Modelling Wa e In e ac ion wi h De o mable S uc u es based on a
Mul i- egion App oach wi hin OpenFOAM
P. J. Ma ´ınez Fe e , L. Qian, Z. Ma, D. Causon, C. Mingham
Cen e o Ma hema ical Modelling and Flow Analysis
Manches e Me opoli an Uni e si y
Ches e S ee , Manches e M1 5GD, Uni ed Kingdom
ABSTRACT
This pape p esen s he de elopmen o a mul i- egion compu a ional
luid-s uc u e dynamics (CFSD) me hod which is in eg a ed in ou i -
ual wa e s uc u e in e ac ion sol e wsiFoam, based on he open-sou ce
OpenFOAM lib a y, in o de o accoun o he hyd o-elas ic e ec s p o-
duced by iolen wa e impac s agains de o mable bodies. This s a -
egy elies en i ely on he ini e olume me hod (FVM) and does no e-
qui e any hi d-pa y sol e s, which ende s i sui able o e icien pa -
allel compu ing. We alida e his no el app oach agains p e ious ex-
pe imen al and nume ical esul s co esponding o a dam b eak o wa e
impac ing on a highly de o mable pla e as well as a lexible wedge en-
e ing wa e a a cons an speed. In gene al, ou p elimina y esul s ag ee
quali a i ely well wi h p e ious da a whils he pe o mance o pa allel
implemen a ion e idences he po en ial o his me hod o be used in u-
u e high pe o ming compu ing (HPC) applica ions.
KEY WORDS: luid s uc u e in e ac ion, hyd oelas ici y, wa e impac
INTRODUCTION
Wa e impac s a e cha ac e is ic o ma ine enginee ing p oblems. Floa -
ing pla o ms, ships and LNG anks can expe ience equen wa e im-
pac s in high sea s a es. The local and s ochas ic na u e o wa e impac
loads due o ee su ace ins abili ies makes his p oblem di icul o s udy
bo h expe imen ally and compu a ionally. Fo example, ai en apmen
may occu when he impac angle be ween he wa e and he s uc u e
is small. Conside ing only luid dynamics aspec s, he ai comp essibil-
i y may play an impo an ole in such condi ions leading o a u he
inc ease in he maximum impac p essu e (Bullock e al. 2007, Lugni
e al. 2010, Ma e al. 2016), which can comp omise he in eg i y o he
s uc u e.
On he o he hand, wa e impac is o en a luid-s uc u e in e ac ion p ob-
lem be ween he ee su ace low and an elas ic s uc u e. A ecen ex-
pe imen ca ied ou using igid (s eel) and elas ic (aluminium) pla es
showed he impo an ole o hyd o-elas ici y du ing he e olu ion o a
lip- h ough e en in a low illed sloshing ank (Lugni e al. 2014). When
he elas ic pla e de o ms agains he incoming wa e, he impac p essu e
inc eases almos by a ac o o 2 wi h espec he case whe e a igid pla e
is used; subsequen p essu e oscilla ions a e also p oduced, hus e eal-
ing a s ong hyd o-elas ic e ec .
The as majo i y o he compu a ional wo ks ela ed o wa e impac s
s ill concen a e on he luid dynamics beha iou on igid s uc u es
hus neglec ing he impo an hyd o-elas ic e ec s. Di e en app oaches
o deal wi h luid-s uc u e in e ac ion (FSI) and o accoun o hyd o-
elas ici y can be ound in he li e a u e (Hou e al. 2012). T adi ionally,
FSI in ol es a combina ion o h ee ools: a ini e olume sol e o he
luid, a ini e elemen sol e o he s uc u e and, inally, a hi d code
o couple he p e ious wo sol e s and manage he da a exchange be-
ween hem. This me hodology su e s om a lo o limi a ions such as
model se up and coupling. Fu he mo e, wi h he con inuous g ow h o
la ge scale simula ions and HPC applica ions, hese s a egies become
apidly obsole e, especially i he da a exchange is no ca ied ou using
as andom-access memo y.
Al hough he ini e elemen me hod is widely used o s uc u al analy-
sis (Ba he & Hahn 1979), he FVM has been gaining popula i y o sol e
compu e s uc u e dynamics (CSD) p oblems due o i s memo y e i-
ciency (Jasak & Welle 2000). The p esen wo k elies on he FVM
p o ided by OpenFOAM o sol e bo h he luid and solid in an uni ied,
pa i ioned amewo k in o de o accoun o hyd oelas ici y phenom-
ena whils allowing he possibili y o use HPC esou ces o la ge scale
applica ions.
The p esen pape is o ganised as ollows. We i s ly desc ibe he nume i-
cal me hods and he implemen a ion o he FSI s a egy o he simula ion
o iolen wa e impac s cha ac e is ic o ocean and coas al enginee ing
applica ions. Then we es wo expe imen al con igu a ions co espond-
ing o (i) a dam b eak o wa e impac ing on an elas ic pla e and (ii) a
lexible wedge en e ing wa e a a cons an speed. Conclusions and u -
he wo k a e p o ided a he end o his documen .
Fig. 1 Mul i- egion i ual wa e s uc u e in e ac ion (WSI) simula ion en i onmen .
NUMERICAL METHOD
Ou nume ical p ocedu es ely on a cell-cen ed, co-loca ed ini e olume
me hod, a ailable in he open-sou ce CFD so wa e OpenFOAM (Jasak
1996). We ha e p e iously de eloped a no el OpenFOAM-based sol e ,
wsiFoam (Ma ´
ınez Fe e e al. 2016a), o he s udy o wa e in e ac-
ion be ween igid s uc u es and loa ing bodies (Ma ´
ınez Fe e e al.
2016b). The aim o wsiFoam is o ga he specialised sol e s, e.g. ully
non-linea po en ial (FNLP), incomp essible Na ie -S okes (INS) and
comp essible Na ie -S okes (CNS) sol e s, a s a egic loca ions o he
compu a ional domain and couple hem h ough special in e aces in o -
de o ge he mos e icien and accu a e desc ip ion o he unde lying
physics in a nume ical wa e ank, as schema ised in Fig. 1.
We sol e he luid wi h he aid o an incomp essible wo-phase p essu e-
based sol e (Rusche 2002), which is based on he olume o luid (VOF)
me hod o desc ibe he wo-phase luid mix u e, i.e. ai and wa e , as-
sumed o be homogeneous and in mechanical equilib ium, i.e. iden ical
eloci y and p essu e. The mass balance equa ion o he wa e olume
ac ion α∈[0,1] is gi en by
∂α
∂ +∇ · Uα+∇ · Ucα(1 −α)=0,(1)
whe e Uis he mix u e eloci y ec o and Uc=min[U,max(U)]. The
densi y o he mix u e is ρ=αρw+(1 −α)ρa;ρwand ρaa e he cons an
pa ial densi ies o wa e and ai , espec i ely. The hi d e m in eq. (1) is
an a i icial comp ession quan i y ha sha pens he in e ace and gua an-
ees bounded alues o αby using he MULES p ocedu e (Welle 2002).
The single momen um equa ion o he homogeneous mix u e is w i en
as
∂ρU
∂ +∇ · (ρUU)−∇ · (µ∇U)=σκ∇α−g·x∇ρ−∇pd,(2)
whe e σdeno es he su ace ension coe icien and κ=∇ · (∇α/|∇α|)
ep esen s he cu a u e o he in e ace. The mix u e iscosi y is gi en
by µ=αµw+(1 −α)µaand he dynamic p essu e is calcula ed as
pd=p−ρg·xwi h gand x he g a i y and posi ion ec o s, espec-
i ely. Finally, he go e ning equa ions (1)–(2) a e linea ised and in e-
g a ed o e each con ol olume o de e mine αand U, espec i ely, and a
p essu e co ec o linea ised equa ion is sol ed o pd. This solu ion p o-
cedu e elies on he seg ega ed p ojec ion algo i hm PIMPLE (Kissling
e al. 2010).
The solid equa ions a e sol ed using he FVM s a egy desc ibed
by Jasak & Welle (2000), Tuko ic & Jasak (2007), Tuko ic e al. (2013).
This app oach cons i u es a as and memo y-e icien al e na i e o well-
es ablished ini e elemen sol e s and can be easily pa allelised o HPC
applica ions. The s uc u e is assumed o be elas ic and comp essible
and he equa ion o he displacemen ec o uw i en wi h espec o he
ini ial, i.e. unde o med, con igu a ion is
ρs
∂2u
∂ 2=∇ · ΣFT,(3)
whe e ρsis he solid densi y, F=I+(∇u)T he de o ma ion g adien
enso , Σ=2µsE+λs (E)I he second Piola-Ki chho s ess enso and
E=(FT·F−I)/2 ep esen s he G een-Lag angian s ain enso . The
Lam´
e coe icien s a e de ined as µs=E/[2(1 +νs)] and λs=νsE/[(1 +
νs)(1 −2νs)], espec i ely, whe e Edeno es he Young’s modulus and νs
he Poisson a io. Fo u he de ails see Tuko ic & Jasak (2007).
Algo i hm 1: Ai ken’s unde - elaxa ion me hod.
begin ime ad ancemen
while |u −us|> do
es ima e in e ace displacemen (Ai ken);
mo e luid mesh (u );
sol e luid equa ions;
upda e solid bounda y condi ions;
sol e solid equa ion (us);
end
upda e he simula ion ime;
end
We u ilise Ai ken’s unde - elaxa ion me hod p esen ed in Algo i hm 1
o gua an ee a s ongly coupled pa i ioned FSI implemen a ion. The
luid and solid meshes a e ea ed as sepa a ed egions sha ing common
in e aces wi h app op ia e bounda y condi ions. In his con igu a ion,
luid and solid egions a e sol ed al e na ely un il dynamic equilib ium
is eached. Ai ken’s me hod is used o es ima e he in e ace displace-
men and mo e he luid mesh acco dingly. Once he luid equa ions a e
calcula ed in his new mesh, he luid o ce is ans e ed o he s uc u al
sol e . The solid sol e calcula es he displacemen o he in e ace and
ans e s i back o he luid side. This i e a i e p ocess epea s un il he
di e ence be ween he luid and solid displacemen s mee s a ole ance
c i e ia. A e his, he ime is upda ed and his p ocedu e epea s.
Fig. 2 Dam b eak impac ing on an elas ic pla e: compu a ional
domain o H0=0.3 m.
DAM BREAK IMPACTING ON AN ELASTIC PLATE
This expe imen has been ca ied ou a he RIAM labo a o y by Liao
e al. (2014, 2015) and consis s o an elas ic pla e si ua ed a he igh
hand side o a wa e ank and a column o wa e a he le hand side o
a iable heigh , i.e. H0=0.2 m, H0=0.3 m and H0=0.4 m, see Fig. 2.
A =0 s a ga e holding he column o wa e opens and he gene a ed
low impac s he pla e, which begins o bend. The dimensions o he ank
a e 0.8×0.6×0.2 m3and he dis ance be ween he elas ic pla e and he
igh wall o he ank is 0.2 m. The pla e ea u es a hickness o 0.004 m
and a heigh o 0.09 m. The wo-dimensional luid mesh is disc e ised
wi h 82 ×59 ×1 cells whils he solid mesh is cons i u ed o 2 ×18 ×1
cells. The mesh is e ined nea he bo om wall and s e ched abo e a
heigh o 0.4 m o sa e compu a ional esou ces. Bounda ies a e se as
ollows: he walls o he ank and he pla e sha e a non-slip condi ion
and he op bounda y emains open o he a mosphe e; he mo ion e ec
o he opening ga e on he wa e column is no aken in o accoun in his
s udy. The p ope ies o he pla e, i.e. he solid, as well as wa e and ai ,
i.e. he luid, a e summa ised in Table 1. Finally, he simula ion is un o
=1 s wi h a ixed ime s ep o 10−4s.
Table 1 Dam b eak impac ing on an elas ic pla e: solid and luid
p ope ies. SI uni s.
P ope y Solid Wa e Ai
Densi y 1161.54 997.0 1.225
K. iscosi y – 0.89 ×10−60.82 ×10−5
Young’s modulus 3.5×106– –
Poisson a io 0.48 – –
Figs. 3–5 show snapsho s aken a di e en imes compa ing he expe i-
men s o Liao e al. (2014, 2015) agains ou nume ical esul s ob ained
wi h wsiFoam. La ge di e ences can be obse ed in Fig. 3 co espond-
ing o H0=0.2 m. A =0.32 s he elas ic pla e do no exhibi a second
mode o de lec ion and he o al displacemen o he pla e emains un-
de es ima ed om ha ime onwa d. These di e ences a e somewha
educed when H0=0.3 m and we ob ain a good quali a i e ag eemen
o H0=0.4 m. The nume ical esul s o Liao e al. (2015) also showed
non-negligible quali a i e disc epancies compa ed o he expe imen s. I
is also wo h men ioning ha we we e no able o cap u e highe o de s
o ib a ion obse ed du ing he expe imen s.
Figs. 6–8 show he ime his o y o he ho izon al displacemen measu ed
a he ip o he elas ic pla e, co esponding o he ed do in he sequence
o images aken om he expe imen s. The di e ences discussed abo e
become e iden o H0=0.2 m, showing a maximum disc epancy o ap-
p oxima ely 2 cm (36%) be ween he ho izon al displacemen measu ed
in he expe imen and he simula ion. Fo H0=0.3 m his di e ence e-
duces o 15% and we ge close o he nume ical solu ion o Liao e al.
(2015), which also unde es ima es he expe imen s. Indeed, bo h nume -
ical solu ions emain qui e simila a e =0.35 s. The las case co e-
sponding o H0=0.4 m shows he bes ag eemen be ween expe imen s
and nume ical simula ions. The cases wi h H0=0.2 m and H0=0.3 m
become mo e di icul o simula e due o he p esence o complex and
u bulen low s uc u es as well as he de elopmen o ai ca i ies a e
he ini ial impac .
Finally, a nega i e displacemen in he o m o a small bump can be ob-
se ed in all he expe imen al cu es and is associa ed o he ini ial im-
pac o he wa e on agains he bo om o he elas ic pla e. As a con-
sequence, he ip o he pla e lexes owa ds he le as shown in Fig. 5 a
=0.25 s. Ou p esen esul s end o unde es ima e his ea ly displace-
=0.52 s
=0.47 s
=0.42 s
=0.37 s
=0.32 s
=0.27 s
Fig. 3 Dam b eak impac ing on an elas ic pla e (H0=0.2 m):
snapsho compa ison be ween he expe imen and simu-
la ion; he blue o ed ainbow pale e ep esen s en ee
su ace iso-con ou s (0.4≤α≤0.6).
men . Liao e al. (2015) ca ied ou wo se s o simula ions, wi h and
wi hou conside ing he in luence o he ga e mo ion, and showed an sig-
ni ican imp o emen in hei nume ical esul s when his in luence was
aken in o accoun , cap u ing accu a ely nega i e displacemen s. The e-
=0.50 s
=0.45 s
=0.40 s
=0.35 s
=0.30 s
=0.25 s
Fig. 4 Dam b eak impac ing on an elas ic pla e (H0=0.3 m):
snapsho compa ison be ween he expe imen and simu-
la ion; he blue o ed ainbow pale e ep esen s en ee
su ace iso-con ou s (0.4≤α≤0.6).
o e we can expec a gene al imp o emen in ou esul s by conside ing
he in luence o he ga e.
=0.39 s
=0.35 s
=0.33 s
=0.31 s
=0.29 s
=0.25 s
Fig. 5 Dam b eak impac ing on an elas ic pla e (H0=0.4 m):
snapsho compa ison be ween he expe imen and simu-
la ion; he blue o ed ainbow pale e ep esen s en ee
su ace iso-con ou s (0.4≤α≤0.6).
WATER ENTRY OF RIGID AND ELASTIC WEDGES
The wa e en y o bodies, and in pa icula wedges, has been widely
s udied, see o ins ance Gu e al. (2014). An ex ensi e analysis compa -
ing simila i y, asymp o ic and nume ical solu ions o igid wedges en-
∆x[m]
-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.2 0.25 0.3 0.35 0.4 0.45 0.5
P esen
Liao e al. (2015)
Expe imen
[s]
Fig. 6 Dam b eak impac ing on an elas ic pla e (H0=0.2 m):
ime his o y o he pla e’s ip ho izon al displacemen .
∆x[m]
-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.2 0.25 0.3 0.35 0.4 0.45 0.5
P esen
Liao e al. (2015)
Expe imen
[s]
Fig. 7 Dam b eak impac ing on an elas ic pla e (H0=0.3 m):
ime his o y o he pla e’s ip ho izon al displacemen .
∆x[m]
-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.2 0.25 0.3 0.35 0.4 0.45 0.5
P esen
Liao e al. (2015)
Expe imen
[s]
Fig. 8 Dam b eak impac ing on an elas ic pla e (H0=0.4 m):
ime his o y o he pla e’s ip ho izon al displacemen .
e ing wa e wi h cons an speed can be ound in he ea ly wo k o Zhao
& Fal insen (1993). This s udy was la e ex ended by Lu e al. (2000),
who conside ed he luid s uc u e in e ac ion o he equi alen elas ic
wedge. In hese wo wo ks he luid was ea ed as a po en ial low and
Fig. 9 Wedge wa e en y: compu a ional domain.
hus he in luence o he ai was no aken in o accoun . Liao e al. (2013)
e isi ed his classic luid s uc u e in e ac ion p oblem and conside ed
bo h igid and elas ic wedges as well as he p esence o ai by sol ing he
Na ie -S okes equa ions o a mul i-phase luid.
Table 2 Wedge wa e en y: solid and luid p ope ies. SI uni s.
P ope y Solid Wa e Ai
Densi y 7800 1000 1.225
K. iscosi y – 1.0×10−60.82 ×10−5
Young’s modulus 2.0×109– –
Poisson a io 0.3 – –
The wo-dimensional compu a ional domain o dimensions 0.95 ×1.5×
1 m3is shown in Fig. 9, which ep esen s only hal o he p oblem due
o symme y. Following Lu e al. (2000), he o al heigh o he wedge is
0.3 m whils he bo om measu es 0.4 m and o ms an angle o 30◦wi h
espec o he calm wa e su ace. When conside ing a igid wedge, FSI is
deac i a ed and we only sol e he luid equa ions. O he wise, he wedge
bo om consis s o a de o mable pla e cons ained a i s ex emi ies and
ea u ing h ee possible hicknesses: 5 mm, 8 mm (shown in Fig. 9) and
11 mm. All he walls o he wedge ha e a non-slip condi ion and he
bo om and le domain bounda ies ha e speci ied a cons an alue o
he eloci y U=(0,1,0) m/s. A mosphe ic p essu e is se a he op
bounda y and he igh domain bounda ies sha e a symme y condi ion.
The p ope ies o he solid and luid ma ch hose speci ied by Liao e al.
(2013), see Table 2. Finally, g a i y e ms a e no conside ed in his
s udy, which is un up o 0.2 s wi h a ixed ime s ep o 10−5s.
The esul s issued om he i s se o simula ions, co esponding o he
igid wedge, allowed us o de e mine he app op ia e mesh disc e isa ion
in o de o gua an ee he con e gence o ou nume ical solu ions. Fig. 10
shows he no malised p essu e dis ibu ion along he wedge bo om o
h ee mesh con igu a ions. Ou nume ical esul s ob ained by sol ing he
mul i-phase Na ie -S okes equa ions a e compa ed agains p e ious sim-
ila i y and analy ical solu ions based on po en ial heo y (Lu e al. 2000).
I can be eadily seen ha he mos e ined mesh, ea u ing 11560 cells,
p o ides he bes ag eemen . This body- i ed mesh is e ined nea he
wedge bo om (∆x≈2.5 mm) and s e ched owa ds he ou e bounda ies
whe e ∆x≈8 cm. In his con igu a ion, he solu ion also shows a small
bump nea he ee-su ace, which is caused by low sepa a ion. This be-
ha iou was p e iously epo ed by Liao e al. (2013), who associa ed i
o g a i a ional e ms. Howe e , we suspec ha low de achmen may be
a consequence o he ai mo ion and nume ical ins abili ies accumula ed
nea he in e ace, as we did no conside ed g a i y in ou simula ions.
2(p−p0)/(ρV2
0) [–]
-1
0
1
2
3
4
5
6
7
8
-1 -0.5 0 0.5 1
3420
6920
11560
Liao e al. (2013)
Simila i y
Analy ic
l/(V0 ) [–]
Fig. 10 Wedge wa e en y ( igid case): dis ibu ion o he p es-
su e ield along he wedge bo om o a di e en numbe
o mesh cells.
The nex se o simula ions accoun ing o an elas ic wedge we e ca ied
ou using he same ine mesh o 11560 cells o he luid domain whils
he solid was disc e ised wi h 160 cells. Th ee cases, co esponding o
di e en pla e hicknesses, we e calcula ed. Figs. 11–13 epo on he
s uc u al de lec ion o he wedge’s bo om middle poin . Ou nume ical
esul s show simila ends compa ed o hose a ailable in he li e a u e.
Maximum de lec ion alues a e eached a nea ly simila imes and di-
minish as he pla e becomes hicke . Ne e heless, he FSI pa i ioned
s a egy in eg a ed in wsiFoam ails o e ie e alues close o hose e-
po ed by Lu e al. (2000) and Liao e al. (2013), specially o 8 mm and
11 mm hicknesses. Fu he analysis is hus equi ed o in es iga e he
solid mechanics sol e s in eg a ed in OpenFOAM in o de o imp o e
he solu ion accu acy.
∆x×10−5[m]
-5
0
5
10
15
20
25
30
35
40
0 0.05 0.1 0.15 0.2
P esen
Liao e al. (2013)
Lu e al. (2000)
[s]
Fig. 11 Wedge wa e en y (elas ic case): ime his o y o he
wedge’s middle poin displacemen o 5 mm hickness.
Finally, Table 3 epo s on he pa allel scalabili y o his FSI app oach. I
∆x×10−5[m]
-2
0
2
4
6
8
10
12
14
16
0 0.05 0.1 0.15 0.2
P esen
Liao e al. (2013)
Lu e al. (2000)
[s]
Fig. 12 Wedge wa e en y (elas ic case): ime his o y o he
wedge’s middle poin displacemen o 8 mm hickness.
∆x×10−5[m]
-2
0
2
4
6
8
10
12
0 0.05 0.1 0.15 0.2
P esen
Liao e al. (2013)
Lu e al. (2000)
[s]
Fig. 13 Wedge wa e en y (elas ic case): ime his o y o he
wedge’s middle poin displacemen o 11 mm hickness.
can be eadily seen ha he e is abou an o de o magni ude o di e ence
be ween he solid and elas ic simula ion imes. We ge a speed up o
2.75 when using 4 co es in he igid case. Howe e , he elas ic case
exhibi s a lowe pe o mance o 1.79 o he same numbe o co es. Such
small alues a e expec ed in OpenFOAM since he numbe o mesh cells
is ela i ely low: 11560 and 160 cells o he luid and solid meshes,
espec i ely. The e o e, g ea e pa allel pe o mance should be achie ed
o la ge scale compu a ions.
Table 3 Wedge wa e en y: simula ion speed up ( m/ ) o he igid
and elas ic cases; e e ence imes co esponding o he se-
quen ial simula ion a e e =3861 s ( igid) and e =
83930 s (elas ic).
Co es 1 2 3 4
Rigid 1.00 1.72 2.33 2.75
Elas ic 1.00 1.52 1.64 1.79
CONCLUSIONS
We ha e de eloped a mul i- egion, luid-s uc u e in e ac ion p ocedu e
based on he open-sou ce lib a y OpenFOAM and in eg a ed i in ou
i ual wa e s uc u e in e ac ion sol e wsiFoam. I o e s he lexibili y
o OpenFOAM as well as an uni ied FVM amewo k o he luid and
solid equa ions, which ende s i sui able o la ge scale compu a ions.
We ha e es ed ou FSI s a egy in cases ea u ing wa e iolen impac s.
Ou i s nume ical esul s, co esponding o a dam b eak impac ing on a
lexible pla e, show a quali a i e good ag eemen agains expe imen s and
p e ious nume ical da a. Howe e , we we e no able o p edic highe
modes o ib a ion wi h ou cu en p ocedu e. La ge di e ences we e
also obse ed o he lowes heigh o wa e . These disc epancies may
be caused by he in luence o he ga e mo ion used in he expe imen s as
poin ed ou by o he in es iga o s.
We ha e also s udied he wa e en y o igid and lexible wedges. On
he one hand, he igid case shows good ag eemen agains analy ical
and simila i y solu ions based on po en ial heo y and also con i ms he
sepa a ion o he low nea he wedge bo om in conco dance wi h p e-
ious wo ks in which a mul i-phase Na ie -S okes model was used. On
he o he hand, ou esul s co esponding o he elas ic case show simila
ends o hose p e iously epo ed. Ne e heless, we could no accu-
a ely p edic he maximum displacemen s alues.
Finally, we pe o med pa allel simula ions o show he capabili ies o his
FSI sol e o HPC applica ions. Pa allel pe o mance was no op imal
o he enginee ing applica ions conside ed in his wo k due o he ela-
i ely small numbe o mesh cells used. Howe e , be e pe o mance is
expec ed o la ge scale p oblems and hus we belie e ha his me hod
is an a ac i e candida e o he HPC o ealis ic ocean and coas al engi-
nee ing scena ios.
Fu he wo k will be ca ied ou o imp o e he accu acy o Open-
FOAM’s solid mechanics sol e s. A gene alised g id in e pola o ech-
nique be ween non-con o mal luid and solid meshes will be inco po-
a ed o gua an ee a be e FSI coupling. Finally, a comp essible Na ie -
S okes sol e will be also conside ed in he u u e in o de o accoun
o he combined e ec s o comp essibili y and hyd oelas ici y du ing
iolen wa e impac s.
ACKNOWLEDGEMENTS
The au ho s acknowledge wi h g a i ude inancial suppo om he En-
ginee ing and Physical Sciences Resea ch Council (EPSRC) unde he
So wa e o he Fu u e (SoFT) ini ia i e and ela ed esea ch g an s
EP/K037889/1, EP/K038168/1 and EP/K038303/1.
REFERENCES
Ba he, K. & Hahn, W. (1979), ‘On ansien analysis o luid-s uc u e
sys ems’, Compu e s &S uc u es 10(1-2), 383–391.
Bullock, G., Obh ai, C., Pe eg ine, D. & B edmose, H. (2007), ‘Violen
b eaking wa e impac s. Pa 1: Resul s om la ge-scale egula wa e
es s on e ical and sloping walls’, Coas al Enginee ing 54, 602–617.
URL: h p://dx.doi.o g/10.1016/j.coas aleng.2006.12.002
Gu, H., Qian, L., Causon, D., Mingham, C. & Lin, P. (2014), ‘Nume ical
simula ion o wa e impac o solid bodies wi h e ical and oblique
en ies’, Ocean Enginee ing 75, 128–137.
URL: h p://dx.doi.o g/10.1016/j.oceaneng.2013.11.021
Hou, G., Wang, J. & Lay on, A. (2012), ‘Nume ical me hods o luid-
s uc u e in e ac ion — a e iewjasa’, Communica ions in Compu a-
ional Physics 12, 337–377.
URL: h p://dx.doi.o g/10.4208/cicp.291210.290411s
Jasak, H. (1996), E o analysis and es ima ion o he ini e olume
me hod wi h applica ions o luid lows, PhD hesis, Uni e si y o
London.
Jasak, H. & Welle , H. G. (2000), ‘Applica ion o he ini e olume
me hod and uns uc u ed meshes o linea elas ici y’, In e na ional
Jou nal o Nume ical Me hods in Enginee ing 48, 267–287.
Kissling, K., Sp inge , J., Jasak, J., Schu z, S., U ban, K. & Piesche,
M. (2010), A coupled p essu e based solu ion algo i hm based on he
olume-o - luid app oach o wo o mo e immiscible luids, in ‘P o-
ceedings o he V Eu opean Con e ence on Compu a ional Fluid Dy-
namics, ECCOMAS CFD’.
Liao, K., Hu, C. & Duan, W. (2013), ‘Two-dimensional nume ical sim-
ula ion o an elas ic wedge wa e en y by a coupled FDM-FEM
me hod’, Jou nal o Ma ine Science and Applica ion 12, 163–169.
URL: h p://dx.doi.o g/10.1007/s11804-013-1181-2
Liao, K., Hu, C. & Sueyoshi, M. (2014), Nume ical simula ion o ee
su ace low impac ing on an elas ic pla e, in ‘29 h In e na ional
Wo kshop on Wa e Wa es and Floa ing Bodies’.
Liao, K., Hu, C. & Sueyoshi, M. (2015), ‘F ee su ace low impac ing
on an elas ic s uc u e: Expe imen e sus nume ical simula ion’, Ap-
plied Ocean Resea ch 50, 192–208.
URL: h p://dx.doi.o g/10.1016/j.apo .2015.02.002
Lu, C., HE, Y. & WU, G. (2000), ‘Coupled analysis o nonlinea in e -
ac ion be ween luid AND s uc u e du ing impac ’, Jou nal o Fluids
and S uc u es 14, 127–146.
URL: h p://dx.doi.o g/10.1006/j ls.1999.0257
Lugni, C., Ba dazzi, A., Fal insen, O. M. & G aziani, G. (2014), ‘Hy-
d oelas ic slamming esponse in he e olu ion o a lip- h ough e en
du ing shallow-liquid sloshing’, Physics o Fluids 26(3), 032108.
URL: h p://dx.doi.o g/10.1063/1.4868878
Lugni, C., B occhini, M. & Fal insen, O. M. (2010), ‘E olu ion o he
ai ca i y du ing a dep essu ized wa e impac . II. The dynamic ield’,
Phys. Fluids 22, 056102.
URL: h p://dx.doi.o g/10.1063/1.3409491
Ma, Z., Causon, D., Qian, L., Mingham, C. & Ma ´
ınez Fe e , P. (2016),
‘Nume ical in es iga ion o ai enclosed wa e impac s in a dep es-
su ised ank’, Ocean Enginee ing 123, 15–27.
URL: h p://dx.doi.o g/10.1016/j.oceaneng.2016.06.044
Ma ´
ınez Fe e , P., Causon, D., Qian, L., Mingham, C. & Ma, Z.
(2016a), ‘A mul i- egion coupling scheme o comp essible and
incomp essible low sol e s o wo-phase low in a nume ical wa e
ank’, Compu e s &Fluids 125, 116–129.
URL: h p://www.sciencedi ec .com/science/a icle/pii/S004579301500376X
Ma ´
ınez Fe e , P. J., Causon, D. M., Qian, L., Mingham, C. G. & Ma,
Z. H. (2016b), Nume ical simula ion o wa e slamming on a lap ype
oscilla ing wa e ene gy de ice, in ‘P oceedings o he 26 h In e na-
ional Ocean and Pola Enginee ing Con e ence’, pp. 672–677.
Rusche, H. (2002), Compu a ional luid dynamics o dispe sed wo-
phase lows a high phase ac ions, PhD hesis, Uni e si y o London.
Tuko ic, Z., I anko ic, A. & Ka ac, A. (2013), ‘Fini e- olume s ess
analysis in mul i-ma e ial linea elas ic body’, In e na ional Jou nal
o Nume ical Me hods in Enginee ing 93, 400–419.
URL: h p://dx.doi.o g/10.1002/nme.4390
Tuko ic, Z. & Jasak, H. (2007), ‘Upda ed Lag angian ini e olume
sol e o la ge de o ma ion dynamic esponse o elas ic body’, T ans-
ac ions o FAMENA 31, 55–70.
Welle , H. (2002), De i a ion, modelling and solu ion o he condi ion-
ally a e aged wo-phase low equa ions, Technical epo , Nabla L d,
No Technical Repo TR/HGW/02.
Zhao, R. & Fal insen, O. (1993), ‘Wa e en y o wo-dimensional bod-
ies’, Jou nal o Fluid Mechanics 246, 593–612.
URL: h p://dx.doi.o g/10.1017/S002211209300028X
