NSGA-II-based S airfoil profile and bidirectional axial flow pump multi-objective optimization method

Abstract

The invention discloses an S airfoil profile and bidirectional axial flow pump multi-objective optimization method based on NSGA-II, and belongs to the field of S airfoil profile optimization. The method comprises the following steps: parameterizing an S airfoil profile to obtain a parameterized S airfoil profile geometric model; performing numerical calculation based on the parameterized S airfoil profile geometric model to obtain airfoil profile performance data; constructing an S airfoil profile multi-objective optimization model based on the performance data of the airfoil profile, and solving the S airfoil profile multi-objective optimization model based on an NSGA-II algorithm to obtain an optimal airfoil profile design scheme; constructing a new impeller model based on the optimal airfoil design scheme; and performing numerical simulation on the new impeller model to obtain a water conservancy performance evaluation result.

Description

Technical Field

[0001] The invention belongs to the technical field of S-airfoil optimization, and in particular relates to a multi-objective optimization method of an S-airfoil and a bidirectional axial flow pump based on NSGA-II. Background Art

[0002] Bidirectional pumps are widely used in horizontal tubular units due to their simple flow channel structure, good inflow state, and convenient operation and management. They are also widely used in agricultural production due to their low head, large flow rate, and bidirectional pumping function to meet the needs of irrigation in the dry season and drainage in the rainy season. However, the number of existing bidirectional pump models is limited, which is difficult to meet the needs of engineering construction and agricultural production activities. In particular, the S-shaped airfoil, a reversible airfoil commonly used in reversible rotating machinery, is still less studied and applied in the field of hydraulic machinery, which limits the further improvement of the performance of bidirectional pumps.

[0003] The existing technologies on S-foils in the field of hydraulic machinery include: Huang Diangui et al. constructed an S-foil with good performance based on the front half of some NACA 4-digit series airfoils, and designed a reversible fan with high forward and reverse efficiency. Xie Rong et al. also used a similar method to design an asymmetric S-foil suitable for reversible fans. Wang Shiming et al. designed a tidal energy turbine based on the S-foil modified from the NACA24112 airfoil, and pointed out that the optimal design angle of attack of the S-foil is between 12° and 16°. In the field of bidirectional pumps, Tang Fangping et al. designed an S-foil suitable for low-lift bidirectional pumps. Compared with the flat airfoil, the S-foil has better efficiency and cavitation performance, and pointed out that the lift coefficient of the bidirectional airfoil is less than 0.8, and the head of the bidirectional pump generally does not exceed 5m. After that, Xie Chuanliu et al. studied the energy performance and cavitation performance of the S-foil and the bidirectional pump.

[0004] Although certain achievements have been made in the research on S-airfoils in the prior art, there are obvious technical defects in the design and optimization of S-airfoils in the prior art, which is mainly manifested in the challenges of improving lift-to-drag ratio and cavitation performance. Therefore, the present invention proposes a multi-objective optimization method of S-airfoils and bidirectional axial flow pumps based on NSGA-II to comprehensively consider these two aspects of performance. Summary of the invention

[0005] In order to solve the above technical problems, the present invention proposes a multi-objective optimization method of an S-shaped airfoil and a bidirectional axial flow pump based on NSGA-II to solve the problems existing in the above prior art.

[0006] To achieve the above object, the present invention provides a method for multi-objective optimization of an S-shaped airfoil and a bidirectional axial flow pump based on NSGA-II, comprising the following steps:

[0007] Parameterizing the S-airfoil to obtain a parameterized S-airfoil geometric model;

[0008] Performing numerical calculation based on the parameterized S-airfoil geometric model to obtain performance data of the airfoil;

[0009] Building an S-shaped airfoil multi-objective optimization model based on the performance data of the airfoil, and solving the S-shaped airfoil multi-objective optimization model based on the NSGA-II algorithm to obtain an optimal airfoil design solution;

[0010] Constructing a new impeller model based on the optimal airfoil design scheme;

[0011] The new impeller model is numerically simulated to obtain hydraulic performance evaluation results.

[0012] Optionally, the parameterized S-airfoil geometric model includes mathematical expressions of camber line and thickness distribution;

[0013] Wherein, the camber line is a fourth-order Bezier curve; and the thickness distribution is a combination of a circular arc and a fourth-order Bezier curve.

[0014] Optionally, the turbulence model used in the numerical calculation based on the parameterized S-airfoil geometric model is the SST k-ω model, and the airfoil boundary layer grid is encrypted to control the wall y + Within 30.

[0015] Optionally, the performance data of the airfoil include: lift coefficient and drag coefficient;

[0016] Wherein, the expressions of lift coefficient and drag coefficient are:

[0017]

[0018] In the formula, C L is the lift coefficient, C D Indicates the drag coefficient, F L and F D are the lift and drag on the airfoil, ρ is the density, b is the span length of the airfoil, u ∞ is the inlet flow velocity, and c is the chord length.

[0019] Optionally, the expression of the S-wing multi-objective optimization model is:

[0020]

[0021] In the formula, max F1 represents the lift-to-drag ratio optimization target, max F2 represents the cavitation performance optimization index, and C Ldrepresents the lift coefficient of the airfoil at the design angle of attack, ori represents the original solution, opt represents the optimized solution, m2, m3, m4 are the horizontal coordinates of the second, third and fourth control points of the camber line, e3 is the vertical coordinate of the third control point of the camber line, α1 and α2 are the angles between the tangent line and the horizontal line at the leading edge and the middle of the camber line, r b and α3 represent the arc radius and arc angle at the leading edge in the thickness distribution, v3, v4, and v5 are the abscissa values ​​of the third, fourth, and fifth control points of the camber line, respectively, and o4 represents the ordinate value of the fourth control point in the thickness distribution.

[0022] Optionally, the parameters of the NSGA-II algorithm are set as: crossover probability is 0.9, initial population size is 28, genetic generation is 60, crossover distribution index is 10, and mutation distribution index is 20.

[0023] Optionally, in the numerical simulation, the impeller is a rotating domain, and the remaining areas are stationary domains, and a mixed plane model is used to process parameter transfer of the flow during dynamic and static coupling between the impeller inlet and the impeller outlet and the guide vane section.

[0024] Compared with the prior art, the present invention has the following advantages and technical effects:

[0025] The present invention adopts a multi-objective optimization method based on the NSGA-II algorithm, and realizes the simultaneous optimization of the lift-to-drag ratio and the cavitation performance for the S-shaped airfoil and the bidirectional axial flow pump design. By parameterizing the S-shaped airfoil geometric model and combining numerical calculation to obtain the airfoil performance data, a multi-objective optimization model is constructed, and the NSGA-II algorithm is used to solve the optimal airfoil design scheme. This scheme not only improves the lift-to-drag ratio of the airfoil and enhances the hydraulic performance of the pump, but also improves the cavitation performance and reduces the cavitation risk of the pump during operation. Furthermore, the hydraulic performance of the new impeller model constructed based on the optimal airfoil design scheme is verified by numerical simulation, ensuring the efficiency and reliability of the design. This method provides an innovative optimization path for the design of a bidirectional axial flow pump, which has important engineering application value and broad market prospects. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] The drawings constituting a part of the present application are used to provide a further understanding of the present application. The illustrative embodiments and descriptions of the present application are used to explain the present application and do not constitute an improper limitation on the present application. In the drawings:

[0027] Figure 1 is a flow chart of an embodiment of the present invention;

[0028] Figure 2 A schematic diagram of the forward and reverse operation of a bidirectional pump according to an embodiment of the present invention;

[0029] Figure 3This is a schematic diagram of the main parameters of the S-shaped airfoil according to an embodiment of the present invention;

[0030] Figure 4 The airfoil parameterization method of an embodiment of the present invention, wherein Figure (a) represents the camber line, Figure (b) represents the thickness distribution, and Figure (c) represents the airfoil fitting;

[0031] Figure 5 The airfoil calculation grid of the embodiment of the present invention;

[0032] Figure 6 is the lift coefficient C of the airfoil of the embodiment of the present invention L Comparison of calculated values ​​with experimental values;

[0033] Figure 7 A water pump computational domain grid according to an embodiment of the present invention;

[0034] Figure 8 The grid independence detection result of the bidirectional pump in the embodiment of the present invention;

[0035] Fig. 9 The figure is a comparison of the external characteristics of a bidirectional pump obtained by numerical calculation and experiment of an embodiment of the present invention, wherein figure (a) represents the forward operation condition, and figure (b) represents the reverse operation condition;

[0036] Fig.10 is the lift coefficient C of the embodiment of the present invention L Pareto frontier of iteration and optimization results, where (a) represents the lift coefficient C L Iteration,Figure (b) shows the Pareto front;

[0037] Fig.11 The shape comparison of the airfoil before and after optimization of the embodiment of the present invention;

[0038] Fig.12 Comparison of lift and drag performance of the optimized front and rear airfoils of the embodiment of the present invention;

[0039] Fig.13 The airfoil surface pressure coefficient distribution at different attack angles according to an embodiment of the present invention;

[0040] Fig.14 The forward and reverse performance comparison of the pump section before and after optimization of the embodiment of the present invention;

[0041] Fig.15 is the total pressure rise coefficient Y on the back of the blade in the embodiment of the present invention p and streamline distribution, where Figure (a) represents the forward operating condition, and Figure (b) represents the reverse operating condition;

[0042] Fig.161 is the distribution of different spanwise cross-sectional velocity coefficients v* of the embodiments of the present invention, wherein Figure (a) represents the forward running condition, and Figure (b) represents the reverse running condition;

[0043] Fig.17 1 is a bidirectional pump cavitation performance curve under the design flow condition of an embodiment of the present invention, wherein Figure (a) represents the forward operation condition, and Figure (b) represents the reverse operation condition;

[0044] Fig.18 1 is the cavitation volume distribution near the inlet edge of the back side of the blade under different cavitation numbers σ of the embodiment of the present invention, wherein Figure (a) represents the forward operating condition, and Figure (b) represents the reverse operating condition. DETAILED DESCRIPTION

[0045] It should be noted that, in the absence of conflict, the embodiments and features in the embodiments of the present application can be combined with each other. The present application will be described in detail below with reference to the accompanying drawings and in combination with the embodiments.

[0046] It should be noted that the steps shown in the flowcharts of the accompanying drawings can be executed in a computer system such as a set of computer executable instructions, and that, although a logical order is shown in the flowcharts, in some cases, the steps shown or described can be executed in an order different from that shown here.

[0047] Embodiment 1

[0048] like Figure 1 As shown, in this embodiment, a method for multi-objective optimization of an S-shaped airfoil and a bidirectional axial flow pump based on NSGA-II is provided, comprising the following steps: parameterizing the S-shaped airfoil to obtain a parameterized S-shaped airfoil geometric model; performing numerical calculation based on the parameterized S-shaped airfoil geometric model to obtain performance data of the airfoil; constructing an S-shaped airfoil multi-objective optimization model based on the performance data of the airfoil, and solving the S-shaped airfoil multi-objective optimization model based on the NSGA-II algorithm to obtain an optimal airfoil design scheme; constructing a new impeller model based on the optimal airfoil design scheme; and performing numerical simulation on the new impeller model to obtain a hydraulic performance evaluation result.

[0049] As a specific implementation of this embodiment, it specifically includes: taking a high specific speed bidirectional pump with a specific speed (forward operation) of about 1200 as the object, further exploring the optimization design method of the bidirectional pump. On the basis of verifying the accuracy of the numerical calculation, based on NSGA-II (the second generation non-dominated sorting genetic algorithm, non-dominated sorting genetic algorithm-II), the reversible airfoil (S airfoil) near the mid-span of the impeller is selected to optimize the airfoil performance under multiple attack angles (+2°, +4° and +6°). During the optimization, the optimal weighted cavitation performance of the airfoil and the maximum lift-to-drag ratio are taken as the goals. Under the premise of ensuring that the head of the bidirectional pump remains unchanged, the lift coefficient under the original airfoil design angle of attack is basically unchanged as a constraint, the maximum thickness and maximum thickness position of the S airfoil are kept unchanged, and the 12 design parameters of the center arc and thickness distribution are optimized and solved. From the Pareto solution set, the S airfoil with better lift-to-drag ratio performance and cavitation performance is selected as the preferred model of the bidirectional pump impeller, and the impeller performance before and after optimization is compared. The research results can provide a reference for the optimal design of reversible rotating machinery.

[0050] 1. Research object: A bidirectional axial flow pump with a specific speed of about 1200 (forward operation) is taken as the object. The impeller of the axial flow pump adopts S-shaped airfoil design. The main design parameters include: impeller diameter D = 0.3m, hub ratio is 0.4, impeller blade number is 4, guide vane number is 5. The design flow rate under forward operation condition is 0.368m 3 / s, the design head is 3.77m, the maximum efficiency is 80.37%; the design flow rate under reverse operation is 0.316m 3 / s, the designed head is 3.66m and the maximum efficiency is 70.37%.

[0051] Figure 2 This is a schematic diagram of the forward and reverse operation of a bidirectional pump. When running in the forward direction, the water flows through the impeller first and then the guide vanes. When viewed from the impeller outlet, the impeller rotates clockwise. When running in the reverse direction, the water flows through the guide vanes first and then the impeller. When viewed from the impeller outlet, the impeller rotates counterclockwise. Since the guide vanes act as reverse pre-rotation during reverse operation, the impeller inflow state is poor. Compared with forward operation, the reverse performance of a bidirectional pump is usually relatively poor.

[0052] 2. S-wing and bidirectional pump optimization design method:

[0053] S-wing parameterization: Figure 3 The main parameters of the S-shaped airfoil are shown in Figure 2. The S-shaped airfoil is a centrally symmetrical structure. Its main structural parameters include: chord length c, maximum camber ratio f max / c and maximum camber position x fmax / c, maximum thickness ratio t max / c and maximum thickness position x tmax / c. The parametric expression of airfoil is the basic condition for airfoil optimization. The superposition of the camber line and thickness distribution is one of the commonly used airfoil parameterization methods. The camber line of the S-shaped airfoil is a reverse symmetrical structure about the midpoint, and the thickness distribution is a front-back symmetrical structure about the midpoint. Therefore, the shape of the entire airfoil can be obtained by describing the camber line and thickness distribution of the front half of the airfoil. Airfoil parameterization methods such as Figure 4 The Bezier curve is used to describe the camber line and thickness distribution. The camber line is a fourth-order Bezier curve, which is controlled by six parameters: α1, m2, m3, e3, m4 and α2. The thickness distribution is a combination of a circular arc and a fourth-order Bezier curve, which is controlled by six parameters: α3, r b , v3, v4, o4 and v5 six parameters control. This method has a wide degree of freedom in airfoil modification and has a better fitting effect on the original airfoil.

[0054] Define the spanwise dimensionless distance r * :

[0055]

[0056] Where r is the radius of the stream surface, r t and r h They are the rim and hub radius respectively, in m.

[0057] x / c and y / c represent the relative horizontal coordinate and the relative vertical coordinate, respectively, x is the horizontal coordinate value, y is the vertical coordinate value; f / c and t / c represent the relative camber and relative thickness of the airfoil, respectively, f represents the camber of the airfoil, t represents the thickness of the airfoil, and the camber and thickness at the leading edge (x=0) are both 0; m2, m3, m4 are the horizontal coordinate values ​​of the second, third, and fourth control points of the camber line, respectively; e3 represents the vertical coordinate value of the third control point of the camber line; α1 and α2 are the angles between the tangent and the horizontal line at the leading edge (x=0) and the middle (x=0.5) of the camber line, respectively; r b and α3 represent the arc radius and arc angle at the leading edge in the thickness distribution; v3, v4, v5 are the horizontal coordinate values ​​of the third, fourth, and fifth control points of the arch line, respectively; o4 represents the vertical coordinate value of the fourth control point in the thickness distribution.

[0058] Select the area near the mid-span of the impeller from the blade wooden model diagram (r * =0.44) section blade shape is optimized, and its main parameters include: chord length c = 0.15m, maximum camber ratio f max / c and maximum camber position x fmax / c are 1.4% and 0.23 respectively, and the maximum thickness ratio t max / c and maximum thickness position x tmax / c is 5% and 0.5 respectively. The maximum thickness and maximum thickness position of the airfoil are kept unchanged, and the camber line and thickness distribution of the S airfoil are optimized. The range of airfoil design parameters is shown in Table 1, which mainly refers to the original airfoil fitting results to ensure a large exploration space while avoiding distortion of the airfoil surface.

[0059] Table 1

[0060]

[0061] Numerical calculation method of airfoil: The performance of airfoil is predicted by solving the Reynolds time-averaged NS equations, and the SST k-ω model is selected as the turbulence model. The SST k-ω model takes into account the transmission of turbulent shear stress and has a high prediction accuracy for the separation of fluid under negative pressure gradient conditions. It combines the advantages of the high accuracy of the k-ω model near the wall and the fast calculation speed of the k-ε model in the fully developed turbulence area. ICEM is used to divide the airfoil into two-dimensional structured grids, encrypt the airfoil boundary layer grid, and control the wall y + Within 30. The maximum angle of attack of the original airfoil lift-to-drag ratio is about +4°, so at +4° angle of attack, the lift coefficient is used as an indicator for grid independence detection. L and the drag coefficient C D The definition is as follows:

[0062]

[0063] In the formula, F L and F D are the lift and drag of the airfoil, in N; ρ is the density, in kg / m 3 ; b is the span length of the airfoil, unit: m; u ∞ is the inlet flow velocity, unit: m / s.

[0064] Keeping the same topology, changing the maximum grid size, five grids with different numbers of grids are generated, with the numbers of grids being 25,400, 37,600, 55,700, 68,700, and 74,400 respectively. ∞ =9m / s, Reynolds number Re = 1.34×10 6 Under these conditions, the calculated values ​​of lift coefficient are 0.295, 0.296, 0.299, 0.298 and 0.299 respectively. When the number of grids reaches about 55,700, the lift coefficient tends to be stable. Finally, the number of calculation grids is determined to be 55,700. The calculation grid of the airfoil is as follows: Figure 5 shown. Figure 6 The calculated values ​​of lift coefficient of airfoil at different angles of attack are compared with the experimental values. The two values ​​are in good agreement and the numerical calculation results have high accuracy.

[0065] S-wing optimization model: lift-to-drag ratio C L / C D It is an important indicator to measure the performance of the airfoil. The hydrodynamic optimization goal of the airfoil is to maintain a high lift-to-drag ratio in a large angle of attack range. The lift-to-drag ratio performance of the airfoil is defined by a weighted method. The maximum weighted value F1 of the lift-to-drag ratio at three angles of attack of +2°, +4° and +6° is taken as the optimization goal. The lift-to-drag ratio optimization goal is

[0066]

[0067] Where a i is the weight factor, i=1, 2, 3 represent +2°, +4° and +6° respectively. A bidirectional pump station has been operating under large flow and low head conditions for a long time, that is, the airfoils of various sections of the pump are mostly working under small angle of attack conditions. Therefore, under the condition of ensuring the lift-to-drag ratio at the design angle of attack, a higher weight is given to the lift-to-drag ratio of the airfoil at small angle of attack, and the weight factors a1, a2, and a3 are 0.3155, 0.4059, and 0.2786 respectively.

[0068] Define the dimensionless pressure coefficient C P

[0069]

[0070] In the formula, P represents pressure, P in is the inlet pressure, unit: Pa.

[0071] Cavitation performance is another important index of hydraulic airfoil. For axial flow pump, the lowest pressure coefficient near the leading edge of airfoil (0~0.1c) usually determines the cavitation number of cavitation initiation, while the pressure coefficient within the range of 0.1c~0.25c of airfoil is closely related to the development of cavitation, and usually determines the critical cavitation number of axial flow pump. Therefore, when optimizing the cavitation performance, the present embodiment takes the maximum weighted value of the minimum pressure coefficient of 0~0.1c and 0.1c~0.25c of airfoil as the goal, and the cavitation performance optimization index F2 is defined as:

[0072]

[0073] In the formula, min(C P ) le Indicates the minimum pressure coefficient on the leading edge of the airfoil, min(C P ) mid It represents the minimum pressure coefficient near the middle of the airfoil, and the weight factors b1, b2, and b3 also have values ​​of 0.3155, 0.4059, and 0.2786.

[0074] In order to ensure that the specific speed of the bidirectional pump is basically the same before and after optimization, the head should not change significantly under the design flow condition. According to the lift method design theory, the head and lift coefficient are positively correlated. Under the design flow condition, the airfoil usually works near the optimal angle of attack. Therefore, the lift coefficient C of the airfoil under the design angle of attack before and after optimization is Ld It should remain basically unchanged. In this embodiment, the lift coefficient value before and after optimization under the constraint design attack angle changes less than 0.02. Finally, the airfoil optimization problem can be described as:

[0075]

[0076] In the formula, the subscript ori represents the original solution, and the subscript opt ​​represents the optimized solution.

[0077] NSGA-II algorithm: In multi-objective optimization problems, the various objectives conflict with each other. The optimal solution cannot be a single solution, but a solution set, called the Pareto optimal solution set, and the image of the corresponding objective function space is called the Pareto frontier. The NSGA-II algorithm is a genetic algorithm based on the concept of Pareto optimality. Its main contents include: fast non-dominated sorting, individual crowding distance operator design and elite retention strategy. The specific process is as follows:

[0078] 1) Randomly generate an initial population A0 with a population size of N, perform non-dominated sorting according to the fitness function values ​​of individuals in population A0, and calculate the crowding distance of individuals in each Pareto frontier;

[0079] 2) For the parent population A t Perform selection, crossover, and mutation operations to generate offspring population B t ;

[0080] 3) The parent population A t and progeny population B t Merge to generate a new population C t , for population C t Perform a fast non-dominated sort operation to obtain a new Pareto frontier;

[0081] 4) Calculate the individual crowding distance according to the Pareto frontier order, and select N individuals according to the crowding ranking to form a new parent population A t+1 ;

[0082] 5) Determine whether the algorithm has reached the termination condition. If so, stop the calculation. Otherwise, increase the number of iterations by 1 and return to step 2 to repeat the above operation.

[0083] 6) The algorithm iteration is completed and the Pareto optimal solution set is obtained.

[0084] This embodiment is based on multidisciplinary optimization design software and adopts NSGA-II algorithm for optimization. The NSGA-II used is set as follows: crossover probability is 0.9, initial population size is 28, genetic generation is 60, crossover distribution index is 10, variation distribution index is 20, and a total of 1680 iterations are performed.

[0085] Bidirectional pump impeller modeling and numerical simulation method: After the airfoil optimization is completed, the airfoil with better lift-to-drag ratio and cavitation performance is selected from the Pareto frontier as the final solution, and the main design parameters of the original impeller (blade density distribution, airfoil placement angle distribution, maximum thickness distribution, etc.) are kept unchanged. The original airfoil is replaced with the preferred airfoil. The airfoil is scaled according to the blade density and maximum thickness of each spanwise section to form a new impeller, which is recorded as IM1. The original impeller is recorded as IM0. The impeller IM0 and IM1 plus the guide vanes form a pump section for numerical simulation. The turbulence model also uses the SST k-ω model, and Turbogrid is used for hexahedral structured mesh division. At the same time, the near-wall mesh is encrypted to ensure its y + In order to fully develop the flow, an extension section is set at the impeller inlet and the guide vane outlet during forward operation, with a length of about 1.5 times the impeller diameter. The entire calculation domain includes the inlet section, impeller, guide vane and outlet section. The calculation domain grid of the water pump is as follows: Figure 7 shown.

[0086] The calculation domain of the water pump is divided into a rotating domain and a stationary domain. The impeller is the rotating domain with an impeller speed of 1450r / min, and the rest of the area is the stationary domain. The "Stage" interface (also called a mixed plane model) is used to handle the parameter transfer of the flow when the impeller inlet and impeller outlet are coupled with the guide vane section. The total pressure inlet and flow outlet conditions are used in the calculation, and the solid wall surfaces such as the inlet section, outlet section, impeller hub, casing and guide vane body are applied with no slip conditions. In forward operation, the design flow rate is 0.368m 3 / s working condition, the grid independence detection is carried out with the lift as the index. The independence detection method is the same as the previous one. Figure 8 This is the result of the bidirectional pump grid independence test. The total number of pump calculation grids is finally determined to be 5.48×10 6 .

[0087] Fig. 9The figure shows the comparison of the external characteristics of the bidirectional pump obtained by numerical calculation and experiment. Due to the simplification of the calculation model and the neglect of the elbow loss, the forward and reverse heads and efficiency of the bidirectional pump obtained by numerical calculation are slightly higher than the experimental values. Within the calculated flow range, the numerical calculation results are basically consistent with the experimental value change trend. The maximum point of numerical calculation error appears near the saddle area, mainly due to the strong instability of the flow near the saddle area. Under the forward and reverse design flow conditions, the error of the head and efficiency obtained by numerical calculation is less than 5%. The numerical simulation has a high prediction accuracy, and the numerical calculation method is relatively reasonable.

[0088] 3. Analysis of optimization results:

[0089] Analysis of S-wing optimization results: The NSGA-Ⅱ algorithm is used to optimize and solve the design space of formula (6). Fig.10 The Pareto frontier of the lift coefficient iteration history and optimization results shows that there are 1196 solutions in the iteration history that satisfy the lift coefficient constraint of formula (6), of which 115 Pareto solutions satisfy the optimization target. The appearance of the Pareto frontier means that the lift-drag performance and cavitation performance of the airfoil are mutually restricted, and it is difficult to achieve the optimal solution at the same time. The original solution in the figure is denoted as Ori. Compared with the original solution, the lift-drag ratio performance index F1 in the Pareto optimal solution is increased by about 27.7% at most, and the cavitation performance index F2 is increased by about 1.7% at most. The multi-objective optimization of the airfoil has achieved good results. In order to further analyze the optimization results, three solutions Sc1, Sc2 and Sc3 are selected from the Pareto frontier for further analysis. Sc1 is the solution with the maximum lift-drag ratio performance index F1, Sc3 is the solution with the maximum cavitation performance index F2, and Sc2 is the solution with improved lift-drag ratio and cavitation performance. Table 2 compares the airfoil parameters of different solutions.

[0090] Table 2

[0091]

[0092] Fig.11 To optimize the front and rear airfoil shape comparison. Schemes Sc1 to Sc3 Maximum camber position x fmax / c are 0.32, 0.22, 0.32 respectively, and the maximum camber of the airfoil f max / c are 1.0%, 0.9% and 1.6% respectively. Compared with the original scheme Ori, the maximum camber of the airfoil of schemes Sc1 and Sc2 is reduced, while the maximum camber of the airfoil of scheme Sc3 is increased. For the S airfoil, within a certain range, the reduction of the maximum camber of the airfoil is conducive to the improvement of the lift-to-drag ratio performance, while the increase of the maximum camber of the airfoil is conducive to the improvement of the cavitation performance. Comparison of airfoil thickness distribution ( Fig.11b) It can be seen that the thickness distribution of the original Ori airfoil increases approximately linearly in the range of 0.02 to 0.4 times the chord length. After optimization, the airfoil thickness of each scheme increases within this range, and the airfoil thickness distribution becomes fuller and smoother, which is conducive to improving the pressure distribution on the airfoil surface.

[0093] Fig.12 Comparison of lift and drag performance of the airfoil before and after optimization. L The curves are basically the same. Except for the negative lift angle of attack of 0° and the large angle of attack of +10°, the lift coefficient changes at other angles of attack are basically within 0.04. In the non-stall angle of attack range, the lift coefficient increases approximately in a straight line. Compared with the original scheme, the lift-to-drag ratio of schemes Sc1 and Sc2 is significantly improved in the range of 0° to +10° angle of attack, and the lift-to-drag ratio at angles of attack of +2°, +4°, and +6° is increased by 45.7%, 16.1%, 37.0% and 15.0%, 9.1%, 36.9% respectively. Scheme Sc3 focuses more on improving cavitation performance. In the range of 1.3° to 5° angle of attack, the lift-to-drag ratio is less than the original scheme. Due to the large changes in the camber line and thickness distribution of the airfoil, the maximum angle of attack of the lift-to-drag ratio of schemes Sc1 to Sc3 changes from around +4° to around +6°.

[0094] Fig.13 The distribution of pressure coefficient on the airfoil surface at different angles of attack. The minimum pressure coefficient on the airfoil surface of each scheme before and after optimization is at the leading edge of the airfoil, and the minimum pressure coefficient shows a gradual increase from scheme Sc1 to Sc3. Except that the minimum pressure coefficient of scheme Sc1 is lower than that of the original scheme at +4° and +6° angles of attack, the minimum pressure coefficient on the airfoil surface of each scheme after optimization is improved, and the cavitation number at the initiation point of airfoil cavitation is reduced. In the range of 0.1c to 0.25c, the minimum pressure coefficients of each scheme before and after optimization are roughly equal. The axial flow pump impeller designed by these four airfoil schemes theoretically has a similar critical cavitation number. In the range of 0 to 0.1c, the pressure coefficient on the lower surface of the airfoil of the original scheme rises more sharply. At angles of attack of +2° and +4°, there is also a large local high pressure area (elliptical area in the figure) in this area. The local distortion of pressure causes an increase in the drag coefficient, which is effectively suppressed in schemes Sc1 to Sc3. Compared with the original scheme, the pressure distribution on the airfoil surface of schemes Sc1 and Sc2 is smoother and the pressure gradient is more reasonable. Due to the particularity of the S-shaped airfoil structure, there is an intersection in the pressure coefficients of the upper and lower surfaces of the airfoil near the middle of the airfoil, and the rear half of the airfoil provides negative lift. Compared with other schemes, the pressure intersection point of scheme Sc3 is significantly advanced, which is also the main reason for its relatively low lift-to-drag ratio.

[0095] Energy performance analysis: Fig.14To compare the forward and reverse performance of the bidirectional pump before and after optimization, the forward head and efficiency are recorded as H1 and η1, and the reverse head and efficiency are recorded as H2 and η2. From the analysis in the previous article, it can be seen that the lift coefficient of the optimized airfoil is equivalent to that of the original airfoil at different attack angles, so the head curve of the optimized model pump section is basically consistent with the original scheme, which meets the optimization requirement that the specific speed of the water pump remains basically unchanged. Within the calculated flow range, the forward and reverse efficiencies of the optimized model pump section are improved. Taking 0.8, 1.0 and 1.2 times the forward and reverse design flow rates as examples, the forward efficiency is increased by 0.6, 0.5 and 2.2 percentage points, and the reverse efficiency is increased by 1.1, 0.2 and 0.4 percentage points. Due to the decrease in the drag coefficient of the airfoil, the impeller has a smaller shaft power after optimization within the calculated flow range, which is also the main reason for its efficiency improvement.

[0096] Impeller flow field analysis: To clarify the reasons for the improvement of bidirectional pump performance, the impeller flow field before and after optimization is further analyzed. Define the total pressure rise coefficient Y p And the velocity coefficient v*:

[0097]

[0098] Where P t Indicates the total pressure, and are the average total pressure and static pressure at the inlet, Pa; v t Indicates the blade tip circumferential speed, m / s.

[0099] The forward and reverse design flow rates are denoted as Q d1 and Q d2 , m 3 / s. Fig.15 is the total pressure rise coefficient and streamline distribution on the back of the blade. Under the conditions of 0.8 and 1.0 times the forward and reverse design flow rates, there are mainly two flow separation areas on the back of the blade: the leading edge of the blade (the elliptical area in the figure) and the blade hub near the trailing edge (the rectangular area in the figure). Under the conditions of 1.2 times the forward and reverse design flow rates, the flow separation on the back of the blade is mainly concentrated at the blade hub near the trailing edge (the rectangular area in the figure). The pressure of the original airfoil rises rapidly near the leading edge. Correspondingly, there is a local high total pressure area near the leading edge of the blade. Under the action of the total pressure gradient, the direction of the water flow is deflected. On the one hand, it induces a backflow pointing to the inlet edge (0.8 times the design flow), and on the other hand, it induces a secondary flow from the hub to the rim. The flow separation at the blade hub is also related to the uneven distribution of total pressure. There is an obvious total pressure gradient from the hub to the rim near the hub, which deflects the water flow toward the rim. Compared with the original scheme, the total pressure distribution on the blade surface of the optimized model is more uniform, the range of the local total pressure high value area at the leading edge of the back of the blade is reduced, the total pressure gradient near the hub is more reasonable, and the separation flow on the back of the blade is effectively suppressed.

[0100] Depend on Fig.15 It can be seen that the flow separation on the back of the blade mainly occurs near the hub and rim, so the intercept r * = 0.15, 0.85 spanwise cross sections are further analyzed. Fig.16 is the velocity coefficient distribution of different spanwise sections. When running in forward and reverse directions, the velocity variation trends in the blade channel before and after optimization under different flow conditions are similar, mainly manifested as follows: * =0.85 section, the local low-speed area at the head of the blade working surface and the local high-speed area in the middle are reduced, and the local low-speed area near the trailing edge of the back of the blade is reduced; * =0.15 section, the local low-speed area at the head of the blade working surface, the local high-speed area in the middle, and the local low-speed area near the trailing edge of the back also show a decreasing trend. In addition, the local low-speed area near the head of the back of the blade in the original scheme under low flow conditions almost disappears after optimization, and the speed change is smoother. In summary, the speed distribution in the blade channel after optimization is more uniform, which is conducive to reducing the reduction of the pump shaft power, thereby improving the hydraulic efficiency of the pump.

[0101] Impeller cavitation performance analysis: Check the cavitation performance of the bidirectional pump under the design flow condition, and use the Zwart model as the cavitation model. During the calculation, the pump inlet pressure is continuously reduced to produce cavitation in the pump, and the critical cavitation number is determined by the efficiency drop of 1 percentage point. The cavitation number σ is defined as

[0102]

[0103] Where P v is the vaporization pressure, Pa; u tip is the peripheral speed at the wheel rim, m / s.

[0104] Fig.17 To design the cavitation performance curve of the bidirectional pump under flow conditions, when running in the forward direction, the critical cavitation number of the IM0 scheme before optimization is 0.442, and the critical cavitation number of the IM1 scheme after optimization is 0.455; when running in the reverse direction, the critical cavitation number of the IM0 scheme before optimization is 0.426, and the critical cavitation number of the IM1 scheme after optimization is 0.432. The difference between the forward and reverse critical cavitation numbers before and after optimization is small, which are 0.013 and 0.006 respectively, that is, the changes in the critical cavitation margin are about 0.34m and 0.16m respectively. The critical cavitation margin of the pump after optimization does not increase much, which can meet the requirements of engineering applications.

[0105] The cavitation distribution in the impeller channel is shown using an isosurface with a cavitation volume fraction of 0.1. Fig.18The cavitation distribution near the inlet edge of the back of the blade under the design flow condition. Under the design flow condition, cavitation initiation occurs near the rim of the inlet edge of the back of the blade. The primary cavitation number of the optimized pump is significantly reduced: under the forward operation condition, the primary cavitation number of the original scheme is around 1.5, while the primary cavitation number of the optimized scheme is around 1.0; under the reverse operation condition, the primary cavitation number of the original scheme is around 2.0, while the primary cavitation number of the optimized model is around 1.5. With the decrease of the cavitation number, the cavitation gradually develops towards the outlet edge and the hub. Compared with the original scheme, the cavitation develops faster towards the outlet edge and slower towards the hub. This is related to the distribution of pressure coefficients on the airfoil surface analyzed in the previous article: the minimum pressure coefficient near the leading edge of the optimized airfoil is larger, which reduces the primary cavitation number. Although the local high pressure area on the back of the original airfoil near 0~0.1c causes an increase in hydraulic loss, it also inhibits the development of cavitation towards the outlet edge to a certain extent.

[0106] 1) Starting from the bidirectional pump airfoil, the present invention proposes a multi-objective optimization design method for the S-airfoil that comprehensively considers the lift-to-drag ratio performance and cavitation performance based on the combination of numerical simulation and NSGA-Ⅱ algorithm. This method can provide a reference for the optimization design of reversible airfoils and reversible rotating machinery.

[0107] 2) In the Pareto solution set, the lift-to-drag ratio performance index of the S-shaped airfoil is improved by about 27.7% at most, and the cavitation performance index is improved by about 1.7% at most. Further analysis of the three solutions in the solution set shows that for the S-shaped airfoil, reducing the maximum camber of the airfoil within a certain range is conducive to improving the lift-to-drag ratio, while increasing the camber of the airfoil is conducive to improving the cavitation performance.

[0108] 3) From the Pareto solution set, the solution with better lift-to-drag ratio and cavitation performance is selected as the optimal airfoil for the bidirectional pump impeller. The hydraulic performance of the bidirectional pump is effectively improved through airfoil optimization. At 0.8, 1.0 and 1.2 times the forward and reverse design flow rates, the forward efficiency is increased by 0.6, 0.5 and 2.2 percentage points, and the reverse efficiency is increased by 1.1, 0.2 and 0.4 percentage points. The internal flow field distribution is more uniform, the critical cavitation number of the pump changes little, and the primary cavitation number decreases significantly. In order to further improve the performance of the bidirectional pump, in the future, based on the optimized airfoil, the blade parameters such as the density of the bidirectional pump blade cascade will be optimized.

[0109] The above are only preferred specific implementations of the present application, but the protection scope of the present application is not limited thereto. Any changes or substitutions that can be easily thought of by a person skilled in the art within the technical scope disclosed in the present application should be included in the protection scope of the present application. Therefore, the protection scope of the present application should be based on the protection scope of the claims.

Claims

1. A multi-objective optimization method for S-shaped airfoil and bidirectional axial flow pump based on NSGA-II, characterized in that: The following steps are involved: Parameterizing the S-airfoil to obtain a parameterized S-airfoil geometric model; Performing numerical calculation based on the parameterized S-airfoil geometric model to obtain performance data of the airfoil; Building an S-shaped airfoil multi-objective optimization model based on the performance data of the airfoil, and solving the S-shaped airfoil multi-objective optimization model based on the NSGA-II algorithm to obtain an optimal airfoil design solution; Constructing a new impeller model based on the optimal airfoil design scheme; The new impeller model is numerically simulated to obtain hydraulic performance evaluation results.

2. The method according to claim 1, characterized in that The parameterized S-airfoil geometric model includes mathematical expressions of camber line and thickness distribution; Wherein, the camber line is a fourth-order Bezier curve; and the thickness distribution is a combination of a circular arc and a fourth-order Bezier curve.

3. The method according to claim 1, characterized in that The turbulence model used in the numerical calculation based on the parameterized S-airfoil geometry model is the SST k-ω model, and the airfoil boundary layer grid is encrypted to control the wall y + Within 30.

4. The method according to claim 3, characterized in that The performance data of the airfoil include: lift coefficient and drag coefficient; Wherein, the expressions of lift coefficient and drag coefficient are: In the formula, C L is the lift coefficient, C D Indicates the drag coefficient, F L and F D are the lift and drag on the airfoil, ρ is the density, b is the span length of the airfoil, u ∞ is the inlet flow velocity, and c is the chord length.

5. The method according to claim 1, characterized in that The expression of the S-wing multi-objective optimization model is: In the formula, max F1 represents the lift-to-drag ratio optimization target, max F2 represents the cavitation performance optimization index, and C Ld represents the lift coefficient of the airfoil at the design angle of attack, ori represents the original solution, opt represents the optimized solution, m2, m3, m4 are the horizontal coordinates of the second, third and fourth control points of the camber line, e3 is the vertical coordinate of the third control point of the camber line, α1 and α2 are the angles between the tangent line and the horizontal line at the leading edge and the middle of the camber line, r b and α3 represent the arc radius and arc angle at the leading edge in the thickness distribution, v3, v4, and v5 are the abscissa values ​​of the third, fourth, and fifth control points of the camber line, respectively, and o4 represents the ordinate value of the fourth control point in the thickness distribution.

6. The method according to claim 1, characterized in that The parameters of the NSGA-II algorithm are set as follows: crossover probability is 0.9, initial population size is 28, genetic generations is 60, crossover distribution index is 10, and mutation distribution index is 20.

7. The method according to claim 1, characterized in that In the numerical simulation, the impeller is a rotating domain, and the remaining areas are stationary domains. A mixed plane model is used to process parameter transfer of flow during dynamic and static coupling between the impeller inlet and the impeller outlet and the guide vane section.

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