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Lecture 3 Rapid Granular Flow Applications

Presented at the University of Salerno: May, 2011. Lecture 3 Rapid Granular Flow Applications. Anthony D. Rosato Granular Science Laboratory ME Department New Jersey Institute of Technology Newark, NJ, USA. Lecture 3 : Rapid Granular Flow Applications. Granular Science Lab - NJIT.

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Lecture 3 Rapid Granular Flow Applications

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  1. Presented at the University of Salerno: May, 2011 Lecture 3 Rapid Granular Flow Applications Anthony D. Rosato Granular Science Laboratory ME Department New Jersey Institute of Technology Newark, NJ, USA

  2. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT Presentation Outline • Application 1: Galton’s Board • Application 2: Vibrated Systems • Application 3: Couette Flows • Application 4: Intruder Dynamics in CouetteFlows • Application 5: Density Relaxation -Continuous Vibrations • Application 6: Tapped Density Relaxation

  3. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT Application 1: Galton’s Board Investigate the behavior of a single particle migrating under gravity through an ordered, planar array of rigid obstacles – a system known as a Galton’s board. Examine subtle connections between the deterministic particle simulations, physical experiments, and discrete dynamical models First step in a larger picture to extract generic dynamical features of granular flows through the analyses of “simple” models

  4. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT Historical Background Sir Francis Galton (1822 – 1911): British scientist, Fellow of the Royal Society; Geographer, meteorologist, tropical explorer, founder of differential psychology, inventor of fingerprint identification, pioneer of statistical correlation and regression, convinced hereditarian, eugenicist, proto-geneticist, half-cousin of Charles Darwin and best-selling author. http://www.mugu.com/galton/start.html Developed “board” to describe biological processes statistically “I have no patience with the hypothesis occasionally expressed, and often implied, especially in tales written to teach children to be good, that babies are born pretty much alike, and that the sole agencies in creating differences between boy and boy, and man and man, are steady application and moral effort. It is in the most unqualified manner that I object to pretensions of natural equality. The experiences of the nursery, the school, the University, and of professional careers, are a chain of proofs to the contrary.” -- Francis Galton, Hereditary Genius

  5. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT Galton’s Board: EXPERIMENTS Rendering of the board depicting the pins, collection slots, traverse, location of the optical timer beams, and detail of the triangular lattice configuration of pins.

  6. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT

  7. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT Experimental Apparatus Schematic of the automatic Galton Board data acquisition system (AGB). Balls fed from the supply hopper through a flexible tube are dropped one at a time using a system of solenoids. The residence time is recorded via an optical sensor (“stop eye”). The exit position is also recorded with an array of 49 custom-built optical cell detectors.

  8. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT Experimental Parameters Materials of the sphere Aluminum Brass Stainless Steel Release Height H (max = 15.53”) Board Tilt Angle q (30o to 70o) – measured from horizontal Measurements Made Residence Times Distribution of Exit Positions Computed Quantities Average downward velocity (cm/sec) Lateral dispersion (cm2/sec)

  9. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT Sampling of Experimental Results Average residence time Tav as a function of release height H for stainless steel spheres.

  10. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT Distribution of Exit Positions for Stainless Steel Spheres

  11. Lecture 3: Rapid Granular Flow Applications - delta-distribution centered at x = 0 and height No - concentration of particles at (x,t) for an infinitely wide board Granular Science Lab - NJIT Lateral Dispersion or Diffusivity Diffusion model [Bridgwater et al., Trans. Instn. Chem. Engrs. 49, 163-169 (1971) ] Solution …

  12. Lecture 3: Rapid Granular Flow Applications - number of particles in the interval [-x, x] Least squares fit of the stainless steel data (solid circles) to the model. Spheres were released from the top of the board set at q = 70o. The origin of the x-axis denotes the center of the board. D = 1.85 cm2/sec Granular Science Lab - NJIT Summary of Dispersion Results

  13. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT Figure 9: Three typical trajectories from the discrete element simulation (q = 70o) obtained by slightly varying the initial positions. Residence times are indicated for each trajectory. The center of the board is located at X = 0.2032 meters. Sample Trajectories Generated by the Simulation

  14. Lecture 3: Rapid Granular Flow Applications Exit distribution of the number of particles for 1/8” spheres at board angle q = 70o from simulation in which e = 0.6 Granular Science Lab - NJIT Simulated Exit Position Distribution

  15. Lecture 3: Rapid Granular Flow Applications Lateral Dispersion Computed from Limiting Slope of Mean Square Displacement (m2) Granular Science Lab - NJIT

  16. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT Comparison of Simulated Results with Experiments

  17. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT Application 2:Vibrated Systems Investigate macroscopic behavior of granular materials subjected to vibrations Gravitationally loaded into a rectangular, periodic cell having an open top and plan floor Vibrations imposed through sinusoidally oscillated floor Compare with kinetic theory predictions Compare with physical experiments Y. Lan, A. Rosato, “Macroscopic behavior of vibrating beds of smooth inelastic spheres, Phys. Fluids7 [8], 1818 (1995)

  18. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT Geometry of Periodic Computational Cell Spheres are smooth (no friction) and inelastic, obeying the soft contact laws of Walton and Braun. Steady state computations performed. Simulation Parameters

  19. Lecture 3: Rapid Granular Flow Applications Averaging layer A layer is ‘identified’ by its center y-coordinate. Designates the long-term average Long term cumulative mean velocity of layer-y taken over the time interval (to, t1). Instantaneous fluctuating (or deviatoric velocity) of the ith particle in layer-y Granular Science Lab - NJIT Steady-State Diagnostics In computing depth profiles, the cell is partitioned into layers of thickness equal to approximately the particle diameter d. Instantaneous layer diagnostic: Mass-weighted average taken over all particles that occupy the layer at time t. Mass hold-up: bulk mass supported by the floor of cross-sectional area A N = # of spheres

  20. Lecture 3: Rapid Granular Flow Applications Granular Temperature depth profile Measure of the kinetic energy per unit mass attributed to the particles’ fluctuating velocity components. Granular Science Lab - NJIT Depth profile of the instantaneous RMS deviatoric velocity Long-term cumulative, mass-weighted average deviatoric velocity depth profile Non-dimensional Granular Temperature depth profile S. Ogawa, “Multi-temperature theory of granular materials”, Proceedings of the US-Japan Seminar on Continuum-Mechanical and Statistical Approaches in the Mechanics of Granular Materials, Tokyo, 1978, pp. 208-217.

  21. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT Comparisons with Kinetic Theory of Richman and Martin

  22. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT

  23. Lecture 3: Rapid Granular Flow Applications Paper lid Simulation Parameters Granular Science Lab - NJIT Comparisons with Experiments of Hunt et al. • Validation against Experiments M. Hunt et al., J. Fluids Eng. 116, pg. 785 (1994). Relatively smooth spheres used in experiment 136 grams of particles used, mt = 5.0

  24. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT

  25. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT

  26. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT Summary of Findings • The behavior of the system depends on the magnitude of the floor acceleration G = aw2/g • High accelerations: Dense upper region supported on a ‘fluidized’ lower-density region near the floor • Granular temperature is maximum near the floor and attenuates (upwards) towards the surface, and the solids fraction depth profiles peaks within the center of the system. • Lower Accelerations: Granular temperature does not decrease monotonically from the floor, and the solids fraction depth profile bulges near the floor. Upper region of the system is highly agitated. • For accelerations less than (approx.) 1.2, the steady-state height of the system remains constant. • For 1.2 < G < 2.0: System undergoes a large vertical expansion. • Computed steady-state granular temperature and solids fraction profiles in good agreement with kinetic theory predictions when the system is sufficiently agitated, and with physical experiments.

  27. Lecture 3: Rapid Granular Flow Applications Continuously shake the vessel up and down. Particles will flow upwards near the walls and downward in the center. Granular Science Lab - NJIT Convection in a Vibrated Vessel of Granular Materials Rough, inelastic spheres obeying the Walton & Braun soft-particle models.

  28. Lecture 3: Rapid Granular Flow Applications Velocity Spheres Granular Science Lab - NJIT Velocity Field – Long-time Average Superimposed Trajectory of Large Intruder Parameters m= mb = 0.8, f = 7 Hz, a/d = 0.5, G = 10 Width = 20d Y. Lan and A. D. Rosato, Phys. Fluids9 (12), 3615-3624 (1997).

  29. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT

  30. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT Average Convection Velocity as a Function of G

  31. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT Long-term velocity field in a computational cell whose lateral walls are smooth (no friction). Notice the downward flow in the center and upward motion adjacent to the walls.

  32. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT Instantaneous velocity fields and sphere center projections for the 100d cell (f = 7 Hz, a/d = 0.5, G = 10) reveals the formation of arches during the downwards motion of the floor. The dashed line represents the equilibrium position of the floor. Although the arches are not very distinct in (b), the corresponding instantaneous velocity field reveals a pattern where groups of particles are moving collectively towards or away from the floor. This has been marked by the arrows in (c) whose directions indicate the general sense of the flow at a time subsequent to that shown in (b).

  33. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT Comparison of trajectory of large intruder in a narrow and wide cell. Notice the re-entrainment in (b), while the intruder is trapped at the surface in (a).

  34. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT Summary of Findings • The onset of convection is controlled by (a/d)G rather than by G alone. • When the lateral walls are frictional, a long-term convective flow develops that is upward in the center of the cell and downward adjacent to the walls. • Reversal in the direction of the long-term convective flow occurs when the side-walls are smooth. • As the cell width (w/d) is increased, a visible pattern in the long-term velocity field is reduced and eventually it ceases to be evident. • Over the time scale of the period of vibration, adjacent internal convection fields with opposed circulations were visible. Averaging over long time scales caused these flow structures not to appear. • However, near the side walls, persistent vortex-like structures were attached, having a length scale that appeared to be of the same order as the height of the static system. • A single, large intruder sphere placed on the floor in the center of the system was carried up to the surface at nearly the same velocity as the mean convection. Upon reaching the surface, it migrated toward the side-walls. There it was either trapped, or re-entrained into the bed, depending on the width of the downward flow field near the wall relative to the particle diameter.

  35. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT Application 3: Couette Flow Upper and lower bumpy walls move at constant velocity in opposite directions. Collisions with flow particles causes them to flow.

  36. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT General Features of the Flow - Steady-state Profiles - Velocity Granular Temperature Solids Fraction Pressure

  37. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT Steady State Average # of collisions/sec ~ 30 for each particle

  38. Lecture 3: Rapid Granular Flow Applications Dimensionless “Peculiar” Velocity vParticle Velocity Effective shear rate = 2U/H Mean Velocity Granular Science Lab - NJIT Granular temperature - kinetic energy of the velocity fluctuations

  39. Lecture 3: Rapid Granular Flow Applications n = 0.45 Granular Science Lab - NJIT Granular Temperature Profiles Mean Velocity Profiles

  40. Lecture 3: Rapid Granular Flow Applications 16 1 14 12 0.9 10 d / 8 Y 0.8 6 4 0.7 2 * P yy 0 0.1 0.3 0.5 0.7 0.9 0.6 Y/H Granular Science Lab - NJIT 0.5 H=16d 0.4 H=32d 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 n Solids Fraction Profiles Pressure - Pyy

  41. Lecture 3: Rapid Granular Flow Applications Secondary Velocity Field Slab used to compute U = 8 d/s H/d = 8 Averaging layer used for profiles H y x z Dx L U 2y/d U Granular Science Lab - NJIT x/d Dy Dx

  42. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT Velocity field for U = 8 d/s (W/d=64)

  43. Lecture 3: Rapid Granular Flow Applications for U = 8 d/s(W/d=64) Auto-Correlation Peak at l = =15 R FFT spectrum analysis Peak atl = 7.5 d Granular Science Lab - NJIT

  44. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT Figure 8: Wavelength l/d of the convection cells as a function of effective shear rate for a fixed shear gap H/d = 8. The solid line is included to show the trend. Wavelength vs. Effective Shear Rate

  45. Lecture 3: Rapid Granular Flow Applications • IntruderProperties • Different size, but same density • Different mass, but same size • Different size, same mass - U Granular Science Lab - NJIT Application 4:Intruder Dynamics in Couette Flows

  46. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT Size and Mass Ratios f= Intruder diameter/Flow particle diameter fm = Intruder mass/Flow particle mass

  47. Lecture 3: Rapid Granular Flow Applications (a)Crossing time Tc (seconds) versus f at U = +/-16, 32, 64 r/s; (b) Average intruder velocity , where S is the distance traveled by the mass center from its initial position near the wall to the mid-plane of the cell. Granular Science Lab - NJIT Intruder Velocity and Mid-Plane Crossing Time

  48. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT Tc = Rise Time for the intruder to reach the middle layer from bottom. = 1.0 = 2.0 As the relative size of the intruder increase, its rise time decreases.

  49. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT Y* Trajectory + Power Spectrum ( =1.0) Ym(f) - Closest distance possible between the center of the intruder and boundary plane

  50. Lecture 3: Rapid Granular Flow Applications Granular Science Lab - NJIT Background: Noise Signals

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