Investigating a new approach to enhance the discharge capacity of labyrinth weirs

The following symbols are utilised in the present study:

A

Inside apex width

An

Notch area

Aw

Sidewall area of the weir

b₁

Bottom width of the notch

B

Labyrinth weir length in the flow direction

C_dc

Compound coefficient of discharge

D

Outside apex width

έ

Cycle efficiency

g

Acceleration of gravity

h

Flow depth over the weir crest (high stage)

h′

Flow depth over the notch (low stage)

H_t

Total head over the weir crest (high stage)

H_t′

Total head over the notch (low stage)

H_t′/P′

Headwater ratio

Lc

Total centreline length of labyrinth weir

lc

Centreline length of weir side wall

Lc-cycle

Centreline length for a single labyrinth weir cycle

Length of the labyrinth weir crest after subtracting the length of notches (ΔL), (⁠⁠)

l′

Length of the notch

N

Number of labyrinth weir cycles

n

Number of notches

P

Weir height

P′

Notch height

Q

Discharge over the weir

R_crest

Radius of crest shape

t_w

Thickness of weir wall

V

Average cross-sectional flow velocity upstream of weir

W

Channel width

w

A single labyrinth weir cycle width

ΔL

The top width for the notch

ΔP

The notch depth

α

Sidewall angle

Climate change and associated extreme weather events are resulting in flood events that occur at higher frequency and magnitude (Fowler & Kilsby 2003; Bruwier et al. 2015; Kvočka et al. 2016). Climate change caused a change in the hydrological information that was relied upon in the existing weir design. However, the return period of the flood is often updated. This situation could lead to increased water depth upstream of a weir. When this occurs, it leads to an increased risk of failure in the structure that might be linked to the high level of water over the weir crest, which is more than the allowable value of the design water head. Therefore, it has become impossible to discharge flood waves regularly and safely as a result of climate change. Therefore, old weirs need to be upgraded to labyrinth weirs because labyrinth weirs play an important role in ensuring the security of human life as well as safety in developing areas and the natural environment.

However, labyrinth weirs are overflow structures folded in plan-view and consist of a series of linear weirs installed non-perpendicularly to the sidewall of a channel, resulting in a longer crest length compared with the channel width. For sites where the weir width is restricted but a high discharge is demanded, the labyrinth weirs are an effective and economical choice (Khode et al. 2011), since they can provide a higher discharge capacity for a given water head. Also, labyrinth weirs have become popular as an alternative rehabilitation measure in the upgrading of existing weirs. For example, in areas where rainfall has resulted in peak flows exceeding the capacity of existing weir systems, the labyrinth weir can provide a practical means of rehabilitation (Vasquez et al. 2007). Labyrinth weirs are useful not only for handling larger flow rates but are also used for energy dissipation and self-aeration.

The hydraulic behaviour of labyrinth weirs has received interest from researchers and engineers. Early studies were conducted by Taylor (1968) that provided information on the hydraulic performance and design of the labyrinth weir. Hay & Taylor (1970) developed design criteria based on Taylor's earlier work. Darvas (1971) developed a set of curves to estimate labyrinth spillway performance. Houston (1982) extensively studied a physical model to assess different labyrinth geometries and approach conditions. Lux (1989) mentioned that the aeration condition behind the nappe flow of the labyrinth weir was identified as aerated, transitional, and suppressed. Tullis et al. (1995) investigated labyrinth weir performance and obtained design curves relating the coefficient of discharge, upstream head, and the height of the weir for various sidewall angles from 6 to 18°. Savage et al. (2004) identified errors of up to ±25% in the proposed method by Tullis et al. (1995). Falvey (2003) showed crest flow conditions consist of four conditions of nappe aeration, including pressure, atmospheric, cavity, and sub-atmospheric. Ouamane & Lempérière (2006) studied the rectangular planform of the labyrinth weir and found the rectangular plan shape to be more efficient than the trapezoidal plan shape. Tullis et al. (2007) conducted physical model studies and derived relationships of head–discharge for labyrinth weirs. Ghare et al. (2008) found that discharge capacity was reduced when the crest length of the labyrinth weir was decreased. A sidewall angle has a significant effect on both the discharge capacity and the labyrinth weir layout. Effective length is increased with a smaller angle, which thereby provides a higher discharge capacity.

Also, Crookston & Tullis (2010) included experimental outcomes for 8° sidewall angles. Khode et al. (2011) showed the relationship between coefficients of discharge and H/P for various sidewall angles. They found that the discharge coefficient decreased as the sidewall angle decreased. Paxson et al. (2011) showed that flow conditions downstream of the labyrinth weir could also influence discharge capacity. Khode et al. (2012) developed the relationship between the coefficient of discharge and H/P for a variety of sidewall angles from 8 to 30°. Crookston et al. (2012) developed different models with a sidewall angle α = 15° and used trapezoidal labyrinth weirs. A significant decrease in cycle efficiency with an increase in the Ac/lc ratio occurred. The apex had the lowest impact on total discharge efficiency, as it had a larger impact on the total cycle length and caused a decrease in sidewall length, which thereby caused a reduction in total flow efficiency. Carollo et al. (2012) studied five groups of physical models and adapted the empirical equation suggested by Ghodsian (2009). Dimensional analysis was used to investigate the discharge over the sharp crest of the labyrinth weir. Kumar et al. (2012) investigated discharge capacity using the sharp crest and curving plan shape weir. The coefficients of discharge equations were proposed and the results showed an increase of approximately 40% of the flow capacity over a curved weir compared with a conventional weir. Gupta et al. (2013) studied the characteristics of the discharge under free-flow conditions for a sharp-crested, contracted triangular planform weir. The discharge efficiency across the triangular planform weir was greater than across the normal weir. The equation of flow was suggested for the given range of data and was within ±5% of the observed data.

Moreover, Gupta et al. (2014) investigated W-planform weirs to determine discharge characteristics. The efficiency of the discharge for W-planform labyrinth weirs was greater than that of the normal weir. The equation of flow was suggested for the given range of data and was within ±5% of the observed data. Gupta et al. (2015) used 24 sharp crests to create a labyrinth weir. A rectangular planform weir was utilised to study the impact of the height of the crest on the performance of discharge with different crest lengths. The discharge efficiency of the rectangular labyrinth weir was shown to be higher than that of the conventional weir. Savage et al. (2016) showed the design curves that were found by Crookston & Tullis (2013) are suitable for headwater ratios (H_t/P) > 2.0. Also, Gebhardt et al. (2017) solved navigation requirement problems in the Ilmenau waterway in northern Germany by using a side labyrinth weir. A relationship between the coefficient of discharge and head over various weir heights was obtained. The labyrinth weir is more reliable to operate and safer. Ghaderi et al. (2020a) conducted a numerical and experimental study to investigate the effects of the geometry parameters of trapezoidal–triangular labyrinth weirs (TTLW) on the energy dissipation, discharge coefficient, and downstream flow regime. The results demonstrated that the experimental results agree with the numerical model. Energy dissipation is reduced with an increase in sidewall angle. Daneshfaraz et al. (2020) used the FLOW-3D model to study the effect of channel-bed slope and different channels on the coefficient of discharge of a labyrinth weir. The findings showed that modifying the channel improved the coefficient of discharge. Ghaderi et al. (2020b) used FLOW-3D software to simulate the hydraulic performance of labyrinth weirs. They used the inclining crest edge of the weir and notches in the weir wall. The outcomes showed that modifying the labyrinth weir geometry improved discharge capacity over the labyrinth weir compared with a conventional weir.

Despite the availability of many previous studies about labyrinth weirs’ geometry, evaluation of the performance of the labyrinth weir is still a primary concern and a serious challenge for designers of this type of weir due to major concerns about flood control. This concern has emerged because of climate change and the development of new methods to estimate extreme floods. These methods have demonstrated the insufficiency of a significant number of weirs in handling increasing numbers of extreme flooding events. Consequently, solutions for increasing the capacity of existing weirs and providing more efficient weirs for new projects are being sought. However, due to a lack of knowledge and exemplary ideas, labyrinth weirs need to be investigated further to improve their performance. However, in the present study, a new shape of the labyrinth weir has been proposed. This shape is developed by modifying a labyrinth weir structure (named a compound labyrinth weir). These modifications are made by using trapezoidal notches on the sidewalls of the labyrinth weir. This work provides new knowledge about the new shape of the labyrinth weir. The results of the present study are important to engineers and designers, especially given the current climate change impacts evidenced by possibly higher storm intensities and peak discharges, and where weir rehabilitation is becoming an issue of concern.

The experiments were carried out in a rectangular flume of length 7 m, width 0.5 m, and height 0.6 m. The flume walls were made from acrylic panels supported by a steel frame. The flume bed was set horizontally. The water was supplied from a storage tank with a volume of 2.5 m³ and recirculated via a 200 mm supply pipe. Two pumps are connected in parallel using a water supply line. The capacity of each pump was 50 L/s giving a total capacity of 100 L/s for both. The flow metre was installed in the main pipeline. The flow metre diameter is 150 mm with an accuracy of 0.5% of the discharge rate. The discharge rates ranged from 10 to 100 L∕s with an accuracy of ± 0.05 to ± 0.5 L/s. A gate valve was utilised in the pipeline to control the flow rate. The flume also contains one regulating gate downstream to control the tailwater elevation. The wave suppressors were provided at upstream of the flume. The wave suppressors were used to control the flow and dissipate the surface disturbances. The water level was measured upstream of the labyrinth weir using a movable pointer gauge with an accuracy of 0.001 m mounted on the flume side rails (allowing longitudinal and transverse movement). As recommended by Dabling (2014), the point gauge was located at a distance of 4 P times the weir height. In the present study, approximately 80 cm was measured from the inlet of the flume to the upstream of the physical models to avoid a water drawdown at the weir edge. Water level measurements varied from 5 to 100 mm for the range of flows tested. For each tested weir geometry, 25 readings of head–discharge were carried out. For each run, the water head over the crest of the labyrinth weir was measured about 2–3 times by a point gauge and the average of the readings was recorded to ensure more accuracy.

In the present study, nine models were used to investigate the optimal notch geometry of the compound labyrinth weir. Table 1 shows the physical model test for the optimal geometry of the trapezoidal notch and Figure 1(a) shows the three-dimensional geometry of the compound labyrinth weir. In addition, eight models are used to study the hydraulic characteristics of compound labyrinth weirs, as shown in Table 2. These models consisted of rang sidewall angles (α) configurations of 6, 8, 10, 12, 15, 20, 35, and 90° (linear weir for comparison). The quarter-round crest experiments were carried out with the curved edge facing upstream, as shown in Figure 1(b). These models were two cycles (N = 2) of compound labyrinth weirs and the total width (W) was 0.5 m. All models were fabricated from acrylic sheets with a thickness (t_w) of 10 mm. The acrylic sheets were cut using a laser machine to obtain precise dimensions and then the parts were assembled using screws. Silicon was used to prevent the leakage of water through all the joints of the models. The walls of the model were fixed on the base of the acrylic with a thickness of 6 mm to assist with installing the weir inside the flume. To avoid the curvature effect, all models were located approximately 1.5 m from the inlet point of the flume. An is the notch area that is defined as An= ΔP (ΔL+b₁)/2. Aw is the sidewall area of the weir that is defined as Aw=lc P.

The current research is based mainly on laboratory experiments to study the effect of the sidewall angle, which contains different notch geometries for each angle. The trapezoidal shape is considered a suitable choice for the notch shape because the trapezoidal shape is best for hydraulic sections. In the present study, to maintain the notches’ shape as a trapezoid, the geometry of the notches was adopted as 0.1 ≤ ΔP/P ≤ 0.3 and 4% ≤ An/Aw ≤ 8%. In other words, if the notches scale is outside this range, it will change to a triangular shape and thus it will be outside the limitations of the present study. Also, the height of the notch from the bottom of the channel to the notch crest should be at least two times the maximum estimated head of water level above the notch crest. This is essential to reducing the velocity of approach (Walkowiak 2006).

Water level data was collected with respect to the weir crest elevation with 0.05 < H_t′/P′ < 0.80 (for the range of flow rate (Q) tested) by measuring the water level over the low stage (h′) and high stage (h) weirs. The measurements were taken only after the flow conditions had reached approximately steady conditions (Crookston 2010), where Q, h, and h′ were measured and recorded. The total heads, H_t and H_t′ were computed by adding the velocity head (V²/2g) to h and h′. The velocity of flow was calculated by dividing the actual discharge value, which was measured by a flow metre, by the cross-sectional area of flow based on the width of the weir and flow height. All the measurements were taken at the centre of the width of the flume.

In the present study, a new approach has been proposed for the geometry of the labyrinth weir. The new proposal focused on making notches on the sidewalls of the labyrinth weir. The benefit of using notches was to decrease the water level upstream of the weir during peak flood events. In addition, this type of labyrinth weir can provide sufficient flow capacity to deliver greater storm events. The low stage weir is designed to pass moderate-flow events, while the higher stage weir is designed to add sufficient capacity to deliver the more extreme weir design flood. Therefore, it was important to determine the optimal geometry of the notches and the geometry effect on the C_dc under free-flow conditions.

According to Figure 3, the impact of notch depth on the C_dc has been observed when the ratio of dropped height increases (ΔP/P from 0.1 to 0.3). C_dc values differed significantly at low H_t′/P′, but C_dc values converged as H_t′/P′ values increased. This difference affects the notch depth in low discharge, but this effect reduces at high discharge. When H_t′/P′ > 0.5, the flow over the compound labyrinth weir becomes submerged; therefore, the effect of notch depth is neglected. When An/Aw = 4% (Figure 3(a)), the C_dc for ΔP/P = 0.3 was about 3 and 7% greater than the C_dc values for ΔP/P = 0.2 and 0.1, respectively. While An/Aw = 6% (Figure 3(b)), the C_dc value for ΔP/P = 0.3 was 4.5 and 8% greater than the C_dc values for ΔP/P = 0.2 and 0.1, respectively. Furthermore, when An/Aw = 8% (Figure 3(c)), the C_dc value for ΔP/P = 0.3 was 5 and 10% higher than the C_dc values for ΔP/P = 0.2 and 0.1, respectively.

Moreover, Figure 4 shows that the difference in C_dc values was significant at low H_t′/P′ and that the C_dc values converged when H_t′/P′ values increased. This difference is attributed to the effect of notch length in low discharge, but this effect decreases with high discharge. When H_t′/P′ > 0.5, the flow over the labyrinth weir may become submerged and, therefore, the effect of the notch length is neglected. When ΔP/P = 0.1 (Figure 4(a)), the average C_dc for (ΔL/Lc = 0.82, An/Aw = 8%) was 4.5 and 8% higher than the C_dc values for (ΔL/Lc = 0.62, An/Aw = 6%) and (ΔL/Lc = 0.42, An/Aw= 4%), respectively. While ΔP/P = 0.2 (Figure 4(b)), the average C_dc for (ΔL/Lc = 0.45, An/Aw = 8%) was about 5 and 9% greater than the C_dc values for (ΔL/Lc = 0.35, An/Aw = 6%) and (ΔL/Lc = 0.25, An/Aw = 4%), respectively. Also, in case ΔP/P = 0.3 (Figure 4(c)), the average C_dc value for (ΔL/Lc = 0.34, An/Aw = 8%) was about 4 and 11% greater than the C_dc values for (ΔL/Lc = 0.27, An/Aw = 6%) and (ΔL/Lc = 0.21, An/Aw = 4%), respectively.

Using a trapezoidal notch at the sidewall of a labyrinth weir has a good effect on the C_dc when using various An/Aw percentages (4, 6, and 8%) and different values of ΔP/P (0.1, 0.2, and 0.3). From the results, the optimum geometry of the notch to yield the optimum C_dc was noted. For the present study, the An/Aw does not exceed 8%, ΔL/Lc does not exceed 0.32, and ΔP/P does not exceed 0.3 when the other parameters are constant. These parameters have also been given a maximum C_dc of 0.74.

Figure 5 shows that C_dc initially behaves similarly for all models when H_t′/P′ are low and the flow only passes through the notch. When H_t′/P′ values are low, the C_dc begins to rise because the nappe flow is still non-aerated with little to no interference nappe, as shown in Figure 6(a). At a low value of H′_t/P′, the flow touches the sidewalls of the compound labyrinth wear. This case is analogous to the linear weir because losses at the weir crest still dominate. Also, obviously, nappe interference is absent. When the flow passes over the entire labyrinth weir, C_dc reaches a maximum value because the flow is compound. The discharge over the notches is aerated and the nappe flow becomes more effective. At the same time, the discharge over the entire weir is non-aerated, and nappe flow does not interfere. Figure 5 also shows that C_dc reaches a maximum value before decreasing and tending towards an asymptotic value when the flow is large values of H_t′/P′. This is different from linear weirs, where C_dc reaches a maximum and approaches C_dc asymptotically after a slight decrease. When H_t′/P′ is large, increasing H_t′/P′ has a marginal effect on C_dc. When the weir is fully submerged, interference effects between the discharges from the individual weir elements become insignificant. The clinging condition finished at H′_t/P′∼0.07 for α = 6, 8, 12, and 20°, whereas it ended at H′_t/P′ ∼0.06 for α = 10 and 15°. In comparison to linear weirs, the results showed that the C_dc decreases significantly after its peak. A significant decrease is because labyrinth weirs are subject to the nappe interference of the falling jets (Crookston & Tullis 2011), which is absent in linear weirs. In addition to the sudden removal of the air cavities behind the nappe, nappe interference becomes less significant as α increases; therefore, C_dc increases as α increases for a given upstream head.

The results showed that when increasing H_t′ over a compound labyrinth weir, the nappe flow condition changes from clinging aerated to partially aerated to finally drowned. However, all aeration conditions do not certainly take place for all geometries of the compound labyrinth weir. The nappe aeration condition is one of the factors that influence the efficiency of flow over a compound labyrinth weir. Aeration conditions also characterise nappe behaviour. The nappe flow may initially be relatively tranquil and then develop into an unstable condition because of the fluctuating pressure produced on the weir wall. This pressure causes nappe vibrations and noise. In general, the compound labyrinth weir is more efficient when the nappe is clinging (Figure 8(a)) than when it is aerated (Figure 8(b)). This difference is because sub-atmospheric pressures develop on the downstream face of the weir. A partially aerated nappe (Figure 8(c)) occurs at higher H_t′/P′ values. In this case, the air cavities behind the nappe are unstable and vary both temporally and spatially. The air cavities fluctuate between labyrinth weir apexes. Also, the air cavities may be completely removed when the nappe flow condition is submerged. This process reappears as the unsteady flow and turbulent levels behind the nappe fluctuate. The air cavities are highly dynamic and cause oscillating pressures on the downstream face of the weir.

The results showed that nappe trajectories are unstable for the partially aerated nappe condition based on flow conditions and weir geometry. For a stable nappe, the partially aerated condition had less effect on the nappe trajectory. In addition, increasing H_t′/P′ causes the nappe to transition from partially aerated to drowned (Figure 8(d)). The drowned nappe appears as a thick nappe without an air cavity.

The clinging condition finished at H_t′/P′ ∼ 0.07 for α = 6, 8, 12, and 20° and at H_t′/P′ ∼ 0.06 for α = 10°, 15°. The compound labyrinth weir with α = 35° was observed to change directly to an aerated or partially aerated nappe condition. The nappe aeration condition was observed when 0.12 < H_t′/P′ < 0.22 for α = 35°. Depending on α, the largest range of aeration nappe was observed for α = 6–20°. The nappe condition changes from aerated to partially aerated at 0.23 ≤ H_t′/P′ ≤ 0.43 for α = 6°. Depending on α, the largest range of partially aerated nappe was observed for α = 8–35°. The nappe drowned condition was observed when H_t′/P′ > 0.43 for α = 6°. Also, when α increased, the nappe drowned condition increased. The range of H_t′/P′ observed for each nappe aeration condition of compound labyrinth weirs is shown in Table 5. This table will assist the designer to avoid using undesirable ranges of H_t′/P′ in the design process.

The purpose of the present study is to improve the discharge capacity of the labyrinth weir. The labyrinth weir reflects an effective method for increasing storage capacity. The results have shown that dimensionless terms of notches area ratio (An/Aw), notches length ratio (ΔL/Lc), and notches depth ratio (ΔP/P) do not exceed 8, 32, and 3%, respectively, when setting the other parameters as constants. These dimensionless terms have been provided with a maximum C_dc of 0.74. For small values of H_t′/P′, C_dc increases with increased H_t′/P′, while C_dc reduces with increased H_t′/P′ and decreases towards a constant value as a function of α when H_t′/P′ is large. This reduction in C_dc is observed in the case of a compound labyrinth weir with a range of sidewall angles (α) (from 6 to 35°) but not in the linear weir. The results also showed that the highest values of the C_dc were with a sidewall angle of 35° and the lowest value of the C_dc was with a sidewall angle of 6° for all examined models. The outcomes demonstrated relationships between C_dc values as a function of H_t′/P′. Empirical equations relating to C_dc and H_t′/P′ were created, and the data have a good correlation representation (R² ≥ 0.98). Empirical equations have been used to predict C_dc values. The set of curves was well-behaved up to H_t′/P′ ≤ 2.0.

Furthermore, the efficiency (ε) is larger when the sidewall angle (α) is small. For each α angle, the efficiency (ε) decreases with increasing H_t′/P′. The compound labyrinth weir has higher efficiency when compared with conventional labyrinth weirs, the percentage of the improvement in efficiency was 10, 8.9, 7.8, 7.2, 6.6, 5, and 4% for α = 6, 8, 10, 12, 15, 20, and 35°, respectively. Moreover, the results demonstrated that nappe flow conditions pass through four phases for ranges of H_t′/P′. These phases were clinging, aerated, partially aerated, and drowned. In addition, the discharge capacity of the compound labyrinth weir was more efficient with a clinging nappe condition than an aerated, partially aerated, or drowned nappe for all tested models. The maximum value of the improved percentage in the C_dc value, when comparing the present results with a traditional labyrinth weir that was done by Willmore (2004), was as large as 20.5% for α = 6°, while the minimum value of the improved percentage was as large as 17.3% for α = 8°. While comparing the present study with Crookston & Tullis (2013), the maximum value of the improved percentage was as large as 18.8% for α = 8°, and the minimum value of the improved percentage was as large as 15.3% for α = 35°.

Although the methods and information for compound labyrinth weirs have been obtained in the present study, we recommend that a compound labyrinth weir design be verified with numerical and/or physical model studies. This information should be taken into account for site-specific conditions that may be outside the limitations of the present study. Also, the results of the present study will give valuable insights into the operation and performance of the compound labyrinth weir.

The authors would like to express their sincere thanks and gratitude to the government and the Ministry of Higher Education and Scientific Research in Iraq for providing financial support for this study. They also express their sincere thanks to the school of engineering at Deakin University for the use of the new test facility. They appreciate the technical support that is provided by laboratory staff at the School of Engineering (Deakin University).

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

The authors declare there is no conflict.