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V1/Vc = 1 and 3; y1/a = 0.33, 1, and 3; D50 = 0.2 and 3 mm; able. The Sheppard and Miller (2006) equation was modified
and a = 2 in., 3 ft, and 33 ft. The results from this evaluation are to create the new S/M equation.
presented in the form of bar charts in Figures 13 though 20.
Negative scour depth predictions were set equal to zero in
Sheppard/Melville (S/M) Equation
the charts.
Figures 13 and 15 show scour depth predictions for scenar- The Sheppard and Miller (2006) and Melville (1997) equa-
ios comprising clear-water to live-bed transition (threshold) tions were melded and slightly modified to form a new equa-
flows (V1/Vc = 1), fine sand (D50 = 0.2 mm), and two different tion referred to here as the S/M equation. The modifications
flow depth to pier width ratios (y1/a = 3, 0.33). Figures 14 and consisted of:
16 are a parallel set of results for live-bed conditions (V1/Vc = 3).
Similarly, Figures 17 through 20 apply to situations with coarser 1. Changing the 1.75 coefficient to 1.2 in the term for f2 in
sediment (D50 = 3 mm). Equation 23,
The equations used in producing the results shown in 2. Changing the value of V1/Vc where local scour is initiated
Figures 13 through 20 span the period from 1949 to 2006 in from 0.47 to 0.4, and
their development and publication. Improvements in the 3. Modifying/simplifying the manner in which the live-bed
understanding of local scour processes and scour hole peak velocity is computed.
development during this time period resulted in improve-
ments to the equilibrium scour predictive equations/methods.
The resulting equation is presented in Table 5.
For example, several of the earlier equations predicted
These changes improved the accuracy of the predictions
negative scour depths for some of the input conditions.
for both laboratory and field data. However, this equation
Also, the differences between the predictions become less
underpredicts some of the measured field data at very low
with time.
velocities (i.e., low values of V1/Vc), most likely due to rela-
Variations in the predictions of local scour for different pier
tively large sediment size distributions (large g). The under-
sizes ("laboratory" to "typical field" to "very large field") are
predictions are illustrated in Figures 21 and 22, which show
reported. Some methods predict scour depth ratios decreas-
before- and after-modification upper bound curves for labo-
ing with increasing pier size; others show constant values of
ratory and field data, respectively. The scour depths in this
scour depth ratio from laboratory to field, with one equation
range of flow velocities are, however, very small and therefore
by Coleman (1971) showing larger normalized scour depths in
are not likely to affect prediction of design scour depths. Also,
the field than in the laboratory.
the reported scour depths in this range of V1/Vc seem large for
These plots help identify those equations that produce un-
the magnitude of the flow velocities (i.e., the accuracy of these
realistic results for prototype-scale piers and thus aid in elimi-
data is questionable).
nating such equations from further consideration. The regime
equations of Inglis (1949), Ahmad (1953) and Chitale (1962)
yield negative scour depths in some cases. The Coleman (1971) Equation in HEC-18
equation yields an unrealistic trend with increasing pier size
No attempt was made to modify the scour equation in the
and therefore was eliminated. Several other equations predict
current version of HEC-18 (Richardson and Davis 2001)
unreasonably high normalized scour depths (Inglis 1949, because it does not properly account for the physics of the
Ahmad 1953, Chitale 1962, Hancu 1971, and Shen et al. 1969) local scour processes. That is, all of the known local scour
and were eliminated. This process left 17 methods/equations mechanisms are not accounted for with the dimensionless
for the final analysis. groups in this equation. The equation does, however, contain
a wide-pier correction factor developed by Johnson and
Torrico (1994). Predicted versus measured scour depth plots
Modifications to Equilibrium Scour
using this equation are shown in Figure 23 (laboratory data)
Predictive Methods
and Figure 24 (field data). In general, the wide-pier correc-
One of the objectives of this study was to determine if any tion decreases the magnitude of the predicted scour depths.
of the predictive equations could be modified to improve their The wide-pier correction does, however, increase the number
accuracy. The overall accuracy of most of the equations could of underpredictions in both the laboratory and field data. The
be improved by adjusting one or more of their coefficients. overall error for the dimensional scour is reduced with the
However, in almost every case this adjustment increased wide-pier correction factor, but the error for the normalized
underprediction, which for design equations is not accept- scour is increased. The wide-pier correction factor also creates
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Table 5. The S/M local scour equations.
Reference Equation Notes No.
ys V1
2.5 f1 f 2 f 3 for 0.4 1.0
a* Vc
V1 Vlp V1
1
ys Vc Vc Vc V1 Vl p
f1 2.2 2.5f 3 for 1.0
a* Vl p Vlp Vc Vc
1 1
Vc Vc
ys V1 Vl p
2.2 f1 for
a* Vc Vc
0.4
y1 a* e ffective diameter
f1 tanh
a* pr ojected width * shape factor
2
Shape factor =1, circu lar
V1
S/M f2 1 1. 2 l n 4 25
Vc
= 0.86 + 0.97 , square
4
a* = flow skew angle in radians
D50
f3 1 .2 0.13
a* a*
0 .4 10.6
D50 D50
Vl p1 5Vc
Vl p 2 0.6 g y1
Vl p1 for Vl p1 Vl p 2
Vl p
Vl p 2 for Vl p2 Vl p1
3
Data
2.5 Sheppard & Miller 2006
S/M
2
ys/a
1.5
1
0.5
0
0 0.2 0.4 0.6 0.8 1
V1/Vc
Figure 21. Measured laboratory data at low velocities
compared to the upper limit of Sheppard and Miller (2006)
and S/M equations.
