Energy Dissipation and Hydraulics of Flow Over Trapezoidal–Triangular Labyrinth Weirs

Article Energy Dissipation and Hydraulics of Flow over Trapezoidal–Triangular Labyrinth Weirs

Amir Ghaderi 1,2,* , Rasoul Daneshfaraz 3, Mehdi Dasineh 3 and Silvia Di Francesco 4

1 Department of Civil Engineering, Faculty of Engineering, University of Zanjan, Zanjan 4537138791, Iran 2 Department of Civil Engineering, University of Calabria, Arcavacata 87036, Italy 3 Department of Civil Engineering, Faculty of Engineering, University of Maragheh, Maragheh 8311155181, Iran; [email protected] (R.D.); [email protected] (M.D.) 4 Engineering Faculty, Niccolò Cusano University, Rome 00166, Italy; [email protected] * Correspondence: [email protected]; Tel.: +98-938-450-351-2

Received: 10 June 2020; Accepted: 12 July 2020; Published: 14 July 2020

Abstract: In this work experimental and numerical investigations were carried out to study the influence of the geometric parameters of trapezoidal–triangular labyrinth weirs (TTLW) on the discharge coefficient, energy dissipation, and downstream flow regime, considering two different orientations in labyrinth weir position respective to the reservoir discharge channel. To simulate the free flow surface, the volume of fluid (VOF) method, and the Renormalization Group (RNG) k-ε model turbulence were adopted in the FLOW-3D software. The flow over the labyrinth weir (in both orientations) is simulated as a steady-state flow, and the discharge coefficient is validated with experimental data. The results highlighted that the numerical model shows proper coordination with experimental results and also the discharge coefficient decreases by decreasing the sidewall angle due to the collision of the falling jets for the high value of H/P (H: the hydraulic head, P: the weir height). Hydraulics of flow over TTLW has free flow conditions in low discharge and submerged flow conditions in high discharge. TTLW approximately dissipates the maximum amount of energy due to the collision of nappes in the upstream apexes and to the circulating flow in the pool generated behind the nappes; moreover, an increase in sidewall angle and weir height leads to reduced energy. The energy dissipation of TTLW is largest compared to vertical drop and has the least possible value of residual energy as flow increases.

Keywords: TTLW labyrinth weir; discharge coeﬃcient; energy dissipation; subcritical ﬂow; FLOW-3D

1. Introduction Weirs are extensively used for passage flood, flow measurement and deviation, and control of water level river sand open channels [1]. Since the volume of the passing flow through the weir is dependent on the length and shape of the weir crest, most research has been devoted to the influence of hydraulic and geometrical parameters on the flow discharge coefficient and the overflow discharge through weirs. To increase the flow, weir length at a certain width can be modified, adopting non-linear weirs, such as triangular, trapezoidal, and circular called labyrinth weirs. They are usually made in one or several cycles. Hydraulic performance of labyrinth weirs has been widely assessed in the past [2–6]. The first serious investigations into the labyrinth weirs were conducted by Taylor [7] and Hay and Taylor [8]; they used the ratio of the discharges of a labyrinth and linear weirs to show the performance of the labyrinth weirs, presenting design curves with an effective head on the weir coincident with the flow depth. Kumar et al. [9] experimentally investigated the labyrinth weir discharge coefficient with the triangular plan, showing that by decreasing the apex weir angle, the length of the interference zone

Water 2020, 12, 1992; doi:10.3390/w12071992 www.mdpi.com/journal/water Water 2020, 12, 1992 2 of 18 increased and the weir discharge coefficient decreased significantly. They also provided relationships to calculate the flow discharge coefficient at different apex angles. Crookston and Tullis [10], following Houston [3], focused on the development of labyrinth weir into the reservoir, observing a better performance of arched labyrinth weirs as a result of the orientation of the weir cycles. Carollo et al. [11], using dimensional analysis, presented relationships to calculate the weir outflow rate from triangular labyrinth weirs. Bilhan et al. [12] experimentally investigated the discharge capacity of labyrinth weirs with and without nappe breakers. They indicated that nappe breakers placed on the trapezoidal labyrinth weirs and circular labyrinth weirs reduced the discharge coefficient up to 4% compared to an un-amended weir. Abbaspuor et al. [13] experimentally and numerically examined hydraulic passing flow through a triangular labyrinth weir. Their results showed that by increasing the angle of the weir apex, the crossflow interference with the lateral fall blades reduced and created less flow vortex. Azimi and Hakim [14] experimentally studied hydraulic passing flow through a rectangular labyrinth weir, demonstrating its suitability for the ratio h0/P < 0.4 (h0 the flow head over the weir and P weir height). Also, under submerged conditions, a rectangular labyrinth weir is more sensitive to the linear weir, and its efficiency is reduced by 10% over the linear weir. The improvements in computational speed and computer storage capacity, along with the advent of suitable turbulence modeling approaches, have made Computational Fluid Dynamics (CFD) a viable complementary investigation tool [15–24]. Sangsefidi et al. [25] numerically studied the effect of the downstream bed level on the discharge coefficient of labyrinth weirs. The results showed that for high-head conditions, the lowering of downstream bed level increases its efficiency, especially in the case of an arced labyrinth weir. Shaghaghian and Sharifi [26] investigated the characteristics of flow in triangular labyrinth weirs using commercial software FLUENT. Carrillo et al. [27] numerically studied the discharge coefficient on free and submerged flows over labyrinth weirs and analyzed the free surface flow profile. The results showed that CFD models are able to provide very good evaluations of the discharge on free and submerged labyrinth weirs for a large sidewall angle. Norouzi et al. [28] and Shafiei et al. [29] investigated the discharge coefficient of labyrinth weirs using support vector machines and adaptive neuro-fuzzy interface system (ANFIS) as well as a hybrid model called “firefly algorithm” via the CFD approach. The firefly algorithm has the ability to find optimized values for non-linear problems with high convergence speed. The results showed that both artificial intelligence and ANFIS models had better accuracy in estimating the discharge coefficient. Despite the above literature review, there is still a strong need for fundamental studies of flow over labyrinth weirs and its corresponding characteristics. The present study aims to investigate the hydraulic flow over the composite form of a trapezoidal–triangular labyrinth weirs (TTLW) located in a reservoir with normal and reverse orientations. The main objectives of this study focus on the effects of the geometric parameters on the discharge coefficient, energy dissipation, and downstream flow regime of TTLW. The paper is organized as follows: Section2 presents the governing equations of free surface flow and main parameters affecting the discharge coefficient and energy dissipation. The description of the laboratory model and the numerical setup are shown in Section3. Subsequently, details of the verification of the numerical modeling versus the experimental (Section4) data are presented. Moreover, flow characteristics and discharge coefficients of a TTLW are described and the effects of the geometric parameters, considering two different orientations with respect to the reservoir, on energy dissipation and hydraulics of flow will be analyzed. The paper closes with a discussion on the results obtained. Water 2020, 12, 1992 3 of 18

2. Materials and Methods

2.1. Effective Parameters in the Discharge Coefficient The weirs discharge coefficient depends on several factors including the hydraulic properties of passing flow the weir crest and the geometric properties of the weir. The general relationship of the weirs discharge is as follows: Water 2020, 12, x FOR PEER REVIEW 2 p 3 of 18 Q = C L 2gH1.5 (1) 3 d = 2 1.5 QCLgHd 2 (1) where Q is the discharge flow, Cd is the discharge3 coefficient, L is the effective length of the weir crest (L = N* (2D + 2l), H is the hydraulic head relative to the crest level upstream of the weir, and g is the where× Q is the discharge flow, Cd is the discharge coefficient, L is the effective length of the weir crest gravity(L = acceleration N* × (2D + 2l), [6 H]. is the hydraulic head relative to the crest level upstream of the weir, and g is the Accordinggravity acceleration to Figure [6].1, the main parameters a ffecting the discharge coefficient of the TTLW are (EquationAccording (2)): the weirto Figure height 1, the (P ),main the parameters number of af labyrinthfecting the weir discharge cycles coefficient (N), the of length the TTLW of each are weir cycle(Equation (l), the width (2)): ofthe each weir cycleheight (w (P),), the the weir number external of laby spanrinth ( Dweir), the cycles weir (N internal), the length span of (d each), the weir angle of cycle (l), the width of each cycle (w), the weir external span (D), the weir internal span (d), the angle the labyrinth weir wall (α), and the longitudinal slope of the weir bed (S0). of the labyrinth weir wall (α), and the longitudinal slope of the weir bed (S0). = α Q =QfHPlNwdSgf (H(,,,,,,,,), P, l, N, w, d, α, S00, g) (2) (2)

Figure 1. The eﬀective parameters of the trapezoidal–triangular labyrinth weir (TTLW). Figure 1. The effective parameters of the trapezoidal–triangular labyrinth weir (TTLW). After performing the dimensional analysis, Equation (2) can be written as follows: After performing the dimensional analysis, Equation (2) can be written as follows: Q QHwl fNS(,,,,,,)0H w l α = f ( ,1.5 , , , N, α, S0)0 = 0(3) (3) L √LgHgH1.5 P PPPP P

In test cases of our work where N = 2 and S0 = 0, it follows that these parameters can be eliminated Infrom test Equation cases of (3): our work where N = 2 and S0 = 0, it follows that these parameters can be eliminated from Equation (3): QHwl= α Q f (,,,)H w l (4) LgH1.5= f ( PPP, , , α) (4) L √gH1.5 P P P Combining Equations (1) and (4) it is possible to derive the following equation: Combining Equations (1) and (4) it is possible to derive the following equation: = Hw lα Cfd (,, ) (5) HPPPw l Cd = f ( , , α) (5) The effect of the dimensionless parameters PpresentedP P in Equation (5) on the TTLW discharge coefficient will be examined in the next sections. The eﬀect of the dimensionless parameters presented in Equation (5) on the TTLW discharge coeﬃ2.2.cient Effective will be Parameters examined in the in theEnergy next Dissipation sections. The energy dissipation rate in the TTLW can be investigated by comparing the total energy dissipation in the labyrinth weir (ΔE) with the maximum possible energy dissipation (ΔEmax). According to the special energy curve, as known, the minimum specific energy (Emin) occurs in critical

Water 2020, 12, 1992 4 of 18

2.2. Effective Parameters in the Energy Dissipation The energy dissipation rate in the TTLW can be investigated by comparing the total energy dissipation in the labyrinth weir (∆E) with the maximum possible energy dissipation (∆Emax). According to the special energy curve, as known, the minimum specific energy (Emin) occurs in critical flow conditions (Froude number (Fr) = 1). Thus, the minimum amount of energy remaining downstream of the TTLW will be as follows:

E1min = Emin = 1.5yc (6) where yc indicates the critical depth of passing ﬂow the labyrinth weir. The maximum possible energy dissipation is therefore given by:

∆Emax = E E = E 1.5yc (7) 0 − 1min 0 − The equation of maximum relative energy dissipation is given as follows:

∆E yc max = 1 1.5 (8) E0 − E0

The relative energy dissipation on the weir can be derived, through a dimensional analysis, based on the relationship of the geometric parameters of the weir and the flow conditions [30,31]. Regardless of the effect of viscosity and surface tension, the labyrinth weir energy dissipation can be stated as follows: f (∆E, E0, yc, P, l, w, α) = 0 (9) Using Buckingham π theorem, the dimensionless parameters affecting the rate of energy dissipation over the TTLW wrote as follows: ∆E yc l l w = f ( , , , , α) (10) E0 E0 w P P

2.3. Numerical Model FLOW-3D® software package was used to numerically simulate the flow pattern. The software solves the three-dimensional Reynolds averaged Navier–Stokes equations (RANS) with the finite volume method [32,33]: ∂ρ ∂(ρuAx) ∂ ρvAy ∂(ρwAz) V + + + = R + R (11) F ∂t ∂x ∂y ∂z SOR DIF ! ∂u 1 ∂u ∂u ∂u 1 ∂P + uAx + vAy + wAz = + Gx + fx (12) ∂t VF ∂x ∂y ∂z −ρ ∂x ! ∂v 1 ∂v ∂v ∂v 1 ∂P + uAx + vAy + wAz = + Gy + fy (13) ∂t VF ∂x ∂y ∂z −ρ ∂y ! ∂w 1 ∂w ∂w ∂w 1 ∂P + uAx + vAy + wAz = + Gz + fz (14) ∂t VF ∂x ∂y ∂z −ρ ∂z where (x, y, z) are the cartesian coordinates, P is the pressure, Vf is the volume fraction of the fluid in each cell, and (u, v, w) are the Cartesian velocity components. The A symbols are the fractional areas associated with the flow direction. The G terms are local accelerations of the fluid and the f terms are related to frictional forces. The ρ symbol indicates the fluid density, RDIF is a turbulent diffusion term, and RSOR is mass source. A turbulence model is necessary to account for the turbulent Reynolds stresses. Here, the Renormalization Group (RNG) k-ε turbulence model was selected. This model has been Water 2020, 12, 1992 5 of 18 applied to similar flow scenarios and is capable of providing accurate and efficient solutions [34–38]. The RNG k-ε model equations are given by:

∂ ∂(ρkui) ∂ ∂k (ρk) + = (αkµe f f ) + Gk + Gb ρε YM + Sk (15) ∂t ∂xi ∂xj ∂xj − −

2 ∂ ∂(ρεui) ∂ ∂k ε ε (ρε) + = (αsµe f f ) + C1s (Gk + G3sGb) + C2ερ Rε + Sε (16) ∂t ∂xi ∂xj ∂x − k k − where k is the turbulent kinetic energy per unit mass, ε is the turbulence dissipation rate per unit mass, Gk is the production of turbulent kinetic energy due to velocity gradient, Gb is turbulent kinetic energy production from buoyancy, and YM is turbulence dilation oscillation distribution [39,40]. In the above equations, αk = αs = 1.39, C1s = 1.42, and C2ε = 1.6 are model constants. All of the constants are derived explicitly in the RNG procedure. The terms Sk and Sε are source terms for k and ε, respectively. Also µeff is the effective viscosity (µeff = µ + µt), where µt is called eddy viscosity or turbulent viscosity. Among the two equation models, the one by Launder and Spalding [41] popularly known as RNG k-ε model, proposes: ρk2 µt = C (17) µ ε

Here Cµ is an empirical constant equal to 0.09. FLOW-3D® uses an advanced algorithm for tracking free-surface flows, called the volume of fluid (VOF) [42]. The VOF method consists of three main components: fluid ratio function, VOF transport equation solution, and boundary conditions at the free surface. The VOF transport equation is expressed by (18): " # ∂F 1 ∂(FAxu) ∂(FAyv) ∂(FAzw) + + + = 0 (18) ∂t VF ∂x ∂y ∂z In the above equation, A is the average ratio of flow area along x, y, and z directions; u, v, and w are average velocities along x, y, z; and F is the fluid ratio function which takes on values between 0 and 1. If F = 0, the cell is completely full of air, and if F = 1, the cell is full of water. The free surface is tracked where F = 0.5.

3. Case Study Laboratory tests were performed in a horizontal rectangular channel with length, width, and height of 5, 0.3, and 0.45 m, respectively. Ten weir models with heights of 0.08, 0.1, and 0.12 m were constructed (Table1). To provide a smooth surface with the least roughness, floors and channel walls were constructed from Plexiglas. However, according to Johnson [43], the effect of channel walls on the discharge coefficient of labyrinth weir can be neglected. To ensure steady flow, weir models were installed 1.5 m downstream of the inlet tank. At the inlet of the channel, there was a screen that eliminated the flow turbulence and the flow slowly entered the laboratory flume. Channel flow was measured by two pumps each at a maximum discharge of 7.5 L/s by two valves connected to two rotameters with an accuracy of 2% installed at the pump outlet (Figure2). In total, 210 experiments ± were performed to investigate the research objectives. Water 2020, 12, 1992 6 of 18 Water 2020, 12, x FOR PEER REVIEW 6 of 18

Figure 2. Schematic view of the test section and flumes. ﬂumes.

The flow flow depth for both the upstream and downstreamdownstream sections was determined by a gauge point mounted on the flume top with an accuracy of 1 mm. This technique was adopted from point mounted on the flume top with an accuracy of ±±1 mm. This technique was adopted from previous studies, for for example, example, Ghad Ghaderieri et et al. al. [44] [44 and] and studies. studies. To Torecord record the theupstream upstream flow flow depth, depth, the gaugethe gauge point point was waspositioned positioned at 3 ath to 3 h4 toh [45]. 4 h [ 45Also]. Also,, the flow the flow depth depth was was recorded recorded downstream downstream of the of the weir at a distance of 0.8 m. Upstream Froude number was calculated as Fr = Q/[gW2(h + P)3]0.5, weir at a distance of 0.8 m. Upstream Froude number was calculated as Fr00 = Q/[gW2(h + P)3]0.5, downstream Froude number as Fr = V /(gy )0.5, Reynolds number as R = (gh3)0.5/υ, critical depth as downstream Froude number as Fr11 = V11/(gy1)10.5, Reynolds number as R = (gh3)0.5/υ, critical depth as yc 2 2 1/3 yc = 2(Q /Gw2 1/3 ) , upstream speed as V = Q/y , and downstream speed as V = Q/Wy . = (Q /Gw ) , upstream speed as V0 = Q0/y0, and0 downstream speed as V1 = Q/1Wy1. 1

Table 1. Geometrical properties of weir models and hydraulic ﬂow parameters in the present study. Table 1. Geometrical properties of weir models and hydraulic flow parameters in the present study. Models Q (l/s) α ( ) l (cm) w (cm) P (cm) l/w w/p Fr Models Q(l/s) α (°)◦ l (cm) w (cm) P (cm) l/w w/p Fr0 0 Range 3–15 3–15 10–30 10–30 27–72 27–72 15 158–12 8–12 1.8–4.8 1.8–4.8 1.87–1.251.87–1.25 0.08–0.16 0.08–0.16

Computational Mesh and Boundary Conditions The weir setup was performed by inserting a sterolithography (STL) file. The spatial domain was meshed using a structured rectangular hexahedral mesh. A containing mesh block was created

Water 2020, 12, 1992 7 of 18

Computational Mesh and Boundary Conditions Water 2020The, 12 weir, x FOR setup PEER was REVIEW performed by inserting a sterolithography (STL) file. The spatial domain7 of was 18 meshed using a structured rectangular hexahedral mesh. A containing mesh block was created for for the whole spatial domain, and then, a nested mesh block with refined cells for the area of interest the whole spatial domain, and then, a nested mesh block with refined cells for the area of interest (in correspondence of the weir) was built (see Figure 3). This technique, i.e., a nested mesh block, was (in correspondence of the weir) was built (see Figure3). This technique, i.e., a nested mesh block, was adopted from previous studies (see, for example, Ghaderi and Abbasi [33], Choufu et al. [46], and adopted from previous studies (see, for example, Ghaderi and Abbasi [33], Choufu et al. [46], and Ghaderi et al. [47]). Considering the geometry dimensions, five computational meshes were tested to Ghaderi et al. [47]). Considering the geometry dimensions, five computational meshes were tested to select the appropriate one. To this aim, a comparison between simulated and measured free surface select the appropriate one. To this aim, a comparison between simulated and measured free surface profiles for the different meshes was performed (Table 2). profiles for the different meshes was performed (Table2).

FigureFigure 3. 3. SketchSketch of of mesh mesh setup. setup.

AccordingAccording to Figure4 4,, which which shows shows the the variation variation of of the the mean mean relative relative error error as aas function a function of the of cellthe cellsizes sizes for freefor surfacefree surface profiles, profiles, we can we see can that see the that simulated the simulated free surface free profiles surface exhibit profiles better exhibit agreement better agreementwith the measured with the freemeasured surface free profiles surface for profiles the finer for cell th sizee finer of 0.30cell cm.size Inof addition,0.30 cm. In the addition, variation the of variationmean relative of mean errors relative can be errors neglected can be by neglected decreasing by the decreasing cell size fromthe cell 0.35 size cm from to 0.30 0.35 cm. cm In to summary, 0.30 cm. Inthe summary, cell size of the 0.35 cell cm size was of selected 0.35 cm andwas there selected were an 598,500d there coarsewere 598,500 cells (0.8 coarse cm in cells all directions) (0.8 cm in andall directions)2,104,958 finer and cells 2,104,958 (0.35 cmfiner in allcells directions). (0.35 cm in To all reduce directions). the effect To of reduce computational the effect mesh of computational on simulation meshresults, on the simulation same mesh results, was adoptedthe same formesh all modelswas adopted of this for research. all models of this research. ComputationsComputations in in the the viscous viscous sub-layer sub-layer were were prev prevented.ented. The The first first node node was was located located near near a a wall wall + + inin order order to to keep keep the the dimensionless dimensionless parameter parameter of of y+y (Equation(Equation (19)) (19)) in the in the range range of 5 of < 5y+< < y30 [48].< 30 The [48]. + ratioThe ratioof turbulent of turbulent and laminar and laminar influences influences in a cell, in a y cell,+, isy defined, is defined as: as:

+ yyuppu * y+y == ∗ (19)(19) υυ where yp is the distance of the first node from the wall, u* is the shear velocity of the wall and υ is the kinematic viscosity.

Water 2020, 12, 1992 8 of 18

Waterwhere 2020yp, 12is, thex FOR distance PEER REVIEW of the ﬁrst node from the wall, u* is the shear velocity of the wall and υ 8is of the 18 kinematic viscosity. Table 2. Mesh sensitivity analysis for the present study. Table 2. Mesh sensitivity analysis for the present study. * MAPE (%) ** RMSE (cm) Coarser Finer Total n XX− ** RMSEn (cm) Test Turbulence Computational 1* MAPEexp (%) num s 1 2 Test Turbulence CellsCoarser Size Cells CellsFiner Size CellsMeshTotal Mesh Computational100× n n ()XX− P Xexp Xnum P exp num No. Model Time (min) 100 1 | − | 1 )2 No. Model Size(cm) (cm) (cm)Size (cm)NumberNumber Time (min) nX1n Xexpexp n 1(Xexp Xnum × 1 1 −

T1 k-ε (RNG) 1.10 0.65 711,758 50 19.39 3.44 T1 k-ε (RNG) 1.10 0.65 711,758 50 19.39 3.44 T2 T2k- k-ε ε(RNG)(RNG) 1.00 1.00 0.55 0.55 1,302,587 1,302,587 85 85 11.12 11.12 1.96 1.96 T3 T3k- k-ε ε(RNG)(RNG) 0.90 0.90 0.45 0.45 1,997,425 1,997,425 120 120 6.07 6.07 0.85 0.85 T4 T4k- k-ε ε(RNG)(RNG) 0.80 0.80 0.35 0.35 2,703,458 2,703,458 175 175 3.75 3.75 0.56 0.56 T5 k-ε (RNG) 0.70 0.30 3,587,624 220 3.32 0.44 T5 k-ε (RNG) 0.70 0.30 3,587,624 220 3.32 0.44 * mean absolute percentage error, ** root mean square error, Xexp: experimental value of X; Xnum: * mean absolute percentage error, ** root mean square error, Xexp: experimental value of X; Xnum: numerical value numericalof X; and n: value total number of X; and of data.n: total number of data.

Figure 4. Variations of the relative error of the free surface profiles versus cell size. Figure 4. Variations of the relative error of the free surface profiles versus cell size. According to the experimental conditions, the inlet boundary condition was set to a discharge According to the experimental conditions, the inlet boundary condition was set to a discharge flow rate (Q; equal to the experimental flow exit discharge) and outlet (O) flow conditions were applied flow rate (Q; equal to the experimental flow exit discharge) and outlet (O) flow conditions were to the downstream boundary. Wall roughness has been neglected due to the small roughness of the applied to the downstream boundary. Wall roughness has been neglected due to the small roughness material of the experimental facility (plexiglass) used for validation. The lower Z (Zmin) and both of the material of the experimental facility (plexiglass) used for validation. The lower Z (Zmin) and of the side boundaries were treated as rigid walls (W). No-slip conditions were applied at the wall both of the side boundaries were treated as rigid walls (W). No-slip conditions were applied at the boundaries, treated as non-penetrative boundaries. An atmospheric boundary condition was set to wall boundaries, treated as non-penetrative boundaries. An atmospheric boundary condition was set the upper boundary of the channel: the flow can enter and leave the domain as null von Neumann to the upper boundary of the channel: the flow can enter and leave the domain as null von Neumann conditions are imposed on all variables, except for pressure, set to zero (i.e., atmospheric pressure). conditions are imposed on all variables, except for pressure, set to zero (i.e., atmospheric pressure). Symmetry boundary conditions (S) were used at the inner boundaries as well. Figure5 shows the Symmetry boundary conditions (S) were used at the inner boundaries as well. Figure 5 shows the computational domain of the present study and the associated boundary conditions. computational domain of the present study and the associated boundary conditions.

Water 2020, 12, 1992 9 of 18 Water 2020, 12, x FOR PEER REVIEW 9 of 18 Water 2020, 12, x FOR PEER REVIEW 9 of 18

Figure 5. Applied boundary conditions. Figure 5. Applied boundaryboundary conditions. conditions. 4. Results and Discussions 4.4. Results Results and and Discussions Discussions 4.1. The Validity of the FLOW-3D Model 4.1.4.1. The The Validity Validity of of the the FLOW-3DFLOW-3D Model During the iteration, the time-step size has been controlled by stability (based on Courant number) DuringDuring thethe iteration,iteration, thethe time-step size hashas beenbeen controlledcontrolled byby stabilitystability (based (based on on Courant Courant and convergence criterion, which leads to time steps between 0.0001 and 0.0014 s. The evolution in number)number) andand convergenceconvergence criterion,criterion, which leads toto timetime stepssteps betweenbetween 0.0001 0.0001 and and 0.0014 0.0014 s. s.The The time was used as a relaxation to the final steady state. The steady-state condition was checked for evolutionevolution in in time time waswas usedused asas aa relaxation to thethe finalfinal steadysteady state. state. The The steady-state steady-state condition condition was was monitoring the kinetic energy of the flow, flow rate at the outlet boundary, and the free surface elevation checkedchecked for for monitoring monitoring thethe kinetickinetic energy of thethe flowflow,, flowflow rate rate at at the the outlet outlet boundary, boundary, and and the the free free at the inlet boundary. It was found that with the 12 s simulation of the flow, the solution became fully surfacesurface elevation elevation atat thethe inletinlet boundary.boundary. It was founfoundd thatthat withwith the the 12 12 s ssimulation simulation of of the the flow, flow, the the converged and the steady-state condition was achieved for all of the considered H/P(H: the hydraulic solutionsolution became became fully fully converged converged and the steady-state condition condition was was achieved achieved for for all all of of the the considered considered head, P: the weir height) values. The flow over the labyrinth weir in both orientations is simulated as a H/PH/P ( H(H: :the the hydraulichydraulic head,head, PP:: thethe weir height) values.values. TheThe flowflow over over the the labyrinth labyrinth weir weir in in both both steady-state of flow (Figure6A), and the discharge coe fficient (Cd) is validated with experimentald data. orientationsorientations is is simulatedsimulated asas aa steady-statesteady-state of flowflow (Figure(Figure 6A),6A), and and the the discharge discharge coefficient coefficient (C (C) dis) is To ensure a good agreement between the numerical and experimental data, the error was determined validatedvalidated withwith experimentalexperimental data. To ensure aa goodgood agreementagreement betweenbetween thethe numerical numerical and and and is presented in Figure6B. experimentalexperimental data, data, the the errorerror waswas determined and isis presentedpresented in in Figure Figure 6B. 6B.

Figure 6. Cont.

Water 2020, 12, 1992 10 of 18 Water 2020, 12, x FOR PEER REVIEW 10 of 18

FigureFigure 6. 6.(A )(A Comparison) Comparison of discharge of discharge coeﬃ cientcoefficient obtained ob fromtained numerical from numerical solution withsolution experimental with data;experimental (B) Determination data; (B) Determination of error percentage. of error percentage.

ItIt can can be be observed observed thatthat simulated discharge discharge coeffi coefficientcient values values at atdifferent different H/P H /ratiosP ratios were were similar similar toto the the experimental experimental data. data. Discharge Discharge coe coefficientfficient C dCincreasedd increased for for a a low low H H/P/P ratio ratio and and decreased decreased for for high ratios.high ratios. A quantitative A quantitative evaluation evaluation of the computedof the computed and measured and measured discharge discharge coeffi cientcoefficient versus versus different Hdifferent/P ratios H/P comparisons ratios comparisons were made were by usingmade by mean using absolute mean absolute relative relative error (MARE). error (MARE). 1 N OP− MARE =×XN ii 100 1  Oi Pi (20) MARE = NOi = 1 −i 100 (20) N Oi × i=1 where Oi and Pi are measured and computed values, respectively, and N is the total number of data. whereTableO 3i givesand P thei are results measured for MARE and computedat different values,H/P ratio respectively, and weir orientation. and N is the total number of data. Table3 gives the results for MARE at di fferent H/P ratio and weir orientation. Table 3. MARE Values for the discharge coefficient versus different H/P ratios on NO and IO orientation. Table 3. MARE Values for the discharge coefficient versus different H/P ratios on NO and IO orientation. Cd-NO- Cd-NO- Cd-IO- Cd-IO- MARE MARE Model H/PC -NO-MeasuredC -NO-Computed Computed CMeasured-IO- Measured ComputedC -IO- Computed (%)- MARE(%)- MARE Model H/P d d d d Measured ValuesValues ValuesValues ValuesValues ValuesValues NO (%)-NOIO (%)-IO 0.1760.176 0.7990.799 0.7890.789 0.804 0.804 0.790 0.7901.29 1.291.70 1.70 0.256 0.779 0.757 0.764 0.772 2.87 1.10 0.256 0.779 0.757 0.764 0.772 2.87 1.10 Trapezoidal–Trapezoidal–0.338 0.716 0.722 0.706 0.719 0.77 1.90 Triangular 0.4170.338 0.6700.716 0.6840.722 0.706 0.662 0.719 0.6420.77 2.021.90 3.00 Triangular0.509 0.606 0.613 0.600 0.585 1.14 2.50 Labyrinth 0.417 0.670 0.684 0.662 0.642 2.02 3.00 Weir (TTLW)Labyrinth0.589 0.575 0.560 0.570 0.559 2.55 2.00 0.6880.509 0.5250.606 0.5100.613 0.600 0.521 0.585 0.5291.14 2.722.50 1.60 Weir 0.589 0.575 0.560Mean 0.570 0.559 2.55 1.912.00 1.97 (TTLW) 0.688 0.525 0.510 0.521 0.529 2.72 1.60 Regarding the overall mean value MAREMean in Table3, there was a good1.91 agreement 1.97 between numerical and laboratory results. The maximum percentage of error for H/P = 0.417 was 3% and the minimumRegarding value the foroverall H/P =mean0.338 value was 0.77%,MARE whichin Table confirms 3, there the was ability a good of theagreement numerical between model to numerical and laboratory results. The maximum percentage of error for H/P = 0.417 was 3% and the predict specifications of flow over TTLW. minimum value for H/P = 0.338 was 0.77%, which confirms the ability of the numerical model to 4.2.predict Hydraulics specifications of Free Flow of flow over TTLW.

4.2.According Hydraulics of to Free Ackers Flow et al. [45] experimental investigations have demonstrated that the weir performance in free flow conditions can be assessed by the discharge coefficient. Figure7 shows the According to Ackers et al. [45] experimental investigations have demonstrated that the weir variations of discharge coefficient C versus H/P for different weir geometries. It can be noted that C performance in free flow conditionsd can be assessed by the discharge coefficient. Figure 7 shows the d decreases by decreasing the sidewall angle due to the collision of the falling jets, for high value of H/P. variations of discharge coefficient Cd versus H/P for different weir geometries. It can be noted that Cd Itdecreases can be noted by decreasing that for the the low sidewall value of angle H/P, thedueC tod value the collision is high, of due the to falling the fact jets, that for the high collision value ofof jets is not so severe. As a result, by increasing the sidewall angle, the decrease rate of C with H/P increases. H/P. It can be noted that for the low value of H/P, the Cd value is high, due to the factd that the collision of jets is not so severe. As a result, by increasing the sidewall angle, the decrease rate of Cd with H/P increases.

Water 20202020,, 1212,, 1992x FOR PEER REVIEW 11 of 18 Water 2020, 12, x FOR PEER REVIEW 11 of 18

(a) (b) (a) (b)

(c)

FigureFigure 7. 7. Experimental Experimental results results obtaobtainedobtainedined for for the effects eﬀects of of weir weir geom geometrygeometryetry on on variations variations of of discharge discharge coecoefficientcoefficientﬃcient C dCd versusd versus H/P, H H/P,/P, (( aa()Pa) )P P == = 8 8 cm,cm, ((bb)P)) PP = 1010 cm, cm, (( (ccc)))P PP === 121212 cm. cm. cm.

AccordingAccording to to Crookston Crookston [[49], 49[49],], nappenappe interactioninteraction occurs when when two twotwo or oror more moremore nappes nappesnappes collide. collide.collide. It Itcan can bebe seen seen that that C dCd decreasesd decreases by by increasing increasing thethe weirweir heights due due to to nappe nappenappe interaction. interaction.interaction. In In fact, fact, the the nappe nappe interactioninteraction increases increases with with discharge, discharge, leading leadingleading to to a a decrease decrease inin in dischargedischarge discharge coefficient coefficient coeﬃcient C CdC ofdd of ofthe the the TTLW. TTLW. TTLW. ForFor TTLW, TTLW, nappenappe nappe interferenceinterference interference originatesoriginates originates atatat thethe upstreamupstream apex apex and andand can cancan produce produceproduce wakes wakeswakes downstream downstream ofof the the apexapex apex (Figure(Figure (Figure8 8).). 8). ItIt It cancan can be bebe stated statedstated that thatthat forfor triangulartriangular shape,shape,shape, downstreamdownstream of ofof the thethe apex, apex,apex, nappe nappenappe interactionsinteractions dodo do notnot not happen.happen. happen.

Figure 8. Cont.

Water 2020, 12, x FOR PEER REVIEW 12 of 18

Water 2020, 12, 1992 12 of 18 WaterWater 2020 2020, 12, 12, x, x FOR FOR PEER PEER REVIEW REVIEW 12 of12 18 of 18

Figure 8. The collision of nappes from adjacent sidewalls of TTLW in (A) normal orientation and (B)

inverse orientation. FigureFigure 8. 8. TheThe collision of of nappes nappes from from adjacent adjacent sidewalls sidewalls of TTLW of TTLW in (A in) normal (A) normal orientation orientation and (B and) Figureinverse 8. orientation. The collision of nappes from adjacent sidewalls of TTLW in (A) normal orientation and (B) (TheB) inverse behavior orientation. of Cd with H/P for the weirs of different orientations, obtained from experimental inverse orientation. and numerical results, is shown in Figure 9. It can be noted that the trend of discharge coefficient for The behavior of Cd with H/P for the weirs of different orientations, obtained from experimental The behavior of Cd with H/P for the weirs of diﬀerent orientations, obtained from experimental normal and inverse orientation is the same and Cd decreases by decreasing the sidewall angle. andand numericalThe numerical behavior results, results, of C isdis with shownshown H/P inin for Figure the weirs9 9.. ItIt can of be different noted that thatorientations, the the trend trend ofobtained of discharge discharge from coefficient coe experimentalﬃcient for for andnormal numerical and inverse results, orientation is shown isin the Figure same 9. and It can Cd decreasesbe noted thatby decreasing the trend theof discharge sidewall angle. coefficient for normal and inverse orientation is the same and Cd decreases by decreasing the sidewall angle. normal and inverse orientation is the same and Cd decreases by decreasing the sidewall angle.

FigureFigure 9.9. Variations ofof dischargedischarge coecoefficientfficient CCdd with HH/P/P inin normalnormal andand inverseinverse orientation.orientation. Figure 9. Variations of discharge coefficient Cd with H/P in normal and inverse orientation. AA comparisoncomparison ofof experimentalexperimental tests TTLW orientationsorientations showedshowed thatthat nono didifferencefference inin dischargedischarge A comparisonFigure 9. Variations of experimental of discharge tests coefficient TTLW orient Cd withations H/P showedin normal that and no inverse difference orientation. in discharge eefficiencyfficiency waswas observedobserved betweenbetween thethe normalnormal andand inverseinverse orientationsorientations (Figure(Figure 10 10).). efficiency was observed between the normal and inverse orientations (Figure 10). A comparison of experimental tests TTLW orientations showed that no difference in discharge efficiency was observed between the normal and inverse orientations (Figure 10).

Figure 10. Comparison of experimental tests TTLW orientations in discharge coeﬃcient (C ). FigureFigure 10. 10. Comparison Comparison of experimental tests tests TTLW TTLW orientations orientations in in discharge discharge coefficient coefficient (C d().Cd d).

Figure 10. Comparison of experimental tests TTLW orientations in discharge coefficient (Cd).

Water 2020, 12, 1992 13 of 18

Water 2020, 12, x FOR PEER REVIEW 13 of 18 4.3. Energy Charachterization and Flow Regime 4.3. Energy Charachterization and Flow Regime The weir downstream flow regime can be evaluated through the Froude number of flow Fr1. The weir downstream flow regime can be evaluated through the Froude number of flow Fr1. Variation of Fr1 with the relative critical depth yc/E0 for experimental results is shown in Figure 11 for Variation of Fr1 with the relative critical depth yc/E0 for experimental results is shown in Figure 11 for differentdifferent sidewall sidewall angles angles and and height height of of the the weirs. weirs. As highlighted highlighted in inth thisis figure, figure, in most in most of the of cases, the cases, the downstreamthe downstream flow flow regime regime ofTTLW of TTLW was was supercritical: supercritical: Froude Froude numbernumber was larger larger than than one one with with an increasean increase of yc/E0 ofat y dic/Eff0 erentat different weir heights.weir heights.

(a) (b)

(c)

FigureFigure 11. Downstream 11. Downstream Froude Froude number number ( (FrFr1)1 )versus versus the the relative relative critical critical depth depth yc/E0, (ay)c P/E =0 ,(8 cm,a)P (b=) P8 cm, (b)P = 1010 cm, cm, (c ()c P)P = 12= 12cm. cm.

By increasing discharge (corresponding to the relative critical depth yc/E0), Fr1 tends to be 1 but in some models with a high sidewall angle Fr1 < 1 (subcritical flow); this is the result of weak hydraulic jump occurrence,By increasing due todischarge circulating (corresponding flows behind to the the relative nappes critical and collision depth yc/ ofE0), supercritical Fr1 tends to be flows 1 but at the in some models with a high sidewall angle Fr1 < 1 (subcritical flow); this is the result of weak hydraulic base of nappes (see Figure 12). Also, in the same discharge values, the submerged flow was earlier jump occurrence, due to circulating flows behind the nappes and collision of supercritical flows at with a low sidewall angle. Waterthe base 2020 ,of 12 ,nappes x FOR PEER (see REVIEW Figure 12). Also, in the same discharge values, the submerged flow was 14earlier of 18 with a low sidewall angle.

FigureFigure 12. The12. The collision collision of of nappes nappes and and circulatingcirculating flows ﬂows in in the the pool pool at atupstream upstream apexes apexes of the of TTLW. the TTLW.

In Figure 13, the relative energy dissipation ∆E/E0 versus the relative critical depth yc/E0 is presented. Energy dissipation results of vertical drops are compared with the results of Rajaratnam and Chamani [50] and Chamani et al. [51], indicating that TTLW approximately dissipates the maximum amount of energy. Increasing the sidewall angle and weir height (from 8 to 12) and subsequently increasing the dimensionless ratio of yc/E0, TTLW reduces more energy with respect to the vertical case. In fact, collision of nappes happens in the upstream apexes of TTLW, generating a localized hydraulic jump condition near it that leads therefore to energy dissipation. Moreover, with the creation of a pool behind the nappes in the upstream apexes of TTLW, circulating flow (in the pool) enhances the turbulent mixing and dissipates a large portion of energy (see Figure 12). It should also be noted that a weak hydraulic jump downstream of TTLW contributes to dissipating a higher and significant portion of energy.

(a) (b)

(∆E/E0 versus yc/E0), (a) P = 8 cm, (b) P = 10 cm, (c) P = 12 cm.

Water 2020, 12, x FOR PEER REVIEW 14 of 18

Water 2020, 12, 1992 14 of 18 Figure 12. The collision of nappes and circulating flows in the pool at upstream apexes of the TTLW.

InIn FigureFigure 1313,, thethe relativerelative energyenergy dissipationdissipation ∆∆EE//EE00 versus the relative critical depth depth yycc//EE0 0isis presented.presented. Energy Energy dissipation dissipation results result ofs of vertical vertical drops drops are are compared compared with with the the results results of Rajaratnamof Rajaratnam and Chamaniand Chamani [50] and [50] Chamani and Chamani et al. [51 et], al. indicating [51], indicating that TTLW that approximately TTLW approximately dissipates dissipates the maximum the amountmaximum of energy.amount Increasingof energy. theIncreasing sidewall the angle sidew andall weir angle height and weir (from height 8 to 12) (from and 8 subsequently to 12) and increasingsubsequently the dimensionlessincreasing the ratiodimensionless of yc/E0, TTLW ratio of reduces yc/E0, TTLW more energy reduces with more respect energy to thewith vertical respect case. to Inthe fact, vertical collision case. of In nappes fact, collision happens of in nappes the upstream happens apexes in the of upstream TTLW, generating apexes of TTLW, a localized generating hydraulic a jumplocalized condition hydraulic near jump it that condition leads therefore near it that to energyleads therefore dissipation. to energy Moreover, dissipation. with theMoreover, creation with of a poolthe creation behind theof a nappes pool behind in the upstreamthe nappes apexes in the ofupstream TTLW, circulatingapexes of TTLW, ﬂow (in circulating the pool) enhancesflow (in the the turbulentpool) enhances mixing the and turbulent dissipates mixing a large and portion dissipates of energy a large (see portion Figure of 12 energy). It should (see Figure also be 12). noted It should that a weakalso be hydraulic noted that jump a weak downstream hydraulic of TTLWjump downstream contributes toof dissipatingTTLW contributes a higher to and dissipating signiﬁcant a higher portion ofand energy. significant portion of energy.

(a) (b)

(c) FigureFigure 13.13. EnergyEnergy dissipationdissipation on TTLW with vertical drop and maximum energy energy dissipation dissipation [31] [31 ] (∆(∆EE/E/E0 0versus versusy ycc//EE00), (a)P) P = 88 cm, cm, ( (bb))P P == 1010 cm, cm, (c ()c )PP = =1212 cm. cm.

For these reasons, values of energy dissipation on TTLW are more than energy dissipation on the vertical drop. Based on Rajaratnam and Chamani [50] and Chamani et al. [51], circulating ﬂow in the pool is the main reason for energy dissipation on the linear weir and vertical drop. The measured residual energy E1 downstream of weir models is normalized with the minimum theoretical minimum residual energy Emin, and results are then presented with the normalized critical depth yc/E0 (Figure 14). This result indicates that TTLW has energetic dissipation performance that leads to value of residual energy very close to the minimum possible value of residual energy. The energy dissipation of TTLW is largest compared to vertical drop and has the least possible value of residual energy as ﬂow increases. Water 2020, 12, x FOR PEER REVIEW 15 of 18

For these reasons, values of energy dissipation on TTLW are more than energy dissipation on the vertical drop. Based on Rajaratnam and Chamani [50] and Chamani et al. [51], circulating flow in the pool is the main reason for energy dissipation on the linear weir and vertical drop. The measured residual energy E1 downstream of weir models is normalized with the minimum theoretical minimum residual energy Emin, and results are then presented with the normalized critical depth yc/E0 (Figure 14). This result indicates that TTLW has energetic dissipation performance that leads to value of residual energy very close to the minimum possible value of residual energy. The

Waterenergy2020 dissipation, 12, 1992 of TTLW is largest compared to vertical drop and has the least possible value15 of of 18 residual energy as flow increases.

(a) (b)

(c)

FigureFigure 14. 14.Comparison Comparison of of the the residual residual energy energy downstream downstream of TTLW of TTLW with a verticalwith a drop,vertical and drop, minimum and

residualminimum energy residual (E1/ Eenergymin versus (E1/Eyminc/ Eversus0), (a)P yc/=E08), cm,(a) P (b =)P 8 cm,= 10 (b cm,) P = (c 10)P cm,= 12 (c cm.) P = 12 cm. 5. Conclusions 5. Conclusions This paper presented and discussed the effects of the geometric parameters on the discharge This paper presented and discussed the effects of the geometric parameters on the discharge coefficient, energy dissipation, and downstream flow regime of trapezoidal–triangular labyrinth weirs coefficient, energy dissipation, and downstream flow regime of trapezoidal–triangular labyrinth (TTLW)weirs (TTLW) located inlocated a reservoir in a reservoir with the normalwith the and norm inverseal and orientations inverse orientations (experimentally (experimentally and numerically and studied).numerically studied). InIn fact, fact, weirs weirs areare veryvery importantimportant structuralstructural measuresmeasures extensively used, used, for for instance, instance, for for water water levellevel and and erosion erosion controlcontrol inin riversrivers [[52–54]52–54] andand thethe optimizationoptimization of their pe performancerformance have have objective objective advantagesadvantages for for hydraulic hydraulic engineeringengineering works.works. ToTo simulate simulate the the free free flow flow surface surface and and turbulence, turbulence, the the volume volume offluid of fluid (VOF) (VOF) method method and theandRNG the k-RNGε model k-ε inmodel FLOW-3D in FLOW-3D software software were used, were respectively. used, respectively. The capability The capability of this numericalof this numerical method tomethod reproduce to reproduce the actual the turbulent actual turbulent flow conditions flow condit in theions analyzed in the analyzed flow over flow labyrinth over labyrinth weir has weir been assessedhas been through assessed comparisons through comparisons with experimental with experi data.mental The resultsdata. The of the results study of can the be study summarized can be assummarized follows: as follows: • ItIt can can be be concluded concluded that that the the FLOW-3D FLOW-3D software software can can accurately accurately predict predict characteristics characteristics of flow of flow over • TTLW.over TTLW. In comparisons In comparisons between between the calculated the calc andulated measured and measured free surface free profiles, surface appropriate profiles, meshappropriate with 2,118,270 mesh with elements 2,118,270 by relative elements error by and relative RMSE error of 3.05%, and RMSE 0.43 cm of and3.05%, maximum 0.43 cm aspect and ratiomaximum 1.07, was aspect selected ratio for1.07, calculations. was selected The for maximum calculations. and The minimum maximum value and MARE minimum errors arevalue 3% andMARE 0.77%. errors are 3% and 0.77%. In the high value of H/P due to the collision of the falling jets, the discharge coefficient decreases • by decreasing the sidewall. While for the low of H/P, the collision of jets is not so severe, hence in high values for Cd. Also, by increasing the sidewall angle, the decrease of Cd with H/P increases. Increasing the weir heights decreases the discharge coefficient due to nappe interactions occurring when two or more nappes collide, and the nappe interaction increases as discharge increases. In low discharge, hydraulics of flow over the TTLW has free flow conditions and for high discharge • it has submerged conditions. In all models, the downstream Froude number is larger than one with an increase of relative critical depth yc/E0 at different weir heights. Water 2020, 12, 1992 16 of 18

TTLW approximately dissipates the maximum amount of energy and increases the sidewall angle, • weir height, and relative critical depth yc/E0, it reduces more energy dissipation. Circulating ﬂows behind the nappes enhances the turbulent mixing and dissipates a large portion of energy in the upstream apexes of TTLW. It must be noted that a weak hydraulic jump downstream of TTLW contributes to the more energy dissipation.

Overall, CFD models may provide very good predictions of the energy dissipation and hydraulic characterization of flow over on labyrinth weirs for a different geometric and hydraulic conditions. These structures establish a unique, complex, and three-dimensional flow field in their vicinity. A hydraulic jump as the formation and location of hydraulic jump on the downstream of the weir can be simulated with numerical solutions. However, the accurate modeling of the erosion downstream of the weir caused by hydraulic jump, in particular in its vicinity, remains still an issue to be faced.

Author Contributions: Conceptualization, A.G., R.D., M.D., and S.D.F.; methodology, A.G., R.D., M.D., and S.D.F.; software, A.G. and M.D.; validation, A.G., M.D., and S.D.F.; formal analysis, A.G., M.D., and S.D.F.; investigation, A.G., R.D., M.D., and S.D.F.; writing—original draft preparation, A.G., R.D., M.D., and S.D.F.; writing—review and editing, A.G., R.D., M.D., and S.D.F.; supervision, R.D.; project administration, A.G.; funding acquisition, S.D.F. All authors have read and agreed to the published version of the manuscript. Funding: This work was partially supported by the Italian Ministry of Education, University and Research under PRIN grant No. 20154EHYW9 “Combined numerical and experimental methodology for fluid structure interaction in free surface flows under impulsive loading”. Acknowledgments: The authors of this paper would like to thank the joint cooperation support of the Departments of the Civil Engineering University of Maragheh, University of Zanjan and Faculty of Engineering of the Niccolò Cusano University in completing the laboratory tests of this research and acquiring all necessary data and information for all hydraulic simulations. The authors wish to thank all who assisted in conducting this work. Conflicts of Interest: The authors declare no conflict of interest.

References

1. Falvey, H. Hydraulic Design of Labyrinth Weirs; American Society of Civil Engineers: Reston, VA, USA, 2003. 2. Mayer, P.G. Bartletts Ferry Project, Labyrinth Weir Model Studies; Project No. E-20-610; Georgia Institute of Technology: Atlanta, GA, USA, 1980. 3. Houston, K.L. Hydraulic Model Study of the UTE Dam Labyrinth Spillway; Report No. GR-82-7; U.S. Bureau of Reclamations: Denver, CO, USA, 1982. 4. Lux, F.; Hinchliff, D.L. Design and construction of labyrinth spillways. In Proceedings of the 15th Congress ICOLD, Lausanne, Switzerland, 24–28 June 1985; Volume 4, pp. 249–274. 5. Lux, F. Design and application of labyrinth weirs. In Design of Hydraulic Structures 89; Alberson, M.L., Kia, R.A., Eds.; Balkema/Rotterdam/Brookfield: Toronto, ON, Canada, 1989. 6. Tullis, B.P.; Amanian, N.; Woldron, D. Design of labyrinth spillways. J. Hydraul. Eng. 1995, 121, 247–255. [CrossRef] 7. Talor, G. The Performance of Labyrinth Weirs. Ph.D. Thesis, University of Nottingham, Nottingham, UK, 1968. 8. Hay, N.; Taylor, G. Performance and design of labyrinth weirs, American Society of Civil Engineering. J. Hydraul. Eng. 1970, 96, 2337–2357. 9. Kumar, S.; Ahmad, Z.; Mansoor, T. A new approach to improve the discharging capacity of sharp crested triangular plan form weirs. J. Flow Meas. Instrum. 2011, 22, 175–180. [CrossRef] 10. Crookston, B.M.; Tullis, B.P. Arced Labyrinth Weirs. J. Hydraul. Eng. 2012, 138, 555–562. [CrossRef] 11. Carollo, G.F.; Ferro, V.; Pampalone, V. Experimental Investigation of the Outflow Process over a Triangular Labyrinth-Weir. ASCE J. Irrig. Drain. Eng. 2012, 138, 73–79. [CrossRef] 12. Bilhan, O.; Emiroglu, M.E.; Miller, C.J. Experimental Investigation of Discharge Capacity of Labyrinth Weirs with and without Nappe Breakers. World J. Mech. 2016, 6, 207–221. [CrossRef] 13. Abbaspuor, B.; Haghiabi, A.H.; Maleki, A.; Poodeh, H.T. Experimental and numerical evaluation of discharge capacity of sharp-crested triangular plan form weirs. Int. J. Eng. Syst. Model. Simul. 2017, 9, 113–119. [CrossRef] 14. Azimi, A.H.; Hakim, S.S. Hydraulics of flow over rectangular labyrinth weirs. Irrig. Sci. 2019, 37, 183–193. [CrossRef] Water 2020, 12, 1992 17 of 18

15. Savage, B.M.; Crookston, B.M.; Paxson, G.S. Physical and numerical modeling of large headwater ratios for a 15 labyrinth spillway. J. Hydraul. Eng. 2016, 142, 04016046. [CrossRef] 16. Daneshfaraz, R.; Joudi, A.R.; Ghahramanzadeh, A.; Ghaderi, A. Investigation of flow pressure distribution over a stepped spillway. Adv. Appl. Fluid Mech. 2016, 19, 811–822. [CrossRef] 17. Ghaderi, A.; Daneshfaraz, R.; Dasineh, M. Evaluation and prediction of the scour depth of bridge foundations with HEC-RAS numerical model and empirical equations (Case Study: Bridge of Simineh Rood Miandoab, Iran). Eng. J. 2019, 23, 279–295. [CrossRef] 18. Daneshfaraz, R.; Minaei, O.; Abraham, J.; Dadashi, S.; Ghaderi, A. 3-D Numerical simulation of water flow over a broad-crested weir with openings. ISH J. Hydraul. Eng. 2019, 25, 1–9. [CrossRef] 19. Biscarini, C.; Di Francesco, S.; Ridolfi, E.; Manciola, P. On the Simulation of Floods in a Narrow Bending Valley: The Malpasset Dam Break Case Study. Water 2016, 8, 545. [CrossRef] 20. Paxson, G.; Savage, B. Labyrinth Weirs: Comparison of Two Popular U.S.A Design Methods and Consideration of Non-Standard Approach Conditions and Geometries. In International Junior Researcher and Engineer Workshop on Hydraulic Structures; Report CH61/06, Div. of Civil Eng.; Matos, J., Chanson, H., Eds.; The University of Queensland: Brisbane, Australia, 2006. 21. Daneshfaraz, R.; Ghahramanzadeh, A.; Ghaderi, A.; Joudi, A.R.; Abraham, J. Investigation of the Effect of Edge Shape on Characteristics of Flow under Vertical Gates. J. Am. Water Works Assoc. 2016, 108, 425–432. [CrossRef] 22. Dabling, M.R.; Tullis, B.P. Modifying the downstream hydrograph with staged labyrinth weirs. J. Appl. Water Eng. Res. 2018, 6, 183–190. [CrossRef] 23. Ghaderi, A.; Abbasi, S.; Abraham, J.; Azamathulla, H.M. Efficiency of Trapezoidal Labyrinth Shaped Stepped Spillways. Flow Meas. Instrum. 2020, 72, 101711. [CrossRef] 24. Macián-Pérez, J.F.; García-Bartual, R.; Huber, B.; Bayon, A.; Vallés-Morán, F.J. Analysis of the Flow in a Typified USBR II Stilling Basin through a Numerical and Physical Modeling Approach. Water. 2020, 12, 227. [CrossRef] 25. Sangsefidi, Y.; Mehraein, M.; Ghodsian, M. Numerical simulation of flow over labyrinth weirs. Sci. Iran. 2015, 22, 1779–1787. 26. Shaghaghian, M.R.; Sharifi, M.T. Numerical modeling of sharp-crested triangular plan form weirs using FLUENT. Indian J. Sci. Technol. 2015, 8, 1–7. [CrossRef] 27. Carrillo, J.M.; Matos, J.; Lopes, R. Numerical modeling of free and submerged labyrinth weir flow for a large sidewall angle. Environ. Fluid Mech. 2019, 20, 1–18. [CrossRef] 28. Norouzi, R.; Daneshfaraz, R.; Ghaderi, A. Investigation of discharge coefficient of trapezoidal labyrinth weirs using artificial neural networks and support vector machines. Appl. Water Sci. 2019, 9, 148. [CrossRef] 29. Shafiei, S.; Najarchi, M.; Shabanlou, S. A novel approach using CFD and neuro-fuzzy-firefly algorithm in predicting labyrinth weir discharge coefficient. J. Braz. Soc. Mech. Sci. Eng. 2020, 42, 44. [CrossRef] 30. Haghiabi, A.H.; Azamathulla, H.M.; Parsaie, A. Prediction of head loss on cascade weir using ANN and SVM. ISH J. Hydraul. Eng. 2017, 23, 102–110. [CrossRef] 31. Mohammadzadeh-Habili, J.; Heidarpour, M.; Samiee, S. Study of energy dissipation and downstream flow regime of labyrinth weirs. Iran. J. Sci. Technol. Trans. Civil Eng. 2018, 42, 111–119. [CrossRef] 32. Flow Science Inc. FLOW-3D V 11.2 User’s Manual; Flow Science Inc.: Santa Fe, NM, USA, 2016. 33. Ghaderi, A.; Abbasi, S. CFD simulation of local scouring around airfoil-shaped bridge piers with and without collar. Sadhan¯ a¯ 2019, 44, 216. [CrossRef] 34. Daneshfaraz, R.; Ghaderi, A. Numerical Investigation of Inverse Curvature Ogee Spillway. Civil Eng. J. 2017, 3, 1146–1156. [CrossRef] 35. Zahabi, H.; Torabi, M.; Alamatian, E.; Bahiraei, M.; Goodarzi, M. Effects of Geometry and Hydraulic Characteristics of Shallow Reservoirs on Sediment Entrapment. Water 2018, 10, 1725. [CrossRef] 36. Sangsefidi, Y.; MacVicar, B.; Ghodsian, M.; Mehraein, M.; Torabi, M.; Savage, B.M. Evaluation of flow characteristics in labyrinth weirs using response surface methodology. Flow Meas. Instrum. 2019, 69, 101617. [CrossRef] 37. Ghaderi, A.; Dasineh, M.; Abbasi, S.; Abraham, J. Investigation of trapezoidal sharp-crested side weir discharge coefficients under subcritical flow regimes using CFD. Appl. Water Sci. 2020, 10, 31. [CrossRef] 38. Ghaderi, A.; Dasineh, M.; Daneshfaraz, R.; Abraham, J. Reply to the discussion on paper: 3-D numerical simulation of water flow over a broad-crested weir with openings. ISH J. Hydraul. Eng. 2019, 1–9. [CrossRef] Water 2020, 12, 1992 18 of 18

39. Yakhot, V.; Orszag, S.A.; Thangam, S.; Gatski, T.B.; Speziale, C.G. Development of turbulence models for shear flows by a double expansion technique. Phys. Fluids A Fluid Dyn. 1992, 4, 1510–1520. [CrossRef] 40. Chero, E.; Torabi, M.; Zahabi, H.; Ghafoorisadatieh, A.; Bina, K. Numerical analysis of the circular settling tank. Drink. Water Eng. Sci. 2019, 12, 39–44. [CrossRef] 41. Launder, B.E.; Spalding, D.B. The numerical computation of turbulent flows. In Numerical Prediction of Flow, Heat Transfer, Turbulence and Combustion; Pergamon: Bergama, Turkey, 1983; pp. 96–116. 42. Hirt, C.W.; Nichols, B.D. Volume of Fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 1981, 39, 201–225. [CrossRef] 43. Johnson, M. Discharge Coefficient Scale Effects Analysis for Weirs. Ph.D. Thesis, Utah State University, Logan, UT, USA, 1996. 44. Ghaderi, A.; Daneshfaraz, R.; Torabi, M.; Abraham, J.; Azamathulla, H.M. Experimental investigation on effective scouring parameters downstream from stepped spillways Water Supply. Water Supply 2020, in press. [CrossRef] 45. Ackers, P.; White, W.R.; Perkins, J.A.; Harrison, A.J. Weirs and Flumes for Flow Measurements; Wiley: Chichester, UK, 1978; p. 327. 46. Choufu, L.; Abbasi, S.; Pourshahbaz, H.; Taghvaei, P.; Tfwala, S. Investigation of flow, erosion, and sedimentation pattern around varied groynes under different hydraulic and geometric conditions: A numerical study. Water 2019, 11, 235. [CrossRef] 47. Ghaderi, A.; Daneshfaraz, R.; Abbasi, S.; Abraham, J. Numerical analysis of the hydraulic characteristics of modified labyrinth weirs. Int. J. Energy Water Res. 2020, 1–12, in press. [CrossRef] 48. Sicilian, J.M.; Hirt, C.W.; Harper, R.P. FLOW-3D. Computational Modeling Power for Scientists and Engineers; Report FSI-87-00-1; Flow Science: Los Alamos, NM, USA, 1987. 49. Crookston, B.M. Labyrinth Weirs. Ph.D. Thesis, Utah State University, Logan, UT, USA, 2010. 50. Rajaratnam, N.; Chamani, M.R. Energy loss at drops. J. Hydraul. Res. 1995, 33, 373–384. [CrossRef] 51. Chamani, M.R.; Rajaratnam, N.; Beirami, M.K. Turbulent jet energy dissipation at vertical drops. J. Hydraul. Eng. 2008, 134, 1532–1535. [CrossRef] 52. Ridolfi, E.; Di Francesco, S.; Pandolfo, C.; Berni, N.; Biscarini, C.; Manciola, P. Coping with extreme events: Effect of different reservoir operation strategies on flood inundation maps. Water 2019, 11, 982. [CrossRef] 53. Manciola, P.; Di Francesco, S.; Biscarini, C. Flood Protection and Risk Management: The Case of Tescio River Basin; IAHS-AISH Publication: Capri, Italy, 2009; Volume 327, pp. 174–183. 54. Di Francesco, S.; Biscarini, C.; Manciola, P. Characterization of a Flood Event through a Sediment Analysis: The Tescio River Case Study. Water 2016, 8, 308. [CrossRef]

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).