Hydraulic characteristics optimization of an airfoil pillar-shaped flume based on the Hicks–Henne function and NSGA-II algorithm

Open channel flow measurement is the basis for water resource allocation and water fee collection in irrigation areas and plays a vital role in agricultural irrigation management. Among many flow measurement facilities, a series of measuring flumes developed based on the Venturi principle have been widely used because of their advantages of low construction cost, reliable measurement accuracy, and wide application range. The principle of flow measurement is that when the water flows through the channel section narrowed by the measuring flume, its flow velocity will increase, the water level will drop, and a critical flow state will be formed near its throat. Under this state, there is a stable and single stage–discharge relationship. However, because of the differences in their hydraulic characteristics corresponding to various facility structures, as well as the improvement of the applicable requirements of the measuring facilities in irrigation area management, research on the improvement and optimization of the flow measurement facilities remains an urgent problem to be solved.

The earliest Venturi flume belongs to the short-throat flume. After comparing several different ratios of throat widths to flume lengths, throat lengths, and arrangements of end wings, the results showed that a greater length of converging and diverging section and a rounding of the throat section would result in less head loss and great accuracy in measurement of flow, but the standard was chosen as a compromise between accuracy and cost (Cone 1917). The Parshall flume improved the throat length and slope of the original Venturi flume, which made critical flow easier to occur and improved the measurement accuracy (Parshall 1926). But the overall structure of the Parshall flume is complex, and the construction requirements are high. Based on this problem, Skogerboe & Hyatt (1967) removed the throat section and designed the standard shape of the cutthroat flume, which makes the flume simple and economical. However, the cutthroat flume will cause large head loss under free flow conditions (Ramamurthy et al. 1988). In order to find flow measurement facilities with wider adaptability, the simplified NACA airfoil equation profile is introduced into the flow measurement field (Wang & Wen 1990), and it has the advantages of smooth flow, small head loss, high critical submergence, and high flow measurement accuracy (Hong et al. 2005; Lu et al. 2006; Liu et al. 2008).

In consideration of the difficulty of building or installing the measuring flume in different channels and the convenience of flow measurement, a cylindrical measuring flume was designed to be fixed in the center of the channel, which is applicable to all symmetric prismatic channels (Hager 1985). Then, the cylindrical diffuser section was modified into a V-shaped tail, which improved the flow pattern and reduced head loss (Liu et al. 2013). Moreover, the principle of optimizing or designing a cylindrical moving flume was discussed by comparing it with other types of the cylindrical moving flume (Li et al. 2020). In addition, Peruginelli & Bonacci (1997) also proposed a moving pier-shaped prism structure in the center of a rectangular channel, which was later called the central baffle flume. The influence of different geometric parameters of the central baffle flume was studied through experiments (Kolavani et al. 2019) and the optimization design results were verified (Bijankhan & Ferro 2019). According to the central layout principle, Liu et al. (2019) conducted comparative experiments and simulations on the application performance of the airfoil pillar-shaped flume in three common channels. The results show that the advantages of smooth flow, small head loss, high flow measurement accuracy, and high submergence are still retained. However, the measuring flume is an additional device relative to the channel. In the actual project, the construction of the channel and the installation of the measuring flume cannot be carried out simultaneously. The installation of the measuring flume will have an impact on the channel flow, causing the backwater upstream and head loss. Especially for the plain irrigation area or the environmental conditions with poor head, the airfoil pillar-shaped flume, like other types of measuring flume, still faces the problem of water blocking.

With the development of demand and the improvement of research technology, numerical simulation technology has been widely used in the field of alternative model computing, which brings more possibilities for innovation. More importantly, the optimization concept has been demonstrated in other fields such as water resource allocation optimization, aerodynamic performance optimization, hydrodynamic performance optimization, and material structure optimization by integrating optimization algorithms and numerical simulation technology (Sadeghi-Tabas et al. 2017; Yang et al. 2018; Yan et al. 2019; Nozari et al. 2021; Yin et al. 2021; Wang et al. 2022). Taking optimization algorithm as the intelligent guide of numerical simulation technology improves the efficiency, fineness of optimization, and rational utilization of computing resources. However, from the above research, it can be seen that this concept is rarely applied to the improvement of flow measurement facilities. At present, the optimization of hydraulic characteristics of measuring flumes is mostly the result of comparative analysis by presetting several limited test groups, which cannot be determined to be the relative optimal or global optimal in a specific area. The reason is that in the traditional method, when designing the test groups for comparison, the design variable values to control the geometric structure of the facility are selected at equal intervals, and then the best group is selected by a unified comparison of the results. However, the optimization algorithm first randomly selects the design variable values and can compare each group of results with the previous results in time. It makes the selection direction of the next design variables close to the area with good performance, and more computing resources can be concentrated in the effective area, so as to improve the efficiency of optimization and the fineness of the geometric structure adjustment of the facility. Therefore, research on intelligent and fine optimization of flow measurement facilities will become an inevitable development trend in irrigation areas.

To summarize, the difficulty of construction or installation of the measuring flume, as well as the impact on the flow of the original channel following the installation of the measuring flume, must be considered concurrently. Based on the airfoil pillar-shaped flume with the advantage of streamlining, the optimization design system of hydraulic characteristics is established by integrating the parameterization method of the airfoil, computational fluid dynamics (CFD) numerical simulation technology, and optimization algorithm in this paper. It can improve hydraulic performance and expand the application scope as much as possible under limited hydraulic conditions. The design concept can provide a certain theoretical reference for the optimization research of similar flumes.

The airfoil structure is the main factor that affects the flow characteristics of the airfoil pillar-shaped flume, so the optimization of hydraulic characteristics is essentially to find the optimal design scheme for the airfoil shape. The link relationship between airfoil shape and hydraulic characteristics must be transformed from physical property problems into mathematical model problems before continuing research. Therefore, choosing an appropriate airfoil expression method is a prerequisite for optimization.

The parameterization of the airfoil is to express the airfoil to be optimized through a finite number of functions, and then adjust the airfoil shape by controlling the variable parameter values of the function. The optimization process is to find the optimal solution of the variable parameters. On the basis of ensuring the smoothness and practicability of the airfoil, this paper selects the most widely used Hicks–Henne function method (Hicks & Henne 1978) in the linear superposition method of analytic functions to express the airfoil. The airfoil shape is controlled by parameterizing and superimposing the variation of the airfoil curvature and thickness, which not only changes the key points of the airfoil profile curve, but also ensures the smoothness of the airfoil surface. The Hicks–Henne function expresses the geometry of the airfoil in three parts: the basic airfoil ⁠, the shape function and the design variable ⁠. The shape function is also called the disturbance function, and the design variable is used as the amplification factor of the shape function, and its role is to constrain the disturbance range.

The MATLAB software is applied to realize the airfoil shape reconstruction module in this study. First, the co-ordinate data corresponding to each point of the basic airfoil is extracted. Then the value of the design variable of each key point is brought into Equation (3) to calculate the disturbance data corresponding to the ordinate, and brought into Equation (2) to realize the airfoil adjustment. Finally, the new airfoil co-ordinate points are converted into a file format recognized by the modeling software.

In this study, the ICEM software is used for modeling and meshing the fluid domain. The ICEM has the function of recording scripts, which can record all relevant commands involved in the normal operation process. After executing the script recording function, call the new airfoil co-ordinate data generated by the airfoil shape reconstruction module to establish the model, divide the fluid domain, generate the mesh block, establish the mapping relationship, define the meshing method and mesh size, and output mesh file, finally generate a script file. When the airfoil is adjusted within a small range, it does not affect the original mapping relationship between the fluid domain model and the block. Just run script file again to get the corresponding model output mesh file automatically.

In addition, for the capture method of free liquid surface, this study adopts the VOF model. The model achieves the tracking of the interphase interface in the computational domain by introducing the variable phase volume fraction, which is very suitable for calculating fluid flows such as air and water that cannot be mixed with each other. The sum of the phase volume fractions of a unit grid is 1, and the free water surface is formed by finding the combination of grids with two phase volume fractions of 0.5.

The FLUENT software also has a secondary development function, it can read two types of commands such as GUI and TUI. The GUI commands are similar to the script recording function of ICEM software. Script recording can be realized by executing the Start Journal function, which is automatically written in Scheme language. The TUI commands are written in C programming language but cannot be recorded and written according to the user interface. The difference between the two is that the GUI command file can be automatically generated, and the familiar operation is simple, but when reading the Journal file again and executing the corresponding GUI command, it is easy to be disturbed by third-party operations and cause confusion in the settings. The TUI commands are relatively concise and without redundant operations, it is not affected by other operations when executing the commands, but users need to be familiar with the relevant setting command lines in order to write related commands. Considering the stability of settings, this module uses the TUI commands to realize automatic settings such as reading mesh files, adding materials, selecting turbulence models, defining boundary conditions, initializing calculations and file output.

As a means to find the optimal solution, optimization algorithm is a key part of the optimization system. It can constantly adjust the search direction in the design space until it finds the shape scheme that can make the hydraulic performance of the airfoil pillar-shaped flume reach the best, which greatly affects the optimization efficiency and the quality of the optimization results. Moreover, the optimization problem is generally a multi-objective problem in practical engineering. The NSGA-II algorithm is selected as the multi-objective problem solving method in this paper, which belongs to the genetic algorithm (GA).

The content of optimization algorithm settings includes objective fitness function design, constraint condition setting and algorithm parameter setting. First, designing a good fitness function according to the objective function is an important part of the optimization system, and it is also the basis for judging the pros and cons of the design scheme. For the airfoil pillar-shaped flume, the optimization objectives of its hydraulic characteristics can mainly be selected from the quantitative parameters of upstream backwater height, Froude number, submerged degree, and head loss. The fitness function can be reasonably designed by the penalty function method according to the physical parameters or flow characteristics of the airfoil pillar-shaped flume and channel. The penalty function method refers to adding a penalty constant to the original objective function to obtain an augmented objective function. Its function is to give a maximum value to the non-feasible point or the point trying to cross the boundary and escape from the feasible region, that is, to convert the constrained optimization problem into the unconstrained optimization problem. After the fitness function is determined, it is necessary to impose constraints on the range of independent variables or dependent variables of each module according to the design requirements to control the degree of change. The parameter settings of the optimization algorithm are factors such as the size of the design population, selection, crossover, mutation characteristics, and convergence criteria, which affect the computational efficiency of finding the optimal solution.

In this paper, the airfoil pillar-shaped flume in the rectangular channel is selected as the research object, aiming at the problem of water blocking in the flow measurement in plain irrigation area, the airfoil shape is optimized with the reduction of the backwater height and the head loss as the hydraulic characteristics optimization objectives. According to the actual situation, the design discharge rate in the optimization process is taken as 30 L/s, and the relevant design parameters of the rectangular channel and the airfoil pillar-shaped flume are shown in Table 1.

The fluid domain of the airfoil pillar-shaped flume is symmetrical, so half of the original fluid domain is modeled in ICEM, and the fluid domain is meshed by the Cartesian co-ordinate method. For the two hydraulic characteristics of upstream backwater height and head loss in this study, the key is to extract the water depth data of the upstream stable section and the downstream stable section. The Cartesian co-ordinate meshing method is not disturbed by the change of the airfoil shape, and it can effectively ensure the uniformity of mesh quality in these two key regions. Set the side lengths to 0.025, 0.02, 0.015, and 0.01 m, respectively, for the mesh independent analysis of the cube mesh size, the results are shown in Table 2. As the mesh size decreases, the relative error between meshes becomes smaller and smaller. Considering the requirements of computational accuracy and computational efficiency, this study finally decided to use a mesh size of 0.015 m.

In this study, the NSGA-II algorithm is used as the guide for discriminative optimization in the operation framework. The content of optimization algorithm setting includes three aspects: objective fitness function design, constraint condition setting, and algorithm parameter setting.

According to the optimization objective and hydraulic parameter characteristics, the fitness function is designed as follows:

The value ranges of the design variables are shown in Table 3. The value standard is to ensure that each co-ordinate point of the basis airfoil can move between the minimum and maximum values, that is, the active area of the corresponding airfoil curve is a rectangle (1 m × 0.1 m).

The parameter values of the NSGA-II algorithm are set according to the optimization requirements and experience, as shown in Table 4.

As can be seen from Figure 6, in terms of shape, compared with the basic airfoil, under the condition of fixed airfoil chord length, the throat position of the optimized airfoil moves backward from the position of 0.3 m to the position of 0.46 m, making the contraction section longer and the curvature of its contour curve smaller, while the corresponding diffusion section is shorter and the curvature of its contour curve larger. It can be seen from Table 5 that the backwater height of the airfoil pillar-shaped flume optimized in terms of hydraulic characteristics is reduced from 5.48 to 4.96 cm, with a relative reduction of 9.49%. The head loss decreased from 5.80 to 5.33 cm, with a relative reduction of 8.10%. According to the analysis, the change trend of the shape is to improve the water contraction effect in time and space as much as possible on the basis of taking into account the diffusion effect of the downstream water flow, which reduces the disturbance and obstruction effect of the flume structure on the upstream water flow, thus retaining more flow velocity head and making the upstream backwater lower than the prototype. To sum up, the analysis proves that the hydraulic characteristics optimization system for the airfoil pillar-shaped flume is reliable.

With the continuous improvement of the requirements for flow measurement facilities in the water transmission and distribution system of the irrigation area, it is necessary to reasonably improve the existing facilities to reduce the impact on the flow and adapt to more severe hydraulic conditions. In order to make the work more systematic, refined, and intelligent, this paper designs an optimization system of hydraulic characteristics based on the airfoil pillar-shaped flume as the research object, which is mainly composed of three modules. Firstly, this paper uses the Hicks–Henne function in the airfoil parameterization method to ensure the smooth transition of the curve while disturbing the adjustment of the basic airfoil curve. Then the flow field of the airfoil pillar-shaped flume is automatically simulated by using the numerical simulation technology and the required hydraulic parameters are extracted. In the optimization process, the NSGA-II algorithm is used to properly constrain the design variables and hydraulic parameters, and the penalty function method is used to process the fitness value to guide the optimization direction in a timely manner.

The optimization design concept of the system is demonstrated concretely by using the example of the airfoil pillar-shaped flume under a discharge rate of 30 L/s. The optimization results show that the backwater height is reduced by 9.49% compared with the prototype, and the head loss is reduced by 8.10% compared with the prototype. Furthermore, the optimized airfoil also performs well in hydraulic characteristics under other discharge conditions. These prove the feasibility of the optimization system. In addition, it is possible to flexibly adjust each module in the system according to the demand and actual situation. For example, other parameterization methods can be used in the shape reconstruction module to represent different flow measurement facilities, different channel fluid domain models can be established in numerical simulation, and other optimization objectives and constraints can be considered in the optimization algorithm.

This research was supported by the National Key Research and Development Program of China (No. 2017YFC1501204), the National Natural Science Foundation of China (No. 51909242), the Program for Science and Technology Innovation Talents in Universities of Henan Province (No. 19HASTIT043), and the Outstanding Young Talent Research Fund of Zhengzhou University (1621323001).

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

The authors declare there is no conflict.