A model for scour around bridge piers caused by flood waves
29 Aug 2016pp 841846
Abstract: The local scour process during flood waves is modeled introducing the idea of an effective work by the flow on the sediment bed around the pier. Dimensional considerations show different possible formulations for the dimensionless effective flow work W* that, in each case, is shown to be a generalization of the flow intensity concept, commonly used in existing scour formulas. A novel experimental installation able to reproduce any hydrograph with high precision in the laboratory flume is used to carry out steady and unsteady flow experiments in order to calibrate and validate the mathematical model. Results confirm the uniqueness of the relationship between the dimensionless, effective flow work W* and the relative scour depth Z*, highlighting the high predictive capacity of scour depth caused by any hydrograph. The proposed model provides good performances allowing a straightforward prediction of maximum scour depth and its evolution.
Topics: Bridge scour (69%), Hydrograph (54%)
1 INTRODUCTION
Bridge pier scour occurs mostly during flood waves,
which in medium and small sized rivers (annual
mean discharge < 1000 m
3
/s) have durations in the
order of hours or days, which is typically shorter
than the required time needed to develop the
equilibrium scour, which in sands is in the order of
weeks. Most existing scour formulas assume a
“worst condition”, considering the peak discharge of
an extreme event (typically a 100yr flood), as a
constant discharge acting for an infinite time on the
streambed. On the one side, this approach tends to
give an overestimation of scour; on the other side
this approach overlooks the possibility that smaller
but longer lasting discharges produce more scour
than the extreme peak discharge acting for a short
time. Estimation of scour caused by flood waves is
not a straightforward process. Existent formulations
are scarce and their solution is iterative due to the
strong dependence of actual scour rate from
antecedent scour depth.
In this study, the concept of an effective flow
work (Bagnold 1966, Lai et al. 2009) for scour depth
prediction is adopted. Following dimensional
analysis, different possible formulations for the
dimensionless flow work are obtained. A novel
installation able to reproduce any hydrograph with
high precision in laboratory flumes is presented, in
order to verify the hypothesis and select the best
performing alternative for the flow work
formulation.
2 THE PROPOSED MODEL
2.1 Dimensional considerations
Scour depth at a cylindrical bridge pier depends on
variables characterizing the fluid, flow, sediment,
and pier. In functional form:
(, ,,,, , , , ,,) 0
ss
f uhgd Dt z
µρ ρ s
=
(1)
where μ = dynamic fluid viscosity; ρ = fluid density;
u = section averaged flow velocity; h = flow depth;
g = gravitational acceleration; d
s
= representative
sediment particle diameter; ρ
s
= density of sediments
particles; σ = standard deviation of the sediment par
ticle sizes; D = pier diameter; t = time; and z = scour
depth. Eq. (1) constitutes the basis of most existent
scour formulas.
Performing a dimensional analysis on Eq. (1), the
following dimensionless relationship can be ob
tained:
2
*
, ', , , , , , 0
'
sssss
u h D ut z
fD
gddddd
ρs
ρ
=
(2)
where D
*
= ((ρ'g)/ν
2
)
1/3
d
s
denotes the dimensionless
sediment particle diameter; ν = kinematic viscosity;
ρ' = relative particle density. Both Eq. (1) and Eq.
(2) are valid only under steady state.
A model for scour around bridge piers caused by flood waves
O. Link, A. Pizarro, C. Castillo
University of Concepción, Edmundo Larenas 215, Concepción, Chile
B. Ettmer
Magdeburg University of Applied Sciences, Breitscheidstraße 2, 39104 Magdeburg, Germany
S. Manfreda
University of Basilicata, Via dell’Ateneo Lucano 10, 85100 Potenza, Italy
ABSTRACT: The local scour process during flood waves is modeled introducing the idea of an effective
work by the flow on the sediment bed around the pier. Dimensional considerations show different possible
formulations for the dimensionless effective flow work W
*
that, in each case, is shown to be a generalization
of the flow intensity concept, commonly used in existing scour formulas. A novel experimental installation
able to reproduce any hydrograph with high precision in the laboratory flume is used to carry out steady and
unsteady flow experiments in order to calibrate and validate the mathematical model. Results confirm the
uniqueness of the relationship between the dimensionless, effective flow work W
*
and the relative scour depth
Z
*
, highlighting
the high predictive capacity of scour depth caused by any hydrograph. The proposed model
provides good performances allowing a straightforward prediction of maximum scour depth and its evolution.
The challenge of the present research is to identify
formulations able to interpret the timedependence
of scour from a nonstationary process of flood. In
this frame, Pizarro (2015) defined the effective di
mensionless flow work at a cylinder as:
3
*
0
( ) 0.5
1
end
t
c
RR
ut u
W dt
ut
d
−
=
∫
(3)
where t
end
= hydrograph duration, t
R
= a reference
time; u
R
= a reference velocity; u(t) = time
dependent section averaged flow velocity; u
c
= criti
cal velocity for incipient motion of sediment parti
cles. For practical purposes, it is assumed that scour
occurs when u(t) ≥ 0.5u
c
, thus:
{
0 ( ) 0.5
1 ( ) 0.5
c
c
ut u
ut u
d
<
=
≥
(4)
The dimensionless flow work represents a gener
alization of the flow intensity concept (u/u
c
) to the
unsteady case. Using such concept, Eq. (2) can be
written as:
** *
, ', , , ,
ss
D
Z fD W
dd
h
ρs
=
(5)
where Z
*
= dimensionless scour depth (= z/z
R
) and
z
R
= a reference length.
Two alternatives are analyzed for the reference
values involved in Z
*
and W
*
:
1) Classical approach: Assuming that the reference
length, time and velocity correspond to pier diame
ter, equilibrium time and critical velocity, respec
tively. The dimensionless flow work assumes the
following form:
3
*
0
1 ()
( ) 0.5
end
t
end
ceq
ut
W t dt
t u
d
= −
∫
(6)
*
z
Z
D
=
(7)
2) New approach: Adopting the mass conservation
law for the estimation of the reference time and ref
erence length and using the characteristic time con
cept (t
c
) for the pier local scour introduced by Guo
(2014),
( )
2
2
c
sc
D
t
d uu
=
−
(8)
a new formulation for the dimensionless flow work
is derived.
Note that Equation (8) can be written using the
same argument of the dimensionless effective flow
work W
*
, i.e.:
( )
22
...
2 ()
( ) 0.5
2
c
sc
c
sR
R
DD
t
d ut u
ut u
du
u
= = =
−
−
(9a)
*
...
() ()
0.5 0.5
RR
R
RR
zt
ut ut
u
uu
= =
−−
(9b)
where z
R
= D
2
/(2d
s
) denotes the reference length and
t
R
*
= z
R
/ u
R
is the new reference time. The reference
velocity is taken as the proposed by Oliveto & Hager
(2002), i.e.: u
R
= (ρ' g d
s
)
0.5
. Thus the dimensionless
effective flow work W
*
and the dimensionless scour
depth Z
*
become:
3
*
0
( ) 0.5
1
( ) ...
end
t
c
en
c
d
R
t
ut u
W t dt
u
d
−
= =
∫
(10a)
*
4
0
( ) 0.5
1
...
end
t
c
RR
ut u
dt
ut
d
−
=
∫
(10b)
*
R
z
Z
z
=
(11)
2.2 Timedependent scour depth and scour rate
The dimensionless flow work can be used to derive
an empirical formulation describing the scour depth
as function of time. In this context, based on the em
pirical results by Franzetti et al. (1982), it is suggest
the following relationship between the dimension
less scour depth and effective flow work:
( )
3
*
2
*
1
1
c
cW
Zc e
−
= −
(12)
where c
1
, c
2
and c
3
are fitting coefficients.
Using the above, it is possible to derive the scour
rate (V
s
), defined as the variation of Z
*
in time (t):
*
s
Z()
V=
t
t
∂
∂
(13)
3 EXPERIMENTAL SETUP
Experiments were conducted within an infloor rec
tangular flume of 26 m long, 1.4 m wide and 0.74 m
deep at the Laboratory of Hydraulic Engineering,
University of Concepción, Chile. A Plexiglas cylin
der with a diameter D = 0.15 m was mounted in the
middle of a sedimentrecess located 20 m down
stream of the flume entrance. The sedimentrecess
had a length of 2 m, a width of 1.4 m and a depth of
0.3 m.
3.1 Measuring techniques
The scourhole radius was measured with an accura
cy of ±0.4 mm using a laser distance sensor (LDS)
located inside the Plexiglas cylinder and aligned in
horizontal and radial direction, so that no refraction
on the cylinder wall was observed. The sensor was
driven in the vertical direction by a stepmotor with
a precision of ±1/50 mm. In the azimuthal direction,
the vertical positioning system was driven by a sec
ond stepmotor with an accuracy of ±1/100°. That al
lowed the turnaround view of the distance sensor in
the scourhole, taking various vertical profiles in dif
ferent azimuthal halfplanes for determination of the
azimuthal halfplane where maximum scour depth
occurred. The measured radius, vertical coordinate,
and the azimuthal coordinate of the sensor position
were registered with a frequency of 70 Hz. Scour
rate, maximum scour depth in time, and final scour
depth were computed in order to verify the proposed
model.
Discharge was controlled with a closed loop con
trol system, taking the measured discharge and flow
depth as outputs. Corrections to account for differ
ences between specified and measured discharges
were made on the frequency of the pump motor,
based on a ProportionalIntegral (PI) controller
(Åström & Hägglund 1995) set up by means of a
programmable logic controller (PLC) and a variable
frequency drive (VFD), as schematized in Figure 1.
Flow depth was controlled by adjusting the tail gate
at the end of the flume, and was measured with ul
trasonic distance sensors (UDS) placed along the
flume. The discharge Q
measured
was measured with an
orifice plate device installed in the recirculation sys
tem with a precision of ±1%.
Figure 1. Schematic view of the experimental installation.
3.2 Bed material
In the experiments, fine sand was used as bed mate
rial. The dimensionless particle diameter D
*
= 9, the
relative density ρ' = 1.65, and the standard deviation
of the sediment grain sizes σ = 1.45. The critical ve
locity for the incipient motion of the sediment parti
cles u
c
was experimentally determined in prelimi
nary runs. Table 1 summarizes the properties of the
bed material.
Table 1. Properties of the bed material.
Property
ρ
s
d
50
σ
u
c
kg/m³
mm

m/s
Quantity
2650
0.36
1.45
0.32
3.3 Hydraulic conditions and experimental series
Experiments on scour were conducted with constant
discharges until equilibrium and with hydrographs
having different duration, and shapes. Maximum
discharge during an experiment corresponded to
95% of Shields critical condition for the initiation of
motion of the sand particles at the undisturbed plane
bed. Thus, the runs corresponded to clear water con
ditions. A base flow of 35 l/s and a section averaged
flow depth of h = 1.4D = 0.21 m, corresponding to a
flow intensity of 0.37, were imposed as initial condi
tion. A total of three experimental series are ana
lyzed: Series A included two scour experiments with
constant discharge with the aim of exploring the re
lationship between W
*
and Z
*
until equilibrium, with
different flow intensities (A1: u/u
c
= 0.91 / A2:
u/u
c
= 0.76). Series B included two scour experi
ments with short duration, i.e. high flow accelera
tion, and triangular and sinusoidal shapes, respec
tively. Series C included three scour experiments
with longer duration than Series B: Stepwise and
sinusoidal hydrographs were produced. Figure 2
shows the hydrographs corresponding to each exper
iment and Table 2 summarizes the hydraulic condi
tions for the experiments.
Table 2. Hydraulic conditions for the experiments.
No.
u
p*
h
p*
t
p*
t
end
u
p*
/u
c
m/s
m
min
min

A1
0.29
0.22
N/A
6400
0.91
A2
0.24
0.23
N/A
8600
0.76
B1
0.28
0.23
12.75
25.5
0.87
B2
0.28
0.23
3.3
20
0.87
C1
0.29
0.22
20
120
0.91
C2
0.29
0.22
100
120
0.91
C3
0.29
0.22
40
90
0.91
* Subscript “p” denotes peak condition.
Note that for all experiments D
*
= 9; ρ' = 1.65;
σ = 1.45; h
p
/d
s
≈ 610; and D/d
s
= 416.67. Thus the
functional relationship presented in Eq. (5) can be
simplified to:
( )
**
Z fW=
. (14)
4 RESULTS
The experimental installation reproduced the desired
hydrographs very well, with standard errors less than
1.21 l/s, i.e less than 1.7%, confirming that the flow
control scheme is adequate for applications in fluvial
hydraulics.
Figure 2. Hydrographs of the experimental series.
4.1 Steady flow
The functional relationship between dimensionless
scour depth Z
*
, and dimensionless flow work W
*
is
explored using the two different formulations of the
dimensionless flow work (Eq. 6 vs Eq. 10b). Such
comparison was possible using experiments carried
out under steadystate conditions (Series A). Figure
3 shows the dimensionless scour depth on dimen
sionless flow work for the two experiments of Series
A, according to the two alternative formulations of
the reference values examined herein.
All formulations of the reference values proposed
show the same tendency, increasing Z
*
with W
*
. For
the new approach measured values clearly collapse
into one single curve, demonstrating that the relation
between Z
*
and W
*
is unique. Thus, in the following,
the reference variables are adopted according to the
new approach.
Using the formulation proposed in Eq. (12) , it
was possible to calibrate the coefficients c
1
, c
2
and c
3
that have been determined using the Matlab nonline
ar curvefit function. The values of the coefficients
are c
1
= 0.0075, c
2
= 0.097, and c
3
= 0.38. The de
termination coefficient obtained is r
2
= 0.99.
4.2 Timedependent scour depth
In the following the proposed model is adopted un
der unsteady conditions. In particular, Figure 4
shows that hydrograph shapes affected the evolution
of scour depth in time, but not its final value. Com
parison of final scour depth in experiments B1 and
B2, as well as in C1, C2 and C3 confirm that the
relation between the dimensionless, effective flow
work and scour depth is unique (same final W
*
value
produces the same Z
*
value).
Figure 3. Evolution of Z
*
on W
*
for Series A.
4.3 Comparison with literature
Scarce data have been published in the literature to
test the model proposed in Eq. (12) for prediction of
scour under flood waves.
In this article, the experimental data by López et
al. (2014), reported Table 3, is used. López et al.’s
(2014) measurements need to be rescaled because of
the differences in their experimental setup. These
dissimilarities are due to dimensionless sediment
particle diameter (In the present study D
*
= 9, while
in López et al.’s (2014) experiments D
*
= 41.5) and
relative roughness (In the present study D/d
s
=
416.67, while in López et al.’s (2014) experiments
D/d
s
= 54.55). In order to take scale effects due to
differences in relative roughness into account, the
predictions of Eq. (12) were corrected using the for
mulation proposed by Lee & Sturm (2009).
Figure 4. Scour depth in time for Series B and C.
0 5000 10000
60
90
120
t [min]
Q [l/s]
0 5000
10000
60
75
90
t [min]
Q [l/s]
0 10 20 30
25
50
75
100
Q [l/s]
t [min]
0
10
20 30
25
50
75
100
Q [l/s]
t [min]
0 30 60 90 120
25
50
75
100
t [min]
Q [l/s]
0 30 60
90120
25
50
75
100
t [min]
Q [l/s]
0
30 60
90
120
25
50
75
100
t [min]
Q [l/s]
A1
B1
A2
B2
C1
C2
C3
0 4
8 12 16
20 24
0
0.01
0.02
0.03
0.04
0.05
t [min]
z [m]
0 50
100
0
0.02
0.04
0.06
0.08
t [min]
z [m]
0 50 100
0
0.02
0.04
0.06
0.08
t [min]
z [m]
0
50 100
0
0.02
0.04
0.06
0.08
t [min]
z [m]
0 4 8
12 16 20
24 28
0
0.01
0.02
0.03
0.04
0.05
t [min]
z [m]
computed
measured
C1
C2
B1
B2
C3
10
4
10
3
10
2
10
1
10
0
10
1
0
0.5
1
1.5
W
*
Z
*
10
6
10
4
10
2
10
0
10
2
10
4
0
5
x 10
3
W
*
Z
*
A1
A2
Classical
approach
New
approach
Table 3. López et al. (2014) data employed in the
present study.
Run
Q
p*
h
p*
u
p*
/u
c
z
l/s
cm 
cm
U1a
69.2
20.4
0.82
12.51
U1b
70.1
20.1
0.85
12.34
U1c
68.8
20.0
0.84
12.08
U2a
69.7
20.0
0.85
11.31
U2b
69.8
20.1
0.85
11.16
U3
69.4
19.8
0.85
13.19
U4
70.2
20.1
0.85
10.74
U5
54.3
18.3
0.73
7.73
U6
54.0
18.2
0.73
7.41
U7
54.6
18.1
0.74
11.30
* Subscript “p” denotes peak condition.
Figure 5 shows the comparison between the ex
perimental results by López et al. (2014) and scour
depth computed with the corrected Eq. (12). Except
run U3, all experiments by López et al. (2014) are
properly interpreted by the proposed model with er
rors less than 25%. Differences between the two
studies are attributable to viscous effects (D
*
).
Figure 5. Comparison with literature data.
5 CONCLUSIONS
The pier scour caused by several different flood hy
drographs was analyzed introducing the idea of an
effective work by the flow on the sediment bed
around the pier. Different possible formulations for
the dimensionless flow work and the corresponding
dimensionless parameters that govern scouring were
tested with laboratory experiments conducted in a
novel installation able to reproduce any hydrograph
with high precision in a flume.
The results highlighted that the effective dimen
sionless work W
*
has a high predictive capacity of
scour caused by any hydrograph, as the relation be
tween the effective dimensionless work
and the di
mensionless scour depth is unique. In particular, the
proposed model provides a good performance allow
ing for a straightforward prediction of maximum
scour depth.
The presented experiments represent a first step
in the study of the interaction of more complex flood
hydrographs with fluvial structures. In the future, we
plan to explore: (1) Hydrographs with multiple peaks
and shapes closer to reality for testing the effects of
flow acceleration on scour depth; (2) The role of
sediment properties on scour, as well as the scour
behavior under flood waves with significantly higher
flow velocities to improve the understanding regard
ing the reliability of the proposed formulation.
ACKNOWLEDGEMENTS
Presented results are part of the research project
Fondecyt 1150997. Financial support by Red Doc
toral REDOC.CTA and by CRHIAM FondapCenter
15130015 is greatly acknowledged. Academic ex
change was possible through the financial support of
the German academic exchange service DAAD, the
Chilean research council CONICYT through grant
PCCI12027, and the Erasmus Mundus project Elarch
Grant Nr. 552129EM120141ITERA MUN
DUSEMA21.
REFERENCES
Åström, K. J. & Hägglund, T. (1995). New tuning methods for
PID controllers. In Proceedings of the 3rd European Con
trol Conference: 245662.
Bagnold, R. (1966). An approach to the sediment transport
problem from general physics. U.S. Geological Survey
Professional Paper. 422J.
Franzetti, S., Larcan, E., and Mignosa, P. (1982). Influence of
scour tests duration on the evaluation of ultimate scour
around circular piers. Proceedings of the International Con
ference on Hydraulic Modeling of Civil Engineering Struc
tures, Coventry, UK: 381–396.
Guo, J. (2014). Semianalytical Model for Temporal Clear
water Scour at Prototype Piers. Journal of. Hydraulic Re
search. 52(3): 366374.
Lai, J., Chang, W., and Yen, C. (2009). Maximum Local Scour
Depth at Bridge Piers under Unsteady Flow. Journal of Hy
draulic Engineering. 135(7): 609–614.
Lee, S. & Sturm, T. (2009). Effect of Sediment Size Scaling on
Physical Modeling of Bridge Pier Scour. Journal of Hy
draulic Engineering. 135(10): 793–802.
López, G., Teixeira, L., OrtegaSánchez, M., and Simarro,
G. (2014). Estimating Final Scour Depth under ClearWater
Flood Waves. Journal of Hydraulic Engineering. 140(3):
328–332.
0
5
10
15
0
5
10
15
z measured [cm]
z computed [cm]
López et al. (2014)
This study
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01 Jan 2015Abstract: Magister en Ciencias de la Ingenieria con mencion en Ingenieria Civil Universidad de Concepcion 2015
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01 Jan 1988
1,815 citations
••
01 Jan 19661,702 citations
"A model for scour around bridge pie..." refers methods in this paper
...In this study, the concept of an effective flow work (Bagnold 1966, Lai et al. 2009) for scour depth prediction is adopted....
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••
Abstract: Scour related to bridge hydraulics received much attention in the past decade, including its relation to flood hydrology and hydraulic processes in addition to steady flow. This paper presents new research on bridge pier and abutment scour based on a large data set collected at ETH Zurich, Switzerland. In total six different sediments were tested, of which three were uniform. Also a large variety of scour elements were considered, from 1 to 60% of the channel width, and flow depths ranging from 1 to about 40% of the channel width. Using similarity arguments and the analogy to flow resistance, an equation for temporal scour evolution is proposed and verified with the available literature data. The agreement of the present scour equation with both the VAW data and the literature data were considered sufficient in terms of river engineering accuracy, provided limitations relating to hydraulic, granulometric, and geometrical parameters are satisfied. These limitations are discussed and refer particularly to e...
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Abstract: Local pier scour experiments were performed in the laboratory to investigate the effect of relative sediment size on pier scour depth using three uniform sediment sizes and three bridge pier designs at different geometric model scales. When the data from a large number of experimental and field investigations are filtered according to a Froude number criterion, the effect of relative sediment size on dimensionless pier scour depth is brought into focus. The choice of sediment size in the laboratory model distorts the value of the ratio of pier width to sediment size in comparison with the prototype which in turn causes larger values of scour depth in the laboratory than in the field. This model distortion due to sediment size is shown to be related to the scaling of the largescale unsteadiness of the horseshoe vortex by studying the relevant time scales of its coherent structure upstream of a bridge pier using acoustic Doppler velocimeter measurements. Observations of sediment movement, probability distributions of velocity components, and phaseaveraging of velocity measured upstream of a bridge pier reveal properties of coherent motions that are discussed in terms of their contribution to the relationship between dimensionless pier scour depth and the ratio of pier width to sediment size over a large range of physical scales.
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...(12) were corrected using the formulation proposed by Lee & Sturm (2009)....
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01 Jan 1995103 citations
"A model for scour around bridge pie..." refers methods in this paper
...…for differences between specified and measured discharges were made on the frequency of the pump motor, based on a ProportionalIntegral (PI) controller (Åström & Hägglund 1995) set up by means of a programmable logic controller (PLC) and a variablefrequency drive (VFD), as schematized in Figure 1....
[...]