Experimental and numerical study the effect of flow splitters on trapezoidal and triangular labyrinth weirs

ABSTRACT Nonlinear weirs are among the major structures used in water transfer. Thus, these structures have been the subject of various studies. This study investigates the effect of flow splitters on discharge performance of trapezoidal and triangular labyrinth weirs of one cycle. This study delves into the effects of flow splitters installed on the crest of trapezoidal and trapezoidal one-cycle labyrinth weirs using numerical methods (Flow-3D) and physical models. In the physical model, two, four, or six vanes were placed on the crest at regular intervals. The results showed that the triangular weir offered 4% higher efficiency than its trapezoidal counterpart and that the vanes had better effects on the trapezoidal weir. The vanes can improve the weir efficiency by up to 11%, and the discharge coefficient was found to be 10.2% higher in the numerical simulation than in the physical model.


Introduction
Weirs serve to measure and regulate water flow in dams and water distribution canals and must be designed with high discharge capacity.Accordingly, labyrinth (nonlinear) weirs are used as an effective and economical solution to increase the throughput.The labyrinth weir increases the discharge compared to a linear weir for the same hydraulic load by extending the crest within a fixed width (Falvey, 2003).
Idrees, Al-Ameri, and Das (2016) determined the discharge coefficient of a one-cycle compound trapezoidal labyrinth weir.It was shown that the discharge coefficient initially increases, only to gradually decline beyond a maximum.The smallest discharge coefficient corresponded to 6° inclination with respect to the sidewall.The statistical results suggested that 20° is the best angle for normal weir orientation, given that it produces a coefficient of variance of below 18.8%.Meanwhile, in the inverse orientation, 35° was the best angle as it corresponded to a 16.7% coefficient of variance In their study on the effects of submerged flow splitters on the discharge coefficient of triangular labyrinth weirs with 45 and 90° apex angles, Hosseini Tashnizi et al. concluded that narrower vanes offer better effects and the 45° weir performed better than the 90° one regardless of orientation (Hosseini Tashnizi, Heidarpour, & Eslamian, 2013).Further, at higher flow rates, flow perturbation was much more intense in the model with a smaller apex angle.Therefore, these vanes can direct the flow perpendicular to the weir, normalizing the flow over the weir and improving its performance.Moreover, at small hydraulic loads, the nappe passing the weir sticks to the downstream side of the wall.Despite the high efficiency of the labyrinth weir, in such circumstances, it can create oscillations under the nappe, which was the reason for the shaking of houses neighboring Avon dam, Australia.The general solution for aeration is to install flow splitter piers near the downstream apex or pouring aggregates downstream of the crest to make a porous line (Falvey, 2003).Flow splitters separate the nappe from the weir, allowing the stream to aerate.The best place for the splitters to be installed is the downstream apex of the labyrinth weir (Crookston, 2010).It must be noted that aeration from underneath the nappe makes the flow splitter vane remarkably effective in enhancing the hydraulic efficiency of the weir and increasing its discharge coefficient.
In an experimental study, Crookston and Tullis investigated nappe interference and local submergence in labyrinth weirs with two and four triangular cycles and different apex angles (Crookston & Tullis, 2012).The discharge coefficient was shown to be higher than that in a linear weir at a low flow rate thanks to little nappe interference, which gradually increases with discharge.This outcome reduces the discharge coefficient and brings it close to the level of broad-crested weirs.By investigating the discharge relations for the nonlinear weir, the authors concluded that the half-cycle crest is more effective than the quarter-cycle one for H t /P < 0.4.Belzner et al. studied piano-key and labyrinth weirs under free-flow and submerged flow conditions in German waterways.This study relied on physical models of rectangular, triangular, and trapezoidal labyrinth weirs, as well as piano-key weirs.The results showed that trapezoidal and rectangular labyrinth weirs are more susceptible to submergence than the piano-key and triangular labyrinth weirs, but the latter offers a lower hydraulic efficiency (Belzner, Merkel, Gebhardt, & Thorenz, 2017).Accordingly, Gebhardt et al. compared side weirs and labyrinth weirs built on the Ilmenau.They investigated an empirical stage-discharge relation for side weirs and trapezoidal, rectangular, and triangular labyrinth weirs under free and submerged flow conditions.Based on the findings of this study, a labyrinth weir discharged more fluid compared to a side weir for smaller hydraulic heads.On the other hand, the upstream head of a labyrinth weir is higher than that of a side weir during flooding.Furthermore, the side weir offers better performance with a submerged flow (Gebhardt, Merkel, Belzner, & Thorenz, 2017).
Using the computational fluid dynamic (CFD) method, Cihan Aydin and Emin Emiroglu (2016) examined the hydrodynamic performance of a side weir in two cycles.Furthermore, grid convergence index (GCI) was used in their research to evaluate the effect on the results of grid refinement.There found to be a good agreement between the empirical and numerical results of their study.The results obtained showed the discharge coefficient to decrease with increase in the Froude number.The best performance was found to be corresponding to a sidewall angle and weir height of 30 degree and 20 cm respectively.
Omer Bilhan, Cihan Aydin, Emiroglu, and Carol (2018) examined circular labyrinth weirs using numerical and experimental methods.The numerical model used CFD analysis with single-phase and two-phase turbulent flow regimes.The results of the numerical models were found to be close to those of the experimental findings, with a difference being 4% of the latter on the average.The distribution of the discharge coefficient for a circular labyrinth weir with Ht = P is similar to that of a trapezoidal labyrinth weir of a large sidewall angle (35 degree for instance) observed by Crookston BM, Tullis, Young, and Chandler (2007).Besides, the use of nappe breaker caused a slight decrease in the discharge performance of the weir, but this effect was deemed negligible.Norouzi, Daneshfaraz, and Ghaderi (2019) studied the discharge coefficient (Cd) of a trapezoidal labyrinth weir by using an artificial neural network and vector machines.They found the MLP model to be more acceptable and closer to experimental results.
Ghaderi, Daneshfaraz, Dasineh, and Francesco (2020) investigated energy dissipation and hydraulics of flow on triangular trapezoidal labyrinth weir (TTLW).According to their results, the energy dissipation of TTLWs was higher than that of the vertical drops.
Ghaderi, Daneshfaraz, Abbasi, and Abraham (2020) investigated the hydraulic features of the modified labyrinth weirs numerically.According to the results, modifying the geometry of labyrinth weirs in low sections H T /P (H T /P < 0.2) results in an increased discharge coefficient.
In light of previous studies, it was necessary to investigate the effects of installing flow splitters on the crest of labyrinth weirs on the discharge coefficient.On the other hand, this study aims to experimentally investigate the coefficient of discharge in one-cycle triangular and trapezoidal labyrinth weirs by placing flow splitters on the crest and determine the best configuration in terms of discharge coefficient for both triangular and trapezoidal labyrinth weirs, comparing the numerical (Flow-3D) and experimental model results and determining the level of difference.

Materials and methods
This study was carried out in the hydraulics laboratory of Khuzestan Water and Power Authority, Iran, on a rectangular flume that measured 7 m long, 60 cm wide, and 60 cm tall and had a maximum flow rate of 55 l.s −1 (Figure 1).
The flow rate was measured by a magnetic flow meter with an accuracy of ±2 l.s −1 at the inlet.Figure 2 illustrates the flume settings.
Following the recommendation of the Bos'research, the hydraulic depth was measured upstream of the weir at a distance three to four times the hydraulic depth over the weir (50 cm upstream of the weir) using a mechanical depth gauge with an accuracy of 0.5 mm (Bos, 1989).Compound labyrinth weirs (Figure 3) were designed and made of plexiglass as recommended by the research (Tullis, Young, & Chandler, 2007).
Table 1 lists the geometric specifications of the compound labyrinth weirs, and Figure 4 illustrates the plan and profile of the labyrinth weir with its geometric specifications.Figure 5 showes threedimensional view of the compound labyrinth weir with splitter vanes.
The experiments were carried out under free-flow conditions.The flow rate was increased in increments, creating different hydraulic load ratios on the weir.In every experiment, the flow rate and water depth were measured after reaching steady-state conditions.Researchers often use a hydraulic conductivity equation and rely on experimental results from physical models to determine the discharge coefficient.In the case of this study, C d depends on the parameters expressed by Equation (1):   Wherein, Q represents discharge, Ht stands for the total hydraulic head, g represents the acceleration of gravity, Nv denotes the number of flow splitters, αv stands for the angle of the flow splitters, N represents the number of weir cycles, B denotes the weir length, Lt denotes the effective weir length, P represents the weir height, W stands for the weir width, ym denotes the upstream depth of the weir, σ represents the surface tension of the weir, ρ denotes the density of water, µ stands for the dynamic viscosity, and S0 represents the slope of the test flume.The Buckingham π method was employed for a dimensional analysis to determine dimensionless parameters of C d (Equation ( 2)).Furthermore, in relation 2, parameters P, Q and ρ were considered to be repeated parameters, with the rest of the parameters being the variable parameters.It is obtained by eliminating the fixed parameter and converting some dimensionless parameters to new parameters.Therefore, Cd can be represented as a dimensionless number.The Reynolds number exceeding 5000 is indicative of a turbulent flow, which means viscosity effects can be ignored.Moreover, surface tension has considerable effects at small depths, but the three-dimensional nature of the flow and its overmixture in present experiments rendered these effects negligible.The flume width and bed inclination were kept constant in all experiments.Therefore, the Froude number (Fr) was directly affected by the flow rate and depth.In the end, C d was obtained as a function of parameters expressed by Equation (3): Where the discharge coefficient depends on the hydraulic load ratio of the weir discharge, the geometry of the compound weir crest, vane orientation on the sidewalls, and the number of guide vanes.Previous studies have shown this function to be fitting as it considers all effective parameters.Table 2 presents the experiments plan in this study for the one-cycle compound labyrinth weir.Velocity and pressure terms were coupled In Table 2, Q denotes the discharge capacity, α V is the angle of Flow Splitters, n v is the number of vanes, and N shows the number of weir cycles.Given the significance of the flow hydraulics, weirs were installed at 3.5 m from the beginning of the flume and were sealed and leveled to prevent errors in hydraulic load measurement.After starting the pump, the flow control valve was opened slowly, increasing the flow rate in the flume.The required flow conditions were adjusted, and the water depth was measured at three points, namely at the upstream, over the crest, and at the downstream after reaching a steady state.This study was carried out on two types of weirs: triangular and trapezoidal.Each type of the weirs were modeled in the software Flow3D, with nine different arrangements of flow splitters and under ten different flow rates.The best-performing arrangement of the splitters was found for each type of the weir, and the results of the simulation were compared to those obtained from experimenting with physical models.

Numerical model flow 3D
Numerical representations of these models were created using Flow-3D.This software tool has extensive applications in analyzing complex fluid dynamics problems, including transient three-dimensional  flow with free surface and complex geometry.The RNG model was used in the present study to derive time-averaged Reynold's equations.Numerical solutions were produced by Flow-3D, and the transient governing equations were solved by the finitevolume method.The software program used the Fractional Area-Volume Obstacle Representation (FAVOR) algorithm to define the geometry by the finite-volume method.The free surface of the flow was determined by the Volume of Fluid (VOF) algorithm implicitly in continuity and momentum equations using pressure and velocity data from the past.Accordingly, 3D models of the physical models were developed using AutoCad, importing the results into Flow-3D for meshing by VOF and FAVOR and determining the boundaries and the computational grid.After introducing the geometry data to the software environment and determining the boundaries of the main and secondary channels, the area of interest was meshed by VOF and FAVOR.The grid size was optimized by a trade-off between accuracy requirements and computation time, and the field grid was created with orthogonal lines.Here, 2,065,012 cells were used in three 2.5 × 2.5 × 2.5 mm 3 mesh blocks to mesh the models for computations.
After the computational grid was ready, the boundary and initial conditions were applied, simulating the water flow (Figure 6).Twenty three-dimensional simulations were performed in this study.After the weir geometry was created and introduced to Flow-3D, it was meshed, and the boundary and input conditions were applied.Table 3 lists the simulations carried out.

Results and discussion
These experiments aimed to look into the effects of the count and orientation of guide vanes installed on triangular and trapezoidal compound weirs.Accordingly, the experimental results were analyzed in two parts: (1) Effects of H t /P on the coefficient of discharge (C d ) (2) Effects of Fr on the coefficient of discharge (C d )

Effects of H t /P on C d
For a more accurate evaluation of the results, the corresponding graphs are investigated in two parts.
Part 1: Here the effects of the number and orientation of vanes on the coefficient of discharge are discussed.Figures 7 and 8 show the results.
As evident from Figure 7 (trapezoidal weir), the best case corresponds to the 60° configuration with two vanes, as it produced the highest coefficient of discharge (C d = 0.79), making the weir 10% more efficient.On the other hand, installing the splitter at 45° reduced the weir performance by 11%.
At angles, the vanes improve the weir performance, but their number is also critical.For example, installing two vanes at 60 and 90° was found to improve the efficiency, whereas installing six vanes at three angles compromised it.
Figure 8 shows that splitter vanes compromised the weir efficiency, as even in the best case, corresponding to the configuration with two vanes installed at 45°, the efficiency was reduced by 7%.The worst impact corresponded to the case with six vanes installed at 90°, where the efficiency was reduced by up to 17%.
Figures 8 and 9 show that the triangular weir offers up to 4% higher efficiency than the trapezoidal weir.
Part 2: Figure 10 illustrates the effects of orientation on the coefficient of discharge.Figure 9 indicates a 5-9% performance edge for the trapezoidal weir over the triangular weir at 60 and 90° vane installation angles.However, the triangular weir offered better performance at 45°.Overall, the vanes can be said to improve the performance of trapezoidal weirs but compromise triangular weirs.Nonetheless, the same vanes exhibited a higher efficiency in triangular weirs compared to trapezoidal ones.In conclusion, vanes are not suitable for all weirs.
The best configurations are compared in Figure 10 in terms of geometry and orientation.
Figure 10 indicates that the best results correspond to two vanes installed at 60°.

In this section, the results are discussed in two parts, for orientation and geometry
Figure 11 shows that the discharge coefficient can be increased by raising the flow velocity up to Fr = 0.5, beyond which increasing the velocity compromises the coefficient of discharge.The best discharge efficiency was achieved at around 0.5 (subcritical).Moreover, configurations with two vanes were found to offer the best performance, and trapezoidal weirs were the most efficient.The graphs investigating the effects of Fr on C d are suggestive of the positive effects of velocity at a low flow rate.However, at higher flow rates, increasing the velocity would reduce the discharge over the weir.

Comparing numerical and physical results
The best vane performance was determined for each weir type by experimentation and simulated using Flow-3D, comparing the software outputs with the physical model results (Figure 12).
According to the comparison of the numerical and physical models in Figure 12, the numerical simulation's estimation of the discharge coefficient is 10.2% higher than the physical model.Regardless, both graphs have similar trends.

Comparison of the results with previous studies
Table 4 presents the details of previous studies in comparison with this one (trapezoidal weir with two vanes installed at 60°).

Conclusion
The results of this study on the effects of splitter vanes installed on two types of labyrinth weirs (trapezoidal and triangular) are as follows.
For the trapezoidal weir, the best case corresponded to the 60° configuration with two vanes, as it produced the highest coefficient of discharge (C d = 0.79), making the weir 10% more efficient.On the other hand, installing the splitter at 45° reduced the weir performance by 11%.At certain angles, the vanes improve the weir performance, but their number is also critical.For example, installing two vanes at 60 and 90° was found to improve the efficiency, whereas installing six vanes at three angles compromised it.Splitter vanes compromised the efficiency of the triangular weir, as even in the best case, corresponding to the configuration with two vanes installed at 45°, the efficiency was reduced by 7%.The worst impact corresponded to the case with 6 vanes installed at 90°, where the efficiency was reduced by up to 17%.The triangular weir was up to 4% more efficient than the trapezoidal one.Trapezoidal weirs had a 5-9% performance edge over triangular weirs at 60 and 90° vane installation angles.However, the triangular weir offered better performance at 45°.The best effects were achieved with two vanes installed on a trapezoidal weir at 60°.
The discharge coefficient can be increased by raising the flow velocity up to Fr = 0.5, beyond which increasing the velocity compromises the coefficient of discharge.The best discharge efficiency was achieved at around 0.5 (subcritical).Moreover, configurations with two vanes were found to offer the best performance, and trapezoidal weirs were the most efficient.The graphs investigating the effects of Fr on C d were suggestive of the positive effects of velocity at low flow rates.However, at higher flow rates, increasing the velocity would reduce the discharge over the weir.
A graph comparison between numerical and physical model results showed the numerical simulation to have estimated the discharge coefficient 10.2% higher than the physical model, but both graphs exhibited a similar trend.
In summary, results of this study showed that the introduction of vanes could lead to an increase of up to 17% in the performance of weirs.

Figure 5 .
Figure 5. Three-dimensional view of the compound labyrinth weir with splitter vanes.

Figure 9 .Figure 10 .
Figure 9. Effects of H t /P on C d at three installation angles.

Figure 11 .
Figure 11.Effects if Fr on C d .

Figure 12 .
Figure 12.Comparing the numerical and physical model results.

Table 1 .
Geometric specifications of compound labyrinth weirs in this study.

Table 2 .
Experiments plan for the one-cycle compound labyrinth weir.
Figure 6.Models run in flow-3D.Figure 7. Effects of Ht/P on Cd in trapezoidal weirs.
Figure 8. Effects of H t /P on C d in triangular weirs.

Table 4 .
Results of the present study in comparison with previous studies.Experimental specifications in previous studies.