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Numerical simulation of sediment mixture deposition part 1 ... - LTHE

Numerical simulation of sediment mixture deposition part 1 ... - LTHE

Table 1 - Definition

Table 1 - Definition of grain size classes and distribution sediment input A constant liquid discharge Q = 0.049 m 3 /s is introduced at the upstream end (longitudinal distance x = 0). Downstream water level (x = 60 m) is fixed at a constant value h = 0.5 m. Sediment mixture The sediment mixture, which ranges from 0.125 mm to 64 mm, is described by size fractions of 9 different classes of sediment. We have adopted the same logarithmic scale as SAFL reports (see for example Cui et al. 1996) for definition of limiting diameters of these classes: D = 2 ψ . small time step is used (less than 15 min). However some errors could be observed in mass balance and the simulation fails after 13 days (312 hours). Because solution of such difficulties is not within the scope of this paper, input flux of sediment is adjusted for the next simulation (simulation B) in order that the final slope, which can be computed a-prori, is a subcritical slope. The total feed rate is reduced to G s = 0.015 kg/s, less than 1/3 of the input in Run 3. The grain size distribution of the input, Table 1, is directly taken from Seal et al. (1997, fig.3). A histogram and corresponding sieve curve of the mixture are also plotted in Figure 7 (bed load). The mean diameter of the mixture is d m = p j d j = 11.6 mm . ∑ j Bed-load input SAFL runs 1 to 3 are characterized by relatively high sediment input. It seems obvious that their authors were guided by the necessity of keeping the duration of the experiments within reasonable limits. However such high loads generate large deposits leading to supercritical flow conditions. With the assumptions of equation (1), it can be computed that feed conditions of Run 3 would lead to an equilibrium profile whose slope is S 0 = 0.014 and Froude number Fr = 1.16. Some theoretical work has been done e.g. by Rahuel (1988) and by Sieben (1997), based on the analysis of characteristic directions, which demonstrate the particular behavior of sediment transport equations in supercritical flow conditions. As mentioned by Cui et al. (1996), SEDICOUP, as a fully coupled model, should be capable of dealing with supercritical flow conditions. We confirm such an assertion, provided that correct boundary conditions are imposed and that space weighting of the corresponding discretized equations conforms to the characteristic directions. A first simulation (simulation A) is however attempted with sediment input of Run 3: G s = 0.047 kg/s. The discretization scheme and weighting are kept with their 'subcritical' configuration. Evolution of bed profiles will be discussed in the next section. Figure 2 shows the evolution of Froude number at different abscissae during the simulation. As can be seen, the software supports a small amount of 'supercriticality' on condition that a Fig. 2. Simulation A - Evolution of Froude number during simulation at different sections. Simulation is started with initial backwater steady-flow conditions. Practical constraints of laboratory experiments concerning the duration of the run are alleviated. The total simulation time is 50 days (1200 hours) with progressive time steps from 3 min (t = 0) to 1.2 hr (from simulation time 2.5 days). 4 Deposit geometry Progress of gravel deposit Figure 3 shows the evolution of bed profiles during the simulation. A front propagates downstream in the first hours in the same manner as in the experiments. A much gentler aggradation is then computed with a progressive evolution towards a constant bed profile with uniform bed-slope. Fig. 3. Profiles of bed elevation h - plain lines: simulation A - dotted lines: simulation B. 420 JOURNAL DE RECHERCHES HYDRAULIQUES, VOL. 38, 2000, NO. 6

Front elevation is very similar to its value in SAFL experiments (figure 5 in Toro-Escobar et al. 1996). From reports of these experiments, and from our results, it does not appear that it is very dependent on sediment discharge. For simulation A, the front edge has similar steepness to that in the SAFL description. Front shape is smoother when sediment load is lower (simulation B). Celerity of the front is nearly constant throughout the run as noticed by Toro-Escobar et al. (1996). It is very sensitive to sediment feed rate: Fig. 4 compares the celerities from SAFL runs and from our simulations. Front celerity as computed during simulation A is also reported in this plot. The small difference of the celerity between this simulation and Run 3 could have been easily cancelled with minor calibration of the transport formula. deposit. (iii) A slow aggradation after deposit with a nearly constant rate before the front reaches the downstream end of the flume (approx. time 260 hr) which can be considered as stabilized at the end of the simulation (t = 1200 hr). The small wiggles observed at the upstream point of the model reflect the very severe numerical conditions at the feed point (large time step ∆t = 1.2 hr and nearly critical flow conditions Fr = 0.79). Fig. 6. Evolution of bed elevation h. Fig. 4. Celerity of front propagation. Similarity collapse of the bed elevation profile is shown on Figure 5. As for SAFL experiments, the same similarity is found for the profiles at different times. As was noticed by Seal et al. (1997, fig.9), a weak concavity of bed profiles is noticed, which increases gradually with time. The equilibrium profile, which is obtained at the end of the simulation, shows a uniform slope (i.e. uniform shear conditions) and uniform grain distribution. At equilibrium the slope is S equ = 0.00071 and the Froude number is Fr = 0.79. Equation (1) translates some kind of "equal mobility". The effective solid discharge capacity of the j th sediment class is proportional to the discharge capacity of single sized sediment of the same diameter ĝ j and to the volumetric fraction β j of sediment j on the bed surface (within the mixing layer): g j = β j ĝ j (5) As ĝ is a function of diameter d j , it becomes obvious that the volumetric fraction of transported material P j is different than the volumetric fraction at the bed-surface: Fig. 5. Similarity collapse of bed elevation profiles upstream of the front. xd, hd = front edge coordinates, h 0 = upstream bed elevation. P j g = ---------- j ≠ β j (5) g j ∑ i This result is well known by geomorphologists and engineers and is confirmed by the results of the simulation. Figure 7a compares size fractions of each class within the mixing layer and of transported material at equilibrium. The same data are shown in Figure 7b in the form of sieve curves. Naturally, curves for bed load correspond to sediment mixture at the input. Towards equilibrium The simulation has been pursued until a steady state is obtained. Then, transporting conditions are uniform within the channel and sufficient to convey downstream the whole sediment load that is introduced upstream. Figure 6 shows the evolution of bed elevation at different points of the (simulated) channel. Three different stages can be detected. (i) Initial deposit before front arrival. (ii) Front Table 2 - Dimensionless shear stress at equilibrium JOURNAL OF HYDRAULIC RESEARCH, VOL. 38, 2000, NO. 6 421

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