Download DISCRETE PARTICLE MODELING OF ENTRAINMENT FROM FLAT

DISCRETE PARTICLE MODELING OF ENTRAINMENT
UNIFORMLY SIZED SEDIMENT BEDS
FROM
FLAT
By Ian McEwan1 and John Heald2
ABSTRACT: The stability of randomly deposited sediment beds is examined using a discrete particle model in
which individual grains are represented by spheres. The results indicate that the threshold shear stress for flat
beds consisting of cohesionless uniformly sized grains cannot be adequately described by a single-valued parameter; rather, it is best represented by a distribution of values. Physically, this result stems from the localized
heterogeneity in the arrangement of surface grains. For uniformly sized beds, geometric similarity exists such
that the critical entrainment shear stress distributions scale directly with grain size. A Shields parameter of 0.06
is commonly used to define ‘‘threshold conditions,’’ and it was found that this corresponds to a point on the
distributions where approximately 1.4% by weight of the surface is mobile. Furthermore the analysis includes
a comparison of the contributions of sheltering to variation in critical entrainment shear stress. It was found that
remote sheltering, induced by prominent upstream grains, has a significant effect in increasing the apparent
critical entrainment shear stress of exposed surface grains.
INTRODUCTION
The concept of an entrainment threshold is at the heart of
sediment transport in theory and practice. It is important to be
able to quantify entrainment as a basis for transport rate predictions, as many equations [e.g., Meyer-Peter and Müller
(1948)] require a threshold shear stress. Shields (1936) is generally considered as the seminal work on entrainment. His
work is traditionally used as a means to calculate a discrete
entrainment threshold; the underlying implication being that
there is a definable mean boundary shear stress below which
sediment motion does not occur and above which surface
grains become mobile.
Yet there is an alternative hypothesis concerning entrainment that challenges, or at least amends, the conventional view
that a single-valued entrainment threshold may be defined.
Einstein (1942) stated, ‘‘It is just as impossible to determine
the limit of initial movement as to determine the maximum
possible flood of a river . . .’’ Paintal (1971a) noted, ‘‘The experiments reported . . . indicate that a distinct condition for the
beginning of movement does not exist.’’ Neill and Yalin (1969)
promoted research toward a probabilistic view of the Shields
curve, and Lavelle and Mofjeld (1987) stated that ‘‘. . . this
statistical view of particle movement disallows the concept of
threshold as a motion–no motion transition.’’ More recently
Kirchener et al. (1990) and Buffington et al. (1992) have both
concluded that there is no distinct threshold of motion within
the size fractions of a mixed-grain-size sediment.
These less conventional ideas are, in fact, quite consistent
with other observations. The original Shields data showed considerable scatter and could be interpreted as representing a
band rather than a well-defined curve (Buffington 1999). The
difficulty of observing the threshold, even for uniformly sized
sediments, is well known, and the scatter in data has often
been attributed to the subjective judgments of different observers. Kramer (1932) identified four categories of motion
that describe the progress from a static bed to one that is fully
mobile. This categorization of stages of motion recognizes implicitly that a distinct threshold condition is not well defined.
1
Reader, Dept. of Engrg., Univ. of Aberdeen, AB24 3UE, U.K. E-mail:
[email protected]
2
Grad. Student, Dept. of Engrg., Univ. of Aberdeen, AB24 3UE, U.K.
Note. Discussion open until December 1, 2001. To extend the closing
date one month, a written request must be filed with the ASCE Manager
of Journals. The manuscript for this paper was submitted for review and
possible publication on January 20, 1999; revised March 22, 2001. This
paper is part of the Journal of Hydraulic Engineering, Vol. 127, No. 7,
July, 2001. 䉷ASCE, ISSN 0733-9429/01/0007-0588–0597/$8.00 ⫹ $.50
per page. Paper No. 20081.
588 / JOURNAL OF HYDRAULIC ENGINEERING / JULY 2001
Paintal (1971a) measured transport at shear stresses that would
generally be considered well below the ‘‘threshold of motion.’’
The reference shear stress method, possibly used by Shields
(1936), avoids the problem of visual identification of the
threshold by defining the reference shear stress to be that
which produces a small finite value of transport rate. In mixedgrain-size sediments, the reference shear stress for a particular
size fraction can be derived from transport rate data obtained
over a range of mean boundary shear stresses. Parker et al.
(1982) described the reference shear stress as a ‘‘surrogate’’
for the threshold shear stress. Therefore, although the reference
shear stress method avoids many of the methodological difficulties in defining a threshold, it remains linked, at a conceptual level, to the notion that the threshold of motion can be
described by a single-valued mean flow parameter.
Although the majority of predictive relationships in sediment transport do employ a distinct threshold condition [e.g.,
Meyer-Peter and Müller (1948)], a number of exceptions to
this are present in the literature. These include Gessler (1965),
Grass (1970, 1983), Paintal (1971b), Sutherland and Irvine
(1973), Hsu and Holly (1992), Nakagawa and Tsujimoto
(1980), and Shen and Cheong (1980). These models represent
the destabilizing fluid forces and the threshold shear stress as
distributions rather than as single values, thus providing a
probabilistic model of sediment entrainment. Conceptually,
they offer a more realistic interpretation of the underlying
physics of sediment transport than the classical deterministic
approach in which the destabilizing forces are represented entirely by mean flow parameters and the stability of the sediment bed is represented by a discrete threshold condition.
However, despite being present in the literature for >3 decades,
probabilistic methods have not been widely adopted for practical purposes. This paper suggests that this is because they
are generally more complex to implement than deterministic
equations without producing any obvious improvement in the
accuracy and reliability of the prediction.
It is well known that sediment transport predictions are subject to significant errors. Reasons for this are many and certainly include poor data quality. If poor parameterization of
the physical processes is a major contributor to these errors,
then methods with a stochastic basis are strong contenders for
producing improvements in prediction reliability, not least because they force more explicit treatment of the physics than
more conventional deterministic methods. However, to date,
the form of the distributions of critical entrainment shear stress
and fluid shear stress that have been used in probabilistic models have usually been assumed, often on the basis of rather
tenuous evidence. Discrete particle modeling provides an additional means to evaluate these distributions. In the longer-
term, this may lead to refinements in probabilistic models and,
consequently, to improvements in sediment transport predictions.
This paper uses a computationally intensive technique (discrete particle modeling) to calculate the form of the critical
entrainment shear stress distribution for three uniformly sized
sediment beds. These distributions form a key component of
earlier stochastic transport models [e.g., Paintal (1971b)].
However, this paper has not used these distributions in sediment transport predictions as there are further fundamental
questions that must be addressed before this can be done. The
discrete particle model is capable of simulating and analyzing
mixed-grain-size sediment beds, but a number of important
general principles are demonstrated by analyzing uniformly
sized sediment beds. Therefore, in the interest of clarity, some
of these general principles are examined in the context of uniformly sized sediment beds and the more complex case of
mixed-grain-size sediment beds are treated in McEwan et al.
(1999).
The fundamental hypothesis underpinning this work is that
the entrainment threshold is not a single value; rather, it takes
the form of a probability distribution as a result of local heterogeneity in the sediment bed geometry (Fenton and Abbott
1977). The motivation of this paper is (1) to present a new
method (discrete particle modeling) for calculating distributions of critical entrainment shear stress; (2) to quantify the
form of these distributions for uniformly sized sediment beds;
and (3) to indicate the relative importance of variations in the
supporting grain arrangement and sheltering in determining the
form of the distribution.
This discrete particle model represents individual sediment
particles as spheres (Jefcoate 1995; Jefcoate and McEwan
1997; McEwan et al. 1999) and is a development of the ‘‘multiparticle simulation method’’ first applied to fluvial sediment
transport by Jiang and Haff (1993). In the present model, sediment beds are formed by releasing particles in sequence, each
from a random position in a horizontal plane located well
above the surface. Each particle then falls under gravitational
and drag forces before it undergoes a series of collisions with
previously deposited particles, eventually settling in a stable
position. The geometry of numerically deposited beds is then
analyzed to reveal the ‘‘critical entrainment shear stresses’’ of
individual particles exposed on the surface. The discrete particle method permits a large number of particles to be handled,
which enables the simulation to develop some aspects of the
localized surface heterogeneity inherent in natural sediment
beds.
The model is descended from previous treatments of the
stability of individual particles [e.g., White (1940), Wiberg and
Smith (1987), James (1990), Ling (1995), Coleman (1967),
and Chin and Chiew (1993)]. In general these analyses are
highly simplified, using some or all of the following assumptions: 2D geometry, regular arrangement on the supports on a
horizontal plane, single supporting grain size, and small number of particles. The current model relaxes all of these assumptions and, in particular, allows the heterogeneity of the
supporting arrangement to be treated, opening the way to the
probabilistic analysis of complex sediment beds.
entrainment thresholds in uniformly sized flat cohesionless
sediment beds (Fenton and Abbott 1977). The model generates
a 3D random packing of discrete, cohesionless spherical particles of a known size distribution, although herein the treatment is restricted to uniformly sized beds. Initially spherical
particles (from now on referred to as ‘‘grains’’) are dropped
onto a horizontal reference plane and are assumed to have
settled when half their volume lies below this plane. This process rapidly forms an irregular foundation layer on which a
deep bed of grains is then deposited. The grains fall under the
action of forces due to added mass, fluid drag, and submerged
weight. The governing equation for grain motion is therefore
(␳s ⫹ ␳f cm)V
du
1
= (␳s ⫺ ␳f)Vg ⫺ ␳f cd A兩u rel兩u rel
dt
2
(1)
in which cm = 0.5 and is the coefficient of added mass; u =
grain’s velocity; ␳s = sediment density (=2,650 kg/m3); ␳f =
fluid density (=1,000 kg/m3); V and A = volume and for crosssectional area, respectively, of the spherical grain; cd = drag
coefficient; and u rel = relative velocity between the grain and
the fluid (Sekine and Kikkawa 1992).
Each grain is released from a random position in a horizontal plane well above the impact surface and falls vertically
from rest, reaching a terminal velocity of fall before its first
collision with the static grains making up the bed surface. The
fluid, through which the grain falls, has zero velocity everywhere. When the falling grain collides with a surface grain,
the exact point of the impact is calculated. A 3D rebound takes
place, and the subsequent trajectory of the grain is calculated
by a numerical integration of (1). The initial conditions for
this trajectory are calculated from the rebound of the grain
after its collision, having applied a coefficient of restitution (e
= 0.5) to the normal component of the relative velocity between the colliding grains. Note that, after the initial collision
with the surface, lateral motion is likely to occur because of
the 3D nature of each collision. Prior to settling, a series of
trajectories and collisions takes place with the surface, so the
bouncing grain progressively loses momentum. The final resting place of the grain is determined when it comes to rest in
(or is captured by) a stable position in contact with three other
previously deposited grains. When a particle has repeatedly
struck the same three grains, their feasibility as a stable resting
place is examined by a moment analysis (the grain’s submerged weight being the only active force) about each axis
joining the centers of the supporting grains. Once it has settled,
a grain’s position is fixed and it remains undisturbed by sequent impacts.
If the grain is stable on the supporting triad (Fig. 1), then
its exact location is calculated at the apex of a tetrahedron
with a vertex at the center of each supporting grain. The position vectors (ri, i = 1, 2, or 3) joining these vertices to the
apex have lengths equal to the sum of the two collinear radii,
as indicated in Fig. 1(b). The depositional model provides a
NUMERICAL MODELING OF COMPLEX
SEDIMENT BEDS
Deposition of Sediment Beds
The discrete element model has been developed with the
long-term aim of understanding and quantifying the fundamental processes that govern sediment transport. In this embodiment, the model is used to examine the effect of variability in relative protrusion and support geometry on
FIG. 1. Force Diagram for Surface Grain Supported by Three Neighbors in Flow
JOURNAL OF HYDRAULIC ENGINEERING / JULY 2001 / 589
macroscopically flat geometrical surface upon which entrainment threshold investigations can be carried out.
Entrainment
Before describing the calculation of grain stability, it is important to state clearly how grain sheltering is treated in the
analysis of the simulated beds. As an aid to doing this, two
types of sheltering that can occur in a sediment bed are distinguished. The first type is termed ‘‘direct sheltering,’’ in
which the particles upon which a particular grain is resting
reduce the grain’s effective area and hence alter the fluid force
acting upon it. Direct sheltering is a function of a grain’s support environment, and the reduction in effective particle area
will remain constant as long as the grain is static in its rest
position. The second type is termed ‘‘remote sheltering,’’ in
which other upstream particles, potentially located a number
of diameters upstream, modify the fluid velocity in the vicinity
of the grain and hence have an influence on the fluid forces
acting on it. Remote sheltering is a consequence of the nonuniformity of the near-bed flow that occurs over complex
roughness. This paper quantifies the probability distributions
for entrainment shear stress in uniformly sized sediments,
identifying the contributions made by the supporting arrangement, direct sheltering, and remote sheltering. To demonstrate
the relative contribution of each factor, results are presented
from three uniformly sized sediment beds showing the effect
on the distributions of critical entrainment shear stress of variation in (1) the supporting arrangement only (i.e., downstream
friction angle); (2) supporting arrangement and direct sheltering; and (3) combined effect of the supporting arrangement
and direct and remote sheltering.
It is instructive to examine the distinction drawn here between direct and remote sheltering in another way. In principle, each grain on the surface of the sediment bed will experience a different flow field because of the uniqueness of its
surroundings (Raudkivi 1998). Intuitively these differences
will manifest themselves in terms of both the mean and the
unsteady components of the velocity field. Simulating direct
sheltering only accounts for the influence of support geometry
and reduction in effective area caused by the supporting grains.
As a result, two grains may have the same critical entrainment
shear stress in terms of direct sheltering but would actually be
subjected to quite different fluid forces because of differences
in the geometry of the bed upstream. In this analysis of remote
sheltering an attempt is made to reconcile these differences by
adjusting the critical entrainment force for a particle to take
account of the upstream geometry of the bed. By necessity,
the modeling of this effect is highly simplistic because the flow
field is complex, consisting of multiple interacting wakes. To
make this analysis tractable this study has not sought to model
the fluid behavior in detail but instead approaches it through
a simplified geometrical sheltering argument that provides an
estimate of the effect.
Modeling Entrainment of Grain
Although friction may have an appreciable role to play in
the stability of natural sediment grains, it is not considered in
the moment analysis of grain stability presented in this paper.
The force that does determine the stability of the supported
grain is the resultant of the drag, lift, and submerged weight.
The analysis calculates the critical value of the force acting in
the streamwise direction, which produces a destabilizing moment sufficient to overcome the stabilizing moment due to the
grain’s submerged weight W. The submerged weight is assumed to act through the center of the exposed grain. The
action of the streamwise destabilizing fluid force F may be
displaced from the center of the exposed grain as a result of
590 / JOURNAL OF HYDRAULIC ENGINEERING / JULY 2001
sheltering. The displacement is denoted by the position vector
⌬ in Fig. 1. The calculation of ⌬ depends on the particular
model of sheltering that is employed, and this will be described in more detail below. In the analysis one assumes that
a grain is entrained through rotation about one of the three
axes aij connecting the respective centers of its supporting
grains—the subscript ij denotes the axis connecting the ith
and jth supporting grains. The value of the destabilizing fluid
force required to initiate rotation about the axis aij is F*ij, and
it is obtained when the net moment Mij in (2) is zero. Entrainment of a grain is assumed to occur about the axis aij for which
the streamwise positive magnitude of F*ij (ij = 12, 23, or 31)
is minimized. The net moment acting on the grain about an
axis aij is given by
Mij = (ri ⫻ W) ⭈ âij ⫹ ((ri ⫹ ⌬) ⫻ F) ⭈ âij,
ij = 12, 23, 31
(2)
in which Mij = net moment acting on the grain about axis aij;
ri = position vector from the center of the ith supporting grain
to the center of the exposed grain; âij = unit vector in the
direction of and colinear with aij; F = fluid force acting on the
exposed grain; and ⌬ = position vector (in a plane normal to
the flow direction) from the center of the exposed grain to the
line of action of the fluid force.
The moment analysis represents incipient rotation around
the saddle between two supporting grains, and this is consistent with the action of a drag force acting in the streamwise
direction. In principle, there would be no difficulty in introducing a vertical component to the force F in the analysis of
grain stability. However, given the complexity of the near-bed
flow in terms of nonuniformity and unsteadiness, the direction
as well as the magnitude of F varies in time and space in an
unknown way. Schmeeckle et al. (unpublished manuscript,
2000) found that ‘‘instantaneous’’ drag forces acting on the
grain correlate well with the instantaneous streamwise velocity
but, conversely, lift forces do not exhibit the same behavior.
Thus the mechanism to produce drag forces on a surface particle seems to be reasonably well known, but the origin of lift
forces is much less well understood (Nelson et al. 2000).
Therefore, to make the analysis tractable, the direction of F
has been fixed in the streamwise direction, thereby neglecting
lift forces. This approach may be justified as follows. Previous
models [e.g., Wiberg and Smith (1987)] generate ‘‘aerodynamic’’ lift forces due to mean near-bed velocity gradients that
are more than an order of magnitude smaller than the drag
forces acting at the same point in the velocity profile. Ling
(1995) has shown that the threshold for rolling arising from a
drag force is much closer to the observed entrainment threshold than the threshold for suspension calculated from shear
generated lift forces. These results are consistent with the
model of entrainment adopted herein. Because the simulated
beds are deposited in zero flow velocity, they are topographically isotropic, so the orientation of F in the horizontal plane
is initially an arbitrary choice, which is then kept constant for
all grains and denotes the streamwise direction. By so fixing
the direction of F, one neglects the effect of transverse velocity
fluctuations as well as lift. However it is considered that, as
far as it is presently known, these effects will be second order
for the size of particles analyzed in this paper. Note that the
fluid force F responsible for entrainment is derived directly
from considerations of grain stability, so no explicit formulation of this force is requireed in these calculations.
Modeling of Sheltering for Grain
The influence of the downstream friction angle is examined
for Case 1 by setting 兩⌬兩 = 0, meaning that the fluid force acts
through the center of the grain without any effects of upstream
sheltering. The analysis has not sought to account for the influence of velocity gradient. Previous models [e.g., Wiberg and
Smith (1987)] have assumed that the velocity profile near the
bed is logarithmic and have integrated this profile to calculate
the magnitude and line of action of the fluid force. However,
the logarithmic velocity profile only applies in a temporally
and spatially averaged sense to the flow above roughness sublayer within which the surface grains are enveloped (Raupach
et al. 1991). In the present context one envisages the entraining
velocities to be induced by turbulent eddies, which are transient and many times larger than a grain. Thus it would be
artificial to account for temporally and spatially averaged velocity gradients on the scale of an individual grain and such
has not been done here.
Cases 2 and 3 account for upstream sheltering and the effects of particle wakes (兩⌬兩 =/ 0). The algorithm is based on
the results of Schmeeckle (1998), which show that the local
velocity modification downstream of a protruding particle persists for approximately 8⫻ its protrusion height. This paper’s
sheltering algorithm has two effects. First, the point of application of the fluid force is displaced toward the top of the
exposed grain, which tends to reduce the necessary critical
entrainment force due to an extended lever arm for its moment.
Second, the apparent critical entrainment force is, in effect,
increased by the reduction because of sheltering in the effective cross-stream area of the grain (direct sheltering) and the
likely reduction in flow velocity within a wake zone (remote
sheltering).
To simulate sheltering, the upstream projected area of the
exposed grain is divided into m small elements each of area
dA. The sheltering factor Si of the element dAi is given by
Si = min
冋 册
␭j
,1
8ej
(3)
j =1, 2, . . . , n
where min [ ] = function that returns the minimum positive
value of its arguments; ␭j = distance from exposed grain to
the jth upstream grain; ej = greatest overlap of cross-stream
projections of the exposed grain and jth upstream grain; and
n = number of upstream grains examined. In the case of direct
sheltering, the set of n grains is limited to those in direct contact with the exposed grain, whereas with remote sheltering,
all grains directly upstream from the element dAi are included
in the examination of the upstream bed topography.
Suppose that, in the absence of sheltering, each element dAi
has a small force dFi acting upon it such that total force on
the grain would be 兺mi=1 dFi. To account for sheltering, the
center of the distribution of Si ⫻ dFi is taken to be the point
of application of the fluid force and its location in a crossstream plane relative to the grain’s center is denoted by the
position vector ⌬ = [⌬x, ⌬y] (Fig. 1). In the Case 1 analysis,
the critical value of the fluid force Fc is equivalent to the
min[F*ij]ij =12,23,31 obtained by solving (2) with ⌬ = 0. In Cases
2 and 3, Fc is found by solving (2) for Mij = 0 and ⌬ ≠ 0 and
m
scaling min[F*ij]ij =12,23,31 by the factor m/兺i=1
Si to account for
the reduction in the effective cross-stream area resulting from
upstream sheltering.
Therefore partially covered particles are not significant because the principal conclusions of this paper are largely based
on the lower tails of the cumulative distribution of ␶cg for the
simulated beds. One obtains the critical entrainment shear
stress for an individual grain ␶cg from its critical destabilizing
force Fc as follows:
␶cg =
4兩Fc兩
␲d 2
(4)
where ␶cg = critical entrainment shear stress required to mobilize the exposed grain; and d = diameter of the grain so that
␲d 2/4 is the grain’s plan area.
Before discussing the results, it is important to relate what
is meant by the term critical entrainment shear stress for a
single grain ␶cg to its more conventional usage describing the
threshold of motion for a grain population ␶cp. In the literature,
the term critical entrainment shear stress usually refers to a
value of the mean boundary shear stress exerted by the flow
on the bed. This quantity is generally averaged with respect
to space and time. Nelson et al. (1995) showed that active
periods of sediment movement correlated strongly with high
streamwise velocities, suggesting that, at least for low transport stage, it is the extremes of the near-bed streamwise velocity distribution that drive the entrainment process. This is
consistent with the entrainment mechanism in this analysis. It
is formulated in terms of a critical entrainment force, which
could have been equally well expressed as a critical entrainment velocity, providing that some assumptions about drag
coefficients were introduced. Instead one choses to express the
critical entrainment force as a stress to compare these results
with conventional measurements of entrainment. The critical
entrainment shear stress that one calculates for a single grain
[(4)] is therefore equivalent to the drag stress described by
Buffington and Montgomery (1999), which is derived from
instantaneous streamwise velocity rather than an instantaneous
Reynolds stress.
SIMULATIONS OF COMPLEX SEDIMENT BEDS:
RESULTS AND DISCUSSION
Three uniformly sized sediment beds have been simulated
with grain diameters of 2, 4, and 8 mm. The simulated beds
have periodic boundaries in both horizontal directions so that
grains on one extremity of the bed are supported by suitable
grains at the opposite edge. Fig. 2 shows a detailed view of a
small section of a simulated bed of uniformly sized 2-mm
grains.
Obtaining Critical Entrainment Shear Stress of Grain
To ensure clarity when referring to critical entrainment shear
stress, the symbols ␶cg and ␶cp are used to refer to the critical
entrainment shear stress of a single grain and the threshold
shear stress of the grain population, respectively. The most
likely grains to be entrained are those with three contacts such
that they do not have other particles resting upon them. Only
these grains are considered as candidates for entrainment in
this analysis. Grains that have other particles resting upon
them are not represented in the results. Such grains will tend
to have higher values of ␶cg than fully exposed grains, as a
result of the additional stabilizing force acting upon them.
FIG. 2. Detailed View of Small Section of Bed of Uniformly Sized
2-mm Grains (Note Variation in Exposure and Topography, Which Produces Significant Variation in Entrainment Shear Stress for Individual
Grains)
JOURNAL OF HYDRAULIC ENGINEERING / JULY 2001 / 591
FIG. 3.
Three Cumulative Distributions Calculated for Case 1 for Uniform Sediment Beds of 2, 4, and 8 mm
Results and Analysis of Uniformly Sized Sediment
Beds for Case 1
Fig. 3 shows three cumulative distributions of critical entrainment shear stress calculated from the simulations of uniformly sized sediment beds for Case 1. The results confirm
that, even for an idealized uniformly sized sediment, there is
substantial variability in the stability of individual particles.
Thus, even for a uniformly sized sediment, the concept of a
single-valued shear stress below which no particles move and
above which all particles move is not valid. Although this has
been recognized in the literature [e.g., Paintal (1971a)] the
continued use of Shields (1936) as a single-valued threshold
criterion implies the existence of a well-defined threshold. According to the results presented here, this threshold is not well
defined but is essentially an arbitrary point located within a
distribution of critical entrainment shear stresses. The distribution of shear stress for the three uniformly sized beds is
consistent with those observations that describe levels of grain
activity as the mean boundary shear stress is increased—the
levels ranging from occasional sporadic movement to general
motion over the whole bed. The distribution of shear stresses
also illustrates why there is such variability in the visual determination of threshold conditions in the laboratory. Even if
the reference shear stress methods are used [e.g., Wilcock
(1993)], the outcome is essentially an arbitrarily chosen point
somewhere in the lower tail of the distribution.
The entrainment of individual grains is often considered in
terms of the destabilizing fluid forces and the stabilizing force
due to submerged weight. This is the essence of the Shields
parameter. Consider the threshold of motion for the idealized
particle shown in Fig. 1. The critical conditions for incipient
motion occur when the moments due to the fluid forces and
submerged weight are balanced. In simple terms this can be
represented in terms of (5) if H and L are considered to be
the lever arms of the fluid force and the submerged weight,
respectively, about an axis of rotation joining the centers of
two supporting grains.
Thus
兩Fc兩H ⫺ 兩W兩L = 0
(5)
FIG. 4. Three Cumulative Distributions for Case 1 Scaled by Factor (␳s ⫺ ␳f)gd; Three Distributions Now Fall onto Single Curve, Which Shows That
Uniformly Sized Sediment Beds Are Geometrically Similar
592 / JOURNAL OF HYDRAULIC ENGINEERING / JULY 2001
min which 兩W兩 = (␳s ⫺ ␳f)gV = submerged weight of the grain.
Combining (4) and (5) and rearranging, one has
␶cg =
2
(␳s ⫺ ␳f)gd
3
冉冊
L
H
(6)
For a grain in a uniformly sized sediment bed, both L and
H scale directly with the grain diameter d. Therefore (L/H )
depends on the geometry of the supporting arrangement and,
for uniformly sized sediment beds, will show no dependency
on grain size. Furthermore one has found that the variation of
(L/H ) takes the form of the distributions shown in Fig. 3.
Because this distribution is independent of grain size, its
form is revealed by a plot of the cumulative distribution of
critical entrainment shear stress scaled by (␳s ⫺ ␳f)gd. Fig. 4
shows the scaled distributions, and as expected, they fall onto
the same curve. The resultant dimensionless distribution is defined by a function P [␪cg], which returns the fraction by weight
of the surface population entrained by the dimensionless shear
stress, ␪cg = ␶cg /(␳s ⫺ ␳f)gd.
The correspondence of the distributions indicates that the
simulated uniformly sized sediment beds are geometrically
similar in a statistical sense. It also suggests that a single dimensionless curve can be used to represent the entrainment of
any uniformly sized sediment, neglecting variations in the
shape and mineralogy (relative density) or natural sediments.
Role of Sheltering in Determining Threshold
Fig. 5 shows the threshold shear stress distributions calculated for Cases 1–3 for the 2-mm uniformly sized sidement
bed. For Case 1, there is no sheltering and the distribution
results from the variation in the supporting arrangement only.
In Case 2, the effect of direct sheltering is included in the
analysis. In Case 3, the effects of the supporting arrangement,
direct sheltering and remote sheltering are all included in the
analysis. The results indicate that direct sheltering in a uniformly sized sediment plays a significant role in increasing the
threshold shear beyond the value due to the supporting grain
arrangement only. However a larger change in the distribution
is obtained when the effects of remote sheltering by upstream
grains are included.
FIG. 5. Compares Critical Entrainment Shear Stress Distributions Calculated for Case 1, Influence of Supporting Grain Arrangement Only; Case 2
Supporting Grain Arrangement and Direct Sheltering; and Case 3 Supporting Grain Arrangement, Direct and Remote Sheltering
FIG. 6.
Three Cumulative Distributions Calculated for Case 3 for Uniform Sediment Beds of 2, 4, and 8 mm
JOURNAL OF HYDRAULIC ENGINEERING / JULY 2001 / 593
FIG. 7.
Three Distributions for Cases 1–3 in Frequency Form [Distributions Are Approximately Lognormal (Table 1)]
TABLE 1. Parameters Describing Dimensionless Lognormal Critical
Shear Stress Distributions
Parameter
Case 1—no sheltering
Case 2—direct sheltering
Case 3—remote sheltering
Mean ␪cg
Standard deviation
␴g
0.351
0.455
0.870
2.52
2.64
2.77
Fig. 6 shows the Case 3 distribution for the three simulated
beds, scaled by (␳s ⫺ ␳f)gd. Again the geometric similarity of
the beds is demonstrated. In essence this is not surprising because remote sheltering, as it is calculated in this analysis,
depends only on the geometry of the bed. Thus, although the
three simulated beds are unique at a grain scale, they are statistically similar when a large number of grains are considered.
To a first approximation, the distributions for Cases 1–3
may be considered to be lognormal. Fig. 7 shows the three
dimensionless distributions plotted in frequency form. Table 1
gives the mean and standard deviation of the distributions obtained by a least-squares fit to the calculated data. As expected,
the mean of the distribution increases as the degree of sheltering increases. Note also that the standard deviation is similar
for each distribution.
Comparisons with Shields (1936)
冉冊
␶0
␳
1/2
d
;
␯
␪=
␶0
(␳s ⫺ ␳f)gd
(7)
in which ␪ = Shields number; R = particle Reynolds number;
u* = shear velocity of the flow; ␶0 = mean boundary shear
stress; and ␯ = kinematic viscosity of the fluid (=10⫺6 m2/s).
Often the threshold condition for a uniformly sized sediment
in hydraulically rough flow is given by
␪cp =
Implications for Transport Rate Prediction
Because the analyses have been performed on static beds,
caution should be exercised in using the results as a basis to
discuss transport rates. Nevertheless some useful information
can be derived from considering how the form of the distribution might influence the parameterization of the transport
rate. Fig. 9 shows the Case 3 analyses for three simulated beds
plotted on logarithmic axes. From geometric similarity, one
concludes that each distribution must have the same slope, and
regression of the intermediate section of the distributions in
Fig. 9 shows this to be close to 3/2.
The bed-load transport rate for a uniformly sized sediment
can be defined
Qb ⬅ GbUb
The Shields curve remains the standard method for analyzing entrainment in uniformly sized sediments. It is therefore
useful to compare the simulation results for Case 3 with the
Shields curve, which has the form (R, ␪) such that
u d
R= * =
␯
the point where ␪cg = 0.06, so the corresponding fraction by
weight entrained is roughly 0.014. Thus the value of the
Shields number often associated with the critical conditions is
represented by a point on the distribution of critical entrainment shear stresses for Case 3 where approximately 1.4% by
weight of the surface population is in motion.
Fig. 8 shows the results of the simulations displayed in
terms of the Shields diagram. Curves show the values of the
Shields parameter for 0.5, 1, 1.4, 2, and 8% by weight of the
surface population in motion.
␶cp
= 0.06
(␳s ⫺ ␳f)gd
(8)
Now in terms of the nondimensionalized cumulative distribution curve for Case 1 (Fig. 4) this condition corresponds to
594 / JOURNAL OF HYDRAULIC ENGINEERING / JULY 2001
(9)
in which Qb = transport rate of grains moving in bed load; Gb
= mass of sediment in motion over a unit area of bed; and Ub
= mean streamwise velocity of the moving grains.
The form of the shear stress distributions in Fig. 9 suggests
that Gb can be approximated by Gb ⬀ (␶0 ⫺ ␶cp)n where n =
3/2. Yalin (1963) developed a bed-load formula through a parameterization of (9). He suggested that Gb ⬀ (␶0 ⫺ ␶cp)n and
proposed that n = 1 but did not provide convincing reasons
for this choice of the exponent. Yalin (1979) contains a fuller
discussion of this. In contrast, the results herein indicate that
Gb ⬀ (␶0 ⫺ ␶cp)n where n = 3/2. The recovery of the 3/2 exponent from the discrete particle modeling is intriguing and
offers some additional insight into the processes that govern
sediment transport. Although one is unable to offer a full explanation on why the exponent should be 3/2, one suggests
that it stems from some organizational or bulk property of the
deposited sediment bed. The implication of this is that the
packing or organization of the sediment bed may have a cru-
FIG. 8. Shields Curve Together with Numerical Results Showing 0.5, 1, 1.4, 2, 4, and 8% by Mass of Surface Population in Motion as Predicted by
Case 3 Remote Sheltering Analyses
FIG. 9.
Three Distributions Plotted on Logarithmic Axes (Note That Slope of Each Distribution Is Approximately 3/2)
cial and, previously, largely neglected role in regulating the
flux of particles entrained from the bed. Clearly further confirmation of this behavior would have implications for parameterizing entrainment rates and, consequently, transport
rates, as it suggests that surface structure, as well as fluid
forces, can be important. To date it is conventionally assumed
that the surface, as a source of sediment, can be represented
entirely in terms of single-value threshold shear stress. However this analysis suggests that perhaps the surface plays a
more active role in determining transport rates than has previously been considered to be the case.
EVALUATION OF DISCRETE PARTICLE MODELING
The results clearly demonstrate that, even for a uniformly
sized sediment, variation in support geometry for surface
grains is a significant determinant of the stability of sediment
surfaces. The form of this distribution for low and moderate
shear stresses must play a role in dictating entrainment rates
during transport. These analyses have been performed on uniformly sized sediment beds in the absence of flow—the influ-
ence of the flow on a particle being represented by the force
that would destabilize the particle from its supports. The ultimate goal of this work is to incorporate these distributions
into a probabilistic calculation for predicting entrainment and
transport rates (Grass 1970, 1983). This step would require the
critical entrainment shear stress distributions analyzed here to
be combined with distributions of fluid forces due to turbulence. Many studies [e.g., Kirchener et al. (1990), Buffington
et al. (1992), and Johnstone et al. (1998)] have examined particle stability in the absence of fluid motion, representing the
fluid forces only in an abstracted form. This has enabled them
to quantify directly the forces that destabilize particles rather
than express them indirectly in the form of some measured
flow parameter. The result is a much clearer picture of the role
of surface heterogeneity in the variation of critical entrainment
shear stresses. However the surface heterogeneity that causes
variable particle stability also produces local disturbances in
the flow, with the result that each surface particle is subjected
to a unique flow regime. These disturbances are difficult to
quantify without resort to empiricism because of the complexity of the near-bed flow. Nevertheless a first attempt has been
JOURNAL OF HYDRAULIC ENGINEERING / JULY 2001 / 595
made at doing so by adjusting the necessary critical entrainment shear stress of a particle to take account of the degree
of sheltering afforded by the grains located upstream of the
particle. The analyses indicate that remote sheltering differs
from no sheltering to a first approximation only in terms of
an increase in the mean of the distribution, the standard deviation showing only a slight change. The conclusion that remote sheltering does not significantly change the shape of the
distribution, only its mean, suggests that particle stability studies can form the basis for constructing critical entrainment
shear stress distributions for developing prediction models
(Grass 1970, 1983; Paintal 1971b). Note, however, that the
present analysis neglects effects due to interacting wakes shed
by grains upstream of the exposed grain and also possible
mixed-grain-size effects.
The numerical model results are limited in three important
ways due to the idealization of the simulation. The first of
these is the way in which the beds are deposited. Only one
form of deposition has been examined, where particles are
dropped from a height and numerically undergo a series of
collisions until they settle in a geometrically feasible position.
Clearly this does not represent the manner in which a natural
sediment surface is formed, so different structures and packing
may be expected to occur. In general terms, this is an area that
has received little attention in the technical literature as sediment beds in flume experiments are also formed in a manner
quite different from natural waterborne deposition. Unfortunately the numerical model does not yet have the capability
to water work its bed, as can be done in a flume experiment.
Thus the examination of how the deposition mechanism affects
the form of the distribution of critical entrainment shear
stresses is not examined in this paper, although it may well
prove to be important.
Second, the particles in the simulations are frictionless
spheres. As such they will not reproduce any of the interlocking that might be expected to occur with natural, angular
grains. Moreover, natural sediments are not spherical, so shape
effects, which are present in natural beds, cannot be accounted
for in the modeling. Thus processes such as imbrication, in
which the orientation of oblate grains influences their threshold shear stress, cannot be modeled (Miller and Byrne 1966;
Li and Komar 1986). Moreover, the degree of sphericity of a
natural sediment may also be size dependent, so, in principle,
the geometric similarity found in the idealized simulation may
not be fully realized in natural sediments.
Third, it is likely that the nature of the entrainment process
will change for a higher stage when a significant fraction of
the bed is likely to be in motion. Under these circumstances
the applicability of the entrainment model, which analyses the
stability of a surface grain supported by three other static
grains, will become increasingly questionable as mean boundary shear stress is increased. Thus the shape of the cumulative
distribution curve calculated for higher shear stresses, when
the surface is likely to be in general movement, will be less
representative of actual conditions; rather, the analysis applies
to low and moderate values of shear stresses for which much
of the surface would be static. This suggests a limitation to
the simulated results in terms of the range of shear stresses to
which they can be applied.
CONCLUSIONS
• The critical entrainment shear stress for cohesionless uniformly sized sediments in flat beds should be represented
by a distribution rather than a single value.
• There is no distinct point in this distribution that may be
reasonably considered as a threshold value. Therefore one
concludes that a distinct threshold for the initiation of motion does not exist.
596 / JOURNAL OF HYDRAULIC ENGINEERING / JULY 2001
• Geometric similarity exists for uniformly sized sediment
beds consisting of spherical grains with the same density.
Thus the form of distribution of critical entrainment shear
stresses is independent of sediment size, at least for fully
turbulent flow.
• A value of 0.06 has traditionally been taken to represent
threshold conditions for a uniformly sized sediment. This
value corresponds to a point on the critical shear stress
distribution for Case 3—remote sheltering when 1.4% of
the surface is mobile.
• The supporting grain arrangement and the influence of
remote sheltering by upstream grains dominate the critical
entrainment shear stress distribution. Direct sheltering by
the supporting arrangement is also important, but it plays
a lesser role in determining the distribution. Distributions
for all three cases are approximately lognormal, with the
mean critical entrainment shear stress increasing with the
degree of sheltering.
• The relationship between the mass entrained per unit area
and the mean boundary shear stress for low to moderate
shear stress can be fairly well approximated by Gb ⬀ (␶0
⫺ ␶cp)n, where n = 3/2. Somewhat speculatively, it is suggested that there is some organizational property of granular beds that gives rise to the 3/2 exponent. This has
implications for bed-load prediction, indicating that surface structure and organization may play a more active
role in transport dynamics than has previously been recognized.
ACKNOWLEDGMENTS
The second writer has been funded by the Bank of Scotland, Edinburgh, and the University of Aberdeen. The Engineering and Physical
Sciences Research Council of the U.K. and the Royal Academy of Engineering of the U.K. also provided financial support for this work. Barry
Jefcoate undertook much of the early development work on this model,
and his contribution to the work is warmly acknowledged. Comments
from Brian Willetts, Roger Bettess, and Mark Schmeekle have been helpful in improving this manuscript.
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