Geology Reference
In-Depth Information
Fig. 2.1
Schematic force
balancing of individual
grains on a horizontal bed
yielding the commonly observed logarithmic velocity
profile over depth, i.e., the law of the wall:
u
¤³
z
uz
()
*
ln
¥
¦µ
(2.1)
k
z
o
Where
u
(
z
) = current velocity with respect to depth,
z
= vertical coordinate representing water depth,
u
*
= current related bed-shear velocity, N = Von Karman's
constant, typically taken as 0.4, and
z
o
= vertical level
with zero velocity, also often referred to as bed rough-
ness. A list of notation and conventional units are pro-
vided at the beginning of this chapter. Figure
2.2
illustrates an example of a logarithmic profile. The
dynamics of the bottom boundary layer where the cur-
rent velocity decreases rapidly with respect of depth is
crucial to sediment entrainment and transport. For plane
bed, the bed roughness (Fig.
2.2
) is a function of sedi-
ment grain size. When bedforms exist, the bed roughness
is related to the geometry of the bedform. The bed shear
velocity is directly related to bed shear stress (
t
b
) as:
Fig. 2.2
An example of a logarithmic current profile, showing the
bed roughness (
z
o
) and the schematic bottom boundary layer
2
tr
w
u
(2.2)
b
*
where
r
w
= density of water (seawater in the case of tidal
environments). The bed shear velocity and bed shear
stress are two of the key parameters describing the fluid-
sediment interaction and are commonly used in comput-
ing sediment transport. Determining bed shear velocity
and bed shear stress can be difficult and often comprises
an essential part of a sediment transport study. By mea-
suring a velocity profile through the water column,
Eq.
2.1
can be used to determine bed shear velocity and
bed shear stress, as well as the bottom roughness.
Another commonly used approach to determine the bot-
tom shear stress, especially for depth-averaged models,
is to relate bottom stress to velocity squared as:
1
2
2
t
r
fu
(2.3)
b
w
