MO HINH SO K £ T HOfP CO - THUY DEM - PFV VA LTNG DUNG VAO M 6 PHONG DONG THAM
TRONG MOI TRU^OnVG DAT BAO HOA
T O N G ANH TUAN B^ mdn Thdy luc - Thiiy vdn Trudng Dgi ttgc Giao Ihong Van Idi
Tdm Idt: Mo hmh so kit hap ca-thiiy DEM-PFV dugc trinh bdy dua tren su kit hgp giira phuang phdp phdn tir rdi rgc (DEM') vd mo hinh ddng chdy trong moi trudng rong (PFV^). So sdnh giiia cdc kit qud md phong sd vdi thuc nghiim vd cdc cong thirc bdn thirc nghiem ve thdm Irong mdi trudng ddt bdo hda ldm ca sd ddnh gid su phii hap cua md hinh DEM-PFV.
Summary: The numerical model of hydro-mechanical coupling DEM-PFV presented in this paper uses a combination ofthe discrete element method (DEM'J for the solid phase and a pore-scale finite volume (PFV^) of the flow problem. Comparison between the numerical and experimental results on granular assemblies shows a good agreement ofthe model DEM-PFV
I. © A T V A N D E
So vdi p h u a n g p h a p phan t u hGu han ( F E M ' ) , mo p h o n g d o n g chay t r o n g cac 16 rong bang p h u a n g p h a p phan t u rdi rac c h o p h e p chiing ta cd the hieu ro h a n cac q u a trinh p h u c tap xay ra trong moi t r u d n g r d n g giifa cac hat, ciing n h u moi quan he giiJa d d n g chay t r o n g cac 16 r o n g [1]
vdi ung x u c a h g c ciia vat lieu d ty le vi m o [2]. P h u o n g phap D E M , phat trien trong vai thap nien gan day [6] va d u g c irng dung kha pho bien hien nay trong nghien cuu u n g x u c a hoc ciia vat lieu hat nhir be tong, d a t , . . . [4, 5, 8], tuy nhien p h u a n g phap nay con kha mdi me d Viet Nam.
M o phong ket h g p c 0 - t h u y gitJa p h u a n g phap D E M vdi dong chay trong m6i trudng r6ng thu hiit s y quan tam ciia cgng d 6 n g khoa hpc trong ITnh v u c dia ky thuat lien quan den cac van de v l thdm [11], xoi ngSm [12], 6n djnh mai d i e [4] va lun c6 k i t [5], ... Cac m6 hinh ket h g p c a - t h i i y chii yeu khac nhau d ky thuat m o phong doi vdi pha long, va co the chia thanh ba nhom sau' (i) M o phong vi m o d y a tren su rdi rac ciia khdng gian rong bang p h u a n g p h a p F E M [6] hoac Lattice - Boltzmann ( L B ) [7], (ii) M o hinh tien t^lc cua dong chay va t u o n g tac r^n - long tren c a sd ap dimg djnh luat Darcy [10], (iii) M o hinh mang 16 rong d u bao tham va ap dung vdi dong chay nhieu pha [8, 9 ] .
Phan thir nhat, bai bao gidi thieu mo hinh so ket h g p c a - thiiy D E M - PFV [3, 4, 5, 11, 1 2 ] - phSn m e m m a ngudn m d d u g c phat trien tai Lab 3SR^, viet bang cac ngon ngit C + + va Python va
T a p chi K H O A H O C G I A O T H O N G V A N T A I S 6 4 7 - 0 8 / 2 0 1 5 9 1
chay tren he dieu hanh Linux. U'ng dung ciia md hinh DEM - PFV vao mo phdng ddng tham troftg mdi trudng d4t bao hda, dong thdi ket qua ciia mo phong so sanh vdi thuc nghiem va cac quan he ban thuc nghi?m duge trinh bay trong phan thii hai.
II. MO HINH KET HOfP CO - THUY DEM - PFV 2.1. Mo phong pha ran - Phirong phap DEM
Phuang phap DEM su dyng mo ta Lagrangian dang ludi ty do, trong dd mdi hat la mgt hinh c5u d6ng nhat cd khdi lugng, ban kfnh va m6men quan tinh. Tai moi budc thdi gian At. luc tuang tac giiia cac hat dugc tinh toan, sau do dinh luat 2 Newton dugc ap dung bang sai phan b|c hai de xac dinh vi tri mdi cua hat cho At tiep theo.
J''
U"
-/„lan(p p--
(a) _ (b) _ ''^^ . . . ("^f
Hinh I. Mo hinh liep xiic ddn deo tuyin tlnh (a): dg cirng phdp tuyen vd tiip tuyen, (h): su goi len nhau;
quy lugt tuang ldc (c): phdp tuyin, (d) • dip luyin vdi <p la gdc ma sal trong ciia hgt Tuong tac giira cac hat dugc gia thiet la dan dec (hinh I (a), (b)). Cac lyc tuong tac f„ va f, theo cac phuang phap tuyen va tiep tuyen, ty le vdi cac chuyen vi tuong doi CC va li 'giira hai hat va phy thugc vao cac do cung phap tuyen k^ va tiep tuyen k, thong qua ban kfnh ciia hat vamo dun dan h6i ciiav|t Ii?u (hinh 1(c), (d)) [12].
Phiro-ng trinh: Vec ta gia t6c chuySn dgng X''=(x'',U)'') cua hat k nhan dugc tir djnh luat 2 Newton:
n'-x''= j a , n d S + j p,gdV = J]!/-''+m''g-F'='''+m''g (1)
6 day: m la khdi lugng cua hat, V'' th6 tich ciia h^t, cr,n la irng suat tac dung len b6 mat ciia hat theo hudng ciia vec ta dan vj n , p^. khdi lugng rieng ciia h^t, g gia t6c trgng trudng;
f^'' lyc tuong tac giira hai hat j va k, F ' * luc tong hgp tac dung len hat k va n^ s6 lugng tiep xiic cua hat k.
Doi vdi 6xN do ty do ciia chuyen dgng tinh tifin va quay, vec ta vi tri viet dudi d?ng ma
92 Tap chi KHOA H O C G I A O T H O N G VAN TAI So 47 - 08/2015
x=j-'(T' +u;)
(2) Trong ddJ la momen quan tinh, T ' la vecta luc, Vfnh^n dugc bang each them vao ma tran khoi lugng M cac thanh phan cua chuyen dgng quay va N la so hat md phong.Vi tri cac hat X,^^ se dugc cap nhat sau moi budc thdi gian At tir md hinh tich phan hien bang phuang phap sai phan hii'u han trung tam b^c hai:
X , . , - 2 X , + X ,
M' - ^ ^ = J^'(T'+W) Luc tuong tac giiia cac hat ciing se dugc cap nh|t sau m6i budc thdi gian At:
X,.^-X, X , « „ 2 = - Ar
(3)
(4)
Thuat toan: DEM sii dyng thuat toan lap th6ng qua hai budc ca ban la tinh toan lyc tiep xiic va tinh toan vj tri cua cac h^t.
2.2. Mo phong pha long - Mo hinh dong chay trong moi trudng rong bao hoa (PFV) Rdi rac khong gian: Kh6ng gian dugc rdi rac dudi dang ludi tam giac thanh timg phan tu hinh tu di$n c6 dinh la tam eua hat (hinh 2(a), (b)).
(a) ^ (b) Hinh 2. Rai rac khong gian (a): 2 chiiu. (b): 3 chieu [12]
Phuvng trinh lien tuc: Toe do thay d6i th6 tfch ciia mot phSn tu tir di?n co quan h? vdi luu lugng q,j trao doi gi&a tir dien vdi cac ph5n tir lan can qua bon mat S,j tir phuang trinh lien tyc;
V', = ±\(^'-s,')dS=±q„
(5)Trong do: (U - v'^) la van toe tirang doi cua chSt long so vdi mot mat ciia tir dien.
T?p clii KHOA HOC GIAO THONG VAN TAI S6 47 - 08/2015
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Dong tham thuy lire: Tir phuang trinh Stokes, dinh luat Darcy va quan h? Hagen - Poiseuille, quan he giira luu lugng tham q.jVdi t6c do thay d6i thS tich ciia mgt phSn tu- tir dien va trudng ap suk {p^.} dugc viet dudi dang [11]:
v ; = Z l » = ' ' » n ^ = K,/p,-p,) (6)
Trong do: K =-^ dac trung cho dong thSm tai mat S,j ciia phSn tir tir dien do gradient ap k
suit (p,-pj).
Luc tu-ffng tac giira cac pha long - ran: Luc tuang tac ldng - rSn dugc suy ra tir trudng ap suat, bao gom: (i) luc day Archimede F ' , (ii) ap lyc m^t f ' va (iii) luc nhdt F '
F* = jpgzndS-\- j p'ndS+ jmdS = F''-' +F''-* +F"-* (7)
dVi dVt dVt
Trong do: n la vec ta dan vi ciia m|t bien. 0 mdi budc thdi gian, toe do thay d6i the tich 16 rong dugc tinh toan, ap suat nhan tir he phuang trinh (6) va sau do cac luc lai dugc c&p nh$t d budc thdi gian tiep theo tir phuang trinh (7).
2.3.-Mo hinh ket hgp cc - thuy DEM - PFV
Phuffng trinh: Ldi giai cua m6 hinh ket hgp DEM - PFV dya tren hai quan h$ dang ma tran tong quat ndi chuyen vi eiia cac hat vdi trudng ap suat:
[G]{P} = [E]{X} + {Q„} + {Q,} (8) [M]{X) = {F'} + {W} + {F'} (9) Trong 66: phuong trinh (8) mo ta bao toan kh6i lugng ciia ddng chay, chat long nhdt khong
nen dugc trong moi trudng rdng bien dang va phuang trinh (9) mo ta chuyen dgng cua pha ran.
Thuat to^n: Trudng ap suat nhan dugc tir phuang trinh (10) b§ng each ap dung tich phan bac mot theo vj tri X cua hat:
[G]{P(0> = [ E / ^ ^ ' " ' ^ ^ | " ^ ^ ' ^ ] + { Q , ( 0 } + { Q „ ( 0 } (10)
Luc tuang tac long - rSn tinh loan theo quan he (11):
{Fn = [S]{P} (11) Cu6i cimg, ti'ch phan ciia mo hinh hien cua DEM se dugc thyc hien bdi quan h$ (9). Mo
hinh thuat toan ciia DEM - PFV dugc gidi thieu trong hinh 3.
9 4 Tap chf K H O A H O C GIAO THONG V ^ N TAI So 4 7 - 08/2015
Hinh 3. Sa do ihugi todn cua md hinh DEM - PFV [12]
III. M O PHONG DONG THAM BANG M O HINH DEM - PFV
Trinh ty mo phdng bao gdm cac budc: (i) tao mau md phong so, (ii) dp dyng cac dieu kien bien, (iii) ap dung gradient thiiy luc va (iv) do dong tham. Hinh 4(a) gidi thieu cac dieu kien bien ve ap suSt, thanh ben la bien cung khong tham nudc. Hinh 4(b) gidi thieu bieu do Voronoi trudng hgp s6 hat mo phdng la N = 200. Hinh 4(c) hien thi trudng ap suSt nhan dugc khi
N = 50000.
Hinh 4. Biiu do Voronoi vd cdc dieu kien bien ve dp sudt, (a): Dieu kien hien, (b) Biiu do Voronoi vd diiu ki4n biin khi i\ = 200. (c): Trudng dp sudt khi N = 50000 [11]
Trudng hgp gradient ap suat tac dyng theo phuang ngang, nhin chung da mo phong dugc quy dao ciia ddng chay va cho thdy tinh hgp ly ciia gia thi^t xkp xi ap suSt la hang so trong timg khoang ciia md hinh DEM - PFV khi so sanh vdi FEM [3] (hinh 5).
T9P chi K H O A H Q C G I A O THONG V ^ N TAI So 47 - 08/2015 95
(a) (b) Hlnh 5. Trudng dp sudt nhgn dugc bdi mo hinh (a): FEM vd (b) • PFV
Ket qua so sanh: K8t qua thim ngi t^i tuyet d6i nh|n dugc tir md phong so sanh vdi thyc nghiem va cae cong thu:c ban thuc nghiem dugc gidi thi^u tren hinh 6(a). He sd thim kh6ng thti nguyen K* dugc chuan hoa th6ng qua thdng sd D* dac trung cho ban kfnh thiiy lyc dugc gidi thi?u tren Hinh 6(b).
Nhin chung, xu hudng ciia cac dudng cong mo phong va thyc nghiem tuang d6ng vdi dudng ket qua thi nghi?m. Gia trj ldn nhSt cua K* = K / D *^ nh|n dugc trong trudng hgp hat deu. Thim giam den gia tri nho nhit khi M « 0.3 cho tit ca cac trudng hgp. Nhin chung, dang tong quat ciia cac dudng cong phii hgp vdi su thay d6i ciia do r6ng nhung kh6ng hoan toan tuong ty nhau. Cy thS, K* thay ddi kh6ng ldn khi 0.7 < M < I, khong tuong ddng vdi thay d6i dg rdng, xu hudng nay phan anh diing cho ca md phong sd va tinh theo cac cdng thirc thuc nghi?m hay ban thyc nghiem.
i \ ''---^^
(ci) , . ^*> .
Hinh 6. So sdnh kit qud thdm cua mo phong/thi^c nghlim/cdng thuc, (a): thdm ngi tgi, (b): thdm khdng thu nguyen dugc chudn hda bdi bdn kinh thuy luc D* [12]; M = m, / (m,+m2),
vdi m, vd mj tuang ung la khdi lugng hgt min vd hgt thd
Ve dinh lugng, quan h? thyc nghi?m co xu hudng dy bao tham cao han gia tri thim do dac trong phong thf nghiem khi M nho. Quan h? thyc nghiem ciia Kozeny - Carman (K-C) chua phan anh dugc quan h? thim va biSu thj kem nhit trong tat ca cac trudng hgp va kk qua thim cao hon 5 lan, trong khi dd cac quan he khac udc lugng thim tuang doi tdt kht 0.2 < M < 0.9.
Quan he thyc nghiem K - C phii hgp doi vdi v|it li?u h^t cd cap phdi deu, tuy nhien, ket qua nghien ciru chi ra rSng quan h? nay khong phii hgp t6t vdi cac trudng hgp cap phoi kh6ng lien 96 Tap chi KHOA HQC GIAO THONG VAN TAI S6 47 - 08/2015
tyc hoac dong nhat.
Du bao ciia mo hinh DEM - PFV la thoa man khi M<0.5 vdi sai sd 50%. Udc lugng tham kem doi vdi 0.7 < M < 0.9 vdi sai so 2.5 Ian. Nhin chung, sai so du bag ciia m6 hinh la ciing dg ldn so vdi thuc nghiem.
IV. KET LUAN
Dy bao ciia md hinh kk hgp ca thiiy DEM - PFV phii hgp tot vdi thuc nghiem, phan anh quy tac anh hudng cua thanh phan cap phdi hat va do r6ng den tinh tham. Sai s6 giua ket qua thyc nghiem va mo phong sd co cimg trj s6 so vdi cac quan he thuc nghiem ciia Terzaghi va Slichter va nhd han nhiSu so vdi gia tri nhan dugc tu quan he thyc nghiem ciia Kozeny - Carman.
Ngay nhan bai lan dau: 8/5/20IS Ngay nhan bai s ^ : 2/6/2015 Ngay chap nhan dang bai: 11/6/2015 Tai lieu ijisini khiio
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[II]. Tong .-i--'! . Catalano E. and Chareyre B, "Pore-Scale Flow Simulations: Model Predictions Compared with Experiments on Bi-Dispersed Granular Assemblies," Oil & Gas Science and Technology - Revue d'IFP Energies nouvelles, vol. 67, no. 5, pp. 743-752, Nov. 2012.
[12]. Tong A Tuan, "Modelisation micromecanique des couplages hydromecaniques et des mecanismes d'erosion interne dans les ouvrages hydrauliques", http://www.theses.fr, !5-Jan-20]4»
Phan bifn: Tr5n Dinh Nghien; Nguyen Dii^c Manh
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