Abstract. 1. Introduction

Transcription

1 Dispersion of pollutants in coastal waters near a river mouth: a simplified numerical model M. Di Natale Dipartimento dilngegneria Civile, Seconda Universita dinapoli, Aversa (CE), Italy Abstract A mathematical model is presented for studying the hydrodynamic dispersion of pollutants discharged in coastal areas near a river mouth during flooding. The hydrodynamic process is represented in a 2D depth integrated system and the solute is assumed non-reactive. The model is numerically solved by finite differences characterized by an appropriate algorithm which is explicit to describe the field of motion and implicit to integrate the transport equation. 1. Introduction Within the framework of pollution problems in coastal waters, areas near river mouths assume particular technical and scientific interest. Due to the possible presence of many pollution sources in the hydrographic basin (eg. domestic sewage, toxic substances from industrial discharges, surface runoff of pesticides and fertilisers from agricultural land etc.), water-courses become carriers of pollutants which, without appropriate treatment plants, are transported to the mouth and then out to sea. The study of such problems is of considerable importance so as to define the extent of the polluted stretch of water and to ascertain the environmental impact upon bordering coastal zones. In some cases, dispersion phenomena in the marine ambient may be enhanced by wave motion or coastal currents. Thus, the most serious conditions may occur when the sea is calm. Moreover, it should be noted that the transport of pollutants is strictly correlated to variation of river flows in time, particularly during flooding. As is well known, during the initial phase of a flood, pollutant concentrations increase due to the washing effect of the first rains on sealed areas in urban zones, on vegetation and agricultural land. The

2 156 Environmental Problems in Coastal Regions subsequent phases, during the rising limb and the recession limb of the hydrograph, are characterized by a reduction in the polluting load. In this paper, a mathematical model is presented to describe the above phenomenon under the hypothesis of a calm sea into which flows a flood wave transporting a polluting load with a time-variable concentration. Hydrodynamic analysis of the problem is characterized by two phases: in thefirstphase, we obtain the equations of the field of motion in the coastal water stretch deriving from the in-flow of the flood Q(t); subsequently, using the equation of advective-diffusive transport, we examine hydrodynamic dispersion of time-variable concentration C(t) in the above-mentioned field of motion. The proposed model is not constrained either by the morphology of the mouth and the sea-bottom, or by Q(t) and C(t) functions which may be completely arbitrary. The introduced approximations concern the hypothesis of shallow water conditions, in which the field of motion and the concentrations refer to a 2D depth integrated scheme. Moreover, buoyancy effects deriving from small density differences betweenfluvialand marine water were ignored. The pollutants were assumed non-reactive in the sense that they do not modify the field of motion, although their chemical and biological transformation in time is made possible by introducing an appropriate decay term in the transport equation. The mathematical model is solved numerically and an application is reported to show its validity in a real case. 2. Hydrodynamic formulation With reference to the scheme in fig. 1, in the most general case of a 3- dimensional field of motion, the equations which describe the hydrodynamic process are [1]: which expresses the continuity condition, and : du dv dw + +%- = o (i) dx dy dz du _gu _gu gu Idp d( du} d du d( du ^^= (2-a) dv dv _5v 0v 1 dp d ( dv} d dv d ( dv (2-b) dv/ dv/ dv/ dv/ 1 dp d ( dv/} d ( dv/\ d ( dvf m} -- i-u +v +w = - + c.. H--g. + e at dx dy dz g dz M * dxj dy{ * dy ) dz{ * fr) dz (2-c)

3 Environmental Problems in Coastal Regions 157 which indicate the dynamic equilibrium. The terms u, v, w and p represent, respectively, the time means of the velocity components and the pressure. By using the Boussinesq's approximation to describe the turbulent stresses, E^ and e^ are horizontal and vertical eddy viscosity coefficients. Supposing L%, Ly and L% are, respectively, the characteristic scales of the lengths in directions x, y and z, in the case in question we obtain [2]: O[LJ»O[L,} O[L,]>O[LJ (3) from which it follows that w (x,y,z,t) = 0 and hence eqn. (2c) becomes: ar-«* (4) which characterizes a hydrostatic distribution of the pressures. z A Figure 1: Hydrodynamic scheme By integrating eqns. (2a-b) along the vertical and introducing the mean velocity U = J udz and V = j vdz, being h = hg -ho -ho (e.g. with reference to component U):, we also obtain

4 158 Environmental Problems in Coastal Regions 3x1 3x 0[LJ o -k dy( * dy Thus, from eqn. (3), we have the following: He.,^-»0 OL (5-a) O a f au a ( au ax (5-b) As regards the terms of resistance, it should be noted that, integrating along the vertical, it is: ^ r r -HQ a av p p TS and ty being the shear stresses on the surface and on the bottom. In practical applications, it is commonly considered that: and TS and %,, can be expressed as [3]: (6) (7-a) (7-b) in which p is the water density and f the non-dimensional friction coefficient which is assumed [4] (f = 1(H -* 10-^)

5 Environmental Problems in Coastal Regions 159 Finally, taking into consideration (4), (5a-b), (6) and (7a-b), the equations of motion and continuity in the 2D depth integrated model are as follows [5], [6]: I bx t/ \-/ 6V gn T_ g^y c^ <9Uh -^+ + = 0 gt gx 9y (io) 3. Mathematical model for turbulent advective-diffusive pollution transport Let C(x,y,t) be the instantaneous concentration (defined as the pollutant mass per unit of volume of the fluid). The solute mass conservation equation is [7]: dc Eqn. (11) obviously refers to the above 2D-depth integrated model and thus the C values (x,y,t) represent the means along the depth. In (11) the molecular diffusion effect was ignored since it is very slight compared with the effect of turbulence [8]. The terms Kx and Ky represent the turbulent dispersion coefficients while 9 (dimension time'*) is a coefficient which characterizes the time decay of the pollutant. Kx and Ky values, which represent one of the most difficult unknowns to estimate, are commonly obtained through the distribution assigned to the velocity profiles u(x,y,z,t) and v(x,y,z,t) and through the eddy diffusivity coefficients D% and Dy which, in accordance with Boussinesq's approximation, describe the diffusive terms in the most complete 3D form of eqn. 01). It is also worth noting that the differential eqn. (11) is a mixed type in the sense that the terms of the left-hand member (hyperbolic part) represent advective transport while the terms of the right-hand member (parabolic part) describe the diffusion effects. The relative importance between advective and diffusion terms is highlighted by the Pec let number: Pe = U;L/K in which U«,, L and K represent, respectively, the characteristic velocity of the fluid, the characteristic length of the flow domain and the dispersion coefficient. When Pe > 1, the advection effect prevails over diffusion and vice versa.

6 160 Environmental Problems in Coastal Regions 4. Numerical solution The solution of the PDE (9 a-b) and (10), which describe the field of motion and the solution of the transport differential equation (11) is obtained numerically by using finite difference methods that are particularly straightforward and efficient for the case in question [9], [10], [11]. In particular for eqns. (9 a-b) and (10), an explicit form is used. The flow domain is characterized by an orthogonal staggered grid for space-time discretizations with mesh sides Ax, Ay (fig. 2) and time step At. The calculation points are fixed by the i, j, m indices: the first and second for spatial coordinates x, y, the third for time t. The finite difference forms, according to the explicit scheme presented, are [12]: gat ( -4 "41 2f (12-a) vr'=vr- ^ (12-b) (12-c) where:

7 Environmental Problems in Coastal Regions 161 Ay i. J+1 Ax 4- Figure 2: Orthogonal staggered grid for spatial discretization The boundary conditions are fixed as follows: - the velocities at the mouth are known V" = VQ ; - the velocity components orthogonal to the shoreline are zero VJ" = 0; -in the open sea U^,V^,^ = U?,V^ along x-direction and U"u, V y,t y = Uf,V»,1$ alongy^lirection. Due to the explicit form of the integration, the space steps Ax, Ay and the time steps At are fixed so as to verify the stability condition CFL [13]: w< jj being water depth in the generic mesh i, j.

8 162 Environmental Problems in Coastal Regions As regards the transport eqn. (11), it should be pointed out that, as is well known, explicit numerical finite difference methods generate significant errors due to the effect of numerical diffusion. The proposed calculation method is therefore implicit with centered differences and is always stable for each Ax, Ay and At value. In such applications, with a view to reducing the dimensions of the arrays, simplified algorithms are used such as the ADI method. This method alternately involves at time step m an implicit scheme along x and an explicit scheme along y, while at time step m+1 an implicit scheme along y and an explicit scheme along x. The solution proposed here is fully implicit and, with reference to fig. 2, the inductive relation is [12]: 'ym+l pm+1 _ ym+\ pm+1 Mj+1 ^ij+1 Mj-1 ^ij-1 At 2Ax 2Ay K x Ay' 03) The system of linear algebraic equations supplied by (13) is characterized by a large, sparse, unsymmetric array. To reduce calculation times, use is made of a particular simplifying algorithm [14] which involves the transformation of the initial arrays into two packed arrays, one to memorize the non-zero elements and the other to fix the real positions of their indices. Boundary conditions are fixed as follows: - concentration values are known at the mouth: C = CQ ; - the concentration flux in the direction orthogonal to the shoreline is null: 8C ^ * - in the open sea the diffusive transport component is null, that is: g^c <^C r = 0 along y-direction and = 0 along x-direction. ox dy 5. Applicative example The results obtained from the application of the proposed numerical model in an applicative example are reported below. The geometric scheme in question is that infig.2, where it is assumed that a = 50 m, b = 600 m and c=1000 m. The water depth at therivermouth is 5 m and the sea-bottom has a uniform 1% slope increasing in the offshore direction. The flood hydrograph and the corresponding time variation of concentrations CQ at the mouth are reported in the histograms in fig. 3 a-b. The following assumptions are made: Ax = Ay = 25 m; moreover, it is considered that K = Kx = Ky and two different values of the dispersion coefficients are chosen, K = 1 nf/s and K = 10 nf/s, in

9 Environmental Problems in Coastal Regions 163 order to highlight the effect of the diffusive component for two different Peclet numbers. L t 3- s 2 - i _ 0 - k ^ t(h) Figure 3-a: Time variations of VQ i? I 1^ '" ^ d, n«0 ' 0,6-0, fe» , J 0 Figure 3-b: Time variations of GO The results are shown in the graphs in fig. 4, 5 and 6 where the spatial patterns of the concentration at fixed values of the time t are reported by means of isoconcentration curves. It should be noted that, at increasing times, the zones with high concentrations move towards the open sea. Such an effect is more evident for the lower K value. Moreover, as is to be expected, for K = 10 m^/s the greater effect of diffusive transport reduces the extent of the areas with high concentrations.

10 164 Environmental Problems in Coastal Regions 6. Conclusions Dispersion of pollutants transported by a river flood to a coastal zone near a mouth may be studied with good approximation by using the proposed mathematical model which is based on a 2D depth integrated representation of the field of motion. The proposedfinitedifference numerical model, which is both reliable and not excessively time-consuming, requires the use of an explicit scheme for defining the flow domain and a fully implicit scheme for estimating the distribution of pollutant concentrations. References 1. Ligget, J.A., Fluid Mechanics, McGraw-Hill, Pedlosky, J., Geophysical Fluid Dynamics, Springer-Verlag, Csanady, G.T., Circulation in the Coastal Ocean, Reidel Pub.Company, Soulsby, R.L., Tidal current boundary layers, Ocean Eng. Science vol.9-a, Abbott, M.B.,Price, W.A.,Coastal Estuarial and Harbour Engineers, Reference Book,FN SPON, Di Natale, M., A 2D stochastic modelfor the hydrodynamic dispersion of pollutants on the sea surface, Second International Conference on Computer Modelling of Seas and Coastal Region II, Cancun-Mexico, Holly, P.M. Jr. Dispersion in river and coastal waters-physical principles, Developments in Hydraulic Engineering 3, Elsevier Pub. New York, Fisher, H.B.,List, E.J., Koh, R.C.Y. Imberger, J.,Brooks, N.H., Mixing in Inland and Coastal Waters, Academic Press.Inc., Sauvaget, P., Dispersion in river and coastal -waters-numerical simulation, Developments in Hydraulic Engineering 3, Elsevier Pub. New York, Kowalik, Z.,Murty, T.S., Numerical Modelling of Ocean Dynamics,Advanced Series on Ocean Engineering, World Scientific Pub., Brebbia, C.A., Computational Hydrodynamics, Butterworths, Koutitas, C.G. Mathematical Models in Coastal Engineeringfeatech Press, Smith, G., Numerical Solution ofpartial Differencial Equations, Oxford, Gupta, S.K., Tanji K.K., Computer program for solution of large, sparse, unsymmetric linear systems of equations, International Journal for Numerical Methods in Engineering, vol. 11, 1987

11 Environmental Problems in Coastal Regions 165 Figure 4 - Visualization of the isoconcentration lines at t = 1 h a) values of C for k = 10 m2/s b) values of C for k = 1 m2/s

12 166 Environmental Problems in Coastal Regions Figure 5 - Visualization of the isoconcentration lines at t = 3 h a) values of C for k = 10 m2/s b) values of C for k = 1 m2/s

13 Environmental Problems in Coastal Regions 167 Figure 6 - Visualization of the isoconcentration lines at t = 6 h a) values of C for k = 10 m2/s b) values of C for k= 1 m2/s