Delefortrie 2010 Squat Prediction in Muddy Navigation Areas

8/6/2019 Delefortrie 2010 Squat Prediction in Muddy Navigation Areas

http://slidepdf.com/reader/full/delefortrie-2010-squat-prediction-in-muddy-navigation-areas 1/13

Squat prediction in muddy navigation areas

G. Delefortriea,n, M. Vantorre b, K. Eloot a,b, J. Verwilligen a, E. Lataire b

a Flanders Hydraulics Research, Berchemlei 115, 2140 Antwerp, Belgiumb Ghent University, Maritime Technology Division, Technologiepark 904, 9052 Ghent, Belgium

a r t i c l e i n f o

Article history:

Received 5 May 2010

Accepted 22 August 2010

Editor-in-Chief: A.I. IncecikAvailable online 15 September 2010

Keywords:

Mud

Sinkage

Trim

Mathematical model

a b s t r a c t

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 2001 to 2004 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. This article

presents a mathematical model to predict the squat in muddy navigation areas.

& 2010 Elsevier Ltd. All rights reserved.

1. 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 bottomtouching as already mentioned by Constantine (1960).

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. (2007). When

initially the ship has a small under keel clearance referred to the

mud layer, she may hit the mud layer due to squat. This mud,

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.

2. State of the art

2.1. General research on squat

Scientific research on squat took off with Constantine (1960)

who discussed the different squat behaviour for subcritical,

critical and supercritical vessel speeds. In the subcritical domain

(F rho1) Tuck (1966) proved that for open water conditions of

constant depth the sinkage and trim of the vessel to be linear with

the parameter

g F rhð Þ ¼F 2rh

ffiffiffiffiffiffiffiffiffiffiffiffiffi1ÀF 2rhq

ð1Þ

In which F rh represents the depth related Froude number

F 2rh ¼V 2

ghð2Þ

This theory was later extended to dredged channels by Beck

et al. (1975). Naghdi and Rubin (1984) offer some reflections on

Tuck’s theory and introduce a new one. An analogous theory has

been developed by Cong and Hsiung (1991).

Ankudinov and Daggett (1996) however are pessimistic about

the complexity of numerical theories. For this reason, several

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/oceaneng

Ocean Engineering

0029-8018/$- see front matter & 2010 Elsevier Ltd. All rights reserved.

doi:10.1016/j.oceaneng.2010.08.003

n Corresponding author. Tel.: +32 3 224 69 62; fax: +32 3 224 60 36.

E-mail addresses: [email protected] (G. Delefortrie),

[email protected] (M. Vantorre), [email protected]

(K. Eloot), [email protected] (J. Verwilligen),

[email protected] (E. Lataire).

Ocean Engineering 37 (2010) 1464–1476

http://-/?-

http://www.elsevier.com/locate/oceaneng

http://dx.doi.org/10.1016/j.oceaneng.2010.08.003

mailto:[email protected]















http://www.elsevier.com/locate/oceaneng

http://-/?-



authors carried out experimental research as Dand (1972) and

Gourlay (2000) – who offers a solution for squat prediction with

random bottom conditions – to endorse their theories. Jiang and

Henn (2003) present a numerical method valid from subcritical to

supercritical speed. An overview of slender body methods is given

in Gourlay (2008).

More practical methods based on experimental research are

presented by Barrass (1979), however his results could not be

validated by Seren et al. (1983). Barrass (2004) gives an overview

of the work he performed on squat. More general overviews are

given by Dumas (1982); Blaauw and Van der Knaap (1983),

Millward (1990) and PIANC (1997) working group 30.

Interesting full scale measurements were carried out byAnkudinov et al. (2000), Stocks et al. (2004), Harting et al.

(2009) and Harting and Reinking (2002) 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

(1985), Martin and Puls (1986), de Koning Gans and Boonstra

(2007) and Eloot et al. (2008).

2.2. Research on squat in muddy areas

The 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.

The oldest results are presented by Sellmeijer and van Oortmerssen

(1983) who also registered undulations in the water mud

interface. The sinkage is less above mud in comparison with the

solid bottom condition and decreases with increasing layer

thickness. The mud density does not seem to have any effect.

Vantorre and Coen (1988) 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. The 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. The amplitude of the elevation can exceed

the mud layer thickness several times.

The latter occurs at a speed, which for inviscid fluids is given

by the theoretical expression

V crit ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8

27 gh1 1À

r1

r2

ð1Àm1Þ3

s ð3Þ

m1 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 1 refers to the water layer, subscript 2 to

the mud layer.

Nomenclature

AEP expanded area ratio of propeller (–)

AR rudder area (m2)

ai regression coefficient, (i ¼0,1) (–)

B ship beam (m)

b mud type, Table 2 (–)

bi regression coefficient, (i ¼0,1,2) (–)c mud type, Table 2 (–)

C B block coefficient (–)

c i regression coefficient, (i ¼0,1,2) (–)

C S dimensionless sinkage, Eq. (4) (–)

C T dimensionless trim, Eq. (5) (–)

D 6000 TEU container ship model (–)

d mud type, Table 2 (–)

di regression coefficient, (i ¼0,r) (–)

DP propeller diameter (–)

E tanker model (–)

e mud type, Table 2 (–)

ei regression coefficient, (i ¼0,1) (–)

f mud type, Table 2 (–)

f 0 regression coefficient (–)F rh depth related Froude number, Eq. (2) (–)

g mud type, Table 2 (–)

g i regression coefficient, (i ¼0,h) (–)

h total depth (m) mud type, Table 2 (–)

h* hydrodynamically equivalent depth (m)

i0 regression coefficient (–)

i1 regression coefficient (–)

ji regression coefficient, (i ¼0, r) (–)

ki regression coefficient, (i ¼0, r) (–)

LPP ship length (m)

P propeller pitch (m)

p0 regression coefficient (–)

q0 regression coefficient (–)

S solid bottom condition (–)

si regression coefficient (i¼0,n) (–)

T ship draft (m)

TEU (number of) twenty feet equivalent unit containers (–)

T P thrust (N)

U 8000 TEU container ship model (–)

ukc under keel clearance (–)

V ship speed (m/s)V T propeller induced speed, Eq. (22) (m/s)

x longitudinal coordinate, positive towards the stern

(m)

z A sinkage aft (m)

z F sinkage fore (m)

a increase parameter, Eq. (31) (–)

g parameter, Eq. (1) (–)

z amplitude of rising (m)

zMAX maximal amplitude of rising (m)

m dynamic viscosity (Pa s)

mcrit critical dynamic viscosity (Pa s)

r density (kg/m3)

r* dimensionless density, Eq. (16) ( –)

P keel penetration parameter, Eq. (21) (–)

Ph keel penetration parameter, Eq. (14) (–)

F fluidization parameter (–)

Fij regression coefficient (i¼0,h) ( j ¼0,r) (–)

F0ij regression coefficient (i¼0,h) ( j ¼0,r) (–)

F00ij regression coefficient (i¼0,h) ( j ¼0,r) (–)

F00000 regression coefficient (–)

Subscripts

n propeller

1 related to water

2 related to mud

G. Delefortrie et al. / Ocean Engineering 37 (2010) 1464–1476 1465



The 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. (1990) 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:

The rigidity of the mud has only a small effect;

The 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.

The trim of the vessel is only significantly affected by rigid

mud. In this case an increase of trim with increasing density

gradient was observed. The sign of the trim changes when

penetrating the mud.

Doctors et al. (1996) showed that for the ship hydrodynamics a

shallow water approach can serve as a quite reliable approxima-

tion for analyzing the case of a viscous lower layer, where themud 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

3.1. Test facilities

The new squat formulae presented in this article are all

derived from experimental research carried out from 2001 until

2004 at the Towing Tank for Manoeuvres in Shallow Water—

cooperation Flanders Hydraulics Research, Ghent University. The

formulae are valid within the range of conditions covered by this

experimental research. The shallow water towing tank (88 m Â 7

m Â 0.5 m) is equipped with a planar motion carriage, a wave

generator and an auxiliary carriage for ship–ship interaction tests.

Thanks to computerized control and data-acquisition, the facil-

ities are operated in a fully automated manner. The carriage runs

24/7 without the need for permanent surveillance.

3.2. Ship models

Most runs in muddy navigation areas were carried out

with (Table 1) a 6000 TEU container carrier (D), which was the

design ship for the Belgian harbours at that time. The mathema-

tical 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 8000 TEU container carrier (U ) and a

scale model of a tanker (E ) to assess the influence of the hull form,

see 4.4.

3.3. 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.

The selected bottom conditions are represented in Table 2. The

density–viscosity combinations were based on measurements of

density and rheology profiles carried out in the outer harbour of

Zeebrugge, Belgium in 1997–1998. A mud layer configuration is

defined by two characters: a letter (b,y,h) denoting the material

characteristics and Figs. 1–3 re-presenting the layer thickness.

Tests carried out above a solid bottom are referred to as ‘‘ S ’’.The gross under keel clearance relative to the tank bottom was

varied between 7% and 32% of draft, yielding À12% to +21% ukc

relative to the mud–water interface. Throughout this paper, the

interface water–mud will be used as a reference for expressing

the under keel clearance, unless specified otherwise.

3.4. Test types

Stationary captive motion model tests were carried out in each

combination of mud layer and realistic under keel clearance at

different speeds, from 2 kn ahead to 10 kn ahead at least in steps

of 2 kn (full scale values).

Table 2

Bottom conditions and tested models.

Mud type Density (kg/m3) Dynamic viscosity (Pa s) Layer thickness

0.75 m ‘‘1’’ 1.50 m ‘‘2’’ 3.00 m ‘‘3’’

‘‘d’’ 1100 0.03 D/E D/E D/E/U

‘‘c’’ 1150 0.06 D D D

‘‘b’’ 1180 0.10 D D D

‘‘f’’ 1200 0.11 – D –

‘‘h’’ 1210 0.19 D/E D/E D

‘‘e’’ 1260 0.29 – D –

‘‘g’’ 1250 0.46 D/E D/E D/E

fresh water 1000 0.0010

sea water 1025 0.0012

‘‘S’’ solid bottom

Table 1

Ship models (even keel).

Model D U E

Scale 1/75 1/80 1/75

LPP (m) 289.8 331.8 286.8

B (m) 40.25 42.8 46.8

T (m) 13.50 14.54 15.5

C B 0.59 0.65 0.82

AR (m2) 60.96 83.13 98.34

# blades 5 6 5

DP (m) 8.145 8.46 7.73

P /DP (–) 0.97 1.00 0.65

AEP (–) 0.8 0.96 0.62




The propeller rate was varied at 6 kn ahead between À60% and

100% 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.

3.5. Measurements

The 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 undula-

tions 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

Fig. 2. Arrangement of the wave meters in the towing tank.

0

0.5

1

1.5

2

-1

dy = 0.28 B

dy = 1.30 B

dy = 2.32 B

(−)+ h2

h2

ζ

(−) LPP

x

-0.5 0 0.5 1 1.5 2

Fig. 3. Undulations of the interface at various lateral distances of the ship D. Mud

f 2, +3.9% ukc, F rh¼0.38, no propeller action. The ship is represented taking squat

into account and sails to the left.Fig. 1. Positioning the tripods in the towing tank.




tripod (see Fig. 1). In total three tripods were assembled in order

to register the undulations of the interface.

The mufo consists of a floater, which is resistant to the artificial

mud. The density of the floater is situated between the water and

the mud density. The position of the floater is therefore similar to

the amplitude of the interface. The floater is attached to a disk,

which reflects a laser beam. The variations of the laser beam are

measured 20 times per second and register the actual position of

the mud layer.The wafo is based upon the principles of the potentiometer.

A constant electric current is sent through a string which has a

homogeneous resistance. The 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.

Three tripods were placed in the towing tank as close as

possible to the passing ship. The lateral distance between the ship

and the mufos is as shown in Fig. 2.

Depending on the ship’s velocity the position of the interface

was measured 10–20 times per second. The 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

4.1. 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.

The maximal amplitude closest to the ship is represented in Fig. 4.

The rising increases with increasing speed, but this increase is

limited once the undulations are behind the ship, see Fig. 5. This 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 0.12 and 0.18 Pa s.

A significant undulation is always observed when the shipnavigates in contact with the mud layer, see Fig. 6. The 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.

The transitory situation is a rising occurring in two phases.

0

0.25

0.5

0.75

1

0

Frh (-)

mud G mud H mud B mud C

mud D amidship aft

( − )

L P P x

0.1 0.2 0.3 0.4 0.5

Fig. 5. Longitudinal position, at which the rising is maximal, in function of ship

speed. Ship D. Thickness of the mud layer: 3 m full scale. No propeller or rudder

action. 9.8% under keel clearance referred to the water mud interface.

11.1

1.21.3

1.41.5

1.61.7

1.81.9

2

0

Frh (-)

mud F mud H mud B mud C mud D

( − )

h 2

M A X + h 2

0.1 0.2 0.3 0.4 0.5

Fig. 6. Maximal rising in function of ship speed. Ship D. Thickness of the mud

layer: 1.5 m full scale. No propeller or rudder action. À 1.1% under keel clearance

referred to the water mud interface.

-0.25

0

0.25

0.5

0.75

1

0

Frh (-)

mud F mud H mud B mud C

mud D amidship aft

( − )

L P P

x

0.1 0.2 0.3 0.4 0.5

Fig. 7. Longitudinal position, at which the rising is maximal, in function of ship

speed. Ship D. Thickness of the mud layer: 1.5 m full scale. No propeller or rudder

action. À1.1% under keel clearance referred to the water mud interface.

1

1.2

1.4

1.6

1.8

2

2.2

-100%

propeller rate (-)

mud G mud H mud B mud C mud D

( − )

h 2

M A X + h 2

-50% 0% 50% 100%

Fig. 8. Maximal rising in function of the propeller rate. Ship D. Thickness of the

mud layer: 3 m full scale. No rudder action. Ship speed: 6 kn full scale. À12%

under keel clearance referred to the water mud interface.

1

1.1

1.2

1.3

1.41.5

1.6

1.7

1.8

1.9

2

0

Frh (-)


h 2

M A X +

h 2

( − )

0.1 0.2 0.3 0.4 0.5

Fig. 4. Maximal rising in function of ship speed. Ship D. Thickness of the mud

layer: 3 m full scale. No propeller or rudder action. 9.8% 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.

The 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 explana-

tion is the higher viscosity of the mud layers in this experimental

program and the assumption of inviscid fluids in Eq. (3).

4.2. Sinkage

The mean sinkage as measured during the tests will be

represented dimensionless as C S

C s ¼ 100 z F þ z A

2LPP ð4Þ

The values z A and z F are positive downwards. Figs. 10 and 11

give an overview of the ship’s sinkage in function of the speed for

different bottom conditions. The 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. The 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. The sinkage will consequently be smaller.

The 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. The same observations were made by Sellmeijer

and Van Oortmerssen (1983) and Vantorre and Coen (1988),

nevertheless the latter mentioned a slight increase of sinkage at

low speeds.

-0.25

0

0.25

0.5

0.75

-100%

propeller rate (-)

mud G mud H mud B mud C

mud D aft amidship

( −

)

L P P

x

-50% 0% 50% 100%

Fig. 9. Longitudinal position, at which the rising is maximal, in function of the propeller rate. Ship D. Thickness of the mud layer: 3 m full scale. No rudder action. Ship

speed: 6 kn full scale. À12.2% under keel clearance referred to the water mud interface.

0

0.02

0.04

0.06

0.080.1

0.12

0.14

0.16

0

Frh (-)

C S

( - )

solid mud F mud G mud H

mud B mud C mud D

0.1 0.2 0.3 0.4 0.5

Fig. 10. Sinkage in function of the ship speed. Ship D. Thickness of the mud layer:

1.5 m full scale. No propeller or rudder action. 3.9% under keel clearance referred

to the water mud interface.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0

Frh (-)

C S

( - )

solid mud F mud G mud Hmud E mud C mud D

0.1 0.2 0.3 0.4 0.5

Fig. 11. Sinkage in function of the ship speed. Ship D. Thickness of the mud layer:

1.5 m full scale. No propeller or rudder action. À 1.1% under keel clearance referred


0

0.01

0.02

0.03

0.04

0.05

0.06

-100%

propeller rate (-)


C S

( - )

-50% 0% 50% 100%

Fig. 12. Sinkage in function of the propeller rate. Ship D. Thickness of the mud

layer: 3 m full scale. No rudder action. À12.2% under keel clearance referred to the

water mud interface.




A working propeller generates an additional longitudinal speed

which changes the pressure balance and thus the squat of the

ship. As shown in Fig. 12 the additional sinkage is more or less

quadratic with the propeller rate.

4.3. Trim

As the sinkage is not constant along the ship’s hull, the ship

will be dynamically trimmed. For slender hulls this generallyresults 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.

The dimensionless total trim C T as measured during the tests

C T ¼ 100 z F À z A

LPP ð5Þ

is represented for different navigation conditions in Figs. 13 and

14, where a negative trim means a larger sinkage abaft. It can be

stated that, in combination with the observations of the undula-

tions of the interface:

A rising will have the largest influence on the trim when it

takes place amidships. The influence will decrease when the

rising moves abaft;

The trim will be smaller when the top of the rising is wider;

In all cases a larger rising causes a larger asymmetry and thus a

larger trim. This is in accordance with the observations made

by Vantorre and Coen (1988).

A change of trim sign when penetrating the mud as reported

by Brossard et al. (1990) is not observed.

As for the sinkage propeller action influences the trimming of

the ship. Propeller action yields a larger dynamic trim, especiallywith propeller action astern, see Fig. 15. This coincides with the

effect of propeller action on the rising of the water mud interface

as shown in Fig. 9.

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0

solid mud F mud G mud Hmud E mud C mud D

C T

( - )

Frh (-)

0.1 0.2 0.3 0.4 0.5

Fig. 14. Trim in function of the ship speed. Ship D. Thickness of the mud layer:

1.5 m full scale. No propeller or rudder action. À1.1% under keel clearance referred


-0.05-0.045

-0.04

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

-100%

propeller rate (-)


C T

( - )

-50% 0% 50% 100%

Fig. 15. Trim as a function of the propeller rate. Ship D. Thickness of the mud

layer: 3 m full scale. No rudder action. À12.2% under keel clearance referred to the

water mud interface.

1

1.2

1.4

1.6

1.8

2

0

Frh (-)

U, -12.2% ukc D, -12.2% ukc D, +10% ukc

U, +10% ukc E, -9.4% ukc E, -4.4% ukc

( − )

h 2

M A X +

h 2

0.1 0.2 0.3 0.4 0.5

Fig. 16. Maximal rising in function of ship speed and ship type. Thickness of the

mud layer D: 0.04 m model scale. No propeller or rudder action.

0

0.05

0.1

0.15

0.2

0.25

0

U, -12.2% ukc D, -12.2% ukc D, +10% ukc

U, +10% ukc E, -9.4% ukc E, -4.4% ukc

C S

( - )

Frh (-)

0.1 0.2 0.3 0.4 0.5

Fig. 17. Sinkage in function of the ship speed and ship type. Thickness of the mud

layer D: 0.04 m model scale. No propeller or rudder action.

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0

solid mud F mud G mud H

mud B mud C mud D

C T

( - )

Frh (-)

0.1 0.2 0.3 0.4 0.5

Fig. 13. Trim in function of the ship speed. Ship D. Thickness of the mud layer:

1.5 m full scale. No propeller or rudder action. 3.9% under keel clearance referred





4.4. Effect of the hull form

Fig. 16 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. 17. The same can be concluded for the trim,

Fig. 18, but the fuller ship has a larger bow sinkage.

5. Mathematical model

5.1. 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 6000 TEU container carrier. The effect of a rotating

propeller will be taken into account. According to Tuck (1966) themean sinkage of the vessel in the subcritical speed range can be

modelled as

C s ¼ s0gðF rhÞ ð6Þ

With s0 a coefficient to be derived from regression analysis. An

analogous relationship is valid for the vessel’s trim

C T ¼ t 0gðF rhÞ ð7Þ

This relationship is also valid in muddy navigation areas, even

when the ship is penetrating a highly viscous mud layer, as shown

in Fig. 19.

Eqs. (6) and (7) can consequently be used to predict the squat

in muddy areas. However some physical awareness is needed.

Tuck (1966) developed his expressions for an open shallow waterenvironment without any mud layer. The 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.

5.2. 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 s0 and t 0 will be formulated as mud dependent

parameters. Furthermore even above a solid bottom some

discrepancies occur in function of the under keel clearance,e.g. Fig. 20.

To take account of very shallow water effects Eqs. (6) and (7)

should thus be reformulated as

C S ¼ a1T

hÀT þa0

gðF rhÞ ð8Þ

C t ¼ b2T

hÀT

2

þ b1T

hÀT þ b0

" #gðF rhÞ ð9Þ

5.3. The hydrodynamically equivalent depth

The 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 h2 the thickness of the mud layer and h1 the height of the

upper lying water layer, the total depth can be written as

h ¼ h1 þh2 ð10Þ

The 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 h1 ð11Þ

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0

T/(h-T) (-)

Linear trendQuadratic trend

s0

t0

s 0 ,

t 0 ( - )

2 4 6 8 10 12

Fig. 20. Regression coefficients s0 (6) and t 0 (7) in function of the under keel

clearance above a solid bottom. Ship D. No propeller or rudder action.

-0.06

-0.04

-0.02

0

0.02

0.040.06

0.08

0

U, -12.2% ukc D, -12.2% ukc D, +10% ukc

U, +10% ukc E, -9.4% ukc E, -4.4% ukc

C T

( - )

Frh (-)

0.1 0.2 0.3 0.4 0.5

Fig. 18. Trim in function of the ship speed and ship type. Thickness of the mud

layer D: 0.04 m model scale. No propeller or rudder action.

CS = 0.5942γ

R2 = 0.9826

CT = -0.1919γ

R2

= 0.9893

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0

C T , S

( - )

0.05 0.1 0.15 0.2 0.25

γ (-)

Fig. 19. Application of expressions (6) and (7). Ship D. Thickness of mud layer G:

3 m full scale. No propeller or rudder action. À 12.2% under keel clearance referred





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 h1 þh2 ¼ h ð12Þ

For intermediate situations a parameter F can be defined, so

that

h* ¼ h1 þFh2rh ð13Þ

Particular values for the parameter F are 0 ( ¼ hard layer of

thickness h2) and 1 ( ¼watery layer of thickness h2), 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 theseconditions the mud layer is stirred and will behave more fluidly.

The assumption the mud layer does not affect the critical

speed means that the parameter gwill always be expressed with h

instead of h*.

5.4. Effect of the mud layer

5.4.1. Without propulsion

The effect of the mud layer on the hydrodynamic force

balance could be modelled using the fluidization parameter.

Figs. 21 and 22 show the effect of using the same concept on the

regression coefficient s0. A reasonable agreement can be observed.

The hydrodynamic equivalent under keel clearances shown inFigs. 21 and 22 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

Ph ¼T Àh1

h1ð14Þ

and the composition of the mud.

The following interpretations can be derived from Figs. 23 and 24:

When penetrating the mud the fluidization parameter is

significantly larger than 1, meaning that for an equal ship’s

speed the sinkage will be smaller due to the presence of the

mud. This can be related to the changed buoyancy, never-

theless a significant density effect cannot be observed. This is

in accordance with Sellmeijer and van Oortmerssen (1983).

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. This is rather relative

as the sinkage will always be smaller for larger under keel

clearances above any bottom condition.

The decrease of the fluidization parameter with increasing

under keel clearance referred to the top of the mud layer is

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

Solid h1 h2 h3

s 0 ( - )

T/(h*-T) (-)

2 4 6 8 10 12 14

Fig. 22. Regression coefficients s0 (6) in function of the hydrodynamic equivalent

under keel clearance above mud h. Ship D. No propeller or rudder action.

-1

-0.5

0

0.5

1

1.5

2

2.5

-0.2

3 m 1.5 m 0.75 m

Φ ( - )

Πh (-)

-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

Fig. 23. 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.

-1

-0.5

0

0.5

1

1.5

2

2.5

-0.2

3 m 1.5 m 0.75 m

Φ ( -

)

Πh (-)

-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2


keel clearance for sinkage prediction above mud layers of high viscosity. Ship D.

No propeller or rudder action.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0

Solid b1 b2 b3

s 0 ( - )

T/(h*-T) (-)

2 4 6 8 10 12

Fig. 21. Regression coefficients s0 (6) in function of the hydrodynamic equivalent

under keel clearance above mud b. Ship D. No propeller or rudder action.




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.

The effect of the mud’s viscosity, as mentioned in 4.1, has thus

a significant influence on the sinkage of the ship. The fluidization

parameter can consequently be written as

F¼ c 1ðr*Þ½Ph þc 2ðmÞ� þc 0 ð15Þ

With r* the dimensionless density

r* ¼r2Àr1

r1

ð16Þ

In Eq. (15) the following restrictions have to be taken into

account:

PhZÀ0:15 ð17Þ

ðPhÀc 2Þr0 ð18Þ

m4mcrit : c 2 ¼ 0 ð19Þ

Eq. (17) states the fluidization parameter will not decrease

infinitely with increasing under keel clearance, while Eq. (18) allowsa constant fluidization parameter when the keel (almost) penetrates

the mud. The parameter c 2 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

F¼ ðd0 þdrr*ÞPþF00 þFh0T

h2þr* F0rþFhr

T

h2

ð20Þ

which uses an alternative formulation for the position of the

ship’s keel referred to the mud layer

P¼T Àh1

h2ð21Þ

No restrictions apply to Eq. (20). Both Eqs. (15) and (20) are

valid within the experimental scope.

5.4.2. With propulsion

A working propeller generates an additional longitudinal speed

V T that can be estimated as a function of the thrust generated by

the propeller

V T ¼ signðT P Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1:5T P

rpD2P

v uut ð22Þ

This additional speed changes the pressure distribution along

the ship’s hull and thus the squat. The factor 1.5 in Eq. (22) has

been determined experimentally for an under keel clearance of

26% above a solid bottom so that the total sinkage in this

condition can be written as

C s ¼ s0½gðV ,hÞ þgðV T ,hÞ� ¼ C S ðV ,hÞ þ C S ðV T ,hÞ ð23Þ

The effect of the under keel clearance on the thrust dependent

term needs however a new set of regression coefficients ei, see

Fig. 25

C S ðV T ,h*Þ ¼ e1T

h*ÀT þe0

gðV T ,hÞ

¼ sngðV T ,hÞ ð24Þ

The total sinkage, including propeller action is then

C S ¼ s0gðV ,hÞ þsngðV T ,hÞ ð25Þ

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. (24) can be written simply as

F¼ f 0 ð26Þ

If the viscosity drops below the critical one, the fluidization

parameter changes to

F¼ g hPh þ g 0 ð27Þ

This fluidization parameter does not increase further once

PhZ0.05. Figs. 26 and 27 show the low density mud D as an

example.

The total trim taking account of propeller action can be written

accordingly

C T ¼ t 0gðV ,hÞ þt ngðV T ,hÞ ð28Þ

Fig. 25 shows the effect of the under keel clearance above a

solid bottom linearly expressed

C T ðV T ,h*Þ ¼ i1T

h*ÀT þi0 gðV T ,hÞ ð29Þ

As for the sinkage a new hydrodynamically equivalent depth is

needed to express the propeller effect in muddy areas

F¼ ð j0 þ jrr*ÞPþF000 þF0

h0

h2

T þr* F0

0rþF0hr

h2

T

ð30Þ

However this is only valid for propeller action ahead. Fig. 15

showed a larger longitudinal speed V T is needed to predict the

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0

Solid d1 d2 d3

s n

( - )

T/(h*-T) (-)

5 10 15 20 25

Fig. 26. Regression coefficients sn (25) in function of the hydrodynamic equivalent

under keel clearance above mud D. Ship D. Effect of propeller action.

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0

Linear trend

s n ,

t n ( - )

2 4 6 8 10 12

T/(h-T) (-)

Fig. 25. Regression coefficients sn (25) and t n (28) in function of the under keel

clearance above a solid bottom. Ship D. Effect of propeller action.




effect of propeller action astern on the trim. For trim Eq. (22)

should be replaced by

V T ¼aUsignðT P Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1:5T P

rpD2

s ð31Þ

With a¼1 for positive values of the propeller’s thrust. For an

astern rotating propeller the increase a depends on the under keel

clearance, see Fig. 28

T P o0 : a¼ k1T

h*ÀT þk0 ð32Þ

When Eq. (32) is used to correct V T , Eq. (29) can be used to

predict the trim for any propeller rate above a solid bottom,

see Fig. 29.

The correction coefficient a depends also on the composition

of the mud layer. The hydrodynamically equivalent depth in

Eq. (32) is a function of a fluidization parameter. For mud

layers below the critical viscosity this parameter can be

determined as

F0 ¼ p0PþF0 000 þ

h2

T

ðF0 0h0 þr*F00

hrÞ ð33Þ

-0.8

-0.6

-0.4-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-0.2

Πh (-)

d1 d2 d3

Φ ( -

)

-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2


keel clearance for sinkage prediction above the light mud layer D. Ship D. Effect of

propeller action.

-2.4

-2.2

-2

-1.8

-1.6

0

α ( - )

T/(h-T) (-)

2 4 6 8 10 12

Fig. 28. Increase of propeller induced longitudinal speed due to astern rotation.

Ship D. Influence of the under keel clearance above a solid bottom.

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0

26% ukc 15% ukc 10% ukc 32% ukc Linear trend

γ (VT) (-)

C T

( V T ) ( - )

0.2 0.4 0.6 0.8 1 1.2

Fig. 29. Evaluation of formula (29). Solid bottom condition, influence of propeller

action, both ahead and astern. Ship D.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

-0.05

C S m o d e l ( - )

CS measurement (-)

R2 = 0.9502

0.05 0.1 0.15 0.20

Fig. 30. Comparison between measured and modelled sinkage values: all runs.

Ship D.

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

-0.06

C T m o d e l

( - )

R2 = 0.9289

CT measurement (-)

-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01

Fig. 31. Comparison between measured and modelled trim values: all runs.

Ship D.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.160.18

0

measurement model

-60%

100%Increasing propeller rate C S

( - )

Frh (-)

0.1 0.2 0.3 0.4 0.5

Fig. 32. Comparison between measured and modelled sinkage values. 10% underkeel clearance above a solid bottom. Ship D. Runs with propeller action are

labelled.




While more viscous mud layers can use a simpler expression

F0 ¼ q0PþF00000 ð34Þ

5.5. Results

The overall comparison between measured and modelled

values is shown in Figs. 30 and 31. Some outliers occur, but the

overall accuracy is fairly well.

Some spot checks in different conditions are shown in Figs. 32–35.

6. Conclusions and recommendations

For a same small under keel clearance referred to the solid

bottom the sinkage will be mostly smaller when a mud layer is

present. This is not always the case for the ship’s trim. The 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 squatfor container carriers has been built taking into account the

bottom conditions and propeller action. Therefore, the principle of

a hydrodynamically equivalent depth has been used. For several

fluidization parameters a different formulation is needed depend-

ing on the viscous characteristics of the mud layers. This can be

linked to a critical viscosity as observed with the undulations of

the water mud interface. As this critical viscosity lies somewhere

between 0.11 and 0.19 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.

Acknowledgements

The 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 T.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.

References

Ankudinov, V., Daggett, L., 1996. Squat predictions for manoeuvring applications.In: Proceedings of MARSIM 1996 (International Conference on MarineSimulation and Manoeuvrability), Copenhagen, Denmark, pp. 467–495.

Ankudinov, V., Daggett, L., Hewlett, J., Jakobsen, B. 2000. Prototype measurement of ship sinkage in confined water. In: Proceedings of MARSIM 2000 (InternationalConference on Marine Simulation and Manoeuvrability), Orlando, USA, p 233.

Barras, C., 1979. The phenomena of ship squat. International Shipbuilding Progress26, 44–47.

Barras, C. 2004. Thirty-two years of research into ship squat. Squat workshop 2004,Elsfleth, Germany, pp. 1–25.

Beck, R., Newman, J., Tuck, E., 1975. Hydrodynamic forces on ships in dredgedchannels. Journal of Ship Research 19 (3), 166–171.

Blaauw, H., Van der Knaap, F., 1983. Prediction of squat of ships sailing in restrictedwater. In: Proceedings of the International Harbour Congress, pp. 81–93.

Brossard, C., Delouis, A., Galichon, P., Granboulan, J., Monadier, P., 1990. Navigabilityin channels subjectto siltation. In: Proceedings of the22nd International CoastalEngineering Conference, Delft, The Netherlands, pp. 3088–3101.

Cong, L., Hsiung, C., 1991. Computing wave resistance, wave profile and sinkageand trim of transom stern ships. Marine and Offshore Operations, 99–112.Constantine, T., 1960. On the movement of ships in restricted waterways. Journal

of Fluid Mechanics 9 (2), 247–257.Dand, I., 1972. On full form ships in shallow water: some methods for the

prediction of squat in subcritical flows. National Physical Laboratory, ShipDivision, Ship Report 160.

de KoningGans, H., Boonstra,H., 2007. Squateffects of very large containershipswithdrift in a harbor environment. In: Proceedings of the International Maritime-portTechnology and Development Conference, Singapore, pp. 613–620.

Delefortrie, G., Vantorre, M., Verzhbitskaya, E., Seynaeve, K., 2007. Evaluation of safety of navigation in muddy areas through real time manoeuvring simulation.

Journal of Waterway, Port, Coastal and Ocean Engineering 133 (2), 125–135.Doctors, L.J., Zilman, G., Miloh, T., 1996. The influence of a bottom mud layer on the

steady-state hydrodynamics of marine vehicles. In: Proceedings of the 21stSymposium on Naval Hydrodynamics, pp. 727 – 742.

Dumas, R., 1982. Le surenfoncement des navires du a leur mouvement. RevueTechnique du Service des Phares et Balises 53, 26–58 In French.

Eloot, K., Verwilligen, J., Vantorre, M., 2008. An overview of squat measurements

for container ships in restricted water. In: Varyani, K.S. (Ed.), International

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0

measurement model

-60%

100%

60%

C T ( - )

Frh (-)

0.1 0.2 0.3 0.4 0.5

Fig. 33. Comparison between measured and modelled trim values. 10% under keel

clearance above a solid bottom. Ship D. Runs with propeller action are labelled.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0

measurement model

-60%

100%Increasing propeller rate

C S

( - )

Frh (-)

0.1 0.2 0.3 0.4 0.5

Fig. 34. Comparison between measured and modelled sinkage values. Ship D.

Thickness of the mud layer ‘‘c’’: 3 m full scale. À7.2% under keel clearance referred

to the water mud interface. Runs with propeller action are labelled.

-0.045

-0.04

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0

measurement model

-60%

100%

60%

C T

( - )

Frh

(-)

0.1 0.2 0.3 0.4 0.5

Fig. 35. Comparison between measured and modelled trim values. Ship D.

Thickness of the mud layer ‘‘c’’: 3 m full scale. À7.2% under keel clearance referred

to the water mud interface. Runs with propeller action are labelled.




Conference on Safety and Operations in Canals and Waterways SOCW 2008,15–16 September 2008 Glasgow, UK, pp. 106–116.

Gourlay, T., 2000. Mathematical and computational techniques for predicting the squatof ships. Ph.D. thesis, University of Adelaide, Department of Applied Mathematics.

Gourlay, T., 2008. Slender-body methods for predicting ship squat. OceanEngineering 35 (2), 191–200.

Harting, A., Reinking, J., 2002. SHIPS: a new method for efficient full-scale shipsquat determination. In: Proceedings of the PIANC 30th InternationalNavigation Congress, Sydney, Australia, pp. 84–91.

Harting, A., Laupichler, A., Reinking, J., 2009. Considerations on the squat of unevenly trimmed ships. Ocean Engineering 36, 193–201.

Jiang, T., Henn, R., 2003. Numerical prediction of ship’s squat and trim in shallowwater. In: Proceedings of MARSIM 2003 (International Conference on MarineSimulation and Manoeuvrability), Kanazawa, Japan.

Martin, H., Puls, D., 1986. Vertical forces, trim moments and changes of draught and trim of ships in shallow water. Schiffbauforschung 25 (3),155–159.

Millward, A., 1990. A preliminary design method for the prediction of squat inshallow water. Marine Technology 27 (1), 10–19.

Naghdi, P., Rubin, M., 1984. On the squat of a ship. Journal of Ship Research 28 (2),107–117.

PIANC, 1997. Approach channels – A guide for design. Final report of the jointWorking Group PIANC and IAPH, in cooperation with IMPA and IALA.Supplement to PIANC Bulletin, No. 95, 108 p.

Sellmeijer, R., van Oortmerssen, G., 1983. The effect of mud on tanker manoeuvres.The Royal Institution of Naval Architects, Spring Meetings 1983, paper no. 7.

Seren, D., Ferguson, A., McGregor, R., 1983. Squat—an examination of two practicalprediction methods. The Naval Architect 5, 228–230.

Stocks, D., Daggett, L., Page, Y., 2004. Maximization of ship draft in the St.LawrenceSeaway. Squat workshop 2004, Elsfleth, Germany.

Tuck, E., 1966. Shallow water flow past slender bodies. Journal of Fluid Mechanics26 (1), 81–95.

Vantorre, M., Coen, I., 1988. On sinkage and trim of vessels navigating above amud layer. The Royal Society of Flemish Engineers, International HarbourCongress.

Von Bovet, G., 1985. Tiefgangs- un Trimmwinkelanderungen von Schiffen underBerucksichtigung endlicher Wassertiefe un Einflub auf die Steuerbarkeit.Schiffbauforschung 24 (1), 3–11 (in German).


Documents

Delefortrie 2010 Squat Prediction in Muddy Navigation Areas