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Computational modeling of flow and sediment transport and deposition in meandering rivers

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Computational modeling of ﬂow and sediment transportand deposition in meandering rivers
Mehrzad Shams
a
, Goodarz Ahmadi
a,*
, Duane H. Smith
b
a
Department of Mechanical and Aeronautical Engineering, Clarkson University, Box 5725 Camp Building, Potsdam, NY 13699-5725, USA
b
National Energy Technology Laboratory, U.S. Department of Energy, Morgantown, WV 26507-0880, USA
Abstract
A computational modeling analysis of the ﬂow and sediment transport, and deposition in meandering-river models was per-formed. The Reynolds stress transport model of the FLUENT
TM
code was used for evaluating the river ﬂow characteristics, in-cluding the mean velocity ﬁeld and the Reynolds stress components. The simulation results were compared with the availableexperimental data of the river model and discussed. The Lagrangian tracking of individual particles was performed, and thetransport and deposition of particles of various sizes in the meandering river were analyzed. Particular attention was given to thesedimentation patterns of diﬀerent size particles in the river-bend model. The ﬂow patterns in a physical river were also studied. AFroude number based scale ratio of 1:100 was used, and the ﬂow patterns in the physical and river models are compared. The resultshows that the mean-ﬂow quantities exhibit dynamic similarity, but the turbulence parameters of the physical river are diﬀerent fromthe model. More strikingly, the particle sedimentation features in the physical and river models do not obey the expected similarityscaling.Published by Elsevier Science Ltd.
1. Introduction
Understanding the mechanisms that control sedimenttransport in rivers is of fundamental importance to theﬁelds of hydraulics, hydrology, and water resources.Flows in river are, generally, three-dimensional, un-steady, in a state of turbulent motion, and involveinteractions of diﬀerent phases. Therefore, accurateanalysis of ﬂow and sediment transport in a meanderingriver is a rather diﬃcult task. While numerous booksand papers on the subject have been published, details of the micro-mechanics of sediment resuspension, trans-port, and deposition are not fully understood.The traditional approaches for studying river ﬂowsare based on ﬁeld measurements, laboratory exper-iments, and simpliﬁed depth-averaged computer models.Field measurements are rather tedious and expensive.There are also problems associated with the laboratorymodel studies. The laboratory models hardly ever satisfythe principal of dynamic similarity with the originalphysical systems, due to disparity of one or more of thedominant nondimensional parameters. Therefore, directextrapolation of the ﬂow condition from laboratorydata to ﬁeld applications is not always possible. At-tempts to relate sediment transport in rivers and corre-sponding model studies are even more questionable.Furthermore, it is rather diﬃcult from ﬁeld measure-ments and laboratory studies to provide a detailedphysical understanding of the micro-mechanics of indi-vidual particle transport and deposition processes.River hydraulics and sediment transport are widelystudied subjects. Extensive reviews of earlier works wereprovided by Chow [1], Vanoni [2], French [3], Chang [4],Yalin [5,6] and Nezu and Nakagawa [7], among others.Here, recent advances in numerical modeling of riverhydraulics are summarized; and this is followed by areview of the recent computational modeling of sedi-ment transport. Recent developments in related ﬁelds of particle resuspension, transport, and deposition in tur-bulent ﬂows are also brieﬂy described.In most earlier studies, a depth-averaging techniquewas used to reduce the river ﬂow to a two-dimensionalproblem. Here, only the works that treat the full three-dimensional ﬂow (not based on depth-averaging) areoutlined. Flow in straight, open channels was considered
Advances in Water Resources 25 (2002) 689–699www.elsevier.com/locate/advwatres
*
Corresponding author. Tel.: +1-315-268-6536; fax: +1-315-268-6438.
E-mail address:
ahmadi@clarkson.edu (G. Ahmadi).0309-1708/02/$ - see front matter Published by Elsevier Science Ltd.PII: S0309-1708(02)00034-9
by a number of investigators. Rastogi and Rodi [8],Alfrink and Rijin [9], and Gibson and Rodi [10] used thetwo-equation and Reynolds stress transport turbulencemodels and computed three-dimensional ﬂows in openchannels. Leschziner and Rodi [11] and Demuren andRodi [12] employed a partially parabolic version of theaveraged Navier–Stokes equations and the
k
–
e
turbu-lence model (while neglecting streamwise diﬀusion) tostudy, respectively, the ﬂow through a 180
bend and ameandering channel.Meselhe et al. [13] employed the fully elliptic form of the governing equations formulated in generalized cur-vilinear coordinates to simulate the ﬂow through ameandering channel with a trapezoidal cross-section.Demuren [14] reported his computational study for ﬂowthrough a natural-like cross-section river. He used a ﬁ-nite volume numerical method for solving the fullReynolds averaged Navier–Stokes (RANS) equationsand the
k
–
e
turbulence model. Ambrosi et al. [15] de-scribed their numerical simulation of the ﬂow at thedelta of the Po River. They used a ﬁnite element ap-proach for their modeling, but restricted their study toa two-dimensional ﬂow analysis. Recently Sinha et al.[16] presented a three-dimensional numerical model forsimulating ﬂow through a natural river. They includedthe large-scale bed roughness and islands using aboundary-ﬁtted mesh and also used a two-point wallfunction approach. Their model predictions for a 4 kmstretch of the Columbia River showed good agreementwith the laboratory and ﬁeld measurements.Evaluation of bed-load transport has a long historystarting with DuBoys [17], Shields [18], and Einstein[19]. Reviews of more recent works were provided byChow [1], French [3], Chang [4], Yalin [5,6] and Nezuand Nakagawa [7], among others. A state-of-the-artreview of river mixing and sediment dispersion waspublished by Elhadi et al. [20]. Most of the earlier eﬀortwas focused on developing semi-empirical expressionfor bed-load transport. A recent application of this ap-proach was presented by Yang et al. [21], who studiedsediment transport in the Yellow River in China.While the semi-empirical models have been used ex-tensively in engineering design, the details of processesthat control the sediment transport are not fully under-stood. Clearly, sediment transport involves two-phase,liquid–solid ﬂows and their interactions. McTigue [22]noted the need for using a two-phase ﬂow model forsediment transport analysis. Accordingly, suspendedparticles are treated as a continuous second phase thatinteracts with the ﬂuid phase. Kobayashi and Seo [23]and Cao et al. [24] presented their analyses of sedimenttransport based on a two-phase ﬂow model that in-cluded a bed-load layer. Cao and Ahmadi [25] per-formed an investigation of sediment-laden ﬂow in anopen channel using a two-phase mixture model, andpresented an approach for evaluating the sedimentconcentration. Recently Lopez and Garcia [26] studiedthe ﬂow through an open channel with vegetation. Theyalso used a two-ﬂuid approach and the
k
–
e
turbulencemodel.Extensive reviews of the literature on earlier workrelated to two-phase ﬂow models were provided by Soo[27], Ishii [28], Hetsroni [29], and Ahmadi [30]. Recently,more-advanced models for turbulent two-phase ﬂowswere reported by Ahmadi and co-workers [31–33].Computational modeling of particulate and two-phaseﬂows were reported by Cao et al. [34] and Cao andAhmadi [35,36], among others.The presented survey shows that the current ad-vanced procedure for analyzing sediment transport usesa dispersion model that treats the particles as a trans-ferable scalar. Even the recently proposed two-ﬂuidmodels cannot account for the full interactions of par-ticles with the turbulence ﬂuctuations in rivers or thedetailed physics of sediment deposition processes. Fur-thermore, the micro-mechanics of resuspension of sedi-mentary particles is not fully understood and has notbeen included in the available models. On the otherhand, there has been considerable recent progress incomputational modeling of particle deposition, trans-port, and resuspension processes in related ﬁelds. Li andAhmadi and coworkers [37–39] developed a computa-tional model for simulating the turbulent deposition of particles in complex passages using a Lagrangian par-ticle trajectory analysis procedure. The approach wasextended and used by Ahmadi and Smith [40,41] foranalyzing particle transport, and deposition during hot-gas ﬁltration. The particle resuspension processes inturbulent ﬂow were recently studied by Soltani andAhmadi [42–45], among others.In this work, the ﬂow and the sediment transport anddeposition in meandering-river models are studied. Ariver model that is identical to the one used in the ex-perimental investigation of Shiono and Muto [46] isstudied ﬁrst. This is followed by the analysis of aphysical-scale meandering-river model with a Froudenumber the same as the laboratory-scale model. Usingthe FLUENT
TM
code, the mean-ﬂow properties, tur-bulence intensities, and sedimentation patterns for boththe laboratory and the physical rivers are evaluated. Thecomputational results for the laboratory-scale model arecompared with the experimental data of Shiono andMuto [46] and discussed. Motions of individual particlesand their deposition patterns in the laboratory andphysical-scale rivers are analyzed using a Lagran-gian particle trajectory analysis procedure. In particu-lar, the sedimentation patterns of particles of diﬀerentsizes under various conditions in the river-bends areanalyzed. These simulation results show that themean velocities satisfy the principle of dynamic simi-larity. The secondary ﬂow pattern, the turbulence in-tensity and, more importantly, the particle deposition
690
M. Shams et al. / Advances in Water Resources 25 (2002) 689–699
patterns, however, seem to defy the expected similarityscaling.
2. Flow simulation
Since the ﬂow in the river is in a state of turbulentmotion, it is important to use an appropriate turbulencemodel for evaluating the mean-ﬂow ﬁeld. The FLU-ENT
TM
code provides options for using either the
k
–
e
orthe Reynolds stress transport model (RSTM), which is asimpliﬁed version of the one developed by Launder et al.[47]. While the
k
–
e
model is widely used in industrialapplications, it suﬀers from several shortcomings. Its useof an isotropic eddy viscosity limits its applicability andcauses the model to be incapable of handling the tur-bulence normal-stress eﬀects. The RSTM, however, ac-counts for the evolution of individual turbulence stresscomponents, and is well suited for handling anisotropicturbulence stresses. In the present study, the RSTM of the FLUENT
TM
code was used in the simulations.(Additional details of the ﬂow simulation featuresand the computational schemes may be found in theFLUENT
TM
User’s Guide [48].)
3. Mean-ﬂow river model
For an incompressible ﬂuid ﬂow, the equations of continuity and balance of momentum for the meanmotion are given as
o
uu
i
o
x
i
¼
0
ð
1
Þ
o
uu
i
o
t
þ
uu
j
o
uu
i
o
x
j
¼
1
q
o
p p
o
x
i
þ
m
o
2
uu
i
o
x
j
o
x
j
oo
x
j
R
ij
ð
2
Þ
where
uu
i
is the mean velocity,
x
i
is the position,
t
is thetime,
p p
is the mean pressure,
q
is the constant massdensity,
m
is the kinematic viscosity, and
R
ij
¼
u
0
i
u
0
j
is theReynolds stress tensor. Here,
u
0
i
¼
u
i
uu
i
is the
i
th ﬂuidﬂuctuation velocity component.The RSTM provides for diﬀerential transport equa-tions for evaluation of the turbulence stress components.i.e.,
oo
t R
ij
þ
uu
k
oo
x
k
R
ij
¼
oo
x
k
m
t
r
k
oo
x
k
R
ij
R
ik
o
uu
j
o
x
k
"
þ
R
jk
o
uu
i
o
x
k
#
C
1
e
k R
ij
23
d
ij
k
C
2
P
ij
23
d
ij
P
23
d
ij
e
ð
3
Þ
where the turbulence production terms are deﬁned as
P
ij
¼
R
ik
o
uu
j
o
x
k
R
jk
o
uu
i
o
x
k
;
P
¼
12
P
ij
ð
4
Þ
with
P
being the ﬂuctuation kinetic energy production.Here
m
t
is the turbulent (eddy) viscosity; and
r
k
¼
1
:
0,
C
1
¼
1
:
8,
C
2
¼
0
:
6 are empirical constants [47].The transport equation for the turbulence dissipationrate,
e
, is given as
o
e
o
t
þ
uu
j
o
e
o
x
j
¼
oo
x
j
m
þ
m
t
r
e
o
e
o
x
j
C
e
1
e
k R
ij
o
uu
i
o
x
j
C
e
2
e
2
k
ð
5
Þ
In Eq. (5),
k
¼
12
u
0
i
u
0
i
is the ﬂuctuation kinetic energy, and
e
is the turbulence dissipation. The values of constantsare
r
e
¼
1
:
3
;
C
e
1
¼
1
:
44
;
C
e
2
¼
1
:
92
:
ð
6
Þ
The RSTM of the FLUENT
TM
code and the standardwall function boundary condition were used for evalu-ating the mean velocity ﬁeld and the Reynolds stresscomponents in the river.
4. Fluctuating velocities simulation
The dispersion of small particles is strongly aﬀectedby the instantaneous ﬂuctuation ﬂuid velocity. Theturbulence ﬂuctuations are random functions of spaceand time. Here, a discrete random walk (DRW) model isused for evaluating the instantaneous velocity ﬂuctua-tions. The values of
u
0
,
v
0
and
w
0
that prevail during thelifetime of the turbulent eddy,
T
e
, are sampled by as-suming that they obey a Gaussian probability distribu-tion. That is, the instantaneous velocity in the
i
thdirection is given as
u
0
i
¼
f
ﬃﬃﬃﬃﬃﬃﬃﬃ
u
0
i
u
0
i
q
ð
7
Þ
In Eq. (7)
f
is a zero-mean, unit-variance, normallydistributed, random number;
ﬃﬃﬃﬃﬃﬃﬃ ﬃ
u
0
i
u
0
i
q
is the local root-mean-square (RMS) ﬂuctuation velocity in the
i
th direc-tion; and the summation convention on
i
is suspended.The characteristic lifetime of the eddy is deﬁned as aconstant given by
T
e
¼
2
T
L
ð
8
Þ
where
T
L
is the eddy turnover time given as
T
L
¼
0
:
15
ð
k
=
e
Þ
. The other option allows for a log–normalrandom variation of eddy lifetime that is given by
T
e
¼
T
L
log
ð
r
Þ ð
9
Þ
where
r
is a uniform random number between 0 and 1.The particle is assumed to interact with the ﬂuid ﬂuc-tuation ﬁeld, which stays ﬁxed over the eddy lifetime.When the eddy lifetime is reached, a new value of theinstantaneous velocity is obtained by introducing a newvalue of
f
in Eq. (7).
M. Shams et al. / Advances in Water Resources 25 (2002) 689–699
691
5. Particle equation of motion
The equation of motion of a small particle, includingthe eﬀects of nonlinear drag and gravitational forces, isgiven byd
u
p
i
d
t
¼
3
m
C
D
Re
p
4
d
2
S u
i
ð
u
p
i
Þ þ
g
i
ð
10
Þ
andd
x
i
d
t
¼
u
p
i
:
ð
11
Þ
Here,
u
p
i
is the velocity of the particle and
x
i
is its po-sition,
d
is the particle diameter,
S
is the ratio of particledensity to ﬂuid density,
g
i
is the acceleration of gravity,and
m
is the mass of the particle.The ﬁrst term on the right-hand side (RHS) of Eq.(10) is the drag force due to the relative slip between theparticle and the ﬂuid. The drag force is, generally, thedominating force. According to Hinds [49], the dragcoeﬃcient,
C
D
, is given as
C
D
¼
24
Re
p
for
Re
p
<
1
ð
12
Þ
and
C
D
¼
24
Re
p
1
þ
16
Re
2
=
3p
for 1
<
Re
p
<
400
;
ð
13
Þ
where
Re
p
is the particle Reynolds number deﬁned as
Re
p
¼
d u
j
u
p
j
m
ð
14
Þ
The relaxation time of the particles is deﬁned as
s
¼
q
d
2
18
l
ð
15
Þ
The relaxation time is nondimensionalized on the basisof the river length-scale and velocity scale:
s
¼
s
U
0
H
ð
16
Þ
where
U
0
is the average inlet velocity and
H
is the riverheight.Eq. (10) includes all the relevant forces and forms thebasis for the discrete second-phase analysis of theFLUENT
TM
code that was used in the present compu-tation. The particle equation of motion used requiresknowledge of the instantaneous turbulent ﬂuid velocityat the location of each particle at every instance of time.As noted before, the mean liquid velocity was evaluatedby the use of the Reynolds stress transport turbulencemodel (RSTM) and the ﬂuctuation velocity componentswere calculated form Eq. (7).
6. Results
Shiono and Muto [46] performed a series of detailedexperimental measurements of the ﬂow conditions in alaboratory-scale meandering river model using a laser-Doppler anemometer. They reported the cases of main-channel and over-bank ﬂows. In the present study, theriver model geometry studied is the same as that inthe experimental measurements of [46] for the ﬂow inthe main channel. The river is assumed to have a lon-gitudinal slope of 0.001. The meandering shape of theriver is modeled by connecting 60
circles with internalradius of 35 cm and straight segment of 37.6 cm long asshown in Fig. 1. Sections 1–3 at the river-bend areidentiﬁed in Fig. 1a, for which details of the secondaryﬂow patterns are studied. At the third bend (Section 3),the riverbed is divided into 10 equal-area zones forparticle deposition analysis. The details of the zonesused for sedimentation analysis in the river-bend areshown in Fig. 1b. The cross-section of the laboratoryriver model, which is 15 cm wide rectangle with a heightof 5 cm, is shown in Fig. 1c.The ﬂow patterns for the physical (natural-scale) riverare also evaluated. The Froude number scaling is usedto relate the parameters of the laboratory model and thephysical river. The Froude number is identiﬁed as
F
¼
V
2
gL
ð
17
Þ
where
V
is the mean velocity,
L
is a length-scale (waterdepth) and
g
is the acceleration of gravity. The physical(natural) river is assumed to be 100 times larger than thelaboratory model. That is, the width and depth of the physical river are 15 and 5 m, respectively. To keepthe Froude number for the laboratory and the physicalriver the same, the velocity ratio then becomes 1:10.A 40
20
400 staggered rectangular grid is gener-ated with the Gambit code for analyzing the ﬂow in theriver (FLUENT
TM
, 1998). Fig. 2 shows the details of the computational grid for the meandering-river model.The grid is mostly uniform; the distance between theﬁrst node and the wall is 1.2 mm, and the longitudinaldistance between the nodes is 3 mm.The free surface was treated as a symmetry boundarycondition. (Thus, the shear stress and all ﬂuxes becomeszero at the free surface.) The SIMPLE algorithm of theFLUENT
TM
Code was also used for solving the dis-cretized equations. As noted before, the RSTM was usedin the simulation, and the computation was continueduntil the solution converged with a total relative error of less than 0.0005.
6.1. Laboratory river model
Laboratory-scale model results for an inlet watervelocity of 0.2 m/s are presented in this section. This
692
M. Shams et al. / Advances in Water Resources 25 (2002) 689–699
corresponds to one of the cases studied experimentallyby Shiono and Muto [46]. The ﬂow Reynolds number,based on the water depth, is about 10
4
. The velocitymagnitude contours on the free surface are shown inFig. 3. Here the ﬂow is from right to left. This ﬁgureshows that the contour patterns are rather complex. Thedetails of the velocity pattern are shown in Fig. 3b. It isobserved that the peak velocity occurs near the innerpart of the bend on the approaching ﬂow section. Thevelocity then decreases away from these regions.To study the nature of the secondary ﬂows in theriver-bend, in-plane ﬂows at Sections 1–3 and the outletsection of the laboratory-scale model were carefullyexamined, and the results are shown in Fig. 4. Thesesections are at diﬀerent bends as illustrated in Fig. 1a.The inner and outer parts of the bend are also identiﬁedin the ﬁgure for clarity. As noted before, the meanvelocity is 0.2 m/s.Fig. 4 shows that there are noticeable vortical mo-tions at the river-bend. At Sections 1 and 3, the vortex atthe outer edge is clockwise, while the one near the bot-tom of the inner edge is counter clockwise. The direc-tion of the vortices reverses at Section 2 and outletsection. As a result, at the bottom of the channel there isalways a movement from the outer edge toward theinner edge. The existence of such motion was conjec-tured by a number of researchers, although many pos-tulated the presence of a single vortical motion at theriver-bends. In their laboratory experiments, Shionoand Muto [46] observed similar multi-vortical mo-tions. Close comparison shows that the general fea-tures of the simulated secondary ﬂow are similar tothose reported [46], but there are some quantitative
Fig. 1. (a) Schematics of the meandering-river model Layout. (b) Details of the zones 1–10 in the third bend. (c) River cross-section.Fig. 2. (a) Computational grid on the free surface for a segment of theriver. (b) Zoomed grid near the bend.
M. Shams et al. / Advances in Water Resources 25 (2002) 689–699
693

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