(1)
Carravetta A., Della Morte R., Greco M.
Hydraulic and Environmental Department G. Ippolito, Via Claudio n. 21. Naples, Italy.
Tel. +39+81+7683459; fax +39+81+5938936; email: arcarrav@unina.it
Abstract: Sediment removal from a fixed bed is a problem having large implications in sewer and open channel design. There is a large experimental evidence that particle motion starts for bed shear velocities much smaller than the ones predicted by Shields theory and that the sediment transport rate is larger than the one on a mobile bed. The experiments of one of the writers confirm these evidences.An innovative quasi-3D model described in the paper, including additional bottom shear stress due to the unsteadiness produced by the large scale vortex, can predict transport, erosion and deposition of sediment on a fixed bed. Model results reported in the paper are in good agreement with experiments.
Keywords: Sediment removal, fixed bed, numerical model, unsteady friction
Numerical models have a large diffusion for the prediction of river and channel evolution by sediment transport, erosion and deposition. Here the results of a numerical model are confronted with the experimental data collected by one of the authors on a laboratory flume. The data is about the removal of a sediment stratum imposed on part of the fixed bed. Relevant problem of sediment dynamics are connected with this very simple situation. Therefore, computer simulations are an important tool also in process interpretation.
Recently, very powerful 3D models have been developed for scientific as well as commercial applications. These models have been applied efficiently in many schemes to predict the flow field and the bed evolution (Ouillon & Dartus, 1997; Fiorotto & Cividin, 1996, Wang & Jia, 1999). Nevertheless, the use of these model requires long work times, coming from both the definition and the computational part of the problem.
Therefore 2DH models are still fashionable when dealing with complex river geometry or sewer systems and are largely used in river hydraulics also within commercial codes. These models are based on vertically averaged flow and transport equations under shallow water hypothesis, requiring gradual variations of depth and width along the river. Despite this limitation, 2DH models seem the only reasonable choice in a design phase of water works or in management problems where a large number of simulations are needed. Unfortunately, model prediction in situations where a 3D flow field is expected, as in presence of obstacles and singularities, can lead to a very poor localization and sizing of the bottom shear stresses, so compromising any attempt to simulate bed evolution (Duan et al., 1999).
To overcome some of the 2DH limitations, the writers have developed an innovative quasi-3D model described in detail in a previous paper (Carravetta et al., 2000).
Shallow water equations:
(1)
where:
|
|
are
solved in a finite volume explicit integration scheme. In (1) g is gravity, x
and y are the horizontal plane coordinates, t is the independent time variable,
h is water depth, zf is bed elevation, U and V are the x and y
components of the depth averaged local velocity,
.
is the bottom stress, calculated using Chezy formula.
A zero equation model (Rodi, 1980) is
used to solve turbulence, assuming
(Elder, 1959), where
is the shear velocity. This formula
is still widely used today (Lambert & Sellin, 1996; Olsen , 1999). The term
is introduced assuming an
additional bottom stress due to the unsteadiness of the flow field at the
computational time scales.
As in LES, if the computational grid is smaller than significant energy containing eddies, those should appear resolved by the model. But depth averaging inhibits this possibility in the vertical direction. So, a model is required for simulate the effect of these eddies in the vertical direction. This is accomplished, here, through the Su term exerted towards the bottom. The corresponding u* value brings a modification also to the nt coefficient, and hence to the side stress between adjoining computational cells. For rational analogy with the modeling of unsteady flow in pressure pipes (Brunone et al., 1995) and according to recent findings based on EIT approximation (Axworty et al., 1999), the last term is evaluated by the formula:
(2)
Then, the bidimensional flow field
is expanded to a fully three dimensional flow field. Logarithmic wall law is
assumed in vector form to get the u and v components of the local velocity
vector; k being the von Karman constant:
(3)
The different direction of the
and
vectors, due to the
additional bottom stresses implied by the Su term, which is not
parallel to the depth averaged velocity
, can lead to gradual rotations with depth of the horizontal component of
the local velocity, as experimentally observed by many authors, and implied by
the presence of secondary currents. Finally, the vertical component of the local
velocity comes out from the divergence equation.
Suspended transport is computed using the three dimensional diffusion equation:
(4)
where c is the local
averaged concentration of sediment, esx ,
esy ,
esz
are the components of the turbulent diffusivity tensor, w’
is the particle settling velocity. In our applications the Reynolds analogy
was assumed. At the bottom the
entrainment or deposition flux is computed as w’ (cz=0.05h – E)
where E is the dimensionless entrainment rate following Garcia & Parker
(1991).
The Novak & Nalluri (1978, 1984) formula, for fixed bed channels, is considered into the model for computing bed-load transport qb. Specific of this formulation is the lack of a lower limit of shear stress to activate bed transport. An effective shear velocity, u*eff, was used in Garcia & Parker and Novak & Nalluri equations, as discussed in detail in the next paragraph.
Finally, Exner equation for sediment continuity is solved for bed elevation zf:
(5)
where lp is the bed porosity.
All equations are solved numerically with the same finite volume method already outlined.
The model used for
the bottom stresses
, where
is given by (2), may produce vastly
larger instantaneous stresses at the channel bottom than provided by the uniform
motion formulas. They may rise when a resolved vortex passes through. These
short duration stress pulses cannot produce the same effect on sediment motion
that steady, long duration stresses do.
Still little is known about the dynamic response of sediment transport to flow acceleration. Recently, Admiraal & Garcia (1999) investigated the effect of flow acceleration on the entrainment into suspension of sediment particles. A delay of the entrainment with respect to the velocity peak was observed, together with a somewhat larger value of the entrainment peak than would be estimated using the Garcia & Parker formula.
Here, a similar behaviour is
supposed to hold not only for entrainment but also for bed transport. More in
detail, it has been assumed that both for bed transport and entrainment into
suspension, the value of the shear velocity, u*eff, to be substituted into Garcia e Parker and Novak
& Nalluri formulas, could be derived by a convolution of the instantaneous u*2
values with a transfer function. The simplest transfer function, i.e. the
“linear reservoir” model, has been assumed. In discrete time increments,
this gives:
(6)
where S is the storage and K is a coefficient having dimension [t-1]. A physical interpretation of (6) could be to assume that ru2*Dt is proportional to the instantaneous impulse on the bed particles, while S is proportional to the bulk average of their momentum.
The model introduced above is capable of reproducing the main features of the experimental data published by Admiraal & Garcia (1999).
The tests were carried out in a 18 m long, 0.75 m wide, 0.60m deep, rectangular flume (Della Morte, 1991; Biggiero & Della Morte, 1996). Sediment mean diameters, d50, used in the experiments varied between 0.5mm to 6 mm, having an average relative density of 2.65. The fixed bed of the flume was in perspex and was covered by grouped particles forming a sediment deposit across the whole width of the flume. Hydraulic measurements included longitudinal mean velocity and turbulence profiles at the mid-plane of the channel.
The deposits were of 1.5 cm thickness. The length L of the deposit was different during each test, between 0.3 m and 3 m. The deposit was located so that its downstream end was always at 9 m from the inlet of the channel. The starting of particle motion was determined with a succession of steady state conditions of increasing flow rates. The sediment bed was stable for low flow rates. Then, a flow rate was imposed corresponding to a low sediment motion. Bed-forms were present during the experiments. According to van Rijn abacus, for d50 = 0.5 mm, bed-forms were identified in ripples region; for d50 > 0.5mm, bed-forms were identified between ripples and dunes.
The first zone of the sediment layer that was completely removed was located downstream. The erosion phenomena was strongly dependent on grain size and deposit length. A digital chronometer was used to measure the time of erosion of the deposit from its initial location. Several times T% were measured for each test, corresponding to different percentages of removal.
Three tests (runs A, B, C) have been selected for model application. Run A and B were performed both with sediment d50=0.5 mm. The length of the deposit was L=2.5 m, for Run A, and L=3.0 m, during Run B. Rate of flow were 0.008 m3/s and 0.080 m3/s, for Run A and B respectively. The measured times of erosion were: T40@600 s and T80@1200 s for Run A; T90@900 s for Run B. Run C was performed with a coarser sediment, d50=3.8 mm, with an initial length of the deposit L=3.0 m, and with a discharge Q=0.101 m3/s. The measured time of erosion was T90@720 s.
Preliminary calculations, performed without including in the model the additional bottom stresses, showed that the transport process is seriously underestimated considering the only Chezy bottom stress. On the contrary, the time scale of the process can be explained, in the model framework, by the presence of the velocity fluctuations exalted by the presence of the sediment layer along the channel.
Figures 1 and 2 represent model results corresponding to experiments B and C. In each Figure two conditions at the mid-plane are plotted: bed and water profile at t=0; bed and water profile at t=T90/2. The presence of bed-forms along the sediment layer is reproduced by the model.
Fig. 1 Model results for Run B – (a) t=0; (b) t=450 s.
Fig. 2 Model results for Run C – (a) t=0; (b) t=360 s.
For Run B and C measured and computed time of sediment removal were very close. Calculated times were T90@1200 s (Run B) and T90@720 s (Run C). During Run A with a much lower discharge computed and measured times were similar for T40. Times corresponding to higher percentages of erosion were greatly underestimated.
The process of erosion of a sediment deposit from a fixed bed is ruled by the presence of three dimensional flow structures determining large fluctuations of the local velocity on the sediment bed. The quasi-3D writer’s model was found to be very efficient in this particular situation because of the particular bottom-stress formula used within the model. Part of the bottom-stresses is attributed to the effect of three-dimensional eddies not resolved in shallow water hypothesis, but considered in the model with an ad hoc additional resistance term.
A complete research program, including a larger number of experimental conditions, is still in progress.
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