Siphon Removal of Cohesionless Materials
Downloaded from ascelibrary.org by University of California, San Diego on 01/14/15. Copyright ASCE. For personal use only; all rights reserved.
S. M. Ullah1; K. A. Mazurek, M.ASCE2; N. Rajaratnam, F.ASCE3; and S. Reitsma4
Abstract: This paper presents the results of an experimental study of the scour created by a siphon flow through a vertical tube of
diameter d of 9.7 to 20.4 mm in a cohesionless sand bed of approximately uniform size D of 0.58 mm. The position of the inlet of the
siphon tube ranged from −101.6 to 6.4 mm relative to the surface of the sand bed and the velocity of flow through the tube U0 varied from
0.34 to 6.41 m / s. Data from this and previous studies are used to show the radial extent of the scour hole and the maximum depth of scour
depend primarily on the densimetric Froude number F0, where F0 = U0 / 冑g共⌬␳ / ␳兲D, g is the gravitational acceleration, ⌬␳ is the
difference in density between the sediment and fluid, and ␳ is the density of the fluid. A theoretical analysis of the problem is used to
further assess the data and it is found that viscous effects are also important to consider. The profile of the scour hole at equilibrium, for
all tube positions, is found to be similar when the scour depth is measured relative to the tube inlet.
DOI: 10.1061/共ASCE兲0733-950X共2005兲131:3共115兲
CE Database subject headings: Siphons; Abatement and removal; Suction; Dredging; Cohesionless sediment; Sand; Scour; Erosion.
Introduction
Siphoning or suctioning of sediments is often utilized in the many
techniques used for hydraulic dredging of sediment beds. An understanding of the factors affecting the removal of sediment by
siphon flows and the prediction of the region of the sediment bed
affected is helpful for developing effective sediment removal
techniques. In this paper, the factors governing the removal of
cohesionless materials by siphon flows by a vertical straight tube
are discussed using both theoretical analysis and experimental
results. This flow most closely models the plain suction dredge,
where a long suction pipe and the suction flow through the pipe is
used to dig and lift the sediment to the water surface 共USEPA
1994兲. Plain suction dredging is a useful technique for removing
contaminated sediments, as there tends to be less resuspension of
the contaminated sediment into the water column as compared to
other types of hydraulic dredges 共Herbich 2000兲. In Canada, the
assessment and removal of contaminated sediments is a particularly important issue for the Great Lakes region.
There have been several studies on sediment removal by
siphon flows 共Slotta 1968; Gladigau 1975; Brahme 1983; Brahme
and Herbich 1986兲. These investigations have generally been
empirical in nature and concerned mainly with assessing the
radial extent of the scour hole created by the suction flow and
1
PhD Provisional Candidate, Dept. of Civil and Environmental
Engineering, Univ. of Alberta, Edmonton AB, Canada T6G 2G7.
2
Assistant Professor, Dept. of Civil and Geological Engineering, Univ.
of Saskatchewan, Saskatoon SK, Canada S7N 5A9.
3
Professor Emeritus, Dept. of Civil and Environmental Engineering,
Univ. of Alberta, Edmonton AB, Canada T6G 2W2
4
Principal, Geosource Energy Inc., 6 Sante Fe Ct., Hamilton ON,
Canada L9C 5Z7.
Note. Discussion open until October 1, 2005. Separate discussions
must be submitted for individual papers. To extend the closing date by
one month, a written request must be filed with the ASCE Managing
Editor. The manuscript for this paper was submitted for review and possible publication on April 1, 2004; approved on November 19, 2004. This
paper is part of the Journal of Waterway, Port, Coastal, and Ocean
Engineering, Vol. 131, No. 3, May 1, 2005. ©ASCE, ISSN 0733-950X/
2005/3-115–122/$25.00.
determining the important parameters for sediment removal.
Slotta 共1968兲 determined the radius of scour for a vertical suction
tube set above a board with a layer of sand grains. Gladigau
共1975兲 and Brahme 共1983兲 studied scour produced in different
sediment beds by flow through a vertical circular tube 共or plain
suction dredge兲 when the tube was set above the bed. In general,
it is observed that a scour hole is formed in which there is a small
mound centered below the tube. The scour holes reached an
“equilibrium” size for which there is no particle movement within
the scour hole after about 75 to 90 min 共Brahme 1983兲. Rehbinder 共1994兲 also did experiments with a siphon tube above a
sand bed, but focused on assessing the critical conditions 共critical
inlet height above the bed兲 for the initial motion of the sediment
particles for a given flow. Rehbinder 共1994兲 performed a theoretical analysis assuming that the flow into the siphon can be considered a sink, where the flow above the bed creates a pressure
distribution that induces flow within the sand bed. The seepage in
the bed creates lift on the surface particles, which was thought to
be the primary influence on particle stability.
In general, the flow into the siphon tube has been considered
as that produced by a sink 共Apgar and Basco 1973; Rehbinder
1994兲. Apgar and Basco 共1973兲 used potential flow theory and
carried out experiments to examine the flow field around a tube
inlet near a flat, horizontal, and smooth rigid bed. They observed
that when the tube position was set close to the boundary the
velocity measured within the first few minutes agreed fairly well
with the theory. The final flow patterns around the inlet are also
similar to the theory, except that velocities measured below the
inlet and near the boundary are larger than what was predicted
共1.5–2 times higher than the calculated values兲. They explained
that the potential flow theory considers the deformation of a fluid
element in two planes, whereas in the actual case this happens in
three planes.
The present study investigates the scour created by a suction
共siphon兲 flow through a tube set vertically above, on, and below
the surface of a cohesionless sand bed. These results can then be
used to help assess the more complicated case of scour by suction
flows in fine-grained sediments and also that associated with a
cross flow. The maximum scour depth, radius of the scour hole,
and scour hole profile are the main characteristics for scour of
JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING © ASCE / MAY/JUNE 2005 / 115
J. Waterway, Port, Coastal, Ocean Eng. 2005.131:115-122.
Downloaded from ascelibrary.org by University of California, San Diego on 01/14/15. Copyright ASCE. For personal use only; all rights reserved.
Fig. 2. Typical scour hole 共oblique view兲
Fig. 1. Experimental setup
interest in this study. Dimensional analysis is first used to relate
the characteristics of flow to the size of the scour hole due to
sediment removal. Potential flow and boundary layer theory are
then used to suggest improvements to the developed relations.
Experimental Setup and Experiments
The experiments were conducted in a large cylindrical plastic
tank of 1 , 219 mm diameter and 1 , 219 mm depth. A smaller cylindrical tank of 550 mm diameter and 705 mm depth was set
within the larger tank so that its edge was at a distance of 100 mm
from the wall of the larger tank. The smaller tank was filled with
a cohesionless sand until the sand was level with its top. The large
cylindrical tank was filled with water, so that the depth of water
above the sand bed in the smaller tank was about 400 mm. A
siphon tube with an inside diameter d of either 9.7, 13.9, or
20.4 mm was set up vertically above the smaller tank by means of
a horizontal traversing system. The tube was connected to a 50 L
enclosed glass jar that acted as a settling chamber for the siphoned material. Flow was recirculated in the system by an 0.5
horsepower jet pump and was measured volumetrically at the
pump outflow. The experimental setup is shown in Fig. 1. For the
experiments, the height of the tube above the sand bed z0 varied
from −101.6 to 6.4 mm, and the velocity of flow at the tube inlet
U0 varied from 0.34 to 6.41 m / s 共z0 and U0 were held constant
during a test兲.
The sand used had mean grain size D of 0.58 mm and a geometric standard deviation ␴g = 冑D84.1 / D15.9 of 1.38, where Di is
the ith percent passing of the material. The specific gravity of the
sand particles was 2.65, and the hydraulic conductivity of the
sand was 2.2⫻ 10−3 m / s. The sand used in the experiment had
been washed and was dust free. To prepare for the experiments,
the smaller tank was filled with the sand by dumping it from a
bucket. The large cylindrical tank was then filled with water and
the sand bed up to 100 mm below the surface was delved by hand
so that there was no trapped air inside the sediment. The bed was
leveled to be even with the top of the smaller cylindrical tank
using a wooden planer. No noticeable surface crust ever developed on the sediments.
Initially, the tests were run for 1 h and, after noting that the
time for the scour hole to reach a state where there was a stable
geometry and no further movement of sand grains—called equilibrium state—was only about 30 s to a few minutes, later tests
were run for approximately 10 to 15 min. Since scouring occurred
in a very rapid manner, measurements of the scour hole during its
growth were not taken. The scour hole profile measurements were
taken at equilibrium after the flow was stopped along one cross
section through the center of the scour hole. The scour depths
were measured relative to the original bed level using a digital
point gauge 共graded to 0.01 mm兲. Details of the experiments can
be found in Ullah 共2003兲.
Observations
In the experiments, there was no removal of the bed material until
the siphon tube was set below the critical height for a few particles on the bed to begin moving, zc. For a tube set at zc above the
bed, a few particles near the edge of the tube would begin to roll
along the bed toward the center of the inlet and then be carried
into the inlet by the flow. As the inlet height above the bed, z0,
was decreased further, there was a critical height for a more general bed motion, zg. If z0 was decreased again, a characteristically
shaped scour hole formed as shown in Fig. 2. A small conical
mound was observed at the center of the scour hole as seen by
Rehbinder 共1994兲.
In the scouring process, most material removal occurred
within about 30– 60 s from the initiation of flow, depending on
the relative position of the inlet from the original bed surface. The
deeper the inlet tube was placed within the bed, the longer the
initial period of high sediment removal rate. During this initial
phase of scouring, the sediment particles around the tube appeared to collapse into the scour hole and the scour hole size
increased progressively in the radial direction. After this first
major amount of scouring, sediment particles were removed only
when few unstable particles roll from the edge of the scour hole
or from inside the scour hole toward the center of the scour hole
in an intermittent manner. This phase normally lasted for
5 – 10 min, after which there was no movement of the sediment
grains inside the scour hole, which was considered equilibrium
state. Typical scour hole profiles are shown in Fig. 3共a兲. In Fig. 3,
r is the radial distance of the scour hole boundary from the centerline and ␧ is the scour depth at equilibrium as measured from
the original bed surface. It is seen in Fig. 3共a兲 that the profiles are
similar in shape for z0 艋 0 共inlet at or above bed兲. When the tube
116 / JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING © ASCE / MAY/JUNE 2005
J. Waterway, Port, Coastal, Ocean Eng. 2005.131:115-122.
Downloaded from ascelibrary.org by University of California, San Diego on 01/14/15. Copyright ASCE. For personal use only; all rights reserved.
Fig. 3. 共a兲 Typical scour hole profiles at equilibrium for varying
inlet height and velocity at the inlet and 共b兲 typical scour hole at
equilibrium when vortex was formed and not formed
is set below the bed, it is seen that the flow scours the material
below the tube in the same shape around the inlet as if the tube
was set on the bed, and the rest of the scour hole profile is linear
up to the surface of the sand bed. The slope of the sand from the
tube level to the surface of the bed ranged from 35 to 40°, with an
average value of 37° from the horizontal.
Some initial tests gave unusually high scour depths and took a
very long period of time to reach equilibrium. For the same experimental setup and flow conditions, different tests would give
widely varying scour depths, and take different periods to reach
equilibrium 共10 to 60 min兲. Two typical scour hole profiles for
these tests are shown in Fig. 3共b兲. After some careful observations, it appeared that circulation in the flow around the tube inlet
caused a vortex to form beneath the inlet, especially at higher
velocities of flow. This vortex typically started to develop after
the initial phase of scour when the major portion of the sediment
was removed. The formation of vortex below the center of the
tube inlet caused the pickup of the sediment to occur in a spiral
manner 共the sand was swirling兲 and resulted in a deeper scour
hole and more sediment removal than if the vortex was not
present. This is apparently the same behavior as the “spiral
threshold pickup” as seen by Gladigau 共1975兲. In the present
study, the vortex was prevented from forming during the experiments by ensuring there was no circulation in the larger tank
which contained the experiments.
Analysis
In a first attempt to analyze the scour data, an effort to develop
appropriate dimensionless parameters was made using dimensional analysis. Fig. 4 gives definition sketches for a scour hole
formed for the tube position z0 艋 0 and z0 ⬎ 0. In Fig. 4共a兲, rb is
the radius of the sour hole at bed level, rt is the scour hole radius
at the level of the inlet, ␧ is the depth of scour measured from the
Fig. 4. Definition sketch for scour hole 共a兲 when z0 艋 0 and 共b兲 when
z0 ⬎ 0
original bed at any radial distance r, ␧r is the scour depth as
measured from the level of the inlet 共or relative scour depth兲, and
␧rm is the maximum scour depth measured from the level of the
inlet 共relative maximum scour depth兲. In Fig. 4共b兲, where the tube
level is set above the original bed, ra is another variable which
has been defined as the distance between center of the tube inlet
and the edge of the scour hole or the “aerial radius.” Let us denote
the characteristic dimensions of the scour hole by a general length
term ᐉ. Based on a physical understanding of this flow, we may
write
ᐉ = f 1兵U0,d,g⌬␳,␳,D,␯其
共1兲
where g⫽is gravitational acceleration; d⫽inside diameter of the
tube, ⌬␳⫽difference in density between the sediment particles
and the fluid; ␳⫽ fluid density; and ␯⫽ kinematic viscosity of the
fluid. Using dimensional considerations, it can be shown that
l/d = f 2
再
U0
冑gD⌬␳/␳ ,
U 0d d
,
␯ D
冎
共2兲
In Eq. 共2兲, the parameter d / D is possibly not important so long
as it is large enough that the sand particles do not choke the flow.
In the present work, d / D was in the range from 16.7 to 35.2 and
the corresponding range for the data of Mazurek and Rajaratnam
共unpublished–referred to later herein兲 was 22.6 to 79.0. The Reynolds number U0d / v reflects the effects of viscosity. The Reynolds number was in the approximate range from 5,000 to 60,000
in the present study and the corresponding range was 5,000 to
23,000 for the earlier work of Mazurek and Rajaratnam. It is
possible that, for this range of Reynolds number, it might be a
secondary parameter which is supported by the analysis presented
later. The most important parameter is perhaps the densimetric
Froude number F0 = U0 / 冑gD⌬␳ / ␳. With these considerations, Eq.
共2兲 reduces to the form
再
l
U0
= f 3 F0 =
冑gD⌬␳/␳
d
冎
共3兲
JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING © ASCE / MAY/JUNE 2005 / 117
J. Waterway, Port, Coastal, Ocean Eng. 2005.131:115-122.
Downloaded from ascelibrary.org by University of California, San Diego on 01/14/15. Copyright ASCE. For personal use only; all rights reserved.
Fig. 6. Variation with F0 of 共a兲 zc and 共b兲 zg
Fig. 5. Variation with F0 of 共a兲 rt 共b兲 ra and 共c兲 ␧rm
To examine the dimensionless relationship developed in Eq.
共3兲, the scour hole radius at equilibrium was plotted against F0.
For the case where the tube is set on or below the bed, the equilibrium scour hole radius at the tube inlet level, rt, from the
present experiments and similar experiments by Mazurek and Rajaratnam 共unpublished兲 are plotted as shown in Fig. 5共a兲. In the
Mazurek and Rajaratnam 共unpublished兲 experiments, U0 varied
from 0.6 to 4.57 m / s, d was 4.3, 7.4, 13.7, or 15 mm, the sand
bed had a D of 0.19 mm and a ␴g = 1.1, and the inlet tube was set
on or below the bed with z0 = 0 to −30.5 mm. It appears that
although rt is a strong function of F0, the two sets of experiments
fall on separate curves, with the rt from the present study always
larger than that found by Mazurek and Rajaratnam. For the data
from the present study
rt
= 0.64F0.41
0
d
共4兲
with correlation coefficient R2 of 0.95. For the present experiments, rt⫽average of the two values measured for the cross section across the scour hole; the two measurements of rt for a cross
section varied by a maximum of 18% for the readings taken on
each side of the scour hole.
For the case when z0 ⬎ 0, when the inlet tube was positioned
above the original bed surface, which is shown in Fig. 4共b兲, it
would appear that Eq. 共3兲 is also applicable. Results from the
present study and those from Brahme 共1983兲 are shown in Fig.
5共b兲 for the aerial radius of the scour hole in terms of d. In
Brahme’s experiments, the inlet tube was set above the bed at a z0
of 12.7, 22.9, or 31.8 mm, and the sand bed had either a D of
0.22 mm and specific gravity of 2.66 共fine sand兲 or a D of
0.40 mm and a specific gravity of 2.64 共medium sand兲. The data
again show a strong functional relationship with F0, but do not
follow the same relationship. For the present study, the aerial
radius varied by a maximum of 16% from the average.
For the maximum depth of scour, the average of the two maximum relative depths obtained on both sides of the tube centerline
was used as ␧rm. Note again that ␧rm is the maximum depth of
scour measured relative to the tube inlet, following Niven and
Khalili 共1998兲. The two depths measured for each cross section
varied by a maximum of 13% relative to the average. Results
from the present study and Mazurek and Rajaratnam, where tube
was set on and below the bed, are used in the analysis. The relation between ␧rm and F0 is shown in Fig. 5共c兲. It was found that
the data from the present experiments for all tube positions was
best fit by
␧rm
= 0.17F0.50
共present study for all tube positions兲
0
d
共5兲
which gave a correlation coefficient R2 of 0.93. Again, the data
from the present study lie separately from that of Mazurek and
Rajaratnam.
The critical heights for first movement of particles zc and zg
were also plotted against F0 as shown in Fig. 6. It is seen that the
data from the present experiments and Mazurek and Rajaratnam
fall on one curve and zc and zg can be found using
zc
= 0.25F0.43
0
d
共6兲
zg
= 0.21F0.42
0
d
共7兲
and
with correlation coefficients of 0.88 and 0.87, respectively.
118 / JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING © ASCE / MAY/JUNE 2005
J. Waterway, Port, Coastal, Ocean Eng. 2005.131:115-122.
冉 冊
1 ⳵ u⬘
vrb ⳵ ␩
Theory for the Scour Hole Radii
Downloaded from ascelibrary.org by University of California, San Diego on 01/14/15. Copyright ASCE. For personal use only; all rights reserved.
From Figs. 5共a–c兲, it appears that the main characteristics of the
scour hole depend strongly on the densimetric Froude number
when a particular cohesionless material is considered. However,
since the different data sets do not collapse on one curve, it
appeared that viscous effects are possibly responsible for this difference. Hence, a theoretical approach was developed to help assess the problem.
Consider a scour hole as it approaches the equilibrium condition,
for the case where the tube inlet is set on or some distance below
the original bed level. Part of the total flow in the tube comes
from the flow above the scoured sediment bed, and the rest comes
from the seepage across the sediment-fluid interface inside the
scour hole. Let us consider a point A located inside the equilibrium scour hole boundary at the tube level 关Fig. 4共a兲兴. Suppose,
the velocity just outside the boundary layer is vrb. Let us assume
that ␯rb is directly proportional to the average velocity through the
scoured out area which can be represented by an annular section
in the plane AOB. Then
Qa
Q
= − k1 2
␲共r2t − d2/4兲
␲rt*
共8兲
where Qa⫽portion of discharge other than the seepage flow; and
Q⫽total flow rate through inlet. The negative sign in Eq. 共8兲
corrects for the fact that the flow is moving radially inward. Here,
k and k1⫽constants and rt* = 冑r2t − d2 / 4. Let us assume that the
physically likely range for d / rt is 0 ⬍ d / rt 艋 1, and we may write
共9兲
r t* = r t − C 1d
where C1⫽constant which corrects for the effect of the diameter
of the inlet tube on the radius of the scour hole. Let us also
assume the boundary layer over the point A 关see Fig. 4共a兲兴 is
laminar and that the velocity distribution in the boundary layer on
the bed is similar to that created by a line sink located on the bed
given as 共Schlichting and Gersten 2000兲
冉
u⬘
␩
= f ⬘共␩兲 = 3 tanh2
冑2 + arctanh
vrb
冑冊
2
3
冑
r t*␯
共− vrb兲
共13兲
where c1⫽constant. Substituting ␩ from Eq. 共12兲 and Eq. 共8兲 into
the above
冉 冊 冑 冉冊
⳵ u⬘
⳵ z⬘
␶0A = ␮
k31
Q
7
r t*␯ ␲
= c1
A
冉 冊
⳵ u⬘
⳵ z⬘
3
共14兲
c1k3/2
1
A
冑64
␳
冑
␶c = ␶c␪sK共␪,i兲 = K⬘共␪兲␳
␯U30d6
7
r t*
= ␶c␪s
共15兲
冑
␯U30d6
r*7
t
共16兲
where ␶c⫽critical shear stress of a particle of the same size,
shape, and density on a horizontal bed with no seepage. Let us
further assume that although there is a seepage flow is present
across the bed, the hydraulic gradient i at point A is small compared to the critical hydraulic gradient, ic and the function K and
the other constants can be replaced by K⬘共␪兲.
Julien 共1998兲 presented equations for estimating ␶c for
different ranges of the dimensionless particle diameter D*
= D关⌬␳g / ␳␯2兴1/3. Using the equation where D* ⬍ 0.3, we can
write
␶c = K⬘共␪兲␳
冑
␯U30d6
7
r t*
= 共␳s − ␳兲Dg ⫻ 0.5 tan ␾
共17兲
where ␳s⫽density of the sediment particles; and ␾⫽angle of repose of the sediment. This can be rewritten as
共10兲
共11兲
=
where ␮⫽dynamic viscosity of the fluid. The critical shear stress
␶c␪s for a particle at A depends both on the seepage through the
bed 共with hydraulic gradient i兲 and the slope of the bed at point A,
␪ 共Chiew and Parker 1994; Cheng and Chiew 1999兲. Let us assume that
r t*
where vrb⫽velocity above the boundary layer at point A;
u⬘⫽velocity in the boundary layer at a distance z⬘ normal to the
bed; and ␩ = z⬘ / ␦⫽dimensionless distance from the bed. The
thickness of the boundary layer is taken as 共Parthasarathy 1969;
Schlichting and Gersten 2000兲
␦=
A
2
冑3 = c1
Next, consider the period of scouring just before equilibrium is
reached. By setting the shear stress on the bed A, ␶0A, equal to the
critical shear stress of the particle in the inclined bed at A, ␶c␪s, we
have
Development of Relation for Equilibrium Scour Hole
Radius at the Tube-Inlet Level
␯rb = − k
=
d
=
冉
K⬘共␪兲
0.5 tan ␾
冊冉 冊
2/7
F40
R0
1/7
共18兲
Again, using Eqs. 共9兲 and 共18兲, we get
冋
F40
rt
= C1 + K⬙共␪兲
d
共0.5 tan ␾兲2R0
册
1/7
共when D* ⬍ 0.3兲 共19兲
where K⬙共␪兲 = 关K⬘共␪兲兴2/7. The equations from Julien 共1998兲 for ␶c
for other ranges of D* can be used in a similar manner to find
冋
0.8
F3.2
rt
0 Rs
= C1 + K⬙共␪兲
d
共0.25 tan ␾兲2R0
册
1/7
共when 0.3 ⬍ D* ⬍ 19兲
共20兲
thus
␩ = z⬘
冑
− vrb
r t*␯
共12兲
冋
F6.8
rt
0
= C1 + K⬙共␪兲
0.8
d
共0.013 tan ␾兲3R1.5
0 Rs
Using the Karman integral momentum theory, we find for a
particle at point A
册
2/21
共when 19 ⬍ D* ⬍ 50兲
共21兲
JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING © ASCE / MAY/JUNE 2005 / 119
J. Waterway, Port, Coastal, Ocean Eng. 2005.131:115-122.
seepage flow is now included in the total flow, Q. Since the bed is
now no longer horizontal and there is a seepage flow, the velocity
over point A should differ to some extent from the vr that has
been expressed in Eq. 共24兲. Suppose, just before equilibrium, that
the component of the radial velocity over point A is vrb. Let us
assume that
Downloaded from ascelibrary.org by University of California, San Diego on 01/14/15. Copyright ASCE. For personal use only; all rights reserved.
vrb = g1共␪,i兲vr
Fig. 7. Variation of rt / d with the theoretical function for z0 = 0 and
z0 ⬍ 0
冋
F40
rt
= C1 + K⬙共␪兲
d
共0.06 tan ␾兲2R0
册
1/7
共when D* ⬎ 50兲
共22兲
where Rs = U0D / ␯. Note that the above discussion is focused on
the equilibrium scour hole formed when the tube is set on or
below the bed of cohesionless materials.
The theory developed in the previous discussion will now be
evaluated using the experimental data of the equilibrium scour
hole radius rt. The angle of repose of the sand ␾ was measured
for the sand in the present study as 34° and assumed to be the
same for sands used by other researchers, since this is a characteristic value for loose sands. The dimensionless particle diameters for all experimental data for all studies were in the range
of 0.3⬍ D* ⬍ 19, thus we will use the functional relation described in Eq. 共20兲. Fig. 7 shows a plot of rt data for the present
study as well as Mazurek and Rajaratnam 共unpublished兲. It is
interesting to see the data sets from both these studies are well
described by one function. This relation for rt / d is described by
the following equation
冋
0.8
F3.2
rt
0 Rs
= 0.32 + 0.45
d
共0.25 tan ␾兲2R0
册
with a correlation coefficient R2 of 0.95. Comparing Eqs. 共20兲 and
共23兲, we can see that C1 is equal to 0.32 and the function K⬙共␪兲
has a constant value of 0.45 within this range of data. It is possible that these values for C1 and K⬙共␪兲 might be valid for a wider
range of D*, but this remains to be checked by further experiments.
Relation for the Aerial Radius of the Equilibrium Scour
Hole for a Tube set above the Bed
For the scour hole data with the inlet tube set above the bed 共z0
⬎ 0兲, the flow to the tube changes from that described above, and
thus a slightly different theoretical function from the above was
developed for this case. For z0 ⬎ 0, from Apgar and Basco 共1973兲,
it appears the flow field agrees fairly well with that potential flow,
therefore for this case potential flow theory was used for assessing the flow velocities. Let us assume that the center of the tube
can be considered a point sink. The radial velocity over any point
on a rigid horizontal bed is
vr = −
Qr
2␲关z20 + r2兴3/2
where g1 depends on the slope of the bed at point A, ␪, and
hydraulic gradient of the seepage flow at point A , i. Just before
the equilibrium, it also may be reasonable to assume that the
streamlines are almost parallel to the bed surface over the point A
and seepage has a negligible effect on further scouring. Therefore,
g1 in Eq. 共25兲 may be considered primarily as a function of ␪.
Since the boundary layer on the bed is likely laminar, the velocity
distribution might be expressed in a form as in Eq. 共10兲. In this
case, the dimensionless distance from the bed is
␩ = z⬘
共24兲
where r⫽radial distance of the point 共Rehbinder 1994兲. Fig. 4共b兲
shows the typical scour hole formed by this flow. In Fig. 4共b兲,
point A is any point on the edge of the scour hole near bed level.
Below the tube inlet, the bed is no longer horizontal and also the
冑
− vrb
= z⬘
共z20 + r2b兲1/2␯
冑
− ␯rb
r a␯
共26兲
where ra = 共z20 + r2b兲0.5 and⫽distance between the point sink at O
and the edge of the scour hole boundary at A; and z⬘⫽direction to
the normal of bed at point A. From trigonometry,
rb = ra sin ␺
共27兲
where ␺⫽angle between the tube centerline and the aerial radius
ra 关Fig. 4共b兲兴.
If again we set the shear stress on the bed at A equal to the
critical shear stress ␶c␪s for the particle to move on the sloped bed,
we find
␶c␪s =
1/7
共23兲
共25兲
关g1共␪兲兴3/2
␳
19.6
冑
␯U30d6r3b
共28兲
r10
a
where ␶c␪s takes into account the effects of bed slope and seepage.
Substituting Eq. 共27兲 into Eq. 共28兲 yields
␶c␪s =
关g1共␪兲兴3/2
␳
19.6
冑
␯U30d6 sin3 ␺
r7a
共29兲
To relate the critical shear stress developed for a horizontal bed to
the scour hole radius, we have then
␶c =
关g1共␪兲兴3/2F共␪,i兲
␳
19.6
冑
␯U30d6 sin3 ␺
r7a
共30兲
where is F⫽some constant that depends on the bed slope ␪ and
the seepage i. In this case, if the hydraulic gradient for seepage
flow is again small compared to its critical hydraulic gradient, F
should be a function of only ␪. For the dimensionless particle size
D* ⬍ 0.3, the following expression can be derived from Julien
共1998兲 in a similar manner to Eq. 共18兲
冉
sin3/2 ␺
ra
= F⬙共␪兲
d
0.5 tan ␾
冊冉 冊
2/7
F40
R0
1/7
共31兲
where F⬙⫽parameter that depends on the bed slope at A, ␪. Eq.
共31兲 has been derived for the scour hole formed by a point sink
located above a cohesionless sediment bed. It is likely that the
diameter of the inlet has an effect on the radius so that a constant
C2 is added to Eq. 共31兲. Thus, using the equations for ␶c presented
by Julien 共1998兲, we have
120 / JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING © ASCE / MAY/JUNE 2005
J. Waterway, Port, Coastal, Ocean Eng. 2005.131:115-122.
Downloaded from ascelibrary.org by University of California, San Diego on 01/14/15. Copyright ASCE. For personal use only; all rights reserved.
Fig. 8. Variation of ra / d with the theoretical function for z0 ⬎ 0
冋
F40 sin3 ␺
ra
= C2 + F⬙共␪兲
d
共0.5 tan ␾兲2R0
册
1/7
共when D* ⬍ 0.3兲 共32兲
Similarly, for other ranges of D* 共Julien 1998兲, we find
冋
0.8
3
F3.2
ra
0 Rs sin ␺
= C2 + F⬙共␪兲
d
共0.25 tan ␾兲2R0
册
1/7
冋
3
F6.8
ra
0 sin ␺
= C2 + F⬙共␪兲
0.8
d
共0.013 tan ␾兲3R1.5
0 Rs
共when 0.3 ⬍ D* ⬍ 19兲
册
共33兲
Fig. 9. 共a兲 Dimensionless scour hole profile for z0 ⬍ 0 and z0 = 0, and
共b兲 for z0 ⬎ 0 data
2/21
共when 19 ⬍ D* ⬍ 50兲
and
冋
F40 sin3 ␺
ra
= C2 + F⬙共␪兲
d
共0.06 tan ␾兲2R0
册
共34兲
1/7
共when D* ⬎ 50兲
共35兲
To find relationship for the aerial radius, ra, of the equilibrium
scour hole and the proposed function, data from the present study
for z0 ⬎ 0 and that from Brahme 共1983兲 are used. Because the
sands used by Brahme 共1983兲 also fall within the range of 0.3
⬍ D* ⬍ 19, we must use the function described in Eq. 共33兲. Fig. 8
shows the plot of dimensionless aerial radius ra with the theory
based function. The relation is fitted with a straight line having
the following equation
冋
0.8
3
F3.2
ra
0 Rs sin ␺
= 0.24 + 0.46
d coefficient R2共0.25
tan ␾兲2R0
with a correlation
of 0.93.
册
1/7
共36兲
equation has been found that can properly fit all data for the
conditions where the tube inlet is below the bed surface; one
equation is required for the part of the profile below the inlet and
another for the part of the profile above the inlet level. After some
trial and error, two separate curves 共a third-order polynomial and
a straight line兲 have been found to fit the data; the equations have
an identical slope at r / rt = 1.2. The equations are as follows:
冉冊
r
␧r
= − 1.12
␧rm
rt
3
冉冊
冉
冉冊
+ 3.61
r
rt
2
冉冊
冊
冉
− 1.71
when 0
r
− 0.78
rt
r
艋 1.2
rt
␧r
r
= 2.12
− 2.11
␧rm
rt
when
共37兲
r
⬎ 1.2
rt
冊
共38兲
with a respective R2 of 0.97 and 0.99. Finally, ␧rm must be determined; it was found that ␧rm / rt ⬵ 0.35 for the experiments.
Scour Hole Profiles at Equilibrium State
Conclusions
Once the radius of the scour hole is determined, we can also
determine the entire profile of the scour hole. It was seen in Fig.
3共a兲 that the scour hole profiles were similar in shape and, if the
scour depth measured relative to the tube inlet was used, the scour
hole profiles for inlet depths below the bed plotted with those set
at the bed. A dimensionless scour hole profile thus can be developed. This was done using the radius of the scour hole at inlet
level rt as the scale for the radial distance, and the maximum
scour depth from the tube inlet level ␧rm as the scale for scour
depth. The data of the present study as well as that of Mazurek
and Rajaratnam 共unpublished兲 are used to develop the dimensionless profiles, as shown in Fig. 9. In Fig. 9, the Mazurek and
Rajaratnam data are marked by an asterisk in the legend of the
figure. For the case when z0 ⬎ 0, rt was predicted using Eq. 共23兲
since it cannot be measured for those experiments. No single
This study presents an investigation of the equilibrium scour hole
geometry and the important parameters for removal of cohesionless sediments by a siphon flow. The use of dimensional analysis
showed that the scour produced by the siphon flow depends
strongly on the densimetric Froude number, although it appears
that viscous effects must also influence scour, as data from different investigators did not collapse on one curve. A theoretical approach, which looked at the stability of a particle inside the scour
hole, suggested the additional parameters that should be considered. Using the theory and experimental data, equations were
developed to predict the radius of the scour hole at equilibrium. It
was also seen that the scour hole profiles at equilibrium are
successfully nondimensionalized using the maximum scour depth
measured from tube inlet for the scour depth, and radius of
JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING © ASCE / MAY/JUNE 2005 / 121
J. Waterway, Port, Coastal, Ocean Eng. 2005.131:115-122.
the scour hole at tube inlet level for the radial distance from the
jet centerline. A dimensionless curve was developed to predict
that profile. In order to verify the results and theory presented
herein, it would be useful to perform similar experiments at much
larger scale.
Downloaded from ascelibrary.org by University of California, San Diego on 01/14/15. Copyright ASCE. For personal use only; all rights reserved.
Acknowledgments
The writers are thankful to R. Clark and L. Pop for helping to
setting up the experimental arrangement and to the Univ. of
Windsor and the Natural Sciences and Engineering Research
Council of Canada for financial support to the second writer for
this project. The writers also acknowledge the helpful comments
of the reviewers.
Notation
The following symbols are used in this paper:
C1 , C2 ⫽ constants;
c1 ⫽ constant;
D ⫽ mean grain diameter;
D* ⫽ dimensionless grain diameter;
Di ⫽ diameter of the particle of which i% the material
is finer;
d ⫽ inside diameter of the tube inlet;
F , F⬙ ⫽ functions;
F0 ⫽ densimetric Froude number;
f 1 , f 2 , f 3 ⫽ functions;
g ⫽ acceleration due to gravity;
g1 ⫽ function;
i ⫽ hydraulic gradient for the seepage flow;
ic ⫽ critical hydraulic gradient for the sediment;
K , K⬘ , K⬙ ⫽ functions;
k , k1 ⫽ constants;
l ⫽ characteristic length;
Q ⫽ flow rate through the tube inlet;
Qa ⫽ portion of discharge which does not include
seepage flow;
R2 ⫽ correlation coefficient;
R0 ⫽ Reynolds number of the flow through the inlet;
Rs ⫽ Reynolds number for the sediment at bed;
rt* ⫽ equivalent radius;
r ⫽ radial distance from the centerline of the inlet;
ra ⫽ aerial radius;
rb ⫽ radius of the equilibrium scour hole at the
surface of the bed;
rt ⫽ radius of the equilibrium scour hole at the level
of the inlet;
U0 ⫽ velocity of flow through the inlet;
u⬘ ⫽ velocity in the boundary layer at a distance of
z ⬘;
vr ⫽ radial component of velocity;
vrb ⫽ flow velocity over sediment bed at a distance r
from the tube centerline;
z⬘ ⫽ perpendicular distance from the bed;
z0 ⫽ height of tube inlet above the surface of the sand
bed;
zc ⫽ height of the tube inlet above the sand bed for a
few single particles to move;
zg ⫽ height of the tube above the sand bed for a
general motion of the sediments;
⌬␳ ⫽ difference in density between the sediment and
fluid;
␦ ⫽ boundary layer thickness;
␧ ⫽ scour depth on any point measured from the
original bed surface;
␧r ⫽ scour depth measured relative to the inlet;
␧rm ⫽ maximum scour depth measured relative to the
tube inlet;
␾ ⫽ angle of repose of the sediment grain;
␩ ⫽ dimensionless height from the bed;
␮ ⫽ dynamic viscosity of the fluid;
␯ ⫽ kinematic viscosity of the fluid;
␪ ⫽ slope of sediment bed at tube level;
␳ ⫽ density of the fluid;
␳s ⫽ density of the sediment particle;
␴g ⫽ geometric standard deviation of bed material;
␶c ⫽ critical shear stress of the sediment particle on a
horizontal bed with no seepage;
␶c␪s ⫽ critical shear stress of the sediment particle in an
inclined plane with seepage;
␶0A ⫽ wall shear stress on the point A of the scour hole
boundary; and
␺ ⫽ angle between the aerial radius and the centerline.
References
Apgar, W. J., and Basco, D. R. 共1973兲. “An experimental and theoretical
study of flow field surrounding a pipe inlet.” TAMU-SG-74-203, Sea
Grant Office, Texas A&M Univ., College Station, Tex.
Brahme, S. B. 共1983兲. “Environmental aspects of suction cutterheads.”
PhD thesis, Texas A&M Univ., College Station, Tex.
Brahme, S. B., and Herbich, J. 共1986兲. “Hydraulic model studies for
suction cutterheads.” J. Waterw., Port, Coastal, Ocean Eng., 112共5兲,
591–606.
Chiew, Y. M., and Parker, G. 共1994兲. “Incipient sediment motion on
nonhorizontal slopes.” J. Hydraul. Res., 32共5兲, 649–660.
Cheng, N. S., and Chiew, Y. M. 共1999兲. “Incipient sediment motion with
upward seepage.” J. Hydraul. Res., 37共5兲, 665–681.
Gladigau, L. N. 共1975兲. “Interactions between sand and water.” Proc., 6th
World Dredging Conference (WODCON I), World Organization of
Dredging Associations, Amsterdam, The Netherlands, 261–294.
Herbich, J. B. 共2000兲. Handbook of dredging engineering, 2nd Ed.,
McGraw-Hill, N.Y.
Julien, P. Y. 共1998兲. Erosion and sedimentation, Cambridge University
Press, N.Y.
Niven, R. K., and Khalili, N. 共1998兲. “In-situ fluidization by a single
internal vertical jet.” J. Hydraul. Res., 36共2兲, 199–288.
Parthasarathy, S. P. 共1969兲. “The transient boundary layer produced by a
sink on a plane wall.” PhD thesis, California Institute of Technology,
Pasadena, Calif.
Rehbinder, G. 共1994兲. “Sediment removal with a siphon at critical flux.”
J. Hydraul. Res., 132共6兲, 845–860.
Schlichting, H., and Gersten, K. 共2000兲. Boundary-layer theory, 8th Ed.,
Springer, N.Y.
Slotta, L. S. 共1968兲. “Flow visualization techniques used in dredge cutterhead evaluation.” Proc., 1968 World Dredging Conf. (WODCON
II), World Organization of Dredging Associations, Amsterdam, The
Netherlands, 56–77.
Ullah, S. M. 共2003兲. “Siphon removal of cohesionless materials.” MASc.
thesis, Univ. of Windsor, Ontario , Canada.
United States Environmental Protection Agency 共USEPA兲 共1994兲. “Assessment and remediation of contaminated sediments 共ARCS兲 program.” EPA/905/R-94/003, USEPA, Great Lakes National Program
Office, Chicago.
122 / JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING © ASCE / MAY/JUNE 2005
J. Waterway, Port, Coastal, Ocean Eng. 2005.131:115-122.