Environmental Engineering Reference
In-Depth Information
5.4.2.3
Riverbed Inertia
Wang (1999) found from experiments that the riverbed deformation obviously lags behind the variation
of flow and sediment carrying capacity. If the riverbed scour by flows is regarded as a kind of accelerating
motion of the bed, the sediment carrying capacity of the flow is a “positive force” and the incoming
sediment rate is a “negative force”, and the idleness toward motion can be thought of as “inertia” of the bed.
The “positive force” causes bed scour whereas the “negative force” causes siltation. Only if the “positive
force” is balanced by the “negative force”, the river bed does not change. For an open channel flow within
hardened banks erosion and sediment deposition can only occur on the channel bed, and the equation of
the bed motion can be expressed as
d
d
Z
= g
g
(5.71)
I
b
b*
b
t
in which
Z
is bed elevation, -d
Z/
d
t
is the rate of bed degradation due to scour,
I
b
is called riverbed inertia,
g
b
is the rate of bed load transport, and
g
b
*
is the bed load carrying capacity of the flow, or the rate of bed
load transport by the flow when in equilibrium. Because the dimensions of
g
b
and -d
Z/
d
t
are [Mass/(Time
u
Length)] and [Length/Time], respectively, the dimension of inertia of the bed is [Mass/Length
2
].
For a flow carrying less sediment than its sediment-carrying capacity, channel bed scour occurs. The
scour rate,
S
r
,
is defined as the weight of sediment scoured from the bed per unit area per time, i.e.
S
r
W
s
/
A
b
T
s
(5.72)
where
W
s
is the weight of sediment scoured from the river bed,
A
b
is the area of channel bed subject to
scour, and
T
s
is the time of scour. Scour occurs in unsteady flow and non-equilibrium sediment-laden
flow. If a flow is stable and uniform, the sediment and flow are in equilibrium and the scoured sediment
is balanced by the depositing sediment, which yields a zero scour rate. Wang (1999) has obtained an
empirical formula for the scour rate under both conditions of clear water flow and sediment-laden flow.
The rate of bed degradation is a function of the scour rate as follows:
d
d
Z
t
S
r
(5.73)
J
(1
p
)
s
in which
p
is the porosity. Combining Eq. (5.73) with Eq. (5.71) yields a relation between the riverbed
inertia and scour rate, which may be regarded as the definition of the riverbed inertia:
gg
gg
=
b*
b
=
J
(1
p
)
b*
b
(5.74)
I
b
s
dZ / dt
S
r
in which J
s
(1-
p
) represents the dry weight of the bed material. For a given composition of bed material,
the bed inertia should be a constant. Therefore, for different values of
g
b
*
and
g
b
, calculation with the
equation yields the same
I
b
value. Indeed, experimental results obtained by Wang (1999) showed that the
values of inertia for each bed composition under different flow conditions and feeding rates of sediment
load are the same. This proves that the river bed inertia is a physical property of the granular bed.
A small bed inertia implies that the bed deforms quickly with the change of flow and its capacity. If
the bed inertia is large, the bed is slow to or does not respond to the flow changes, and it takes a long
distance for the transport rate
g
b
to reach its equilibrium
g
b
*
. If the bed material is composed of a wide
range of sediment sizes, the inertia of the bed is large because an armoring layer will develop during the
erosion process. In the experiments with sediment with a sorting coefficient larger than 15, the bed
inertia is between 45 and 50 t/m
2
. For mountain rivers, the bed material usually has a large sorting
coefficient and the bed inertia can be roughly taken as 50 t/m
2
for convenience of calculation (Wang,
1999). For a riverbed composed with light grains, such as coal or other materials with small specific
weight, the bed inertia is small. It seems that the inertia is proportional to the specific weight ratio (J
s
-
J)/J
.
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