PIAS Manual  2025
Program for the Integral Approach of Shipdesign
Internal flooding in case of damage, through pipe lines and compartment connections
When a ship becomes damaged, the flooding need not be confined to the immediately ruptured compartments, but may also extend to other compartments due to the presence of pipelines, ducts or other forms of compartment connections. To this end, PIAS is equipped with a number of tools and mechanisms that will be discussed in this chapter.

Background from tools for ship-internal connections in PIAS

If a compartment is damaged in such a way that it is open to sea water then it will obviously be flooded, which can also extend further into the vessel through all kinds of connections between compartments. In stability regulations, the word progressive flooding is sometimes used for this, but we rather avoid this word because it suggests that the flooding continues until it is fatal, which of course is not necessarily the case. Obviously, this process can be modelled in PIAS and computed. There are two facilities for this purpose:

  • The first dates back to ±1990 and is called Complex intermediate stages of flooding. This works on the basis of non-uniform filling percentages per compartment, supplemented where necessary by virtual compartment connections. This allows to specify whether there was a connection between two compartments, however, it had no geometry (although it had an optional threshold height, called a ‘critical point’). There was also only a single connection possible between two compartments. In Define compartment connections, a table of such connections can be defined, which is used in damage case generation to co-generate such complex intermediate stages with. This Probdam function has been extended in 2018 and that will be the last modification to this ‘complex stages’ system. Although it will not be disappear, its further development has stopped, as it has been replaced by consecutive flooding.
  • The second system was introduced in 2023 and is called Consecutive flooding. It was developed based on the specification New inter-compartment flooding mechanism in PIAS, which was drafted in 2018, in collaboration with a number of key users op PIAS. Consecutuve Flooding works on the basis of the actual geometry of pipes and connections, and can also calculate flooding in time domain.

The choice between these two systems can be made in Config (or through Project Setup in the upper bar of the PIAS window), Please also refer for this setting to Calculation damage stability according to the method of, which also contains a list of PIAS features and functions which are disabled in combination with Consecutive Flooding.

This chapter discusses the following:

Flooding through ducts and pipes: Consecutive Flooding, after 2022

This system is discussed in the four sections below, viz.:

  1. The basic operation of the conventional intermediate stages of flooding method in consecutive flooding; With conventional intermediate stages of flooding ("Fractional").
  2. The basic operation of the time domain computation in consecutive flooding; Damage stability in time domain.
  3. The background of the definition of some pipeline properties in Layout, see Hydrodynamic parameters from pipes and piping systems.
  4. Finally, a short section where the different settings are summarized, in Summary of settings for Consecutive Flooding.

This order may appear to be a bit unnatural — because we should first define, before being able to use — but is nevertheless deliberately chosen this way. By the way, before doing any calculations, one will of course have to define the ducts and pipes. This is done integrated with bulkheads, decks and compartments, in Layout, as described in Pipe lines and piping systems.

With conventional intermediate stages of flooding ("Fractional")

This method was conceptualized given two facts:

  • Standard stability regulations apply the concept of “intermediates stages of floodings” of fixed percentages of flooding, i.e. 25%, 50%, 75% and 100%.
  • Not all compartments are always flooded with the same percentages, i.e. with small connections between compartments the flooding of the connected compartments may lag behind the flooding of the ruptured compartment.

In the elder “compartment connection” method of PIAS the latter was facilitated by so-called “complex stages of flooding”, which support individual percentages of flooding for different compartments. This offered full freedom, however at the cost of significant manual input labour. For the numerous damage cases of probabilistic damage stability this is not practical, so module Probdam offered a specific feature to generate these complex stages, where the binary concepts of “open” and “pipe” (as discussed in Damage cases generation including "progressive flooding") offered a flexibility sufficient for the majority of cases, but not for all cases. So, in the consecutive flooding system a novel subsystem for unequal intermediate stages of flooding has been created which is a) flexible, b) based on generation so does not require much user input, and c) works for all PIAS damage stability calculation modules.

This subsystem maintains the notion of “percentual stage of flooding”, because a) this is a fundamental concept in present damage stability regulations, b) therefore this concept is familiar to authorities and classification societies and c) the concept is easy to understand. In order to have a shorthand word for this concept it was labelled “fractional”, because essentially it fills compartments by ‘fractions’ of the final volume. So that fraction is the unit, which enables us to introduce an integer “delay” in the flooding of connected compartments. Assume, for the time being, that the percentages of flooding are 0, 25, 50, 75 and 100%, so one fraction corresponds with 25%. If we use a delay of zero (so, no delay), then the flooding of a connected compartment will obviously be the same as for the ruptured compartment:

Fraction ruptured compartmentFraction connected compartment
1 (=25%)1 (=25%)
2 (=50%)2 (=50%)
3 (=75%)3 (=75%)
4 (=100%)4 (=100%)


With a delay of 1, there will be a single fraction delay:

Fraction ruptured compartmentFraction connected compartment
1 (=25%)0 (=0%)
2 (=50%)1 (=25%)
3 (=75%)2 (=50%)
4 (=100%)3 (=75%)
4 (=100%)4 (=100%)

The last row is added because the filling should always end up with all flooded compartments filled to their final levels.


And with a delay≥4:

Fraction ruptured compartmentFraction connected compartment
1 (=25%)0 (=0%)
2 (=50%)0 (=0%)
3 (=75%)0 (=0%)
4 (=100%)0 (=0%)
4 (=100%)1 (=25%)
4 (=100%)2 (=50%)
4 (=100%)3 (=75%)
4 (=100%)4 (=100%)


This was an example of a single connection between two compartments. But also cascaded connections are supported by nature, take for instance this configuration of three compartments and two connections (with each a delay of 1):

flooding_frac1.png
Three serially connected compartments
Fraction comp1 Fraction comp2 Fraction comp3
1 0 0
2 1 0
3 2 1
4 3 2
4 4 3
4 4 4


Also more complicated topologies are allowed. The delay factors for the different paths might contradict, but that poses not a real dilemma, because the smallest fraction determines the actual flooding, which is a sound logical consequence of the underlying assumptions. For example this case:

flooding_frac2.png
Four connected compartments
Fraction comp1 Fraction comp2 Fraction comp3 Fraction comp4
1 0 0 0
2 1 1 1
3 2 2 2
4 3 3 3
4 4 4 4

Initially the fraction sequence for Comp3 appeared us to be 0,0,1,2,3,4 but through Comp2 and Comp4 the compartment fills up faster.
Delay factors are given for each pipe segment (which is a connection between two joints or compartments, without any branch inbetween, see Segment list)). That enables a pipe topology and delay factors such as:

flooding_frac3.png
Many connected compartments, with varying delay factors

Which will lead to eleven different stages of flooding (including the final stage).
That's the basic idea. And how realistic will modelling method be? Just as realistic as the whole presumption of fixed percentages of flooding, as imposed by all major damage stability rules. For added flexibility PIAS goes one step further, by allowing multiple (to a maximum of three) delay factors per pipe segment. In most cases a single delay factor will suffice, but even more complex scenarios than presented in this section can be modelled with combinations of factors.

Finally, a remark on the percentages of flooding as used in the examples of this section. As demontration, multiples of 25% have been used, but you will certainly be aware that in PIAS the number of intermediate stages of flooding is user-defined, as well as the percentage of each stage. So, where we have used multiples of 25% here, in reality for this percentage the user-defined flooding percentages will be applied.

Damage stability in time domain

A calculation in the time domain involves analysing, for a whole series of small time steps, how fluids flow through pipes and openings and how that effects ship's position and stability. That is essentially a model which is based on physics, and therefore needs much less explanation than the fractional method, which is a bit artificial. Nevertheless, some choices and assumptions have been applied here and there, which are discussed in Basis of damage stability in time domain.

Hydrodynamic parameters from pipes and piping systems

The piping geometry and connectivity can be defined in combination with the internal (i.e. compartment) geometry, with module Layout. This is discussed in Pipe lines and piping systems. In order to keep data of the same categories as much together as possible, the flow-related choices and parameters will be discussed in this section.

Fluid flow resistance factors

Frictional resistance from pipes lines

Frictional resistance through pipes is in the essence a complex issue. In practice, however, there are a number of practical methods and parameters in use, and in PIAS we have chosen to implement a selection:

  • The (cross-sectional) shape. Choice of round or square, the most common shapes.
  • The cross-sectional dimension. If round then diameter, if square then edge length. In meters, as commonly used in PIAS.
  • The (dimensionless) resistance coefficient. Three common methods are implemented in PIAS:
    • According to IMO resolution MSC.362(92), where the frictional resistance per meter length is 0.02 ÷ hydraulic diameter.
    • With a user-specified Darcy-Weisbach coefficient, where the frictional resistance per meter of length is that coefficient ÷ hydraulic diameter.
    • With a user-specified resistance coefficient per meter of length.

Fluid outlet energy loss

The resistance of fluid flowing through a pipe consists of two components. One is the frictional or possibly pressure resistance in the pipe, and the other one is due to the energy loss of the fluid outflow at the pipe outlet. The latter one simply follows from the fact that the fluid travels with some velocity through the pipe, and hence carries kinetic energy. After outflow (into a reservoir, or into the open air) that velocity vanishes, so the energy content also has gone. This is inevitable, also in a frictionless world. This phenomenon should be included — once, not twice. Unfortunately the different IMO regulations are not consistent in this respect. There are three IMO regulations involved, res. A266 (1973), MSC. 245(83) (2007) and MSC. 362(92) (2013). In A266 and 362 this outflow loss is included implicitly, while in 245 a energy loss factor should be included explicitly, in combination with frictional resistance coefficients of the piping configuration.

PIAS offers you this choice. It can be given in a user setting, in Layout, labelled ‘Processing of outlet losses’ (please see General piping settings), with binary choice ‘Explicitly user-defined by means of resistance coefficients’ or ‘Implicitly taken into account’. Please ascertain to set this switch in accordance with the resistance coefficients you assign to the pipe line components, and with this background in mind:

  • In A266 en 362 the resistance coefficient formula is \( 1 / \sqrt{ 1 + \sum{K} } \). This factor 1 in the denominator probably represents the pipe outlet energy loss. In PIAS this is implemented so that it is only included for pipe ends that exit into a compartment or into seawater, and where actual outflow occurs (i.e., not inflow).
  • In MSC.245 the formula reads \( 1 / \sqrt{\sum{K} } \), where the result may not be taken smaller than 1. To give the user maximum freedom, that limitation to 1 is intentionally not included in PIAS. The user himself must ensure that the resistance coefficients are specified conscientiously (and, incidentally, only applies the factor of 1 for pipe ends where actual outflow occurs).
  • If you find it confusing that multiple types of formulas are used in the various IMO rules, and that the background of this factor 1 is not even properly explained then you will find an ally in SARC.

Damage stability criteria to be applied

When assessing the damage stability with a cross flooding device, the question may arise which stability criteria to apply; those for final stage or those for intermediate stages of flooding? PIAS' consecutive flooding system has two mechanisms for this, one for the fractional intermediate stages and one for the time domain calculation.

Choice of stability criteria with the time domain method

In the essence, the application of intermediate or final criteria depends on the time span of the filling process. Some regulations (or accompanying explanatory notes) therefore contain a maximum time period until equalization. This implies that if the ship does not equalize within that time, the cross-flooding arrangement is deemed ineffective, so stability in the non-equalized condition is considerd to be a final stage, which should meet the final stage stability criterla. Well, the whole concept of equalization might apply to two simple interconnected tanks on either side of centerline, but for a slightly more complex tank arrangement, it is an abstraction that is not so easy to relate to reality. Nevertheless, the underlying idea is easy to transform into a universal algorithm:

  • If the ship comes to rest within the time allowed, then the last time step, which represents the final position, is tested against the final-stage criteria, and earlier time steps from which the damage stability is calculated to the intermediate-stage criteria.
  • If the ship does not come to rest within the time allowed, the damage stability calculations for all time steps are tested against the criteria for the final stage of flooding.

To this end, that maximum time allowed must be given, which can be done in the general settings for damage stability, see General settings damage stability.

Choice of stability criteria with the fractional method

Above, we have seen that with the time domain calculation, the criterion choice can be elegantly linked to an equalisation time, but for a calculation with conventional intermediate stages of flooding the time information is missing. To allow the user to influence this, PIAS has a facility to specify whether a pipe is large or small. These concepts ‘large’ and ‘small’ have no numerical relationship to the pipe size, but only to the choice of stability criteria to be applied, in this fashion:

  • Whith ‘large’ the cross flooding is assumed to be quick, so intermediate stages are thus checked against the (user-specified) criteria for intermediate stages of flooding. If those are not defined, then the criteria for final stage are used, as there is nothing else.
  • With ‘small’, the flooding process is considered to occur slowly, so that also the intermediate stages are tested against the stability criteria for the final stage.

The size, large or small, of a cross section can be specified as a general piping setting (see General piping settings) with exceptions per network where applicable.

Layered definition of flow-related parameters

For the purpose of damage stability calculations, many properties of pipe components can be specified, such as dimensions and coefficients. Because many pipes will be similar it can happen in practice that one is typing in the same numbers over and over again, and nobody likes that. In order to increase the ease of use, PIAS is therefore equipped with the feature to specify some of those parameters in ‘layers&rsquo. So, a parameter can explicitly be ‘not specified’, which will make PIAS to look for the corresponding parameter in a higher layer.

  • The pipe resistance coefficient, as discussed in Frictional resistance from pipes lines, is initially taken to be as specified for that pipe section (which is in a segment). If not specified then the default given in the general pipeline settings (see General piping settings) will be used. If it is not specified either then the pipe is considered free from resistance.
  • The component resistance coefficient is similar: 1) from the component and 2) the default from the general pipeline settings.
  • The resistance coefficient of a connection follows the same system as that from a component.
  • For the other resistance parameters (shape and size), those as specified for that individual pipe, component or connection are taken in the first instance. If these are not specified, then the parameters as defined with their network are taken. If that is not specified either, then PIAS looks one level higher, at the default shape and dimensions of the system where this thing belongs to. If those are not specified either, then the thing is assumed to have no resistance. There is deliberately no global default for shape and dimensions, because in general there will be no standard pipe size for the ship as a whole. For a particular system, e.g. ballast or sounding pipe system, on the other hand such a standard may exist.
  • A time-domain calculation is performed with a fixed time interval, in seconds. This can be specified globally at the general settings for damage stability calculations (see Time domain calculation time step), but it is also possible to set a so-called  ‘overruling time interval’ per piping network. If this is used, it prevails. This mechanism offers the user the possibility to apply a longer interval in networks with small pipes than with large ducts.

Clearly, this system is designed to minimize user input. So, one can specify a default at a higher level (e.g. at the level of a piping system) and leave all the items below it set to ‘not specified’, hence corresponding to the default. Only the exceptions then actually have to be given individually.

Summary of settings for Consecutive Flooding

As can be read in this chapter, Consecutive Flooding is controlled by quite a few settings and parameters. Because these are given in various places in PIAS, the impression may arise that there was no sharp plan behind this when developing the software, but this is by no means the case. Some parameters simply belong to a physical thing — a pipe or a valve — and hence to its definition in Layout, while others are calculation parameters which belong to the general settings for damage stability calculations, or to a specific piping network. Anyway, to provide the user with an overview, the several settings are summarised in the table below.

TODO make summary in table

Complex stages of flooding (before 2023)

Preliminary remark: as mentioned in the introduction to this chapter, PIAS now has two systems that can be used to account internal flooding between compartments. The subject of this section's system, complex intermediate stages, was in development until ±2020, after which development has focused on the more advanced Consecutive flooding system, see Flooding through ducts and pipes: Consecutive Flooding, after 2022.

This section a number of distinct facilities will be discussed:

  • When necessary PIAS takes into account intermediate stages of flooding. Normally these stages are equal within a damage case for all damaged compartments. With this option a mechanism is available to define the intermediate stages of flooding more specifically, especially according to IMO regulations for seagoing passenger vessels.
  • Special kinds of openings can be defined for which it will not be assumed that the vessel will immediately sinks when flooded, but for which the procedure described at the previous bullet will be adopted. Two types of such openings are available: internal openings, which connect the compartment with another compartment, and external openings, which connect a compartment with the sea.
  • Calculation of cross-flooding times.
  • The use of this function for the calculation of Ro-Ro ferries with water on deck (abbreviated to STAB90+50).

In each module for damage stability of PIAS, with the exception of the computation of floodable lengths, damage cases are defined at least by defining which compartments will be flooded for that case. This is discussed at Input and edit damage cases, where in the menu bar the [Flooding stages] function is included. When this fuction is used the following option menu appears:

Specify calculation type, number of intermediate stages and other parameters

The first option in this menu concerns the calculation type. There are two types of calculation:

  • Non-uniform intermediate stages of flooding. This type must be used if intermediate stages of flooding (expressed as percentage of the final stage) are not equal for all compartments. For this calculation type, at the second line the number of intermediate stages can be defined, with a maximum of 12. The final stage of flooding should not be defined because it is included automatically.
  • Time calculation for cross-flooding arrangements. With this option for each time step it is determined how much water enters a compartment, and what time a complete flooding of a compartment requires. For this calculation type also the time step and maximum number of time steps must be specified. The time step is used as an integration step in the calculation and this step must not be too large.

Specify intermediate stages and critical points

Calculation type `Non-uniform intermediate stages of flooding'

After selecting this option the following input screen appears which contains all damaged compartments:

Damage case ABC
CompartmentConnected withVia critical point
LengthBreadthHeightSB&PS
DEF-----
PQRSeawater10.1238.1236.123Yes
STUPQR23.1238.1236.123No
XYZSTU43.1238.1236.123No

A critical point defines an internal opening between two damaged compartments. The compartment will only then be flooded (with the percentage of flooding in a certain intermediate stage) when the level of liquid of the compartment in ‘Connected with’ is higher than the critical point. When for a critical point the column ‘SB&PS’ is set to ‘Yes’, than that point exists on SB and on PS (with an equal breadth from CL). The same mechanism is applicable to critical points if ‘Connected with’ is set to ‘Seawater’.

The mechanism contains three limitations:

  • Weathertight openings cannot be taken into account, but of course weathertight openings may be specified as usual in Hulldef .
  • A compartment which can be flooded through a critical point may not contain any liquid in intact condition.
  • With the combination of critical points and intermediate stages of flooding the following mechanism applies:
    • If the calculation is made without ‘global equal liquid level’ the procedure is as might be expected, that is, that every compartment has its own percentage, and its own level of filling. The compartment which can only be flooded through a critical point will only be flooded if the liquid level of the corresponding compartment exceeds the height of the critical point.
    • A calculation with ‘global equal liquid level’ is ligically inconsistent with the concept of ‘critical point’. Therefore, for the question whether a compartiment is flooded through a critical point is solely determined at the final stage of flooding, and this condition (flooded yes/no) is also used at intermediate stages, regardless the actual liquid level at the critical point.
  • If unequal percentages of flooding have been defined here, then the switch ‘Equal liquid level’, as dissussed in Intermediate stages with global equal liquid level will be ignored.

With the text cursor on a specific compartment, and pressing <Enter>, the next input screen appears in which the percentage of filling for a certain intermediate stage of flooding for that compartment can be defined, for example:

Compartment XYZ
Stage NumberPercentage of floodingWater on deckStab.crit.final
125NoNo
250NoNo
375NoNo
4100NoNo
5100NoNo
6100NoYes

A calculation with all compartments filled with 100% does not have to be defined, because this is the final stage of flooding which is calculated automatically.

Water on deck

According to the rules of the ‘Agreement concerning specific stability requirements for Ro-Ro passenger ships undertaking regular scheduled international voyages between or to or from designated ports in North West Europe and the Baltic Sea’ (Circular letter 1891), as adopted on 27-28 February 1996. A.k.a. as ‘Stockholm agreement’ and by EU directive 2003/25/EC 2003 also applicable to amongst others the Mediterranean. The core of the regulations is an additional amount of water on deck, dependant on the residual freeboard. To include the effects of water on deck:

  • If necessary, specify the significant wave height, as discussed in Significant wave height for SOLAS STAB90+50 (RoRo).
  • Define all spaces above deck as compartments.
  • Define the deckline ( Deck line).
  • Specify the correct permeability for damage stability for the compartments above deck.
  • Include the relevant deck compartments in all damage cases.
  • For each damage case define a complex stage of flooding, where all compartments below deck are flooded by 100%, and all above deck compartments are marked with ‘Yes’ in the column ‘Water on deck’.
  • At each damage case two calculations are made: One with the upperdeck compartments damaged, without extra water on deck, and one with the upperdeck compartments intact, with a fixed amount of water (which moves with heel and trim). At the last calculation in the last column marked ‘%’ the height of the extra amount of water can be read.

Calculation type `Time calculation for cross-flooding arrangements'

This calculation type is aimed at the very simple case of a compartment connected to sea water via a pipe or hole which has fluid flow resistance. Much more complex systems of pipes and connections can be addressed with Consecutive Flooding, see Background from tools for ship-internal connections in PIAS.

After chosing this calculation, a list of compartments is presented, where for each compartment can be specified:

  • Whether the compartment is flooded through a cross flooding arrangement. If that is not the case the compartment is always filled for 100% (or, in other words, the water level inside is always equal to the sea water level).
  • If the compartment is flooded through a cross-flooding arrangement, also the product of cross-section area S (in m2) and a dimensionless speed reduction factor F must be specified. These parameters, as well as the calculation method for F, are further elucidated in IMO res. MSC.362(92). This PIAS computation of cross-flooding times is discussed with the menu where it can be invoked, in Calculate cross-flooding times.

Output

In the output of the deterministic damage stability (with Loading) with complex intermediate stages, the percentage of flooding is not printed in the heading but in the table with weights per compartment. For the intermediate stages of flooding of the computations of maximum allowable VCG' (as discussed in Maximum VCG' damaged tables and diagrams) only the number of the intermediate stage is printed.

Underpinning of (damage) stability computations during flooding

The method of damage stability calculations is largely fixed by rules and conventions, and obviously this forms the basis for the implementation in PIAS. However, there are also a number of issues that are less clearly elaborated — such as the question of what exactly is the amount of fluid corresponding to a certain percentage at an intermediate stage of filling, or how to deal with a small internal opening when calculating the stability curve. The choices as made for such issues in PIAS are discussed below, for the two available systems:

The elder method is based on intermediate stages of flooding, and the newer one also includes a sub-method on that basis. However, the two do differ slightly, which is discussed in Difference in principles at intermediate stages.

By the way, while searching through decades-old documentation for the program approval by classification societies, we still came across the phrase that the damage stability calculation of PIAS includes the free-to-trim effect. A bit overdone to repeat that message, but just to be sure: it still does, also with Consecutive Flooding.

Underpinning at Consecutive Flooding (after 2022)

Here we discuss the effects that the internal connections and its components have on the calculations of intact and damaged stability, when using the system of Consecutive Flooding.

Basis of damage stability with fractions ("intermediate stages of filling")

The whole assumption behind the idea of a fraction (a generalization of an intermediate stage of flooding, see With conventional intermediate stages of flooding ("Fractional")) is that the immediately affected compartments will be flooded through a small damage. After all, if the damage were large, the ingressed water would spread rapidly, and the intermediate stage would be so short that it would have no effect on ship's position and stability. So, then the intermediate stage would actually not exist. Based on this physics-based reasoning, a distinction is made between large and small damages.

To assess stability in damaged condition, the worst-case scenario will have to be considered and since it is not known in advance how large the damage will be, cases with both a large and small damage are calculated.

In the event of large damage, seawater can flow freely in and out of the affected compartments, so that even during roll the water level in those compartments is equal to the sea water level. Because this all happens so quickly, intermediate stages do not actually emerge.

In the case of a small damage, on the other hand, the water flows through the hole so slowly that the intermediate stages can take a long time, and thus should be considered separately. However, if the water flows slowly, then during rolling it does not have time to flow in and out significantly. So, in this case the volume of water in a compartment can be assumed to be constant for all heeling angles.

The percentages of the stages of flooding can be set by the user, this is discussed in Define stages of flooding. Suppose intermediate stages of 25, 50 and 75% are given, then the complete damage stability evaluation will consist of:

DamageStageWater in compartimentVerified against stability criteria for
LargeFinalFreely flowing in and outFinal stage
SmallFinalConstant, as at equilibrium angle (call that W)Final stage
SmallIntermediate75% from WIntermediate stage
SmallIntermediate50% from WIntermediate stage
SmallIntermediate25% from WIntermediate stage


So, this intermediate-stage system is governed by the size of the damage. However, one could argue that the size of further internal openings (or pipelines or other connections) will also play a role. Indeed, this effect on the computation of the GZ-curve is discussed in Basis of larger angle stability (GZ-curve). Additionally, the sizes of the internal connections also determine the choice of the damage stability criteria to be applied. Also in this area PIAS gives influence to the user, which is discussed in Choice of stability criteria with the fractional method.

Basis of damage stability in time domain

The whole idea behind the time domain method is that the vessel gradually fills up through openings and pipes and so on. With each time step subdivided in these sub-steps:

  1. For time step t, tank fillings and densities are known, as well as which tanks are through a damage connected to the sea, so that ship's position — draft, trim and heeling angle — can be determined.
  2. With this, all liquid levels are known, and thus pressure differences between tanks, and between tanks and seawater can be can be determined.
  3. Since resistance coefficients are also known, Bernoulli's law can be used to determine the fluid velocities in all pipes and openings. And since all cross-sectional areas are also known, the fluid quantities (the flow rates) can also be determined.
  4. These flow rates are added to or subtracted from the compartments the pipes are connected to, creating new tank fillings and a new time step t+1.
  5. With this, the process jumps back to its first step.

This is somewhat simplified — e.g. additional analyses take place, such as checking whether the points of a pipe segment are all below the liquid level, if not the segment flow is blocked — the process, which the user can control by e.g.:

  • The time step, given in seconds, amongst others in Config, see Time domain calculation time step. Obviously, the accuracy of the calculation increases with decreasing time step duration. Using very small time steps, however, leads to longer computation time and more bulky output. An optimal time step cannot be given; at SARC we do try to keep it to a maximum of a few hundred steps.
  • The maximum number of time steps, see Time domain maximum number of time steps. This prevents a calculation taking very long; if this maximum is exceeded then it stops. On the other hand, then one doesn't have a finalized result, so one will still have to restart the calculation with a different time step. This maximum can therefore best be set quite high, it is just intended to cut off extreme cases.

In principle, at each of the time steps, the (damage) stability could be calculated, but that would lead to abundant output. Therefore, there are handles that allow the user to limit those time steps, see Time domain calculation time step and Minimum weight difference for a GZ calculation. At the time of such a stability calculation, the fluid content (generally consisting of a mixture of intact content and ingressed sea water) is held constant, and a stability calculation is made according to the same principles as with “fractions” (see Basis of larger angle stability (GZ-curve)). Actually a rather simple mechanism. ALthough there are still a few more details worth mentioning:

  • For a piping component, the single point of its position is determinative, both for calculating pressure differences in the fluid and for determining whether or not a threshold (or a demi-bulkhead) will overflow. The dimensions of the component will not have an effect on these two aspects. The dimensions, in combination with the resistance coefficient, only play a role in determining resistance faced by the fluid flow. As also explained in the examples of Modelling specific things from the real world.
  • The inertia of the vessel and its contents is not taken into account. That means that if e.g. a threshold is overflooded, the water flows instantaneously over it. Nor is the momentum of the ingressed water incorporated. As a result, the vessel does not enter a harmonic motion due to the damage and flooding.
  • Different types of liquids are supposed to mix. So, oil will not not float on water.
  • At the end of the filling process, the system is at rest; then no further fluids flow between sea and/or compartments. At least, it is so because the system is pressure-driven, i.e. at a pressure difference fluids start to flow and mix. However, there is also a long-term effect dispersion that causes a mixture to dissolve in seawater more and more over time. So that after a long time, the mixture is completely replaced by seawater. If the end of the filling process results in a mixture of seawater and original compartment contents, then because of worst case consideration, an additional “time”-step will be added with the damaged compartments (in equilibrium) filled with seawater. Damage stability will also be calculated for that step, from which the time is indicated by an ∞ in the output.
  • The relevant IMO resolutions also contain a formula to determine the cross-flooding time. However, difference could arise between these IMO formulae and PIAS. For the reason that the IMO resolutions apply an approximation method, while the time-stepwise computation of PIAS will in general yield a more accurate results. This application is covered by section 4 van MSC.362 (92): “As an alternative to the provisions in sections 2 and 3, and for arrangements other than those shown in appendix 2, direct calculation using computational fluid dynamics, time-domain simulations or model testing may also be used”.

Basis of larger angle stability (GZ-curve)

In Basis of damage stability with fractions ("intermediate stages of filling"), it has been plausibly shown that a small damage results in the volume of the compartment located behind the damage can be assumed constant during heel, whereas with a large damage the fluid can flow in and out freely during heel. Exactly the same reasoning applies to internal connections, holes and pipes. This treats internal connectionss in the same way as the damage, albeit that with the damage, since only one of these is assumed, a worst case scenario with combinations of small and large damages can be drawn up. Internal connections, on the other hand, can be numerous and it would take large amounts of computing time to start calculating all kinds of combinations of large and small from them. Therefore, for the internal connections, it was chosen to have the demarcation between ‘large’ and ‘small’ set by the user, which was discussed atconfig,config_damage_stability_CFminarea}. In short, if the cross-sectional area of a connection is larger than this value, then the fluid flows freely through that connection during heel, and otherwise not.

By the way, it will be obvious that due to sudden fluid transfer between heeling angles, the GZ curve will not always be nice and smooth. Traditionally, the GZ (plus associated draft and trim) is calculated at a fixed series of angles (of, e.g. 5°, 10°, 20°, 30° etcetera) but this does not model discontinuities in the GZ curve properly. After all, it will make a difference when fluid flows over an internal threshold at 21° or at 29°. Therefore, in Consecutive Flooding the GZ calculations are done at many more angles, in the order of every degree. To limit the output in loading and damage calculations, only the angles set at Angles of inclination for stability calculations are printed from these.

Effects on intact stability

The application of Consecutive Flooding is not limited to damage stability; also in intact conditions a compartment will be flooded through a submerged opening connected by a piping system. While that could be an interesting possibility, the practical importance of this is so small that it will not be included in PIAS for now.

Underpinning at Complex Stages (before 2023)

The method of calculation in intermediate stages of flooding is simple in the essence, with the following steps:

  • For each set inclination angle, for the final stage of flooding the ship's floating position (= draft and trim) is determined, as well as the weight of ingressed sea water in each compartment. -At each intermediate stage of flooding of X%, linear interpolation is performed per compartment between the weight in intact condition (0% intermediate stage) and in fully damaged condition (100% intermediate stage). With those weights per compartment, the floating position is calculated at that gradient. And the righting moment.
  • In this way, the term “intermediate percentual stage of flooding” is understood in a literal sense; it is precisely the percentual interpolation between 0% and 100%. This way provides continuity between 0 and 1% and between 99 and 100%.
  • This calculation scheme is complicated a bit by a disturbing regulatory element, which is that in general for the determination of tank volumes the stability rules assume a permeability (μ) of 98%, while for damage stability the μ never needs to be taken higher than 95%. Particularly annoying and physically untenable, but a fact of life, which PIAS hides away by also interpolating the μ linearly between 0% and 100% stages of flooding, so that also in this respect continuity is achieved at 0→1% and at 99→100%.
  • If a damaged compartment and an adjacent compartment are connected by a threshold or something similar then that can prevent the connected compartment from flooding. This can be specified as a so-called critical point — please refer for that to Specify intermediate stages and critical points. In intermediate stages of flooding, such a critical point behaves like a yes/no switch; at each angle of inclination, the final stage of flooding determines whether the critical point becomes overflowed, and if so, the connected compartment is filled in intermediate stages with the usual percentages, as if that whole critical point did not exist. And if the critical point does not overflow at 100% then the connected compartment is not filled at all in intermediate stages.

This scenario will not always represent reality, but it is the best fit for the dogma of the “fixed percentual intermediate stage of floodingt”. Those who do not agree with this approach will have to choose a more adequate scenario, e.g. a calculation in the time domain.

Two more details are worth knowing about the calculation method:

  • It was mentioned above that with each inclination angle, location is determined, as well as the righting moment. The righting lever — GZ or GN'sin(φ) — is obviously that moment divided by the displacement. For that displacement, PIAS has a choice, see Righting levers denominator.
  • As described, floating position GZ are determined at fixed inclination angles. The equilibrium angle is not calculated directly, it is determined by (nonlinear) interpolation on the GZ curve, viz. that angle at which GZ is zero.

Difference in principles at intermediate stages

Both the elder (pre-2023) damage stability calculation system, and the Consecutive Flooding system contain calculations with intermediate stages of flooding. Perhaps one would expect their results to be the same, but that will by not always be the case. This is not so surprising, as both methods have their own calculation bases. The details of these have been discussed in previous sections, but in summary they boil down to this:

  • The elder system is entirely based on the idea of continuity: at the final stage of flooding, for each compartment at each heeling angle is known how much seawater it contains, and in intact condition likewise, so that the water quantity at each intermediate stage can be interpolated between them. This has the advantage of consistency: after all, a 1% intermediate stage will be (almost) equal to the intact state, and a 99% stage to the final stage of flooding. Hence, the case ‘grows’ gradually and traceable from intact through the intermediate stages to the final stage of flooding.
  • The Consecutive Flooding system is more based on physical reasoning, looking at the physically expected effect under the various scenarios. Here, the 99% stage need not be (nearly) the same as the final stage of flooding at all. Indeed, two 100% stages occur here, one being an intermediate stage and the other the final stage.

One might ask which method is best. In general, that question cannot be answered. After all, the first method has the advantage of elegance, adding that many thousands of such calculations have been approved by classification societies and Shipping Inspectorates over the past decades. While the second method has the advantage that diverse configurations with varying sizes of damages, openings, pipe lines, holes and connections rest on a certain physical foundation. Because Consecutive Flooding supports just such a large variety, it was necessary to switch to the second method for that.

Effect of internal openings on the GZ-curve

How does PIAS deal with an internal threshold or pipe which is submerged, and hence allows transfer of water, at an angle of heel which is beyond the static equilibrium? Take for example the GZ-curve as sketched below, where at angle P the upper edge of a partial bulkhead overflows, leading to the filling of an adjacent compartment and hence a deteriorated stability. It will be undebatable that the GZ will initially follow curve A, until angle P is reached, where a greater amount of ingressed water will lead to reduced curve B. However, the question is what happens on the “way back”, i.e. with decreasing angle of heel? The water will not fully flow back over the bulkhead, so a curve more or less as indicated by C can be expected. And the subsequent question is which curve to use for the verification of GZ against stability criteria, A+B or C+B?

flooding_GZbranchesA.png
GZ-curve with internal opening submerged at angle P

In PIAS the past decades A+B has always been used — numerous calculations have been issued at classification societies and shipping inspections, and approved — based on the reasoning that the notion “way back” is never properly addressed, neither in literature nor in regulations. A few more arguments can be made in favour of this choice:

  • The example above is expressive, but counter examples also exist. Take the GZ-curve as sketched below, with the partial bulkhead now immersed at an angle P which is much larger. If the vessel is subject to IMO's Intact Stability Code then the maximum heel for criteria evaluation is 50° — the IMO weather criterion — while angle P is much larger than 50° now. So, this loading condition meets all stability criteria long before P is reached, and a reduced C-branch will not be applicable.
  • Will 50° then be the determining angle? In many cases not, because dynamic stability equality (area A=B from the weather criterion) may have been reached at a much smaller angle. So, the possible branching of the GZ-curve should be related to the applicable stability criteria, one way or another.
  • Assume now that at the same large angle P not an internal opening spills over, but instead an external opening (e.g. a ventilation inlet), which sinks the ship. Then beyond P, the GZ curve will vanish, so also branch B. If one would argue that with an internal opening branch C should be taken, then the same reasoning should be applied to external ones. However, with branch B also C has vanished, so using this branch will render the whole GZ-curve non-existent. Nobody — user, researcher, authority nor classification society — has ever suggested such a ‘solution’, because it would be unrealistic.
flooding_GZbranchesB.png
GZ-curve with internal opening submerged at a large angle

Supported by these arguments, it was — in 2022, during the re-evaluation at the introduction of consecutive flooding — chosen to keep the computation method for this subject in PIAS as it always has been. Please understand that this is an implementation choice, not the irrevocable result of the modelling method in PIAS. So alternative choices could be made, if there would be a reason for that, such as a generally accepted convention. Other reasons could be clear and unambiguous guidance by rules or regulations or unified interpretations from institutions, such as IMO, IACS or national authorities.