Onset of sediment transport in mono and bidisperse beds under turbulent shear flow
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Abstract
In this paper, we present a study on the onset of sediment motion of particles under turbulent shear flow. We use a fully resolved coupled lattice Boltzmann–discrete element solver to study the behavior of monodisperse and bidisperse beds of spheres. For monodisperse beds, we find that the onset of sediment motion is in agreement with the predictions with models from the literature. For bidisperse beds, segregation phenomena at a short timescale are found that lead to increased entrainment of larger particles. This shows that, in addition to longterm segregation phenomena such as riverbed armoring, segregation also occurs at much shorter timescales.
Keywords
Lattice Boltzmann method Discrete element method Sediment transport Bidisperse bed Shields curve1 Introduction
Starting from the perspective of a single particle, the local bed morphology clearly plays a crucial role. The probability of a particle to be picked up by the flow significantly decreases when other particles are in close vicinity [1, 10]. Dancey et al. [10] found that the value of \(\Theta _{\text {c}}\) is highly sensitive to the planar packing density \(\lambda \) (area covered by particles over total area). Already a single nearby particle decreases, the chance of entrainment [1]. This underlines the necessity of detailed information on the flow field on the subparticle scale to understand the relevant mechanisms.
Resolved numerical modeling of particle beds under shear flow is computationally demanding, especially in the case of turbulent flow. Therefore, not many studies in that field exist. Pan and Banerjee [25] and Shao et al. [29] investigated channel flow with buoyant particles, but put more emphasis on the turbulence structure in the fluid than on particle dynamics. Kidanemariam and Uhlmann performed simulations of particle beds under laminar [21] and turbulent [20] flow and focused on the time evolution of the beds. All articles mentioned above addressed monodisperse systems, and to the best of the authors’ knowledge, no detailed simulations on polydisperse systems are found in the literature.
In this paper, we present fully resolved simulations of mono and bidisperse beds of spheres under turbulent shear flow. After explanation of the numerical methods and a validation example, we analyze the behavior of the bed with respect to motion threshold and identify a fast mechanism for segregation that could be of relevance in situations where the flow velocity varies frequently. To conclude, we give an outlook how to integrate such smallscale simulations with high resolution into less detailed ones on larger scales via an LBDEM magnification lens.
2 Computational methods
In this section, the simulation methods employed in the present study are explained. All algorithms were implemented by creating a coupling framework between the lattice Boltzmann code Palabos [15] and the discrete element method code LIGGGHTS [22]. This allows us to take advantage of robust and welltested software platforms. The coupling code, LBDEMcoupling ^{1} [27, 28] was released under a free license.
2.1 The lattice Boltzmann method
A lattice Boltzmann simulation is performed in repeating two steps: First, a new set of populations \(\{ f_i^* \}\) is computed using the righthand side of Eq. 2 (collision step). Then, the populations are redistributed according to the velocity vectors \(\mathbf {c}_i\): Each population \(f_i^* \left( \mathbf {x}\right) \) is shifted to the position \(\mathbf {x} + \mathbf {c}_i\). This is given by the lefthand side of Eq. 2 (streaming step). The simulation then continues with a new set of offequilibrium populations at each grid node.
2.2 The discrete element method
2.3 Coupling methodology
2.4 Corona boundary condition
3 Validation
Figure 2a , b shows comparisons of position and velocity over time between simulation and experiment. We see very good agreement for all Reynolds numbers except for the lowest one (\(\mathrm {Re}= 1.5\)), where the final stage is not depicted accurately. It is known that the LBM cannot properly compute the forces between moving objects when their distance becomes smaller than the grid spacing, so that lubrication corrections—the research topic of Ten Cate et al. [33]—might be necessary. Since in our simulations, no lubrication forces were considered, the slowest approach was not modeled accurately. The acceleration and maximum velocity, however, are correct. We therefore conclude that our implementation is able to properly depict the interactions between a moving sphere and a surrounding fluid.
4 Setup
Material properties of the particles used in all simulations
Property  Value 

Density  \(\rho _p = 2000\,\hbox {kg/m}^{3}\) 
Coefficient of friction  \(\mu _{\text {f}} = 0.45\) 
Coefficient of rolling friction  \(\mu _{\text {r}} = 0.02\) 
Coefficient of restitution  \(\alpha = 0.2\) 
Young’s modulus  \(Y = 10^6\,\hbox {Pa}\) 
Shear modulus  \(G = 3.57 \times 10^5\,\hbox {Pa}\) 
Poisson ratio  \(\nu = 0.4\) 

monodisperse (small) 2170 spheres with a diameter of \(d_{\text {p}} = 1\,\hbox {mm}\) and \({h_{\text {bed}} = 5.5\,\hbox {mm}}\)

monodisperse (large) 273 spheres with \(d_p = 2 \,\hbox {mm}\) and \(h_{\text {bed}}=6\,\hbox {mm}\)

bidisperse \(n_{\text {small}}=623\) spheres with \(d_{\text {p,small}}=1\,\hbox {mm}\) and \(n_{\text {large}}=213\) spheres with \(d_{\text {p,large}} = 2\,\hbox {mm}\) with \(h_{\text {bed}}=6\,\hbox {mm}\)
With respect to the particle diameter, the domain size was \(20^3 d_{\text {p}}\). This is comparable to the boxes used in some other studies [21, 25, 29]. However, Kidanemariam and Uhlmann [20] used a domain with streamwise extent of \(\approx 300 d_{\text {p}}\) and still reported influence of the box size. Therefore, we assume that our box was large enough to obtain qualitative findings, but cannot exclude box size influence on quantitative results.
5 Results and discussion
5.1 Monodisperse beds
In Fig. 4a, the fraction of moving particles over time is shown for the simulations with \(d_{\text {p}} = 1 \hbox {mm}\). Once transport was established, the fraction did not vary much anymore. Only the smallest considered fluid velocity \(u_{\text {top}} = 0.4 \,\hbox {m/s}\) could not sustain particle motion. Figure 5 shows the locations of the four cases in the Shields plot together with a case with \(d_{\text {p}} = 2 \hbox {mm}\), \(u_{\text {top}} = 1 \,\hbox {m/s}\), relative to the Shields curve. While the small grains were set in motion for all except the lowest fluid velocity, the large particles could not be moved in any of the investigated cases.
Interestingly, the parameter setting corresponding to \(u_{\text {top}} = 0.6 \,\hbox {m/s}\) lies below the Shields curve, but nonetheless features movement in our simulations. This can be explained by looking at the angle of repose. Shields used different types of sand and gravel with angular to subangular grain morphology in his experiments. Such materials are typically characterized by an angle of repose of \(35^{\circ } \le \phi _{\text {gravel}} \le 40^{\circ }\). According to Zhou et al. [41], spheres with rolling and sliding friction as used in the simulation typically show an angle of repose between \(23^{\circ }\) (\(d_{\text {s}} = 1 \hbox {mm}\)) and \(26 ^{\circ }\) (\(d_{\text {s}} = 2 \hbox {mm}\)). To take this parameter into account, we employed a semianalytic model proposed by Dey [11] and added the corresponding curves for \(\phi = 20^{\circ }\) and \(\phi = 25^{\circ }\) in Fig. 5. The case with \(u_{\text {top}} = 0.6 \,\hbox {m/s}\) lies between them, while the case with \(d_{\text {p}} = 2 \hbox {mm}\) is very close to the \(\phi = 20^{\circ }\) curve. Dey’s model predicts no movement for the large grains and movement for \(u_{\text {top}} = 0.6 \,\hbox {m/s}\), \(d_{\text {s}} = 1 \hbox {mm}\) for the small particles, just as found in the simulations.
5.2 Bidisperse bed
Closer inspection revealed that the most pronounced drop corresponding to \(u_{\text {top}} = 1.0 \,\hbox {m/s}\) was accompanied by a change in particle motion. Figure 6 shows the fractions of moving small, large, and all particles over time. For \(u_{\text {top}} = 0.8 \,\hbox {m/s}\), the large particles showed almost no movement during the whole simulation, while \(f_{\text {m,small}}\), and with it \(f_{\text {m}}\), decreased over time from 0.04 to 0.02. For \(u_{\text {top}} = 1.0 \,\hbox {m/s}\), however, a different situation was found: After approximately \(0.5\,\hbox {s}\), the large particles started moving. Once their motion was fully established, that of the of small particles drastically decreased until \(f_{\text {m,small}} \approx f_{\text {m,large}} \approx f_{\text {m}}\). At that point, particles of both sizes were in continuous motion.
Reynolds number \(\mathrm {Re}_{\text {f}}\) and actual and critical shear stress \(\Theta \) and \(\Theta _{\text {c}}\) for the two highest velocities computed with the average diameter \(D_{\text {m}}\)
\( u_{\mathrm{top}}\,\hbox {m/s}\)  \(\mathrm {Re}_{f}\)  \(\Theta \)  \({\Theta }_\mathrm{c}\) 

0.8  23.57  0.0287  0.0341 
1  25.98  0.0384  0.035 
Our findings are consistent with observed segregation phenomena in bidisperse beds known as riverbed armoring: a slow process in which the bed segregates in a top region with large particles and a region below with smaller particles [5, 19]. Recently, it has been found that such granular creep is very similar to kinetic sieving processes in dry granular flows [14]. The segregation phenomena we observed yield a similar outcome, but occur at much shorter time scales.
6 Conclusions and outlook
Connection between \(\lambda \) and \(\Theta _{\text {c}}\) from Dancey et al. [10]
\(\lambda \)  0.026  0.26  0.45  0.65  0.91  0.29 
\(\Theta _\mathrm{c}\)  0.008  0.0108  0.037  0.048  0.074  0.021 
\(\Theta _\mathrm{c} / \Theta _\mathrm{c}(\lambda =0.45)\)  0.216  0.292  1  1.297  2.0  0.56 
The transition from the presented, small test cases to realsized riverbeds is not straight forward. Fully resolved LBDEM simulations of processes spanning multiple time and length scales are prohibitively demanding for even the largest supercomputers. Therefore, we are aiming at magnification lens concepts: A fullscale simulation using continuum models is conducted. In this coarse simulation, a fully resolved, smallscale simulation is embedded and receives flow events, for example, via a corona boundary condition similar to the one presented in Sect. 2.4. The results of the smallscale simulation can then either be postprocessed individually or fed back to trigger events in the largescale model. To achieve this, appropriate boundary conditions need to be developed and validated.
Footnotes
Notes
Acknowledgements
Open access funding provided by Johannes Kepler University Linz.
Compliance with ethical standards
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
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