Experimental Study on the Distribution of Boundary Shear Stress at an Overfall

Abstract

Overfall flow, characterized by high Froude numbers and intense turbulence, generates boundary shear stress on vertical surfaces, which is considered the direct cause of headcut erosion. This study aims to analyze the hydraulic characteristics of nappe flow over a vertical or near-vertical overfall. Detailed experiments using hot-film anemometry were conducted in an indoor flume to examine the shear stress distribution on vertical surfaces under varying flow rates, overfall heights, and backwater depths. The results show that when the jet dynamic pressure head is less than the backwater depth, the dimensionless relative shear stress and relative depth relationship can be fitted with a beta probability density function. When the dynamic pressure head exceeds the backwater depth, the distribution follows a cubic polynomial form. Dimensional analysis and flow trajectory calculation methods were used to establish shear stress distribution formulas, with determination coefficients of 0.829 and 0.652, and the mean absolute percentage error (MAPE) between the measured and predicted values being 0.106 and 0.081, respectively. The findings provide valuable insights into the effects of complex flow structures on shear stress and offer essential support for the development of scour models for overfall structures.

1. Introduction

As water flow encounters suddenly drop in elevation, it forms a free-falling jet, known as overfall flow. The water flow from the upstream side of the drop accelerates significantly under the influence of gravity, transitioning from a subcritical to a supercritical state, which causes the water surface (level) to lower and the streamline to bend sharply at the drop, resulting in a more complex flow pattern. As the water flows over the drop, the impinging jet impacts the bed surface and creates reverse vortices. These vortices exert shear stress on the vertical wall, resulting in scour at the bottom of the overfall, which can ultimately lead to instability and collapse of the overfall, prompting advance migration. Thus, the shear stress exerted by overfall flow on the vertical wall is considered a direct cause of headcut erosion. The phenomenon of headcut erosion is commonly observed in processes such as dam failures, soil gully erosion, and the erosion of cohesive sediment deposits in reservoir areas. Among these, the jet flow associated with overfall represents a typical flow structure in the headcut erosion process. At present, there remains a significant gap in systematic research regarding the characteristics of wall shear stress induced by overfall flow on vertical surfaces. Robinson et al. conducted a series of large-scale flume experiments (with overfall heights ranging from approximately 0.9 m to 1.5 m) to investigate the advancement rate of steep drops under various influencing factors, and explored the variation of shear stress distribution on the vertical wall of the overfall with changes at a certain backwater depth, proposing a dimensionless maximum shear stress equation for the vertical wall to predict the headcut erosion rate (Bennett & Casalí) [1,2]. Su Y et al. measured the wall shear stress of gas–liquid two-phase bubbly flow in a horizontal pipe with a diameter of 35 mm using a hot-film probe, obtaining the distribution data for wall shear stress around the circumference in the fully developed section [3,4].

Several scholars have also proposed computational equations for the wall shear stress in scour pits caused by jet flow at drops, which affect the downstream bed surface. For example, Stein associated the maximum shear stress acting on the bed surface in the impact zone with the maximum velocity by introducing a friction coefficient [5,6]. Wei et al. proposed a predictive equation for the maximum stress on the left and right walls of the scour pit [7]. When the downstream water depth of the overfall is relatively high, or when the downstream bed surface has high erosion resistance, scour pits often do not form, leading to significant errors when using shear stress distribution formulas within scour pits to predict headcut erosion rates. Wang et al. conducted a series of indoor scour tests to evaluate the effects of microtopography on gully erosion processes, gully morphology, and the rate of gully retreat, using different gully head heights [8]. Zhang et al. analyzed the morphological parameters of 7 gully headcut scour holes in the Yuanmou Dry-hot Valley region, Southwest China, and examined three categories of potential influencing factors. The study identified the key factors influencing the morphological changes of the scour holes [9]. Dey proposed that the intensity of headcut erosion can be determined by the height of the drop and the flow rate [10]. Bennett found through flume experiments at various drop heights (5 to 50 mm) that as the drop height increases, the scour rate, maximum scour depth, and sediment yield all gradually increase [2,11]. Dong et al. conducted flow scour experiments with two different flow rates under similar initial topographic conditions to investigate the influence of gravitational and other dynamic factors on the headcut retreat process [12].

In terms of numerical simulation of the overfall, Zhu et al. conducted three-dimensional numerical simulations of overfall flow resulting from dam failures, analyzing the shear stress distribution and flow velocity distribution patterns on the dam drop under different operating conditions [13]. He also carried out experiments on jet flow at drops with heights ranging from 0.04 m to 0.14 m, studying the flow field structure and velocity distribution of the jet flow, which provided a theoretical foundation for the development of overtopping breach of earthen dams (Zhu et al.) [14]. In a more fundamental context, Parker & Izumi developed a theoretical framework for purely erosional cyclic and solitary steps formed by flow over cohesive beds, revealing the conditions under which self-organized step-like morphologies evolve [15]. Their findings highlight the intrinsic link between flow structure and bed evolution, providing important insights into headcut dynamics. Headcut erosion not only has a significant impact on sediment yield in the Loess Plateau watershed but is also one of the effective methods for sediment flushing in reservoirs. In the Xiaolangdi and Sanmenxia reservoirs, headcut or retrogressive erosion was induced by lowering the water level, which led to a substantial recovery of storage capacity (Yang et al.; Yu & Wang) [16,17].

In summary, there have been limited model experiments focusing on the mechanisms of headcut erosion, particularly regarding the experimental research on shear stress distribution patterns (Wang et al.) [18]. This study utilizes a hot-film anemometer to investigate the shear stress distribution of overfall flow on vertical walls under varying flow rates, overfall heights, and backwater depth conditions. Additionally, based on boundary shear stress measurements, a functional relationship between relative depth and relative shear stress values has been established. This provides a reference for studying the shear stress on vertical walls caused by complex flow structures and provides important insights into the mechanisms for the study of the headcut erosion model.

2. Overview of the Experiment

2.1. Experimental Setup

The experiments were conducted in a variable-slope glass-walled flume with a length of 18 m, a width of 0.8 m, and a height of 0.5 m. The inlet of the flume is equipped with a centrifugal pump that allows for adjustable flow rates, while a gate at the downstream end controls the backwater depth below the overfall model. The bottom of the downstream end of the flume is connected to a tailwater pool, linked to a recirculation flume system, allowing for a continuous flow of water throughout the experiment. Additionally, an electromagnetic flowmeter is installed at the front of the flume to measure the flow rate provided by the pump.

A schematic diagram of the experimental setup is shown in Figure 1. An overfall model was constructed using acrylic at a distance of 10 m from the flume inlet, matching the width of the flume and measuring 2.4 m in length. In the figure, the overfall model is marked with a red outline for visual emphasis. A mild slope was set up 2 m upstream of the vertical wall of the overfall to create a gradual variation of water depth. On the vertical surface of the overfall, circular holes with a diameter of 4 mm were installed at different heights to accommodate embedded hot-film anemometer probes. To access the probes from the outside, a small window was cut into the tempered glass on one side of the overfall section. To study the shear stress distribution on the vertical wall of the overfall under varying backwater conditions, a measuring scale was placed 2 m downstream from the overfall to record water depth data. A circulation pump was installed in the tailwater pool to pump a part of the water from the pool to a calibration pipeline and then back to the tailwater pool. The acrylic calibration pipe, with a diameter of 44 mm and a length of 2 m, is drilled with a small hole for inserting the hot-film anemometer probe and four holes for connecting pressure taps to measure the hydraulic gradient.

The core component of a hot-film anemometer is the hot-film sensor, typically composed of a thin metal film (such as nickel or platinum) deposited on an insulating substrate. When current flows through the film, it heats up, and when placed in a fluid, it exchanges heat with the flow. At thermal equilibrium, the heat loss due to convection correlates with fluid velocity, leading to a voltage signal (Li et al.) [19]. Additionally, since the upper surface of the probe is flush with the solid boundary wall during installation, this measured velocity corresponds to the velocity gradient in the boundary layer, establishing a relationship with wall shear stress. In this study, the stress measurement principle of the hot-film anemometer is based on the relationship between the voltage value measured by the hot-film probe and the shear stress at the boundary. Previous research has shown that using hot-film anemometers to analyze wall shear stress is reliable, confirming the effectiveness of this method for measuring wall shear stress (Su B et al.; Su Y et al.) [20,21].

Before and after the experiments, wall shear stress at known flow velocity levels was measured in the acrylic circular pipe to calibrate the hot-film anemometer. During the experiments, the hot-film anemometer was used to measure voltage signals at different positions along the vertical height at the overfall. By converting the voltage signals into shear stress values, the hot-film anemometer was effectively utilized for stress measurement. The configuration of the hot-film anemometer system is shown in Figure 2. The sensing element, which is the core component of the hot-film sensor, is located at the center of the diagram. When current passes through the sensing element, it generates heat and warms the element. In the diagram, Q₁ represents the heat dissipated from the surface of the sensing element to the surrounding environment through convection, Q₂ represents the heat conducted down to the base of the sensing element, and Q₃ represents the heat radiated outward. One end of the hot-film sensor probe is connected to the measurement point, while the other end connects to a MiniCTA (the main unit of the hot-film anemometer), which is further linked to an A/D board (analog-to-digital converter) and ultimately to a computer, allowing for operation and reception of voltage signals.

The calibration process involves measuring the pressure gradient in the calibration acrylic pipe at different flow velocity levels to calculate the wall shear stress $\tau$ within the pipe, thereby completing the calibration. The formula for calculating boundary shear stress is as follows:

$$\tau =\rho g{\displaystyle \frac{d}{4}}J,$$

(1)

where $d$ is the pipe diameter and $J$ is the hydraulic gradient between the measurement points ($J=\mathsf{\Delta }h/\mathsf{\Delta }x$, where $\mathsf{\Delta }h$ is the head difference and $\mathsf{\Delta }x$ is the distance between the measuring points of pressure pipe); $\rho$ denotes the fluid density; $g$ is the gravitational acceleration. By adjusting different levels of flow rate, a specific range of shear stress values is obtained to calibrate the probe. For each flow condition, the Reynolds number is calculated based on the average velocity and pipe diameter. All calibration flows correspond to fully developed turbulent conditions (R_e > 4000). For each measurement point, the measurement frequency is set to 0.5 kHz, with a measurement duration of 10 s, resulting in a sample size of 5000. The average wall shear stress for each measurement is obtained using the following formula:

$$\overline{\tau }={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{\tau }_i}$$

(2)

In the formula, ${\tau }_i$ represents the instantaneous value of wall shear stress and N is the total number of samples.

The relationship between the output voltage E of the hot-film anemometer and the boundary flow velocity $u$ is given by

$$E^2=a+b{u}^{0.5}$$

(3)

In the above equation, a represents the static hot-film voltage constant, which corresponds to the baseline voltage when the boundary flow velocity is zero; b is the sensor sensitivity coefficient, which is related to the material, structure of the hot-film sensor, and the properties of the surrounding fluid.

In turbulent flow, shear stress is typically proportional to the square of the flow velocity. For turbulent flow in a pipe, shear stress can be approximated as

$$\tau \propto \rho u^2$$

(4)

By combining Equations (3) and (4), it can be derived that the one-quarter power of the shear stress $\tau$ is proportional to the square of the voltage E measured by the hot-film anemometer. During the calibration process, different flow rates were adjusted, resulting in multiple sets of shear stress and voltage data at varying flow velocity levels. The shear stress calculation formula was then derived by curve fitting, with the determination coefficient R² generally exceeding 0.85, as shown in Figure 3.

2.2. Experimental Conditions

In this study, experiments were conducted at two different overfall heights, with two modified overfall models installed into the flume respectively. For each height, three different flow rate tests were carried out. Based on the combination of drop height and flow rate, six experimental conditions were created. For each condition, shear stress values were measured at different positions along the vertical wall of the overfall. The specific experimental condition data are shown in

Table 1. Experimental conditions.

Condition Number	Overfall Height (cm)	Flow Rate (m3/h)	Number of Backwater Depth Intervals	Measuring Point Number
1-1	33	10	10	5
1-2	33	30	10	5
1-3	33	50	8	5
2-1	43	10	10	9
2-2	43	30	10	9
2-3	43	50	10	9

.

Figure 4 shows a schematic diagram of the overfall model and the actual overfall model device. The experiment aimed to analysis the shear stress distribution at different positions along the vertical wall. For the first overfall height, five measurement holes were vertically spaced from the flume bottom at distances of 2 cm, 6.6 cm, 13.2 cm, 19.8 cm, and 26.4 cm, respectively. For the second overfall height, nine measurement holes were spaced at vertical distances of 4.3 cm, 8.6 cm, 12.9 cm, 17.2 cm, 21.5 cm, 25.8 cm, 30.1 cm, 34.4 cm, and 38.7 cm, respectively. The measurement points were evenly distributed along the vertical wall and shear stress values were measured using a hot-film anemometer to analyze shear stress variations under different experimental conditions. To account for the influence of downstream water depth on the wall shear stress distribution, shear stress values were measured at each measurement point whenever the backwater increased by 1/10 of the overfall height during the experiment.

2.3. Experimental Procedure and Measurement Method

To use the hot-film anemometer for shear stress measurements, the hot-film anemometry probe needs to be calibrated both before and after each run of the experiment. The specific operating procedure is as follows:

(1) Activate the water pump in the tailwater pool, allowing the water to circulate through the calibration pipe. Simultaneously, vent the piezometer tubes connected to the calibration pipe to prepare for calibration.

(2) Record the output voltage of the probe at different flow levels across 10 pump settings, and measure the pressure gradient from the piezometer, while also recording the water temperature in the tailwater pool.

(3) Remove the hot-film probe from the calibration loop and insert it into the small holes on the vertical wall of the overfall model.

(4) Set the upstream flow rate in the flume, and starting from the low backwater level, record the output voltage at each measurement point on the vertical wall below the downstream water surface, also recording the temperature. Then, gradually raise the backwater level and re-record the probe voltage at each measurement point below the downstream water surface until the backwater level reaches the same overfall height, completing the measurements for all overfall positions.

(5) Remove the probe from the model and reinsert it into the calibration loop, repeating steps 1–2 to recalibrate the probe.

(6) Plot the relationship between shear stress and voltage using the average shear stress and voltage levels, and establish predictive equations.

(7) If the R² coefficients of the predictive equations established from both calibrations is high, average the results of the two predictive equations, and use the corrected predictive equation to convert the measured voltage values into shear stress values.

3. Characteristics of Shear Stress Distribution

The experiment measured the shear stress values at various positions on the overfall for two overfall heights and three flow rates, with different backwater depths. Data analysis of the experimental results indicates that the upstream flow rate Q, downstream backwater depth $B_w$, and overfall height $H$ all have a certain impact on both the maximum value and distribution of wall shear stress. Figure 5 and Figure 6 respectively illustrate the variation trends of shear stress values at different points on the vertical wall as the backwater depth increases for overfall heights of 33 cm and 43 cm.

As shown in Figure 5 and Figure 6, there are significant differences in the distribution of shear stress at these measurement points under different flow conditions. For both two different overfall heights, as the downstream backwater depth increases, the shear stress on the vertical wall first increases and then decreases. This trend is consistent across different flow conditions, indicating that within certain ranges of backwater depth, the shear stress reaches a peak value and then decreases as the backwater depth continues to increase. However, the position and magnitude of the peak shear stress vary with changes in flow rate and overfall height. Notably, it can be observed that the shear stress measured at higher points on the vertical wall is generally lower. This is likely due to the relatively reduced flow velocity and impact force in the upper region of the drop when the downstream depth is higher.

With the upstream inflow remaining constant, as the backwater depth increases, the peak shear stress typically occurs near the medium backwater level, particularly evident at flow rates of 30 m³/h and 50 m³/h. By comparing the experimental results for different overfall heights under the same flow conditions, it can be observed that as the overfall height increases, the overall measured shear stress values also increase. For the higher overfall height (43 cm), both the peak shear stress and overall levels are significantly higher than those for a lower overfall height (33 cm).

Based on the experimental data at the two heights, it was found that at an overfall height of 33 cm, the positions of the measurement points where the maximum shear stress happened at the three flow rates are 6.6 cm, 13.2 cm, and 6.6 cm, respectively, concentrated in the middle and lower region of the overfall. When the overfall height is increased to 43 cm, the position with the highest frequence that the maximum shear stress is measured is at 4.3 cm. Furthermore, regardless of whether the weir height is 33 cm or 43 cm, higher flow rates (50 m³/h) result in higher peak shear stress and overall shear stress levels. This is primarily due to the increased flow rate, which leads to greater flow kinetic energy, resulting in greater shear stress. In both figures, when the flow rate is 10 m³/h, the shear stress variation is relatively mild with lower peak values, while at a flow rate of 50 m³/h, the shear stress variation becomes more pronounced and the peak shear stress significantly increases.

Table 2. The calculation of the peak-to-average ratio of shear stress at an overfall height of 33 cm.

		H = 33 cm
Backwater Level		10 m3/h	30 m3/h	50 m3/h
33		1.20	1.40	1.67
29.7		1.29	1.42	1.48
26.4		1.32	1.39	1.51
23.1		1.23	1.55	1.63
19.8		1.04	1.31	1.41
16.5		1.46	1.24	1.64
13.2		1.08	1.02	1.28
9.9		1.29	1.14	1.39
Average PAR		1.24	1.31	1.50

and

Table 3. The calculation of the peak-to-average ratio of shear stress at an overfall height of 43 cm.

			H = 43 cm
Backwater Level		10 m3/h	30 m3/h	50 m3/h
43		1.19	1.13	1.09
38.7		1.15	1.16	1.20
34.4		1.16	1.21	1.19
30.1		1.11	1.19	1.20
25.8		1.14	1.21	1.23
21.5		1.18	1.23	1.17
17.2		1.18	1.26	1.21
12.9		1.03	1.19	1.16
8.6		1.07	1.04	1.01
Average PAR		1.13	1.18	1.16

show the PAR (peak-to-average ratio) of shear stress for each measurement point under different backwater levels. It can be observed that for an overfall height of 33 cm, the average PAR of shear stress at flow rates of 10 m³/h, 30 m³/h, and 50 m³/h are 1.24, 1.31, and 1.50, respectively. This indicates that as the flow rate increases, the shear stress distribution becomes more uneven and exhibits greater variability. The variation in the peak-to-mean ratio is most pronounced under the flow rate of 50 m³/h, with a maximum value of 1.67 and a minimum value of 1.28. This indicates that at higher flow rates, the kinetic energy of the water flow increases, leading to greater flow instability, which results in a larger difference between the peak and average shear stress values. In contrast, when the overfall height is 43 cm, the changes in the peak-to-average ratio are relatively stable, with less variation in response to different flow rates. When the flow rates are 30 m³/h and 50 m³/h, the PAR fluctuates within a range of 1.09 to 1.26, with average values of 1.18 and 1.16, respectively. Overall, the peak-to-average ratio for the 33 cm overfall height remains roughly between 1 and 1.7, while for the 43 cm overfall height, it is between 1 and 1.3. This indicates that under the higher overfall condition (43 cm), the shear stress distribution along the wall is more uniform.

Figure 7 shows the distribution of shear stress values from bottom to top at different measurement points under three flow rates when the overfall height is 33 cm. As shown in the figure, when the back water level is relatively high, the shear stress shows a noticeable peak at the middle measurement point (approximately 15 cm), especially under the 50 m³/h flow condition. Additionally, at a flow rate of 50 m³/h, the peak shear stress is higher and more concentrated, indicating that the increase in flow rate enhances the kinetic energy of the water, resulting in greater impact force at specific measurement points. In contrast, under the lower flow rate of 10 m³/h, the shear stress distribution is relatively smooth, with no distinct peak, suggesting that the impact of the water on the wall is relatively uniform. From the figure, it is evident that the distribution of shear stress along the measurement points exhibits nonlinear characteristics, indicating that the flow’s effect on the vertical wall is not uniform but is influenced by a combination of factors such as flow rate, measurement point position, and flow regime.

If the ratio of measurement point height to the backwater depth is used as the relative height, and the ratio of the measured shear stress value to the critical shear stress value at the corresponding flow rate is defined as the relative shear stress, a box plot of the relationship between relative shear stress and relative height is drawn (as shown in Figure 8). It can be observed that at the relative height of 0.2, the relative shear stress is the most dispersed. The maximum relative shear stress at various relative heights reached approximately 7. As a comparison, Robinson reported a maximum stress of 5 to 15 times the average stress according to his experiment [22]. Therefore, the magnitude of measured relative shear stress in this research is reasonable. Overall, the distribution of the data points appears quite scattered, with no obvious pattern. Therefore, it is difficult to fit an accurate calculation formula under the above definition method. This phenomenon may be due to the diversity and complexity of the experimental conditions, which significantly affected the measurement results. A more detailed analysis of this issue will be provided in the following sections.

Based on the above discussion, the following conclusions can be drawn: (i) On the vertical face of the overfall, the shear stress is relatively larger in the middle and lower sections. As the overfall heights increases, the location of the maximum shear stress on the overfall moves downward. (ii) As the backwater depth increase, the shear stress at each point shows a trend of first increasing and then decreasing along the vertical face. When the backwater depth approaches the middle section, the shear stress at each point reaches their maximum. (iii) As the flow rate increases, the required backwater depth for each point to reach the maximum shear stress also increases. This pattern indicates that there is a critical value during the increase of the downstream backwater depth of the overfall. When the backwater level reaches a certain height, the stress exerted by the flow on the vertical face of the overfall reaches its peak. Beyond this height, the shear stress on the vertical face gradually weakens. Furthermore, based on the above findings, it can also be inferred that the failure of an overfall during retrogressive erosion is very likely to originate from the bottom basis of the overfall. The shear stress exerted by the flow on the vertical boundary exceeds the critical shear stress of sediment at the overfall, leading to lateral scouring at the bottom, which eventually causes the instability of the overall and slab or cantilever failures.

4. Formulas of Shear Stress Distribution

In recent years, the overfall flow has attracted the attention of many scholars due to its complex hydrodynamic characteristics, and numerous researchers have conducted in-depth analyses through numerical simulations (Wu et al.; Zhang et al.; Qin et al.; Sun et al.) [23,24,25,26]. When the jet flow impacts the downstream water surface, a large amount of air is entrained, forming aerated flow (Yang & Zhang; Xu et al.; Zhang et al.) [27,28,29]. Studies have shown that after the falling nappe enters the downstream pool, the potential energy causes the formation of hydraulic vortices in the drop structure’s backwater zone, leading to the aeration of the flow and the formation of a water–air two-phase flow (water–air mixture). As the backwater depth increases, the influence of the vortex decreases, while an increase in upstream flow rate significantly enlarges the impact zone of the vortex.

After the overfall flow enters the downstream pool, it becomes a submerged jet and then splits into two directions: one part flows directly downstream, while the other forms a vortex between the jet flow and the vertical wall, which affects the distribution of shear stress on the boundary. This study introduces the dynamic head $D_p$ of the submerged jet to approximate the scale of the vortex influence, and the value of $D_p$ can be calculated using the following formula:

$$D_p=V_j^2/2g$$

(5)

In the formula, $V_j$ represents the jet velocity at the downstream water surface, which can be calculated using the following equation:

$$V_j=\sqrt{V_i^2+2g(H-B_w)}$$

(6)

where $V_i$ represents the upstream flow velocity over the overfall. In this study, the parameter $L_e$ is used to represent the influence range of the vortex, $L_e=\min (B_w,D_p)$, and the schematic diagram is shown below.

At the same time, this study combines the dimensional analysis method with the calculation method of the jet trajectory in ski-jump energy dissipation theory to establish the relationship between the dimensionless relative shear stress and the relative depth. The relative depth is defined as $\eta =(B_w-h)/L_e$, where h is the height of the measurement point from the bottom. The dimensionless relative shear stress ${\tau }_{*}$ is the ratio of the experimentally measured shear stress $\tau$ to the maximum shear stress on the vertical wall, expressed as ${\tau }_{*}=\tau /{\tau }_{\max }$; the calculation formula for ${\tau }_{\max }$ is as follows (Robinson et al.; Robinson & Hanson) [30,31]:

$${\tau }_{\max }=k{D}_a{\left({\displaystyle \frac{q^2}{g{D}_a^3}}\right)}^{p_1}{\left({\displaystyle \frac{H}{D_a}}\right)}^{p_2}{\left({\displaystyle \frac{B_w}{D_a}}\right)}^{p_3}{\left({\displaystyle \frac{X_p}{D_a}}\right)}^{p_4}$$

(7)

In this formula, $q$ represents the upstream unit discharge, $m^2/s$; $X_p$ is the distance from the point of maximum pressure on the downstream bed surface to the vertical wall of the overfall, $X_p=D_s-S-\left({\displaystyle \frac{N_{\mathrm{w}}}{2}}\right)$, where D_s is the horizontal distance from the point of impact of the jet to the overfall structure (see Figure 9), S represents eccentricity, per Schauer and Eustis [32], N_w is the nappe width; $H$ is the overfall height, $m$; and $D_a$ is the upstream flow depth. The parameters $k$, $p_1$, $p_2$, $p_3$, $p_4$ are calibration coefficients that need to be determined.

Based on the experimental results, it was found that when the vortex influence range is large, the distribution of the relative shear stress value with respect to the relative depth exhibits a smooth, symmetric profile. This trend can be reasonably approximated using a cubic polynomial. Additionally, using the experimental data from this study, the relationship between the relative shear stress value and relative depth under large vortex scale conditions ($D_p$≥$B_w$) was plotted, as shown in Figure 10. The fitted formula is Equation (8). Figure 11 presents a comparison between the predicted values from this formula and the measured values, with a coefficient of determination (R²)of 0.829, the mean absolute percentage error (MAPE) between the true and predicted values is 0.106, indicating a good fit between the formula and the experimental data.

$${\tau }_{*}=-0.187{\eta }^3+0.223{\eta }^2-0.059\eta +0.014$$

(8)

When the vortex influence range is small ($D_p$ < $B_w$), the distribution of the dimensionless relative shear stress along the vertical wall exhibits a highly asymmetric profile. This behavior can be approximately fitted using the $\beta$ probability density function. In river dynamics, the $\beta$ probability density function was first used to describe the lateral distribution of depth-averaged flow velocities in natural rivers (Seo & Baek) [33], and in recent years, it has also been used to describe the vertical velocity distribution of density currents (Cantero-Chinchilla et al.; Wang et al.) [34,35]. Given the similar asymmetry and skewness observed in our shear stress profiles under small vortex conditions, the Beta function provides a physically meaningful and flexible form for representing the distribution. The original form of the $\beta$ probability density function is as follows:

$$f(x;\alpha ,\beta )={\displaystyle \frac{1}{B(\alpha ,\beta )}}{x}^{\alpha -1}{\left(1-x\right)}^{\beta -1}$$

(9)

in which $\alpha$ and $\beta$ are parameters greater than 0; x is a random variable defined over the interval [0, 1]; and $B(\alpha ,\beta )$ represents the $\beta$ function, with its result calculated as ${\displaystyle \frac{\mathsf{\Gamma }(\alpha )\mathsf{\Gamma }(\beta )}{\mathsf{\Gamma }(\alpha +\beta )}}$.

By analogy with the above form, the formula expressing the relationship between relative shear stress and relative depth can be written as

$${\tau }_{*}=\sigma {\left({\displaystyle \frac{{\eta }^a}{{\eta }^a+1}}\right)}^m{\left({\displaystyle \frac{1}{{\eta }^a+1}}\right)}^n$$

(10)

where $m$ and $n$ are the fitted parameters of the formula, with values of 41.17 and 36.76 for the data set in this research, respectively. The value of parameter a is 0.091. $\sigma$ is the coefficient of the formula, which can be calculated using the following equation: $\sigma ={(m+n)}^{(m+n)}/m^m/n^n$.

The relationship between relative shear stress values and relative depth, as well as the comparison between the predicted values from the formula and the measured values, are shown in Figure 12 and Figure 13. The experimental data cover measurement results at various heights and positions along the vertical wall under different backwater conditions. For the small vortex scale conditions, the R² value is 0.652, MAPE between the true and predicted values is 0.081. Therefore, the fitted formula provides a physically interpretable and moderately accurate fit to the shear stress distribution for exploring the distribution relationship between relative shear stress values and relative depth, offering valuable insights and references.

Based on the predictive formulas for relative depth and relative shear stress values proposed in Equations (8) and (10), analysis of the experimental results reveals that under large vortex scale conditions (see Figure 10), the peak of relative shear stress occurs within a range of relative depth from 0.6 to 0.8, indicating that under large vortex conditions, the peak shear stress is located in the upper-middle region of the flow and exhibits a relatively smooth distribution. Under small vortex scale conditions (see Figure 12), the relative shear stress variation curve shows significant asymmetry around the peak (where the relative shear stress equals 1). Below the location of peak stress, i.e., on the upper half of the curve, the relative shear stress gradually increases as the height of the measurement point increases, while above this location, the relative shear stress rapidly decreases with increasing height.

The analysis indicates that, under large vortex conditions, the maximum shear stress occurs at a relative depth of around 0.7, whereas in the case of small vortices, the maximum shear stress is found at a relative depth of approximately 3–4. Additionally, the vortex scale has a significant impact on the distribution of relative shear stress. In the case of large vortices, the shear stress distribution is relatively uniform; however, under small vortex conditions, the shear stress increases rapidly from water surface to the peak location and then decreases with much smaller rate towards the bottom, exhibiting a stronger nonlinear characteristic, likely due to the presence of turbulent structures. The measurements of boundary shear stress distribution in this study can provide a reference for future numerical model calculations of headcut erosion. The proposed formulas offer strong support for understanding the relationship between relative depth and relative shear stress values.

To further interpret the underlying physical mechanisms of the observed shear stress patterns, a discussion on vortex–wall interaction is provided. The results suggest that the distribution of wall shear stress is closely related to the spatial scale and position of vortices induced by the submerged jet. Under small vortex scale conditions, the vortex structures are mainly confined to the upper region of the flow, resulting in the peak shear stress occurring at a higher position on the vertical wall. Conversely, under large vortex scale conditions, the vortex cores are located closer to the lower wall region, leading to a downward shift in the location of peak shear stress and a smoother distribution profile. These characteristics are consistent with classical descriptions of turbulent separated flows, where jet impingement, recirculation, and boundary layer evolution jointly determine the shear intensity along solid boundaries (Robinson & Hanson; Schauer & Eustis;) [32,33]. The present results thus provide a complementary experimental perspective on the flow–boundary interaction in the context of submerged jet impingement over vertical surfaces.

5. Conclusions

The presence of a drop is one of the characteristics of headcut erosion. When the flow reaches the drop, the sudden change in terrain transforms the flow into a jet, causing a severe impact on the bottom basis of the overfall boundary. This leads to significant shear stress variations on the vertical boundary, which in turn contributes to the development of headcut erosion. In this study, an indoor flume physical model experiment was conducted, using hot-film anemometers to measure the boundary shear stress on the vertical boundary of a drop. The distribution patterns of shear stress on the vertical boundary were investigated for two different overfall heights. This study has found that, under constant flow discharge, the magnitude of the shear stress on the vertical wall initially increases and then decreases as the downstream backwater depth increases. Additionally, the increase in overfall heights and flow rate also significantly affects the shear stress values on vertical boundary. Based on the experimental results, it can be further inferred that the failure of the overfall structure may originate at the bottom basis. The shear stress exerted by the flow on the vertical face exceeds the critical shear stress for the sediment incipient motion at the drop, leading to lateral scouring at the bottom basis of the overfall boundary, which causes the failure of the overfall. This result provides significant supplementary insights for the study of the mechanisms of headcut erosion and for hydraulic engineering design. It is especially valuable as a reference for predicting shear stress distribution under complex flow conditions.

In addition, this study established the relationship between relative depth and relative shear stress on the vertical boundary of overfall. This study found that when the influence of the vortex is significant, the relationship between relative shear stress and relative depth approximates a cubic polynomial form distribution. The peak of relative shear stress occurs in the range of 0.6 to 0.8, with the maximum shear stress appearing at a relative depth of approximately 0.7. When the influence of the vortex is small, the relative shear stress is significantly affected at higher positions, with the maximum shear stress occurring near a relative depth of 3 to 4. This relationship can be fitted using a beta probability density function. Dimensional analysis and jet range calculation methods were used to establish shear stress distribution formulas, with determination coefficients of 0.829 and 0.652, and the mean absolute percentage error (MAPE) between the measured and predicted values being 0.106 and 0.081, respectively. This establishes a theoretical foundation for investigating the scouring mechanisms associated with headcut erosion.

Overall, this work enhances our understanding of the interaction between submerged jet flows and vertical boundaries under overfall conditions. It offers both experimental evidence and predictive tools for evaluating the spatial distribution of wall shear stress. The proposed models, based on a generalized definition of relative depth incorporating both backwater depth and jet dynamic pressure, are applicable to a range of submerged overfall scenarios. When applying the proposed models, it is important to ensure that the actual flow conditions are hydraulically similar to the smooth, supercritical overfall flows examined in this study. However, their use should be limited to the validated experimental ranges (relative depth 0–1 for large vortices and up to 8 for small vortices), and caution should be exercised in practical applications with differing hydraulic or geometric characteristics.