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Granular hydrodynamics and pattern formation in vertically oscillated granular disk layers

Published online by Cambridge University Press:  01 February 2008

JOSÉ A. CARRILLO ,
THORSTEN PÖSCHEL  and
CLARA SALUEÑA
JOSÉ A. CARRILLO
Affiliation:
ICREA (Institució Catalana de Recerca i Estudis Avançats) and Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain
THORSTEN PÖSCHEL
Affiliation:
Charité, Augustenburger Platz 1, 10439 Berlin, Germany
CLARA SALUEÑA
Affiliation:
Departament de Enginyeria Mecànica-ETSEQ, Universitat Rovira i Virgili, E-43007 Tarragona, Spain
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Abstract

The goal of this study is to demonstrate numerically that certain hydrodynamic systems, derived from inelastic kinetic theory, give fairly good descriptions of rapid granular flows even if they are way beyond their supposed validity limits. A numerical hydrodynamic solver is presented for a vibrated granular bed in two dimensions. It is based on a highly accurate shock capturing state-of-the-art numerical scheme applied to a compressible Navier–Stokes system for granular flow. The hydrodynamic simulation of granular flows is challenging, particularly in systems where dilute and dense regions occur at the same time and interact with each other. As a benchmark experiment, we investigate the formation of Faraday waves in a two-dimensional thin layer exposed to vertical vibration in the presence of gravity. The results of the hydrodynamic simulations are compared with those of event-driven molecular dynamics and the overall quantitative agreement is good at the level of the formation and structure of periodic patterns. The accurate numerical scheme for the hydrodynamic description improves the reproduction of the primary onset of patterns compared to previous literature. To our knowledge, these are the first hydrodynamic results for Faraday waves in two-dimensional granular beds that accurately predict the wavelengths of the two-dimensional standing waves as a function of the perturbation's amplitude. Movies are available with the online version of the paper.

JFM classification

Granular media: Granular media Waves/Free-surface Flows: Faraday waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 597 , 25 February 2008 , pp. 119 - 144
Copyright
Copyright © Cambridge University Press 2008

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References

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Carrillo et al. supplementary movie

Movie 1. Transient stage of an Event-Driven Molecular Dynamics simulation of a layer of 6 particles vibrated vertically under gravity at a frequency of 3.5 Hz and an amplitude of 5.7 diameters. Particles are perfectly smooth and the coefficient of restitution is 0.75. The bottom plate is highlighted in green. From the initial condition and within a few cycles of oscillation, one observes the clear development of what is known as the Faraday instability in granular matter.  Note that there is a neat gap  between the particles and the bottom wall prior to the development of the instability, which turns less prominent when the peaks form and particles start to move along the wall to fill the valleys. Even so, the density can be shown to go to zero at the bottom (see movie 4) and thus  one can conclude that the particle layer "levitates".

Download Carrillo et al. supplementary movie(Video)
Video 6.4 MB

Carrillo et al. supplementary movie

Movie 1. Transient stage of an Event-Driven Molecular Dynamics simulation of a layer of 6 particles vibrated vertically under gravity at a frequency of 3.5 Hz and an amplitude of 5.7 diameters. Particles are perfectly smooth and the coefficient of restitution is 0.75. The bottom plate is highlighted in green. From the initial condition and within a few cycles of oscillation, one observes the clear development of what is known as the Faraday instability in granular matter.  Note that there is a neat gap  between the particles and the bottom wall prior to the development of the instability, which turns less prominent when the peaks form and particles start to move along the wall to fill the valleys. Even so, the density can be shown to go to zero at the bottom (see movie 4) and thus  one can conclude that the particle layer "levitates".

Download Carrillo et al. supplementary movie(Video)
Video 6.4 MB

Carrillo et al. supplementary movie

Movie 2. The transient stage of a hydrodynamic simulation of the same system above, using  a WENO shock-capturing scheme. The top and bottom walls are perfectly reflecting walls (tangential motion is not affected by the presence of the boundary). The homogeneity of the initial condition must be broken by imposing some perturbation in order to observe the instability, otherwise the homogeneous solution tends to perpetuate. The granular Navier-Stokes equations are solved in the reference frame where the bottom plate is at rest.

Download Carrillo et al. supplementary movie(Video)
Video 380.6 KB

Carrillo et al. supplementary movie

Movie 2. The transient stage of a hydrodynamic simulation of the same system above, using  a WENO shock-capturing scheme. The top and bottom walls are perfectly reflecting walls (tangential motion is not affected by the presence of the boundary). The homogeneity of the initial condition must be broken by imposing some perturbation in order to observe the instability, otherwise the homogeneous solution tends to perpetuate. The granular Navier-Stokes equations are solved in the reference frame where the bottom plate is at rest.

Download Carrillo et al. supplementary movie(Video)
Video 343.6 KB

Carrillo et al. supplementary movie

Movie 3.  The density field along two cycles of the driving period shows the fully developed Faraday instability obtained by means of the two different simulation methods. Top: The Molecular Dynamics movie is a phase average over 50 cycles of the particle positions, after which statistical noise is still visible in the dilute phase. Bottom: The hydrodynamic solution is not averaged.

Download Carrillo et al. supplementary movie(Video)
Video 315 KB

Carrillo et al. supplementary movie

Movie 3.  The density field along two cycles of the driving period shows the fully developed Faraday instability obtained by means of the two different simulation methods. Top: The Molecular Dynamics movie is a phase average over 50 cycles of the particle positions, after which statistical noise is still visible in the dilute phase. Bottom: The hydrodynamic solution is not averaged.

Download Carrillo et al. supplementary movie(Video)
Video 414.5 KB

Carrillo et al. supplementary movie

Movie 4. The evolution of the packing fraction (solid line) and the adimensional thermal energy profiles (dashed line) as a function of height at the location of a valley/cusp. The parameters are the same as in the movie above. The rising of the thermal energy accompanies the propagation of the transient shock wave generated after each impact of the material with the bottom wall. Observe that in the hydrodynamic simulation (bottom) the packing fraction is never close to zero at zero height, value which is actually reached in the particle simulation (top). The fact that there is always some material stuck to the bottom in the hydrodynamic simulation reduces the flying time of the layer and thus the wall impacts earlier. This fact explains the anticipated production of shock waves in the hydrodynamic simulation, very explicit in the evolution of the thermal energy profile.

Download Carrillo et al. supplementary movie(Video)
Video 1.1 MB

Carrillo et al. supplementary movie

Movie 4. The evolution of the packing fraction (solid line) and the adimensional thermal energy profiles (dashed line) as a function of height at the location of a valley/cusp. The parameters are the same as in the movie above. The rising of the thermal energy accompanies the propagation of the transient shock wave generated after each impact of the material with the bottom wall. Observe that in the hydrodynamic simulation (bottom) the packing fraction is never close to zero at zero height, value which is actually reached in the particle simulation (top). The fact that there is always some material stuck to the bottom in the hydrodynamic simulation reduces the flying time of the layer and thus the wall impacts earlier. This fact explains the anticipated production of shock waves in the hydrodynamic simulation, very explicit in the evolution of the thermal energy profile.

Download Carrillo et al. supplementary movie(Video)
Video 419.8 KB

Carrillo et al. supplementary movie

Movie 5. The linear momentum shown in arrows over the density field. The detail shows that the morphology of the pattern and the convective motion are extremely similar. The only difference -the opening and closing of the gap in the Molecular Dynamics simulation (top), is a true feature observed in real vibrated sand which is not observed in the hydrodynamic solution (bottom). This points at the inadequacy of the boundary condition implemented on the bottom wall.

Download Carrillo et al. supplementary movie(Video)
Video 4 MB

Carrillo et al. supplementary movie

Movie 5. The linear momentum shown in arrows over the density field. The detail shows that the morphology of the pattern and the convective motion are extremely similar. The only difference -the opening and closing of the gap in the Molecular Dynamics simulation (top), is a true feature observed in real vibrated sand which is not observed in the hydrodynamic solution (bottom). This points at the inadequacy of the boundary condition implemented on the bottom wall.

Download Carrillo et al. supplementary movie(Video)
Video 5 MB