TECHNICAL PAPER

Experimental evaluation of ship squat in shallow waters

Mohammadreza Fathi Kazerooni • Mohammad Saeed Seif

Received: 12 April 2013 / Accepted: 19 October 2013

� The Brazilian Society of Mechanical Sciences and Engineering 2013

Abstract Enlargement of ship size in recent decades and

no change in the harbors and approach channels have

resulted in global attention toward navigation in shallow

and confined waters. A phenomenon which restricts ship

navigation in shallow waters is reduction of under-keel

clearance in terms of sinkage and dynamic trim, which is

called squatting. Due to the complexity of flow around the

ship hull, one of the best methods for predicting the ship

squat is the experimental approach based on systematic

model tests in the towing tank. In this study, model tests for

tanker ship model and traditional Persian Gulf and Oman

Sea vessel called dhow had been performed in the towing

tank and the squat of the models were measured and ana-

lyzed. Based on the experimental results, suitable formulae

for the prediction of these types of ship squat in fairways

are obtained.

Keywords Ship � Squat � Shallow fairway �Model test �Towing tank

1 Introduction

Ship squat is defined as variations in ship vertical position.

This is usually simultaneous with sinkage and dynamic

trim change. So in this condition, bow and stern sink in

water, and the amount of sinkage is different for aft and

fore. Ship squat is a common phenomenon that occurs in

shallow waters. As a ship moves in water, the fluid sur-

rounding the body flows into the lateral and bottom

direction of the vessel and the flow pattern around the hull

changes. Any change in flow pattern brings about changes

in pressure distribution compared to hydrostatic pressure.

Asymmetric distribution of pressure causes the ship trim-

ming by fore or aft and draft variations. For ships with

common speeds, the vessel usually sinks in the water and

changes in draft are negligible [1]. In high-speed crafts in

which the ship hull separates from the water surface, this

phenomenon results in hydrodynamic lift forces on the

vessel. A combination of this vertical motion and trim

angle variation in calm water is called ship squat [2].

Therefore even in deep waters, ship squats. But the amount

of squat is usually negligible and is dependent on ship hull

form and forward speed [1].

The increment of ship size during the 1960s has caused

the under ship clearance to be critical for navigation in

shallow waters and approach channels. In the shallow

water condition, ship squat is distinctly also a function of

water depth. In shallow waters, the under-keel clearance is

important and the pressure drop is more than in the deep

water case. So, the vessel sinks in water until buoyancy

force equals the hydrodynamic pressure forces. By analogy

to Venturi effect in fluid mechanics, the ship’s squat will be

proportional to ship speed and water depth, and according

to Bernoulli equation dynamic pressure is proportional to

speed squared. Compared to deep water, squat amount is

much more proportional to water depths than the hull form

and ship speed in shallow waterways.

Ship squat has been known for a long time. But acci-

dents have occurred to ships due to ship squat even in

recent years. As an example, sinkage of passenger ship

Herald of Free Enterprise in Zeebruge harbor, Belgium,

Technical Editor: Celso Kazuyuki Morooka.

M. F. Kazerooni � M. S. Seif (&)

Center of Excellence in Hydrodynamics & Dynamics of Marine

Vehicles, Department of Mechanical Engineering, Sharif

University of Technology, Tehran, Iran

e-mail: seif@sharif.edu

123

J Braz. Soc. Mech. Sci. Eng.

DOI 10.1007/s40430-013-0114-y

1987, led to the death of 200 passengers [2]. Another

example is the grounding of QE2, 1992, with a financial

loss evaluated at 20 million pounds. The cause of the

accident was the flooding of the tanks in the bow due to the

damage made by extreme squat and draft increment in the

ship fore [2]. A more recent report of grounding due to ship

squat is of Iran Noor tanker in Ningbo, China, 2004. All of

these cases strongly demand the need for precise guidelines

of predicting ship squat in shallow waters.

In the technical literature, various methods are described

for squat prediction: the rule of thumb predicting methods,

experimental data gathered from model tests, full-scale

trials and theoretical and semi-theoretical methods. Ship

squat prediction started with studies on ship sinkage at low

speeds [3]. In earlier studies, the effect of free surface was

simply neglected. Therefore, these models could be used at

low speeds where ship wave making is not the main con-

cern. Ship navigation in restricted waterways has been

considered [4]. Factors affecting ship squat are described

and a guideline is presented for squat estimation. Further-

more, experimental expressions and diagrams for calcula-

tion of ship squat calculation have been presented, but they

are limited to special models in special waterways [5].

With the sudden increase in ship sizes in the 1960s and

1970s due to the global oil crisis, ship squat prediction in

shallow waters has become demanding and a significant

number of studies have focused on the calculation of ship

squat in shallow water. So far, in the case of slender ships

moving in shallow water, the water depth is small com-

pared to ship length and the flow around the hull would be

definitely two-dimensional. Two-dimensional flow condi-

tions imply that the longitudinal speed of the flow particles

dominates the vertical speed. Tuck [6] calculated the flow

around a slender hull in shallow water of constant depth

which is laterally open. This solution is not capable of

calculating the wave making resistance but it can be used

to calculate the dynamic trim and sinkage except for case

that ship speed is near to the propagation speed of gravity

waves in shallow water [6]. Flow simulation past slender

bodies in shallow water is an important issue, and sinkage

and trim can be evaluated by this method [6, 7]. Changes in

ship trim and sinkage during ship squat is a nonlinear effect

and the slender body theory is a good method to predict

these nonlinearities [8]. Ship motions in shallow water and

squat have been studied by slender body theory [9]. Line-

arized solutions of slender body theory are obtained and it

is emphasized that nonlinear effects such as viscosity might

be influential in small values of under-keel clearance. In

full hull forms, neglecting the nonlinear terms causes the

slender body theory to be far from reality. Mathematical

and computational techniques are applicable to ship squat

evaluation [10] and performance in shallow water [11].

Ship vertical motion is calculated by CFD techniques using

commercial software Fluent. Dynamic mesh generation

and application of suitable turbulence models may be

necessary in these methods.

Squat can be evaluated by full-scale measurements

onboard ships [12]. DGPS can measure ship vertical

position in fore and aft even by sufficient accuracy and

could be used for squat measurements of full-scale ships.

The main problem of squat measurement on full-scale

ships is the measurement of ship sinkage in water and

elevation of the generated surface wave in shallow water.

However, in full-scale measurements, model tests and

computational methods should be used for deterministic

squat estimation method.

There are changes in maneuvering hydrodynamic

derivatives because ship squats are considerable [13].

When a ship enters shallow water, maneuvering hydrody-

namic derivatives change significantly. These coefficients

are totally different in the cases that squat occurs. Ship trim

may be influential and should be taken into account.

Analysis of the time history of ship sinkage and trim

angle in shallow water shows that ship squat is an unsteady

phenomenon in nature, and dynamic changes of the

parameters should be considered [14].

Forces and moments exerted on ships in shallow water

can be studied through model tests [15]. Ship sinkage, trim,

boundary layer, wake and free surface elevation can be

studied by model tests in shallow water towing tanks.

However, ship squat is not a precisely determinable phe-

nomenon and the difficulties of flow analysis around ship

hull mathematically clarify the need for experimental

approaches. So, empirical formulae have always drawn

attentions to predict ship squat. Model tests have been held

all around the world and among these efforts the results

obtained by Eryuzhu and Hausser in 1978, Eryuzhu et al.

1994 and Barras 1979 are remarkable [1]. As expected,

accurate squat prediction of a ship needs a large amount of

calculations. Therefore, it is not possible to use a precise

method in operational cases, and decision about the accu-

racy of the methods depends on the applications.

Finally, ship squat prediction is a probabilistic phe-

nomenon due to nonlinearities in the problem. Random

model is used to evaluate under-keel clearance [16] and the

statistical method is presented for large draft ship squat

passing St. Lawrence waterway. It is shown that the per-

formance of statistical models has been more accurate than

empirical models in practice [17]. In recent years, the

Monte Carlo method has been used to determine ship under-

keel clearance in harbors. A probabilistic model of under-

keel clearance has been used to determine the probability of

ship grounding. The method is based on uncertainties in

depth, draft, navigation and water level [18].

The disasters that occurred to ships through shallow

water intensify the importance of squat prediction and

J Braz. Soc. Mech. Sci. Eng.

123

serious need for guidelines. In references, various methods

have been suggested for squat evaluation [1]. The rule of

thumb methods, experimental data obtained from model

tests and full-scale measurements, and theoretical and

semi-theoretical methods are suggested. The simplest rule

of thumb is the M930 rule [19]. This rule just informs

people that squat may occur and implies without any

details that squat depends on ship speed. Ortlepp has pre-

sented a method of squat evaluation based on Archimedes

principle in deep waters [20]. His results show that squat is

intensely a function of water depth and depending on water

depth the amount of squat is two to seven times the squat in

deep water for a given speed. Ortlepp’s results in deep

water are gathered from experiments with trim gauges

installed on a particular vessel, Irving Glen. However, trim

indicators used in the past also showed the draft increment

due to elevation of surface waves. Admiralty Manual of

Navigation represents three methods to predict ship squat

[21]. It is totally obvious that the methods are not accurate

and no application limit is mentioned for them. These

methods are suitable for urgent cases where the ship

motion characteristics and waterway topologies are not

well known. For emergency applications, the most suitable

method is the one presented by Barras, in which an addi-

tional factor is considered for ship hull geometry [2].

There are too many experimentally based methods for

ship squat prediction based on hull form parameters as

length, draft and block coefficient. Most of these methods

are oriented from model tests in shallow water towing

tanks. In the present study, model tests were carried out in

the Sharif University of Technology towing tank and the

results have been used to find out squat behavior. The

results can be expanded to full scale by taking into account

possible scale effects. Scale effects in model tests for

resistance test and longitudinal forces are well known. But

there has been little research for the studying of scale

effects in vertical and transverse forces. However, it is

expected that scale effect is small in the vertical motion

case [19]. A series of systematic tests were performed in

the laboratory for a tanker and a traditional Persian Gulf

and Oman Sea vessel known as dhow. The results were

compared with the studies published by Milward and

Norrbin [1]. The compatibility of results emphasize that the

data obtained can be used for squat prediction of vessels in

the vicinity of Iranian harbor.

2 Main factors

Important factors affecting ship squat are forward speed,

block coefficient, hull form, deadrise angle and water

depth. The most important factor in ship squat is forward

speed. Squat is approximately proportional to square speed.

For example, if the ship speed halves, squat would decrease

to 25 % of the initial amount. Of course, the relative speed

of ship and marine current should be used in squat pre-

diction. The relative speed should be considered in the

presence of currents and tides in the fairway.

Ship squat is proportional to block coefficient directly

and the ratio of squat to ship draft is higher in the ships

with large block coefficients. Therefore, tanker ships are

expected to have more squat than the passenger ferries or

other slender hulls.

High deadrise angle of the hull form can decrease the ship

squat. Ships with no deadrise angle in the parallel middle

body such as tankers are acutely subjected to squat.

Ship squat is a function of hull form and also body lines.

If the vessel has flat surfaces, especially at the bottom, these

flat surfaces would be influenced by pressure drops and

squat can be increased. This means that tanker ships and

barge shape vessels are critical from the squat point of view.

Decrement in water depth at a given speed would result

in more ship squat. Under keel clearance (UKC) is defined

as below in order to determine the water depth.

UKC ¼ h� T

Tð1Þ

Figure 1 shows the parameters used in this definition.

Another prominent factor in squat evaluation is the

blockage factor. Blockage factor is defined as the ratio of

ship wetted cross section to channel wetted cross section as

shown in Fig. 2.

S ¼ b� T

B� Hð2Þ

In the formula above, b is the ship breadth, T ship draft,

B channel beam and H is water depth.

If the ship navigates in the fairway with no lateral

restriction, equivalent channel breadth is used that varies as

much as 8.25 times the ship breadth in ultra large tankers to

Fig. 1 Definition of ship draft and water depth in shallow fairway

J Braz. Soc. Mech. Sci. Eng.

123

9.5 times the ship breadth in general cargoes to 11.75 times

of the ship breadth in containerships [2]. This width is the

significant width of waterway that the flow around the ship

is affected by lateral boundaries.

For evaluation of shallow water effects, it is common to

define a Froude number according to water depth as follows:

Fnh ¼Vffiffiffiffiffi

ghp ð3Þ

When the ship moves in shallow waters at high speeds,

the risk of grounding at fore or aft is probable. For ships

with fair hull forms such as container ships, grounding

usually occurs by aft, but for bluff hull shapes such as

tankers grounding occurs by fore. However, the above-

mentioned statement is only applicable under an even keel

(no heel, no trim) condition at initial equilibrium.

3 Motion regimes in shallow water

Ship motion regime in shallow water up to channel Froude

number 0.85 is called the subcritical region. Most ships sail

through shallow water at subcritical speeds. According to

experimental results, the ship resistance increases dramat-

ically with channel Froude numbers and reaches a peak at

approximately 0.85. So in this region, the ship resistance in

shallow water is much more than in deep water. Therefore

if the engine power remains constant, the vessel would

experience speed loss in shallow water. As a result, to

maintain the speed in subcritical shallow water, more

engine power is needed. Further, the diagrams of trim by

fore and sinkage with respect to depth Froude number

increase in this region. The ship trims more and sinks in the

water till it reaches an obvious peak. According to tech-

nical literature, this peak occurs approximately in channel

Froude number 0.85–0.9 [1].

The flow regime around the ship hull moving in shallow

water with channel Froude numbers more than 0.85 is

called the supercritical region. The vessel hydrodynamic

behavior in this region is reverse to subcritical flow regime.

In this region, the resistance ratio (resistance in shallow

water/resistance in deep water) is less than a unit. So, the

ship resistance in supercritical regime is less than in deep

water. The trim angle persists to an almost steady amount

and the ship bows up. However, the sinkage would be

negative in this region. The vessel lifts up and the draft

decreases. The supercritical region is not so much of

interest because most ships enter the shallow water in the

subcritical regime. The majority of squat studies have been

conducted in the subcritical regime.

4 Experimental modeling of laboratory tests

4.1 Shallow water towing tank

The main dimensions of the towing tank depend on the

dimensions of the ship model that would be used. There-

fore, the dimensions of the towing tanks around the world

that could be used for shallow water tests and approach

channels modeling vary. Shallow water towing tanks are

not accessible within the country. So, some facilities have

been added to the towing tank in the Sharif University of

Technology for the easy performance of shallow water

tests (Fig. 3).

The measuring unit is attached to the center of gravity of

the models as shown in Fig. 4. During the tests, the vertical

motion of the model at the center of gravity is measured,

that is, the heave motion in dynamic mode and sinkage of

the model in the steady-state condition. Rotation around

the transverse axis is measured, which is the trim in steady-

state motion with forward speed or pitch angle in the

Fig. 3 Artificial bottom installed in the towing tank

Fig. 2 Ship in channel at static equilibrium

J Braz. Soc. Mech. Sci. Eng.

123

dynamic motion mode. The accuracy of the sensor used for

sinkage measurement is 0.08 mm and the sensor for mea-

surement of dynamic trim 0.1�. The model is fixed in all

directions except in the vertical plane, including sinkage

and trim motion.

All the measurements performed in the towing tank are

done for steady-state condition. Therefore, it is essential

that all the parameters in the towing tank tests reach a

steady level and persist for an adequate time interval. Also

it is not necessary to install the artificial bottom among all

the tank lengths but it is obvious that the length of the

artificial bottom should be as enough for all the parameters

to reach the steady-state condition in shallow water. As an

example in the Sharif University of Technology towing

tank with 25 m length, all the test parameters reach steady

amount in a 10 m artificial bottom. Figure 3 shows the

artificial bottom installed in the towing tank.

4.2 Ship model selection

The model used to perform tests depends on various

parameters. Studying the ship squat in the vicinity of har-

bors consists of two separate problems. The first is to study

the squat characteristic of a certain ship in harbor, and the

other one is to study the squat of ships in general in order to

obtain a criterion for ship squatting in Iranian harbors. For

the first problem, the ship model is well defined, but for the

second one, it is necessary to obtain methods to select the

ship models in such a way so as to cover vast ship types

entering the Iranian harbors. This method concludes in a

database that can be used for prediction of ship squat. So

the selection of ship models is very important. In this

research a wide study of ships entering Iranian harbors has

been done. Simply the ships can be divided into two cat-

egories based on their length and tonnage. Large tankers

and container ships that have lengths of more than 110 m

and small and medium size cargo and passenger ships with

the lengths of less than 80 m. Because enormous motion

characteristics of the ships depend on the size and weight,

this division is reasonable and covers the ships entering

Iranian harbors so well.

In order to make a true prediction of ship squat in Ira-

nian harbors, the model ships are selected so as to cover

majority of ships entering the harbors. The ships entering

Iranian harbors are in two different classes based on ship

size. The main ports of the country and oil terminals are

able to berth large ships although in small local harbors

cargo ships of medium and small sizes are navigating.

Technically the squat behavior of these two kinds of ships

is different. On the other hand, it had been necessary to

consider two different size models to study the effect of

hull form on squat characteristics of ships. According to

above discussion finally it had been decided to use a model

of tanker ship as described in Table 1 and a traditional

Persian Gulf cargo vessel called dhow, as in Table 2.

Figures 5 and 6 show the models that have been used.

5 Data acquisition and analysis

The model tests are performed in different ranges of water

depth and ship speed. In each run, the sensors of sinkage

and dynamic trim measurement are attached to the model.

The water depth is adjusted until the desired value of UKC

is obtained; then the model is towed to reach the desired

speed. The speed will remain fixed and the measurements

are done in the condition of constant forward speed. The

data acquisition is done in a certain time. Finally a time

history of sinkage and dynamic trim is obtained. There is a

Fig. 4 Arrangement of measuring unit to model CG

Table 1 Tanker and model specifications

Tanker Model tanker

Length (m) 176 1

Breadth (m) 31 0.17

Draft (m) 9 0.05

Displacement 41,523 tons 7 kg

Fn 0.06–0.27

Table 2 Dhow and model specifications

Dhow Dhow model

Length (m) 60 0.71

Breadth (m) 19 0.225

Draft (m) 4.22 0.05

Displacement 3,575 tons 5 kg

Fn 0.06–0.27

J Braz. Soc. Mech. Sci. Eng.

123

great difference between the time history of the measured

signals in shallow water and the time history of the mea-

sured signals in deep water. In deep water tests, the mea-

sured signal remains approximately constant during a

considerable portion of the test. However, when the vessel

enters the shallow water it experiences a decaying oscil-

latory motion around a certain limit. This oscillating

motion is very similar to motions of vessels in a seaway.

Therefore, according to seakeeping literature, the motion at

the center of gravity of a vessel is defined as heave motion.

The oscillations damp rapidly and the vessel maintains a

steady vertical displacement at the center of gravity which

is defined as sinkage, and a steady angle of rotation around

the transverse axis which is defined as dynamic trim. This

oscillatory motion has a period equal to a vessel’s natural

period of heave. An example of time history of heave

motion is shown in Fig. 7.

From Fig. 7, it is obvious that the maximum speed of

test in a towing tank depends on the water depth as well. In

order to precisely measure the sinkage and dynamic trim, it

is essential that the model’s oscillatory motions be damped.

The length of the shallow water towing tank is limited.

Therefore, the model’s speed should be below a certain

limit to give the oscillations enough time to decay. The

shallow water model test with forward speeds more than

1 m/s2 are neglected for this purpose.

Since the probability of grounding is so important in

practice, in the model tests the maximum vertical motion of

the ship is defined as ship squat (in bow or aft which is

larger). In each case, the sinkage at the center of gravity of

the model and the dynamic trim are measured, and the

squat is calculated as follows:Fig. 6 Dhow model

Fig. 5 Tanker model

Fig. 7 Example of time history of sinkage in shallow water

J Braz. Soc. Mech. Sci. Eng.

123

squat mð Þ ¼ sinkage in CG mð Þþ longitudinal distance of point to CG mð Þþ dynamic trim ðradiansÞ

ð4Þ

In usual speeds, sinkage is always downward. According

to the ship speed and the hull form, when the dynamic trim

is by aft, the point is defined as AP; and when it is by fore,

the point is located at FP.

It should be mentioned that every test run is executed

twice, and the average value of sinkage and dynamic trim

is used in the calculation of squat. In the cases that the

measured values in two tests are, to a high degree, different

from one another, the test is repeated for the third time and

two closer values are averaged as final amount of the

measured parameter.

One of the main aspects of the model tests in the

towing tank is the possibility of extending the results to

the full scale. The flow around the model should be

similar to the flow around the ship. Similarity of gravity

effects is provided through the Froude number in the

tests. Similarity of viscous effects is not applicable.

Therefore, it is essential to at least maintain the same

regime of boundary layer in the model and in full scale.

In full scale, the boundary layer is almost turbulent. It

seems that the level of turbulence is higher in the model

tests in shallow water so that there will be no need for

turbulence simulators. However in small forward speeds,

the difference of results in model test and full scale may

be more.

The period of time between two successive tests can

affect the results. This interval should be long enough that

vortexes disappear in the towing tank. It is observed in the

tests that more time is needed when performing model tests

in shallow water.

6 Experimental results and validation

There are so many experimental methods for predicting

ship squat in literature. The vast majority of these

methods are based on experimental measurements on ship

models in towing tanks. Several of these model tests were

carried out in the Sharif University of Technology towing

tank and the results are in good agreement with the

published data. In most cases, the results can be expanded

to full scale.

In order to evaluate the squat of ships entering Iranian

harbors, model tests in Marine Engineering Laboratory are

conducted and the results are compared to experimental

formulae based on model tests. To obtain the lower limit of

ship squat, a comparison with Norrbin formula is so suit-

able [1]:

S ¼ 0:01888CB

B

L

T

hV2 ð5Þ

In this formula S is maximum ship squat predicted by

Norrbin formula, CB the block coefficient of the hull form,

B ship breadth in m, L ship length in m, T ship draft in

meters, h water depth in meters, and V is ship speed relative

to flow velocity in kmph.

In order to predict the higher limit for ship squat, the

formula presented by Milward is applicable [1]. This for-

mula can be used up to both the end of the subcritical flow

regime in shallow water and the channel Froude number of

0.4 subcritical region.

S ¼15:0 CBB

L

� �

� 0:55� �

F2nh

1� 0:9Fnh

L ð6Þ

In the following diagrams, the results obtained from

model tests in different speeds and water depths are

compared to Norrbin and Milward formulae. It is clear that

the Norrbin formula is in lower level of value compared to

model tests, which consequently means that Norrbin

formula gives a lower limit for ship squat. Also, the scale

effects of the models are prominent and some of the

differences may be due to the scale effects (Figs. 8, 9, 10).

The results presented in Fig. 11 show that the Norrbin’s

criteria are not accurate in higher forward speeds. In such

occasions the model’s test results should be used directly.

Furthermore, in all diagrams one could see that the model

Fig. 8 Tanker squat, Froude number 0.064

Fig. 9 Tanker squat, Froude number 0.128

J Braz. Soc. Mech. Sci. Eng.

123

test results are far behind the Norrbin’s formula for lower

UKC values. In these extreme cases, the flow blockage is

more and the flow speed outside the boundary layer

increases significantly so that the pressure drops are more

obvious and the model sinkage increases (Figs. 12, 13).

The higher limit of squat for large cargo ships can be

calculated by the Milward formula. However, the squat of

tanker model measured from model tests is within the two

limits given by Milward and Norrbin. So, such formulae

are applicable to ship entrance in Iranian harbors for low

speeds. In higher speeds the simulation methods based on

model tests and computational fluid mechanics are sug-

gested. The results obtained from model tests in the towing

tank of Sharif University of Technology are very good

references in these cases.

The tanker squat calculated from Norrbin and Milward

approximation is compared to model test results. At lower

waterway depths, the Norrbin approximation is accurate, and

for higher water depths so is the Milward approximation.

For squat prediction of traditional dhows, the applica-

tion of formulae presented in literature is not so logical.

Generally, these formulae are based on experimental

methods gathered from measurement on ship models and

full scale. Such relations are practical only for hull forms

similar to forms included in the tests done. Because the

dhow’s hull form is so far from typical ship hull forms,

calculations based on these methods will result in inaccu-

rate evaluation of ship squat. Bearing this discussion in

mind, model tests were performed in the towing tank for

prediction of dhow squat and a comprehensive database is

gathered from the model tests in different water depths and

forward speed, as shown in the following diagrams.

Regarding these diagrams, one can obtain an acceptable

prediction of dhow squat in Iranian shallow waters

(Figs. 14, 15, 16, 17, 18).

As it is seen, the prediction of dhow squat from typical

squat formulae is not so accurate. Therefore, a formula for

prediction of dhow squat is suggested based on experi-

mental results in the towing tank. Figure 19 shows the

dhow squat and its exponential fitting curve.

The formula for dhow squat is calculated as below from

the experimental results. This equation is validated to

model test results up to channel Froude number of 0.5.

Fig. 10 Tanker squat, Froude number 0.176

Fig. 11 Tanker squat, Froude number 0.223

Fig. 12 Tanker squat, Froude number 0.271

Fig. 13 Tanker squat, Froude number 0.319

Fig. 14 Dhow squat, Froude number 0.057

J Braz. Soc. Mech. Sci. Eng.

123

S

T¼ 0:03018CB

B

Le4:875Fnh ð7Þ

7 Affecting parameters analysis

7.1 Effect of depth Froude number on squat

The measured squat is plotted versus depth Froude number

in Fig. 20 for tanker and dhow model. According to these

results, the ship squat reaches a peak of Fnh = 0.6 for

tanker model and Fnh = 0.48 for dhow model in the sub-

critical region. Therefore, the curvature of the ship’s hull

can reduce the depth Froude number in which the maxi-

mum peak occurs in the subcritical region. In order to

obtain the most satisfactory navigation condition, this fact

should be taken into account in shallow water. Further-

more, in higher depth Froude numbers, for example, Fro-

ude numbers more than 0.6, obvious humps and hollows

are observed in the experiments. For safe navigation in

shallow water, it is recommended to navigate with forward

speeds of hollows in this curve.

7.2 Effect of trim and sinkage on ship squat

As shown in Fig. 20, humps and hollows are seen in the

squat of the vessels at different depth Froude numbers.

Fig. 15 Dhow squat, Froude number 0.114

Fig. 16 Dhow squat, Froude number 0.171

Fig. 17 Dhow squat, Froude number 0.227

Fig. 18 Dhow squat, Froude number 0.269

Fig. 19 Dhow squat versus channel Froude number

Fig. 20 Squat versus depth Froude number

J Braz. Soc. Mech. Sci. Eng.

123

These humps and hollows due to interaction of ship sink-

age and trim change in shallow water. As an example, the

squat of tanker model, heave motion, and dynamic trim are

plotted together in Fig. 21.

In some Froude numbers, the heave motion and dynamic

trim angle interaction is positive which results in more ship

squat. For example, in Fnh = 0.48 a minimum is observed

in the heave motion but the maximum in trim angle curve

removes the effect and no certain peak is seen in the squat

curve. Another cancellation effect occurs in Fh = 0.86,

0.95, 1.00. As it is seen in the Froude numbers more than

0.8, the interaction between dynamic trim and heave

motion is always cancellation, but at lower depth Froude

numbers, these interactions are magnified.

8 Regression model

In order to evaluate the amount of tanker and dhow squat

for different water depths and speeds, a regression analysis

has been conducted and a regression formula is obtained

from the model tests results. The formulae give the amount

of squat as a function of UKC and motion Froude number

for model. Figures 22 and 23 show the normal probability

plot of the Squat/T, considering parameters involved in

regression analysis. The interaction of these two factors is

taken into account.

S

T¼ �0:133288þ 0:0404538 UKCþ 2:18262 Fn

� 0:656682 UKC� Fn; for tanker ð8ÞS

T¼ �0:0612834þ 0:0271849 UKCþ 0:839984 Fn

� 0:278239 UKC� Fn; for dhow ð9Þ

The R2 factors for Eqs. (8) and (9) are 91.81 and 86.4 %,

respectively, which prove the extracted equations are

satisfactory in practice for this range of results.

Most tanker ships and dhows navigating in the vicinity

of Iranian harbors have hull forms similar to the models

used in these experiments, so that the above equations can

be used to estimate these types of vessel squats in shallow

fairways. It goes without saying that these formulae are

based on a limited number of model tests, which means one

should exercise caution using them. First of all, the

geometry of the hull should fit well within the specifica-

tions of the hulls tested in this paper. Second, the UKC and

speed of the vessel must be within the range as well. If the

regression formulae are used in other cases, there may be a

serious error in the calculations. However, obtaining more

Fig. 21 Squat, heave, and

dynamic trim versus depth

Froude number for tanker model

Fig. 22 Normal probability plot of measured squat/T for tanker

Fig. 23 Normal probability plot of measured squat/T for dhow

J Braz. Soc. Mech. Sci. Eng.

123

precise regression formulae demands more towing tank

tests. By introducing the forward speed Froude number and

UKC in the Eqs. (8) and (9), the vessel squat can be

approximated in practice.

9 Conclusion

In this study, squat phenomena have been investigated

experimentally. The results obtained can be used as an

appropriate database for tanker ship and dhow squat pre-

diction in shallow waters. According to the test results,

squat magnitude is very high for the models studied.

In the model tests, it is shown that in the limiting con-

ditions, the squat of this vessel can reach thirty percent of

ship draft, which is seriously dangerous for tanker navi-

gation in shallow water due to grounding risks. Further-

more, in this study, a well-defined method for predicting

ship squat is introduced and facilities are provided in the

towing tank. In addition, it is shown that ship squat vari-

ation with depth Froude number has distinct humps and

hollows according to Fig. 20. Therefore, in a limited range

of speeds and depths the fluctuations in ship forward speed

can reduce the amount of squat significantly. Different

formulae have been published in navigation guidelines

across the world. Most of these relations are very simple

and hull form is not considered as a result of which the

actual ship’s hull form becomes far from the hull forms

tested which subsequently leads to less accurate squat

prediction through such relations. In the current paper, the

results of model tests of dhow are compared to the pub-

lished data which means that empirical or theoretical for-

mulae presented in the literature are not applicable in these

cases. The CFD approach may be an alternative to the

model tests in these cases. In cargo ship operations a few

centimeters of draft increment may result in considerable

payload and benefit. In these cases, accurate squat simu-

lations such as the procedure described in this paper will be

highly beneficial.

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