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 3l 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 D33C 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 hwh wh , q h u u f u h , h dQ t Pn h c k wh k u hckh 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 hckh 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 onPn u , w h 0onPn 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 limt n , 0 dQ ddt, I n th Qn dp ddt 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 0,5ui , 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