A Review of the Prediction of Squat

in Shallow Water
A. Millward
( Uni versity of Li verpool)

Recent evidence has shown that the effect of a ship moving in shallow water and the resultant
squat are not well understood. This paper reviews the general problem of a ship in shallow
water and illustrates the corresponding resistance, trim and sinkage at both sub-critical and
also at super-critical speeds.
The paper then reviews the various methods of predicting the squat of a ship in shallow
water in the sub-critical range, which is applicable to most ships. It is suggested that the simple
rule-of-thumb methods are, at best, unreliable and the paper gives examples of empirical
methods which have been tested against various sets of data and seem to give more
representative answers. A summary of the notation used is given at the end of the paper.

i. I N T R O D U C T I O N . The phenomenon of squat has been known about in

principle for many years yet accidents to ships are still occurring where it is
evident that squat was a factor which had not been taken into account properly
— the most public of these in recent years have been the sinking of the ferry
Herald of Free Enterprise at Zeebrugge in 1987 and the grounding of the QE.2 off
Massachusetts in 1992. It is therefore the purpose of this paper to review the
phenomenon of squat, what causes it and to give guidance on how it may be
predicted.
As a ship moves through the water, the fluid ahead of it is displaced both to the
side and underneath the hull, giving a flow pattern around the hull which is most
evident to the observer in the form of the wave pattern around the ship. This flow
pattern around the ship results in a change in the distribution of pressures around
the hull from the ordinary hydrostatic pressures exerted when the ship is
stationary. This change in the pressures can result in the ship altering its fore and
aft trim and, in addition, moving vertically in the water. Usually for lower speed
ships this vertical movement is downwards, a sinkage, and is small, but in some
of the smaller high speed vessels — particularly planing hulls — the dynamic forces
developed by the motion through the water can be sufficiently large at higher
speeds that the hull rises up above its static waterline so that the sinkage is, in
fact, negative. The combination of the vertical movement or sinkage and the
change in trim with forward motion together are given the label ' squat'. Thus
all vessels squat as they move, even in deep water, although the magnitude of the
squat is usually small, and is a function of the shape of the hull and the forward
speed through the water.
As ships increased in size through the 1960s the clearance under a ship as it
entered the approach channel to a harbour became a limiting factor. It was found
77
78 A. MILLWARD VOL. 49
that the clearance under the keel was smaller than had been expected suggesting
that squat was clearly also a function of the water depth. The reason for this is
illustrated in Fig. i, which shows a schematic diagram of a ship in shallow water.

n/////////
Fig. 1. A large ship in shallow water

The area under the hull is very much more restricted than in deep water,
resulting in a change in the flow pattern under the hull leading to even lower
pressures exerted on the bottom of the hull. These pressures cause the hull to
sink lower into the water until the increased buoyancy force balances the
pressure forces, resulting in a new sinkage and trim or squat compared with deep
water. By analogy with the Venturi effect in fluid mechanics it can be expected
that the amount of squat in shallow water will be dependent on the depth of water
and also on the speed of the ship. Indeed, if the analogy is taken further, since
in basic fluid mechanics, pressure is related to speed squared (by Bernoulli's
equation), it might be expected that squat may also be related to speed squared
or something similar. Thus it is clear that, in addition to the shape of the hull
and the speed of the ship which seem to govern the small amount of squat
occurring in deep water, the depth of water is also likely to be a very significant
factor.
The Merchant Shipping Notices are required to be carried on British ships and,
since one of these is concerned with the squat phenomenon, there is, at first
sight, no reason why the officers in charge of a ship should not know about the
effects of squat. The relevant Notice (no. M930)1 states:
relatively high speeds in very shallow water must be avoided due to the danger of
grounding because of squat. An increase in draught of well over ten percent has been
observed at speeds of about ten knots, but when speed is reduced squat rapidly
diminishes. It has also been found that additional squat due to interaction can occur
when two ships are passing each other.
Although this does seem to appreciate that speed is a major factor the other
guidance is very vague and seems to indicate that squat is a fixed proportion of
the draught at a given speed irrespective of the type and size of ship or the depth
of water.
Since the publishing of that Notice in 1980 two major incidents have again
brought the squat phenomenon to public attention: the first was the capsize of
the ferry Herald of Free Enterprise in 1987 with the loss of nearly 200 lives, the
second was the grounding of the QE2 in 1992 with an estimated cost of around
£20 million in repairs and lost revenue to the company. The investigation into
the loss of the ferry showed that water entered the open bow doors because of
the squat combined with the enhanced bow wave, both effects being caused by
NO. I PREDICTION OF SQUAT 79

high speed in shallow water, so flooding the car deck and leading to lateral
instability and capsize. This cause was identified by the author to the Department
of Transport within three days of the tragedy and was subsequently confirmed by
an investigation at British Maritime Technology.2 In the case of the grounding of
the QE2, the subsequent investigation again showed that the effect of squat in
shallow water was a contributory factor coupled with questionable accuracy of
the charts used. In both cases the ships were travelling at high speed relative to
the depth of water they were in and it is evident that the true amount of squat
was not appreciated at the time. Thus it is clear that more guidance on the
magnitude of squat is needed.
2. THE EFFECT OF SHALLOW WATER. Since a ship travels on the surface
of the sea it creates a wave pattern which travels with the ship. It is therefore
useful to appreciate that, in water where there is a limited depth, there is a
maximum speed at which a free standing wave can travel. This maximum speed
or critical speed can be expressed by the equation:

Vc = Vgh (i)
where g is the acceleration due to gravity and h is the depth of water. This is not
to say that a ship cannot travel above this critical speed because the wave pattern
around the ship is not free standing but connected to the ship but it can be
expected that the wave pattern and other flow phenomena may change depending
on whether a ship is travelling at super-critical or sub-critical speeds. It does,
however, suggest that any change is most likely to be related directly to the speed
and depth of water rather than to the size of the ship. As a result, when
investigating the effect of shallow water, it is often more informative to express
speed in terms of a Froude number related to the depth of water instead of the
length of the ship; that is:

insteadof f
i »
where V is the speed of the ship, h is the depth of water and L is the length of
the ship. A graph of the variation of resistance, trim and sinkage plotted against
this depth Froude number Fnh is shown in Fig. 2 with the resistance expressed
as the ratio of the resistance in shallow water to the resistance in deep water. It
can be seen that for all three parameters there is a noticeable difference between
the sub-critical and super-critical region with the change taking place generally
from about o-8g < Fnh < I - I .
2.1 Sub-critical. In the sub-critical range, which is where most ships operate,
the graph shows that the resistance ratio rises sharply, depending on the hull
length to water depth ratio, reaching a peak at a Froude number Fnh of about o-8^.
Expressed in terms of resistance this means that, for this part of the speed range,
the resistance of a ship in shallow water is greater than in deep water so that, if
the power to the engines is maintained at the same setting, then the ship will slow
down in shallow water. Alternatively, in order to maintain the same speed in
shallow water as in deep water a higher power setting would be required.
8o A. MILLWARD VOL. 49
Resistance
15 Ratio

10

Trim Angle
(Degrees) *o
0-5

10 1-5
Depth Froude Number Fnh

0-5

10 Sinkage
(%LPP)
Fig. 2. Resistance, trim and sinkage in shallow water (L/h = 3)

Over the same range, the upward trim by the bow increases while the sinkage
increases downwards again reaching a peak at a Froude number of around 0-85
to 0-9.
2.2 Super-critical. In the super-critical range, the graph shows that the effects
are quite different: the resistance ratio is less than one, meaning that the
resistance itself is actually slightly lower than in deep water, the trim is fairly
steady and bow up while the sinkage has actually become negative so that the hull
is riding higher in the water than when stationary.
Since most ships move at sub-critical speeds in shallow water, this is the main
area of interest, particularly as far as squat is concerned. There are also other
effects caused by shallow water so that the overall result is:
NO. I PREDICTION OF SQUAT 8l

(i) The resistance is increased compared with deep water so that either the ship
will slow down or a higher power setting is required to maintain speed. This
is caused primarily by an increase in the wave resistance so that the wave
heights around the bow and stern particularly are not necessarily the same
as in deep water - a factor which should be borne in mind if one is
attempting to estimate the amount of squat by observation of the waterline.
Corrections for the effect of the depth of water on the effective speed of a
ship can be found in classical textbooks such as Comstock.4
(ii) The hull will sink deeper into the water. This sinkage increases rapidly with
speed and a first simple approximation appears to be an increase as the speed
squared. The actual sinkage may be different from the observed apparent
sinkage because of the change in wave heights around the hull, particularly
at the bow and stern, as discussed above.
(iii) The trim also changes — in this example the trim was positive (bow up). In
particular, if the ship is not trimmed precisely level in the first instance it
is likely that the effect of shallow water will be to exaggerate that trim,
(iv) The rudder setting for going straight ahead will change. Studies of both the
flow over the rudder5 and the inflow to the propeller6 show that, even in
deep water, a ship can be expected to require a small helm angle to counter
the side force developed by the interaction of the hull, rudder and
propeller. The flow pattern will be altered by the presence of the sea
bottom in shallow water, which will be experienced as a change in helm
setting. The manoeuvring ability is also likely to be affected to some extent.
(v) The changed inflow to the propeller caused by the shallow water is likely
to result in increased propeller vibration at settings where it would not
normally be experienced.

3. P R E D I C T I O N OF SQUAT. While it is interesting to know the overall

effects of shallow water on a ship, particularly at the sub-critical speeds where
most ships operate, the evidence of the incidents involving the Herald of Free
Enterprise and the QE2 suggest that it would also be wise to be able to predict the
amount of squat in shallow water. There are various methods available in the
technical literature, varying from a simple rule-of-thumb, through methods using
empirical data either from model tests or full-size ships, to theoretical or semi-
theoretical methods. As might be expected, the complexity of the calculation
needed also increases in this progression so that there is no 'correct' method
— it depends on the degree of complexity which is acceptable together with an
assessment of the level of accuracy which is required.
3.1 Rule-of-thumb methods. Perhaps the simplest rule-of-thumb method is that
indicated by the Merchant Shipping Notice M930 which has been quoted earlier
and states that:
relatively high speeds in very shallow water must be avoided due to the danger of
grounding because of squat. An increase in draught of well over ten percent has been
observed at speeds of about ten knots, but when speed is reduced squat diminishes
rapidly.
82 A. MILLWARD VOL. 49

Apart from alerting people to the fact that squat exists, this piece of information
is not very helpful since it seems to imply that squat is somehow related to the
absolute speed of the ship, although the relationship is unspecified, but is not
dependent on the water depth nor on the shape of the ship. Neither does the
Notice indicate how the squat might be estimated at speeds other than ten knots.
Ortlepp7 has shown a method for obtaining the squat in deep water, based
apparently on Archimedes' principle, but states that the squat in shallow water
is twice that in deep water. The results for a tanker hull in several depths of water
are shown in Fig. 3 and it can be seen that, even for this limited investigation,

Legend
6
8
A 10
B 12
Sinking 8
Ratio
ss/sd

a B
A
A A

© ° © ©0©
©

0-1 0-15 0-2

Froude Number Fn
Fig. 3. A comparison of the sinkage in shallow water and deep water

the shallow water squat varies from about two to seven depending on the depth
of water. Thus, apart from the important point that the squat in shallow water
is greater than in deep water, Ortlepp's result is not particularly useful either.
Ortlepp's deductions of the squat in deep water appear to be based on
measurements of trim and sinkage as indicated by the trim indicators installed on
a particular ship, the Irving Glen. No indication was given of how these trim
indicators worked but it is worth noting that some devices, which effectively
measure the height of water alongside the ship at a particular point, would be
affected by the wave pattern and wave height around the ship which varies with
speed. This is particularly important for measurements in shallow water where
the wave heights can be noticeably different from those in deep water. This
NO. I PREDICTION OF SQUAT 83
8
suggests that the results reported by Dennis, based on visual observations of
immersion of the ship at the stern, also may not be accurate. In this case it is
likely that the squat is not as large as has been deduced but that is at least an error
on the side of safety since the actual squat will be less than the squat deduced
from the measurements.
Three methods of estimating the squat are shown in the Admiralty Manual of
Navigation.9 These are:
Squat = 10% of draught (3)
Squat = 03 metres for each £ knots of forward speed (4)
V2
Squat (metres) = (knots) (g)
100
It is suggested in the Admiralty Manual that the value of squat should be calculated
by all three methods and the worst case taken. The estimate resulting from
Eq. 3 is essentially the same as that given in the Merchant Shipping Notice M930
except it implies that this value applies at all speeds and is therefore potentially
dangerous. The estimate from Eq. 4 is a little better since it shows a variation
with speed although it is indicated that the variation is linearly proportional to
speed. However it has already been shown in Fig. 2 that squat does not vary
linearly with speed so that Eq. 4 must also be suspect. Equation £ does at least
show that the squat varies with speed squared and is therefore more realistic. It
does, however, imply that all ships will squat the same amount regardless of the
size of the ship, its shape or the depth of the water and therefore clearly leaves
a lot to be desired. A better prediction can be gained from the simplified formula
given by Barrass (Appendix A1) which does incorporate a factor for the shape of
the ship.
3.2 Empirical methods. There is a large number of empirical methods given in
the technical literature which result in formulae for predicting squat based on the
main parameters of the hull such as the length, draught and block coefficient.
These empirical methods are usually based on data obtained from measurements
of the squat of ship models in a towing tank. Inherently there is no reason why
data from model tests should not be useful when scaled up to represent the full
size ship. Although some assumptions have to be made in the techniques used for
scaling measurements of resistance from model to full scale, these have been well
authenticated over the years and are, in any case, relevant to the forces in the
direction of motion; that is, horizontal forces. There is no evidence in any case
that these assumptions would have any significant effect on the deduction of squat
which is concerned with vertical forces.
A number of existing empirical formulae for predicting the squat in shallow
water were given by Blauw and van der Knaap10 and were later used by
Millward1' for comparison with a range of model data for a wide variety of ship
types from patrol boats to tankers. As can be seen from Table 1 some of these
methods were for a restricted range of water depths and were sometimes
developed from experiments on a limited range of ship types. The corresponding
formulae are given in the Appendix and the original sources are listed in the
references for completeness. Millward used his own data to deduce general
84 A. MILLWARD VOL. 49

TABLE I . SUMMARY OF EMPIRICAL SQUAT PREDICTION METHODS

Author Ref. Validity-Width -Depth Ship

Fuhrer and Romisch 18 0032 <AS/AC< 0-43 119 <h/T < 229 all types
Soukhomel and Zass 19 Unrestricted Unrestricted all types
Eryuzlu and Hausser 20 Unrestricted 108 <h/T < 2-8 VLCCs
Barrass 17 Unrestricted r i < h/T < 1•J °S < C B < °9
Dand 12 Unrestricted i'o < h/T < 1S 08 <CB < 09
Millward 11 AjAc < o-o 8 Vl£<h/T< 6 all types

formulae and then tested all the methods against data for the ferry Herald of Free
Enterprise. The conclusion was that his own formulae predicted the Herald of Free
Enterprise situation better than almost all the other methods and, if used on other
cases, would be likely to err on the side of safety if at all; that is, the formulae
would predict a larger squat than would be observed in practice. A slightly
surprising result was that the simpler formulae given by Barrass were almost as
good even though they were being used outside their range of water depths as
shown in Table i. The formulae obtained by Barrass are given in Appendix A i
and the formulae deduced by Millward are:
. = K ' 2 - " CB.B/Lvp)-o-4.6]I*nh L^

and i,, = "^ . —tUL

1 —(

where the squat 5 is the squat at the midships or the bow as indicated by the
subscript. These formulae can be used in the Froude number Fnh range between
about 0-4 and o-8 — the higher end of the sub-critical speed range.
3.3 Semi-empirical or theoretical methods. A number of other methods have been
devised based either on semi-empirical methods, such as that by Dand, 12 ' 13 or
more theoretical methods such as that obtained by Tuck. 14 ' 15 Both of these
methods require a considerable quantity of detailed information on the shape of
the hull and have therefore not been tested over a wide variety of hull shapes.
Other methods are also being developed, often related to predicting the
resistance of a ship, but would have the same problem that a detailed knowledge
of the ship's hull shape is required and access to a fairly powerful computer would
be necessary. In the case of Dand's work, the method has been written as a
computer program and the results expressed in graphical form. The range of
validity is r o < h/T < !•$, o-i < Fnh < o-6 for vessels with o-8 < CB < 09 as
shown in Table 1.
4. C O N C L U S I O N S . The evidence reviewed in this present paper suggests that
most simple rule-of-thumb methods of predicting squat are ineffective except to
indicate that the phenomenon exists and can be positively misleading if used to
attempt to predict the amount of squat.
At present empirical methods do exist for predicting squat and the evidence
suggests that the predictions of squat obtained from them are likely to be
NO. I PREDICTION OF SQUAT 8c.

reasonably accurate (giving an answer to within about i o percent of the actual

squat) and, if in error, are likely to err on the side of safety in that they predict
a larger value for the squat than will actually occur. These methods use the
leading hull parameters together with the speed of the ship and the depth of water
and can be evaluated using a calculator. If a more accurate prediction is needed,
then it is recommended that a method should be used which is restricted to the
type of ship and the range of depths of water being considered.
More complex methods have been developed which use a more detailed
representation of the hull shape although there is no clear indication that these
methods have been tested over a wide enough range of hull shapes to be certain
that the answers given will be accurate. At some stage in the future, however,
it is likely that this will be the appropriate method of predicting squat where the
detailed information on the individual hull etc. will be held on disc and the
calculations made using an on-board computer or alternatively made onshore and
transmitted to the ship.
A more detailed review can be found in Collinson.16
£. NOTATION USED IN THIS PAPER AND THE APPENDIX.

Symbol Definition

K Cross-sectional area of channel

K Cross-sectional area of ship
B Beam of ship on the waterline
CB Block coefficient W/LBT
f» Froude number based on ship length — Eq. 2
Froude number based on water depth — Eq. 2
L Length of ship on the waterline
PP
Length of ship between perpendiculars
S Squat of ship
T Draught of ship
V Ship's speed
vc Critical wave speed —Eq. i
b Width of channel
g Acceleration due to gravity
h Depth of water
V Volume displacement of ship
Suffixes

b Bow
mid Midships
max Maximum

REFERENCES
1
Department of Trade (1980). Interaction between ships. Merchant Shipping Notice No. M93O.
2
Dand, I. W. (1989). Hydrodynamic aspects of the sinking of the ferry Herald of Free
Enterprise. Transactions Royal Institution of Naval Architects, vol. 131, p p . 14J—I6J.
86 A. MILLWARD VOL. 49
3
Millward, A. and Bevan, M. G. (1990). Data on the effect of shallow water on high speed
round bilge ship forms. University of Liverpool, Department of Mechanical Engineering. Report
TF/002/90.
4
Comstock, J. P. (1967). Principles of Naval Architecture. Society of Naval Architects and
Marine Engineers.
5
Willis, C. J., Crapper, G. D. and Millward, A. (1994). A numerical study of the
hydrodynamic forces developed by a marine rudder. Journal of Ship Research, vol. 38, no. 3, pp.
182-192.
6
Willis, C. J. (1988). Steady sideforce on a ship due to a marine propeller. University of
Liverpool, Ph.D. Thesis.
7
Ortlepp (1989). Natural squat. Canadian Maritimes Sailing Aids, vol. IV, Peltro Ltd, Canada.
8
Dennis, B. C. (1993). Squat in depths of less than twice the draft. Seaways, pp. 27, October.
9
Admiralty Manual of Navigation (1987). HMSO, vol. 1, pp. 308.
10
Blaauw, H. G. and van der Knaap, F. M. C. (1983). The prediction of squat of ships sailing
in restricted water. Proceedings 8th International Harbour Congress, Antwerp, Belgium. Ardon
International Ltd, Publication No. 302, pp. 1—13.
11
Millward, A. (1990). A preliminary design method for the prediction of squat in shallow
water. Marine Technology, vol. 27, no. 1, pp. 10—19.
12
Dand, I. W. and Ferguson, A. M. (1973). The squat of full ships in shallow water.
Transactions Royal Institution of Naval Architects, vol. 115, pp. 237—2££.
13
Dand, I. W. (1972). Full form ships in shallow water: some methods for the prediction of
squat in sub-critical flows. NPL Ship Division Report 160.
14
Tuck, E. O. (1978). Hydrodynamic problems of ships in restricted waters. Annual Review
of Fluid Mechanics, vol. 10, pp. 33—46.
15
Tuck, E. O. andTaylor, P. J. (1970). Shallow water problems in ship hydrodynamics. Proc.
8th Symposium on Naval Hydrodynamics, Pasadena, pp. 627—6^9, August.
16
Collinson, R. G. (1994). A review of methods of predicting the squat of a ship in shallow
water. University of Liverpool, Department of Mechanical Engineering, M.Sc (Eng) Thesis.
17
Barrass, C. B. (1979). The phenomenon of ship squat. International Shipbuilding Progress,
vol. 26, pp. 44-47.
18
Fuhrer, M. and Romisch, K. (1977). Effects of modern ship traffic on inland and ocean
waterways and their structures. Proc. PIANC XXIV, Leningrad, pp. 79-93.
19
Soukhomel, G. I. and Zass, V. M. (1978). Abaissement du navire en marche. Navires, Ports
et Chantiers, p p . 18—23.
20
Eryuzlu, N . E . and Hausser, R . ( 1 9 7 8 ) . E x p e r i m e n t a l investigation i n t o s o m e aspects of
large vessel navigation in restricted waterways. Proceedings Symposium on Aspects of Navigability,
Delft, Netherlands, vol. 2, pp. 1—IJ.

KEY WORDS
1. Hydrodynamics. 2. Squat. 3. Shallow water. 4. Ship handling.

APPENDIX

A 1. B a r r a s s 1 7
T h e f o r m u l a e a r e b a s e d o n m o d e l a n d p r o t o t y p e e x p e r i m e n t s f o r I - I < h/T < I-J

S ,208
max = 7CB

where y = o-i 33 for full size ships and y = o-i 2 1 for models. For ships in water where
only the depth of water is restricted an effective channel width was given as:
where A = h b C
c - eff ^ W = !CB + J
NO. I PREDICTION OF SQUAT 87

Simplified formulae are:

V2
for confined waters, 5 = 2 CB
100

V2
and for open waters, S = CB
100

where V is in knots although the result for S is in metres.

A 2. Fuhrer and Romisch18
The method calculates the sinkage at the bow and the stern at the critical speed using
the expression:
r.ocg
= °-2 \—}—
L
L w
Kent
where Sb crit and Sscrlt are the sinkage or squat at the bow and stern respectively. The
squat at any other speed can be found from:

where VcrU is the so-called critical speed, which is not related to the critical wave speed
Vv, and has to be calculated as shown in Ref. 10.
Three speed ranges have been defined:

(i) Lpp<ib also ^f>-

(ii) Lpp^3b also ^f<-

Ac 6

when ^ =

(L \ 0 ' 55
with /?=o-2 4 - ^

(iii)
Lpp
Ih I
when ^-i(

A 3. Soukhomel and Zass19

This method identified two separate water depth/draft ratios (h/T) from experiments
in laterally unrestricted water:

(i) For h/T> 1-4 S=i2-96iF2-

W
(ii) For h/T^ 14 5= n-96kV2
88 A. MILLWARD VOL. 49
where

k = 00143 ( ^ f ) " " 1 for iS<!f<9

The maximum sinkage, which was always at the stern, was calculated from:
Smax
S max=i-ioS for 9>Lpp/B>j

S
max = i^o S for £ > Ipp/B ^ 3 - 5 .

A 4. Eryuzlu and Hausser20

This formula was derived from experiments on self-propelled models of VLCCS
in open, effectively unconfined water (31 < b/B < 42) with a restricted depth
(ro8 < h/T < 278). The maximum sinkage, which was always at the bow, was found to
be

S -o-,, B(-T(-V^
V Wgh