Experimental and numerical investigation of flow over a spillway bend with different combinations of permeable spur dikes

In many water related projects, water-retaining structures such as gates and dams are usually required to be built to meet the function of flow interception, flood control and reservoir storage adjustment. A significant head difference between upstream and downstream of the constructed project is thus formed, leading to huge energy contained in the water flowing over the weir or dam. Building a spillway is a commonly used engineer measure to safely release the flow from the reservoir to the downstream river (Parsaie & Haghiabi 2019; Kocaer & Yarar 2020) and dissipate the tremendous flow energy generated by crossing weirs or dams (Khatibi et al. 2014; Rajaei et al. 2020; Wu et al. 2020). However, in an actual project, the spillway is often built with a bend section (Figure 1) owing to the limits of topography, geology, construction conditions and so on. The water flowing into the bend section generally turns into a state of complex transverse circulation flow because of the centrifugal force (Blanckaert & De Vriend 2004). It is the transverse circulation flow in the bend that gives rise to such common issues as uneven distribution of water surface, transverse shock wave and so on. It sometimes even exerts a negative effect on the energy dissipation downstream of the bend, resulting in severe cavitation erosion, scour of the riverbed downstream and undermining of the spillway foundation (Eghlidi et al. 2020).

Various researchers in recent years have studied the flow characteristics in the spillway bend and have been dedicated to seeking kinds of auxiliary energy dissipaters for reducing water surface difference, improving the flow characteristics, and enhancing energy dissipation efficiency within the bend, such as a guide wall (Zhang et al. 2016; Lian et al. 2019), diversion structure, sloped ridge (Zhou et al. 2014), and solid spur dike (Jamieson et al. 2013). Safarzadeh et al. (2016) investigated the flow and scoured patterns in a 90° bend with two submerged or non-submerged T-shaped spur dikes using the SSIIM (Sediment Simulation In Intakes with Multiblock) numerical model. Sadegh & Parsaie (2016) studied the effect of geometries of guide walls on the flow pattern and rating curve of the Balaroud dam spillway's (Iran) using numerical simulation and laboratory experiment, obtaining the optimal shape of the guide walls. Although the measures mentioned above have been incorporated into existing engineering measures, they are often applicable to only one discharge or working condition because of their poor versatility. There is still a large water level difference on both sides of the bend with the above measures, and the energy dissipation rate is also not very satisfactory. Therefore, it is of great necessity to explore a new and effective measure to improve the flow characteristics in the spillway bend.

The permeable spur dike is a new type of river training structure that has developed rapidly since the 1970s. Compared with the conventional impermeable spur dike (the solid spur dike) the permeable spur dike possesses specific permeable capability due to the permeable holes. Although part of flow in front of the permeable spur dike has to bypass it due to its lateral blocking, most of the flow could straightly run to downstream by crossing through the permeable holes, thus greatly weakening the circulation flow in the spillway bend and resulting in the decrease of water surface difference on both sides of the bend. Besides, the permeable spur dike is characteristic of simple structure, reliable operation, easy maintenance, better stability and less investment relative to traditional impermeable spur dike. However, literature review shows that to date, few researchers have carried out experiments or simulations on the application of permeable spur dikes in a spillway bend for the improvement of the flow characteristics and energy dissipation efficiency.

The main purpose of this paper is to experimentally and numerically investigate water surface uniformity, flow characteristics, turbulent kinetic energy, and energy dissipation rate in the bend section of a mild-slope spillway in which two or three permeable spur dikes are installed at 1/4, 1/2 or 3/4 of the bend section. The digital water level point gauge and acoustic Doppler velocimeter were used for measurements of water depth and velocity in the spillway bend. The RNG k-ε turbulent model in FLOW-3D software was applied to conduct numerical simulation and the numerical results was validated by comparison with physical model test results. The results obtained in this study could serve as a reference for the application of permeable spur dikes in the bend section of the spillway.

A series of experiments on the influence of different combinations of permeable spur dikes (detailed introduction later) on the flow characteristics in the spillway bend was performed at the water conservancy laboratory of Shandong agricultural university. The experiment was carried out in a curved flume under clear water conditions. The dimension of the curved flume is designed at a certain scale with reference to a real spillway project. According to the dimension of the real spillway project, the axis radius of the real spillway bend project is 2.4 times the width of the spillway, and the slope of spillway is 0.02, so the same scale was adopted in the laboratory model with the project. The curved flume with a central axis radius of 1200 mm and a bottom slope of 0.02 was supported by six small jacks. The bottom slope elevation of the curved flume can be accurately controlled by fine-tuning the elevation controller of the jack. A straight upstream flume of 3200 mm long and a straight downstream flume of 1000 mm long are respectively connected with both ends of the curved flume to guarantee the smooth transition when the water flows into and out of the curved flume. A flow steadying grid is placed at the front end of the upstream straight flume to steady the extremely turbulent flow from the pump. All the cross-sections of the flumes were rectangle with a net width of 500 mm. The permeable spur dikes used in the experiment were 200 mm long, 60 mm tall and 10 mm thick, along the length of which the round permeable holes were arranged uniformly and symmetrically and total 30. The flumes and permeable spur dikes were all made with PVC plastic sheets. Figure 2 exhibits the layout of the spillway bend with permeable spur dikes and the profile of permeable spur dike.

Yang et al. (2019) investigated the effect of installing angles of a single permeable spur dike on the water surface uniformity in a spillway bend and concluded that the water surface uniformity could get significant improvement when the installing angles of the single permeable dike were respectively 60, 75 and 75° at 1/4, 1/2 and 3/4 of the concave bank of the bend. As shown in Figure 2, the installing angle of the permeable spur dike refers to the included angle between the permeable spur dike and the tangent line at the intersection of the permeable spur dike and sidewall. Based on the experimental results of Yang et al. (2019), five groups of experiments related to different combinations of installing locations of the permeable spur dikes were performed in this study (Table 1).

Measurements in the experiments primarily focused on water depth, flow velocity and discharge. The experimental measurement points related to water depth and velocity are shown in Figure 2. The discharge required in the experiment is directly displayed on the electromagnetic flowmeter screen in real-time through the manual adjustment of the water valve. The water depths for different cases were measured by digital water level point gauge with a precision of 0.01 mm. Instantaneous three-dimensional velocity was collected with two side-facing Nortek Vectrino (four-beam) acoustic Doppler velocimeters (ADV). The ADVs were mounted on a movable carriage, which provided convenient adjustment of the ADVs in the lateral and vertical directions so that the ADVs can collect the three-dimensional velocity at any measurement point. The carriage with ADVs would be moved to the next cross-section as soon as the velocity collection at one cross-section was completed. The probes of the ADVs were toward the flume wall and were fully submerged in the flow. The sample volume of the ADVs was approximately 50 mm from the central emitting transducer (Jamieson et al. 2013). A sampling frequency of 100 Hz and a sampling duration of 120 s were adopted according to the recommendation in García et al. (2005). The time-averaged velocity at any measurement point can be obtained by the post-processing software WinADV. The ADVs and digital water level point gauge are shown in Figure 3.

In this study, the FLOW-3D numerical software, which takes the specially developed VOF technique for free face flow, was employed as a solver of the Navier-Stokes equation to simulate the 3D flow characteristics in the spillway bend with spur dikes.

There are three main turbulence models included in FLOW-3D, which are, k-e, k-w, and renormalized group (RNG) k-ε turbulent models, respectively. According to Karami et al. (2017), the equation of the RNG k-ε turbulent model is similar to that of the standard k-ε turbulent model, except that the values of coefficients of the two models are different. The constants in the standard k-ε equation are generally obtained by laboratory experiment, while the coefficients in the RNG k-ε equation are derived by theory. Generally, the RNG k-ε turbulent model has broader applicability than the standard k-ε model. Consequently, the RNG k-ε turbulent model was employed in this study for investigating the flow patterns in the spillway chute with spur dikes.

As illustrated in Figure 6, contrast of the simulated and experimental data (with their measurement points depicted in Figure 2) shows that the simulated water depth and longitudinal velocity are in good agreement with experimental data. Table 2 shows the results of error analysis of experimental and simulated data for all cases, which indicated that the results of error analysis are all within the allowable range. From Figure 6 we can also see that backwaters are formed in front of the first permeable spur dike in the bend, and the water depth in front of the first permeable spur dike is noticeably greater than that at the back side of the spur dike. Also, along the bend axis, an obvious decrease in the longitudinal velocity occurs within a certain distance in front of the permeable dikes, which is consistent with laboratory experiments. Therefore, the RNG k-ε model is able to simulate with reasonable accuracy the flow characteristics in the spillway bend where regulation structures such as permeable spur dikes are installed.

Figure 7 shows the water surface uniformity in the bend for all cases according to the formula proposed by Zhang et al. (2016).

Compared with no permeable spur dike installed in the bend (case C5), a significant improvement could be seen from Figure 7 on the water surface uniformity after the permeable spur dikes are installed in the spillway bend (cases C1–C4). The water surface uniformity increases with the increase of the unit discharge in the spillway bend and it reaches the highest when three permeable spur dikes are installed in the bend. For instance, when the unit discharge in the bend is 240 m²/h, the water surface uniformity for cases C1, C2, C3 and C4 are respectively 83.65, 84.6, 86.49 and 90.13%, whereas it is only 73.77% for case 5.

It is interesting to note from Figure 7 that there is a considerable improvement on the water surface uniformity whenever there is a permeable spur dike installed at 3/4 of the spillway bend (cases C2, C3, C4). It is easily conceivable that by altering the flow direction the permeable spur dikes deflect part of the flow upstream to the center of the bend and even to the convex bank, thus improving the water surface uniformity in the bend. However, the spur dikes possess limited ranges deflecting the flow, and they only act on the deflection of the flow within a certain distance upstream of them and have little influence on the flow downstream. That is, the permeable spur dike installed at 1/4 of the bend can only be responsible for the deflection of flow within a certain distance in front of 1/4 of the bend. By contrast, the permeable spur dike installed at 3/4 of the bend can deflect flow within a certain distance in front of 3/4 of the bend. It seems that the spur dike installed at 3/4 of the bend has a more extensive impact range controlling the flow deflection than that installed at 1/4 of the bend. So, it can be reasonably speculated that it is the permeable spur dike installed at 3/4 of the bend that plays a relatively critical role in the improvement of water surface uniformity. What is more, a more substantial flow deflection occurs when two or three permeable spur dikes are installed at different locations of the bend, leading to a more distinct improvement effect on water surface uniformity.

However, the slight decrease in the water surface uniformity for case 2 when the unit discharge in the bend is relatively large should be noted. This is mainly ascribed to the relatively sizeable installing distance, which exceeds the range that the following permeable spur dike deflects the flow, between the two permeable spur dikes. The flow crossing through the first permeable spur dike and being not influenced by the second spur dike continues to flow to the concave bank, hence leading to a slight decrease in the water surface uniformity.

Results obtained in prior studies employing experiments and simulations indicated that visible and stable circulating zones developed upstream and downstream of a solid spur dike (Koken 2011; Koken & Constantinescu 2014). According to Safarzadeh et al. (2016), the circulating zone could be completely changed from vertical to horizontal circulation by increasing the length or changing the shape of solid spur dikes. However, in this study, no circulation zone is formed upstream and downstream of the permeable spur dikes due to its permeability. Figure 8 exhibits the time-averaged flow streamlines for different cases at three horizontal profiles, which are respectively 1 cm (k = 1), 3 cm (k = 3) and 5 cm (k = 5) from the bottom plate of the bend. The typical velocity vector field and the 3D flow trajectories around the permeable spur dike are shown in Figure 9.

As can be seen from Figure 8, the flow direction around the permeable spur dikes has a distinct depth-varying nature and this nature is also related to the location of the permeable spur dikes in the bend. The first permeable spur dike in the bend (at 1/4 of the bend) has little effect on the flow direction. At the near-bed profile (k = 1), as the flow reaches immediately upstream of the first permeable spur dike, most of the streamlines are able to smoothly cross through the permeable spur dike, whereas for streamlines at k = 1 upstream of the other permeable spur dikes in the bend, the streamlines adjacent to the concave bank show a noticeable tendency of bending toward the tip of the spur dike, and the streamlines at the tip of the spur dike tend to bypass the spur dike and manage to merge with the flow in the center of the bend. Also, the extent to which the streamlines bend towards the center of the bend varies with the location of the spur dike at k = 1. The streamlines upstream of the last spur dike bend more seriously than that of the streamlines upstream of the spur dike preceding it (i.e. three permeable spur dikes are installed in the bend, Figure 8(j)). What is worth noting is that with the increase of the vertical distance of the streamlines from the bottom plate (k = 3 and k = 5), the phenomenon that the streamlines upstream of the permeable spur dike bend towards the tip of the spur dike gradually mitigate, and it even disappears entirely at k = 5, making the streamlines capable of directly crossing through the permeable spur dikes rather than bypassing them (Figure 8(c), 8(f), 8(i) and 8(l)).

The flow upstream of the permeable spur dike could be divided into three distinctly different flow modes according to the laboratory observation by dye tracing, which is further demonstrated by the results of numerical simulation displayed in Figure 9(b). The trajectories in Figure 9(b) are colored by z coordinate values of the model to distinguish the flow in different modes more clearly. The first mode occurs approximately at the top of the permeable spur dike. In this mode, the flow upstream of the spur dike is completely above and slightly below the top of the permeable spur dike and usually possesses relatively high momentum. While flowing over the permeable spur dike, the flow immediately above the top of the spur dike roughly forms an arch shape, part of the kinetic energy of the flow is progressively converted to potential energy, resulting in a rise in water level and a decrease in flow velocity. The second mode happens within the region adjacent to the upstream of the permeable spur dike body. Most of the flow in this mode crosses straight through the spur dike body to the downstream, except for a small part of flow near the tip of the spur dike which may bypass the spur dike. It is interesting to note that due to the inertia, the flow directly through the permeable spur dike still maintains a short distance approximately parallel to the axis line of the permeable holes, and then the flow begins to exhibit a tendency bending toward the center of the bend (Figure 8(b), 8(e) and 8(h)). The flow in the second mode is completely different from that in the bend without spur dikes, and it serves as the dominant flow that strongly disrupts the original circulation flow in the bend. The third mode mainly locates near the bottom plate upstream of the spur dike. As detailed in Figure 9(a), when the near-bed flow reaches immediately upstream of the spur dike, the absence of the permeable holes at the bottom of the spur dike forces the flow to be switched to the direction nearly parallel to the permeable spur dike and then bypass the spur dike, ending up merging into the flow in the center of the bend.

Figure 10 exhibits the contours of time-averaged longitudinal velocity measured by ADVs at the profiles of 1 cm (k = 1) and 3 cm (k = 3) from the bottom plate of the bend for all tested cases. Three velocity zones, zone I, zone II and zone III, were divided in the contours according to different locations in the bend for ease of analysis and comparison of the longitudinal velocity variation after the permeable spur dikes were installed in the bend. As illustrated in Figure 10, zone I refers to the region between permeable spur dikes at the concave bank, zone II is located in the convex bank area opposite the permeable spur dike, and zone III is related to downstream of the bend.

As can be seen from Figure 10, the time-averaged longitudinal velocity in the bend with no spur dike installed is more considerable than that in the bend with spur dikes, especially at the exit section of the spillway bend. Downstream of the spillway bend for case C5, the longitudinal velocity at the convex bank is much larger than that at the concave bank at k = 1 profile and is characteristic of relatively large and severely uneven transverse distribution, which adversely affects the energy dissipation of the spillway bend. At the exit section, the maximum longitudinal velocity could reach 1.85 times of the approaching velocity (time-averaged longitudinal velocity at upstream straight section) and the maximum longitudinal velocity nonuniform coefficient (the ratio of the velocity of the convex bank to that of the concave bank at the same cross-section) can reach 1.43. What needs to be explained is the significant decrease in the longitudinal velocity at the k = 3 profile at the convex bank downstream of the spillway for case 5. This is mainly attributed to the relatively shallow water depth which cannot fully submerge the probe of ADV, leading to the measured velocity being slightly less than the actual velocity to some degree. However, the downward trend of the longitudinal velocity at the k = 3 profile is correct on the whole, which will be discussed in the following.

Different degrees of decrease in the longitudinal velocity in each zone happens after the permeable spur dikes are installed in the bend and the most significant decrease occurs in zone I due to the lateral blocking of the spur dikes. For example, at k = 1, the minimum longitudinal velocity in zone I for cases C1, C2, C3 and C4 respectively decreases by 28.72, 30.96, 21.62, and 22.7%, compared with the longitudinal velocity at the corresponding position for case C5 (no permeable spur dike in the bend). Observation from the contour plots of the three zones for all the cases with permeable spur dikes shows that there is a vast difference between the longitudinal velocity in the three zones. Zone I is dominantly the low-velocity zone, zone III is basically the high-velocity zone, and the velocity in zone II is between that of zone I and zone III. With the increasing vertical distance from the bottom plate of the spillway bend, the area of the low-velocity zone is progressively reduced and the area of the medium-velocity zone and the high-velocity zone remain basically unchanged. It is apparent that upstream of the permeable spur dike, the absence of permeable holes at the bottom of the permeable spur dike leads to the stagnation and direction change of most of the flow near the bottom plate, thus producing a considerably larger area of the low-velocity zone near the bottom plate. With the increasing vertical distance from the bottom plate, the presence of permeable holes on the spur dike body allows the flow to continue to run to downstream of the spur dikes, resulting in the significant increase in the velocity downstream of the spur dikes and accordingly the reduction in the area of the low-velocity zone.

With permeable spur dikes installed in the spillway bend, the minimum longitudinal velocity in the spillway bend occurs immediately downstream of the tip of the first spur dike in zone I and the maximum longitudinal velocity occurs at the convex bank of the exit section in zone III. The longitudinal velocity is significantly larger than that for other cases with permeable spur dikes, and the velocity amplifications (the ratio of average longitudinal velocity to approach velocity) in zone III for cases C1, C2, C3 and C4 are 1.6, 1.34, 1.46 and 1.42, respectively.

The distributions of turbulent kinetic energy at different horizontal profiles for various tested cases are displayed in Figure 11. ⁠, and respectively represent fluctuations of velocity components and the overbar denotes time-averaging. As illustrated in Figure 11, the presence of permeable spur dikes causes concentration of the TKE zone at the concave bank of the spillway bend, except for the dispersed TKE zone next to the bottom plate. The TKE zone initiates immediately upstream of the first permeable spur dike, and dominantly develops downstream of the spur dike. These zones where TKE develops downstream of spur dikes are much more extended than that upstream of spur dikes. This phenomenon is mainly attributed to the permeable holes on spur dikes, which exacerbates the disturbance of flow through permeable holes. This disturbance cannot disappear within a short distance, bringing about extension of the TKE zone.

Near the bottom plate (k = 1, Figure 11(a), 11(d), 11(g) and 11(j)), large areas of thick and long horizontal belts with mild TKE occur proximity to the convex bank and downstream of the spillway bend. Downstream of permeable spur dikes, small patches with slightly more intense TKE appear and a distinctly decreasing trend of TKE from core to periphery of these small patches could be observed. The areas and locations of the belts and patches differ for various tested cases.

With increasing vertical distance from the bottom plate (k = 3 and k = 5), the distribution of the TKE zone changes significantly. As depicted in Figure 11, at k = 1, the thick and long horizontal belts proximity to the convex bank and downstream of the spillway bend are much more extensive compared with the small patches of TKE downstream of the permeable spur dikes. By contrast, at k = 3 and k = 5, the belts have totally disappeared and the small patches of TKE downstream of the permeable spur dikes become more extensive, especially for the patch downstream of the first spur dike, whose area at k = 3 is approximately 1.5 times at k = 1. Also, at the core of these small patches, i.e. the parts immediately next to the back surface of the permeable spur dikes, TKE gets larger with increasing vertical distance, and TKE at the core of these patches at k = 3 is about 30% larger than that at k = 1.

Horizontally, the small patches of TKE downstream of the permeable spur dikes are approximately trapezoidal distribution, which means that the span of the TKE zone at the concave bank of the bend is significantly smaller than that at the tip of the spur dikes. With increasing vertical distance from the bottom plate (k = 3), the TKE zone at the outermost periphery of the patches at the tip of the spur dikes gradually evolves into a horizontally elongated belt with a mild TKE, which extends in a curved belt nearly parallel to the concave bank to the end of the spillway bend. At k = 5, the horizontally elongated belt considerably shrinks and even breaks.

For more convenient and intuitive observation of the vertical distribution of TKE upstream and downstream of permeable spur dikes, Figure 12 exhibits the TKE distribution and 2D streamlines in several typical vertical profiles perpendicular to permeable spur dikes. As shown in Figure 12, vertically, the TKE downstream of permeable spur dikes exhibits a typical parabolic distribution, the TKE downstream of spur dikes first increases and then decreases with increasing vertical distance from the bottom plate.

Energy dissipation is one of the considerations when placing an auxiliary energy dissipator to prevent potential scour and cavitation in the spillway bend (Eghlidi et al. 2020). Zhang & Chanson (2018) established physical models to examine the effect of step edge and cavity geometry on energy loss performance in stepped chutes. In this work, the energy dissipation rate in the spillway bend with permeable spur dikes was calculated according to the measured longitudinal velocity and water depth in laboratory experiments.

The calculated energy dissipation rate for various tested cases are listed as follows:

Comparisons of the energy dissipation rates for various tested cases from Table 3 reveal that the permeable spur dikes have a substantial contribution to the energy dissipation in the spillway bend. Compared with the case with no permeable spur dike installed in the bend (case 5), the energy dissipation rate in the spillway bend with permeable spur dikes increases by at least 13.49%. When there are three permeable spur dikes installed in the spillway bend, the energy dissipation rate reaches the highest at 21.08%. Besides, higher energy dissipation rates are produced for the cases with the first permeable spur dike (installed at 1/4 of the bend) in the bend. The prime reason is that for the case with no permeable spur dike installed in the spillway bend, the flow velocity reaches the highest approximately at the apex (1/2) of the bend, and then begins to decrease due to the blocking of the sidewall. By contrast, for the cases with the first permeable spur dike installed at 1/4 of the bend, the decrease in flow velocity initiates at 1/4 of the bend due to the lateral blocking of the spur dike, thus resulting in a better energy dissipation effect. It can therefore be concluded that the first permeable spur dike at 1/4 of the spillway bend plays a critical role in energy dissipation.

The combination of permeable spur dikes installed in the different locations of a spillway bend can be a novel approach to modify the flow characteristics and improve the energy dissipation rate in the spillway bend in theory. In order to investigate the water surface uniformity, flow characteristics, turbulent kinetic energy and energy dissipation rate in the bend section of a mild-slope spillway after two or three permeable spur dikes are installed at 1/4, 1/2 or 3/4 of the bend section, and laboratory experiments and numerical simulation were employed in this study. The numerical simulation was carried out using the FLOW-3D software. The main results of this study are as follows:

The comparison between the numerical results and experimental data revealed that the RNG k-ε turbulent model in the FLOW-3D software is capable of simulating the flow in the spillway bend in which the permeable spur dikes are installed and the simulated depth and longitudinal velocity are in good agreement with the laboratory experimental data.

The permeable spur dikes installed in the spillway bend appreciably contributes to the improvement on the water surface uniformity. The water surface uniformity increases with the increase of the unit discharge in the spillway bend and it reaches the highest when three permeable spur dikes are installed in the bend under current conditions. The permeable spur dike installed at 3/4 of the spillway bend has the most significant impact on the improvement of water surface uniformity.

No circulation zone is formed upstream and downstream of the permeable spur dikes due to the permeability, which is different from the previous study. The flow direction around the permeable spur dike has a distinct depth varying nature and this nature is also related to the location of the spur dike in the spillway bend. The flow upstream of the permeable spur dikes could be divided into three distinctly different flow modes according to the laboratory observation by dye tracing.

Three velocity zones were divided according to the magnitude of the longitudinal velocity in different areas of the spillway bend in which the permeable spur dikes are installed. Different degrees of decrease in the time-averaged longitudinal velocity in each zone happens after the permeable spur dikes are installed in the spillway bend and the most significant decrease occurs in the low-velocity zone due to the lateral blocking of the spur dikes to the flow. With the increase of the vertical distance of the horizontal profile from the bottom plate of the spillway bend, the area of the zone is progressively reduced, and the area of the medium-velocity zone and the high-velocity zone remain basically unchanged. The minimum longitudinal velocity in the spillway bend occurs immediately downstream of the tip of the first spur dike, and the maximum longitudinal velocity occurs at the convex bank of the exit section.

The presence of permeable spur dikes causes the concentration of the TKE zone at the concave bank of the spillway bend, except for the TKE zone immediately next to the bottom plate. The zones where TKE develops downstream of the spur dikes are much more extended and higher than that upstream of the spur dikes. TKE downstream of the permeable spur dike exhibits a typical parabolic distribution and reaches the highest in the middle of the backside of the permeable spur dike. However, TKE proximity to the convex bank gradually decreases and eventually totally disappears with the increasing vertical distance from the bottom plate.

The permeable spur dikes have a substantial contribution to the energy dissipation in the spillway bend. The energy dissipation rate in the spillway bend with installed permeable spur dikes increases by at least 13.49%. When there are three permeable spur dikes installed in the spillway bend, the energy dissipation rate reaches the highest at 21.08%. The first permeable spur dike at 1/4 of the spillway bend plays a critical role in energy dissipation.

This work is supported by the National Natural Science Foundation of China/Yalong River Joint Fund Project (NO. U1765205), Jiangsu Colleges and Universities Advantageous Discipline Construction Project (Water Conservancy Project) (NO. YS11001) and the Natural Science Foundation Project for Youth Scientists of Shandong Province (Grant No. ZR2020QE286).

All relevant data are included in the paper or its Supplementary Information.