Modelling Hydrodynamical Processes of the Pacific Ocean Littoral and the Amur River (the Far Eastern...

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)

Modelling Hydrodynamical Processes of the Pacific Ocean Littoral and the Amur River (the Far Eastern...

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