Research ArticleNumerical Simulations for Nonlinear Waves Interaction withMultiple Perforated Quasi-Ellipse Caissons

Xiaozhong Ren1 and Yuxiang Ma2

1Dalian Jiucheng Municipal Design Co Ltd Dalian 116000 China2State Key Laboratory of Coastal and Offshore Engineering Dalian University of Technology Dalian 116024 China

Correspondence should be addressed to Yuxiang Ma yuxmadluteducn

Received 9 April 2015 Revised 15 June 2015 Accepted 16 June 2015

Academic Editor Ming Zhao

Copyright copy 2015 X Ren and Y Ma This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A three-dimensional numerical flume is developed to study cnoidal wave interaction with multiple arranged perforated quasi-ellipse caissonsThe continuity equation and the Navier-Stokes equations are used as the governing equation and the VOFmethodis adopted to capture the free surface elevation The equations are discretized on staggered cells and then solved using a finitedifference method The generation and propagation of cnoidal waves in the numerical flume are tested first And the ability ofthe present model to simulate interactions between waves and structures is verified by known experimental results Then cnoidalwaves with varying incident wave height and period are generated and interact withmultiple quasi-ellipse caissonswith andwithoutperforation It is found that the perforation plays an effective role in reducing wave runuprundown andwave forces on the caissonsThe wave forces on caissons reduce with the decreasing incident wave period The influence of the transverse distance of multiplecaissons on wave forces is also investigated A closer transverse distance between caissons can produce larger wave forces But whenrelative adjacent distance LD (L is the transverse distance andD is the width of the quasi-ellipse caisson) is larger than 3 the effectof adjacent distance is limited

1 Introduction

Perforated caissons which can not only weaken the wavereflection but also reduce the wave forces [1] are now com-mon protection structures in coastal engineering especiallyin the construction of vertical breakwaters and wharfs

During the past decades many studies were carriedout to investigate the performances of perforated struc-tures but mainly on perforated circular and rectangularcaissons Huang et al [2] presented a comprehensive reviewon the hydraulic performance and wave loadings of per-foratedslotted coastal structures Using the eigenfunctionexpansion method Darwiche et al [3] theoretically inves-tigated the interaction of linear waves with a cylindricalbreakwater surrounding a semiporous circular cylinder andfound that the semiporous structure can significantly reducewave load Yip and Chwang [4] proposed a perforated wallbreakwater with an internal horizontal plate and studied thehydrodynamic performance of this type of structure theoret-ically Suh et al [5] developed an analytical model to predict

irregular wave reflection of a perforated wall caisson and theability of the model was tested by an experiment Li et al[6] presented analytical results on the reflection of obliquelyincident waves by breakwaters with a partially perforatedwalland then further examined the wave absorbing performanceof the breakwater with double partially perforated front walls[7] Later Liu et al [8] developed a theoretical model tostudy the hydrodynamic performance of a perforated wallbreakwater with a submerged horizontal porous plate usingthe linear potential theory Using the same method thereflection of obliquely incident waves by an infinite array ofpartially perforated caisson was also investigated [9] Usingthe eigenfunction expansion method Sankarbabu et al [10]theoretically examined the hydrodynamic performance of adual cylindrical breakwater which was formed by a row ofcaissons each of which consists of a porous outer cylindercircumscribing an impermeable inner cylinder

Neelamani et al [11] conducted an experiment on waveinteraction with a vertical cylinder encircled by a perfo-rated square caisson and found that the wave forces and

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 895673 14 pageshttpdxdoiorg1011552015895673

2 Mathematical Problems in Engineering

moments on the cylinder are influenced by the porosity ofthe outer caisson Vijayalakshmi et al [12] experimentallystudied wave runup and rundown on a twin perforatedcircular cylinder and then developed a predictive formulafor the wave runup and rundown on the perforated cylinderbased on the experimental results Liu et al [13] performedan experimental and theoretical study on wave forces onpartially perforated rectangular caissons by normally inci-dent irregular waves Lee and Shin [14] experimentallytested wave reflection of partially perforated wall caissonstructures with single and double chambers Lee et al[15] conducted a three-dimensional experiment to studythe evolution of stem waves on partially perforated wallsand found that the perforation can effectively reduce waveheight

As aspects on numerical simulations Chen et al [16]developed a numerical model based on a two-dimensionalVOF method combined with the 119896-120576 model and used themodel to study wave interactions with perforated rectangularcaisson breakwaters Later this model was extended to inves-tigate wave forces on a perforated caisson with a top coverLiu et al [17] used the semianalytical scaled boundary finiteelement method to deal with the short-crested wave inter-action with a concentric cylindrical structure with double-layered perforated walls Later Liu and Lin [18] extendedthe numerical model to study short-crested wave interactionwith a concentric cylindrical system Recently Aristodemoet al [19] developed a numerical model based on the SPHmethod to calculate wave pressures acting on vertical andslotted coastal structures

The type of quasi-ellipse caisson is a relatively new struc-ture For open deep-water wharves quasi-ellipse caissonswere proven to be more effective and economical than thecircular and rectangular caissons [20] It was used to supportgravity dolphin wharves in Dalian Ore Terminal Phase IIProject Dalian China and the ore wharf of DongjiakouPort Tsingtao China Compared with the researches onthe circular and rectangular types investigations on thehydrodynamics of quasi-ellipse caissons are rare especiallyfor the perforated ones Wang et al [21] developed a 3Dnumerical model based on the VOF method to study theinteractions between waves and a single quasi-ellipse caissonIt was found that both wave height and caisson size havesignificant influences on the wave forces on the caissonHowever in real project caissons are always arranged ingroups Therefore the efforts on the interactions betweenwaves and multiple quasi-ellipse caissons with and withoutperforation should be carried outThis is the main aim of thepresent study In this study the numerical model developedby Wang et al [21] is extended to study normally incidentcnoidal waves interactionwithmultiple quasi-ellipse caissonsand the influence of perforation

Following the introduction the detailed setup of a 3Dnumerical model is described in Section 2 Then the modelis validated in Section 3 Simulations on interactions betweenwaves and multiple caissons with and without perforationand the results are under discussion in Section 4 Lastly someconclusions are advanced in Section 5

2 Numerical Model

In this section the governing equations and numerical algo-rithm are introduced in detail especially for the treatmentof the boundary conditions Figure 1 shows the schematicdrawing of the present model where 119871 is the tank length 119861is the tank width 119889 is the still water depth 119871

1is the length

of the working zone of the tank and 1198712is the length of the

sponger layer The origin of the Cartesian coordinate system119900119909119910119911 is set in the corner of the bottomwith the119909-axis positivein the direction of incoming wave propagation and the 119911-axispositive upwards as shown in Figure 1

21 Government Equations In this study the flow is assumedto be incompressibleThe continuity equation and theNavier-Stokes equations are adopted as the governing equations Andthe water surface is traced by the volume of fluid (VOF)method

Continuity equation is as follows

120597

120597119909(120579119906) +

120597

120597119910(120579V) +

120597

120597119911(120579119908) = 0 (1)

Navier-Stokes equations are as follows

120597119906

120597119905+ 119906

120597119906

120597119909+ V

120597119906

120597119910+119908

120597119906

120597119911

= minus1120588

120597119901

120597119909+ ]119896(1205972119906

1205971199092 +1205972119906

1205971199102 +1205972119906

1205971199112)

120597V120597119905

+ 119906120597V120597119909

+ V120597V120597119910

+119908120597V120597119911

= minus1120588

120597119901

120597119910+119892+ ]

119896(1205972V

1205971199092 +1205972V

1205971199102 +1205972V

1205971199112)

120597119908

120597119905+ 119906

120597119908

120597119909+ V

120597119908

120597119910+119908

120597119908

120597119911

= minus1120588

120597119901

120597119911+ ]119896(1205972119908

1205971199092 +1205972119908

1205971199102 +1205972119908

1205971199112)

(2)

where 119906 V and 119908 are the velocity components in the 119909119910 and 119911 directions respectively 119905 represents the time 119892 isthe gravitational acceleration 120588 is the fluid density and isequal to 1000 kgm3 119901 is the pressure ]

119896is the coefficient

of kinematic viscosity and is set to 10 times 10minus6m2s 120579 is thevolume fraction of the partial cell and its value is determinedafter the computational discretization

To capture the free surfaces a function 119865(119909 119910 119909 119905) isintroducedThe value of 119865 in a cell represents the occupationratio by fluid 119865 = 1 for a cell full with fluid whereas 119865 = 0

for a cell without fluid Cells with 119865 values between 0 and 1indicate that the cells are partially filled with fluidThese cellsare intersected by the free surface or contain bubbles whichare smaller than the cells [22 23] The time dependence of 119865can be handled by

120597

120597119905(120579119865) +

120597

120597119909(120579119906119865) +

120597

120597119910(120579V119865) +

120597

120597119911(120579119908119865) = 0 (3)

Mathematical Problems in Engineering 3

z

D1

D2S1 S2

S3

S4

S5

S6

d

x

B

B1

B2

L

L1 L2y

oA1

A2

C1

C2

Figure 1 Schematic drawing of the numerical tank and the coordinate system

Δz k

y Δxi

VCijk Fijk

z

Δyj

xo

pijk

(a)

o

y

Vij+12k

AFijk

ATijk

z

x

AR ij

ku i+

12jk

wijk+12

(b)

Figure 2 Parameter definition of staggered cells (a) at the center of a cell (b) at the sides of a cell

22 Discretization Scheme A finite difference method ischosen to solve the governing equations The staggered cellis adopted for the computational discretizationThe locationsof the variations 119901 119865 119906 V and 119908 used in the computationare illustrated in Figure 2 Quantities 119860119877

119894119895119896 119860119879119894119895119896

119860119865119894119895119896

and 119881119862

119894119895119896 denote the fractions open to flow for the right

top and front cell faces and the cell volume respectivelyThe discretization scheme of the continuity equation is

written as

1119881119862119894119895119896

(119906119899+1119894119895119896

119860119877119894119895119896

minus 119906119899+1119894minus1119895119896119860119877

119894minus1119895119896

Δ119909119894

+V119899+1119894119895119896

119860119865119894119895119896

minus V119899+1119894119895minus1119896119860119865

119894119895minus1119896

Δ119911119896

+119908119899+1119894119895119896

119860119879119894119895119896

minus 119908119899+1119894119895119896minus1119860119879

119894119895119896minus1

Δ119910119895

) = 0

(4)

The time derivative is discretized by the forward timedifference scheme for the momentum equations the finitedifference approximation for a typical cell in 119909 direction isgiven as follows

119906119899+1119894119895119896

= 119906119899

119894119895119896+

Δ119905

Δ119909[minus (119901119899+1119894+1119895119896 minus119901

119899+1119894119895119896

) minus FUXminus FUY

minus FUZ+VISX] (5)

where FUX FUY and FUZ are the convection flux termsin the three directions and a combination of the centraldifference scheme and the upwind scheme is employed todiscretize them For example the discretized form for FUXis defined as

FUX = (119906120597119906

120597119909)

119894+1119895119896=

119906119899

119894119895119896

Δ119909120572

Δ119909119894+1

Δ119909119894

(119906119899

119894119895119896minus119906119899

119894minus1119895119896)

+Δ119909119894

Δ119909119894+1

(119906119899

119894+1119895119896 minus119906119899

119894119895119896) + 120572 sdot sign (119906

119899

119894119895119896)

sdot [Δ119909119894+1

Δ119909119894

(119906119899

119894119895119896minus119906119899

119894minus1119895119896)

+Δ119909119894

Δ119909119894+1

(119906119899

119894+1119895119896 minus119906119899

119894119895119896)]

Δ119909120572= Δ119909119894+1 +Δ119909

119894+120572 sdot sign (119906

119899

119894119895119896) (Δ119909119894+1 minusΔ119909

119894)

sign (119906) =

1 119906 gt 0

minus1 119906 lt 0

(6)

where120572 is a parameter to control the difference scheme120572 = 0

indicates the central difference scheme is used and 120572 = 1

denotes the first-order upwind scheme is invoked In this

4 Mathematical Problems in Engineering

study 120572 = 05 is used For the viscosity term VISX thesecond-order central difference scheme is applied

VISX = []119896(1205972119906

1205971199092 +1205972119906

1205971199102 +1205972119906

1205971199112)]

119894+1119895119896

=]119896

Δ119909119894+12Δ119909119894+1Δ119909119894

[Δ119909119894(119906119899

119894+1119895119896 minus119906119899

119894119895119896)

minusΔ119909119894+1 (119906119899

119894119895119896minus119906119899

119894minus1119895119896)]

+]119896

(Δ119910119895+12 + Δ119910

119895minus12) Δ119910119895+12Δ119910119895minus12[Δ119910119895minus12 (119906

119899

119894119895+1119896

minus119906119899

119894119895119896) minusΔ119910

119895+12 (119906119899

119894119895119896minus119906119899

119894119895minus1119896)]

+]119896

(Δ119911119896+12 + Δ119911

119896minus12) Δ119911119896+12Δ119911119896minus12[Δ119911119896minus12 (119906

119899

119894119895119896+1

minus119906119899

119894119895119896) minusΔ119911

119896+12 (119906119899

119894119895119896minus119906119899

119894119895119896minus1)]

(7)

Additionally the convection flux and viscosity terms inthe other two directions can be obtained in the same way

23 Velocity-Pressure Coupling and Free Surface CapturingAt each time step the velocity field and pressure in eachcontrol volume must be solved For each cell the velocitiesat the time step 119905

119899+1are approximately obtained from the

momentum equations using the values at 119905119899 Then the

pressure and velocities must be iterated until the continuityequation is satisfied to the required accuracy In this studythe iteration accuracy is set to 120576 lt 0005

For the cells which are fully occupied by fluid thevelocity-pressure coupling is completed in the followingprocess Let 119878

119899+1 represent the left term of the continuousequation at the time step 119905

119899+1 The continuity equation is sat-

isfied when 119878119899+1 approaches zero This process is completed

by changing the pressure and can be defined as

120575119901 = minus119878119898

119899+1120597119878119898119899+1120597119901

(8)

where the subscript 119898 represents the 119898th iteration and theterm 120597119878

119898

119899+1120597119901 can be defined as

120597119878

120597119901=

Δ119905

119881119862119894119895119896

1

Δ119909119894

(1

Δ119909119894+12

119860119877119894119895119896

+1

Δ119909119894minus12

119860119877119894minus1119895119896)+

1Δ119910119895

(1

Δ119910119895+12

119860119865119894119895119896

+1

Δ119910119895minus12

119860119865119894119895minus1119896)+

1Δ119911119896

(1

Δ119911119896+12

119860119879119894119895119896

+1

Δ119911119896minus12

119860119879119894119895119896minus1)

(9)

For the cells containing a free surface the free surfaceboundary condition is satisfied by setting the surface cell

pressure 119901119899119894119895119896

equal to the value obtained by a linear interpo-lation between the surface pressure 119901

119904and the pressure of the

corresponding cell fully occupied by fluid below the surfacecell 119901

119899

120575119901 = (1minus119889119888

119889119891

)119901119899+

119889119888

119889119891

119901119904minus119901119899

119894119895119896 (10)

where 119889119888is the distance between the two interpolation cells

and 119889119891is the distance from the below cell to the free surface

After the velocity field and pressure have been obtainedby the above iteration process in each computational celloccupied by fluid the volume of fluid function 119865 is thencomputed for the new time level to give the new fluid con-figuration The fact that 119865 is a step function requires specialcare in computing the fluxes to preserve the sharp definitionof a free surface here a donor-acceptor flux approximation ofthe SOLA-VOF method [22 23] is used to determine the 119865

function at each interface

24 Boundary Conditions As shown in Figure 1 the wavemaker is set on the left side of the numerical tank (119878

1) To

generate nonlinear shallow waves and mitigate re-reflectedwaves at the wave maker the numerical absorbing cnoidalwave maker proposed byWang et al [24] is adopted Namelythe paddle velocity is defined

119880(1198961015840 119905 + 1) =

1205780 (1198961015840 119905) 119888

119889 + 1205780 (1198961015840 119905)

+[1205780 (119896

1015840 119905) minus 120578 (119896

1015840 119905)] 119888

119889 + 1205780 (1198961015840 119905) minus 120578 (1198961015840 119905)

(11)

where 120578(1198961015840 119905) is the actual wave elevation at the 1198961015840th cell of thewave maker 120578

0(1198961015840 119905) is the theoretical wave elevation and 119888

is the celerity of the cnoidal waveA sponge layer (as shown in Figure 1) is set to absorb

the outgoing waves and the reduction coefficient of particlevelocities is defined as

120583 (119909) =

radic1 minus (119909 minus 11987111198712

) 1198711 le 119909 le 119871

1 119909 lt 1198711

(12)

Besides the sponge layer the Sommerfeld radiationcondition is adopted at the right boundary of the flumeAdditionally sidewall boundaries (119878

5 1198786) and the bottom

boundary (1198783) of the numerical tank are treated as free-

slip rigid walls that is the normal velocities at the free-slip boundaries and the normal derivative of the tangentialvelocities are both set to zero

3 Validation of the Model

Before using the present model to study the interactionof waves and quasi-ellipse caissons its performance forsimulating waves and their interaction with structures shouldbe validated In this section the ability of the numericalmodel is verified using theoretical and existing experimentalresults

Mathematical Problems in Engineering 5

0 120 15030 60 90

minus1

0

1

2

3

x = 100m

120578(m

)

t (s)

(a)

0 120 15030 60 90

minus1

0

1

2

3

120578(m

)

t (s)

x = 300m

(b)

0 120 15030 60 90

x = 500m

minus1

0

1

2

3

120578(m

)

t (s)

(c)

0 120 15030 60 90

x = 700m

minus1

0

1

2

3

120578(m

)

t (s)

(d)

Figure 3 The simulated wave surface elevations at different locations along the tank for a case with 119867 = 3m 119879 = 10 s (the red dash line inthe panel of 119909 = 100m is the corresponding theoretical solution)

31 Wave Generation and Propagation Tests of the wavegeneration of the numerical wave tank are examined firstCnoidal waves with wave height 119867 varying from 10m to40m and period 119879 from 80 s to 130 s are considered Forthese simulations the sizes of the numerical tank are set asfollows 119871 = 800m 119861 = 260m 119889 = 20m 119871

1= 600m

1198712= 200m Δ119909 = 2m Δ119910 = 3m and Δ119911 = 2m Figure 3

shows the simulated wave surface elevations along the wavetank for a cnoidal wave with incident height 119867 = 3m andperiod119879 = 10 s It is found that the simulatedwaves can fit thetargeted wave very well and the surface elevations are quiteuniform and stable Increasing propagation distance due totriad nonlinear interactions the wave crests become moreandmore sharp and the troughs are flattened consisting withthe shape characteristics of shallow water waves [25 26] Thesnapshots of the simulation case are illustrated in Figure 4which shows that the waves are quite uniform along the 119910

direction and the sponge layer works well for the dampingof wave energy The present model also works well for othertests and the results are not shown here for brevity

32 Wave Pressures on a Perforated Rectangular Caisson Tofurther investigate the validation of themodel simulations onthe interactions between waves and a perforated rectangular

caisson are carried out Then the computed results arecompared with an existing experiment

The experiment was conducted in a wave flume locatedin the State Key Laboratory of Coastal and Offshore Engi-neering Dalian University of Technology China The flumeis 56m long 07m wide and 10m deep The detailed layoutof the experiments can be seen in the work of Liu et al [13]Here only schematic diagrams of the perforated rectangularcaisson model and the locations of pressure sensors areillustrated (as shown in Figure 5) The model caisson was03m long 068m wide and 07m height It was made ofPlexiglass plates of 10 cm thickness The water depth 119889 infront of the structure was 04m The caisson was dividedinto four uniform boxes named A B C and D by partitionwalls For each box the front plate was perforated by fourrectangular holes from a distance 20 cm above the bottomplate and the back wall was solid The length and width ofthe holes were 011m and 0074m respectively As marked inFigure 5 three pressure sensors (1 2 and 3) were placedon the front walls of the B box and another three sensors (45 and 6) were fixed on the back wall of the C

In the experiment cnoidal waves with incident waveheight 6 cm and wave period 12 s were generated by the wavepaddle For the simulation the mesh sizes of the numerical

6 Mathematical Problems in Engineering

260195

13065

0 0200

400600

800

Y (m) X (m)

120578(m

)

0

25

(a) 119905 = 400 s

260195

13065

0 0200

400600

800

Y (m) X (m)

120578(m

)

0

25

195400

600

(b) 119905 = 1200 s

Figure 4 Snapshots of the numerical tank for the simulation of waves with119867 = 3m 119879 = 10 s

AB

CD

3

2

1

170

170

170

170

680300

700

200400

SWL

(a)

A

SWL

B C D

400

200

680

700

6

5

4

(b)

Figure 5 Sketch of the perforated caisson model (unit mm) and the placement of pressure sensors (a) oblique front view (b) back view

tank are set as follows Δ119909 = 3 cm Δ119910 = 405 cm andΔ119911 = 425 cm The comparisons between the simulated andexperimental results are shown in Figure 6 It is found that thecomputed pressure matches well with experimental resultsunder the water surface (sensors 1 2 4 and 5) For thelocations close to the water surface (sensors 3 and 6) thecomputation slightly overestimates the results at the troughsof the pressure time history The discrepancy may be mainlydue to the complex flow characteristics near the water surfacecaused by strong interactions between the waves and thestructure

The above discussion indicates that the present three-dimensional numerical model can be used to investigate theinteractions betweenwaves and coastal structures In the nextsection the model will be used to study the interactionsbetween waves and quasi-ellipse caissons

4 Waves Interaction with PerforatedQuasi-Ellipse Caissons

41 Description of Quasi-Ellipse Caissons A quasi-ellipsecaisson is made up by two hollow semicylinders which areconnected by two rectangular plates to form a chamber [20]Figure 7 shows a cross section of a quasi-ellipse 119863 denotes

the diameter the semicylinders and119861 is the length of the rect-angular plates In this study the values of 119863 and 119861 are fixedfor all simulations 119863 = 20m and 119861 = 12m For multiplecaissons119863

119888represents the distance between caissons In this

study the caissons are perforated by rectangular holes with3m in height and 4m in length and the positions of the holesare marked in Figure 8

As mentioned in the previous section the computationdomain of the model is divided by regular cuboid cellsHowever for quasi-ellipse caissons some cells at the bound-aries cannot be fully occupied by the solid boundaries (seeFigure 9) To solve this problem the partial cell methodwhich is also used to descript the free surface is adopted Asshown in Figure 9 solid straight lines are used to representthe intersection lines between the caisson boundary and thefluid for simplicity

42 Numerical Results For the simulations of wave interac-tion with quasi-ellipse caissons the sizes of the numericaltank are set as follows 119871 = 800m 119861 = 260m 119889 = 20m1198711= 600m 119871

2= 200mΔ119909 = 2mΔ119910 = 3m andΔ119911 = 2m

Figure 10 shows the distribution of surface elevations aroundthe multiple quasi-ellipse caissons in a wave period and theadjacent distance between the caissons 119863

119888is set equal to 2119863

Mathematical Problems in Engineering 7

0 1 2 3 4 5minus04

minus02

00

02

04

066

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

065

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

064

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

063

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

062

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

06

t (s)

p(k

Pa)

1

Figure 6 Comparisons of the time series of wave pressures on the perforated rectangular caisson (open circles) experimental results (solidlines) numerical results

D

Fminus F+D

2

X

Z

B

(a)

X

Dc FminusXM D

F+XM

(b)

Figure 7 Sketch of a quasi-ellipse caisson (a) and multiple quasi-ellipse caissons (b)

(40m) For this simulation the height of the incident cnoidalwave is 3m with a period of 10 s As wave crest approachesthe front faces of the caissons the runup of waves is veryhigh and partially caused by significant reflection of thecaissons The difference between the front and rear faces inthis situation is pronounced suggesting that the wave forceson the caissons are very large It is noticed that the surface

elevations between the caissons are also remarkable Theresults for the perforated caissons are shown in Figure 11Thesurface elevations close to the caissons are reduced comparedto that without perforation indicating that the perforationcan reduce the wave runuprundown on the caissons Thesurface elevations between the caissons are also significantlyweakened

8 Mathematical Problems in Engineering

30

m

2m

4m

3m

SWL4m 4m

2m 1

m SWL

Front view Side view

1m

Figure 8 Sketch of a perforated quasi-ellipse caisson

(a) Solid quasi-ellipse caisson (b) Perforated quasi-ellipse caisson

Figure 9 Sketch of the mesh division of a cross section for the numerical model

The perforation on caissons can enhance the waterexchange wave energy is partially dissipated by the perfo-rated wall and some wave energy is transferred to the fluidinside the chamber of the caissons (see Figure 12) and hencethe wave reflection and runup are reduced As the wavetroughs approach the middle of the caissons (Figure 12(a))the difference of water levels between inside and outside ofthe chambers is pronounced which causes the water flowto the outside Therefore the wave height in the middle ofthe caisson is reduced For the time when the wave crestsare close to this region the wave flow direction is reversed(Figure 12(c)) and the wave height outside the chambersis decreased The intensive water exchange between insideand outside of the chambers suggests that wave energy isdissipated significantly

The previous discussion suggests the perforation cansignificantly reduce the wave force of the caissons In thissection the wave forces and the influence factor will beinvestigated for multiple caissons To achieve these targetscnoidal waves with different incident height and periodare generated and the interactions between the waves anda caisson without peroration are simulated first and thenthe multiple placed caissons Following the definition ofwave force used for an isolated square structure [21] thenondimensional wave force on the single placed caisson canbe written as

119865119909119878

=119865119909

120588119892 (119861 + 119863) 1198892 [tanh (119896119889) 119896119889] (13)

where 119896 is the wave number of the incident wave and 119865119909is the

maximum wave force in the 119909 direction and can be obtainedby integrating the surface pressure on the structure

The wave forces on the single placed caisson are shownin Figure 13 As expected higher incident waves (119867119889 =

015) can produce larger wave forces With decreasing ofwave period (ie 119896 is increased) wave force on the caissonis increased

For convenient illustration and comparison the waveforces on a single placed caisson without perforation areused as benchmarkThen a nondimensional force which canreflect the influence of multiple placement is defined as

119877119909=

10038161003816100381610038161198651199091198721003816100381610038161003816

10038161003816100381610038161198651199091198781003816100381610038161003816

(14)

where 119865119909119872

is maximum force in 119909 direction for the centralone of the caisson group (as illustrated in Figure 7)

The nondimensional forces 119877119909on the middle caisson

without perforation are shown in Figure 14 It is found thatthe values of 119877

119909are larger than 1 for all the simulated cases

indicating that the presence of adjacent caissons can increasethe wave force With decreasing incident wave period (ieincreasing 119896119863) 119877

119909decreases Additionally 119877

119909for the cases

in the condition with 119863119888119863 = 2 is significantly larger than

those cases with larger transverse distance 119863119888 But the 119877

119909

differences between the cases in the conditions 119863119888119863 = 3

and 119863119888119863 = 4 are much smaller especially for the cases

with 119896119863 gt 10 By increasing incident wave nonlinearity

Mathematical Problems in Engineering 9

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(a) 119905 = 45119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(c) 119905 = (45 + 05)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(d) 119905 = (45 + 075)119879

Figure 10 Contours of water surface elevations around multiple impermeable quasi-ellipse caissons in a wave period (119879 = 10 s119867 = 30m)unit (m)

119877119909can also be enhanced but the increment is limited For the

tests with perforation the influence of incident wave periodand adjacent distance on the forces has a similar trend (seeFigure 15)

To directly reveal the influence of the perforation the119877119909difference between the impermeable caisson and the

perforated one is defined as

119876119909=10038161003816100381610038161003816119877119909119904

minus119877119909119901

10038161003816100381610038161003816 (15)

where 119877119909119904

is the relative wave for the caisson withoutperforation and 119877

119909119901is for the perforated one Figure 16

demonstrates the variations of 119876119909for all the simulated cases

For tests with the smaller incident wave height (119867119889 = 005)the reduction of wave forces by the perforation increaseswith the decrease of wave period The distance between thecaissons 119863

119888also has influence on the reduction wave forces

But if the adjacent distance is larger enough the influenceis limited for smaller incident waves For incident waveswith higher nonlinearity (ie 119867119889 = 015) the effect ofperforation appears smaller than the cases with the smallerincident wavesThewave period effect for the conditionswith119863119888119863 = 30 and 40 follows the trend of the cases with

the smaller incident height However the tendency for thecondition 119863

119888119863 = 20 is much complex When 119896119863 lt 10

the perforationworks better than those conditionswith largertransverse distances

5 Conclusions

A three-dimensional numerical wave flume was establishedusing the continuity equation and the Navier-Stokes equa-tions The VOF method was adopted to capture the freesurfaces The ability of generation of cnoidal waves and its

10 Mathematical Problems in Engineering

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(a) 119905 = 45119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(c) 119905 = (45 + 05)119879

140 175 210 245 280

200

180

160

140

120

100

80

60

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(d) 119905 = (45 + 075)119879

Figure 11 Contours of water surface elevations around multiple perforated quasi-ellipse caissons in a wave period (119879 = 10 s 119867 = 30m)unit (m)

interaction with a perforated rectangle caisson were verifiedusing the existing experimental data Then the presentmodel was used to investigate interactions between cnoidalwaves with varying incident height and period and multipleperforated quasi-ellipse caissons The influence of the per-foration on the wave field and the velocity flow around thecaissons was examined The numerical results demonstratethat the wave runuprundown on the caissons was reducedsignificantly by perforation and hence the wave forces arealso decreased The reduction of wave runuprundown ismainly due to the extensivewater exchange inside and outsidethe caisson chamber It is also found that the effect of thepresence of adjacent caissons on wave forces decreases withdecreasing incident wave period for both multiple placedcaissons with and without perforation The wave forces forcloser transverse adjacent distance are larger than those

conditions with a farther distance but when relative adjacentdistance 119863

119888119863 (119863

119888is the transverse distance and 119863 is the

width of the quasi-ellipse caisson) is larger than 3 the effectof adjacent distance is limited Additionally the effect ofthe perforation increases with the reduction of incidentwave period The performance of the perforation is betterwhen incident wave height is small suggesting that wavenonlinearity also plays an important role in the process ofinteraction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(a) 119905 = 45119879

200

180

160

140

120

100

80

60

Y(m

)

140 175 210 245 280

X (m)

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(c) 119905 = (45 + 05)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(d) 119905 = (45 + 075)119879

Figure 12 The flow vector fields corresponding to the wave surface elevations of Figure 11

014

012

010

008

006

004

00207 08 09 10 11 12 13 14 15

FxS

kD

Hd = 005

Hd = 015

Figure 13 Variations of wave forces on an isolated caisson without perforation

12 Mathematical Problems in Engineering

06 08 10 12 14 16095

100

105

110

115

120

DcD = 2

DcD = 3

DcD = 4

Rx

kD

(a)

06 08 10 12 14 16095

100

105

110

115

120

Rx

kD

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 14 Influence of transverse distance 119871 on the nondimensional wave forces 119877119909of the caisson without perforation (a) cases with119867119889 =

005 (b) cases with119867119889 = 015

06 08 10 12 14 16075

080

085

090

095

100

Rx

kD

DcD = 2

DcD = 3

DcD = 4

(a)

06 08 10 12 14 16

Rx

kD

075

080

085

090

095

100

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 15 Influence of transverse distance 119871 on the nondimensional wave forces 119877119909of the caisson without perforation (a) cases with119867119889 =

005 (b) cases with119867119889 = 015

Mathematical Problems in Engineering 13

06 08 10 12 14 16

025

020

010

015

Qx

kD

DcD = 2

DcD = 3

DcD = 4

(a)Qx

kD

06 08 10 12 14 16

025

020

010

015

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 16 Variations of 119876119909with respect to incident wave period for cases in conditions with different transverse distance 119871 (a) cases with

119867119889 = 005 (b) cases with119867119889 = 015

Acknowledgments

This research is supported financially by the National NaturalScience Foundation (Grant nos 51422901 and 51221961) andA Foundation for the Author of National Excellent DoctoralDissertation of PR China (Grant no 201347)

References

[1] G E Jarlan ldquoA perforated vertical wall breakwaterrdquo The Dockand Harbour Authority vol 41 no 486 pp 394ndash398 1961

[2] Z Huang Y Li and Y Liu ldquoHydraulic performance andwave loadings of perforatedslotted coastal structures a reviewrdquoOcean Engineering vol 38 no 10 pp 1031ndash1053 2011

[3] M K M Darwiche A N Williams and K-H Wang ldquoWaveinteraction with semiporous cylindrical breakwaterrdquo Journal ofWaterway Port Coastal and Ocean Engineering vol 120 no 4pp 382ndash403 1994

[4] T L Yip and A T Chwang ldquoPerforated wall breakwater withinternal horizontal platerdquo Journal of EngineeringMechanics vol126 no 5 pp 533ndash538 2000

[5] K D Suh J C Choi B H Kim W S Park and K S LeeldquoReflection of irregular waves from perforated-wall caissonbreakwatersrdquo Coastal Engineering vol 44 no 2 pp 141ndash1512001

[6] Y C Li H J Liu B Teng and D P Sun ldquoReflection of obliqueincident waves by breakwaters with partially perforated wallrdquoChina Ocean Engineering vol 16 no 3 pp 329ndash342 2002

[7] Y C Li G H Dong H J Liu and D P Sun ldquoThe reflectionof oblique incident waves by breakwaters with double-layeredperforated wallrdquo Coastal Engineering vol 50 no 1-2 pp 47ndash602003

[8] Y Liu Y-C Li and B Teng ldquoWave interactionwith a perforatedwall breakwater with a submerged horizontal porous platerdquoOcean Engineering vol 34 no 17-18 pp 2364ndash2373 2007

[9] Y Liu Y-C Li and B Teng ldquoThe reflection of oblique wavesby an infinite number of partially perforated caissonsrdquo OceanEngineering vol 34 no 14-15 pp 1965ndash1976 2007

[10] K Sankarbabu S A Sannasiraj and V Sundar ldquoHydrodynamicperformance of a dual cylindrical caisson breakwaterrdquo CoastalEngineering vol 55 no 6 pp 431ndash446 2008

[11] S Neelamani G Koether H Schuttrumpf M Muttray andH Oumeraci ldquoWave forces on and water-surface fluctuationsaround a vertical cylinder encircled by a perforated squarecaissonrdquo Ocean Engineering vol 27 no 7 pp 775ndash800 2000

[12] K Vijayalakshmi S Neelamani R Sundaravadivelu and KMurali ldquoWave runup on a concentric twin perforated circularcylinderrdquo Ocean Engineering vol 34 no 2 pp 327ndash336 2007

[13] Y Liu Y Li B Teng J Jiang and B Ma ldquoTotal horizontal andvertical forces of irregular waves on partially perforated caissonbreakwatersrdquo Coastal Engineering vol 55 no 6 pp 537ndash5522008

[14] J-I Lee and S Shin ldquoExperimental study on the wave reflectionof partially perforated wall caissons with single and doublechambersrdquo Ocean Engineering vol 91 pp 1ndash10 2014

[15] J-I Lee Y-T Kim and S Shin ldquoExperimental studies onwave interactions of partially perforated wall under obliquelyincident wavesrdquo The Scientific World Journal vol 2014 ArticleID 954174 14 pages 2014

[16] X F Chen Y C Li Y X Wang G H Dong and X BaildquoNumerical simulation of wave interaction with perforatedcaisson breakwatersrdquoChina Ocean Engineering vol 17 no 1 pp33ndash43 2003

[17] J Liu G Lin and J Li ldquoShort-crested waves interaction with aconcentric cylindrical structure with double-layered perforatedwallsrdquo Ocean Engineering vol 40 pp 76ndash90 2012

14 Mathematical Problems in Engineering

[18] J Liu and G Lin ldquoNumerical modelling of wave interactionwith a concentric cylindrical system with an arc-shaped porousouter cylinderrdquo European Journal of MechanicsmdashBFluids vol37 pp 59ndash71 2013

[19] F Aristodemo D D Meringolo P Groenenboom A Lo Schi-avo P Veltri and M Veltri ldquoAssessment of dynamic pressuresat vertical and perforated breakwaters through diffusive SPHschemesrdquo Mathematical Problems in Engineering vol 2015Article ID 305028 10 pages 2015

[20] J Bai and J Hu ldquoResearch on and design of new structure ofelliptical caissoned pierrdquo Port and Waterway Engineering vol11 pp 25ndash30 2006 (Chinese)

[21] Y Wang X Ren and G Wang ldquoNumerical simulation ofnonlinear wave force on a quasi-ellipse caissonrdquo Journal ofMarine Science and Application vol 10 no 3 pp 265ndash271 2011

[22] C W Hirt and B D Nichols ldquoVolume of fluid (VOF) methodfor the dynamics of free boundariesrdquo Journal of ComputationalPhysics vol 39 no 1 pp 201ndash225 1981

[23] Y Wang and T Su ldquoNumerical simulation of liquid sloshing incylindrical containersrdquo Acta Aerodynamica Sinica vol 9 no 1pp 112ndash119 1991 (Chinese)

[24] Y Wang J Zang and D Qiu ldquoNumerical model of cnoidalwave flumerdquo China Ocean Engineering vol 13 no 4 pp 391ndash398 1999

[25] S Elgar and R T Guza ldquoObservations of bispectral of shoalingsurface gravity wavesrdquo Journal of Fluid Mechanics vol 161 pp425ndash448 1985

[26] G Dong Y Ma M Perlin X Ma B Yu and J Xu ldquoExperi-mental study of wavemdashwave nonlinear interactions using thewavelet-based bicoherencerdquo Coastal Engineering vol 55 no 9pp 741ndash752 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

moments on the cylinder are influenced by the porosity ofthe outer caisson Vijayalakshmi et al [12] experimentallystudied wave runup and rundown on a twin perforatedcircular cylinder and then developed a predictive formulafor the wave runup and rundown on the perforated cylinderbased on the experimental results Liu et al [13] performedan experimental and theoretical study on wave forces onpartially perforated rectangular caissons by normally inci-dent irregular waves Lee and Shin [14] experimentallytested wave reflection of partially perforated wall caissonstructures with single and double chambers Lee et al[15] conducted a three-dimensional experiment to studythe evolution of stem waves on partially perforated wallsand found that the perforation can effectively reduce waveheight

As aspects on numerical simulations Chen et al [16]developed a numerical model based on a two-dimensionalVOF method combined with the 119896-120576 model and used themodel to study wave interactions with perforated rectangularcaisson breakwaters Later this model was extended to inves-tigate wave forces on a perforated caisson with a top coverLiu et al [17] used the semianalytical scaled boundary finiteelement method to deal with the short-crested wave inter-action with a concentric cylindrical structure with double-layered perforated walls Later Liu and Lin [18] extendedthe numerical model to study short-crested wave interactionwith a concentric cylindrical system Recently Aristodemoet al [19] developed a numerical model based on the SPHmethod to calculate wave pressures acting on vertical andslotted coastal structures

The type of quasi-ellipse caisson is a relatively new struc-ture For open deep-water wharves quasi-ellipse caissonswere proven to be more effective and economical than thecircular and rectangular caissons [20] It was used to supportgravity dolphin wharves in Dalian Ore Terminal Phase IIProject Dalian China and the ore wharf of DongjiakouPort Tsingtao China Compared with the researches onthe circular and rectangular types investigations on thehydrodynamics of quasi-ellipse caissons are rare especiallyfor the perforated ones Wang et al [21] developed a 3Dnumerical model based on the VOF method to study theinteractions between waves and a single quasi-ellipse caissonIt was found that both wave height and caisson size havesignificant influences on the wave forces on the caissonHowever in real project caissons are always arranged ingroups Therefore the efforts on the interactions betweenwaves and multiple quasi-ellipse caissons with and withoutperforation should be carried outThis is the main aim of thepresent study In this study the numerical model developedby Wang et al [21] is extended to study normally incidentcnoidal waves interactionwithmultiple quasi-ellipse caissonsand the influence of perforation

Following the introduction the detailed setup of a 3Dnumerical model is described in Section 2 Then the modelis validated in Section 3 Simulations on interactions betweenwaves and multiple caissons with and without perforationand the results are under discussion in Section 4 Lastly someconclusions are advanced in Section 5

2 Numerical Model

In this section the governing equations and numerical algo-rithm are introduced in detail especially for the treatmentof the boundary conditions Figure 1 shows the schematicdrawing of the present model where 119871 is the tank length 119861is the tank width 119889 is the still water depth 119871

1is the length

of the working zone of the tank and 1198712is the length of the

sponger layer The origin of the Cartesian coordinate system119900119909119910119911 is set in the corner of the bottomwith the119909-axis positivein the direction of incoming wave propagation and the 119911-axispositive upwards as shown in Figure 1

21 Government Equations In this study the flow is assumedto be incompressibleThe continuity equation and theNavier-Stokes equations are adopted as the governing equations Andthe water surface is traced by the volume of fluid (VOF)method

Continuity equation is as follows

120597

120597119909(120579119906) +

120597

120597119910(120579V) +

120597

120597119911(120579119908) = 0 (1)

Navier-Stokes equations are as follows

120597119906

120597119905+ 119906

120597119906

120597119909+ V

120597119906

120597119910+119908

120597119906

120597119911

= minus1120588

120597119901

120597119909+ ]119896(1205972119906

1205971199092 +1205972119906

1205971199102 +1205972119906

1205971199112)

120597V120597119905

+ 119906120597V120597119909

+ V120597V120597119910

+119908120597V120597119911

= minus1120588

120597119901

120597119910+119892+ ]

119896(1205972V

1205971199092 +1205972V

1205971199102 +1205972V

1205971199112)

120597119908

120597119905+ 119906

120597119908

120597119909+ V

120597119908

120597119910+119908

120597119908

120597119911

= minus1120588

120597119901

120597119911+ ]119896(1205972119908

1205971199092 +1205972119908

1205971199102 +1205972119908

1205971199112)

(2)

where 119906 V and 119908 are the velocity components in the 119909119910 and 119911 directions respectively 119905 represents the time 119892 isthe gravitational acceleration 120588 is the fluid density and isequal to 1000 kgm3 119901 is the pressure ]

119896is the coefficient

of kinematic viscosity and is set to 10 times 10minus6m2s 120579 is thevolume fraction of the partial cell and its value is determinedafter the computational discretization

To capture the free surfaces a function 119865(119909 119910 119909 119905) isintroducedThe value of 119865 in a cell represents the occupationratio by fluid 119865 = 1 for a cell full with fluid whereas 119865 = 0

for a cell without fluid Cells with 119865 values between 0 and 1indicate that the cells are partially filled with fluidThese cellsare intersected by the free surface or contain bubbles whichare smaller than the cells [22 23] The time dependence of 119865can be handled by

120597

120597119905(120579119865) +

120597

120597119909(120579119906119865) +

120597

120597119910(120579V119865) +

120597

120597119911(120579119908119865) = 0 (3)

Mathematical Problems in Engineering 3

z

D1

D2S1 S2

S3

S4

S5

S6

d

x

B

B1

B2

L

L1 L2y

oA1

A2

C1

C2

Figure 1 Schematic drawing of the numerical tank and the coordinate system

Δz k

y Δxi

VCijk Fijk

z

Δyj

xo

pijk

(a)

o

y

Vij+12k

AFijk

ATijk

z

x

AR ij

ku i+

12jk

wijk+12

(b)

Figure 2 Parameter definition of staggered cells (a) at the center of a cell (b) at the sides of a cell

22 Discretization Scheme A finite difference method ischosen to solve the governing equations The staggered cellis adopted for the computational discretizationThe locationsof the variations 119901 119865 119906 V and 119908 used in the computationare illustrated in Figure 2 Quantities 119860119877

119894119895119896 119860119879119894119895119896

119860119865119894119895119896

and 119881119862

119894119895119896 denote the fractions open to flow for the right

top and front cell faces and the cell volume respectivelyThe discretization scheme of the continuity equation is

written as

1119881119862119894119895119896

(119906119899+1119894119895119896

119860119877119894119895119896

minus 119906119899+1119894minus1119895119896119860119877

119894minus1119895119896

Δ119909119894

+V119899+1119894119895119896

119860119865119894119895119896

minus V119899+1119894119895minus1119896119860119865

119894119895minus1119896

Δ119911119896

+119908119899+1119894119895119896

119860119879119894119895119896

minus 119908119899+1119894119895119896minus1119860119879

119894119895119896minus1

Δ119910119895

) = 0

(4)

The time derivative is discretized by the forward timedifference scheme for the momentum equations the finitedifference approximation for a typical cell in 119909 direction isgiven as follows

119906119899+1119894119895119896

= 119906119899

119894119895119896+

Δ119905

Δ119909[minus (119901119899+1119894+1119895119896 minus119901

119899+1119894119895119896

) minus FUXminus FUY

minus FUZ+VISX] (5)

where FUX FUY and FUZ are the convection flux termsin the three directions and a combination of the centraldifference scheme and the upwind scheme is employed todiscretize them For example the discretized form for FUXis defined as

FUX = (119906120597119906

120597119909)

119894+1119895119896=

119906119899

119894119895119896

Δ119909120572

Δ119909119894+1

Δ119909119894

(119906119899

119894119895119896minus119906119899

119894minus1119895119896)

+Δ119909119894

Δ119909119894+1

(119906119899

119894+1119895119896 minus119906119899

119894119895119896) + 120572 sdot sign (119906

119899

119894119895119896)

sdot [Δ119909119894+1

Δ119909119894

(119906119899

119894119895119896minus119906119899

119894minus1119895119896)

+Δ119909119894

Δ119909119894+1

(119906119899

119894+1119895119896 minus119906119899

119894119895119896)]

Δ119909120572= Δ119909119894+1 +Δ119909

119894+120572 sdot sign (119906

119899

119894119895119896) (Δ119909119894+1 minusΔ119909

119894)

sign (119906) =

1 119906 gt 0

minus1 119906 lt 0

(6)

where120572 is a parameter to control the difference scheme120572 = 0

indicates the central difference scheme is used and 120572 = 1

denotes the first-order upwind scheme is invoked In this

4 Mathematical Problems in Engineering

study 120572 = 05 is used For the viscosity term VISX thesecond-order central difference scheme is applied

VISX = []119896(1205972119906

1205971199092 +1205972119906

1205971199102 +1205972119906

1205971199112)]

119894+1119895119896

=]119896

Δ119909119894+12Δ119909119894+1Δ119909119894

[Δ119909119894(119906119899

119894+1119895119896 minus119906119899

119894119895119896)

minusΔ119909119894+1 (119906119899

119894119895119896minus119906119899

119894minus1119895119896)]

+]119896

(Δ119910119895+12 + Δ119910

119895minus12) Δ119910119895+12Δ119910119895minus12[Δ119910119895minus12 (119906

119899

119894119895+1119896

minus119906119899

119894119895119896) minusΔ119910

119895+12 (119906119899

119894119895119896minus119906119899

119894119895minus1119896)]

+]119896

(Δ119911119896+12 + Δ119911

119896minus12) Δ119911119896+12Δ119911119896minus12[Δ119911119896minus12 (119906

119899

119894119895119896+1

minus119906119899

119894119895119896) minusΔ119911

119896+12 (119906119899

119894119895119896minus119906119899

119894119895119896minus1)]

(7)

Additionally the convection flux and viscosity terms inthe other two directions can be obtained in the same way

23 Velocity-Pressure Coupling and Free Surface CapturingAt each time step the velocity field and pressure in eachcontrol volume must be solved For each cell the velocitiesat the time step 119905

119899+1are approximately obtained from the

momentum equations using the values at 119905119899 Then the

pressure and velocities must be iterated until the continuityequation is satisfied to the required accuracy In this studythe iteration accuracy is set to 120576 lt 0005

For the cells which are fully occupied by fluid thevelocity-pressure coupling is completed in the followingprocess Let 119878

119899+1 represent the left term of the continuousequation at the time step 119905

119899+1 The continuity equation is sat-

isfied when 119878119899+1 approaches zero This process is completed

by changing the pressure and can be defined as

120575119901 = minus119878119898

119899+1120597119878119898119899+1120597119901

(8)

where the subscript 119898 represents the 119898th iteration and theterm 120597119878

119898

119899+1120597119901 can be defined as

120597119878

120597119901=

Δ119905

119881119862119894119895119896

1

Δ119909119894

(1

Δ119909119894+12

119860119877119894119895119896

+1

Δ119909119894minus12

119860119877119894minus1119895119896)+

1Δ119910119895

(1

Δ119910119895+12

119860119865119894119895119896

+1

Δ119910119895minus12

119860119865119894119895minus1119896)+

1Δ119911119896

(1

Δ119911119896+12

119860119879119894119895119896

+1

Δ119911119896minus12

119860119879119894119895119896minus1)

(9)

For the cells containing a free surface the free surfaceboundary condition is satisfied by setting the surface cell

pressure 119901119899119894119895119896

equal to the value obtained by a linear interpo-lation between the surface pressure 119901

119904and the pressure of the

corresponding cell fully occupied by fluid below the surfacecell 119901

119899

120575119901 = (1minus119889119888

119889119891

)119901119899+

119889119888

119889119891

119901119904minus119901119899

119894119895119896 (10)

where 119889119888is the distance between the two interpolation cells

and 119889119891is the distance from the below cell to the free surface

After the velocity field and pressure have been obtainedby the above iteration process in each computational celloccupied by fluid the volume of fluid function 119865 is thencomputed for the new time level to give the new fluid con-figuration The fact that 119865 is a step function requires specialcare in computing the fluxes to preserve the sharp definitionof a free surface here a donor-acceptor flux approximation ofthe SOLA-VOF method [22 23] is used to determine the 119865

function at each interface

24 Boundary Conditions As shown in Figure 1 the wavemaker is set on the left side of the numerical tank (119878

1) To

generate nonlinear shallow waves and mitigate re-reflectedwaves at the wave maker the numerical absorbing cnoidalwave maker proposed byWang et al [24] is adopted Namelythe paddle velocity is defined

119880(1198961015840 119905 + 1) =

1205780 (1198961015840 119905) 119888

119889 + 1205780 (1198961015840 119905)

+[1205780 (119896

1015840 119905) minus 120578 (119896

1015840 119905)] 119888

119889 + 1205780 (1198961015840 119905) minus 120578 (1198961015840 119905)

(11)

where 120578(1198961015840 119905) is the actual wave elevation at the 1198961015840th cell of thewave maker 120578

0(1198961015840 119905) is the theoretical wave elevation and 119888

is the celerity of the cnoidal waveA sponge layer (as shown in Figure 1) is set to absorb

the outgoing waves and the reduction coefficient of particlevelocities is defined as

120583 (119909) =

radic1 minus (119909 minus 11987111198712

) 1198711 le 119909 le 119871

1 119909 lt 1198711

(12)

Besides the sponge layer the Sommerfeld radiationcondition is adopted at the right boundary of the flumeAdditionally sidewall boundaries (119878

5 1198786) and the bottom

boundary (1198783) of the numerical tank are treated as free-

slip rigid walls that is the normal velocities at the free-slip boundaries and the normal derivative of the tangentialvelocities are both set to zero

3 Validation of the Model

Before using the present model to study the interactionof waves and quasi-ellipse caissons its performance forsimulating waves and their interaction with structures shouldbe validated In this section the ability of the numericalmodel is verified using theoretical and existing experimentalresults

Mathematical Problems in Engineering 5

0 120 15030 60 90

minus1

0

1

2

3

x = 100m

120578(m

)

t (s)

(a)

0 120 15030 60 90

minus1

0

1

2

3

120578(m

)

t (s)

x = 300m

(b)

0 120 15030 60 90

x = 500m

minus1

0

1

2

3

120578(m

)

t (s)

(c)

0 120 15030 60 90

x = 700m

minus1

0

1

2

3

120578(m

)

t (s)

(d)

Figure 3 The simulated wave surface elevations at different locations along the tank for a case with 119867 = 3m 119879 = 10 s (the red dash line inthe panel of 119909 = 100m is the corresponding theoretical solution)

31 Wave Generation and Propagation Tests of the wavegeneration of the numerical wave tank are examined firstCnoidal waves with wave height 119867 varying from 10m to40m and period 119879 from 80 s to 130 s are considered Forthese simulations the sizes of the numerical tank are set asfollows 119871 = 800m 119861 = 260m 119889 = 20m 119871

1= 600m

1198712= 200m Δ119909 = 2m Δ119910 = 3m and Δ119911 = 2m Figure 3

shows the simulated wave surface elevations along the wavetank for a cnoidal wave with incident height 119867 = 3m andperiod119879 = 10 s It is found that the simulatedwaves can fit thetargeted wave very well and the surface elevations are quiteuniform and stable Increasing propagation distance due totriad nonlinear interactions the wave crests become moreandmore sharp and the troughs are flattened consisting withthe shape characteristics of shallow water waves [25 26] Thesnapshots of the simulation case are illustrated in Figure 4which shows that the waves are quite uniform along the 119910

direction and the sponge layer works well for the dampingof wave energy The present model also works well for othertests and the results are not shown here for brevity

32 Wave Pressures on a Perforated Rectangular Caisson Tofurther investigate the validation of themodel simulations onthe interactions between waves and a perforated rectangular

caisson are carried out Then the computed results arecompared with an existing experiment

The experiment was conducted in a wave flume locatedin the State Key Laboratory of Coastal and Offshore Engi-neering Dalian University of Technology China The flumeis 56m long 07m wide and 10m deep The detailed layoutof the experiments can be seen in the work of Liu et al [13]Here only schematic diagrams of the perforated rectangularcaisson model and the locations of pressure sensors areillustrated (as shown in Figure 5) The model caisson was03m long 068m wide and 07m height It was made ofPlexiglass plates of 10 cm thickness The water depth 119889 infront of the structure was 04m The caisson was dividedinto four uniform boxes named A B C and D by partitionwalls For each box the front plate was perforated by fourrectangular holes from a distance 20 cm above the bottomplate and the back wall was solid The length and width ofthe holes were 011m and 0074m respectively As marked inFigure 5 three pressure sensors (1 2 and 3) were placedon the front walls of the B box and another three sensors (45 and 6) were fixed on the back wall of the C

In the experiment cnoidal waves with incident waveheight 6 cm and wave period 12 s were generated by the wavepaddle For the simulation the mesh sizes of the numerical

6 Mathematical Problems in Engineering

260195

13065

0 0200

400600

800

Y (m) X (m)

120578(m

)

0

25

(a) 119905 = 400 s

260195

13065

0 0200

400600

800

Y (m) X (m)

120578(m

)

0

25

195400

600

(b) 119905 = 1200 s

Figure 4 Snapshots of the numerical tank for the simulation of waves with119867 = 3m 119879 = 10 s

AB

CD

3

2

1

170

170

170

170

680300

700

200400

SWL

(a)

A

SWL

B C D

400

200

680

700

6

5

4

(b)

Figure 5 Sketch of the perforated caisson model (unit mm) and the placement of pressure sensors (a) oblique front view (b) back view

tank are set as follows Δ119909 = 3 cm Δ119910 = 405 cm andΔ119911 = 425 cm The comparisons between the simulated andexperimental results are shown in Figure 6 It is found that thecomputed pressure matches well with experimental resultsunder the water surface (sensors 1 2 4 and 5) For thelocations close to the water surface (sensors 3 and 6) thecomputation slightly overestimates the results at the troughsof the pressure time history The discrepancy may be mainlydue to the complex flow characteristics near the water surfacecaused by strong interactions between the waves and thestructure

The above discussion indicates that the present three-dimensional numerical model can be used to investigate theinteractions betweenwaves and coastal structures In the nextsection the model will be used to study the interactionsbetween waves and quasi-ellipse caissons

4 Waves Interaction with PerforatedQuasi-Ellipse Caissons

41 Description of Quasi-Ellipse Caissons A quasi-ellipsecaisson is made up by two hollow semicylinders which areconnected by two rectangular plates to form a chamber [20]Figure 7 shows a cross section of a quasi-ellipse 119863 denotes

the diameter the semicylinders and119861 is the length of the rect-angular plates In this study the values of 119863 and 119861 are fixedfor all simulations 119863 = 20m and 119861 = 12m For multiplecaissons119863

119888represents the distance between caissons In this

study the caissons are perforated by rectangular holes with3m in height and 4m in length and the positions of the holesare marked in Figure 8

As mentioned in the previous section the computationdomain of the model is divided by regular cuboid cellsHowever for quasi-ellipse caissons some cells at the bound-aries cannot be fully occupied by the solid boundaries (seeFigure 9) To solve this problem the partial cell methodwhich is also used to descript the free surface is adopted Asshown in Figure 9 solid straight lines are used to representthe intersection lines between the caisson boundary and thefluid for simplicity

42 Numerical Results For the simulations of wave interac-tion with quasi-ellipse caissons the sizes of the numericaltank are set as follows 119871 = 800m 119861 = 260m 119889 = 20m1198711= 600m 119871

2= 200mΔ119909 = 2mΔ119910 = 3m andΔ119911 = 2m

Figure 10 shows the distribution of surface elevations aroundthe multiple quasi-ellipse caissons in a wave period and theadjacent distance between the caissons 119863

119888is set equal to 2119863

Mathematical Problems in Engineering 7

0 1 2 3 4 5minus04

minus02

00

02

04

066

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

065

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

064

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

063

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

062

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

06

t (s)

p(k

Pa)

1

Figure 6 Comparisons of the time series of wave pressures on the perforated rectangular caisson (open circles) experimental results (solidlines) numerical results

D

Fminus F+D

2

X

Z

B

(a)

X

Dc FminusXM D

F+XM

(b)

Figure 7 Sketch of a quasi-ellipse caisson (a) and multiple quasi-ellipse caissons (b)

(40m) For this simulation the height of the incident cnoidalwave is 3m with a period of 10 s As wave crest approachesthe front faces of the caissons the runup of waves is veryhigh and partially caused by significant reflection of thecaissons The difference between the front and rear faces inthis situation is pronounced suggesting that the wave forceson the caissons are very large It is noticed that the surface

elevations between the caissons are also remarkable Theresults for the perforated caissons are shown in Figure 11Thesurface elevations close to the caissons are reduced comparedto that without perforation indicating that the perforationcan reduce the wave runuprundown on the caissons Thesurface elevations between the caissons are also significantlyweakened

8 Mathematical Problems in Engineering

30

m

2m

4m

3m

SWL4m 4m

2m 1

m SWL

Front view Side view

1m

Figure 8 Sketch of a perforated quasi-ellipse caisson

(a) Solid quasi-ellipse caisson (b) Perforated quasi-ellipse caisson

Figure 9 Sketch of the mesh division of a cross section for the numerical model

The perforation on caissons can enhance the waterexchange wave energy is partially dissipated by the perfo-rated wall and some wave energy is transferred to the fluidinside the chamber of the caissons (see Figure 12) and hencethe wave reflection and runup are reduced As the wavetroughs approach the middle of the caissons (Figure 12(a))the difference of water levels between inside and outside ofthe chambers is pronounced which causes the water flowto the outside Therefore the wave height in the middle ofthe caisson is reduced For the time when the wave crestsare close to this region the wave flow direction is reversed(Figure 12(c)) and the wave height outside the chambersis decreased The intensive water exchange between insideand outside of the chambers suggests that wave energy isdissipated significantly

The previous discussion suggests the perforation cansignificantly reduce the wave force of the caissons In thissection the wave forces and the influence factor will beinvestigated for multiple caissons To achieve these targetscnoidal waves with different incident height and periodare generated and the interactions between the waves anda caisson without peroration are simulated first and thenthe multiple placed caissons Following the definition ofwave force used for an isolated square structure [21] thenondimensional wave force on the single placed caisson canbe written as

119865119909119878

=119865119909

120588119892 (119861 + 119863) 1198892 [tanh (119896119889) 119896119889] (13)

where 119896 is the wave number of the incident wave and 119865119909is the

maximum wave force in the 119909 direction and can be obtainedby integrating the surface pressure on the structure

The wave forces on the single placed caisson are shownin Figure 13 As expected higher incident waves (119867119889 =

015) can produce larger wave forces With decreasing ofwave period (ie 119896 is increased) wave force on the caissonis increased

For convenient illustration and comparison the waveforces on a single placed caisson without perforation areused as benchmarkThen a nondimensional force which canreflect the influence of multiple placement is defined as

119877119909=

10038161003816100381610038161198651199091198721003816100381610038161003816

10038161003816100381610038161198651199091198781003816100381610038161003816

(14)

where 119865119909119872

is maximum force in 119909 direction for the centralone of the caisson group (as illustrated in Figure 7)

The nondimensional forces 119877119909on the middle caisson

without perforation are shown in Figure 14 It is found thatthe values of 119877

119909are larger than 1 for all the simulated cases

indicating that the presence of adjacent caissons can increasethe wave force With decreasing incident wave period (ieincreasing 119896119863) 119877

119909decreases Additionally 119877

119909for the cases

in the condition with 119863119888119863 = 2 is significantly larger than

those cases with larger transverse distance 119863119888 But the 119877

119909

differences between the cases in the conditions 119863119888119863 = 3

and 119863119888119863 = 4 are much smaller especially for the cases

with 119896119863 gt 10 By increasing incident wave nonlinearity

Mathematical Problems in Engineering 9

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(a) 119905 = 45119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(c) 119905 = (45 + 05)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(d) 119905 = (45 + 075)119879

Figure 10 Contours of water surface elevations around multiple impermeable quasi-ellipse caissons in a wave period (119879 = 10 s119867 = 30m)unit (m)

119877119909can also be enhanced but the increment is limited For the

tests with perforation the influence of incident wave periodand adjacent distance on the forces has a similar trend (seeFigure 15)

To directly reveal the influence of the perforation the119877119909difference between the impermeable caisson and the

perforated one is defined as

119876119909=10038161003816100381610038161003816119877119909119904

minus119877119909119901

10038161003816100381610038161003816 (15)

where 119877119909119904

is the relative wave for the caisson withoutperforation and 119877

119909119901is for the perforated one Figure 16

demonstrates the variations of 119876119909for all the simulated cases

For tests with the smaller incident wave height (119867119889 = 005)the reduction of wave forces by the perforation increaseswith the decrease of wave period The distance between thecaissons 119863

119888also has influence on the reduction wave forces

But if the adjacent distance is larger enough the influenceis limited for smaller incident waves For incident waveswith higher nonlinearity (ie 119867119889 = 015) the effect ofperforation appears smaller than the cases with the smallerincident wavesThewave period effect for the conditionswith119863119888119863 = 30 and 40 follows the trend of the cases with

the smaller incident height However the tendency for thecondition 119863

119888119863 = 20 is much complex When 119896119863 lt 10

the perforationworks better than those conditionswith largertransverse distances

5 Conclusions

A three-dimensional numerical wave flume was establishedusing the continuity equation and the Navier-Stokes equa-tions The VOF method was adopted to capture the freesurfaces The ability of generation of cnoidal waves and its

10 Mathematical Problems in Engineering

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(a) 119905 = 45119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(c) 119905 = (45 + 05)119879

140 175 210 245 280

200

180

160

140

120

100

80

60

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(d) 119905 = (45 + 075)119879

Figure 11 Contours of water surface elevations around multiple perforated quasi-ellipse caissons in a wave period (119879 = 10 s 119867 = 30m)unit (m)

interaction with a perforated rectangle caisson were verifiedusing the existing experimental data Then the presentmodel was used to investigate interactions between cnoidalwaves with varying incident height and period and multipleperforated quasi-ellipse caissons The influence of the per-foration on the wave field and the velocity flow around thecaissons was examined The numerical results demonstratethat the wave runuprundown on the caissons was reducedsignificantly by perforation and hence the wave forces arealso decreased The reduction of wave runuprundown ismainly due to the extensivewater exchange inside and outsidethe caisson chamber It is also found that the effect of thepresence of adjacent caissons on wave forces decreases withdecreasing incident wave period for both multiple placedcaissons with and without perforation The wave forces forcloser transverse adjacent distance are larger than those

conditions with a farther distance but when relative adjacentdistance 119863

119888119863 (119863

119888is the transverse distance and 119863 is the

width of the quasi-ellipse caisson) is larger than 3 the effectof adjacent distance is limited Additionally the effect ofthe perforation increases with the reduction of incidentwave period The performance of the perforation is betterwhen incident wave height is small suggesting that wavenonlinearity also plays an important role in the process ofinteraction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(a) 119905 = 45119879

200

180

160

140

120

100

80

60

Y(m

)

140 175 210 245 280

X (m)

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(c) 119905 = (45 + 05)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(d) 119905 = (45 + 075)119879

Figure 12 The flow vector fields corresponding to the wave surface elevations of Figure 11

014

012

010

008

006

004

00207 08 09 10 11 12 13 14 15

FxS

kD

Hd = 005

Hd = 015

Figure 13 Variations of wave forces on an isolated caisson without perforation

12 Mathematical Problems in Engineering

06 08 10 12 14 16095

100

105

110

115

120

DcD = 2

DcD = 3

DcD = 4

Rx

kD

(a)

06 08 10 12 14 16095

100

105

110

115

120

Rx

kD

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 14 Influence of transverse distance 119871 on the nondimensional wave forces 119877119909of the caisson without perforation (a) cases with119867119889 =

005 (b) cases with119867119889 = 015

06 08 10 12 14 16075

080

085

090

095

100

Rx

kD

DcD = 2

DcD = 3

DcD = 4

(a)

06 08 10 12 14 16

Rx

kD

075

080

085

090

095

100

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 15 Influence of transverse distance 119871 on the nondimensional wave forces 119877119909of the caisson without perforation (a) cases with119867119889 =

005 (b) cases with119867119889 = 015

Mathematical Problems in Engineering 13

06 08 10 12 14 16

025

020

010

015

Qx

kD

DcD = 2

DcD = 3

DcD = 4

(a)Qx

kD

06 08 10 12 14 16

025

020

010

015

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 16 Variations of 119876119909with respect to incident wave period for cases in conditions with different transverse distance 119871 (a) cases with

119867119889 = 005 (b) cases with119867119889 = 015

Acknowledgments

This research is supported financially by the National NaturalScience Foundation (Grant nos 51422901 and 51221961) andA Foundation for the Author of National Excellent DoctoralDissertation of PR China (Grant no 201347)

References

[1] G E Jarlan ldquoA perforated vertical wall breakwaterrdquo The Dockand Harbour Authority vol 41 no 486 pp 394ndash398 1961

[2] Z Huang Y Li and Y Liu ldquoHydraulic performance andwave loadings of perforatedslotted coastal structures a reviewrdquoOcean Engineering vol 38 no 10 pp 1031ndash1053 2011

[3] M K M Darwiche A N Williams and K-H Wang ldquoWaveinteraction with semiporous cylindrical breakwaterrdquo Journal ofWaterway Port Coastal and Ocean Engineering vol 120 no 4pp 382ndash403 1994

[4] T L Yip and A T Chwang ldquoPerforated wall breakwater withinternal horizontal platerdquo Journal of EngineeringMechanics vol126 no 5 pp 533ndash538 2000

[5] K D Suh J C Choi B H Kim W S Park and K S LeeldquoReflection of irregular waves from perforated-wall caissonbreakwatersrdquo Coastal Engineering vol 44 no 2 pp 141ndash1512001

[6] Y C Li H J Liu B Teng and D P Sun ldquoReflection of obliqueincident waves by breakwaters with partially perforated wallrdquoChina Ocean Engineering vol 16 no 3 pp 329ndash342 2002

[7] Y C Li G H Dong H J Liu and D P Sun ldquoThe reflectionof oblique incident waves by breakwaters with double-layeredperforated wallrdquo Coastal Engineering vol 50 no 1-2 pp 47ndash602003

[8] Y Liu Y-C Li and B Teng ldquoWave interactionwith a perforatedwall breakwater with a submerged horizontal porous platerdquoOcean Engineering vol 34 no 17-18 pp 2364ndash2373 2007

[9] Y Liu Y-C Li and B Teng ldquoThe reflection of oblique wavesby an infinite number of partially perforated caissonsrdquo OceanEngineering vol 34 no 14-15 pp 1965ndash1976 2007

[10] K Sankarbabu S A Sannasiraj and V Sundar ldquoHydrodynamicperformance of a dual cylindrical caisson breakwaterrdquo CoastalEngineering vol 55 no 6 pp 431ndash446 2008

[11] S Neelamani G Koether H Schuttrumpf M Muttray andH Oumeraci ldquoWave forces on and water-surface fluctuationsaround a vertical cylinder encircled by a perforated squarecaissonrdquo Ocean Engineering vol 27 no 7 pp 775ndash800 2000

[12] K Vijayalakshmi S Neelamani R Sundaravadivelu and KMurali ldquoWave runup on a concentric twin perforated circularcylinderrdquo Ocean Engineering vol 34 no 2 pp 327ndash336 2007

[13] Y Liu Y Li B Teng J Jiang and B Ma ldquoTotal horizontal andvertical forces of irregular waves on partially perforated caissonbreakwatersrdquo Coastal Engineering vol 55 no 6 pp 537ndash5522008

[14] J-I Lee and S Shin ldquoExperimental study on the wave reflectionof partially perforated wall caissons with single and doublechambersrdquo Ocean Engineering vol 91 pp 1ndash10 2014

[15] J-I Lee Y-T Kim and S Shin ldquoExperimental studies onwave interactions of partially perforated wall under obliquelyincident wavesrdquo The Scientific World Journal vol 2014 ArticleID 954174 14 pages 2014

[16] X F Chen Y C Li Y X Wang G H Dong and X BaildquoNumerical simulation of wave interaction with perforatedcaisson breakwatersrdquoChina Ocean Engineering vol 17 no 1 pp33ndash43 2003

[17] J Liu G Lin and J Li ldquoShort-crested waves interaction with aconcentric cylindrical structure with double-layered perforatedwallsrdquo Ocean Engineering vol 40 pp 76ndash90 2012

14 Mathematical Problems in Engineering

[18] J Liu and G Lin ldquoNumerical modelling of wave interactionwith a concentric cylindrical system with an arc-shaped porousouter cylinderrdquo European Journal of MechanicsmdashBFluids vol37 pp 59ndash71 2013

[19] F Aristodemo D D Meringolo P Groenenboom A Lo Schi-avo P Veltri and M Veltri ldquoAssessment of dynamic pressuresat vertical and perforated breakwaters through diffusive SPHschemesrdquo Mathematical Problems in Engineering vol 2015Article ID 305028 10 pages 2015

[20] J Bai and J Hu ldquoResearch on and design of new structure ofelliptical caissoned pierrdquo Port and Waterway Engineering vol11 pp 25ndash30 2006 (Chinese)

[21] Y Wang X Ren and G Wang ldquoNumerical simulation ofnonlinear wave force on a quasi-ellipse caissonrdquo Journal ofMarine Science and Application vol 10 no 3 pp 265ndash271 2011

[22] C W Hirt and B D Nichols ldquoVolume of fluid (VOF) methodfor the dynamics of free boundariesrdquo Journal of ComputationalPhysics vol 39 no 1 pp 201ndash225 1981

[23] Y Wang and T Su ldquoNumerical simulation of liquid sloshing incylindrical containersrdquo Acta Aerodynamica Sinica vol 9 no 1pp 112ndash119 1991 (Chinese)

[24] Y Wang J Zang and D Qiu ldquoNumerical model of cnoidalwave flumerdquo China Ocean Engineering vol 13 no 4 pp 391ndash398 1999

[25] S Elgar and R T Guza ldquoObservations of bispectral of shoalingsurface gravity wavesrdquo Journal of Fluid Mechanics vol 161 pp425ndash448 1985

[26] G Dong Y Ma M Perlin X Ma B Yu and J Xu ldquoExperi-mental study of wavemdashwave nonlinear interactions using thewavelet-based bicoherencerdquo Coastal Engineering vol 55 no 9pp 741ndash752 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

z

D1

D2S1 S2

S3

S4

S5

S6

d

x

B

B1

B2

L

L1 L2y

oA1

A2

C1

C2

Figure 1 Schematic drawing of the numerical tank and the coordinate system

Δz k

y Δxi

VCijk Fijk

z

Δyj

xo

pijk

(a)

o

y

Vij+12k

AFijk

ATijk

z

x

AR ij

ku i+

12jk

wijk+12

(b)

Figure 2 Parameter definition of staggered cells (a) at the center of a cell (b) at the sides of a cell

22 Discretization Scheme A finite difference method ischosen to solve the governing equations The staggered cellis adopted for the computational discretizationThe locationsof the variations 119901 119865 119906 V and 119908 used in the computationare illustrated in Figure 2 Quantities 119860119877

119894119895119896 119860119879119894119895119896

119860119865119894119895119896

and 119881119862

119894119895119896 denote the fractions open to flow for the right

top and front cell faces and the cell volume respectivelyThe discretization scheme of the continuity equation is

written as

1119881119862119894119895119896

(119906119899+1119894119895119896

119860119877119894119895119896

minus 119906119899+1119894minus1119895119896119860119877

119894minus1119895119896

Δ119909119894

+V119899+1119894119895119896

119860119865119894119895119896

minus V119899+1119894119895minus1119896119860119865

119894119895minus1119896

Δ119911119896

+119908119899+1119894119895119896

119860119879119894119895119896

minus 119908119899+1119894119895119896minus1119860119879

119894119895119896minus1

Δ119910119895

) = 0

(4)

The time derivative is discretized by the forward timedifference scheme for the momentum equations the finitedifference approximation for a typical cell in 119909 direction isgiven as follows

119906119899+1119894119895119896

= 119906119899

119894119895119896+

Δ119905

Δ119909[minus (119901119899+1119894+1119895119896 minus119901

119899+1119894119895119896

) minus FUXminus FUY

minus FUZ+VISX] (5)

where FUX FUY and FUZ are the convection flux termsin the three directions and a combination of the centraldifference scheme and the upwind scheme is employed todiscretize them For example the discretized form for FUXis defined as

FUX = (119906120597119906

120597119909)

119894+1119895119896=

119906119899

119894119895119896

Δ119909120572

Δ119909119894+1

Δ119909119894

(119906119899

119894119895119896minus119906119899

119894minus1119895119896)

+Δ119909119894

Δ119909119894+1

(119906119899

119894+1119895119896 minus119906119899

119894119895119896) + 120572 sdot sign (119906

119899

119894119895119896)

sdot [Δ119909119894+1

Δ119909119894

(119906119899

119894119895119896minus119906119899

119894minus1119895119896)

+Δ119909119894

Δ119909119894+1

(119906119899

119894+1119895119896 minus119906119899

119894119895119896)]

Δ119909120572= Δ119909119894+1 +Δ119909

119894+120572 sdot sign (119906

119899

119894119895119896) (Δ119909119894+1 minusΔ119909

119894)

sign (119906) =

1 119906 gt 0

minus1 119906 lt 0

(6)

where120572 is a parameter to control the difference scheme120572 = 0

indicates the central difference scheme is used and 120572 = 1

denotes the first-order upwind scheme is invoked In this

4 Mathematical Problems in Engineering

study 120572 = 05 is used For the viscosity term VISX thesecond-order central difference scheme is applied

VISX = []119896(1205972119906

1205971199092 +1205972119906

1205971199102 +1205972119906

1205971199112)]

119894+1119895119896

=]119896

Δ119909119894+12Δ119909119894+1Δ119909119894

[Δ119909119894(119906119899

119894+1119895119896 minus119906119899

119894119895119896)

minusΔ119909119894+1 (119906119899

119894119895119896minus119906119899

119894minus1119895119896)]

+]119896

(Δ119910119895+12 + Δ119910

119895minus12) Δ119910119895+12Δ119910119895minus12[Δ119910119895minus12 (119906

119899

119894119895+1119896

minus119906119899

119894119895119896) minusΔ119910

119895+12 (119906119899

119894119895119896minus119906119899

119894119895minus1119896)]

+]119896

(Δ119911119896+12 + Δ119911

119896minus12) Δ119911119896+12Δ119911119896minus12[Δ119911119896minus12 (119906

119899

119894119895119896+1

minus119906119899

119894119895119896) minusΔ119911

119896+12 (119906119899

119894119895119896minus119906119899

119894119895119896minus1)]

(7)

Additionally the convection flux and viscosity terms inthe other two directions can be obtained in the same way

23 Velocity-Pressure Coupling and Free Surface CapturingAt each time step the velocity field and pressure in eachcontrol volume must be solved For each cell the velocitiesat the time step 119905

119899+1are approximately obtained from the

momentum equations using the values at 119905119899 Then the

pressure and velocities must be iterated until the continuityequation is satisfied to the required accuracy In this studythe iteration accuracy is set to 120576 lt 0005

For the cells which are fully occupied by fluid thevelocity-pressure coupling is completed in the followingprocess Let 119878

119899+1 represent the left term of the continuousequation at the time step 119905

119899+1 The continuity equation is sat-

isfied when 119878119899+1 approaches zero This process is completed

by changing the pressure and can be defined as

120575119901 = minus119878119898

119899+1120597119878119898119899+1120597119901

(8)

where the subscript 119898 represents the 119898th iteration and theterm 120597119878

119898

119899+1120597119901 can be defined as

120597119878

120597119901=

Δ119905

119881119862119894119895119896

1

Δ119909119894

(1

Δ119909119894+12

119860119877119894119895119896

+1

Δ119909119894minus12

119860119877119894minus1119895119896)+

1Δ119910119895

(1

Δ119910119895+12

119860119865119894119895119896

+1

Δ119910119895minus12

119860119865119894119895minus1119896)+

1Δ119911119896

(1

Δ119911119896+12

119860119879119894119895119896

+1

Δ119911119896minus12

119860119879119894119895119896minus1)

(9)

For the cells containing a free surface the free surfaceboundary condition is satisfied by setting the surface cell

pressure 119901119899119894119895119896

equal to the value obtained by a linear interpo-lation between the surface pressure 119901

119904and the pressure of the

corresponding cell fully occupied by fluid below the surfacecell 119901

119899

120575119901 = (1minus119889119888

119889119891

)119901119899+

119889119888

119889119891

119901119904minus119901119899

119894119895119896 (10)

where 119889119888is the distance between the two interpolation cells

and 119889119891is the distance from the below cell to the free surface

After the velocity field and pressure have been obtainedby the above iteration process in each computational celloccupied by fluid the volume of fluid function 119865 is thencomputed for the new time level to give the new fluid con-figuration The fact that 119865 is a step function requires specialcare in computing the fluxes to preserve the sharp definitionof a free surface here a donor-acceptor flux approximation ofthe SOLA-VOF method [22 23] is used to determine the 119865

function at each interface

24 Boundary Conditions As shown in Figure 1 the wavemaker is set on the left side of the numerical tank (119878

1) To

generate nonlinear shallow waves and mitigate re-reflectedwaves at the wave maker the numerical absorbing cnoidalwave maker proposed byWang et al [24] is adopted Namelythe paddle velocity is defined

119880(1198961015840 119905 + 1) =

1205780 (1198961015840 119905) 119888

119889 + 1205780 (1198961015840 119905)

+[1205780 (119896

1015840 119905) minus 120578 (119896

1015840 119905)] 119888

119889 + 1205780 (1198961015840 119905) minus 120578 (1198961015840 119905)

(11)

where 120578(1198961015840 119905) is the actual wave elevation at the 1198961015840th cell of thewave maker 120578

0(1198961015840 119905) is the theoretical wave elevation and 119888

is the celerity of the cnoidal waveA sponge layer (as shown in Figure 1) is set to absorb

the outgoing waves and the reduction coefficient of particlevelocities is defined as

120583 (119909) =

radic1 minus (119909 minus 11987111198712

) 1198711 le 119909 le 119871

1 119909 lt 1198711

(12)

Besides the sponge layer the Sommerfeld radiationcondition is adopted at the right boundary of the flumeAdditionally sidewall boundaries (119878

5 1198786) and the bottom

boundary (1198783) of the numerical tank are treated as free-

slip rigid walls that is the normal velocities at the free-slip boundaries and the normal derivative of the tangentialvelocities are both set to zero

3 Validation of the Model

Before using the present model to study the interactionof waves and quasi-ellipse caissons its performance forsimulating waves and their interaction with structures shouldbe validated In this section the ability of the numericalmodel is verified using theoretical and existing experimentalresults

Mathematical Problems in Engineering 5

0 120 15030 60 90

minus1

0

1

2

3

x = 100m

120578(m

)

t (s)

(a)

0 120 15030 60 90

minus1

0

1

2

3

120578(m

)

t (s)

x = 300m

(b)

0 120 15030 60 90

x = 500m

minus1

0

1

2

3

120578(m

)

t (s)

(c)

0 120 15030 60 90

x = 700m

minus1

0

1

2

3

120578(m

)

t (s)

(d)

Figure 3 The simulated wave surface elevations at different locations along the tank for a case with 119867 = 3m 119879 = 10 s (the red dash line inthe panel of 119909 = 100m is the corresponding theoretical solution)

31 Wave Generation and Propagation Tests of the wavegeneration of the numerical wave tank are examined firstCnoidal waves with wave height 119867 varying from 10m to40m and period 119879 from 80 s to 130 s are considered Forthese simulations the sizes of the numerical tank are set asfollows 119871 = 800m 119861 = 260m 119889 = 20m 119871

1= 600m

1198712= 200m Δ119909 = 2m Δ119910 = 3m and Δ119911 = 2m Figure 3

shows the simulated wave surface elevations along the wavetank for a cnoidal wave with incident height 119867 = 3m andperiod119879 = 10 s It is found that the simulatedwaves can fit thetargeted wave very well and the surface elevations are quiteuniform and stable Increasing propagation distance due totriad nonlinear interactions the wave crests become moreandmore sharp and the troughs are flattened consisting withthe shape characteristics of shallow water waves [25 26] Thesnapshots of the simulation case are illustrated in Figure 4which shows that the waves are quite uniform along the 119910

direction and the sponge layer works well for the dampingof wave energy The present model also works well for othertests and the results are not shown here for brevity

32 Wave Pressures on a Perforated Rectangular Caisson Tofurther investigate the validation of themodel simulations onthe interactions between waves and a perforated rectangular

caisson are carried out Then the computed results arecompared with an existing experiment

The experiment was conducted in a wave flume locatedin the State Key Laboratory of Coastal and Offshore Engi-neering Dalian University of Technology China The flumeis 56m long 07m wide and 10m deep The detailed layoutof the experiments can be seen in the work of Liu et al [13]Here only schematic diagrams of the perforated rectangularcaisson model and the locations of pressure sensors areillustrated (as shown in Figure 5) The model caisson was03m long 068m wide and 07m height It was made ofPlexiglass plates of 10 cm thickness The water depth 119889 infront of the structure was 04m The caisson was dividedinto four uniform boxes named A B C and D by partitionwalls For each box the front plate was perforated by fourrectangular holes from a distance 20 cm above the bottomplate and the back wall was solid The length and width ofthe holes were 011m and 0074m respectively As marked inFigure 5 three pressure sensors (1 2 and 3) were placedon the front walls of the B box and another three sensors (45 and 6) were fixed on the back wall of the C

In the experiment cnoidal waves with incident waveheight 6 cm and wave period 12 s were generated by the wavepaddle For the simulation the mesh sizes of the numerical

6 Mathematical Problems in Engineering

260195

13065

0 0200

400600

800

Y (m) X (m)

120578(m

)

0

25

(a) 119905 = 400 s

260195

13065

0 0200

400600

800

Y (m) X (m)

120578(m

)

0

25

195400

600

(b) 119905 = 1200 s

Figure 4 Snapshots of the numerical tank for the simulation of waves with119867 = 3m 119879 = 10 s

AB

CD

3

2

1

170

170

170

170

680300

700

200400

SWL

(a)

A

SWL

B C D

400

200

680

700

6

5

4

(b)

Figure 5 Sketch of the perforated caisson model (unit mm) and the placement of pressure sensors (a) oblique front view (b) back view

tank are set as follows Δ119909 = 3 cm Δ119910 = 405 cm andΔ119911 = 425 cm The comparisons between the simulated andexperimental results are shown in Figure 6 It is found that thecomputed pressure matches well with experimental resultsunder the water surface (sensors 1 2 4 and 5) For thelocations close to the water surface (sensors 3 and 6) thecomputation slightly overestimates the results at the troughsof the pressure time history The discrepancy may be mainlydue to the complex flow characteristics near the water surfacecaused by strong interactions between the waves and thestructure

The above discussion indicates that the present three-dimensional numerical model can be used to investigate theinteractions betweenwaves and coastal structures In the nextsection the model will be used to study the interactionsbetween waves and quasi-ellipse caissons

4 Waves Interaction with PerforatedQuasi-Ellipse Caissons

41 Description of Quasi-Ellipse Caissons A quasi-ellipsecaisson is made up by two hollow semicylinders which areconnected by two rectangular plates to form a chamber [20]Figure 7 shows a cross section of a quasi-ellipse 119863 denotes

the diameter the semicylinders and119861 is the length of the rect-angular plates In this study the values of 119863 and 119861 are fixedfor all simulations 119863 = 20m and 119861 = 12m For multiplecaissons119863

119888represents the distance between caissons In this

study the caissons are perforated by rectangular holes with3m in height and 4m in length and the positions of the holesare marked in Figure 8

As mentioned in the previous section the computationdomain of the model is divided by regular cuboid cellsHowever for quasi-ellipse caissons some cells at the bound-aries cannot be fully occupied by the solid boundaries (seeFigure 9) To solve this problem the partial cell methodwhich is also used to descript the free surface is adopted Asshown in Figure 9 solid straight lines are used to representthe intersection lines between the caisson boundary and thefluid for simplicity

42 Numerical Results For the simulations of wave interac-tion with quasi-ellipse caissons the sizes of the numericaltank are set as follows 119871 = 800m 119861 = 260m 119889 = 20m1198711= 600m 119871

2= 200mΔ119909 = 2mΔ119910 = 3m andΔ119911 = 2m

Figure 10 shows the distribution of surface elevations aroundthe multiple quasi-ellipse caissons in a wave period and theadjacent distance between the caissons 119863

119888is set equal to 2119863

Mathematical Problems in Engineering 7

0 1 2 3 4 5minus04

minus02

00

02

04

066

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

065

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

064

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

063

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

062

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

06

t (s)

p(k

Pa)

1

Figure 6 Comparisons of the time series of wave pressures on the perforated rectangular caisson (open circles) experimental results (solidlines) numerical results

D

Fminus F+D

2

X

Z

B

(a)

X

Dc FminusXM D

F+XM

(b)

Figure 7 Sketch of a quasi-ellipse caisson (a) and multiple quasi-ellipse caissons (b)

(40m) For this simulation the height of the incident cnoidalwave is 3m with a period of 10 s As wave crest approachesthe front faces of the caissons the runup of waves is veryhigh and partially caused by significant reflection of thecaissons The difference between the front and rear faces inthis situation is pronounced suggesting that the wave forceson the caissons are very large It is noticed that the surface

elevations between the caissons are also remarkable Theresults for the perforated caissons are shown in Figure 11Thesurface elevations close to the caissons are reduced comparedto that without perforation indicating that the perforationcan reduce the wave runuprundown on the caissons Thesurface elevations between the caissons are also significantlyweakened

8 Mathematical Problems in Engineering

30

m

2m

4m

3m

SWL4m 4m

2m 1

m SWL

Front view Side view

1m

Figure 8 Sketch of a perforated quasi-ellipse caisson

(a) Solid quasi-ellipse caisson (b) Perforated quasi-ellipse caisson

Figure 9 Sketch of the mesh division of a cross section for the numerical model

The perforation on caissons can enhance the waterexchange wave energy is partially dissipated by the perfo-rated wall and some wave energy is transferred to the fluidinside the chamber of the caissons (see Figure 12) and hencethe wave reflection and runup are reduced As the wavetroughs approach the middle of the caissons (Figure 12(a))the difference of water levels between inside and outside ofthe chambers is pronounced which causes the water flowto the outside Therefore the wave height in the middle ofthe caisson is reduced For the time when the wave crestsare close to this region the wave flow direction is reversed(Figure 12(c)) and the wave height outside the chambersis decreased The intensive water exchange between insideand outside of the chambers suggests that wave energy isdissipated significantly

The previous discussion suggests the perforation cansignificantly reduce the wave force of the caissons In thissection the wave forces and the influence factor will beinvestigated for multiple caissons To achieve these targetscnoidal waves with different incident height and periodare generated and the interactions between the waves anda caisson without peroration are simulated first and thenthe multiple placed caissons Following the definition ofwave force used for an isolated square structure [21] thenondimensional wave force on the single placed caisson canbe written as

119865119909119878

=119865119909

120588119892 (119861 + 119863) 1198892 [tanh (119896119889) 119896119889] (13)

where 119896 is the wave number of the incident wave and 119865119909is the

maximum wave force in the 119909 direction and can be obtainedby integrating the surface pressure on the structure

The wave forces on the single placed caisson are shownin Figure 13 As expected higher incident waves (119867119889 =

015) can produce larger wave forces With decreasing ofwave period (ie 119896 is increased) wave force on the caissonis increased

For convenient illustration and comparison the waveforces on a single placed caisson without perforation areused as benchmarkThen a nondimensional force which canreflect the influence of multiple placement is defined as

119877119909=

10038161003816100381610038161198651199091198721003816100381610038161003816

10038161003816100381610038161198651199091198781003816100381610038161003816

(14)

where 119865119909119872

is maximum force in 119909 direction for the centralone of the caisson group (as illustrated in Figure 7)

The nondimensional forces 119877119909on the middle caisson

without perforation are shown in Figure 14 It is found thatthe values of 119877

119909are larger than 1 for all the simulated cases

indicating that the presence of adjacent caissons can increasethe wave force With decreasing incident wave period (ieincreasing 119896119863) 119877

119909decreases Additionally 119877

119909for the cases

in the condition with 119863119888119863 = 2 is significantly larger than

those cases with larger transverse distance 119863119888 But the 119877

119909

differences between the cases in the conditions 119863119888119863 = 3

and 119863119888119863 = 4 are much smaller especially for the cases

with 119896119863 gt 10 By increasing incident wave nonlinearity

Mathematical Problems in Engineering 9

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(a) 119905 = 45119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(c) 119905 = (45 + 05)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(d) 119905 = (45 + 075)119879

Figure 10 Contours of water surface elevations around multiple impermeable quasi-ellipse caissons in a wave period (119879 = 10 s119867 = 30m)unit (m)

119877119909can also be enhanced but the increment is limited For the

tests with perforation the influence of incident wave periodand adjacent distance on the forces has a similar trend (seeFigure 15)

To directly reveal the influence of the perforation the119877119909difference between the impermeable caisson and the

perforated one is defined as

119876119909=10038161003816100381610038161003816119877119909119904

minus119877119909119901

10038161003816100381610038161003816 (15)

where 119877119909119904

is the relative wave for the caisson withoutperforation and 119877

119909119901is for the perforated one Figure 16

demonstrates the variations of 119876119909for all the simulated cases

For tests with the smaller incident wave height (119867119889 = 005)the reduction of wave forces by the perforation increaseswith the decrease of wave period The distance between thecaissons 119863

119888also has influence on the reduction wave forces

But if the adjacent distance is larger enough the influenceis limited for smaller incident waves For incident waveswith higher nonlinearity (ie 119867119889 = 015) the effect ofperforation appears smaller than the cases with the smallerincident wavesThewave period effect for the conditionswith119863119888119863 = 30 and 40 follows the trend of the cases with

the smaller incident height However the tendency for thecondition 119863

119888119863 = 20 is much complex When 119896119863 lt 10

the perforationworks better than those conditionswith largertransverse distances

5 Conclusions

A three-dimensional numerical wave flume was establishedusing the continuity equation and the Navier-Stokes equa-tions The VOF method was adopted to capture the freesurfaces The ability of generation of cnoidal waves and its

10 Mathematical Problems in Engineering

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(a) 119905 = 45119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(c) 119905 = (45 + 05)119879

140 175 210 245 280

200

180

160

140

120

100

80

60

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(d) 119905 = (45 + 075)119879

Figure 11 Contours of water surface elevations around multiple perforated quasi-ellipse caissons in a wave period (119879 = 10 s 119867 = 30m)unit (m)

interaction with a perforated rectangle caisson were verifiedusing the existing experimental data Then the presentmodel was used to investigate interactions between cnoidalwaves with varying incident height and period and multipleperforated quasi-ellipse caissons The influence of the per-foration on the wave field and the velocity flow around thecaissons was examined The numerical results demonstratethat the wave runuprundown on the caissons was reducedsignificantly by perforation and hence the wave forces arealso decreased The reduction of wave runuprundown ismainly due to the extensivewater exchange inside and outsidethe caisson chamber It is also found that the effect of thepresence of adjacent caissons on wave forces decreases withdecreasing incident wave period for both multiple placedcaissons with and without perforation The wave forces forcloser transverse adjacent distance are larger than those

conditions with a farther distance but when relative adjacentdistance 119863

119888119863 (119863

119888is the transverse distance and 119863 is the

width of the quasi-ellipse caisson) is larger than 3 the effectof adjacent distance is limited Additionally the effect ofthe perforation increases with the reduction of incidentwave period The performance of the perforation is betterwhen incident wave height is small suggesting that wavenonlinearity also plays an important role in the process ofinteraction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(a) 119905 = 45119879

200

180

160

140

120

100

80

60

Y(m

)

140 175 210 245 280

X (m)

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(c) 119905 = (45 + 05)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(d) 119905 = (45 + 075)119879

Figure 12 The flow vector fields corresponding to the wave surface elevations of Figure 11

014

012

010

008

006

004

00207 08 09 10 11 12 13 14 15

FxS

kD

Hd = 005

Hd = 015

Figure 13 Variations of wave forces on an isolated caisson without perforation

12 Mathematical Problems in Engineering

06 08 10 12 14 16095

100

105

110

115

120

DcD = 2

DcD = 3

DcD = 4

Rx

kD

(a)

06 08 10 12 14 16095

100

105

110

115

120

Rx

kD

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 14 Influence of transverse distance 119871 on the nondimensional wave forces 119877119909of the caisson without perforation (a) cases with119867119889 =

005 (b) cases with119867119889 = 015

06 08 10 12 14 16075

080

085

090

095

100

Rx

kD

DcD = 2

DcD = 3

DcD = 4

(a)

06 08 10 12 14 16

Rx

kD

075

080

085

090

095

100

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 15 Influence of transverse distance 119871 on the nondimensional wave forces 119877119909of the caisson without perforation (a) cases with119867119889 =

005 (b) cases with119867119889 = 015

Mathematical Problems in Engineering 13

06 08 10 12 14 16

025

020

010

015

Qx

kD

DcD = 2

DcD = 3

DcD = 4

(a)Qx

kD

06 08 10 12 14 16

025

020

010

015

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 16 Variations of 119876119909with respect to incident wave period for cases in conditions with different transverse distance 119871 (a) cases with

119867119889 = 005 (b) cases with119867119889 = 015

Acknowledgments

This research is supported financially by the National NaturalScience Foundation (Grant nos 51422901 and 51221961) andA Foundation for the Author of National Excellent DoctoralDissertation of PR China (Grant no 201347)

References

[1] G E Jarlan ldquoA perforated vertical wall breakwaterrdquo The Dockand Harbour Authority vol 41 no 486 pp 394ndash398 1961

[2] Z Huang Y Li and Y Liu ldquoHydraulic performance andwave loadings of perforatedslotted coastal structures a reviewrdquoOcean Engineering vol 38 no 10 pp 1031ndash1053 2011

[3] M K M Darwiche A N Williams and K-H Wang ldquoWaveinteraction with semiporous cylindrical breakwaterrdquo Journal ofWaterway Port Coastal and Ocean Engineering vol 120 no 4pp 382ndash403 1994

[4] T L Yip and A T Chwang ldquoPerforated wall breakwater withinternal horizontal platerdquo Journal of EngineeringMechanics vol126 no 5 pp 533ndash538 2000

[5] K D Suh J C Choi B H Kim W S Park and K S LeeldquoReflection of irregular waves from perforated-wall caissonbreakwatersrdquo Coastal Engineering vol 44 no 2 pp 141ndash1512001

[6] Y C Li H J Liu B Teng and D P Sun ldquoReflection of obliqueincident waves by breakwaters with partially perforated wallrdquoChina Ocean Engineering vol 16 no 3 pp 329ndash342 2002

[7] Y C Li G H Dong H J Liu and D P Sun ldquoThe reflectionof oblique incident waves by breakwaters with double-layeredperforated wallrdquo Coastal Engineering vol 50 no 1-2 pp 47ndash602003

[8] Y Liu Y-C Li and B Teng ldquoWave interactionwith a perforatedwall breakwater with a submerged horizontal porous platerdquoOcean Engineering vol 34 no 17-18 pp 2364ndash2373 2007

[9] Y Liu Y-C Li and B Teng ldquoThe reflection of oblique wavesby an infinite number of partially perforated caissonsrdquo OceanEngineering vol 34 no 14-15 pp 1965ndash1976 2007

[10] K Sankarbabu S A Sannasiraj and V Sundar ldquoHydrodynamicperformance of a dual cylindrical caisson breakwaterrdquo CoastalEngineering vol 55 no 6 pp 431ndash446 2008

[11] S Neelamani G Koether H Schuttrumpf M Muttray andH Oumeraci ldquoWave forces on and water-surface fluctuationsaround a vertical cylinder encircled by a perforated squarecaissonrdquo Ocean Engineering vol 27 no 7 pp 775ndash800 2000

[12] K Vijayalakshmi S Neelamani R Sundaravadivelu and KMurali ldquoWave runup on a concentric twin perforated circularcylinderrdquo Ocean Engineering vol 34 no 2 pp 327ndash336 2007

[13] Y Liu Y Li B Teng J Jiang and B Ma ldquoTotal horizontal andvertical forces of irregular waves on partially perforated caissonbreakwatersrdquo Coastal Engineering vol 55 no 6 pp 537ndash5522008

[14] J-I Lee and S Shin ldquoExperimental study on the wave reflectionof partially perforated wall caissons with single and doublechambersrdquo Ocean Engineering vol 91 pp 1ndash10 2014

[15] J-I Lee Y-T Kim and S Shin ldquoExperimental studies onwave interactions of partially perforated wall under obliquelyincident wavesrdquo The Scientific World Journal vol 2014 ArticleID 954174 14 pages 2014

[16] X F Chen Y C Li Y X Wang G H Dong and X BaildquoNumerical simulation of wave interaction with perforatedcaisson breakwatersrdquoChina Ocean Engineering vol 17 no 1 pp33ndash43 2003

[17] J Liu G Lin and J Li ldquoShort-crested waves interaction with aconcentric cylindrical structure with double-layered perforatedwallsrdquo Ocean Engineering vol 40 pp 76ndash90 2012

14 Mathematical Problems in Engineering

[18] J Liu and G Lin ldquoNumerical modelling of wave interactionwith a concentric cylindrical system with an arc-shaped porousouter cylinderrdquo European Journal of MechanicsmdashBFluids vol37 pp 59ndash71 2013

[19] F Aristodemo D D Meringolo P Groenenboom A Lo Schi-avo P Veltri and M Veltri ldquoAssessment of dynamic pressuresat vertical and perforated breakwaters through diffusive SPHschemesrdquo Mathematical Problems in Engineering vol 2015Article ID 305028 10 pages 2015

[20] J Bai and J Hu ldquoResearch on and design of new structure ofelliptical caissoned pierrdquo Port and Waterway Engineering vol11 pp 25ndash30 2006 (Chinese)

[21] Y Wang X Ren and G Wang ldquoNumerical simulation ofnonlinear wave force on a quasi-ellipse caissonrdquo Journal ofMarine Science and Application vol 10 no 3 pp 265ndash271 2011

[22] C W Hirt and B D Nichols ldquoVolume of fluid (VOF) methodfor the dynamics of free boundariesrdquo Journal of ComputationalPhysics vol 39 no 1 pp 201ndash225 1981

[23] Y Wang and T Su ldquoNumerical simulation of liquid sloshing incylindrical containersrdquo Acta Aerodynamica Sinica vol 9 no 1pp 112ndash119 1991 (Chinese)

[24] Y Wang J Zang and D Qiu ldquoNumerical model of cnoidalwave flumerdquo China Ocean Engineering vol 13 no 4 pp 391ndash398 1999

[25] S Elgar and R T Guza ldquoObservations of bispectral of shoalingsurface gravity wavesrdquo Journal of Fluid Mechanics vol 161 pp425ndash448 1985

[26] G Dong Y Ma M Perlin X Ma B Yu and J Xu ldquoExperi-mental study of wavemdashwave nonlinear interactions using thewavelet-based bicoherencerdquo Coastal Engineering vol 55 no 9pp 741ndash752 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

study 120572 = 05 is used For the viscosity term VISX thesecond-order central difference scheme is applied

VISX = []119896(1205972119906

1205971199092 +1205972119906

1205971199102 +1205972119906

1205971199112)]

119894+1119895119896

=]119896

Δ119909119894+12Δ119909119894+1Δ119909119894

[Δ119909119894(119906119899

119894+1119895119896 minus119906119899

119894119895119896)

minusΔ119909119894+1 (119906119899

119894119895119896minus119906119899

119894minus1119895119896)]

+]119896

(Δ119910119895+12 + Δ119910

119895minus12) Δ119910119895+12Δ119910119895minus12[Δ119910119895minus12 (119906

119899

119894119895+1119896

minus119906119899

119894119895119896) minusΔ119910

119895+12 (119906119899

119894119895119896minus119906119899

119894119895minus1119896)]

+]119896

(Δ119911119896+12 + Δ119911

119896minus12) Δ119911119896+12Δ119911119896minus12[Δ119911119896minus12 (119906

119899

119894119895119896+1

minus119906119899

119894119895119896) minusΔ119911

119896+12 (119906119899

119894119895119896minus119906119899

119894119895119896minus1)]

(7)

Additionally the convection flux and viscosity terms inthe other two directions can be obtained in the same way

23 Velocity-Pressure Coupling and Free Surface CapturingAt each time step the velocity field and pressure in eachcontrol volume must be solved For each cell the velocitiesat the time step 119905

119899+1are approximately obtained from the

momentum equations using the values at 119905119899 Then the

pressure and velocities must be iterated until the continuityequation is satisfied to the required accuracy In this studythe iteration accuracy is set to 120576 lt 0005

For the cells which are fully occupied by fluid thevelocity-pressure coupling is completed in the followingprocess Let 119878

119899+1 represent the left term of the continuousequation at the time step 119905

119899+1 The continuity equation is sat-

isfied when 119878119899+1 approaches zero This process is completed

by changing the pressure and can be defined as

120575119901 = minus119878119898

119899+1120597119878119898119899+1120597119901

(8)

where the subscript 119898 represents the 119898th iteration and theterm 120597119878

119898

119899+1120597119901 can be defined as

120597119878

120597119901=

Δ119905

119881119862119894119895119896

1

Δ119909119894

(1

Δ119909119894+12

119860119877119894119895119896

+1

Δ119909119894minus12

119860119877119894minus1119895119896)+

1Δ119910119895

(1

Δ119910119895+12

119860119865119894119895119896

+1

Δ119910119895minus12

119860119865119894119895minus1119896)+

1Δ119911119896

(1

Δ119911119896+12

119860119879119894119895119896

+1

Δ119911119896minus12

119860119879119894119895119896minus1)

(9)

For the cells containing a free surface the free surfaceboundary condition is satisfied by setting the surface cell

pressure 119901119899119894119895119896

equal to the value obtained by a linear interpo-lation between the surface pressure 119901

119904and the pressure of the

corresponding cell fully occupied by fluid below the surfacecell 119901

119899

120575119901 = (1minus119889119888

119889119891

)119901119899+

119889119888

119889119891

119901119904minus119901119899

119894119895119896 (10)

where 119889119888is the distance between the two interpolation cells

and 119889119891is the distance from the below cell to the free surface

After the velocity field and pressure have been obtainedby the above iteration process in each computational celloccupied by fluid the volume of fluid function 119865 is thencomputed for the new time level to give the new fluid con-figuration The fact that 119865 is a step function requires specialcare in computing the fluxes to preserve the sharp definitionof a free surface here a donor-acceptor flux approximation ofthe SOLA-VOF method [22 23] is used to determine the 119865

function at each interface

24 Boundary Conditions As shown in Figure 1 the wavemaker is set on the left side of the numerical tank (119878

1) To

generate nonlinear shallow waves and mitigate re-reflectedwaves at the wave maker the numerical absorbing cnoidalwave maker proposed byWang et al [24] is adopted Namelythe paddle velocity is defined

119880(1198961015840 119905 + 1) =

1205780 (1198961015840 119905) 119888

119889 + 1205780 (1198961015840 119905)

+[1205780 (119896

1015840 119905) minus 120578 (119896

1015840 119905)] 119888

119889 + 1205780 (1198961015840 119905) minus 120578 (1198961015840 119905)

(11)

where 120578(1198961015840 119905) is the actual wave elevation at the 1198961015840th cell of thewave maker 120578

0(1198961015840 119905) is the theoretical wave elevation and 119888

is the celerity of the cnoidal waveA sponge layer (as shown in Figure 1) is set to absorb

the outgoing waves and the reduction coefficient of particlevelocities is defined as

120583 (119909) =

radic1 minus (119909 minus 11987111198712

) 1198711 le 119909 le 119871

1 119909 lt 1198711

(12)

Besides the sponge layer the Sommerfeld radiationcondition is adopted at the right boundary of the flumeAdditionally sidewall boundaries (119878

5 1198786) and the bottom

boundary (1198783) of the numerical tank are treated as free-

slip rigid walls that is the normal velocities at the free-slip boundaries and the normal derivative of the tangentialvelocities are both set to zero

3 Validation of the Model

Before using the present model to study the interactionof waves and quasi-ellipse caissons its performance forsimulating waves and their interaction with structures shouldbe validated In this section the ability of the numericalmodel is verified using theoretical and existing experimentalresults

Mathematical Problems in Engineering 5

0 120 15030 60 90

minus1

0

1

2

3

x = 100m

120578(m

)

t (s)

(a)

0 120 15030 60 90

minus1

0

1

2

3

120578(m

)

t (s)

x = 300m

(b)

0 120 15030 60 90

x = 500m

minus1

0

1

2

3

120578(m

)

t (s)

(c)

0 120 15030 60 90

x = 700m

minus1

0

1

2

3

120578(m

)

t (s)

(d)

Figure 3 The simulated wave surface elevations at different locations along the tank for a case with 119867 = 3m 119879 = 10 s (the red dash line inthe panel of 119909 = 100m is the corresponding theoretical solution)

31 Wave Generation and Propagation Tests of the wavegeneration of the numerical wave tank are examined firstCnoidal waves with wave height 119867 varying from 10m to40m and period 119879 from 80 s to 130 s are considered Forthese simulations the sizes of the numerical tank are set asfollows 119871 = 800m 119861 = 260m 119889 = 20m 119871

1= 600m

1198712= 200m Δ119909 = 2m Δ119910 = 3m and Δ119911 = 2m Figure 3

shows the simulated wave surface elevations along the wavetank for a cnoidal wave with incident height 119867 = 3m andperiod119879 = 10 s It is found that the simulatedwaves can fit thetargeted wave very well and the surface elevations are quiteuniform and stable Increasing propagation distance due totriad nonlinear interactions the wave crests become moreandmore sharp and the troughs are flattened consisting withthe shape characteristics of shallow water waves [25 26] Thesnapshots of the simulation case are illustrated in Figure 4which shows that the waves are quite uniform along the 119910

direction and the sponge layer works well for the dampingof wave energy The present model also works well for othertests and the results are not shown here for brevity

32 Wave Pressures on a Perforated Rectangular Caisson Tofurther investigate the validation of themodel simulations onthe interactions between waves and a perforated rectangular

caisson are carried out Then the computed results arecompared with an existing experiment

The experiment was conducted in a wave flume locatedin the State Key Laboratory of Coastal and Offshore Engi-neering Dalian University of Technology China The flumeis 56m long 07m wide and 10m deep The detailed layoutof the experiments can be seen in the work of Liu et al [13]Here only schematic diagrams of the perforated rectangularcaisson model and the locations of pressure sensors areillustrated (as shown in Figure 5) The model caisson was03m long 068m wide and 07m height It was made ofPlexiglass plates of 10 cm thickness The water depth 119889 infront of the structure was 04m The caisson was dividedinto four uniform boxes named A B C and D by partitionwalls For each box the front plate was perforated by fourrectangular holes from a distance 20 cm above the bottomplate and the back wall was solid The length and width ofthe holes were 011m and 0074m respectively As marked inFigure 5 three pressure sensors (1 2 and 3) were placedon the front walls of the B box and another three sensors (45 and 6) were fixed on the back wall of the C

In the experiment cnoidal waves with incident waveheight 6 cm and wave period 12 s were generated by the wavepaddle For the simulation the mesh sizes of the numerical

6 Mathematical Problems in Engineering

260195

13065

0 0200

400600

800

Y (m) X (m)

120578(m

)

0

25

(a) 119905 = 400 s

260195

13065

0 0200

400600

800

Y (m) X (m)

120578(m

)

0

25

195400

600

(b) 119905 = 1200 s

Figure 4 Snapshots of the numerical tank for the simulation of waves with119867 = 3m 119879 = 10 s

AB

CD

3

2

1

170

170

170

170

680300

700

200400

SWL

(a)

A

SWL

B C D

400

200

680

700

6

5

4

(b)

Figure 5 Sketch of the perforated caisson model (unit mm) and the placement of pressure sensors (a) oblique front view (b) back view

tank are set as follows Δ119909 = 3 cm Δ119910 = 405 cm andΔ119911 = 425 cm The comparisons between the simulated andexperimental results are shown in Figure 6 It is found that thecomputed pressure matches well with experimental resultsunder the water surface (sensors 1 2 4 and 5) For thelocations close to the water surface (sensors 3 and 6) thecomputation slightly overestimates the results at the troughsof the pressure time history The discrepancy may be mainlydue to the complex flow characteristics near the water surfacecaused by strong interactions between the waves and thestructure

The above discussion indicates that the present three-dimensional numerical model can be used to investigate theinteractions betweenwaves and coastal structures In the nextsection the model will be used to study the interactionsbetween waves and quasi-ellipse caissons

4 Waves Interaction with PerforatedQuasi-Ellipse Caissons

41 Description of Quasi-Ellipse Caissons A quasi-ellipsecaisson is made up by two hollow semicylinders which areconnected by two rectangular plates to form a chamber [20]Figure 7 shows a cross section of a quasi-ellipse 119863 denotes

the diameter the semicylinders and119861 is the length of the rect-angular plates In this study the values of 119863 and 119861 are fixedfor all simulations 119863 = 20m and 119861 = 12m For multiplecaissons119863

119888represents the distance between caissons In this

study the caissons are perforated by rectangular holes with3m in height and 4m in length and the positions of the holesare marked in Figure 8

As mentioned in the previous section the computationdomain of the model is divided by regular cuboid cellsHowever for quasi-ellipse caissons some cells at the bound-aries cannot be fully occupied by the solid boundaries (seeFigure 9) To solve this problem the partial cell methodwhich is also used to descript the free surface is adopted Asshown in Figure 9 solid straight lines are used to representthe intersection lines between the caisson boundary and thefluid for simplicity

42 Numerical Results For the simulations of wave interac-tion with quasi-ellipse caissons the sizes of the numericaltank are set as follows 119871 = 800m 119861 = 260m 119889 = 20m1198711= 600m 119871

2= 200mΔ119909 = 2mΔ119910 = 3m andΔ119911 = 2m

Figure 10 shows the distribution of surface elevations aroundthe multiple quasi-ellipse caissons in a wave period and theadjacent distance between the caissons 119863

119888is set equal to 2119863

Mathematical Problems in Engineering 7

0 1 2 3 4 5minus04

minus02

00

02

04

066

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

065

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

064

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

063

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

062

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

06

t (s)

p(k

Pa)

1

Figure 6 Comparisons of the time series of wave pressures on the perforated rectangular caisson (open circles) experimental results (solidlines) numerical results

D

Fminus F+D

2

X

Z

B

(a)

X

Dc FminusXM D

F+XM

(b)

Figure 7 Sketch of a quasi-ellipse caisson (a) and multiple quasi-ellipse caissons (b)

(40m) For this simulation the height of the incident cnoidalwave is 3m with a period of 10 s As wave crest approachesthe front faces of the caissons the runup of waves is veryhigh and partially caused by significant reflection of thecaissons The difference between the front and rear faces inthis situation is pronounced suggesting that the wave forceson the caissons are very large It is noticed that the surface

elevations between the caissons are also remarkable Theresults for the perforated caissons are shown in Figure 11Thesurface elevations close to the caissons are reduced comparedto that without perforation indicating that the perforationcan reduce the wave runuprundown on the caissons Thesurface elevations between the caissons are also significantlyweakened

8 Mathematical Problems in Engineering

30

m

2m

4m

3m

SWL4m 4m

2m 1

m SWL

Front view Side view

1m

Figure 8 Sketch of a perforated quasi-ellipse caisson

(a) Solid quasi-ellipse caisson (b) Perforated quasi-ellipse caisson

Figure 9 Sketch of the mesh division of a cross section for the numerical model

The perforation on caissons can enhance the waterexchange wave energy is partially dissipated by the perfo-rated wall and some wave energy is transferred to the fluidinside the chamber of the caissons (see Figure 12) and hencethe wave reflection and runup are reduced As the wavetroughs approach the middle of the caissons (Figure 12(a))the difference of water levels between inside and outside ofthe chambers is pronounced which causes the water flowto the outside Therefore the wave height in the middle ofthe caisson is reduced For the time when the wave crestsare close to this region the wave flow direction is reversed(Figure 12(c)) and the wave height outside the chambersis decreased The intensive water exchange between insideand outside of the chambers suggests that wave energy isdissipated significantly

The previous discussion suggests the perforation cansignificantly reduce the wave force of the caissons In thissection the wave forces and the influence factor will beinvestigated for multiple caissons To achieve these targetscnoidal waves with different incident height and periodare generated and the interactions between the waves anda caisson without peroration are simulated first and thenthe multiple placed caissons Following the definition ofwave force used for an isolated square structure [21] thenondimensional wave force on the single placed caisson canbe written as

119865119909119878

=119865119909

120588119892 (119861 + 119863) 1198892 [tanh (119896119889) 119896119889] (13)

where 119896 is the wave number of the incident wave and 119865119909is the

maximum wave force in the 119909 direction and can be obtainedby integrating the surface pressure on the structure

The wave forces on the single placed caisson are shownin Figure 13 As expected higher incident waves (119867119889 =

015) can produce larger wave forces With decreasing ofwave period (ie 119896 is increased) wave force on the caissonis increased

For convenient illustration and comparison the waveforces on a single placed caisson without perforation areused as benchmarkThen a nondimensional force which canreflect the influence of multiple placement is defined as

119877119909=

10038161003816100381610038161198651199091198721003816100381610038161003816

10038161003816100381610038161198651199091198781003816100381610038161003816

(14)

where 119865119909119872

is maximum force in 119909 direction for the centralone of the caisson group (as illustrated in Figure 7)

The nondimensional forces 119877119909on the middle caisson

without perforation are shown in Figure 14 It is found thatthe values of 119877

119909are larger than 1 for all the simulated cases

indicating that the presence of adjacent caissons can increasethe wave force With decreasing incident wave period (ieincreasing 119896119863) 119877

119909decreases Additionally 119877

119909for the cases

in the condition with 119863119888119863 = 2 is significantly larger than

those cases with larger transverse distance 119863119888 But the 119877

119909

differences between the cases in the conditions 119863119888119863 = 3

and 119863119888119863 = 4 are much smaller especially for the cases

with 119896119863 gt 10 By increasing incident wave nonlinearity

Mathematical Problems in Engineering 9

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(a) 119905 = 45119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(c) 119905 = (45 + 05)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(d) 119905 = (45 + 075)119879

Figure 10 Contours of water surface elevations around multiple impermeable quasi-ellipse caissons in a wave period (119879 = 10 s119867 = 30m)unit (m)

119877119909can also be enhanced but the increment is limited For the

tests with perforation the influence of incident wave periodand adjacent distance on the forces has a similar trend (seeFigure 15)

To directly reveal the influence of the perforation the119877119909difference between the impermeable caisson and the

perforated one is defined as

119876119909=10038161003816100381610038161003816119877119909119904

minus119877119909119901

10038161003816100381610038161003816 (15)

where 119877119909119904

is the relative wave for the caisson withoutperforation and 119877

119909119901is for the perforated one Figure 16

demonstrates the variations of 119876119909for all the simulated cases

For tests with the smaller incident wave height (119867119889 = 005)the reduction of wave forces by the perforation increaseswith the decrease of wave period The distance between thecaissons 119863

119888also has influence on the reduction wave forces

But if the adjacent distance is larger enough the influenceis limited for smaller incident waves For incident waveswith higher nonlinearity (ie 119867119889 = 015) the effect ofperforation appears smaller than the cases with the smallerincident wavesThewave period effect for the conditionswith119863119888119863 = 30 and 40 follows the trend of the cases with

the smaller incident height However the tendency for thecondition 119863

119888119863 = 20 is much complex When 119896119863 lt 10

the perforationworks better than those conditionswith largertransverse distances

5 Conclusions

A three-dimensional numerical wave flume was establishedusing the continuity equation and the Navier-Stokes equa-tions The VOF method was adopted to capture the freesurfaces The ability of generation of cnoidal waves and its

10 Mathematical Problems in Engineering

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(a) 119905 = 45119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(c) 119905 = (45 + 05)119879

140 175 210 245 280

200

180

160

140

120

100

80

60

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(d) 119905 = (45 + 075)119879

Figure 11 Contours of water surface elevations around multiple perforated quasi-ellipse caissons in a wave period (119879 = 10 s 119867 = 30m)unit (m)

interaction with a perforated rectangle caisson were verifiedusing the existing experimental data Then the presentmodel was used to investigate interactions between cnoidalwaves with varying incident height and period and multipleperforated quasi-ellipse caissons The influence of the per-foration on the wave field and the velocity flow around thecaissons was examined The numerical results demonstratethat the wave runuprundown on the caissons was reducedsignificantly by perforation and hence the wave forces arealso decreased The reduction of wave runuprundown ismainly due to the extensivewater exchange inside and outsidethe caisson chamber It is also found that the effect of thepresence of adjacent caissons on wave forces decreases withdecreasing incident wave period for both multiple placedcaissons with and without perforation The wave forces forcloser transverse adjacent distance are larger than those

conditions with a farther distance but when relative adjacentdistance 119863

119888119863 (119863

119888is the transverse distance and 119863 is the

width of the quasi-ellipse caisson) is larger than 3 the effectof adjacent distance is limited Additionally the effect ofthe perforation increases with the reduction of incidentwave period The performance of the perforation is betterwhen incident wave height is small suggesting that wavenonlinearity also plays an important role in the process ofinteraction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(a) 119905 = 45119879

200

180

160

140

120

100

80

60

Y(m

)

140 175 210 245 280

X (m)

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(c) 119905 = (45 + 05)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(d) 119905 = (45 + 075)119879

Figure 12 The flow vector fields corresponding to the wave surface elevations of Figure 11

014

012

010

008

006

004

00207 08 09 10 11 12 13 14 15

FxS

kD

Hd = 005

Hd = 015

Figure 13 Variations of wave forces on an isolated caisson without perforation

12 Mathematical Problems in Engineering

06 08 10 12 14 16095

100

105

110

115

120

DcD = 2

DcD = 3

DcD = 4

Rx

kD

(a)

06 08 10 12 14 16095

100

105

110

115

120

Rx

kD

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 14 Influence of transverse distance 119871 on the nondimensional wave forces 119877119909of the caisson without perforation (a) cases with119867119889 =

005 (b) cases with119867119889 = 015

06 08 10 12 14 16075

080

085

090

095

100

Rx

kD

DcD = 2

DcD = 3

DcD = 4

(a)

06 08 10 12 14 16

Rx

kD

075

080

085

090

095

100

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 15 Influence of transverse distance 119871 on the nondimensional wave forces 119877119909of the caisson without perforation (a) cases with119867119889 =

005 (b) cases with119867119889 = 015

Mathematical Problems in Engineering 13

06 08 10 12 14 16

025

020

010

015

Qx

kD

DcD = 2

DcD = 3

DcD = 4

(a)Qx

kD

06 08 10 12 14 16

025

020

010

015

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 16 Variations of 119876119909with respect to incident wave period for cases in conditions with different transverse distance 119871 (a) cases with

119867119889 = 005 (b) cases with119867119889 = 015

Acknowledgments

This research is supported financially by the National NaturalScience Foundation (Grant nos 51422901 and 51221961) andA Foundation for the Author of National Excellent DoctoralDissertation of PR China (Grant no 201347)

References

[1] G E Jarlan ldquoA perforated vertical wall breakwaterrdquo The Dockand Harbour Authority vol 41 no 486 pp 394ndash398 1961

[2] Z Huang Y Li and Y Liu ldquoHydraulic performance andwave loadings of perforatedslotted coastal structures a reviewrdquoOcean Engineering vol 38 no 10 pp 1031ndash1053 2011

[3] M K M Darwiche A N Williams and K-H Wang ldquoWaveinteraction with semiporous cylindrical breakwaterrdquo Journal ofWaterway Port Coastal and Ocean Engineering vol 120 no 4pp 382ndash403 1994

[4] T L Yip and A T Chwang ldquoPerforated wall breakwater withinternal horizontal platerdquo Journal of EngineeringMechanics vol126 no 5 pp 533ndash538 2000

[5] K D Suh J C Choi B H Kim W S Park and K S LeeldquoReflection of irregular waves from perforated-wall caissonbreakwatersrdquo Coastal Engineering vol 44 no 2 pp 141ndash1512001

[6] Y C Li H J Liu B Teng and D P Sun ldquoReflection of obliqueincident waves by breakwaters with partially perforated wallrdquoChina Ocean Engineering vol 16 no 3 pp 329ndash342 2002

[7] Y C Li G H Dong H J Liu and D P Sun ldquoThe reflectionof oblique incident waves by breakwaters with double-layeredperforated wallrdquo Coastal Engineering vol 50 no 1-2 pp 47ndash602003

[8] Y Liu Y-C Li and B Teng ldquoWave interactionwith a perforatedwall breakwater with a submerged horizontal porous platerdquoOcean Engineering vol 34 no 17-18 pp 2364ndash2373 2007

[9] Y Liu Y-C Li and B Teng ldquoThe reflection of oblique wavesby an infinite number of partially perforated caissonsrdquo OceanEngineering vol 34 no 14-15 pp 1965ndash1976 2007

[10] K Sankarbabu S A Sannasiraj and V Sundar ldquoHydrodynamicperformance of a dual cylindrical caisson breakwaterrdquo CoastalEngineering vol 55 no 6 pp 431ndash446 2008

[11] S Neelamani G Koether H Schuttrumpf M Muttray andH Oumeraci ldquoWave forces on and water-surface fluctuationsaround a vertical cylinder encircled by a perforated squarecaissonrdquo Ocean Engineering vol 27 no 7 pp 775ndash800 2000

[12] K Vijayalakshmi S Neelamani R Sundaravadivelu and KMurali ldquoWave runup on a concentric twin perforated circularcylinderrdquo Ocean Engineering vol 34 no 2 pp 327ndash336 2007

[13] Y Liu Y Li B Teng J Jiang and B Ma ldquoTotal horizontal andvertical forces of irregular waves on partially perforated caissonbreakwatersrdquo Coastal Engineering vol 55 no 6 pp 537ndash5522008

[14] J-I Lee and S Shin ldquoExperimental study on the wave reflectionof partially perforated wall caissons with single and doublechambersrdquo Ocean Engineering vol 91 pp 1ndash10 2014

[15] J-I Lee Y-T Kim and S Shin ldquoExperimental studies onwave interactions of partially perforated wall under obliquelyincident wavesrdquo The Scientific World Journal vol 2014 ArticleID 954174 14 pages 2014

[16] X F Chen Y C Li Y X Wang G H Dong and X BaildquoNumerical simulation of wave interaction with perforatedcaisson breakwatersrdquoChina Ocean Engineering vol 17 no 1 pp33ndash43 2003

[17] J Liu G Lin and J Li ldquoShort-crested waves interaction with aconcentric cylindrical structure with double-layered perforatedwallsrdquo Ocean Engineering vol 40 pp 76ndash90 2012

14 Mathematical Problems in Engineering

[18] J Liu and G Lin ldquoNumerical modelling of wave interactionwith a concentric cylindrical system with an arc-shaped porousouter cylinderrdquo European Journal of MechanicsmdashBFluids vol37 pp 59ndash71 2013

[19] F Aristodemo D D Meringolo P Groenenboom A Lo Schi-avo P Veltri and M Veltri ldquoAssessment of dynamic pressuresat vertical and perforated breakwaters through diffusive SPHschemesrdquo Mathematical Problems in Engineering vol 2015Article ID 305028 10 pages 2015

[20] J Bai and J Hu ldquoResearch on and design of new structure ofelliptical caissoned pierrdquo Port and Waterway Engineering vol11 pp 25ndash30 2006 (Chinese)

[21] Y Wang X Ren and G Wang ldquoNumerical simulation ofnonlinear wave force on a quasi-ellipse caissonrdquo Journal ofMarine Science and Application vol 10 no 3 pp 265ndash271 2011

[22] C W Hirt and B D Nichols ldquoVolume of fluid (VOF) methodfor the dynamics of free boundariesrdquo Journal of ComputationalPhysics vol 39 no 1 pp 201ndash225 1981

[23] Y Wang and T Su ldquoNumerical simulation of liquid sloshing incylindrical containersrdquo Acta Aerodynamica Sinica vol 9 no 1pp 112ndash119 1991 (Chinese)

[24] Y Wang J Zang and D Qiu ldquoNumerical model of cnoidalwave flumerdquo China Ocean Engineering vol 13 no 4 pp 391ndash398 1999

[25] S Elgar and R T Guza ldquoObservations of bispectral of shoalingsurface gravity wavesrdquo Journal of Fluid Mechanics vol 161 pp425ndash448 1985

[26] G Dong Y Ma M Perlin X Ma B Yu and J Xu ldquoExperi-mental study of wavemdashwave nonlinear interactions using thewavelet-based bicoherencerdquo Coastal Engineering vol 55 no 9pp 741ndash752 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

0 120 15030 60 90

minus1

0

1

2

3

x = 100m

120578(m

)

t (s)

(a)

0 120 15030 60 90

minus1

0

1

2

3

120578(m

)

t (s)

x = 300m

(b)

0 120 15030 60 90

x = 500m

minus1

0

1

2

3

120578(m

)

t (s)

(c)

0 120 15030 60 90

x = 700m

minus1

0

1

2

3

120578(m

)

t (s)

(d)

Figure 3 The simulated wave surface elevations at different locations along the tank for a case with 119867 = 3m 119879 = 10 s (the red dash line inthe panel of 119909 = 100m is the corresponding theoretical solution)

31 Wave Generation and Propagation Tests of the wavegeneration of the numerical wave tank are examined firstCnoidal waves with wave height 119867 varying from 10m to40m and period 119879 from 80 s to 130 s are considered Forthese simulations the sizes of the numerical tank are set asfollows 119871 = 800m 119861 = 260m 119889 = 20m 119871

1= 600m

1198712= 200m Δ119909 = 2m Δ119910 = 3m and Δ119911 = 2m Figure 3

shows the simulated wave surface elevations along the wavetank for a cnoidal wave with incident height 119867 = 3m andperiod119879 = 10 s It is found that the simulatedwaves can fit thetargeted wave very well and the surface elevations are quiteuniform and stable Increasing propagation distance due totriad nonlinear interactions the wave crests become moreandmore sharp and the troughs are flattened consisting withthe shape characteristics of shallow water waves [25 26] Thesnapshots of the simulation case are illustrated in Figure 4which shows that the waves are quite uniform along the 119910

direction and the sponge layer works well for the dampingof wave energy The present model also works well for othertests and the results are not shown here for brevity

32 Wave Pressures on a Perforated Rectangular Caisson Tofurther investigate the validation of themodel simulations onthe interactions between waves and a perforated rectangular

caisson are carried out Then the computed results arecompared with an existing experiment

The experiment was conducted in a wave flume locatedin the State Key Laboratory of Coastal and Offshore Engi-neering Dalian University of Technology China The flumeis 56m long 07m wide and 10m deep The detailed layoutof the experiments can be seen in the work of Liu et al [13]Here only schematic diagrams of the perforated rectangularcaisson model and the locations of pressure sensors areillustrated (as shown in Figure 5) The model caisson was03m long 068m wide and 07m height It was made ofPlexiglass plates of 10 cm thickness The water depth 119889 infront of the structure was 04m The caisson was dividedinto four uniform boxes named A B C and D by partitionwalls For each box the front plate was perforated by fourrectangular holes from a distance 20 cm above the bottomplate and the back wall was solid The length and width ofthe holes were 011m and 0074m respectively As marked inFigure 5 three pressure sensors (1 2 and 3) were placedon the front walls of the B box and another three sensors (45 and 6) were fixed on the back wall of the C

In the experiment cnoidal waves with incident waveheight 6 cm and wave period 12 s were generated by the wavepaddle For the simulation the mesh sizes of the numerical

6 Mathematical Problems in Engineering

260195

13065

0 0200

400600

800

Y (m) X (m)

120578(m

)

0

25

(a) 119905 = 400 s

260195

13065

0 0200

400600

800

Y (m) X (m)

120578(m

)

0

25

195400

600

(b) 119905 = 1200 s

Figure 4 Snapshots of the numerical tank for the simulation of waves with119867 = 3m 119879 = 10 s

AB

CD

3

2

1

170

170

170

170

680300

700

200400

SWL

(a)

A

SWL

B C D

400

200

680

700

6

5

4

(b)

Figure 5 Sketch of the perforated caisson model (unit mm) and the placement of pressure sensors (a) oblique front view (b) back view

tank are set as follows Δ119909 = 3 cm Δ119910 = 405 cm andΔ119911 = 425 cm The comparisons between the simulated andexperimental results are shown in Figure 6 It is found that thecomputed pressure matches well with experimental resultsunder the water surface (sensors 1 2 4 and 5) For thelocations close to the water surface (sensors 3 and 6) thecomputation slightly overestimates the results at the troughsof the pressure time history The discrepancy may be mainlydue to the complex flow characteristics near the water surfacecaused by strong interactions between the waves and thestructure

The above discussion indicates that the present three-dimensional numerical model can be used to investigate theinteractions betweenwaves and coastal structures In the nextsection the model will be used to study the interactionsbetween waves and quasi-ellipse caissons

4 Waves Interaction with PerforatedQuasi-Ellipse Caissons

41 Description of Quasi-Ellipse Caissons A quasi-ellipsecaisson is made up by two hollow semicylinders which areconnected by two rectangular plates to form a chamber [20]Figure 7 shows a cross section of a quasi-ellipse 119863 denotes

the diameter the semicylinders and119861 is the length of the rect-angular plates In this study the values of 119863 and 119861 are fixedfor all simulations 119863 = 20m and 119861 = 12m For multiplecaissons119863

119888represents the distance between caissons In this

study the caissons are perforated by rectangular holes with3m in height and 4m in length and the positions of the holesare marked in Figure 8

As mentioned in the previous section the computationdomain of the model is divided by regular cuboid cellsHowever for quasi-ellipse caissons some cells at the bound-aries cannot be fully occupied by the solid boundaries (seeFigure 9) To solve this problem the partial cell methodwhich is also used to descript the free surface is adopted Asshown in Figure 9 solid straight lines are used to representthe intersection lines between the caisson boundary and thefluid for simplicity

42 Numerical Results For the simulations of wave interac-tion with quasi-ellipse caissons the sizes of the numericaltank are set as follows 119871 = 800m 119861 = 260m 119889 = 20m1198711= 600m 119871

2= 200mΔ119909 = 2mΔ119910 = 3m andΔ119911 = 2m

Figure 10 shows the distribution of surface elevations aroundthe multiple quasi-ellipse caissons in a wave period and theadjacent distance between the caissons 119863

119888is set equal to 2119863

Mathematical Problems in Engineering 7

0 1 2 3 4 5minus04

minus02

00

02

04

066

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

065

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

064

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

063

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

062

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

06

t (s)

p(k

Pa)

1

Figure 6 Comparisons of the time series of wave pressures on the perforated rectangular caisson (open circles) experimental results (solidlines) numerical results

D

Fminus F+D

2

X

Z

B

(a)

X

Dc FminusXM D

F+XM

(b)

Figure 7 Sketch of a quasi-ellipse caisson (a) and multiple quasi-ellipse caissons (b)

(40m) For this simulation the height of the incident cnoidalwave is 3m with a period of 10 s As wave crest approachesthe front faces of the caissons the runup of waves is veryhigh and partially caused by significant reflection of thecaissons The difference between the front and rear faces inthis situation is pronounced suggesting that the wave forceson the caissons are very large It is noticed that the surface

elevations between the caissons are also remarkable Theresults for the perforated caissons are shown in Figure 11Thesurface elevations close to the caissons are reduced comparedto that without perforation indicating that the perforationcan reduce the wave runuprundown on the caissons Thesurface elevations between the caissons are also significantlyweakened

8 Mathematical Problems in Engineering

30

m

2m

4m

3m

SWL4m 4m

2m 1

m SWL

Front view Side view

1m

Figure 8 Sketch of a perforated quasi-ellipse caisson

(a) Solid quasi-ellipse caisson (b) Perforated quasi-ellipse caisson

Figure 9 Sketch of the mesh division of a cross section for the numerical model

The perforation on caissons can enhance the waterexchange wave energy is partially dissipated by the perfo-rated wall and some wave energy is transferred to the fluidinside the chamber of the caissons (see Figure 12) and hencethe wave reflection and runup are reduced As the wavetroughs approach the middle of the caissons (Figure 12(a))the difference of water levels between inside and outside ofthe chambers is pronounced which causes the water flowto the outside Therefore the wave height in the middle ofthe caisson is reduced For the time when the wave crestsare close to this region the wave flow direction is reversed(Figure 12(c)) and the wave height outside the chambersis decreased The intensive water exchange between insideand outside of the chambers suggests that wave energy isdissipated significantly

The previous discussion suggests the perforation cansignificantly reduce the wave force of the caissons In thissection the wave forces and the influence factor will beinvestigated for multiple caissons To achieve these targetscnoidal waves with different incident height and periodare generated and the interactions between the waves anda caisson without peroration are simulated first and thenthe multiple placed caissons Following the definition ofwave force used for an isolated square structure [21] thenondimensional wave force on the single placed caisson canbe written as

119865119909119878

=119865119909

120588119892 (119861 + 119863) 1198892 [tanh (119896119889) 119896119889] (13)

where 119896 is the wave number of the incident wave and 119865119909is the

maximum wave force in the 119909 direction and can be obtainedby integrating the surface pressure on the structure

The wave forces on the single placed caisson are shownin Figure 13 As expected higher incident waves (119867119889 =

015) can produce larger wave forces With decreasing ofwave period (ie 119896 is increased) wave force on the caissonis increased

For convenient illustration and comparison the waveforces on a single placed caisson without perforation areused as benchmarkThen a nondimensional force which canreflect the influence of multiple placement is defined as

119877119909=

10038161003816100381610038161198651199091198721003816100381610038161003816

10038161003816100381610038161198651199091198781003816100381610038161003816

(14)

where 119865119909119872

is maximum force in 119909 direction for the centralone of the caisson group (as illustrated in Figure 7)

The nondimensional forces 119877119909on the middle caisson

without perforation are shown in Figure 14 It is found thatthe values of 119877

119909are larger than 1 for all the simulated cases

indicating that the presence of adjacent caissons can increasethe wave force With decreasing incident wave period (ieincreasing 119896119863) 119877

119909decreases Additionally 119877

119909for the cases

in the condition with 119863119888119863 = 2 is significantly larger than

those cases with larger transverse distance 119863119888 But the 119877

119909

differences between the cases in the conditions 119863119888119863 = 3

and 119863119888119863 = 4 are much smaller especially for the cases

with 119896119863 gt 10 By increasing incident wave nonlinearity

Mathematical Problems in Engineering 9

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(a) 119905 = 45119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(c) 119905 = (45 + 05)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(d) 119905 = (45 + 075)119879

Figure 10 Contours of water surface elevations around multiple impermeable quasi-ellipse caissons in a wave period (119879 = 10 s119867 = 30m)unit (m)

119877119909can also be enhanced but the increment is limited For the

tests with perforation the influence of incident wave periodand adjacent distance on the forces has a similar trend (seeFigure 15)

To directly reveal the influence of the perforation the119877119909difference between the impermeable caisson and the

perforated one is defined as

119876119909=10038161003816100381610038161003816119877119909119904

minus119877119909119901

10038161003816100381610038161003816 (15)

where 119877119909119904

is the relative wave for the caisson withoutperforation and 119877

119909119901is for the perforated one Figure 16

demonstrates the variations of 119876119909for all the simulated cases

For tests with the smaller incident wave height (119867119889 = 005)the reduction of wave forces by the perforation increaseswith the decrease of wave period The distance between thecaissons 119863

119888also has influence on the reduction wave forces

But if the adjacent distance is larger enough the influenceis limited for smaller incident waves For incident waveswith higher nonlinearity (ie 119867119889 = 015) the effect ofperforation appears smaller than the cases with the smallerincident wavesThewave period effect for the conditionswith119863119888119863 = 30 and 40 follows the trend of the cases with

the smaller incident height However the tendency for thecondition 119863

119888119863 = 20 is much complex When 119896119863 lt 10

the perforationworks better than those conditionswith largertransverse distances

5 Conclusions

A three-dimensional numerical wave flume was establishedusing the continuity equation and the Navier-Stokes equa-tions The VOF method was adopted to capture the freesurfaces The ability of generation of cnoidal waves and its

10 Mathematical Problems in Engineering

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(a) 119905 = 45119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(c) 119905 = (45 + 05)119879

140 175 210 245 280

200

180

160

140

120

100

80

60

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(d) 119905 = (45 + 075)119879

Figure 11 Contours of water surface elevations around multiple perforated quasi-ellipse caissons in a wave period (119879 = 10 s 119867 = 30m)unit (m)

interaction with a perforated rectangle caisson were verifiedusing the existing experimental data Then the presentmodel was used to investigate interactions between cnoidalwaves with varying incident height and period and multipleperforated quasi-ellipse caissons The influence of the per-foration on the wave field and the velocity flow around thecaissons was examined The numerical results demonstratethat the wave runuprundown on the caissons was reducedsignificantly by perforation and hence the wave forces arealso decreased The reduction of wave runuprundown ismainly due to the extensivewater exchange inside and outsidethe caisson chamber It is also found that the effect of thepresence of adjacent caissons on wave forces decreases withdecreasing incident wave period for both multiple placedcaissons with and without perforation The wave forces forcloser transverse adjacent distance are larger than those

conditions with a farther distance but when relative adjacentdistance 119863

119888119863 (119863

119888is the transverse distance and 119863 is the

width of the quasi-ellipse caisson) is larger than 3 the effectof adjacent distance is limited Additionally the effect ofthe perforation increases with the reduction of incidentwave period The performance of the perforation is betterwhen incident wave height is small suggesting that wavenonlinearity also plays an important role in the process ofinteraction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(a) 119905 = 45119879

200

180

160

140

120

100

80

60

Y(m

)

140 175 210 245 280

X (m)

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(c) 119905 = (45 + 05)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(d) 119905 = (45 + 075)119879

Figure 12 The flow vector fields corresponding to the wave surface elevations of Figure 11

014

012

010

008

006

004

00207 08 09 10 11 12 13 14 15

FxS

kD

Hd = 005

Hd = 015

Figure 13 Variations of wave forces on an isolated caisson without perforation

12 Mathematical Problems in Engineering

06 08 10 12 14 16095

100

105

110

115

120

DcD = 2

DcD = 3

DcD = 4

Rx

kD

(a)

06 08 10 12 14 16095

100

105

110

115

120

Rx

kD

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 14 Influence of transverse distance 119871 on the nondimensional wave forces 119877119909of the caisson without perforation (a) cases with119867119889 =

005 (b) cases with119867119889 = 015

06 08 10 12 14 16075

080

085

090

095

100

Rx

kD

DcD = 2

DcD = 3

DcD = 4

(a)

06 08 10 12 14 16

Rx

kD

075

080

085

090

095

100

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 15 Influence of transverse distance 119871 on the nondimensional wave forces 119877119909of the caisson without perforation (a) cases with119867119889 =

005 (b) cases with119867119889 = 015

Mathematical Problems in Engineering 13

06 08 10 12 14 16

025

020

010

015

Qx

kD

DcD = 2

DcD = 3

DcD = 4

(a)Qx

kD

06 08 10 12 14 16

025

020

010

015

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 16 Variations of 119876119909with respect to incident wave period for cases in conditions with different transverse distance 119871 (a) cases with

119867119889 = 005 (b) cases with119867119889 = 015

Acknowledgments

This research is supported financially by the National NaturalScience Foundation (Grant nos 51422901 and 51221961) andA Foundation for the Author of National Excellent DoctoralDissertation of PR China (Grant no 201347)

References

[1] G E Jarlan ldquoA perforated vertical wall breakwaterrdquo The Dockand Harbour Authority vol 41 no 486 pp 394ndash398 1961

[2] Z Huang Y Li and Y Liu ldquoHydraulic performance andwave loadings of perforatedslotted coastal structures a reviewrdquoOcean Engineering vol 38 no 10 pp 1031ndash1053 2011

[3] M K M Darwiche A N Williams and K-H Wang ldquoWaveinteraction with semiporous cylindrical breakwaterrdquo Journal ofWaterway Port Coastal and Ocean Engineering vol 120 no 4pp 382ndash403 1994

[4] T L Yip and A T Chwang ldquoPerforated wall breakwater withinternal horizontal platerdquo Journal of EngineeringMechanics vol126 no 5 pp 533ndash538 2000

[5] K D Suh J C Choi B H Kim W S Park and K S LeeldquoReflection of irregular waves from perforated-wall caissonbreakwatersrdquo Coastal Engineering vol 44 no 2 pp 141ndash1512001

[6] Y C Li H J Liu B Teng and D P Sun ldquoReflection of obliqueincident waves by breakwaters with partially perforated wallrdquoChina Ocean Engineering vol 16 no 3 pp 329ndash342 2002

[7] Y C Li G H Dong H J Liu and D P Sun ldquoThe reflectionof oblique incident waves by breakwaters with double-layeredperforated wallrdquo Coastal Engineering vol 50 no 1-2 pp 47ndash602003

[8] Y Liu Y-C Li and B Teng ldquoWave interactionwith a perforatedwall breakwater with a submerged horizontal porous platerdquoOcean Engineering vol 34 no 17-18 pp 2364ndash2373 2007

[9] Y Liu Y-C Li and B Teng ldquoThe reflection of oblique wavesby an infinite number of partially perforated caissonsrdquo OceanEngineering vol 34 no 14-15 pp 1965ndash1976 2007

[10] K Sankarbabu S A Sannasiraj and V Sundar ldquoHydrodynamicperformance of a dual cylindrical caisson breakwaterrdquo CoastalEngineering vol 55 no 6 pp 431ndash446 2008

[11] S Neelamani G Koether H Schuttrumpf M Muttray andH Oumeraci ldquoWave forces on and water-surface fluctuationsaround a vertical cylinder encircled by a perforated squarecaissonrdquo Ocean Engineering vol 27 no 7 pp 775ndash800 2000

[12] K Vijayalakshmi S Neelamani R Sundaravadivelu and KMurali ldquoWave runup on a concentric twin perforated circularcylinderrdquo Ocean Engineering vol 34 no 2 pp 327ndash336 2007

[13] Y Liu Y Li B Teng J Jiang and B Ma ldquoTotal horizontal andvertical forces of irregular waves on partially perforated caissonbreakwatersrdquo Coastal Engineering vol 55 no 6 pp 537ndash5522008

[14] J-I Lee and S Shin ldquoExperimental study on the wave reflectionof partially perforated wall caissons with single and doublechambersrdquo Ocean Engineering vol 91 pp 1ndash10 2014

[15] J-I Lee Y-T Kim and S Shin ldquoExperimental studies onwave interactions of partially perforated wall under obliquelyincident wavesrdquo The Scientific World Journal vol 2014 ArticleID 954174 14 pages 2014

[16] X F Chen Y C Li Y X Wang G H Dong and X BaildquoNumerical simulation of wave interaction with perforatedcaisson breakwatersrdquoChina Ocean Engineering vol 17 no 1 pp33ndash43 2003

[17] J Liu G Lin and J Li ldquoShort-crested waves interaction with aconcentric cylindrical structure with double-layered perforatedwallsrdquo Ocean Engineering vol 40 pp 76ndash90 2012

14 Mathematical Problems in Engineering

[18] J Liu and G Lin ldquoNumerical modelling of wave interactionwith a concentric cylindrical system with an arc-shaped porousouter cylinderrdquo European Journal of MechanicsmdashBFluids vol37 pp 59ndash71 2013

[19] F Aristodemo D D Meringolo P Groenenboom A Lo Schi-avo P Veltri and M Veltri ldquoAssessment of dynamic pressuresat vertical and perforated breakwaters through diffusive SPHschemesrdquo Mathematical Problems in Engineering vol 2015Article ID 305028 10 pages 2015

[20] J Bai and J Hu ldquoResearch on and design of new structure ofelliptical caissoned pierrdquo Port and Waterway Engineering vol11 pp 25ndash30 2006 (Chinese)

[21] Y Wang X Ren and G Wang ldquoNumerical simulation ofnonlinear wave force on a quasi-ellipse caissonrdquo Journal ofMarine Science and Application vol 10 no 3 pp 265ndash271 2011

[22] C W Hirt and B D Nichols ldquoVolume of fluid (VOF) methodfor the dynamics of free boundariesrdquo Journal of ComputationalPhysics vol 39 no 1 pp 201ndash225 1981

[23] Y Wang and T Su ldquoNumerical simulation of liquid sloshing incylindrical containersrdquo Acta Aerodynamica Sinica vol 9 no 1pp 112ndash119 1991 (Chinese)

[24] Y Wang J Zang and D Qiu ldquoNumerical model of cnoidalwave flumerdquo China Ocean Engineering vol 13 no 4 pp 391ndash398 1999

[25] S Elgar and R T Guza ldquoObservations of bispectral of shoalingsurface gravity wavesrdquo Journal of Fluid Mechanics vol 161 pp425ndash448 1985

[26] G Dong Y Ma M Perlin X Ma B Yu and J Xu ldquoExperi-mental study of wavemdashwave nonlinear interactions using thewavelet-based bicoherencerdquo Coastal Engineering vol 55 no 9pp 741ndash752 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

260195

13065

0 0200

400600

800

Y (m) X (m)

120578(m

)

0

25

(a) 119905 = 400 s

260195

13065

0 0200

400600

800

Y (m) X (m)

120578(m

)

0

25

195400

600

(b) 119905 = 1200 s

Figure 4 Snapshots of the numerical tank for the simulation of waves with119867 = 3m 119879 = 10 s

AB

CD

3

2

1

170

170

170

170

680300

700

200400

SWL

(a)

A

SWL

B C D

400

200

680

700

6

5

4

(b)

Figure 5 Sketch of the perforated caisson model (unit mm) and the placement of pressure sensors (a) oblique front view (b) back view

tank are set as follows Δ119909 = 3 cm Δ119910 = 405 cm andΔ119911 = 425 cm The comparisons between the simulated andexperimental results are shown in Figure 6 It is found that thecomputed pressure matches well with experimental resultsunder the water surface (sensors 1 2 4 and 5) For thelocations close to the water surface (sensors 3 and 6) thecomputation slightly overestimates the results at the troughsof the pressure time history The discrepancy may be mainlydue to the complex flow characteristics near the water surfacecaused by strong interactions between the waves and thestructure

The above discussion indicates that the present three-dimensional numerical model can be used to investigate theinteractions betweenwaves and coastal structures In the nextsection the model will be used to study the interactionsbetween waves and quasi-ellipse caissons

4 Waves Interaction with PerforatedQuasi-Ellipse Caissons

41 Description of Quasi-Ellipse Caissons A quasi-ellipsecaisson is made up by two hollow semicylinders which areconnected by two rectangular plates to form a chamber [20]Figure 7 shows a cross section of a quasi-ellipse 119863 denotes

the diameter the semicylinders and119861 is the length of the rect-angular plates In this study the values of 119863 and 119861 are fixedfor all simulations 119863 = 20m and 119861 = 12m For multiplecaissons119863

119888represents the distance between caissons In this

study the caissons are perforated by rectangular holes with3m in height and 4m in length and the positions of the holesare marked in Figure 8

As mentioned in the previous section the computationdomain of the model is divided by regular cuboid cellsHowever for quasi-ellipse caissons some cells at the bound-aries cannot be fully occupied by the solid boundaries (seeFigure 9) To solve this problem the partial cell methodwhich is also used to descript the free surface is adopted Asshown in Figure 9 solid straight lines are used to representthe intersection lines between the caisson boundary and thefluid for simplicity

42 Numerical Results For the simulations of wave interac-tion with quasi-ellipse caissons the sizes of the numericaltank are set as follows 119871 = 800m 119861 = 260m 119889 = 20m1198711= 600m 119871

2= 200mΔ119909 = 2mΔ119910 = 3m andΔ119911 = 2m

Figure 10 shows the distribution of surface elevations aroundthe multiple quasi-ellipse caissons in a wave period and theadjacent distance between the caissons 119863

119888is set equal to 2119863

Mathematical Problems in Engineering 7

0 1 2 3 4 5minus04

minus02

00

02

04

066

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

065

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

064

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

063

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

062

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

06

t (s)

p(k

Pa)

1

Figure 6 Comparisons of the time series of wave pressures on the perforated rectangular caisson (open circles) experimental results (solidlines) numerical results

D

Fminus F+D

2

X

Z

B

(a)

X

Dc FminusXM D

F+XM

(b)

Figure 7 Sketch of a quasi-ellipse caisson (a) and multiple quasi-ellipse caissons (b)

(40m) For this simulation the height of the incident cnoidalwave is 3m with a period of 10 s As wave crest approachesthe front faces of the caissons the runup of waves is veryhigh and partially caused by significant reflection of thecaissons The difference between the front and rear faces inthis situation is pronounced suggesting that the wave forceson the caissons are very large It is noticed that the surface

elevations between the caissons are also remarkable Theresults for the perforated caissons are shown in Figure 11Thesurface elevations close to the caissons are reduced comparedto that without perforation indicating that the perforationcan reduce the wave runuprundown on the caissons Thesurface elevations between the caissons are also significantlyweakened

8 Mathematical Problems in Engineering

30

m

2m

4m

3m

SWL4m 4m

2m 1

m SWL

Front view Side view

1m

Figure 8 Sketch of a perforated quasi-ellipse caisson

(a) Solid quasi-ellipse caisson (b) Perforated quasi-ellipse caisson

Figure 9 Sketch of the mesh division of a cross section for the numerical model

The perforation on caissons can enhance the waterexchange wave energy is partially dissipated by the perfo-rated wall and some wave energy is transferred to the fluidinside the chamber of the caissons (see Figure 12) and hencethe wave reflection and runup are reduced As the wavetroughs approach the middle of the caissons (Figure 12(a))the difference of water levels between inside and outside ofthe chambers is pronounced which causes the water flowto the outside Therefore the wave height in the middle ofthe caisson is reduced For the time when the wave crestsare close to this region the wave flow direction is reversed(Figure 12(c)) and the wave height outside the chambersis decreased The intensive water exchange between insideand outside of the chambers suggests that wave energy isdissipated significantly

The previous discussion suggests the perforation cansignificantly reduce the wave force of the caissons In thissection the wave forces and the influence factor will beinvestigated for multiple caissons To achieve these targetscnoidal waves with different incident height and periodare generated and the interactions between the waves anda caisson without peroration are simulated first and thenthe multiple placed caissons Following the definition ofwave force used for an isolated square structure [21] thenondimensional wave force on the single placed caisson canbe written as

119865119909119878

=119865119909

120588119892 (119861 + 119863) 1198892 [tanh (119896119889) 119896119889] (13)

where 119896 is the wave number of the incident wave and 119865119909is the

maximum wave force in the 119909 direction and can be obtainedby integrating the surface pressure on the structure

The wave forces on the single placed caisson are shownin Figure 13 As expected higher incident waves (119867119889 =

015) can produce larger wave forces With decreasing ofwave period (ie 119896 is increased) wave force on the caissonis increased

For convenient illustration and comparison the waveforces on a single placed caisson without perforation areused as benchmarkThen a nondimensional force which canreflect the influence of multiple placement is defined as

119877119909=

10038161003816100381610038161198651199091198721003816100381610038161003816

10038161003816100381610038161198651199091198781003816100381610038161003816

(14)

where 119865119909119872

is maximum force in 119909 direction for the centralone of the caisson group (as illustrated in Figure 7)

The nondimensional forces 119877119909on the middle caisson

without perforation are shown in Figure 14 It is found thatthe values of 119877

119909are larger than 1 for all the simulated cases

indicating that the presence of adjacent caissons can increasethe wave force With decreasing incident wave period (ieincreasing 119896119863) 119877

119909decreases Additionally 119877

119909for the cases

in the condition with 119863119888119863 = 2 is significantly larger than

those cases with larger transverse distance 119863119888 But the 119877

119909

differences between the cases in the conditions 119863119888119863 = 3

and 119863119888119863 = 4 are much smaller especially for the cases

with 119896119863 gt 10 By increasing incident wave nonlinearity

Mathematical Problems in Engineering 9

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(a) 119905 = 45119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(c) 119905 = (45 + 05)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(d) 119905 = (45 + 075)119879

Figure 10 Contours of water surface elevations around multiple impermeable quasi-ellipse caissons in a wave period (119879 = 10 s119867 = 30m)unit (m)

119877119909can also be enhanced but the increment is limited For the

tests with perforation the influence of incident wave periodand adjacent distance on the forces has a similar trend (seeFigure 15)

To directly reveal the influence of the perforation the119877119909difference between the impermeable caisson and the

perforated one is defined as

119876119909=10038161003816100381610038161003816119877119909119904

minus119877119909119901

10038161003816100381610038161003816 (15)

where 119877119909119904

is the relative wave for the caisson withoutperforation and 119877

119909119901is for the perforated one Figure 16

demonstrates the variations of 119876119909for all the simulated cases

For tests with the smaller incident wave height (119867119889 = 005)the reduction of wave forces by the perforation increaseswith the decrease of wave period The distance between thecaissons 119863

119888also has influence on the reduction wave forces

But if the adjacent distance is larger enough the influenceis limited for smaller incident waves For incident waveswith higher nonlinearity (ie 119867119889 = 015) the effect ofperforation appears smaller than the cases with the smallerincident wavesThewave period effect for the conditionswith119863119888119863 = 30 and 40 follows the trend of the cases with

the smaller incident height However the tendency for thecondition 119863

119888119863 = 20 is much complex When 119896119863 lt 10

the perforationworks better than those conditionswith largertransverse distances

5 Conclusions

A three-dimensional numerical wave flume was establishedusing the continuity equation and the Navier-Stokes equa-tions The VOF method was adopted to capture the freesurfaces The ability of generation of cnoidal waves and its

10 Mathematical Problems in Engineering

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(a) 119905 = 45119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(c) 119905 = (45 + 05)119879

140 175 210 245 280

200

180

160

140

120

100

80

60

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(d) 119905 = (45 + 075)119879

Figure 11 Contours of water surface elevations around multiple perforated quasi-ellipse caissons in a wave period (119879 = 10 s 119867 = 30m)unit (m)

interaction with a perforated rectangle caisson were verifiedusing the existing experimental data Then the presentmodel was used to investigate interactions between cnoidalwaves with varying incident height and period and multipleperforated quasi-ellipse caissons The influence of the per-foration on the wave field and the velocity flow around thecaissons was examined The numerical results demonstratethat the wave runuprundown on the caissons was reducedsignificantly by perforation and hence the wave forces arealso decreased The reduction of wave runuprundown ismainly due to the extensivewater exchange inside and outsidethe caisson chamber It is also found that the effect of thepresence of adjacent caissons on wave forces decreases withdecreasing incident wave period for both multiple placedcaissons with and without perforation The wave forces forcloser transverse adjacent distance are larger than those

conditions with a farther distance but when relative adjacentdistance 119863

119888119863 (119863

119888is the transverse distance and 119863 is the

width of the quasi-ellipse caisson) is larger than 3 the effectof adjacent distance is limited Additionally the effect ofthe perforation increases with the reduction of incidentwave period The performance of the perforation is betterwhen incident wave height is small suggesting that wavenonlinearity also plays an important role in the process ofinteraction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(a) 119905 = 45119879

200

180

160

140

120

100

80

60

Y(m

)

140 175 210 245 280

X (m)

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(c) 119905 = (45 + 05)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(d) 119905 = (45 + 075)119879

Figure 12 The flow vector fields corresponding to the wave surface elevations of Figure 11

014

012

010

008

006

004

00207 08 09 10 11 12 13 14 15

FxS

kD

Hd = 005

Hd = 015

Figure 13 Variations of wave forces on an isolated caisson without perforation

12 Mathematical Problems in Engineering

06 08 10 12 14 16095

100

105

110

115

120

DcD = 2

DcD = 3

DcD = 4

Rx

kD

(a)

06 08 10 12 14 16095

100

105

110

115

120

Rx

kD

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 14 Influence of transverse distance 119871 on the nondimensional wave forces 119877119909of the caisson without perforation (a) cases with119867119889 =

005 (b) cases with119867119889 = 015

06 08 10 12 14 16075

080

085

090

095

100

Rx

kD

DcD = 2

DcD = 3

DcD = 4

(a)

06 08 10 12 14 16

Rx

kD

075

080

085

090

095

100

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 15 Influence of transverse distance 119871 on the nondimensional wave forces 119877119909of the caisson without perforation (a) cases with119867119889 =

005 (b) cases with119867119889 = 015

Mathematical Problems in Engineering 13

06 08 10 12 14 16

025

020

010

015

Qx

kD

DcD = 2

DcD = 3

DcD = 4

(a)Qx

kD

06 08 10 12 14 16

025

020

010

015

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 16 Variations of 119876119909with respect to incident wave period for cases in conditions with different transverse distance 119871 (a) cases with

119867119889 = 005 (b) cases with119867119889 = 015

Acknowledgments

This research is supported financially by the National NaturalScience Foundation (Grant nos 51422901 and 51221961) andA Foundation for the Author of National Excellent DoctoralDissertation of PR China (Grant no 201347)

References

[1] G E Jarlan ldquoA perforated vertical wall breakwaterrdquo The Dockand Harbour Authority vol 41 no 486 pp 394ndash398 1961

[2] Z Huang Y Li and Y Liu ldquoHydraulic performance andwave loadings of perforatedslotted coastal structures a reviewrdquoOcean Engineering vol 38 no 10 pp 1031ndash1053 2011

[3] M K M Darwiche A N Williams and K-H Wang ldquoWaveinteraction with semiporous cylindrical breakwaterrdquo Journal ofWaterway Port Coastal and Ocean Engineering vol 120 no 4pp 382ndash403 1994

[4] T L Yip and A T Chwang ldquoPerforated wall breakwater withinternal horizontal platerdquo Journal of EngineeringMechanics vol126 no 5 pp 533ndash538 2000

[5] K D Suh J C Choi B H Kim W S Park and K S LeeldquoReflection of irregular waves from perforated-wall caissonbreakwatersrdquo Coastal Engineering vol 44 no 2 pp 141ndash1512001

[6] Y C Li H J Liu B Teng and D P Sun ldquoReflection of obliqueincident waves by breakwaters with partially perforated wallrdquoChina Ocean Engineering vol 16 no 3 pp 329ndash342 2002

[7] Y C Li G H Dong H J Liu and D P Sun ldquoThe reflectionof oblique incident waves by breakwaters with double-layeredperforated wallrdquo Coastal Engineering vol 50 no 1-2 pp 47ndash602003

[8] Y Liu Y-C Li and B Teng ldquoWave interactionwith a perforatedwall breakwater with a submerged horizontal porous platerdquoOcean Engineering vol 34 no 17-18 pp 2364ndash2373 2007

[9] Y Liu Y-C Li and B Teng ldquoThe reflection of oblique wavesby an infinite number of partially perforated caissonsrdquo OceanEngineering vol 34 no 14-15 pp 1965ndash1976 2007

[10] K Sankarbabu S A Sannasiraj and V Sundar ldquoHydrodynamicperformance of a dual cylindrical caisson breakwaterrdquo CoastalEngineering vol 55 no 6 pp 431ndash446 2008

[11] S Neelamani G Koether H Schuttrumpf M Muttray andH Oumeraci ldquoWave forces on and water-surface fluctuationsaround a vertical cylinder encircled by a perforated squarecaissonrdquo Ocean Engineering vol 27 no 7 pp 775ndash800 2000

[12] K Vijayalakshmi S Neelamani R Sundaravadivelu and KMurali ldquoWave runup on a concentric twin perforated circularcylinderrdquo Ocean Engineering vol 34 no 2 pp 327ndash336 2007

[13] Y Liu Y Li B Teng J Jiang and B Ma ldquoTotal horizontal andvertical forces of irregular waves on partially perforated caissonbreakwatersrdquo Coastal Engineering vol 55 no 6 pp 537ndash5522008

[14] J-I Lee and S Shin ldquoExperimental study on the wave reflectionof partially perforated wall caissons with single and doublechambersrdquo Ocean Engineering vol 91 pp 1ndash10 2014

[15] J-I Lee Y-T Kim and S Shin ldquoExperimental studies onwave interactions of partially perforated wall under obliquelyincident wavesrdquo The Scientific World Journal vol 2014 ArticleID 954174 14 pages 2014

[16] X F Chen Y C Li Y X Wang G H Dong and X BaildquoNumerical simulation of wave interaction with perforatedcaisson breakwatersrdquoChina Ocean Engineering vol 17 no 1 pp33ndash43 2003

[17] J Liu G Lin and J Li ldquoShort-crested waves interaction with aconcentric cylindrical structure with double-layered perforatedwallsrdquo Ocean Engineering vol 40 pp 76ndash90 2012

14 Mathematical Problems in Engineering

[18] J Liu and G Lin ldquoNumerical modelling of wave interactionwith a concentric cylindrical system with an arc-shaped porousouter cylinderrdquo European Journal of MechanicsmdashBFluids vol37 pp 59ndash71 2013

[19] F Aristodemo D D Meringolo P Groenenboom A Lo Schi-avo P Veltri and M Veltri ldquoAssessment of dynamic pressuresat vertical and perforated breakwaters through diffusive SPHschemesrdquo Mathematical Problems in Engineering vol 2015Article ID 305028 10 pages 2015

[20] J Bai and J Hu ldquoResearch on and design of new structure ofelliptical caissoned pierrdquo Port and Waterway Engineering vol11 pp 25ndash30 2006 (Chinese)

[21] Y Wang X Ren and G Wang ldquoNumerical simulation ofnonlinear wave force on a quasi-ellipse caissonrdquo Journal ofMarine Science and Application vol 10 no 3 pp 265ndash271 2011

[22] C W Hirt and B D Nichols ldquoVolume of fluid (VOF) methodfor the dynamics of free boundariesrdquo Journal of ComputationalPhysics vol 39 no 1 pp 201ndash225 1981

[23] Y Wang and T Su ldquoNumerical simulation of liquid sloshing incylindrical containersrdquo Acta Aerodynamica Sinica vol 9 no 1pp 112ndash119 1991 (Chinese)

[24] Y Wang J Zang and D Qiu ldquoNumerical model of cnoidalwave flumerdquo China Ocean Engineering vol 13 no 4 pp 391ndash398 1999

[25] S Elgar and R T Guza ldquoObservations of bispectral of shoalingsurface gravity wavesrdquo Journal of Fluid Mechanics vol 161 pp425ndash448 1985

[26] G Dong Y Ma M Perlin X Ma B Yu and J Xu ldquoExperi-mental study of wavemdashwave nonlinear interactions using thewavelet-based bicoherencerdquo Coastal Engineering vol 55 no 9pp 741ndash752 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0 1 2 3 4 5minus04

minus02

00

02

04

066

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

065

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

064

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

063

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

062

p(k

Pa)

t (s)

0 1 2 3 4 5minus04

minus02

00

02

04

06

t (s)

p(k

Pa)

1

Figure 6 Comparisons of the time series of wave pressures on the perforated rectangular caisson (open circles) experimental results (solidlines) numerical results

D

Fminus F+D

2

X

Z

B

(a)

X

Dc FminusXM D

F+XM

(b)

Figure 7 Sketch of a quasi-ellipse caisson (a) and multiple quasi-ellipse caissons (b)

(40m) For this simulation the height of the incident cnoidalwave is 3m with a period of 10 s As wave crest approachesthe front faces of the caissons the runup of waves is veryhigh and partially caused by significant reflection of thecaissons The difference between the front and rear faces inthis situation is pronounced suggesting that the wave forceson the caissons are very large It is noticed that the surface

elevations between the caissons are also remarkable Theresults for the perforated caissons are shown in Figure 11Thesurface elevations close to the caissons are reduced comparedto that without perforation indicating that the perforationcan reduce the wave runuprundown on the caissons Thesurface elevations between the caissons are also significantlyweakened

8 Mathematical Problems in Engineering

30

m

2m

4m

3m

SWL4m 4m

2m 1

m SWL

Front view Side view

1m

Figure 8 Sketch of a perforated quasi-ellipse caisson

(a) Solid quasi-ellipse caisson (b) Perforated quasi-ellipse caisson

Figure 9 Sketch of the mesh division of a cross section for the numerical model

The perforation on caissons can enhance the waterexchange wave energy is partially dissipated by the perfo-rated wall and some wave energy is transferred to the fluidinside the chamber of the caissons (see Figure 12) and hencethe wave reflection and runup are reduced As the wavetroughs approach the middle of the caissons (Figure 12(a))the difference of water levels between inside and outside ofthe chambers is pronounced which causes the water flowto the outside Therefore the wave height in the middle ofthe caisson is reduced For the time when the wave crestsare close to this region the wave flow direction is reversed(Figure 12(c)) and the wave height outside the chambersis decreased The intensive water exchange between insideand outside of the chambers suggests that wave energy isdissipated significantly

The previous discussion suggests the perforation cansignificantly reduce the wave force of the caissons In thissection the wave forces and the influence factor will beinvestigated for multiple caissons To achieve these targetscnoidal waves with different incident height and periodare generated and the interactions between the waves anda caisson without peroration are simulated first and thenthe multiple placed caissons Following the definition ofwave force used for an isolated square structure [21] thenondimensional wave force on the single placed caisson canbe written as

119865119909119878

=119865119909

120588119892 (119861 + 119863) 1198892 [tanh (119896119889) 119896119889] (13)

where 119896 is the wave number of the incident wave and 119865119909is the

maximum wave force in the 119909 direction and can be obtainedby integrating the surface pressure on the structure

The wave forces on the single placed caisson are shownin Figure 13 As expected higher incident waves (119867119889 =

015) can produce larger wave forces With decreasing ofwave period (ie 119896 is increased) wave force on the caissonis increased

For convenient illustration and comparison the waveforces on a single placed caisson without perforation areused as benchmarkThen a nondimensional force which canreflect the influence of multiple placement is defined as

119877119909=

10038161003816100381610038161198651199091198721003816100381610038161003816

10038161003816100381610038161198651199091198781003816100381610038161003816

(14)

where 119865119909119872

is maximum force in 119909 direction for the centralone of the caisson group (as illustrated in Figure 7)

The nondimensional forces 119877119909on the middle caisson

without perforation are shown in Figure 14 It is found thatthe values of 119877

119909are larger than 1 for all the simulated cases

indicating that the presence of adjacent caissons can increasethe wave force With decreasing incident wave period (ieincreasing 119896119863) 119877

119909decreases Additionally 119877

119909for the cases

in the condition with 119863119888119863 = 2 is significantly larger than

those cases with larger transverse distance 119863119888 But the 119877

119909

differences between the cases in the conditions 119863119888119863 = 3

and 119863119888119863 = 4 are much smaller especially for the cases

with 119896119863 gt 10 By increasing incident wave nonlinearity

Mathematical Problems in Engineering 9

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(a) 119905 = 45119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(c) 119905 = (45 + 05)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(d) 119905 = (45 + 075)119879

Figure 10 Contours of water surface elevations around multiple impermeable quasi-ellipse caissons in a wave period (119879 = 10 s119867 = 30m)unit (m)

119877119909can also be enhanced but the increment is limited For the

tests with perforation the influence of incident wave periodand adjacent distance on the forces has a similar trend (seeFigure 15)

To directly reveal the influence of the perforation the119877119909difference between the impermeable caisson and the

perforated one is defined as

119876119909=10038161003816100381610038161003816119877119909119904

minus119877119909119901

10038161003816100381610038161003816 (15)

where 119877119909119904

is the relative wave for the caisson withoutperforation and 119877

119909119901is for the perforated one Figure 16

demonstrates the variations of 119876119909for all the simulated cases

For tests with the smaller incident wave height (119867119889 = 005)the reduction of wave forces by the perforation increaseswith the decrease of wave period The distance between thecaissons 119863

119888also has influence on the reduction wave forces

But if the adjacent distance is larger enough the influenceis limited for smaller incident waves For incident waveswith higher nonlinearity (ie 119867119889 = 015) the effect ofperforation appears smaller than the cases with the smallerincident wavesThewave period effect for the conditionswith119863119888119863 = 30 and 40 follows the trend of the cases with

the smaller incident height However the tendency for thecondition 119863

119888119863 = 20 is much complex When 119896119863 lt 10

the perforationworks better than those conditionswith largertransverse distances

5 Conclusions

A three-dimensional numerical wave flume was establishedusing the continuity equation and the Navier-Stokes equa-tions The VOF method was adopted to capture the freesurfaces The ability of generation of cnoidal waves and its

10 Mathematical Problems in Engineering

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(a) 119905 = 45119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(c) 119905 = (45 + 05)119879

140 175 210 245 280

200

180

160

140

120

100

80

60

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(d) 119905 = (45 + 075)119879

Figure 11 Contours of water surface elevations around multiple perforated quasi-ellipse caissons in a wave period (119879 = 10 s 119867 = 30m)unit (m)

interaction with a perforated rectangle caisson were verifiedusing the existing experimental data Then the presentmodel was used to investigate interactions between cnoidalwaves with varying incident height and period and multipleperforated quasi-ellipse caissons The influence of the per-foration on the wave field and the velocity flow around thecaissons was examined The numerical results demonstratethat the wave runuprundown on the caissons was reducedsignificantly by perforation and hence the wave forces arealso decreased The reduction of wave runuprundown ismainly due to the extensivewater exchange inside and outsidethe caisson chamber It is also found that the effect of thepresence of adjacent caissons on wave forces decreases withdecreasing incident wave period for both multiple placedcaissons with and without perforation The wave forces forcloser transverse adjacent distance are larger than those

conditions with a farther distance but when relative adjacentdistance 119863

119888119863 (119863

119888is the transverse distance and 119863 is the

width of the quasi-ellipse caisson) is larger than 3 the effectof adjacent distance is limited Additionally the effect ofthe perforation increases with the reduction of incidentwave period The performance of the perforation is betterwhen incident wave height is small suggesting that wavenonlinearity also plays an important role in the process ofinteraction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(a) 119905 = 45119879

200

180

160

140

120

100

80

60

Y(m

)

140 175 210 245 280

X (m)

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(c) 119905 = (45 + 05)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(d) 119905 = (45 + 075)119879

Figure 12 The flow vector fields corresponding to the wave surface elevations of Figure 11

014

012

010

008

006

004

00207 08 09 10 11 12 13 14 15

FxS

kD

Hd = 005

Hd = 015

Figure 13 Variations of wave forces on an isolated caisson without perforation

12 Mathematical Problems in Engineering

06 08 10 12 14 16095

100

105

110

115

120

DcD = 2

DcD = 3

DcD = 4

Rx

kD

(a)

06 08 10 12 14 16095

100

105

110

115

120

Rx

kD

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 14 Influence of transverse distance 119871 on the nondimensional wave forces 119877119909of the caisson without perforation (a) cases with119867119889 =

005 (b) cases with119867119889 = 015

06 08 10 12 14 16075

080

085

090

095

100

Rx

kD

DcD = 2

DcD = 3

DcD = 4

(a)

06 08 10 12 14 16

Rx

kD

075

080

085

090

095

100

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 15 Influence of transverse distance 119871 on the nondimensional wave forces 119877119909of the caisson without perforation (a) cases with119867119889 =

005 (b) cases with119867119889 = 015

Mathematical Problems in Engineering 13

06 08 10 12 14 16

025

020

010

015

Qx

kD

DcD = 2

DcD = 3

DcD = 4

(a)Qx

kD

06 08 10 12 14 16

025

020

010

015

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 16 Variations of 119876119909with respect to incident wave period for cases in conditions with different transverse distance 119871 (a) cases with

119867119889 = 005 (b) cases with119867119889 = 015

Acknowledgments

This research is supported financially by the National NaturalScience Foundation (Grant nos 51422901 and 51221961) andA Foundation for the Author of National Excellent DoctoralDissertation of PR China (Grant no 201347)

References

[1] G E Jarlan ldquoA perforated vertical wall breakwaterrdquo The Dockand Harbour Authority vol 41 no 486 pp 394ndash398 1961

[2] Z Huang Y Li and Y Liu ldquoHydraulic performance andwave loadings of perforatedslotted coastal structures a reviewrdquoOcean Engineering vol 38 no 10 pp 1031ndash1053 2011

[3] M K M Darwiche A N Williams and K-H Wang ldquoWaveinteraction with semiporous cylindrical breakwaterrdquo Journal ofWaterway Port Coastal and Ocean Engineering vol 120 no 4pp 382ndash403 1994

[4] T L Yip and A T Chwang ldquoPerforated wall breakwater withinternal horizontal platerdquo Journal of EngineeringMechanics vol126 no 5 pp 533ndash538 2000

[5] K D Suh J C Choi B H Kim W S Park and K S LeeldquoReflection of irregular waves from perforated-wall caissonbreakwatersrdquo Coastal Engineering vol 44 no 2 pp 141ndash1512001

[6] Y C Li H J Liu B Teng and D P Sun ldquoReflection of obliqueincident waves by breakwaters with partially perforated wallrdquoChina Ocean Engineering vol 16 no 3 pp 329ndash342 2002

[7] Y C Li G H Dong H J Liu and D P Sun ldquoThe reflectionof oblique incident waves by breakwaters with double-layeredperforated wallrdquo Coastal Engineering vol 50 no 1-2 pp 47ndash602003

[8] Y Liu Y-C Li and B Teng ldquoWave interactionwith a perforatedwall breakwater with a submerged horizontal porous platerdquoOcean Engineering vol 34 no 17-18 pp 2364ndash2373 2007

[9] Y Liu Y-C Li and B Teng ldquoThe reflection of oblique wavesby an infinite number of partially perforated caissonsrdquo OceanEngineering vol 34 no 14-15 pp 1965ndash1976 2007

[10] K Sankarbabu S A Sannasiraj and V Sundar ldquoHydrodynamicperformance of a dual cylindrical caisson breakwaterrdquo CoastalEngineering vol 55 no 6 pp 431ndash446 2008

[11] S Neelamani G Koether H Schuttrumpf M Muttray andH Oumeraci ldquoWave forces on and water-surface fluctuationsaround a vertical cylinder encircled by a perforated squarecaissonrdquo Ocean Engineering vol 27 no 7 pp 775ndash800 2000

[12] K Vijayalakshmi S Neelamani R Sundaravadivelu and KMurali ldquoWave runup on a concentric twin perforated circularcylinderrdquo Ocean Engineering vol 34 no 2 pp 327ndash336 2007

[13] Y Liu Y Li B Teng J Jiang and B Ma ldquoTotal horizontal andvertical forces of irregular waves on partially perforated caissonbreakwatersrdquo Coastal Engineering vol 55 no 6 pp 537ndash5522008

[14] J-I Lee and S Shin ldquoExperimental study on the wave reflectionof partially perforated wall caissons with single and doublechambersrdquo Ocean Engineering vol 91 pp 1ndash10 2014

[15] J-I Lee Y-T Kim and S Shin ldquoExperimental studies onwave interactions of partially perforated wall under obliquelyincident wavesrdquo The Scientific World Journal vol 2014 ArticleID 954174 14 pages 2014

[16] X F Chen Y C Li Y X Wang G H Dong and X BaildquoNumerical simulation of wave interaction with perforatedcaisson breakwatersrdquoChina Ocean Engineering vol 17 no 1 pp33ndash43 2003

[17] J Liu G Lin and J Li ldquoShort-crested waves interaction with aconcentric cylindrical structure with double-layered perforatedwallsrdquo Ocean Engineering vol 40 pp 76ndash90 2012

14 Mathematical Problems in Engineering

[18] J Liu and G Lin ldquoNumerical modelling of wave interactionwith a concentric cylindrical system with an arc-shaped porousouter cylinderrdquo European Journal of MechanicsmdashBFluids vol37 pp 59ndash71 2013

[19] F Aristodemo D D Meringolo P Groenenboom A Lo Schi-avo P Veltri and M Veltri ldquoAssessment of dynamic pressuresat vertical and perforated breakwaters through diffusive SPHschemesrdquo Mathematical Problems in Engineering vol 2015Article ID 305028 10 pages 2015

[20] J Bai and J Hu ldquoResearch on and design of new structure ofelliptical caissoned pierrdquo Port and Waterway Engineering vol11 pp 25ndash30 2006 (Chinese)

[21] Y Wang X Ren and G Wang ldquoNumerical simulation ofnonlinear wave force on a quasi-ellipse caissonrdquo Journal ofMarine Science and Application vol 10 no 3 pp 265ndash271 2011

[22] C W Hirt and B D Nichols ldquoVolume of fluid (VOF) methodfor the dynamics of free boundariesrdquo Journal of ComputationalPhysics vol 39 no 1 pp 201ndash225 1981

[23] Y Wang and T Su ldquoNumerical simulation of liquid sloshing incylindrical containersrdquo Acta Aerodynamica Sinica vol 9 no 1pp 112ndash119 1991 (Chinese)

[24] Y Wang J Zang and D Qiu ldquoNumerical model of cnoidalwave flumerdquo China Ocean Engineering vol 13 no 4 pp 391ndash398 1999

[25] S Elgar and R T Guza ldquoObservations of bispectral of shoalingsurface gravity wavesrdquo Journal of Fluid Mechanics vol 161 pp425ndash448 1985

[26] G Dong Y Ma M Perlin X Ma B Yu and J Xu ldquoExperi-mental study of wavemdashwave nonlinear interactions using thewavelet-based bicoherencerdquo Coastal Engineering vol 55 no 9pp 741ndash752 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

30

m

2m

4m

3m

SWL4m 4m

2m 1

m SWL

Front view Side view

1m

Figure 8 Sketch of a perforated quasi-ellipse caisson

(a) Solid quasi-ellipse caisson (b) Perforated quasi-ellipse caisson

Figure 9 Sketch of the mesh division of a cross section for the numerical model

The perforation on caissons can enhance the waterexchange wave energy is partially dissipated by the perfo-rated wall and some wave energy is transferred to the fluidinside the chamber of the caissons (see Figure 12) and hencethe wave reflection and runup are reduced As the wavetroughs approach the middle of the caissons (Figure 12(a))the difference of water levels between inside and outside ofthe chambers is pronounced which causes the water flowto the outside Therefore the wave height in the middle ofthe caisson is reduced For the time when the wave crestsare close to this region the wave flow direction is reversed(Figure 12(c)) and the wave height outside the chambersis decreased The intensive water exchange between insideand outside of the chambers suggests that wave energy isdissipated significantly

The previous discussion suggests the perforation cansignificantly reduce the wave force of the caissons In thissection the wave forces and the influence factor will beinvestigated for multiple caissons To achieve these targetscnoidal waves with different incident height and periodare generated and the interactions between the waves anda caisson without peroration are simulated first and thenthe multiple placed caissons Following the definition ofwave force used for an isolated square structure [21] thenondimensional wave force on the single placed caisson canbe written as

119865119909119878

=119865119909

120588119892 (119861 + 119863) 1198892 [tanh (119896119889) 119896119889] (13)

where 119896 is the wave number of the incident wave and 119865119909is the

maximum wave force in the 119909 direction and can be obtainedby integrating the surface pressure on the structure

The wave forces on the single placed caisson are shownin Figure 13 As expected higher incident waves (119867119889 =

015) can produce larger wave forces With decreasing ofwave period (ie 119896 is increased) wave force on the caissonis increased

For convenient illustration and comparison the waveforces on a single placed caisson without perforation areused as benchmarkThen a nondimensional force which canreflect the influence of multiple placement is defined as

119877119909=

10038161003816100381610038161198651199091198721003816100381610038161003816

10038161003816100381610038161198651199091198781003816100381610038161003816

(14)

where 119865119909119872

is maximum force in 119909 direction for the centralone of the caisson group (as illustrated in Figure 7)

The nondimensional forces 119877119909on the middle caisson

without perforation are shown in Figure 14 It is found thatthe values of 119877

119909are larger than 1 for all the simulated cases

indicating that the presence of adjacent caissons can increasethe wave force With decreasing incident wave period (ieincreasing 119896119863) 119877

119909decreases Additionally 119877

119909for the cases

in the condition with 119863119888119863 = 2 is significantly larger than

those cases with larger transverse distance 119863119888 But the 119877

119909

differences between the cases in the conditions 119863119888119863 = 3

and 119863119888119863 = 4 are much smaller especially for the cases

with 119896119863 gt 10 By increasing incident wave nonlinearity

Mathematical Problems in Engineering 9

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(a) 119905 = 45119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(c) 119905 = (45 + 05)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(d) 119905 = (45 + 075)119879

Figure 10 Contours of water surface elevations around multiple impermeable quasi-ellipse caissons in a wave period (119879 = 10 s119867 = 30m)unit (m)

119877119909can also be enhanced but the increment is limited For the

tests with perforation the influence of incident wave periodand adjacent distance on the forces has a similar trend (seeFigure 15)

To directly reveal the influence of the perforation the119877119909difference between the impermeable caisson and the

perforated one is defined as

119876119909=10038161003816100381610038161003816119877119909119904

minus119877119909119901

10038161003816100381610038161003816 (15)

where 119877119909119904

is the relative wave for the caisson withoutperforation and 119877

119909119901is for the perforated one Figure 16

demonstrates the variations of 119876119909for all the simulated cases

For tests with the smaller incident wave height (119867119889 = 005)the reduction of wave forces by the perforation increaseswith the decrease of wave period The distance between thecaissons 119863

119888also has influence on the reduction wave forces

But if the adjacent distance is larger enough the influenceis limited for smaller incident waves For incident waveswith higher nonlinearity (ie 119867119889 = 015) the effect ofperforation appears smaller than the cases with the smallerincident wavesThewave period effect for the conditionswith119863119888119863 = 30 and 40 follows the trend of the cases with

the smaller incident height However the tendency for thecondition 119863

119888119863 = 20 is much complex When 119896119863 lt 10

the perforationworks better than those conditionswith largertransverse distances

5 Conclusions

A three-dimensional numerical wave flume was establishedusing the continuity equation and the Navier-Stokes equa-tions The VOF method was adopted to capture the freesurfaces The ability of generation of cnoidal waves and its

10 Mathematical Problems in Engineering

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(a) 119905 = 45119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(c) 119905 = (45 + 05)119879

140 175 210 245 280

200

180

160

140

120

100

80

60

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(d) 119905 = (45 + 075)119879

Figure 11 Contours of water surface elevations around multiple perforated quasi-ellipse caissons in a wave period (119879 = 10 s 119867 = 30m)unit (m)

interaction with a perforated rectangle caisson were verifiedusing the existing experimental data Then the presentmodel was used to investigate interactions between cnoidalwaves with varying incident height and period and multipleperforated quasi-ellipse caissons The influence of the per-foration on the wave field and the velocity flow around thecaissons was examined The numerical results demonstratethat the wave runuprundown on the caissons was reducedsignificantly by perforation and hence the wave forces arealso decreased The reduction of wave runuprundown ismainly due to the extensivewater exchange inside and outsidethe caisson chamber It is also found that the effect of thepresence of adjacent caissons on wave forces decreases withdecreasing incident wave period for both multiple placedcaissons with and without perforation The wave forces forcloser transverse adjacent distance are larger than those

conditions with a farther distance but when relative adjacentdistance 119863

119888119863 (119863

119888is the transverse distance and 119863 is the

width of the quasi-ellipse caisson) is larger than 3 the effectof adjacent distance is limited Additionally the effect ofthe perforation increases with the reduction of incidentwave period The performance of the perforation is betterwhen incident wave height is small suggesting that wavenonlinearity also plays an important role in the process ofinteraction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(a) 119905 = 45119879

200

180

160

140

120

100

80

60

Y(m

)

140 175 210 245 280

X (m)

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(c) 119905 = (45 + 05)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(d) 119905 = (45 + 075)119879

Figure 12 The flow vector fields corresponding to the wave surface elevations of Figure 11

014

012

010

008

006

004

00207 08 09 10 11 12 13 14 15

FxS

kD

Hd = 005

Hd = 015

Figure 13 Variations of wave forces on an isolated caisson without perforation

12 Mathematical Problems in Engineering

06 08 10 12 14 16095

100

105

110

115

120

DcD = 2

DcD = 3

DcD = 4

Rx

kD

(a)

06 08 10 12 14 16095

100

105

110

115

120

Rx

kD

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 14 Influence of transverse distance 119871 on the nondimensional wave forces 119877119909of the caisson without perforation (a) cases with119867119889 =

005 (b) cases with119867119889 = 015

06 08 10 12 14 16075

080

085

090

095

100

Rx

kD

DcD = 2

DcD = 3

DcD = 4

(a)

06 08 10 12 14 16

Rx

kD

075

080

085

090

095

100

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 15 Influence of transverse distance 119871 on the nondimensional wave forces 119877119909of the caisson without perforation (a) cases with119867119889 =

005 (b) cases with119867119889 = 015

Mathematical Problems in Engineering 13

06 08 10 12 14 16

025

020

010

015

Qx

kD

DcD = 2

DcD = 3

DcD = 4

(a)Qx

kD

06 08 10 12 14 16

025

020

010

015

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 16 Variations of 119876119909with respect to incident wave period for cases in conditions with different transverse distance 119871 (a) cases with

119867119889 = 005 (b) cases with119867119889 = 015

Acknowledgments

This research is supported financially by the National NaturalScience Foundation (Grant nos 51422901 and 51221961) andA Foundation for the Author of National Excellent DoctoralDissertation of PR China (Grant no 201347)

References

[1] G E Jarlan ldquoA perforated vertical wall breakwaterrdquo The Dockand Harbour Authority vol 41 no 486 pp 394ndash398 1961

[2] Z Huang Y Li and Y Liu ldquoHydraulic performance andwave loadings of perforatedslotted coastal structures a reviewrdquoOcean Engineering vol 38 no 10 pp 1031ndash1053 2011

[3] M K M Darwiche A N Williams and K-H Wang ldquoWaveinteraction with semiporous cylindrical breakwaterrdquo Journal ofWaterway Port Coastal and Ocean Engineering vol 120 no 4pp 382ndash403 1994

[4] T L Yip and A T Chwang ldquoPerforated wall breakwater withinternal horizontal platerdquo Journal of EngineeringMechanics vol126 no 5 pp 533ndash538 2000

[5] K D Suh J C Choi B H Kim W S Park and K S LeeldquoReflection of irregular waves from perforated-wall caissonbreakwatersrdquo Coastal Engineering vol 44 no 2 pp 141ndash1512001

[6] Y C Li H J Liu B Teng and D P Sun ldquoReflection of obliqueincident waves by breakwaters with partially perforated wallrdquoChina Ocean Engineering vol 16 no 3 pp 329ndash342 2002

[7] Y C Li G H Dong H J Liu and D P Sun ldquoThe reflectionof oblique incident waves by breakwaters with double-layeredperforated wallrdquo Coastal Engineering vol 50 no 1-2 pp 47ndash602003

[8] Y Liu Y-C Li and B Teng ldquoWave interactionwith a perforatedwall breakwater with a submerged horizontal porous platerdquoOcean Engineering vol 34 no 17-18 pp 2364ndash2373 2007

[9] Y Liu Y-C Li and B Teng ldquoThe reflection of oblique wavesby an infinite number of partially perforated caissonsrdquo OceanEngineering vol 34 no 14-15 pp 1965ndash1976 2007

[10] K Sankarbabu S A Sannasiraj and V Sundar ldquoHydrodynamicperformance of a dual cylindrical caisson breakwaterrdquo CoastalEngineering vol 55 no 6 pp 431ndash446 2008

[11] S Neelamani G Koether H Schuttrumpf M Muttray andH Oumeraci ldquoWave forces on and water-surface fluctuationsaround a vertical cylinder encircled by a perforated squarecaissonrdquo Ocean Engineering vol 27 no 7 pp 775ndash800 2000

[12] K Vijayalakshmi S Neelamani R Sundaravadivelu and KMurali ldquoWave runup on a concentric twin perforated circularcylinderrdquo Ocean Engineering vol 34 no 2 pp 327ndash336 2007

[13] Y Liu Y Li B Teng J Jiang and B Ma ldquoTotal horizontal andvertical forces of irregular waves on partially perforated caissonbreakwatersrdquo Coastal Engineering vol 55 no 6 pp 537ndash5522008

[14] J-I Lee and S Shin ldquoExperimental study on the wave reflectionof partially perforated wall caissons with single and doublechambersrdquo Ocean Engineering vol 91 pp 1ndash10 2014

[15] J-I Lee Y-T Kim and S Shin ldquoExperimental studies onwave interactions of partially perforated wall under obliquelyincident wavesrdquo The Scientific World Journal vol 2014 ArticleID 954174 14 pages 2014

[16] X F Chen Y C Li Y X Wang G H Dong and X BaildquoNumerical simulation of wave interaction with perforatedcaisson breakwatersrdquoChina Ocean Engineering vol 17 no 1 pp33ndash43 2003

[17] J Liu G Lin and J Li ldquoShort-crested waves interaction with aconcentric cylindrical structure with double-layered perforatedwallsrdquo Ocean Engineering vol 40 pp 76ndash90 2012

14 Mathematical Problems in Engineering

[18] J Liu and G Lin ldquoNumerical modelling of wave interactionwith a concentric cylindrical system with an arc-shaped porousouter cylinderrdquo European Journal of MechanicsmdashBFluids vol37 pp 59ndash71 2013

[19] F Aristodemo D D Meringolo P Groenenboom A Lo Schi-avo P Veltri and M Veltri ldquoAssessment of dynamic pressuresat vertical and perforated breakwaters through diffusive SPHschemesrdquo Mathematical Problems in Engineering vol 2015Article ID 305028 10 pages 2015

[20] J Bai and J Hu ldquoResearch on and design of new structure ofelliptical caissoned pierrdquo Port and Waterway Engineering vol11 pp 25ndash30 2006 (Chinese)

[21] Y Wang X Ren and G Wang ldquoNumerical simulation ofnonlinear wave force on a quasi-ellipse caissonrdquo Journal ofMarine Science and Application vol 10 no 3 pp 265ndash271 2011

[22] C W Hirt and B D Nichols ldquoVolume of fluid (VOF) methodfor the dynamics of free boundariesrdquo Journal of ComputationalPhysics vol 39 no 1 pp 201ndash225 1981

[23] Y Wang and T Su ldquoNumerical simulation of liquid sloshing incylindrical containersrdquo Acta Aerodynamica Sinica vol 9 no 1pp 112ndash119 1991 (Chinese)

[24] Y Wang J Zang and D Qiu ldquoNumerical model of cnoidalwave flumerdquo China Ocean Engineering vol 13 no 4 pp 391ndash398 1999

[25] S Elgar and R T Guza ldquoObservations of bispectral of shoalingsurface gravity wavesrdquo Journal of Fluid Mechanics vol 161 pp425ndash448 1985

[26] G Dong Y Ma M Perlin X Ma B Yu and J Xu ldquoExperi-mental study of wavemdashwave nonlinear interactions using thewavelet-based bicoherencerdquo Coastal Engineering vol 55 no 9pp 741ndash752 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(a) 119905 = 45119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(c) 119905 = (45 + 05)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(d) 119905 = (45 + 075)119879

Figure 10 Contours of water surface elevations around multiple impermeable quasi-ellipse caissons in a wave period (119879 = 10 s119867 = 30m)unit (m)

119877119909can also be enhanced but the increment is limited For the

tests with perforation the influence of incident wave periodand adjacent distance on the forces has a similar trend (seeFigure 15)

To directly reveal the influence of the perforation the119877119909difference between the impermeable caisson and the

perforated one is defined as

119876119909=10038161003816100381610038161003816119877119909119904

minus119877119909119901

10038161003816100381610038161003816 (15)

where 119877119909119904

is the relative wave for the caisson withoutperforation and 119877

119909119901is for the perforated one Figure 16

demonstrates the variations of 119876119909for all the simulated cases

For tests with the smaller incident wave height (119867119889 = 005)the reduction of wave forces by the perforation increaseswith the decrease of wave period The distance between thecaissons 119863

119888also has influence on the reduction wave forces

But if the adjacent distance is larger enough the influenceis limited for smaller incident waves For incident waveswith higher nonlinearity (ie 119867119889 = 015) the effect ofperforation appears smaller than the cases with the smallerincident wavesThewave period effect for the conditionswith119863119888119863 = 30 and 40 follows the trend of the cases with

the smaller incident height However the tendency for thecondition 119863

119888119863 = 20 is much complex When 119896119863 lt 10

the perforationworks better than those conditionswith largertransverse distances

5 Conclusions

A three-dimensional numerical wave flume was establishedusing the continuity equation and the Navier-Stokes equa-tions The VOF method was adopted to capture the freesurfaces The ability of generation of cnoidal waves and its

10 Mathematical Problems in Engineering

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(a) 119905 = 45119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(c) 119905 = (45 + 05)119879

140 175 210 245 280

200

180

160

140

120

100

80

60

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(d) 119905 = (45 + 075)119879

Figure 11 Contours of water surface elevations around multiple perforated quasi-ellipse caissons in a wave period (119879 = 10 s 119867 = 30m)unit (m)

interaction with a perforated rectangle caisson were verifiedusing the existing experimental data Then the presentmodel was used to investigate interactions between cnoidalwaves with varying incident height and period and multipleperforated quasi-ellipse caissons The influence of the per-foration on the wave field and the velocity flow around thecaissons was examined The numerical results demonstratethat the wave runuprundown on the caissons was reducedsignificantly by perforation and hence the wave forces arealso decreased The reduction of wave runuprundown ismainly due to the extensivewater exchange inside and outsidethe caisson chamber It is also found that the effect of thepresence of adjacent caissons on wave forces decreases withdecreasing incident wave period for both multiple placedcaissons with and without perforation The wave forces forcloser transverse adjacent distance are larger than those

conditions with a farther distance but when relative adjacentdistance 119863

119888119863 (119863

119888is the transverse distance and 119863 is the

width of the quasi-ellipse caisson) is larger than 3 the effectof adjacent distance is limited Additionally the effect ofthe perforation increases with the reduction of incidentwave period The performance of the perforation is betterwhen incident wave height is small suggesting that wavenonlinearity also plays an important role in the process ofinteraction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(a) 119905 = 45119879

200

180

160

140

120

100

80

60

Y(m

)

140 175 210 245 280

X (m)

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(c) 119905 = (45 + 05)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(d) 119905 = (45 + 075)119879

Figure 12 The flow vector fields corresponding to the wave surface elevations of Figure 11

014

012

010

008

006

004

00207 08 09 10 11 12 13 14 15

FxS

kD

Hd = 005

Hd = 015

Figure 13 Variations of wave forces on an isolated caisson without perforation

12 Mathematical Problems in Engineering

06 08 10 12 14 16095

100

105

110

115

120

DcD = 2

DcD = 3

DcD = 4

Rx

kD

(a)

06 08 10 12 14 16095

100

105

110

115

120

Rx

kD

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 14 Influence of transverse distance 119871 on the nondimensional wave forces 119877119909of the caisson without perforation (a) cases with119867119889 =

005 (b) cases with119867119889 = 015

06 08 10 12 14 16075

080

085

090

095

100

Rx

kD

DcD = 2

DcD = 3

DcD = 4

(a)

06 08 10 12 14 16

Rx

kD

075

080

085

090

095

100

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 15 Influence of transverse distance 119871 on the nondimensional wave forces 119877119909of the caisson without perforation (a) cases with119867119889 =

005 (b) cases with119867119889 = 015

Mathematical Problems in Engineering 13

06 08 10 12 14 16

025

020

010

015

Qx

kD

DcD = 2

DcD = 3

DcD = 4

(a)Qx

kD

06 08 10 12 14 16

025

020

010

015

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 16 Variations of 119876119909with respect to incident wave period for cases in conditions with different transverse distance 119871 (a) cases with

119867119889 = 005 (b) cases with119867119889 = 015

Acknowledgments

This research is supported financially by the National NaturalScience Foundation (Grant nos 51422901 and 51221961) andA Foundation for the Author of National Excellent DoctoralDissertation of PR China (Grant no 201347)

References

[1] G E Jarlan ldquoA perforated vertical wall breakwaterrdquo The Dockand Harbour Authority vol 41 no 486 pp 394ndash398 1961

[2] Z Huang Y Li and Y Liu ldquoHydraulic performance andwave loadings of perforatedslotted coastal structures a reviewrdquoOcean Engineering vol 38 no 10 pp 1031ndash1053 2011

[3] M K M Darwiche A N Williams and K-H Wang ldquoWaveinteraction with semiporous cylindrical breakwaterrdquo Journal ofWaterway Port Coastal and Ocean Engineering vol 120 no 4pp 382ndash403 1994

[4] T L Yip and A T Chwang ldquoPerforated wall breakwater withinternal horizontal platerdquo Journal of EngineeringMechanics vol126 no 5 pp 533ndash538 2000

[5] K D Suh J C Choi B H Kim W S Park and K S LeeldquoReflection of irregular waves from perforated-wall caissonbreakwatersrdquo Coastal Engineering vol 44 no 2 pp 141ndash1512001

[6] Y C Li H J Liu B Teng and D P Sun ldquoReflection of obliqueincident waves by breakwaters with partially perforated wallrdquoChina Ocean Engineering vol 16 no 3 pp 329ndash342 2002

[7] Y C Li G H Dong H J Liu and D P Sun ldquoThe reflectionof oblique incident waves by breakwaters with double-layeredperforated wallrdquo Coastal Engineering vol 50 no 1-2 pp 47ndash602003

[8] Y Liu Y-C Li and B Teng ldquoWave interactionwith a perforatedwall breakwater with a submerged horizontal porous platerdquoOcean Engineering vol 34 no 17-18 pp 2364ndash2373 2007

[9] Y Liu Y-C Li and B Teng ldquoThe reflection of oblique wavesby an infinite number of partially perforated caissonsrdquo OceanEngineering vol 34 no 14-15 pp 1965ndash1976 2007

[10] K Sankarbabu S A Sannasiraj and V Sundar ldquoHydrodynamicperformance of a dual cylindrical caisson breakwaterrdquo CoastalEngineering vol 55 no 6 pp 431ndash446 2008

[11] S Neelamani G Koether H Schuttrumpf M Muttray andH Oumeraci ldquoWave forces on and water-surface fluctuationsaround a vertical cylinder encircled by a perforated squarecaissonrdquo Ocean Engineering vol 27 no 7 pp 775ndash800 2000

[12] K Vijayalakshmi S Neelamani R Sundaravadivelu and KMurali ldquoWave runup on a concentric twin perforated circularcylinderrdquo Ocean Engineering vol 34 no 2 pp 327ndash336 2007

[13] Y Liu Y Li B Teng J Jiang and B Ma ldquoTotal horizontal andvertical forces of irregular waves on partially perforated caissonbreakwatersrdquo Coastal Engineering vol 55 no 6 pp 537ndash5522008

[14] J-I Lee and S Shin ldquoExperimental study on the wave reflectionof partially perforated wall caissons with single and doublechambersrdquo Ocean Engineering vol 91 pp 1ndash10 2014

[15] J-I Lee Y-T Kim and S Shin ldquoExperimental studies onwave interactions of partially perforated wall under obliquelyincident wavesrdquo The Scientific World Journal vol 2014 ArticleID 954174 14 pages 2014

[16] X F Chen Y C Li Y X Wang G H Dong and X BaildquoNumerical simulation of wave interaction with perforatedcaisson breakwatersrdquoChina Ocean Engineering vol 17 no 1 pp33ndash43 2003

[17] J Liu G Lin and J Li ldquoShort-crested waves interaction with aconcentric cylindrical structure with double-layered perforatedwallsrdquo Ocean Engineering vol 40 pp 76ndash90 2012

14 Mathematical Problems in Engineering

[18] J Liu and G Lin ldquoNumerical modelling of wave interactionwith a concentric cylindrical system with an arc-shaped porousouter cylinderrdquo European Journal of MechanicsmdashBFluids vol37 pp 59ndash71 2013

[19] F Aristodemo D D Meringolo P Groenenboom A Lo Schi-avo P Veltri and M Veltri ldquoAssessment of dynamic pressuresat vertical and perforated breakwaters through diffusive SPHschemesrdquo Mathematical Problems in Engineering vol 2015Article ID 305028 10 pages 2015

[20] J Bai and J Hu ldquoResearch on and design of new structure ofelliptical caissoned pierrdquo Port and Waterway Engineering vol11 pp 25ndash30 2006 (Chinese)

[21] Y Wang X Ren and G Wang ldquoNumerical simulation ofnonlinear wave force on a quasi-ellipse caissonrdquo Journal ofMarine Science and Application vol 10 no 3 pp 265ndash271 2011

[22] C W Hirt and B D Nichols ldquoVolume of fluid (VOF) methodfor the dynamics of free boundariesrdquo Journal of ComputationalPhysics vol 39 no 1 pp 201ndash225 1981

[23] Y Wang and T Su ldquoNumerical simulation of liquid sloshing incylindrical containersrdquo Acta Aerodynamica Sinica vol 9 no 1pp 112ndash119 1991 (Chinese)

[24] Y Wang J Zang and D Qiu ldquoNumerical model of cnoidalwave flumerdquo China Ocean Engineering vol 13 no 4 pp 391ndash398 1999

[25] S Elgar and R T Guza ldquoObservations of bispectral of shoalingsurface gravity wavesrdquo Journal of Fluid Mechanics vol 161 pp425ndash448 1985

[26] G Dong Y Ma M Perlin X Ma B Yu and J Xu ldquoExperi-mental study of wavemdashwave nonlinear interactions using thewavelet-based bicoherencerdquo Coastal Engineering vol 55 no 9pp 741ndash752 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(a) 119905 = 45119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(c) 119905 = (45 + 05)119879

140 175 210 245 280

200

180

160

140

120

100

80

60

Y(m

)

X (m)

36

3

24

18

12

06

0

minus06

minus12

minus18

minus24

minus3

(d) 119905 = (45 + 075)119879

Figure 11 Contours of water surface elevations around multiple perforated quasi-ellipse caissons in a wave period (119879 = 10 s 119867 = 30m)unit (m)

interaction with a perforated rectangle caisson were verifiedusing the existing experimental data Then the presentmodel was used to investigate interactions between cnoidalwaves with varying incident height and period and multipleperforated quasi-ellipse caissons The influence of the per-foration on the wave field and the velocity flow around thecaissons was examined The numerical results demonstratethat the wave runuprundown on the caissons was reducedsignificantly by perforation and hence the wave forces arealso decreased The reduction of wave runuprundown ismainly due to the extensivewater exchange inside and outsidethe caisson chamber It is also found that the effect of thepresence of adjacent caissons on wave forces decreases withdecreasing incident wave period for both multiple placedcaissons with and without perforation The wave forces forcloser transverse adjacent distance are larger than those

conditions with a farther distance but when relative adjacentdistance 119863

119888119863 (119863

119888is the transverse distance and 119863 is the

width of the quasi-ellipse caisson) is larger than 3 the effectof adjacent distance is limited Additionally the effect ofthe perforation increases with the reduction of incidentwave period The performance of the perforation is betterwhen incident wave height is small suggesting that wavenonlinearity also plays an important role in the process ofinteraction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(a) 119905 = 45119879

200

180

160

140

120

100

80

60

Y(m

)

140 175 210 245 280

X (m)

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(c) 119905 = (45 + 05)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(d) 119905 = (45 + 075)119879

Figure 12 The flow vector fields corresponding to the wave surface elevations of Figure 11

014

012

010

008

006

004

00207 08 09 10 11 12 13 14 15

FxS

kD

Hd = 005

Hd = 015

Figure 13 Variations of wave forces on an isolated caisson without perforation

12 Mathematical Problems in Engineering

06 08 10 12 14 16095

100

105

110

115

120

DcD = 2

DcD = 3

DcD = 4

Rx

kD

(a)

06 08 10 12 14 16095

100

105

110

115

120

Rx

kD

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 14 Influence of transverse distance 119871 on the nondimensional wave forces 119877119909of the caisson without perforation (a) cases with119867119889 =

005 (b) cases with119867119889 = 015

06 08 10 12 14 16075

080

085

090

095

100

Rx

kD

DcD = 2

DcD = 3

DcD = 4

(a)

06 08 10 12 14 16

Rx

kD

075

080

085

090

095

100

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 15 Influence of transverse distance 119871 on the nondimensional wave forces 119877119909of the caisson without perforation (a) cases with119867119889 =

005 (b) cases with119867119889 = 015

Mathematical Problems in Engineering 13

06 08 10 12 14 16

025

020

010

015

Qx

kD

DcD = 2

DcD = 3

DcD = 4

(a)Qx

kD

06 08 10 12 14 16

025

020

010

015

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 16 Variations of 119876119909with respect to incident wave period for cases in conditions with different transverse distance 119871 (a) cases with

119867119889 = 005 (b) cases with119867119889 = 015

Acknowledgments

This research is supported financially by the National NaturalScience Foundation (Grant nos 51422901 and 51221961) andA Foundation for the Author of National Excellent DoctoralDissertation of PR China (Grant no 201347)

References

[1] G E Jarlan ldquoA perforated vertical wall breakwaterrdquo The Dockand Harbour Authority vol 41 no 486 pp 394ndash398 1961

[2] Z Huang Y Li and Y Liu ldquoHydraulic performance andwave loadings of perforatedslotted coastal structures a reviewrdquoOcean Engineering vol 38 no 10 pp 1031ndash1053 2011

[3] M K M Darwiche A N Williams and K-H Wang ldquoWaveinteraction with semiporous cylindrical breakwaterrdquo Journal ofWaterway Port Coastal and Ocean Engineering vol 120 no 4pp 382ndash403 1994

[4] T L Yip and A T Chwang ldquoPerforated wall breakwater withinternal horizontal platerdquo Journal of EngineeringMechanics vol126 no 5 pp 533ndash538 2000

[5] K D Suh J C Choi B H Kim W S Park and K S LeeldquoReflection of irregular waves from perforated-wall caissonbreakwatersrdquo Coastal Engineering vol 44 no 2 pp 141ndash1512001

[6] Y C Li H J Liu B Teng and D P Sun ldquoReflection of obliqueincident waves by breakwaters with partially perforated wallrdquoChina Ocean Engineering vol 16 no 3 pp 329ndash342 2002

[7] Y C Li G H Dong H J Liu and D P Sun ldquoThe reflectionof oblique incident waves by breakwaters with double-layeredperforated wallrdquo Coastal Engineering vol 50 no 1-2 pp 47ndash602003

[8] Y Liu Y-C Li and B Teng ldquoWave interactionwith a perforatedwall breakwater with a submerged horizontal porous platerdquoOcean Engineering vol 34 no 17-18 pp 2364ndash2373 2007

[9] Y Liu Y-C Li and B Teng ldquoThe reflection of oblique wavesby an infinite number of partially perforated caissonsrdquo OceanEngineering vol 34 no 14-15 pp 1965ndash1976 2007

[10] K Sankarbabu S A Sannasiraj and V Sundar ldquoHydrodynamicperformance of a dual cylindrical caisson breakwaterrdquo CoastalEngineering vol 55 no 6 pp 431ndash446 2008

[11] S Neelamani G Koether H Schuttrumpf M Muttray andH Oumeraci ldquoWave forces on and water-surface fluctuationsaround a vertical cylinder encircled by a perforated squarecaissonrdquo Ocean Engineering vol 27 no 7 pp 775ndash800 2000

[12] K Vijayalakshmi S Neelamani R Sundaravadivelu and KMurali ldquoWave runup on a concentric twin perforated circularcylinderrdquo Ocean Engineering vol 34 no 2 pp 327ndash336 2007

[13] Y Liu Y Li B Teng J Jiang and B Ma ldquoTotal horizontal andvertical forces of irregular waves on partially perforated caissonbreakwatersrdquo Coastal Engineering vol 55 no 6 pp 537ndash5522008

[14] J-I Lee and S Shin ldquoExperimental study on the wave reflectionof partially perforated wall caissons with single and doublechambersrdquo Ocean Engineering vol 91 pp 1ndash10 2014

[15] J-I Lee Y-T Kim and S Shin ldquoExperimental studies onwave interactions of partially perforated wall under obliquelyincident wavesrdquo The Scientific World Journal vol 2014 ArticleID 954174 14 pages 2014

[16] X F Chen Y C Li Y X Wang G H Dong and X BaildquoNumerical simulation of wave interaction with perforatedcaisson breakwatersrdquoChina Ocean Engineering vol 17 no 1 pp33ndash43 2003

[17] J Liu G Lin and J Li ldquoShort-crested waves interaction with aconcentric cylindrical structure with double-layered perforatedwallsrdquo Ocean Engineering vol 40 pp 76ndash90 2012

14 Mathematical Problems in Engineering

[18] J Liu and G Lin ldquoNumerical modelling of wave interactionwith a concentric cylindrical system with an arc-shaped porousouter cylinderrdquo European Journal of MechanicsmdashBFluids vol37 pp 59ndash71 2013

[19] F Aristodemo D D Meringolo P Groenenboom A Lo Schi-avo P Veltri and M Veltri ldquoAssessment of dynamic pressuresat vertical and perforated breakwaters through diffusive SPHschemesrdquo Mathematical Problems in Engineering vol 2015Article ID 305028 10 pages 2015

[20] J Bai and J Hu ldquoResearch on and design of new structure ofelliptical caissoned pierrdquo Port and Waterway Engineering vol11 pp 25ndash30 2006 (Chinese)

[21] Y Wang X Ren and G Wang ldquoNumerical simulation ofnonlinear wave force on a quasi-ellipse caissonrdquo Journal ofMarine Science and Application vol 10 no 3 pp 265ndash271 2011

[22] C W Hirt and B D Nichols ldquoVolume of fluid (VOF) methodfor the dynamics of free boundariesrdquo Journal of ComputationalPhysics vol 39 no 1 pp 201ndash225 1981

[23] Y Wang and T Su ldquoNumerical simulation of liquid sloshing incylindrical containersrdquo Acta Aerodynamica Sinica vol 9 no 1pp 112ndash119 1991 (Chinese)

[24] Y Wang J Zang and D Qiu ldquoNumerical model of cnoidalwave flumerdquo China Ocean Engineering vol 13 no 4 pp 391ndash398 1999

[25] S Elgar and R T Guza ldquoObservations of bispectral of shoalingsurface gravity wavesrdquo Journal of Fluid Mechanics vol 161 pp425ndash448 1985

[26] G Dong Y Ma M Perlin X Ma B Yu and J Xu ldquoExperi-mental study of wavemdashwave nonlinear interactions using thewavelet-based bicoherencerdquo Coastal Engineering vol 55 no 9pp 741ndash752 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(a) 119905 = 45119879

200

180

160

140

120

100

80

60

Y(m

)

140 175 210 245 280

X (m)

(b) 119905 = (45 + 025)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(c) 119905 = (45 + 05)119879

200

180

160

140

120

100

80

60140 175 210 245 280

Y(m

)

X (m)

(d) 119905 = (45 + 075)119879

Figure 12 The flow vector fields corresponding to the wave surface elevations of Figure 11

014

012

010

008

006

004

00207 08 09 10 11 12 13 14 15

FxS

kD

Hd = 005

Hd = 015

Figure 13 Variations of wave forces on an isolated caisson without perforation

12 Mathematical Problems in Engineering

06 08 10 12 14 16095

100

105

110

115

120

DcD = 2

DcD = 3

DcD = 4

Rx

kD

(a)

06 08 10 12 14 16095

100

105

110

115

120

Rx

kD

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 14 Influence of transverse distance 119871 on the nondimensional wave forces 119877119909of the caisson without perforation (a) cases with119867119889 =

005 (b) cases with119867119889 = 015

06 08 10 12 14 16075

080

085

090

095

100

Rx

kD

DcD = 2

DcD = 3

DcD = 4

(a)

06 08 10 12 14 16

Rx

kD

075

080

085

090

095

100

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 15 Influence of transverse distance 119871 on the nondimensional wave forces 119877119909of the caisson without perforation (a) cases with119867119889 =

005 (b) cases with119867119889 = 015

Mathematical Problems in Engineering 13

06 08 10 12 14 16

025

020

010

015

Qx

kD

DcD = 2

DcD = 3

DcD = 4

(a)Qx

kD

06 08 10 12 14 16

025

020

010

015

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 16 Variations of 119876119909with respect to incident wave period for cases in conditions with different transverse distance 119871 (a) cases with

119867119889 = 005 (b) cases with119867119889 = 015

Acknowledgments

This research is supported financially by the National NaturalScience Foundation (Grant nos 51422901 and 51221961) andA Foundation for the Author of National Excellent DoctoralDissertation of PR China (Grant no 201347)

References

[1] G E Jarlan ldquoA perforated vertical wall breakwaterrdquo The Dockand Harbour Authority vol 41 no 486 pp 394ndash398 1961

[2] Z Huang Y Li and Y Liu ldquoHydraulic performance andwave loadings of perforatedslotted coastal structures a reviewrdquoOcean Engineering vol 38 no 10 pp 1031ndash1053 2011

[3] M K M Darwiche A N Williams and K-H Wang ldquoWaveinteraction with semiporous cylindrical breakwaterrdquo Journal ofWaterway Port Coastal and Ocean Engineering vol 120 no 4pp 382ndash403 1994

[4] T L Yip and A T Chwang ldquoPerforated wall breakwater withinternal horizontal platerdquo Journal of EngineeringMechanics vol126 no 5 pp 533ndash538 2000

[5] K D Suh J C Choi B H Kim W S Park and K S LeeldquoReflection of irregular waves from perforated-wall caissonbreakwatersrdquo Coastal Engineering vol 44 no 2 pp 141ndash1512001

[6] Y C Li H J Liu B Teng and D P Sun ldquoReflection of obliqueincident waves by breakwaters with partially perforated wallrdquoChina Ocean Engineering vol 16 no 3 pp 329ndash342 2002

[7] Y C Li G H Dong H J Liu and D P Sun ldquoThe reflectionof oblique incident waves by breakwaters with double-layeredperforated wallrdquo Coastal Engineering vol 50 no 1-2 pp 47ndash602003

[8] Y Liu Y-C Li and B Teng ldquoWave interactionwith a perforatedwall breakwater with a submerged horizontal porous platerdquoOcean Engineering vol 34 no 17-18 pp 2364ndash2373 2007

[9] Y Liu Y-C Li and B Teng ldquoThe reflection of oblique wavesby an infinite number of partially perforated caissonsrdquo OceanEngineering vol 34 no 14-15 pp 1965ndash1976 2007

[10] K Sankarbabu S A Sannasiraj and V Sundar ldquoHydrodynamicperformance of a dual cylindrical caisson breakwaterrdquo CoastalEngineering vol 55 no 6 pp 431ndash446 2008

[11] S Neelamani G Koether H Schuttrumpf M Muttray andH Oumeraci ldquoWave forces on and water-surface fluctuationsaround a vertical cylinder encircled by a perforated squarecaissonrdquo Ocean Engineering vol 27 no 7 pp 775ndash800 2000

[12] K Vijayalakshmi S Neelamani R Sundaravadivelu and KMurali ldquoWave runup on a concentric twin perforated circularcylinderrdquo Ocean Engineering vol 34 no 2 pp 327ndash336 2007

[13] Y Liu Y Li B Teng J Jiang and B Ma ldquoTotal horizontal andvertical forces of irregular waves on partially perforated caissonbreakwatersrdquo Coastal Engineering vol 55 no 6 pp 537ndash5522008

[14] J-I Lee and S Shin ldquoExperimental study on the wave reflectionof partially perforated wall caissons with single and doublechambersrdquo Ocean Engineering vol 91 pp 1ndash10 2014

[15] J-I Lee Y-T Kim and S Shin ldquoExperimental studies onwave interactions of partially perforated wall under obliquelyincident wavesrdquo The Scientific World Journal vol 2014 ArticleID 954174 14 pages 2014

[16] X F Chen Y C Li Y X Wang G H Dong and X BaildquoNumerical simulation of wave interaction with perforatedcaisson breakwatersrdquoChina Ocean Engineering vol 17 no 1 pp33ndash43 2003

[17] J Liu G Lin and J Li ldquoShort-crested waves interaction with aconcentric cylindrical structure with double-layered perforatedwallsrdquo Ocean Engineering vol 40 pp 76ndash90 2012

14 Mathematical Problems in Engineering

[18] J Liu and G Lin ldquoNumerical modelling of wave interactionwith a concentric cylindrical system with an arc-shaped porousouter cylinderrdquo European Journal of MechanicsmdashBFluids vol37 pp 59ndash71 2013

[19] F Aristodemo D D Meringolo P Groenenboom A Lo Schi-avo P Veltri and M Veltri ldquoAssessment of dynamic pressuresat vertical and perforated breakwaters through diffusive SPHschemesrdquo Mathematical Problems in Engineering vol 2015Article ID 305028 10 pages 2015

[20] J Bai and J Hu ldquoResearch on and design of new structure ofelliptical caissoned pierrdquo Port and Waterway Engineering vol11 pp 25ndash30 2006 (Chinese)

[21] Y Wang X Ren and G Wang ldquoNumerical simulation ofnonlinear wave force on a quasi-ellipse caissonrdquo Journal ofMarine Science and Application vol 10 no 3 pp 265ndash271 2011

[22] C W Hirt and B D Nichols ldquoVolume of fluid (VOF) methodfor the dynamics of free boundariesrdquo Journal of ComputationalPhysics vol 39 no 1 pp 201ndash225 1981

[23] Y Wang and T Su ldquoNumerical simulation of liquid sloshing incylindrical containersrdquo Acta Aerodynamica Sinica vol 9 no 1pp 112ndash119 1991 (Chinese)

[24] Y Wang J Zang and D Qiu ldquoNumerical model of cnoidalwave flumerdquo China Ocean Engineering vol 13 no 4 pp 391ndash398 1999

[25] S Elgar and R T Guza ldquoObservations of bispectral of shoalingsurface gravity wavesrdquo Journal of Fluid Mechanics vol 161 pp425ndash448 1985

[26] G Dong Y Ma M Perlin X Ma B Yu and J Xu ldquoExperi-mental study of wavemdashwave nonlinear interactions using thewavelet-based bicoherencerdquo Coastal Engineering vol 55 no 9pp 741ndash752 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

06 08 10 12 14 16095

100

105

110

115

120

DcD = 2

DcD = 3

DcD = 4

Rx

kD

(a)

06 08 10 12 14 16095

100

105

110

115

120

Rx

kD

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 14 Influence of transverse distance 119871 on the nondimensional wave forces 119877119909of the caisson without perforation (a) cases with119867119889 =

005 (b) cases with119867119889 = 015

06 08 10 12 14 16075

080

085

090

095

100

Rx

kD

DcD = 2

DcD = 3

DcD = 4

(a)

06 08 10 12 14 16

Rx

kD

075

080

085

090

095

100

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 15 Influence of transverse distance 119871 on the nondimensional wave forces 119877119909of the caisson without perforation (a) cases with119867119889 =

005 (b) cases with119867119889 = 015

Mathematical Problems in Engineering 13

06 08 10 12 14 16

025

020

010

015

Qx

kD

DcD = 2

DcD = 3

DcD = 4

(a)Qx

kD

06 08 10 12 14 16

025

020

010

015

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 16 Variations of 119876119909with respect to incident wave period for cases in conditions with different transverse distance 119871 (a) cases with

119867119889 = 005 (b) cases with119867119889 = 015

Acknowledgments

This research is supported financially by the National NaturalScience Foundation (Grant nos 51422901 and 51221961) andA Foundation for the Author of National Excellent DoctoralDissertation of PR China (Grant no 201347)

References

[1] G E Jarlan ldquoA perforated vertical wall breakwaterrdquo The Dockand Harbour Authority vol 41 no 486 pp 394ndash398 1961

[2] Z Huang Y Li and Y Liu ldquoHydraulic performance andwave loadings of perforatedslotted coastal structures a reviewrdquoOcean Engineering vol 38 no 10 pp 1031ndash1053 2011

[3] M K M Darwiche A N Williams and K-H Wang ldquoWaveinteraction with semiporous cylindrical breakwaterrdquo Journal ofWaterway Port Coastal and Ocean Engineering vol 120 no 4pp 382ndash403 1994

[4] T L Yip and A T Chwang ldquoPerforated wall breakwater withinternal horizontal platerdquo Journal of EngineeringMechanics vol126 no 5 pp 533ndash538 2000

[5] K D Suh J C Choi B H Kim W S Park and K S LeeldquoReflection of irregular waves from perforated-wall caissonbreakwatersrdquo Coastal Engineering vol 44 no 2 pp 141ndash1512001

[6] Y C Li H J Liu B Teng and D P Sun ldquoReflection of obliqueincident waves by breakwaters with partially perforated wallrdquoChina Ocean Engineering vol 16 no 3 pp 329ndash342 2002

[7] Y C Li G H Dong H J Liu and D P Sun ldquoThe reflectionof oblique incident waves by breakwaters with double-layeredperforated wallrdquo Coastal Engineering vol 50 no 1-2 pp 47ndash602003

[8] Y Liu Y-C Li and B Teng ldquoWave interactionwith a perforatedwall breakwater with a submerged horizontal porous platerdquoOcean Engineering vol 34 no 17-18 pp 2364ndash2373 2007

[9] Y Liu Y-C Li and B Teng ldquoThe reflection of oblique wavesby an infinite number of partially perforated caissonsrdquo OceanEngineering vol 34 no 14-15 pp 1965ndash1976 2007

[10] K Sankarbabu S A Sannasiraj and V Sundar ldquoHydrodynamicperformance of a dual cylindrical caisson breakwaterrdquo CoastalEngineering vol 55 no 6 pp 431ndash446 2008

[11] S Neelamani G Koether H Schuttrumpf M Muttray andH Oumeraci ldquoWave forces on and water-surface fluctuationsaround a vertical cylinder encircled by a perforated squarecaissonrdquo Ocean Engineering vol 27 no 7 pp 775ndash800 2000

[12] K Vijayalakshmi S Neelamani R Sundaravadivelu and KMurali ldquoWave runup on a concentric twin perforated circularcylinderrdquo Ocean Engineering vol 34 no 2 pp 327ndash336 2007

[13] Y Liu Y Li B Teng J Jiang and B Ma ldquoTotal horizontal andvertical forces of irregular waves on partially perforated caissonbreakwatersrdquo Coastal Engineering vol 55 no 6 pp 537ndash5522008

[14] J-I Lee and S Shin ldquoExperimental study on the wave reflectionof partially perforated wall caissons with single and doublechambersrdquo Ocean Engineering vol 91 pp 1ndash10 2014

[15] J-I Lee Y-T Kim and S Shin ldquoExperimental studies onwave interactions of partially perforated wall under obliquelyincident wavesrdquo The Scientific World Journal vol 2014 ArticleID 954174 14 pages 2014

[16] X F Chen Y C Li Y X Wang G H Dong and X BaildquoNumerical simulation of wave interaction with perforatedcaisson breakwatersrdquoChina Ocean Engineering vol 17 no 1 pp33ndash43 2003

[17] J Liu G Lin and J Li ldquoShort-crested waves interaction with aconcentric cylindrical structure with double-layered perforatedwallsrdquo Ocean Engineering vol 40 pp 76ndash90 2012

14 Mathematical Problems in Engineering

[18] J Liu and G Lin ldquoNumerical modelling of wave interactionwith a concentric cylindrical system with an arc-shaped porousouter cylinderrdquo European Journal of MechanicsmdashBFluids vol37 pp 59ndash71 2013

[19] F Aristodemo D D Meringolo P Groenenboom A Lo Schi-avo P Veltri and M Veltri ldquoAssessment of dynamic pressuresat vertical and perforated breakwaters through diffusive SPHschemesrdquo Mathematical Problems in Engineering vol 2015Article ID 305028 10 pages 2015

[20] J Bai and J Hu ldquoResearch on and design of new structure ofelliptical caissoned pierrdquo Port and Waterway Engineering vol11 pp 25ndash30 2006 (Chinese)

[21] Y Wang X Ren and G Wang ldquoNumerical simulation ofnonlinear wave force on a quasi-ellipse caissonrdquo Journal ofMarine Science and Application vol 10 no 3 pp 265ndash271 2011

[22] C W Hirt and B D Nichols ldquoVolume of fluid (VOF) methodfor the dynamics of free boundariesrdquo Journal of ComputationalPhysics vol 39 no 1 pp 201ndash225 1981

[23] Y Wang and T Su ldquoNumerical simulation of liquid sloshing incylindrical containersrdquo Acta Aerodynamica Sinica vol 9 no 1pp 112ndash119 1991 (Chinese)

[24] Y Wang J Zang and D Qiu ldquoNumerical model of cnoidalwave flumerdquo China Ocean Engineering vol 13 no 4 pp 391ndash398 1999

[25] S Elgar and R T Guza ldquoObservations of bispectral of shoalingsurface gravity wavesrdquo Journal of Fluid Mechanics vol 161 pp425ndash448 1985

[26] G Dong Y Ma M Perlin X Ma B Yu and J Xu ldquoExperi-mental study of wavemdashwave nonlinear interactions using thewavelet-based bicoherencerdquo Coastal Engineering vol 55 no 9pp 741ndash752 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

06 08 10 12 14 16

025

020

010

015

Qx

kD

DcD = 2

DcD = 3

DcD = 4

(a)Qx

kD

06 08 10 12 14 16

025

020

010

015

DcD = 2

DcD = 3

DcD = 4

(b)

Figure 16 Variations of 119876119909with respect to incident wave period for cases in conditions with different transverse distance 119871 (a) cases with

119867119889 = 005 (b) cases with119867119889 = 015

Acknowledgments

This research is supported financially by the National NaturalScience Foundation (Grant nos 51422901 and 51221961) andA Foundation for the Author of National Excellent DoctoralDissertation of PR China (Grant no 201347)

References

[1] G E Jarlan ldquoA perforated vertical wall breakwaterrdquo The Dockand Harbour Authority vol 41 no 486 pp 394ndash398 1961

[2] Z Huang Y Li and Y Liu ldquoHydraulic performance andwave loadings of perforatedslotted coastal structures a reviewrdquoOcean Engineering vol 38 no 10 pp 1031ndash1053 2011

[3] M K M Darwiche A N Williams and K-H Wang ldquoWaveinteraction with semiporous cylindrical breakwaterrdquo Journal ofWaterway Port Coastal and Ocean Engineering vol 120 no 4pp 382ndash403 1994

[4] T L Yip and A T Chwang ldquoPerforated wall breakwater withinternal horizontal platerdquo Journal of EngineeringMechanics vol126 no 5 pp 533ndash538 2000

[5] K D Suh J C Choi B H Kim W S Park and K S LeeldquoReflection of irregular waves from perforated-wall caissonbreakwatersrdquo Coastal Engineering vol 44 no 2 pp 141ndash1512001

[6] Y C Li H J Liu B Teng and D P Sun ldquoReflection of obliqueincident waves by breakwaters with partially perforated wallrdquoChina Ocean Engineering vol 16 no 3 pp 329ndash342 2002

[7] Y C Li G H Dong H J Liu and D P Sun ldquoThe reflectionof oblique incident waves by breakwaters with double-layeredperforated wallrdquo Coastal Engineering vol 50 no 1-2 pp 47ndash602003

[8] Y Liu Y-C Li and B Teng ldquoWave interactionwith a perforatedwall breakwater with a submerged horizontal porous platerdquoOcean Engineering vol 34 no 17-18 pp 2364ndash2373 2007

[9] Y Liu Y-C Li and B Teng ldquoThe reflection of oblique wavesby an infinite number of partially perforated caissonsrdquo OceanEngineering vol 34 no 14-15 pp 1965ndash1976 2007

[10] K Sankarbabu S A Sannasiraj and V Sundar ldquoHydrodynamicperformance of a dual cylindrical caisson breakwaterrdquo CoastalEngineering vol 55 no 6 pp 431ndash446 2008

[11] S Neelamani G Koether H Schuttrumpf M Muttray andH Oumeraci ldquoWave forces on and water-surface fluctuationsaround a vertical cylinder encircled by a perforated squarecaissonrdquo Ocean Engineering vol 27 no 7 pp 775ndash800 2000

[12] K Vijayalakshmi S Neelamani R Sundaravadivelu and KMurali ldquoWave runup on a concentric twin perforated circularcylinderrdquo Ocean Engineering vol 34 no 2 pp 327ndash336 2007

[13] Y Liu Y Li B Teng J Jiang and B Ma ldquoTotal horizontal andvertical forces of irregular waves on partially perforated caissonbreakwatersrdquo Coastal Engineering vol 55 no 6 pp 537ndash5522008

[14] J-I Lee and S Shin ldquoExperimental study on the wave reflectionof partially perforated wall caissons with single and doublechambersrdquo Ocean Engineering vol 91 pp 1ndash10 2014

[15] J-I Lee Y-T Kim and S Shin ldquoExperimental studies onwave interactions of partially perforated wall under obliquelyincident wavesrdquo The Scientific World Journal vol 2014 ArticleID 954174 14 pages 2014

[16] X F Chen Y C Li Y X Wang G H Dong and X BaildquoNumerical simulation of wave interaction with perforatedcaisson breakwatersrdquoChina Ocean Engineering vol 17 no 1 pp33ndash43 2003

[17] J Liu G Lin and J Li ldquoShort-crested waves interaction with aconcentric cylindrical structure with double-layered perforatedwallsrdquo Ocean Engineering vol 40 pp 76ndash90 2012

14 Mathematical Problems in Engineering

[18] J Liu and G Lin ldquoNumerical modelling of wave interactionwith a concentric cylindrical system with an arc-shaped porousouter cylinderrdquo European Journal of MechanicsmdashBFluids vol37 pp 59ndash71 2013

[19] F Aristodemo D D Meringolo P Groenenboom A Lo Schi-avo P Veltri and M Veltri ldquoAssessment of dynamic pressuresat vertical and perforated breakwaters through diffusive SPHschemesrdquo Mathematical Problems in Engineering vol 2015Article ID 305028 10 pages 2015

[20] J Bai and J Hu ldquoResearch on and design of new structure ofelliptical caissoned pierrdquo Port and Waterway Engineering vol11 pp 25ndash30 2006 (Chinese)

[21] Y Wang X Ren and G Wang ldquoNumerical simulation ofnonlinear wave force on a quasi-ellipse caissonrdquo Journal ofMarine Science and Application vol 10 no 3 pp 265ndash271 2011

[22] C W Hirt and B D Nichols ldquoVolume of fluid (VOF) methodfor the dynamics of free boundariesrdquo Journal of ComputationalPhysics vol 39 no 1 pp 201ndash225 1981

[23] Y Wang and T Su ldquoNumerical simulation of liquid sloshing incylindrical containersrdquo Acta Aerodynamica Sinica vol 9 no 1pp 112ndash119 1991 (Chinese)

[24] Y Wang J Zang and D Qiu ldquoNumerical model of cnoidalwave flumerdquo China Ocean Engineering vol 13 no 4 pp 391ndash398 1999

[25] S Elgar and R T Guza ldquoObservations of bispectral of shoalingsurface gravity wavesrdquo Journal of Fluid Mechanics vol 161 pp425ndash448 1985

[26] G Dong Y Ma M Perlin X Ma B Yu and J Xu ldquoExperi-mental study of wavemdashwave nonlinear interactions using thewavelet-based bicoherencerdquo Coastal Engineering vol 55 no 9pp 741ndash752 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Mathematical Problems in Engineering

[18] J Liu and G Lin ldquoNumerical modelling of wave interactionwith a concentric cylindrical system with an arc-shaped porousouter cylinderrdquo European Journal of MechanicsmdashBFluids vol37 pp 59ndash71 2013

[19] F Aristodemo D D Meringolo P Groenenboom A Lo Schi-avo P Veltri and M Veltri ldquoAssessment of dynamic pressuresat vertical and perforated breakwaters through diffusive SPHschemesrdquo Mathematical Problems in Engineering vol 2015Article ID 305028 10 pages 2015

[20] J Bai and J Hu ldquoResearch on and design of new structure ofelliptical caissoned pierrdquo Port and Waterway Engineering vol11 pp 25ndash30 2006 (Chinese)

[21] Y Wang X Ren and G Wang ldquoNumerical simulation ofnonlinear wave force on a quasi-ellipse caissonrdquo Journal ofMarine Science and Application vol 10 no 3 pp 265ndash271 2011

[22] C W Hirt and B D Nichols ldquoVolume of fluid (VOF) methodfor the dynamics of free boundariesrdquo Journal of ComputationalPhysics vol 39 no 1 pp 201ndash225 1981

[23] Y Wang and T Su ldquoNumerical simulation of liquid sloshing incylindrical containersrdquo Acta Aerodynamica Sinica vol 9 no 1pp 112ndash119 1991 (Chinese)

[24] Y Wang J Zang and D Qiu ldquoNumerical model of cnoidalwave flumerdquo China Ocean Engineering vol 13 no 4 pp 391ndash398 1999

[25] S Elgar and R T Guza ldquoObservations of bispectral of shoalingsurface gravity wavesrdquo Journal of Fluid Mechanics vol 161 pp425ndash448 1985

[26] G Dong Y Ma M Perlin X Ma B Yu and J Xu ldquoExperi-mental study of wavemdashwave nonlinear interactions using thewavelet-based bicoherencerdquo Coastal Engineering vol 55 no 9pp 741ndash752 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of