Geoscience Reference
In-Depth Information
5.1.2.2 1-D equilibrium sediment transport equations
The assumption of local equilibrium transport described in Section 2.6.1 ignores the
temporal and spatial lags of sediment transport and sets the actual sediment transport
rate to be equal to the equilibrium (capacity) one at each cross-section:
Q tk =
Q t k (
U , h ,
τ
, B , d k , p bk ,
γ s ,
γ...) (
k
=
1, 2,
...
, N
)
(5.37)
In the equilibrium transport model, the change in bed area due to size class k is
calculated by
p m )
A bk
+
Q tk
(
1
=
0
(5.38)
t
x
The total change in bed area is calculated by Eq. (5.31) and the bed-material
gradations by Eqs. (5.32) and (5.33).
It should be noted that the local equilibrium assumption does not mean that the
sediment transport in the entire channel is at equilibrium. Conversely, the sediment
transport capacities at two consecutive cross-sections may be different under varying
flow and sediment conditions, and thus the channel bed between these two cross-
sections may change according to Eq. (5.38).
5.1.2.3 Characteristics of equilibrium and non-equilibrium
transport models
The equilibrium sediment transport model is simple but may lead to a numerical
difficulty near the inlet with constrained sediment loading. Fig. 5.5 shows the sed-
iment discharge profiles determined by the equilibrium transport model on a finite
difference mesh in cases of erosion ( Q t 0 =
0) and deposition ( Q t 0 =
2 Q t ). Here, Q t 0
is the sediment discharge loaded at the inlet ( x
is assumed constant in the
entire channel. The sediment discharge at cross-section 1 is specified by the boundary
condition (constraint), while the sediment discharge at cross-section 2 is determined
using Eq. (5.37). If the sediment is strongly over- or under-loaded, the sediment dis-
charges at these two cross-sections will be significantly different, and strong deposition
or erosion will be computed in the first reach. The smaller the grid spacing, the larger
the deposition or erosion rate calculated in this reach. This is physically unreasonable,
and may cause numerical instability. Therefore, the application of the equilibrium sed-
iment transport model should be limited to situations with near-equilibrium loading
at the inlet.
For uniform sediment under steady flow conditions, the non-equilibrium transport
equation (5.34) with constant L t and Q t and without side discharge has an analytical
solution:
=
0), and Q t
exp
x
L t
Q t
=
Q t + (
Q t 0
Q t )
(5.39)
Fig. 5.6 illustrates the sediment discharge profiles determined by Eq. (5.39) for the
 
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