Volume 41

2014

An NRC Research Press Journal

Une revue deNRC 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

ARTICLE

Transverse mixing of pollutants in streams: a reviewH. 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 inherentproperty of water to dissolve the substance, the particulate is well mixed in water. To monitor the extent of pollution in a streamit is essential to know how the pollutants mix in the river. It is observed that vertical mixing of pollutants is a very rapid processin the vertical directions and longitudinal mixing occurs very far from source of pollutant, which is generally out of reach ofobservations. Thus intermediate or transverse mixing zone is considered very important for water quality modeling. This paperis an attempt to summarize the phenomenon behind pollutant transport, reduction of three-dimensional advection–dispersionequation 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 del’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 mélangentdans la rivière. Il a été remarqué que le mélange vertical des polluants est un processus très rapide et que le mélange longitudinalsurvient très loin de la source de pollution, laquelle est généralement hors de portée des observations. La zone de mélangeintermédiaire ou transversal est considérée comme étant très importante pour la modélisation de la qualité de l’eau. Le présentarticle 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 despolluants. [Traduit par la Rédaction]

Mots-clés : mélange en rivière, mélange transversale, courants secondaires, équation advection–dispersion, modèles numériques.

IntroductionWater is a source for many industrial and domestic needs,

which include from drinking to cooling of reactors. Need of purewater for these tasks, is as important as management of waterquality. When accidents happen and pollutants are spilled intothe water, they get mixed in the water and pollute it. Understand-ing the mechanism of mixing of pollutants is necessary for themanagement of pollutants control in streams. When any pollut-ant is spilled into the river, due to the dominance of verticalturbulence and least dimension of any river system, the pollutantgets mixed along the depth very quickly. The process of the pol-lutant getting mixed along the depth is termed as vertical mixingand longitudinally it spreads up to 50–100 times of the depth(Yotsukura and Sayre 1976). After getting mixed into the verticaldirection under the action of transverse turbulence and variationin vertical profiles of transverse velocity, pollutant starts to spreadalong the river width and mixes along the cross section of theriver. This transverse spreading and mixing of pollutant under theaction of transverse shear and turbulence is termed as transversemixing. If the source of pollutant is unsteady in nature, due towhich pollutant further travels downstream under the action oflongitudinal gradients and this mixing of pollutant under theaction of longitudinal gradients is termed as longitudinal mixing.As the vertical mixing is a very rapid process and is dominant nearthe source of pollutant spill, whereas the longitudinal mixing is ofimportance very far from the source so that it makes it unimportantto model for environmental concern. This paper is a compendiumregarding various processes involved in pollutant mixing in a riverand the process of transverse mixing and factors affecting it.

Physical processes of pollutant transportIf 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 dyetracer mixing involves various physical processes, which are clas-sified as advection, diffusion, and dispersion. These processes willbe discussed briefly.

AdvectionAdvection can be defined as the process where tracer cloud

moves bodily in the flowing stream under the action of imposedcurrent. 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 flowingmedia. To quantify the intensity of advection, the term advectiveflux is defined as follows (Fischer et al. 1979; Elhadi et al. 1984;Rutherford 1994):

“The amount of tracer/substance transported per unit time perunit area perpendicular to the current is termed as advective fluxand is the product of velocity and transported concentration oftracer/substance.”

Mathematically, advective flux can be formulized as (Elhadiet al. 1984):

(1) Tad � 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.

Here, Tad is the advective transport per unit time per unit area; Uis the velocity perpendicular to unit area; and C is the concentra-tion of substance/tracer transported.

Molecular diffusionThe spreading of tracer due to the motion of molecules of fluid

is called as molecular diffusion. In laminar flow the transport oftracer is mainly governed by the molecular diffusion and the mo-tion of molecules is random but occurs from higher concentrationto lower concentration perpendicular to a unit area. Fick’s law ofdiffusion 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 � �em�C�n

Here, Tm is the transport by molecular diffusion per unit area perunit time; n is the direction normal to the unit area; and em is themolecular diffusion coefficient.

Turbulent diffusionIn 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 ofthe substance while the fluctuating component, which is respon-sible for the eddying motion in the flow, spreads and transportsthe tracer. This spreading and transport of tracer cloud due to theturbulence generated eddies is called as turbulent diffusion. If ui

′

is the fluctuating component of velocity in a certain ith directionand c= is the fluctuating component of tracer concentration, thenthe transport of tracer is given by (Fischer et al. 1979; Elhadi et al.1984; Rutherford 1994):

(3) Tt � u ′c ′

Here, Tt is the transport of substance by turbulent diffusion perunit time per unit area and the bar indicates the temporal aver-aging. The turbulent diffusion is also random like moleculardiffusion and occurs from higher concentration to lower concen-tration. Due to this likeliness, the turbulent diffusion can also beformulated similar to the molecular diffusion, in the form of thegradient based Fick’s diffusion law as follows:

(4) Tt � ���C�n

Here, � is the eddy diffusivity. Since similar equations could bewritten in other directions as in general the eddy diffusivitiesdepend on local concentration gradients, hence, varies from loca-tion to location.

DispersionIn natural channels, the velocity varies across the channel di-

mensions, i.e., near the wall the velocity will be zero and willachieve the maxima away from the wall region. Consider Fig. 1 forinstance, when a dye patch is introduced in the flow it is a pointmass. But, as the non-uniformly distributed velocity profile inter-acts with tracer mass, it gets elongated laterally and also spreadsin the longitudinal direction. Since, the velocity tends to decreaseas the wall area is approached, thus sides of tracer near to walladvects slower than the central portion of tracer mass, whichdistorts 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 diffusivetransport of tracer mass.

The processes of transport are still advection and diffusion butthe 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 andshear generated by this profile is called as dispersive transport orsimply as dispersion. Dispersion is assumed to be proportional tothe mean concentration gradient, thus, it is assimilated with theturbulent diffusion as it is also a gradient based phenomenon.Hence, mathematically the combined effect of turbulent diffusionand dispersion can be written as (Elhadi et al. 1984):

(5) Tdisp � Tt � �e�C�n

Here, Tdisp is the dispersive transport and e is the dispersion coef-ficient.

Three-dimensional mixing of pollutant in flowingstream

In any natural stream, when any neutrally buoyant substance isintroduced due to the dominance of vertical turbulence and leastdimension of any river system, the pollutant gets mixed along thedepth 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 muchdominant near the source of tracer release, hence it is also calledas “near field mixing”. At the commencement of the transversemixing, 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 (Rutherford1994).

Sharma and Ahmad 473

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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 theaction of longitudinal gradients and this mixing of pollutant un-der the action of longitudinal gradients is termed as longitudinalmixing. The mixing in this zone is one dimensional in nature astracer has now become fully mixed along vertical and lateral di-rections. Due to the farther reach of this zone from source oftracer release, this mixing is called as “far-field mixing” (Fischeret al. 1979; Rutherford 1994). The three dimensional unsteadytracer transport equation is as follows (Fischer et al. 1979;Rutherford 1994):

(6)�C�t

��(uC)

�x�

�(vC)�y

��(wC)

�zÇ

1

���x�(ex � �xx)

�C�x� �

��y�(ey � �yy)

�C�y � �

��z�(ez � �zz)

�C�z �

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 localdispersion 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 advectiveand diffusive transport of substance. As the three-dimensionaladvection–dispersion equation shows the true nature of pollutanttransport in any stream but certain restrictions are associatedwith it like misalignment of coordinate system, generation ofextra curvature terms in curvilinear coordinate systems, spatialvariation of flow, river bank impingement of tracers, dead zonesin rivers, etc. Hence it is compatible to reduce the dimensionalityof the equation. Thus, for the ease of calculations, omission of thetransport component in any particular direction can be done byaveraging in that pre-described direction and dimensionality ofthree-dimensional equation can be reduced to two or one.

Two-dimensional mixing of pollutant in flowingstream 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 invertical direction is very rapid process and is predominant in theregion near to the effluent source. Thus, averaging is done in thevertical direction and the terms which contain differential of ver-tical axis are omitted as the quantities have become uniformalong the vertical axis. The final format of equation (for the com-plete derivation see Appendix A) is as follows:

(7)�(hC)

�t�

�(huC)�x

���x�hEx

�C�x� �

��y�hEy

�C�y �

Here, Ex and Ey are the depth averaged dispersion coefficients inlongitudinal and lateral directions. The above equation is called asdepth-averaged advection dispersion equation and is known to bethe governing equation of transverse mixing. If the tracer sourceis continuously supplying the tracer, i.e., steady tracer flow, andassuming that longitudinal diffusion of tracer has not started yetfor the uniformly flowing stream, thus time differentiation ofeq. (7) will be zero and term “h” will come outside of the differen-tial making the third term of eq. (7), as longitudinal diffusion hasnot begun yet. Thus, the equation here is found in many of theliterature (Lau and Krishnappan 1977; Lau and Krishnappan 1981;Demetrocopolous 1994; Ahmad 2008; etc.) and is known as steadystate transverse mixing equation:

(8) u�C�x

� Ey�2C

�y2

It is clearly understandable that mechanisms accounting for thelateral movement of substance are transverse turbulent diffusionand transverse dispersion. But in the majority of the literature(Lau and Krishnappan 1977; Lau and Krishnappan 1981; Boxall andGuymer 2003; Boxall et al. 2003; Albers and Steffler 2007; etc.)reveals that transverse dispersion, i.e., non-uniform distributionof transverse velocity in vertical direction, is a major phenome-non responsible for the lateral mixing of solutes. Since manyfactors 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 diffusionAccording to the Prandtl’s mixing length theory (Rodi 1980;

Nezu and Nakagawa 1993):

(9) �zz � ℓm2 |�U

�z |

Here, ℓm is the mixing length; z is the vertical distance; and U isthe average velocity of flow. This hypothesis predicted the resultsvery well for plane shear flows. Now, an analogy was drawn justsimilar to eq. (9) and was written as follows (Rutherford 1994):

(10) �yy � ℓt2|�U

�y |

Here, y is the transverse distance; ℓt is the transverse turbulencelength scale; and �yy is the transverse turbulent diffusion coeffi-cient. Since in plane shear flow the velocity across the stream doesnot vary, hence, the transverse diffusivity automatically vanishes.But, investigators (Lau and Krishnappan 1977; Lau and Krishnappan1981; Holley and Nerat 1983; Boxall et al. 2003; Boxall and Guymer2003; etc.,) have observed that transverse dispersion does occur inthe channels. Hence, similar to Prandtl’s mixing length theory,i.e., turbulent diffusivity is equal to product of a turbulent lengthscale and turbulent velocity scale (Pope 2000; Tennekes andLumley 1972; etc.), i.e.:

(11) �yy � ℓtut

Here, ut is the velocity scale of turbulence. But the major flaw witheq. (11) is that we have accounted for the transverse turbulencescale for the calculation of diffusion coefficient but there is noclear way to calculate and evaluate this scale length as there is noplausible reason to explain the rotation of bed generated verticaleddies to transverse direction (Rutherford 1994). Thus to maintainthe relation of transverse diffusion with the flow parameters forthe lt = h, i.e., depth of flow was chosen and for velocity scale ut =u�, shear velocity was chosen as it is the main agent which gener-ates eddying motion in flow.

Transverse dispersionDispersion as discussed before is a phenomenon that occurs due

to non-uniformity of velocity profile. It is also a known fact thatthe variation of transverse velocity along the vertical is identicalto the variation of longitudinal velocity. Thus, the tracer near tothe water surface moves faster in comparison to the tracer presentnear the bed. This causes generation of vertical gradients andhence promoting vertical mixing. This increment in transversemixing due to the non-uniformity in velocity profile is called as

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transverse dispersion. There are two main components that con-tribute in the transverse dispersion, they are meander currentsand secondary currents.

Meander currentsIn any natural channel the main flow meanders from one bank

to another bank by depending on the channel cross section. Thesemeandering currents move uniformly along the depth and thusare responsible for transverse advection of tracer. Considering,Fig. 1., at the initial section, say Section A, plume is containedwithin the flow, q, and in the remaining section there is no tracerconcentration and clearly the tracer cloud is near to the true leftbank of the channel. As the flow moves forward, the tracer advectsand reaches the section B. From, Fig. 1., it can be clearly seen thatthe channel is shallower near the left bank and due to the bedfriction the velocity is lower near the left bank. To avoid large bedfriction the main flow shifts towards the right bank and hencecarries the plume towards the right bank of the natural channel.

Further advection of tracer causes the plume cloud to reachsection C where the depth at the left bank again becomes deeperthan the right bank, thus causing main flow to meander towardsthe left bank and hence making the plume to contract towards theleft bank.

Now, in eq. (7) we assume that there is no transverse componentof velocity, but clearly between the section A and section B thereis a positive component of velocity acting on the tracer plume andhence expanding it, but in between section B and section C theshifting of currents from the right bank to left bank causes cre-ation of negative transverse velocity component, thus contractingthe plume. Due to the variation of transverse velocity across thewidth, some transverse mixing (represented as e in Fig. 1.) occurs.Although, these meandering currents are not a very importantcomponent of transverse mixing phenomenon, however if consid-ering channels of irregular cross-sections these currents have tobe considered otherwise, as reported by Holley et al. (1972); Elhadiet al. (1984), etc, an over-estimation of diffusion coefficients ornegative diffusion coefficients are observed.

Secondary currentsIn 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 thetransverse velocity generate currents prominent in directionsother 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; Holleyand Nerat 1983; Boxall and Guymer 2003; Boxall et al. 2003; etc.) itis evident that secondary currents in the bends tends to dispersethe tracer apart in the opposite directions at the different depths

causing the transverse mixing to happen rapidly in bends. It wasobserved that transverse mixing in a natural river is a prominentfunction of secondary currents and due to irregular cross-sectionsand bends in the natural channels the equilibrium between thevertical velocity shear of transverse velocity and vertical turbulentdiffusion is generally not able to co-exist. Transverse mixingcoefficient is greatly affected by secondary currents. Transversemixing coefficient was studied by many investigators throughexperimental and numerical studies, thus a few investigator’sstudies are reported in Table 1. From the upcoming section vari-ous models available for the study of transverse mixing equationwill be discussed in detail.

Analytical models for analysis of transverse mixingIt is evident that for pollutant transport modeling, modeling of

transverse mixing is an important phenomenon. Thus to modelthe transverse mixing process, solution of transverse mixingequation (eq. (7)) has to be obtained. Thus to obtain the solution ofthe transverse mixing equation there are two methods, viz.:

Constant coefficient methodConsider eq. (7) for the source supplying the tracer at constant

steady rate. Thus the temporal differential, i.e., �C/�t will becomezero and hence eq. (7) will get transformed in the following for-mat:

(12)�(huC)

�x�

��x�hEx

�C�x�

Ç

1

���y�hEy

�C�y �

Ç

2

Now it is a known fact that longitudinal mixing usually does notinterfere with the transverse mixing, hence the term (1) in eq. (12)becomes zero, thus eq. (12) reduces to the following format:

(13)�(huC)

�x�

��y�hEy

�C�y �

Now as the method suggests, it is assumed that variation in depth,Ey is negligible and flow is uniform throughout the flow regime asthe channel is assumed to be straight, thus flow depth, Ey and u aretaken out of their respective brackets. Now, the eq. (13) reduces to

u�C�x

� Ey�2C

�y2

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. InvestigatorsRange of transversemixing coefficient (ey/u�h) Remarks

1 Holley and Abraham (1973) 0.15–0.16 Without groins0.4–0.5 With groins

2 Beltaos (1980) 0.22–0.42 For meandering channel3 Holley Jr. and Nerat (1983) 0.5–2.5 River measurement4 Demetracopoulos and Stefan (1983) 0.24–4.65 Temperature profile measurement5 Webel and Schatzmann (1984) 0.175–0.177 Smooth bed

0.130–0.159 Rough bed6 Bruno et al. (1990) 0.21–1.89 Buoyant tracer7 Chau (2000) 0.14–0.21 Smooth bed

0.14–0.21 Sandpaper0.13–0.19 Steel mesh0.15–0.24 Small stones

8 Seo et al. (2006) 0.23–1.21 For meandering river9 Baek and Seo (2010) 0.21–0.91 Validation of stream tube routing method

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analytical solution for a few sets of tracer sources, which arehelpful in analyzing the transverse mixing process.

Vertical line sourceAssume a steady vertical line source at x = 0, y = yo. Now at both

banks the transverse tracer flux is zero, thus it comply:

(14)�C�y

� 0 at y � 0 and y � B

Thus for this vertical line source analytical solution for un-bounded flow condition is given as

(15) C(x, y) �M.

H�4�Eyuxexp��

u�y � yo�2

4Exx�

where M is the mass inflow rate. Since eq. (8) and eq. (14) are bothlinear in nature, thus method of images can be applied to solvethe 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 oftransverse mixing equation where transverse velocity is zero. Thetransformed equation in x-q coordinate is given by

(16)�C�x

���q�h2uEy

�C�q�

Now investigators like Yotsukura and Cobb (1972), Yotsukuraand Sayre (1976), Lau and Krishnappan (1977), etc. have showedthat for practical purposes the term h2uEy can be taken as constantand equals the average value across the cross section with respectto q, that is

(17) Dy � h2uEy

This term Dy is known as diffusion factor. Thus eq. (16) will reduceto the following format:

(18)�C�x

� Dy�2C

�q2

Now the boundary conditions for eq. (18) will be that there is noflux of pollutant across the flow boundaries and hence

(19)�C�q

� 0 at q � 0 and q � Q

Here, Q is the stream discharge. Cumulative discharge methoddoes take into account the channel curvature, effect of changesin depth and width, etc. Yotsukura and Cobb (1972) and Lipsettand Beltaos (1978) provided an analytical solution for eq. (18)and assimilating the boundary conditions of eq. (19) with itfor a plane vertical source of length (� = y2 – y1) and it is asfollows:

(20)CC∞

�1

2��12�erf2 �

�2 � erf2 �

�2 � erf1 �

�2 � erf1 �

�2�

��m�1

∞

erf2 � 2m �

�2 � erf2 � 2m �

�2 � erf1 � 2m �

�2

�erf1 � 2m �

�2 � erf2 � 2m �

�2 � erf2 � 2m �

�2

�erf1 � 2m �

�2 � erf1 � 2m �

�2

Here, = y/B (here B is the width of the stream); = 2Dyx/Q2; C∞ isthe 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 thatdeviation of concentration lies around 5% of its mean value(Fischer et al. 1979).

Slug injectionFor 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 developedby Shen (1978), which is as follows:

(21) C(x � ut, q, t) �1

2Q��Ext�m�0

∞

�m cosm�Qa

Q

× exp��(x � ut)2

4Ext� Ey

m2�2u2h2tQ �cos�m�q

Q �Here, �m = 1, when m = 0 and �m = 2 when m ≠ 0; Q a = cumulativedischarge at the location of the point source.

Deficiencies of constant mixing coefficient methodThe constant transverse mixing coefficient model allows the

mixing equation to be solved analytically and greatly enhancesthe problem solving. In spite of this fact, the thing that restrictsthe use of the constant mixing coefficient method is the complexbathymetry 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 ratherconstant but vary across the width, hence the main assumption isinvalidated from the real time observations. As the depth andvelocity 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 shifttowards a more realistic and heuristic approach to analyze thefield problem of transverse mixing. Thus, it gave rise to a variablemixing 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

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transverse mixing coefficients which are not realistic like Holleyet al. (1972) who observed the negative value of mixing coefficientin the area of high separation when calculated while consideringthe mixing coefficient to be constant along and across the flowfield. Also in a realistic scenario the velocity, depth, and crosssection vary in the flow region, which the constant mixing coeffi-cient method does not take into account. It is a known fact that ingeneral the bathymetry of channel is not known in advance andnor 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(�, ) � � �0

m hv�d

Here, � and are the longitudinal and transverse coordinates inan orthogonal curvilinear system; � is parallel with the localdepth-averaged longitudinal velocity and is transverse acrossthe flow; q is the cumulative discharge; m is the metric coefficientwhich accounts for channel curvature and lies between 0.8–1.2 formeandering channels and 1.0 for straight channels; v� is the depthaveraged velocity. By convention q = 0 is taken for left bank and q =Q for right bank. For an unsteady tracer source in steady flow, thetracer transport equation is as follows (Rutherford 1994):

(23)�C�t

�v�

m�

�C��

�1

hm�m

���m

m�

hE�

�C�� �

v�

m�

��q�m�h2v�E

�C�q�

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)�C��

���q�m�h2v�E

�C�q�

Now, defining factor of diffusion as

(25) Dy � m�h2v�E

Since, at the banks the longitudinal velocity and depth of flow isvery low, thus the factor of diffusion varies across the channel. Forconvenience Dy can be written as

(26) Dy � �H2UEy

Here, � is a dimensionless shape factor that lies in the range of1.0–3.6 (Beltaos 1980) and is given by

(27) � �1B �

y�0

B

� hH�2 u

Udy

Now, several investigators (such as Yotsukura and Cobb 1972;Sayre 1979; Almquist and Holley 1985, etc.) assumed that variationof Dy can be taken to be uniform and constant over the flow field,thus taking eq. (18) with the boundary conditions defined ineq. (19) analytical solution of eq. (18) for unbounded flow withsource placed at x = 0 and q = qo is given by

(28) C(x, q) �M

�4�Dyxexp��

(q � qo)2

4Dyx�

By utilizing method of images the effect of boundaries can beaccommodated in the solution.

Numerical solution of transverse mixing equationAnalytical 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 ofmass 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 thesimple analytical solution, they provide a much wider picture ofprocess than doing preliminary calculations using simple analyticalsolutions. Though there are many numerical solutions, a few areprovided 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)�C�x

� V�C�

� D�2C

�2

Here,

(30) V � �1

Q2

�Dy

�

and

(31) D �1

Q2Dy

Stone and Brian (1963) discretized the eq. (29) to obtain thefollowing finite difference analogue:

(32)1

�x�g(Ci�1,j � Ci,j) ��2

(Ci�1,j�1 � Ci,j�1) � m(Ci�1,j�1 � Ci,j�1)� �Vi,j

��a(Ci,j�1 � Ci,j) ��2

(Ci,j � Ci,j�1) � b(Ci�1,j�1 � Ci,�1,j)

� d(Ci�1,j � Ci,�1,j�1)� �Di,j

2(�)2[(Ci,j�1 � 2Ci,j � Ci,j�1) � (Ci�1,j�1 � 2Ci�1,j � Ci�1,j�1)]

In this scheme, Stone and Brian (1963) used weights orweighing coefficients, a, �/2, b, and d to approximate the deriv-ative �C/�; and to approximate the derivative �C/�x, they usedg, �/2, and m as weighing coefficients. They implied the follow-ing conditions on weighing coefficients which they have tosatisfy:

(33) a ��2

� b � d � 1

and

(34) g ��2

� m � 1

Sharma and Ahmad 477

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Lau and Krishnappan (1981) did not use any weighing factors forapproximating the second derivative �2C/�2. Crank-Nicolsonscheme was utilized by Lau and Krishnappan (1981) to discretizeequation eq. (29). By optimizing the weighing coefficients to sat-isfy eq. (33) and eq. (34), though they give solutions which nearlymatches the analytical solution, they obtained the following val-ues of weighing coefficients:

(35) g �23

; � � m �16

; a � b � d ��2

�14

They suggested using the aforesaid values of weighing coefficientswhen the values of V and D are not constant. If the concentrationdistribution at any station say i is known, then to obtain the valueof concentration distribution at any unknown level say i + 1 has tobe calculated for known boundary condition and zero flux condi-tions at side walls, thus Lau and Krishnappan (1981) proposed thefollowing matrix notation for eq. (32):

(36) �q2 �r2

�p3 q3 �r3

�p4 q4 �r4

Ì Ì Ì�rM�1

�pM �qM

�Ci�1,2

Ci�1,3

Ci�1,4

Ci�1,M�1

Ci�1,M

� �S2

S3

S4

SM�1

SM

Here,

qj �g

�x� Di,j

1

(�)2� Vi,j

1�

(d � b), (j � 2, 3...M)

pj � ��2

1�x

� Di,j1

2(�)2� Vi,j

1�

d, (j � 2, 3...M)

rj � �m�x

� Di,j1

2(�)2� Vi,j

1�

b, (j � 2, 3...M � 1)

(37) Sj � Ci,j�1��2

1�x

� Di,j1

2(�)2� Vi,j

�2��

� Ci,j� g�x

� Di,j1

(�)2� Vi,j

�2��

� Ci,j�1� m�x

� Di,j1

2(�)2� Vi,j

a��, (j � 2, 3...M)

The equations created during the solution were solved with thehelp 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 transportof pollutant in natural river. They utilized the stream-tube con-cept to derive their method called as MABOCOST (mixing analysisbased on concept of stream tube) by solving the following equa-tion:

(38)�C�t

�Umx

�C�x

���qh2U2Ey

Q2

�C�q � �C �

Here, � is the decay constant of pollutant and is any source orsink present in pollutant mixing zone. First of all Luk et al. (1990)decided the selection of numerical grid in which they selected thesimplest asymmetrical, first order explicit scheme that is quitestable, non-diffusive, and non dispersive when courant number(Cr) is set as unity. Courant number is defined as follows:

(39) Cr �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 areusing is stable only when Courant number is unity, hence replac-ing Cr = 1 in eq. (39), they obtained the following equation to setthe grid size and time step for finite difference grid that will beused to solve eq. (38):

(40) u�t � �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 theyargued that since the grid was constructed on the basis that ittakes a unit time step for pollutant to advect from one element toanother, hence they obtained the following equation by advectingthe pollutant to next element in temporal and spatial grid:

(41) C(i, j, t � �t) � C(i, j � 1, t)

Here, i is the number of stream tubes and j is the number ofelement 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 � �t) � C(i, j, t) ��t� �

p�1

m �l�xlh�yDy�C�p

�ql2

�

(l�xl)p�(h�y)p � h�y��Cp

�ql2

� �p�1

k �r�xrh�yDy�C�p

�qr2

�

(r�xr)p�(h�y)p � h�y��Cp

�qr2

Here, Dy = H2U2Ey/Q2 is the dispersion/diffusion factor; � is thevolume of a mesh element; l and r are the fractions of overlappingarea between an element and its adjacent element on the left andright sides, respectively; �x is the elemental length; �y is theelemental width; �C is the concentration excess; and �n is thefraction 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 ofthe left and right adjacent elements, respectively. Overbars repre-sent quantities averaged between an element and its adjacentelements. 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 � �t) � C(i, j, t) × exp(��t)

Finally Luk et al. (1990) added the, if possible, source or sink termsfor 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 inMABOCOST did not affect the quality of results in a considerablemanner.

478 Can. J. Civ. Eng. Vol. 41, 2014

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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)�C�x

� Dy�2C

�q2

Here,

(45) Dy � h2uey

Since according to stream-tube model there should be no flux ofpollutants across the flow, hence Ahmad (2008) provided the fol-lowing boundary conditions:

(46)�C�x

� 0 at q � 0 and q � Q

To compare the variable ey and constant ey methods, Ahmad (2008)took the average value of ey/u�h = 0.225 for constant mixing coef-ficient method. Ahmad (2008) solved the steady state transportequation numerically by incorporating the upwind scheme of fi-nite difference, finally obtained the following equation for a pe-culiar node P (i, j):

(47) asjCi

j�1 � apjCi

j � anjCi

j�1 � awj Ci�1

j � b j

Here as, an, ap, and aw are the coefficients of concentration; Cij is the

concentration at node (i,j). Also,

(48a) asj � a3

j

(48b) apj � �a1

j � a2j � a3

j�

(48c) anj � a2

j

(48d) awj � a1

j

(48e) b j � 0

(48f) a1j �

hjuj�y2

�x

(48g) a2j � h j � h j�1

2ey

j � eyj�1

2

(48h) a3j � h j � h j�1

2ey

j � eyj�1

2

Ahmad (2008) observed that while solving the system of equationgenerated for different i and j from eq. (47), a tri-diagonal matrixwas formed. Thomas algorithm or tri-diagonal matrix algorithm(TDMA) was utilized by Ahmad (2008) to solve the tri-diagonalmatrix iteratively and line by line until a converging value ofconcentration was obtained.

Effect of secondary currents on the transversemixing

It is clear from the previous section which states that secondarycurrents positively affects the process of transverse mixing by

generating secondary currents which helps flow to spread tracerlaterally in a faster manner. Fischer (1969) was one of the first tostudy the effect of bend generated secondary currents on theprocess of dispersion. He also observed that predicted mixingcoefficient and observed coefficients almost varied within the fac-tor of two and the Ey to HU� � 10 times to what it was observed instraight 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 weregenerated and caused the enhanced mixing of dye. Chang (1971) inhis work concluded that phenomenon of transverse mixing inmeandering channel is closely related to the generation and decayof bend generated secondary currents. Holley et al. (1972) alsoconsidered effects of bend on transverse dispersion and observedthat as the curvature of bends change, it tends to generate a nettransverse velocity.

Holley and Abraham (1973) observed that there was rise in valueof lateral mixing coefficient due to the generation of secondarycurrents in flow by groins. Ward (1974) concluded that increase indepth or change in radius of curvature does affect the structure ofturbulence and secondary currents, thus, affects the transversemixing phenomenon. Krishnappan and Lau (1977) observed thatvariation of depth and secondary currents played important rolesin transverse dispersion. They also observed that transverse dis-persion and convective transport of pollutant were both of equalmagnitude and were equally liable for pollutant transport acrossthe width of river.

Krishnappan and Lau (1981) deduced that regardless of aspectratio 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 thetransverse 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 inthe meanders caused considerable lateral spreading of pollutants.Boxall and Guymer (2003) conducted experimentation in mean-dering channel and observed that after dye was mixed verticallydue to variation of transverse velocities in vertical direction, dyemixing was observed to speed up due to the action of strongsecondary currents of bend.

Seo et al. (2006) studied the transverse mixing phenomenonunder the slug test conditions and they observed that along thebend it was observed that the core of tracer cloud shifted towardsthe outer bank while traversing downstream due to the secondarycurrents. It was observed that enhanced transverse mixing wasobtained due to the presence of noticeable secondary currents inthe flow. Dow et al. (2009) observed that due to secondary currentsgenerated 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 sawthat as the bend starts, the flow has high intensity of bend gener-ated secondary currents and thus due to these secondary currentsthe lateral spreading of tracer was observed to increase.

Effect of aspect ratio on the transverse mixingLau and Krishnappan (1977) experimentally observed the effect

of altering the aspect ratio on secondary currents by (1) changingwidth 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 stressdistribution, secondary currents are generated which enhancedthe transverse mixing. Webel and Schatzmann (1984) argued thatexcept 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

lateral mixing was faster in the discordant bed than in the con-cordant bed. While plotting the normalized transverse dispersioncoefficient with aspect ratio (width–depth ratio), it was observedby Seo et al. (2006) reported that with the increase in the aspectratio the normalized transverse mixing coefficient also increasedas the strength of secondary currents was observed to increasewith the increase in aspect ratio.

Effect of bed roughness on the transverse mixingChang (1971) suggested that increasing roughness enhanced the

transverse mixing due to increment in bed friction while Lau andKrishnappan (1977) observed that observations relied on a singlecurve and inferring that friction factor had a little effect on theparameter. 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 frictionwhile parameter Ey/u�h was observed to decrease with increasingfriction.

Demuren and Rodi (1986) conducted experiments on the widechannel (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 withrough bed was predominantly led by bed generated turbulencerather than secondary currents. For limiting case of buoyancy flux

(Bo/hu�3), Bruno et al. (1990) observed that transverse mixing ratewas higher for rough bed than smooth bed. Chau (2000) also ob-served that under a constant aspect ratio by varying the bottomroughness did not change the dimensionless transverse mixingcoefficient by a considerable amount.

Effect of variability of the transverse mixingcoefficient on transverse mixing

Holley et al. (1972) assumed three cases in variability of trans-verse mixing coefficient across the cross section and observedconsiderable difference between the concentration profiles ob-tained by all three cases and inferred that assumed variation of Eyhas an effect on the Ey computed from measured concentrationprofile. Lau and Krishnappan (1981) considered five cases for thevariation of transverse mixing coefficient across the cross sectionof the natural channel and observed in three of the cases thatwhere the values are allowed to vary across the cross section, thesimulated 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 ratherthan constant coefficient method as it provided a more realisticview of the process. Ahmad (2008) observed that near the bankwhere shear velocity and flow depth are less, there is significantdifference between constant and variable transverse mixingmethod. Ahmad (2008) provided reasoning for this discrepancy asnear the bank average and constant mixing coefficient results inmore 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 mixingcoefficient 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 oftransverse mixing in steady state stream.

Effect of ice cover on the transverse mixingEngmann and Kellerhals (1974) studied the effect of ice cover on

transverse mixing and observed that due to the ice cover thelateral mixing coefficients were reduced by �50% and intuitivelydepicted that due to the generation of ice cover on the watersurface, convection due to spiral motions and mixing due to tur-bulence were damped, hence reducing the mixing and mixingcoefficients. Beltaos (1980) observed that transverse mixing coef-ficient was increased in the ice covered condition by a margin of2.5 times which was opposite to the observations of Engmann andKellerhals (1974). Lau (1985) measured the transverse mixing coef-ficients in four different reaches for open water and ice coveredconditions and found that there was no considerable differencesbetween transverse mixing coefficients measured in open waterand ice covered conditions. Zhang and Zhu (2011) recently studiedthe effect of ice cover on transverse mixing in an unregulatedriver. They observed that Ey/U�H was reduced by 21% for ice coverconditions compared to the open water conditions due to de-crease in discharge in winter.

Effect of buoyancy on the transverse mixingPrych (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 longitudinaland 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 theneutrally buoyant tracer follows. Bruno et al. (1990) conductedexperimentations in straight rectangular channel to establish ef-

Fig. 2. Effect of sinuosity (Sn) on Ey/U�B (Elhadi et al. 1984).

Fig. 3. Effect of sinuosity (Sn) on Ey/U�H (Elhadi et al. 1984).

480 Can. J. Civ. Eng. Vol. 41, 2014

Published by NRC Research Press

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 buoyancyeffects were reduced and the value of Cv also reduced. After x= = 13the value of Cv became constant implying the zone to be of com-plete transverse mixing. For limiting values of Bo/hu�3 they ob-served that transverse mixing was higher in case of narrow outletsand 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 needsTransverse mixing is a relatively more important phenomenon

than vertical and longitudinal mixing. Thus, literature has givenmore stress to evaluate the lateral mixing coefficient. But despiteso much research, there are certain gaps that have to be filled,such as

1. Although most of the studies suggested that increasing aspectratio increases the rate of mixing, but a few studies have alsorevealed that this dependency lies only in the near wall region(�2–3 times of water depth) and in core region aspect ratiodoes 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 ofbuoyancy lies in their mixing process. Up to this instance intime 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 influencestransverse mixing rate because studies are reporting contra-dictory results, thus, more elaboration is needed to know theeffect of ice cover on river mixing.

ConclusionsSpilling 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 mixingand 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 asliterature suggests it to be more accurate. Secondary currentswere observed to affect the transverse mixing more than anyother factors. Whilst regarding effects of roughness, aspect ratio,and ice cover on transverse mixing results were contradictory andmore research has to be done on it.

AcknowledgementAuthors 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 thisreview paper. Also, first author would like to specially thankMr. Vikrant Vishal, Research Scholar, University of Alberta, Al-berta, Canada, for assistance he has provided while paper wasbeing written.

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Appendix AConsider eq. (6):

�c�t

��(uc)�x

��(vc)�y

��(wc)

�z�

��x�(ex � �xx)

�c�x�

���y�(ey � �yy)

�c�y� �

��z�(ez � �zz)

�c�z�

Now, putting together terms like ex + �xx as kx, ey + �yy as ky, andez + �zz as kz, which is called as dispersion coefficient accountingfor diffusion caused due to shear generated turbulence eq. (6)transforms to the following:

(A1)�c�t

��(uc)�x

��(vc)�y

��(wc)

�z�

��x�kx

�c�x� �

��y�ky

�c�y�

���z�kz

�c�z�

Consider a source of pollution giving a constant supply of pol-lutant release to uniformly flowing water, thus the first term ofeq. (A1), i.e., time derivative, will vanish and as the flow is uniformthe only term that will exist will be ��uc�/�x, other terms like��vc�/�y and ��wc�/�z will also vanish. Hence, eq. (A1) will transformto the following format:

(A2)�(uc)�x

���x�kx

�c�x� �

��y�ky

�c�y� �

��z�kz

�c�z�

Now, to achieve the equation governing the transverse mixingprocess, eq. (A2) has to be depth-averaged:

(A3) �a

b

�(uc)�x

dz � �a

b

��x�kx

�c�x�dz � �

a

b

��y�ky

�c�y�dz

� �a

b

��z�kz

�c�z�dz

Since, transverse mixing starts after the completion of verticalmixing, thus, third dispersion term on the right hand side ofeq. (A3) representing vertical mixing will become zero. Hence, theequation will convert into the following format:

(A4) �a

b

�(uc)�x

dz � �a

b

��x�kx

�c�x�dz � �

a

b

��y�ky

�c�y�dz

Consider the term on the left hand side of eq. (A4)

(A5) �a

b

�(uc)�x

dz ≈�(huC)

�x

Similarly, integrating other terms we obtain the following equation:

(A6)�(huC)

�x�

��x�hEx

�C�x� �

��y�hEy

�C�y �

which is called the steady state transverse mixing equation, gov-erning the process of transverse mixing.

Here,

(A7) C �1h �

a

b

cdz

(A8) Ex �1h �

a

b

kxdz

(A9) Ey �1h �

a

b

kydz

482 Can. J. Civ. Eng. Vol. 41, 2014

Published by NRC Research Press