Numerical study of long wave runup on a conical island

D uring tho Flores Island evcnt, tw o villages locatcd on the Southern sid c of the circular Babi Island, w hose d iam eter is approxim atcly 2 km, w cre vvashcd aw ay by the tsunam i at

VN U J o u rn a l of Sciencc, L arth Scicnccs 24 (2008) 79-86

Numerical study of long wave runup on a conical island

P h u n g D a n g H ieư*

Ccntcrfor Marinc and Occan-Atmosphcrc Intcraction Research

R cccivcd 5 Jn n u ary 2008; rccoivcd in rev ised form 10 July 2008

A b stra ct A n u m eric aỉ model b ascd on thc 2D shallovv w a tc r e q u a tio n s was d c v c lo p c d u sin g the

F inite V olum c M cth o d T h e m odcl w as ap p lic d to th e stu d y o f long vvave p ro p a g a tio n an d ru n u p

o n a co n ical is ỉa n d The s im u la tc d r c s u lts b y th c m o d c l w c rc c o m p a r c d w ith p u b lis h e d

ex p c rim cn tal d a ta a n d an n ly zc d to u n d c rsta n d m o rc about th e in tcractio n p roccsscs bctwc?en the

long vvavcs a n d conical isla n d in tc rm s of w a tc r p ro íile and w a v c ru n u p hcight 'ITic rc su lts o f the

stu d y c o n íirm c d th c cffccts of c d g c vvaves o n thc r u n u p h c ig h t at th c lcc sid e of th c island.

Kcyiuords: C onical island; R u n u p ; rin ite volum e m e th o d ; S haỉlow w a tc r m odcl.

1 Introduction

Sim ulation of tvvo-dim ensional cvolution

and ru n u p of long w av es on a sloping beach

is a classical problcm of hydrodynam ics It is

u su ally related vvith th e c alc u la tio n of Coastal

effects Oi long w av es such as tido and

tsunam i M any research ers h av e contributcd

signiíicantly c íío rts to th e d cv elo p m en t of

m odels capable of so lving the problcm

Notable studies can bc mentioncd Shuto and

Goto (1978) developed a num erical m odcl

based on íinite difference m cth o d (FDM) on a

staggered schem e |9j H ibbcrt an d Peregrine

(1979) [2] proposed a m odel solving the

shallovv vvater eq u atio n in the conservation

form using the Lax-W ondroff schem e and

allovving for possiblc calculation of w ave

brcaking Hovvever, thcir m odcl had not bcen

capable to calculate vvave ru n u p an d obtain

* Tel.: 84-914365198.

E-mail: phungdanghieu@vkttv.cdu.vn

physically realistic solutions Subsoquontly,

Kobayashi et al (1987, 1989, 1990, 1992) [3, 4,

5, 6] re íin e d th e m o d e l for p ractical USG, by

a d d in g dissipation term s in the íinite- differencc equations, w h at is novv the m ost

p o p u la r m ethod for solving the shallovv

w atcr oquations Liu et al (1995) [7] m odeled the ru n u p of solitary w av e on a d rc u la r island by FDM Titov an d Synolakis (1995, 1998) [11, 12] proposed m odels to calculate long w av e ru n u p on a sloping bcach and cừcular island using FDM Wei Gt al (2006) [13] developed a m odcl based on thc shallovv

w atcr eq u atio n s u sin g tho íinite volum e

m ethod to sim ulate solitary vvaves ru n u p on

a sloping bcach and a cừcular island Sim ulated rcsu lts obtaincd by VVei et al agreed notably vvith laboratory experimcntal data [13]

M em orablc tsu n am i in Indonesia and Japan caused m illions of dollars in dam ages

an d killed th o u sa n d s of peoplc O n Decem ber

12, 1992, a 7.5-m agnitudc earth q u ak e off

79

80 Phung Dang ỉ ỉieu / VN U Ịounial ọ f Science, r.artli Sciences 24 (200S) 79-86

Flores Island, Indonesia, killed nearly 2500

people and w ash cd aw ay e n tữ c villages

(Briggs et al., 1995) Ị1Ị O n JuUy 12, 1993, a

7.8-m agnitude earth q u ak e oíf O kushiri

Island, Japan, triggcrod a d cv astatin g tsunam i

w ith rGCorded ru n u p as h ig h as 30 m This

tsunam i rcsulted ứì larger p ro p crty dam age

than any 1992 tsunam is, and it com pletely

in undatcd an village w ith overland flow

Estừnatcd propcrty d am ag e w as 600 million

u s dollars Rccently, the h ap p en cd at

Dccember 26, 2004 S um atra-A ndam an

tsunam i-earthquake in the Indian Ocoan vvith

9.3-m agnitude a n d an opicenter off thc wcst

coast oí Sum atra, Indonesia had killed m ore

than 225,000 pcople in cleven countrics and

resulted in m ore th an 1,100,000 people

hom cless Inundation of Coastal areas w as

created by vvavcs up to 30 m etcrs in height

TTiis was tho ninth-deadliest natural disastor in

m odorn history Indonesia, Sri Lanka, India,

Thailand, and M yanm ar w cre hardost hit

Field survcys of tsu n am i dam agc on both

unexpcctcdly Iarge ru n u p heights, especially

on tho back or leo sido of the islands,

rcspcctively to the incidont tsunam i dừection

D uring tho Flores Island evcnt, tw o villages

locatcd on the Southern sid c of the circular

Babi Island, w hose d iam eter is approxim atcly

2 km, w cre vvashcd aw ay by the tsunam i

attacking from thc north Sim ilar p henom cna

occurrcd on th e pear-sh ap ed O kushiri Island,

w hich is approxim ately 20 km long and 10

km w ide (Liu ct al., 1995) [7Ị

In this stu d y , the interaction of long

vvaves and a conical island is investigated

using a num orical m odol bascd on the

shallovv w ater cq u atio n an d finite volum e

m ethod The stu d y is to sim ulatc tho

processes of vvave p ropagation and ru n u p on

thc island in o rd cr to u n d e rsta n d m ore thc

ru n u p p h en o m en a on conical islands

S u p p o rtin g to the sim u lated rcsults by tho

m odel, the cxperim ental d ata p ro p o se d by Briggs el al (1995) Ị1 ] vvere used

2 N u m erical m odel

2.7 Govcrning cqaation

T h e present stu d y conbiders tvvo- dim cnsional (2D) d c p th -in tc g ra tc d shallovv

w ator equations in tho C artesian coordinate

system ( x , y ) T he conservation form of the

non-linear shallow w atcr e q u a tio n s is vvritten

as [13]:

-— = s

w h ere u is the vcctor of conscrvcd variablos;

F, G is the flux vectors, respcctively, in the

X an d 1/ directions; an d s is tho source term Tho explicit form «f thoso voctors is cxplaincd

as follows:

u = H u F = H u 2 + ị g H 2

H v H u v

G =

H v

H u v

H v 2 + ị g H :

s =

0

g H Ệ - - d\j p

(2)

vvhere g : gravitational acceleration; p : vvater density; h : still vvater d c p th ; H : total vvater dopth, H = h + T| in w hich ii(.v,Ị/,f) is the

displaccm ent of w a ter su ríacc from tho still

w atcr level; Tx , T y: bottom shcar stross givon by

Tx = p C f U y Ị i r + V 2 ,

Ty = ọ C f V y j u 2 + v 2 , Cf = S"

(3)

H1 /3

w h cre n : M anning coc'fficient for tho suríacu

roughness

Phung Dang l licu / VNU lơunial o f Scicncc, i.nrtlĩ Sciciĩces 24 (2008) 79-S6 81

2.2 Numcrical schcmc

The fmito v o lu m c íorm ulation iinposes

conservation law s in a control volum e

Intcgration of Eq (1) ovcr a cell w ith thc

application of the GrceiVs theorem , gives:

í n f r d Q + ỉ r (F” * + G "* ) í í r = í o s d n ' (4)

vvhcre Q : ccll d o m a in ; f : b o u n d ary of Q ;

{ n x*n ỵ ) : n o rm al outvvard vector of the

boundary

T aking ti m e in tcg ratio n of Eq (4) over

d u ration At írom t-ị to t2, w c h a v e

J u (x,y, f2 )dQ - u (.Y, \J,í, )dn

+J*J (Fỉix + G n v)d r = ỊdtỊ Sdn

Tho prcscnt m odel usos un iío rm cells

vvith dim cnsion A.V, Ai/, thus, tho integrated

governing c q u a tio n s (5) \vith a tim o step At

can be ap p ro x im ate d vvith a h alí tim e stcp

avcragc for tho in teríac c Auxos and sourco

torm to bocomo:

At_

Ax r->kf\j2 "1 Ạ i^A.tl/2

— J Af Í5i./

vvhere i, ị are in d iccs at tho ccll center; k

denotes the c u rre n t tim e stcp; the half indices

í + 1 /2 , í - 1 /2 a n d ý + 1 / 2 , / - 1 / 2 indicate

tho cell intcríaces; and Ả:+ 1 /2 denotes tho

average w ithin a tim o stcp betvvoen k and

k + 1 Note that, in Eq (6) the variables u

and source term s arc cell-avcragcd valuos

(vve use this m c a n in g from novv on)

To solve Eq (6), w e n eed to estim ate thc

num erical íluxcs ĩ ^ / ỉ ị , ĩ ị - v ỉ ị an d G ^ l y 2f

at thc cell in teríaces In this study, vve

use the G odunov-typo schem e for this purposo

According to the G o d u n o v -ty p c scherrte, tho

num crical íluxcs at a cell intcríaco could be

*V+1/2,/ 1 /2./ 7

Ay (6)

obtained by solving a local Riem ann problem

at tho interíacc

Since đircct solutions are not available for

tw o or threc dim cnsional Ricm ann problcm s, thc present m odel uscs the sccond-order splitting schem e of Strang (1968) [10] to sep arate Eq (6) into tvvo one-dim cnsionai equations, vvhich are intcgrated scquontially as:

vvhere X and y den o te the intcgration operators in the X an d y dừections, respectively Tho cquation in the V direction

is íirst in tcg rated over a half tim c stcp and this is íollovved by integration of a full tim e

step in the y direction These arc expressed as:

Ĩ ] [ k + ì f 2 ) r ỊẢ r » 1 4 p i ♦ 1/ 4 "I

(8)

*T<S.C*-l/4

TT^I) = ii*+i/2) u / - u / M - r *’.1'2 1

W./+1/2 " ^./-1/2

+*(SJ,)Í.?'2

w here the astcrisk (*) indicatcs partial solutions at the respective tim e increm ents

vvithin a tim e stop and Sx , S v arc the source

term s in the X dừ ection and 1/ directions lntcgration in thc X dừ ection ovcr the rem ain in g half tim c stop advanccs the solution to the next tim e step:

TT^+l I ĩ(fc+l) ^ í ~ ♦ 3/ 4 xjỉ:+3/4 ~1

u «./ “ Ui./ 9Av|_rí-l/2.i »1/2./J

(10)

+ ^ < S , ) Ĩ T2 ' -'x ) i.ja + 3 / 4

T h e partial solutions ư ^ , U jy i/2) and u< ^> , p rovidc the interíace flux torm s ừi equations (8), (9) and (10) through a Ricmann solver in onc-dim ensional problem s In this study, w e usc the HLL approxim ate Riemann solvcr for the estim ation of num erical íluxes For tho w et and dry cell treatm ent, w c use the

82 Phung Dmig llieu / VNU Ịounínl o f Sàence, Enrth Sciences 24 (2008) 79-86

m inim um w et depth, the cell is assum ed to be

dry if its w ater d ep th loss th an the m inim um

vvct depth (in this stu d y vvc choose m inim um

vvct dcpth of 10'5m)

3 S im u latio n resu lts an d d iscu ssio n

3.1 Expcrimeiĩtal condition

A num erical expcrim ent is carried out for

the condition sim ilar to the experim ent done

by Briggs et al (1995) [1] In this expcrim ent,

there vvas a conical island Setup in a w ave

basin having th e dim cnsion of 30 m vvide and

25 m long Tho conical island has thc shape of

a truncated cone w ith diam etors of 7.2 m at

the base and 2.2 m at tho crest The island is

0.625 m high and has a side slope of 1:4 The

suríacc of the island and basin has a sm ooth

concrete íinish Thoro is absorbing m aterials

placcd at tho four sidcw alls to reduce w ave

roílection T h e vvater d e p th is h=0.32 m A

solitary vvave w ith tho hcight of A / h = 0.2

was gcncratGd for tho exporimental observation

Fig 1 shovvs the sketch of the cxperim ent and

w avc gauge location for w ater suríace

m casurcm ent Five tim e-sories data of w atcr

suríace elevation vvere collccted for the

com parison

Dr =2-2m

h' 0625m

- = 0.2

h

/ i- 0 3 2 m

- !■ _ ■

D b =7.2m

B * 30rt»

C:t

• G1 L=25m —

[n Fig 1, the vvave gauge GI is Setup for the m casu rem en t of the incident vvaves; w av e gauges G6 and G9 are for thc vvaves in the shoaling area; and thc vvave gauges G I 6 an d G22 are respcctively, for vvaves on th e right side and lee sidc of the island T he locations

of th e five w a v c gaugcs are given in Tablc 1

in relation vvith tho center of the island

T ab le 1 L ocation of w a v e g a u g e s

G a u g e n u m x - x c (m ) \ j - y c (m)

( x ( , y c ): c o o rd in a te o f th c cen ter o f th e island

3.2 Numerical simulation and discussion

In thc num erical sim ulation, a com putation

d o m ain is sctu p sim ilar to the expcrim ent The m csh is regular vvith grid size of 0.1 m in both X an d \J directions At four sides of th e

co m p u tatio n dom ain, radiation b o u n d ary conditions arc u sed in o rdcr to allovv w avcs

to go íreely th ro u g h tho sidc bo u n d ary A solitary vvave is g cn cratcd as the initial condition at a line parallcl w ith the \J direction, and located at the distance of 12.96 m from the center of thc island The M anning cocffíciont

is sct to be constant n= 0.016 Tho initial

solitary vvave is created by using the íollovving equation:

T|(.r) = i4sech2

(12)

Fig 1 Sketch of the experinìgnt.

whcrc x s is thc ccntcr of the solitary wavc.

The num erical rcsults of w ator suríace elevation at fivo w ave-gauge locations and

ru n u p h eig h t on the island arc recordod for

Phung Dang Hicu / VN U Ịoumal ofScicìĩce, Earth Scicìices 24 (2008) 79-86 83

validation of the sim ulation Fig 2a show s the

tim e proíile of vvater suríaco elev atio n at tho

vvave gaugc G l In this íigure, it is sccn that

the incident solitary w av e sim u lated by thc

m odel agrees very well w ith th e experim ental

data This gives us a coníidence in com parison

of tim e series of vvater suríace elevation at other

locations in tho com putation dom ain, as vvell

as in com parison of vvavc ru n u p on thc island

In the Fig 2b a n d 2c, at th e w av c gaugcs

G6 and G9, it is seen th at thc solitary vvavc is wcll sim ulated on the shoaling rcgion, the

w ave com es to the location aítcr about 4 soconds ừ o m thc initial timc At íirst, thc num crical results an d expcrim ental data agree vcry w cll/ aíter that, there are som e discrepancy appcared This deílection can bc explaincd d u e to thc reílection from tho side

bo u n d arics in thc experim ent donc by Briggs

ct al, m u ch largcr than that in thc sim ulation

Time (sec)

Time (sec)

Time (sec)

Fig 2 C o m p a riso n of vvater s u ría c e e lc v a tio n at locations G l, G6, G9: so lỉd th in line: sim u la te d by com nion shallow w a te r c q u a tio n ; so lid th ick linc: s im u la te d by a d d in g B ơussincsq te rm to the shallovv w a tc r cquation.

84 Phung Dang Hieu / VNU Ịơunwỉ o f Science, Earth Scioĩces 24.(2008) 79-86

Time (sec)

Time (sec)

Fig 3 C o m p ariso n of w a tc r su ría c e elevation at locations G16 a n d G22: so lid th in linc: sim u la te d by coĩìim on shallow vvator cq u atio n ; solid thick linc: sim ulatecỉ by a d d in g B oussinesq tc rm to th e shallovv w a tc r equation.

It can bc coníirm cd írom thc íigure that,

the numorical rcsults very soon becom c stablc

having non-fluctuation w h en tho w ave goes

írecly out of the experim cnt dom ain

Inversely, the expcrim cntal data havc a long

tail of d isturbance and could not be calm aíter

20s (see Fig 2j at w avo gaugos G6 and G9;

and Fig 3, at vvave gaugos G16 and G22) This

Auctuation is d u e to the vvave encrgy

dissipation not enough at the sides of tho

experim ent basin Hovvever, the form and

height of thc arriving solitary w avc at all

locations are vvell m atched bctw ecn

experừnental and num erical results This is

very im portant to allovv latcr com parison of

w ave ru n u p on tho island

From Fig 2 and Fig 3, it is also seen that,

the w ave hcight at the lee side (gaugc G22,

Fig 3b) of th e island is still very high in

com parison w ith tho height at thc íront side

(gaugc G6, G9, Fig 2b, 2c) of the island, and

m uch biggcr th an that at the right side (gauge

G16, Fig 3a) of the island T hese results give

us a confidcncc in co níirm ing thai tho vvave height at lce sid e of an circular island can be large also In Fig 2 and Fig 3, tvvo sots of num crical results are plottcd O ne is sim ulated by the com m on non-linear shallow vvatcr equation (NSW ), and thc othcr is simulatGd by a d d in g the Boussinosq dispersion term [8] into thc NSW From the íiguros, it is co n íirm cd that tho m odcl using the Boussinesq app ro x im atio n can givc sim ulated rcsults m u ch b etter than th c com m on NSW bascd m odel T hus, for thc practical p u rp o se of sim ulation non-lincar long w av c problem , tho Boussinesq

ap proxim ation torm s should bc considorcd Fig 4 show s th e sn ap sh o t of vvatcr suríacc displaccment on tho computation domain From the íigure, w e can sec that, aítor tho solitary

w av c comcs to the island, thc w av c roíraction

ap p cars d u e to the variation of w ater depth

B chlnd the island, the cdgo vvavcs comc from

tw o sides of the islan d d u e to w avcs bcn d in g

a ro u n d the island an d m atching togcther at

Phuììg Dang iìicu / VNU Ịounuìỉ ọ f Science, Larth S c ì c ỉ ì c c s 24 (2008) 79-86 85

tho leeside of the island Then, they íorm an

aroa of very high w ave ru sh in g up to thc lcc

side coast of the island This m echanism can

bc cxplainod for tho unexpectedly largc

ru n u p heights on thứ leosidtí of tho Babi and

O kushiri Islands d u c to the tsunam i

Fig 5 is tho com parison of w avc ru n u p

aro u n d the island, betvveen num crical

sim ulation and experim ent The horixontal

axis in tho íiguro indicatcs tho anglc betvveen

the line drasving from thc conter of the island

to the point of ru n u p m casu rem cn t and tho \J

diroction Thi} anglo of 0 degree m cans th at

the m easuring point is at tho right side of tho

island and on Ihc line through tho centcr of

the island and norm al to the incident w ave

diroction (i.e parallol to tho \J direction) It is

shovvn from tho íigurc that, thc ru n u p is

highcst at the ío resid e of thc island, tho

m axim um sim ulatcd ru n u p hcight is somevvhat less than oxperim ontal data At the lecside of the island, thorc is an area vvith

ru n u p h ig h cr than both sides of the island The num erical results of ru n u p hcight in this area are also sm aller th an cxperim ental data These m ight be d u c to the fact that tho com putational m esh not fine en ough to capture highly non-linear interactions of edge

w aves at the Ieeside In overall, tho num crical

m odcl can sim ulatc woll tho ru n u p height at

m any locations aro u n d tho island Especially, thc tendcncy of tho ru n u p variation and

ru n u p location are well sim ulatcd by the prescnt num erical m odcl This m eans that, the model dovelopcd in this study has potential íeatures to apply to the study of practical problem s rolatcd w ith long w aves, such as

in u n d atio n of tsu n am i on Coastal areas

Fig 4 S n ap sh o ts o f the w a tc r su ríac c d isp lace m e n t d u c to the so lita ry w ave.

Angle (deg)

Fig 5 R unup of w a te r a ro u n d th e islan d d u c to th e solitary w av e (270 deg.: at ío rc sid c in thc n o rm al directio n of w a v e p ro p a g atio n ; 90 dcg.: at thc leeside of the island; 0 dcg.: at th c rig h t sid c o f th c island;

an d 180 dcg.: at thc left sid c of th e island).

86 Phung D ang Hicu / VNU Ịountal ọ f Sôence, Larth Scieììces 24 (2008) 79-86

4 Conclusions

A 2D num erical m odel based on thc

shallow w ater cq u atio n has been successíully

developed for tho sim ulation of long vvave

propagation, d eío rm atio n and ru n u p on the

conical island The num erical results

sim ulated by NSW m odel and by Boussinesq

m odel revealed that by a d d in g Boussinesq

tcrm s to the NSW m odel, sim ulated results oí

long w ave pro p ag atio n and d eíorm ation can

bc signiíicantly im proved Thercíore, it is

approxim ation should be considered in a

practical p roblcm related w ith long vvaves

Tho m odcl also has potcntial íeatures to

apply to the study of practical problem s

related to long vvaves, such as inundation of

tsunam i on Coastal arcas

Simulated rosults in this study also

contìrm that tho arca bchind an island can be

attacked by big w aves Corning from the

opposite sidc of the island d u e to non-linear

intcraction of edge vvaves rcsulted from

rcíraction processes

Acknowledgments

This p ap er w as com pleted w ithin thc

fram ew ork of Fundam cntal Research Project

304006 íu n d c d by V ietnam M inistry of

Science and Technology

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[9] N S h u to , c G oto, N um orical sim u latio n t>f

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21 (1978) 13.

[10] G S tran g , O n th c co n stru c tio n an d c o m p a riso n

of d iffc rcn c c schem es, SI A M (Soc Int A p p l

M ath.) Ịounuỉl o f Numcrícnỉ Aĩiaỉỵsis 5 (1968) 506.

[11] v v T ito v , C.E Synolakis, M o d o lin g o í b rc ak in g and n o n -b re a k in g long-vvavc cvolution a n d

ru n u p u s in g VTCS-2, louninỉ o f Waterwaì/, ỉyort,

Coastal ntid Occmi r.ĩiỊỊÌncerÌHg 121 (1995) 308.

[12] v v T ỉtov, C.E S ynolakis, N u m ericaỉ m o d elin g

of tid al vvave ru n u p , Ịoum aỉ o f Watenoa\f, Port,

Coastal (ì)ĩd Occtììì Eìiginecriìíg 124 (1998) 157.

[13] Y W ci/ x z M ao, K.p C h cu n g , VVcll-balanccd

íinỉtc-volumc model for long-wavc runup

Ịountnỉ o f Wntcrwn\f, Port, Coastal rnd Occaìì Eìỉgbìccrhĩg 132 (2(X)6) 114.