Ci a ion: Es eban, G.A.; Ezku a, X.;
Bidagu en, I.; Albaina, I.; Izquie do, U.
Compu a ional S udy o O e opping
Phenomenon o e Cylind ical
S uc u es Including Mi iga ion
S uc u es. J. Ma . Sci. Eng. 2024,12,
1441. h ps://doi.o g/
10.3390/jmse12081441
Academic Edi o : Claudio Lugni
Recei ed: 11 July 2024
Re ised: 1 Augus 2024
Accep ed: 15 Augus 2024
Published: 20 Augus 2024
Copy igh : © 2024 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
Jou nal o
Ma ine Science
and Enginee ing
A icle
Compu a ional S udy o O e opping Phenomenon o e
Cylind ical S uc u es Including Mi iga ion S uc u es
Gus a o A. Es eban * , Xabie Ezku a, Iñigo Bidagu en , Iñigo Albaina and U ko Izquie do
Ene gy Enginee ing Depa men , Uni e sidad del País Vasco/Euskal He iko Unibe si a ea (UPV/EHU), Pza.
Ingenie o To es Que edo 1, E-48013 Bilbao, Spain; [email p o ec ed] (X.E.); i.bidagu [email p o ec ed] (I.B.);
[email p o ec ed] (I.A.); [email p o ec ed] (U.I.)
*Co espondence: [email p o ec ed]
Abs ac : Wa e o e opping occu ing in o sho e wind enewable ene gy s uc u es such as ension
leg pla o ms (TLPs) o semi-subme sible pla o ms is a phenomenon ha is wo h s udying and
p e en ing in o de o ex end he emaining use ul li e o he co esponding acili ies. The beha iou o
his phenomenon has been ex ensi ely epo ed o linea coas al de ences like seawalls. Howe e , no
e e enced s udy has ea ed he case o cylind ical s uc u es ypical o hese applica ions o a simila
ex en . The aim o he p esen s udy is o de ine an empi ical exp ession ha po ays he ela i e
o e opping a e o e a e ical cylinde including a a ie y o bull-nose ype mi iga ion s uc u es o
educe he o e opping a e in he same ashion as o he linea s uc u es cha ac e is ic o sho eline
de ences. Hyd odynamic in e ac ion was s udied by means o an expe imen ally alida ed nume ical
model applied o a non-impulsi e egula wa e egime and he esul s we e compa ed wi h he case o
a plain cylinde o e alua e he expec ed imp o emen in he o e opping pe o mance. Fou di e en
ypes o pa ape s we e added o he c es o he base cylinde , wi h di e en pa ape heigh and
ho izon al ex ension, o see he in luence o he geome y on he mi iga ion e iciency. Compu a ional
esul s con i med he e ec i i y o he p oposed solu ion in he o e opping educ ion, hough he
singula i y o each pa ape geome y did no lead o an ou s anding di e ence be ween he analysed
op ions. Consequen ly, he esul ing o e opping dec ease in all he p oposed geome ies could
be modelled by a unique speci ic Weibull- ype unc ion o he ela i e eeboa d, which go e ned
he phenomenon, showing a ne educ ion in compa ison wi h he cylinde wi hou he geome ic
modi ica ions. In addi ion, he ela ionship be ween he educed ela i e o e opping a e and he
mean low hickness o e he e ical cylinde c es was s udied as an al e na i e me hodology o
assess he po en ial damage caused by o e opping in eal s uc u es wi hou complex olume ic
measu emen s. The collec ion o compu a ional esul s was i ed o a use ul unc ion, allowing o
he de ini ion o he o e opping discha ge once he mean low hickness was known.
Keywo ds: wa e o e opping; o e opping discha ge; cylind ical s uc u e; nume ical wa e lume;
bull-nose; pa ape ; wa e e u n wall
1. In oduc ion
O e opping is a nonlinea phenomenon ha occu s when a wa e o a ac ion o a
wa e su passes a ixed o loa ing s uc u e and gene a es a laye o wa e o e i s c es
ha can cause looding o su ace damage. I is a well-es ablished phenomenon, and i is
conside ed in he design phase o ixed and loa ing s uc u es in he ma ine ene gy sec o
because i can damage he s uc u es o acili ies moun ed on op, o he han modi ying he
expec ed oscilla ion esponse o he inciden wa e ain a ec ing he ope a ion o he wind
u bine [
1
–
3
]. The s anda d measu es aken agains o e opping a e de eloped o de ence
s uc u es like b eakwa e s, dikes o seawalls, which a e implemen ed in ma ine s uc u es
exposed o wa es such as sho eline p omenades, po s o ha bou s.
J. Ma . Sci. Eng. 2024,12, 1441. h ps://doi.o g/10.3390/jmse12081441 h ps://www.mdpi.com/jou nal/jmse
J. Ma . Sci. Eng. 2024,12, 1441 2 o 22
In o de o de elop e ec i e designs o p o ec i e s uc u es, he e has been a con inu-
ous e o o s udy, unde s and and cha ac e ise o e opping. The s udy o o e opping
was ca ied ou independen ly by di e en esea ch g oups du ing i s i s s ages, bu he
de elopmen o he CLASH p ojec in oduced a common da abase and a s anda dised
me hodology o s udy o e opping [
4
]. A e he comple ion o he p ojec , he Eu O op
manual [
5
] was published in 2007, ga he ing all he in o ma ion om di e en da abases
in a single collec ion and p oposing he co esponding ep esen a i e uni ied analy ical
p edic i e models. The Eu O op manual became he main e e ence o o e opping, and a
second edi ion was published in 2018 [6] wi h new indings and imp o emen s.
The me hodology employed by mos g oups ha s udy o e opping is o measu e
he o e opping discha ge o e a scaled physical model es ed in an expe imen al wa e
lume o a nume ical wa e lume (NMF) by using compu a ional luid dynamics (CFD)
calcula ions. The elemen s equi ed o bo h he expe imen al and he nume ical p ocedu es
a e a wa emake sys em, a wa e heigh measu ing sys em and a p ocedu e o measu e he
o e opped wa e olume.
On he one hand, he expe imen al p ocedu e implies wa e gene a ion by a pulsa ing
paddle wi h pis on- ype linea mo ion and/o a ecip oca ing oscilla ion o a lap a ound
a hinge. The gene a ed wa e a one end o he lume de elops and p opaga es owa ds
he speci ic s uc u e unde s udy and he opposi e end o he lume whe e wa e e lec ion
may occu [
7
]. To minimise he e ec o e lec ion, passi e and ac i e abso p ion sys ems
a e no mally used. Some imes he e ec i e ime in e als o he es s a e limi ed o he
pe iods whe e a e lec ed wa e a els backwa ds o he gene a ion sys em and back again
o he es ing a ea a e e- e lec ion on he wa emake [
8
]. The e a e di e en sys ems o
measu e he gene a ed wa es by moni o ing he ee su ace posi ion o in e he inciden
wa e heigh , pe iod and wa eleng h. Wi h ega d o o e opping cha ac e isa ion, a
calib a ed ank is placed a he end o he main s uc u e o measu e and accumula e he
o e opped liquid olume o mass. As an example, ep esen a i e ecen expe imen al
es s in lumes measu e he o e opping discha ge o e speci ic s uc u es occu ing in he
case o sunami- ypes wa es gene a ed in he sea o onsho e om glacie cal ing [9–11].
On he o he hand, in he case o he compu a ional app oach, he simula ions a e pe -
o med by nume ically sol ing he conse a ion laws o luid mo ion ha can be exp essed
in ei he he Eule ian o he Lag angian app oach. The Eule ian app oach disc e ises he
wo king olume in o a ine mesh o med by cells con aining nodes whe e he solu ions
o he equa ions a e nume ically calcula ed ollowing he ini e olume me hod. On he
o he hand, he Lag angian app oach manages luid pa icles o blobs, o which luid
equa ions a e applied, wi h models such as smoo hed pa icle hyd odynamics (SPH) [
12
].
Di e en ep esen a i e la e examples o he applica ion o he Eule ian app oach based
on he olume o luid (VOF) me hod [
13
] (based on Reynolds-a e aged Na ie –S okes
(RANS)) ha e succeeded in desc ibing and e alua ing he o e opping discha ge o e
s uc u es o s udy he in luence o he associa ed hyd aulic load and e osion, geome y
design in luence, damaged/upg aded s a e o b eakwa e s and co esponding modi ica ion
o he o e opped laye hickness and i s eloci y [
14
–
18
]. Mo eo e , he SPH me hod has
been p o en o be a success ul al e na i e o s udying o e opping on di e en seawalls,
decks and dams [19,20].
The di e en s uc u e op ions and he co esponding pa ame e s o mi iga e he
o e opping phenomenon a e epo ed in [
5
]. In sea ch o o e opping p edic ion by ep e-
sen a i e models, an de Mee e al. [
21
] p oposed o use a euni ied Weibull- ype unc ion
alid o he whole ange om ze o eeboa d condi ions, di e en wa e b eaking/non-
b eaking condi ions, di e se na u e o he o esho e, composi e s uc u es and a ia ion
o he slope. The las upda es on he issue we e inco po a ed in he second edi ion o he
Eu O op manual [
6
]. Special men ion should be made o Zanu igh and Fo men in o hei
enowned s udy abou ex eme and admissible o e opping discha ges using a special neu-
al ne wo k [
22
] and he impac o c own walls wi h pa ape s [
23
,
24
]. Some o he esea ch
wo ks ega ding o e opping add ess he o ces ac ing on he de ence s uc u es due o
J. Ma . Sci. Eng. 2024,12, 1441 3 o 22
wa e impac s, which a e s udied o ensu e he physical in eg i y o he s uc u e; esea ch
by Van Doo slae and De Rouck [
25
,
26
] and hose by Ma inelli and
Cas ellino e al. [27–29]
a e wo h highligh ing in ela ion o he e alua ion o wa e e u n walls.
Wi h ega d o he di e en solu ions applied o educe o e opping discha ge unde
ex eme sea s a es, i is common p ac ice o modi y he o esho e by in oducing b eak-
wa e s, slopes, p omenades o wa e un-ups [
18
,
30
–
32
]. These solu ions a e speci ically
designed o p e en inland looding, and a e ou o he applicabili y o he e ical s uc-
u es ins alled in o sho e loa ing sys ems. Fo ha eason, his s udy ocuses on common
geome ic modi ica ions o he main s uc u e acing he wa e on , like cham e s, ecu ed
walls o pa ape s [
33
–
39
]. Mo e speci ically, among all hese ypes o geome y, pa ape s
ha e been chosen in his s udy because hey ep esen he main op ion applied o e ical
walls and hey a e epo ed o wo k p ope ly. In his case, he educ ion o o e opping is
based on he low de lec ion p o oked by he pa ape ha sends back he up- ushing wa e
olume in he di ec ion o he wa e ha is coming in.
The p ocedu e ollowed in his campaign is based on nume ical simula ions o he
o e opping discha ge o e a ixed e ical cylinde wi h pa ape ings moun ed on op,
o compa ison wi h esul s ob ained in p e ious esea ch [
40
], whe e he o e opping
discha ge was de ined o a plain cylinde in he same wa e condi ions and base geome y
(i.e., diame e o ee-boa d ange). The compu a ional con igu a ion in he CFD model
used he e ollows he same RANS-VOF a angemen as he one al eady alida ed wi h
he co esponding expe imen al campaign [
40
]. This way, he expec ed o e opping a e
educ ion is e alua ed o analyse whe he he p oposed s uc u al modi ica ions a e a iable
al e na i e o p e en o e dimensioning he main cylind ical s uc u es in eal acili ies in
sea ch o o e opping pe o mance o he inal s uc u e con igu a ion.
An addi ional seconda y aspec ea ed he e is he s udy o he ela ion be ween he
ela i e o e opping discha ge and nondimensionalised mean low hickness o he wa e
shee lowing o e he s uc u e. This app oach would o e a p ac ical me hodology o
assess o e opping discha ges by a simple measu emen o wa e ele a ion on he c es
o he pla o m ins ead o moun ing complex wa e collec ion s uc u es o measu e he
o e opped olume. The ins an aneous heigh o he low hickness could be measu ed
wi h a wa e p obe in a simple and cos -e ec i e way.
This s udy is he basic s ep and undamen al basis be o e ackling he p oblem o
o e opping subjec ed o loa ing s uc u es ypical o o sho e ene gy echnologies wi h
hese cylinde -based geome ic cha ac e is ics like TLP o semi-subme sible pla o ms.
2. Theo e ical Backg ound
The o al o e opping discha ge o e a s uc u e may con ain h ee di e en wa e
o e opping con ibu ions: g een wa e o e opping, whi e wa e o e opping and o e -
opping due o sp ay. G een wa e o e opping is cha ac e ised by he non-impulsi e
egime whe e he wa e heigh is enough o he wa e o o m a con inuous laye o e
he s uc u e. Con a ily, whi e wa e o e opping occu s in he impulsi e egime when
wa es b eak agains he a ea o he s uc u e acing he incoming wa e on , p oducing
la ge masses o wa e mixed wi h en ained ai ha mo e o e he s uc u e because o
he momen um and ine ia o he wa e mo ing mass. Sp ay o e opping accoun s o he
in e ac ion be ween wa e c es s and wind, i s con ibu ion being conside ed negligible in
no mal enginee ing con ex s.
Wa e o e opping is a ec ed by a mul i ude o p ope ies ha con ain bo h he
cha ac e is ics o he sea s a e and he s uc u e’s geome y and o esho e. The p ima y
ac o s conside ed in he analysis o sea s a e e ec s on o e opping include wa e pe iod
(T), wa e heigh (H), wa e dep h (h) and wa e s eepness (s) a he base o he s uc u e.
Addi ionally, geome ic a ibu es o he s uc u e a e conside ed, such as he be m, he
o esho e slope and, specially, he c es eeboa d (R
c
), which is associa ed wi h he heigh
o he s uc u e (H
c
). The c es eeboa d is known o be a c i ical pa ame e in he s udy o
wa e o e opping. In he case o he use o pa ape s as o e opping mi iga ion s uc u es,
J. Ma . Sci. Eng. 2024,12, 1441 4 o 22
addi ional key pa ame e s in luencing he phenomenon a e he pa ape heigh (h
) and he
ho izon al ex ension o he pa ape (B ) (Figu e 1).
J. Ma . Sci. Eng. 2024, 12, x FOR PEER REVIEW 4 o 22
di ionally, geome ic a ibu es o he s uc u e a e conside ed, such as he be m, he o e-
sho e slope and, specially, he c es eeboa d (Rc), which is associa ed wi h he heigh o
he s uc u e (Hc). The c es eeboa d is known o be a c i ical pa ame e in he s udy o
wa e o e opping. In he case o he use o pa ape s as o e opping mi iga ion s uc u es,
addi ional key pa ame e s in luencing he phenomenon a e he pa ape heigh (h ) and
he ho izon al ex ension o he pa ape (B ) (Figu e 1).
Figu e 1. Main pa ame e s ha a ec he o e opping phenomenon in a e ical wall including a
pa ape . (O iginal elabo a ion based on Figu e 7.21 o Eu O op manual [6].)
Desc ibing wa e o e opping is challenging because o i s nonlinea na u e and,
hus, i is no well ep esen ed by simple analy ical models. The e alua ion o wa e o e -
opping o a speci ic s uc u e ypically in ol es he u ilisa ion o empi ical exp essions
de i ed om CFD simula ions o expe imen s wi h scaled physical models. Bo h me hods
a e pa icula ly aluable in assessing wa e o e opping, as his phenomenon is in luenced
by a a ie y o ac o s wi h in e connec ed impac s ha a e di icul o p edic . The empi -
ical app oach con ains a simple ep esen a ion o he unde lying physics o he phenom-
enon, o en p esen ed as an equa ion ha co ela es dimensionless pa ame e s speci ic o
he p ocess (e.g., o e opping discha ge, wa e a ibu es and s uc u al geome ic ac o s).
The main objec i e pa ame e in his analysis is he mean o e opping low discha ge
q (m2/s), which accoun s o he a e age liquid olume pe uni ime and ans e sal ho -
izon al leng h o s uc u e passing o e , which is usually p esen ed in i s nondimensional
o m (by di iding by √𝑔𝐻𝑚03), known as he ela i e o e opping a e. The signi ican
wa e heigh (Hm0) equals ou imes he oo mean squa e displacemen o he ee su ace,
which is equi alen o he a e age heigh o he 1/3 highes wa es o a se ies o i egula
wa es, and g is he accele a ion o g a i y.
The dependence o he ela i e o e opping a e is no mally de ined as a unc ion o
he c es eeboa d Rc, being he ele a ion di e ence be ween he s uc u e c es and he
s ill wa e le el (SWL), which is gi en, again, in a nondimensionalised o m (by di iding
by he signi ican wa e heigh Hm0) known as he ela i e eeboa d.
To assess he wa e o e opping accu a ely, i is necessa y o iden i y he wa e–s uc-
u e in e ac ion egime because di e en p ocesses and ou comes may occu in di e en
scena ios. Speci ically, when dealing wi h e ical walls wi h s uc u al modi ica ions,
o e opping can occu unde wo dis inc egimes: non-impulsi e and impulsi e condi-
ions. Non-impulsi e condi ions happen when wa e heigh s a e ela i ely modes com-
pa ed wi h he dep h a he base o he s uc u e, and he wa e s eepness emains low,
s aying a om he poin o b eaking. In such a si ua ion, o e opping wa es gene a e
smoo hly e ol ing loads, and wa e lows gen ly o e he s uc u e. In con as , impulsi e
condi ions happen a e ical o s eep walls when wa e heigh s signi ican ly exceed local
Figu e 1. Main pa ame e s ha a ec he o e opping phenomenon in a e ical wall including a
pa ape . (O iginal elabo a ion based on Figu e 7.21 o Eu O op manual [6]).
Desc ibing wa e o e opping is challenging because o i s nonlinea na u e and, hus,
i is no well ep esen ed by simple analy ical models. The e alua ion o wa e o e opping
o a speci ic s uc u e ypically in ol es he u ilisa ion o empi ical exp essions de i ed
om CFD simula ions o expe imen s wi h scaled physical models. Bo h me hods a e
pa icula ly aluable in assessing wa e o e opping, as his phenomenon is in luenced by
a a ie y o ac o s wi h in e connec ed impac s ha a e di icul o p edic . The empi ical
app oach con ains a simple ep esen a ion o he unde lying physics o he phenomenon,
o en p esen ed as an equa ion ha co ela es dimensionless pa ame e s speci ic o he
p ocess (e.g., o e opping discha ge, wa e a ibu es and s uc u al geome ic ac o s).
The main objec i e pa ame e in his analysis is he mean o e opping low discha ge
q (m
2
/s), which accoun s o he a e age liquid olume pe uni ime and ans e sal ho i-
zon al leng h o s uc u e passing o e , which is usually p esen ed in i s nondimensional
o m (by di iding by
qgHm03
), known as he ela i e o e opping a e. The signi ican
wa e heigh (H
m0
) equals ou imes he oo mean squa e displacemen o he ee su ace,
which is equi alen o he a e age heigh o he 1/3 highes wa es o a se ies o i egula
wa es, and g is he accele a ion o g a i y.
The dependence o he ela i e o e opping a e is no mally de ined as a unc ion o
he c es eeboa d R
c
, being he ele a ion di e ence be ween he s uc u e c es and he
s ill wa e le el (SWL), which is gi en, again, in a nondimensionalised o m (by di iding
by he signi ican wa e heigh Hm0) known as he ela i e eeboa d.
To assess he wa e o e opping accu a ely, i is necessa y o iden i y he wa e–
s uc u e in e ac ion egime because di e en p ocesses and ou comes may occu in di e -
en scena ios. Speci ically, when dealing wi h e ical walls wi h s uc u al modi ica ions,
o e opping can occu unde wo dis inc egimes: non-impulsi e and impulsi e condi ions.
Non-impulsi e condi ions happen when wa e heigh s a e ela i ely modes compa ed
wi h he dep h a he base o he s uc u e, and he wa e s eepness emains low, s aying
a om he poin o b eaking. In such a si ua ion, o e opping wa es gene a e smoo hly
e ol ing loads, and wa e lows gen ly o e he s uc u e. In con as , impulsi e condi ions
happen a e ical o s eep walls when wa e heigh s signi ican ly exceed local wa e dep hs
and show a high wa e s eepness. In his scena io, he wa es c ash agains he s uc u e,
gi ing ise o a as , e ical and je -like low o wa e ha jumps o e he s uc u e due o
J. Ma . Sci. Eng. 2024,12, 1441 5 o 22
momen um and ine ia. The likelihood o ha ing impulsi e o non-impulsi e o e opping
is disc imina ed by employing he “impulsi eness” cons an de ined in Equa ion (1) [6]:
h∗=h2
Hm0Lm−1,0
(1)
whe e
Lm−1,0 =g(Tm−1,0)2/
2
π
is he deep-wa e wa eleng h co esponding o he wa e
ene gy pe iod Tm−1,0. F om h∗>0.23 onwa ds, wa es a e conside ed non-impulsi e.
Ano he e ec ha may ha e a decisi e in luence is he exis ence o a sloping o esho e
a a shallow o in e media e dep h because i may cause shoaling and/o b eaking du ing i s
p opaga ion owa ds he s uc u e suscep ible o o e opping. All hese di e en esponses
co esponding o di e en egimes a e summa ised in Figu e 2.
J. Ma . Sci. Eng. 2024, 12, x FOR PEER REVIEW 5 o 22
wa e dep hs and show a high wa e s eepness. In his scena io, he wa es c ash agains
he s uc u e, gi ing ise o a as , e ical and je -like low o wa e ha jumps o e he
s uc u e due o momen um and ine ia. The likelihood o ha ing impulsi e o non-im-
pulsi e o e opping is disc imina ed by employing he “impulsi eness” cons an de ined
in Equa ion (1) [6]:
ℎ∗=ℎ2
𝐻𝑚0𝐿𝑚−1,0
(1)
whe e 𝐿𝑚−1,0=𝑔(𝑇𝑚−1,0)2/2𝜋 is he deep-wa e wa eleng h co esponding o he wa e
ene gy pe iod 𝑇𝑚−1,0. F om ℎ∗>0.23 onwa ds, wa es a e conside ed non-impulsi e.
Ano he e ec ha may ha e a decisi e in luence is he exis ence o a sloping o e-
sho e a a shallow o in e media e dep h because i may cause shoaling and/o b eaking
du ing i s p opaga ion owa ds he s uc u e suscep ible o o e opping. All hese di e -
en esponses co esponding o di e en egimes a e summa ised in Figu e 2.
In addi ion, i is no iceable in Figu e 2, on he one hand, he di e ence be ween he
ela i e o e opping discha ge p oduced o e a linea e ical wall (g een line) and he
one p oduced o e a cylind ical s uc u e ( ed line) and, on he o he hand, he educ ion
in oduced by a pa ape (blue line) wi h espec o he e e ence e ical wall (g een line).
The objec i e he e is o accoun o he o e opping educ ion expec ed o a cylind ical
s uc u ed wi h simila pa ape s moun ed below he c own o he cylinde unde he same
non-impulsi e wa e egime.
Figu e 2. O e iew o di e en o e opping egimes on e ical walls and cylinde s [6,21,40–42].
The widesp ead empi ical model o p edic ing wa e o e opping akes he o m o
a Weibull unc ion (2) p oposed by an de Mee e al. [6]:
𝑞
√𝑔𝐻𝑚03=𝑎·exp[−(𝑏 𝑅𝑐
𝐻𝑚0)𝑐]
(2)
Pa ame e 𝑅𝑐/𝐻𝑚0, known as he ela i e eeboa d, is he go e ning pa ame e o Equa-
ion (2), oge he wi h he signi ican wa e heigh . Equa ion (2) is he nondimensionalised
Figu e 2. O e iew o di e en o e opping egimes on e ical walls and cylinde s [6,21,40–42].
In addi ion, i is no iceable in Figu e 2, on he one hand, he di e ence be ween he
ela i e o e opping discha ge p oduced o e a linea e ical wall (g een line) and he
one p oduced o e a cylind ical s uc u e ( ed line) and, on he o he hand, he educ ion
in oduced by a pa ape (blue line) wi h espec o he e e ence e ical wall (g een line).
The objec i e he e is o accoun o he o e opping educ ion expec ed o a cylind ical
s uc u ed wi h simila pa ape s moun ed below he c own o he cylinde unde he same
non-impulsi e wa e egime.
The widesp ead empi ical model o p edic ing wa e o e opping akes he o m o a
Weibull unc ion (2) p oposed by an de Mee e al. [6]:
q
qgHm03=a·exp−bRc
Hm0c(2)
Pa ame e
Rc/Hm0
, known as he ela i e eeboa d, is he go e ning pa ame e o
Equa ion (2)
,
oge he wi h he signi ican wa e heigh . Equa ion (2) is he nondimensionalised exp ession
co e ing he whole ange o he ela i e eeboa d. Pa ame e s
a
,
b
and
c
depend on he
case s udy, including he di e en wa e egimes men ioned be o e, and a e de e mined
J. Ma . Sci. Eng. 2024,12, 1441 6 o 22
om empi ical esul s ob ained ei he expe imen ally o nume ically. This is an impo an
equa ion when e alua ing whe he ole able o e opping limi is su passed o no unde a
ce ain ha sh sea s a e. E en i only he ole able limi s a e well de ined o coas de ences
in [
6
] ( o example, a la ge yach in a po wi h some acili ies ins alled on deck has a limi
o q< 5 L/s pe m o H
m0
> 5 m), as i is no ye publicly es ablished o loa ing o sho e
wind u bines, his design c i e ion will be o high ele ance owa ds he implemen a ion
o he echnology o make i comme cially compe i i e.
In he case o s udying egula wa es, he esul ing o e opping discha ge is known
o o e es ima e he esul s ob ained wi h he Rayleigh dis ibu ion, cha ac e ised by he
co esponding signi ican wa e heigh ; a ac o be ween 2.3 and 2.8 is epo ed in [
43
].
Consequen ly, he esul s ob ained wi h egula wa es can be aken as conse a i e alues
o he co esponding ones ob ained wi h he spec a cha ac e ised by he same signi ican
wa e heigh s in such a compa ison.
The e e ence equa ions o p edic o e opping wi h no in luence o he o esho e
and unde non-impulsi e egula wa es o a e ical wall [
6
] (3) and a e ical cylinde
wi hou any pa ape [40] (4) a e as ollows:
q
qgHm03=0.047·exp"−2.35 Rc
H1.3#(3)
Q
pgH3D=0.0478·exp"−2.74 Rc
H1.9#(4)
The in oduc ion o a pa ape o he geome y is expec ed o esul in a d op in he
o e opping low discha ge o e he s uc u e, and o co espondingly modi y he p e ious
exp essions. Howe e , he e ec i eness o he mi iga ion s uc u e is highly ela ed o he
alue o he ela i e eeboa d. Wi hin a low ela i e eeboa d egime (R
c
/H
≤
0.5), he
pa ape becomes easily subme ged unde he o e opped wa e low, whe eas, in a high
ela i e eeboa d egime (R
c
/H> 1.0), he e is a maximum pe o mance o he pa ape
because o he esul ing e ec i e seawa d p ojec ion o he incoming wa e. Be ween hese
wo di e en ia ed egimes, he e is an in e media e egime (1.0
≥
R
c
/H> 0.5) wi h a
medium e iciency in he o e opping educ ion.
Consequen ly, he bes way o accoun o he educ ion due o he in oduc ion o a
pa ape is de ined in he Eu O op manual [
6
] as a piecewise eg ession (Figu e 2). This same
app oach has been ollowed o he cylind ical case accoun ing o he in e media e and low
ela i e eeboa d ange co e ed in his piece o esea ch wi h he ollowing
i ing unc ion
:
Q
pgH3D=
a·exph−bRc
Hci,Rc
H≤0.5
d·exp−eRc
H ,Rc
H>0.5
(5)
O he han he o e opping low a e, he o e opping low hickness
δ
is also s udied
in his campaign. This pa ame e de ines he hickness o he liquid laye passing o e he
main s uc u e du ing an o e opping e en . In he pa icula case o he cylinde , i is
e alua ed a he cen e o he c es ( he axis o he cylinde ). The a e age low hickness
δ
is compu ed by a e aging he ins an aneous signal o he a iable o e he ime in e al
aken by he in e ac ion unde s udy ∆ :
δ=1
∆ Z∆
0δ( )d (6)
This pa ame e will be compu ed, and i s ela ionship wi h he o e opping discha ge
add essed in Sec ion 4.
J. Ma . Sci. Eng. 2024,12, 1441 7 o 22
3. Aims and Me hods
The objec i e o he p esen s udy is o de e mine he ela ionship be ween he geome -
ic cha ac e is ics o he cylinde wi h he selec ed op ions o pa ape and wa e o e opping
discha ge. As a esul , a model based on p edic i e equa ions o ixed e ical cylinde
wi h pa ape s will be p oposed.
In his campaign, a nume ical model ha was used and expe imen ally alida ed
in a p e ious campaign [
40
] was used o he analysis. Tha esea ch s udy conduc ed a
nume ical calcula ion o wa e o e opping on a ixed e ical cylinde a ec ed by egula
wa es, and he co esponding expe imen al analysis o i s alida ion. In he cu en s udy,
he expe imen ally alida ed nume ical model was used o analyse he consequence o he
speci ic geome y modi ica ion o he pa ape moun ed on he op pa o he cylinde .
3.1. Tes P og amme
All he simula ions we e execu ed wi h egula wa es, conside ing wa e heigh s
om H= 0.065 m o 0.160 m and pe iod T= 1.3 s. The pe iod was no a ied because i s
in luence has been epo ed o be negligible on he dimensionless ep esen a ion o he
o e opping discha ge [
40
]. In ac , no dependence on he pe iod o wa eleng h was s a ed
in he empi ical models ep esen ed in Figu e 2and he co esponding well-es ablished
Equa ions (2)–(4). The alues we e de e mined h ough he applica ion o F oude simili ude
wi h a educ ion scale ac o o 1:100 co esponding o wa e heigh s o 6.5 m o 16 m and a
pe iod o 13 s o ex eme sea s a e. These wa e condi ions a e ypical o Beau o numbe s
in he ange 9–11, which co espond o a iolen s o m. A cons an wa e dep h o h= 0.32
m o he model was main ained in all he expe imen s.
The cylinde showed a diame e o D= 0.11 m, which was 11 m in ull scale. Fou
di e en cylinde heigh s we e simula ed, co e ing a ange o eeboa d R
c
om 0.02 o
0.07 m in he empi ical model (Table 1), by aking in o accoun he design pa ame e s o he
ballas cylind ical columns o he NATULUS-10 loa ing o sho e wind u bine [
44
]. The
wa e heigh H was a ied wi hin he a o emen ioned ange o each indi idual eeboa d
case in o de o manage a uni o mly dis ibu ed ange o ela i e eeboa d R
c
/H alues
wi h o e lapping ansi ion ames be ween cases. A b oad ange was s udied o he
ela i e eeboa d be ween 0.14 and 0.78, amoun ing o 24 alues (6 alues o wa e heigh
o each eeboa d case).
Table 1. Wa e se s used in his campaign.
Rc(m) Rc/H
Case 1 0.020 0.143, 0.167, 0.200, 0.222, 0.267, 0.308
Case 2 0.037 0.229, 0.282, 0.319, 0.367, 0.431, 0.489
Case 3 0.053 0.333, 0.381, 0.427, 0.458, 0.533, 0.593
Case 4 0.070 0.500, 0.538, 0.583, 0.636, 0.700, 0.778
I is cus oma y in he s udy o pa ape s o de ine he heigh o he pa ape h
as a unc ion
o he c es eeboa d R
c
, and he ho izon al ex ension o he pa ape B
as a unc ion o he
heigh o he pa ape h
[
24
,
33
]. The s udies used as e e ences o geome ic a ios in pa ape s
in sea ch o be e o e opping pe o mance [
34
] ecommend beginning he s udy o pa ape s in
he ange o h
/B
∈
[0.5, 1]. The e o e, he ou geome ies s udied he e we e chosen o s udy
bo h limi a ios wi h wo di e en pa ape heigh s (Table 2and Figu e 3).
Table 2. Geome ies s udied in his campaign.
h (m) B (m)
Geome y 1 0.021 0.021
Geome y 2 0.021 0.042
Geome y 3 0.028 0.028
Geome y 4 0.028 0.056
J. Ma . Sci. Eng. 2024,12, 1441 8 o 22
J. Ma . Sci. Eng. 2024, 12, x FOR PEER REVIEW 8 o 22
Table 2. Geome ies s udied in his campaign.
h (m)
B (m)
Geome y 1
0.021
0.021
Geome y 2
0.021
0.042
Geome y 3
0.028
0.028
Geome y 4
0.028
0.056
Figu e 3. Visualisa ion o pa ape geome ies. The o al heigh ep esen ed is he maximum c es
eeboa d. The axial-symme ic c oss-sec ion is ep esen ed.
Each o he ou pa ape geome ies was exposed o he same wa e campaign,
amoun ing o a o al o 96 compu a ional cases. The ange o “impulsi eness” conside ed
du ing he expe imen s s a ed a 0.243 and wen up o 0.466, all o hem g ea e han
he limi alue o 0.23 and consequen ly belonging o he non-impulsi e egime. “Impul-
si eness” and b eaking wa es we e closely ela ed. And, because he objec i e was o
cha ac e ise he non-impulsi e egime, he es campaign co e ed he non-b eaking e-
gime. In his case, he b eake index, 𝐻/ℎ, showed a minimum alue o 0.203 and a max-
imum o 0.500, which implied ha he wa es we e no limi ed by wa e dep h. Mo eo e ,
he alue o he dep h o wa eleng h a io o h/λ = 0.159 used in his campaign was a
om he ange o shallow wa e s (h/λ < 0.05), whe e shoaling phenomenon may occu and
he ea men should accoun o a ce ain o esho e in luence.
The U sell numbe (Equa ion (7)) is a dimensionless numbe calcula ed as he quo-
ien o he nonlinea e m o ini e heigh and he linea e m o small heigh o he ana-
ly ical solu ion o he wa e ele a ion [45] and i is de ined as:
𝑈𝑟=𝐻λ2/ℎ3
(7)
The U sell numbe p o ides anges o alidi y o di e en wa e app oxima ion
me hods. In cases whe e he U sell numbe is low, 𝑈𝑟≪1, simpli ied linea wa e heo y
can be applied. Howe e , he U sell numbe ange o his expe imen al campaign was
be ween 3.85 and 23.36. Fo his ange o he U sell numbe , he second o de S okes’ wa e
heo y could be applied acco ding o he alidi y limi gi en by:
𝑈𝑟<8𝜋2
3≅26.3
(8)
This is he eason why S okes’ wa e heo y was implemen ed in he ini ial and
bounda y condi ions o he compu a ional model ins ead o he Ai y linea heo y.
The o e opping discha ge was compu ed as he mean olume ic low a e ha
c ossed he e ical c oss-sec ional plane o e he c es o he s uc u e con aining he axis
o he cylinde , no mal o he ad ancing di ec ion o he wa e. Fo he sake o di ec com-
pa ison wi h he base case, he egion o he plane abo e he diame e o he cylinde was
conside ed, wi hou including he ex ension caused by he addi ion o he pa ape , which
was assumed o be a non-usable po ion o he op ho izon al pla o m. The ins an aneous
olume ic low a e c ossing he egion was in eg a ed, yielding he o al olume Vo. An
example o a compu a ional es can be obse ed in Figu e 4.
Figu e 3. Visualisa ion o pa ape geome ies. The o al heigh ep esen ed is he maximum c es
eeboa d. The axial-symme ic c oss-sec ion is ep esen ed.
Each o he ou pa ape geome ies was exposed o he same wa e campaign, amoun -
ing o a o al o 96 compu a ional cases. The ange o “impulsi eness” conside ed du ing
he expe imen s s a ed a 0.243 and wen up o 0.466, all o hem g ea e han he limi
alue o 0.23 and consequen ly belonging o he non-impulsi e egime. “Impulsi eness”
and b eaking wa es we e closely ela ed. And, because he objec i e was o cha ac e ise
he non-impulsi e egime, he es campaign co e ed he non-b eaking egime. In his case,
he b eake index,
H/h
, showed a minimum alue o 0.203 and a maximum o 0.500, which
implied ha he wa es we e no limi ed by wa e dep h. Mo eo e , he alue o he dep h
o wa eleng h a io o h/
λ
= 0.159 used in his campaign was a om he ange o shallow
wa e s (h/
λ
< 0.05), whe e shoaling phenomenon may occu and he ea men should
accoun o a ce ain o esho e in luence.
The U sell numbe (Equa ion (7)) is a dimensionless numbe calcula ed as he quo ien
o he nonlinea e m o ini e heigh and he linea e m o small heigh o he analy ical
solu ion o he wa e ele a ion [45] and i is de ined as:
U =Hλ2/h3(7)
The U sell numbe p o ides anges o alidi y o di e en wa e app oxima ion
me hods. In cases whe e he U sell numbe is low,
U ≪
1, simpli ied linea wa e heo y
can be applied. Howe e , he U sell numbe ange o his expe imen al campaign was
be ween 3.85 and 23.36. Fo his ange o he U sell numbe , he second o de S okes’ wa e
heo y could be applied acco ding o he alidi y limi gi en by:
U <8π2
3
∼
=26.3 (8)
This is he eason why S okes’ wa e heo y was implemen ed in he ini ial and
bounda y condi ions o he compu a ional model ins ead o he Ai y linea heo y.
The o e opping discha ge was compu ed as he mean olume ic low a e ha
c ossed he e ical c oss-sec ional plane o e he c es o he s uc u e con aining he
axis o he cylinde , no mal o he ad ancing di ec ion o he wa e. Fo he sake o
di ec compa ison wi h he base case, he egion o he plane abo e he diame e o he
cylinde was conside ed, wi hou including he ex ension caused by he addi ion o he
pa ape , which was assumed o be a non-usable po ion o he op ho izon al pla o m. The
ins an aneous olume ic low a e c ossing he egion was in eg a ed, yielding he o al
olume Vo. An example o a compu a ional es can be obse ed in Figu e 4.
The alues o he mean o e opping low a e Qand he co esponding mean o e -
opping discha ge qwe e de e mined by aking in o accoun he o al ime in e al
∆
when
wa e o e opped he s uc u e and he cylinde diame e , D, as:
q=Q
D=Vo
∆ ·D(9)
Then, his alue was nondimensionalised in sea ch o gene alisa ion by using wa e heigh
H o ob ain he ela i e o e opping discha ge q/pgH3.
J. Ma . Sci. Eng. 2024,12, 1441 9 o 22
J. Ma . Sci. Eng. 2024, 12, x FOR PEER REVIEW 9 o 22
The alues o he mean o e opping low a e Q and he co esponding mean o e -
opping discha ge q we e de e mined by aking in o accoun he o al ime in e al ∆
when wa e o e opped he s uc u e and he cylinde diame e , D, as:
𝑞=𝑄
𝐷=𝑉𝑜
Δ ⋅𝐷
(9)
Then, his alue was nondimensionalised in sea ch o gene alisa ion by using wa e heigh
H o ob ain he ela i e o e opping discha ge (𝑞/√𝑔𝐻3).
(a)
(b)
Figu e 4. Expo ed low a e signals: (a) ins an aneous low a e, (b) accumula ed o e opped ol-
ume. (Case: Rc/H = 0.538, H = 0.13 m, Geome y 4.)
In addi ion, he e olu ion low hickness δ wi h ime was de e mined a he cen e o
he op su ace o he cylinde (Figu e 5). The mean alue was compu ed a e wa ds by
in eg a ion (Equa ion (6)). Fo bo h analyses, a ime in e al ∆ o 5 pe iods, implying 5
o e opping e en s, we e compu ed o all cases.
(a)
(b)
Figu e 5. Expo ed low hickness signals: (a) ins an aneous low hickness, (b) in eg a ed alue o e
ime. (Case: Rc/H = 0.538, H = 0.13 m, Geome y 4.)
The de ini ion o he ela ionship be ween he o e opping wa e olume and he
o e opping low hickness may enable expe imen al de e mina ion o o e opping wi h-
ou using a speci ic de ice o accumula e and ead he o e opped wa e olume by em-
ploying wa e p obes loca ed on op o he main s uc u e.
Figu e 4. Expo ed low a e signals: (a) ins an aneous low a e, (b) accumula ed o e opped olume.
(Case: Rc/H= 0.538, H= 0.13 m, Geome y 4).
In addi ion, he e olu ion low hickness δwi h ime was de e mined a he cen e o
he op su ace o he cylinde (Figu e 5). The mean alue was compu ed a e wa ds by
in eg a ion (Equa ion (6)). Fo bo h analyses, a ime in e al
∆
o 5 pe iods, implying 5
o e opping e en s, we e compu ed o all cases.
J. Ma . Sci. Eng. 2024, 12, x FOR PEER REVIEW 9 o 22
The alues o he mean o e opping low a e Q and he co esponding mean o e -
opping discha ge q we e de e mined by aking in o accoun he o al ime in e al ∆
when wa e o e opped he s uc u e and he cylinde diame e , D, as:
𝑞=𝑄
𝐷=𝑉𝑜
Δ ⋅𝐷
(9)
Then, his alue was nondimensionalised in sea ch o gene alisa ion by using wa e heigh
H o ob ain he ela i e o e opping discha ge (𝑞/√𝑔𝐻3).
(a)
(b)
Figu e 4. Expo ed low a e signals: (a) ins an aneous low a e, (b) accumula ed o e opped ol-
ume. (Case: Rc/H = 0.538, H = 0.13 m, Geome y 4.)
In addi ion, he e olu ion low hickness δ wi h ime was de e mined a he cen e o
he op su ace o he cylinde (Figu e 5). The mean alue was compu ed a e wa ds by
in eg a ion (Equa ion (6)). Fo bo h analyses, a ime in e al ∆ o 5 pe iods, implying 5
o e opping e en s, we e compu ed o all cases.
(a)
(b)
Figu e 5. Expo ed low hickness signals: (a) ins an aneous low hickness, (b) in eg a ed alue o e
ime. (Case: Rc/H = 0.538, H = 0.13 m, Geome y 4.)
The de ini ion o he ela ionship be ween he o e opping wa e olume and he
o e opping low hickness may enable expe imen al de e mina ion o o e opping wi h-
ou using a speci ic de ice o accumula e and ead he o e opped wa e olume by em-
ploying wa e p obes loca ed on op o he main s uc u e.
Figu e 5. Expo ed low hickness signals: (a) ins an aneous low hickness, (b) in eg a ed alue o e
ime. (Case: Rc/H= 0.538, H= 0.13 m, Geome y 4).
The de ini ion o he ela ionship be ween he o e opping wa e olume and he
o e opping low hickness may enable expe imen al de e mina ion o o e opping wi hou
using a speci ic de ice o accumula e and ead he o e opped wa e olume by employing
wa e p obes loca ed on op o he main s uc u e.
3.2. Nume ical Model
The comme cial so wa e STAR CCM+ ( 17.02) was used o c ea e and implemen
he nume ical model, which was an applica ion o he ini e olume me hod, whe e he
compu a ional ield was di ided in small cells and low equa ions we e nume ically sol ed
o each cell.
J. Ma . Sci. Eng. 2024,12, 1441 16 o 22
J. Ma . Sci. Eng. 2024, 12, x FOR PEER REVIEW 16 o 22
(a)
(b)
(c)
(d)
Figu e 11. Rela i e o e opping a e as a unc ion o ela i e eeboa d o he pa ape geome ies:
(a) Geome y 1, (b) Geome y 2, (c) Geome y 3 and (d) Geome y 4.
Table 4. Fi ing pa ame e s o he piecewise Weibull equa ion o each geome y.
a
b
c
d
e
Geome y 1
0.0449
3.07
1.9
0.0062
1.74
6.9
Geome y 2
0.0703
4.74
1.1
0.0051
1.73
7.8
Geome y 3
0.0578
3.85
1.3
0.0066
1.79
6.2
Geome y 4
0.0543
3.76
1.4
0.0048
1.71
8.2
The o e opping educ ion esul ing om he ou pa ape op ions ollowed a simila
pa e n, wi h a mode a e educ ion in he low ela i e eeboa d egime, which became
e y p onounced in he in e media e ela i e eeboa d egime (Rc/H > 0.5). I we compa e
he esul s ob ained o he ou geome ies in he same g aph (Figu e 12a), one can ob-
se e ha Geome ies 2 and 4 we e sligh ly be e han Geome ies 1 and 3, poin ing ou
he ad an age o using a a io o he heigh o he pa ape o i s ho izon al ex ension o
h /B = 0.5 o e he alue o 1.0. In addi ion, he inc ease by one- hi d (33%) o he leng hs
o he pa ape when compa ing Geome y 1 wi h Geome y 3 o , sepa a ely, Geome y 2
wi h Geome y 4 did no lead o any signi ican a ia ion in he o e opping.
Howe e , one should bea in mind ha he inhe en dispe sion in he da a se s o he
nume ically p edic ed o e opping a e was e y signi ican in his ype o es due o he
highly nonlinea na u e o he o e opping phenomenon and he inc easingly high s a is-
ical unce ain y o he o e opping discha ge o highe ela i e eeboa ds wi h smalle
Figu e 11. Rela i e o e opping a e as a unc ion o ela i e eeboa d o he pa ape geome ies:
(a) Geome y 1, (b) Geome y 2, (c) Geome y 3 and (d) Geome y 4.
J. Ma . Sci. Eng. 2024, 12, x FOR PEER REVIEW 17 o 22
o e opping. Thus, he p e iously add essed dis inc i e beha iou ell well wi hin he
admi ed s a is ical unce ain y i a common unique eg ession was used o he ou ge-
ome ies (Figu e 12b). The co esponding dispe sion esul ing om he i ing p ocedu e
was pe o med acco ding o e y well-es ablished s anda ds in he ield [21]. The co a i-
ance ma ix o he i ing cons an s was used o de ine he bounding cu es con aining he
da a wi h a ce ain con idence le el. Assuming ha he unce ain y o he i ed pa ame-
e s ollowed a no mal dis ibu ion, he exceedance lines we e calcula ed as 𝑎𝑖+1.64·σ
and 𝑎𝑖−1.64·σ, wi h 𝑎𝑖 being he 𝑖 h i ing pa ame e and σ i s s anda d de ia ion.
This was equi alen o using a 90% con idence in e al band in he calcula ion o he i ing
pa ame e s.
(a)
(b)
Figu e 12. Rela i e o e opping a e as a unc ion o ela i e eeboa d: (a) di e en eg essions o
each geome y; (b) common eg ession o all geome ies.
The esul ing Weibull piecewise exp ession ep esen ing he o e opping educ ion
o cases h /B ∈ [0.5, 1] om he i ing o he indi idual compu a ional poin s (Figu e 12b)
was as ollows:
𝑄
√𝑔𝐻3𝐷=
{
0.0558·exp[−(3.76𝑅𝑐
𝐻)1.4], 𝑅𝑐
𝐻≤0.5
0.00544·exp[−(1.73𝑅𝑐
𝐻)7.4], 𝑅𝑐
𝐻>0.5
}
(17)
The coe icien o de e mina ion o his unique eg ession line showed a easonable alue
o 𝑅2=0.8517 and all he poin s we e well wi hing he ange o 90% con idence in e al.
The unce ain y o each i ing pa ame e is gi en in Table 5.
Table 5. Fi ing pa ame e s o each geome y.
a
b
c
d
e
0.0558
3.76
1.4
0.00544
1.73
7.4
σ(a)
σ(b)
σ(c)
σ(d)
σ(e)
σ( )
0.0107
0.586
0.29
0.00231
0.0853
1.1
Figu e 13 shows a good co ela ion o he indi idual compu a ional esul s wi h he
co esponding alues gi en by he p e ious eg ession line de ined in Equa ion (17),
whe e all he poin s ell inside he band, limi ing he ange o 10 imes and 0.1 imes he
calcula ed alues.
Figu e 12. Rela i e o e opping a e as a unc ion o ela i e eeboa d: (a) di e en eg essions o
each geome y; (b) common eg ession o all geome ies.
J. Ma . Sci. Eng. 2024,12, 1441 17 o 22
Howe e , one should bea in mind ha he inhe en dispe sion in he da a se s
o he nume ically p edic ed o e opping a e was e y signi ican in his ype o es
due o he highly nonlinea na u e o he o e opping phenomenon and he inc easingly
high s a is ical unce ain y o he o e opping discha ge o highe ela i e eeboa ds
wi h smalle o e opping. Thus, he p e iously add essed dis inc i e beha iou ell well
wi hin he admi ed s a is ical unce ain y i a common unique eg ession was used o
he ou geome ies (Figu e 12b). The co esponding dispe sion esul ing om he i ing
p ocedu e was pe o med acco ding o e y well-es ablished s anda ds in he ield [
21
].
The co a iance ma ix o he i ing cons an s was used o de ine he bounding cu es
con aining he da a wi h a ce ain con idence le el. Assuming ha he unce ain y o he
i ed pa ame e s ollowed a no mal dis ibu ion, he exceedance lines we e calcula ed
as
ai+
1.64
·σ
and
ai−
1.64
·σ
, wi h
ai
being he
i
h i ing pa ame e and
σ
i s s anda d
de ia ion. This was equi alen o using a 90% con idence in e al band in he calcula ion
o he i ing pa ame e s.
The esul ing Weibull piecewise exp ession ep esen ing he o e opping educ ion
o cases h
/B
∈
[0.5, 1] om he i ing o he indi idual compu a ional poin s (Figu e 12b)
was as ollows:
Q
pgH3D=
0.0558·exp−3.76Rc
H1.4,Rc
H≤0.5
0.00544·exp−1.73Rc
H7.4,Rc
H>0.5
(17)
The coe icien o de e mina ion o his unique eg ession line showed a easonable
alue o
R2=
0.8517 and all he poin s we e well wi hing he ange o 90% con idence
in e al.
The unce ain y o each i ing pa ame e is gi en in Table 5.
Table 5. Fi ing pa ame e s o each geome y.
a b c d e
0.0558 3.76 1.4 0.00544 1.73 7.4
σ(a) σ(b) σ(c) σ(d) σ(e) σ( )
0.0107 0.586 0.29 0.00231 0.0853 1.1
Figu e 13 shows a good co ela ion o he indi idual compu a ional esul s wi h
he co esponding alues gi en by he p e ious eg ession line de ined in Equa ion (17),
whe e all he poin s ell inside he band, limi ing he ange o 10 imes and 0.1 imes he
calcula ed alues.
As a as he mean low hickness
δ
was conce ned, he indi idual cases ob ained
by compu ing he indi idual signals (Figu e 10c,d) coming om all he es s and ge-
ome ies we e analysed wi h a non-dimensional app oach. The o e all endency o he
non-dimensional low hickness
δ/H
gene a ed by he o e opping phenomenon ollowed
a ma ked exponen ial dependency on he ela i e eeboa d
Rc/H
(Figu e 14). In he
same way as o he o e opping discha ge, i was di icul o dis inguish be ween he
phenomenon gene a ed by he di e en pa ape geome ies (Figu e 14a), and he use o
a unique exp ession esul ed in being a easonable app oach, whe e he majo i y o he
nume ical esul s ell wi hin he 5% exceedance band.
He e, he di ec ela ionship be ween he ela i e a e age low hickness and he
ela i e eeboa d sugges ed in [40] by ollowing an exponen ial endency was ce i ied:
δ
H=0.1515·exp−4.16 Rc
H(18)
J. Ma . Sci. Eng. 2024,12, 1441 18 o 22
This i ing exp ession showed a easonable alue o he coe icien o de e mina ion
o R2= 0.9661.
I can be obse ed how he pa ape - ype mi iga ion s uc u e induced a ne educ ion
o he low hickness in compa ison wi h he plain cylinde . The quan i a i e educ ion
could be e alua ed by simul aneous conside a ion wi h he low hickness a ibu ed o he
cylind ical case, wi h no in luence o o esho e and non-impulsi e condi ions [37]:
δ
H=0.1769·exp−4.11 Rc
H(19)
This esul con i med he iabili y o de i ing he o e opping educ ion by compu ing
he educ ion o he low hickness [
40
]. In Figu e 15, a di ec linea ela ionship be ween
he wo a iables is e y well co ela ed.
J. Ma . Sci. Eng. 2024, 12, x FOR PEER REVIEW 18 o 22
Figu e 13. Co ela ion be ween compu a ional and es ima ed esul s o he o e opping a e.
As a as he mean low hickness 𝛿 was conce ned, he indi idual cases ob ained by
compu ing he indi idual signals (Figu e 10c,d) coming om all he es s and geome ies
we e analysed wi h a non-dimensional app oach. The o e all endency o he non-dimen-
sional low hickness 𝛿/𝐻 gene a ed by he o e opping phenomenon ollowed a ma ked
exponen ial dependency on he ela i e eeboa d 𝑅𝑐/𝐻 (Figu e 14). In he same way as
o he o e opping discha ge, i was di icul o dis inguish be ween he phenomenon
gene a ed by he di e en pa ape geome ies (Figu e 14a), and he use o a unique ex-
p ession esul ed in being a easonable app oach, whe e he majo i y o he nume ical
esul s ell wi hin he 5% exceedance band.
(a)
(b)
Figu e 14. Nondimensionalised mean low hickness as a unc ion o ela i e eeboa d o he di -
e en pa ape geome ies (a) and o e all endency (b).
He e, he di ec ela ionship be ween he ela i e a e age low hickness and he el-
a i e eeboa d sugges ed in [40] by ollowing an exponen ial endency was ce i ied:
Figu e 13. Co ela ion be ween compu a ional and es ima ed esul s o he o e opping a e.
J. Ma . Sci. Eng. 2024, 12, x FOR PEER REVIEW 18 o 22
Figu e 13. Co ela ion be ween compu a ional and es ima ed esul s o he o e opping a e.
As a as he mean low hickness 𝛿 was conce ned, he indi idual cases ob ained by
compu ing he indi idual signals (Figu e 10c,d) coming om all he es s and geome ies
we e analysed wi h a non-dimensional app oach. The o e all endency o he non-dimen-
sional low hickness 𝛿/𝐻 gene a ed by he o e opping phenomenon ollowed a ma ked
exponen ial dependency on he ela i e eeboa d 𝑅𝑐/𝐻 (Figu e 14). In he same way as
o he o e opping discha ge, i was di icul o dis inguish be ween he phenomenon
gene a ed by he di e en pa ape geome ies (Figu e 14a), and he use o a unique ex-
p ession esul ed in being a easonable app oach, whe e he majo i y o he nume ical
esul s ell wi hin he 5% exceedance band.
(a)
(b)
Figu e 14. Nondimensionalised mean low hickness as a unc ion o ela i e eeboa d o he di -
e en pa ape geome ies (a) and o e all endency (b).
He e, he di ec ela ionship be ween he ela i e a e age low hickness and he el-
a i e eeboa d sugges ed in [40] by ollowing an exponen ial endency was ce i ied:
Figu e 14. Nondimensionalised mean low hickness as a unc ion o ela i e eeboa d o he di e en
pa ape geome ies (a) and o e all endency (b).
J. Ma . Sci. Eng. 2024,12, 1441 19 o 22
J. Ma . Sci. Eng. 2024, 12, x FOR PEER REVIEW 19 o 22
δ
𝐻=0.1515·exp(−4.16𝑅𝑐
𝐻)
(18)
This i ing exp ession showed a easonable alue o he coe icien o de e mina ion
o R2 = 0.9661.
I can be obse ed how he pa ape - ype mi iga ion s uc u e induced a ne educ ion
o he low hickness in compa ison wi h he plain cylinde . The quan i a i e educ ion
could be e alua ed by simul aneous conside a ion wi h he low hickness a ibu ed o
he cylind ical case, wi h no in luence o o esho e and non-impulsi e condi ions [37]:
δ
𝐻=0.1769·exp(−4.11𝑅𝑐
𝐻)
(19)
This esul con i med he iabili y o de i ing he o e opping educ ion by compu-
ing he educ ion o he low hickness [40]. In Figu e 15, a di ec linea ela ionship be-
ween he wo a iables is e y well co ela ed.
(a)
(b)
Figu e 15. Dependence o he ela i e o e opping a e on he nondimensionalised mean low hick-
ness: (a) i ing line and (b) quali y o he i ing.
The ela ionship be ween nondimensionalised low hickness and ela i e o e op-
ping a e in his case was:
𝑄
√𝑔𝐻3𝐷=−0.00229+0.5009·δ
𝐻
(20)
He e, in he linea eg ession, a nega i e independen e m was allowed (Figu e 15a),
add essing he ac ha , e en a a e y low o e opping discha ge close o ze o, a ce ain
wa e laye exis ed o e he c es wi h e y low low eloci y. This e ec could be no iced
by simul aneously analysing he low hickness and he co esponding o e opping a e,
as shown in Figu e 10. The co ela ion o he p edic ed alues by linea eg ession wi h
he co esponding alues coming om he nume ical calcula ion was easonably accep a-
ble (Figu e 15b), wi h a coe icien o de e mina ion o R2 = 0.9661, by aking in o accoun
he high dispe sion a ibu ed o his ype o phenomenon.
5. Conclusions
In he p esen s udy, a compu a ional model was used o cha ac e ise he o e op-
ping educ ion p oduced by di e en geome ic op ions o o e opping mi iga ion s uc-
u es ins alled a he op o a e e ence ixed cylinde . The selec ed mi iga ion s uc u e
was a pa ape wi h a a iable heigh o ex ension a io o h /B ∈ [0.5, 1].
Figu e 15. Dependence o he ela i e o e opping a e on he nondimensionalised mean low
hickness: (a) i ing line and (b) quali y o he i ing.
The ela ionship be ween nondimensionalised low hickness and ela i e o e opping
a e in his case was: Q
pgH3D=−0.00229 +0.5009·δ
H(20)
He e, in he linea eg ession, a nega i e independen e m was allowed (Figu e 15a),
add essing he ac ha , e en a a e y low o e opping discha ge close o ze o, a ce ain
wa e laye exis ed o e he c es wi h e y low low eloci y. This e ec could be no iced
by simul aneously analysing he low hickness and he co esponding o e opping a e,
as shown in Figu e 10. The co ela ion o he p edic ed alues by linea eg ession wi h he
co esponding alues coming om he nume ical calcula ion was easonably accep able
(Figu e 15b), wi h a coe icien o de e mina ion o R
2
= 0.9661, by aking in o accoun he
high dispe sion a ibu ed o his ype o phenomenon.
5. Conclusions
In he p esen s udy, a compu a ional model was used o cha ac e ise he o e opping
educ ion p oduced by di e en geome ic op ions o o e opping mi iga ion s uc u es
ins alled a he op o a e e ence ixed cylinde . The selec ed mi iga ion s uc u e was a
pa ape wi h a a iable heigh o ex ension a io o h /B ∈[0.5, 1].
The compu a ional model ollowed he same con igu a ion used o cha ac e ise a plane
cylinde in p e ious esea ch, which was used as he basis o he p esen s udy. The wa e
cha ac e is ics we e he same as o he e e ence cylinde , belonging o he non-impulsi e
egime wi hou in luence o he o esho e in he in e media e wa e dep h ange.
A o al o 96 indi idual compu a ional cases we e un wi h ou di e en pa ape s, wi h
di e en ela i e eeboa d alues anging om 0.143 o 0.778. In he same way as o linea
de ence e ical walls, he inclusion o a pa ape led o a compa a i e ne educ ion o he
o e opping discha ge, and he esul was success ully modelled by a Weibull piecewise unc ion.
This exp ession epo ed ha he o e opping educ ion p oduced by he pa ape became e y
no iceable in he in e media e ela i e eeboa d egime, o alues o Rc/H> 0.5.
The nondimensionalised mean low hickness was calcula ed a he cen e o he c es
o he s uc u e by indica ing a di ec ela ionship wi h he o e opping discha ge. This low
hickness pa ame e was educed in ela ion o he e e ence cylinde wi hou a pa ape
when compa ing he same ela i e eeboa d cases. A di ec ela ionship be ween he
nondimensionalised mean low hickness and he ela i e o e opping a e was de e mined,
showing a good al e na i e o indi ec es ima ion o he o e opping discha ge wi hou
needing a complex quan i ica ion o o e opped olume in eal ull scale s uc u es.
J. Ma . Sci. Eng. 2024,12, 1441 20 o 22
Among he di e en geome ic op ions, he selec ion o g ea e pa ape ex ension B
in compa ison wi h he pa ape heigh h
seemed o p oduce a sligh ly highe pe o mance
in o e opping educ ion. Howe e , his di e ence did no p o e a meaning ul dis inc ion
conside ing he s a is ical dispe sion o he esul s, and a unique o mula ion was a ibu ed
o he g oup o pa ape s conside ed in his s udy.
This s udy may be conside ed he basis o u u e cha ac e isa ion o he o e opping
phenomenon in a mo e complex en i onmen whe e cylind ical componen s used in
loa ing o sho e s uc u es a e exposed o ex eme wa es and, as a consequence, mo ing
wi h he six deg ees o eedom bu cons ained by he moo ing sys em a he same ime.
Au ho Con ibu ions: Concep ualisa ion, me hodology and supe ision, G.A.E.; so wa e, in es iga ion,
da a cu a ion and w i ing—o iginal d a p epa a ion, X.E.; o mal analysis, X.E. and G.A.E.; esou ces and
isualisa ion, I.B. and I.A.; w i ing— e iew and edi ing G.A.E. and U.I.; p ojec adminis a ion, G.A.E. and
U.I. All au ho s ha e ead and ag eed o he published e sion o he manusc ip .
Funding: This wo k was ca ied ou wi hin he amewo k o he ITSAS-REM Resea ch G oup
(IT-1514-22) unded by he Basque Go e nmen .
Acknowledgmen s: Technical and human suppo p o ided by IZO-SGI, SGIke (UPV/EHU,
MICINN, GV/EJ, ERDF and ESF) is g a e ully acknowledged, as along wi h he suppo p o ided by
he Join Resea ch Labo a o y on O sho e Renewable Ene gy (JRL-ORE).
Con lic s o In e es : The au ho s decla e no con lic o in e es .
Re e ences
1.
Taglia ie o, B.; Ka imi ad, M.; Al oma e, C.; Gö eman, M.; Ma ínez-Es é ez, I.; Capasso, S.; Domínguez, J.M.; Viccione, G.;
Gómez-Ges ei a, M.; C espo, A.J.C. Nume ical Valida ions and In es iga ion o a Semi-Subme sible Floa ing O sho e Wind
Tu bine Pla o m In e ac ing wi h Ocean Wa es Using an SPH F amewo k. Appl. Ocean Res. 2023,141, 103757. [C ossRe ]
2.
Zhang, Y.; Xu, H.; Law, Y.; San o, H.; Magee, A. Hyd odynamic Analysis and Valida ion o he Floa ing DeepCwind Semi-
Subme sible unde 3-h I egula Wa e wi h he HOS and CFD Coupling Me hod. Ocean Eng. 2023,287, 115701. [C ossRe ]
3.
Zhou, Y.; Xiao, Q.; Liu, Y.; Incecik, A.; Pey a d, C.; Li, S.; Pan, G. Nume ical Modelling o Dynamic Re-sponses o a Floa ing
O sho e Wind Tu bine Subjec o Focused Wa es. Ene gies 2019,12, 3482. [C ossRe ]
4.
an de Mee , J.W.; Ve haeghe, H.; S eendam, G.J. The new wa e o e opping da abase o coas al s uc u es. Coas . Eng. 2009,56,
108–120. [C ossRe ]
5.
Pullen, T.; Allsop, N.W.H.; B uce, T.; Ko enhaus, A.; Schü ump , H.; an de Mee , J.W. Eu o op, Wa e O e opping o Sea De ences
and Rela ed S uc u es: Assessmen Manual; Ku a o ium ü Fo schung im Küs eningenieu wesen, Boyens Medien GmbH & Co.
KG,: Heide im Hols ein, Ge many, 2007.
6.
Van de Mee , J.W.; Allsop, N.W.H.; B uce, T.; De Rouck, J.; Ko ehaus, A.; Pullen, T.; Schümp ump , H.; T och, P.; Zanu igh,
B. Eu O op Manual on Wa e O e opping o Sea De ences and Rela ed S uc u es an O e opping Manual La gely Based on
Eu opean Resea ch, Bu o Wo ldwide Applica ion, 2nd ed. 2018. A ailable online: www.o e opping-manual.com (accessed on
10 Augus 2024).
7.
Izquie do, U.; Es eban, G.A.; Blanco, J.M.; Albaina, I.; Peña, A. Expe imen al alida ion o a CFD model using a na ow wa e
lume. Appl. Ocean Res. 2019,86, 1–12. [C ossRe ]
8.
Chen, X.; Ho land, B.; Al oma e, C.; Suzuki, T.; Uij ewaal, W. Fo ces on a e ical wall on a dike c es due o o e opping low.
Coas . Eng. 2015,95, 94–104. [C ossRe ]
9.
Lu, X.; Zhou, G.G.D.; Cui, K.F.E.; Tang, H.; Xie, Y. O e opping olume o impulse wa es in glacie lakes: Expe imen al and
nume ical in es iga ion using igid dams. Eng. Geol. 2022,306, 106763. [C ossRe ]
10.
Wuppukondu , A.; Baldock, T.E. Physical and nume ical modelling o ep esen a i e sunami wa es p opaga ing and o e opping
in con e ging channels. Coas . Eng. 2022,174, 104120. [C ossRe ]
11.
Zago, V.; Dal ymple, R.A.; Almashan, N.; Bilo a, G.; Al-Hou i, D.; Neelamani, S. Cha ac e iza ion and modeling o g eenwa e
o e opping o a sea-le el deck. Ocean Eng. 2023,275, 114131. [C ossRe ]
12.
Dang, B.L.; Nguyen-Xuan, H.; Abdel Wahab, M. Nume ical s udy on wa e o ces and o e opping o e a -ious seawall
s uc u es using ad anced SPH-based me hod. Eng. S uc . 2021,226, 111349. [C ossRe ]
13.
Hi , C.W.; Nichols, B.D. Volume o Fluid (VOF) Me hod o he Dynamics o F ee Bounda ies. J. Compu . Phys. 1981,39, 201–225.
[C ossRe ]
14.
an Be geijk, V.M.; Wa mink, J.J.; Hulsche , S.J.M.H. The e ec s o ansi ions in co e ype and heigh on he wa e o e opping
load on g ass-co e ed lood de ences. Appl. Ocean Res. 2022,125, 103220. [C ossRe ]
15.
Buccino, M.; Di Leo, A.; Tuozzo, S.; Cò do a Lopez, L.F.; Clab ese, M.; Den ale, F. Wa e o e opping o a e ical seawall in a su
zone: A join analysis o nume ical and labo a o y da a. Ocean Eng. 2023,288, 116144. [C ossRe ]
J. Ma . Sci. Eng. 2024,12, 1441 21 o 22
16.
Sha i ian, M.; Moghim, M.N.; Chamani, M.R. Nume ical simula ion o wa es o e opping o e impe meable sloped seadikes.
Ocean Eng. 2022,266, 113100. [C ossRe ]
17.
S agni i, M.; La a, J.L.; Musumeci, R.E.; Fo i, E. Nume ical modeling o wa e o e opping o damaged and up-g aded ubble-
mound b eakwa e s. Ocean Eng. 2023,280, 114798. [C ossRe ]
18.
I ías Ma a, M.; an Gen , M.R.A. Nume ical modelling o wa e o e opping discha ges a ubble mound b eakwa e s using
OpenFOAM®.Coas . Eng. 2023,181, 104274. [C ossRe ]
19.
Vannes e, D.F.A.; Al oma e, C.; Suzuki, T.; T och, P.; Ve waes , T. Compa ison o nume ical models o wa e o e opping and
impac on a sea wall. Coas . Eng. P oc. 2014,1, 5. [C ossRe ]
20.
Zhou, M.; Shi, Z.; Peng, C.; Peng, M.; Cui, K.F.E.; Li, B.; Zhang, L.; Zhou, G.G.D. Two-phase modelling o e osion and deposi ion
p ocess du ing o e opping ailu e o landslide dams using GPU-accele a ed ED-SPH. Compu . Geo ech. 2024,166, 105944.
[C ossRe ]
21.
an de Mee , J.; B uce, T. New Physical Insigh s and Design Fo mulas on Wa e O e opping a Sloping and Ve ical S uc u es. J.
Wa e w. Po Coas . Ocean Eng. 2014,140, 04014025. [C ossRe ]
22.
Zanu igh, B.; Fo men in, S.M.; an de Mee , J.W. P edic ion o ex eme and ole able wa e o e opping dis-cha ges h ough an
ad anced neu al ne wo k. Ocean Eng. 2016,127, 7–22. [C ossRe ]
23.
Zanu igh, B.; Fo men in, S.M. Reduc ion o he wa e o e opping discha ge a dikes in p esence o c own walls wi h bullnoses.
Coas . Eng. P oc. 2018,1, 110. [C ossRe ]
24.
Fo men in, S.M.; Palma, G.; Zanu igh, B. In eg a ed assessmen o he hyd aulic and s uc u al pe o mance o c own walls on
op o smoo h be ms. Coas . Eng. 2021,168, 103951. [C ossRe ]
25.
Van Doo slae , K.; De Rouck, J. Wa e o e opping o e sea dikes and impac o ces on s o m walls. In P oceedings o he Nin h
In e na ional Con e ence on Coas al and Po Enginee ing in De eloping Coun ies—PIANC COPEDEC IX, Río de Janei o, B azil,
16–21 Oc obe 2016.
26.
Van Doo slae , K.; Romano, A.; De Rouck, J.; Ko enhaus, A. Impac s on a s o m wall caused by non-b eaking wa es o e opping
a smoo h dike slope. Coas . Eng. 2017,120, 93–111. [C ossRe ]
27.
Ma inelli, L.; Ruol, P.; Volpa o, M.; Fa a e o, C.; Cas ellino, M.; De Gi olamo, P.; F anco, L.; Romano, A.; Samma co, P.
Expe imen al in es iga ion on non-b eaking wa e o ces and o e - opping a he ecu ed pa ape s o e ical b eakwa e s. Coas .
Eng. 2018,141, 52–67. [C ossRe ]
28.
Cas ellino, M.; Samma co, P.; Romano, A.; Ma inelli, L.; Ruol, P.; F anco, L.; De G iloamo, P. La ge impulsi e o ces on ecu ed
pa ape s unde non-b eaking wa es. A nume ical s udy. Coas . Eng. 2018,136, 1–15. [C ossRe ]
29.
Cas ellino, M.; Romano, A.; La a, J.L.; Losada, I.J.; De Gi olamo, P. Con ined-c es impac : Fo ces dimensional analysis and
ex ension o he Goda’s o mulae o ecu ed pa ape s. Coas . Eng. 2021,163, 103814. [C ossRe ]
30.
Suzuki, T.; Al oma e, C.; Veale, W.; Ve waes , T.; T ouw, K.; T och, P.; Zijlema, M. E icien and obus wa e o e opping es ima ion
o impe meable coas al s uc u es in shallow o esho es using SWASH. Coas . Eng. 2017,122, 108–123. [C ossRe ]
31.
an Gen , M.R.A.; Wol e s, G.; Capel, A. Wa e o e opping discha ges a ubble mound b eakwa e s includ-ing e ec s o a c es
wall and a be m. Coas . Eng. 2022,176, 104151. [C ossRe ]
32.
E emad-Shahidi, A.; Koosheh, A.; an Gen , M.R.A. On he mean o e opping a e o ubble mound s uc u es. Coas . Eng. 2022,
177, 104150. [C ossRe ]
33.
Van Doo slae , K.; De Rouck, J. Reduc ion on wa e o e opping on a smoo h dike by means o a pa ape . Coas . Eng. P oc. 2011,1,
6. [C ossRe ]
34.
Ko enhaus, A.; Pea son, J.; B uce, T.; Allsop, N.W.H.; an de Mee , J.W. In luence o pa ape s and ecu es on wa e o e opping
and wa e loading o complex e ical walls. In P oceedings o he Coas al S uc u es 2003, Po land, OR, USA, 26–30 Augus
2003; pp. 369–381. [C ossRe ]
35.
De Chowdhu y, S.; Anand, K.V.; Sannasi aj, S.A.; Sunda , V. Nonlinea wa e in e ac ion wi h cu ed on sea-walls. Ocean Eng.
2017,140, 84–96. [C ossRe ]
36.
Oh, S.-H.; Jang, S.-C.; Lee, J. Wa e O e opping and Loading o he Recu ed Pa ape s on he C es o Rubble Mound B eakwa e ; Thomas
Tel o d L d.: London, UK, 2018; pp. 979–985. [C ossRe ]
37.
T ung, L.H.; Linh, D.T.; Tho, T.X.; Duy, N.T.; Tung, T.T. Wa e o e opping and splash-up a seawalls wi h bullnose. Vie nam. J. Sci.
Technol. 2020,20, 333–342. [C ossRe ]
38.
Kisacik, D.; Ta akcioglu, G.O.; Cappie i, L. Adap a ion measu es o seawalls o wi hs and sea-le el ise. Ocean Eng. 2022,250,
110958. [C ossRe ]
39.
Pea son, J.; B uce, T.; Ko enhaus, A.; an de Mee , J. E ec i eness o ecu e walls in educing wa e o e opping on seawalls
and b eakwa e s. Coas . Eng. 2004,4, 4404–4416. [C ossRe ]
40.
Es eban, G.A.; A is ondo, A.; Izquie do, U.; Blanco, J.M.; Pé ez-Mo an, G. Expe imen al analysis and nume ical simula ion o
wa e o e opping on a ixed e ical cylinde unde egula wa es. Coas . Eng. 2022,173, 104097. [C ossRe ]
41. F anco, L. Ve ical b eakwa e s: The I alian expe ience. Coas . Eng. 1994,22, 31–55. [C ossRe ]
42.
Allsop, W.; B uce, T.; Pea son, J.; Besley, P. Wa e o e opping a e ical and s eep seawalls. P oc. Ins . Ci . Eng. Ma i . Eng. 2005,
158, 103–114. [C ossRe ]
J. Ma . Sci. Eng. 2024,12, 1441 22 o 22
43.
Hubne , R.G.; F agassa, C.; Pai a, M.D.S.; Oleinik, P.H.; Gomes, M.D.N.; Rocha, L.A.; San os, E.D.D.; Machado, B.N.; Isoldi, L.A.
Nume ical Analysis o an O e opping Wa e Ene gy Con e e Subjec ed o he Incidence o I egula and Regula Wa es om
Realis ic Sea S a es. J. Ma . Sci. Eng. 2022,10, 1084. [C ossRe ]
44.
Gal án, J.; Sánchez-La a, M.J.; Mendikoa, I.; Pé ez-Mo án, G.; Na a, V.; Rod íguez-A ias, R. NAUTI-LUS-DTU10 MW Floa ing
O sho e Wind Tu bine a Gul o Maine: Public Nume ical Models o an Ac- i ely Ballas ed Semisubme sible. In P oceedings o
he WindEu ope Con e ence 2018 wi hin he Global Wind, Hambu g, Ge many, 25–28 Sep embe 2018; Volume 1102. [C ossRe ]
45. U sell, F. The long-wa e pa adox in he heo y o g a i y wa es. Ma h. P oc. Camb. Philos. Soc. 1953,49, 685–694. [C ossRe ]
46.
Chak aba i, S.K. O sho e S uc u e Modeling. In Applied Ocean Resea ch, Ad anced Se ies on Ocean Enginee ing; Wo ld Scien i ic
Publishing: Singapo e, 1994. [C ossRe ]
47. Fen on, J.D. A i h-o de S okes heo y o s eady wa es. J. Wa e w. Po Coas . Ocean Eng. 1985,111, 216–234. [C ossRe ]
48.
Wind , C.; Da idson, J.; Schmi , P.; Ringwood, J. On he assessmen o nume ical wa e make s in CFD simula ions. J. Ma . Sci.
Eng. 2019,7, 47. [C ossRe ]
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