# Comparative Investigation on Hydrodynamic Performance of Pump-Jet Propulsion Designed by Direct and Inverse Design Methods

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## Abstract

**:**

## 1. Introduction

## 2. Design Methods of Pump-Jet Propulsion

#### 2.1. Direct Design of Pump-Jet Model

#### 2.1.1. Lifting Design Method

- (1)
- The flow in the rotor is regarded as potential flow, and there is no radial velocity component;
- (2)
- The distribution of velocity loop is constant along the radius;
- (3)
- There is no induced velocity in the axial direction.

_{u}

_{1}and v

_{u}

_{2}are the circumferential components of the absolute velocity at the inlet and outlet of the rotor; u

_{1}and u

_{2}are the circumferential velocities of the rotor inlet and outlet. Based on the assumption of the independence of the cylindrical layer, the fluid at the inlet on the same cylindrical section has no circumferential pre-swirl, that is v

_{u}

_{1}= 0. Then the velocity circulation of the blade at radius r is:

_{t}, the fluid gravity acceleration is g, and ω is the angular velocity of rotor rotation.

_{l}is the lift, and F

_{d}is the resistance, given by:

_{∞}is the average of the relative velocities, C

_{y}is the lift coefficient, C

_{D}is the drag coefficient, and A is the maximum projected area. Then the circumferential component of the resultant force is given by

_{m}is the axial velocity, and hydraulic efficiency can also be expressed as ƞ

_{h}= H/H

_{t}. Thus, it can be obtained from Formula (4) and Formula (6) as

_{y}and attack angle, and the result can be finally achieved by iterative calculation. The related design process for the lifting method is shown in Figure 4.

#### 2.1.2. Lifting-Line Method

- (1)
- The fluid is regarded as ideally incompressible;
- (2)
- Treating the inflow as steady and axisymmetric;
- (3)
- The rotor wake is regarded as non-contracting, and its influence on the shape of the vortex is not considered;
- (4)
- The radial induced speed is not considered;
- (5)
- It is assumed that the circulation at the hub diameter is 0, but for the rotor with a larger hub, a later correction is required.

_{a}and V

_{t}, the axial and circumferential components of the inlet flow at the rotor are v

_{a}and v

_{t}, respectively, the resultant force F is divided into lift and drag, the pitch angle of the airfoil profile is the placement angle of the profile with ${\psi}^{*}$ means, the angle of attack is α, β and β

_{i}are the airfoil inlet angle and induced velocity angle, respectively. Assuming that the circulation and the resultant speed v* are known, the rotor thrust and torque can be expressed in the following integral form:

_{D}is used to express the drag coefficient of the airfoil profile.

_{in}represents the rotor inlet flow velocity, G is the dimensionless circulation, i

_{a}and i

_{t}are the axial and tangential induction factors, respectively. Moreover, x and x

_{0}are the dimensionless radial coordinates and x

_{h}is the dimensionless radius at the hub of the rotor.

#### 2.1.3. Optimization Following Direct Method Design

_{i}

_{1}and ω

_{i}

_{2}are the weights. To improve the propulsion performance at a higher sailing speed, the weights were 0.4 and 0.6, respectively.

#### 2.2. Inverse Design of Pump-Jet Model

_{b}is tangent to the blade surface and orthogonal to the blade normal vector $\nabla S$, namely, ${W}_{b}\xb7\nabla S=0$. Expanding the relative speed, we can express this relationship as

## 3. Numerical Method and Computational Setup

#### 3.1. Governing Equation

#### 3.2. Turbulence Model

_{1}represents the weighting function; F

_{2}represents the mixing function; S represents the curl amplitude, ${a}_{1}$ is empirical coefficient which is 0.31. The constant values are α

_{1}= 5/9, α

_{2}= 0.44, σ

_{ω}

_{1}= 1, σ

_{ω}

_{2}= 1.1682, β

_{1}= 0.075, β

_{2}= 0.0828, β* = 0.075, σ

_{ω}

_{1}= 2, σ

_{k}

_{1}= 2, σ

_{k}

_{2}= 1.

#### 3.3. Pump-Jet Basic Parameters and Models

_{t}= 0.324 m, and the total length of the pump-jet was 0.30 m, while the propulsion has 9 rotor blades and 11 stator blades. Considering the gap flow between the rotor and the inner wall of the duct, 1 mm was chosen as the minimum gap between the rotor tip and the inner wall of the duct. Comparing the direct design method, the blade designed by the inverse design method becomes more slant, while the stator placement grew larger.

#### 3.4. Mesh Generation and Boundary Conditions

^{−5}.

_{T}represents thrust coefficient, K

_{Q}represents torque coefficient, and η is propulsion efficiency. By comparing the thrust coefficient, torque coefficient, and propulsion efficiency with the different number of grid cells of the computational domain under the specific working condition, the most appropriate number of grid cells was concluded.

#### 3.5. Numerical Method Validation

_{t}is the thrust coefficient and K

_{q}is the torque coefficient. The propulsion efficiency η is defined as η = K

_{t}/K

_{q}*J/(2π). Moreover, it can be seen from the figure that the numerical results generally agree well with the experimental data with an error of less than ten percent, which implies that the numerical approach employed in this study is stable and reliable.

## 4. Results and Discussion

#### 4.1. Analysis of Pump-Jet Performance by Direct Design Method

#### 4.1.1. Comparison of Lifting Method and Lifting-Line Method

_{p}in the figure represents the pressure coefficient at a certain position, which is defined as

_{x}/U, V

_{y}/U, and V

_{z}/U are the dimensionless parameters of tangential velocity, radial velocity, and axial velocity, respectively. As shown in Figure 23 and Figure 24, each velocity component at the rotor inlet shows a periodic oscillation distribution with peaks and valleys whose number is the same as the number of blades. The amplitude of the axial velocity oscillation and the velocity value is larger and the distribution is more uniform. When axial positions are between the rotor and stator, the value of the axial velocity component is larger, especially in the position where the span is large, while the flow is more uniform and the fluctuation becomes smaller. At the stator exit, the fluid is further accelerated in the channel and the axial velocity increases, but the tangential and radial velocities decrease significantly. The number of peaks and valleys also become inconsistent with the stator blades number.

#### 4.1.2. Optimized Pump-Jet Model Performance

_{x}and the radial velocity V

_{y}are smaller than that before optimization, which shows that the optimized internal flow field becomes more stable and the turbulent kinetic energy is smaller. At the outlet position, the radial and tangential velocity of the pump-jet is lower after optimization, and the axial velocity component is larger, indicating that the propulsion produces a larger thrust and more effective for fluid work.

#### 4.2. Analysis of Pump-Jet Performance by Inverse Design Method

## 5. Conclusions

- For the direct design method, compared to the lifting-line method, the pump-jet propeller designed by lifting method has higher efficiency under the working condition of J < 1.5, which is roughly 5% higher on average, while the range of the high-efficiency operating conditions (η > 60%) of the two methods is similar. The pump-jet designed by the lifting method has a weaker accelerating effect on the mainstream, which produces lower thrust and torque. Moreover, the cavitation is less likely to occur on the blades and the pressure distribution is more uniform, which indicates that the work exerted on the fluid is more uniform and the fluid merges with the mainstream to a greater degree after acceleration so that the matching stator is less prone to cavitation. The turbulent kinetic energy of the internal flow field is also lower. Thus, for small and medium-sized underwater vehicles, the pump-jet designed by the lift method is more suitable.
- After optimizing, the rotor hub ratio changed from 0.4 to 0.45; The rotor placement angle was changed from 64.9° to 69.66°; The tip clearance was changed from 1 to 1.9 mm; The gap of rotor-stator increased from 26.8 to 33.49 mm. In addition, the stator placement angle increased from 77° to 82°. In terms of the hydrodynamic performance, the weighted average efficiency of the two working conditions J = 1.06 and J = 1.07 is 5.372%, which is higher than that before the optimization. After the optimization, the highest efficiency of the pump-jet increased by 5.14%, with the thrust increasing roughly 224.8 N, while the thrust coefficient and efficiency curve both deviate to the direction of the higher speed, with the optimal working condition point slightly shifting. The efficiency drop is slightly slowed down at a higher advance coefficient, which broadens the high-efficiency range. Moreover, the blade has a stronger effect on fluid rotation and acceleration so that the energy obtained by the fluid is higher, with the thrust relatively increasing at a higher advance coefficient. For the stator part, it is more likely to produce cavitation at the outlet edge than that before optimization, but part of it converts the kinetic energy at the rotor outlet into the high-pressure energy.
- The pump-jet obtained by the inverse design method has a greater thrust, with the negative thrust generated by the stator shifting to a higher speed at the operating point, implying that the matching of the rotor and stator is better. The maximum efficiency is 78.56%, which is 5.94% higher than that obtained by the direct design method. Furthermore, the high-efficiency range is wider, so that efficiency value is more stable at a medium speed. Compared to the pump-jet obtained by the direct method, the inverse method has a larger effect on the rotation acceleration of the fluid, so that the fluid achieves greater energy, with greater thrust and velocity. As the advance coefficient increases, the adaptive advance speed range is larger. However, the fluid accelerated by the rotor makes the stator flow field more unstable and the turbulent kinetic energy is greater. As the fluid further develops, the periodicity in the channel weakens as the internal flow field becomes more complicated, which indicates that the turbulent kinetic energy is higher, causing relatively larger energy loss. On the whole, the pump-jet obtained by the inverse design method has better balance and anti-cavitation performance, especially at higher speeds.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 7.**Load of the rotor and stator blades. (

**a**) Load of the rotor, (

**b**) Load of the stator blades.

**Figure 8.**Pump-jet propulsion model designed by direct method: (

**a**) lifting method; (

**b**) lifting-line method; (

**c**) optimal model.

**Figure 14.**Hydrodynamic performance of pump-jet designed by lifting method and lifting-line method: (

**a**) Lifting method; (

**b**) lifting-line method.

**Figure 16.**Tangential pressure distribution of the rotor designed by lifting method: (

**a**) v = 7 m/s (J = 0.78), (

**b**) v = 8.5 m/s (J = 0.95), (

**c**) v = 10 m/s (J = 1.12), and (

**d**) v = 11.5 m/s (J = 1.28).

**Figure 17.**Tangential pressure distribution of the rotor designed by lifting-line method: (

**a**) v = 7 m/s (J = 0.78), (

**b**) v = 8.5 m/s (J = 0.95), (

**c**) v = 10 m/s (J = 1.12), and (

**d**) v = 11.5 m/s (J = 1.28).

**Figure 18.**Tangential pressure distribution of stator matched with lifting method: (

**a**) v = 7 m/s (J = 0.78), (

**b**) v = 8.5 m/s (J = 0.95), (

**c**) v = 10 m/s (J = 1.12), and (

**d**) v = 11.5 m/s (J = 1.28).

**Figure 19.**Tangential pressure distribution of stator matched with lifting-line method: (

**a**) v = 7 m/s (J = 0.78), (

**b**) v = 8.5 m/s (J = 0.95), (

**c**) v = 10 m/s (J = 1.12), and (

**d**) v = 11.5 m/s (J = 1.28).

**Figure 20.**Axial section pressure distribution of rotor designed by lifting method: (

**a**) 20% rotor channel, (

**b**) 40% rotor channel, (

**c**) 70% rotor channel, and (

**d**) 90% rotor channel.

**Figure 21.**Axial section pressure distribution of rotor designed by lifting-line method: (

**a**) 20% rotor channel, (

**b**) 40% rotor channel, (

**c**) 70% rotor channel, and (

**d**) 90% rotor channel.

**Figure 23.**Velocity distribution of pump-jet designed by lifting method (J = 1.12) (

**a1–a3**are rotor inlet,

**b1–b3**are between rotor and stator,

**c1–c3**are stator outlet).

**Figure 24.**Velocity distribution of pump-jet designed by lifting-line method (J = 1.12) (

**a1–a3**are rotor inlet,

**b1–b3**are between rotor and stator,

**c1–c3**are stator outlet).

**Figure 26.**Tangential pressure distribution of rotor after optimization: (

**a**) v = 7 m/s (J = 0.78), (

**b**) v = 8.5 m/s (J = 0.95), (

**c**) v = 10 m/s (J = 1.12), and (

**d**) v = 11.5 m/s (J = 1.28).

**Figure 27.**Tangential pressure distribution of stator after optimization: (

**a**) v = 7 m/s (J = 0.78), (

**b**) v = 8.5 m/s (J = 0.95), (

**c**) v = 10 m/s (J = 1.12), and (

**d**) v = 11.5 m/s (J = 1.28).

**Figure 28.**Axial section pressure distribution of rotor after optimization: (

**a**) 20% rotor channel, (

**b**) 40% rotor channel, (

**c**) 70% rotor channel, and (

**d**) 90% rotor channel.

**Figure 29.**Velocity distribution of pump-jet after optimization (

**a1–a3**are rotor inlet,

**b1–b3**are between rotor and stator,

**c1–c3**are stator outlet).

**Figure 31.**Tangential pressure distribution of rotor designed by inverse design method: (

**a**) v = 7 m/s (J = 0.78), (

**b**) v = 8.5 m/s (J = 0.95), (

**c**) v = 10 m/s (J = 1.12), and (

**d**) v = 11.5 m/s (J = 1.28).

**Figure 32.**Tangential pressure distribution of stator designed by inverse design method: (

**a**) v = 7 m/s (J = 0.78), (

**b**) v = 8.5 m/s (J = 0.95), (

**c**) v = 10 m/s (J = 1.12), and (

**d**) v = 11.5 m/s (J = 1.28).

**Figure 33.**Axial section pressure distribution of rotor designed by inverse design method (

**a**) 20% rotor channel, (

**b**) 40% rotor channel, (

**c**) 70% rotor channel, and (

**d**) 90% rotor channel.

**Figure 34.**Velocity distribution of pump-jet designed by inverse design method (

**a1–a3**are rotor inlet,

**b1–b3**are between rotor and stator,

**c1–c3**are stator outlet).

Parameter | Setting Value |
---|---|

The maximum number of iterations | 25 |

Number of particle groups | 20 |

Inertia weight | 0.9 |

Global increment | 0.9 |

Particle increment | 0.9 |

Maximum search speed | 0.1 |

Run failure threshold | 1.0 × 10^{30} |

Run failure target value | 1.0 × 10^{30} |

Optimization Variable | Initial Value | Upper and Lower Boundary |
---|---|---|

Rotor placement angle | 64.9° | [55, 70] |

Stator placement angle | 77° | [70, 85] |

Rotating stator shaft spacing | 26.8 mm | [16, 36] |

Tip clearance | 1 mm | [1, 3] |

Hub ratio | 0.4 | [0.3, 0.5] |

Parts | Y Plus |
---|---|

rotor blades | 4.5 |

rotor wall | 20.5 |

stator blades | 2.9 |

stator wall | 16.8 |

Physical Parameter | Definition |
---|---|

Advance Coefficient | $J=U/\left(nD\right)$ |

Rotating System Thrust Coefficient | ${K}_{{T}_{r}}={T}_{r}/\left(\rho {n}^{2}{D}^{4}\right)$ |

Static System Thrust Coefficient | ${K}_{{T}_{s}}={T}_{s}/\left(\rho {n}^{2}{D}^{4}\right)$ |

Rotating System Torque Coefficient | ${K}_{{Q}_{r}}={Q}_{r}/\left(\rho {n}^{2}{D}^{5}\right)$ |

Total Thrust Coefficient | ${K}_{T}={K}_{{T}_{r}}+{K}_{{T}_{s}}$ |

Total Torque Coefficient | ${K}_{Q}={K}_{{Q}_{r}}$ |

Propulsion Efficiency | $\eta =\frac{J}{2\pi}\xb7\frac{{K}_{T}}{{K}_{Q}}$ |

J | Number of Grids | K_{T} | K_{Q} | η |
---|---|---|---|---|

1.12 | 1,768,631 | 0.235 | 0.0628 | 66.58% |

2,364,892 | 0.241 | 0.0629 | 68.17% | |

2,873,435 | 0.234 | 0.0631 | 64.98% | |

3,371,002 | 0.239 | 0.0631 | 67.37% | |

3,812,531 | 0.240 | 0.0633 | 67.46% | |

4,371,656 | 0.241 | 0.0635 | 67.52% |

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## Share and Cite

**MDPI and ACS Style**

Zhou, Y.; Wang, L.; Yuan, J.; Luo, W.; Fu, Y.; Chen, Y.; Wang, Z.; Xu, J.; Lu, R.
Comparative Investigation on Hydrodynamic Performance of Pump-Jet Propulsion Designed by Direct and Inverse Design Methods. *Mathematics* **2021**, *9*, 343.
https://doi.org/10.3390/math9040343

**AMA Style**

Zhou Y, Wang L, Yuan J, Luo W, Fu Y, Chen Y, Wang Z, Xu J, Lu R.
Comparative Investigation on Hydrodynamic Performance of Pump-Jet Propulsion Designed by Direct and Inverse Design Methods. *Mathematics*. 2021; 9(4):343.
https://doi.org/10.3390/math9040343

**Chicago/Turabian Style**

Zhou, Yunkai, Longyan Wang, Jianping Yuan, Wei Luo, Yanxia Fu, Yang Chen, Zilu Wang, Jian Xu, and Rong Lu.
2021. "Comparative Investigation on Hydrodynamic Performance of Pump-Jet Propulsion Designed by Direct and Inverse Design Methods" *Mathematics* 9, no. 4: 343.
https://doi.org/10.3390/math9040343