Tabakalı körfezlerdeki su kütlesi hareketlerinin belirlenmesi ve İzmit Körfezi`ne uygulanması

Abstract

Bu çalışmada, körf eslerdeki tabakalı akımlar için üç boyutlu bir matematik model geliştirilmiştir. Yeraltı suyunun körfeze girişi de gözönüne alınmıştır. Altı bölüm halinde sunulan bu çalışmanın birinci bölümünde çalışmanın amacı ve kapsamı açıklanmıştır. îkinci bölümde, su kütlesi hareketleri için gelişti rilmiş matematik modellerin temeli olan Navieı - Stokes, kütlenin korunumu, tuz dengesi ve durum denklemi ile ilgili bilgiler verilmiştir. Denklemlerin elde edilme sinde kullanılan Boussinesq ve hidrostatik basınç dağılışı yaklaşımları ile bazı kabul, sadeleştirme çalışmaları ele alınmıştır. Hareket, süreklilik ve tuz dengesi denklemlerinin İzmit Körfezine ait datalarla mertebe analizi yapılmıştır. Üçüncü bölüm iki alt bölüm halinde sunulmuştur. Birinci alt bölümde kıyılarda tatlı su - tuzlu su girişimi ile ilgili çalışmalar incelenmiştir. ikinci alt bölümde ise tabakalı akımların matematiksel modellenmesi ile ilgili dünyada ve ülkemizde yapılmış çalışmalar özetlenmiş, kritiği yapılmıştır. Dördüncü bölümde, tatlı su-tuzlu su girişimini gözlemek amacıyla gerçekleştirilen hidrolik model deneyleri, deney fotoğrafları ve elde edilen sonuçlar yer almakta dır. Daha sonra geliştirilen matematik modele ait çözüm algoritması ve temel denklemlerin sonlu farklarla yazılmış ifadeleri verilmiştir. Ayrıca körfezlerdeki tabakalı akımların üç boyutlu matematik modeli için hazırlanan FORTRAN-77 dilinde kodlanmış KÖRFEZ-3D bilgisayar programı tanıtılmıştır. Geliştirilen matematik modelin İzmit Körfezine uygulanması düşünülmüş olduğundan beşinci bölümde İzmit Körfezinin hidrografik, oseanografik, topoğrafik, Jeolojik özellikleri, iklimi, yeraltı ve yüzeysel su kaynakları, çevresindeki endüstrileri ile ilgili bilgiler bulunmakta dır. Ayrıca, sınır şartlarının belirlenmesi amacıyla İzmit Körfezi ve çevresindeki ovalara ait yeraltı suyu ölçümleri ile Körfezdeki sıcaklık, tuzluluk ve hız ölçümleri değerlendirilmiştir. İzmit Körfezindeki tabakalı akımın nümerik çözümü, geliştirilen KÖRFEZ-3D bilgisayar programı yardımı ile yapılmıştır. Altıncı bölümde, bu çalışmada elde edilen sonuçlar açıklanmıştır. İzmit Körfezine ait İS yatay ve 29 düşey düzlemde akım hızları ve tuzluluk eğrileri sunulmuş, su kütlesi hareketlerinin yorumu yapılıp ölçümlerle uyumu gösterilmiştir.

In recent years, bays, estuaries and open coasts have been polluted by direct and indirect waste water disharge as a result of growth of population and industrial development. Hydrodynami c properties of these type of water resources must be very well known in order to conserve and use them optimally. This is possible with hydraulic and mathematical modelling. In hydraulic modelling some effective parameters which are important in phenomena can not be realized in laboratory conditions resulting generally with unrealistic results. In addition hydraulic model works need to spend longer time and higher cost because of the limited physical and capital capacity of laboratories. Field measurements of flow characteristics should be carried out extensively in a long time period in order to obtain a safe design. But extensive measuring programmes are quite difficult with regard to time and cost. Moreover it is impossible to measure various alternatives if it is necessary to understand the phenomena. For this reason mathematical models are more suitable. In this thesis, which consists of six chapters, three dimensional mathematical model has been developed for stratified flow in bays which take into account of the entrance of groundwater into the bay. In the first chapter, the objectives and contents are given. In the second chapter, necessary informations about the basic equations developed for water body motion, namely Navies-Stokes, mass conservation, salt balance state equations are given. Boussinesg and hydrostatic pressure distribution approximations are used for derivation of equations. These approximations: i) The Boussinesg approximation: The density variation in a water body is much smaller than the density itself; therefore, constant density can be used buoyancy force. xxii) The hydrostatic pressure distribution approximation: The vertical acceleration of water particles is very small relative to the gravity acceleration in bags and currents systems. For this reason effects of vertical accelerations can be neglected. Models which are used to solve equations can be grouped as follows: 1) Vertically averaged two-dimensional models: These are used when vertical variation of flow parameters is very small and when only lateral distributions are needed. These type of models consist of set of equations which are obtained by taking vertically averages of variables. In vertically averaged models vertical turbulence transport is neglected and it is assumed that horizontal velocity distribution is uniform over the depth. These models can be applied to twoor more stratified flows but density is constant in each strata. 2) Laterally averaged two-dimensional models: Vertical structure of flow parameters is used when mixing water bodies which have different density are taken into account. These types of models can be obtained by taking average of all variables over the wi dth. 3) Three dimensional models: First type of models can not give information about vertical variation of flow parameters because of the vertically taken average parameters. In the second type of models, it is not possible to see the variation over the width. For these reason three dimensional models are most developed types. In this investigation, rigid lid approximation has been also used together with Boussinesg and hydrostatic pressure approximations. In rigid lid approximation it is assumed that water particles do not move in vertical direction in free sea surface. With this assumption all free surface waves are eliminated. In all layers shear stresses caused by viscosity are neglected with comparision to turbulent shear stresses except in the viscous sublayer. Moleculer diffusion effects are also neglected with comparison to turbulence diffusion. In addition all motions are assumed to be steady: With above assumptions and some definition and eliminations the motion, continuity and salt balance equations can be written as follows: xxiCD (2) (3) (4) By applications of these equations to îzmit Bay data, it is seen that convective and eddy viscosity terms in the equations can be neglected because they are smaller than other terms and the equations take the following final forms: 0 = -gft -£- J* Sdz+fv (5) 0 = -gft ° J Sdz-fu (6) o The above cited assumptions have no impact on the salt balance equation, i.e. it remains unchanged, so that all it terms should be conserved. Under these considerations, equations to be used in mathematical modelling take the following forms: 0 = -gft dx J Sdz+fv (7) 0 = -gft d J` Sdz-fu (8) o XX11_§H_+_£-+-£_ = o (9) ax oy irz u-^-+v-^-+w^-= K -£Ş-+K -2İŞ-+K -2-î- (10) OX ây &Z x ^2 y ^2 z ^2 The third chapter has been presented in two subsections. In the first subsection salt water-fresh water interaction in coastlines has been investigated. Some of the previous works on this topics are given by Ghyben-Herzberg (1889), Hubbert (1940), Glover (19B9), Cooper (1959), Mc Whorter and Sunada (1977). In the second subsection some previous work which has been carried out in Turkey and in the world about mathematical modelling of stratified flows has been summarized and critisized. Important works are due to Lee and Liggett (1970), Liggett and Lee (1977), Grubert and Abbott (1972), Hyden (1974), Abbott vd., (1975), Leendertse, Alexandre, Liu (1973), Leendertse, Liu (1975), Blumberg (1977), Vasiliev and Kvon (1977), Oey, Mellor, Hires (1985), Bloss, Lehfeldt, Patterson (1988). The works on the Izmit Bay of the Turkish investigators are considered quite extensively. These works are due to SWECO-BMB (1976), Akkaya vd. (1983), Albek and Yenigün (1986), Sur (1988). In the fourth chapter works carried out for designing the mathematical model have been given. In order to show salt water-fresh water interaction, test photos and test conclusions have been given. Viscous fluid analogy method (Hele-Shaw) has been selected. Hele-Shaw instrument consists of two glass or plexiglass plates which are very close to each other and are vertical and it is possible to neglect their thickness with comprasion to their lengths. A set of experiments have been carried out in Hele-Shaw instruments to show salt water-fresh water interaction. Sieved and cleaned sand has been placed in Hele-Shaw instrument. Salt water has been utilized to represent sea water whose salinity has been measured by salinometer. To obtain a steady motion salt and fresh water tanks have been fed continuously. Both tanks have a weir in order to keep constant the water levels. In order to follow the movement of the fresh water, both in sandy bottom and in the salty water, it was coloured with potassium permanganate. Fresh water advancement, meeting with salt water and the formation of the interface between fresh and salt waters are photographi ed. The experiences were stopped after that a steady state was obtained. In the second subsection of fourth, chapter the solving algoritm has been explained. Numerical solutions have been used because it is impossible to obtain exact solution of the given differential xxiiiequations. In this work, the finite difference method has been used as numerical method. x, y, z axies are shown in figure 1; in the first order derivatives according to all the space variables and in the second order derivative according to x the backward form and; in the second order derivatives according to a and z the central form have been utilized. Figure 1 : The coordinate system used in math, model In cross sections implicit schemes have been used. Velocity components and salinities in the first cross section and at the free surface have been given as boundary conditions. The vertical velocity at the surface is zero: w = 0 for z = 0 (11 ) Because of the rigid lid approximation as a free surface boundary condition. The second free surface boundary condition defines that the pressure at all points of the surface is atmospheric. On the side surfaces of cross section and at the bottom no-slip condition has been used. For whole solid surface velocity components are zero except that the ground water exist. u = v = w = 0 (12) Ground water velocity has also been as a boundary condition. In the same chapter the computer program called KÖRFEZ-3D written in FORTRAN-77 language for the mathematical model solution have been also explained. Some information about the İzmit Bay hydrographi c, ocenographic, topographic and geological properties, its climate and surface and ground water resources and indurtries in its environment have been given because the developed mathematical model will be applied to the tzmit Bay. The Izmit Bay is stuated in the Northeast of Marmara Sea. Its length i s 50 km., width is 2-10 km. and its surface area is 310 km. From the oceanographi c point of view îzmit Bay has three different sections which are East, Middle and West (see Figure 2). Eastern xxivFigure 2 : izmit Bay section has 16 km. length and it is the smallest and shallowest section of îzmit Bay- The middle section' is the biggest one and has length of 20 km. and area of 166 km. In this section average depth in north coast is 60 m. which reaches 180 m. in the south coast. Between middle and western section there is a step with 3 km. width and 55 m. depth. After that sill the depth the Izmit Bay. West width to the Marmara Sea is 5.5 km. Marmara Sea is between the Black Sea which has low salinity and Aegean Sea which has high salinity, so that a density stratification can be Marmara Sea. îzmit Bay is also a part of the Marmara Sea and for this reason the same stratification is also found in this Bay. increases very rapidly in west section has 100 km surface area and the inlet In the same chapter, in order to predict boundary conditions an evaluation of the previous field measurements which are made in îzmit Bay and its environment have been given. These measurements are the ground water level data for surrounding area of İzmit Bay and temperature, salinity and velocity data for İzmit Bay. In Nato-Tu waters project some curves showing variation of temperature, salinity and dissolved oxygen for the period of 1984-85 May have been given. By using these curves and interpolating if it is necessary, salinity, temperature and corresponding sigma-t values have been found for each depth. As known, sigma-t which dominates the water body movements, depends on the temperature and salinity. Water body movements have been determined along the length and width of îzmit Bay (north, middle and south of Bay) by sigma-t. In this chapter, while making application of the mathematical model for îzmit Bay, the bay has been idealized in three dimensional form in plan and cross section as close as possible to actual boundaries. The cross-section which contains the measurement section 1 and 2 is taken as the entrance section of the Bay to the Marmara Sea and is adopted as the initial cross-section of the mathematical scheme. In this cross section, horizontal velocity components (u, v) and salinity at each point have been determined by interpolating and extrapolating in plan xxvand profile by using the salinity and velocity measurements which had been taken at the depths of 10 m. and BO m. in December, 1984- Vertical profiles of horizontal velocity components have been drawn according to the no-slip cpndition in which all velocity components are zero on solid boundary. While obtaining velocity profiles it has been considered that there is an interface at the depth of 22.5 m (roughly between 20 m. and 25 m. ). Horizontal velocity components and salinities at the free surface have been calculated through the field measurements by intepolating and exceptionally in necessary cases by extrapolating. While solving the basic equations by the method of finite differences. Grid size has been selected as Ax=1600 m., Ay=800m. and Az=10m. In the sixth chapter, results obtained have been presented as velocity profiles and salinity curves in 15 plans and in 29 cross sections. The i nterprectati on of water body movements has been made and it has been shown that the results fit quite well with the field measurements. XXVI