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Modelling HydrodynamicalProcesses of the Pacific Ocean Littoral
and the Amur River (the Far Eastern Region of Russia)K. A. Chekhonin
Resheach Institute of Computer Sience, Khabarovsk,Russian Federation
Computer modelling hydrodynamical processes of the littoral oceanic zone is a very important problem, solution of which permits the country s sea-ports to be designed most optimally, sea structures to be erected
(water-protecting works, tidal power stations, navigable channels etc). Moreover numerical modelling of this class problems makes it possible to observe the influence of the already existing sea constructions and industrial centres located in the littoral zone upon hydrodynamical process character and water pollution by industrial wastes. In general case the coastal configuration line and that of the floor topography of the littoral zone are
extremely complicated, all processes being of3-dimentional nature. These circustations complicate the problem's numerical Solution (Volzinger N. E.- 1977, Chekhonin K. A.- 1989, Chekhonin K. A.- 1995,Kawahara M.- 1986,
Antonzev S. N.- 1986, Chekhonin K. A.- 1993).
Mathematical modelling hydrodynamical processes of the Pacific Ocean littoral
According to the shallow water wave theory, the basic equations are derived from the Navier-Stokes equations assuming hydrostatic pressure distribution averaged over turbulent time scale. The governing equation for the one layer current model is expressed as follows[1,2,5]: – equation of continuity
2,1,0
juHt ji
(1) – motion equation
;2,1,,j jiSgfuuu iijjiijij (2) – concentration equation
.2,1,0kkk
jiCСHDСHuCHt
kj
keffjjj (3)
Here iu – velocity vector components; jiji uff – Coriolis force components, f – Coriolis parameter
ij
aai Rw ,,
- Edington epsilon function, – density; ij– stress tenzor deviator in the form ,2,,,, jiuu ijjieffji (4)
tteff ,, – molecular and turbulent viscosity factors, teff DDD –effective diffusion (dispersion) coefficient; g – gravity acceleration, kC – concentration of the k-th impurity in water, C 0- internal production chemical reactions, - water elevation level, iS– frictional force
kkiaaS
i
kkiBi
Si
Bii
wwwH
R
uuuH
R
S
(5)
and Rwind velocity, wind friction coefficient, air density and bottom friction coefficient respectively.
Boundary Conditions.Numerical solving of the problem is performed under the following boundary conditions on for:– we specify velocity component
ii Uu on (6)
– on the free surface we have
ontngt ijjiijiˆ
S
(7)
– for equation (1) we prescribe water elevation level
ˆ on (8)
on (9)
kk CC
uS
– for concentration we set
on (10) 2S
.jk
jeffik
ik nCHDnCHuq
or we specify impurity concentration flux
S
Where is the Kronecker delta function, is the unit normal to the boundary,
SSSSS uu ,
.21 SSS
1S
ijjn
Layers Model.If impurity density, for instance, water salinity or industrial wastes, influences upon sea water density,
the domain in the direction of Oz axis decomposes into n layers of the same density. For this case we have:
– density distribution ,10
kC (11)
(12) ,
1
13
1
1
1
1
1
l
ь
llm
л
kml
m
ml hxhhggpp
Convective and diffusive components of the equations (2, 3) containing vertical derivatives will be in the following form:
,1
2
1
32
1
333
l
il
il
li uuuu
huu (13)
– for friction stress between layers
,1int iiliii h
S (14) ,2
112
lli
li
ll
li Vuuh
,
2
11
2
lli
li
ll
li Vuuh
where ,j
21i
li
li
l uuV
– pressure distribution
,21,,,1 inl
– for vertical velocity 2,1,3 juhu l
jl
jl
(15)
– for concentration equation
,1
2
1
32
1
333
ll
lCvCv
hCu
(16)
,2
1
3
2
1
3
3333
ll
l x
C
x
C
h
DCD (17)
Free Surface. Fluid flow simulation that involve deforming domains, in the presence of one or more moving
boundaries or fluid – fluid interfaces (layers), continue to present unique challenges, and form one of the frontiers of computational science. Free – surface flows in particular involve the motion of the fluid interface which is unknown at the outset of the simulation. Thus, both the domain and the flow field are parts of the solution. The requirement of accurate and robust tracking of the domain deformation, together with the need for proper representation of artificial flow boundaries and interface effects, are some of the difficult issues encountered in free – surface flow simulations.
Two dramatically different approaches, that of interface – tracking and interface – capturing, have emerged, and both have their proponents. An interface – tracking method always places computational nodes at the moving interface and adjust the computational mesh to the movement of those nodes. An interface – capturing method lets the computational mesh be stationary, and simply records which computational cells, or elements, are filed with fluid empty, or contain the interface. Each method presents advantages and disadvantages, and neither can be discounted. We present a set of method, based on the interface – tracking approach, which is being used in simulations of free – surface flows in modeling hydrodynamic processes of the Pacific Ocean Littoral and the Amur River. The cinematic conditions requires that the free surface is a material surface; i.e. the fluid particles with are at some time on the free surface always stay on it. This condition is used to describe the motion of the free surface. For a 3D problem, on the free – surface
tyxHz ,, 0
zyx uy
ux
ut
where ux,uy,uz is the fluid velocity components are evaluated at the free surface.The computations are based on the finite element formulation, which takes automatically into account
the motion of the free surface. The free surface height is governed by a cinematic free – surface condition, which is solved with a stabilized formulation. The meshes consist of quadrangular isoparametric second – order element in 2D (2,5D) and prism elements in 3D. The mesh update is achieved with general or special – purpose mesh moving schemes.
Variational Formulation Problem.The stabilized Space – time formulation of Equations (2) and (3) can then be written as follows:
given
nh
nh cu , , find nhh Vu , nhh
nhh VVc , such that nhh Vw and nhh Qq :
0
,
:
,
,,1
,:
41
31
021
0
11
dSuy
ux
uty
qu
x
qu
dSyy
q
xx
qdSu
yu
xu
tq
dPngradcDw
dQgradcDccut
cgradWDwu
t
w
dccwdQceDwedQccut
cw
dPw
dQufuut
uqwwu
t
w
duuwdQuwedQfuut
uw
hz
hh
y
hh
x
hhh
y
hh
x
S
nel
e
hhhh
S
nel
ez
hh
y
hh
x
hh
S
kkeff
h
P
hkeff
khk
hhkkk
effhh
h
Q
n
e
hhkh
hkn
hhk
keff
h
Q
khk
hkh
Q
hh
P
hhhhh
hhhhh
Q
n
e
nh
nh
nhhhh
Q
hhh
Q
e
e
hn
en
nel
nnh
hn
en
nel
nnn
Here for each space – time slab Qn, we define the following finite –
dimensional trial solution (Vh)n, (Qh)n spaces for the velocity concentration and the water elevation:
unh
unhhm
nhhhh PonwPonUuRHuuV
n0,,1 (21)
nhhh
nh RHqqQ 1 ,
nlh RH
by using first – order polinomials in time and second in space. The interpolation function spaces are discontinuous in time and continuous in space
nn
h tu0
lim
,
dtddQn
htn IQ
,
dtddpn
htn I SP
,
The stabilization parameter
hstart
hk cc
0
,
hhhh oruu 0000
4321 ,,, for equations (18)-(20) can be defined in (Chekhonin K. A.- 1995).
The solution to Equation (18)-(20) is obtained for all of the space - time slabs Q0, Q1, …, Qn-1 sequentially, and the computations star with
represents the finite – dimensional function space constructed over the space – time slab Qn
Fig 1a Finite element mesh the domain Fig. 1b. Floor topografhy
vq
Fig. 2 Velocity field on the gulf surface. gFig. 1c. The floor sechri along A–A line–fresh water flow
Non-flowing water condition was specified along the shore boundary. The numerical calculations of the velocity field is shown in Fig. 2, 3.
The presence of fresh-water outlet was the peculiarity of the hydrodynamical process in the gulf. It results in existing density gradients both in the gulf and in the water salinity level. This water salinity level in the 1-3 layers of the gulf is shown in Fig. 3.
Fig. 3a. Water salinity in percentage on the gulf surface
Fig.3b. Water salinity at the delth of 100 m Fig. 3c. Water salinity at the of 220 m
The peculairity of the hydrodynaical proceses in the Amur and Ussuri gulfs nealy Vladivostok City is shown in Fig. 4.
Fig.4a. Velocity field on the Amur and Ussuri gulfs nealy Vladivostok City
Amur Gulf
Vladivostok City
Ussuri Gulf
Fig.4b. Hudrodynamical Processes on The Amur and Ussuri gulf surface
2. Modelling a hydrodynamic flow of the Amur river in Khabarovsk environs – the equation of continuity
2,1,0
ix
M
t i
i(24)
kkiijэjijj
i MMfMMMut
M 2;1,,0coscos5.0 2
kjihx
hgi
i (25)
– the equation for concentrating industrial waste
2,1,
jCDCu
t
Cijmjijj
i , Nk ,1 (26)
Here
2,1, iHUM ii (27)
is outlet of water per unit of a river bed width, iu , H
hare velocity and depth of water,
, is water rise relative
to the average level, tэ is effective viscosity,
g acceleration of gravity, ACht2 turbulent viscosity,
jiijeeA 22 – second invariant ,5,0 ,, ijjiij uue hC ,15,0 is the element length defined here as the maximum
of the edge lengths for the element, ,sin1 sin2 is level of a bottom slope in x and y directions,
While solving equations (24–26) we use the following boundary conditions:
– on the boundary 1S with the water outlet iQ we have
respectively.
1, SxQM iii (28)
2S we have– on the free surface of the river outlet
iji
j
j
iýi n
x
M
x
M ˆ
, (29)
- the equation of motion
where jn , is normal vector to a boundary. It is evident that on the boundary part 2S an additional condition with
a set level of water rise can be put ˆ. (30)
Boundary conditions are also necessary for equation (24). They assume
ii CC ˆ (31) or
jijm nCDq . (32)
Fig.6. Topology Amur River
Fig.5. The domain
Fig.7. Finite element mesh the domain Amur River Fig. 8.1 Velocity field
Fig. 8.2 Horizontal velocity Fig. 8.3 Vertical velocity
Fig.9. Velocity field Fig. 10. The industrial wastes distribution on the river surface at the time moment
Fig. 11. Finite element mesh mouth Amur River Fig.12. Velocity field on the surface Estuary
Fig.13a. Water salinity in percentage on the surface Estuary Amur River
Fig.13b. Water salinity at the delpth of som Mouth Amur River
Numbering of area
Finite element net of area
Area depth (bathymetric chart)
Velocity vector
Concentration of detrimental impurities (part 1.1)
Concentration of detrimental impurities (part 1.2)
Concentration of detrimental impurities (part 1.3)
Concentration of detrimental impurities (part 2.1)
Concentration of detrimental impurities (part 2.2)
Concentration of detrimental impurities (part 2.3)
Concentration of detrimental impurities (part 3.1)
Concentration of detrimental impurities (part 3.2)
Concentration of detrimental impurities (part 3.3)
Tests