Xây dựng mô hình mô phỏng dòng chảy và truyền chất hai chiều, ứng dụng cho vùng vịnh hạ long

DAI HOC QUOC GIA HA NQI

Trudng Dai hoc Khoa hoc Tir nhien

XkY DUNG MO HINH MO PHONG DONG CHAY VA TRUYEN
CHAT HAI CHKU, VTNG DUNG CHO YUNG VINH HA LONG
MQ SO: Q T . 0 6 . 3 4

Chu Iri 3S kcii: TS Phung Dang Hie'u
CQC thcinh vien thorn gio: ThS Phqm Hodng Lam
CN Trdn Dvcc Trvt

OAI H O C Q U O C GIA HA NO:
TRUNG T.Ar/ THONG TIN THU ' " ^ ^

DT / 64^
HA NOI - 2005


BAO CAO TOM TAT

D £ TAI

1. Ten de tai: Xay dung m6 hinh mo phdng ddng chay va truydn chat hai chieu,
Ihig dung cho viing vinh Ha Long
2. Ma sd: QT.06.34
3. Chu tri de tai: TS Phiing Dang Hieu
4. Cac can bo tham gia:
1) ThS Pham Hoang Lam
2) CN Trdn Dixc Trur
5. Muc tieu va not dung nghien curu:

1) Muc tieu:
- Xay dung dugc mo hinh so m6 phdng ddng chay va truyen chat dang hai chidu
sir dung h6 phuong trinh nude nong phi tuyen va phuong trinh truydn tai khuech tan.
- LSig dung tinh loan thir nghiem cho viing vinh Ha Long.
2) Npi dung:
a) xay dung mo hinh so mo phong ddng chay va truydn chat hai chieu
- Nghien cihi thuat toan thich hop sai phan giai he phuong trinh nude nong hai
chi6u. Lap chuong trinh may tinh.
- Nghien cuu thuat toan thich hop sai phan giai he phuong trinh truyen chat. Lap
chuong trinh may tinh.
- Tinh toan va kiem nghiem mo hinh so theo cac di6u kien ly thuyet va thuc te.
b) Ung dung mo hinh tinh toan thir nghiem cho viing vinh Ha Long
- Thu thap va so hoa trudng dp sau viing vinh Ha Long; thu thap cac sd lieu v6
thuy trieu va gid viing bien ven bd Quang Ninh.
- ung dung mo hinh tinh toan thir nghiem cho viing vinh Ha Long theo cac kich
ban.
- Phan tich ket qua, danh gia mo hinh va viet bao cao long ket, nop san pham,
nghiem thu.
6. Cac ket qua dat dugc:
1) Da utig dung phuong phap CIP (Constraint Interpolation Profile) hien dai cd dp
chinh xac bac ba va phat trie'n mot phuong phap sai phan an vao viec sai phan hoa giai
11


h6 phuong trinh nude ndng phi tuy^n cho bai toan ddng chay. Viet thanh cong chuong
trinh may tinh m6 phdng ddng chay sinh ra do gid va do thuy trilu. Tinh toan thu
nghidm va ki^m chung md hinh cho cac dieu kien ly thuyet cd nghiem giai tich va
kilm chiing trong trudng hop dieu kien thuc te ciia bai toan truydn trieu Bien Dong.
K^t qua cho tha^y phuong phap CIP sir dung d day cho bai toan mo phdng ddng chay la
r^t hop ly va cd d6 chinh xac kha cao. Mo hinh chiing to kha nang ung dung tot cho

cac viing bi^n nong ven bd.
2) Da xay dung thanh cong chuong trinh truyin tai va khuech tan vat cha't su
dung phuong phap th^ tich huu han vdi so do TVD (Total Variation Diminishing) cd
dp chinh xac bac hai. Viet thanh cong chuong trinh may tinh va kiem chung mo hinh
cho cac bai toan ly thuyet cd nghiem giai tich. Ket qua cho thay phuong phap TVD siJ
dung d day cd dp chinh xac cao hon nhieu so vdi mot sd phuong phap sai phan truyen
thdng hay dugc sir dung nhu UPWIND bac mot hay Lax-Wendrof. Mo hinh cho phep
md phdng qua trinh truydn tai khuech tan vat cha't dudi tac dpng cua ddng chay trung
binh.
3) Da thu thap va sd hoa dia hinh viing vinh Ha Long tren ludi vuong gdc trong
toa dp D6 Cac vdi budc ludi 250 m x 250 m. Thu thap va phan tich nhung sd lieu v6
tdc dp gid va hudng gid viing ven bd Quang Ninh, sd lieu thiiy tridu. LSig dung mo
hinh m6 phdng he thdng ddng chay gay ra do gid trung binh mija Dong va miia He va
do tac dpng ciaa thuy trieu trong cac chu ky trilu len va xudng Idn. Mo phdng Ian
truyen vat cha't dugc thuc hien tren n6n ddng chay long hop ca gid va thuy tri^u vdi gia
thiet chat chi thi bin vung khong phan huy va cd sir cd mo trudng thai ra tai mot sd
di^m gia dinh. Cac ket qua cho ra cac bu'c tranh phan bd trudng ddng chay va phan bd
nong dp cha't 6 nhilm khi cd tac dpng ciia cac yeu td dpng lire la gid va thuy trieu.
4) Ndi chung md hinh xay dung trong di tai da dap ihig dugc yeu cau v6 mo
phdng ddng chay va truyen chat vdi dp chinh xac dang tin cay va cd kha nang urng
dung de nghien cihi cho cac viing bien ven bd khac. Ket qua ung dung tinh toan cho
viing vinh Ha Long, Quang Ninh cho thay ro kha nang ung dung ciia md hinh da phat
trien.
5) Dang dugc mot bai bao tren tap chi khoa hpc Dai hoc Qudc Gia.
7. Tinh hinh kinh phi cua de tai
Tdng kinh phi dugc cap: 20.000.000 ddng
Da nhan:

20.000.000 ddng

Da thanh toan:

20.000.000 dong

111


XAC NHAN
CHU TRI D £

CUA BAN CHU NHlfiM KHOA

TAX

(^>^>t^-^^

TS Phung Ddng Hieu

PGS. TS Pham Van Hudn
XAC NHAN CUA TRU^ONG

OHO HIEU TRUONU

GS.TS.^^rfin/ ^ / ^ / ^

IV


ABSTRACT
1. Project title: Development of a numerical 2-D model based on the shallow

water equation and convection-diffusion equation for the application to the Halong Bay
2. Code number: QT.06.34
3. Project Leader: Dr. Phung Dang Hieu
4- Members:

1) M. Sc. Pham Hoang Lam
2) B.Sc. Tran Due Tru

5. Aims and contents of project:
1) Aims:
- Development of a numerical 2-D model based on the shallow water equation
and convection-diffusion equation.
- Application of the model to the study of tide-induced currents and windinduced currents, the transportation of tracer in the Halong Bay.
2) Contents:
a) Development of a numerical 2-D model
- Development and application of a finite difference scheme for solving the
shallow water equation. Development of a computer program.
- Development and application of a finite volume TVD scheme for solving the
conveetion-diffution equation. Development of a computer program.
- Verification of the numerical model with some analytical and practical
conditions.
b) Application of the model to the study of tide-induced currents and windinduced currents, the transportation of tracer in the Halong Bay
- Collection and digitization of the bathymetry of Halong Bay; Collection of wind
and tidal data in the near-shore area of Quang Ninh.
- Application of the model to the simulation of tide-induced, wind-induced
currents in the Halong Bay. Simulation of the propagation and distribution of pollution
(tracer) based on some scenarios.
- Analysing and assessing the simulated results in order to give conclusions and
writing the final report.
6. Results:

1) The CIP (Constraint Interpolation Profile) method and a implicit finite
difference scheme have been applied to the developement of a numerical model based
on the shallow water equation. A numerical computer program has been successfully
developped for the simulation of wind and tide induced currents. Computation has been
carried out using the numerical model for the caseses of analiytical and practical


problems. Computed results show that the application of CIP method for the simulation
of shallow water currents is fairly accurate. The numerical model can be applied to the
near-shore shalow seas.
2) A numerical model based on the convection-diffusion equation using the TVD
(Total Variation Diminishing) scheme has been successfully developed. A computer
program has developed. Some numerical tests with analytical conditions show that the
TVD method used here is more accurate than the conventional methods such as
Upwind first order or Lax-Wendrof. The numerical model is capable of simulation of
the advection and diffusion processes of suspended materials under the driven action of
the depth averaged currents.
3) Collection and digitization of the bathymetry in the Halong have been done for
an orthogonal mesh with the same mesh size in x and y direction 250 m. The wind and
tidal data in the near shore area of Quang Ninh also were collected. Then, the
numerical model was applied to the simulation of wind induced currents resulted by
the averaged wind in the sommer and winter; and by the tide. Simulation of
propagation and distribution of suspended materials initiated at some locations under
the depth averaged currents induced separately by wind and tide, and by both wind and
tide was also carried out. Simulated resuhs showed resonable pictures of the water
current and pollution concentration distribution.
4) In general, the numerical model developed in this study meets the requirement
on the simulation of depth averaged currents and pollution propogation with an
acceptable accuracy and the model can be applied to other areas of shallow sea.
5) Results of the study were published as an article on the Journal of Science,

VNU.

VI


MUC LUC
Mdd^u

2

Chuong 1. Md hinh md phdng ddng chay hai chilu
1.1 H6 phuong trinh ca ban va cac di6u kien bien
1.2 Phuong phap sai phan theo so dd CIP cho he phuong trinh nude nong
1.3 Kiem chung md hinh sd'theo cac bai toan ly thuyet va thuc te
1.4 Nhan xet

4
4
5
10
16

Chuong 2. Mo hinh mo phdng truydn chat hai chi6u
2.1 He phuong trinh coban vacac didu kien bien
2.2 He phuong trinh dudi dang rdi rac sir dung phuong phap TVD
2.3 Kiem chung mo hinh sd theo cac bai toan ly thuyet

17
17
17

18

Chuong 3: Ung dung md hinh cho vung vinh Ha Long
3.1 Gidi thieu viing nghien cuxi
3.2 Sdlieu sijrdung trong nghien cuu
3.3 Ket qua mo phdng

26
26
35
35

Ket luan va kien nghi

51

Tai lieu tham khao

53

Phuluc

54

Tom tat cac cong trinh NCKH cua ca nhan (Miu 1)

55

Scientific Project (Mlu 2)

56

Phieu dang ky ket qua nghien cuu KH-CN

57


MODAU
O nhidm moi trudng bien hien dang la va'n dl dugc quan tarn ciia hSu het cac
qudc gia cd bien tren the gidi. Vdi nhu c^u phat trien kinh te hudng vao tiem nang ciia
bi^n da ngay cang gia tang ap lire v6 6 nhilm moi trudng bien ndi chung va d nhilm
mdi trudng vung ven bien ndi rieng. Chinh vi le dd, viec phat trien kinh te ciia mdt
viing ludn dugc ddi hoi dat trong bdi canh phat tri^n ben vung. Trong dd viec danh gia
tac ddng ciia boat dpng kinh te dan sinh d6'n thay doi moi trudng ddi hoi phai dugc
thuc hidn mdt each nghiem tiic de cd nhirng chien luge phat trien cung nhu qui hoach
phat trien dudi sir quan ly tdng th^ sao cho vira dap ung dugc nhu cau phat trien tat yeu
cua kinh te xa hoi, vira dam bao giam thieu dugc tac hai cho moi trudng tir nhien vd'n
cd.
Hiu het cac qudc gia phat trien va dang phat trien cd bien (nhu: My, Nhat ban,
Anh, Ha Ian, Mexico, Trung Qudc, v.v.) d6u da va dang thuc hien cac dir an khao sat,
danh gia va du bao bien dpng moi trudng nham dua ra cac canh bao, cac thong tin trp
giiip cho he thdng ra chinh sach, phue vu cho qui hoach phat trien kinh te cua cac viing
bi^n va ven bien quan trong (Peterlin va nnk, 2005 [10]; Lozano va nnk, 2005 [7]; Lau,
2005 [8]). Trong dd viec md phdng va danh gia muc dp Ian truyen chat d nhi6m trong
mot viing nude cu the la mot trong nhung mat xich quan trpng trong cac chuong trinh
quan ly tich hgp phuc vu cho viec xir dung hop ly tai nguyen va phat trien ben vimg d
h^u het cac qudc gia tren the gidi trong dd c6 Viet Nam.
Cac cong cu md phdng thuy dpng lire va Ian truydn vat cha't noi chung va cac chat
d nhilm ndi rieng (COD, BOD, TSS, ...) da va dang dupe rat nhieu trung tam nghien
cull cd uy tin tren the gidi phat trien va cho ra cac san phfa thuong mai cd gia tri cao

dap ung dugc cac nhu cau ung dung trong quan ly va sir dung hgp ly ngudn nude cung
nhu trg giiip cho qui hoach phat trien kinh te. Cac san pham cong cu phan mem dai
dien nhu SMS (ciia My), MINLAKE (cua My), MIKE 21 (cua Dan mach). Dd la
nhung cong cu hiru ich de ting dung tren thuc te.
0 nude ta cac chuyen gia ve mo hinh hoa va md phong loan hpc tren may tinh
cac qua trinh thuy dpng lire hpc va Ian truyen cac cha't d nhidm da va dang dan tiep can
dugc nhimg thanh tuu khoa hpc hien dai tien tien tren the gidi. He phuong trinh nude
nong phi tuyen hai chi6u da dugc nhi^u tac giai sir dung de md phdng bai toan ddng
chay hai chi^u trung binh theo dp sau nhu bai toan thuy trieu, bai toan ddng chay do
gio. Tuy nhien cac tac gia chu yeu don gian hoa he phuong trinh thanh dang tuyen tinh
hoa nhu Nguyln Ngpc Thuy (1969), Dang Cong Minh (1975) [1, 2] hoac giai phuong
trinh nude nong d^y dii nhung vdi cac so dd toan don gian nhu UPWIND bac 1 hoac
sai phan trung tam cd dp chinh xac khong cao. Ddi vdi bai toan truyen tai vat chat thi
va'n dt con khd hon la vdi cac so dd toan don gian thi viec ton tai cac nghiem sd khong
that (thi du nghiem am, hoac nhiiu sd) la va'n de thudng gap. Do vay viec phat trien


mdt md hinh md phdng ddng chay va chuydn cha't hai chi6u dang tin cay cho viing ven
bd la v^n di cin thiet phuc vu cho nghien cuu ciing nhu qui hoach moi trudng trong
cac viing nude ven bd.
Muc tieu ciia dd tai la phat trien va ung dung thanh cong cac so dd sd tien tien cd
dd chinh xac cao vao xay dung mo hinh sd md phdng dugc ddng chay va truyen chat
hai chi^u trong viing ven bd dudi tac dpng ciia gid va thuy tridu. Sau dd ung dung tinh
toan thii nghiem cho viing vinh Ha Long.
Ngoai phSn md ddu, ket luan va kien nghi, tai li6u tham khao, npi dung bao cao
cua d^ tai dugc trinh bay trong ba chuong:
Chuong 1 : Md hinh ddng chay hai chidu. Trong chuong nay trinh bay co sd ly
thuyet, he phuong trinh xua^t phat, cac dieu kien bien ciia md hinh md phdng ddng chay
hai chi6u cd tinh den anh hudng ciia gid. Trinh bay tdm luge phuong phap sai phan cd
dd chinh xac cao CIP (Constrained Interpolation Profile); phuong phap sai phan an

dang SMAC va each ung dung rdi rac hoa giai he phuong trinh nude nong phi tuyen.
Cac bai toan kiem nghiem md hinh dudi dang ly thuyet va thuc tiln cung dupe trinh
bay nham khang dinh tinh on dinh, dung dan cua md hinh so.
Chuong 2 : Md hinh truyin cha't hai chidu. Trong chuong nay trinh bay he
phuong trinh md ta qua trinh truyen tai va khuech tan vat chat hai chieu dudi tac dpng
cua ddng chay nen. Md ta cac dilu kien bien. Trinh bay phuong phap sai phan dang
TVD cd dp chinh xac bac hai va ling dung rdi rac hoa giai he phuong trinh xuat phat.
Md hinh sd dugc kiem chung thdng qua cac bai toan co ban cd nghiem giai tich nham
kiem nghiem kha nang mo phdng qua trinh truydn tai vat chat ciia md hinh.
Chuong 3: Ung dung md hinh cho viing vinh Ha Long. Chuong nay trinh bay tom
lupe viing nghien cun, di6u kien gid, thuy treu, dia hinh. Lira chpn cac sd lieu dau vao
de md phdng. Cac ket qua mo phdng Ian truyen vat chat gia dinh khong bi phan huy
theo thdi gian theo cac kich ban dudi tac dpng ciia gid va thuy trieu. Cac nhan xet v6
ket qua nghien cun.
De tai nay dupe hoan thanh vdi sir hd trp kinh phi tir phia Dai hpc Qudc gia
(DHQG) Ha Ndi, sir giiip do ciia Ban Khoa hpc & Cong nghe, DHQG, Phong Khoa
hpc Cdng nghe, trudng Dai hpc Khoa hpc Tir nhien, sir ung ho nhiet tinh ciia Hoi ddng
Khoa hpc Trai da't, DHQG HN, ciia Ban chu nhiem khoa Khi tugng Thuy van va Hai
duong hpc, cung nhu su ddng gop y kien eiia cac nha khoa hpc, cac ddng nghiep tham
gia di tai va tap the can bd cua Bp mdn Hai duong hpc. Nhan day chiing tdi xin chan
thanh cam on nhimg giiip do qui bau dd.


CHUONG 1:
MO HINH MO PHONG DONG CHAY HAI CHltv
LI H £ PHUONG TRINH CO BAN VA CAC Difiu KI$N BifiN
He phuong trinh xuat phat diroc sir dung trong phat trien md hinh la he phuong
trinh nude ndng phi tuyen, bao gom :
Phuong trinh bao toan dpng lugng
du

du
du dC
— + u— + v
fv = -S^Adt
dx
dy
dx p{h + 0
dv
dv
dv ^
dC
1
— + u — + v — + fu = -2 -^- +
dt

dx

dy

dy

(LI)

k'rlh^^^

(1.2)

p{h + 0

Phuong trinh bao toan khdi lupng

dC , d(h + Ou , d(h + Ov
dx

dt

trong dd

dy

(

du^

I

dx.1 dy

dx

d (
+—
dy

dx

(1.3)

du]

(1.4)

dv]

(1.5)

(
•'

^

'J,

—

w,v: cac thanh phan van tdc trung binh dp sau theo cac hudng ox va. oy\ f: tham sd
Coriolis; (^ : dao dpng muc nude; p : mat dp nude; h : dp sau nude; u^: he sd nhdt rdi.
T^Ty la cac thanh phan ung suat gio tren mat theo true x va true y, chung dupe xac
dinh theo cdng thuc sau:
T: =P^C,^\W\W/,

T].=P^C\W\W^,^

(1.6)

trong dd p^ la mat dp khong khi; w la vector van tdc gid. He sd C, dugc xac dinh
theo Smith & Banke (1975) nhu sau :
(0.63 + 0.066|W^|)xlO^

C.

(2.28+ 0.033(1^^^1-20)) X 10"

|fF|<20m/5
\w\>2t)mls

Cac thanh phan ung suat ma sat day dupe xac dinh theo cdng thuc sau:
CLU^JU

+ V

h
r^
I 2
2
Tj = p Cf,vyju
+ v^

He sd ma sat Q dugc xac dinh theo cdng thuc
C,

(h + cy

Ai^ 0.0264

(1.7)


H6 phuong trinh tren cho phep md phdng ddng chay cd chu ky dai trong viing
nude ndng bao gom ca ddng chay phat sinh do thiiy triSu va do gid.
Cac diiu kien bien dugc sir dung trong md hinh nhu sau:

Bi6n cung: sir dung dieu kien khong tha'm, ture la thanh phan van tdc theo phuong
phap tuyen vdi mat bi6n bi triet tieu.
Bidn long: dieu kien bien phat xa tir do cho sdng dai dugc sir dung. Khi dd gia tri
van tdc thang gdc vdi bien dugc xac dinh theo cdng thirc
a - ^

(1.8)

Bi6n Idng cho dao done thuy trieu: dao dpng muc nude dugc tinh theo cac hang
sd dieu hoa cho tren bien.

1.2 PHUONG PHAP SAI PHAN THEO SO DO CIP CHO H £ PHUONG TRINH
NU6C NONG
Va'n d6 kha quan trpng trong viec giai sd he phuong trinh nude ndng phi tuyen la
xa'p xi dugc tot thanh phan binh luu phi tuyen. Thuc te giai sd cho thay neu xap xi cac
thanh ph^n phi tuyen nay bang cac so do sai phan trung tam cd dp chinh xac bac hai thi
thudng dan den cac nhilu gia trong ket qua gia sd trong trudng hgp tdn tai cac gradient
Idn ciia muc nude hay ddng chay. De khac phuc va'n d^ nay mot trong nhung phep sai
phan thudng dupe sir dung do la phuong phap upwind bac mot hay so dd Quick bac ba
ket hop vdi so do nhdt nhan tao.
Gan day mot sd phuong phap sai phan mdi ket hgp giira hai phuong phap
Lagrangian va Euler cd dp chinh xac cao da dupe nhdm tac gia Yabe va Aoki (1991)
[11] cong bd cd ten la CIP (Constrained Interpolation Profile). Phuong phap nay da
dugc chiing minh la co dp chinh xac bac ba va xap xi rat tot cho cac thanh phan truyen
tai phi tuyen. Thi du ve mot sd nghien cun sir dung phuong phap CIP da xuat ban nhu
nghien cihi tuong tac sdng va cong trinh ngam (Hieu va Tanimoto, 2006) [9] hay
nghien ciin thuy trieu Quang Ninh (Hieu, 2006) [3]. Trong nghien cun nay, phuong
phap CIP dugc sir dung de xap xi thanh phan binh luu phi tuyen trong phuong trinh
chuyen dpng cua he phuong trinh nude nong.
Dudi day trinh bay tdm tat tu tudng va npi dung ciia phuong phap CIP (chi tiet

xem Yabe va Aoki, 1991) [11].
1.2.1 Tdm tdt phuong phap CIP
Tren thuc te hau het cae qua trinh vat ly thudng diln ra trong mdi trudng lien tue
nhung giai sd cac qua trinh dd ta phai thuc hien rdi rac hda mi6n tinh toan. Muc tieu co
ban CLia thuat giai sd la khdi phuc nhung thong tin bi mat (hay bi bd qua) giira cac
diem rdi rac dd. Da sd cac so do sd trudc day d6u khong quan tam den nghiem thuc
phan bd ben trong 6 ludi va do dd mii'c dp chi tiet quan tam la d miJc kich thudc ludi
chia (Ax,Ay). Phuong phap CIP do Yabe va Aoki (1991) [11] trinh bay da cd gang di


xay dung phan bd nghidm trong d ludi sao cho g^n dung nhat vdi phan bd nghiem thuc
cua phuong trinh md ta vdi mot sd phep ap dat cu the. De di6n ta phuong phap CIP ta
xet mdt phuong trinh truydn tai dang sau :
dt

(1.9)

dx

Khi van tdc u la hang sd thi phuong trinh (1.9) md ta chuyen ddng tinh tien don
gian ciia trudng / vdi tdc dp u . Dang phan bd ban dau (xem dudng liin net tren hinh
1.1 a) di chuyen thanh dudng dut net trong trudng hgp bieu diln lien tuc (xem hinh
1.1a). Tai thdi diem nay nghiem tai cac diem ludi dugc ky hieu la cae cham trdn va nd
gidng nhu nghiem dung tai cac di^m dd. Tuy nhi6n neu ta loai bd dudng durt net nhu
tr6n hinh Lib thi thdng tin ve hinh dang (profile) ciia nghiem ben trong d ludi bi mat
va raft khd tudng tugng ra dung profile nghiem, khi dd mot each rat tu nhien ta cd the
tudng tugng ra dang profile ciia nghifm nhu dudng lien net tren hinh Lie. Nhu the
khuech tan sd cd the phat sinh khi ta xay dung dang profile cua nghiem bang phep ndi
suy tuyen tinh mac dii ta da dua tren cac nghiem diing tai cac diem ludi (hinh Lie).
Qua trinh ndi suy nay dugc thuc hien trong so dd Upwind (ngugc ddng) bac 1. Mat

khac neu ta xap xi npi suy bang ham bac hai thi se gap phai ket qua la cac gia tri ndi
suy vugt qua gia tri that, qua trinh npi suy nay dupe thuc hien trong cac so dd LaxWendroff hay trong so do Leith.

uAt

gradient

Hinh 1.1: Nguyen ly cua phucmg phap CIP. (a) duong li^n net la duong ban dau, duong dut net
la nghiem dung sau budc thcfi gian At; (b) nghiem tai tung die'm rdi rac; (c) khi noi suy tuyen tinh,
xuat hifin khuech tan so; (d) so do CIP: dao ham khong gian cung di chuydn va profile ciia nghiem
trong o ludi dugc khoi phuc.

Dieu gi da lam nghiem kem chinh xac di? Dd la do ta da bd qua co che phan bd
nghiem phia trong d ludi va ta da di theo cac nghiem tron. Do dd ta thay phuong phap
dua dang phan bd nghiem thuc vao trong 6 ludi la rat quan trpng. Phuong phap CIP da
dua ra each xap xi phan bd nghiem thuc trong 6 ludi nhu sau :


Trudc h6't l£y dao ham phuong trinh (1.9) theo bien x ta thu dugc :

^ +3 = - ^ ^
dt

dx

(1.10)

dx

Trong dd g = df Idx la dao ham khong gian ciia / . Trong trudng hgp don gian

nh^t u = const thi phuong trinh (1.10) gidng nhu (1.9) la md ta chuye'n ddng cua dao
ham khdng gian theo van tdc u. Bang each siir dung phuong trinh nay ta cd the duoi
theo cac tien trien thdi gian cua / va g dua tren phuong trinh co sd (1.9). Neu g tinh
dupe va di chuyen nhu trinh bay bang mui ten tren hinh Lid, thi ta de dang sir dung
cac dao ham g nay de hinh dung ra phan bd nghiem va dudng profile se gan dung vdi
dudng ban d^u (nghiem diing) hon rat nhidu
Neu ca hai gia tri ciia / va g dupe cho trudc tai hai diem ludi thi dang profile
cua nghiem cd the dupe npi suy bang mdt da thuc bac ba (Nakamura va nnk, 2001):
F:{x) = a,X'+b,X'+g:X

+ f;

(1.11)

trong dd
g:+g:,
Ax'

2{f:-f:^)
""'
Ax,'

(1.12)

2g:+g:^)
Ax,

(1.13)

+ '''

3(/;-r)
Ax.'
^ .

=^,up-X,

iup = /-sgn(w,)
X = x-x,
sgn(w) la ham lay da'u ciia w . x la tpa dp phia ngupc ddng cua diem xet sau budc
thdi gian A/. Nhu vay profile tai budc thdi gian (« + l) d^ dang thu dugc bang each
dich chuyen profile di mot doan w^A/ (tuong tu nhu phuong phap Lagrangian) vi the
/"*' =F"{x, -u,At) va g;^' =dF"{x, - w , A / ) / ^ . Do dd ta cd :
fr'=ci,^!^b,^'+g:^,+f:

(LI4)

gr'=3a,^,^+2Z),^,+g;

(1.15)

Trong dd Nhu the phuong phap CIP sir dung phap tinh tien Lagrangian tren ndn ludi Euler
va do dd nd thupc vao dang phuong phap Semi-Lagrangian. Thirc te tinh toan sir dung
phuong phap CIP da cho thay vdi phuong phap CIP hien cd the sir dung vdi sd Courant
- Friedrichs - Lewy (CFL) kha Idn ma so d6 van rat on dinh.
1.2.1 Ung dung phuong phap CIP cho he phuong trinh nude ndng
Vdi he phuong trinh nude ndng ta co the bieu di6n tong quat dudi dang nhu sau:
7

^^u^^^^^G
dt
dx
dy

(1.16)

trong dd G bao gdm cac sd hang cdn lai ciia phuong trinh chuyen ddng (gradient muc
nude, ung su^ft gid, ung sua't day, lire Coriolis, lire nhdt).
Sir dung thuat toan tach ta tim nghiem ciia phuong trinh (1.16) theo hai budc
(Yabe va Aoki, 1991):
Budc A: Pha truyen tai (Advection phase)
df
df
df ^
-^ + u^^ + v^^ = 0
dt
dx
dy
Budc B: Pha khdng truy6n tai (Non-Advection phase)

.. . , .
(1.17)

^ =G
(1.18)
dt
Sau khi budc A dupe giai cho pha truyen tai thi budc B dugc tinh toan cho pha
khong truydn tai dua tren cac gia tri vira tinh tir budc A. Trong budc A ta sir dung

nghidm giai tich dja phuong ciia phuong trinh (1.17) dudi dang Lagrangian:
f{x,,y,J

+ At) = f{x^,^y^„J)

(1.19)

vdi x^„ =x, + ^ , y^„ =y, +;;, ^ = -w,A/, rj = -v,At
Nhu the phuong trinh (1.19) cd the thu dupe bieu thurc hien theo cdng thure (1.14)
va bang each diing so do CIP nay luan chuyen cho timg hudng ox va oy cho bai toan
hai chidu.
Sau khi tim dugc f"*^' = f{x,,y

,t+ At) a budc 1 thi gia tri ciia / tai budc thdi

gian n +1 that bao ham ca cac ddng gdp ciia cac yeu td khdng thupc pha truyen tai se
dugc giai theo phuong trinh (1.18) la:
/ r ' =fr'' +GAt

(1.20)

Ddi vdi ham gradient g ciia / (tii'c la — = g^,— = g ) cung duoc thuc hien
dx
dy
tuong tu nhu ham / nhung sir dung cong thuc CIP phuong trinh (1.15).
Trong nghien cuu nay de tang tinh on dinh eiia bai toan trong budc giai phuong
trinh (1.19) va phuong trinh lien tuc da sir dung phep sai phan an nhu sau:
Trudc het ta lira chpn vi tri cac bien dat tren he ludi so le: cac bien w, v dugc dat
tai cae canh ciia d ludi chu* nhat con bien ^ dupe xac dinh tai tam ciia d ludi (nhu so dd
Akagawa-C). Phuong trinh (1.1) va (1.2) dugc viet lai dudi dang don gian nhu sau:

^ = -F(u,v,h,...)-g^
dt
dx

(1.21)


(1.22)

— = -G(M,v,/i,...)-g^
dt
dy

trong dd ham F va G chu:a dung cac thanh ph^n cdn lai trong phuong trinh (1.1) va
(L2).
Ta sir dung phep sai phan tuong tu nhu phuong phap SMAC (Simplified Marker
and Cell Method) cho phuong trinh (1.21) va (1.22). Ddi vdi budc thdi gian cho trudc
n trudng van tdc du doan cho budc thdi gian tiep theo dupe xac dinh hien thdng qua
cac phuong trinh ddng lupng gpi la budc 1
,("+!)•

dC
g dx

w" A-At

dt;"

.("-^D* =- ,v"
, " + Ar

dy

(1.23)

F{u,v,h,...)

(1.24)

-G(M.V,/?,...)

Chi sd dau sao d tren cho biet trudng van tdc du doan ban dau (predicted velocity)
d budc thdi gian « +1. Trudng van tdc du doan nay sai lech so vdi trudng van tdc thuc
do cd sir thay ddi ciia dao dpng muc nude A^ d thdi diem mdi, do dd van tdc d thdi
di^m n +1 dupe xac dinh tir trudng van tdc du doan ban di\x\

V

„<-.•,,,, _ ^ ^ j

(1.25)

dAC
dy )

(1.26)

= V

(n+D*

+ A^

(1.27)

r^'=r+A<-

va

Tir phuong trinh bao toan khdi lugng thay neu nhan cac ve cua phuong trinh
(1.25), (1.26) vdi dai lugng {hA-Q va lay dao ham tuong ung theo x va theo y roi
cdng lai ta thu dugc phuong trinh Poisson cho gia sd At^ nhu sau:
dv ( - 1 ) - / ^ N

r^^in.vr H
dt

trong d d

dx

^gAt^l-in'-^
^ ^dx\

dy

dx

d(

^dAC

oy

oy

(1.28)

H = {h + (;")

Neu la'y xap xi sai phan cho dao ham thdi gian eiia ^ va sir dung phuong trinh
(1.27) ta thu dugc phuong trinh sau:
d_

At

^

\dx\

dx

dy

H

dA£_
dy

cu

H
dx

c^-'"^"'//'
dy

(1.29)

Sai phan hoa phuong trinh (1.29) cho cac dao ham khdng gian tren ludi so le vdi
^n la dp tang muc nude AC ta thu dugc mot he phuong trinh dai sd tuyen tinh vdi ma
tran ddi xiing cd he sd tren dudng cheo chinh xac dinh duong. He phuong trinh dai sd
tuyen tinh nay cd the giai lap bang phuong phap SOR hay CG rat hieu qua. Trong


nghiSn cun nay da sir dung phuong phap BiCGSTAB (Bi-Conjugate Gradient ciia van
derVort(1992).
Vdi phep sai phan nhu tren, ta cd the tdm luge qui trinh tim nghiem nhu sau:
Budc 1: tinh toan trudng van tdc du doan ban dau sir dung phuong trinh (8) va (9).
Trong dd cac thanh ph^n binh luu phi tuyen, nhdt, coriolis va ma sat dupe xac dinh
theo cac so dd tuy chon, d day da sir dung phuong phap xap xi CIP cua Yabe va Aoki
(1991) [3].
Budc 2: giai he phuong trinh dai sd tuyen tinh cho A^ tir phuong trinh (1.29) vdi
cac gia tri H va w'"^"', v*"^"* da biet tir budc 1.
Budc 3: cac gia tri w"*', v"^' va 4'"*' d budc thdi gian can tim dupe xac dinh tir
phuong trinh (1.25), (1.26) va (1.27).
Budc 4: gan cac gia tri budc thdi gian n +1 cho budc n rdi tiep tuc qui trinh cho
cac budc thdi gian tiep theo.
1.3 KI^M C H U N G M 6 HINH SO THEO B A I TOAN L V THUYET VA THlTC TE
Tr6n thuc te tinh toan nhi6u khi gap nhung trudng hgp tdn tai cac gradient muc

nude Idn, thi du nhu cac front nude tran len cac viing dat thap (nude dang d cac bai
tri^u, sdng nude dang do bao v.v.), do dd viec kiem tra xem kha nang mo phdng cua
mo hinh toan mo ta d tren cd sir dung phuong phap CIP ddi vdi cac trudng hpp cd
gradient Idn ciia muc nude la rat can thiet. De thuc hien dieu nay bai toan sdng vo dap
dupe sir dung lam dieu kien kiem tra thu* nhat.
Muc tieu ciia md hinh la mo phong ddng chay trong viing ven bd do dd viec kiem
tra kha nang md phdng sdng dai ciia mo hinh cung la van di dugc dat ra. Yeu cau quan
trpng ciia md hinh sd la phai md ta dupe tuong tac phi tuyen ciia sdng dai va hon nira
phai bao toan nang lupng ciia cac sdng dai nay d miic dp chap nhan dupe. De thuc hien
kiem nghiem cho muc tieu dat ra thi bai toan sdng truyen trong kenh thing kin mot
d£u, trong kenh gia thiet khdng co ma sat, khong cd lire coriolis, sdng dung hoan toan
dugc gia thiet va sir dung lam bai toan kiem tra thii hai.
Bai toan thu: ba diing de kiem nghiem kha nang md phdng thiiy trieu vdi dieu kien
thuc te ciia Bien Ddng. Tinh toan dugc thuc hien cho truyen trieu tren Bien Dong va
ket qua dugc so sanh vdi sd lieu muc nude thuc do d mot sd tram hai van ven bien.
1.3.1 Kiem chung vdi bai toan vd dap
Trudc tien, md hinh sd dupe kiem nghiem qua viec mo phdng sd bai toan vo
dap. Day la mot trong nhung bai toan kinh dien diing de kiem tra cac mo hinh sd rat
hun hieu. Bai toan dupe gia thiet la co hai viing nude d thdi diem ban dau cd dp cao cot
nude la /?, d phia trai va K dphia phai (h^>h._). TiJc thi, vach ngan giira hai viing nude
dupe rut bd, do tac dung eiia lire trpng trudng nude tir phia trai tran sang phi'a phai. Bai
toan nay cd nghiem giai tich cho sir bien dang va di chuyen ciia cot nude. Viec md
10


phdng qua trinh nay la kha khd ddi vdi md hinh nude ndng khi sir dung phuong phap
sai phan hiiu han do ton tai gradient Idn tai diem ngan each hai viing nude.
Di kiim tra so dd toan trong md hinh sd d day da sir dung viing nude chu nhat cd
dd dai 1000 m, dp cao h^ =3m d phia trai va /z2=lm d phia phai, hai viing nude cd dp
rdng bang nhau la 500m (xem hinh 1.2). Ket qua md phdng sd va nghiem giai tich cho

thdi di^m 50 giay sau khi dap ngan giira hai viing nude bi vo, dugc trinh bay tren hinh
2. Hinh 2 cho tha'y cac nghiem sd sir dung cac so do khac nhau cho cac ket qua rat
khac nhau va khac vdi nghiem giai tich, trong dd nghiem sd cua phuong phap sai phan
in six dung trong nghien cun nay ket hgp vdi so do CIP (xem hinh 1.3) cho ket qua gdn
triing vdi phan bd cua nghiem giai tich. Hon nira cae nhidu gia do so dd sd khdng xuat
hien tai cac diem front cua cot nude. Diiu nay cho ta mdt co sd tin cap de tin tudng vao
su 6n dinh va kha nang ung dung ciia md hinh cho cac bai toan thuc te.

£3
c
E
o
Q

100

200

300

400
500
600
Khoang each ngang (m)

700

800

900

1000

Hinh 1.2: Phan bo'cot nU6c tai thdi diem ban dau.

Nghiem
Nghiem
Nghiem
Nghiem

giai tich
so (Linear)
so (Upwind)
so (CIP)

c

E
o

s
O 1

100

200

300

400

500
600
Khoang each ngang (m)

700

800

900

Hinh 1.3: So sanh ket qua mo phong so'va nghiem giai tich
trUdng hop bai toan vd dap sau khoang thoi gian 50 giay.
11

1000


1.3.2 Kiem chumg vdi trudng hgp sdng dai dumg trong kenh thang
Do bai toan thuy trieu la mdt bai toan md phdng Ian truydn sdng dai nen nhat
thi^^t phai ki^m nghi6m md hinh toan ddi vdi trudng hgp sdng dung. Day la mot trong
nhirng bai toan kiem nghiem don gian nhung lai ra't hihi hieu trong viec kiem tra tinh
bao toan nang lugng eiia so do sd. Bai toan dat ra la khi cd mot sdng dai truyen vao
mot kenh thang cd dp sau khdng doi, kin mot dau. Gia thiet phan xa hoan toan va
khdng cd tac dung ciia luc coriolis thi do ket hgp ciia sdng tdi va sdng phan xa, trudng
sdng dung se xu£t hien trong kenh vdi cac diem nut va diem bung phan bd theo qui
luat. Neu suy giam sd cua md hinh sd la nhd thi tai cac diem bung ta phai thu dupe cac
dao ddng cd bidn dd it nha't la ga'p hai Ian bien dp eiia sdng tdi. Tai cac diem nut cd cac
dao ddng nhd. Neu tinh phi tuyd'n dupe md phdng tot thi cac dao ddng tai diem nut
khdng bi triet tieu ma se xuat hien song thir ca'p cd t^n sd ga'p hai \in t^n sd ciia song
tdi.

Vdi bai toan kiem tra thur hai nay, kenh din dugc chpn cd dp dai /=1000 m, dp
rdng b =50m, dp sau h =10m, sdng tdi cd chu ky r=30s, bien dp A =20cm. Viing kenh
dugc chia thanh cac ludi hinh chir nhat 2,5m x 2,5m. Bien tai cudi kenh dugc chpn la
bien khdng tha'm (phan xa toan phdn). Thdi gian tich phan dupe chpn la 250s du de
sdng truyen het hai Ian chieu dai kenh. Cac ket qua md phdng dao dpng muc nude tai
cac di^m bung va diem nut (each nhau xen ke bang mot phan tu dp dai sdng L x T^fgh
dupe trinh bay tren cac hinh 1.4 va hinh 1.5.
- Dao dpng tai bien
Dao dong tai diem bung

T

r

100

150
Thoi gian (s)

Hinh 1.4: Dao dpng mUc nu6c tai diem bung (diem ngay phia tru6c tu6ng diing) va dao
dong song t6i
Hinh 1.4 trinh bay dao dpng ciia muc nude tai diem bung ngay phia trudc tudng
diing va dao dpng cua sdng tdi tai bien vao dau kenh. Tren hinh ve ta thay, sau mot
khoang thdi gian khoang 100 giay song truyen tdi tudng va hien tuong phan xa xuat
hien. Dao dpng tai diem sat tudng co bien dp gap hai Ian bien dp sdng tdi va on djnh
sau mot khoang thdi gian rat ngdn. Sau khoang 200 giay, sdng phan xa tir tudng ket
hgp vdi sdng tdi tai bien va tao thanh mot song ket hpp, luc nay dao dpng ciia sdng tai

12

bidn khdng cdn la dao dpng cua rieng sdng tdi nira, ta cd the thay rd sir bien ddi ciia
dudng dtJt net trdn hinh 1.4 cua dao ddng tai bien.
Dao dpng tai bien

0.4

Dao dpng tai diem nut
^

02--

-0.2 •
-0.4

-1—

50

T

•

150

100

200

250

Thai gian (s)

Hinh 1.5: Dao dong mUc nu6c tai diem nut (each tUdng mot khoang 5L/4) va dao dong tai
bien.
Hinh 1.5 trinh bay dao dpng cua muc nude tai diem each tudng mdt khoang 5/4
dd dai sdng tdi ^, tuong ung vdi diem gdn diem niit. Ta tha'y sau khi sdng tdi di qua
mot khoang thdi gian ngin co 130 giay thi sdng phan xa tir tudng quay lai gap sdng tdi
tai dilm nay va tao thanh sdng ket hpp cd dao ddng nho hon rat nhidu dao ddng cua
sdng tdi. Dae biet ta cd the quan sat thay dao dpng cd tan sd cao tai diem nay do sir ket
hpp phi tuy6'n giira sdng tdi va sdng phan xa chinh vi vay cac dao dpng muc nude van
tdn tai d diem niit.
Nhu vay md hinh sd cd the mo phdng tot trudng hpp Ian truyen song dai va md
phdng dugc cac tuong tac giira cae sdng vdi sir suy giam sd khong dang ke. Ket qua
nay cho phep ta tin tudng vao md hinh de cd the ap dung cho cac bai toan truyen sdng
dai tren thuc te.
1.3.3 Kiem chumg vdi bai toan thuy trieu Bien Ddng
Md hinh sd da kiem nghiem d tren dugc ap dung tinh toan thir cho truyen trieu
trong Bien Ddng. Viing bien dupe chia thanh cac ludi hinh vudng vdi Ax =1/5 dp
Ay =1/5 dp, cac true dupe chpn xap xi hudng theo kinh tuyen va vi tuyen tren he toa dp
Di Cac. Bien ciing dupe xap xi vdi didu kien thong lupng bang khdng. Cac bien long
dupe cho cac dao dpng muc nude theo cac hang sd dieu hoa ciia 4 sdng chinh. Cd ba
bien long dugc sir dung cho mo hinh la eo Dai Loan, eo Basi va eo Malaea. Cae hang
sd di6u hoa tren cac eo bien nay dugc lay theo bang thuy trieu Admiralty Tidal Tables
[6]. Cac toa dp diem la'y hang sd dilu hoa va cac bien dupe trinh bay trong bang 1.1.

13


Bang 1.1: Cdc bifin long va cac hang s6 6iiu hoa

VTdp

Kinh do

M2

S2

Kl

01

H(cm)

G"

H(cin)

G°

H(cm)

G"

H(cm)

G"

DM Loan

23033,

119033,

87

326

23

15

25

278

21

236

Basi

21^58'

120M5'

26

188

11

214

23

262

18

224

Malaea

rir

108''59'

26

125

5

199

14

66

16

10

Hinh 1.6 trinh bay trudng van tdc ddng trung binh tai luc 10 gid ngay 12 thang 2 nam
1988.

Hinh 1.6: TrUcJng dong ehay trieu mo phong tren bien Dong
(luc 0 gid ngay 12/2/1988; mien tinh dUdc chia thanh 1116 theo kinh dp v6i bu6c lu6i 1/5
do va 116 6 theo vi do v6i bu6c lu6i 1/5 dp; go'c (0,0) tai toa dp (0,8'N;98,8°E))

14


Tinh toan
ThiJc do

.
ft'

'/1

.1 V

•/ I

,"/ I

•/ f

"ft

,7 I

'A

'1

f\ ./s^. A .7 \ ;/
_ _, _

S -1

50

100

150

200

250

300

350

400

Thdi gian (gio)

Hinh 1.7: Dao dpng mUc nu6c tai khu vUc Hon Da'u (bit dau tir 0 gid ngay 01/1/1988; Tpa
dp: VI dp 20"41'N, kinh dp 106'^49'E)

300

600

Thdi gian (gid)

Hinh 1.8: Dao dong mUc nu6c do trieu tai Qui Nhdn (bit dau tijf 0 gid 15/5/2002; Toa dp:
VI dp 13*'45'N, kinh dp 109n3'E )
Tren cac hinh 1.7 va 1.8 trinh bay so sanh dao dpng muc nude giira tinh toan theo
md hinh va thuc do tai hai tram hai van Hdn Dau va Qui Nhon. Tren hinh ve ta thay
dao dpng muc nude tinh toan kha phii hgp vdi sd lieu thuc do. Dao dpng muc nude the
hien rd tinh nhat trieu diu a khu virc Hon Dau va xu the thay doi ciia dao dpng muc
nude tinh toan rat phii hgp vdi thuc te. Tren hinh ve so sanh ta cung thay cd sir sai khac
nho vi bien dp tri6u giira md phdng sd va sd lieu thuc do, nguyen nhan co the giai
thich do vi tri lay ket qua md phdng sai lech kha Idn so vdi vi tri co sd lieu vi ludi la
1/5 dp va hon nira do ludi md phdng eon kha thd nen chua md ta dupe het nhimg anh
hudng phi tuyen cua viing nude ndng va sir thay ddi dia hinh phu'e tap len dao dpng
muc nude trilu. Ndi chung md hinh da mo phdng dupe kha chinh xac dao dpng ciia
tri^u tren Bien Ddng.

15


1.4 NHi^N XET
Cac k6t qua ki^m chung md hinh cho mot sd bai toan ly thuyet va thuc te da
chung td phuong phap sd sir dung de phat trien md hinh la rat phii hgp. Md hinh cd kha

nang md phdng tot sdng dai trong vimg ven bd va ddng chay hai chi6u trong viing bien
ndng. Ket qua kiem chung d tren cho ta su tin tudng vao kha nang ung dung ciia md
hinh cho cac bai toan thuc l6.

16


CHirONG 2:
MO HINH MO PHONG TRUYEN CHAT HAI CHl£u
2.1 H$ PHl/ONG TRINH CO BAN VA CAC DI^U K I £ N BIEN
He phuong trinh co ban dugc sir dung de phat trien md hinh md phdng Ian truyen
vat cha't trong nude la he phuong trinh truyen tai va khuech tan vat chat dudi dang hai
chi^u. H^ phuong trinh dugc viet nhu sau :
dC

dt

dC
dC d (^ dC\
d
A-U— + v — = — D. — +

dx

dy dx\

dx)

dy

' dy

^Q-oC

(2.1)

Trong dd C(x,y,r) la ndng dp vat cha't trong nude; D^, Dy la he sd khuech tan
theo phuong cac true tpa dd; Q la ham ngudn; a la he sd phan huy.
Cac dieu kien bien dupe sir dung la: tai bien cung cho rang thdng lupng truyen tai
vudng gdc vdi bien bang khdng; tai bien md truyen tai tu do va khuech tan tu do.

2.2 H £ PHl/ONG TRINH DUdi DANG R O I RAC SIT DUNG PHl/ONG PHAP TVD
He phuong trinh (2.1) tuy don gian song de thu dupe nghiem sd hgp ly va ed dp
chinh xac cao lai khdng phai la van de don gian chut nao. Thuc te tinh toan cho thay
neu sijr dung cac so dd toan thong thudng cd dp chinh xac hon bac 1 cho thanh ph^n
truy6n tai thi trong khdng gian nghiem thudng ton tai cae nhilu gia tao ra cac nghiem
am hay cac nghiem vugt qua cao so vdi nghiem dung. De giai quyet van de nay thi mdt
sd tac gia da di sir dung so do ngugc ddng bac 1 de giai. Tuy nhien phuong phap ngupc
dong (Upwind) bac 1 nay lai mdc phai sir khuech tan sd rat Idn lam cho nghiem sd
thudng nhd hon nhidu so vdi thuc va viing phan bd thi rdng hon. De khac phuc van de
nay trong nghien cun nay chiing toi di sir dung so do FLUX-LIMITING cd dang TVD
vdi dp chinh xac bac 2 nhung khong gay nhi^u sd. Dudi day trinh bay so dd toan nay.
De don gian xet phuong trinh truyen tai mot chilu dang bao toan sau
^ +^^^ =0
dt

(2.2)

dx

Khi dd bieu thuc sai phan cd the viet dudi dang sau
cr'=c:-^[F:^,,,-F,%,,)

(2.3)

So dd Flux-Limiting di xac dinh cac ham Flux F^^i ^ va F,%:-^ theo cong thu:c
sau :
17
OAI HOC Q U O C G I A uh Nr>'
TRUNG ^•-^^ ^H ^ ; G 'ir .Hi: -.;.


F,:,n = ^ - ? ( / . , -/,) + ^ ( - ^ ) ( / , . -/,)
2

2

2

(2.4)

Ax

Vdi cc = sign{ii^^^,2)\ 0(r) la ham gidi han (Limiter) thda man bieu thu:c
<^{r) = 0 neu r<0
0(r) = min(2i-,l) neu 0 < r < l

(2.5)

0(r) = min(r, 2) neu r > 1

Trong dd tham sd r dupe xac dinh theo bieu thure

Ddi vdi trudng hpp cd thanh ph^n khuech tan tham gia vao phuong trinh xuat
phat thi sau khi giai phuong trinh (2.3) theo so dd TVD rdi se them bieu thu'e sai phan
trung tam bac hai thdng thudng cho thanh phan khuech tan. Cdn ddi vdi bai toan hai
chieu thi ta sir dung phuong phap tach, tire la ta di giai luan chuyen cho hai hudng x va
hudng y dudi dang bai toan mot chieu. Ket qua eiia phep giai cho chilu thii: nhat se la
ban dau cho phep giai d ehilu thur hai. Phuong phap trinh bay d tren tuy don gian
nhung lai ra't hiru hieu. Dudi day se trinh bay mot sd kiem chung cho bai toan truyen
tai va khuech tan vat cha't cd nghiem giai tich.
2.3 KI^M C H C N G M 6 HINH SO THEO CAC BAI TOAN L^ THUYET
Tren thuc te qua trinh truyen tai va khuech tan vat chat lien quan den hai di6u
kien ban dau chinh: mot la phan bd vat chat cd gradient Idn cua nong dp theo khdng
gian tren mot pham vi hep (d gin ngudn thai - tuong ung vdi bai toan shock); hai la
phan bd vat chat cd gradient khdng Idn va phan bd trong pham vi rpng (d xa ngudn thai
- tuong ung vdi bai toan smooth). Do dd mot so d6 toan tot phai md phdng dupe tot ca
hai trudng hpp tren, cac nghiem sd khdng vi pham ban chat vat ly ciia hien tugng dugc
md ta (chang han vat cha't bi am, hay nhilu sd gia).
Dudi day se trinh bay cae kiem nghiem mo hinh vdi cae bai toan shock va smooth
neu tren cho cac trudng hgp mot chilu va hai ehilu.
2.3.1 Bai toan mot chieu
a) Bai toan Smooth
Dieu kien ban dau ciia bai toan la eo mot ngudn vat chat phan bd dudi dang tron
tren nin ddng chay cd van tdc U =0.75m/s. Gia sir he sd khuech tan bang khdng. Ta di
m6 phong sir di chuyen cua ngudn vat chat nay. Hinh 2.1 trinh bay phan bd ban dau
ciia vat chat. Vdi dieu kien cho thi do khdng cd khuech tan nen dang phan bd n6ng dp
ban dIu phai giii* nguyen va chi tinh tien di theo trudng van tdc. Do dd ta d6 dang tim
dupe nghiem giai tich.

10
8
^
•

U=0.75m/s;
he s6 khuech tan D = 0.

6

^

5 4
u
2

0
20

-2

40

60

80

100

120

140

160

180

200

x(m)

Hinh 2.1: Phan b6' n6ng d6 6 thai die'm ban diu co dang ircm
(van toe U=0.15m/s\ he s6' khue'ch lan D=0.)

200

Hinh 2.2: phan bo n6ng dp theo thai gian tinh theo so d6 Upwind
(tnJcfng hop phan bd' ban dau tran)

—

Lax-Wendrof

o Giai tich

x(m)

Hinh 2.3: phan bo ndng 66 theo thcfi gian tinh theo so do Lax-Wendrof
(truong hop phan bo ban dau tron)

Ket qua md phdng so sanh vdi nghiem giai tich trinh bay tren hinh 2.2 khi sir
dung so dd Upwind thdng thudng, hinh 2.3 ket qua sir dung so d6 Lax-Wendroff va
hinh 2.4 la cua so dd Flux-Limiting TVD. Ta thay vdi so dd Upwind suy giam sd rat
19