^{1}

^{1}

Due to great velocity gradients among the outgoing flow, it is much common to form large-scale reverse flow with oblique movements outside the conventional separated stilling basin. Aimed at above problems, this paper proposes to remove the longitudinal splitter wall and then physically and numerically investigate the corresponding influence upon the compound stilling basin. The standard k-

As an effective method of energy dissipation, a hydraulic jump is widely applied among the domain of low water head projects, such as ogee weirs and sluice gates [

Recently, a new kind of energy dissipation concept (three-dimensional (3D) hydraulic jump) has been proposed and applied into low water head domain. Compared to a classic hydraulic jump, the 3D hydraulic jump would not only maintain the traditional transverse vortex of two-dimensional (2D) hydraulic jump, but also force the discharging flow to laterally diffuse to promote mutual impingement, even resulting in vertical vortex with large-scale lateral movements. The investigations about flaring gate pier, expanding hydraulic jump, and symmetric water jets have attained some progress and shown great energy dissipation effect [

Kordi and Abustan [

It can be noticed from the above statements that the expanding hydraulic jump can significantly increase the energy loss and decrease down postdepth. But the precedent investigations are mainly based on the expanding stilling basin with expanding cross section. Such arrangement is much difficult to achieve due to the increased amount of engineering excavation, especially for low head spillways among the alpine valley areas. Almost none of the academic researchers have utilized the conventional arrangement of flood releasing structure in a typical low head projects. In fact, it is quite common that a bottom outlet is closely arranged next to a surface outlet among the typical low head flood releasing structure. Thus, due to structure differences, the bottom outlet and the surface outlet may jointly operate to generate quite different inflows. If such feature can be utilized, a 3D hydraulic jump with much lateral diffusion would also be formed. However, a longitudinal splitter wall would be arranged to divide the downstream region as a bottom outlet stilling basin and a surface outlet stilling basin in the traditional arrangement of stilling basins. Such arrangement is intended to avoid mutual interferences when both outlets are jointly running.

Aimed at the above arrangement, this paper proposes to remove the downstream splitter wall to form a compound stilling basin; thus both outlets can jointly share the downstream stilling basin. In view of the enlarged cross section, the discharging inflow would greatly diffuse to form large-scale reverse flow in the compound stilling basin, consequently enhancing mutual impingement and shear friction among the flow. In the present paper, a physical model is particularly arranged to investigate the influence of splitter wall upon the downstream stilling basin. Furthermore, as a complement, the numerical simulation by employing the standard k-

Based on a common practical engineering case, the Lotus Temple Hydropower Station in the Gourd River region, Gansu Province, China, is physically modeled at the Hydraulic Model Test Hall of Wuhan University at a geometrical scale of 1:25. As shown in Figure

Plane layout

Surface outlet profile (Section 2-2)

Bottom outlet profile (Section #1-1)

In the original design, a splitter wall of 12.5m×2m (depth × width) is longitudinally arranged in the downstream region, thus dividing the downstream region as the bottom outlet stilling basin and the surface outlet stilling basin (shown in Figure

Like some other hydraulic projects, the main purpose for this hydropower station is to generate electricity and irrigate. Thus, to ensure the normal operation for turbines, the upstream water level should be maintained as stable as 1147m (the designed water level). Such requirement is quite common among the practical engineering cases. But due to the randomly varying reservoir inflows, both kinds of outlets may be forced to flexibly operate to meet that requirement in the long time. Thus, as shown in Table

Running condition for the hydropower station.

Condition | Reservoir inflow (m^{3}^{−1}) | Upstream water level (m) | Description |
---|---|---|---|

1 | 215 | 1147 | Open the bottom outlet and close the surface outlet |

2 | 226 | 1147 | Close the bottom outlet and open the surface outlet |

3 | 441 | 1147 | Both open the surface and bottom outlets |

The whole physical model, about 14m×4m (length × width), mainly covers the upstream reservoir region, the dam body-stilling basin region, and the downstream river bed region. A 50cm diameter pipeline is arranged to pump water from an underground tank into the upstream reservoir. A valve and an intelligent electromagnetic flow meter (Jiangsu Runyi Instrument Co., ltd, accuracy to be ±0.1L/s) are both arranged along the pipeline to adjust and record the reservoir inflows. To control the downstream tail water, a tailgate is arranged at the downstream region. In addition, the upstream topography and downstream beaches are all modeled with evenly arranged sectional panels. For clear observation, the dam body is made up of the plexiglass to conform to the prototype curve. Considering the difference between adjacent outlets, a point gauge with an accuracy of ±0.1mm is longitudinally arranged along Sections 1-1 and 2-2.

In addition, to validate the influence of splitter wall upon the downstream stilling basin, two different schemes are particularly arranged to investigate mutual differences. Scheme 1 is the original design plan and can be called the separated stilling basin (as shown in Figure

The commercial software Flow 3D is adopted here to simulate the 3D flow field among the numerical model. Compared to other similar computational fluid dynamics (CFD) software, the Flow 3D is much more convenient for gird generation due to the unique Fractional Area-Volume Obstacle Representation (Favor) technology. The continuity equation and Reynolds Average Navier Stokes (RANS) equations are applied here to numerically model the present case. For an incompressible, Newtonian fluid flow, these equations can be expressed as

where_{i} are the velocity components in the_{i} directions in the Cartesian coordinate,_{i} and_{F} respectively represent the fraction area and fraction volume open to the flow in the_{i} directions; t is the time;_{i} are the gravitational force in the corresponding_{i} directions;_{i} stand for Reynolds stress that can be calculated by a turbulence closure model.

With the development of CFD technology, a variety of turbulence closure models have been proposed and applied into the numerical simulation. In the present study, the standard k-

The SKE is probably one of the most widely applied two-equation turbulence closure models because of its simplicity and shorter computation time. Assuming the flow to be fully turbulent, the SKE model ignores the influence of molecular viscosity. In addition, it assumes the Reynolds stress should be proportional to the time-average strain at a local point and overlooks the historical effect in the streamwise direction. Thus, the SKE model is only suitable for fully turbulent flow and shows poor accuracies in the separation flow or large curvature flow. Owing to the development of computation fluid dynamic (CFD), many academic researchers have achieved great progress about the model turbulence. Taking the turbulent vortex into account, the RNG model accurately deduces each coefficient of turbulent energy equation (k equation) and turbulent dissipation rating equation (

The computational domain should be as similar as possible to the physical model. In consideration of the tiny influence of the retaining dam on both sides upon the discharging flow, the numerical model is slightly simplified for convenience of mesh generalization. As shown in Figure

Schematic for 3D solid model

Solid model with mesh

The 3D model is subdivided as three different blocks in view of velocity gradients in each region, among which Blocks 1, 2, and 3 successively represent the upstream reservoir region, the dam body-stilling basin region, and the downstream river bed region. In view of similar geometric sizes in all directions, the unique gird generation technology of Fractional Area-Volume Obstacle Representation (FAVOR) is adopted to define the numerical model with hexahedral cells. Due to the sharp variations of hydraulic parameters in the stilling basin, Block 2 is compacted with much finer grids (as shown in Figure

The upstream boundary is located at 50m upstream the dam axis to make sure there is a fully developed flow into the reservoir. Correspondingly, the downstream boundary is located at 100m downstream the stilling basin to reserve enough room to adjust the outgoing flow. In consideration of the approach velocity into the reservoir, the upstream boundary condition (

Usually, all the above mentioned turbulence models are quite accurate and valid only to fully turbulent flows. But near the wall, the flow is almost laminar and the turbulent stress hardly works, especially for the viscous sublayer region [^{+}. y^{+}=yv^{+} < 200~400) and then relates the viscous layer to the log law region with empirical formulas. Compared with the near wall modeling method, the wall function method does not need to specifically compact the grid near the wall but saves much more computation time with higher efficiency [

In view of the unique grid generation technology in the Flow 3D software, as shown in Figure ^{+} in all grid systems ranges from 46.71 to 109.15 and confirms the requirement of the wall function method. The Reynolds number among the discharging flow in the stilling basin reaches 2.4~4.8E+08 and the hydraulic diameter ranges between 17.9 and 29.9m in all conditions.

Grid details in each block.

Block | Grid 1 | Grid 2 | Grid 3 | Near wall distance (m) | dimensionless distance y+ |
---|---|---|---|---|---|

Grid size (m) | Grid size (m) | Grid size (m) | |||

Upstream reservoir block | 0.8×0.8×0.8 | 0.8×0.8×0.8 | 0.8×0.8×0.8 | 0.1 | 46.71 |

Dam body-stilling basin block | 0.3×0.3×0.3 | 0.25×0.25×0.25 | 0.2×0.2×0.2 | 0.1 | 94.63 |

downstream river bed block | 0.5×0.5×0.5 | 0.5×0.5×0.5 | 0.5×0.5×0.5 | 0.1 | 109.15 |

There are two main methods to trace the free surface among the numerical software, one is the Marker and Cell (MAC) method and the other is the volume of fluid (VOF) method. The MAC method mainly utilizes the uniformly distributed marking points to trace the movements inside the control volume. But the calculation accuracy of such method is strongly dependent on the density of marking points. Thus, such method is more suitable for two-dimensional model in view of the huge computational workload in the 3D numerical model [

The VOF method utilizes a filling process to explain the compositions inside each control volume. Assuming that both kinds of phases (water and air) would not interpenetrate into each other, a volume fraction (F) is adopted to represent how much percentage the computational cell is occupied by the phase. Compared to the MAC method, the VOF method is much more applicable for 3D complicated flow field in view of small computation memory [

where u, v, and w are the fluid velocity components in the x, y, and z directions.

The generalized geometric reconstruction for unstructured grids is one of the most popular schemes to capture the free surface among the VOF applications. During the calculation process, the first step is to deduce the position of the linear interface between the air and the water on the basis of information about the volume fraction F and its derivatives. Then the second step is to determine the advecting amount of water through each face based on the normal and tangential velocity distribution on the cell. The last step is to determine the volume fraction in each cell with the balance of mass fluxes calculated in previous step [

The finite-volume method is used to solve the RANS equations with a single-fluid approach [

To accelerate the numerical simulation, some water is initially arranged inside the blocks to reduce reciprocating perturbations (hot start). The total simulation process is performed on a PC with four parallel 3.20 GHz i5 processors. Stamou et al. [

A grid convergence study is performed to validate the influence of grid sizes upon the numerical results. Following the procedure of Roache [

where _{2} and_{3} are the computed velocities from the medium and fine grid solutions;

where e represents the differences between different grids;

In the current paper, the numerical results about the stilling basin (Block 2) are the most concerned. As shown in Table _{1}, Δh_{2}, and Δh_{3}) are 0.3m, 0.25m, and 0.20m, respectively. The computed velocities are adopted along the lateral section (Y=31m, Y=64m, and Y=95m) among horizontal section (Z=1127m) to validate the numerical uncertainties, among which each lateral section is averagely divided into 9 points.

Taking the velocity results from the RNG model in Condition 3, for example, the discretization error estimations for different grid systems are, respectively, shown in Tables

Discretization error estimation in measured point (x,y,z)=(10.5, 31,1127) for RNG in Condition 3.

Parameters | Values |
---|---|

r_{12}, r_{23} | 1.20, 1.25 |

u_{1}, u_{2}, u_{3} | 8.47, 8.54, 8.59 |

P | 2.68 |

GCI | 0.89% |

GCI distribution among each section for RNG in Condition 3.

Section | Number of points | | | | Maximum |
---|---|---|---|---|---|

Y=31m | 9 | 1.77 | 11.68 | 4.59 | 1.41% |

Y=64m | 9 | 3.8 | 8.05 | 5.41 | 0.66% |

Y=95m | 9 | 4.48 | 10.98 | 6.94 | 1.35% |

As an alternative method to simulate the flow field, the numerical results should be validated before being applied into the practical engineering cases. Thus, the discharge flow rates and computed velocities along partial sections are specially adopted herein to facilitate a mutual comparison with the experimental results. The discharge flow rates in each scheme are all revealed in Table

Discharge flow rates among each scheme.

Scheme | Condition | Physical | SKE | RNG | RKE | LES | ||||
---|---|---|---|---|---|---|---|---|---|---|

Q(m^{3} | Q(m^{3} | | Q(m^{3} | | Q(m^{3} | | Q(m^{3} | | ||

Separated stilling basin | 1 | 222.5 | 218.4 | 1.84 | 219.2 | 1.48 | 215.4 | 3.19 | 214.5 | 3.60 |

2 | 233.4 | 228.3 | 2.19 | 230.2 | 1.37 | 229.1 | 1.84 | 225.1 | 3.56 | |

3 | 452.6 | 443.5 | 2.01 | 449.1 | 0.77 | 440.5 | 2.67 | 435.9 | 3.69 | |

| ||||||||||

Compound stilling basin | 1 | 223.1 | 219.4 | 1.66 | 220.1 | 1.34 | 216.2 | 3.09 | 213.9 | 4.12 |

2 | 233.9 | 229.3 | 1.97 | 231.6 | 0.98 | 228.5 | 2.31 | 226.8 | 3.04 | |

3 | 453.2 | 445.1 | 1.79 | 450.9 | 0.51 | 441.2 | 2.65 | 437.1 | 3.55 |

Furthermore, referred to the discharging flow of bottom layer (0.5m above the slab) in the physical model, elevations about the computed velocities along the lateral sections of Y=10m, 20m, 30m, 40m, 50m, 60m, 70m, 80m, and 90m are performed to reveal the mean square error (MSE) and the mean absolute relative error (MARE). The MSE and MARE can be calculated with following equations.

where

MSE and MARE values in Condition 1 among the compound stilling basin scheme.

SKE | RNG | RKE | LES | |||||
---|---|---|---|---|---|---|---|---|

Y | MSE | MARE | MSE | MARE | MSE | MARE | MSE | MARE |

/m | /m^{2}^{2} | /% | /m^{2}^{2} | /% | /m^{2}^{2} | /% | /m^{2}^{2} | /% |

10 | 0.025 | 0.648 | 0.015 | 0.569 | 0.035 | 0.796 | 0.041 | 0.855 |

20 | 0.075 | 4.256 | 0.123 | 3.854 | 0.085 | 4.584 | 0.105 | 5.643 |

30 | 0.142 | 4.379 | 0.135 | 4.153 | 0.156 | 4.585 | 0.154 | 5.532 |

40 | 0.121 | 3.975 | 0.109 | 3.985 | 0.125 | 4.269 | 0.168 | 5.075 |

50 | 0.107 | 3.529 | 0.082 | 3.258 | 0.135 | 3.963 | 0.125 | 4.956 |

60 | 0.101 | 3.257 | 0.075 | 3.876 | 0.096 | 3.814 | 0.095 | 4.584 |

70 | 0.096 | 3.145 | 0.061 | 2.908 | 0.076 | 3.679 | 0.075 | 4.345 |

80 | 0.067 | 3.178 | 0.043 | 2.695 | 0.053 | 3.846 | 0.049 | 4.87 |

90 | 0.046 | 3.085 | 0.023 | 2.589 | 0.049 | 3.614 | 0.045 | 4.691 |

Mean | 0.087^{(2)} | 3.272^{(2)} | 0.074^{(1)} | 3.099^{(1)} | 0.090^{(3)} | 3.683^{(3)} | 0.095^{(4)} | 4.506^{(4)} |

MSE and MARE values in Condition 2 among the compound stilling basin scheme.

SKE | RNG | RKE | LES | |||||
---|---|---|---|---|---|---|---|---|

Y | MSE | MARE | MSE | MARE | MSE | MARE | MSE | MARE |

/m | /m^{2}^{2} | /% | /m^{2}^{2} | /% | /m^{2}^{2} | /% | /m^{2}^{2} | /% |

10 | 0.026 | 0.858 | 0.025 | 0.748 | 0.036 | 0.985 | 0.023 | 0.855 |

20 | 0.235 | 4.656 | 0.198 | 4.364 | 0.354 | 5.876 | 0.376 | 5.543 |

30 | 0.364 | 3.965 | 0.285 | 4.125 | 0.465 | 6.565 | 0.453 | 5.743 |

40 | 0.276 | 3.645 | 0.252 | 3.583 | 0.358 | 5.643 | 0.446 | 4.758 |

50 | 0.212 | 3.532 | 0.167 | 3.282 | 0.245 | 4.767 | 0.342 | 4.543 |

60 | 0.169 | 2.976 | 0.155 | 2.657 | 0.155 | 3.543 | 0.229 | 4.643 |

70 | 0.152 | 2.523 | 0.125 | 1.533 | 0.152 | 2.123 | 0.153 | 3.534 |

80 | 0.125 | 1.459 | 0.09 | 1.035 | 0.124 | 2.543 | 0.19 | 3.53 |

90 | 0.09 | 1.75 | 0.08 | 1.536 | 0.142 | 2.645 | 0.142 | 2.954 |

Mean | 0.183^{(2)} | 2.818^{(2)} | 0.153^{(1)} | 2.540^{(1)} | 0.226^{(3)} | 3.854^{(3)} | 0.262^{(4)} | 4.011^{(4)} |

MSE and MARE values in Condition 3 among the compound stilling basin scheme.

SKE | RNG | RKE | LES | |||||
---|---|---|---|---|---|---|---|---|

Y | MSE | MARE | MSE | MARE | MSE | MARE | MSE | MARE |

/m | /m^{2}^{2} | /% | /m^{2}^{2} | /% | /m^{2}^{2} | /% | /m^{2}^{2} | /% |

10 | 0.019 | 0.875 | 0.014 | 0.785 | 0.024 | 0.893 | 0.025 | 0.865 |

20 | 0.17 | 1.453 | 0.112 | 0.965 | 0.154 | 2.564 | 0.123 | 2.455 |

30 | 0.263 | 2.563 | 0.247 | 1.973 | 0.468 | 3.142 | 0.466 | 5.732 |

40 | 0.214 | 2.032 | 0.228 | 1.545 | 0.324 | 3.521 | 0.435 | 4.72 |

50 | 0.287 | 1.762 | 0.225 | 1.642 | 0.285 | 2.543 | 0.353 | 4.433 |

60 | 0.159 | 1.232 | 0.145 | 0.942 | 0.185 | 2.212 | 0.153 | 2.436 |

70 | 0.107 | 1.962 | 0.046 | 0.825 | 0.154 | 2.233 | 0.105 | 2.545 |

80 | 0.055 | 0.823 | 0.025 | 0.754 | 0.053 | 1.121 | 0.181 | 2.258 |

90 | 0.021 | 0.756 | 0.021 | 0.685 | 0.022 | 1.832 | 0.125 | 2.987 |

Mean | 0.144^{(2)} | 1.495^{(2)} | 0.118^{(1)} | 1.124^{(1)} | 0.185^{(3)} | 2.229^{(3)} | 0.218^{(4)} | 3.159^{(4)} |

Comprehensively observing Tables ^{2}^{2} and 5.732%, respectively. At the downstream region (Y=70~90m), both MSE and MARE values in all models gradually decrease almost to zero. All these results can prove the numerical simulation to be reliable.

Combined with Tables

The lateral velocity distributions among the horizontal section of upper layer (Z=1136m) are depicted in Figures

Velocity distribution in Condition 1.

Y=20m

Y=30m

Y=45m

Y=60m

Y=80m

Y=95m

Y=105m

Velocity distribution in Condition 2.

Y=20m

Y=30m

Y=45m

Y=60m

Y=80m

Y=95m

Y=105m

Velocity distribution in Condition 3.

Y=20m

Y=30m

Y=45m

Y=60m

Y=80m

Y=95m

Y=105m

Observing velocity distributions between two different schemes in detail, it can be noticed the velocity distributions in the dam body region (Y=20m) are almost the same between two schemes in all conditions, further indicating the little influence of splitter wall upon the dam body. But there exhibits great differences between both schemes in the downstream stilling basin. Due to the limited diffusion room inside the left part of the separated stilling basin (Scheme 1), the high-velocity discharging flow from the bottom outlet can only longitudinally move without any lateral diffusion, resulting in relatively hysteretic velocity decay in the upstream region of stilling basin. Thus, as shown in Figures

Details of maximum velocity among outgoing flow (Y=105m) in both schemes.

Condition | Separated stilling basin | Compound stilling basin | ||||
---|---|---|---|---|---|---|

Left | Right | Gradients | Left | Right | Gradients | |

1 | 6.45 | -1.16 | 7.61 | 1.20 | 0.73 | 0.47 |

2 | 0.03 | 2.31 | 2.28 | 1.27 | 2.01 | 0.74 |

3 | 6.59 | 1.81 | 4.78 | 2.09 | 3.17 | 1.08 |

But such situation of highly uneven velocity outgoing flow can be significantly improved in the compound stilling basin (Scheme 2). As shown in Figures

In general, compared to the separated stilling basin, high energy dissipation region in the compound stilling basin is significantly moved upstream in view of the removed splitter wall. As shown in Table

The flow pattern in the stilling basin presents great differences between both schemes. In view of the splitter wall, both adjacent stilling basins are quite independent in the separated stilling basin scheme. The pressurized jet from the bottom outlet in Condition 1 would not dive down but almost horizontally move, thus extending so far away. A repelled downstream hydraulic jump with fluctuating water can be clearly observed in the bottom outlet stilling basin, directly resulting in velocity decay to be hysteretic. In addition, in view of the suddenly expanding flow section outside the bottom outlet stilling basin, the high-velocity outgoing flow would greatly diffuse towards the right side and form large-scale reverse flow. Similarly, there can also observe lateral diffusion and local reverse flow outside the surface outlet stilling basin in Condition 2. There are only some slight improvements for the outgoing flow in Condition 3 but obliquely moving outgoing flow can still exist in view of the great velocity gradients (shown in Table

But due to the removed splitter wall, both kinds of outlets can share the downstream compound stilling basin. Thus, the pressurized jet from the bottom outlet in Condition 1 would greatly offset to form a typical 3D hydraulic jump with large-scale reverse flow in the right side of the compound stilling basin, significantly enhancing mutual impingement and shear friction among the discharging flow. Correspondingly, the discharging flow becomes much stable in the downstream region with much uniform lateral velocity distributions (as shown in Figure

To trace the movements of discharging flow inside the stilling basin, the longitudinal and horizontal sections are particularly adopted to reveal the velocity vectors. In view of the differences between each condition, the longitudinal sections would adopt Sections 1-1, 2-2 and 2-2 in all conditions and the horizontal section adopts the discharging flow of middle layer (Z=1132m). In addition, the RNG model are applied among the separated and compound stilling basin to figure out differences between both schemes and the computed velocity vectors from the SKE model are utilized to further verify the numerical accuracy.

As shown in Figure

Separated stilling basin in Condition 1 (Section 1-1)

Separated stilling basin in Condition 2(Section 2-2)

Separated stilling basin in Condition 3 (Section 2-2)

Compound stilling basin in Condition 1 (Section 1-1)

Compound stilling basin in Condition 2 (Section 2-2)

Compound stilling basin in Condition 3 (Section 2-2)

Compound stilling basin in Condition 1 (Section 1-1)

Compound stilling basin in Condition 2 (Section 2-2)

Compound stilling basin in Condition 3 (Section 2-2)

Separated stilling basin in Condition 1

Separated stilling basin in Condition 2

Separated stilling basin in Condition 3

Compound stilling basin in Condition 1

Compound stilling basin in Condition 2

Compound stilling basin in Condition 3

Compound stilling basin in Condition 1

Compound stilling basin in Condition 2

Compound stilling basin in Condition 3

However, the discharging inflow in the compound stilling basin presents great differences. In view of the removed splitter wall, both kinds of outlets can share the stilling basin, thus the lateral flow section in the upstream region is greatly enlarged. As shown in Figures

Comprehensively observing Figures

Hydraulic jump length and mainstream depth.

Condition | Separated stilling basin | Compound stilling basin | ||
---|---|---|---|---|

Experiment | Experiment | RNG model | Standard model | |

11 | 25-60(5.25) | 25-73(3.56) | 25-75(3.48) | 25-78(3.41) |

22 | 25-65(3.36) | 23-72(2.15) | 20-70(2.23) | 23-68(2.09) |

33 | 20-68(3.25) | 22-72(3.20) | 20-70(3.25) | 20-70(3.35) |

Notification: the values in parentheses are the mainstream depth above the stilling basin slab.

The turbulent kinetic energy (

To make a convenience for mutual comparison among both schemes, the colour bar among all sections is set as 0~10m^{2}/s^{2}. As shown in Figures ^{2}/s^{2}. But the flow inside the adjacent surface outlet stilling basin is almost motionless. Therefore, there exist great gradients among the outgoing flow and the maximum ^{2}/s^{2} to 2.4 m^{2}/s^{2} from the left to the right, which is quite corresponding to the situation of insufficient energy dissipation and obliquely moving outgoing flow. The similar situation of uneven outgoing flow can also be observed in Figure ^{2}/s^{2} to 2.1 m^{2}/s^{2} from the left to the right outgoing flow.

Longitudinal turbulent kinetic energy distribution (Unit: m^{2}/s^{2}).

Condition 1 for separated stilling basin scheme

Condition 2 for separated stilling basin scheme

Condition 3 for separated stilling basin scheme

Condition 1 for compound stilling basin scheme

Condition 2 for compound stilling basin scheme

Condition 3 for compound stilling basin scheme

Horizontal turbulent kinetic energy distribution (Unit: m^{2}/s^{2}).

Condition 1 for separated stilling basin scheme

Condition 2 for separated stilling basin scheme

Condition 3 for separated stilling basin scheme

Condition 1 for compound stilling basin scheme

Condition 2 for compound stilling basin scheme

Condition 3 for compound stilling basin scheme

But in view of the removed splitter wall in the compound stilling basin, discharging inflow in all conditions present great differences. Compared to the separated stilling basin, the maximum ^{2}/s^{2} and 6.5m^{2}/s^{2} in Condition 1 and Condition 2, but the sweep regions are significantly broadened to laterally cover the whole stilling basin (as shown in Figures

Summarizing the details between Figures ^{2}/s^{2}, 2.1 m^{2}/s^{2}, and 1.8 m^{2}/s^{2} in the compound stilling basin scheme in three conditions, which is successively 76.9%, 32.0%, and 70.7% lower than that of the separated stilling basin scheme. Gradients between the left and right outgoing flow almost decrease to zero in the compound stilling basin scheme, far lower than that of the separated stilling basin scheme.

Maximum turbulent kinetic energy among the outgoing flow in Y=105 section (unit: m^{2}/s^{2}).

Condition | Separated stilling basin | Compound stilling basin | ||||
---|---|---|---|---|---|---|

Left | Right | Gradients | Left | Right | Gradients | |

11 | 7.8 | 2.4 | 5.4 | 2.0 | 1.8 | 0.2 |

22 | 2.5 | 2.0 | 0.5 | 2.1 | 1.7 | 0.4 |

33 | 5.8 | 2.1 | 3.7 | 1.8 | 1.7 | 0.1 |

Figure

Longitudinal water level distribution.

Condition 1

Condition 2

Condition 3

The discharging inflow in Condition 1 gradually increase in the downstream region of the separated stilling basin and the highest water level even reaches 1139.44m, slightly higher than the splitter wall top (Z=1138.5m). The similar situation of increasing water level in the separated stilling basin can also be observed in Condition 2 and Condition 3 with maximum water level, respectively, to be 1137.57m and 1138.23m. But in view of the removed splitter wall, lateral cross section is significantly enlarged in the compound stilling basin. Thus, discharging inflow can greatly offset to decrease down the water level. As shown in Figures

In the practical engineering applications, the index of energy dissipation rate, representing total water head differences between upstream and downstream sections, is mainly adopted to weigh the actual energy dissipation effect among the flow in the stilling basin. Energy dissipation rates for each condition are shown in Table

Energy dissipation rates in each scheme.

Scheme | Condition | Inflow section | Outgoing flow section | Energy dissipation rate | ||||
---|---|---|---|---|---|---|---|---|

Potential energy/m | Average velocity/m^{−1} | Total head/m | Potential energy/m | Average velocity/m^{−1} | Total head/m | |||

Separated stilling basin | 1 | 17 | 0.73 | 17.03 | 8.92 | 4.85 | 10.12 | 0.41 |

2 | 17 | 0.87 | 17.04 | 7.2 | 2.07 | 7.42 | 0.56 | |

3 | 17 | 1.22 | 17.08 | 8.62 | 3.23 | 9.15 | 0.46 | |

Compound stilling basin | 1 | 17 | 0.75 | 17.03 | 7.59 | 1.74 | 7.74 | 0.55 |

2 | 17 | 0.9 | 17.04 | 7.15 | 1.45 | 7.26 | 0.57 | |

3 | 17 | 1.23 | 17.08 | 8.75 | 2.49 | 9.07 | 0.47 |

Experimental and numerical investigations are conducted to show the influence of splitter wall upon the downstream stilling basin. Among the computed results from four kinds of turbulence models, the renormalization group k-

In view of the removed splitter wall, discharging flow from the bottom outlet in Condition 1 (or surface outlet in Condition 2) would greatly diffuse to cover the whole compound stilling basin, thus resulting in large-scale reverse flow in the upstream region of the stilling basin. Similarly, in view of structure differences between both kinds of outlets, there would form velocity gradients among discharging inflow from adjacent outlets in Condition 3, thus inducing mutual impingement and shear friction between nearby flow. Due to the slightly moved upstream energy dissipation region in all conditions, the outgoing flow becomes much uniform in the compound stilling basin with the phenomenon of oblique movements being eliminated.

In general, compared to the separated stilling basin scheme, the hydraulic jump length in the compound stilling basin scheme reaches about 1.2-1.6 times longer; the mainstream depth decrease as low as 0.65; the maximum velocity and average velocity of the outgoing flow, respectively, drop more than 30% and 20%. In addition, the velocity gradients between the left and the right outgoing flow decreases over 65% with turbulent kinetic energy gradients almost down to zero.

The all data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This work was funded by National Natural Science Foundation of China (no. 51479145).