(2)
H. Knoblauch1, R. Wanker2, G.
Goekler1, M. Polz1
1Department
of Hydraulic Structures and Water Resources Management, Technical University of
Graz, Stremayrgasse 10/II, 8010 Graz, Austria
Tel.:
++43-316-873-8362 Fax: ++43-316-873-8357
E-mail:
helmut.knoblauch@TUGraz.at
2AVL-List-List
GmbH, Hans-List-Platz 1,8010 Graz, Austria
Tel.:
++43-316-787-1906 Fax: ++43-316-787-777
E-mail:
roland.wanker@avl.com
Abstract:
This paper deals with the sedimentation and transport processes taking place in
a rectangular horizontal channel in the laboratory. These model tests have
served as a basis for the numerical simulation of sedimentation processes and
the various factors involved. An Euler/Euler model for the numerical simulation
of sedimentation effects and sediment transport is presented. The model solves
the transient Navier-Stokes equations with the k-e
turbulence model. The main advantage of the model is its validity in regions of
high and low sediment concentration. Bed changes are accounted for directly
without any adaption of the computational grid. The effects of deposition and
reentrainment of particles are investigated in a laboratory flume. The results
from the numerical model are in good accordance with those coming from the
physical model tests.
The complexity of studying sediment transport processes in natural river environments has led to laboratory investigations for extending our knowledge in this field. A preliminary study phase served to provide general information on the transport and sedimentation processes taking place in a rectangular horizontal channel. The attention was focussed on the effects of different particle sizes and concentrations. These tests provided the basis for checking the accuracy of the numerical simulation.
In recent years
several models for the numerical simulation of sedimentation effects have been
published in literature. The most common procedure for modeling sedimentation
involves splitting the problem into
a hydrodynamic model and into a sediment transport model [1], [2], [3], [4].
These two sub-models are coupled via more or less empirical equations describing
mass exchange between suspended load, bed load and the deposited sediment
itself. The flow models are just valid for very low sediment concentrations in
the fluid, because they neglect momentum exchange between fluid and particles.
Furthermore bed changes cannot be accounted for directly.
In this paper a method is presented that allows to overcome the difficulties mentioned above by utilising the 3D AVL-FIRE multi-phase CFD-code. Simulation results are validated with experiments from literature as well as with experiments that were recently carried out by Polz [5].
Calculation of
sedimentation is performed using the two-fluid model. The two-fluid model
conservation equations are obtained through the ensemble averaging process. As a
result, the local and instantaneous flow details have to be reintroduced by
modelling the turbulence and interfacial effects. Averaging itself is necessary
to obtain a mathematical description of the problem in such a form that allows
practical calculations.
The
actual amount of water (w) and sand (s) that is present in a computational cell
is presented by a volume fraction aw,s.
For the sedimentation problem water is assumed to be the continuous phase, sand
to be the dispersed phase. In each cell the compatibility condition must be
fulfilled:
1
= aw
+ as
(1)
For each phase
the conservation equations of mass and momentum have to be solved. Interaction
between the two phases (momentum exchange) is introduced via exchange terms.
The conservation
of mass in each phase is accounted for by using both a continuity equation for
water and sand:
(2)
Since there is no mass exchange between the two phases the right-hand
side of both equations is zero.
Assuming uniform
pressure field for both phases the momentum conservation equations can be
written as follows (i = w,s):
(3)
where the shear stress due to viscosity equals:
(4)
and the Reynolds stress equals:
(5)
The Reynolds stress in the momentum equations is calculated using the Boussinesq approximation:
(6)
The turbulent viscosity is defined as:
(7)
A standard k-e-model
that was originally developed for single phase flow [6], is used to account for
turbulence effects. In the following there is a list of assumptions that were
used in modelling the closure terms in the turbulence kinetic energy equation
for Euler/Euler simulations:
l
The assumptions
that lead to the closure of the k-e-model
in single phase flow are valid.
l Turbulence intensity of the dispersed phase is assumed to equal the continuous phase turbulence level.
l
The interfacial
interaction between the two phases is neglected.
The momentum
exchange term Mi in eqn
(3) needs detailed modelling for the problem of sedimentation. At the moment
drag and turbulent dispersion forces are included.
The formulation
for the drag force was substituted based on the drag force for a single
spherical particle.
(8)
The number of particles Np in one computational cell is given
by the volume of sand in one cell divided by the volume of one particle:
(9)
The total drag force FD,tot in one cell is the drag force for one particle multiplied by the number of particles.
(10)
Therefore the momentum exchange term for one cell can generally be
written as:
(11)
where ts
is the so called "particle relaxation
time":
(12)
The factor f generally depends on the drag coefficient CD, on
the particle Reynolds number ReP (14) as well as on the volume
fraction of sand.
For one isolated spherical particle f is given as:
(13)
where
(14)
As long as sand particles can be assumed to behave as isolated solid
spheres, the appropriate drag coefficient form is:
(15)
Equation (15) and eqn (13) are not valid for dense particle flows.
Gidaspow [7] suggests to use the relation published by Ergun [8] in
regions of very high particle concentrations aw
< 0.8. Crowe et al. [9] and Laux [10] come to similar conclusions.
(16)
Although in literature there are many additional approaches to account
for the drag force in dense particle flows [11], Ishii and Zuber [12], Gibilaro
et al. [13], the combination of the equations presented above has proven to be a
useful and practical approach.
The turbulent
dispersion force accounts for particle diffusion due to turbulent mixing
process. We assumed a const. value for cTD=0.1.
(17)
A parameter study demonstrated that the variation of cTD
between 0.05 and 0.2 had no significant influence on the results of the
simulation.
In order to
account for the effect of granular friction between particles the approach
presented by Laux [10] was implemented.
(18)
In order to
validate the model capabilities a straight laboratory flume was set up. The main
advantage of the straight flume is the uniform flow profile which can be
realized. Therefore the simulation with a two dimensional numerical model is
possible which reduces significantly the computational effort. Furthermore the
boundary conditions are known very well.
Fig. 1 Schematic drawing of the experimental set-up
Figure 1 shows a
schematic drawing of the experimental set-up. The test section has a length of
10m. The bed is made of rigid plates in order to allow bedforms, reentrainment
and redeposition of particles. Since the front plate of the flume is made of
glass, the visual investigation of all transport effects is possible during the
experiment. See Polz [5] for details of the experimental set-up and procedure.
Sediment is
supplied with a batching station above the water surface at the upstream end of
the channel. The mean velocity is 0.65 m/s and the water depth 0.25m.
Experiments were carried out with different sediment types, different water
velocities and different amounts of discharged material. In this paper
sedimentation behaviour and bed movement is presented for a "fine"
sand (d50=0.6mm) and a "rough" sand d50=1.6mm).
Sand is added for three minutes at the beginning of the experiment at a rate of
5.4kg/min. At 5, 10, 20 and 25 minutes the bedform is recorded using a digital
photo camera.
Figures 2 and 3
show the experimental results for both experiments at the four distinct time
steps. One can clearly see that the solid material is settling to the bottom of
the flume very fast downstream of the batching station by forming a bed. As
expected the bed is moving downstream with increasing time. The bed velocity
decreases with increasing particle diameter. After 25 minutes the axial position
of the bed is 6.25m for the fine particles and 5.5m for the rough
particles.
Fig. 2 Comparison of experiment and simulation: particle size d50=0.6mm
Fig. 3 Comparison of experiment and simulation: particle size d50=1.6mm
Although the
difference is rather small this effect is reproducible for additional
experiments, whereas the actual form and length of the deposited bed is not
reproducible. In general we found that the length of the bed varies between 1
and 1.5m, but slightly increases with increasing time of the experiment. All
experiments showed that for the given boundary conditions the maximum height of
bed is approximately 8cm.
In addition to
the experimental results Figures 2 and 3 also display contour plots of the
volume fraction of sand as
as a result of the numerical simulations that were carried out with the same
boundary conditions as the physical model. For all calculations the bed's wall
roughness is assumed to be 1mm. The sand discharge was modelled by an additional
inlet boundary as indicated in the figures. A simulation run takes approximately
5 hours on a typical mid-range workstation.
The agreement
between experiment and simulation is generally good, especially considering the
variations that occur when the experiments are repeated. The model predicts the
movement of the bed correctly. The velocity decreases with increasing particle
diameter and the axial positions predicted for the different time steps are
close to the experimental results. On the one hand the numerical model predicts
the bed-lengths a bit smaller than the experiments, on the other hand the
predicted maximum height of the beds is 10cm.
The Euler/Euler
model of the CFD-Code FIRE was successfully adopted for the simulation of
sedimentation and sediment transport. Comparisons of the numerical and the
experimental results show good agreement. The deviations between the results
can be related to the assumptions made during the model set-up.
The numerical
model accounts for one distinct particle diameter. Therefore one cannot expect
perfect agreement between experiment and simulation. Currently our model is
being adopted to a multi-phase model that will allow the simulation of more than
one particle size. The resulting bed forms depend mainly on the formulation of
the momentum exchange terms and of particle/particle interaction.
Further comparative studies will focus on factors such as particle size and shape. These parameters will be introduced into the sedimentation module to improve the ability of predicting sediment processes.
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Wu, W., Rodi, W. & Wenka, T. 3D Numerical Modeling of Flow and Sediment
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