A Study on the Shape of Parabolic Aeration Facilities with Local Steepness in Slow Slope Chutes

Abstract

For flood discharge structures with high water heads, aeration facilities are usually installed in engineering to promote water flow aeration and prevent cavitation damage to the overflow surface. Actual engineering has shown that as the slope of the discharge channel bottom decreases or water level changes lead to a decrease in the Froude number, the cavity morphology after conventional aeration facilities or allotype aerators is poor. This article proposes a curved aeration facility scheme based on the idea of locally increasing the bottom slope to reduce the impact angle, which is formed by the convex parabolic bottom plate and concave parabolic bottom plate. The convex parabolic bottom plate is tangent to a flat bottom plate behind the offset, and the concave parabolic bottom plate is tangent to the downstream. The jet landing point is controlled at the junction of the convex parabolic bottom plate and the concave parabolic bottom plate, and the lower jet trajectory is in line with the parabolic bottom plate. The corresponding parabolic bottom plate calculation formulas were theoretically derived, and the design method of the shape parameters of the aeration facility was provided. Through specific engineering case studies, it was found that: (1) As the Z_AC/Z_AG value increases, point C becomes closer to point G, the slope of the water tongue landing point C becomes steeper, and the cavity is less likely to return water. (2) When the position of the water tongue landing point is 0.5–0.8 times the height of the water tongue impact point, there is almost no water accumulation in the calculated cavity. At this time, the platform length L_AB = 0.5L_AF, the convex parabolic section length L_BC = (0.45–0.6) L_AG, the concave parabolic section length L_CD = (0.43–0.11) L_AG, the convex parabolic section calculation formula is z (x) = −A₁x₂ (A₁ = 0.0059–0.00564), and the concave parabolic section calculation formula is A₂x₂ − B₂x₂ (A₂ = 0.003347–0.01927).This solved the problem of aeration and corrosion reduction under small bottom slope, large-unit discharge, and low Froude number engineering conditions.

1. Introduction

In high head flood discharge structures, the water flow velocity is high. Low-pressure areas can easily form in the contraction part of the inlet section of the flood discharge tunnel, the side wall behind the gate groove, the turning section of the tunnel, the top of the overflow weir, the uneven part of the discharge groove, and the energy dissipation ridge. When the pressure in these areas drops below the saturated vapor pressure of the water, the gas core in the water will quickly expand and form bubbles. These bubbles will flow with the water flow and quickly collapse if the pressure rises again [1]. The collapse process of bubbles will generate great impact force and high temperature, causing serious damage to the walls of flood discharge structures, namely cavitation damage [2]. Aeration and erosion reduction measures are usually adopted to prevent cavitation damage on the flow surface [3,4]. At present, aeration facilities can be divided into conventional aeration facilities and small bottom slope aeration facilities [5]. A conventional aeration facility consists of several transverse and vertical grooves around the chute walls that are connected to an air supply. Air is entrained into the water flow through the air cavity neighboring the water flow surface owing to the difference in atmospheric pressure. The cavity jet length, air entrainment coefficient, and air diffusion evolution near the wall are the main parameters used to describe the aerator efficiency [6]. For example, Pfister and Hager’s [7,8] investigation focuses on the flow structure and the air transport downstream of chute aerators: Minimum F_r values for efficient air entrainment by aerators are F_r = 6 for offsets, F_r = 5 for deflector aerators with the deflector angle = 6°, and F_r = 4 for the deflector angle = 11°. Steep deflectors are more efficient than those with small values of the deflector angle. Aerators with a floor slope of 30° on steep chutes operate more efficiently than those on flat chutes. Therefore, as the bottom plate angle of the discharge channel decreases, the efficiency of conventional aeration facilities significantly decreases. This is mainly due to the serious return water in the cavity under the action of small bottom slopes, which blocks the ventilation holes and leads to the failure of aeration facilities [9]. In recent years, domestic scholars have proposed some small bottom slope aeration facilities based on engineering practice, namely irregular aerators. These special-shaped aerators have the same structural parameters as conventional aerators. By controlling the lateral distribution of water tongue landing points and reducing the impact angle between the water tongue and the bottom plate behind the aerator, the shape of the aerator is changed to some extent, eliminating cavity backflow. For example, Pang et al. [10] proposed a U-shaped groove aerator; Sun et al. [11] proposed a concave aerator in conjunction with the Xiao wan spillway tunnel; Wang et al. [12] proposed a V-shaped groove aerator; Deng et al. [6] proposed a wedge-shaped aerator arranged at the end of the pressurized inlet of the small bottom slope open channel. When the bottom slope of the discharge channel is small, the application range of these irregular aerators is small, and cavity water accumulation is still inevitable. According to research results, the key factor causing cavity water accumulation is the impact angle between the jet water tongue and the bottom plate. When the impact angle decreases to a certain extent, the water accumulation disappears. The critical impact angle given by Zhang Liheng et al. [9] is 9°. Based on this, some scholars have proposed “aerator + local steep slope” [13], such as Sun Zhenxing, Wang Fangfang, and others [14] proposed a “planar V-shaped ridge + local steep slope” aeration facility to solve the problem of aeration and erosion reduction in flood discharge cavities under conditions of gentle bottom slopes and large water level fluctuations; Liu et al. [15] proposed a “bottom plate bending aerator” by fitting the local bottom plate curve with the lower jet trajectory. The aerated facility has a continuous and simple two-dimensional structure, which not only effectively suppresses cavity backflow, but also has a good flow state. However, due to the lack of specific design methods for shape parameters, the aerated facility still needs to continuously adjust the local bottom plate shape behind the offset in specific engineering applications, which is time-consuming and laborious. Based on the research results of Liu et al. [15], this article proposes a curved aeration facility scheme based on the idea of locally increasing the bottom slope to reduce the impact angle of the jet flow. The parabolic curve of the bottom plate is theoretically derived, and a design method for the shape parameters of the aeration facility is provided. The reliability and rationality of this design method are verified through engineering examples. Through engineering examples, it is verified that the parabolic aerated facility proposed in this article effectively eliminates the problem of cavity backflow and solves the problem of cavity backflow in small bottom slope aerated facilities.

2. The Principle of the Double Parabolic Aerator Shape Design

The impact angle of the jet flow behind offset is the most critical factor causing cavity backflow. When the impact angle decreases to a certain extent, the accumulated water in the cavity disappears [16]. Based on this idea, as shown in Figure 1, this study transforms the local bottom slope behind the offset into a body structure composed of the flat bottom plate L_AB, the convex parabolic bottom plate L_BC, and the concave parabolic bottom plate L_CD. The lower edge OC of the jet flow impacts on the connection point C of the convex bottom plate and the concave parabolic bottom plate, where the local slope is the largest. By designing the parabolic curves reasonably, the lower jet trajectory fits to the bottom plate, effectively reducing the jet impact angle, eliminating cavity backflow, and ensuring sufficient aeration flow.

Establish three Cartesian coordinate systems, xOz, x₁Bz₁, and x₂Cz₂, with the vertex O, B, and C as the origin, as shown in Figure 1. In the figure, the deflector height is t and the deflector angle is γ, and the jet take-off angle is θ_L. The bottom slopes of the upstream and downstream discharge channels are, respectively, α₁and α₂. β₁ is the angle between the jet landing point C and the x-axis, β₂ is the tangent angle of the convex parabolic the bottom plate at point C, β is the impact angle between the lower jet trajectory and the convex parabolic bottom plate, and β = β₁ − β₂. The endpoint A at the bottom of the offset intersects with the lower jet trajectory at point F, and B is the point on the segment AF. Create a convex parabolic segment BC that is tangent to line segment AF at point B, tangent to the lower jet trajectory at point C, and then create a concave parabolic segment CD that is tangent to the convex parabolic segment at point C, and tangent to the downstream bottom plate at point D. This is the parabolic aerator design scheme.

Next, determine the relevant design parameters for the jet landing point C that fits the body shape.

3. The Lower Jet Trajectory

To construct the relationship between the lower edge of the jet water tongue behind the ridge and the local bottom plate, it is necessary to understand the trajectory of the jet water tongue movement.

Jet trajectories are typically described by the physically based point-mass parabola versus the take-off conditions, thereby assuming constant density and neglecting jet disintegration and aerodynamic interaction. However, the effective take-off angle of a jet trajectory is not identical with the boundary angle of the terminal structure due to pressure rearrangement. Therefore, take-off angles are usually derived from prototype observation or from physical model tests, thereby assuming a parabolic trajectory geometry and fitting the take-off angles, depending on the deflector’s geometry and the hydraulic conditions [17].

In the case of Froude number F_r < 7, under the condition of a small bottom slope, the jet distance is short, the influence of aeration resistance is small, and the effect of gravity is significant, so only the correction of the jet take-off angle needs to be made [18]. According to mass point dynamics, the lower jet trajectory equation reads:

$$z(x)=x\tan {\theta }_L-\frac{x^2}{2F_r^{2}h{\cos }^2{\theta }_L}$$

(1)

Here, the approach flow Froude number is F_r; the approach flow depth is h; the jet take-off angle is θ_L.

There have been many achievements in the study of the take-off angles of the lower jet trajectory, and significant achievements have been made, such as Chanson [19], Steiner et al. [20], Rutschmann and Hager [21], Wu and Ruan [22], Pfister [23], and Long Qiang et al. [24]. The shape of the parabolic aerator is achieved by designing the reasonable convex concave parabolic curve to achieve the lower jet trajectory which fits the bottom plate. The accuracy of the take-off angle of the lower jet trajectory directly affects the shape of the cavity, and also is a key parameter for the successful design of parabolic aerated facilities. In recent years, turbulence numerical simulation has been widely used in hydraulic engineering. Deng and Xu [25], Luo and Diao [26], and Gao et al. [27] simulated the aeration flow in tunnels and spillways. The results show that the numerical simulation results are in good agreement with the experimental data, which shows that turbulence numerical simulation is feasible for simulating the hydraulic characteristics of the Large Discharge Spillway Tunnel. Based on this, the numerical simulation method is used to calculate the take-off angle of the lower jet trajectory.

4. Calculation Formulas of the Parabolic Bottom Plate

As shown in Figure 1, the bottom plate structure behind the offset is divided into three parts: the first part is the platform bottom plate, the second part is the convex parabolic bottom plate, and the third part is the concave parabolic bottom plate. Through theoretical analysis, the impact point G is a key parameter for constructing the relationship between the lower edge of the jet water tongue and the local bottom plate. Therefore, the relationship between the platform section and the horizontal jet distance is established, and the relationship between the lower edge of the water tongue and the convex concave bottom plate is established. The specific process is as follows:

4.1. The Length of the Platform Bottom Plate

Put Z = −Δ into Equation (1), from which can be obtained the intersection point of the lower jet trajectory and the platform extension line:

$$L_{AF}=0.5F_r^{2}h\sin 2{\theta }_L\left(1\pm \sqrt{1+\frac{2\Delta }{F_r^{2}h{\sin }^2{\theta }_L}}\right)$$

(2)

In the formula: Δ is the total height, m; if θ_L > 0, take “+”, if θ_L < 0, take “−”.

Therefore, the length of the platform bottom plate is written as:

$$L_{AB}=k{L}_{AF}$$

(3)

where k is the coefficient to be determined, and 0.45 ≤ k < 0.50.

4.2. Calculation Formulas of the Convex Parabolic Bottom Plate

In the coordinate system x₁Bz₁, the formula for calculating the convex parabolic bottom plate is:

$$z(x_1)=a_1{x}_1^2+b_1{x}_1+c_1$$

(4)

In the formula, a₁, b₁, and c₁ are undetermined coefficients.

When x₁ = 0 and z₁ = 0, c₁ = 0.

Taking the derivative of calculating Equation (4) yields the following Equations (5) and (6):

$${\left.\frac{d{\mathrm{z}}_1}{d{x}_1}\right|}_{x_1=0}=2a_1{x}_1+b^1=0$$

(5)

$${\left.\frac{d{\mathrm{z}}_1}{d{x}_1}\right|}_{x_1=L_{BC}}=2a_1{L}_{BC}+b_1=-\tan {\beta }_2$$

(6)

The calculation formula of the convex parabolic the bottom plate is:

$$z(x_1)=\frac{-\tan {\beta }_2}{2L_{BC}}{x}_1^2$$

(7)

In the formula: L_AF < L_BC + L_AB < L_AG.

β₂ = β₁ − β

In the formula: β is the actual impact angle between the lower jet trajectory and the bottom plate; if β = 0, it indicates that the lower jet trajectory is completely tangent to the bottom plate. β₁ is obtained from Equation (8); based on β₂ = β₁ − β, β₂ can be obtained.

$$\tan {\beta }_1=-{\left.\frac{dz}{dx}\right|}_{x=L_{AC}}$$

(8)

According to Equation (7), the length of L_BC depends crucially on the vertical distance Z_AC from the point C to the platform bottom plate. The value of Z_AC is closely related to the elevation of the original impact point G, as the elevation of the jet landing point C will not be lower than the original impact point G. Therefore,

$$Z_{AC}=n{Z}_{AG}$$

(9)

In the formula, n ≤ 1.0 is used to ensure that the point C position is as far away as possible and the elevation is low, and n is taken as high as possible.

After the Z_AC value is determined, the L_BC value can be determined.

4.3. Calculation Formula of the Concave Parabolic Bottom Plate

In the second coordinate system x₂Cz₂, the formula of the concave parabolic the bottom plate is obtained as follows:

$$z(x_2)=\frac{-\tan {\alpha }_2+\tan {\beta }_2}{2L_{CD}}{x}_2^2-\tan {\beta }_2{x}_2$$

(10)

Here: L_CD > L_AG − L_AC.

The calculation formulas (3), (7), (9), and (10) are the general equations for parabolic bottom plates for an aerated facility. This equation is applicable to conventional two-dimensional aerated facilities.

5. Engineering Application

The ecological drainage (emptying) tunnel on the left bank of the particular hydropower station has a total length of approximately 1260 m. The outlet of the pressure tunnel is connected to the working gate chamber, and the working gate chamber is connected to the non-pressure tunnel section. The longitudinal slope of the non-pressure section bottom plate is i = 0.1, and the verified flood level is H = 3894.18 m, Q = 983 m³/s.

5.1. The Shape Design Based on This Paper’s Approach

The original first aerator in the non-pressure section of the ecological drainage tunnel was located at the 0 + 920.20 m section of the gate chamber. Due to the gentle slope of the bottom of the spillway, the stability of the cavity of conventional aeration facilities is poor, and is prone to the formation of water accumulation. Water accumulation will reduce the length of the cavity, causing a decrease in ventilation volume. In severe cases, it can block the ventilation holes and cause poor air intake. Aeration facilities have instead become artificially convex bodies, forming cavitation sources and causing cavitation and erosion damage. This article adopts a combination of “the small deflector + the offset + parabolic bottom plate” aerators, which is a parabolic aerator. The small deflector height is 0.3 m, with a slope of 1:10, and the offset height is 0.7 m. The reason for using a combination of “the small deflector + the offset + parabolic bottom plate” aerators is that the small deflector + the offset causes little disturbance to the original water surface and easily forms a stable cavity. The parabolic bottom plate can reduce the impact angle between the jet water tongue and the original bottom plate, effectively eliminating cavity backflow. This combination setting fully utilizes the advantages of various body structures, overcomes their shortcomings, and achieves the goals of aerating and reducing corrosion.

Given the hydraulic parameters under the verified water level, use the parabolic bottom plate calculation formula in the third subsection of the text to design the parameters of each part of the bottom plate. In the design process, not only should the convex parabolic bottom plate be tangent to the platform bottom plate, but also the concave parabolic section should be tangent to the downstream bottom plate. Therefore, iterative calculation methods should be used in the calculation and design process.

5.1.1. Known Conditions

Known hydraulic parameters in front of the deflector: the water depth h = 6.1 m, Froude number F_r = 4.163, total height of the aerator Δ = 1.0 m, the small deflector height is 0.3 m, with a slope of 1:10, and the offset height is 0.7 m, upstream slope i₁ = 0, downstream bottom slope i₂ = 0.1; numerical simulation is used to simulate the combination of “the small deflector + the offset” combination aerator and calculate the take-off angle θ_L = 4 °of the lower jet trajectory.

5.1.2. The Control Parameters Calculation

①

L_AF length

Calculate the position of the intersection point between the lower jet trajectory and the platform extension line using Equation (2):

L_AF = 23.611 m;

②

L_AB length

According to Equation (3), let k = 0.5; then, the length of the platform segment is:

L_AB = 11.806 m;

③

L_AG length

Calculate the jet distance L_AG when the jet flow impacts the original bottom plate, L_AG = 40.20 m;

④

Z_AG height

Calculate the height of the impact point G where the jet flow impacts the original bottom plate relative to the platform section: Z_AG = 4.02 m.

5.1.3. The Design Parameters Calculation

①

Calculate Z_AC

After determining the Z_AC value using Equations (3) and (7), a reasonable body shape can be obtained. According to Equation (9), let n = 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and calculate the Z_AC values listed in

Table 1. Type design based on this paper’s approach.

Design Shape		Convex Parabolic Bottom Plates			Concave Parabolic Bottom Plates			β2/°
	ZAC /m	LAB/m	LBC /m	Calculation Formulas	LCD /m	Calculation Formulas	ZCD /m
M0.3	1.206	11.806	13.861	−0.00628x2	37.292	0.000994x2 − 0.1741x	5.108	9.876
M0.4	1.608	11.806	16.305	−0.00605x2	25.140	0.001936x2 − 0.1972x	3.736	11.163
M0.5	2.010	11.806	18.461	−0.00590x2	17.601	0.003347x2 − 0.2177x	2.796	12.289
M0.6	2.412	11.806	20.416	−0.00579x2	12.163	0.005608x2 − 0.2363x	2.045	13.301
M0.7	2.814	11.806	22.214	−0.00571x2	7.909	0.009704x2 − 0.2534x	1.397	14.225
M0.8	3.216	11.806	23.887	−0.00564x2	4.394	0.019277x2 − 0.26926x	0.811	15.078
M0.9	3.618	11.806	25.460	−0.00558x2	1.380	0.066795x2 − 0.28420x	0.265	15.873

, represented by M_0.3, M_0.4, M_0.5;

②

Calculate L_BC and β₂

According to L_AF − L_AB < L_BC < L_AG − L_AB, within this range, in descending order, β₁ is calculated by initially assuming the L_BC value, substituting it into Equation (8);

③

Calculation formula of the convex parabolic segment

Substitute the β₂ value and the L_BC value into Equation (7) to obtain the calculation formula for the convex parabolic bottom plate;

④

Calculate L_CD

Within the range of L_CD > L_AG − L_AC, in ascending order, the L_CD value that is inputted into Equation (10) to calculate Z′_CD is preliminarily assumed;

⑤

Calculation formula of the concave parabolic section

Take β₂, L_BC, α_2. Substitute into Equation (10) to obtain the calculation formula for the concave parabolic bottom plate.

5.2. The Influence of the Jet Landing Point C Position on the Bottom Plate Shape Parameters

List the parameters of each part of the bottom plate obtained above in Table 1, and plot the relationship between Z_AC/Z_AG and the relative length L_BC/L_AG, the relative length L_CD/L_AG, and the relative total length L_AD/L_AG, as shown in Figure 2. As the Z_AC/Z_AG value increases, the relative length L_BC/L_AG gradually increases, the relative length L_CD/L_AG significantly decreases, and the relative total length L_AD/L_AG gradually decreases. This indicates that the larger the Z_AC/Z_AG value, the closer point C is to point G, the steeper the slope of the jet landing point C, and the less likely the cavity is to backflow.

5.3. Numerical Simulation

To visually observe the shape of the cavity, turbulent numerical simulation was used to simulate and calculate these seven types of body shapes.

5.3.1. Computational Model

The numerical simulation calculation area includes the upstream reservoir, pressure tunnel section, gate chamber section, and non-pressure section, with a total length of about 1260 m, as shown in Figure 3. The RNG k-ε turbulence model is suitable for complex flows such as strongly turbulent water flow, jet collision, and separated flow. It considers the anisotropy of turbulence and corrects turbulence viscosity, which can repair complex turbulence simulations to a certain extent, and especially for situations with large flow amplitudes, the RNG k-ε model has more advantages, so this article adopts the anisotropic RNG k-ε turbulence model [25]. A coupled solution was obtained using the PISO algorithm, using unsteady flow to approximate constant flow for iterative calculation.

In order to accurately observe the shape of the cavity, the aerator section is the focus of simulation calculations. Due to its clear interface tracking, conservation, interface reconstruction techniques, wide applicability, and ease of implementation, the VOF method is particularly suitable for capturing air/water interfaces in simulations [26,27]. In this paper, the VOF [28,29,30] method is used to calculate the water/air interface. The basic principle of the VOF method is to define a fluid volume function F in each grid unit in the computational domain. When F = 1, it means that the grid unit is completely filled with a certain fluid phase. When F = 0, it indicates that there is no fluid phase in the grid cell. When 0 < F < 1, it indicates that the grid cell is filled with at least two fluid phases, and the cell is referred to as the interface between the fluids. The value of F can be obtained by solving Equation (11), which needs to satisfy the following transport equation:

$$\frac{\partial F}{\partial t}+\frac{\partial uF}{\partial x}+\frac{\partial vF}{\partial y}=0$$

(11)

In the formula, F is the ratio of the fluid volume within a certain unit to the total volume of that unit.

5.3.2. Grid Division and Boundary Conditions

Due to the fast convergence speed, high accuracy, and high computational efficiency of structured grids, the overall model of ecological drainage tunnels adopts structured grid division.

However, the grid size has a direct impact on the accuracy of capturing convective features. If the grid is too large, it may not be possible to capture the subtle structural changes at the lower edge of the jet water tongue behind the ridge; if the grid is too small, although it can improve accuracy, it will increase computation time and resource consumption. In order to reduce time and improve calculation speed, only the aeration facility section is encrypted. The grid size of this area along the X and Z directions is 0.1 m, and the grid size in the Y direction is 0.1 m, in order to obtain a clearer cavity shape. The connecting calculation areas are set as gradient grids, and the total number of grids is kept below 1.2 million; Figure 4 shows the local grid division of the aerator section. In the case of a known flow rate, the inlet boundary condition of the upstream reservoir area adopts a pressure inlet boundary, setting a characteristic water level for the verification flood level, corresponding to a flow rate of 983 m³/s. The outlet adopts a pressure outlet, and the ventilation port and the top surface of the reservoir are in contact with the atmosphere. The pressure inlet is given at standard atmospheric pressure, and the wall surface adopts a no-slip boundary condition. The viscous bottom layer is treated using the wall function method. The constant discrimination condition for numerical simulation calculation is that the error between the inlet flow rate and the outlet flow rate does not exceed ±5%.

5.3.3. Numerical Simulation Calculation Results

Calculate Cavity Shape

The numerical simulation first calculates “the small deflector + the offset“ combination aerator, and the calculated cavity shape is shown in Figure 5. Then, seven reasonable shapes designed in Section 4.1 and Section 4.2 are simulated and calculated. The calculated cavity shape of the longitudinal profile of the discharge channel bottom plate and the bottom plate are shown in Figure 6. (1) Due to the dual effects of small bottom slope and gravity, traditional aerators have severe cavity backflow and poor flow state. The seven reasonable types of aerators calculated using the design method in this article have stable cavities and better flow state than traditional aerating facilities. When Z_AC/Z_AG ≤ 0.5, there is a small amount of water accumulation in the cavity, and when Z_AC/Z_AG ≥ 0.5, there is almost no water accumulation in the cavity. (2) As the Z_AC/Z_AG value increases, the maximum net cavity length shows a trend of first increasing and then gradually decreasing. (3) The maximum net cavity length on the bottom plate of the discharge channel shows a shape of “convexity on the middle and concavity on both sides”, which is mainly due to the fact that in the two-dimensional continuous aeration facility the middle jet flow has a high velocity, while the jet flow on both sides has a low velocity.

Distribution Characteristics of Bottom Plate Pressure

Time-average pressure distribution of the local bottom plates is shown in Figure 7 and the calculated value of time-average pressure at the centerline of the bottom plate is shown in Figure 8. Under different Z_AC/Z_AG values, we plot the relationship between relative convex parabolic segment L_AC/L_AG and relative impact force peak position L_A-IFP/L_AG, and the relationship between relative total length L_AD/L_AG and relative centrifugal force peak position L_A-CFP/L_AG is shown in Figure 9. (1) The pressure change in the platform bottom plate is not significant, with a range of −2 KPa < p < 0 KPa. When Z_AC/Z_AG = 0.6, the negative pressure is minimum, and when Z_AC/Z_AG = 0.9, the negative pressure is maximum. (2) The pressure distribution in the convex concave parabolic section shows a “bimodal” pattern. The maximum pressure value generated when the jet flow impacts the convex parabolic bottom plate is the impact force peak, and the maximum pressure value generated when the centrifugal force acts on the concave parabolic bottom plate is the centrifugal force peak. At the same Z_AC/Z_AG value, the pressure significantly increases to the impact force peak, gradually decreases and fluctuates within a certain range, then gradually increases to the centrifugal force peak and significantly decreases and tends to stabilize. (3) As the Z_AC/Z_AG value increases, the impact force peak first increases and then decreases, and the maximum net cavity length also increases first and then decreases, while the centrifugal force peak gradually decreases. Possible causes of this phenomenon are as follows: One reason is that as the Z_AC/Z_AG value increases, the concave parabolic section gradually decreases and rapid stream, resulting in the flow unable to discharge smoothly, causing the upstream jet flow to impact the bottom plate in advance, and the impact force peak and the centrifugal force peak to gradually coincide. Secondly, although the take-off angle of the lower jet trajectory is accurately calculated in the process of body shape design, the Froude number Fr calculation uses the average velocity in front of the aerator, resulting in a difference between the designed point C and the impact force peak position. Overall, the position of the impact force peak fluctuates back and forth within a certain range of the point C.

But regardless of the reason, when Z_AC/Z_AG ≥ 0.5, the calculated net cavity can still maintain a certain length and almost no water accumulation after the aerator.

6. Discussion

Through the above research, it has been found that under small bottom slopes and low Froude number engineering conditions, the “aerator + local steep slope” aerated facility can effectively suppress or reduce cavity backflow, and obtain a relatively stable aerated cavity. However, whether it is a special-shaped aerator or a combination of “aerator + local steep slope” aerator, when the bottom plate of the discharge channel decreases or the Froude number decreases, the accumulated water in the cavity cannot be eliminated due to the dual effects of gravity and small bottom slope, indicating a relatively weak adaptability. The double parabolic aerator is designed for engineering conditions such as small low slopes or flat bottom slopes. The basic principle of the design is to locally increase the bottom slope behind the offset, reduce the impact angle between the jet water tongue and the bottom plate, and keep the lower edge of the jet water tongue as close as possible to the local bottom plate. Locally increasing the bottom slope only changes the shape of the local bottom plate and does not change the size of the original bottom slope, which is independent of the size of the original bottom slope. It is suitable for a wider and stronger range of bottom slopes. In addition, both the irregular aerator and the “aerator + local steep slope” measures are determined through repeated engineering tests, which is time-consuming and laborious. The reason for this is that there is no design method with body shape parameters for reference, and it all depends on engineering experience. The double parabolic aerated facility has its own design method for body shape parameters, which can be used as a preselection scheme for designing the optimal body shape parameters of the aerated facility, saving time and effort.

7. Conclusions

To solve the problem of aeration and erosion reduction in discharge structures under conditions of small bottom slope, low Froude number, and large single-width flow rate, this paper proposes a new double parabolic aeration facility based on the concept of local steepening. The convex parabolic section is tangent to the flat section behind the ridge, and the concave parabolic section is tangent to the downstream bottom plate. The water tongue landing point is controlled at the junction of the convex concave parabolic curve, and the lower edge of the water tongue is in contact with the parabolic bottom plate. The calculation formula for the parabolic bottom plate has been theoretically derived, and a design method for the shape parameters of the aeration facility has been provided. Through engineering examples, it has been found that the larger the Z_AC/Z_AG value, the closer the C point is to the G point, and the less likely the cavity behind the offset is to retain water. When the water tongue landing point is 0.5–0.8 times the height of the water tongue impact point, there is almost no water accumulation in the calculated cavity. Therefore, the larger the Z_AC/Z_AG value, the better, under the condition of meeting the downstream connection conditions of the project. The research results of this article make up for the lack of experience in the design of aerated structures, and promote the quantification of aerated corrosion reduction in discharge structures, which has important engineering value. In addition, to increase the volume of the aerated cavity, systematic large-scale model experiments will be carried out in subsequent work to optimize the structural parameters of the double parabolic aerated facility and provide a design method for the optimal aerated facility body parameters.