# A Fast Simulation Method for Damaged Ship Dynamics

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Model

**f**

_{ext}and

**m**

_{ext}, consist of restoring, radiation, and wave actions that are Froude-Krylov forces and moments and diffraction [26]. The so-call blended approach is adopted: It consists in linear diffraction and radiation forces (based on potential theory [27]) and includes all pertinent non-linearities in Froude-Krylov and hydrostatic forces.

_{i}, is modelled as a lumped mass (3), whose position and velocity are

**r**

_{i}and

**u**, respectively.

_{i}**r**

_{i}, is restrained on a discrete three-dimensional path, depending on the amount of flooded water and on the free surface inclination in the longitudinal and lateral direction. The free surface of the floodwater is treated as a flat surface. It can have different inclinations from the ship roll and pitch angles. The inclinations of the free surface are evaluated by the analogy with the pressure distribution in a liquid within a tank that is uniformly accelerated [29]. It holds:

**a**

_{L}= a

_{Lx}

**i**+ a

_{Ly}

**j**+ a

_{Lz}

**k**). It is equal to:

**a**

_{Li}= a

_{Lix}

**I**+ a

_{Liy}

**J**+ a

_{Liz}

**K**). The effect of the transversal acceleration component, a

_{Liy}, is appreciated in roll motion, while the longitudinal acceleration component, a

_{Lix}, will contribute in pitch motion, yielding to the so called “effective”

**g**

_{e}as follows:

_{e}(see for example Figure 1) and θ

_{e}:

## 3. Flooding Law Modelling

_{0}(t) – H

_{t}(t);

_{0}, and the flooded water height, H

_{t}, vary in time, according to the ship dynamics (see Figure 2).

_{D}, is 0.65, in analogy with [14]. Additional details for Section 2 and Section 3 can be found in [20,21].

_{e}, the friction coefficient variable with time, ${\delta}_{f}\left(t\right)$, together with the flooding law modelling, are the novel enhancements to the model presented in the previous research [21].

## 4. Case Studies

#### 4.1. Flooding Simulations on the Barge Model

_{e}, is calculated from Equation (7) while the longitudinal inclination equals the pitch angle of the barge, θ. The dissipation of the energy of standing waves in rectangular rooms, expressed by Equation (8), is herein implemented accounting for the variable height of water in the compartment, h = h(t). This allows for modelling of the friction effects during the transient stage of flooding to be conducted.

#### 4.2. Roll Decay Simulations and Roll Response in Beam Waves on the Ferry Model

_{4}/ka, in beam waves is presented in Figure 9. The wave frequency is made non-dimensional dividing by $\sqrt{L/g}$. The experimental data (diamond marker) show a two-peak behaviour usually observed in an anti-roll tank [32]. The numerical simulation (round marker) matches the general trend of the roll response curve in particular around 0.633 Hz (in the non-dimensional form, it is 2.38), i.e., the lower frequency peak. Moreover, looking at the phase diagram, in analogy with anti-roll tanks, the numerical simulation succeeds in modelling the maximum damping effects of the flooded tank at the non-dimensional frequency of 2.64, with a phase between the roll motion, ϕ, and free surface inclination, ϕ

_{e}, of –90.6°. Although the implemented dynamic approach on the free surface fails in modelling the second peak of the roll response amplitude operator (RAO), it features both the two frequency modes, as observed from the roll decay simulations. Therefore, by the FFT analysis of the simulated roll decay, it is still possible to predict, with a certain accuracy, the exciting frequencies where roll motion amplifies.

#### 4.3. Motion Responses in Beam and Head Waves on the DTMB5415 Model

- Non-linear damping.
- Internal viscous effects.
- Longitudinal dynamic effects on the free surface.
- Water exchange through the opening.
- Wave actions.

_{w}= 1/80 to avoid wave heights that exceed the hull freeboard. This is a limit of the applicability of the implemented code also in the intact condition [21].

_{w}= 1/50. The roll response amplitude, η

_{4}, is obtained from FFT analysis of the simulated roll, referring to the first harmonic (the same applies also for the heave and pitch responses, η

_{3}and η

_{5}, in Figure 12 and Figure 13, respectively). It is possible to notice that the simulated results are fairly close to the experimental ones except for the peak value at resonance. It is important to recall that the choice of a smaller wave steepness makes the non-linear damping actions less effective. Therefore, additional simulations are carried out for the wave steepness, H/λ

_{w}= 1/50 (same as the experimental data), limited to the wave frequencies up to the peak value. In Figure 11, it can be observed that the simulated data with the larger steepness shows values closer to the experimental ones [25], although in a limited frequency range. Nevertheless, the numerical model is not fully effective in depicting the whole damping contribution of the damaged compartment. It is worth pointing out that in the simulations, the modelling of the water exchange through the opening affects the inertia and the restoring actions on the ship, rather than the water dynamics. Thus, the hydrodynamic problem through the damage opening, affecting the dynamics of the damaged hull, is only partially modelled.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Symbol | Description |

a_{L} | acceleration of the lumped mass (m/s^{2}) |

a_{Lx} | acceleration of the lumped mass horizontal component (m/s^{2}) |

a_{Ly} | acceleration of the lumped mass transversal component (m/s^{2}) |

a_{Lz} | acceleration of the lumped mass vertical component (m/s^{2}) |

f_{ext} | external forces (N) |

g_{e} | apparent gravity (m/s^{2}) |

H_{0} | height of the undisturbed sea water (m) |

I | matrix of inertia (kg m^{2}) |

m | body mass (kg) |

m_{ext} | external moments (Nm) |

Q(t) | flow rate (m^{3}/s) |

r_{i} | position vector of the lamped mass (m) |

u | velocity vector (m/s) |

ω | angular velocity vector (rad/s) |

Symbol | GREEK ALPHABET |

ϕ | instantaneous roll angle (rad) |

θ | instantaneous pitch angle (rad) |

${\delta}_{f}$ | dissipation of the energy of standing waves in rectangular tanks (-) |

λ_{w} | wave length (m) |

Symbol | TANK PROPERTIES |

l | length of a rectangular tank (m) |

b | breadth of a rectangular tank (m) |

h | water height in the tanks (m) |

k | k = π/b tank wave number (1/m) |

$\omega $ | $\omega =\sqrt{\frac{g\pi}{b}\mathrm{tanh}\frac{\pi h}{b}}$ natural frequency of sloshing (rad/s) |

H_{t} | the height of the flooded water in the compartment (m) |

C_{D} | discharge coefficient (-) |

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**Figure 3.**General arrangement and compartment details of the barge model. Flooding of the compartment, R11, is investigated.

**Figure 4.**Roll behaviour of the damaged barge during the transient stage of flooding: experimental vs. numerical results.

**Figure 6.**Transient flooding draft of the damaged barge: experimental vs. numerical results (zero is the initial draft).

**Figure 9.**Roll RAO for the flooded ship in beam waves (

**a**) (numerical vs. experimental) and phase angles (

**b**) (numerical only).

Symbol | Description | Model Scale |
---|---|---|

L_{OA} | Length over all | 4.000 m |

B_{OA} | Breadth over all | 0.800 m |

D | Depth | 0.800 m |

m | Total mass | 657 kg |

T | Draft | 0.253 m |

GM_{02} | Initial metacentric height of the intact ship | 0.0274 m |

h_{b} | Damage opening height from the compartment bottom | 50 mm |

l_{0} | Dimension of the square damage opening | 200 mm |

Symbol | Description | Ship Scale | Model Scale |
---|---|---|---|

L_{BP} | Length between perpendicular | 77.55 m | 3.525 m |

B | Breadth | 17.930 m | 0.815 m |

T | Draft | 3.960 m | 0.180 m |

Δ | Displacement | 1810 t | 0.170 t |

KG | Vertical centre of gravity | 6.886 m | 0.313 m |

GM_{0} | Transversal metacentric height of the intact ship | 5.808 m | 0.264 m |

ω_{0} | Natural roll frequency | 1.173 rad/s | 5.507 rad/s |

k_{xx} | Roll radius of inertia in air | 5.450 m | 0.247 m |

k_{yy} | Pitch radius of inertia in air | 24.384 m | 1.108 m |

k_{zz} | Yaw radius of inertia in air | 24.499 m | 1.113 m |

L_{t} | Length of the compartment | 16.940 m | 0.767 m |

B_{t} | Width of the compartment | 17.930 m | 0.815 m |

d_{f} | Flooded water depth | 3.674 m | 0.167 m |

d_{f}/B_{t} | Aspect ratio | 0.205 | 0.205 |

W_{t} | Flooded water weight | 688.9 t | 64.7 kg |

Symbol | Description | Ship Scale | Model Scale |
---|---|---|---|

L_{BP} | Length between perpendicular | 142.200 m | 2.788 m |

B | Breadth on waterline | 19.082 m | 0.374 m |

T | Draft | 6.150 m | 0.120 m |

D | Depth | 12.470 m | 0.244 m |

Δ | Displacement | 8635 t | 63.5 kg |

KG | Vertical centre of gravity | 7.555 m | 1.375 m |

GM_{0} | Transversal metacentric height of the intact ship | 1.938 m | 0.038 m |

k_{xx} | Roll radius of inertia in water | 6.932 m | 0.136 m |

k_{yy} | Pitch radius of inertia in air | 36.802 m | 0.696 m |

k_{zz} | Yaw radius of inertia in air | 36.802 m | 0.696 m |

L_{t} | Length of the compartment | 24.360 m | 0.478 m |

B_{t} | Width of the compartment | 19.458 m | 0.382 m |

W_{t} | Flooded water weight | 2638.9 t | 19.4 |

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**MDPI and ACS Style**

Acanfora, M.; Begovic, E.; De Luca, F.
A Fast Simulation Method for Damaged Ship Dynamics. *J. Mar. Sci. Eng.* **2019**, *7*, 111.
https://doi.org/10.3390/jmse7040111

**AMA Style**

Acanfora M, Begovic E, De Luca F.
A Fast Simulation Method for Damaged Ship Dynamics. *Journal of Marine Science and Engineering*. 2019; 7(4):111.
https://doi.org/10.3390/jmse7040111

**Chicago/Turabian Style**

Acanfora, Maria, Ermina Begovic, and Fabio De Luca.
2019. "A Fast Simulation Method for Damaged Ship Dynamics" *Journal of Marine Science and Engineering* 7, no. 4: 111.
https://doi.org/10.3390/jmse7040111