Energy Dissipation Rate and Conjugate Depth After Hydraulic Jump for Counterflow Underflow Energy Dissipation in Spillways

Abstract

To address the energy dissipation requirements of hydraulic engineering projects with medium-low water heads and medium-high unit discharges, counterflow-type underflow energy dissipation can significantly enhance the energy dissipation efficiency through the head-on collision of flows from spillways on both sides. In this study, the spillway of the Lieshen Reservoir was used as the prototype. Since gravity dominates the flow in spillways, we established a 1:15 physical model based on the Froude similarity criterion, and conducted numerical simulations using the volume of fluid method coupled with the realizable k-ε turbulence model. Furthermore, the hydraulic characteristics of counterflow energy dissipation under different flow rates and stilling basin length conditions were analyzed. The results show that the counterflow energy dissipation rate first increases before decreasing with increasing stilling basin length, and the maximum energy dissipation rate can exceed 85%; however, the change in the stilling basin depth has a small impact on the energy dissipation rate, especially under relatively high flow rates; furthermore, an empirical formula for the conjugate depth after a hydraulic jump suitable for counterflow energy dissipation with Froude number in the range of 2.0 < Fr₁ < 9.7 and stilling basin depth of 0.5–1.5 m is proposed, with the relative error between its predicted and simulated values being less than 6%. Based on the analysis of the water depth outer envelope curve at the outlet section of the stilling basin, it is suggested that the sidewall height be set to 0.6–0.8 times the conjugate depth after the hydraulic jump.

1. Introduction

As an indispensable and crucial component of water-conservation projects, discharge structures perform the function of safe water discharge. The discharged flow from these structures carries high kinetic energy, which needs to be converted into other forms of energy and dissipated in a timely manner, to avoid causing severe scouring damage to the downstream riverbed. It may even pose significant threats to the riverbed banks and adjacent structures. Flood discharge and energy dissipation have become extremely important technical challenges for the design of engineering projects such as hydropower. As a vital guarantee for the safe operation of water conservancy hubs during flood seasons, spillways ensure the safety of dams and water conservancy hubs by discharging flood waters that exceed the storage and regulation capacities of reservoirs [1,2]. The spillway structure comprises several components, including the approach channel, dam crest, chute, and energy dissipator at the downstream end [3]. The selection of energy dissipation methods for spillways should be based on factors such as the project water head, flow rate, geological conditions, and downstream river channel conditions. The main energy dissipation methods include underflow, surface flow, ski-jump, stepped, diffusion, and drop energy. In practical engineering, a single energy dissipation method may fail to satisfy complex conditions, and composite methods that combine multiple energy dissipators are often adopted [4]. Underflow energy dissipation is typically adopted for hydraulic engineering projects featuring a medium-low water head (head height < 30–50 m), with a medium-large discharge rate of 10–50 m³/s), shallow downstream water depth, weak riverbed anti-scouring capacity, and the requirement of strictly controlling the downstream scouring range. This includes small and medium-sized sluices, low-head hydropower stations, and irrigation hubs. Underflow energy dissipation has the advantages of a stable energy dissipation effect and minimal downstream disturbance; however, it requires a considerable engineering effort (involving the construction of stilling basins, aprons, and other structures) [5]. For the energy dissipation of the Lieshen Reservoir in this study, innovations were made based on the traditional underflow energy dissipation method, and counterflow energy dissipation on both sides was adopted.

Studies on spillway energy dissipation have always centered around the objectives of “safety, efficiency, economy, and ecology.” Improving the energy dissipation efficiency and reducing engineering costs are core objectives, which require optimizing the structure of energy dissipators or innovating energy dissipation methods to enhance the kinetic energy dissipation efficiency from the traditional 60–70% to over 80% [6]. To ensure the energy dissipation effect, engineering quantities and costs should be reduced by optimizing the structural dimensions such as reducing the stilling basin length and the hydraulic jump height, simplifying the flip-bucket form [7]. To achieve these objectives, a closed-loop research methodology integrating “mechanism-model-verification”, which encompasses theoretical analysis, experimental observation, and numerical simulation, has been developed.

The hydraulic jump energy dissipation theory is the core theory of underflow energy dissipation. A hydraulic jump is an intense turbulent phenomenon generated during the transition from supercritical to subcritical flow in an open channel. It dissipates energy through the interaction between the main and bottom recirculation flows, water body fragmentation, and turbulent dissipation. Since the Italian physicist Bidone (1825) [8] first conducted experimental and analytical studies on the hydraulic jump phenomenon, many studies have been conducted on this phenomenon. In 1828, the French scholar Jean-Baptiste Belanger derived the well-known hydraulic jump equation based on experiments and the constant flow momentum equation [9]. The empirical formula of Bazin is widely used in engineering applications:

$$h_2=0.5h_1\left(\sqrt{1+8F{r}_1^2}-1\right)$$

(1)

where h₁ and h₂ are the conjugate depths before and after the hydraulic jump, respectively; Fr₁ is the Froude number before the hydraulic jump.

Widely used classic formulas for hydraulic jump length include the Elevatorski equation $L_j=6.9\left(h_2-h_1\right)$, which is applicable to $F{r}_1=4.5~10$; the Chen Chunting equation $L_j=9.4\left(h_2-h_1\right)F{r}_1^{-0.13}$, which is applicable to $F{r}_1=2.5~14$; and the Chow equation $L_j=10.3h_1{\left(F{r}_1-1\right)}_{0.81}$, which is applicable to $F{r}_1=2~12$ [10,11,12].

L_j refers to the hydraulic jump length, which is a critical parameter for evaluating the energy dissipation effect and structural design of the stilling basin.

As a key hydraulic phenomenon marking the transition from supercritical to subcritical flow, the study of water jump has spanned nearly two centuries, laying the theoretical foundation for flood discharge and energy dissipation design. Early research focused on establishing and modifying the classical water jump equation. For example, Liu combined projectile principles with experimental data and proposeda semi-theoreticaland a semi-empirical formula for the water jump length in rectangular horizontal open channels [13]. With the advancement of computational fluid dynamics (CFD), numerical simulation has become the core method for studying multidimensional flow in spillways, the dynamics of water jumps, and the performance evolution of energy dissipation structures [14,15,16,17]. Scholars have employed numerical methods to conduct in-depth investigations into various complex flows, such as the formation of water jumps and the dissipation of turbulent kinetic energy within dissipation basins under conditions of multiple horizontal submerged jets [18,19], and the significant influence of step roughness on energy dissipation and water jump characteristics in stepped spillways [20]. Recent studies have further refined the approach. For example, numerical simulations of water jumps under high-flow conditions were validated using multi-scale physical models and prototype data [21,22,23]. Fu et al. [24] focused on the influence of particle characteristics on the velocity of the reverse flow of the hydraulic jump and proposed an empirical formula.

Current industry research and practice focus on optimizing energy dissipation efficiency and flow stability through structural innovation. Regarding stepped spillways, pool-type stepped structures have been proven to achieve higher energy dissipation rates and superior flow stability compared to traditional platform-type stepped structures by enhancing turbulence and improving flow field distribution [25,26,27]. Simultaneously, downstream auxiliary energy dissipation measures, such as optimizing the angle of porous baffles, have also been proven to effectively regulate the length of the hydraulic jump and enhance energy dissipation efficiency [28]. These studies generally indicate that the key parameters governing dissipation efficiency are often the Froude number and specific flow parameters, while structural dimension optimization must be matched accordingly [29].

Although research on water jumps and energy dissipation under single-channel conditions has been relatively thorough, with innovative structures continually emerging, studies on the hydraulic characteristics of the specific energy dissipation pattern—where bidirectional flows collide due to arranged spillways—remain relatively scarce. Most existing studies focus on single-sided inflow patterns, failing to fully clarify how the coupling mechanism between velocity and depth differences on both sides, as well as the geometric parameters of the energy dissipation basin (such as basin length and depth), affects the counterflow pattern, energy dissipation efficiency, and post-surge water depth. This gap hinders the theoretical optimization of spillway designs—structures capable of adapting to complex topography and potentially reducing excavation volumes. Therefore, this study aims to systematically investigate the hydraulic characteristics of energy dissipation in spillways through a combination of numerical simulation and experimental validation. This approach seeks to address existing research deficiencies and provide design guidance for relevant engineering practices.

Existing studies on spillway energy dissipation primarily focus on unidirectional hydraulic jumps, stepped spillways [30], or submerged jet dissipation [31]. Although impinging jets and counterflow configurations have been studied in closed conduits or plunge pools, their application to free-surface underflow energy dissipation in stilling basins, particularly with bidirectional hydraulic jumps, remains unexplored [32,33]. In addition, existing hydraulic jump theories cannot predict the conjugate depth under opposed-flow conditions.

2. Spillway Model Research Methodology

2.1. Project Overview

Figure 1 shows Lieshen Reservoir is the first completed rock-filled concrete gravity dam in Chongqing Municipality. It is located in Lieshen Village, Zhushan Town, Liangping District (within the Baili Bamboo Sea Scenic Area), 25.0 km away from Liangping District and 6.0 km from Zhushan Town. The reservoir is situated on the Qijian River, a tributary of the Longxi River in the Yangtze River Basin, and lies in a subtropical humid monsoon climate zone. The multi-year average precipitation is 1245.9 mm, the multi-year average temperature is 16.6 °C, the multi-year average maximum wind speed reaches 9.6 m/s, and the dominant wind direction is northeast (NE). The catchment area above the dam site is 5.05 km², with a total storage capacity of 590,000 m³.

Owing to the small tolerance coefficient of the reservoir, measures were taken to reduce the drawdown zone height, improve the normal water level and flood discharge capacity, facilitate water-releasing structure connections, and simplify operation and maintenance. The water discharge system innovatively arranges open-type WES ogee weirs on both sides of the dam, adopting a two-way counterflow underflow energy dissipation scheme (Figure 2). By symmetrically arranging the spillways, the discharged flows collide head-on, significantly enhancing the internal energy dissipation. This design increases the energy dissipation rate to more than 1.8× that of conventional designs and utilizes a V-shaped canyon terrain [34,35].

2.2. Experimental Model Design

To verify the reliability of the numerical simulation, we followed the Froude gravity similarity criterion strictly and constructed a 1:15 scale physical model (Figure 3).

The model accurately reproduced the spatial morphologies of the reservoir area, rockfill dam body, spillway, asymmetric flow channel, and simplified the discharge flow channel. Key hydraulic parameters such as the curve radius of the spillway side channel and the curvature of the reverse arc segment were subjected to fine calibration, and the model structure was manufactured using high-precision machining.

The experimental system comprised a water supply system, a model main body, and a measurement system. The water supply system combined a variable-frequency speed-regulating pump set and a stable-pressure water tank, ensuring that the inlet flow fluctuation < ±2%.

2.3. Numerical Calculation Method

The realizable k − ε model introduces flow field curvature and rotation correction terms, enhancing turbulent viscosity calculation rationality. It considers near-wall flow and rotating flow dynamic characteristics, maintaining numerical stability and reducing wall function dependence.

Continuity Equation:

$${\displaystyle \frac{\partial \left(\rho u_i\right)}{\partial t}}=0$$

(2)

Momentum Equation:

$${\displaystyle \frac{\partial \left(\rho u_i\right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho u_i{u}_j\right)}{\partial x_j}}=\rho f_i-{\displaystyle \frac{\partial p}{\partial x_i}}+{\displaystyle \frac{\partial }{\partial x_i}}\left[\left(\mu +{\mu }_t\right)\left({\displaystyle \frac{\partial u_i}{\partial x_j}}+{\displaystyle \frac{\partial u_j}{\partial x_i}}\right)\right]$$

(3)

ρ is the density of the liquid; u_i denote the velocity components in the x, y, and z coordinate directions, respectively; t represents time; p is the hydrostatic pressure; μ is the dynamic viscosity; and μ_t is the turbulent viscosity.

The governing equations are the turbulent kinetic energy (k) and dissipation rate (ε) Equations (4) and (5), respectively:

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

(4)

$${\displaystyle \frac{\partial (\rho \epsilon )}{\partial t}}+{\displaystyle \frac{\partial (\rho \epsilon u_i)}{\partial x_i}}={\displaystyle \frac{\partial }{\partial x_i}}\left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_{\epsilon }}}\right){\displaystyle \frac{\partial \epsilon }{\partial x_j}}\right]+\rho C_{1}S\epsilon -\rho C_2{\displaystyle \frac{{\epsilon }^2}{k+\sqrt{\mu \epsilon /\rho }}}$$

(5)

where G_k is the turbulent kinetic energy production term induced by mean velocity gradients; G_b is the turbulent kinetic energy production term induced by buoyancy; S is the magnitude of the mean strain rate tensor ($S=\sqrt{2S_{ij}{S}_{ij}}$, where $S_{ij}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial u_j}{\partial x_i}}+{\displaystyle \frac{\partial u_i}{\partial x_j}}\right)$ is the mean strain rate tensor); σ_k, σ_ε, and C₂ are model constants with values of σ_k = 1.0, σ_ε = 1.2, C₂ = 1.9; and C₁ is the dynamic parameter, C₁ = max [0.43, ${\displaystyle \frac{\eta }{\eta +5}}$], where $\eta =S{\displaystyle \frac{k}{\epsilon }}$.

Compared with the standard and RNG k − ε models, the realizable k-ε model introduces an improved turbulent viscosity formula and incorporates key computational terms related to flow curvature and rotational effects. Previous studies have shown that this model exhibits superior numerical convergence and computational accuracy when simulating complex flow problems, such as multiphase jet flows [36,37] and circular jet flows [38].

Numerical methods for gas–liquid two-phase flows include Eulerian and Lagrangian frameworks. The Eulerian-based VOF method is widely used to capture complex multiphase interfaces by solving the phase-fraction transport equation [39,40,41,42]. Unstructured grids are used to discretize the phase interfaces, and the pressure correction method addresses pressure–velocity decoupling in the Navier–Stokes equations.

In this study, we used ANSYS Fluent 2022 R1, a commercial CFD software widely validated in hydraulic engineering applications, to simulate the gas–liquid two-phase flow of the spillway. We did not develop any custom code for this study; instead, we utilized the software’s built-in VOF (Volume of Fluid) model for multiphase flow simulation, with the turbulence closure implemented via the realizable k − ε model (consistent with our earlier theoretical derivation).

The pressure implicit with the split of operators’ algorithm designed for unsteady flow fields is suitable for solving transient two-phase flow interface deformations. This study adopts the VOF method coupled with the realizable k − ε turbulence model, constructing a coupled solution framework for two-phase flow interface topological evolution and turbulent transport equations.

The computational domain model and mesh are shown in Figure 4, computational mesh for the Lieshen Reservoir spillway numerical model, with refined grid resolution applied to high-gradient flow regions (spillway crest, stilling basin). This mesh design ensures computational accuracy in critical hydraulic zones while maintaining efficiency in low-variability regions. Inlets 1 and 2 were set as pressure inlets with values determined by the flow conditions (velocities and water levels in

Table 1. Characteristic hydraulic design parameters of Lieshen Reservoir spillway.

Working Condition (Nh)	Peak Flood Discharge (m3/s)	Maximum Water Level (m)	Experimental Flow Rate (m3/h)
5	25	725.23	103
30	50	725.54	206
200	75	725.83	309

). The outlet (the blue area in Figure 4) and upper boundary of the fluid domain were set as pressure outlets under standard atmospheric pressure. The other boundaries were no-slip solid walls, and the outlet lower-wall roughness height was 0.3 m.

To ensure the reliability of numerical simulation results, grid independence verification was conducted prior to formal calculations to eliminate the influence of grid density on key hydraulic parameters. Based on the geometric model of the spillway and stilling basin, three sets of unstructured grids (coarse, medium, and fine) were generated. Grid refinement was focused on regions with intense flow variations—such as the spillway chute and the collision zone of the stilling basin—while grids in regions with gentle flow were appropriately coarsened to balance accuracy and efficiency. The cell counts of the three grid sets are as follows: 1,859,357 for the coarse grid, 2,263,480 for the medium grid, and 2,673,754 for the fine grid.

A typical operating condition (flow rate Q = 75 m³/s, stilling basin length L = 13 m, depth hk = 1 m) was selected for verification, with the energy dissipation rate η as the evaluation index. Calculations were performed under identical boundary conditions, and the results are presented in

Table 2. Grid Independence Verification Results.

Grid Type	Number of Grid Cells	Energy Dissipation Rate η	Relative Error of Energy Dissipation Rate (%)
Coarse Grid	1,859,357	77.80	6.55
Medium Grid	2,263,480	82.36	1.07
Fine Grid	2,673,754	83.25	--

: The energy dissipation rate of the coarse grid is 77.80%, and its relative error compared to the fine grid (energy dissipation rate: 83.25%) reaches 6.55%—exceeding the reasonable range of 5%, which indicates insufficient resolution. By contrast, the energy dissipation rate of the medium grid is 82.36%, and its relative error relative to the fine grid is only 1.07%, which is below the 3% engineering acceptable threshold. Considering both accuracy and efficiency, the medium grid scheme was adopted in this study. This scheme can accurately reproduce the flow collision, vortex evolution, and hydraulic jump characteristics of counterflow energy dissipation, thus ensuring the reliability of the simulation results. The specific grid is shown in Figure 2.

2.4. Numerical Calculation Results of the Experimental Model

2.4.1. Experimental Operating Conditions

Three experimental flow rates (103, 206, and 309 m³/h), corresponding to the three working conditions outlined in Table 1. A comparative analysis of hydraulic characteristics was carried out by means of monitoring cross-sections arranged along the flow direction. To eliminate the influence of hydraulic fluctuation characteristics on water surface elevation, a multi-stage averaging method was adopted for data normalization in the experiment. Each monitoring cross-section was equally divided into 5 vertical lines, where the maximum water depth h_max and minimum water depth h_min were measured, respectively. The arithmetic mean value h_avg was then calculated and defined as the characteristic water depth. Finally, the mean value of the characteristic water depths of the 5 vertical lines was computed, which was used as the representative water depth parameter of the monitoring cross-section.

2.4.2. Simulation Results and Validation

A transient numerical calculation was adopted with a time step of 0.1 s. Figure 5 shows the x − z plane flow patterns at four intervals. Given the fluctuating nature of water flow, we calculate the average values after excluding obviously abnormal data values. Figure 5 presents the comparison between measured and calculated water depths, and these results verify the rationality of the numerical calculation method.

3. Analysis of Energy Dissipation Rate for Prototype Spillway

After verifying the feasibility of the simulation method, a simulation of the prototype flow was performed. The focus was on the energy dissipation effect and post-jump water depth under different flow rates and basin lengths, analyzing the variation law of the energy dissipation rate and exploring the hydraulic jump calculation formula for opposed underflow energy dissipation.

3.1. Operating and Inlet Boundary Conditions

The energy dissipation rate is analyzed for five flow rates (25, 50, 75, 100, and 150 m³/s), nine stilling basin lengths (6, 8, 13, 15, 17, 18, 19, 22, and 30 m) and two basin depths (0.5 and 1 m). According to the open-channel uniform flow theory, the water depth at the spillway inlet is designed to adopt a normal water depth matching the corresponding flow rate, and its hydraulic relationship is established as a mathematical expression through the Manning-Strickler formula:

$$v_{in}={\displaystyle \frac{1}{\mathrm{n}}}{R}_h^{2/3}\cdot i^{1/2}$$

(6)

v_in is the Inlet velocity; n is the roughness coefficient; R_h is the hydraulic radius; and i specifies the channel’s slope. This analytical method provides a theoretical basis for the boundary conditions at the spillway inlet. The operating conditions are summarized in

Table 3. Experimental inlet flow parameters.

Slope Side	Flow Rate (m3/h)	Inlet Velocity (m/s)	Inlet Water Depth (m)
Gentle	103	0.798	0.065
	206	0.682	0.110
	309	1.066	0.118
Steep	103	0.607	0.068
	206	0.843	0.118
	309	0.922	0.146

.

3.2. Flow Regime of Stilling Basin

Under extreme conditions, the stilling basin exhibits flow aeration, expansion, and severe fluctuations, with a calculated water level close to the original sidewall top elevation. To ensure safety, the prototype basin sidewalls were raised by 3 m.

Figure 6 shows the simulated flow regimes for different stilling basin lengths L (x − z plane at y = 2.75 m), where red indicates water and blue indicates air. The flow fluctuates violently; with a short basin length and high flow rate, and the water level exceeds the sidewall height. As the flow rate increases, the flow impingement intensifies, and the maximum impingement height is near the gentle slope side. Flows collide dissipating energy, increasing the water depth, and decreasing the velocity in the basin with back-flow at the gentle slope end (a shorter basin length leads to a longer back-flow). At low flow rates, the water surface stabilizes with increasing basin length; at high flow rates, the water surface fluctuates more, and the flow is turbulent.

3.3. Energy Dissipation Rate Analysis

The equation for energy dissipation rate is

$$\eta ={\displaystyle \frac{\left(E_1+E_2\right)-E_3}{E_1+E_2}}\times 100\%$$

(7)

where E₁, E₂, and E₃ are the total energies at sections inlet1, inlet2, and outlet, respectively, i.e., $E=h+{\displaystyle \frac{\alpha v^2}{2g}}$; h is the section water depth; v is the section average velocity; α is the kinetic energy correction coefficient (=1), this assumption applies to fully developed turbulence, where the cross-sectional velocity distribution is relatively uniform and acceptable for engineering accuracy requirements [43]; g is the gravitational acceleration. As Figure 7 shows, the initial energy refers to the energy at the two inlets, and the outlet section was set at a position that exits the stilling basin.

Due to bidirectional impingement, the flow pattern is unstable and the outlet energy varies with time. Therefore, the outlet energy is set as the average energy over n time instants (${\overline{E}}_3={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{E}_3}$). To determine the energy at each time instant, the water phase at the outlet section is divided into four segments, L₁, L₂, L₃, and L₄, with the central water depth of each segment being h₁, h₂, h₃, and h₄, respectively, as shown in Figure 8. The area of each water phase segment is calculated using Si = Li × hi. Summing up the areas of all water phase segments and dividing by the stilling basin length L gives the average water depth $\overline{h}={\displaystyle \frac{1}{L}}{\displaystyle \sum _{i=1}^n{S}_i}$. The cross-sectional average velocity is $\overline{v}=Q/\overline{S}$, and the cross-sectional energy is obtained as $E_3=\overline{h}+{\displaystyle \frac{\alpha {\overline{v}}^2}{2g}}$.

To quantitatively evaluate the influence of the stilling basin depth on energy dissipation efficiency, this study compares designed basin depths of 1.0 m and 0.5 m under five typical operating conditions while maintaining a basin length of 13 m. The results (Figure 9) indicate that the influence of the stilling basin depth variation on the energy dissipation rate is significantly dependent on the flow rate: under low-flow conditions (25 and 50 m³/s), increasing the basin depth increases the energy dissipation rate by approximately 4.7–4.8%; however, increasing the flow rate sharply weakens this influence, and under high-flow conditions (100 and 150 m³/s), the energy dissipation rate increase is less than 0.6%. This shows that under medium- and high-flow conditions, the energy dissipation rate is insensitive to changes in the basin depth.

Figure 10 shows the variation in the energy dissipation rate with the stilling basin length and flow rate (basin depth = 1 m). Flow rate has a weaker influence than basin length. For a basin length of 22 m, the maximum energy dissipation rate difference is 4.98% (85.81% at 100 m³/s, and 80.83% at 25 m³/s). The energy dissipation rate first increases and then decreases with increasing basin length: the optimal stilling basin length is 17 m for 25 m³/s and 19 m for 100 m³/s. At a short basin length (6 m), a low turbulence intensity leads to a low energy dissipation rate; at 13–19 m, a stable turbulent vortex system forms, reaching the peak energy dissipation rate (optimal interval); beyond 19 m, the flow roller breaks into discrete vortex clusters, reducing the efficiency.

Different flow rate groups show asymmetric responses: low flow rates (25 m³/s) and short basin lengths (8 m) have intense shear vortices, whereas high flow rates (100–150 m³/s) and long basin lengths (30 m) have an attenuated energy dissipation rate due to energy redistribution over a longer distance. Medium flow rates (50–75 m³/s) have gentle fluctuations. High flow rates have short critical basin lengths and more significant negative energy-redistribution effects.

3.4. Analysis of Conjugate Depth After Hydraulic Jump

3.4.1. Variation with Stilling Basin Depth

This section analyzes the conjugate depths after the hydraulic jump formed in the stilling basin at different stilling basin depths.

Table 4. Net conjugate depth (m) after hydraulic jump.

Discharge Rate (m3/s)	Stilling Basin Length (m)	Stilling Basin Depth
		0.5 m	1.0 m	1.5 m
25	6	2.11	2.02	2.03
	13	1.38	1.40	1.45
	22	1.08	0.91	0.97
100	6	6.28	6.49	6.36
	13	4.48	5.59	4.47
	22	3.51	4.51	3.49

shows the data of net conjugate depths under different stilling basin lengths and flow conditions, with the stilling basin depth deducted. The net conjugate depths after the hydraulic jump are consistent, showing that the basin depth has little influence on the energy dissipation rate and conjugate depths after the hydraulic jump, compared to the flow rate and stilling basin length.

3.4.2. Variation with Flow Rate and Stilling Basin Length

Water bodies on both sides of the spillway impinge and collide in the stilling basin resulting in large conjugate depths after the hydraulic jump compared to unidirectional energy dissipation. Figure 11 shows the conjugate water depths obtained from the simulations under non-aerated concentration for different basin lengths and flow rates, revealing that the conjugate depths decrease as the stilling basin length increases up to 22 m, after which the depths show minimal change.

4. Analysis for Axisymmetric Spillway

4.1. Numerical Results

To expand the application scenarios of impingement-type energy dissipation, the spillway was adjusted to an axisymmetric structure, and its energy dissipation effect and conjugate depths after the hydraulic jump were analyzed. The slope of the steep slope side of the axisymmetric structure was set to 0.44.

Figure 12 shows axisymmetric flow regimes that allow for a more intuitive observation of the flow inside a stilling basin (Q = 75 m³/s). Figure 13a compares energy dissipation rates of axisymmetric and non-axisymmetric spillways. For basin lengths of 6–22 m, the energy dissipation rate increases with basin length, slowing down at lengths > 16 m (optimal design length). The axisymmetric spillways have higher energy dissipation rates and stable hydraulic jumps and uniform turbulent shear layers are formed. Non-axisymmetric flow has local high-velocity zones and an uneven energy distribution, reducing the efficiency by 5–10%. Figure 13b shows the relationship between the stilling basin length and conjugate depth after the hydraulic jump for axisymmetric and non-axisymmetric spillways. Both configurations show depth decreases with increasing basin length. Increasing the basin length improves the efficiency of the dissipation of flow energy and reduces the conjugate depth. The axisymmetric spillway conjugate depths were generally higher than those of the non-axisymmetric spillway. This indicates that the symmetric structure constrains the flow more, making the energy dissipation of hydraulic jumps relatively “conservative” and the water depth attenuation slower. By contrast, the channel geometry of the non-axisymmetric structure made the flow turbulence and energy dissipation more intense, resulting in a faster decrease in the conjugate depths.

It demonstrates that the structural symmetry affects the characteristics of hydraulic jumps by influencing the development of flow turbulence and energy dissipation paths in the channel. Energy efficiency dissipation can be optimized by tuning the symmetry of the spillway and stilling basin geometry.

4.2. Outlet Cross-Section Water Depth Determination

4.2.1. Fitting of Empirical Formula

The conjugate depth after the hydraulic jump is a core parameter in the spillway energy dissipation design and is directly related to the stilling basin sidewall height, structural stability, and energy dissipation efficiency. The existing theoretical equations for hydraulic jumps are derived assuming unidirectional energy dissipation, making it difficult to accurately calculate the conjugate depth after a hydraulic jump for impingement energy dissipation. The non-aerated conjugate depths after the hydraulic jump for axisymmetric impingement energy dissipation are summarized in

Table 5. Conjugate depth (m) after hydraulic jump for axisymmetric spillways.

Stilling Basin Length (m)	Flow Rate (m3/s)
	25	50	75	100	150
6	3.12	5.17	6.49	8.18	10.97
8	3.03	4.58	5.92	7.52	10.09
10	2.89	4.33	5.48	6.91	9.38
13	2.62	3.89	4.92	6.13	8.29
15	2.23	3.72	4.69	5.78	7.68
17	2.11	3.51	4.52	5.31	7.19
18	2.12	3.33	4.41	5.13	7.11
19	2.13	3.22	4.31	5.02	6.98
22	2.02	3.03	3.92	4.91	6.51
30	1.91	2.82	3.61	4.42	5.81

.

Based on Bazin’s empirical formula, a multifactor correction term is introduced to construct a revised formula for the conjugate depth after a hydraulic jump suitable for impingement energy dissipation:

$$h_2=h_1[(\sqrt[0.7]{2.5+6F{r}_1^2}-2){\left({\displaystyle \frac{2b_1}{b_2}}\right)}^{{\displaystyle \frac{2}{5}}}-(1-h_k)]$$

(8)

where h₁ is the pre-jump water depth (m); h₂ is the post-jump water depth (m); Fr₁ is the pre-jump Froude number, and after arrangement, 2.0 < $F{r}_1$ < 9.7; b₁ is the inlet width of the single-side stilling basin (m); b₂ is the outlet width of the stilling basin, (during trial calculation; b₂ can be preliminarily estimated by the calculation formula of the traditional stilling basin (for unidirectional flow)); h_k is the depth of the stilling basin; and it is recommended that 0.5 m ≤ h_k ≤ 1.5 m.

The relative error between the empirical formula prediction value and the simulated value is defined as

$$\mathit{RE}={\displaystyle \frac{\left|X_1-X_2\right|}{X_2}}\times 100\%$$

(9)

Here, X₁ represents the conjugate water depth after hydraulic jump calculated by the empirical formula, and X₂ denotes the conjugate water depth after hydraulic jump obtained from numerical simulation.

The relative errors between the predicted values of the conjugate depth after the hydraulic jump from Equation (8) and the numerical simulation are within 6%, as shown in Figure 14, meeting the engineering design accuracy requirements.

4.2.2. Water Depth Envelope at Stilling Basin Outlet

In this study, the outer envelope curve of the aerated water depth at the outlet of a stilling basin for axisymmetric impingement energy dissipation was summarized, providing a basis for the determination of the sidewall height. A mid-section was taken from the middle of the stilling basin and positioned at 0. From this mid-section, six cross-sections were taken towards the left and right inlets. The relative positions from the starting point were (0.1, 0.3, 0.5, 0.7, 0.9, and 1). The water depth outer envelope curve of each cross section at different flow rates is summarized in

Table 6. Water depth outer envelope curve at stilling basin outlet.

Flow Rate	Max. Water Depth	Left Half-Width of Stilling Basin						Mid-Section	Right Half-Width of Stilling Basin
(m3/s)	(m)	−1	−0.9	−0.7	−0.5	−0.3	−0.1	0	0.1	0.3	0.5	0.7	0.9	1
25	2.62	1.94	2.00	2.03	2.14	2.21	2.43	2.62	2.41	2.13	2.14	2.01	1.93	1.84
50	3.72	2.42	2.61	3.12	3.52	3.42	3.48	3.72	3.51	3.42	3.41	3.08	2.53	2.28
75	4.54	2.81	2.98	3.38	3.91	4.02	4.31	4.54	4.42	3.98	3.96	3.51	2.98	2.65
100	5.43	3.22	3.52	4.11	4.68	5.29	5.42	5.43	5.32	5.23	4.72	4.18	3.41	3.05
150	7.02	3.89	4.08	5.08	6.47	6.76	6.88	7.02	6.78	6.91	6.43	5.20	4.22	3.78

.

The water depth outer envelope curves at the stilling basin outlet cross-section exhibit a typical horizontal arch-shaped distribution. Under high-flow conditions (150 m³/s), the water depth shows an obvious peak value and a large variation range reflecting more intense flow turbulence and energy dissipation at the stilling basin outlet. Under low-flow conditions (25 m³/s), the water depth is relatively low and stable, demonstrating the gentle nature of flow transition. This confirms that flow rate is a key factor affecting the hydraulic characteristics of the stilling basin outlet. Large changes in water depth under high flow rates impose strict requirements on spillway system design (structural anti-scour performance and shape optimization) and operation control (gate operation and water level control). Figure 15 shows the proportional relationship curve of the water depth outer envelope curve at the outlet cross section; from −0.6 to +0.6, the proportion exceeds 80% providing a basis for the design of the sidewall height.

5. Conclusions

This study investigates the impingement-type underflow energy dissipation of spillways using simulations and model experimental methods. The influences of flow rate and stilling basin length on the energy dissipation effect is analyzed and an equation for conjugate depths after a hydraulic jump suitable for impingement energy dissipation is obtained. An outlet water depth outer envelope curve is determined to support practical engineering projects. The conclusions are as follows:

(1) Water depth comparisons between a 1:15 scale experimental model of a spillway and simulations show that the measured values are consistent with the simulations, with an error range of 0.67–13.91%, confirming the rationality of the simulations and parameter settings.

(2) The energy dissipation rate exhibits a nonlinear “first increase then decrease” variation with the stilling basin length, and the range of 13–19 m is identified as the optimal interval. Within this range, the flow forms stable turbulent vortex systems and complete hydraulic jump rollers; for instance, at a flow rate of 50 m³/s, the energy dissipation rate reaches 82.14% when the basin length is 19 m, which is a 21.3% increase compared to that at a basin length of 6 m. When the basin length exceeds 19 m, the hydraulic jump rollers break into discrete vortex clusters, resulting in a decline in the energy dissipation rate (e.g., at a flow rate of 100 m³/s, the energy dissipation rate decreases by 12.3% when the basin length is 30 m compared to that at 19 m). The post-jump water depth tends to stabilize when the basin length exceeds 22 m, providing a basis for determining the minimum effective length of the stilling basin.

(3) The energy dissipation rate of the symmetric spillway is consistently higher than that of the asymmetric spillway (the difference reaches 6.35% when the stilling basin length is 8 m), which is attributed to the fact that the symmetric flow pattern enables the formation of stable hydraulic jumps and uniform shear layers. However, the asymmetric structure exhibits a larger reduction amplitude of post-jump water depth (8.7% higher than that of the symmetric structure within the stilling basin length range of 6–22 m). Thus, the selection of structure type can be determined based on the requirements of either “stability priority” or “enhanced energy dissipation”.

(4) The energy dissipation rate of the stilling basin with a depth of 1 m is higher than that with a depth of 0.5 m. Moreover, the increase amplitude of the energy dissipation rate under small flow rate (Q = 25 m³/s, 3.66%) is significantly larger than that under large flow rate (Q = 100 m³/s, 1.24%), which is because the high-inertia flow weakens the constraint of basin depth on turbulence. When the basin depth doubles, the post-jump water depth increases by 0.39–0.88 m, requiring simultaneous optimization of the sidewall height. When the height difference (between the two sides of the spillway) increases from 2.2 m to 6 m, the pre-jump water depth decreases (for Q = 75 m³/s, the pre-jump water depth on the steep slope side decreases by 32.5%, which is due to the enhanced potential energy suppressing backwater), while the post-jump water depth increases (for Q = 100 m³/s, the post-jump water depth increases by 14.0%). This phenomenon is consistent with the conjugate law of hydraulic jump in single-side energy dissipation.

(5) Based on numerical simulations and experimental data, an empirical formula for post-hydraulic jump water depth applicable to axisymmetric and asymmetric counterflow-type underflow energy dissipation scenarios was established. This formula comprehensively considers factors such as flow rate, stilling basin length, stilling basin depth, and structural symmetry, achieving a prediction error within 6%. It can serve as a reference for the design of similar projects.