UFL/COEL90/019
LOCAL STRUCTURE INDUCED SEDIMENT SCOUR
By
D. Max Sheppard
Department of Coastal and Oceanographic Engineering
and
Alan Wm. Niedoroda
Hunter Services
March 1990
LOCAL STRUCTURE INDUCED SEDIMENT SCOUR
D. Max Sheppard'
and
Alan Wm. Niedoroda2
March 1990
INTRODUCTION
When a structure is placed in the vicinity of the water bottom it
will alter the local flow field. This in turn will modify the
bottom shear stress near the structure and can affect the local
sediment transport (erosion/accretion). In general, the shear
stress is increased resulting in local erosion or scour. The
scour that results from the flow modification due to the
structure is called local structure induced scour and is the
topic of this chapter.
The extent and volume of scour depends on the shape and size of
the structure, it's location relative to the bottom, the nature
of the primary flow, and the sediment parameters. The flow field
in the vicinity of even the most simple of structures is complex
and impossible to analyze analytically for situations of
practical significance. Researchers in this field have attempted
to obtain a general understanding of the physics of these
processes through flow visualization in laboratory experiments
and by analyzing laboratory and field data. The study that
resulted in this publication collected and analyzed published
laboratory and field data uncovered in an extensive computer and
manual literature search. New empirical equations for
dimensionless maximum scour depth as functions of independent
dimensionless groups involving structure, sediment and flow
variables were developed. A comparison of these equations with
others in the literature is presented in appendix B. These
equations form the basis of the computer program that accompanies
this chapter.
1 Department of Coastal & Oceanographic Engineering,
University of Florida, Gainesville, Florida, 32611.
2 Hunter Services, Gainesville, Florida, 32602.
In spite of the vast number of technical publications on this
subject (see the bibliography in Appendix A) there are still many
practical situations that have not been investigated or at least
not to the point of producing a useable solution. Some of the
more important aspects of the structure induced scour problem
that need further work are described in appendix C.
This chapter deals with the specific structural shapes where
sufficient data exists to predict scour depths and volumes. The
structural elements treated here are vertical cylinders,
horizontal cylinders, vertical elongated cylinders/piers, and
vertical rectangular crosssection piers. In addition, an
attempt has been made to compute scour depths and volumes for
vertical cylinder groups even though little quantitative
information exists for this situation. It should also be pointed
out that since most of the available data on structure induced
scour was collected in the laboratory, the range of the important
dimensionless groups is less than desirable for use in predicting
scour in the field. For example, it is not possible to achieve
the same flow Reynolds numbers in the laboratory as those
experienced in the field. The computer program checks the values
of these parameters to see if they fall within the general range
of the data. If the input data is such that one or more of the
parameters is out of bounds the program changes one of the
variables until the parameters are all within range and then
computes the scour depth and volume. The output file gives the
modified input conditions with the corresponding values of scour
depth and volume. The program then determines if the input
variables are such that they fall within the extended or
extrapolated range (see figures in appendix G). If the data is
within this extrapolated range of validity then the scour depth
and volume are computed. If the data is out of this range then
the velocity is reduced until it comes within range and the scour
information computed and written to the output file along with
the modified velocity. Thus, even if the input data is out of
the extrapolated range the results of the above two computations
will be helpful in estimating the actual scour.
DESCRIPTION OF PROCESSES
The process of structure induced scour is somewhat different for
waves than for steady currents. The general conclusion by
several investigators (e.g. Eadie (1986)) is that the largest
scour depths occur with steady currents. Waves alone generate
scour but with lesser depths than "equivalent" currents. The
addition of waves to currents will accelerate the rate of scour
but will have little effect on the maximum scour depth. Since
most circumstances of practical significance will involve both
waves and currents this simplifies the problem of computing scour
depth and volume. That is, one need not be too concerned with
the duration of the storm since the presence of waves will, in
In spite of the vast number of technical publications on this
subject (see the bibliography in Appendix A) there are still many
practical situations that have not been investigated or at least
not to the point of producing a useable solution. Some of the
more important aspects of the structure induced scour problem
that need further work are described in appendix C.
This chapter deals with the specific structural shapes where
sufficient data exists to predict scour depths and volumes. The
structural elements treated here are vertical cylinders,
horizontal cylinders, vertical elongated cylinders/piers, and
vertical rectangular crosssection piers. In addition, an
attempt has been made to compute scour depths and volumes for
vertical cylinder groups even though little quantitative
information exists for this situation. It should also be pointed
out that since most of the available data on structure induced
scour was collected in the laboratory, the range of the important
dimensionless groups is less than desirable for use in predicting
scour in the field. For example, it is not possible to achieve
the same flow Reynolds numbers in the laboratory as those
experienced in the field. The computer program checks the values
of these parameters to see if they fall within the general range
of the data. If the input data is such that one or more of the
parameters is out of bounds the program changes one of the
variables until the parameters are all within range and then
computes the scour depth and volume. The output file gives the
modified input conditions with the corresponding values of scour
depth and volume. The program then determines if the input
variables are such that they fall within the extended or
extrapolated range (see figures in appendix G). If the data is
within this extrapolated range of validity then the scour depth
and volume are computed. If the data is out of this range then
the velocity is reduced until it comes within range and the scour
information computed and written to the output file along with
the modified velocity. Thus, even if the input data is out of
the extrapolated range the results of the above two computations
will be helpful in estimating the actual scour.
DESCRIPTION OF PROCESSES
The process of structure induced scour is somewhat different for
waves than for steady currents. The general conclusion by
several investigators (e.g. Eadie (1986)) is that the largest
scour depths occur with steady currents. Waves alone generate
scour but with lesser depths than "equivalent" currents. The
addition of waves to currents will accelerate the rate of scour
but will have little effect on the maximum scour depth. Since
most circumstances of practical significance will involve both
waves and currents this simplifies the problem of computing scour
depth and volume. That is, one need not be too concerned with
the duration of the storm since the presence of waves will, in
most cases, assure that the maximum scour for the given
conditions will be reached.
When a steady current flows over a bottom as shown in Figure 1,
the boundary layer (water layer affected by the boundary)
encompasses the entire depth of the flow. This results in a
significant decrease in velocity with depth. When this flow
impacts a vertical cylinder the flow is brought to rest
(stagnates) along the leading edge of the cylinder producing a
"stagnation pressure". The stagnation pressure at any level is
proportional to the square of the free stream velocity at that
level. Since the velocity decreases with depth the stagnation
pressure along the leading edge of the cylinder decreases even
more dramatically (due to the velocity squared dependency)
resulting in a strong vertical pressure gradient. The pressure
gradient in turn generates a vortex with a horizontal axis as
shown in Figure 1. When viewed from above this vortex has the
appearance of a horseshoe and thus is called a horseshoe vortex.
The bottom shear stress and the near bottom turbulence generated
by this secondary flow is the main scour mechanism for steady
flow around blunt vertical structures.
A second, somewhat independent, scour producing flow process
exists due to flow acceleration around the structure and due to
flow separation on the structure. The flow moving around a
cylinder accelerates until it reaches the maximum breath of the
cylinder. This accelerated flow results in an increased bottom
shear stress which in turn can produce a scour depression. Once
the flow passes the maximum width of the cylinder it experiences
an increasing pressure with distance (adverse pressure gradient).
The fluid adjacent to the cylinder is slowed by the increasing
pressure and comes to rest at a point (line) called the point of
separation. Beyond the separation point the time mean flow is in
the opposite direction and is more turbulent and disorganized
than the upstream flow. The nature of the flow in the wake
region (region between the separation streamlines; see Figure 1)
depends on the Reynolds number based on the cylinder diameter for
the flow. For a range of Reynolds numbers an "organized"
heading of vortices known as the Karman Vortex Street occurs in
the wake. These vortices increase the bottom shear stress in
their vicinity and assist in maintaining sediment in suspension
thus promoting scour. Most researchers agree, however, that for
most steady flow situations around blunt vertical structures the
primary scour mechanism is the horseshoe vortex.
Shallow water wave induced flow is almost uniform in depth with a
very thin boundary layer near the bottom (see Figure 2). The
flow is unsteady and complex but since the flow is (near) uniform
the pressure gradient resulting from the variation in stagnation
pressure does not exist. The horseshoe vortex is minimal and
confined to the very thin boundary layer. The mechanism of flow
separation and wake formation discussed above applies here as
well. As the wave progresses and the flow direction reverses,
vortices and turbulence in the wake are swept back and forth
across the structure thus creating a complex, turbulent flow
field near the structure. Acceleration of the flow around the
cylinder along with the vorticies and turbulence generated by
flow separation are the primary sources of increased bottom shear
stress and scour for structures subjected to shallow water waves
only.
The discussion thus far has concentrated on the flow field and
the bottom shear stress (i.e. the shear stress exerted on the
bottom by the moving fluid). The processes by which sediment is
placed and maintained in motion are complex but at present it is
assumed that if the bottom shear stress exceeds a certain value
the sediment entrainment in the flow will occur. If the sediment
is made up of a range of particle sizes and densities the
critical shear stress (shear stress needed to initiate particle
motion) will vary from particle to particle. Thus for a given
bottom shear stress the smaller less dense particles may be in
motion in suspension (suspended transport), the medium size and
density particles may be moving along the bottom (bed load
transport) with the even larger and more dense particles
remaining stationary on the bottom. For a given water density,
viscosity, grain size, and density the critical shear stress can
be obtained from the modified Shields curve shown in Figure 3.
If the flow is fully developed and steady the bottom shear stress
can be related to the depth average velocity by the expressions:
u, h
U = 2.5u, In (3.31 )
u, ks
when < 5.0
U = 2.5u,. n ( 22
c 2.72z2
u. ks
when 5.0 < k 70.0
U = 2.5u, ln ([ 1oh
c ks
u ks
when > 70.0
U
Uc = critical depth average velocity
u, = i = critical time mean friction velocity
y pw
ks = roughness height of bed (bottom)
h = mean water depth
u = = water dynamic viscosity / mass density
p
= kinematic viscosity
zo = the turbulent roughness parameter (obtained from
the plot in Figure 4.
For more information on these expressions, see Sleath (1984).
The modified Shields curve, Figure 3, the above equations, and
the zo curve, Figure 4, are all incorporated into the scour
program and used to compute the critical depth average velocity.
Evolution of the scour hole near a vertical cylinder due to a
steady current can be described as follows. First consider the
"clear water scour" case. Clear water scour means that the
current is not sufficient to generate the critical bottom shear
stress away from the structure. The flow intensification and
enhanced turbulence adjacent to the structure does locally
produce bottom shear stresses above the critical entrainment
values. Thus, sediment is scoured near the structure and not
replaced from upstream. Assume that the sediment consists of a
cohesionless quartz sand with a relatively narrow range of
diameters and densities. Scour will continue until the scour
hole is sufficiently large to alter the flow and reduce the
bottom shear stress near the structure to below the critical
value. Scour will not proceed beyond this maximum depth unless
the flow, sediment or structure conditions change.
Next consider the case where the depth average velocity is
sufficient to exceed the critical bottom shear stress away from
the structure. This situation will be referred to as "live bed
scour". Once sediment motion occurs along the bottom, the bottom
boundary condition for the flow changes from a no slip condition
to one with the velocity of the sediment. This alters the
velocity distribution and the bottom shear stress. In addition,
there is a constant stream of sediment flowing into and out of
where
the scour hole. When the sediment flow into the scour hole is
equivalent to the flow out the equilibrium or maximum scour depth
has been reached. Experimental data indicates that there is at
least, a local maximum in the nondimensional maximum scour depth
(scour depth divided by structure diameter) just prior to
reaching the live bed scour condition. This is illustrated in
the sketch in Figure 5.
The flow around horizontal structures on or near the bottom is
similar to that for vertical structures but with some significant
differences. The case of a steady current over a horizontal
cylinder is shown in Figure 6. Flow separation occurs as in the
case of a vertical cylinder but the wake is unsymmetrical due to
the vertical velocity gradient. Once a gap between the cylinder
and the bottom exists the flow in this constriction will be
accelerated and the bottom shear stress increased. As for other
orientations, the scour hole will increase until the bottom shear
stress falls below the critical value. Mao (1986) solved for the
potential (inviscid, irrotational) flow around a horizontal
cylinder (pipeline). The scour depth was increased until flow at
the bottom in the scour hole reached the "free stream" value.
The free stream value was assumed to be that necessary to produce
the critical value of bottom shear stress. The approach is
promising but insufficient results were presented to provide
useful information. This potential flow problem was solved as
part of this study using a finite element analysis. The problem
was set up so as to allow a parameter study to be made. That is,
set up to allow different initial gaps between the cylinder and
the bottom. This would, in general, require the generation of a
new grid and starting over for each case considered. A
description of this analysis along with figures of the grid and
flow are included in appendix C.
When there are multiple vertical structures in close proximity to
each other there can be flow interaction which results in
additional scour. The existence and/or extent of this
interaction depends on the shape, size, spacing and orientation
of the structures. As the spacing between the structures is
reduced the group begins to "act" like a single porous structure
with the associated increased and somewhat homogeneous turbulence
within the structure along with a drop in pressure due to
blockage of the flow. The level of turbulence and the magnitude
of the bottom shear stress reaches a maximum at some spacing of
the individual structures. As the spacing is reduced beyond this
critical value the flow and turbulence is reduced and in the
limit, of course, goes to zero.
There is a range of spacings where the group is in effect a large
porous structure. The structure induced scour for this large
"group" structure is called dishpan scour after its dish shape
(see Figure 7). Shallow water offshore oil platforms have
experienced dishpan scour and reported this in the literature but
in a very qualitive way, Posey (1971), Chow (1977) and Chow
(1978).
METHOD OF ANALYSIS
The flow and sediment transport processes described above defy a
purely analytical treatment at this time. Flow visualization
studies and some carefully executed experiments have, however,
resulted in a reasonable understanding of the processes involved
and the important variables and dimensionless parameters. Armed
with this descriptive understanding of the important mechanisms a
dimensional analysis can be performed to obtain the pertinent
independent dimensionless groups for the problem. Such analyses
have been performed by several investigators (e.g. Baker (1978),
Eadie (1986)) each with similar results. A dimensional analysis
was performed in this study (see Appendix D) with results similar
to those obtained earlier. Data from a number of investigators
were reanalyzed in this study using the parameters developed in
this analysis in an attempt to obtain the best surface fit
possible. This analysis (as well as most of the others) resulted
in a large number of pertinent independent groups. Many of these
could not be considered since values for the variables in these
groups were not measured or at least not reported. Different
investigators used different parameters, however, and for the
most part only fit their own data. In this study three
parameters dimensionlesss groups) were settled on after numerous
attempts with two, three and four groups and combinations of
groups. The details of the physics of the problem were strongly
considered in selecting the final parameters. The parameters
chosen are:
de
Y =  = Dimensionless Maximum Scour Depth
D
X1 = 1 = Sediment Transport Regime Number
Uc
X = = Structure Aspect Ratio
X3 = Froude Number based on water depth
where de = maximum scour depth
D = diameter of structure
0 = depth average velocity
Uc = critical depth average velocity
h = water depth
g = acceleration of gravity.
Plots in the literature and plots made in this study suggested
that a cubic surface in four dimensions with all cross terms had
the right properties to fit the data. Least squares cubic
surface fit routines in four and five dimensions were developed
to analyze the data. The nineteen term cubic expressions
produced by the four dimension analysis are contained in the
accompanying scour program for analyzing those structural shapes
with sufficient data in the literature to produce reliable
coefficients. These equations have the following form:
2 3
Y = K1 + K2X1 + K3X1 + K4X1
2 3
+ K5X2 + K6X2 + K7X2
SK8X3 + K9X2 + K X3
+ KIlX1X2 + KI2XlX3 + KI3X2X3
2 2
14 1 2 12 13 1323
2 2
+ K XX +K XX
16 1 3 17 1 3
2 2
+ K18X2X3 + K9X2X3
where K1..K19 = coefficients determined by the least square
surface fit routine.
The data used for steady flow around vertical cylinders consisted
of laboratory results by Baker (1978), Shen (1966), Jain (1979)
and Chabert (1956) and field data by Arkhipov (1984). These data
as well as data used for the other vertical structural shapes are
given in Appendix E. Data for scour near vertical and horizontal
cylinders subjected to waves only was obtained from a recent
paper by Sumer etal (1989). This data is also included in
Appendix E.
When analyzing a complex structure or group of structures the
following philosophy must be adopted. First the structure must
be separated into its components. Some thought must be given as
to the proximity of these components and to how the flow field
associated with the individual components will interact. At this
point an equivalent model structure or group of structures
(constructed of shapes and orientations contained in the computer
program) must be created. Experience using the scour program
will improve the user's ability to model complex structures with
structure producing equivalent scour. Example problems are
presented in the scour program documentation that lead the user
from beginning to end in a calculation of maximum scour depths
and volume of sediment removed for actual DNR problems.
in a very qualitive way, Posey (1971), Chow (1977) and Chow
(1978).
METHOD OF ANALYSIS
The flow and sediment transport processes described above defy a
purely analytical treatment at this time. Flow visualization
studies and some carefully executed experiments have, however,
resulted in a reasonable understanding of the processes involved
and the important variables and dimensionless parameters. Armed
with this descriptive understanding of the important mechanisms a
dimensional analysis can be performed to obtain the pertinent
independent dimensionless groups for the problem. Such analyses
have been performed by several investigators (e.g. Baker (1978),
Eadie (1986)) each with similar results. A dimensional analysis
was performed in this study (see Appendix D) with results similar
to those obtained earlier. Data from a number of investigators
were reanalyzed in this study using the parameters developed in
this analysis in an attempt to obtain the best surface fit
possible. This analysis (as well as most of the others) resulted
in a large number of pertinent independent groups. Many of these
could not be considered since values for the variables in these
groups were not measured or at least not reported. Different
investigators used different parameters, however, and for the
most part only fit their own data. In this study three
parameters dimensionlesss groups) were settled on after numerous
attempts with two, three and four groups and combinations of
groups. The details of the physics of the problem were strongly
considered in selecting the final parameters. The parameters
chosen are:
de
Y =  = Dimensionless Maximum Scour Depth
D
X1 = 1 = Sediment Transport Regime Number
Uc
X = = Structure Aspect Ratio
X3 = Froude Number based on water depth
where de = maximum scour depth
D = diameter of structure
0 = depth average velocity
FIGURES
Side View
Figure 1
Surface
 ~!1 ~ .'hTI~~
Vortex
SHorsesh(
Top View
Initial Flow Field for a Steady Current Around a Vertical
Cylinder
Instantaneous
Velocity Profile
Due to Shallow
Water Wave
j>
Vortex Formed During
Previous Half Cycle
of Wave
.... '.r..r .,.. _. 1'
Instantaneous .
View of Streamlines
TOP VIEW
SIDE VIEW
Figure 2
Initial Flow Field for a Vertical Cylinder in a Wave Only
Environment
0
N
0)

II
mI
oS sgzo
S.=4 (s)v zo
4v
Figure 3 Modified Shields Parameter
Figure 4
0.04
0.04 1 1 i ll  I r m 1 1 1 1
0.03 
zo
0.02
Hydraulically
Smooth
0.011
0.01 I I  ; I 1 I I I 1 1
1 10 100 1000
U. ks
v
V
Turbulent Roughness Parameter Zo as a Function Reynolds
Number
I
CD
Maximum Scour at
LJ / Threshold Velocity
0
C.
2 )
Clear
S Waer LiveBed Scour
Scour
MEAN VELOCITY, U
Figure 5 Sketch of Equilibrium Scour Depth as a Function of Mean
Approach Flow Velocity
Small KC Number
L
kV
Large KC Number
v
L
L  I
Figure 6 Structure Induced Scour Near a Horizontal Cylinder for
Two Different Values of KeuleganCarpenter Number
Original Bottom
Scour
Dishpan
Scour
Figure 7 Dishpan Scour For a Group of Vertical Cylinders
APPENDICES
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10.0
6= h= 10 ft .
S8.0 =h= 5ft
z
4.0 5=h=l ft
"1
!
UL
2.0
0.0
0.0 5.0 10.0 15.0 20.0
CYLINDER DIAMETER, D (ft)
Figure G5 Extrapolated Ranges Of Validity Of StructureInduced
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Appendix B  Recommendations for Future Work
Appendix B
Recommendations For Future Work
Two areas of critical need were identified during the course
of this investigation. One deals with local scour depth and
volume, the other with global or dishpan scour depth and volume.
The majority of scour data reported in the literature is for
steady flow around vertical pilelike structures but even here
the data are sparse. This study synthesized laboratory and field
data for a variety of structural shapes and produced a predictive
equation for local scour depths that far exceeds previous
equations in range of applicability and accuracy. In addition,
for the first time, bounds on the use of a predictive scour
equation were established. These bounds provide a way of
determining where additional data is needed. For example, the
physics of local scour suggest that for a given structure,
sediment, and sediment distribution there should be an upper
limit to scour depth and volume as the depth mean velocity is
increased. The upper limits on velocities used in laboratory
experiments thus far have, for the most part, been controlled by
scaled practical limits of river flow velocities since much of
this work was done for scour near bridge piers. Thus the
existence of upper bounds on scour has not been established. For
most permitting situations encountered by DNR the geometries and
environmental conditions are so complex that accurate prediction
of flow velocities is very difficult. If the velocities where
maximum scour depths occur are within the range anticipated for
severe storm events then using the maximum depths and volumes for
the given structure and sediment conditions would be appropriate.
If on the other hand, the limiting scour depth is larger than
that anticipated under severe storm events better ways of
predicting flow velocities are needed. These scour depth bounds
need to be determined.
The second problem area is also one of importance to DNR due
to its potential impact on the stability of the beach/dune
system. If a beach structure is supported by a number of
vertical pilelike components, as is usually the case, there is,
in addition to local scour near each member, a global or dishpan
scour. Dishpan scour gets its name from its dish like shape and
extends beyond the structure in all directions a distance of
about half the structure diameter and with measured depths up to
15 feet. The collection of piles can be thought of as a single
"porous" structure with the dishpanshaped scour hole being the
scour associated with this large composite. The total scour is
then the sum of the dishpan scour and the local scour for the
individual piles. Dishpan scour is important both from the
standpoint of structural integrity and for the loss of sand from
the beach/dune system due to the large quantities of sediment
involved. Unfortunately, this phenomena is not well understood
and at present no technique exists for its quantification. Not
only is there a void in data for dishpan scour (only some few
papers giving rough estimates of scour hole size and depth with
no information on the environmental conditions causing the scour)
to date no one has even suggested a methodology for approaching
the problem. A pilot or exploratory study is needed to answer
such fundamental questions as; can the processes causing dishpan
scour be produced in laboratory scale experiments? if so can the
results be extrapolated to prototype scale conditions?, are
prototype field studies technically and economically feasible?
can an idealized analytical approach provide some insight into
the mechanisms causing the scour and help in designing laboratory
and/or field experiments?
Appendix C  Finite Element Analysis of Horizontal Members
Appendix D  Dimensional Analysis
Dimensional Analysis
The quantities that are important in structure induced scour
around vertical cylindrical piles are listed below along with
their symbols and dimensions.
Quantity Symbol Dimensions
1. Equilibrium scour depth de L
2. Cylinder diameter D L
3. Sediment diameter ds L
4. Mass density of sediment pg ML3
5. Mass dinsity of water Pw ML3
6. Water depth h L
7. Absolute viscosity of water p ML1T1
8. Depth mean average velocity U LT1
9. Depth mean average critical
velocity corresponds to the
threshold Shield's parameter U LT1
10. Wave length X L
11. Acceleration due to gravity g LT2
where
M = mass
L = length
T = time.
It is assumed that the waves will be depth limited. Thus,
specification of the water depth and wave length uniquely
determines the wave height and eliminates the need to include it
in the analysis.
Using the Buchingham n theorem we obtained the following
independent dimensionless groups:
de
l D
2 DpU
Ps
I3 p '
4 
ds
5TI =
D
5 F2
h
T6 =
x
TI =
7 D
8 
Uc
By manipulating and combining the above groups, we can obtain the
following independent groups that are physically more meaningful.
TI' = nI 1 = 
1 8
C
1 UD
TI' =T 
2 2 v
h
T' = hT
3 6 D
T = [q4T6]1
TI' =
U
Sediment Transport Regime number,
Pile Reynolds number,
Structure Aspect Ratio,
Froude Number based on water depth,
wave length to structure diameter ratio.
Appendix E  Data Analyzed and Curve Fit Procedures
Appendix E
Data Analyzed And Curve Fit Procedures
A least squares curve fit program for a cubic equation in four
dimensional space was used to analyze the data. The cubic
equation includes all of the cross product terms and thus has
nineteen terms. It has the following form:
Y = K + K X +
1 2 1
+ K X +
5 2
+ K X +
8 3
2
K X
3 1
2
K X
6 2
2
K X
9 3
3
+ K X
4 1
3
+ K X
+KX
7 2
+ K X
10 3
+ K XX
11 1 2
2
+ K XX
14 1 2
2
+ K XX
16 1 3
2
+ K XX
18 2 3
+ K X X +
12 1 3
2
+ K XX
15 1 2
2
+ K XX
17 1 3
2
+ K XX
19 2 3
where K ...K = coefficients determined by the least square
1 19
surface fit routine. The data from five different investigators
were reanalyzed to obtain the values of the dimensionless groups
used in this study. The reanalyzed data is given in Table E1.
K XX
13 2 3
Table E1. Scour Data for Vertical Cylinders
Subjected to Steady Currents
BAKER'S DATA
de/DIA U/Uc 1
1.254 0.306
1.402 0.255
1.615 0.170
1.615 0.064
1.615 0.017
1.459 0.217
1.459 0.294
1.270 0.387
1.361 0.562
0.738 0.802
1.008 0.730
1.320 0.646
1.197 0.679
1.344 0.640
1.164 0.688
0.992 0.230
1.016 0.089
1.246 0.183
1.205 0.264
1.131 0.345
Re DEPTH/DIA FROUDE D/SED.DIA W.DEPTH(CM)
2828 8.681 0.214
3036 8.661 0.230
3383 8.661 0.257
3817 8.661 0.289
4007 8.661 0.304
4962 8.661 0.376
5274 8.661 0.400
5655 8.661 0.429
6367 8.661 0.483
1613 4.331 0.061
2204 4.331 0.084
2885 4.331 0.109
2616 4.331 0.099
2938 4.331 0.111
2544 4.331 0.096
12560 2.165 0.238
14850 2.165 0.282
19291 2.165 0.366
20609 2.165 0.391
21928 2.165 0.416
27.7
27.7
27.7
27.7
27.7
27.7
27.7
27.7
27.7
55.5
55.5
55.5
55.5
55.5
55.5
110.9
110.9
110.9
110.9
110.9
11.00
11.00
11.00
11.00
11.00
11.00
11.00
11.00
11.00
11.00
11.00
11.00
11.00
11.00
11.00
11.00
11.00
11.00
11.00
11.00
SHEN'S DATA
de/DIA U/Uc Re DEPTH/DIA FROUDE D/SED.DIA W.DEPTH(CM)
1.020 0.585 56022 0.746 0.416 635.0 11.37
1.180 0.344 49634 1.438 0.272 635.0 21.92
0.820 0.175 28840 0.770 0.213 635.0 11.73
0.880 0.155 40376 0.760 0.300 635.0 11.58
1.100 0.952 68447 0.782 0.502 635.0 11.92
1.148 2.235 114378 0.754 0.844 635.0 11.49
1.080 2.649 137299 0.770 0.949 635.0 11.73
1.300 1.632 93585 1.014 0.612 635.0 15.45
1.180 1.036 77486 1.040 0.472 635.0 15.85
0.980 0.692 69247 0.994 0.403 635.0 15.15
1.040 0.240 50753 1.026 0.291 635.0 15.64
0.924 0.002 40300 0.986 0.239 635.0 15.03
0.880 0.038 40860 1.422 0.199 635.0 21.67
1.120 0.277 48363 1.350 0.266 635.0 20.57
1.380 0.478 55373 1.380 0.306 635.0 21.03
1.200 0.764 72397 1.360 0.370 635.0 20.73
1.180 0.858 70594 1.398 0.382 635.0 21.31
1.220 0.350 56914 1.728 0.258 635.0 26.33
0.750 0.128 48360 1.760 0.214 635.0 26.82
1.088 0.121 58181 1.156 0.290 331.3 17.62
CHABERT & ENGELDINGER'S DATA
de/DIA U/Uc 1 Re DEPTH/DIA FROUDE D/SED.DIA W.DEPTH(CM)
1.250 0.044 85000 2.000 0.607 33.3 20.00
1.233 0.044 127500 1.333 0.607 50.0 20.00
1.740 0.044 38000 2.000 0.768 16.7 10.00
1.310 0.044 76000 1.000 0.768 33.3 10.00
1.960 0.121 33000 4.000 0.471 33.3 20.00
1.700 0.121 66000 2.000 0.471 66.7 20.00
1.353 0.121 99000 1.333 0.471 100.0 20.00
1.900 0.278 20000 3.940 0.288 96.2 19.70
1.220 0.278 40000 1.970 0.288 192.3 19.70
0.993 0.278 60000 1.313 0.288 288.5 19.70
1.800 0.085 21000 7.000 0.227 96.2 35.00
1.200 0.085 42000 3.500 0.227 192.3 35.00
0.913 0.085 63000 2.333 0.227 288.5 35.00
1.150 0.101 37000 1.000 0.374 192.3 10.00
0.887 0.101 55500 0.667 0.374 288.5 10.00
0.869 0.007 39000 0.385 0.429 260.0 5.00
ARKHIPOV'S DATA
de/DIA U/Uc 1 Re DEPTH/DIA FROUDE D/SED.DIA W.DEPTH(CM)
0.417 0.283 1806000 1.167 0.062 26250.0 490.00
0.451 0.062 2933600 1.630 0.097 9650.0 629.00
0.830 0.135 4796000 2.589 0.103 10000.0 1139.00
0.942 0.446 4633100 2.584 0.102 27062.5 1119.00
1.020 0.809 7889000 3.147 0.131 19600.0 1542.00
0.820 1.452 3141500 0.295 0.347 15250.0 90.00
0.852 1.783 3904000 0.426 0.359 15250.0 130.00
1.246 2.018 5246000 1.049 0.307 15250.0 320.00
1.311 2.273 5490000 0.918 0.344 15250.0 280.00
1.410 2.541 6588000 1.377 0.337 15250.0 420.00
1.705 3.533 8296000 1.311 0.434 15250.0 400.00
JAIN'S DATA
de/DIA U/Uc 1 Re DEPTH/DIA FROUDE D/SED.DIA W.DEPTH(CM)
1.654 0.760 25400 2.008 0.500 203.2 10.20
1.949 1.640 38100 2.008 0.750 203.2 10.20
2.244 2.520 50800 2.008 1.000 203.2 10.20
1.693 0.028 25400 2.008 0.500 33.9 10.20
1.713 0.263 33020 2.008 0.650 33.9 10.20
1.693 0.458 38100 2.008 0.750 33.9 10.20
1.929 0.652 43180 2.008 0.850 33.9 10.20
2.264 0.944 50800 2.008 1.000 33.9 10.20
2.539 1.332 60960 2.008 1.200 33.9 10.20
1.909 0.253 25400 2.008 0.500 20.3 10.20
1.437 0.073 31496 2.008 0.620 20.3 10.20
1.476 0.989 38100 2.008 0.750 20.3 10.20
2.028 0.975 50800 2.008 1.000 20.3 10.20
2.106 0.962 60960 2.008 1.200 20.3 10.20
1.713 1.000 41656 4.862 0.527 20.3 24.70
2.224 0.996 71628 4.252 0.969 20.3 21.60
1.850 0.994 40132 4.744 0.514 20.3 24.10
1.181 0.977 50800 1.004 0.500 406.4 10.20
1.476 0.967 76200 1.004 0.750 406.4 10.20
1.565 0.953 101600 1.004 1.000 406.4 10.20
1.299 1.005 50800 1.004 0.500 67.7 10.20
1.211 1.001 66040 1.004 0.650 67.7 10.20
1.220 1.019 76200 1.004 0.750 67.7 10.20
1.368 1.019 86360 1.004 0.850 67.7 10.20
1.516 1.019 101600 1.004 1.000 67.7 10.20
1.713 1.019 121920 1.004 1.200 67.7 10.20
1.575 1.015 50800 1.004 0.500 40.6 10.20
1.388 1.015 62992 1.004 0.620 40.6 10.20
1.368 1.015 76200 1.004 0.750 40.6 10.20
1.467 1.014 101600 1.004 1.000 40.6 10.20
1.565 1.014 121920 1.004 1.200 40.6 10.20
Appendix F  Comparison of Emperical Scour Prediction Formulas
1. Equation used in this study
Y = 0.29 0.49 ( + 0.15 ( 1)2
o.oo00 ( ) 0.14 (A) + 0.091 ()2
0.0068 A) + 3.2 5.0
+ 0.21 ) (h)
UcD
+0.55 ( ) + 0.72 ()
0.018 ) 0.044 ) )2 (h)
0.24( 1) )
+0.12 () 0.
 0.093 ( 1)2 (
(11 h) ( )
D1 N \Vh
2. CSU's Equation
de 2.0 ()0.65 ( )0.43
3. Jain and Fischer's Equations
for u > 0.2
de= 2.0 0.25 .5
for "clear water scour" U < 0
/ \ 0.25 0.3
de = 1.84 ) (05
4. University of Auckland's Equations
for D > 18
Dso
de = K
D
where K is a function of gradation of sediments
for D < 18
Dso
d = 0.45K ( D 0.53
D Dso)
Table 1. Scour Depth Prediction Equations and
Average
Percent
Difference
9.6
30.7
92.6
37.1
Maximum
Percent
Difference
40.9
133.0
735.8
277.7
Minimum
Percent
Difference
0.05
0.40
0.93
0.00
+2.3 7
Results of Comparison Test.
Average Maximum Minimum
Percent Percent Percent
Difference Difference Difference
5. Froelich's Equation 32.9 246.4 0.27
for livebed scour (i.e. U > Uc)
de 0 h 0.46 U 0.20 0.08
De= 0.32 (h)" ) U Dj +1
6. Arkhipov's Equation 24.1 147.3 0.25
de C (U)0H' (h
where
C, a and / are functions of () presented in a graph.
7. Laursen's Equation 49.9 285.1 0.93
de 1.5 (h0.3
8. Baker's (1980) Equation 180.0 718.0 2.56
d = 2.0 tanh [2.0L 1.0]
where
U
N=
[( 1) gd,]
and
Uc
[( 1) gd,]
9. Baker's (1981),Equation 104 360 1.08
D 2 tanh () f() f2f3
where
0 0.0 < 0.5
U,
U, Uc
1.0 1.0 < 1
Uc
f2 and fa depend on structure shape and flow
orientation, and
f2 = f3 = 1.0 for vertical cylinders.
Table 1. Scour Depth Prediction Equations and Results of Comparison Test.
(cont.)
APPENDIX G  RANGES OF VALIDITY OF SCOUR DEPTH EQUATION
Appendix G
Ranges Of Validity Of Scour Depth Equation
Figures G1G5 are provided to illustrate the effect of the
variation of physical quantities like depth mean velocity, cylinder
diameter and water depth on the dimensionless scour depth for
typical values of sediment diameter and density. The lightly
shaded area of the surface in Figure G2 shows the input conditions
under which the dimensionless groups will be within the range of
the data used to generate the scour depth equation. Input
conditions that fall within this range are said to be "within the
range of validity" of the equation. The scour depths computed in
this range of conditions will be the most reliable.
When the environmental conditions (velocity, grain size,
density etc.) and/or structure dimensions yields values of the
independent dimensionless groups (Sediment Regime Number, Reynold's
Number, Structure Aspect Ratio and Froude Number) that are beyond
the domain established by the 98 data points, the "surface" must
be extrapolated. Examination of the data and the surfaces in
figures G1G3 (and other similar figures) led to the conclusion
that the surfaces (and thus the equation) follow the trend of the
data for some distance beyond the bounds set by the data. This
region is indicated as the darker shaded area in Figure G2. These
three dimensional plots are good for visualizing trends but are
difficult to use when actual values must be taken from the curves,
thus two dimensional plots of the projections of these surfaces in
the horizontal plane are given in Figures G4 and G5. A quick look
at the appropriate plot will let the user know if the input data
is within the "range of validity", or if not, if it is within the
"extrapolated range of validity".
The computer program that accompanies this report will test
to see if the input data is such that the dimensionless groups 1)
fall within the range of validity, and 2) fall within the extended
range of validity of the equation. If the conditions are outside
the range of validity the input conditions are adjusted until the
conditions are within bounds and the scour information (depth and
volume) computed. If the conditions are outside the extrapolated
range of validity again the conditions are adjusted until they are
within these bounds and the scour information computed.
N
Figure G1 Surface Plot Using StructureInduced Scour Equation
For Vertical Cylinders. Sediment Diameter = 0.25 mm,
Sediment Mass Density = 165 lb /ft'.
Water Depth = 5 ft.
0 /i I I I IC <''
a<" \CI' I I I IIIIIIIPIII1IIII K ~
^ ^ .^ ^ ^ ^ ^ ^
";^ < X IIIII IIIIIIII < IIIIII 4 I
Ls 4 
Fiur G ufc ltUigSrcueIdcd ScorEuto
Extrapolated
Range
Within Range of
Parameters
Figure G2 Surface Plot Using StructureInduced Scour Equation
For Vertical Cylinders. Sediment Diameter = 0.25 mm,
Sediment Mass Density = 165 lb /ft3.
Water Depth = 10 ft.
0
CC
C)
LU
j
0
CO
2z
LU
<3
1.5
1.0.
0.5
0.0.
u<" *%
^ o
0L
3o
Figure G3 Surface Plot Using StructureInduced Scour Equation
For Vertical Cylinders. Sediment Diameter = 0.25 mm,
Sediment Mass Density = 165 lb /ft3.
Water Depth = 20 ft.
10.0
8.0
6.0
4.0
2.0
0.0
0.0
5.0 10.0 15.0
20.0
CYLINDER DIAMETER, D (ft)
Figure G4 Ranges Of Validity Of StructureInduced Scour Equation
Sediment Diameter = 0.25 mm, Sediment Mass Density =
165 lb /ft3. Water Depths = 1, 5, 10, 20 ft.
