# On Boundary Conditions for Damage Openings in RoPax-Ship Survivability Computations

## Abstract

**:**

## 1. Introduction

_{P}of 10.0 s.

## 2. General Description of the RoPax Vessel

## 3. Observations from Model Tests and Simulations

#### 3.1. Introduction

#### 3.2. Difference in Capsize Rate of a Damaged RoPax Ship in Beam Seas between Free-Drifting and Soft-Moored Condition in Model Tests

_{S}of 7.5 m 15 times out of 20 (75%). For the soft-moored ship the capsize boundary was found to be at H

_{S}4.5–5.0 m. In regular waves the capsize started from the wave height H

_{W}6.0 m for the free-drifting vessel and from H

_{W}4.5 m for the soft-moored one.

_{S}3.5 m 12 times out of 17 (70%). For the soft-moored ship the capsize boundary was found to be at H

_{S}2.5–3.0 m. In regular waves the capsize started from H

_{W}6.0 m for the free-drifting vessel and from H

_{W}4.5 m for the soft-moored one, exactly as with the higher GM value [10].

#### 3.3. Effect of Ship Motion on Floodwater Flow through a Damage Opening during a Wave Cycle

_{YDMG}) shows the horizontal velocity of the center of the damage opening on the ship side. It mostly consists of the slowly varying ship drift, together with sway and roll motions oscillating with the wave period. The velocity is positive towards lee, and negative towards the incoming waves. The two solid blue curves show the water height on the vehicle deck at the centerline of the damage opening, which is mostly a result of the wave elevation at the damage opening, the ship heave motion and the vertical motion of the damage opening caused by the ship roll motion. The higher and darker blue curve shows the elevation near the damage opening on the ship side, whereas the lighter blue curve shows it in the middle between the ship side and the center casing. The thin gray curve above all other curves gives the ship’s heeling angle, which shows values of a few degrees towards the damaged side and incoming waves. The red curve showing the heave, the two solid blue curves, the black curve showing the horizontal velocity of the damage opening, and also the gray curve have correct phase differences between them. Based on the ship motion cycles in Figure 3, we can make the following observations.

- (1)
- We assume the flow to be unsteady due to rapidly varying pressure heads on both sides of the opening and also due to the horizontal acceleration of the damage opening on the ship side itself. For this, the dynamic orifice equation by Lee [3] can be extended and applied.
- (2)
- In the case of a transom stern, the speed of the ship leads to a lower water level at the transom. Thus, in analogy, for a ship drifting in beam seas, depending on the combination of sway and drift speeds, the average water level on the ship side should be slightly lower on the wave side and slightly higher on the lee side. This should have a small effect on the pressure head just outside the damage opening, which can be taken into account in modeling the inflow through the damage opening.

## 4. Steps towards Better Inflow/Outflow Boundary Condition for the Flow on the Vehicle Deck of a Damaged RoPax Ship in Waves

#### 4.1. The Bernoulli Equation vs. the Dynamic Orifice Equation

#### 4.2. The Inflow/Outflow Flow Mechanism on the Vehicle Deck as Observed in Model Tests

- (1)
**In regular beam waves,**the wave crests hit the ship side and water flows through the damage opening onto the vehicle deck of the RoPax ship and, of course, also into the damaged compartments below. Once the crest has passed the damaged ship side, the water on the vehicle deck flows back along the downwardly inclined deck towards the damage opening and further through the opening out of the vehicle deck. With the next regular wave, this process is repeated anew.- (2)
**In irregular beam seas,**the highest wave crests bring water onto the vehicle deck and into the damaged compartments below. In between these high wave crests there are lower wave crests and wave troughs, which do not bring any water onto the vehicle deck, as the water elevation at the damage opening does not reach the vehicle deck level at the opening. During these relatively long periods between higher wave crests, the floodwater mostly flows along the inclined vehicle deck out of the damage opening back to the sea.

#### 4.3. Outflow through the Damage Opening on the Heeled Vehicle Deck

#### 4.4. Formulations for Floodwater Discharge at the Damage Opening

#### 4.5. Improved Boundary Condition for the Damage Opening to an Open Deck

- (1)
**The roll or heeling angle is negative or zero**. The floodwater flows from the damage opening on the starboard side towards the center casing.The flow speed at the damage opening is determined with Bernoulli Equation or with Dynamic Orifice Equation, and the change in the linear momentum due to the water inflow in the opening is taken into account as a boundary condition in the numerical solution of the SWE on the vehicle deck. As water outside the damage opening can be assumed to have practically zero speed, this formulation is a quite proper and suitable approximation. This, along with BE, is the original model in HSVA Rolls.- (2)
**The roll or heeling angle is positive**. The floodwater flows from the inner parts of the vehicle deck downward along the inclined deck to the damage opening and further out to the sea below.In this case the floodwater on the deck as a shallow-water layer can develop a significant speed towards the damage opening. This can be seen in model tests with a RoPax ship with side damage in beam seas. The measured flow speed data obtained with a test rig can be found in Appendix A. In numerical simulations the flow speed on the inclined deck can be determined with SWE. The speed at the damage opening can be taken as a combination of the speed given by the SWE and that given by the BE or DOE based on the water level difference in the opening.With this boundary condition in the numerical solution, the floodwater flowing down along the inclined vehicle deck can be better taken into account than solely with BE or DOE. The modeling should be particularly important in irregular seas, in which there are long periods of floodwater outflow between the occasional higher wave crests that bring water onto the vehicle deck.

## 5. Comparison of the Computed Results with FLARE Benchmark Test Experimental Data

- The use of the BE as a boundary condition is easy, and the ship heeling process is in this case quite linear. Using the classical value for the discharge coefficient (0.6) leads to too short a time to capsize. With reduction of the value of the discharge coefficient, the TTC could be prolonged. This may lead to practical results, but it does not reflect the prevailing physics too well.
- The use of the DOE as a boundary condition requires time-integration of the floodwater discharge through the damage opening simultaneously with the time-integration of the ship motions in the simulation program. The programmed boundary condition tested gives the discharge in the opening also as a function of the ship horizontal and transverse acceleration and speed at the damage opening. Thus, the lateral ship motions influence the flow in the opening, which appears to lead to a less smooth development of the ship roll angle than in the case of the BE. In transient cases the DOE is a much more physically correct boundary condition than the BE. The use of the DOE delays capsizing in comparison with the BE. However, as several ship motion components influence the flow in the damage opening, the flow can also be more easily distorted if the ship motions are not accurately predicted.
- The use of the SWEDOE requires time-integration of the discharge through the damage opening and the input of the flow speed on the vehicle deck into the boundary condition. Also in this formulation, the lateral ship motions are taken into account. In most cases the inclusion of the outflow speed on the inclined deck in the boundary condition reduced the net inflow onto the vehicle deck, delayed capsize, and had a prolonging effect on the time to capsize TTC. Thus, the SWEDOE curve showing the development of the roll angle over time is similar to the DOE curve but more gradual. In some cases, the SWEDOE formulation postpones capsize considerably, when large amounts of floodwater flow out of the vehicle deck.

- The boundary conditions at DOE and SWEDOE yield better values for the Time to Capsize TTC than the BE.
- In the lower sea state with H
_{S}3.5 m, the boundary conditions of DOE and SWEDOE yield better values for the survival rate than the BE. The survival rate given by BE is too low. - In the higher sea state with H
_{S}7.5 m, the boundary conditions of DOE and SWEDOE yield too high survival rates. In addition, in this case, the BE gives a too low survival rate. - Thus, the use of the DOE and SWEDOE are certainly steps in the direction of a better boundary condition for the damage openings, but the formulations used in this brief study do not yet lead to very satisfactory results.

## 6. Comparison of the Computed Results with FLARE Flood Mitigation Test Experimental Data

- The boundary conditions at DOE and SWEDOE yield much better values (77.7–93.2%) for the Time to Capsize TTC than the BE (38.0–54.4%).
- In the lower sea state with H
_{S}3.5 m, the boundary conditions of DOE and SWEDOE yield better values for the survival rate than the BE. The survival rate given by BE is too low. - In the higher sea state with H
_{S}5.0 m, all boundary conditions BE, DOE, and SWEDOE yield zero survival rates, like the experiments do. - The use of the DOE and SWEDOE are certainly steps in the direction of a better boundary condition for the damage opening to the vehicle deck. In this damage case the results simulated using DOE or SWEDOE are already quite satisfactory.

## 7. Discussion

## 8. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Model Tests for Floodwater Outflow

_{1}–S

_{4}. The rig was heeled to 0°, 5°, 10°, 15°, and 20° inclination angles to provide the deck with a slope, as in a heeling ship. Three different initial water level heights in the tank were used in the tests.

**Figure A1.**Test rig for outflow tests. The instrumented four sections are denoted S

_{1}...S

_{4}, from left to right.

_{1}is shown in Figure A3 with the solid red curve. The solid blue curve shows the water level in the tank. The two dashed curves RWS1 and RWS2 show the water level at section S

_{1}. The dashed curve V1-BE shows the flow speed at the section S

_{1}according to the Bernoulli model based on the difference in the water level in the tank and the average given by the two water level sensors in the section S

_{1}.

**Figure A3.**Flow speed V

_{1}and water elevations RWS1 and RSW2 at the first section S

_{1}for the test run 009.

_{1}show the rise of the water front in section S

_{1}at ca. 1.0–1.2 s (in f.sc.) before the flow sensor shows the rise in the flow velocity. This delay is assumed to be the reaction time of the flow sensor, and it is visible in all measurements. The fluid starts at rest when the gate is suddenly opened, and after the initial delay in the flow sensor the flow reaches its measured full speed rapidly, but not instantly. Depending on the case, this rise-time to full flow speed amounts to ca. 1.0–1.3 s, which is something the BE completely ignores.

_{4}as in the preceding figure, the additional dashed red curve (velocity V

_{4}computed as BC) in Figure A4 shows also the outflow velocity computed with the typical boundary condition for the damage opening based on BE. That is, based only on the water elevation difference inside (at S

_{4}) and outside of the opening for outflow at right. The water elevation outside is zero, when the free surface lies below the deck level. The measured flow speed value (V

_{4}, shown by the violet curve) at S

_{4}lies initially higher than the red computed value but decreases faster. The velocity V

_{4}computed as a boundary condition shows higher values as long as there is any water on the deck, because the water elevation outside is zero. Thus, the classical boundary condition based on BE describing the (inviscid) flow through the damage opening (a) does not in general take the flow speed on the vehicle deck into account, and (b) may show too high flow values, when there is a thin layer of water on a horizontal deck.

**Figure A4.**Flow speed V

_{4}and water elevations RWS7 and RSW8 at the fourth section S

_{4}for the test run 009. In addition, the red dashed curve shows the flow speed V

_{4}computed with BE as the typical boundary condition at the damage opening.

_{4}computed as a boundary condition with BE, and (2) based on the measured flow speed V

_{4}. These two curves coincide only in this particular case. If the initial water depth in the tank is lower than in this test case 009, the viscous reduction in the flow speed is more pronounced, and the BE overpredicts the discharge.

**Figure A5.**Flow speed V

_{4}and water elevations RWS7 and RSW8 at the fourth section S

_{4}for the test run 009 as earlier. In addition, the water discharge volumes Vol

_{4}based on the flow speed V

_{4}computed with BE and based on the measured flow speed V

_{4}are shown. These two curves coincide only in this particular case.

**Figure A6.**Flow speed V

_{4}and water elevations RWS7 and RSW8 at the fourth section S

_{4}for the test run 064. The water discharge volumes Vol

_{4}based on the flow speed V

_{4}computed with BE and based on the measured flow speed V

_{4}are also shown.

_{1}–S

_{4}for all test conditions. Each maximum value is the average maximum flow speed of five test runs. For easier comparison of the flow speeds, the flow speed in section S

_{1}at zero heeling angle has been given the nominal value of 100 % and the other speeds are scaled accordingly. Note that the nominal water speeds between the different initial water depths in the tank differ.

**Table A1.**Comparison of maximum flow speeds at sections S

_{1}–S

_{4}at different initial water depths in the tank and heeling angles of the test rig. For each water depth, the V

_{1}sensor at zero heeling angle has the nominal value of 100%.

Water Depth [m] | Heeling Angle [°] | Flow Speed at Sections S_{1}–S_{4} | |||
---|---|---|---|---|---|

V_{1}[%] | V_{2}[%] | V_{3}[%] | V_{4}[%] | ||

0.7 | 0 | 100 | 105 | 94 | 77 |

0.7 | 5 | 169 | 162 | 160 | 142 |

0.7 | 10 | 197 | 217 | 204 | 193 |

0.7 | 15 | 222 | 261 | 139 | 224 |

0.7 | 20 | 268 | 300 | 297 | 255 |

Water Depth[m] | Heeling Angle[°] | Flow Speed at Sections S_{1}–S_{4} | |||

V_{1}[%] | V_{2}[%] | V_{3}[%] | V_{4}[%] | ||

2.1 | 0 | 100 | 86 | 85 | 78 |

2.1 | 5 | 113 | 112 | 92 | 88 |

2.1 | 10 | 118 | 125 | 97 | 103 |

2.1 | 15 | 125 | 135 | 117 | 114 |

2.1 | 20 | 130 | 144 | 136 | 123 |

Water Depth[m] | Heeling Angle[°] | Flow Speed at Sections S_{1}–S_{4} | |||

V_{1}[%] | V_{2}[%] | V_{3}[%] | V_{4}[%] | ||

4.2 | 0 | 100 | 80 | 75 | 52 |

4.2 | 5 | 102 | 95 | 80 | 70 |

4.2 | 10 | 105 | 101 | 66 | 78 |

4.2 | 15 | 108 | 105 | 91 | 81 |

4.2 | 20 | 110 | 107 | 100 | 85 |

_{4}near the outflow boundary, the outflow speed (255%) with a heeling angle of 20° is ca. 3.5 times the speed with a zero heeling angle (77%).

#### Conclusions on the Tests with the Inclined Rig

**Phase difference:**In the experiments, the inertia of the floodwater mass delayed any change in the flow speed or in the flow direction, whereas the typical boundary condition for inflow/outflow through the damage opening based on the Bernoulli equation does not do this. When the ship is floating in waves, the inflow to and outflow from damaged compartments are continuously changing. In the numerical simulations, the lack of inertia in the Bernoulli model can cause a phase difference in the inflow and outflow through the damage opening in comparison with the experiments. This may influence the excitation of the ship’s rolling motion in waves.

**Increase in the net water discharge through the damage opening:**The inertia of the floodwater has a slowing effect on the continuously changing flow speed through the damage opening. The use of the BE for the inflow/outflow through any damage opening in the numerical models for damaged ships can lead to a somewhat too high floodwater discharge through the opening and a too short predicted time to flood or capsize.

**Outflow from inclined decks:**The tests carried out with the rig show that the floodwater on the damaged vehicle deck of a RoPax or on another deck in any ship can develop considerable outflow speeds, when the ship heels and the deck gets an inclination angle. This speed of the mostly shallow water flow on the inclined deck can be clearly higher than the outflow speed described by the Bernoulli equation as a boundary condition for the damage opening. Thus, the flow speed on the deck should be determined and taken into account in the boundary condition for the damage opening in the numerical simulations. Omitting the outflow speed calculation and using the classical BE as a boundary condition can lead to a too high net accumulation of floodwater on the vehicle deck and thus to a too short time to flood or time to capsize.

## Note

1 | Maritime Safety Research Centre of the University of Strathclyde, UK. |

## References

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**Figure 3.**Measured time histories of the horizontal velocity of the damage opening VY-DMG, the heave motion, the relative water elevations RW1011 and RW89 on the vehicle deck at the centerline of the damage opening, and two wave elevations (Wave2, Wave3) are shown for a part of the test run 265 in beam seas.

**Figure 4.**View on the vehicle deck of the ship model through a ship-fixed camera (1/3). Floodwater can be seen flowing massively in through the damage opening on the starboard, wave side of the vessel. On the deck, there are sensors for measurement of the flow speed and water elevation. The red tufts on the deck for flow direction visualization are still all pointing towards the damage opening, as a result of floodwater flown out just before the present wave came in.

**Figure 5.**View on the vehicle deck of the ship model (2/3). Floodwater has reached its momentary maximum extent, and it starts to flow back towards the damage opening. The red tufts for flow visualization are still pointing towards the center casing and towards ship fore and aft directions, as a result of the floodwater that just flowed in.

**Figure 6.**View on the vehicle deck of the ship model (3/3). Floodwater is flowing back out of the damaged opening. The red tufts for flow visualization are pointing again towards the damage opening, as a result of the floodwater flowing out. As the view is given by a ship-fixed camera, the ship heeling angle is not really visible, but can be inferred from the floodwater accumulating towards the starboard side of the deck.

**Figure 8.**Computed and experimental time histories of the roll angle of the ship in regular beam waves with 6.0 m wave height, shown until capsize. The dashed curves at the center show the computed results obtained using the boundary conditions BE, DOE, and SWEDOE.

**Figure 9.**Computed and experimental time histories of the roll angle of the ship in irregular beam seas with H

_{S}3.5 m and T

_{P}10.0 s, shown until capsize. The computed curves show the results obtained using the boundary condition SWEDOE.

**Figure 10.**Computed and experimental time histories of the roll angle of the ship in irregular beam seas with H

_{S}5.0 m and T

_{P}10.0 s, shown until capsize. The computed curves show the results obtained using the boundary condition SWEDOE.

MT RoPax—HSVA Model No: 5460/5539 | Symbol | Unit | Ship |
---|---|---|---|

Length overall | L_{OA} | m | 162.00 |

Length between perpendiculars | L_{PP} | m | 146.72 |

Breadth at the waterline | B_{WL} | m | 28.00 |

Draught at the aft perpendicular | T_{A} | m | 6.10/6.30 |

Draught at the forward perpendicular | T_{F} | m | 6.10/6.30 |

Depth to trailer deck | D | m | 9.20 |

Displaced volume (bare hull) | ∇_{BH} | m^{3} | 16,799.4 |

Block coefficient | C_{B} | - | 0.6522 |

Intact transverse GM | GM | m | 1.425–3.40 |

FLARE MT RoPax Benchmark Damage Case | |||||
---|---|---|---|---|---|

GM | H_{S} | BE | DOE | SWEDOE | EXP |

1.425 m | 3.5 m | 28.3% | 53.1% | 63.1% | 100% |

3.250 m | 7.5 m | 36.7% | 96.3% | 96.6% | 100% |

FLARE MT RoPax Benchmark Damage Case | |||||
---|---|---|---|---|---|

GM | H_{S} | BE | DOE | SWEDOE | EXP |

1.425 m | 3.5 m | 0/20 | 3/20 | 4/20 | 7/20 |

3.250 m | 7.5 m | 0/20 | 17/20 | 18/20 | 5/20 |

FLARE MT RoPax MSRC Damage Case 2 for Flooding Mitigation | |||||
---|---|---|---|---|---|

GM | H_{S} | BE | DOE | SWEDOE | EXP |

3.40 m | 3.5 m | 38.0% | 77.7% | 93.2% | 100% |

3.40 m | 5.0 m | 54.4% | 87.4% | 82.5% | 100% |

FLARE MT RoPax MSRC Damage Case 2 for Flooding Mitigation | |||||
---|---|---|---|---|---|

GM | H_{S} | BE | DOE | SWEDOE | EXP |

3.40 m | 3.5 m | 0/20 | 2/20 | 3/20 | 4/20 |

3.40 m | 5.0 m | 0/20 | 0/20 | 0/20 | 0/20 |

Boundary Condition for the Damage Opening on Ship Side | Type | ||
---|---|---|---|

Effect Modeled | BE | DOE | SWEDOE |

Pressure or water height difference | ✓ | ✓ | ✓ |

Flow speed in the opening | ✓ | ✓ | |

Horizontal drift + sway velocity of the ship | ✓ | ✓ | |

Horizontal acceleration of the damage opening | ✓ | ✓ | |

Floodwater inertia in the opening | ✓ | ✓ | |

Shallow water speed on deck in front of the opening | ✓ | ||

Simple formulation with no memory effect | ✓ | ||

Time-integration of the flow speed in the opening | ✓ | ✓ | |

Water viscosity on deck or in the damage opening |

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## Share and Cite

**MDPI and ACS Style**

Valanto, P. On Boundary Conditions for Damage Openings in RoPax-Ship Survivability Computations. *J. Mar. Sci. Eng.* **2023**, *11*, 643.
https://doi.org/10.3390/jmse11030643

**AMA Style**

Valanto P. On Boundary Conditions for Damage Openings in RoPax-Ship Survivability Computations. *Journal of Marine Science and Engineering*. 2023; 11(3):643.
https://doi.org/10.3390/jmse11030643

**Chicago/Turabian Style**

Valanto, Petri. 2023. "On Boundary Conditions for Damage Openings in RoPax-Ship Survivability Computations" *Journal of Marine Science and Engineering* 11, no. 3: 643.
https://doi.org/10.3390/jmse11030643