# A Unified Internal Flow Model with Fluid Momentum for General Application in Shipflooding and Beyond

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## Abstract

**:**

## 1. Introduction

## 2. A Unified Internal Flow Model including Fluid Inertia

#### 2.1. Introduction

#### 2.2. Construction of Virtual Pipes

#### 2.3. Solver Description

- Determine, for each cell, the centroid of the free surface area and the centroids of wetted openings;
- Construct the virtual pipes in each cell;
- Initialize the virtual pipe flow (average) velocity based on the cell-averaged momentum—see Equation (9);
- Solve the linear system $A\overrightarrow{y}=\overrightarrow{b}$ for the change in the pipe flow velocities based on the pressure differences over each opening, as expressed in Equation (6);
- Check the solution against the constraints of the cell capacity (full or empty) and, if needed, exchange free nodes for connected nodes;
- Update the cell fluid volumes and the cell-averaged momentum through Equation (9).

#### 2.4. Solver Equations, Fluid Forcing, and Change of Momentum

#### 2.5. Energy Losses over an Opening

#### 2.6. Entrapped Air

## 3. Applications and Validation

#### 3.1. A Single Tank Outflow Experiment to Measure the Discharge Coefficient

^{2}. At the start of the simulation, the compartment was completely full. A vertical sliding door was used to open the box quickly. The water level was measured at two locations (corners) in the box at 200 Hz, showing nearly identical results.

#### 3.2. A Two Compartment Down-Flooding Experiment

#### 3.3. Behaviour of an Oscillating Water Column (OWC) for Wave Energy Harvesting

#### 3.4. Flooding of a Cruise Ship in Calm Water Conditions

^{3}; the volume of the floodable compartments amounts to 48,012 m

^{3}.

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Example of network development over time. The connected nodes are shown as solid circles, the free nodes as open circles, and the virtual pipes as dashed lines.

**Figure 4.**Experimental results. (

**a**) Velocity through the orifice and normalized wetted area of the orifice, (

**b**) derived Reynolds number, and (

**c**) measured and theoretical discharge coefficient over time.

**Figure 5.**

**Top**: Comparison of water height H with respect to the orifice bottom edge, and

**Bottom**: Average discharge velocity.

**Figure 6.**Two-compartment down-flooding experimental setup. The downflood opening, positioned in the middle of the deck, measured 40 × 40 mm. The breach in the upper compartment measured 80 × 80 mm.

**Figure 7.**ComFLOW CFD results, 3D and 2D view, with model scale velocity. Snapshots at $T=$ 0.577, 1.545, and 3.594 s.

**Figure 8.**Comparison of water height. Experimental, ComFlow (CFD), TOR simulation, and UIF simulation.

**Figure 10.**

**Left:**2D cross section of the OWC scale experiment adopted from [4], with the assumed Bernoulli streamline and

**Right:**3D simulation setup utilizing three compartments, two internal openings, and one vent opening at the top (all indicated in red).

**Figure 11.**Comparison of pressure in caisson-2 between experiment (case C from reference [5]) and simulations.

**Figure 12.**UIF and TOR simulation while approaching the resting condition. Water level H w.r.t. bottom of caisson-2.

**Figure 13.**Cruise ship side view (non-damaged port-side) and drawing of internal space on all decks. Openings (red) with external breach (green).

Test | $\mathit{\zeta}$ [m] | a [m] | ${\mathit{p}}_{1}$ [Pa] | ${\mathit{p}}_{2}$ [Pa] |
---|---|---|---|---|

Experiment—starting (in) | 0.306 | 0.107 | 1227 | 1288 |

Experiment—end/resting (0) | 0.366 | 0.079 | 1013 | 1077 |

Simulation—starting (in) | 0.306 | 0.107 | 1227 | 1288 |

Simulation—end | 0.366 | 0.135 | 1013 | 1077 |

Simulation—resting (0) | 0.366 | 0.079 | 1013 | 1072 |

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## Share and Cite

**MDPI and ACS Style**

van ’t Veer, R.; Berg, J.v.d.; Boonstra, S.
A Unified Internal Flow Model with Fluid Momentum for General Application in Shipflooding and Beyond. *J. Mar. Sci. Eng.* **2023**, *11*, 1175.
https://doi.org/10.3390/jmse11061175

**AMA Style**

van ’t Veer R, Berg Jvd, Boonstra S.
A Unified Internal Flow Model with Fluid Momentum for General Application in Shipflooding and Beyond. *Journal of Marine Science and Engineering*. 2023; 11(6):1175.
https://doi.org/10.3390/jmse11061175

**Chicago/Turabian Style**

van ’t Veer, Riaan, Joris van den Berg, and Sander Boonstra.
2023. "A Unified Internal Flow Model with Fluid Momentum for General Application in Shipflooding and Beyond" *Journal of Marine Science and Engineering* 11, no. 6: 1175.
https://doi.org/10.3390/jmse11061175