BRITISH M A R I T I M E T E C H O L O G Y
MSA
MARINE SAFETY AGENCYMarine Safety Agency Spring Place 105 Commercial Road
Southampton S015 l E G Tel: 0703 329100
TABLE OF CONTENTS
Section
3o
Page
INTRODUCTION 3 SCOPE OF THE STUDY 4
EXPERIMENT DETAILS 5
3.1 The Model ^ 2J1 Additional Features ^
3.2.1 Static Heel 6 3.2.2 Steady Wind 6 3.2.3 Reduced Size of Damage Opening 7
3.2.4 Sudden Damage and Flooding " • • • • 7
3.2.5 Obstructions in Flooded Compartment 8
3.2.6 Vehicles on Deck 8 3.2.7 Structural Sponsons 8 3.2.8 Inflated Buoyancy Bags 8 3.2.9 Additional Flare 3 J GZ and KG Measurement 9 3.3.1 GZ Measurement 9 3.3.2 KG Measurement 9 3.3.3 Hydrostatics Calculations 10 3.4 Experiment Technique 10 3.5 Data Collection and Processing 11
3.6 Error Analysis 1^ RESULTS OBTAINED AND DISCUSSION 14
4.1 General
4.2 Repeat Phase 1 Runs 1^ 4 J Effect of Static Ust 16 4.4 Effect of a Steady Wind 16 4.5 Effect of a Reduced Damage Opening 16
4.6 Sudden Influx of Water 17 4.6.1 Roll Moment Response
4.6.2 Free Model Response 18 4.7 Effect of Compartment Obstructions 18
4.8 Effect of Deck Obstructions and Permeability 19
4.9 Effect of Structural Sponsons 19
4.10 Effect of Inflated Air Bags 20
4.11 Effect of Flare 21 5. THEORETICAL STUDY 22
5.1 Purpose 22 5.2 Basic Assumptions 23
5.3 Basic Simulation Model 23 5.3.7 Basic Structure 24 5.3.2 Equations of Motion 25 5.3.3 Dynamics Module 26 5.3.4 The Mass Matrix 26 5.3.5 The Gravity/Buoyancy Force Module 26
5.3.6 Hydrodynamics Module 27 5.3.7 Hull Form Definition 27 5.3.8 Hydrostatics Module 28 5.3.9 Auxiliary Modules 29 5.4 Representation of Flooded Compartment 30
5.5 W a t ^ on Deck and the Damage Opening 30
5.6 Casing Position 32 5.7 Wave Synthesis 32 5.8 Program Testing and Developing 33
5.5.7 Box Vessel 33 5.8.2 RoRo ferry (Ship'A') 34
5.9 Results Obtained and Discussion 35 5. P. 7 Effect of Wave Period 35 5.9.2 Effect of Casing Position 36 5.9.3 Circumstances Leading to Capsize 36
5.9.4 Effect of Wave Height 37
6. GENERAL DISCUSSION 38 6.1 Effectiveness of the Remedial Measures 38
6.2 Experiment and Theory 39
7. CONCLUSIONS 41 8. REFERENCES 42
PHASE 2 EXPERIMENTS WITH A FLOODABLE FERRY MODEL
1. INTRODUCTION
In reference 1 a series of model experiments is described in which a floodable model of a ro-ro passenger ferry was tested in waves with various conditions of flooded stability. As a result of these experiments, and a similar series carried out on another hull form at the Danish Maritime Institute, it was possible to assess the suitability of proposed safety legislation under SOLAS '90 for this type of vessel.
The model tests of reference 1 were for a model of a RoRo ferry, referred to in this report as ship 'A*, in its basic form having no vehicles on deck and no engines, generators, etc. in the flooded compartment. Various remedial measures to prevent capsize of such vessels had been proposed and so it was decided by the UK Department of Transport's Marine Directorate that a further series of studies should be conducted. These would investigate not only the effects of permeability in both the flooded compartment and the vehicle deck, but also the effectiveness of various remedial measures in preventing capsize.
Accordingly British Maritime Technology Ltd. (BMT) were, among others, commissioned to undertake a physical and theoretical smdy using the same physical model as that described in reference 1.
This report describes the work done, presents the results and draws some conclusions. It also presents predictions using a mathematical model of capsize and sets these against the results of the experiments.
In addition to this report a set of video tapes (in U-matic format) is available showing most runs, together with an Annex giving the Run Log and plots of measured results from all runs.
2. SCOPE OF THE STUDY
In broad terms, the Phase 2 experiments were to provide experimental confirmation (or
otherwise) that calculated improvements to statical stability due to the fitting of certain devices (reference 2) would produce corresponding increases in safety under dynamic conditions in waves. The effect of various permeabilities in both the flooded compartment and the vehicle deck were also to be studied as was the effect of a sudden flooding of the compartment. A theoretical flooding and capsize model was also to be developed and results from this were to be compared with experiment.
The detailed scope of work allocated to BMT was as follows:
1. Carry out additional tasks in the Phase 1 Condition including:
• re-run of selected runs; • effect of static heel; • effect of wind;
• effect of 25% damage opening;
• effect of sudden damage and influx of water; • effect of obstructions in the flooded compartment; • effect of vehicles on deck.
2. Determine the effect of structural sponsons on capsize behaviour. 3. Determine the effect of inflated buoyancy bags on capsize behaviour. 4. Determine the effect of flare on capsize behaviour.
5. Develop a mathematical model of capsize behaviour.
The study was carried out with the above scope, using design data in references 2 and 3 for the sponsons, air bags and flare. The means by which the various changes/additions to the model were accomplished are described below; all were designed and developed as part of this project.
3. EXPERIMENT DETAILS 3.1 The Model
The model as built for the Phase 1 tests is shown in Figure 1. It represented a ro-ro passenger ferry with the principal particulars given in Table 1.
Length overall m 131.9
1
Length between perpendiculair SP m 126.1
Subdivision length Ls m 129.9
Breadth, moulded B m 22.7
Design draught T m 5.7
Subdivision draught Ts m 5.7
Depth to main deck, amidships m 7.3
Block coefficient CB 0.525
Prismatic coefficient Cp 0.556
Midship section coefficient Cx 0.943
Displacement volume at T 8592
Number of rudders 2 (one bow, one stem)
Aft rudder area/(Lpp.T) 0.0221
Number of stem propellers 3
Number of bow propellers 3
Aft propeller diameter m 3.3 m
Aft propeller type CPP
Installed power (mer) kW 20,142
Lateral windage area at T m2 2,400
Transverse windage area at T m2 706
TABLE 1
Other details of the model, built to a scale of 1:42.0333, are given in reference 1 although its body plan is shown m Figure 2 of this report for completeness. For the experiments described below, the model was mn at a slightly heavier displacement mass equivalent to 9142 tonnes fiill scale.
For the Phase 2 tests all decks above the vehicle deck were removed (Decks 'C' and 'E') to allow better visual photographic access to the behaviour of any water trapped on the vehicle deck.
At the request of the Department of Transport, all buoyant foam on the upper decks (representing intemal accommodation) was also removed so that the model would no longer be prevented firom a complete capsize.
As a result of these modifications the mass deck was re-designed to make up the weight lost and improvements were also made to its jacking arrangements to allow easier height adjustments.
Additional water-tight hatches were cut in the vehicle deck to allow fixed ballast to be added, adjusted or removed without removing the vehicle deck, and the whole ofthe under-deck area was cleaned and resealed. Additional pumping/venting arrangements were provided under the vehicle deck to remove any loose water that might find its way below. All flexible seals were renewed and the remainder of the model was generally refurbished.
For these tests, adjustment of the flooded freeboard was made by the 'Variable Compartment Length* method described in reference 1; the 'Variable Displacement' method was not employed. All casings, trim tanks, etc. on 'G' deck were modelled as in the Phase 1 experiment series.
3.2 Additional Features
In order to carry out all the experiments within the scope of the study, it was necessary to develop means of representing additional features (such as sponsons, air bags, etc.) on the model. In this section the means by which these were achieved are described.
3.2.1 Static Heel
Initial static heel bias was achieved simply by moving ballast weights on the mass deck until the required steady heel angle, measured on a clinometer, was achieved.
3.2.2 Steady Wind
A steady wind, blowing on the beam of the vessel has three dominant effects: • It can cause a heel away from the wind on a high-sided vessel.
• It will increase wave encounter period due to down-wind drift (assuming wmd and waves are co-linear).
o It can cause wave set-up and spray, both of which may increase water on deck. The first two of these effects were represented m the experiments; the third was not. The means by which the heel and drift effects of wind were modelled followed directly from the technique described in reference 1. Wind loadings corresponding to the wind speeds given in Table 2 were calculated using wind hmnel data held in-house at BMT for the ship under consideration.
SiGNincANT W A V E HEIGHT (m) W I N D SPEED (knots) 0.5 10.0 1.0 13.5 2.5 20.0 5.0 26.0 = ^ = a ^ = ^ = ^ ^ ^ ^ ^ ^ ^ ^ ^ TABLE 2
Wind loadings were applied to the model by means of two light lines attached at bow and stem at a height equivalent to about 6.0 metres above the waterline, corresponding to the esthnated height of the centre of wind pressure. Each Ime was passed over a low-friction pulley held by a member of the experiment team, one on each side of the towing tank. Weights, corresponding at model scale to half the wind loading, were attached to the free end of each line.
As the model drifted down the tank, the experimenters holding the weighted Imes moved at the same speed as the model to ensure that line tension (and hence the simulated wmd loading) remained constant.
3.2.3 Reduced Size of Damage Opening
The area ofthe damage opening was reduced to 25% of its Phase 1 value for some tests. The 15- slope to the damage opening was retained and the triangular hole m the vehicle deck was correspondingly reduced in size while maintaining the same side slopes. As in all other tests, the damage opening continued to the top of the model superstmcture.
The resultant opening is shown in Figure 3 and it was achieved on the model by means of a new hatch in the hull and additional side-pieces in the aluminium superstmcture sides. A small plate was also fitted into the damage opening in the vehicle deck to reduce the size of the deck penetration.
3.2.4 Sudden Damage and Flooding
To represent sudden ingress of water below the waterline to an otherwise intact vessel, a special hatch was made to fit into the hull damage opening. In this a cu-cular hole, representing one of 1.96 metres diameter at ftill scale, was cut. The penetration damage in the vehicle deck was closed with a sealed plate and the damage opening in the superstmcture was closed with another sealed plate.
The circular hole beneath the waterline was sealed using a flat lever with an 'O' ring seal which fitted over the hole. The other end of the lever was pivoted on the model superstmcture and a tensioned elastic cord was attached at its upper end. This sealing lever was held in place using a piece of light line in such a way as to ensure a watertight seal over the hole in the hull (see Figure 4).
To activate the mechanism, without imposing an external load on the model, the restraining line was burnt through, upon which the sealing lever would rotate about its pivot under the action of the elastic cord. This action would open the flooding hole and water would be admitted under pressure to the floodable compartment. The movement of the sealing arm (which was made of a light alloy) was rapid and, as it was in a vertical fore-and-aft plane, exerted minimal impulse to the model.
Time histories of the captive model's roll moment response to the sudden influx of water were measured on the GZ apparatus in calm water. Further experiments were carried out to measure the free model's response in calm water and waves.
3.2.5 Obstructions in Flooded Compartment
The flooded compartment represents the engine and generator rooms on the ship and therefore would be filled with machinery which would affect the flow m and out of floodwater. These were represented on the model by blocks of foam of roughly the same size as the machinery normally present in such spaces, arranged fore and aft in a manner similar to the main engines.
In order to retain the 1.0 metre and 0.25 metre flooded freeboards, the volume permeability of the flooded compartment had to remain the same. To achieve this, appropriate quantities of the rigid foam normally present in the compartment (to vary its length thus providmg various flooded ft-eeboards) were removed to compensate for the additional 'engine' pieces. This meant that although the volume permeability of the compartment was unaltered, the distribution of permeability was changed. In particular the 'engine* blocks were arranged on the double bottom and across the damage opening so that flow in and out of the compartment was impeded.
3.2.6 Vehicles on Deck
Dummy vehicles were manufactured from rigid plastic foam and stuck to the vehicle deck with contact adhesive for easy removal. Care was taken to ensure an agreed figure for volume permeability over the first 1.5 metres above the deck of 90%, and vehicle 'wheels' were modelled accordingly (see Figure 5). Models of large conunercial vehicles were placed in the centre bay of the vehicle deck with smaller vehicles (cars, etc.) being placed in the outer bays between the casings and the bulwarks.
3.2.7 Structural Sponsons
Structural sponsons were assumed to be those presented in reference 3 and shown in Figure 6. They extended over part of the length of the model and increased its beam amidships by 6.3%. Figure 7 shows the model, fitted with sponsons in the GZ Apparatus and it is seen that they extended both above and below the at-rest waterline. They were fixed to the model with a suitable adhesive and blended in to the shape of the hull with a flexible filler. 3.2.8 Inflated Buoyancy Bags
The buoyancy bags were made of rigid foam conforming to their inflated form as in references 2 and 3. Wooden mounting plates (representing the bag chambers) were glued and
screwed to the superstructure so that the bottoms of the bags were just above the at-rest waterline (see Figure 8). In all 24 bags were fitted, 13 on the intact starboard side and 2 less on the port side due to the damage opening. Figure 9 shows the model at rest when fitted with the buoyancy bags.
3.2.9 Additional Flare
Additional flare was added to the model in accordance with the proposed design in reference 3. The flare extended outward from the bull to increase the beam by 4.6% locally over a region of parallel middle body on either side of midships. Above the level of the subdivision deck the flare became tumbl^ome wbich extended most of the way to tbe top of the model's superstructure. Figure 10 shows the extent of the flare fitted while Figure 11 shows the model at rest complete with flare pieces added.
As can be seen, the flare was represented as additional side-pieces on the model, although m reality they would have formed part of the shape of the vessel. The flare additions were made of wood glued to the hull with their edges blended into the hull as necessary with flexible filler.
33 GZ and K G Measurement 3.3.1 GZ Measurement
As in the Phase 1 experiments, the model GZ values in various conditions of floodmg, as well as üie time histories of roll moment after a sudden influx of water, were all measured on the BMT GZ Apparatus.
This is shown in Figure 12 and has been described m reference 1. A roll moment flexure, mounted to the top of the model, is connected to a counter balanced vertical carriage which responds to any change in sinkage of the model as it is heeled. A motor-driven yoke allows any heel angle to about ±50° to be set, and, once the model has settled, the restoring roll moment is measured. Knowmg the model displacement mass allows the GZ value to be calculated. Typical measurements for an intact GM of 1.75 metres full scale are shown in Figures 13 to 15 for a flooded fireeboard equivalent to 0.25 metres full scale.
The changes in range and maximum righting lever brought about by the various remedial measures may be noted; the increase in range and GZ^^ due to the fitting of air bags is particularly marked.
3.3.2 KG Measurement
The position ofthe vertical centre of gravity and the dry model mass were checked frequently throughout the experiment series using the rocking cradle method described in reference 1. The purpose of these frequent checks was both to ensure that no unwanted loose water bad entered the model (and was thus adversely affecting KG) and to allow adjustments to be made for any additional weight. This arose from several of the remedial devices fitted to the model, most notably the dummy air bags whose weight was not insubstantial.
Where possible additional weight was allowed for by adjusting the weights on the mass deck, but in some cases some of the permanent below-deck ballast was removed. In all cases displacement was maintained at a constant value with the range of KG unaltered,
3.3.3 Hydrostatics Calculations
Calculations ofthe static stability of the model were made using AutoHYDRO 3,16 software, TTiese were done to provide a cross-check on the measurements and to give GMfluij values for the various experiment conditions. Agreement between calculation and measurement is discussed elsewhere in this report, but it was sufficiently satisfactory to enable calculated values of GMflyjj to be used in the analysis of the experiment results.
3.4 Experiment Technique
The experiments were run in the number 2 towmg tank at BMT. Tank, wavemaker and test conditions were therefore identical to those used in Phase 1. Once again JONSWAP wave spectra with repeat periods of 13.5 minutes were used with y = 3.3.
As in Phase 1 wave heights and periods from the BMT PC-Global Wave Statistics software package were used as shown in Table 3.
Significant Wave Height (m)
Period from PC-GWS (sec)
= = = ^ ^ — =
Modal Period Assumed (sec) 0.5 5.5 7.095 1.0 5.5 7.095 2.5 6.5 8.385 5.0 6.5 8.385 TABLE 3 The technique used for each run followed that used in Phase 1 and comprised the following main steps:
1. Set up model in required condition, re-ballast as necessary, check GZ, mass and KG. 2. Place model in tank at beach end and tow slowly, behind the carriage, to the
wavemaker end of the tank.
3. Steady the model upright (or with damage heeled away ft-om the water and on-coming waves to prevent early ingress of unwanted loose water before the run proper began) in calm water at the end of the carriage remote from the wavemaker.
4, Start wavemaker at set point in the wave time history,
5, Wait for 60 seconds, holding the model to prevent water coming on deck, 6, Gently allow model to come to the upright and release.
7. Follow the model as it drifts down the tank until it capsizes, reaches the end of the tank or a period of 10 mmutes (model scale) has elapsed.
8. Retrieve model, retum to start position and drain.
9. Make any adjustments to wave setting or KG and retum to Step 2.
At a given wave condition the capsize boundary was sought by making small adjustments to KG and re-mnning. Some guidance as to the correct setting was obtained from tbe Phase 1 result, but, as will be shown, this was not always a good guide due to the random capsize behaviour of the model.
As no permanent buoyancy was fixed in the upper superstmcture of the model, it was prone to roll on to its back when it capsized. Due to the problems this posed in its retrieval, the model was generally caught by the experiment team as it began its final rapid roll to capsize. As in Phase 1 a mn log was maintained fbr all mns and the indexing system used in the earlier tests was retained. However, as will be seen, a further index was required to d^cribe behaviour when air bags were fitted when the decks were awash with large quantities of water, but no capsize resulted. The indices used are given in Table 4.
Index Behaviour
1 No water on deck
2 Occasional water on deck
3 Frequent water on deck; model developed a list 4 List increasing throughout the mn
5 Capsize
6 Deck awash, but no capsize
T A B L E 4
3.5 Data Collection and Processing
Instmmentation, on both the model and the carriage was the same as that used in Phase 1. This consisted of the following model-bome mstmmentation:
• roll/pitch gyroscope
• roll and yaw rate gyroscopes
• surge, sway and heave accelerometers.
These were mounted on the mass deck and connected to the carriage-based instmmentation by a ribbon cable supported by a hand-held out-rigger to keep the cable slack at all times, thereby applying no force to the model.
Carriage-based instrumentation consisted of signal-conditioning equipment and power supplies together with a Hewlett Packard 9816 micro-computer for data acquisition. All data for each run was stored on disk for off-line analysis, samples having been obtained from all channels at a 4 Hz frequency after filtering.
As in the earlier experiments the instrumentation was completed by a twin-wire capacitance wave probe mounted some 8 metres downstream of the wavemaker and fixed to the tank wall. Its signal was recorded by the carriage-borne instrumentation via a trailing cable.
3.6 Error Analysis
In reference 1 sources of error and estimates of their magnitude, were discussed. The same procedures in KG, mass and GZ measurement were followed to minimise systematic errors, while the calculation of hydrostatics and stability were done using the same model oftset file as in Phase 1. Water leakage in the model was largely eliminated by improved sealing and any water taken on-board was removed after each run by improved pumping arrangements. The run procedure mentioned above was rigidly adherwl to in each run. By these means the effects of major systematic errors were believed to have been eliminated, or at least minimised.
Estimates for absolute and relative random errors have not changed since Phase 1 and are given in Table 5.
Parameter Absolute Error Relative Error (%)
Mass, M ±0.05 kg ±0.04 KG ± 1 mm ±0.42 Wave height, Hg ± 1 mm ± 0 . 8 4 - ± 8 . 4 Freeboard, F ±0.5 mm ±2.1 - ±27.7 Roll angle, ^ ± 0 . 5 " ±1.0 © 5 0 ° Pitch angle, 6 ±0.5° ± 5 0 ® 1° P ±0.1°/sec ± 1 0 q ±0.1°/sec ± 1 0 KM* ± 1 mm ±0.38
Note: All values measured except those marked * which were calculated. TABLE 5
From these values it is possible to estimate the error in the calculated solid GM of the vessel, by compounding the random errors in KG and KM, thus:
G M \ ' K M \ * [ K G ,
This leads to an error of ±0.57% in GM or ±0.011 m full scale for a GM of 2.0 metres. Relative Errors in range, G Z ^ ^ and times to capsize remain as estimated for Phase 1 at ± 4 % , ± 5 % and ± 1 % to +2% respectively. However, as flooded GM values, GMf, were calculated using the hydrostatics software, their relative accuracy, especially for the lower flooded freeboards, is assumed to be improved and a value of ± 5 % would appear to be satisfactory.
The uncertainty with which the capsize boundary curve is defmed remains. In spite of usmg a central barrier for most experiments (see section 4,2), it was not easy to ensure repeatability of capsize in all cases. This is explored further below in discussion on both the experimental and theoretical results and at present it would seem inappropriate to assign an error value to this boundary.
4. RESULTS OBTAINED AND DISCUSSION 4.1 General
In this section the results obtained in the model experiments are presented and discussed. The essence of much of the discussion is a comparison with some sort of benchmark set of results, in this case those obtained in Phase 1,
Where possible, the transition between 'capsize' and 'no capsize' has been used as a basis for those comparisons, combined with the non-dimensionalisation used in Phase 1 in which measured significant wave height Hg was non-dimensionalised with respect to flooded freeboard F and flooded metacentric height GMf was non-dimensionalised with respect to beam B, draught T, and block coefficient Cg. This gave the following two parameters:
HJ F and 10 GM^.C^.T I (2)
Cases where the transition between 'capsize' and 'no capsize' was not "clear have been discussed by consideration of the individual results.
As it was not possible, given the financial and temporal limits placed on the study, to repeat all wave height/flooded freeboard combinations it was decided at the outset to limit these to the following equivalent full scale values:
Flooded freeboard: 0.25 m, 1.0 m Significant wave height: 2.5 m, 5.0 m
However, in some of the Phase 1 repeat runs, wave heights equivalent to 0.5 m and 1.0 m were also used.
In what follows therefore, an attempt is made to compare results obtained with what has gone before, note various features that were observed and explain what happened and why. 4.2 Repeat Phase 1 Runs
Non-dimensional Phase 1 results, from reference 1, are shown in Figures 16 and 17, replotted to a different scale. As may be seen (and as pointed out in reference 1) there was no clearly-defined demarcation between 'capsize' and 'no capsize' in such a plotting, but rather a region in which capsize was likely, a region in which no capsize was likely and an intermediate region in which capsize may or may not occur.
Results obtained from a limited number of repeat runs in the Phase 1 condition are shown in Figures 18 and 19. Comparison of Figures 16 and 18 shows that reasonably good agreement of the two sets of results was obtained when the damage opening faced into the waves. The region in which capsize may or may not occur is common to both sets of results.
From damage facing away from the waves, agreement is apparently less satisfactory. However, far fewer runs were carried out in this condition and, of those that were, several of the 'no capsize' runs exhibited a pronounced heel toward the damage at the end of the allotted time for the run.
One of the reasons for the non-repeatability of the capsize behaviour of the model was due to the observed tendency of water, once admitted to the vehicle deck, to pass across the deck and collect on the side remote from the damage opening. This caused the model to heel thereby lifting the damage, and the exposed subdivision deck edge, clear of the water. This had two effects:
o It prevented the trapped water draining off the deck through the damage opening. o It gradually caused less and less water to go onto the deck (in some cases) so that an
equilibrium heel, with damage out of the water, resulted.
As a result of this, capsize would be prevented. If, on the other hand, the action of the first few waves caused water to collect on the side of the damage, the model would heel toward the damage and capsize would often result.
The ease with which water could pass across the vehicle deck was occasioned by the fact that the model was fitted with side, (rather than a centreline) casings. These hardly obstructed the passage of water across the deck.
To explore this apparently important effect of the casing position, a number of repeat runs at nominally the same GMflujj, flooded freeboard and significant wave height, were carried out. In all cases the damage was toward the waves and the runs were started with the model upright and using the standard technique given in section 3.4 above.
In all some 22 repeat runs were carried out at a flooded freeboard of 1.0 metre, a GMfjyjj of 1.74 m and a significant wave height of around 3 metres. For 10 of these runs a short barrier 40.7. metres long, 1,26 m wide and 1.81 m high at full scale was positioned on tbe deck centreline opposite the damage hole. Its location is shown in Figure 20 and it may also be seen in some of the photographs of the model m waves (Figure 49 for example). The barrier was sufficient to prevent water moving across the deck in any quantity, although large quantities of water on deck could pass over the barrier.
The results obtained may be seen near the value of 10 GMnuid.Cb,T//B2 = 0,10 in Figure 18, Of the 12 runs without the barrier, 6 resulted in a capsize with the model capsizing to port, the damage side. Of the remaining runs 3 developed a list to port, but did not result in capsize, while the remaining 3 resulted in a list to starboard and no capsize.
Ofthe 10 runs with the centre barrier installed, 9 resulted in a capsize (always to port) and, in the remaining run, the model almost capsized at the time recorded for this event in the other runs. With the centre barrier in place, capsize times were variable as Figure 21 shows, but in almost every run the model appeared as if it was about to capsize at a full scale time of 16 to 22 minutes.
The effect of the central barrier on capsize was profound, its presence making it almost certain that the run would end in a capsize for the parameter set chosen. This suggests strongly that the presence of a centreline casing would have a marked effect on whether a capsize occurred or not. Side casings with openings lessen the chance of capsize without eliminating the possibility altogether.
It is perhaps of interest to note that in other similar studies (references 4 and 5) the ferry models had centreline casings and little if any of the uncertainty of capsize, experienced with the model used in this study, was encountered.
43 Effect of Static List
Retaining the central barrier, a 4° static list was induced on the model with a flooded freeboard of 1.0 metre when upright, reducing, at the damage, to about 0.21 metres for a list to port and increasing to 1.8 metres for a list to starboard. The results are shown in Figures 22 to 25, from which it is clear that a list to port virmally guaranteed capsize whereas a list to starboard virtually ensured that no capsize would take place.
Comparison of Figures 23 and 17 shows that, for the damage facing away, from the waves, the reduction in freeboard due to the 4" port list considerably lowers the capsize boundary in spite of the sheltering effect due to the vessel's orientation to the waves.
4.4 Effect of a Steady Wind
The results obtained in loadings corresponding to the steady winds of Table 2 are shown in non-dimensional form in Figures 26 and 27. Again there is no clearly-defined 'capsize' zone, there being the odd run in which the model developed a heel to starboard, lifting the damage out of the water and preventing capsize.
In general however it would appear that wind had a minor effect on capsize, the apparent boundary zone between 'capsize' and 'no capsize' being little changed from those apparent in Figures 16 and 17.
It may be mentioned that, for the damage facing the waves, wind would be expected to induce a heel to starboard thereby reducing the chance of capsize. I f this were so, then the effect would be expected to be greatest at the lowest flooded freeboard. The results show this not to be the case for every run, it appearing almost as likely that a capsize as no capsize would occur.
4.5 Effect of a Reduced Damage Opening
Results obtained with the area of the damage opening reduced to 25% of its Phase 1 value are shown in Figures 28 and 29. The centre barrier was in place for these runs.
It is clear from Figure 28 that, with the damage facing the waves, some slight improvement was obtained, the model showing rather more 'no capsize' points above the boundary zone of Figure 16. Any effect was, however, small.
With the damage facing away from the waves, Figure 29 shows the effect of the smaller opening to be quite marked. The model showed a pronounced reluctance to capsize and a definite improvement was obtained.
The reason for this behaviour may be due to the observation that, with damage facing away from the waves, the main method by which water gains access to the vehicle deck is by a 'pumping' action through the hole in the deck as the model heaves and rolls. The reduced size of the opening means therefore that less water gets on the deck. In this condition it was observed that the model often developed a list to starboard thereby further reducing the chance of capsize.
With the damage facing the waves, water can gain access to the deck both by the pumping action referred to above, and by over-topping through the damage openmg. Over-topping will be reduced due to the smaller opening and so a slight improvement in capsize behaviour results. Times to capsize are compared to the basic condition in Figure 30.
4.6 Sudden Influx of Water
As mentioned above, the tests in which water was suddenly admitted to the flooded compartment were carried out in two parts. Initially a series of time histories of the roll moment L, induced by the entry of the floodwater were measured. Secondly a series of free model tests in calm water and waves was carried out in which the model's transient response (primarily in roll and sway acceleration) was measured. The compartment was open to flooding for a 1.0 metre fiill scale flooded freeboard and was vented to the atmosphere. 4.6.1 Roll Moment Response
The purpose of these tests, carried out with the model captive on the GZ Apparatus, was to determine whether there was any evidence to suggest that transient asymmetric flooding, as discussed in reference 6 could be observed. The model was initially floating at its design draught.
Figure 31 shows results obtained for an intact GM of 1.0 metres with the model upright. The instant at which the damage hole was unsealed is marked and it is seen that an initial positive (starboard) roll moment was induced. This was no doubt due to the inflow passmg across the compartment and impinging on its far side. This gradually settled as the compartment flooded and ultimately, after about 20 seconds into the run the model had settled at its new equilibrium draught with virtually zero roll moment again.
The eff'ects of static heels of 2° and 4" to starboard at the same GM are shown in Figures 32 and 33. In both cases it is seen that an initial tendency to roll to starboard was replaced by a gradually-increasing negative (to port) roll moment as the compartment filled. This apparently anomalous behaviour is explained by the fact that, when heeled, the flooded compartment and flexure cease to be in the same vertical plane and so the added weight of floodwater causes an anti-clockwise (negative) moment to port. The relationship of the flexure and the model when heeled may be seen in Figure 7.
Similar behaviour was observed for an intact GM of 2.0 metres; Figures 34 and 35 show two examples.
These results indicated the transient nature of the initial flooding in a compartment with virtually 100% permeability. With zero heel the initial tendency to roll to starboard is clear, although this transient was short-lived and whether the model (or ship) would respond significantly remained to be seen. Presumably in a compartment with a much lower permeability (a ftiUy-equipped engine room for example) the transients would last longer and might have a greater effect on a ship.
4.6.2 Free Model Response
The free model response in roll is shown in Figures 36 and 37. These results were obtained in calm water and show the initial roll to starboard (positive) immediately after the hole opened. The model then developed a mean heel to port due, presumably, to the effect ofthe floodwater moving rapidly to the port side of the flooded compartment.
The effect of an initial 2' heel to starboard is seen in Figure 37 from which the initial transient starboard heel is clear, the model gradually settling back to its initial heel of 2" as the compartment filled.
It should be remembered when assessing Figures 36 and 37 that the roll angle was measured to an accuracy of ±0.5° (see Table 5) which is a significant proportion of the measured angles. In spite of this the free-model results in calm water serve to indicate the effects of sudden flooding.
Further tests were run in 5.0 m (significant) waves with a GM^uj^j of 0.36 m fiill scale and a flooded freeboard of 1.0 metre. No damage opening was present in the vehicle deck or the superstructure side. The effect of the instantaneous flooding was to cause the model to settle as the compartment flooded, but otherwise there was no other effect and the model showed no tendency whatsoever to capsize or to behave in any way other than that of an intact model with a large fiill tank amidships.
4.7 Effect of Compartment Obstructions
Results from the experiments carried out with obstructions in the flooded compartment are shown in Figures 38 and 39. The model was fitted with the central barrier for these tests and it appears that two distinct cases emerge for damage facing the waves.
Results at a flooded freeboard of 1.0 metre (the lowest level of results at an Hg/F value of around 3.0 in Figure 38) show capsize/no capsize values consistent with the boundary zone in Figure 16. However, values obtained at a flooded freeboard of 0.25 metres (Hg/F values of around 12.0 and 21.0 in Figure 38) show, with the limited results available, an apparent improvement in capsize behaviour with several 'no capsize' values occurring in what would be regarded as the 'capsize' region. Furthermore, results for damage facing away from the waves suggest again that a slight improvement may have resulted at the lower freeboards. The reason for any improvement in performance (if such exists) is not immediately apparent. It is possible that the changed distribution of permeability, by having an effect on the freedom of flow into and out of the compartment, resulted in more water being held in the compartment as the model moved. This would act as an increase in added mass and would therefore be expected to affect the model's motions. By making the model more sluggish,
or by changing the phasing of its motion relative to the waves, the quantity of water reaching the vehicle deck could have been changed, resulting in a better capsize performance. Although this is to a certain extent conjecture (and could be pursued with the mathematical model in the future), it was clear from these experiments that changing the distribution of volume permeability can have an effect on capsize behaviour. This is perhaps not entirely surprising when the dynamic, rather than the static, situation of a flooded vessel in waves is considered. Anything that impedes the movement of floodwater in the vessel, as well as its added mass, must have an effect on motions - and, presumably, capsize.
4.8 Effect of Deck Obstructions and Permeability
Results obtained with dummy vehicles, but no central barrier, on the subdivision deck are shown in Figures 40 and 41. Once again it is clear that the effect of changing deck permeability and providing obstructions to water movement on the subdivision deck is profound.
At a flooded freeboard of 0.25 metres full scale, capsize was virtually eliminated, for damage both facing and away firom the waves. Inde^, for damage facing away from the waves the model capsized only once in 9 runs in significant wave heights up to 5 metres.
Part of the reason for this behaviour lay in the observed fact that the dununy vehicles tended to trap any water which moved across the vehicle deck to the starboard side. By 'holding' this loose water to starboard the vessel heeled to starboard, less water gained access to the the vehicle deck and capsize was prevented.
This trapping behaviour was especially prevalent for a flooded freeboard of 0.25 metres and with damage facing the waves. For a 1.0 metre flooded fireeboard it is possible that less water initially managed to reach the vehicle deck and any that did in the early stages of the run was able to get trapped as easily to port as to starboard. Once trapped to port the resultant heel would help more water to come aboard and ultimately would cause a capsize. Therefore the greater freeboard, from the limited results obtained, gave, paradoxically, the greater chance of capsize.
4.9 Effect of Structural Sponsons
GZd Measured computed values for the model fitted with structural sponsons are shown in Figure 42. TTiey were obtained for an intact GZ of 1.75 metres full scale and at the two flooded freeboards used in the tests. The results show the negligible flooded stability with a flooded freeboard of 0.25 metres and the considerably greater range and maximum value at a freeboard of 1.0 metre. The calculations were carried out using model dimensions both for the hull and the flooded compartment.
Results obtained with the model, with its central barrier fitted, are shown in Figures 43 and 44. It is immediately apparent that the sponsons were effective when the damage faced away firom the waves. Figure 44 shows no capsizes in spite of low (and nominally negative) GMf values being used at large wave heights and flow flooded fireeboards. One run, at a fluid GM of -0.85 metres did result in a capsize. It is outside the range of the plot and so is not shown
on Figure 44, but proved to be the only time a capsize occurred with the damage away from the waves.
Rather more capsizes occurred when the damage was facing the waves (Figure 43). Once again there seems to be two distinct sets of results corresponding to the two flooded freeboards used. The upper set of points in Figure 43 were obtained at 0.25 metres freeboard and indicate a reluctance to capsize brought about mainly by the model developing a steady heel to starboard. At the greater freeboard, steady heels to port were more common and as a consequence more capsizes were obtained.
It would appear therefore that, on the basis of this evidence, sponsons are beneficial. It must be stated however that it was found difficult in some runs to obtain a good leak-proof seal round the starboard sponson and, for one run at least, water was able to get on the vehicle deck from this cause, giving rise to a beneficial starboard heel. This occurred for the damage away from the waves and a flooded freeboard of 0.25 metres and, as soon as it was observed, the model was re-sealed and the run repeated. Of the remaining runs with the re-sealed model, 90% resulted in a beneficial heel to starboard.
It is not therefore considered that unwanted leakage could have been the cause of the model's behaviour and that the structural sponsons were of some benefit.
4.10 Effect of Inflated Air Bags
There was no doubt that fitting inflatable air bags was beneficial. In no case was a capsize ever a possibility, in spite of large quantities of water being taken on deck.
Figure 45 shows the measured and calculated GZ values obtained at an intact GM of 1.75 m full scale and flooded freeboards of 1.0 and 0.25 metres. The increased range and maximum GZ are striking when compared to results in Figure 42. No adjustment to heel was made to allow for the loss of 2 airbags over the damage and so the curves show a non-zero GZ when upright.
Figures 46 and 47 show the results obtained with the model which confirm that no capsize occurred. Figure 48 shows the model in the tank with damage facing the waves, a flooded GM of 0.85 metres full scale and a flooded freeboard equivalent to 0.25 metres full scale. The floodwater draining from the flooded compartment can be seen quite clearly.
However, there was a price to be paid for this improvement. It was noted that roll motion, especially at the greater intact GM values, was particularly severe, the roll accelerations being very high and the model correspondingly stiff. This is hardly surprising in view of the significant increase in effective beam caused by the air bags and the corresponding increase in flooded GZ. High roll accelerations could cause problems both to passengers and cargo during" rescue; cargo could break loose and passengers become injured.
Another feature of the airbags was the tendency of the model to take such large quantities of water on to the vehicle deck that the model soon became awash. Figure 49 shows the model in this condition and the water on deck can be seen clearly. Whether such large amounts of water on the vehicle deck would pose a problem during the rescue period is debatable. Clearly damage to the vehicles would result and many of them would be moved about with
consequently damaging effects on stability. If, however, the additional transverse stability afforded by air bags is sufficient to overcome these problems, there is of course the possibility that sufficient water could be taken on deck to sink the vessel. Nothing in the model tests suggested that there was the slightest possibility of this happening within the full scale time limit of one hour from the exposure of the damage to waves.
4.11 Effect of Flare
Figure 50 shows calculated and measured GZ values for the model fitted with dummy flare. Comparison with Figure 42 shows the range of damage stability and maximum GZ to be similar to those obtained with sponsons.
Figures 51 and 52 show the results obtained with the model in this condition and agam it would appear from Figure 52 that, with damage facing away from the waves, flare was beneficial although not to such an extent as sponsons or air bags.
With damage facing the waves (Figure 51) the results are less clear. At a flooded freeboard equivalent to 1.0 metre full scale (the lowest set of points in the Figure) it appears, from comparison with Figures 16 and 18 that little is to be gained from flare. • But at a flooded ft-eeboard equivalent to 0.25 metres full scale, steady heels to starboard prevailed, in spite of the central barrier, and capsizals were few. Figure 53 shows the model in such a condition with a substantial amount of water collected on the starboard side of the vehicle deck. The damaged vehicle deck edge is well clear ofthe water in the photograph and water can be seen draining out of the flooded compartment.
Figure 54 shows the model in waves with a flooded freeboard equivalent to 0.25 metres. In both photographs water can be seen entering or leaving the damage opening. Figure 55 shows the model about to capsize to port.
These results suggested that the effect of flare was similar to that of strucmral sponsons, and far less than tiiat of tiie inflated air bags. Again tiie results showed tiie effect of water on deck causing a beneficial heel to starboard in spite of tiie presence of tiie central barrier. This was especially noticeable at tiie lowest flooded freeboard where, perhaps due to a change in heave and roll motions brought about by tiie presence of flare (altiiough tiiis seems very unlikely) significant amounts of water were taken on deck, over tiie centre barrier and oyer to starboard. There tiiey caused a heel and were trapped by tiie central barrier which prevented their drainage when the model rolled to port.
5. THEORETICAL STUDY 5.1 Purpose
The experimental results discussed in section 4 above (and also the Phase 1 results discussed in reference 1) revealed that, in spite of repeated precautions being taken, the model under test did not always behave as expected. A non-dimensional scheme, developed by semi-empirical reasoning, had failed to collapse all results, although some success had been achieved.
The importance of the casing position has been demonstrated in section 4.2 and the persistence of the model in heeling away from the damage (and as a result of this not capsizing) has been a recurring theme.
Finally, it has been shown by observation of the model (and the video records made of the model tests) that water on the vehicle deck was not always likely to lead to capsize and the water draining off the vehicle deck and draining out of the flooded compartment was at least as important as water gaining access to these two important parts of the model.
On completion of the Phase 1 tests, there was a general feeling, shared by the client and the research teams, that some sort of dieoretical model was necessary. By the intelligent use of such a model, in concert with the experiments, it was believed that a greater insight into the behaviour of a flooded ship in waves would be obtained.
Observations had shown that:
• The motions of the flooded model were not unduly severe.
• Although water sloshing on the vehicle deck occurred it was not unduly severe and a quasi-static approach to water on deck might be a promising line of approach. • Water entry/drainage to the vehicle deck was of crucial importance.
• The dynamics of capsize took place rapidly after a long period in which a steady heel gradually increased.
« Heave and roll were the two dominant motions; pitch, yaw and sway also occurred, but to a much lesser extent. (This observation was based on the fact of the model starting beam-on to the waves).
• Damage facing the waves was more likely to lead to capsize.
The purpose of the computer model described below was therefore to make use of these observations and, above all, to gain ftirther insight into the behaviour of a flooded ship in waves. It was becoming clear from the model tests that purely static considerations were inadequate to understand how the model behaved and held the danger of a mind-set m which the results could be forced into a format predicated by pre-conceptions of what the answer should be.
Insight rather than precise prediction, qualitative rather than quantitative results were seen to be the prime goals of the computer model. It was hoped at the outset that it would indicate which areas required any further research so that, far ftom replacing model tests, it would provide indications of where more focused studies could be aimed in the future. Above all
it was seen as providing a model, based on dynamic rather than purely static considerations, of the behaviour of a flooded ship in waves.
5.2 Basic Assumptions
The following basic assumptions were made in building the computer model:
1. A 6 degree-of-fteedom time domain motion sunulation model, already in use at BMT, could be used, with flooding aspects added to the gravity vector.
2. Wave reflection effects could be neglected, as could the effects of spray.
3. Waves would be long-crested and uni-directional regular or irregular in form, as in the towing tank.
4. Down- and up-flooding ftom the vehicle deck and flooded compartment could be neglected or represented simply by a coefficient.
5. The behaviour of water-on-deck could be represented by quasi-static methods. 6. The flooded compartment could be treated as completely open to the sea (this is
discussed further below).
7. The lost-buoyancy approach was more appropriate that the added weight method. 8. Water gained access to the vehicle deck through the damage opening in the vessel's
side and vehicle deck when the outside water level exceeded that of the deck at the innermost limit of damage penetration.
9. Viscous damping and added mass terms were not dependent on ftequency. (This is also discusswl further below).
10. Yaw and pitch motions were of second order and could be neglected.
Several of these assumptions were made with a view to simplifying the computer model enough to be able to make a start. As will be seen, results obtained with the model in its present form suggest ways in which some of the above assumptions may be removed. 53 Basic Simulation Model
This section describes the simulation model as developed, together with a description of the algorithms and numerical techniques used in its construction. The basic structure follows that in reference 7.
5.3.1 Basic Structure
First the basic structure is considered. At the heart of the model is the numerical solution of Newton's Second Law of Motion:
F = Ma (3) where F is a force vector
a is an acceleration vector M is a 6 X 6 mass matrix. This is solved by re-writmg equation (3) as:
a = M-^F (4) where M"^ is the inverse of the mass matrix. If F and M are known, the acceleration vector
•a' may be calculated at each increment of time and, from this vector, velocity and position vectors may also be calculated by integration.
Note that we assumed the mass M to remain constant in equation 3. Although this remained the case for the model as built, it implies that any variable mass effects due to water on deck or in the flooded compartment, are negligible. As force is the time rate of change of momentum:
F = 6(Mv)/6t = v5M/6t + M6v/öt (5) We are essentially neglecting the first term on the right of equation (5). This will hold good
if öM/6t is small which should be the case until capsize, at which point its effect will also in all probability be small and perhaps of less interest.
The problem arises in equation (4) in the proper determination of the components of the vector F. These consist of three forces and three moments, each one in turn consistmg of several components. These are, in general, components from:
o hydrostatic forces • hydrodynamic forces • gravity forces
• dynamic forces
• other forces arising from, say, winds or waves.
The basic structure of the model is shown in Figure 56. It is seen to be modular in form with the data paths between modules inviolate, while the modules themselves may be altered as required.
Integration of the equations of motion was carried out using a simple Eulerian integration:
where f ( ) denotes some function of time t At denotes a time step
X ( + 1 denotes an integrated value of f(t) at time t-l-1
and the results were printed out at every time step. The results listing had a variable number of columns so that selected variables could be printed. Results from these listings were subsequently transferred to a plotting programme to plot vessel tracks and other variables as time histories. A separate listing gave body forces and moments at each time step if required. Having introduced the basic structure of the programme, it is now necessary to look in detail at the various modules of which it is comprised.
5.3.2 Equations of Motion
The full sidegrees-of-freedom equations of motion, equation (3), are well-known. The x-equation may be written as follows, assuming an axis origin which does not coincide with the centre of gravity (reference 8):
m(u -I- qw - rv) - XQ(q2 + r^) -I- yQ(rq - f ) -I- ZQ(PT + iO =
X H + Xw + X A + (W - B)SinÖ (7) The other five equations have a similar form, and where we have, in the usual notation:
u , V, w = linear velocities in x, y, z axes using the axis system in Figure 57;
Ü, V, w = linear acceleration in the same directions;
P, q, r, P» 4, f = angular velocities and accelerations in roll, pitch and yaw about the x, y, z axes;
^xx' lxy» lyz' ~ moments/products of inertia;
X, Y, Z = forces in axis directions (surge, sway and heave); L, M , N - moments about three axes (roll, pitch and yaw); W = weight of vessel plus water on deck;
B = buoyant force acting on damaged vessel;
X f i . YB» ^B ~ co-ordinates of centre of buoyancy;
^G' yo» ~ co-ordinates of centre of gravity;
<}>,e, \l/ = angular displacements in roll, pitch and yaw;
Subscripts : H = hydrodynamic W = waves A = wind, etc. 5.3.3 Dynamics Module
Many of the terms on the left hand side of equations (7) etc. can be collected together in a vector of forces arising from inertial terms. This is conveniently contained in a subroutme and recalculated each timestep.
The components of the dynamics vector have the typical form:
[-m^qw + myrv + m[\Q(q^ + r^) + y^tq - ZQPT]
etc.
where specific mass and inertia terms were collected together in groups such as: ""x = - hx = • Lp
my = m - Jyy " ^yy " > StC. with the identities:
Jyz = lyz + Mf = ^yz ^q' ^tc
using the notation that means the partial derivative of X with respect to u. Terms such as X^, Y ^ are the added mass terms in the x and y directions while terms such as L ^ , M a are the added inertias about the x and y axes.
5.3.4 The Mass Matrix
The complete mass matrix, including added mass terms is given by:
0 0 -m^-X^ 0 0 -fnzo - mxo -0 0 "^G - Zp -mXf. - Z„ a q -L -Zfjn - -Jzx ZfJn - M , - M , -Xfjn - m^ -Jy. -y(/n - Xfjn - -Jy^
5.3.5 The Gravity/Buoyancy Force Module
To compute the force vector due to any imbalance between weight force and buoyant force, the gravity vector is required. This is:
-{W - B)smd
(W - B)cose sin(|)
{W - B)cos8 cos<t»
(yfW - y^B) cosQ cos^ - {z^W - z^)cose sin<t)
-{XgW - JCgB)cos8 cos(J) - {ZgW - z^)sin0
- x f i ) cose sin<l) - (y^W - y^) sinS
5.3.6 Hydrodynamics Module
The hydrodynamics module computes the hydrodynamic force vector. A very simple hydrodynamic model was used in the study. This did not involve any cross-coupling terms between motions such as would occur in, for example, manoeuvring activities. The simple model employed was known to be crude and capable of considerable refinement. For the purposes of this exercise, coefficients were adjusted until plausible motion damping, compared to the model results, was obtained. The lack of frequency-dependence of the coefficients is clearly something that could be improved in a future version of the model. The hydrodynamics vector was therefore given by:
y,v|v| Zj|w|w
M,q^ N,r\r\
5.3.7 Huil Form Definition
The geometry of the vessel hull, both above and below water, was defmed as a set of offsets in the X, y and z directions. This allowed conventional lines plans to be used to define the vessel and avoided the complication of panelisation had a surface panel method of defmition been employed.
5.3.8 Hydrostatics Module
As mentioned above, the buoyancy forces and moments were computed at each timestep in order to determine the position, in body axes, of the centre of buoyancy. This was done in two stages:
• The vessel was assumed to be intact at the start of a run with the flooded compartment's length, breadth and depth defined in body axes. Before the time-stepping began the equilibrium draught and trim (and hence centre of buoyancy co-ordinates) of the flooded vessel were calculated using the lost buoyancy method. This was then used as the initial static condition of the vessel, the buoyant force being made equal to the intact weight of the vessel.
o The increment/decrement in buoyant force and moments was computed at each timestep, together with the values of (xg, yg, zg) and the results incorporated m the equations of motion.
In statical stability calculations, to calculate the position of the metacentre and the rigbtmg lever ZQZ (or 'GZ'), two approaches are traditionally adopted. One assumes the disturbance from an equilibrium position to be small, which allows the righting couple to be expressed as:
Lg = pgGMvsin(^ (8) for the case of roll motion.
If the angular disturbance fi'om the equilibrium is large, then equation (8) does not apply. This is because the underwater shape of the vessel changes markedly as the angular disturbance increases so that, in order that weight and buoyancy forces remain equal, the vessel must sink/rise and trim.
Most large-angle stability calculations have an ancestry dating back to pre-computer days. In these the vessel is rotated about a fixed axis and corrections are applied to preserve equilibrium. In some cases changes in trim are conveniently ignored.
In the present case we are not dealing with a static, but a dynamic situation; weight and buoyancy force need not be in equilibrium at a given time - indeed in a dynamic situation it is unlikely that they will be.
So in the program the concept of a metacentric height was dropped. The instantaneous co-ordinates of the centre of buoyancy were calculated at each timestep by means of the 'in' and 'out' wedges at each station, the shape of the wedges being determined by the angular and
linear position of the vessel in space together with the instantaneous water depth, port and starboard, at each station due to the wave system.
This was done in two stages: the additional 'wedge volume' for the intact hull was computed and from this was removed the "wedge volume' for the flooded compartment to leave the 'wedge volume' for the damaged hull.
Longitudinal, lateral and vertical moments of the 'wedge volume' were calculated in addition to the volumes themselves and these were used, in conjunction with the centre-of buoyancy position of the hull at rest to compute the new centre of buoyancy position at each timestep. 5.3.9 Auxiliary Modules
In addition to the main computation modules outlined above, several auxiliary modules were written to aid in the general computation. These are now outlined briefly.
5.3.9.1 Kinematic Equations
To compute the angular position of the vessel the following relationships were used (reference 9):
<^> = p + q sin <^ tan Ö + r cos ^ tan Ö (9)
è = q cos </» - r sin <^ (10) ^ = q sin / cos 0 + r cos <^> / cos Ö (11)
These were included in the general integration at each timestep. 5.3.9.2 Change of Axis System
On several occasions it was necessary to change from space to body-centred co-ordinates and back again. This was achieved by transformations of the type:
^1 ^3
«2 "3.
(12) ipace
where I j , I2, m|, m2, etc. are the direction cosines of the appropriate axis system. Changes in linear position in space or body systems were also dealt with as a simple translation.
5.3.9.3 Matrix Inversion Routine
A standard library subroutine was used to invert the mass matrix. This used a Gauss-Siedel elimination process.
5.3.9.4 Output Modules
The data output was handled by a subroutine which sent data to two files. The first contained all the motion variables u, v, w, p, q and r as well as the position variables x, y, z, B and ^, wave amplitude and water volume on deck for all timesteps. The second file contained all the force and moment data from each module at each timestep.
The X, y and data was then read into a track-plotting program to plot out the computed track of the vessel, while another auxiliary program allowed time histories of any output variable to be plotted. Only the data firom the first file was archived on disk for all runs. 5.4 Representation of Flooded Compartment
One of the basic assumptions underlying the computer model was that the flooded compartment, although allowing for its permeability /t^, was nevertheless fully open to the sea. This convenient simplification meant that the model, in its present form, assumes the compartment to be fiill (to the flooded waterline) at the start ofthe run and to play no fiirther part in the proceedings. It also tacitly assumes that the flooded compartment has no bottom. Such assumptions are something of a simplification. In reality it is unlikely that the bottom ofthe flooded compartment will be missing and so, as the ship heaves and rolls, it will have to move the water that lies within the compartment. This will have an effect on the behaviour of the vessel and will primarily affect the acceleration vector.
It would not be appropriate to represent the water in the flooded compartment as a transient increase in displacement weight W as this would affect the gravity vector in the model and cause the model incorrectly to sink/rise.
It would be possible to include the mass/inertia of the floodwater in the added mass matrix so that its effect on acceleration would be allowed for without compromising the gravity vector. This modification could be incorporated in the next version of the model.
A further consequence ofthe assumption that the flooded compartment is completely open to the sea is that no modelling of transient filling/emptying of the compartment is necessary. Observations ofthe physical model's behaviour (see Figures 48 and 53 for example) show that as the water head inside the compartment is instantaneously greater or less than that outside, water will begin to flow out from or flow in to the compartment through the damage opening. This is a ftirther consequence of the fact that the flooded compartment has a bottom whereas the computer model, in its present form, assumes that it does not.
Therefore subsequent versions of the computer model should allow for the compartment bottom to be modelled with its resultant effects on added mass and flow in and out of the flooded space.
5.5 Water on Deck and the Damage Opening
As mentioned above, the movement of water on deck was modelled on the assumption that it moved in a quasi-static manner. In other words, no attempt was made to model sloshing or time lags due to flow across the deck, the loose water being assumed to take up an
instantaneous position which kept its free surface horizontal. Observations of the model (see Figures 49 and 53 for example) show that this was not an unreasonable assumption.
The steps involved in computing the amount of water on deck and its effect on the vessel were as follows:
1. Define the damage opening in the side of the vessel and the vehicle deck For simplicity this was defmed by rectangular openings whose width was the mean of the deck penetration opening, whose depth was to the double bottom, whose height was unrestricted and whose penetration was 20% ofthe beam, as m the physical model. 2. Test for those occasions when the local wave height, at the position of the mner edge
of the deck penetration in space, exceeds the instantaneous vertical position of the edge of the deck penetration in space.
3. Store instantaneous ordinates (in body axes) ofthe water over the deck, until the local wave height drops below the instantaneous level of the inner edge of the deck penetration.
4. Using the instantaneous water velocity in the wave crest, compute hdw far across the deck the previous 'water slice' ordinate would have moved m one time step and use this m the ultimate determination of the 'slice' volume by trapezoidal mtegration. The width of the slice is taken as that of the damage opening.
5. Allow for end effects of wave slice by interpolation.
6. Observation of the physical model showed that the amount of water taken on board might be augmented by forcing of water up through the penetration hole in the vehicle deck and might be reduced by reflection from any obstruction on deck. The ability to modify the computed volume taken on was therefore required and this was provided by multiplying the computed volume taken on board by a constant a which was generally chosen to be between 0.95 and 1.5. In addition the volume was modified by the assumed deck volume permeability HQ.
7. Once the wave slice was determined, it was added to the volume of water on deck This was assumed to be evenly distributed over a length 1^ along the vehicle deck (Figure 53 shows this to be a not unreasonable assumption) which allowed its mean cross sectional area a^ to be determined.
8. The depth of the water on deck, either at the deck edge where the superstrucmre side began, at the casing or on the deck itself (all dependent on the volume on deck and the roll angle) was calculated by simple geometry. If the head in way of the damage opening exceeded the local head outside, water drained out using the expression for a flooded weir (reference 10):
Q = C/D,'h, I ^ (13)
P25104/00 31
where: volume flow in m^/sec
a discharge coefficient, taken as 0.6 width of the damage opening
head difference between inside and outside.
5.6
5.7
The change in volume of the water on deck 6(vol) was given at each timestep by:
where 6t is the timestep.
When the head of water outside the vessel was greater than that inside at the damage opening, water was assumed to enter the vehicle deck unimpeded.
9. The additional weight of the water on deck and its moment about the three body axes were used to compute the new displacement weight W and the new centre of gravity position. The change in vessel mass and inertia were also accounted for in the dynamics equation (4).
In this way a model of the action of loose water on deck, which accommodated drain-off as well as water entry, was developed. Suitable values of a and /IQ were to be obtained either by trial and error or by standard practice, while a value of C j of 0.6 was taken from previous smdies for the Department of Transport Marine Directorate carried out by BMT.
Casing Position
The importance of the casing position has been highlighted in the discussion of the model test results. It was therefore of importance to include a casing on the vehicle deck to impede the movement of water across that deck.
A long impermeable barrier was assumed which could be positioned anywhere across the vehicle deck. This formed the boundary of movement for the volume of water on deck and therefore determined the heads of this water, as the ship rolled, at the inboard or outboard extremities.
Wave Synthesis
Both regular and irregular long crested uni-directional waves can be synthesised in the program. They are sinusoidal in form with both having a random initial phase angle in the general wave elevation expression:
5(vol) = Q5t (14)
?(yo,t) = a sin (kyo - 8)t -I- a j ) (15) where: ? is the instantaneous wave elevation
a is wave amplitude
yo is the instantaneous position in space axes
t is time
