THREE-DIMENSIONAL NUMERICAL MODELING FOR THE MOTION OF STONES WITH DIFFERENT SIZES AND SHAPES IN STREAMS

Size: px

Start display at page:

Download "THREE-DIMENSIONAL NUMERICAL MODELING FOR THE MOTION OF STONES WITH DIFFERENT SIZES AND SHAPES IN STREAMS"

Kerry Skinner
5 years ago
Views:

1 Proceedng of the 10th Intl. Conf.on Hydrocence & Engneerng, Nov. 4-7, 01, Orlando, Florda, U.S.A. THREE-DIMENSIONAL NUMERICAL MODELING FOR THE MOTION OF STONES WITH DIFFERENT SIZES AND SHAPES IN STREAMS Tomoo Fukuda 1, Sho Fukuoka and Tatuhko Uchda 3 ABSTRACT The movng mechanm of tone wth dfferent ze and hape n tream requred to evaluate the edment tranport rate and bed varaton n gravel bed rver. One of the author made a real cale experment of tone movng n tream and meaured movng traectore of tone. In th paper, a three-dmenonal numercal model preented whch mulate the moton of a cloud of tone wth dfferent ze and hape n tream. The valdaton of the model checked by the real cale experment. And the mechanm of tone movng n tream dcued by ung the reult of the real cale experment and the mulaton. It demontrated that hape of tone are mportant for etmatng ther moton n tream, and the moton of tone contanng ut a few percent of volume denty n tream have gnfcant effect on momentum tranport of the flow n the vertcal drecton. 1. INTRODUCTION The movng mechanm of tone n tream requred to etmate the edment tranport rate and bed varaton n gravel bed rver. Generally, bed materal n gravel bed rver are wdely dtrbuted n ze and n hape. Therefore, regardng the mechanm of tone movng n tream, t mportant to elucdate effect of ze and hape of tone on collon between tone and nteracton between tone and water. Fukuoka et al. (005) made a real cale experment of a cloud of tone movng n tream wth a 45m long concrete channel and meaured movng traectore of colored tone by ung mage analye (herenafter referred to a the real cale experment). It ha been ndcated that tone hape have gnfcant effect on ther rregular altaton and mgraton. In order to elucdate the dynamc mechanm of tone movng n tream, the author develop a three-dmenonal numercal model, n whch the moton of each haped tone mulated wth the Lagrangan method and flud moton around tone are mulated drectly wth the Euleran method. 1 Graduate tudent, Department of Scence and Engneerng, Chuo Unverty Kauga, Bunkyo-ku, Tokyo, , Japan (t-fukuda66@cvl.chuo-u.ac.p) Profeor, Reearch and Development Intatve, Chuo Unverty, Kauga, Bunkyo-ku, Tokyo, , Japan (fuku@tamacc.chuo-u.ac.p) 3 Aocate Profeor, Reearch and Development Intatve, Chuo Unverty, Kauga, Bunkyo-ku, Tokyo, , Japan (utda@tamacc.chuo-u.ac.p)

2 Proceedng of the 10th Intl. Conf.on Hydrocence & Engneerng, Nov. 4-7, 01, Orlando, Florda, U.S.A. tone Dfferent color ndcate dfferent denty Fgure 1 Combned tone model Fgure Concept of the flud analy Many prevou tude on a numercal computaton of tone movng n tream had attempted to evaluate hydrodynamc force baed on emprcal coeffcent (e.g. Gotoh et al., 000). However, drag coeffcent of varou haped tone movng n the cloud ha remaned unknown. On the other hand, numercal method have been propoed to evaluate the nteracton between flud flow and arbtrary haped old obect n whch flud moton around tone and hydrodynamc force are calculated drectly by ung mall computatonal cell (e.g. Uhma et al., 008). A rgd body compoed of tetrahedron element ha been ued a an arbtrary haped obect (Suzuk et al., 003). Although the combned tetrahedron element model utable to evaluate rgd body properte uch a ma, gravty center and moment of nerta, proper numercal procedure are requred to detect poton of contact and collon between obect. Matuhma et al. (004) propoed the tone model of combned phere to evaluate contact force ealy by rad and the dtance between two phere. However combned phere model ha dffculte n calculaton of rgd body properte due to overlapped phere. The author developed the three-dmenonal numercal model for multphae flow wth tone of dfferent ze and hape movng n tream n whch the combned phere are ued a tone model (Fgure 1) and hydrodynamc force of tone are calculated drectly. In the model, tone moton are mulated a rgd body. We evaluated rgd body properte of tone cont of combned phere by numercal ntegraton wth uffcently mall cell around and n a tone. The mot dtnctve feature of the preent numercal model are n the ablty to nvetgate dynamc nteracton between a cloud of rregular haped tone and water n detal whch mpoble to meaure n expermental channel and feld.. NUMERICAL MODEL In the preent numercal model, flud moton are mulated wth the Eauleran method and tone moton are mulated wth the Lagrangan method. To take nto account the effect of old phae on lqud phae, flud moton are mulated by governng equaton of multphae flow (Fgure ). Flud force actng on tone are evaluated drectly by ntegratng the equaton for multphae flow moton. The Dcrete Element Method (Cundall and Strack, 1979) appled to mulate moton of arbtrary haped tone. Arbtrary hape of tone are made by combnaton of everal mall phere (Fgure 1). The moton of the tone are mulated by momentum and angular momentum equaton for a rgd body..1 Governng Equaton of Flud Moton

3 Proceedng of the 10th Intl. Conf.on Hydrocence & Engneerng, Nov. 4-7, 01, Orlando, Florda, U.S.A. Flud moton mulated by one-flud model for multphae flow ung Smagornky model a the ubgrd turbulence model: Calculaton of Flud Moton Contnuty Equaton, Momentum Equaton Calculaton of flud moton wth one-flud model of mult phae flow n whch tone are dealt wth contnuum dfferng from water Calculaton of the preure dtrbuton around each tone wth flud calculaton cell maller than a tone. In calculaton of flud moton, phycal properte (denty) and velocte n the part of tone are et by tone data n each tme tep. The effect of the old phae to the lqud phae evaluated a the tree of old part actng on the lqud part. Set of tone poton and velocty n mult-phae flow Calculaton of flud force actng on tone Calculaton of Stone Moton Momentum Equaton Smulaton of tone done by tranlatonal and angular momentum equaton of rgd body (Modelng of Shaped Stone) Shaped Stone Compoed of Combned Sphere Evaluaton of the Rgd Body Properte Rgd body properte are calculated wth numercal ntegraton by makng uffcent mall cell n tone (External Force Actng on Stone) Gravty Force, Flud Force, Contact Force between Stone (Flud Force) Flud force actng on tone are calculated wth volume ntegraton of force actng on the tone part n multphae flow. (Contact Force) Contact force actng on the gravty center of tone are calculated by ntegraton of contact force of each phere. Fgure 3 Concept of mulaton model for tone movng n tream u x 0 (1) u t u u x F 1 P x u t x x (), t C S SS, 1 f, f f l 1 f S 1 u x g u x (3) (4) u u 1 u (5) where u :denty-weghted average n flud velocty, P : normal tre, : calculaton grd ze, f :flud and old volume fracton n flud calculaton cell, :old volume fracton n flud calculaton cell except ga part, :phycal property (denty, coeffcent of dynamc vcoty ), uffx l, and g denote lqud phae, old phae and ga phae, repectvely. Contnuty equaton and momentum equaton are olved by the SMAC cheme wth taggered grd. Water urface evaluated wth volume fracton of flud f n flud calculaton cell baed on the VOF method (Hrt and Nchol, 1981). f fu t x 0 l l (6) 3

4 Proceedng of the 10th Intl. Conf.on Hydrocence & Engneerng, Nov. 4-7, 01, Orlando, Florda, U.S.A. Stone Surface Flud Calculaton Cell Sub-Cell Fgure 4 Evaluaton of tone volume wth ub-cell Fgure 5 Cell mage for calculaton of rgd body properte Sold moton n flud flow recognzed and calculated by u and phycal property whch are et n conderng old volume fracton n flud calculaton cell by eq. (4) and eq. (5). Sold velocty u n eq. (5) calculated equaton below: u r ωr (7) G f where r G :tranlatonal velocty vector of tone (velocty at the gravty center), ω :rotatonal velocty vector of tone, r f :poton vector from gravty center to the evaluaton pont of u. The old volume fracton n flud calculaton cell n eq. (4) and eq. (5) calculated wth ung ub-cell plttng a flud calculaton cell (hown n Fgure 4).. Modelng of Arbtrary Shaped Stone Arbtrary hape of tone are made by the combnaton of everal mall phere to detect contact and collon pont between tone a een n Fgure 1. On the other hand, t dffcult to evaluate geometre of lapped part of phere. We propoe the numercal ntegraton for calculatng rgd body properte of tone by ung mall cell n tone (hown n Fgure 5). The rgd body properte of tone n moton are not changed, therefore the calculaton of rgd body properte need to be done ut once at the begnnng of the mulaton..3 Governng Equaton of Stone Moton n Stream Irregular haped tone moton are mulated by tranlatonal and rotatonal equaton of moton for the rgd body: ω r I 1 1 r R r Mr r G N ω I ω Mg F r f F c (8), N N f N (9) c where r G :tranlatonal acceleraton vector of tone, uffx f, c : flud force, contact force between tone, repectvely, uffx r :component n local coordnate ytem of each tone, R :tranlatonal tenor from the global coordnate ytem to the local coordnate ytem and I :tenor of momentum nerta. After the calculaton of rotatonal velocty n local coordnate ytem ω r n eq. (9), rotatonal velocty ω n global coordnate ytem olved by coordnate tranformaton. In calculaton of 4

5 Proceedng of the 10th Intl. Conf.on Hydrocence & Engneerng, Nov. 4-7, 01, Orlando, Florda, U.S.A. coordnate tranformaton from the local coordnate ytem of the tone to the global coordnate ytem, the quaternon ued n tead of tranlatonal tenor R (Uhma et al., 008)..4 Evaluaton of Flud Force Flud force actng on the tone are evaluated wth ntegratng preure and hear tre term of eq. () n the tone volume (Uhma et al., 008): F f, t x P u d (10) x x rcp,1 F cp,1 Gravty Center of a Stone Contact detecton and calculaton of contact force are conducted wth each mall phere compong tone N P u k d (11) xlxl f, krf, t xk where F f, : component of flud force, N f, : component of torque caued by flud force, r f, : poton vector from the gravty center of tone to the flud calculaton cell, : an area occuped by a tone Lev-Cvta ymbol. and k.5 Evaluaton of Contact Force Fgure 6 Evaluaton of contact force Contact force actng between tone are calculated by mall phere formng a haped tone(hown n Fgure 6). The force and torque actng on tone are calculated by the ummaton of the contact force: F F c cp, n, N r, n F c cp cp, n (1) where F c, N c : contact force and torque, F cp, n :contact force actng on each phere, r cp, n : poton vector from the gravty center of tone to the contact pont. 5

Percentage (%) Z(m) Proceedng of the 10th Intl. Conf.on Hydrocence & Engneerng, Nov. 4-7, 01, Orlando, Florda, U.S.A.

3 0. 0.1 0-0.1-0.6-0.4-0. 0 0. 0.4 0.6 Y(m) Fgure 7 Shape of the expermental channel Fgure 8 Channel cro ecton 100 80 60 Exp. Cal.

Calculaton cell Sze 0.0 m for One Stone Movng Smulaton for a Cloud of Stone Smulaton 0.01 m t :Tme Step for Flud Calclaton 1.0 10-4 C :Smagornky Coeffcent 0.

9 10-4 kg/m 3 w po:poon' Rato 0.33 :Vcoty Coeffcent of Stone 8.9 10-4 Pa h:coeffcent of Dahpot 0.

6 Percentage (%) Z(m) Proceedng of the 10th Intl. Conf.on Hydrocence & Engneerng, Nov. 4-7, 01, Orlando, Florda, U.S.A. z y x 45m 0m Meaurng Secton of Movng Stone n the Experment (Gla Wall Secton) 4m Y(m) Fgure 7 Shape of the expermental channel Fgure 8 Channel cro ecton Exp. Cal Gran Sze (mm) Fgure 9 Gran ze dtrbuton Fgure 10 Shape of tone Table 1 Parameter ued n the mulaton x, y, z :Flud Calculaton cell Sze x, y, z :Flud Calculaton cell Sze 0.0 m for One Stone Movng Smulaton for a Cloud of Stone Smulaton 0.01 m t :Tme Step for Flud Calclaton C :Smagornky Coeffcent w :Denty of Water 1,000 kg/m 3 t:tme Step for DEM Calclaton :Denty of Stone,650 kg/m 3 E :Elatc Modulu Pa :Vcoty Coeffcent of Water kg/m 3 w po:poon' Rato 0.33 :Vcoty Coeffcent of Stone Pa h:coeffcent of Dahpot 0.11 Contact force between tone are calculated by the Dcrete Element Method (Cundall and Strack, 1979) and the prng contant and coeffcent of dahpot are calculated by equaton (13)- (15) (Gotoh, 004): k n 4 r1 r 9 1 r r E 1 po 1 3 en (13) 0 k k n 1 1 po (14) where uffx n :a component of drecton from center of phere to the contact pont, uffx :orthogonal two drecton on the tangental plane, r 1, r :rad of contactng two phere m :ma of contactng phere. m 1, c m m 1 n h k n m1 m, c cn 0 (15) 6

7 Lateral Dtance Y(m) Dtance from Channel Bottom Z(m) Proceedng of the 10th Intl. Conf.on Hydrocence & Engneerng, Nov. 4-7, 01, Orlando, Florda, U.S.A. 3. APPLICATION TO REAL SCALE EXPERIMENT 3.1 The Condton of the Numercal Analy Fgure 7 how the expermental channel wth 45m long and 1:0 bed gradent (Fukuoka et al., 005). In the experment, movng traectore of tone were meaured n the gla wall ecton. Fgure 8 how the channel cro ecton. In the numercal analy, the coordnate ytem defned a x ax n the longtudnal drecton and z ax n the vertcal drecton of the channel bed, a hown n Fgure 7. We mulated by the model for 38m long from uptream end to the downtream end of the channel. In order to reproduce coare aggregate of the expermental channel bed expoed by rollng tone, mulaton channel bed compoed of phere wth rregular rad and heght varyng n the range of 0.0m. The dcharge of 0.5m 3 / gven at the uptream end of the channel, and zero preure ntenty gven at the downtream end. Fgure 9 how the gran ze dtrbuton of uppled tone nto the mulaton channel. The dameter of the rregularly haped tone are gven a correpondng phere dameter wth the ame volume. In the mulaton, 4 dfferent type of tone-hape whch are compoed of 8-9 mall phere are prepared a hown n Fgure 10. Stone uppled at the uptream end of the mulaton channel are 360kg n 4 econd a well a the experment. In the real cale experment, the moton of one tone n tream wa alo nvetgated. Exp.(D=0~30mm) Exp.(D=30~40mm) Exp.(D=40~60mm) Exp.(D=60~90mm) Exp.(D=90~10mm) Cal. Shaped Stone(D=105mm) Cal. Sphere Stone(D=105mm) Longtudnal Dtance X (m) Fgure 11 The Traectore of one tone (x-z plane) Exp.(D=0~30mm) Exp.(D=30~40mm) Exp.(D=40~60mm) Exp.(D=60~90mm) Exp.(D=90~10mm) Cal. Shaped Stone(D=105mm) Cal. Sphere Stone(D=105mm) Longtudnal Dtance X (m) Effect of the tone hape on one tone moton are etablhed by the comparon of mulaton reult of No.1 tone n Fgure 10 and a phercal tone. Table 1 how the parameter ued n the mulaton. The tme tep of dcrete element method et one hundredth of tme tep of flow calculaton whch keep mulated moton of tone tably. 3. One Stone Movng Fgure 1 Traectore of one tone (x-y plane) 7

8 Proceedng of the 10th Intl. Conf.on Hydrocence & Engneerng, Nov. 4-7, 01, Orlando, Florda, U.S.A. Fgure 11 and Fgure 1 how the mulated traectore of one tone movng n a tream wth a haped tone and a phere on x-z plane and x-y plane, repectvely. Thee fgure alo how meaured traectore of a tone meaured by the experment. The mulated haped tone altate frequently, and flow down a wde range acro the channel a well a the expermental reult. On the other hand, mulated phere do not altate o frequently, and flow down ut only the center of the channel, whch reveal the mportance of hape of tone on the tone moton. The reaon of the dfference of traectore between the haped and phere tone are from that the frontal area of the haped tone larger than that of the phere tone wth the ame volume, and the haped tone can altate due to the collon wth the channel bottom n rregular manner for the dfferent hape from phere. It concluded that tone hape are gnfcant effect on ther moton n tream. 3.3 A cloud of Stone Movng The numercal model appled to the movement of a cloud of tone n the real cale experment (Fukuoka et al., 005). Fgure 13 the nap hot of the mulated tone movng n the tream. In the experment, all of uppled tone reached downtream end of the channel n about 0 econd. However n the mulated reult, ome mall tone remaned at the upplyng ecton n 0 econd after tone upply. Th becaue flud cell ze of calculaton a lttle bt larger than the utable ze for calculatng flud moton around mall phere. Therefore, we nvetgate nteracton between movng tone and tream at uffcent downtream ecton. Fgure 14 how the comparon of expermental and numercal vertcal dtrbuton of tone velocty. The fgure alo how a mulated water velocty dtrbuton. The mulated tone velocte ndcate almot the ame value a the expermental value whch become almot unform n vertcal drecton. Fgure 15 how vertcal dtrbuton of extence probablte of tone. The mulated extence probablte of tone for D < 60mm (D=30mm-60mm) are larger around the bottom and maller at the upper poton than thoe of the experment. However, for D>60mm (D=60mm-90mm, 90mm- 10mm), the mulated probablte have good agreement wth thoe of the experment. Flow 4m 6m Water Surface Yellow :D=90-10m Green :D=60-90m Lght Blue :D=40-60m Red :D=30-40m Blue :D=0-30m Meaurng Secton 8m The channel wall not hown n the fgure. Fgure 13 Snap hot of the meaurng tone ecton n the analy 8

9 Dtance from Channel Bottom Z(m) Dtance from Channel Bottom (m) Dtance from Chanel Bottom(m Dtance from Channel Bottom Z(m) Dtance from Chanel Bottom(m Dtance from Channel Bottom Z(m) Dtance from Channel Bottom Z(m) Dtance from Chanel Bottom Z(m) Proceedng of the 10th Intl. Conf.on Hydrocence & Engneerng, Nov. 4-7, 01, Orlando, Florda, U.S.A. 水路中央部の底面からの距離 Z(m) D=0-30mm D=30-40mm D=40-60mm Exp. D=60-90mm D=90-10mm Flow Velocty D=0-30mm D=30-40mm Cal. D=40-60mm D=60-90mm D=90-10mm 流速, 礫の流下速度 (m/) Velocty Velocty (m/) (m/) Fgure 14 Vertcal dtrbuton of tone velocty 0.5 Exp. Cal. 0.5 Exp. Cal. 0.5 Exp. Cal Extence Probablty of Stone (%) Probabrty of tone(%) D=30-60mm Extence Probablty of Stone (%) Extence Probablty of Stone (%) Probavrty of Stone(%) D=60-90mm Probavrty of Stone(%) D=90-10mm Fgure 15 Vertcal dtrbuton of extence probablte of tone tone velocty 3.4 Dcuon of Movng Mechanm of Stone n Stream The tone movng mechanm dcued by the mulaton and the experment. At frt, force actng on the tone and tone movng confguraton are nvetgated. Fgure 16 and 17 how mulated vertcal dtrbuton of volume denty of tone, force actng on tone n unt volume n x and z drecton, repectvely. In Fgure 17, total force n x and z drecton change to the oppote drecton around mddle of the depth. The drecton of the force actng on tone uptream-upward drecton n the lower part of the depth and downtream drecton n the upper part of the depth. Fgure 18 how tone traectore relatve to the vewer movng wth the average tone velocty.65m/. Stone move to the uptream-upward drecton and fall to the downtream drecton to the average tone velocty. 9

10 Dtance from Channel Bottom Z (m) Dtance from Channel Bottom Z(m) Dtance from Channel Bottom Z(m) Dtance from Channel Bottom Z(m) 水路底面からの高さ Z(m) Proceedng of the 10th Intl. Conf.on Hydrocence & Engneerng, Nov. 4-7, 01, Orlando, Florda, U.S.A Volume 体積濃度 Denty (%) of Stone (%) Fgure 16 Vertcal dtrbuton of volume denty of tone Total Force Contact Force FcF c Flud Force F Ff f Gravty Force Mg Total Force Contact Force Fc c Flud Force FfF f Gravty Force Mg F/((σ/ρ-1)ρVg) X-Drecton F/((σ/ρ-1)ρVg) Z-Drecton Fgure 17 Vertcal dtrbuton of force actng on tone Relatve Dtance from the Flowng Down Vew wth the Average Stone Velocty (m) Fgure 18 Stone Traectore relatve to the vewer movng wth tone velocty Next, the mechanm of tone-water multphae flow dcued. Fgure 19 how x and z drectonal force actng on a unt volume evaluated at the center of the channel on tme average. Contact force F c calculated by eq. (1) are evaluated at gravty center of tone. Contact force n the x-drecton larger near the bottom. On the other hand, momentum exchange by the tone larger around the mddle part of the depth. Apparent hear tre and normal tre dtrbuton are hown n Fgure 0. Here, the apparent tree are calculated by the ntegraton of the force hown n Fgure 19 from the water urface to the bottom wth the aumpton of lttle varaton of tmeaveraged multphae flow tructure n x and y drecton. In other word, the hear tre and normal tre are decrbed by net momentum exchange n x and z drecton acro the z ourface, repectvely. From thee fgure, the rato of contact force to the total hear tre larger than the rato of contact force n normal tre. The above ndcate that the hydrodynamc force actng on tone n x-drecton much larger than that n z-drecton, and the tream-we momentum of tone are tranported by collon. It notced that depte ut only 1-% of the tone volume denty, the rato of contact force and momentum exchange between tone to the total hear tre become almot 70% around the bed. In other word, tone wth a few percent of volume denty n tream play a gnfcant effect on vertcal momentum tranport. 10

11 Dtance from Channel Bottom Z (m) Dtance from Channel Bottom Z (m) Dtance from Channel Bottom Z (m) Dtance from Channel Bottom Z (m) Dtance from Channel Bottom Z (m) Dtance from Channel Bottom Z (m) Dtance from Channel Bottom Z (m) Dtance from Channel Bottom Z (m) 水路底面からの高さ Z(m) 水路底面からの高さ Z(m) 水路底面からの高さ Z(m) X - 軸方向力 0(10 4 N) X 軸方向力 (10 4 N) Proceedng of the 10th Intl. Conf.on Hydrocence & Engneerng, Nov. 4-7, 01, Orlando, Florda, U.S.A. -1, , ,000 10,000 15,000 Force n X Drecton (N/m 3 ) Force n X-Drecton (N/m 3 ) Force n Force Z Drecton n Z-Drecton (N/m 3 ) (N/m 3 ) 全合力 Total Force 全合力衝突力 Contact Force 衝突力重力 Force Acted between Flud Momentum Exchange by Stone 重力流体間の力流体間の力石による運動量交換石による運動量交換 Momentum Exchange u Contact Force: F c u by Stone: x Volume Fracton of Stone n a Flud Calculaton Cell: Force Acted between Flud and Momentum Exchange by Flud: P u t 1 x x x u u x Fgure 19 Vertcal dtrbuton of force n tone-tream multphae flow Shear Stre (Pa) Shear Stre zx (Pa) Total 全応力 Stre Contact between 衝突による応力 Stone Force Acted between Flud and 流体間の応力 Momentum Exchange by Flud Momentum 石の運動量交換を応 Exchange by Stone 力として表示したもの ,000 1,500,000, ,000 1,500,000,500 Normal Stre 等方応力 (Pa) Normal Stre (Pa) Stre n the Cae of Hydrotatc Preure Fgure 0 Vertcal dtrbuton of tre 4. CONCLUSION The three-dmenonal numercal model for flow wth haped tone wa developed and appled to the reult of real cale experment regardng tone movng n tream. The model gave good explanaton to the reult of real cale experment, uch a tone traectore, vertcal dtrbuton of velocte and extence probablte of tone. From the nvetgaton of one tone movng n tream, t wa clarfed that expermental traectory wth frequent altaton and lateral mgraton could not be reproduced n the mulaton unle the rregular tone hape condered. It wa demontrated that the moton of tone wth ut a few percent of volume denty n tream play a gnfcant role of vertcal momentum tranport from the conderaton of dynamc mechanm of a cloud of tone movng n tream. REFERENCES Cundall,P.A. and Strack,O.D.L. (1979) A dcrete numercal model for granular aemble, Geotechnque, Vol.9, No.1, pp

12 Proceedng of the 10th Intl. Conf.on Hydrocence & Engneerng, Nov. 4-7, 01, Orlando, Florda, U.S.A. Gotoh,H., Yeganeh,A. and Saka,T. (000) Couplng of Multphae-Flow Model and Dtnct Element Method for Smulaton of Sedment Tranport under Hgh Bottom Shear, Proceedng of JSCE, Vol. 649/II-51, pp.17-6, n Japanee. Gotoh,H. (004) Computatonal Mechanc of Sedment Tranport, Morkta Shuppan Co.,Ltd., n Japanee. Fukuoka,S., Watanabe,A., Shnohara,Y., Yamahta,S. and Satou,K. (005) Movement Mechanm for a Ma of Gravel Flow wth Hgh Speed and Etmaton of Chanel Bed Abraon, Proceedng, Advance n Rver Engneerng, JSCE, Vol.16, pp.91-96, n Japanee. Hrt,C.W. and Nchol,B.D. (1981) Volume of Flud(VOF)Method for the Dynamc of Free Boundare, J.Comput.Phy., 39, pp Kahma,T., Takguch,S., Hamaak,H. and Myake,Y. (001) Turbulence Structure of Partcle- Laden Flow n a Vertcal Plane Channel Due to Vortex Sheddng, JSME Sere B, Vol.44, No.4 pp Suzuk,K., Kubota,J. and Ohtubo,H. (003) Voxel baed collon detecton algorthm for rgd body mulaton, J. Appled Mechanc, JSCE, Vol.6, pp , n Japanee. Matuhma,T., Katagr,J., Ueug,K., Tuchyama,A. and Nakano,T. (009) 3D Shape Characterzaton and Image-Baed DEM Smulaton of the Lunar Sol Smulant FJS-1 J.Aeropace Engneerng, Vol., No. 1, pp Uhma,S., Fukutan,A. and Makno,O. (008) Predcton Method for Movement wth Collon of Arbtrary-Shaped Obect n 3D Free-Surface Flow Proceedng, JSCE, B Vol.64 No. pp , n Japanee. 1

Method Of Fundamental Solutions For Modeling Electromagnetic Wave Scattering Problems

Method Of Fundamental Solutions For Modeling Electromagnetic Wave Scattering Problems Internatonal Workhop on MehFree Method 003 1 Method Of Fundamental Soluton For Modelng lectromagnetc Wave Scatterng Problem Der-Lang Young (1) and Jhh-We Ruan (1) Abtract: In th paper we attempt to contruct