A numerical simulation on seepage-induced collapse of caisson breakwater using a stabilized ISPH method

Transcription

1 A numercal smulaton on seepage-nduced collapse of casson breakater usng a stablzed ISPH method *Tetsuro Goda 1) and Mtsuteru Asa ) 1), ) Department of Cvl and Structural Engneerng, Kyushu Unversty, 744 Motooka, Nsh-ku, Fukuoka cty, Fukuoka , Japan ) asa@doc.kyushu-u.ac.p ABSTRACT The clarfcaton on the collapse mechansm of a casson breakater has been an urgent task n order to reduce damages caused by next mllennum tsunams. In ths study, ppng destructon of a mound nduced by seepage flo s taken nto consderaton solely. A stablzed ISPH method proposed by Asa (01) s adopted th some modfcaton assocated th the extended Darcy la (Akbar 013, 014) to smulate both seepage flo and surface-flo phenomenon. The numercal results n pezo ater head sho quanttatve agreement th hydraulc expermental results obtaned by Kasama (013). Furthermore, ntaton of the mound collapse behavor can be predcted by the crtcal hydraulc gradent calculated from our numercal solutons. 1. INTRODUCTION The huge tsunam nduced by the Tohoku-Kanto earthquake caused great damages to the port structures ncludng breakaters. In order to reduce the expected damages nduced by next mllennum tsunams, the clarfcaton on the collapse mechansm of breakaters has been an urgent task. A varety of research on a casson-typed breakater has been done, and the follong three causes have been found: () horzontal force due to the ater-level dfference beteen the front and back of a casson, () scour nduced by tsunam overtoppng n the rear of a casson, () seepage-nduced ppng caused by the bearng capacty degradaton of a mound. The nteracton of these three causes has not been clarfed yet because of the complexty assocated th the falure of breakaters. The development of analyss methods for each cause ll lead to a robust analyss scheme hch s capable of handlng these dfferent phenomena smultaneously. In ths paper, e only focus on the cause (), seepage-nduced ppng caused by the bearng capacty degradaton of a mound, as a fundamental study. 1) Graduate Student ) Assocate Professor

2 In ths study, e reformulated a stablzed ISPH method (Asa 01) by referrng to the governng equatons employed n (Akbar 013, 014) for solvng porous flo. Moreover, e mplemented a valdaton test by usng the analyss method. The obtaned results ll be compared th the experments, hch s conducted by Kasama (013), focused on seepage flo actng on a casson-typed breakater.. The governng equatons for surface-flo and seepage flo In ths secton, the governng equatons for solvng surface-flo ll be descrbed. Then, the unfed equaton hch s capable of handlng surface-flo and seepage flo contnuously ll be also ntroduced..1 The governng equatons for surface-flo In general, surface-flo s modeled as a Netonan flud and the Naver-Stokes equaton and contnuty equaton are appled to prescrbe ts moton. D Dt P g v v 1 T (1) D Dt v 0 () here v, P, g are the flud velocty, the flud pressure, and the gravtatonal acceleraton respectvely. Besdes, ν and ρ are the coeffcent of knematc vscosty and the densty of ater. Here, ν T represents the coeffcent of eddy vscosty.. The unfed governng equaton for both surface-flo and seepage flo The unfed governng equaton, hch can handle both surface-flo and seepage flo contnuously, s ntroduced by Akbar (013, 014) as follos: C r ( ) Dv D 1 1 P E ( ) Dt g vd a( ) vd b( ) vd vd (3) Here, s the densty calculated from the mass of pore ater excludng the sold parts, such as the medum partcles and so on. ε s the porosty of the medum. The and ρ satsfy the correlaton hen the saturated medum s assumed. Moreover, ν E s the effectve knematc vscosty gven by ν E (ε) =(ν +ν T )/ε, hle v D s called as the Darcy velocty or the mean velocty prescrbed by the follong equaton v D =εv. Here, C r (ε) s the nerta coeffcent suggested by Van Gent (1995). 1 C : Inerta coeffcent (4) r Furthermore, a(ε), b(ε) are the lnear and non-lnear coeffcent respectvely. There are varous ays to defne these to emprcal coeffcents. Hoever, there has been no

3 general agreement yet accordng to (Akbar 013, 014). In ths research, the follong coeffcents proposed by Irmay (1958) are employed n our formulaton. a b 1 : Lnear coeffcent (5) c 3 D 50 1 : Non-lnear coeffcent (6) c 3 D 50 D 50 n above equatons represents the average dameter of the medum. There are also many proposals for the lnear porous parameter α c and the non-lnear porous parameter β c n Eq. (5) and Eq. (6). By referrng to (Akbar 013, 014), α c and β c are set to 100 and 1.1 n ths study. 3. The formulaton of a stablzed ISPH method A stablzed ISPH method, formulated by Asa (01), s one of the partcle methods for ncompressble flud. The characterstc pont of ths method s the relaxaton coeffcent hch plays an key role to ntroduce the effect of the densty-nvarance condton nto the general Pressure Posson Equaton. In ths secton, the stablzed ISPH method reformulated based on the unfed governng equatons ll be summarzed. (For detals see Mormoto 014.) 3.1 The basc concept of SPH method Here, the fundamental concept of the scheme ll be summarzed. A physcal scalar functon ( x, t) at an arbtral samplng pont x may be approxmated as x, t W x y, h y, tdy V (7) here W s called as the smoothng kernel functon. In ths research, the cubc B-splne curve proposed by Monaghan (1985) s adopted. For the SPH numercal analyss, the ntegral equaton (7) s rertten usng the summaton notaton, consderng the contrbuton of neghbor-dscretzed partcles n the range of the smoothng length h. m ( x, t) : W r, hx, t (8) Here, the subscrpts and ndcate the partcle numbers, hle ρ and m are the representatve densty and the representatve mass related to the partcle respectvely. The trangle blankets ndcate the SPH nterpolaton calculated by referrng to the neghbor partcles nsde the supported doman.

4 3. Pressure Posson Equaton consderng seepage flo Assumng that slght compressblty s alloed; the densty-nvarance condton s not needed to be satsfed nstantaneously, hle consderably smaller ncremental partcle densty than the actual densty dfference may be gven by multplyng the coeffcent, α. n n (9) Here, α ( 0 1) s called as the relaxaton coeffcent, and s set to 0.01 n ths research. Then, the Pressure Posson Equaton may be descrbed as n1 P Cr t n * v D (10) t Accordng to the above formulaton, good dstrbuton n pressure s obtaned by referrng to the velocty dstrbuton n each tme step. Furthermore, the error related to densty may be elmnated gradually by the effect of the densty-deference term. Snce densty s almost kept constant durng the numercal computaton, ths analyss scheme can also yeld farly good conservaton of volume. 4. The physcal quantty of a mound partcle In ths research, the follong three physcal quanttes related to a mound partcle, hch are the pezo ater head, hydraulc gradent and coeffcent of permeablty, need to be evaluated. Note that the mound partcles are not related to the SPH computaton, and ust handled as reference ponts for observng physcal quanttes. These physcal quanttes are calculated on ater partcles frst. Then, these obtaned physcal quanttes are nterpolated nto a mound marker follong the SPH approxmaton. Moreover, the threshold for ppng destructon s also ntroduced. 4.1 The physcal quanttes for a mound The pezo ater head s gven accordng to ts defnton as Pezo P z (11) g here, z s the heght from the settled datum. Note that the densty s defned as on the frst term n the rght hand sde of Eq. (11). Moreover, the hydraulc gradent s defned as the gradent of the pezo ater head as follo: Pezo Pezo Pezo m W r h ˆ, ˆ ˆ (1)

5 The coeffcent of permeablty s gven by v k (13) All these three physcal quanttes are related to ater partcle. Then, the physcal quantty on the mound marker s estmated by referrng to the physcal quantty on ater partcle ater mound usng the eghtng functon. m mound ater Wr, h (14) ater 4. The threshold for udgng ppng destructon For udgng ppng destructon, the hydraulc gradent on the mound marker, hch s gven by nterpolatng the hydraulc gradent of ater partcle nto the mound marker usng the SPH approxmatng functon Eq. (14), s compared th the crtcal hydraulc gradent proposed by Terzagh (1948). The crtcal hydraulc gradent s defned as c ' G 1 s 1 e (15) here, and ' are the unt eght of ater and the submerged unt eght of the sol respectvely. G s s the specfc gravty of the mound partcle, and s set to.03 referrng to the experment conducted by Kasama (013). e s the vod rato, hch s defned as e=ε/(1-ε). As a fundamental study, the condton of the mound marker s consdered to change from the fxed-condton to the moveable-condton hen the norm of the gradent of the pezo ater head on the mound marker exceeds the crtcal hydraulc gradent defned by Eq. (15). 5. Numercal test As an obectve of ths numercal test, the hydraulc experment conducted by Kasama (013) s selected. In ths secton, the analyss model and results ll be descrbed. 5.1 The expermental model and analyss model The expermental model s shon n Fg.1. The ater level of both outsde and nsde a port as beng kept at the same level durng the experment by the effect of the nstalled dranage functon. There are 5 ater-pressure gauges to measure the pezo ater head. In the experment, the four dfferent cases n ater-level dfference Δh, 40mm, 80mm, 10mm and 145mm, ere conducted. In partcular, ppng destructon as observed hen the ater-level dfference s 145mm. Therefore, only one case th Δ h=145mm s only taken nto consderaton n ths study.

6 Outsde a port 185 Insde a port Water-level dfference Δh 195 : Water-pressure gauge Rubble mound Fg.1 Expermental model Unt:[mm] Sde ve Δh dth:190 Unt : [mm] nflo Top ve dranage dranage nflo dranage dranage Fg. Analyss model dranage Meanhle, the analyss model s presented n Fg.. For the analyss model, 5 reference ponts are also provded at the same locatons as the expermental model and the measured pezo ater n numercal analyss ll be compared th the expermental result. The total number of partcles s approxmately 900 thousand, as ell as the dameter of partcles s set to 1cm. The tme ncrement s 0.001sec. 5. Analyss result of the dstrbuton n pezo ater head Fg.3 shos the dstrbuton n pezo ater head hen the seepage flo nsde the mound s almost steady. The datum, for calculatng pezo ater head usng Eq. (11), s set to the same heght as the ater level nsde the port by referrng to the experment. Note that the dstrbuton n pezo ater head n Fg.3 s the nterpolated dstrbuton on the mound markers. Fg.4 llustrates the measured pezo ater head at 5 aterpressure gauges. As can be seen n fg.4, there are slght dfference n pezo ater head beteen the expermental and analytcal results. Ths error may be consdered to be caused from the dffculty to keep the ater at the settled level n both experment and analyss.

7 Pezo ater head [mm] [mm] Fg.3 Dstrbuton n pezo ater head Experment Analyss Fg.4 Measured pezo ater head at the 5 ater-pressure gauges 5.3 Analyss result of ppng destructon Fg.5 depcts the result on the proecton of ppng destructon usng the crtcal hydraulc gradent. Note that the movement of the casson s not taken nto consderaton n ths study. As shon n Fg.5, t s confrmed that ppng destructon starts up n the rear of the mound and under the casson, as ell as the shape of ppng destructon as reproduced qualtatvely. Hoever, the development of ppng phenomenon n ths analyss s consderably faster than the experment. The reasons may be as follos; () the casson does not play a role to suppress the mound due to no contact force consdered beteen the casson and mound partcles. () the movable mound partcle s not handled n a proper ay, partcularly n ts moton. () the contact beteen mound partcles s not taken nto account n ths study. 6. Concluson Water-pressure gauge numbers In ths research, t s confrmed that good dstrbuton n pezo ater head, hch corresponds quanttatvely to the experment mplemented by Kasama (013), as obtaned by usng a stablzed ISPH method based on the unfed governng equatons. Moreover, the shape of ppng destructon of the mound s reproduced qualtatvely, though there are stll some requred modfcatons to be done.

8 Analyss Experment Red partcle - destructon Fg.5 Proecton of ppng destructon ACCKNOWLEGEMENT Ths ork as supported by JSPS KAKENHI Grant Numbers 15K1484 and Ths ork as also partally supported by the ``Advanced Computatonal Scentfc Program'' of the Research Insttute for Informaton Technology, Kyushu Unversty and ``Jont Usage/Research Center for Interdscplnary Large-scale Informaton Infrastructures(Proect ID:h160048)'' n Japan.

9 REFERENCES Akbar, H. (013), Movng partcle method for modelng ave nteracton th porous structures, Coastal Engneerng 74, pp Akbar, H. (014), Modfed movng partcle method for modelng ave nteracton th mult layered porous structures, Coastal Engneerng., 89, Asa, M., Aly, M.A., Sonoda, Y. and Saka, Y. (01), A Stablzed Incompressble SPH Method by Relaxng the Densty Invarance Condton, Journal of Appled Mathematcs., Volume 01, Artcle ID , 4pages. Irmay, S. (1958), On the theoretcal dervaton of Darcy and Forchhemer formula, Am. Geophys. Unon 39, Kasama, K., Zen, K. and Kasuga, Y. (013), Model experment for the nstablty of casson-type composte breakater under tsunam condton, Proceedng of Coastal engneerng, JSCE, 013. Monaghan, J.J. and Lattanzo, J.C. (1985), A refned partcle method for astrophyscal problems, Astronomy and Astrophyscs, Vol.149, No.1, p Mormoto, T., Asa, M., Kasama, K., Fusaa, K. and Imoto, Y. (014), Unfed numercal analyss beteen the extended Darcy s la and Naver-Stokes Equaton by usng stablzed ISPH method, Journal of Japan Socety of Cvl Engneers, Ser. A (Appled Mechancs), Vol.70, pp. I_13-I_1. Terzagh, K. and Peck, R.B. (1948) Sol mechancs n engneerng practce, Ne York, John ley & sons, nc., London, Chapman & Hall, Ltd. Van Gent, M.R.A. (1995), Wave nteracton th permeable costal structures, Delft Unversty of Technology, Delft, The Netherlands