Scaling analysis of pier-scouring processes
1
2
Nian-Sheng Cheng1, Yee-Meng Chiew2 and Xingwei Chen3
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ABSTRACT
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This study presents a scaling analysis of the time development of clear-water scour
7
depth at bridge piers. It shows that the widely-used exponential formula can be
8
theoretically derived with scaling arguments. The derivation provides connections
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between the physical pier-scouring process and two empirical constants used in the
10
formula. The dependence of the two constants on sediment coarseness is calibrated
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using published laboratory data with scour duration ranging from 49 to 1094 hours.
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The analysis presented is restricted to the conditions including steady flow, clear-
13
water scour, narrow circular cylindrical pier and uniform sediment with low
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coarseness.
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Keywords: Scour; Bridge piers; Sediment transport; Time effects; Time development.
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1
School of Civil & Environmental Engrg., Nanyang Technological University, Nanyang Avenue, Singapore
639798. Email: [email protected] (corresponding author)
2
School of Civil & Environmental Engrg., Nanyang Technological University, Nanyang Avenue, Singapore
639798. Email: [email protected]
3College
of Geographical Science, Fujian Normal University, Cangshan, Fuzhou, 350007, China. Email:
[email protected]
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Introduction
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The temporal evolution of clear-water scour depth around bridge piers has been
20
intensively studied in the past decades. Examples of the published works are those
21
by Raudkivi and Ettema (1985), Hoffmans and Verheij (1997), Melville and Chiew
22
(1999) and Oliveto and Hager (2002). To describe the temporal variation of the local
23
scour depth under clear-water conditions, an exponential function could be used, as
24
shown in Table 1, which summarizes various formulas that involve exponential
25
functions. This study focuses on the exponential function of the following form
  t n 
ds
 1  exp  C   
dse
  T  
(1)
26
where ds is the maximum scour depth at time t, dse is the equilibrium value of ds, T is
27
a time scale, C is a coefficient and n is an exponent. Eq. (1) is often used as an
28
empirical representation of the temporal development of clear-water scour depth
29
around bridge piers. However, it could be developed analytically based on scaling
30
arguments. This is described in detail in the following section.
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32
Scaling consideration
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The consideration presented here is restricted to the following conditions:
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(1) The pier is circular cylindrical;
35
(2) The channel bed comprises uniform cohesionless sediment;
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(3) The sediment size is not coarse in comparison with the pier width, with d50/Dp <
37
0.04, where d50 is the sediment median diameter and Dp is the pier diameter;
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(4) The pier is relatively narrow with h/Dp > 1.4 (Melville and Coleman 1997),
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where h is the flow depth;
40
(5) Clear-water scour, which occurs at u*/u*c < 1, where u* is the undisturbed shear
41
velocity on the approach bed and u*c is the critical shear velocity for incipient
42
sediment motion; and
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(6) The viscous effect is negligible.
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First, two extreme scenarios are discussed. They are the very beginning stage
45
of the scour process and the final stage of scouring as the scour hole approaches the
46
equilibrium state. At the onset of scour, the volume of the scour hole Vs is
47
characterized in terms of the sediment size, say d50. For example, when the first
48
sediment particle is picked up from the sediment bed around a pier, Vs would be
49
equivalent to (/6)d503. When the first few layers of particles are entrained, the
50
scour hole may not be clearly shaped but Vs could be still measured in the order of
51
d503. On the other hand, when approaching the equilibrium stage, the volume of the
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scour hole Vse will be in the order of Asedse, where Ase is the planar area of the
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channel bed that covers the scour hole. If Ase is proportional to dse2, Vse would be in
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the order of dse3.
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At the intermediate stage of the scour development, the volume of the scour
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hole Vs could be measured in the order of Asds. In a typical three-dimensional space,
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As could be scaled with ds2, i.e. As  ds2, and thus Vs  ds3. However, this scaling may
58
not hold generally. For example, if the rate of As approaching Ase is greater than that
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of ds2 approaching dse2, then As  dsp with p < 2. Otherwise, if the rate of As
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approaching Ase is less than that of ds2 approaching dse2, then As  dsp with p > 2.
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Similarly, if the rate of change of Vs is greater than that of ds3, then Vs  dsq with q < 3,
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and if the rate of change of Vs is less than that of ds3, then Vs  dsq with q > 3.
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The above analysis can be further illustrated by considering the development
64
of a two-dimensional clear-water scour beneath a submerged pipeline where the
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scour hole grows only in the longitudinal and vertical direction. Such a type of scour
66
is commonly encountered in laboratory studies with a submarine pipeline conducted
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in a narrow flume (e.g. Chiew 1991). With this type of scour, if the longitudinal
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dimension of the scour hole is proportional to ds, then As  ds and Vs  ds2. Therefore,
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for this case, As increases at a rate faster than that of ds2, and Vs increases at a rate
70
faster than that of ds3. This is understandable because this type of scour hole only
71
evolves in the longitudinal and vertical directions. However, it should be noted that
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scour around a bridge pier is much more complicated and relevant information is not
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readily available for a general description of the variation of As and Vs with ds.
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Without loss of generality, it is assumed here that Vs  dsq or ds  Vs1/q, where q
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characterizes the dependence of the scour volume on the scour depth, and ds scales
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with d50 at the beginning of scour and dse at the stage of equilibrium.
In the intermediate phase of a local scour, it can be assumed that Vs increases
77
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linearly with increasing t . By noting that ds  Vs1/q, one gets
ds  tn
(2)
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where n = 1/q. Based on the foregoing analysis, n = 1/3 if the rate of change of Vs is
80
the same as that of ds3. This n-value is the same as that suggested by Franzetti et al.
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(1982) in their exponential formula of scour evolution (see Table 1). However, n
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would increase for the case of the rate of change of Vs being greater than that of ds3,
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and vice-versa.
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Next, possible scales applicable to ds and tn are examined. To scale ds, certain
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characteristic lengths are needed. By noting that the volume of the scour hole at the
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beginning of scour scales with d503, it is reasonable to consider d50 as a normalizing
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length scale when t is small. In comparison, when t is large such that ds is close to dse,
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the dimension of the scour hole would be in the order of dse. Therefore, ds  d50 for
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small t and ds  dse for large t. To facilitate subsequent analyses, ds is replaced with
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the scour depth defect, dse-ds. Furthermore, the time scale for the flow around a
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cylinder can be characterized with Dp/U (e.g. Franzetti et al. 1982; Melville and
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Chiew 1999). By scaling t with T (= Dp/U) and (dse - ds) with d50 (for small t) and dse
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(for large t), one gets the scaling relationships as follows:
t*  f  
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(for small t*)
where t* = (t/T)n and  = (dse – ds)/d50, and
t*  g   
(for large t*)
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where  = (dse – ds)/dse.
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Using Eq. (3) and differentiating t* with respect to ds,
dt*
1 df  
(for small t*)

dds
d50 d
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(4)
(5)
Using Eq. (4) and differentiating t* with respect to ds,
dt*
1 dg   
(for large t*)

dds
dse d
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(3)
(6)
Both df/d and dg/d are negative because dds/dt* > 0 and thus dt*/dds > 0.
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In the above consideration, t* is taken to be either small or large. However,
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for the intermediate time zone in which t* is neither very small nor very large, an
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interpolation can be derived by applying both Eqs. (5) and (6). As a result,
1 df   1 dg   

d50 d
dse d
(7)
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The above interpolation is similar to that applied for studying the velocity
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distribution in the inertial sublayer near the wall (Kundu et al. 2004). Multiplying
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both sides of Eq. (7) with (dse – ds),

df  
d

dg   
(8)
d
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By noting that each side of Eq. (8) only involves a different variable, and  and  are
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not necessarily interrelated, both sides should be equal to a constant. If taking

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df  
d
 c1
(9)
where c1 = a negative constant because df/d < 0, and integrating Eq. (9),
 dse  ds
t
   c1 ln 
T
 d50
n

  c2

(10)
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where c2 = constant. Similarly, taking the right-hand-side of Eq. (8) to be c1 and
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integrating,
 dse  ds 
t
  c3
   c1 ln 
T
 dse 
n
(11)
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Since ds = 0 when t = 0, c3 = 0. Comparing Eqs. (10) and (11), one gets c2 =
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c1ln(d50/dse). Using Eq. (11),
 1  t n 
ds
 1  exp    
dse
 c1  T  
(12)
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which is the same as Eq. (1) if C is taken as -1/c1. From the derivation, it follows that
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Eq. (12) is applicable for the intermediate time zone. However, Eq. (12) also implies
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that ds approaches zero when t is small and dse when t is large.
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Eq. (12) appears very similar to those reported previously by Franzetti et al.
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(1982), Hoffmans and Verheij (1997) and Sumer and Fredsoe (2002). However,
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coefficient C, exponent n and time scale T have been evaluated differently in those
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studies. For example, Franzetti et al. (1982) obtained two empirical values, n = 1/3
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and C = 0.021-0.042, while Sumer and Fredsoe (2002) used n = 1 and C = 1 and
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proposed that T is related to the Shields parameter. In the study by Hoffmans and
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Verheij (1997), the exponential formula is applied to several cases of clear-water
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scour with fixed C and T but varying n.
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With the scaling argument, n could be viewed as a value that characterises
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the evolution of the scour hole in a three-dimensional space or the resulting
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dependence of the scour volume on scour depth. Different types of local scour
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clearly differ from each other in terms of both the local flow field and sediment
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entrainment processes. However, each particular clear-water scour, e.g., that around
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a circular cylindrical pier, could be characterized largely by its inherent vortex
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structure, which dominates the scouring processes and also remains similar even
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under different flow conditions (Ettema et al. 2011; Melville and Coleman 1997). This
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suggests that n could be taken as a constant for clear-water scour around a circular
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cylindrical pier.
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To obtain C, one may first consider what happens if Dp is comparable to and
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even smaller than d50. For such cases, it is expected that ds is very small. This
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suggests that C may increase with increasing d50 in comparison with Dp or sediment
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coarseness d50/Dp as defined by Melville and Chiew (1999). In the following,
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published experimental data will be used to explore to what extent n varies and how
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C is related to d50/Dp.
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Data sources and calibration
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In the derivation presented above, both small and large t are considered. Therefore,
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it is necessary to calibrate Eq. (12) using long time data series that adequately
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quantify the evolution of the scour depth in all stages of development. To this end,
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three sources of data are employed, namely, Miller (2003), Alabi (2006) and Lanca et
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al. (2010). In all these studies, the test duration for a single run lasted from 49 to
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1094 hours, which is much longer than those reported in many other studies. Only
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two sets of Miller’s (2003) data are selected for the present analysis because the rest
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were either not in the range of narrow pier or affected by the presence of fine
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suspended sediment in the tests. Altogether 10 datasets are utilized to perform the
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curve-fitting with Eq. (12). They are summarized in Table 2, which also includes the
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values of the three parameters, dse, n and C, derived from the curve-fit. Fig. 1 shows
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the comparison of Eq. (12) with each set of the data. Some cases do not reach the
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equilibrium state, which may cause uncertainties in the determination of the
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parameters, in particular the value of C. This is because n is not strongly affected by
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data points that represent low rates of scour when the scour hole approaches the
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equilibrium state.
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correlates to the principal phase of the pier-scouring process (Ettema 1980). This
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issue has also been discussed by Simarro-Grande and Martin-Vide (2004).
Actually, n intrinsically measures the rate of scour which
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To understand what factors could possibly affect n, two dimensionless
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parameters, sediment coarseness d50/Dp and flow shallowness h/Dp, as defined by
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Melville and Chiew (1999), are used for analysis. The results show that the
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dependence of n on d50/Dp is expectedly much clearer than that on h/Dp because
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only narrow piers (i.e. h/Dp > 1.4) are considered here. The variation of n with d50/Dp,
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which is plotted in Fig. 2, shows that n is almost constant (n  0.22) for 0.006 <
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d50/Dp < 0.038 and increases with decreasing d50/Dp for d50/Dp < 0.006. Raudkivi and
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Ettema (1985) observed that for 0.008 < d50/Dp < 0.033 (denoted by the two vertical
168
bars in Fig. 2), sediment grains are entrained mainly by the downflow from the
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groove around the upstream perimeter of the pier. When d50/Dp < 0.008, the
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sediment size is considered fine, for which grains were observed to be entrained
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both by the downflow and the horseshoe vortex (Ettema et al. 1998; Lee and Sturm
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2009; Raudkivi and Ettema 1985). Based on this argument, one may hypothesize
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that the horseshoe vortex plays a significant role in the entrainment of sediment
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particles during the scouring process if a large value of n can be derived from the
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observed variation in the scour depth.
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Also superimposed in Fig. 2 are the n-values for d50/Dp < 0.04, which are
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derived from the curve-fit with the data reported by Ettema (1980). By varying the
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pier-sediment-flow configuration, Ettema conducted systematic tests under clear-
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water pier-scour conditions. Fig. 2 shows that the n-values obtained with Ettema’s
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data are close to the others in spite of his short-duration tests. Furthermore, an
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additional n-value is calculated based on the observed variation of the scour volume
182
with scour depth (rather than the curve-fit with the time series of scour depth). The
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measurement of the scour volume was reported by Link et al. (2008), who
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monitored the geometrical development of a scour hole using a laser distance sensor.
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Their result shows that Vs/Vse can be approximated as (ds/dse)2 for the condition of
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Dp = 0.2m, d50 = 0.26mm, and h = 0.3m. This implies that n = ½ for d50/Dp = 0.0013.
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Fig. 2 shows that n increases significantly when d50/Dp is small, but the change is only
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supported by the few data points available for d50/Dp < 0.002. The small value of
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d50/Dp is difficult to be realized in a typical laboratory setup, but can be often
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encountered in field. Therefore, the n-variation for low ratios of d50/Dp can be
191
further examined with field data.
192
Fig. 3 shows that C increases monotonically with increasing sediment
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coarseness d50/Dp, where the best-fit curve can be expressed as C = 52(d50/Dp)1.5.
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This result implies that taking C to be a constant, as was assumed by Franzetti et al.
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(1982), only applies to a very limited range of d50/Dp. The behavior that may be
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deduced from Fig. 3 is to be expected from the earlier discussion in this note. The
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increasing function reveals that a much longer time is needed to reach equilibrium
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for small values of C. In other words, in situations when the bed sediment particles
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are very small when compared to the pier diameter, say d50/Dp < 0.005, the time
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needed to reach equilibrium is very long. This can be explained by considering two
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circular cylindrical bridge piers with the same Dp subjected to the same shear
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velocity ratio, u*/u*c and undisturbed approach flow depth under clear-water
203
conditions. If one of the piers is founded in a fine sediment bed while the other in a
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much coarser one, the d50/Dp of the former pier is correspondingly much smaller
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than the latter, with a consequential lower and higher C, respectively. Using the
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graph in Fig. 3, one will expect the scour hole associated with the fine sediment will
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take a much longer time to reach equilibrium than that with the coarse one. This is
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true because the strength of the dominant scour mechanism, namely the downflow
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and horseshoe vortex of both the piers is similar since the primary variables, i.e., pier
210
diameter, flow depth, flow velocity, etc. that affect it are similar. However, since the
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finer sediment particles are subjected to the same scour mechanism as its coarser
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counterpart, it is only reasonable to surmise that the former can be entrained more
213
readily by the fluctuating component of the pier-induced vorticity, even at the later
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stages of scour hole development, rendering a continual degradation of the scour
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hole resulting in a lengthening of the time needed to reach equilibrium. On the
216
other hand, the coarser sediment particles likely are more resistant to the pier-
217
induced flow in the scour hole.
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Limitations
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This study presents a scaling analysis of the time development of scour depth under
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a series of restricted conditions. Discrepancies would exist when the proposed
223
formula, Eq. (12), is applied under more complicated conditions, for example, for
224
high flow intensity of u* > u*c, or with a sediment bed of non-uniform mixture. When
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the flow velocity is much greater than its critical value for incipient sediment motion,
226
the scour depth reaches its equilibrium within a short duration, and also exhibits
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large fluctuations induced by bed forms propagating through the scour hole. In such
228
a case, C would be much greater than that applicable for clear-water scour and also
229
vary depending on the ratio of u*/u*c. If a sediment bed consists of non-uniform
230
grains, it may take a shorter time to reach the equilibrium scour depth and thus a
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larger C would be expected. For this case, both d50 and the geometric standard
232
deviation, which characterises the sediment mixture, need to be considered in the
233
determination of C and n.
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In the scaling analysis, the variation of the scour depth is considered with
235
reference to the equilibrium value, dse. However, the dse applied here is not provided
236
independently. Instead, it is obtained, together with C and n, by fitting Eq. (12) to
237
each dataset. Though the dependence of C and n on d50/Dp has been empirically
238
established, how to fix dse is not examined in the present study. As a result, if Eq. (12)
239
is used to calculate the scour depth, dse must be obtained first through curve-fit. This
240
makes impossible an independent calculation of the scour depth solely based on Eq.
241
(12).
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Conclusions
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Using scaling arguments, the exponential formula is re-examined and theoretically
246
derived for the description of the temporal development of the maximum scour
247
depth around bridge piers. This provides an impetus toward an improved
248
understanding of the otherwise purely empirical approach in the study of the
249
temporal development of clear-water pier-scour depth. The results show that the
250
two constants included in the formula are closely related to the physical processes
251
associated with the development of a pier-scour hole. The long-duration data series
252
available in published literature are then used to substantiate possible dependence
253
of the two constants on the sediment coarseness. The results show that both
254
coefficient C and exponent n in the exponential function are related to the sediment
255
coarseness d50/Dp. The analysis presented here is restricted to the conditions
256
including laboratory scale model, steady flow, clear-water scour, narrow circular
257
cylindrical pier and uniform sediment with low coarseness. This note concerns only
258
the change in the scour depth with reference to the equilibrium state.
259
260
References
261
262
Alabi, P. D. (2006). "Time development of local scour at a bridge pier fitted with a
collar." Ph.D Thesis, University of Saskatchewan, Saskatchewan, Canada.
263
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Chiew, Y. M. (1991). "Prediction of maximum scour depth at submarine pipelines."
Journal of Hydraulic Engineering-ASCE, 117(4), 452-466, 10.1061/(asce)07339429(1991)117:4(452).
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Ettema, R. (1980). "Scour at bridge piers." Ph.D Thesis, University of Auckland, New
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Ettema, R., Constantinescu, G., and Melville, B. (2011). "Evaluation of bridge scour
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Ettema, R., Melville, B. W., and Barkdoll, B. (1998). "Scale effect in pier-scour
experiments." Journal of Hydraulic Engineering-ASCE, 124(6), 639-642,
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Franzetti, S., Larcan, E., and Mignosa, P. (1982). "Influence of tests duration on the
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Hoffmans, G. J. C. M., and Verheij, H. J. (1997). Scour manual, A.A. Balkema,
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Lanca, R., Fael, C., and Cardoso, A. (2010). "Assessing equilibrium clear water scour
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Lee, S. O., and Sturm, T. W. (2009). "Effect of Sediment Size Scaling on Physical
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Link, O., Pfleger, F., and Zanke, U. (2008). "Characteristics of developing scour-holes
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Melville, B. W., and Coleman, S. E. (1997). Bridge scour, Water Resources
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Miller, W. (2003). "Model for the time rate of local sediment scour at a cylindrical
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Simarro-Grande, G., and Martin-Vide, J. P. (2004). "Exponential expression for time
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321
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Sumer, B. M., and Fredsøe, J. (2002). The mechanics of scour in the marine
environment, World Scientific, River Edge, N.J.
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Notation
327
The following symbols are used in this paper:
328
As
= scour area;
329
Ase
= equilibrium value of As;
330
C
= coefficient;
331
c1, c2, c3 = constants;
332
Dp
= pier diameter;
333
d50
= sediment median diameter;
334
ds
= maximum scour depth at time t;
335
dse
= equilibrium value of ds;
336
h
= flow depth;
337
n
= exponent;
338
T
= time scale;
339
t
= time;
340
t*
= (t/T)n;
341
U
= undisturbed depth-averaged velocity;
342
Uc
= critical depth-averaged velocity for incipient sediment motion;
343
u*
= undisturbed shear velocity;
344
u*c
= critical shear velocity for incipient sediment motion;
345
Vs
= scour volume;
346
Vse
= equilibrium value of Vs;
347
p
= exponent;
348
q
= exponent;
349
σg
= standard geometric deviation;
350

= (dse – ds)/d50; and
351

= (dse – ds)/dse.
352
353