# Effects of Energy Dissipation Pier Arrangements on the Hydraulic Characteristics of Segmented Pier-Type Step Energy Dissipator Structures

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Construction of Numerical Models

#### 2.1.1. Turbulence Model

_{k}denotes the turbulent kinetic energy generation term due to the mean flow velocity gradient; μ

_{t}denotes the turbulent viscosity coefficient; μ

_{eff}is the effective viscosity; C

_{1ε}and C

_{2ε}are the turbulent model coefficients, taking the values of 1.44 and 1.92, respectively; α

_{κ}and α

_{ε}denote the inverse of the Planck number of k and ε, taking the values of 1.0 and 1.3, respectively; C

_{μ}is the empirical coefficient, which takes the value of 0.0845; η

_{0}and β are the model constants, which take the values of 4.38 and 0.015, respectively. See Table 1 for details.

#### 2.1.2. Model Building

#### 2.1.3. Mesh Division and Boundary Setting

#### 2.1.4. Mesh Sensitivity Analysis

^{3}/h. We then carried out a simulation for six different mesh sizes of 0.004 m, 0.005 m, 0.006 m, 0.007 m, 0.008 m, and 0.009 m. The results are shown in Figure 3.

#### 2.2. Experimental Validation of Numerical Model

^{3}/h, using a numerical simulation, we obtained values for the water depth, flow rate, and pressure along the distance of the double-row arrangement and compared these with the measured values of the corresponding physical tests. The pressure value was measured at the level of the fourth and fifth steps. Test results are presented in Figure 6 and show good agreement of the simulated and measured values for water depth, flow velocity, and pressure along the course, with maximum relative discrepancies of 7.09%, 6.54%, and 6.46%, respectively, under the double-row arrangement. These results confirm the feasibility of the simulation method.

## 3. Results and Discussion

#### 3.1. Flow Analysis

^{3}/h in the y = 7 cm section. A zero water volume fraction represents air. In the first stage, the water flow is basically the same for steps of all sizes and is relatively stable; the water surface line is wavy, and the water flow starts to mix with gas near the sixth step. In the straight section under the first stage, different degrees of water jumps occur in each body type. When the energy dissipation pier is arranged in the straight section of the step, the point at which the water jumps is shifted backwards to the waterward side of the pier. This accentuates the water jumping effect in the straight section and results in a flow of water more fully mixed with air. The water level also rises after jumping in the staggered arrangement, resulting in better air-doping.

^{3}/h, in the longitudinal section, without the energy dissipation pier (y = 5 cm), and with the energy dissipation pier (y = 7 cm). At the end of the flat section, the longitudinal section with the pier (y = 7 cm) produces a large water jump, while the longitudinal section without the pier (y = 5 cm) has no water jump; however, due to the obstructing effect of the pier on the water flow in the flat section, the water jump occurs in the downstream position of the first-stage cyclone, when a large amount of air is mixed in, increasing the air mixing effect of the water flow so that, when the water enters the second stage, no non-air mixing area is produced, effectively reducing the cavitation hazard of the stage; the water flow in the second- and third-stage section exhibits basically the same doping effect, effectively reducing the risk of cavitation damage.

^{3}/h, with two different longitudinal sections, as follows: y = 5 cm for the second row of the energy dissipation pier section, and y = 7 cm for the first row. Water jumps of different degrees can be seen in both sections; those of the y = 7 cm longitudinal section are higher than those of the y = 5 cm longitudinal section, indicating a better gas-doping effect, which is due to the obstruction of the water flow of the first row of energy-dissipating piers so that the water flow is higher when passing through the first row of piers, while the water depth is greater after the second row. Comparing the effects of gas blending in the second and third stages of the two sections, it can be seen that, due to the higher water jump in the y = 7 cm longitudinal section, there is a higher doping concentration along the second stage, because the air-doping escapes along the way and the second straight section does not produce a larger water jump, so the two sections of the third stage of the air-doping effect produce basically the same effect. The staggered arrangement, therefore, enables the entire stage section to be air-doped; however, the air-doping effect does vary from one longitudinal section to another.

#### 3.2. Flow Rate Analysis

^{3}/h. On the virtual bottom plate formed by the convex angle of the steps, for steps of all sizes, the upper-part flow rate is greater than that of the middle and lower parts, and the highest rates of water flow are mainly seen near the free liquid surface. In the first stage, the flow velocity distribution is basically the same for each type of step dissipator, with an average flow velocity of about 2.0 m/s. In the flat and straight section under the first stage, the segmented pier-type step dissipator forms a water jump behind the pier due to the obstructing effect of the pier on the water flow, resulting in a lower flow velocity in the second stage than the traditional step dissipator. The action of the second stage under the flat section of the energy dissipation pier produces another water jump, resulting in a further reduction in the water flow velocity into the third stage. The action of multistage energy dissipation piers can reduce the flow velocity to below 1.3 m/s, thus lowering the risk of water scouring damage downstream.

#### 3.3. Pressure Analysis

^{3}/h. On the horizontal surface of the steps, from the concave angle to the convex angle position, the pressure first increases and then decreases. On each step surface, a higher positive pressure mainly occurs near the convex angle, at about 0.3 times the length of the step position. This is because the water impact on the horizontal surface of the steps at the impact point causes a larger positive pressure; after the impact of the steps, some of the flow continues downstream, while the rest of the water turns to the vertical side of the steps and climbs upwards when it meets them, resulting in a gradual decrease in pressure, which reaches a minimum near the downstream side of the steps. In addition, the climbing water near the vertical side is guided by the mainstream and, thus, flows downstream, forming a vortex. Negative pressure mainly occurs on the vertical surface of each step, at approximately 0.6–1 times the height of the step. In the straight section of the step, the pressures are all positive. Here, the step water impacting the flat section creates a higher pressure; elsewhere, the pressure distribution is uniform.

#### 3.4. Energy Dissipation Analysis

_{1}and E

_{2}are the total energy of the 1-1 section and the 2-2 section, that is, the upstream and downstream sections; Δh is the difference in height between the two sections; ΔE is the energy difference; α

_{1}and α

_{2}are the flow rate coefficients of the upstream and downstream sections, which generally take 1; v

_{1}and v

_{2}are the average flow rates of upstream and downstream sections; the η is the dissipation rate.

## 4. Conclusions

- (1)
- The staggered arrangement of energy-dissipating piers causes water to jump back. After the jump back, the water depth increases and this improves the air-doping effect of the water flow by increasing the air-doping volume, additionally lowering the risk of cavitation damage in this stage. The staggered arrangement results in a better doping concentration than the double-row arrangement; however, the double-row arrangement is better able to reduce the cavitation hazard. Both arrangements exhibit a full section stage-doping, but the staggered arrangement results in different doping concentrations in different longitudinal sections.
- (2)
- The water flow velocity in the segmented energy dissipation pier structure is high at the surface and lower towards the bottom; the water jump generated by the flat section of the pier reduces the flow velocity along the pier and reduces scour damage downstream. The staggered arrangement improves the flow structure and stabilises the flow; the lower flow velocity at the downstream outlet results in reduced scour damage downstream.
- (3)
- The existence of the flat section of the energy dissipation pier in the segmented pier-type structure means that the piers face the water surface, resulting in high positive water pressure upstream, while high negative pressure builds at the top of the pier and the backwater surface. This negative pressure is higher with the staggered arrangement, and this should be borne in mind by engineers involved with real-world structure projects.
- (4)
- Compared with the traditional no-pier design, the segmented pier-added step energy dissipation structure increases the energy dissipation rate by more than 5% on average and produces an improved energy dissipation effect; however, the choice of a double-row or staggered arrangement has little or no effect on the energy dissipation overall.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

ρ | Density factor derived from weighted average of volume fractions (-) |

μ | Molecular viscosity factor derived from a weighted average of volume fractions (-) |

G_{k} | Turbulent energy generation term due to mean flow gradient (-) |

μ_{t} | Turbulent viscosity coefficient (-) |

μ_{eff} | Effective viscosity (-) |

C_{1ε}, C_{2ε} | Turbulence model coefficients of 1.44 and 1.92, respectively (-) |

α_{κ}, α_{ε} | The inverse of the Planter number for k and ε, taken as 1.0 and 1.3, respectively (-) |

C_{μ} | Experience factor, 0.0845 (-) |

η_{0}, β | Model constants, taking values of 4.38 and 0.015, respectively (-) |

Q | Flow (m^{3}/h) |

E_{1} | 1-1 Total energy of the cross-section (J) |

E_{2} | 2-2 Total energy of the cross-section (J) |

∆E | Energy difference (J) |

α_{1}, α_{2} | Flow coefficient at upstream and downstream sections, generally taken as 1 (-) |

v_{1} | Average cross-sectional flow rate upstream (m/s) |

v_{2} | Average cross-sectional flow rate downstream (m/s) |

η | Energy dissipation rate (-) |

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**Figure 2.**Schematic diagram of the arrangement of energy dissipation pier. (

**a**) Double-row arrangement, (

**b**) staggered arrangement, and (

**c**) traditional steps.

**Figure 6.**Comparison of simulated and measured values of Q = 30 m

^{3}/h. (

**a**) Water depth along the course, (

**b**) flow rate, and (

**c**) pressure.

**Figure 7.**Flow pattern of each body type step water flow at Q = 30 m

^{3}/h. (

**a**) Traditional steps, (

**b**) double-row arrangement, and (

**c**) staggered arrangement.

**Figure 8.**Flow pattern of different longitudinal sections in a double row arrangement. (

**a**) y = 5 cm, and (

**b**) y = 7 cm.

**Figure 9.**Flow pattern of different longitudinal sections in staggered arrangement. (

**a**) y = 5 cm, and (

**b**) y = 7 cm.

**Figure 10.**Flow velocity diagram of each body type step section at Q = 30 m

^{3}/h. (

**a**) Traditional steps, (

**b**) double-row arrangement, and (

**c**) staggered steps.

**Figure 12.**Segmented pier-type step dissipation pressure distribution at Q = 30 m

^{3}/h. (

**a**) Double-row arrangement and (

**b**) staggered arrangement.

Parameters | Range of Variations |
---|---|

Q | 20 m^{3}/h, 30 m^{3}/h, 40 m^{3}/h, 50 m^{3}/h, 60 m^{3}/h |

C_{1ε} and C_{2ε} | 1.44 and 1.92 |

α_{κ} and α_{ε} | 1.0 and 1.3 |

C_{μ} | 0.0845 |

η_{0} and β | 4.38 and 0.015 |

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**MDPI and ACS Style**

Feng, Z.; Li, Y.; Tian, Y.; Li, Q.
Effects of Energy Dissipation Pier Arrangements on the Hydraulic Characteristics of Segmented Pier-Type Step Energy Dissipator Structures. *Water* **2022**, *14*, 3590.
https://doi.org/10.3390/w14223590

**AMA Style**

Feng Z, Li Y, Tian Y, Li Q.
Effects of Energy Dissipation Pier Arrangements on the Hydraulic Characteristics of Segmented Pier-Type Step Energy Dissipator Structures. *Water*. 2022; 14(22):3590.
https://doi.org/10.3390/w14223590

**Chicago/Turabian Style**

Feng, Ziwei, Yongye Li, Yu Tian, and Qian Li.
2022. "Effects of Energy Dissipation Pier Arrangements on the Hydraulic Characteristics of Segmented Pier-Type Step Energy Dissipator Structures" *Water* 14, no. 22: 3590.
https://doi.org/10.3390/w14223590