BAI BAO KHOA HOC
NGHIEN Ctru CAU TRUC DONG CHAY XUNG QUANH HINH TRV T R 6 N SU'DVNG "LAGRANGIAN COHERENT STRUCTURE"
f V0 Huy C6ng*
T6m tat: TYong nghiin cihi ndy, cdu triic dong chcy xung quanh hinh tru da duac nghien citu d^a trin phucmg phdp "Lagrangian Coherent Structure " (LCS). LCS da chiing minh nhiiu tfu diim so voi cdc phuang phdp trudc do Id dimg vec ta, cdc dudng dong mice xoay hay cdc dudng ddng di thi hiin cdu tnic dong chdy. Vdi LCS, cdu triic dong chdy dugc thi hien ddy dii hon vd ngu&i nghien ciru co thi dinh luang duac cdc bien cua cdc vung xoay dphia sau hinh tri^, diiu nay khdng thy:c hi$n dugc vdi cdc phuang phdp thudng diing trudc do.
Tic khda: "Lagrangian Coherent Structure", hinh tru, ciu tnic dong ch^y, xoay.
nghien ciiu eau triic dong chay chi tiet han vk toan diSn hon. Trong nghien ciiu nay nhihig uu diem noi b£it cua LCS so voi phuang phap trudc dSy sS dugc ph^n tich d^y dii chi tiSt.
2, PHirofNG PHAP "LAGRANGIAN COHERENT STRUCTURE"
2.1. Gioi thi^u vl phirong phap "Lagrangian Coherent structure''
LCS dugc biet dSn nhu la duang ranh gi6i An phan chia tnrdng dong chay thanh cac viing khac nhau va cac phan tu vat chat dong chay dugc xem nhu la Idiong di qua cac duang ranh giai nay. Vi d\i nhu cac duong mau do va xanh d tren hinh 1 la nhung dudng "iC^ backward- time vd forward-time'\ Nhung dudng nay cd tinh chdt nhu sau; khi chiing ta quan sat 2 dilm di chuyen d 2 ben dudng "forward-time"
(dudng net lien, mau xanh) thi 2 diem nay se phan ky theo thuan chieu thdi gian (theo chieu miii ten nho), xem hinh 1(b). Ngupc Iai, khi chiing ta quan sat 2 diem d hai ben dudng '^backward-time " (dudng net dirt, mau do) thi 2 diem nay se di chuyen phan ky theo ngirgc chieu thdi gian (ngugc chieu miii ten nho), xem hinh 1(c). NhijTig dudng ranh gidi nay hi dn di khi thi hi?n cdu true dong chay bang trudng van toe. Cac dudng LCS co the coi nhu la "bg khung xuang" ciia dong chay. D I tinh toan LCS, theo Shadden et al., (2005), ta cdn tim I. D^T V A N D E
Dong chay xung quanh hinh tru la m\ic tieu nghiSn cuu cua rdt nhieu cong trinh khoa hgc bdi tinh img dung pho bien cua no. Chiing ta co thi de dang bat gap cac ket cau co dang hinh tru trong cu6c song vi du nhu la tru cdu, 6ng khoi, c^t di§n, nha cao tang, v.v... S\r xuat hi^n ciia cac ket cau hinh try nay se lam thay ddi d6ng chay xung quanh n6, dac biSt cd thi tao ra vimg xoay phia sau dan din su xao trgn d6ng chay hoac tao ra cac luc can tac dung len hinh tru. Cac xoay nay se bat dau xuat hi?n sau hinh try khi he so Reynolds cua dong chay Ion han 47 va lam tSng sy xao trdn d ddng sau hinh try. Vi?c nghien ciiu cau tnic dong chay ma d day la cac xoay nay da dugc thuc hien bdi rdt nhieu cong trinh khoa hpc. Tuy nhien, hdu nhu tat ca cac nghien ciiu trudc day dlu thi hi^n cac xoay nay dua tren dudng ddng miic do xody {''vorticity contour''). Nghien ciiu nay se ap dyng mgt phuang phap khac do la
"Lagrangian Coherent Structure" (LCS) dk thi hi$n cau tnic dong chay. Vdi LCS, ngudi nghien ciiu se tim thay cac dac diem mdi ma trudc day no bi an di khi diing cac phucmg phap thong thudng. Dieu nay giiip cho vi^c
' Khoa Xdy dung Thuy lai - Thiiy di^n, Trucmg Dai hoc Bach Khoa - Dgi hQc Da Ndng.
KHOA Hpc KY T H U A T THUY Lgi VA M 6 | TRlTdNG • S6 57 ( 6 / 2 0 1 7 )
"Finite-Time Lyrqnmov ExportenT^ (FTLE). Thong sd nay thS lu$n mirc d$ phan t ^ cua c ^ phan tir vSt ch&, va tji noi c<5 FTLE ldm thi cSc phlin tit se phSn tan nhieu. Tiong trudng FILE, tjp hgp cac vj tri mi FTLE e6 gia trj 16n dugc xem nhu la cac ducmg c^u tnic LCS. Chi ti& vk LCS cflng nhu cich tinh toan cd thi tham khdo cac cdng trinh nghien c6u eua Shadden et al., (2005).
(•)
•4
(b) ' d (c)Hinh 1. (a) Minh hga duong cdu true LCS (mdu do: "LCS backward-time"; mdu xanh:
"LCSforward-time"}; (b): tinh chdt cua LCS forward-time; (c) tinh chdt cua
LCS backward-time
LCS dugc tinh toan d\ia trSn trudng vec to v$n t6c dong chiy theo th« gian cia vung nghien cftu. Trong nghito ciiu niy, tac gii da su dyng phfc m&n Matlab va tinh toan LCS theo cic cong thiic cHa Shadden et al, (2005).
22. Kiim djnh chinmg trinh tinh toin LCS Chuong trmh tinh toin LCS dugic tic gia vik trfin ngfin ngO Matlab vi vi$c kiem djnh chuong trinh niy dugc thvc hi$n cho bii toan don gian.
D6 la tinh toto LCS tit truimg v6c to ciia
"Double Gyre" dugc cho bdi phuong trinh:
^A;;;,0=^sin(ff/(x,r))siii()ry) (1) f{x,y) = a(t)x'+b(t)x
vdi: a(r) = esin(oO (2) A(i) = l-£-sin(i»0
tien vung tinh [0, 2]x [0, 1] va trudng v6c to vin t6c cho bdi cong thirc:
„ = - ^ ; v = ^ (3)
dy ax
Trudng vec ta vdi cac thong so khac nhau tai cac thdi dilm khdc nhau dugc thS hiSn nhu tren hinh 2.
,(f \\y:X^ m \ 'tm
Hinh2. Trudng vgn toe cira "Double-gyre" khi A= 0.1, a>=2n/10 and £=0.25 tgi cdc thai diim: (a) t=0, (b) t=2.5, (c) 1=7.5
Trudng gia tri FTLE dugc tinh toan va thS hien tren hinh 3. Trong trudng FTLE, nhiing dudng mau do dugc xem la cac dudng LCS. So sanh ket qua tinh toan ciia tac gia (hinh 3) voi
ket qua ciia Jakobsson (2012) (tren hinh 4) cho thiy chuong trinh tinh toan LCS cua tie gii hoan toan du dg tin cay dS tinh toan LCS cho dong chay ximg quanh hinh tru.
mnh3. Truang FTLE cua "Double-gyre" tgi t = 0, A =0.1, co = 27r/10, £=0.25; (a) "LCS forward-time ", (b) "LCS backward-time ". (Kit qud cira tdc gid)
KHOA H()C KY THUAT THUY Lgi VA H6| TRUdHC • SO 57 (6/2017)
Hinh 4. Trudng FTLE ciia "Double-gyre" tgi t=0, A= 0.1, 0=2n/lO, e=0.25; (a) "LCSforward- time", (b) "LCS backward-time". (Kit qud cung cdp bai Jakobsson, 2012)
3. CAu TRUC DONG CHAY XUNG QUANH HINH TRU
Nhu da trinh bdy d tren, LCS dugc tinh todn dua tren trudng vec ta ddng chay. De co dugc trudng vec ta xung quanh hinh try, mo hinh sd thuy luc Fluent d3 dugc tac gid ap dung. Fluent la mo hinh so dugc danh gid cao trong cac nghien ciiu ve thuy luc. Sau khi cd dugc trudng vec ta dong chdy, tac gid tinh todn LCS dua tren ehuang tiinh Matlab da dugc kiSm dinh d phan tren.
3.1. Thiet lap md hinh s6 trong Fluent Fluent dya trSn phuang phdp thi tich huu han d6 gidi h^ phuang trinh ca ban. Phuang trinh bao todn khoi lugng c6 dang;
= 0 (4)
^ ^ V . f p v )
tru. Bien hai ben dugc bo tri cdch hinh tru mgt khodng bang 10 lan dudng kinh. Vi^c bo tri cac bien vdi khodng each nhu vdy dl trdnh anh hudng ciia bien dSn ket cau dong chdy xung quanh hinh try (Meneghini et al. 2001). Bien vao la ddng deu Uo vdi dang bien "velocity inlet" con bien ra Id bien "pressure outlet"
Day la c|p bien dugc ngudi sii dung Fluent sir dyng nhiSu khi mo phdng ddng chdy qua cdc vat cdn (Vu et al., 2015). Bien "pressure outlet" CO the cho ph^p hien tugng "back-flow"
nen cac xody nude khi di ra khoi bien ciia ra dugc mo phong chinh xde.
Bien tren
Trong do p la khoi lugng ri^ng, v la van t6c.
Phuang trinh bdo toan dpng lugng co dang:
^ ( p v ) + V-(/3vv) = -V/j + V-(?) + /?i + F (5) Trong dop la dp suat, rla tensor iing sudt, vd F la ngoai luc.
Cac phuang trinh dugc giai theo phuong phap "semi-implicit pressure linked equations"
(SIMPLE). Mo hinh r6i dugc ap dyng la SST k- w va ly do sir dung md hinh nay dugc giai thich trong Vuetal., (2015).
Mo hinh toan 2D ciia dong chay qua hinh try dugc the hi?n tren hinh 5. Khoang each tir bien vao va bien ra cua mo hinh din tam hinh try ldn lugt bdng 8 va 24 ldn dudng kinh hinh
Hinh 5. Sa do dieu kien bien cho mo hinh.
Mien ludi tinh todn cho mo hinh dugc the hien tren hinh 6. Todn bp mien tinh toan gom 193920 phan tii vdi so lugng phan tir tren chu vi hinh try la 240 phan tii. Kich thudc phdn tir thay doi tir 0.5mm (tren hinh try) den 2.5 mm (d cac bien). Chi tiet ludi xung quanh hinh try dugc thi hien tren hinh 6(b). Cach chia ludi nay da dugc ap dung thanh cong trong cac nghien ciru cua cac tdc gia trudc nhu Vu et al., (2015). Budc thdi gian dugc thiet tap trong mo phong la Is.
KHOA Hpc K? THUAT THUY Lpl VA M O I TRLTdNG - S6 57 ( 6 / 2 0 1 7 )
Hinh 6. (a) Chia luai miln tinh todn, (b) Chia ludi xung quanh Mnh try.
Cii cdch 6 diim ludi thi co mot diim ludi duac ve.
Trong cac nghien ciiu ddng chiy qua hinh tm 3.2.1. Vu Oiem cua LCS so v&i cac phinmg sit dymg mfi hinh s5, he s6 \\fc cin hoac ap l^c phap khdc.
Htn huih trv thudng dugc dimg dfi kifim tra dS Khi hfi s6 Reynolds cila ddng chiy ldn hon chinh xic cua mfi hinh. Trong nghifin ciiu nay 47, cic xoiy se hinh thanh sau hinh tnj. Cac tic gia cung su dvng h | s6 luc cin di kiim djnh xoiy se lan lugt tach ra khdi hinh tru trong khi mo hinh bang each so sinh vdi cic kit qui da cie xoay tiq) theo dugc Mnh thanh, dieu niy dugc cfing b6. H? s6 luc cin tic dung len hinh dugc biit din nhu li "Karman vortex streef.
try dugc tmh theo cong thuc:
c = - 2 ^
" PUID
(6) Trong do:
Uo: vgn tic tai bien vio Fit: liic cin tic dung Ifin hinh trv D: dudng kinh hinh try
Bang 1. So sanh kiim dinh mfi hinh Reynold
60
100
200
1000
Cic nghien cuu Tritton(1959)(Re=60,5) Kit qui tic gia Lam et al., (2008) Meneghini et al., (2001)_
Kit qua tac gia Lam et al., (2008) Meneghini et al., (2001) Ket qua tac gia BrazaetaL, (1986) Ket qua tac gia
1.47
o
1.468 1.36 1.37 1.366 1.32 1.3 1.33 1.21 1.259 Bang 1 the hien he so luc can tCr mo hinh tinh so sanh vdi cac nghien ciru khdc khi he so Reynolds thay doi doi vdi dong chay tdng. Ket qua cho thay mo hinh tinh hoan toan chinh xac va dong chay xung quanh hinh tru dugc mo phong chinh xac.
3.2. Xac dinh cau triic dong chay dya tren LCS
Cdc xoay ndy dugc thi hi?n chii yeu dya trSn dudng ddng muc xody "vorticity contour". O nghien cuu nay tdc gid dd sii dung phuong phap LCS dl thi hi?n nhilu d5c dilm mdi trong cau tnic ddng chay md vi^c diing tnrdng vdn toe hay dudng d6ng mdc xody khong thi hien dugc.
Hinh 7 thi hi?n cdu tnic ddng chdy xung quanh hinh tru dya tr^n cdc phucmg phdp khdc nhau.
Hmh 7(a) thi hien trudng vdn t6c ciia dong chay. Viec xac dinh cdu tnic dong chdy dira tren tnidng van tdc co nhieu han che bdi ngudi nghien cuu chi cd thi nhin thdy hudng vd dp ldn cua d6ng chdy, ma khong thdy dugc vi tri ciia cac tam xody. Hinh 7(b) thi hi?n bang cac dudng dong muc xody. Tam cua cac xoay nude co the xdc dinh ro nhung be rpng hay pham vi ciia vimg xoay thi kho xdc dinh. Ngudi nghien cuu c6 thS chgn mpt gia tri ngudng ndo do cua dudng d6ng mirc xoay de the hien pham vi vimg xoay nhung theo each do thi ket qud hoan todn phu thugc vdo y chu quan ciia ngudi ngliien ciru. Day la mgt han che ciia viec sir dung dudng dong miic xoay cho vi?c nghien ciiu cau tnic dong chay. Ngugc lai, khi dimg LCS de xac dinh pham vi vimg xoay, ta khong can chpn bat ky gia tri nguong nao. Ta co the xac dinh dugc ro tam ciia cac xoay hay pham vi ciia cac xoay phia sau hinh try mot each true quan, dieu nay se dugc nghien ciru k9 trong phan 3.2.2 va 3.2.3.
KHOA H p c K f T H U A T THUY Lgi VA WOl TRlrtTNG • Sd 5 7 ( 6 / 2 0 1 7 )
(b)
( 0 )
(d)
Hinh 7. Cdu triic dong chdy phia sau hinh tri^; (a) Su d^ng trudng vec ta, (b) sii dung dudng dong mice xody, (c) su dung "LCS backward-time "; and (d) so sdnh ket qud giiia (b) and (c).
Ngodi ra d viing gan hinh try, viSc sir dung dudng ddng miic xoay khdng the hien dugc su tach bi^t giu'a cac xoay nude. Dac bi^t, nd khong thi hien dugc dudng HSn k i t gifia vi tri tdm cac xody nude. Khi sir dung LCS, vi tri dong chdy "tdch" ra khdi be m^t hinh tru dugc the hien m6t cdch true quan. Vi tri nay anh hudng ldn den su phan bo dp luc tren hinh try va cGng quyet dinh dSn b l r6ng ciia pham vi viing xody nude sau hinh try. Cdc nghien ciiu trudc ddy dd dua vao su thay doi cua h? sd
"shear stress" tren be mat hinh try de xdc dinh vi tri "tach" nay nSn co qud trinh tinh toan phiic tgp. Vdi LCS, vi tri tach ndy da dugc thi hi^n mpt each true quan, td do cd the xdc dinh mot each d l dang.
Mpt uu diem khac noi bat cua LCS chinh la giiip xdc djnh "vet" hay "dudng di" cua cdc phan tii (vi dy nhu chat chi thi mdu) quanh hinh try. Theo tinh chdt ciia LCS (Shadden et al.
2005), cac phan tii nay khi di chuyen s6 co xu hudng bdm dgc theo dudng LCS. Hinh 8 thi hi?n su so sanh cdu tnic LCS vdi kit qud thi nghi?m cua Perry et al., (1982). Perry da sir dyng chat chi thj mau de the hien cac xoay nude d phia sau hinh try. So sdnh giiia hinh 7(c) va 8 cho thay dudng di ciia cac phan tu chi thi mau (the hien cac xoay nude) va cdu tnic LCS hoan toan tuang tu nhau. Dieu nay da thi hien dugc tinh uu vi?t ciia phuang phap LCS, ngudi
nghien ciiu khong can sir dung nhiing thi '^s " o ~ s lo "is nghi?m ton kem nhung van xac dinh dugc su di "^ '>
chuyin cua cdc phdn tir chdt mau va biit dugc Hinh 9. Dinh nghia bi rong cua miin xody cdu tnic dong chay. p/,/^ ^au hinh tru
Hinh 8. Kit qud thi nghidm chdt chi thi tndu the hi^n xody nude phia sau hinh tru
(Perry etal 1982) 3.2.2 Xdc dfnh bi r^ng viing cdxody Viec nghien cuu phgm vi cua viing xody phia sau hinh try c6 y nghia quan trpng bdi trong viing nay se cd sy xao trpn manh so vdi cac vimg ben ngodi vd van toe dong chay trong vimg ndy ciing yeu hon. Dieu nay dac bi^t co y nghia khi nghien ciiu su xdo trpn, hay khuech tan cua cac phan tir vat chat. LCS sg giiip ta xdc dinh dugc dudng bien ngan each giiia viing co cac xoay nay vd viing khong xoay. Hinh 9 the hien be rgng (Bw) cua viing xoay lay tai vi tri each hinh try bdng 5 lan dudng kinh hinh try.
KHOA Hpc KY THUAT THUY Lpl VA M O I TRl^TNG - S6 57 (6/2017)
Sv thay ddi ciia Bw/D Idii h$ s6 Reynold (Re) thay ddi dugc the hien ro trfin hinh 10. Vdi h$ s6 Re nho (=60), bi rgng vimg xody nhd.
Tuy nhiSn d ddy cd su tang nhanh cua Bw/D khi Re tdng tir 60 din 200. Sau d6, Bw/D tdng rdt nhd khi Re tang tir 200 din 1000.
Hinh 10. Be rong cua viing co xody nude xdc dinh tgi vi trix= 5D.
3.2.3 Xdc dfnh chieu ddi cua vung khdi tao xody
d vung sdt ngay phia sau hinh try, cdc xody nude lien tuc hinh thanh tir 2 ben Wnh try, sau d6 tdch ra roi di chuyen ve phia sau. Chieu ddi cua vung khdi tao cdc xody nay co anh hudng den tan so hinh thanh cac xoay, do do se anh hudng din miie dO su xdo lr6n phia sau hinh try.
Tan so hinh tharih xoay ti le nghich vdi chieu ddi cua vimg khdi tao xoay. Schaefer and Eskinazi (1959) da dinh nghia chiiu dai viing khdi tao xody dya tren diem gid dinh phia sau hinh try noi ma su dao dpng cua van t6c dat gid tri ldn nhat. Trong nghien cuu nay tac gia dd su dyng mpt djnh nghia mdi de xdc dinh chiiu ddi vung khdi tao do la dua tren LCS. Theo do, vimg khdi tao xoay dugc tinh tii tam hinh try den diem giao nhau xa nhat ciia "LCS backward -time" va "LCS forward-time" nhu hinh 11.
Hinh 12 the hien su thay doi kich thudc viing khdi tao xoay khi he so Re thay d6i. Chiiu dai Lw CO su giam dot nggt khi Re tang tCr 60 din 200. Tir 200 trd di, su giam cua Lw la nho hon.
Hay noi each khac sir thay doi ciia he so Re trong pham vi tir 200 den 1000 anh hudng rdt nho den kich thudc vimg khdi tao xoay.
'\y^\\
Hinh 11. Dfnh nghta chiiu dai vung khoi tgo xody Lw. Mdu do: "LCS backward-time ";
mdu xanh: "LCSforward-time"
Hinh 12. Chieu ddi vimg khai tgo xodyph^
thuoc vdo hi so Reynold Viec xdc dinh kich thudc viing khdi tao xody dya tren djnh nghia mdi nay ciing dugc so sdnh vdi cac nghien cdu trudc, cy thi nhu ciia Nishioka vd Sato (1978), Schaefer va Eskinazi (1959), trong do cac tac gia nay dd sd dyng diem CO v$n t6c giao dpng ldn nhdt dl tinh viing khdi tgo xody nhu da dl cap d trudc. Hinh 13 the hi?n su so sanh ndy. Kit qud cho thdy chllu dai viing khdi tao xoay dua tren cdc dinh nghTa khae nhau nhung tuong d6i gidng nhau. Tuy nhien vdi LCS cdch xdc dinh true quan hon va de dang hon.
X NisliiokaandSaio(1978) + SchaeferaiidEskinazi(l959) fc-Presenlresulls
Hinh 13. Chieu ddi viing khdi tgo xody xdc dinh theo 2 phuomg phdp cita Nishioka va Sato
(1978), Schaefer vd Eskinazi (1959);
vd cua tdc gid.
24 KHOA HQC K? T H U A T THUY Lgi VA MOI TRUdNG - S6 5 7 ( 6 / 2 0 1 7 )
4. K^TLUAN Theo quan diim cua tic gia, LCS s€ la cfing cu Trong nghifin ciiu niy tic gii <S ip dving hiiu hieu di nghien ctiu ciu tnic ddng chiy. LCS phuong phap LCS di n ^ e n cihi ciu tnic ciia khong nhiing dugc sit dyog trong ddng chiy tang ddng chiy d phia sau hinh trv. Phuong phip mi no cung dtrgc apdymg nhiiu ci trong ddng r6i.
LCS dS the hi$n dugc nhihig uu diim so vdi cic Vi dv nhu li ddng chiy tich ra khdi cinh may bay phuong phap trudc dfi. Dvta vio LCS ta cd thi vi vung xio trdn phia sau no ciing dugc nghifin xac dinh cic bifin ciu triic ddng chiy m$t each ciiu bing cich sii dving LCS. Ngoii nghien ciiu rfi rang vi cd thi xic djnh dugc dudng di ciia cau tnic ddng chay, LCS cd the dvt doin dudng di cic phan tur ed khdi lugng nhe nhu chat chi thi cua cac vat chit nhe trong ddng chiy, hoic nghien miu, cic vat nhd trdi ndi mi khdng c ^ lam thi ciiu v€ tran diu tren bien,... V i ^ xic djnh gidi ban nghi6m. ViSc xic dinh rd ring cic bidn niy cho kich thudc vi khii lugng ciia cic vat trdi ndi mi phep ta nghiSn ciiu dugc anh hudng cua h$ sd chiing cd the dvr doan dugc dudng di bing LCS se Reynolds 16n cau triic ddng chiy. la hudng nghien ciiu tiep theo cua tic gii TAI LI$U THAM KHAO
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Shadden, S. C, Lckien, F., and Marsden, J. E. (2005). "Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows."
Physica D: Nonlinear Phenomena, 212(3-4), 271-304.
Tritton, D. J. (1959). "Experiments on the flow past a circular cylinder at low Reynolds numbers."
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Vu, H. C, Ahn, J., and Hwang, J. H. (2015). "Numerical simulation of flow past two circular cylinders in tandem and side-by-side arrangement at low Reynolds numbers." KSCE Joumal of Civil Engineering, 1-11.
Abstract:
STUDY OF FLOW AROUND CIRCULAR CYLINDER USING
"LAGRANGIAN COHERENT STRUCTURE"
In this work, the flow structure around circular cylinder was investigated quantitatively using new method that is Lagrangian coherent structure, (LCS). LCS showed advantages compared with previous methods which used velocity and vorticity contour lo show flow structure. Based on LCS, flow structure was investigated in detail. Especially, the boundaries of wake structure can be
quantified, which can not be obtained by using other methods.
Keywords: Lagrangian coherent stmcture, circular cylinder, ilow stmcture, vorticity.
BBTnhdn bdi: 03/3/2017 Phan bien xong: 22/4/2017
KHO* HQC K? THUiT THliV Lpi VA Mdi TRlTdNO • SO 57 (6/2017)
