DEVELOPMENT OF FLUID-SEDIMENT-SEABED INTERACTION MODEL AND ITS APPLICATION. Tomoaki Nakamura 1 and Norimi Mizutani 1

Transcription

1 DEVELOPMENT OF FLUID-SEDIMENT-SEABED INTERACTION MODEL AND ITS APPLICATION Tomoak Nakamura 1 ad Norm Mzuta 1 To provde a computatoal framework for smulatg flud-sedmet-seabed teractos, a three-dmesoal coupled flud-sedmet-seabed teracto model was developed ad appled to seabed respose uder regular waves ad local scourg from tsuam ru-up. For the seabed respose uder the regular waves, the valdty of the model was verfed from comparso wth a aalytcal soluto. For the seabed respose uder the tsuam-duced local scourg, umercal results showed that the model had suffcet computatoal capablty to smulate the seabed respose durg the oset of the local scourg, ad suggested that the model s a useful tool for aalyzg ad evaluatg a geeral complex flud-sedmet-seabed teracto pheomeo a mare evromet. Keywords: flud-sedmet-seabed teracto; umercal model; fte elemet method; couplg scheme; tsuamduced local scourg INTRODUCTION Complex mult-physcs felds formed by dyamc teractos betwee flud flows, seabed profle evoluto, ad seabed respose are atural a mare evromet. Modelg of such flud-sedmetseabed teractos s essetal to accurately smulate ad aalyze spatal ad temporal chage the surface profle of the seabed ad resultg respose of the seabed uder wave acto. To deal wth ths type of teracto pheomeo, umercal models applcable to flud-sedmet teracto (e.g., Lee ad Mzuta 2006, Lu ad García 2008, Nakamura ad Ym 2011) ad those applcable to fludseabed teracto (e.g., Jag et al. 2000, Takahash et al. 2002, Nakamura et al. 2008) have bee developed depedetly. However, o aalytcal ad umercal tools whch cosder the physcs of the etre coupled flud-sedmet-seabed system are curretly avalable. To provde a overarchg computatoal framework for smulatg flud-sedmet-seabed teractos, a three-dmesoal coupled flud-sedmet-seabed teracto model ad a correspodg umercal code are developed ths study. To verfy the valdty of the model, t s appled to seabed respose uder regular waves. To demostrate the computatoal capablty of the model, t s also appled to hydraulc expermets o tsuam-duced local scourg. THREE-DIMENSIONAL COUPLED FLUID-SEDIMENT-SEABED INTERACTION MODEL The three-dmesoal coupled flud-sedmet-seabed teracto model s composed of a ma solver ad three modules. The ma solver s a large-eddy smulato model based o cotuty ad mometum equatos for compressble vscous ar-water mult-phase flow that cosders seepage flow porous meda ad the profle evoluto of a sedmet bed. The frst module s a volume-of-flud (VOF) module based o the mult-terface advecto ad recostructo solver (MARS; Kuug 2000) for ar-water terface trackg. The secod module s a sedmet trasport (ST) module for computg the surface profle evoluto of a sedmet bed duced by bed-load ad suspeded sedmet trasport, ad the dstrbuto of suspeded sedmet cocetrato that cosders all trasport processes of pckup, advecto, dffuso, ad settlg. The thrd module s a fte elemet (FE) module based o a fte elemet method computg the u-p approxmato of the Bot equato for coupled sol-water aalyss of the sedmet bed. I the followg sectos, the FE module ad a couplg procedure betwee the ma solver ad the modules are preseted detal. Explaatos of the ma solver ad the VOF ad ST modules ca be foud Nakamura ad Ym (2011). Fte Elemet Module I ths module, the followg u-p approxmato of the Bot equato, whch s derved from the complete form of the Bot equato assumg that the accelerato of relatve water dsplacemet w s smaller tha that of sol skeleto u, s used after Takahash et al. (2002): u p g (1) j, j, m kk k p u p g 0 (2) t K g s w, w w w, 1 Departmet of Cvl Egeerg, Nagoya Uversty, Furo-cho, Nagoya , Japa 1

2 2 COASTAL ENGINEERING 2014 where u s the dsplacemet vector of sol skeleto; p s the pore-water pressure; j s the effectve stress tesor, whch s defed as postve compresso; j s the stra tesor; x s the posto vector; t s the tme; m s the porosty of sol; s the desty of sol (= (1 m) s + m w ); s s the desty of sol partcles; w s the desty of water; k s s the hydraulc coductvty of sol; K w s the apparet bulk modulus of water; g s the vector of the gravtatoal accelerato (= [0 0 g] T ); ad g s the gravtatoal accelerato. As a costtutve equato, the followg Hooke law for sotropc elastc materals oresstat to tesle force s adopted for smplcty: j kkj 2 j (3) whch ad are the Lamé costats, whch are gve as E (4) 1 12 E G (5) 21 uder compresso, ad = = 0 uder teso. I these equatos, E s the modulus of elastcty, G s the shear modulus of elastcty, ad s the Posso rato. After Sadhu s formulato (e.g., Sadhu ad Wlso 1969), the cremetal form of the fte elemet equato s derved as UU UU UP M 0dU 0 0 du K K du df PU PU PP PP d d d d (6) M 0 P C C P 0 K P Q from Eqs. (1) ad (2), ad computed usg a fte elemet method. I ths equato, du ad dp are the cremetal forms of the global sol dsplacemet vector U ad pore-water pressure vector P, ad M UU, M PU, C PU, C PP, K UU, K UP, K PP, df, ad dq are wrtte as UU T M N N dv (7) k g PU T s M G N (8) PU T T C N m B dv (9) PP T m C N N dv Kw (10) UU T K B DB dv (11) UP T K B mn dv (12) PP T ks K G G dv wg (13) T df dˆ N t ds t (14) dq T N dqˆ ds q (15) whch N ad N are the shape fuctos of du ad dp; D s the stress-stra matrx; B s the dsplacemet-stra matrx (= LN); G = LN ; dt ˆ s the cremetal form of exteral force vector o a ut area of the tracto boudary t ; dq ˆ s the cremetal form of water dscharge o the atural boudary q ; m s the vector otato of Kroecker s delta j ; ad L ad L are the matrx ad vector otatos of the Laplace operator. For stable computato of the spatal dscretzato, the twety-ode quadratc ad eght-ode lear soparametrc brck elemets were used as N ad N. For the tme tegrato, the Newmark ad Crak-Ncolso schemes were adopted for du ad dp. I addto, the modfed Newto-Raphso scheme was used for the olear costtutve equato. Couplg Procedure I the model, the VOF ad ST modules are coected to the ma solver usg a two-way couplg procedure mplemeted at every tme step to esure flud-sedmet teracto. To esure the cotuty of seepage flow velocty ad pore-water pressure o the surface of a sedmet bed, the FE module s coected to the ma solver usg a oe-way couplg procedure proposed by Mzuta et dv

3 COASTAL ENGINEERING al. (1996). To take to accout the surface profle evoluto of the bed duced by sedmet trasport, the FE module s also coected to the ST module usg a same procedure as the smulato of embakmet costructo ad excavato. Specfcally, ode force equvalet to the weght of added elemets acts dowward for deposto, whle ode force equvalet to the weght of deleted elemets acts upward for eroso. Fgure 1 shows the couplg procedure, whch s mplemeted as follows: 1. The flow velocty v ad pressure p at the th tme step are computed usg the ma solver. 2. The VOF fucto F s computed usg the VOF module. 3. The suspeded sedmet cocetrato C ad the heght of the surface of the sedmet bed above a referece level z s are computed usg the ST module. 4. The seepage flow velocty ad pore-water pressure o the surface of the bed are terpolated from v ad p, ad the values of du ad dp are computed usg the FE module The ma module s mplemeted aga to compute the values of v 1 ad p at the ext ( + 1)th tme step from v, p, F, ad z s. 6. Ths procedure s repeated utl the value of reaches the umber of specfc tme steps. Fgure 1. Couplg procedure. SEABED RESPONSE UNDER REGULAR WAVES Dyamc respose of the seabed uder regular waves was computed ad compared wth a aalytcal soluto derved by Yamamoto (1977). Fgure 2 shows a schematc of a computatoal doma. As show ths fgure, the horzotal flat seabed was set a pt wth a legth of 6.0 m ad a depth of 0.3 m, ad regular waves wth a perod T of 1.0 s, a heght H of 0.03 m at the ceter of the seabed (x = 5.0 m), ad a stll water depth h of 0.3 m were geerated. The values of the meda gra sze of sol partcles d 50, porosty of the seabed m, desty of the sol partcles s, shear modulus of elastcty G, ad Posso rato were set at 0.45 mm, 0.4, kg/m 3, 10 8 N/m 2, ad 0.33, ad the hydraulc coductvty of the seabed k s was determed from the followg equatos (Bear 1972): m gd50 ks (16) m w Fgure 2. Computatoal doma for the seabed respose uder the regular waves.

4 4 COASTAL ENGINEERING 2014 (a) Pore-water pressure P. (a) Pore-water pressure P. (b) Horzotal dsplacemet. (b) Horzotal dsplacemet. (c) Vertcal dsplacemet. Fgure 3. Comparso wth the aalytcal soluto for K w = N/m 2. (c) Vertcal dsplacemet. Fgure 4. Comparso wth the aalytcal soluto for K w = N/m 2. whch w s the kematc molecular vscosty of water. The apparet bulk modulus of water K w was chaged to N/m 2 ad N/m 2. The other parameters were the same as Nakamura ad Ym (2011). Note that the ST module were ot used because t was assumed Yamamoto (1977) that the surface profle of the seabed dd ot chage. Fgures 3 ad 4 show a comparso of pore-water pressure P, horzotal dsplacemet of the seabed skeleto, ad vertcal dsplacemet of the seabed skeleto at the ceter of the seabed (x = 5.0 m). As show these fgures, the ampltudes of P ad decrease the egatve z drecto, whle that of at z = 0.4 m s larger tha that ear the surface ad bottom of the seabed. Furthermore, the phase of P gradually delays the egatve z drecto for K w = N/m 2 ulke K w = N/m 2. From Fgs. 3 ad 4, t s also foud that the umercal results are cosstet wth the aalytcal soluto terms of the atteuato ad phase of P ad the dampg of ad. As a result, t was demostrated that the valdty of the FE module ad the couplg procedure was verfed agast the aalytcal soluto.

5 COASTAL ENGINEERING SEABED RESPONSE UNDER TSUNAMI-INDUCED LOCAL SCOURING Computatoal Doma ad Codto To exame the computatoal capablty of the model, t was appled to hydraulc expermets o tsuam-duced local scourg aroud a lad structure (Nakamura et al. 2008). Fgure 5 shows a schematc of a computatoal doma modeled from a 1/20 scale expermetal setup. As show ths fgure, a mpermeable seawall ad a sad substrate wth a meda gra sze d 50 of 0.2 mm were set o a mpermeable bed, ad a mpermeable structure wth a 0.14-m-wde square cross-secto was fxed at 0.28 m ladward from the ladward surface of the seawall. Because of the lack of avalable expermetal data, the values of the porosty of the substrate m, desty of sad partcles s, shear modulus of elastcty G, Posso rato, ad apparet bulk modulus of water K w were assumed to be 0.4, kg/m 3, 10 8 N/m 2, 0.33, ad N/m 2, ad the hydraulc coductvty k s was estmated to be m/s from Eq. (16). The other parameters were the same as the prevous secto. I ths study, a represetatve wave codto was selected from over 150 cases the hydraulc expermets. The stll water depth h was set at m, ad a sgle leadg log-perod wave wth a durato of 6.0 s ad a heght of m at W1 (see Fg. 5) was geerated toward the substrate. Note that the computatoal accuracy of the ma solver, VOF module, ad ST module was cofrmed agast expermetal data terms of water surface elevato, pore-water pressure, ad fal surface profle of the substrate. For detals, see Nakamura ad Mzuta (2012). Fgure 5. Computatoal doma for the seabed respose durg the oset of the tsuam-duced local scourg. Numercal Results ad Dscusso Fgures 6, 7, ad 8 show pore-water pressure P, mea effectve stress m ad relatve effectve stress rato (RESR; = 1 m / m0 ; m0 : tal value of m ), whch the value s oe of the odmesoal dces for evaluatg lquefacto, wth lquefacto occurrg whe = 1. From Fgs. 6(a) ad 7(a), at whch the ru-up tsuam begs to touch the seaward surface of the structure, t s observed that the value of P creases ad that of m decreases ear the surface of the substrate at the ladward sde of the seawall. Ths decrease m, as show Fg. 8(a), causes a crease, suggestg the oset of lquefacto. Coversely, the value of decreases at the seaward sde of the structure because of a crease m. After that, as dcated Fg. 6(b), the value of P creases ear the surface of the substrate at the seaward sde of the structure. However, aroud the seaward corer of the structure, the P value remas close to zero ad the load o the substrate decreases due to local scourg, resultg a decrease m ad a resultat crease (see Fgs. 7(b) ad 8(b)). Ths result suggests that lquefacto occurs durg the local scourg. Smlar pheomeo was observed Tok et al. (2003). After the tsuam subsdes, as show Fgs. 6(c), 7(c), ad 8(c), t s observed that although the value of P returs to zero, the value remas hgh. From these results, t s foud that the model has suffcet computatoal capablty for assessg dyamc respose of a sedmet bed durg the oset of tsuam-duced local scourg. Ths also

6 6 COASTAL ENGINEERING 2014 (a) t = 7.2 s. (b) t = 7.7 s. (c) t = 15.2 s. Fgure 6. Dstrbuto of the pore-water pressure P. (a) t = 7.2 s. (b) t = 7.7 s. (c) t = 15.2 s. Fgure 7. Dstrbuto of the mea effectve stress m. suggests that the model s a useful tool for aalyzg ad evaluatg a geeral complex flud-sedmetseabed teracto pheomeo a mare evromet.

7 COASTAL ENGINEERING (a) t = 7.2 s. (b) t = 7.7 s. (c) t = 15.2 s. Fgure 8. Dstrbuto of the relatve effectve stress rato (RESR). CONCLUDING REMARKS A three-dmesoal flud-sedmet-seabed teracto model that cosstetly cosdered the physcs of the etre coupled flud-sedmet-seabed system was developed ad appled to seabed respose uder regular waves ad local scourg from tsuam ru-up. The model s composed of a ma solver based o a large-eddy smulato model for compressble vscous ar-water mult-phase flow, a volume-of-flud module for ar-water terface trackg, a sedmet trasport module for computg the surface profle of a sedmet bed ad suspeded sedmet cocetrato, ad a fte elemet module for coupled sol-water aalyss of the bed. The three modules were coected to the ma solver usg a couplg procedure that esured flud-sedmet-seabed teractos. For the seabed respose uder the regular waves, t was foud that the valdty of the fte elemet module ad the couplg procedure was verfed from comparso wth a aalytcal soluto terms of pore-water pressure ad dsplacemet of the seabed skeleto. For the seabed respose uder the tsuam-duced local scourg, t was foud that the model had suffcet computatoal capablty to smulate pore-water pressure, mea effectve stress, ad relatve effectve stress rato durg the oset of the local scourg. From these results, t was cofrmed that the model s a useful tool for aalyzg ad evaluatg a geeral complex flud-sedmet-seabed teracto pheomeo a mare evromet. ACKNOWLEDGMENTS Ths research was partly supported by the Japaese Mstry of Lad, Ifrastructure, Trasport ad Toursm uder the Research ad Developmet of Rver ad Sedmet Cotrol Techology (Coastal Techology Feld; prcpal vestgator: Norm Mzuta). REFERENCES Bear, J Dyamcs of Fluds Porous Meda, Amerca Elsever Pub. Co., New York, 764 p. Jag, Q., S. Takahash, Y. Murash, ad M. Isobe A VOF-FEM model for the teracto amog waves, sols ad structures, Proceedgs of Coastal Egeerg, JSCE, 47, ( Japaese). Kuug, T MARS for multphase calculato, Computatoal Flud Dyamcs Joural, 9(1), 1-10.

8 8 COASTAL ENGINEERING 2014 Lee, K. H., ad N. Mzuta Local scour ear a vertcal submerged breakwater ad developmet of ts tme doma aalyss, Aual Joural of Coastal Egeerg, JSCE, 53, ( Japaese). Lu, X., ad M. H. García Three-dmesoal umercal model wth free water surface ad mesh deformato for local sedmet scour, Joural of Waterway, Port, Coastal, ad Ocea Egeerg, ASCE, 134(4), Mzuta, N., W. G. McDougal, ad A. M. Mostafa BEM-FEM combed aalyss of olear teracto betwee wave ad submerged breakwater, Proceedgs of 25th Iteratoal Coferece o Coastal Egeerg, ASCE, Orlado, FL, Nakamura, T., Y. Kuramtsu, ad N. Mzuta Tsuam scour aroud a square structure, Coastal Egeerg Joural, JSCE, 50(2), Nakamura, T., ad S. C. Ym A olear three-dmesoal coupled flud-sedmet teracto model for large seabed deformato, Joural of Offshore Mechacs ad Arctc Egeerg, ASME, 133(3), Nakamura, T., ad N. Mzuta Sedmet trasport model cosderg pore-water pressure surface layer of seabed ad ts applcato to local scourg due to tsuam, Joural of Japa Socety of Cvl Egeers, Seres B2 (Coastal Egeerg), JSCE, 68(2), I_216-I_220 ( Japaese). Sadhu, R. S., ad E. L. Wlso Fte elemet aalyss of seepage elastc meda, Joural of the Egeerg Mechacs Dvso, ASCE, 95, EM3, Takahash, S., K. Suzuk, Y. Murash, ad M. Isobe U-π form VOF-FEM program smulatg wave-sol teracto: CADMAS-GEO-SURF, Proceedgs of Coastal Egeerg, JSCE, 49, ( Japaese). Tok, S., H. Yeh, F. Kato, ad S. Sato Tsuam scour aroud a cylder, Joural of Flud Mechacs, 496, Yamamoto, T Wave duced stablty seabeds, Proceedg of Coastal Sedmets 77, ASCE,