Three-dimensional Effects on Gap-Resonances in Twin-Hull Vessels in

Time-Harmonic Vertical Oscillations

GIULIANO VERNENGO

Department of Electric, Electronic, Telecommunication Engineering and Naval Architecture

(DITEN),

Polytechnic School of the University of Genova,

Via all’Opera Pia 11 A, 16145, Genova,

ITALY

Abstract: - Three-dimensional effects induced by dimensional ratios on the gap resonances happening in twin

hull vessels oscillating in forced vertical motion have been analyzed. They can lead to relevant consequences,

such as the amplification of the inner radiated waves or the generation of standing waves in between the demi-

hulls, that can have a direct effect on the operating profile of the vessel. The response of twin hull vessels in

waves can be strongly affected by these resonant phenomena. Also, some of these behaviors can be exploited in

the framework of wave energy conversion systems. The present analysis is carried out by using an open-source,

linear, Boundary Element Method (BEM), based on the Green function approach. Mathematical backgrounds

of the added mass and damping coefficients computation for a floating body under harmonic vertical oscillation

are provided as well as details of the numerical discretization used in the BEM. A panel mesh sensitivity study

is carried out and the numerical prediction is validated by comparison against available experimental data,

another CFD solution obtained by a high-fidelity viscous solver based on the open-source libraries Open-

FOAM and approximate analytic formulations. The effect of the beam ratio and the length-to-beam ratio on the

resonant phenomena has been analyzed. This has been achieved by systematic variations of the geometric

dimensions of the hull, focusing on the trends of the hydrodynamic coefficients, the amplitude of the radiated

waves, and the location of the resonant frequencies over the analyzed range.

Key-Words: - Gap resonance, Catamaran; Piston Mode, Wave Trapping, Sloshing frequencies, Boundary

Element Method (BEM), Wave energy.

Received: March 19, 2023. Revised: February 9, 2024. Accepted: March 8, 2024. Published: May 2, 2024.

1 Introduction

Gap-induced resonances are extremely relevant to

characterize the seakeeping response of floating

structures, particularly at zero-forward speed. Such

type of phenomena happen in the presence of a

confined free water surface that can either be within

the same body or between two adjacent bodies.

The first concern with the so-called moonpools,

i.e. an opening in the hull giving access to the water

creating a confined free surface. This is a typical

design solution e.g. in drilling ships, ships for the

installation of marine risers, and diving support

vessels. The latter situation is instead typical of

independent ships operating side-by-side such as

e.g. barges, Floating Liquefied Natural Gas (LNG)

Bunkering Terminal or Bunkering Shuttles serving

LNG carriers, and connected twin hull surface

piercing bodies such as multi-hull vessels like

catamarans or Small Waterplane Area Twin Hulls

(SWATHs).

Based on the exciting frequency of the motion

and of the geometric features of the hulls different

gap resonant phenomena can occur, as represented

in Fig. 1.

The inner free surface in between the hulls

shows two waving modes called piston and sloshing

mode, respectively, happening at identifiable

frequencies of oscillation. The first appears as a

heavily amplified peak of the free surface oscillation

concerning its normal behavior occurring at a single,

specific, frequency. The sloshing mode generates a

standing wave in the free surface in the gap, whose

length depends on the oscillation frequency, that

instead can be seen at different frequencies. The last

resonance phenomenon, classified as trapping mode,

results in the cancellation of the radiated wave

outside of the gap. The correct prediction of the

frequencies at which these phenomena will appear is

of vital importance to correctly understand the

possible limits of the operating profile of the many

floating structures.

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(a)

(b)

(c)

Fig. 1: Schematic representation of the most relevant resonant phenomena happening in multi-hull vessels in

waves. A demi-hull is shown, and the symmetry plane of the vessel is indicated by a vertical dashed line. Left

to right: Piston mode (a), Wave trapping (b), and Sloshing mode (c)

The occurrence of such a complex, non-linear,

pressure-driven phenomenon can be captured in the

context of potential theories, typically at the cost of

overestimating the involved forces. This is mainly

due to the absence of viscous effects and the lack of

strongly non-linear free surface developments with

possible fragmentation and wave breaking, a

phenomenon that can be modeled by means e.g. of

viscous Reynolds Averaged Navier Stokes (RANS),

[1], [2] and by particle methods, [3], [4].

[5], developed an analytic solution to study the

excitation of the free surface between two-

dimensional sections. Potential flow-based methods

have been widely used to analyze the piston mode

resonance in rectangular barges with moonpools [6],

in the presence of rigid ice sheets [7], side-by-side

barges [8], [9], considering the effect of the finite

water depth [10] and mooring lines, [11]. Two-

dimensional studies on the piston and sloshing

resonances have been carried out by [12] and by

[13] accounting for coupled ship resonances.

Recently, the resonance effects of moonpools with

recesses have been studied by [14] and by [15]. In

the framework of potential flow theories,

suppression methods have been developed to deal

with over-prediction of resonant phenomena such as

the so-called rigid lid technique by [16], the flexible

lid developed by [17], or the rigid damping method

by [18]. These techniques have been applied to

study resonances e.g. in side-by-side ships (see

among the other [19], [20], [21], [22], [23]).

Viscous effects in gap flows have been accounted

for by coupling a Boundary Element Method (BEM)

to a vortex tracking method [24] by using a Volume

of Fluid method [25], by Spectral Wave Explicit

Navier-Stokes approach [26] or by RANS

computations [2], [27], [28], [29].

The sensitivity of gap resonance of catamaran

hull at zero speed under forced, harmonic, and

vertical oscillations concerning geometric

parameters has been studied. Such a multi-hull

configuration is of particular interest for many

applications, including e.g. those related to wave

energy conversion. The analysis is carried out in the

framework of potential theory by using the open-

source linear BEM, [30]. This solver has been used

to study the performance of both single and arrays

of Wave Energy Converters (WECs), [31], [32],

[33] and it has been shown to provide results

comparable to those obtained by other approaches

[34]. This numerical method has also been used in

the framework of multi-fidelity optimization of an

unconventional SWATH with double-canted struts,

[35].

The mathematical backgrounds and the

numerical implementation of the BEM are briefly

presented. A mesh coarsening study on the panel

mesh density has been carried out before a

preliminary validation of the hydrodynamic added

mass and damping coefficients against available

experimental data. The systematic study carried out

by varying the dimensional ratios of the catamaran

focuses on the identification of the gap resonant

phenomenon, highlighting the effect of the selected

geometric parameters on the occurrence and the

type of resonance.

2 Mathematical Backgrounds of

Frequency Domain Analysis of

Oscillating Bodies

The problem of a body oscillating in forced motion

at the free surface is described in the frequency

domain within the framework of a linear potential

theory. An ideal, incompressible, and irrotational

fluid is assumed over a fluid domain , leading to

null viscosity ,  and null vorticity

, respectively, being V the fluid

velocity. Thanks to these hypotheses, instead of

using the Navier-Stokes equations, the problem

could be solved by Euler and Bernoulli equations

defining a velocity potential function  such that:







(1)

Inner wave

amplification

Gap

Outer wave

cancellation

Gap

Standing inner

wave

Gap

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Such a velocity potential function satisfies the

Laplace equation Eq. (3a) over , plus several ad-

hoc boundary conditions. Neumann-type boundary

condition 

 on the body surface SB,

stated that the normal velocity of the fluid vanishes

at the body surface. Since a so-called radiation

problem is under investigation, namely imposed

body motions in calm water are considered, the term

 represents the velocity of the rigid body

induced by the  motion. Considering the specific

problem of a body oscillating in a forced, vertical,

sinusoidal motion η󰇛󰇜󰇛󰇜, the potential

function for the so-called radiation problem

becomes Φη󰇗󰇛󰇜, being η󰇗 the time derivative

of η. The conventional notation for ship motions is

used so that k=3 identifies the heave (vertical)

motion.

Considering the vertical harmonic oscillation

η, being  and  their amplitude and

frequency, respectively, the velocity η󰇗 and the

acceleration η󰇘 of the imposed motion are simply

defined by derivation concerning time as in Eq. (2).

η󰇗

η󰇘

(2)

So the rigid body vertical velocity induced by

the imposed heave motion, projected onto the

normal to the hull surface n, is .

Being , with  the time independent

potential function, the body boundary condition then

holds as in Eq. (3b). A similar boundary condition

representing null normal velocity at the horizontal

bottom of the domain z=zBottom, 

 , is not

necessarily for the proposed study since deep water

case is investigated, .

In addition, dynamic, 

 , and

kinematic, 

 

 , free surface boundary

conditions, linearized at the undisturbed free surface

level z=0, are imposed. In particular they are

combined to exclude the term , i.e. the radiated

wave elevation, which is part of the solution of the

problem, from the formulation itself to enable a

linear solution of the system of equations, reaching

the boundary condition at the free surface stated in

Eq. (3c). A far-field, Sommerfeld-type, radiation

condition ensuring that the potential nullify far away

from the body, i.e. for an ideal distance , Eq.

(3d) is also needed.











 







  



 

(3)

The unknown potential function is found by

solving a linear system of equations built on the

above-mentioned boundary conditions. The time-

independent  component of hydrodynamic

radiation force induced by the  motion is then

computed by integrating the dynamic pressure,

excluding terms higher than first order from

Bernoulli equation. For the forced heave motion, it

results:







(4)

Considering the model for rigid, floating body

motions, hence introducing the three-dimensional

added mass and damping coefficients  and ,

respectively, i.e. the components of the

hydrodynamic forces in phase with the acceleration

and with the velocity, and accounting for Eq. (2),

the time-independent, heave hydrodynamic force

can be written as:

󰇛󰇜

(5)

The occurrence of the gap-resonances related

phenomena can be identified by the analysis of the

trends of these coefficients over a suitable range of

oscillating frequencies.

2.1 Numerical Solution by Green Function

based BEM

The hydrodynamic solution is found by using a

three-dimensional BEM based on a Green function

approach, suitable for offshore structures and

floating bodies of generic shapes. According to [36],

the first type Green function is written in the

following form:

󰇛󰇜

󰇛󰇜

󰇛󰇜

(6)

Being r the horizontal distance between the

source point 󰇛󰇜 and the field point

󰇛󰇜, -Z the vertical distance between the

image source point 󰆒󰇛󰇜, mirrored

concerning the mean free surface, and the field point

M, R the distance between the source and field

points and R1 the distance between the image source

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point and the field point.  is the deep water Green

function and  represents the so-called free surface

term. 󰇛󰇜 is the  order Bessel function of the

first type and, from the deep-water dispersion

relation, 

. Nemoh is based on a particular

choice of the free surface term, [30], [37], which

involves a finite integral of the complex exponential

integral in the variable angle , defined as 



, within the range 󰇟󰇠. The hull is

represented by using N panels, either quadrilateral

or triangular ones, arranged in an unstructured

mesh. Considering that an unknown source 󰇛󰇜 is

placed on the  panel, the discrete form of the

induced velocity potential at a field point M can be

resumed as:

󰇛󰇜

󰇛󰇜󰇛󰇜



 

(7)

By using Eq. (7) and the boundary conditions of

Eq. (3), a linear system of equations can be written

in the unknown source strengths on each panel of

the hull. Once the strength of the source distribution

is found, all the other flow characteristics can be

computed.

3 Benchmark Hull Shape

A round bilge catamaran hull has been selected as a

reference shape for the present study, [38]. It is a

cylindrical catamaran with a constant semi-circular

cross-section as displayed in Fig. 2, where the

characteristic dimensions of the hull, i.e. the demi-

hulls separation 2b, the demi-hull beam 2a, the hull

length L and the hull beam B, are shown too. The

lateral positions of the buoys used for measuring the

wave height both in between the demi-hulls and on

the outer free surface are shown in the cross-section

in Fig. 3. All the buoys are located at the

longitudinal center of the hull. For such a hull,

experimental values of the added mass and the

damping coefficients are available.

Fig. 2: Catamaran hull used for the systematic

analysis of the resonant phenomena. Hull length, L,

hull beam, B, demi-hull separation, 2b, and demi-

hull beam, 2a, are highlighted

Fig. 3: Position of the probes used to measure the

wave height

4 Results of the Analysis

A sensitivity analysis on the panel mesh density has

been carried out and results are shown in the

following Section 4.1. The validation of the

numerical method by comparison against

experimental data is presented in Section 4.2.

Finally, results of the systematic variations of both

the demi-hulls separation ratio  and the length-to-

beam ratio  of the catamaran hull are presented in

Section 4.3.

4.1 Panel Mesh Sensitivity

Effect of panel mesh density on the hydrodynamic

coefficients has been investigated. The reference

hull ratios are equal to L/a=10 and b/a=1.5. The

three-panel meshes shown in Fig. 4 have been

analyzed, namely a coarse mesh made of 400

panels, a medium-mesh made of 770, and a fine

mesh made of 1200 elements. A structured panel

mesh has been used for the whole hull surface but

for the aft and forward closures where unstructured

triangular panels have been used too. Aspect ratio

equal to AR=1 has been used for the coarser mesh

configuration while both the medium and fine

meshes show rectangular panels with AR=0.5. The

predicted heave-added mass and damping of the

three meshes are compared in Fig. 5. They have

been computed over the same range of oscillating

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frequencies. Both coefficients are presented in non-

dimensional form as  and . Results

from the medium and the fine meshes are

equivalent. The coarser panel mesh shows slightly

over-predicted peaks of both the added mass and the

damping coefficients. Despite this difference, all the

curves are in very good agreement, proving the

robustness of the BEM prediction. In addition, since

the computational time required by the three meshes

almost doubles at each mesh density step, the

medium-size mesh has been chosen for all the

further computations.

Fig. 4: Three mesh configurations used for the

sensitivity analysis on the catamaran hull with

L/a=5 and b/a=1.5. Left to right: coarse, medium,

and fine panel mesh

Fig. 5: Heave 3D added mass and damping

coefficient computed by the BEM on the three mesh

configurations used for the sensitivity analysis on

the catamaran hull with L/a = 10 and b/a = 1.5

4.2 Validation of the BEM Prediction by

Comparison Against Experimental

Measurements

The experimental tests were carried out to reproduce

2D flow conditions by using two endplates at the

catamaran aft and forward ends, [38]. The

catamaran hull was located in the middle of the tank

length, with the generating axis of the cylinders

perpendicular to the sides of the tank so that the

radiated waves could freely move alongside the tank

itself.

(a) b/a=1.5

(b) b/a=1.5

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(d) b/a=2.0

(e) b/a=4.0

(f) b/a=4.0

Fig. 6: Non-dimensional 3D heave added mass and

damping coefficients for the catamaran hull having

B/L = 18. Top to bottom: b/a = 1.5; 2.0; 4.0. Left

column: added mass coefficients. Right column:

damping coefficients. BEM (filled black squares),

RANSE (dashed line), and EFD (empty green

squares) results are shown

Fig. 6 compares the experimental and the

numerical results for the hull with the highest

length-to-beam ratio, namely L/a=18. This is the

higher aspect ratio configuration then reducing as

much as possible the three-dimensional effects.

Three beam-to-length ratios have been analyzed,

corresponding to b/a=1.5; 2.0; and 4.0, respectively.

Results obtained by using a high-fidelity, viscous,

2D RANS solver based on openFOAM libraries,

[39] are shown too. Such a RANS approach has

been validated in several seakeeping-related

problems involving both oscillating SWATH

sections, [40] and ship motion prediction, [41].

The experimental measurements and the

numerical predictions are in good agreement over

the entire range of analyzed frequencies for both the

added mass and damping coefficients, respectively.

The location of the peaks of the hydrodynamic

coefficients is well predicted. However, either the

experimental measurements or the RANS

predictions show lower values of these peaks of the

responses due to the lack of any viscous correction

in the BEM results.

Considering the inversion of the trends added

mass coefficients crossing the horizontal axis, the

piston mode gap resonance is experienced by all

three designs. This phenomenon happens at

decreasing frequencies as the separation increases,

ranging from 

 for b/a=1.5 to 

 for

b/a=4.0. At the piston mode resonance, the damping

coefficient is maximum meaning that the energy

dissipated by the radiated wave is the highest. Close

to the piston mode gap resonance, at a slightly

higher frequency the hull experiences the so-called

wave trapping. This phenomenon appears as a

cancellation of the outer radiated wave and can then

be identified by the nullification of the damping

coefficient. Consistently, the lower trapping

frequency is found at the higher hull separation.

At the higher separation ratio, i.e. b/a=4.0, two

sloshing frequencies are discovered by analyzing the

trend of the added mass coefficient. Numerical

results obtained by using the BEM are consistent

with those obtained from Eq. (8) defining a standing

wave derived from the dispersion relation for

infinite depth.





(8)

According to this simplified formulation, the

sloshing frequencies should be  and 

, for n=1 and n=2, respectively.

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The other sudden variation of the response

found by the BEM at 

 is referred to an

irregular frequency. This is a numerical issue of the

BEM that of course is not seen neither in the

experiments nor by the RANSE. This is a well-

known drawback of the Green function approach

first highlighted by [42], [43]. It has then been

described e.g. by [44] in analogy to the sloshing that

happens on the fictitious free surface contained

inside the hull. For the proposed study, it is

unnecessary to use irregular frequency suppression

methods, but it is enough to identify this numerical

behavior of the BEM.

[45], provides an approximate formulation for

irregular frequencies of a simple rectangular

pontoon only based on its length and beam, L and B,

respectively, and on the depth T. By using Eq. (9)

on the single demi-hull of the catamaran the first

irregular frequency, corresponding to the variation

of the trend of the coefficients, is found.







󰇧



󰇨

(9)

Specific features of the radiated wave patterns

are triggered at each gap resonance identified by

analyzing the trends of the hydrodynamic

coefficients. However, there have been no available

data to verify the prediction of the radiated waves.

Then, an approximate theoretical formulation has

been used to carry out such a validation on the

radiated waves. Such a theoretical approach is

rigorously valid in the framework of a linear theory,

neglecting the second-order interactions between the

hull surface and the waves, meaning that the hull

would have vertical sides, i.e. a rectangular section,

and that a mean wet hull surface can be considered.

Assuming that (a) the potential-flow damping is

related to the outgoing waves, (b) the energy is

carried out of the hull by the outgoing waves, (c) the

BEM can capture the generation of such waves, the

work of the damping force can be equated to the

power associated to the radiated waves. Considering

a hull heaving at a given frequency , the work

exerted by the damping forces over an oscillation

period 

 can be written as in Eq. (10):

󰇗







󰇛󰇜







(10)

Being  the amplitude of the heaving motion

󰇛󰇜.

Fig. 7: Comparison of the damping coefficient B33 computed by using the BEM (black curve) and the analytic

formulation (red curve) based on the radiated waves, shown on the top side of each sub-figure

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The power of the radiated outgoing waves over

an oscillation period can be written based on the

mean energy flux 

, being 

 the group

celerity, as in the following Eq. (11):

















(11)

where

 and  are the amplitude of the radiated

waves at the inner free surface and at the outer free

surface. These waves are here assumed to be

cylindrical. The multiplying factor 2 has been left

out of the square brackets to highlight that this

energetic contribution needs to be doubled due to

the twin-hull. By equating Eq. (10) to Eq. (11) and

considering , an approximate, analytic

formulation for the (potential) heave damping

coefficient can be found as:











(12)

Fig. 7 displays the amplification of the inner

wave for the design with b/a=1.5, measured at the

four buoys (Fig. 3), for an harmonic oscillation

corresponding to the piston mode resonant

frequency. The wave height reaches a maximum

value of  at the inner buoy A, that is much

higher than the average value of  or than

the amplitude reached at the outer free surface

where . The amplitude of the radiated

wave pattern is higher at the inner free surface close

to the center of the hulls while decaying towards the

aft and forward ends of the catamaran due to the

three-dimensional effects, as shown in Fig. 7. The

inner free surface elevation is far higher than that of

the outer free surface at this first gap resonance

frequency.

Eq. (13) has been proposed to find the piston

mode frequencies of a moonpool [6]:



󰇡󰇡

󰇢󰇢

(13)

Being b the beam of the moonpool,

corresponding in this case to the distance between

the two demi-hulls, h the depth and H the beam of

the whole body, corresponding to the beam of the

whole catamaran. As further verification, Table 1

reports the comparison between the piston mode

frequencies computed by using Eq. (13) and those

predicted by the BEM. An acceptable agreement is

obtained. A general over-prediction of the piston

mode frequencies is experienced by the BEM that

decreases as the hull separations increase. The

maximum percentage difference  is

found for the lower transverse hull separation.

Table 1. Non-dimensional piston mode frequencies

ω2a/g for the hull having L/B = 18.

b/a

Eq. (13)

BEM



1.5

0.709

0.81

14.2%

2.0

0.614

0.68

10.2%

4.0

0.455

0.46

1.10%

Fig. 7 and Fig. 8 display an example of wave

elevation and wave pattern, respectively, at the

wave trapping frequency for the hull with b/a=2.0.

At such oscillation frequency the minimum

amplitude of the external radiated waves, that are

almost completely canceled, is reached. This is

consistent with the trend of the damping coefficient

that is very close to zero at . In fact, B33

does not depend on the viscous effects since it is a

force coefficient derived in the framework of a

potential theory. It is depending on the energy

dissipated by the generation of the radiated waves.

If no waves are generated, no (potential-flow)

damping is created, and vice versa. The cancellation

is related to the outer wave field that is the most

responsible for the energy dissipation, then

contributing to a great extent to the damping

coefficient. The small values of the B33 coefficient

are related to both the inner wave, which is anyway

present, and again to the three-dimensional

characteristic of the wave pattern.

Effects of oscillations at the sloshing

frequencies are highlighted in Fig. 7 for the hull

configuration with b/a=4.0. This gap resonance

occurs at relatively higher frequencies compared to

the other two previously described phenomena. This

means that it generally happens for faster

oscillations. Local amplification of the inner wave is

shown in Fig. 8 at the two discovered sloshing

frequencies. The characteristics of the wave pattern

in between the two hulls resemble those of standing

waves whose frequencies are proportional to the

oscillation frequency of the body. The wave

elevation of the free surface is shown in Fig. 8 for

the first and the second sloshing frequencies,

respectively. At the lower sloshing frequency, the

inner wave pattern shows three wave crests in the

transverse direction while five crests are identified

at the higher sloshing frequency. Consistently with

, this means that longer

waves are generated in the first case while shorter

waves are created in the second one.

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Fig. 8: Top view of the real part of the three-dimensional radiated wave patterns at different separation ratios.

Top left, clockwise: piston mode, trapping mode, 1st sloshing mode, 2nd sloshing mode

4.3 Sensitivity Analysis of the Effects of the

Hull Geometric Ratios

A systematic variation of two relevant geometric

ratios of the catamaran have been carried out to

investigate their effect on the occurrence of the gap

resonances. A separation ratio and a slenderness

ratio, b/a and L/a, respectively, have been

considered. The first is defined as the ratio between

the demi-hull separation distance and the beam of

the demi-hull. The latter is defined as the ratio

between the hull length and the beam of the demi-

hull. Nine combinations have been accounted for,

by using the following values:

󰇟󰇠

󰇟󰇠

(14)

Each hull design is then identified by a couple

of values of (b/a; L/B). The added mass and

damping coefficients for each design undergoing

forced heave motion have been computed by the

BEM over a range of oscillation frequencies

corresponding to 

󰇟󰇠. To focus on

the trends of these responses, the results have been

interpolated over a continuous domain. In particular,

a Gaussian Process regression-based fitting method

(see for instance [45], [46]) has been used to create

the response surfaces corresponding to A33 and B33

at a given L/a ratio. This class of methods has been

recently used e.g. in the context of single and multi-

fidelity optimization [35], [47], [48], [49] and big

data regression, [50] and it is here applied to reveal

the trends of the hydrodynamic coefficients to

concerning changes of the geometric ratios.

Fig. 9 highlights a trend on both the added

masses and damping that is common to all three

ratios L/a. The first negative peak of , close

to the piston mode gap resonance, is decreased and

shifted to lower frequencies as the separation ratio

increases. The maximum value of such peaks of the

added mass increases as the length-to-beam ratio

increases. This has a twofold meaning. On a side,

the effect of the forces that are synchronized with

the acceleration of the body becomes less relevant

increasing the separation between the demi-hulls,

and as expected, such an effect is strongly decreased

by the three-dimensional effects, i.e. for the

maximum L/a.

Considering again the trends of the added mass

coefficient, two sloshing mode gap resonances are

identified in the upper half of the range of the

separation ratio. The maximum variation of 

due to this gap resonance corresponds to the second

sloshing mode frequency at the intermediate length-

to-beam ratio L/a=14.

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All the peaks of the responses, except that of the

second sloshing mode frequency, are increased at

the highest L/a, especially those of . This

means that the gap resonances happen at the same

oscillation frequency, regardless the value of the

slenderness ratio but with a different amplitude. The

b/a ratio instead is more responsible for the type of

gap resonance, since the piston mode resonance is

revealed in the lower half of its range while the

sloshing mode resonance appears for higher values.

Consistently with the results obtained by other

authors [6], the piston mode resonance frequency

predicted by the BEM is inversely proportional to

the distance between the demi-hulls, as displayed in

Fig. 10. It slightly decreases as the length-to-beam

ratio increases. The inner wave amplitude is heavily

amplified, up to four times the amplitude of the

oscillation for [b/a=1.5; L/a=18]. So, as expected,

the piston mode is greatly affected by the distance

between the demi-hulls. As regards the wave

trapping phenomenon, there are no simplified

formulations based e.g. on dimensional ratios of the

body.

The wave trapping frequency decreases as the

gap of the hull increases being 





󰇟󰇠 for b/a=[1.5; 2.0; 4.0],

respectively. Since this phenomenon strongly

depends on non-linear effects, this trend is possibly

due to the different interactions between the demi-

hulls. Moreover, it seems that this cancellation of

the external radiated wave could occur at several

frequencies in the case of the larger separation

b/a=4.0 for which  tends to zero at least

three times, for 

󰇟󰇠. This last

result however should be further verified by using a

fully viscous, non-linear, method.

Fig. 9: Non-dimensional 3D heave added mass and damping coefficients in the continuous range b/a = [1.5;

4.0]. Top to bottom: L/a = 10; 14; 18. Left column: non-dimensional added mass coefficient. Right column:

non-dimensional damping coefficient

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Fig. 10: Non-dimensional piston mode frequency (solid lines, left y axis) and non-dimensional inner wave

amplitude at Probe A (dashed lines, right y axis) as function of b/a and L/a ratios

5 Conclusion

A numerical analysis of the gap resonances

happening in twin hull vessels undergoing forced,

harmonic, vertical motion at zero forward speed has

been proposed. An open-source linear Boundary

Element Method has been applied to this aim. The

mathematical framework and the numerical

approach used to predict these resonant phenomena

have been described. A preliminary sensitivity

analysis of the numerical prediction concerning the

panel mesh density has been carried out finding the

best trade-off between accuracy and computational

time. The numerical method has then been validated

by comparison against available experimental

towing tank results. A satisfactory agreement has

been found over a wide range of oscillation

frequencies between the BEM prediction, the

experimental results, and previous results from the

high-fidelity viscous RANSE solver.

The influence of the demi-hull separation ratio

and the hull length-to-beam ratio have been

investigated by systematic numerical analysis. Nine

hull configurations have been studied in terms of

added mass and damping coefficients focusing on

the occurrence of significant gap-resonances. Both

the piston mode and the sloshing resonance

frequencies predicted by the BEM are like those

provided by approximate formulations available in

the literature, proving that the method can recognize

such a phenomenon. Wave trapping frequencies

have been identified by the trends of the

hydrodynamic coefficients. Such a resonant

phenomenon happens at lower frequencies as the

gap between the demi-hulls increases. The obtained

results highlighted a general major effect of the

separation ratio of the hull concerning the

slenderness ratio. This, in conclusion, means that the

three-dimensional effects at the afterward and

forward ends of the hull are less relevant than the

transverse size of the gap on the occurrence of these

resonant phenomena.

That information might be of particular

relevance in at least two cases: on one side, if

seakeeping is concerned, to avoid specific wave

frequencies to minimize e.g. unwanted wave

behaviors in between the demi-hulls and, on the

opposite side, if wave energy devices want to be

created based on twin hull configuration to capture

the maximum possible wave elevation of the piston

mode.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

Giuliano Vernengo carried out the simulation,

analysed the results, wrote and reviewed the

manuscript.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The author has no conflicts of interest to declare.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

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