Storm Surge and Tsunami Simulator in Oceans and Coastal Areas

Transcription

1 Proc. Int. Conf. on Montorng, Predcton and Mtgaton of Water-Related Dsasters, Dsaster Preventon Research Insttute, Kyoto Unv. (2005) Storm Surge and Tsunam Smulator n Oceans and Coastal Areas Taro Kaknuma and Takash Tomta Marne Envronment and Engneerng Department, Port and Arport Research Insttute, Nagase, Yokosuka, Kanagawa , Japan ABSTRACT: Ths paper descrbes a numercal model for storm surge and tsunam smulaton consderng three-dmensonal characterstcs of flow. The developed model, STOC (Storm surge and Tsunam smulator n Oceans and Coastal areas), conssts of three parts: 1) 3D model, 2) multlevel model and 3) connecton model. The 3D model s appled to a narrow area surrounded by a wder area, whch s covered by the multlevel model. By adoptng the 3D model locally we can economcally and accurately evaluate complcated flow around coastal structures or over varous topographes, whch are smoothly expressed wth a porous model ntroduced nto the set of contnuty and Naver-Stokes equatons. STOC was appled to the 1868 Me-Sanrku tsunam, resultng n representaton of a remarkably three-dmensonal velocty around the open mouth of tsunam breakwaters as well as the total pressure ncludng both hydrostatc and dynamc pressures on the superstructure of submerged breakwater. 1 INTRODUCTION Seawater moton from offshore to coastal zones, whch ncludes phenomena of varous scales on both space and tme, should be solved effcently n numercal calculaton. For ths purpose a hybrd model, whch conssts of a 3D model, a multlevel model and a connecton model, has been developed to smulate three-dmensonal flow due to tsunams or storm surges. By local applcaton of the 3D model to narrower areas surrounded by wder areas whch are covered by the multlevel model, we can represent three-dmensonal characterstcs of flud moton around structures over steep topography economcally and accurately. The governng equatons are the contnuty and Naver-Stokes equatons for ncompressble fluds, where a porous model as presented by Sakakyama and Kama (1992) s adopted to descrbe confguratons of structure and seabed smoothly. We solve ths set of equatons usng numercal schemes based on the fnte dfference method by Masamura et al. (2001), who connected a 3D model to a hydrostatc one-layer model. In the present study we have connected the 3D model, where pressure s not assumed to be hydrostatc, to the hydrostatc but multlevel model, nto whch two addtonal models are ntroduced: 1) the sesmc deformaton model (Mansnha and Smyle, 1971) to generate tsunams, and 2) Myers model to descrbe both pressure decrease and wnd drag force caused on the sea surface by tropcal cyclones. Applyng the multlevel model to a wde area of ocean, we can represent vertcal dstrbuton of velocty n storm surges, as well as not only nonlnearty but also dspersvty of tsunams. The new hybrd model was appled to the 1868 Me-Sanrku tsunam to examne complcated flow around the tsunam breakwaters of Kamash Port, Japan, resultng n evaluaton of total pressure ncludng both hydrostatc and dynamc pressures on the structures. For dsaster preventon, many thngs are mportant to be stocked by ndvduals as well as ther organzatons, e.g. food, faclty, commodty, knowledge, nformaton, atttude of mnd, etc. Accordngly we call ths numercal calculaton model Storm surge and Tsunam smulator n Oceans and Coastal areas,.e., STOC. 2 NUMERICAL MODEL 2.1 Governng equatons In the numercal smulator, STOC, we solve the followng set of contnuty and Naver-Stokes equatons,.e., γ u x = 0, (1) u ε t γ u u C = ε ρ 0 p ρ ρ 0 ε ρ 0 g x u γ ν T v, (2)

2 2 where x descrbes Cartesan coordnate system, (x, y, z); u s velocty n the drecton of x, (u, v, w); ρ s densty; ρ 0 s reference densty; p s pressure; ε s porosty; γ s transmssvty n the drecton of x ; g s gravtatonal acceleraton; ν T s total vscosty ncludng knematc vscosty and eddy vscosty; C s Corols term. These porosty and transmssvty are ntroduced to express confguratons of sea bottoms or structure faces smoothly as n the porous model presented by Sakakyama and Kama (1992). We solve these governng equatons usng numercal schemes based on the fnte dfference method by Masamura et al. (2001), who connected a 3D model to a hydrostatc one-layer model. A spatally staggered mesh s adopted,.e., the dffuson terms are dscretzed wth the second-order central scheme, whle the advecton terms are expanded wth a hybrd scheme where the frst-order upwnd scheme s combned wth the second-order central scheme usng weghtngs for stablty. In the temporal drecton a leapfrog method s utlzed to stagger calculaton tme-steps of velocty and those of both water surface dsplacement and pressure. 2.2 Structure of numercal smulator, STOC Multlevel model STOC conssts of three parts as shown n Fg. 1: 1) 3D model, 2) multlevel model and 3) connecton model. In wde-area calculaton we use the multlevel model named STOC-ML, whch solves vertcally ntegrated contnuty and Naver-Stokes equatons n each level wth assumpton of hydrostatc pressure. Two physcal models are ntroduced nto STOC-ML: 1) the sesmc deformaton model (Mansnha and Smyle, 1971) to generate tsunams due to submarne earthquakes, and 2) Myers model to descrbe both pressure decrease and wnd drag force caused by tropcal cyclones D model The 3D model labeled as STOC-NS s appled locally to a narrower area, where pressure s solved wthout assumpton of hydrostatc pressure, such that we can economcally and accurately evaluate complcated dstrbuton of velocty and pressure around coastal structures over steep topography Connecton model For smooth connecton between multlevel areas and 3D areas, a connecton model should be appled to overlap regons, where physcal varables are defned by nterpolaton of those n the two areas. The nterpolated varables are selected among water surface elevaton, velocty, momentum, pressure, eddy vscosty coeffcent, etc. In the connecton model we can choose between two connectng methods,.e., two-way method and one-way method. The two-way method solves both the multlevel model and the 3D model n an overlappng area smultaneously at each tme-step to consder nteracton of phenomena n the multlevel area and the 3D area. When more accuracy than speed s requred n calculaton, t s recommended applyng ths two-way method. On the other hand, n the one-way method, we perform calculaton over the whole doman throughout the target perod usng the multlevel model, after whch we solve the 3D model wth nterpolated varables on outer boundares of connecton areas. Thus the results obtaned by the multlevel model are reflected n the 3D calculaton but the reverse s not done. Ths one-way method s adequate to mmedate calculaton, where the 3D model can run at dfferent tme-steps from those of the multlevel model, even f the 3D calculaton requres much shorter ntervals of tme-steps correspondng to ts mnute meshes. In the present calculaton we use the one-way method, where nterpolaton s conducted for only water surface elevaton. Specfcally, water surface dsplacement nsde connecton areas s compensated by η = kη NS (1 k)η ML, (3) where η NS and η ML are water surface dsplacements calculated at the last tme-step by STOC-NS and STOC-ML, respectvely. The weghtng k changes gradually from 0 to 1 through the connecton area from the multlevel area to the 3D area for consstency of physcal varables, resultng n stablty of numercal calculaton. Ths compensated value, η, s used n calculaton at the followng tme-steps.

3 3 Fgure 1. Dagram of STOC. 2.3 Calculaton method of water surface dsplacement After calculaton n cells at the current tme-step, we should know poston of water surface at the next tme-step. In STOC-NS and STOC-ML we use the vertcally ntegrated equaton of contnuty,.e., γ v dz = 0, (4) where η s water surface dsplacement and h s stll water depth. In runup or nundaton regons, a bucket-brgade method s used,.e., there exsts water where water depth s larger than some reference value h mn, e.g. h mn = m. 3 VERIFICATION OF STOC-NS η γ z t x η h γ u dz x We have confrmed accuracy of STOC by comparng ts results wth those of other numercal models or hydraulc experments for varous benchmark tests. Several results obtaned by STOC-NS are shown here. Fgure 2(a) shows the calculaton result by STOC-NS of velocty n horzontal cavty flow wth a sde wall sldng at a constant speed of u 0 n the postve drecton of x where y/d = 1.0. Every calculaton result n ths paper has been obtaned under nonslp condton on any walls and bottoms. In Fg. 2(b) we can see correspondence between the dstrbuton along the lne where x/d = 0.5 of velocty n the drecton of x, u, obtaned by STOC-NS and that gven by another commercal numercal model, α -FLOW. Fgure 3 shows calculaton results by STOC-NS of a drft current entaled by a wnd blowng n the postve drecton of x. In ths problem we have theoretcal solutons of water surface gradent and horzontal velocty,.e., y η h y η 3r f a = U U 2gh hr f a z z and u 3 = 1 1 U U, (5) 4K z h h where η s water surface dsplacement; u s horzontal velocty along the lne where x = 500m. In the calculaton for Fg. 3, wnd speed on sea surface U = 20m/s; frcton coeffcent on sea surface f a = ; densty rato of atmosphere and seawater r = ; knematc eddy vscosty coeffcent K z = 0.01m 2 /s; stll water depth h = 10.1m. Ths benchmark test s for verfcaton of wnd-drven currents n storm-surge calculaton.

4 4 (a) Water surface elevaton. (a) Velocty vectors. (b) Horzontal velocty (x = 500m). (b) Velocty n the drecton of x, u (x/d = 0.5). Fgure 2. Calculaton results of 2D cavty flow. Fgure 3. Water surface elevaton and horzontal velocty n wnd-drven current: Calculaton results by STOC-NS (ο) n comparson wth the correspondent theoretcal solutons by Eq. (5) ( ). 4 NUMERICAL SIMULATION 4.1 Calculaton condtons STOC can treat the whole area from offshore, where submarne earthquakes are generated, to coastal zones, whch nclude coastal structures. In ths study we apply the set of STOC-ML, STOC-NS and the one-way connecton method to a full-scale case, where the 1868 Me-Sanrku tsunam attacks Kamash Port, Japan. Fgure 4 shows the bathymetry of Kamash Port protected by two breakwaters, B1 and B2, between whch a submerged breakwater s sandwched. The brd s eye vew of the breakwaters s brefly shown n Fg. 5. The tsunam s generated n STOC-ML, whose number of levels s one, by use of the sesmc deformaton model (Mansnha and Smyle, 1971) wth fault parameters ncludng length 210km, wdth 50km, depth 1km, dslocaton 12.5m, slp 58 degrees, dp 20 degrees and strke 156 degrees (Sato ed., 1997), and correcton coeffcent of dslocaton The area covered by STOC-NS s ndcated n Fg. 4 as a one-klometer square around the open mouth of tsunam breakwaters. Ths 3D calculaton area, where horzontal mesh sze s equally 12.5m, whle vertcal mesh sze s varable from 1m to 10m, has bult-n the connecton area overlappng the STOC-ML s area from each sde of the 3D calculaton area towards ts center by 10 meshes. Calculaton was conducted under nonslp condton on every wall and bottom, as mentoned above. 4.2 Calculaton results and dscussons Water surface elevaton Water surface elevaton at Pont A n Fg. 4, over the center of the submerged breakwater, s shown n Fg. 6, where the sold lne ndcates the result gven by the calculaton wth only STOC-ML whose number of levels s one, namely, a one-layer model, whle the crcles show the correspondent result obtaned by STOC-NS wth the one-way connecton method after applyng STOC-ML. The frst peak of water surface elevaton passes the center of submerged breakwater when t = 28.4mn n both these cases. The two results are almost consstent except about the bottoms of trough. In the present paper we focus attenton on the frst perod of tsunam Velocty dstrbuton Fgure 7 shows velocty pattern n the vertcal secton ncludng C-C lne nsde the 3D calculaton area drawn n Fg. 4. At ths tme, when the water surface dsplacement changes from negatve to postve, the flow s

5 5 Fgure 4. Plan vew of Kamash Port, Japan. Fgure 6. Water surface elevaton at Pont A. Fgure 5. Sketch of breakwaters. Fgure 7. Velocty vectors n C-C secton (t = 27.0mn). dammed n front of the breakwater and rased over the submerged breakwater but under the offshore stream drven by the frst draw-down. On the other hand, Fg. 8 shows velocty patterns n the horzontal sectons when t = 28.7mn, showng nflux behnd the breakwaters, such that remarkably three-dmensonal flow wth both vertcal and horzontal crculatons are observed around the open mouth of breakwaters. Thus STOC can provde three-dmensonal flud moton, whch s a blue rose for one-layer long wave models Pressure on superstructure of submerged breakwater The submerged breakwater, see Fg. 5, has ts superstructure, on whch tsunam force ntegrated spatally over ths superstructure shows the maxmum value when t = 28.7mn, ust after the peak tme of water dsplacement. The calculaton result of total pressure on ths superstructure s shown n Fg. 9, where z = 27.5m and t = 28.7mn. Dfference between onshore-sde pressure, p on, and offshore-sde pressure, p off, n the drecton perpendcular to the superstructure sdes,.e., p = p off p on, works as net pressure on the structure. Dstrbuton of water level over the submerged breakwater at the same tme s shown n Fg. 10, from whch we obtan dstrbuton of hydrostatc pressure. Accordngly we can evaluate the dynamc pressure determned as an excess over hydrostatc pressure, that s, the total pressure s a sum of hydrostatc and dynamc pressures. Fgure 11 shows the net dynamc pressure,.e., p = p off p on, ndcatng Pa at the center of the submerged breakwater where z = 27.5m, whch accounts for about 58 percent of the net total pressure. Thus STOC can provde three-dmensonal dstrbuton of tsunam force on structures not ust consderng hydrostatc pressure but takng nto account dynamc pressure as well, whch s advantage of applcaton of STOC-NS wthout assumpton of hydrostatc pressure. 5 CONCLUSIONS The numercal smulator, STOC, whch conssts of STOC-NS, STOC-ML and the connecton model, has been developed to descrbe seawater moton from offshore to coastal zones due to tsunams or storm surges

6 6 (a) z = 17.5m. (b) z = 35.0m. Fgure 8. Velocty vectors n horzontal sectons (t = 28.7mn). Fgure 10. Dstrbuton of water level around open mouth of breakwaters (t = 28.7mn). Fgure 9. Total pressures on onshore and offshore faces of superstructure and net total pressures on superstructure of submerged breakwater (z = 27.5m, t = 28.7mn, p = p off p on ). Fgure 11. Dynamc pressures on onshore and offshore faces of superstructure and net dynamc pressures on superstructure of submerged breakwater (z = 27.5m, t = 28.7mn, p = p off p on ). ncludng nundaton onto beaches. In comparson wth theoretcal, expermental and numercal data, the calculaton results show accuracy of STOC n the varous test cases. In the present paper STOC was appled to tsunam calculaton to represent the three-dmensonal phenomena around the open mouth of tsunam breakwaters. We found the net dynamc pressure makng up more than half of the net total pressure on the superstructure of submerged breakwater. STOC, whch can provde threedmensonal dstrbuton of flud force on structures ncludng not only hydrostatc but also dynamc pressures, has an edge on tradtonal long wave models based upon the assumpton of hydrostatc pressure. 6 ACKNOWLEDGMENTS Sncere grattude s extended to Mr. M. Akyama and Mr. A. Shmada, Fu Research Insttute Corporaton, for benefcal help to program codng. 7 REFERENCES Mansnha, L., and Smyle, D.E. (1971). "The Dsplacement Felds of Inclned Faults." Bulletn of the Sesmologcal Socety of Amerca, Vol. 61, No. 5, pp Masamura, K., Fuma, K., Goto, C., Ida, K., and Shgemura, T. (2001). "Numercal Analyss of Tsunam by Usng 2D/3D Hybrd Model." Journal of Hydraulcs, Coastal and Envronmental Engneerng, 670/II-54, pp (n Japanese) Sakakyama, T., and Kama, R. (1992). "Numercal Smulaton of Nonlnear Wave Interactng wth Permeable Breakwaters." Proceedngs of 23rd Internatonal Conference on Coastal Engneerng, pp Sato, Y. ed. (1997). Handbook of Earthquake Dslocaton Parameters n Japan. Kama Insttute Publshng Co., Ltd., Tokyo, No. 52. (n Japanese)