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1 INTRODUCTION
Moonpools are a vertical opening in the hull that
provides sheltered access to the sea and facilitates the
deployment of underwater equipment and operations,
making it a common feature of research, service, and
drilling vessels. The free surface inside the moonpool
exhibits dynamic behaviour distinct from that of the
surrounding sea surface, both in calm water and under
wave excitation. This phenomenon not only affects the
hydrodynamic environment within the opening but
also increases the vessel’s resistance, as water is drawn
into the moonpool while proceeding. Partial mitigation
can be achieved through structural modifications, such
as installation of damping plates. However, such
structures are particularly prone to resonance
problems, caused by external wave action and/or
wave-induced vessel motions [1]. The water motion
inside the moonpool is characterized by two resonant
modes. In shorter moonpools, the piston mode
prevails, with the entrapped water oscillating
vertically in a nearly solid-body manner. In longer
moonpools, the sloshing mode becomes more
pronounced, resembling the behaviour observed in
tanks [1]. From a design perspective, determining the
hydrodynamic characteristics of the moonpool is
essential, since excessive internal wave elevation may
impair safety and limit the functionality of the opening.
Research on moonpool hydrodynamics has been
developing for several decades, combining
experimental, analytical, and numerical approaches.
Early work by Fukuda [2] provided extensive model
tests of circular and square moonpools, leading to an
empirical formula for estimating resonance
frequencies. Matusiak [3] further advanced the
theoretical framework by deriving the governing
equations for vertical water motion in both open and
covered moonpools, highlighting nonlinearities due to
changes in the oscillating water mass. Molin offered a
Determination of Hydrodynamic Coefficients
of Moonpool from Free Decay and Forced Oscillation
Tests Including the Influence of Damping Plates
E. Ciba, M. Jurczyński & P. Dymarski
Gdańsk University of Technology, Gdańsk, Poland
ABSTRACT: The paper presents Reynolds-Averaged Navier–Stokes (RANS) simulations of a hull with a
moonpool, aiming to determine its hydrodynamic coefficients and to evaluate the effect of damping plates.
Simulations included the response of the floating body in head waves as well as free decay and forced oscillation
tests. From these tests, added mass and damping coefficients of the moonpool were determined and analysed.
Special attention was given to the application of additional horizontal damping plates installed at the moonpool
entrance. The results demonstrate that the non-dimensional damping coefficient exhibits an approximately linear
relationship with the degree of inlet area blockage caused by the plates. Furthermore, the influence of the
excitation wave amplitude on the amplification factor was investigated, revealing that the factor decreases with
increasing wave amplitude. These findings provide new insights into the hydrodynamic behaviour of moonpools
and may support the design of offshore vessels and floating platforms equipped with such structures.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 20
Number 2
June 2026
DOI: 10.12716/1001.20.02.21
476
theoretical formulation of piston and sloshing modes
[4], later refined with extensions for recesses [5] and
finite water depth [6]. These contributions laid the
foundations for modern predictive models, though
their reliance on potential flow assumptions limits their
applicability in strongly nonlinear regimes.
Several authors have sought to bridge theory and
practice. Gaillarde and Coteleer [7] combined empirical
observations with numerical simulations using the
ComFlow solver and Volume of Fluid (VOF) method,
capturing phenomena such as flow separation and
vortex formation at the moonpool inlet. Yang et al. [8]
compared potential flow calculations for moonpools
with and without a cofferdam against the theoretical
formulas of Molin and Fukuda, confirming their
validity for geometry without cofferdam. Gordon [9]
proposed both linear and nonlinear piston-mode
models and implemented them in practical Excel
worksheets, providing a tool for rapid engineering
estimates. More recent studies have shifted towards
coupled ship–moonpool dynamics and mitigation
strategies. Ravinthrakumar et al. [10] performed model
tests on square and rectangular moonpools in
simplified hulls under regular and irregular waves.
Reiersen et al. [11] demonstrated that moonpools with
rounded or square inlets, and with vertical plates, can
significantly reduce pitch motions of floating
structures. Michima and Kawabe [12] also emphasized
operability aspects by linking drillship performance
with moonpool sloshing effects.
In parallel, advanced numerical approaches have
been ap-plied. Fredriksen et al. [13] compared semi-
nonlinear and fully nonlinear hybrid methods for
calculating piston-mode motion in the moonpool and
rigid-body motions, showing that the semi-nonlinear
approach tends to overestimate the results. Authors
reported that for the considered case, the amplification
factor of the moonpool wave relative to the incoming
wave was about 2–2.5 near resonance, whereas linear
theory predicts a value of up to 10 [13]. Liu et al. [14]
investigated the impact of moonpool geometry on
drillship motion performance through combined
potential flow theory and experimental validation.
Saghi et al. [15] employed RANS-CFD to study
trapezoidal moonpools in barge-type floating wind
platforms, focusing on their potential integration with
an Oscillating Water Column (OWC) wave energy
capture system. Finally, several studies have addressed
effects of moonpool localization and design
sensitivities. Garad et al. [16] showed that the
longitudinal position of a moonpool influences internal
wave elevation and that ship draft affects the
moonpool’s natural oscillation period.
Together, these studies reveal a consistent
trajectory: from empirical and potential-flow
formulations toward nonlinear, CFD-based, and
coupled vessel–moonpool investigations. The
literature underlines both the complexity of internal
water motion and its practical implications for vessel
operability, resistance, and safety. Previous RANS-
CFD studies on moonpools have primarily focused on
flow analysis and the mitigation of resonant
oscillations. However, the literature lacks systematic
investigations of free decay and forced oscillation tests
specifically aimed at determining hydrodynamic
coefficients, as well as assessments of the influence of
horizontal plates on these characteristics. Such an
approach has been applied in the context of floating
platforms [17], but not for moonpools, which is
addressed in the present work.
2 GEOMETRY AND DESIGN DETAILS
The numerical analyses were performed for a 1:34.5
scale model, 4 m in length. The model featured both
square and rectangular moonpools with vertical walls
and was based on the geometry previously studied and
described by Ravinthrakumar et al. [10]. The
dimensions of the model and the moonpool, as well as
its location within the hull, are shown in Fig. 1.
Figure 1. Dimensions of the model and moonpool location
within the hull [10]
Subsequently, the effect of an additional horizontal
damping plate installed at the moonpool inlet was
investigated for the square moonpool cases (0.2 m ×
0.2 m and 0.4 m × 0.4 m). For each configuration, six
simulations were carried out corresponding to
different degrees of inlet area blockage. The blockage
ratio A/A0 was defined, where A0 is the original
moonpool area and A is the area reduced by damping
plates. In both cases, four damping plates were
installed around the perimeter of the moonpool, and
their widths are summarized in Tab. 1. A side view of
the moonpool, the hull, and the damping plates is
presented in Fig. 2.
Table 1. Width of damping plates corresponding to selected
blockage ratios
Blockage ratio A/A0
1.00
0.81
0.64
0.49
0.36
0.25
Width of plates LP (0.2 m x 0.2 m) [m]
0.00
0.01
0.02
0.03
0.04
0.05
Width of plates LP (0.4 m x 0.4 m) [m]
0.00
0.02
0.04
0.06
0.08
0.10
Figure 2. Side view of the moonpool, the hull, and the
damping plates
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3 METHODOLOGY
The Reynolds-Averaged Navier–Stokes (RANS)
simulations were carried out using the CFD software
STAR-CCM+. The Volume of Fluid (VOF) model was
employed to capture the free surface, while turbulence
was represented by the k-ε model. For simulations
involving body motion, the Dynamic Fluid Body
Interaction (DFBI) approach with overset mesh
technique was applied. Three types of numerical
analyses were carried out. First, simulations of the
floating body motion in regular head waves with an
internal moonpool were performed and compared
with experimental data available in the literature [10].
In addition, free decay oscillation tests of the water
inside the moonpool were conducted to determine the
natural frequencies and damping characteristics.
Finally, forced oscillation tests were carried out, where
the water motion was induced by applying a time-
varying pressure boundary condition on the upper
closing surface of the moonpool.
A mesh and time-step sensitivity study was
conducted, involving three mesh densities and four
time-step values, as summarized in Tab. 2. Mesh and
time-step validation was performed for the fixed hull
with a square moonpool of 0.2 m × 0.2 m subjected to
regular head waves (case MP1). The results obtained
for different time steps (t1 – t4) and mesh densities (m1
– m3) are compared in Fig. 3. The results indicated low
sensitivity to mesh refinement but significant
sensitivity to the time-step size. Based on these
findings and available computational resources, case t3
– m2 was selected for further simulations.
Table 2. Width of damping plates corresponding to selected
blockage ratios
Id.
Time step (t), [s]
Number of grid
elements (m)
1
0.0100
1.38 ∙ 107
2
0.0020
1.97 ∙ 107
3
0.0010
4.86 ∙ 107
4
0.0005
-
Figure 3. Mesh and time-step sensitivity study – comparison
of numerical results of surface elevation for the fixed hull
with a 0.2 m × 0.2 m moonpool subjected to regular head
waves
4 RESULT AND DISCUSSION
Result presented in this section are obtained from two
distinct numerical setups, depending on the objective
of each analysis. Comparison with experimental data
(section 4.1.) was obtained from simulations with a
freely floating hull subjected to regular head waves. In
contrast, the results of free decay tests and forced
oscillation tests (sections 4.2. and 4.3.) were obtained
using a fixed-hull model in order to isolate the intrinsic
hydrodynamic behavior of the moonpool.
4.1 Comparison with experimental data on a regular wave
Simulations were performed for the hull with a square
moonpool of 0.2 m × 0.2 m, denoted as MP1 in
Ravinthrakumar et al. [10]. The analyses considered the
floating body subjected to regular head waves of height
H=0.04 m and varying frequencies. Consequently, the
parameter H/λ, used by the authors, was variable in the
present study and ranged from 0.01 to 0.04. The
obtained results were compared with the literature
data for moonpool surface elevation as well as the
heave and pitch motions of the hull, with all results
presented at model scale.
a) Moonpool response (MP1)
b) Heave response
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c) Pitch response
Figure 4. Responses in head waves for MP1 from [10]
compared with RANSE CFD results
Figures 4. a-c) present the comparisons: moonpool
surface elevation (Fig. 4. a), heave response (Fig. 4. b),
and pitch response (Fig. 4. c), including data from both
the original study [10] for different experimental cases
(irregular wave H/λ = 1/100, 1/80, 1/60; regular wave :
H/λ = 1/30, 1/60) and WAMIT simulations, along with
the carried out RANS-CFD calculations. The WAMIT
predictions, based on linear potential flow theory in the
frequency domain, are in good agreement with the
experimental data, accurately capturing both the wave
periods and response amplitudes. In contrast, the
RANS-CFD results reproduce the wave periods
reasonably well, but the heave amplitudes are
underestimated while the pitch amplitudes are slightly
overestimated. These differences arise because the
RANS-CFD model includes viscous damping and
nonlinear free-surface effects, which generally reduce
the response amplitude. For small waves and low
motion amplitudes (typical of laboratory conditions),
viscous effects may be less dominant, which partly
explains why the potential flow solver can provide a
closer match to experimental observations. Therefore,
while RANS-CFD provides a more physically complete
representation of the flow, WAMIT predictions happen
to match the experimental amplitudes more closely
under these specific conditions.
4.2 Free decay tests
Subsequently, free decay tests were performed for the
considered cases. To this end, an initial free surface
level inside the moonpool was set higher than that
outside the hull, and the time history of its decay was
measured. The motion was assumed to follow a linear
damping model, expressed by Eq. 1:
( )
( )
0
cos
t
t Ae t
−
=+
(1)
where: A- initial oscillation amplitude,
- free surface
elevation inside the moonpool,
- damping coefficient,
- oscillation frequency,
0- phase shift
The dimensionless damping coefficient was
determined from four successive amplitudes according
to Eq. 2. The calculated values of surface elevation are
shown in Fig. 5. The value of 1 m in the free decay test
is used as the reference level and represents the water
level height measured from the bottom of the
computational domain. For cases MP1 (0.2 m × 0.2 m)
and MP2 (0.4 m × 0.4 m), the decay was approximately
linear, whereas case MP3 exhibited strongly nonlinear
behavior.
1
23
0
1
ln
2
ii
ii
aa
aa
+
++
−
==
−
(2)
Figure 5. Free decay tests – changes in water surface level
over time
The average oscillation periods for each case are
listed in Tab. 3, together with a comparison to the
theoretical predictions [6]. However, the results for
MP1 and MP2 are not fully linear. When estimating the
damping coefficient using the least-squares method, it
was found that the fitting could only be satisfactorily
achieved within a limited range (Fig. 6.).
Table 3. Comparison of free decay periods in moonpools
(MP1–MP3) obtained from RANS-CFD simulations and
analytical formulation [6]
Free decay period [s]
Case
RANS-CFD
Molin et al. [6]
MP1 (0.2 m x 0.2 m)
1.08
1.07
MP2 (0.4 m x 0.4 m)
1.21
1.19
MP3 (0.4 m x 2.0 m)
1.39
1.31
Figure 6. Surface elevation in moonpool over time - fitting of
a linear damping function within a limited time range
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Instantaneous damping coefficients and
frequencies were determined for successive oscillation
periods. It can be observed that the instantaneous
damping coefficient decreases with successive
oscillations in both cases (Fig. 7.). The oscillation
frequency is initially lower, then increases, and finally
stabilizes at a constant level (Fig. 8.).
Figure 7. Instantaneous damping coefficients over successive
oscillation periods
Figure 8. Oscillation frequencies over successive oscillation
periods
Additional computations were carried out for
moonpools MP1 and MP2 equipped with damping
plates. The resulting time histories for different
blockage ratios are shown in Fig. 9. With increasing
plate width, the natural oscillation frequency
decreases, as illustrated in Fig. 10. An average
instantaneous damping coefficient was also estimated
based on the first four amplitudes and is presented as
a function of the inlet blockage ratio in Fig. 11.
Figure 9. Free decay test of moonpool MP1 with damping
plate for varying inlet blockage ratio
Figure 10. Effect of inlet blockage on the natural oscillation
frequency of the moonpool
Figure 11. Average non-dimensional damping coefficient as
a function of the inlet blockage ratio
4.3 Forced oscillation tests
Forced oscillation tests allow for the determination of
hydrodynamic coefficients over a broader frequency
range compared to the free decay method, which is
limited to the natural frequency. The forced oscillation
simulations were carried out by prescribing a time-
varying pressure boundary condition on the upper
surface of the moonpool, while recording the free-
surface elevation inside the moonpool. The resulting
time histories were approximated with a sinusoidal
function (Eq. 3), from which the free-surface velocity
, and acceleration
were obtained.
( )
( )
0
sin
A
tt
= +
(3)
In the next step, the hydrodynamic coefficients
were determined using the Solver tool in Microsoft
Excel by minimizing the difference between the
solution of the forced oscillation (Eq. 4) and the applied
excitation force.
( ) ( )
sinF t m a b c
= + + +
(4)
where: F- force calculated as the ratio of the imposed
pressure to the moonpool surface area [N], m - water
mass inside the moonpool at equilibrium [kg], a -
added mass [kg], b - damping coefficient [kg/s], c -
stiffness coefficient [N/m]
Due to the computational effort (each excitation
frequency requires a separate simulation), the analysis
was performed only for the 0.2 m × 0.2 m moonpool
480
(MP1), both without damping plates and with plates of
2 cm and 5 cm, corresponding to the inlet blockage
ratios A/A0 = {1.0, 0.64, 0.25}. The non-dimensional
added mass coefficient Ca33, the non-dimensional
damping coefficient κ, and the amplification factor ζ/ζ0
were also determined (Fig. 12. a-c). Additionally, the
effect of excitation amplitude on the derived
hydrodynamic coefficients was investigated for MP1
configuration without a damping plate at the excitation
period of 0.8 s (Fig. 13. a-c).
a) Non-dimensional added mass coefficient
b) Non-dimensional damping coefficient
c) Amplification factor
Figure 12. Hydrodynamic coefficients as a function of
oscillation frequency and inlet opening ratio (a - non-
dimensional added mass coefficient, b - non-dimensional
damping coefficient, c - Amplification factor)
a) Non-dimensional added mass coefficient
b) Non-dimensional damping coefficient
c) Amplification factor
Figure 13. Effect of excitation amplitude on hydrodynamic
coefficients (a - non-dimensional added mass coefficient, b -
non-dimensional damping coefficient, c - Amplification
factor)
5 CONCLUSIONS
This study demonstrated the application of RANS-
CFD methods to determine the hydrodynamic
coefficients of moonpools. Numerical simulations of a
fixed hull with a square moonpool subjected to regular
waves showed good agreement with experimental
data available in the literature, confirming the
reliability of the adopted approach. Free decay tests
carried out for moonpools of different dimensions
reproduced the natural periods predicted by theory
and highlighted the essentially nonlinear character of
the oscillations. A local fitting procedure revealed that
the non-dimensional damping coefficient decreases
with oscillation amplitude, whereas the oscillation
frequency increases and tends to stabilize after the
initial transients.
The influence of damping plates at the moonpool
entrance was also investigated. The simulations
481
showed that increasing the blockage ratio reduces the
oscillation frequency while enhancing damping, with
comparable values of the dimensionless damping
coefficient obtained for both the 0.2 m × 0.2 m and 0.4
m × 0.4 m moonpools. Forced oscillation analyses
provided a broader picture, indicating that the added
mass remains nearly constant across frequencies,
whereas the damping coefficient peaks at resonance.
Finally, the influence of excitation amplitude was
assessed, demonstrating that both added mass and
damping coefficients increase with amplitude, while
the amplification factor decreases.
The results confirm the suitability of the applied
methodology for estimating damping coefficients and
natural frequencies of moonpools. Importantly, these
findings have practical relevance for both design and
operation. For designers, knowledge of how the inlet
clearance affects the oscillation characteristics provides
a useful criterion for optimizing moonpool geometry to
mitigate unwanted resonance effects. From the
operational perspective, awareness of the dependence
between inlet clearance, natural frequencies, and
damping levels can guide decision-making regarding
allowable environmental conditions for safe use of the
moonpool, as well as inform operational limitations
when resonance is expected. Consequently, the
developed approach may contribute to both improved
safety margins and enhanced efficiency of offshore
operations involving moonpools.
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