Hydrodynamics and Free-Flow Characteristics of Piano Key Weirs with Different Plan Shapes

: This paper focuses on Piano Key Weirs (PKWs) as an effective solution for improving the discharge capacity of spillway systems. The ﬂow behavior in inlet and outlet keys is experimentally studied to analyze the discharge capacity of PKWs with different plan shapes (i.e., rectangular, trapezoidal, and triangular). The results show that in outlet keys, the ﬂow aeration regimes extend to higher values of headwater ratios ( H o / P ) by increasing the length magniﬁcation ratio ( B / w ) and apex width ratio ( A / w ). In addition, the local submergence length is a decreasing function of A / w , especially in high ﬂow heads. While the total interference length enlarges by reducing A / w in lower H o / P values ( H o / P < 0.5), a reverse trend is observed in higher headwater ratios. PKW performance may also be impacted by the ﬂow contraction and recirculation zone in inlet keys, which intensify in higher values of H o / P , B / w , and A / w . According to the obtained results, while the discharge coefﬁcient is a decreasing function of A / w in H o / P > 0.4, it may have a reverse trend in lower head conditions. In addition, a trapezoidal PKW has the highest discharge efﬁciency in a wide range of the studied domain ( H o / P > 0.25 and B / w ≥ 2). It can improve the discharge efﬁciency by around 5%, while its body volume is almost 7% smaller than the traditional rectangular PKW. However, for low-length and high-head conditions ( B / w = 1 and H o / P > 0.5), the efﬁciency a rectangular PKW exceeds that of the other shapes.


Introduction
Spillways play an essential role in the safety and service of water reservoirs; they are designed to safely pass floods, especially after considering the growing climate change effects leading to more extreme flood events [1].A survey of US Army Corps of Engineers indicates that ~30% of existing dams are currently unsafe due to having inadequate spillway capacity [2].This shortcoming has been the cause of one-third of dam failures that has resulted in huge life and financial losses [3].As a recent example, in May 2020, the collapse of Edenville Dam in Michigan, USA forced thousands of people to evacuate [4].
In response to the required rehabilitation of dam reservoirs and their aged and inefficient structures, innovative solutions have been developed by researchers to allow the structures to meet the safe passage of floods and increased water demands.A nonlinear weir with a folded crest in plan view improves the discharge capacity of spillway systems by extending the crest length for a given channel width.It can be an economical and effective solution for those sites where the structure width is limited while a large discharge capacity is required [5].In addition, a nonlinear weir can reduce the upstream water head and land submergence compared to conventional water storage diversion systems [6].
Moreover, as low-head control structures, they can also be used in river restoration projects to provide low-velocity regions that are potentially conducive to fish passage [7].
Labyrinth weirs are a classic type of nonlinear weir with vertical walls with different plan shapes such as rectangular [8], trapezoidal [9], triangular [10], semicircular [11], duckbill [12], or arced [13].Labyrinth weirs have been extensively used in water engineering projects (around 100 constructed prototypes across the globe) [14].At the beginning of the current century, an innovative and efficient modification, known as Piano Key Weir (PKW), was proposed and included sloped floors and cantilevered apexes into a rectangular labyrinth weir [15].
By comparing a rectangular labyrinth and piano key weirs with the same length, Anderson and Tullis (2012) showed that the PKW discharge coefficient was ~8% larger, which is associated with an increased wetted perimeter, causing a reduction in head loss and making room for the passing flow in the inlet and outlet keys, respectively [16].More importantly, compared to a traditional labyrinth weir, a PKW accommodates a smaller footprint length (B b in Figure 1).Thus, it allows a significant increase in the crest length (and discharge capacity) for a given spillway footprint base [16][17][18].These features may overcome both restrictions in the length and width of a spillway footprint and may even allow for the implementation of PKWs in challenging situations such as the top of concrete dams or steep, narrow streams.The mentioned advantages have made PKWs an efficient alternative for reservoir management projects (construction of ~30 PKWs during the last decade) [14].
Water 2021, 13, x FOR PEER REVIEW 4 of 21

Effective Parameters
The PKW discharge can be determined using the following equation [16]: where Q = weir discharge, Cd = discharge coefficient, L = total length of PKW, g = acceleration due to gravity, and Ho = total upstream head = h + V 2 /2g (h is the approach flow depth over the weir, and V is the average cross-sectional velocity of approaching flow).The independent parameters affecting the PKW discharge can be presented as: where f is a functional symbol, P = weir height, B = weir length in streamwise direction, w = cycle width, Bi = inlet key (or downstream) overhang length, Bo = outlet key (or upstream) overhang length, A = inlet key (or downstream) apex width, D = outlet key (or upstream) apex width, N = number of keys, t = weir wall thickness, ρ = fluid density, μ = dynamic viscosity, σ = surface tension, So = longitudinal bed slope, and Se represents the crest shape.Besides These parameters, their dependent parameters are shown in Figure 1.Using the dimensional analysis, one can derive: φ , , , , , , , , , , , ,

H B B B w A A t C N S S P w P B B w D P R W (3)
where φ is another functional symbol, and R = ρVd/μ and W = V(ρd/σ) 0.5 are the Reynolds and Weber numbers of the approach flow, respectively (Figure 1 presents V and d as dependent parameters).Since the viscosity effects are insignificant (turbulent flow with R > 25000), R is eliminated from Equation (3) [47,48].In lower values of the flow depth, surface tension may affect Froude-scaled models [49].Pfister et al. (2013) and Erpicum et al. (2016) specifically recommended h ≥ 0.03 m to avoid these effects on the head-discharge relationship of PKWs [50,51].The above recommendations have been satisfied in the collection of data; therefore, W (in the range of 7−50) was also removed from Equation (3).In addition, the geometric parameters of w/P, Bi/B, Bo/B, A/D, t/P, N, So, and Se were constant in this study (note that the values of w/P, A/D, t/P, N, and Se are different between the base and other models (mentioned in the next section)).Thus, removing these parameters from Equation (3) gives: Regarding the rapid development of PKWs, there has been an apparent link between research and industry projects.In addition to several publications in scientific journals, three international conferences (i.e., PKW2011 (Liège, Belgium), PKW2013 (Paris, France), and PKW2017 (Qui Nhơn, Vietnam)) were held in the last decade, specifically focusing on advancements in PKWs [19][20][21].From a selection of the prominent research studies, Lempérière (2013), Laugier et al. (2011), Leite Ribeiro et al. (2012), Kabiri-Samani and Javaheri (2012), Anderson and Tullis (2013), Cicero and Delisle (2013), and Machiels et al. (2014) conducted parametric studies on rectangular PKWs (including but not limited to headwater ratio (H o /P), weir length magnification ratio (L/W or B/w), wall thickness ratio (t/P), inlet-to-outlet width ratio (A/D), inlet-to-outlet overhang ratio (B i /B o ), crest shape, and cycle width ratio (w/P)) and provided some recommendations for the selection of their involved parameters [17,[22][23][24][25][26][27].While the majority of constructed prototypes and conducted research models (including the above-mentioned) have featured a rectangular shape, the PKW literature shows that the plan shape significantly affects its performance.By comparing a trapezoidal PKW to a traditional rectangular one, Safarzadeh and Noroozi (2017) indicated that a trapezoidal PKW can be approximately 20% more efficient than an ordinary rectangular one with an identical magnification ratio (L/W ≈ 5) [28].This point has been proven by the experimental studies of Mehboudi et al. (2016) [29].Cicero et al. (2013) showed that the gain in discharge may be limited to 5% in high head conditions (H o /P > 0.8) [30].
The literature review also indicates that previous studies have mostly focused on the determination of the head-discharge relationship of PKWs, with relatively few studies examining the effects of flow physics and hydrodynamics on a PKW.The experiments by Machiels et al. (2011) on a large-scale rectangular PKW (P = 52.5 cm) revealed that due to the sharp-corner entrance, a recirculation zone may form in the inlet key of a rectangular PKW [31].The simulations of Li et al. (2020) and Safarzadeh and Noroozi (2017) confirmed the recirculation zone formation in a rectangular PKW, which enlarges and results in higher heads, leading to a reduction in the inlet key effective area [28,32].Denys et al. (2017) and Denys (2019) investigated the specifications of the recirculation zone in more detail [33,34].The three-dimensional numerical simulations showed that a PKW creates two different recirculation zones near its base and crest with the nearly vertical and horizontal rotation axes, respectively.It is worth noting that the 5-15% efficiency gains of the inclusion of an upstream nose [35,36] may be due to limiting the recirculation zone development caused by a more gradual transition at the entrance to the inlet keys.Denys and Basson (2020) showed that due to the formation of the recirculation zone, the upstream face of the rectangular PKW sidewall experiences significant pressure fluctuations (with an average amplitude of ~10% of the total hydrostatic pressure) [37].According to Kumar et al. (2021), a sediment particle accelerates along the inlet key due to the flow contraction and an increase in shear stress [38].Regardless of the limited ranges of the above-mentioned studies, they imply that the understanding of the complex flow structure (fundamental research) and the analysis of its possible relation to the discharge capacity (applied research) may clarify PKW benefits and limitations.
Since much of the passing flow over a PKW discharges into outlet keys [39], nappe behavior in this area should be also taken into account when analyzing their performance.Although there is significant information regarding nappe behavior in labyrinth weirs (i.e., aeration conditions, nappe interference and instability, and nappe breaker effects), less data have been provided for PKWs.Through his studies of ~40 rectangular PKW models, Machiels (2012) showed that as the nappe interference zone rises above the crest elevation, the local submergence reduces the inlet key discharge and velocity [40].By increasing H o /P, the control section in the outlet keys moves in the downstream direction (also occurs by increasing A/D).Crookston and Tullis (2012) and Crookston and Tullis (2013) studied aeration, instability, and nappe interference in trapezoidal labyrinth weirs [41,42].Defining different nappe aeration conditions, they stated that when the nappe clings to the downstream surface of the weir, some undesirable conditions such as vibrations, noise, and pressure fluctuations may occur due to subatmospheric pressures under the nappe.Focusing on the flow characteristics over a semicircular labyrinth weir, Bilhan et al. (2018) showed that the subatmospheric pressures can be significantly diminished due to the use of nappe breakers [11].Bilhan et al. (2018) then estimated the discharge coefficient of trapezoidal labyrinth weirs with/without nappe breakers [43].The experimental studies of Mehboudi et al. (2017) indicated that trapezoidal PKWs experience aerated and drowned flow regimes when H o /P < 0.18 and H o /P > 0.35, respectively [44].The corresponding values for rectangular PKWs were reported as H o /P < 0.15 and H o /P > 0.2 by Kabiri-Samani and Javaheri (2012) [25].According to Vermeulen et al. (2017), although air pockets can exist on the downstream normal walls and sidewalls of a PKW, they is less prone to the resonance phenomenon (due to having folded and sloped walls) compared to a flap gate [45].Besides the use of nappe breakers and crest roughness, Lombaard (2020) recommended artificial aeration to reduce the intensity of the fluctuations if nappe oscillations are expected for low head conditions (H o /P ≤ 0.1) [46].In high head conditions, the pressure fluctuations in outlet keys diminish due to the momentum of the accelerated flow.
Due to the multiplicity of effective parameters and the complexity of flow patterns, the performance of PKWs merits more investigation.The literature review has shown that there is still a strong necessity for fundamental and applied research on PKWs, which can improve the comprehension of flow structure around these weirs, and subsequently, set up efficient rules for their design.In the broad ranges of the two main effective parameters (i.e., headwater ratio and weir length magnification ratio), this research evaluates the flow behavior around PKWs for different plan shapes (i.e., rectangular, trapezoidal, and triangular).Its principal objectives are:

•
to gain new insights into PKW flow physics through studying water surface profiles and velocity fields in inlet keys and nappe aeration and interference in outlet keys; • to estimate the discharge capacity of PKWs with different plan shapes and analyze their performance based on the findings and observations of the complex flow structure.

Effective Parameters
The PKW discharge can be determined using the following equation [16]: where Q = weir discharge, C d = discharge coefficient, L = total length of PKW, g = acceleration due to gravity, and H o = total upstream head = h + V 2 /2g (h is the approach flow depth over the weir, and V is the average cross-sectional velocity of approaching flow).
The independent parameters affecting the PKW discharge can be presented as: where f is a functional symbol, P = weir height, B = weir length in streamwise direction, w = cycle width, B i = inlet key (or downstream) overhang length, B o = outlet key (or upstream) overhang length, A = inlet key (or downstream) apex width, D = outlet key (or upstream) apex width, N = number of keys, t = weir wall thickness, ρ = fluid density, µ = dynamic viscosity, σ = surface tension, S o = longitudinal bed slope, and S e represents the crest shape.Besides These parameters, their dependent parameters are shown in Figure 1.
Using the dimensional analysis, one can derive: where ϕ is another functional symbol, and R = ρVd/µ and W = V(ρd/σ) 0.5 are the Reynolds and Weber numbers of the approach flow, respectively (Figure 1 presents V and d as dependent parameters).Since the viscosity effects are insignificant (turbulent flow with R > 25000), R is eliminated from Equation (3) [47,48].In lower values of the flow depth, surface tension may affect Froude-scaled models [49].Pfister et al. (2013) and Erpicum et al. (2016) specifically recommended h ≥ 0.03 m to avoid these effects on the head-discharge relationship of PKWs [50,51].The above recommendations have been satisfied in the collection of data; therefore, W (in the range of 7−50) was also removed from Equation (3).
In addition, the geometric parameters of w/P, B i /B, B o /B, A/D, t/P, N, S o , and S e were constant in this study (note that the values of w/P, A/D, t/P, N, and S e are different between the base and other models (mentioned in the next section)).Thus, removing these parameters from Equation (3) gives: Water 2021, 13, 2108 5 of 21 The present study analyzes the effects of these parameters on the performance of PKWs.

Experimental Method
To conduct the experiments, a recirculating flume (10 m long, 2 m wide, and 0.9 m deep) was implemented at the Hydraulic Laboratory of Tarbiat Modares University, Tehran, Iran.As shown in Figure 2, the channel was contracted to a width of 0.75 m using a converging transition.The tested weirs were located within a 1-meter distance from the contracted channel downstream end, where inflow disturbances were effectively damped, and flow under riverine approach conditions were achieved [52].The discharges (in the range of 0.020−0.185m 3 /sec) were determined using two calibrated portable ultrasonic flow meters (TFM3100-F1) with an accuracy of ±1% attached to the inflow pipes.They were recorded during 5-7 min time steps, and the mean values were used for further analyses [53].Water surface elevations were measured using a Mitutoyo digital point gauge (±0.1 mm accuracy).To calculate the total hydraulic heads, water surface elevations were measured at a minimum of 3h upstream from the weir, where the upstream water surface curvature (due to the drawdown effect) is negligible [54].Flow velocities were measured using Nortek−Vectrino Acoustic Doppler Velocimetry (ADV) with ±0.5% accuracy, in which the sampling duration was 2 minutes at a frequency of 200 Hz.Using WinADV software [55], phase space threshold despiking along with the SNR > 15 and CC > 0.7 was used to filter the contaminated time series (SNR is the signal-to-noise ratio, and CC is the correlation coefficient).Side-looking and down-looking probes were used to measure the flow velocities in different regions around PKW, especially near the vertical walls and the sloped floors (Figure 3).The flow visualizations were recorded using dye injection and a camera with a 14.1 Mega Pixel resolution.
Water 2021, 13, x FOR PEER REVIEW 5 of 21 φ , , The present study analyzes the effects of these parameters on the performance of PKWs.

Experimental Method
To conduct the experiments, a recirculating flume (10 m long, 2 m wide, and 0.9 m deep) was implemented at the Hydraulic Laboratory of Tarbiat Modares University, Tehran, Iran.As shown in Figure 2, the channel was contracted to a width of 0.75 m using a converging transition.The tested weirs were located within a 1-meter distance from the contracted channel downstream end, where inflow disturbances were effectively damped, and flow under riverine approach conditions were achieved [52].The discharges (in the range of 0.020−0.185m 3 /sec) were determined using two calibrated portable ultrasonic flow meters (TFM3100-F1) with an accuracy of ±1% attached to the inflow pipes.They were recorded during 5-7 min time steps, and the mean values were used for further analyses [53].Water surface elevations were measured using a Mitutoyo digital point gauge (±0.1 mm accuracy).To calculate the total hydraulic heads, water surface elevations were measured at a minimum of 3h upstream from the weir, where the upstream water surface curvature (due to the drawdown effect) is negligible [54].Flow velocities were measured using Nortek−Vectrino Acoustic Doppler Velocimetry (ADV) with ±0.5% accuracy, in which the sampling duration was 2 minutes at a frequency of 200 Hz.Using WinADV software [55], phase space threshold despiking along with the SNR > 15 and CC > 0.7 was used to filter the contaminated time series (SNR is the signal-to-noise ratio, and CC is the correlation coefficient).Side-looking and down-looking probes were used to measure the flow velocities in different regions around PKW, especially near the vertical walls and the sloped floors (Figure 3).The flow visualizations were recorded using dye injection and a camera with a 14.1 Mega Pixel resolution.The 3D rendering of the tested PKWs is presented in Figure 4.The base model was fabricated with P = 325 mm, t = 20 mm, and N = 2 (using High-Density Polyethylene (HDPE) sheeting and a computer numerically controlled (CNC) machine).The non-dimensional geometric parameters of this model were considered to be similar to the tested model by Anderson and Tullis (2013) [17], except for that the base model had a uniform short parapet wall (with the height of Pp = 10 mm) with a half-round crest shape (with the The 3D rendering of the tested PKWs is presented in Figure 4.The base model was fabricated with P = 325 mm, t = 20 mm, and N = 2 (using High-Density Polyethylene (HDPE) sheeting and a computer numerically controlled (CNC) machine).The non-dimensional geometric parameters of this model were considered to be similar to the tested model by Anderson and Tullis (2013) [17], except for that the base model had a uniform short parapet wall (with the height of P p = 10 mm) with a half-round crest shape (with the radius of R crest = t/2 = P p = 10 mm).The nine other models (Tri-B1 to Rec-B3 in Figure 4) were fabricated using a 3D printer having P = 200 mm, t = 12 mm, N = 3 and flat and quarter-round crest shapes for the normal and side walls (R crest = t = 12 mm).Table 1 indicates the specifications of all of the tested weirs along with the measured parameters (denoted by the symbol).
Water 2021, 13, x FOR PEER REVIEW 6 of 21 radius of Rcrest = t/2 = Pp = 10 mm).The nine other models (Tri-B1 to Rec-B3 in Figure 4) were fabricated using a 3D printer having P = 200 mm, t = 12 mm, N = 3 and flat and quarter-round crest shapes for the normal and side walls (Rcrest = t = 12 mm).Table 1 indicates the specifications of all of the tested weirs along with the measured parameters (denoted by the ■ symbol).

Down Looking Probe
Side Looking Probe

Flow Behavior in Inlet Keys
The complex flow structure around PKWs might be better understood by studying water surface profiles.The longitudinal profiles in the inlet key may have a dual behavior.As shown in Figure 5a, in the upstream part of the inlet key, the water surface drops due to the contraction of the subcritical flow (caused by the weir walls and upward floors).the supercritical inflow in the terminal portions then responds to the contraction as a slight rise in the water surface.From Figure 5a, the comparison of profiles in the inlet key and the sidewall centerlines indicates that the water surface decreases near the sidewalls due to the drawdown effects of the lateral spilling flow.Similar trends for the water surface profile were observed by Anderson and Tullis (2012) and Machiels (2012) [16,40].

Flow Behavior in Inlet Keys
The complex flow structure around PKWs might be better understood by studying water surface profiles.The longitudinal profiles in the inlet key may have a dual behavior.As shown in Figure 5(a), in the upstream part of the inlet key, the water surface drops due to the contraction of the subcritical flow (caused by the weir walls and upward floors).the supercritical inflow in the terminal portions then responds to the contraction as a slight rise in the water surface.From Figure 5(a), the comparison of profiles in the inlet key and the sidewall centerlines indicates that the water surface decreases near the sidewalls due to the drawdown effects of the lateral spilling flow.Similar trends for the water surface profile were observed by Anderson and Tullis (2012) and Machiels (2012) [16,40].
From Figure 5(b), the water surface drop in the inlet key is more significant in larger A/w values (especially in high head conditions).This is because the initial portion of the inlet key, through which the inflow enters into the weir, is wider in a triangular shape (shown in Figure 6); thus, the inflow contraction lessens.According to Figure 5(c), a larger drop occurs for larger values due to having a narrower space between the weir walls.The interactions between the two parameters A/w and B/w in Figure 5(d) illustrate that the water surface drop occurring at higher values is more severe for a larger A/w.Flow field around the PKWs was also studied to improve understanding of their hydrodynamics and flow mechanisms.Figure 7 shows the velocity profiles in the inlet key centerline of the base model (u = time-averaged velocity in x-direction; V = average cross- From Figure 5b, the water surface drop in the inlet key is more significant in larger A/w values (especially in high head conditions).This is because the initial portion of the inlet key, through which the inflow enters into the weir, is wider in a triangular shape (shown in Figure 6); thus, the inflow contraction lessens.According to Figure 5c, a larger drop occurs for larger values due to having a narrower space between the weir walls.The interactions between the two parameters A/w and B/w in Figure 5d illustrate that the water surface drop occurring at higher values is more severe for a larger A/w.

Flow Behavior in Inlet Keys
The complex flow structure around PKWs might be better understood by studying water surface profiles.The longitudinal profiles in the inlet key may have a dual behavior.As shown in Figure 5(a), in the upstream part of the inlet key, the water surface drops due to the contraction of the subcritical flow (caused by the weir walls and upward floors).the supercritical inflow in the terminal portions then responds to the contraction as a slight rise in the water surface.From Figure 5(a), the comparison of profiles in the inlet key and the sidewall centerlines indicates that the water surface decreases near the sidewalls due to the drawdown effects of the lateral spilling flow.Similar trends for the water surface profile were observed by Anderson and Tullis (2012) and Machiels (2012) [16,40].
From Figure 5(b), the water surface drop in the inlet key is more significant in larger A/w values (especially in high head conditions).This is because the initial portion of the inlet key, through which the inflow enters into the weir, is wider in a triangular shape (shown in Figure 6); thus, the inflow contraction lessens.According to Figure 5(c), a larger drop occurs for larger values due to having a narrower space between the weir walls.The interactions between the two parameters A/w and B/w in Figure 5(d) illustrate that the water surface drop occurring at higher values is more severe for a larger A/w.Flow field around the PKWs was also studied to improve understanding of their hydrodynamics and flow mechanisms.Figure 7 shows the velocity profiles in the inlet key centerline of the base model (u = time-averaged velocity in x-direction; V = average cross- Flow field around the PKWs was also studied to improve understanding of their hydrodynamics and flow mechanisms.Figure 7 shows the velocity profiles in the inlet key centerline of the base model (u = time-averaged velocity in x-direction; V = average cross-sectional velocity of approaching flow).As shown, the velocity profiles have a dual behavior that is similar to water surface profiles.When the flow enters the inlet key, the velocity increases in the streamwise direction due to the accelerating drawdown effects and the inflow contraction.However, the longitudinal velocity reduces in the inlet key terminal portions (higher x/B values) because of the sloped floor and the reduction in the inflow rates in the downstream direction as flow discharges into outlet keys.From Figure 7a, the flow velocity increases by an increase in H o /P to pass a larger discharge over the weir.However, the dimensionless velocity (u/V) in Figure 7b is a decreasing function of H o /P, which can be attributed to the large relative inlet key flow contraction and the associated head loss.The difference between the u/V profiles of the two different headwater ratios (H o /P = 0.1 and 0.2) is at its maximum in the middle portions of the inlet key (x/B = 0.37), which is where the inflow contraction occurs.However, according to Figure 8, this difference is negligible for a triangular plan shape (Tri-B2 model), where the inflow contraction is not significant.From Figure 9, it can be found that u/V, and subsequently, the flow momentum in the streamwise direction can be enhanced by increasing A/w and B/w (more weir length).
, 13, x FOR PEER REVIEW 8 of 21 sectional velocity of approaching flow).As shown, the velocity profiles have a dual behavior that is similar to water surface profiles.When the flow enters the inlet key, the velocity increases in the streamwise direction due to the accelerating drawdown effects and the inflow contraction.However, the longitudinal velocity reduces in the inlet key terminal portions (higher x/B values) because of the sloped floor and the reduction in the inflow rates in the downstream direction as flow discharges into outlet keys.From Figure 7(a), the flow velocity increases by an increase in Ho/P to pass a larger discharge over the weir.However, the dimensionless velocity (u/V) in Figure 7(b) is a decreasing function of Ho/P, which can be attributed to the large relative inlet key flow contraction and the associated head loss.The difference between the u/V profiles of the two different headwater ratios (Ho/P = 0.1 and 0.2) is at its maximum in the middle portions of the inlet key (x/B = 0.37), which is where the inflow contraction occurs.However, according to Figure 8, this difference is negligible for a triangular plan shape (Tri-B2 model), where the inflow contraction is not significant.From Figure 9, it can be found that u/V, and subsequently, the flow momentum in the streamwise direction can be enhanced by increasing A/w and B/w (more weir length).Water 2021, 13, x FOR PEER REVIEW 8 of 21 sectional velocity of approaching flow).As shown, the velocity profiles have a dual behavior that is similar to water surface profiles.When the flow enters the inlet key, the velocity increases in the streamwise direction due to the accelerating drawdown effects and the inflow contraction.However, the longitudinal velocity reduces in the inlet key terminal portions (higher x/B values) because of the sloped floor and the reduction in the inflow rates in the downstream direction as flow discharges into outlet keys.From Figure 7(a), the flow velocity increases by an increase in Ho/P to pass a larger discharge over the weir.However, the dimensionless velocity (u/V) in Figure 7(b) is a decreasing function of Ho/P, which can be attributed to the large relative inlet key flow contraction and the associated head loss.The difference between the u/V profiles of the two different headwater ratios (Ho/P = 0.1 and 0.2) is at its maximum in the middle portions of the inlet key (x/B = 0.37), which is where the inflow contraction occurs.However, according to Figure 8, this difference is negligible for a triangular plan shape (Tri-B2 model), where the inflow contraction is not significant.From Figure 9, it can be found that u/V, and subsequently, the flow momentum in the streamwise direction can be enhanced by increasing A/w and B/w (more weir length).A separation (or recirculation) zone was identified by Machiels et al. (2011) in the inlet keys of a rectangular PKW [31].To determine its variations, the velocities were measured in different planes around different PKWs (P1 in Tri-B1, P1 in Tri-B2, and P1 and P4 in Rec-B2, as shown in Table 2).However, the collected data may not visualize the separation zone because of having a CC < 0.7 (probably due to the high level of flow unsteadiness and weir wall disruption in the transmission of acoustic waves).Nevertheless, from Figure 10, the high diffusion of the injected dye confirms the presence of the separation zone in the downstream section of a rectangular PKW.It may diminish at the smaller A/w values, where the sidewalls act as a nose to guide the inflow.In addition, the recirculation vortices may appear as some undulations on the water surface (not observed for smaller A/w and Ho/P values).A separation (or recirculation) zone was identified by Machiels et al. (2011) in the inlet keys of a rectangular PKW [31].To determine its variations, the velocities were measured in different planes around different PKWs (P1 in Tri-B1, P1 in Tri-B2, and P1 and P4 in Rec-B2, as shown in Table 2).However, the collected data may not visualize the separation zone because of having a CC < 0.7 (probably due to the high level of flow unsteadiness and weir wall disruption in the transmission of acoustic waves).Nevertheless, from Figure 10, the high diffusion of the injected dye confirms the presence of the separation zone in the downstream section of a rectangular PKW.It may diminish at the smaller A/w values, where the sidewalls act as a nose to guide the inflow.In addition, the recirculation vortices may appear as some undulations on the water surface (not observed for smaller A/w and H o /P values).A separation (or recirculation) zone was identified by Machiels et al. (2011) in th inlet keys of a rectangular PKW [31].To determine its variations, the velocities were meas ured in different planes around different PKWs (P1 in Tri-B1, P1 in Tri-B2, and P1 and P in Rec-B2, as shown in Table 2).However, the collected data may not visualize the sepa ration zone because of having a CC < 0.7 (probably due to the high level of flow unstead iness and weir wall disruption in the transmission of acoustic waves).Nevertheless, from Figure 10, the high diffusion of the injected dye confirms the presence of the separation zone in the downstream section of a rectangular PKW.It may diminish at the smaller A/w values, where the sidewalls act as a nose to guide the inflow.In addition, the recirculation vortices may appear as some undulations on the water surface (not observed for smalle A/w and Ho/P values).A separation (or recirculation) zone was identified by Machiels et al. (2011) in th inlet keys of a rectangular PKW [31].To determine its variations, the velocities were meas ured in different planes around different PKWs (P1 in Tri-B1, P1 in Tri-B2, and P1 and P in Rec-B2, as shown in Table 2).However, the collected data may not visualize the sepa ration zone because of having a CC < 0.7 (probably due to the high level of flow unstead iness and weir wall disruption in the transmission of acoustic waves).Nevertheless, from Figure 10, the high diffusion of the injected dye confirms the presence of the separatio zone in the downstream section of a rectangular PKW.It may diminish at the smaller A/w values, where the sidewalls act as a nose to guide the inflow.In addition, the recirculatio vortices may appear as some undulations on the water surface (not observed for smalle A/w and Ho/P values).A separation (or recirculation) zone was identified by Machiels et al. ( 2011) in th inlet keys of a rectangular PKW [31].To determine its variations, the velocities were meas ured in different planes around different PKWs (P1 in Tri-B1, P1 in Tri-B2, and P1 and P4 in Rec-B2, as shown in Table 2).However, the collected data may not visualize the sepa ration zone because of having a CC < 0.7 (probably due to the high level of flow unstead iness and weir wall disruption in the transmission of acoustic waves).Nevertheless, from Figure 10, the high diffusion of the injected dye confirms the presence of the separation zone in the downstream section of a rectangular PKW.It may diminish at the smaller A/w values, where the sidewalls act as a nose to guide the inflow.In addition, the recirculation vortices may appear as some undulations on the water surface (not observed for smalle A/w and Ho/P values).A separation (or recirculation) zone was identified by Machiels et al. (2011) in the inlet keys of a rectangular PKW [31].To determine its variations, the velocities were meas ured in different planes around different PKWs (P1 in Tri-B1, P1 in Tri-B2, and P1 and P4 in Rec-B2, as shown in Table 2).However, the collected data may not visualize the sepa ration zone because of having a CC < 0.7 (probably due to the high level of flow unstead iness and weir wall disruption in the transmission of acoustic waves).Nevertheless, from Figure 10, the high diffusion of the injected dye confirms the presence of the separation zone in the downstream section of a rectangular PKW.It may diminish at the smaller A/w values, where the sidewalls act as a nose to guide the inflow.In addition, the recirculation vortices may appear as some undulations on the water surface (not observed for smalle A/w and Ho/P values).Beyond the PKW structure, the ADV data may have a high quality (e.g., CC > 0.9 for planes P5 and P6 in the Rec-B2 model, as shown in Table 2).Figure 11 presents the velocity vectors in the vicinity of the crest of the Rec-B2 and base models.By increasing Ho/P, the accelerated inflow obeys the general channel flow direction more.However, at a smaller distance from the crest, the inflow more tends to stay at the sidewalls.The contours of the angle between the velocity vectors and the weir sidewall (Θ) are presented in Figure 12 for the two mentioned rectangular models (α = 0°).As shown, Θ reduces by increasing Ho/P and by moving away from the crest in both the vertical and horizontal directions (less weir effects on the flow).Figure 13 shows Θ variations along the sidewall centerline of the triangular and rectangular PKWs.Generally, Θ increases with x/B in the sidewall initial portion; however, it shows a reverse trend in its terminal portion (due to the increased flow acceleration in the streamwise direction).In addition, since α has a significant contribution in Θ, a triangular plan has a larger Θ compared to a rectangular one (closer to the ideal condition; Θ = 90°).Moreover, Θ increases by decreasing B/w because a lower flow momentum allows the inflow to turn more readily and spill over sidewalls (refer to Figure 9(a)).Beyond the PKW structure, the ADV data may have a high quality (e.g., CC > 0.9 for planes P5 and P6 in the Rec-B2 model, as shown in Table 2).Figure 11 presents the velocity vectors in the vicinity of the crest of the Rec-B2 and base models.By increasing H o /P, the accelerated inflow obeys the general channel flow direction more.However, at a smaller distance from the crest, the inflow more tends to stay at the sidewalls.The contours of the angle between the velocity vectors and the weir sidewall (Θ) are presented in Figure 12 for the two mentioned rectangular models (α = 0 • ).As shown, Θ reduces by increasing H o /P and by moving away from the crest in both the vertical and horizontal directions (less weir effects on the flow).Figure 13 shows Θ variations along the sidewall centerline of the triangular and rectangular PKWs.Generally, Θ increases with x/B in the sidewall initial portion; however, it shows a reverse trend in its terminal portion (due to the increased flow acceleration in the streamwise direction).In addition, since α has a significant contribution in Θ, a triangular plan has a larger Θ compared to a rectangular one (closer to the ideal condition; Θ = 90 • ).Moreover, Θ increases by decreasing B/w because a lower flow momentum allows the inflow to turn more readily and spill over sidewalls (refer to Figure 9a).Beyond the PKW structure, the ADV data may have a high quality (e.g., CC > 0.9 for planes P5 and P6 in the Rec-B2 model, as shown in Table 2).Figure 11 presents the velocity vectors in the vicinity of the crest of the Rec-B2 and base models.By increasing Ho/P, the accelerated inflow obeys the general channel flow direction more.However, at a smaller distance from the crest, the inflow more tends to stay at the sidewalls.The contours of the angle between the velocity vectors and the weir sidewall (Θ) are presented in Figure 12 for the two mentioned rectangular models (α = 0°).As shown, Θ reduces by increasing Ho/P and by moving away from the crest in both the vertical and horizontal directions (less weir effects on the flow).Figure 13 shows Θ variations along the sidewall centerline of the triangular and rectangular PKWs.Generally, Θ increases with x/B in the sidewall initial portion; however, it shows a reverse trend in its terminal portion (due to the increased flow acceleration in the streamwise direction).In addition, since α has a significant contribution in Θ, a triangular plan has a larger Θ compared to a rectangular one (closer to the ideal condition; Θ = 90°).Moreover, Θ increases by decreasing B/w because a lower flow momentum allows the inflow to turn more readily and spill over sidewalls (refer to Figure 9(a)).The discharge locally changes along the crest of a PKW (unit discharge = q = magnitude of velocity component normal to the wall × flow depth).From Figure 5a, the uniform flow depth over the upstream normal wall (the same water surface levels at x/B ≈ 0 for the sidewall and outlet key centerlines) is clearly larger than that over the side walls, and their difference progressively enlarges with an increase in H o /P.Thus, higher values of u compared to v (which are the normal velocity components on the normal and side walls of a rectangular PKW, respectively) in Figure 14 reveal that the unit discharge on the upstream normal wall (=u × h) can be several times larger than that of the sidewall (=v × h) in such a relatively high H o /P.It is worth noting that the low-quality velocity data in the downstream section prevented the determination of q on the whole PKW crest.In addition, despite a lower q, the total discharge over the sidewalls is larger than that of the normal walls because the sidewalls are usually much longer than the normal walls.The discharge locally changes along the crest of a PKW (unit discharge = q = magnitude of velocity component normal to the wall × flow depth).From Figure 5(a), the uniform flow depth over the upstream normal wall (the same water surface levels at x/B ≈ 0 for the sidewall and outlet key centerlines) is clearly larger than that over the side walls, and their difference progressively enlarges with an increase in Ho/P.Thus, higher values of u compared to v (which are the normal velocity components on the normal and side walls of a rectangular PKW, respectively) in Figure 14 reveal that the unit discharge on the upstream normal wall (= u × h) can be several times larger than that of the sidewall (= v × h) in such a relatively high Ho/P.It is worth noting that the low-quality velocity data in the downstream section prevented the determination of q on the whole PKW crest.In addition, despite a lower q, the total discharge over the sidewalls is larger than that of the normal walls because the sidewalls are usually much longer than the normal walls.The discharge locally changes along the crest of a PKW (unit discharge = q = magnitude of velocity component normal to the wall × flow depth).From Figure 5(a), the uniform flow depth over the upstream normal wall (the same water surface levels at x/B ≈ 0 for the sidewall and outlet key centerlines) is clearly larger than that over the side walls, and their difference progressively enlarges with an increase in Ho/P.Thus, higher values of u compared to v (which are the normal velocity components on the normal and side walls of a rectangular PKW, respectively) in Figure 14 reveal that the unit discharge on the upstream normal wall (= u × h) can be several times larger than that of the sidewall (= v × h) in such a relatively high Ho/P.It is worth noting that the low-quality velocity data in the downstream section prevented the determination of q on the whole PKW crest.In addition, despite a lower q, the total discharge over the sidewalls is larger than that of the normal walls because the sidewalls are usually much longer than the normal walls.

Flow Behavior in Outlet Keys
The flow mechanisms in outlet keys are also mapped by studying the nappe aeration and interference.By increasing Ho/P, the nappe aeration regime generally transforms from a clinging regime to an aerated regime, then transforms to partially aerated regime, and finally, it transforms to drowned regime [40,42].According to experimental observations, in the clinging regime, the nappe adheres to the downstream face of the sidewalls (not

Flow Behavior in Outlet Keys
The flow mechanisms in outlet keys are also mapped by studying the nappe aeration and interference.By increasing H o /P, the nappe aeration regime generally transforms from a clinging regime to an aerated regime, then transforms to partially aerated regime, and finally, it transforms to drowned regime [40,42].According to experimental observations, in the clinging regime, the nappe adheres to the downstream face of the sidewalls (not necessarily the overhangs) due to the subatmospheric pressures on them [11].By increasing the head, the flow momentum separates the nappe from the weir wall (aerated regime).In the partially aerated regime, one or more unstable air cavities (in size and location) move along the weir sidewalls, leading to an uncertainty in the determination of the PKW discharge capacity (addressed in the next section).Finally, in the drowned regime, the thick overflow nappe prevents air pocket formation.
Figure 15 indicates that H o /P ≈ 0.1 can be considered as an approximate threshold for the aerated regime of the PKWs (which may be sensitive to weir scale and crest shape).In addition, by increasing A/w (changing the plan shape from triangular to trapezoidal and then to rectangular again), the drowned regime starts with higher H o /P values.This is because the sharper apex corners (smaller values of α) facilitate nappe splitting and aeration.However, at higher B/w and A/w values, the nappe aeration regime is less impacted by the weir geometry.It is also shown that for lower values of B/w, the drowned regime occurs at smaller headwater ratios because these geometries are more susceptible to the nappe interference (discussed later in the current section).
ing the head, the flow momentum separates the nappe from the weir wall (aerated regime).In the partially aerated regime, one or more unstable air cavities (in size and location) move along the weir sidewalls, leading to an uncertainty in the determination of the PKW discharge capacity (addressed in the next section).Finally, in the drowned regime, the thick overflow nappe prevents air pocket formation.
Figure 15 indicates that Ho/P ≈ 0.1 can be considered as an approximate threshold for the aerated regime of the PKWs (which may be sensitive to weir scale and crest shape).In addition, by increasing A/w (changing the plan shape from triangular to trapezoidal and then to rectangular again), the drowned regime starts with higher Ho/P values.This is because the sharper apex corners (smaller values of α) facilitate nappe splitting and aeration.However, at higher B/w and A/w values, the nappe aeration regime is less impacted by the weir geometry.It is also shown that for lower values of B/w, the drowned regime occurs at smaller headwater ratios because these geometries are more susceptible to the nappe interference (discussed later in the current section).
According to Figure 15, when B/w = 3 and A/w = 0, the thresholds of the partially aerated and drowned regimes are Ho/P = 0.42 and 0.58, respectively.The corresponding values were reported as 0.29 and 0.39 by Crookston and Tullis (2013) for a labyrinth weir with relatively similar dimensionless parameters (B/w = 2.8 and A/w = 0.06) [42].Therefore, it can be presumed that due to the presence of the inlet key (or downstream) overhangs, a PKW experiences better aeration conditions compared to a labyrinth weir.Studying trapezoidal PKWs with small overhangs (Bi/B = Bo/B = 0.1), Mehboudi et al. (2017) reported values of Ho/P = 0.18 and 0.35, as the general thresholds of the partially aerated and drowned regimes for all of their 36 models were in the ranges of 1.22 ≤ B/w ≤ 2.86 and 0.19 ≤ A/w ≤ 0.42 [44].Although theses researchers did not report the variations of nappe aeration with PKW geometry, their general thresholds are smaller than the corresponding headwater ratios for the whole range of the present study (1 ≤ B/w ≤ 3 and 0 ≤ A/w ≤ 0.5), as shown in Figure 15.This may be due to the larger overhangs of the current study models (Bi/B = Bo/B = 0.25).Again, it worth noting that the scale effects may significantly affect the nappe behavior, especially for the above-mentioned Ho/P threshold values for Ho/P.According to Figure 15, when B/w = 3 and A/w = 0, the thresholds of the partially aerated and drowned regimes are H o /P = 0.42 and 0.58, respectively.The corresponding values were reported as 0.29 and 0.39 by Crookston and Tullis (2013) for a labyrinth weir with relatively similar dimensionless parameters (B/w = 2.8 and A/w = 0.06) [42].Therefore, it can be presumed that due to the presence of the inlet key (or downstream) overhangs, a PKW experiences better aeration conditions compared to a labyrinth weir.Studying trapezoidal PKWs with small overhangs (B i /B = B o /B = 0.1), Mehboudi et al. (2017) reported values of H o /P = 0.18 and 0.35, as the general thresholds of the partially aerated and drowned regimes for all of their 36 models were in the ranges of 1.22 ≤ B/w ≤ 2.86 and 0.19 ≤ A/w ≤ 0.42 [44].Although theses researchers did not report the variations of nappe aeration with PKW geometry, their general thresholds are smaller than the corresponding headwater ratios for the whole range of the present study (1 ≤ B/w ≤ 3 and 0 ≤ A/w ≤ 0.5), as shown in Figure 15.This may be due to the larger overhangs of the current study models (B i /B = B o /B = 0.25).Again, it worth noting that the scale effects may significantly affect the nappe behavior, especially for the above-mentioned H o /P threshold values for H o /P.
Nappe interference also occurs in PKW outlet keys including either a region of local submergence, a standing wave, or both [40,41].As shown in Figure 16, considering the intersection of the weir crest elevation with the water surface profile, the parameter of Water 2021, 13, 2108 13 of 21 interference length (B int ) as the summation of the local submergence length (B sub ) and the standing wave length (B wav ) are used to characterize the extent of nappe interference (measured on the centerline of outlet keys).It is notable that a standing wave is not formed in a triangular PKW (Figure 16c), most likely due to the large width of the downstream portion of outlet keys.In addition, the flow features in the outlet keys are often dynamic and highly unstable, especially in high head conditions (time-mean values of the measured parameters were recorded for further analyses).
conditions, the standing wave is lower than the weir crest (Bwav = 0), while it exceeds the crest in higher flow heads (Bwav is also added to Bint).
According to Figure 17(a), the local submergence length of triangular PKWs is larger than that of the rectangular ones.This is because the initial part of the outlet key (susceptible location for local submergence occurrence) is narrower in a triangular PKW (Figure 6).This point also justifies why in the interference lengths of triangular PKWs Figure 17(b) are larger than those of rectangular PKWs in low head conditions, where the overflow is concentrated on the initial portions of the PKWs.However, a triangular shape may have a smaller Bint/B at higher Ho/P values (more pronounced in lower B/w values corresponding to larger sidewall angles).This is because in high head conditions, the overflow concentrates on the terminal portions of the PKWs, where a triangular shape has larger discharge area.From Figure 17, it can also be seen that although Bsub and Bint may increase due to an increase in B (larger discharge capacity), their growth rates are not as fast as that of B. As a result, the dimensionless parameters Bsub/B and Bint/B are decreasing functions of B/w.This result is in agreement with the performance of trapezoidal labyrinth weirs [41].nted and analyzed in this o/P in very low head conre 15).In this regime, an er the nappe and consed of Cd stops at the beginh Ho/P may be due to the nd the inflow contraction hich reduce the effective er Θ values in Figure 12).(from the present study) demonstrates that a halfconditions (~ Ho/P < 0.15).f a half-round crest shape Ho/P value (which can be e and the Rec-B2 models erence in A/D happens to ous studies [17, 25,27], the inlet-to-outlet width ratio locity.On the other hand, pe interference ratio [53]

Discharge Coefficient and Efficiency
The variations of the PKW discharge coefficient are presented and analyzed in this section.According to Figure 18, Cd is an increasing function of Ho/P in very low head conditions (Ho/P < 0.1 corresponding to the clinging regime in Figure 15).In this regime, an increase in Ho/P intensifies the subatmospheric pressures under the nappe and consequently leads to an increase in Cd.However, the increasing trend of Cd stops at the beginning of the aerated regime.Its monotonic decreasing trend with Ho/P may be due to the local submergence enlargement in the outlet keys (Figure 17) and the inflow contraction in the inlet keys (shown as water surface drop in Figure 5), which reduce the effective crest length and weakens the performance of the sidewalls (lower Θ values in Figure 12).
From Figure 18, the comparison between the base model (from the present study) and the studied geometry by Anderson and Tullis (2013) [17] demonstrates that a halfround crest shape is more efficient than a flat one in low head conditions (~ Ho/P < 0.15) Based on the results of Cicero and Delisle (2013) [26], the gain of a half-round crest shape is limited to ⁓20 percent, and it diminishes by increasing the Ho/P value (which can be confirmed by the mentioned comparison).In addition, the base and the Rec-B2 models from the present study show similar performance since the difference in A/D happens to be countered by the w/P difference.Based on the results of previous studies [17,25,27], the Cd of a rectangular PKW increases by ~ 10% by increasing the inlet-to-outlet width ratio from A/D = 1 to ⁓1.25, leading to a reduction in the inlet key velocity.On the other hand Cd decreases in lower w/P values due to an increase in the nappe interference ratio [53] which can be seen by comparing the data of Anderson  water surface profile in inlet key centerline).
The variations of B int /B and B sub /B are presented in Figure 17 (B wav /B can be also determined by subtracting these two parameters).As expected, both parameters generally increase with an increase in H o /P.However, their increasing trends may stop (or reverse) with smaller values of A/w and B/w when 0.3 < H o /P < 0.6.This drop may be due to the aeration regime transition (Figure 15).By starting the partially aerated regime, which results in a reduction in the nappe aeration, the nappe appears to become closer to the weir sidewalls.Thus, the nappe interference reduces.It should be noted that the mentioned drop diminishes with larger values of B/w and A/w (minor effects of aeration regime transition).The jump noted in the B int /B curve of the Rec-B1 model is because in the low head conditions, the standing wave is lower than the weir crest (B wav = 0), while it exceeds the crest in higher flow heads (B wav is also added to B int ).

Discharge Coefficient and Efficiency
The variations of the PKW discharge coefficient are presented and analyzed in this section.According to Figure 18, Cd is an increasing function of Ho/P in very low head conditions (Ho/P < 0.1 corresponding to the clinging regime in Figure 15).In this regime, an increase in Ho/P intensifies the subatmospheric pressures under the nappe and consequently leads to an increase in Cd.However, the increasing trend of Cd stops at the beginning of the aerated regime.Its monotonic decreasing trend with Ho/P may be due to the local submergence enlargement in the outlet keys (Figure 17) and the inflow contraction in the inlet keys (shown as water surface drop in Figure 5), which reduce the effective crest length and weakens the performance of the sidewalls (lower Θ values in Figure 12).
From Figure 18, the comparison between the base model (from the present study)  According to Figure 17a, the local submergence length of triangular PKWs is larger than that of the rectangular ones.This is because the initial part of the outlet key (susceptible location for local submergence occurrence) is narrower in a triangular PKW (Figure 6).This point also justifies why in the interference lengths of triangular PKWs Figure 17b are larger than those of rectangular PKWs in low head conditions, where the overflow is concentrated on the initial portions of the PKWs.However, a triangular shape may have a smaller B int /B at higher H o /P values (more pronounced in lower B/w values corresponding to larger sidewall angles).This is because in high head conditions, the overflow concentrates on the terminal portions of the PKWs, where a triangular shape has larger discharge area.From Figure 17, it can also be seen that although B sub and B int may increase due to an increase in B (larger discharge capacity), their growth rates are not as fast as that of B. As a result, the dimensionless parameters B sub /B and B int /B are decreasing functions of B/w.This result is in agreement with the performance of trapezoidal labyrinth weirs [41].

Discharge Coefficient and Efficiency
The variations of the PKW discharge coefficient are presented and analyzed in this section.According to Figure 18, C d is an increasing function of H o /P in very low head conditions (H o /P < 0.1 corresponding to the clinging regime in Figure 15).In this regime, an increase in H o /P intensifies the subatmospheric pressures under the nappe and consequently leads to an increase in C d .However, the increasing trend of C d stops at the beginning of the aerated regime.Its monotonic decreasing trend with H o /P may be due to the local submergence enlargement in the outlet keys (Figure 17) and the inflow contraction in the inlet keys (shown as water surface drop in Figure 5), which reduce the effective crest length and weakens the performance of the sidewalls (lower Θ values in Figure 12).Besides the variations in the apex width, the change in the sidewall angle should also be considered when analyzing the dual effects of A/w on Cd. Figure 6 indicates that for a given width, an increase in A/w enlarges the apex width with a normal orientation to the approaching flow (improving factor for Cd), while this change aligns the weir sidewalls to the flow (worsening factor for Cd).Thus, there is a balance between the performance of normal wall and sidewall crests.According to Figure 19, at low Ho/P values, a trapezoidal PKW (A/w = 0.25) has higher Cd compared to a triangular one (A/w = 0).This is because the trapezoidal plan shape has larger apexes (as the best-orientated weir parts to the approaching flow), while having a lower sidewall angle does not limit its performance in low head conditions.However, the Cd of a triangular PKW overtakes that of the other shapes in higher headwater ratios (Ho/P > 0.4).In these conditions, the accelerated flow in the stream-wise direction cannot quickly turn and pass over the PKW sidewalls.Thus, an increase in A/w (leading to a higher α) can improve the sidewall orientation to the approaching flow (higher Θ values in Figure 13) and increases its contribution in the discharging flow.Another reason for the Cd reduction in higher A/w values may be the separation zone enlargement in the inlet keys (Figure 10), leading to a reduction in the effective flow area.
Since the flow features in the outlet keys are improved by increasing B/w (smaller values of Bsub/B and Bint/B in Figure 17), the decreasing trend of Cd with B/w is related to the inlet key features, including the increased inflow contraction (larger water surface drop in Figure 5(d)), the enhanced longitudinal flow inertia (larger u/V values in Figure 9(a)), and subsequently, the weaker performance of the sidewalls (smaller Θ values in Figure 13).According to Figure 19, for the low values of A/w and B/w, there is a jump in Cd when 0.3 < Ho/P < 0.4.This is attributed to the transition from an aerated to a partially From Figure 18, the comparison between the base model (from the present study) and the studied geometry by Anderson and Tullis (2013) [17] demonstrates that a half-round crest shape is more efficient than a flat one in low head conditions (~H o /P < 0.15).Based on the results of Cicero and Delisle (2013) [26], the gain of a half-round crest shape is limited to ~20 percent, and it diminishes by increasing the H o /P value (which can be confirmed by the mentioned comparison).In addition, the base and the Rec-B2 models from the present study show similar performance since the difference in A/D happens to be countered by the w/P difference.Based on the results of previous studies [17, 25,27], the C d of a rectangular PKW increases by ~10% by increasing the inlet-to-outlet width ratio from A/D = 1 to ~1.25, leading to a reduction in the inlet key velocity.On the other hand, C d decreases in lower w/P values due to an increase in the nappe interference ratio [53], which can be seen by comparing the data of Anderson and Tullis (2013) and Li et al. (2020) [17,32].This point can be validated by comparing the Rec-B2 model to the mentioned geometry of Machiels (2012) [40], which has a smaller C d regardless of having a lower B/w value (the decreasing trend of C d with B/w is subsequently presented in the current section).
Besides the variations in the apex width, the change in the sidewall angle should also be considered when analyzing the dual effects of A/w on C d .Figure 6 indicates that for a given width, an increase in A/w enlarges the apex width with a normal orientation to the approaching flow (improving factor for C d ), while this change aligns the weir sidewalls to the flow (worsening factor for C d ).Thus, there is a balance between the performance of normal wall and sidewall crests.According to Figure 19, at low H o /P values, a trapezoidal PKW (A/w = 0.25) has higher C d compared to a triangular one (A/w = 0).This is because the trapezoidal plan shape has larger apexes (as the best-orientated weir parts to the approaching flow), while having a lower sidewall angle does not limit its performance in low head conditions.However, the C d of a triangular PKW overtakes that of the other shapes in higher headwater ratios (H o /P > 0.4).In these conditions, the accelerated flow in the stream-wise direction cannot quickly turn and pass over the PKW sidewalls.Thus, an increase in A/w (leading to a higher α) can improve the sidewall orientation to the approaching flow (higher Θ values in Figure 13) and increases its contribution in the discharging flow.Another reason for the C d reduction in higher A/w values may be the separation zone enlargement in the inlet keys (Figure 10), leading to a reduction in the effective flow area.The discharge efficiency of PKWs (ε = Cd × L/W) is also presented and analyzed [5].According to Figure 20, while ε has a decreasing trend with Ho/P, it can be improved by increasing B/w and A/w; thus, the decrease in Cd can be restituted by increasing the weir length.However, Figure 21 demonstrates that the efficiency gains of increasing B/w and Since the flow features in the outlet keys are improved by increasing B/w (smaller values of B sub /B and B int /B in Figure 17), the decreasing trend of C d with B/w is related to the inlet key features, including the increased inflow contraction (larger water surface drop in Figure 5d), the enhanced longitudinal flow inertia (larger u/V values in Figure 9a), and subsequently, the weaker performance of the sidewalls (smaller Θ values in Figure 13).According to Figure 19, for the low values of A/w and B/w, there is a jump in C d when 0.3 < H o /P < 0.4.This is attributed to the transition from an aerated to a partially aerated regime (smaller air pocket under the nappe leading to a smaller nappe interference).However, this jump diminishes the larger values of B/w and A/w that correspond to the minor effects of the aeration regime transition.
Equation ( 5) is selected to describe the discharge coefficient of the PKWs [5], where a 1 to a 4 are empirical coefficients (obtained using the least-squares method).These coef-assuming that the PKW global cost is proportional to its body volume [57], the trapezoidal shape may be selected as the optimal one.For a specific project, this point can be confirmed after a detailed cost estimation on the different elements of PKWs (i.e., weir foundation, wedge shape base, overhang, and vertical wall).From Table 4, C has almost a constant growth rate with respect to B/w ([i.e., the body volume of a PKW having B/w = n is almost n times that of a PKW with B/w = 1).].However, according to Figure 22, the discharge growth rate does not have a linear trend, and it slows down by increasing B/w.Therefore, there is no optimal solution for B/w from the hydraulic point of view, and it can be determined after considering site-specific constraints and costs.From Table 4, C has almost a constant growth rate with respect to B/w ([i.e., the body volume of a PKW having B/w = n is almost n times that of a PKW with B/w = 1).].However, according to Figure 22, the discharge growth rate does not have a linear trend, and it slows down by increasing B/w.Therefore, there is no optimal solution for B/w from the hydraulic point of view, and it can be determined after considering site-specific constraints and costs.

Conclusion
This paper studies the flow behavior ofaround PKWs with different plan shapes in the wide ranges of Ho/P and B/w.The results led to the following conclusions: • The water surface drop and recirculation zone, occurring in PKW inlet keys respectively due to the inflow contraction and the sharp-corner entrance, respectively, are more severe at higher values of Ho/P, B/w, and A/w.At higher Ho/P values and at larger distances from the weir crest, the passing flow is less affected by the PKW structure.However, by reducing the weir length and apex width, the weir structure effects on the passing flow retrieve; • While the aeration regimes vary with the weir geometry, Ho/P = 0.1 can be considered as the general threshold for the aerated regime of the tested PKWs (sensitive to weir scale and crest shape).In addition, by increasing B/w and A/w, the flow aeration regimes extend to higher Ho/P values.By comparing the present study and previous studies [42], a PKW has better aeration conditions compared to a

Conclusions
This paper studies the flow behavior around PKWs with different plan shapes in the wide ranges of H o /P and B/w.The results led to the following conclusions:

•
The water surface drop and recirculation zone, occurring in PKW inlet keys respectively due to the inflow contraction and the sharp-corner entrance, are more severe at higher values of H o /P, B/w, and A/w.At higher H o /P values and at larger distances from the weir crest, the passing flow is less affected by the PKW structure.However, by reducing the weir length and apex width, the weir structure effects on the passing flow retrieve;

Figure 2 .
Figure 2. Overview of implemented test facilities: (a) the 3D view from downstream; (b) the plan view; (c) the side view.

Figure 2 .
Figure 2. Overview of implemented test facilities: (a) the 3D view from downstream; (b) the plan view; (c) the side view.

Figure 4 .
Figure 4.A Three-dimensional view of the tested models (1 key, downstream view).

Figure 4 .
Figure 4.A Three-dimensional view of the tested models (1 key, downstream view).

Figure 4 .
Figure 4.A Three-dimensional view of the tested models (1 key, downstream view).

Figure 7 .
Figure 7. Velocity profiles in the inlet key centerline of the base model: (a) dimensional value; (b) dimensionless value.(V = 0.139 and 0.293 m/s for Ho/P = 0.1 and 0.2, respectively).

Figure 7 .
Figure 7. Velocity profiles in the inlet key centerline of the base model: (a) dimensional value; (b) dimensionless value.(V = 0.139 and 0.293 m/s for H o /P = 0.1 and 0.2, respectively).

Figure 7 .
Figure 7. Velocity profiles in the inlet key centerline of the base model: (a) dimensional value; (b) dimensionless value.(V = 0.139 and 0.293 m/s for Ho/P = 0.1 and 0.2, respectively).

Water 2021 , 21 Figure 12 .
Figure 12.Contours of Θ for the base model (upper figures) and the Rec-B2 model (lower figures).

Figure 12 .
Figure 12.Contours of Θ for the base model (upper figures) and the Rec-B2 model (lower figures).

Water 2021 , 21 Figure 12 .
Figure 12.Contours of Θ for the base model (upper figures) and the Rec-B2 model (lower figures).

Figure 17 .
Figure 17.Variations of (a) local submergence length and (b) interference length.

21 Figure 18 .
Figure 18.Comparison between the results of the present and previous studies (all geometries have a rectangular plan shape and identical overhang ratio of Bi/B = Bo/B = 0.25).

Figure 18 .
Figure 18.Comparison between the results of the present and previous studies (all geometries have a rectangular plan shape and identical overhang ratio of B i /B = B o /B = 0.25).

Table 4 .
Volume of PKW bodies (C) in the experimental scale with three keys.

Figure 21 .
Figure 21.Relative efficiency of PKWs versus: (a) H o /P and A/w (subscripts of B3 and B1 respectively refer to B/w = 3 and 1), respectively;); (b) H o /P and B/w (subscripts of Tri, Tra, and Rec respectively refer to A/w = 0, 0.25, and 0.5, respectively).

Table 4 .
Volume of PKW bodies (C) in the experimental scale with three keys.

Table 1 .
Different data measurements (denoted by the ■ Symbol) for PKW Models with various apex width ratios (A/w) and length magnification ratios (B/w).

) α (°) Inlet Keys Outlet Keys Head-Discharge Data Water Surface Profiles Velocit y Data Nappe Aeration Interference Length
Table cates the specifications of all of the tested weirs along with the measured paramete noted by the ■ symbol).

Table 1 .
Different data measurements (denoted by the ■ Symbol) for PKW Models with various apex width ratios (A/w and length magnification ratios (B/w).

Table 1 .
Different data measurements (denoted by the Symbol) for PKW Models with various apex width ratios (A/w) and length magnification ratios (B/w).aSpecifications of the base model:

Table 2 .
Specifications of velocity measurement planes

Table 2 .
Specifications of velocity measurement planes.
Water 2021, 13, x FOR PEER REVIEW 9 of 2

Table 2 .
Specifications of velocity measurement planes Water 2021, 13, x FOR PEER REVIEW 9 of 2

Table 2 .
Specifications of velocity measurement planes Water 2021, 13, x FOR PEER REVIEW 9 of 2

Table 2 .
Specifications of velocity measurement planes Water 2021, 13, x FOR PEER REVIEW 9 of 2

Table 2 .
Specifications of velocity measurement planes

Table 3 .
Empirical constants of the discharge coefficient equation.