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Methodological Proposal for the Hydraulic Design of Labyrinth Weirs

Methodological Proposal for the Hydraulic Design of Labyrinth Weirs water Article Methodological Proposal for the Hydraulic Design of Labyrinth Weirs 1 2 , 3 Erick Dante Mattos-Villarroel , Waldo Ojeda-Bustamante * , Carlos Díaz-Delgado , 3 4 1 Humberto Salinas-Tapia , Jorge Flores-Velázquez and Carlos Bautista Capetillo Campus Siglo XXI, Autonomous University of Zacatecas “Francisco García Salinas”, Road Zacatecas-Guadalajara Km 6, Zacatecas 98160, Mexico Research and Postgraduate Division, Chapingo Autonomous University, Chapingo, Texcoco 56230, Mexico Inter-American Institute of Technology and Water Sciences, Autonomous University of the State of Mexico, Road Toluca-Atlacomulco Km 14.5, Toluca 50200, Mexico Postgraduate College, Coordination of Hydrosciences, Road Mexico-Texcoco Km 36.5, Texcoco 62550, Mexico * Correspondence: w.ojeda@riego.mx Abstract: A labyrinth weir allows for higher discharge capacity than conventional linear weirs, especially at low hydraulic heads. In fact, this is an alternative for the design or rehabilitation of spillways. It can even be used as a strategy in problems related to dam safety. A sequential design method for a labyrinth weir is based on optimal geometric parameters and the results of discharge flow analysis using Computational Fluid Dynamics and the experimental studies reported in the literature. The tests performed were for weirs with values of H /P  0.8 and for angles of the cycle sidewall of 6  a  20 . The results of the discharge coefficient are presented as a family of curves, which indicates a higher discharge capacity when H /P  0.17. Four aeration conditions are identified with higher discharge capacity when the nappe is adhering to the downstream face of the weir wall and lower discharge capacity when the nappe is drowned. Unstable flow was present when 12  a  20 , with a greater presence when the nappe was partially aerated and drowned. The interference of the nappe is characterized and quantified, reaching up to 60% of the length between the apex, and a family of curves is presented as a function of H /P in this respect. Finally, a spreadsheet and a flowchart are proposed to support the design of the labyrinth type weir. Citation: Mattos-Villarroel, E.D.; Ojeda-Bustamante, W.; Díaz-Delgado, Keywords: labyrinth weir; Computational Fluid Dynamics (CFD); spillways discharge capacity; C.; Salinas-Tapia, H.; spillway weir design Flores-Velázquez, J.; Bautista Capetillo, C. Methodological Proposal for the Hydraulic Design of Labyrinth Weirs. Water 2023, 15, 722. https://doi.org/10.3390/w15040722 1. Introduction Labyrinth weirs are polygonal hydraulic structures (Figure 1) used to increase dis- Academic Editor: Diana De Padova charge capacity, for a fixed width, reducing hydraulic head relative to linear weirs. Their Received: 14 November 2022 hydraulic performance characteristics make them an efficient and cost-effective alternative Revised: 2 February 2023 for spillway weir design or rehabilitation. This structure makes it possible to increase the Accepted: 5 February 2023 storage volume in a reservoir and to control the water level. Figure 1 shows the geometrical Published: 11 February 2023 parameters of a labyrinth weir, where W is the width of the channel (m), w is the width of the weir cycle (m), t is the width of the weir wall (m), D is the external length of the apex (m), A is the internal length of the apex (m), l is the length of the cycle sidewall (m), a is the angle of the cycle sidewall with respect to the flow direction ( ), B is the distance Copyright: © 2023 by the authors. between apexes (m), P is the height of the weir (m), h is the piezometric head (m), H is the Licensee MDPI, Basel, Switzerland. 3 1 total head (m), and Q is the design flow (m s ). This article is an open access article The design of a labyrinth weir is laborious because its discharge capacity is simultane- distributed under the terms and ously affected by several factors, including the approach conditions and the geometry of conditions of the Creative Commons the weir [2]. According to Bilhan, Emiroglu, and Miller (2016) [3], optimizing the geometry Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ variables involved in the design of a labyrinth weir is an engineering challenge in which the 4.0/). following must be determined: (a) the configuration of the cycles, (b) the shape of the crest, Water 2023, 15, 722. https://doi.org/10.3390/w15040722 https://www.mdpi.com/journal/water Water 2023, 15, 722 2 of 29 and (c) the orientation of the weir. For any type of weir, its geometry, particularly the shape of the crest profile, influences the value of the discharge coefficient. Willmore (2004) [4] in- dicated that half-round crest profiles are more efficient than sharp-crest and quarter-round profiles, because they allow the nappe to remain adhered to the wall of the structure for small heads. Typically, the cycle’s apex facilitates the concrete construction of a labyrinth weir. From a hydraulic perspective, a labyrinth weir with a smooth upstream transition is relatively more efficient than the abrupt transition presented by a trapezoidal labyrinth Water 2023, 15, x FOR PEER REVIEW 2 of 32 weir [5]. In this regard, Tullis and Young (2005) [6] reported an increase in discharge efficiency at Brazos Dam (Wako, TX, USA) by creating a smooth transition between the flow and the circular apexes of the labyrinth spillway cycles. Figure 1. Geometric parameters of a labyrinth type weir (Mattos-Villarroel et al., 2021 [1]). Figure 1. Geometric parameters of a labyrinth type weir (Mattos-Villarroel et al., 2021 [1]). The design of a labyrinth weir is laborious because its discharge capacity is simulta- Families of discharge coefficient curves (C ) for the design of labyrinth weirs have neously affected by several factors, including the approach conditions and the geometry been determined from physical models of various prototype structures, examples of which of the weir [2]. According to Bilhan, Emiroglu, and Miller (2016) [3], optimizing the geom- are as follows: Avon and Woronova [2]; Harrezza, Dungo, Keddara, Alijó, Gema, and São etry variables involved in the design of a labyrinth weir is an engineering challenge in Domingo [7]; Hyrum [8]; Ute [9]; Lake Brazos [6]; and Lake Townsend [10]. The evolution which the following must be determined: (a) the configuration of the cycles, (b) the shape of commonly used and documented design methods can be summarized in the following of the crest, and (c) the orientation of the weir. For any type of weir, its geometry, partic- sequence: (a) Hay and Taylor (1970) [11]; (b) Darvas (1971) [2]; (c) Hinchliff and Houston ularly the shape of the crest profile, influences the value of the discharge coefficient. Will- (1984) m[o 12 re]; (2(d) 004) Lux [4] in (1989) dicated [13 th]; at (e) half Magalh -round cr ães est and profiLor les a ena re m (1989) ore efficient [7]; (f) thT an ullis, sharAmanian, p-crest and W an aldr d qon uarte (1995) r-roun [14 d pr ]; o and files, (g) beCr cau ookston se they and allow T ullis the na (2012) ppe to [re 15 m ].ain However adhered, to the thr e esults wall of of the structure for small heads. Typically, the cycle’s apex facilitates the concrete con- Idrees and Al-Ameri (2022) [16] showed that common design equations did not take into struction of a labyrinth weir. From a hydraulic perspective, a labyrinth weir with a smooth account all the parameters that affect the performance of a labyrinth weir such as geometry upstream transition is relatively more efficient than the abrupt transition presented by a and flow conditions. trapezoidal labyrinth weir [5]. In this regard, Tullis and Young (2005) [6] reported an On the other hand, physical and numerical modeling are used as complementary tools increase in discharge efficiency at Brazos Dam (Wako, TX, USA) by creating a smooth to improve the design of hydraulic structures. Indeed, Computational Fluid Dynamics (CFD) transition between the flow and the circular apexes of the labyrinth spillway cycles. makes it possible to obtain, by means of numerical techniques, adequate results for the Families of discharge coefficient curves (Cd) for the design of labyrinth weirs have equations that predict the behavior of any flow. In general, the equations to be solved with been determined from physical models of various prototype structures, examples of CFD are the Navier–Stokes equations. Thus, numerical modeling facilitates the hydraulic which are as follows: Avon and Woronova [2]; Harrezza, Dungo, Keddara, Alijó, Gema, evaluation of structures, such as weirs, to obtain information on pressure fields and velocities and São Domingo [7]; Hyrum [8]; Ute [9]; Lake Brazos [6]; and Lake Townsend [10]. The under ev d o ilut ffeiro en nt of g com eom m et orn ic lya use ndd h a yn dd r a du olcumen ic cond ted iti o dn esi s.gn It m iseth im op do sr ca tan n t be to sum hig m ha liriz ghed t t h in e th us ee of CFD a fo slla owi too nl g iseq n th uence e gen : ( ear)a Ha tioy nao nfdi n Ta fo yl rm or a (t 1i9 o7n 0)t o [11 su ]; p (b p )o Da rt rv de ac s is (1 io 97 n1 -) m [2 a]k ; i (n c)g Hin and chl th iff e a d ne d s ign Houston (1984) [12]; (d) Lux (1989) [13]; (e) Magalhães and Lorena (1989) [7]; (f) Tullis, of hydraulic structures. In particular, the performance of CFD in the modeling of flow over Amanian, and Waldron (1995) [14]; and (g) Crookston and Tullis (2012) [15]. However, linear weirs, particularly the labyrinth type, stands out, as has been demonstrated by several the results of Idrees and Al-Ameri (2022) [16] showed that common design equations did researchers [1,17–22]. Ben Said et al. (2022) [23] using CFD proved that the discharge capacity not take into account all the parameters that affect the performance of a labyrinth weir of a labyrinth weir can increase as its downstream channel bed level decreases. Samadi et al. such as geometry and flow conditions. (2022) [24] experimentally and numerically investigated the effects of geometric parameters On the other hand, physical and numerical modeling are used as complementary on the efficiency in triangular and trapezoidal labyrinth weirs. tools to improve the design of hydraulic structures. Indeed, Computational Fluid Dynam- ics (CFD) makes it possible to obtain, by means of numerical techniques, adequate results 1.1. Discharge Flow Characteristics for the equations that predict the behavior of any flow. In general, the equations to be The behavior and characteristics of the discharge flow presented by the complex solved with CFD are the Navier–Stokes equations. Thus, numerical modeling facilitates geometry of the labyrinth weir must be considered in its hydraulic analysis and design. the hydraulic evaluation of structures, such as weirs, to obtain information on pressure fields and velocities under different geometric and hydraulic conditions. It is important This analysis must consider the aeration conditions, the instability of the discharge flow, to highlight the use of CFD as a tool in the generation of information to support decision- making and the design of hydraulic structures. In particular, the performance of CFD in the modeling of flow over linear weirs, particularly the labyrinth type, stands out, as has been demonstrated by several researchers [1,17–22]. Ben Said et al. (2022) [23] using CFD proved that the discharge capacity of a labyrinth weir can increase as its downstream Water 2023, 15, 722 3 of 29 and the collision between the nappes that occurs between the walls of the weir cycles. It should be noted that each of these factors individually affects the value of the discharge coefficient C and the efficiency of the weir. However, the main challenge is to know the joint effect of all these factors on the discharge coefficient. 1.1.1. Aeration Conditions The efficiency and discharge capacity of the labyrinth weir also depends on the aeration regime of the nappe, which is influenced by the shape of the crest, the level of the hydraulic head, the height of the weir, the flow path over the crest, the turbulence, and the pressure under the nappe. The increase in discharge flow over the weir generates different aeration conditions; these have been identified by some researchers and have already been included in de- sign curves, through the value of the discharge coefficient [25,26]. Lux and Hinchliff (1985) [25] identified three types of aeration: aerated, transitional (partially aerated), and suppressed (drowned). However, for Falvey (2003) [27], there are four types of aera- tion conditions: cavity, atmospheric, sub-atmospheric, and pressure. On the other hand, Crookston and Tullis (2012) [26] also reported four types of aeration: clinging, aerated, par- tially aerated, and drowned, and indicated that aeration conditions can produce pressure fluctuations at the sidewalls of the weir, low frequency sound, and vibrations. In addition, previous studies point to the labyrinth weir as an excellent aeration control structure. Ex- amples of this are the work of Hauser (1996) [28], who describes the methods for the design of this type of weir, taking into consideration the aeration conditions during discharge. Wormleaton and Soufani (1998) [29] and Wormleaton and Tsan (2000) [30] found that a rectangular labyrinth weir has better aeration efficiency than a triangular labyrinth weir and that the latter is better than a linear one. 1.1.2. Nappe Instability Under certain flow conditions and weir geometry, the nappe becomes unstable, and vibrations occur in the hydraulic structure. Several researchers have conducted studies on nappe instability. Crookston and Tullis (2012) [26] defined nappe instability as a nappe with an oscillating trajectory. These authors, supported by experimental observations, reported that the flow streamlines under this condition are helical, adjacent, and parallel to the weir walls. Furthermore, these nappes occur momentarily with changes of aeration condition, most frequently during the aerated and partially aerated condition, when their presence causes vibrations that threaten the safety of the structure. It is recommended that these conditions should be considered during the design of the weir and are to be avoided. According to Casperson (1995) [31], vibrations are easily felt by touch and the sound can be heard over a kilometer away. For Naudascher and Rockwell (2017) [32], the vibrations are attributed to inadequate aeration below the discharge flow and indicate that an unventilated air pocket behind the nappe can amplify the instability of the weir. Similarly, according to their research, the three-dimensional characteristic of the flow during discharge, at the point of detachment and the height of fall, may be a significant parameter in the presence of vibrations. Falvey (2003) [27] even pointed out that the vibration of the nappe occurs when the weir operates at low hydraulic heads, in the range 0.01 h/P 0.06. Some researchers recommend the use of splitters, placed vertically and normal to the flow to reduce vibrations [12,33]. However, because several splitters are required, this solution was not recommended by Falvey (2003) [27]. To eliminate vibrations, the Metropolitan Water, Sewerage, and Drainage Board (1980) [34] conducted studies on a physical model of the Avon Dam, located in Australia, where they increased the crest roughness. However, with only a 15 mm increase in crest height, the discharge decreased by 2%. 1.1.3. Nappe Interference Nappe interference refers to the collision between nappes at the upstream apexes of the weir. This hydraulic phenomenon decreases the efficiency of the weir. The length of Water 2023, 15, 722 4 of 29 the interference depends on the width apex, the shape of the crest, the weir height, the hydraulic head, and the aeration conditions. Indlekofer and Rouve (1975) [35] studied the nappe interference in single-cycle tri- angular weirs whose sidewalls are perpendicular to the channel walls, also known as corner weirs. These authors identified a perturbed region, where the nappes from each weir sidewall collide, and a second region, where streamlines are perpendicular to the weir wall, similar to the discharge over a linear weir. Finally, they found an empirical relationship between the average discharge coefficient of the disturbed area, the theoretical crest length of the disturbed region, and the discharge to compare the efficiency of a corner weir with a linear weir. Lux (1989) [13] recommended that the ratio of the apex width to the weir cycle width should be less than 0.0765, so that the effects of the collisions of the nappes do not generate large reductions in the performance of the trapezoidal labyrinth weir. Falvey (2003) [27] developed an empirical model that considered the data relationship of Indlekofer and Rouve (1975) [35], and then developed a second equation based on the analysis of experimental data from a labyrinth weir. However, this author did not indicate which of the two proposed equations was the most appropriate, but the significant influence of the nappe interference on aeration in triangular labyrinth weirs was highlighted. Based on the quantity of movement, Osuna 2000 [36] analyzed the hydraulic behav- ior of a weir, located inside a channel and oblique to the upstream flow direction, and proposed an equation that allows for the calculation of the flow per unit length of the weir. The results indicated that the direction of flow at the outlet of the weir depends on the ratio of the thickness of the nappe contracted over the weir and the height of free- surface water, upstream from a point sufficiently far from the weir. In the last decade, Granell and Toledo (2010) [37] adapted the mathematical model of Osuna (2000) [37] to labyrinth weirs in order to obtain the nappe’s collision length. 1.1.4. Drowning It has been observed that the drowning phenomenon in labyrinth weirs has a behavior similar to that in the operation of linear weirs [38]. However, Taylor (1968) [39] concluded that drowning effects are less significant for labyrinth weirs than for linear weirs. The linear weir drowning analysis developed by Villemonte (1947) [40] was the most applied method for labyrinth weirs under drowning conditions. However, at the beginning of the century, Tullis et al. (2007) [38] developed a dimensionless relationship for drowning heads in labyrinth weirs, obtaining an average error of up to 0.9%. Figure 2 shows the function describing drowning at the weir, which was divided into three sections to better explain its behavior and is represented by Equations (1)–(3). 4 2 H H H H d d d = 0.3320 + 0.2008 + 1; 0   1.53 (1) H H H H T T T T H H H d d = 0.9379 + 0.2174; 1.53   3.5 (2) H H H T T T H = H ; 3.5  (3) H* is the upstream total head when the weir is drowned (m), H is the upstream total head when the weir is not drowned (m), and H is the downstream total head of the weir in the drowning state (m). Based on the above, and due to the complexity in the hydraulic analysis of a labyrinth weir, the objective of this work is focused on proposing a sequential design that facilitates the conception of a labyrinth weir, based on the results of numerical modeling in CFD and by considering the relationships between the geometric variables of the weir and the simultaneous effect of the hydraulic phenomena that occur on the nappe. Water 2023, 15, x FOR PEER REVIEW 5 of 32 Water 2023, 15, 722 5 of 29 𝐻 = 𝐻 ; 3.5 ≤ (3) Figure 2. Submergence limits (adapted from Tullis et al., 2007 [38]). Figure 2. Submergence limits (adapted from Tullis et al., 2007 [38]). 2. Materials and Methods H* is the upstream total head when the weir is drowned (m), HT is the upstream total 2.1. Description of the Physical Model head when the weir is not drowned (m), and Hd is the downstream total head of the weir The construction of the conceptual model and its evaluation in CFD are based on in the drowning state (m). the experimental prototype reported by Crookston and Tullis (2012) [15]. The physical Based on the above, and due to the complexity in the hydraulic analysis of a labyrinth model consisted of a trapezoidal labyrinth weir with a quarter-round crest profile made of weir, the objective of this work is focused on proposing a sequential design that facilitates high-density polyethylene, located within a rectangular channel. The description of the the conception of a labyrinth weir, based on the results of numerical modeling in CFD and dimensions and geometric characteristics of the labyrinth weir are specified in Table 1. by considering the relationships between the geometric variables of the weir and the sim- ultaneous effect of the hydraulic phenomena that occur on the nappe. Table 1. Geometric characteristics of the trapezoidal labyrinth weir [15]. 2. Materials and Methods a ( ) N L (m) A (m) w (m) P (m) W (m) Crest Profile 2.1. Description of the Physical Model 15 2 4 0.038 0.617 0.305 1.235 Quarter-round The construction of the conceptual model and its evaluation in CFD are based on the experimental prototype reported by Crookston and Tullis (2012) [15]. The physical model 2.2. Numerical Solution Method consisted of a trapezoidal labyrinth weir with a quarter-round crest profile made of high- 2.2.1. Computational Fluid Dynamics (CFD) density polyethylene, located within a rectangular channel. The description of the dimen- Numerical modeling was performed using the ANSYS-FLUENT simulation software [41,42]. sions and geometric characteristics of the labyrinth weir are specified in Table 1. This software is considered to be a general purpose CFD code, which has been widely used in recent years, due to the advancement of technology, mainly computational technology. ANSYS- Table 1. Geometric characteristics of the trapezoidal labyrinth weir [15]. FLUENT allows for the modeling of flows in one phase to flows in multiple phases (multiphase) α (° ) N L (m) A (m) w (m) P (m) W (m) Crest Profile in both closed and open domains. The numerical simulation was performed with two-phase flow 15 2 4 0.038 0.617 0.305 1.235 Quarter-round models, incompressible conditions, and a free-surface interface. ANSYS-FLUENT implements the approach to surface analysis using the volume of fluid 2.2. N (VOF) umerica scheme. l Solution M The VOF ethod scheme is ideal for applications involving free-surface flows. It involves defining a volume fraction function for each of the fluids in the entire domain. 2.2.1. Computational Fluid Dynamics (CFD) The governing equations for all fluid fractions were solved using the two-phase (air- Numerical modeling was performed using the ANSYS-FLUENT simulation software water) flow model. A two-phase flow model was used to detect under pressure; air inertia [41,42]. This software is considered to be a general purpose CFD code, which has been and air–water interaction effects were neglected in the numerical model. The VOF method widely used in recent years, due to the advancement of technology, mainly computational was used to improve the accuracy of the free-surface simulations. technology. ANSYS-FLUENT allows for the modeling of flows in one phase to flows in For incompressible fluids, when the density of water is constant, the mass continuity multiple phases (multiphase) in both closed and open domains. The numerical simulation equation of motion of the fluid in Cartesian coordinates is given as Equation (4) [43]: ¶ ¶ ¶ (u A ) + v A + (w A ) = 0. (4) x y z ¶x ¶y ¶z Water 2023, 15, 722 6 of 29 where A , A , and A are the fractional areas open to the fluid in directions x, y, and x y z z, respectively. Additionally, u, v, and w are the velocity components in the directions x, y, and z, respectively. The Navier–Stokes equations, with the velocity components as momentum equations, are used to determine the 3D fluid motion in Cartesian coordinates (Equations (5)–(7)) [43]. ¶u 1 ¶u ¶u ¶u 1 ¶( p) + u A + v A + w A = + G + f (5) x y z x x ¶t V ¶x ¶y ¶z r ¶y ¶v 1 ¶v ¶v ¶v 1 ¶( p) + u A + v A + w A = + G + f (6) x y z y y ¶t V ¶x ¶y ¶z r ¶y ¶w 1 ¶w ¶w ¶w 1 ¶ p ( ) + u A + v A + w A = + G + f (7) x y z z z ¶t V ¶x ¶y ¶z r ¶z where V is the fraction volume open to the fluid; G , G , and G are the acceleration F x y z components of the body, and f , f , and f are the viscous acceleration components in the x y z directions x, y, and z, respectively. To define fluid configurations, the volume of fluid function (VOF) represents a volume of fluid per unit volume [44], as indicated in Equation (8): ¶F 1 ¶ ¶ ¶ FA u + (FA u) + FA v + (FA w) + j = F + F (8) x y z D S ¶t V ¶x ¶y ¶z x where F is the diffusion term, applied only in the turbulent mixture of a two-phase flow, and F is a source term. To determine the discharge and pressure characterization at the weir crest, the two- phase model (air and water) was applied. In addition, the standard k–# model was used to analyze the effects of turbulence, which solves two transport equations with Reynolds stresses for the turbulent kinetic energy (k) and the dissipation rate (#). The k–# turbulence model is a semi-empirical model with low computational cost; several researchers have demonstrated its advantages and excellent results for the simulation of confined, internal, and free-surface flows [19,43,45–47]. Equations (9) and (10) were used to obtain the turbulent kinetic energy and dissipation rate. The k–# model is described below [48]. ¶k ¶ku ¶ ¶k + = Dk + G # (9) e f f k ¶t ¶x ¶x ¶x i i i ¶# ¶#u ¶ ¶# # # + = D# + C G C (10) e f f 1# 2# ¶t ¶x ¶x ¶x k k k i i i where G is the generation of turbulent kinetic energy due to mean velocity gradients; DDk and D# are the effective diffusivity for the turbulent kinetic energy (k) and the e f f e f f dissipation rate (#), respectively, determined by Equations (11) and (12): Dk = n + n (11) e f f t D# = n + (12) e f f where n = C is the turbulent kinematic viscosity at each point; s is the Prandtl number t m # for # and takes the value 1.3; and the constants C , C , and C have values of 1.44, 1.92, 1# 2# m 2 2 and 0.09, respectively. G = 2n S is the turbulent kinetic energy result, and S is the strain i j i j rate tensor. For the solution of the equations, the Semi-Implicit Method for Pressure Linkek Equa- tions (SIMPLE) algorithm was used, which approaches convergence through a series of intermediate pressure and velocity fields satisfying continuity [49]. The higher order Water 2023, 15, x FOR PEER REVIEW 7 of 32 where 𝐺 is the generation of turbulent kinetic energy due to mean velocity gradients; 𝑘 and 𝐷 𝜀 are the effective diffusivity for the turbulent kinetic energy (𝑘 ) and the 𝑓𝑓𝑒 𝑓𝑓𝑒 dissipation rate (𝜀 ), respectively, determined by Equations (11) and (12): 𝐷 𝑘 = 𝜈 + 𝜈 (11) 𝑓𝑓𝑒 𝑡 𝐷 𝜀 = 𝜈 + (12) 𝑓𝑓𝑒 where 𝜈 = 𝐶 is the turbulent kinematic viscosity at each point; 𝜎 is the Prandtl num- 𝑡 𝜇 𝜀 ber for 𝜀 and takes the value 1.3; and the constants 𝐶 , 𝐶 , and 𝐶 have values of 1.44, 1𝜀 2𝜀 𝜇 2 2 1.92, and 0.09, respectively. 𝐺 = 2𝜈 𝑆 is the turbulent kinetic energy result, and 𝑆 is 𝑘 𝑡 the strain rate tensor. For the solution of the equations, the Semi-Implicit Method for Pressure Linkek Equations (SIMPLE) algorithm was used, which approaches convergence through a series of intermediate pressure and velocity fields satisfying continuity [49]. The higher order Upwind spatial discretization system was also used, which ensures stable schemes by min- imizing numerical diffusion errors [49]. Both schemes are integrated in ANSYS-FLUENT. The ANSYS-FLUENT Geometry module was used to build the conceptual models. Water 2023, 15, 722 7 of 29 The Meshing module was used to generate the mesh, where they were spatially discretized using predominantly hexahedral meshes. The advantage of this type of mesh is the reduc- tion in the number of cells and the improvement in the convergence of the solution [50]. Upwind spatial discretization system was also used, which ensures stable schemes by mini- The computational meshes were refined in the vicinity of the weir wall (Figure 3B), where mizing numerical diffusion errors [49]. Both schemes are integrated in ANSYS-FLUENT. the turbulence is dissipated, and its behavior has a significant effect on the results. The ANSYS-FLUENT Geometry module was used to build the conceptual models. The In the simulation of free-surface flows, it is important to define the boundary condi- Meshing module was used to generate the mesh, where they were spatially discretized using tions appropriately at the inlet and outlet of the model domain. In this work, the boundary predominantly hexahedral meshes. The advantage of this type of mesh is the reduction conditions implemented in ANSYS-FLUENT were applied at the inlet and outlet of the in the number of cells and the improvement in the convergence of the solution [50]. The domain, with the fluid velocity as inlet and atmospheric pressure as outlet. The approach computational meshes were refined in the vicinity of the weir wall (Figure 3B), where the of the boundary conditions (Figure 3A) and definition of the fluid properties are specified turbulence is dissipated, and its behavior has a significant effect on the results. in Table 2. Figure 3. (A) Boundary conditions (Mattos-Villarroel et al., 2021 [1]). (B) Refined mesh. Figure 3. (A) Boundary conditions (Mattos-Villarroel et al., 2021 [1]). (B) Refined mesh. Table 2. Boundary and initial conditions. In the simulation of free-surface flows, it is important to define the boundary condi- tions appropriately at the inlet and outlet of the model domain. In this work, the boundary Boundary and Initial Conditions Solution Method conditions implemented in ANSYS-FLUENT were applied at the inlet and outlet of the Domain: inlet Velocity domain, with the fluid velocity as inlet and atmospheric pressure as outlet. The approach Domain: outlet Atmospheric pressure of the boundary conditions (Figure 3A) and definition of the fluid properties are specified Domain: weir, sidewalls, and channel platform Solid, stationary, and non-slip. in Table 2. Table 2. Boundary and initial conditions. Boundary and Initial Conditions Solution Method Domain: inlet Velocity Domain: outlet Atmospheric pressure Domain: weir, sidewalls, and channel platform Solid, stationary, and non-slip. Viscosity model k–# standard Multiphasic model Volume of fluid (VOF) Pressure–velocity coupling SIMPLE Spatial discretization scheme Upwind 2.2.2. Grid Convergence Index (GCI) Within the simulation, the shape and density of the mesh or grid in the analysis have a very significant importance and influence the total number of elements, the computing time, and the accuracy of the analysis. In the present study, 40 different scenarios with 7 grid sizes of different densities were analyzed (Table 3). In each grid, the number of elements was decreased until we obtained an adequate convergence of calculations, and the independence results of the mesh were obtained. 𝑖𝑗 𝑖𝑗 𝐷𝐷 Water 2023, 15, 722 8 of 29 Table 3. Simulated scenarios for each grid. Grid Scenario Grid Scenario I 1–10 V 26–30 II 11–15 VI 31–35 III 16–20 VII 36–40 IV 21–25 It is important that, before calculating any discretization error estimate, it must be guaranteed that the convergence of the iterative process presents a decrease of at least three orders of magnitude in the normalized residuals for each solved equation [51]. In the present study, the convergence and discretization errors were verified at each time step to control the convergence of the solution of the time-dependent problems and, thereby, guarantee an adequate solution to the equations describing the phenomenon. The recommended method for discretization error estimation is the Richardson ex- trapolation method. Roache (1994) [52] proposed a way of reporting the results of grid convergence studies with the Grid Convergence Index (GCI), which is based on the Richard- son Extrapolation method, a method that has been extensively evaluated in case studies using CFD [51,53,54]. GCI indicates the percentage by which the calculated value deviates from the asymptotic numerical value and how much the solution would change with further refinement of the mesh. Thus, a small value of GCI indicates that the calculation is within the asymptotic range [55]. To estimate the order of convergence and verify that the solutions are within the asymptotic range of convergence, Roache (1994) [52] recommended using three different grid sizes. Equation (13) shows how to determine the CGI of a fine or coarse grid [56]: F j j F j jr s i+1,1 s i+1,1 GC I = ; GC I = (13) f ine coarse p p (r 1) (r 1) where F is a security factor (taking a value of 3 for comparisons of two grids and 1.25 for comparisons of three or more grids [56]), r is the mesh refinement ratio, p is the order of convergence, and e is the error relative to the control variable f . The representative model cell size (l) was estimated using Equation (14) for three- dimensional models. " # 1 l = (DV ) (14) i=1 In this equation, DV is the volume of the i-th cell and N is the number of cells. Therefore, i C to perform the calculation, at least three different grid sizes must be selected to determine the value of the control variables f ( f , f , f ) considered important for the simulation objective. 1 2 3 Then, for l < l < l , the mesh refinement factors were determined as r = l l , 1 2 3 21 2 1 r = l /l and the order of convergence p was calculated with Equation (15) [52]. 32 3 2 f f 3 2 ln f f 2 1 p = (15) ln r Equation (16) enables verification that the solutions are within the asymptotic range of convergence [55]: GC I 1 (16) GC I r Seven different grids are presented in this study. To perform the GCI analysis, the hydraulic head (h , h , h ) was considered as a control variable to estimate discretiza- 1 2 3 3 1 3 1 tion errors. The flows used were as follows: Q = 0.2036 m s , Q = 0.2003 m s , I II 3 1 3 1 3 1 3 1 Q = 0.2137 m s , Q = 0.2170 m s , Q = 0.2149 m s , Q = 0.2036 m s , and III IV V VI 3 1 Q = 0.1915 m s for the grids: I, II, III, IV, V, VI, and VII, respectively. The GCI cal- VII Water 2023, 15, x FOR PEER REVIEW 10 of 32 𝑓 𝑓 3− 2 𝑓 𝑓 (15) 2− 1 𝑝 = ln 𝑟 Equation (16) enables verification that the solutions are within the asymptotic range of convergence [55]: 𝐶𝐺𝐼 ≈ 1 (16) 𝐶𝐺𝐼 𝑟 Seven different grids are presented in this study. To perform the GCI analysis, the hydraulic head (h1, h2, h3) was considered as a control variable to estimate discretization 3 −1 3 −1 errors. The flows used were as follows: QI = 0.2036 m s , QII = 0.2003 m s , QIII = 0.2137 3 −1 3 −1 3 −1 3 −1 3 −1 m s , QIV = 0.2170 m s , QV = 0.2149 m s , QVI = 0.2036 m s , and QVII = 0.1915 m s for the grids: I, II, III, IV, V, VI, and VII, respectively. The GCI calculations of the numerical Water 2023, 15, 722 9 of 29 solutions are summarized in Table 4; the asymptotic ranges of convergence obtained are approximately equal to 1. Therefore, the numerical solutions are within the asymptotic range. In the present work, the hydraulic head was achieved with a maximum error of up culations of the numerical solutions are summarized in Table 4; the asymptotic ranges of to 2.80%, corresponding to grid II. convergence obtained are approximately equal to 1. Therefore, the numerical solutions are within Table 4.the Estasymptotic imation of grrange. id convIn ergthe ence pr in esent dex (CG work, I). the hydraulic head was achieved with a maximum error of up to 2.80%, corresponding to grid II. Asymptotic h1 h2 h3 Richardson Ex- GCI21 GCI32 Grid r p Ꜫ21 Ꜫ32 Range of Conver- Table 4. Estimation of grid convergence index (CGI). (m) (m) (m) trapolate (m) (%) (%) gence h h h Richardson Extrapolate GCI GCI Asymptotic Range of 1 2 3 21 32 I 1.60 1.80 0.1622 0.1637 0.1672 0.1611 0.0092 0.0214 0.87 2.00 0.99 Grid r p e e 21 32 (m) (m) (m) (m) (%) (%) Convergence II 1.58 1.78 0.1182 0.1197 0.1231 0.1170 0.0127 0.0284 1.25 2.80 0.99 I 1.60 1.80 0.1622 0.1637 0.1672 0.1611 0.0092 0.0214 0.87 2.00 0.99 III 1.58 1.83 0.1302 0.1312 0.1335 0.1294 0.0077 0.0175 0.74 1.69 0.99 II 1.58 1.78 0.1182 0.1197 0.1231 0.1170 0.0127 0.0284 1.25 2.80 0.99 IV III 1 1.58 .59 11.83 .76 0 0.1302 .1412 0.1312 0.1420.1335 3 0.1448 0.1294 0.1403 0.0077 0.007 0.0175 8 0.017 0.74 6 0.77 1.69 1.73 0.99 0.99 IV 1.59 1.76 0.1412 0.1423 0.1448 0.1403 0.0078 0.0176 0.77 1.73 0.99 V 1.65 1.79 0.1482 0.1495 0.1527 0.1473 0.0088 0.0214 0.75 1.83 0.99 V 1.65 1.79 0.1482 0.1495 0.1527 0.1473 0.0088 0.0214 0.75 1.83 0.99 VI 1.58 1.97 0.1502 0.1519 0.1561 0.1490 0.0113 0.0276 0.96 2.35 0.99 VI 1.58 1.97 0.1502 0.1519 0.1561 0.1490 0.0113 0.0276 0.96 2.35 0.99 VII 1.64 1.89 0.1622 0.1631 0.1654 0.1616 0.0055 0.0141 0.45 1.13 0.99 VII 1.64 1.89 0.1622 0.1631 0.1654 0.1616 0.0055 0.0141 0.45 1.13 0.99 Based on Richardson’s extrapolation for the two finer grids, an estimate of the hy- Based on Richardson’s extrapolation for the two finer grids, an estimate of the hy- draulic head value for zero-grid spacing was obtained. The graph in Figure 4 shows the draulic head value for zero-grid spacing was obtained. The graph in Figure 4 shows the hydraulic head with variable grid spacing; as the grid spacing reduced, the hydraulic head hydraulic head with variable grid spacing; as the grid spacing reduced, the hydraulic head approached an asymptotic value of zero mesh spacing. approached an asymptotic value of zero mesh spacing. Figure 4. Hydraulic head approaching an asymptotic zero-grid spacing value. 2.3. Evaluation of the Computational Model Numerically, 40 scenarios were modeled. Table 5 details the design flow applied and the geometric characteristics of the simulated weirs for each scenario. For the evaluation of the computational model, the results of the numerically obtained discharge coefficient were compared with the experimental values reported by Crookston and Tullis (2012) [15], corresponding to scenarios 1–10. The discharge coefficient (C ) was calculated from the height of the measured hydraulic head on the weir and the general weir equation (Equation (17)). 3/2 Q = 2gC L H (17) 3 1 where Q is the design flow (m s ), C is the discharge coefficient (dimensionless), g is the acceleration due to gravity (m s ), L is the characteristic length of the weir (m) (defined as the total length referenced at the center of the weir crest wall thickness), and H is the total hydraulic head (m). 𝑙𝑛 Water 2023, 15, 722 10 of 29 Table 5. Simulation scenarios for labyrinth weirs. a P L Q cycle Scenario w/P N Crest Profile Apex Shape ( ) (m) (m) (m s ) 0.0190, 0.0532, 0.0919, 0.1309, 0.1681, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 15 0.305 2.00 2.024 1 CR Trapezoidal 0.2036, 0.2373, 0.2697, 0.3013, 0.3325 0.0780, 0.1240, 0.2003, 11, 12, 13, 14, 15 6 0.305 3.64 2.024 1 MR Circular 0.2703, 0.3558 0.0429, 0.0750, 0.2137, 16, 17, 18, 19, 20 8 0.305 3.07 2.024 1 MR Circular 0.2873, 0.3744 0.0367, 0.0532, 0.2170, 21, 21, 23, 24, 25 10 0.305 2.65 2.024 1 MR Circular 0.2891, 0.3712 0.0380, 0.0671, 0.1450, 26, 27, 28, 29, 30 12 0.305 2.33 2.024 1 MR Circular 0.2149, 0.3728 0.0532, 0.1308, 0.2036, 31, 32, 33, 34, 35 15 0.305 2.02 2.024 1 MR Circular 0.2697, 0.3325 0.0290, 0.0517, 0.1247, 36, 37, 38, 39, 40 20 0.305 1.59 2.024 1 MR Circular 0.1915, 0.3364 Note: CR: Quarter-round; MR: Half-round; L : Cycle length (m). cycle Statistical criteria were applied to evaluate the performance of the model, such as Pearson’s coefficient of determination (R ), the relative percentage error (Er), and the mean absolute error (MAE) (Equations (18)–(20)). These criteria assess the agreement between the results of the physical experiment and those from the CFD models. Cov(num, ex p) R = (18) 2 2 s s num ex p Y Y num,i ex p,i M AE = (19) Y Y num ex p Er(%) =  100 (20) ex p Cov(num, ex p) is the covariance of the numerical and experimental results, s is the ex p variance of the experimental results, s is the variance of the numerical results, Y are num num the numerical values, and Y are the experimental values. exp The mean absolute error of the discharge coefficient (MAE) was 0.0128, and the relative percentage error varied from 1.89% to 4.92%. In Figure 5, the relative percentage errors of the discharge coefficient as a function of the ratio H /P are presented. The calculated coefficient of determination was R = 0.984, which confirms the agree- ment between the numerically obtained discharge coefficients and the experimental values. In the graph in Figure 6, the numerical results are compared with the experimental results; the dotted diagonal line corresponds to a perfect fit. In this study, the ANOVA statistical analysis was performed, and there was no signifi- cant difference between the discharges coefficients obtained numerically and experimen- tally for a significance level of 5% (0.05). 2.4. Proposed Sequential Design Method for a Labyrinth Weir Figure 7 details the variables and geometric characteristics considered in the proposed sequential design method for the labyrinth weir. The design method is basically subdivided into four stages: (i) initial data require- ments, (ii) definition of the number of cycles and the angle a to make the weir hydraulically efficient, (iii) calculation of the geometric variables, and (iv) analysis of the submerged weir developed according to Tullis et al., (2007) [38]. Water 2023, 15, x FOR PEER REVIEW 12 of 32 𝐶𝑜𝑣 (𝑛𝑢𝑚 , ) is the covariance of the numerical and experimental results, 𝜎 is 𝑝𝑥𝑒 the variance of the experimental results, 𝜎 is the variance of the numerical results, Ynum 𝑛𝑢𝑚 Water 2023, 15, x FOR PEER REVIEW 12 of 32 are the numerical values, and Yexp are the experimental values. The mean absolute error of the discharge coefficient (MAE) was 0.0128, and the rela- tive percentage error varied from 1.89% to 4.92%. In Figure 5, the relative percentage er- rors of the discharge coefficient as a function of the ratio HT/P are presented. 𝐶𝑜𝑣 (𝑛𝑢𝑚 , ) is the covariance of the numerical and experimental results, 𝜎 is 𝑝𝑥𝑒 the variance of the experimental results, 𝜎 is the variance of the numerical results, Ynum 𝑛𝑢𝑚 are the numerical values, and Yexp are the experimental values. The mean absolute error of the discharge coefficient (MAE) was 0.0128, and the rela- Water 2023, 15, 722 11 of 29 tive percentage error varied from 1.89% to 4.92%. In Figure 5, the relative percentage er- rors of the discharge coefficient as a function of the ratio HT/P are presented. Figure 5. Relative percentage error of the discharge coefficient as a function of HT/P (Mattos-Villar- roel et al., 2021 [1]). The calculated coefficient of determination was R = 0.984, which confirms the agree- ment between the numerically obtained discharge coefficients and the experimental val- ues. In the graph in Figure 6, the numerical results are compared with the experimental Figure 5. Relative percentage error of the discharge coefficient as a function of HT/P (Mattos-Villar- Figure 5. Relative percentage error of the discharge coefficient as a function of H /P results; the dotted diagonal line corresponds to a perfect fit. roel et al., 2021 [1]). (Mattos-Villarroel et al., 2021 [1]). The calculated coefficient of determination was R = 0.984, which confirms the agree- ment between the numerically obtained discharge coefficients and the experimental val- ues. In the graph in Figure 6, the numerical results are compared with the experimental results; the dotted diagonal line corresponds to a perfect fit. Figure 6. Comparison of numerically obtained discharge coefficients with experimental values Figure 6. Comparison of numerically obtained discharge coefficients with experimental values (Mattos-Villarroel et al., 2021 [1]). (Mattos-Villarroel et al., 2021 [1]). In (i) th Stage is study, one: th Th e is As NO tage VA co n st sa is tis tstica of d l e atn er am lyisi ns inwa g th s e pe drf ato arme nece ds ,s a an ry d fth orer th e ewa des sin go n si an g- d comprises information previously obtained from topographic and hydrological analysis, namely: nificant difference between the discharges coefficients obtained numerically and experi- mentally for a significance level of 5% (0.05). (a) The design flow (Q), which represents the design discharge for a given return period; Figure 6. Comparison of numerically obtained discharge coefficients with experimental values (b) The upstream head of the weir (H ), which depends on the channel width (W) and is (Mattos-Villarroel et al., 2021 [1]). limited by the freeboard; (c) The downstream head of the weir (H ) is calculated from the drop height and the flow In this study, the ANOVA statistical danalysis was performed, and there was no sig- velocity at the foot of the weir; nificant difference between the discharges coefficients obtained numerically and experi- (d) The weir height (P) corresponds to the height of the storage volume or the Ordinary mentally for a significance level of 5% (0.05). Maximum Water Level obtained from the topography and the operation of the basin. The width of the weir or the channel (W) is generally restricted by the topography of the study area. 𝑒𝑥𝑝 𝑒𝑥𝑝 Water 2023, 15, x FOR PEER REVIEW 13 of 32 2.4. Proposed Sequential Design Method for a Labyrinth Weir Water 2023, 15, 722 12 of 29 Figure 7 details the variables and geometric characteristics considered in the pro- posed sequential design method for the labyrinth weir. Figure 7. Geometric variables of a labyrinth weir. Figure 7. Geometric variables of a labyrinth weir. (ii) Stage two: The topography of the study area allows for selecting the angle a of The design method is basically subdivided into four stages: (i) initial data require- the weir sidewall (6  a  20 ). A large angle can be chosen when the number of cycles N ments, (ii) definition of the number of cycles and the angle α to make the weir hydrau- and the length between the cycle apexes (B) are limited by topography. The selection of lically efficient, (iii) calculation of the geometric variables, and (iv) analysis of the sub- the angle a is also a function of the ratio H /P (0.05  H /P  0.8). For certain values of merged weir developed according to TulliT s et al., (2007)T [38]. H /P and a, the flow becomes unstable and is a phenomenon to be avoided in the design T (i) Stage one: This stage consists of determining the data necessary for the design and of the weir, for the safety of the hydraulic structure. Subsequently, the discharge coefficient comprises information previously obtained from topographic and hydrological analysis, is calculated. Its calculation is a function of a and the ratio H /P, and the value of the namely: discharge coefficient will determine the discharge capacity of the weir. (a) The design flow (Q), which represents the design discharge for a given return period; Then, the cycle width w is calculated as a function of the weir height P. For (b) The upstream head of the weir (HT), which depends on the channel width (W) and is Taylor (1968) [39], the ratio w/P (known as the vertical aspect) should not be less than limited by the freeboard; 2 because it would contribute to reducing weir efficiency. On the other hand, Tullis et al. (c) The downstream head of the weir (Hd) is calculated from the drop height and the (1995) [14] recommended that w/P should not be greater than 4. Considering both criteria, flow velocity at the foot of the weir; it is recommended that the ratio w/P should be equal to 3 or, in other words, the cycle width (d) The weir height (P) corresponds to the height of the storage volume or the Ordinary w should be 3 times the weir height P, to ensure that the weir is hydraulically efficient. Maximum Water Level obtained from the topography and the operation of the basin. Another important variable influencing the design is the number of cycles (N). The The width of the weir or the channel (W) is generally restricted by the topography of number of cycles is calculated as the ratio of the weir width (W) to the cycle width (w). the study area. For ease of design, it is recommended that this be a multiple of 0.5, and so the cycle width (ii) Stage two: The topography of the study area allows for selecting the angle α of should be recalculated as w = W/N, with the restriction that the ratio w/P is within the the weir sidewall (6 ≤ α ≤ 20°). A large angle can be chosen when the number of cycles N range 2  w/P  4. and (iii) the leng Stage th three: betweeIn n th this e cycle stage, ape the xes geometric (B) are lim variables ited by toof pothe graphy weir . Tar he e se calculated lection of the angle α is also a function of the ratio HT/P (0.05 ≤ HT/P ≤ 0.8). For certain values of HT/P as follows: and α, the flow becomes unstable and is a phenomenon to be avoided in the design of the The length of the weir (L). The selection of the angle a will determine the length of the weir, for the safety of the hydraulic structure. Subsequently, the discharge coefficient is weir. Its calculation is a function of the discharge coefficient, the hydraulic head, and the design flow. The width of the weir wall (t ) and the internal apex rope (C ) must both be equal w c to P/8. The internal and external apex arc (Arc , Arc ) are both functions of t and a. int ext w Water 2023, 15, 722 13 of 29 The length of the cycle wall (l ), as a function of L, N, Arc , and Arc . c int ext The length of the platform (B) is a function of L, N, Arc , Arc , a, and t . ext w int In this stage, the weir efficiency (# ) and the cycle (#”) are also determined, both as a function of L, w, N, and C (a). The weir efficiency is also a function of the discharge coefficient of a linear weir; its calculation method is described by Crooskton (2010) [5]. Subsequently, the nappe interference length (B ) is calculated from the ratio H /P int T and the angle a. Finally, the type of aeration of the nappe is determined according to the value of H /P and the selected angle a. (iv) Stage Four: The last stage of the design method includes the dimensionless head rela- tionships for the drowned weir, which were developed and described by Tullis et al. (2007) [38]. The following section describes the results obtained from the studies applied to the discharge flow considered in the proposed sequential design method for the weir; it describes the equations for each variable in detail. 3. Results 3.1. Discharge Coefficient, Weir, and Cycle Efficiency In this study, the discharge coefficients of circular apex weirs are presented as a function of the ratio H /P, whose weir cycle sidewall angles vary from 6 to 20 and are compared with the discharge coefficients reported by Crookston and Tullis (2012) [15] for trapezoidal labyrinth weirs. Both weirs have a half-round crest. Water 2023, 15, x FOR PEER REVIEW 15 of 32 The discharge coefficients C (a) of each weir with a circular apex are presented graphi- cally in Figure 8 for H /P  0.8. Figure 8. Curves of C (a) as a function of H /P for different values of a for labyrinth weirs. Figure 8. Curves of Cd(α) as a function of HT/P for different values of α for labyrinth weirs. d T The values of the discharge coefficient in Figure 8 were used to obtain a mathematical The values of the discharge coefficient in Figure 8 were used to obtain a mathematical model of a regressive type through a fifth-degree polynomial equation (Equation (22)) as a model of a regressive type through a fifth-degree polynomial equation (Equation (22)) as function of H /P. Weir design methods and curves are mostly generated from empirical a function of HT/P. Weir design methods and curves are mostly generated from empirical equations derived from laboratory experiments [2,5–9]. For example, several researchers equations derived from laboratory experiments [2,5–9]. For example, several researchers reported polynomial equations obtained by non-linear regression to have good fitting [14]. reported polynomial equations obtained by non-linear regression to have good fitting Statistical analysis was used to determine the accuracy of Equation (21) compared to the [14]. Statistical analysis was used to determine the accuracy of Equation (21) compared to C (a) results obtained from numerical data. The calculated Pearson’s coefficient was very the Cd(α) results obtained from numerical data. The calculated Pearson’s coefficient was good, varying from 0.999 to 1 for weirs with angles from 6 to 20 . Therefore, Equation very good, varying from 0.999 to 1 for weirs with angles from 6 to 20°. Therefore, Equation (21) provides sufficient accuracy to determine C (a). Equations (22)–(27) correspond to the (21) provides sufficient accuracy to determine Cd(α). Equations (22)–(27) correspond to the coefficients of Equation (21) as a function of the angle . The accuracy of predictive Equa- coefficients of Equation (21) as a function of the angle α. The accuracy of predictive Equa- tions (22)–(27) was also evaluated with the numerical results using Pearson’s determination tions (22)–(27) was also evaluated with the numerical results using Pearson’s determina- tion coefficient, obtaining values of 1 for the case of Equations (22)–(26) and 0.996 for Equation (27). Therefore, reliable results can be obtained using coefficients for Equation (21). 5 4 3 2 𝐻 𝐻 𝐻 𝐻 𝐻 𝑇 𝑇 𝑇 𝑇 𝑇 (21) 𝐶 (𝛼 ) = 𝑎 ( ) + 𝑏 ( ) + 𝑐 ( ) + 𝑑 ( ) + 𝑒 ( ) + 𝑓 𝑃 𝑃 𝑃 𝑃 𝑃 𝑎 = 42.99 + 48.93 (0.1926𝛼 ) − 24.14 (0.1926𝛼 ) + 7.60 (0.3852𝛼 ) (22) − 15.95 (0.3852𝛼 ) ( ) ( ) ( ) ( ) (23) 𝑏 = −61.88 − 65.87 0.2241𝛼 − 4.273 0.2241𝛼 − 22.5 0.4482𝛼 + 5.11 0.4482𝛼 (24) 𝑐 = 47.39 + 36.05 (0.2408𝛼 ) + 14.27 (0.2408𝛼 ) + 13.57 (0.4816𝛼 ) + 3.893 (0.4816𝛼 ) 𝑑 = −20.19 − 11.21 (0.2396𝛼 ) − 4.43 (0.2396𝛼 ) − 4.327 (25) (0.4792𝛼 ) − 1.013 (0.4792𝛼 ) 𝑒 = 3.853 + 2.084 (0.2076𝛼 ) − 0.7578 (0.2076𝛼 ) + 0.5083 (26) (0.4152𝛼 ) − 0.7128 (0.4152𝛼 ) −5 4 −3 3 −2 2 𝑓 = −5.158 × 10 𝛼 + 2.591 × 10 𝛼 − 4.62 × 10 𝛼 + 0.3487𝛼 − 0.3085 (27) The maximum values that can be obtained for the discharge coefficients occur when HT varies between 0.10 and 0.17 times the height of the weir. Table 6 shows the maximum values of the discharge coefficient for each weir as a function of HT/P. 𝑠𝑖𝑛 𝑐𝑜𝑠 𝑠𝑖𝑛 𝑐𝑜𝑠 𝑠𝑖𝑛 𝑐𝑜𝑠 𝑠𝑖𝑛 𝑐𝑜𝑠 𝑠𝑖𝑛 𝑐𝑜𝑠 𝑠𝑖𝑛 𝑐𝑜𝑠 𝑠𝑖𝑛 𝑐𝑜𝑠 𝑠𝑖𝑛 𝑐𝑜𝑠 𝑠𝑖𝑛 𝑐𝑜𝑠 𝑠𝑖𝑛 𝑐𝑜𝑠 Water 2023, 15, 722 14 of 29 coefficient, obtaining values of 1 for the case of Equations (22)–(26) and 0.996 for Equation (27). Therefore, reliable results can be obtained using coefficients for Equation (21). 5 4 3 2 H H H H H T T T T T C (a) = a + b + c + d + e + f (21) P P P P P a = 42.99 + 48.93cos(0.1926a) 24.14sin(0.1926a) + 7.60cos(0.3852a) 15.95sin(0.3852a) (22) b = 61.88 65.87cos(0.2241a) 4.273sin(0.2241a) 22.5cos(0.4482a) + 5.11sin(0.4482a) (23) c = 47.39 + 36.05cos(0.2408a) + 14.27sin(0.2408a) + 13.57cos(0.4816a) + 3.893sin(0.4816a) (24) d = 20.19 11.21cos(0.2396a) 4.43sin(0.2396a) 4.327cos(0.4792a) 1.013sin(0.4792a) (25) e = 3.853 + 2.084cos(0.2076a) 0.7578sin(0.2076a) + 0.5083cos(0.4152a) 0.7128sin(0.4152a) (26) 5 4 3 3 2 2 f = 5.158 10 a + 2.59110 a 4.6210 a + 0.3487a 0.3085 (27) The maximum values that can be obtained for the discharge coefficients occur when H varies between 0.10 and 0.17 times the height of the weir. Table 6 shows the maximum values of the discharge coefficient for each weir as a function of H /P. Table 6. Maximum values of the discharge coefficient. (a) H /P C (a) 6 0.10 0.736 8 0.11 0.762 10 0.13 0.771 12 0.14 0.784 15 0.15 0.803 20 0.17 0.833 The discharge coefficients of the circular apex weirs are compared with the trapezoidal apex weirs. The graph in Figure 9 shows the increase in the discharge coefficient of the circular apex weir with respect to the trapezoidal apex, which is 10% when a = 20 to 46% when a = 6 , and the slope of the family of curves is significantly higher for  10 . The weir efficiency (Equation (28)) allows for the hydraulic behavior of a labyrinth weir to be compared with a conventional linear weir, and the advantages obtained by increasing its length can then be determined [23]. C (a) L 0 d e = (28) C W d(90 ) The cycle efficiency indicator proposed by Willmore (2004) [4] allows for the design of the labyrinth type weir to be optimized and facilitates decision-making. Its calculation is essentially useful for small heads and is obtained from Equation (29). ciclo e = C (a) (29) The graphs in Figure 10 represent the efficiency of the labyrinth weir and the cycle as a function of H /P. Both graphs show that the maximum efficiency occurs for small values of H /P and increases with the decreasing sidewall angle. T Water 2023, 15, x FOR PEER REVIEW 16 of 32 Table 6. Maximum values of the discharge coefficient. (𝜶 ) HT/P 𝑪 (𝜶 ) 6° 0.10 0.736 8° 0.11 0.762 10° 0.13 0.771 12° 0.14 0.784 15° 0.15 0.803 20° 0.17 0.833 The discharge coefficients of the circular apex weirs are compared with the trapezoi- dal apex weirs. The graph in Figure 9 shows the increase in the discharge coefficient of Water 2023, 15, 722 15 of 29 the circular apex weir with respect to the trapezoidal apex, which is 10% when α = 20° to 46% when α = 6°, and the slope of the family of curves is significantly higher for α ≤ 10°. Water 2023, 15, x FOR PEER REVIEW 17 of 32 The efficiency of the labyrinth weir, represented by the family of curves in Figure 10A, has an accelerated reduction in its value for α ≤ 10° when HT/P > 0.1. Figure 10B represents the cycle efficiency of the weir; the dotted line passes through the values of Figure 9. Relationship of the discharge coefficient of the circular apex weir to the trapezoidal apex Figure 9. Relationship of the discharge coefficient of the circular apex weir to the trapezoidal apex HT/P where the maximum efficiency of the cycle is present and coincides with the maxi- w weir eir.. The The dotted dotted line line indicates indicates the the inflection inflection point point on on each each curve. curve. mum values of Cd(α). The weir efficiency (Equation (28)) allows for the hydraulic behavior of a labyrinth weir to be compared with a conventional linear weir, and the advantages obtained by increasing its length can then be determined [23]. 𝐶 (𝛼 ) 𝐿 Ꜫ′ = (28) 𝐶 𝑊 𝑑 (90°) The cycle efficiency indicator proposed by Willmore (2004) [4] allows for the design of the labyrinth type weir to be optimized and facilitates decision-making. Its calculation is essentially useful for small heads and is obtained from Equation (29). 𝑐𝑜𝑖𝑐𝑙 Ꜫ′′ = 𝐶 (𝛼 ) (29) The graphs in Figure 10 represent the efficiency of the labyrinth weir and the cycle as a function of HT/P. Both graphs show that the maximum efficiency occurs for small values of HT/P and increases with the decreasing sidewall angle. (A) (B) Figure 10. (A) Weir efficiency. (B) Cycle efficiency. Figure 10. (A) Weir efficiency. (B) Cycle efficiency. 3.2. Nappe Aeration Conditions The efficiency of the labyrinth weir, represented by the family of curves in Figure 10A, has an With acce th ler e aitn ecr d ea red se uc in ti o th ne in hydr its v aa uli luc e h fo ea r d o n1 t0 he w weir, hen f H ou/r Ptypes > 0.1. o Ffi g au er ra etio 10n B were repre s ide enn ts- ttified he cy c(lFi e e gure ffici e1 n 1 c)y : n oa f ppe the w ae dih r;er th ed e d to ot t th ed e lwa inellp o as f ste h se th weir, roug h aer tha eted val , u pa esrtia of H lly / aP erw ated her,e a tn hd e maximum efficiency of the cycle is present and coincides with the maximum values of C (a). drowned. 3.2. Nappe Aeration Conditions With the increase in the hydraulic head on the weir, four types of aeration were identified (Figure 11): nappe adhered to the wall of the weir, aerated, partially aerated, and drowned. Table 7 presents the ranges of H /P that correspond to the aeration conditions ob- served for each weir and is included in the family of discharge coefficient design curves (Figure 12). Water 2023, 15, x FOR PEER REVIEW 17 of 32 The efficiency of the labyrinth weir, represented by the family of curves in Figure 10A, has an accelerated reduction in its value for α ≤ 10° when HT/P > 0.1. Figure 10B represents the cycle efficiency of the weir; the dotted line passes through the values of HT/P where the maximum efficiency of the cycle is present and coincides with the maxi- mum values of Cd(α). (A) (B) Figure 10. (A) Weir efficiency. (B) Cycle efficiency. 3.2. Nappe Aeration Conditions With the increase in the hydraulic head on the weir, four types of aeration were iden- Water 2023, 15, 722 16 of 29 tified (Figure 11): nappe adhered to the wall of the weir, aerated, partially aerated, and drowned. Water 2023, 15, x FOR PEER REVIEW 18 of 32 Figure 11. Nappe aeration conditions: (A) clinging, (B) aerated, (C) partially aerated, and (D) Figure 11. Nappe aeration conditions: (A) clinging, (B) aerated, (C) partially aerated, and (D) drowned. drowned. Table 7. Ranges of nappe aeration conditions. Table 7 presents the ranges of HT/P that correspond to the aeration conditions ob- H /P served for each weir and is included in the family of discharge coefficient design curves a( ) Flow Clinging (Figure 12). Flow Aerated Flow Partially Aerated Flow Drowned 6 <0.165 0.165–0.270 0.270–0.487 >0.487 Table 7. Ranges of nappe aeration conditions. 8 <0.200 0.200–0.350 0.350–0.500 >0.500 10 <0.265 0.265–0.350 0.350–0.540 >0.540 HT/P 12 <0.300 0.300–0.410 0.410–0.550 >0.550 Flow Cling- Flow Flow Flow 15 <0.325 0.325–0.400 0.400–0.600 >0.600 𝜶 (°) 20 <0.450 0.450–0.500 ing Aerated 0.500–0.600 Partially Aerated >0.600 Drowned 6° <0.165 0.165–0.270 0.270–0.487 >0.487 3.3. Nappe 8° Instability<0.200 0.200–0.350 0.350–0.500 >0.500 10° <0.265 0.265–0.350 0.350–0.540 >0.540 Nappe instability occurs when the nappe has an oscillating trajectory accompanied by 12° <0.300 0.300–0.410 0.410–0.550 >0.550 turbulent helical flow adjacent and parallel to the side walls of the weir cycle (Figure 13). 15° <0.325 0.325–0.400 0.400–0.600 >0.600 Under these conditions, vibration is generated in the weir and may represent a safety hazard for the hydraulic structure. 20° <0.450 0.450–0.500 0.500–0.600 >0.600 Water 2023, 15, x FOR PEER REVIEW 19 of 32 Water 2023, 15, 722 17 of 29 The results of this work confirm the presence of unstable flow located downstream of the weir, i.e., between the side walls of the weir. The values of H /P and the aeration Water 2023, 15, x FOR PEER REVIEW 19 of 32 conditions where instability is generated are presented in Table 8 and are included in the discharge coefficient design curves (Figure 14). Figure 12. Identification of aeration zones. 3.3. Nappe Instability Nappe instability occurs when the nappe has an oscillating trajectory accompanied by turbulent helical flow adjacent and parallel to the side walls of the weir cycle (Figure 13). Under these conditions, vibration is generated in the weir and may represen t a safety Figure 12. Identification of aeration zones. hazard for the hydraulic structure. Figure 12. Identification of aeration zones. 3.3. Nappe Instability Nappe instability occurs when the nappe has an oscillating trajectory accompanied by turbulent helical flow adjacent and parallel to the side walls of the weir cycle (Figure 13). Under these conditions, vibration is generated in the weir and may represent a safety hazard for the hydraulic structure. Figure 13. Helical streamlines parallel to the weir wall. Figure 13. Helical streamlines parallel to the weir wall. Table 8. The Ranges resulof ts nappe of this instability work coand nfirm aeration the pr conditions. esence of unstable flow located downstream of the weir, i.e., between the side walls of the weir. The values of HT/P and the aeration a( ) Instability Aeration Condition conditions where instability is generated are presented in Table 8 and are included in the 6 - - discharge coefficient design curves (Figure 14). 8 - - 10 - - Figure 13. Helical streamlines parallel to the weir wall. 12 0.56  HT/P  0.8 Drowned. 15 0.49  HT/P  0.8 Partially aerated, and drowned. The results of this work confirm the presence of unstable flow located downstream 20 0.40  HT/P  0.8 Clinging, aerated, partially aerated, and drowned. of the weir, i.e., between the side walls of the weir. The values of HT/P and the aeration conditions where instability is generated are presented in Table 8 and are included in the discharge coefficient design curves (Figure 14). Water 2023, 15, x FOR PEER REVIEW 20 of 32 Table 8. Ranges of nappe instability and aeration conditions. 𝜶 (°) Instability Aeration Condition 6° - - 8° - - 10° - - 12° 0.56 ≤ HT/P ≤ 0.8 Drowned. 15° 0.49 ≤ HT/P ≤ 0.8 Partially aerated, and drowned. Water 2023, 15, 722 Clinging, aerated, partially aerated, an 18dof 29 20° 0.40 ≤ HT/P ≤ 0.8 drowned. Fig Figure ure 14. 14. I Identification dentification of of instability instability zones. zones. 3.4. Nappe Interference 3.4. Nappe Interference The effect of the collision between nappes on the reduction of weir efficiency is The effect of the collision between nappes on the reduction of weir efficiency is sig- significant. Therefore, its analysis and behavior are characterized and considered in the nificant. Therefore, its analysis and behavior are characterized and considered in the de- design of the weir. The CFD simulations carried out made it possible to visualize the sign of the weir. The CFD simulations carried out made it possible to visualize the for- formation of air contrail, accompanied by standing waves or hydraulic jumps between weir mation of air contrail, accompanied by standing waves or hydraulic jumps between weir cycles (Figure 15A), which decrease with the increasing hydraulic head and presence of Water 2023, 15, x FOR PEER REVIEW 21 of 32 cycles (Figure 15A), which decrease with the increasing hydraulic head and presence of drowning in the weir. In weirs where a  10, the local drowning at the apexes is generated drowning in the weir. In weirs where α ≤ 10, the local drowning at the apexes is generated earlier than in the presence of larger angles (Figure 15B). Furthermore, depending on the earlier than in the presence of larger angles (Figure 15B). Furthermore, depending on the aeration condition, turbulent flow may also occur (Figure 16B). aeration condition, turbulent flow may also occur (Figure 16B). Figure 15. Effects of the nappe interference. (A) Air contrail and standing waves and (B) local Figure 15. Effects of the nappe interference. (A) Air contrail and standing waves and (B) local drowning. drowning. In order to quantify the size of nappe interference, perpendicular measurements (Bint) were made from the upstream apex to the point (downstream) where the nappe from the sidewall intersects (Figure 16A). The term Lint denotes the projection of Bint on the weir crest that is affected by this phenomenon. Figure 16. (A) Definition of lengths Bint and Lint and (B) local drowning and turbulence. The graph in Figure 17 is presented as a family of curves, which are the results of the interference length (Bint) in relation to the length of B (perpendicular distance between Water 2023, 15, x FOR PEER REVIEW 21 of 32 Figure 15. Effects of the nappe interference. (A) Air contrail and standing waves and (B) local drowning. In order to quantify the size of nappe interference, perpendicular measurements (Bint) were made from the upstream apex to the point (downstream) where the nappe from the Water 2023, 15, 722 19 of 29 sidewall intersects (Figure 16A). The term Lint denotes the projection of Bint on the weir crest that is affected by this phenomenon. Figure 16. (A) Definition of lengths Bint and Lint and (B) local drowning and turbulence. Figure 16. (A) Definition of lengths B and L and (B) local drowning and turbulence. int int The graph in Figure 17 is presented as a family of curves, which are the results of the In order to quantify the size of nappe interference, perpendicular measurements (B ) int interference length (Bint) in relation to the length of B (perpendicular distance between were made from the upstream apex to the point (downstream) where the nappe from the sidewall intersects (Figure 16A). The term L denotes the projection of B on the weir int int Water 2023, 15, x FOR PEER REVIEW 22 of 32 crest that is affected by this phenomenon. The graph in Figure 17 is presented as a family of curves, which are the results of the interference length (B ) in relation to the length of B (perpendicular distance between int upstream and downstream apexes) for values of HT/P ≤ 0.8. The graph allows for the pre- upstream and downstream apexes) for values of H /P  0.8. The graph allows for the diction of the length of Bint and indicates that its value can vary from 20% to 60% of the prediction of the length of B and indicates that its value can vary from 20% to 60% of the int length of B in the drowning condition. length of B in the drowning condition. Figure 17. Percentage of the nappe interference in relation to distance B (6° ≤ α ≤ 20°). Figure 17. Percentage of the nappe interference in relation to distance B (6   20 ). To facilitate the use of the graph in Figure 17 in the prediction of the nappe interfer- ence length, the family of curves has been modeled through a second-degree equation, as a function of HT/P (Equation (30)). 𝐵 𝐻 𝐻 𝑛𝑡𝑖 𝑇 𝑇 (30) = 𝑚 ( ) + 𝑛 + 𝑜 𝐵 𝑃 𝑃 The coefficients of Equation (30) are calculated with Equations (31)–(33) as a function of the angle α. −4 4 −3 3 2 𝑚 = 1.284 × 10 𝛼 − 6.583 × 10 𝛼 + 0.115𝛼 − 0.764𝛼 + 1.978 (31) −4 4 −3 3 2 (32) 𝑛 = −1.095 × 10 𝛼 + 5.648 × 10 𝛼 − 0.01𝛼 + 0.722𝛼 − 1.781 −6 4 −4 3 2 𝑜 = 6.004 × 10 𝛼 − 3.349 × 10 𝛼 + 0.006𝛼 − 0.046𝛼 + 0.110 (33) 3.5. Application of the Proposed Method As a case study, information from Houston (1982) [9] on the Ute Dam weir in Logan, New Mexico was studied. The design procedure followed the method proposed by Hay and Taylor (1979) [11], i.e., the original design was for a 10-cycle weir based on the design curves of Hay and Taylor (1970) [11]. However, it did not pass the desired design discharge within the max- imum elevation of the reservoir. A 14-cycle weir, designed according to the criteria of the laboratory channel tests conducted by the Bureau of Reclamation, satisfactorily met the required discharge and water surface elevation [9]. To avoid instability and oscillations of the nappe and provide aeration, two dividers were placed at the crest of each cycle, 3.35 m upstream from the downstream apex of the cycle. The weir design was based on a fam- ily of dimensionless ratio curves L/W, in the graph Q/QN versus HT/P, where Q/QN is the discharge magnification and QN is the discharge over a linear weir. The results reported by Houston (1982) [9] on the weir design are summarized in Table 9 and Figure 18. Water 2023, 15, 722 20 of 29 To facilitate the use of the graph in Figure 17 in the prediction of the nappe interference length, the family of curves has been modeled through a second-degree equation, as a function of H /P (Equation (30)). B H H int T T = m + n + o (30) B P P The coefficients of Equation (30) are calculated with Equations (31)–(33) as a function of the angle a. 4 4 3 3 2 m = 1.284 10 a 6.583 10 a + 0.115a 0.764a + 1.978 (31) 4 4 3 3 2 n = 1.095 10 a + 5.648 10 a 0.01a + 0.722a 1.781 (32) 6 4 4 3 2 o = 6.004 10 a 3.349 10 a + 0.006a 0.046a + 0.110 (33) 3.5. Application of the Proposed Method As a case study, information from Houston (1982) [9] on the Ute Dam weir in Logan, New Mexico was studied. The design procedure followed the method proposed by Hay and Taylor (1979) [11], i.e., the original design was for a 10-cycle weir based on the design curves of Hay and Taylor (1970) [11]. However, it did not pass the desired design discharge within the maximum elevation of the reservoir. A 14-cycle weir, designed according to the criteria of the laboratory channel tests conducted by the Bureau of Reclamation, satisfactorily met the required discharge and water surface elevation [9]. To avoid instability and oscillations of the nappe and provide aeration, two dividers were placed at the crest of each cycle, 3.35 m upstream from the downstream apex of the cycle. The weir design was based on a family of dimensionless ratio curves L/W, in the graph Q/Q versus H /P, where Q/Q is the N T N discharge magnification and Q is the discharge over a linear weir. The results reported by Houston (1982) [9] on the weir design are summarized in Table 9 and Figure 18. Table 9. Dimensions of the labyrinth weir of the Ute Dam [8]. Concept Symbol Value-Unit Observations (i) Initial data 3 3 Design flow Q 15,574 m /s Initially, the design discharge was 16,042 m /s. Weir width W 256 m - Weir height P 9.14 m - Upstream total head H 5.79 m - (ii) Geometric variables and non-dimensional relationships 0.05  H /P  1 (upper range is expanded from Head water ratio H /P 0.63 0.5 to 1 to use the design curves) Flow magnification Q/Q 2.4 - Angle of sidewall a 12.1475 - Length magnification L/W 4 2  L/W  8. Vertical aspect ratio w/P 2 2  w/P  5 Cycle width w 18.29 m - Number of cycles N 14 - Weir length L 1024.24 m - Sidewall length l 34.76 m - Length between apexes B 33.99 m - Apex A 1.82 m - Crest radius R 0.30 - Crest Upper crest width t 0.61 m - Lower crest width t 1.52 m - w2 Water 2023, 15, x FOR PEER REVIEW 23 of 32 Table 9. Dimensions of the labyrinth weir of the Ute Dam [8]. Concept Symbol Value-Unit Observations (i) Initial data 3 3 Design flow Q 15,574 m /s Initially, the design discharge was 16,042 m /s. Weir width W 256 m - Weir height P 9.14 m - Upstream total head HT 5.79 m - (ii) Geometric variables and non-dimensional relationships 0.05 ≤ HT/P ≤ 1 (upper range is expanded from 0.5 to 1 to Head water ratio HT/P 0.63 use the design curves) Flow magnification Q/QN 2.4 - Angle of sidewall 𝛼 12.1475° - Length magnification L/W 4 2 ≤ L/W ≤ 8. Vertical aspect ratio w/P 2 2 ≤ w/P ≤ 5 Cycle width w 18.29 m - Number of cycles N 14 - Weir length L 1024.24 m - Sidewall length lc 34.76 m - Length between apexes B 33.99 m - Apex A 1.82 m - Water 2023, 15, 722 21 of 29 Crest radius RCrest 0.30 - Upper crest width tw−1 0.61 m - Lower crest width tw−2 1.52 m - Figure 18. Shape and dimensions of the 14-cycle labyrinth weir [9]. Figure 18. Shape and dimensions of the 14-cycle labyrinth weir [9]. The design sequence shown in Table 10 and in the flowchart in Figure 19 was used as an example of the application of the proposed design methodology for the Ute Dam The design sequence shown in Table 10 and in the flowchart in Figure 19 was used as Water 2023, 15, x FOR PEER REVIEW 25 of 32 weir. an example of the application of the proposed design methodology for the Ute Dam weir. Figure 19. Flowchart for the design procedure. Figure 19. Flowchart for the design procedure. 4. Discussion 4.1. Discussion of Discharge Coefficient, Weir, and Cycle Efficiency The magnitude of the discharge coefficient Cd helps us to understand the hydraulic behavior of a weir and is essential when making decisions during weir design, where its value depends on geometry, aeration conditions, and flow behavior during the discharge. When the ratio HT/P < 0.2, higher values of Cd(α) up to 0.833 are presented; this is when the nappe is adhered to the weir wall. During the transition of a nappe adhered to the wall becoming partially aerated, there is an accelerated decrease in Cd(α) values at weirs with angles varying from 6° to 10°. The reduction is less abrupt when α ≥ 12° and, as the head HT on the weir increases, the value of Cd(α) decreases. For values of HT/P < 0.1, the 8° and 10° weirs exhibit similar behavior in Cd(α), and the Cd(α) of the 12° weir is slightly higher Water 2023, 15, 722 22 of 29 Table 10. Spreadsheet for the design of the labyrinth weir. Concept Symbol Value-Unit Equations and Limits (i) Input data Design flow Q 15,574 m /s - Weir width W 256 m - Weir height P 9.14 m - Upstream total head H 5.79 m - (ii) Definition of and the number of cycles (N) Head water ratio H /P 0.63 0.05  H /P  0.8 T T Angle of sidewall a 11.5 6   20 Nappe stability - Stable Stable/Unstable: Table 8 and Figure 14 Labyrinth weir discharge coefficient C (a) 0.483 C (a) = f ( H T/P, a), Equations (21)–(27) d d Cycle width w 27.42 m w = 3P Number of cycles N 9 N = W/w New cycle width w 28.44 m w = W/N Vertical aspect ratio w/P 3.11 2  w/P  4 (iii) Calculation of geometric variables, weir and cycle efficiencies, nappe interference and aeration condition Geometric variables h i 0.5 1.5 Total centerline length of weir L 783.20 m L = 1.5Q/ C (a) H T (2g) Wall width t 1.14 m  P/8 w t Internal apex rope C 1.14 m C = t c c w Internal apex arc Arc 1.60 m Arc = t p(90 a)/(180cosa) int int w External apex arc Arc 1.16 m Arc = t p(2cosa + 1)(90 )/(180 cos ) ext ext w Centerline length of sidewall l 42.14 m lc = L/(2N) ( Arc + Arc )/2 c int ext B = [L/(2N) ( Arc + Arc )/2]cosa + 2t + ext w int Length of apron B 44.28 m t [1 sena(1 + cosa)]/cosa (or input data) Weir and cycle efficiency Magnification ratio M 3.17 M = L/(wN) Linear weir coefficient discharge C  0.754 C  = 1/[8.609 + 22.65H T/P + 1.812/ H T/P] + 0.6375 [5] d (90 ) d(90 ) 00 00 Cycle efficiency e 0.74 e = C (a) M 0 0 Weir efficiency e 1.96 e = C (a) M/C d(90 ) Nappe interference length and aeration condition Nappe interference length B 10.89 m Equations (30)–(33) int Aeration condition - Drowned Table 7 and Figure 12 (iv) Submergence (Tullis et al., 2007 [38]) Downstream total head H 1.22 m - Head ratio H /H 0.21 - Submergence upstream total head H* 5.84 m Equations (1)–(3) and Figure 2 Submergence level S 0.20 S = H /H*; 0  S  1 1.5 Submerged weir discharge coefficient C 0.476 C = C (a)( H / H T) dsum dsum d 4. Discussion 4.1. Discussion of Discharge Coefficient, Weir, and Cycle Efficiency The magnitude of the discharge coefficient C helps us to understand the hydraulic behavior of a weir and is essential when making decisions during weir design, where its value depends on geometry, aeration conditions, and flow behavior during the discharge. When the ratio H /P < 0.2, higher values of C (a) up to 0.833 are presented; this is when T d the nappe is adhered to the weir wall. During the transition of a nappe adhered to the wall becoming partially aerated, there is an accelerated decrease in C (a) values at weirs with angles varying from 6 to 10 . The reduction is less abrupt when a  12 and, as the head H on the weir increases, the value of C (a) decreases. For values of H /P < 0.1, the 8 and T T 10 weirs exhibit similar behavior in C (a), and the C (a) of the 12 weir is slightly higher d d than that of the 10 weir. The higher angle weirs have better discharge capacities. However, lower angle weirs have the advantage of having a longer weir crest length. The increase of the discharge coefficient of the circular apex weir, with respect to the trapezoidal apex weir, is immediate from H /P  0.1. In addition, the dotted line in Figure 9 indicates the inflection point of each curve, where the discharge coefficients acquire their maximum value (Table 6). In effect, the slopes increase until the nappe is no Water 2023, 15, 722 23 of 29 longer aerated and presents local drowning at the weir apex. When the weirs work in a drowned manner, efficiency decreases, projecting curves with slightly descending slopes at the end. The weir and cycle efficiency values decrease when the nappe is no longer adhered to the wall and occurs earlier in weirs where  10. In addition, when the weir begins to drown, the efficiencies generate minimum values, stabilizing from H /P > 0.8. In Figure 10A, the immediate reduction in weir efficiency occurs when local drowning at the apex upstream of the weir becomes present. On the other hand, in Figure 10B, it has been observed that the reduction of the cycle efficiency for a  10 is almost immediate after presenting its maximum value; this phenomenon is due not only to the presence of local drowning, but also to the change of aeration regime of the nappe. 4.2. Discussion of Nappe Aeration Conditions According to the values of the discharge coefficient, it has been identified that the weir is more efficient when the nappe is adhered to the wall. In fact, when the flow is aerated, the discharge coefficient decreases and sub-atmospheric pressures occur behind the nappe. When the flow is in a transitional or partially aerated state, the air cavities under the nappe are removed. Finally, when the weir begins to be drowned, it is characterized by presenting a thicker nappe without the presence of air cavities. The weir is also at its minimum efficiency, remaining constant from H /P > 0.8. In the latter case, the behavior of the weir is equivalent to that of the linear weir. Depending on the aeration condition, turbulent flow has been observed on the walls of the channel. For 20 weirs, aeration conditions tend to occur under turbulent flow when the nappe is adhered to the weir wall, while partially aerated conditions occur for 15 to 20 weirs, and drowned conditions for 12 to 20 weirs, as shown on Table 9. Turbulent flow can also occur between the walls of the cycle as the weir head increases, with greater occurrence under drowned conditions and lesser occurrence when the nappe is adhered to the weir wall. All the weirs have the nappe adhered to the wall when H /P  0.16. On the other hand, it has been observed that, with an increase of the angle a, the presence of this regime increases up to H /P  0.45. However, when a = 20 , the opposite occurs for the case of the aerated regime, i.e., its presence is lower when the angle a increases. The value of the discharge coefficient presents a rapid decrease for angles that vary from 6 to 10 , and this is when the transition from clinging flow to aerated flow occurs. When 15   20 , the weir has a greater range of flow clinging to the wall, in contrast to the aerated flow condition that is briefly produced by changing to the partially aerated regime. The drowning condition is generated for larger heads, i.e., when H /P > 0.49 (a = 6 ). 4.3. Discussion of Nappe Instability and Interference From a  12 , the presence of turbulent flow and helical streamlines was detected on the wall cycles and accompanied by changes in the aeration condition. According to the simulations carried out, the instability is more prevalent when the nappe is partially aerated or drowned than when it is clinging or aerated. On the other hand, the effect of collision between nappes on the reducing of weir efficiency was demonstrated and, therefore, its behavior was characterized for the labyrinth weir design. The length of the nappe interference is a function of the hydraulic load and the sidewall angle of the cycles. Figure 20 shows that weirs with  12 tend towards a stable value of crest length affected by the nappe interference, for similar values of H /P. On the other hand, it should be noted that weirs where a < 10 have a shorter crest length affected by the nappe interference. Water 2023, 15, x FOR PEER REVIEW 27 of 32 Water 2023, 15, 722 24 of 29 On the other hand, it should be noted that weirs where α < 10° have a shorter crest length affected by the nappe interference. Figure Figure 20. 20. Le Length ngth of of t the he cr crest est af affect fected ed by by the the nappe nappe interfer interference. ence. 4.4. Discussion of Application of the Proposed Method 4.4. Discussion of Application of the Proposed Method The geometric differences of the weir and the crest presented in the Houston (1982) [9] The geometric differences of the weir and the crest presented in the Houston (1982) report and the one analyzed in this work are evident. The shape of the crest influences the [9] report and the one analyzed in this work are evident. The shape of the crest influences behavior of the nappe as aeration, in the nappe interference, and, most importantly, flow the behavior of the nappe as aeration, in the nappe interference, and, most importantly, instability in the discharge. The weir crest of the Ute Dam has a quarter-rounded shape. flow instability in the discharge. The weir crest of the Ute Dam has a quarter-rounded However, a half-round shape, as recommended in the proposed design, helps the flow to shape. However, a half-round shape, as recommended in the proposed design, helps the remain adhered to the weir wall, which increases its efficiency; if the flow separates then flow to remain adhered to the weir wall, which increases its efficiency; if the flow sepa- efficiency is lost [5,42]. The design of the Ute Dam weir followed the procedure of Hay rates then efficiency is lost [5,42]. The design of the Ute Dam weir followed the procedure and Taylor (1970) [11]. This means that it considered the w/P ratio to be equal to 2, which of Hay and Taylor (1970) [11]. This means that it considered the w/P ratio to be equal to 2, resulted in a weir of 14 cycles, and a total length of 1024.24 m was obtained. The latter was which resulted in a weir of 14 cycles, and a total length of 1024.24 m was obtained. The determined from the flow magnification and the dimensionless L/W ratio, with a sidewall latter was determined from the flow magnification and the dimensionless L/W ratio, with angle of 12.15 to design the discharge required for a certain reservoir level. The triangular a sidewall angle of 12.15° to design the discharge required for a certain reservoir level. shape of the downstream wall of the weir caused a greater length of nappe interference The triangular shape of the downstream wall of the weir caused a greater length of nappe to be produced, which translated into a lower discharge capacity. The design method interference to be produced, which translated into a lower discharge capacity. The design proposed here considered that the w/P ratio  3, assuming a conservative value between method proposed here considered that the w/P ratio ≈ 3, assuming a conservative value the limits of w/P reported by Hay and Taylor (1972) [11] and Tullis et al. (1995) [14], for the between the limits of w/P reported by Hay and Taylor (1972) [11] and Tullis et al. (1995) weir to be efficient. The number of cycles was reduced to nine, and the total length of the [14], for the weir to be efficient. The number of cycles was reduced to nine, and the total weir was reduced to 783.20 m, as determined by the general discharge equation for weirs, length of the weir was reduced to 783.20 m, as determined by the general discharge equa- thus discharging the required design discharge. However, the length of platform B was tion for weirs, thus discharging the required design discharge. Howeve r, the length of increased from 10.29 m to 44.28 m, and a maximum sidewall angle of 11.5 was chosen to platform B was increased from 10.29 m to 44.28 m, and a maximum sidewall angle of 11.5° avoid generating flow instability. was chosen to avoid generating flow instability. The Bureau of Reclamation spillway design [9] has two dividers in each cycle to reduce The Bureau of Reclamation spillway design [9] has two dividers in each cycle to re- the instability and oscillations of the nappe. However, this method is not recommended [26] duce the instability and oscillations of the nappe. However, this method is not recom- due to the number of dividers required, incurring the possible danger of failure of the mended [26] due to the number of dividers required, incurring the possible danger of hydraulic structure. The proposed sequential design method indicates the ranges of H /P failure of the hydraulic structure. The proposed sequential design method indicates the and the aeration conditions in which the instability originates, which is an important ranges of HT/P and the aeration conditions in which the instability originates, which is an indicator at the time of design. important indicator at the time of design. The sequential design method proposed in the present work is a complete method The sequential design method proposed in the present work is a complete method because it considers the flow behavior during discharge and the possible instability of the because it considers the flow behavior during discharge and the possible instability of the nappe. They are integrated in the design table in Table 10 and in the flowchart in Figure 19. nappe. They are integrated in the design table in Table 10 and in the flowchart in Figure However, it is limited for values of H /P (from 0.5 to 0.8), sidewall angles from 6 to 20 , and 19. Ho for weve weirs r, located it is limin ited a channel. for values of HT/P (from 0.5 to 0.8), sidewall angles from 6° to 20°, a In ndthe for design weirs lo of caa ted labyrinth in a chan weir nel. , it is undoubtedly advisable to perform physical modeling, together with numerical modeling, to validate the hydraulic performance of the labyrinth weir. The design method, design graphs, and charts are limited to the geometries and hydraulic conditions analyzed in this study. Water 2023, 15, 722 25 of 29 5. Conclusions The design procedure of a circular apex labyrinth weir is presented based on its geometric characteristics and the discharge flow behavior. To generate design parameters, the experimental results of the discharge coefficient reported in the literature, were first validated, and verified in CDF and later incorporated into the proposed design method for labyrinth weirs. The proposed design procedure applies to weirs where H /P  0.8 and 6  a  20 . The values of the discharge coefficient are presented as a family of curves as function of H /P and using a mathematical model of a regressive type (through a fifth-degree polynomial equation found for this purpose). The results indicate a higher discharge capacity of the weir while increasing the angle a. The contrast between the discharge coefficients of circular apex weirs with those of a trapezoidal apex indicate an increase in their value of up to 46% (a = 6 ) in relation to the trapezoidal apex weir. The cycle and weir efficiency are presented as a tool in the design procedure. Both parameters indicate that the maximum values occur for H /P  0.17 and the efficiencies are higher with the reduction of the angle a. Four aeration conditions were identified (clinging, aerated, partially aerated, and drowned) with ranges of H /P for each condition. The relationship between the discharge coefficient and the aeration condition is evident: when the nappe is adhered to the wall, the weir has a higher discharge coefficient value. In addition, its presence is greater when a increases, and the opposite occurs when the nappe is aerated. Nappe instability occurs when 12  a  20 and it is accompanied by changes in aeration conditions; there is a greater presence when the flow is partially aerated and drowned. Similarly, ranges of H /P were identified when instability occurred. It is necessary not to incur the instability ranges when designing the weir to avoid possible damage to the hydraulic structure. The length of the crest affected by the nappe interference was characterized and quantified. For this purpose, a family of curves B /B is presented herein as a function int of H /P, and a mathematical model was found for its estimation. This model is a second- degree equation. The results show that the length of B reaches a maximum of 60% of the int length of B. A flowchart implemented in a spreadsheet is also presented as a tool to guide the design process of a labyrinth weir, considering its geometric variables and the phenom- ena that occur in the discharge flow. Additionally, the drowning study carried out by Tullis et al. (2007) [38] is also considered. The proposed sequential method for the design of a labyrinth weir represents a contri- bution to the improvement of the hydraulic performance of weirs of this type of hydraulic structure. It should be noted that this proposal takes into consideration parameters such as the following: (a) flow stability during discharge, (b) aeration condition of the nappe, (c) the nappe interference length, (d) weir and cycle efficiencies, and (e) weir drowning [38], which have been traditionally ignored in traditional design methods or have been studied indepen- dently [1,6,10–12]. In addition, Tullis et al. (1995) [14] and Crookston and Tullis (2012) [15] generated spreadsheets to help in the weir design. However, they did not include the nappe aeration and its instability conditions, as well as the length of the nappe interference, which influences the efficiency of the weir operation and the safety of the hydraulic structure. Finally, although the methods and tools presented in this study were highly effective when used in the design and study of a labyrinth weir, it is recommended that physical and numerical modeling be performed to validate the hydraulic performance of a specific pre-designed hydraulic structure with the proposed sequential method. Author Contributions: Conceptualization, W.O.-B. and C.D.-D.; methodology, E.D.M.-V. and W.O.-B.; software, E.D.M.-V. and J.F.-V.; validation, E.D.M.-V., H.S.-T. and W.O.-B.; formal analysis, C.D.-D. and H.S.-T.; investigation, E.D.M.-V.; resources, C.B.C.; writing—original draft preparation, E.D.M.-V. and W.O.-B.; writing—review and editing, H.S.-T., C.D.-D. and C.B.C.; visualization, E.D.M.-V. and Water 2023, 15, 722 26 of 29 J.F.-V.; supervision, W.O.-B.; project administration, W.O.-B.; funding acquisition, W.O.-B. and C.B.C. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Data Availability Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest. Abbreviations A internal length apex. Arc internal apex arc. int Arc external apex arc. ext A , A , A fractional area in the x, y, z direction, respectively. x y z a adjustment factor to obtain the discharge coefficient. B length of apron. B nappe interference length. int b adjustment factor to obtain the discharge coefficient. c adjustment factor to obtain the discharge coefficient. C discharge coefficient. C submerged weir discharge coefficient. d-sum C (a) labyrinth weir discharge coefficient C linear weir discharge coefficient. d (90 ) C internal apex rope. C , C , C constants of the turbulent k-" model. e e m 1 2 Cov covariance. D external apex length. Dk effective diffusivity for turbulent kinetic energy. eff D# effective diffusivity for dissipation rate. eff d adjustment factor to obtain the discharge coefficient. e adjustment factor to obtain the discharge coefficient. F diffusion term. F source term. F security factor. f adjustment factor to obtain the discharge coefficient. f control variable. f , f , f viscous acceleration in x, y, z direction, respectively. x y z G turbulent kinetic energy generation due to mean velocity gradients. G , G , G acceleration of the body in the x, y, z direction, respectively. x y z g acceleration gravity. H downstream total head. H upstream total head. H* upstream total head of the drowned weir. h piezometric head. k turbulent kinetic energy. L characteristic length of the weir. L cycle length. cycle L length of the crest affected by the nappe interference. int l centerline length of sidewall. M magnification ratio. m adjustment coefficient to obtain the length B . int N number of cycles. N number of cells. n adjustment coefficient to obtain the length B . int o adjustment coefficient to obtain the length B . int P weir height. p order of convergence. Q design flow. Water 2023, 15, 722 27 of 29 Q flow of a linear weir. R fradius of the weir crest. crest r fmesh refinement ratio. S fsubmergence level. S fstrain rate tensor. ij t weir wall width. t upper crest width. w1 t lower crest width. u velocity component in the x direction. V fraction volume. v velocity component in y direction. v turbulent kinematic viscosity. W channel width. w cycle width. Y experimental results. exp Y numerical results. num z velocity component in z direction. a angle of sidewall. DV volume of the i-th cell. g representative cell size. e dissipation rate. e relative error. e weir efficiency. e” cycle efficiency. s variance of the experimental results. exp s variance of the numerical results. num s Prandtl number. References 1. Mattos-Villarroel, E.D.; Flores-Velazquez, J.; Ojeda-Bustamante, W.; Iñiguez-Covarrubias, M.; Díaz-Delgado, C.; Salinas-Tapia, H. Hydraulic analysis of a compound weir (triangular-rectangular) simulated with computational fluid dynamics (CFD). Technol. Cienc. Agua 2021, 12, 112–162. [CrossRef] 2. Darvas, L.A. Discussion of performance and design of labyrinth weirs, by Hay and Taylor. J. Hydraul. Eng. ASCE 1971, 97, 1246–1251. [CrossRef] 3. Bilhan, O.; Emiroglu, E.; Miller, C.J. 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Available online: https://www.grc.nasa.gov/www/wind/valid/tutorial/spatconv.html (accessed on 4 November 2021). 56. Roache, P.J. Quantification of uncertainty in computational fluid dynamics. Ann. Rev. Fluid Mech. 1997, 29, 123–160. [CrossRef] Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Water Multidisciplinary Digital Publishing Institute

Methodological Proposal for the Hydraulic Design of Labyrinth Weirs

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Abstract

water Article Methodological Proposal for the Hydraulic Design of Labyrinth Weirs 1 2 , 3 Erick Dante Mattos-Villarroel , Waldo Ojeda-Bustamante * , Carlos Díaz-Delgado , 3 4 1 Humberto Salinas-Tapia , Jorge Flores-Velázquez and Carlos Bautista Capetillo Campus Siglo XXI, Autonomous University of Zacatecas “Francisco García Salinas”, Road Zacatecas-Guadalajara Km 6, Zacatecas 98160, Mexico Research and Postgraduate Division, Chapingo Autonomous University, Chapingo, Texcoco 56230, Mexico Inter-American Institute of Technology and Water Sciences, Autonomous University of the State of Mexico, Road Toluca-Atlacomulco Km 14.5, Toluca 50200, Mexico Postgraduate College, Coordination of Hydrosciences, Road Mexico-Texcoco Km 36.5, Texcoco 62550, Mexico * Correspondence: w.ojeda@riego.mx Abstract: A labyrinth weir allows for higher discharge capacity than conventional linear weirs, especially at low hydraulic heads. In fact, this is an alternative for the design or rehabilitation of spillways. It can even be used as a strategy in problems related to dam safety. A sequential design method for a labyrinth weir is based on optimal geometric parameters and the results of discharge flow analysis using Computational Fluid Dynamics and the experimental studies reported in the literature. The tests performed were for weirs with values of H /P  0.8 and for angles of the cycle sidewall of 6  a  20 . The results of the discharge coefficient are presented as a family of curves, which indicates a higher discharge capacity when H /P  0.17. Four aeration conditions are identified with higher discharge capacity when the nappe is adhering to the downstream face of the weir wall and lower discharge capacity when the nappe is drowned. Unstable flow was present when 12  a  20 , with a greater presence when the nappe was partially aerated and drowned. The interference of the nappe is characterized and quantified, reaching up to 60% of the length between the apex, and a family of curves is presented as a function of H /P in this respect. Finally, a spreadsheet and a flowchart are proposed to support the design of the labyrinth type weir. Citation: Mattos-Villarroel, E.D.; Ojeda-Bustamante, W.; Díaz-Delgado, Keywords: labyrinth weir; Computational Fluid Dynamics (CFD); spillways discharge capacity; C.; Salinas-Tapia, H.; spillway weir design Flores-Velázquez, J.; Bautista Capetillo, C. Methodological Proposal for the Hydraulic Design of Labyrinth Weirs. Water 2023, 15, 722. https://doi.org/10.3390/w15040722 1. Introduction Labyrinth weirs are polygonal hydraulic structures (Figure 1) used to increase dis- Academic Editor: Diana De Padova charge capacity, for a fixed width, reducing hydraulic head relative to linear weirs. Their Received: 14 November 2022 hydraulic performance characteristics make them an efficient and cost-effective alternative Revised: 2 February 2023 for spillway weir design or rehabilitation. This structure makes it possible to increase the Accepted: 5 February 2023 storage volume in a reservoir and to control the water level. Figure 1 shows the geometrical Published: 11 February 2023 parameters of a labyrinth weir, where W is the width of the channel (m), w is the width of the weir cycle (m), t is the width of the weir wall (m), D is the external length of the apex (m), A is the internal length of the apex (m), l is the length of the cycle sidewall (m), a is the angle of the cycle sidewall with respect to the flow direction ( ), B is the distance Copyright: © 2023 by the authors. between apexes (m), P is the height of the weir (m), h is the piezometric head (m), H is the Licensee MDPI, Basel, Switzerland. 3 1 total head (m), and Q is the design flow (m s ). This article is an open access article The design of a labyrinth weir is laborious because its discharge capacity is simultane- distributed under the terms and ously affected by several factors, including the approach conditions and the geometry of conditions of the Creative Commons the weir [2]. According to Bilhan, Emiroglu, and Miller (2016) [3], optimizing the geometry Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ variables involved in the design of a labyrinth weir is an engineering challenge in which the 4.0/). following must be determined: (a) the configuration of the cycles, (b) the shape of the crest, Water 2023, 15, 722. https://doi.org/10.3390/w15040722 https://www.mdpi.com/journal/water Water 2023, 15, 722 2 of 29 and (c) the orientation of the weir. For any type of weir, its geometry, particularly the shape of the crest profile, influences the value of the discharge coefficient. Willmore (2004) [4] in- dicated that half-round crest profiles are more efficient than sharp-crest and quarter-round profiles, because they allow the nappe to remain adhered to the wall of the structure for small heads. Typically, the cycle’s apex facilitates the concrete construction of a labyrinth weir. From a hydraulic perspective, a labyrinth weir with a smooth upstream transition is relatively more efficient than the abrupt transition presented by a trapezoidal labyrinth Water 2023, 15, x FOR PEER REVIEW 2 of 32 weir [5]. In this regard, Tullis and Young (2005) [6] reported an increase in discharge efficiency at Brazos Dam (Wako, TX, USA) by creating a smooth transition between the flow and the circular apexes of the labyrinth spillway cycles. Figure 1. Geometric parameters of a labyrinth type weir (Mattos-Villarroel et al., 2021 [1]). Figure 1. Geometric parameters of a labyrinth type weir (Mattos-Villarroel et al., 2021 [1]). The design of a labyrinth weir is laborious because its discharge capacity is simulta- Families of discharge coefficient curves (C ) for the design of labyrinth weirs have neously affected by several factors, including the approach conditions and the geometry been determined from physical models of various prototype structures, examples of which of the weir [2]. According to Bilhan, Emiroglu, and Miller (2016) [3], optimizing the geom- are as follows: Avon and Woronova [2]; Harrezza, Dungo, Keddara, Alijó, Gema, and São etry variables involved in the design of a labyrinth weir is an engineering challenge in Domingo [7]; Hyrum [8]; Ute [9]; Lake Brazos [6]; and Lake Townsend [10]. The evolution which the following must be determined: (a) the configuration of the cycles, (b) the shape of commonly used and documented design methods can be summarized in the following of the crest, and (c) the orientation of the weir. For any type of weir, its geometry, partic- sequence: (a) Hay and Taylor (1970) [11]; (b) Darvas (1971) [2]; (c) Hinchliff and Houston ularly the shape of the crest profile, influences the value of the discharge coefficient. Will- (1984) m[o 12 re]; (2(d) 004) Lux [4] in (1989) dicated [13 th]; at (e) half Magalh -round cr ães est and profiLor les a ena re m (1989) ore efficient [7]; (f) thT an ullis, sharAmanian, p-crest and W an aldr d qon uarte (1995) r-roun [14 d pr ]; o and files, (g) beCr cau ookston se they and allow T ullis the na (2012) ppe to [re 15 m ].ain However adhered, to the thr e esults wall of of the structure for small heads. Typically, the cycle’s apex facilitates the concrete con- Idrees and Al-Ameri (2022) [16] showed that common design equations did not take into struction of a labyrinth weir. From a hydraulic perspective, a labyrinth weir with a smooth account all the parameters that affect the performance of a labyrinth weir such as geometry upstream transition is relatively more efficient than the abrupt transition presented by a and flow conditions. trapezoidal labyrinth weir [5]. In this regard, Tullis and Young (2005) [6] reported an On the other hand, physical and numerical modeling are used as complementary tools increase in discharge efficiency at Brazos Dam (Wako, TX, USA) by creating a smooth to improve the design of hydraulic structures. Indeed, Computational Fluid Dynamics (CFD) transition between the flow and the circular apexes of the labyrinth spillway cycles. makes it possible to obtain, by means of numerical techniques, adequate results for the Families of discharge coefficient curves (Cd) for the design of labyrinth weirs have equations that predict the behavior of any flow. In general, the equations to be solved with been determined from physical models of various prototype structures, examples of CFD are the Navier–Stokes equations. Thus, numerical modeling facilitates the hydraulic which are as follows: Avon and Woronova [2]; Harrezza, Dungo, Keddara, Alijó, Gema, evaluation of structures, such as weirs, to obtain information on pressure fields and velocities and São Domingo [7]; Hyrum [8]; Ute [9]; Lake Brazos [6]; and Lake Townsend [10]. The under ev d o ilut ffeiro en nt of g com eom m et orn ic lya use ndd h a yn dd r a du olcumen ic cond ted iti o dn esi s.gn It m iseth im op do sr ca tan n t be to sum hig m ha liriz ghed t t h in e th us ee of CFD a fo slla owi too nl g iseq n th uence e gen : ( ear)a Ha tioy nao nfdi n Ta fo yl rm or a (t 1i9 o7n 0)t o [11 su ]; p (b p )o Da rt rv de ac s is (1 io 97 n1 -) m [2 a]k ; i (n c)g Hin and chl th iff e a d ne d s ign Houston (1984) [12]; (d) Lux (1989) [13]; (e) Magalhães and Lorena (1989) [7]; (f) Tullis, of hydraulic structures. In particular, the performance of CFD in the modeling of flow over Amanian, and Waldron (1995) [14]; and (g) Crookston and Tullis (2012) [15]. However, linear weirs, particularly the labyrinth type, stands out, as has been demonstrated by several the results of Idrees and Al-Ameri (2022) [16] showed that common design equations did researchers [1,17–22]. Ben Said et al. (2022) [23] using CFD proved that the discharge capacity not take into account all the parameters that affect the performance of a labyrinth weir of a labyrinth weir can increase as its downstream channel bed level decreases. Samadi et al. such as geometry and flow conditions. (2022) [24] experimentally and numerically investigated the effects of geometric parameters On the other hand, physical and numerical modeling are used as complementary on the efficiency in triangular and trapezoidal labyrinth weirs. tools to improve the design of hydraulic structures. Indeed, Computational Fluid Dynam- ics (CFD) makes it possible to obtain, by means of numerical techniques, adequate results 1.1. Discharge Flow Characteristics for the equations that predict the behavior of any flow. In general, the equations to be The behavior and characteristics of the discharge flow presented by the complex solved with CFD are the Navier–Stokes equations. Thus, numerical modeling facilitates geometry of the labyrinth weir must be considered in its hydraulic analysis and design. the hydraulic evaluation of structures, such as weirs, to obtain information on pressure fields and velocities under different geometric and hydraulic conditions. It is important This analysis must consider the aeration conditions, the instability of the discharge flow, to highlight the use of CFD as a tool in the generation of information to support decision- making and the design of hydraulic structures. In particular, the performance of CFD in the modeling of flow over linear weirs, particularly the labyrinth type, stands out, as has been demonstrated by several researchers [1,17–22]. Ben Said et al. (2022) [23] using CFD proved that the discharge capacity of a labyrinth weir can increase as its downstream Water 2023, 15, 722 3 of 29 and the collision between the nappes that occurs between the walls of the weir cycles. It should be noted that each of these factors individually affects the value of the discharge coefficient C and the efficiency of the weir. However, the main challenge is to know the joint effect of all these factors on the discharge coefficient. 1.1.1. Aeration Conditions The efficiency and discharge capacity of the labyrinth weir also depends on the aeration regime of the nappe, which is influenced by the shape of the crest, the level of the hydraulic head, the height of the weir, the flow path over the crest, the turbulence, and the pressure under the nappe. The increase in discharge flow over the weir generates different aeration conditions; these have been identified by some researchers and have already been included in de- sign curves, through the value of the discharge coefficient [25,26]. Lux and Hinchliff (1985) [25] identified three types of aeration: aerated, transitional (partially aerated), and suppressed (drowned). However, for Falvey (2003) [27], there are four types of aera- tion conditions: cavity, atmospheric, sub-atmospheric, and pressure. On the other hand, Crookston and Tullis (2012) [26] also reported four types of aeration: clinging, aerated, par- tially aerated, and drowned, and indicated that aeration conditions can produce pressure fluctuations at the sidewalls of the weir, low frequency sound, and vibrations. In addition, previous studies point to the labyrinth weir as an excellent aeration control structure. Ex- amples of this are the work of Hauser (1996) [28], who describes the methods for the design of this type of weir, taking into consideration the aeration conditions during discharge. Wormleaton and Soufani (1998) [29] and Wormleaton and Tsan (2000) [30] found that a rectangular labyrinth weir has better aeration efficiency than a triangular labyrinth weir and that the latter is better than a linear one. 1.1.2. Nappe Instability Under certain flow conditions and weir geometry, the nappe becomes unstable, and vibrations occur in the hydraulic structure. Several researchers have conducted studies on nappe instability. Crookston and Tullis (2012) [26] defined nappe instability as a nappe with an oscillating trajectory. These authors, supported by experimental observations, reported that the flow streamlines under this condition are helical, adjacent, and parallel to the weir walls. Furthermore, these nappes occur momentarily with changes of aeration condition, most frequently during the aerated and partially aerated condition, when their presence causes vibrations that threaten the safety of the structure. It is recommended that these conditions should be considered during the design of the weir and are to be avoided. According to Casperson (1995) [31], vibrations are easily felt by touch and the sound can be heard over a kilometer away. For Naudascher and Rockwell (2017) [32], the vibrations are attributed to inadequate aeration below the discharge flow and indicate that an unventilated air pocket behind the nappe can amplify the instability of the weir. Similarly, according to their research, the three-dimensional characteristic of the flow during discharge, at the point of detachment and the height of fall, may be a significant parameter in the presence of vibrations. Falvey (2003) [27] even pointed out that the vibration of the nappe occurs when the weir operates at low hydraulic heads, in the range 0.01 h/P 0.06. Some researchers recommend the use of splitters, placed vertically and normal to the flow to reduce vibrations [12,33]. However, because several splitters are required, this solution was not recommended by Falvey (2003) [27]. To eliminate vibrations, the Metropolitan Water, Sewerage, and Drainage Board (1980) [34] conducted studies on a physical model of the Avon Dam, located in Australia, where they increased the crest roughness. However, with only a 15 mm increase in crest height, the discharge decreased by 2%. 1.1.3. Nappe Interference Nappe interference refers to the collision between nappes at the upstream apexes of the weir. This hydraulic phenomenon decreases the efficiency of the weir. The length of Water 2023, 15, 722 4 of 29 the interference depends on the width apex, the shape of the crest, the weir height, the hydraulic head, and the aeration conditions. Indlekofer and Rouve (1975) [35] studied the nappe interference in single-cycle tri- angular weirs whose sidewalls are perpendicular to the channel walls, also known as corner weirs. These authors identified a perturbed region, where the nappes from each weir sidewall collide, and a second region, where streamlines are perpendicular to the weir wall, similar to the discharge over a linear weir. Finally, they found an empirical relationship between the average discharge coefficient of the disturbed area, the theoretical crest length of the disturbed region, and the discharge to compare the efficiency of a corner weir with a linear weir. Lux (1989) [13] recommended that the ratio of the apex width to the weir cycle width should be less than 0.0765, so that the effects of the collisions of the nappes do not generate large reductions in the performance of the trapezoidal labyrinth weir. Falvey (2003) [27] developed an empirical model that considered the data relationship of Indlekofer and Rouve (1975) [35], and then developed a second equation based on the analysis of experimental data from a labyrinth weir. However, this author did not indicate which of the two proposed equations was the most appropriate, but the significant influence of the nappe interference on aeration in triangular labyrinth weirs was highlighted. Based on the quantity of movement, Osuna 2000 [36] analyzed the hydraulic behav- ior of a weir, located inside a channel and oblique to the upstream flow direction, and proposed an equation that allows for the calculation of the flow per unit length of the weir. The results indicated that the direction of flow at the outlet of the weir depends on the ratio of the thickness of the nappe contracted over the weir and the height of free- surface water, upstream from a point sufficiently far from the weir. In the last decade, Granell and Toledo (2010) [37] adapted the mathematical model of Osuna (2000) [37] to labyrinth weirs in order to obtain the nappe’s collision length. 1.1.4. Drowning It has been observed that the drowning phenomenon in labyrinth weirs has a behavior similar to that in the operation of linear weirs [38]. However, Taylor (1968) [39] concluded that drowning effects are less significant for labyrinth weirs than for linear weirs. The linear weir drowning analysis developed by Villemonte (1947) [40] was the most applied method for labyrinth weirs under drowning conditions. However, at the beginning of the century, Tullis et al. (2007) [38] developed a dimensionless relationship for drowning heads in labyrinth weirs, obtaining an average error of up to 0.9%. Figure 2 shows the function describing drowning at the weir, which was divided into three sections to better explain its behavior and is represented by Equations (1)–(3). 4 2 H H H H d d d = 0.3320 + 0.2008 + 1; 0   1.53 (1) H H H H T T T T H H H d d = 0.9379 + 0.2174; 1.53   3.5 (2) H H H T T T H = H ; 3.5  (3) H* is the upstream total head when the weir is drowned (m), H is the upstream total head when the weir is not drowned (m), and H is the downstream total head of the weir in the drowning state (m). Based on the above, and due to the complexity in the hydraulic analysis of a labyrinth weir, the objective of this work is focused on proposing a sequential design that facilitates the conception of a labyrinth weir, based on the results of numerical modeling in CFD and by considering the relationships between the geometric variables of the weir and the simultaneous effect of the hydraulic phenomena that occur on the nappe. Water 2023, 15, x FOR PEER REVIEW 5 of 32 Water 2023, 15, 722 5 of 29 𝐻 = 𝐻 ; 3.5 ≤ (3) Figure 2. Submergence limits (adapted from Tullis et al., 2007 [38]). Figure 2. Submergence limits (adapted from Tullis et al., 2007 [38]). 2. Materials and Methods H* is the upstream total head when the weir is drowned (m), HT is the upstream total 2.1. Description of the Physical Model head when the weir is not drowned (m), and Hd is the downstream total head of the weir The construction of the conceptual model and its evaluation in CFD are based on in the drowning state (m). the experimental prototype reported by Crookston and Tullis (2012) [15]. The physical Based on the above, and due to the complexity in the hydraulic analysis of a labyrinth model consisted of a trapezoidal labyrinth weir with a quarter-round crest profile made of weir, the objective of this work is focused on proposing a sequential design that facilitates high-density polyethylene, located within a rectangular channel. The description of the the conception of a labyrinth weir, based on the results of numerical modeling in CFD and dimensions and geometric characteristics of the labyrinth weir are specified in Table 1. by considering the relationships between the geometric variables of the weir and the sim- ultaneous effect of the hydraulic phenomena that occur on the nappe. Table 1. Geometric characteristics of the trapezoidal labyrinth weir [15]. 2. Materials and Methods a ( ) N L (m) A (m) w (m) P (m) W (m) Crest Profile 2.1. Description of the Physical Model 15 2 4 0.038 0.617 0.305 1.235 Quarter-round The construction of the conceptual model and its evaluation in CFD are based on the experimental prototype reported by Crookston and Tullis (2012) [15]. The physical model 2.2. Numerical Solution Method consisted of a trapezoidal labyrinth weir with a quarter-round crest profile made of high- 2.2.1. Computational Fluid Dynamics (CFD) density polyethylene, located within a rectangular channel. The description of the dimen- Numerical modeling was performed using the ANSYS-FLUENT simulation software [41,42]. sions and geometric characteristics of the labyrinth weir are specified in Table 1. This software is considered to be a general purpose CFD code, which has been widely used in recent years, due to the advancement of technology, mainly computational technology. ANSYS- Table 1. Geometric characteristics of the trapezoidal labyrinth weir [15]. FLUENT allows for the modeling of flows in one phase to flows in multiple phases (multiphase) α (° ) N L (m) A (m) w (m) P (m) W (m) Crest Profile in both closed and open domains. The numerical simulation was performed with two-phase flow 15 2 4 0.038 0.617 0.305 1.235 Quarter-round models, incompressible conditions, and a free-surface interface. ANSYS-FLUENT implements the approach to surface analysis using the volume of fluid 2.2. N (VOF) umerica scheme. l Solution M The VOF ethod scheme is ideal for applications involving free-surface flows. It involves defining a volume fraction function for each of the fluids in the entire domain. 2.2.1. Computational Fluid Dynamics (CFD) The governing equations for all fluid fractions were solved using the two-phase (air- Numerical modeling was performed using the ANSYS-FLUENT simulation software water) flow model. A two-phase flow model was used to detect under pressure; air inertia [41,42]. This software is considered to be a general purpose CFD code, which has been and air–water interaction effects were neglected in the numerical model. The VOF method widely used in recent years, due to the advancement of technology, mainly computational was used to improve the accuracy of the free-surface simulations. technology. ANSYS-FLUENT allows for the modeling of flows in one phase to flows in For incompressible fluids, when the density of water is constant, the mass continuity multiple phases (multiphase) in both closed and open domains. The numerical simulation equation of motion of the fluid in Cartesian coordinates is given as Equation (4) [43]: ¶ ¶ ¶ (u A ) + v A + (w A ) = 0. (4) x y z ¶x ¶y ¶z Water 2023, 15, 722 6 of 29 where A , A , and A are the fractional areas open to the fluid in directions x, y, and x y z z, respectively. Additionally, u, v, and w are the velocity components in the directions x, y, and z, respectively. The Navier–Stokes equations, with the velocity components as momentum equations, are used to determine the 3D fluid motion in Cartesian coordinates (Equations (5)–(7)) [43]. ¶u 1 ¶u ¶u ¶u 1 ¶( p) + u A + v A + w A = + G + f (5) x y z x x ¶t V ¶x ¶y ¶z r ¶y ¶v 1 ¶v ¶v ¶v 1 ¶( p) + u A + v A + w A = + G + f (6) x y z y y ¶t V ¶x ¶y ¶z r ¶y ¶w 1 ¶w ¶w ¶w 1 ¶ p ( ) + u A + v A + w A = + G + f (7) x y z z z ¶t V ¶x ¶y ¶z r ¶z where V is the fraction volume open to the fluid; G , G , and G are the acceleration F x y z components of the body, and f , f , and f are the viscous acceleration components in the x y z directions x, y, and z, respectively. To define fluid configurations, the volume of fluid function (VOF) represents a volume of fluid per unit volume [44], as indicated in Equation (8): ¶F 1 ¶ ¶ ¶ FA u + (FA u) + FA v + (FA w) + j = F + F (8) x y z D S ¶t V ¶x ¶y ¶z x where F is the diffusion term, applied only in the turbulent mixture of a two-phase flow, and F is a source term. To determine the discharge and pressure characterization at the weir crest, the two- phase model (air and water) was applied. In addition, the standard k–# model was used to analyze the effects of turbulence, which solves two transport equations with Reynolds stresses for the turbulent kinetic energy (k) and the dissipation rate (#). The k–# turbulence model is a semi-empirical model with low computational cost; several researchers have demonstrated its advantages and excellent results for the simulation of confined, internal, and free-surface flows [19,43,45–47]. Equations (9) and (10) were used to obtain the turbulent kinetic energy and dissipation rate. The k–# model is described below [48]. ¶k ¶ku ¶ ¶k + = Dk + G # (9) e f f k ¶t ¶x ¶x ¶x i i i ¶# ¶#u ¶ ¶# # # + = D# + C G C (10) e f f 1# 2# ¶t ¶x ¶x ¶x k k k i i i where G is the generation of turbulent kinetic energy due to mean velocity gradients; DDk and D# are the effective diffusivity for the turbulent kinetic energy (k) and the e f f e f f dissipation rate (#), respectively, determined by Equations (11) and (12): Dk = n + n (11) e f f t D# = n + (12) e f f where n = C is the turbulent kinematic viscosity at each point; s is the Prandtl number t m # for # and takes the value 1.3; and the constants C , C , and C have values of 1.44, 1.92, 1# 2# m 2 2 and 0.09, respectively. G = 2n S is the turbulent kinetic energy result, and S is the strain i j i j rate tensor. For the solution of the equations, the Semi-Implicit Method for Pressure Linkek Equa- tions (SIMPLE) algorithm was used, which approaches convergence through a series of intermediate pressure and velocity fields satisfying continuity [49]. The higher order Water 2023, 15, x FOR PEER REVIEW 7 of 32 where 𝐺 is the generation of turbulent kinetic energy due to mean velocity gradients; 𝑘 and 𝐷 𝜀 are the effective diffusivity for the turbulent kinetic energy (𝑘 ) and the 𝑓𝑓𝑒 𝑓𝑓𝑒 dissipation rate (𝜀 ), respectively, determined by Equations (11) and (12): 𝐷 𝑘 = 𝜈 + 𝜈 (11) 𝑓𝑓𝑒 𝑡 𝐷 𝜀 = 𝜈 + (12) 𝑓𝑓𝑒 where 𝜈 = 𝐶 is the turbulent kinematic viscosity at each point; 𝜎 is the Prandtl num- 𝑡 𝜇 𝜀 ber for 𝜀 and takes the value 1.3; and the constants 𝐶 , 𝐶 , and 𝐶 have values of 1.44, 1𝜀 2𝜀 𝜇 2 2 1.92, and 0.09, respectively. 𝐺 = 2𝜈 𝑆 is the turbulent kinetic energy result, and 𝑆 is 𝑘 𝑡 the strain rate tensor. For the solution of the equations, the Semi-Implicit Method for Pressure Linkek Equations (SIMPLE) algorithm was used, which approaches convergence through a series of intermediate pressure and velocity fields satisfying continuity [49]. The higher order Upwind spatial discretization system was also used, which ensures stable schemes by min- imizing numerical diffusion errors [49]. Both schemes are integrated in ANSYS-FLUENT. The ANSYS-FLUENT Geometry module was used to build the conceptual models. Water 2023, 15, 722 7 of 29 The Meshing module was used to generate the mesh, where they were spatially discretized using predominantly hexahedral meshes. The advantage of this type of mesh is the reduc- tion in the number of cells and the improvement in the convergence of the solution [50]. Upwind spatial discretization system was also used, which ensures stable schemes by mini- The computational meshes were refined in the vicinity of the weir wall (Figure 3B), where mizing numerical diffusion errors [49]. Both schemes are integrated in ANSYS-FLUENT. the turbulence is dissipated, and its behavior has a significant effect on the results. The ANSYS-FLUENT Geometry module was used to build the conceptual models. The In the simulation of free-surface flows, it is important to define the boundary condi- Meshing module was used to generate the mesh, where they were spatially discretized using tions appropriately at the inlet and outlet of the model domain. In this work, the boundary predominantly hexahedral meshes. The advantage of this type of mesh is the reduction conditions implemented in ANSYS-FLUENT were applied at the inlet and outlet of the in the number of cells and the improvement in the convergence of the solution [50]. The domain, with the fluid velocity as inlet and atmospheric pressure as outlet. The approach computational meshes were refined in the vicinity of the weir wall (Figure 3B), where the of the boundary conditions (Figure 3A) and definition of the fluid properties are specified turbulence is dissipated, and its behavior has a significant effect on the results. in Table 2. Figure 3. (A) Boundary conditions (Mattos-Villarroel et al., 2021 [1]). (B) Refined mesh. Figure 3. (A) Boundary conditions (Mattos-Villarroel et al., 2021 [1]). (B) Refined mesh. Table 2. Boundary and initial conditions. In the simulation of free-surface flows, it is important to define the boundary condi- tions appropriately at the inlet and outlet of the model domain. In this work, the boundary Boundary and Initial Conditions Solution Method conditions implemented in ANSYS-FLUENT were applied at the inlet and outlet of the Domain: inlet Velocity domain, with the fluid velocity as inlet and atmospheric pressure as outlet. The approach Domain: outlet Atmospheric pressure of the boundary conditions (Figure 3A) and definition of the fluid properties are specified Domain: weir, sidewalls, and channel platform Solid, stationary, and non-slip. in Table 2. Table 2. Boundary and initial conditions. Boundary and Initial Conditions Solution Method Domain: inlet Velocity Domain: outlet Atmospheric pressure Domain: weir, sidewalls, and channel platform Solid, stationary, and non-slip. Viscosity model k–# standard Multiphasic model Volume of fluid (VOF) Pressure–velocity coupling SIMPLE Spatial discretization scheme Upwind 2.2.2. Grid Convergence Index (GCI) Within the simulation, the shape and density of the mesh or grid in the analysis have a very significant importance and influence the total number of elements, the computing time, and the accuracy of the analysis. In the present study, 40 different scenarios with 7 grid sizes of different densities were analyzed (Table 3). In each grid, the number of elements was decreased until we obtained an adequate convergence of calculations, and the independence results of the mesh were obtained. 𝑖𝑗 𝑖𝑗 𝐷𝐷 Water 2023, 15, 722 8 of 29 Table 3. Simulated scenarios for each grid. Grid Scenario Grid Scenario I 1–10 V 26–30 II 11–15 VI 31–35 III 16–20 VII 36–40 IV 21–25 It is important that, before calculating any discretization error estimate, it must be guaranteed that the convergence of the iterative process presents a decrease of at least three orders of magnitude in the normalized residuals for each solved equation [51]. In the present study, the convergence and discretization errors were verified at each time step to control the convergence of the solution of the time-dependent problems and, thereby, guarantee an adequate solution to the equations describing the phenomenon. The recommended method for discretization error estimation is the Richardson ex- trapolation method. Roache (1994) [52] proposed a way of reporting the results of grid convergence studies with the Grid Convergence Index (GCI), which is based on the Richard- son Extrapolation method, a method that has been extensively evaluated in case studies using CFD [51,53,54]. GCI indicates the percentage by which the calculated value deviates from the asymptotic numerical value and how much the solution would change with further refinement of the mesh. Thus, a small value of GCI indicates that the calculation is within the asymptotic range [55]. To estimate the order of convergence and verify that the solutions are within the asymptotic range of convergence, Roache (1994) [52] recommended using three different grid sizes. Equation (13) shows how to determine the CGI of a fine or coarse grid [56]: F j j F j jr s i+1,1 s i+1,1 GC I = ; GC I = (13) f ine coarse p p (r 1) (r 1) where F is a security factor (taking a value of 3 for comparisons of two grids and 1.25 for comparisons of three or more grids [56]), r is the mesh refinement ratio, p is the order of convergence, and e is the error relative to the control variable f . The representative model cell size (l) was estimated using Equation (14) for three- dimensional models. " # 1 l = (DV ) (14) i=1 In this equation, DV is the volume of the i-th cell and N is the number of cells. Therefore, i C to perform the calculation, at least three different grid sizes must be selected to determine the value of the control variables f ( f , f , f ) considered important for the simulation objective. 1 2 3 Then, for l < l < l , the mesh refinement factors were determined as r = l l , 1 2 3 21 2 1 r = l /l and the order of convergence p was calculated with Equation (15) [52]. 32 3 2 f f 3 2 ln f f 2 1 p = (15) ln r Equation (16) enables verification that the solutions are within the asymptotic range of convergence [55]: GC I 1 (16) GC I r Seven different grids are presented in this study. To perform the GCI analysis, the hydraulic head (h , h , h ) was considered as a control variable to estimate discretiza- 1 2 3 3 1 3 1 tion errors. The flows used were as follows: Q = 0.2036 m s , Q = 0.2003 m s , I II 3 1 3 1 3 1 3 1 Q = 0.2137 m s , Q = 0.2170 m s , Q = 0.2149 m s , Q = 0.2036 m s , and III IV V VI 3 1 Q = 0.1915 m s for the grids: I, II, III, IV, V, VI, and VII, respectively. The GCI cal- VII Water 2023, 15, x FOR PEER REVIEW 10 of 32 𝑓 𝑓 3− 2 𝑓 𝑓 (15) 2− 1 𝑝 = ln 𝑟 Equation (16) enables verification that the solutions are within the asymptotic range of convergence [55]: 𝐶𝐺𝐼 ≈ 1 (16) 𝐶𝐺𝐼 𝑟 Seven different grids are presented in this study. To perform the GCI analysis, the hydraulic head (h1, h2, h3) was considered as a control variable to estimate discretization 3 −1 3 −1 errors. The flows used were as follows: QI = 0.2036 m s , QII = 0.2003 m s , QIII = 0.2137 3 −1 3 −1 3 −1 3 −1 3 −1 m s , QIV = 0.2170 m s , QV = 0.2149 m s , QVI = 0.2036 m s , and QVII = 0.1915 m s for the grids: I, II, III, IV, V, VI, and VII, respectively. The GCI calculations of the numerical Water 2023, 15, 722 9 of 29 solutions are summarized in Table 4; the asymptotic ranges of convergence obtained are approximately equal to 1. Therefore, the numerical solutions are within the asymptotic range. In the present work, the hydraulic head was achieved with a maximum error of up culations of the numerical solutions are summarized in Table 4; the asymptotic ranges of to 2.80%, corresponding to grid II. convergence obtained are approximately equal to 1. Therefore, the numerical solutions are within Table 4.the Estasymptotic imation of grrange. id convIn ergthe ence pr in esent dex (CG work, I). the hydraulic head was achieved with a maximum error of up to 2.80%, corresponding to grid II. Asymptotic h1 h2 h3 Richardson Ex- GCI21 GCI32 Grid r p Ꜫ21 Ꜫ32 Range of Conver- Table 4. Estimation of grid convergence index (CGI). (m) (m) (m) trapolate (m) (%) (%) gence h h h Richardson Extrapolate GCI GCI Asymptotic Range of 1 2 3 21 32 I 1.60 1.80 0.1622 0.1637 0.1672 0.1611 0.0092 0.0214 0.87 2.00 0.99 Grid r p e e 21 32 (m) (m) (m) (m) (%) (%) Convergence II 1.58 1.78 0.1182 0.1197 0.1231 0.1170 0.0127 0.0284 1.25 2.80 0.99 I 1.60 1.80 0.1622 0.1637 0.1672 0.1611 0.0092 0.0214 0.87 2.00 0.99 III 1.58 1.83 0.1302 0.1312 0.1335 0.1294 0.0077 0.0175 0.74 1.69 0.99 II 1.58 1.78 0.1182 0.1197 0.1231 0.1170 0.0127 0.0284 1.25 2.80 0.99 IV III 1 1.58 .59 11.83 .76 0 0.1302 .1412 0.1312 0.1420.1335 3 0.1448 0.1294 0.1403 0.0077 0.007 0.0175 8 0.017 0.74 6 0.77 1.69 1.73 0.99 0.99 IV 1.59 1.76 0.1412 0.1423 0.1448 0.1403 0.0078 0.0176 0.77 1.73 0.99 V 1.65 1.79 0.1482 0.1495 0.1527 0.1473 0.0088 0.0214 0.75 1.83 0.99 V 1.65 1.79 0.1482 0.1495 0.1527 0.1473 0.0088 0.0214 0.75 1.83 0.99 VI 1.58 1.97 0.1502 0.1519 0.1561 0.1490 0.0113 0.0276 0.96 2.35 0.99 VI 1.58 1.97 0.1502 0.1519 0.1561 0.1490 0.0113 0.0276 0.96 2.35 0.99 VII 1.64 1.89 0.1622 0.1631 0.1654 0.1616 0.0055 0.0141 0.45 1.13 0.99 VII 1.64 1.89 0.1622 0.1631 0.1654 0.1616 0.0055 0.0141 0.45 1.13 0.99 Based on Richardson’s extrapolation for the two finer grids, an estimate of the hy- Based on Richardson’s extrapolation for the two finer grids, an estimate of the hy- draulic head value for zero-grid spacing was obtained. The graph in Figure 4 shows the draulic head value for zero-grid spacing was obtained. The graph in Figure 4 shows the hydraulic head with variable grid spacing; as the grid spacing reduced, the hydraulic head hydraulic head with variable grid spacing; as the grid spacing reduced, the hydraulic head approached an asymptotic value of zero mesh spacing. approached an asymptotic value of zero mesh spacing. Figure 4. Hydraulic head approaching an asymptotic zero-grid spacing value. 2.3. Evaluation of the Computational Model Numerically, 40 scenarios were modeled. Table 5 details the design flow applied and the geometric characteristics of the simulated weirs for each scenario. For the evaluation of the computational model, the results of the numerically obtained discharge coefficient were compared with the experimental values reported by Crookston and Tullis (2012) [15], corresponding to scenarios 1–10. The discharge coefficient (C ) was calculated from the height of the measured hydraulic head on the weir and the general weir equation (Equation (17)). 3/2 Q = 2gC L H (17) 3 1 where Q is the design flow (m s ), C is the discharge coefficient (dimensionless), g is the acceleration due to gravity (m s ), L is the characteristic length of the weir (m) (defined as the total length referenced at the center of the weir crest wall thickness), and H is the total hydraulic head (m). 𝑙𝑛 Water 2023, 15, 722 10 of 29 Table 5. Simulation scenarios for labyrinth weirs. a P L Q cycle Scenario w/P N Crest Profile Apex Shape ( ) (m) (m) (m s ) 0.0190, 0.0532, 0.0919, 0.1309, 0.1681, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 15 0.305 2.00 2.024 1 CR Trapezoidal 0.2036, 0.2373, 0.2697, 0.3013, 0.3325 0.0780, 0.1240, 0.2003, 11, 12, 13, 14, 15 6 0.305 3.64 2.024 1 MR Circular 0.2703, 0.3558 0.0429, 0.0750, 0.2137, 16, 17, 18, 19, 20 8 0.305 3.07 2.024 1 MR Circular 0.2873, 0.3744 0.0367, 0.0532, 0.2170, 21, 21, 23, 24, 25 10 0.305 2.65 2.024 1 MR Circular 0.2891, 0.3712 0.0380, 0.0671, 0.1450, 26, 27, 28, 29, 30 12 0.305 2.33 2.024 1 MR Circular 0.2149, 0.3728 0.0532, 0.1308, 0.2036, 31, 32, 33, 34, 35 15 0.305 2.02 2.024 1 MR Circular 0.2697, 0.3325 0.0290, 0.0517, 0.1247, 36, 37, 38, 39, 40 20 0.305 1.59 2.024 1 MR Circular 0.1915, 0.3364 Note: CR: Quarter-round; MR: Half-round; L : Cycle length (m). cycle Statistical criteria were applied to evaluate the performance of the model, such as Pearson’s coefficient of determination (R ), the relative percentage error (Er), and the mean absolute error (MAE) (Equations (18)–(20)). These criteria assess the agreement between the results of the physical experiment and those from the CFD models. Cov(num, ex p) R = (18) 2 2 s s num ex p Y Y num,i ex p,i M AE = (19) Y Y num ex p Er(%) =  100 (20) ex p Cov(num, ex p) is the covariance of the numerical and experimental results, s is the ex p variance of the experimental results, s is the variance of the numerical results, Y are num num the numerical values, and Y are the experimental values. exp The mean absolute error of the discharge coefficient (MAE) was 0.0128, and the relative percentage error varied from 1.89% to 4.92%. In Figure 5, the relative percentage errors of the discharge coefficient as a function of the ratio H /P are presented. The calculated coefficient of determination was R = 0.984, which confirms the agree- ment between the numerically obtained discharge coefficients and the experimental values. In the graph in Figure 6, the numerical results are compared with the experimental results; the dotted diagonal line corresponds to a perfect fit. In this study, the ANOVA statistical analysis was performed, and there was no signifi- cant difference between the discharges coefficients obtained numerically and experimen- tally for a significance level of 5% (0.05). 2.4. Proposed Sequential Design Method for a Labyrinth Weir Figure 7 details the variables and geometric characteristics considered in the proposed sequential design method for the labyrinth weir. The design method is basically subdivided into four stages: (i) initial data require- ments, (ii) definition of the number of cycles and the angle a to make the weir hydraulically efficient, (iii) calculation of the geometric variables, and (iv) analysis of the submerged weir developed according to Tullis et al., (2007) [38]. Water 2023, 15, x FOR PEER REVIEW 12 of 32 𝐶𝑜𝑣 (𝑛𝑢𝑚 , ) is the covariance of the numerical and experimental results, 𝜎 is 𝑝𝑥𝑒 the variance of the experimental results, 𝜎 is the variance of the numerical results, Ynum 𝑛𝑢𝑚 Water 2023, 15, x FOR PEER REVIEW 12 of 32 are the numerical values, and Yexp are the experimental values. The mean absolute error of the discharge coefficient (MAE) was 0.0128, and the rela- tive percentage error varied from 1.89% to 4.92%. In Figure 5, the relative percentage er- rors of the discharge coefficient as a function of the ratio HT/P are presented. 𝐶𝑜𝑣 (𝑛𝑢𝑚 , ) is the covariance of the numerical and experimental results, 𝜎 is 𝑝𝑥𝑒 the variance of the experimental results, 𝜎 is the variance of the numerical results, Ynum 𝑛𝑢𝑚 are the numerical values, and Yexp are the experimental values. The mean absolute error of the discharge coefficient (MAE) was 0.0128, and the rela- Water 2023, 15, 722 11 of 29 tive percentage error varied from 1.89% to 4.92%. In Figure 5, the relative percentage er- rors of the discharge coefficient as a function of the ratio HT/P are presented. Figure 5. Relative percentage error of the discharge coefficient as a function of HT/P (Mattos-Villar- roel et al., 2021 [1]). The calculated coefficient of determination was R = 0.984, which confirms the agree- ment between the numerically obtained discharge coefficients and the experimental val- ues. In the graph in Figure 6, the numerical results are compared with the experimental Figure 5. Relative percentage error of the discharge coefficient as a function of HT/P (Mattos-Villar- Figure 5. Relative percentage error of the discharge coefficient as a function of H /P results; the dotted diagonal line corresponds to a perfect fit. roel et al., 2021 [1]). (Mattos-Villarroel et al., 2021 [1]). The calculated coefficient of determination was R = 0.984, which confirms the agree- ment between the numerically obtained discharge coefficients and the experimental val- ues. In the graph in Figure 6, the numerical results are compared with the experimental results; the dotted diagonal line corresponds to a perfect fit. Figure 6. Comparison of numerically obtained discharge coefficients with experimental values Figure 6. Comparison of numerically obtained discharge coefficients with experimental values (Mattos-Villarroel et al., 2021 [1]). (Mattos-Villarroel et al., 2021 [1]). In (i) th Stage is study, one: th Th e is As NO tage VA co n st sa is tis tstica of d l e atn er am lyisi ns inwa g th s e pe drf ato arme nece ds ,s a an ry d fth orer th e ewa des sin go n si an g- d comprises information previously obtained from topographic and hydrological analysis, namely: nificant difference between the discharges coefficients obtained numerically and experi- mentally for a significance level of 5% (0.05). (a) The design flow (Q), which represents the design discharge for a given return period; Figure 6. Comparison of numerically obtained discharge coefficients with experimental values (b) The upstream head of the weir (H ), which depends on the channel width (W) and is (Mattos-Villarroel et al., 2021 [1]). limited by the freeboard; (c) The downstream head of the weir (H ) is calculated from the drop height and the flow In this study, the ANOVA statistical danalysis was performed, and there was no sig- velocity at the foot of the weir; nificant difference between the discharges coefficients obtained numerically and experi- (d) The weir height (P) corresponds to the height of the storage volume or the Ordinary mentally for a significance level of 5% (0.05). Maximum Water Level obtained from the topography and the operation of the basin. The width of the weir or the channel (W) is generally restricted by the topography of the study area. 𝑒𝑥𝑝 𝑒𝑥𝑝 Water 2023, 15, x FOR PEER REVIEW 13 of 32 2.4. Proposed Sequential Design Method for a Labyrinth Weir Water 2023, 15, 722 12 of 29 Figure 7 details the variables and geometric characteristics considered in the pro- posed sequential design method for the labyrinth weir. Figure 7. Geometric variables of a labyrinth weir. Figure 7. Geometric variables of a labyrinth weir. (ii) Stage two: The topography of the study area allows for selecting the angle a of The design method is basically subdivided into four stages: (i) initial data require- the weir sidewall (6  a  20 ). A large angle can be chosen when the number of cycles N ments, (ii) definition of the number of cycles and the angle α to make the weir hydrau- and the length between the cycle apexes (B) are limited by topography. The selection of lically efficient, (iii) calculation of the geometric variables, and (iv) analysis of the sub- the angle a is also a function of the ratio H /P (0.05  H /P  0.8). For certain values of merged weir developed according to TulliT s et al., (2007)T [38]. H /P and a, the flow becomes unstable and is a phenomenon to be avoided in the design T (i) Stage one: This stage consists of determining the data necessary for the design and of the weir, for the safety of the hydraulic structure. Subsequently, the discharge coefficient comprises information previously obtained from topographic and hydrological analysis, is calculated. Its calculation is a function of a and the ratio H /P, and the value of the namely: discharge coefficient will determine the discharge capacity of the weir. (a) The design flow (Q), which represents the design discharge for a given return period; Then, the cycle width w is calculated as a function of the weir height P. For (b) The upstream head of the weir (HT), which depends on the channel width (W) and is Taylor (1968) [39], the ratio w/P (known as the vertical aspect) should not be less than limited by the freeboard; 2 because it would contribute to reducing weir efficiency. On the other hand, Tullis et al. (c) The downstream head of the weir (Hd) is calculated from the drop height and the (1995) [14] recommended that w/P should not be greater than 4. Considering both criteria, flow velocity at the foot of the weir; it is recommended that the ratio w/P should be equal to 3 or, in other words, the cycle width (d) The weir height (P) corresponds to the height of the storage volume or the Ordinary w should be 3 times the weir height P, to ensure that the weir is hydraulically efficient. Maximum Water Level obtained from the topography and the operation of the basin. Another important variable influencing the design is the number of cycles (N). The The width of the weir or the channel (W) is generally restricted by the topography of number of cycles is calculated as the ratio of the weir width (W) to the cycle width (w). the study area. For ease of design, it is recommended that this be a multiple of 0.5, and so the cycle width (ii) Stage two: The topography of the study area allows for selecting the angle α of should be recalculated as w = W/N, with the restriction that the ratio w/P is within the the weir sidewall (6 ≤ α ≤ 20°). A large angle can be chosen when the number of cycles N range 2  w/P  4. and (iii) the leng Stage th three: betweeIn n th this e cycle stage, ape the xes geometric (B) are lim variables ited by toof pothe graphy weir . Tar he e se calculated lection of the angle α is also a function of the ratio HT/P (0.05 ≤ HT/P ≤ 0.8). For certain values of HT/P as follows: and α, the flow becomes unstable and is a phenomenon to be avoided in the design of the The length of the weir (L). The selection of the angle a will determine the length of the weir, for the safety of the hydraulic structure. Subsequently, the discharge coefficient is weir. Its calculation is a function of the discharge coefficient, the hydraulic head, and the design flow. The width of the weir wall (t ) and the internal apex rope (C ) must both be equal w c to P/8. The internal and external apex arc (Arc , Arc ) are both functions of t and a. int ext w Water 2023, 15, 722 13 of 29 The length of the cycle wall (l ), as a function of L, N, Arc , and Arc . c int ext The length of the platform (B) is a function of L, N, Arc , Arc , a, and t . ext w int In this stage, the weir efficiency (# ) and the cycle (#”) are also determined, both as a function of L, w, N, and C (a). The weir efficiency is also a function of the discharge coefficient of a linear weir; its calculation method is described by Crooskton (2010) [5]. Subsequently, the nappe interference length (B ) is calculated from the ratio H /P int T and the angle a. Finally, the type of aeration of the nappe is determined according to the value of H /P and the selected angle a. (iv) Stage Four: The last stage of the design method includes the dimensionless head rela- tionships for the drowned weir, which were developed and described by Tullis et al. (2007) [38]. The following section describes the results obtained from the studies applied to the discharge flow considered in the proposed sequential design method for the weir; it describes the equations for each variable in detail. 3. Results 3.1. Discharge Coefficient, Weir, and Cycle Efficiency In this study, the discharge coefficients of circular apex weirs are presented as a function of the ratio H /P, whose weir cycle sidewall angles vary from 6 to 20 and are compared with the discharge coefficients reported by Crookston and Tullis (2012) [15] for trapezoidal labyrinth weirs. Both weirs have a half-round crest. Water 2023, 15, x FOR PEER REVIEW 15 of 32 The discharge coefficients C (a) of each weir with a circular apex are presented graphi- cally in Figure 8 for H /P  0.8. Figure 8. Curves of C (a) as a function of H /P for different values of a for labyrinth weirs. Figure 8. Curves of Cd(α) as a function of HT/P for different values of α for labyrinth weirs. d T The values of the discharge coefficient in Figure 8 were used to obtain a mathematical The values of the discharge coefficient in Figure 8 were used to obtain a mathematical model of a regressive type through a fifth-degree polynomial equation (Equation (22)) as a model of a regressive type through a fifth-degree polynomial equation (Equation (22)) as function of H /P. Weir design methods and curves are mostly generated from empirical a function of HT/P. Weir design methods and curves are mostly generated from empirical equations derived from laboratory experiments [2,5–9]. For example, several researchers equations derived from laboratory experiments [2,5–9]. For example, several researchers reported polynomial equations obtained by non-linear regression to have good fitting [14]. reported polynomial equations obtained by non-linear regression to have good fitting Statistical analysis was used to determine the accuracy of Equation (21) compared to the [14]. Statistical analysis was used to determine the accuracy of Equation (21) compared to C (a) results obtained from numerical data. The calculated Pearson’s coefficient was very the Cd(α) results obtained from numerical data. The calculated Pearson’s coefficient was good, varying from 0.999 to 1 for weirs with angles from 6 to 20 . Therefore, Equation very good, varying from 0.999 to 1 for weirs with angles from 6 to 20°. Therefore, Equation (21) provides sufficient accuracy to determine C (a). Equations (22)–(27) correspond to the (21) provides sufficient accuracy to determine Cd(α). Equations (22)–(27) correspond to the coefficients of Equation (21) as a function of the angle . The accuracy of predictive Equa- coefficients of Equation (21) as a function of the angle α. The accuracy of predictive Equa- tions (22)–(27) was also evaluated with the numerical results using Pearson’s determination tions (22)–(27) was also evaluated with the numerical results using Pearson’s determina- tion coefficient, obtaining values of 1 for the case of Equations (22)–(26) and 0.996 for Equation (27). Therefore, reliable results can be obtained using coefficients for Equation (21). 5 4 3 2 𝐻 𝐻 𝐻 𝐻 𝐻 𝑇 𝑇 𝑇 𝑇 𝑇 (21) 𝐶 (𝛼 ) = 𝑎 ( ) + 𝑏 ( ) + 𝑐 ( ) + 𝑑 ( ) + 𝑒 ( ) + 𝑓 𝑃 𝑃 𝑃 𝑃 𝑃 𝑎 = 42.99 + 48.93 (0.1926𝛼 ) − 24.14 (0.1926𝛼 ) + 7.60 (0.3852𝛼 ) (22) − 15.95 (0.3852𝛼 ) ( ) ( ) ( ) ( ) (23) 𝑏 = −61.88 − 65.87 0.2241𝛼 − 4.273 0.2241𝛼 − 22.5 0.4482𝛼 + 5.11 0.4482𝛼 (24) 𝑐 = 47.39 + 36.05 (0.2408𝛼 ) + 14.27 (0.2408𝛼 ) + 13.57 (0.4816𝛼 ) + 3.893 (0.4816𝛼 ) 𝑑 = −20.19 − 11.21 (0.2396𝛼 ) − 4.43 (0.2396𝛼 ) − 4.327 (25) (0.4792𝛼 ) − 1.013 (0.4792𝛼 ) 𝑒 = 3.853 + 2.084 (0.2076𝛼 ) − 0.7578 (0.2076𝛼 ) + 0.5083 (26) (0.4152𝛼 ) − 0.7128 (0.4152𝛼 ) −5 4 −3 3 −2 2 𝑓 = −5.158 × 10 𝛼 + 2.591 × 10 𝛼 − 4.62 × 10 𝛼 + 0.3487𝛼 − 0.3085 (27) The maximum values that can be obtained for the discharge coefficients occur when HT varies between 0.10 and 0.17 times the height of the weir. Table 6 shows the maximum values of the discharge coefficient for each weir as a function of HT/P. 𝑠𝑖𝑛 𝑐𝑜𝑠 𝑠𝑖𝑛 𝑐𝑜𝑠 𝑠𝑖𝑛 𝑐𝑜𝑠 𝑠𝑖𝑛 𝑐𝑜𝑠 𝑠𝑖𝑛 𝑐𝑜𝑠 𝑠𝑖𝑛 𝑐𝑜𝑠 𝑠𝑖𝑛 𝑐𝑜𝑠 𝑠𝑖𝑛 𝑐𝑜𝑠 𝑠𝑖𝑛 𝑐𝑜𝑠 𝑠𝑖𝑛 𝑐𝑜𝑠 Water 2023, 15, 722 14 of 29 coefficient, obtaining values of 1 for the case of Equations (22)–(26) and 0.996 for Equation (27). Therefore, reliable results can be obtained using coefficients for Equation (21). 5 4 3 2 H H H H H T T T T T C (a) = a + b + c + d + e + f (21) P P P P P a = 42.99 + 48.93cos(0.1926a) 24.14sin(0.1926a) + 7.60cos(0.3852a) 15.95sin(0.3852a) (22) b = 61.88 65.87cos(0.2241a) 4.273sin(0.2241a) 22.5cos(0.4482a) + 5.11sin(0.4482a) (23) c = 47.39 + 36.05cos(0.2408a) + 14.27sin(0.2408a) + 13.57cos(0.4816a) + 3.893sin(0.4816a) (24) d = 20.19 11.21cos(0.2396a) 4.43sin(0.2396a) 4.327cos(0.4792a) 1.013sin(0.4792a) (25) e = 3.853 + 2.084cos(0.2076a) 0.7578sin(0.2076a) + 0.5083cos(0.4152a) 0.7128sin(0.4152a) (26) 5 4 3 3 2 2 f = 5.158 10 a + 2.59110 a 4.6210 a + 0.3487a 0.3085 (27) The maximum values that can be obtained for the discharge coefficients occur when H varies between 0.10 and 0.17 times the height of the weir. Table 6 shows the maximum values of the discharge coefficient for each weir as a function of H /P. Table 6. Maximum values of the discharge coefficient. (a) H /P C (a) 6 0.10 0.736 8 0.11 0.762 10 0.13 0.771 12 0.14 0.784 15 0.15 0.803 20 0.17 0.833 The discharge coefficients of the circular apex weirs are compared with the trapezoidal apex weirs. The graph in Figure 9 shows the increase in the discharge coefficient of the circular apex weir with respect to the trapezoidal apex, which is 10% when a = 20 to 46% when a = 6 , and the slope of the family of curves is significantly higher for  10 . The weir efficiency (Equation (28)) allows for the hydraulic behavior of a labyrinth weir to be compared with a conventional linear weir, and the advantages obtained by increasing its length can then be determined [23]. C (a) L 0 d e = (28) C W d(90 ) The cycle efficiency indicator proposed by Willmore (2004) [4] allows for the design of the labyrinth type weir to be optimized and facilitates decision-making. Its calculation is essentially useful for small heads and is obtained from Equation (29). ciclo e = C (a) (29) The graphs in Figure 10 represent the efficiency of the labyrinth weir and the cycle as a function of H /P. Both graphs show that the maximum efficiency occurs for small values of H /P and increases with the decreasing sidewall angle. T Water 2023, 15, x FOR PEER REVIEW 16 of 32 Table 6. Maximum values of the discharge coefficient. (𝜶 ) HT/P 𝑪 (𝜶 ) 6° 0.10 0.736 8° 0.11 0.762 10° 0.13 0.771 12° 0.14 0.784 15° 0.15 0.803 20° 0.17 0.833 The discharge coefficients of the circular apex weirs are compared with the trapezoi- dal apex weirs. The graph in Figure 9 shows the increase in the discharge coefficient of Water 2023, 15, 722 15 of 29 the circular apex weir with respect to the trapezoidal apex, which is 10% when α = 20° to 46% when α = 6°, and the slope of the family of curves is significantly higher for α ≤ 10°. Water 2023, 15, x FOR PEER REVIEW 17 of 32 The efficiency of the labyrinth weir, represented by the family of curves in Figure 10A, has an accelerated reduction in its value for α ≤ 10° when HT/P > 0.1. Figure 10B represents the cycle efficiency of the weir; the dotted line passes through the values of Figure 9. Relationship of the discharge coefficient of the circular apex weir to the trapezoidal apex Figure 9. Relationship of the discharge coefficient of the circular apex weir to the trapezoidal apex HT/P where the maximum efficiency of the cycle is present and coincides with the maxi- w weir eir.. The The dotted dotted line line indicates indicates the the inflection inflection point point on on each each curve. curve. mum values of Cd(α). The weir efficiency (Equation (28)) allows for the hydraulic behavior of a labyrinth weir to be compared with a conventional linear weir, and the advantages obtained by increasing its length can then be determined [23]. 𝐶 (𝛼 ) 𝐿 Ꜫ′ = (28) 𝐶 𝑊 𝑑 (90°) The cycle efficiency indicator proposed by Willmore (2004) [4] allows for the design of the labyrinth type weir to be optimized and facilitates decision-making. Its calculation is essentially useful for small heads and is obtained from Equation (29). 𝑐𝑜𝑖𝑐𝑙 Ꜫ′′ = 𝐶 (𝛼 ) (29) The graphs in Figure 10 represent the efficiency of the labyrinth weir and the cycle as a function of HT/P. Both graphs show that the maximum efficiency occurs for small values of HT/P and increases with the decreasing sidewall angle. (A) (B) Figure 10. (A) Weir efficiency. (B) Cycle efficiency. Figure 10. (A) Weir efficiency. (B) Cycle efficiency. 3.2. Nappe Aeration Conditions The efficiency of the labyrinth weir, represented by the family of curves in Figure 10A, has an With acce th ler e aitn ecr d ea red se uc in ti o th ne in hydr its v aa uli luc e h fo ea r d o n1 t0 he w weir, hen f H ou/r Ptypes > 0.1. o Ffi g au er ra etio 10n B were repre s ide enn ts- ttified he cy c(lFi e e gure ffici e1 n 1 c)y : n oa f ppe the w ae dih r;er th ed e d to ot t th ed e lwa inellp o as f ste h se th weir, roug h aer tha eted val , u pa esrtia of H lly / aP erw ated her,e a tn hd e maximum efficiency of the cycle is present and coincides with the maximum values of C (a). drowned. 3.2. Nappe Aeration Conditions With the increase in the hydraulic head on the weir, four types of aeration were identified (Figure 11): nappe adhered to the wall of the weir, aerated, partially aerated, and drowned. Table 7 presents the ranges of H /P that correspond to the aeration conditions ob- served for each weir and is included in the family of discharge coefficient design curves (Figure 12). Water 2023, 15, x FOR PEER REVIEW 17 of 32 The efficiency of the labyrinth weir, represented by the family of curves in Figure 10A, has an accelerated reduction in its value for α ≤ 10° when HT/P > 0.1. Figure 10B represents the cycle efficiency of the weir; the dotted line passes through the values of HT/P where the maximum efficiency of the cycle is present and coincides with the maxi- mum values of Cd(α). (A) (B) Figure 10. (A) Weir efficiency. (B) Cycle efficiency. 3.2. Nappe Aeration Conditions With the increase in the hydraulic head on the weir, four types of aeration were iden- Water 2023, 15, 722 16 of 29 tified (Figure 11): nappe adhered to the wall of the weir, aerated, partially aerated, and drowned. Water 2023, 15, x FOR PEER REVIEW 18 of 32 Figure 11. Nappe aeration conditions: (A) clinging, (B) aerated, (C) partially aerated, and (D) Figure 11. Nappe aeration conditions: (A) clinging, (B) aerated, (C) partially aerated, and (D) drowned. drowned. Table 7. Ranges of nappe aeration conditions. Table 7 presents the ranges of HT/P that correspond to the aeration conditions ob- H /P served for each weir and is included in the family of discharge coefficient design curves a( ) Flow Clinging (Figure 12). Flow Aerated Flow Partially Aerated Flow Drowned 6 <0.165 0.165–0.270 0.270–0.487 >0.487 Table 7. Ranges of nappe aeration conditions. 8 <0.200 0.200–0.350 0.350–0.500 >0.500 10 <0.265 0.265–0.350 0.350–0.540 >0.540 HT/P 12 <0.300 0.300–0.410 0.410–0.550 >0.550 Flow Cling- Flow Flow Flow 15 <0.325 0.325–0.400 0.400–0.600 >0.600 𝜶 (°) 20 <0.450 0.450–0.500 ing Aerated 0.500–0.600 Partially Aerated >0.600 Drowned 6° <0.165 0.165–0.270 0.270–0.487 >0.487 3.3. Nappe 8° Instability<0.200 0.200–0.350 0.350–0.500 >0.500 10° <0.265 0.265–0.350 0.350–0.540 >0.540 Nappe instability occurs when the nappe has an oscillating trajectory accompanied by 12° <0.300 0.300–0.410 0.410–0.550 >0.550 turbulent helical flow adjacent and parallel to the side walls of the weir cycle (Figure 13). 15° <0.325 0.325–0.400 0.400–0.600 >0.600 Under these conditions, vibration is generated in the weir and may represent a safety hazard for the hydraulic structure. 20° <0.450 0.450–0.500 0.500–0.600 >0.600 Water 2023, 15, x FOR PEER REVIEW 19 of 32 Water 2023, 15, 722 17 of 29 The results of this work confirm the presence of unstable flow located downstream of the weir, i.e., between the side walls of the weir. The values of H /P and the aeration Water 2023, 15, x FOR PEER REVIEW 19 of 32 conditions where instability is generated are presented in Table 8 and are included in the discharge coefficient design curves (Figure 14). Figure 12. Identification of aeration zones. 3.3. Nappe Instability Nappe instability occurs when the nappe has an oscillating trajectory accompanied by turbulent helical flow adjacent and parallel to the side walls of the weir cycle (Figure 13). Under these conditions, vibration is generated in the weir and may represen t a safety Figure 12. Identification of aeration zones. hazard for the hydraulic structure. Figure 12. Identification of aeration zones. 3.3. Nappe Instability Nappe instability occurs when the nappe has an oscillating trajectory accompanied by turbulent helical flow adjacent and parallel to the side walls of the weir cycle (Figure 13). Under these conditions, vibration is generated in the weir and may represent a safety hazard for the hydraulic structure. Figure 13. Helical streamlines parallel to the weir wall. Figure 13. Helical streamlines parallel to the weir wall. Table 8. The Ranges resulof ts nappe of this instability work coand nfirm aeration the pr conditions. esence of unstable flow located downstream of the weir, i.e., between the side walls of the weir. The values of HT/P and the aeration a( ) Instability Aeration Condition conditions where instability is generated are presented in Table 8 and are included in the 6 - - discharge coefficient design curves (Figure 14). 8 - - 10 - - Figure 13. Helical streamlines parallel to the weir wall. 12 0.56  HT/P  0.8 Drowned. 15 0.49  HT/P  0.8 Partially aerated, and drowned. The results of this work confirm the presence of unstable flow located downstream 20 0.40  HT/P  0.8 Clinging, aerated, partially aerated, and drowned. of the weir, i.e., between the side walls of the weir. The values of HT/P and the aeration conditions where instability is generated are presented in Table 8 and are included in the discharge coefficient design curves (Figure 14). Water 2023, 15, x FOR PEER REVIEW 20 of 32 Table 8. Ranges of nappe instability and aeration conditions. 𝜶 (°) Instability Aeration Condition 6° - - 8° - - 10° - - 12° 0.56 ≤ HT/P ≤ 0.8 Drowned. 15° 0.49 ≤ HT/P ≤ 0.8 Partially aerated, and drowned. Water 2023, 15, 722 Clinging, aerated, partially aerated, an 18dof 29 20° 0.40 ≤ HT/P ≤ 0.8 drowned. Fig Figure ure 14. 14. I Identification dentification of of instability instability zones. zones. 3.4. Nappe Interference 3.4. Nappe Interference The effect of the collision between nappes on the reduction of weir efficiency is The effect of the collision between nappes on the reduction of weir efficiency is sig- significant. Therefore, its analysis and behavior are characterized and considered in the nificant. Therefore, its analysis and behavior are characterized and considered in the de- design of the weir. The CFD simulations carried out made it possible to visualize the sign of the weir. The CFD simulations carried out made it possible to visualize the for- formation of air contrail, accompanied by standing waves or hydraulic jumps between weir mation of air contrail, accompanied by standing waves or hydraulic jumps between weir cycles (Figure 15A), which decrease with the increasing hydraulic head and presence of Water 2023, 15, x FOR PEER REVIEW 21 of 32 cycles (Figure 15A), which decrease with the increasing hydraulic head and presence of drowning in the weir. In weirs where a  10, the local drowning at the apexes is generated drowning in the weir. In weirs where α ≤ 10, the local drowning at the apexes is generated earlier than in the presence of larger angles (Figure 15B). Furthermore, depending on the earlier than in the presence of larger angles (Figure 15B). Furthermore, depending on the aeration condition, turbulent flow may also occur (Figure 16B). aeration condition, turbulent flow may also occur (Figure 16B). Figure 15. Effects of the nappe interference. (A) Air contrail and standing waves and (B) local Figure 15. Effects of the nappe interference. (A) Air contrail and standing waves and (B) local drowning. drowning. In order to quantify the size of nappe interference, perpendicular measurements (Bint) were made from the upstream apex to the point (downstream) where the nappe from the sidewall intersects (Figure 16A). The term Lint denotes the projection of Bint on the weir crest that is affected by this phenomenon. Figure 16. (A) Definition of lengths Bint and Lint and (B) local drowning and turbulence. The graph in Figure 17 is presented as a family of curves, which are the results of the interference length (Bint) in relation to the length of B (perpendicular distance between Water 2023, 15, x FOR PEER REVIEW 21 of 32 Figure 15. Effects of the nappe interference. (A) Air contrail and standing waves and (B) local drowning. In order to quantify the size of nappe interference, perpendicular measurements (Bint) were made from the upstream apex to the point (downstream) where the nappe from the Water 2023, 15, 722 19 of 29 sidewall intersects (Figure 16A). The term Lint denotes the projection of Bint on the weir crest that is affected by this phenomenon. Figure 16. (A) Definition of lengths Bint and Lint and (B) local drowning and turbulence. Figure 16. (A) Definition of lengths B and L and (B) local drowning and turbulence. int int The graph in Figure 17 is presented as a family of curves, which are the results of the In order to quantify the size of nappe interference, perpendicular measurements (B ) int interference length (Bint) in relation to the length of B (perpendicular distance between were made from the upstream apex to the point (downstream) where the nappe from the sidewall intersects (Figure 16A). The term L denotes the projection of B on the weir int int Water 2023, 15, x FOR PEER REVIEW 22 of 32 crest that is affected by this phenomenon. The graph in Figure 17 is presented as a family of curves, which are the results of the interference length (B ) in relation to the length of B (perpendicular distance between int upstream and downstream apexes) for values of HT/P ≤ 0.8. The graph allows for the pre- upstream and downstream apexes) for values of H /P  0.8. The graph allows for the diction of the length of Bint and indicates that its value can vary from 20% to 60% of the prediction of the length of B and indicates that its value can vary from 20% to 60% of the int length of B in the drowning condition. length of B in the drowning condition. Figure 17. Percentage of the nappe interference in relation to distance B (6° ≤ α ≤ 20°). Figure 17. Percentage of the nappe interference in relation to distance B (6   20 ). To facilitate the use of the graph in Figure 17 in the prediction of the nappe interfer- ence length, the family of curves has been modeled through a second-degree equation, as a function of HT/P (Equation (30)). 𝐵 𝐻 𝐻 𝑛𝑡𝑖 𝑇 𝑇 (30) = 𝑚 ( ) + 𝑛 + 𝑜 𝐵 𝑃 𝑃 The coefficients of Equation (30) are calculated with Equations (31)–(33) as a function of the angle α. −4 4 −3 3 2 𝑚 = 1.284 × 10 𝛼 − 6.583 × 10 𝛼 + 0.115𝛼 − 0.764𝛼 + 1.978 (31) −4 4 −3 3 2 (32) 𝑛 = −1.095 × 10 𝛼 + 5.648 × 10 𝛼 − 0.01𝛼 + 0.722𝛼 − 1.781 −6 4 −4 3 2 𝑜 = 6.004 × 10 𝛼 − 3.349 × 10 𝛼 + 0.006𝛼 − 0.046𝛼 + 0.110 (33) 3.5. Application of the Proposed Method As a case study, information from Houston (1982) [9] on the Ute Dam weir in Logan, New Mexico was studied. The design procedure followed the method proposed by Hay and Taylor (1979) [11], i.e., the original design was for a 10-cycle weir based on the design curves of Hay and Taylor (1970) [11]. However, it did not pass the desired design discharge within the max- imum elevation of the reservoir. A 14-cycle weir, designed according to the criteria of the laboratory channel tests conducted by the Bureau of Reclamation, satisfactorily met the required discharge and water surface elevation [9]. To avoid instability and oscillations of the nappe and provide aeration, two dividers were placed at the crest of each cycle, 3.35 m upstream from the downstream apex of the cycle. The weir design was based on a fam- ily of dimensionless ratio curves L/W, in the graph Q/QN versus HT/P, where Q/QN is the discharge magnification and QN is the discharge over a linear weir. The results reported by Houston (1982) [9] on the weir design are summarized in Table 9 and Figure 18. Water 2023, 15, 722 20 of 29 To facilitate the use of the graph in Figure 17 in the prediction of the nappe interference length, the family of curves has been modeled through a second-degree equation, as a function of H /P (Equation (30)). B H H int T T = m + n + o (30) B P P The coefficients of Equation (30) are calculated with Equations (31)–(33) as a function of the angle a. 4 4 3 3 2 m = 1.284 10 a 6.583 10 a + 0.115a 0.764a + 1.978 (31) 4 4 3 3 2 n = 1.095 10 a + 5.648 10 a 0.01a + 0.722a 1.781 (32) 6 4 4 3 2 o = 6.004 10 a 3.349 10 a + 0.006a 0.046a + 0.110 (33) 3.5. Application of the Proposed Method As a case study, information from Houston (1982) [9] on the Ute Dam weir in Logan, New Mexico was studied. The design procedure followed the method proposed by Hay and Taylor (1979) [11], i.e., the original design was for a 10-cycle weir based on the design curves of Hay and Taylor (1970) [11]. However, it did not pass the desired design discharge within the maximum elevation of the reservoir. A 14-cycle weir, designed according to the criteria of the laboratory channel tests conducted by the Bureau of Reclamation, satisfactorily met the required discharge and water surface elevation [9]. To avoid instability and oscillations of the nappe and provide aeration, two dividers were placed at the crest of each cycle, 3.35 m upstream from the downstream apex of the cycle. The weir design was based on a family of dimensionless ratio curves L/W, in the graph Q/Q versus H /P, where Q/Q is the N T N discharge magnification and Q is the discharge over a linear weir. The results reported by Houston (1982) [9] on the weir design are summarized in Table 9 and Figure 18. Table 9. Dimensions of the labyrinth weir of the Ute Dam [8]. Concept Symbol Value-Unit Observations (i) Initial data 3 3 Design flow Q 15,574 m /s Initially, the design discharge was 16,042 m /s. Weir width W 256 m - Weir height P 9.14 m - Upstream total head H 5.79 m - (ii) Geometric variables and non-dimensional relationships 0.05  H /P  1 (upper range is expanded from Head water ratio H /P 0.63 0.5 to 1 to use the design curves) Flow magnification Q/Q 2.4 - Angle of sidewall a 12.1475 - Length magnification L/W 4 2  L/W  8. Vertical aspect ratio w/P 2 2  w/P  5 Cycle width w 18.29 m - Number of cycles N 14 - Weir length L 1024.24 m - Sidewall length l 34.76 m - Length between apexes B 33.99 m - Apex A 1.82 m - Crest radius R 0.30 - Crest Upper crest width t 0.61 m - Lower crest width t 1.52 m - w2 Water 2023, 15, x FOR PEER REVIEW 23 of 32 Table 9. Dimensions of the labyrinth weir of the Ute Dam [8]. Concept Symbol Value-Unit Observations (i) Initial data 3 3 Design flow Q 15,574 m /s Initially, the design discharge was 16,042 m /s. Weir width W 256 m - Weir height P 9.14 m - Upstream total head HT 5.79 m - (ii) Geometric variables and non-dimensional relationships 0.05 ≤ HT/P ≤ 1 (upper range is expanded from 0.5 to 1 to Head water ratio HT/P 0.63 use the design curves) Flow magnification Q/QN 2.4 - Angle of sidewall 𝛼 12.1475° - Length magnification L/W 4 2 ≤ L/W ≤ 8. Vertical aspect ratio w/P 2 2 ≤ w/P ≤ 5 Cycle width w 18.29 m - Number of cycles N 14 - Weir length L 1024.24 m - Sidewall length lc 34.76 m - Length between apexes B 33.99 m - Apex A 1.82 m - Water 2023, 15, 722 21 of 29 Crest radius RCrest 0.30 - Upper crest width tw−1 0.61 m - Lower crest width tw−2 1.52 m - Figure 18. Shape and dimensions of the 14-cycle labyrinth weir [9]. Figure 18. Shape and dimensions of the 14-cycle labyrinth weir [9]. The design sequence shown in Table 10 and in the flowchart in Figure 19 was used as an example of the application of the proposed design methodology for the Ute Dam The design sequence shown in Table 10 and in the flowchart in Figure 19 was used as Water 2023, 15, x FOR PEER REVIEW 25 of 32 weir. an example of the application of the proposed design methodology for the Ute Dam weir. Figure 19. Flowchart for the design procedure. Figure 19. Flowchart for the design procedure. 4. Discussion 4.1. Discussion of Discharge Coefficient, Weir, and Cycle Efficiency The magnitude of the discharge coefficient Cd helps us to understand the hydraulic behavior of a weir and is essential when making decisions during weir design, where its value depends on geometry, aeration conditions, and flow behavior during the discharge. When the ratio HT/P < 0.2, higher values of Cd(α) up to 0.833 are presented; this is when the nappe is adhered to the weir wall. During the transition of a nappe adhered to the wall becoming partially aerated, there is an accelerated decrease in Cd(α) values at weirs with angles varying from 6° to 10°. The reduction is less abrupt when α ≥ 12° and, as the head HT on the weir increases, the value of Cd(α) decreases. For values of HT/P < 0.1, the 8° and 10° weirs exhibit similar behavior in Cd(α), and the Cd(α) of the 12° weir is slightly higher Water 2023, 15, 722 22 of 29 Table 10. Spreadsheet for the design of the labyrinth weir. Concept Symbol Value-Unit Equations and Limits (i) Input data Design flow Q 15,574 m /s - Weir width W 256 m - Weir height P 9.14 m - Upstream total head H 5.79 m - (ii) Definition of and the number of cycles (N) Head water ratio H /P 0.63 0.05  H /P  0.8 T T Angle of sidewall a 11.5 6   20 Nappe stability - Stable Stable/Unstable: Table 8 and Figure 14 Labyrinth weir discharge coefficient C (a) 0.483 C (a) = f ( H T/P, a), Equations (21)–(27) d d Cycle width w 27.42 m w = 3P Number of cycles N 9 N = W/w New cycle width w 28.44 m w = W/N Vertical aspect ratio w/P 3.11 2  w/P  4 (iii) Calculation of geometric variables, weir and cycle efficiencies, nappe interference and aeration condition Geometric variables h i 0.5 1.5 Total centerline length of weir L 783.20 m L = 1.5Q/ C (a) H T (2g) Wall width t 1.14 m  P/8 w t Internal apex rope C 1.14 m C = t c c w Internal apex arc Arc 1.60 m Arc = t p(90 a)/(180cosa) int int w External apex arc Arc 1.16 m Arc = t p(2cosa + 1)(90 )/(180 cos ) ext ext w Centerline length of sidewall l 42.14 m lc = L/(2N) ( Arc + Arc )/2 c int ext B = [L/(2N) ( Arc + Arc )/2]cosa + 2t + ext w int Length of apron B 44.28 m t [1 sena(1 + cosa)]/cosa (or input data) Weir and cycle efficiency Magnification ratio M 3.17 M = L/(wN) Linear weir coefficient discharge C  0.754 C  = 1/[8.609 + 22.65H T/P + 1.812/ H T/P] + 0.6375 [5] d (90 ) d(90 ) 00 00 Cycle efficiency e 0.74 e = C (a) M 0 0 Weir efficiency e 1.96 e = C (a) M/C d(90 ) Nappe interference length and aeration condition Nappe interference length B 10.89 m Equations (30)–(33) int Aeration condition - Drowned Table 7 and Figure 12 (iv) Submergence (Tullis et al., 2007 [38]) Downstream total head H 1.22 m - Head ratio H /H 0.21 - Submergence upstream total head H* 5.84 m Equations (1)–(3) and Figure 2 Submergence level S 0.20 S = H /H*; 0  S  1 1.5 Submerged weir discharge coefficient C 0.476 C = C (a)( H / H T) dsum dsum d 4. Discussion 4.1. Discussion of Discharge Coefficient, Weir, and Cycle Efficiency The magnitude of the discharge coefficient C helps us to understand the hydraulic behavior of a weir and is essential when making decisions during weir design, where its value depends on geometry, aeration conditions, and flow behavior during the discharge. When the ratio H /P < 0.2, higher values of C (a) up to 0.833 are presented; this is when T d the nappe is adhered to the weir wall. During the transition of a nappe adhered to the wall becoming partially aerated, there is an accelerated decrease in C (a) values at weirs with angles varying from 6 to 10 . The reduction is less abrupt when a  12 and, as the head H on the weir increases, the value of C (a) decreases. For values of H /P < 0.1, the 8 and T T 10 weirs exhibit similar behavior in C (a), and the C (a) of the 12 weir is slightly higher d d than that of the 10 weir. The higher angle weirs have better discharge capacities. However, lower angle weirs have the advantage of having a longer weir crest length. The increase of the discharge coefficient of the circular apex weir, with respect to the trapezoidal apex weir, is immediate from H /P  0.1. In addition, the dotted line in Figure 9 indicates the inflection point of each curve, where the discharge coefficients acquire their maximum value (Table 6). In effect, the slopes increase until the nappe is no Water 2023, 15, 722 23 of 29 longer aerated and presents local drowning at the weir apex. When the weirs work in a drowned manner, efficiency decreases, projecting curves with slightly descending slopes at the end. The weir and cycle efficiency values decrease when the nappe is no longer adhered to the wall and occurs earlier in weirs where  10. In addition, when the weir begins to drown, the efficiencies generate minimum values, stabilizing from H /P > 0.8. In Figure 10A, the immediate reduction in weir efficiency occurs when local drowning at the apex upstream of the weir becomes present. On the other hand, in Figure 10B, it has been observed that the reduction of the cycle efficiency for a  10 is almost immediate after presenting its maximum value; this phenomenon is due not only to the presence of local drowning, but also to the change of aeration regime of the nappe. 4.2. Discussion of Nappe Aeration Conditions According to the values of the discharge coefficient, it has been identified that the weir is more efficient when the nappe is adhered to the wall. In fact, when the flow is aerated, the discharge coefficient decreases and sub-atmospheric pressures occur behind the nappe. When the flow is in a transitional or partially aerated state, the air cavities under the nappe are removed. Finally, when the weir begins to be drowned, it is characterized by presenting a thicker nappe without the presence of air cavities. The weir is also at its minimum efficiency, remaining constant from H /P > 0.8. In the latter case, the behavior of the weir is equivalent to that of the linear weir. Depending on the aeration condition, turbulent flow has been observed on the walls of the channel. For 20 weirs, aeration conditions tend to occur under turbulent flow when the nappe is adhered to the weir wall, while partially aerated conditions occur for 15 to 20 weirs, and drowned conditions for 12 to 20 weirs, as shown on Table 9. Turbulent flow can also occur between the walls of the cycle as the weir head increases, with greater occurrence under drowned conditions and lesser occurrence when the nappe is adhered to the weir wall. All the weirs have the nappe adhered to the wall when H /P  0.16. On the other hand, it has been observed that, with an increase of the angle a, the presence of this regime increases up to H /P  0.45. However, when a = 20 , the opposite occurs for the case of the aerated regime, i.e., its presence is lower when the angle a increases. The value of the discharge coefficient presents a rapid decrease for angles that vary from 6 to 10 , and this is when the transition from clinging flow to aerated flow occurs. When 15   20 , the weir has a greater range of flow clinging to the wall, in contrast to the aerated flow condition that is briefly produced by changing to the partially aerated regime. The drowning condition is generated for larger heads, i.e., when H /P > 0.49 (a = 6 ). 4.3. Discussion of Nappe Instability and Interference From a  12 , the presence of turbulent flow and helical streamlines was detected on the wall cycles and accompanied by changes in the aeration condition. According to the simulations carried out, the instability is more prevalent when the nappe is partially aerated or drowned than when it is clinging or aerated. On the other hand, the effect of collision between nappes on the reducing of weir efficiency was demonstrated and, therefore, its behavior was characterized for the labyrinth weir design. The length of the nappe interference is a function of the hydraulic load and the sidewall angle of the cycles. Figure 20 shows that weirs with  12 tend towards a stable value of crest length affected by the nappe interference, for similar values of H /P. On the other hand, it should be noted that weirs where a < 10 have a shorter crest length affected by the nappe interference. Water 2023, 15, x FOR PEER REVIEW 27 of 32 Water 2023, 15, 722 24 of 29 On the other hand, it should be noted that weirs where α < 10° have a shorter crest length affected by the nappe interference. Figure Figure 20. 20. Le Length ngth of of t the he cr crest est af affect fected ed by by the the nappe nappe interfer interference. ence. 4.4. Discussion of Application of the Proposed Method 4.4. Discussion of Application of the Proposed Method The geometric differences of the weir and the crest presented in the Houston (1982) [9] The geometric differences of the weir and the crest presented in the Houston (1982) report and the one analyzed in this work are evident. The shape of the crest influences the [9] report and the one analyzed in this work are evident. The shape of the crest influences behavior of the nappe as aeration, in the nappe interference, and, most importantly, flow the behavior of the nappe as aeration, in the nappe interference, and, most importantly, instability in the discharge. The weir crest of the Ute Dam has a quarter-rounded shape. flow instability in the discharge. The weir crest of the Ute Dam has a quarter-rounded However, a half-round shape, as recommended in the proposed design, helps the flow to shape. However, a half-round shape, as recommended in the proposed design, helps the remain adhered to the weir wall, which increases its efficiency; if the flow separates then flow to remain adhered to the weir wall, which increases its efficiency; if the flow sepa- efficiency is lost [5,42]. The design of the Ute Dam weir followed the procedure of Hay rates then efficiency is lost [5,42]. The design of the Ute Dam weir followed the procedure and Taylor (1970) [11]. This means that it considered the w/P ratio to be equal to 2, which of Hay and Taylor (1970) [11]. This means that it considered the w/P ratio to be equal to 2, resulted in a weir of 14 cycles, and a total length of 1024.24 m was obtained. The latter was which resulted in a weir of 14 cycles, and a total length of 1024.24 m was obtained. The determined from the flow magnification and the dimensionless L/W ratio, with a sidewall latter was determined from the flow magnification and the dimensionless L/W ratio, with angle of 12.15 to design the discharge required for a certain reservoir level. The triangular a sidewall angle of 12.15° to design the discharge required for a certain reservoir level. shape of the downstream wall of the weir caused a greater length of nappe interference The triangular shape of the downstream wall of the weir caused a greater length of nappe to be produced, which translated into a lower discharge capacity. The design method interference to be produced, which translated into a lower discharge capacity. The design proposed here considered that the w/P ratio  3, assuming a conservative value between method proposed here considered that the w/P ratio ≈ 3, assuming a conservative value the limits of w/P reported by Hay and Taylor (1972) [11] and Tullis et al. (1995) [14], for the between the limits of w/P reported by Hay and Taylor (1972) [11] and Tullis et al. (1995) weir to be efficient. The number of cycles was reduced to nine, and the total length of the [14], for the weir to be efficient. The number of cycles was reduced to nine, and the total weir was reduced to 783.20 m, as determined by the general discharge equation for weirs, length of the weir was reduced to 783.20 m, as determined by the general discharge equa- thus discharging the required design discharge. However, the length of platform B was tion for weirs, thus discharging the required design discharge. Howeve r, the length of increased from 10.29 m to 44.28 m, and a maximum sidewall angle of 11.5 was chosen to platform B was increased from 10.29 m to 44.28 m, and a maximum sidewall angle of 11.5° avoid generating flow instability. was chosen to avoid generating flow instability. The Bureau of Reclamation spillway design [9] has two dividers in each cycle to reduce The Bureau of Reclamation spillway design [9] has two dividers in each cycle to re- the instability and oscillations of the nappe. However, this method is not recommended [26] duce the instability and oscillations of the nappe. However, this method is not recom- due to the number of dividers required, incurring the possible danger of failure of the mended [26] due to the number of dividers required, incurring the possible danger of hydraulic structure. The proposed sequential design method indicates the ranges of H /P failure of the hydraulic structure. The proposed sequential design method indicates the and the aeration conditions in which the instability originates, which is an important ranges of HT/P and the aeration conditions in which the instability originates, which is an indicator at the time of design. important indicator at the time of design. The sequential design method proposed in the present work is a complete method The sequential design method proposed in the present work is a complete method because it considers the flow behavior during discharge and the possible instability of the because it considers the flow behavior during discharge and the possible instability of the nappe. They are integrated in the design table in Table 10 and in the flowchart in Figure 19. nappe. They are integrated in the design table in Table 10 and in the flowchart in Figure However, it is limited for values of H /P (from 0.5 to 0.8), sidewall angles from 6 to 20 , and 19. Ho for weve weirs r, located it is limin ited a channel. for values of HT/P (from 0.5 to 0.8), sidewall angles from 6° to 20°, a In ndthe for design weirs lo of caa ted labyrinth in a chan weir nel. , it is undoubtedly advisable to perform physical modeling, together with numerical modeling, to validate the hydraulic performance of the labyrinth weir. The design method, design graphs, and charts are limited to the geometries and hydraulic conditions analyzed in this study. Water 2023, 15, 722 25 of 29 5. Conclusions The design procedure of a circular apex labyrinth weir is presented based on its geometric characteristics and the discharge flow behavior. To generate design parameters, the experimental results of the discharge coefficient reported in the literature, were first validated, and verified in CDF and later incorporated into the proposed design method for labyrinth weirs. The proposed design procedure applies to weirs where H /P  0.8 and 6  a  20 . The values of the discharge coefficient are presented as a family of curves as function of H /P and using a mathematical model of a regressive type (through a fifth-degree polynomial equation found for this purpose). The results indicate a higher discharge capacity of the weir while increasing the angle a. The contrast between the discharge coefficients of circular apex weirs with those of a trapezoidal apex indicate an increase in their value of up to 46% (a = 6 ) in relation to the trapezoidal apex weir. The cycle and weir efficiency are presented as a tool in the design procedure. Both parameters indicate that the maximum values occur for H /P  0.17 and the efficiencies are higher with the reduction of the angle a. Four aeration conditions were identified (clinging, aerated, partially aerated, and drowned) with ranges of H /P for each condition. The relationship between the discharge coefficient and the aeration condition is evident: when the nappe is adhered to the wall, the weir has a higher discharge coefficient value. In addition, its presence is greater when a increases, and the opposite occurs when the nappe is aerated. Nappe instability occurs when 12  a  20 and it is accompanied by changes in aeration conditions; there is a greater presence when the flow is partially aerated and drowned. Similarly, ranges of H /P were identified when instability occurred. It is necessary not to incur the instability ranges when designing the weir to avoid possible damage to the hydraulic structure. The length of the crest affected by the nappe interference was characterized and quantified. For this purpose, a family of curves B /B is presented herein as a function int of H /P, and a mathematical model was found for its estimation. This model is a second- degree equation. The results show that the length of B reaches a maximum of 60% of the int length of B. A flowchart implemented in a spreadsheet is also presented as a tool to guide the design process of a labyrinth weir, considering its geometric variables and the phenom- ena that occur in the discharge flow. Additionally, the drowning study carried out by Tullis et al. (2007) [38] is also considered. The proposed sequential method for the design of a labyrinth weir represents a contri- bution to the improvement of the hydraulic performance of weirs of this type of hydraulic structure. It should be noted that this proposal takes into consideration parameters such as the following: (a) flow stability during discharge, (b) aeration condition of the nappe, (c) the nappe interference length, (d) weir and cycle efficiencies, and (e) weir drowning [38], which have been traditionally ignored in traditional design methods or have been studied indepen- dently [1,6,10–12]. In addition, Tullis et al. (1995) [14] and Crookston and Tullis (2012) [15] generated spreadsheets to help in the weir design. However, they did not include the nappe aeration and its instability conditions, as well as the length of the nappe interference, which influences the efficiency of the weir operation and the safety of the hydraulic structure. Finally, although the methods and tools presented in this study were highly effective when used in the design and study of a labyrinth weir, it is recommended that physical and numerical modeling be performed to validate the hydraulic performance of a specific pre-designed hydraulic structure with the proposed sequential method. Author Contributions: Conceptualization, W.O.-B. and C.D.-D.; methodology, E.D.M.-V. and W.O.-B.; software, E.D.M.-V. and J.F.-V.; validation, E.D.M.-V., H.S.-T. and W.O.-B.; formal analysis, C.D.-D. and H.S.-T.; investigation, E.D.M.-V.; resources, C.B.C.; writing—original draft preparation, E.D.M.-V. and W.O.-B.; writing—review and editing, H.S.-T., C.D.-D. and C.B.C.; visualization, E.D.M.-V. and Water 2023, 15, 722 26 of 29 J.F.-V.; supervision, W.O.-B.; project administration, W.O.-B.; funding acquisition, W.O.-B. and C.B.C. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Data Availability Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest. Abbreviations A internal length apex. Arc internal apex arc. int Arc external apex arc. ext A , A , A fractional area in the x, y, z direction, respectively. x y z a adjustment factor to obtain the discharge coefficient. B length of apron. B nappe interference length. int b adjustment factor to obtain the discharge coefficient. c adjustment factor to obtain the discharge coefficient. C discharge coefficient. C submerged weir discharge coefficient. d-sum C (a) labyrinth weir discharge coefficient C linear weir discharge coefficient. d (90 ) C internal apex rope. C , C , C constants of the turbulent k-" model. e e m 1 2 Cov covariance. D external apex length. Dk effective diffusivity for turbulent kinetic energy. eff D# effective diffusivity for dissipation rate. eff d adjustment factor to obtain the discharge coefficient. e adjustment factor to obtain the discharge coefficient. F diffusion term. F source term. F security factor. f adjustment factor to obtain the discharge coefficient. f control variable. f , f , f viscous acceleration in x, y, z direction, respectively. x y z G turbulent kinetic energy generation due to mean velocity gradients. G , G , G acceleration of the body in the x, y, z direction, respectively. x y z g acceleration gravity. H downstream total head. H upstream total head. H* upstream total head of the drowned weir. h piezometric head. k turbulent kinetic energy. L characteristic length of the weir. L cycle length. cycle L length of the crest affected by the nappe interference. int l centerline length of sidewall. M magnification ratio. m adjustment coefficient to obtain the length B . int N number of cycles. N number of cells. n adjustment coefficient to obtain the length B . int o adjustment coefficient to obtain the length B . int P weir height. p order of convergence. Q design flow. Water 2023, 15, 722 27 of 29 Q flow of a linear weir. R fradius of the weir crest. crest r fmesh refinement ratio. S fsubmergence level. S fstrain rate tensor. ij t weir wall width. t upper crest width. w1 t lower crest width. u velocity component in the x direction. 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Journal

Water – Multidisciplinary Digital Publishing Institute

Published: Feb 11, 2023

Keywords: labyrinth weir; Computational Fluid Dynamics (CFD); spillways discharge capacity; spillway weir design

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