A hybrid DEM CFD approach for solid-liquid flows(2)

density, respectively. The modified Tait EOS is inde-

pendent of entropy and hence the energy equation is not required[11]. For all simulations presented in this

paper, =

B3.0×108 and n=7 are assumed, as sugge- sted by Brujan[11].

1.2 Governing equations for particles

The particle motion is governed by the Newton’s second law of motion as follows:

mdvi

idt

=fci+ffpi+mig (10)IdZi

=Ti (11)

in which mi and Iiare the mass and the moment of inertia of the particle, respectively, vi and Zi are the linear and angular velocities of the particle, Ti is the torque arising from the tangential components of the contact force,fciis the particle-particle and parti- cle-wall contact force and ffpiis the fluid-particle in- teraction force.

The particle-wall interactions in this study are treated in the same way as the particle-particle intera- ctions except that a wall is assumed to be a particle with infinite mass and diameter. The interactions bet- ween particles are modelled using the classical contact mechanics, in which the Hertz theory is employed to analyze the normal interaction, and the theory of Mindlin and Deresiewicz[13] is used to model the tan- gential interaction. The evolution of contacts can thus be determined explicitly by using realistic physical properties, such as the Young’s modulus and the Poisson’s ratio. This is superior to the spring-dashpot type contact models used in the DEM codes, in which artificial contact stiffnesses are used, and the link with the actual physical properties of the particles is obscu- re.

1.3 Numerical strategy

In the present numerical scheme, the computatio- nal domain is discretised into a number of fluid cells. All quantities such as the fluid pressure and the velo-

city are averaged in the fluid cells. As suggested by Tsuji et al.[14], the size of the fluid cells should be smaller than the macroscopic motion of bubbles, but larger than the particle size. A fluid cell size of 3-5 times the particle diameter was recommended by Tsuji et al.[14] and was followed by many others[9,10]. The numerical scheme employed here is identical to that given by Kafui et al.[9]. At each time step, the equa- tions of motion for particles are first solved to obtain the particle positions and velocities. Then the porosity of each computational cell can be calculated. Using the known particle positions and velocities, the fluid- phase hydrodynamic equations are then solved to ob- tain the fluid velocity and pressure fields, from which the drag forces acting on particles can then be compu- ted, as well as the fluid-particle interaction force. The forces acting on particles and each fluid cell are then updated in order to determine the positions and veloci- ties of particles at the next time step. This procedure is repeated until a specified time (or a specific number of time steps) is reached.

Fig.1 Schematic diagram of the sphere settling in water

Fig.2 Evolutions of vertical position of the sphere

The time step 't in the discrete particle model is determined by the smallest particle diameter dmin,leading to a critical time step 'tc given by

[9]

'tc=

(12)

Fig.3 Snapshots of the sphere positions and the vertical velocity of the flow around the sphere at a given instant

whereO=0.1631Q+0.876605,G,UpandQare the shear modulus, the density and the Poisson’s ratio of the particle, respectively. In our simulations, the time step is specified as 't=D'tc and the value of the constant D is normally less than 0.5, depending on the problem considered.

The maximum allowable time step for a CFD solver is limited by the Courant condition and the vis- cosity stability criterion. In general, the time step dete- rmined in the DEM part of the code has been found to be much smaller than the maximum determined by using the CFL criterion in CFD part[9], thus the global time step is determined in the DEM part. 2. Numerical examples

2.1 Settling of a single spherical particle in water

To evaluate the reliability and accuracy of the proposed DEM/CFD program, we first carried out a three dimensional simulation of the sedimentation of a single spherical particle in water. The simulation con- ditions are identical to those employed in the experi- ments of Li[15]. This experiment provides an effective validation for the proposed method since the sphere is released from zero velocity under a quiescent ambient fluid condition, which makes the initial condition for the simulation easy to enforce. A steel sphere of 0.0095 m in diameter is used in the experiment. The properties of the solid material include the solid den-

Up7 780 kg/m3, the Young’s modulus E=sity =

200 GPa, and the Poisson’s ratio Q=0.33. The water density is 1 000 kg/m3 and the dynamic viscosity of the water is 10??3 kg/ms. In the experiment, a steel sphere immerged in water is released with zero initial velocity (see Fig.1). The resulting trajectories of the impact and rebound particles are compared with the experimental data of Li[15].

The computational domain is 0.040 m×0.040 m×

y and z-directions, respecti- 0.055 m in the x-, -vely. Gravity acts in the z-direction with the gravita- gz–9.81 m/s2. In the simulatio- tionalacceleration =

ns, walls are treated as impermeable with no slip

boundaries. The computational domain is discretised with a grid of 5×5×6 fluid cells. The time step is spe- cified as 't=D'tc and the value of the constant Dis set to 0.5 and 'tc is calculated using Eq.(12).

Figure 2 shows the evolutions of the vertical po- sition h for the solid sphere, in which the experime- ntal result of Li[15] is also displayed, and our numeri- cal results are shown to be in agreement with the ex- perimental data at an acceptable level. It is clear that once the sphere is released, it accelerates under gra- vity until touches the container bottom, at which the sphere boundary will contact the wall and then rebou- nd. Thereafter, the sphere repeatedly contacts and re- bounds in water until reaches a rest position due to the damping of water.

The numerical simulation using the proposed model captures not only the dynamics of the particles but also the evolution of the surrounding flow field during the falling and rebounding process. Figure 3 shows the snapshots of the sphere position and the vertical velocity field of the flow around the sphere at different instants. The first instants of contact between the sphere and the container bottom is about t=0.1 s (see Fig.3(b)). Figure 3(c) shows an instant of the sphere rebound after the contact. The simulated results indicate that the present DEM/CFD method can captu- re the sedimentation process in a viscous liquid inclu- ding the trajectory of a particle as it accelerates from zero velocity after release, collides with a wall, rebou- nds and falls again until it comes to rest.

2.2Two equal spheres settling in a closed container

filled with liquids

In order to further validate the presented method, we perform a 3-D simulation of the motion of two equal spheres sedimenting in a closed container filled with a viscous liquid and reproduce the drafting, kis- sing and tumbling phenomenon as a very important mechanism that controls the particle microstructure in flows of a Newtonian fluid.

The dimensions of the container are 0.080 m× 0.080m×0.400minthex-, -y and z

-directions,

Fig.4Snapshots of the process of drafting-kissing-tumbling. Color in slice indicates the values of vertical velocity field of the flow

around the spheres at a given instant

respectively. Gravity acts in the z-direction with the gravitational accelerationgz=–9.81 m/s2. The diame- ter of the two spheres isd=0.015 m. The mass densi- ties of the spheres and the liquid areUp=2 800 kg/m3andUf=1 000 kg/m3, respectively. The dynamic vis- cosity of the liquid is 10??3 kg/ms. In the simulations, the walls are treated as impermeable with no slip boundaries. The computational domain is discretised with a grid of 7×7×35 fluid cells. The time step is spe- cified as 't=D'tc and the value of the constant Dis set to 0.5 and 'tc is calculated using Eq.(12).