## A method for intermediate flooding and sinking simulation of a damaged floater in time domain (2)

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## Highlights

Flooding and sinking procedures of a damaged floater is defined.

A method for intermediate flooding and sinking simulation of the damaged floater in time domain is proposed.

Theoretical background for the simulation is established.

We perform the intermediate flooding and sinking simulation.

The time to be equilibrium or the time to sink is estimated from the simulation. ## Keywords

Intermediate flooding
Simulation
Damaged floater
Quasi-static equilibrium
Time domain

## 1. Introduction

A floater such as a commercial vessel and an offshore plant receives gravity and buoyancy, which have the same magnitudes but act on the opposite directions, respectively. Due to these two forces, the floater can be restored to its equilibrium position even though external disturbances exist. If the floater is damaged by an accident, there is a possibility that the damaged floater is flooding. That is, if the restoring force is not enough, the floater starts to sink. Fig. 1 shows several examples of accidents of the floaters by flooding. Fig. 1

As the size of a floater has been growing recently, the safety of the floater is considered more important than the past. When the floater is damaged, it is urgent to predict whether the floater is sinking or not and how long it can float on the water. At this time, uncertainties should be taken into account by international regulations such as SOLAS (Safety Of Life At Sea), MARPOL (International Convention for the Prevention of Marine Pollution from Ships), and ICLL (International Convention on Load Line). These regulations mainly concentrate on the final stage (sinking) of the flooding procedure. However, several accidents have shown that it is also important to understand the intermediate stage of the flooding procedure because the intermediate stage can be more hazardous than the final stage. Fig. 2 shows main stages of the flooding procedure . As shown in this figure, the motion of the floater at the intermediate stage fluctuates more than at the final stage. Therefore, it is important to evaluate the motion of the floater not only at the final stage but also at the intermediate stage of the flooding procedure. Furthermore, if the floater cannot have an equilibrium position, it starts to sink. Therefore, it is necessary to evaluate whether the floater will sink or not and to estimate the time to sink in the case of sinking. Fig. 2

By such necessities, fast, stable, and acceptable analysis in time domain is prerequisite to the flooding simulation. The static method is not enough to be applied to the intermediate stage because of its simplicity. Besides, the CFD (Computational Fluid Dynamics) method is too slow and sophisticated for the flooding simulation. Therefore, the quasi-static method has been usually adopted for the flooding simulation. The simulation should be able to give results such as the motion (position and attitude) of the floater as time goes, the time to sink, and so on. These results can be used to determine a strategy of rescue or abandonment of the floater and to investigate causes of the flooding .

This paper is organized as follows. Section 2 explains related studies. Section 3 presents the flooding and sinking procedures of the damaged floater. Section 4 shows theoretical background of the intermediate flooding and sinking simulation. Section 5 shows overall process and implementation of the intermediate flooding and sinking simulation. Section 6 describes simulation examples and their results. Finally, Section 7 gives conclusions and suggests future works.

## 2. Related studies

So far, there have been several related studies about the flooding simulation.

Park et al.  performed the intermediate flooding simulation for a barge-type damaged ship which the damaged hole is located in the bottom of the ship. The ship motion was calculated by using the quasi-static method. However, they did not consider the effect of the air compression and various locations of damaged holes. In addition, they simulated only the intermediate stage of the flooding simulation.

Ruponen  and Dankowski  used the quasi-static method to perform the intermediate flooding simulation. They calculated the equilibrium position of a damaged ship at every time step considering incoming water from damaged holes. They also compared their results with the experimental data. However, they simulated only the intermediate stage of the flooding simulation. In addition, Ruponen used the water plane-based method when calculating the water volume in the damaged compartments . The water volume calculated from the water plane-based method can have some errors for compartments having the complicated shape because the water plane area is approximated in the method . That is, he calculated the water volume in the compartment by integrating the derivative of the water volume in the compartment (=the approximated water plane area * the derivative of the water height in the compartment). And Dankowski used the interpolation method to calculate the water volume from the water height in the compartment under a certain attitude of the ship. These methods require the iterative process to reduce an error caused by approximation or interpolation.

Lee et al.  calculated the motion of both intact and damaged ships by using the CFD method. They compared simulation results with the experimental data. However, they did not consider the increased weight of incoming water through a damaged hole and the effect of air compression. The air compression influences the flooding speed of water because the air compression due to incoming water can prevent further flooding progress. They mainly focused on the motion itself when the damaged compartments are already filled with sea water at the intermediate flooding stage.

As mentioned above, most of studies have some limitations on the flooding simulation. To perform more realistic flooding simulation, the quasi-static method was also adopted and considerations on the increased weight due to incoming water, the effect of the air compression, and various locations of damaged holes were made in this study. Moreover, the water volume in the damaged compartments was calculated by using a polyhedron integral method which can calculate exactly the volume regardless of the complexity of the compartments and the attitude of the floater with an error range of 0.1%. In this method, the water volume can be accurately calculated by integrating surrounding faces of the compartment and water plane Table 1 shows the comparison of this study with the related studies.

Table 1. Comparison of this study with related studies.

 Park et al.  Ruponen ,  Dankowski , , ,  Lee et al.  This study Analysis method Quasi-static Quasi-static Quasi-static CFD Quasi-static Consideration on incoming water O O O X O Consideration on air compression X O O X O Location of damaged holes Fixed Various Various Fixed Various Sinking simulation for the final stage X X X X O Volume calculation – Water plane-based method Interpolation method – Polyhedron integral method

## 3. Flooding and sinking procedures of the damaged floater

Once a floater is damaged or an opening is exposed under the sea level, the damaged floater is flooding due to damaged compartments or openings. At this time, Fig. 3 shows the flooding procedure of the damaged floater. The sea water starts to come into the damaged compartments (Fig. 3-1). Due to the increased weight, the equilibrium position of the floater is changed (Fig. 3-2). Due to the change of hydrostatic pressure, more water comes into the damaged compartments until the inside pressure is same as the outside pressure (Fig. 3-3). Finally the floater stops its movement at the equilibrium position (Fig. 3-4). To simulate this state, we need to perform the intermediate simulation. Fig. 3

Sometimes the damaged can sink according to the magnitude of damage. At this time, Fig. 4 shows the sinking procedure of the damaged floater. If it is failed to find the equilibrium position of the damaged floater, the sinking procedure starts (Fig. 4-1). The equilibrium position of the damaged floater is determined by comparing the moment arms between the center of gravity and the center of buoyancy (Fig. 4-2). If the damaged floater cannot satisfy the equilibrium condition for any position, the damaged floater keeps rotating and will be sunken (Fig. 4-3). To simulate fully this state, we need to perform the sinking simulation. Fig. 4

## 4. Theoretical background of the intermediate flooding and sinking simulation

In this section, physical background to perform the intermediate flooding and sinking simulation is described. The key issue is how we deal with the water flow and the air flow in the damaged compartments of a floater. Basically, the water flow and the air flow are determined by Bernoulli׳s equation with energy loss. Several assumptions and governing equations of the water flow and the air flow are introduced here.

### 4.1. Assumptions

Ruponen , Dankowski , and Papanikolaou  used several assumptions for the intermediate flooding simulation. These assumptions are also used in this study. First, several factors such as applied forces and time duration when the damage is made are neglected. Although these factors affect the damaged floater, it is difficult to know their effects on the floater and the duration of the transient stage is short relatively, as show in Fig. 2. Second, the sea state is considered as calm sea like in Ruponen׳s study . According to EU Research Project HARDER, 90% of collision accidents happen in calm seas or wave height less than 2.0 m . And it is known that the roll motion of the damaged floater is almost same both in calm sea and in sea state with the significant wave height of 2.0 m. Therefore, it is reasonable to assume that the sea is calm. Third, the equilibrium state is calculated with the quasi-static method by considering water and free surface effect. Next, the water flow and the air flow are inviscid and irrotational. The water flow is incompressible. Besides, sloshing and inertia of incoming water are also neglected. These effects have been neglected for fast and efficient simulation in many studies including this study. Finally, according to previous studies such as Ruponen  and Dankowski , flux over the opening is assumed to be constant when the size of the opening is small. The size of the opening used in this study is enough small compared to the size of the floater. Thus this assumption can be reasonable.

### 4.2. Governing equation of water flow

The water flow can be determined from Bernoulli׳s equation, as shown in Eq. (1).∫dpρ+12u2+gh=Cwhere p is the air pressure in kPa, ρ is the density of fluid in kg/m3, u is the velocity of flow in m/s, g is the gravitational acceleration in m/s2, h is the reference height in m, and C is a constant on the same streamline in m2/s2.

Fig. 5 shows the water flow between two compartments. Fig. 5

In this figure, pi and pj are the air pressure of the compartments i and jHi and Hj are the water height of the compartments i and j, Hi and Hk are the height of the points i and k, respectively. The water flow at the point i can be represented as Eq. (2).∫dpρw+12uw,i2+g(Hi−Hk)=Ciwhere ρw is the density of water and uw,i is the velocity of water at the point i.

Similarly, the water flow at point k can be represented as Eq. (3).∫dpρw+12uw,k2+g(Hj−Hk)=Ckwhere uw,k is the velocity of water at the point k.

And then, the magnitudes of Ci and Ck are same because both points i and k exist on the same streamline A. Consequently, in the case of the water flow, the streamline A can be expressed by Eq. (4).∫ikdpρw+12(uw,k2−uw,i2)+g(Hj−Hi)=0

It is assumed that the velocity of water at the point i (uw,i) is negligible far from the opening (point k). This assumption is validated by previous studies such as Ruponen  and Dankowski . Therefore, Eq. (4) can be stated as Eq. (5).∫ikdpρw+12uw,k2+g(Hj−Hi)=0

Fig. 6 shows the flow expansion through an opening. The flow leads to a certain loss in energy. This energy loss can be considered by adding the empirical term which is proportional to the square of the velocity of water at the opening (uw,k), as shown in Eq. (6).12kLuw,k2where kL is the non-dimensional coefficient of energy loss . Fig. 6

Besides, ρw is assumed to be constant in case of the water flow. As a result, Eq. (7) can be derived from Eq. (5).∫ikdp+12ρwuw,k2+ρwg(Hj−Hi)+12kLρwuw,k2=0

Eq. (7) is composed of four terms. Physical meaning of the first term is the difference of the air pressure between the compartments i and j at the point k, as shown in Eq. (8).∫ikdp=pj−pi

The second term means the kinetic energy at the point k. The equation for mass flow of water at the point k can be stated as Eq. (9).ṁw,k=ρwAw,kuw,kwhere ṁw,k is the mass flow of water at the point k in kg/s and Aw,k is the area of the opening in m2.

Solving the second term in terms of the mass flow of water at the point k, then second term turns into Eq. (10).12ρwuw,k2=12ρw(ṁw,kρwAw,k)2

Next, the third term means the difference of hydrostatic pressure between the compartments i and j at the point k. If the water height of the compartment is lower than the height of the opening, there is not hydrostatic pressure at the opening in the compartment. Therefore, when Hi or Hj is higher than Hk, each hydrostatic pressure can be calculated by Eqs. (11)(12).ρwg[max(Hi−Hk),0]ρwg[max(Hj−Hk),0]

Therefore, summations of air pressure and hydrostatic pressure can be expressed by Eqs. (13)(14). Generally, these were called effective pressure.Pi=(pi+ρwg[max(Hi−Hk),0])Pj=(pj+ρwg[max(Hj−Hk),0])

And then, the first and third terms can be stated as Eq. (15).(pj+ρwg[max(Hj−Hk),0])−(pi+ρwg[max(Hi−Hk),0])=Pj−Pi

Similarly, the fourth term also can be solved in terms of the mass flow of water at the point k, then the fourth term can be stated as Eq. (16).12kLρwuw,k2=12kLρw(ṁw,kρwAw,k)2

Substituting Eqs. (10)(15)(16) into Eq. (7), then Eq. (17) can be obtained.12ρw(kL+1)(ṁw,kρwAw,k)2=Pi−Pj

After defining a dimensional coefficient of energy loss (Kw,k in kg−1 m−1), as shown in Eq. (18),Kw,k=kL+1ρwAw,k2

Eq. (17) can be written as Eq. (19).12Kw,kṁw,k2=Pi−Pj

The absolute value can be used to define the direction of the water flow. Therefore, Eq. (19) can be stated as Eq. (20). Eq. (20) is the modified Bernoulli׳s equation for the water flow. The mass flow of water can be calculated by solving this equation.12Kw,kṁw,k|ṁw,k|=Pi−Pj

### 4.3. Governing equation of air flow

The air flow can be also determined from Bernoulli׳s equation, as shown in Eq. (1)Fig. 7 shows the air flow between two compartments. Fig. 7

In this figure, Hq and Hr are the height of the points q and r. The air flow at the point q can be represented as Eq. (21).∫dpρa+12ua,q2+gHq=Cqwhere ρa is the density of air and ua,q is the velocity of air at the point q.

Similarly, the air flow at the point r can be represented as Eq. (22).∫dpρa+12ua,r2+gHr=Crwhere ua,r is the velocity of air at the point r.

And then, the magnitudes of Cq and Cr are same because both points q and r exist on the same streamline B. Consequently, in the case of the air flow, the streamline B can be expressed by Eq. (23).∫qrdpρa+12(ua,r2−ua,q2)+g(Hr−Hq)=0

In case of the air flow, the potential energy is small compared to other terms of Eq. (23). In addition, it is assumed that the velocity of air at the point q (ua,q) is negligible far from the upper opening (point r). Therefore, the streamline B can be express by Eq. (24).∫qrdpρa+12ua,r2=0

If the flooding is isothermal, Boyle׳s law can be applied to calculate the air pressure. Eq. (25) shows Boyle׳s law.p0ρ0=pρawhere ρ0 and p0 are the atmospheric air density and the pressure, respectively. Thus, ρa can be expressed by Eq. (26).ρa=pρ0p0

Therefore, substituting Eq. (26) into Eq. (24) makes Eq. (27).p0ρ0∫qrdpp+12ua,r2=0

This energy loss can be considered by adding the empirical term which is proportional to the square of the velocity of air at the opening (ua,r), as shown in Eq. (28) .12kLua,r2

As a result, Eq. (29) can be derived from Eq. (27).p0ρ0∫qrdpp+12ua,r2+12kLua,r2=0

Eq. (29) is composed of three terms. Physical meaning of the first term is the difference of air pressure between the compartments i and j at the point r. The first term of Eq. (29) can be written as Eq. (30).p0ρ0∫qrdpp=p0ρ0ln(pipj)

The physical meaning of the second term is the kinetic energy. The equation for the mass flow of air at the point r can be stated as Eq. (31).ṁa,r=ρaAa,rua,rwhere ṁa,r is the mass flow of air at the point r in kg/s and Aa,r is the area of the upper opening (point r).

Solving the second term in terms of the mass flow of air at the point r, then the second term turns into Eq. (32).12ua,r2=12(ṁa,rρa,rAa,r)2

Similarly, the third term also can be solved in terms of the mass flow of air at the point r, then the third terms can be stated as Eq. (33).12kLua,r2=12kL(ṁa,rρa,rAa,r)2

Substituting Eqs. (30)(32)(33) into Eq. (29), then Eq. (34) can be obtained.12(kL+1)(ṁa,rρa,rAa,r)2=p0ρ0ln(pipj)

After defining a dimensional coefficient of energy loss (Ka,r in kg−1 m−1), as shown in Eq. (35),Ka,r=kL+1ρa,kAa,k2

Eq. (34) can be written as Eq. (36).12Ka,rṁa,r2=ρa,rp0ρ0ln(pipj)

The absolute value can be used to define the direction of the air flow. Eq. (36) can be expressed by Eq. (37).12Ka,rṁa,r|ṁa,r|=ρa,rp0ρ0ln(pipj)

When considering the direction of the air flow, the air flow progress from higher pressure to lower pressure. Therefore, Eq. (38) is satisfied.ρa,rp0ρ0=min(pi,pj)

Therefore, Eq. (37) can be stated as Eq. (39).12Ka,rṁa,r|ṁa,r|=min(pi,pj)ln(pipj)

Furthermore, Eq. (41) can be obtained by applying Taylor serious, as expressed in Eq. (40).yln(x/y)=y[(xy−1)−(xy−1)22⋯]≈x−y12Ka,rṁa,r|ṁa,r|=pi−pj

Eq. (41) is the modified Bernoulli׳s equation of the air flow. The mass flow of air can be calculated by solving this equation.

## 5. Overall process and implementation of the intermediate flooding and sinking simulation

In this section, overall process of the intermediate flooding and sinking simulation is presented. To perform the simulation, a prototype program was developed in this study. The implementation of the program is also described.

### 5.1. Overall process of the intermediate flooding and sinking simulation

Fig. 8 shows the overall process of the intermediate flooding and sinking simulation. As shown in this figure, this simulation is done by several steps. In the first step (Fig. 8-1), data of the damaged floater such as a hull form, compartments, openings, and damaged holes are prepared. Geometric models for the hull form and the compartments are used to calculate the underwater volume or water volume inside the compartment. In the second step (Fig. 8-2), the mass flow of water and air are calculated by using governing equations (Eq. (20) for water and Eq. (41) for air). In third step (Fig. 8-3), the weight of increased water and its center of gravity (COG) are calculated. In the fourth step (Fig. 8-4), the current weight and the COG of the floater are calculated by considering the initial weight and the COG of the floater and the weight of increased water and its COG. In the fifth step (Fig. 8-5), the changed buoyancy and the center of buoyancy (COB) of the floater are calculated by considering the initial buoyancy and the COB of the floater and the changed draft due to the weight of increased water. At this time, the polyhedron integral method is used to calculate accurately the volume and the COB. In the sixth step (Fig. 8-6), the quasi-static equilibrium position of the floater is calculated and it is determined whether the damaged floater is sinking or not. If the equilibrium position is not be found, the floater cannot be equilibrium state. It means that the floater is in sinking state and will sink. Thus, we go to the second step (Fig. 8-2) and try to find the equilibrium state again. If the equilibrium position of the floater is found, it is checked whether the mass flow of water and air is zero or not in the seventh step (Fig. 8-7). If the mass flow is not zero, it means that the floater is in intermediate flooding state and will be flooded. Thus, we go to the second step (Fig. 8-2) and try to find the equilibrium state again. If the mass flow is zero, it means that the floater reaches the final state. The state can be one of the equilibrium state or the sinking state. Thus, the simulation finishes and the estimated time to be equilibrium or sink is calculated as one of output of the simulation (Fig. 8-8). Fig. 8

### 5.2. Implementation of the intermediate flooding and sinking simulation

To perform the intermediate flooding and sinking simulation, a prototype program was developed in this study. The developed a program consists of five parts: ribbon style menu, model builder (tree view), 3D simulation view, property editor, and timeline. The C# program language and the WPF (Window Presentation Foundation) were used to implement the program. With this program, the motion of the damaged floater and the time to sink can be checked or estimated for the given scenario of damage. Fig. 9 shows a screenshot of the developed program. Fig. 9

## 6. Application of the intermediate flooding and sinking simulation

This section presents some examples to show the applicability of the proposed method in this study. First, the intermediate flooding and sinking simulation for a simple box was performed. Then, the simulation was made for more realistic example of a barge-type damaged floater.

### 6.1. Simple box example

Two boxes selected as a simple example haves two compartments. They are all laid on the ground. One compartment of the boxes is filled with the water and the other compartment is filled with the air. However, the first box (Box 1) has one opening, the second box (Box 2) has two openings. There are the water flow for both boxes. In the case of the second box, there is also the air flow. Table 2 shows the initial condition of each box. And Fig. 10 shows the geometrical modeling of these two boxes before the simulation.

Table 2. Initial condition of the simple box example.

Box 1Box 2
Compartment 1Compartment 2Compartment 1Compartment 2
Water height (m) 0 7 0 7
Air pressure (atm) 1 1
Height of the opening (m) 2 2, 7.5 Fig. 10

For two boxes, the intermediate flooding and sinking simulation was performed with the prototype program according to the overall process of the simulation in Fig. 8. The equilibrium position for two boxes was well found. Table 3 shows the water height and the air pressure of each box at the final state. In the first box, there is no the air flow. Thus, there was the difference in water heights and air pressures between two compartments after the simulation. On the other hand, there was no difference in water heights and air pressures between two compartments due to the air flow between them. Fig. 11 shows the visualization of the simulation results.

Table 3. Simulation result of the simple box example.

Box 1Box 2
Compartment 1Compartment 2Compartment 1Compartment 2
Water height (m) 0.56 6.44 3.5 3.5
Air pressure (atm) 1.07 0.63 1 1 Fig. 11

### 6.2. Example of a barge-type floater

The next example is about a barge-type damaged floater. That is, the damaged floater was simplified as a rectangular parallelepiped in this example. The time step for the simulation was 0.01 s and the explicit Euler method was used in this example as the integration scheme.

#### 6.2.1. Principal particulars of the floater

Fig. 12 shows the floater (ship) and its simplified model. The barge-type simplified floater was used for the intermediate flooding and sinking simulation in this study. It has 7 compartments: 3 compartments at the port side, 3 compartments at the starboard side, and 1 compartment at the center. Table 4 shows principle particulars of the barge-type floater. Fig. 12

Table 4. Principle particulars of the barge-type floater.

ItemUnitValue
Length overall m 120
Breadth m 21
Depth m 13
Displacement (total weight) ton 9040
No of compartments 7

#### 6.2.2. Definition of damage cases

Before the simulation, two damage cases were defined according to the location of a damaged hole. Openings are located at ceiling of each compartment and between compartments. Thus, the air pressure of each compartment is 1 atm. The openings between compartments are installed above 6.5 m from the bottom of the floater. In addition, it was assumed that the all openings and the damaged hole are rectangles whose length and breadth are all 3 m. It is assumed that the area of the opening for the water flow (Aw,k) and the air flow (Aa,k), and the non-dimensional coefficient of energy loss (kL) are 9 m2, 0.09 m2, and 0.7, respectively . And the damaged hole is located at the port-bottom of the floater in Case 1 and at the center-bottom in Case 2. Table 5 shows the data on the openings and the damaged hole of the floater.

Table 5. Data on the openings and the damaged hole.

TypeItemUnitValue
Openings Number of openings 13 (7 at ceiling and 6 between compartments)
Length m 3
Breadth m 3
Area of the opening for the water flow (Aw,k) m2 9
Area of the opening for the air flow (Aa,k) m2 0.09
Non-dimensional coefficient of energy loss (kL) 0.7
Damaged hole Number of damaged hole 1
Location of damaged hole Port-bottom (Case 1)/Center-bottom (Case 2)

Fig. 13Fig. 14 show the configuration of the openings and the damaged hole of Cases 1 and 2, respectively. Fig. 13 Fig. 14

#### 6.2.3. Simulation results for the cases

For two cases, the intermediate flooding and sinking simulation was performed with the prototype program according to the overall process of the simulation in Fig. 8. In Case 1, the damaged floater inclines to the port side and starts to submerge because of the incoming water through the damaged hole. As more water comes into the compartments, the damaged floater becomes submerged deeper. Finally, the floater stops its movement at the equilibrium position 430 s later because the inside effective pressure and the outside effective pressure at the damaged hole become same due to the air compression of air pocket. At the final state, the angle of heel reaches 90° as shown in Fig. 17 and the immersion is 10 m from the free surface as shown in Fig. 18. That is, the floater will be floating on the water for Case 1. Fig. 15 shows the visualization of the simulation result of Case 1. Fig. 15

In Case 2, the damaged floater starts to submerge without roll motion because of the incoming water through the damaged hole. As more water comes into the compartments, the damaged floater becomes submerged deeper and finally the floater starts to sink at 938 s because the restoring force isn’t enough. The angle of heel does not change during the simulation as shown in Fig. 17. The immersion increase as time goes as shown in Fig. 18. That is, the floater will sink for Case 2. Fig. 16 shows the visualization of the simulation result of Case 2. Fig. 16

Fig. 17 shows the change of the angle of heel in time domain. As shown in this figure, the floater has angle of heel of 90° for Case 1 and no angle of heel for Case 2. In Case 1, it can be seen that the angle of heel is changed during flooding. The flooding progress of Case 1 is faster than that of Case 2. Fig. 17

Fig. 18 shows the change of the immersion in time domain. As shown in this figure, the floater has the immersion of 10 m from free surface at the final state for Case 1. This means that the floater reaches the equilibrium state and are floating on the water. In Case 2, it can be seen that the immersion is changed during flooding. This means that the floater does not find the equilibrium position and is sinking. From this simulation, the estimate time to be equilibrium for Case 1 was 430 s and the estimated time to sink for Case 2 was 938 s, as shown in Fig. 18. Fig. 18

Fig. 19 shows the change of the transverse metacentric height (GM) in time domain. The GM can be determined by the vertical center of buoyancy (KB), the vertical center of gravity (KG), and the transverse metacentric radius (BM), as shown in Eq. (42).GM=KB−KG+BM=KB−KG+IT∇where, IT is the moment of inertia of the water plane about the longitudinal axis in m4 and ∇ is the displacement volume in m3. Fig. 19

As shown in this figure, as the floater becomes immersed deeper, the GM of the floater becomes larger till at about 200 s. After then, the GM decreases since the BM decreases due to the increase of the displacement volume and the KG increases due to the incoming water. The floater loses its righting arm (or righting moment) at 276 s. The GM does not change after the equilibrium of the floater in Case 1. In Case 2, it can be also seen that as the floater becomes immersed deeper, the GM becomes smaller, and finally the floater loses its righting arm at 938 s.

Fig. 20 shows the change of the water mass in time domain. As shown in this figure, as the floater becomes immersed deeper, more water comes into the compartments. In Case 1, inflow of the water stops about 430 s when the floater reaches to equilibrium condition. This means that the compressed air pocket blocks inflow of the water into the compartments after the equilibrium. In Case 2, it can be seen that inflow of water keeps increased. This means that the compressed air pocket cannot block inflow of the water because the floater keeps sinking. Fig. 20

## 7. Conclusions and future works

In this study, a method for intermediate flooding and sinking simulation of a damaged floater in time domain was proposed in this study. To do that, the flooding and sinking procedures of the damaged floater was established and the governing equations for the simulation were derived. Then, overall process for the simulation was made and the prototype program was developed here. To check the feasibility and applicability of the proposed method, it was applied to a simple box problem. Finally, it was applied to intermediate flooding simulation of a barge-type damaged floater. As a result, the floater for Case 1 could reach the equilibrium state with the angle of heel of 90° and the immersion of 10 m. On the other hand, the floater for Case 2 could not reach the equilibrium and sank with no angle of heel. From this simulation, the estimate time to be equilibrium for Case 1 was 430 s and the estimated time to sink for Case 2 was 938 s.

As future works, the improvement of this simulation will be made. First, the validation of simulation results will be made with some experiments or accident reports of floaters. And the simulation will be applied to a more realistic example of a damaged floater which has a complicated hull form and a number of compartments. In addition, a study on the dynamic effect such as sloshing and inertia of incoming water will be made. Finally, the simulation for fluids having different density in the compartment will be performed.

## Conflict of Interest

The author has confirmed that there is no conflict of interest.

## Acknowledgements

This work was partially supported by (a) Brain Korea 21 Plus Program (Education and Research Center for Creative Offshore Plant Engineers of Seoul National University) funded by the Ministry of Education, Republic of Korea, (b) The National Research Foundation of Korea (NRF) grant funded by the Ministry of Education, Science and Technology (No. 2016R1A2B4016253), and (c) Research Institute of Marine Systems Engineering of Seoul National University, Republic of Korea.

## References

Peer review under responsibility of Society for Computational Design and Engineering.

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