A large scale tsunami run-up simulation and numerical evaluation of fluid force during tsunami by using a particle method

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1 A large scale tsunam run-up smulaton and numercal evaluaton of flud force durng tsunam by usng a partcle method *Mtsuteru Asa 1), Shoch Tanabe 2) and Masaharu Isshk 3) 1), 2) Department of Cvl Engneerng, Unversty of Kyushu, Fukuoka, JAPAN 3) Graduate School of Scence and Engneerng, Ehme Unversty, Matsuyama, JAPAN 1) asa@doc.kyushu-u.ac.jp ABSTRACT In ths study, the ncompressble Smoothed Partcle Hydrodynamcs (ISPH) s utlzed as a 3D tsunam run-up smulator. Frstly, a modelng tool for ground level ncludng structure and sea floor was developed to generate smulaton model. The geometrcal modelng utlzed aerosurvey data for the ground level and bathymetry data for sea floor. A 50 mllon partcles model for a small cty was generated by our developed modelng tool for the partcle smulatons. The numercal solutons were compared wth a dsaster nvestgaton report. The accuracy of flud force evaluated by the ISPH has been nvestgated by comparng wth an expermental data. 1. INTRODUCTION The huge tsunam caused by the Great Tohoku Earthquake devastated the coastal area on March 11, After the Tohoku Earthquake, the Japanese government and each local government has been dscussng the next dsaster preventon and mtgaton method aganst mllennum Tsunam. In order to construct safe and secure coastal structures, t s necessary to generate an accurate smulator, whch can predct not only the flood area but also the structural damage. Therefore, we are developng a 3D tsunam smulator based on a partcle method. In ths study, the ncompressble Smoothed Partcle Hydrodynamcs (ISPH) s utlzed as a 3D tsunam run-up smulator. Frstly, a modelng tool for ground level ncludng structure and sea floor was developed to generate smulaton model. The geometrcal modelng utlzed aerosurvey data for the ground level and bathymetry data for sea floor. The numercal solutons were compared wth a dsaster nvestgaton report. In the next step, the accuracy of flud force evaluated by the ISPH has been nvestgated by comparng wth an expermental data. In ths comparson, the boundary treatment of the partcle smulaton has been dscussed to get accurate solutons. 1) Assocate Professor 2) Graduate Student 1559

2 2. A stablzed ISPH In ths secton, a stablzed ISPH (Asa 2012), whch ncludes a modfed source term n the pressure Posson equaton, for ncompressble flow s summarzed. Then, a conventonal turbulence model Smagornsky model s ntroduced nto the ISPH. 2.1 Governng equatons The mass and momentum equatons of the flows are gven as: D u 0 (1) Dt Du p u τ F 0 (2) Dt where and are densty and knematc vscosty of flud, u and p are the velocty and pressure vectors of flud, respectvely. F s external force, and t ndcates tme. In the most general ncompressble flow approach, the densty s assumed by a constant value wth ts ntal value Modfcaton n the source term of pressure Posson equaton The man concept n an ncompressble SPH method s solvng a dscretzed pressure Posson equaton at every tme step to evaluate the pressure value mplctly. In the sense of physcal and theoretcal observaton, densty should keep ts ntal value for ncompressble flow. However, durng numercal smulaton, the partcle densty may change slghtly from the ntal value because the partcle densty s strongly dependent on partcle locatons n the SPH method. If the partcle dstrbuton can keep almost unformty, the dfference between physcal and partcle densty may be vanshngly small. In other words, accurate SPH results n ncompressble flow need to keep the unform partcle dstrbuton. For ths purpose, the dfferent source term n pressure Posson equaton can be derved usng the partcle densty. The SPH nterpolatons are ntroduced nto the orgnal mass conservaton law before the perfect compressblty condton s appled. n1 * n1 1 u (3) 0 t Then, the pressure Posson equaton reformulated as: 0 0 * 2 n1 * p u (4) 2 t t where s relaxaton coeffcent, u s temporal velocty and trangle bracket < > means * SPH approxmaton. Note that ths relaxaton coeffcent s strongly dependent on the tme ncrement and the partcle resoluton. Then, the reasonable value can be estmated by the smple hydrostatc pressure test usng the same settngs on ts tme ncrement and the resoluton. 2.3 Introducton of an eddy vscosty by Smagornsky model 1560

3 A smple Large Eddy Smulaton has been mplemented by ntroducng the Smagornsky sub-grd model. The sub-grd stress tensor s modeled through the tradtonal Boussnesq eddy vscosty assumpton as IJ T 2 2 S IJ k IJ (5) 3 T 2 C s S (6) T n whch n the Smagornsky model s an eddy vscosty, C s s Smagornsky constant, s constant and t taken as smoothng compact support n ths scheme. The local stran rate S s calculated as (Voleau 2007). The turbulent knetc energy ncorporated n the pressure term, and then the current vscous term s gven by SPH assumpton as follows: u j j rjw m j uj j j r ( ) (7) 2 2 j The numercal examples have llustrated that the proposed fnte elements could be very useful for geometrcally nonlnear analyss as well as free vbraton 3. Tsunam Smulaton By usng the stablzed ISPH tool, a tsunam smulaton has been conducted wth a realstc geometrcal model n 3D. 3.1 Tsunam smulaton at Taro In ths smulaton, tsunam run-up behavor s smulated wth the real geometrcal map generated from a mult-resoluton aeral survey data. The tsunam s nputted on the rght boundary edge n our model shown n Fg.1, and the nput of tsunam s assumed from the nvestgaton of Tohoku earthquake. The heght s 3m, and ts velocty s 10m/s. The constant tsunam s gven durng our tsunam smulaton for 5 mnutes n the real tme. Fg 2 shows the snapshot of tsunam smulaton after 20s, 2m20s, and 4m20s. In the fgures, contour color ndcates the velocty magntude n the horzontal drecton of the fgure. These run-up behavors show a good agreement wth the damage nvestgaton results. The computaton was manly carred out usng the facltes at Research Insttute for Informaton Technology at Kyushu Unversty. It spends about 3days for the 5mnutes behavors. 1561

4 10m/s wth 3m heght Fg. 1 Taro cty and tsnam nflow condton 0m00s 2m30s 5m00s Fg. 2 Taro cty and tsnam nflow condton 4. Valdaton of flud mpact force by comparng wth an expermental test In ths secton, the accuracy and effcences of our proposed method are valdated by comparson between a numercal soluton and expermental results. The necessty of our proposed boundary treatment, whch s descrbed n prevous secton, s dscussed at the same tme, and the relaton between the dfference of the boundary condtons and evaluaton of flud mpact force are also nvestgated. Next, the degree of the flud mpact force acted on real sze and shape of brdge grders s estmated. 1562

5 4.1 Detals of the valdaton test The analyss model and the detal of the model for grders are shown n Fg.3 and Fg.4 respectvely. Ths experment was carred out by Nakao et al. (2011), and the flud mpact force s evaluated whle the wave acts on the model of grders. The shape of the model for grders s rectangle and upsde down trapezod. The partcle dstance d0= 0.5cm, tme ncrement dt= 0.001s and the total number of partcles s about 8mllons. Fg. 3 Valdaton model (unt: mm) Fg. 4 Detal of the Brdge grder models 4.2 Valdaton results At fast, Fg.5 shows the result of horzontal force n rectangle model and (a) s gven the slp condton and (b) s gven the noslp condton respectvely. So far, n the conventonal method, the accuracy of flud mpact force could not be valdated suffcently because of the problem such as the declne of maxmum force, whch orgnates from penetraton of water partcle nto the sold boundary. Accordng to the result shown n Fg.5 ths smulaton applyng our proposed method s confrmed to be useful and ths smulaton result matches the expermental one at the practcal level. In addton, from the detal of the result, the maxmum force gven the noslp condton s closer to the expermental result. Ths reason may be concerned wth the tp shape of the wave. In the slp condton, the tp shape of the wave s more acute angle than the experment, on the other hand, the one n the no-sp s rounded, and snce the latter wave attacks the grder model as a mass of water, t s consdered that the maxmum force gven the noslp condton s more accurate to the experment

6 Fg. 5 Horzontal force n rectangle model From the result of the moment to attack the grder model, the moment n the slp condton almost matches the expermental result. Ths may be related to the velocty (Table.1) and shape of the wave. Referrng to Table.1, the wave velocty n the slp condton s faster than the expermental one, so t s thought that the moment to attack the grder model s also faster, but the result s almost equal to the experment. It can be because that the ncrease of water level s delayed by the acute-angled shape of the wave. In the no-slp condton, the velocty of the wave s slower than the expermental one, because t becomes a flow as the frcton acts on the boundary and the waterway s long enough to cause the effect of that frcton. Table.1 Comparson of the wave velocty The velocty of wave (unt: m/s) Experment Smulaton (slp condton) Smulaton (no-slp condton) Next, Fg.6 shows the result of horzontal and vertcal force n upsde down trapezod model and (a) s gven the slp condton and (b) s gven the no-slp condton respectvely. Referrng to Fg.11, the trend smlar to the rectangle model s obtaned, that s, the maxmum flud mpact force n the no-slp condton s closer to the experment and the moment to attack the model for grders n the slp condton almost matches the expermental result. From the above results, the accuracy and effcences of our proposed method were valdated by comparson between a numercal soluton and expermental results. It showed that the applcaton of our proposed method to the model havng ncompatble step-shaped boundary s also utlzed n a practcal level to evaluate the flud mpact force. As how to gve the boundary condtons, t s not concluded whch condton s the best for the smulaton, but t seems to be mddle or 1564

7 somewhere close to the mddle of the two condtons. Further dscusson s requred n ths verfcaton. (a) Slp condton (b) No-slp condton Fg. 6 Horzontal and vertcal force n upsde down trapezod model 4. Valdaton of flud mpact force by comparng wth an expermental test A stablzed ncompressble smoothed partcle hydrodynamcs s proposed to smulate free surface flow. The modfcaton s appeared n the source term of pressure Posson equaton. The ISPH has been appled to a tsunam run-up smulaton wth a hgh performance computaton, and then the accuracy of the proposed model has been verfed wth an expermental test for the estmaton of tsunam wave force. In the valdaton test, treatment of the boundary condton has been carefully nvestgated to get accurate soluton especally for estmatng the flud mpact force. 1565

8 REFERENCES Asa M, Aly Adbelraheem M., Sonoda Y and Saka Y. (2012) A stablzed ncompressble SPH method by relaxng the densty nvarance condton. Journal of Appled Mathematcs,Vol.2012, Artcle ID Voleau D and Issa R. (2007) Numercal modelng of complex turbulent free-surface flows wth the SPH method: an overvew, Internatonal Journal for Numercal Methods n Fluds, 53(2), pp cadappa, D.C., Sanjayan, J.G. and Setunge, S. (2001), Complete traxal stress-stran curves of hgh-strength concrete, J. Mat. Cvl Eng., ASCE, 13(3),