Advances in River Sediment Research – Fukuoka et al. (eds)© 2013 Taylor & Francis Group, London, ISBN 978-1-138-00062-9

Application of steady flows for simulating the local scour depth
under time varying flows
T. Esmaeili & T. Sumi
Department of Urban Management, Graduate School of Engineering, Kyoto University, Kyoto, Japan

W.Y. Chang
National Center for High-Performance Computing, Science Based Industrial Park, Hsinchu, Taiwan

A. Vakili
Department of Computer Engineering, Islamic Azad Unversity, Aq Qala Center, Aq Qala, Iran

ABSTRACT: Local scouring in the vicinity of bridge piers which are implemented in the natural
streams is the subsequent of 3D complex flow patterns. During the local scour process the accelerated
flow develops the local scour depth and if it is not predicted correctly the bridge failure occurs. Local
scouring around bridge piers is a strictly time dependent phenomenon and subsequently the evolution of
local scour depth under the unsteady flow condition is essential for both structure safety and ecologicalissues. In the present study, development of local scour depth around a circular pier under time varying
flows of the three different types of step-wise hydrographs has been simulated by using a fully 3D numerical program SSIIM along with applying a superposition scheme. Results show good agreement between
predicted numerical results and experimental data whereas the error rates of the maximum simulated local
scour depths were less than 10%.

1

INTRODUCTION

Due to the diverse economic and social costs thatcan be imposed by bridge failure the evaluationof the maximum local scour depth around bridgepiers is an important topic in river engineering.
Streambed scour is the leading cause of bridge
failure in the United States (Murillo 1987). The
costs associated with restoring damaged structures are substantial, but the indirect costs associated with the disruption of traffic can be morethan four times the direct costs (Rhodes & Trent
1999). Implementation of bridges’ pier foundation in a deeper depth than the maximum potential local scour hole was accepted as a conservative
scour countermeasure in most of practical cases
although, that would require expensive and deeppiers in the streambeds (Esmaeili et al. 2011).
The three-dimensional flow field around a
bridge pier is extremely complex due to separation and generation of multiple vortices. The complexity of the flow field is further compoundedby the dynamic interaction between the flow andthe moveable boundary during the developmentof a scour hole (Raudkivi 1986). Local scouring around bridge piers can be a severe problem,
especially when a high flow occurs in the river(Esmaeili et al. 2009). When this undesirable event
occurs, a scour hole develops around bridge piersas, in general, the local scour depth increases withincreasing flow (Dehghani et al. 2012). If thesevere scour finally exposes the pier foundation in
the scour hole, the safety and stability of the bridge
will be in danger. Therefore, there is a great needto develop good prediction methods for pier scour
evolution especially under the time varying flowsof the flood hydrograph.
Many studies have been carried out to develop
the relationships for predicting the maximum local
scour depth at bridge piers under clear-water scourand other idealized conditions, such as steadyflow, uniform sediment, simplified geometry,etc. Although these relations used extensively fordesign purposes but, most of them are based on
equilibrium condition using the peak flood flow,which are only achieved after a long time period,so that in the realistic cases the peak flood flows
usually may not last long enough to develop equilibrium scour depth. Thus, the equilibrium scour
depth in clear water may yield considerably greater
values of scour depth than may occur if the flow isof shorter duration (Dehghani et al. 2009, Changet al. 2004). For a known time-to-peak value ofdesign flood hydrograph, smaller scour depths

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may be obtained which reduce the total cost of
construction (Yanmaz 2006). Figure 1 shows thebridge failure during the time varying flow of aflood event in a rural area with a limited accessroad network.
Some researchers have focused on the scourdepth evolution with uniform sediment understeady flow condition (e.g. Yanmaz & Altinbilek
1991, Melville & Chiew 1999). Mia & Nago (2003)
proposed a method for computing the time variation of local scour depth. Their method considered the development of bed shear stress aroundthe pier according to bed-load sediment transport
theory. Kothyari et al. (1992) studied the temporal variation of scour depth around bridge piers
under clear-water conditions by using unsteadyflow that was approximated by a stepwise discharge hydrograph. Nowadays, modeling 3D flow
field and bed deformations is practically feasible
because of the considerable reduction in computational time and costs.
A 3D flow model, EllipSys 3D, was developed by
Roulund et al. (2005) and coupled with a morphologic model to simulate the scour process arounda circular pier in uniform sediment. Nagata et al.
(2005) developed a 3D numerical model for simulating the flow and bed deformation around river
hydraulic structures such as bridge piers. Recently,
the SSIIM program has been used by Bihs andOlsen (2008) and the time scaled scour hole development around a circular pier was simulated.
Moreover, evaluation of changes in river geomorphology (i.e. variation of river bed in thevicinity of river hydraulic structures) caused byvarious unsteady flow regimes is one of the interests of researchers focusing on river ecology issues.
Furthermore, assessing the variation of bed softness and subsequently the habitat structures ofaquatic animals as well as dynamic changes ofthe sandbars geometry as a result of time varying
flows are essential for discussing about real timescaled river ecology.
In the present study, the SSIIM program wasused to simulate the scour depth evolution arounda circular bridge pier by applying the concept of

Figure 1. A) Flow pattern in the vicinity of a bridgepiers during a flood. B) Failure of the bridge during theflood.

superposition procedure which was introduced by
Chang et al. (2004). The SSIIM program, implements fully 3D numerical model developed by
Olsen (2011), is based on the finite volume method
solving the Navier-Stokes equations with the various turbulence models for 3D flows, and solving
sediment continuity equation for bed variations(Olsen et al. 1999). This program was made forusing in river/environmental/hydraulic and sedimentation engineering but, the main objective wassediment transportation. The main advantage of
SSIIM program compared to other Computational
Fluid Dynamics (CFD) programs is simulation of
sediment transport with moveable bed in complex
geometry.
2

NUMERICAL MODEL SSIIM

2.1

Three dimensional flow model

The SSIIM program solves the Navier-Stokes equations with the k-E model on a three dimensional general non-orthogonal coordinates. These equations
are discretized with a control volume approach.
The SIMPLE or SIMPLEC methods can be used
for solving the pressure-poisson equation. An
implicit solver is used, producing the velocity fieldin the computational domain. The velocities areused when solving the convection-diffusion equations for different sediment sizes (Olsen 2011).
The Navier-Stokes equations for non-compressible and constant density flow can be modeled as:
tui
0
txi
tui
tt

uj

(1)
tui 1 t
tx j l tx
tx j

 P`

ij

l u `i u `j(2)where the first term on the left side of the Equation 2
is the transient term. The next term is the convective term. The first term on the right-hand side is
the pressure term and the second term on the rightside of the equation is the Reynolds stress term. In
order to evaluate this term, a turbulence model isrequired. SSIIM program can use different turbulence models, such as the standard k-E model or thek-W model by Wilcox (2000). However, the default
turbulence model is the standard k-E model.
The free surface is modeled as a fixed-lid, withzero gradients for all variables. The locationsof the fixed lid and its movement are as a function of time, which can be computed by different
algorithms. The 1D backwater computation is thedefault algorithm and it is invoked automatically.
Pressure and Bernoulli algorithm can be used forboth steady and unsteady computations. The algo-906

rithm is based on the computed pressure field. It
uses the Bernoulli equation along the water surface
to compute the water surface location based on
one fixed point that does not move. The algorithm
is fairly stable, so that it can also be used in connection with computation of sediment transport andbed changes (Olsen 2011).
For the wall boundary treatment, it is assumed
that the velocity profile follows a certain empirical function called a wall law. It is a semi-analytical
function to model the turbulence near the wall inthe boundary layer and consequently the CFD
model will not need to resolve the turbulence offlow in boundary layer. As a result, it would notnecessary too many grid points near the wall:

For calculating the suspended load in Equation 4, Equation 5 is solved incorporated with theformula by van Rijn (1987) for computing the equilibrium sediment concentration close to the bed as
the bed boundary. In order to solve Equation 4 andEquation 5, conditions of z and C should be given
at inflow and outflow boundaries. For the inlet
boundary, due to the clear water scour conditions,z  C  0 can be given and for the outlet boundary,far away from the pier, t tx  tz/t  0 can begiven due to the uniform flow.
The concentration formula has the following
expression:1 ¤ 30 y ³U lnux g ¥¦ ks ´µCbed  0.015(3)where the shear velocity is denoted ux, g is a constant equal to 0.4, y is the distance to the wall andks is the roughness equivalent to a diameter of particles on the bed.
2.2 Morphological model
The sediment transport process in rivers is describedby the following equation, Exner’s equation, whichis the sediment continuity equation integrated over
the water depth:
(

h)

tzb
tt

ttqqtx
tx

tqty
ty

0

tc
tt

tc
uj
tx j

tc ³
tc
t ¤
'T

w
tx j ´µtz tx j ¥¦

qs,i

iT
Sc

zf

¯a uic dz(5)where Sc is the Schmidt number representing theratio of diffusion coefficient to eddy viscosity coefficient vt and set to 1.0 as default.

(7)

(8)

qb
D501.5

(6)

d 0.3
0.1
a § (l¶sw )g¨·
lwi 2 ¸©

where qs,i (i  1,2) are components of suspendedload sediment transport in x- and y-directions,
respectively.
For calculating the bed load in Equation 4, thefollowing relation proposed by van Rijn’s formula(1987) is used:

( ls

w )g

lw
1.5

§n n c ¶¨·© nc ¸ 0.0530.1§¤ ( l³¶w )g
D50 0.3 ¨¥ s
·
´
2
µ ·¸
¨©¦ lwiwhere w is the fall velocity of sediment particles and'T is the diffusion coefficient and can be expressedin the following way:
'T 

1.5

where, Cbed is the sediment concentration, d is thesediment particle diameter, a is a reference level set
equal to the roughness height, n is the bed shear
stress, nc is the critical bed shear stress for movement of sediment particles according to Shield’s
curve, lw and ls are the density of water and sediment, i is the viscosity of the water and g is theacceleration of gravity.
Once Equation 5 is solved, the suspended load
can be calculated as follows:(4)

where zb is the bed elevation, h is the porosity of bed
material, and qtx and qty are components of totalload sediment transport in x- and y-directions,
respectively.
In this 3D CFD program, the suspended load
can be calculated with the convection-diffusion
equation for the sediment concentration, which isexpressed as follows:§ n nc ¶¨·© nc ¸(9)where D50 is the mean size of sediment. Then, thecomponents of bed-load sediment transport in xand y-directions can be calculated as follows:
qbx

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qb cos(] b ); qby

qb sin(i (] b )(10)where Ab is the direction of bed-load sedimenttransportation.
3

depth increases steadily during the rising period of
the hydrograph and the significant amount of final
scour depth will be obtained at the end of rising
period of hydrograph.

MODEL PROPERTIES AND STUDY
CASE
4The experimental results of Chang et al. (2004)
were used for calibration and validation of model.
In their study, the experiments were carried out
in a 36 m long, 1 m wide and 1.1 m deep flume.
A false floor was set in the flume with a recess of
2.8 m long and 0.3 m deep. A hollow cylindricalpier made of transparent plexiglas with a diameterof 0.1 m was located at the center of the recess. The
recess was filled with the desired sediment size andwas flattened before each experiment run. Changet al. (2004) & Dehghani et al. (2012) concluded
that the recession limb of hydrograph plays muchless effective role than rising limb in scour depth
evolution. Subsequently, three different stepwise
discharge hydrographs were used for simulation of
the unsteady flow in channel entrance (Fig. 2).
Figure 2 shows that the hydrographs A, B and C
have the same time to peak (tP) and different peakflow discharge (QP). So, due to the different flow
volume, the induced bed shear stress and sedimenttransport capacity for each of the aforementionedhydrographs will be different.
The boundary condition of discharge and waterlevel during first hour is same for all types of stepwisehydrographs and equal to 0.034 (m3/s) and 0.15 (m).
The temporal variation of the bed surface profilearound the pier was recording using two 3 cm longcharge coupled device with 2 mm diameter lenses,
placed in the hollow plexiglas pier. Uniform sediments were used for unsteady flow condition andthe median size of sediment is 0.71 mm with mg of
1.2. The experimental results showed that the scour

Figure 2. Stepwise discharge hydrographs used formodeling (Chang et al., 2004).

NUMERICAL SIMULATIONS

4.1 Domain description and model calibration
Making an appropriate grid is a very important
process in the preparation of input data for SSIIM
program. The size and alignment of the cells willstrongly influence the accuracy, the convergenceand the computational time (Olsen 2011). For getting more precise results, it is better to refine themesh around bridge piers (Dehghani et al. 2009).
A preprocessing code was used for dividing flow
field area into regions of different sizes. In thisstudy, the length of channel is divided by six sections and in each section the various cell sizes wereused. The distribution of mesh in half of the channel length is presented in Figure 3.
In the first region, in x- and y-directions, the sizeof cells is 5 cm r 1 cm and in the second region cells
are 2.5 cm r 1 cm. The central part of the flume is
3 m long and the size of cells is 1 cm r 1 cm. The
total number of cells in the computational domain
is 546000.
For the inflow boundary condition, the velocity distribution was specified while the gradient ofpressure is given zero. At the outflow boundary, thevertical gradient of velocity is zero and the hydrostatic pressure distribution is specified according
to the water depth. For the solid boundary, walllaws introduced by Schlichting (1979) were used
for the side walls and the bed. As previously mentioned, the experimental conditions are the samefor the first hour of all stepwise hydrographs. In
order to model calibration, by running the SSIIM
program, variation of computed scour depth

Figure 3. a) Plan view of computational mesh used fornumerical modeling (not scale) and b) Corresponding
vertical grid distribution.

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against time under steady flow condition of firststep of stepwise hydrographs was compared withthe experimental results and when numerical andexperimental results adapted well, then the corresponding time step and roughness were accepted
as the calibrated time step and roughness. This
process has been illustrated in Figure 4. The time
step and roughness were obtained 5 s and 5.2 mm
respectively.
Dehghani et al. (2012) found that when the ratioof peak-flow discharge to the discharge of first stepincreases, the discrepancy between the experimental and numerical model results further magnifies.
They also attributed the reason to sharp increaseof the side slope of scour hole and in such a condition the scour depth may increase too quickly tocause the side slope of scour hole to exceed therepose angle of sediment in the moveable areaaround the pier. Moreover, they concluded thatnumerical program could not respond immediately
to the change of discharge as well as boundary
condition. So due to the considerable difference
between the peak-flow discharge and the dischargeof first step, recalibration of the sediment relatedparameters (i.e. repose angle of sediment as well as
bed rouhness) will be necessary in the cases withsharp increase of discharge.
4.2

depth S1 at time TA becomes the scour depth at the
beginning of the following segment C’D’. In thisway, the memory of segment A’B’ is brought into
segment C’D’. Similarly, for discharge Q1 of the
stepwise hydrograph, the scour-depth evolution
along segment C’D’ is the same as the corresponding segment CD along C1 curve for steady flow Q1.
Likewise, the same procedures are applied to theremaining segments of the stepwise hydrograph.
In the present study, the curves Ci in Figure 5afor steady flow discharges are obtained by SSIIM
program.
After the model calibration, the temporal variation of scour depth around circular bridge pierunder boundary condition of each step of stepwisehydrographs has been conducted by employing
the SSIIM program. Figure 6 illustrates the scour
depth development under each step’s boundary
condition.
Figure 7 demonstrates the local scour depth
evolution under the unsteady flow condition ofhydrographs by using the aforementioned superposition procedure. As can be clearly observed

Procedure for simulating the scour
depth evolution under time varying
flows & discussion

The concept of superposition procedure which
introduced by Chang et al. (2004) was used for
simulation of scour depth evolution under timevarying flows of stepwise hydrographs. As shownin Figure 5b, the scour depth evolves from 0
to S1 in segment A’B’ for Q0, under the stepwisehydrograph, is in the same way as segment AB
along the C0 curve for steady flow Q0 shown in
Figure 5a. At the end of segment A’B’, the scour

Figure 4. Evolution of local scour depth against time
under steady flow condition of first step of stepwisehydrographs for calibrating the model.

Figure 5. Illustrations of superposition scheme
for computing scour depth evolution under stepwisehydrograph.

Figure 6. Temporal variation of local scour depth aroundcircular pier under steady flow condition of each step ofA) hydrograph A, B) hydrograph B, C) hydrograph C.

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Figure 7. Variation of computed scour depth
against time; A) hydrograph A, B) hydrograph B, C)
hydrograph C.

from Figure 7, there is a good agreement between
the experimental and numerical results under theboundary condition of all types of hydrographs.
Furthermore, Results demonstrate that the difference between the maximum simulated local scour
depth and measured one at the end of hydrographs
A, B and C is less than 10%. So it is concluded
that this superposition scheme in connection withnumerical model results, as an approach, could provide the accurate results to predict the local scour
depth evolution around bridge piers under timevarying flows by employing steady flows. The main
application of this approach would be designingof temporal bridges with a limited service period
or main bridges in transportation networks when
the safe and cost-effective designing have a direct
relation with the implementation depth of thefoundation.
It should be noted that, high peak flow discharges during a flood will have an important rolein river geomorphology formation by eroding anddepositing of sediments in the main channel stream
and flood plains. Prevalent examples about the roleof high flow discharge (i.e. floods), in changing
the geomorphology of the rivers, are erosion anddeposition of sediments in sand bar areas as wellas river meanders that strongly influence the habitat structures and subsequently the biodiversity ineach river segment. Takemon (1997) derived thatdevelopment of transition zone in the upper partof a bar structure in a riffle, when sedimentation
increases during the floods, may result in increasing the porosity of the substrate and well drainage
of underflow water in streams. Also, Reice (1994)
concluded that impacts of frequent disturbance instreams such as floods and their effects on creating
environmental patches for populations is important for diversity of aquatic species. Though thehigh flow discharges during the time varying flows

Figure 8. Final bed topography by employing the stepwised boundary condition of hydrograph B.

have the dominant roles for changing the geometryof whole habitat structures, consequent low flow
discharges are able to have a function for initiating the second small scale erosion and deposition
in previously scoured or deposited areas. So thesecond stage grain size redistribution and new bed
softness would be considered as the low-flow discharges effects on river geomorphology during theunsteady flows.
As a result, from the eco-hydraulic point of view,
it will be useful to simulate the local scour geometry evolution under time varying flows because of
their role in eroding and depositing of sediments
around the bridge piers which will be a potential
habitat for some aquatic animals. Figure 8 shows
the bed changes in the vicinity of the circular pierunder the boundary condition of the 3rd hour ofhydrograph B. This figure clearly demonstrates the
local scour geometry and also bed changes patternaround the pier.
5

CONCLUSIONSIn this study the evolution of scour depth wassimulated by a 3D numerical program and a superposition scheme was employed for connecting theresults of steady flow condition to the time varying flows of 3 different types of hydrographs. The
method was based on employing the scour depth
evolution under the steady discharge corresponding to each step of the step-wise hydrograph.
The following outcomes were obtained from thisstudy:
1. In this study, the superposition concept alongwith the satisfactory ability of the 3D numerical program SSIIM for providing the accurate
scour depth development under steady flows
corresponds to the discharge of each step of910

stepwised hydrograph is the basis of the connection between the steady flows and timevarying flows of hydrographs as an approach.
Comparing the simulated results with theexperimental data shows good agreement. In
this way the complex condition of time varying flows of hydrographs will break into simple
case of steady flows. This approach can be useful for safe and cost effective designing of not
only bridge piers, but also other river structures(i.e. groynes, abutments) which are periodically exposed to time varying flows of different
floods.
2. The result accuracy of the numerical programis tightly depends on the calibration and adjustment of the model. In this study by employing the superposition concept, the error rateof the calculated scour depth at the end of thehydrographs was less than 10%.
3. The Main aspect of the present study is to showthat the amount of predicted local scour under
unsteady hydrographs is less than the equilibrium local scour depth under a steady flow,which is related to the peak flow discharge.
As for the engineering applications for temporal hydraulic structures, the proposed method
may be capable of scour assessment in the fieldsituations, in which the hydrographs should
be appropriately stepwised beside of considering the type of hydrograph for adjusting themodel.
4. Although this superposition scheme could predict the local scour depth under time varying
flows of hydrographs, the development of local
scouring geometry during the rising period of thehydrograph cannot be provided by this scheme.
Owing to the fact that geomorphology changes
in main channel stream in where some hydraulic structures are implemented, may affect the
habitat structures of some aquatic organisms,
it is necessary to have a concept for monitoring the stepwise changes in the bed topographyduring the unsteady flows for future studies.
In this way, it will be possible to monitor thetime-scaled topography changes of a designatedlocation in alluvial streams (i.e. in the vicinityof hydraulic structures such as bridge piers or
groynes) and considering both the structure
safety issues and ecological challenging aspectsof hydraulic structures construction in one
frame work.
ACKNOWLEDGEMENTFirst author would like to give his sincere gratitude
to Assoc. Prof. Yasuhiro TAKEMON for his useful comments on the paper.

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