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International Journal of Civil Engineering and Technology (IJCIET)
Volume 10, Issue 03, March 2019, pp. 3274-3294, Article ID: IJCIET_10_03_329
Available online at http://www.iaeme.com/ijciet/issues.asp?JType=IJCIET&VType=10&IType=03
ISSN Print: 0976-6308 and ISSN Online: 0976-6316
© IAEME Publication Scopus Indexed
NUMERICAL INVESTIGATION OF FLOW
PATTERN AND SCOUR CHARACTERISTICS
AROUND ELECTRICAL TOWER
FOUNDATIONS
Gamal M. Abdel-Aal, Maha R. Fahmy, Amany A. Habib and Mohamed Galal
Elbagoury
Water and Water Structures Engineering Department, Faculty of Engineering, Zagazig, Egypt
ABSTRACT
The presence of some high voltage towers in flood stream is one of the most
important problems that may lead to the collapse of these towers. The main reason for
collapse is the soil erosion around the tower foundation during flood. The shape of
the foundation is a vital factor in scouring process. This research is focused on
studying different shapes of a tower foundation and its effect on the maximum scour
depth. A sediment scour model has been investigated by using Flow 3D V 11.2
Program. The numerical simulation results of the maximum scour depth surrounding
a single square pile model have been assured using prior experimental findings and
showed good agreement. After that, different four shapes of footing and five values of
the inclination angle for pyramid and cone footing have been investigated. The results
of cuboid footing have been used as a reference to compare with different shapes.
Seventy-two numerical runs have been carried out considering the wide range of
Froude number ranging from 0.26 to 0.50 under clear water condition. It is found
that, for pyramid and cone footing, the lager the inclination angle, the smaller the
scour depth will be and vice versa. The cone footing is better than the other footing
shapes. An empirical equation has been developed by using the nonlinear regression
to predict the relative maximum scour depth around the footing.
Keywords: Scour, Vortex, Footing, Flood, Flow-3D.
Cite this Article: Gamal M. Abdel-Aal, Maha R. Fahmy, Amany A. Habib and
Mohamed Galal Elbagoury, Numerical Investigation of Flow Pattern and Scour
Characteristics Around Electrical Tower Foundations. International Journal of Civil
Engineering and Technology, 10(3), 2019, pp. 3274-3294
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Gamal M. Abdel-Aal, Maha R. Fahmy, Amany A. Habib and Mohamed Galal Elbagoury
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1. INTRODUCTION
Local scour around electric towers footing during flood may cause failure of these towers.
Once the scour depth around the footing is sufficiently deep, the foundation may become
unsettled or even broke down. Hence, for the safe and economic design of these towers, it has
become essential to foresee the scour depth around such towers with greater accuracy. The
prognosis of the scour depth around towers footing during a flood is important to determine
the depth of the foundation of such towers.
Several experimental studies have been carried out to investigate the local scour around
various shapes of piers. The studies showed that, the streamlined piers gave the minimum
scour depth. Ismael et al. (2015) examined the local scour around different shapes of piers
like downstream round-nosed, upstream round-nosed and circular bridge piers. The results
confirmed that, the downstream round-nosed pier was an efficient to decrease the depth of
scour. Khan et al. (2017) analyzed the scour around different shapes of the pier (circular and
square) and different sizes. In fixed experimental conditions of flow, sediment properties and
pier geometry the scour obtained from the square shaped model was greater than the scour
obtained from the circular model. By increasing the size of the pier the scour increased. Three
equations have been developed by using multi linear regression, genetic function and
artificial neural network. By comparing the experimental data with the previous equations, it
was found that the genetic function model worked better than the rest of the models. Al-
Shukur and Obeid (2016) investigated the effect of several shapes of bridge pier on local
scour to obtain the perfect shape that gave the least scour depth. The used shapes were
rectangular, circular, chamfered, octagonal, hexagonal, elliptical, joukowsky, oblong, sharp
nose and streamline. The experiments concluded that, the least scour depth was obtained
from the streamline shape while the largest scour depth was obtained from the rectangular
shape. Vijayasree et al. (2017) investigated the flow pattern and local scour around several
shapes of pier like rectangular, triangular-nosed, trapezoidal-nosed, oblong and lenticular.
The results confirmed that, the sharp nose with the curved body was perfect for the bridge
pier since there was less scour depth around the pier. Li and Tao (2017) investigated the
effect of pier streamlining on local scour under clear water scour conditions. It was noticed
that, the streamlined piers gave minimum scour depth from oblong piers. Fael et al. (2016)
investigated the impact of pier shape on the scour depth surrounding the single pier. Several
pier shapes were examined such as circular, rectangular (square and round nosed), oblong
and pile groups. The results confirmed that, the shape factor could be regarded as 1.0, for
rectangular round nosed and oblong cross section piers, and as 1.2, for rectangular square
nosed and packed pile group cross section piers.
The horseshoe vortex was the main cause of the development of scouring around the pier
(Muzzammil and Gangadhariah 2003; Vijayasree et al. 2017). Muzzammil and Gangadhariah
(2003) investigated the features of horseshoe vortex around a cylindrical pier. By the
evolution of the scour hole, the horseshoe vortex gradually sank into the scour hole while its
size increased. At the initial stages the vortex strength and velocity increased while being
decreased at later stages. Unger and Hager (2007) explored the temporal development of the
down flow and horseshoe vortex at circular pier. The vertical jet and the horseshoe vortex
were the main reasons for the scouring process. Dey and Raikar (2007) noticed and analyzed
the horseshoe vortex during the development of a scour hole around cylindrical pier. With the
evolution of the scour hole, the horseshoe vortex shape became ellipse and its size became
larger. Zhao et al. (2012) studied the mechanism of local scour around cuboid-shape sub-sea
caissons. The height of the caisson models under study was less than or equal to their
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horizontal dimensions. The results showed that the effect of horseshoe vortex was less than
the velocity at the sharp edge of the caisson.
Khwairakpam et al. (2012) investigated the local scour surrounding a circular pier under
clear water conditions. An empirical equation had been developed to predict the parameters
of a scour hole (depth, length, width, area and volume) around a circular pier.
Few studies have been conducted to investigate the scour depth around conical pier. Givi
et al. (2011) used the FLUENT program to investigate the flow pattern and scour depth
around cylindrical pier and four conical piers with several slopes. It was found that, the scour
depth around conical pier was less than that of cylindrical pier. By increasing the slope of
conical pier the scour depth decreased. Aghaee-Shalmani and Hakimzadeh (2015)
investigated the scour around the conical pier with different lateral slopes under steady
current. It was found that, by increasing the conical pier angle the scour depth decreased
compared with cylindrical pier.
Zhao and Huhe (2006) investigated numerically the mechanism of scour and the turbulent
of flow surrounding a circular pier using Large Eddy Simulation. Zhao et al. (2010)
investigated experimentally and numerically the mechanism of local scour surrounding
submerged vertical pile under steady flow. It was found that, the scour depth that had been
estimated by numerical model was less than the experimentally measured scour by about 10
to 20%. Khosronejad et al. (2012) studied the scour around different shapes of pier such as
cylindrical, square and diamond under clear-water condition by using experimental and
numerical models. Baykal et al. (2015) studied the flow and scour around cylindrical pile due
to the steady flow by using three dimension numerical models. Nagata et al. (2005) created a
3D numerical model that had been used to simulate flow and scour geometry around
hydraulic structures. Experimental results of spur dike and cylindrical pier were compared to
the findings of the proposed numerical model. The comparison proved that, the numerical
model represented flow and scour surrounding these structures with great accuracy. Ghiassi
and Abbasnia (2013) developed 3D numerical model to simulate the flow pattern and the bed
deformation around a bridge pier and groyne used proposed equation. Elsaeed (2011)
compared the previous experimental data with numerical model results for the scour depth
around a square pile by using SSIIM program. Jia et al. (2017) simulated the local scour
around cylindrical pier using CCHE3D software. It was found that, the down flow and the
turbulent kinetic energy around the pier were the main factors in the scouring process. The
strong down flow transported the turbulent kinetic energy to the bed and leaded to an increase
in shear stress, so scour occurred. Salaheldin et al. (2004) used 3D numerical model
FLUENT to simulate the turbulent flow surrounding circular piers in clear water conditions.
The numerical model results had been compared with the previous experimental data and
showed good agreement. Huang et al. (2009) used the FLUENT program to investigate the
effect of scale on turbulence flow and sediment scour around the pier.
Several investigators have carried out a numerical study to simulate the local scour
around piers by using Flow 3D Program and comparing the numerical results with
experimental results. The studies showed that, the Flow 3D program could simulate the
scouring process around piers with high accuracy. Alemi and Maia (2016) investigated
numerically the local scour around the cylindrical pier under clear water scour conditions
using SSIIM and FLOW 3D codes. The numerical results had been compared with the
previous experimental data. The results showed that, the two CFD codes could accurately
predict the scour at the upstream side and lateral sides of the pier but not the downstream side
of the pier. The scour depth at down-stream the pier was under-predicted by SSIIM code
while it was over-predicted by FLOW 3D code. Amini and Parto (2017) compared the
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previous experimental data with numerical results for the characteristics of a scour hole
around different arrangements of two piles by using Flow 3D program. The results concluded
that the scour hole around groups of piles could be simulated by using Flow 3D Program.
Wang et al. (2017) tested experimentally and numerically (Flow 3-D) the influence of using
sacrificial piles at upstream the pile to reduce the scour depth. The result concluded that, the
numerical model was an effective way to investigate the phenomena of scouring around piles.
Zhang et al. (2017) investigated numerically by using a Flow 3-D program the scour hole
characteristics around three piles with different arrangements. During the comparison of the
three models standard k-e model, RNG model and LES model, it was found that the RNG
model was more applicable than the others in the indication of scour process phenomena.
Omara et al. (2018) investigated numerically the scouring process around vertical and
inclined piers using the FLOW-3D program. The findings of numerical model in terms of
flow velocity, water depth, scour depth and shear stress compared with various sets of
previous experimental and numerical data. The results showed that, the numerical model gave
prediction of scour depth surrounding piers with great accuracy.
Several empirical equations are available to estimate the equilibrium depth of scour
around piers for non-cohesive soil. Mohamed et al. (2006) compared four empirical equations
such as HEC-18 (Richardson and Davis 2001), Melville and Sutherland (1988), Jain and
Fischer (1979), and Laursen and Toch (1956) with field data collected from bridges located
in India, Canada and Pakistan. The comparison showed that, the HEC-18 formula
(Richardson and Davis 2001) was the best from the other selected formula. Gaudio et al.
(2010) compared six empirical equations such as Breusers et al. (1977), Jain and Fischer
(1979), Froehlich (1988), Kothyari et al. (1992), Melville (1997) and HEC-18 formula
(Richardson and Davis 2001) used to predict the scour depth around the pier with synthetic
and original field data. The comparison results proved that the HEC-18 formula (Richardson
and Davis 2001) was better than the other selected formula in both clear-water and live-bed
scour. Qi et al. (2016) compared three empirical common equations such as Melville and
Sutherland (1988) equations, Chinese equations (Dongguang et al. 1993) and HEC-18
equations (Richardson and Davis 2001) with laboratory and field data. The results showed
that, the Chinese equations (Dongguang et al. 1993) gave satisfactory results with field data.
The HEC-18 equations (Richardson and Davis 2001) gave good result with laboratory data.
The Melville and Sutherland (1988) equations gave over-estimated the scour depth for
laboratory and field data. From the previous comparisons, HEC-18 equations (Richardson
and Davis 2001) and Chinese equations (Dongguang et al. 1993) have been selected to be
compared with current numerical results.
Kalaga and Yenumula (2016) discussed the different types of foundations used in
transmission line structures such as piers, spread, direct embedment, pile, micro-piles and
anchor foundations. To the authors’ knowledge, no efforts have been made to study the scour
around electrical tower foundations. The major objectives of this research are: (1) to confirm
the accuracy of numerical model in the prediction of scour around piers, (2) to investigate the
scour depth around new proposed shapes of towers footing under clear-water condition, (3) to
investigate the effect of new proposed shapes of towers footing on the down flow and
horseshoe vortex, (4) to develop an empirical equation to predict the maximum scour depth
around proposed footing.
2. DIMENSIONAL ANALYSIS
Dimensional analysis based on Buckingham theory is used to develop a functional
relationship between the maximum scour depth around the tower footing and the other
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relevant scour variables. The different shapes of tower footing (cuboid, cylindrical, pyramid,
and cone) are shown in Figure. 1. The maximum scour depth ds can be expressed as follows:
( )
(
1
)
where ds is the maximum scour depth around footing, B is the flume width, d1 is the lower
width or diameter of footing, h is the height of footing above channel bed, is the inclination
angle of cone or pyramid footing with vertical axis, Ks is the shape correction coefficient, y is
the upstream flow depth, V is the upstream mean velocity, Q is the flow rate, is the density
of water,s is the density of sand particles, g is the gravitational acceleration, and d50 is the
mean diameter of sand layer.
Applying the Buckingham theorem with y, V, as repeating variables, Eq. (1) can be
written in dimensionless form as:
(
)
(
2
)
There the ds/y is the relative maximum scour depth, F is the upstream Froude number,
d1/y is the relative lower width or diameter of footing and h/y is the relative height of footing.
(a) (b)
Figure. 1 Different shapes of tower footing (a) cuboid and cylindrical footing, (b) pyramid and cone footing
3. NUMERICAL WORK
3.1. Numerical Model Scale
The foundation of the tower consisted of a base mat and a square or circular pier (Kalaga and
Yenumula 2016) as shown in Figure. 2. The width or diameter of the pier depends on the
concrete bearing capacity and the value of the load, Therefore they are variable values. In this
study, fixed dimensions of the pier are selected with a width of 60 cm and a height of 50 cm.
These dimensions are the most common in the construction field. A scale of 1:5 is chosen to
estimate the numerical model dimensions.
Figure. 2 Concrete footing for lattice transmission towers (Kalaga and Yenumula 2016)
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3.2 Meshing and Geometry of Model
A sufficient distance is provided before and behind the pier to ensure that the flow returns to
the undisturbed pattern about 6 and 12 times the pier diameter respectively (Sarker 1998). In
this sense, the length of the numerical model is set as 30d1, with a fixed bed length of 7d1 at
the inlet to prevent the scour at the inlet. The footing is placed at distances 15.5d1 in x
direction and 0.5B in y direction from the origin point to the center of the footing.
The mesh block has non-uniform cells that become finer close to the footing where the
area of scour is existed as shown in Figure. 3. For accurate and efficient results, the size ratio
between adjacent cells and cell aspect ratios should not exceed 1.25 and 3.0 respectively. For
all geometric configurations the number of cells is about 1856512 cells.
In x direction, the total model length in this direction is 3.60 m. Four mesh planes are
installed at distances 0.00, 1.68, 2.04 and 3.60 m respectively from the origin point. From
first to second mesh plane, the cell size decreasing gradually from 0.0095 m to 0.005 m.
Constant cell size 0.005 m from second to third mesh plane where the area of scour is existed.
From third to fourth mesh plane, the cell size increasing gradually from 0.005 m to 0.0095 m.
In y direction, the total model length in this direction is 0.66 m. Four mesh planes are
installed at distances 0.00, 0.15, 0.51 and 0.66 m respectively from the origin point. From
first to second mesh plane, the cell size decreasing gradually from 0.0095 m to 0.005 m.
Constant cell size 0.005 m from second to third mesh plane where the area of scour is existed.
From third to fourth mesh plane, the cell size increasing gradually from 0.005 m to 0.0095 m.
In z direction, the total model length in this direction is 0.25 m. Three mesh planes are
installed at distances -0.15, 0.00 and 0.10 m respectively from the origin point. From first to
second mesh plane, the cell size decreasing gradually from 0.014 m to 0.0032 m. From
second to third mesh plane, the cell size increasing gradually from 0.0032 m to 0.01 m.
Figure. 3 Meshing of footing model in FLOW-3D
3.3. Boundary Condition
The boundary conditions for the mesh block of the numerical model have been defined
carefully to simulate the experimental flow conditions accurately as shown in Figure. 4. The
upstream boundary is defined as volume flow rate with different discharge (Q = 12, 13, 15,
16, 18, 21, 24 and 26 l/sec). The downstream boundary is defined as outflow. The right side,
the left side, and the bottom boundary are defined as a wall. The top boundary is defined as
specified pressure with standard atmospheric pressure value. The fluid is defined as a fluid
region with initial depth (y = 8 cm) and initial velocity in x direction.
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Figure. 4 Numerical model and boundary conditions
The critical velocity could be estimated from the logarithmic Eq. (3) of the velocity
profile as used by Melville (1997):
(
) (3)
Where Vc is the critical velocity, U*c is the critical shear velocity, y is the flow depth and
d50 is the mean diameter of soil.
The shear velocities were determined from Eq. (4), which was illustrated by Melville
(1997) as a useful approximation to the Shields diagram for quartz sediments in water at
20C:
(4)
In which U*c is in m/sec and d50 is in mm, and valid for the range of 1 mm < d50 < 100
mm.
3.4. Numerical Model Validation
A comparison between the numerical model and previous experimental results (Moussa
2018) has been investigated to achieve the accuracy of the numerical model. The experiments
had been carried out in a straight open flume with a vertical square pile (6 cm x 6 cm). The
flume was consisted of a rectangular cross section with a width 0.66 m, a depth of 0.65 m and
a length of 16.2 m, which contained a 20 cm deep layer of fine sand with a mean particle
diameter of 1.4 mm. The time for each experiment was one hour, at which, 85% of the
equilibrium scour depth was achieved based on preliminary experiments.
The numerical model has been set-up in Flow 3D program with a diameter of sand = 1.4
mm and mass density = 2650 kg/cm3. The device needs ten days to complete the calculation
of numerical model using core i7 processor. Ten numerical runs are carried out with different
discharge and flow depth to achieve the accuracy of numerical model as shown in
Table 1. By comparison the previous experiments (Moussa 2018) and present numerical
results, it is noticed that the numerical model gives a good agreement with an error by about
± 8.0% as shown in
Table 1 and Figure. 5.
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Table 1 Comparison between experimental and numerical results
Run Q (l/s) y (m) V (m/s) F Exp. Num.
Error ds (cm) ds (cm)
1 15.00 0.08 0.28 0.32 4.60 4.55 1.09
2 20.16 0.12 0.25 0.23 4.00 3.91 2.25
3 20.16 0.10 0.31 0.31 5.80 5.72 1.38
4 20.16 0.08 0.38 0.43 7.00 7.14 -2.00
5 24.85 0.14 0.27 0.23 4.00 4.32 -8.00
6 24.85 0.10 0.38 0.38 7.20 7.48 -3.89
7 24.85 0.08 0.47 0.53 9.20 9.35 -1.63
8 30.38 0.14 0.33 0.28 7.10 6.79 4.37
9 30.38 0.10 0.46 0.46 10.60 9.82 7.36
10 30.38 0.08 0.58 0.65 12.00 11.40 5.00
± 8.0
Figure. 5 Relative maximum scour depth between experimental and numerical results
4 ANALYSIS AND DISCUSSIONS
4.1. Local Scour Mechanism
To illustrate the mechanism of local scour around electric tower foundation during flood,
cuboid footing (12 cm x 12 cm) has been investigated. The details of the numerical runs for
cuboid footing are listed in Table 2. Figure. 6 shows the flow streamlines around the cuboid
footing at the initial stage of scour at time 80 sec.
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Num
eric
al d
s/y
Measured ds/y
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Figure. 7 shows the flow streamlines behind the cuboid footing at the initial stage of scour at
time 80 sec and the equilibrium state of scour. Figure. 8 shows the scour contour map around
the cuboid footing at the equilibrium state. Figure. 9 illustrates the relationship between the
relative maximum scour depth and Froude number for cuboid footing.
The scour occurs due to presence of footing in front of the flow, which acts as obstruction
to change the direction of flow to down which is called the down flow as shown in Figure. 6.
The down flow is the main cause to create the scour hole, which acts as a vertical jet to
remove the grains from the bed. Due to the separation of the flow at the edges of the footing
with the effect of down flow, the flow changes its direction in the scour hole creating the
helical flow which is called the horseshoe vortex as shown in Figure. 6. Both of the down
flow and the horseshoe vortex lead to an increase in the bed shear stress on the soil. Once the
shear stress is more than the critical shear stress, the grains on the bed surface can be
removed.
The dead zone behind the footing gains velocity in the opposite direction of the flow as a
result of accelerating the flow at the rear edges of the footing, creating a vortex in this area
which is called wake vortex as shown in
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Figure. 7a. The wake vortex acts as a little storm lifting the grains from the bed and form a scour hole
downstream of the footing. The effect of down flow, horseshoe vortex and wake vortex
Figure. 7) becomes weaker gradually in time, thus lower bed shear stress occurs. Once the
value of the shear stress is lower than the critical shear stress, the grains on the bed surface
cannot be removed. As a result, the scour depth reaches to a state of equilibrium as shown in
Figure. 8.
Table 2 Numerical data for cuboid footing
Run d1
(cm) Q (l/s) y (cm)
V
(m/s) V/Vc F
ds
(cm)
1 12.0 12.0 7.85 0.23 0.51 0.26 4.97
2 12.0 13.0 7.76 0.25 0.56 0.29 6.26
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3 12.0 15.0 7.65 0.30 0.66 0.34 8.70
4 12.0 16.0 7.62 0.32 0.71 0.37 9.50
5 12.0 18.0 7.58 0.36 0.80 0.42 11.00
6 12.0 24.0 8.43 0.43 0.95 0.47 13.90
Figure. 6 Flow streamlines show the down flow and the horseshoe vortex in front of the cuboid
footing at time 80 sec
Figure. 7 Flow streamlines show the wake vortex behind the cuboid footing at (a) time 80 sec and (b)
equilibrium state
Figure. 8 Contour map shows the change in bed elevation around cuboid footing at the equilibrium
state
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Figure. 9 Relationship between ds/y and F for different shapes of footing
4.2. Effect of Different Shapes of Footing
The pyramid footing is the most common in the tower construction field, so the effect of
different inclination angles ( = 5º, 10º, 15º, 20º and 25º) for pyramid footing on the
maximum scour depth has been investigated. The details of the numerical runs for different
pyramid footing are listed in Table 3. Figure. 9 illustrates the relationship between the
relative maximum scour depth and Froude number for pyramid and cuboid footing. Figure.
10 shows the flow streamlines around the pyramid footing with angle = 25º at the initial
stage of scour at time 80 sec. Figure. 11 shows the scour contour map around the pyramid
footing with angle = 25º at the equilibrium state.
By comparing the behavior of flow and the scour contour maps around the different
pyramid footing with cuboid footing, it is found that, the inclination angle in pyramid footing
directs part of the flow upwards. By increasing the value of this angle, the flow upwards
increases and the down flow decreases. Once the down flow decreases the horseshoe vortex
decreases (see Figure. 6 and Figure. 10). As a result, the shear stress on soil surface decreases
and then the scour decreases around the pyramid footing (see Figure. 11). The pyramid
footing with different inclination angles in all operating conditions records a reduction in the
relative maximum scour depth as shown in Table 3.
Figure. 10 Flow streamlines show the down flow and the horseshoe vortex in front of the pyramid
footing ( = 25º) at time 80 sec
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Figure. 11 Contour map shows the change in bed elevation around pyramid footing ( = 25º) at the
equilibrium state
Table 3 Numerical data for pyramid footing
Run º Q (l/s) y (cm) V (m/s) V/Vc F ds (cm) ds/y
(%)
7 5 12.0 7.81 0.23 0.52 0.27 4.55 11.88
8 5 13.0 7.55 0.26 0.58 0.30 6.44 4.22
9 5 15.0 7.42 0.31 0.69 0.36 8.16 7.33
10 5 21.0 8.45 0.38 0.82 0.41 11.20 7.48
11 5 24.0 8.66 0.42 0.92 0.46 13.00 6.28
12 5 26.0 8.62 0.46 0.99 0.50 13.70 9.37
13 10 12.0 7.75 0.23 0.52 0.27 4.48 15.43
14 10 13.0 7.70 0.26 0.57 0.29 5.78 10.66
15 10 14.0 7.36 0.29 0.65 0.34 7.08 11.52
16 10 15.0 7.09 0.32 0.72 0.38 8.30 10.25
17 10 24.0 8.97 0.41 0.88 0.43 12.10 10.65
18 10 25.0 8.61 0.44 0.96 0.48 12.40 14.71
19 15 12.0 7.77 0.23 0.52 0.27 3.85 26.63
20 15 15.0 8.12 0.28 0.62 0.31 6.42 16.76
21 15 18.0 8.39 0.33 0.71 0.36 7.86 20.81
22 15 21.0 8.75 0.36 0.79 0.39 10.10 13.93
23 15 24.0 9.11 0.40 0.86 0.42 11.70 12.60
24 15 26.0 8.97 0.44 0.95 0.47 12.40 16.21
25 20 12.0 7.82 0.23 0.52 0.27 3.21 37.69
26 20 13.0 7.84 0.25 0.56 0.29 4.08 34.37
27 20 18.0 8.68 0.31 0.69 0.34 7.00 26.30
28 20 21.0 8.86 0.36 0.78 0.39 9.40 18.92
29 20 24.0 8.98 0.40 0.88 0.43 11.00 18.68
30 20 26.0 8.95 0.44 0.96 0.47 11.80 20.35
31 25 12.0 7.73 0.24 0.52 0.27 2.60 51.26
32 25 15.0 8.01 0.28 0.63 0.32 4.97 37.05
33 25 18.0 8.52 0.32 0.70 0.35 6.27 35.55
34 25 21.0 8.66 0.37 0.80 0.40 8.60 27.46
35 25 24.0 8.97 0.41 0.88 0.43 9.30 31.34
36 25 26.0 8.93 0.44 0.96 0.47 11.10 25.20
To illustrate the effect of sharp edges in the cuboid and pyramid footing on the scour
depth, cylindrical footing has been investigated. The details of the numerical runs for
cylindrical footing are listed in Table 4. Figure. 12 shows the flow streamlines around the
cylindrical footing at the initial stage of scour at time 80 sec.
Gamal M. Abdel-Aal, Maha R. Fahmy, Amany A. Habib and Mohamed Galal Elbagoury
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Figure. 13 shows the scour contour map around the cylindrical footing at the equilibrium
state. Figure. 14 illustrates the relationship between the relative maximum scour depth and
Froude number for cylindrical and cuboid footing.
By comparing the behavior of flow and the scour contour maps around the cylindrical footing with
cuboid footing, it is found that, the smooth body of the cylindrical footing reduces the obstruction of
the flow during separation, which leads to a decrease in the effect of down flow. As a result, the
relative maximum scour depth decreases (see
Figure. 13). The cylindrical footing in all operating conditions records a reduction in the
relative maximum scour depth as shown in Table 4.
Figure. 12 Flow streamlines show the down flow and the horseshoe vortex in front of the cylindrical
footing at time 80 sec
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Figure. 13 Contour map shows the change in bed elevation around cylindrical footing at the
equilibrium state
Figure. 14 Relationship between ds/y and F for different shapes of footing
Table 4 Numerical data for cylindrical footing
Run Q (l/s) y (cm) V
(m/s) V/Vc F
ds
(cm) ds/y
(%)
37 12.0 7.78 0.23 0.52 0.27 3.91 25.20
38 15.0 8.08 0.28 0.62 0.32 6.11 21.49
39 18.0 8.35 0.33 0.72 0.36 8.01 19.77
40 21.0 8.60 0.37 0.81 0.40 9.20 22.91
41 24.0 8.77 0.41 0.90 0.45 10.60 22.97
42 26.0 8.71 0.45 0.99 0.49 11.70 22.22
Cone footing has been suggested for studying, as it combines the characteristics of both
the pyramid footing and the cylindrical footing. The effect of different inclination angles ( =
5º, 10º, 15º, 20º and 25º) for cone footing on the maximum scour depth has been investigated.
The details of the numerical runs for cone footing are listed in Table 5. Figure. 14 illustrates
the relationship between the relative maximum scour depth and Froude number for different
cone footing. Figure. 15 shows the flow streamlines around the cone footing with angle =
25º at the initial stage of scour at time 80 sec. Figure. 16 shows the scour contour map around
the cone footing with angle = 25º at the equilibrium state.
By comparing the behavior of flow and the scour contour maps around the different cone
footing with cuboid footing and pyramid footing, it is found that, the inclination angle in cone
footing directs part of the flow upwards. By increasing the value of this angle, the flow
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0.25 0.30 0.35 0.40 0.45 0.50
ds/
y
F
cuboid cylindrical cone 5 degree cone 10 degree cone 15 degree cone 20 degree cone 25 degree
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upwards increases and the down flow decreases as in the pyramid footing. The smooth body
for the cone footing reduces the obstruction of the flow during separation, which leads to a
decrease in the down flow when it is compared with the pyramid footing at the same angle
(see Figure. 10 and Figure. 15). Once the down flow decreases the horseshoe vortex
decreases (see Figure. 6, Figure. 10 and Figure. 15). As a result, the shear stress on soil
surface decreases and then the scour decreases around the cone footing (see Figure. 16). The
cone footing with different inclination angles in all operating conditions records a reduction
in the relative maximum scour depth as shown in Table 5.
Figure. 15 Flow streamlines show the down flow and the horseshoe vortex in front of the cone
footing ( = 25º) at time 80 sec
Figure. 16 Contour map shows the change in bed elevation around cone footing ( = 25º) at the
equilibrium state
Table 5 Numerical data for cone footing
Run º Q (l/s) y (cm) V (m/s) V/Vc F ds (cm) ds/y
(%)
43 5 12.0 7.67 0.24 0.53 0.27 3.49 35.89
44 5 15.0 7.89 0.29 0.64 0.33 5.69 29.59
45 5 18.0 8.51 0.32 0.70 0.35 7.21 25.98
46 5 21.0 8.60 0.37 0.81 0.40 8.80 26.23
47 5 24.0 8.59 0.42 0.92 0.46 10.50 24.68
48 5 26.0 8.63 0.46 0.99 0.50 10.70 29.19
49 10 12.0 7.72 0.24 0.52 0.27 3.00 43.84
50 10 15.0 7.85 0.29 0.64 0.33 5.10 37.47
51 10 18.0 8.37 0.33 0.72 0.36 7.06 29.02
52 10 21.0 8.65 0.37 0.80 0.40 8.09 31.83
53 10 24.0 8.78 0.41 0.90 0.45 9.90 27.98
54 10 26.0 8.96 0.44 0.95 0.47 10.10 31.81
55 15 12.0 7.75 0.23 0.52 0.27 2.49 52.83
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56 15 15.0 8.13 0.28 0.62 0.31 4.50 41.49
57 15 18.0 8.41 0.32 0.71 0.36 6.21 37.18
58 15 21.0 8.68 0.37 0.80 0.40 7.73 34.67
59 15 24.0 9.05 0.40 0.87 0.43 9.60 28.66
60 15 26.0 8.87 0.44 0.97 0.48 10.10 32.20
61 20 12.0 7.74 0.23 0.52 0.27 2.16 59.33
62 20 15.0 8.25 0.28 0.61 0.31 3.65 51.33
63 20 18.0 8.37 0.33 0.71 0.36 5.69 42.79
64 20 21.0 8.66 0.37 0.80 0.40 7.05 40.50
65 20 24.0 9.00 0.40 0.88 0.43 8.80 34.88
66 20 26.0 9.17 0.43 0.93 0.45 8.88 39.18
67 25 12.0 7.74 0.24 0.52 0.27 2.02 62.01
68 25 15.0 8.08 0.28 0.62 0.32 3.39 56.46
69 25 18.0 8.31 0.33 0.72 0.36 5.25 47.69
70 25 21.0 8.55 0.37 0.81 0.41 6.56 45.26
71 25 24.0 9.03 0.40 0.87 0.43 8.05 40.25
72 25 26.0 9.15 0.43 0.93 0.45 9.10 37.76
5. VERIFICATION
The results of numerical models for cuboid (d1= 6.0 cm and 12.0 cm) and cylindrical (d1 =
12.0 cm) footing have been compared with other equations such as HEC-18 equation
(Richardson and Davis 2001) and Chinese Equation (Dongguang et al. 1993). Table 6
illustrates the selected equations to predict the local scour depth around the footing. The
scour depths for present and previous data are divided by 0.85 to reach an equilibrium scour
depth. A comparison of the relative scour depth that has been measured by numerical models
and other predicted equations shown in Figure. 17. It is found that, Chinese Equation
(Dongguang et al. 1993) gives strong agreement with the present numerical results.
Figure. 17 Comparison of relative scour depth measured by flow 3D and other predicted equations
Table 6 Pier scour equations
Name Equation Notes Reference
HEC-18
(
)
K1 is the shape factor,
K2 is the flow skew angle factor,
K3 is the dune factor and
K4 is the correction factor for armoring by
bed material size.
Richardson
and Davis
(2001)
0.2
0.5
0.8
1.1
1.4
1.7
2.0
2.3
0.2 0.5 0.8 1.1 1.4 1.7 2.0 2.3
Pre
dic
ted
ds/
y
Measured ds/y
HEC-18 Eq.
65-2 Chinese Eq.
+20%
-20%
Gamal M. Abdel-Aal, Maha R. Fahmy, Amany A. Habib and Mohamed Galal Elbagoury
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65-2
(
)
( )
( )
( )
Dongguang
et al.
(1993)
6. STATISTICAL REGRESSION
The scour depth value increases in both of the cuboid footing and the pyramid footing by
comparing it with cylindrical footing and cone footing respectively. The reasons for the
increase of the scour depth are the sharp edges. The sharp edges are expressed by shape
correction coefficient Ks as shown in Eq. (5). The shape correction coefficient for cuboid and
pyramid footing is calculated from Eq. (5) as shown in Table 7. Different values of shape
correction coefficient are shown in Table 8.
An empirical equation (6) has been developed by using a technique of nonlinear
regression analysis to predict the relative maximum scour depth around different shapes of
footing. For cuboid and cylindrical footing, the inclination angle equals zero = 0):
( )
( )
(
5
)
( )
( )
(
6
)
Figure. 18 illustrates a comparison between the predicted values for maximum depth of a
scour hole by Eq. (6) and numerical results for all numerical model tests. It is found that, the
results indicate a good agreement between the numerical and predicted values of ds/y where,
R2 = 0.95.
Table 7 Shape coefficient for cuboid and pyramid footing
Shape 1 Shape 2 Ks
Cuboid Cylindrical 1.29
Pyramid ( = 5º) Cone ( = 5º) 1.29
Pyramid ( = 10º) Cone ( = 10º) 1.32
Pyramid ( = 15º) Cone ( = 15º) 1.31
Pyramid ( = 20º) Cone ( = 20º) 1.35
Pyramid ( = 25º) Cone ( = 25º) 1.27
1.30
Table 8 Correction coefficient based on the shape of the footing
Footing Shape coefficient Ks
Cylindrical and Cone 1.00
Cuboid and Pyramid 1.30
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Figure. 18 Comparison between the measured and predicted data for all numerical data
7. CONCLUSIONS
The following results have been created from this research:
1. The numerical model results give a good agreement with an error by about ±
8.0%.
2. The cylindrical, pyramid and cone footing record a reduction in the relative
maximum scour depth.
3. In both of pyramid footing and cone footing, the larger the inclination angle, the
smaller the scour depth will be.
4. At the same inclination angle in both of pyramid footing and cone footing, the
cone footing is better than the others.
5. Chinese Equation (65-2) gives well acceptance with the present numerical model
results.
6. An empirical equation is developed by regression analysis to predict the relative
maximum scour depth around different footing.
8 LIST OF SYMBOLS
B Flume width
ds Maximum scour depth around footing
d1 Lower width or diameter of footing
d2 Upper width or diameter of footing
h Height of footing above channel bed
Inclination angle of cone or pyramid footing
with vertical axis
Ks Shape coefficient
t Sand layer thickness
y Upstream flow depth
F Upstream Froude number
V Upstream mean velocity
Vc Critical velocity of bed material
U*c Critical shear velocity
Q Flow rate
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Pre
dic
ted
ds/
y
Measured ds/y
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Density of water
s Density of sand particles
g Gravitational acceleration
d50 Mean diameter of the sand layer
ds/y Relative maximum scour depth
ds/y Reduction in relative maximum scour depth
d1/y Relative lower width or diameter of footing
h/y Relative height of footing
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