# Computational Fluid Dynamics Study of Magnus Force on an Axis-Symmetric, Disk-Type AUV with Symmetric Propulsion

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modeling the Equations of Motion

_{0}, y

_{0}, and z

_{0}axes is referenced by the flow domain and the vehicle motion trajectory.

_{zz}are the moments and products of inertia of the AUH; u and v denote the velocities in surge and sway, respectively; and r denotes the yaw rate.

## 3. RANS Solver (ANSYS-CFX)

#### 3.1. Mesh Generation

^{+}≤ 200 and Δy

^{+}≤ 5 using the k–ε and SST turbulence models, respectively.

#### 3.2. Boundary Conditions

#### 3.3. Computer Simulation

^{–5}. The details of the computational parameters applied to the CFD study on the spinning AUH in uniform flow are given in Table 2.

## 4. Simulation Results of Magnus Force

#### 4.1. Formulation of Hydrodynamic Performance

_{D}and F

_{L}denote the drag and Magnus force, respectively; C

_{D}and C

_{L}denote the hydrodynamic coefficients of drag and Magnus force, respectively; S is the area of wetted surface of the hull projection in the x–y plane, S = πL

^{2}/4; L is the max hull diameter; and V is the freestream flow speed. F

_{D}and F

_{L}were calculated using the ANSYS-CFX RANS solver.

#### 4.2. Distribution of Pressure and Streamlines

#### 4.3. CFD Prediction of Hydrodynamic Coefficients

## 5. Hydrodynamic Performance of the High-Speed Propeller

#### 5.1. Mesh Generation

#### 5.2. Hydrodynamic Performance of the Propeller

_{T}, torque coefficients, K

_{Q}, efficiency, ƞ, and advance ratio, J, as follows:

_{a}denotes the velocity of the currents, n denotes the rotating speed of the propeller, and D is the propeller diameter.

_{T}ratio to J

^{2}as in Equation (9) was obtained, i.e., the propeller loading coefficient, which presents the AUH propeller thrust per unit area of the propeller plane in this study. Hence, the intersection point between the curve of K

_{T}and the curve of K

_{T}about J

^{2}is the value of the advance ratio J that the propeller is working on the AUH. The intersection of the two lines is a suitable mechanical power energy-saving matching point, as shown in Figure 19.

## 6. A Comparison Using Three Different Methods to Escape Currents

^{5}J of energy.

_{p}(W) is the required power of the propeller, and P

_{H}(W) is the power of the system and sensors required for payload operation.

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**The mesh types in the computational fluid domain: (

**a**) Boundary layer thickness; (

**b**) sliding meshes in a cuboid computational fluid domain.

**Figure 6.**The computational fluid domain used to simulate turbulent flow over the spinning AUH and the computational fluid dynamics (CFD) meshing results.

**Figure 7.**The boundary conditions for the spinning AUH: (

**a**) Top view in the x–z plane; (

**b**) side view in the x–y plane.

**Figure 8.**Streamlines produced by ANSYS-CFX CFD calculations of turbulent flow over the AUH with spin ratios of 0 and 5.83, using both the k–ε and shear stress transport (SST) k–ω turbulence models.

**Figure 9.**Pressure distribution of the AUH hull form with spin ratios of 0 and 5.83, using both the k–ε and SST k–ω turbulence models.

**Figure 10.**Variations in Magnus force and drag for different spinning speeds with inflow velocities of 1 and 2 knots.

**Figure 11.**The variation of drag with an inflow velocity of 1 knot and Magnus force with an inflow velocity of 2 knots as spinning speed increases.

**Figure 15.**The CFD meshes of the propeller: (

**a**) The sliding mesh technology applied to the propeller surface meshed around a hexahedron-mesh static computing domain; (

**b**) the tetrahedral mesh generated around the propeller surface in the rotational computing domain; (

**c**) the front view on the tetrahedral mesh around the propeller; (

**d**) the perspective view on the tetrahedral meshes of the blade and hub surfaces.

**Figure 16.**The pressure distribution of propellers at different rotating speeds: (

**a**) 0 rpm; (

**b**) 1900 rpm; (

**c**) −1900 rpm.

Design Parameters | Symbol | Physical Unit | Value |
---|---|---|---|

Diameter | L | m | 1.0 |

Design depth | h | m | 0–1000 |

Height | H | m | 0.43–0.48 |

Mass | m | kg | 50–150 |

Design velocity | V | kn | 1 |

Center of gravity | G(x_{G}, y_{G}, z_{G}) | mm | (0, 0, 0) |

Center of buoyancy | B(x_{B}, y_{B}, z_{B}) | mm | (0, 0, −39.5) |

Parameters | Setting |
---|---|

Computing performance | 4 Intel Xeon CPUs, 72 cores, 36 threads, 2.1 GHz, 256 GB of RAM |

No. of meshing elements | 2.05 million |

Mesh type | Structured, sliding meshes |

Turbulence models | k–ε, k–ω (SST) |

Advection scheme | ANSYS-CFX high resolution |

Residual convergence | RMS residual < 10^{–5} |

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**MDPI and ACS Style**

Chen, C.-W.; Jiang, Y.
Computational Fluid Dynamics Study of Magnus Force on an Axis-Symmetric, Disk-Type AUV with Symmetric Propulsion. *Symmetry* **2019**, *11*, 397.
https://doi.org/10.3390/sym11030397

**AMA Style**

Chen C-W, Jiang Y.
Computational Fluid Dynamics Study of Magnus Force on an Axis-Symmetric, Disk-Type AUV with Symmetric Propulsion. *Symmetry*. 2019; 11(3):397.
https://doi.org/10.3390/sym11030397

**Chicago/Turabian Style**

Chen, Chen-Wei, and Yong Jiang.
2019. "Computational Fluid Dynamics Study of Magnus Force on an Axis-Symmetric, Disk-Type AUV with Symmetric Propulsion" *Symmetry* 11, no. 3: 397.
https://doi.org/10.3390/sym11030397