# Ship Bow Wings with Application to Trim and Resistance Control in Calm Water and in Waves

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Methodology

#### Solver Desription

**u**), and an indicator function ($a$). The governing equations integrated over control volume $\Omega $ with a boundary of $\partial \Omega $ are obtained by the following expression:

## 3. Numerical Setup

- (i)
- the extrapolated value$${\phi}_{ext}^{21}=\frac{{r}_{21}^{p}{\phi}_{1}-{\phi}_{2}}{{r}_{21}^{p}-1},$$
- (ii)
- the approximate relative error$${e}_{a}^{21}=\left|\frac{{\phi}_{2}-{\phi}_{1}}{{\phi}_{1}}\right|,$$
- (iii)
- the extrapolated relative error$${e}_{ext}^{21}=\left|\frac{{\phi}_{ext}^{21}-{\phi}_{1}}{{\phi}_{ext}^{21}}\right|,$$
- (iv)
- the fine GCI$$GC{I}_{fine}^{21}=\frac{1.25{e}_{ext}^{21}-{\phi}_{1}}{{r}_{21}^{p}-1}.$$

## 4. Results and Discussion

#### 4.1. Dynamic Trim and Sinkage of the Bare Hull

#### 4.2. Effect of a Wing Arranged in Front of the Bow in Calm Water Conditions

#### 4.3. Effect of the Foil Inclination in Calm Water Conditions

#### 4.4. Effect of the Wing in the Presence of Regular Waves

_{5}) and heave (ξ

_{3}). The heave was nondimensionalized based on the wave amplitude (A), while pitch used wave number (k) times wave amplitude (A). The figure includes both experimental and computational results. First, it was noted that the agreement between the computational results and the measurements was fair. Larger deviation was seen in the heave amplitude for the shortest wavelength in the case of the bare hull. Aside from that, the CFD calculations were in accordance with the experimental results for the bare hull. Regarding the comparison for when the foil was present, it was noted that, although the CFD results followed the experimental trends, there was almost-constant deviation from the measurements. Since in the simulations only the foil was considered, this deviation was attributed to the different configurations considered in the experiment and the simulations.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\mathrm{Fn}$ | Ship Froude Number |

V | Ship Advance Speed |

${V}_{nom}$ | Ship Nominal Advance Speed (1.42 m/s) |

T | Ship Draft |

Ls | Ship Length |

s, c | Wing Span and chord length |

Lw | Wave Length |

f | Wave Frequency |

k | Wave Number |

A | Wave Amplitude |

${R}_{p}$ | Mean Resistance |

ξ_{3}_{,} ξ_{5} | Heave, Pitch |

${A}_{\phi}$ | Amplitude of Parameter φ |

$\overline{\phi}$ | Mean Value of Parameter φ |

## References

- Triantafyllou, M.S.; Triantafyllou, G.S.; Yue, D.K.P. Hydrodynamics of Fishlike Swimming. Annu. Rev. Fluid Mech.
**2000**, 32, 33–53. [Google Scholar] [CrossRef] - Wu, X.; Zhang, X.; Tian, X.; Li, X.; Lu, W. A review on fluid dynamics of flapping foils. Ocean Eng.
**2020**, 195, 106712. [Google Scholar] [CrossRef] - Bøckmann, E.; Steen, S. Model test and simulation of a ship with wavefoils. Appl. Ocean Res.
**2016**, 57, 8–18. [Google Scholar] [CrossRef] - Bøckmann, E.; Steen, S. Experiments with actively pitch-controlled and spring-loaded oscillating foils. Appl. Ocean Res.
**2014**, 48, 227–235. [Google Scholar] [CrossRef] - Filippas, E.S.; Papadakis, G.P.; Belibassakis, K.A. Free-surface effects on the performance of flapping-foil thruster for augmenting ship propulsion in waves. J. Mar. Sci. Eng.
**2020**, 8, 357. [Google Scholar] [CrossRef] - Belibassakis, K.A.; Politis, G.K. Hydrodynamic performance of flapping wings for augmenting ship propulsion in waves. Ocean Eng.
**2013**, 72, 227–240. [Google Scholar] [CrossRef] - Belibassakis, K.; Filippas, E.; Papadakis, G. Numerical and Experimental Investigation of the Performance of Dynamic Wing for Augmenting Ship Propulsion in Head and Quartering Seas. J. Mar. Sci. Eng.
**2021**, 10, 24. [Google Scholar] [CrossRef] - Papadakis, G. Development of a Hybrid Compressible Vortex Particle Method and Application to External Problems including Helicopter Flows. Ph.D. Thesis, National Technical University of Athens, Athens, Greece, 2014. [Google Scholar]
- Ntouras, D.; Papadakis, G. A Coupled Artificial Compressibility Method for Free Surface Flows. J. Mar. Sci. Eng.
**2020**, 8, 590. [Google Scholar] [CrossRef] - Mavrakos, A.S.; Konispoliatis, D.N.; Ntouras, D.G.; Papadakis, G.P.; Mavrakos, S.A. Hydrodynamics of Moonpool-type Floaters: A theoretical and a CFD formulation. Energy
**2021**, 15, 570. [Google Scholar] [CrossRef] - Hirt, C.W.; Nichols, B.D. Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries. J. Comput. Phys.
**1981**, 39, 201–225. [Google Scholar] [CrossRef] - Kunz, R.F.; Boger, D.A.; Stinebring, D.R.; Chyczewski, T.S.; Lindau, J.W.; Gibeling, H.J.; Venkateswaran, S.; Govindan, T.R. A preconditioned Navier-Stokes method for two-phase flows with application to cavitation prediction. Comput. Fluids
**2000**, 29, 849–875. [Google Scholar] [CrossRef] - Venkateswaran, S.; Lindau, J.; Kunz, R.; Merkle, C. Preconditioning algorithms for the computation of multi-phase mixture flows. In Proceedings of the 39th Aerospace Sciences Meeting and Exhibit, Reno, NV, USA, 8–11 January 2001. [Google Scholar] [CrossRef][Green Version]
- Ntouras, D.; Manolas, D.; Papadakis, G.; Riziotis, V. Exploiting the limit of BEM solvers in moonpool type floaters. J. Phys. Conf. Ser.
**2020**, 1618, 052059. [Google Scholar] [CrossRef] - Perić, R.; Vukčević, V.; Abdel-Maksoud, M.; Jasak, H. Optimizing wave-generation and wave-damping in 3D-flow simulations with implicit relaxation-zones. Coast. Eng.
**2021**, 171, 104035. [Google Scholar] [CrossRef] - Biedron, R.; Vatsa, V.; Atkins, H. Simulation of Unsteady Flows Using an Unstructured Navier-Stokes Solver on Moving and Stationary Grids. In Proceedings of the 23rd AIAA Applied Aerodynamics Conference, Toronto, ON, Canada, 6–9 June 2005; pp. 1–17. [Google Scholar] [CrossRef][Green Version]
- Zhao, Y.; Tai, J.; Ahmed, F. Simulation of micro flows with moving boundaries using high-order upwind FV method on unstructured grids. Comput. Mech.
**2001**, 28, 66–75. [Google Scholar] [CrossRef] - Celik, I.B.; Ghia, U.; Roache, P.J.; Freitas, C.J.; Coleman, H.; Raad, P.E. Procedure for Estimation and Reporting of Uncertainty Due to Discretization in CFD Applications. J. Fluids Eng.
**2008**, 130, 078001. [Google Scholar] [CrossRef][Green Version] - Nichols, D.S. Development of a Free Surface Method Utilizing an Incompressible Multi-Phase Algorithm to Study the Flow about Surface Ships and Underwater Vehicles. Ph.D. Thesis, Mississippi State University, Starkville, MS, USA, 2002. [Google Scholar]

**Figure 1.**Overview of the computational domain. (

**a**) Side view and (

**b**) top view of the grid. The forcing zones (Z1, Z2, Z3) are noted on the right figure.

**Figure 4.**(

**a**) Free surface elevation and (

**b**) y

^{+}values on the ship hull in calm water conditions for draft T = 0.135 m and Fn = 0.25.

**Figure 6.**Comparison of the CFD data and experimental predictions for the two drafts for (

**a**) trim angle and (

**b**) sinkage.

**Figure 7.**Experimental configuration of the dynamic wing arranged at the bow of the ferry ship model tested in the tank of LMSH at NTUA.

**Figure 11.**Comparison of the predicted resistance for the various angles of attack. The results concern the nominal speed (V = 1.42 m/s) for the heavier condition (T = 0.135 m). The experimental data were corrected by subtracting the resistance of the skegs and vane.

**Figure 12.**Comparison of the predicted trim angle for the various angles of attack. The results concern the nominal speed (V

_{nom}= 1.42 m/s) for the heavier condition (T = 0.135 m).

**Figure 13.**Comparison of the predicted sinkage for the various angles of attack. The results concern the nominal speed (V

_{nom}= 1.42 m/s) for the heavier condition (T = 0.125 m).

**Figure 14.**Numerical results of (

**a**) model resistance and (

**b**) pitch and heave motions with and without the foil over one encounter period. In the left figure, the dotted line corresponds to the mean value of the model’s resistance. Wave frequency: f = 0.55 Hz. Wavelength to ship length: L

_{w}/L

_{s}= 1.62.

**Figure 15.**Numerical results of (

**a**) model resistance and (

**b**) pitch and heave motions with and without the foil over one encounter period. In the left figure, the dotted line corresponds to the mean value of the model’s resistance. Wave frequency: f = 0.67 Hz. Wavelength to ship length: L

_{w}/L

_{s}= 1.09.

**Figure 16.**Numerical results of (

**a**) model resistance and (

**b**) pitch and heave motions with and without the foil over one encounter period. In the left figure, the dotted line corresponds to the mean value of the model’s resistance. Wave frequency: f = 0.75 Hz. Wavelength to ship length: L

_{w}/L

_{s}= 0.87.

**Figure 17.**Comparison between numerical and experimental results for regular waves with and without foil. The figures present the response amplitude operators for (

**a**) nondimensional heave and (

**b**) normalized pitch, where A is the wave amplitude, and k is the wave number.

**Figure 18.**Iso-surface of the density field for ρ

_{m}= 500 kg/m

^{3}colored by the vertical coordinate. The two figures correspond to the heavier draft (T = 0.135 m) for an incident wave frequency of 0.67 Hz; (

**a**) presents the bare hull case and (

**b**) depicts the hull equipped with the static foil.

**Figure 19.**Pressure field at the symmetry plane (y = 0 m) for the two configurations: (

**a**) presents the base hull case, while figure (

**b**) depicts the hull equipped with the static foil. The two figures correspond to the heavier draft (T = 0.135 m) for an incident wave frequency of 0.67 Hz.

Grid-Time-Step | Resistance (kp) | Trim (deg) | Sinkage (cm) |
---|---|---|---|

G3-T_{ref}/2500 | 0.704 | 0.0977 | −0.386 |

G2-T_{ref}/2500 | 0.687 | 0.0954 | −0.388 |

G1-T_{ref}/2500 | 0.682 | 0.0922 | −0.392 |

G2-T_{ref}/500 | 0.686 | 0.0959 | −0.387 |

G2-T_{ref}/250 | 0.686 | 0.0958 | −0.387 |

h_{1} | h_{2} | h_{3} | r_{21} | r_{32} | p | ${\mathit{\phi}}_{\mathit{e}\mathit{x}\mathit{t}}^{21}$ | ${\mathit{e}}_{\mathit{a}}^{21}$ | ${\mathit{e}}_{\mathit{e}\mathit{x}\mathit{t}}^{21}$ | $\mathit{G}\mathit{C}{\mathit{I}}_{\mathit{f}\mathit{i}\mathit{n}\mathit{e}}^{21}$ |
---|---|---|---|---|---|---|---|---|---|

0.0455 | 0.0616 | 0.0860 | 1.35 | 1.40 | 1.53 | 0.674 | 0.73% | 1.26% | 1.56% |

**Table 3.**Contribution of the individual components of the full configuration on the model’s resistance. Three different Froude numbers were examined for the T = 0.135 m draft.

Froude No. | Total [kp] | Hull [kp] | Skegs [kp] | Wing [kp] |
---|---|---|---|---|

0.25 | 0.829 | 0.66 | 0.077 | 0.092 |

0.22 | 0.653 | 0.52 | 0.061 | 0.072 |

0.176 | 0.431 | 0.34 | 0.043 | 0.048 |

**Table 4.**Contribution of the individual components of the full configuration on the mode’s resistance. Three different Froude numbers were examined for the T = 0.125 m draft.

Froude No. | Total [kp] | Hull [kp] | Skegs [kp] | Wing [kp] |
---|---|---|---|---|

0.25 | 0.788 | 0.62 | 0.073 | 0.095 |

0.22 | 0.614 | 0.48 | 0.059 | 0.075 |

0.176 | 0.413 | 0.32 | 0.042 | 0.051 |

**Table 5.**Experimental results for the static wing in various AoAs. All results regard the nominal speed of the vessel (Fn = 0.25 or V

_{nom}= 1.42 m/s) in the case of the heavier condition (T = 0.135 m).

Angle of Attack [deg] | Resistance [kp] | Trim [deg] | Sinkage [cm] |
---|---|---|---|

−3.1 | 1.031 | −0.159 | −0.198 |

−1.6 | 1.007 | −0.039 | −0.291 |

1.6 | 1.028 | 0.264 | −0.494 |

0 | 1.042 | 0.132 | −0.401 |

**Table 6.**Time-step independency study for the case study with regular waves. The table presents the mean value $\overline{\left(\xb7\right)}$ and amplitudes (A) of the heave (${\xi}_{3}$), pitch (${\xi}_{5}$), and resistance (${A}_{p}$).

dt [s] | $\overline{{\mathit{\xi}}_{3}}$ | $\overline{{\mathit{\xi}}_{5}}$ | $\overline{{\mathit{R}}_{\mathit{p}}}$ | ${\mathit{A}}_{{\mathit{\xi}}_{3}}$ | ${\mathit{A}}_{{\mathit{\xi}}_{5}}$ | ${\mathit{A}}_{{\mathit{R}}_{\mathit{p}}}$ |
---|---|---|---|---|---|---|

0.003 | 1.93 | 2.27 | 1.18 | 1.93 | 2.27 | 1.18 |

0.002 | 1.79 | 2.19 | 1.10 | 1.79 | 2.19 | 1.10 |

0.001 | 1.75 | 2.14 | 1.08 | 1.75 | 2.14 | 1.08 |

**Table 7.**Mean resistance per encounter cycle for the bare hull and foil configurations at considered wave excitations.

Wavelength to Ship Length L_{w}/L_{s} | Resistance [kp] Bare Hull | Resistance [kp] with Foil | Gain (%) |
---|---|---|---|

0.87 | 1.153 | 0.883 | +23.5 |

1.09 | 1.165 | 0.801 | +31.2 |

1.16 | 1.177 | 0.774 | +34.2 |

1.62 | 0.854 | 0.747 | +12.5 |

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**MDPI and ACS Style**

Ntouras, D.; Papadakis, G.; Belibassakis, K.
Ship Bow Wings with Application to Trim and Resistance Control in Calm Water and in Waves. *J. Mar. Sci. Eng.* **2022**, *10*, 492.
https://doi.org/10.3390/jmse10040492

**AMA Style**

Ntouras D, Papadakis G, Belibassakis K.
Ship Bow Wings with Application to Trim and Resistance Control in Calm Water and in Waves. *Journal of Marine Science and Engineering*. 2022; 10(4):492.
https://doi.org/10.3390/jmse10040492

**Chicago/Turabian Style**

Ntouras, Dimitris, George Papadakis, and Kostas Belibassakis.
2022. "Ship Bow Wings with Application to Trim and Resistance Control in Calm Water and in Waves" *Journal of Marine Science and Engineering* 10, no. 4: 492.
https://doi.org/10.3390/jmse10040492