Developing Serpent-Type Wave Generators to Create Solitary Wave Simulations with BEM

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China OceanEng．，Vo1．27，No．5，PP．671—682◎ 2013 Chinese Ocean Engineering Society and Springer-Verlag Berlin HeidelbergDOI 10．1007／s13344—013—0056—2， ISSN 0890—5487Developing Serpent-Type W ave Generators to Create Solitary W aveSimulations with BEM W en—Kai W ENG ．Ruey—Syan SHIHb— and Chung—Ren CHOUDepartment ofHarbour＆River Engineering,National Taiwan Ocean University，Keelung 20224，ChinaDepartment ofConstruction and Spatial Design，Tungnan University，New Taipei 22202，China(Received 19 January 2011；received revised form 16 May 2013；accepted 28 June 2013)ABSTRACTDeveloping serpent—type wave generators to generate solitary waves in a 3D—basin was investigated in this study.

Based on the Lagrangian description with time—marching procedures and finite differences of the time derivative，a 3Dmulfir，le directional wave basin with multidirectional piston wave generators was developed to simulate ocean waves byusing BEM with quadrilateral elements．and to simulate wave-caused problems with fully nonlinear water surfaceconditions．The simulations of perpendicular solitary waves were conducted in the first instance to verify this scheme.

Furtherm ore，the comparison of the waveform variations confirm s that the estimation of 3D solitary waves is a feasiblescheme.

Key words：boundary element method；quadrilateral element；3D wave basin；time domain；solitary wave；nonlinearW口Ve1．IntroductionNumerical wave basin is used to evaluate wave impact on coastal structures．The variations ofoceanic physical characteristics must be accurately predicted to reproduce certain aspects of near-shorewave activity．Numerous investigations on 3D nulnerical models regarding the simulation of nonlinearwaves have been enthusiastically established which thallks to the rapid development of personalcomputers in the last two decades．The development of a 3D numerical wave basin in practicalapplications that use personal computers has been an arduous task due to the considerable quantities ofarithmetic units；therefore．simulations of fuly nonlinear waves in 3D models were in straitenedcircumstances．Until recently,these dificulties have been overcome by innovation and greatadvancements in computer science．These innovations and advancements shortened the arithmetic timeof modeling．To overcome the problem previously mentioned，parallel computing techniques on PCclusters were adopted in the present study for calculations and will be discussed in a later section.

By using a number of wavemakers as an absorption facility,a method for active absorption ofmultidirectional waves in a 3D numerical wave tank model(NWT、was presented by Skourup andSch~ifer f1 998)based on a traditional 2D active absorption method．i．e．2D．AWACS(Active WlaveThe work was financialy supported by the Science Council under the Project Nos．NSC一95—2221-E一019—075-MY3 fCRC1 andNSC一97—2221一E一236-011-(RSS).

1 Coresponding author．Email：rsshih###mail．tnu．edu．tw672 Wen-KaiWENG eta1．／ChinaOceanEng．，27(5)，2013，671—682Absorption Control System)．Uni—as wel as multi—directional wave can be generated by a serpent typewave generator according to linear wave maker theory(Dean and Darlymple，1993)for each segment。

Practical application of this theory includes generators such as“snake—type”，“serpent-type”，or‘segmented”wave generator in a physical laboratory．Unidirectional and multi—directional waves canbe generated spatially by sinusoidal motion and from the basic‘‘snake principle’’of the segments of aserpent-type wave generator,respectively．The incident waves were generated by prescribing motionsas a series of piston wave makers.

The directional wave maker theory with sidewall reflections developed by Dalrymple wasexperimentaly confirmed by Mansard and Miles(1 994)．Similar experimental investigations on theapplicability and reproducibility of the multi-face generators with small segments were made byHiraishi et a1．(1 995)．An optimization method to improve the uniform ity of monochromatic obliquewaves in a wave basin was adopted by Matsumoto and Hanzawa(1996)using the non-linear leastsquare form ulation to determ ine individual paddle motions of a multi—directional wave maker．Shih eta1．(2009)made the improvements on the oblique planar wave train in a basin generated by multipleirregular wave generators and numericaly investigated those improvements adopting the boundaryelement method(BEM)．Simulations of perpendicular waves were conducted in the first instance toverify the scheme，and proceeded with the generations and propagations of oblique waves in a largeangle．Li et a1．(201 l1 developed a simulation method to generate waves by two—sided segm entedwave—makers．This paper introduces the two-sided segmented wave—makers and the wave simulationmethods on long-crested waves，short—crested waves，and irregular waves．However,development of3D NWT remains rather difcult，because an extremely long CUP time(tremendous computing time)and extensive memory capacity are required for a fully nonlinear numerical model by means of 3DBEM to solve fully nonlinear problems directly in time domain，and the influence matrix needs to besetup andinverted at eachtime step asthenodes onthefree surfacemovetonew positions.

This study investigated the application of 3D BEM to the serpent—type wave generator for solitarywave generations．The generations of a perpendicular moving solitary wave were presented throughprescribing adequate snak elike motions at the input boundary of the wave maker．As an alternative tophysical wave basin，a 3D algorithm based on BEM with linear tetragonal elements was developed tomodel the generation of a solitary wave，and boundary values were updated at each time step by aforward difference time marching procedure.

2．Numerical M odeling and Simulations2．1 Theoretieal Development and Boundary ConditionsThe 3D numerical wave basin with constant water depth h contains a segmented wave generatorthat occupies one wal1 of the wave basin(Fig．1)．Th e Cartesian coordinates ，Y，z)were employed，the region of which was boun ded by the free water surface，厂l，the segmented wave generator,／-I2，theright and left lateral imperm eable sidewalls，厂3 and／'4，the imperm eable across wall from the generators，．
and an impermeable botom， ．Under the assumption of an inviscid，incompressible，andhomogeneous(irotationa1)fluid within the region Q(力，the solution of velocity potential ( ，Y，z；f)Wen-KaiWENGeta1．／ChinaOceanEng．，27( ，2013，671—682is given to satisfy the folowing Laplace equation：a a a。

+ +Fig．1．Definition sketch of 3D—wave basin withsegmented wave generator.

673(1)Based on the Lagrangian description and time-marching procedure with finite diferencing of thetime derivative，the undisturbed free surface，the fully nonlinear kinematic and dynamic boundaryconditions Can be obtained(see Shih et a1．，2009).

The boundaries of the three vertical walls，referred to as f ，F4，and r ，an d the stationaryboaom，厂6，are impermeable，and the no—flow boundary conditions are prescribed as：： 0， on ，r4，厂5，aI1d ，d(2)where is the unit outward norm al vector．In accordan ce with the continuity of the velocities of thewave paddle and that of adherent water particles，the velocities are extrapolated at the interface withoutthe distinction betw een the wave paddle and fluid．The boun dary condition on the wave—paddles isobtainedthrough：划 (刈)，on厂2， (3)where k represents the series number of wave paddles.

The boundary condition in Eq．(3)for the generation of a solitary wave Can be derived from theBoussinesq equation expressed as：u( ，)= ．sech2 J, 3~ ~3C(t-to)-xcos0 l； (4)C=~／g(h+H)， (5)where H is the wave height；g，C，h，and 0 are the gravitational acceleration，wave celerity，still waterdepth，an d the angle of incident wave，respectively；te is a characteristics time scale，which is defined asha】fthe time 0fthe stroke.

2．2 Algorithm and Diferencing SchemeThe boundary value problem for the velocity potential was solved by the boun dary integralequation based on Green’s second identity；the velocity potential within the region was solved by using674 Wen-KaiWENG eta1
． ／ChinaOceanEng．，27(5)，2013，671—682the velocity potential ( ， ，4z；t)and its normal derivative a ( ， ， ；f)／锄 on thex,yz；tH{ G 蝣，) (6)I 1 inside the fluid domaina(z)={1／2 on the smooth boundary1 0 outside the fluid domainG( ， ) 1 (7)= ( )= -1
2Or
； (8)= ( ，Y，z)； (9)= ( ， ， )； (10)r=I — I=√( 一 ) +( 一 ) +( 一z) ． (1)Linear quadrilateral element was adopted in this study,which was defined by its four comerpoints，and the location of each node within the element would take a process of the conformalmapping from the Cartesian coordinates ，Y，z)to the local coordinates( ， ，7)，as shown in Fig．2.

， ，and 77 are the three directions corresponding to local coordinates五Y，and Z． Thus，the velocity( ， ， ，f)=耋 ( ， ， ，f)， (12)： (1一 1一 = (1+ 1一= (1+ 1+ =丢(1—43(1+ (13)2·3 Integral Formulation and DiscretizationThe domain of a 3D model was bounded by six diferent boundaries；therefore，Eq．(6)cal berearranged and written as：aa~(x，Y，z， )+∑L ( ， ， ，t)TdA=∑『r。 ( ， ， ，t)qdA． (14)The integral representation of the solution for the Green function may be written for thediscretized boundaries， =1—6)，and linear quadrilateral element as：where脆 一 f WENG et a1．／China Ocean Eng．，27(5)，2013，671—682 675( )， ，f)+∑6 NP
一
4 h ( ， ， ，f)： 壹彰 ，( ， ， ，f) (15) ，z 一 ∑ 【l
= l ，；l s=l P 1， 1 lh S=
8
1 [1fI 1： Or．Gl dg = 1 flfl lIGI d 7= 1_4)；(s=l-4).

．，(16)(17)Fig．2．Definition sketch of conformal mapping process from triangular unit to dimensionless quadrate。l。ment·The above equations can be writen in a dissolution form and are contingent on domain integralfrom utilizing a Gauss integration to proceed with the calculations．And the equations will be discussedunder two circumstances，i．e．when and f ：wheniCj：= 一 喜耋 专 IGI8n On 智 吒 。
喜砉 1 Iwhere= 而 (s=l-4)；：1_4)，( =1-4)；(18)(19)(20)g，=0y Oz
一 妻苗 =毒嵩一妻高 ：妻 一妻岛； c2 f一一 一 p? 一．=一 一。 、‘1， a7 aa7’6 a a7 a a7’6。 aa a ’
On = ㈡ + f~,鱼OnJ1j， (2) 锄 ，where represents the distance betw een the source i and the evaluated point over integratedsegmentj using Gaussian integration，and and are the weighting functions·when坷 ：As a resultof Or／Ov=0，we acquire：S
= 0 l_4)． (23)676 Wen-KaiWENG eta1．／ChinaOceanEng．，27(5)，2013，671—682The integrals ofEqs．(18)and(19)exist as singularities as i approachesj，in which r correspondsto f，i．e．rilm approaches to zero．The singularities occur when the base point and field point coincideand are not integrable；thus，the regu lar Gaussian does not give accurate results under this circumstanceWe，therefore，should remedy the situation by rearranging the polynomial approximation for g asfollows.

An eigenvalue canbefoundinEq．(22)wheni-j，whichwasprocessedbythefollowingtreatment(Fig．3)，and the four nodes of each quadrilateral element being marked as P1，P2，P3 and e4，were firstsegmented into two triangular elements and expressed as 、P 3 and 、P ～．When the occurrenceof i-j happened on P1．each node within the unit was to be transformed to homogeneous coordinate byconformal mapping process．The trian gular units eventually concluded with the mapping of the abovetriangular unit to dimensionless quadrate element.

PtPFig．3．Unit transformed to homogeneous coordinate by conformal mapping process岛 喜耋 (1一 一 + 1 l I (24)g 2 善耋 (1+ 一 ) IG ； (25)g 3 n n wm(1+专 + ) ，； (26)g 4 喜耋 (1一 +／~,n-饥) IG (27)where= 1_4)； (28)·
=善岛一簧 =善岛一善岛 ’ 簧 一 Oy" Oxgl ， c29 ’g2 ’g3 ’
represents the distance between the source i and the evaluated point( ，r／m)over the integratedsefment，using Gaussian inte~ration.

Wen-KaiWENG eta1．／ChinaOceanEng．，27(5)，2013，671—682 6772·4 Boundary Conditions on Wave PaddlesThe 3D numerical wave basin contains a series of segmented wave generators that occupy onewall of 40 segmented pseudo wavemakers．The oscillating motion of the arrayed wave makers in aph) sical wave basin may be a rather dimcult job for the numerical model due to the discontinuousphase motion between each wave board．This Cal be solved by considering each node as a hinge thatlinks the detached segments(Fig．4)．Waves are generated separately by a series of serpent-type wavegenerators，particularly the possibility of the continuity of the produ【ced wave crest lines．Formulti-directional wave generation，oblique waves catl be generated by setting diferent phases betw eenadjacent segments．A num erical serpentiform motion iS also established by prescribing the velocitiesbased on the fluid particles velocities of water wave theory with a series of piston．type wavegenerators.

Fig·4·Phase motion between each wave board with hingethat links the detached segmented wave boards
.

The boundary condition for the generation of solitary wave can be derived from the Boussinesqequation expressed in Eqs．(4)and(5)．For periodical wave generation，the boundary condition 力onthe series generators is given by：U(x， )= 口( )o-sin(at-kxcos ． f30)For short-crested wave generation，the boundary condition is expressed as：U(x，)= ( ) sin(Gt～XCOS )+ a(A) sin(o-2t-xcos02) (31)and(厂)= sinh(kh)cosh(kh)+kh2sinh (kh) ’ (32)where ， ， h，0 and 0[∽ denote the angular frequency,the incident wave height．the wave num ber,the water depth，the angle of incident wave，and the transfer function
， respectively．In Eq．(3 1)，thefootnotes“1”and ‘‘2”denote the first and second conditions of the two intersecting waves
． Thegenerations of oblique regular waves and short-crested waves wil be a rudimentary debate in thefolowing discussion.

678 Wen-KaiWENG eta1．／ChinaOceanEng．，27(5)，2013，671—6822．5 Parallel Computation oil A PC ClusterTo simulate wave deformation more authentically,meshes were segmented more delicately on thewater surface．The quantity ofnodes on the free water surface requires 121x41 nodes．and further,theinclusion of the other five surfaces makes a total amount of 59 1 4 nodes．As a result，an arrangement forthe ma仃ix of order 5914x5914 was required when calculating．We wrote the square matrix of order5914．The influence matrix must be set up and solved for every time st D since the computationaldomain continuously changes．and the computation requires substantial CPU time and iterative solvers.

Therefore，the modeling of 3D NWT using a single personal computer(PC)is unfeasible，massiveparallel computing systems and PC clusters are used．Efective parallel programming has becomecritical to the development of 3D numerical mode1．The PC cluster is established by parallelconnections of numerous PCs，and in this study,eight PCs were used．The computation was carried outusing message passing interface(MPI)parallel language and MPICH software developed by theAmerican Argonne National Laboratory and Mississippi State University．MPICH is a free．high-performance，and widely portable implementation of MPI．A standard for message-passing betweendistributed-memory applications used in parallel computing was developed and authored by Mathematicsand Computer Science Division(Argone National Laboratory)and Department of Computer Science，University of Illinois at Urbana-Champaign．In computation，iterations were perform ed collaborativelyby diferent PCs simultaneously．The data were transferred to the designated PC to complete thecomputation．Each PC cluster we use comprises eight interconnected dual Pentium 4 CPUsworkstations，with which the parallel matrix factorization algorithm was developed to solve theresulting large matrix and the original inverse matrix was partitioned into eight submatrices for parallelcomputation.

3．Numerical Examples and Discussion3．1 Generation of Oblique W avesThe generation，propagation，and deform ation of a solitary wave in a 3D wave basin with a constantwater depth h were simulated．The size of the basin is 40 mx20 mx0．5 m and 40 mx20 mx 1．0 m，andthe wave height was／4-0．2 m and the time interval was At：t
c
|100．The boundaries，f ～f ，weredivided into Ⅳ_～ discrete elements，where NI=1681，Nz=42，N3-82，N4=82，N5=82 and N6 441.

Although the wave field was produced based on the snake principle，the spatially sinusoidal motion ofa serpent-type wave mak er would produce any desired type of waves propagating obliquely to theplane．However,the finite width of the basin with fuly reflecting sidewals resulted in the wavegenerations influenced by sidewall reflections．The unidirectional solitary wave propagation was firstcarried out for simplicity,and to verify the validity of the numerical scheme.

The variation of the wave height for oblique wave was verifed by the generation of periodicalregular waves．Fig．5 shows the spatial distributions of velocity in x-y plane to the generations ofoblique regular wave in diferent time steps using Eq．(30)．In a multi—directional 3D wave basincOmposed of multi—face serpent—type wave generators with individual controlled paddles，the motion of

Wen-KaiWENGeta1．／ChinaOceanEng．，27(5)，2013，671—682 681amplitude of the solitary wave should be slightly decreased as a result of the fractional energy transferto the dispersive tail.

The conservative property can be crosschecked through the following formulae：M=It(t) ( ’n)d／-"； (33))= g Jr∽ d厂；( )= 1 ∽ · 厂 ；(34)(35)(f)=Ek(f)+E(f)， (36)where M，E，Ek，and denote the total mass of the waveform，the total energy,the kinetic andpotential energy,respectively.

Fig．9 shows the total energy E is approximately equal to 2．73 and remains unchanged when thetime interval At=to／100．The same phenomenon is also found for the kinetic energy Ek and potentialenergy ，but E，Ek，and slightly increase or decrease after t=-2．3 S．Furthermore，the total mass，M rises to 8．8 m。after the solitary wave is formed；nevertheless
， the total mass decreases along with thepropagation of the wave．This may be due to dimensions of time interval and the repetition of thecalculation，because the model was established wjIh a time marching procedure．Therefore，theattenuation of mass may be the cause of numerical errors．Although a parallel computing system wasutilized，it stil took a long time for the calculation to complete．Since the meshes on the water surfacewere more delicately segmented，the segmentation of time spacing At with element size As was likelyto cause large instability and erors of the scheme．Thus，the calculation process of convergence andstability stil1 needs to be improved for better results in the conservation ofmass and energy432lO0．0 O．52O- 11．O 1．5 2．0Time，tc2．O1．61．2O．80．4O．02．5 0．0 0．5 1．O 1．5 2．0 2．5Time，tcO．O O．5 1．O 1．5 2．OTime，to16l284O2．5 O．O 0．5 1．O 1．5 2．O 2．5Tim e，tcFig．9．Total mass，energy，potential energy and kinetic energy with H／h：0．05， =1 m，and△户td1004．ConclusionThe numerical simulation of 3D wave making problem of serpent—type wave generators to682 Wen-KaiWENG eta1
． ／ChinaOceanEng．，27(5)，2013，671—682generate solitary waves with fully nonlinear water surface condition in the basin are presented．Theunidirectional solitary wave propagation was first carried out to verify the validity of the numericalscheme．This study provided proof to the validity of Boussinesq equations for the generation of a 3Dsolitary wave．A solitary wave accompanied by a co-propagating wave in its tail was detected．Thisstudy also shows：the amplitude of the CO—propagating oscillatory tail trailing behind the wave isrelated to the amplitude of the solitary wave；the oscilation of the preceding trailing wave was on aconspicuous increase along with the enlargement of H／h；numerical errors—caused side effects Can bereduced by the retrenchment of the time interval；a reduction of time step interval could improve theaccuracy and stability．The method developed in this paper can be extended to oblique wavegeneration.

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