Research Article Interaction of Submerged Breakwater by a ...
Transcript of Research Article Interaction of Submerged Breakwater by a ...
Research ArticleInteraction of Submerged Breakwater by a SolitaryWave Using WC-SPH Method
Afshin Mansouri1 and Babak Aminnejad2
1 Departmentof Civil Engineering The Islamic Azad University of Roudehen Branch Tehran Iran2Department of Civil Engineering Faculty of Civil Engineering The Islamic Azad University of Roudehen Branch Tehran Iran
Correspondence should be addressed to Afshin Mansouri afshinmansoorigmailcom
Received 3 January 2014 Revised 8 March 2014 Accepted 11 March 2014 Published 13 April 2014
Academic Editor Agostino Bruzzone
Copyright copy 2014 A Mansouri and B Aminnejad This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
Interaction of a solitary wave and submerged breakwater is studied in a meshless Lagrangian approach For this purpose a two-dimensional smoothed particle hydrodynamics (SPH) code is developed Furthermore an extensive set of simulations is conductedIn the first step the generated solitary wave is validated Subsequently the interaction of solitary wave and submerged breakwateris investigated thoroughly Results of the interaction of solitary wave and a submerged breakwater are also shown to be in goodagreementwith published experimental studies Afterwards the effects of the inclination and length of breakwater aswell as distancebetween two breakwaters are evaluated on damping ratio of breakwater
1 Introduction
Due to development of ports and marine protection of theseashores against the sea waves must be considered as animportant issue in the field of marine structure In thiscontext breakwaters have been used as appropriate toolsin previous decades A wide range of breakwaters has beenintroduced so far Submerged breakwater is one of themwhich has been utilized in many countries Generally someadvantages of submerged breakwater can be mentioned asfollows
(1) simple structure
(2) cost effectiveness
(3) safety of marine transport vessels
(4) short construction period
(5) preserving beauty of the coast (touristy aspect)
(6) high strength of the structures against sea waves
(7) preventing coastal erosion area with scattering ofwave energy
In addition the performance of submerged breakwatersdepends on geometry material properties and environ-mental characteristics of surrounding breakwater [1] Inthe following paragraphs several studies dealing with thesubmerged breakwaters are reviewed
Effects of overtopping submerged obstacle on wavecharacteristics were investigated by Hara et al [2] Theyused a numerical model as well as a regression method tocalculate the rate of the wave breaking Hayakawa et al[3] also studied flow field around a trapezoidal submergedbreakwater numerically They compared the obtained 2Dand 3D numerical results with an experimental findingSolitary wave interaction with submerged multibodies wasconsidered by Wan Decheng and Guoxiong [4] theoreticallyAn experimental investigation of the wave transmission onsubmerged breakwaters was also considered by Armono andHall [5] An experimental comparison of wave field behinda submerged trapezoidal breakwater and double submergedtrapezoidal breakwater was performed by Yoshida et al[6] In their study the ratio between breakwaters and thewave height has been studied Interaction of a solitary waveand permeable rectangular submerged breakwater had beenevaluated by Huang et al [7] They measured the wave
Hindawi Publishing CorporationModelling and Simulation in EngineeringVolume 2014 Article ID 524824 9 pageshttpdxdoiorg1011552014524824
2 Modelling and Simulation in Engineering
transmission as well as the amount of dragon the break-water Stamos et al [8] performed an experimental studyand considered performance of absorption and reflectionof waves from hemicylindrical and rectangular submergedbreakwaters Johnson et al [9] conducted an experimentaland theoretical study of currents and waves around trape-zoidal submerged breakwater Muni-Reddy and Neelamani[10] presented a numerical and experimental investigationfor measurement of wave breaking and force on a protectedoffshore base with trapezoidal breakwater Chen et al [11]experimentally studied transformation of waves with variousprofiles between a trapezoidal submerged breakwater andsea wall Christou et al [12] presented numerical simulationof regular nonlinear waves interaction with rectangularsubmerged breakwaters for investigation of transmitted andreflected waves profile Zaman et al [13] investigated theamount of deformation of waves passing over the parabolicbreakwater and the extent of waves reflected area Losadaet al [14] investigated transmission reflection turbulenceand breakage of regular and irregular waves overtoppingof rubble mound breakwaters Cao et al [15] numericallystudied hydrodynamic properties of wave propagation whichsurrounded two impermeable trapezoidal submerged break-waters
In the area of breakwater computations smoothed par-ticle hydrodynamics (SPH) can also be implemented as anovel meshless numerical method that is being developedfor the study of near shore waves and navy needs In thiscontext several numerical models have been developed withSPH method for modeling free surface flows Dalrymple etal [16] Gotoh et al [17] and Shao [18] have presented somenumerical techniques in conjunction with SPH which havebeen widely used Furthermore Dalrymple and Rogers [19]showed that SPH method offers a variety of advantages forfluid modeling especially those with a free surface Yimet al [20] investigated waversquos interaction with porous rigidsemisubmerged rectangular breakwater with RANS and SPHmethods Cherfils et al [21] also considered flow surroundinga horizontal submerged plate as a breakwater near shoreswithSPHmethodDidier andNeves [22] studiedwave speed waveheight and breakage of wave over sloped sea wall for a regionof the Portuguese with SPH method
Furthermore to simulate water waves with SPH manydifferent approaches have been presented so far Shao [23]offered a coupled incompressible smoothed particle hydro-dynamics (ISPH) porous medium method to simulate freesurface profile Liu et al [24] optimized 3D SPH methodfor application of water wave modeling Generally interestedreader can refer to some articles [25 26] for finding reviewliterature of SPH development
Due to necessity of obtaining appropriate comparisonbetween the different factors affecting the performance ofsubmerged breakwater and lack of widespread study onsubmerged breakwaters with SPHmethod a relatively exten-sive parametric study on trapezoidal submerged breakwatersis presented For this purpose various geometrical aspectsof breakwater are considered and a comparative study isperformed Furthermore it is being tried to introducean appropriate combination of numerical techniques in SPH
for accurate simulation of solitary wave-breakwaters interac-tion
2 Governing Equations
In smooth particle hydrodynamics fluid flow can be rep-resented by 119873 discrete particles Each of these particlesmay be specified by an individual mass Moreover particlesmove according to the equations of Newtonian mechanicsincluding a pressure force For each of these particles theequations are based on Lagrangian form of the Navier-Stokesequations
119889120588
119889119905= minus120588nablaV
119889V119889119905= minus1
120588nabla119901 + 119892 + Π
(1)
where V 119901 and 120588 are the velocity pressure and densityrespectively 119892 is the gravitational acceleration and Π refersto the diffusion terms
3 SPH Formulation
Navier-Stokes equations solution in SPHmethod is by defin-ing a kernel function119882(119909 minus 1199091015840 ℎ) around every 1199091015840 locationof an SPH particle The maximum of this function occurs at119909 = 119909
1015840 The features of kernel function must be similar toDirac Delta function Therefore behavior of kernel functionis completely dependent on value of 119903 = 119909 minus 1199091015840 The valueℎ is the parameter which defines the ldquosizerdquo of influence areaof kernel function and is called the smoothing length Thekernel is normalized to unity in the following way
intΩ
119882(119909 minus 1199091015840 ℎ) 119889119909
1015840= 1 (2)
and we also havelimℎrarr0
119882(119909 minus 1199091015840 ℎ) = 120575 (119909 minus 119909
1015840) (3)
We should regard the SPH kernel as a defining region ofinfluence of the SPH particle It will turn out with only SPHparticleswhich liewithin each otherrsquos regions of influence andare able to interact with each other
Furthermore kernel function is implemented to approx-imate field variables at any point in a computational domainFor instance an estimate value of a function 119891(119909) at thelocation 119909 is given in a continuous form by an integral of theproduct of the function and a kernel function119882(119909 minus 1199091015840 ℎ)
⟨119891 (119909)⟩ = intΩ
119891 (1199091015840)119882(119909 minus 119909
1015840 ℎ) 119889119909
1015840 (4)
where the angle brackets ⟨⟩ denote a kernel approximationIn addition to the abovementioned characteristics kernelfunction must satisfy several other features such as positivitycondition [27] Moreover (4) can be also approximated by asummation as follows
⟨119891 (119909)⟩ cong sum
119895
119898119895
120588119895
119891 (119909119895)119882(
10038161003816100381610038161003816119909 minus 119909101584010038161003816100381610038161003816 ℎ) (5)
Modelling and Simulation in Engineering 3
where119898119895120588119895is the volume associated with the particle 119895 This
equation can be used if the function 119891(119909) is only known at119873discrete points
The kernel function description can be finalized bydefining a specific kernel function Numerous possibilitiesexist A large number of kernel function types are discussedin literature ranging from polynomial to Gaussian Themost common is the B-spline kernel that was proposed byMonaghan and Lattanzio [28]
119908 (119903 ℎ) = 120572119863
1 minus3
21199022+3
411990230 le 119902 le 1
1
4(2 minus 119902)
3
1 le 119902 le 2
0 119902 ge 2
(6)
which is considered in the present paper and 119902 = 119903ℎ and 120572119863
is equal to 10120587ℎ2 in two dimensional cases
4 SPH Implementation onGoverning Equations
Now based on the SPH formulation mentioned in previoussection governing equations will be discretized In this con-text the continuity equation for particle 119894 can be consideredin the SPH formulation as
119889120588119894
119889119905= minus
119873
sum
119895=1
119898119895(V119895minus V119895) sdot nabla119882
119894119895 (7)
where the sum extends over all neighboring particles and119882is the smoothing kernel evaluated at the distance betweenparticles 119894 and 119895
The SPH formulation of the momentum equations basedon artificial viscosity can also be obtained through theparticle approximation procedure as in
119889V119894
119889119905= minus
119873
sum
119895=1
119898119895(119875119894
1205882
119894
+
119875119895
1198752
119895
+ Π119894119895)nabla119882
119894119895+ 119892 (8)
where Π119894119895is the viscosity term To introduce viscous dissi-
pation an artificial viscosity concept which has been devel-oped using Monaghan [29] is implemented This term issymmetric conserves momentum and introduces a shearviscosity into the momentum equation The SPH form of themomentum equation is written as
Π119886119887=
minus120572119862119886119887120583119886119887
120588119886119887
119881119886119887sdot 119903119886119887lt 0
0 for other values(9)
where
120583119886119887=ℎV119886119887sdot 119903119886119887
1199032
119886119887+ 1205782
(10)
120588119886119887=1
2(120588119886+ 120588119887) 119862
119886119887=1
2(119862119886+ 119862119887)
1205782= 001ℎ
2
(11)
120572 is also a constant Monaghan [29] suggests that a value of120572 = 001 can be used for most computations
5 Numerical Details
51 XSPH Technique Tomodify the particlersquos velocity XSPHtechnique is used and particle 119894 is moved based on thistechnique In XSPH scheme it is formulated as follows
119889119903119894
119889119905= V119894+ 120576sum
119895
119898119895
120588119894119895
V119894119895119882119894119895 (12)
Velocity of particle 119894 is recalculated in such manner thatthe velocity of particle 119894 and the average velocity of the closestneighborhood particles interacting with particle 119894 are takeninto account
52 Time Stepping Procedure In the present paper predictor-corrector time stepping algorithm is implemented In thepredictor-corrector formulation variation of particles prop-erties (such as velocity) is estimated as [17] Consider
V119899+12119894
= V119899119894+Δ119905
2119865119899+12
119894 (13)
And by implementing forces at half step the values ofvelocity can be calculated at the end of any step which isformulated as follows [17]
V119899+1119894= 2V119899+12119894
minus V119899119894 (14)
53 Density Filter In weakly compressible SPH formulationlarge pressure oscillation has been observed These pressureinaccuracies can be modified using some procedures Inaddition to pressure modification the free surface profilewill be also improved Therefore MLS (moving least square)density filter which is a first order density correction is usedto reproduce the linear variation of the density field
120588119886= sum
119887
120588119887119882
MLS119886119887
119898119887
120588119887
= sum
119887
119898119887119882
MLS119886119887 (15)
The corrected kernel is also evaluated as follows
119882MLS119886119887
= 119882MLS119886119887
(119903119886) = 120573 (119903
119886) sdot (119903119886minus 119903119887)119882119886119887 (16)
More details can be found in many references in theliterature [30 31]
54 Boundary Conditions Two rows of stationary waterparticles at the boundary (Figure 1) remain fixed or moveaccording to an externally imposed function for say wavemaker boundaries When a fluid particle approaches theboundary the density and hence pressure exerted by theboundary particles increase This generates the necessaryrepulsive force on the water particles A major deficiency ofthis method is the generation of numerical boundary layersWhenwater particles get close to the boundary particles theytend to stick to the boundary thus they generate a boundarylayer which is not physically observed
4 Modelling and Simulation in Engineering
dx dx
dx2 dz2
Figure 1 Interaction betweenfluid particles and boundary particles
6 Solitary Wave Generation
Solitary wave is generated by Prescribe Piston Wave Maker[5] Value of paddle displacement is applied in the followingequation [22]
119883119901(119905) =
119867
119896[tanh (119883 (119905)) + tanh(119896
119889120582)]
119883 (119905) =119896
119889(119888119905 minus 119883
119901(119905) minus 120582)
(17)
where119867 is the wave height 119896 = radic((31198674119889)) is wave numberand 119888 = radic(119892(119889 + 119867)) is speed Wave speed can also beobtained from the following equation [22]
119906119901(119905) =
119888119867
119889
1
cosh2119883 (119905) + (119867119889) (18)
61 Validation of Generated Solitary Wave According toMaiti and Sen [32] simulation which is a benchmark test thelength and flat length of the tank water depth and slope ofthe right hand side of the tank were respectively consideredto be 10m 97m 03m and 45 degree (Figure 2)Themotionof wave paddle is calculated for 119867119889 = 01 Therefore waveheight and 120582 are 003 and 6882
The result of solitary wave generation is illustrated inFigure 3 Simulation is conducted from 0 till 10 sec It isobserved that the wave height is equal to 003m and obtainedfree surface profile is relatively accurate for further studiesHowever some inconsistencies are visible at second 4 Inaddition average error of SPH simulation (point to point) isequal to 189 percent
7 Results and Discussion
In this section various aspects of solitary wave-breakwaterinteraction will be studied At first step free surface related toa specific breakwater under interaction with a solitary waveis computed using SPH to ensure that our numerical setupis suitable for investigation of more complicated problemsAfterward various geometrical characteristics of breakwatersare also discussed Formore detailed studies two breakwatersare considered and effects of distance between these breakwa-ters will be evaluated
71 Validation For further evaluation of SPH accuracy thefree surface profile generated in solitary wave-breakwater
H
d
R
L
120573
Figure 2 Computational domain
0
002
004
0 2 4 6 8 10 12Wav
e hei
ght
(m)
Maiti and Sen (1999)SPH
minus002X (m)
Figure 3 Comparison of generated solitary wave with Sagnitarsquosresults
Table 1 Wave flume solitary wave and breakwater features
Value Characteristics03m Wave height (119867)10m Water depth (119889)45m (119871
0)
70m (119871overal)08m Height of breakwater (ℎ)04m Breakwaterrsquos beam1 2 Slope of breakwaterrsquos wall
interaction is compared with experimental work of Grilliet al [33] To perform this comparison a computationaldomain is defined (Table 1) Furthermore the characteristicsof wave flume solitary wave and breakwater are presentedResults of this simulation are illustrated at Figure 4 Featuresof the presented simulation are also reported in Table 2 Freesurface profiles in Figure 4(a) through 4(d) are at 10 1181 13and 1392 sec respectively RMS of obtained solutions whichare calculated relative to experimental findings seem to bereasonable Therefore it can be declared that our numericalsetup can model solitary wave-breakwater interaction withsatisfying precision
72 Investigation of Breakwater Inclination A subject studiedhere is considering the effects of breakwaterrsquos slope on1198671198671015840ratio For this purpose computational domain as well assolitary wave is considered to be the same as previous sectionSlope of breakwater at aft part (120573
2) which is shown in
Figure 5 is varied systematically All considered angles are
Modelling and Simulation in Engineering 5
0
05
0 1 2 3 4
Wav
e hei
ght
(m)
X (m)minus05
minus1minus2minus3minus4
Experimental dataSPH
(a)
0
05
0 1 2 3 4
Experimental data
Wav
e hei
ght
(m)
X (m)minus05
minus1minus2minus3minus4
SPH
(b)
0
05
0 2 4
Wav
e hei
ght
(m)
X (m)minus2minus4
Experimental dataSPH
(c)
Experimental dataSPH
0
05
0 1 2 3 4
Wav
e hei
ght
(m)
X (m)minus05
minus1minus2minus3minus4
(d)
Figure 4 Comparison of SPHrsquos free surface profile with experimen-tal data [33]
Loveral
L0 L1
L2
1205731
1205732
d
h
Figure 5 Computational domain
Table 2 Computational error
Time (sec) Figure RMS10 Figure 4(a) 00361181 Figure 4(b) 003413 Figure 4(c) 00491392 Figure 4(d) 0053
presented in Table 3 Furthermore geometrical characteris-tics of breakwater are also proposed in this table
Table 3 Breakwaterrsquos characteristics
1205732
1205731
ℎ 1198711
0 tanminus1(05) 04 215 tanminus1(05) 04 245 tanminus1(05) 04 260 tanminus1(05) 04 275 tanminus1(05) 04 290 tanminus1(05) 04 2
0010203
0 20 40 60 80 100X (m)
minus01minus02minus03W
ave h
eigh
t (m
)
Outcome wave (120573 = 0)
Incoming wave
Figure 6 Free surface profile for 1205732= 0 degree
Free surface profile related to each considered conditionis illustrated in Figures 6 7 8 9 10 and 11 In each caseobtained free surface profile is compared with generatedsolitary wave without considering breakwater effect Exactevaluation of breakwaterrsquos slope on the free surface profileis presented in Table 4 It is obvious that wherever 120573
2= 90
maximumwave reduction can be achieved Correspondinglywhen 120573
2decreased to zero minimum breakwater effect
occurs Generally Figure 12 also shows a representative frameof SPH solution
73 Investigation of Breakwater Length To perform furtherparametric studies the effects of breakwaterrsquos length onfree surface can also be evaluated In this context previouscomputational characteristics are kept and only the lengthratio L1L2 is changed systematically Considered situationsare presented in Table 5 Figures 13 and 16 show the obtainedresults at various length ratios In all cases breakwatergenerates a hollow in free surface at breakwater positionwhich leads to a reduction in wave crest Generally basedon Table 6 it is observed that L1L2 ratio decreased sobreakwater performance will be improved In addition itcan be concluded that smaller length ratio leads to morereduction of wave amplitude To complete this section a viewof SPH solution is also presented in Figure 17
74 Investigation of Distance between Two Submerged Break-waters In this part influences of distance variation betweentwo breakwaters are evaluated Distance between two break-waters (119883) is considered to be less than wavelength at thebeginning of simulations Consequently this distance willincrease until the gap becomes larger than the wavelengthIn addition wave height water depth and length of waveflume are considered to be 119867 = 03m 12m and 70msuccessively Breakwater is located 30 meters from the leftGeometrical characteristics of breakwaters are also presented
6 Modelling and Simulation in Engineering
Table 4 Effect of breakwater slope on HH1015840 ratio
Values 1205732= 0 120573
2= 15 120573
2= 45 120573
2= 60 120573
2= 75 120573
2= 90
HH1015840 1078973 1185936 1297549 132081 1470258 1503094
Table 5 Geometrical characteristics
11987111198712
1205731
1205732
1198712
1198711
9 tanminus1 (05) tanminus1(05) 02 18633 tanminus1(05) tanminus1(05) 03 1942 tanminus1(05) tanminus1(05) 05 21367 tanminus1(05) tanminus1(05) 06 22
0010203
0 20 40 60 80 100
Wav
e hei
ght
(m)
X (m)minus01minus02
Outcome wave (120573 = 15)
Incoming wave
Figure 7 Free surface profile for 1205732= 15 degree
in Table 7 Figure 18 shows the gap effects between twoconsidered breakwaters It is observed that when wavereaches the first breakwater the wave height is equal to025m However second breakwater decreases the waveheight in correspondence with considered distance betweentwo breakwaters In this context two parameters 119867 (waveheight above first breakwater) and 1198671015840 (wave height abovesecond breakwater) are introduced Breakwater influencecoefficient is also defined as 1198671198671015840 Precise values of definedparameter are presented in Table 8 It is obviously clear thatfor 119883 = 03m maximum wave reduction is achievedFurthermore it can be concluded that optimum gap betweenbreakwaters is affected by wave length In reality when gapbetween breakwaters is smaller than wave length effects ofsecond breakwater will be decreased Additionallymaximumeffectiveness of breakwater is observedwhenever the distanceis equal to wave length A representation of our SPH solutionis also depicted in Figure 19
8 Results
The present paper focuses on the breakwater performanceunder a solitary wave Furthermore we have tried to imple-ment meshless and Lagrangian approach of smoothed par-ticle hydrodynamics method to study considered physicsIn this context a two-dimensional SPH code is developedTo consider diffusion terms an artificial viscosity model isincluded Particles movement is also calculated using XSPHvariant and predictor-corrector scheme which is investigatedas time stepping algorithm To reinitialize density field MLSdensity filter is utilized
To start computations generated solitary wave is verifiedby comparing previous studies Afterwards a breakwater
001020304
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus01minus02
Outcome wave (120573 = 45)
Incoming wave
Figure 8 Free surface profile for 1205732= 45 degree
0010203
0 20 40 60 80 100
Outcome wave (120573 = 60)
Incoming wave
Wav
e hei
ght
(m)
X (m)minus01minus02
Figure 9 Free surface profile for 1205732= 60 degree
00204
0 20 40 60 80 100
Wav
e hei
ght
(m)
X (m)minus02
minus04
Outcome wave (120573 = 75)
Incoming wave
Figure 10 Free surface profile for 1205732= 75 degree
0
05
0 20 40 60 80 100
Wav
e hei
ght
(m)
X (m)minus05
Outcome wave (120573 = 90)
Incoming wave
Figure 11 Free surface profile for 1205732= 90 degree
Figure 12 A representative frame of SPH solution
Modelling and Simulation in Engineering 7
Table 6 Exact evaluation of length ratio effect on free surface reduction
Values 11987111198712= 9 119871
11198712= 633 119871
11198712= 42 119871
11198712= 367
HH1015840 1094116 1105263 1111112 1222223
0
02
04
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus02
Incoming waveOutcoming wave (120574 = 90)
Figure 13 Free surface profile with ratio 9 in length
0010203
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus01
Incoming waveOutcome wave (120574 = 633)
Figure 14 Free surface profile at length ratio 633
Table 7 Breakwater characteristics
(119883) Left sideslope
Right sideslope
Breakwaterheight
Lowerlength
02 tanminus1(05) tanminus1(05) 04 203 tanminus1(05) tanminus1(05) 04 204 tanminus1(05) tanminus1(05) 04 205 tanminus1(05) tanminus1(05) 04 2
Table 8 Effects of breakwaters distance
Damping ratio 119883 = 02 119883 = 03 119883 = 04 119883 = 05
HH1015840 1247285 142718 13381 1337985
is also included in computational domain To validate ourSPH solutions obtained free surface profile is compared withexisting experimental results and it is observed that our SPHsolutions have a reasonable accuracy In the second phasevarious geometrical aspects of breakwater are also studied
Generally effects of implementing two breakwaters incli-nation of breakwater and breakwaterrsquos length on free surfacedepression are evaluated It is found that when gap betweenbreakwaters is smaller than wave length effects of secondbreakwater will be decreased Maximum effectiveness ofbreakwater also occurred whenever the distance is equal towave length In addition when the inclination of breakwaterincreased free surface deformation will be modified Fur-thermore it is observed that length ratio of breakwater caneffectively influence free surface profile (Figures 14 and 15)
0010203
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus01
Incoming waveOutcome wave (120574 = 42)
Figure 15 Free surface profile at length ratio 42
0010203
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus01
Incoming waveOutcome wave (120574 = 367)
Figure 16 Free surface profile at length ratio 367
Figure 17 A view of SPH solution
0
02
04
020 40 60 80 100W
ave h
eigh
t (m
)
X (m)minus02
Incoming waveOutcome wave (X = 02)
Outcome wave (X = 03)
Outcome wave (X = 04)
Figure 18 Simulated free surface profiles at different distancesbetween breakwaters
Figure 19 Particle representation of SPH solution
8 Modelling and Simulation in Engineering
To perform more precise computation extending SPHsolutions to three-dimensional case will be in attendanceVarious simple breakwater geometries will also be introducedin near future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Tokura and T Ida ldquoSimulation of wave dissipation mech-anism on submerged structure using fluid-structure couplingcapability in LS-DYNArdquo in Proceedings of the 5th European LS-DYNA Conference 2005
[2] M Hara T Yasuda and Y Sakakibara ldquoCharacteristics of asolitary wave breaking caused by a submerged obstaclerdquoCoastalEngineering vol 1 no 23 1992
[3] N Hayakawa T Hosoyamada S Yoshida and G TsujimotoldquoNumerical simulation of wave fields around the submergedbreakwater with SOLA-SURF methodrdquo in Proceedings of the26th International Conference on Coastal Engineering (ICCErsquo98) pp 843ndash852 June 1998
[4] W Decheng and W Guoxiong ldquoNumerical simulation of asolitary wave interaction with submerged multi-bodiesrdquo ActaMechanica Sinica vol 14 no 4 pp 304ndash305 1998
[5] H D Armono and K R Hall ldquoWave transmission on sub-merged breakwaters made of hollow hemispherical shape arti-ficial reefsrdquo in Canadian Coastal Conference pp 313ndash322 June2003
[6] A Yoshida S Yan M Yamashiro and I Irie ldquoWave FieldBehind a Double-Submerged Breakwaterrdquo in Proceedings of the12th International Offshore and Polar Engineering Conferencepp 785ndash790 May 2002
[7] C-J Huang H-H Chang and H-H Hwung ldquoStructuralpermeability effects on the interaction of a solitary wave anda submerged breakwaterrdquo Coastal Engineering vol 49 no 1-2pp 1ndash24 2003
[8] D G Stamos M R Hajj and D P Telionis ldquoPerformanceof hemi-cylindrical and rectangular submerged breakwatersrdquoOcean Engineering vol 30 no 6 pp 813ndash828 2003
[9] H K Johnson T V Karambas I Avgeris B Zanuttigh DGonzalez-Marco and I Caceres ldquoModelling of waves andcurrents around submerged breakwatersrdquo Coastal Engineeringvol 52 no 10-11 pp 949ndash969 2005
[10] M G Muni-Reddy and S Neelamani ldquoWave interaction withcaisson defenced by an offshore low-crested breakwaterrdquo Jour-nal of Coastal Research no 39 pp 1767ndash1770 2006
[11] H-B Chen C-P Tsai and C-C Jeng ldquoWave transformationbetween submerged breakwater and seawallrdquo Journal of CoastalResearch no 50 pp 1069ndash1074 2007
[12] M Christou C Swan and O T Gudmestad ldquoThe interactionof surface water waves with submerged breakwatersrdquo CoastalEngineering vol 55 no 12 pp 945ndash958 2008
[13] M H Zaman H Togashi and R E Baddour ldquoDeformationof monochromatic water wave trains propagating over a sub-merged obstacle in the presence of uniform currentsrdquo OceanEngineering vol 35 no 8-9 pp 823ndash833 2008
[14] I J Losada J L Lara R Guanche and J M Gonzalez-OndinaldquoNumerical analysis of wave overtopping of rubble moundbreakwatersrdquoCoastal Engineering vol 55 no 1 pp 47ndash62 2008
[15] Y-G Cao C-B Jiang and Y-C Bai ldquoNumerical study onflow structure near two impermeable trapezoid submergedbreakwaters on slop bottomsrdquo Journal of Hydrodynamics vol22 no 5 pp 190ndash196 2010
[16] R A Dalrymple O Knio D T Cox M Gesteira and S ZouldquoUsing a lagrangian particle method for deck overtoppingrdquo inProceedings of the 4th International SymposiumWaves pp 1082ndash1091 ASCE September 2001
[17] H Gotoh S Shao and T Memita ldquoSPH-LES model for numer-ical investigation of wave interaction with partially immersedbreakwaterrdquo Coastal Engineering Journal vol 46 no 1 pp 39ndash63 2004
[18] S Shao ldquoIncrompressible SPH simulation of wave breaking andovertopping with turbulence modellingrdquo International Journalfor Numerical Methods in Fluids vol 50 no 5 pp 597ndash6212006
[19] R A Dalrymple and B D Rogers ldquoNumerical modeling ofwater waves with the SPHmethodrdquoCoastal Engineering vol 53no 2-3 pp 141ndash147 2006
[20] S C Yim D Yuk A Panizzo M di Risio and P L-F LiuldquoNumerical simulations ofwave generation by a vertical plungerusing RANS and SPH modelsrdquo Journal of Waterway PortCoastal andOcean Engineering vol 134 no 3 pp 143ndash159 2008
[21] J M Cherfils L Blonce G Pinon and E Rivoalen ldquoSimulationof water wavemdashcoastal structure interactions by smoothedparticle hydrodynamicsrdquo in Proceedings of the 8th InternationalConference onHydrodynamics Nantes France September 2008
[22] E Didier and M G Neves ldquoWave over topping of a typicalcoastal structure of the portuguese coast using A SPH modelrdquoJournal of Coastal Research vol 56 pp 496ndash500 2009
[23] S Shao ldquoIncompressible SPH flow model for wave interactionswith porous mediardquo Coastal Engineering vol 57 no 3 pp 304ndash316 2010
[24] C Liu J Zhang and Y Sun ldquoThe optimization of SPHmethod and its application in simulation of water waverdquo inProceedings of the IEEE 7th International Conference on NaturalComputation (ICNC rsquo11) pp 2327ndash2331 July 2011
[25] PW ClearyM Prakash J Ha N Stokes and C Scott ldquoSmoothparticle hydrodynamics status and future potentialrdquo Progress inComputational Fluid Dynamics vol 7 no 2-4 pp 70ndash90 2007
[26] M B Liu and G R Liu ldquoSmoothed particle hydrodynamics(SPH) an overview and recent developmentsrdquo Archives ofComputational Methods in Engineering vol 17 no 1 pp 25ndash762010
[27] G R Liu and M B Liu Smoothed Particle HydrodynamicsmdashAMeshfree Particle Methods World Scientific Publishing 2007
[28] J JMonaghan and J C Lattanzio ldquoA refined particlemethod forastrophysical problemsrdquo Astronomy and Astrophysics vol 149pp 135ndash143 1985
[29] J J Monaghan ldquoSmoothed particle hydrodynamicsrdquo AnnualReview of Astronomy and Astrophysics vol 30 no 1 pp 543ndash574 1992
[30] A Colagrossi and M Landrini ldquoNumerical simulation ofinterfacial flows by smoothed particle hydrodynamicsrdquo Journalof Computational Physics vol 191 no 2 pp 448ndash475 2003
[31] A Crespo Application of smoothed particle hydrodynamicsmodel SPHysics to free surface hydrodynamics [Doctoral thesis]2008
Modelling and Simulation in Engineering 9
[32] S Maiti and D Sen ldquoComputation of solitary waves duringpropagation and runup on a sloperdquo Ocean Engineering vol 26no 11 pp 1063ndash1083 1999
[33] S T Grilli M A Losada and F Martin ldquoCharacteristics ofsolitary wave breaking induced by breakwatersrdquo Journal ofWaterway Port Coastal ampOcean Engineering vol 120 no 1 pp74ndash92 1994
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2 Modelling and Simulation in Engineering
transmission as well as the amount of dragon the break-water Stamos et al [8] performed an experimental studyand considered performance of absorption and reflectionof waves from hemicylindrical and rectangular submergedbreakwaters Johnson et al [9] conducted an experimentaland theoretical study of currents and waves around trape-zoidal submerged breakwater Muni-Reddy and Neelamani[10] presented a numerical and experimental investigationfor measurement of wave breaking and force on a protectedoffshore base with trapezoidal breakwater Chen et al [11]experimentally studied transformation of waves with variousprofiles between a trapezoidal submerged breakwater andsea wall Christou et al [12] presented numerical simulationof regular nonlinear waves interaction with rectangularsubmerged breakwaters for investigation of transmitted andreflected waves profile Zaman et al [13] investigated theamount of deformation of waves passing over the parabolicbreakwater and the extent of waves reflected area Losadaet al [14] investigated transmission reflection turbulenceand breakage of regular and irregular waves overtoppingof rubble mound breakwaters Cao et al [15] numericallystudied hydrodynamic properties of wave propagation whichsurrounded two impermeable trapezoidal submerged break-waters
In the area of breakwater computations smoothed par-ticle hydrodynamics (SPH) can also be implemented as anovel meshless numerical method that is being developedfor the study of near shore waves and navy needs In thiscontext several numerical models have been developed withSPH method for modeling free surface flows Dalrymple etal [16] Gotoh et al [17] and Shao [18] have presented somenumerical techniques in conjunction with SPH which havebeen widely used Furthermore Dalrymple and Rogers [19]showed that SPH method offers a variety of advantages forfluid modeling especially those with a free surface Yimet al [20] investigated waversquos interaction with porous rigidsemisubmerged rectangular breakwater with RANS and SPHmethods Cherfils et al [21] also considered flow surroundinga horizontal submerged plate as a breakwater near shoreswithSPHmethodDidier andNeves [22] studiedwave speed waveheight and breakage of wave over sloped sea wall for a regionof the Portuguese with SPH method
Furthermore to simulate water waves with SPH manydifferent approaches have been presented so far Shao [23]offered a coupled incompressible smoothed particle hydro-dynamics (ISPH) porous medium method to simulate freesurface profile Liu et al [24] optimized 3D SPH methodfor application of water wave modeling Generally interestedreader can refer to some articles [25 26] for finding reviewliterature of SPH development
Due to necessity of obtaining appropriate comparisonbetween the different factors affecting the performance ofsubmerged breakwater and lack of widespread study onsubmerged breakwaters with SPHmethod a relatively exten-sive parametric study on trapezoidal submerged breakwatersis presented For this purpose various geometrical aspectsof breakwater are considered and a comparative study isperformed Furthermore it is being tried to introducean appropriate combination of numerical techniques in SPH
for accurate simulation of solitary wave-breakwaters interac-tion
2 Governing Equations
In smooth particle hydrodynamics fluid flow can be rep-resented by 119873 discrete particles Each of these particlesmay be specified by an individual mass Moreover particlesmove according to the equations of Newtonian mechanicsincluding a pressure force For each of these particles theequations are based on Lagrangian form of the Navier-Stokesequations
119889120588
119889119905= minus120588nablaV
119889V119889119905= minus1
120588nabla119901 + 119892 + Π
(1)
where V 119901 and 120588 are the velocity pressure and densityrespectively 119892 is the gravitational acceleration and Π refersto the diffusion terms
3 SPH Formulation
Navier-Stokes equations solution in SPHmethod is by defin-ing a kernel function119882(119909 minus 1199091015840 ℎ) around every 1199091015840 locationof an SPH particle The maximum of this function occurs at119909 = 119909
1015840 The features of kernel function must be similar toDirac Delta function Therefore behavior of kernel functionis completely dependent on value of 119903 = 119909 minus 1199091015840 The valueℎ is the parameter which defines the ldquosizerdquo of influence areaof kernel function and is called the smoothing length Thekernel is normalized to unity in the following way
intΩ
119882(119909 minus 1199091015840 ℎ) 119889119909
1015840= 1 (2)
and we also havelimℎrarr0
119882(119909 minus 1199091015840 ℎ) = 120575 (119909 minus 119909
1015840) (3)
We should regard the SPH kernel as a defining region ofinfluence of the SPH particle It will turn out with only SPHparticleswhich liewithin each otherrsquos regions of influence andare able to interact with each other
Furthermore kernel function is implemented to approx-imate field variables at any point in a computational domainFor instance an estimate value of a function 119891(119909) at thelocation 119909 is given in a continuous form by an integral of theproduct of the function and a kernel function119882(119909 minus 1199091015840 ℎ)
⟨119891 (119909)⟩ = intΩ
119891 (1199091015840)119882(119909 minus 119909
1015840 ℎ) 119889119909
1015840 (4)
where the angle brackets ⟨⟩ denote a kernel approximationIn addition to the abovementioned characteristics kernelfunction must satisfy several other features such as positivitycondition [27] Moreover (4) can be also approximated by asummation as follows
⟨119891 (119909)⟩ cong sum
119895
119898119895
120588119895
119891 (119909119895)119882(
10038161003816100381610038161003816119909 minus 119909101584010038161003816100381610038161003816 ℎ) (5)
Modelling and Simulation in Engineering 3
where119898119895120588119895is the volume associated with the particle 119895 This
equation can be used if the function 119891(119909) is only known at119873discrete points
The kernel function description can be finalized bydefining a specific kernel function Numerous possibilitiesexist A large number of kernel function types are discussedin literature ranging from polynomial to Gaussian Themost common is the B-spline kernel that was proposed byMonaghan and Lattanzio [28]
119908 (119903 ℎ) = 120572119863
1 minus3
21199022+3
411990230 le 119902 le 1
1
4(2 minus 119902)
3
1 le 119902 le 2
0 119902 ge 2
(6)
which is considered in the present paper and 119902 = 119903ℎ and 120572119863
is equal to 10120587ℎ2 in two dimensional cases
4 SPH Implementation onGoverning Equations
Now based on the SPH formulation mentioned in previoussection governing equations will be discretized In this con-text the continuity equation for particle 119894 can be consideredin the SPH formulation as
119889120588119894
119889119905= minus
119873
sum
119895=1
119898119895(V119895minus V119895) sdot nabla119882
119894119895 (7)
where the sum extends over all neighboring particles and119882is the smoothing kernel evaluated at the distance betweenparticles 119894 and 119895
The SPH formulation of the momentum equations basedon artificial viscosity can also be obtained through theparticle approximation procedure as in
119889V119894
119889119905= minus
119873
sum
119895=1
119898119895(119875119894
1205882
119894
+
119875119895
1198752
119895
+ Π119894119895)nabla119882
119894119895+ 119892 (8)
where Π119894119895is the viscosity term To introduce viscous dissi-
pation an artificial viscosity concept which has been devel-oped using Monaghan [29] is implemented This term issymmetric conserves momentum and introduces a shearviscosity into the momentum equation The SPH form of themomentum equation is written as
Π119886119887=
minus120572119862119886119887120583119886119887
120588119886119887
119881119886119887sdot 119903119886119887lt 0
0 for other values(9)
where
120583119886119887=ℎV119886119887sdot 119903119886119887
1199032
119886119887+ 1205782
(10)
120588119886119887=1
2(120588119886+ 120588119887) 119862
119886119887=1
2(119862119886+ 119862119887)
1205782= 001ℎ
2
(11)
120572 is also a constant Monaghan [29] suggests that a value of120572 = 001 can be used for most computations
5 Numerical Details
51 XSPH Technique Tomodify the particlersquos velocity XSPHtechnique is used and particle 119894 is moved based on thistechnique In XSPH scheme it is formulated as follows
119889119903119894
119889119905= V119894+ 120576sum
119895
119898119895
120588119894119895
V119894119895119882119894119895 (12)
Velocity of particle 119894 is recalculated in such manner thatthe velocity of particle 119894 and the average velocity of the closestneighborhood particles interacting with particle 119894 are takeninto account
52 Time Stepping Procedure In the present paper predictor-corrector time stepping algorithm is implemented In thepredictor-corrector formulation variation of particles prop-erties (such as velocity) is estimated as [17] Consider
V119899+12119894
= V119899119894+Δ119905
2119865119899+12
119894 (13)
And by implementing forces at half step the values ofvelocity can be calculated at the end of any step which isformulated as follows [17]
V119899+1119894= 2V119899+12119894
minus V119899119894 (14)
53 Density Filter In weakly compressible SPH formulationlarge pressure oscillation has been observed These pressureinaccuracies can be modified using some procedures Inaddition to pressure modification the free surface profilewill be also improved Therefore MLS (moving least square)density filter which is a first order density correction is usedto reproduce the linear variation of the density field
120588119886= sum
119887
120588119887119882
MLS119886119887
119898119887
120588119887
= sum
119887
119898119887119882
MLS119886119887 (15)
The corrected kernel is also evaluated as follows
119882MLS119886119887
= 119882MLS119886119887
(119903119886) = 120573 (119903
119886) sdot (119903119886minus 119903119887)119882119886119887 (16)
More details can be found in many references in theliterature [30 31]
54 Boundary Conditions Two rows of stationary waterparticles at the boundary (Figure 1) remain fixed or moveaccording to an externally imposed function for say wavemaker boundaries When a fluid particle approaches theboundary the density and hence pressure exerted by theboundary particles increase This generates the necessaryrepulsive force on the water particles A major deficiency ofthis method is the generation of numerical boundary layersWhenwater particles get close to the boundary particles theytend to stick to the boundary thus they generate a boundarylayer which is not physically observed
4 Modelling and Simulation in Engineering
dx dx
dx2 dz2
Figure 1 Interaction betweenfluid particles and boundary particles
6 Solitary Wave Generation
Solitary wave is generated by Prescribe Piston Wave Maker[5] Value of paddle displacement is applied in the followingequation [22]
119883119901(119905) =
119867
119896[tanh (119883 (119905)) + tanh(119896
119889120582)]
119883 (119905) =119896
119889(119888119905 minus 119883
119901(119905) minus 120582)
(17)
where119867 is the wave height 119896 = radic((31198674119889)) is wave numberand 119888 = radic(119892(119889 + 119867)) is speed Wave speed can also beobtained from the following equation [22]
119906119901(119905) =
119888119867
119889
1
cosh2119883 (119905) + (119867119889) (18)
61 Validation of Generated Solitary Wave According toMaiti and Sen [32] simulation which is a benchmark test thelength and flat length of the tank water depth and slope ofthe right hand side of the tank were respectively consideredto be 10m 97m 03m and 45 degree (Figure 2)Themotionof wave paddle is calculated for 119867119889 = 01 Therefore waveheight and 120582 are 003 and 6882
The result of solitary wave generation is illustrated inFigure 3 Simulation is conducted from 0 till 10 sec It isobserved that the wave height is equal to 003m and obtainedfree surface profile is relatively accurate for further studiesHowever some inconsistencies are visible at second 4 Inaddition average error of SPH simulation (point to point) isequal to 189 percent
7 Results and Discussion
In this section various aspects of solitary wave-breakwaterinteraction will be studied At first step free surface related toa specific breakwater under interaction with a solitary waveis computed using SPH to ensure that our numerical setupis suitable for investigation of more complicated problemsAfterward various geometrical characteristics of breakwatersare also discussed Formore detailed studies two breakwatersare considered and effects of distance between these breakwa-ters will be evaluated
71 Validation For further evaluation of SPH accuracy thefree surface profile generated in solitary wave-breakwater
H
d
R
L
120573
Figure 2 Computational domain
0
002
004
0 2 4 6 8 10 12Wav
e hei
ght
(m)
Maiti and Sen (1999)SPH
minus002X (m)
Figure 3 Comparison of generated solitary wave with Sagnitarsquosresults
Table 1 Wave flume solitary wave and breakwater features
Value Characteristics03m Wave height (119867)10m Water depth (119889)45m (119871
0)
70m (119871overal)08m Height of breakwater (ℎ)04m Breakwaterrsquos beam1 2 Slope of breakwaterrsquos wall
interaction is compared with experimental work of Grilliet al [33] To perform this comparison a computationaldomain is defined (Table 1) Furthermore the characteristicsof wave flume solitary wave and breakwater are presentedResults of this simulation are illustrated at Figure 4 Featuresof the presented simulation are also reported in Table 2 Freesurface profiles in Figure 4(a) through 4(d) are at 10 1181 13and 1392 sec respectively RMS of obtained solutions whichare calculated relative to experimental findings seem to bereasonable Therefore it can be declared that our numericalsetup can model solitary wave-breakwater interaction withsatisfying precision
72 Investigation of Breakwater Inclination A subject studiedhere is considering the effects of breakwaterrsquos slope on1198671198671015840ratio For this purpose computational domain as well assolitary wave is considered to be the same as previous sectionSlope of breakwater at aft part (120573
2) which is shown in
Figure 5 is varied systematically All considered angles are
Modelling and Simulation in Engineering 5
0
05
0 1 2 3 4
Wav
e hei
ght
(m)
X (m)minus05
minus1minus2minus3minus4
Experimental dataSPH
(a)
0
05
0 1 2 3 4
Experimental data
Wav
e hei
ght
(m)
X (m)minus05
minus1minus2minus3minus4
SPH
(b)
0
05
0 2 4
Wav
e hei
ght
(m)
X (m)minus2minus4
Experimental dataSPH
(c)
Experimental dataSPH
0
05
0 1 2 3 4
Wav
e hei
ght
(m)
X (m)minus05
minus1minus2minus3minus4
(d)
Figure 4 Comparison of SPHrsquos free surface profile with experimen-tal data [33]
Loveral
L0 L1
L2
1205731
1205732
d
h
Figure 5 Computational domain
Table 2 Computational error
Time (sec) Figure RMS10 Figure 4(a) 00361181 Figure 4(b) 003413 Figure 4(c) 00491392 Figure 4(d) 0053
presented in Table 3 Furthermore geometrical characteris-tics of breakwater are also proposed in this table
Table 3 Breakwaterrsquos characteristics
1205732
1205731
ℎ 1198711
0 tanminus1(05) 04 215 tanminus1(05) 04 245 tanminus1(05) 04 260 tanminus1(05) 04 275 tanminus1(05) 04 290 tanminus1(05) 04 2
0010203
0 20 40 60 80 100X (m)
minus01minus02minus03W
ave h
eigh
t (m
)
Outcome wave (120573 = 0)
Incoming wave
Figure 6 Free surface profile for 1205732= 0 degree
Free surface profile related to each considered conditionis illustrated in Figures 6 7 8 9 10 and 11 In each caseobtained free surface profile is compared with generatedsolitary wave without considering breakwater effect Exactevaluation of breakwaterrsquos slope on the free surface profileis presented in Table 4 It is obvious that wherever 120573
2= 90
maximumwave reduction can be achieved Correspondinglywhen 120573
2decreased to zero minimum breakwater effect
occurs Generally Figure 12 also shows a representative frameof SPH solution
73 Investigation of Breakwater Length To perform furtherparametric studies the effects of breakwaterrsquos length onfree surface can also be evaluated In this context previouscomputational characteristics are kept and only the lengthratio L1L2 is changed systematically Considered situationsare presented in Table 5 Figures 13 and 16 show the obtainedresults at various length ratios In all cases breakwatergenerates a hollow in free surface at breakwater positionwhich leads to a reduction in wave crest Generally basedon Table 6 it is observed that L1L2 ratio decreased sobreakwater performance will be improved In addition itcan be concluded that smaller length ratio leads to morereduction of wave amplitude To complete this section a viewof SPH solution is also presented in Figure 17
74 Investigation of Distance between Two Submerged Break-waters In this part influences of distance variation betweentwo breakwaters are evaluated Distance between two break-waters (119883) is considered to be less than wavelength at thebeginning of simulations Consequently this distance willincrease until the gap becomes larger than the wavelengthIn addition wave height water depth and length of waveflume are considered to be 119867 = 03m 12m and 70msuccessively Breakwater is located 30 meters from the leftGeometrical characteristics of breakwaters are also presented
6 Modelling and Simulation in Engineering
Table 4 Effect of breakwater slope on HH1015840 ratio
Values 1205732= 0 120573
2= 15 120573
2= 45 120573
2= 60 120573
2= 75 120573
2= 90
HH1015840 1078973 1185936 1297549 132081 1470258 1503094
Table 5 Geometrical characteristics
11987111198712
1205731
1205732
1198712
1198711
9 tanminus1 (05) tanminus1(05) 02 18633 tanminus1(05) tanminus1(05) 03 1942 tanminus1(05) tanminus1(05) 05 21367 tanminus1(05) tanminus1(05) 06 22
0010203
0 20 40 60 80 100
Wav
e hei
ght
(m)
X (m)minus01minus02
Outcome wave (120573 = 15)
Incoming wave
Figure 7 Free surface profile for 1205732= 15 degree
in Table 7 Figure 18 shows the gap effects between twoconsidered breakwaters It is observed that when wavereaches the first breakwater the wave height is equal to025m However second breakwater decreases the waveheight in correspondence with considered distance betweentwo breakwaters In this context two parameters 119867 (waveheight above first breakwater) and 1198671015840 (wave height abovesecond breakwater) are introduced Breakwater influencecoefficient is also defined as 1198671198671015840 Precise values of definedparameter are presented in Table 8 It is obviously clear thatfor 119883 = 03m maximum wave reduction is achievedFurthermore it can be concluded that optimum gap betweenbreakwaters is affected by wave length In reality when gapbetween breakwaters is smaller than wave length effects ofsecond breakwater will be decreased Additionallymaximumeffectiveness of breakwater is observedwhenever the distanceis equal to wave length A representation of our SPH solutionis also depicted in Figure 19
8 Results
The present paper focuses on the breakwater performanceunder a solitary wave Furthermore we have tried to imple-ment meshless and Lagrangian approach of smoothed par-ticle hydrodynamics method to study considered physicsIn this context a two-dimensional SPH code is developedTo consider diffusion terms an artificial viscosity model isincluded Particles movement is also calculated using XSPHvariant and predictor-corrector scheme which is investigatedas time stepping algorithm To reinitialize density field MLSdensity filter is utilized
To start computations generated solitary wave is verifiedby comparing previous studies Afterwards a breakwater
001020304
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus01minus02
Outcome wave (120573 = 45)
Incoming wave
Figure 8 Free surface profile for 1205732= 45 degree
0010203
0 20 40 60 80 100
Outcome wave (120573 = 60)
Incoming wave
Wav
e hei
ght
(m)
X (m)minus01minus02
Figure 9 Free surface profile for 1205732= 60 degree
00204
0 20 40 60 80 100
Wav
e hei
ght
(m)
X (m)minus02
minus04
Outcome wave (120573 = 75)
Incoming wave
Figure 10 Free surface profile for 1205732= 75 degree
0
05
0 20 40 60 80 100
Wav
e hei
ght
(m)
X (m)minus05
Outcome wave (120573 = 90)
Incoming wave
Figure 11 Free surface profile for 1205732= 90 degree
Figure 12 A representative frame of SPH solution
Modelling and Simulation in Engineering 7
Table 6 Exact evaluation of length ratio effect on free surface reduction
Values 11987111198712= 9 119871
11198712= 633 119871
11198712= 42 119871
11198712= 367
HH1015840 1094116 1105263 1111112 1222223
0
02
04
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus02
Incoming waveOutcoming wave (120574 = 90)
Figure 13 Free surface profile with ratio 9 in length
0010203
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus01
Incoming waveOutcome wave (120574 = 633)
Figure 14 Free surface profile at length ratio 633
Table 7 Breakwater characteristics
(119883) Left sideslope
Right sideslope
Breakwaterheight
Lowerlength
02 tanminus1(05) tanminus1(05) 04 203 tanminus1(05) tanminus1(05) 04 204 tanminus1(05) tanminus1(05) 04 205 tanminus1(05) tanminus1(05) 04 2
Table 8 Effects of breakwaters distance
Damping ratio 119883 = 02 119883 = 03 119883 = 04 119883 = 05
HH1015840 1247285 142718 13381 1337985
is also included in computational domain To validate ourSPH solutions obtained free surface profile is compared withexisting experimental results and it is observed that our SPHsolutions have a reasonable accuracy In the second phasevarious geometrical aspects of breakwater are also studied
Generally effects of implementing two breakwaters incli-nation of breakwater and breakwaterrsquos length on free surfacedepression are evaluated It is found that when gap betweenbreakwaters is smaller than wave length effects of secondbreakwater will be decreased Maximum effectiveness ofbreakwater also occurred whenever the distance is equal towave length In addition when the inclination of breakwaterincreased free surface deformation will be modified Fur-thermore it is observed that length ratio of breakwater caneffectively influence free surface profile (Figures 14 and 15)
0010203
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus01
Incoming waveOutcome wave (120574 = 42)
Figure 15 Free surface profile at length ratio 42
0010203
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus01
Incoming waveOutcome wave (120574 = 367)
Figure 16 Free surface profile at length ratio 367
Figure 17 A view of SPH solution
0
02
04
020 40 60 80 100W
ave h
eigh
t (m
)
X (m)minus02
Incoming waveOutcome wave (X = 02)
Outcome wave (X = 03)
Outcome wave (X = 04)
Figure 18 Simulated free surface profiles at different distancesbetween breakwaters
Figure 19 Particle representation of SPH solution
8 Modelling and Simulation in Engineering
To perform more precise computation extending SPHsolutions to three-dimensional case will be in attendanceVarious simple breakwater geometries will also be introducedin near future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Tokura and T Ida ldquoSimulation of wave dissipation mech-anism on submerged structure using fluid-structure couplingcapability in LS-DYNArdquo in Proceedings of the 5th European LS-DYNA Conference 2005
[2] M Hara T Yasuda and Y Sakakibara ldquoCharacteristics of asolitary wave breaking caused by a submerged obstaclerdquoCoastalEngineering vol 1 no 23 1992
[3] N Hayakawa T Hosoyamada S Yoshida and G TsujimotoldquoNumerical simulation of wave fields around the submergedbreakwater with SOLA-SURF methodrdquo in Proceedings of the26th International Conference on Coastal Engineering (ICCErsquo98) pp 843ndash852 June 1998
[4] W Decheng and W Guoxiong ldquoNumerical simulation of asolitary wave interaction with submerged multi-bodiesrdquo ActaMechanica Sinica vol 14 no 4 pp 304ndash305 1998
[5] H D Armono and K R Hall ldquoWave transmission on sub-merged breakwaters made of hollow hemispherical shape arti-ficial reefsrdquo in Canadian Coastal Conference pp 313ndash322 June2003
[6] A Yoshida S Yan M Yamashiro and I Irie ldquoWave FieldBehind a Double-Submerged Breakwaterrdquo in Proceedings of the12th International Offshore and Polar Engineering Conferencepp 785ndash790 May 2002
[7] C-J Huang H-H Chang and H-H Hwung ldquoStructuralpermeability effects on the interaction of a solitary wave anda submerged breakwaterrdquo Coastal Engineering vol 49 no 1-2pp 1ndash24 2003
[8] D G Stamos M R Hajj and D P Telionis ldquoPerformanceof hemi-cylindrical and rectangular submerged breakwatersrdquoOcean Engineering vol 30 no 6 pp 813ndash828 2003
[9] H K Johnson T V Karambas I Avgeris B Zanuttigh DGonzalez-Marco and I Caceres ldquoModelling of waves andcurrents around submerged breakwatersrdquo Coastal Engineeringvol 52 no 10-11 pp 949ndash969 2005
[10] M G Muni-Reddy and S Neelamani ldquoWave interaction withcaisson defenced by an offshore low-crested breakwaterrdquo Jour-nal of Coastal Research no 39 pp 1767ndash1770 2006
[11] H-B Chen C-P Tsai and C-C Jeng ldquoWave transformationbetween submerged breakwater and seawallrdquo Journal of CoastalResearch no 50 pp 1069ndash1074 2007
[12] M Christou C Swan and O T Gudmestad ldquoThe interactionof surface water waves with submerged breakwatersrdquo CoastalEngineering vol 55 no 12 pp 945ndash958 2008
[13] M H Zaman H Togashi and R E Baddour ldquoDeformationof monochromatic water wave trains propagating over a sub-merged obstacle in the presence of uniform currentsrdquo OceanEngineering vol 35 no 8-9 pp 823ndash833 2008
[14] I J Losada J L Lara R Guanche and J M Gonzalez-OndinaldquoNumerical analysis of wave overtopping of rubble moundbreakwatersrdquoCoastal Engineering vol 55 no 1 pp 47ndash62 2008
[15] Y-G Cao C-B Jiang and Y-C Bai ldquoNumerical study onflow structure near two impermeable trapezoid submergedbreakwaters on slop bottomsrdquo Journal of Hydrodynamics vol22 no 5 pp 190ndash196 2010
[16] R A Dalrymple O Knio D T Cox M Gesteira and S ZouldquoUsing a lagrangian particle method for deck overtoppingrdquo inProceedings of the 4th International SymposiumWaves pp 1082ndash1091 ASCE September 2001
[17] H Gotoh S Shao and T Memita ldquoSPH-LES model for numer-ical investigation of wave interaction with partially immersedbreakwaterrdquo Coastal Engineering Journal vol 46 no 1 pp 39ndash63 2004
[18] S Shao ldquoIncrompressible SPH simulation of wave breaking andovertopping with turbulence modellingrdquo International Journalfor Numerical Methods in Fluids vol 50 no 5 pp 597ndash6212006
[19] R A Dalrymple and B D Rogers ldquoNumerical modeling ofwater waves with the SPHmethodrdquoCoastal Engineering vol 53no 2-3 pp 141ndash147 2006
[20] S C Yim D Yuk A Panizzo M di Risio and P L-F LiuldquoNumerical simulations ofwave generation by a vertical plungerusing RANS and SPH modelsrdquo Journal of Waterway PortCoastal andOcean Engineering vol 134 no 3 pp 143ndash159 2008
[21] J M Cherfils L Blonce G Pinon and E Rivoalen ldquoSimulationof water wavemdashcoastal structure interactions by smoothedparticle hydrodynamicsrdquo in Proceedings of the 8th InternationalConference onHydrodynamics Nantes France September 2008
[22] E Didier and M G Neves ldquoWave over topping of a typicalcoastal structure of the portuguese coast using A SPH modelrdquoJournal of Coastal Research vol 56 pp 496ndash500 2009
[23] S Shao ldquoIncompressible SPH flow model for wave interactionswith porous mediardquo Coastal Engineering vol 57 no 3 pp 304ndash316 2010
[24] C Liu J Zhang and Y Sun ldquoThe optimization of SPHmethod and its application in simulation of water waverdquo inProceedings of the IEEE 7th International Conference on NaturalComputation (ICNC rsquo11) pp 2327ndash2331 July 2011
[25] PW ClearyM Prakash J Ha N Stokes and C Scott ldquoSmoothparticle hydrodynamics status and future potentialrdquo Progress inComputational Fluid Dynamics vol 7 no 2-4 pp 70ndash90 2007
[26] M B Liu and G R Liu ldquoSmoothed particle hydrodynamics(SPH) an overview and recent developmentsrdquo Archives ofComputational Methods in Engineering vol 17 no 1 pp 25ndash762010
[27] G R Liu and M B Liu Smoothed Particle HydrodynamicsmdashAMeshfree Particle Methods World Scientific Publishing 2007
[28] J JMonaghan and J C Lattanzio ldquoA refined particlemethod forastrophysical problemsrdquo Astronomy and Astrophysics vol 149pp 135ndash143 1985
[29] J J Monaghan ldquoSmoothed particle hydrodynamicsrdquo AnnualReview of Astronomy and Astrophysics vol 30 no 1 pp 543ndash574 1992
[30] A Colagrossi and M Landrini ldquoNumerical simulation ofinterfacial flows by smoothed particle hydrodynamicsrdquo Journalof Computational Physics vol 191 no 2 pp 448ndash475 2003
[31] A Crespo Application of smoothed particle hydrodynamicsmodel SPHysics to free surface hydrodynamics [Doctoral thesis]2008
Modelling and Simulation in Engineering 9
[32] S Maiti and D Sen ldquoComputation of solitary waves duringpropagation and runup on a sloperdquo Ocean Engineering vol 26no 11 pp 1063ndash1083 1999
[33] S T Grilli M A Losada and F Martin ldquoCharacteristics ofsolitary wave breaking induced by breakwatersrdquo Journal ofWaterway Port Coastal ampOcean Engineering vol 120 no 1 pp74ndash92 1994
International Journal of
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International Journal of
Modelling and Simulation in Engineering 3
where119898119895120588119895is the volume associated with the particle 119895 This
equation can be used if the function 119891(119909) is only known at119873discrete points
The kernel function description can be finalized bydefining a specific kernel function Numerous possibilitiesexist A large number of kernel function types are discussedin literature ranging from polynomial to Gaussian Themost common is the B-spline kernel that was proposed byMonaghan and Lattanzio [28]
119908 (119903 ℎ) = 120572119863
1 minus3
21199022+3
411990230 le 119902 le 1
1
4(2 minus 119902)
3
1 le 119902 le 2
0 119902 ge 2
(6)
which is considered in the present paper and 119902 = 119903ℎ and 120572119863
is equal to 10120587ℎ2 in two dimensional cases
4 SPH Implementation onGoverning Equations
Now based on the SPH formulation mentioned in previoussection governing equations will be discretized In this con-text the continuity equation for particle 119894 can be consideredin the SPH formulation as
119889120588119894
119889119905= minus
119873
sum
119895=1
119898119895(V119895minus V119895) sdot nabla119882
119894119895 (7)
where the sum extends over all neighboring particles and119882is the smoothing kernel evaluated at the distance betweenparticles 119894 and 119895
The SPH formulation of the momentum equations basedon artificial viscosity can also be obtained through theparticle approximation procedure as in
119889V119894
119889119905= minus
119873
sum
119895=1
119898119895(119875119894
1205882
119894
+
119875119895
1198752
119895
+ Π119894119895)nabla119882
119894119895+ 119892 (8)
where Π119894119895is the viscosity term To introduce viscous dissi-
pation an artificial viscosity concept which has been devel-oped using Monaghan [29] is implemented This term issymmetric conserves momentum and introduces a shearviscosity into the momentum equation The SPH form of themomentum equation is written as
Π119886119887=
minus120572119862119886119887120583119886119887
120588119886119887
119881119886119887sdot 119903119886119887lt 0
0 for other values(9)
where
120583119886119887=ℎV119886119887sdot 119903119886119887
1199032
119886119887+ 1205782
(10)
120588119886119887=1
2(120588119886+ 120588119887) 119862
119886119887=1
2(119862119886+ 119862119887)
1205782= 001ℎ
2
(11)
120572 is also a constant Monaghan [29] suggests that a value of120572 = 001 can be used for most computations
5 Numerical Details
51 XSPH Technique Tomodify the particlersquos velocity XSPHtechnique is used and particle 119894 is moved based on thistechnique In XSPH scheme it is formulated as follows
119889119903119894
119889119905= V119894+ 120576sum
119895
119898119895
120588119894119895
V119894119895119882119894119895 (12)
Velocity of particle 119894 is recalculated in such manner thatthe velocity of particle 119894 and the average velocity of the closestneighborhood particles interacting with particle 119894 are takeninto account
52 Time Stepping Procedure In the present paper predictor-corrector time stepping algorithm is implemented In thepredictor-corrector formulation variation of particles prop-erties (such as velocity) is estimated as [17] Consider
V119899+12119894
= V119899119894+Δ119905
2119865119899+12
119894 (13)
And by implementing forces at half step the values ofvelocity can be calculated at the end of any step which isformulated as follows [17]
V119899+1119894= 2V119899+12119894
minus V119899119894 (14)
53 Density Filter In weakly compressible SPH formulationlarge pressure oscillation has been observed These pressureinaccuracies can be modified using some procedures Inaddition to pressure modification the free surface profilewill be also improved Therefore MLS (moving least square)density filter which is a first order density correction is usedto reproduce the linear variation of the density field
120588119886= sum
119887
120588119887119882
MLS119886119887
119898119887
120588119887
= sum
119887
119898119887119882
MLS119886119887 (15)
The corrected kernel is also evaluated as follows
119882MLS119886119887
= 119882MLS119886119887
(119903119886) = 120573 (119903
119886) sdot (119903119886minus 119903119887)119882119886119887 (16)
More details can be found in many references in theliterature [30 31]
54 Boundary Conditions Two rows of stationary waterparticles at the boundary (Figure 1) remain fixed or moveaccording to an externally imposed function for say wavemaker boundaries When a fluid particle approaches theboundary the density and hence pressure exerted by theboundary particles increase This generates the necessaryrepulsive force on the water particles A major deficiency ofthis method is the generation of numerical boundary layersWhenwater particles get close to the boundary particles theytend to stick to the boundary thus they generate a boundarylayer which is not physically observed
4 Modelling and Simulation in Engineering
dx dx
dx2 dz2
Figure 1 Interaction betweenfluid particles and boundary particles
6 Solitary Wave Generation
Solitary wave is generated by Prescribe Piston Wave Maker[5] Value of paddle displacement is applied in the followingequation [22]
119883119901(119905) =
119867
119896[tanh (119883 (119905)) + tanh(119896
119889120582)]
119883 (119905) =119896
119889(119888119905 minus 119883
119901(119905) minus 120582)
(17)
where119867 is the wave height 119896 = radic((31198674119889)) is wave numberand 119888 = radic(119892(119889 + 119867)) is speed Wave speed can also beobtained from the following equation [22]
119906119901(119905) =
119888119867
119889
1
cosh2119883 (119905) + (119867119889) (18)
61 Validation of Generated Solitary Wave According toMaiti and Sen [32] simulation which is a benchmark test thelength and flat length of the tank water depth and slope ofthe right hand side of the tank were respectively consideredto be 10m 97m 03m and 45 degree (Figure 2)Themotionof wave paddle is calculated for 119867119889 = 01 Therefore waveheight and 120582 are 003 and 6882
The result of solitary wave generation is illustrated inFigure 3 Simulation is conducted from 0 till 10 sec It isobserved that the wave height is equal to 003m and obtainedfree surface profile is relatively accurate for further studiesHowever some inconsistencies are visible at second 4 Inaddition average error of SPH simulation (point to point) isequal to 189 percent
7 Results and Discussion
In this section various aspects of solitary wave-breakwaterinteraction will be studied At first step free surface related toa specific breakwater under interaction with a solitary waveis computed using SPH to ensure that our numerical setupis suitable for investigation of more complicated problemsAfterward various geometrical characteristics of breakwatersare also discussed Formore detailed studies two breakwatersare considered and effects of distance between these breakwa-ters will be evaluated
71 Validation For further evaluation of SPH accuracy thefree surface profile generated in solitary wave-breakwater
H
d
R
L
120573
Figure 2 Computational domain
0
002
004
0 2 4 6 8 10 12Wav
e hei
ght
(m)
Maiti and Sen (1999)SPH
minus002X (m)
Figure 3 Comparison of generated solitary wave with Sagnitarsquosresults
Table 1 Wave flume solitary wave and breakwater features
Value Characteristics03m Wave height (119867)10m Water depth (119889)45m (119871
0)
70m (119871overal)08m Height of breakwater (ℎ)04m Breakwaterrsquos beam1 2 Slope of breakwaterrsquos wall
interaction is compared with experimental work of Grilliet al [33] To perform this comparison a computationaldomain is defined (Table 1) Furthermore the characteristicsof wave flume solitary wave and breakwater are presentedResults of this simulation are illustrated at Figure 4 Featuresof the presented simulation are also reported in Table 2 Freesurface profiles in Figure 4(a) through 4(d) are at 10 1181 13and 1392 sec respectively RMS of obtained solutions whichare calculated relative to experimental findings seem to bereasonable Therefore it can be declared that our numericalsetup can model solitary wave-breakwater interaction withsatisfying precision
72 Investigation of Breakwater Inclination A subject studiedhere is considering the effects of breakwaterrsquos slope on1198671198671015840ratio For this purpose computational domain as well assolitary wave is considered to be the same as previous sectionSlope of breakwater at aft part (120573
2) which is shown in
Figure 5 is varied systematically All considered angles are
Modelling and Simulation in Engineering 5
0
05
0 1 2 3 4
Wav
e hei
ght
(m)
X (m)minus05
minus1minus2minus3minus4
Experimental dataSPH
(a)
0
05
0 1 2 3 4
Experimental data
Wav
e hei
ght
(m)
X (m)minus05
minus1minus2minus3minus4
SPH
(b)
0
05
0 2 4
Wav
e hei
ght
(m)
X (m)minus2minus4
Experimental dataSPH
(c)
Experimental dataSPH
0
05
0 1 2 3 4
Wav
e hei
ght
(m)
X (m)minus05
minus1minus2minus3minus4
(d)
Figure 4 Comparison of SPHrsquos free surface profile with experimen-tal data [33]
Loveral
L0 L1
L2
1205731
1205732
d
h
Figure 5 Computational domain
Table 2 Computational error
Time (sec) Figure RMS10 Figure 4(a) 00361181 Figure 4(b) 003413 Figure 4(c) 00491392 Figure 4(d) 0053
presented in Table 3 Furthermore geometrical characteris-tics of breakwater are also proposed in this table
Table 3 Breakwaterrsquos characteristics
1205732
1205731
ℎ 1198711
0 tanminus1(05) 04 215 tanminus1(05) 04 245 tanminus1(05) 04 260 tanminus1(05) 04 275 tanminus1(05) 04 290 tanminus1(05) 04 2
0010203
0 20 40 60 80 100X (m)
minus01minus02minus03W
ave h
eigh
t (m
)
Outcome wave (120573 = 0)
Incoming wave
Figure 6 Free surface profile for 1205732= 0 degree
Free surface profile related to each considered conditionis illustrated in Figures 6 7 8 9 10 and 11 In each caseobtained free surface profile is compared with generatedsolitary wave without considering breakwater effect Exactevaluation of breakwaterrsquos slope on the free surface profileis presented in Table 4 It is obvious that wherever 120573
2= 90
maximumwave reduction can be achieved Correspondinglywhen 120573
2decreased to zero minimum breakwater effect
occurs Generally Figure 12 also shows a representative frameof SPH solution
73 Investigation of Breakwater Length To perform furtherparametric studies the effects of breakwaterrsquos length onfree surface can also be evaluated In this context previouscomputational characteristics are kept and only the lengthratio L1L2 is changed systematically Considered situationsare presented in Table 5 Figures 13 and 16 show the obtainedresults at various length ratios In all cases breakwatergenerates a hollow in free surface at breakwater positionwhich leads to a reduction in wave crest Generally basedon Table 6 it is observed that L1L2 ratio decreased sobreakwater performance will be improved In addition itcan be concluded that smaller length ratio leads to morereduction of wave amplitude To complete this section a viewof SPH solution is also presented in Figure 17
74 Investigation of Distance between Two Submerged Break-waters In this part influences of distance variation betweentwo breakwaters are evaluated Distance between two break-waters (119883) is considered to be less than wavelength at thebeginning of simulations Consequently this distance willincrease until the gap becomes larger than the wavelengthIn addition wave height water depth and length of waveflume are considered to be 119867 = 03m 12m and 70msuccessively Breakwater is located 30 meters from the leftGeometrical characteristics of breakwaters are also presented
6 Modelling and Simulation in Engineering
Table 4 Effect of breakwater slope on HH1015840 ratio
Values 1205732= 0 120573
2= 15 120573
2= 45 120573
2= 60 120573
2= 75 120573
2= 90
HH1015840 1078973 1185936 1297549 132081 1470258 1503094
Table 5 Geometrical characteristics
11987111198712
1205731
1205732
1198712
1198711
9 tanminus1 (05) tanminus1(05) 02 18633 tanminus1(05) tanminus1(05) 03 1942 tanminus1(05) tanminus1(05) 05 21367 tanminus1(05) tanminus1(05) 06 22
0010203
0 20 40 60 80 100
Wav
e hei
ght
(m)
X (m)minus01minus02
Outcome wave (120573 = 15)
Incoming wave
Figure 7 Free surface profile for 1205732= 15 degree
in Table 7 Figure 18 shows the gap effects between twoconsidered breakwaters It is observed that when wavereaches the first breakwater the wave height is equal to025m However second breakwater decreases the waveheight in correspondence with considered distance betweentwo breakwaters In this context two parameters 119867 (waveheight above first breakwater) and 1198671015840 (wave height abovesecond breakwater) are introduced Breakwater influencecoefficient is also defined as 1198671198671015840 Precise values of definedparameter are presented in Table 8 It is obviously clear thatfor 119883 = 03m maximum wave reduction is achievedFurthermore it can be concluded that optimum gap betweenbreakwaters is affected by wave length In reality when gapbetween breakwaters is smaller than wave length effects ofsecond breakwater will be decreased Additionallymaximumeffectiveness of breakwater is observedwhenever the distanceis equal to wave length A representation of our SPH solutionis also depicted in Figure 19
8 Results
The present paper focuses on the breakwater performanceunder a solitary wave Furthermore we have tried to imple-ment meshless and Lagrangian approach of smoothed par-ticle hydrodynamics method to study considered physicsIn this context a two-dimensional SPH code is developedTo consider diffusion terms an artificial viscosity model isincluded Particles movement is also calculated using XSPHvariant and predictor-corrector scheme which is investigatedas time stepping algorithm To reinitialize density field MLSdensity filter is utilized
To start computations generated solitary wave is verifiedby comparing previous studies Afterwards a breakwater
001020304
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus01minus02
Outcome wave (120573 = 45)
Incoming wave
Figure 8 Free surface profile for 1205732= 45 degree
0010203
0 20 40 60 80 100
Outcome wave (120573 = 60)
Incoming wave
Wav
e hei
ght
(m)
X (m)minus01minus02
Figure 9 Free surface profile for 1205732= 60 degree
00204
0 20 40 60 80 100
Wav
e hei
ght
(m)
X (m)minus02
minus04
Outcome wave (120573 = 75)
Incoming wave
Figure 10 Free surface profile for 1205732= 75 degree
0
05
0 20 40 60 80 100
Wav
e hei
ght
(m)
X (m)minus05
Outcome wave (120573 = 90)
Incoming wave
Figure 11 Free surface profile for 1205732= 90 degree
Figure 12 A representative frame of SPH solution
Modelling and Simulation in Engineering 7
Table 6 Exact evaluation of length ratio effect on free surface reduction
Values 11987111198712= 9 119871
11198712= 633 119871
11198712= 42 119871
11198712= 367
HH1015840 1094116 1105263 1111112 1222223
0
02
04
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus02
Incoming waveOutcoming wave (120574 = 90)
Figure 13 Free surface profile with ratio 9 in length
0010203
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus01
Incoming waveOutcome wave (120574 = 633)
Figure 14 Free surface profile at length ratio 633
Table 7 Breakwater characteristics
(119883) Left sideslope
Right sideslope
Breakwaterheight
Lowerlength
02 tanminus1(05) tanminus1(05) 04 203 tanminus1(05) tanminus1(05) 04 204 tanminus1(05) tanminus1(05) 04 205 tanminus1(05) tanminus1(05) 04 2
Table 8 Effects of breakwaters distance
Damping ratio 119883 = 02 119883 = 03 119883 = 04 119883 = 05
HH1015840 1247285 142718 13381 1337985
is also included in computational domain To validate ourSPH solutions obtained free surface profile is compared withexisting experimental results and it is observed that our SPHsolutions have a reasonable accuracy In the second phasevarious geometrical aspects of breakwater are also studied
Generally effects of implementing two breakwaters incli-nation of breakwater and breakwaterrsquos length on free surfacedepression are evaluated It is found that when gap betweenbreakwaters is smaller than wave length effects of secondbreakwater will be decreased Maximum effectiveness ofbreakwater also occurred whenever the distance is equal towave length In addition when the inclination of breakwaterincreased free surface deformation will be modified Fur-thermore it is observed that length ratio of breakwater caneffectively influence free surface profile (Figures 14 and 15)
0010203
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus01
Incoming waveOutcome wave (120574 = 42)
Figure 15 Free surface profile at length ratio 42
0010203
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus01
Incoming waveOutcome wave (120574 = 367)
Figure 16 Free surface profile at length ratio 367
Figure 17 A view of SPH solution
0
02
04
020 40 60 80 100W
ave h
eigh
t (m
)
X (m)minus02
Incoming waveOutcome wave (X = 02)
Outcome wave (X = 03)
Outcome wave (X = 04)
Figure 18 Simulated free surface profiles at different distancesbetween breakwaters
Figure 19 Particle representation of SPH solution
8 Modelling and Simulation in Engineering
To perform more precise computation extending SPHsolutions to three-dimensional case will be in attendanceVarious simple breakwater geometries will also be introducedin near future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Tokura and T Ida ldquoSimulation of wave dissipation mech-anism on submerged structure using fluid-structure couplingcapability in LS-DYNArdquo in Proceedings of the 5th European LS-DYNA Conference 2005
[2] M Hara T Yasuda and Y Sakakibara ldquoCharacteristics of asolitary wave breaking caused by a submerged obstaclerdquoCoastalEngineering vol 1 no 23 1992
[3] N Hayakawa T Hosoyamada S Yoshida and G TsujimotoldquoNumerical simulation of wave fields around the submergedbreakwater with SOLA-SURF methodrdquo in Proceedings of the26th International Conference on Coastal Engineering (ICCErsquo98) pp 843ndash852 June 1998
[4] W Decheng and W Guoxiong ldquoNumerical simulation of asolitary wave interaction with submerged multi-bodiesrdquo ActaMechanica Sinica vol 14 no 4 pp 304ndash305 1998
[5] H D Armono and K R Hall ldquoWave transmission on sub-merged breakwaters made of hollow hemispherical shape arti-ficial reefsrdquo in Canadian Coastal Conference pp 313ndash322 June2003
[6] A Yoshida S Yan M Yamashiro and I Irie ldquoWave FieldBehind a Double-Submerged Breakwaterrdquo in Proceedings of the12th International Offshore and Polar Engineering Conferencepp 785ndash790 May 2002
[7] C-J Huang H-H Chang and H-H Hwung ldquoStructuralpermeability effects on the interaction of a solitary wave anda submerged breakwaterrdquo Coastal Engineering vol 49 no 1-2pp 1ndash24 2003
[8] D G Stamos M R Hajj and D P Telionis ldquoPerformanceof hemi-cylindrical and rectangular submerged breakwatersrdquoOcean Engineering vol 30 no 6 pp 813ndash828 2003
[9] H K Johnson T V Karambas I Avgeris B Zanuttigh DGonzalez-Marco and I Caceres ldquoModelling of waves andcurrents around submerged breakwatersrdquo Coastal Engineeringvol 52 no 10-11 pp 949ndash969 2005
[10] M G Muni-Reddy and S Neelamani ldquoWave interaction withcaisson defenced by an offshore low-crested breakwaterrdquo Jour-nal of Coastal Research no 39 pp 1767ndash1770 2006
[11] H-B Chen C-P Tsai and C-C Jeng ldquoWave transformationbetween submerged breakwater and seawallrdquo Journal of CoastalResearch no 50 pp 1069ndash1074 2007
[12] M Christou C Swan and O T Gudmestad ldquoThe interactionof surface water waves with submerged breakwatersrdquo CoastalEngineering vol 55 no 12 pp 945ndash958 2008
[13] M H Zaman H Togashi and R E Baddour ldquoDeformationof monochromatic water wave trains propagating over a sub-merged obstacle in the presence of uniform currentsrdquo OceanEngineering vol 35 no 8-9 pp 823ndash833 2008
[14] I J Losada J L Lara R Guanche and J M Gonzalez-OndinaldquoNumerical analysis of wave overtopping of rubble moundbreakwatersrdquoCoastal Engineering vol 55 no 1 pp 47ndash62 2008
[15] Y-G Cao C-B Jiang and Y-C Bai ldquoNumerical study onflow structure near two impermeable trapezoid submergedbreakwaters on slop bottomsrdquo Journal of Hydrodynamics vol22 no 5 pp 190ndash196 2010
[16] R A Dalrymple O Knio D T Cox M Gesteira and S ZouldquoUsing a lagrangian particle method for deck overtoppingrdquo inProceedings of the 4th International SymposiumWaves pp 1082ndash1091 ASCE September 2001
[17] H Gotoh S Shao and T Memita ldquoSPH-LES model for numer-ical investigation of wave interaction with partially immersedbreakwaterrdquo Coastal Engineering Journal vol 46 no 1 pp 39ndash63 2004
[18] S Shao ldquoIncrompressible SPH simulation of wave breaking andovertopping with turbulence modellingrdquo International Journalfor Numerical Methods in Fluids vol 50 no 5 pp 597ndash6212006
[19] R A Dalrymple and B D Rogers ldquoNumerical modeling ofwater waves with the SPHmethodrdquoCoastal Engineering vol 53no 2-3 pp 141ndash147 2006
[20] S C Yim D Yuk A Panizzo M di Risio and P L-F LiuldquoNumerical simulations ofwave generation by a vertical plungerusing RANS and SPH modelsrdquo Journal of Waterway PortCoastal andOcean Engineering vol 134 no 3 pp 143ndash159 2008
[21] J M Cherfils L Blonce G Pinon and E Rivoalen ldquoSimulationof water wavemdashcoastal structure interactions by smoothedparticle hydrodynamicsrdquo in Proceedings of the 8th InternationalConference onHydrodynamics Nantes France September 2008
[22] E Didier and M G Neves ldquoWave over topping of a typicalcoastal structure of the portuguese coast using A SPH modelrdquoJournal of Coastal Research vol 56 pp 496ndash500 2009
[23] S Shao ldquoIncompressible SPH flow model for wave interactionswith porous mediardquo Coastal Engineering vol 57 no 3 pp 304ndash316 2010
[24] C Liu J Zhang and Y Sun ldquoThe optimization of SPHmethod and its application in simulation of water waverdquo inProceedings of the IEEE 7th International Conference on NaturalComputation (ICNC rsquo11) pp 2327ndash2331 July 2011
[25] PW ClearyM Prakash J Ha N Stokes and C Scott ldquoSmoothparticle hydrodynamics status and future potentialrdquo Progress inComputational Fluid Dynamics vol 7 no 2-4 pp 70ndash90 2007
[26] M B Liu and G R Liu ldquoSmoothed particle hydrodynamics(SPH) an overview and recent developmentsrdquo Archives ofComputational Methods in Engineering vol 17 no 1 pp 25ndash762010
[27] G R Liu and M B Liu Smoothed Particle HydrodynamicsmdashAMeshfree Particle Methods World Scientific Publishing 2007
[28] J JMonaghan and J C Lattanzio ldquoA refined particlemethod forastrophysical problemsrdquo Astronomy and Astrophysics vol 149pp 135ndash143 1985
[29] J J Monaghan ldquoSmoothed particle hydrodynamicsrdquo AnnualReview of Astronomy and Astrophysics vol 30 no 1 pp 543ndash574 1992
[30] A Colagrossi and M Landrini ldquoNumerical simulation ofinterfacial flows by smoothed particle hydrodynamicsrdquo Journalof Computational Physics vol 191 no 2 pp 448ndash475 2003
[31] A Crespo Application of smoothed particle hydrodynamicsmodel SPHysics to free surface hydrodynamics [Doctoral thesis]2008
Modelling and Simulation in Engineering 9
[32] S Maiti and D Sen ldquoComputation of solitary waves duringpropagation and runup on a sloperdquo Ocean Engineering vol 26no 11 pp 1063ndash1083 1999
[33] S T Grilli M A Losada and F Martin ldquoCharacteristics ofsolitary wave breaking induced by breakwatersrdquo Journal ofWaterway Port Coastal ampOcean Engineering vol 120 no 1 pp74ndash92 1994
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
4 Modelling and Simulation in Engineering
dx dx
dx2 dz2
Figure 1 Interaction betweenfluid particles and boundary particles
6 Solitary Wave Generation
Solitary wave is generated by Prescribe Piston Wave Maker[5] Value of paddle displacement is applied in the followingequation [22]
119883119901(119905) =
119867
119896[tanh (119883 (119905)) + tanh(119896
119889120582)]
119883 (119905) =119896
119889(119888119905 minus 119883
119901(119905) minus 120582)
(17)
where119867 is the wave height 119896 = radic((31198674119889)) is wave numberand 119888 = radic(119892(119889 + 119867)) is speed Wave speed can also beobtained from the following equation [22]
119906119901(119905) =
119888119867
119889
1
cosh2119883 (119905) + (119867119889) (18)
61 Validation of Generated Solitary Wave According toMaiti and Sen [32] simulation which is a benchmark test thelength and flat length of the tank water depth and slope ofthe right hand side of the tank were respectively consideredto be 10m 97m 03m and 45 degree (Figure 2)Themotionof wave paddle is calculated for 119867119889 = 01 Therefore waveheight and 120582 are 003 and 6882
The result of solitary wave generation is illustrated inFigure 3 Simulation is conducted from 0 till 10 sec It isobserved that the wave height is equal to 003m and obtainedfree surface profile is relatively accurate for further studiesHowever some inconsistencies are visible at second 4 Inaddition average error of SPH simulation (point to point) isequal to 189 percent
7 Results and Discussion
In this section various aspects of solitary wave-breakwaterinteraction will be studied At first step free surface related toa specific breakwater under interaction with a solitary waveis computed using SPH to ensure that our numerical setupis suitable for investigation of more complicated problemsAfterward various geometrical characteristics of breakwatersare also discussed Formore detailed studies two breakwatersare considered and effects of distance between these breakwa-ters will be evaluated
71 Validation For further evaluation of SPH accuracy thefree surface profile generated in solitary wave-breakwater
H
d
R
L
120573
Figure 2 Computational domain
0
002
004
0 2 4 6 8 10 12Wav
e hei
ght
(m)
Maiti and Sen (1999)SPH
minus002X (m)
Figure 3 Comparison of generated solitary wave with Sagnitarsquosresults
Table 1 Wave flume solitary wave and breakwater features
Value Characteristics03m Wave height (119867)10m Water depth (119889)45m (119871
0)
70m (119871overal)08m Height of breakwater (ℎ)04m Breakwaterrsquos beam1 2 Slope of breakwaterrsquos wall
interaction is compared with experimental work of Grilliet al [33] To perform this comparison a computationaldomain is defined (Table 1) Furthermore the characteristicsof wave flume solitary wave and breakwater are presentedResults of this simulation are illustrated at Figure 4 Featuresof the presented simulation are also reported in Table 2 Freesurface profiles in Figure 4(a) through 4(d) are at 10 1181 13and 1392 sec respectively RMS of obtained solutions whichare calculated relative to experimental findings seem to bereasonable Therefore it can be declared that our numericalsetup can model solitary wave-breakwater interaction withsatisfying precision
72 Investigation of Breakwater Inclination A subject studiedhere is considering the effects of breakwaterrsquos slope on1198671198671015840ratio For this purpose computational domain as well assolitary wave is considered to be the same as previous sectionSlope of breakwater at aft part (120573
2) which is shown in
Figure 5 is varied systematically All considered angles are
Modelling and Simulation in Engineering 5
0
05
0 1 2 3 4
Wav
e hei
ght
(m)
X (m)minus05
minus1minus2minus3minus4
Experimental dataSPH
(a)
0
05
0 1 2 3 4
Experimental data
Wav
e hei
ght
(m)
X (m)minus05
minus1minus2minus3minus4
SPH
(b)
0
05
0 2 4
Wav
e hei
ght
(m)
X (m)minus2minus4
Experimental dataSPH
(c)
Experimental dataSPH
0
05
0 1 2 3 4
Wav
e hei
ght
(m)
X (m)minus05
minus1minus2minus3minus4
(d)
Figure 4 Comparison of SPHrsquos free surface profile with experimen-tal data [33]
Loveral
L0 L1
L2
1205731
1205732
d
h
Figure 5 Computational domain
Table 2 Computational error
Time (sec) Figure RMS10 Figure 4(a) 00361181 Figure 4(b) 003413 Figure 4(c) 00491392 Figure 4(d) 0053
presented in Table 3 Furthermore geometrical characteris-tics of breakwater are also proposed in this table
Table 3 Breakwaterrsquos characteristics
1205732
1205731
ℎ 1198711
0 tanminus1(05) 04 215 tanminus1(05) 04 245 tanminus1(05) 04 260 tanminus1(05) 04 275 tanminus1(05) 04 290 tanminus1(05) 04 2
0010203
0 20 40 60 80 100X (m)
minus01minus02minus03W
ave h
eigh
t (m
)
Outcome wave (120573 = 0)
Incoming wave
Figure 6 Free surface profile for 1205732= 0 degree
Free surface profile related to each considered conditionis illustrated in Figures 6 7 8 9 10 and 11 In each caseobtained free surface profile is compared with generatedsolitary wave without considering breakwater effect Exactevaluation of breakwaterrsquos slope on the free surface profileis presented in Table 4 It is obvious that wherever 120573
2= 90
maximumwave reduction can be achieved Correspondinglywhen 120573
2decreased to zero minimum breakwater effect
occurs Generally Figure 12 also shows a representative frameof SPH solution
73 Investigation of Breakwater Length To perform furtherparametric studies the effects of breakwaterrsquos length onfree surface can also be evaluated In this context previouscomputational characteristics are kept and only the lengthratio L1L2 is changed systematically Considered situationsare presented in Table 5 Figures 13 and 16 show the obtainedresults at various length ratios In all cases breakwatergenerates a hollow in free surface at breakwater positionwhich leads to a reduction in wave crest Generally basedon Table 6 it is observed that L1L2 ratio decreased sobreakwater performance will be improved In addition itcan be concluded that smaller length ratio leads to morereduction of wave amplitude To complete this section a viewof SPH solution is also presented in Figure 17
74 Investigation of Distance between Two Submerged Break-waters In this part influences of distance variation betweentwo breakwaters are evaluated Distance between two break-waters (119883) is considered to be less than wavelength at thebeginning of simulations Consequently this distance willincrease until the gap becomes larger than the wavelengthIn addition wave height water depth and length of waveflume are considered to be 119867 = 03m 12m and 70msuccessively Breakwater is located 30 meters from the leftGeometrical characteristics of breakwaters are also presented
6 Modelling and Simulation in Engineering
Table 4 Effect of breakwater slope on HH1015840 ratio
Values 1205732= 0 120573
2= 15 120573
2= 45 120573
2= 60 120573
2= 75 120573
2= 90
HH1015840 1078973 1185936 1297549 132081 1470258 1503094
Table 5 Geometrical characteristics
11987111198712
1205731
1205732
1198712
1198711
9 tanminus1 (05) tanminus1(05) 02 18633 tanminus1(05) tanminus1(05) 03 1942 tanminus1(05) tanminus1(05) 05 21367 tanminus1(05) tanminus1(05) 06 22
0010203
0 20 40 60 80 100
Wav
e hei
ght
(m)
X (m)minus01minus02
Outcome wave (120573 = 15)
Incoming wave
Figure 7 Free surface profile for 1205732= 15 degree
in Table 7 Figure 18 shows the gap effects between twoconsidered breakwaters It is observed that when wavereaches the first breakwater the wave height is equal to025m However second breakwater decreases the waveheight in correspondence with considered distance betweentwo breakwaters In this context two parameters 119867 (waveheight above first breakwater) and 1198671015840 (wave height abovesecond breakwater) are introduced Breakwater influencecoefficient is also defined as 1198671198671015840 Precise values of definedparameter are presented in Table 8 It is obviously clear thatfor 119883 = 03m maximum wave reduction is achievedFurthermore it can be concluded that optimum gap betweenbreakwaters is affected by wave length In reality when gapbetween breakwaters is smaller than wave length effects ofsecond breakwater will be decreased Additionallymaximumeffectiveness of breakwater is observedwhenever the distanceis equal to wave length A representation of our SPH solutionis also depicted in Figure 19
8 Results
The present paper focuses on the breakwater performanceunder a solitary wave Furthermore we have tried to imple-ment meshless and Lagrangian approach of smoothed par-ticle hydrodynamics method to study considered physicsIn this context a two-dimensional SPH code is developedTo consider diffusion terms an artificial viscosity model isincluded Particles movement is also calculated using XSPHvariant and predictor-corrector scheme which is investigatedas time stepping algorithm To reinitialize density field MLSdensity filter is utilized
To start computations generated solitary wave is verifiedby comparing previous studies Afterwards a breakwater
001020304
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus01minus02
Outcome wave (120573 = 45)
Incoming wave
Figure 8 Free surface profile for 1205732= 45 degree
0010203
0 20 40 60 80 100
Outcome wave (120573 = 60)
Incoming wave
Wav
e hei
ght
(m)
X (m)minus01minus02
Figure 9 Free surface profile for 1205732= 60 degree
00204
0 20 40 60 80 100
Wav
e hei
ght
(m)
X (m)minus02
minus04
Outcome wave (120573 = 75)
Incoming wave
Figure 10 Free surface profile for 1205732= 75 degree
0
05
0 20 40 60 80 100
Wav
e hei
ght
(m)
X (m)minus05
Outcome wave (120573 = 90)
Incoming wave
Figure 11 Free surface profile for 1205732= 90 degree
Figure 12 A representative frame of SPH solution
Modelling and Simulation in Engineering 7
Table 6 Exact evaluation of length ratio effect on free surface reduction
Values 11987111198712= 9 119871
11198712= 633 119871
11198712= 42 119871
11198712= 367
HH1015840 1094116 1105263 1111112 1222223
0
02
04
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus02
Incoming waveOutcoming wave (120574 = 90)
Figure 13 Free surface profile with ratio 9 in length
0010203
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus01
Incoming waveOutcome wave (120574 = 633)
Figure 14 Free surface profile at length ratio 633
Table 7 Breakwater characteristics
(119883) Left sideslope
Right sideslope
Breakwaterheight
Lowerlength
02 tanminus1(05) tanminus1(05) 04 203 tanminus1(05) tanminus1(05) 04 204 tanminus1(05) tanminus1(05) 04 205 tanminus1(05) tanminus1(05) 04 2
Table 8 Effects of breakwaters distance
Damping ratio 119883 = 02 119883 = 03 119883 = 04 119883 = 05
HH1015840 1247285 142718 13381 1337985
is also included in computational domain To validate ourSPH solutions obtained free surface profile is compared withexisting experimental results and it is observed that our SPHsolutions have a reasonable accuracy In the second phasevarious geometrical aspects of breakwater are also studied
Generally effects of implementing two breakwaters incli-nation of breakwater and breakwaterrsquos length on free surfacedepression are evaluated It is found that when gap betweenbreakwaters is smaller than wave length effects of secondbreakwater will be decreased Maximum effectiveness ofbreakwater also occurred whenever the distance is equal towave length In addition when the inclination of breakwaterincreased free surface deformation will be modified Fur-thermore it is observed that length ratio of breakwater caneffectively influence free surface profile (Figures 14 and 15)
0010203
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus01
Incoming waveOutcome wave (120574 = 42)
Figure 15 Free surface profile at length ratio 42
0010203
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus01
Incoming waveOutcome wave (120574 = 367)
Figure 16 Free surface profile at length ratio 367
Figure 17 A view of SPH solution
0
02
04
020 40 60 80 100W
ave h
eigh
t (m
)
X (m)minus02
Incoming waveOutcome wave (X = 02)
Outcome wave (X = 03)
Outcome wave (X = 04)
Figure 18 Simulated free surface profiles at different distancesbetween breakwaters
Figure 19 Particle representation of SPH solution
8 Modelling and Simulation in Engineering
To perform more precise computation extending SPHsolutions to three-dimensional case will be in attendanceVarious simple breakwater geometries will also be introducedin near future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Tokura and T Ida ldquoSimulation of wave dissipation mech-anism on submerged structure using fluid-structure couplingcapability in LS-DYNArdquo in Proceedings of the 5th European LS-DYNA Conference 2005
[2] M Hara T Yasuda and Y Sakakibara ldquoCharacteristics of asolitary wave breaking caused by a submerged obstaclerdquoCoastalEngineering vol 1 no 23 1992
[3] N Hayakawa T Hosoyamada S Yoshida and G TsujimotoldquoNumerical simulation of wave fields around the submergedbreakwater with SOLA-SURF methodrdquo in Proceedings of the26th International Conference on Coastal Engineering (ICCErsquo98) pp 843ndash852 June 1998
[4] W Decheng and W Guoxiong ldquoNumerical simulation of asolitary wave interaction with submerged multi-bodiesrdquo ActaMechanica Sinica vol 14 no 4 pp 304ndash305 1998
[5] H D Armono and K R Hall ldquoWave transmission on sub-merged breakwaters made of hollow hemispherical shape arti-ficial reefsrdquo in Canadian Coastal Conference pp 313ndash322 June2003
[6] A Yoshida S Yan M Yamashiro and I Irie ldquoWave FieldBehind a Double-Submerged Breakwaterrdquo in Proceedings of the12th International Offshore and Polar Engineering Conferencepp 785ndash790 May 2002
[7] C-J Huang H-H Chang and H-H Hwung ldquoStructuralpermeability effects on the interaction of a solitary wave anda submerged breakwaterrdquo Coastal Engineering vol 49 no 1-2pp 1ndash24 2003
[8] D G Stamos M R Hajj and D P Telionis ldquoPerformanceof hemi-cylindrical and rectangular submerged breakwatersrdquoOcean Engineering vol 30 no 6 pp 813ndash828 2003
[9] H K Johnson T V Karambas I Avgeris B Zanuttigh DGonzalez-Marco and I Caceres ldquoModelling of waves andcurrents around submerged breakwatersrdquo Coastal Engineeringvol 52 no 10-11 pp 949ndash969 2005
[10] M G Muni-Reddy and S Neelamani ldquoWave interaction withcaisson defenced by an offshore low-crested breakwaterrdquo Jour-nal of Coastal Research no 39 pp 1767ndash1770 2006
[11] H-B Chen C-P Tsai and C-C Jeng ldquoWave transformationbetween submerged breakwater and seawallrdquo Journal of CoastalResearch no 50 pp 1069ndash1074 2007
[12] M Christou C Swan and O T Gudmestad ldquoThe interactionof surface water waves with submerged breakwatersrdquo CoastalEngineering vol 55 no 12 pp 945ndash958 2008
[13] M H Zaman H Togashi and R E Baddour ldquoDeformationof monochromatic water wave trains propagating over a sub-merged obstacle in the presence of uniform currentsrdquo OceanEngineering vol 35 no 8-9 pp 823ndash833 2008
[14] I J Losada J L Lara R Guanche and J M Gonzalez-OndinaldquoNumerical analysis of wave overtopping of rubble moundbreakwatersrdquoCoastal Engineering vol 55 no 1 pp 47ndash62 2008
[15] Y-G Cao C-B Jiang and Y-C Bai ldquoNumerical study onflow structure near two impermeable trapezoid submergedbreakwaters on slop bottomsrdquo Journal of Hydrodynamics vol22 no 5 pp 190ndash196 2010
[16] R A Dalrymple O Knio D T Cox M Gesteira and S ZouldquoUsing a lagrangian particle method for deck overtoppingrdquo inProceedings of the 4th International SymposiumWaves pp 1082ndash1091 ASCE September 2001
[17] H Gotoh S Shao and T Memita ldquoSPH-LES model for numer-ical investigation of wave interaction with partially immersedbreakwaterrdquo Coastal Engineering Journal vol 46 no 1 pp 39ndash63 2004
[18] S Shao ldquoIncrompressible SPH simulation of wave breaking andovertopping with turbulence modellingrdquo International Journalfor Numerical Methods in Fluids vol 50 no 5 pp 597ndash6212006
[19] R A Dalrymple and B D Rogers ldquoNumerical modeling ofwater waves with the SPHmethodrdquoCoastal Engineering vol 53no 2-3 pp 141ndash147 2006
[20] S C Yim D Yuk A Panizzo M di Risio and P L-F LiuldquoNumerical simulations ofwave generation by a vertical plungerusing RANS and SPH modelsrdquo Journal of Waterway PortCoastal andOcean Engineering vol 134 no 3 pp 143ndash159 2008
[21] J M Cherfils L Blonce G Pinon and E Rivoalen ldquoSimulationof water wavemdashcoastal structure interactions by smoothedparticle hydrodynamicsrdquo in Proceedings of the 8th InternationalConference onHydrodynamics Nantes France September 2008
[22] E Didier and M G Neves ldquoWave over topping of a typicalcoastal structure of the portuguese coast using A SPH modelrdquoJournal of Coastal Research vol 56 pp 496ndash500 2009
[23] S Shao ldquoIncompressible SPH flow model for wave interactionswith porous mediardquo Coastal Engineering vol 57 no 3 pp 304ndash316 2010
[24] C Liu J Zhang and Y Sun ldquoThe optimization of SPHmethod and its application in simulation of water waverdquo inProceedings of the IEEE 7th International Conference on NaturalComputation (ICNC rsquo11) pp 2327ndash2331 July 2011
[25] PW ClearyM Prakash J Ha N Stokes and C Scott ldquoSmoothparticle hydrodynamics status and future potentialrdquo Progress inComputational Fluid Dynamics vol 7 no 2-4 pp 70ndash90 2007
[26] M B Liu and G R Liu ldquoSmoothed particle hydrodynamics(SPH) an overview and recent developmentsrdquo Archives ofComputational Methods in Engineering vol 17 no 1 pp 25ndash762010
[27] G R Liu and M B Liu Smoothed Particle HydrodynamicsmdashAMeshfree Particle Methods World Scientific Publishing 2007
[28] J JMonaghan and J C Lattanzio ldquoA refined particlemethod forastrophysical problemsrdquo Astronomy and Astrophysics vol 149pp 135ndash143 1985
[29] J J Monaghan ldquoSmoothed particle hydrodynamicsrdquo AnnualReview of Astronomy and Astrophysics vol 30 no 1 pp 543ndash574 1992
[30] A Colagrossi and M Landrini ldquoNumerical simulation ofinterfacial flows by smoothed particle hydrodynamicsrdquo Journalof Computational Physics vol 191 no 2 pp 448ndash475 2003
[31] A Crespo Application of smoothed particle hydrodynamicsmodel SPHysics to free surface hydrodynamics [Doctoral thesis]2008
Modelling and Simulation in Engineering 9
[32] S Maiti and D Sen ldquoComputation of solitary waves duringpropagation and runup on a sloperdquo Ocean Engineering vol 26no 11 pp 1063ndash1083 1999
[33] S T Grilli M A Losada and F Martin ldquoCharacteristics ofsolitary wave breaking induced by breakwatersrdquo Journal ofWaterway Port Coastal ampOcean Engineering vol 120 no 1 pp74ndash92 1994
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Modelling and Simulation in Engineering 5
0
05
0 1 2 3 4
Wav
e hei
ght
(m)
X (m)minus05
minus1minus2minus3minus4
Experimental dataSPH
(a)
0
05
0 1 2 3 4
Experimental data
Wav
e hei
ght
(m)
X (m)minus05
minus1minus2minus3minus4
SPH
(b)
0
05
0 2 4
Wav
e hei
ght
(m)
X (m)minus2minus4
Experimental dataSPH
(c)
Experimental dataSPH
0
05
0 1 2 3 4
Wav
e hei
ght
(m)
X (m)minus05
minus1minus2minus3minus4
(d)
Figure 4 Comparison of SPHrsquos free surface profile with experimen-tal data [33]
Loveral
L0 L1
L2
1205731
1205732
d
h
Figure 5 Computational domain
Table 2 Computational error
Time (sec) Figure RMS10 Figure 4(a) 00361181 Figure 4(b) 003413 Figure 4(c) 00491392 Figure 4(d) 0053
presented in Table 3 Furthermore geometrical characteris-tics of breakwater are also proposed in this table
Table 3 Breakwaterrsquos characteristics
1205732
1205731
ℎ 1198711
0 tanminus1(05) 04 215 tanminus1(05) 04 245 tanminus1(05) 04 260 tanminus1(05) 04 275 tanminus1(05) 04 290 tanminus1(05) 04 2
0010203
0 20 40 60 80 100X (m)
minus01minus02minus03W
ave h
eigh
t (m
)
Outcome wave (120573 = 0)
Incoming wave
Figure 6 Free surface profile for 1205732= 0 degree
Free surface profile related to each considered conditionis illustrated in Figures 6 7 8 9 10 and 11 In each caseobtained free surface profile is compared with generatedsolitary wave without considering breakwater effect Exactevaluation of breakwaterrsquos slope on the free surface profileis presented in Table 4 It is obvious that wherever 120573
2= 90
maximumwave reduction can be achieved Correspondinglywhen 120573
2decreased to zero minimum breakwater effect
occurs Generally Figure 12 also shows a representative frameof SPH solution
73 Investigation of Breakwater Length To perform furtherparametric studies the effects of breakwaterrsquos length onfree surface can also be evaluated In this context previouscomputational characteristics are kept and only the lengthratio L1L2 is changed systematically Considered situationsare presented in Table 5 Figures 13 and 16 show the obtainedresults at various length ratios In all cases breakwatergenerates a hollow in free surface at breakwater positionwhich leads to a reduction in wave crest Generally basedon Table 6 it is observed that L1L2 ratio decreased sobreakwater performance will be improved In addition itcan be concluded that smaller length ratio leads to morereduction of wave amplitude To complete this section a viewof SPH solution is also presented in Figure 17
74 Investigation of Distance between Two Submerged Break-waters In this part influences of distance variation betweentwo breakwaters are evaluated Distance between two break-waters (119883) is considered to be less than wavelength at thebeginning of simulations Consequently this distance willincrease until the gap becomes larger than the wavelengthIn addition wave height water depth and length of waveflume are considered to be 119867 = 03m 12m and 70msuccessively Breakwater is located 30 meters from the leftGeometrical characteristics of breakwaters are also presented
6 Modelling and Simulation in Engineering
Table 4 Effect of breakwater slope on HH1015840 ratio
Values 1205732= 0 120573
2= 15 120573
2= 45 120573
2= 60 120573
2= 75 120573
2= 90
HH1015840 1078973 1185936 1297549 132081 1470258 1503094
Table 5 Geometrical characteristics
11987111198712
1205731
1205732
1198712
1198711
9 tanminus1 (05) tanminus1(05) 02 18633 tanminus1(05) tanminus1(05) 03 1942 tanminus1(05) tanminus1(05) 05 21367 tanminus1(05) tanminus1(05) 06 22
0010203
0 20 40 60 80 100
Wav
e hei
ght
(m)
X (m)minus01minus02
Outcome wave (120573 = 15)
Incoming wave
Figure 7 Free surface profile for 1205732= 15 degree
in Table 7 Figure 18 shows the gap effects between twoconsidered breakwaters It is observed that when wavereaches the first breakwater the wave height is equal to025m However second breakwater decreases the waveheight in correspondence with considered distance betweentwo breakwaters In this context two parameters 119867 (waveheight above first breakwater) and 1198671015840 (wave height abovesecond breakwater) are introduced Breakwater influencecoefficient is also defined as 1198671198671015840 Precise values of definedparameter are presented in Table 8 It is obviously clear thatfor 119883 = 03m maximum wave reduction is achievedFurthermore it can be concluded that optimum gap betweenbreakwaters is affected by wave length In reality when gapbetween breakwaters is smaller than wave length effects ofsecond breakwater will be decreased Additionallymaximumeffectiveness of breakwater is observedwhenever the distanceis equal to wave length A representation of our SPH solutionis also depicted in Figure 19
8 Results
The present paper focuses on the breakwater performanceunder a solitary wave Furthermore we have tried to imple-ment meshless and Lagrangian approach of smoothed par-ticle hydrodynamics method to study considered physicsIn this context a two-dimensional SPH code is developedTo consider diffusion terms an artificial viscosity model isincluded Particles movement is also calculated using XSPHvariant and predictor-corrector scheme which is investigatedas time stepping algorithm To reinitialize density field MLSdensity filter is utilized
To start computations generated solitary wave is verifiedby comparing previous studies Afterwards a breakwater
001020304
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus01minus02
Outcome wave (120573 = 45)
Incoming wave
Figure 8 Free surface profile for 1205732= 45 degree
0010203
0 20 40 60 80 100
Outcome wave (120573 = 60)
Incoming wave
Wav
e hei
ght
(m)
X (m)minus01minus02
Figure 9 Free surface profile for 1205732= 60 degree
00204
0 20 40 60 80 100
Wav
e hei
ght
(m)
X (m)minus02
minus04
Outcome wave (120573 = 75)
Incoming wave
Figure 10 Free surface profile for 1205732= 75 degree
0
05
0 20 40 60 80 100
Wav
e hei
ght
(m)
X (m)minus05
Outcome wave (120573 = 90)
Incoming wave
Figure 11 Free surface profile for 1205732= 90 degree
Figure 12 A representative frame of SPH solution
Modelling and Simulation in Engineering 7
Table 6 Exact evaluation of length ratio effect on free surface reduction
Values 11987111198712= 9 119871
11198712= 633 119871
11198712= 42 119871
11198712= 367
HH1015840 1094116 1105263 1111112 1222223
0
02
04
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus02
Incoming waveOutcoming wave (120574 = 90)
Figure 13 Free surface profile with ratio 9 in length
0010203
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus01
Incoming waveOutcome wave (120574 = 633)
Figure 14 Free surface profile at length ratio 633
Table 7 Breakwater characteristics
(119883) Left sideslope
Right sideslope
Breakwaterheight
Lowerlength
02 tanminus1(05) tanminus1(05) 04 203 tanminus1(05) tanminus1(05) 04 204 tanminus1(05) tanminus1(05) 04 205 tanminus1(05) tanminus1(05) 04 2
Table 8 Effects of breakwaters distance
Damping ratio 119883 = 02 119883 = 03 119883 = 04 119883 = 05
HH1015840 1247285 142718 13381 1337985
is also included in computational domain To validate ourSPH solutions obtained free surface profile is compared withexisting experimental results and it is observed that our SPHsolutions have a reasonable accuracy In the second phasevarious geometrical aspects of breakwater are also studied
Generally effects of implementing two breakwaters incli-nation of breakwater and breakwaterrsquos length on free surfacedepression are evaluated It is found that when gap betweenbreakwaters is smaller than wave length effects of secondbreakwater will be decreased Maximum effectiveness ofbreakwater also occurred whenever the distance is equal towave length In addition when the inclination of breakwaterincreased free surface deformation will be modified Fur-thermore it is observed that length ratio of breakwater caneffectively influence free surface profile (Figures 14 and 15)
0010203
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus01
Incoming waveOutcome wave (120574 = 42)
Figure 15 Free surface profile at length ratio 42
0010203
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus01
Incoming waveOutcome wave (120574 = 367)
Figure 16 Free surface profile at length ratio 367
Figure 17 A view of SPH solution
0
02
04
020 40 60 80 100W
ave h
eigh
t (m
)
X (m)minus02
Incoming waveOutcome wave (X = 02)
Outcome wave (X = 03)
Outcome wave (X = 04)
Figure 18 Simulated free surface profiles at different distancesbetween breakwaters
Figure 19 Particle representation of SPH solution
8 Modelling and Simulation in Engineering
To perform more precise computation extending SPHsolutions to three-dimensional case will be in attendanceVarious simple breakwater geometries will also be introducedin near future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Tokura and T Ida ldquoSimulation of wave dissipation mech-anism on submerged structure using fluid-structure couplingcapability in LS-DYNArdquo in Proceedings of the 5th European LS-DYNA Conference 2005
[2] M Hara T Yasuda and Y Sakakibara ldquoCharacteristics of asolitary wave breaking caused by a submerged obstaclerdquoCoastalEngineering vol 1 no 23 1992
[3] N Hayakawa T Hosoyamada S Yoshida and G TsujimotoldquoNumerical simulation of wave fields around the submergedbreakwater with SOLA-SURF methodrdquo in Proceedings of the26th International Conference on Coastal Engineering (ICCErsquo98) pp 843ndash852 June 1998
[4] W Decheng and W Guoxiong ldquoNumerical simulation of asolitary wave interaction with submerged multi-bodiesrdquo ActaMechanica Sinica vol 14 no 4 pp 304ndash305 1998
[5] H D Armono and K R Hall ldquoWave transmission on sub-merged breakwaters made of hollow hemispherical shape arti-ficial reefsrdquo in Canadian Coastal Conference pp 313ndash322 June2003
[6] A Yoshida S Yan M Yamashiro and I Irie ldquoWave FieldBehind a Double-Submerged Breakwaterrdquo in Proceedings of the12th International Offshore and Polar Engineering Conferencepp 785ndash790 May 2002
[7] C-J Huang H-H Chang and H-H Hwung ldquoStructuralpermeability effects on the interaction of a solitary wave anda submerged breakwaterrdquo Coastal Engineering vol 49 no 1-2pp 1ndash24 2003
[8] D G Stamos M R Hajj and D P Telionis ldquoPerformanceof hemi-cylindrical and rectangular submerged breakwatersrdquoOcean Engineering vol 30 no 6 pp 813ndash828 2003
[9] H K Johnson T V Karambas I Avgeris B Zanuttigh DGonzalez-Marco and I Caceres ldquoModelling of waves andcurrents around submerged breakwatersrdquo Coastal Engineeringvol 52 no 10-11 pp 949ndash969 2005
[10] M G Muni-Reddy and S Neelamani ldquoWave interaction withcaisson defenced by an offshore low-crested breakwaterrdquo Jour-nal of Coastal Research no 39 pp 1767ndash1770 2006
[11] H-B Chen C-P Tsai and C-C Jeng ldquoWave transformationbetween submerged breakwater and seawallrdquo Journal of CoastalResearch no 50 pp 1069ndash1074 2007
[12] M Christou C Swan and O T Gudmestad ldquoThe interactionof surface water waves with submerged breakwatersrdquo CoastalEngineering vol 55 no 12 pp 945ndash958 2008
[13] M H Zaman H Togashi and R E Baddour ldquoDeformationof monochromatic water wave trains propagating over a sub-merged obstacle in the presence of uniform currentsrdquo OceanEngineering vol 35 no 8-9 pp 823ndash833 2008
[14] I J Losada J L Lara R Guanche and J M Gonzalez-OndinaldquoNumerical analysis of wave overtopping of rubble moundbreakwatersrdquoCoastal Engineering vol 55 no 1 pp 47ndash62 2008
[15] Y-G Cao C-B Jiang and Y-C Bai ldquoNumerical study onflow structure near two impermeable trapezoid submergedbreakwaters on slop bottomsrdquo Journal of Hydrodynamics vol22 no 5 pp 190ndash196 2010
[16] R A Dalrymple O Knio D T Cox M Gesteira and S ZouldquoUsing a lagrangian particle method for deck overtoppingrdquo inProceedings of the 4th International SymposiumWaves pp 1082ndash1091 ASCE September 2001
[17] H Gotoh S Shao and T Memita ldquoSPH-LES model for numer-ical investigation of wave interaction with partially immersedbreakwaterrdquo Coastal Engineering Journal vol 46 no 1 pp 39ndash63 2004
[18] S Shao ldquoIncrompressible SPH simulation of wave breaking andovertopping with turbulence modellingrdquo International Journalfor Numerical Methods in Fluids vol 50 no 5 pp 597ndash6212006
[19] R A Dalrymple and B D Rogers ldquoNumerical modeling ofwater waves with the SPHmethodrdquoCoastal Engineering vol 53no 2-3 pp 141ndash147 2006
[20] S C Yim D Yuk A Panizzo M di Risio and P L-F LiuldquoNumerical simulations ofwave generation by a vertical plungerusing RANS and SPH modelsrdquo Journal of Waterway PortCoastal andOcean Engineering vol 134 no 3 pp 143ndash159 2008
[21] J M Cherfils L Blonce G Pinon and E Rivoalen ldquoSimulationof water wavemdashcoastal structure interactions by smoothedparticle hydrodynamicsrdquo in Proceedings of the 8th InternationalConference onHydrodynamics Nantes France September 2008
[22] E Didier and M G Neves ldquoWave over topping of a typicalcoastal structure of the portuguese coast using A SPH modelrdquoJournal of Coastal Research vol 56 pp 496ndash500 2009
[23] S Shao ldquoIncompressible SPH flow model for wave interactionswith porous mediardquo Coastal Engineering vol 57 no 3 pp 304ndash316 2010
[24] C Liu J Zhang and Y Sun ldquoThe optimization of SPHmethod and its application in simulation of water waverdquo inProceedings of the IEEE 7th International Conference on NaturalComputation (ICNC rsquo11) pp 2327ndash2331 July 2011
[25] PW ClearyM Prakash J Ha N Stokes and C Scott ldquoSmoothparticle hydrodynamics status and future potentialrdquo Progress inComputational Fluid Dynamics vol 7 no 2-4 pp 70ndash90 2007
[26] M B Liu and G R Liu ldquoSmoothed particle hydrodynamics(SPH) an overview and recent developmentsrdquo Archives ofComputational Methods in Engineering vol 17 no 1 pp 25ndash762010
[27] G R Liu and M B Liu Smoothed Particle HydrodynamicsmdashAMeshfree Particle Methods World Scientific Publishing 2007
[28] J JMonaghan and J C Lattanzio ldquoA refined particlemethod forastrophysical problemsrdquo Astronomy and Astrophysics vol 149pp 135ndash143 1985
[29] J J Monaghan ldquoSmoothed particle hydrodynamicsrdquo AnnualReview of Astronomy and Astrophysics vol 30 no 1 pp 543ndash574 1992
[30] A Colagrossi and M Landrini ldquoNumerical simulation ofinterfacial flows by smoothed particle hydrodynamicsrdquo Journalof Computational Physics vol 191 no 2 pp 448ndash475 2003
[31] A Crespo Application of smoothed particle hydrodynamicsmodel SPHysics to free surface hydrodynamics [Doctoral thesis]2008
Modelling and Simulation in Engineering 9
[32] S Maiti and D Sen ldquoComputation of solitary waves duringpropagation and runup on a sloperdquo Ocean Engineering vol 26no 11 pp 1063ndash1083 1999
[33] S T Grilli M A Losada and F Martin ldquoCharacteristics ofsolitary wave breaking induced by breakwatersrdquo Journal ofWaterway Port Coastal ampOcean Engineering vol 120 no 1 pp74ndash92 1994
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
6 Modelling and Simulation in Engineering
Table 4 Effect of breakwater slope on HH1015840 ratio
Values 1205732= 0 120573
2= 15 120573
2= 45 120573
2= 60 120573
2= 75 120573
2= 90
HH1015840 1078973 1185936 1297549 132081 1470258 1503094
Table 5 Geometrical characteristics
11987111198712
1205731
1205732
1198712
1198711
9 tanminus1 (05) tanminus1(05) 02 18633 tanminus1(05) tanminus1(05) 03 1942 tanminus1(05) tanminus1(05) 05 21367 tanminus1(05) tanminus1(05) 06 22
0010203
0 20 40 60 80 100
Wav
e hei
ght
(m)
X (m)minus01minus02
Outcome wave (120573 = 15)
Incoming wave
Figure 7 Free surface profile for 1205732= 15 degree
in Table 7 Figure 18 shows the gap effects between twoconsidered breakwaters It is observed that when wavereaches the first breakwater the wave height is equal to025m However second breakwater decreases the waveheight in correspondence with considered distance betweentwo breakwaters In this context two parameters 119867 (waveheight above first breakwater) and 1198671015840 (wave height abovesecond breakwater) are introduced Breakwater influencecoefficient is also defined as 1198671198671015840 Precise values of definedparameter are presented in Table 8 It is obviously clear thatfor 119883 = 03m maximum wave reduction is achievedFurthermore it can be concluded that optimum gap betweenbreakwaters is affected by wave length In reality when gapbetween breakwaters is smaller than wave length effects ofsecond breakwater will be decreased Additionallymaximumeffectiveness of breakwater is observedwhenever the distanceis equal to wave length A representation of our SPH solutionis also depicted in Figure 19
8 Results
The present paper focuses on the breakwater performanceunder a solitary wave Furthermore we have tried to imple-ment meshless and Lagrangian approach of smoothed par-ticle hydrodynamics method to study considered physicsIn this context a two-dimensional SPH code is developedTo consider diffusion terms an artificial viscosity model isincluded Particles movement is also calculated using XSPHvariant and predictor-corrector scheme which is investigatedas time stepping algorithm To reinitialize density field MLSdensity filter is utilized
To start computations generated solitary wave is verifiedby comparing previous studies Afterwards a breakwater
001020304
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus01minus02
Outcome wave (120573 = 45)
Incoming wave
Figure 8 Free surface profile for 1205732= 45 degree
0010203
0 20 40 60 80 100
Outcome wave (120573 = 60)
Incoming wave
Wav
e hei
ght
(m)
X (m)minus01minus02
Figure 9 Free surface profile for 1205732= 60 degree
00204
0 20 40 60 80 100
Wav
e hei
ght
(m)
X (m)minus02
minus04
Outcome wave (120573 = 75)
Incoming wave
Figure 10 Free surface profile for 1205732= 75 degree
0
05
0 20 40 60 80 100
Wav
e hei
ght
(m)
X (m)minus05
Outcome wave (120573 = 90)
Incoming wave
Figure 11 Free surface profile for 1205732= 90 degree
Figure 12 A representative frame of SPH solution
Modelling and Simulation in Engineering 7
Table 6 Exact evaluation of length ratio effect on free surface reduction
Values 11987111198712= 9 119871
11198712= 633 119871
11198712= 42 119871
11198712= 367
HH1015840 1094116 1105263 1111112 1222223
0
02
04
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus02
Incoming waveOutcoming wave (120574 = 90)
Figure 13 Free surface profile with ratio 9 in length
0010203
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus01
Incoming waveOutcome wave (120574 = 633)
Figure 14 Free surface profile at length ratio 633
Table 7 Breakwater characteristics
(119883) Left sideslope
Right sideslope
Breakwaterheight
Lowerlength
02 tanminus1(05) tanminus1(05) 04 203 tanminus1(05) tanminus1(05) 04 204 tanminus1(05) tanminus1(05) 04 205 tanminus1(05) tanminus1(05) 04 2
Table 8 Effects of breakwaters distance
Damping ratio 119883 = 02 119883 = 03 119883 = 04 119883 = 05
HH1015840 1247285 142718 13381 1337985
is also included in computational domain To validate ourSPH solutions obtained free surface profile is compared withexisting experimental results and it is observed that our SPHsolutions have a reasonable accuracy In the second phasevarious geometrical aspects of breakwater are also studied
Generally effects of implementing two breakwaters incli-nation of breakwater and breakwaterrsquos length on free surfacedepression are evaluated It is found that when gap betweenbreakwaters is smaller than wave length effects of secondbreakwater will be decreased Maximum effectiveness ofbreakwater also occurred whenever the distance is equal towave length In addition when the inclination of breakwaterincreased free surface deformation will be modified Fur-thermore it is observed that length ratio of breakwater caneffectively influence free surface profile (Figures 14 and 15)
0010203
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus01
Incoming waveOutcome wave (120574 = 42)
Figure 15 Free surface profile at length ratio 42
0010203
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus01
Incoming waveOutcome wave (120574 = 367)
Figure 16 Free surface profile at length ratio 367
Figure 17 A view of SPH solution
0
02
04
020 40 60 80 100W
ave h
eigh
t (m
)
X (m)minus02
Incoming waveOutcome wave (X = 02)
Outcome wave (X = 03)
Outcome wave (X = 04)
Figure 18 Simulated free surface profiles at different distancesbetween breakwaters
Figure 19 Particle representation of SPH solution
8 Modelling and Simulation in Engineering
To perform more precise computation extending SPHsolutions to three-dimensional case will be in attendanceVarious simple breakwater geometries will also be introducedin near future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Tokura and T Ida ldquoSimulation of wave dissipation mech-anism on submerged structure using fluid-structure couplingcapability in LS-DYNArdquo in Proceedings of the 5th European LS-DYNA Conference 2005
[2] M Hara T Yasuda and Y Sakakibara ldquoCharacteristics of asolitary wave breaking caused by a submerged obstaclerdquoCoastalEngineering vol 1 no 23 1992
[3] N Hayakawa T Hosoyamada S Yoshida and G TsujimotoldquoNumerical simulation of wave fields around the submergedbreakwater with SOLA-SURF methodrdquo in Proceedings of the26th International Conference on Coastal Engineering (ICCErsquo98) pp 843ndash852 June 1998
[4] W Decheng and W Guoxiong ldquoNumerical simulation of asolitary wave interaction with submerged multi-bodiesrdquo ActaMechanica Sinica vol 14 no 4 pp 304ndash305 1998
[5] H D Armono and K R Hall ldquoWave transmission on sub-merged breakwaters made of hollow hemispherical shape arti-ficial reefsrdquo in Canadian Coastal Conference pp 313ndash322 June2003
[6] A Yoshida S Yan M Yamashiro and I Irie ldquoWave FieldBehind a Double-Submerged Breakwaterrdquo in Proceedings of the12th International Offshore and Polar Engineering Conferencepp 785ndash790 May 2002
[7] C-J Huang H-H Chang and H-H Hwung ldquoStructuralpermeability effects on the interaction of a solitary wave anda submerged breakwaterrdquo Coastal Engineering vol 49 no 1-2pp 1ndash24 2003
[8] D G Stamos M R Hajj and D P Telionis ldquoPerformanceof hemi-cylindrical and rectangular submerged breakwatersrdquoOcean Engineering vol 30 no 6 pp 813ndash828 2003
[9] H K Johnson T V Karambas I Avgeris B Zanuttigh DGonzalez-Marco and I Caceres ldquoModelling of waves andcurrents around submerged breakwatersrdquo Coastal Engineeringvol 52 no 10-11 pp 949ndash969 2005
[10] M G Muni-Reddy and S Neelamani ldquoWave interaction withcaisson defenced by an offshore low-crested breakwaterrdquo Jour-nal of Coastal Research no 39 pp 1767ndash1770 2006
[11] H-B Chen C-P Tsai and C-C Jeng ldquoWave transformationbetween submerged breakwater and seawallrdquo Journal of CoastalResearch no 50 pp 1069ndash1074 2007
[12] M Christou C Swan and O T Gudmestad ldquoThe interactionof surface water waves with submerged breakwatersrdquo CoastalEngineering vol 55 no 12 pp 945ndash958 2008
[13] M H Zaman H Togashi and R E Baddour ldquoDeformationof monochromatic water wave trains propagating over a sub-merged obstacle in the presence of uniform currentsrdquo OceanEngineering vol 35 no 8-9 pp 823ndash833 2008
[14] I J Losada J L Lara R Guanche and J M Gonzalez-OndinaldquoNumerical analysis of wave overtopping of rubble moundbreakwatersrdquoCoastal Engineering vol 55 no 1 pp 47ndash62 2008
[15] Y-G Cao C-B Jiang and Y-C Bai ldquoNumerical study onflow structure near two impermeable trapezoid submergedbreakwaters on slop bottomsrdquo Journal of Hydrodynamics vol22 no 5 pp 190ndash196 2010
[16] R A Dalrymple O Knio D T Cox M Gesteira and S ZouldquoUsing a lagrangian particle method for deck overtoppingrdquo inProceedings of the 4th International SymposiumWaves pp 1082ndash1091 ASCE September 2001
[17] H Gotoh S Shao and T Memita ldquoSPH-LES model for numer-ical investigation of wave interaction with partially immersedbreakwaterrdquo Coastal Engineering Journal vol 46 no 1 pp 39ndash63 2004
[18] S Shao ldquoIncrompressible SPH simulation of wave breaking andovertopping with turbulence modellingrdquo International Journalfor Numerical Methods in Fluids vol 50 no 5 pp 597ndash6212006
[19] R A Dalrymple and B D Rogers ldquoNumerical modeling ofwater waves with the SPHmethodrdquoCoastal Engineering vol 53no 2-3 pp 141ndash147 2006
[20] S C Yim D Yuk A Panizzo M di Risio and P L-F LiuldquoNumerical simulations ofwave generation by a vertical plungerusing RANS and SPH modelsrdquo Journal of Waterway PortCoastal andOcean Engineering vol 134 no 3 pp 143ndash159 2008
[21] J M Cherfils L Blonce G Pinon and E Rivoalen ldquoSimulationof water wavemdashcoastal structure interactions by smoothedparticle hydrodynamicsrdquo in Proceedings of the 8th InternationalConference onHydrodynamics Nantes France September 2008
[22] E Didier and M G Neves ldquoWave over topping of a typicalcoastal structure of the portuguese coast using A SPH modelrdquoJournal of Coastal Research vol 56 pp 496ndash500 2009
[23] S Shao ldquoIncompressible SPH flow model for wave interactionswith porous mediardquo Coastal Engineering vol 57 no 3 pp 304ndash316 2010
[24] C Liu J Zhang and Y Sun ldquoThe optimization of SPHmethod and its application in simulation of water waverdquo inProceedings of the IEEE 7th International Conference on NaturalComputation (ICNC rsquo11) pp 2327ndash2331 July 2011
[25] PW ClearyM Prakash J Ha N Stokes and C Scott ldquoSmoothparticle hydrodynamics status and future potentialrdquo Progress inComputational Fluid Dynamics vol 7 no 2-4 pp 70ndash90 2007
[26] M B Liu and G R Liu ldquoSmoothed particle hydrodynamics(SPH) an overview and recent developmentsrdquo Archives ofComputational Methods in Engineering vol 17 no 1 pp 25ndash762010
[27] G R Liu and M B Liu Smoothed Particle HydrodynamicsmdashAMeshfree Particle Methods World Scientific Publishing 2007
[28] J JMonaghan and J C Lattanzio ldquoA refined particlemethod forastrophysical problemsrdquo Astronomy and Astrophysics vol 149pp 135ndash143 1985
[29] J J Monaghan ldquoSmoothed particle hydrodynamicsrdquo AnnualReview of Astronomy and Astrophysics vol 30 no 1 pp 543ndash574 1992
[30] A Colagrossi and M Landrini ldquoNumerical simulation ofinterfacial flows by smoothed particle hydrodynamicsrdquo Journalof Computational Physics vol 191 no 2 pp 448ndash475 2003
[31] A Crespo Application of smoothed particle hydrodynamicsmodel SPHysics to free surface hydrodynamics [Doctoral thesis]2008
Modelling and Simulation in Engineering 9
[32] S Maiti and D Sen ldquoComputation of solitary waves duringpropagation and runup on a sloperdquo Ocean Engineering vol 26no 11 pp 1063ndash1083 1999
[33] S T Grilli M A Losada and F Martin ldquoCharacteristics ofsolitary wave breaking induced by breakwatersrdquo Journal ofWaterway Port Coastal ampOcean Engineering vol 120 no 1 pp74ndash92 1994
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Modelling and Simulation in Engineering 7
Table 6 Exact evaluation of length ratio effect on free surface reduction
Values 11987111198712= 9 119871
11198712= 633 119871
11198712= 42 119871
11198712= 367
HH1015840 1094116 1105263 1111112 1222223
0
02
04
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus02
Incoming waveOutcoming wave (120574 = 90)
Figure 13 Free surface profile with ratio 9 in length
0010203
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus01
Incoming waveOutcome wave (120574 = 633)
Figure 14 Free surface profile at length ratio 633
Table 7 Breakwater characteristics
(119883) Left sideslope
Right sideslope
Breakwaterheight
Lowerlength
02 tanminus1(05) tanminus1(05) 04 203 tanminus1(05) tanminus1(05) 04 204 tanminus1(05) tanminus1(05) 04 205 tanminus1(05) tanminus1(05) 04 2
Table 8 Effects of breakwaters distance
Damping ratio 119883 = 02 119883 = 03 119883 = 04 119883 = 05
HH1015840 1247285 142718 13381 1337985
is also included in computational domain To validate ourSPH solutions obtained free surface profile is compared withexisting experimental results and it is observed that our SPHsolutions have a reasonable accuracy In the second phasevarious geometrical aspects of breakwater are also studied
Generally effects of implementing two breakwaters incli-nation of breakwater and breakwaterrsquos length on free surfacedepression are evaluated It is found that when gap betweenbreakwaters is smaller than wave length effects of secondbreakwater will be decreased Maximum effectiveness ofbreakwater also occurred whenever the distance is equal towave length In addition when the inclination of breakwaterincreased free surface deformation will be modified Fur-thermore it is observed that length ratio of breakwater caneffectively influence free surface profile (Figures 14 and 15)
0010203
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus01
Incoming waveOutcome wave (120574 = 42)
Figure 15 Free surface profile at length ratio 42
0010203
0 20 40 60 80 100Wav
e hei
ght
(m)
X (m)minus01
Incoming waveOutcome wave (120574 = 367)
Figure 16 Free surface profile at length ratio 367
Figure 17 A view of SPH solution
0
02
04
020 40 60 80 100W
ave h
eigh
t (m
)
X (m)minus02
Incoming waveOutcome wave (X = 02)
Outcome wave (X = 03)
Outcome wave (X = 04)
Figure 18 Simulated free surface profiles at different distancesbetween breakwaters
Figure 19 Particle representation of SPH solution
8 Modelling and Simulation in Engineering
To perform more precise computation extending SPHsolutions to three-dimensional case will be in attendanceVarious simple breakwater geometries will also be introducedin near future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Tokura and T Ida ldquoSimulation of wave dissipation mech-anism on submerged structure using fluid-structure couplingcapability in LS-DYNArdquo in Proceedings of the 5th European LS-DYNA Conference 2005
[2] M Hara T Yasuda and Y Sakakibara ldquoCharacteristics of asolitary wave breaking caused by a submerged obstaclerdquoCoastalEngineering vol 1 no 23 1992
[3] N Hayakawa T Hosoyamada S Yoshida and G TsujimotoldquoNumerical simulation of wave fields around the submergedbreakwater with SOLA-SURF methodrdquo in Proceedings of the26th International Conference on Coastal Engineering (ICCErsquo98) pp 843ndash852 June 1998
[4] W Decheng and W Guoxiong ldquoNumerical simulation of asolitary wave interaction with submerged multi-bodiesrdquo ActaMechanica Sinica vol 14 no 4 pp 304ndash305 1998
[5] H D Armono and K R Hall ldquoWave transmission on sub-merged breakwaters made of hollow hemispherical shape arti-ficial reefsrdquo in Canadian Coastal Conference pp 313ndash322 June2003
[6] A Yoshida S Yan M Yamashiro and I Irie ldquoWave FieldBehind a Double-Submerged Breakwaterrdquo in Proceedings of the12th International Offshore and Polar Engineering Conferencepp 785ndash790 May 2002
[7] C-J Huang H-H Chang and H-H Hwung ldquoStructuralpermeability effects on the interaction of a solitary wave anda submerged breakwaterrdquo Coastal Engineering vol 49 no 1-2pp 1ndash24 2003
[8] D G Stamos M R Hajj and D P Telionis ldquoPerformanceof hemi-cylindrical and rectangular submerged breakwatersrdquoOcean Engineering vol 30 no 6 pp 813ndash828 2003
[9] H K Johnson T V Karambas I Avgeris B Zanuttigh DGonzalez-Marco and I Caceres ldquoModelling of waves andcurrents around submerged breakwatersrdquo Coastal Engineeringvol 52 no 10-11 pp 949ndash969 2005
[10] M G Muni-Reddy and S Neelamani ldquoWave interaction withcaisson defenced by an offshore low-crested breakwaterrdquo Jour-nal of Coastal Research no 39 pp 1767ndash1770 2006
[11] H-B Chen C-P Tsai and C-C Jeng ldquoWave transformationbetween submerged breakwater and seawallrdquo Journal of CoastalResearch no 50 pp 1069ndash1074 2007
[12] M Christou C Swan and O T Gudmestad ldquoThe interactionof surface water waves with submerged breakwatersrdquo CoastalEngineering vol 55 no 12 pp 945ndash958 2008
[13] M H Zaman H Togashi and R E Baddour ldquoDeformationof monochromatic water wave trains propagating over a sub-merged obstacle in the presence of uniform currentsrdquo OceanEngineering vol 35 no 8-9 pp 823ndash833 2008
[14] I J Losada J L Lara R Guanche and J M Gonzalez-OndinaldquoNumerical analysis of wave overtopping of rubble moundbreakwatersrdquoCoastal Engineering vol 55 no 1 pp 47ndash62 2008
[15] Y-G Cao C-B Jiang and Y-C Bai ldquoNumerical study onflow structure near two impermeable trapezoid submergedbreakwaters on slop bottomsrdquo Journal of Hydrodynamics vol22 no 5 pp 190ndash196 2010
[16] R A Dalrymple O Knio D T Cox M Gesteira and S ZouldquoUsing a lagrangian particle method for deck overtoppingrdquo inProceedings of the 4th International SymposiumWaves pp 1082ndash1091 ASCE September 2001
[17] H Gotoh S Shao and T Memita ldquoSPH-LES model for numer-ical investigation of wave interaction with partially immersedbreakwaterrdquo Coastal Engineering Journal vol 46 no 1 pp 39ndash63 2004
[18] S Shao ldquoIncrompressible SPH simulation of wave breaking andovertopping with turbulence modellingrdquo International Journalfor Numerical Methods in Fluids vol 50 no 5 pp 597ndash6212006
[19] R A Dalrymple and B D Rogers ldquoNumerical modeling ofwater waves with the SPHmethodrdquoCoastal Engineering vol 53no 2-3 pp 141ndash147 2006
[20] S C Yim D Yuk A Panizzo M di Risio and P L-F LiuldquoNumerical simulations ofwave generation by a vertical plungerusing RANS and SPH modelsrdquo Journal of Waterway PortCoastal andOcean Engineering vol 134 no 3 pp 143ndash159 2008
[21] J M Cherfils L Blonce G Pinon and E Rivoalen ldquoSimulationof water wavemdashcoastal structure interactions by smoothedparticle hydrodynamicsrdquo in Proceedings of the 8th InternationalConference onHydrodynamics Nantes France September 2008
[22] E Didier and M G Neves ldquoWave over topping of a typicalcoastal structure of the portuguese coast using A SPH modelrdquoJournal of Coastal Research vol 56 pp 496ndash500 2009
[23] S Shao ldquoIncompressible SPH flow model for wave interactionswith porous mediardquo Coastal Engineering vol 57 no 3 pp 304ndash316 2010
[24] C Liu J Zhang and Y Sun ldquoThe optimization of SPHmethod and its application in simulation of water waverdquo inProceedings of the IEEE 7th International Conference on NaturalComputation (ICNC rsquo11) pp 2327ndash2331 July 2011
[25] PW ClearyM Prakash J Ha N Stokes and C Scott ldquoSmoothparticle hydrodynamics status and future potentialrdquo Progress inComputational Fluid Dynamics vol 7 no 2-4 pp 70ndash90 2007
[26] M B Liu and G R Liu ldquoSmoothed particle hydrodynamics(SPH) an overview and recent developmentsrdquo Archives ofComputational Methods in Engineering vol 17 no 1 pp 25ndash762010
[27] G R Liu and M B Liu Smoothed Particle HydrodynamicsmdashAMeshfree Particle Methods World Scientific Publishing 2007
[28] J JMonaghan and J C Lattanzio ldquoA refined particlemethod forastrophysical problemsrdquo Astronomy and Astrophysics vol 149pp 135ndash143 1985
[29] J J Monaghan ldquoSmoothed particle hydrodynamicsrdquo AnnualReview of Astronomy and Astrophysics vol 30 no 1 pp 543ndash574 1992
[30] A Colagrossi and M Landrini ldquoNumerical simulation ofinterfacial flows by smoothed particle hydrodynamicsrdquo Journalof Computational Physics vol 191 no 2 pp 448ndash475 2003
[31] A Crespo Application of smoothed particle hydrodynamicsmodel SPHysics to free surface hydrodynamics [Doctoral thesis]2008
Modelling and Simulation in Engineering 9
[32] S Maiti and D Sen ldquoComputation of solitary waves duringpropagation and runup on a sloperdquo Ocean Engineering vol 26no 11 pp 1063ndash1083 1999
[33] S T Grilli M A Losada and F Martin ldquoCharacteristics ofsolitary wave breaking induced by breakwatersrdquo Journal ofWaterway Port Coastal ampOcean Engineering vol 120 no 1 pp74ndash92 1994
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
8 Modelling and Simulation in Engineering
To perform more precise computation extending SPHsolutions to three-dimensional case will be in attendanceVarious simple breakwater geometries will also be introducedin near future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Tokura and T Ida ldquoSimulation of wave dissipation mech-anism on submerged structure using fluid-structure couplingcapability in LS-DYNArdquo in Proceedings of the 5th European LS-DYNA Conference 2005
[2] M Hara T Yasuda and Y Sakakibara ldquoCharacteristics of asolitary wave breaking caused by a submerged obstaclerdquoCoastalEngineering vol 1 no 23 1992
[3] N Hayakawa T Hosoyamada S Yoshida and G TsujimotoldquoNumerical simulation of wave fields around the submergedbreakwater with SOLA-SURF methodrdquo in Proceedings of the26th International Conference on Coastal Engineering (ICCErsquo98) pp 843ndash852 June 1998
[4] W Decheng and W Guoxiong ldquoNumerical simulation of asolitary wave interaction with submerged multi-bodiesrdquo ActaMechanica Sinica vol 14 no 4 pp 304ndash305 1998
[5] H D Armono and K R Hall ldquoWave transmission on sub-merged breakwaters made of hollow hemispherical shape arti-ficial reefsrdquo in Canadian Coastal Conference pp 313ndash322 June2003
[6] A Yoshida S Yan M Yamashiro and I Irie ldquoWave FieldBehind a Double-Submerged Breakwaterrdquo in Proceedings of the12th International Offshore and Polar Engineering Conferencepp 785ndash790 May 2002
[7] C-J Huang H-H Chang and H-H Hwung ldquoStructuralpermeability effects on the interaction of a solitary wave anda submerged breakwaterrdquo Coastal Engineering vol 49 no 1-2pp 1ndash24 2003
[8] D G Stamos M R Hajj and D P Telionis ldquoPerformanceof hemi-cylindrical and rectangular submerged breakwatersrdquoOcean Engineering vol 30 no 6 pp 813ndash828 2003
[9] H K Johnson T V Karambas I Avgeris B Zanuttigh DGonzalez-Marco and I Caceres ldquoModelling of waves andcurrents around submerged breakwatersrdquo Coastal Engineeringvol 52 no 10-11 pp 949ndash969 2005
[10] M G Muni-Reddy and S Neelamani ldquoWave interaction withcaisson defenced by an offshore low-crested breakwaterrdquo Jour-nal of Coastal Research no 39 pp 1767ndash1770 2006
[11] H-B Chen C-P Tsai and C-C Jeng ldquoWave transformationbetween submerged breakwater and seawallrdquo Journal of CoastalResearch no 50 pp 1069ndash1074 2007
[12] M Christou C Swan and O T Gudmestad ldquoThe interactionof surface water waves with submerged breakwatersrdquo CoastalEngineering vol 55 no 12 pp 945ndash958 2008
[13] M H Zaman H Togashi and R E Baddour ldquoDeformationof monochromatic water wave trains propagating over a sub-merged obstacle in the presence of uniform currentsrdquo OceanEngineering vol 35 no 8-9 pp 823ndash833 2008
[14] I J Losada J L Lara R Guanche and J M Gonzalez-OndinaldquoNumerical analysis of wave overtopping of rubble moundbreakwatersrdquoCoastal Engineering vol 55 no 1 pp 47ndash62 2008
[15] Y-G Cao C-B Jiang and Y-C Bai ldquoNumerical study onflow structure near two impermeable trapezoid submergedbreakwaters on slop bottomsrdquo Journal of Hydrodynamics vol22 no 5 pp 190ndash196 2010
[16] R A Dalrymple O Knio D T Cox M Gesteira and S ZouldquoUsing a lagrangian particle method for deck overtoppingrdquo inProceedings of the 4th International SymposiumWaves pp 1082ndash1091 ASCE September 2001
[17] H Gotoh S Shao and T Memita ldquoSPH-LES model for numer-ical investigation of wave interaction with partially immersedbreakwaterrdquo Coastal Engineering Journal vol 46 no 1 pp 39ndash63 2004
[18] S Shao ldquoIncrompressible SPH simulation of wave breaking andovertopping with turbulence modellingrdquo International Journalfor Numerical Methods in Fluids vol 50 no 5 pp 597ndash6212006
[19] R A Dalrymple and B D Rogers ldquoNumerical modeling ofwater waves with the SPHmethodrdquoCoastal Engineering vol 53no 2-3 pp 141ndash147 2006
[20] S C Yim D Yuk A Panizzo M di Risio and P L-F LiuldquoNumerical simulations ofwave generation by a vertical plungerusing RANS and SPH modelsrdquo Journal of Waterway PortCoastal andOcean Engineering vol 134 no 3 pp 143ndash159 2008
[21] J M Cherfils L Blonce G Pinon and E Rivoalen ldquoSimulationof water wavemdashcoastal structure interactions by smoothedparticle hydrodynamicsrdquo in Proceedings of the 8th InternationalConference onHydrodynamics Nantes France September 2008
[22] E Didier and M G Neves ldquoWave over topping of a typicalcoastal structure of the portuguese coast using A SPH modelrdquoJournal of Coastal Research vol 56 pp 496ndash500 2009
[23] S Shao ldquoIncompressible SPH flow model for wave interactionswith porous mediardquo Coastal Engineering vol 57 no 3 pp 304ndash316 2010
[24] C Liu J Zhang and Y Sun ldquoThe optimization of SPHmethod and its application in simulation of water waverdquo inProceedings of the IEEE 7th International Conference on NaturalComputation (ICNC rsquo11) pp 2327ndash2331 July 2011
[25] PW ClearyM Prakash J Ha N Stokes and C Scott ldquoSmoothparticle hydrodynamics status and future potentialrdquo Progress inComputational Fluid Dynamics vol 7 no 2-4 pp 70ndash90 2007
[26] M B Liu and G R Liu ldquoSmoothed particle hydrodynamics(SPH) an overview and recent developmentsrdquo Archives ofComputational Methods in Engineering vol 17 no 1 pp 25ndash762010
[27] G R Liu and M B Liu Smoothed Particle HydrodynamicsmdashAMeshfree Particle Methods World Scientific Publishing 2007
[28] J JMonaghan and J C Lattanzio ldquoA refined particlemethod forastrophysical problemsrdquo Astronomy and Astrophysics vol 149pp 135ndash143 1985
[29] J J Monaghan ldquoSmoothed particle hydrodynamicsrdquo AnnualReview of Astronomy and Astrophysics vol 30 no 1 pp 543ndash574 1992
[30] A Colagrossi and M Landrini ldquoNumerical simulation ofinterfacial flows by smoothed particle hydrodynamicsrdquo Journalof Computational Physics vol 191 no 2 pp 448ndash475 2003
[31] A Crespo Application of smoothed particle hydrodynamicsmodel SPHysics to free surface hydrodynamics [Doctoral thesis]2008
Modelling and Simulation in Engineering 9
[32] S Maiti and D Sen ldquoComputation of solitary waves duringpropagation and runup on a sloperdquo Ocean Engineering vol 26no 11 pp 1063ndash1083 1999
[33] S T Grilli M A Losada and F Martin ldquoCharacteristics ofsolitary wave breaking induced by breakwatersrdquo Journal ofWaterway Port Coastal ampOcean Engineering vol 120 no 1 pp74ndash92 1994
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Modelling and Simulation in Engineering 9
[32] S Maiti and D Sen ldquoComputation of solitary waves duringpropagation and runup on a sloperdquo Ocean Engineering vol 26no 11 pp 1063ndash1083 1999
[33] S T Grilli M A Losada and F Martin ldquoCharacteristics ofsolitary wave breaking induced by breakwatersrdquo Journal ofWaterway Port Coastal ampOcean Engineering vol 120 no 1 pp74ndash92 1994
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of