Granular jet composed of elliptical particles impacting a fixed target
Graphical Abstract
Introduction
Recently, some experimental studies reported that granular materials could behave like a perfect fluid when the size of the granular particles is small enough. For example, when a granular material is shot from a tube by pressurized gas and impacts a fixed target, similar to liquid, a sheet-like or cone-like formation, is produced in the wake zone [1]. Numerical simulations were carried out to explain such kind of behavior. Huang et al. [2] used kinetic analysis and energy dissipation to explain the opening angle of the shock, Ψ, when the jet impacts a fixed target. They classified the flow field near the target into three zones; a stagnation zone, a fast speed zone and a wake zone. The stagnation zone forms close to the target, where the particles are almost immobile. The fast speed zone is located out of the stagnation zone, where the particles are densely packed with shear interaction. In the wake zone (also called free zone), the particles are almost free-moving with low density and without collision. The effect of surface friction and coefficients of restitution were discussed by Guttenberg [3] and Huang et al. [4]. While Sano & Hayakawa carried out three-dimensional simulations to show the structure in the stagnation zone [5], [6]. So far, all these numerical simulations mentioned above used spherical particles for three-dimensional problems or circular particles for two-dimensional problems. In this article, elliptical particles were used to replace circular particles in the Discrete Element Method (DEM) simulation. All simulations were carried out employing our DEM program, ÅDEM.
Several methods were developed to describe the shape of a non-circular (spherical) particle [7], [8], such as glued-particle method (GPM) [9], polyhedra method [10] and super-ellipsoid method [11], [12]. In this article, we used the GPM, which is also called multi-sphere particle, to connect several overlapping circular particles to form an elliptical particle [9], [13]. In order to make the results from elliptical and circular particles comparable, the areas of these particles were set to be the same. Two different aspect ratios for the elliptical particle were selected, as ϑ = 2.0 and 4.0, where ϑ = ra/rb. Here ra and rb are the semi-major axis and semi-minor axis respectively, as shown in Fig. 1. Theoretically, as more slave particles are employed to comprise a complex particle, the better the result can be obtained [7]. Seven circular particles, p = 7, were used to compose each elliptical particle in all simulations about ϑ = 4.0, as shown in Fig. 1 (c). To validate the selection for the composition number, p = 7, is enough for such problems, parts of those simulations were repeated by using the model of Fig. 1 (d), with p = 13.
Lot of researchers [8], [14], [15], [16] used DEM coupling with the CFD technique to simulate liquid–solid flows since the pioneering work of Tusji et al. [17]. The surrounding fluid plays an important role when the particles are small or the viscosity of the fluid is high. Since the diameters of those particles used in this paper were about millimeter in size, the flow field was mainly controlled by the transport of momentum via direct frictional and collisional inelastic contacts between the particles [18]. Furthermore, using a single phase can lead to a better understanding of the effect due to the geometrical profile. This strategy used was the same as some previous works [2], [3], [5], [6].
The structure of this article is as follows: After the introduction of computational models in Section 2, some results are presented in Section 3. Based on the comparison between elliptical and circular particles, conclusions are drawn in the last section.
Section snippets
Methodology
As illustrated in Fig. 1, an elliptical particle can be modelled by connecting several overlapping circular particles using GPM. One circular particle is called a host particle, at which the mass center of the elliptical particle is located. While, the rest of the circular particles are called slave particles. The geometry radius of the i −th circular particle is given by, . The position and radii of all the circles composing the ellipse of aspect ratio ϑ = 2.0 , shown in Fig. 1 (b)
Results and discussion
Side views of both circular and elliptical particles impacting the different sized targets are presented in Fig. 4. For an individual particle in the wake zone, the individual scattering angle is defined as, The distribution curves of the scattering angle are plotted in Fig. 6. Since the limited number of particles in the wake zone, the statistical samples were collected from several adjacent instantaneous recorded moments of time steps when the flow field near the target gets
Conclusions
It was reported that granular materials perform like perfect fluids for the problem of granular jet impacting a fixed target, if the size of the granular particles is small enough. A theory points out that this phenomenon is affected by kinetic dissipation [2]. To validate this theory, elliptical particles with different aspect ratios were used to replace circular particles by using GPM, which is a common and simple way to approximate a particle with a complex profile. The validity and
Letters
- A
area
scale parameter in MB distribution, Eq. (9)
- C
coordination number
- D
diameter
- d
distance between two particles
- e
coefficient of restitution
- e′
coefficient of restitution obtained from simulation
- F
magnitude of force
- F
force in vector form
- fθ
scattering angle distribution
- fu
velocity probability density function
- I
momentum of inertia
- Kn
effective stiffness coefficient in the model given by Eq. (1)
- kn
damping coefficient,
- L
distance from center of mass to the contact point in vector form
- m
mass
- p
composition
Acknowledgments
This work is partially supported by National Natural Science Foundation of China Nos.(11472083 and 91434112).
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