Study on Cascade Density of the Impeller Based on Response Surface Analysis

Abstract

In order to improve the cavitation performance of hydraulic machinery in the design stage, the response surface analysis method is used to study the influence of cascade density on the hydraulic characteristics and cavitation characteristics of a three-dimensional rotating hydrofoil. For this method, an axial flow impeller with a specific speed of 750 is selected as the experimental object. Nine groups of three-dimensional rotating hydrofoils with different cascade densities were designed by the D-optimal quadratic sampling method. The impeller and guide vane were structured and meshed by TurboGrid. The inlet and outlet extension sections were structured and meshed by ICEM. Grid independence analysis was carried out. The hydraulic characteristics and cavitation characteristics of nine groups of hydrofoils were calculated by CFD numerical simulation. The calculation results of nine groups of schemes were analyzed from three angles of external characteristics, flow pattern and cavitation performance, and the better schemes under each analysis angle were obtained. By means of response surface analysis, more external characteristics and cavitation characteristics of different cascade densities will be filled. Finally, combined with the response target, the influence of cascade density on each target is analyzed and the comprehensive selection range of hub and shroud is given. This method can improve the rationality of sampling, and select a more suitable range of cascade density by designing fewer impellers.

1. Introduction

Cavitation is a common phenomenon in hydraulic machinery. When severe cavitation occurs, a large number of bubbles will be generated, blocking the impeller channel, resulting in a decrease in head and efficiency. The collapse of the cavitation bubble will not only induce vibration and noise, but also form an impinging jet to destroy the surface of blades, which will seriously threaten the operational stability of the pump. Therefore, cavitation performance should be improved as much as possible in the design stage. For the unsteady cavitation problem, domestic and foreign scholars have carried out a great deal of research on the hydrofoil, and have made great progress. The time-averaged pressure values at different attack angles and different positions are often used as parameters to study the cavitation evolution process of three-dimensional twisted hydrofoils.

Yu et al. [1] analyzed the effects of inlet incidence angle and cascade density on the hydraulic efficiency and internal flow characteristics of axial flow pump impellers. It was found that increasing the cascade density or reducing the inlet incidence angle is beneficial to improve the internal flow of the axial flow pump, reduce the pressure change and energy loss on the pressure surface of the blade and help to improve the hydraulic efficiency of the impeller. Ning et al. [2] analyzed the influence of the number of blades on the hydraulic performance and internal flow field of the new open pump under the same cascade density, which provides a theoretical basis for the selection of the number of composite impeller blades. Lin et al. [3] experimentally investigated the effect of fluid—structure interaction on unsteady cavitating flows around flexible and stiff National Advisory Committee for Aeronautics 0015 hydrofoils in a low-pressure cavitation tunnel, and observed that the fluid—structure interaction can significantly affect the cavitation-induced vibration of the flexible hydrofoil. Zhang et al. [4] experimentally study the cavitating fluid—structure interaction of composite hydrofoils with different ply angles. Podnar et al. [5] studied the influence of a bulb runner blade hydrofoil shape on flow characteristics around the blade, and observed that cavitation length is dependent on Reynolds number and the cavitation number. Custodio et al. [6] examined cavitation characteristics and hydrodynamic forces of hydrofoils with bioinspired, wavy leading edges experimentally in a water tunnel, and recorded cavitation patterns by directly imaging the hydrofoil surface. Gu et al. [7] used numerous turbulent viscosity correction approaches to improve the turbulence model, and conducted numerical simulations in conjunction with unsteady cavitation of a hydrofoil with 3° angle of attack, as well as application evaluation. The mechanism of hydrofoil cavitation instability and shedding was revealed, providing a theorem. Liu et al. [8] investigated the influence of the cavitation behavior response of three composite hydrofoils of carbon-fiber reinforced plastic with different ply angles. Park et al. [9] studied the effect of leading-edge droop in the oscillating hydrofoil with cavitation numerically, and compared the hydrodynamic performance of the baseline hydrofoil with that of the fixed droop and the variable droop hydrofoil. Conesa et al. [10] studied the effectiveness of a slotted hydrofoil in minimizing the cavitation phenomenon, to improve the overall water take-off performance of an amphibious aircraft, and performed a multi-objective optimization with a response surface model to simultaneously minimize the cavitation and maximize the hydrodynamic efficiency of the hydrofoil. Dular et al. [11] explained the phenomenon that cavitation on two-dimensional hydrofoils with swept leading edges always displays some three-dimensional effects by numerical simulation and experimental measurements. Ghadimi et al. [12] numerically assessed hydrodynamic performance of two NACA hydrofoils under cavitation and non-cavitation conditions, and observed that lift and drag forces of the two-dimensional Naca 6612 hydrofoil are larger than those of the Naca 0012 hydrofoil in both cavitation and non-cavitation conditions. Zhi et al. [13] numerically investigated the unstable vaporous cavities around a surface-piercing hydrofoil using the large-eddy simulation coupled with the Schnerr—Sauer cavitation model, and the results could contribute to the novel hydrofoil designs and their flow control. Wang et al. [14] presented a method of water injection to the flow field using distributed holes on the suction surface of a hydrofoil to control cavitation flow, which will facilitate development of engineering designs. Sun et al. [15] numerically studied the cavitating flow fields of a NACA 66 rigid hydrofoil and flexible hydrofoil, and used dynamic mode decomposition to capture the flow field modal characteristics. Gao et al. [16] used a combination of numerical simulation, orthogonal experiment and grey correlation analysis to optimize the design of auxiliary blades in multistage centrifugal pumps. Zhou et al. [17] studied the influence of the end clearance of the impeller on the performance and internal flow field of a high-speed electrical submersible pump under different working conditions, and gave a reasonable recommended value of the end clearance, which is beneficial to the optimization design and engineering application of the high-speed electrical submersible pump.

The cascade density (l/t) is defined as the ratio of the blade chord length l to the cascade spacing t. Increasing the cascade density (l/t) indicates that the total area of the impeller blade increases, and the pressure difference between the pressure surface and the suction surface decreases, which can improve the cavitation performance of the pump [18]. Traditional experimental designs (full factor experimental design, partial factor experimental design, response surface design) are suitable for linear models, but in some cases, the model is nonlinear. The D-optimal quadratic sampling method addresses these limitations of traditional design. In this paper, the cascade density of the hydrofoil at the hub and tip clearance is changed by the D-optimal quadratic sampling method [19], and the middle 13 sections are interpolated by the linear method. Based on these methods, nine groups of three-dimensional rotating hydrofoils with different cascade densities were designed. The hydraulic characteristics and cavitation characteristics of three-dimensional twisted hydrofoils with different cascade densities were analyzed by CFD numerical simulation. The response surface analysis method is used to give the optimal cascade density of the impeller with the specific speed under the optimization objectives of head, efficiency and net positive suction head. Finally, the comprehensive value range of the cascade density of the hub and shroud of the impellers under the specific speed impeller is given. The research process of this paper is shown in Figure 1.

2. Calculation Model and Numerical Method

2.1. Calculation Model and Control Parameters

In this paper, CFD numerical simulation is used to analyze the influence of cascade density on the hydraulic characteristics and cavitation characteristics of an axial flow pump section. The calculation area includes the inlet extension section, impeller chamber, guide vane chamber and 60° outlet elbow. The calculation model diagram is shown in Figure 2:

For the consideration of other factors, other parameters such as blade placement angle, chord length, blade thickness and inlet angle of attack are solved according to the relevant principles of impeller design when the cascade density is determined, and then different three-dimensional hydrofoils are drawn. It is worth mentioning that doing so loses the meaning of the control variables, but for practical projects such as coastal pumping stations, when designing the impeller, the cascade density should be available at the same time as other matching parameters, so that it has practical significance.

$${\mathrm{C}}_{\mathrm{yp}}\frac{1}{\mathrm{t}}=\frac{2\Delta {\mathrm{V}}_{\mathrm{u}}}{{\mathrm{W}}_{\mathrm{m}}}\frac{1}{1+\frac{\mathrm{tg}\mathsf{\lambda }}{{\mathrm{tg}\mathsf{\beta }}_{\mathrm{m}}}}$$

(1)

$\frac{1}{\mathrm{t}}$ is the cascade density, ${\mathrm{C}}_{\mathrm{yp}}$ is the lift coefficient of the airfoil in the cascade, ${\mathrm{W}}_{\mathrm{m}}$ is the geometric average of the relative velocity w₁ of the liquid flow before entering the cascade and the relative velocity w₂ of the liquid flow after leaving the cascade, $\mathsf{\lambda }$ is the angle between the lift and drag of the hydrofoil, ${\mathsf{\beta }}_{\mathrm{m}}$ is the blade placement angle [18].

D-optimal quadratic is used for scheme design [20]. Based on this method, nine samples of cascade density at different hubs and shrouds are extracted. The cascade density of different schemes is shown in

Table 1. Cascade density at hub and shroud of different impeller schemes.

Samples	Hub	Shroud
1	0.918	0.900
2	1.122	0.810
3	1.122	0.900
4	1.020	0.990
5	1.122	0.990
6	1.020	0.900
7	1.020	0.810
8	0.918	0.990
9	0.918	0.810

:

For these different 9 impellers, this paper uses Cfturbo (version 2020) software for parametric design, and the results are shown in Figure 3:

Figure 3a shows the impellers for nine cascade density samples. In order to reflect the difference, they are placed in the same picture. In order to visually and clearly show the differences in blades, the blades for nine cascade density samples are displayed in a semi-transparent manner, and each of the three groups is shown in Figure 3b, Figure 3c and Figure 3d, respectively. The numbers 1–9 in the figures are the sample numbers. Figure 3b is the impeller shape of scheme 1 to 3. Figure 3c is the impeller shape of scheme 4 to 6. Figure 3d is the impeller shape of scheme 7 to 9.

2.2. Numerical Simulation Method

The numerical simulation is based on the Reynolds time-averaged N—S control equation and the RNG k—ε turbulence model, which is intended to simulate the flow pattern. The discrete equations are solved by the separation solver, which is the SIMPLEC algorithm [21]. The flow pattern in the pump satisfies the mass conservation and momentum conservation equations. In this paper, the mixture is regarded as a single fluid by using the European single fluid method, and its density is related to the gas phase volume fraction α_v:

$${\rho }_m=(1-a_v){\rho }_l+a_v{\rho }_v$$

(2)

The momentum of the mixed fluid can be calculated as follows:

$${\rho }_m{u}_{mi}=(1-a_v){\rho }_l{u}_{li}+a_v{\rho }_v{u}_{vi}$$

(3)

The two phases have the same pressure, and the velocity difference between the two phases can be considered in the model by additional source terms. Only the continuity equation and momentum equation of the mixed fluid can be solved. The formula is:

$$\frac{\partial \overline{u_{mj}}}{\partial x_j}=0$$

(4)

In the formula: m denotes the relevant field quantity of the mixed fluid, u_mj denotes the time-averaged velocity in all directions.

$$\frac{\partial \overline{u_{mi}}}{\partial t}+\overline{u_{mj}}\frac{\partial \overline{u_{mi}}}{\partial x_j}=\overline{f_i}-\frac{1}{{\rho }_m}\frac{\partial \overline{p}}{\partial x_i}+\frac{1}{{\rho }_m}\frac{\partial }{\partial x_j}(\overline{{\tau }_{ji}}-{\rho }_m\overline{{u^{\prime }}_{mi}{u^{\prime }}_{mj}})$$

(5)

At present, the widely used cavitation models are Schnerr—Sauer cavitation model, Kunz cavitation model and ZGB (Zwart—Gerber—Belamri) cavitation model [22,23,24,25]. The Zwart—Gerber—Belamri cavitation model is the most widely used cavitation model at present, and it is also the model used in this paper. The mass change rate equation of a single bubble is:

$$\frac{d{m}_B}{dt}={\rho }_V\frac{d{V}_B}{dt}=4\pi R_B^2{\rho }_V\sqrt{\frac{2(P_V-P)}{3{\rho }_l}}$$

(6)

The gas volume fraction formula is:

$$a_V=N_B{V}_B=\frac{4}{3}\pi R_B^2{N}_B$$

(7)

In the formula, N_B is the amount of cavitation in unit volume, and R_B is the radius of cavitation bubbles (generally $1\times {10}^{-6}$ m).

The total interphase mass transport rate formula in unit volume is:

$$m_l=N_B\frac{\partial m_B}{\partial t}=\frac{3a_V{\rho }_V}{R_B}\sqrt{\frac{2\left(P_V-P\right)}{3{\rho }_l}}$$

(8)

In addition, the formula of the bubble in the collapse process is:

$$R_e=F\frac{3a_V{\rho }_V}{R_B}\sqrt{\frac{2\left|P_V-P\right|}{3{\rho }_l}}sign\left(P_V-P\right)$$

(9)

The main difference of the Zwart—Gerber—Belamri cavitation model is to replace α_V with α_nuc. The Zwart—Gerber—Belamri cavitation model equation is:

$$m=\left\{\begin{array}{ll}-F_{vap}\frac{3a_{nuc}\left(1-a_V\right){\rho }_V}{R_B}\sqrt{\frac{2\left(P_V-P\right)}{3{\rho }_l}},& P\le P_V\\ F_{cond}\frac{3a_V{\rho }_V}{R_B}\sqrt{\frac{2\left(P_V-P\right)}{3{\rho }_l}},& P\le P_V\end{array}\right.$$

(10)

In the formula, $a_{nuc}$ is the volume fraction of the bubble core, which is generally 5 × 10⁻⁴.

F_vap is the empirical correction coefficient of vaporization, generally taken as 50.

F_cond is the empirical correction coefficient of condensation, which is generally taken as 0.001.

In addition, the reason why the evaporation coefficient and the condensation coefficient in the model are different is that the evaporation process is much faster than the condensation process.

2.3. Grid Division and Independence Analysis

In order to ensure that the number of grid y+ values is small, a structured grid is used to divide the calculation area. The impeller and guide vane are divided by TurboGrid and then the periodic changes of the grid are carried out by ICEM to reduce the calculation error caused by too many interfaces. For the slightly simpler structure of the inlet extension section and the outlet extension section, ICEM is directly used for structural division. Taking sample 1 in Table 1 as an example, the grid division of the impeller with cascade density in sample 1 and the remaining parts of the pump section are as shown in Figure 4, Among them, Figure 4a is the grid of the inlet extension section, Figure 4b is the grid of the Outlet extension section, and Figure 4c is the grid of impeller, Figure 4d is the grid of guide vane.

For CFD fluid calculations, the quality of the grid has a great influence on the accuracy of the calculation results. In view of this, different sizes of grids were chosen to divide the model, finally obtaining 12 grids with the number of grids of 0.097, 0.221, 0.728, 3.349, 4.128, 5.011, 6.341, 7.040, 8.179 10.052 million, respectively. The total efficiency and the device head are used as the characteristic parameters of the grid independence analysis to determine how many grids have no effect on the model calculation results. The relationship between the calculation results and the number of grids is shown in Figure 5:

Figure 5 shows the efficiency of different grid numbers. When the number of grids exceeds 6 million, the efficiency and the head is almost unchanged, and the error is within ±2%. The grid of the final calculation file in this paper is about 7 million (impeller grid number: 1,813,824). The grid number model meets the calculation requirements.

2.4. Boundary Condition Setting

The blades of the axial flow pump are twisted, and the three-dimensional flow in the impeller channel is complex. As shown in Figure 6, an axial flow pump with a specific speed of 750 is selected as the research object in this paper; the impeller diameter is 300 mm. The calculation domain of the pump device comprises the inlet straight pipe section, impeller chamber, guide vane chamber and 60° outlet elbow. As shown in Figure 7, in order to simulate the internal flow of the pump unit better, the inlet is set at the inlet section of the extension section. The total pressure inlet condition which is adopted is 1 atm. Similarly, an extension section whose outlet section is set as the outlet of the calculated flow field is added after the outlet channel. The outlet is cut off and mass flow outflow is adopted.

To better capture the flow near the wall, the wall needs to be processed to ensure the accuracy of the simulation [26]. As shown in Figure 8, the structured grid is used to ensure that Y plus of impeller is less than 10, and meet the requirements of the RNG k—ε turbulence model. The inlet and outlet flow channels, impeller casing and guide vane body of the pump device are all set as static wall surfaces, and the non-slip condition is applied. The rotating interface needs to be specifically defined in the numerical simulation. The ‘Stage’ interface is used to deal with the parameter transfer of the dynamic and static coupling flow between the impeller and the inlet duct and the guide vane body. The interface between the guide vane and the outlet is a static interface.

3. Scheme Design

This paper mainly studies the comprehensive performance and cavitation performance of each impeller under different cascade density. Therefore, the external characteristics and cavitation characteristics of the pump device under each cascade density are calculated. For the external characteristics, 11 calculation schemes of 0.3Q~1.3Q are designed. For the cavitation characteristics, in order to study the critical cavitation margin of the axial flow pump, the internal cavitation characteristics of the axial flow pump are numerically calculated. In this paper, CFX software is used. In the cavitation calculation, the calculation results without cavitation are loaded as the initial conditions, and the cavitation numerical simulation is carried out by changing the pressure of the inlet conditions. Fourteen calculation schemes of different inlet pressures, namely 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.5, 0.55, 0.65, 0.7, 0.8, 0.9 and 1 atm, were designed. The detailed schemes are shown in Figure 9.

4. Analysis of Calculation Results

4.1. External Characteristic Analysis

Based on the steady-state calculation, the comprehensive performance of the pump under different impeller schemes is analyzed, and the calculation results are shown in Figure 10, the vertical line in the figure is the design condition.

From Figure 10a, it can be seen that the head of the pump under scheme 1 is the highest, and the head of the pump under scheme 9 is the lowest; under the design flow condition (Q = 315 L/s), the maximum head of scheme 1 is 8.59 m, and the minimum head of scheme 9 is 5.41 m. The lowest head of the pump saddle area under scheme 1, scheme 7 and scheme 8 occurs at Q = 140 L/s, but the lowest head of the pump saddle area under other schemes occurs at Q = 175 L/s. From the perspective of pump head, the pump head under scheme 1 is the highest, scheme 5 is the second, and scheme 9 is the worst.

It can be seen from Figure 10b that the efficiency of the pump is the greatest under scheme 1, and the efficiency of the pump is the smallest under scheme 9. Under the design flow condition (Q = 315 L/s), the maximum efficiency of scheme 1 is 88.21%, and the minimum efficiency of scheme 9 is 81.00%. Under the condition of low flow rate (Q = 105 L/s), the efficiency of scheme 1 is 35.65%, and the efficiency of scheme 9 is 20.21%. Under the condition of large flow rate (Q = 385 L/s), the efficiency of scheme 1 is 84.78%, and the efficiency of scheme 9 is 74.43%. From the perspective of pump efficiency, the efficiency of the pump under scheme 1 is the highest, followed by scheme 5, and scheme 9 is the worst.

The energy test of the pump (scheme 1) was carried out on the high-precision water conservancy test bench. Figure 11 shows the comparison between the experimental results and the numerical simulation results. It can be seen that the trend of the numerical simulation results is in good agreement with the experimental results. The error in the high efficiency area is within 1%, and the error in the saddle area is within 3%, indicating that the numerical simulation results are reliable.

4.2. Flow Analysis

In order to compare and analyze the influence of cascade density on the flow pattern of pump impeller, scheme 1, scheme 5 and scheme 9 are selected as the research objects, and the internal flow characteristics of impeller under three working conditions (Q = 245 L/s, Q = 315 L/s, Q = 385 L/s) are analyzed.

Figure 12 shows the streamlines and pressure contours of the blade surface at Q = 245 L/s. It is not difficult to find that under the three kinds of leaf density, the streamline distribution of the blade pressure surface is smooth and there is no poor flow pattern. The pressure at the inlet of the blade is greater than that at the outlet edge, and the pressure at the hub is basically less than that at the rim. There is a streamline turbulence zone near the hub at the inlet edge of the suction surface of the blade, and the pressure at the inlet edge is smaller than that at the outlet edge. Compared with scheme 5 and scheme 9, the range of turbulence zone and low-pressure zone on the suction surface of scheme 1 is the smallest.

Figure 13 shows the streamlines and pressure contours of the impeller blade surface at Q = 315 L/s. It is not difficult to find that under the design flow condition, no matter what kind of cascade density, the flow pattern on the pressure surface and suction surface of the blade is good, and there is no adverse flow pattern such as backflow and turbulence. Under scheme 1, there is a high-pressure zone at the hub of the inlet side of the blade pressure surface, and a low-pressure zone at the rim of the inlet side, but the pressure distribution at the inlet side of the suction surface is just the opposite. Under other schemes, the pressure distribution on the blade surface is consistent with the small flow condition.

Figure 14 is the streamline and pressure cloud map of the impeller blade surface under the condition of Q = 385 L/s. It is not difficult to find that under scheme 1 there is a small recirculation zone and a low-pressure zone at the inlet flange of the blade pressure surface. In other schemes, the streamline of the blade surface is smooth, and no recirculation occurs. The pressure distribution on the suction surface of the blade is different from other working conditions, and the pressure at the inlet edge is significantly greater than that at the outlet edge.

From the perspective of the flow field performance in the impeller, it can be found that the impeller flow pattern of scheme 1 is the best under the small flow condition and the design condition. However, under large flow conditions, the impeller flow patterns of Scheme 5 and Scheme 9 are better.

$$NPS{H}_a=\frac{P_{in}}{\rho g}+\frac{V_{in}^2}{2g}+\frac{P_c}{\rho g}$$

(11)

In the formula: P_in is the pump inlet pressure, V_in is the pump inlet velocity, P_c is the water vaporization pressure.

The cavitation performance curve is shown in Figure 15. When the NPSHa decreases to a certain value, the efficiency and head begin to decrease. In engineering, the operating point when the head decreases by 3% or the efficiency decreases by 1% is generally taken as the critical cavitation point, and the corresponding NPSH is the necessary NPSHr (the net positive suction head required by the pump to operate without experiencing damaging cavitation and a dramatic reduction in pumping production). In order to judge the results of each scheme in the cavitation performance curve, the histogram of NPSHr is used to visually display the cavitation performance of each case as shown in Figure 16:

It can be clearly seen from the NPSHr that the cavitation performance of cases 3 and 5 is better, while the cavitation performance of cases 1, 8 and 9 is poor. From the perspective of the scheme, the most obvious reason for the phenomenon of cavitation is that the higher necessary cavitation margin for cases 1, 8 and 9 is due to the smaller density of cascade at the hub, which leads to the occurrence of hub cavitation. As shown in Figure 17, the critical cavitation conditions of cases 1 and 9 are used to analyze the cavitation at the hub.

From the perspective of mechanism, it can be seen from Formula 1 that a small cascade density at the hub will lead to a high lift coefficient, resulting in large flow rate and small pressure. When the pressure is less than the vaporization pressure, cavitation occurs. In order to prove this point, the pressure distribution of the blade surface is studied. The distribution and development of the cavitation area can be easily understood by analyzing the static pressure load distribution of the blade, and analyzing the power capacity of the impeller under different cavitation conditions by studying the difference of ground pressure between the blade suction surface and pressure surface. In order to compare the static pressure distributions of blades under different critical cavitation conditions of superior and inferior schemes, the static pressure load distributions of cases 1, 3 and 9 are analyzed in this paper. The distance from hub to shroud is defined as span normalized distance, which is 1, 0 is at the hub, and 1 is at the blade tip. In this paper, near the hub (span 0.1), middle of the blade (span 0.5), and near the blade tip (span 0.9) are selected as analysis objects. Figure 18 shows the location distribution of analysis objects.

The static pressure load of the blade near the hub of cases 1, 3 and 9 was analyzed. The distance from leading edge to trailing edge is defined as dimensionless distance (streamwise) 1, and 0 is the position of the blade leading edge and 1 is the position of the trailing edge. The static pressure load of blades near the hub (span 0.1) is shown in Figure 19.

In the Figure 19, the position marked by the bottom line is cavitation pressure. If it is lower than this value, cavitation occurs. The cavitation locations of the three schemes are all at the leading edge, and there is an obvious difference in length. The length of cavitation bubble group of case 3 is shorter than that of cases 1 and 9.

Figure 20 is the head transformed by the static pressure load pressure difference between the upper and lower surfaces of the blade. In order to facilitate the observation of the head distribution, the design head is shifted up and down in the diagram to analyze the distribution of the head. It can be concluded from the diagram that the area of case 3 within the upper and lower boundaries is wider, while case 1 and case 9 are obviously shorter than case 3. In particular, case 9 has been significantly lower than the design head at 0.55 in streamwise.

In general, the cascade density at the hub has a great influence on the cavitation performance, especially when the smaller cascade density at the hub will lead to a higher lift coefficient, resulting in too large a flow rate and too small a pressure, which is prone to cause cavitation.

5. Optimization Analysis

In order to analyze the influence of blade cascade density on the performance of a rotating three-dimensional twisted hydrofoil, the response surface method polynomial function is used to fit the design space. Taking the cascade density of hub and shroud as the standard order of continuous factors, and the impeller efficiency, NPSHr and pump efficiency as the response items, the response isosurface diagram is shown in Figure 21, Figure 22 and Figure 23:

From Figure 21, it can be found that NPSHr decreases with the increase in cascade density at hub and shroud. As an important parameter to measure cavitation, the smaller the better. Therefore, from the perspective of cavitation, the greater the cascade density at the hub and shroud, the better. This is consistent with the above cavitation performance analysis.

It can be found from Figure 22 that the efficiency of the pump section increases with the increase in the cascade density at the hub and the decrease in the cascade density at the shroud. From the analysis of the whole pump device, the greater the density of the cascade at the hub, the better, and the smaller the density of the cascade at the shroud, the better.

It can be found from Figure 23 that there is no obvious positive and negative relationship between cascade density and the efficiency of the impeller. It is not even consistent with the efficiency zone of the pump section. In order to comprehensively select the appropriate cascade density, this paper sets the corresponding threshold to filter the combination area of cascade density. Among them, the range of NPSHr is less than 0.66, the pump section efficiency is more than 87.6%, and the impeller efficiency is more than 90.5%. The final result is shown in the vertical isosurface (Figure 19).

The white region in Figure 24 is a set of regions that satisfy the above selection, and is also the cascade density combination recommended in this paper with a specific speed of 750.

6. Conclusions

(1)

The influence of cascade density on the hydraulic performance and cavitation performance of an axial flow pump section is analyzed by CFD numerical simulation. The accuracy of numerical simulation is ensured by grid independence analysis. The hydraulic characteristics and cavitation characteristics of a three-dimensional twisted hydrofoil under different cascade densities are analyzed;

(2)

From the perspective of flow pattern and external characteristics, the impeller flow pattern of scheme 1 is the best under small flow condition and design condition. However, under the condition of large flow rate, the impeller flow state of scheme 5 and scheme 9 is better; from the perspective of cavitation, the density of the cascade at the hub has a great influence on the cavitation performance. Especially when the density of the cascade at the hub is small, the lift coefficient is high, resulting in too large a flow velocity and too small a pressure, which is prone to cause cavitation;

(3)

Through the method of response surface analysis, the effective response surface of a single target parameter and cascade density is obtained, and the influence of cascade density on a three-dimensional hydrofoil is analyzed. According to the corresponding range of comprehensive target parameters, the range of cascade density of the comprehensive target parameters is obtained.