(1)
Abbas Yeganeh-Bakhtiary1, Hitoshi Gotoh2
and Tetsuo
Sakai3
1 JSPS Postdoctoral Fellow, Graduate School of Civil Engineering, Kyoto University, Yoshida Hon-machi, Sakyo-ku, Kyoto 606-8501, Japan. Voice: +81-75-753-5099,
Fax: +81-75-761-0646, E-mail: yeganeh@coast.kuciv.kyoto-u.ac.jp
2 Associate Professor, Graduate School of Civil Engineering, Kyoto University, ditto.
Voice & Fax: +81-75-753-5098, E-mail: gotoh@coast.kuciv.kyoto-u.ac.jp
3 Professor, Graduate School of Civil Engineering, Kyoto University, ditto.
Voice & Fax: +81-75-753-5097, E-mail: sakai@coast.kuciv.kyoto-u.ac.jp
Abstract:
A two-phase flow model is presented for the simulation of sediment transport at
high bottom shear stress with focusing on characteristics of the movable bed
layer. The model is constructed with Euler-Lagrange coupling of fluid and
sediment phases. The flow phase is computed by solving the Reynolds Averaged
Navier-Stokes (RANS) equation in conjunction with the k-e
turbulence model. The sediment transport is described as the motion of a
granular material under the action of steady flow, on the basis of the Distinct
Element Method (DEM), with accounting for the very frequent interparticle
collision of the moving particles at the hyper-concentrated layer. The
characteristics of the mean-flow velocity on a movable bed are well reproduced
by the present model, and the internal structure of flow is discussed based on
the result of the flow turbulent properties. In addition, the characteristics of
mean velocity profile of sediment phase is studied and discussed in light of the
interphase momentum transfer. Finally, the model result as movable bed transport
has been compared with the result of the single-particle-motion model of
irregular successive saltation over fixed bed.
Keywords:
two-phase
flow, granular material model, hyper-concentrated flow, rans, k-e turbulence model, dem
Experimental studies reveal that the micro-mechanism of sediment transport at low shear stress is different from that of the high shear stress, which induces a movable bed transport layer. For example, Yeganeh (1997) described that the mean-flow velocity under low-to-moderate bottom shear tends to a two-layer type profile, while at high bottom shear the so-called three-layer type profile appears. The different characteristics in the mean-flow velocity is attributed to the fact that at high bottom shear stress, sediment transport takes place in a hyper-concentrated layer with a thickness of several bed grains, namely sheet flow mode, in which the frequent interparticle collision is the predominant micro-mechanism of transport.
On the other hand, two-phase flow model provides more deep information on the hydrodynamics of sediment transport. Wiberg and Smith (1989) presented a two-phase flow model by coupling of the irregular successive saltation with mixing length model. Then, Gotoh et al. (1994) extended the two-phase flow concept by coupling of the irregular successive saltation with the k-e turbulence model. The afore-mentioned models simulate the sediment transport at low-to-moderate bottom shear stress at which the fluid-particle interaction is the dominant micro-mechanism. Yeganeh et al. (2000) implemented an Euler-Lagrange coupling model with assuming the single-particle motion for moving sediments at high bottom shear stress. By comparing with experiments, they concluded on the limitation of single-particle motion for sediment transport because of neglecting the existence of predominant particle-particle interaction in sheet flow layer. In this study, we have updated our pervious model with introducing the sediment transport as a motion of granular material, in which the predominant micro-mechanism of transport at high bottom shear stress, namely the particle-particle interaction of moving particles is taken into account in detail. Moreover, the result of the present model is compared with that of the single-particle motion, and the different characteristics of the two cases have been discussed in light of the micro-mechanism of sediment motion.
The fluid phase is computed with the vertically two-dimensional Reynolds Averged Navier-Stokes equation in conjunction with the k-e - turbulence model under a steady flow condition, (for more details see Yeganeh et al., 2000).
The sediment particles are modeled as disk particles with the uniform diameters, and their motions are numerically traced in a vertically two-dimensional plane. At each contacting point of particles, spring-dashpot system is introduced to express the particle-particle interaction. The equation of motion of the ith particle is as follows:
(1)
(2)
(3)
in which CM=added mass coefficient (=0.5); A2, A3=two- and three-dimensional geometrical coefficients; d=diameter of particle; upi, vpi=velocity of particles in x and y directions; fn, fs=normal and tangential forces acting between ith and jth particles at local coordinate system n-s; CD=drag coefficient; aij=contacting angle between ith and jth particles; upi, vpi=velocity components of particle in x and y directions; and wpi=rotational angle of the ith particle.
Between each contacting particles the spring-dashpot system should be activated in both normal and tangential directions of the n-s local coordinate, as shown in Fig.1. The estimation method of forces acting between two contacting particles is mentioned by Gotoh and Sakai (1998).
Both sides of the calculating domain are periodic to save the calculation time. At the bottom, the wall function is implemented as the boundary condition, namely the logarithmic law holds between the wall and its adjacent grid point. The test particle is 0.5 cm in diameter and 2.65 in specific gravity. The origin of the vertical coordinate is placed at the bottom of the constituting particles at the initial state of the calculation.
Fig. 1 Interaction system between contacting particles, Gotoh and Sakai (1998).
Figure 2
depicts the simulation result of mean-flow velocity on a movable bed and that
with saltation. The mean-flow velocity shows tendency towards the so-called
three-layer type profile, while the saltation flow is clearly tends to a
two-layer type one. Three-layer type profile indicates the coexistence of
different modes of transport in the movable bed transport case, i.g., sheet flow and
saltation. That part in velocity profile follows rather a power law at y/d<10,
apparently shows the sheet flow layer thickness as suggested by Sumer
et al. (1996). The linear part with milder gradient above sheet flow layer
reflects the effects of saltating grains on the mean-flow; whereas, the flow
region with no sediment shows a linear gradient of 1/k log10e.
(a) Movable bed transport (b) Saltation transport
Fig. 2 Mean-flow velocity,two-and three-layer type profile.
On the other hand, the two-layer type
profile exhibits two linear distribution with different gradients corresponding
to saltation layer and flow with no sediment region with the same gradient as
the movable bed transport.
Figure 3 shows the Reynolds stress as well as distribution of turbulent energy and kinematic eddy viscosity for both of a movable bed transport and a saltation flow. Reynolds stress of both cases shows a linear distribution up to the region where the effect of saltating particle is not significant, then it deviates to an upward concave profile with reaching to a peak value.
(a) Reynolds stress distribution
(b) Turbulent energy
(c) Kinematic eddy viscosity
Fig. 3 Reynolds stress and fluid turbulent properties of movable bed transport and saltation flow.
For the case of movable bed transport, however,
Reynolds stress increases, to some extent, in the neighborhood of the original
bed surface or at y/h=0.0; then it decreases with going deep region of depositing
sediment layer. In the sheet flow layer, an upward convex profile of the
Reynolds stress can be observed, which is identical to the Reynolds stress
distribution of the flow over a rigid vegetation as pointed out by Tsujimoto et
al. (1992).
The turbulent energy of the movable bed transport and saltating flow show a peak at two different elevations. In the case of saltating flow, it coincides with the elevation of the maximum concentration of saltating particle, (see Fig. 3.b). Also rather a linear distribution can be detected for the turbulent energy in the depositing sediment layer of movable bed transport. Moreover, the vertical distribution of kinematic eddy viscosity of the movable bed transport shows the existence of two peaks. The first peak at y/h=0.5 locates inside of the saltation layer and the second one exists in the sheet flow layer. While for the flow with saltating particles one peak in the eddy viscosity is observed.
Mean velocity and concentration of moving particle for both cases of movable bed transport and saltation are shown in Fig.4. Close observation of mean velocity of movable bed transport inside of the sheet flow layer indicates on a laminar motion of moving particles with an upward convex shape of distribution. With moving upward after reaching to a inflection point the velocity profile changes smoothly its shape to the upward concave one. On the other hand, mean velocity of saltating particles tends to an upward concave profile similar to upper part of the profile of movable bed transport. A peak in concentration of saltating particles can be observed at a point over the fixed bed, while for the case of movable bed transport maximum concentration takes place in the lower section of sediment depositing layer.
(a) Movable bed transport (b) Saltation transport
Fig. 4 Mean-flow velocity and averaged particle velocity as well as concentration.
In this study, the predominant micro-mechanism
in movable bed transport is simulated on the basis of the momentum transfer,
namely fluid-particle and particle-particle interaction system. Hence the
characteristics of sub-layer transport is attributed to the vertical momentum
transfer between fluid and sediment phases. In the sheet flow layer, the
vertical motion of moving particles is inactive due to the high concentration of
moving particles, which induces a frequent interparticle collision; therefore,
the particle momentum in this layer is preserved and the velocity gradient is
rather large. With the smooth shift to zero velocity being considered at the
bottom of the high-density moving layer, the velocity profile shows an upward
convex one. In contrast, in the saltation layer fewer particles are picked up by
the direct action of fluid, but vertical motion of particle is significantly
active to exchange momentum between the upper and lower layers, hence the
velocity gradient decreases. Interphase momentum transfer is promoted, because
of the frequent interparticle collision decreases than that of the sheet flow
layer. Thus, the velocity profile follows an upward concave one.
In this study an Euler-Lagrange two-phase flow model is presented for the simulation of bed-load transport at high bottom shear stress by fully coupling of the RANS model with a granular material model. The hydrodynamics of the movable bed transport is simulated with expressing both the interparticle collision and the fluid-particle interaction.
The characteristics of mean-flow velocity as the
three-layer type profile is well reproduced, which indicates the coexistence of
different modes of sediment particle motion. The presence of sub-layers in the
movable bed transport case is explained in light of the differences in the
vertical momentum transfer. The upward concave profile in the saltation layer is
a result of the active interphase momentum exchange, which caused by the
momentum transfer owing to the inter-layer mixing.
Acknowledgements
The first author expresses his gratitude to Japan Society for the Promotion of Science (JSPS) for providing financial support to this research study.
References
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