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A balanced treatment of secondary currents, turbulence and dispersion in a depth-integrated hydrodynamic and bed deformation model for channel bends

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A balanced treatment of secondary currents, turbulence and dispersion in adepth-integrated hydrodynamic and bed deformation model for channel bends
Lorenzo Begnudelli
a,
*
, Alessandro Valiani
b
, Brett F. Sanders
c
a
Department of Civil Environmental Engineering, University of Trento, Trento 38100, Italy
b
Department of Engineering, University of Ferrara, Ferrara 44100, Italy
c
Department of Civil and Environmental Engineering, University of California, Irvine, CA 92697, United States
a r t i c l e i n f o
Article history:
Received 25 May 2009Received in revised form 6 October 2009Accepted 7 October 2009Available online 28 October 2009
Keywords:
Bed morphologySediment transportChannel bendsDepth-integrated modelFinite volume method
a b s t r a c t
Thisworkdealswiththeformulationandnumerical implementationofatwo-dimensionalmathematicaland numerical model describing open channel hydrodynamics, sediment and/or scalar transport and riv-erbed evolution in curved channels. It is shown that a well balanced 2D model can predict ﬂow features,sediment and scalar concentration, and bed elevation with an accuracy that is suitable for practical riverengineering. The term ‘‘balanced” implies that important physical processes are modeled with a similardegree of complexity and exhaustiveness. The starting point of the model formulation is the assumptionof self-similarity of vertical velocity proﬁles (horizontal velocities in the longitudinal and transversedirections), that are scaled by shear velocity and streamline curvature, both resolved by the model.The former is scaled by a bed-resistance coefﬁcient that must be estimated or calibrated – as usual –onaapplication-speciﬁc basis, andthelatter is computed byanew, grid-based but grid orientation inde-pendent, scheme that acts on the discrete solution. All processes, including bottom shear, momentumdispersion, scalar dispersion, turbulent diffusion, bed load, and suspended load, are modeled using phys-icallybased,averagedvaluesofempiricalorsemi-empiricalconstants,andconsistentlywiththeassumedvelocity proﬁles (and bed-generated turbulence). Bed deformation modeling can be implemented witheitheranequilibriumornon-equilibriumformulationoftheExnerequation,dependingontheadaptationlength scale, which must be taken under consideration if signiﬁcantly larger than the length scale of thespatialdiscretization. Thegoverningequationsaresolvedbyhigh-resolution, unstructured-gridGodunovmethod, which is elsewhere tested and shown to be reliable and second-order accurate. Application of the model to laboratory test cases, using standard parameter values and previously reported bed-resis-tance coefﬁcients, gives results comparable to many 2D and 3D models previously applied to the samecases,mostpartofwhichbeneﬁtfromcase-speciﬁcparametertuning.Thereareobviouslyintrinsiclimitsto the descriptive ability of 2D models in river modeling, but the results of this study point to the utilityand cost-effectiveness of a well-designed 2D model.
2009 Elsevier Ltd. All rights reserved.
1. Introduction
Numerical modeling of ﬂow and bed deformation in alluvialchannels is of primary importance in river engineering to simulatethe morphological changes and to predict the impact of hydraulicstructures on channel stability, navigability and habitat. To be suc-cessful in channel bends, numerical models must account forimportant three-dimensional (3D) ﬂow features, in particular, aspiral motion where ﬂuid parcels near the free surface deviate to-wards the outer bank, while parcels near the bed deviate towardstheinnerbank[8,9,16,26–28,33,34,46,47,67,97].Thecomponentof
thisspiral motioninthetransversedirectionaretermedsecondarycurrents and are responsible for a redistribution of longitudinalvelocitytowardstheouterbank,aswellasthenettransportofsed-iment towards the inner bank [43,46,47,52]. The 3D structure of
channelﬂowinbendshasmotivatedmany3Dmodeldevelopmentstudies to predict sediment transport and bed deformations[1,51,53,61,64,74,91].However,thecomputationalcostof3Dmod-
els is high and often prohibitive for engineering analysis and de-sign studies. Moreover, the physical knowledge of most part of sediment transport processes is not established concerning 3D as-pects like pick-up ﬂuxes evaluation, signiﬁcant local bed slope ef-fects on particle entrainment, sediment hiding by different grainsizes,effectsofsedimentsuspensionduetoturbulentburstsratherthantomeanﬂowconvection, andso on. This has motivatedmanyresearchers to formulate depth-integrated, two-dimensional (2D)models [2,24,25,29,43–47,52,54,57,60,75,77,94–96]. The challenge
0309-1708/$ - see front matter
2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.advwatres.2009.10.004
*
Corresponding author. Tel.: +39 0461 882629.
E-mail address:
lorenzo.begnudelli@ing.unitn.it (L. Begnudelli).Advances in Water Resources 33 (2010) 17–33
Contents lists available at ScienceDirect
Advances in Water Resources
journal homepage: www.elsevier.com/locate/advwatres
of 2D modeling is to reasonably depict 3D ﬂow features that affectﬂow,sedimenttransport,andmorphologicalchange.Therearetwoaspects to this challenge. The ﬁrst is to reconstruct 3D (horizontal)velocities from 2D ﬂow attributes such as the depth, depth-aver-aged velocity, shear velocity, and 2D streamline curvature. Thiscan be accomplished by adopting one of several different verticalvelocity proﬁles for the longitudinal and transverse directions, en-abling key ﬂow features such as the near-bed velocity to be esti-mated. The second is to account for the impact of 3D ﬂowfeatures on 2D ﬂow attributes which is addressed using so-calleddispersive terms that arise when convective non-linearities in the3D equations are vertically integrated [37,43,46–48,60]. The as-sumed vertical velocity proﬁles dictate the expression of theseterms, and are therefore dependent on 2D ﬂow attributes. 2D ﬂowmodels may also need to consider turbulence terms (i.e., Reynoldsstresses) particularly in the context of recirculating ﬂows[2,7,37,44,54,57]. An overview of 1D, 2D, and 3D modeling ap-proaches for river dynamics is presented by Wu [93].There are several examples of 2D ﬂow, mass transport, andmorphological models that carefully consider Reynolds stresseswith turbulence transport models, but ignore dispersive terms,and do not perform well in channel bends [54,57,73,94]. Hence,dispersive terms must be considered in modeling studies of chan-nel bends. However, the appropriate level of approximation is notclear.BernardandSchneider[15]andBernard[14]presentamodel
that considers the transport of secondary-current vorticity, whichincreases due to streamline curvature and is dissipated by a num-beroffactors.Thisisthoughttobeimportantinchannelswithvar-iable curvature where there is a lag between curvature andsecondarycurrents. Asimpleralternativeistoassumealocal equi-librium between transverse pressure gradients and radial acceler-ation (e.g. [58]). When this approximation is adopted, it isreasonable to address Reynolds stresses with a similar level of approximation.Forexample,turbulenceintensitiesanddissipationmechanisms can be assumed to be locally controlled by bed shear,not transported as in 2D
k
—
type models (e.g. [5,63]). Finally,upon formulation of a model that similarly approximates disper-siveandturbulencetermsanumericalsolverisrequiredwithgoodstability and low dissipation properties over a wide range of ﬂowconditions. For example, Abad et al. [2] adopted the model byBernard[14]whichismathematicallyformulatedforchannelbendapplication, but the numerical solver limits the model to a re-stricted set of ﬂow conditions, since the secondary ﬂow correctiononthebedloaddoesnotseemsoaccurateforhigh-curvaturecases,whichin the opinion of Abad et al. [2] is due to the curvature eval-uation method used in the paper.Inthisstudywe developa balanced2Dmodel for hydrodynam-ics, mass transportandbedmorphologyforchannelbendsthatap-plies asimilar level of approximationrelative to theformulationof secondary currents, dispersive terms and turbulence, as describedabove, utilizes 3D ﬂow structure information to model sedimenttransport, and adopts a stable and robust numerical solver. Theindependent validationof the simple treatment of dispersion, bothfor the passive scalar contaminant transport and the sedimenttransport dynamics in curved ﬂows, is considered to be a reliablesupport to a self consistent mechanical scheme of the real physics.We are motivated by the diversity of previously reported modelswhichvaryconsiderablyincomplexitybutrarelyoffersigniﬁcantlybetterresults. Indeed, giventheuncertaintyinsedimentdynamics,the beneﬁt of hydrodynamic model complexityshouldbe carefullyscrutinized.Hence,wearemotivatedtostreamlinemodelformula-tion, making it as simple as possible, and to minimize the numberof parameters that must be speciﬁed by user, and the associatedcalibration requirements. This is accomplished by adopting estab-lished values of model parameters wherever possible, and relyingheavily on a case-speciﬁc bed-resistance coefﬁcient which engi-neersareaccustedtoestimatingorcalibratingonasite-speciﬁcba-sis. For example, the velocity proﬁles (vertical distribution of longitudinal andtransversevelocities) arescaled byshearvelocity,which is turn related to the chosen bed-resistance parameter, andstreamline curvature, which is computed by a new grid-basedschemethatactsonthediscretesolution.Asisdescribedinthefol-lowing sections, all processes of importance are modeled consis-tent with the assumed velocity proﬁles (and bed-generatedturbulence) including bottom shear, momentum dispersion, scalardispersion, turbulent diffusion, bed load, and suspended load. Themodel is evaluated using a set of laboratory channel bend testproblems with validation data. In addition, predictions by severalother 2D and 3D models are compared and provide insight intothe merit of the proposed formulation, its utility for engineeringanalysis and design purposes and its limitations. Accuracy relativeto the overall simplicity of the model is emphasized. Having inmind practical river engineering applications, we also stress theimmense challenge (perhaps even futility) of a 3D description of highly variable near-bed dynamics that bear on vorticity andpick-up capacity. According to the present experience, a well-de-signed and carefully veriﬁed 2D scheme that adopts analyticalmodels for critical 3D ﬂow features is able to reproduce the essen-tial aspects of the ﬂow ﬁeld, solute and sediment transport, andbed deformation for a manageable computational cost.
2. Flow model
By assuming a hydrostatic pressure distribution, the Reynolds-Averaged Navier–Stokes equations are integrated over the depthof a river (
z
direction) to obtain the following system of equationsthat conserve mass and balance momentum in the horizontaldirections,
x
and
y
,
@
U
@
t
þ
@
E
@
x
þ
@
G
@
y
¼
S
ð
1
Þ
where
U
¼ð
h
uh
v
h
Þ
T
;
h
is the depth,
u
and
v
are the depth-averaged velocity in the
x
and
y
directions, respectively, and
E
,
G
and
S
are deﬁned as,
E
¼
h
uh
u
2
þ
12
gh
2
þ
h
ð
T
xx
þ
D
xx
Þ
h
u
v
þ
h
ð
T
yx
þ
D
yx
Þ
264375
ð
2
Þ
G
¼
h
v
h
u
v
þ
h
ð
T
xy
þ
D
xy
Þ
h
v
2
þ
12
gh
2
þ
h
ð
T
yy
þ
D
yy
Þ
264375
ð
3
Þ
S
¼
0
gh
@
z
b
@
x
c
D
uU
gh
@
z
b
@
y
c
D
v
U
264375
ð
4
Þ
where
g
is the acceleration due to gravity;
z
b
is the bed elevation;
U
¼
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
u
2
þ
v
2
p
;
c
D
is the bed drag coefﬁcient, related to theManning’s coefﬁcient
n
M
from the relation
c
D
¼
gn
2
M
h
1
=
3
and tothe Chézy coefﬁcient
C
by
c
D
¼
gC
2
;
T
xx
;
T
xy
;
T
yx
;
T
yy
are depth-integratedReynoldsstresses;
D
xx
;
D
xy
;
D
yx
,
D
yy
aredispersionterms.
The most important source of momentum dissipation in rivermodeling is bottom shear which is modeled with a quadratic draglawabove.Aspatiallydistributedresistancecoefﬁcientcaninprin-ciplebeusedwithoutcomplicationintheabovemodel, butforriv-er modeling the estimation of such distributions is rarely possiblewithout detailed knowledge of bed forms. Exceptions include thecase of ﬂood inundation modeling where channel and overbanksurfaces are distinctly different. Nevertheless, note that bottomshear stress is computed locally by the model as
s
o
¼
q
c
D
U
2
, andthe shear velocity is simply
u
¼ð
c
D
Þ
1
=
2
U
. The latter forms a key
18
L. Begnudelli et al./Advances in Water Resources 33 (2010) 17–33
component of our Reynolds stress and dispersion formulations de-scribed next.
2.1. Reynolds stress formulation
Depth-integrated Reynolds stresses account for the transfer of energy to recirculating ﬂows and should be modeled to predictplanform circulation [37]. Most models adopt the Boussinesq
approximation to describe Reynolds stresses in terms of an eddyviscosity
m
t
and gradients in depth-averaged velocities (e.g.[2,30,52,57]). This gives the Reynolds stresses as [55],
T
xx
¼
2
ð
m
t
þ
m
Þ
@
u
@
x
ð
5
Þ
T
xy
¼
T
yx
¼ ð
m
t
þ
m
Þ
@
u
@
y
þ
@
v
@
x
ð
6
Þ
T
yy
¼
2
ð
m
t
þ
m
Þ
@
v
@
y
ð
7
Þ
where
m
t
istheturbulenceeddyviscosityand
m
isthemolecularvis-cosity which can be ignored in environmental applications as it isseveral orders of magnitude smaller [36].
Researchers have adopted models of varied complexity for theeddy viscosity. The simplest option is a constant eddy viscosity[37];algebraicclosureschemesoftheform
m
t
u
h
areonlyslightlymore complicated, while
k
—
turbulent transport models requiresolution of additional scalar transport equations [5,63]. Barbarusti
et al. [7] report that the importance of turbulent transport modelsrelative to recirculation zone modeling diminishes in relativelyshallow conditions where bottom shear dominates turbulenceand dampens turbulence caused by lateral shear. In this limit, theeddy viscosity predicted by
k
—
models converges to that of thealgebraicmodelunderuniformﬂowconditions[5,63].Furthermore,
Barbarusti et al. [6] report experimental work on recirculationzones and subsequent modeling [7] shows that depth-integratedmodels of 2D recirculation zones are insensitive to the eddyviscosity when
c
D
W
=
h
>
0
:
1, where
W
represents the width of therecirculationzone measured in the transverse direction.Forthisstudyanalgebraicturbulencemodelisadoptedbecauseitstrikesanattractivebalancebetweenmodelcomplexityandper-formance. As indicated earlier, the distribution of shear stress iscomputedtoaccountforbedshear, soitistrivial tocomputeadis-tribution of
m
t
¼
a
u
h
where
a
is a proportionality constant, set inthis study to 0.08 [7,63]. A consequence of this approximation is
that poor results are to be expected in applications with recircula-tion zones if
c
D
W
=
h
>
0
:
1, in which case inclusion of turbulencetransport in the model should be considered.
2.2. Dispersion formulation
Whereas Reynolds stresses account for energy transfer to circu-latingﬂows,dispersiontermsaccountforthetransferofenergyoutof circulating ﬂows and are most important in channel bends[37,43,46–48,60]. Dispersion terms can be expressed as
D
xx
¼
1
h
Z
h
0
ð
u
ð
z
Þ
u
Þ
2
dz
ð
8
Þ
D
xy
¼
D
yx
¼
1
h
Z
h
0
ð
u
ð
z
Þ
u
Þð
v
ð
z
Þ
v
Þ
dz
ð
9
Þ
D
yy
¼
1
h
Z
h
0
ð
v
ð
z
Þ
v
Þ
2
dz
ð
10
Þ
where
u
ð
z
Þ
and
v
ð
z
Þ
are the
x
and
y
components of the horizontalvelocity distribution, respectively. We note that
u
ð
z
Þ
and
v
ð
z
Þ
arealso functions of
x
;
y
, and
t
, as well as
u
and
v
, but this dependenceis not shown to simplify the presentation.
The dispersion stresses can be evaluated analytically once avelocity distribution is adopted. For the streamwise componentprevious researchers have adopted power-law distributions[37,45,92], log-law distributions [30], and linear models
[30,41,92], while linear models have been used extensively for
the transverse component [41,45]. A power-law is adopted here
for the longitudinal component and a linear model is adopted forthe transverse component. Indicating with
u
‘
and
u
t
the velocitycomponents in longitudinal and transverse direction, respectively,the longitudinal velocity is then given by [58],
u
‘
ð
z
Þ
U
¼
m
þ
1
m z h
1
=
m
ð
11
Þ
where
m
¼
j
U
=
u
¼
j
ð
8
=
f
Þ
1
=
2
¼
j
C
=
g
1
=
2
¼
j
n
M
g
1
=
2
=
h
1
=
6
and
j
¼
0
:
41isthevonKarman’sconstant.Thetransversevelocityisgivenby[58],
u
t
ð
z
Þ
v
s
¼
2
z h
1
;
v
s
¼
2
m
þ
12
j
2
mhRU
ð
12
Þ
where
v
s
representsthetransversevelocityatthefreesurfaceand
R
is the local radius of curvature. An important assumption of thetransverse velocity proﬁle is that transverse pressure gradientsare in equilibrium with the radial acceleration [58]. This is a goodassumption when the radius of curvature remains constant in thestreamwise direction, but questionable when the radius of curva-turevaries.BernardandSchneider[15]derivedatransportequationfor streamwise vorticity that considers vorticity generation (due tocurvature) and dissipation mechanisms to estimate the dispersionterms.ThisisimplementedinthemodelSTREMR [14]whichhasre-cently been modiﬁed further by Abad et al. [2]. Another approachproposed by Ghamry and Stefﬂer [40,41] involves the solution of
vertically averaged and moment (VAM) equations. In this approacha velocity distribution is adopted for
u
;
v
, and
w
, the vertical com-ponent of velocity, as well as pressure
p
. Each of these distributionsare modeled with parameters (i.e., moments) and separate equa-tions are solved for each. For example, the
x
component of velocityismodeledas
u
¼
u
o
þ
u
1
ð
2
z
=
h
1
Þ
andevolutionequationsarede-visedfromadepth-averagingprocesstopredict
u
o
and
u
1
.Inall,thismethod requires the solution of a coupled system of 10 equations.The streamwise vorticity and the VAM methods represent alterna-tives to Eq. (12) that promise a more complete process description.However, eachof theserequiresthesolutionof additional transportequations which mandates greater computational effort. In addi-tion, given that a relatively simple algebraic turbulent closurescheme was adopted, for consistency a dispersion treatment of asimilar level of complexity may be justiﬁed.
Integrationof Eqs. (8)–(10) usingthesevelocityproﬁlesleadsto[92],
D
‘‘
¼
U
2
m
ð
2
þ
m
Þ
;
D
‘
t
¼
D
t
‘
¼
v
s
U
1
þ
2
m
sign
½
R
;
D
tt
¼
v
2
s
3
ð
13
Þ
wherethesubscripts
‘
and
t
havebeenusedtoemphasizethattheseexpressions apply to a coordinate system aligned with the longitu-dinal and transverse coordinate systems, respectively. Further,
R
istaken to be positive when currents bend clockwise along thestreamwise direction, while negative
R
implies the opposite. Totransformthedispersiontermsrelativeto
x
and
y
directions, arota-tional transformation can be applied as follows,
D
xx
D
xy
D
yx
D
yy
¼
M
ð
u
Þ
D
‘‘
D
‘
t
D
t
‘
D
tt
M
T
ð
u
Þ
;
M
ð
u
Þ ¼
cos
u
sin
u
sin
u
cos
u
ð
14
Þ
where
u
is the angleof the depth-averagedhorizontal velocityvec-tor measuredcounter-clockwise fromthe
x
axis and
M
ð
u
Þ
accountsfor a counter-clockwise rotational transformation by an angle
u
,and
M
T
represents the transpose of
M
.
L. Begnudelli et al./Advances in Water Resources 33 (2010) 17–33
19
We remark that this dispersion formulation requires no addi-tional parameters. The required input includes
U
;
m
;
u
and
R
,and it is straightforward to compute all of these except
R
basedusing model predictions of
h
;
u
;
v
and the assumed resistanceparameter. Curvature evaluation is a unique problem facing 2Dmodels.In3D,the3Dﬂowfeaturesareresolvedsothereisnoneedto reconstruct velocities. In 1D, no attempt is made to model ﬂowdirection. Focusing on 2D models, strategies to locally estimate
R
have been devised for idealized cases where ﬂow is aligned withthe curvilinear grid [23,34,43,45–47,73,79]. These cases have in-volved channels of constant width with bends of constant radiusor whose centerline is expressed by a sine-generated curve, sothe evaluation of the radius of curvature is straightforward. Themodel presented by Struiksma et al. [77] is applied to ﬁeld cases[76] but it requires as input a piecewise constant value of
R
. Manyresearchers have not even reported the method of evaluating
R
[30,31,40,52,57]. Abad et al. [2] present a more general, grid- and
rotation-insensitive method to evaluate
R
that appears suited topractical applications. However, Abad et al. [2] report that it isnot accurate in high-curvature cases. A new method is developedin this study and it is presented later in Section 6.Despite the relative simplicity and potential importance of dis-persion in channel bends, many researchers have presented ﬂowand sediment transport models that have either ignored disper-sion, or claimed to have captured it through a manipulation of the Reynolds stresses [54,57,73,94]. One potential disadvantage
to such models is the introduction of additional parameters thatmust be tuned to bring model predictions in line with observa-tions. On the other hand, it is also possible that these stresses aresimply too small in comparison to bottom shear to signiﬁcantlyimpact ﬂow conditions. Numerical tests presented later are de-signed, in part, to shed insight into this issue. Moreover, Godu-nov-based schemes utilized in this study provide an excellentframework to assess the importance of dispersion and Reynoldsstress terms because of their stability in the limit that these termsvanish.Previousnumericalmodelingstudiesofthisissuemayhaveutilized unrealistically large eddy diffusivities, for stability pur-poses, which masked the contribution of dispersion.
3. Scalar transport model
Scalar transport must be considered to account for suspendedsediment andto predict the transport of dissolvedsubstances suchas contaminants, which are often sorbed to sediments. Scalartransport models are also required to solve
k
—
type turbulenttransport equations, though this is not done in this study. For gen-erality, the scalar transport equations are presented here in termsof
N
arbitrary scalars,
c
i
;
i
¼
1
;
. . .
;
N
(e.g., suspended sediment,passive tracers, temperature, contaminant concentration) and areobtainedbyverticallyintegratingthe3DReynolds-Averagedtrans-portequations.Writteninadifferentialformsimilartothe2Dﬂowequations, the transport equations appear as,
@
C
@
t
þ
@
E
C
@
x
þ
@
G
C
@
y
¼
S
C
ð
15
Þ
where
C
¼½
h
c
1
h
c
2
h
c
N
T
is the vector of the depth-inte-grated concentrations;
S
C
¼½
s
1
s
2
s
n
T
is a generalizedsource/sink term vector that accounts for non-conservative pro-cesses;
E
C
and
G
C
are the depth-integrated ﬂuxes in
x
and
y
direc-tion, respectively [3],
E
C
¼
h
u
c
1
h
ð
T
x
c
1
þ
D
x
c
1
Þ
h
u
c
2
h
ð
T
x
c
2
þ
D
x
c
2
Þ
h
u
c
N
h
ð
T
x
c
N
þ
D
x
c
N
Þ
2666437775
;
G
C
¼
h
v
c
1
h
ð
T
y
c
1
þ
D
y
c
1
Þ
h
v
c
2
h
ð
T
y
c
2
þ
D
y
c
2
Þ
h
v
c
N
h
ð
T
y
c
N
þ
D
y
c
N
Þ
2666437775
ð
16
Þ
where
T
accounts for Reynolds stresses and
D
accounts for disper-sion. Note that
c
i
represents a depth-averaged scalar concentration.
Reynolds stresses account for scalar transport caused by turbu-lent velocityﬂuctuationsand Taylor [80] showed that these can bemodeled as a Fickian process in terms of a turbulent diffusivity,
e
t
that is orders of magnitude larger than molecular diffusivities.Deﬁning a turbulent Schmidt number
r
t
¼
m
t
=
e
t
, the Reynoldsstresses can be modeled as,
T
x
c
i
T
y
c
i
¼
m
t
r
t
@
c
i
@
x
@
c
i
@
y
" #
ð
17
Þ
where the eddy viscosity is computed from the algebraic turbulentclosure previously described and the Schmidt number is as of yetundeﬁned. Scalar dispersion like momentumdispersion is propertyofreduced-dimensionmodels,butscalardispersioncanbemodeledas a Fickianprocess [32,81,82]. Elder [32] formulated the longitudi-
nal dispersion coefﬁcient for a wide rectangular channel as:
K
‘
¼
1
h
Z
h
0
ð
u
‘
U
Þ
dz
Z
z
0
dz
e
z
Z
z
0
ð
u
‘
U
Þ
dz
ð
18
Þ
and using
e
z
ð
z
Þ¼
j
u
z
1
z h
and
u
‘
ð
z
Þ¼
U
þð
u
=
j
Þð
1
þ
log
z
=
h
Þ
,where
u
‘
is the longitudinal velocity, obtained
K
‘
¼
a
‘
u
h
;
a
l
5
:
86
ð
19
Þ
Adopting instead a power-law (Eq. (11)), the ﬁnal result becomesnearly identical.
Transverse dispersion can be evaluated in a similar manner byconsidering thetransversevelocityproﬁle, followingthesame rea-soningadoptedbyElder [32]for longitudinal dispersion. Whenthetransverse dispersion is deﬁned to be normal to the depth-aver-aged velocity, the depth-averaged transverse velocity is zero andthe dispersion coefﬁcient can be computed as,
K
t
¼
1
h
Z
h
0
u
t
dz
Z
z
0
dz
e
z
Z
z
0
u
t
dz
ð
20
Þ
Fischer [35] adopted the tranverse velocity proﬁle of Rozovskii[67] to obtain a transverse dispersion coefﬁcient,
K
t
. Using thelinear distribution of velocity of Odgaard [58], Eq. (12) ,
K
t
is givenby,
K
t
¼
a
t
u
h
;
a
t
¼
16
j
v
s
u
2
¼ ð
2
m
þ
1
Þ
2
24
j
7
hR
2
ð
21
Þ
Yen[98]reportsthat
m
variesfrom4to12, and
m
¼
7hasbeenuti-lized by a number of researchers to describe turbulent boundarylayers. Using Eq. (21), the values of
a
t
corresponding to
m
¼
7 anda range of channel curvatures become,
R
=
h
¼
50
)
a
t
¼
1
:
9
R
=
h
¼
100
)
a
t
¼
0
:
48
R
=
h
¼
200
)
a
t
¼
0
:
12
ð
22
Þ
Such values are in general agreement with laboratory observations,forexampleWard[90]reportedvaluesof
a
t
rangingfrom0.2to1.7.A similar approach has been followed by Holly and Usseglio-Polatera [42], who ignored the turbulent diffusion and simply set
a
t
¼
0
:
23 to account for transverse dispersion. However, this elim-inatesthesensitivityof
a
t
to
R
whichshouldberetainedforgeneralpurpose river modeling.
Now, a rotational transformation is again used to obtain a dis-persion coefﬁcient tensor for the case where ﬂow is directed withan angle
u
measured counter-clockwise from the
x
axis,
K
¼
K
xx
K
xy
K
yx
K
yy
¼
M
ð
u
Þ
K
‘
00
K
t
M
T
ð
u
Þ ð
23
Þ
and the dispersive ﬂuxes follow as,
20
L. Begnudelli et al./Advances in Water Resources 33 (2010) 17–33
D
x
c
i
D
y
c
i
¼
K
xx
K
xy
K
yx
K
yy
@
c
i
@
x
@
c
i
@
y
" #
ð
24
Þ
Note that the combined effects of Reynolds stresses and disper-sion can be modeled as follows,
T
x
c
i
þ
D
x
c
i
T
y
c
i
þ
D
y
c
i
¼
K
0
xx
K
0
xy
K
0
yx
K
0
yy
" #
@
c
i
@
x
@
c
i
@
y
" #
ð
25
Þ
where
K
0
¼
K
þð
m
=
r
t
Þ
I
and
I
represents the unit tensor.
Implementationoftheaboveexpressionsforturbulentdiffusionanddispersionispossiblewithlocalknowledgeof
U
;
m
;
u
;
R
,and
r
t
. All of these except
r
t
were used in closing the momentumbal-ance, and here
r
t
is set to unity. A number of researchers haveviewed
r
t
as a ﬁtting parameter to bring numerical model predic-tions in line withobservations. For example, Ye and McCorquodale[94] adoptedthe2Dturbulencemodel of Rastogi and Rodi [63] but
ignoreddispersive terms and comparedmodel predictions to labo-ratory observations of ﬂow and mass transport by Chang [22] whoexperimented using a ﬂume with two 90
bends in alternatingdirections. Results show that the 2D model captures depth-aver-aged velocities reasonable well, but performs poorly relative tomasstransportasaconsequenceofsecondaryﬂow.Hence,anunre-alistically small Schmidt number,
r
c
¼
0
:
15, was required to com-pensate for the lack of dispersion. Duan [30] incorporatedmomentum dispersion into a 2D model and revisited the Chang[22]experiment;thiswasfoundtoimprovethevelocityprediction.However, Duan [30] also considered the mass transport observa-tions of Chang [22] but ignored the effect of scalar dispersion. Tobring predictions in line with observations, the Schmidt numberwasviewedasaﬁttingparameterandanunrealisticallysmallvalue,
r
c
¼
0
:
02,wasagainrequired.Theseandotherstudiessuggestthat,insomesituations,itmaybepossibletoobtainreasonablygoodpre-dictions of water depth and velocity in meandering channels with2D models by considering Reynolds stresses and ignoring momen-tum dispersion. However, mass transport in meandering channelscannot be realistically predicted by models that ignore dispersion.
3.1. Magnitude of Reynolds stresses and dispersion terms inmomentum equation
In order to better understand the relative importance of Rey-nolds stresses and dispersion terms in momentum equation (1),it is useful to evaluate the different terms as follows.For the Reynolds stresses we have (Eqs. (5)–(7))
hT
x
i
x
j
m
t
H U R
a
C H R
½
U
2
H
½
O
ð
10
1
Þ
O
ð
10
2
Þ
H R
½
U
2
H
ð
26
Þ
where
½
U
;
½
H
, and
½
R
are representative values of velocity, ﬂowdepth, and radius of curvature, respectively, and
m
t
¼
a
U
H
, where
a
is set in this study to 0.08 [7,63].
For the dispersion terms we have, considering the transversedispersion (Eq. (13)):
hD
tt
¼
13
h
v
2
s
ð
2
m
þ
1
Þ
2
12
j
4
m
2
H R
2
½
U
2
H
O
ð
10
1
Þ
h i
H R
2
½
U
2
H
ð
27
Þ
and considering the longitudinal direction (Eq. (13)):
hD
‘‘
¼
hU
2
m
ð
2
þ
m
Þ ½
U
2
H
m
ð
2
þ
m
Þ ½
O
ð
10
1
Þ
O
ð
10
2
Þ½
U
2
H
ð
28
Þ
Therefore, assuming
m
to range from 4 to 12, we can comparethe magnitude of the three terms for the three following cases:
H
=
R
½
O
ð
10
3
Þ
(mild curvature),
H
=
R
½
O
ð
10
2
Þ
(strong curva-ture), and
H
=
R
½
O
ð
10
1
Þ
(verystrong curvature, laboratory chan-nels). The values are reported in Table 1.According to this quite simpliﬁed analysis, transverse disper-sion is equally or slightly less important than Reynolds stressesfor mild curvatures and can probably be ignored for very mild cur-vatures, while it is equally or slightly more important than Rey-nolds stresses for strong curvatures. For very strong curvatures,than can be found mostly in laboratory channels, they becomethe most important term. On the other hand, the longitudinal dis-persion does not depends on the curvature, being generally 1 or 2orders of magnitude smaller than momentum ﬂux
U
2
H
.
4. Bed deformation model
The evolution of bed elevation is modeled by solving a massconservation equation for sediment, the Exner equation, whichcan be written as,
ð
1
/
Þ
@
z
b
@
t
þ
r
q
b
¼
M
D
M
E
ð
29
Þ
where
/
represents the porosity of the bed,
q
b
represents the near-bottomvolumetricﬂuxofsedimentperunitwidthinthehorizontaldirection (bed load),
r
represents the 2D divergence operator,
M
D
represents the volumetric ﬂux of sediment from the suspended-load layer to the bed-load layer due to settling (deposition), and
M
E
represents the volumetric ﬂux from the bed-load layer to thesuspended-load layer caused by turbulence (entrainment). To solveEq. (29), one option is to adopt empirical models for
q
b
and
M
E
, torelate
M
D
to a suspended sediment concentration by volume
c
through a settling velocity,
w
s
, and to solve Eq. (15) to predict sus-pended sediment, e.g.,
c
1
¼
c
with
s
1
¼
M
E
M
D
. Another approachthat avoids use of a transport equation for suspended sediment isknown as an equilibrium formulation. The essence of the equilib-rium formulation is that horizontal volumetric ﬂux of suspendedsediment per unit width
q
s
is based only on local ﬂow conditionsand bed properties. Furthermore, it is assumed that the rate of entrainment and deposition adjusts instantaneously to account forchanges in ﬂow or bed properties that affect
q
s
. Mathematically,thisiswrittenintermsofthedivergenceofthesuspendedsedimentﬂux as follows,
r
q
s
¼
M
E
M
D
ð
30
Þ
In view of Eq. (15), this approach assumes that the accumulation,turbulent diffusion and dispersion terms are small in comparisonto the those appearing in Eq. (30). When Eq. (29) is added to Eq.
(30) the result is,
ð
1
/
Þ
@
z
b
@
t
þ
r
ð
q
b
þ
q
s
Þ ¼
0
ð
31
Þ
The applicability of the equilibrium assumption can be testedby considering the length (or time) scales (
L
s
or
T
s
¼
L
s
=
u
) of theadaptation of the sediment suspensions to changes in bed condi-tions. The adaptationlengthscale
L
s
isbasedonlongitudinal trans-port and is deﬁned as [62, Chapter 10, 91, Chapter 2],
L
s
¼
h
u
c
w
s
ð
32
Þ
Table 1
Order of magnitude of Reynolds stresses and dispersion terms in momentumequations (1) for different values of the radius of curvature.
H
=
R
½
O
ð
10
3
Þ
H
=
R
½
O
ð
10
2
Þ
H
=
R
½
O
ð
10
1
Þ½
hT
ij
=
½
U
2
H
½
O
ð
10
4
Þ
O
ð
10
5
Þ ½
O
ð
10
3
Þ
O
ð
10
4
Þ ½
O
ð
10
2
Þ
O
ð
10
3
Þ½
hD
tt
=
½
U
2
H
½
O
ð
10
5
Þ ½
O
ð
10
3
Þ ½
O
ð
10
1
Þ½
hD
‘‘
=
½
U
2
H
½
O
ð
10
1
Þ
O
ð
10
2
Þ ½
O
ð
10
1
Þ
O
ð
10
2
Þ ½
O
ð
10
1
Þ
O
ð
10
2
Þ
L. Begnudelli et al./Advances in Water Resources 33 (2010) 17–33
21

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