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Adaptive time step in simulation of progressive flooding
Ocean Engineering 78 (2014) 35–44
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Adaptive time step in simulation of progressive flooding Pekka Ruponen n Napa Ltd., P.O. Box 470, FI-00181 Helsinki, Finland
art ic l e i nf o
a b s t r a c t
Article history: Received 28 August 2013 Accepted 29 December 2013 Available online 18 January 2014
Time accurate simulation of progressive flooding inside a ship often requires a short time step. However, after the initial phases, as the flooding progresses to undamaged compartments, the flow rates decrease and much longer time step could be used. Yet the collapsing of non-watertight structures may cause additional phases of fast flooding, where a shorter time step is required. In this paper the use of an adaptive time step in flooding simulation is discussed, and a new approach to this problem is presented. The time step is automatically adjusted during the simulation on the basis of the maximum absolute velocity of the free surface levels in the flooded compartments. The method is tested with small case studies, presenting typical flooding conditions in passenger ships. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Damage stability Flooding simulation Adaptive time step Progressive flooding
1. Introduction Progressive flooding inside a damaged passenger ship can be a slow process, especially if there are several closed doors that will only leak under a moderate pressure head of the floodwater. However, in the beginning of flooding the flow rates are typically very large. This is evident in the results of previous studies on progressive flooding, such as van0 t Veer and de Kat (2000) and Ruponen et al. (2010). During the past decades, several time-domain flooding simulation tools have been developed, e.g. Spanos and Papanikolaou (2001), Jasionowski (2001), Ruponen (2006, 2007), Santos and Guides Soares (2009), Ypma and Turner (2010), Schreuder et al. (2011) and Dankowski (2012, 2013). These methods use Bernoulli0 s equation for calculation of the water flow through the openings. The general approach in these simulation tools is to use a constant time step. The possibility to adjust the time step during the simulation in order to achieve shorter computation times has not yet been studied thoroughly. In computational fluid dynamics (CFD) the use of an adaptive time step is more common, e.g. Turek (1999), Berrone and Marro (2009) and Volker and Rang (2010). The CFD methods have also been used for flooding analyses with time step size limitation based on the Courant number, Gao et al. (2010). However, due to very slow computation times, the CFD tools are not feasible for any practical design work and analyses. Therefore, the further development of simulation methods that are using Bernoulli0 s equation is considered to be very important. In general, the flooding of a ship can be divided into three separate stages that are illustrated in Fig. 1. First there is transient flooding as water enters the damaged compartments. Typically this causes transient heeling. After a while the flow rates decrease and water may progress to undamaged compartments through
openings or non-watertight structures. This stage is referred to as progressive flooding, and it can be very long. Eventually a steady state is achieved, unless the ship sinks or capsizes before that. The optimal time step that provides sufficient accuracy and efficient computations may vary during the simulation. During the transient flooding, a relatively short step is usually needed in order to capture the fast changes in both flooding and ship motions. Later, during the stage of progressive flooding, a longer step may be applied without sacrificing the accuracy. However, if a long time step is used, it is also necessary to check that the effective pressure heads on closed doors are not resulting in collapsing, since this may cause a short phase of fast flooding, at least locally. The maximum allowed value of the time step depends also on the motions of the ship. However, usually also heeling changes rather slowly during the phase of progressive flooding, at least in a moderate seaway. It is also noteworthy that the increase of the time step is not reasonable for simulation of damaged ship motions in high waves, where the rolling is significant and thus limits the applicable time step. In this paper a new approach to this problem is presented. The time step is automatically adjusted during the simulation on the basis of the maximum absolute velocity of the free surface levels in the flooded compartments. The adaptive time step method is tested with two simple scenarios that demonstrate the problems in simulation of progressive flooding. In addition, the method is applied for a realistic damage scenario with extensive progressive flooding inside a large passenger ship. Finally the pros and cons of adaptive time-stepping in flooding simulation are discussed. 2. Pressure-correction method for flooding simulation 2.1. Background
n
Tel.: þ 358 50 5766078; fax: þ 358 922813800. E-mail address: pekka.ruponen@napa.fi
0029-8018/$ - see front matter & 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.oceaneng.2013.12.014
The presented method on the use of adaptive time step in flooding simulation has been implemented in the pressure-correction method,
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P. Ruponen / Ocean Engineering 78 (2014) 35–44
floodwater
free discharge
submerged flow
floodwater
ΔH
time
heeling
ΔH heeling transient
progressive flooding
u
steady state
u
Fig. 1. Schematic representation of different stages of flooding.
described in detail in Ruponen (2006, 2007). This simulation tool has been successfully validated against model tests, Ruponen et al. (2007), and full-scale data, Ruponen et al. (2010). The pressure-correction algorithm can also be used for solving air compression inside the flooded compartments, Ruponen (2007) and Ruponen et al. (2013), but for the sake of simplicity, in the present study it is assumed that air pressure is constant. However, it is noteworthy that simulation of air compression usually requires a shorter time step, Ruponen (2007) and Dankowski (2013). The applied simulation method is based on implicit time integration with a pressure-correction algorithm. This has proven to be an efficient and accurate approach for calculation of extensive progressive flooding to several compartments. In the following, the algorithm is briefly presented. The principles of the applied simulation method are also described, although the presented application of adaptive time step is valid for any method for the solution of the progressive flooding when Bernoulli0 s equation is used. 2.2. Governing equations At each time step the conservation of mass must be satisfied in each flooded room. The equation of continuity is Z Z ∂ρ dΩ ¼ ρv UdS ð1Þ Ω ∂t S where ρ is density, v is the velocity vector and S is the surface that bounds the control volume Ω. The normal vector of the surface points outwards from the control volume. The velocities in the openings are calculated by applying Bernoulli0 s equation for a streamline from point A that is in the middle of a flooded room to point B in the opening: Z B dp 1 2 1 þ ðuB u2A Þ þ gðhB hA Þ þ kL u2B ¼ 0 ð2Þ 2 2 A ρ where p is air pressure, u is flow velocity and h is the water height from the common reference level. All losses in the duct are represented by the non-dimensional pressure-loss coefficient kL. Consequently, by denoting ΔH ¼hB hA and assuming that uA ¼ 0 and constant air pressure, the volumetric flow through an opening with area dS is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dQ ¼ C d U 2g UΔH U dS ð3Þ where the semi-empirical discharge coefficient is 1 C d ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 1 þ kL
ð4Þ
The effective pressure head for free discharge and fully submerged flow are illustrated in Fig. 2. For large openings the flow rate Q is integrated over the submerged area of the opening, and the flow condition may contain both free discharge and submerged flow. It is assumed that the water levels in the flooded compartments are parallel to the sea level, and thus the possible sloshing effects are ignored. However, it is believed that the presented method for
Fig. 2. Effective pressure head ΔH for different flow conditions.
adaptive time step can be extended to cover also the simplified methods for sloshing. The free surface area for each flooded room is calculated from the geometry for each time step with the floating position of the ship taken into account. During the pressure-correction iteration the free surface areas are assumed to be constant.
2.3. Initial and boundary conditions The loading condition of the ship before the damage defines the initial condition for the flooding simulation. The applied method does not handle mixtures of different liquids but loaded water in damaged tanks is treated similarly to floodwater, and thus for example draining of a swimming pool can be simulated. A collision with another ship obviously affects the ship motions, e.g. Tabri et al. (2010). However, it is a widely used simplification in damage stability calculations to assume that the collision or grounding force does not affect the ship motions or transient flooding. Consequently, the flooding starts with the ship floating freely with a damage opening in the hull surface. The surrounding sea is treated as a ghost cell that sets the boundary condition for flooding. The hydrostatic pressure for sea level is calculated on the basis of the floating position for each time step.
2.4. Time discretization In the pressure-correction method the simulation within a time step is iterative, based on the linearized Bernoulli0 s equation. The algorithm iterates the pressures in the flooded rooms until both Bernoulli0 s equation for each opening and continuity for each room is satisfied with sufficient accuracy. After that, the floating position of the ship is calculated on the basis of the distribution of floodwater in the compartments, accounting the free surface effect on all flooded rooms. In the simulation algorithm the instantaneous free surface area Sfs is assumed to be constant within the time step, and the time derivative of the volume of water in a flooded room can be presented as dV w dH w ¼ Sf s dt dt
ð5Þ
The permeability μ is taken into account, so that Sf s ¼ μStot
ð6Þ
where Stot is the total area of the cross-section of the room at the floodwater level.
P. Ruponen / Ocean Engineering 78 (2014) 35–44
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3.2. Time step change indicator The new approach is somewhat similar to the one presented by Berrone and Marro (2009) for unsteady Navier–Stokes problems. In the case of progressive flooding the time error indicator η is considered to depend on the maximum absolute time derivative of the water height in the flooded rooms
water height
door collapses
net inflow starts again
_ w;i jÞ η ¼ maxðjH
flooding starts
ð10Þ
The default time step Δt0 is considered to be suitable for the beginning of flooding, and therefore, the indicator η is normalized by the maximum absolute time derivative of water level heights in the beginning of flooding
s ¼ maxðjH_ w;i ðt ¼ 0ÞjÞ
time
ð11Þ
Fig. 3. Examples of flooding conditions where implicit Euler method must be used.
The default time step is acceptable if the following double inequality is satisfied:
Normally the time derivatives for the pressure-correction algorithm are calculated by using the three-level implicit method
η 1 αr r s α
nþ1
_ w 3H w H
4H nw þ H nw 1 2Δt
ð7Þ
However, at the steps, where there is potential discontinuity in the flooding rate, the implicit Euler method is used nþ1
_ w Hw H
H nw Δt
ð8Þ
The discontinuities can occur e.g. when flooding to a new compartment starts or a closed door collapses. These are illustrated in Fig. 3.
3. Adaptive time step 3.1. Background
where α is a given parameter in the range (0,1). If the indicator exceeds the upper bound a shorter time step is needed. A longer time step can be used if the indicator is smaller than the lower bound. In the presented case studies the value α¼0.5 was used. The thresholds for changing the time step, α and 1/α, can also be independent. Moreover, even longer time step might be feasible when η/s o oα. As mentioned previously, when the time step is adjusted, the time derivatives need to be calculated with the implicit backward Euler method instead of the normal three time level method. Thus it is not favorable to adjust the time step very frequently in order to avoid increased error in the calculation of the floating position. Consequently, the increase of time step is allowed only after a short period tbuffer since the previous change. In the presented case studies this period is 10-times the initial time step. 3.3. Effects on the calculation of ship motions
The pressure-correction method is iterative, and thus there is no need to limit the time step in order to prevent the change of flooding direction in the openings during the time step. For explicit time integration this kind of limitation is necessary, especially with longer time steps, as presented in Ruponen et al. (2012). In principle the adjustment of the time step can be done with two different approaches: either gradually changing or by a sudden larger change. The latter approach is used in this study since it keeps the equations in a simpler format, especially in the calculation of the time derivatives. The simplest way is to change the time step suddenly when certain conditions are met. In practice this means that the new time step is Δt ¼ C Δt Δt 0
ð12Þ
ð9Þ
where Δt0 is the initial time step and the coefficient CΔt depends on the flooding process at the time. In the previous study on the use of adaptive time step in flooding simulation, Ruponen (2009), the time step was increased when the maximum absolute net flow in the flooded compartments was below a predefined limit value. However, this approach does not take into account the geometry of the compartments. Even a small net flow to a room with a small free surface area, such as a staircase or a lift trunk, can cause rather rapid changes in the water level. Thus in the present study the criterion for adjusting the time step is based on the maximum absolute time derivative of the water height.
The length of the time step affects also the calculation of the ship motions. If the motions are considered to be fully quasistationary, the time step can be selected purely on the basis of the flooding process. However, even a simple approach with dynamic roll motion and linear model for roll damping restrict the use of adaptive time step. A simple approach is to allow the increase of the time step only when the roll velocity is small. In the present study this is taken into account by setting a limit of 0.011/s for the moving absolute maximum of the roll velocity over the past 20 time steps. Obviously, if other motions, such as trim, are rapid, they have to be included in the limitations to the maximum allowed time step. 3.4. Algorithm Unlike in Navier–Stokes problems, for progressive flooding the adjustments can be done abruptly. Thus the similar restrictions to the size of the time step change as in Berrone and Marro (2009) are not necessary. The initial time step Δt0 is considered to be suitable for the beginning flooding, i.e. it is mainly dependent on the size of the damage openings and phenomena during the transient flooding. However, a shorter time step may also be needed later during the stage of progressive flooding as there can be short phases of fast internal flooding after collapsing of closed doors. The applied algorithm for adaptive time step is described in the following.
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P. Ruponen / Ocean Engineering 78 (2014) 35–44
_ 4f _ if (jfj crit ) then if (Δt 4 Δtmin ) then if (Z=s 41=a) then Δt ¼ Δtmin else if (Δt 4 Δt0 ) then Δt ¼ Δt0 else no change end if else no change end if else if (t 4 tadj þ tbuffer ) then if (Z=s 4 1=a) then if (Δt 4 Δtmin ) then Δt ¼ Δtmin else no change end if else if (Z=s o a) then if (Dt o Δt0 ) then Δt ¼ Δt0 else if (Dt o Dtmax ) then Δt ¼ Δtmax else no change end if else if (Δt o Δt0 or Δt 4Δt0 ) then Δt ¼ Δt0 else no change end if end if else no change end if
hydrostatic pressure exceeds the critical threshold of 2.9 m. The height of the rooms is 3.0 m. Thus the flooding process contains several subsequent phases of both fast and slow flooding rates. The floating position is fixed. The modeled openings and the flooded rooms are illustrated in Fig. 4. The area of the damage opening is 0.3 m2 and the area of the doors is 2.0 m2. The sea level height is 0.25 m above the room
4.5 4.0
water height [m]
3.5 3.0 2.5 2.0 1.5 R1 R2 R3 R4
1.0 0.5 0.0
0
2
4
6
8
10
12
14
16
18
time [min]
0.18
4. Case studies 4.1. Flooding through successive compartments The developed algorithm is tested with small case studies. The first one involves flooding though successive rooms that are connected by closed doors. The doors will collapse after the
floodwater max(dHw/dt)
800
0.16
600
0.12
500
0.10
400 300
door R3-R4
0.14
door R2-R3
700
door R1-R2
total volume of floodwater [m3]
900
0.08 0.06
200
0.04
100
0.02
0
0
2
4
6
8
10
12
14
16
0.00 18
max absolute water level velocity [m/s]
Fig. 5. Reference results for the water level heights in the first case study.
time [min] Fig. 6. Reference results for total volume of floodwater and maximum absolute free surface velocity in the first case study.
Fig. 4. Openings and flooded rooms for the first test case.
P. Ruponen / Ocean Engineering 78 (2014) 35–44
4.2. Down-flooding through a staircase Progressive flooding may involve rooms of different sizes. The second case study concentrates on down-flooding through a staircase with small free surface area compared to the other flooded rooms. Sea level height is 0.25 m below the top of the upper deck and the height of both decks is 2.5 m. In the beginning, the flooding trough the damage opening is fast. Eventually the leakage threshold of the closed door is reached and floodwater progresses to the staircase. The flooding rate is small but the water level rises fast since the free surface area in the staircase is small. The final phases involve leakage and collapsing of the closed door from the staircase to the lower deck and both the flooding rate and the time derivative of the hydrostatic pressure height are small. The rooms and openings are shown in Fig. 9 and the flooding phases are illustrated in Fig. 10. Similarly to the first case, the floating position is fixed. The reference results are calculated with a constant time step Δt ¼0.2 s. The water level heights in the rooms are shown in Fig. 11 and the maximum absolute net flow and the maximum absolute time derivative of the free surface are presented in Fig. 12 as functions of time. At about 2.5 min the staircase is flooded. The water level in this room with relatively small free surface area rises very rapidly. At about 4 min the door from the staircase to the lower deck collapses, causing a sudden drop in the water level in the staircase due to the fast flooding through the collapsed door. Soon thereafter the water levels on the lower deck and in the staircase rise slowly at the same pace. Eventually at 45 min the room on the lower deck is filled up and the water level in the staircase starts to rise rapidly until the final equilibrium is reached at about 47 min. Most of the time the flooding is slow but locally there are rapid changes of the water level height in the staircase. Thus accurate simulation requires a short time step during these phases.
time step [s]
R2-R3
R3-R4
relative computation time
R1-R2
2.5 2.0 1.5 1.0 0.5 0.0
10
10%
3.0
0
2
4
6
8 10 time [min]
12
14
16
8%
8
6%
6
4%
4
2%
2
0%
18
max difference in flooding events
0.5 s
1.0 s
2.0 s
0.5...2.0 s
max difference [s]
height (3.25 m above door bottom). Consequently, flooding is significantly reduced before the collapsing door enables progressive flooding to another room. The reference data is obtained by using a very short constant time step of 0.1 s. The water level heights in the rooms are shown in Fig. 5 and the total volume of floodwater and the maximum absolute free surface velocity are presented in Fig. 6. The collapse of each closed door causes a short phase of very fast flooding that lasts less than a minute. Thereafter, flooding is slow until the next door collapses. The developed algorithm for adaptive time step is tested with this flooding scenario by using an initial time step of 1.0 s. Thus the time step is adjusted between 0.5 s and 2.0 s. The time history for the time step is presented in Fig. 7. The collapse of each closed door causes rapid flooding and the time step is shortened to the minimum value of 0.5 s. As the flooding then slows down again, the time step is increased back to 1.0 s and after a while to the maximum value of 2.0 s. Simulations with different constant time steps and with an adaptive time step are compared to the reference results, Table 1. With an increased constant time step the times when the doors collapse become more inaccurate. When the time step is adjusted according to the presented algorithm between 0.5 s and 2.0 s, the accuracy of the results is nearly as good as with a constant step of 1.0 s. Yet the computation time is almost as short as with a constant step of 2.0 s, Fig. 8. It is noteworthy that even though adaptive time step improves the accuracy, a longer time step will inevitably mean that the exact event of the collapse is captured more inaccurately. However, this error is of the magnitude of one time step, and thus it is not significant since the applied limits for collapsing pressure heads are always rather approximate.
39
0
time step
Fig. 7. Adaptive time step in the first case study, showing the times when the closed doors collapse.
Fig. 8. Comparison of computation times for the first case study with different time steps against the reference case with Δt¼ 0.1 s.
Table 1 Comparison of flooding events and difference to the reference results with different time steps for the first case study. Flooding event
Door R1-R2 collapses Door R2-R3 collapses Door R3-R4 collapses
Time step 0.1 s
0.5 s
1.0 s
2.0 s
0.5–2.0 s
Reference
Time
Diff
Time
Diff
Time
Diff
Time
Diff
177.4 405.6 664.0
177.5 405.2 663.0
0.1 0.4 1.0
179.0 407.5 664.5
1.6 1.9 0.5
180.0 413.0 672.0
2.6 7.4 8.0
180.0 407.5 665.5
2.6 1.9 1.5
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P. Ruponen / Ocean Engineering 78 (2014) 35–44
2.0
I
III
0.20
1.5
0.15 1.0
0.5
lower deck filled up
stairs flooded
max absolute net flow [m3/s]
door collapses
II
0.25
max(Q ) max(dH /dt)
0.10
0.05
IV 0.0
0
10
Fig. 10. Flooding phases of the second case study.
30
0.00 50
40
time [min] Fig. 12. Maximum absolute net flow rate and time derivative of hydrostatic pressure head as functions of time for the second case study with a constant time step Δt¼ 0.2 s.
4.0 3.5
3.0 door collapses
3.0
lower deck filled up
2.5
2.5 tween deck
2.0
time step [s]
water height [m]
20
max absolute water level velocity [m/s]
Fig. 9. Flooded compartments and openings.
1.5 1.0
Deck 1 Deck 2 Stairs
0.5 0.0
2.0 1.5 1.0 0.5
0
10
20 30 time [min]
40
50
Fig. 11. Reference results for the water level heights in the second case study; the vertical lines show the transitions between the phases as defined in Fig. 10.
0.0
0
10
20
30
40
50
time [min] Fig. 13. Adaptive time step in the second case study, showing the times when the closed door collapses and the lower deck is filled up.
Applying a longer time step causes inaccuracy in the time when the undamaged rooms are flooded and the closed door collapses. Thus for this kind a flooding scenarios the automatically adjusted time step is a very suitable solution for improving the computing efficiency without significant sacrifice in the accuracy. The applied time step is presented in Fig. 13 as a function of time. The times for different flooding events and the corresponding difference to the reference values with various time steps are presented in Table 2, and the relative computation times are shown in Fig. 14. The adaptive time step provides more accurate
results than a constant time step, although the differences are very small. However, it should be noted that mainly due to the implicit time integration with the pressure-correction method, the results are fairly accurate even with longer constant time step. The effect of the time step on the computation time is very similar to the first case study, but the maximum difference in the occurrence of the flooding events is smaller, even though the total time-to-flood is longer.
P. Ruponen / Ocean Engineering 78 (2014) 35–44
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Table 2 Comparison of flooding events and difference to the reference results with different time step for the second case study. Flooding event
Time step
Staircase flooded Lower deck flooded Door to lower deck collapses
0.2 s
0.5 s
Reference
Time
Diff
Time
Diff
Time
Diff
Time
Diff
150.6 176.0 215.8
150.0 175.5 215.0
0.6 0.5 0.8
151.0 177.0 216.0
0.4 1.0 0.2
152.0 178.0 218.0
1.4 2.0 2.2
151.0 176.0 216.0
0.4 0.0 0.2
25%
1.0 s
2.0 s
0.5–2.0 s
10
20%
8
15%
6
10%
4
5%
2
0%
max difference [s]
relative computation time
max difference in flooding events
0 0.5 s
1.0 s
2.0 s
0.5...2.0 s
time step
Deck 2
Fig. 14. Comparison of computation times for the second case study with different time steps against the reference case with.
damage
WT comp.
1
2
3
4
4.3. Progressive flooding of a cruise ship The last case study concentrates on a more realistic flooding case. The 63,000 GT passenger ship design, Luhmann (2009), was developed in the FLOODSTAND project. The damage scenario is a long and narrow gash in the starboard side of the aft part of the ship. This kind of damage could result e.g. from a collision to a pier. Total of three watertight compartments are flooded. The arrangement of the flooded compartments and the openings are shown in Figs. 15 and 16. It is assumed that all watertight doors are closed and fully tight. Most of the fire doors are considered to be open. However, five fire doors and the lift doors are initially closed, see Fig. 16, and these can leak and collapse under the pressure of floodwater. The applied leaking and collapsing model is based on full-scale tests in the FLOODSTAND project, SLF 54/INF.8/Rev.1, IMO (2011). The ship is considered to have an initial metacentric height of GM0 ¼ 2.5 m. The trim and the draft are considered to be quasistationary, but the uncoupled roll motion ϕ(t) is calculated from the following equation: _ þ M st ðϕ; tÞ þ M ext ðϕ; V w ; tÞ ¼ 0 € þBxx ϕ ðI xx þAxx Þϕ
ð13Þ
where Ixx and Axx are the mass moment of inertia for roll and added mass, Mst is the static righting moment and Mext is the heeling moment caused by the floodwater Vw. The roll damping is assumed to be linear, so that Bxx ¼ 2ξωϕ ðI xx þ Axx Þ
ð14Þ
The natural roll frequency ωϕ is based on approximated natural period of Tϕ ¼20.0 s and the linear roll damping coefficient is ξ ¼0.05. The total mass moment of inertia (Ixx þ Axx) is obtained
Deck 1 Fig. 15. Cruise ship design, Luhmann, 2009, and general arrangement of the flooded compartments.
from the equation for the natural roll period Tϕ ¼
2π ¼ 2π ωϕ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðI xx þ Axx Þ ΔU GM 0
ð15Þ
where Δ is the initial displacement. The damage is located on Deck 2, just below the waterline. The breach has a length of 22.4 m and the total area is 22.95 m2. The damaged rooms are the void in the compartment 2, the workshop and the hotel stores in the compartments 3 and 4. The stern thruster room and the two rooms for electric motors on Deck 1 remain dry due to the closed WT doors. The final floating position of the ship with the flooded compartments and status of the doors is shown in Fig. 17. Initially, the flow rates are fast but the closed fire doors restrict progressive flooding. In the compartment 4 there is down-flooding to Deck 1 through the staircase. Floodwater is spread fast on Deck 2 in the compartment 3 due to the open fire doors. As the void in the compartment 1 is filled up, the draft is increased and the ship
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P. Ruponen / Ocean Engineering 78 (2014) 35–44
Fig. 16. Illustration of the flooded compartments and the modeled openings for the third case study; the staircase that allows down-flooding to the lower deck is highlighted.
0.0 -0.5
roll angle [°]
-1.0 -1.5 -2.0 -2.5 -3.0 Δt = 0.2 s Δt = 2.0 s Δt = 4.0 s Δt = 1.0 s .. 4.0 s
-3.5 -4.0 -4.5
0
2
4
6
8
10
12
time [min] Fig. 18. Development of heel in the third case study.
volume of floodwater [m3]
3000 2500 2000 1500 1000 Δt = 0.2 s Δt = 2.0 s Δt = 4.0 s Δt = 1.0 s .. 4.0 s
500 0
0
2
4
6
8
10
12
time [min] Fig. 17. Final floating position and the status of the doors for the third case study.
Fig. 19. Development of the total mass of floodwater in the third case study.
P. Ruponen / Ocean Engineering 78 (2014) 35–44
time step is acceptable and this is retained until the final equilibrium is reached at 52 min. The times when the two closed doors collapse and the time of the maximum roll angle with different time steps are presented in Table 3. The relative computation times and the maximum difference in these three flooding events are shown in Fig. 21. The use of an adaptive time step provides as accurate results as a constant time step of 1.0 s but the computation time is reduced by 65%. With a modern laptop this means that the computation time is reduced from 84 s to 30 s.
5. Conclusions Use of an adaptive time step enables capturing the details of progressive flooding more accurately without significant increase in the computation time. In the presented damage case in a large cruise ship the computation time was decreased by 65% without any notable differences in the results. The applied pressurecorrection method for calculation of progressive flooding is numerically very stable due to the implicit scheme. Thus reasonable results can be achieved even with long time steps. It is believed that the adaptive time step has even more significant effects if explicit time integration is used in the flooding simulation algorithm. The developed method of adaptive time step can be used to increase the calculation accuracy, especially in the early stages of flooding, and during the possible short phases of fast flooding that may be encountered later during the progressive flooding when closed doors can collapse. The benefits of adaptive time-stepping are evident for damage cases involving long stages of slow progressive flooding. On the other hand, the approach is not that
25%
5.0
relative computation time
4.0 time step [s]
100
max difference in flooding events
3.0
2.0 door on Deck 2 collapses 1.0
20%
80
15%
60
10%
40
5%
20
max difference [s]
trims aftwards. Thus the damage opening is further submerged and the flow rates slowly increase. The closed fire door on Deck 1 from the staircase in the compartment 3 collapses under the floodwater pressure. After a while also the fire door from the same staircase to the elevator lobby on Deck 2 collapses, allowing faster equalization of the asymmetric flooding in the compartment no. 4. The converter rooms and the lift trunks are flooded through the leaking doors but these doors do not collapse since the pressure heads remain small. After about 52 min a stable final equilibrium is reached. The time histories of the heeling angle and the total volume of floodwater with different time steps are presented in Figs. 18 and 19, respectively. The use of an adaptive time step provides almost exactly the same results as in the reference case with a constant time step of 0.2 s. With a constant time step of 2.0 s the flooding is slightly faster during the phase of transient heeling. With a time step of 4.0 s the difference is very significant and the transient heeling angle is underestimated by nearly 11. The transient flooding excites the roll motion and a time step of 4.0 s is too long to accurately model this. As a result, there is notable error in the pressure heads on the openings and the inaccuracies in the floating position are reflected in the calculation of the flooding process. After 12 min the progressive flooding is very slow and there are no notable differences in the results. The adaptive time step is shown in Fig. 20 as a function of time. The collapse of the door on Deck 1 results in fast flooding and the step is shortened. Thereafter, the flooding rates slow down, and as soon as the roll velocity is small enough the maximum step of 4.0 s is used. At about 4 min the roll velocity increases above the applied limit of 0.01º/s, and the time step is shortened to the default value of 2.0 s. At about 9 min the roll velocity allows longer time step, but the maximum water level velocity is slightly above the limit and the default time step is restored. At 10 min the longer
43
door on Deck 1 collapses 0.0
0
2
4
6
8
10
0%
12
1.0 s
2.0 s
4.0 s
1.0...4.0 s
0
time step
time [min] Fig. 20. Adaptive time step in the third case study; the shaded areas mark the time spans when the time step adjustment was limited by fast roll motion.
Fig. 21. Comparison of computation times for the third case study with different time steps against the reference case with Δt ¼0.2 s.
Table 3 Comparison of flooding events and difference to the reference results with different time step for the third case study. Flooding event
Door on Deck 1 collapses Maximum roll angle Door on Deck 2 collapses
Time step 0.2 s
1.0 s
2.0 s
4.0 s
1.0–4.0 s
Reference
Time
Diff
Time
Diff
Time
Diff
Time
Diff
17.6 365.0 423.0
19.0 355.0 413.0
1.4 10.0 10.0
22.0 348.0 410.0
4.4 17.0 13.0
24.0 288.0 348.0
6.4 77.0 75.0
20.0 355.0 417.0
2.4 10.0 6.0
44
P. Ruponen / Ocean Engineering 78 (2014) 35–44
feasible for simulation of flooding and damaged ship motions in waves. Already in the previous study, Ruponen (2009), it was shown that the increase of the time step will also increase the required number of iterations for converged solution with the pressurecorrection algorithm. This is logical since the changes in the flooding condition are larger within a longer time step. Thus the benefit of a longer time step is partly lost due to the increased number of iterations. A longer time step will also inevitably mean increased numerical error. Consequently, for damage cases with very slow progressive flooding through leaking doors, the adaptive time step is likely not increasing the accuracy since the stages with the longest allowed time step cover most of the time to flood. The presented algorithm for adaptive time-stepping is a general one and it can be extended to include even longer time step in certain conditions. The applied criteria for changing the time step are based on a limited set of flooding scenarios, and some adjustment might be needed for more optimal results in totally different cases. The trend is towards performance based damage stability analyses, and all improvements in the computation times of flooding simulation are considered to be very important. The use of adaptive time-stepping will ensure that the initial stage of fast flooding is calculated accurately enough and the simulation of the long stage of progressive flooding does not take too much computation time. The benefits are notable especially with a large amount of damage scenarios to be analyzed. References Berrone, S., Marro, M., 2009. Space–time adaptive simulations for unsteady Navier– Stokes problems. Comput. Fluids 38, 1132–1144. Dankowski, H., 2012. An explicit progressive flooding simulation method. In: Proceedings of the 11th International Conference on Stability of Ships and Offshore Structures STAB 2012, Athens, Greece, pp. 411–423. Dankowski, H., 2013. A Fast and Explicit Method for Simulating Flooding and Sinkage Scenarios of Ships (Doctoral Dissertation). Technischen Universität Hamburg-Harburg, Hamburg, Germany, p. 107 Gao, Z., Vassalos, D., Gao, Q., 2010. Numerical simulation of water flooding into a damaged vessel0 s compartment by volume of fluid method. J. Ocean Eng. 37, 1428–1442.
IMO, 2011. SLF54/INF.8/Rev. 1, Modelling of leaking and collapsing of closed nonwatertight doors, submitted by Finland, 28 October, 2011. Jasionowski, A., 2001. An Integrated Approach to Damage Ship Survivability Assessment (Ph.D. thesis). University of Strathclyde Luhmann, H., 2009. Concept Ship Design B, FLOODSTAND deliverable D1.1b. Available from: 〈http://floodstand.aalto.fi〉. Ruponen, P., 2006. Pressure-correction method for simulation of progressive flooding and internal airflows. Ship Technol. Res.—Schiffstechnik 53, 63–73. Ruponen, P., 2007. Progressive flooding of a damaged passenger ship, dissertation for the degree of doctor of science in technology. TKK Dissertations 94, 124. Ruponen, P., 2009. On the application of pressure-correction method for simulation of progressive flooding. In: Proceedings of the 10th International Conference on Stability of Ships and Ocean Vehicles STAB 2009, St. Petersburg, Russia, 22–26 June, 2009, pp. 271–279. Ruponen, P., Sundell, T., Larmela, M., 2007. Validation of a simulation method for progressive flooding. Int. Shipbuild. Prog. 54 (4), 305–321. Ruponen, P., Kurvinen, P., Saisto, I., Harras, J., 2010. Experimental and numerical study on progressive flooding in full-scale, transactions of royal institute of naval architects. Int. J. Marit. Eng., A-197–A-207 Ruponen, P., Larmela, M., Pennanen, P., 2012. Flooding prediction onboard a damaged ship. In: Proceedings of the 11th International Conference on Stability of Ships and Ocean Vehicles STAB2012, Athens, Greece, 23–28 June, 2012, pp. 391–400. Ruponen, P., Kurvinen, P., Saisto, I., Harras, J., 2013. Air compression in a flooded tank of a damaged ship. J. Ocean Eng. 57, 64–71. Santos, T.A., Guides Soares, C., 2009. Numerical assessment of factors affecting the survivability of damaged ro-ro ships in wave. J. Ocean Eng. 36 (11), 797–809 (August 2009). Schreuder, M., Hogström, P., Ringsberg, J.W., Johnson, E., Janson, C.-E., 2011. A method for assessment of the survival time of a ship damaged by collision. J. Ship Res. 55 (2), 86–99. Spanos, D., Papanikolaou, A., 2001. On the stability of fishing vessels with trapped water on Deck. Ship Technol. Res.—Schiffstechnik 48, 124–133. Tabri, L., Varsta, P., Matusiak, J., 2010. Numerical and experimental motion simulations of nonsymmetric ship collisions. J. Mar. Sci. Technol. 15, 87–101. Turek, S., 1999. Efficient Solvers for Incompressible Flow Problems—An Algorithmic and Computational Approach, 1999. Springer Verlag, Berlin Heidelberg p. 352. van0 t Veer, R., de Kat, J.O., 2000. Experimental and numerical investigation on progressive flooding and sloshing in complex compartment geometries. In: Proceedings of the 7th International Conference on Stability of Ships and Ocean Vehicles STAB 2000, Launceston, Tasmania, Australia, 7–11 February, 2000, pp. 305–321. Volker, J., Rang, J., 2010. Adaptive time step control for the incompressible Navier– Stokes equations. Comput. Methods Appl. Mech. Eng. 199 (2010), 514–524. Ypma, E., Turner, T., 2010 An approach to the validation of ship flooding simulation models. In: Proceedings of the 11th International Ship Stability Workshop, Wageningen, The Netherlands.
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Adaptive time step in simulation of progressive flooding Pekka Ruponen n Napa Ltd., P.O. Box 470, FI-00181 Helsinki, Finland
art ic l e i nf o
a b s t r a c t
Article history: Received 28 August 2013 Accepted 29 December 2013 Available online 18 January 2014
Time accurate simulation of progressive flooding inside a ship often requires a short time step. However, after the initial phases, as the flooding progresses to undamaged compartments, the flow rates decrease and much longer time step could be used. Yet the collapsing of non-watertight structures may cause additional phases of fast flooding, where a shorter time step is required. In this paper the use of an adaptive time step in flooding simulation is discussed, and a new approach to this problem is presented. The time step is automatically adjusted during the simulation on the basis of the maximum absolute velocity of the free surface levels in the flooded compartments. The method is tested with small case studies, presenting typical flooding conditions in passenger ships. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Damage stability Flooding simulation Adaptive time step Progressive flooding
1. Introduction Progressive flooding inside a damaged passenger ship can be a slow process, especially if there are several closed doors that will only leak under a moderate pressure head of the floodwater. However, in the beginning of flooding the flow rates are typically very large. This is evident in the results of previous studies on progressive flooding, such as van0 t Veer and de Kat (2000) and Ruponen et al. (2010). During the past decades, several time-domain flooding simulation tools have been developed, e.g. Spanos and Papanikolaou (2001), Jasionowski (2001), Ruponen (2006, 2007), Santos and Guides Soares (2009), Ypma and Turner (2010), Schreuder et al. (2011) and Dankowski (2012, 2013). These methods use Bernoulli0 s equation for calculation of the water flow through the openings. The general approach in these simulation tools is to use a constant time step. The possibility to adjust the time step during the simulation in order to achieve shorter computation times has not yet been studied thoroughly. In computational fluid dynamics (CFD) the use of an adaptive time step is more common, e.g. Turek (1999), Berrone and Marro (2009) and Volker and Rang (2010). The CFD methods have also been used for flooding analyses with time step size limitation based on the Courant number, Gao et al. (2010). However, due to very slow computation times, the CFD tools are not feasible for any practical design work and analyses. Therefore, the further development of simulation methods that are using Bernoulli0 s equation is considered to be very important. In general, the flooding of a ship can be divided into three separate stages that are illustrated in Fig. 1. First there is transient flooding as water enters the damaged compartments. Typically this causes transient heeling. After a while the flow rates decrease and water may progress to undamaged compartments through
openings or non-watertight structures. This stage is referred to as progressive flooding, and it can be very long. Eventually a steady state is achieved, unless the ship sinks or capsizes before that. The optimal time step that provides sufficient accuracy and efficient computations may vary during the simulation. During the transient flooding, a relatively short step is usually needed in order to capture the fast changes in both flooding and ship motions. Later, during the stage of progressive flooding, a longer step may be applied without sacrificing the accuracy. However, if a long time step is used, it is also necessary to check that the effective pressure heads on closed doors are not resulting in collapsing, since this may cause a short phase of fast flooding, at least locally. The maximum allowed value of the time step depends also on the motions of the ship. However, usually also heeling changes rather slowly during the phase of progressive flooding, at least in a moderate seaway. It is also noteworthy that the increase of the time step is not reasonable for simulation of damaged ship motions in high waves, where the rolling is significant and thus limits the applicable time step. In this paper a new approach to this problem is presented. The time step is automatically adjusted during the simulation on the basis of the maximum absolute velocity of the free surface levels in the flooded compartments. The adaptive time step method is tested with two simple scenarios that demonstrate the problems in simulation of progressive flooding. In addition, the method is applied for a realistic damage scenario with extensive progressive flooding inside a large passenger ship. Finally the pros and cons of adaptive time-stepping in flooding simulation are discussed. 2. Pressure-correction method for flooding simulation 2.1. Background
n
Tel.: þ 358 50 5766078; fax: þ 358 922813800. E-mail address: pekka.ruponen@napa.fi
0029-8018/$ - see front matter & 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.oceaneng.2013.12.014
The presented method on the use of adaptive time step in flooding simulation has been implemented in the pressure-correction method,
36
P. Ruponen / Ocean Engineering 78 (2014) 35–44
floodwater
free discharge
submerged flow
floodwater
ΔH
time
heeling
ΔH heeling transient
progressive flooding
u
steady state
u
Fig. 1. Schematic representation of different stages of flooding.
described in detail in Ruponen (2006, 2007). This simulation tool has been successfully validated against model tests, Ruponen et al. (2007), and full-scale data, Ruponen et al. (2010). The pressure-correction algorithm can also be used for solving air compression inside the flooded compartments, Ruponen (2007) and Ruponen et al. (2013), but for the sake of simplicity, in the present study it is assumed that air pressure is constant. However, it is noteworthy that simulation of air compression usually requires a shorter time step, Ruponen (2007) and Dankowski (2013). The applied simulation method is based on implicit time integration with a pressure-correction algorithm. This has proven to be an efficient and accurate approach for calculation of extensive progressive flooding to several compartments. In the following, the algorithm is briefly presented. The principles of the applied simulation method are also described, although the presented application of adaptive time step is valid for any method for the solution of the progressive flooding when Bernoulli0 s equation is used. 2.2. Governing equations At each time step the conservation of mass must be satisfied in each flooded room. The equation of continuity is Z Z ∂ρ dΩ ¼ ρv UdS ð1Þ Ω ∂t S where ρ is density, v is the velocity vector and S is the surface that bounds the control volume Ω. The normal vector of the surface points outwards from the control volume. The velocities in the openings are calculated by applying Bernoulli0 s equation for a streamline from point A that is in the middle of a flooded room to point B in the opening: Z B dp 1 2 1 þ ðuB u2A Þ þ gðhB hA Þ þ kL u2B ¼ 0 ð2Þ 2 2 A ρ where p is air pressure, u is flow velocity and h is the water height from the common reference level. All losses in the duct are represented by the non-dimensional pressure-loss coefficient kL. Consequently, by denoting ΔH ¼hB hA and assuming that uA ¼ 0 and constant air pressure, the volumetric flow through an opening with area dS is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dQ ¼ C d U 2g UΔH U dS ð3Þ where the semi-empirical discharge coefficient is 1 C d ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 1 þ kL
ð4Þ
The effective pressure head for free discharge and fully submerged flow are illustrated in Fig. 2. For large openings the flow rate Q is integrated over the submerged area of the opening, and the flow condition may contain both free discharge and submerged flow. It is assumed that the water levels in the flooded compartments are parallel to the sea level, and thus the possible sloshing effects are ignored. However, it is believed that the presented method for
Fig. 2. Effective pressure head ΔH for different flow conditions.
adaptive time step can be extended to cover also the simplified methods for sloshing. The free surface area for each flooded room is calculated from the geometry for each time step with the floating position of the ship taken into account. During the pressure-correction iteration the free surface areas are assumed to be constant.
2.3. Initial and boundary conditions The loading condition of the ship before the damage defines the initial condition for the flooding simulation. The applied method does not handle mixtures of different liquids but loaded water in damaged tanks is treated similarly to floodwater, and thus for example draining of a swimming pool can be simulated. A collision with another ship obviously affects the ship motions, e.g. Tabri et al. (2010). However, it is a widely used simplification in damage stability calculations to assume that the collision or grounding force does not affect the ship motions or transient flooding. Consequently, the flooding starts with the ship floating freely with a damage opening in the hull surface. The surrounding sea is treated as a ghost cell that sets the boundary condition for flooding. The hydrostatic pressure for sea level is calculated on the basis of the floating position for each time step.
2.4. Time discretization In the pressure-correction method the simulation within a time step is iterative, based on the linearized Bernoulli0 s equation. The algorithm iterates the pressures in the flooded rooms until both Bernoulli0 s equation for each opening and continuity for each room is satisfied with sufficient accuracy. After that, the floating position of the ship is calculated on the basis of the distribution of floodwater in the compartments, accounting the free surface effect on all flooded rooms. In the simulation algorithm the instantaneous free surface area Sfs is assumed to be constant within the time step, and the time derivative of the volume of water in a flooded room can be presented as dV w dH w ¼ Sf s dt dt
ð5Þ
The permeability μ is taken into account, so that Sf s ¼ μStot
ð6Þ
where Stot is the total area of the cross-section of the room at the floodwater level.
P. Ruponen / Ocean Engineering 78 (2014) 35–44
37
3.2. Time step change indicator The new approach is somewhat similar to the one presented by Berrone and Marro (2009) for unsteady Navier–Stokes problems. In the case of progressive flooding the time error indicator η is considered to depend on the maximum absolute time derivative of the water height in the flooded rooms
water height
door collapses
net inflow starts again
_ w;i jÞ η ¼ maxðjH
flooding starts
ð10Þ
The default time step Δt0 is considered to be suitable for the beginning of flooding, and therefore, the indicator η is normalized by the maximum absolute time derivative of water level heights in the beginning of flooding
s ¼ maxðjH_ w;i ðt ¼ 0ÞjÞ
time
ð11Þ
Fig. 3. Examples of flooding conditions where implicit Euler method must be used.
The default time step is acceptable if the following double inequality is satisfied:
Normally the time derivatives for the pressure-correction algorithm are calculated by using the three-level implicit method
η 1 αr r s α
nþ1
_ w 3H w H
4H nw þ H nw 1 2Δt
ð7Þ
However, at the steps, where there is potential discontinuity in the flooding rate, the implicit Euler method is used nþ1
_ w Hw H
H nw Δt
ð8Þ
The discontinuities can occur e.g. when flooding to a new compartment starts or a closed door collapses. These are illustrated in Fig. 3.
3. Adaptive time step 3.1. Background
where α is a given parameter in the range (0,1). If the indicator exceeds the upper bound a shorter time step is needed. A longer time step can be used if the indicator is smaller than the lower bound. In the presented case studies the value α¼0.5 was used. The thresholds for changing the time step, α and 1/α, can also be independent. Moreover, even longer time step might be feasible when η/s o oα. As mentioned previously, when the time step is adjusted, the time derivatives need to be calculated with the implicit backward Euler method instead of the normal three time level method. Thus it is not favorable to adjust the time step very frequently in order to avoid increased error in the calculation of the floating position. Consequently, the increase of time step is allowed only after a short period tbuffer since the previous change. In the presented case studies this period is 10-times the initial time step. 3.3. Effects on the calculation of ship motions
The pressure-correction method is iterative, and thus there is no need to limit the time step in order to prevent the change of flooding direction in the openings during the time step. For explicit time integration this kind of limitation is necessary, especially with longer time steps, as presented in Ruponen et al. (2012). In principle the adjustment of the time step can be done with two different approaches: either gradually changing or by a sudden larger change. The latter approach is used in this study since it keeps the equations in a simpler format, especially in the calculation of the time derivatives. The simplest way is to change the time step suddenly when certain conditions are met. In practice this means that the new time step is Δt ¼ C Δt Δt 0
ð12Þ
ð9Þ
where Δt0 is the initial time step and the coefficient CΔt depends on the flooding process at the time. In the previous study on the use of adaptive time step in flooding simulation, Ruponen (2009), the time step was increased when the maximum absolute net flow in the flooded compartments was below a predefined limit value. However, this approach does not take into account the geometry of the compartments. Even a small net flow to a room with a small free surface area, such as a staircase or a lift trunk, can cause rather rapid changes in the water level. Thus in the present study the criterion for adjusting the time step is based on the maximum absolute time derivative of the water height.
The length of the time step affects also the calculation of the ship motions. If the motions are considered to be fully quasistationary, the time step can be selected purely on the basis of the flooding process. However, even a simple approach with dynamic roll motion and linear model for roll damping restrict the use of adaptive time step. A simple approach is to allow the increase of the time step only when the roll velocity is small. In the present study this is taken into account by setting a limit of 0.011/s for the moving absolute maximum of the roll velocity over the past 20 time steps. Obviously, if other motions, such as trim, are rapid, they have to be included in the limitations to the maximum allowed time step. 3.4. Algorithm Unlike in Navier–Stokes problems, for progressive flooding the adjustments can be done abruptly. Thus the similar restrictions to the size of the time step change as in Berrone and Marro (2009) are not necessary. The initial time step Δt0 is considered to be suitable for the beginning flooding, i.e. it is mainly dependent on the size of the damage openings and phenomena during the transient flooding. However, a shorter time step may also be needed later during the stage of progressive flooding as there can be short phases of fast internal flooding after collapsing of closed doors. The applied algorithm for adaptive time step is described in the following.
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P. Ruponen / Ocean Engineering 78 (2014) 35–44
_ 4f _ if (jfj crit ) then if (Δt 4 Δtmin ) then if (Z=s 41=a) then Δt ¼ Δtmin else if (Δt 4 Δt0 ) then Δt ¼ Δt0 else no change end if else no change end if else if (t 4 tadj þ tbuffer ) then if (Z=s 4 1=a) then if (Δt 4 Δtmin ) then Δt ¼ Δtmin else no change end if else if (Z=s o a) then if (Dt o Δt0 ) then Δt ¼ Δt0 else if (Dt o Dtmax ) then Δt ¼ Δtmax else no change end if else if (Δt o Δt0 or Δt 4Δt0 ) then Δt ¼ Δt0 else no change end if end if else no change end if
hydrostatic pressure exceeds the critical threshold of 2.9 m. The height of the rooms is 3.0 m. Thus the flooding process contains several subsequent phases of both fast and slow flooding rates. The floating position is fixed. The modeled openings and the flooded rooms are illustrated in Fig. 4. The area of the damage opening is 0.3 m2 and the area of the doors is 2.0 m2. The sea level height is 0.25 m above the room
4.5 4.0
water height [m]
3.5 3.0 2.5 2.0 1.5 R1 R2 R3 R4
1.0 0.5 0.0
0
2
4
6
8
10
12
14
16
18
time [min]
0.18
4. Case studies 4.1. Flooding through successive compartments The developed algorithm is tested with small case studies. The first one involves flooding though successive rooms that are connected by closed doors. The doors will collapse after the
floodwater max(dHw/dt)
800
0.16
600
0.12
500
0.10
400 300
door R3-R4
0.14
door R2-R3
700
door R1-R2
total volume of floodwater [m3]
900
0.08 0.06
200
0.04
100
0.02
0
0
2
4
6
8
10
12
14
16
0.00 18
max absolute water level velocity [m/s]
Fig. 5. Reference results for the water level heights in the first case study.
time [min] Fig. 6. Reference results for total volume of floodwater and maximum absolute free surface velocity in the first case study.
Fig. 4. Openings and flooded rooms for the first test case.
P. Ruponen / Ocean Engineering 78 (2014) 35–44
4.2. Down-flooding through a staircase Progressive flooding may involve rooms of different sizes. The second case study concentrates on down-flooding through a staircase with small free surface area compared to the other flooded rooms. Sea level height is 0.25 m below the top of the upper deck and the height of both decks is 2.5 m. In the beginning, the flooding trough the damage opening is fast. Eventually the leakage threshold of the closed door is reached and floodwater progresses to the staircase. The flooding rate is small but the water level rises fast since the free surface area in the staircase is small. The final phases involve leakage and collapsing of the closed door from the staircase to the lower deck and both the flooding rate and the time derivative of the hydrostatic pressure height are small. The rooms and openings are shown in Fig. 9 and the flooding phases are illustrated in Fig. 10. Similarly to the first case, the floating position is fixed. The reference results are calculated with a constant time step Δt ¼0.2 s. The water level heights in the rooms are shown in Fig. 11 and the maximum absolute net flow and the maximum absolute time derivative of the free surface are presented in Fig. 12 as functions of time. At about 2.5 min the staircase is flooded. The water level in this room with relatively small free surface area rises very rapidly. At about 4 min the door from the staircase to the lower deck collapses, causing a sudden drop in the water level in the staircase due to the fast flooding through the collapsed door. Soon thereafter the water levels on the lower deck and in the staircase rise slowly at the same pace. Eventually at 45 min the room on the lower deck is filled up and the water level in the staircase starts to rise rapidly until the final equilibrium is reached at about 47 min. Most of the time the flooding is slow but locally there are rapid changes of the water level height in the staircase. Thus accurate simulation requires a short time step during these phases.
time step [s]
R2-R3
R3-R4
relative computation time
R1-R2
2.5 2.0 1.5 1.0 0.5 0.0
10
10%
3.0
0
2
4
6
8 10 time [min]
12
14
16
8%
8
6%
6
4%
4
2%
2
0%
18
max difference in flooding events
0.5 s
1.0 s
2.0 s
0.5...2.0 s
max difference [s]
height (3.25 m above door bottom). Consequently, flooding is significantly reduced before the collapsing door enables progressive flooding to another room. The reference data is obtained by using a very short constant time step of 0.1 s. The water level heights in the rooms are shown in Fig. 5 and the total volume of floodwater and the maximum absolute free surface velocity are presented in Fig. 6. The collapse of each closed door causes a short phase of very fast flooding that lasts less than a minute. Thereafter, flooding is slow until the next door collapses. The developed algorithm for adaptive time step is tested with this flooding scenario by using an initial time step of 1.0 s. Thus the time step is adjusted between 0.5 s and 2.0 s. The time history for the time step is presented in Fig. 7. The collapse of each closed door causes rapid flooding and the time step is shortened to the minimum value of 0.5 s. As the flooding then slows down again, the time step is increased back to 1.0 s and after a while to the maximum value of 2.0 s. Simulations with different constant time steps and with an adaptive time step are compared to the reference results, Table 1. With an increased constant time step the times when the doors collapse become more inaccurate. When the time step is adjusted according to the presented algorithm between 0.5 s and 2.0 s, the accuracy of the results is nearly as good as with a constant step of 1.0 s. Yet the computation time is almost as short as with a constant step of 2.0 s, Fig. 8. It is noteworthy that even though adaptive time step improves the accuracy, a longer time step will inevitably mean that the exact event of the collapse is captured more inaccurately. However, this error is of the magnitude of one time step, and thus it is not significant since the applied limits for collapsing pressure heads are always rather approximate.
39
0
time step
Fig. 7. Adaptive time step in the first case study, showing the times when the closed doors collapse.
Fig. 8. Comparison of computation times for the first case study with different time steps against the reference case with Δt¼ 0.1 s.
Table 1 Comparison of flooding events and difference to the reference results with different time steps for the first case study. Flooding event
Door R1-R2 collapses Door R2-R3 collapses Door R3-R4 collapses
Time step 0.1 s
0.5 s
1.0 s
2.0 s
0.5–2.0 s
Reference
Time
Diff
Time
Diff
Time
Diff
Time
Diff
177.4 405.6 664.0
177.5 405.2 663.0
0.1 0.4 1.0
179.0 407.5 664.5
1.6 1.9 0.5
180.0 413.0 672.0
2.6 7.4 8.0
180.0 407.5 665.5
2.6 1.9 1.5
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P. Ruponen / Ocean Engineering 78 (2014) 35–44
2.0
I
III
0.20
1.5
0.15 1.0
0.5
lower deck filled up
stairs flooded
max absolute net flow [m3/s]
door collapses
II
0.25
max(Q ) max(dH /dt)
0.10
0.05
IV 0.0
0
10
Fig. 10. Flooding phases of the second case study.
30
0.00 50
40
time [min] Fig. 12. Maximum absolute net flow rate and time derivative of hydrostatic pressure head as functions of time for the second case study with a constant time step Δt¼ 0.2 s.
4.0 3.5
3.0 door collapses
3.0
lower deck filled up
2.5
2.5 tween deck
2.0
time step [s]
water height [m]
20
max absolute water level velocity [m/s]
Fig. 9. Flooded compartments and openings.
1.5 1.0
Deck 1 Deck 2 Stairs
0.5 0.0
2.0 1.5 1.0 0.5
0
10
20 30 time [min]
40
50
Fig. 11. Reference results for the water level heights in the second case study; the vertical lines show the transitions between the phases as defined in Fig. 10.
0.0
0
10
20
30
40
50
time [min] Fig. 13. Adaptive time step in the second case study, showing the times when the closed door collapses and the lower deck is filled up.
Applying a longer time step causes inaccuracy in the time when the undamaged rooms are flooded and the closed door collapses. Thus for this kind a flooding scenarios the automatically adjusted time step is a very suitable solution for improving the computing efficiency without significant sacrifice in the accuracy. The applied time step is presented in Fig. 13 as a function of time. The times for different flooding events and the corresponding difference to the reference values with various time steps are presented in Table 2, and the relative computation times are shown in Fig. 14. The adaptive time step provides more accurate
results than a constant time step, although the differences are very small. However, it should be noted that mainly due to the implicit time integration with the pressure-correction method, the results are fairly accurate even with longer constant time step. The effect of the time step on the computation time is very similar to the first case study, but the maximum difference in the occurrence of the flooding events is smaller, even though the total time-to-flood is longer.
P. Ruponen / Ocean Engineering 78 (2014) 35–44
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Table 2 Comparison of flooding events and difference to the reference results with different time step for the second case study. Flooding event
Time step
Staircase flooded Lower deck flooded Door to lower deck collapses
0.2 s
0.5 s
Reference
Time
Diff
Time
Diff
Time
Diff
Time
Diff
150.6 176.0 215.8
150.0 175.5 215.0
0.6 0.5 0.8
151.0 177.0 216.0
0.4 1.0 0.2
152.0 178.0 218.0
1.4 2.0 2.2
151.0 176.0 216.0
0.4 0.0 0.2
25%
1.0 s
2.0 s
0.5–2.0 s
10
20%
8
15%
6
10%
4
5%
2
0%
max difference [s]
relative computation time
max difference in flooding events
0 0.5 s
1.0 s
2.0 s
0.5...2.0 s
time step
Deck 2
Fig. 14. Comparison of computation times for the second case study with different time steps against the reference case with.
damage
WT comp.
1
2
3
4
4.3. Progressive flooding of a cruise ship The last case study concentrates on a more realistic flooding case. The 63,000 GT passenger ship design, Luhmann (2009), was developed in the FLOODSTAND project. The damage scenario is a long and narrow gash in the starboard side of the aft part of the ship. This kind of damage could result e.g. from a collision to a pier. Total of three watertight compartments are flooded. The arrangement of the flooded compartments and the openings are shown in Figs. 15 and 16. It is assumed that all watertight doors are closed and fully tight. Most of the fire doors are considered to be open. However, five fire doors and the lift doors are initially closed, see Fig. 16, and these can leak and collapse under the pressure of floodwater. The applied leaking and collapsing model is based on full-scale tests in the FLOODSTAND project, SLF 54/INF.8/Rev.1, IMO (2011). The ship is considered to have an initial metacentric height of GM0 ¼ 2.5 m. The trim and the draft are considered to be quasistationary, but the uncoupled roll motion ϕ(t) is calculated from the following equation: _ þ M st ðϕ; tÞ þ M ext ðϕ; V w ; tÞ ¼ 0 € þBxx ϕ ðI xx þAxx Þϕ
ð13Þ
where Ixx and Axx are the mass moment of inertia for roll and added mass, Mst is the static righting moment and Mext is the heeling moment caused by the floodwater Vw. The roll damping is assumed to be linear, so that Bxx ¼ 2ξωϕ ðI xx þ Axx Þ
ð14Þ
The natural roll frequency ωϕ is based on approximated natural period of Tϕ ¼20.0 s and the linear roll damping coefficient is ξ ¼0.05. The total mass moment of inertia (Ixx þ Axx) is obtained
Deck 1 Fig. 15. Cruise ship design, Luhmann, 2009, and general arrangement of the flooded compartments.
from the equation for the natural roll period Tϕ ¼
2π ¼ 2π ωϕ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðI xx þ Axx Þ ΔU GM 0
ð15Þ
where Δ is the initial displacement. The damage is located on Deck 2, just below the waterline. The breach has a length of 22.4 m and the total area is 22.95 m2. The damaged rooms are the void in the compartment 2, the workshop and the hotel stores in the compartments 3 and 4. The stern thruster room and the two rooms for electric motors on Deck 1 remain dry due to the closed WT doors. The final floating position of the ship with the flooded compartments and status of the doors is shown in Fig. 17. Initially, the flow rates are fast but the closed fire doors restrict progressive flooding. In the compartment 4 there is down-flooding to Deck 1 through the staircase. Floodwater is spread fast on Deck 2 in the compartment 3 due to the open fire doors. As the void in the compartment 1 is filled up, the draft is increased and the ship
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P. Ruponen / Ocean Engineering 78 (2014) 35–44
Fig. 16. Illustration of the flooded compartments and the modeled openings for the third case study; the staircase that allows down-flooding to the lower deck is highlighted.
0.0 -0.5
roll angle [°]
-1.0 -1.5 -2.0 -2.5 -3.0 Δt = 0.2 s Δt = 2.0 s Δt = 4.0 s Δt = 1.0 s .. 4.0 s
-3.5 -4.0 -4.5
0
2
4
6
8
10
12
time [min] Fig. 18. Development of heel in the third case study.
volume of floodwater [m3]
3000 2500 2000 1500 1000 Δt = 0.2 s Δt = 2.0 s Δt = 4.0 s Δt = 1.0 s .. 4.0 s
500 0
0
2
4
6
8
10
12
time [min] Fig. 17. Final floating position and the status of the doors for the third case study.
Fig. 19. Development of the total mass of floodwater in the third case study.
P. Ruponen / Ocean Engineering 78 (2014) 35–44
time step is acceptable and this is retained until the final equilibrium is reached at 52 min. The times when the two closed doors collapse and the time of the maximum roll angle with different time steps are presented in Table 3. The relative computation times and the maximum difference in these three flooding events are shown in Fig. 21. The use of an adaptive time step provides as accurate results as a constant time step of 1.0 s but the computation time is reduced by 65%. With a modern laptop this means that the computation time is reduced from 84 s to 30 s.
5. Conclusions Use of an adaptive time step enables capturing the details of progressive flooding more accurately without significant increase in the computation time. In the presented damage case in a large cruise ship the computation time was decreased by 65% without any notable differences in the results. The applied pressurecorrection method for calculation of progressive flooding is numerically very stable due to the implicit scheme. Thus reasonable results can be achieved even with long time steps. It is believed that the adaptive time step has even more significant effects if explicit time integration is used in the flooding simulation algorithm. The developed method of adaptive time step can be used to increase the calculation accuracy, especially in the early stages of flooding, and during the possible short phases of fast flooding that may be encountered later during the progressive flooding when closed doors can collapse. The benefits of adaptive time-stepping are evident for damage cases involving long stages of slow progressive flooding. On the other hand, the approach is not that
25%
5.0
relative computation time
4.0 time step [s]
100
max difference in flooding events
3.0
2.0 door on Deck 2 collapses 1.0
20%
80
15%
60
10%
40
5%
20
max difference [s]
trims aftwards. Thus the damage opening is further submerged and the flow rates slowly increase. The closed fire door on Deck 1 from the staircase in the compartment 3 collapses under the floodwater pressure. After a while also the fire door from the same staircase to the elevator lobby on Deck 2 collapses, allowing faster equalization of the asymmetric flooding in the compartment no. 4. The converter rooms and the lift trunks are flooded through the leaking doors but these doors do not collapse since the pressure heads remain small. After about 52 min a stable final equilibrium is reached. The time histories of the heeling angle and the total volume of floodwater with different time steps are presented in Figs. 18 and 19, respectively. The use of an adaptive time step provides almost exactly the same results as in the reference case with a constant time step of 0.2 s. With a constant time step of 2.0 s the flooding is slightly faster during the phase of transient heeling. With a time step of 4.0 s the difference is very significant and the transient heeling angle is underestimated by nearly 11. The transient flooding excites the roll motion and a time step of 4.0 s is too long to accurately model this. As a result, there is notable error in the pressure heads on the openings and the inaccuracies in the floating position are reflected in the calculation of the flooding process. After 12 min the progressive flooding is very slow and there are no notable differences in the results. The adaptive time step is shown in Fig. 20 as a function of time. The collapse of the door on Deck 1 results in fast flooding and the step is shortened. Thereafter, the flooding rates slow down, and as soon as the roll velocity is small enough the maximum step of 4.0 s is used. At about 4 min the roll velocity increases above the applied limit of 0.01º/s, and the time step is shortened to the default value of 2.0 s. At about 9 min the roll velocity allows longer time step, but the maximum water level velocity is slightly above the limit and the default time step is restored. At 10 min the longer
43
door on Deck 1 collapses 0.0
0
2
4
6
8
10
0%
12
1.0 s
2.0 s
4.0 s
1.0...4.0 s
0
time step
time [min] Fig. 20. Adaptive time step in the third case study; the shaded areas mark the time spans when the time step adjustment was limited by fast roll motion.
Fig. 21. Comparison of computation times for the third case study with different time steps against the reference case with Δt ¼0.2 s.
Table 3 Comparison of flooding events and difference to the reference results with different time step for the third case study. Flooding event
Door on Deck 1 collapses Maximum roll angle Door on Deck 2 collapses
Time step 0.2 s
1.0 s
2.0 s
4.0 s
1.0–4.0 s
Reference
Time
Diff
Time
Diff
Time
Diff
Time
Diff
17.6 365.0 423.0
19.0 355.0 413.0
1.4 10.0 10.0
22.0 348.0 410.0
4.4 17.0 13.0
24.0 288.0 348.0
6.4 77.0 75.0
20.0 355.0 417.0
2.4 10.0 6.0
44
P. Ruponen / Ocean Engineering 78 (2014) 35–44
feasible for simulation of flooding and damaged ship motions in waves. Already in the previous study, Ruponen (2009), it was shown that the increase of the time step will also increase the required number of iterations for converged solution with the pressurecorrection algorithm. This is logical since the changes in the flooding condition are larger within a longer time step. Thus the benefit of a longer time step is partly lost due to the increased number of iterations. A longer time step will also inevitably mean increased numerical error. Consequently, for damage cases with very slow progressive flooding through leaking doors, the adaptive time step is likely not increasing the accuracy since the stages with the longest allowed time step cover most of the time to flood. The presented algorithm for adaptive time-stepping is a general one and it can be extended to include even longer time step in certain conditions. The applied criteria for changing the time step are based on a limited set of flooding scenarios, and some adjustment might be needed for more optimal results in totally different cases. The trend is towards performance based damage stability analyses, and all improvements in the computation times of flooding simulation are considered to be very important. The use of adaptive time-stepping will ensure that the initial stage of fast flooding is calculated accurately enough and the simulation of the long stage of progressive flooding does not take too much computation time. The benefits are notable especially with a large amount of damage scenarios to be analyzed. References Berrone, S., Marro, M., 2009. Space–time adaptive simulations for unsteady Navier– Stokes problems. Comput. Fluids 38, 1132–1144. Dankowski, H., 2012. An explicit progressive flooding simulation method. In: Proceedings of the 11th International Conference on Stability of Ships and Offshore Structures STAB 2012, Athens, Greece, pp. 411–423. Dankowski, H., 2013. A Fast and Explicit Method for Simulating Flooding and Sinkage Scenarios of Ships (Doctoral Dissertation). Technischen Universität Hamburg-Harburg, Hamburg, Germany, p. 107 Gao, Z., Vassalos, D., Gao, Q., 2010. Numerical simulation of water flooding into a damaged vessel0 s compartment by volume of fluid method. J. Ocean Eng. 37, 1428–1442.
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