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An Iterative Method for the Solution of Dispersion Equation in Shallow Water

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An Iterative Method for the Solution of Dispersion Equation in Shallow Water Transactions on Ecology and the Environment vol 49, © 2001 WIT Press, www.witpress.com, ISSN 1743-3541 An iterative method for the solution of dispersion equation in shallow water H. Karahan Department of Civil Engineering, Panzukkale University, Denizli, Turkey Abstract In order to keep the pollution levels below permissible limits and also to forecast the final concentrations to be reached, current patterns and dispersion characteristics are required for water quality problems in lakes and coastal regions. In this case, it is necessary to solve the dispersion equations that include transport, diffusion and reaction processes to find the distribution and concentrations of pollutant. In this study, a numerical model was developed for simulating pollutant dispersion in well-mixed lakes and reservoirs. The model is based on solving dispersion equation using the Kewton-Raphson algorithm and results of hydrodynamic model. The developed model can be used for long-term distribution of conservative or non-conservative pollutants in lakes and reservoirs. 1 Introduction Contaminants discharged into natural and artificial lakes and coastal regions are both carried by the currents in the environment and also diffuse in the environment by the effect of molecular and turbulent diffusion and become sparse. Simultaneously with these processes the non-conservative materials change physically, chemically and biochemically. In natural water environments, for the conservation and improvement of water quality, two different methods like taking long-term measurements or using mathematical models calibrated with a limited number of observations may be used. Because of the dynamic inputs like wind, tide, temperature, atmospheric pressure changes which are effective in the formation of the currents Transactions on Ecology and the Environment vol 49, © 2001 WIT Press, www.witpress.com, ISSN 1743-3541 in these environments, expensive long-term measurement programs should be applied. For this reason, mathematical studies become widespread parallel to the improvements in the computer technologes in the recent years. To determine the contaminant material emission in natural aquatic environments, dispersion equation should also be solved as well as momentum and continuity equations whch determine the hydrodynamic structure of the environment. There are mathematical models developed by various investigators ([l]-141). In most of the present mathematical models to hearise the time vaIying non- linear partial Merential equation system, velocity components and water levels are computed by ,ftaggering in time and space. Dunng this computation; a quantity like [f(t)] is linearised by being written as f(t-At)*f(t). This assumpon small. gives satisfactory results when the At calculation steps are chosen But choosing the time between two calculation steps small increase the computation time [5]. In the presented study the dispersion equation expressing the pollution dspersion in the natural water environments is solved by using a consecutive solution algorithm. Because of the fact that all the quantities are calculated at the same time, the time step At between two solutions can be chosen large and solution for hgh Courant numbers becomes possible. So, computation time decreases, seasonal and yearly changes can be easily computed and the sensitivity of the results increase. 2 Basic equations In the coastal zones where there is no strabfication, well mixed and vertical water motions can be neglected by horizontal water motions, the continuity, momentum and dispersion equations can be made two dimensional by using depth averaged values ( [l], [6] ). The Qspersion equation: may be written ( [l], [71) Transactions on Ecology and the Environment vol 49, © 2001 WIT Press, www.witpress.com, ISSN 1743-3541 In th~sequation: U, V : depth-averaged velocity components in X and y directions, respectively : water surface elevation above or below mean depth 17 d : mean depth H : total depth of water column (d + q) P : pollutant concentration Dx, : Wsion coefficient in X and y duections, respectively 41 K : reaction matrix S : source or sink term t : time 3 Numerical solution method When equation (1) is rearranged as a function in xdmction for "nn'liquidpoints which are on j-axis, by writing dispersion equation at (i, j) point$ the following non-linear differential equmon system with "n" unknowns of whch the unknowns are contaminant concentrations "P" may be obtained: Equation (2) can be written in vectoral form as: Equation (3) is stated as Taylor series for and by using the first two terms in vectoral form: Transactions on Ecology and the Environment vol 49, © 2001 WIT Press, www.witpress.com, ISSN 1743-3541 448 Water-Pollution 17 can be written: This equation may be rearranged as: to enable consecutive calculation. Here: .. dx, dx, ax, J is the Jacobean matrix of function of with n variables = 0 If Xk+lis the solution of the given equation system F(Xk+J is written and this value is applied to Equation (6): can be written [S]. Equation (7) is a linear algebraic equation system of which the unknown is Ax vector. The coefficients matrix of thrs equation system is tridiagonal and by separating the coefficients matrix to multipliers in the form A = LU the unknowns vector dX is obtained and by writingXk+,= & + AY the calculation is Transactions on Ecology and the Environment vol 49, © 2001 WIT Press, www.witpress.com, ISSN 1743-3541 repeated untd sufficient convergence is reached. Thls algorithm is called Newton-Raphson method. When the algorithm described above is repeated for all 7"axes along ''X" direction the P contaminant concentrations for all points are obtained. Similarly, when equation (1) is rearranged as a function in y-direction for "PI" liquid points whch are on "inaxis; by writing dispersion equation at (i, j) points, the non-linear partial differential equation system with "n " unknowns of which "P" the unknowns are is obtained. By solving hsequation as iubcated above, the contamiDant concentrations "P" for the liquid points on '7" axis are obtained. When this operation is repeated for the '7" axises along "yy"dmction, ''PP" contaminant concentrations at all points are obtained So by using simultaneous values the dispersion equation can be solved and contaminant concentrations for the whole environment can be obtained +-+-+-+-+-+ IIIIII j+l+-+-+-+- +-+ - : U velocity IIIIII I : V velocity j +-+-+-+-+-+ + : Water level (5) and III/Il pollutant-concentration j-1 +-+-+-+-+-+ IIIIII +-+- +-+-+-+ i-l i i+ l T Figure 1: Grid system used in the model. 4 Application When a constant continuous discharge is given from a point in the middle of southern coast on the example whose geometry and bathymetry is given on Figure 2, the distribution and concentration of the contaminant in the environment is investigated for various wind directions and velocities. The hydrodynamic values needed for the solution of dispersion equation (velocities and elevations) are obtained by the model presented in references [9] and [10]. The developed dxqxrsion model can use hydrodynamic model results or observation results. In Figure 3, the current pattern produced by a north wind blowing at 5 m/s speed is shown. The contribution of the currents on pollutant dispersion and msion is explained in reference [5]. In Figures 4,5 and 6 the variation of the pollution against time is shown. The model can be used for any kind of geometry Transactions on Ecology and the Environment vol 49, © 2001 WIT Press, www.witpress.com, ISSN 1743-3541 and boundary condition without constraint but in th~spaper a simple geometry is chosen to see and evaluate the results easily. Figure 2: Water depths and geometry for the sample solution Simulated depth-averaged velocities (&-l800 ;Cr=17.828 ) Figure 3: Velocity field for At = 1800 seconds and Courant number = 17.828 Transactions on Ecology and the Environment vol 49, © 2001 WIT Press, www.witpress.com, ISSN 1743-3541 X (gnd mmh) Figure 4:Pollution's dispersion 10 days after the beginning of discharge. (Contours are drawn in 0.1 m@ steps). X (gid numbers) Figure 5: Pollution's dispersion 20 days after the beginning of discharge. (Contours are drawn in 0.1 mg/l steps). Transactions on Ecology and the Environment vol 49, © 2001 WIT Press, www.witpress.com, ISSN 1743-3541 X (grld numbers) Figure 6: Pollution's dispersion 30 days after the beginrung of discharge. (Contours are drawn in 0.1 mgll steps). 5 Conclusions It has been shown in reference [9] that when At calculation steps are chosen sufficiently small, the approaclj that depends on the basis of linearisation of a non-linear quantity like r(Q]in the momentum, continuity and dispersion equations by writing in the form f(t-dt) * f#, gives results in sufficient correctness for Courant numbers less than 5. Though hsapproach gives results for obtaining the short term current patterns in a very short time accordmg to the improvements in the computer technology, long computation times are needed to obtain long term current patterns and solve water pollution problems. To distinguish thls problem, in the study presented in reference [l01 for the purpose of obtaining current velocities and water elevations in natural water enviro~unents.the unknowns in momentum and continuity equations are solved simultaneously by solving a consecutive solution algorithm. It has been shown that there is no si@icant difference between the results obtained for Courant numbers hgh as 30-35 and the results obtained for the Courant numbers less than 5. In this study, the consecutive solution algorithm used in the solution of the momentum and continuity equations is applied to the dispersion equation. Results that are very compatible with the study given in reference [l l] in which Transactions on Ecology and the Environment vol 49, © 2001 WIT Press, www.witpress.com, ISSN 1743-3541 the dtspersion equation was solve& have been obtained by writing a nonlinear quantity in the form of f(t-At) * f(r).
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