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Pollutant dispersion in water currents under windactionAngel N. Menéndez a b & Carlos E. Laciana ba INA (National Institute for Water) , Autopista Ezeiza-Cañuelas, Tramo J.Newbery, km1,620, 1804, Ezeiza, Argentinab Faculty of Engineering , University of Buenos Aires , Av. Las Heras 2214, 1127, BuenosAires, ArgentinaPublished online: 26 Apr 2010.

To cite this article: Angel N. Menéndez & Carlos E. Laciana (2006) Pollutant dispersion in water currents under windaction, Journal of Hydraulic Research, 44:4, 470-479, DOI: 10.1080/00221686.2006.9521698

To link to this article: http://dx.doi.org/10.1080/00221686.2006.9521698

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Journal of Hydraulic ResearchVol. 44, No. 4 (2006), pp. 470–479

© 2006 International Association of Hydraulic Engineering and Research

Pollutant dispersion in water currents under wind action

Dispersion de polluant par des courants engendrés par le ventANGEL N. MENÉNDEZ,INA (National Institute for Water), Autopista Ezeiza-Cañuelas, Tramo J.Newbery, km 1,6201804 Ezeiza, Argentina and Faculty of Engineering, University of Buenos Aires, Av. Las Heras 2214, 1127 Buenos Aires, Argentina

CARLOS E. LACIANA, Faculty of Engineering, University of Buenos Aires, Av. Las Heras 2214, 1127 Buenos Aires, Argentina

ABSTRACTA closed expression for the dimensionless dispersion coefficient due to differential advection in the presence of wind is obtained, through integrationof the associated velocity and concentration vertical profiles. Tsuruya’s model is used for the velocity distribution, while the concentration profile isobtained from a simplified pollutant transport equation explicitly deduced. The expression reduces to Elder’s value (≈5.9) for the particular case ofno wind. Higher values arise for winds acting in the same direction as the flow; and vice versa, at least for a practical range of wind velocities. Forthe case in which the wind acts along a direction non-coplanar with the mean flow direction, it is proposed to apply the analysis independently to thecoplanar direction and its normal, where the mean current is, then, null. For this normal direction, the dimensionless dispersion coefficient value isaround 24.9. In this way, the theory can be applied to 2D-horizontal pollutant transport modeling.

RÉSUMÉUne expression fermée pour le coefficient sans dimensions de dispersion dû à l’advection différentielle en présence de vent est établie, par l’intégrationde profils verticaux associés de la vitesse et de la concentration. Le modèle de Tsuruya est employé pour la distribution de vitesses, alors que le profilde concentration est obtenu à partir d’une équation de transport simplifiée de polluant explicitement établie. L’expression se réduit à la valeur de Elder(≈5.9) pour le cas particulier sans vent. Des valeurs plus élevées apparaissent pour des vents agissant dans la même direction que l’écoulement ; etvice versa, au moins pour une certaine gamme de vitesses de vent. Pour le cas du vent non-coplanaire avec la direction de l’écoulement moyen, onpropose d’appliquer l’analyse, indépendamment, dans la direction coplanaire et sa normale où le courant moyen est pris nul. Pour la direction normale,la valeur du coefficient sans dimensions de dispersion est autour 24.9. De cette façon, la théorie peut être appliquée pour le transport de polluant en2D-horizontal.

Keywords: Pollutant dispersion, dispersion coefficient, wind-driven water currents, wind-driven dispersion.

1 Introduction

As it is well known, when dealing with a depth-integrated anal-ysis of the transport of a passive weightless substance by awater current, the action of differential advection along the ver-tical direction, in combination with vertical turbulent diffusion,induces an effective longitudinal diffusion which adds to the tur-bulent longitudinal diffusion (Fischer, 1967, 1973). Moreover,this extra longitudinal diffusion, usually called dispersion, isgenerally much more important than the turbulent longitudinaldiffusion itself.

Elder (1959), in a foundational paper, determined theoreti-cally in a closed form, and verified experimentally the coefficientof dispersion associated to an “equilibrium” vertical veloc-ity profile, i.e., the one that develops in a wide rectangularchannel in uniform flow. In this case, the shear stress lead-ing to a vertical velocity gradient and, hence, to differentialadvection, is caused by the retarding effect of the channel

Revision received November 17, 2005/Open for discussion until August 31, 2007.

470

bottom. The result should also be valid for a quasi-uniformflow, i.e., the one that develops over a slowly varying cross-section form or, more rigurously, for which the relative variationof the cross-section form is negligible over spatial scales ofthe order of the longitudinal scale of the flow (Menéndez,2003).

When the flow is subjected to wind action, an additional shearstress appears at the water surface, leading to a deformation ofthe velocity profile and, hence, to a change in the dispersioncoefficient. The objective of the paper is to obtain a closed expres-sion for the coefficient of dispersion in the presence of wind,containing, as a particular case, Elder result.

The paper proceeds through the following steps:

• The law for the velocity profile in the presence of wind isrevisited.

• The shear stress and eddy diffusivity distributions across thevertical direction are obtained.

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• The transport equation for the concentration “anomaly” (i.e.,the concentration with the mean value subtracted) is deducedthrough a series of simplifying assumptions.

• The concentration anomaly profile is obtained through inte-gration of the transport equation.

• The dispersion coefficient is calculated according to its defini-tion, which implies an integration of the product of the velocityand concentration anomalies.

This derivation procedure follows, essentially, the same stepsas Elder (1959), but they are more explicitly shown for the sakeof clarity.

With the so obtained expression, results are calculated fordifferent conditions, within practical ranges of the parameters.

The theory is developed for the coplanar case (wind direc-tion in the same vertical plane as the water current), as validatedmodels for the vertical velocity profile exist for this situation.However, a generalization to the non-coplanar case is in order towiden the practical applicability of the results, mainly to include2D-horizontal pollutant transport modeling. This generalizationis postulated at the end.

2 Velocity profile

There are many laws proposed to model the velocity profilegenerated by wind action acting in the same plane as the flowdirection. A detailed study (Bombardelli, 1998; Bombardelli andMenéndez, 1999), comparing their performances against exper-imental data, concluded that the laws proposed by Reid (1957)and Tsuruya (1985) were the most general, i.e., they could repro-duce the whole set of available experimental profiles. Due toits simplicity, Tsuruya’s law was adopted for the present study.Its closure as a model was discussed and validated elsewhere(Bombardelli, 1998; Bombardelli and Menéndez, 1999).

Tsuruya’s (1985) law for the horizontal componentu, as afunction of the vertical coordinatez (measured upwards from thewater surface), can be written as:

u(z) = u∗s

κln

(zos + h

zos − z

)+ u∗f

κln

(zof + h + z

zof

)(1)

whereu∗s is the shear velocity on the water surface produced bythe wind action,u∗f the shear velocity on the bottom surface,h

the water depth,zos the effective water surface roughness,zof theeffective bottom surface roughness andκ von Karman constant.Apositive value of the shear velocity means a shear stress pointingin the positivex direction, and vice versa.

Note that, according to Eq. (1),u vanishes at the bottomsurface (z = −h), as expected. On the water surface thevelocity is

us ≡ u(z = 0) = αosu∗s

κ+ αof

u∗f

κ(2)

where

αos ≡ ln

(1 + h

zos

); αof ≡ ln

(1 + h

zof

)(3)

In the absence of wind, Eq. (1) reduces to a form of the classicallogarithmic law for an equilibrium velocity profile (White, 1974).

The depth-averaged velocityU turns out to be

U ≡ 1

h

∫ 0

−h

u(z)dz = βosu∗s

κ+ βof

u∗f

κ(4)

where

βos ≡ 1 − αoszos

h, βof ≡ −1 + αof

(1 + zof

h

)(5)

The surface shear velocity is an input. Hence, Eq. (4) can beused to obtain the bottom shear velocity if the depth-averagedvelocity is provided (e.g., from a depth-averaged model):

u∗f = κU − βosu∗s

βof(6)

which, in the absence of wind(u∗s = 0), reduces to

u∗f = κ

βofU (7)

In the fully rough hydraulic regime the bottom shear stressτf ≡ ρu∗f |u∗f |, whereρ is the fluid density, is usually para-meterized using Manning roughness coefficientn, τf = ρg(n2/

h1/3)U|U|, withg the acceleration of gravity. Hence, from Eq. (7)it comes out the following relation betweenzof andn:

βof = κh1/6

√g

1

n(8)

Taking into account that, for this hydraulic regime, the effec-tive roughness heightks relates tozof through zof = ks/30(Schlichting, 1979), Eq. (8) can be compared with the muchsimpler Williamson relation (Henderson, 1966):

ks = (26n)6 (9)

This is done in Fig. 1, for the particular caseh = 5 m, down tothe limit of the fully rough regime. It is observed that they givevery similar results down to aboutn ≈ 0.020. Beyond that limit,Eq. (8) should be considered a physically more sound expression.For practical applications, where the Manning coefficient is themost commonly used parameter to characterize bottom rough-ness, the relationship betweenn andzof expressed in Eq. (8) willbe considered valid even in the presence of wind.

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

0.010 0.012 0.014 0.016 0.018 0.020 0.022 0.024 0.026 0.028 0.030

n

z of/h

Williamson Tsuruya

Figure 1 Relation between the effective bottom roughness and Manningcoefficient.

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3 Shear stress profile

The momentum equation (White, 1974) in the flow direction,x,under shallow-water conditions (hydrostatic pressure) and usingReynolds averaging, can be written as:

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y+ w

∂u

∂z+ g

∂h

∂x

= gx + 1

ρ

(∂τxx

∂x+ ∂τxy

∂y+ ∂τxz

∂z

)(10)

wheret is the time coordinate,y the lateral horizontal coordinate,z the vertical coordinate,v andw the ensemble-averaged velocitycomponents in they andz directions, respectively,gx the compo-nent of the acceleration of gravity in the flow direction, andτxx,τxy andτxz, the components in thex direction of the Reynoldsstresses on surfaces pointing locally toward the positivex, y andz directions, respectively (viscous stresses are neglected).

If the time variation is sufficiently slow to assume that theflow regime is quasi-steady(∂/∂t ≈ 0) and the horizontal (x, y

plane) variations are assumed much weaker than the vertical ones(∂/∂x ≈ 0, ∂/∂y ≈ 0, w ≈ 0), Eq. (10) reduces to

∂τxz

∂z= −ρgx (11)

Equation (11) shows that the shear stress distributes linearly alongthe vertical. Its integration from the water surface down to anarbitrary pointz gives:

τxz

ρ(z) = τs

ρ− gIoz (12)

whereτs is the wind shear stress acting on the water surface andIo

is the bottom slope, assumed to be small enough to approximatethe projection factor of the acceleration of gravity on thex direc-tion. Forz = −h, i.e., at the bottom surface, Eq. (12) indicatesthat the shear stress acting on the bottom,τf , is given by

τf

ρ= τs

ρ+ gIoh (13)

Note that forτs = 0, i.e., no wind, Eq. (13) reduces tothe well-known expression for uniform flow (Henderson, 1966).Equation (13) indicates that the bottom shear stress is higher, rel-ative to its uniform flow value, when the wind acts in the flowdirection, and vice versa. Moreover,τf can become negative fora sufficiently strong upstream directed wind.

Equations (12) and (13) can be combined in order to obtainan expression for the shear stress profile in terms of the shearstresses at both ends:

τxz

ρ(z) = − z

h

τf

ρ+

(1 + z

h

) τs

ρ(14)

Equation (14) can also be expressed in terms of the shearvelocities:

τxz

ρ= − z

hu∗f |u∗f | +

(1 + z

h

)u∗s|u∗s| (15)

4 Vertical eddy diffusivity profile

The vertical eddy viscosityez is defined as

ez ≡ (τxz/ρ)

∂u/∂z(16)

Introducing Eqs (1) and (15) into Eq. (16) the followingexpression is obtained

ez = − zhu∗f |u∗f | + (

1 + zh

)u∗s|u∗s|

u∗fκ

1zof+h+z

+ u∗sκ

1zos−z

(17)

It will be assumed that the vertical eddy diffusivity can beapproximated by the vertical eddy viscosity, given by Eq. (17).As an illustration, Fig. 2 shows, then, the vertical distributionof the vertical eddy diffusivity, together with the correspond-ing velocity profile from Eq. (1), for the following particularconditions: h = 5 m, U = 0.50 m/s, n = 0.025, W (windvelocity) = 20 km/h,Ho (wave height) = 0.10 m, from whichthe following values arise:u∗s = 0.0086 m/s (using a drag coef-ficient of 0.002) ,zos/h = Ho/(15h) = 0.0013 (rough watersurface; Kondo, 1976),zof/h = 0.00050 (from Eq. (8)), andu∗f = 0.029 m/s (from Eq. (6)). It is observed that the result-ing eddy diffusivity vanishes both at the bottom and at the watersurface.

5 Transport equation

The transport equation for the concentrationc(x, y, z) of a dis-solved pollutant (Holleyet al., 1970) in a unidirectional watercurrent along thex axis can be written as:

∂c

∂t+ u

∂c

∂x= ∂

∂x

(ex

∂c

∂x

)+ ∂

∂y

(ey

∂c

∂y

)+ ∂

∂z

(ez

∂

∂z

)(U) (A) (Dx) (Dy) (Dz) (18)

where ex, ey and ez are the eddy diffusivities in the three(main) spatial directions. Note that the terms in Eq. (18) havebeen labeled as unsteady(U), advective(A), dispersive-x (Dx),dispersive-y (Dy) and dispersive-z (Dz).

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

z/h

u/U

0.0E+00

1.0E-03

2.0E-03

3.0E-03

4.0E-03

5.0E-03

6.0E-03e z

/hU

Velocity Eddy diffusivity

Figure 2 Vertical distribution of vertical eddy diffusivity.

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Pollutant dispersion in water currents under wind action473

Equation (18) is complemented with the boundary conditions

∂c

∂z= 0 for z = 0 and z = −h (19)

which means no pollutant flux across the water and bottomsurfaces.

Similarly to Eq. (4), the depth-averaged concentrationC isdefined as

C ≡ 1

h

∫ 0

−h

c(z)dz (20)

where only thez dependence is explicitly acknowledged. Hence,the anomaly profilesu′′ andc′′ for velocity and concentration canbe defined according to

c(z) = C + c′′(z), u(z) = U + u′′(z) (21)

where, obviously,∫ 0

−h

c′′(z)dz = 0 and∫ 0

−h

u′′(z)dz = 0 (22)

If Eq. (21) is replaced into Eq. (18) and the depth-averagedoperator is applied to the resulting expression, the followingequation is obtained:

∂C

∂t+ 1

h

∫ 0

−h

∂c′′

∂tdz+ U

∂C

∂x+ U

1

h

∫ 0

−h

∂c′′

∂xdz

(U1) (U2) (A1) (A2)

+ ∂C

∂x

1

h

∫ 0

−h

u′′ dz + 1

h

∫ 0

−h

u′′ ∂c′′

∂xdz

(A3) (A4)

.

=∂

∂x

(1

h

∫ 0

−h

ex dz∂C

∂x

)+ ∂

∂x

[1

h

∫ 0

−h

ex

∂c′′

∂xdz

](Dx1) (Dx2)

+ ∂

∂y

(1

h

∫ 0

−h

ey dz∂C

∂y

)+ ∂

∂y

(1

h

∫ 0

−h

ey

∂c′′

∂ydz

)(Dy1) (Dy2)

+ 1

h

(ez

∂c′′

∂z

) ∣∣∣∣0−h

(Dz)

(23)

Note that theA3-term vanishes due to Eq. (22). TheU2-termalso vanishes for the same reason, as the temporal derivative canbe taken out of the integral (no time variation of the bottom isassumed). In addition, theDz-term is also null taking into accountEqs (19) and (21).

Now, the following assumption will be made:

H1: The parameters which depend on the hydrodynamics(h, U, u′′, ex, ey, ez) vary significantly over much longerhorizontally spatial and temporal scales than the concen-tration(C, c′′).

HypothesisH1 is easily fulfilled close to a point discharge ina fully developed flow (but far from the near field). It meansthat terms containing horizontal spatial derivatives of the formerquantities can be neglected compared with similar terms includ-ing the latter ones. This hypothesis will be used in the followingto simplify Eq. (23).

In the first place, the derivative in theA2-term can be takenout of the integral, which then vanishes due to Eq. (22).

TheA4-term can be written as

1

h

∫ 0

−h

u′′ ∂c′′

∂xdz ≈ 1

h

∂

∂x

∫ 0

−h

u′′c′′dz (24)

If the apparent longitudinal diffusivityEd due to differentialadvection is defined by the relation

1

h

∫ 0

−h

u′′c′′dz ≡ −Ed∂C

∂x(25)

then theA4-term reduces to

1

h

∫ 0

−h

u′′ ∂c′′

∂xdz ≈ −1

h

∂

∂x

(hEd

∂C

∂x

)≈ −Ed

∂2C

∂x2(26)

TermsDx1 andDy1 can be simplified if the depth-averagedhorizontal eddy diffusivities are defined through the followingexpressions:

Ex ≡ 1

h

∫ 0

−h

ex(z)dz (27)

Ey ≡ 1

h

∫ 0

−h

ey(z)dz (28)

One more hypothesis will be introduced in order to simplifyEq. (23), namely:

H2: The longitudinal gradient of the concentration anomalyc′′ issmall compared with the longitudinal gradient of the meanconcentrationC: |∂c′′/∂x| � |∂C/∂x|

Hence, termsDx2 andDy2 can be neglected when compared withtheir respective predecessors.

With the above simplifications, Eq. (23) reduces to

∂C

∂t+ U

∂C

∂x= (Ex + Ed)

∂2C

∂x2+ Ey

∂2C

∂y2(29)

Now, using hypothesisH1, Eq. (18) can be rewritten as

∂c

∂t+ u

∂c

∂x= ex

∂2c

∂x2+ ey

∂2c

∂y2+ ∂

∂z

(ez

∂c

∂z

)(30)

Subtracting Eq. (29) from Eq. (30), the following equation isobtained:

∂c′′

∂t+ U

∂c′′

∂x+ u′′

(∂C

∂x+ ∂c′′

∂x

)

= (ex − Ex)∂2C

∂x2+ ex

∂2c′′

∂x2− Ed

∂2C

∂x2

+ (ey − Ey)∂2C

∂y2+ ey

∂2c′′

∂y2+ ∂

∂z

(ez

∂c′′

∂z

)(31)

Another hypothesis will be introduced, so as to simplifyEq. (31), namely:

H3: Vertical gradients of concentration are much larger than hor-izontal gradients (meaning that the study zone is not in thenear-field region):|∂/∂z| � |∂/∂x|, |∂/∂y|

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Using hypothesesH2 andH3, Eq. (31) reduces to

∂c′′

∂t+ U

∂c′′

∂x+ u′′ ∂C

∂x= ∂

∂z

(ez

∂c′′

∂z

)(32)

Equation (32) is an expression just a little more general than theone obtained by Holleyet al. (1970).

Further simplifications are still possible in Eq. (32). Let us callT andL to the time and longitudinal spatial scales, respectively,of concentration changes (out of the near-field zone, i.e., afterthe initial dilution) andH to the vertical spatial scale (associatedto the flow depth). Hence, the order of magnitude of the fourterms of Eq. (32), once the second one is normalized to 1, are,respectively:

St−1 1β

ε

1

αPe(33)

whereSt = L/UT: Strouhal number;Pe= UH/ez: Peclet num-ber;α = H/L: aspect ratio;β = u′′/U: scale of relative velocityanomaly;ε = c′′/C: scale of relative concentration anomaly.

As ez ∼ u∗f H (see Fig. 2), thenPe ∼ U/u∗ � 1. On theother hand, out of the near-field zone it holds thatα � 1. Thefollowing extra assumptions will be made:

H4: The aspect ratio is much smaller than the ratio between thebottom shear and depth-averaged velocities:αPe � 1

H5: The relative concentration anomaly is much smaller than therelative velocity anomaly:ε � β

H6: The time scale of unsteadiness of the problemT is at leastof the order of the flow time scaleL/U: St >̃ 1

Based on hypothesesH4–H6, it is concluded that the last twoterms of Eq. (32) are dominant. Hence, Eq. (32) can be reduced,to a first order, to the following simplified form:

u′′ ∂C∂x

= ∂

∂z

(ez

∂c′′

∂z

)(34)

The boundary conditions forc′′ arise directly from Eqs (19)and (21):

∂c′′

∂z= 0 for z = 0 and z = −h (35)

Equation (34) is the same as that obtained by Elder (1959)using a much compact derivation.

Note that, using Eqs (22), (34) and (35), it comes out thatc′′ ≡ 0 if dC/dx = 0, i.e.,anomalies in the concentration pro-file occurs only if there is a longitudinal mean concentrationgradient.

6 Concentration profile

Equation (34) can be used to obtain the vertical profile of theconcentration anomalyc′′. A first integration of Eq. (34) leads to

∂c′′

∂ξ= h2 ∂C

∂xe−1z

∫ ξ

0u′′(η)dη (36)

where

ξ = − z

h(37)

Introducing the dimensionless quantity

c∗ = c′′

h(∂C/∂x)(38)

and performing the remaining integration in Eq. (36), thefollowing expression results

c∗(ξ) = h

∫ ξ

0e−1z (σ)

( ∫ σ

0u′′(η)dη

)dσ + co (39)

where the integration constantco arises from Eq. (22), which nowmeans∫ 1

0c∗(ξ)dξ = 0 (40)

Using Eqs (1) and (4), the inner integral in Eq. (39) can beexpressed as∫ ξ

0u′′(ξ′)dξ′ = A + Bξ − u∗f

κ(1 + ξ0f − ξ) ln(1 + ξ0f − ξ)

− u∗s

κ(ξ0s + ξ) ln(ξ0s + ξ) (41)

where

A ≡ u∗s

κξ0s ln ξ0s + u∗f

κ(ξ0f + 1) ln(ξ0f + 1) (42)

B ≡ u∗s

κ[(ξ0s + 1) ln(ξ0s + 1) − ξ0s ln ξ0s]

− u∗f

κ[(ξ0f + 1) ln(ξ0f + 1) − ξ0f ln ξ0f ] (43)

ξos ≡ zos

hξof ≡ zof

h(44)

Now, Eq. (17) for the vertical eddy diffusivity can berewritten as

e−1z = 1

hκufs(ξ + a)

[u∗s

ξ + ξ0s+ u∗f

ξ0f + 1 − ξ

](45)

where

ufs ≡ u∗f |u∗f | − u∗s|u∗s| (46)

a ≡ u∗s|u∗s|ufs

(47)

When Eqs (41) and (45) are introduced into Eq. (39), thefollowing expression for the dimensionless concentration results:

c∗(ξ) = 1

κufs

(u∗sG1 − u∗su∗f

κ(G2 + G6)

− u2∗s

κG3 + u∗f G4 − u2∗f

κG5

)+ co (48)

where

G1(ξ) =∫ ξ

0

(A + Bξ′)(a + ξ′)(ξ0s + ξ′)

dξ′ (49)

G2(ξ) =∫ ξ

0

(1 + ξ0f − ξ′)(a + ξ′)(ξ0s + ξ′)

ln(1 + ξ0f − ξ′)dξ′ (50)

G3(ξ) =∫ ξ

0

ln(ξ0s + ξ′)a + ξ′ dξ′ (51)

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Pollutant dispersion in water currents under wind action475

G4(ξ) =∫ ξ

0

(A + Bξ′)(a + ξ′)(1 + ξ0f − ξ′)

dξ′ (52)

G5(ξ) =∫ ξ

0

ln(1 + ξ0f − ξ′)a + ξ′ dξ′ (53)

G6(ξ) =∫ ξ

0

(ξ0s + ξ′) ln(ξ0s + ξ′)(1 + ξ0f − ξ′)(ξ0s + ξ′)

dξ′ (54)

The integration constantco can now be obtained usingcondition (40):

co = − 1

κufs

(u∗sF1 − u∗su∗f

κ(F2 + F6) − u2∗s

κF3

+ u∗f F4 − u2∗f

κF5

)(55)

where the integralsFi, with 1 ≤ i ≤ 6, are given by

Fi =∫ 1

0Gi(ξ)dξ (56)

Taking into account that

Gi(ξ) = (1 − ξ)dGi

dξ− d

dξ[(1 − ξ)Gi] (57)

Eq. (56) transforms into

Fi =∫ 1

0(1 − ξ)

dGi(ξ)

dξdξ − [(1 − ξ)Gi(ξ)]1

0 (58)

The last term of the right-hand side of Eq. (58) is null becauseGi(0) = 0, as can be readily seen from Eqs (49) to (54). Hence,Eq. (58) can be expressed as

F1 =∫ 1

0

(1 − ξ)(A + Bξ)

(a + ξ)(ξ0s + ξ)dξ (59)

F2 =∫ 1

0

(1 − ξ)(1 + ξ0f − ξ)

(a + ξ)(ξ0s + ξ)ln(1 + ξ0f − ξ)dξ (60)

F3 =∫ 1

0

(1 − ξ) ln(ξ0s + ξ)

a + ξdξ (61)

F4 =∫ 1

0

(1 − ξ)(A + Bξ)

(a + ξ)(1 + ξ0f − ξ)dξ (62)

F5 =∫ 1

0

(1 − ξ)

a + ξln(1 + ξ0f − ξ)dξ (63)

F6 =∫ 1

0

(1 − ξ)(ξ0s + ξ)

(1 + ξ0f − ξ)(a + ξ)ln(ξ0s + ξ)dξ (64)

Elder’s (1959) case is obtained from Eqs (48) and (55) byassuming absence of wind(u∗s = 0), and working to the zerothorder inξof , taking into account thatξof ∨ 1:

c∗E(ξ) = − 1

κ2

∫ ξ

0

ln(1 − ξ′)ξ′ dξ′ + coE (65)

coE = − 1

κ2

∫ 1

0

(ξ − 1)

ξln(1 − ξ)dξ = 1

κ2

(1 − π2

6

)(66)

As an illustration, Fig. 3 presents the concentration anomalyprofile as given by Eq. (48) for the same particular case of Fig. 2.

-1.0

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

-8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 10.0

c*, u"/U

z/h

With wind No wind Velocity

Figure 3 Vertical distribution of concentration anomaly.

The corresponding profile in the absence of wind(u∗s = 0) is alsoshown. It is observed that the wind acting in the same directionas the water current causes an intensification of the concentrationanomaly.

It is interesting to remark that the calculation of theGi integrals(49)–(54), which were made numerically to obtain Fig. 3, are onlynecessary if the concentration anomaly is required, but it can beavoided if only the dispersion coefficient is needed (see below).

7 Longitudinal dispersion coefficient

Knowing the velocity and concentration anomaly profiles, it isnow possible to obtain the longitudinal dispersion coefficientthrough its definition (25), which can be rewritten as

Ed = −h

∫ 1

0u′′(ξ)c∗(ξ)dξ (67)

The dimensionless dispersion coefficient is defined as

E∗d ≡ Ed

u∗f h(68)

Then, Eq. (67) leads to

E∗d = − 1

u∗f

∫ 1

0u′′(ξ)c∗(ξ)dξ (69)

Introducing Eqs (1), (4), (48), and (55) into Eq. (69), thefollowing expression is obtained:

E∗d = γ

κ2u∗f ufs

{u∗sF(1, 1, 1) − u∗su∗f

κ[F(2, 1, 1)

+ F(6, 1, 1)] − u2∗s

κF(3, 1, 1) + u∗f F(4, 1, 1)

− u2∗f

κF(5, 1, 1)

}

+ 1

κ2ufs

{−u∗sF(1, 2, 1) + u∗su∗f

κ[F(2, 2, 1)

+ F(6, 2, 1)] + u2∗s

κF(3, 2, 1) − u∗f F(4, 2, 1)

+ u2∗f

κF(5, 2, 1)

}

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476 Menéndez and Laciana

+ u∗s

κ2u∗f ufs

{u∗sF(1, 1, 2) − u∗su∗f

κ[F(2, 1, 2)

+ F(6, 1, 2)] − u2∗s

κF(3, 1, 2) + u∗f F(4, 1, 2)

− u2∗f

κF(5, 1, 2)

}(70)

where

γ = u∗s[(ξ0s + 1) ln(ξ0s + 1) − ξ0s ln ξ0s − 1]− u∗f [(ξ0f + 1) ln(ξ0f + 1) − ξ0f ln ξ0f − 1] (71)

F(i, 1, 1) = −∫ 1

0Gi dξ (72)

F(i, 1, 2) =∫ 1

0Gi ln(ξ0s + ξ)dξ (73)

F(i, 2, 1) =∫ 1

0Gi ln(ξ0f + 1 − ξ)dξ (74)

with 1 ≤ i ≤ 0.6. Equations (72)–(74) can be rewritten in theform:

F(i, µ, ν) =∫ 1

0[(µ − 1)(χ(ξ) ln χ(ξ) − χ(1) ln χ(1))

− (ν − 1)(ψ(ξ) ln ψ(ξ) − ψ(1) ln ψ(1))

+ ξ − 1]dGi

dξdξ (75)

where

χ ≡ ξ0f + 1 − ξ, ψ ≡ ξ0s + ξ (76)

and(µ, ν) = (1, 1), (1, 2), (2, 1). The integrals in Eq. (75) mustbe solved numerically (e.g., using Simpson’s rule).

Equation (70) is the general expression for the dispersion coef-ficient in the presence of wind. It is straightforward to verify thatEq. (70) reduces to Elder expression|E∗

d| ≈ 5.9 (Elder, 1959)for the case of no wind; this is shown in the Appendix A. Notethat, as already commented in the previous section, the inte-grals (75) do not depend on theGis but on their derivatives,which are the integrands of Eqs (49)–(54); hence, the integra-tions in Eqs (49)–(54) are not necessary to obtain the dispersioncoefficient.

The particular conditions used to obtain Figs 2 and 3 are nowtaken as a reference, around which the different parameters arevaried in order to analyze what is the corresponding change inthe dispersion coefficient. For the sake of clarity, the conditionsfor the reference state are stated again:h = 5 m,U = 0.50 m/s,n = 0.025,W (wind velocity) = 20 km/h,Ho (wave height) =0.10 m, from which the following values arise:u∗s/κU ≈ 0.043,u∗f /κU ≈ 0.15,ξos = 0.0013, andξof = 0.00050. The variationranges, trying to represent a spectrum of practical conditions,are the following: 1≤ h (m) ≤ 20, 0.10 ≤ U (m/s) ≤ 1.50,0.015≤ n ≤ 0.050,−100≤ W (km/h)≤ 100, 0.05 ≤ Ho (m)≤1.00, from which it results−0.2 ≤ u∗s/κU ≤ 0.2, 0.08 ≤u∗f /κU ≤ 0.4, 10−4 ≤ ξos ≤ 10−2, and 10−5 ≤ ξof ≤ 5 × 10−3.

Figure 4 shows the variation ofE∗d with u∗s/κU. As expected,

dispersion increases with wind speed acting in the same directionas the water current, and vice versa. Of course, Elder’s value is

0

5

10

15

20

25

30

-0.35 -0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20

us*/Uk

Ed*

Present Elder

Figure 4 Variation of dispersion coefficient with surface shear stress.

0

2

4

6

8

10

12

14

16

0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

u*f/kU

Ed*

Present Elder

Figure 5 Variation of dispersion coefficient with bottom shear stress.

ξ

0

2

4

6

8

10

12

14

0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

Ed*

Present Elder

0s

Figure 6 Variation of dispersion coefficient with surface roughness.

obtained foru∗s = 0. Though the dispersion coefficient decreasesbelow Elder’s value foru∗s < 0, within the defined variationrange, it is expected that it will eventually increase again forhigher negative wind velocities due to symmetry considerations;but this observation only has a theoretical value.

In Fig. 5, the variation ofE∗d with u∗f /κU is presented. The

trend is that the dispersion coefficient approaches Elder’s valuewhen the bottom shear stress increases relative to the wind shearstress, as expected.

The variation ofE∗d with ξ0s is shown in Fig. 6. The increase

in water surface roughness produces lower dispersion, due to a

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Pollutant dispersion in water currents under wind action477

ζ

0

2

4

6

8

10

12

14

0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

Ed*

Present Elder

0f

Figure 7 Variation of dispersion coefficient with bottom roughness.

0

10

20

30

40

50

60

70

-100 -80 -60 -40 -20 0 20 40 60 80 100

W (km/h)

Ed*

Elder Present for 0.25 m/s Present for 0.5 m/s

Figure 8 Variation of dispersion coefficient with wind velocity.

decrease in the concentration anomaly. However, it is observedthat the sensitivity to this parameter is relatively low.

Finally, Fig. 7 presents the variation ofE∗d with ξ0f . The trend

is similar to the previous case, due to the same reason, but aslightly higher sensitivity to this parameter is observed. In anycase, it seems that the influence ofξ0s andξ0f on E∗

d could beneglected in practical applications.

As a more practical way of presenting results, Fig. 8 showsE∗

d as a function of wind velocity (in km/h), forξ0s = 0.001,ξ0f = 0.0001, and for two values of the flow velocityU (0.25and 0.50 m/s). Obviously, the trends are similar to Fig. 4.

8 Non-coplanar case

A heuristic generalization of the above theory is proposed forthe case in which the wind and flow directions are not copla-nar, i.e., the 2D-horizontal situation. It is postulated that theabove analysis can be applied independently to two perpendicularplanes: one coplanar with the mean current and one normal to it,where the mean current is, then, null. For each plane, the corre-sponding component of the wind velocity is used. This approachwas originally proposed in Bombardelli (1998), and reported byBombardelli and Menéndez (1999).

The expressions for the null-current case(U = 0) constitute arelatively simple specialization of the above obtained equations.According to Eq. (6), the following relationship between the shearvelocities holds

u∗f = −βos

βofu∗s (77)

Introducing Eq. (77) into Eq. (70) it is obtained that

|E∗d| = 24.9 (78)

i.e., a value about 4.2 times bigger than Elder’s value.

9 Conclusions

A closed expression has been obtained for the dimensionless dis-persion coefficient due to differential advection in the presenceof wind. The value of this dispersion coefficient is larger thanElder’s value (≈5.9), associated to the particular case of no wind,for winds acting in the same direction as the flow; and vice versa,at least for a practical range of wind velocities.

For the case in which the wind acts along a direction non-coplanar with the mean flow direction, it is proposed to apply theanalysis independently to the coplanar direction and its normal,where the mean current is, then, null. For this normal direction,the dimensionless dispersion coefficient value is around 24.9. Inthis way, the theory can be applied to 2D-horizontal pollutanttransport modeling.

Appendix A: Elder’s limit

Let first assume thatu∗s = 0. Then, Eqs (42), (43), (46), (47),(70), and (71) reduce to

A = u∗f

κ(ξ0f + 1) ln(ξ0f + 1) (A1)

B = − u∗f

κ[(ξ0f + 1) ln(ξ0f + 1) − ξ0f ln ξ0f ] (A2)

ufs = u∗f |u∗f | (A3)

a = 0 (A4)

E∗d = γ

κ2u∗f |u∗f |{F(4, 1, 1) − u∗f

κF(5, 1, 1)

}

+ 1

κ3

u∗f

|u∗f |F(5, 2, 1) (A5)

γ = −u∗f [(ξ0f + 1) ln(ξ0f + 1) − ξ0f ln ξ0f − 1] (A6)

Now, if it is further assumed thatξ0f → 0, then

ξ0f ln ξ0f → 0 (A7)

and Eqs (A1), (A2), and (A6) transform into

A → 0 (A8)

B → 0 (A9)

γ → u∗f (A10)

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478 Menéndez and Laciana

Using Eqs (A8) and (A9) in Eq. (52) we obtain

G4 → 0 (A11)

From Eq. (A11) and Eq. (72) it turns out that

F(4, 1, 1) → 0 (A12)

Additionally, using Eq. (A4), Eq. (53) reduces to

G5 →∫ ξ

0

ln(1 − ξ)

ξdξ (A13)

With the simplifications (A10) and (A12), and taking intoaccount Eqs (73) and (74), Eq. (A5) can now be written as

E∗d → u∗f

|u∗f |1

κ3F (A14)

where

δF ≡ F(5, 2, 1) − F(5, 1, 1) =∫ 1

0G5(1 + ln(1 − ξ))dξ

(A15)

which can be rewritten as

δF =∫ 1

0G5

dI

dξdξ (A16)

with

I =∫ ξ

0[1 + ln(1 − ξ)]dξ = (ξ − 1) ln(1 − ξ) (A17)

Taking into account Eqs (A13) and (A17), Eq. (A16) can betransformed into

δF = G5I|10 −∫ 1

0

dG5

dξI dξ =

∫ 1

0

(1 − ξ)

ξ[ln(1 − ξ)]2 dξ

(A18)

or, replacing Eq. (A18) into Eq. (A14),

E∗d = u∗f

|u∗f |1

κ

∫ 1

0

(1 − ξ)

ξ[ln(1 − ξ)]2 dξ ≈ 5.9 sign(u∗f )

(A19)

which is Elder’s expression, except for the consideration of thepossibility thatu∗f be negative.

Acknowledgement

This work was partly supported with grant number 1609, Uni-versity of Buenos Aires.

Notation

A, B = ConstantsC = Depth-averaged concentration

c(x, y, z) = Concentration of a dissolved pollutantc′′ = Concentration profile anomalyco = Integration constant

c∗E(ξ) = Elder’s concentration profilecoE = Elder’s integration constantex = Eddy diffusivity along flow directioney = Eddy diffusivity along lateral directionez = Vertical eddy viscosity and diffusivity

Ed = Apparent longitudinal diffusivityE∗

d ≡ Ed/(u∗f h) = Dimensionless dispersion coefficientEx, Ey = Depth-averaged horizontal eddy

diffusivitiesFi, F(i, 1, 1),

F(i, 1, 2), F(i, 2, 1) = Definite integralsGi(ξ) = Indefinite integrals

g = Acceleration of gravitygx = Component of acceleration of gravity

in the flow directionH = Vertical spatial scaleHo = Wave heighth = Water depthIo = Bottom slopeks = Effective roughness heightL = Longitudinal spatial scale of

concentration changen = Manning roughness coefficient

Pe = UH/ez = Peclet numberSt = L/UT = Strouhal number

T = Time scale of concentration changet = Time coordinate

U = Depth-averaged velocityu∗f = Shear velocity on the bottom surfaceu∗s = Shear velocity on the water surfaceu′′ = Velocity profile anomalyv = Velocity component in they direction

W = Wind velocityw = Velocity component in thez directionx = Flow directiony = Lateral horizontal coordinatez = Vertical coordinate

zos = Effective water surface roughnesszof = Effective bottom surface roughnessκ = Von Karman constantτf = Bottom shear stressρ = Fluid density

τxx, τxy, τxz = Components in thex direction of theReynolds stresses on surfaces pointinglocally towards thex, y andz

directions, respectivelyτs = Wind shear stress acting on the water

surfaceα = H/L = aspect ratio

β = u′′/U = Scale of relative velocity anomalyγ = Constants

ε = c′′/C = Scale of relative concentrationanomaly

References

1. Fischer, H.B. (1967). “The Mechanics of Dispersionin Natural Streams”.J. Hydraul. Div. ASCE93(HY6),187–216.

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Pollutant dispersion in water currents under wind action479

2. Fischer, H.B. (1973). “Longitudinal Dispersion and Turbu-lent Mixing in Open-channel Flow”.Annu. Rev. Fluid Mech.5, 59–78.

3. Elder, J.W. (1959). “The Dispersion of Marked Fluid inTurbulent Shear Flow”.J. Fluid Mech.5, 544–560.

4. Menéndez, A.N. (2003). “Selection of Optima Mathemat-ical Models for Fluvial Problems”.Third IAHR Sympo-sium on River, Coastal and Estuarine Morphodynamics,Barcelona, Spain, September, 2003.

5. Bombardelli, F.A. (1998). “Un modelo cuasi-tridimen-sional para simular flujos en aguas poco profundas bajo laacción del viento”. Master Thesis, Faculty of Engineering,University of Buenos Aires, Buenos Aires, Argentina.

6. Bombardelli, F.A. and Menéndez, A.N. (1999). APhysics-Based Quasi-3D Numerical Model for the Simu-lation of Wind-Induced Shallow Water Flows. InternationalWater Resources Engineering, ASCE, Seattle, WA, USA.

7. Reid, R.O. (1957).Modification of the Quadratic Bottom-Stress Law for Turbulent Channel Flow in the Presence

of Surface Wind-Stress. Technical Memorandum No. 93,Beach Erosion Board, Corps of Engineers, USA.

8. Tsuruya, H. (1985). “Turbulent Structure of CurrentsUnder the Action of Wind Shear”.J. Hydrosci. Hydraul.Engng.13(1), 23–43.

9. White, F.M. (1974).Viscous Fluid Flow. McGraw-Hill,New York.

10. Schlichting, H. (1979). Boundary-Layer Theory.McGraw-Hill, New York.

11. Henderson, F.M. (1966).Open Channel Flow. MacmillanPublishing Co., Inc., New York.

12. Kondo, J. (1976). “Parameterization of Turbulent Trans-port in the Top Meter of the Ocean”.J. Phys. Oceanogr.6,712–720.

13. Holley, E.R., Harleman, D.R.F. and Fischer, H.B.(1970). “Dispersion in Homogeneous Flow”.J. Hydraul.Div. ASCE96(HY8), 1691–1709.

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