Numbering of area

Report
Modelling Hydrodynamical
Processes 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

  i H   u j  0, j  1,2
t


– motion equation
(1)

 u j   j u i u j   f i   i g   j  ij  S i , i , j  1, 2

(2)
;
– concentration equation

t
H   C k
  j u j H    С k ij  j D effk H    j С k   C 0k i , j  1, 2 .
(3)
Here ui – velocity vector components; f i  f  ij u j – Coriolis force components, f – Coriolis parameter
- Edington epsilon function,  – density;–stress
tenzor deviator in the form
ij
 i j   eff u i , j  u j , i  , i , j  2 ,
(4)
eff    t ,  , t – molecular and turbulent viscosity factors, D  D  D –effective diffusion (dispersion)
eff
t
coefficient; g – gravity acceleration, C k – concentration of the k-th impurity in water, C - internal production
0
– frictional
force
Si
chemical reactions, - water elevation level,
S i   iB   iS
R
u u u 
H    i k k
R 
 iS  a a wi wk wk 
H   
 iB 
(5)
wi , Ra , a and R wind 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
on Su
(6)
u i U i
– on the free surface we have
ti   g ij   i j n j  tˆi
on S
(7)
– for equation (1) we prescribe water elevation level
  ˆ
on
S
(8)
– for concentration we set
k
C C
k
on
(9)
S1
or we specify impurity concentration
flux

 


q k  ui H   C k ni  Deff H     j C k n j
Where  ij is the Kronecker delta function,
Su  S  S , Su  S 

.
on
S2
n j is the unit normal to the boundary,
S1 S 2 S .
(10)
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
   0 1   C k  ,
(11)
– pressure distribution
l 1
 m m 1 k  

l
p  p g     g  h   h     x3   h l  ,
m 1
л 1
ь 1




l
l 1
1
m
(12)
Convective and diffusive components of the equations (2, 3) containing vertical derivatives will be in
the following form:
 3 u i u 3 
l
1
1
1 
l
l 
2
 l u i u 3 
 u i u 3  2  , l  1, n , i  1, 2 ,
h 

(13)
– for friction stress between layers
Siint   i  i 
 i 

2 h l 
where
2

1 

l 1





  l uil 1  uil   V l  ,



,
i
2
i
i
l
h
2 hl 
 l   l 1 uil  uil 1  V l  ,
V l 

u

l 1
i
 uil   ij
2
(14)
,
– for vertical velocity
u 3l    j h l u lj , j  1, 2
(15)
– for concentration equation
1
1
1 
l
l 
2  Cv  2 ,


Cv
3
3

hl 
1
1
l
l 

2




D
C
C 2 
  

3 D33C  l3 
,

h  x3 

x
3 



 3 Cu3  
(16)
(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
z  H     x, y, t 



 ux
 uy 
 uz  0
t
x
y
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
h 
h 
u n , c n, find u h  ,V h n c h  V h n , h  V h n such that wh  V h n and q h  :Q h n

  u h  h  h
Q w  t  u u 
n

 nel n
1   wh
   1   
   t
e 1 Q e
n

  wh h dP,
      



     
f dQ   e wh :  u h , h dQ   wh n  u h n  u h n d 

Qn
n

  u h  h  h  

  


 u hwh    wh , q h  
 u u  f    u h , h  dQ 


  t

  




 Pn h

      


  c


 

k
wh  k  u hckh  c0k dQ   e wh : Deff
e ckh dQ   wh n ckh h  ckh h d 
 t

Qh
Qn
n

 nel n
 wh  h h
  c h  

k
    2 
 u w   Deff
gradWk   k  u hckh  c0k    Deff gradckh
e 1 Q e
 t
  t

n
h k
k 
 w Deff gradc ndP,
 



 Pn h
 


h
h
nel
  h

h 
h 
q 
 u x 
 u y 
 u z dS  
x
y
e 1
 t


h
S
nel

e 1

Se


Se
 dQ 

 q h  h q h  h 
dS 
 3 

y y 
 x x
h
h
h
h
h

h q
h q  
h 
h 
h



 4  u x 
 u y 
 u x 
 u y 
 u z  dS  0


x
y  t
x
y


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:
V   u u  H R  , u
Q   q q  H R  ,
h
h
h
m
1h
n
n
h
h
n
h
1h
n
h
 U h onPn u , w h  0onPn u

(21)
H lh Rn 
represents the finite – dimensional function space constructed over the space – time slab Qn
by using first – order polinomials in time and second in space. The interpolation function spaces are
discontinuous in time and continuous in space
,
u 
h 
n
 limt n   
,
 0
 dQ    ddt,
I n  th
Qn
 dp    ddt
I n Sth
Pn
The stabilization parameter 1 ,  2 ,  3 ,  4
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
, star with
the computations
c 
h
 c start
u 
 u 0h or  h
h 
k 0
h 
0
 

0
  0h
Fig 1a Finite element mesh the domain
Fig. 1b. Floor topografhy
qv
Fig. 1c. The floor sechri along A–A line
Fig. 2 Velocity field on the gulf surface. g 
–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 13 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.
Amur Gulf
Vladivostok City
Ussuri Gulf
Fig.4a. Velocity field on the Amur and Ussuri
gulfs nealy Vladivostok City
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

t

M i
xi
 0, i  1, 2
(24)
- the equation of motion
M i
  j u j M i    j  э j M i  fM i M k M k  g h     0.5  h   2 cos cos   0 i, j, k  1; 2
i


t
xi


(25)
– the equation for concentrating industrial waste
Ci
  j u j Ci    j Dm  j Ci , j  1, 2 , k  1, N
t
Here
M i  U i H   ,
(26)
i  1, 2
(27)
is outlet of water per unit of a river bed width, u i , H hare velocity and depth of water,  , is water rise relative
2
to the average level,  э    t is effective viscosity, g acceleration of gravity, t   Ch A turbulent viscosity,
A2  2eij e ji  – second invariant eij  0,5ui , j  u j ,i , C  0,15, h is the element length defined here as the maximum
of the edge lengths for the element, 1  sin  ,  2  sin  is level of a bottom slope in
respectively.
While solving equations (24–26) we use the following boundary conditions:
– on the boundary S1 with the water outlet Qi we have
M i  Qi , xi  S1
– on the free surface of the river outletweShave
2


M 
M i

j 
n j  ˆi ,
 i   ý 

xi 

  x j

x and
y directions,
(28)
(29)
where n j , is normal vector to a boundary. It is evident that on the boundary part
a set level of water rise can be put
  .ˆ
S2
an additional condition with
(30)
Boundary conditions are also necessary for equation (24). They assume
Ci  Cˆ i
or

(31)

q   Dm  j Ci  n j .
(32)
Fig.5. The domain
Fig.6. Topology Amur River
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

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