## Abstract

In the present study, a new nonlinear weir called the T-Shaped Weir (TSW), which is a combination of the Labyrinth Weir (LW) and the Piano Key Weir (PKW), was introduced, and its hydraulic performance was compared with the PKW. Based on the presence of the vertical walls at the inlet key, outlet key, or both keys, the TSW weirs were classified as type A, B, or C weirs, respectively. The flow pattern of different TSW cases was analyzed, and the discharge coefficient curves were provided. Furthermore, to accurately study the hydrodynamics of the tested weirs, 3D numerical simulations were performed using the FLOW-3D software. The results showed that inserting a vertical wall at the upstream of the outlet keys (C-TSW type) has a negligible effect on the hydraulic performance of the PKW. A maximum increase of 16% occurred in the discharge coefficient of the B-TSW in comparison to the PKW, and up to a head to height ratio (H_{t}/p) of 0.45, the effect of the vertical wall on increasing the performance of the B-TSW was maintained. Based on the experimental and numerical tests, the optimal height ratio of the vertical wall (P_{d}/P) in B-TSW with highest discharge capacity was determined to be equal to 0.4.

## HIGHLIGHTS

A new nonlinear weir called the T-Shaped Weir (TSW), which is a combination of the Labyrinth Weir (LW) and the Piano Key Weir (PKW), is introduced.

To investigate the hydrodynamics of the tested weirs in more detail, 3D numerical models are developed on the CFD-software FLOW-3D.

By testing different vertical wall sizes, the optimal size of the vertical wall is determined for B-TSW weir.

## INTRODUCTION

*et al.*2016). Based on their plan shape, weirs are classified as linear and nonlinear ones. The nonlinear weirs, such as curved weirs in the plan and the Labyrinth weirs, increase the overflow capacity of the structure by increasing the flow passage length (L

_{t}). The labyrinth weirs consist of zigzag sequences of the linear weirs. Despite the lower discharge coefficient (C

_{d}) of the nonlinear weirs compared to the linear one, according to Equation (1), in the nonlinear weirs the total length of the weir multiplied by the discharge coefficient (C

_{d}*L

_{t}) is higher than the linear weirs. Consequently, their hydraulic performance is three to four times greater than that of the linear weirs (Tullis

*et al.*2007).where Q is weir flow discharge, C

_{d}, L

_{t}, and H

_{t}are the discharge coefficient, total crest length, and total head (sum of the velocity and hydrostatic heads) over the weir, respectively. Recently, some attempts have been made to use intelligent methods for the prediction of the discharge capacity of weirs (Zahiri

*et al.*2013; Parsaie

*et al.*2018; Ghasemlounia & Saghebian 2021).

*et al.*2019). Figure 1 shows the 3D view of the labyrinth and PK weirs along with the geometrical parameters. As can be seen from Figure 1, contrary to the labyrinth weirs, they have sloped inlet and outlet to downstream and upstream, respectively. Depending on the slope of the inlet and outlet keys, they may have upstream and downstream overhangs. In Figure 1, W

_{i}, W

_{o}, B

_{i}, and B

_{o}are the inlet key width, outlet key width, upstream overhang length, and downstream overhang length, respectively. In addition, B is the length of the side crest, P is the total height of the weir, B

_{b}, and T are the foundation width and wall thickness, respectively. The total crest length (L

_{t}) and the width of a cycle (W

_{u}) are defined as follows:

Anderson & Tullis (2012) compared the hydraulic performance of the labyrinth and rectangular PKWs. They showed that the PKW was more efficient than the geometrically identical rectangular labyrinth weir. The higher performance of the PKW was partly attributed to the reduction in entrance losses associated with the geometry of the PKW inlet key.

To improve the performance of the PKWs, some parametric studies are performed in the previous studies. Kabiri-Samani & Javaheri (2012) performed an experimental study to investigate the effect of dimensionless parameters on the discharge capacity of PKWs. Ribeiro *et al.* (2012) proposed a simplified equation to calculate the ratio of the PKW discharge to that of a linear sharp crested weir (r) for identical H_{t}. According to Ribeiro *et al.* (2012), the outlet key of the PKWs plays a significant role in high overflow conditions. Cicero *et al.* (2013) studied the trapezoidal pianokey weirs (TPKW). The discharge capacity of the trapezoidal PKW was about 5 to 25 percent higher than the conventional rectangular PKWs.

Results of previous studies show that 3D numerical models are a promising tool to simulate the detailed hydrodynamics and discharge capacity of PK weirs with an accuracy of about ±10% (Pralong *et al.* 2011b; Lefebvre *et al.* 2013). According to Paxson & Savage (2006), numerical simulation using the FLOW-3D model, for a given labyrinth geometry configuration, was shown to provide results equivalent to those obtained using the previous design methods (5% deviation). Cicero *et al.* (2013) performed numerical simulations of flow over PKW and TPKW using the FLOW-3D model. The numerical results were in rather good agreement with the physical model results. Bremer & Oertel (2017) used FLOW-3D software to study the effect of wall thickness on the discharge coefficient of the PKW. Pralong *et al.* (2011a) performed sensitivity analysis and proved that the turbulence model has no effect on the predicted discharge capacity of the PKW. Zahiri *et al.* (2014) performed numerical modelling of flow over compound sharp crested weirs. Safarzedeh & Noroozi (2017) compared the hydraulics and 3D ﬂow features of the rectangular and trapezoidal piano key weirs with the same developed crest lengths. They found that the higher efficiency of TPKWs is partly attributed to flow characteristics along the inlet keys: a three dimensional flow field over the solid and porous broad crested weirs.

## INTRODUCING THE T-SHAPED WEIR

Usually weirs work under free flow condition, but in some cases the submerged condition may also occur. Submergence of nonlinear weirs is classified into general and local conditions. General submergence refers to tail water rising above the weir crest caused by a downstream control (Crookston & Tullis 2012). The local submergence is attributed to the complex flow features over the weir under higher heads. When the inlet cycle discharge exceeds the free-flow capacity of the outlet cycle, the upstream part of the outlet cycle is drowned. This phenomenon results in locally elevated tail water for the approaching flow along the outlet cycles. By increasing the approach flow, nappes of the side crests collide. The nappes' collision results in rising water surface elevation along the outlet cycle above the weir crest. Consequently, the side flow into the outlet cycles over the side crests discharge under submerged condition. Submergence of the weir increases the upstream head over the weir and according to Equation (1) the discharge coefficient decreases (Safarzadeh & Noroozi 2017).

Anderson & Tullis (2011) compared the hydraulics of rectangular labyrinth weirs with sloping ramped floors (hereafter called ramps) and rectangular PKWs with the same horizontal shape. The released water from the upstream apex of the ramps of labyrinth and piano key weirs flows as a jet toward the bottom of the outlet key. This phenomenon reduces the local submergence in both weirs. However, the experimental results show that the PKW is more efficient than the labyrinth weir with ramps. This phenomenon is attributed to the effects of the PKW upstream key overhangs on reduction of the inlet flow contraction and energy loss associated with entering flow to the inlet keys.

Parapet walls are vertical extensions placed over the crest of a standard PKW. The use of a parapet wall has been shown to increase the discharge capacity of the PKW by increasing the height of the inlet keys (Ribeiro *et al.* 2009; Machiels *et al.* 2011; Pralong *et al.* 2011b). It should be noted that the higher efficiency of the PKW with parapet walls is achieved by increasing the total height of the weir up to an optimal height.

According to previous studies, each of the PK and labyrinth weirs has pros and cons. The aim of the present study was to use the advantages of both weirs to reduce the local submergence of the outlet key and achieve a weir with a higher discharge coefficient. For this purpose, a new nonlinear weir, which is in fact a geometrical combination of the labyrinth and the PK weirs was introduced. Figure 2 illustrates the concept of the T-shaped weir and its details. In the geometry of the proposed weir, by keeping the weir height (P) constant, the advantages of the PK and labyrinth weirs were used so that the width of foundation effect line of the proposed weir was considered the same as that of the PKW, and the inlet and outlet keys were designed as a combination of the keys of the PK and labyrinth weirs. The main specifications of the T-shaped weir are:

- 1.
Vertical walls at the upstream crests create short drops at the beginning of the outlet keys. This geometrical modification deepens the upstream part of the key and reduces the likelihood of the local submergence.

- 2.
Vertical walls at the end of the inlet keys increase the volume of the key and modify the flow pattern to control or eliminate the interference of side nappes along the outlet keys.

- 3.
The parapet wall increases the weir height to improve the performance of the PKW. While in the T-shaped weir, without changing the weir height, vertical walls are inserted to the upstream and downstream of the keys.

- 4.
The footprint of the T-shaped weir is similar to the PKW, but the bottom slope of the alveoli is milder.

The new proposed weir was named the T-shaped weir (TSW) because of its similarity to the letter T. According to Figure 3, based on the presence of the vertical walls at the inlet key, outlet key, or both keys, the T-shaped weirs are classified as type A, B, and C weirs, respectively. As an assumption, the A-type weir was called TWS.

Parameter . | Description . |
---|---|

P | Weir height |

P_{d} | Vertical wall height at inlet key |

P_{u} | Vertical wall height at outlet key |

L | Crest length (L = N(W_{i}+ W_{o} + 2B)) |

W | Total weir width |

W_{i} | Inlet key width |

W_{o} | Outlet key width |

B | Weir sidewall length |

B_{b} | Footprint length |

B_{i} | Inlet key overhang length |

B_{o} | Outlet key overhang length |

T | Wall thickness |

N | Cycle number |

Parameter . | Description . |
---|---|

P | Weir height |

P_{d} | Vertical wall height at inlet key |

P_{u} | Vertical wall height at outlet key |

L | Crest length (L = N(W_{i}+ W_{o} + 2B)) |

W | Total weir width |

W_{i} | Inlet key width |

W_{o} | Outlet key width |

B | Weir sidewall length |

B_{b} | Footprint length |

B_{i} | Inlet key overhang length |

B_{o} | Outlet key overhang length |

T | Wall thickness |

N | Cycle number |

## LABORATORY EQUIPMENT AND DESIGNING THE TESTS

Laboratory experiments were carried out in a straight, horizontal flume 10 m long, 1 m wide and 0.8 m high, with glass sides and steel bed (Figure 4). Water was supplied by means of a pump delivering up to 100 ls^{-1} into an upstream head tank. The flow rate was measured by an electromagnetic flow meter with an accuracy of ±1 ls^{-1}. Three successive perforated metal plates with synthetic membrane were installed along the head tank ensuring uniform approach ﬂow conditions. Surface waves at the flume entrance were eliminated by using a 1 m long polystyrene plate floating at the water surface downstream of the channel inlet. In order to test the weirs for high water heads, the width of the flume is gradually reduced to 0.5 m using specific convergent structures and parallel walls made of PVC plates (Figure 4).

A 20 cm high base was installed on the floor at a distance 3 m from the downstream end of the flume. The base prevents tail water effects on the outlet keys. Furthermore, it facilitates easy installation, leveling and removal of each tested weir. A long ramp with gentle slope was installed upstream of the base in order to provide good approach flow from the flume bed to the tested weirs. Two glass walls on both sides of the internal 0.5 m wide channel allow observation of the flow patterns at the location of the tested weirs. Water head measurements have been performed using a digital point gauge with a precision of ±0.01 mm, installed on a rolling carriage above the flume.

Weirs are made of 9 mm thickness white PVC sheets (Figure 5). The geometric characteristics of the tested weirs are listed in Table 2. Upon installation of the weirs on the flume and their complete sealing, the experiments were performed by measuring the developed heads on the weirs for upstream flow discharges. Each test for any specific upstream discharge was repeated three times and the discharge coefficient value was determined using Equation (1) for average head-discharge values.

Model . | P . | N . | L . | W . | W_{i}
. | W_{o}
. | B . | B_{i}
. | B_{o}
. | T . | P/ W_{u}
. | P_{d}/P
. | P_{u}/P
. |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

PKW | 16 | 5 | 260.8 | 50 | 8.3 | 6.6 | 15.4 | 10.3 | 10.3 | 0.9 | 0.96 | 0 | 0 |

B-TSW2/5 | 2/5 | 0 | |||||||||||

C-TSW1/3 | 0 | 1/3 | |||||||||||

C-TSW1/4 | 0 | 1/4 |

Model . | P . | N . | L . | W . | W_{i}
. | W_{o}
. | B . | B_{i}
. | B_{o}
. | T . | P/ W_{u}
. | P_{d}/P
. | P_{u}/P
. |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

PKW | 16 | 5 | 260.8 | 50 | 8.3 | 6.6 | 15.4 | 10.3 | 10.3 | 0.9 | 0.96 | 0 | 0 |

B-TSW2/5 | 2/5 | 0 | |||||||||||

C-TSW1/3 | 0 | 1/3 | |||||||||||

C-TSW1/4 | 0 | 1/4 |

### Governing equations and numerical package

*et al.*2020):Where

*ρ*is the water density, is the fractional volume open to flow in the FAVOR method, and

*A*represents the fractional area open to flow in the

_{i}*i*th direction. In momentum conservation equations,

*p*shows pressure, while () and () represent body and viscous accelerations for the

*i*th direction, respectively. The viscosity-induced acceleration involves both the molecular () and turbulent eddy ( viscosities. Turbulence models are used to express the turbulent viscosity in terms of the mean flow variables.

The renormalized group theory models (RNG based k-*ɛ* model) was used to model the Reynolds stresses and turbulence closure. This model is known to describe flows having strong shear regions more accurately, and consequently it has wider applicability than the standard k-*ɛ* model (Safarzadeh & Mohajeri 2018).

The Fractional Area/Volume Obstacle Representation (FAVOR) method is used to inspect the geometry in the finite volume mesh. FAVOR appoints the obstacles in a calculation cell with a fractional value in the range of 0 to 1 as obstacle fills in the cell. Fluid surface shape is illustrated by the volume-of-fluid (VOF) method. This method is more flexible and efficient than other methods for treating complicated free surface flows. The pressure and velocity are coupled implicitly by using the time-advanced pressures and time-advanced velocities in the momentum and continuity equations, respectively. FLOW-3D solves these semi-implicit equations iteratively using relaxation techniques. In this paper, the generalized minimum residual method (GMRES) technique has been used as a pressure implicit solver. In the following section, more detail is given about the spatial discretization of the computational domain, boundary conditions and convergence criteria.

## EXPERIMENTAL RESULTS

Figure 6 shows the variations of discharge per unit width (q = Q/W_{u}) of the PKW against the head over the weir. According to Figure 6, the discharge per unit width increases linearly with the head over the weir. The results are in good agreement with experimental data reported by Machiels *et al.* (2014) for P/W_{u} = 1.

Figure 7 shows the 3D water surface shape over the conventional piano key weir for different approach discharges. Development of local submergence areas at the beginning and downstream of the outlet key were apparent. These zones are shown by eccentric contour lines inside the outlet key. By increasing the discharge, the interference of nappe flow from the side crests was intensified (Figure 7(b)), and the two submergence areas were connected. Therefore, depending on the discharge, the outlet key was filled up to a level lower than the crest. In this case, due to the high velocity flow from the upstream crest into the outlet key, a cavity was formed at the beginning of the outlet key. Further increase in the upstream discharge intensified the side nappe interference in the downstream part of the outlet key, and the water level increased to a level above the weir crest level. In this case, a static wave was formed inside the outlet key, and a significant part of the key was fully submerged. The experimental observations showed that with further increase in the discharge, the static wave at the end of the outlet key turned into a hydraulic jump. Finally, for very high discharges as well as for H_{t}/*P* > 0.6, the whole volume of the keys was filled and the PKW practically acts as a broad-crested weir.

Figure 8 shows a comparison of the discharge coefficients of the PKW and C-TSW. By adding the upstream vertical wall with a height of one fourth of the total weir height (C-TSW1/4 weir), for low heads, the discharge coefficient slightly increased in comparison to the conventional PKW. For higher heads, discharge coefficient data of C-TSW1/4 and PKW are consistent.

By increasing the height of the upstream vertical wall of the outlet key (C-TSW1/3), the discharge coefficient for low heads decreased in comparison to that of the C-TSW1/4. At higher heads, the same discharge coefficient curves were obtained for all three weirs.

Vertical walls at the upstream crests create short drops at the beginning of the outlet keys. This geometrical modification deepens the upstream part of the key and reduces the likelihood of local submergence. However, the vertical wall may have negative effects on the weir performance in two ways. First, the vertical wall acts as an obstacle against the flow approaching the outlet key, which causes higher energy loss than the PKW. Second, the vertical wall reduces the bed slope of the outlet key ramp in C-TSW and decreases the discharge capacity of the key in comparison to the PKW.

In the C-TSW1/4, the positive effect of the vertical wall on increasing the depth of the upstream part of the key was more than the negative effects mentioned above. It should be noted that, with a further increase in the height of the vertical wall, the negative effects overcome the positive effects. In the next sections, these points are investigated in more details using the results of numerical simulation.

In Figure 9, the discharge coefficients of the conventional PKW and the B-TSW2/5 are compared. Unlike the C-TSW, inserting the vertical wall downstream of the inlet key significantly improved the hydraulic performance of the weir. In this figure, the improvement percentage in the discharge coefficient is also shown with a bar graph. The maximum difference in the discharge coefficients was 16% for H_{t}/*P* = 0.2 and up to H_{t}/*p* = 0.45, the effect of the vertical wall on increasing the performance of the B-TSW was maintained. For H_{t}/*p* >0.45, the discharge coefficient curves of the PKW and B-TSW2/5 coincide.

### Details of numerical simulations

Crookston *et al.* (2017) performed numerical simulation of flow over the PKW using Flow-3D software. Results showed that modeling of a single unit key of the weir considering the symmetry boundary condition at transverse faces of the computational domain yields results very similar to the experimental ones. In the present study, 3D flow over the tested weirs was simulated using the Flow-3D model for H/*P* = 0.4. As shown in Figure 10, the numerical model has a single unit key composed of an outlet key along with two half-key on both sides. A calculation block with Cartesian meshing is used to discretize the computational domain. AutoCAD^{®} software was used to create the geometry of the models by inserting an STL (stereolithography) file. The computational domain, including boundary conditions, is shown in Figure 10. In the Xmin plane, the boundary condition of stagnation pressure was applied. With this algorithm, Flow-3D is able to model various flow heights beginning at a stagnation pressure state corresponding to various H/P values. The Symmetry boundary conditions were used for the Zmax, Xmax,Ymin and Ymax planes, indicating that identical flows occur on the other side of the boundary. The Zmin boundary, which is located adequately far from the weir location, was considered the outflow boundary. The computed steady state flow rate at this boundary equals the overflow capacity of the weir. The Zmin plane was placed at a sufficient distance below the bottom of the reservoir. Consequently, the jet-like flow of the outlet keys can pass freely through this boundary, with no submergence. It should be mentioned that due to the presence of the bed solid component, the FAVOR algorithm considers a part of the Zmin plane, located in the reservoir, as a non-penetrative boundary. The Xmax plane is located far from the weir and it would not affect the weir flow. As an initial condition, a static water column with a volume of fraction equal to one (VOF = 1) was considered inside the field so that it was extended longitudinally from the upper boundary of the block to the weir downstream and the initial water level was equal to the water level applied at the input boundary and considered as the boundary condition. The static water column covers the entire width of the calculation block.

The domain was discretized using one uniform mesh block. The mesh independency test, conducted using the grid convergence index (GCI) method (Roache 1994), suggested that the optimum number of grids are (282 × 31 × 113) in the x, y, and z directions, respectively.

## MODEL VERIFICATION

To assure the numerical convergence of simulations, the simulation time is set to 14 seconds. Also, for controlling the satisfaction of the continuity equation, time series of the inlet discharge is compared to the outflow discharge from the Z_{min} boundary, downstream of the weir. Analyses showed that a 10-second period is sufficient for model convergence (Figure 11).

The upstream boundary distance from the weir was 2 m and temporal variations of the water surface level at various stations downstream of the inlet boundary showed that this length was sufficient for an undisturbed approach flow to be established. Simulation of 14 seconds of transient outflow for any PK weir requires 10 hours of calculation time using an 8-core 4.2 GHz computer.

For validation, the computed longitudinal water surface profile along the center line of the inlet key of PKW is compared with the experimental data for H/*P* = 0.4. In Figure 12, the profile resulting from the numerical simulation perfectly fit with the experimental data. Furthermore, simulated discharges (Q) for different water heads over the tested weirs (H) were used to calculate the discharge coefficients using Equation (1). In Figure 13, discharge coefficients obtained from numerical simulation are compared with experimental ones.

_{d}.

_{Exp}) and the numerical discharge coefficient, (C

_{d}.

_{Num}) were used and the values of 0.75, 2.98, and 0.016 were obtained, respectively. According to the obtained results and considering Figure 13, it can be seen that there is a good agreement between the numerical results and the experimental data. Safarzadeh

*et al.*(2017) used the same statistical measures to evaluate the goodness of different soft computing methods in prediction of flow around a single groyne.

## RESULTS OF NUMERICAL SIMULATIONS

Simulated flow fields are visualized in terms of the general flow pattern and streamlines passing through different parts of the weirs, using Tecplot software. Figure 14 shows the 3D general flow pattern for H/*P* = 0.4 case in three different weir types. In PKW and C-TSW1/4 models, much of the nappes cascading from the side crests are colliding at the downstream part of the outlet key and the water surface rises higher than the crest level. The colliding area in PKW is longer than the C-TSW1/4. But in the B-TSW2/5 model, the adjacent nappes do not collide and the flow disturbance at the downstream of the outlet key is much lower than the two other weirs. In this weir, the nappes cascading from the side crests cling to the side walls. The nappes impinge over the slope ramp of the outlet key, at a level sufficiently lower than the crest elevation. As a result of the B-TSW2/5 model the discharge capacity of the outlet key is higher than the two other weirs.

This phenomena is attributed to the layout of the weir. The downstream vertical wall in B-TSW reduces the velocity along the inlet key and more uniformly distributes the flow over the side crest. In the following section, these flow features will be discussed in more detail.

For a more accurate study, flow streamlines passing the vertical plates near the sidewalls are shown in Figure 15. As can be seen from Figure 15, nappe flows from the upstream crest into the outlet key (Lj) with different lengths were formed in three different models so that in the C-TSW, a shorter nappe flow was formed in comparison to two other weirs. In the B-TSW, due to the lack of outlet key submergence, the flow passing the upstream crest is more easily discharged into the discharge key in comparison to the PKW. In the C-TSW, the upstream vertical wall acts as a sharp-crested weir and while deviating the flow upward, causes flow to fall from the upstream crest vertically into the outlet key. In addition, because of the low elevation of the outlet key bed, no local submergence occurred upstream and the flow passed over the upstream crest more easily and discharged faster into the outlet key in comparison to two other weirs, due to the two above-mentioned reasons. This in turn relatively improved the hydraulic performance of the C-TSW1/4 in comparison to the PKW.

Flow study downstream of the inlet keys of the tested weirs indicated a significant difference between the B-TSW and the two other weirs. In the PKW as well as the C-TSW, the flow over the downstream crests was in the form of a free jet, and as shown in Figure 15, while passing over the downstream crests, flow was also deviated toward the outlet key and caused the interference of the nappe jets from the side crests at the end of the outlet key. However, in the B-TSW, flow was discharged at the downstream crest in a clinging form and prevented nappe interfering at the end of the outlet key. This was evident from the streamlines upstream of the vertical wall at the end of the inlet key of the B-TSW, as shown in Figure 15. According to Figure 15, the presence of a vertical wall at the end of the inlet key caused water surface rise at the end of the key, and the downstream crest acted as a sharp-crested weir. Comparison of the streamlines on the side crests of the PKW and B-TSW showed that better flow passage over the downstream crest of the B-TSW, as well as the lack of local submergence of the outlet key of the weir, resulted in better distribution and lower density of streamlines on the side crests.

Because of better hydraulic performance of the B-TSW in comparison to other weirs, in order to determine the optimal drop height of the inlet key, numerical modeling was performed for H/*P* = 0.4 and different heights at (of) the inlet key. The discharge results versus drop height to weir height ratio (P_{d}/*P* = 0.4) are shown in Figure 16. As can be seen from Figure 16, flow discharge passing the weir increased by P_{d}/P and reached its maximum value at P_{d}/*P* = 2.5 and then decreased. This means that the use of a drop in the inlet key generally improved the hydraulic performance of the weir; however, as is evident from Figure 19, there is an optimal value of the drop relative to the weir height.

In order to accurately study the differences in the performance of the studied weirs, the baffle was used to determine the passing discharge from different sections of each weir crest in the single-key case. Figure 17 shows the location of one of the baffles on the weir crest. Generally, nine baffles were used on each weir crest and simulations were done for H/*P* = 0.4.

A flux surface is a diagnostic feature in the FLOW-3D numerical model for computing fluid flow rates. It can be used to obtain time-dependent information about the flow in different parts of the domain. A typical flux surface is a 100% porous baffle with no flow loss, so it does not affect flow in any way. This feature gives the opportunity to determine the specific discharge along the crest of the PKW and separate the percentage contribution of the inlet, outlet, and side crests to the discharge capacity of the weir. In order to do so, nine consecutive flux surfaces were placed along the crest of each weir and a steady mass flow rate was calculated for every baffle.

Figure 18 shows the discharge distribution per unit width of the crest length for half of the weir crest length using the results of the single-key models. As can be seen from Figure 18, in the half-part of the inlet crest, performance of the B-TSW2/5 was significantly better than for the other weirs, and the performance of the other weirs was almost the same. By moving on the initial 0.2 m of the side crest, the B-TSW2/5 still performed better; however, by getting closer to the center of the side crest, the discharge per unit width of the B-TSW2/5 decreased, and on the final 0.2 m of the side crest length as well as the whole crest of the outlet key, the discharge per unit width of all the weirs was the same. Figure 18 also showed that the weir with better performance at the intersection of the inlet key crest and the side crest, or in other words, the weir with a more uniform flow distribution at the mentioned intersection can pass greater discharge and has a better hydraulic performance.

To better analyze Figure 18, discharge percentage passing different parts of the single-key weirs’ crest was obtained by using the numerical modeling results, as listed in Table 3. As can be seen from Table 3, the discharge capacity of the weir and, therefore, its hydraulic performance increased by the discharge distribution percentage of the side crest and the inlet key crest. The results shown in Table 3 and Figure 18 indicate that the B-TSW2/5 had a better performance than the other weirs, both at the intersection of the inlet key crest and the side crest and the total discharge passing over the side crest and the inlet key crest.

Type . | Inlet key's crest . | Side crest . | Outlet key's crest . | Sum of the side crest and single key's crest . |
---|---|---|---|---|

PKW | 17.26% | 66.09% | 16.65% | 83.35% |

B-TSW1/8 | 16.99% | 66.64% | 16.37% | 83.63% |

B-TSW1/4 | 16.82% | 67% | 16.18% | 83.82% |

B-TSW1/3 | 16.67% | 67.24% | 16.09% | 83.91% |

B-TSW2/5 | 18.88% | 65.34% | 15.78% | 84.22% |

B-TSW1/2 | 16.42% | 67.64% | 15.94% | 84.06% |

B-TSW2/3 | 16.15% | 67.85 | 16% | 8 |

Type . | Inlet key's crest . | Side crest . | Outlet key's crest . | Sum of the side crest and single key's crest . |
---|---|---|---|---|

PKW | 17.26% | 66.09% | 16.65% | 83.35% |

B-TSW1/8 | 16.99% | 66.64% | 16.37% | 83.63% |

B-TSW1/4 | 16.82% | 67% | 16.18% | 83.82% |

B-TSW1/3 | 16.67% | 67.24% | 16.09% | 83.91% |

B-TSW2/5 | 18.88% | 65.34% | 15.78% | 84.22% |

B-TSW1/2 | 16.42% | 67.64% | 15.94% | 84.06% |

B-TSW2/3 | 16.15% | 67.85 | 16% | 8 |

To better analyze the results and to more accurately study the reasons behind the difference in the hydraulic performance of the studied weirs for different P_{d}/P ratios presented in Figure 16, the numerical modeling results for H/*P* = 0.4 were imported from the FLOW-3D numerical model to the Tecplot numerical model. The obtained 2D and 3D streamlines and 2D and 3D water surface profiles on different parts of the weirs are presented in Figures 19–22.

Figure 19 shows the water surface profile in the middle of the inlet key of the studied weirs. As can be seen from Figure 19, the water surface profile of B-TSW2/5 was in the plunging form, without interference and water surface rise, indicating its better performance. The water surface profile of other weirs was formed as two separate profiles at the beginning and end of the outlet key and the middle of the outlet key, which is located near the middle of the side crest and plays an important role in flow discharge from the outlet key, played an insignificant role in the flow discharge. This in turn caused increase the outlet flow, and nappe flow interference at the beginning and end of the side crest and led to the local submergence of the end of the outlet key. The local submergence prevents easy flow discharge and, therefore, can decrease the weir's performance and discharge coefficient.

To more accurately study the flow discharge from the outlet key, the water surface profile of flow approaching the weirs in different sections is shown in Figure 20. The results showed that the beginning parts of the outlet key of all the studied weirs had a similar performance. By approaching the end part of the outlet key, the performance of the B-TSW2/5 improved because of less nappe interference and maximum use of the outlet key volume. As shown in Figure 21, the less the angle of outlet flow from the side crest relative to the vertical axis of the crest, the less the flow interference and therefore, the greater the discharge capacity of the weir would result. Water surface profiles on the weirs indicated that, in the B-TSW2/5, because of the flow energy loss due to the weir geometry, the flow discharged different parts of the outlet key in the clinging form, which led to the maximum usage of the outlet key volume for the flow discharging. Therefore, the interference of the nappes at the end of the outlet key decreased and consequently, the passing discharge and the hydraulic performance increased.

Figure 22 shows the streamlines at different layers for three types of studied weirs for different P_{d}/P. It should be noted that the streamlines started from the field entrance at various levels and, for comparison purposes, the same number of streamlines were released from the inlet in all models and at each elevation. Upon collision of the approaching streamlines in the lower layer to the vertical wall of the inlet key of the B-TSW2/5, laminar flow was formed with lower outlet velocity and therefore, flow discharged the crest edge in the clinging form. Due to a smaller drop height in the inlet key of the B-TSW1/8, the flow discharged the vertical crest edge with a higher velocity, which led to flow interference and local submergence at the beginning of the outlet key. With further increase in the drop height, flow velocity and energy decreased by colliding with the vertical part of the inlet key. Therefore, a part of the flow was deviated to the side crest and discharged and, consequently, local submergence occurred and the weir performance decreased. At the middle layer, there was a significant difference between the motion pattern of the water particles around the studied weirs.

In all three types of B-TSWs, the flow concentration on the side crest decreased and the flow discharged uniformly from different parts of the side crest and vertical edge of the inlet key.

However, in the B-TSW2/5, the mid-layer streamlines’ distribution on different parts of the weir crest was more uniform than the other two former weirs. Comparison of the side crest lengths of the studied weirs having no streamlines confirmed the improved performance of the B-TSW2/5. In the mid-layer, similar to the lower layer, the velocity reduction and laminar streamlines in the B-TSW2/5 were evident in comparison to the other two weirs, which in turn led to a reduced streamline interference in the inlet key and, consequently, local submergence in the outlet key of the B-TSW2/5 reduced in comparison to the other two weirs. This is clear in the mid-layer plan. Comparison of the streamlines at different heights of the studied weirs showed that formation of a drop in the input key acted as a sharp-crested weir and up to an optimal height (2/5 in the present study) led to a more uniform distribution of streamlines and by further increase in the drop height, the weir performance decreased.

## CONCLUSIONS

In the present study, a new weir called the T-shaped weir (TSW), which is a combination of the piano key weir (PKW) and labyrinth weir, was designed to reduce the local submergence in the outlet key and consequently improve the performance of the available weirs. The physical models of the PKW and TSW were made and some experimental tests were performed for a wide range of hydraulic conditions. By using the numerical simulation and experimental modeling, the flow over the weirs was studied and after describing the general flow structure, the discharge coefficient curves were extracted and analyzed. Based on the performed analyses, the following results were obtained:

- 1.
By inserting a vertical wall at the upstream of the outlet key (C-TSW), the negative and positive factors affected the hydraulic performance of the weir simultaneously, and practically, in this case, the discharge coefficient of the weir had an insignificant improvement in comparison to the conventional PKW.

- 2.
Due to the better flow distribution over the side crests and downstream crest of the B-TSW, the interference of nappes in the outlet key greatly decreased and local submergence was prevented. This modified the flow pattern and increased the discharge coefficient of the B-TSW in comparison to the conventional PKW.

- 3.
The optimal vertical wall height to weir height ratio (P

_{d}/P) in the B-TSW model was determined as 0.4. - 4.
Due to the more uniform flow distribution over the inlet key crest and side crest and also the flow energy loss due to the weir geometry, flow in the B-TSW2/5 discharged in the form of a clinging nappe, which led to the maximum usage of the outlet key volume for flow discharge, and consequently, the interference of the nappes at the end of the outlet key decreased, which increased the passing discharge and improved its hydraulic performance.

## CONFLICT OF INTEREST

The authors declare that they have no conflict of interest.

## DATA AVAILABILITY STATEMENT

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

Labyrinth and Piano Key Weirs III – PKW 2017. CRC Press, London, UK, pp. 101–108