(1)
Zhong De-yu1,
Liu Jin-mei1,
Zhang Hong-wu1
and Wang
Guang-qian1
1.
Hydraulic and Coastal Engineering Institute, Tsinghua University, Beijing 100084
Abstdract:
A PDF (probability density function) model for entrainment rate of bed material
into suspension is presented in this paper. The transport equation for PDF is
introduced to obtain the statistical properties of sediment moving near the
riverbed. Regarding the different effects of ejection and sweeping of turbulence
burst on the movement of sediment, the entrainment rate is obtained. Since the
distribution of the upward flux of sediment is derived, it permits to calculate
the variation of the entrainment flux perpendicular to riverbed after sediment
resting on bed surface is picked up. The reference concentration of suspended
load is also obtained in this paper. Comparisons show that there is a
satisfactory agreement between present model and observation.
Keywords: suspended load, entrainment rate, probability density function, turbulence burst
In alluvial river,
suspended load consecutively exchanges its material with the bed: on the one
hand, particles in suspension are deposited to the bed due to gravity, on the
other hand, the particles in bed surface are entrained into suspension. If the
exchange taking place between suspended load and the bed material is not
equilibrium, i.e., the deposition flux is not balanced out by entrainment flux,
variation of bed level occurs. Since the net flux determines the erosion and
deposition rate, and servers as a lower boundary condition for the concentration
transport equation of suspended sediment, determination of these fluxes,
therefore, is required in 1D, as well as 2D and 3D mathematical modeling of
morphological processes.
In the turbulence diffusion theory for suspended load, it is generally accepted that the deposition flux can be expressed as the product of effective settling velocity and the local concentration of suspended sediment near the bed surface. As to entrainment rate, there are two different definitions. One is defined to be the turbulence flux passing through the interface between bed load and suspended load. The definition is obtained from depth-averaged convection diffusion equation for suspended sediment concentration. Because the diffusion flux is expressed in terms of the time-averaged correlation of fluctuation concentration and vertical velocity, such definition met difficulty in closing the turbulence correlation. In the work of Garcia and Parker (1991), by assuming that the entrainment rate is equal to the deposition flux when suspended sediment is transported in equilibrium state, they derived an empirical relation for entrainment rate. The dimensionless entrainment function derived in their work, in nature, is of equivalence to equilibrium bed concentration. The other definition considers the entrainment rate to be the number or the volume of bed material picked up by flow per unit time from unit area of bed surface (Cao 1997). In the study of Cao (1997), the entrainment of bed material into suspension was attributed to the ejection of turbulence burst in the near bed region of open channel flow. The conceptual model reveals certain essence of entrainment rate. But it is questionable that all grains picked up by flow behave as suspended load. Experiments on sediment motion in near bed region (Kaftori, etc. 1995) show that not all particles entrained by ejection of burst are in suspension, but certain of them move in the pattern of saltation as bed load. The fraction occupied by suspended load in total load is a function of Shields parameter, the larger the Shields parameter, the more the fraction in suspension. This fact may be used to explain why the relation proposed by Cao (1997) has a decreased empirical parameter with the size of sediment increasing.
Theoretically
speaking, the characteristics of sediment movement when sediment is entrained
from riverbed can be obtained by solving the motion equations of particles.
However, the number of sediment in suspension is so large even in low
concentrated sediment laden flow, therefore, the massive calculations involved
make it impossible to solve the motion equations of sediment particles, at least
impossible in practical level. Furthermore, the motions of sediment particles
are influenced by eddies as small as the same dimension of the particles them
own, large scaled mathematical modeling in engineering does not permit to
provide detailed information about eddies with same dimension as suspended
particles. Therefore, an approach which can reflect how the forces acting on
particles affect the motions and at the same time no complex calculation
required is necessary in the study of entrainment of bed material into
suspension. The kinetic theory, or probability density function (PDF) model
fulfills the requirement. This model gives the distribution of probability
density from the transport equation of PDF without solving motion equations of
particles. This is what we need in the study the entrainment rate. In addition,
from PDF model, the distribution of upward and downward flux perpendicular to
riverbed can be obtained. The distribution makes it impossible to calculate the
variation of the entrainment flux in vertical direction after bed material is
picked up by flow.
In current work, firstly the transport equation, or the kinetic equation for PDF of particles was derived; secondly the distribution of PDF along vertical direction of channel flow was obtained with the consideration of the effect owing to turbulence burst on sediment motion in near bed region; then the entrainment rate was gained. The equilibrium bed concentration of suspended sediment was also given in this paper, and comparison showed satisfactory agreement between present theory and observations.
The motion equation of a particle moving in the turbulent flow can be expressed as:
(1)
and
(2)
where
position vector of particle;
instantaneous velocity vector of particle;
the sum of the forces acting on a particle;
mass of a particle with diameter
.
Since the flow around the particle being turbulence, the Eq. (1) and Eq. (2) are of the random differential equations. The transport equation for PDF of particle corresponding to the motion equations is
(3)
where
instantaneous PDF of particles;
i-component of the vector
and
, respectively;
i-component of
.
Since the flow taken into account being turbulence, the variables in Eq. (3) are decomposed into the time averaged part and fluctuation part:
(4)
and
(5)
Substituting Eq. (4) and (5) into (3) and applying Reynolds’s averaged method, thus the time averaged kinetic equation for PDF is
(6)
There is no available result on closing the term on the right hand of Eq. (6) yet, when forces in addition to drag force are considered. For first approximation, this term is not considered in the present work. The neglect of this correlation term implies that only mean forces acting on particles are taken into account. The approximation is the same as Wang and Ni (1991) made in their research. Then the equation is reduced to
(7)
PDFs OF SEDIMENT MOVING IN THE NEAR BED REGION
Consider a steady and uniform
two-dimensional open-channel flow. The bed of the channel is assumed to be
covered with uniform sediment with diameter
. The solution of Eq. (7) is derived as
(8)
where
is the number density of particles
at
;
，
。
is
(9)
Then integration
must be a constant, for uniform flow along streamwise is assumed. For simplicity
it is set to be equal to zero. Hence
(10)
is the sum of gravity, buoyant force, lift force, drag force and other forces.
It can be expressed as
(11)
where
,
=density of fluid and particles, respectively;
,
,
=lift force, drag force and other forces acting on particles.
Turbulence
burst near the bed plays an important role on sediment movement there. During
the ejection phase of burst, particles resetting on bed are lifted up by upward
flow experiencing acceleration. In this period, particles are moving from low
velocity region to higher one and trying to catch up with fluid, therefore there
must considerable relative velocity between sediment and fluid. On the contrary,
in sweeping phase of burst, particles are carried by high-speed fluid moving
downward and mixing with lower speed fluid, therefore, during sweeping phase,
the relative velocity must be minor. This has been demonstrated by the
experiment. According to the experimental results of Nino and Garcia (1996), it
reveals that the ensemble averaged mean streamwise relative velocities in
ejection phase and sweeping phase are different. It shows that during ejection
phase of burst, the stremwise relative velocity is obvious, and the mean
streamwise relative velocity
approximately equals to the
streamwise turbulence intensity of fluid
; whereas during the sweeping phase, it appears that
. Considering this fact and bearing in mind that lift force is proportional to
, when particles are lifted up by flow, the lift force must have significant
effect on upward motion of sediment; when particles carried by sweeping fluid
moving downward, the lift force nearly has no influence on particles. Hence
during upward motion, forces acting on particle are
(12)
whereas during
downward motion, the total forces on particle denoted by
are
(13)
where superscripts “+” and “-” denote upward motion and downward motion respectively. Consequently, the PDFs are different between upward and downward motion of sediment. They are
(14)
where
are the numbers of particles at
; and
.
Corresponding to PDFs of sediment moving near the bed, the upward and downward fluxes are, respectively
(15)
and
(16)
where
are the volumetric concentrations
.
The functions
need to be determined explicitly. During upward motion of particles, Summer
(1984) suggested that the lift force, drag force and other forces can be
regarded as a comprehensive lift force. Herein we assume it having the form as
(17)
where
is the lift coefficient. The lift coefficient
is a function of
(Summer 1984), but its distribution along
has not been obtained. In this
paper, it was regarded as a constant parameter in integration.
During upward motion, with the
assumption that
, therefore, the forces acting on particles in upward moving particles is
(18)
during downward motion,
(19)
If
being not taken into account, when
, we obtained that
(20)
where
drag coefficient;
settling velocity of sediment in still water. Hence Eq. (19) can be written as
(21)
where
downward velocity of sediment. Therefore
and
are
(22)
and
(23)
in which
and
. Correspondingly the upward and downward fluxes are, respectively
(24)
and
(25)
Because the bed sediment being
entrained into suspension is the grains, which directly experience the shear of
flow on bed surface, hence
. Correspondingly, the reference concentration
therefore equals to
, the maximum volumetric concentration of bed material on bed surface. As to
, it must be the local concentration at
, denoted by
. Hence, Eq. (24) and (25) can be written as
(26)
and
(27)
The turbulence intensity of particle
was assumed to be equal to the friction velocity, i.e.
. Thus
and
can be simplified into
(28)
and
(29)
In order to obtain the entrainment
flux of bed material into suspension, firstly the interface of bed load and
suspended load must be defined. Detailed observations on the movement of bed
load shows that there are three different motion patterns for bed load: sliding,
rolling, and jumping (Chien et al. 1983). The maximum thickness of bed load
layer has the same dimension with the average height of saltation grains. The
level of interface, therefore, must be located at several grains diameter above
the bed. But Van Rijn (1984a, b) argued that the reference level should be
located at half ripple height above the average bed when bed form appears. But
to present model, the difficulty in definition of the interface is not so
serious, since the profile of entrainment rate along vertical direction given by
Eq. (27) makes it possible to give the entrainment rate at any elevation above
riverbed, as long as the parameter
is properly defined.
Through being a significant parameter
in river mechanics, only a few groups of experiments on pick-up function of bed
sediment were reported (Van Rijn, 1984c ), extensive experimental research has
not been carried out due to the difficulty in measurement near river bed without
disturbance both the fluid flowing and sediment moving. Therefore, present PDF
model cannot be directly verified. However, many efforts have been contributed
to the equilibrium bed concentration, or the reference concentration of
suspended sediment (Einstein 1950, Engleund and Fredsoe 1976, Van Rijn 1984b,
etc.), which can be obtained from
, and the results can bed used to verify present PDF model. The validation of
the verification with equilibrium bed concentration data lies in the fact that
the upward or entrainment flux is independent of the concentration, which means
that, no mater wheter the sediment is under equilibrium transport or not, the
entrainment rate of sediment with certain diameter keeps the same under the
action of river flow with certain intensity.
From Eq. (28) and Eq. (29), we
obtained the equilibrium bed concentration
being
(30)
In this paper, the same experiment data had been used by Zyserman and Fredsoe (1994) in their work were employed to verify present model. The equilibrium bed concentration was chosen to be the concentration two particles diameter above the bed as Einstein (1950) proposed. Such that Eq. (30) was written as
(31)
As shown in
Figure 1, present model agrees satisfactorily with experiment data and the
empirical relation obtained by Zyserman and Fredsoe (1994). In calculation of Eq.
(29),
and
.
Fig. 1 Comparison of
present model with experiment data used by Zyserman and Fresdose. The empirical
relation proposed by Zyserman and Fredsoe also was dipicted.
Entrainment rate of bed material into suspension
is a key parameter in the study of sediment transport. In this paper, a PDF
model for it was established and verified.
Many researches on the turbulence-particles interaction in the
near bed region of open channel flow reveal that the turbulence burst plays an
importance role on sediment movement(related literatures can be found in the
papers of many researchers, for instance, Kaftori etc. 1995, Nino etc. 1996).
Furthermore, experimental research(Nino 1996) also showed that the movement
chracteristicses of sediment are different from burst ejection to sweeping,
which indicated that the forces on sediment, corresponding to different phases
of tubulence burst, are not the same. Hence, a model has the ability to identify
the diffence is necessary in the study of the entrainment rate. In order to
avoid complicated and massive calculations, a PDF model was introduced in this
paper. The model reported in this paper has the special feartures as follows:
(1) effects on sediment with respect to different forces can be easily taken
into account without solving the motion equations of sediment; (2) the
distruibution of upward flux along water depth was obtained. This feature makes
present model can predict the variation of the upward flux along
direction after sediment entrained from river bed, which fits to different
definations of the reference level of suspended load.
The equilibrium bed concentration, or
the reference concentration for suspended load was also derived in this paper.
The comparison showed satisfactory agreement between present relation and the
experiment. The bed concentration (Eq. (31)) indicates that when
,
approaches limited value, which
agrees with previous researches (Garcia and Parke,
and Fredsoe, etc.)
in this paper was considered to be a empirical papameter, because the present
knowledge on lift coefficient is limited so that the theoretical relation and
the description of its variation along vertical direction are not available by
now. But it is expectable that
increases when the reference level
increased. The reason is that
decreases with increased distance to bed suface.
Acknowledgements
The study reported in this paper was supported
by Basic research foundation of hydraulic engineering department of Tsinghua
Universtiy, and Natural Science Foundation (No. 59879006 and No. 59890200). The
authors also want to express their thanks to J. A. Zyserman and J. Fredsoe for
their kindly help in providing the data for verificaiton.
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