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natural grain size [mm]
0
50
100
150
200
250
300
35 0
100
80
60
40
natural grain size distribution
step 1: geometrical scaling 1:35
step 2: coarsening according to Zarn
step 3: elimination of the fine fraction
20
0
0
1
2
3
4
5
6
7
8
9
10
modeled grain size [mm]
Fig. 5 Natural grain size distribution (upper x -axes) and converted grain size distribution in
model scale (lower x -axes)
transport and a higher deposition, if the sediment input is not adapted. In order to
guarantee correct transport processes, it is here proposed that the ratio between the
sediment input at each time step of the load curve and the theoretical maximum
transport capacity is kept identical in nature and the model ( 4 ). Consequently, the
equilibrium slope at each time step equals in nature scale and in model scale.
Bed load input nature
Max. transport capacity nature ΒΌ
Bed load input model
Max. transport capacity model
(4)
The theoretical maximum transport capacity in the nature and the model has
been calculated with the formula of Smart and Jaggi ( 1983 ) for steep channels with
a gradient between 3% and 20%. Due to the coarser material, an approx. 50%
smaller maximum transport capacity results in the model. Thus, the sediment input
per time step, and hence the total bed load, is diminished by half in comparison to
the original input. Figure 6 shows the original input for the natural material (a) and
the reduced input for the coarsened material (b), both in natural scale. The discharge
underlies no change.
As a consequence of the reduced sediment input in the model, both, the sediment
volume per time step and the total volume that exceed the transport capacity (Fig. 6
(II)), are smaller than in nature (Fig. 6 (I)). This leads to a lower deposition height
and water level. Overtopping would thus occur later in the model than in nature. To
compensate for this effect, a compensation of the reduction of the sediment load is
needed and the volume must be increased. For this purpose, we propose to extend
the duration of the model discharge and sediment hydrograph by an extension
factor, guaranteeing a correct Froude-scaled total bed load. The extension factor g
is calculated from the relation of the total bed load of original and reduced input of
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