Pe o mance Op imiza ion o a Floa ing B eakwa e Model Using SPH
Me hod, wi h a P ac ical Applica ion
D. Alejand o Rueda Du an
Final Repo o he Mas e Thesis
Di ec ed By
D . Ge mán Rod íguez Rod íguez
D . Alejand o Jacobo Cab e a C espo
UNIVERSIDAD DE LAS PALMAS DE GRAN CANARIA
FACULTAD DE CIENCIAS DEL MAR
LAS PALMAS DE GRAN CANARIA
2013
Pe o mance Op imiza ion o a Floa ing B eakwa e Model Using SPH Me hod wi h, a
P ac ical Applica ion
Alejand o Ruedaa, Ge mán Rod ígueza, A.J.C. C espob
aDepa amen o de Física, Uni e sidad de Las Palmas de G an Cana ia, 35017 - Las Palmas de G an
Cana ia, Spain (g od iguez@d is.ulpgc.es).
bEn i onmen al Physics Labo a o y, Uni e sidad de Vigo (Campus de Ou ense) - Facul ade de Ciencias,
As Lagoas, 32004 - Ou ense- Spain (alexbexe@u igo.es).
ABSTRACT
The use o a loa ing b eakwa e , as an en i onmen ally iend s uc u e, is sugges ed as a coas al
p o ec ion s uc u e o he EGSAM pie in San a Ma a Bay, Ca ibbean Colombian coas . A
la ge numbe o loa ing b eakwa e s uc u es exis ing in he li e a u e a e examined and
compa ed o de ine he s uc u e o be used in he s udied case. The selec ion o he mo e
adequa e s uc u e is mainly based on wo aspec s, hei dimensions and e iciency, in e ms o
he wa e ene gy ansmission. The e iciency assessmen , as a unc ion o he inciden wa e
pe iod and he dis ance be ween he loa ing b eakwa e and he pie , has been ca ied ou by
means o he smoo hed pa icle hyd odynamic (SPH) me hod, a ee-mesh nume ical me hod.
Resul s e eal ha SPH me hod p ope ly simula es he in e ac ion be ween wa e wa es and a
coas al p o ec ion s uc u es. Ne e heless, 3D nume ical expe imen s equi e conside able
compu a ional cos , whe eas he use o 2D expe imen s alle ia es hese d awbacks bu include
un ealis ic e ec s, mainly due o he p esence o closed, non dissipa i e, bounda ies. The
p oposed s uc u e o he EGSAM pie e eals conside able ad an ages in con as wi h
con en ional coas al p o ec ion s uc u es, which mus be e alua ed in a wa e channel, as pa o
he design p ocess.
Key Wo ds: Floa ing B eakwa e , Smoo hed Pa icle Hyd odynamic, Wa es and Coas al
S uc u es
1
1. INTRODUCTION
A la ge numbe o coas al s uc u es o p o ec human goods agains wind-wa e a ack,
including po acili ies bo h comme cial and ec ea ional, ha e been buil du ing ecen
decades along he Colombian coas s, especially on he No h o he Ca ibbean coas .
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 led enginee s o look o mo e
so and “en i onmen iendly” coas al p o ec ion s uc u es, while ensu ing p ope sa e y
le els and unc ionali y.
The indisc imina e use o ubble-mound b eakwa e du ing he las o y yea s, as he only
al e na i e o p o ec small pie s, ma inas and beaches om e osion, has gene a ed impo an
en i onmen al p oblems, mainly associa ed wi h he in e up ion o al e a ion o li o al d i ,
e osion and pollu ion. This ac has s imula ed in e es and esea ch ac i i ies o he
de elopmen o e icien s uc u es helping o p ese e he coas al en i onmen . In
pa icula , he use o loa ing b eakwa e s (FB he ea e ), an old idea (i.e. De la Sala, 1873),
has been e isi ed and picked up as an al e na i e o p o ec coas al a eas and o gene a ing
shel e ed a eas. The exis ence o imp o ed acili ies o implemen mo e ealis ic physical
models, as well as he de elopmen o nume ical models and compu a ional capabili ies has
suppo ed ad ances in his ield.
2
A clea example o he need o p o ec a po acili y is ha o he San a Ma a Coas Gua d
S a ion - EGSAM, uni loca ed inside he bay o San a Ma a, which is dedica ed o
sa egua d human li e a sea and con ol illegal ac i i ies in he no h o he Ca ibbean coas
o Colombia. This pie buil in 2005 was designed wi h he op imum cha ac e is ics o
moo ing, gi ing se ice and main enance o he as boa assigned o his mili a y uni .
Howe e , due o he change in sedimen a e o he Manzana es Ri e , which a e six
seasons o hea y ain alls has inc eased i s a e o sedimen supply and disposal in he zone,
leading o a p ocess o acc e ion unde he pie o he EGSAM and inducing a change in he
b eaking zone posi ion owa ds he be hing a ea. As a esul , he s uc u e is now pa ially
useless.
The aims o his s udy is o explo e he possibili y o imp o ing ope a ing condi ions in he
EGSAM pie by using a FB, unde ce ain es ic ions conce ning he ac ion o wa e
ene gy ansmission. Fo his, nume ical expe imen s, based on he use o he Smoo hed
Pa icle Hyd odynamics (SPH) me hodology, a e pe o med o simula e monoch oma ic
inciden wa es and examine he FB e iciency. The inal goal is o sugges some e icien FB
s uc u e o p o ec he s udy case acili y.
The emainde o he hesis is s uc u ed as ollows. Main geog aphical and clima ic
condi ions o he s udy a ea a e in oduced in sec ion 2. A b ie e iew o he de elopmen
and main cha ac e is ics o FB is p esen ed in sec ion 3. Theo e ical basis o he SPH
me hodology, as well as expe imen al se -up a e desc ibed in sec ion 4. P elimina y esul s
de i ed om nume ical simula ion conce ning wa e ene gy ansmission ough he selec ed
3
loa ing b eakwa e geome y a e discussed in sec ion 5. Finally, conclusions a e
summa ized in sec ion 6.
2. STUDY AREA
San a Ma a bay is loca ed in he Sou hwes e n side o he Ca ibbean Sea (11°14’44.98N,
74º13’11.11W), app oxima ely, see Fig. 1.
Figu e 1. Loca ion map o he San a Ma a bay.
Acco ding o he inal epo o wea he condi ions o Colombian Ca ibbean po s,
p esen ed by he Cen e o Hyd og aphic and Oceanog aphic Resea ches (CIOH) in 2010,
a a la ge scale, i s wea he is in luenced by luc ua ions in he Azo es high p essu e sys em
he In e - opical Con e gence Zone posi ions. Du ing he d y season (Decembe o Ap il)
The Azo es high p essu e sys em mo es owa d lowe la i udes leading o an inc ease in he
wind in ensi y, eaching alues o e o 3.0 m/s. The p edominan di ec ion is de e mined by
he e ec o he ade winds, mainly in he d y season, lowing om no h and no heas
di ec ions du ing 26% and 15% o he ime, espec i ely (See Table 1). The modal mean
wa e di ec ion in he zone is om NE, while mo e se e e sea s a es come om ENE (See
N
N

4
La . 11°14´05.74N
Long. 74°13’16.09W
Table 1). Due o e ch es ic ions he emainde di ec ional sec o s p esen a e y low o
null equency o occu ence.
Da a p esen ed in able 1 ha e been de i ed om a hindcas ing s udy de eloped by CIOH
o he Colombian Ca ibbean a ea co e ing a pe iod o 32 yea s, by using he hi d
gene a ion WAM model.
Table 1. Mean wind di ec ion and wa e condi ions in San a Ma a bay.
A ecen ly acqui ed ba hyme y (Colombian Na y Oceanog aphic and hyd og aphic
Resea ch Cen e , CIOH, 2012), as well as a ecen ae ial pho o o he s udy a ea a e shown
in Figu e 2(a) y 2(b), o acili a e iden i ica ion o he majo geog aphic ea u es o he zone.
A e age empe a u e is abou 28 ° C, and ela i e humidi y is close o 79%. A e age ain all
le els a e in he o de o 98 mm/mon h, bu can change d as ically du ing he ainy season,
eaching alues close o 330mm/mon h, in he su oundings o he Sie a Ne ada de San a
NE 5.0 -7.0
7.0 -9.0
5.0 -7.1
3.0 -3.5
ENE
Wa e
Di ec ion
Signi ican
Wa e
Heigh
Wa e
Pe iod
1.5 - 2.0
2.0 -2.5
Di ec ion %
N26
NE 15
E 4
SE 3
S10
SO 9
O 8
NO 8
Calm 14
San a Ma a
Wind Di ec ion
5
Ma a, gi ing ise o a signi ican inc ease o he a e age low o he i e s in he a ea, see
Figu e 3(a-d).
Figu e 2. (a) EGSAM Ba hyme y (dep h in m), (b) San a Ma a Ae ial Pho o
(a) Yea 2006 (b) Yea 2009
(c) Yea 2011 (d) Yea 2012
Figu e 3. E ec s o Manzana es Ri e sedimen a e discha ge a iabili y (2006 o 2012).
EGSAM Pie
Manzana es
Ri e
N
N
N
N
6
2.1 FLOATING BREAKWATERS: A SHORT REVIEW
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”.
The e iciency o FB is commonly de e mined by means o he ela ionship be ween he
inciden wa e heigh (Hi) and he ansmi ed wa e (H ), known as he ansmission
coe icien (K ):
𝐾𝑡=𝐻𝑡
𝐻𝑖 (1)
Assume a FB wi h ec angula p ism (Poon on ype) o m and conside ing a wo-
dimensional iew along he wa e p opaga ion axis, such as depic ed in Figu e 4. The wa e
column in he su ounding a ea can be di ided in o h ee sec ions; (1) a zone be o e he
s uc u e; (2) o he jus below; and (3) o he a e he FB. Wa e ene gy app oaching o he
s uc u e is a unc ion o Hi, while he ene gy ac ion ansmi ed owa ds he a ea o be
7
p o ec ed will depend on H . Wa e ene gy ansmission will also depend, among o he
ac o s, on he wa eleng h, λ, he FB heigh , Z , d a , D , and wid h W, as well as o he
wa e dep h, h.
Figu e 4. Geome ic cha ac e is ics o a simple FB.
The use o loa ing b eakwa e s as coas al p o ec ion s uc u es da e back, a leas o 1811,
when a loa ing s uc u e was ins alled o p o ec he Plymou h Bay (Massachuse s). In ha
case, a loa ing b eakwa e wi h i on ames o nine ee wide and six ee high, ixed o he
bo om wi h ancho s, was used. Used ma e ials we e wood and i on, which made i s li e
ime e y sho due o co osion and biological ac i i y. The e a e eco ds documen ing a
leas h ee a emp s o use his p o ec ion me hod in he bays o Plymou h and B igh on
be ween 1811 and 1844, De la Sala (1873). The shape o he s uc u es used was e y
di e en , bu all o hem had in common he use o i on ames o con ain wooden boxes,
and e en i on pipes in wooden boxes (see Figu e 5), so ha he inal esul was no di e en
o ha o hei p edecesso s.
I was no un il Wo ld Wa II when he FB use was di e si ied and began i s eal
op imiza ion om "Phoenix" and "Bomba don" models, de eloped by he Allies o he
1)
2)
D
Z
M.S.L.
H
H
i
λ
W
z
x
h
3)
14
Figu e 7. Ske ch o he in luence domain o a pa icle.
The app oxima ion (2), in disc e e no a ion, leads o he ollowing app oxima ion o he
unc ion a he pa icle (in e pola ion poin ) a:
𝐴𝑎=∑𝑚𝑏𝐴𝑏
𝜌𝑏𝑊𝑎𝑏𝑏 (3)
whe e he summa ion is o e all he pa icles (b) wi hin he egion o compac suppo o he
ke nel unc ion, ixed by h. The mass and densi y a e deno ed by 𝑚𝑏 and 𝜌𝑏 espec i ely and
𝑊𝑎𝑏 =𝑊(𝑟𝑎−𝑟𝑏,ℎ) is he weigh ing unc ion o ke nel be ween wo pa icles a and b.
• Ke nel unc ion
The pe o mance o SPH models depends on he choice o he weigh ing unc ion, which
should sa is y se e al condi ions such as:
o Posi i i y: 𝑊(𝑟𝑎−𝑟𝑏,ℎ)≥0 in he domain Ω
o Compac suppo : 𝑊(𝑟𝑎−𝑟𝑏,ℎ)= 0 ou he domain Ω
o No maliza ion:∫𝑊(𝐫𝒂−𝐫𝑏,ℎ)𝑑𝐫𝑏= 1
Ω
o Del a unc ion beha io : limℎ→0𝑊(𝐫𝒂−𝐫𝑏,ℎ)𝑑𝐫𝑏=𝛿(𝐫𝒂−𝐫𝑏)
o Also 𝑊𝑎𝑏 mus be mono onically dec easing wi hin c easing dis ance om pa icle a.
Poin ´
In luence
d
Domain Ω

15
Ke nels depend on he smoo hing leng h, h, and he non-dimensional dis ance be ween
pa icles gi en by 𝑞=𝑟𝑎𝑏 ℎ
� whe e 𝑟𝑎𝑏 is he dis ance be ween pa icles a and b
�𝑟𝑎𝑏=𝑟𝑎−𝑟𝑏�.
The Lag angian o m o he Na ie -S okes se equa ion is w i en as ollows:
• Momen um equa ion
The momen um conse a ion equa ion in a con inuum ield is:
𝐷𝑣
𝐷𝑡=−1
𝜌𝜵𝑃+𝐠+𝜣
(4)
whe e is he eloci y, P y ρ a e p essu e and densi y, g = (0,0,-9.81)ms-2 is he g a i a ional
accele a ion and Θ e e s o he di usion e ms.
The p essu e e m is exp essed in SPH no a ion as:
−1
𝜌𝜵𝑃=−�𝑚𝑏�𝑃𝑎
𝜌𝑎2+𝑃𝑏
𝜌𝑏2�𝜵𝑎𝑊𝑎𝑏
𝑏
(5)
whe e 𝑃𝑏 y 𝜌𝑏 a e p essu e and densi y co esponding o a pa icle a and b and 𝑊𝑎𝑏=
𝑊(𝑟𝑎−𝑟𝑏,ℎ) is he ke nel unc ion.
Then, he SPH momen um equa ion, Monaghan (1992) becomes:
𝑑𝐯𝒂
𝑑𝑡 =−�𝑚𝑏�𝑃𝑎
𝜌𝑎2+𝑃𝑏
𝜌𝑏2�𝜵𝑎𝑊𝑎𝑏+𝐠
𝑏
(6)
Di e en app oaches, based on a ious exis ing o mula ions o he di usi e e ms, can be
conside ed in he SPH me hod o desc ibe he momen um equa ion.
• Viscosi y
Two di e en op ions o di usion can be used in SPH: a i icial o lamina iscosi y.
16
o A i icial Viscosi y
The a i icial iscosi y p oposed by Monaghan (1992) has been used e y o en due o i s
simplici y. In SPH no a ion Eq. (6) can be w i en as:
𝑑𝐯𝒂
𝑑𝑡 =−∑𝑚𝑏�𝑃𝑎
𝜌𝑎
2+𝑃𝑏
𝜌𝑏
2+Π𝑎𝑏�𝜵𝑎𝑊𝑎𝑏+𝐠
𝑏 (7)
whe e Πab is he iscosi y e m:
Π𝑎𝑏=�−𝛼𝑐𝑎𝑏𝜇𝑎𝑏
𝜌𝑎𝑏 ,𝑖𝑓 𝐯𝑎𝑏·𝐫𝑎𝑏< 0
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
(8)
wi h
𝜇𝑎𝑏=ℎ𝐯𝑎𝑏·𝐫𝑎𝑏
𝐫𝑎𝑏
2+𝜂2 (9)
whe e 𝜌𝑎𝑏=1
2(𝜌𝑎+𝜌𝑏), 𝑐𝑎𝑏=1
2(𝑐𝑎+𝑐𝑏); 𝜂2= 0.01ℎ2; α is a ee pa ame e ha can be
changed acco ding o each p oblem.
o Lamina Viscosi y
The momen um conse a ion equa ion wi h lamina iscous s esses is gi en by:
𝐷𝐯
𝐷𝑡=−1
𝜌𝜵𝑃+𝐠+𝜐0𝛻2𝐯 (10)
whe e he lamina s ess e m simpli ies Mo is e al., (1997), Lo and Shao (2002) o:
(𝜈0𝛻2𝐯)𝑎=∑𝑚𝑏�4𝜈0𝐫𝑎𝑏𝜵𝑎𝑊𝑎𝑏
(𝜌𝑎+𝜌𝑏)|𝐫𝑎𝑏|2�𝐯𝑎𝑏𝑏 (11)
whe e 𝜈0 is he kine ic iscosi y o lamina low (0.893·10-6 m2/s). So, in SPH no a ion, Eq.
(11) can be w i en as:
𝑑𝐯𝑎
𝑑𝑡 =−∑𝑚𝑏�𝑃𝑎
𝜌𝑎
2+𝑃𝑏
𝜌𝑏
2�𝜵𝑎𝑊𝑎𝑏𝑏+ +𝐠+∑𝑚𝑏�4𝜈0𝐫𝑎𝑏𝜵𝑎𝑊𝑎𝑏
(𝜌𝑎+𝜌𝑏)|𝐫𝑎𝑏|2�𝐯𝑎𝑏𝑏 (12)
• Con inui y Equa ion
Changes in he luid densi y a e calcula ed in DualSPHysics using:
𝑑𝜌𝑎
𝑑𝑡 =∑𝑚𝑏𝐯𝑎𝑏𝜵𝑎𝑊𝑎𝑏𝑏 (13)
17
ins ead o using a weigh ed summa ion o mass e ms (Monaghan, 1992), since i is known
o esul in an a i icial densi y dec ease nea luid in e aces.
• Equa ion o S a e
The luid in he SPH o malism can be ea ed as weakly comp essible. This acili a es he
use o an equa ion o s a e o de e mine luid p essu e, which is much as e han sol ing an
equa ion such as he Poisson´s equa ion. Following (Monaghan e al., 1999; Ba chelo ,
1974), he ela ionship be ween p essu e and densi y is assumed o ollow he exp ession:
𝑃=𝐵��𝜌
𝜌0�𝛾−1� (14)
whe e B is a cons an ela ed o he modulus o comp essibili y o he luid, 𝜌0=
1000.0 𝐾𝑔𝑚3
� being he e e ence densi y, usually a he ee su ace and γ is a cons an
be ween 1 and 7 (al hough 7 is used in mos o he oceanic applica ions).
The speed o sound c, depends on he de i a i e o p essu e wi h espec o densi y:
𝑐2(𝜌)=𝜕𝑃
𝜕𝜌=𝐵𝛾
𝜌0�𝜌
𝜌0�𝛾−1=𝐵𝛾
𝜌0𝛾𝜌𝛾−1 (15)
𝑐02=𝑐2(𝜌0) = 𝜕𝑃
𝜕𝜌|𝜌=𝜌0=𝐵𝛾
𝜌0 (16)
whe e c0 is he speed o sound a he e e ence densi y; 𝐵=𝑐02𝜌0𝛾
⁄. Addi ionally, B
p o ides a limi o he maximum allowed change o densi y in each simula ion.
3.2 Box Model Design and code alida ion
The p oblem o s udy p esen s a 3D na u e. Howe e , o educe he compu a ional ime,
se e al expe imen s we e ca ied ou in a 2D domain. The main goal o hese 2D simula ions
has been o assess he use ulness and possible limi a ions o he me hodology o explo e he
18
p oblem unde s udy, especially in e ms o he ba hyme y. A e hese, he inal s eps o he
s udy ha e been de eloped by using a 3D domain. The ollowing dimensions ha e been
used o he box model design: 300m long, 100m wide (3D) and 45m dep h (see Figu e 8).
The wa emake in he nume ical ank is a pis on- ype wa e gene a o . Sepa a ion be ween
pa icles has been es ablished as 0.25m. The numbe o pa icles used in he 2D model is
121.812 and 7.264.747 in 3D expe imen s.
Figu e 8. Con igu a ion o he 2D nume ical ank.
I is impo an o ema k he hypo hesis unde lying he expe imen s: (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 alongsho e di ec ion in 2D, (c) due o shoaling e ec s, he
wa e p opaga ion di ec ion is almos pe pendicula o he beach. Hence, all simula ions we e
ca ied ou wi h he FB loca ed pa allel o pie line, d) he s uc u es we e designed using a
e e ence dep h o h = 20m and a pe iod T = 7s. Fu he mo e ideal and weakly comp essible
luid and i o acional low a e assumed, as well as he applicabili y o lineal wa e heo y.
The FB s uc u e used o alida e he applicabili y o he me hodology o ackle he p oposed
s udy, has been ha p oposed by B uce (1985). A o al o 90 simula ions we e ca ied ou in
19
2D wi h he aid o wo G aphic P ocessing Uni s (GPU) wi h he suppo om he
En i onmen al Physics Labo a o y o he Uni e si y o Vigo.
The i s p oblem e idenced while ca ying ou 2D expe imen has been he la ge
impo ance acqui ed by wa e e lec ion, since in his case he box model beha es like a
closed box. P og essi e inpu o wa e ene gy o he nume ical wa e ank and successi e
e lec ions will gi e ise o ha bo agi a ion p oblems which, o some inciden wa e pe iods
can gi e ise o esonance condi ions. Na u ally, hese phenomena a e equen in he na u e
when wa e ene gy p opaga es in o o owa ds a closed o semi-enclosed basing. Howe e ,
in he p esen simula ions, hese a e mainly an a i ac gene a ed by using a 2D box model.
No e ha in a 2D model, he la e al bounda ies, especially hose cons i u ed by he
wa emake and he pie ully close he domain o s udy, such ha ene gy canno be p ope ly
dissipa ed, as occu s in he beach placed behind he pie in na u e (Fig. 1).
The alida ion p ocedu e consis ed o six di e en simula ion expe imen s: a) wi hou
ba hyme y, b) ba hyme y wi h cons an slope o 9%, c) ba hyme y wi h cons an slope o
18%, d) a ba hyme y designed o s okes, e) San a Ma a's ac ual ba hyme y and, )
ba hyme y wi h cons an slope o 16%, wi hou pie , X = 350m and a 50m leng h beach.
The la e case was conside ed o enhance wa e ene gy dissipa ion, educing e lec ion
e ec s. Poin s o measu e wa e heigh ha e been placed a 1.5, 25, 165 and 200m sepa a ed
om he pie . The alues wa e heigh s we e nume ical measu ed by using pos -p ocessing
ools o he DualSPHysics code.

20
Each expe imen has been pe o med o 160s o physical ime. Since a e age wa e pe iod
in San a Ma a is a ound T = 7s, simula ions we e pe o med o pe iods T = 6, 7 and 8 s.
The FB e iciency was e alua ed a ou di e en dis ances (50, 75, 100 and 150 m) om he
EGSAM pie . The exis ence o ha bo oscilla ions induced by inciden and e lec ed wa es
in simple geome ies can be p edic ed by means o he Me ian o mula (ie., So ensen, 1993),
which in wo-dimensional condi ions is gi en by:
𝑇𝑛=2𝐿
𝑛�𝑔ℎ (16)
whe e 𝑇𝑛 is he na u al pe iod o oscilla ion, L is he leng h o he domain, n he numbe o
nodes, g he g a i a ional accele a ion and h he basin dep h. Fu he mo e, he c i ical a io
o esonance occu ence be ween he wa eleng h λ and he domain leng h L, in enclosed and
semi-enclosed basins a e gi en in Table 2 (So ensen, 1993).
Table 2. C i ical alues o λ/L inducing esonance condi ions.
Open-ended Basin C i ical Pe iods
0.25
0.75
1.25
1.75
2.25
Closed Basin C i ical Pe iods
0.5
1
1.5
2
2.5
To compa e heo e ical e lec ion and ansmission coe icien wi h he alues obse ed in
he nume ical wa e ank, he ollowing equa ions can be de i ed:
𝐻𝑅
𝐻𝑖=�𝐸𝑅
𝐸𝐼�12
�=�1−sinh�4∗𝜋 (ℎ−𝐷𝑟)/𝜆�
sinh4𝜋ℎ/𝜆�12
� (17)
𝐻𝑡
𝐻𝑖=�𝐻𝑖2−𝐻𝑅2�12
�=�sinh�4∗𝜋 (ℎ−𝐷𝑟)/𝐿�
sinh4𝜋ℎ/𝐿�12
� (18)
whe e, 𝐸𝑅 y 𝐸𝑖 a e he e lec ed and ansmi ed ene gy, espec i ely.
21
3.3 Floa ing B eakwa e s Assessmen
The selec ion o he FB s uc u e o be e alua ed has been done a e an in-dep h li e a u e
e iew. Fou een cases, including s uc u al in o ma ion o he FB and p ope ly desc ibed
wa e condi ions in physical, nume ical o ac ual cases, we e conside ed o u he analysis,
by aking in o accoun he use ulness o he in o ma ion p o ided (see Table 3).
The nume ical implemen a ion o he FB s uc u es in he model has been de eloped by
using dimensionless ela ionships be ween geome ical dimensions o he s uc u e,
ba hyme y and wa e pa ame e s gi en in Table 3. This me hodology makes possible
compa isons be ween s uc u es wi h di e en dimensions. The e iciency o each one o he
selec ed s uc u es has been assessed by using he a e age wa e and ba hyme ic condi ions
eco ded in San a Ma a Bay.
The e iciency o hese ou een selec ed s uc u es was compa ed using a decision ma ix in
which he ansmission coe icien K and olume o he s uc u e a e e alua ed o selec ha
o e ing a be e balance be ween e iciency and geome ical dimensions, o cos .
Table 3. FB Me hodology Design
Re e ences Thesis Code Place
Heigh FB
/D a
Z /D
Wide /
D a
W/D
Wa e Heigh /
Dep h
H/h
Dep h/ Wa e
Lengh
h/λ
Wide/ Wa e
Lengh
W/λ
D a /
Dep h
D /h
Wa e Heigh /
Wa e Lengh
Hi/λ
Dimen ional
Dep h
2πh/λ
B ebne (1968) Case 1 Physical Model 2,221844 7,5647092 0,200 0,345 0,383 0,147 0,069 2,167
B uce (1985) Case 2 Olympia Ha bo (Washing on) 1,5714286 60,156 0,196 0,165 0,140 0,031 1,231
To um (1987) Case 3 Physical Model 1,4285714 4,6869141 0,123 0,223 0,091 0,088 0,027 1,401
Manuel (1995) Case 4 Physical Model 1,3333333 20,032 0,620 0,248 0,200 0,020 3,895
Mu ali (1997) Case 5 Physical Model 1,4347826 2,826087 0,200 0,108 0,141 0,460 0,022 0,680
Sannasi aj (1998) Case 6 Physical and Nume ical Model 4 4 0,019 0,510 0,087 0,043 0,010 3,203
Allyn (2004) Case 7 Physical Model 1,2352941 2,2941176 0,076 0,326 0,127 0,170 0,025 2,046
Fous e (2007) Case 8 Nume ical Model 1,25 2,25 0,123 0,396 0,178 0,200 0,049 2,488
Ma inelli (2008) Case 9 Physical Model 2,3333333 6,6666667 0,170 0,246 0,105 0,064 0,042 1,545
Elchahal (2009) Case 10 Nume ical Model 1,1315789 0,6644737 0,050 0,401 0,101 0,380 0,020 2,518
Wang (2010) Case 11 Nume ical Model 1,1111111 1,6 0,150 0,168 0,135 0,450 0,025 0,731
Yoon (2011) Case 12 Physical and Nume ical Model 2,6666667 6,6666667 0,086 0,116 0,106 0,136 0,010 0,731
He (2012) Case 13 Physical Model 2,259887 8,7570621 0,044 0,225 0,388 0,197 0,010 1,414
Loukogeo gaki (2012) Case 14 Nume ical Model 1,8181818 4,5454545 0,037 0,158 0,250 0,347 0,006 0,995
22
The h ee mo e e icien s uc u es, in e ms o wa e ene gy educ ion and dimensions, we e
conside ed o decide he geome ical shape o he FB o be sugges ed as an ini ial op imal
in as uc u e o he s udy a ea, bo h in 2D and 3D. Fo his, 75 nume ical expe imen s we e
ca ied ou in 2D.
4. RESULTS AND DISCUSSION
A i s obse ed esul is ha in 2D simula ions he wa e e lec ion om he on and
backside bounda ies plays a dominan ole, due o he lack o ene gy dissipa ing a eas.
Fu he mo e, i has been obse ed ha wa e condi ions become s able a e some ansien
pe iod. Thus, only a inal pa o he o al wa e da a eco d measu ed has been conside ed o
examine he esul ing condi ions associa ed o each case. (See igu e 9).
Figu e 9. F ac ion o wa e da a se ies selec ed o be analyzed.
When a wa e ain p opaga es owa ds beach and mee s he FB, pa o he inciden ene gy
is e lec ed. Ano he pa o he ac ion ansmi ed h ough he s uc u e can be e lec ed a
he sloped bo om, a he deck, and a he on bounda y o he box model. In his sense, o
all he s udied cases, when he FB was placed a a dis ance o 50, 75 and 100m om he pie ,
he box model beha ed like a semi-enclosed basin, wi h he e lec ion be ween FB and he
pie been he mos impo an e ec . When he FB was loca ed 150m apa om he pie , in
he middle o he box model domain, he e lec ion e ec was he same bo h be ween he
23
pis on and he s uc u e, as be ween he pie and he FB, so ha in his condi ion he box
model beha es in a simila way o a closed basin.
All he simula ions ha e e ealed ha wa e ene gy ansmission inc eases wi h he pe iod o
he inciden wa es. In o he wo ds, e iciency o FB dec eases wi h he leng h o he inciden
wa es.
In he i s simula ion, ca ied ou wi hou ba hyme y (see Figu e 10a) i has been ound ha
basin oscilla ions a e conside able when he s uc u e is sepa a ed 75m om he pie , o
pe iods o 8 and 6 s, inducing oscilla ing modes close o T3 and T1, espec i ely. When he
wa e pe iod is 7s, his phenomenon was mainly obse ed wi h he FB s uc u e placed a
100 and 150m a om he deck.
Adding a bo om slope o 9%, simula ions o Figu e 10b e eal ha o sho e wa e pe iod,
6s, he e ec o basin oscilla ion educes because o his phenomenon is pa icula ly ele an
o long wa es, and is in ensi ied o longe pe iods. This enhancemen is pa icula ly no able
o T=8s, due o he ma ch wi h he c i ical pe iod gi en by he Me ian exp ession,
(So ensen 1993).
By inc easing he slope (see Figu e 11a), i is obse ed ha he ansmission coe icien
mode a ely educes. This is p obably due o he inc ease in wa e ene gy dissipa ion by
bo om ic ion. Howe e , he gene al beha io emains simila and con inues along
alida ion es by using 2D nume ical ank.
30
Figu e 15. Case 15: FB E iciency Compa ison (2D s. 3D)
5. CONCLUSIONS
• The SPH me hod is adequa e o simula e wi h accu acy he in e ac ion be ween wa e wa es
and a coas al p o ec ion s uc u e, including he e ec o an i egula ba hyme y.
• Howe e , physical phenomena, such as wa e e lec ion, gene a ed in 2D nume ical wa e
anks, a e non ealis ic and mus be ca e ully conside ed. Ne e heless, esul s de i ed wi h
2D models pe mi s o assess e iciency o di e en s uc u es wi h a low compu a ional cos .
• The e iciency, in e ms o he ansmission coe icien , o he loa ing b eakwa e s uc u e
selec ed o he s udy case, e alua ed unde almos eal condi ions, o inciden wa es o 7s,
is conside able high.
• I is shown ha a s uc u e wi h an a ea o 39 m2, o wi h a olume o 1560 m3, can p o ec
adequa ely he pie wi h low en i onmen al impac s.
• The p oposed s uc u e o he EGSAM pie e eals conside able ad an ages in con as wi h
con en ional coas al p o ec ion s uc u es, which mus be e alua ed in a wa e channel, as
pa o he design p ocess.
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0100 200 300
K
Case 15 -2D
Case 15 - 3D

31
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