Ocean Engineering 37 (2010) Contents lists available at ScienceDirect. Ocean Engineering

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1 Ocean Engineering 37 () Contents lists available at ScienceDirect Ocean Engineering journal homepage: Squat prediction in muddy navigation areas G. Delefortrie a,n, M. Vantorre b, K. Eloot a,b, J. Verwilligen a, E. Lataire b a Flanders Hydraulics Research, Berchemlei 5, 4 Antwerp, Belgium b Ghent University, Maritime echnology Division, echnologiepark 94, 95 Ghent, Belgium article info Article history: Received 5 May Accepted August Editor-in-Chief: A.I. Incecik Available online 5 September Keywords: Mud Sinkage rim Mathematical model abstract Common squat prediction formulae to assess the navigation safety usually do not take into account the bottom condition. Nevertheless, the presence of a fluid mud layer is not an uncommon condition in confined areas where accurate squat predictions are necessary. From to 4 an extensive experimental research program was carried out to measure the manoeuvring behaviour of deep drafted vessels in muddy areas. A part of the program focused on the undulations of the water mud interface and their relationship to the ship s squat. Mostly the sinkage of the ship is damped due to the presence of the mud layer, but a larger trim can occur due to the water mud interface undulations. his article presents a mathematical model to predict the squat in muddy navigation areas. & Elsevier Ltd. All rights reserved.. Introduction Squat, defined as the sinkage and trim of vessels due to their own forward speed, is of particular importance in shallow water areas. Small under keel clearances cause large return currents which lead to important sinkages and higher risks of bottom touching as already mentioned by Constantine (96). In shallow navigation areas the presence of a soft fluid mud layer on the bottom is not exceptional, but its effect is mostly neglected in the formulation of squat. As a consequence pilots and scientists may disagree on the safety of navigation. Mostly pilots have to rely on the high frequency echo to determine the water depth. As the latter detects the top of the mud layer and not the solid (or nautical) bottom level, they may still be able to navigate safely through a muddy navigation area, even in case the ship is navigating at a zero (or even negative) under keel clearance according to the echo sounder. In cases where common squat formulae would predict grounding of a ship navigating in a shallow fairway at a rather high speed, the presence of a mud layer may prevent such grounding. Indeed a limited or even negative under keel clearance referred to a mud layer does not necessarily lead to impracticable manoeuvres as mentioned by Delefortrie et al. (7). When initially the ship has a small under keel clearance referred to the mud layer, she may hit the mud layer due to squat. his mud, n Corresponding author. el.: ; fax: addresses: Guillaume.Delefortrie@mow.vlaanderen.be (G. Delefortrie), Marc.Vantorre@UGent.be (M. Vantorre), Katrien.Eloot@mow.vlaanderen.be (K. Eloot), Jeroen.Verwilligen@mow.vlaanderen.be (J. Verwilligen), Evert.Lataire@UGent.be (E. Lataire). having a larger density than water, will affect the buoyancy of the ship and will probably smoothen the squat effect. However, to be sure about the mud effect additional research had to be carried out, because the literature offers very limited results on this topic.. State of the art.. General research on squat Scientific research on squat took off with Constantine (96) who discussed the different squat behaviour for subcritical, critical and supercritical vessel speeds. In the subcritical domain (F rh o) uck (966) proved that for open water conditions of constant depth the sinkage and trim of the vessel to be linear with the parameter gðf rh Þ¼ F rh qffiffiffiffiffiffiffiffiffiffiffiffiffi F rh In which F rh represents the depth related Froude number F rh ¼ V gh his theory was later extended to dredged channels by Beck et al. (975). Naghdi and Rubin (984) offer some reflections on uck s theory and introduce a new one. An analogous theory has been developed by Cong and Hsiung (99). Ankudinov and Daggett (996) however are pessimistic about the complexity of numerical theories. For this reason, several ðþ ðþ 9-88/$ - see front matter & Elsevier Ltd. All rights reserved. doi:.6/j.oceaneng..8.3

2 G. Delefortrie et al. / Ocean Engineering 37 () Nomenclature AEP expanded area ratio of propeller ( ) A R rudder area (m ) a i regression coefficient, (i¼,) ( ) B ship beam (m) b mud type, able ( ) b i regression coefficient, (i¼,,) ( ) c mud type, able ( ) C B block coefficient ( ) c i regression coefficient, (i¼,,) ( ) C S dimensionless sinkage, Eq. (4) ( ) C dimensionless trim, Eq. (5) ( ) D 6 EU container ship model ( ) d mud type, able ( ) d i regression coefficient, (i¼,r) ( ) D P propeller diameter ( ) E tanker model ( ) e mud type, able ( ) e i regression coefficient, (i¼,) ( ) f mud type, able ( ) f regression coefficient ( ) F rh depth related Froude number, Eq. () ( ) g mud type, able ( ) g i regression coefficient, (i¼,h) ( ) h total depth (m) mud type, able ( ) h* hydrodynamically equivalent depth (m) i regression coefficient ( ) i regression coefficient ( ) j i regression coefficient, (i¼, r) ( ) k i regression coefficient, (i¼, r) ( ) L PP ship length (m) P propeller pitch (m) p regression coefficient ( ) q regression coefficient ( ) S solid bottom condition ( ) s i regression coefficient (i¼,n) ( ) ship draft (m) EU (number of) twenty feet equivalent unit containers ( ) P thrust (N) U 8 EU container ship model ( ) ukc under keel clearance ( ) V ship speed (m/s) V propeller induced speed, Eq. () (m/s) x longitudinal coordinate, positive towards the stern (m) z A sinkage aft (m) z F sinkage fore (m) a increase parameter, Eq. (3) ( ) g parameter, Eq. () ( ) z amplitude of rising (m) z MAX maximal amplitude of rising (m) m dynamic viscosity (Pa s) m crit critical dynamic viscosity (Pa s) r density (kg/m 3 ) r* dimensionless density, Eq. (6) ( ) P keel penetration parameter, Eq. () ( ) P h keel penetration parameter, Eq. (4) ( ) F fluidization parameter ( ) F ij regression coefficient (i¼,h) (j¼,r) ( ) Fij regression coefficient (i¼,h) (j¼,r) ( ) F ij regression coefficient (i¼,h) (j¼,r) ( ) F regression coefficient ( ) Subscripts n propeller related to water related to mud authors carried out experimental research as Dand (97) and Gourlay () who offers a solution for squat prediction with random bottom conditions to endorse their theories. Jiang and Henn (3) present a numerical method valid from subcritical to supercritical speed. An overview of slender body methods is given in Gourlay (8). More practical methods based on experimental research are presented by Barrass (979), however his results could not be validated by Seren et al. (983). Barrass (4) gives an overview of the work he performed on squat. More general overviews are given by Dumas (98); Blaauw and Van der Knaap (983), Millward (99) and PIANC (997) working group 3. Interesting full scale measurements were carried out by Ankudinov et al. (), Stocks et al. (4), Härting et al. (9) and Härting and Reinking () among others. Most discussions focus on ships sailing in open water or in rectangular shaped canals without drift angle or propulsion. In some cases the drift angle was considered, as by Von Bovet (985), Martin and Puls (986), de Koning Gans and Boonstra (7) and Eloot et al. (8)... Research on squat in muddy areas he research on squat in muddy areas is a topic that has not been tackled thoroughly. Only three research institutes carried out experimental research focussing on the hydrodynamic forces. he oldest results are presented by Sellmeijer and van Oortmerssen (983) who also registered undulations in the water mud interface. he sinkage is less above mud in comparison with the solid bottom condition and decreases with increasing layer thickness. he mud density does not seem to have any effect. Vantorre and Coen (988) showed that three speed ranges can be detected for the behaviour of the water mud interface: At low speed a small sinkage near the fore body is detected, which disappears amidships and turns into an elevation abaft; At a certain speed value the sinkage at the entrance changes suddenly into an elevation. he section at which the jump occurs moves abaft with increasing speed; If the speed increases more, the rising of the interface occurs behind the stern. he amplitude of the elevation can exceed the mud layer thickness several times. he latter occurs at a speed, which for inviscid fluids is given by the theoretical expression sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 V crit ¼ 7 gh r ð m r Þ 3 ð3þ m being the blockage of the ship in the water layer, meaning the ratio of the ship s immersed cross sectional area and the canal s cross section. Subscript refers to the water layer, subscript to the mud layer.

3 466 G. Delefortrie et al. / Ocean Engineering 37 () he sinkage of the vessel is related to these speed ranges, but experimental results do not always follow the theory. At low speeds the mud layer causes a very slight increase of sinkage while at higher speed a sinkage decrease with mud layer is observed together with an increase of trim. Brossard et al. (99) described that the sinkage is identical as in the solid bottom condition when the ship s keel does not penetrate the mud layer. An effect is observed at negative under keel clearances: he rigidity of the mud has only a small effect; he density gradient significantly affects the sinkage: the higher the gradient, the smaller the sinkage. It is assumed that the buoyancy is an important factor; Adding currents leads to further reduction of the sinkage. he trim of the vessel is only significantly affected by rigid mud. In this case an increase of trim with increasing density gradient was observed. he sign of the trim changes when penetrating the mud. Doctors et al. (996) showed that for the ship hydrodynamics a shallow water approach can serve as a quite reliable approximation for analyzing the case of a viscous lower layer, where the mud viscosity can be interpreted as an effective reduction in the total depth of the water. In spite of these results, until now no sufficiently correct models have been presented to predict the squat in muddy areas. 3. Experimental setup 3.. est facilities he new squat formulae presented in this article are all derived from experimental research carried out from until able Ship models (even keel). Model D U E Scale /75 /8 /75 L PP (m) B (m) (m) C B A R (m ) # blades D P (m) P/D P ( ) AEP ( ) at the owing ank for Manoeuvres in Shallow Water cooperation Flanders Hydraulics Research, Ghent University. he formulae are valid within the range of conditions covered by this experimental research. he shallow water towing tank (88 m 7 m.5 m) is equipped with a planar motion carriage, a wave generator and an auxiliary carriage for ship ship interaction tests. hanks to computerized control and data-acquisition, the facilities are operated in a fully automated manner. he carriage runs 4/7 without the need for permanent surveillance. 3.. Ship models Most runs in muddy navigation areas were carried out with (able ) a 6 EU container carrier (D), which was the design ship for the Belgian harbours at that time. he mathematical model will be based on the measurements carried out with this ship. Additional runs in a selection of conditions were carried out with a scale model of a 8 EU container carrier (U) and a scale model of a tanker (E) to assess the influence of the hull form, see Bottom conditions Mud was simulated by a mixture of two types of chlorinated paraffin and petrol, so that both density and viscosity could be controlled within certain ranges. For environmental reasons, the tank was divided into three compartments: a test section, a mud reservoir and a water reservoir. Bottom and tank walls were protected with a polyethylene coating. No viscosity or density gradients were included. he selected bottom conditions are represented in able. he density viscosity combinations were based on measurements of density and rheology profiles carried out in the outer harbour of Zeebrugge, Belgium in A mud layer configuration is defined by two characters: a letter (b,y,h) denoting the material characteristics and Figs. 3 re-presenting the layer thickness. ests carried out above a solid bottom are referred to as S. he gross under keel clearance relative to the tank bottom was varied between 7% and 3% of draft, yielding % to +% ukc relative to the mud water interface. hroughout this paper, the interface water mud will be used as a reference for expressing the under keel clearance, unless specified otherwise est types Stationary captive motion model tests were carried out in each combination of mud layer and realistic under keel clearance at different speeds, from kn ahead to kn ahead at least in steps of kn (full scale values). able Bottom conditions and tested models. Mud type Density (kg/m 3 ) Dynamic viscosity (Pa s) Layer thickness.75 m.5 m 3. m 3 d.3 D/E D/E D/E/U c 5.6 D D D b 8. D D D f. D h.9 D/E D/E D e 6.9 D g 5.46 D/E D/E D/E fresh water. sea water 5. S solid bottom

4 G. Delefortrie et al. / Ocean Engineering 37 () he propeller rate was varied at 6 kn ahead between 6% and % of the maximal propeller rate. Some runs were carried out with a drift or rudder angle or with astern speeds, but these runs will not be discussed in this article Measurements he sinkage of the ship was measured at four positions on the hull: starboard side fore and aft and portside fore and aft. A positive trim angle is measured when the sinkage at the bow is larger than at the stern. As the literature already mentioned the occurrence of undulations of the water mud interface measures were taken to register them. A device which follows the level of the mud layer (mufo) and one which follows the water level (wafo) are assembled on a ζ + h h x L PP dy =.8 B dy =.3 B dy =.3 B Fig.. Positioning the tripods in the towing tank. Fig. 3. Undulations of the interface at various lateral distances of the ship D. Mud f, +3.9% ukc, F rh ¼.38, no propeller action. he ship is represented taking squat into account and sails to the left. Fig.. Arrangement of the wave meters in the towing tank.

5 468 G. Delefortrie et al. / Ocean Engineering 37 () tripod (see Fig. ). In total three tripods were assembled in order to register the undulations of the interface. he mufo consists of a floater, which is resistant to the artificial mud. he density of the floater is situated between the water and the mud density. he position of the floater is therefore similar to the amplitude of the interface. he floater is attached to a disk, which reflects a laser beam. he variations of the laser beam are measured times per second and register the actual position of the mud layer. he wafo is based upon the principles of the potentiometer. A constant electric current is sent through a string which has a homogeneous resistance. he voltage is therefore also constant. A second electrode is a tube made of stainless steel. When the water level in the tube changes a proportional change of voltage will be measured. hree tripods were placed in the towing tank as close as possible to the passing ship. he lateral distance between the ship and the mufos is as shown in Fig.. Depending on the ship s velocity the position of the interface was measured times per second. he tripods were placed in the middle of the tank so that the position of the interface could be measured before, while and after the ship was passing. 4. Observations 4.. Undulations of the water mud interface Fig. 3 gives an example of the measured undulations of the water mud interface, which seem to behave as a Kelvin pattern. he maximal amplitude closest to the ship is represented in Fig. 4. ζ MAX + h h Frh (-) mud G mud H mud B mud C mud D Fig. 4. Maximal rising in function of ship speed. Ship D. hickness of the mud layer: 3 m full scale. No propeller or rudder action. 9.8% under keel clearance referred to the water mud interface. he rising increases with increasing speed, but this increase is limited once the undulations are behind the ship, see Fig. 5. his is especially the case with low density mud layers (c, d). When the vessel navigates above the mud layer the rising will increase faster with the velocity when the density and viscosity of the mud layer are closer to water. With thinner mud layers the rising becomes only significant once the viscosity drops below a certain critical value, which is located between. and.8 Pa s. A significant undulation is always observed when the ship navigates in contact with the mud layer, see Fig. 6. he rising is mostly located amidships for higher density mud layers. For lower density and viscosity the rising is located abaft, as shown in Fig. 7. he transitory situation is a rising occurring in two phases. h ζ MAX + h mud F mud H mud B mud C mud D Fig. 6. Maximal rising in function of ship speed. Ship D. hickness of the mud layer:.5 m full scale. No propeller or rudder action..% under keel clearance referred to the water mud interface. x L PP mud F mud H mud B mud C mud D amidship aft Fig. 7. Longitudinal position, at which the rising is maximal, in function of ship speed. Ship D. hickness of the mud layer:.5 m full scale. No propeller or rudder action..% under keel clearance referred to the water mud interface.. x L PP ζ MAX + h h Frh (-) mud G mud H mud B mud C mud D amidship aft. -% -5% % 5% % propeller rate (-) mud G mud H mud B mud C mud D Fig. 5. Longitudinal position, at which the rising is maximal, in function of ship speed. Ship D. hickness of the mud layer: 3 m full scale. No propeller or rudder action. 9.8% under keel clearance referred to the water mud interface. Fig. 8. Maximal rising in function of the propeller rate. Ship D. hickness of the mud layer: 3 m full scale. No rudder action. Ship speed: 6 kn full scale. % under keel clearance referred to the water mud interface.

6 G. Delefortrie et al. / Ocean Engineering 37 () x L PP % -5% % 5% % propeller rate (-) mud G mud H mud B mud C mud D aft amidship Fig. 9. Longitudinal position, at which the rising is maximal, in function of the propeller rate. Ship D. hickness of the mud layer: 3 m full scale. No rudder action. Ship speed: 6 kn full scale..% under keel clearance referred to the water mud interface. C S (-) solid mud F mud G mud H mud B mud C mud D Fig.. Sinkage in function of the ship speed. Ship D. hickness of the mud layer:.5 m full scale. No propeller or rudder action. 3.9% under keel clearance referred to the water mud interface. C S (-) solid mud F mud G mud H mud E mud C mud D Fig.. Sinkage in function of the ship speed. Ship D. hickness of the mud layer:.5 m full scale. No propeller or rudder action..% under keel clearance referred to the water mud interface. Reversed propeller action in case of navigating ahead yields a relatively large rising near the propeller, see Figs. 8 and 9. In this case the pattern of the undulations is rather random. he rising does not seem to start abaft the ship in the given experimental speed range, although some experimental speeds were higher than those predicted by Eq. (3). A possible explanation is the higher viscosity of the mud layers in this experimental program and the assumption of inviscid fluids in Eq. (3). C S (-) % 4.. Sinkage he mean sinkage as measured during the tests will be represented dimensionless as C S C s ¼ z F þz A L PP -5% % 5% % propeller rate (-) mud G mud H mud B mud C mud D Fig.. Sinkage in function of the propeller rate. Ship D. hickness of the mud layer: 3 m full scale. No rudder action..% under keel clearance referred to the water mud interface. he values z A and z F are positive downwards. Figs. and give an overview of the ship s sinkage in function of the speed for different bottom conditions. he following can be observed: When the ship navigates above the mud layer the rising of the interface is significantly larger for mud layers with a viscosity below a critical viscosity. When the under keel clearance is small, this can eventually result in contact between the vessel and the mud layer. he mud will yield an increase of buoyancy, which results in a decrease of the sinkage; If the ship s keel penetrates the mud layer, the large rising amidships, which occurs for higher density mud layers, will cause an increase of buoyancy. he sinkage will consequently be smaller. he sinkage, for a same small under keel clearance referred to the solid bottom, is always larger above a solid bottom than above a muddy bottom. he same observations were made by Sellmeijer and Van Oortmerssen (983) and Vantorre and Coen (988), nevertheless the latter mentioned a slight increase of sinkage at low speeds. ð4þ

7 47 G. Delefortrie et al. / Ocean Engineering 37 () A working propeller generates an additional longitudinal speed which changes the pressure balance and thus the squat of the ship. As shown in Fig. the additional sinkage is more or less quadratic with the propeller rate rim As the sinkage is not constant along the ship s hull, the ship will be dynamically trimmed. For slender hulls this generally results in a larger sinkage at the stern, while full body ships mostly have a larger sinkage at the bow. When the ship navigates in a muddy area, the trim will be influenced as well and its absolute value will usually increase, due to the extra asymmetry in the buoyancy caused by the rising of the interface. he dimensionless total trim C as measured during the tests C ¼ z F z A ð5þ L PP is represented for different navigation conditions in Figs. 3 and 4, where a negative trim means a larger sinkage abaft. It can be stated that, in combination with the observations of the undulations of the interface: A rising will have the largest influence on the trim when it takes place amidships. he influence will decrease when the rising moves abaft; he trim will be smaller when the top of the rising is wider; C (-) C (-) solid mud F mud G mud H mud B mud C mud D Fig. 3. rim in function of the ship speed. Ship D. hickness of the mud layer:.5 m full scale. No propeller or rudder action. 3.9% under keel clearance referred to the water mud interface solid mud F mud G mud H mud E mud C mud D Fig. 4. rim in function of the ship speed. Ship D. hickness of the mud layer:.5 m full scale. No propeller or rudder action..% under keel clearance referred to the water mud interface. In all cases a larger rising causes a larger asymmetry and thus a larger trim. his is in accordance with the observations made by Vantorre and Coen (988). A change of trim sign when penetrating the mud as reported by Brossard et al. (99) is not observed. As for the sinkage propeller action influences the trimming of the ship. Propeller action yields a larger dynamic trim, especially with propeller action astern, see Fig. 5. his coincides with the effect of propeller action on the rising of the water mud interface as shown in Fig. 9. C (-) % -5% % 5% % propeller rate (-) mud G mud H mud B mud C mud D Fig. 5. rim as a function of the propeller rate. Ship D. hickness of the mud layer: 3 m full scale. No rudder action..% under keel clearance referred to the water mud interface. ζ MAX + h h U, -.% ukc D, -.% ukc D, +% ukc U, +% ukc E, -9.4% ukc E, -4.4% ukc Fig. 6. Maximal rising in function of ship speed and ship type. hickness of the mud layer D:.4 m model scale. No propeller or rudder action. C S (-) U, -.% ukc D, -.% ukc D, +% ukc U, +% ukc E, -9.4% ukc E, -4.4% ukc Fig. 7. Sinkage in function of the ship speed and ship type. hickness of the mud layer D:.4 m model scale. No propeller or rudder action.

8 G. Delefortrie et al. / Ocean Engineering 37 () Effect of the hull form Fig. 6 shows the maximal rising of the interface for the three ships above or in contact with the thickest mud layer D. For both container carriers the trend is more or less the same, but for the tanker the rising seems to reach a maximum at low speed. For the fuller ship the rising tends to occur fore or amidships, an increase of speed will cause a shift towards the stern of the ship, but not necessarily an increase of amplitude. For a same Froude number the dimensionless sinkage will be larger for the fuller ship, while both container carriers follow the same trend, see Fig. 7. he same can be concluded for the trim, Fig. 8, but the fuller ship has a larger bow sinkage. C,S (-) C S =.594γ R =.986 C = -.99γ R = γ (-) 5. Mathematical model 5.. Definitions A mathematical model will be built to predict the ship s squat when sailing straight ahead based on the measurements carried out with the 6 EU container carrier. he effect of a rotating propeller will be taken into account. According to uck (966) the mean sinkage of the vessel in the subcritical speed range can be modelled as C s ¼ s gðf rh Þ ð6þ With s a coefficient to be derived from regression analysis. An analogous relationship is valid for the vessel s trim C ¼ t gðf rh Þ ð7þ his relationship is also valid in muddy navigation areas, even when the ship is penetrating a highly viscous mud layer, as shown in Fig. 9. Eqs. (6) and (7) can consequently be used to predict the squat in muddy areas. However some physical awareness is needed. uck (966) developed his expressions for an open shallow water environment without any mud layer. he critical speed, which is related to the return current, will certainly be affected by the presence of a mud layer. Nonetheless Eqs. (6) and (7) will be used as a starting point to develop new expressions. 5.. Effect of the water depth Instead of trying to find a new expression for the subcritical speed in function of the muddy environmental conditions the coefficients s and t will be formulated as mud dependent C (-) U, -.% ukc D, -.% ukc D, +% ukc U, +% ukc E, -9.4% ukc E, -4.4% ukc Fig. 8. rim in function of the ship speed and ship type. hickness of the mud layer D:.4 m model scale. No propeller or rudder action. Fig. 9. Application of expressions (6) and (7). Ship D. hickness of mud layer G: 3 m full scale. No propeller or rudder action..% under keel clearance referred to the water mud interface. s, t (-) Linear trend Quadratic trend /(h-) (-) Fig.. Regression coefficients s (6) and t (7) in function of the under keel clearance above a solid bottom. Ship D. No propeller or rudder action. parameters. Furthermore even above a solid bottom some discrepancies occur in function of the under keel clearance, e.g. Fig.. o take account of very shallow water effects Eqs. (6) and (7) should thus be reformulated as C S ¼ a h þa gðf rh Þ ð8þ " # C t ¼ b þb h h þb gðf rh Þ ð9þ 5.3. he hydrodynamically equivalent depth he effect of the presence of the mud layer can be modelled with Eqs. (8) and (9) using a hydrodynamically equivalent depth h* instead of the real depth h. With h the thickness of the mud layer and h the height of the upper lying water layer, the total depth can be written as h ¼ h þh ðþ he bottom material can vary from water over soft mud to consolidated mud. If the mud has large viscosity and density values, like sand or clay, the material will hardly move when a ship passes by and its top can be considered as the actual seabed. In this case the hydrodynamically equivalent depth h* is h* ¼ h ðþ s t

9 47 G. Delefortrie et al. / Ocean Engineering 37 () On the other hand, if the material is very fluid the mud layer cannot be considered as a solid bottom. In the limit condition of two equivalent water layers, the hydrodynamically equivalent depth is h* ¼ h þh ¼ h ðþ For intermediate situations a parameter F can be defined, so that h* ¼ h þfh rh ð3þ Particular values for the parameter F are (¼ hard layer of thickness h ) and (¼watery layer of thickness h ), F represents consequently the degree of watery behaviour of the bottom layer and is therefore called the fluidization parameter. Intuitively the fluidization parameter of the mud covering the seabed depends on the following aspects: the rheological properties (e.g. viscosity) of the mud: a decrease of the latter means a more fluid mud layer and will logically result in an increased fluidization parameter; the under keel clearance referred to the mud water interface: the fluidization parameter increases when the ship s keel is located closer to the mud or penetrates the mud. In these conditions the mud layer is stirred and will behave more fluidly. he assumption the mud layer does not affect the critical speed means that the parameter g will always be expressed with h instead of h* Effect of the mud layer Without propulsion he effect of the mud layer on the hydrodynamic force balance could be modelled using the fluidization parameter. Figs. and show the effect of using the same concept on the regression coefficient s. A reasonable agreement can be observed. he hydrodynamic equivalent under keel clearances shown in Figs. and have been determined with a fluidization parameter that takes into account the position of the ship s keel referred to the top of the mud, expressed by P h ¼ h ð4þ h and the composition of the mud. he following interpretations can be derived from Figs. 3 and 4: When penetrating the mud the fluidization parameter is significantly larger than, meaning that for an equal ship s s (-).8 Solid b b b /(h*-) (-) Fig.. Regression coefficients s (6) in function of the hydrodynamic equivalent under keel clearance above mud b. Ship D. No propeller or rudder action. s (-).9 Solid h h h /(h*-) (-) Fig.. Regression coefficients s (6) in function of the hydrodynamic equivalent under keel clearance above mud h. Ship D. No propeller or rudder action. Φ (-) m.5 m.75 m Π h (-) Fig. 3. Fluidization parameter to determine the hydrodynamic equivalent under keel clearance for sinkage prediction above mud layers of low viscosity. Ship D. No propeller or rudder action. Φ (-) m.5 m.75 m Π h (-) Fig. 4. Fluidization parameter to determine the hydrodynamic equivalent under keel clearance for sinkage prediction above mud layers of high viscosity. Ship D. No propeller or rudder action. speed the sinkage will be smaller due to the presence of the mud. his can be related to the changed buoyancy, nevertheless a significant density effect cannot be observed. his is in accordance with Sellmeijer and van Oortmerssen (983). On the other hand, if the keel does not penetrate the mud layer, the sinkage will become larger compared to a solid bottom and this for a same total depth. his is rather relative as the sinkage will always be smaller for larger under keel clearances above any bottom condition. he decrease of the fluidization parameter with increasing under keel clearance referred to the top of the mud layer is

10 G. Delefortrie et al. / Ocean Engineering 37 () more or less equal for all mud compositions. On the other hand, the decrease will start at a larger under keel clearance for mud layers having a low viscosity, which can be ascribed to the buoyancy effect of the higher risings in these conditions. he effect of the mud s viscosity, as mentioned in 4., has thus a significant influence on the sinkage of the ship. he fluidization parameter can consequently be written as F ¼ c ðr*þ½p h þc ðmþšþc ð5þ With r* the dimensionless density r* ¼ r r r ð6þ In Eq. (5) the following restrictions have to be taken into account: P h Z :5 ð7þ ðp h c Þr m4m crit : c ¼ ð8þ ð9þ Eq. (7) states the fluidization parameter will not decrease infinitely with increasing under keel clearance, while Eq. (8) allows a constant fluidization parameter when the keel (almost) penetrates the mud. he parameter c takes into account the higher undulations for mud layers having a viscosity below the critical one. An analogous expression can be built for the trim of the vessel. In this case the fluidization parameter can be written as F ¼ðd þd r r*þpþf þf h þr* F r þf h hr ðþ h which uses an alternative formulation for the position of the ship s keel referred to the mud layer P ¼ h h ðþ No restrictions apply to Eq. (). Both Eqs. (5) and () are valid within the experimental scope With propulsion A working propeller generates an additional longitudinal speed V that can be estimated as a function of the thrust generated by the propeller vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u :5 P V ¼ signð P Þt rpd ðþ P his additional speed changes the pressure distribution along the ship s hull and thus the squat. he factor.5 in Eq. () has been determined experimentally for an under keel clearance of 6% above a solid bottom so that the total sinkage in this condition can be written as C s ¼ s ½gðV,hÞþgðV,hÞŠ ¼ C S ðv,hþþc S ðv,hþ ð3þ he effect of the under keel clearance on the thrust dependent term needs however a new set of regression coefficients e i, see Fig. 5 C S ðv,h*þ¼ e h* þe gðv,hþ ¼ s n gðv,hþ ð4þ he total sinkage, including propeller action is then C S ¼ s gðv,hþþs n gðv,hþ ð5þ Also a new hydrodynamically equivalent depth is needed to assess the influence of the thrust in muddy navigation areas. For mud layers having a high viscosity the fluidization parameter defining h* in Eq. (4) can be written simply as F ¼ f ð6þ If the viscosity drops below the critical one, the fluidization parameter changes to F ¼ g h P h þg ð7þ his fluidization parameter does not increase further once P h Z.5. Figs. 6 and 7 show the low density mud D as an example. he total trim taking account of propeller action can be written accordingly C ¼ t gðv,hþþt n gðv,hþ ð8þ Fig. 5 shows the effect of the under keel clearance above a solid bottom linearly expressed C ðv,h*þ¼ i h* þi gðv,hþ ð9þ As for the sinkage a new hydrodynamically equivalent depth is needed to express the propeller effect in muddy areas F ¼ðj þj r r*þpþf h þf h þr* h F r þf hr ð3þ However this is only valid for propeller action ahead. Fig. 5 showed a larger longitudinal speed V is needed to predict the s n, t n (-) s n (-) Linear trend /(h-) (-) Fig. 5. Regression coefficients s n (5) and t n (8) in function of the under keel clearance above a solid bottom. Ship D. Effect of propeller action. Solid d d d /(h*-) (-) Fig. 6. Regression coefficients s n (5) in function of the hydrodynamic equivalent under keel clearance above mud D. Ship D. Effect of propeller action.

11 474 G. Delefortrie et al. / Ocean Engineering 37 () Φ (-). d d d Π h (-) C S model (-) R = C S measurement (-) Fig. 7. Fluidization parameter to determine the hydrodynamic equivalent under keel clearance for sinkage prediction above the light mud layer D. Ship D. Effect of propeller action. α (-) /(h-) (-) Fig. 8. Increase of propeller induced longitudinal speed due to astern rotation. Ship D. Influence of the under keel clearance above a solid bottom. C (V ) (-) γ (V ) (-) 6% ukc 5% ukc % ukc 3% ukc Linear trend Fig. 9. Evaluation of formula (9). Solid bottom condition, influence of propeller action, both ahead and astern. Ship D. Fig. 3. Comparison between measured and modelled sinkage values: all runs. Ship D. C model (-) R = C measurement (-) Fig. 3. Comparison between measured and modelled trim values: all runs. Ship D. C S (-) measurement model Increasing propeller rate % -6% Fig. 3. Comparison between measured and modelled sinkage values. % under keel clearance above a solid bottom. Ship D. Runs with propeller action are labelled. effect of propeller action astern on the trim. For trim Eq. () should be replaced by s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :5 P V ¼ ausignð P Þ rpd ð3þ With a¼ for positive values of the propeller s thrust. For an astern rotating propeller the increase a depends on the under keel clearance, see Fig. 8 P o : a ¼ k h* þk ð3þ When Eq. (3) is used to correct V, Eq. (9) can be used to predict the trim for any propeller rate above a solid bottom, see Fig. 9. he correction coefficient a depends also on the composition of the mud layer. he hydrodynamically equivalent depth in Eq. (3) is a function of a fluidization parameter. For mud layers below the critical viscosity this parameter can be determined as F ¼ p PþF þ h ðf h þr*f hr Þ ð33þ

12 G. Delefortrie et al. / Ocean Engineering 37 () measurement model 6. Conclusions and recommendations C (-) % % -6% Fig. 33. Comparison between measured and modelled trim values. % under keel clearance above a solid bottom. Ship D. Runs with propeller action are labelled..4. measurement model For a same small under keel clearance referred to the solid bottom the sinkage will be mostly smaller when a mud layer is present. his is not always the case for the ship s trim. he squat of the ship can be related to the observed undulations of the water mud interface, that become dominant once the viscosity of the mud layer is below a critical one. A mathematical model predicting fairly well the ship s squat for container carriers has been built taking into account the bottom conditions and propeller action. herefore, the principle of a hydrodynamically equivalent depth has been used. For several fluidization parameters a different formulation is needed depending on the viscous characteristics of the mud layers. his can be linked to a critical viscosity as observed with the undulations of the water mud interface. As this critical viscosity lies somewhere between. and.9 Pa s a linear interpolation between the formulae should be applied within this viscosity range. Future efforts will be undertaken to include more parameters (drift angle,y) in the mathematical model and to investigate how the mud layer affects the critical speed regimes.. C S (-) While more viscous mud layers can use a simpler expression F ¼ q PþF 5.5. Results % Increasing propeller rate -6% Fig. 34. Comparison between measured and modelled sinkage values. Ship D. hickness of the mud layer c : 3 m full scale. 7.% under keel clearance referred to the water mud interface. Runs with propeller action are labelled. C (-) measurement model 6% % -6% Fig. 35. Comparison between measured and modelled trim values. Ship D. hickness of the mud layer c : 3 m full scale. 7.% under keel clearance referred to the water mud interface. Runs with propeller action are labelled. ð34þ he overall comparison between measured and modelled values is shown in Figs. 3 and 3. Some outliers occur, but the overall accuracy is fairly well. Some spot checks in different conditions are shown in Figs Acknowledgements he data presented in this article were obtained during the research project Determination of the nautical bottom in the harbour of Zeebrugge: Nautical implications, which was carried out co-operatively by Ghent University and Flanders Hydraulics, commissioned by.v. Noordzee & Kust (Ostend, Belgium) in the frame of the optimisation of the maintenance dredging contract for the harbour of Zeebrugge, financed by the Department Maritime Access, a division of the Mobility and Public Works department of the Flemish Government. 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