A hybrid DEM CFD approach for solid-liquid flows(3)

At t=0 s the two spheres are placed in the con- tainer above each other at zc=0.36 m and zc=0.38 m, respectively. The upper and lower spheres have the same horizontal position with xc=yc=0.04 m. The fluid and the spheres are initially at rest.

Figure 4 displays a sequence of the numerically simulated process of drafting-kissing-tumbling. After the two spheres are released in the container at t=0 s, they fall downwards under gravity and there is a wake with low pressure at the back of a sedimenting sphere. Since the upper sphere is caught in the wake of the lower sphere, it experiences a reduced drag from the surrounding fluid and thus falls faster than the lower sphere. This is called drafting, after the well-known bicycle racing strategy based on the same principle. Figures 4(a)-4(c) show the snapshots of the drafting process. The increased speed of falling impels the upper sphere into a kissing contact with the lower sphere. Kissing particles form a long body and the vertical alignment during this stage is unstable and the sphere move quickly around each other towards a more horizontal alignment. Figures 4(c)-4(e) show the snapshots of the kissing process. At the end of the kissing stage the spheres move apart from each other, referred to as the tumbling stage. This is illustrated in Figs.4(e)-4(g).

2.3Sedimentation of suspension composed of 10 000

particles

Finally, we consider the sedimentation of suspe- nsion composed of 10 000 particles in a closed con- tainer. This test problem differs significantly from those considered above since a much larger number of particles are involved. The aim of this simulation is to show that the proposed method can handle much more complex flows with large number of moving particles. In suspensions with particles of different densities, the particles often settle differently in the vertical dire- ction, leading to the well-known density-driven sepa- ration phenomenon. A 3-D simulation of the density- driven separation is performed in this section using the proposed DEM/CFD. The dimensions of the con- tainer are 0.100 m × 0.100 m×0.160 m in thex-,y-andz-directions, respectively. Gravity acts in thez-direction with the gravitational acceleration gz=–9.81 m/s2. The container is fully filled with water and 10 000 particles of 0.005 m in diameter are randomly distributed in the water, among which 5 000 particles have the density of1 200 kg/m3(i.e., heavy particles) and the other 5 000 particle have the density of 800 kg/m3 (i.e., light particles). The Young’s modu- lus of the particles is 8.7u109 Pa and the Poisson’s ratio is 0.3. The mass density of the water is

1 000 kg/m and the dynamic viscosity of the water is 10–3 kg/ms. In our simulations, the walls are treated as impermeable with no slip boundaries. Particles are initially in the stationary state, but are set to move under gravity and the liquid-particle interaction forces once they are released. Figure 5 shows the particle profiles at various time instants during the sedimenta- tion, with the initial packing patterns of the particles in the container showing in Fig.5(a). It can be seen that, once the particles are released, the heavy parti- cles move towards the bottom of the container while the light ones float upwards (Figs.5(b) and 5(c)). Eve- ntuallytheseparticlesareseparatedwithheavy

ones

Fig.5 Snapshots of sedimentation of a suspension of heavy (black) and light (gray ) particles

settled onto the bottom of the container while light ones floating at the upper region of the container. It can be noticed that some heavy particles are entrapped inside the layer of the light one, this happens because the high particle concentration prevents some indivi- dual particle to sift through the gap between counte- rpart particles, instead, they are entrained with them.

centre of particle i. The corresponding evolutions of mass centre and velocity are shown in Fig.6. It can be seen from the left column of Fig.6 that the mass cen- tres for the two particle groups are located in the mi- ddle height of the container (i.e., 0.08 m). Once the particles are released, the centre of mass for the light particles gradually raises to a plateau, while that for the heavy ones sinks until a steady value is reached. The rates of the changes in the mass centres for both particle groups are similar. This is further confirmed in the right column of Fig.6, which shows the time evolutions of the average particle velocity for the two particle groups. It is clear that the vertical average velocities are suddenly increased once they are relea- sed. The heavy particles travel at a negative velocity, i.e., downwards, while the light particles move at a positive velocity, i.e. upwards. The velocity starts to decrease as they congregate at the bottom and upper regions of the container. And the velocity eventually reaches zero as the separation process completes. It is interesting to notice that the separation rates for the two particle groups considered, especially with the chosen particle densities, are essentially identical. This is due to the fact that the density difference bet- ween these two particle groups and the fluid are ide- ntical at 200 kg/m3.

3. Conclusion

A hybrid DEM-CFD method for the numerical simulation of particulate flows is presented. In the proposed approach, the motions of the solid particles obtained by using the DEM while the motion of the liquid is obtained by using the weakly compressible volume-averaged Navier-Stokes equations of conti- nuity and momentum with an additional stiff equation of state. 3-D Numerical simulations of the sedimenta- tion of a single spherical particle and the drafting, kis- sing and tumbling of two particles are carried out to validate the method. The separation of a binary suspe- nsion is simulated to demonstrate that the present DEM-CFD method is capable of modelling complex

flows with large number of moving particles. The

Fig.6 Evolutions of mass centres (a) and velocities (b) of two

particle groups

To quantify the sedimentation process, the ave- rage centre of mass, zav, and average vertical velo- city, wav, of these two particle groups are calculated using zav=N

??1

i=1

and wav=N

??1

|w, where

ii=1

is the number of heavy or light particles and wi and zi are the vertical velocity and position of the mass

numerical results show that the presented method pro- vides a robust and efficient approach to simulate solid-liquid flows. References

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