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TECHNICAL PAPERS

# Complex Flow Dynamics in Dense Granular Flows—Part II: Simulations

[+] Author and Article Information
Piroz Zamankhan

Laboratory of Computational Fluid & BioFluid Dynamics, Lappeenranta University of Technology, Lappeenranta 53851, Finland and Power and Water University of Technology, School of Energy Engineering, P.O. Box 16765-1719, Tehran, Iranqpz002000@yahoo.com

Jun Huang

Laboratory of Computational Fluid & BioFluid Dynamics, Lappeenranta University of Technology, Lappeenranta 53851, Finland

J. Appl. Mech 74(4), 691-702 (Sep 19, 2006) (12 pages) doi:10.1115/1.2711219 History: Received January 02, 2005; Revised September 19, 2006

## Abstract

By applying a methodology useful for analysis of complex fluids based on a synergistic combination of experiments, computer simulations, and theoretical investigation, a model was built to investigate the fluid dynamics of granular flows in an intermediate regime, where both collisional and frictional interactions may affect the flow behavior. In Part I, experiments were described using a modified Newton’s Cradle device to obtain values for the viscous damping coefficient, which were scarce in the literature. This paper discusses detailed simulations of frictional interactions between the grains during a binary collision by employing a numerical model based on finite element methods. Numerical results are presented of slipping, and sticking motions of a first grain over the second one. The key was to utilize the results of the aforementioned comprehensive model in order to provide a simplified model for accurate and efficient granular-flow simulations with which the qualitative trends observed in the experiments can be captured. To validate the model, large scale simulations were performed for the specific case of granular flow in a rapidly spinning bucket. The model was able to reproduce experimentally observed flow phenomena, such as the formation of a depression in the center of the bucket spinning at high frequency of $100rad/s$. This agreement suggests that the model may be a useful tool for the prediction of dense granular flows in industrial applications, but highlights the need for further experimental investigation of granular flows in order to refine the model.

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## Figures

Figure 1

(a) A steep depression observed in flowing fine grains through a small hole in silos. (b) Surface shape at high rotation rate reported in Baxter and Yeung (6).

Figure 2

(a) Two monosized colliding, rough spheres with diameter of σ at the beginning of the approach period. The sphere on the left with the axial and tangential velocity components of Vx, and Vy, respectively, is brought into contact with the sphere on the right, which is initially stationary. They touch initially at a single point C. (b) 3D finite element mesh for the spheres as shown in (a). More than 4×105 tetrahedral elements were used in the numerical treatments. The elements at the vicinity of point of initial contact are magnified in the inset (c). The spatial distribution of nodes used in the numerical treatments. Notice the fine zone in the vicinity of contact area where the gradient of stresses and strains are high.

Figure 3

(a) Two elastic balls are deformed in the vicinity of their point of first contact, C, under the action of the normal and tangential forces due to collision. The contact area is generally finite though small compared with the dimensions of the balls. Notice that the size of gray color contact area is exaggerated. (b) Contour plot of the computed instantaneous normal stress, σxx, on a cutting xy plane passing through the centers of the balls. (Inset) The contact area is magnified to provide a better visualization. (Right) Contour plot of σxx on a cutting yz plane perpendicular to the line of the centers. The position of the cutting yz plane is shown in (a). (c) Contour plot of the computed instantaneous shear stress, σxy, on a cutting xy plane passing through the centers of the balls. (Inset) The contact area is magnified to provide a better visualization. (Right) Contour plot of σxy on a cutting yz plane perpendicular to r(ij): (d) Contour plot of the computed instantaneous effective stress, σeff, on a cutting xy plane passing through the centers of the balls with the contact area is magnified in the inset to provide a better visualization. (Right) Contour plot of σeff on a cutting yz plane perpendicular to r(ij).

Figure 4

(a) Computed instantaneous contours of the normal stress, σxx, on a cutting xy plane at t=0.06ms. (b) Computed instantaneous contours shear stress, σxy, on the same cutting plane as in (a), at t=0.06ms and μ=0. (c) Computed instantaneous contours shear stress, σxy as in (b) for μ=0.4. (d) The tangential component of impact velocities of the colliding balls versus time. The diamonds represent the tangential impact velocity of the initially moving ball and the gradients are those of the initially stationary ball.

Figure 5

(a) Computed instantaneous contours of the normal stress σxx on a cutting xy plane at t=0.06ms. (b) Computed instantaneous contours shear stress σxy on the same cutting plane as in (a) at t=0.06ms. Notice that in this case, a normal contact is sheared by a tangential force, which is insufficient to cause failure. (c) The tangential component of impact velocities of the colliding balls versus time. The diamonds represent the tangential impact velocity of the initially moving ball and the gradients are those of the initially stationary ball.

Figure 6

(a) The arrangement of the super-ball and the wall with some nomenclatures. (b) 3D finite element mesh used in numerical treatments. Here, more than 6×104 hexahedral elements were used. (c) The thrown superball at low speed having a clockwise spin. The initial velocity components of the superball (before collision) were Vx=0.92m∕s, Vy=−0.24m∕s, and Vz=0. Here, the surface of the ball is color coded using the local magnitude of Vy. (d) The incident of the superball on the front side of the flat wall. The configuration was taken at the end of approaching period when the normal pressure at the initial point of contact reached to the maximum value. (e) The backwards spinning superball. Notice the spin reversal which is clearly illustrated in (c) and (e). In this simulation the back side of the wall was fixed.

Figure 8

(a) The computed angular velocities, ωz, versus time for the colliding balls. The diamonds and the gradients represent the results of finite element based model for the initially moving and the initially stationary balls, respectively. The solid lines are the obtained results using the simplified model. As expected, the angular velocity of the identical balls at the end of restitution period is equal to each other. (b) The computed normal velocities, Vx, versus time for the colliding balls. The diamonds, the gradients and the solid lines have the same meaning as those in (a). (c) The computed tangential velocities, Vy, versus time for the colliding balls. The diamonds, the gradients and the solid lines have the same meaning as those in (a). (d) Variations of the normal stress, σxx, and the shear stress, σxy, at the point of initial contact with time for the initially moving ball. The diamonds and the gradients represent the normal and the shear stresses, respectively, obtained using the finite element based model. The lower solid line represents the predictions of the simplified model for the normal stress of the point of contact on the surface of the initially moving ball. The upper line represents the predictions of the simplified model for the shear stress at the point of contact.

Figure 9

(a) Time series of x, y, and z components of impact force, represented by diamonds, squares, and circles, respectively. As well as the magnitude of impact force on the darker particle, depicted in (b), the magnitude of impact force is shown with a solid line. (b) Closeup of a cluster of ten particles.

Figure 10

A cutaway view of the spinning bucket. Note that gravity acts in the negative z direction.

Figure 11

(a) Initial configuration of the spherical balls with a flat free surface before the spinning begins. (b) Configuration of the spherical balls after five complete rotations at the rate of ω0=1001∕s. (c) Contour plot of the time smoothing of the volume averaged solids after five complete rotations of the bucket. (d) A typical instantaneous configuration of spherical balls in a cutaway view of the spinning bucket after several rotations at the rate of ω0=1001∕s. To provide a clear picture of depression the central part of the bucket is magnified in the inset. Notice that the diameter of identical particles is much smaller than that shown in the inset. (e) An instantaneous configuration of the spherical balls. The balls are color coded using the local value of the normal component of Π. (f) Time smoothing of the volume averaged of normal component of Pαβ. (Inset) Time smoothing of the volume averaged of normal component of Pαβ in the central part of bucket using a finer scale.

Figure 12

(a) A typical instantaneous configuration of spherical balls in a cutaway view of the spinning bucket after several rotations at the rate of ω0=501∕s. Notice the formation of a cusp in the central region. (b) Contour plot of the time-smoothing of the volume averaged solids after six complete rotations of the bucket at the rate of ω0=501∕s.

Figure 7

(a) The computed angular velocities, ωz, versus time for the colliding balls. The diamonds and the gradients represent the results of finite element based model for the initially moving and the initially stationary balls, respectively. The solid lines are the obtained results using the simplified model. As expected, the angular velocity of the identical balls at the end of restitution period is equal to each other. (b) The computed normal velocities, Vx, versus time for the colliding balls. The diamonds, the gradients and the solid lines have the same meaning as those in (a). (c) The computed tangential velocities, Vy, versus time for the colliding balls. The diamonds, the gradients and the solid lines have the same meaning as those in (a). (d) Variations of the normal stress, σxx, and the shear stress, σxy, at the point of initial contact with time for the initially moving ball. The diamonds and the gradients represent the normal and the shear stresses, respectively, obtained using the finite element based model. The lower solid line represents the predictions of the simplified model for the normal stress of the point of contact on the surface of the initially moving ball. The agreement between the results of two models is quite satisfactory. The upper line represents the predictions of the simplified model for the shear stress at the point of contact. The maximum shear stress predicted by the simplified model is almost twice as big as that predicted by the finite element based model. However, as can be seen from Fig. 4, the position of the maximum shear stress calculated using the finite element based model is not at the initial point of contact.

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