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Article

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

1
Urban Water Group, San Diego State University, San Diego, CA 92182, USA
2
Water Engineering Research Institute, Faculty of Civil and Environmental Engineering, Tarbiat Modares University, Tehran P.O. Box 14115-114, Iran
3
Faculty of Engineering, Kharazmi University, Tehran 15719-14911, Iran
4
Water Engineering and Hydraulic Structures, Faculty of Civil and Environmental Engineering, Tarbiat Modares University, Tehran P.O. Box 14115-114, Iran
*
Author to whom correspondence should be addressed.
Water 2021, 13(15), 2108; https://doi.org/10.3390/w13152108
Submission received: 13 June 2021 / Revised: 28 July 2021 / Accepted: 29 July 2021 / Published: 31 July 2021
(This article belongs to the Special Issue Advances in Spillway Hydraulics: From Theory to Practice)

Abstract

:
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 (Ho/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 Ho/P values (Ho/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 Ho/P, B/w, and A/w. According to the obtained results, while the discharge coefficient is a decreasing function of A/w in Ho/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 (Ho/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 Ho/P > 0.5), the efficiency a rectangular PKW exceeds that of the other shapes.

1. 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 (Bb 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].
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 (Ho/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 (Bi/Bo), 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 (Ho/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 Ho/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 Ho/P < 0.18 and Ho/P > 0.35, respectively [44]. The corresponding values for rectangular PKWs were reported as Ho/P < 0.15 and Ho/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 (Ho/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.

2. Effective Parameters

The PKW discharge can be determined using the following equation [16]:
Q = 2 3 C d L 2 g H o 1.5
where Q = weir discharge, Cd = discharge coefficient, L = total length of PKW, g = acceleration due to gravity, and Ho = total upstream head = h + V2/2g (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:
Q = f ( H o ,   P ,   B ,   w ,   B i ,   B o ,   A ,   D ,   N ,   t ,   g ,   ρ ,   μ ,   σ ,   S o ,   S e )
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:
C d = φ ( R ,   W ,   H o P ,   B w ,   w P ,   B i B ,   B o B ,   A w ,   A D ,   t P ,   N ,   S o ,   S e )
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:
C d = φ ( H o P ,   B w ,   A w )
The present study analyzes the effects of these parameters on the performance of PKWs.

3. 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 0.020−0.185 m3/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 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).

4. Results and Discussion

4.1. 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].
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 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 Ho/P to pass a larger discharge over the weir. However, the dimensionless velocity (u/V) in Figure 7b 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).
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 9a).
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 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.

4.2. 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 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 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.
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 interference length (Bint) as the summation of the local submergence length (Bsub) and the standing wave length (Bwav) 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).
The variations of Bint/B and Bsub/B are presented in Figure 17 (Bwav/B can be also determined by subtracting these two parameters). As expected, both parameters generally increase with an increase in Ho/P. However, their increasing trends may stop (or reverse) with smaller values of A/w and B/w when 0.3 < Ho/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 Bint/B curve of the Rec-B1 model is because in the low head 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 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 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].

4.3. 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 half-round 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 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 Cd regardless of having a lower B/w value (the decreasing trend of Cd 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 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 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 Cd when 0.3 < Ho/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 a1 to a4 are empirical coefficients (obtained using the least-squares method). These coefficients along with the coefficient of determination (R2) and the average and maximum errors (Errorave and Errormax) are presented in Table 3. The minimum R2 of 0.992 and the maximum error of 2.6% confirm that there is a strong agreement between the measured and calculated Cd values of the tested models (plotted on Figure 19).
C d = a 1 ( H o P ) 3 + a 2 ( H o P ) 2 + a 3 ( H o P ) + a 4
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 A/w diminish at higher Ho/P values (more intense nappe interference). For example, by increasing the magnification ratio from B/w = 1 to 3, the efficiency can be improved up to 80% when Ho/P = 0.2, while the gains are limited to ~25% in high head conditions. It is also shown in Figure 21a that the increase in efficiency caused by increasing B/w is more noticeable for lower values of A/w (lower inflow contraction). According to Figure 21b, although the efficiency a rectangular PKW is the most for Ho/P > 0.5 and B/w = 1, a trapezoidal PKW has the most-efficient plan shape when Ho/P > 0.25 and B/w ≥ 2 ([corresponding to the majority of PKW prototypes across the world [56]).]. In addition, Table 4 shows that the body volume (C) of a trapezoidal shape is less than that of a rectangular one. Thus, 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.

5. Conclusions

This paper studies the flow behavior around 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, 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 labyrinth weir;
  • The parameter Bsub/B increases in lower A/w values. Although the standing wave is not formed in a triangular PKW, its Bint/B is larger than that of a rectangular one in low head conditions (Ho/P < 0.5). However, a reverse trend may happen in high head conditions (more pronounced in lower values of B/w). It was shown that the parameters Bsub/B and Bint/B have a decreasing and increasing trend with respect to B/w and Ho/P, respectively;
  • Although the discharge coefficient has a certain decreasing trend with respect to Ho/P and B/w, it shows a dual trend with the plan shape. Compared to a triangular shape, the higher Cd of a trapezoidal PKW at Ho/P < 0.4 implies that Cd may have an increasing trend with respect to A/w in low head conditions (more data on Ho/P < 0.15 are needed to also include a rectangular shape in this conjecture and fully confirm it). However, it has a decreasing trend with the apex width ratio of Ho/P > 0.4. According to the obtained results, PKW efficiency gains monotonically diminish by increasing Ho/P. For example, by changing Ho/P = 0.2 to 0.7, the benefits of increasing the magnification ratio from B/w = 1 to 3 reduces to less than a third (around 25% gain in discharge capacity);
  • While the rectangular plan shape is traditionally considered in PKW prototypes and scientific models, its efficiency is the most only for low-length PKWs (B/w = 1) under high head conditions (Ho/P > 0.5). By assuming that the cost of PKWs is proportional to their body volume, a trapezoidal shape can be selected as the optimal one. Compared to a rectangular shape, a trapezoidal PKW provides a higher discharge efficiency (~5%) with a smaller body volume (~7%) in the broad ranges of Ho/P > 0.25 and B/w ≥ 2, which correspond to the majority of worldwide prototypes. A detailed cost estimation on the different elements of PKWs is needed to confirm this result for a specific project.

Author Contributions

Y.S., methodology, investigation, formal analysis, writing—original draft; H.T.-D., formal analysis, writing—review and editing, supervision; M.G., methodology, resources, writing—review and editing, supervision; M.M., formal analysis, writing—review and editing; R.Z., methodology, investigation. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data presented in this study are available upon request from the corresponding author.

Conflicts of Interest

The authors declare no conflict of interest.

Notations

The following symbols are used in this paper.
Ainlet key (or downstream) apex width
Bweir length in streamwise direction
Bbfootprint length
Biinlet key (or downstream) overhang length
Bintinterference length
Booutlet key (or upstream) overhang length
Bsublocal submergence length
Bwavstanding wave length
Cweir body volume
CCcorrelation coefficient for velocity data measurements
Cddischarge coefficient
Doutlet key (or upstream) apex width
dflow depth
gacceleration due to gravity
happroach flow depth over the weir
Hototal upstream head
Ltotal length of PKW
Lkeylength of a single key
Nnumber of keys
Pweir heigh
Qweir discharge
qunit discharge
RReynolds number
Rcrestradius of rounded crest
Secrest shape representative
SNRsignal-to-noise ratio
Solongitudinal bed slope
tweir wall thickness
UVelocity vector
u, vtime-averaged velocity in the respective x and y-direction
Vaverage cross-sectional velocity of approaching flow
Wweir (or channel) width
WWeber number
wcycle width
x, y, zlongitudinal, transverse, and vertical directions, respectively
αsidewall angle
εdischarge efficiency
μdynamic viscosity
ρfluid density
σsurface tension
Θangle between velocity vectors and weir sidewall

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Figure 1. PKW schematic: (a) plan view; (b) section view A-A.
Figure 1. PKW schematic: (a) plan view; (b) section view A-A.
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Figure 2. Overview of implemented test facilities: (a) the 3D view from downstream; (b) the plan view; (c) the side view.
Figure 2. Overview of implemented test facilities: (a) the 3D view from downstream; (b) the plan view; (c) the side view.
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Figure 3. Velocity measurements near the vertical walls and sloped floors (upstream view, base model (weir height (P) = 325 mm)).
Figure 3. Velocity measurements near the vertical walls and sloped floors (upstream view, base model (weir height (P) = 325 mm)).
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Figure 4. A Three-dimensional view of the tested models (1 key, downstream view).
Figure 4. A Three-dimensional view of the tested models (1 key, downstream view).
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Figure 5. Water surface profiles in different models with various headwater ratios (Ho/P): (a) base model; (b) Tri-B2 and Rec-B2 models; (c) Tra-B1, Tra-B2, and Tra-B3 models; (d) Tri-B1, Rec-B1, Tri-B3, and Rec-B3 models.
Figure 5. Water surface profiles in different models with various headwater ratios (Ho/P): (a) base model; (b) Tri-B2 and Rec-B2 models; (c) Tra-B1, Tra-B2, and Tra-B3 models; (d) Tri-B1, Rec-B1, Tri-B3, and Rec-B3 models.
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Figure 6. Comparison of the width of rectangular (A/w = 0.5) and triangular (A/w = 0) plan shapes.
Figure 6. Comparison of the width of rectangular (A/w = 0.5) and triangular (A/w = 0) plan shapes.
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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. 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).
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Figure 8. Dimensionless velocity profiles in inlet key centerlines (V = 0.140, 0.293, 0.197, and 0.454 m/s at the base model with Ho/P = 0.1 and 0.2 and the Tri-B2 model with Ho/P = 0.2 and 0.5, respectively).
Figure 8. Dimensionless velocity profiles in inlet key centerlines (V = 0.140, 0.293, 0.197, and 0.454 m/s at the base model with Ho/P = 0.1 and 0.2 and the Tri-B2 model with Ho/P = 0.2 and 0.5, respectively).
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Figure 9. Dimensionless velocity profiles in the inlet key centerline: (a) Ho/P = 0.5 (V = 0.337 and 0.454 m/s at the Tri-B1 and Tri-B2 models, respectively); (b) Ho/P = 0.2 (V = 0.293 and 0.197 m/s at the base and Tri-B2 models, respectively).
Figure 9. Dimensionless velocity profiles in the inlet key centerline: (a) Ho/P = 0.5 (V = 0.337 and 0.454 m/s at the Tri-B1 and Tri-B2 models, respectively); (b) Ho/P = 0.2 (V = 0.293 and 0.197 m/s at the base and Tri-B2 models, respectively).
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Figure 10. Experimental observations: (a) dye trace in Rec-B2 (Ho/P = 0.65, left-to-right flow direction); (b) dye trace in Tri-B2 (Ho/P = 0.65); (c) water surface undulations in Rec-B3 (Ho/P = 0.147).
Figure 10. Experimental observations: (a) dye trace in Rec-B2 (Ho/P = 0.65, left-to-right flow direction); (b) dye trace in Tri-B2 (Ho/P = 0.65); (c) water surface undulations in Rec-B3 (Ho/P = 0.147).
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Figure 11. Velocity vectors in (a) Rec-B2 model; (b) base model.
Figure 11. Velocity vectors in (a) Rec-B2 model; (b) base model.
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Figure 12. Contours of Θ for the base model (upper figures) and the Rec-B2 model (lower figures).
Figure 12. Contours of Θ for the base model (upper figures) and the Rec-B2 model (lower figures).
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Figure 13. Variations of Θ along the PKW sidewall centerline (Ho/P = 0.7).
Figure 13. Variations of Θ along the PKW sidewall centerline (Ho/P = 0.7).
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Figure 14. Velocity contours in Rec-B2 (Ho/P = 0.7): (a) u-component above the upstream normal wall (P6, upstream view); (b) v-component above the side wall (P5, side view).
Figure 14. Velocity contours in Rec-B2 (Ho/P = 0.7): (a) u-component above the upstream normal wall (P6, upstream view); (b) v-component above the side wall (P5, side view).
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Figure 15. Variations of flow aeration regime with Ho/P, A/w, and B/w.
Figure 15. Variations of flow aeration regime with Ho/P, A/w, and B/w.
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Figure 16. Nappe interference characteristics: (a) Rec-B1, Ho/P = 0.30; (b) Tra-B2, Ho/P = 0.30; (c) Tri-B2, Ho/P = 0.49; ( Water 13 02108 i005 water surface profile in outlet key centerline; Water 13 02108 i006 water surface profile in inlet key centerline).
Figure 16. Nappe interference characteristics: (a) Rec-B1, Ho/P = 0.30; (b) Tra-B2, Ho/P = 0.30; (c) Tri-B2, Ho/P = 0.49; ( Water 13 02108 i005 water surface profile in outlet key centerline; Water 13 02108 i006 water surface profile in inlet key centerline).
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Figure 17. Variations of (a) local submergence length and (b) interference length.
Figure 17. Variations of (a) local submergence length and (b) interference length.
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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. 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).
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Figure 19. Variations of the discharge coefficient (Cd): (a) B/w = 1; (b) B/w = 2; (c) B/w = 3.
Figure 19. Variations of the discharge coefficient (Cd): (a) B/w = 1; (b) B/w = 2; (c) B/w = 3.
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Figure 20. Efficiency curves.
Figure 20. Efficiency curves.
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Figure 21. Relative efficiency of PKWs versus: (a) Ho/P and A/w (subscripts of B3 and B1 respectively refer to B/w = 3 and 1), respectively;); (b) Ho/P and B/w (subscripts of Tri, Tra, and Rec respectively refer to A/w = 0, 0.25, and 0.5, respectively).
Figure 21. Relative efficiency of PKWs versus: (a) Ho/P and A/w (subscripts of B3 and B1 respectively refer to B/w = 3 and 1), respectively;); (b) Ho/P and B/w (subscripts of Tri, Tra, and Rec respectively refer to A/w = 0, 0.25, and 0.5, respectively).
Water 13 02108 g021
Figure 22. PKW discharge (Q) and body volume (C) versus B/w: (Ho/P = 0.5, A/w = 0.25).
Figure 22. PKW discharge (Q) and body volume (C) versus B/w: (Ho/P = 0.5, A/w = 0.25).
Water 13 02108 g022
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).
ModelB/wA/wL (m)α (°)Inlet Keys Outlet KeysHead-Discharge Data
Water Surface ProfilesVelocity DataNappe AerationInterference LengthStanding Wave Characteristics
Base a2.050.503.7500
Tri-B1 b1.000.001.61326.6
Tra-B11.000.251.85114.7
Rec-B11.000.502.1780
Tri-B22.000.003.02314.0
Tra-B22.000.253.3277.3
Rec-B22.000.503.6780
Tri-B33.000.004.4919.5
Tra-B33.000.254.8194.8
Rec-B33.000.505.1780
a Specifications of the base model: w/Po = 1.19, Bi/B = Bo/B = 0.25, A/D = 1.25, t/Po = 15.75, N = 2, So = 0, and half-round crest shape [having P = 325 mm (including Pp = 10 mm)]. b Specifications of the models Tri-B1 to Rec-B3: w/P = 1.25, Bi/B = Bo/B = 0.25, A/D = 1, t/P = 16.6, N = 3, So= 0, and the quarter-round and flat crest shapes for side and normal walls, respectively [having P = 200 mm (Pp = 0)].
Table 2. Specifications of velocity measurement planes.
Table 2. Specifications of velocity measurement planes.
ModelPlane No.Orientationx/By/wz/PNode NumbersSketch
BaseP1Horizontal0.00–0.800.50–0.740.86230 Water 13 02108 i001
P2Horizontal0.00–0.800.50–0.740.95430
Tri-B1P1Horizontal0.18–0.440.67–0.860.75056 Water 13 02108 i002
Tri-B2P1Horizontal0.18–0.440.61–0.870.75068 Water 13 02108 i003
Rec-B2P1Horizontal−0.05–0.650.55–0.700.750120 Water 13 02108 i004
P2Horizontal0.00–0.800.55–0.731.10030
P3Horizontal0.00–0.800.55–0.731.20030
P4Vertical0.15–0.450.2940.55–0.85128
P5Vertical0.15–0.250.7261.13–1.2877
P6Vertical0.0000.78–0.941.13–1.3381
Table 3. Empirical constants of the discharge coefficient equation.
Table 3. Empirical constants of the discharge coefficient equation.
ModelB/wA/wHo/Pa1a2a3a4R2Errorave (%)Errormax (%)
Base2.050.500.10‒0.45−0.16481.6960−1.62800.72530.99940.30.8
Tri-B11.000.000.35‒1.00−0.02130.4052−0.81650.81370.99750.51.3
Tra-B11.000.250.35‒1.00−0.41181.2077−1.34520.91060.99910.20.8
Rec-B11.000.500.20‒1.00−0.33880.8687−0.93680.71230.99820.61.8
Tri-B22.000.000.15‒0.85−1.13161.9932−1.40440.70730.99361.32.3
Tra-B22.000.250.15‒0.85−0.61281.4386−1.31580.72140.99660.81.8
Rec-B22.000.500.15‒0.85−1.28092.5798−1.90190.76250.99980.30.9
Tri-B33.000.000.15‒0.80−0.69191.5103−1.28620.62480.99960.41.4
Tra-B33.000.250.15‒0.80−1.24692.5371−1.90930.73690.99930.72.0
Rec-B33.000.500.15‒0.80−1.36022.7110−1.96340.70540.99890.91.7
Table 4. Volume of PKW bodies (C) in the experimental scale with three keys.
Table 4. Volume of PKW bodies (C) in the experimental scale with three keys.
B/wC × 103 (m3)
A/w = 0.00A/w = 0.25A/w = 0.50
19.310.311.2
218.620.121.7
327.830.032.2
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Sangsefidi, Y.; Tavakol-Davani, H.; Ghodsian, M.; Mehraein, M.; Zarei, R. Hydrodynamics and Free-Flow Characteristics of Piano Key Weirs with Different Plan Shapes. Water 2021, 13, 2108. https://doi.org/10.3390/w13152108

AMA Style

Sangsefidi Y, Tavakol-Davani H, Ghodsian M, Mehraein M, Zarei R. Hydrodynamics and Free-Flow Characteristics of Piano Key Weirs with Different Plan Shapes. Water. 2021; 13(15):2108. https://doi.org/10.3390/w13152108

Chicago/Turabian Style

Sangsefidi, Yousef, Hassan Tavakol-Davani, Masoud Ghodsian, Mojtaba Mehraein, and Reza Zarei. 2021. "Hydrodynamics and Free-Flow Characteristics of Piano Key Weirs with Different Plan Shapes" Water 13, no. 15: 2108. https://doi.org/10.3390/w13152108

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