T.S.Yang1, Q.S.Li2, Z.H.Wan3
[1] Dept. of Hydraulic and Hydro-Power Engineering,
Tsinghua University, Beijing 100084, China
Tel: 86-10-62788524 E-mail: yangts@mail.tsinghua.edu.cn
2 Dept. of Hydraulic and Hydro-power Engineering,
Tsinghua University, Beijing 100084, China
3 China Institute of Water
Resources and Hydro-Power Research, Beijing 100044, China
of the deposited sediment have important influences on incipient motion. Authors
developed a relationship between the cohesive force and these two factors, and
established an incipient motion formula for fine sediment particles.
Experimental data collected from various resources agree with this theoretical
result.
Keywords: fine sediment particles, incipient motion, cohesion, water depth,
shear stress
The mechanism of incipient motion of fine
sediment particles under large water depth is one of the key problems for reservoirs and harbors in which
water depths are over 10m. Engineers are interest in the dynamical behaviors of
fine particles deposited under large water depth in order to forecast local
scour and deposition.
Shields (1936) is the first person to express the critical stress for incipient motion of sediment in a dimensionless form. In ‘50s Lane (1953) applied the concept of drag force for incipient motion of sediment into design of stable channels. Later there have been numerous additions, and modifications of Shields work since its original investigation. One aspect of the improvement for Shields curve referred to the incipient motion of non-uniform sand (Gessler 1971; Li and komer 1986; James 1990; Kuhnle 1993; Wilcock 1993). Buffington and Montgomery (1997) carried out a systematic analysis of eight decades of incipient motion studies with special reference to gravel-beded rivers.
For fine sediment particles the cohesive effect rather than the gravity plays a dominant role to control the incipient motion conditions. An acceptable suggestion is that DLVO theory on interactions of micro-particles in colloidal solution can be used to explain the aggregation and flocculation of cohesive sediment particles (Luckham 1999; Mantz 1983; Xia 1980). Generally speaking, the Van der Waals force is one of major factors influencing cohesion effect, and the cohesive force depends on the separation between two particles. It implies that the densities or dry unit weight γ′ of fine sediment particles deposition should be considered as an important factor for incipient motion (Tang 1963; Yang and Wang 1995).
Fig.
1 Comparison of three formulas for incipient motion of
fine sediment particles, d=0.004mm
Another opinion pays attention to water depth and consider it as a major factor causing cohesion effect (Zhang et al. 1989). Zhang pointed out that the thin water film closely wrapping particle surface possesses strange mechanical properties similar to solid materials, and can merely transmit static water pressure unilaterally.
Based upon own comprehension for the origin of cohesion forces Zhang (1961), Dou (1960), and Tang (1963) independently established own formulas to describe the incipient motion of fine sediment particles. Former two of them include water depth H; Tang's formula charge cohesion only to the deposition density but water depth. The perplexing thing is that these three formulas give results close to each other only under the conditions of small depth; but they deviate with water depth increasing Fig.1 presents a comparison of three formulas for incipient motion velocity Uc, and the sediment particle size d = 0.004mm.
With the focuses on whether and how the flow depth influences the incipient motion of fine sediment particles, this paper presents a description of the experiments and a preliminary physico-chemistry explanation, then established an incipient motion formula for fine sediment particles. Since it is hard to practically conduct experiments of open-channel flow with a great depth in laboratory, a pressurized pipeline is actually used to simulate the effects of water depth on the incipient motion of fine particles.
Fig.2 presents the sketch of the
experimental setup with a test section G, a thyristor C, a direct-current motor
F, a pump P, tank
, regulating valve
and a pipe line system. By
adjusting the voltage output of thyristor C and the opening of valve
, water pressure and flow discharge in the pipeline could be changed.Two
pressures gauges
and
were installed at both ends of the test section G in order to measure the water
pressures H1 and H2 at the upstream and downstream ends respectively. The
value of flow discharge Q was measures and displays with an
electron-magnetic meter E and a transforming meter D. At the initial stage of
each test run, the pipeline would be slowly filled up with clean water through
two injection holes
for moderating the disturbances on bed surface. The test section G of 2320mm
length consisted of three parts: the observation part, the upstream and
downstream transitions. Both two transitions parts were smoothly
connected to the pipe line of 50mm diameter and the observation part of 500 mm
length with its rectangular cross section of 90mm×90mm.
The bottom of observation part was actually a movable container filled with
sediment sample. All the three parts of test section G were made of plexiglass.
Three
kinds of sediment samples used for tests were Grade 1 with diameter d50=10.0mm collected from the navigation pass
of Hangzhou Bay near Yangtze River Estuary in southeast China, Grade 2 with d50=6.8 mm from the floodplain near Huayuankou
Hydrologic Station on the Lower Yellow River in northeast China; Grade 3 with d50=8.0 mm was the mixture of Grades 1 and 2.
Totally 75 test runs were carried out
for 3 kinds of samples. The test conditions were listed in the Table 1. The
relationships of critical shear stress
,
, and water depth H are presented in Fig.3-a and Fig.3-b respectively. It is
obvious that both of dry unit weight
and water flow depth H is significant factors to the incipient motion of fine
sediment particles.
Table 1 Varying range of test conditions
|
Silts No. |
Runs of tests |
Particle size d50(mm) |
Unit dry weight r' (g/cm3) |
Water depth H (m) |
Water discharge Q (l/s) |
|
Grade1 |
25 |
0.0100 |
0.800~1.164 |
3.31~13.11 |
2.69~10.09 |
|
Grade2 |
17 |
0.0068 |
0.650~0.966 |
2.20~11.59 |
0.75~8.72 |
|
Grade3 |
33 |
0.0080 |
0.640~0.882 |
3.59~11.45 |
1.36~5.22 |
|
total |
75 |
|
0.650~1.164 |
2.20~13.11 |
0.75~10.09 |
Fig. 3a The
relationships of measured critical shear
stress
and unit dry weight
Fig.
3b The relationships of measured critical shear
stress
and water depth H
The average separation
between two neighboring fine particles may be expressed by
(1)
Where V, m,
stand for the total given volume of
deposition, the total number of particles in this volume, and the unit weight of
the particles, respectively; d and c for the size and the shape coefficient of
particles, respectively. The attractive energy between two particles
may written as
(2)
Where C is the Hamaker constant and n
the exponent (Wang 1990). Consequently the Van Der Waals force
can be expressed as
(3)
Where
is dependent on the dry unit weight
and the properties of ion resolved in water. A combination of the expression of
with Eq. (1) yields
or
(4)
Let
represent the stable value of dry
unit weight
, then
, the special value of
corresponding to
, can be written as
(5)
Van Der Waals force
can simply be obtained：
(6)
Fig. 4 The
relation between contact area
and
increase of average separation
The colloidal chemistry indicates that
bound water layer closely wrapping particle’s surface transmits water pressure
in anisotropy as above mentioned. The cohesive force FH for two fine
sediment particles caused by water pressure should be in proportion to the water
depth H, and the contact area of bound water layers between two particles
, and the unit weight of water
. That is
(7)
The contact area with circular shape
is inversely proportional to the
average separation
thereby proportional to dry unit weight
:
～
(8)
According to the DLVO theory (Luckham
1999), however, the repulsive energy caused by the electric double layers of
surfaces of fine particles would keep itself away from others as the average
separation of the neighboring particles is quite small. Suppose the rate
is proportional to
, thus following relation would be
drawn :
～
(9)
On the other hand the rate
is related to the contact area
, as shown in Fig.4, here D presents the diameter including the bound water
layers for particles O1 and O2;
for the original contact area of
this layer. An increase in variable
would result in a decrease in the
original average separation
by an amount of p1p2=
,
Consequently the contact area
becomes
(10)
Then the following is results
(11)
Where
(12)
In order to use the Taylor expansion, Eq. (11) is written as
(13)
Omitting 3rd-order and higher-order terms of the expansion series yields
(14)
Eq. (14)
indicates that
or (
)~(
) Because the term [(A/2)+(1/8). (w1+w2)] is proportional to
, which is consistent with Eq. (1) if the inverse relationship between
and γ ′ is considered. A
combination of Eq. (9) with above-mentioned description yields
or
(15)
Where k2 is a coefficient. Integrating Eq. (15) produces
(16)
is the special contact area corresponding to
. As shown in Fig.5, let
represent the thickness of bound
water layer and r the radius of the contact area with circular shape, then
may be expressed as
Omitting the 2nd-order terms of
yields
(17)
Substituting this expression into Eq. (16) resulted in
(18)
Combining Eq. (18) with Eq. (7), an expression for the cohesive force caused by water pressure FH can be obtained
(19)
Fig.5
The special
contact area
corresponding to
The Van
Der Waals force actually is of field force type; i.e. the force is in the
centripetal direction (Wang 1990). Therefore the cohesive force
always takes the direction to resist the shear stress of water flow over the bed
surface. On the other hand the bound water layers around particle surfaces, as
above-mentioned, transmit water pressure in anisotropy, and make particles
closely contacted, hence the cohesive force
is of the opposite direction of the
shearing force of water flow as well. So that total critical shear stress
may be written
(20)
Where J1 and J2 are coefficients for
and
respectively, and
the coefficient of particle shape.
Substituting Eq. (6) and Eq. (19) into Eq. (20) yields
(21)
With
the critical shear stress
may be simply written as
(22)
Shear stress
is in N/m2, parameters Am
and Bm in N/m and N/m2 respectively. Both of the water
depth H and the size of fine sediment particle d are in m.
The Least Square Method (LSM) was
adopted to estimate parameters Am, Bm, k1, and
k2 in Eq. (22) by using data from both of tests and fields. To
determinate Am and k1 totally 17 sets of experimental data
of cohesive sands were selected, including 7 sets from the Report on Hangzhou
Bay (Huhe and Yang, 1994) and 10 sets from the Report on Zhongshan Harbor (Yang,
M.Q. et al. 1995). The resulted values Am=17.2×10-6(N/m) and k1=2.19 were obtained, and high
correlation coefficient
=0.99. The estimation of parameters Bm and k2 was carried out with totally 75 sets of measured data of present
research. The resulted values Bm=9.6×10-6(N/m2) and k2=3.94
were obtained, and correlation coefficient
=0.93.
Magnitudes of correlation coefficients R1 and R2 showed a quite good estimation work, hence parameter estimation for Eq. (22) was successful.
All of 40 sets of data used for
verification of incipient motion formula Eq. (22) are independent of those data
used for parameter estimation. They are: (1) 12 sets from Fodda Estuary and
Vilaine Estuary (Migniot 1968); (2) 11 sets from Lianyungang Habour (Huang
1990); (3) 9 sets from Shanghai Harbour (Huang 1990); (4) 8 sets from Hangzhou
Bay (Huhe and Yang 1994). Verification result was shown in Fig.6.
and
represented measured and calculated
values of the critical shear stress
, respectively. A good agreement can be seen.
The experiment with total of 75 sets
for three kinds of graded fine sediment particles showed that both of water
depth H and the dry unit weight
of fine sediment deposition have
important influences on the critical shear stress
of fine sediment particles. The
influence of dry unit weight
on incipient motion can be
microscopically related to the stochastic average separation
between neighboring particles, and
the influence of water depth H on it microscopically related the contact area
of bound water layers between
neighboring particles. According to the colloidal chemistry theory the
relationship between cohesive forces and factors
and H was discussed, and an
incipient motion formula Eq. (22) including water depth H and dry unit weight
was established. Results calculated
by using Eq.(22) showed a good agreement with measured data collected from
various literatures and reports. Because of the suggested herein formula Eq.(22)
is expressed as the critical shear stress, so it can be used for the
open-channel flow cases.
Furthermore, authors observed that in
the most cases incipient motion of fine sediment particles often experiences a
process from gradual change to catastrophes, which includes emergence of Defects
of bed surface (D-stage), Expansion of defects (E-stage) and Collapse of bed surface
(C-stage). During D-E-C process most of fine particles are clustering. The
Scanning Electronic Microscope Photo. of deposition sample showed a complex
hierarchy from flocculation with scale of
mm to particles aggregations with
scale of
mm in which the fractal structures with stochastic geometric
self-similar regulation was clearly observed (Yang et al.1997). A further study
on incipient motion of cohesive sediment deposition might be focused on the
incipient motion of clustering particles based upon the analysis of these
fractal structures. The explanation of the D-E-C process should pay attention to
interactive “bonds” of neighboring fine sediment particles and the
interactive enhancement between the local defect of bed surface and turbulence
flow.
Acknowledgement
This work was supported by National Science Foundation of China (NSFC),and authors would like to thank Prof. Yang M.Q. and Prof. Yuan M. S. for their useful helps.
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Appendix
Notation
A presents the cross-section area of particle including own bound water layer.
and
present coefficients in Eq. (22).
D presents the diameter of a fine sediment particle including own bound water layer, and d the diameter of a particle.
and
present cohesive forces caused by Van Der Waals force and by water pressure, respectively.
present average separation of neighboring fine sediment particles neighboring.
and
present the unit weights of water and of fine sediment deposition, respectively.
and
present the dry unit weight and the stable dry unit weight of fine sediment deposition, respectively.
presents the thickness of bound water layer.
presents the critical shear stress of water flow on bed surface.
presents the contact area between bound water layers of two fine sediment particles.
presents the contact area of bound water of two fine sediment particle layers referring to stable dry unit weight.