# The One-Way FSI Method Based on RANS-FEM for the Open Water Test of a Marine Propeller at the Different Loading Conditions

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Governing Equations of the Flow Around the Propeller

_{i}denotes the body forces presented as forces per unit volume and in the present study assumed that ${f}_{i}=0$. Moreover, u, $\rho $, and P are fluid velocity vectors, density, and pressure, respectively. The Boussinesq assumption is considered to represent the Reynolds stress for incompressible flows, which is commented below:

#### 2.2. Governing the Structural Equations

#### 2.3. Modeling and Computational Setup

#### 2.3.1. Open Water Test Characteristic

#### 2.3.2. Applied Boundry Condition and Dynamic Motions Method

## 3. Results and Discussion

#### 3.1. CFD Validation

#### Hydrodynamic Analysis of the vp1304 Propeller

#### 3.2. Fluid–Structure Interaction Validation

#### 3.2.1. Finite-Element Method

_{1}, ϕ

_{2}). Moreover, weight functions (δu, δv, δW

_{1}, δw

_{2}, δw

_{3}) are approximated:

_{i}) are substituted in the differential equations’ weak form [32]. These functions are nodal parameters (x and y) in which x and y are nodal displacements. At the finite element methods based on displacement, the displacement’s manner in the element boundaries is not separated; unlike the strains, that the manner of strain is continuous only within one element. The point here is, choose between the linear or quadratic elements. Indeed, the strains have a constant value in linear elements, but in quadratic elements, the strains are nonlinear with more accurate strain or stress results than linear elements. According to Barlow [33], strains and stresses can be solved without limitation in the element, just for points, including defined nodes.

#### 3.2.2. One-Way Coupling Approach

- Decrease the complexity of the numerical solution by dividing it into two parts;
- Create two distinct mesh generation schemes depending on the grid dimension needed;
- The ability to use one hydrodynamic solution for many structural sets, using different materials, thickness and different structural design;
- Lower numerical solution cost and time rather than a two-way approach, the solution time is evaluated in the present study illustrated in Table 10;
- High-fidelity results for the cases with low deflection;
- One-way coupling is more useful for cases with large domain and multiphase systems like investigations on marine vessels, propellers, etc.

#### 3.3. Structural Behavior of Propellers’ Blade

#### 3.4. Propellers’ Structural Behavior in Rotation

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Arbitrary mesh interface | AMI |

Boundary element method | BEM |

Computational fluid dynamic | CFD |

Fluid–structure interaction | FSI |

Finite element method | FEM |

International Towing Tank Conference | ITTC |

Large eddy simulation | LES |

Multi-reference frame | MRF |

Pressure implicit with splitting of operators | PISO |

Potsdam propeller test case | PPTC |

Vortex lattice method | VLM |

Volume of fluid | VOF |

Reynolds-averaged Navier–Stokes | RANS |

Rigid/quasi-static | RQS |

Semi-implicit method for pressure-linked equations | SIMPLE |

Turbulent kinetic energy | TKE |

Wetting time equation | WQ |

## Nomenclature

${K}_{t}$ | Thrust coefficient |

${K}_{q}$ | Torque coefficient |

${V}_{a}$ | Advance velocity |

$\eta $ | Efficiency |

J | Advance coefficient |

E | Elasticity |

$\rho $ | Density |

υ | Poissons’ ratio |

ω | Angular frequency |

S | von Mises stress |

${P}_{i}$ | Pressure gauge (i = 1–2–3) |

ε | Strain |

F | Force |

T,t | Time |

e | Error percentage |

rev | Propeller revolution |

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**Figure 1.**Numerical solution domain used for computational fluid dynamics (CFD) part of the present study.

**Figure 5.**Gauge pressure variations of one rotation (t = 0.12 s-t = 0.18 s) for different advanced coefficients.

**Figure 7.**Present method (by considering the wedge impact) verification versus two-way coupling [24].

Propeller Model | Vp1304 |
---|---|

Diameter | 0.25 m |

Hub coefficient | 0.3 |

Number of blades | 5 |

pitch coefficient (r/R = 0.7) | 1.635 |

A_{E}/A_{0} | 0.779 |

Material | Al-Alloy |
---|---|

Elasticity | 120 Gpa |

Poisson’s ratio | 0.34 |

Mass density | 7400 kg/m^{3} |

Parameter | Unit | Model | Real |
---|---|---|---|

Density (water) | kg m^{−3} | 999.0 | 1025 |

Kinematic viscosity (water) | m^{2} s^{−1} | 1.139 × 10^{−6} | 1.188 × 10^{−6} |

Revolution (propeller) | s^{−1} | 15 | 4.33 |

Quality | Base Grid | Cell.NUM | $\frac{{\mathit{k}}_{\mathit{t}}}{{({\mathit{k}}_{\mathit{t}})}_{\mathit{e}\mathit{x}\mathit{c}\mathit{e}\mathit{l}\mathit{l}\mathit{e}\mathit{n}\mathit{t}}}$ | $\frac{{\mathit{k}}_{\mathit{q}}}{{({\mathit{k}}_{\mathit{q}})}_{\mathit{e}\mathit{x}\mathit{c}\mathit{e}\mathit{l}\mathit{l}\mathit{e}\mathit{n}\mathit{t}}}$ | NUM |
---|---|---|---|---|---|

Coarse | 0.11 | 245,210 | 1.1 | 1.11 | (I) |

Mid | 0.09 | 315,402 | 1.05 | 1.055 | (II) |

Mid-fine | 0.08 | 335,183 | 1.025 | 1.024 | (III) |

Fine | 0.064 | 425,060 | 1.015 | 1.013 | (IIII) |

Excellent | 0.0325 | 835,205 | ≈1 | ≈1 | (V) |

J [-] | ω [rps] | V_{a} [m/s] | Number |
---|---|---|---|

0.266 | 15 | 1 | I |

0.533 | 15 | 2 | II |

0.8 | 15 | 3 | III |

1.06 | 15 | 4 | IIII |

1.23 | 15 | 5 | V |

1.6 | 15 | 6 | VI |

Gauge | Distance from the Center | |

p-1 | 0.04 m | |

p-2 | 0.08 m | |

p-3 | 0.12 m |

Characters | Wedge Length | Wedge Thickness | Deadrise Angle |
---|---|---|---|

value | 0.3 m | 0.002 m | 20° |

Characters | Material | E [Gpa] | $\mathit{\rho}$ [kg/m^{3}]
| ν [-] |
---|---|---|---|---|

value | Aluminum | 68 | 2700 | 0.3 |

Case (FEM) | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

Time step(s) | 0.07 | 0.075 | 0.08 | 0.085 | 0.09 | 0.095 | 0.1 |

Rotation Angle | 18° | 45° | 72° | 99° | 126° | 153° | 180° |

Stress (pa) | 6.01 × 10^{6} | 5.92 × 10^{6} | 5.47 × 10^{6} | 5.59 × 10^{6} | 5.91 × 10^{6} | 9.16 × 10^{6} | 5.90 × 10^{6} |

Strain (m) | 4.7 × 10^{−5} | 4.67 × 10^{−5} | 4.34 × 10^{−5} | 4.4 × 10^{−5} | 4.6 × 10^{−5} | 8.4 × 10^{−5} | 4.6 × 10^{−5} |

8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |

0.105 | 0.11 | 0.115 | 0.12 | 0.125 | 0.13 | 0.135 | 0.14 |

207° | 234° | 261° | 288° | 315° | 342° | 369° | 396° |

8.87 × 10^{5} | 8.40 × 10^{5} | 5.20 × 10^{6} | 5.70 × 10^{6} | 5.80 × 10^{6} | 5.80 × 10^{6} | 5.70 × 10^{6} | 5.70 × 10^{6} |

8.2 × 10^{−6} | 7.8 × 10^{−6} | 4.11 × 10^{−5} | 4.5 × 10^{−5} | 4.6 × 10^{−5} | 4.59 × 10^{−5} | 4.56 × 10^{−5} | 4.56 × 10^{−5} |

16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 |

0.145 | 0.15 | 0.155 | 0.16 | 0.165 | 0.17 | 0.175 | 0.18 |

423° | 450° | 477° | 504° | 531° | 558° | 585° | 612° |

5.70 × 10^{6} | 5.60 × 10^{6} | 5.68 × 10^{6} | 5.63 × 10^{6} | 5.60 × 10^{6} | 5.55 × 10^{6} | 5.40 × 10^{6} | 5.30 × 10^{6} |

4.51 × 10^{−5} | 4.44 × 10^{−5} | 4.45 × 10^{−5} | 4.44 × 10^{−5} | 4.41 × 10^{−5} | 4.36 × 10^{−5} | 4.28 × 10^{−5} | 4.17 × 10^{−5} |

Advance Velocity | V = (1, 2, 3, 4, 5) | m/s |
---|---|---|

FEM solution time + gathering datasheet | 2 + 1 | hour |

CFD solution time | 24 | hour |

Cumulative time (present method) | 27 | hour |

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**MDPI and ACS Style**

Masoomi, M.; Mosavi, A. The One-Way FSI Method Based on RANS-FEM for the Open Water Test of a Marine Propeller at the Different Loading Conditions. *J. Mar. Sci. Eng.* **2021**, *9*, 351.
https://doi.org/10.3390/jmse9040351

**AMA Style**

Masoomi M, Mosavi A. The One-Way FSI Method Based on RANS-FEM for the Open Water Test of a Marine Propeller at the Different Loading Conditions. *Journal of Marine Science and Engineering*. 2021; 9(4):351.
https://doi.org/10.3390/jmse9040351

**Chicago/Turabian Style**

Masoomi, Mobin, and Amir Mosavi. 2021. "The One-Way FSI Method Based on RANS-FEM for the Open Water Test of a Marine Propeller at the Different Loading Conditions" *Journal of Marine Science and Engineering* 9, no. 4: 351.
https://doi.org/10.3390/jmse9040351