VNU Jo u rn a l of Sciencc, Larth Scicnccs 24 (2008) 79-86 Numerical study of long wave runup on a conical island P h u n g D a n g Hieư* Ccntcrfor Marinc and Occan-Atmosphcrc Intcraction Research R cccivcd Jnnuary 2008; rccoivcd in revised form 10 July 2008 A b stract A n u m ericaỉ model bascd on thc 2D shallovv w a tc r eq u a tio n s was dcv clo p cd u sin g the F inite V olum c M cthod T h e m odcl w as ap p licd to th e stu d y o f long vvave p ro p a g atio n and ru n u p o n a conical isỉa n d The s im u la tc d rc su lts b y th c m o d c l w c rc c o m p a rc d w ith p u b lis h e d ex p crim cn tal d a ta a n d annlyzcd to u n d c rsta n d m orc about th e intcraction proccsscs bctwc?en the long vvavcs a n d conical islan d in tcrm s of w atcr p ro íile and w a v c ru n u p hcight 'ITic rc su lts o f the stu d y co n íirm cd thc cffccts of c d g c vvaves o n thc ru n u p h cig h t at thc lcc sid e of thc island Kcyiuords: C onical island; R unup; rin ite volum e m eth o d ; S haỉlow w a tc r m odcl physically realistic solutions Subsoquontly, Introduction Kobayashi et al (1987, 1989, 1990, 1992) [3, 4, 5, 6] re íin e d th e m o d el for practical USG, by adding dissipation term s in the íinitedifferencc equations, w hat is novv the most pop u lar m ethod for solving the shallovv w atcr oquations Liu et al (1995) [7] m odeled the ru n u p of solitary w ave on a drcu lar island by FDM Titov and Synolakis (1995, 1998) [11, 12] proposed m odels to calculate long w ave 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 equations using 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 rcsults obtaincd by VVei et al agreed notably vvith laboratory experimcntal data [13] M em orablc tsunam i in Indonesia and Japan caused millions of dollars in dam ages Sim ulation of tvvo-dimensional cvolution and ru n u p of long w aves on a sloping beach is a classical problcm of hydrodynam ics It is u su ally related vvith th e calculation of Coastal effects Oi long w aves such as tido and tsunam i M any researchers have contributcd signiíicantly cíío rts to th e dcvelopm ent of m odels capable of solving 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 and Peregrine (1979) [2] proposed a m odel solving the shallovv vvater equation 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 and obtain an d killed thousands of peoplc O n December 12, 1992, a 7.5-m agnitudc earthquake off * Tel.: 84-914365198 E-mail: phungdanghieu@vkttv.cdu.vn 79 80 Phung Dang ỉ ỉieu / VNU Ịounial ọ f Science, r.artli Sciences 24 (200S) 79-86 Flores Island, Indonesia, killed nearly 2500 people and w ashcd aw ay en tữ c villages (Briggs et al., 1995) Ị1Ị O n JuUy 12, 1993, a 7.8-m agnitude earthquake oíf O kushiri Island, Japan, triggcrod a dcvastating tsunam i w ith rGCorded ru n u p as h ig h as 30 m This tsunam i rcsulted ứì larger propcrty dam age than any 1992 tsunam is, and it completely inundatcd an village w ith overland flow Estừnatcd propcrty dam age w as 600 million u s dollars Rccently, the h appencd at Dccember 26, 2004 Sum atra-A ndam an tsunam i-earthquake in the Indian Ocoan vvith 9.3-m agnitude an 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 than 1,100,000 people homcless Inundation of Coastal areas was 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 tsunam i dam agc on both Babi and O kushiri Islands show ed unexpcctcdly Iarge ru n u p heights, especially on tho back or leo sido of the islands, rcspcctively to the incidont tsunam i dừection During tho Flores Island evcnt, tw o villages locatcd on the Southern sidc of the circular Babi Island, w hose diam eter is approxim atcly km, w cre vvashcd aw ay by the tsunam i attacking from thc north Similar phenom cna occurrcd on th e pear-shaped O kushiri Island, which is approxim ately 20 km long and 10 km w ide (Liu ct al., 1995) [7Ị In this stud y , the interaction of long vvaves and a conical island is investigated using a num orical m odol bascd on the shallovv w ater cquation and finite volum e m ethod The study is to sim ulatc tho processes of vvave propagation and ru n u p on thc island in o rd cr to u n d erstan d m ore thc runu p phenom ena on conical islands S upporting to the sim ulated rcsults by tho m odel, the cxperim ental d ata pro p o sed by Briggs el al (1995) Ị1 ] vvere used N um erical m odel 2.7 Govcrning cqaation T he present stu d y conbiders tvvodim cnsional (2D) d cpth-intcgratcd shallovv w ator equations in tho C artesian coordinate system ( x , y ) The conservation form of the non-linear shallow w atcr eq u atio n s is vvritten as [13]: ỔF dG „ -— =s 0) dt dx dij w here u is the vcctor of conscrvcd variablos; F, G is the flux vectors, respcctively, in the X and 1/ directions; and s is tho source term Tho explicit form «f thoso voctors is cxplaincd as follows: ■H ■ u= Hu Hu F= Hu2 + ịg H Huv Hv (2 ) Hv G = Huv s= * Õx p Hv2 + ị g H : g Hd\jỆ - pvvhere g : gravitational acceleration; p : vvater density; h : still vvater dcpth; H : total vvater dopth, H = h + T| in w hich ii(.v,Ị/,f) is the displaccm ent of w ater suríacc from tho still w atcr level; Tx , T y : bottom shcar stross givon by Tx = p C f U y Ị i r + V , (3) Ty = ọ C f V y j u + v 2, Cf = S"1 /3 H w hcre n : M anning coc'fficient for tho suríacu roughness 81 Phung Dang l licu / VNU lơunial of Scicncc, i.nrtlĩ Sciciĩces 24 (2008) 79-S6 2.2 Numcrical schcmc The fmito v olum c íorm ulation iinposes conservation law s in a control volume 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 ain ; f : boundary of Q ; { n x*n ỵ ) : norm al outvvard vector of the boundary Taking ti m e intcgration of Eq (4) over duration At írom t-ị to t2, w c h a v e J u (x,y, f2)dQ h u (.Y,\J,í, )dn h +J*J (Fỉix + G n v)d r =ỊdtỊ Sdn (5) h «1 Tho prcscnt m odel usos uniíorm cells vvith dim cnsion A.V, Ai/, thus, tho integrated governing cq u atio n s (5) \vith a timo step At can be approxim ated vvith a halí tim e stcp avcragc for tho in teríacc Auxos and sourco torm to bocomo: At_rc**i/2 + *V+1/2,/ 1/2./ Ax Ay (6) r->kf\j2 "1 Ại^A.tl/2 — J Af Í 5i./ vvhere i, ị are indiccs at tho ccll center; k denotes the cu rren t tim e stcp; the half indices í + /2 , í - /2 an d ý + /2 , / - / indicate tho cell intcríaces; and Ả:+ /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 can in g from novv on) To solve Eq (6), w e need to estim ate thc num erical íluxcs ĩ ^ / ỉ ị , ĩ ị - v ỉ ị and G ^ l y 2f at thc cell interíaces In this study, vve use the G odunov-typo scheme for this purposo According to the G odunov-typc scherrte, tho num crical íluxcs at a cell intcríaco could be obtained by solving a local Riemann problem at tho interíacc Since đircct solutions are not available for two or threc dim cnsional Ricm ann problcms, thc present m odel uscs the sccond-order splitting schem e of Strang (1968) [10] to separate Eq (6) into tvvo one-dim cnsionai equations, vvhich are intcgrated scquontially as: u*}1 = XAI/2YA'X AÍ/2U*/ (7) vvhere X and y denote the intcgration operators in the X and y dừections, respectively Tho cquation in the V direction is íirst intcgrated 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+ìf2) r ỊẢ r »14 p i ♦1/ "I (8) *-l/4 *T