A Comparative Analysis for Predicting Ship Squat in Shallow Water

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A Comparative Analysis for Predicting Ship Squat in Shallow Water
1Department of Naval Architecture and Marine Engineering, Bangladesh University of Engineering and
Technology (BUET), BUET Central Road, Dhaka-1000, Bangladesh
---------------------------------------------------------------------***---------------------------------------------------------
Abstract:
This paper focuses on predicting the sea-keeping characteristics of ships, with a specific emphasis on the "Squat Effect" when
ships operate with limited underkeel clearance. The severity of this problem becomes more pronounced at higher speeds. The
main objective of this investigation is to predict the squat phenomenon concerning different ship shapes and speeds. It has
been observed that both sinkage and trim associated with the squat effect increase as the square of the ship's speed. To
expedite the computation of ship squat using essential ship particulars, several reliable programs are developed based on
various squat formulae. These programs serve as efficient tools for quickly estimating ship squat values. Moreover, the study
compares the ship squat results obtained by different researchers, facilitating insights into the accuracy and reliability of
various prediction methods. By consolidating and analyzing this collective knowledge, the paper aims to enhance the
understanding of the squat effect and its implications for ships navigating with limited underkeel clearance at varying speeds.
Keywords: Squat effect; ship speed; bow squat; block coefficient; boundary layer thickness; shear stress.
NOMENCLATURE
Lpp Length between perpendiculars RLB Ratio of length of ship and breadth of
ship
B Breadth of ship RhT Ratio between depth of water and
depth of ship
D Draught of ship Volumetric displacement of ship
T Depth of ship Density of fluid
h Depth of channel Rex Reynolds number
Cb Block coefficientof ship Sb Bow squat
Vs Velocity of the vessel in m/s Boundary layer thickness
Vk Velocity of vessel in knots Shear stress
Fnh Froude number, based on depth
1. INTRODUCTION
In the ever-expanding shipping industry worldwide,
achieving minimum voyage and port turnaround times
has become a top priority for ship operations. However,
as ships increase their speed, they encounter
deteriorating sea-keeping characteristics caused by
various motions and phenomena like heaving, pitching,
yawing, and more. Another phenomenon which is often
overlooked in the study of sea-keeping characteristics is
the "Squat Effect".
The phenomenon of squats is typified by the alteration in
a vessel's position, both in terms of sinkage and trim,
brought about by its forward motion. When the ship
propels ahead, it generates a relative speed disparity
between itself and the encompassing water, culminating
in an upsurge in dynamic pressure coupled with a
subsequent decline in static pressure. This resultant
velocity configuration gives rise to a modification in
hydrodynamic pressure distribution along the vessel,
reminiscent of the Bernoulli principle, where
equilibrium between kinetic and potential energy is
maintained. This dynamic manifests in a downward
vertical force and a rotational effect around the lateral
axis, leading to distinct magnitudes at the bow and stern
sections.
The exploration of the phenomenon known as "Squat"
within naval architecture traces its origins to the
preceding century, and its significance has progressively
heightened, particularly within the context of
investigating high-speed occurrences and their
pertinence to seafaring vessels, especially in shallower
waterways. Most documented findings are rooted in
empirical investigations.
Kazi Naimul Hoque1*
, Awlad Alam Mahmud1
International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 10 Issue: 10 | Oct 2023 www.irjet.net p-ISSN: 2395-0072

A pivotal stride in comprehending "Squat" was taken by
Tuck (1966), who embarked on an innovative study,
employing matched asymptotic expansions to establish
approximate solutions [1]. Through this endeavor, he
derived equations for wave resistance, vertical forces,
and pitching moments, encompassing both subcritical
scenarios (where the depth Froude number is below 1.0)
and supercritical circumstances (where the depth
Froude number surpasses 1.0) for ship velocities. Tuck
also laid the foundation for non-dimensional coefficients
governing sinkage and trim, uncovering a noteworthy
pattern: sinkage dominates in subcritical speeds, while
trim takes precedence in supercritical velocities. His
findings revealed a commendable concurrence with
outcomes obtained from model experiments.
Building on the groundwork laid by Tuck, Beck et al.
(1975) extended the scope of investigation to encompass
channels that have been dredged, featuring shallower
peripheral areas adjacent to the deeper central channel
[2]. They engaged in solving boundary value
predicaments to foresee variations in sinkage, trim, and
ship resistance for ship speeds that are categorized as
subcritical within the central channel, as well as
subcritical or supercritical within the peripheral regions.
Their conclusions brought to light a noteworthy
revelation: the shallower peripheral sections exert a
substantial influence on sinkage, trim, and wave
resistance, a phenomenon particularly evident in
confined waterways and at elevated ship speeds. This
effect is especially pronounced when the depths of the
peripheral areas surpass those of the interior channel.
The study by Beck et al. (1975) demonstrated a strong
correlation across an array of different vessel types [2].
PIANC (1997) released a compendium of empirical
equations concerning squat effects, addressing a
spectrum of ship and channel setups [3]. Within this
array, the advisory inclination gravitated towards
favoring the ICORELS (1980), Barrass (1979), and
Eryuzlu et al. (1978, 1994) formulae, deemed to
encapsulate typical squat outcomes [4-7]. Notably,
Barrass (1981) proposed a bow squat formula,
substantiating its accuracy through validation against
measurements taken at full scale [8]. Furthermore,
distinctive equations for both bow and stern squat were
deduced by diverse researchers through rigorous
physical model experiments, spanning all three channel
configurations.
In a separate study, Demirbilek and Sargent (1999)
brought forth a noteworthy observation, revealing
substantial divergence among these assorted formulae
[9]. Their analysis underscored that adopting the more
cautious or pessimistic forecasts, those which anticipate
larger squat magnitudes, might be prudent, especially
when considering heightened risks of bottom contact.
2. SQUAT EFFECT
When a ship advances through water, it displaces water
in front of it. To maintain a continuous flow of water, this
displaced volume must return along the sides and
beneath the ship. The relative velocity of this returning
flow is slightly greater than the ship's speed, resulting in
a decrease in static pressure, causing the ship to sink
vertically into the water.
In addition to the vertical sinking, the ship typically
trims forward or aft. The overall reduction in the static
clearance under the keel, whether forward or aft, is
known as “Ship Squat”. If a ship moves too quickly in
shallow waters, where the static clearance under the
keel is, for example, 1.0 to 1.5 meters, it could be
grounded either at the bow or stern due to excessive
squat. For full-form vessels like supertankers, grounding
typically occurs at the bow. Conversely, for fine-form
vessels like passenger liners or container ships,
grounding usually happens at the stern, assuming they
are on an even keel when not in motion. However, it's
important to note that recent trends in ship design, with
shorter length and wider breadth, have led to reported
groundings near the midship bilge strakes during slight
rolling motions.
Additionally, the boundary layer also plays a role in
situations with limited under-keel clearance. For a ship
with a length of around 300 meters, the boundary layer
at the stern can extend 2 to 2.5 meters, potentially
exceeding the available under-keel clearance in shallow
channels. The interaction between the boundary layer
and the channel bottom can affect the ship's operational
behavior.
Overall, ship squats are a significant and potentially
damaging phenomenon that primarily impacts the bow
of a vessel, reducing propulsion efficiency,
compromising performance, and causing the ship to sink
deeper into the water.
3. BASIC FORMULATION
This paper focuses on investigating various empirical
formulae proposed by numerous researchers for
calculating squat values. Each formula is carefully
studied, and subsequently, a dedicated program is
developed using the FORTRAN 95 language. These
programs are designed to compute the squat values for
different ship speeds (Vk) and block coefficients (Cb). To
facilitate the calculations, the following ship particulars
are assumed:

Draught of the ship, D = 13.5 m
Depth of water, h = 16 m
Each formula's computed values are presented in tabular
form in this study. Additionally, graphical
representations are provided, with squat (Sb) on the
ordinate and ship speed (Vk) on the abscissa.
Furthermore, a thorough comparison among the
predicted values from different formulae is presented
both in tabular and graphical formats, allowing for a
comprehensive analysis of their performance.
The ICORELS (International Commission for the
Reception of Large Ships) formula (1980) for bow squat
S
b
is defined as [4],
The PIANC (1997) noted that the “2.4” constant is
sometimes replaced with a smaller value of “1.75” for full
form ships with larger Cb [3].
Millward (1990) undertook a series of physical model
experiments utilizing towed scale models, encompassing
diverse vessel types, all in unrestricted channels
characterized by widths roughly twice the vessel's length
overall (LPP) [10]. Millward’s derived formula likely
embraces a cautious perspective, leaning towards safety
by tending to forecast significant squat values [10].
Notably, his experiments were confined to a restricted
range of ship lengths, limiting the applicability of his
squat predictions to the newer and longer vessels. The
expression for the maximum bow squat (Sb) from
Millward's formulation is as follows [10],
Millward (1992) rearranged his test results and
presented them in a format [11]. The formula for bow
squat is given by [11],
Norrbin (1986) developed a formula for bow squat S
b
based on the work of Tuck and Taylor (1970) for a ship
in an unrestricted channel [12]. His predictions satisfied
the constraint that F
nh
< 0.4 and is thus somewhat
limited in its application. It is given by:
It is noted that two of the factors in the equation for S
b
are equivalent to the standard non-dimensional ratios
R
LB
and R
hT
.
The paramount non-dimensional factor is the depth
Froude number, denoted as Fnh, signifying the vessel's
resistance against movement in shallow waters. In
practical terms, most ships possess inadequate power to
surmount Fnh values surpassing 0.6 for tankers and 0.7
for container ships. A significant portion of empirical
equations necessitates Fnh to remain below the threshold
of 0.7. In all instances, it's imperative that the Fnh value
adheres to Fnh < 1, representing a critical limit to
effective speed. Mathematically, the dimensionless Fnh is
formulated as follows:
The boundary layer thickness ( ) also seems to have an
influence on ships motion. The general character of
boundary layer may be estimated based on flat plate
boundary layer theory. For a turbulent boundary layer
past a flat plate, the boundary layer thickness grows
downstream from the leading edge according to the
approximate empirical relation:
An empirical formulation for the wall shear stress, due
to an unrestricted turbulent boundary layer flowing past
a smooth flat plate i,
4. RESULTS AND DISCUSSION
The provided figures offer a comprehensive insight into
the effects of vessel speed and Cb on bow squat,
boundary layer thickness, and shear stress. Figures 1(a)
to 1(d) illustrate the variations in vessel's bow squat
with ship speed for different Cb values ranging from 0.5
to 0.9. Similarly, Figures 2(a) to 2(d) provide a
comparative analysis of bow squat across various
formulae given by ICORELS (1980), Millward (1990),
Millward (1992), and Norrbin (1986), encompassing Cb
values from 0.6 to 0.9 [4, 10-12]. Furthermore, Figures 3
and Figure 4 depict the changes in boundary layer
thickness (𝛿) and shear stress (τ) concerning distance
from the vessel's leading edge (x) at different vessel
speeds (Vk).
Length between perpendiculars of the ship,
Lpp = 320 m
Breadth of the ship, B = 55 m

Figure 1(a) illustrates the variation of bow squat of a
vessel, expressed in meters, as a function of the vessel's
velocity in knots. This depiction encompasses a
spectrum of Cb values, ranging from 0.5 to 0.9, as
dictated by the formulation provided by ICORELS (1980)
[4]. Across all instances, the ascent of bow squat exhibits
a notably steep incline in conjunction with the
augmentation of ship velocity. In the scenario of a Cb
value of 0.5, the surge in bow squat is marginal,
achieving a pinnacle of unity when the speed attains 14
knots. Conversely, with a Cb of 0.9, the magnitude of bow
squat elevates to a peak of 2 m at the identical speed of
14 knots. The remaining curves are situated between
these boundaries, aligning with Cb values of 0.5 and 0.9.
Figure 1(b) shows the variation of bow squat of a vessel
in meters as a function of ship speed in knots for
different values of Cb ranging from 0.5 to 0.9, according
to the formulation given by Millward (1990) [10]. The
curve shows that the bow squat increases significantly
with the increase of ship speed. The maximum value of
bow squat is around 4 m for Cb =0.9 and around 1.5 m
for Cb =0.5 corresponding to a speed of 14 knots.
The remaining curves fall between these two extreme
curves for Cb =0.5 and Cb =0.9 in which the increase is
gradual for all the cases.
Figure 1(c) illustrates the variation of bow squat of a
vessel in meters, relative to ship speed in knots, for
different Cb values varying from 0.5 to 0.9 based on
Millward's (1992) formulation [11]. In contrast to the
notable increase observed in Millward (1990), the
curves here show a less dramatic growth in bow squat.
The maximum bow squat value occurs at a speed of 14
knots, slightly exceeding 0.5 m for Cb = 0.5, and
approximately 2 m for Cb = 0.9. The other curves fall
within this range and share similar characteristics with
the other depicted curves.
Figure 1(d) depicts the variability of vessel's bow squat
in meters, plotted against ship speed in knots, across
distinct Cb values ranging from 0.5 to 0.9 using the
formula devised by Norrbin (1986) [12]. Unlike the
steeper trends observed in other bow squat curves with
increasing ship speed, the curves here exhibit a more
gradual increase in bow squat. This formula, designed
for ships in unrestricted channels, has limited
applicability and is constrained by Fnh < 0.4. The peak
bow squat occurs at a speed of 9 knots, amounting to 0.7
m for Cb =0.9 and approximately 0.4 m for Cb =0.5. The
remaining curves fall within the range defined by these
two extreme curves.
Figure 2(a) represents the comparison of bow squat (for
Cb =0.6) in meters against ship speed in knots among
various formulae developed by ICORELS (1980),
Millward (1990), Millward (1992) and Norrbin (1986)
[4, 10-12]. All the curves for bow squat increase rapidly
with the rise of ship speed. The curve for Millward
(1990) is sharper than the other curves giving a
maximum value of just above 2 m, while the curve for
Millward (1992) shows a slight increase in bow squat
having a maximum value of just above 1 m for the
corresponding speed of 14 knots. This means that the
effect of bow squat is more significant in case of former
one than the latter. The other two curves for ICORELS
(1980) and Norrbin (1986) coincide with each other in
which ICORELS (1980) extends up to 14 knots giving a
peak value of just under 1.5 m while the curve for
Norrbin (1986) cannot extend beyond 9 knots due to the
limitations and Fnh < 0.4, giving a peak value of around
0.5 m at those speed. The effect of bow squat is less
significant for these two curves.
Figure 2(b) provides a comparison of bow squat (for Cb =
0.7) in meters against ship speed in knots, considering
formulae developed by ICORELS (1980), Millward
(1990), Millward (1992), and Norrbin (1986) [4, 10-12].
All the bow squat curves demonstrate a pronounced
increase with increasing ship speed. Notably, the curve
corresponding to Millward (1990) displays a steeper
increase, reaching a peak value just below 3 m at 14
knots, signifying a more impactful bow squat effect. The
remaining three curves for ICORELS (1980), Millward
(1992), and Norrbin (1986) closely align, yielding a peak
value of about 1.5 m at a speed of 14 knots. However, the
curve for Norrbin (1986) is limited by its applicability
and cannot extend beyond 9 knots due to Fnh < 0.4
constraints, resulting in a peak value of 0.5 m. The effect
of bow squat is less pronounced for these latter curves in
contrast to the more prominent impact observed with
Millward (1990).

Figure 1. Variation of bow squat (Sb) with velocity of ship (Vk) for different Cb by (a) ICORELS (1980), (b) Millward (1990),
(c) Millward (1992), and (d) Norrbin (1986).
Figure 2(c) displays a comparative analysis of bow squat
(for Cb = 0.8) in meters against ship speed in knots,
encompassing multiple formulations including ICORELS
(1980), Millward (1990), Millward (1992), and Norrbin
(1986) [4, 10-12]. All the bow squat curves exhibit a
notable increase with the increase of ship speed. The
curve representing Millward (1990) displays a more
pronounced incline, reaching a peak value of
approximately 3.5 m at 14 knots, underscoring its
significant impact. In contrast, the curve for Millward
(1992) shows a less steep incline, giving a maximum
value of about 2 m. This highlights the greater
significance of the bow squat effect in the former
formula compared to the latter. The remaining two
curves, ICORELS (1980) and Norrbin (1986), coincide,
with ICORELS' (1980) peak value just above 1.5 m at a
speed of 14 knots. On the other hand, Norrbin (1986)
reaches a peak value of around 0.5 m at a speed of 9
knots, reflecting its limited applicability. These two
curves offer more reliable values for mitigating the squat
effect.

Table 1: Variation of Sb with Vk for different Cb by ICORELS (1980).
Velocity
of Ship
(knots)
Cb = 0.5 Cb = 0.6 Cb = 0.7 Cb = 0.8 Cb = 0.9
bow squat
(meter)
bow squat
(meter)
bow squat
(meter)
bow squat
(meter)
bow squat
(meter)
1 0.005 0.006 0.007 0.008 0.008
2 0.019 0.023 0.026 0.03 0.034
3 0.043 0.051 0.06 0.068 0.077
4 0.076 0.091 0.107 0.122 0.137
5 0.12 0.144 0.168 0.192 0.216
6 0.174 0.209 0.244 0.279 0.314
7 0.248 0.286 0.336 0.384 0.432
8 0.318 0.382 0.445 0.507 0.573
9 0.407 0.491 0.573 0.655 0.737
10 0.515 0.618 0.721 0.824 0.927
11 0.637 0.764 0.891 1.019 1.146
12 0.777 0.932 1.087 1.243 1.398
13 0.938 1.126 1.313 1.501 1.689
14 1.124 1.349 1.574 1.799 2.024
Table 2: Variation of Sb with Vk for different Cb by Millward (1990).
Velocity
of Ship
(knots)
Cb = 0.5 Cb = 0.6 Cb = 0.7 Cb = 0.8 Cb = 0.9
bow squat
(meter)
bow squat
(meter)
bow squat
(meter)
bow squat
(meter)
bow squat
(meter)
1 0.004 0.006 0.007 0.008 0.01
2 0.017 0.023 0.029 0.035 0.041
3 0.04 0.054 0.069 0.083 0.097
4 0.075 0.101 0.127 0.153 0.179
5 0.122 0.165 0.208 0.25 0.293
6 0.184 0.249 0.313 0.377 0.442
7 0.264 0.355 0.447 0.539 0.631
8 0.362 0.489 0.615 0.741 0.868
9 0.484 0.653 0.821 0.99 1.159
10 0.632 0.853 1.074 1.294 1.515
11 0.813 1.096 1.38 1.663 1.947
12 1.032 1.391 1.751 2.111 2.471
13 1.297 1.747 2.201 2.654 3.106
14 1.619 2.184 2.749 3.313 3.875

Table 3: Variation of Sb with Vk for different Cb by Millward (1992).
Table 4: Variation of Sb with Vk for different Cb by Norrbin (1986).
Velocity
of Ship
(knots)
Cb = 0.5 Cb = 0.6 Cb = 0.7 Cb = 0.8 Cb = 0.9
bow squat
(meter)
bow squat
(meter)
bow squat
(meter)
bow squat
(meter)
bow squat
(meter)
1 0.004 0.005 0.007 0.008 0.009
2 0.015 0.021 0.026 0.032 0.038
3 0.034 0.047 0.06 0.073 0.085
4 0.061 0.084 0.107 0.13 0.152
5 0.097 0.133 0.168 0.204 0.24
6 0.141 0.193 0.245 0.297 0.349
7 0.194 0.265 0.337 0.409 0.481
8 0.256 0.352 0.447 0.542 0.637
9 0.33 0.452 0.575 0.697 0.819
10 0.415 0.569 0.723 0.877 1.031
11 0.513 0.704 0.894 1.085 1.275
12 0.626 0.859 1.091 1.323 1.556
13 0.756 1.037 1.318 1.598 1.879
14 0.906 1.243 1.579 1.915 2.252
Velocity
of Ship
(knots)
Cb = 0.5 Cb = 0.6 Cb = 0.7 Cb = 0.8 Cb = 0.9
bow squat
(meter)
bow squat
(meter)
bow squat
(meter)
bow squat
(meter)
bow squat
(meter)
1 0.005 0.006 0.007 0.008 0.009
2 0.019 0.023 0.027 0.031 0.035
3 0.044 0.052 0.061 0.07 0.078
4 0.077 0.093 0.108 0.124 0.139
5 0.121 0.145 0.169 0.193 0.218
6 0.174 0.209 0.244 0.278 0.313
7 0.237 0.284 0.332 0.379 0.426
8 0.309 0.371 0.433 0.495 0.557
9 0.392 0.47 0.548 0.626 0.705

Figure 2. Comparison of bow squat (Sb) for different velocity of vessel (Vk) among various formulae developed by ICORELS
(1980), Millward (1990), Millward (1992) and Norrbin (1986) at (a) Cb = 0.6, (b) Cb = 0.7, (c) Cb = 0.8, and (d) Cb = 0.9.
Figure 2(d) represents the comparison of bow squat (for
Cb =0.9) in meters against ship speed in knots among
various formulae developed by ICORELS (1980),
Millward (1990), Millward (1992) and Norrbin (1986)
[4, 10-12]. For all these curves, bow squat increases
sharply with the rise of ship speed. The curve for
Millward (1990) is sharper than the other curves, giving
a maximum value of bow squat of around 4 m at a speed
of 14 knots. The curve for Millward (1992) is less sharp
than the previous one, giving a peak value of around 2 m.
So, the curve for Millward (1992) is more reliable in
comparison with Millward (1990) in minimizing the
effect of bow squat. The remaining two curves
corresponding to ICORELS (1980) and Norrbin (1986)
align closely, presenting a peak value of 2 m at a speed of
14 knots for ICORELS (1980). However, the curve
associated with Norrbin (1986) is constrained by its
applicability and cannot extend beyond a speed of 9
knots due to limitations tied to Fnh < 0.4. Consequently, it
yields a peak value of around 0.7 m at these speeds. The
bow squat effect depicted by these curves is
comparatively less pronounced when contrasted with
the curves outlined by Millward (1992) and Millward
(1990).
Figure 1 (a to d) and Figure 2 (a to d) explore how vessel
speed and Cb influence bow squat. As ship speed
increases, bow squat exhibits a steep rise across all

scenarios. Notably, higher Cb values lead to increased
bow squat, reaching peaks at 2 m (Cb = 0.9).
Formulations by Millward (1990) and Millward (1992)
demonstrate significant increases in bow squat, while
Norrbin's formula (1986) yields more subdued results.
Millward (1992) proves to be a reliable means of
mitigating bow squat's effects. The corresponding values
of bow squat with ship speed at different Cb values,
developed by ICORELS (1980), Millward (1990),
Millward (1992), and Norrbin (1986), are detailed in
Table 1 to Table 4.
Figure 3. Variation of boundary layer thickness (𝛿) as
function of distance from the leading edge of vessel (x)
for different values of velocity of the vessel (Vk).
Figure 3 presents a graphical representation illustrating
the dynamic alterations in boundary layer thickness (𝛿)
concerning its relationship with the distance measured
from the leading edge of the vessel (x). This visualization
encompasses a spectrum of distinct velocity values for
the vessel (Vk). Evidently, the boundary layer thickness
experiences a progressive augmentation as the
measurement distance from the vessel's leading-edge
increases. Of noteworthy significance is the observation
that the most substantial magnitude of boundary layer
thickness for any designated distance emerges when the
vessel is operating at its lowest speed. Intriguingly, this
maximum thickness value diminishes as the vessel's
speed increases. In essence, there exists an inverse
correlation between the vessel's speed and the
magnitude of its boundary layer thickness, as
demonstrated by the trends depicted in the graph.
Figure 3 showcases the dynamic relationship between
boundary layer thickness (𝛿) and the distance from the
vessel's leading edge (x) at varying velocities (Vk). A
consistent pattern emerges as x increases, 𝛿 also grows.
The maximum thickness occurs at the lowest vessel
speed, decreasing as speed rises. This reveals an inverse
correlation between vessel speed and boundary layer
thickness. The variation of boundary layer thickness (𝛿)
with respect to distance from the leading edge of vessel
(x) for different values of Vk are detailed in Table 5.
Figure 4. Variation of Shear stress ( ) as function of
distance from the leading edge of vessel (x) for different
values of velocity of the vessel (Vk).
Figure 4 provides an illustrative presentation
highlighting the nuanced changes in shear stress (τ)
concerning its interplay with the distance traversed from
the leading edge of the vessel (x). This depiction
encompasses a spectrum of diverse velocity values
attributed to the vessel (Vk). Evidently, a discernible
pattern emerges: the shear stress experiences a
progressive attenuation as one moves away from the
vessel's leading edge. A pivotal observation of note is the
marked occurrence of the highest shear stress value
when the vessel is operating at its peak speed. This
maximum value surfaces when the vessel is at its utmost
velocity. Particularly noteworthy is the revelation that
the rate of this decrease in shear stress is particularly
prominent when the vessel is moving at higher
velocities, especially in proximity to the leading edge
(represented by smaller x values). This emphasizes that
the attenuation of shear stress is more pronounced when
the vessel's speed is greater. Conversely, at lower
velocities, the shear stress values tend to be of trivial
significance. This discrepancy in shear stress magnitude
between low and high velocities signifies the complex
interplay between vessel speed and the resultant shear
stress, which is acutely dependent on the vessel's
position relative to its leading edge.

Table 5: Variation of boundary layer thickness (𝛿) as function of distance from the leading edge of vessel (x) for different
values of Vk.
boundary
layer
thickness
for
Vk = 3
knots
boundary
layer
thickness
for
Vk = 7
knots
boundary
layer
thickness
for
Vk = 11
knots
20 0.291 0.249 0.232 0.221 0.213 0.207 0.202
50 0.639 0.546 0.508 0.484 0.467 0.454 0.443
80 0.956 0.817 0.76 0.724 0.699 0.679 0.663
110 1.257 1.074 0.998 0.952 0.918 0.892 0.871
140 1.545 1.321 1.228 1.17 1.129 1.097 1.071
170 1.825 1.56 1.45 1.382 1.333 1.296 1.265
200 2.098 1.793 1.667 1.589 1.533 1.489 1.454
230 2.365 2.021 1.879 1.791 1.728 1.679 1.639
260 2.627 2.245 2.087 1.989 1.919 1.865 1.821
290 2.884 2.465 2.292 2.184 2.107 2.048 1.999
320 3.138 2.682 2.494 2.377 2.293 2.228 2.175
Table 6: Variation of shear stress ( ) as function of distance from the leading edge of vessel (x) for different values of Vk.
Distance from
leading edge
(meter)
shear
stress in
(Pa) for
Vk = 1
knots
shear
stress in
(Pa) for
Vk = 3
knots
shear
stress in
(Pa) for
Vk = 5
knots
shear
stress in
(Pa) for
Vk = 7
knots
shear
stress in
(Pa) for
Vk = 9
knots
shear
stress in
(Pa) for
Vk = 13
knots
20 1.307 10.055 25.964 48.501 77.347 112.278 153.12
50 1.147 8.821 22.778 42.55 67.857 98.503 134.333
80 1.072 8.248 21.299 39.787 63.451 92.106 125.61
110 1.025 7.881 20.352 38.017 60.629 88.01 120.024
140 0.99 7.614 19.663 36.73 58.576 85.029 115.959
170 0.963 7.406 19.125 35.725 56.973 82.703 112.787
200 0.941 7.236 18.686 34.905 55.666 80.805 110.199
230 0.922 7.093 18.316 34.215 54.565 79.208 108.02
260 0.906 6.97 17.998 33.621 53.618 77.833 106.145
290 0.892 6.862 17.72 33.101 52.788 76.628 104.502
320 0.88 6.766 17.472 32.639 52.051 75.558 103.042
boundary
layer
thickness
for
Vk = 1
knots
boundary
layer
thickness
for
Vk = 5
knots
boundary
layer
thickness
for
Vk = 9
knots
boundary
layer
thickness
for
Vk = 13
knots
Distance from
leading edge
(meter)
shear
stress in
(Pa) for
Vk = 11
knots

Figure 4 elucidates the intricate interplay between shear
stress (τ) and distance from the vessel's leading edge (x),
considering different Vk values. Notably, shear stress
declines progressively as x increases. The highest shear
stress occurs at the peak vessel speed, especially in
proximity to the leading edge. At high velocities, shear
stress attenuation is more pronounced. Conversely,
lower velocities yield lower shear stress values. This
underlines the complex relationship between vessel
speed and shear stress, dependent on the vessel's
position relative to its leading edge. The relationship
between shear stress (τ) and the distance from the
leading edge of the vessel (x) is detailed in Table 6 for
various Vk values.
In summary, the combined information presented in the
Figures and Tables underscores the substantial influence
of vessel velocity and block coefficient on bow squat,
boundary layer thickness, and shear stress. These
parameters play pivotal roles in the realm of ship
hydrodynamics.
5. CONCLUSIONS
The key findings from this research can be summarized
as follows:
 Bow squat:
o Bow squat increases significantly with higher
vessel speeds across all Cb values.
o Higher Cb values lead to greater bow squat, with
peak values observed at Cb = 0.9.
o Formulations by Millward (1990) and Millward
(1992) show more pronounced increases in bow
squat compared to ICORELS (1980) and Norrbin
(1986).
o Millward (1992) appears to be a reliable
formula for mitigating the bow squat effect.
 Boundary layer thickness:
o Boundary layer thickness increases with
distance from the vessel's leading edge.
o The highest boundary layer thickness occurs at
lower vessel speeds and decreases as speed
increases.
o There is an inverse correlation between vessel
speed and boundary layer thickness.
 Shear stress:
o Shear stress decreases as one moves away from
the vessel's leading edge.
o The highest shear stress values are observed at
the highest vessel speeds, especially near the
leading edge.
o Shear stress attenuation is more pronounced at
higher speeds.
Overall, these findings highlight the complex interplay
between vessel speed, block coefficient, and their effects
on bow squat, boundary layer thickness, and shear stress
in ship hydrodynamics. Understanding these
relationships is crucial for optimizing vessel design and
navigation in various maritime conditions.
ACKNOWLEDGEMENT
The authors extend their heartfelt appreciation to all
individuals who have offered invaluable assistance, both
in direct and indirect capacities, across the diverse
stages of this research endeavor. Additionally, the
authors wish to convey their gratitude to the
Department of Naval Architecture and Marine
Engineering (NAME) at BUET, not only for granting
access to the Journal Library but also for providing with
a selection of highly beneficial papers that aided to this
work significantly.
REFERENCES
[1] Tuck, E. O. (1966), Shallow-water flows past slender
bodies, JFM, vol.26, no. 1, pp.81-95.
[2] Beck, R. F., Newman, J. N., and Tuck, E. O. (1975),
Hydrodynamic forces on ships in dredged channels,
J. Ship Research, vol. 19, no. 3, pp.166-171.
[3] PIANC. (1997), Approach channels: A Guide for
design. Final Report of the Joint PIANC-IAPH
Working Group II-30 in cooperation with IMPA and
IALA, Supplement to Bulletin No. 95, June.
[4] ICORELS (International Commission for the
Reception of Large Ships). (1980) Report on
Working Group IV, PIANC Bulletin No. 35,
Supplement.
[5] Barrass, C. B. (1979), The phenomenon of ship squat,
International Shipbuilding Progress, vol. 26, pp. 44-
47.
[6] Eryuzlu, N. E., and Hausser, R. (1978), Experimental
investigation into some aspects of large vessel
navigation in restricted waterways. Proceedings of
the Symposium of Aspects of Navigability of
Constraint Waterways Including Harbor Entrances,
vol. 2, pp. 1-15.
[7] Eryuzlu, N. E., Cao, Y. L., and D’Agnolo, F. (1994),
Underkeel requirements for large vessels in shallow
waterways. 28
t
International Navigation Congress,
PIANC, Paper S11-2, Sevilla, pp. 17-25.
[8] Barrass, C. B. (1981), Ship squat – A reply. The Naval
Architect, pp. 268-272.
[9] Demirbilek, Z., and Sargent, F. (1999), Deep-draft
coastal navigation entrance channel practice, Coastal

and Hydraulic Engineering Technical Note, pp. 1-11,
March.
[10] Millward, A. (1990), A preliminary design method
for the prediction of squat in shallow water. Marine
Technology, vol. 27, no. 1, pp.10-19.
[11] Millward, A. (1992), A comparison of the theoretical
and empirical prediction of squat in shallow water,
International Shipbuilding Progress, vol.39, no.417,
pp. 69-78.
[12] Norrbin, N. H. (1986), Fairway design with respect
to ship dynamics and operational requirements,
SSPA Research Report No. 102. Gothenburg,
Sweden: SSPA Maritime Consulting.

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