3-D Numerical Modelling of Transport Processes in Bay of Fethiye, Turkey
INTRODUCTION
In the literature, it is possible to find a great number of one, two and three dimensional numerical models which have been used for the estimation of circulation patterns and the water quality in coastal water bodies. In a one dimensional model, flows are assumed be uniform in the cross section. In a two dimensional horizontal model, the depth integrated transport equations are used to predict the variations of variables in the horizontal plane. Two dimensional depth averaged models are successfully used to simulate the tidal circulation. Tidal flushing of small and well-mixed embayments has been well described by numerical models ( ,1992;
and , 1997). The distribution of contaminants in enclosed water bodies can be determined only if the velocity field in time is known in detail. Deep enclosed water bodies can exhibit complex vertical and horizontal circulation structures depending on tides, freshwater inflows, density stratification and geometry ( ., 1998) Wind induced currents have a strong three dimensional character and two dimensional models are poorly adapted to the simulation of circulation induced by the wind ( and , 2000). The crux of the difference between the two and three dimensional models is mainly the parameterization of the bottom stress. In many applications, it is important to predict the vertical structure of the flow due to salinity or temperature variations, which require a three dimensional model ( and , 2001). The volume of water in an enclosed water body may show seasonal variations, causing some water areas in the system to dry out or visa versa. Therefore, simulation of wetting and drying processess is a quite crucial in modelling ( , 2001). The developed three dimensional baroclinic model ( -3) which has hydrodynamic, transport and turbulence model components, can simulate the transport processes due to tidal or nontidal forcing. The Boussinesq approximation, i.e. the density differences are neglected unless the differences are multiplied by the gravity, is the only simplifying assumption in the model. The transport model component simulates the spatial and temporal distributions of water temperature and salinity, and pollutant diffu
( , 1997) also in the prediction of three dimensional thermal discharge flows.
Developed model has been implemented to Fethiye Bay located at the Mediterranean Sea coast of Turkey. In the Mediterranean Sea tidal ranges are small, typically in the order of 0.2 to 0.3 meters. Therefore, the dominant forcing for the water exchange is due to the wind action.
LAWRENCE SMITH
SCOTT
EDINGER
BALAS ÖZHAN
BALAS ÖZHAN
JI
HIDROTAM
STANSBY
et al.
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sion. As the turbulence model, a two- equation k-å turbulence model is used to determine the kinetic
energy of the turbulence and its dissipation rate. The k-å turbulence models are successfully applied to predict the wind and density induced circulations
THREE DIMENSIONAL NUMERICAL MODEL
An unsteady three-dimensional baroclinic circulation model (HIDROTAM-3) has been developed to simulate the transport processes in coastal water bodies. The model predictions were verified by using several experimental and analytical results published in the literature and its successful use for a variety of real life cases was demonstrated ( , 2001; and , 2002). Governing model equations in the three dimensional Cartesian coordinate system are given in Table 1.
In Table1, Equations (1-4) are hydrodynamic equations where, : Horizontal coordinates; : Vertical coordinate;
:Time; : Velocity components in x,y,z directions at any grid locations in space; , , : Eddy viscosity coefficients in x,y and z directions, respectively;
: Reference density; : Gravitational acceleration and : Pressure.
The numerical model includes thermohaline forcing.
Temperature and salinity variations are calculated by solving the three dimensional convection-diffusion equation which is written as given by Equation 5 where, D ,D and D : Turbulent diffusion coefficient in x,y and z directions respectively and Q:
Temperature (T) or salinity (S).
The conservation equation for a pollutant constituent is given by Equation 6 where : Pollutant concentration; : First order decay rate of the pollutant and S : Source term.
The density of sea water is a function of its salt content, its temperature and to a much lesser degree, its pressure. The rate of disappearance of pathogenic bac
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x,y z
t u,v,w
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g p
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x y z
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x y z
s
ñ : Coriolis coefficient;
ñ(x,y,z,t): In situ water density;
p
teria and viruses due to die- Journal of Coastal Research SI 39 1529 - 1532 ICS 2004 (Proceedings) Brazil ISSN 0749-0208
L. Balas † and A. Kücükosmanoglu ††
†Civil Engineering Dept.
Gazi University Ankara, 06570, Turkey lbalas@gazi.edu.tr
BALAS, L. and
- , .
An implicit baroclinic three dimensional numerical model (HIDROTAM-3) has been developed to simulate the transport processes in coastal waters. Model has hydrodynamic, transport and turbulence model components, and it can simulate transport processes due to the tidal or nontidal forcing which may be barotropic and baroclinic. The two equation k- modeling has been tested in the turbulence modeling. It is a composite finite difference - finite element model. In the horizontal plane, second order central finite differences on a staggered scheme and in the vertical Galerkin Method with linear shape functions are used. Implicit equations are solved by Crank Nicholson Method, which has second order accuracy in time also. Developed model whose predictions are compared with the analytical solutions, has been applied to Bay of Fethiye located on the Mediterranean Coast of Turkey, where the dominant forcing for the water exchange is due to the wind action. To control the predictions of the numerical model, a field study has been carried out in the Bay and continuous measurements of velocity have been taken throughout the water depth at different locations. Model predictions are in good agreement with the field data.
KÜÇÜKOSMANOÐLU, A., 2006. 3-D Numerical modelling of transport processes in Bay of Fethiye, Turkey. Journal of Coastal Research, SI 39 (Proceedings of the 8th International Coastal Symposium),1529 1532. Itajaí, SC Brazil, ISSN 0749-0208
ADDITIONAL INDEX WORDS:Baroclinic, turbulence, hydrodynamic.
ABSTRACT
†† Civil Engineering Dept.
Gazi University Ankara, 06570, Turkey akucukosmanoglu@gazi.edu.tr
Journal of Coastal Research Special Issue 39, 2006,
off approximately follows first order kinetics. The die off constant , is computed in terms of T , the time required for 90 percent of the initial bacteria to die, and equals 2.3/T ( , 2001). The two e
, 2000; and
, 2001). To consider the large scale turbulence caused by the horizontal shear, horizontal eddy viscosity is simulated by the Smagorinsky algebraic subgrid scale turbulence model
( and , 2000). :
(7)
Horizontal eddy diffusivities are approximately equal to the horizontal eddy viscosities. On the other hand, the vertical diffusivity, D , is expressed as:
(8)
where, P is the turbulent Prandtl or Schmidth number and v is the vertical eddy viscosity coefficient.
Solution scheme is a composite finite element-finite difference scheme ( and , 2000). The governing equations are solved by Galerkin Weighted Residual Method in the vertical plane and by finite difference approximations in the horizontal plane, without any coordinate transformation. The water depths are divided into the same number of layers following the bottom topography. So, the vertical layer thickness is proportional to the local water depth. A detailed presentation of the finite difference approximations and Galerkin Method of finite elements to the governing hydrodynamic equations are given in and (2000).
HIDROTAM-3 has been implemented to Fethiye Bay
located at the Mediterranean Sea coast of Turkey. The town of Fethiye located inland of the lagoon is one of the most developed coastal resorts along the Turkish coastline and Bay of Fethiye which is located along the sailing route, is busy almost all over the year. M2 tide is the dominant tidal constituent for the area. The typical tidal range in the area is in the order of 0.2 to 0.3 meters. Therefore, the dominant forcing for the water exchange is due to the wind action. Water depths in the Bay are plotted in Figure 1. At two stations shown in Figure 1, continuous velocity measurements were taken for 27 days. The wind speeds and directions during the measurement period are shown in Figure 3. In the model simulations, Fethiye Bay is subjected to recorded wind action. The wind analysis shows that the critical wind direction for the area is WNW-WSW direction.
The grid system used in the model has a square mesh size of 100x100 m. No significant density stratification was recorded at the site. Therefore water density is taken as a constant.
friction coefficient is performed and C =0.0026 provided the best match with the measurements.
To control the predictions of the numerical model, filed data is used. Current measurements are taken throughout the water depth at Station I located at about 5 m. and at Station II located at about 10 m. The horizontal flow patterns at the surface and near the bottom layer after 10 days of simulation are shown in Figure 3 and in Figure 4 respectively and the measured and computed velocity profiles over the water depth are compared in Figure 5 for both of the Stations after 10 days of simulation. For Station I, after 10 days of simulation, the root mean square error is 0.21 cm/s and bias is 0.023 cm/s, and for Station II, the root mean square error is 0.27 cm/s and bias is 0.031 cm/s.
Water near the free surface is driven by the dominant wind shear stress and is transported in the direction of the
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f
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quation k-å turbulence model is used for the turbulence modelling (BALASand
MODEL APPLICATION TO FETHIYE BAY
Horizontal eddy viscosities are calculated by the sub-grid scale turbulence model and the vertical eddy viscosity is calculated by the k- turbulence model. The sea bottom is treated as a rigid boundary. A sensitivity study of model predictions to bottom
Figure 1. Water depths in Fethiye Bay (m) where: +: Station I,
•:Station II.
Figure 2. Wind speeds and directions during the measurement period.
Table 1.Governing model equations.
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¶
÷÷ø ¶ ççè ö æ
¶
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¶ D ¶
D y
+ v y u 2 +1 y + v x y u x 0.01
=
2
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nh
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r z
z
n
Journal of Coastal Research Special Issue 39, 200, 6 Balas and Kucukosmanoglu
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Figure 4. Current pattern near the bottom at the end of 10 days of simulation.
Figure 3. Current pattern near the surface at the end of 10 days of simulation.
Journal of Coastal Research Special Issue 39, 2006,
3-D Numerical Modelling of Transport Processes 1531
wind. A counter flow in the reverse direction exists near the bottom layer. As a result, the model has been successfully applied to Bay of Fethiye (Turkey) where the flow pattern in the Bay is mainly driven by the wind force. The model was also capable of predicting complex circulation patterns.
HIDROTAM-3, the three dimensional baroclinic numerical model which has hydrodynamic, transport and turbulence model components, is presented. Model has been successfully applied to Bay of Fethiye where the flow pattern in the Bay is mainly driven by the wind force. Implementation of the numerical model to Bay of Fethiye has shown an encouraging agreement with the measurements, and the long term simulation of 27 days provided realistic results. Model is capable of predicting complex circulation patterns. It can be used as a practical tool in diverse coastal applications including the induced circulation patterns.
L. and E., 2000. An implicit three dimensional numerical model to simulate transport processes in coastal water bodies.
34, 307-339
L. and E., 2001. Applications of a 3-D numerical model to circulations in coastal waters.
43(2), 99-120.
L., 2001. Simulation of pollutant transport in Marmaris
Bay, 15(4), 565-578.
L. and ÖZHAN, E., 2002. Three dimensional modelling of stratified coastal waters,
, 56, 75-87.
J.E.; E.M. and V.S., 1998.
Modeling flushing and mixing in a deep estuary, , 102, 345-353.
P.S.; W.C. and S.R., 1992.
Model for estimating tidal flushing of small embayments, 118(6), 635-654.
Z.G.; M.R. and J.M., 2001. Wetting
and drying simulation of estuarine processes, , 53, 683-700.
R. and C.F., 1997. Mixing in the tidal
environment, , 123, 332-340.
P.K., 1997. Semi implicit finite volume shallow water flow and solute transport solver with k- turbulence model,
, 25, 285-313.
CONCLUSIONS
LITERATURE CITED
BALAS, ÖZHAN,
BALAS, ÖZHAN,
BALAS, BALAS,
EDINGER, BUCHAK, KOLLURU,
LAWRENCE, BOICOURT, RIVES,
JI, MORTON, HAMRICK,
SMITH, SCOTT, STANSBY,
Int.Journal of Numerical Methods in Fluids,
Coastal Engineering Journal,
Chineese Ocean Engineering Journal,
Estuarine Coastal and Shelf Science
Water, Air and Soil Pollution
Journal of Waterway, Port, Coastal, and Ocean Engineering,
Estuarine, Coastal and Shelf Science
Journal of Hydraulic Research
International Journal for Numerical Methods in Fluids
Figure 5. Velocity profiles over the depth at the end of 10 days of simulation, a) at Station I b) at Station II where solid line:simulation, dot:measurements.
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