Proc. Int. Symp. on Fluval and Coastal Dsasters -Copng wth Extreme Events and Regonal Dversty-, Dsaster Preventon Research Insttute, Kyoto Unv. (2005), pp. 221-228 3D numercal smulaton of tsunam runup onto a complex beach T. Kaknuma Tsunam Research Center, Port and Arport Research Insttute, Kanagawa, Japan ABSTRACT: Two 3D numercal models are appled to tsunam phenomena ncludng runup onto a complex beach. Ther governng equatons are the contnuty and Reynolds-averaged Naver-Stokes equatons for ncompressble fluds n porous meda. In the frst model water surface dsplacement s determned by the vertcally ntegrated equaton of contnuty, whle n the second by the 3D-VOF method. Seabed topography can be smoothly expressed wth the porous model. These two models reproduce the exstng hydraulc experment, whch treated the 1993 Hokkado Nanse-Ok earthquake tsunam n Okushr Island, where the extreme runup of about 32 m was dscovered. The calculaton results of water surface dsplacement generally show correspondence wth the expermental data at off-peak tmes of water level when no breakng wave exsts. The results calculated by these 3D models are also compared wth those by the 2D long-wave model. The hghest runup obtaned by the VOF model ndcates the full-scale value of about 30.6 m. 1 INTRODUCTION When a submarne earthquake or a land slde occurs, the effects appear n varous ways. Especally seawater threatens humanknd near seashore as a tsunam. In coastal zones, the seawater flow shows complcated moton because of affectors to tsunams, e.g. bottom confguraton n both land and sea areas, rver flow, astronomcal tde, densty stratfcaton, etc. Around nearshore structures, flud moton due to tsunams devates from smple flow, resultng n these structures under three-dmensonally dstrbuted forces, whose mechansm should be known because t concerns weakness or nverson of the structures. We cannot evaluate, however, such a 3D dstrbuton usng the tradtonal nonlnear long-wave models. Moreover, from offshore to coastal zones, phenomena, whch have varous scales on space and tme, should be solved effcently and economcally. For these purposes a hybrd model whch conssts of 3D models, multlevel models and connecton models has been developed by Kaknuma & Tomta (2005). The governng equatons are the contnuty and Reynolds-averaged Naver-Stokes equatons for ncompressble fluds wth a porous model descrbng seabed topography smoothly. The numercal schemes are the fnte dfference methods wth orthogonal grd systems. The multlevel model s used for wde-area calculaton as shown n Fgure 1. We assume hydrostatc pressure n these areas, to whch a fault model and a typhoon model are ntroduced. The connecton model s appled to overlap regons for smooth connecton between the multlevel model and the 3D model, where we solve pressure wthout assumpton of hydrostatc pressure. By local applcaton of ths 3D model to narrower areas surrounded by multlevel areas, we can represent 3D characterstcs of flow effcently and economcally around structures, over steep topography and so on. For dsaster preventon, many thngs are requred to be stocked, e.g. food, faclty, commodty, knowledge, nformaton, atttude of mnd, etc. Ths numercal calculaton model s called as Storm surge and Tsunam smulator n Oceans and Coastal areas, STOC. In ths paper, only the 3D part of STOC was appled to smulate tsunam runup. In STOC we have two 3D models: when we calculate water surface dstrbuton, we use 1) Vertcally ntegrated equaton of contnuty or 2) 3D-VOF method. The frst model s labeled as STOC-IC, whle the second one as STOC-VF. We can choose between two. Calculaton results were obtaned to reproduce the tsunam runup measured n the large-scale tank (L 205 m, D 6 m and W 3.4m). Ths experment was performed by Matsuyama & Tanaka (2001) to represent the extreme runup of about 32 m heght dscovered at the tp of a narrow gulley wthn a small cove near the vllage of Mona n Okushr Island, Japan, due to the 1993 Hokkado Nanse-Ok earthquake tsunam. 2 3D NUMERICAL SIMULATOR FOR FLUID MOTION 2.1 Governng Equatons In the numercal smulator, STOC, we solve the set of contnuty and Reynolds-averaged Naver-Stokes equatons,.e., 0 ε t 0 ( γ u x ( εu ) ( γ u u ) C t x ε p ρ = ε g ρ x ρ x ) = 0, (1) u γ ν e x u x (2)
, Fgure 1. Dagram of STOC Fgure 2. An example of calculaton result by STOC-VF: Water surface of an magnary tsunam where x descrbes the Cartesan coordnate system, (x, y, z); u s velocty n the drecton of x, (u, v, w); ρ s densty; ρ 0 s a reference densty; p s pressure; ε s porosty; γ s transmssvty n the drecton of x ; g s gravtatonal acceleraton, (0,0, g); ν e s total vscosty ncludng dynamc vscosty and eddy vscosty; C s Corols term. A porous model s ntroduced to express the smooth shape of sea bottoms or structure faces. We solve these governng equatons usng a fnte dfference method. In space, a staggered mesh s adopted, where for the dffuson terms the secondorder central scheme s used, whle for the advecton terms a hybrd scheme,.e., the frst-order upwnd scheme combned wth the second-order central scheme, s utlzed for stablty. In tme, a leapfrog method s used n STOC-IC, staggerng calculaton tme of both water surface dsplacement and pressure and that of velocty by t, whle SMAC method s used n STOC-VF. 2.2 Calculaton Methods of Water Surface Dsplacement 1) STOC-IC After calculaton n each cell at the current tme-step, we should know the poston of water surface at the next tme-step. In STOC-IC we use the vertcally Integrated equaton of Contnuty,.e.,
Table 1. Wave gauge postons (a) Bathymetry n the flume (b) Detaled topography near the maxmum-runup valley Fgure 3. Bathymetry and topography n the experment η γ z t x η h γ u dz x y γ v dz = 0, (3) where η s water surface dsplacement; h s stll water depth. In runup regons, Bucket-brgade method s used; that s there exsts water at the places where the water depth s larger than some reference value. Unfortunately ths model s not applcable to cases where water surface elevaton s descrbed by a multvalued functon of x, because Equaton 3 s a vertcally-ntegrated-type equaton. 2) STOC-VF On the other hand, STOC-VF s a 3D-VOF method (Arkawa et al., 2005), whch s an extended verson of the 2D-VOF method (CDIT, 2001), usng a convecton equaton on F,.e., η h y Fgure 4. Incdent wave condton ( εf ) ( γ u F ) t x = (source). (4) Usng ths method, whch works also when the water surface elevaton s descrbed by a multvalued functon of x, we can represent breakng-wave phenomena of flud moton as shown n Fgure 2. Ths fgure shows an magnary tsunam attackng buldngs n a town protected by a breakwater, a submerged breakwater and a seawall. 3 NUMERICAL SIMULATION OF TSUNAMI RUNUP ONTO A COMPLEX BEACH 3.1 Calculaton Condton We tred tsunam calculaton usng both STOC-IC and STOC-VF. The calculaton problem s to reproduce the Mona runup n the 1/400 length-scale laboratory experment by Matsuyama & Tanaka
Table 2. Calculaton condton (2001), whose results were also compared wth the calculaton results by Yoneyama et al. (2002). Fgure 3 shows the coastal topography used n the experment. There are reflectve and vertcal sdewalls where y = 0 and 3.4 (m). The wave gauges of the experment were set at the 13 ponts shown n Table 1, among whch three gauges of Nos. 5, 7 and 9 are ponted n Fgure 3(a). The ncdent wave s gven as shown n Fgure 4 from offshore, where the water depth d = 13.5 cm. The calculaton condtons are shown n Table 2. Note that t s not easy to compare the CPU tme for the calculaton wth STOC-IC and that wth STOC- VF because t changes accordng to Courant number and the numbers of grd ponts are dfferent. 3.2 Calculaton Results 1) Water level calculated by STOC-IC and STOC- VF under the slp-bed condton Fgures 5(a), 5(b) and 5(c) show tme varaton of water level at Channels 5, 7 and 9, respectvely. In these fgures the calculaton results by both STOC- IC and STOC-VF are compared wth the expermental data. In STOC-VF, ν e, whch ncludes eddy vscosty, s equal to zero, whle n STOC-IC an LES model s nstalled. Although there s such dfference n treatment of turbulence between these two models, the results by STOC-VF and STOC-IC are smlar except when t = 15 16.5 (s) at Channel 9, where STOC-IC cannot represent breakng phenomena wthout wave-breakng model. 2) Water level calculated by STOC-IC under the non-slp-bed condton In Fgure 5, n comparson wth the expermental data, the calculaton results of water level under the slp-bed condton show hgher frst-peak values, whch appear as the frst local maxmal values n the experment when t 15.0 s. Secondly we have tred STOC-IC under the non-slp-bed condton for the same ncdent waves. The tme varaton of water level on the non-slp bottom s shown n Fgure 5 wth the pnk-colored lnes. These calculaton results get too lower water level than that observed n the experment. Ths reason should be that we do not perform dscretzaton of numercal meshes n the vertcal drecton nsde the bottom boundary layer because t s effcent to use meshes whose vertcal wdth s much larger than the vertcally changng rate of velocty nsde the boundary layer. Such doubtful accuracy may arse n deep water, as well as n very shallow water because we cannot use so mnute meshes as to dvde the shallow water depth. We should examne bottom-frcton models ncludng emprcal frcton laws, such that we ntroduce some adequate frcton model nto STOC to obtan more reasonable solutons also n feld-scale calculaton.
(a) Tme varaton of water level at Channel 5 (b) Tme varaton of water level at Channel 7 (c) Tme varaton of water level at Channel 9 Fgure 5. Calculaton results of tme varaton of water level by STOC-IC and STOC-VF
Fgure 6. Calculaton results by STOC-VF of velocty vectors and sobarc curves n the vertcal secton parallel to the axs y (In ths secton the tsunam shows the hghest runup when t = 16.5 s.) Accordng to Fgure 5, the 2D-calculaton results by Yasuda & Watanabe (2005) based on the nonlnear long-wave equatons ncludng frcton terms wth relatve roughness show some tme-lag of peakvalue appearance n comparson wth the expermental data. Ths tendency s not so strong as but smlar to the results calculated by STOC-IC under the non-slp-bed condton, where the peak values appear later than those of the expermental data. It should be noted that the calculaton results obtaned by both STOC-IC and STOC-VF under the slp-bed condton capture the accurate tme of peak-value appearance n the experment successfully. 3) Runup due to the Okushr tsunam calculated by STOC-VF under the slp-bed condton In ths subsecton we assume that the sea bottom s a slp bed. We can treat runup of waves usng STOC- IC or STOC-VF. In ths paper we show calculaton results of runup obtaned by STOC-VF. Fgure 6 shows the results, whch were calculated by STOC-VF, of velocty vectors and sobarc curves n the vertcal secton parallel to the axs y at the tme when t = 16.5 s. In ths fgure the tsunam shows the hghest runup. We can evaluate 3D flow consderng effect of not only hydrostatc but also dynamc pressure usng STOC-VF as well as STOC-IC. In order to know how dynamc pressure affects tsunams over a complex topography, t s worthwhle, for example, to compare calculaton results by such non-hydrostatc models as STOC-IC and STOC-VF, whch solve Posson equaton of pressure, wth those by hydrostatc models ncludng nonlnear long-wave models. Precse evaluaton of pressure s mportant especally when we study nteracton among tsunams, structures, drftage and ground, whch requres future work. Fgure 7 shows wave profles of the Okushr tsunam calculated by STOC-VF. After draw-down, see Fgure 7(a), the tsunam travels towards onshore. Around Pont A, dffracted waves meet together behnd Muen Island. We can see the valley or gulley, where the tsunam s gong to show the maxmum heght of runup, n the top center of Fgure 7(a). In Fgure 7(b) the tsunam reaches the land. In Fgure 7(c) we can see reflected waves from the coasts. At Pont B n Fgure 7(d), a breakng wave s generated by superposton of waves, whch have come from two dfferent coasts, showng the largest value of wave heght through the whole computaton perod except over the land area. The maxmum runup appears at Pont C n Fgure 7(e). Ths runup heght s about 7.65 cm n the model scale, accordng to alttude of the ntersectons of water surface n consderaton of the VOF functon and vertcal faces of meshes ncludng porosty to descrbe the confguraton of seabed smoothly. Thus the result says that the tsunam got the runup heght of about 30.6 m n the prototype scale. Ths value s closer to the dscovered value n the feld,.e., about 32 m, than the calculaton results by Yasuda & Watanabe (2005), who utlzed the 2D nonlnear long-wave models wth the same sze of horzontal meshes as the present calculaton, resultng n about 28.0 m of runup heght usng square meshes and about 29.0 m usng trangular mproved meshes. It s anmatons, whch are prepared by onng fgures, e.g. Fgures 7(a) 7(f), that are expected to deepen our understandng of tsunam phenomena owng to vsual and dynamc presentaton of numercal calculaton results, as well as to be helpful for resdents to take refuge quckly and adequately. 4 CONCLUSIONS Two 3D numercal models bult n STOC,.e., STOC-IC and STOC-VF, have been appled to seawater moton due to the 1993 Okushr tsunam.
Fgure 7. Runup due to the Okushr tsunam calculated by STOC-VF
Ther governng equatons are the contnuty and Reynolds-averaged Naver-Stokes equatons for ncompressble-flud moton n porous meda. Water surface dsplacement s determned by the vertcally ntegrated equaton of contnuty n STOC-IC, whle by the 3D-VOF method n STOC-VF. In numercal computaton wth these two models, we need nether long-wave assumpton, rrotatonal assumpton nor hydrostatc assumpton, such that we can take nto account vertcal component of velocty ncludng vortcty, as well as dynamc pressure. The calculaton results by STOC-IC and STOC-VF are compared wth both the exstng expermental data by Matsuyama & Tanaka (2001) and the 2Dcalculaton results by Yasuda & Watanabe (2005) wth the nonlnear long-wave models, leadng to thefollowng results: 1) The results of water level calculated by STOC-IC and STOC-VF under the slp-bed condton are smlar when there exsts no breakng wave. 2) These two models generally gve correspondent results of water level wth the expermental data at off-peak tmes of the water level. In comparson wth the expermental data, the results of water level calculated by STOC-IC and STOC- VF show hgher frst-peak values on the slp seabed. 3) The results calculated by STOC-IC under the non-slp-bed condton show too lower water level than that observed n the experment. Some bottom-frcton model s requred. 4) The calculaton results of water level obtaned by both STOC-IC and STOC-VF under the slp-bed condton capture the accurate tme of peak-value appearance n the experment successfully, whle the peak values calculated by both STOC-IC under the non-slp-bed condton and the 2D nonlnear long-wave model wth relatve roughness of the seabed appear later than those of the expermental data. 5) The hghest runup s about 7.65 cm, whch corresponds to 30.6 m n the prototype scale. Ths value, whch was obtaned by STOC-VF, s larger than that evaluated by use of the 2D nonlnear long-wave model adoptng the same sze of numercal meshes. ACKNOWLEDGMENTS Sncere grattude s extended to Mr. M. Akyama and Mr. A. Shmada, Mzuho Informaton & Research Insttute, for benefcal help to codng of the program. It was a good opportunty to attend at the 3rd Internatonal Workshop on Long-Wave Runup Models, whch was held at Wrgley Marne Scence Center, Unversty of Southern Calforna n 2004 (http://ww w.cee.cornell.edu/longwave) for development of the present numercal model. REFERENCES Arkawa, T., Yamada, F. & Akyama, M. 2005. Study of the applcablty of tsunam wave force n a threedmensonal numercal wave flume. Annual J. Coastal Eng. JSCE. 52: 46-50 (n Japanese). CDIT (Coastal Development Insttute of Technology). 2001. CADMAS-SURF. CDIT Lbrary. 12: 296p. (n Japanese). Kaknuma, T. & Tomta, T. 2005. Development of storm surge and tsunam smulator n oceans and coastal areas. Proc. 29th Int. Conf. on Coastal Eng.: 1552-1564. Matsuyama, M. & Tanaka, H. 2001. An expermental study of the hghest run-up n the 1993 Hokkado Nanseok earthquake tsunam. Proc. Int. Tsunam Symposum 2001: 879-889. Yasuda, H. & Watanabe, Y. 2005. Numercal analyss of tsunam runup onto a complex 3 dmensonal beach. Proc. Hokkado Chapter of the JSCE Hokkado. 61: CD-ROM (n Japanese). Yoneyama, N., Matsuyama, M. & Tanaka, H. 2002. Numercal analyss for locally hgh runup of 1993 Hokkado Nanse-ok sesmc tsunam. J. Hydraulc, Coastal and Envronmental Eng. JSCE. 705/II 59: 139-150 (n Japanese).