Investigation of flow measurement mechanism and hydraulic characteristics of the NACA airfoil pillar-shaped flume with different wing lengths in a rectangular channel Open Access

Flow measurement in open channels is the key means for optimal allocation of water resources, flood discharge forecasting, and water-saving irrigation (Dabrowski & Polak 2012; Carollo et al. 2016; Jesson et al. 2017). The measuring flume is a special facility for measuring the flow through various channels. Flumes that are designed according to critical flow theory are widely used in agricultural irrigation and sewage monitoring because of their advantages such as simple structure, large measuring range, and small head loss (Xiao et al. 2016). These flumes can be exemplified by long throat flumes, short throat flumes, such as the Parshall flume (Li et al. 2004; Thornton et al. 2009; Cox et al. 2013), no throat flumes, such as the U-shaped channel parabolic flume (Hu et al. 2014), and cut-throat flumes (Manekar et al. 2007; Torres & Merkley 2008; Das et al. 2017). In terms of effective factors (cost, practicality, stability, and reliability), long throat flumes are more suitable for medium-sized channel flow measurement, while the hydraulic structures, such as sluice and culvert, are more suitable for large channel flow measurement, and cylindrical flumes are more suitable for measuring a massive amount of flow in small channels. In addition, cylindrical measuring flumes have the advantages of simple structure, convenient production, and installation. In addition, these flumes can be made into a fixed type or a mobile type so that they are very suitable for small channels (Ferro 2018).

Hager (1985) proposed a portable flume for measuring flow in rectangular, trapezoidal, and U-shaped channels. In traditional venturi channels, both sides of the channel are commonly contracted to achieve a narrow cross-section, while Hager suggested placing a vertical cylinder on the channel axis to form the contraction. Samani (2017) conducted laboratory experiments and evaluated the hydraulic characteristics of several simple mobile circular flumes. Later, following the experimental results of Hager and Samani et al., Ferro (2002) applied dimensional analysis and self-similarity theory to determine the stage-discharge relationship for a flume with a horizontal bottom and without submergence effects.

It can be concluded from the literature that a cylindrical measuring flume, as a portable venturi flume, is widely used for flow measurement. However, in order to measure the flow in a small channel, the water measuring device not only needs to have portability and high measurement accuracy, but also needs smaller backwater height and larger critical submergence degree. Note that streamlined flow measuring flumes can meet the above requirements at the same time. In addition, most of the above-mentioned types of measuring flumes cannot be applied to established channels. Therefore, it is of practical significance to study streamlined flow measuring flumes.

The National Advisory Committee for Aeronautics (NACA) conducted a systematic study on airfoils in the 1930s and provided a general analytical expression for airfoil thickness distribution. Based on this expression, Liu et al. (2014) proposed the airfoil pillar-shaped flume and carried out model experiments on this flume. The experiments proved that this airfoil has more efficient hydraulic characteristics and can be used for flow measurement in rectangular channels. In addition, it is known that computational fluid dynamics has the advantages of low cost, short time, and easy access to flow field data. Thus, it has become one of the most indispensable methods for hydraulics analyses such as study of the hydraulic characteristics of open channels (Nazari-Sharabian et al. 2020a, 2020b), measuring flumes (Ramamurthy & Tadayon 2008; Qi et al. 2014; Li et al. 2020), and weirs (Aydin 2012; Ziaei et al. 2019; Saadatnejadgharahassanlou et al. 2020). Therefore, in the present work, NACA airfoil pillar-shaped flumes with different wing lengths are further studied by combining laboratory experiments and numerical simulations to provide a theoretical reference for applications of this flume in rectangular channels.

The NACA airfoil pillar-shaped flume can be considered as a special venturi measuring flume because it can be installed or built as an airfoil-shaped cylinder in the center of the channel to make the channel section contracted. It is known that when the water flow sufficiently increases in the flume section the critical flow can be generated, which is not affected by the downstream flow conditions. Therefore, by measuring the upstream water depth, the flow rate can be calculated using a single stable relationship between water level and flow rate.

In this work, six NACA airfoil-shaped flumes (Figure 2) with r = 2, 3, 4, 5, 6, and 7 were studied, for which the specific dimensions are provided in Table 1.

Laboratory experiments were conducted to investigate the effect of different wing lengths (C = 15.0, 22.5, 30.0, 37.5, 45.0, and 52.5 cm) and flow rates (from 7 to 27 L/s) on hydraulic characteristics and measurement accuracy of the NACA airfoil-shaped flume. The working conditions of experiments are presented in Table 2. The experiments were carried out in a plexiglass channel located at the hydraulic laboratory of the Water Engineering Department in Zhengzhou University, Zhengzhou, China. The experimental setup of the NACA airfoil-shaped flume is shown in Figure 3.

All dimensions of NACA airfoil-shaped flumes are fitted with channel centerlines. Measuring points are set every 0.5 m in the upstream section and the canal section after the hydraulic jump. In the contraction section and the hydraulic jump section, measuring points are set every 0.1 m. Finally, five typical sections were selected for analysis: the upstream canal section, the inlet section of the flume, the throat section, the outlet section of the flume, and the downstream canal section. To ensure that the upstream water flow of the flume is stable, the inlet section of the flume is set to 3.0 m away from the inlet of the channel. A centrifugal pump is used to supply water from the downstream reservoir to the upstream reservoir and an electromagnetic flowmeter is installed between the pump and the valve for real-time monitoring of the flow through the flume. For a given discharge value, the water depth at all sections of the rectangular channel is measured.

As shown in Figure 4, correlation analysis of K/B_c and h/B_c of six NACA airfoil-shaped flumes with different r clearly shows that K/B_c and h/B_c have an excellent correlation.

The VOF method (Hirt & Nichols 1981) is employed to describe the free-surface motion and the standard k-ɛ three-dimensional turbulence model (Saadatnejadgharahassanlou et al. 2020) is used to calculate the turbulence variables and to analyze the internal flow field because this turbulence model can calculate sufficiently accurate results for open channel flow problems. This model calculates the turbulent kinetic energy and turbulent dissipation rate by solving partial differential equations, which are provided in Equations (15) and (16), respectively.

In order to completely and accurately simulate the real internal flow field, similar to the actual size of the NACA airfoil pillar-shaped flumes and the experimental setup, three-dimensional geometrical models were developed using the DesignModeler software. The center point of the bottom of the flume inlet section is taken as the origin of the coordinate system, the water flow direction is assumed in the positive direction of the X-axis and the gravity direction is set to the negative direction of the Z-axis. Five typical sections are selected for in-depth analyses of the velocity distribution, as shown in Figure 5(a), where the geometric model is illustrated as well.

The grid division and its quality directly affect the calculation accuracy and efficiency of the numerical simulation. The purpose of this study is to more accurately simulate the flow state of the channel and reduce the calculation time as much as possible. Using five different sizes of grids to simulate the same working condition, as shown in Table 3, the hexahedral grid was used to discretize the entire fluid domain and the grid size was set to 0.01 × 0.01 × 0.01 m (M4). For the boundary conditions, the water inlet was set as the velocity inlet and the air inlet was set as the pressure inlet. In addition, the outlet was set as the pressure outlet, the wall was set to no-slip condition and the roughness height was set to 0.000011 m. The gravity was set to 9.81 m/s² and the air pressure was 101,325 Pa. The mesh generation and boundary conditions of the NACA airfoil pillar-shaped flume are shown in Figure 5.

After constructing the measuring flume, the variation of the water surface height along the channel can be directly affected by the water surface line. By measuring the water depth at 17 sections of six NACA airfoil pillar-shaped flumes in the rectangular channel, and under different working conditions, the variation of the water surface line through the channel can be obtained. In order to verify the reliability of numerical simulation for further analyses, the numerical solutions of the water surface line are compared with the experimental data. Figure 6 shows the comparison of the measured and simulated results of the free water surface height in the channel centerline of the NACA airfoil pillar-shaped flumes. Figure 7 shows the comparison of water depth of the numerical results and the experimental data (r = 4). The maximum error between the experimental data and the numerical results is 7.86%, the minimum error is 0.46%, and the average error is 4.83%. These errors indicate that the results of this numerical simulation can be used for in-depth analyses of the internal flow field.

The flow pattern can directly reflect the movement state of the water flowing through the flume. Note that the Fluent software can clearly simulate the flow pattern diagram of the flume such as the flow diagram, Figure 8(a), the streamline diagram, Figure 8(b), and the longitudinal section diagram, Figure 9.

In the NACA airfoil pillar-shaped flume, the upstream water flow is relatively smooth and the flow direction is basically parallel. After entering the contraction section, the water surface declines slowly, the streamline is bent, caused by the column, and the upstream water flow is divided into two parts. In addition, the throat cross-section shrinks significantly leading to a sharp decrease in water surface level and a sharp fluctuation of the water flow. Due to inertial force, the water flow continuously diffuses to both sides, where the water depth is higher near both walls than near the center. Subsequently, the water surface after the diffusion section of the measuring flume slowly rises and smoothly connects with the downstream water surface and the horizontal distribution of the water surface gradually recovers and becomes balanced. It can be seen in Figure 9 that the water flow patterns of the six NACA airfoil pillar-shaped flumes under the condition of free flow have produced two hydraulic phenomena, hydraulic drop and hydraulic jump. Under the same working conditions, as the value of r increases, the water surface drops more slowly and the distance required for the hydraulic jump is longer.

Equation (17) can be used for calculating the discharge when r is in the range of 2–7. In addition, Table 4 compares the measured discharge (Q₁) and the calculated discharge (Q₂) for all working conditions. The error distribution between the calculated flow and the measured flow is shown in Figure 10. As the average error is 1.72%, evidently the flow discharge calculated by Equation (17) matches the measured flow well. Thus, it can be used to accurately calculate the flow of the NACA airfoil pillar-shaped flume.

Froude number (Fr) is a dimensionless number that can be used as a criterion to judge the open channel flow pattern (Karakouzian et al. 2019). Physically, it compares the inertial and gravity forces of water flows. When Fr= 1, it means that the magnitude of the inertial force is equal to the gravity force and the flow is called critical flow. When Fr> 1, it means that the inertial force is larger than the gravity force and the water flow is in the state of a jet stream. Also, Fr< 1 indicates that the inertial force is less than the gravity force and the water flow is in a slow flow state. Note that the flume measurement is based on the critical flow principle. In the measuring process, the water flow changes from slow flow to jet stream state and then to slow flow state. Figure 11 shows the variation of Fr along the course of the six NACA airfoil pillar-shaped flumes. As can be seen, the six NACA airfoil pillar-shaped flumes have realized the flow pattern conversion between slow flow and rapid flow states and when r increases, a larger distance is required to complete the conversion. Furthermore, Fr in a certain area in front of the airfoil pillar approaches zero and as r increases, the area of this part increases.

Generally, the flow upstream of the measuring flume is required to have Fr less than 0.5 to ensure the water surface is stable and to facilitate the accurate measurement of water level. Figure 11 illustrates the variation trend of Fr at the upstream sections of the six measuring flumes under different working conditions. It can be seen from Figure 12 that, for the same flow rate, Fr continuously decreases with an increase in r, while for the same r, Fr tends to increase with an increase in flow rate.

The velocity distribution can reflect the basic law of water flow movement in the NACA airfoil pillar-shaped flume. The flow velocity distribution in the inlet section (a), throat section (b), and outlet section (c) of six NACA airfoil pillar-shaped flumes with different r for the flow rate of 15 L/s are shown in Figure 13. As can be seen, the lateral flow velocity of the upstream section, from the center to the sidewall of the channel, is a symmetrical distribution, which is first increasing and then decreasing. Due to the influence of the airfoil pillar, the flow velocity in the center area of the channel is close to zero, and the area increases as r decreases. In addition, Figure 13 illustrates that as r increases, the upstream water flow rate of NACA airfoil pillar-shaped flumes decreases continuously, which is caused by different degrees of cross-section contraction. On the throat section, due to the reduction of the cross-section, the flow velocity further increases compared to the upstream section. It can also be seen that the flow velocity at the throat section increases when r increases, while the rate is decreasing. In addition, the flow velocity near the sidewall of the channel is also less than the flow velocity near the airfoil pillar. At the outlet section, it can be seen that, similar to the inlet section, the water flow is still divided into two streams and the water depth becomes smaller. The maximum velocity is located in the middle and upper parts of the flow. The water depth on both sides of the channel is obviously lower than the water depth in the center of the channel, and this phenomenon is more noticeable when r is small.

Note that, as a portable measuring flume, the cylindrical measuring flume is most suitable for small channels. In addition, due to the effect of side contraction or vertical contraction of the measuring flume, the upstream water level has to be larger than the normal depth of the original channel. Thus, the backwater phenomenon is likely to happen. Therefore, after installing the measuring flume, it is necessary to be sure if during measuring flow the backwater height of the channel may cause water to overflow the channel. Note that, in this work, the backwater height (H_b) is defined as the difference between the upstream water depth after installing the NACA airfoil pillar-shaped flume, and the original water depth.

It can be concluded from Figure 14 that, for the same r, the backwater height increases with an increase in the flow rate and, for the same flow rate, the backwater height increases as r increases. When Q = 27 L/s, the backwater heights of the NACA airfoil pillar-shaped flume with r = 2 and 7 are 5.27 and 7.72 cm, respectively. When Q = 7 L/s, the backwater heights with r = 2 and 7 are 1.97 and 2.50 cm, respectively. Therefore, it is recommended to use the NACA airfoil pillar-shaped flume with a small r value for small channel height, especially for large flow conditions.

In addition, it can be seen from Figure 14 that there is a good correlation between the backwater height and the flow rate. Note that the calculation formulas of the backwater height of the NACA airfoil pillar-shaped flume with different r values were obtained by fitting all data (Table 5). Through these formulas, the influence of the installation of the measuring flume on the channel depth can be predicted to facilitate the selection of a suitable NACA airfoil pillar-shaped flume.

The critical submergence degree refers to the ratio of the downstream and upstream water depths. Note that for a certain contraction ratio, flow rate, and channel bottom slope, the critical submergence degree is used when the downstream water depth of the measuring trough just begins to affect the upstream water depth. This parameter reflects the range of water depths that can ensure the flume to be free flow in downstream. In this paper, the downstream water depth was adjusted by a gate at the end of the channel and the critical submergence degree of the NACA airfoil pillar-shaped flume was measured when the critical discharge condition was reached.

Table 6 shows the critical submergence degree of the NACA airfoil pillar-shaped flume under various working conditions. When r = 2, the critical submergence range of the NACA airfoil pillar-shaped flume is from 0.76 to 0.84. When r = 7, the critical submergence degree of the NACA airfoil pillar-shaped flume is in the range of 0.88–0.89. Variation range of the critical submergence degree of the NACA airfoil pillar-shaped flumes with respect to r is shown in Figure 15. This figure illustrates that the critical submergence degree is positively correlated with r so that it increases when r increases. Therefore, to provide larger ranges of free flow, the NACA airfoil pillar-shaped flume with larger r values should be used.

The viscosity of water is the root cause of the energy loss of water flow. Due to the existence of viscosity, each layer of the water flow will generate frictional resistance during the flow process, causing the loss of energy of the moving liquid. There are two kinds of local hydraulic phenomena, the water level drop and water level jump (Figure 16), which can cause large energy losses. In the process of measuring flow, the shape and the size of the solid boundary change when the water flows excessively in the flume and the streamline is bent. Note that these characteristics represent a non-uniform abrupt flow. In addition, the head loss and local head loss will occur along the course. However, in the place of local head loss, the main flow is separated from the boundary and a vortex forms in the separation area. In the vortex region, the turbulence intensifies and a constant energy exchange exists between the mainstream and the vortex region. Also, a lot of mechanical energy is consumed by the friction and violent collision between the particles. Therefore, the local head loss is much larger than the head loss along the same length of a flow segment. In this study, the length of the measuring flume is relatively short. Thus, the head loss is neglected along the flow segment.

Figure 17 illustrates the variation trend of the head loss of NACA airfoil pillar-shaped flumes at critical outflow under all working conditions. Figure 17 shows that when r is constant, the head loss increases with an increase in the flow rate. While, when the flow rate is constant, the head loss shows a decreasing trend as r increases. For plain irrigation areas with poor water head conditions, it is recommended to use NACA airfoil pillar-shaped flumes with r in the range of 5–7.

Open channel flow measurement is mostly used for measuring flow in different applications such as the optimal allocation of water resources, flood discharge forecasting, and water-saving irrigation. In this study, experimental and numerical simulations have been carried out on six NACA airfoil pillar-shaped flumes with different wing lengths to further discuss the abilities of these flumes in flow measurement. For the numerical calculations, the standard k-ɛ three-dimensional turbulence model and VOF method were used. In addition, the Buckingham Pi theorem along with the incomplete self-similarity theory were used for the theoretical flow calculation. Following conclusions can be drawn as a result of this study:

• (1)

  The standard k-ɛ model and the VOF method can accurately simulate the flow in the flume. The average error of the experimental and numerical water surface line is 4.83%. It provides a theoretical basis for further research on the internal flow field distribution and the optimization and promotion of NACA airfoil pillar-shaped flumes.

• (2)

  Dimensional analysis and self-similarity theory are used to derive the relationship between water level and flow and a formula suitable for different wing lengths is obtained, in which R² is 0.9981. The average error between the calculated and measured flows is 1.72%, which meets the requirements of open channel flow measurement.

• (3)

  The flow pattern, Froude number, flow velocity distribution, backwater height, critical submergence degree, and energy loss of six NACA airfoil pillar-shaped flumes were compared. It was found that the backwater height is inversely proportional to r. However, the critical submergence and energy loss are directly proportional to r. In terms of portability and backwater height, NACA airfoil pillar-shaped flumes with smaller r values should be considered. For plain irrigation areas or channels with poor water head conditions, it is recommended to use NACA airfoil pillar-shaped flumes with r in the range of 5–7. This provides a theoretical reference for the selection of NACA airfoil pillar-shaped flumes in small rectangular channels.

The authors gratefully acknowledge the support provided by the National Key Research and Development Program of China (No. 2017YFC1501204), the Program for Science and Technology Innovation Talents in Universities of Henan Province (No. 19HASTIT043), the Outstanding Young Talent Research Fund of Zhengzhou University (1621323001), the Program for Guangdong Introducing Innovative and Enterpreneurial Teams (No. HG-GCKY-01-002), National Natural Science Foundation of China (No. 51909242, 52009125), Key scientific research projects of colleges and universities in Henan Province (No. 21A570007), China Postdoctoral Science Foundation funded project (No. 2020TQ0285).

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