow Structure around the Gate

#### 4.3. Development of anti-Vortex Elements

#### 4.4. Performance of the Vortex Breaker for Different Flow Conditions

_{1}and submergence ratio S. Different flow conditions are addressed in Table 6 to achieve this goal. Case 1 was the validation case used in the experimental and numerical studies in this study. Case 2 and Case 3 were produced for different Froude numbers keeping the submergence ratio as constant in order to evaluate the effect of the inlet Froude number. In order to see the effect of the submergence ratio, which is another key parameter for the SHJ flow, Case 4 and Case 5 were produced to have different submergence factors for identical Froude numbers. The results show that the proposed design can mitigate vortex effects downstream of the submerged gate for different flow conditions.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Sketch of the turbulent flow through a submerged vertical gate: (

**a**) lateral view; (

**b**) top view.

**Figure 2.**Snapshot of the experimental set-up. A supplementary video file is available for the animation of the experimental study.

**Figure 3.**Vertical distributions of dimensionless mean velocity components and Reynolds stresses at x/y

_{1}= 5. Velocity components are compared in (

**a**,

**b**); Reynolds stresses are compared in (

**c**–

**f**).

**Figure 4.**Block-structured computational mesh:

**(a)**three-dimensional view of the mesh in the computational domain;

**(b)**front view of the mesh in the roller region;

**(c)**zoomed-in view of the gate lip.

**Figure 5.**Comparison of velocity profiles at x/y

_{1}= 5 and z/b = 0.4 for different mesh resolutions.

**Figure 6.**(

**a**) Comparison of vertical distributions of dimensionless time-averaged horizontal velocity components at z/b = 0.4. The solid line represents numerical results and symbols represent experimental measurements. (

**b**) Near-wall profile of time-averaged horizontal velocity component.

**Figure 7.**Visualization of time-averaged three-dimensional flow structure using streamlines colored with pressure in (

**a**) and vorticity magnitude in the remaining images: (

**a**) side view; (

**b**) top view; (

**c**) front view; (

**d**) three-dimensional view.

**Figure 9.**Vortex breakers mounted on the downstream face of the gate: (

**a**) vertical solid baffle; (

**b**) horizontal solid baffle; (

**c**) vertical porous baffle; (

**d**) horizontal porous baffle. The control volume is depicted as transparent volume.

**Figure 10.**Visualization of vortex structures downstream of the submerged gate flow using the Q-criteria: (

**a**) no baffle; (

**b**) with horizontal porous baffle for $\epsilon =0.3$ and $L/{L}_{rsj}=0.31$.

**Figure 11.**Time variations of (

**a**) lift forces acting on the gate lip and (

**b**) drag forces acting on the downstream face of the gate.

Q (L/s) | y_{1} (mm) | U1 (m/s) | F_{1} | R | y_{2} (mm) | y_{t} (mm) | S |
---|---|---|---|---|---|---|---|

21 | 25 | 2.1 | 4.24 | 98,824 | 137.93 | 230 | 0.67 |

Grid | Number of cells (× 10^{6}) | ${\left({\mathit{y}}^{+}\right)}_{\mathit{m}\mathit{i}\mathit{n}}$ | ${\left({\mathit{y}}^{+}\right)}_{\mathit{a}\mathit{v}\mathit{e}\mathit{r}\mathit{a}\mathit{g}\mathit{e}}$ | ${\left({\mathit{y}}^{+}\right)}_{\mathit{m}\mathit{a}\mathit{x}}$ | ${\left({\mathit{x}}^{+}\right)}_{\mathit{m}\mathit{i}\mathit{n}}$ | ${\left({\mathit{x}}^{+}\right)}_{\mathit{a}\mathit{v}\mathit{e}\mathit{r}\mathit{a}\mathit{g}\mathit{e}}$ | ${\left({\mathit{x}}^{+}\right)}_{\mathit{m}\mathit{a}\mathit{x}}$ |
---|---|---|---|---|---|---|---|

Mesh 1 | 0.7 | 2.20 | 53.05 | 176.09 | 0.25 | 23.10 | 83.36 |

Mesh 2 | 1.85 | 0.84 | 26.95 | 91.86 | 0.22 | 13.42 | 45.65 |

Mesh 3 | 2.60 | 0.93 | 27.38 | 90.09 | 0.14 | 12.82 | 41.50 |

Mesh 4 | 4.10 | 1.15 | 28.16 | 94.74 | 0.04 | 9.86 | 35.42 |

$\mathbf{Porosity}\text{}\left(\mathit{\epsilon}\right)$ | $\mathit{D}\left({\mathit{m}}^{-2}\right)$ | $\mathit{F}\left({\mathit{m}}^{-1}\right)$ |
---|---|---|

0.31 | 2.76 × 10^{9} | 5.24 × 10^{4} |

0.504 | 7.1 × 10^{7} | 4.78 × 10^{3} |

0.802 | 6.26 × 10^{5} | 3.54 × 10^{2} |

**Table 4.**Comparison of vortex damping performance of each design in Figure 9. N/A—not applicable.

Design | $\mathbf{Porosity}\text{}\left(\mathit{\epsilon}\right)$ | ${\left|\overline{\mathit{\omega}}\right|}_{\mathit{a}\mathit{v}}$ | Dissipation Ratio |
---|---|---|---|

No baffle | N/A | 39.066 | - |

Vertical solid | N/A | 34.417 | 11.90% |

Horizontal solid | N/A | 36.898 | 13.25% |

Vertical porous | 0.3 | 36.898 | 5.55% |

Vertical porous | 0.5 | 36.905 | 5.53% |

Vertical porous | 0.8 | 34.816 | 10.88% |

Horizontal porous | 0.3 | 32.504 | 16.80% |

Horizontal porous | 0.5 | 34.417 | 11.90 |

Horizontal porous | 0.8 | 36.210 | 7.31 |

**Table 5.**Comparison of vortex damping performance of horizontal porous baffle for different lengths. Here, ${L}_{rsj}$ is calculated from Equation (4).

${\mathit{L}}_{\mathit{v}\mathit{b}}/{\mathit{L}}_{\mathit{r}\mathit{s}\mathit{j}}$ | ${\left|\overline{\mathit{\omega}}\right|}_{\mathit{a}\mathit{v}}$ | Dissipation Ratio |
---|---|---|

0.10 | 34.423 | 16.80% |

0.15 | 32.504 | 11.89% |

0.31 | 28.594 | 26.80% |

Case | Q (lt/s) | y_{1} (mm) | U_{1} (m/s) | F_{1} | Re | S | ${\left|\overline{\mathit{\omega}}\right|}_{\mathit{a}\mathit{v}}$ | Vortex Dissipation Ratio (%) | |
---|---|---|---|---|---|---|---|---|---|

No Baffle | Breaker | ||||||||

Case 1 | 21 | 25 | 2.1 | 4.24 | 9.9 × 10^{4} | 0.67 | 39.066 | 28.594 | 26.81 |

Case 2 | 15 | 25 | 1.5 | 3.03 | 7.1 × 10^{4} | 0.67 | 34.838 | 25.880 | 25.71 |

Case 3 | 25 | 25 | 2.5 | 5.05 | 1.2 × 10^{5} | 0.67 | 37.764 | 27.947 | 26.00 |

Case 4 | 21 | 25 | 2.1 | 4.24 | 9.9 × 10^{4} | 0.96 | 30.402 | 23.782 | 21.78 |

Case 5 | 21 | 25 | 2.1 | 4.24 | 9.9 × 10^{4} | 0.50 | 42.627 | 29.787 | 30.12 |

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**MDPI and ACS Style**

Demirel, E.; Aral, M.M.
A Design for Vortex Suppression Downstream of a Submerged Gate. *Water* **2020**, *12*, 750.
https://doi.org/10.3390/w12030750

**AMA Style**

Demirel E, Aral MM.
A Design for Vortex Suppression Downstream of a Submerged Gate. *Water*. 2020; 12(3):750.
https://doi.org/10.3390/w12030750

**Chicago/Turabian Style**

Demirel, Ender, and Mustafa M. Aral.
2020. "A Design for Vortex Suppression Downstream of a Submerged Gate" *Water* 12, no. 3: 750.
https://doi.org/10.3390/w12030750