Model Dispersie

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Hydrologkal Sciences - Journal - des Sciences Hydrologiques, 32, 1, 3/1987

Dispersion of pollutants in riversW. CZERNUSZENKOInstitute of Meteorology and Water Man agemen t,ul. Podlesna 61, 01-673 Warsaw, PolandABSTR AC T The problem under consideration is the mixingof conservative, nonbuoyant pollutants released into ariver. Such pollutants are mixed with the water of ariver by different mechanisms, i.e. molecular diffusion,turbulent diffusion, advection and dispersion. Thesemechanisms are explained. One of the most important isdispersion, which is derived and described in the paper.The mixing process in rivers is represented by theadvection-dispersion equation. Analysing this process inirregular rivers is usually impossible because of thelateral advection term in the dispersion equation.However, this term disappears in a special curvilinearcoordinate system. S uch a system is introduced and amethod of solution of the dispersion equation is presented.Two examples of the simulation of a mixing process incurvilinear channels are given.La dispersion des polluants dans les riviresR E S UME Le problme que l'on considre ici est celui dumlange des polluants conservatoires non-flottants,rejets dans une rivire. D'habitude la description de ceprocessus dans les rivires non amnages est impossiblepar suite d'un terme d'advection latrale dans l'quationde dispersion. C e terme disparait dans un systme spcialde coordonnes curvilignes. C e systme est prsent avecune mthode de solution de l'quation de dispersionpropose. O n a donn deux exemples de simulation duprocessus de mlange dans un canal courviligne.

INTRODUCTIONThe problem of natural mixing in rivers is commonly treated astwo-dimensional. In order to obtain a two-dimensional equationrepresenting the variation of the depth-averaged concentration inboth the transverse and longitudinal directions, the three-dimensional advection-diffusion equation is integrated over the waterdepth. It is generally assumed that Fickian diffusion theory can beapplied to the dispersion process in rivers. This gives the two-*Based on a paper presented at the Anglo-Polish Hydrological Workshop, Jablonna, Poland, 17-21 September 1984. (See report inHydrological Sciences Journal, vol.30, no.l, p.165.)Open for discussion until 1 September 1987.

59


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60 W. Czemuszenkodimensional advection-dispersion equation which is usually used fordescribing the pollution concentration field in rivers. The equationmay be applied in the case of conservative and nonbuoyant pollutantswhich are vertically well-mixed in rivers. Further consideration islimited to the mixing associated with the ambient flow conditionsrather than to any initial mixing which might be caused by energy inthe discharged pollution.

BASIC EQUATIONSA description of the mixing of pollutants in ariver or openchannel is most frequently based on the mass conservation law. Thegeneral form of this law in adifferential form is:

-r- + div q = 0 (1)

where f = the mass density, and q = the mass flux density vector.Mixing in a river sufficiently far from a pollution source will

be considered. A pollutant in ariver is displaced from aplaceofhigher concentration to aplace oflower concentration. In thecase of turbulent flow the displacement occurs due to moleculardiffusion, turbulent diffusion and advection. These three massdensity fluxes are usually expressed in the forms:

q = - K grad C (2)-m mq t = - K tgrad C (3)

q = C (4)-awhere :K m = coefficient of molecular diffusion;K t = coefficient of turbulent diffusion;C =mean concentration at apoint (it isalso the mass density term

in equation (1)); andu =mean velocity at apoint.The two-dimensional case of pollution spreading in rivers is nowunder consideration. The several two-dimensional fluxes can beobtained by averaging quantities defined earlier. Thus in two-dimensional approximations:

- mass density is:C h = J* C dy (5)

(where two overbars means the depth-average operation)- pure advection flux density splits into two fluxes:

u C h = uC +u y C y h (.6)


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w h e r e :Dispersion of pollutan ts in rivers 61

u y ( x , y , z ) = ( x , y , z ) - u ( x , z ) (7 )C y ( x , y , z ) = C ( x , y , z ) - C ^ x . z ) (8 )- turbulent diffusion flux density. h = - K grad C h (9)- t t- molecular diffusion flux density:~q h = - K grad C h ( 10 )-m mThus the total two-dimensional mass flux density can be expressed

9total = H C B + l / C y + 3 t + g i n j h (11)

The term in curved brackets is called the dispersion flux density,D (see, e.g. Holley, 1969). According to equation (1) the law ofconservation of mass in a two-dimensional approximation gives:

~ hC + div I uC h + D h I = 0 (12)

In applying Fickian diffusion theory to the dispersion process inturbulent open-channel flow, an exact analogy with the moleculardiffusion process is assumed:

D = - E . dC/dx. (i = x, z) (13)i l l

where E j = coefficient of dispersion in the i-direction.Equation (12) now takes the form:

d , d , _ 6 d , oC d . _ dC ,., ..Trrh C + - ^ h u C + - ^ h w C = ^ - hE7r- + hE^r- (14)at dx dz dx x dx dz z d z

where :u, w = longitudinal and lateral components of depth-averagedvelocity; andC = depth-averaged concentration.In equation (14) all over-bars are neglected for simplicity.


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6 2 W . CzernuszenkoNUMER ICAL SOLUTIONThe lateral component of velocity, w, occurring in the aboveequation, greatly affects the mixing process, mainly for unregulatedrivers with irregular transverse cross-sections. It is usuallyimpossible to take this velocity into account in computations due tolack of data. Measurements of this component of velocity are notbeing ma de, and its calculation is associated with a very difficultproblem of solution of the N avier-Stokes equation. These difficulties can be overcome by using an orthogonal curvilinear coordinatesystem which is composed of mutually orthogonal longitudinal andtransverse coordinates. The longitudinal coordinate should alignclosely with the depth-averaged velocity vector. In this coordinatesystem w = 0, and the dispersion equation takes the form:

(15)3C \+ U Sx"')/

1m mx z

/~ m hE Id z z dC\ 3x m 3x'\ x

+ 3 m x h E z 3C3z m 3zz

where:m x, m z = metric coefficients; andx', z' = longitudinal and lateral curvilinear coordinates.The metric coefficients, m x and m z, are introduced to calculate thedistance between two points in the curvilinear coordinate system.Distances along a longitudinal coordinate curve from point A topoint B are given by: L^ B = /? m x dx', and along a transversecoordinate curve from C to D by: L C D = I m z d z ' T h e values ofm x and m z may vary from point to point and are in general functionsof both x and z. For small distances these coefficients areconstant, and then:

L A Q = m (x'(B) - x'C A)) (16)AB xLC D = m z ( x ' ( D ) _ x*(C)) (17>

The above relations are used to calculate m x and m z for a numericalgrid.In engineering practice it is reasonable to assume that the depth

and the metric coefficient, m z, does not vary very much in thelongitudinal direction, nor does the coefficient, m x, in the lateraldirection. If so, equation (15) takes the form:

(18)

The above equation together with the initial condition:C (0,x,z) = C (x,z) (19)o

3C u 3C3t m 3 x'X

_ 1 _ 3_m 3x'X

!* 9m 3x'X hm 3z'zz 3c

m 3zz

and the boundary condition:


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Dispersion of pollu tan ts in rivers 633 C / 3 z = 0 f o r z = 0 a n d z = B ( 2 0 )

is solved by using the fractional step method (Yanenko, 1971). Thenumerical solution consists of three steps:Step 1. Solution of the pure advection equation:

3C u 3C+ = 0 (21)3t m dx'x

Step 2. Solution of the longitudina l diffusion equation :3C E x 3 c3 t m 3m m 3 x 'x x ( 2 2 )

Step 3. Solution of the lateral diffusion equation :( 2 3 )C 1 3 mz 3C3 t hm 3 z ' m 3 z 'z z

The distribution of concentration obtained from one step is used asthe initial concentration in the next step.

The advection equation is solved by the two-point higher ordermethod developed by Holly & Preissmann (1977). This method isconditional-stable (Courant condition) and it introduces slighterrors in phase and amplitude.The diffusion equations are solved by use of an explicit difference numerical scheme. It is not a precise scheme, with atruncation error of the order of (At, Ax ) , but it was used especially for showing how the metric coefficients and the bed configurationaffect the pollutant dispersion. The diffusion equation is approximated by a forward difference in the time direction and twice thecentral difference in the space direction. For diffusion in the xdirection it gives the scheme:

E At / \n+l n x / n n n \ ,*C . , = C . , + C . , , - 2 C , , + C . , (24)J' J ' m 2Ax ' 2\ J 1 ' k J , k J- 1 k/x

where subscripts j and k refer to the x-coordinate and k the z-coordinate respectively; and the superscript n refers to the timecoordinate.

The difference equation (24) is conditionally stable with thecondition:

E AtX 0 . 5 (25)

m 2 Ax'2

In the case of the lateral diffusion equation, the numerical schemeis more complicated:


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64 W. CzernuszenkoE At

k k 2Az' k-1 k \ c - n - - 2 C k + c k ! ^ >m k m k 7

E Atz 12Ax'2 mk-l \

k-1 k+1 n _k k-1 n h, k+1 + h. k-1m, kk "k

n(\-~ K - m- hk \ h\ m, kk

1 k+1 k k-1+

\(26)

where m k = 0.5 (m^-i + m k ) , and the subscript j is omitted forsimplicity.One can see how the depth and the metric coefficient distributionsinfluence the concentration field from equation (26) . When varia

tions of depth in a river are not large and the assumption that h isconstant can be made, then equation (26) takes the form:

E Atn+1 n zC. = C, +Az' 2 m k mk-l 'k+1

k-1 n n '"k-l2C + C, n k k-1 m (27)

In regular rivers the transverse component of velocity is rathersmall and the values of the metric coefficient are about unity. Inthis case , equation (27) takes a very simple form. C omparing theabove equations, one can see differences between a straight, regularchannel and an irregular one.

NUMERICAL SIMULATIONCalculations of the field of pollution concentration were carriedout for experimental channels in which measurements of the velocityfield and the elevation of the free water surface were made. C hannelA (Fig.l) has a fixed uneven bed having a large scale configurationas in a natural river bed, but channel B has a rectangular crosssection. The plan view geometry of these channels is the same.

The values of the metric coefficients depend on the geometry ofthe curvilinear coordinate system. The preparation of the distribution of cumulative discharge, q c = / hudz (see Yotsukura &Sayre, 1976) for several measuring cross sections is required toestablish the geometry of a set of longitudinal coordinate curves.The transverse coordinate curves are perpendicular to the longitudinal ones. The values of the metric coefficient are calculated foreach cell of a numerical grid from equations (16) and (1 7 ). In thecase of a circular shape for the longitudinal coordinates, the metriccoefficient, m x, may be calculated from:

m = 1x + z/R (28)


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Dispersion of pollu tants in rivers 65where :z = transverse coordinate (the inner bank of the river is chosen as

the x-axis); andR = radius of curvature.The value of the metric coefficient, m , is difficult to express inany analytical form.

CROSS-SECTION

VELOCITYDISTRIBUTION

0 5.98 MINU TESo

VELOCI TY DIStRIBOTION

OUTLET: 21.8 GRAM

Fig. 1 Conc entration distribu tions in meandering channels for diffe ren t times(instantaneous source).

Simulations of the dispersion of polluted water introduced intochannels A and B, both instantaneously and continuously, werecarried out. In the case of channel A, the important role in mixingis governed by the irregular cross sections. For a river dischargeof 0.463 m 3s - 1 (average velocities of about 0.4 m s -1) the range ofchanges of m z is from 0.4 to 4. The iso-concentration lines are


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66 W . Czemuszenkoshown in Fig.l. One can see the influence of the channel curvatureon the dispersion process. The concentration distributions are notat all similar to Gaussian ones. In the case of channel B, with thesame average velocities of 0.4 m s , the concentration distributions are more regular and the range of changes of m z is smaller,from 0.7 to 1.1. This range of changes may be used as a measure ofirregularity of a river cross section.

The rates of decrease of maximum concentration (percentage ofinitial concentration) in channels A and B are shown in Fig.2. In

40

I 30

zg 10

CHANNEL A

CHANNEL B

TIME , m lnFig. 2 Ma ximum po lluta nt conc entration in tw o differe nt channels (instantaneoussource).

the more regular channel B, concentrations are lower because ofgreater depths and velocities at the pollutant outlet bank.

In the case of a continuous source, the field of pollutionconcentration is easy to achieve from the solution of equation (18)by its integration with respect to time. The simulation of dispersion in this case was carried out at an average velocity of about0.2 m s - 1 . The distributions of concentration in a few crosssections of channels A and B are shown in Fig.3. The concentrationsin channel A are much higher than those in channel B because ofgreater irregularities in channel A. The metric coefficient, m z,changes from 3.8 to 0.5 for channel A, and for channel B, from 1.45to 0.9.

CONCLUSIONS(a) The advection-dispersion equation may be solved in the case

of irregular rivers with irregular cross sections where lateralvelocities exist. In this case a special curvilinear coordinatesystem should be used.

(b) The advection-dispersion equation in a curvilinear coordinate system has two additional parameters viz. the metric coef-


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Dispersion of pollu tants in rivers 67CHANNEL B

Fig. 3source),

) 1 2 3 0 1 2 3DISTANCE FROM OUTLET BANK , m

Concentration distributions at a number of channel cross-sections (continuous

ficients m x and m z The influence of m on the pollution concentration distribution in rivers is rather small and can be neglected,but the role of m z is significant.

REFERENCESHolley, E. R. (1969) Unified view of diffusion and dispersion.

J. Hydraul. Div. ASCE 95 (HY2).Holly, F.M. & Preissmann, A. (1977) Accurate calculation oftransport in two dimensions. J. Hydraul. Div. ASCE 103 (HY11)Yanenko, N.N. (1971) The lethod of Fractional Steps. SpringerVerlag, New York.Yotsukura, N . & Sayre, W.W. (1976) Transverse mixing in naturalchannels. Wat. Resour. Res. 12(4).

Received 30 Decem ber 1985; accepted 26 August 1986.


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