# Numerical Prediction of Ship Resistance Based on Volume of Fluid Implicit Multi-Step Method

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Ship Geometry and Condition

## 3. Numerical Modeling

#### 3.1. Governing Equations

#### 3.2. Physics Modeling

#### 3.2.1. VOF Method

_{b}is the body force vector, ${S}_{i}^{a}$ is the phase momentum source term, E and H denote the total energy and total enthalpy, respectively, and S

_{E}is the customized energy source term.

#### 3.2.2. Mesh Generation

## 4. Results and Discussion

#### 4.1. Verification Study

_{1}, N

_{2}, and N

_{3}, and the mesh and time step [28] refinement factor r

_{21}are defined as follows:

_{21}is the scaling of the grid and time. For the apparent order, p is the order of discretization, and is calculated using Equation (23):

#### 4.1.1. Spatial Convergence Study

_{1}= 4,777,034, N

_{2}= 2,151,616, and N

_{3}= 1,027,928.

#### 4.1.2. Temporal Convergence Study

_{21}remains constant, with a minimum time step of 0.005 s. Table 4 shows the time discretization errors of the main parameters, where Δt

_{1}= 0.005 s, Δt

_{2}= 0.01 s, and Δt

_{3}= 0.02 s.

#### 4.2. Validation Study

_{2}, taking into account the computational accuracy and costs. The accuracy of the numerical methods and physical models was verified by comparing the numerical results with the experimental values, to obtain the relative error of the numerical calculations. The results of the validation analysis are shown in Table 5, where EFD is the experimental value.

_{2}grid were close to those of N

_{1}, the N

_{2}configuration was used in the subsequent CFD simulations to reduce the solution time.

_{2}and Δt

_{2}for the flow field analysis.

#### 4.3. Resistance Prediction and Flow Field Analysis

#### 4.3.1. Computational Cost

_{2}grid configuration, the time steps Δt

_{1}and Δt

_{2}were chosen for the single-step method in the CFD simulations. The implicit multi-step method, based on Δt

_{2}, calculated the calm water resistance and flow field for inner iteration numbers of 1, 2, and 4, where the number of inner iterations is the number of iterations performed in a loop. Table 6 and Figure 9 illustrate the results of the ship resistance calculation using different VOF methods and a comparison of the computational costs, where case 1 is a single-step method with a time step of 0.01 s; case 2 is a single-step method with a time step of 0.005 s; and cases 3, 4, and 5 are implicit multi-step methods with internal iterations of 1, 2, and 4, respectively, and all with a time step of 0.01 s.

#### 4.3.2. Free Surface

## 5. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**A diagram of the computational domain and mesh for resistance prediction. (

**a**) The layout of the computational domain. (

**b**,

**c**) The mesh near the hull.

**Figure 7.**Comparison of wave pattern of KCS model at Fr = 0.260 [23].

**Figure 8.**Comparison of wave profiles of KCS model at Fr = 0.260 [23].

**Figure 11.**Comparison of wave profiles of KCS model using different VOF methods [23].

Main Particulars | Symbols and Units | Values |
---|---|---|

Scale ratio | λ | 31.6 |

Length between the perpendiculars | L_{PP} (m) | 7.2786 |

Length of waterline | L_{WL} (m) | 7.3577 |

Beam of waterline | B_{WL} (m) | 1.019 |

Draft | D (m) | 0.6013 |

Displacement | Δ (m^{3}) | 1.649 |

Block coefficient | C_{B} | 0.6505 |

Wetted area (with rudder) | S (m^{2}) | 9.512 |

Longitudinal center of buoyance | L_{CB} (%L_{PP}) | −1.48 |

Radius of gyration | K_{xx}/B | 0.4 |

Radius of gyration | K_{yy}/L_{PP}, K_{zz}/L_{PP} | 0.25 |

Parameters | Symbols and Units | Values |
---|---|---|

Density | ρ (kg/m^{3}) | 999.5 |

Kinematic viscosity | v (m^{2}/s) | 1.27 × 10^{−6} |

Speed | U (m/s) | 2.196 |

Froude number | Fr | 0.26 |

Reynolds number | Re | 1.26 × 10^{7} |

Resistance Coefficient | Trim | Sinkage | |
---|---|---|---|

r_{21} | 1.30 | 1.30 | 1.30 |

r_{23} | 1.28 | 1.28 | 1.28 |

φ_{1} | 0.0036944 | −0.1740020 deg | −0.0135945 m |

φ_{2} | 0.0036944 | −0.1716800 deg | −0.01359466 m |

φ_{3} | 0.0036180 | −0.1762450 deg | −0.01376446 m |

ε_{32} | −7.64 × 10^{−5} | −4.57 × 10^{−3} deg | −1.70 × 10^{−4} m |

ε_{21} | −3.43 × 10^{−8} | 2.32 × 10^{−3} deg | −1.50 × 10^{−7} m |

s | 1 | −1 | 1 |

q | 0.000929% | 1.334467% | 0.001103% |

p_{a} | 6.15 × 10^{−1} | 3.49 × 10^{−2} | 5.61× 10^{−1} |

${\phi}_{\mathrm{ext}}^{21}$ | 3.13 × 10^{1} | 2.67 × 10^{0} | 2.86 × 10^{1} |

${\mathrm{e}}_{\mathrm{ext}}^{21}$ | 2.00 × 10^{−9} | 1.27 × 10^{−2} | 6.00 × 10^{−9} |

${\mathrm{GCI}}_{\mathrm{fine}}^{21}$ | 0.0000003% | 1.6103827% | 0.0000007% |

Resistance Coefficient | Trim | Sinkage | |
---|---|---|---|

r_{21} | 2.00 | 2.00 | 2.00 |

r_{23} | 2.00 | 2.00 | 2.00 |

φ_{1} | 0.0036809 | −0.169457 deg | −0.0136624 m |

φ_{2} | 0.0036944 | −0.17168 deg | −0.01359466 m |

φ_{3} | 0.0037140 | −0.170599 deg | −0.0135868 m |

ε_{32} | 1.95 × 10^{−5} | 1.08 × 10^{−3} deg | 7.86 × 10^{−6} m |

ε_{21} | 1.35 × 10^{−5} | −2.22 × 10^{−3} deg | 6.77 × 10^{−5} m |

s | 1 | −1 | 1 |

${\mathrm{e}}_{\mathrm{a}}^{21}$ | 0.367314% | 1.311837% | 0.495813% |

p_{a} | 5.32 × 10^{−1} | 1.04 × 10^{0} | 3.11 × 10^{0} |

${\phi}_{\mathrm{ext}}^{21}$ | 3.65 × 10^{−3} | −1.67 × 10^{−1} | −1.37 × 10^{−2} |

${\mathrm{e}}_{\mathrm{ext}}^{21}$ | 8.31 × 10^{−3} | 1.26 × 10^{−2} | 6.50 × 10^{−4} |

${\mathrm{GCI}}_{\mathrm{fine}}^{21}$ | 1.030297% | 1.552207% | 0.081352% |

Mesh Type | Resistance Coefficient | Error | Trim (deg) | Error | Sinkage (m) | Error |
---|---|---|---|---|---|---|

N_{1} | 0.0036944 | 0.446 | −0.1740020 | −2.960 | −0.01359451 | 2.478 |

N_{2} | 0.0036944 | 0.447 | −0.1716800 | −1.586 | −0.01359466 | 2.477 |

N_{3} | 0.0036180 | 2.507 | −0.1762450 | −4.287 | −0.01376446 | 1.259 |

EFD | 0.003711 | −0.169 | −0.01394 |

Case | Method | Resistance Coefficient | Error | Trim (deg) | Sinkage (m) |
---|---|---|---|---|---|

case 1 | SS_0.01 s | 0.0036944 | 0.447 | −0.1716800 | −0.01359466 |

case 2 | SS_0.005 s | 0.0036809 | 0.812 | −0.1694570 | −0.01366240 |

case 3 | IM(1)_0.01 s | 0.0036944 | 0.447 | −0.1716700 | −0.01366019 |

case 4 | IM(2)_0.01 s | 0.0036780 | 0.890 | −0.1730820 | −0.01367844 |

case 5 | IM(4)_0.01 s | 0.0036784 | 0.880 | −0.1727190 | −0.01366914 |

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**MDPI and ACS Style**

Wang, Y.; Rao, H.; Liu, Z.; Liu, K.; Zhou, B.; Zhang, G.
Numerical Prediction of Ship Resistance Based on Volume of Fluid Implicit Multi-Step Method. *J. Mar. Sci. Eng.* **2023**, *11*, 2181.
https://doi.org/10.3390/jmse11112181

**AMA Style**

Wang Y, Rao H, Liu Z, Liu K, Zhou B, Zhang G.
Numerical Prediction of Ship Resistance Based on Volume of Fluid Implicit Multi-Step Method. *Journal of Marine Science and Engineering*. 2023; 11(11):2181.
https://doi.org/10.3390/jmse11112181

**Chicago/Turabian Style**

Wang, Yu, Honghua Rao, Zhengyuan Liu, Kaihua Liu, Bo Zhou, and Guiyong Zhang.
2023. "Numerical Prediction of Ship Resistance Based on Volume of Fluid Implicit Multi-Step Method" *Journal of Marine Science and Engineering* 11, no. 11: 2181.
https://doi.org/10.3390/jmse11112181