# Study on the Hydraulic Characteristics of the Trapezoidal Energy Dissipation Baffle Block-Step Combination Energy Dissipator

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Construction of Mathematical Models

#### 2.1. Turbulence Model

_{i}and u

_{j}are velocity components; x

_{i}and x

_{j}are coordinate components; μ is the molecular viscosity coefficient; μ

_{t}is the turbulent viscosity coefficient, taken as 0.0845; σ

_{k}and σ

_{ε}are the Prandtl numbers corresponding to the turbulent kinetic energy k and the turbulent kinetic energy dissipation rate ε, respectively; G

_{k}is the turbulent kinetic energy generation term due to the average velocity gradient; C

_{1ε}and C

_{2ε}are constant terms, C

_{1ε}= 1.42 and C

_{2ε}= 1.68.

#### 2.2. Air Entrainment Related Models

_{T}is the turbulence length scale; ρ

_{m}is the mixed-phase density; g

_{n}is the gravitational normal component to the water surface; σ is the surface tension coefficient; S

_{a}is the volume of gas admixed into the mesh per unit time; K

_{air}is the scaling factor and the default value is 0.5.

_{ai}is the velocity of motion of the air phase [10,24]; D

_{c}is the diffusion coefficient; S

_{a}is the air admixture source term in Equation (8); V

_{c}is the mesh volume. The average density of the two-phase flow is calculated using the following equation:

_{a}is the air density; ρ

_{b}is the average density of the two phases of water and air; ρ

_{w}is the density of water.

_{r}is the slip velocity, and K is the interphase drag coefficient, calculated by the following equation:

_{p}is the cross-sectional area of the bubble; C

_{d}is the custom resistance coefficient; ρ

_{c}is the density of the continuous phase; μ

_{c}is the dynamic viscosity of the continuous phase; V

_{p}is the volume of a single bubble; R

_{p}is the bubble radius.

#### 2.3. Meshing and Boundary Conditions

#### 2.4. Sensitivity Analysis of Grids

## 3. Experimental Validation of Mathematical Models

#### 3.1. Experimental Systems

_{1}= 1 m and the width was W = 0.2 m. The step stage consisted of six steps, the slope ratio was 1:3, the length of a single step was L

_{2}= 0.18 m, the height was h = 0.06 m, and the width was s = 0.2 m. The model arrangement of two experimental schemes is shown in Figure 4: type I for the traditional step energy dissipator scheme; type II for the trapezoidal energy dissipation baffle block-step combination energy dissipator scheme. In the horizontal plane of the step at the convex angle position, the trapezoidal energy dissipation baffle block was placed. The trapezoidal energy dissipation baffle block lower bottom length was l

_{a1}= 0.02 m, the upper bottom length was l

_{a2}= 0.01 m, the width was l

_{b}= 0.02 m, and the height was l

_{c}= 0.02 m.

_{k}/h to characterize the flow rate, where h

_{k}is the critical water depth:

^{2}/s; g is the gravity acceleration constant, m/s

^{2}. This paper selected four working conditions with ζ of 0.714, 0.936, 1.134, and 1.316 for the experiment.

#### 3.2. Model Validation

_{c}, where d is the elevation head of the water flow and dc is the total elevation of the step dissipator. The horizontal coordinate was position x

_{1}from the upstream of the step (see Figure 6 for the schematic diagram). As shown in Figure 5, the simulation calculation results were in good agreement with the experimental results. The maximum relative error along the water depth was 8.1%, proving that the simulation method was reasonable and the calculation results reliable.

## 4. Results and Discussion

#### 4.1. Air Entrainment Characteristics

_{c}from the diversion channel was measured for both types of steps. Considering the steps as rough bodies, their roughness is expressed as k

_{*}= hcosα, and the friction coefficient f is expressed as:

_{c}/h and f is shown in Figure 8. The figure shows that the initial air entrainment point of both type I and type II gradually moves backward with the increase in the flow rate. The initial air entrainment point of type II is more forward than type I. The difference L

_{c}/h between the initial air entrainment point of the two step energy dissipators gradually increases as the flow rate increases. Therefore, the trapezoidal energy dissipation baffle block-step combination energy dissipator, compared with the traditional step energy dissipator, can make the water flow in advanced aeration and slow down the increased flow so the initial air entrainment point location moves backward. The correlation equation between the initial air entrainment point location and the flow rate for two step-type dissipators was fitted using simulated data, as follows:

#### 4.2. Pressure Distribution

_{2}indicates the horizontal surface distance from the concave angle of the step, z

_{1}indicates the vertical surface distance from the concave angle of the step, and y indicates the different longitudinal profiles of the step. From Figure 9a, it can be seen that in the region of 0–0.6 L

_{2}, the pressure variation pattern in the horizontal plane is basically the same, with an obvious “V” shape. This is in conformity with the results obtained by [5]. In the longitudinal section without the energy dissipation baffle block (y = 0.1 m), the maximum pressure values of type I and type II are the same. However, the minimum pressure value of type I is smaller than that of type II, and the position of the peak and trough of type I appears backward. For type II without the energy dissipation baffle block longitudinal section (y = 0.1 m) and with the energy dissipation baffle block longitudinal section (y = 0.075 m), the peak and trough appear at the same location. The minimum pressure value is the same, but the maximum pressure value is greater with the energy dissipation baffle block longitudinal section (y = 0.075 m). At 0.6–1 L

_{2}, the pressure of type I decreases uniformly; the pressure of type II decreases slowly in the region of 0.6–0.8 L

_{2}. After 0.8 L

_{2}, the pressure of the longitudinal section without the energy dissipation baffle block (y = 0.1 m) decreases rapidly and generates negative pressure in the region of 0.93–1 L

_{2}. In contrast, the longitudinal section with the energy dissipation baffle block (y = 0.075 m) makes the pressure increase from 0.8 L

_{2}to the headwater surface of the energy dissipation baffle block, due to the obstruction of the water flow by the energy dissipation baffle block. From the cloud diagram of the step pressure distribution (Figure 10), it can be seen that by arranging the energy dissipating baffle blocks at the convex corner of the step, the pressure distribution on the horizontal surface of the step can be made more uniform. The impact damage of water flow on the horizontal surface of the step can be reduced.

#### 4.3. Flow Velocity Distribution

_{2}is the water depth perpendicular to the virtual bottom plate, z

_{max}is the maximum water depth of the section, V is the section flow velocity, and V

_{c}is the critical flow velocity. It can be seen that the flow velocity of the two types of step dissipators tends to increase gradually from the convex angle of the step to the water surface and decreases near the water surface position. This is in conformity with the results obtained by [19]. The flow velocity variation of type I from the convex corner of the step to the water surface is slight. The flow velocity at all water depths is greater than that of type II, so type II is more fully dissipating energy. Type II in the longitudinal section with the trapezoidal energy dissipation baffle block (y = 0.075 m) flow velocity variation is largest. In the longitudinal section without the trapezoidal energy dissipation baffle block (y = 0.1 m), the flow velocity variation is smaller; in 0.2–0.4, the water depth appears to be an abnormal region of higher flow velocity. The flow around the blunt-body principle suggests that the trapezoidal energy dissipation baffle block blocks the flow in the mainstream area, so the energy dissipation baffle block side of the water flow velocity increases, resulting in an abnormally larger flow velocity area. The backwater surface of the energy dissipation baffle block produces backflow and the flow velocity is lower, which makes the section flow velocity vary in a wide range.

#### 4.4. Energy Dissipation Rate Analysis

_{0}is the total upstream energy and E

_{1}is the total downstream energy. The calculation results are shown in Figure 15.

## 5. Conclusions

- (1)
- Compared with the traditional step energy dissipator, the trapezoidal energy dissipation baffle block-step combination energy dissipator can increase the energy dissipation rate by about 6.67%, with better energy dissipation characteristics. The use of a trapezoidal energy dissipation baffle block to improve the hydraulic characteristics of the step energy dissipator is feasible.
- (2)
- The trapezoidal energy dissipation baffle block-step combination energy dissipator has a better air entrainment effect. Compared with the traditional step energy dissipator, the trapezoidal energy dissipation baffle block-step combination energy dissipator can advance the initial aeration point by one step and increase the air entrainment volume of the flow. We also proposed the calculation formula of the initial aeration point of the trapezoidal energy dissipation baffle block-step combination energy dissipator.
- (3)
- The trapezoidal energy dissipation baffle block-step combination energy dissipator has a lower risk of cavitation. By adding trapezoidal energy dissipation baffle blocks at the convex corner of the step, the pressure variation law of the horizontal and vertical surfaces of the traditional step is changed, which reduces the extreme value of the pressure on the horizontal surface of the step and reduces the distribution area of the negative pressure on the vertical surface of the step, thus reducing the risk of cavitation of the step. The top and the backwater surface of the trapezoidal energy dissipation baffle block have negative pressure, which can easily cause cavitation damage, and certain protective measures will be needed for the actual project.
- (4)
- The trapezoidal energy dissipation baffle block-step combination energy dissipator has a lower flow velocity. The trapezoidal energy dissipation baffle block-step combination energy dissipator mainstream cross-sectional flow velocity still follows the law of a small bottom layer and large surface layer. However, compared with the traditional step energy dissipator, the trapezoidal energy dissipation baffle block-step combination energy dissipator mainstream section flow velocity change amplitude is higher, the flow velocity is lower, the concave angle roll area low-flow-velocity range is higher, and the energy dissipation effect is better. The trapezoidal energy dissipation baffle block-step combination energy dissipator makes the vortex structure more distributed in the vicinity of the trapezoidal energy dissipation baffle block, which helps the energy dissipation of the step energy dissipator.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A_{p} | the cross-sectional area of the bubble (m^{2}) |

C_{1ε} | constant terms (-) |

C_{2ε} | constant terms (-) |

C_{d} | the custom resistance coefficient (-) |

c | the air admixture density (-) |

D_{c} | the diffusion coefficient (-) |

d | the elevation head of the water flow (m) |

d_{c} | the total elevation of the step dissipator (m) |

f | friction coefficient (-) |

G_{k} | the turbulent kinetic energy generation term (kg/m/s^{−3}) |

g_{n} | the gravitational normal component to the water surface (m/s^{2}) |

h | step height (m) |

h_{k} | critical water depth (m) |

K | interphase resistance coefficient (-) |

K_{air} | scale factor (-) |

K* | roughness (m) |

L_{1} | length of diversion channel (m) |

L_{2} | step length (m) |

L_{c} | initial air entrainment point distance (m) |

L_{T} | turbulence length scale (m) |

l_{a1} | energy dissipating baffle block bottom length (m) |

l_{a2} | energy dissipating baffle block upper length (m) |

l_{b} | energy dissipating baffle block width (m) |

l_{c} | energy dissipating baffle block height (m) |

q | single wide flow (m^{3}/s) |

R_{p} | air bubble radius (m) |

S_{a} | volume of gas blended into the grid per unit time (-) |

t | time (s) |

U_{ai} | velocity of motion of gas phase (m/s) |

U_{r} | slip speed (m/s) |

u_{i} | i-direction velocity component (m/s) |

u_{j} | j-direction velocity component (m/s) |

V | cross-sectional flow rate (m/s) |

V_{b} | grid volume (-) |

V_{c} | critical flow rate (m/s) |

V_{p} | volume of a single bubble (-) |

W | width of diversion channel (m) |

x_{1} | location upstream from the step (m) |

x_{2} | distance from the horizontal plane of the concave angle of the step (m) |

x_{i} | i-directional coordinate components (-) |

x_{j} | j-directional coordinate components (-) |

y | steps in different longitudinal sections (m) |

z_{1} | distance from the vertical plane of the concave angle of the step (m) |

z_{2} | water depth perpendicular to the virtual substrate (m) |

z_{max} | maximum water depth at section (m) |

α | slope (-) |

ζ | dimensionless parameters (-) |

η | energy dissipation (-) |

μ | molecular viscosity coefficient (-) |

μ_{c} | power viscosity of continuous phase (N∙s/m^{2}) |

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

ρ | density (kg/m^{3}) |

ρ_{a} | density of air (-) |

ρ_{b} | average density of water and gas phases (kg/m^{3}) |

ρ_{c} | density of continuous phase (-) |

ρ_{m} | mixed-phase density (kg/m^{3}) |

ρ_{w} | density of water (kg/m^{3}) |

σ | surface tension coefficient (-) |

σ_{k} | the Prandtl number corresponding to the k (-) |

σ_{ε} | the Prandtl number corresponding to the ε (-) |

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**Figure 3.**Model experimental setup and profile arrangement: (

**a**) experimental model; (

**b**) experimental model layout section.

**Figure 4.**Schematic diagram of the experimental model arrangement: (

**a**) Type I: traditional step energy dissipator; (

**b**) Type II: trapezoidal energy dissipation baffle block-step combination energy dissipator.

**Figure 7.**Step air entrainment concentration: (

**a**) Type I, ζ = 0.714; (

**b**) Type II, ζ = 0.714; (

**c**) Type I, ζ = 0.936; (

**d**) Type II, ζ = 0.936; (

**e**) Type I, ζ = 1.134; (

**f**) Type II, ζ = 1.134; (

**g**) Type I, ζ = 1.316; (

**h**) Type II, ζ = 1.316.

**Figure 9.**Pressure distribution on the horizontal and vertical surfaces of the step: (

**a**) horizontal surface pressure distribution graph; (

**b**) vertical surface pressure distribution diagram.

**Figure 10.**Cloud diagram of the pressure distribution on the step surface of different types: (

**a**) Type I, y = 0.1 m; (

**b**) Type II, y = 0.1 m; (

**c**) Type II, y = 0.075 m.

**Figure 13.**Flow velocity vector and cloud diagrams of longitudinal sections for different types of steps: (

**a**) Type I, y = 0.1 m; (

**b**) Type II, y = 0.1 m; (

**c**) Type II, y = 0.075 m.

**Figure 14.**The Q-equivalent surface of two types of step energy dissipators: (

**a**) Type I; (

**b**) Type II.

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**MDPI and ACS Style**

Tian, Y.; Li, Y.; Sun, X.
Study on the Hydraulic Characteristics of the Trapezoidal Energy Dissipation Baffle Block-Step Combination Energy Dissipator. *Water* **2022**, *14*, 2239.
https://doi.org/10.3390/w14142239

**AMA Style**

Tian Y, Li Y, Sun X.
Study on the Hydraulic Characteristics of the Trapezoidal Energy Dissipation Baffle Block-Step Combination Energy Dissipator. *Water*. 2022; 14(14):2239.
https://doi.org/10.3390/w14142239

**Chicago/Turabian Style**

Tian, Yu, Yongye Li, and Xihuan Sun.
2022. "Study on the Hydraulic Characteristics of the Trapezoidal Energy Dissipation Baffle Block-Step Combination Energy Dissipator" *Water* 14, no. 14: 2239.
https://doi.org/10.3390/w14142239