Equilibrium Local Scour Depth Equations for Single, Simple Structures
In the clear-water scour range ( 0.4 
V
 1.0 )
Vc
ys
= 2.5 F1 F2 F3
D*
In the live-bed scour range below the live-bed peak velocity ( 1.0 <
  V

 Vl p V
-1 
 
 V -V
ys
V
c
c
 + 2.5F3  c
= F1  2.2 
V
V
D*
  lp 
 l p -1
 V
  V -1 
 c
  c 
and in the live-bed scour range above the live bed velocity (
ys
= 2.2 F1 ,
D*
where
 y 0.4 
F1  tanh  0*   ,
 D  
2

  V   
F2  1-1.2 ln     ,
  Vc   



 D* 


D 

,
50 

F3 
-0.13 
  * 1.2
*
 0.4  D  +10.6  D  
  D50 
 D50  
1
V Vlp
)

Vc Vc


 ,



V Vlp
)
>
Vc Vc
V1 = 5Vc ,
V2 = 0.6 g y0 , and
.
 V for V1 > V2
Vl p = live bed peak velocity =  1
.
V
for
V
>
V
 2
2
1
The effective diameter, D*, is defined as:
D* = K  K s w
,
where
Ks = shape factor (equations for computing shape factors for common pier shapes are
given in Table Error! No text of specified style in document.-1),
Ka = flow skew angle factor, i.e., the projected width of the structure (can be
approximated for the shapes in Table Error! No text of specified style in
document.-1 by the expression:
Kα =
w cos  α  + L sin  α 
,
w
,
w = structure width, and
 = flow skew angle.
In using the equations in Table Error! No text of specified style in document.-1, note
the general constraints given near the top of the table and the more restrictive constraints
associated with some of the equations.
.
2
Table Error! No text of specified style in document.-1
Shape factor for common pier shapes.
Structure Shape (Plan View)
w
General Constraints 10.0     0.1;
L
Shape Coefficient, Ks
0 

2
w
w
w
w
If   < 0.1 set   = 0.1; If   > 10.0 set   = 10.0
L
L
L
L
Ks  1.0


K s  0.9  1.21    
4

4
0     / 4 
4



K s  0.9   2.63K1  2.37       0.38
4


w
K1  1.23  0.13  
L
0.3
 / 4     / 2


K s  0.9   2.63K 2  2.37     
4



 1 
K1  1.23  0.13 

 w 
  L  
3
0.3
4
Table Error! No text of specified style in document.-1
Continued
Structure Shape (Plan View)
w
General Constraints 1.0     0.1;
L
0 

2
w
If  
L
Shape Coefficient, Ks
w
< 0.1 set   = 0.1
L
0     / 4


K s  K 3  2.63 1.0  K 3     
4

w

L
4
0.3
K 3  0.17  0.83 
w
L
 0.3
4
0.3
0     / 4


K s  K 5   2.37  2.63K 5    
w
K 5  2.11  3.47  
L
L
 0.3 set
 / 4     / 2


K s  K 3  2.63  K 4  K 3     
4



 1 
K 4  0.5  0.5 

  w  
  L  
w
In K 3 if


4
0.3
In K 5 if
4
w
L
 0.5 set
 / 4     / 2
w
L
 0.5


K s  K 5  2.63  K 6  K 5     
4



 1 
K 6  0.3  0.6 

 w 
  L  
4
0.3
4
LIST OF SYMBOLS
b
pile width (diameter)
bcol
column width
bm
model pile width
bp
prototype pile width
bpc
pile cap width
*
d
dimensionless particle diameter
D*col
effective diameter of the column
D*col(min)
minimum value for the column effective diameter when the base of the column is
buried
D*col(max)
maximum value for the column effective diameter
D*col(f)
attenuated effective diameter for the column
D* pc(max)
maximum value for the pile cap effective diameter
D*m
effective diameter of the model pier tested
D*p
effective diameter of the prototype pier
D*pc
effective diameter of the pile cap
D*pg
effective diameter of the pile group
D16
sediment size for which 16 percent of bed material is finer
D50
median sediment diameter
D84
sediment size for which 84 percent of bed material is finer
D90
sediment size for which 90 percent of bed material is finer
f
weighted average value for the pile cap extension, (3f1+f2)/4
f1
distance between the leading edge of the column and the leading edge of the pile cap
f2
distance between the side edge of the column and the side edge of the pile cap
F1
represents the functional dependence of the scour depth at a simple pier on the ratio
of water depth to pier width (y0/D*)
F2
represents the functional dependence of the scour depth at a simple pier on the ratio
of depth averaged velocity to sediment critical velocity (V/Vc)
F3
represents the functional dependence of the scour depth at a simple pier on the ratio
of pier width to median sediment diameter (D*/D50)
Fr
Froude number, V / gy0
5
Frc
critical Froude number = Vc / gy0
g
acceleration of gravity = 32.17 ft/s2
Hcol
distance between the bed (adjusted for general scour, aggradation/degradation, and
contraction scour) and the bottom of the column
Hpc
distance between the bed (adjusted for general scour, aggradation/degradation,
contraction scour, and column scour) and the bottom of the pile cap
H pc
distance from the original bed and the sand water interface
Hpg
distance between the bed (adjusted for general scour, aggradation/degradation,
contraction scour, column scour, and pile cap scour) and the top of the pile group
H pg
distance from the bed to the top of the pile group after adjusting the bed for the scour
caused by the column and pile cap, ys(col+pc)
k
bed roughness height
Kbpc
buried pile cap attenuation coefficient
Kbpg
buried pile group attenuation coefficient
Kcp
peak value of normalized clearwater scour depth
Kd
sediment-size factor
Kh
coefficient that accounts for the height of the pile group above the adjusted bed
KI
flow intensity factor
Klp
dimensionless scour depth magnitude at the live bed peak scour
Km
number of piles in the direction of unskewed flow
Ks
shape factor
Ksp
pile spacing coefficient
Ks(pile)
individual pile group pile shape factor
Ks(pile group) pile group shape factor
K1
factor for shape of pier nose
K2
factor for angle of attack of flow
K
flow skew angle coefficient
K
factor for the gradation of sediment
L
length of pier
lcol
column length
Lm
model scale length
Lp
prototype scale length
lpc
pile cap length
6
q
discharge per unit width
RR
relative roughness of the bed, k / D50
Rep
pier Reynolds number, Vb / 
s
distance between centerlines of adjacent piles in a pile group
sm
distance between centerlines of adjacent piles in line with the flow in a pile group
sn
distance between centerlines of adjacent piles perpendicular to the flow in a pile
group
sg
ratio of sediment density to the density of water s / 
t
time
te
reference time
t90
time to reach 90% of equilibrium scour depth
T
pile cap thickness
Tr
dimensionless parameter in van Rijn’s bed form equations
V
mean depth averaged velocity
Vc
critical depth averaged velocity
Vlp
depth averaged velocity at the live bed peak scour depth
V0
depth averaged velocity upstream of the pier
Wp
projected width of the piles in the pile group
Wpi
projected width of a single unobstructed pile
y1(max)
limiting water depth for which the column scour is impacted
y2(max)
limiting water depth for which the pile cap scour is impacted
y3(max)
limiting water depth for which the pile group scour is impacted
ys
equilibrium scour depth
ys m
model equilibrium scour depth
ys p
prototype equilibrium scour depth
ys(col)
scour due to the column in a complex pier
ys(col)(max) maximum column scour depth
ys(pc)
scour due to the pile cap in a complex pier
ys(pc)(max)
maximum pile cap scour depth
ys(pg)
scour due to the pile group in a complex pier
yo
water depth adjusted for general scour, aggradation/degradation, and contraction
scour
7
yo
water depth adjusted for general scour, aggradation/degradation, contraction scour
and scour caused by the column and pile cap, ys(col+pc)
yst
time dependent scour depth

flow skew angle in degrees

dynamic viscosity of water

kinematic viscosity of water

mass density of water
s
mass density of sediment mineral (example: s for quartz sand = 165 lb/ft3)

measure of the sediment gradation =

stream bed shear stress
c
critical stream bed shear stress
u
upstream bed shear stress
0
maximum bed shear stress in local scour hole

factor for pier shape
≡
symbol for “identically equal to” (or “defined as”)
8
D84 / D16