![. Experimental Conditions of Ni~ no and Garc ıa [1998]](https://www.researchgate.net/profile/M-Nabi/publication/260456362/figure/tbl1/AS:667032286679045@1536044298217/Experimental-Conditions-of-Ni-no-and-Garc-ia-1998_Q320.jpg)
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Detailed simulation of morphodynamics: 2. Sediment pickup,
transport, and deposition
M. Nabi,
1
H. J. de Vriend,
2,3
E. Mosselman,
2,3
C. J. Sloff,
2,3
and Y. Shimizu
1
Received 26 January 2012; revised 30 April 2013; accepted 5 May 2013.
[1] The paper describes a numerical model for simulating sediment transport with eddy-
resolving 3-D models. This sediment model consists of four submodels : pickup, transport
over the bed, transport in the water column and deposition, all based on a turbulent flow
model using large-eddy simulation. The sediment is considered as uniform rigid spherical
particles. This is usually a valid assumption for sand-bed rivers where underwater dune
formation is most prominent. Under certain shear stress conditions, these particles are
picked up from the bed due to an imbalance of gravity and flow forces. They either roll and
slide on the bed in a sheet of sediment or separate from the bed and get suspended in the
flow. Sooner or later, the suspended particles settle on the bed again. Each of these steps is
modeled separately, yielding a physics-based process model for sediment transport, suitable
for the simulation of bed morphodynamics. The sediment model is validated with
theoretical findings such as the Rouse profile as well as with empirical relations of sediment
bed load and suspended load transport. The current model shows good agreement with these
theoretical and empirical relations. Moreover, the saltation mechanism is simulated, and the
average saltation length, height, and velocity are found to be in good agreement with
experimental results.
Citation: Nabi, M., H. J. de Vriend, E. Mosselman, C. J. Sloff, and Y. Shimizu (2013), Detailed simulation of morphodynamics : 2.
Sediment pickup, transport, and deposition, Water Resour. Res.,49, doi:10.1002/wrcr.20303.
1. Introduction
[2] Morphology is the consequence of sediment trans-
port and sedimentation in the river [Church, 2006], and
therefore, the formation of morphological structures with-
out sediment is not possible. The formation of ripples,
dunes, bars, and other alluvial features are governed by
flow-induced sediment motion, the structure of the bed, and
sediment-bed interactions. Given the small particle size,
turbulence at Kolmogorov scales is relevant to sediment
motion. The lack of accurate physics-based models of sedi-
ment transport at these scales makes the prediction of
small-scale alluvial processes such as ripples and dunes for-
mation difficult. Rolling and saltation of sediment grains
over the bed in turbulent flow are inherently complex prob-
lems. Turbulence produces near-bed velocities and pressure
fluctuations that give rise to fluctuations in the forces on
sediment particles, due to which sediment may be picked
up and transported by the fluid. On the other hand, the
presence of sediment particles in the flow may generate or
suppress turbulence, thus creating a dynamic feedback sys-
tem [Bridge and Best, 1998].
[3] In the past, sediment transport has mainly been
described in terms of bulk fluxes related to mean flow prop-
erties. Some of these sediment transport formulae include a
threshold of motion expressed in terms of a critical shear
stress following Shields [1936]. Wiberg and Smith [1987],
Dey [1999], and Wu and Chou [2003] addressed the issue
of incipient motion as a function of the mean bed shear
stress via a deterministic analysis of noncohesive sediment
particles on a loose sediment bed. In contrast to this deter-
ministic view, the stochastic view is that turbulent stress
variations are responsible for the initial dislodgment of
sediment. Einstein and El-Samni [1949], Nelson et al.
[1995], and Papanicolaou et al. [2002] dealt with sediment
entrainment based on stochastic approaches.
[4]Schmeeckle and Nelson [2003] applied direct numer-
ical simulation (DNS) for bed-load transport. The difficulty
arising in this kind of simulations is how to define the lift
force applied on the sediment particles. The phenomenon
of particle lift is still not completely understood and
adequate experimental results are not available to deter-
mine a reliable quantitative relationship. Saffman [1965]
considered the lift force on a small sphere in an unbounded,
linear shear flow. He assumed that the particle and shear
Reynolds numbers are much smaller than unity and that the
particle Reynolds number is much smaller than the shear
Reynolds number. Harper and Chang [1968] extended
Companion paper to Nabi et al. [2012] doi :10.1029/2012WR011911
and Nabi et al. [2013] doi: 10.1002/20457.
1
Laboratory of Hydraulic Research, Hokkaido University, Sapporo,
Japan.
2
Section of Hydraulic Engineering, Delft University of Technology,
Delft, Netherlands.
3
Department of River Dynamics, Morphology & Water Transport, Del-
tares, Delft, Netherlands.
Corresponding author: M. Nabi, Laboratory of Hydraulic Research,
Graduate School of Engineering, Hokkaido University, Sapporo, Japan.
(M.Nabi@eng.hokudai.ac.jp)
©2013. American Geophysical Union. All Rights Reserved.
0043-1397/13/10.1002/wrcr.20303
1
WATER RESOURCES RESEARCH, VOL. 49, 1–17, doi :10.1002/wrcr.20303, 2013
Saffman’s analysis to arbitrary three-dimensional bodies in
linear shear flow. Drew [1978] performed a similar analysis
for a small sphere in two-dimensional strain flow.
McLaughlin [1991] extended Saffman’s analysis to situa-
tions in which the particle Reynolds number is not neces-
sarily small compared to the shear Reynolds number.
Kurose and Komori [1999] studied the lift and drag forces
on a rotating sphere in a linear shear flow. They applied
DNS and investigated the effects of both shear and rota-
tional speed on the drag and lift forces for particle Reyn-
olds numbers 1 Rep500. They showed that their DNS
results are in good agreement with McLaughlin’s results.
[5]Dey [1999] presented a model to compute the thresh-
old shear stress for noncohesive nonuniform sediment
motion on a horizontal sedimentary bed, under a unidirec-
tional steady uniform stream flow. This work was extended
by Dey and Debnath [2000] for a horizontal and stream-
wise sloping sediment bed. Experimental data on thresh-
olds of sediment motion were used to calibrate their model,
using the lift coefficient as a free parameter. McEwan and
Heald [2001] examined the stability of randomly deposited
sediment beds using a discrete particle model in which
individual grains were represented by spheres. The results
indicated that the threshold shear stress for flat beds con-
sisting of cohesionless uniformly sized grains cannot be
adequately described by a single-valued parameter ; rather,
it is best represented by a distribution of values. Papanico-
laou et al. [2002] conducted theoretical studies on the sto-
chastic aspect of the problem of incipient motion. They
showed that the sediment transport may be underpredicted
if the incipient motion criteria are based exclusively on
time-averaged bed shear stress. Moreover, they investi-
gated the role of near-bed turbulent fluctuations and bed
packing density on incipient sediment motion. Wu and
Chou [2003] incorporated the probabilistic features of the
turbulent fluctuations and bed grain geometry and investi-
gated the rolling and lifting probabilities for sediment
entrainment. They extended the theoretical formulation of
the entrainment probabilities in smooth-bed flows by using
the lognormally distributed instantaneous velocity and uni-
formly distributed initial grain position, along with a rela-
tion between lift coefficient and particle Reynolds number.
Their results show that a reliable probability for the critical
state of sediment entrainment cannot be found.
[6] A notable study of particle motion was conducted by
Escauriaza and Sotiropoulos [2011]. They carried out a nu-
merical study of particle motion in a Lagrangian frame-
work. They initially placed a number of spherical particles
on the bed upstream of a vertically mounted circular cylin-
der in a rectangular open channel. The detached-eddy simu-
lation (DES) approach was used to simulate the flow field.
The trajectory and momentum equations for the sediment
particles were solved simultaneously with the flow equa-
tions, assuming one-way coupling between sediment and
flow, and neglecting particle-to-particle interactions. Their
computed results show that the transport of particles is
highly intermittent and exhibits fundamentally all the fea-
tures of bed-load sediment transport observed in laboratory
experiments and field measurements.
[7] In most of the existing models for sediment motion,
a mean bed shear stress is applied. The velocity distribution
is considered to follow a logarithmic profile and the effect
of turbulent structures, as well as fluctuations, on sediment
motion is implicit. Moreover, even if the flow part of the
model is solved accurately, the sediment motion mecha-
nism is not solved completely, and the interaction of sedi-
ment (pickup and deposition) with the bed is not taken into
account. The deficiency of the existing sediment models is
the reason to develop a sediment model in which all stages
of motion and their effect on the bed morphodynamics are
considered.
[8] We developed a high-resolution 3-D numerical model
for morphodynamic processes on small temporal and spatial
scales, based on large-eddy simulation, particle-based trans-
port of sediment, and adaptation of the computational do-
main to bed level changes by means of adaptive grid
refinement and immersed boundary techniques. In this paper,
we present the model for sediment motion, which consists of
submodels for pickup, transport over the bed, transport in
the water column, and deposition. The sediment is modeled
in a deterministic way considering the sediment as rigid
spherical particles in the turbulent flow. Depending on the
shear stresses, the particles roll and slide as a sheet layer
over the bed or separate from the bed while jumping or
being suspended by the flow. We test the sediment model
against well-established analytical and empirical relations.
We find that the model correctly reproduces the settling ve-
locity of a single particle in stagnant water, the angle of
repose after avalanching, and the Rouse profile for the con-
centration of suspended sediments. Computed sediment
transport rates are within the range of values predicted by
empirical transport formulae and agree in particular with the
formulae of Meyer-Peter and M€
uller [1948] and Wiberg and
Smith [1987]. Furthermore, the average saltation length,
height, and velocity are calculated, and a good agreement is
found with the results of Ni~
no and Garc
ıa [1998].
[9] We applied the model to uniform small spherical par-
ticles, which is a suitable approximation for sand beds
where the random orientation of the particles leads to an
averaged spherical form. This approximation may not be
true for gravel beds, because the larger elliptical particles
of these beds develop a preferential orientation, depending
on the stage of armoring [Allen, 1982].
[10] Two companion papers present the methods and the
validation of the underlying hydrodynamic model [Nabi
et al., 2012] and the associated model for changes in bed
level as a result of sediment pickup, transport, and deposi-
tion (M. Nabi et al., Detailed simulation of morphodynam-
ics: 3. Ripples and dunes, submitted to Water Resources
Research, 2013).
2. Model Description
[11] The model for the sediment motion consists of four
submodels, namely, pickup, transport over the bed, trans-
port in the water column, and deposition. If the force on a
particle, exerted by the flow and gravitation forces, exceeds
a critical value, the particle moves. The particle either goes
in suspension or rolls or slides on the bed. In the case of
suspension, the particle may interact with the bed again af-
ter deposition. This interaction can take two forms : either
the particle stays on the bed (viscous damping) or separates
from the bed again (elastic rebounding). Each of the four
submodels relates the motion of sediment to exerted forces
NABI ET AL.: SEDIMENT PICKUP, TRANSPORT, AND DEPOSITION
2
and geometrical properties. Figure 1 shows a flowchart of
the applied algorithms for a single particle. In section 3, we
explain the forces exerted on a spherical particle, and in
section 4, we explain the four submodels of sediment
motion, considering the geometrical properties.
3. Forces on a Single Particle
[12] A particle immersed in a fluid is subject to gravity
and fluid forces. In the next sections, these forces are
formulated.
3.1. Drag Force
[13] The drag force is caused by the pressure gradient
and viscous skin friction. As both are proportional to the
relative flow velocity squared, the drag force for a spherical
particle can be expressed as [Michaelides, 2006]
~
Fdrag ¼CD
8d2~
uf~
vp
j~
uf~
vpjð1Þ
where is the mass density of the fluid, dis the particle di-
ameter, C
D
is the drag coefficient, ~
vpis the particle veloc-
ity, and ~
ufis the flow velocity in the center of the particle,
if the particle would be absent. Morsi and Alexander
[1972] conducted an experimental study to investigate the
relation between the drag coefficient C
D
and the particle
Reynolds number RepRep¼j
~
uf~
vpjd=
, with denot-
ing the kinematic viscosity. Their experiments showed that
C
D
can be approximated as a function of particle Reynolds
number only. Kurose and Komori [1999] showed the drag
force to be a function of the Reynolds particle number and
the particle rotation speed. They showed that the effect of
the rotation speed on the drag force is relatively small ;
therefore, we neglect it in the present study. We adopt the
empirical equation for the drag coefficient, C
D
, for a spheri-
cal particle proposed by Morsi and Alexander [1972]
CD¼a0þa1
Repþa2
Re2
pð2Þ
where a0;a1, and a
2
are coefficients dependent on Rep. The
values of a
0
,a
1,
and a
2
for a range of particle Reynolds
numbers are given in Morsi and Alexander [1972].
3.2. Lift Force
[14]Rubinow and Keller [1961] and Saffman [1965]
used matched asymptotic expansions to analyse the lift
force acting on a rotating sphere in a linear unbounded
shear flow with particle Reynolds numbers much lower
than unity. Saffman’s expression for the lift force coeffi-
cient CL;Sa on a sphere is given by
CL;Sa ¼12:92
eð3Þ
where erepresents the ratio of the square root of the shear
Reynolds number to the particle Reynolds number, which
is restricted to large values in Saffman’s analysis e1ðÞ.
McLaughlin [1991] extended Saffman’s analysis to remove
the restriction given to large values of e.Mei [1992]
obtained the following relation from McLaughlin’s results.
CL
CL;Sa ¼0:443Jð4Þ
where C
L
is the lift coefficient and Jis a nondimensional
quantity arising from integration in McLaughlin’s analysis.
The lift force F
lift
can be determined as
Figure 1. Flowchart of algorithms to simulate the motion of a single sediment particle.
NABI ET AL.: SEDIMENT PICKUP, TRANSPORT, AND DEPOSITION
3
~
Flift ¼CL
8d2j~
uf~
vpj2ð5Þ
[15] Saffman’s result is obtained for J¼2:225. Mei
[1992] fitted a curve through the data from the table in
McLaughlin [1991] to obtain a relation for Jfor
0:1e20:
J¼JeðÞ0:6765 1 þtanh 2:5log10 eþ0:191ðÞ½
fg
0:667 þtanh 6 e0:32ðÞ½
fg
ð6Þ
[16] Moreover, the DNS results of Kurose and Komori
[1999] for the lift force are in good agreement with those
obtained by McLaughlin [1991]. In the present study, we
apply McLaughlin’s results (Mei’s relations) to estimate
the lift force on a sediment particle.
3.3. Submerged Weight
[17] Gravity and buoyancy forces are of major impor-
tance to the motion of sediment particles. Gravity and
buoyancy forces have opposite directions, and their sum
constitutes the submerged weight. The submerged weight
of a spherical particle is given by
~
FG¼
6d3p
~
gð7Þ
where pis the density of the sediment and ~
gis the gravita-
tional acceleration.
3.4. Other Forces
[18] More forces, such as added mass and Basset forces,
affect particle motion. Because a particle and a fluid cannot
exist in the same physical space simultaneously, the parti-
cle, no matter it is accelerating or decelerating, must repel
some amount of surrounding fluid to pass through. There-
fore, added mass or virtual mass is added to the particle to
account for the corresponding inertia. To demonstrate this
in a simplified model, some volume of a moving fluid with
a moving particle can be used. The added mass force can
be calculated by [Auton, 1987]
~
Fadd ¼CmVp
D~
uf
Dt d
~
vp
dt
ð8Þ
where V
p
is the volume of the particle and C
m
is the added
mass coefficient equal to 0.5 for spherical particles [Auton,
1987].
[19] The Basset force results from interaction of the par-
ticle with its own wake. The Basset force is often relatively
small [Armenio and Fiorotto, 2001]; therefore, it is
neglected in the present study.
4. Submodels for Sediment Movement
[20] We developed submodels for pickup, transport in
the water column, deposition, and transport over the bed.
We explain these four submodels of sediment motion in the
next sections.
4.1. Sediment Pickup
[21] A sand bed can be approximated by interlocking
layers of spherical particles. If the shear stress of the flow
on the bed exceeds a certain value, the particles begin to
rotate and may move from their positions. This stress level
is called the critical shear stress, and the initiation of
motion of particles is called incipient motion.
[22] Figure 2 shows the bed schematically. Each individ-
ual spherical particle is resting on three other closely packed
spherical particles. It is the most stable three-dimensional
configuration for spherical particles [Dey, 1999]. All par-
ticles are assumed to have the same size. Depending on the
orientation of the three bed particles with respect to the
direction of exerted forces, the solitary particle tends to roll
either over the valley between two supporting particles, over
the summit of a single particle or somewhere in between.
[23] The incipient motion of a spherical sediment parti-
cle is determined by the resultant of the forces acting on
the particle as shown in Figure 3. The submerged weight
always acts in the downward direction. The drag force is
parallel to the flow direction, and the lift force is in the
direction normal to the bed at the pickup point. The motion
of spherical particles begins mostly by rolling, sometimes
by sliding. For simplicity, incipient motion by sliding is
ignored.
Figure 2. The configuration of an individual particle on the (left) three packed bed particles and a tet-
rahedron formed joining the (right) centers of the four particles.
NABI ET AL.: SEDIMENT PICKUP, TRANSPORT, AND DEPOSITION
4
[24] When a solitary particle is about to dislodge from its
original position, the balance of moments about the point
of contact Mrequires
X~
r~
F¼0ð9Þ
or
Fn:xþFt:y¼0ð10Þ
where xand yare the tangential and normal lever arms, and
F
t
and F
n
are magnitudes of the tangential and normal
forces, respectively. The latter follow from
Ft¼~
Ftotal:
~
tð11Þ
Fn¼~
Ftotal:~
nð12Þ
[25] Here ~
nand~
tare the unit normal and tangential vec-
tors. F
total
is defined as the resultant of all acting forces
~
Ftotal ¼~
FGþ~
Fdrag þ~
Flift
. By dividing F
t
by the area of
the particle, the exerted tangential particle shear stress
can be found.
¼Ft
2d2=4ð13Þ
[26] The lever arms are determined according to Figure
2. T1;T2, and T
3
are the contact points of the three bed par-
ticles with the considered particle. The orientation of the
acting forces determines whether this particle tends to roll
over the valley, over the summit of a single particle, or
somewhere in between. By considering the longest and
shortest paths, the tangential lever arm will be
xmin ¼T0T0¼d
4ffiffiffi
3
pð14Þ
xmax ¼T0T1¼d
2ffiffiffi
3
pð15Þ
[27] A particle can choose different paths according to
the direction of the acting forces. As the number of the
exposed particles is very large, the average particle path
corresponds with an average tangential lever arm of
x¼xmax þxmin
2¼ffiffiffi
3
p
8dð16Þ
[28] The normal lever arm yis equal to the distance
between the center of the solitary particle and the plane
T1T2T3, which passes through the contact points. The nor-
mal lever arm yis not related to the direction of motion. By
considering Figure 3, ycan be found as
y¼d
ffiffiffi
6
pð17Þ
[29] The acting point of moment changes its position during
the rotation of the sphere. Figure 4 shows a particle during dis-
lodge and rotation. The outer circles represent the physical ge-
ometry of the spheres, and the inner circles represent the
effective averaged diameter from equations (16) and (17),
equal to 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x2þy2
p.If’is the angle of the tangential
unit vector with the horizontal plane, the moment can be
defined as
M¼Id2’
dt2ð18Þ
where Iis the inertial moment ; for a solid sphere,
I¼md2=10. By projecting the forces on the line passing
through the sphere centers and the line normal to it through
the center of the moving sphere (~
n0and~
t0in Figure 4), the
moment at point Mcan be written as
M¼r:Ftsin’þFncos’ðÞð19Þ
in which ris the radius of rotation ðr¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x2þy2
pÞ. Insert-
ing equation (19) into equation (18) yields,
d2’
dt2¼Asin’þBcos’ð20Þ
Figure 3. Forces on an individual particle exerted by (left) fluid and gravity and the projection of the
total force on (right) normal and tangential directions.
NABI ET AL.: SEDIMENT PICKUP, TRANSPORT, AND DEPOSITION
5
where
A¼10Ftr
md2ð21Þ
B¼10Fnr
md2ð22Þ
[30] Equation (20) is nonlinear and has no direct analyti-
cal solution, but it is possible to find a solution for d’=dt,
which can be considered as the radial velocity of the mov-
ing particle related to the center of the fixed particle. By
applying the initial condition at ’¼’0as d’=dtjt¼0¼0,
this solution reads
d’
dt¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2Acos’cos’0
ðÞþ2Bsin’sin’0
ðÞ
pð23Þ
[31] A complete dislodge of the sediment particle from
the bed occurs when the rotating particle separates from the
bed particle. In that case, the reaction force of particle 2
tends to zero. By projecting the tangential and normal forces
on the line passing through the particle centers in Figure 4,
Fnsin’Ftcos’¼0ð24Þ
[32] Hence
’¼tan1Ft
Fn
ð25Þ
[33] As can be concluded from Figure 4, a particle sepa-
rates from the bed if the angle of particle separation falls in
the range of =3’2=3. If the angle, ’, does not fall
within this range, the particle cannot separate and will roll
or slide along the bed. The model for transport over the bed
will be discussed in section 4.4. We consider here the case
in which =3’2=3.
[34] By substituting ’into equation (23), the radial ve-
locity can be found. The tangential velocity can be deter-
mined from:
vtan ¼2rd’
dtð26Þ
where v
tan
is the particle velocity in the direction of~
t0. The
components of ~
vtan in x,y, and zdirections will be the ini-
tial velocities of the particle at the moment of its pickup.
[35] Most existing formulations for pickup are empirical
based on experimental data and lack rigorous theoretical
description. The numerical simulation of the amount of
pickup requires implementation of a discrete-element
method (DEM) inside the bed which is computationally
demanding. The huge computational costs in three-
dimensional cases are the reason why we derive a parame-
terized formulation for the instantaneous pickup rate from
the bed. According to this analysis, the particles are consid-
ered to be initially in rest. Due to lift and drag forces, the
particles can accelerate. The acceleration is defined by a.
To move over a distance equal to the grain diameter, d,a
time t¼ffiffiffiffiffiffiffiffiffiffi
2d=a
pis required. According to Newton’s sec-
ond law, the acceleration is the sum of the forces divided
by the mass of the particle,
~
a¼X~
Ftotal
mpð27Þ
where ~
Ftotal is the resultant force on a single particle
defined by ~
Ftotal ¼~
FGþ~
Fdrag þ~
Flift and m
p
is the mass
of the particle. For uniform channel flow, the flow is deter-
mined by the friction velocity, u. Both the lift and drag
forces can be expressed by the shear stress b¼u2
ðÞas
~
Flift ~ bd2and ~
Fdrag ~ bd2. The force balance can
therefore be written as
X~
Fd2~ ~ cr
ðÞ ð28Þ
where cr ðpÞgd.
[36] From here on, we neglect the direction of the force,
implying that we do not care about the direction of motion
of the individual particle. As we are looking for the bulk
quantity of bed entrainment, E, we consider a large number
Figure 4. Presentation of forces and directions of the acting forces (left) before moving and (right) dur-
ing incipient motion.
NABI ET AL.: SEDIMENT PICKUP, TRANSPORT, AND DEPOSITION
6
of small spherical particles. The exact position of a single
particle is therefore of minor importance. Then, the initial
acceleration of the particle follows from
aXF
pd3¼cr
pd¼gp
p
cr
ð29Þ
where denotes the dimensionless shear stress, defined as
¼
p
gd ð30Þ
[37] Substitution of this acceleration into the timescale
definition t¼ffiffiffiffiffiffiffiffiffiffi
2d=a
pgives
tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
p
d
cr
g
sð31Þ
[38] This means that the timescale is inversely propor-
tional to the square root of the excess bed shear stress. The
erosion rate then becomes
Ep
d
t¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
p
gd
cr
sð32Þ
or
E¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
p
gd
cr
sð33Þ
[39] The coefficient is a tuning parameter and contains
such factors as the packing of the bed and the shape of the
particles. Equation (33) is similar to the pickup formula
proposed by De Ruiter [1982, 1983] for uniform sediment,
reading
E¼RuiterpFp
p
gd
cr
cr0tan
"#
0:5
ð34Þ
in which Fis a pickup probability function,
cr0is the in-
stantaneous critical shear stress at a horizontal bed, is the
angle of repose, and Ruiter is a constant for uniform spheri-
cal particles. The angle of repose is constant. The critical
shear stress at a horizontal bed can be determined by the
balance of forces. The lift force can be neglected if it is
small compared with the drag and gravity forces (which is
not always true). Considering the drag parallel to the bed,
and considering equations (16) and (17), the following bal-
ance of forces applies in the critical state :
Fcr
d
ffiffiffi
6
p¼p
6d3gffiffiffi
3
p
8dð35Þ
[40] The particle shear stress is the acting force on a par-
ticle per unit effective surface area. The effective surface
area is the projection of the particle surface in the direction
of acting force, namely, 2 d2=4ðÞ. Hence the critical parti-
cle shear stress is
cr ¼cr
p
dg ¼ffiffiffi
2
p
80:177 ð36Þ
which means that the dimensionless critical particle shear
stress for a flat bed is also constant. Note that the critical
particle shear stress in the current model is generally not
constant, but rather a function of the bed slope and the
directions of the forces. Also note that the critical particle
shear stress in equation (36) is not the same as the Shields
[1936] critical shear stress. The Shields diagram is stochas-
tic in its nature as it is based on the mean bed shear stress
(¼=ðpÞdg, where is averaged bed shear stress)
and it contains only a single line which separates the
motion from no-motion conditions. The original diagram in
the publication of Shields [1936] presents a range of values
at which either motion or rest is possible. This stress is
almost half the critical fluctuating shear stress, because the
effect of turbulent velocity fluctuations is not taken into
account. The peaks of the fluctuating shear stress amount to
about twice the mean value [De Ruiter, 1982].
[41] We performed a large number of simulations with
different bulk velocities and particle diameters and aver-
aged the simulated particle shear stress over the bed area
and compared the result with the mean bed shear stress (as
used in Shields diagram). Figure 5 shows a comparison
between the averaged simulated particle shear stress and
the mean bed shear stress. We found a nearly linear relation
between them. The constant of proportionality for this lin-
ear relation turns out to be independent of the bulk velocity
and is only a function of the particle diameter. The best fit
for this constant is found to be
¼0:1855
p= 1
gd3
362
2
43
5
0:1091
0:1043 ð37Þ
Figure 5. The averaged simulated particle shear stress
(black) and Shields bed shear stress (red) for different parti-
cle diameters and bulk velocities.
NABI ET AL.: SEDIMENT PICKUP, TRANSPORT, AND DEPOSITION
7
[42] The linear relation between the mean bed shear stress
and the mean particle shear stress implies that we can use
equation (34) based on particles shear stress (for uniform
sediment), since the ratio ð
crÞ=0
cr remains constant.
[43] According to Van Rijn [1984a], the constant in
equation (34) is equal to 0.016. This value of Ruiter has
been derived from experiments and is particularly suitable
for larger particles >1000 mðÞ[Van Rijn, 1984a]. There-
fore, the coefficient of proportionality in equation (33)
needs to be calibrated for small particles. In the current
submodel, the pickup rate formula proposed by Nakagawa
and Tsujimoto [1980] is used for this calibration. This for-
mula was validated against a range of physical observa-
tions. Moreover, the method of Nakagawa and Tsujimoto
[1980] yielded one of the best predictions of the pickup
function for fine sediment, as reported by Van Rijn
[1984a]. This approach was also effectively used by Onda
and Hosoda [2004], Giri and Shimizu [2006], and Shimizu
et al. [2009] in models of bedform development. Naka-
gawa and Tsujimoto [1980] express the dimensionless
pickup rate by
P
S¼PSffiffiffi
d
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p= 1
g
r¼F01
cr
3
ð38Þ
in which F
0
is an experimental constant. The variables
and
cr are the dimensionless shear stress and the critical
shear stress for incipient motion, respectively, both aver-
aged over time and space. According to the experimental
studies of Nakagawa and Tsujimoto [1980], suitable values
for the constants in equation (38) are F0¼0:03 and
cr ¼0:035. The factor P
S
can be considered as the proba-
bility density of sand particle dislodgment per unit time
from the bed area occupied by one sediment particle [Naka-
gawa and Tsujimoto, 1980].
[44] Equation (38) is based on unidirectional flow over a
flat bed. Nagata et al. [2005] used Tsujimoto’s pickup rate
formula to find the volumetric rate of sediment pickup
from a specific area on the bed. Applying this approach to a
cellcovering part of the bed yields
_
VS¼A3
A2
PSSd ð39Þ
[45] Here _
VSis the volume of sediment pickup per unit
time, Sis the area of the cell on the bed-surface, and A
2
and
A
3
are shape coefficients for spherical sand grains with
two- and three-dimensional geometrical properties, namely
=4 and =6 respectively.
[46] In order to validate and calibrate the erosion rate in
equation (33), flow computations are made for different
discharges over a flat bed, yielding different bed shear
stress levels. The computed results are averaged over time
and space to find the average entrainment rate and the aver-
age shear stress. The instantaneous volumetric pickup rate
over area Scan be written as
_
VE¼ES
pð40Þ
[47] The computed average entrainment rates _
VEare
compared with the entrainment rate according to equation
(39). Figure 6 shows that the computed average entrain-
ment (based on the mean shear stress) agrees well with that
derived from equation (39). The best fit for factor in
equation (33) is found to be 3:67 104.
[48] The volume of sediment pickup in a time interval
twill be
VE¼_
VEt¼ES
p
tð41Þ
[49] The number of particles involved can be determined
by dividing the pickup volume by the volume of a single
particle V
p
.
npickup ¼VE
Vpð42Þ
[50] The number of particles picked up may be large, so
that treating them one by one is beyond the capacity of exist-
ing computational resources. Hence we use a multiplication
factor, usually smaller than 1 (for fine sediment), to decrease
the number of particles for computational efficiency. In this
way, we consider a smaller number of particles, but with the
same total volume of pickup: we assume each particle to
carry the mass of a number of its neighbors. The mass car-
ried by a particle can be calculated as
m0p¼mp
Cfrac ð43Þ
where m
p
is the mass of a physical particle, m0pis the mass
carried by the computational particle, and C
frac
is the multi-
plication factor. After the particle has been lifted, it may be
transported by the flow. The next section describes the
equations of motion of particles in flow.
4.2. Sediment Transport in the Water Column
[51] After pickup, the particles are transported by the
forces exerted by the flow. According to Maxey and Riley
Figure 6. Comparison between the averaged computed
entrainment and the entrainment rate from the relation of
Nakagawa and Tsujimoto [1980].
NABI ET AL.: SEDIMENT PICKUP, TRANSPORT, AND DEPOSITION
8
[1983], the velocity of an individual sediment particle in
the flow is described by
pVp
d
~
vp
dt¼p
Vp~
gþVp
D~
uf
Dt þ1
2Vp
D~
uf
Dt d
~
vp
dt
þ3
4
CD
dVpj~
uf~
vpj~
uf~
vp
þ3
4
CL
dVpj~
uf~
vpj2
top j
~
uf~
vpj2
bottom
~
n
ð44Þ
where ~
ufrepresents the fluid velocity at the particle location,
~
vpis the velocity of the sediment particle, dis the particle di-
ameter, C
L
is the lift coefficient, C
D
is the drag coefficient,
D=Dt ¼@
@tþ~
u:r;Vpis the particle volume, ~
gis the gravity
vector, and and pare the densities of the fluid and parti-
cle, respectively. The first term in the right-hand side of
equation (44) represents the gravity force. The second and
third terms are the effect of pressure and added mass, respec-
tively [Maxey and Riley, 1983]. The drag coefficient C
D
can
be found from equation (2), but the lift coefficient C
L
is diffi-
cult to determine. There is limited knowledge regarding the
effect of solid boundaries on the particles. The effect of
shear stress has been studied for very simple linear cases
with Rep1. Moreover, the fluid velocity needs to be
determined at the top and the bottom of the particle. In the
model, the size of a particle is 1 or 2 orders of magnitude
smaller than the size of a computational grid cell, which
makes the difference between the interpolated velocities at
the top and the bottom of a particle insignificant. We there-
fore replace the lift force term in equation (44) by the theo-
retical and experimental relations introduced by McLaughlin
[1991] and Mei [1992], which are given in equations (4) and
(6). These equations parameterize the total lift force, includ-
ing Magnus effect and wall drift force.
[52] A major simplification in the present study is that
we neglect the interactions between the suspended particles
as well as the effect of particles on the fluid flow. This sim-
plification is valid if the sediment concentration in the mix-
ture of water and sediment is low.
[53] Equation (44) is solved numerically using an
implicit scheme to avoid instabilities. The position of each
particle is then determined as
d
~
xp
dt¼~
vpð45Þ
where ~
xpis the position vector of the particle. The particles
move according to equations (44) and (45), until they settle.
The mechanism of deposition is discussed in the next
section.
[54] The simulation of sediment transport is the most
time-consuming part in the current sediment model as the
number of suspended particles can be large. Hence, the
sediment transport is parallelized by OpenMP on shared
memory computers. It is the novelty of the model that with
a simple parallelization, without message passing interface
(MPI), by using multicore personal computers, it can still
handle the calculations of large eddy simulation with parti-
cle motions in a relatively time-efficient way. This is real-
ized by the combination of an efficient hydrodynamic
model and an efficient sediment motion model.
4.3. Sediment Deposition
[55] Previous studies give conflicting assessments as to
whether depositing grains rebound partially elastically or
are viscously damped [Schmeeckle et al., 2001]. All
researchers who modeled the collision of saltating grains in
water use the same simple model : the incoming velocity of
the impacting grain is divided into two components; nor-
mal and tangential to the colliding surface. The outgoing
tangential velocity is equal to the incoming tangential ve-
locity times a coefficient t, and the outgoing normal ve-
locity is equal to the incoming normal velocity multiplied
by a factor n.
[56] What happens exactly when depositing particles hit
the bed is not clearly known yet. Particles can interact with
the bed in different ways. Their motion may be viscously
damped, they may be reflected or they may move tangen-
tially to the bed. Schmeeckle et al. [2001] showed that
appropriate physical scaling of this problem requires simili-
tude of a collision Stokes number :
St ¼mpvp
6r2
ð46Þ
where m
p
is the mass of a single particle, is the dynamic
viscosity of the water, v
p
is the particle velocity when it
hits the bed, and ris the relative particle radius
1=r¼1=rpþ1=rst
. The variables r
p
and r
st
are the radii
of the moving particle and the stationary particles located
on the bed, respectively. The Stokes number (St) is a mea-
sure of the inertia of the particle relative to the viscous
pressure force exerted on it by the fluid. Hence, if St 1,
the viscous pressure stops the particle before significant
elastic energy can be stored in the deformation of the par-
ticles. In this case, there will be no initial rebound velocity.
At moderate values of St, the elastic deformation becomes
significant enough to develop an initial rebound velocity
but the rebounding particle will be arrested by negative
pressure and cavitation in the gap between it and the bed.
At still higher values St 1ðÞ, rebounding particles sepa-
rate completely.
[57] Figure 7, from an experimental study by Schmeeckle
et al. [2001], shows a graph of the transition from viscously
damped to partially elastic collisions. According to this fig-
ure, the transition between damped and undamped particle
collisions occurs at about the transition from medium to
coarse sand. The corresponding critical sediment diameter
size is 2.7 mm. Therefore, for sediment larger than sand
(>2 mm), saltation impacts are almost always partially
elastic, whereas no significant normal rebound occurs for
sand and smaller particles.
[58] According to this information, for bed-load trans-
port of sand (<2 mm), the normal elastic coefficient nis
zero, because the collisions are viscously damped. The
tangential elastic coefficient ttakes a value of about 0.9
[Schmeeckle et al., 2001]. For particles larger than sand,
the critical Stokes number ranges between 39 and 105. At
Stokes numbers higher than 105, partially elastic colli-
sions occur, and the new velocities can be calculated as
follows
vn;new ¼nvn;old ð47Þ
NABI ET AL.: SEDIMENT PICKUP, TRANSPORT, AND DEPOSITION
9
vt;new ¼tvt;old ð48Þ
[59] Here, v
n
and v
t
are the normal and tangential veloc-
ities of the particle during deposition, respectively.
Schmeeckle et al. [2001] experimentally determined nat a
value of 0.65. For Stokes numbers less than 39, the particle
motion is viscously damped. In the range between 39 and
105, the behavior is not clear because of the negative pres-
sure and cavitation between the particle and the bed.
Schmeeckle and Nelson [2003] applied equations (47) and
(48) for Stokes number less than 105, but set the normal
rebound coefficient at zero n¼0ðÞ. This approach is also
taken in the present submodel for sediment deposition.
4.4. Sediment Transport Over the Bed
[60] If the angle, ’, calculated from equation (25) does
not fall within the range of =3’2=3, the particle
cannot separate and will roll or slide along the bed. This
can also be observed in flume experiments and in the field :
a significant fraction of sediment transport occurs in this
way. An extreme manifestation of this phenomenon is flu-
idization of the bed under high shear stresses.
[61] This kind of sediment transport cannot be modeled
in the same way as the transport of particles in the fluid
described in section 4.2. Therefore, we need another sub-
model, describing sediment transport in layers over the
upper part of the bed. Here we develop a simple bed-load
submodel, considering the bed load as a sheet moving from
one computational cell to the other. Figure 8 shows this
submodel schematically for one cell on the bed. The sedi-
ment sheet can only move in the direction of the exerted
force. Assuming the force, exerted on the particle, to be
constant over a time step t, it yields
at¼Ft
mpð49Þ
where a
t
is the acceleration due to force F
t
and m
p
is the
mass of a solitary particle. We suppose that the motion of
each solitary particle starts from rotation as shown in Fig-
ure 4. If F
t
exceeds the critical level, but is not sufficient to
separate the particle from the bed, the particle will roll or
slide after it reaches point Pin Figure 4. This point corre-
sponds with angle /2. The initial angular velocity can be
determined according to equation (23). The center of
rotation of the particle is the center of particle 2 located in
the bed. Thus, the tangential velocity v0;tan can be
determined as
v0;tan ¼d!ð50Þ
where !is the radial velocity of the center of the moving
particle.
[62] We suppose that the time necessary to rotate the
particle to the summit of the supporting particle (point P)is
negligible. Thus, the displacement can be calculated as
follows
¼0:5att2þv0;tantð51Þ
[63] It can be projected in xand zdirections
x¼:txð52Þ
z¼:tzð53Þ
where t
x
and t
z
are the components of the tangential unit
vector~
t(in the direction of ~
Ft), respectively.
[64] The amount of sediment pickup is calculated by
equation (33). After displacement of the sediment layer, the
amount of sediment given to a neighboring cell can be
determined as the ratio of the area displaced to the neighbor
cell to the total area multiplied by the amount of sediment
Figure 7. Graph of the transition from viscously damping
to partially elastic collisions at typical bed load saltation
velocities as a function of grain size and transport stage
b=c, where bis the boundary shear stress and cis the
Shields critical shear stress. Taken from Schmeeckle et al.
[2001].
Figure 8. Model for rolling and sliding of sediment after
pickup if sediment particles cannot leave the bed. A layer
of sediment on the cell moves in the direction of the tan-
gential force and slides to the neighbor cells.
NABI ET AL.: SEDIMENT PICKUP, TRANSPORT, AND DEPOSITION
10
picked up. For example, the sediment passed on to the east-
ern cell in Figure 8 can be determined by
PS;east ¼xzzðÞ
xzPSð54Þ
[65] This submodel describes the part of the bed-load
transport without saltation. The combination with the sub-
model for transport of saltating and suspended particles in
the water column provides the total sediment motion on
and above the bed.
5. Numerical Implementation
5.1. Pickup
[66] Three major forces are considered for the sediment
particles during pickup: drag force, lift force, and gravity.
The drag, lift, and gravity forces are calculated by equa-
tions (1), (5), and (7), respectively. The flow velocity at the
particle level which is required for the drag and lift is inter-
polated in a way similar to the ghost-cell technique in Nabi
et al. [2012]. The normal vector of the bed at the sediment
level is extended to form an imaginary point. The flow is
interpolated trilinearly for the imaginary point, and the
flow velocity at the particle level is interpolated by a log-
law function from the imaginary point.
[67] As the forces are calculated, they are projected to
tangential and normal directions using equations (11) and
(12), and the moment acting on a particle is calculated. In
the case of a positive moment, the particle rotates. The
angle of separation and the initial velocities are calculated
by equations (25) and (26), respectively. If the angle of sep-
aration falls in the required range =3’2=3ðÞ, the
particle separates and goes into the flow, for which the sub-
model for transport in the water column governs its subse-
quent motion. If the separation angle is not in the required
range, the submodel for transport over the bed shifts a layer
of sediment on the cell area to the neighboring cells in the
direction of the flow. The volume of the sediment pickup
and hence the number of particles picked up is determined
by equations (41) and (42). As the forces are considered to
be uniform within each cell area on the bed, the particles
are distributed randomly in the cells to avoid following a
unique path.
5.2. Transport in the Water Column
[68] In the case of separation of particles from the bed in
the pickup stage, the flow acts on the particle in a transport
mode. Equations (44) and (45) are applied for the particle
velocities and trajectories, respectively. The drag, lift, and
gravity are calculated with the similar procedure as for the
pickup phase, except that the flow velocities are interpo-
lated by a sixth-order Lagrangian interpolation. The loca-
tions of the particles are monitored in each time step, and if
a particle comes into contact with the bed, the deposition
submodel starts.
5.3. Transport Over the Bed
[69] Transport over the bed occurs if the forces on the
sediment particles do not allow separation from the bed. A
sheet of sediment moves in the direction of the flow from
the cell to its neighbors. The volume of sediment (the thick-
ness of the sheet) is calculated by equation (41) for pickup.
5.4. Deposition
[70] In case a particle comes into contact with the bed, as
mentioned above, it may either be damped viscously or
rebound and return into the water column. In the case of
rebounding, the normal vector at the contact point is calcu-
lated, and the normal and the tangential velocities are
found. The tangential velocity is kept constant, and the nor-
mal velocity is multiplied by a negative value smaller than
unity (elasticity factor). If the condition does not satisfy the
rebounding process, the particle is added to the deposited-
particles number for the cell where the particle comes into
contact with the bed. This number identifies the volume of
sediment deposition in that cell. After this calculation, the
volume is used to deform the bed, which is discussed in the
companion paper on the development of ripples and dunes
(M. Nabi et al., submitted manuscript, 2013).
6. Numerical Validation
6.1. Particle Settling Velocity
[71] Basically, the settling velocity is not a particle prop-
erty, but rather a behavioral property. The terminal settling
velocity (w
s
) of a particle occurs when the fluid drag force
on the particle is in equilibrium with the gravity force. Con-
sidering equations (1) and (7) for the drag and gravity
forces, the fall velocity takes the following form [Swamee
and Ojha, 1991].
ws¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4
3
p= 1
dg
CD
v
u
u
tð55Þ
[72] In order to show how this constant velocity is
reached, a particle of 0.25 mm diameter is released in the
numerical model from a stationary state in a steady flow.
Figure 9a shows the computed time evolution of the settling
velocity and the drag coefficient. The velocity increases
and approaches a constant value. The drag coefficient
decreases and also approaches a constant value. In equation
(55), the drag coefficient C
D
is a function of the settling ve-
locity w
s
, making this relationship implicit. It can be solved
iteratively to determine the values of the mentioned varia-
bles. Figure 9b shows the result for the same sediment di-
ameter. Figure 9 clearly shows that the numerical results
agree well with the analytical solution.
6.2. Sediment Avalanching
[73] During the formation of dunes, the angle of the lee
side may become relatively steep. If this angle is larger
than the angle of repose, it is possible that the sediment
slides along the lee side to the bottom. This process is
called avalanching. Avalanching occurs because of insta-
bility of the sediment pack caused by gravity. If the direc-
tion of the gravity force falls outside the stability region of
the particles, it tends to pull the particles downward. If
opposing hydraulic forces are absent or not strong enough,
avalanching occurs. It has to be noticed that in the current
model, the avalanching is based on the motion of solitary
particles, and the effect of geometrical interactions between
NABI ET AL.: SEDIMENT PICKUP, TRANSPORT, AND DEPOSITION
11
particles, which leads to pack avalanching, is not taken into
account. This model is hence suitable for noncohesive uni-
form sediment, as the avalanching occurs in the form of
small bulks.
[74] In the present model, avalanching occurs automati-
cally (by balance of forces), and no extra restrictions are
applied. Yet, as grain-grain interactions within the bed are
not considered, this model is tested for its capability to
reproduce the empirical values of the angle of repose. As a
test case, we start from a steep sediment mound of 75under
stagnant water. The initial slope of 75is much larger than
the angle of repose. Figure 10 shows the initial condition of
the bed. Although there is no ambient flow, the mound col-
lapses. The particles move downhill, the height of the mound
decreases, and the width at the toe increases. Figure 11
shows the avalanching as a function of time. The slope of
the deformed bed decreases until it reaches the angle of
repose. Then the downhill transport of sediment stops and
the bed reaches a steady state. Our model results in an angle
of repose of 30, as can be concluded from Figure 11. Exper-
imental measurements give an angle of repose between 26
and 34for sands [Chanson, 2004].
6.3. Bed Load Sediment Transport
[75] The aim of this section is to compare the sediment
transport rate resulting from the present model with empiri-
cal relations from previous studies. Since more than a cen-
tury, researchers have been developing methods to calculate
the bulk bed-load flux. One of the simplest bed-load trans-
port formulae was developed by Meyer-Peter and M€
uller
[1948]. They conducted experiments for a sediment bed
with varying grain size d¼0:03 2:9cmðÞand with differ-
ent sediment densities ðs¼1:34:2g=cm 3Þ. Their empir-
ical relation for the bed load flux at high shear stress is
q
b¼8
cr
3=2ð56Þ
[76] Equation (56) represents a general bed-load for-
mula, where
cr ¼0:047 is an average value of nondimen-
sional critical shear stress, is the nondimensional shear
stress, and q
bis the nondimensional bed-load transport rate,
defined as
q
b¼qb
dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p= 1
gd
rð57Þ
[77] However, research in the past two decades has
shown that the constant 8 and even the exponent 3/2 needs
to be reassessed for nonsmooth beds. Different values were
obtained for example by Hunziker [1995] or Wong and
Parker [2006] for rippled beds. These differences are dis-
cussed by Tritthart et al. [2011]. If we simulate sediment
transport over a smooth bed, the bed-load transport formula
by Meyer-Peter and M€
uller [1948] gives proper results.
[78]Wiberg and Smith [1987] pointed out that the
observed variation in the transport coefficient is well cap-
tured by a simple dependence on the Shields stress
ðÞ,
yielding a generalized bed-load transport relation :
q
b¼
cr
nð58Þ
where n¼3=2 and
¼1:6ln
ðÞþ9:89:640:166
ð59Þ
[79] Einstein and others used a different approach to esti-
mate the sediment transport rates. This approach takes into
account the effect of the fluctuating flow field on the proba-
bility of the entrainment of particles within a population
[Einstein, 1942; Brown, 1950], and it also considers the
armoring effects and the intensity of turbulent fluctuations
near the bed. Under equilibrium conditions the number of
grains deposited must be equal to the number of grains
eroded, which, along with experimental data fitting, gives
q
b¼40K3
ð60Þ
Figure 9. The fall velocity of a particle with diameter d¼0.25 mm by (a) simulation and (b) iteration
of equation (55).
NABI ET AL.: SEDIMENT PICKUP, TRANSPORT, AND DEPOSITION
12
where
K¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
3þ362
p= 1
gd3
v
u
u
tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
362
p= 1
gd3
v
u
u
tð61Þ
[80]Van Rijn [1984b] proposed a different formula
which can be used to estimate bed-load transport rates with
mean sediment sizes in the range between 0.2 and 2 mm.
This formula is given as
q
b¼0:053 T2:1
d0:3
ð62Þ
where dand Tare the dimensionless mean particle dia-
meter and transport stage parameter, respectively, and
defined as
d¼d
gp= 1
2
2
43
5
1=3
ð63Þ
T¼
cr
cr ð64Þ
[81] Here
cr is the critical shear stress from the Shields
diagram.
[82] In our numerical model, the sediments are trans-
ported as single particles in a Lagrangian framework. To
estimate the bed-load transport rate (or the total load trans-
port rate), the total number of sediment particles passing
through a plane normal to the streamwise direction is
counted. As the model moves the particles at each time
step t, the passing volume is the volume of sediment
which passes the intersection in a time interval tover the
width of the channel. Dividing the volume by tand the
width, it gives the volumetric transport rate per unit width :
q¼npassVp
btð65Þ
in which n
pass
is the number of passing sediment particles,
bis the width of the channel, and qis the total load trans-
port rate yielding from the numerical model.
[83] The sediment transport is simulated for several flow
rates and grain sizes (yielding different values of the
Shields stress). Figure 12 shows a comparison of the sedi-
ment transport rates from the present model with Meyer-
Peter and M€
uller’s, Wiberg and Smith’s, van Rijn’s, and
Einstein’s formulae. It is important to note, however, that
the simulations were done with a completely flat bed and
that we used the zero velocity level as a calibration factor
to tune the transport rate. The zero velocity level is found
to be 0.25 dunder the top of the particles on the bed, which
agrees well with Van Rijn [1984b]. Figure 12 shows that
the simulation results agree well with Meyer-Peter and
Figure 10. Three-dimensional views of the computed avalanche. It begins from a steep angle and the
angle decreases until it reaches the angle of repose.
Figure 11. Cross sections of the bed during avalanching.
NABI ET AL.: SEDIMENT PICKUP, TRANSPORT, AND DEPOSITION
13
M€
uller’s and Wiberg and Smith’s formulae and have the
same order of magnitude as the predictions from the other
empirical relations.
6.4. Sediment Concentration in a Straight Channel
[84] By considering sediment with a constant settling ve-
locity in uniform stationary flow, the suspended sediment
concentration in the vertical direction satisfies the Rouse
distribution [Rouse, 1937]. This theoretical distribution can
be applied in situations where sediment concentration c
a
at
one particular elevation above the bed z¼acan be quanti-
fied. It is given by the expression :
c¼ca
hz
z
a
ha
ws
uð66Þ
where uis shear velocity, zis the distance from the bed,
is von K
arm
an’s constant, with a value of approximately
0.4, and his the water depth. The Rouse profile is valid for
nonhyperconcentrated water-sediment mixtures. The Rouse
parameter (the exponent in equation (66)) denotes the ratio
of the downward settling velocity to the shear velocity rep-
resenting the upward velocities caused by turbulent fluctua-
tions. The settling velocity can be found from equation
(55), and the shear velocity ucan be found as [Van Rijn,
1993]
u¼ub
ln h
z0
þz0
h1
hi ð67Þ
where u
b
is the bulk velocity and z
0
is the level of zero
intercept near the bed. The value of z
0
is related to the
effective roughness height k
s
and defined as z0¼ks=30
with ks¼2:5d. The bulk velocity follows directly from the
flow discharge via ub¼Q=A, where Ais the cross-
sectional area.
[85] In order to validate the sediment transport submo-
del, suspended sediment in a straight channel with uniform
flow is simulated and the vertical sediment concentration
profile is compared with the Rouse distribution. After the
particles have reached statistically steady motion, the num-
ber of particles is determined in thin layers parallel to the
bed and the total volume of particles is divided by the total
volume of the layer. Figure 13 compares the concentration
profile thus obtained with the theoretical Rouse distribution
for a grain size of 150 mm and a bulk velocity of 0.9 m/s.
The simulated concentration profile is found to be in a
good agreement with the theoretical profile.
6.5. Sediment Saltation
[86] A number of numerical tests are conducted to assess
the capacity of the proposed particles-based numerical
model to simulate the saltation mechanism under steady
Figure 12. Sediment transport computed by the present model and compared with some parameterized
relations for different grain sizes with (a) 300 mm and (b) 650 mm.
Figure 13. Comparison between theoretical and simu-
lated Rouse profiles for sediments with diameter 150 mm
and a bulk velocity of 0.9 m/s.
NABI ET AL.: SEDIMENT PICKUP, TRANSPORT, AND DEPOSITION
14
discharge conditions. The results are compared with a se-
ries of experiments conducted by Ni~
no and Garc
ıa [1998]
in a rectangular flume with a length of 18.6 m, a width of
0.297 m, and a slope of 0.0009.
[87] The flume bed was fixed and composed of uniform
sand particles (diameter of 0.53 mm). The flow depth in the
experiments ranged between 28 and 50 mm. The Reynolds
number of the flow varied from about 7800 to about
21,500. The flow was subcritical, and the values of the
Froude number were in the range from about 0.5 to about
0.6. Table 1 shows the experimental conditions.
[88] Here we simulated the experiments using the present
sediment motion model. The numerical conditions are iden-
tical to the experimental conditions of Table 1, except for
the flume length and width. A square part of flume, with a
length and width of 0.114 m, is considered. To avoid side-
wall effects, the boundaries in streamwise and transverse
directions are set to periodic. A flat bed with a uniform sedi-
ment size of 0.53 mm is considered. In order to generate sig-
nificant turbulence structures an initial perturbation is given
to the bed. The presence of irregularities on the bed has a
major role in maintaining the successive saltation process
[Sekine and Kikkawa, 1992]. The imposed heights of the bed
perturbations are in the order of the grain size.
[89] The sediment transport is turned on after the flow
has reached its statistically homogenous steady state. Sub-
sequently, the locations of the pickup, deposition, and the
maximum height from the bed are stored and averaged in
time and space. Moreover, the time between pickup and
deposition is also stored and averaged. Knowing the dura-
tion of the saltation event, T
s
, the associated mean stream-
wise particle velocity U
s
is obtained as
Table 1. Experimental Conditions of Ni~
no and Garc
ıa [1998]
Experiment h(m) u
(m/s) h/d
p
Fr Re
S11 0.0285 0.0207 53.8 0.515 7750
S12 0.0347 0.0240 65.5 0.541 10955
S13 0.0398 0.0266 75.1 0.561 13941
S14 0.0430 0.0282 81.1 0.572 15971
S15 0.0463 0.0299 87.4 0.583 18188
S16 0.0509 0.0321 96.0 0.598 21484
Figure 14. Comparison between experimental and simulated (a) saltation length, (b) saltation height,
and (c) saltation velocity.
NABI ET AL.: SEDIMENT PICKUP, TRANSPORT, AND DEPOSITION
15
Us¼Ls
Tsð68Þ
where L
s
denotes the saltation length. Then, we determined
ensemble averages of saltation height, length, and stream-
wise particle velocity, which are made dimensionless in the
form of hs¼Hs=d;
s¼Ls=d, and us¼Us=u; where H
s
is saltation height with respect to the top of the bed grains.
The dimensionless saltation length and height, obtained
from the computational results, are plotted in Figures 14a
and 14b as a function of =
cr together with those
obtained by Ni~
no and Garc
ıa [1998], Ni~
no et al. [1994],
Abbott and Francis [1977], and Lee and Hsu [1994]. Simi-
larly, results for the dimensionless streamwise saltation ve-
locity, obtained from the present numerical results, are
plotted in Figure 14c as a function of the ration =
cr, to-
gether with those corresponding to gravel saltation [Ni~
no et
al., 1994]; the experimental results of Francis [1973],
Fernandez-Luque and van Beek [1976], and Lee and Hsu
[1994]; and the theoretical relation of Ashida and Michiue
[1972].
[90] The saltation height from our computational results
agrees well with the experimental measurements. However
an overprediction for the low values of =
cr can be
observed in Figure 14a. This is probably caused by a differ-
ence between the bed topography in the experimental setup
and in our calculations, as the turbulence structures can be
strongly affected by the initial bed perturbations. A similar
behavior can be observed in Figure 14b for the low values
of =
cr. However, our computational results fall inside
the range of standard deviation of the experimental meas-
urements, as they are in good agreement with the results of
Ni~
no and Garc
ıa [1998].
7. Conclusions
[91] We have developed a model for the simulation of
sediment motion in turbulent flow. The sediment is consid-
ered to consist of uniform sand-size spherical particles, and
its motion is modeled in a Lagrangian framework, which
allows involving new concepts that are better suited for rel-
atively small spatial and temporal scales. This approach
gives a better insight into the physical transport phenomena
and makes it possible to simulate details of the sediment
motion, such as jumping, sliding, and rolling.
[92] The motion of sediment generally consists of three
stages: (1) the particles begin to move (pickup), (2) they
get transported (moving in the water column or rolling or
sliding over the bed), and (3) they are deposited at other
locations. These stages are all governed by gravitational
and flow-induced forces, which are simulated using theo-
retical and empirical relations.
[93] The model has been validated for various well-
documented aspects, such as the settling velocity of a sin-
gle particle in stagnant water, the angle of repose after ava-
lanching, the bulk flux and the concentration profile in
uniform flow, and the saltation characteristics. In all cases,
the model results agree well with known formulae and rela-
tions. This model is to be used in high-resolution morpho-
dynamic simulations. This means that the present process-
oriented validation is necessary, but not sufficient. Nabi et
al. (submitted manuscript, 2013) combine the present sub-
models for sediment pickup, transport, and deposition with
a model for evolution of the river bed. They validate the
combined system against experimental data on the forma-
tion and migration of ripples and dunes.
[94]Acknowledgments. The work presented herein was carried out
as part of the work package ‘‘River Morphology’’ of Delft Cluster project
4.30 ‘‘Safety against flooding,’’ financed by the Netherlands government
and supported by Rijkswaterstaat Waterdienst. The authors are grateful to
Sanjay Giri for his fruitful suggestions.
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