Investigation on Optimization Design of High-Thrust-Efficiency Pump Jet Based on Orthogonal Method

Abstract

Compared with traditional propellers, the pump jet has the characteristics of low noise, large thrust, and high efficiency, which is widely used in underwater ships, unmanned underwater vehicles, and underwater rescue equipment. It is difficult to design and evaluate the performance of a pump jet because of its complex structure and the many changes in its component parameters. In this paper, high-efficiency pump jet design and optimization methods are presented. Based on the center-of-gravity accumulation line control, a design method is proposed to design the pump jet rotor. The thrust performance and hydraulic performance of the pump jet are obtained through numerical simulation, and the influence of various parameters on performance is studied. Finally, based on a single-factor analysis and orthogonal optimization, a multi-parameter orthogonal optimization of pump jet was carried out. The angle of attack, chord length, camber angle, and number of blades were selected to establish a four-factor and three-level orthogonal table. The thrust efficiency was taken as the optimization goal. Compared to the original pump jet, the optimized pump jet thrust coefficient and thrust efficiency are significantly improved at various speed conditions, with the maximum thrust efficiency increased by 7.23%, meeting the design requirements. This research provides a theoretical basis and technical guidance for the structural optimization design of a pump jet propulsion system.

1. Introduction

Underwater propulsion technology is one of the core technologies of marine core equipment and is the power source for underwater vehicle navigation. With the development and progress of detection technology, underwater vehicles face more severe challenges in maneuverability and safety. Compared with traditional underwater propulsion technology, pump jet propulsion has the advantages of low noise, high efficiency, and large thrust [1]. Pump jet propulsor has become a hot topic in the research field of propulsion technology. The pump jet has a complex structure, consisting of three parts: the rotor, the duct, and the stator [2]. In order to increase the speed, the stator and rotor are generally covered with ducts on the structure. According to the positional relationship between the stator and the rotor, pump jets can be divided into two types: the front stator pump jet and the rear stator pump jet. The front stator can improve inflow conditions and reduce noise; the rear stator can recover wake energy and improve efficiency [3]. However, due to the complexity and emerging nature of the internal flow of pump jets, a unified and effective design method for pump jets has not yet been formed. Therefore, developing and improving an advanced pump jet design theory and method is of great value and significance for submarines and underwater vehicles.

At present, there is little public information on pump jet propulsion, with the main research focusing on experimental research and CFD simulation calculations. There is relatively less information on design research. The lift line method is used to design the pump jet rotor as a finite two-dimensional airfoil. The lift line of the blade airfoil is used to replace the airfoil. By calculating the circulation and vorticity on the lift line, the lift line design of different blade heights is completed, and then the blades are thickened to complete the blade design. This theory was proposed and improved by Goldstein et al. [4,5,6], and has been well-verified in design, and the optimal circulation theory is still a good method for the design of pump jet vanes [7,8,9,10,11].

The lifting method, which is a deepening of the lift line theory based on one- and two-element design theories, was originally applied to the design of axial flow pump blades, which is still widely used in the design of axial flow pumps [12,13,14]. And the design of the rotor by the lifting method is completely based on the assumption of the independence of the cylindrical layer. The key to the lifting method design is to select appropriate airfoil parameters and airfoil stacking parameters. However, the controllable parameters of the airfoil method design are limited and obvious, including the airfoil type, maximum airfoil thickness, airfoil stagger angle, angle of attack, chord length, camber angle, airfoil accumulation line, etc. [15]. Cox et al. [16,17] applied this theory to underwater propulsion systems. Domestic Wang Guoqiang et al. [18,19] used the lifting method to predict the steady and unsteady hydrodynamic performance of the ducted propeller, and studied the cavitation performance of the propeller. Dong Shitang [20] applied the lifting method to study the boundary problems of propellers under full three-dimensional conditions, improving the accuracy of the lifting method in dealing with boundary problems; Xiong Ying and Tan Tingshou [21] established a propeller design theory suitable for the wake-adapted lifting method, which is still widely used in the design of the pump jet.

The research on pump jet optimization is mostly focused on the effects of a single factor. However, optimization methods from related application fields can serve as a theoretical basis to guide the optimization of the pump jet system. Lin Lu et al. [22,23] studied the open water characteristics and tip clearance of E779A on pump jet performance through experiments and CFD calculations, and pointed out that, with the increase in tip clearance, the tip leakage vortex develops, which eventually leads to a decrease in cavitation performance and open water efficiency. Sang Jun Ahn et al. [24] studied rotors with connecting rings. The cavitation performance on the suction surface of the rotor blades decreases due to the reduction in axial speed. The rotor with connecting rings reduces the torque while ensuring the same thrust and efficiency, providing a new design idea.

The above studies all focus on a certain factor that affects the pump jet efficiency, but do not consider the impact of multiple factors coupled with each other on the design results. The orthogonal optimization method can use fewer computing resources to study the influence of multi-parameters, which is widely used in the optimization of the vane pump. The effect of orthogonal optimization mainly depends on three factors: influencing parameters, parameter levels, and orthogonal optimization tables. After the above three factors are selected, the test sample can be quickly determined and the optimization is completed. Usually, the final result of the optimization is better than the result of the orthogonal table sample. R.Bontempo et al. [25,26] established the model of the airfoil of the pump nozzle through theoretical derivation and extracted the key parameters. They optimized and calculated the four parameters through the orthogonal method. The propulsion efficiency of the pump jet can be improved by increasing the pipe thickness, curvature, and wing chord, and reducing the angle of attack. Xu et al. [27] established a five-factor four-level orthogonal table and pointed out that the outlet blade stagger angle and blade leading edge position have the greatest impact on efficiency and cavitation performance. The subsequent selection of appropriate parameters increased the hydraulic efficiency of the centrifugal pump by 3.09% and reduced pump cavitation by 1.45 m. Liu et al. [28] selected five optimization factors, the hub inlet stagger angle, rim inlet stagger angle, rim outlet stagger angle, and control coefficients of hub and rim profiles, and increased the pump pressure rise by 12.8 kPa at the design flow rate. Yuan et al. [29] took the weighted average efficiency near the design flow rate as the optimization target, and the optimized efficiency was 4.68% higher than the original pump efficiency. Zheng et al. [30] designed an orthogonal experiment with four factors and three levels, and studied the effects of the blade number, airfoil, hub ratio, and distance between the impeller and guide vane, and found that the factors that have the greatest impact on efficiency are the number of blades and hub ratio. Finally, the impeller efficiency was increased by 5.7% and the pressure pulsation was reduced.

In order to achieve a high-efficiency propulsion of the pump jet, the theory and method of the hydraulic design of the pump jet were developed. The pump jet was designed by the baseline model; the thrust performance and hydraulic performance of the pump jet were studied by numerical calculation. The thrust efficiency was optimized as the objective function. Compared to the original pump jet thrust coefficient and thrust efficiency, the optimized pump jet thrust coefficient and thrust efficiency are significantly improved at various speed conditions, with a thrust efficiency increased by 7.23%.

2. Pump Jet Design Method

2.1. Blade Configuration Method

The pump jet design theory was developed based on the axial flow pump lifting method. Based on two-dimensional airfoil stacking for the blade configuration, it is important to select appropriate design parameters according to the requirements. The airfoil can determine the lift and drag, and further affects the hydraulic and thrust performance of the pump jet. The NACA65 foil is selected in this work due to its superior performance characteristics, particularly in terms of the lift-to-drag ratio, which is crucial for optimizing the hydraulic and thrust performance of the pump jet. The blade three-dimensional shape can be controlled by the stagger angle (γ), camber angle (φ), and chord length (l). Figure 1 shows the airfoil shape, stagger angle, camber angle, and chord length.

Firstly, the stagger angle of the blade is determined. The velocity triangle at the blade section’s center of gravity is selected to replace the velocity distribution of the entire airfoil. Then, we select a cylindrical layer with a thickness of dr at a height of r, and unfold it to obtain a plane inline blade cascade. Given an attack angle of α, its force analysis diagram and velocity triangle are shown in Figure 2:

The calculation formula of inflow angle β_∞ can be obtained from the velocity triangle at the center of gravity of the airfoil:

$$\tan {\beta }_{\infty }={\displaystyle \frac{Vm}{U-C{u}_2/2}}$$

(1)

The blade stagger angle γ is given by adding the inflow angle and the angle of attack:

$$\gamma =\alpha +{\beta }_{\infty }$$

(2)

In Figure 2, dF_l is the lift, and dF_d is the resistance, given by:

$$\mathrm{d}{F}_l=C_l\rho {\displaystyle \frac{{W_{\infty }}^2}{2}}A=C_l\rho {\displaystyle \frac{{W_{\infty }}^2}{2}}l\mathrm{d}r$$

(3)

$$\mathrm{d}{F}_d=C_d\rho {\displaystyle \frac{{W_{\infty }}^2}{2}}A=C_d\rho {\displaystyle \frac{{W_{\infty }}^2}{2}}l\mathrm{d}r$$

(4)

where C_l is the lift coefficient and C_d is the drag coefficient.

The resultant force dF on the blade element is given by:

$$\mathrm{d}F={\displaystyle \frac{\mathrm{d}{F}_l}{\cos \alpha }}=C_l\rho {\displaystyle \frac{{W_{\infty }}^2}{2}}\cdot l\mathrm{d}r{\displaystyle \frac{1}{\cos \alpha }}$$

(5)

The resultant force can be decomposed into the axial thrust d_T of the impeller and the resistance dF_M of the impeller’s circumferential direction:

$$\mathrm{d}T=\mathrm{d}F\cdot \cos (\alpha +{\beta }_{\infty })$$

(6)

$$\mathrm{d}{F}_M=\mathrm{d}F\cdot \sin (\alpha +{\beta }_{\infty })$$

(7)

$$\mathrm{d}M=r\mathrm{d}{F}_M$$

(8)

For this element, the impeller does work externally:

$$\mathrm{d}{P}_{in}=Zr\omega \mathrm{d}{F}_M$$

(9)

The energy added by this micro-element fluid is:

$$\mathrm{d}{P}_{inc}=\rho g{H}_{t}dQ=\rho g{H}_t{V}_{m}tZ\mathrm{d}r$$

(10)

According to the law of conservation of energy, the work carried out by the impeller is equal to the energy added by the micro-element fluid:

$$C_L(r){\displaystyle \frac{l}{t}}={\displaystyle \frac{2(C{u}_2-C{u}_1)}{W_{\infty }}}\cdot {\displaystyle \frac{1}{1+\tan \alpha /\tan \beta }}$$

(11)

After selecting the blade density l/t, the lift coefficient can be calculated according to the Euler formula. The lift coefficient is the lift coefficient of the blade when the angle of attack is α. In fact, the two coefficients cannot correspond at the beginning, so it is necessary to iterate it to obtain the suitable angle of attack and lift coefficient. There is no corresponding database for the blade lift coefficient, but the relationship between the lift and drag coefficient of a single airfoil and the angle of attack can be found in the airfoil aerodynamic test data. This method has been tested and verified by NASA. The lift coefficient of a single airfoil needs to be corrected to the blade lift coefficient. The correction formula is as follows:

$$C_L=m{L}_n{C}_l$$

(12)

L_n is the correction factor of the flat blade cascade, and m is the correction factor of the airfoil cascade to the flat blade cascade.

$$\left\{\begin{array}{l}m=6.7{\displaystyle \frac{{\delta }_{\max }}{l}}+0.72,l/t<0.95\\ m=1,l/t>0.95\end{array}\right.$$

(13)

The chord length is determined by the density of the blade cascade, and generally refers to the calculation results of the blade cascade density of the axial flow pump at the same specific speed. The camber angle reflects the curvature of the airfoil. The larger the camber angle, the greater the curvature angle of the airfoil.

2.2. Blade Stacking Method

After determining the airfoils with different blade heights, a three-dimensional blade configuration is required by stacking the blades. There are three common stacking points, the leading edge point, center-of-gravity point, and trailing edge point, corresponding to the three stacking methods. Among them, the center-of-gravity stacking line can give the mass distribution of the blade along the blade height direction, which is convenient for studying the force of the blade, so this paper uses the center-of-gravity point stacking method to perform the three-dimensional blade configuration.

The center-of-gravity stacking line is projected to the axial and circumferential planes of the blade, and two sweeping curves can be obtained in the two planes, as shown in Figure 3 and Figure 4. Among them, the curve of the blade’s axial projection controls the forward and backward sweep by changing the position of the center of gravity of the airfoil at different blade heights, thereby driving the entire airfoil to sweep forward or backward. The curve of the blade’s circumferential projection controls the center of gravity in the circumferential direction, and the bending direction of this curve is consistent with the forward and backward bending of the blade in the circumferential direction.

For the curve that controls the front and rear sweep in the z–r plane, the values in the r direction and z direction are determined by Formula (14), respectively; z_s represents the z-direction co-ordinate of the wheel rim; and the curve that controls the steering bend in the x–y plane is determined by Formula (15):

$$\left\{\begin{array}{l}z=z_s\left[a\sin \theta +\left(1-a\right)\left(1-\cos \theta \right)\right]\\ r=\left(r_s-r_h\right)\left[a\left(1-\cos \theta \right)+\left(1-a\right)\sin \theta \right]\end{array}\right.,0\le \theta \le {\displaystyle \frac{\pi }{2}},0\le a\le 1$$

(14)

$$\left\{\begin{array}{l}x=c\sin 2\theta \\ y=(rs-rh){\displaystyle \frac{1-\cos 2\theta }{2}}\end{array}\right.,0\le \theta \le {\displaystyle \frac{\pi }{2}},-10\le c\le 10$$

(15)

where z_s, a, and c are the adjustable parameters studied, r_s is the rim radius, and r_h is the hub radius. The influence of specific parameters on the pump jet performance will be introduced in detail later.

2.3. Design Results

According to the actual working conditions, the given rated speed V_s is 8 kn, about 4.1 m/s; the rated speed is 1450 r/min, and the actual thrust is about 120 N.

$$T=\rho Q\left(\mu -\alpha \right)V_s$$

(16)

$$H={\displaystyle \frac{\left({\mu }^2-\beta \right){V_s}^2}{2g}}$$

(17)

where α is the momentum influence coefficient, β is the kinetic energy influence coefficient, and V_s is the pump jet inlet speed.

The integral expressions of momentum influence coefficient α and kinetic energy influence coefficient β are:

$$\alpha ={\displaystyle \frac{1}{V_s}}{\displaystyle \frac{{\displaystyle {\iint }_{A_i}{V_z}^{2}d{A}_i}}{{\displaystyle {\iint }_{A_i}{V}_{z}d{A}_i}}}$$

(18)

$$\beta ={\displaystyle \frac{1}{{V_s}^2}}{\displaystyle \frac{{\displaystyle {\iint }_{A_i}{V_z}^{3}d{A}_i}}{{\displaystyle {\iint }_{A_i}{V}_{z}d{A}_i}}}$$

(19)

where V_z is the velocity at the inlet flow section at different radii, and A_i is the inlet flow area.

The design parameters can be calculated through Formula (16) and (17), where flow rate Q = 0.06 m³/s, head H = 2.7 m, maximum diameter of pump jet impeller D = 180 mm, calculated specific speed n_s = 615, and the preliminary design parameters are shown in

Table 1. Pump jet design parameters.

Design Parameter	Value
Speed Vs/kn	8
Thrust T/N	120
Speed n/(r/min)	1450
Flow Q/(m3/s)	0.06
Head H/m	2.7
Impeller diameter D/mm	180
Specific speed ns	615

.

According to the design experience, the number of blades corresponding to an axial flow pump jet with a specific speed of 600 is 5. At the same time, a larger hub ratio should be selected, so the hub ratio was determined to be 0.4.

After the design parameters and inflow angle are determined, the calculation process of the stagger angle is as follows: Initially, the angle of attack is assumed to be 1°. The appropriate airfoil and blade density are selected, and these values are substituted into Formula (11) to calculate the lift coefficient. The calculated lift coefficient is then compared with the given lift characteristic curve. If the calculated lift coefficient does not match the given lift characteristic curve, the angle of attack value is adjusted and recalculated. This process is repeated until the calculated lift coefficient and the corresponding angle of attack are the same as or similar to the lift coefficient and the corresponding angle of attack on the given lift characteristic curve. Once they match, the iteration is ended, and the angle of attack for each blade section is determined. Finally, the angle of attack is superimposed onto the inflow angle to obtain the final blade stagger angle.

In order to realize the automatic selection and design configuration of the pump jet, the software is written based on MATLAB-GUI (R2020a). The input of the program is the design conditions, namely, the rotation speed, speed, and resistance. During the design process, the blade is evenly divided into 21 sections along the blade height. The above calculation procedure is carried out for each section to determine the airfoil stagger angle for each section. Then, the distribution of the blade stagger angles from the hub to the shroud is obtained. After that, the sweep mode of the airfoil stacking line was selected to obtain the three-dimensional airfoil data. The final result of the program design is trimmed and the blade surface is smoothed, and the designed rotor is shown in Figure 5.

In the pump jet, the stator plays the role of rectifying, recovering wake energy, and generating a small amount of thrust. It is not the core work component, so it has a simple structure and small distortion. The streamline method is used here to design the blades. The stator inlet stagger angle is the same as the rotor outlet inflow angle, so that the fluid can enter the stator smoothly, and the outlet stagger angle is 90° to recover the energy of the rotating fluid as much as possible. The number of stators is generally greater than the number of rotors, and the number of blades is relatively prime, so the number of stator blades is determined to be 7. The stator designed according to these principles is shown in Figure 6.

This article mainly studies the impact of the rotor on the pump jet performance; the duct is not the main research object of this article, so the duct only needs to meet the basic functions. The duct is selected as a contraction duct, that is, an acceleration duct, which can increase the flow rate of the liquid inside the pump jet, thus increasing the thrust and thrust efficiency. The duct section is similar to an airfoil and is used to reduce the resistance loss caused by the duct itself. The thickness of the duct in this article meets the 719-airfoil thickness distribution, and the cross-section is shown in Figure 7.

3. Computational Domain and Numerical Method

3.1. Computational Domain

This article selects the rotary body model and fluid domain settings as shown in Figure 8. The length of the rotary body (including the pump jet section) is L; in order to simulate the real underwater environment and minimize the influence of the wall, a water area of L is set up in front of the rotary body, and then a water area of L is also set up behind it. As a whole, it presents a cylindrical fluid domain with a length of 3 L and a diameter of 15 D.

3.2. Computational Mesh

The open water domain is divided by the tetrahedral mesh, and the overall mesh quality is greater than 0.7. The meshes near the wall are refined to capture the detailed flow structure, which are shown in Figure 9. The duct mesh surface is shown in Figure 10a. The impeller domain and guide vane domain are divided by the hexahedral structured mesh, which has the advantages of a short generation period, high mesh quality, and small discrete error, as shown in Figure 10b.

3.3. Mesh Independence Test

The mesh number has influence on the numerical calculation results. Too few meshes will reduce the calculation accuracy, while too many meshes will consume additional computational resources. Therefore, it is necessary to conduct the mesh-independent test. The fluid domain in this article is mainly divided into the open water and pump jet. The open water has a larger number of meshes and the mesh drawing method is different from that of the rotor and stator. Therefore, the mesh-independence verification is also divided into two parts.

Table 2. Calculation results of different grids in the open water area.

Number	Mesh1	Mesh2	Mesh3	Mesh4	Mesh5	Mesh6
Open water	722,208	995,483	1,333,768	1,947,248	2,931,808	3,884,328
Pump jet area	1,727,778	1,727,778	1,727,778	1,727,778	1,727,778	1,727,778
Relative resistance	1	0.985812	0.975463	0.961081	0.951509	0.948059

shows the change in relative resistance with the number of meshes at the design speed. As the number of meshes increases, the calculated value of the relative resistance gradually stabilizes. When the number of meshes increases to more than 2.93 million, the relative change value of resistance is less than 0.4%, so mesh5 for the open water is chosen in the following calculation.

Table 3. Calculation results of different grids in the pump jet area.

Number	Mesh7	Mesh8	Mesh9	Mesh10	Mesh11
Open water	2,931,808	2,931,808	2,931,808	2,931,808	2,931,808
Rotor	238,170	441,980	751,920	1,298,100	1,464,960
Stator	331,722	667,436	1,005,368	1,698,060	1,994,272
Pump jet area	569,892	1,109,416	1,757,288	2,996,160	3,459,232
Relative resistance	1	1.017042	1.028312	1.030201	1.030866

shows the change in relative thrust efficiency with the number of meshes at the design speed. As the number of meshes increases, the calculated value of relative thrust efficiency gradually stabilizes. When the number of pump jet meshes increases to more than 1.75 million, the change in relative efficiency is less than 0.3%. Therefore, mesh9 for the pump jet section is chosen in the following calculation. In addition, the average value of y+ on the pump blades is less than 10, so the mesh quality is acceptable.

3.4. Numerical Method

The software that conducts the numerical calculation for the pump jet is ANSYS CFX 2020 R1. The shear stress transport model (SST k-ω), which introduces the influence of shear stress propagation on turbulent viscosity, is set as the turbulence model.

For the boundary conditions of the computational domain, the inlet boundary of the external flow field was set to the normal speed as the speed inlet, and the outlet was set as the static pressure. No slip boundary condition is adopted at the underwater robot and partial surface of the duct. There are three interfaces in the pump jet section: the dynamic and static interfaces between the mover and the open water area, and the rotor and the stator, using the frozen rotor mode; there is a static interface between the stator and the open water area. The convergence condition is set as 10⁻⁵.

The above calculation mesh is selected to calculate the thrust characteristics of the pump jet. The calculated results are shown in Figure 11, and the thrust efficiency can reach 68.85%.

In order to facilitate the description of its performance, dimensionless speeds are introduced here, namely, the advance coefficient J, the dimensionless thrust coefficient K_T, the dimensionless torque coefficient K_M, and the thrust efficiency η_T; the relevant definition is as follows:

$$J={\displaystyle \frac{v_s}{nD}}$$

(20)

$$K_T={\displaystyle \frac{T}{\rho n^2{D}^4}}$$

(21)

$$K_M={\displaystyle \frac{M}{\rho n^2{D}^5}}$$

(22)

$${\eta }_T={\displaystyle \frac{J}{2\pi }}{\displaystyle \frac{K_T}{K_M}}$$

(23)

The pump jet head H and hydraulic efficiency η can be calculated as follows:

$$H={\displaystyle \frac{P_{out}-P_{in}}{\rho g}}$$

(24)

$$\eta ={\displaystyle \frac{\rho gQH}{2\pi nM}}$$

(25)

where P_out is the pressure at the pump jet outlet, P_in is the pressure at the pump jet inlet, ρ is the density of water, g is the acceleration of gravity, Q is the flow rate, n is the rotation speed, and M is the torque.

4. Orthogonal Optimization

4.1. Airfoil Parameters

4.1.1. Angle of Attack

In the pump jet design process, the airfoil stagger angle of the blade is determined by the sum of the inflow angle and the angle of attack. An appropriate angle of attack can result in the lift and drag characteristics of the airfoil being in a high-efficiency zone. Based on the original pump jet impeller designed in this study, five groups of attack angles, −2°, −1°, 0°, 1°, and 2°, were selected as variables, at the same time, ensuring that the remaining design parameters remain unchanged.

Figure 12 shows the changes in pump jet thrust efficiency at different angles of attack. It can be seen from the figure that, as the angle of attack increases, the maximum thrust efficiency increases, and, at the same time, the maximum thrust efficiency point moves towards the direction of the large advance coefficient. Under the condition of a low advance speed coefficient, the thrust efficiency is similar. Under the condition of a high advance speed coefficient, the high angle of attack pump jet thrust efficiency is larger and the high efficiency range is expanded. From the perspective of actual work requirements, a large attack angle pump jet can provide a greater thrust. Without considering the hydraulic and cavitation noise performance, the pump jet should choose a larger attack angle during the design process.

Figure 13 shows the changes in the pump jet head under different angles of attack: as the angle of attack increases, when the flow rate is greater than 0.05 m³/s, the head increases; when the flow rate is less than 0.05 m³/s, the head decreases. And when the angle of attack is +1° and +2°, observing the head–flow curve, we can find that it has obvious “S”-shaped characteristics, which means that, at a low flow rate, the flow inside the pump jet is unstable and a hump phenomenon may occur, which has a negative impact on the operation of pump machinery.

Figure 14 shows the changes in the hydraulic efficiency of the pump jet under different angles of attack: as the angle of attack increases, the maximum hydraulic efficiency of the pump jet almost remains unchanged, but the optimal efficiency point moves toward the direction of a large flow rate. Under medium and low flow rates, a small-angle-of-attack impeller has a higher hydraulic efficiency, and, under high flow rates, a large-angle-of-attack impeller has a higher hydraulic efficiency.

Under large flow rates, the axial speed inside the impeller increases, and the inflow angle at the blade inlet increases, forming a negative attack angle at the inlet. The working conditions of the blades become worse, resulting in a rapid decline in hydraulic efficiency.

Based on the above analysis, it can be seen that appropriately increasing the angle of attack can not only obtain a greater thrust and thrust efficiency, but can also expand the high thrust efficiency operating range; considering the needs of the hydraulic performance, a large angle of attack can improve the maximum hydraulic efficiency and the corresponding head of the impeller. But the angle of attack should not be too large in order to avoid unfavorable phenomena such as the design point deviation and the impeller running into the hump area under small flow rates.

4.1.2. Chord Length

In this section, the chord length of the blade at different leaf heights is scaled proportionally, and the study investigates the thrust performance and energy characteristics of pump jet impellers with five different chord lengths: 80% chord length, 90% chord length, 100% chord length (which is the chord length of the original pump jet impeller blade), 110% chord length, and 120% chord length.

Figure 15 shows the change in pump jet thrust efficiency with an advance speed coefficient. The results show that impellers with smaller chord lengths can meet the thrust requirements while exhibiting a higher thrust efficiency and a wider high-efficiency speed range. The maximum thrust efficiency corresponds to a higher speed, and its impeller structure is more compact, which helps reduce the weight of the overall underwater vehicle and has greater advantages in terms of underwater vehicle endurance.

Figure 16 shows the changes in the pump jet head under different chord lengths. It can be seen from the figure that, under medium- and low-flow conditions, the head increases with the increase in blade chord length. This is because increasing the chord length expands the work area; at medium and low flow rates, the increase in work capacity dominates the increase in hydraulic loss caused by friction, so the head increases.

At high-flow conditions, the increase in hydraulic losses due to friction dominates, so the head decreases as the chord length increases. In addition, the head of a pump jet with a small chord length changes slowly at small-flow conditions, and the S-shaped trend is obvious, indicating that a pump jet with a small chord length is more likely to enter the hump range than a pump jet with a large chord length.

Figure 17 shows the changes in pump hydraulic efficiency under different chord lengths. It can be seen from the figure that, at medium- and low-flow conditions, the efficiency increases slightly with the increase in the chord length, while, at high-flow conditions, the efficiency decreases significantly with the increase in the chord length, mainly due to the hydraulic loss caused by friction being dominant. As the chord length decreases, the optimal point of the impeller’s hydraulic efficiency moves toward the direction of a large flow rate, and the impeller with a smaller chord length has a relatively larger high-efficiency zone range.

Based on the above analysis, it can be seen that the increase in blade chord length does not change the airfoil and its stacked three-dimensional structure, so it has little effect on the thrust coefficient. However, increasing the chord length increases the length of the flow channel and increases the torque coefficient, which means that the required input power increases, resulting in a decrease in thrust efficiency. Therefore, considering the thrust performance and practical application scenarios, a small chord length can ensure thrust while reducing energy consumption and improving the endurance of underwater vehicles. Considering the hydraulic performance of the pump jet, the increase in chord length leads to an increase in both working capacity and hydraulic friction loss. A large chord length is beneficial to the efficiency and head of medium- and low-flow conditions, while a small chord length is beneficial to the improvement of the head and efficiency of high-flow conditions. The chord length of the blade should not be too small, otherwise the blade will easily enter the hump area when running at small flow condition, affecting the normal operation of the blade.

4.1.3. Camber Angle

Taking the original blade camber angle distribution as the baseline and maintaining the pattern of the camber angle distribution unchanged, five different camber angle variations of −4°, −2°, 0°, 2°, and 4° are selected at each blade height to obtain five research models. And these five models were calculated through numerical simulation to study the influence of the camber angle on the thrust characteristics and energy characteristics of the pump jet.

Figure 18 shows the change in pump jet thrust efficiency with an advance speed coefficient under different camber angles. At low speeds, the thrust efficiency of the pump jet increases slightly as the camber angle decreases; at high speeds, the thrust efficiency of the pump jet increases as the camber angle increases. When the maximum thrust efficiency is close, the larger the camber angle is, the higher the maximum thrust efficiency is, and the efficient range of the thrust efficiency expands with the increase in camber angle.

Figure 19 shows the changes in pump jet head under different camber angles. It can be seen from the figure that increases the camber angle of the airfoil can increase the pump jet head, especially at high-flow conditions. This is because the increase in camber angle increases the work area of the airfoil, and the impact of friction loss limits the lift at low-flow conditions. This effect decreases at high-flow conditions, so the head increase is more significant.

Figure 20 shows the changes in pump jet hydraulic efficiency under different camber angles. It can be seen from the figure that, under medium- and low-flow conditions, the hydraulic efficiency remains basically unchanged under different camber angles; under high-flow conditions, the hydraulic efficiency increases significantly as the camber angle increases. And, at large-flow conditions, the larger the camber angle is, the slower the hydraulic efficiency decreases, and the impeller’s efficient operation area becomes larger.

Based on the above analysis, it can be seen that the camber angle affects the thrust performance and hydraulic performance of the pump jet. The increase in camber angle will increase the thrust coefficient and torque coefficient at the same time, but has no obvious impact on the maximum thrust efficiency and maximum hydraulic efficiency. Choosing a larger camber angle in the design is more conducive to working conditions under large flow conditions and makes the efficient working area wider.

4.2. Orthogonal Table

According to the single-factor study in Section 4.1, the main factors affecting the thrust performance and hydraulic performance of the pump jet are the angle of attack, chord length, and camber angle. And, considering that the number of rotor blades of the pump jet is based on the changing pattern of the number of blades of the axial flow pump at the same specific speed, which is different from the optimal number of blades under the actual working conditions of the pump jet, the fourth factor here selects the number of blades, each factor selects three levels, and we construct a four-factor, three-level orthogonal table, and perform orthogonal optimization design. A reasonable selection of the angle of attack can result in the lift and drag characteristics of the airfoil being in a high-efficiency zone; three levels can be selected within the range of −1°~1°. An excessive chord length may reduce the pump jet thrust efficiency, therefore, three levels of 0.8 L, 0.9 L, and L are selected. The value range of the camber angle is from −2° to 2° due to the single-factor analysis. The number of blades has a significant impact on the performance of the impeller; based on five blades, three levels are selected.

Table 4. Orthogonal optimization factor level table.

Level	Angle of Attack/° (A)	Chord Length (B)	Camber Angle/° (C)	Number of Blades (D)
1	−1	0.8 L	−2	4
2	0	0.9 L	0	5
3	1	L	2	6

shows the orthogonal parameter and level of the pump jet in this work.

Finally, an orthogonal table (L₉) with four parameter factors and three levels is established. There are nine individuals in this orthogonal table, which are evenly distributed in the range of the overall 81 (3⁴) cases.

Table 5. Orthogonal table.

Individual No.	Factor
	A/°	B	C/°	D
1	−1	0.8 L	−2	4
2	−1	0.9 L	0	5
3	−1	L	2	6
4	0	0.8 L	0	6
5	0	0.9 L	2	4
6	0	L	2	5
7	1	0.8 L	2	5
8	1	0.9 L	−2	6
9	1	L	0	4

shows the details of the individuals in this orthogonal table.

5. Results and Discussion

5.1. Orthogonal Analysis

The target parameter of orthogonal optimization is thrust efficiency, which requires the highest thrust efficiency and the corresponding speed to not be lower than the design speed. Through the numerical calculation method introduced above, nine groups of test plans in the orthogonal table are calculated to obtain the thrust efficiency at the design speed and the hydraulic efficiency and head at the design flow rate. The calculation results are shown in

Table 6. Orthogonal optimization test results.

Individual No.	Thrust Efficiency/%	Flow/m3/s	Head/m	Hydraulic Efficiency/%
1	0.731	0.06	2.05	0.826
2	0.698	0.06	2.47	0.830
3	0.631	0.06	2.65	0.800
4	0.693	0.06	2.71	0.816
5	0.750	0.06	2.62	0.826
6	0.692	0.06	2.62	0.818
7	0.752	0.06	2.94	0.818
8	0.689	0.06	2.87	0.810
9	0.744	0.06	2.79	0.826

. It can be seen from the results listed in Table 6 that, for hydraulic efficiency, there is little change at the three levels of parameters, and the efficiency fluctuates by only 3%, while the thrust efficiency fluctuates within a range of 12.1%, so the analysis only focuses on the thrust analysis of efficiency.

In the orthogonal method, to evaluate the influence of every specific parameter, the average value $\overline{Ki}$ of the thrust efficiency is defined as follows:

$$\overline{K_i}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{j=1}^N{K}_j}$$

where i is the level of the parameter, N is the individual number of the corresponding parameter at level i, and K_j is the thrust efficiency of tested pumps with level i. The average value of efficiency is defined as the same formula.

The range R describes the influence weight of each parameter, and can be defined as follows:

$$R=\max (\overline{K_i})-\min (\overline{K_i}),i=1,2,3,4$$

Table 7. Range analysis for thrust efficiency.

Thrust Efficiency	Factor
	A	B	C	D
$\overline{K1}$	0.6865	0.7252	0.7037	0.7417
$\overline{K2}$	0.7116	0.7123	0.7116	0.7141
$\overline{K3}$	0.7284	0.689	0.7112	0.6706
R	0.0419	0.0362	0.0079	0.0711
Order	2	3	4	1

shows the range analysis of the impact of various factors on thrust efficiency. Therefore, the influence order on the pump head can be concluded as follows: D > A > B > C. As parameter A (angle of attack) increases, the pump jet thrust efficiency increases; as parameter B (chord length) increases, the thrust efficiency decreases; and, as parameter D (the number of blades) increases, the thrust efficiency decreases. Therefore, within the design scope, design A3B1C2D1 has the maximum thrust efficiency.

5.2. Orthogonal Optimization

With the comprehensive consideration of the design requirement and optimization, the levels of four parameters are chosen as A3B1C2D1.

Figure 21a shows the comparison of the thrust coefficient of the pump jet before and after optimization. It can be seen from the figure that the thrust coefficient of the optimized pump jet is larger than that of the original pump jet at all speeds, indicating that the optimized pump jet has a greater thrust and can adapt to a higher speed. Figure 21b shows the comparison of the torque coefficient of the pump injection before and after optimization. It can be seen from the figure that the torque coefficient of the optimized pump jet is slightly higher than that of the original pump jet. The torque coefficient increases more at high speeds, which is related to the significant increase in thrust.

Figure 22 shows the comparison of the thrust efficiency of the pump jet before and after optimization. It can be seen from the figure that the thrust efficiency of the optimized pump jet is improved compared to the original pump jet at various speeds. The maximum thrust efficiency is increased from 68.85% to 76.08%, improved by 7.23%. This higher efficiency means that a greater proportion of the input power is converted into useful thrust, reducing energy losses and improving the overall system energy performance. As a result, the underwater vehicle can achieve longer endurance times, as the optimized pump jet system operates more efficiently, thereby conserving energy and extending operational periods. Furthermore, the optimized design led to an increase in thrust output, which enhances the vehicle’s maneuverability and speed. The increased thrust allows the vehicle to overcome higher resistance levels, making it capable of navigating more challenging underwater environments.

5.3. Flow Pattern Analysis

5.3.1. Simulation Result of Pressure

To further investigate the optimal mechanism, the flow pattern in the pump jet is revealed. Figure 23 shows the pressure distribution at the blade pressure surface for the original pump jet and optimal pump jet. It can be seen from the figure that the pressure distribution is more uniform in the optimal pump jet, and the flow separation does not easily occur as the pressure gradient is smaller, which makes the flow pattern more stable and improves the energy performance.

5.3.2. Simulation Result of Velocity

The velocity distribution of different spans under the design flow rate are compared. Figure 24 shows the velocity distribution in the impeller at different blade heights for the original pump jet and the optimal pump jet under the design flow rate. As shown at span 0.5, the velocities in the flow passage for the original pump jet and the optimal pump jet are both fluent. As described at span 0.9, the local high-speed zone of the optimized pump jet impeller blade is significantly reduced, and the flow field changes more evenly, which can effectively inhibit the flow separation.

6. Conclusions

In this work, a blade design method based on center-of-gravity stacking line control was proposed, and the pump jet rotor was designed. Based on a single-factor analysis and orthogonal optimization, a multi-parameter optimization design was carried out for the pump jet. Four optimization parameters including the angle of attack, chord length, camber angle and number of blades are selected in the orthogonal table. Three levels for the optimization parameters are determined by experience. A high-thrust-efficiency pump jet was obtained by orthogonal optimization. Conclusions can be drawn as follows:

(1)

Increasing the angle of attack significantly improves the thrust performance and hydraulic performance of the pump, but it may cause abnormal flow at a small flow rate. The chord length has little effect on the thrust performance. The camber angle has little effect on the maximum thrust efficiency and hydraulic efficiency, but a larger camber angle is beneficial to large flow conditions and has a wider efficient working area.

(2)

According to a range analysis of the orthogonal method, as the angle of attack increases, the pump jet thrust efficiency increases; as the chord length increases, the thrust efficiency decreases; and, as the number of blades increases, the thrust efficiency decreases. The influence level of optimization parameters on pump jet thrust efficiency is sorted as follows: number of blades > angle of attack > chord length > camber angle.

(3)

After the optimization, the thrust coefficient and thrust efficiency of the optimized pump jet are improved under different speed conditions, and the maximum thrust efficiency of the pump jet increased by 7.23%. The flow pattern in the optimal pump jet has been improved. The pressure gradient of the optimal pump jet becomes more fluent than that of original pump jet, which improves the energy performance.

Above all, this paper has conducted an in-depth study on the pump jet rotor structure, but has conducted less research on other structures, such as the duct and stator. In the future, the duct profile, stator-blade-related parameters, and stator–rotor co-ordination relationship would be undergoing further study to improve the hydraulic performance and thrust performance of the pump jet.