VOL. 12, NO. 6 WATER RESOURCES RESEARCH DECEMBER 1976 Dispersionof Water Pollutantsin a Turbulent ShearFlow GOUR-TSYH YEH AND YUN-JuI TSAI Stone and Webster Engineering Corporation, Boston, Massachusetts 02107 A steady statetwo-dimensional turbulentdiffusion equationdescribing the concentration distribution of a substance_from a line source in a shear flow field is solved analytically. A similar formulationmay be developed for any other kindsof sources. In the studythe velocity and eddydiffusivity are assumed to be variables given by the powerlaw approximations, and the depthof the water bodyis assumed finite,with no-fluxboundaryconditions appliedat the water surface and bottom. This represents a first steptoward analyticalwater quality modeling which realistically includes the effects of both the finiteness in water depth and the nonhomogeneity in velocityand diffusivity. Results from the present model are compared with those obtained from the finite depthconstant coefficient modeland from the infinitedepthconstant coefficient model. They show significant and realistic differences in the prediction of concentration patterns. The effectsof nonhomogeneous velocity and diffusivity are cancelled out by the effect of boundaryreflection far away from the source. INTRODUCTION Within the area of applicationsof the advectivediffusion equation to the water quality modelingthere is a great variety of models in terms of coordinate systems to be employed, the numberof dimensions to be considered, the methods of speci- fying boundary and initial conditions, and the functions to describe the velocityand eddy diffusivity. As far asthe methods of describing the velocity and diffusiv- ity and boundaryconditions are concerned, the earliest efforts in analyticalmodeling of water quality assumed unrealistically constan.,t velocity and diffusivity andan infinite or semi-infinite water body [Diachishim,1963;Edinger and Polk, 1969; Wnek and Fochtman,1972]. Progress.has been made to remove the assumption of an infinite or semi-infinite water body and to allow the effects of boundaries to enterthe picture by assuming a finite depth of the water body but with constant velocity and diffusivity [Kullenburg,1971; Cleary and Adrian, 1973; Yeh, 1973], though mathematical solutions forsuch problems have long been available [Carslaw and Jaeger, 1959]. On the other hand, the unrealistic assumption of constantvelocity and dif- fusivity has been replaced by more reasonable power law functions but with infinite or Semi-infinite geometry in the modeling of air pollution [Yih, 1952;Yehand Brutsaert,1971; Tsai and De Harpporte, 1973]. However, whereas the infinite or semi-infinite geometryassumptions are justified in the at- mospheric modeling that is high above the earth, the appli- cation of these methods to an obviously shallow waterbodyis questionable. The present paper presents a model that in- corporates the effects of finite boundaries and variable velocity and diffusivity in the flow field. The vertical dimension of the water body is considered finite with depth H as shown in Figure 1. The velocityand eddy diffusivity are considered as functions of the distance from the bottom of the water body. This realistic approach to the finite boundaries and variable velocity and diffusivity represents a step toward improving the analyticalmodeling of water quality. MODEL FORMULATION The basic two-dimensional equation that describes the con- centration of a substance from a line source in a steady unidi- rectional flow is •9c a Kz + S(x z) (1) U ax t9z • ' Copyright ¸ 1976 by the American Geophysical Union. where c is the concentration, x and z are coordinates measured in the directionof the flow and verticallyupward, S is a source function,and Kz is the eddy diffusivity. Equation(1) is derived from the conservation of mass on a differential element of volume withthediffusive transport in theflowdirection as- sumednegligible in comparison with the advective transport. A theoretical model is available to determinethe velocity u and the eddy diffusivityKz as functionsof the coordinatez from the solutionof boundarylayer equations [Schubauer and Tchen, 1961]. However, the resulting functions are such that they make the analytical solution of (1), under appropriate boundary conditions, extremely difficult if not impossible. Fortunately,it hasbeenshown that these theoretical functions can be approximated by the power law [Schubauer and Tchen, 19611 and tt = a2 m (2) K, = bz n (3) wherea, b, m, and n are constants depending on the Reynolds number and bottom roughness of the water body. For ex- ample, tn is approximately equal to ;[ for a large Reynolds number and smoothsurface. Values of a, b, m, and n have been evaluated for atmospheric modeling [Brutsaert and Yeh, 1970a, b]. Similar effortsshouldbe made for the water quality modelingby usingthe result of the presentpaper. No-flux boundary conditions will be imposed at the water surface and its bottom, and K t9c =0 z=0 (4) Ztgz K t9c =0 z=H (5) Zcgz and the incomingfluid will be assumed to have constantcon- centration c•, c = c• x = 0 (6) Equations (1)-(6) form a closed consistent uniquesystem. Its solution can be obtained by using Green's function [Yeh, 1973], fl 0 = dzo G(x, z; Xo, zo)S(xo, Zo) dxo (7) 1265