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Dispersion of pollutants in an asymmetric flow through a channel
Pergamon
DISPERSION
Vol. 32, No. 9, pp. 1501-1510, 1994 Copyright @ 1994Elsevier Science Ltd Printed in Great Britain. All rights reserved O&20-7225/94$7.00+ 0.00
ht. J. Engng Sci.
002lk?Z25(93)EOO27-3
OF POLLUTANTS IN AN ASYMMETRIC FLOW THROUGH A CHANNEL B. S. MAZUMDERt
and R. XIA
Illinois State Water Survey, University of Illinois at Urbana-Champaign, (Communicated
IL 61820, U.S.A.
by S.-I. PAI)
Abstract-The longitudinal dispersion of a pollutant due to asymmetric flow through a twodimensional (2-D) channel has been studied for two different inputs: (i) instantaneous uniform line source and (ii) oscillatorysource. This study reveals the process whereby the dispersion reaches a stationary state after the release of pollutant as an instantaneous line source; and an estimate is obtained for the axial distance along which fluctuations in concentration decay in the case of low
frequencyrelease.
INTRODUCTION The longitudinal dispersion of soluble matter in fluid flow, after the great work of Taylor [l, 21 has a wide application in the field of environmental fluid mechanics (Fischer et al., [3]). In his classic papers, Taylor pointed out that in a moving fluid, solute is more slowly dispersed by molecular or turbulent diffusion alone than the dispersion due to the “shear effect” caused by combined effects of convection and lateral diffusion. Aris [4] presented a method of moment analysis for the cloud of contaminant in the streamwise direction and studied the asymptotic behaviour of the second moment about the mean. The problem of dispersion phenomena has been extensively studied by many researchers using theoretical analysis, hydraulic experiments and field observations (Chatwin [5,6], Gill and Sankarasubramanian [7], Smith [8], Barton [9], Mukherjee and Mazumder ([lo, ll]), and others). Barton [9] resolved certain technical difficulties in the Aris method of moments and obtained the solutions of the second and third moment equations of the distribution of solute, valid for all time. All the investigations mentioned above were restricted to the case when the discharge was uniform over the cross-section of tube or channel. Carrier [12] studied the longitudinal dispersion of a solute concentration in a steady flow through a pipe when the injected solute is prescribed as a harmonic function of time at a fixed cross-section of the pipe. The same problem was studied by Chatwin [13] who showed that for high frequencies the concentration pattern is transported downstream at the maximum fluid velocity, but for low frequencies it is transported at the discharge velocity. Barton [14] solved the eigenvalue problem arising from the same problem numerically, taking account the effect of longitudinal diffusion. The conclusions of Chatwin were reexamined. The main objective of the present paper is to study the longitudinal dispersion of contaminants in a non-uniform flow through a 2-D channel. Our analytical solutions are restricted to the cases when: (1) the injected material is initially uniform over the cross-section and (2) the injected material varies harmonically with time at a certain cross-section of the channel. Wang et al. [15] analysed the dispersion of a pollutant arising from an instantaneous point source in a 2-D channel with a non-uniform velocity distribution over the cross-section. The non-uniformity in velocity distribution is commonly found in natural streams because of the uneven geometry of the river bed. There is at least one maximum velocity occurring not necessarily at the centre of the channel, but it depends on the cross-sectional topography of the river bottom. The motivation of the present study stems mainly from the important application tpresent address:Stat-MathDivision,Indian StatisticalInstitute,203 B. T. Road, Calcutta-700035, India. El32:9-M
1501
B. S. MAZUMDER and R. XIA
1502
of a simulation of an accidental spill or a controlled release of a pollutant, e.g. the periodic discharge of outfalls from chemical plants in rivers or canals having a non-uniform velocity distribution over the cross-section. Although the processes controlling the dispersion of dissolved and suspended pollutants are numerous and complicated, the present study considers the dispersion process in a simple geometrical domain where better understanding of the basic scientific problems involved would be valuable in predicting dispersion in such real flows.
MATHEMATICAL
FORMULATIONS
AND
SOLUTIONS
Consider a flow in a shallow, 2-D rectangular channel of width d, in Cartesian coordinate system with origin at the wall (y ’ = 0), XI-axis along perpendicular to the flow. The axial velocity of the river flow satisfying the the boundaries can be described by a simple dimensionless expression (see
which we employ a the flow and y ‘-axis no-slip conditions at Wang et al., [15]) as
u(y) = uOy(l - y)ewAy
(I)
where y(=y’/d) is a dimensionless variable, u. is the discharge speed, and the parameter A represents the non-uniformity of the velocity distribution over the cross-section. The velocity profile is similar to the flow field studied by Fukuoka and Sayre [16] for the uneven river bottom. Figure 1 shows the plots of velocity distribution for five different values of A. When a slug is released in the above mentioned flow, the concentration C(t’, x’, y’) of the diffusing substance satisfies the convective-diffusion equation of the form
(2) where t’ is time, k, and ky are the eddy diffusion coefficients assumed constant. Introducing dimensionless variables in equation (2) one gets: (3)
A = -2
0.6
-
8.4
-
7
e
.0.2
8.6
6.4
Y Fig. 1. Velocity distribution.
0.6
1
1503
Dispersion of pollutants
where y = y’fd, x =x1/d, t = k,,t’fd2, Pe = du,/k,,, DO= k,/k,,. Here Pe is the P&let number which measures the relative characteristic times of the diffusion process (d2/ky) to the convective process (d/u*). Case I. Instantaneous uniform line source The longitudinal dispersion of diffusing material is studied in a non-uniform flow through a channel when the initial cloud of contaminant is uniform across the cross-section. The initial and boundary conditions for the pollutant input are
C(0, x, y) = NY) ac
-=0
(4a)
at y=O,l
aY C is finite at all points X and x” ~40
x”C+O
(n = 0, 1,2, . . .)
as ]~(+a 1 -
If0
Cd.xdy=l
(k)
-0)
The initial condition (4a) corresponds to an instantaneous uniform line source at x = 0 across the cross-section. Following Ark’ [4] method of moments, we define the nth moment of concentration through y at time t as x”C(t, x, Y) d.x
wY)=J-_;
(5)
and the nth moment of the concentration over the cross-section is K(f) = j-’ G(t, y) dy = c,,
(6)
0
Using (5) and (6) in (3), one gets the following moment equations
--ac, at
a2c n = Don(n - 1)Cn-2 + Fe nu(y)C,_, ay2
(7)
with the conditions G(0, Y) = CAY 1,
$=O
at y=O,l
(8)
and at
=
D&z
- l)Cn_, f Pe nu(y)C,...,
with the condition M*(O)= 6
(10) where the overbar denotes the cross-sectional mean, and MOrepresents the total amount of diffusing substance (assumed one). Then the nth central moment of the dist~bution can be defined as (x-x,)“C&dy where X8=-
1
1
M0 iii0 -= xCdwdy=z
M
(11)
B. S. MAZUMDER and R. XIA
1504
is the centroid of the slug. The first moment xs measures the location of the centre of gravity of the slug moves with the mean velocity of the fluid particle, which was initially located at the source, and the second moment (v2) can be related to the dispersion of diffusing substance about its mean position. The variance is given by V*(f)
M2 Y&-
=
-
Xf
(13)
0
The skewness and kurtosis factors are defined as p3 = v,/v; - 3
p?. = v,1G2,
(14)
Though the skewness and kurtosis are also the important factors during the initial stage, the present study is concentrated only on the dispersion effect. The moment equations (7) and (8) can be solved by the Aris method of separation of variables as modified by Barton [9]. According to Barton [9], we consider the following eigenvalue problem
d$
-=0
at
dy
(15)
y=O,l
These give us a discrete set of eigenvalues ~(y}=~cos~~, i=l,2 ,..., so that
pi = (ilc)’ and corresponding
eigenfunctions
$=o f%=l =O
if
i=j
if
itlj
(16)
We au~ent this set of eigenfunctions by setting fo = 1, p. = 0 in order to get a complete set. As shown by Barton ([9], equation 3.16), v2(f) is given by v&) = M2(0) + 2 Do + Pe2 C G$.E]r _ 2Pe2 7 (S&,‘(1 L , I
_ e-I+‘)
(17)
The variance v*(f) is plotted against t for various values of A in Figs 2(a, b) when Pe = 100, 1000, Do = 1 and &&(O)= 0. The longitudinal dispersion coefficient D,, which indicates the degree of dispersion effect, at any time is given by ($1’
L&=i2=Do+Pe2zi
Y.I (l-
e-‘“i’)
(18)
Thus, the rate of change of variance is pro~rtion~ to the sum of the ratio of eddy diffusion coefficients (assumed Do = 1) and the apparent dispersion coefficient D, = ??(I
_ e-M) I
09)
The second term of the riot-hod side of (18) represents the interaction between the convection and lateral diffusion. As there is no effect of the first term in the convection, only the apparent dispersion coefficient D, is discussed. As r+ co, the equation (18) can be written as
This result is consistent
with the asymptotic
theory of Chatwin ([5], equation
3.8). Figures
Dispersion of pollutants
e
e.1
6.2
1505
0.3
8.4
8.6
I
I
I
t
1
4
: 3
I
-
I
A = -2
WI
-
-1
:
0:
/
5:
t
J
-2 f, 8
0.1
0.2
8.3
8.4
8.6
t Fig. 2. Plots of variance (log y.) against time (t) for (a) Pe = 100 and (b) Pe = 1000.
3(a, b) shows the variations of apparent (Taylor) dispersion coefficient D, with time t for various values of A = 0,*l,*2. From the figures it is seen that the apparent dispersion coetiicient D, increases for all time with A < 0 and then asymptotes to constant values for respective A's,whereas D, decreases with A > 0.From Fig. 1, it is clearly seen that the velocity increases with negative skew for A < 0 and it decreases with positive skew for A > 0.Therefore, the increases of D, corresponds to the increase of u and the decrease of D, corresponds to A > 0 (that is, decreases of velocity u). The dispersion coefficient can be seen to reach steady state when the time approaches approx. 0.3 of the characteristic time T, = d'/k, of transverse
1506
EL S. MAZUMDER
and R. XIA
CX iE-3) I
1
I
I
6
(a)
I
,
A = -2
4
3 a* 2
1
-1 0
0
8
I
I
8.1
e.2
,
-I
I
1
0.3
0.4
8.5
0.3
8.4
0.5
t
(X
lE-6) 16
12
3 a* 6
3
0
e
8.1
8.2
t Fig. 3. Plots of dispersion coefficient (D,) against time (I) for (a) A 5 0, (b) A 2 0.
mixing. Shortly after the instantaneous uniform release of pollutant, the dispersion takes place in both the lon~tudinal and lateral directions due to the shear effect and eddy diffusion. The longitudinal dispersion of diffusing substance at different distances from the boundary (y = 0) are not the same due to the non-uniformity of the velocity profile. The transverse concentration gradient is large in the zone where the velocity gradient is large. Hence, the lateral dispersion of diffusing substance is large.
Dispersion of pollutants
Iso?
Case II. Oscillatory source The longitudinal dispersion of pollutant substance in the non-uniform flow is discussed when
the injected material at the section (X= 0) is initially prescribed as a harmonic fusion of time, and it is assumed that the concentration C is uniform over the cross-section so that at x = 0, C = C, + C, exp(iot)
(21)
where C, and C1 are constants. As there is no effect of longitudinal eddy diffusion on the flow shear, we pay our attention only to the lateral diffusion effect. Therefore, neglecting the longitudinal diffusion in equation (3), the dimensionless convective-effusion equation can be written as
with boundary conditions XT
at y = 0, 1
dy--0
(23)
where g(y) = y(1 - y)e- Ay. Now, the equation (22) has the solution of the form
where a! = wd’lk, is the dimensionless frequency parameter; A and f(y) are the eigenvalue and eigenfunction in the eigenvalue problem
d”f = i4 dy
7
- Ag(y)lf(y)
W4
at y=O,l
@b)
with boundary conditions
df dy -=0
If Aj, i = 1,2, . * . are the eigenvalues and 5 (i = 1,2, . . , ) are corresponding eigenfunctions, then it can be shown that fi, f2,. . . will form a complete set. Thus, we may take C = C, + c A, exp[ia(t - A~~/Pe)Jf(y) P
(26)
where the constants A, can be determined so that the equation (21) is satisfied. Since the velocity distribution g(y) > 0 everywhere, following the analysis of Chatwin [13], it can be shown that the eigenvalue APsatisfies the inequality Im(h,) < 0
(27)
so that all the solutions of equation (25a) satisfying the condition (25b) are spatially decaying. The eigenvalues APis now ordered in ascending magnitude of their negative imaginary parts so that 0 C -Im(A,) < -Im(A,) < - - Hence, from equation (26), for a large value of x, C=CO+AOexp [*IQ( t- E)lfb(Y) =CO+AOexp[io(t’
-F)]h(y)
cw
B. S. MAZUMDER and R. XIA
1508
Thus for large x’, the concentration pattern is convected at a velocity uO/Re(AO) and decays in an axial distance of order -uO/w Im(&,). For low frequency (a XCl), following the series expansion method due to Carrier [12], we can assume A = boo + aho1 + (u2A02+ * . . f=ho(y)
+ cufol(Y) + ~*fo*(Y) + * * *
(2%
Substituting these equations (29) into equation (25a) and equating the coefficients of a like power of (Y,we get an infinite set of differential equation as:
d*fm
-=
o
(304
dy* d*f,, - = iP - bdy)lf~(y) dy*
(3Ob)
d*$x - = i[(l - Acady)l.fdy)- hfmg(y)l dy*
(3Oc)
with the conditions dfoo -c-c-=() dy
dfo,
dh2
dy
dy
at
y=O,l
WW
The solution of equation (30a) satisfying the conditions (30d) is (30b) twice and using the boundary conditions (30d), one gets A3
foe
=
1. Integrating
equation
A#0
Aoo= (A + 2)ePA + (A - 2)’
&(Y) = i[Y*/2 + Aoo{(4a3- a2)yepAy + (6a4 - 2a3)edAy + a2y2e-Ay}] - iky,
(31) A #O
(32)
where a = l/A, k = (A - 2)/{(A + 2)emA + (A - 2)) and for A = 0, A@J=6, Substituting A ZO,
in(Y)
= w%Y2
-
2Y3+ Y”)
(31) and (32), one can get the solution of (3Oc) after lengthy calculations hoI =
-Z(a)
(33) as for (34)
where F(a) = [3A&(a1 - a2e-2A) + 12A,(a3 - a4e-“) - 6k + 2]/12a0 a0 = a2(2a - 1) - a’(1 + 2a)emA,
a =1/A
al = a73 - 15a + 15a2) a* = a73 + 15a + 15a2) a3 = 24a5 - 6a4 + k(2a3 - 6a4) a4 = (0.5~~ + 3a3 + 12a4 + 24a5) - k(1 + 4a + 6a2)a2 and for A = 0, AoI= -i/35
(35)
1509
Dispersion of pollutants
Therefore,
from equation (29), it is found that h() = hQ(J- i&(a)
AZ0
+ Q(ff2),
(36)
ia/35+o(rx2), A=0
h0=6where Aoois given by equation (31). Thus, the fluctuations in concentration ForA#O,
(37)
decay in the axial distance of the order -uo/w Im(&).
decay distance a (d2u0/k,)~-*~(a) where 4(a) = l/F(a).
The concentration
(38)
pattern is convected with a discharge speed as
transport speed = uJRe(&,) = uo[(A +2)epA+(A - 2)]/A3
(39)
and for A = 0, decay distance = 35(d2u,Jky)o-2 transport speed = u0/6
(40) (41)
The results (40) and (41) for A = 0,agree well with that of Chatwin [I31 and Barton [14f. From Table 1 it is seen that the decay distance in concentration lluctuation decreases consistently with the increase of A > 0 from zero, which corresponds to the decrease of velocity u in the channel, whereas for A < 0,it shows some anomalous behaviour. The decay distance slightly increases, then decreases and again increases for A < -2.This anomalous behaviour may be explained by the fact that the dispersion due to convection at different distance from the wall is not uniform. The ffow velocity having the high shear causes the high dispersion in the lateral direction. When the velocity u is just deviated from the symmetric (A = 0),the decay distance of concentration fluctuation decreases because the lateral dispersion is more than the dispersion due to maximum velocity at the centre. However, when the velocity is significantly larger for A < -3 in one side of the channel, the more pollutants are carried away with the velocity, Hence the decay distance in the concentration fluctuation will increase for the velocity A < -3. From Table 1, it is also seen that the concentration pattern is convected with increasing discharge speed, when the velocity increases. Table 1. Vafues of $~(a) and IIRe
-5 -4 -3 -2 -1 0
57.8558 39.6371 31.4294 30.3069 36.0478 35.m 13.2612 4.1016 1.5648 0.7260 0.3898
3.6179 1.8002 0.9291 0.5000 0.2817 0.1667 0.1036 0.0677 0.0462 0.0330 0.0244
CONCLUSIONS In this paper, we have presented an analytical solution to study the dispersion a shallow channel with non-uniform velocity distribution; and it is confined to the slug is initially unifo~ over the cross-section and it varies harmonically proposed velocity profile can be generalized to fit real velocity distribution
of pollutants in the cases when with time. The in the natural
1510
B. S. MAZUMDER
and R. XIA
stream. An estimation can be made for the axial distance along which the fluctuations in concentration decay in the case of the low frequency input.
REFERENCES [l] G. I. TAYLOR, hoc. R. Sot. Land. A219,186 (1953). [2] G. I. TAYLOR, Proc. R. Sot. Land. A223,446 (1954). [3] H. B. FISCHER, E. J. LIST, R. C. Y. KOH, J. IMBERGER and N. H. BROOKS, Waters. Academic Press, New York (1979). f4] R. ARIS, Proc. R. Sot. Lortd. A235,67 (1956). f5] P. C. CHATWIN, J. Fluid Mech. 43,321 (1970). 161 P. C. CHATWIN. _f. Hvdr. Div. ASCE 106.71 (19801, i7j W. N. GILL and k. SANKARASUBRAGANiAN, i)roc. R. Sot. Land. A316,341 [8] R. SMITH, J. Fluid Mech. 105,469 (1981). [9] N. G. BARTON, J. Fluid Mech. Us,205 (1983). IlO] A. MUKHERJEE and B. S. MAZUMDER, Accu Mechanica 58,137 (1986). 111 A. MUKHERJEE and B. S. MAZUMDER, Acta Mechanica 74,107 (1988). :12] G. F. CARRIER, Q. Appl. Math. 14, 108 (1956). ‘131 P. C. CHATWIN, .I. Fiu~ Mech. 58,657 (1973). ‘141 N. B. BARTON, J. Fluid Mech. 136,243 (1983). 1151 S. T. WANG, A. F. McMILLAN and B. H. CHEN, War. Res. 12,389 (1978). 161 S. FUKUOKA and W. W. SAYRE, J. Hydr. Div. ASCE 99,195 (1973). (Received 26 January 1993; accepted 22 November 1993)
Mixing in Inland and Coastal
(1970).
DISPERSION
Vol. 32, No. 9, pp. 1501-1510, 1994 Copyright @ 1994Elsevier Science Ltd Printed in Great Britain. All rights reserved O&20-7225/94$7.00+ 0.00
ht. J. Engng Sci.
002lk?Z25(93)EOO27-3
OF POLLUTANTS IN AN ASYMMETRIC FLOW THROUGH A CHANNEL B. S. MAZUMDERt
and R. XIA
Illinois State Water Survey, University of Illinois at Urbana-Champaign, (Communicated
IL 61820, U.S.A.
by S.-I. PAI)
Abstract-The longitudinal dispersion of a pollutant due to asymmetric flow through a twodimensional (2-D) channel has been studied for two different inputs: (i) instantaneous uniform line source and (ii) oscillatorysource. This study reveals the process whereby the dispersion reaches a stationary state after the release of pollutant as an instantaneous line source; and an estimate is obtained for the axial distance along which fluctuations in concentration decay in the case of low
frequencyrelease.
INTRODUCTION The longitudinal dispersion of soluble matter in fluid flow, after the great work of Taylor [l, 21 has a wide application in the field of environmental fluid mechanics (Fischer et al., [3]). In his classic papers, Taylor pointed out that in a moving fluid, solute is more slowly dispersed by molecular or turbulent diffusion alone than the dispersion due to the “shear effect” caused by combined effects of convection and lateral diffusion. Aris [4] presented a method of moment analysis for the cloud of contaminant in the streamwise direction and studied the asymptotic behaviour of the second moment about the mean. The problem of dispersion phenomena has been extensively studied by many researchers using theoretical analysis, hydraulic experiments and field observations (Chatwin [5,6], Gill and Sankarasubramanian [7], Smith [8], Barton [9], Mukherjee and Mazumder ([lo, ll]), and others). Barton [9] resolved certain technical difficulties in the Aris method of moments and obtained the solutions of the second and third moment equations of the distribution of solute, valid for all time. All the investigations mentioned above were restricted to the case when the discharge was uniform over the cross-section of tube or channel. Carrier [12] studied the longitudinal dispersion of a solute concentration in a steady flow through a pipe when the injected solute is prescribed as a harmonic function of time at a fixed cross-section of the pipe. The same problem was studied by Chatwin [13] who showed that for high frequencies the concentration pattern is transported downstream at the maximum fluid velocity, but for low frequencies it is transported at the discharge velocity. Barton [14] solved the eigenvalue problem arising from the same problem numerically, taking account the effect of longitudinal diffusion. The conclusions of Chatwin were reexamined. The main objective of the present paper is to study the longitudinal dispersion of contaminants in a non-uniform flow through a 2-D channel. Our analytical solutions are restricted to the cases when: (1) the injected material is initially uniform over the cross-section and (2) the injected material varies harmonically with time at a certain cross-section of the channel. Wang et al. [15] analysed the dispersion of a pollutant arising from an instantaneous point source in a 2-D channel with a non-uniform velocity distribution over the cross-section. The non-uniformity in velocity distribution is commonly found in natural streams because of the uneven geometry of the river bed. There is at least one maximum velocity occurring not necessarily at the centre of the channel, but it depends on the cross-sectional topography of the river bottom. The motivation of the present study stems mainly from the important application tpresent address:Stat-MathDivision,Indian StatisticalInstitute,203 B. T. Road, Calcutta-700035, India. El32:9-M
1501
B. S. MAZUMDER and R. XIA
1502
of a simulation of an accidental spill or a controlled release of a pollutant, e.g. the periodic discharge of outfalls from chemical plants in rivers or canals having a non-uniform velocity distribution over the cross-section. Although the processes controlling the dispersion of dissolved and suspended pollutants are numerous and complicated, the present study considers the dispersion process in a simple geometrical domain where better understanding of the basic scientific problems involved would be valuable in predicting dispersion in such real flows.
MATHEMATICAL
FORMULATIONS
AND
SOLUTIONS
Consider a flow in a shallow, 2-D rectangular channel of width d, in Cartesian coordinate system with origin at the wall (y ’ = 0), XI-axis along perpendicular to the flow. The axial velocity of the river flow satisfying the the boundaries can be described by a simple dimensionless expression (see
which we employ a the flow and y ‘-axis no-slip conditions at Wang et al., [15]) as
u(y) = uOy(l - y)ewAy
(I)
where y(=y’/d) is a dimensionless variable, u. is the discharge speed, and the parameter A represents the non-uniformity of the velocity distribution over the cross-section. The velocity profile is similar to the flow field studied by Fukuoka and Sayre [16] for the uneven river bottom. Figure 1 shows the plots of velocity distribution for five different values of A. When a slug is released in the above mentioned flow, the concentration C(t’, x’, y’) of the diffusing substance satisfies the convective-diffusion equation of the form
(2) where t’ is time, k, and ky are the eddy diffusion coefficients assumed constant. Introducing dimensionless variables in equation (2) one gets: (3)
A = -2
0.6
-
8.4
-
7
e
.0.2
8.6
6.4
Y Fig. 1. Velocity distribution.
0.6
1
1503
Dispersion of pollutants
where y = y’fd, x =x1/d, t = k,,t’fd2, Pe = du,/k,,, DO= k,/k,,. Here Pe is the P&let number which measures the relative characteristic times of the diffusion process (d2/ky) to the convective process (d/u*). Case I. Instantaneous uniform line source The longitudinal dispersion of diffusing material is studied in a non-uniform flow through a channel when the initial cloud of contaminant is uniform across the cross-section. The initial and boundary conditions for the pollutant input are
C(0, x, y) = NY) ac
-=0
(4a)
at y=O,l
aY C is finite at all points X and x” ~40
x”C+O
(n = 0, 1,2, . . .)
as ]~(+a 1 -
If0
Cd.xdy=l
(k)
-0)
The initial condition (4a) corresponds to an instantaneous uniform line source at x = 0 across the cross-section. Following Ark’ [4] method of moments, we define the nth moment of concentration through y at time t as x”C(t, x, Y) d.x
wY)=J-_;
(5)
and the nth moment of the concentration over the cross-section is K(f) = j-’ G(t, y) dy = c,,
(6)
0
Using (5) and (6) in (3), one gets the following moment equations
--ac, at
a2c n = Don(n - 1)Cn-2 + Fe nu(y)C,_, ay2
(7)
with the conditions G(0, Y) = CAY 1,
$=O
at y=O,l
(8)
and at
=
D&z
- l)Cn_, f Pe nu(y)C,...,
with the condition M*(O)= 6
(10) where the overbar denotes the cross-sectional mean, and MOrepresents the total amount of diffusing substance (assumed one). Then the nth central moment of the dist~bution can be defined as (x-x,)“C&dy where X8=-
1
1
M0 iii0 -= xCdwdy=z
M
(11)
B. S. MAZUMDER and R. XIA
1504
is the centroid of the slug. The first moment xs measures the location of the centre of gravity of the slug moves with the mean velocity of the fluid particle, which was initially located at the source, and the second moment (v2) can be related to the dispersion of diffusing substance about its mean position. The variance is given by V*(f)
M2 Y&-
=
-
Xf
(13)
0
The skewness and kurtosis factors are defined as p3 = v,/v; - 3
p?. = v,1G2,
(14)
Though the skewness and kurtosis are also the important factors during the initial stage, the present study is concentrated only on the dispersion effect. The moment equations (7) and (8) can be solved by the Aris method of separation of variables as modified by Barton [9]. According to Barton [9], we consider the following eigenvalue problem
d$
-=0
at
dy
(15)
y=O,l
These give us a discrete set of eigenvalues ~(y}=~cos~~, i=l,2 ,..., so that
pi = (ilc)’ and corresponding
eigenfunctions
$=o f%=l =O
if
i=j
if
itlj
(16)
We au~ent this set of eigenfunctions by setting fo = 1, p. = 0 in order to get a complete set. As shown by Barton ([9], equation 3.16), v2(f) is given by v&) = M2(0) + 2 Do + Pe2 C G$.E]r _ 2Pe2 7 (S&,‘(1 L , I
_ e-I+‘)
(17)
The variance v*(f) is plotted against t for various values of A in Figs 2(a, b) when Pe = 100, 1000, Do = 1 and &&(O)= 0. The longitudinal dispersion coefficient D,, which indicates the degree of dispersion effect, at any time is given by ($1’
L&=i2=Do+Pe2zi
Y.I (l-
e-‘“i’)
(18)
Thus, the rate of change of variance is pro~rtion~ to the sum of the ratio of eddy diffusion coefficients (assumed Do = 1) and the apparent dispersion coefficient D, = ??(I
_ e-M) I
09)
The second term of the riot-hod side of (18) represents the interaction between the convection and lateral diffusion. As there is no effect of the first term in the convection, only the apparent dispersion coefficient D, is discussed. As r+ co, the equation (18) can be written as
This result is consistent
with the asymptotic
theory of Chatwin ([5], equation
3.8). Figures
Dispersion of pollutants
e
e.1
6.2
1505
0.3
8.4
8.6
I
I
I
t
1
4
: 3
I
-
I
A = -2
WI
-
-1
:
0:
/
5:
t
J
-2 f, 8
0.1
0.2
8.3
8.4
8.6
t Fig. 2. Plots of variance (log y.) against time (t) for (a) Pe = 100 and (b) Pe = 1000.
3(a, b) shows the variations of apparent (Taylor) dispersion coefficient D, with time t for various values of A = 0,*l,*2. From the figures it is seen that the apparent dispersion coetiicient D, increases for all time with A < 0 and then asymptotes to constant values for respective A's,whereas D, decreases with A > 0.From Fig. 1, it is clearly seen that the velocity increases with negative skew for A < 0 and it decreases with positive skew for A > 0.Therefore, the increases of D, corresponds to the increase of u and the decrease of D, corresponds to A > 0 (that is, decreases of velocity u). The dispersion coefficient can be seen to reach steady state when the time approaches approx. 0.3 of the characteristic time T, = d'/k, of transverse
1506
EL S. MAZUMDER
and R. XIA
CX iE-3) I
1
I
I
6
(a)
I
,
A = -2
4
3 a* 2
1
-1 0
0
8
I
I
8.1
e.2
,
-I
I
1
0.3
0.4
8.5
0.3
8.4
0.5
t
(X
lE-6) 16
12
3 a* 6
3
0
e
8.1
8.2
t Fig. 3. Plots of dispersion coefficient (D,) against time (I) for (a) A 5 0, (b) A 2 0.
mixing. Shortly after the instantaneous uniform release of pollutant, the dispersion takes place in both the lon~tudinal and lateral directions due to the shear effect and eddy diffusion. The longitudinal dispersion of diffusing substance at different distances from the boundary (y = 0) are not the same due to the non-uniformity of the velocity profile. The transverse concentration gradient is large in the zone where the velocity gradient is large. Hence, the lateral dispersion of diffusing substance is large.
Dispersion of pollutants
Iso?
Case II. Oscillatory source The longitudinal dispersion of pollutant substance in the non-uniform flow is discussed when
the injected material at the section (X= 0) is initially prescribed as a harmonic fusion of time, and it is assumed that the concentration C is uniform over the cross-section so that at x = 0, C = C, + C, exp(iot)
(21)
where C, and C1 are constants. As there is no effect of longitudinal eddy diffusion on the flow shear, we pay our attention only to the lateral diffusion effect. Therefore, neglecting the longitudinal diffusion in equation (3), the dimensionless convective-effusion equation can be written as
with boundary conditions XT
at y = 0, 1
dy--0
(23)
where g(y) = y(1 - y)e- Ay. Now, the equation (22) has the solution of the form
where a! = wd’lk, is the dimensionless frequency parameter; A and f(y) are the eigenvalue and eigenfunction in the eigenvalue problem
d”f = i4 dy
7
- Ag(y)lf(y)
W4
at y=O,l
@b)
with boundary conditions
df dy -=0
If Aj, i = 1,2, . * . are the eigenvalues and 5 (i = 1,2, . . , ) are corresponding eigenfunctions, then it can be shown that fi, f2,. . . will form a complete set. Thus, we may take C = C, + c A, exp[ia(t - A~~/Pe)Jf(y) P
(26)
where the constants A, can be determined so that the equation (21) is satisfied. Since the velocity distribution g(y) > 0 everywhere, following the analysis of Chatwin [13], it can be shown that the eigenvalue APsatisfies the inequality Im(h,) < 0
(27)
so that all the solutions of equation (25a) satisfying the condition (25b) are spatially decaying. The eigenvalues APis now ordered in ascending magnitude of their negative imaginary parts so that 0 C -Im(A,) < -Im(A,) < - - Hence, from equation (26), for a large value of x, C=CO+AOexp [*IQ( t- E)lfb(Y) =CO+AOexp[io(t’
-F)]h(y)
cw
B. S. MAZUMDER and R. XIA
1508
Thus for large x’, the concentration pattern is convected at a velocity uO/Re(AO) and decays in an axial distance of order -uO/w Im(&,). For low frequency (a XCl), following the series expansion method due to Carrier [12], we can assume A = boo + aho1 + (u2A02+ * . . f=ho(y)
+ cufol(Y) + ~*fo*(Y) + * * *
(2%
Substituting these equations (29) into equation (25a) and equating the coefficients of a like power of (Y,we get an infinite set of differential equation as:
d*fm
-=
o
(304
dy* d*f,, - = iP - bdy)lf~(y) dy*
(3Ob)
d*$x - = i[(l - Acady)l.fdy)- hfmg(y)l dy*
(3Oc)
with the conditions dfoo -c-c-=() dy
dfo,
dh2
dy
dy
at
y=O,l
WW
The solution of equation (30a) satisfying the conditions (30d) is (30b) twice and using the boundary conditions (30d), one gets A3
foe
=
1. Integrating
equation
A#0
Aoo= (A + 2)ePA + (A - 2)’
&(Y) = i[Y*/2 + Aoo{(4a3- a2)yepAy + (6a4 - 2a3)edAy + a2y2e-Ay}] - iky,
(31) A #O
(32)
where a = l/A, k = (A - 2)/{(A + 2)emA + (A - 2)) and for A = 0, A@J=6, Substituting A ZO,
in(Y)
= w%Y2
-
2Y3+ Y”)
(31) and (32), one can get the solution of (3Oc) after lengthy calculations hoI =
-Z(a)
(33) as for (34)
where F(a) = [3A&(a1 - a2e-2A) + 12A,(a3 - a4e-“) - 6k + 2]/12a0 a0 = a2(2a - 1) - a’(1 + 2a)emA,
a =1/A
al = a73 - 15a + 15a2) a* = a73 + 15a + 15a2) a3 = 24a5 - 6a4 + k(2a3 - 6a4) a4 = (0.5~~ + 3a3 + 12a4 + 24a5) - k(1 + 4a + 6a2)a2 and for A = 0, AoI= -i/35
(35)
1509
Dispersion of pollutants
Therefore,
from equation (29), it is found that h() = hQ(J- i&(a)
AZ0
+ Q(ff2),
(36)
ia/35+o(rx2), A=0
h0=6where Aoois given by equation (31). Thus, the fluctuations in concentration ForA#O,
(37)
decay in the axial distance of the order -uo/w Im(&).
decay distance a (d2u0/k,)~-*~(a) where 4(a) = l/F(a).
The concentration
(38)
pattern is convected with a discharge speed as
transport speed = uJRe(&,) = uo[(A +2)epA+(A - 2)]/A3
(39)
and for A = 0, decay distance = 35(d2u,Jky)o-2 transport speed = u0/6
(40) (41)
The results (40) and (41) for A = 0,agree well with that of Chatwin [I31 and Barton [14f. From Table 1 it is seen that the decay distance in concentration lluctuation decreases consistently with the increase of A > 0 from zero, which corresponds to the decrease of velocity u in the channel, whereas for A < 0,it shows some anomalous behaviour. The decay distance slightly increases, then decreases and again increases for A < -2.This anomalous behaviour may be explained by the fact that the dispersion due to convection at different distance from the wall is not uniform. The ffow velocity having the high shear causes the high dispersion in the lateral direction. When the velocity u is just deviated from the symmetric (A = 0),the decay distance of concentration fluctuation decreases because the lateral dispersion is more than the dispersion due to maximum velocity at the centre. However, when the velocity is significantly larger for A < -3 in one side of the channel, the more pollutants are carried away with the velocity, Hence the decay distance in the concentration fluctuation will increase for the velocity A < -3. From Table 1, it is also seen that the concentration pattern is convected with increasing discharge speed, when the velocity increases. Table 1. Vafues of $~(a) and IIRe
-5 -4 -3 -2 -1 0
57.8558 39.6371 31.4294 30.3069 36.0478 35.m 13.2612 4.1016 1.5648 0.7260 0.3898
3.6179 1.8002 0.9291 0.5000 0.2817 0.1667 0.1036 0.0677 0.0462 0.0330 0.0244
CONCLUSIONS In this paper, we have presented an analytical solution to study the dispersion a shallow channel with non-uniform velocity distribution; and it is confined to the slug is initially unifo~ over the cross-section and it varies harmonically proposed velocity profile can be generalized to fit real velocity distribution
of pollutants in the cases when with time. The in the natural
1510
B. S. MAZUMDER
and R. XIA
stream. An estimation can be made for the axial distance along which the fluctuations in concentration decay in the case of the low frequency input.
REFERENCES [l] G. I. TAYLOR, hoc. R. Sot. Land. A219,186 (1953). [2] G. I. TAYLOR, Proc. R. Sot. Land. A223,446 (1954). [3] H. B. FISCHER, E. J. LIST, R. C. Y. KOH, J. IMBERGER and N. H. BROOKS, Waters. Academic Press, New York (1979). f4] R. ARIS, Proc. R. Sot. Lortd. A235,67 (1956). f5] P. C. CHATWIN, J. Fluid Mech. 43,321 (1970). 161 P. C. CHATWIN. _f. Hvdr. Div. ASCE 106.71 (19801, i7j W. N. GILL and k. SANKARASUBRAGANiAN, i)roc. R. Sot. Land. A316,341 [8] R. SMITH, J. Fluid Mech. 105,469 (1981). [9] N. G. BARTON, J. Fluid Mech. Us,205 (1983). IlO] A. MUKHERJEE and B. S. MAZUMDER, Accu Mechanica 58,137 (1986). 111 A. MUKHERJEE and B. S. MAZUMDER, Acta Mechanica 74,107 (1988). :12] G. F. CARRIER, Q. Appl. Math. 14, 108 (1956). ‘131 P. C. CHATWIN, .I. Fiu~ Mech. 58,657 (1973). ‘141 N. B. BARTON, J. Fluid Mech. 136,243 (1983). 1151 S. T. WANG, A. F. McMILLAN and B. H. CHEN, War. Res. 12,389 (1978). 161 S. FUKUOKA and W. W. SAYRE, J. Hydr. Div. ASCE 99,195 (1973). (Received 26 January 1993; accepted 22 November 1993)
Mixing in Inland and Coastal
(1970).