E alua ion o wa e ene gy ansmission
h ough a loa ing b eakwa e using he SPH
me hod
A. Rueda1, A. J. C. C espo2 & G. Rod íguez1
1Depa amen o de Física, Uni e sidad de Las Palmas
de G an Cana ia, Spain
2En i onmen al Physics Labo a o y, Uni e sidad de Vigo,
(Campus de Ou ense), Spain
Abs ac
Ene gy ansmission h ough a box-shaped loa ing b eakwa e (FB) is
examined, unde simpli ied condi ions, by using he smoo hed pa icle
hyd odynamics (SPH) me hod, a mesh- ee pa icle nume ical app oach. The
e iciency o he s uc u e is assessed in e ms o he coe icien o ansmission
as a unc ion o he wa e pe iod and he loca ion o he loa ing b eakwa e
ela i e o he zone o be p o ec ed. P elimina y esul s conce ning wa e ene gy
ansmission e eals a clea imp o emen o he e iciency as wa e pe iod
dec eases and an impo an ole o he ba hyme y.
Keywo ds: loa ing b eakwa e s, smoo hed pa icle hyd odynamics, wa e ene gy
ansmission.
1 In oduc ion
A la ge numbe o p oblems in coas al enginee ing in ol e wa e-s uc u e
in e ac ion p ocesses whe e wa e p ope ies a e modi ied by some ype o man-
made s uc u e. In pa icula , ixed b eakwa e s a e commonly used o p o ec
coas al acili ies, such as ha bo s, agains wa es. Howe e , despi e such
s uc u es success ully p o ec coas al zones agains wa es, mainly due o
easons conce ning he p ese a ion o he coas al en i onmen and o aes he ic
cha ac e , he e is an inc easing s ong nega i e public eac ion o he
emplacemen o classical ubble-mound b eakwa e s along he coas . This has
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doi:10.2495/ 3FSI1 0021
led enginee s o look o mo e so and “en i onmen iendly” coas al p o ec ion
s uc u es.
Floa ing b eakwa e s (FB) can p o ide an al e na i e coas al p o ec ion
solu ion wi h low en i onmen al impac , because i s main pu pose is o educe
he wa e ene gy ansmission o a equi ed le el, p o iding a dynamic
equilib ium o he sho eline o p ese e exis ing o a i icially nou ished beaches,
as well as o a oid s agna ion zones, by allowing wa e low ci cula ion below
hei bo om ip and he sea bed. A concise de ini ion o loa ing b eakwa e was
p o ided by Hales [1]: “The basic pu pose o any Floa ing b eakwa e is o
p o ec a pa o sho eline, a s uc u e, a ha bo , o moo ed essels om
excessi e inciden wa e ene gy. A e passi e sys ems; i.e., no ene gy is p oduced
by he de ice o achie e wa e a enua ion. The inciden wa e ene gy is e lec ed,
dissipa ed, ansmi ed, o subjec ed o a combina ion o hese mechanisms. The
in e e ence o a loa ing b eakwa e wi h sho e p ocesses, biological exchange,
and wi h ci cula ion and lushing cu en s essen ial o he main enance o wa e
quali y is minimal”.
Floa ing b eakwa e s can o e a sensi i e, low cos , and highly e sa ile
enginee ing solu ion, since hei loca ion can be a ied and hei cos is no
dependen on he dep h o wa e o he idal ange. Fu he mo e, hey can be used
as mul i-pu pose acili ies. FB a e commonly used o p o ec ma ine s uc u es,
ma inas and ha bo s om wa e a acks, ecen ad ances has simula ed hei use
in many o he ields, such us: coas al and sho e line p o ec ion, Renewable
ene gy p oduc ion, Aquacul u e, Leisu e –Tou ism and design acili ies om
Aqua ic spo s.
In gene al, loa ing b eakwa e s can be used unde a conside able numbe o
geomo phological and oceanog aphic condi ions. B uce [2] enume a es he
ollowing p incipal ad an ages:
 FB may be he only solu ion whe e poo ounda ions will no suppo
bo om-connec ed b eakwa e s.
 FB ins alla ions a e less expensi e han ubble-mound b eakwa e s.
 FB p esen s a minimum o in e e ence wi h wa e ci cula ion.
 FB is easily mo ed and can usually be ea anged in o new layou wi h
minimum e o .
 FB has a low p o ile and p esen s a minimum in usion on he ho izon,
pa icula ly o a eas wi h high ide anges.
Howe e , i is wo h no ing ha loa ing b eakwa e s in gene al, may ha e
se ious disad an ages, wi h he mos signi ican as ollows (Hales [1]):
 The design o a loa ing b eakwa e sys em mus be ca e ully ma ched o he
si e condi ions (bo om changes, wind e ch, e c.) wi h due ega d o he
longe wa es which may a i e om in equen s o ms.
 The loa ing b eakwa e can ail o mee i s design objec i es by ansmi ing
a la ge wa e han can be ole a ed wi hou necessa ily su e ing s uc u al
damage.
 A majo disad an age is ha loa ing b eakwa e s mo e in esponse o wa e
ac ion and hus a e mo e p one o s uc u al- a igue p oblems.
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The e a e many di e en ypes o FB. An excellen e iew on his opic was
p esen ed by Hales [1]. In o ma ion abou mo e ecen ly de eloped ypes o FB
can be ound in PIANC [15], Tadayon [10], Peña e al. [12], among o he s.
Such as commen ed abo e, he main pu pose o a FB is o educe he wa e
ene gy ansmission o a equi ed le el wi hou p oducing a ull blockage o he
ene gy app oaching he zone o in e es . Some pa o he inciden ene gy is
dissipa ed by damping and ic ion, as well as h ough he gene a ion o eddies a
he edges o he b eakwa e . In gene al, he s uc u e spli s inciden wa e ene gy
– , in o ansmi ed – , e lec ed – , and dissipa ed ene gy – Ed. Thus, a
balance o ene gy lux equi es ha
 (1)
So ha di iding bo h sides o (1) by  and aking in o accoun ha wa e
ene gy is p opo ional o he wa e heigh squa ed yields
󰇡
󰇢󰇡
󰇢󰇡
󰇢1 (2)
whe e, ,  and  a e he inciden , ansmi ed, e lec ed, and dissipa ed
wa e heigh s, espec i ely. Equa ion (2) can be ew i en as
1 (3)
whe e , , and , a e, espec i ely, he ansmission, e lec ion and
dissipa ion coe icien s, gi en by
󰇡
󰇢
 󰇡
󰇢
 󰇡
󰇢
 (4)
Na u ally, op imal esul s a e ob ained when ansmission is minimized, by
maximizing he e lec ion and dissipa ion e ec s. Thus, e iciency o a FB is
usually e alua ed by means o he ansmission coe icien .
Du ing a la ge pe iod o ime, ad ances o FB beha io and e iciency we e
achie ed almos exclusi ely by means o expe imen al s udies, including bo h
physical models and ield expe imen s, such us: Chen and Wiegel [19], To um e
al. [14] and B uce [2].
Since he las decade o he pas cen u y nume ical simula ion s udies has
inc easingly become a common app oach o sol e e y complex p oblems in he
luid-s uc u e in e ac ion ield. G id o mesh based nume ical me hods such as
he ini e di e ence me hods (FDM) and he ini e elemen me hods (FEM) ha e
been widely applied o s udy he in e ac ion be ween wa es and FB (i.e.
Williams and Abul-Azm [3]; Williams e al. [4]).
Despi e he success o hei use, g id-based nume ical me hods su e om
di icul ies in dealing wi h ee su ace p oblems. Compu a ional mesh- ee
me hods in gene al, and he smoo hed pa icle hyd odynamics (SPH) me hod, in
pa icula , alle ia e no ably hese d awbacks. Consequen ly, i ep esen s an
in e es ing me hodology o explo e he e iciency o a FB unde he ac ion o
wa es (i.e. Shao [9]).
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This pape aims o con ibu e o he exis ing knowledge on hyd odynamic
in e ac ion o wa es and loa ing b eakwa e s, by explo ing he wa e ene gy
ansmission ough a well-known box-shaped s uc u e B uce [2] in e ms o he
wa e inciden pe iod and he ela i e FB loca ion in ela ion o he zone o be
p o ec ed, as well as he e ec o he ba hyme y, by using he SPH me hod. In
pa icula he open-sou ce code DualSPHysics (www.dual.sphysics.o g) has
been used o simula ed he ocean wa es and FB e iciency.
The pape is s uc u ed as ollows. Expe imen al se -up and he basis o SPH
me hodology a e p esen ed in sec ion 2. P elimina y esul s conce ning wa e
ene gy ansmission ough he ype o loa ing b eakwa e selec ed a e discussed
in sec ion 3. Conclusions a e summa ized in sec ion 4.
2 Me hodology
The loa ing b eakwa e used in he p esen s udy has a simple box-shape
s uc u e, such as ha sugges ed by B uce [2] and ins alled in he Olympia
ha bo (Washing on). The case o s udy s uc u e was buil by using he
ela ionship be ween geome ical and oceanog aphic pa ame e s gi es in Table 1.
Fi s line o he able includes he o iginal condi ions, while dimensionless
ela ionship and case s udy condi ions a e gi en in he second and hi d lines
espec i ely. Wi h his me hodology i is possible make a compa isons be ween
di e en scale s uc u es.
Table 1: Dimensional me hodology.
O iginal Dimensions (m) B uce (1985)
Wa e
Heigh
Hi
Wa e
Leng h
L
Deep
h Pe iod
T D a
D
ROF
Heigh
Z
Wid h
W
1,19 38,91 7,62 4,50 1,07 1,68 6,40
Dimensionless Rela ionship
Z /D W/D H/h h/λ w/λ D/h Hi/λ
1,571 6,000 0,156 0,196 0,165 0,140 0,031
Tes Case S uc u e (m)
2πh/λ λ h T D ZR W
1,8 71,0 20,0 7,0 2,8 4,4 16,8
In he p esen wo k, a ec angula 2D loa ing body is conside ed and he
ollowing assump ions a e made: (a) he FB has a posi ion which is ixed in
space, so ha he possibili y o ene gy adia ion is elimina ed, and (b) he FB is
in ini ely long in a longsho e di ec ion. Fu he mo e o ep oduce he na u al FB
beha io condi ions, ideal and weakly comp essible luid, and i o a ional low
a e assumed, as well as he applicabili y o lineal wa e heo y.
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2.1 Expe imen al se -up
The ba hyme y and he loca ion o he s uc u e o be p o ec ed by he FB a e
shown in Fig. 1. Simula ions we e pe o med by conside ing a 2D compu a ional
domain 300m long and 45m deep. The s uc u e o be p o ec ed is a dock o
small c a s loca ed in a place whe e he wa e column dep h is 5.5m.
Figu e 1: DualSPHysics Box Model – es case.
Pa ame e s used o de ine he FB s uc u e a e depic ed in Fig. 2. Whe e H is
he wa e heigh , λ is he wa eleng h, Z is he FB heigh , D is he d a , W wid h
and h he dep h.
Figu e 2: Desc ip ion and pa ame e iza ion o he FB.
The es case was de eloped using 20m dep h and 7s o wa e pe iod as
e e ence. FB e iciency was es ed o ou di e en dis ances o deck (50, 75,
100 and 150m) o each one o he e alua ed pe iods (8, 7 and 6s such as shown
in Figu e 3. Hi and H we e eco ded a wo poin s loca ed a 1.5m and 130m
away om he dock, whe e he dep hs a e 5.5m and 29m, espec i ely, (poin s 2
and 4 o he ou poin s o measu emen se led along he domain (Fig. 3)).
20
10
0
10
20
30
45
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a b
c d
Figu e 3: Loca ions o FB inc easing dis ance o deck (a. 50m, b. 75m,
c. 100m, d. 150m).
2.2 Smoo hed pa icle hyd odynamics model
The es case was simula ed by using he SPH (Smoo hed Pa icle
Hyd odynamics) model de eloped by esea che s a he Johns Hopkins
Uni e si y (US), he Uni e si y o Vigo (Spain), he Uni e si y o Manches e
(UK). The code named DualSPHysics p o ides good accu acy o di e en
coas al hyd aulics phenomena in 2D (Gómez-Ges ei a e al. [5]; Dal ymple
and Roge s [6]; C espo e al. [7]) and also in 3D (Gómez-Ges ei a and
Dal ymple [8]; C espo e al. [11]).
SPH is a Lag angian mesh- ee me hod. The SPH equa ions desc ibe he
mo ion o he in e pola ing poin s, which can be hough o as pa icles. A each
pa icle, physical magni udes such as mass, eloci y, densi y and p essu e a e
compu ed. Some weigh unc ions, o ke nels, de e mine he in ensi y o he
in e ac ion be ween adjacen luid olumes (pa icles). Di e en ke nels should
ul il he ollowing ma hema ical cons ain s: posi i i y, compac suppo ,
no maliza ion, mono onically dec easing, and del a unc ion beha io . The
smoo hing leng h, h, de e mines he dis ance o in e ac ion be ween wo
neighbou ing pa icles.
DualSPHysics code sol es he equa ions o luid dynamics by:
Momen um equa ion (Monaghan [17])

 ∑


 (3)
1 2 3 4
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whe e is eloci y, Pb and ρb a e he p essu e and densi y o pa icle a and b,
Wab=W( a- b,h) is he weigh unc ion o ke nel, g = (0,0,-9.81)ms-2 is he
g a i a ional accele a ion.
Con inui y equa ion

 ∑ (4)
Equa ion o s a e (Monaghan [17])
󰇣󰇡
󰇢1󰇤 (5)
whe e B is a cons an associa ed wi h he comp essibili y module, ρ0=1000.0
Kg/m3 he e e ence densi y, γ is a poly ophic cons an , wi h alues om 1 o 7.
󰇛󰇜
|
 (6)
whe e c0 is he speed o sound a he e e ence densi y and he cons an B is equal
o 
⁄.
In hese simula ions, luid pa icles we e ini ially placed on a s agge ed g id
(dx = dz = 0.25 m). A smoo hing leng h, h = 0.45 m, was conside ed, being he
o al numbe o pa icles np = 121.812. A pis on gene a es wa es using heo y
o Dal ymple and Dean [18].
3 Resul s and discussion
Twel e di e en simula ions we e ca ied ou , by using h ee wa e pe iods and
ou FB-deck dis ances; he FB e iciency was e alua ed o each one o he
wel e cases. Nume ical esul s a e shown in able 2.
Table 2: Geome ical pa ame e s and coe icien s o ansmission as a
unc ion o wa e pe iod and FB-deck dis ance.
HB: Wa e Heigh Be o e FB. HD: Wa e Heigh in Deck. XFB: Dis ance om deck o FB. hFB: Deep Unde nea h
FB.
Values o he coe icien o ansmission as a unc ion o wa e pe iod and he
dis ance o he FB o he deck a e gi en in Table 2. The alues om HB and HD
co esponding wi h he measu emen s poin s 2 and 4 om he igu e 3, i e es s
we e ca ied ou o each pe iod, 4 wi h he di e en posi ions o FB and ano he
Case/
Pa ame e
T=8s
wi hou
FB
FB
50m
FB
75m
FB
100m
FB
150m
T=7s
wi hou
FB
FB
50m
FB
75m
FB
100m
FB
150m
T=6s
wi hou
FB
FB
50m
FB
75m
FB
100m
FB
150m
HMB 3,51 5,17 3,69 6,68 8,84 4,47 5,96 4,90 4,29 5,49 4,54 5,01 5,78 5,12 5,96
HMM 5,92 2,99 3,02 4,78 5,80 6,06 2,52 1,97 2,29 1,78 3,58 1,41 1,62 0,99 0,91
XRM 50 75 100 150 50 75 100 150 50 75 100 150
hR 11 14 18 27 11 14 18 27 11 14 18 27
ZR 4,4 4,4 4,4 4,4 4,4 4,4 4,4 4,4 4,4 4,4 4,4 4,4
W16,8 16,8 16,8 16,8 16,8 16,8 16,8 16,8 16,8 16,8 16,8 16,8
D2,8 2,8 2,8 2,8 2,8 2,8 2,8 2,8 2,8 2,8 2,8 2,8
K 1,68655 0,58 0,82 0,72 0,66 1,35 0,42 0,40 0,53 0,32 0,79 0,28 0,28 0,19 0,15
ERROR
(+/-)
0,25
(+/-)
0,25
(+/-)
0,25
(+/-)
0,25
(+/-)
0,25
(+/-)
0,25
(+/-)
0,25
(+/-)
0,25
(+/-)
0,25
(+/-)
0,25
(+/-)
0,25
(+/-)
0,25
(+/-)
0,25
(+/-)
0,25
(+/-)
0,25
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one wi hou s uc u e, o check he simula ed o iginal condi ions o wa e heigh s.
The e o da a se ies we e calcula ed om he dis ance o in e ac ion o he SPH
pa icles. I can be obse ed om igu e 4. Tha o any dis ance be ween he FB
and he deck he K alue dec eases wi h he wa e pe iod. These esul s,
indica ing an imp o emen o he FB e iciency as he pe iod dec eases,
ag ee wi h he expe imen al obse a ions made by se e al au ho s (i.e. To um e
al. [14]; Ma inelli e al. [13]).
Figu e 4: FB e iciency s. dis ance om deck and pe iod T.
20
10
0
10
20
30
45
Leng h m
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The pa e n o a ia ion o he coe icien o ansmission as a unc ion o he
dis ance be ween he FB posi ion and he deck, o a gi en pe iod, is clea ly
mo e complex. Fo a wa e ain o 6s pe iod, he e iciency is conside ably high
bu unde goes a ela i e dec ease as he FB app oaches o he deck, especially
be ween 100 and 75 m. Howe e , while he alue o K exhibi s a simila
inc easing beha io o he cases o 7s and 8s o la ge dis ances, i changes
d as ically o sho e dis ances, wi h a ela i e inc ease o he ansmi ed wa e
heigh . P elimina y esul s in his sense indica e ha hese changes in K wi h he
dis ance could be ela ed o he educ ion o he wa e dep h below he FB ip as
i is displaced owa ds he coas and o he associa ed shoaling e ec . Howe e , a
con i ma ion o hese esul s equi e he analysis o addi ional simula ions wi h
simple ba hyme ic condi ions (cons an slope), which a e being ca ied ou .
4 Conclusions
The e iciency o a box-shaped loa ing b eakwa e is examined in e ms o he
pe iod o he inciden wa e ain and by a ying he dis ance be ween he FB and
he s uc u e o be p o ec ed. The e iciency o he FB inc eases as he pe iod o
he inciden wa e dec eases, independen ly o he dis ance be ween he FB and
he deck. The e iciency o he FB ends o ge wo se as he dis ance be ween
bo h s uc u es educes. Howe e he obse ed pa e ns o a ia ion in his case
a e conside ably mo e complex.
Acknowledgemen s
This esea ch wo k was ca ied ou unde suppo and collabo a ions o he
Colombian Na y, The Uni e si y o he Las Palmas de G an Cana ia, he
En i onmen al Physics Labo a o y om he Uni e si y o Vigo and he
Founda ion Ca olina, Spain.
Re e ences
[1] Hales, Z.L., Floa ing B eakwa e s: S a e o he A , U.S A my Co ps o
Enginee s. Technical Repo No 81-1 Cap. 1. pp. 23–45, 1981.
[2] B uce, L., Floa ing B eakwa e Design, J. Wa e way, Po , Coas al, Ocean
Eng. 111, pp. 304–318, 1985.
[3] Williams A.N., and Abul-Azm A.G., Dual pon oon loa ing b eakwa e .
Ocean Enginee ing. 24(5), pp. 465–78, 1997.
[4] Williams, A., N., Lee, H.S., and Huang, Z., Floa ing pon oon b eakwa e s.
Ocean Enginee ing. 27, pp. 221–240, 2000.
[5] Gómez-Ges ei a, M., D. Ce quei o, A.J.C. C espo and R.A. Dal ymple.
G een wa e o e opping analyzed wi h a SPH model. Ocean Enginee ing,
32, pp. 223–238, 2005.
[6] Dal ymple, R. A., Roge s, B., Nume ical modeling o wa e wa es wi h he
SPH me hod. Coas al Enginee ing, 53, pp. 141–147, 2006.
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