## Abstract

A water-retaining curtain (WRC) has become a useful facility in selective withdrawal and sedimentation control, but the force analysis of a curved curtain is still lacking. Based on flume experimental tests and numerical simulations, this paper analyzes the variation laws of pressure difference and thrust of WRC. The results show that under the uniform inflow condition, the distribution of pressure difference on the WRC is relatively even, and the maximum value is located at the upper part of the curtain. When arc length–height ratio increases, the location of maximum pressure difference gets lower. In addition, the variation law of thrust of WRC conforms to the classical resistance equation. The drag coefficient is found to fit a power function of the water-retaining ratio, a second-degree polynomial function of arc length–height ratio, and linear function of inclination ratio. The results also yield a simplified forecasting formula of thrust of WRC which is proposed and verified using flume simulations and a real reservoir model test. The newly developed formula systematically considers the water-retaining height, arc length and inclination degree, providing a rapid and accurate algorithm to predict the thrust, and lays a theoretical foundation for practical application.

## INTRODUCTION

To utilize the water resources, there has been a long history of hydraulic construction all over the world. Traditionally, the rigid hydraulic structures are mainly made of rigid materials such as concrete, steel, cement and rock (Gu & Ouyang 2011; Lin 2015). They have many advantages including high strength, good corrosion resistance, structure stability, and long life. With the development of materials strength and engineering demands, flexible structures have recently been adopted in some hydraulic projects, among which the water-retaining curtain (WRC) is frequently used (Vermeyen 2000; US Department of the Interior 2012). The WRC is made of impermeable flexible materials such as polypropylene fiber geotextile and membrane which has strong mechanical strength and anti-impact ability (Eloy *et al.* 2008; Wang 2010). The main uses of WRC in hydraulic engineering are in the selective withdrawal of reservoirs and sediment trapping, as described below.

To regulate the water temperature discharged from stratified reservoirs, rigid facilities of selective withdrawal are conventionally used including multi-level outlets, stop log gate, side-type orifice intake and surface pumps (Bradford 2000; Zhang & Gao 2010; Wu *et al.* 2013; Gao *et al.* 2014; Soleimani *et al.* 2016). As a flexible facility of selective withdrawal, the WRC is increasingly used to optimize the outflow temperature and water quality due to the advantages of being able to be constructed in an impounded reservoir, being convenient to be managed, and of much lower cost in comparison with a rigid structure (Asaeda *et al.* 1996, 2001; Vermeyen 1997; Bradford 2000; Bartholow *et al.* 2001). For example, WRCs were implemented to change the outflow temperature to meet the downstream ecological requirements in the Lewiston Reservoir, Holston River and Whiskeytown Reservoir (Boles 1985; Bohac 1989; Vermeyen 2000). Through a WRC flume experiment, Shammaa & Zhu (2010) developed practical relationships to predict the withdrawal composition and the interface height. Lian *et al.* (2016) showed that the outflow temperature was affected by the water-retaining height of a WRC and vertical temperature distribution in the reservoir.

Erosion control of river channels, estuaries and coastal beaches is an important issue all over the world. To solve this problem, many traditional anti-scouring structures have been used, such as artificial bars, seawalls, groins, beach nourishment, breakwaters and vanes (Li & Yu 2009). In recent years, the flexible WRC is proposed and validated in sediment trapping (Li & Yu 2009; Wang *et al.* 2015). The top of a WRC is connected with floaters and the bottom is fixed to the riverbed with strings and anchors, and the curtain is self-adapting to orient with the water flow. Li & Yu (2009) have experimentally analyzed the variation of a floating curtain's oblique angle to the dynamic flow. Wang *et al.* (2014) and Zhu *et al.* (2009) showed that the floating WRC could play a role in flexibility, suspension, movability and diversion, and kept itself stable in the water flow. In summary, a WRC is efficient, and economically and environmentally viable to be constructed and managed in sediment trapping compared with solid structures.

As is clearly shown from the above research and engineering experience, a WRC has shown its advantages including easy construction and modification, convenience in maintenance and low cost. To ensure the safe operation of a WRC, the structure stability of a WRC is noteworthy, and its thrust is a crucial index (Aydin & Demirel 2012; Gallegos *et al.* 2012; Bocchiola & Rosso 2014). A WRC is being designed in the Sanbanxi Reservoir of China to optimize the outflow temperature, and a force analysis of WRC is an imperative focus in the engineering process. Flow around a submerged body is a hot topic in fluid dynamics, and there has been much research on the hydrodynamic characteristics of circular cylinder, square cylinder, river bridge, and pipeline (Fiabane *et al.* 2011; Cheng *et al.* 2012; Turcotte *et al.* 2016; Lou *et al.* 2017). Focusing on a curtain, Wang *et al.* (2014) and Zhu *et al.* (2009) have conducted an experimental study on the tension and stability of a WRC, Wang *et al.* (2015) investigated the inclination angle and pull force of the floating curtain, but the curtain used in the research was rigid and straight rather than curved with only the inclination angle being adjustable. In total, previous research mainly focused on the effectiveness of a curtain in water resource management, and therefore further force analysis of a curved curtain is imperative.

Based on the flume experimental tests and numerical simulations, this paper aims to investigate the pressure difference on the curtain, obtain the change law of drag coefficient and thrust under different curtain forms, and provides technical support for the application. This paper is divided into five sections: the introduction; research methods, including experimental tests and numerical simulations; results wherein the flow field and pressure difference on the WRC are analyzed, and the change law of thrust and drag coefficient is investigated; discussions based on acquired results; and finally, conclusions drawn from the main findings of this research.

## METHODS

To quantify the thrust of WRC, the classical resistance equation of cross flow is followed as the theoretical basis for this study. Through experimental tests, the thrust of a WRC is to be obtained, and the numerical simulations can further extract the pressure differences on the curtain.

### Preliminary analysis of thrust of WRC

The WRC is subjected to a thrust caused by the water flow, and the thrust comes from the water pressure difference between the upstream and downstream of the curtain, as shown in Figure 1. Depending on a particular engineering application, the curtain can be either set at the top or bottom part of the cross section of a channel. This paper, however, focuses on a bottom-mounted WRC, and the thrust under a uniform inflow condition is studied, as shown in Figure 1. In addition, fresh water is considered in the study, but the effect of sediment on the thrust is ignored.

*et al.*2001; Munson

*et al.*2009; Ghadimi & Reisinezhad 2012; Jalonen & Järvelä 2014), where

*f*is the thrust per width (N m

^{−1});

*F*is the time-averaged thrust of WRC (N);

*B*is the flow width (m);

*v*is the inflow velocity (m s

_{in}^{−1});

*ρ*is the fluid density (kg m

^{−3});

*C*is the time-averaged dimensionless drag coefficient; and

_{d}*h*is the water-retaining height of the WRC (m).

The dimensionless drag coefficient is the key to determining the thrust of WRC, and is thought to be influenced by the curtain forms. Three main influencing factors are considered, which represent the water-retaining degree, curvature and inclination degree of a WRC, respectively, as follows:

Water-retaining ratio:

Arc length–height ratio:

Inclination ratio:

*H*is the water depth (m);

*l*is the arc length of curtain (m);

_{a}*l*is the inclination length of curtain (m), as illustrated in Figure 1.

_{i}### Experimental test

#### Model description

Experimental tests are adopted to study hydraulic theories (Jamali *et al.* 2005; Shammaa & Zhu 2010; Sun *et al.* 2016). In this study, a re-circulating rectangular water flume is built, as shown in Figure 2. The flume is 12.5 m long and 2 m wide, and the water depth is 1 m. The inflow velocity and forms of WRC are set according to the model scenarios, as presented in the ‘*Model scenarios*’ section below. The WRC used in the experiments is made of impermeable nylon fabric, and is fixed to the force sensors through fine strings and pulleys. As a flexible material, the curtain is curved in both vertical and lateral directions. Considering the topography of the reservoir, river and estuary, the lateral span of WRC is always much greater than the vertical size. Thus, this paper mainly studies the thrust on a vertically curved curtain. In the span direction, the overall curvature of curtain is kept small although the curvature in some edge areas may be large. The water-retaining height, arc length and inclination length of the curtain can be adjusted by using different forms of curtain. The schematic and arrangement of experimental test are shown in Figure 2.

#### Test instrument and method

The main instruments used in the experiment include acoustic Doppler velocimetry (ADV), force sensors, dynamic strain gauge, water level sensor system, flow meter and protractor. In the experimental tests, the inflow and outflow flow rates and water level are measured by flow meters and water level sensor system, respectively. The longitudinal water velocity along the channel is measured by ADV at multiple locations 0.05 m below the water surface, as shown in Figure 2(a).

There are six force-measuring points on the curtain comprising three bottom points and three top points, and they are connected with force sensors using fine lines and pulleys. Laterally, the force-measuring points are uniformly distributed to make the radiation small, as shown in Figure 2(b). The tensile forces of force-measuring points are measured by force sensors, the angles between fine lines and flow direction are measured through a protractor, and the thrust of WRC is calculated based on the tensile forces and their angles, as shown in Figure 2(a).

### Numerical simulations

#### Model description

The experimental tests can measure the flow field and thrust of WRC, and a numerical simulation based on the common *Fluent* (version 14.5) is set up to further extract the distribution of pressure difference across the curtain (Niedzwiedzka *et al.* 2017). The modeling area and boundary conditions of the numerical model are the same as for the experimental tests. The computational domain is 12.5 m long and 1.2 m high. The water depth is set to 1 m, and the gas-liquid multiphase flow is simulated in the model to ensure the accuracy of the pressure field. Unstructured quadrilateral meshes are generated and used. According to the magnitude of modeling scope (water: 25 × 1 m; air: 25 × 0.2 m) and research priority, considering the computation quantity and accuracy, the average mesh sizes of water and air are determined as 0.02 and 0.04 m, respectively. The numerical values of drag coefficients fit well with observed values in the physical model test, the MAREs are smaller than 5%, indicating that the mesh size is acceptable.

As a flexible body, the WRC is characterized as a large geometric deformation with micro-strain. Similar to the previous research, the experimental tests in this study have shown that the curtain presents a stable geometric deformation under steady flow and tension conditions (Cheng *et al.* 2014; Wang *et al.* 2014). In the numerical model, the WRC is presented with stable wall units, and the coupling effect between curtain and water flow is ignored. The material parameters of WRC are set based on the geotextile characteristics (Frost & Lee 2001; Shi & Li 2016). The bottom boundary of the water channel is set as a wall boundary, and the bottom material is set in line with the experimental test. The upstream and downstream water boundary conditions are set as velocity boundaries, and the air boundary conditions are set as pressure boundaries, as shown in Figure 3.

#### Governing equations

*t*is the time (s);

*x*and

_{i}*x*are the co-ordinates (m);

_{j}*u*and

_{i}*u*are the velocity components (m s

_{j}^{−1});

*P*is the fluid pressure (Pa);

*μ*is the dynamic viscosity coefficient (m

^{2}s

^{−1}).

*k*-

*ɛ*turbulence model is applied in the 2D model. It is based on the transport equations for turbulent kinetic energy (

*k*; m

^{2}s

^{−2}) and its dissipation rate (

*ɛ*; m

^{2}s

^{−3}), and is expressed as follows: where is the turbulent kinetic energy per unit mass,

*ɛ*is the turbulent energy dissipation rate per unit mass, is the kinematic turbulent viscosity, constant is 0.09, is the production of turbulent kinetic energy from the mean flow, and constants , , and are 1.44, 1.92, 1.0 and 1.3, respectively.

*a*∈ (0,1) is the volume fraction of water, and 1–

*a*represents the volume fraction of air. A cell is full of water (air) at

*a*= 1 (0). The mixture properties of fluid, such as density and viscosity, are determined by the volume fraction of water and air, i.e.

The finite volume method (FVM) forms the core of the numerical approach by discretization in the solution domain. FVM derives directly from the integral form of the conservation laws for fluid motion and possesses the conservation properties. The pressure-based unsteady implicit method is used in the model solution, and the Pressure Implicit with Splitting of Operators (PISO) scheme is employed in the pressure-velocity coupling. The computational time step is set as 0.0001 seconds. The model simulation time for each scenario is 1,800 seconds, and the CPU (central processing unit) run time is 40–80 hours depending on the flow intensity using a Core i7-3770 3.4 GHz desktop machine. When the changes of time-averaged thrust of WRC and water velocity above WRC are both less than 0.1%, the calculation is judged as convergence. Through numerical simulation, the flow field is simulated, and the time-averaged water velocity and pressure is extracted. Using water pressure upstream and downstream of the curtain, the pressure difference and thrust of the curtain are obtained.

### Model scenarios

Currently, a WRC is being designed in the Sanbanxi Reservoir of China to regulate the outflow temperature, and the force estimation of WRC is needed considering different curtain forms. The model scenarios of experimental and numerical simulations are set considering the engineering designation of WRC and hydrological data of Sanbanxi Reservoir (Lian *et al.* 2016; Powerchina ZhongNan Engineering Corporation Limited 2016). The base scenario focuses on the flood releasing at the design standard. In the prototype, the water depth upstream of the dam is approximately 150 m at the flood control level. The outflow discharge of the design standard is 9,600 m^{3} s^{−1}, and the corresponding average velocity of the selected cross section is 0.26 m s^{−1} (see Appendix, available with the online version of this paper). In the base scenario, the water-retaining proportion of curtain is 80%, and the arc length-height ratio of the curtain is 1.1. The model boundaries of base scenario are converted from the prototype data based on the gravity similarity theory (Munson *et al.* 2009; Lian *et al.* 2015), as shown in Table 1. As the water velocity in the prototype is small, the main factor influencing the thrust of the WRC is the average velocity instead of turbulence. Hence, the similarity of average velocity is primarily considered in the model setup, and the similarity of turbulence is currently ignored.

Water depth (m) | Inflow velocity (m s^{−1}) | Water-retaining ratio R _{h} | Arc length–height ratio R _{a} | Inclination ratio R _{i} | Bottom roughness | |
---|---|---|---|---|---|---|

Scale ratio | 1 | 1 | 1 | |||

Prototype | 150 | 0.26 | 0.8 | 1.1 | 0 | 0.035 |

Laboratory and numerical models | 1 | 0.02 | 0.8 | 1.1 | 0 | 0.015 |

Water depth (m) | Inflow velocity (m s^{−1}) | Water-retaining ratio R _{h} | Arc length–height ratio R _{a} | Inclination ratio R _{i} | Bottom roughness | |
---|---|---|---|---|---|---|

Scale ratio | 1 | 1 | 1 | |||

Prototype | 150 | 0.26 | 0.8 | 1.1 | 0 | 0.035 |

Laboratory and numerical models | 1 | 0.02 | 0.8 | 1.1 | 0 | 0.015 |

*Note:**λ _{L}λ_{v}* = the length and velocity scale of model, respectively.

In the base scenario, four boundary variables are considered, including inflow velocity *v _{in}*, water-retaining ratio , arc length–height ratio and inclination ratio . When one boundary variable changes, the others remain constant. The details of the model scenarios are summarized in Table 2. In the experimental tests and numerical simulations of model scenarios, the boundary conditions are constant, and the time-averaged flow velocity, water pressure, thrust of WRC can be obtained.

Number | Description | Inflow velocity v (m s_{in}^{−1}) | Water-retaining ratio R _{h} | Arc length–height ratio R _{a} | Inclination ratio R _{i} |
---|---|---|---|---|---|

A0 | Base scenario | 0.02 | 0.8 | 1.1 | 0 |

A1 | Varying inflow velocities v _{in} | 0.005 | 0.8 | 1.1 | 0 |

A2 | 0.0075 | 0.8 | 1.1 | 0 | |

A3 | 0.01 | 0.8 | 1.1 | 0 | |

A4 | 0.03 | 0.8 | 1.1 | 0 | |

A5 | 0.05 | 0.8 | 1.1 | 0 | |

A6 | Varying water-retaining ratios R _{h} | 0.02 | 0.9 | 1.1 | 0 |

A7 | 0.02 | 0.6 | 1.1 | 0 | |

A8 | 0.02 | 0.4 | 1.1 | 0 | |

A9 | Varying arc length–height ratios R _{a} | 0.02 | 0.8 | 1 | 0 |

A10 | 0.02 | 0.8 | 1.05 | 0 | |

A11 | 0.02 | 0.8 | 1.2 | 0 | |

A12 | 0.02 | 0.8 | 1.3 | 0 | |

A13 | Varying inclination ratios R _{i} | 0.02 | 0.8 | 1.1 | 0.25 |

A14 | 0.02 | 0.8 | 1.1 | 0.5 | |

A15 | 0.02 | 0.8 | 1.1 | 1 | |

A16 | 0.02 | 0.8 | 1.1 | 1.5 |

Number | Description | Inflow velocity v (m s_{in}^{−1}) | Water-retaining ratio R _{h} | Arc length–height ratio R _{a} | Inclination ratio R _{i} |
---|---|---|---|---|---|

A0 | Base scenario | 0.02 | 0.8 | 1.1 | 0 |

A1 | Varying inflow velocities v _{in} | 0.005 | 0.8 | 1.1 | 0 |

A2 | 0.0075 | 0.8 | 1.1 | 0 | |

A3 | 0.01 | 0.8 | 1.1 | 0 | |

A4 | 0.03 | 0.8 | 1.1 | 0 | |

A5 | 0.05 | 0.8 | 1.1 | 0 | |

A6 | Varying water-retaining ratios R _{h} | 0.02 | 0.9 | 1.1 | 0 |

A7 | 0.02 | 0.6 | 1.1 | 0 | |

A8 | 0.02 | 0.4 | 1.1 | 0 | |

A9 | Varying arc length–height ratios R _{a} | 0.02 | 0.8 | 1 | 0 |

A10 | 0.02 | 0.8 | 1.05 | 0 | |

A11 | 0.02 | 0.8 | 1.2 | 0 | |

A12 | 0.02 | 0.8 | 1.3 | 0 | |

A13 | Varying inclination ratios R _{i} | 0.02 | 0.8 | 1.1 | 0.25 |

A14 | 0.02 | 0.8 | 1.1 | 0.5 | |

A15 | 0.02 | 0.8 | 1.1 | 1 | |

A16 | 0.02 | 0.8 | 1.1 | 1.5 |

^{2}) are adopted, the former is used to determine the dispersion between measured and calculated values, and the latter is used to present goodness-of-fit equation, as presented in Equations (8) and (9). where

*n*is the total number of observations, is the

*i*th observed value,

*X*

_{i}^{cal}is the

*i*th calculated or fitted value,

*X*

^{mean}is the mean of observed values, SST is the sum of squares for total, SSR is the sum of squares for regression, and SSE is the sum of squares for error.

## RESULTS

Through experimental tests and numerical simulations, the hydrodynamic characteristics are investigated, the pressure difference and thrust of WRC are extracted, and the results are described in detail in the following sections.

### Flow field

Taking the base scenario (A0) as an example, the flow field near the WRC is extracted and analyzed. In the upstream of the curtain, the flow field hardly changes along the channel. Near the curtain, the water column in the lower layer upwells, the water column is rapidly contracted to the flow area above WRC, and the flow field looks like a jet flow, as shown in Figure 4. Downstream of the curtain, owing to the upwelling of water column, the jet flow zone initially contracts and then gradually expands, and there is a large recirculation zone below the jet flow zone. The flow field under the setup of the curtain is similar with that of flow over a submerged cylinder, sill and pipelines (Sabag *et al.* 2000; Jamali & Haddadzadegan 2010; Mudgal & Pani 2012; Inverno *et al.* 2016).

The flow velocity above the curtain tends to be very large, and can impose a significant influence on the arrangement of hydraulic structures. For the flow above the curtain, the longitudinal velocity of the water surface along the channel is extracted, as shown in Figure 5. The flow velocity above the curtain rapidly increases initially and then slowly decreases, and the maximum flow velocity appears at the location about 0.3 m downstream the curtain. Hence, instead of exactly above the curtain, the maximum flow velocity of water surface appears somewhere downstream of the curtain, which should be noticed in practice.

### Pressure difference and thrust of WRC

#### Effect of inflow velocity

Through numerical simulation, the longitudinal pressure differences on the WRC are extracted, as shown in Figure 6. Under the scenarios of different inflow velocities *v _{in}*, vertical distributions of longitudinal pressure difference are, in general, similar. From the bottom to the top of the curtain, the pressure difference first slowly increases and then rapidly decreases near the top of the curtain. The maximum pressure difference is located at the upper part of the curtain (approximately 95% from the bottom). In terms of magnitude, the maximum pressure difference is approximately 1.04 times of the average value, and the coefficient of dispersion is 0.09. In summary, the distribution of pressure difference on the WRC is relatively homogeneous. The reason is that the inflow velocity is uniform, thus making the distribution of pressure difference even.

The calculated and measured thrusts of WRC and the corresponding drag coefficients are shown in Figure 7. The thrust of the curtain rapidly grows as *v _{in}* increases. Drag coefficients under different inflow velocities are in the range 77.5–82. The MARE between calculated and measured drag coefficients is 2.1%. When the inflow velocity is large, the drag coefficients are all close to 79, but when the inflow velocity is small, the measured drag coefficients show some dispersion. This may be due to the frictional forces between the lines and pulleys which can cause large errors when inflow velocity and resulting thrust are small. In general, the variation range of drag coefficient under different inflow velocities is small, as shown in Figure 7(b).

#### Effect of water-retaining ratio

Under scenarios of different water-retaining ratios *R _{i}*, the vertical distributions of longitudinal pressure difference are similar, as shown in Figure 8. The locations of maximum pressure difference are at about 95% of the upper curtain. In terms of magnitude, when the water-retaining ratio increases from 0.4 to 0.9, the pressure difference on the curtain gets much larger. The water-retaining ratio imposes a large impact on the magnitude of pressure difference. A larger water-retaining height will significantly contract the water flow, and make the flow field near the curtain change more drastically, as a result, the pressure difference caused by water flow rapidly increases.

The calculated and measured thrusts of WRC and the corresponding drag coefficients are shown in Figure 9. The thrust and drag coefficient rapidly increase as the water-retaining ratio increases, and their changes are in agreement with the variation rule of power fitting. The MARE between calculated and measured drag coefficients is 2.8%. The drag coefficients under different water-retaining ratios are in the range 5.5–333.2, which means that the water-retaining ratio has a different influence on the drag coefficient. With the increase in the water-retaining ratio, not only does the impact area of curtain increase, but also the flow area above WRC gets smaller and the flow field near the curtain changes more drastically, so the thrust and drag coefficient caused by water flow rapidly increases. Using both the calculated and measured values of drag coefficients, a fitting formula is obtained to quantify the influence of the water-retaining ratio on the drag coefficient, as shown in Figure 9(b).

#### Effect of arc length–height ratio

Under different arc length–height ratios, *R _{a}*, the vertical distributions of longitudinal pressure difference are distinguishable, as shown in Figure 10. From the bottom to the top of the curtain, when

*R*is larger than 1.1, the pressure difference increases first and then decreases. However, when

_{a}*R*is smaller than 1.1, the pressure difference presents a trend of decrease–increase–decrease upwards. Within the range of

_{a}*R*from 1 to 1.3, the location of maximum pressure difference moves down from 95% to 75% height of the curtain from the bottom. The reason for the change of the pressure difference distribution may be as follows. The longitudinal pressure difference is the product of pressure difference and its longitudinal vector. Owing to the curved shape of the curtain, the longitudinal vector of water pressure in the upper part of curtain is smaller than that in the middle part (approximately 1). When

_{a}*R*increases, the curvature of WRC increases, the longitudinal vector in the middle part keeps unchanged, and the longitudinal vector in the upper part gets smaller. Thus, under the combined influence of water pressure and longitudinal vector, the maximum pressure difference will move down.

_{a}In terms of magnitude, the pressure difference increases under a larger *R _{a}*. When

*R*increases, the curtain is bending backwards, the flow field near the curtain changes more drastically, and the pressure difference gets larger.

_{a}The calculated and measured thrusts of the curtain and the corresponding drag coefficients are shown in Figure 11. The thrust and drag coefficient nonlinearly grow as *R _{a}* increases, and its change accords with the variation rule of second-degree polynomial fitting. Using both the calculated and measured values of drag coefficients, a fitting formula is obtained to quantify the influence of arc length–height ratio on the drag coefficient, as shown in Figure 11(b).

#### Effect of inclination ratio

Under scenarios of different curtain inclination ratios *R _{i}*, the vertical distributions of longitudinal pressure difference are similar, as shown in Figure 12. The locations of maximum pressure difference on the WRC are at about 95% of the curtain. In terms of magnitude, as the inclination ratio increases from 0 to 1.5, the pressure difference on the curtain gets smaller. When the curtain inclination ratio

*R*increases, the flow field near the curtain changes more moderately, and hence the pressure difference reduces.

_{i}The calculated and measured thrusts and the corresponding drag coefficients are shown in Figure 13. As the flow field near the curtain changes more moderately, the thrust and drag coefficient synchronously decrease when *R _{i}* increases, and their changes are in accord with variation rule of linear fitting. Using both the calculated and measured values of drag coefficients, a linear fitting formula is obtained to quantify the influence of inclination ratio on the drag coefficient, as shown in Figure 13(b).

### Formula of thrust of WRC

A fast quantization method of thrust is very useful and necessary for the application of WRC, especially in case of emergency, such as sudden heavy rains or large inflow events. Based on the results above, the influence the curtain has on the thrust and drag coefficient is investigated, and a simplified formula of thrust is proposed and verified using flume simulations and a real reservoir model test, which are described below.

#### Proposition of formula

From the results presented above, the influence of different boundary variables on the drag coefficient of WRC is obtained, and their sensitivity analysis is shown in Table 3. Evidently, the influence degree of various variables is water-retaining ratio > inclination ratio > arc length–height ratio > inflow velocity, and the drag coefficient is barely affected by the inflow velocity. In combination with the results of flow field, the flow characteristics over a WRC are similar to the classical problem of cross flow, and the thrust of the curtain matches with the resistance equation (Equation (1)).

Parameter | Change | Drag coefficient and the relative change |
---|---|---|

Baseline | – | 79.5 |

Inflow velocity | +50% | 78.0 (−2%) |

−50% | 78.0 (−2%) | |

Water-retaining ratio | +12.5% | 319.8 (+02%) |

−25% | 16.8 (−79%) | |

Arc length–height ratio | +10% | 68.3 (−14%) |

−10% | 82.5 (+4%) | |

Inclination ratio | +0.5 | 73.3 (−8%) |

+1 | 61.2 (−23%) |

Parameter | Change | Drag coefficient and the relative change |
---|---|---|

Baseline | – | 79.5 |

Inflow velocity | +50% | 78.0 (−2%) |

−50% | 78.0 (−2%) | |

Water-retaining ratio | +12.5% | 319.8 (+02%) |

−25% | 16.8 (−79%) | |

Arc length–height ratio | +10% | 68.3 (−14%) |

−10% | 82.5 (+4%) | |

Inclination ratio | +0.5 | 73.3 (−8%) |

+1 | 61.2 (−23%) |

*R*, a second-degree polynomial function with arc length–height ratio

_{h}*R*and linear function with inclination ratio

_{a}*R*, the fitting equation is shown in Figures 9(b), 11(b) and 13(b). Systematically considering the variation trend of thrust and dimensionless drag coefficient under different curtain forms, a fitting calculation is applied to find the simplified approximate formula. Three main characteristic variables representing the form parameter of WRC are used, including arc length–height ratio

_{i}*R*, water-retaining ratio

_{a}*R*and inclination ratio

_{h}*R*. Together with the resistance equation (Equation (10)), the simplified forecasting formula of drag coefficient and thrust is shown below. where

_{i}*v*is the inflow velocity (m s

_{in}^{−1}),

*H*is the water depth (m),

*h*is the water-retaining height (m), is the water-retaining ratio,

*R*∈ (0.4,0.9),

_{h}*l*is the arc length of curtain (m), is the arc length–height ratio,

_{a}*R*∈ (1,1.3),

_{a}*l*is the inclination length of curtain (m), and is the inclination ratio,

_{i}*R*

_{i}∈ (0,1.5).

#### Verification using flume simulations

Firstly, the proposed forecasting formula of thrust is verified using flume simulations described above. The thrusts under various boundary conditions are calculated using flume numerical simulations and forecasting formula. Boundary conditions are set as various combinations of inflow velocities (0.005, 0.001, 0.02, 0.05 m s^{−1}), arc length–height ratios (1.1, 1.2, 1.3), water-retaining ratios (0.2, 0.4, 0.6) and inclination ratios (0, 0.5, 1), a total of 108 sets. Calculated thrusts are shown in Figure 14. Results from forecasting formula coincide well with the results from flume simulations with MARE and R^{2} being 13% and 0.97, respectively.

#### Verification using a real reservoir model test

Secondly, taking Sanbanxi Reservoir as an example, the forecasting formula of thrust is verified using a real reservoir model test. A hydraulic physical model of the Sanbanxi Reservoir was built, and a detailed description of the physical model is presented in the Appendix (available with the online version of this paper). The water-retaining ratios of WRC are set to 45%, 67% and 76%, and the outflow discharges are 4,870, 9,600 and 14,100 m^{3} s^{−1}, respectively. The thrusts are obtained using Sanbanxi Reservoir experiments and forecasting formula, as shown in Figure 15. Results calculated from the forecasting formula match well with the results from the Sanbanxi Reservoir experiment, the values for MARE and R^{2} are 14% and 0.96, respectively. In total, the flume simulations and Sanbanxi Reservoir experiments show a good practicability and accuracy of the proposed forecasting formula.

## DISCUSSION

This research has studied the influence of curtain forms on the pressure difference and thrust of WRC, and proposed a forecasting formula of thrust and drag coefficient. The results are further discussed on the following aspects.

The first point regards the application of the forecasting formula of thrust. On one hand, the study of this paper is based on the experimental research of a rectangular flume. While in the practical engineering, the shape of flow cross section of most deep canyon reservoirs and natural river channels tends to be irregular. In the calculation of dimensionless drag coefficient *C _{d}*, the water-retaining ratio should be the area ratio of the curtain to the cross section, rather than the ratio of water-retaining height to the water depth. In addition, this formula is obtained under the condition of uniform inflow velocity, and the influence of inflow velocity irregularity on the result should be noted. Qualitatively, when averaged velocity is the same, the thrust is expected to be smaller if the inflow velocity in the upper layers gets larger. In the follow-up study, a force analysis of WRC under different inflow distributions should be developed.

The second point is about the designation of form of WRC. This paper has quantified the influence of arrangement form of the curtain on the thrust of WRC, the thrust of WRC is significantly smaller when water-retaining height decreases or inclination length increases. However, from an engineering viewpoint, the water-retaining effect under a curtain which is vertical and higher would be better, i.e., the degree of improvement of water temperature and quality of vertical curtain is better. Thus in the designation and implementation of a curtain, both the thrust and engineering aspects should be considered to determine the arrangement form of the curtain. The internal stress of WRC is another noteworthy issue concerning the safety and stability of the structure. Regarding hydrodynamic force on the curtain, the thrust of the curtain increases by 9% when the arc length–height ratio increases from 1.1 to 1.3. But with respect to structural internal stress, the curvature of the curtain decreases under a smaller arc length–height ratio, and in turn the internal tensile force of the curtain would be very large even though the hydrodynamic force is small. Thus, a longer arc length will decrease the structural internal stress and increase the robustness of the curtain system. The arc length should be determined comprehensively considering the hydrodynamic force and structural internal stress. In a further study, a finite element analysis of the internal stress of the curtain would be conducted to provide more accurate technical support for the curtain system stability.

The third point is about the application and generalization of WRC. So far, the curtain has been used in selective withdrawal from reservoirs and sedimentation control, and its application can be further extended in other fields. On the one hand, the curtain can be used to store freshwater in estuaries, coasts and rivers, acting as a newly developed coastal reservoir (Yang & Lin 2011; Liu *et al.* 2013) or traditional rubber dam. On the other hand, the curtain can also be used as a good carrier of water purifiers or biofilms in the river channel to improve water quality (Carpenter & Helbling 2017; Li *et al.* 2017). With the development of material strength and construction technology, the flexible curtain would be economic, environmentally sound and conveniently constructed, managed, dismantled and moved in the two examples above.

Meanwhile, there are certain limitations and prospects to our study. Firstly, because the control of a flexible curtain in the experimental tests is harder than that of a rigid structure, there are unavoidably some systematic errors in the experimental tests, i.e. there is spanwise radian in edge areas of the curtain, and there also exists frictional forces between the lines and pulleys which can influence the measured thrust of WRC, especially when the thrust is small. Secondly, this paper mainly studied the effect of vertical radian on the thrust, and the effect of radian in a spanwise direction on the thrust can be further studied, and a 3-D model can be developed. Thirdly, the fitting calculation is mainly applied to find the simplified approximate formula, and the physical mechanism about the influence of different factors should be further investigated.

## CONCLUSIONS

In this study, both numerical modeling and laboratory experiments are used to investigate the influence of curtain forms on the pressure difference and thrust of WRC, and a forecasting formula is proposed and verified. This study shows the following: There is a large recirculation zone behind the WRC, and the maximum flow velocity of the water surface appears at a certain location downstream of the curtain; the distribution of pressure difference on the curtain is relatively uniform, and the maximum value is located at the upper part of the curtain. When arc length–height ratio increases, the location of maximum pressure difference moves downwards; the variation of thrust of WRC conforms to the classical resistance equation. The drag coefficient changes little under different inflow velocities. Increasing water-retaining height and arc length, and reducing curtain inclination ratio will result in an increase of the drag coefficient. The drag coefficient is a power function with the water-retaining ratio, a second-degree polynomial function with the arc length–height ratio and a linear function with the inclination ratio; and finally, forecasting formula of thrust of WRC is proposed and verified using both flume simulations and Sanbanxi Reservoir experiments. This formula has systematically considered the water-retaining height, arc length and inclination degree, which is practical and convenient for the prediction of thrust.

The results presented in this paper can be further utilized in fields related to other materials like membrane, sheet and plate. With the increasing use of flexible materials in hydraulic engineering and other fields, this research can be extended to include flexible structures in future research.

## ACKNOWLEDGEMENTS

This research was supported by the National Key Research and Development Program of China (2016YFC0401708), Program of Introducing Talents of Discipline to Universities (B14012), and National Natural Science Foundation of China (51609167). The authors acknowledge the assistance of anonymous reviewers.

## REFERENCES

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*Numerical and Experimental Investigation on the Interaction Between Moving Fluid and Flexible Bodies*

*Thesis*