numerical investigation of flow pattern and scour ... · scour. khan et al. (2017) analyzed the...

\http://www.iaeme.com/IJCIET/index.asp 3274 [email protected]

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

http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=10&IType=03

Gamal M. Abdel-Aal, Maha R. Fahmy, Amany A. Habib and Mohamed Galal Elbagoury

http://www.iaeme.com/IJCIET/index.asp 3275 [email protected]

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

Numerical Investigation of Flow Pattern and Scour Characteristics Around Electrical Tower

Foundations


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



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


Foundations


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)



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.


Foundations


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.



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


Foundations


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



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


Foundations


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



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


Foundations


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.



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


Foundations


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



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


Foundations


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%



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


Foundations


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



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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