Structural Design and Energy Dissipation Characteristics of a Horizontal Opposing Jet Energy Dissipator

Abstract

To address the limitations of traditional energy dissipation technologies, such as difficulty in arranging energy dissipators due to narrow river valleys and complex geological conditions and the low energy dissipation efficiency of existing air jet collision methods, this study proposes a novel structural form of a horizontal opposing jet energy dissipator. Water is diverted to the open area downstream of the reservoir hub via diversion pipelines, and energy dissipation is achieved through horizontal opposing collision of jets in the air. Focusing on this new energy dissipator, numerical simulations combined with physical experiments were conducted to investigate its energy dissipation characteristics, with the dimensionless parameters l/d (collision distance/pipeline diameter) and Reynolds number (Re) as the main variables. The results indicate that two opposing jets formed a crown-shaped water jet after horizontal collision in the air. The rising height in the Z-direction and expanding width in the Y-direction of the crown-shaped water jet exhibit a negative correlation with l/d and a positive correlation with Re. Energy dissipation was achieved through jet collision, mixing, friction, diffusion, aeration, and fragmentation in the air. This energy dissipation method improved the energy dissipation rate by extending the collision time and mixing length of jets in the air. The primary factors influencing the energy dissipation rate were l/d and Re. Under the study conditions, the energy dissipation rate of jet collision in the air ranged from 16.25% to 39.54%. The energy dissipation efficiency exhibits a negative correlation with l/d and a positive correlation with Re. This study provides a new approach for energy dissipation in hydraulic engineering.

1. Introduction

Energy dissipation of hydraulic structures is crucial for the overall safety and stability of water conservancy projects [1]. Traditional energy dissipation methods include bottom flow energy dissipation, trajectory jet energy dissipation, surface flow and bucket flow energy dissipation, and energy dissipation in spillway tunnels [2]. However, the discharged flow from traditional methods typically features high water heads, large flow velocities, and concentrated water jets, carrying substantial mechanical energy (especially kinetic energy) that significantly increases the energy of normal flow in downstream channels. This may exceed the bearing capacity of downstream hydraulic structures, causing damage and even threatening their safe operation [3,4,5,6]. According to incomplete statistics, nearly one-third of built discharge and energy dissipation structures have suffered varying degrees of damage, some of which are severe [7]. Moreover, in regions with narrow river valleys, high water heads, large flow rates, and complex geological conditions, arranging traditional energy dissipators for high dams and large reservoir projects is challenging. Energy dissipation design and safe operation have long been research hotspots in the water conservancy field [8]. To address these issues, air jet collision energy dissipation has gained increasing attention in recent years, which can be divided into two types: vertical collision (upper and lower jets) and horizontal collision (left and right jets) [9].

The air collision energy dissipation method using surface and deep holes primarily aims to reduce the concentrated water entry of jets. By arranging surface and deep holes with vertical differential elevation, layered and multi-hole collision energy dissipation is achieved, and a plunge pool is constructed downstream to form a combined energy dissipator of dam spillways and plunge pools [10,11,12]. The Ertan Hydropower Station adopted this method, significantly reducing downstream riverbed scouring and achieving excellent energy dissipation effects [13,14,15]. The Fengtan Gravity Arch Dam effectively reduced the depth of the scouring pit through air collision and longitudinal aeration extension [16]. Guo et al. [13] studied the relationship between collision angle, collision point position, surface hole depression angle, and middle hole trajectory angle during discharge at the Ertan Hydropower Station. Through experiments, Diao and Yang [17] concluded that the main function of jet collision is to disperse water flow. Sun et al. [18] derived formulas for calculating collision velocity, collision efficiency, and collision energy loss. Lei et al. [19] investigated the variation in flow characteristics and dynamic water pressure on the bottom plate under different flow rate ratios of surface and deep holes.

Sun et al. [20,21,22] established mathematical models to estimate the impact of left right jet collision on the equilibrium scouring depth of riverbed bedrock and derived formulas for velocity vectors and collision energy dissipation efficiency. Taylor [23] focused on the formation of thin liquid films by two colliding jets, established a theoretical model for film radius, and revealed the balance mechanism between inertial force and surface tension. Choo and Kang [24] systematically explored the velocity distribution characteristics of liquid films formed by two low-speed impinging jets and the influence of collision angle and jet velocity. Bush and Hasha [25] investigated the flow structure generated by laminar jet collision, clarifying its formation mechanism, influencing factors, and stability characteristics. Chen et al. [26] studied the formation and fragmentation of liquid films by jet impact, identified different flow patterns, and revealed the dynamic mechanism of impact waves. Lian et al. [27,28] proposed a numerical simulation method for planar and vertical jet collision modes, predicting the atomization characteristics of complex air jet collisions in high dam spillways.

Currently, the main limitation of air jet collision energy dissipation is its low efficiency. Therefore, this study proposes a novel horizontal opposing jet energy dissipator to address the challenges of arranging traditional energy dissipators in canyon high dam hubs with complex geological conditions and improve the low energy dissipation efficiency of existing air jet collision methods. Combining numerical simulations, experimental research, and theoretical analysis, the energy dissipation characteristics of this dissipator are investigated, aiming to provide a new approach for the design of energy dissipators in canyon high dam hubs with complex geological conditions.

2. Structural Form and Energy Dissipation Principle of the Horizontal Opposing Jet Energy Dissipator

2.1. Structural Form

The horizontal opposing jet energy dissipator is a type of downstream energy dissipator (Figure 1). By separating the energy dissipator from the discharge structure, water from the reservoir is diverted to the open downstream area via pipelines for horizontal opposing collision energy dissipation in the air. This design eliminates restrictions from terrain and geological conditions, enabling a flexible arrangement. A horizontal opposing collision module is installed at the outlet of the downstream jet pipeline (Figure 2). The main variables in this study are l/d (collision distance/pipeline diameter) and Re (Reynolds number).

2.2. Flow Characteristics and Energy Dissipation Path Analysis

The movement of two horizontally opposing colliding jets (see Figure 3) can be divided into four stages:

• Horizontal jet stage: Two jets are emitted horizontally along the same axis from their respective pipeline outlets with the same flow rate and opposite directions, moving as horizontal direct jets with an initial velocity v. During this stage, aeration occurs due to friction between the jet surface and air.
• Collision and mixing stage: The two jets move to the midpoint between the two outlets and collide horizontally, and the jet droplets slide, squeeze, and mix with each other. When passing through the vertical plane at the midpoint of the outlets, the flux of the two jets is equal in magnitude and opposite in direction, which can be regarded as the overall zero-flux plane (see Figure 4).
• Crown-shaped water jet formation stage: The mixed flow rubs and squeezes symmetrically on both sides of the overall zero-flux plane, dissipating jet energy. Meanwhile, the remaining kinetic energy in the horizontal direction is converted into radial energy in the vertical plane, causing the flow to expand radially and form a crown-shaped water jet. After the two horizontal jets collide head-on to form this crown-shaped water jet, its radial expansion and fragmentation process is accompanied by the generation and dissipation of numerous vortices, which constitutes one of the core mechanisms of energy dissipation. Chang et al. [29] analyzed using the Ω-vortex method and normalized helicity H_n and found that the stability of vortex structures is directly related to energy loss—reductions in flow separation vortices and wake vortices significantly decrease energy dissipation.
• Falling and secondary energy dissipation stage: The upper part of the crown-shaped water jet and the flow reaching the radial boundary fall freely into the plunge pool under gravity and air resistance. The lower part of the crown-shaped water jet moves downward in the vertical plane under radial compression, expansion, and gravity until it enters the plunge pool. The energy of all vertical flow movements is further dissipated through mixing after falling into the plunge pool.

3. Physical Experiments and Numerical Simulation Validation

This study focused on the energy dissipation characteristics of two jets colliding horizontally in the air, forming a crown-shaped water jet, and falling into the plunge pool. Due to the difficulty in measuring the time-averaged velocity of the crown-shaped water jet before it falls into the plunge pool through physical experiments alone, a combination of physical experiments, numerical simulations, and theoretical analysis was adopted.

3.1. Physical Experiment System

To ensure the accuracy of the simulation results of the horizontal opposing energy dissipator, physical model experiments were conducted for validation. The experimental system consisted of a water inlet tank, a flow stabilizer, a centrifugal pump (Shanghai Kaiquan Pump (Group) Co., Ltd., Shanghai, China), pipelines, a needle valve, an electromagnetic flowmeter (Taiyuan Jinyi Thermal Instrument Manufacturing Co., Ltd., Taiyuan, China) a surge chamber, a piezometer tube, a water outlet tank, and an open channel (see Figure 5). Water was driven by a centrifugal pump, flowed through pipelines, a needle valve, an electromagnetic flowmeter, a surge chamber, and experimental pipelines, and finally collided horizontally above the water outlet tank. The crown-shaped water jet formed after collision fell into the plunge pool and flowed back to the water inlet tank through the open channel, forming a closed circulation loop.

The water inlet tank, water outlet tank, and open channel were made of organic glass. The dimensions of the water inlet tank were 80 cm × 80 cm × 80 cm, the water outlet tank was 100 cm × 100 cm × 120 cm, and the open channel was 200 cm × 6 cm × 8 cm. High-precision seamless steel pipes with an inner diameter of 1.9 cm and an outer diameter of 2.5 cm were used as experimental pipelines. A variable-frequency centrifugal pump with a rated flow rate of 8 m³/h was used to provide power. An electromagnetic flowmeter with a range of 0.5–2 m³/h and an accuracy of 0.0001 m³/h was used for flow measurement. To ensure stable flow and reduce air content during the experiment, a flow stabilizer was installed in the water inlet tank, and a surge chamber was installed on the experimental pipeline.

This study analyzed the statistically averaged hydraulic characteristics of the flow field. As shown in Figure 6, a coordinate system was established with the midpoint of the line connecting the horizontal jet outlets as the origin, X as the horizontal coordinate, Y as the transverse coordinate, and Z as the vertical coordinate. Figure 7 shows the instantaneous shape of the crown-shaped water jet formed by the collision of two jets when Re = 29,591 and l/d = 3.6842. Through physical model experiments, the spatial characteristics of the crown-shaped water jet, including its rising height h in the Z-direction and expanding width w in the Y-direction, were observed to verify the accuracy of the numerical simulation results.

3.2. Numerical Simulation

To clarify the spatial characteristics of the crown-shaped water jet, numerical simulations were conducted to study the rising height and expanding width of the crown-shaped water jet in the Z-directions and Y-directions, respectively. The effectiveness and accuracy of the RNG k-ε model in simulating the height and width of the crown-shaped water jet were verified through physical experimental results.

A gas–liquid two-phase flow model was adopted in this study: First, a geometric model was constructed using SpaceClaim 2021R1 software, and then calculations were performed in Fluent 2021R1 software using the RNG k-ε (Renormalization Group k-ε) turbulence model and VOF (Volume of Fluid) model based on the PISO (Pressure Implicit with Splitting of Operators) algorithm. The horizontal opposing jets in this study featured a strong vortex field with anisotropic turbulence, high strain rate, and large streamline curvature, so the RNG k-ε turbulence model was selected for calculations [30]. The governing equations are as follows:

Continuity equation:

$${\displaystyle \frac{\partial \rho }{\partial t}}+{\displaystyle \frac{\partial (\rho u_i)}{\partial x_i}}=0$$

(1)

Momentum equation:

$${\displaystyle \frac{\partial (\rho u_i)}{\partial t}}+{\displaystyle \frac{\partial (\rho u_i{u}_j)}{\partial x_{\mathrm{j}}}}=-{\displaystyle \frac{\partial \rho }{\partial x_i}}+{\displaystyle \frac{\partial }{\partial x_{\mathrm{j}}}}[(\mu +{\mu }_t)({\displaystyle \frac{\partial u_i}{\partial x_{\mathrm{j}}}}+{\displaystyle \frac{\partial u_j}{\partial u_i}})]$$

(2)

k equation:

$${\displaystyle \frac{\partial (\rho k)}{\partial t}}+{\displaystyle \frac{\partial (\rho k{u}_i)}{\partial x_i}}={\displaystyle \frac{\partial }{\partial x_{\mathrm{j}}}}[{\sigma }_k(\mu +{\mu }_t){\displaystyle \frac{\partial k}{\partial x_{\mathrm{j}}}}]+G_k-\rho \epsilon$$

(3)

ε equation:

$${\displaystyle \frac{\partial (\rho \epsilon )}{\partial t}}+{\displaystyle \frac{\partial (\rho \epsilon u_i)}{\partial x_i}}={\displaystyle \frac{\partial }{\partial x_{\mathrm{j}}}}[{\sigma }_{\epsilon }(\mu +{\mu }_t){\displaystyle \frac{\partial \epsilon }{\partial x_{\mathrm{j}}}}]+C_{1\epsilon }{\displaystyle \frac{\epsilon }{k}}{G}_k-C_{2\epsilon }\rho {\displaystyle \frac{{\epsilon }^2}{k}}$$

(4)

In the equations, ρ is the density, kg/m³; t is the time, s; u_i and u_j are velocity components, m/s; μ is the molecular dynamic viscosity, Pa·s; μ_t = ρv_t, μ_t is the turbulent dynamic viscosity, Pa·s; v_t is the turbulent kinematic viscosity, m²/s, which can be derived from the turbulent kinetic energy k and turbulent dissipation rate ε; σ_k and σ_ε are the Prandtl numbers for k and ε, respectively, both taken as 0.7179; C_u is an empirical coefficient reflecting the correlation between turbulent viscosity and turbulent kinetic energy, taken as 0.085; C_1ε and C_2ε are empirical constants for the dissipation rate equation, taken as 1.42 and 1.68, respectively [31].

Meshing was performed using Fluent Meshing integrated in ANSYS 2021, with polyhedral structured meshes adopted. To ensure that the numerical simulation results accurately reflect the actual situation, a mesh independence test was conducted. Taking the working conditions of Re = 25,892 and l/d = 5.7895 as an example, polyhedral structured meshes with mesh sizes (m) of 0.084d, 0.168d, 0.337d, 0.674d, and 1.347d were used to analyze the expansion width w (in the Y-direction) of the crown-shaped water jet formed by the aerial collision of the two jets. As shown in Figure 8, with the continuous reduction in m, w decreased gradually, and the rate of change also diminished. When m was further reduced to 0.084d, w remained essentially unchanged with any further decrease in m. Additionally, the relative error between the numerical simulation results obtained with mesh sizes of 0.084d and 0.168d was only 0.96%. It can thus be concluded that when the mesh size was less than 0.168d, the mesh size was no longer the main factor influencing the numerical simulation results. Considering both the simulation time and the accuracy of the simulation results comprehensively, the mesh size used in this simulation was finally determined to be 0.168d.

3.3. Validation of the Numerical Simulation Results

In this study, statistical indicators, including the correlation coefficient (R²) and root mean square error (RMSE), were used to verify the accuracy of the numerical simulation, and slopes K and K′ were used to verify the reliability. The calculation methods are as follows:

$$R^2={\displaystyle \frac{{[{\displaystyle {\sum }_{i=1}^n(P_i-\overline{P})(O_i-\overline{O})}]}^2}{{\displaystyle {\sum }_{i=1}^n{(P_i-\overline{P})}^2{\displaystyle {\sum }_{i=1}^n{(O_i-\overline{O})}^2}}}}$$

(5)

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle {\sum }_{i=1}^n{(P_i-\overline{O})}^2}}$$

(6)

$$K={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n{O}_i\times P_i}}{{\displaystyle {\sum }_{i=1}^n{P}_i^2}}}$$

(7)

$$K^{\prime }={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n{O}_i\times P_i}}{{\displaystyle {\sum }_{i=1}^n{O}_i^2}}}$$

(8)

where n is the number of data points; P_i is the simulated data; $\overline{P}$ is the average value of the simulated data; O_i is the experimental data; $\overline{O}$ is the average value of the experimental data. Simulation results are considered highly reliable when R² > 0.6 and 0.85 < K, K′ < 1.15 [32].

The rising height h in the Z-direction and expanding width w in the Y-direction of the crown-shaped water jet of the horizontal opposing jet energy dissipator under different l/d working conditions were simulated and verified with experimental values when Re = 18,495, 22,194, 25,892, 29,591, and 33,290 (see Figure 7 for h and w).

3.3.1. Validation of Rising Height in the Z-Direction

Figure 9 shows a comparison between the numerical simulation results and experimental values of h/d (rising height normalized by pipeline diameter) under five different Re values. It can be seen that h/d decreased with increasing l/d for all Re values, and the numerical simulation results are in good agreement with the experimental values. The relative errors and statistical indicators of h/d are shown in

Table 1. Relative errors and statistical indicators of h/d.

Re	R2	RMSE	K	K′
18,495	0.9973	0.0666	1.0255	0.9747
22,194	0.9590	0.0623	0.9970	1.0027
25,892	0.9341	0.0666	1.0002	0.9995
29,591	0.8658	0.0942	1.0111	0.9889
33,290	0.9210	0.1200	1.0003	0.9994

. The minimum R² of the numerical simulation results was 0.8658, and the maximum was 0.9973 (approaching 1). The minimum RMSE was 0.0623, and the maximum was 0.1200 (relatively small), indicating that the numerical simulation results are consistent with the physical model experimental values and the simulation accuracy is high. Additionally, R² > 0.6, the minimum K was 0.9970, the maximum was 1.0255, the minimum K′ was 0.9747, and the maximum was 1.0027. Both K and K′ were within the range of 0.85–1.15, indicating high reliability of the numerical simulation. Therefore, it is feasible to simulate the rising height of the crown-shaped water jet in the Z-direction of the horizontal opposing jet energy dissipator using Fluent software.

3.3.2. Validation of Expanding Width in the Y-Direction

Figure 10 shows a comparison between the numerical simulation results and experimental values of w/d (expanding width normalized by pipeline diameter) under five different Re values. It can be seen that w/d decreased with increasing l/d for all Re values, and the numerical simulation results are in good agreement with the experimental values. The relative errors and statistical indicators of w/d are shown in

Table 2. Relative errors and statistical indicators of w/d.

Re	R2	RMSE	K	K′
18,495	0.9741	0.2949	1.0068	0.9928
22,194	0.8903	0.2725	0.9935	1.0063
25,892	0.9051	0.1793	1.0012	0.9987
29,591	0.9785	0.0942	0.9999	1.0000
33,290	0.8709	0.1331	1.0027	0.9973

. The minimum R² of the numerical simulation results was 0.8709, and the maximum was 0.9785 (approaching 1). The minimum RMSE was 0.0942, and the maximum was 0.2949 (relatively small), indicating that the numerical simulation results are consistent with the physical model experimental values and the simulation accuracy is high. Additionally, R² > 0.6, the minimum K was 0.9935, the maximum was 1.0068, the minimum K′ was 0.9928, and the maximum was 1.0063. Both K and K′ were within the range of 0.85–1.15, indicating high reliability of the numerical simulation. Therefore, it is feasible to simulate the expanding width of the crown-shaped water jet in the Y-direction of the horizontal opposing jet energy dissipator using Fluent software.

This section mainly introduced the physical experiments and numerical simulations of the horizontal opposing jet energy dissipator. It verified the simulated values of the rising height (in the Z-direction) and expansion width (in the Y-direction) of the crown-shaped water jet under different l/d working conditions at five different Reynolds numbers (Re) through experiments. The accuracy and reliability of the simulated values versus the experimental values were compared and analyzed using the correlation coefficient (R²), root mean square error (RMSE), and slopes (K and K′). The results showed that the numerical simulation values were in good agreement with the experimental values, indicating that it is feasible to simulate the hydraulic characteristics of the horizontal opposing jet energy dissipator using the VOF method and RNG k-ε model in Fluent software.

4. Study on the Energy Dissipation Characteristics of the Horizontal Opposing Jet Energy Dissipator

The energy dissipation rate is one of the important indicators for evaluating the energy dissipation effect of horizontal opposing jet energy dissipators. In hydraulic engineering, the design and operation of discharge structures need to consider how to minimize the scouring damage of water flow to hydraulic structures and their downstream channels. Reducing the energy of water flow by improving the energy dissipation rate can effectively reduce the impact force and protect the safety of hydraulic structures and channels. Therefore, in practical applications, comparing and analyzing the energy dissipation rates under various working conditions is of great significance for evaluating the energy dissipation effect and assisting in decision-making and optimal design.

4.1. Analysis of Influencing Factors on the Energy Dissipation Rate

The energy dissipation rate η of the horizontal opposing jet energy dissipator can be expressed as a function of the following dimensionless parameters:

$$\eta =f(l/d,\mathrm{Re},N,\theta )$$

(9)

where l/d is the ratio of collision distance to pipeline diameter; Re is the Reynolds number; N is the number of jets; θ is the plane angle of the two opposing jets, i.e., the projection angle of the two jets on the horizontal plane.

In this experiment, d = 19 mm, N = 2, θ = 180 degrees; the maximum average flow velocity before opposing collision was 1.76 m/s. The volume compression of water was negligible under conventional operating conditions, so the influence of water density change was ignored; the viscosity of water was not the main influencing factor and was temporarily not considered; the outlet type and conditions were unchanged and temporarily not considered. Thus, the above formula can be simplified as:

$$\eta =f(l/d,\mathrm{Re})$$

(10)

That is, under the experimental conditions, the energy dissipation rate of the horizontal opposing jet energy dissipator is only related to l/d and Re. It is worth noting that Chang et al. [33] demonstrated that wall roughness is also a critical factor influencing fluid energy dissipation—with an increase in equivalent sand-grain roughness, vortex structures in the flow field tend to aggregate more easily, the entropy production rate rises significantly, and ultimately leads to a decline in equipment efficiency. In this study, high-precision seamless steel pipes (with smooth walls) were used for the test pipelines, and the influence of roughness was not considered temporarily. However, this factor can be regarded as a direction for future research; by optimizing the smoothness of the inner wall of the pipeline or the shape of the jet outlet, the energy dissipation efficiency can be further improved.

4.2. Construction of a Theoretical Calculation Formula for the Horizontal Opposing Collision Energy Dissipation Rate

After colliding, the two jets fall into the downstream plunge pool. Analyzing the horizontal opposing collision point, it can be simplified to the confluence of two jets. As shown in Figure 11, for two converging jets with flow rates Q₁ and Q₂, according to the law of conservation of energy, the total energy input by the liquid per unit time from Sections 1-1 and 2-2 should be equal to the total energy output from Section 3-3 plus the energy loss of the two water flows. Sections 1-1 and 2-2 are 53 cm away from their respective pipeline jet outlets (greater than 20d), and Section 3-3 is the section before the crown-shaped water jet fall into the plunge pool, 35 cm away from the pipeline centerline.

$$\rho g{Q}_1(Z_1+{\displaystyle \frac{P_1}{\rho g}}+{\displaystyle \frac{{\alpha }_1{v}_1^2}{2g}})+\rho g{Q}_2(Z_2+{\displaystyle \frac{P_2}{\rho g}}+{\displaystyle \frac{{\alpha }_2{v}_2^2}{2g}})=\rho g{Q}_3(Z_3+{\displaystyle \frac{P_3}{\rho g}}+{\displaystyle \frac{{\alpha }_3{v}_3^2}{2g}})+\rho g{Q}_1{h}_{w1-3}+\rho g{Q}_2{h}_{w2-3}$$

(11)

where Q₁ + Q₂ = Q₃. Substituting into Equation (11) and rearranging provides

$$Q_1\left[(Z_1+{\displaystyle \frac{P_1}{\rho g}}+{\displaystyle \frac{{\alpha }_1{v}_1^2}{2g}})-(Z_3+{\displaystyle \frac{P_3}{\rho g}}+{\displaystyle \frac{{\alpha }_3{v}_3^2}{2g}})-h_{w1-3}\right]+Q_2\left[(Z_2+{\displaystyle \frac{P_2}{\rho g}}+{\displaystyle \frac{{\alpha }_2{v}_2^2}{2g}})-(Z_3+{\displaystyle \frac{P_3}{\rho g}}+{\displaystyle \frac{{\alpha }_3{v}_3^2}{2g}})-h_{w2-3}\right]=0$$

(12)

For Equation (12), i.e., Q₁ · A + Q₂ · B = 0 (where A and B denote the bracket terms corresponding to Q₁ and Q₂, respectively), two critical premises must be satisfied

• The flow rates Q₁ and Q₂ in the split branches are positive Q₁ > 0, Q₂ > 0, as negative flow rates are physically implausible in practical hydraulic scenarios;
• Q₁ and Q₂ are independent variables with no linear dependence (their magnitudes are determined by the boundary conditions and pipeline parameters of each individual branch, rather than being constrained by a linear relationship).

Under the conditions that Q₁ > 0, Q₂ > 0, and Q₁ is independent of Q₂, the linear combination of these two positive terms equals zero if and only if their respective coefficients (i.e., A and B) are zero individually. Specifically, if A ≠ 0 or B ≠ 0, the product of a positive flow rate and a non-zero term would yield either a positive or negative value. In such cases, the sum of these two terms cannot consistently equal zero, which violates the mathematical constraint of non-negative linear combination for independent positive variables.

$$(Z_1+{\displaystyle \frac{P_1}{\rho g}}+{\displaystyle \frac{{\alpha }_1{v}_1^2}{2g}})-(Z_3+{\displaystyle \frac{P_3}{\rho g}}+{\displaystyle \frac{{\alpha }_3{v}_3^2}{2g}})-h_{w1-3}=0$$

(13)

$$(Z_2+{\displaystyle \frac{P_2}{\rho g}}+{\displaystyle \frac{{\alpha }_2{v}_2^2}{2g}})-(Z_3+{\displaystyle \frac{P_3}{\rho g}}+{\displaystyle \frac{{\alpha }_3{v}_3^2}{2g}})-h_{w2-3}=0$$

(14)

Thus, it can be obtained that

$$Z_1+{\displaystyle \frac{P_1}{\rho g}}+{\displaystyle \frac{{\alpha }_1{v}_1^2}{2g}}=Z_3+{\displaystyle \frac{P_3}{\rho g}}+{\displaystyle \frac{{\alpha }_3{v}_3^2}{2g}}+h_{w1-3}$$

(15)

$$Z_2+{\displaystyle \frac{P_2}{\rho g}}+{\displaystyle \frac{{\alpha }_2{v}_2^2}{2g}}=Z_3+{\displaystyle \frac{P_3}{\rho g}}+{\displaystyle \frac{{\alpha }_3{v}_3^2}{2g}}+h_{w2-3}$$

(16)

$$E_1=Z_1+{\displaystyle \frac{P_1}{\rho g}}+{\displaystyle \frac{{\alpha }_1{v}_1^2}{2g}}$$

(17)

$$E_2=Z_2+{\displaystyle \frac{P_2}{\rho g}}+{\displaystyle \frac{{\alpha }_2{v}_2^2}{2g}}$$

(18)

$$E_3=Z_3+{\displaystyle \frac{P_3}{\rho g}}+{\displaystyle \frac{{\alpha }_3{v}_3^2}{2g}}$$

(19)

Then the total energy loss before the water flow falls into the plunge pool is

$$h_w=h_{1-3}+h_{2-3}=E_1+E_2-2E_3$$

(20)

The energy dissipation rate is

$$\eta ={\displaystyle \frac{E_1+E_2-2E_3}{E_1+E_2}}\times 100\%$$

(21)

This energy dissipation rate only represents the percentage of energy consumed after the two jets form a water jet through horizontal opposing collision, excluding the energy dissipation rate when the two jets fall into the plunge pool after collision.

4.3. Energy Dissipation Rate of Horizontal Opposing Collision

Due to the uneven and inconsistent flow velocity at various points on the same horizontal plane of the crown-shaped water jet formed by horizontal opposing collision, numerical simulation was used to obtain the time-averaged velocity on the plane 35 cm away from the pipeline centerline before the crown-shaped water jet falls into the plunge pool (see

Table 3. Summary of the time-averaged velocities at Section 3-3.

Re	l/d = 3.6842	l/d = 4.7368	l/d = 5.7895	l/d = 6.8421	l/d = 7.8947
18,495	2.6588	2.6523	2.6707	2.6678	2.6707
22,194	2.6581	2.6633	2.6638	2.6964	2.6917
25,892	2.6808	2.6792	2.7048	2.7200	2.7159
29,591	2.7117	2.6831	2.7238	2.7191	2.7554
33,290	2.6925	2.7030	2.7516	2.7620	2.7908

), and then E₃ was calculated.

Substituting the above data into Equation (21), the opposing collision energy dissipation rate η was obtained when Re = 18,495, 22,194, 25,892, 29,591, and 33,290 and l/d = 3.6842, 4.7368, 5.7895, 6.8421, and 7.8947 (see

Table 4. Summary of the opposing collision energy dissipation rate η.

Re	l/d = 3.6842	l/d = 4.7368	l/d = 5.7895	l/d = 6.8421	l/d = 7.8947
18,495	0.1721	0.1756	0.1635	0.1648	0.1625
22,194	0.2347	0.2310	0.2301	0.2105	0.2125
25,892	0.2841	0.2842	0.2697	0.2607	0.2622
29,591	0.3290	0.3423	0.3214	0.3229	0.3038
33,290	0.3954	0.3898	0.3668	0.3611	0.3468

, Figure 12 and Figure 13).

As can be seen in Table 3, Figure 12 and Figure 13, under the studied conditions, η had a linearly decreasing relationship with l/d; that is, when Re remained constant, η decreased gradually as l/d increased. In contrast, η had a linearly increasing relationship with Re; that is, when l/d remained constant, η increased gradually as Re increased. With a further increase in Re, η would have increased further.

5. Discussion

5.1. Analysis of the Energy Dissipation Mechanism of Horizontal Opposing Jets

For traditional jet energy dissipation (free jet), when the fluid is injected into still air, the velocity difference between the jet and the surrounding air forms a shear layer, generating and breaking vortices, and converting large-scale kinetic energy into small-scale turbulent kinetic energy for dissipation [34]. This energy dissipation method mainly eliminates energy in the plunge pool, which is prone to forming deep scouring pits, potentially endangering the safe operation of the dam [35].

For vertical air collision energy dissipation of upper and lower jets, kinetic energy dissipation is achieved through three mechanisms: vertical momentum opposing collision, turbulent mixing, and water jet fragmentation. Three-dimensional shear turbulence is generated due to the velocity difference, and vortex splitting and fragmentation dissipate large-scale kinetic energy; collision impact causes bidirectional diffusion of the water jet and a significant increase in surface area, breaking it into fine water jets that dissipate energy through air friction and entraining air to form a gas–liquid two-phase flow, reducing momentum transfer efficiency for further energy dissipation [18,36].

For horizontal air collision energy dissipation of left and right jets, kinetic energy dissipation is achieved through three mechanisms: horizontal momentum opposing collision, turbulent mixing, and water jet fragmentation. High-speed jets collide to form a combined water jet, and intense shear turbulence is generated due to the velocity difference and impact. While vortex splitting and fragmentation occur, the surface area of the combined water jet increases, entraining a large amount of air to form a gas–liquid two-phase flow, which not only reduces momentum transfer efficiency but also further dissipates energy through bubble bursting [22].

The horizontal opposing jet energy dissipator causes two jets to collide, squeeze, deform, mix, rub, diffuse, and aerate in the air with the same flow rate and opposite directions along the same horizontal axis, ultimately forming a crown-shaped water jet. The horizontal opposing collision of the two jets increases the mixing length and collision time of the jets in the air, resulting in a more intense horizontal momentum opposing collision, longer turbulent mixing length, and larger spatial size of the formed crown-shaped water jet. This also increases the contact area between the crown-shaped water jet and the air, accelerating the fragmentation of the crown-shaped water jet into water bundles, sheets, and droplets, thereby accelerating energy dissipation. Compared with existing jet energy dissipation methods, the horizontal opposing jet energy dissipator has a richer energy dissipation mechanism and more effective energy dissipation.

5.2. Variation Characteristics of the Energy Dissipation Rate of Horizontal Opposing Jets

Liu et al. [36] calculated the energy dissipation rate of vertical collision of upper and lower jets at the Xiaowan Hydropower Station. The Re values of the two jets were 1.05 × 10⁸ and 1.92 × 10⁸, and the collision energy dissipation rate ranged from 19.02% to 27.17% under different working conditions. Diao and Yang [37] calculated the energy dissipation rate of air collision of surface and middle hole water jets. The Re values of the two jets were both 3.00 × 10⁵, and the collision energy dissipation rate was 19.8% when using the total kinetic energy of the water jet before collision as the total energy, and 5.0% when using the upstream and downstream water level difference as the total energy. Tian and Liao [38] calculated the air collision energy dissipation rate of upper and lower water jets under the arrangement of large differential elevation high and low sills in the surface holes of the Xiakou Arch Dam. The Re values of the side surface hole and middle surface hole jets were both 1.76 × 10⁸, and the collision energy dissipation rate ranged from 2.7% to 9.0% under different collision angle working conditions.

The introduction of the horizontal opposing jet energy dissipator extended the collision time and mixing length of the two jets in the air, resulting in a larger range of the formed crown-shaped water jet, which is more suitable for high-velocity jet energy dissipation. When the Reynolds number was only 1.85 × 10⁴–3.33 × 10⁴, the energy dissipation rate of the air collision part of the two jets reached 16.25–39.54%. The energy dissipation rate is 3.55–12.37% higher than that of the energy dissipation by the vertical collision of upper and lower water jets. Based on the relationship between the Reynolds number and energy dissipation rate revealed in this study, it can be predicted that when the Reynolds number is larger, the energy dissipation rate of horizontal opposing jet collision will be higher. This energy dissipator not only improves the energy dissipation rate of the air collision of the two jets but also increases the radial size of the water jet in the horizontal plane after collision, further reducing the density per unit area of the water jet, which is conducive to reducing the impact force of the water jet on the plunge pool when falling and further improving the overall energy dissipation rate.

5.3. Differences in Turbulence-Aeration Characteristics Between Prototype and Model and Their Impacts on Energy Dissipation Efficiency

With the prototype Reynolds number (Re ≈ 10⁸) and the model Reynolds number (Re ≈ 10⁴), based on the scaling law ε∝v³/L (where ε is the turbulent dissipation rate and L is the characteristic length), the prototype exhibits more pronounced turbulence anisotropy, higher intensity of vortex generation and breakdown, and more small-scale vortices, leading to improved energy dissipation efficiency. Aeration in the model is limited by low Reynolds number (Re) and surface tension, leading to differences from that in the prototype. The prototype features a higher jet velocity, which causes a sharp increase in the turbulence intensity of the shear layer, breaking bubbles into micron scale. Consequently, the aeration concentration of the prototype will be higher than that of the model, reducing the momentum transfer efficiency of the gas–liquid two-phase flow and further promoting energy dissipation [10,11].

6. Conclusions

To address the limitations of traditional energy dissipation technologies—such as difficulty in arranging energy dissipators due to narrow river valleys and complex geological conditions—and the low energy dissipation efficiency of existing air jet collision methods, this study proposed a novel structural form of a horizontal opposing jet energy dissipator. Water is diverted to the open area downstream of the reservoir hub via diversion pipelines for horizontal opposing collision energy dissipation in the air. A combination of numerical simulations and physical experiments was used to study this energy dissipation form, and the following conclusions were drawn:

• The main influencing factors of the rising height in the Z-direction and expanding width in the Y-direction of the crown-shaped water jet were l/d and Re. The rising height in the Z-direction decreased with increasing l/d and increased with increasing Re; the expanding width in the Y-direction decreased with increasing l/d and increased with increasing Re.
• The main influencing factors of the energy dissipation rate η of the horizontal opposing jet energy dissipator were l/d and Re. Under the study conditions, η exhibited a linear decreasing relationship with l/d and a linear increasing relationship with Re. According to the experimental data, with further increases in Re, the energy dissipation rate η will continue to increase. Therefore, this energy dissipator will have a higher energy dissipation rate in high-dam energy dissipation with high water heads and large flow rates.
• Owing to the limitations of experimental equipment and computational power, this study only investigates the energy dissipation characteristics of the horizontal opposing jet energy dissipator with two jets under normal flow velocity conditions. The energy dissipation efficiency under high flow velocity, multiple jets, and opposing jets with different angles will be addressed in the next phase of research.