1. Volume 41
2014
An NRC Research
Press Journal
Une revue de
NRC Research
Press
www.nrcresearchpress.com
Canadian Journal of
Civil Engineering
Revue canadienne de
génie civil
In cooperation with the
Canadian Society for Civil Engineering
Avec le concours de la
Société canadienne de génie civil

2. ARTICLE
Transverse mixing of pollutants in streams: a review
H. Sharma and Z. Ahmad
Abstract: Spilling or release of foreign particles in the flowing water is considered as pollution of water, and due to the inherent
property of water to dissolve the substance, the particulate is well mixed in water. To monitor the extent of pollution in a stream
it is essential to know how the pollutants mix in the river. It is observed that vertical mixing of pollutants is a very rapid process
in the vertical directions and longitudinal mixing occurs very far from source of pollutant, which is generally out of reach of
observations. Thus intermediate or transverse mixing zone is considered very important for water quality modeling. This paper
is an attempt to summarize the phenomenon behind pollutant transport, reduction of three-dimensional advection–dispersion
equation to two-dimensional equation, and factors causing and affecting transverse mixing of pollutants.
Key words: river mixing, transverse mixing, secondary currents, advection–dispersion equation, numerical models.
Résumé : Le déversement ou l’émission de particules étrangères dans l’eau mouvante est considéré comme une pollution de
l’eau et, en raison de la propriété inhérente de l’eau de dissoudre des substances, les particules sont bien mélangées dans l’eau.
Pour surveiller l’étendue de la pollution dans un ruisseau, il est essentiel de connaître la manière dont les polluants se mélangent
dans la rivière. Il a été remarqué que le mélange vertical des polluants est un processus très rapide et que le mélange longitudinal
survient très loin de la source de pollution, laquelle est généralement hors de portée des observations. La zone de mélange
intermédiaire ou transversal est considérée comme étant très importante pour la modélisation de la qualité de l’eau. Le présent
article se veut une tentative de résumer le phénomène a` la base du transport des polluants, la réduction de l’équation tridimen-
sionnelle d’advection–dispersion a` une équation bidimensionnelle et les facteurs causant et affectant le mélange transversal des
polluants. [Traduit par la Rédaction]
Mots-clés : mélange en rivière, mélange transversale, courants secondaires, équation advection–dispersion, modèles numériques.
Introduction
Water is a source for many industrial and domestic needs,
which include from drinking to cooling of reactors. Need of pure
water for these tasks, is as important as management of water
quality. When accidents happen and pollutants are spilled into
the water, they get mixed in the water and pollute it. Understand-
ing the mechanism of mixing of pollutants is necessary for the
management of pollutants control in streams. When any pollut-
ant is spilled into the river, due to the dominance of vertical
turbulence and least dimension of any river system, the pollutant
gets mixed along the depth very quickly. The process of the pol-
lutant getting mixed along the depth is termed as vertical mixing
and longitudinally it spreads up to 50–100 times of the depth
(Yotsukura and Sayre 1976). After getting mixed into the vertical
direction under the action of transverse turbulence and variation
in vertical profiles of transverse velocity, pollutant starts to spread
along the river width and mixes along the cross section of the
river. This transverse spreading and mixing of pollutant under the
action of transverse shear and turbulence is termed as transverse
mixing. If the source of pollutant is unsteady in nature, due to
which pollutant further travels downstream under the action of
longitudinal gradients and this mixing of pollutant under the
action of longitudinal gradients is termed as longitudinal mixing.
As the vertical mixing is a very rapid process and is dominant near
the source of pollutant spill, whereas the longitudinal mixing is of
importance very far from the source so that it makes it unimportant
to model for environmental concern. This paper is a compendium
regarding various processes involved in pollutant mixing in a river
and the process of transverse mixing and factors affecting it.
Physical processes of pollutant transport
If a small quantity of any tracer like dye is injected into the
river, the tracer cloud not only changes its shape but also in-
creases in the volume at the same time as it is carried away down-
stream by the stream flow. The phenomenon of transport and dye
tracer mixing involves various physical processes, which are clas-
sified as advection, diffusion, and dispersion. These processes will
be discussed briefly.
Advection
Advection can be defined as the process where tracer cloud
moves bodily in the flowing stream under the action of imposed
current. Advection helps in transporting any dissolved or sus-
pended tracer away from the fixed source in the downstream,
hence showing that it is an important phenomenon to be consid-
ered while modeling the transport of pollutant in the flowing
media. To quantify the intensity of advection, the term advective
flux is defined as follows (Fischer et al. 1979; Elhadi et al. 1984;
Rutherford 1994):
“The amount of tracer/substance transported per unit time per
unit area perpendicular to the current is termed as advective flux
and is the product of velocity and transported concentration of
tracer/substance.”
Mathematically, advective flux can be formulized as (Elhadi
et al. 1984):
(1) Tad ϭ UC
Received 19 December 2013. Accepted 15 March 2014.
H. Sharma and Z. Ahmad. Department of Civil Engineering, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, 247 667, India.
Corresponding author: Himanshu Sharma (e-mail: smile4anshu@gmail.com).
472
Can. J. Civ. Eng. 41: 472–482 (2014) dx.doi.org/10.1139/cjce-2013-0561 Published at www.nrcresearchpress.com/cjce on 31 March 2014.

3. Here, Tad is the advective transport per unit time per unit area; U
is the velocity perpendicular to unit area; and C is the concentra-
tion of substance/tracer transported.
Molecular diffusion
The spreading of tracer due to the motion of molecules of fluid
is called as molecular diffusion. In laminar flow the transport of
tracer is mainly governed by the molecular diffusion and the mo-
tion of molecules is random but occurs from higher concentration
to lower concentration perpendicular to a unit area. Fick’s law of
diffusion describes the transport of tracer due to molecular mo-
tion based diffusion by a gradient based law, which is as follows
(Fischer et al. 1979; Elhadi et al. 1984; Rutherford 1994):
(2) Tm ϭ Ϫem
ѨC
Ѩn
Here, Tm is the transport by molecular diffusion per unit area per
unit time; n is the direction normal to the unit area; and em is the
molecular diffusion coefficient.
Turbulent diffusion
In any turbulence dominated flow, instantaneous velocity
of a fluid particle can be broken down into a temporally aver-
aged steady component and a fluctuating component. The time-
averaged component is responsible for the advective transport of
the substance while the fluctuating component, which is respon-
sible for the eddying motion in the flow, spreads and transports
the tracer. This spreading and transport of tracer cloud due to the
turbulence generated eddies is called as turbulent diffusion. If ui
is the fluctuating component of velocity in a certain ith direction
and c= is the fluctuating component of tracer concentration, then
the transport of tracer is given by (Fischer et al. 1979; Elhadi et al.
1984; Rutherford 1994):
(3) Tt ϭ u c
Here, Tt is the transport of substance by turbulent diffusion per
unit time per unit area and the bar indicates the temporal aver-
aging. The turbulent diffusion is also random like molecular
diffusion and occurs from higher concentration to lower concen-
tration. Due to this likeliness, the turbulent diffusion can also be
formulated similar to the molecular diffusion, in the form of the
gradient based Fick’s diffusion law as follows:
(4) Tt ϭ Ϫ␧
ѨC
Ѩn
Here, ␧ is the eddy diffusivity. Since similar equations could be
written in other directions as in general the eddy diffusivities
depend on local concentration gradients, hence, varies from loca-
tion to location.
Dispersion
In natural channels, the velocity varies across the channel di-
mensions, i.e., near the wall the velocity will be zero and will
achieve the maxima away from the wall region. Consider Fig. 1 for
instance, when a dye patch is introduced in the flow it is a point
mass. But, as the non-uniformly distributed velocity profile inter-
acts with tracer mass, it gets elongated laterally and also spreads
in the longitudinal direction. Since, the velocity tends to decrease
as the wall area is approached, thus sides of tracer near to wall
advects slower than the central portion of tracer mass, which
distorts the tracer mass. This causes the total longitudinal spread-
ing to be greater than what would result if diffusion acted alone.
Also as the dye spreads laterally, this also generates a lateral con-
centration gradient, which helps in enhancement of diffusive
transport of tracer mass.
The processes of transport are still advection and diffusion but
the non-uniformity of velocity profile helps in enhancing the pro-
cess of mixing of tracer mass. The transport and mixing of sub-
stance under the action of non-uniformity of velocity profile and
shear generated by this profile is called as dispersive transport or
simply as dispersion. Dispersion is assumed to be proportional to
the mean concentration gradient, thus, it is assimilated with the
turbulent diffusion as it is also a gradient based phenomenon.
Hence, mathematically the combined effect of turbulent diffusion
and dispersion can be written as (Elhadi et al. 1984):
(5) Tdisp ϩ Tt ϭ Ϫe
ѨC
Ѩn
Here, Tdisp is the dispersive transport and e is the dispersion coef-
ficient.
Three-dimensional mixing of pollutant in flowing
stream
In any natural stream, when any neutrally buoyant substance is
introduced due to the dominance of vertical turbulence and least
dimension of any river system, the pollutant gets mixed along the
depth very quickly (Fischer et al. 1979; Rutherford 1994). The na-
ture of mixing is three dimensional in nature and is very compli-
cated. Since the vertical mixing is the process which is very much
dominant near the source of tracer release, hence it is also called
as “near field mixing”. At the commencement of the transverse
mixing, vertical mixing usually ends, thus the mixing in this re-
gion becomes two dimensional. Since the field of this two-
dimensional mixing is not too far from the source of tracer
Fig. 1. Transverse dispersion by meander currents (Rutherford
1994).
Sharma and Ahmad 473
Published by NRC Research Press

4. release, hence is termed as “mid-field mixing” (Fischer et al. 1979;
Rutherford 1994). If the source of pollutant is unsteady in nature,
due to which pollutant further travels downstream under the
action of longitudinal gradients and this mixing of pollutant un-
der the action of longitudinal gradients is termed as longitudinal
mixing. The mixing in this zone is one dimensional in nature as
tracer has now become fully mixed along vertical and lateral di-
rections. Due to the farther reach of this zone from source of
tracer release, this mixing is called as “far-field mixing” (Fischer
et al. 1979; Rutherford 1994). The three dimensional unsteady
tracer transport equation is as follows (Fischer et al. 1979;
Rutherford 1994):
(6)
ѨC
Ѩt
ϩ
Ѩ(uC)
Ѩx
ϩ
Ѩ(vC)
Ѩy
ϩ
Ѩ(wC)
Ѩz
Ç
1
ϭ
Ѩ
Ѩxͫ(ex ϩ ␧xx)
ѨC
Ѩxͬϩ
Ѩ
Ѩyͫ(ey ϩ ␧yy)
ѨC
Ѩyͬϩ
Ѩ
Ѩzͫ(ez ϩ ␧zz)
ѨC
Ѩz ͬ
2
Here, t is the time; u, v, and w are the components of instanta-
neous velocity in x, y, and z directions; ex, ey, and ez are the local
dispersion coefficients in x, y, and z directions; and ␧xx, ␧yy, and ␧zz
are the turbulent diffusion coefficients in x, y, and z directions.
Terms 1 and 2 in the equation respectively represent the advective
and diffusive transport of substance. As the three-dimensional
advection–dispersion equation shows the true nature of pollutant
transport in any stream but certain restrictions are associated
with it like misalignment of coordinate system, generation of
extra curvature terms in curvilinear coordinate systems, spatial
variation of flow, river bank impingement of tracers, dead zones
in rivers, etc. Hence it is compatible to reduce the dimensionality
of the equation. Thus, for the ease of calculations, omission of the
transport component in any particular direction can be done by
averaging in that pre-described direction and dimensionality of
three-dimensional equation can be reduced to two or one.
Two-dimensional mixing of pollutant in flowing
stream and equation of transverse mixing
In general flow systems, the width of rivers are very large com-
pared to their depths and also in any flow system there is domi-
nance of vertical turbulence in flow. Thus, mixing of solute in
vertical direction is very rapid process and is predominant in the
region near to the effluent source. Thus, averaging is done in the
vertical direction and the terms which contain differential of ver-
tical axis are omitted as the quantities have become uniform
along the vertical axis. The final format of equation (for the com-
plete derivation see Appendix A) is as follows:
(7)
Ѩ(hC)
Ѩt
ϩ
Ѩ(huC)
Ѩx
ϭ
Ѩ
Ѩx͑hEx
ѨC
Ѩx͒ϩ
Ѩ
Ѩy͑hEy
ѨC
Ѩy͒
Here, Ex and Ey are the depth averaged dispersion coefficients in
longitudinal and lateral directions. The above equation is called as
depth-averaged advection dispersion equation and is known to be
the governing equation of transverse mixing. If the tracer source
is continuously supplying the tracer, i.e., steady tracer flow, and
assuming that longitudinal diffusion of tracer has not started yet
for the uniformly flowing stream, thus time differentiation of
eq. (7) will be zero and term “h” will come outside of the differen-
tial making the third term of eq. (7), as longitudinal diffusion has
not begun yet. Thus, the equation here is found in many of the
literature (Lau and Krishnappan 1977; Lau and Krishnappan 1981;
Demetrocopolous 1994; Ahmad 2008; etc.) and is known as steady
state transverse mixing equation:
(8) u
ѨC
Ѩx
ϭ Ey
Ѩ2
C
Ѩy2
It is clearly understandable that mechanisms accounting for the
lateral movement of substance are transverse turbulent diffusion
and transverse dispersion. But in the majority of the literature
(Lau and Krishnappan 1977; Lau and Krishnappan 1981; Boxall and
Guymer 2003; Boxall et al. 2003; Albers and Steffler 2007; etc.)
reveals that transverse dispersion, i.e., non-uniform distribution
of transverse velocity in vertical direction, is a major phenome-
non responsible for the lateral mixing of solutes. Since many
factors affect these two processes, i.e., diffusion and dispersion,
basic mechanisms will be briefly discussed in the following sec-
tion.
Mechanisms causing transverse mixing
Transverse turbulent diffusion
According to the Prandtl’s mixing length theory (Rodi 1980;
Nezu and Nakagawa 1993):
(9) ␧zz ϭ ℓm
2
|ѨU
Ѩz
|
Here, ℓm is the mixing length; z is the vertical distance; and U is
the average velocity of flow. This hypothesis predicted the results
very well for plane shear flows. Now, an analogy was drawn just
similar to eq. (9) and was written as follows (Rutherford 1994):
(10) ␧yy ϭ ℓt
2
|ѨU
Ѩy
|
Here, y is the transverse distance; ℓt is the transverse turbulence
length scale; and ␧yy is the transverse turbulent diffusion coeffi-
cient. Since in plane shear flow the velocity across the stream does
not vary, hence, the transverse diffusivity automatically vanishes.
But, investigators (Lau and Krishnappan 1977; Lau and Krishnappan
1981; Holley and Nerat 1983; Boxall et al. 2003; Boxall and Guymer
2003; etc.,) have observed that transverse dispersion does occur in
the channels. Hence, similar to Prandtl’s mixing length theory,
i.e., turbulent diffusivity is equal to product of a turbulent length
scale and turbulent velocity scale (Pope 2000; Tennekes and
Lumley 1972; etc.), i.e.:
(11) ␧yy ϭ ℓtut
Here, ut is the velocity scale of turbulence. But the major flaw with
eq. (11) is that we have accounted for the transverse turbulence
scale for the calculation of diffusion coefficient but there is no
clear way to calculate and evaluate this scale length as there is no
plausible reason to explain the rotation of bed generated vertical
eddies to transverse direction (Rutherford 1994). Thus to maintain
the relation of transverse diffusion with the flow parameters for
the lt = h, i.e., depth of flow was chosen and for velocity scale ut =
u‫,ء‬ shear velocity was chosen as it is the main agent which gener-
ates eddying motion in flow.
Transverse dispersion
Dispersion as discussed before is a phenomenon that occurs due
to non-uniformity of velocity profile. It is also a known fact that
the variation of transverse velocity along the vertical is identical
to the variation of longitudinal velocity. Thus, the tracer near to
the water surface moves faster in comparison to the tracer present
near the bed. This causes generation of vertical gradients and
hence promoting vertical mixing. This increment in transverse
mixing due to the non-uniformity in velocity profile is called as
474 Can. J. Civ. Eng. Vol. 41, 2014
Published by NRC Research Press

5. transverse dispersion. There are two main components that con-
tribute in the transverse dispersion, they are meander currents
and secondary currents.
Meander currents
In any natural channel the main flow meanders from one bank
to another bank by depending on the channel cross section. These
meandering currents move uniformly along the depth and thus
are responsible for transverse advection of tracer. Considering,
Fig. 1., at the initial section, say Section A, plume is contained
within the flow, q, and in the remaining section there is no tracer
concentration and clearly the tracer cloud is near to the true left
bank of the channel. As the flow moves forward, the tracer advects
and reaches the section B. From, Fig. 1., it can be clearly seen that
the channel is shallower near the left bank and due to the bed
friction the velocity is lower near the left bank. To avoid large bed
friction the main flow shifts towards the right bank and hence
carries the plume towards the right bank of the natural channel.
Further advection of tracer causes the plume cloud to reach
section C where the depth at the left bank again becomes deeper
than the right bank, thus causing main flow to meander towards
the left bank and hence making the plume to contract towards the
left bank.
Now, in eq. (7) we assume that there is no transverse component
of velocity, but clearly between the section A and section B there
is a positive component of velocity acting on the tracer plume and
hence expanding it, but in between section B and section C the
shifting of currents from the right bank to left bank causes cre-
ation of negative transverse velocity component, thus contracting
the plume. Due to the variation of transverse velocity across the
width, some transverse mixing (represented as e in Fig. 1.) occurs.
Although, these meandering currents are not a very important
component of transverse mixing phenomenon, however if consid-
ering channels of irregular cross-sections these currents have to
be considered otherwise, as reported by Holley et al. (1972); Elhadi
et al. (1984), etc, an over-estimation of diffusion coefficients or
negative diffusion coefficients are observed.
Secondary currents
In the natural channel there are some sections in which the
transverse velocity is highly non-uniform, e.g., bends in the chan-
nels, flow around obstructions, etc. These non-uniformities in the
transverse velocity generate currents prominent in directions
other than the primary direction of flow, i.e., longitudinal direc-
tion, hence are termed as secondary currents. From the literature,
(Fischer 1969; Holley et al. 1972; Lau and Krishnappan 1981; Holley
and Nerat 1983; Boxall and Guymer 2003; Boxall et al. 2003; etc.) it
is evident that secondary currents in the bends tends to disperse
the tracer apart in the opposite directions at the different depths
causing the transverse mixing to happen rapidly in bends. It was
observed that transverse mixing in a natural river is a prominent
function of secondary currents and due to irregular cross-sections
and bends in the natural channels the equilibrium between the
vertical velocity shear of transverse velocity and vertical turbulent
diffusion is generally not able to co-exist. Transverse mixing
coefficient is greatly affected by secondary currents. Transverse
mixing coefficient was studied by many investigators through
experimental and numerical studies, thus a few investigator’s
studies are reported in Table 1. From the upcoming section vari-
ous models available for the study of transverse mixing equation
will be discussed in detail.
Analytical models for analysis of transverse mixing
It is evident that for pollutant transport modeling, modeling of
transverse mixing is an important phenomenon. Thus to model
the transverse mixing process, solution of transverse mixing
equation (eq. (7)) has to be obtained. Thus to obtain the solution of
the transverse mixing equation there are two methods, viz.:
Constant coefficient method
Consider eq. (7) for the source supplying the tracer at constant
steady rate. Thus the temporal differential, i.e., ѨC/Ѩt will become
zero and hence eq. (7) will get transformed in the following for-
mat:
(12)
Ѩ(huC)
Ѩx
ϭ
Ѩ
Ѩx͑hEx
ѨC
Ѩx͒Ç
1
ϩ
Ѩ
Ѩy͑hEy
ѨC
Ѩy͒Ç
2
Now it is a known fact that longitudinal mixing usually does not
interfere with the transverse mixing, hence the term (1) in eq. (12)
becomes zero, thus eq. (12) reduces to the following format:
(13)
Ѩ(huC)
Ѩx
ϭ
Ѩ
Ѩy͑hEy
ѨC
Ѩy͒
Now as the method suggests, it is assumed that variation in depth,
Ey is negligible and flow is uniform throughout the flow regime as
the channel is assumed to be straight, thus flow depth, Ey and u are
taken out of their respective brackets. Now, the eq. (13) reduces to
u
ѨC
Ѩx
ϭ Ey
Ѩ2
C
Ѩy2
Equation above is the same as eq. (8), hence is called as govern-
ing equation of constant coefficient method. Now given next are
Table 1. Transverse mixing coefficient observed by various investigators.
Site No. Investigators
Range of transverse
mixing coefficient (ey/u‫ء‬h) Remarks
1 Holley and Abraham (1973) 0.15–0.16 Without groins
0.4–0.5 With groins
2 Beltaos (1980) 0.22–0.42 For meandering channel
3 Holley Jr. and Nerat (1983) 0.5–2.5 River measurement
4 Demetracopoulos and Stefan (1983) 0.24–4.65 Temperature profile measurement
5 Webel and Schatzmann (1984) 0.175–0.177 Smooth bed
0.130–0.159 Rough bed
6 Bruno et al. (1990) 0.21–1.89 Buoyant tracer
7 Chau (2000) 0.14–0.21 Smooth bed
0.14–0.21 Sandpaper
0.13–0.19 Steel mesh
0.15–0.24 Small stones
8 Seo et al. (2006) 0.23–1.21 For meandering river
9 Baek and Seo (2010) 0.21–0.91 Validation of stream tube routing method
Sharma and Ahmad 475
Published by NRC Research Press

6. analytical solution for a few sets of tracer sources, which are
helpful in analyzing the transverse mixing process.
Vertical line source
Assume a steady vertical line source at x = 0, y = yo. Now at both
banks the transverse tracer flux is zero, thus it comply:
(14)
ѨC
Ѩy
ϭ 0 at y ϭ 0 and y ϭ B
Thus for this vertical line source analytical solution for un-
bounded flow condition is given as
(15) C(x, y) ϭ
M
.
H͙4␲Eyux
expͫϪ
u͑y Ϫ yo͒2
4Exx ͬ
where M is the mass inflow rate. Since eq. (8) and eq. (14) are both
linear in nature, thus method of images can be applied to solve
the problem in bounded flow conditions.
Vertical line source over a width (plane source)
For prismatic channels of arbitrary cross-sectional shape,
Yotsukura and Cobb (1972) developed a method called as cumula-
tive discharge model, which provides the analytical solution of
transverse mixing equation where transverse velocity is zero. The
transformed equation in x-q coordinate is given by
(16)
ѨC
Ѩx
ϭ
Ѩ
Ѩqͫh2
uEy
ѨC
Ѩqͬ
Now investigators like Yotsukura and Cobb (1972), Yotsukura
and Sayre (1976), Lau and Krishnappan (1977), etc. have showed
that for practical purposes the term h2uEy can be taken as constant
and equals the average value across the cross section with respect
to q, that is
(17) Dy ϭ h2
uEy
This term Dy is known as diffusion factor. Thus eq. (16) will reduce
to the following format:
(18)
ѨC
Ѩx
ϭ Dy
Ѩ2
C
Ѩq2
Now the boundary conditions for eq. (18) will be that there is no
flux of pollutant across the flow boundaries and hence
(19)
ѨC
Ѩq
ϭ 0 at q ϭ 0 and q ϭ Q
Here, Q is the stream discharge. Cumulative discharge method
does take into account the channel curvature, effect of changes
in depth and width, etc. Yotsukura and Cobb (1972) and Lipsett
and Beltaos (1978) provided an analytical solution for eq. (18)
and assimilating the boundary conditions of eq. (19) with it
for a plane vertical source of length (␻ = y2 – y1) and it is as
follows:
(20)
C
C∞
ϭ
1
2␻
΄
1
2ͭerfͩ␩2 ϩ ␩
͙2␰
ͪϩ erfͩ␩2 Ϫ ␩
͙2␰
ͪϪ erfͩ␩1 ϩ ␩
͙2␰
ͪϪ erfͩ␩1 Ϫ ␩
͙2␰
ͪͮ
ϩ͚mϭ1
∞
erfͩ␩2 ϩ 2m Ϫ ␩
͙2␰
ͪϩ erfͩ␩2 Ϫ 2m Ϫ ␩
͙2␰
ͪϪ erfͩ␩1 ϩ 2m Ϫ ␩
͙2␰
ͪ
Ϫerfͩ␩1 Ϫ 2m Ϫ ␩
͙2␰
ͪϩ erfͩ␩2 ϩ 2m ϩ ␩
͙2␰
ͪϩ erfͩ␩2 Ϫ 2m ϩ ␩
͙2␰
ͪ
Ϫerfͩ␩1 ϩ 2m ϩ ␩
͙2␰
ͪϪ erfͩ␩1 Ϫ 2m ϩ ␩
͙2␰
ͪ
΅
Here, ␩ = y/B (here B is the width of the stream); ␰ = 2Dyx/Q2; C∞ is
the fully cross-sectional mixed concentration of pollutants, m de-
notes the number of images, and erf denotes the error function.
To choose the value of m, the value should be large enough so that
deviation of concentration lies around 5% of its mean value
(Fischer et al. 1979).
Slug injection
For neutrally buoyant tracer released as a slug into any rectan-
gular stream with constant longitudinal velocity, the concentra-
tion distribution is provided by the analytical solution developed
by Shen (1978), which is as follows:
(21) C(x Ϫ ut, q, t) ϭ
1
2Q͙␲Ext ͚mϭ0
∞
␶m cosͩm␲Qa
Q
ͪ
× expͫϪ
(x Ϫ ut)2
4Ext
Ϫ Ey
m2
␲2
u2
h2
t
Q ͬcos͑m␲q
Q ͒
Here, ␶m = 1, when m = 0 and ␶m = 2 when m ≠ 0; Qa = cumulative
discharge at the location of the point source.
Deficiencies of constant mixing coefficient method
The constant transverse mixing coefficient model allows the
mixing equation to be solved analytically and greatly enhances
the problem solving. In spite of this fact, the thing that restricts
the use of the constant mixing coefficient method is the complex
bathymetry and velocity distributions. Since, the major assump-
tion of the model is that the depth, velocity, and transverse dis-
persion coefficient are constant along and across the channel,
however in the natural channels the depth and velocity are rather
constant but vary across the width, hence the main assumption is
invalidated from the real time observations. As the depth and
velocity vary across the width it will definitely make the trans-
verse dispersion coefficient vary. Thus, from these problems aris-
ing from the invalid basic assumptions it is a requirement to shift
towards a more realistic and heuristic approach to analyze the
field problem of transverse mixing. Thus, it gave rise to a variable
mixing coefficient approach, which will be discussed further.
Stream tube model (variable coefficient method)
Since the flaws discussed above in the constant mixing model
are of such an extent that they give rise to the magnitude of
476 Can. J. Civ. Eng. Vol. 41, 2014
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7. transverse mixing coefficients which are not realistic like Holley
et al. (1972) who observed the negative value of mixing coefficient
in the area of high separation when calculated while considering
the mixing coefficient to be constant along and across the flow
field. Also in a realistic scenario the velocity, depth, and cross
section vary in the flow region, which the constant mixing coeffi-
cient method does not take into account. It is a known fact that in
general the bathymetry of channel is not known in advance and
nor is the value of transverse mixing coefficient.
Thus, Yotsukura and Cobb (1972) and Yotsukura and Sayre (1976)
introduced a method called as “cumulative discharge method”
which was based on orthogonal curvilinear system and is defined as
(22) q(␣, ␤) ϭ ͵␤ϭ0
␤
m␤hv␣d␤
Here, ␣ and ␤ are the longitudinal and transverse coordinates in
an orthogonal curvilinear system; ␣ is parallel with the local
depth-averaged longitudinal velocity and ␤ is transverse across
the flow; q is the cumulative discharge; m␤ is the metric coefficient
which accounts for channel curvature and lies between 0.8–1.2 for
meandering channels and 1.0 for straight channels; v␣ is the depth
averaged velocity. By convention q = 0 is taken for left bank and q =
Q for right bank. For an unsteady tracer source in steady flow, the
tracer transport equation is as follows (Rutherford 1994):
(23)
ѨC
Ѩt
ϩ
v␣
m␣
ѨC
Ѩ␣
ϭ
1
hm␣m␤
Ѩ
Ѩ␣ͩm␤
m␣
hE␣
ѨC
Ѩ␣ͪϩ
v␣
m␣
Ѩ
Ѩq͑m␣h2
v␣E␤
ѨC
Ѩq͒
Here, E␣ and E␤ are the dispersion coefficients in ␣ and ␤ direc-
tions, which are in longitudinal and transverse direction.
Now, Yotsukura and Sayre (1976) assumed that source of pollu-
tion is a steady source, thus eq. (23) will reduce to
(24)
ѨC
Ѩ␣
ϭ
Ѩ
Ѩq͑m␣h2
v␣E␤
ѨC
Ѩq͒
Now, defining factor of diffusion as
(25) Dy ϭ m␣h2
v␣E␤
Since, at the banks the longitudinal velocity and depth of flow is
very low, thus the factor of diffusion varies across the channel. For
convenience Dy can be written as
(26) Dy ϭ ␺H2
UEy
Here, ␺ is a dimensionless shape factor that lies in the range of
1.0–3.6 (Beltaos 1980) and is given by
(27) ␺ ϭ
1
B ͵yϭ0
B
͑h
H͒
2 u
U
dy
Now, several investigators (such as Yotsukura and Cobb 1972;
Sayre 1979; Almquist and Holley 1985, etc.) assumed that variation
of Dy can be taken to be uniform and constant over the flow field,
thus taking eq. (18) with the boundary conditions defined in
eq. (19) analytical solution of eq. (18) for unbounded flow with
source placed at x = 0 and q = qo is given by
(28) C(x, q) ϭ
M
͙4␲Dyx
expͫϪ
(q Ϫ qo)2
4Dyx
ͬ
By utilizing method of images the effect of boundaries can be
accommodated in the solution.
Numerical solution of transverse mixing equation
Analytical solution, although roughly provides an approximate
picture of the flow and process of transverse mixing, are not suffi-
ciently accurate as they are evolved due to over-simplification of
mass conservation equation. Thus, to provide more precise and gen-
eralized solution, numerical methods are employed. Though, nu-
merical methods require more effort and data to operate than the
simple analytical solution, they provide a much wider picture of
process than doing preliminary calculations using simple analytical
solutions. Though there are many numerical solutions, a few are
provided below:
Lau and Krishnappan (1981)
Lau and Krishnappan (1981) utilized the method developed by
Stone and Brian (1963) to solve the advection–diffusion type equa-
tions of the following format:
(29)
ѨC
Ѩx
ϩ V
ѨC
Ѩ␩
ϭ D
Ѩ2
C
Ѩ␩2
Here,
(30) V ϭ Ϫ
1
Q2
ѨDy
Ѩ␩
and
(31) D ϭ
1
Q2
Dy
Stone and Brian (1963) discretized the eq. (29) to obtain the
following finite difference analogue:
(32)
1
⌬xͫg(Ciϩ1,j Ϫ Ci,j) ϩ
␪
2
(Ciϩ1,jϩ1 Ϫ Ci,jϩ1) ϩ m(Ciϩ1,jϩ1 Ϫ Ci,jϩ1)ͬϩ
Vi,j
⌬␩ͫa(Ci,jϩ1 Ϫ Ci,j) ϩ
␧
2
(Ci,j Ϫ Ci,jϩ1) ϩ b(Ciϩ1,jϩ1 Ϫ Ci,ϩ1,j)
ϩ d(Ciϩ1,j Ϫ Ci,ϩ1,jϩ1)ͬϭ
Di,j
2(⌬␩)2
[(Ci,jϩ1 Ϫ 2Ci,j ϩ Ci,jϩ1) ϩ (Ciϩ1,jϩ1 Ϫ 2Ciϩ1,j ϩ Ciϩ1,jϩ1)]
In this scheme, Stone and Brian (1963) used weights or
weighing coefficients, a, ␧/2, b, and d to approximate the deriv-
ative ѨC/Ѩ␩; and to approximate the derivative ѨC/Ѩx, they used
g, ␪/2, and m as weighing coefficients. They implied the follow-
ing conditions on weighing coefficients which they have to
satisfy:
(33) a ϩ
␧
2
ϩ b ϩ d ϭ 1
and
(34) g ϩ
␪
2
ϩ m ϭ 1
Sharma and Ahmad 477
Published by NRC Research Press

8. Lau and Krishnappan (1981) did not use any weighing factors for
approximating the second derivative Ѩ2
C/Ѩ␩2
. Crank-Nicolson
scheme was utilized by Lau and Krishnappan (1981) to discretize
equation eq. (29). By optimizing the weighing coefficients to sat-
isfy eq. (33) and eq. (34), though they give solutions which nearly
matches the analytical solution, they obtained the following val-
ues of weighing coefficients:
(35) g ϭ
2
3
; ␪ ϭ m ϭ
1
6
; a ϭ b ϭ d ϭ
␧
2
ϭ
1
4
They suggested using the aforesaid values of weighing coefficients
when the values of V and D are not constant. If the concentration
distribution at any station say i is known, then to obtain the value
of concentration distribution at any unknown level say i + 1 has to
be calculated for known boundary condition and zero flux condi-
tions at side walls, thus Lau and Krishnappan (1981) proposed the
following matrix notation for eq. (32):
(36)
΄
q2 Ϫr2
Ϫp3 q3 Ϫr3
Ϫp4 q4 Ϫr4
Ì Ì Ì
ϪrMϪ1
ϪpM ϪqM
΅΄
Ciϩ1,2
Ciϩ1,3
Ciϩ1,4
Ciϩ1,MϪ1
Ciϩ1,M
΅ϭ
΄
S2
S3
S4
SMϪ1
SM
΅
Here,
qj ϭ
g
⌬x
ϩ Di,j
1
(⌬␩)2
ϩ Vi,j
1
⌬␩
(d Ϫ b), (j ϭ 2, 3...M)
pj ϭ Ϫ
␪
2
1
⌬x
ϩ Di,j
1
2(⌬␩)2
ϩ Vi,j
1
⌬␩
d, (j ϭ 2, 3...M)
rj ϭ Ϫ
m
⌬x
ϩ Di,j
1
2(⌬␩)2
Ϫ Vi,j
1
⌬␩
b, (j ϭ 2, 3...M Ϫ 1)
(37) Sj ϭ Ci,jϪ1ͫ␪
2
1
⌬x
ϩ Di,j
1
2(⌬␩)2
ϩ Vi,j
␧
2⌬␩ͬ
ϩ Ci,jͫ g
⌬x
Ϫ Di,j
1
(⌬␩)2
Ϫ Vi,j
␧
2⌬␩ͬ
ϩ Ci,jϩ1ͫm
⌬x
ϩ Di,j
1
2(⌬␩)2
Ϫ Vi,j
a
⌬␩ͬ, (j ϭ 2, 3...M)
The equations created during the solution were solved with the
help of Gauss-Seidal method by Lau and Krishnappan (1981).
Luk et al. (1990)
Luk et al. (1990) developed a two-dimensional model to solve the
transient two-dimensional mixing equation to simulate transport
of pollutant in natural river. They utilized the stream-tube con-
cept to derive their method called as MABOCOST (mixing analysis
based on concept of stream tube) by solving the following equa-
tion:
(38)
ѨC
Ѩt
ϩ
U
mx
ѨC
Ѩx
ϭ
Ѩ
Ѩq
ͩh2
U2
Ey
Q2
ѨC
Ѩq
ͪϩ ␣C ϩ ␤
Here, ␣ is the decay constant of pollutant and ␤ is any source or
sink present in pollutant mixing zone. First of all Luk et al. (1990)
decided the selection of numerical grid in which they selected the
simplest asymmetrical, first order explicit scheme that is quite
stable, non-diffusive, and non dispersive when courant number
(Cr) is set as unity. Courant number is defined as follows:
(39) Cr ϭ
u⌬t
⌬x
Here, ⌬t and ⌬x are the time step and longitudinal grid size. Since,
Luk et al. (1990) found that the finite difference scheme they are
using is stable only when Courant number is unity, hence replac-
ing Cr = 1 in eq. (39), they obtained the following equation to set
the grid size and time step for finite difference grid that will be
used to solve eq. (38):
(40) u⌬t ϭ ⌬x
Luk et al. (1990) utilized split operator scheme to solve the eq. (38).
They first solved for the advection part of the eq. (38) and they
argued that since the grid was constructed on the basis that it
takes a unit time step for pollutant to advect from one element to
another, hence they obtained the following equation by advecting
the pollutant to next element in temporal and spatial grid:
(41) C(i, j, t ϩ ⌬t) ϭ C(i, j Ϫ 1, t)
Here, i is the number of stream tubes and j is the number of
element in the stream tube. By following the step given by eq. (41),
Luk et al. (1990) allowed concentration of pollutant to get dis-
persed laterally to each adjacent element by utilizing the follow-
ing relationship:
(42) C(i, j, t ϩ ⌬t) ϭ C(i, j, t) ϩ
⌬t
␪ ͚pϭ1
m
͑l⌬xlh⌬yDy⌬C͒p
⌬ql
2
ϩ
(l⌬xl)pԽ(h⌬y)p Ϫ h⌬yԽ⌬Cp
⌬ql
2
ϩ ͚pϭ1
k
͑r⌬xrh⌬yDy⌬C͒p
⌬qr
2
ϩ
(r⌬xr)pԽ(h⌬y)p Ϫ h⌬yԽ⌬Cp
⌬qr
2
Here, Dy = H2U2Ey/Q2 is the dispersion/diffusion factor; ␪ is the
volume of a mesh element; l and r are the fractions of overlapping
area between an element and its adjacent element on the left and
right sides, respectively; ⌬x is the elemental length; ⌬y is the
elemental width; ⌬C is the concentration excess; and ⌬n is the
fraction of cumulative discharge between the centres of two ele-
ments. The subscript p represents the quantities for the pth adja-
cent element, and the subscripts l and r represent the quantity of
the left and right adjacent elements, respectively. Overbars repre-
sent quantities averaged between an element and its adjacent
elements. To simulate the decay of concentration of pollutant,
Luk et al. (1990) considered the kinetics of chemical decay of pol-
lutants to be of first order and applied the following step to sim-
ulate the observations:
(43) C(i, j, t ϩ ⌬t) ϭ C(i, j, t) × exp(␣⌬t)
Finally Luk et al. (1990) added the, if possible, source or sink terms
for the current time step to each individual element. These pro-
cesses were repeated for all the time steps to obtain the concen-
tration profiles at each time step for all elements. Luk et al. (1990)
by performing various experiments observed that by interchang-
ing the order of adding various processes that were being added in
MABOCOST did not affect the quality of results in a considerable
manner.
478 Can. J. Civ. Eng. Vol. 41, 2014
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9. Ahmad (2008)
Ahmad (2008) developed a finite volume model to study the
transverse mixing in steady state streams. Ahmad (2008) inte-
grated numerically the steady state transverse mixing equation.
Ahmad (2008) transformed the equation by taking the cross-
sectional average into the following format:
(44)
ѨC
Ѩx
ϭ Dy
Ѩ2
C
Ѩq2
Here,
(45) Dy ϭ h2
uey
Since according to stream-tube model there should be no flux of
pollutants across the flow, hence Ahmad (2008) provided the fol-
lowing boundary conditions:
(46)
ѨC
Ѩx
ϭ 0 at q ϭ 0 and q ϭ Q
To compare the variable ey and constant ey methods, Ahmad (2008)
took the average value of ey/u‫ء‬h = 0.225 for constant mixing coef-
ficient method. Ahmad (2008) solved the steady state transport
equation numerically by incorporating the upwind scheme of fi-
nite difference, finally obtained the following equation for a pe-
culiar node P (i, j):
(47) as
j
Ci
jϩ1
ϩ ap
j
Ci
j
ϩ an
j
Ci
jϩ1
ϭ aw
j
Ciϩ1
j
ϩ bj
Here as, an, ap, and aw are the coefficients of concentration; Ci
j
is the
concentration at node (i,j). Also,
(48a) as
j
ϭ a3
j
(48b) ap
j
ϭ ͑a1
j
ϩ a2
j
ϩ a3
j
͒
(48c) an
j
ϭ a2
j
(48d) aw
j
ϭ a1
j
(48e) bj
ϭ 0
(48f) a1
j
ϭ
hjuj⌬y2
⌬x
(48g) a2
j
ϭ ͩhj
ϩ hjϩ1
2
ͪͩey
j
ϩ ey
jϩ1
2
ͪ
(48h) a3
j
ϭ ͩhj
ϩ hjϩ1
2
ͪͩey
j
ϩ ey
jϩ1
2
ͪ
Ahmad (2008) observed that while solving the system of equation
generated for different i and j from eq. (47), a tri-diagonal matrix
was formed. Thomas algorithm or tri-diagonal matrix algorithm
(TDMA) was utilized by Ahmad (2008) to solve the tri-diagonal
matrix iteratively and line by line until a converging value of
concentration was obtained.
Effect of secondary currents on the transverse
mixing
It is clear from the previous section which states that secondary
currents positively affects the process of transverse mixing by
generating secondary currents which helps flow to spread tracer
laterally in a faster manner. Fischer (1969) was one of the first to
study the effect of bend generated secondary currents on the
process of dispersion. He also observed that predicted mixing
coefficient and observed coefficients almost varied within the fac-
tor of two and the Ey to HU‫ء‬ ϳ 10 times to what it was observed in
straight reach of the channel. Yotsokura et al. (1970) also con-
cluded that in natural river systems where due to change in chan-
nel cross-sections, meandering, etc., secondary currents were
generated and caused the enhanced mixing of dye. Chang (1971) in
his work concluded that phenomenon of transverse mixing in
meandering channel is closely related to the generation and decay
of bend generated secondary currents. Holley et al. (1972) also
considered effects of bend on transverse dispersion and observed
that as the curvature of bends change, it tends to generate a net
transverse velocity.
Holley and Abraham (1973) observed that there was rise in value
of lateral mixing coefficient due to the generation of secondary
currents in flow by groins. Ward (1974) concluded that increase in
depth or change in radius of curvature does affect the structure of
turbulence and secondary currents, thus, affects the transverse
mixing phenomenon. Krishnappan and Lau (1977) observed that
variation of depth and secondary currents played important roles
in transverse dispersion. They also observed that transverse dis-
persion and convective transport of pollutant were both of equal
magnitude and were equally liable for pollutant transport across
the width of river.
Krishnappan and Lau (1981) deduced that regardless of aspect
ratio sinuosity, which is a representative of curvature of channel,
also plays a role in the enhancement of transverse mixing coeffi-
cient due to the generation of secondary currents (Fig. 2 and 3).
Holley and Nerat (1983) measured the transverse mixing coeffi-
cient in rivers and observed that secondary currents enhance the
transverse spreading of pollutant. Demuren and Rodi (1986) ex-
perimentally studied the process of dispersion in curved and me-
andering channels and they observed that secondary motions in
the meanders caused considerable lateral spreading of pollutants.
Boxall and Guymer (2003) conducted experimentation in mean-
dering channel and observed that after dye was mixed vertically
due to variation of transverse velocities in vertical direction, dye
mixing was observed to speed up due to the action of strong
secondary currents of bend.
Seo et al. (2006) studied the transverse mixing phenomenon
under the slug test conditions and they observed that along the
bend it was observed that the core of tracer cloud shifted towards
the outer bank while traversing downstream due to the secondary
currents. It was observed that enhanced transverse mixing was
obtained due to the presence of noticeable secondary currents in
the flow. Dow et al. (2009) observed that due to secondary currents
generated by expansion and contraction of river cross sections,
protruding of stones, etc., enhanced transverse mixing was ob-
tained. Baek and Seo (2010) on the analysis of observations saw
that as the bend starts, the flow has high intensity of bend gener-
ated secondary currents and thus due to these secondary currents
the lateral spreading of tracer was observed to increase.
Effect of aspect ratio on the transverse mixing
Lau and Krishnappan (1977) experimentally observed the effect
of altering the aspect ratio on secondary currents by (1) changing
width of flow region and keeping flow depth constant and (2)
changing depth of flow and keeping width constant. They ob-
served that due to the change in turbulence structure and stress
distribution, secondary currents are generated which enhanced
the transverse mixing. Webel and Schatzmann (1984) argued that
except near the vicinity of walls, transverse mixing is indepen-
dent of the aspect ratio. Biron et al. (2004) while conducting ex-
perimentations on concordant and discordant beds observed that
Sharma and Ahmad 479
Published by NRC Research Press

10. lateral mixing was faster in the discordant bed than in the con-
cordant bed. While plotting the normalized transverse dispersion
coefficient with aspect ratio (width–depth ratio), it was observed
by Seo et al. (2006) reported that with the increase in the aspect
ratio the normalized transverse mixing coefficient also increased
as the strength of secondary currents was observed to increase
with the increase in aspect ratio.
Effect of bed roughness on the transverse mixing
Chang (1971) suggested that increasing roughness enhanced the
transverse mixing due to increment in bed friction while Lau and
Krishnappan (1977) observed that observations relied on a single
curve and inferring that friction factor had a little effect on the
parameter. Webel and Schatzmann (1984) observed that by non-
dimensionalizing lateral mixing coefficient with flume velocity
(Ey/Uh), it was observed that it increases by increasing bed friction
while parameter Ey/u‫ء‬h was observed to decrease with increasing
friction.
Demuren and Rodi (1986) conducted experiments on the wide
channel (aspect ratio, B/h = 20) and two narrow channels (B/h ≈ 5);
one having smooth bed and other having rough bed. They ob-
served that the process of lateral mixing for the channel with
rough bed was predominantly led by bed generated turbulence
rather than secondary currents. For limiting case of buoyancy flux
(Bo/hu‫ء‬3), Bruno et al. (1990) observed that transverse mixing rate
was higher for rough bed than smooth bed. Chau (2000) also ob-
served that under a constant aspect ratio by varying the bottom
roughness did not change the dimensionless transverse mixing
coefficient by a considerable amount.
Effect of variability of the transverse mixing
coefficient on transverse mixing
Holley et al. (1972) assumed three cases in variability of trans-
verse mixing coefficient across the cross section and observed
considerable difference between the concentration profiles ob-
tained by all three cases and inferred that assumed variation of Ey
has an effect on the Ey computed from measured concentration
profile. Lau and Krishnappan (1981) considered five cases for the
variation of transverse mixing coefficient across the cross section
of the natural channel and observed in three of the cases that
where the values are allowed to vary across the cross section, the
simulated results matched well with analytical solution and ob-
servations, while for two cases the matching was not proper. Thus,
Lau and Krishnappan (1981) suggested using the model where quan-
tities affecting dispersion are allowed to vary. Demetracopoulos
(1994) also seconded the observations of Lau and Krishnappan
(1981) and suggested use of variable coefficient method rather
than constant coefficient method as it provided a more realistic
view of the process. Ahmad (2008) observed that near the bank
where shear velocity and flow depth are less, there is significant
difference between constant and variable transverse mixing
method. Ahmad (2008) provided reasoning for this discrepancy as
near the bank average and constant mixing coefficient results in
more mixing near the bank hence there is large difference be-
tween the constant and variable transverse mixing method,
whilst in the main channel region the variable transverse mixing
coefficient is more than the constant transverse mixing coeffi-
cient hence the difference between the two methods reduces.
Ahmad (2008) hence recommended that utilizing variable coeffi-
cient method would give a better and more realistic picture of
transverse mixing in steady state stream.
Effect of ice cover on the transverse mixing
Engmann and Kellerhals (1974) studied the effect of ice cover on
transverse mixing and observed that due to the ice cover the
lateral mixing coefficients were reduced by ϳ50% and intuitively
depicted that due to the generation of ice cover on the water
surface, convection due to spiral motions and mixing due to tur-
bulence were damped, hence reducing the mixing and mixing
coefficients. Beltaos (1980) observed that transverse mixing coef-
ficient was increased in the ice covered condition by a margin of
2.5 times which was opposite to the observations of Engmann and
Kellerhals (1974). Lau (1985) measured the transverse mixing coef-
ficients in four different reaches for open water and ice covered
conditions and found that there was no considerable differences
between transverse mixing coefficients measured in open water
and ice covered conditions. Zhang and Zhu (2011) recently studied
the effect of ice cover on transverse mixing in an unregulated
river. They observed that Ey/U‫ء‬H was reduced by 21% for ice cover
conditions compared to the open water conditions due to de-
crease in discharge in winter.
Effect of buoyancy on the transverse mixing
Prych (1970) concluded that enhanced lateral spreading of buoy-
ant tracer is due to the buoyancy generated secondary currents.
He also concluded that standard deviation growth in longitudinal
and lateral directions after a short distance downstream ap-
proaches the similar variation of standard deviations of concen-
tration distribution in longitudinal and lateral directions as the
neutrally buoyant tracer follows. Bruno et al. (1990) conducted
experimentations in straight rectangular channel to establish ef-
Fig. 2. Effect of sinuosity (Sn) on Ey/U‫ء‬B (Elhadi et al. 1984).
Fig. 3. Effect of sinuosity (Sn) on Ey/U‫ء‬H (Elhadi et al. 1984).
480 Can. J. Civ. Eng. Vol. 41, 2014
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11. fect of buoyancy on the process of transverse mixing. Bruno et al.
(1990) observed that initially Cv was high, which infers that spread-
ing in lateral direction was due more to buoyancy driven second-
ary currents, but as the tracer moved downstream the buoyancy
effects were reduced and the value of Cv also reduced. After x= = 13
the value of Cv became constant implying the zone to be of com-
plete transverse mixing. For limiting values of Bo/hu‫ء‬3 they ob-
served that transverse mixing was higher in case of narrow outlets
and heavier effluents than broader outlets and lighter effluents,
respectively. Bruno et al. (1990) finally concluded that their exper-
imental results show a good match with the observations of Prych
(1970).
Research needs
Transverse mixing is a relatively more important phenomenon
than vertical and longitudinal mixing. Thus, literature has given
more stress to evaluate the lateral mixing coefficient. But despite
so much research, there are certain gaps that have to be filled,
such as
1. Although most of the studies suggested that increasing aspect
ratio increases the rate of mixing, but a few studies have also
revealed that this dependency lies only in the near wall region
(ϳ2–3 times of water depth) and in core region aspect ratio
does not affects the mixing rate. Thus, more elaborative stud-
ies must be done to show how aspect ratio affects the mixing.
2. It is a well-known fact that pollutants which are either dis-
charged or are spilled accidently do not usually meet the cri-
terion of neutrally buoyant tracer, some amount of effect of
buoyancy lies in their mixing process. Up to this instance in
time only two studies have been completed, thus, more elab-
orate study should be done to investigate the effect of buoy-
ancy on the transverse mixing.
3. It is also not clear how the ice-cover over the river influences
transverse mixing rate because studies are reporting contra-
dictory results, thus, more elaboration is needed to know the
effect of ice cover on river mixing.
Conclusions
Spilling of pollutant and its mixing in any riverine system is an
important phenomenon to be monitored. Initially pollutant mix-
ing is three dimensional in nature and its dimensionality de-
creases along the length of river. Due to rapidity of vertical mixing
and longevity of longitudinal mixing reach, only transverse mix-
ing is important and to be monitored for water quality modeling.
It is hereby concluded that out of the constant and variable mix-
ing model, it is more realistic to use variable mixing method as
literature suggests it to be more accurate. Secondary currents
were observed to affect the transverse mixing more than any
other factors. Whilst regarding effects of roughness, aspect ratio,
and ice cover on transverse mixing results were contradictory and
more research has to be done on it.
Acknowledgement
Authors hereby want to thank the Department of Science and
Technology for supporting the project entitled “Enhanced trans-
verse mixing of pollutants in streams with submerged vanes.”
(Grant No. DST-471-CED), whose study led to the creation of this
review paper. Also, first author would like to specially thank
Mr. Vikrant Vishal, Research Scholar, University of Alberta, Al-
berta, Canada, for assistance he has provided while paper was
being written.
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Appendix A
Consider eq. (6):
Ѩc
Ѩt
ϩ
Ѩ(uc)
Ѩx
ϩ
Ѩ(vc)
Ѩy
ϩ
Ѩ(wc)
Ѩz
ϭ
Ѩ
Ѩxͫ(ex ϩ ␧xx)
Ѩc
Ѩxͬ
ϩ
Ѩ
Ѩyͫ(ey ϩ ␧yy)
Ѩc
Ѩyͬϩ
Ѩ
Ѩzͫ(ez ϩ ␧zz)
Ѩc
Ѩzͬ
Now, putting together terms like ex + ␧xx as kx, ey + ␧yy as ky, and
ez + ␧zz as kz, which is called as dispersion coefficient accounting
for diffusion caused due to shear generated turbulence eq. (6)
transforms to the following:
(A1)
Ѩc
Ѩt
ϩ
Ѩ(uc)
Ѩx
ϩ
Ѩ(vc)
Ѩy
ϩ
Ѩ(wc)
Ѩz
ϭ
Ѩ
Ѩx͑kx
Ѩc
Ѩx͒ϩ
Ѩ
Ѩy͑ky
Ѩc
Ѩy͒
ϩ
Ѩ
Ѩz͑kz
Ѩc
Ѩz͒
Consider a source of pollution giving a constant supply of pol-
lutant release to uniformly flowing water, thus the first term of
eq. (A1), i.e., time derivative, will vanish and as the flow is uniform
the only term that will exist will be Ѩ͑uc͒/Ѩx, other terms like
Ѩ͑vc͒/Ѩy and Ѩ͑wc͒/Ѩz will also vanish. Hence, eq. (A1) will transform
to the following format:
(A2)
Ѩ(uc)
Ѩx
ϭ
Ѩ
Ѩx͑kx
Ѩc
Ѩx͒ϩ
Ѩ
Ѩy͑ky
Ѩc
Ѩy͒ϩ
Ѩ
Ѩz͑kz
Ѩc
Ѩz͒
Now, to achieve the equation governing the transverse mixing
process, eq. (A2) has to be depth-averaged:
(A3) ͵a
b
Ѩ(uc)
Ѩx
dz ϭ ͵a
b
Ѩ
Ѩx͑kx
Ѩc
Ѩx͒dz ϩ ͵a
b
Ѩ
Ѩy͑ky
Ѩc
Ѩy͒dz
ϩ ͵a
b
Ѩ
Ѩz͑kz
Ѩc
Ѩz͒dz
Since, transverse mixing starts after the completion of vertical
mixing, thus, third dispersion term on the right hand side of
eq. (A3) representing vertical mixing will become zero. Hence, the
equation will convert into the following format:
(A4) ͵a
b
Ѩ(uc)
Ѩx
dz ϭ ͵a
b
Ѩ
Ѩx͑kx
Ѩc
Ѩx͒dz ϩ ͵a
b
Ѩ
Ѩy͑ky
Ѩc
Ѩy͒dz
Consider the term on the left hand side of eq. (A4)
(A5) ͵a
b
Ѩ(uc)
Ѩx
dz ≈
Ѩ(huC)
Ѩx
Similarly, integrating other terms we obtain the following equation:
(A6)
Ѩ(huC)
Ѩx
ϭ
Ѩ
Ѩx͑hEx
ѨC
Ѩx͒ϩ
Ѩ
Ѩy͑hEy
ѨC
Ѩy͒
which is called the steady state transverse mixing equation, gov-
erning the process of transverse mixing.
Here,
(A7) C ϭ
1
h ͵a
b
cdz
(A8) Ex ϭ
1
h ͵a
b
kxdz
(A9) Ey ϭ
1
h ͵a
b
kydz
482 Can. J. Civ. Eng. Vol. 41, 2014
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