Particle Flow Instabilities

Transcription

1 University of Colorado, Boulder CU Scholar Chemical & Biological Engineering Graduate Theses & Dissertations Chemical & Biological Engineering Spring Particle Flow Instabilities Peter Paul Mitrano University of Colorado Boulder, Follow this and additional works at: Part of the Process Control and Systems Commons Recommended Citation Mitrano, Peter Paul, "Particle Flow Instabilities" (2014). Chemical & Biological Engineering Graduate Theses & Dissertations This Dissertation is brought to you for free and open access by Chemical & Biological Engineering at CU Scholar. It has been accepted for inclusion in Chemical & Biological Engineering Graduate Theses & Dissertations by an authorized administrator of CU Scholar. For more information, please contact

2 PARTICLE FLOW INSTABILITIES By Peter Paul Mitrano B.S., University of New Hampshire, 2009 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirement for the degree of Doctor of Philosophy Department of Chemical and Biological Engineering 2014

3 This thesis entitled: Particle Flow Instabilities written by Peter Paul Mitrano has been approved for the Department of Chemical and Biological Engineering Christine M. Hrenya Paul D. Beale Vicente Garzó Arthi Jayaraman Charles B. Musgrave August 2014 The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline.

4 Mitrano, Peter Paul (Ph.D., Chemical and Biological Engineering) Particle Flow Instabilities Dissertation Directed by Professor Christine M. Hrenya Previous work has indicated that inelastic grains undergoing homogeneous cooling may be unstable, giving rise to the formation of velocity vortices and particle clusters for sufficiently large systems. Such instabilities are observed in industrial coal and biomass gasifiers and are known to influence gas-solid contact area, mixing dynamics, and heat/mass transfer rates. However, the driving mechanisms that lead to vortices and clusters are not well understood. Discrete-particle simulations provide a well-established method for understanding such mechanisms but are not a feasible technique for predicting the behavior of large-scale systems. Kinetic-theory-based hydrodynamic (continuum) models offer an effective means of describing such flows, and instabilities present a stringent test of such models due to the transient, threedimensional nature of instabilities and the large range of time and length scales over which these mechanisms occur. This work begins with the study, via a combination of hydrodynamic (continuum) models and discrete-particle simulations, of a relatively simple flow and includes additional complexities in a stepwise manner to assess various driving mechanisms. Comparisons with discrete-particle simulations, which offer detailed, well-established (but computationally limited) descriptions of particle flows, indicate the ability of hydrodynamic (continuum) models to accurately incorporate each mechanism. Specifically, the critical length scale for velocity vortices and/or particle clusters are studied in systems of moderate dissipation and particle concentration, extreme iii

5 dissipation, frictional particles, high gradients, polydisperse particles, and gas-solid flows. This effort begins with a granular flow of monodisperse, frictionless spheres. The results indicate the validity of the hydrodynamic model, derived from the revised Enskog equation, in predicting the onset of instabilities for moderately dense and dissipative flows. Granular flows of extreme dissipation are then studied to determine the accuracy of the standard Sonine polynomial approximation. Discrepancies are observed for extreme levels of dissipation. A Sonine approximation with a modified zeroth-order contribution is shown to remedy this disagreement. Frictional particle interactions introduce a new mechanism for the dissipation of granular energy that is not relevant for frictionless interactions. Counterintuitive findings, which stem from the interplay of rotation and translation, for the influence of friction on instabilities are presented. While monodisperse particles flows allow for a straightforward choice of the hydrodynamic quantities present in the balance equations, polydispersity allows for separate species energy balances or a single mixture-temperature-based energy balance. The results indicate that a mixture-temperature-based energy balance is successful. Granular flows containing high gradients are studied. These flows introduce an important challenge to hydrodynamic descriptions because the constitutive quantities depend on first-order closures. The results indicate the ability of Navier-Stokes-order theories beyond the expected range of applicability. The introduction of a fluid phase leads to a new source of instability: fluid-solid drag. A model that incorporates the fluid force into the starting kinetic equation is iv

6 assessed via comparison with direct numerical simulation. Additionally, we compare the relative importance of the drag force in such systems, and find that this mechanism is relevant in such flows. v

7 Dedication To William Bill Phillips

8 Acknowledgements I am very lucky to have had the opportunity to work with the outstanding group of people who have collectively performed this work. The early contributions to this project by Mike Pacella and Daniel Cromer allowed me an advantageous starting point, while the work and guidance of Steven Dahl provided me the technical foundation necessary to succeed. It s abundantly clear to me that without the contributions of these individuals, the work of the initial chapters of this dissertation would have been a far lengthier struggle, and the latter chapters would likely not exist. The collaborations in which I have had the pleasure to engage have greatly enhanced the quality of this work. The work of Vicente Garzó has been critical to my success, and I am extremely thankful for the opportunity to study with him and for his hospitality. The collaborative efforts of Xiaolong Yin and Xiaoqi Li have been equally rewarding and have offered, along with Vicente, an ideal mix of technical approaches that greatly augmented this effort. Additionally, the tutoring of Sofiane Benyahia and Janine Galvin were tremendously valuable. The independent efforts of John Zenk were especially admirable given how quickly he became a respected member of our research group at CU. The contributions of John and fellow undergraduate researchers Andrew Hilger and Chris Ewasko were far beyond my expectation. Each of them made strong, individual contributions to this work, and I m thankful to have had the opportunity to work with them. vii

9 In my early years at Colorado, the countless hours of study, collaborative homework efforts, and basketball with JA Murray were invaluable. There is no way I could have developed my studies and game to the same extent without him. I m thankful to all the teachers and professors that have encouraged my passion for mathematics and science throughout my education. Greta Mills played a critical role in helping me first embrace this interest along with John Donnelly, Carl Merhbach, Mr. Bill Hammond, and Dr. Sally Hair at Hanover High School. The relentless excitement of Prof. Russell Carr, who is simply my favorite professor, the always-interesting insight of Prof. P.T. Vasudevan, who told me that no one ever regrets getting a Ph.D. when I was pondering my future, and the technical prowess of Prof. Nivedita Gupta, who first introduced me to fluid dynamics, were invaluable to my educational pursuits at the chemical engineering department at UNH, and I thank each one of them. My mother pointed me to chemical engineering when I d never heard of the discipline. I wanted to take a year off after high school, but she convinced me to continue on the path that led me here. I simply wouldn t be here without her. I m thankful for her and a wonderful collection of aunts and uncles, J., A., Mo, Joe, John, and Liz along with my Aunt Marie and Uncle Michael. I m grateful for the support and love of my grandparents, Mary Kelly and Grandpa Duke. I dearly love my brother, Chris, who has always been there for me, and I admire and appreciate my father. I m thankful for his constant reminders to just get through the grind. It s difficult for me to describe how thankful I am to have worked with Christine Hrenya. I knew after speaking with her for about 20 minutes during my recruiting visit that I wanted to study with her. Her enthusiasm has been an incessant inspiration to me, viii

10 and her level of technical expectation has been a constant motivation to improve. Her extreme level of expertise combined with an impeccable ability to explain technical concepts is a gift to those she mentors. She is well renowned as one of the premiere scientists in the field, yet she may be a better mentor than researcher. The decision to study under Christine is one of the best decisions I ve ever made. ix

11 Table of Contents 1. Introduction Motivation Velocity Vortices and Particle Clusters Mechanisms Relevant to the Formation of Instabilities System Description Modeling Approaches Granular Flow Discrete Element Method (DEM) Kinetic-Theory-Based Hydrodynamic (Continuum) Modeling Gas-Solid Flow Direct Numerical Simulation (DNS) Discrete Element Method- Computational Fluid Dynamics (DEM-CFD) Two-Phase Hydrodynamic (Continuum) Modeling Dissertation Objectives Assessing Ability of Hydrodynamics to Describe Particulate Flows Granular: Moderate Concentration and Dissipation (Chapter 2; [47]) Granular: Extreme Dissipation (Chapter 3; [48]) Granular: Frictional Particle Collisions (Chapter 4; [25]) Granular: Polydisperse Particles (Chapter 5; [50]) Granular: High Gradients (Chapter 6; [54]) Gas-solid: Thermal Drag (Chapter 7) Identification of Dominant Mechanisms for Instability Granular: Frictional Particle Collisions (Chapter 4; [25]) Gas-solid: Thermal Drag (Chapter 7; [55]) Gas-solid: Mean Drag (Chapter 8) References Granular Flows: Effect of Moderate Concentration and Dissipation Introduction Computational Methods System of Interest Discrete Element Method Parameter Space Considered Detection of Instability Hydrodynamic (Continuum) Description Results and Discussion Determination of the Critical Length Scale in DEM Critical Length Scale and Type of Instability Time to Instability in DEM Concluding Remarks References Granular Flows: Effect of Extreme Dissipation Introduction Computational Methods Simulation Method x

12 3.2.2 Stability Detection: Fourier Analysis Hydrodynamic Description Results Concluding Remarks References Granular Flows: Effect of Frictional Particle Collisions Introduction Computational Methods Results Concluding Remarks References Granular Flows: Effect of Polydispersity Introduction Computational Methods Results Concluding Remarks References Granular Flows: Effect of High-Gradients Introduction Computational Methods Transient Simulation of Hydrodynamic Model Linear Stability Analysis of Hydrodynamic Model Discrete Element Method Simulations Results Concluding Remarks References Gas-Solid Flows: Effect of Thermal Drag Introduction Computational Methods System Description Granular DEM Simulations Gas-solid Direct Numerical Simulations Two-Fluid Model (TFM) simulations Verification of DNS Computations and Instability Detection Results Concluding Remarks References Gas-Solid Flows: Effect of Mean Drag Introduction Computational Methods System of Interest Two-Fluid Model (TFM) simulations Gas-Solid Direct Numerical Simulation (DNS) xi

13 8.3 Results Concluding Remarks References Summary and Future Work Conclusions Recommendations for Future Work References xii

14 List of Figures Figure 1.1 Clustering in a circulating fluidized bed (CFB): (a) depiction of the freeboard section of a riser with a cluster located at top right (circled) and (b) camera image of particle clustering instability taken in the cross section of the freeboard with cluster at the middle left (circled). Dark circles are particles. The mark on the bottom right is 150 microns in width Figure 1.2 Visualizations from a granular, discrete-element method simulation of a binary mixture: (a) stable, coarse-grained velocity field at an early time, (b) stable particle positions at an early time, (c) unstable, coarse-grained velocity field at a later time, and (d) cluster systems at a later time. A cell size of one-fifth of the domain size is used for local velocity averaging Figure 1.3 Visualization of before (left), during (middle), and after (right) a particleparticle collision with a restitution coefficient e= Figure 1.4 Depictions of various approaches to modeling gas-solid flows: (a) direct numerical simulation (DNS) in which the detailed motion of the fluid is resolved around each particle, (b) discrete element method-computational fluid dynamics (DEM-CFD) in which individual particles are tracked while averaged fluid motion is solved on a grid larger than inter-particle spacing, and (c) hydrodynamic (continuum) modeling of both gas and solid phases. Blue and black arrows indicate the fluid and particle velocity, respectively. Dashed lines in (b) and (c) represent the numerical grid used to solve the continuum balances Figure 2.1 Snapshot of particles extracted from DEM simulations of the clustering instability in (a) the HCS with e=0.6,! =0.05, and N=40,000 and (b) a SSF with e=0.6,! =0.2, and N=2064, where N refers to the total number of particles Figure 2.2 (Color Online) Snapshots of particle positions, shown as dots (grey online), and spatially-averaged local-velocity vectors, shown as arrows (red online), extracted from DEM simulations of (a) homogeneous concentration and velocity fields at 10 collisions per particle (cpp) and (b) homogenous concentration and inhomogeneous velocity fields at 210 cpp with e=0.98,! =0.05, and N=40, Figure 2.3 DEM snapshot of particle positions in a three-dimensional HCS showing a two-dimensional slice of thickness= L/10 for (a) 1 cpp and (b) 50 cpp with e=0.6,! =0.05, and N=40, Figure 2.4 Illustration of the use of Haff s law to detect instability in the HCS for e=0.8,! =0.3: (a) stable behavior for L/d=7.04 and (b) unstable behavior for L/d=9.09. DEM data is shown as a solid line (blue online) and Haff s law is shown as a dashed line Figure 2.5 DEM determination of L Vortex / d at 800 cpp for e=0.8 and! =0.3. The length scales associated with the 0 and 1 probabilities (as predicted by the linear fit) represent the lower ( L Vortex / d ) and upper bound, respectively, of the transition range. This range is represented as vertical error bars in Figures 2.8 and xiii

15 Figure 2.6 Sensitivity of instability probability to the (a) number of replicates (seeds) and (b) duration of simulation (total number of collisions) for a system with e=0.8 and! = Figure 2.7 A spatially averaged velocity field at the time of instability detection (80 cpp) for e=0.8,! =0.3, and L/d=9.56 with the domain split into square cells with sides of length L/5 for purposes of averaging Figure 2.8 Comparison of the critical length scale ( L Vortex / d ) for (a) e=0.8 and (b) e=0.7. The error bars correspond to the transition length scale range obtained from DEM simulations (see Section and Figure 2.5). Theoretical predictions are obtained from the linear stability analysis of Garzó [23] Figure 2.9 Comparison of the critical length scale ( L Vortex / d ) for (a)! =0.1 and (b)! =0.4. The error bars correspond to the transition length scale range obtained from DEM simulations (see Section and Fig. 2.5). Theoretical predictions are obtained from a linear stability analysis of Garzó [23] Figure 2.10 Dimensionless time until the onset of the instability (t*) extracted from DEM Figure 3.1 Spatially averaged velocity fields from DEM simulations with e=0.4,! =0.05, and number of particles N=2096 at (a) 5 collisions per particle (cpp) and (b) 90 cpp, and (c) corresponding Fourier momentum spectra (P) normalized to the value at k=4!. Square cells with length L/5 were averaged for visualization in (a) and (b) Figure 3.2 Snapshots of particle positions (showing only a slice of thickness L/10) extracted from three-dimensional DEM simulations with e=0.6,! =0.2, N=2000 at (a) 2 cpp, (b) 40 cpp, (c) 800 cpp, and (d) corresponding Fourier mass spectra (R). 55 Figure 3.3 Critical clustering length scale (L Cluster /d) as a function of solids fraction for (a) e=0.25 and (b) e=0.4. The solid and dashed lines correspond to the modified and standard theories, respectively. The data points correspond to DEM Figure 3.4 Critical length scale for vortex and cluster instabilities (i.e., L Vortex /d and L Cluster /d, respectively) plotted as a function of solids fraction for e=0.25. The solid and dashed lines correspond to the modified and standard theories, respectively Figure 4.1 Visualizations from a discrete element method simulation of a HCS with normal restitution coefficient e=0.7, solids fraction φ =0.3, tangential restitution coefficient β =-0.7, and L/d=18, where L is the domain length and d is the particle diameter, after 100 collisions per particle: (a) coarse-grained velocity field with cell size L/5 (vortex instability) and (b) particle positions within a L/10 domain slice (clustering instability) Figure 4.2 DEM data for critical dimensionless length scale for vortex, LVortex / d (blue) and cluster instabilities, LCluster / d (red) as a function of tangential restitution coefficient for e=0.9 and φ =0.3. Lines represent critical dimensionless length scales in the frictionless limit ( β =-1) Figure 4.3 Translational tangential restitution coefficient extracted from DEM simulations as a function of β for e=0.9 and φ =0.3 in a domain with L/d=4 (red) and L/d=10 (blue) xiv

16 Figure 4.4 The ratio of rotational kinetic energy (RE) to translational kinetic energy (KE) as a function of β for e=0.9 and φ =0.3. Data points are averages from DEM simulations run for 1600 collision per particle with 124 particles (blue) and 990 particles (red). The line is the theoretical prediction [19] Figure 4.5 LVortex / d (blue) and LCluster / d (red), normalized to the respective frictionlessparticle DEM prediction (see Fig. 4.2), as a function of the tangential restitution coefficient, as predicted by DEM (data points) and theory (lines) for e=0.9 and φ =0.3. Values greater than 1 refer to attenuation of instabilities relative to the frictionless case Figure 4.6 Theoretical non-dimensional cooling rate normalized to the smooth-particle value as a function of tangential restitution coefficient for = Figure 5.1 Visualizations from a DEM simulation of an equimolar mixture (x 1 = 0.5) with m 1 /m 2 = 2, d 1 /d 2 = 3, = 0.2, and e = 0.7 of (a) stable, coarse-grained velocity field at five collisions per particle (or cpp ), (b) stable particle positions at five cpp, (c) unstable, coarse-grained velocity field at 400 cpp, and (d) cluster systems at 400 cpp. A cell size of L/5 is used for local velocity averaging Figure 5.2 Critical length scale for velocity vortices as a function of the mass ratio m 1 /m 2 with x1 = 0.1, d 1 /d 2 = 1 for (a) = 0.1 and (b) = 0.2. The data points correspond to DEM simulations, while the lines are the theoretical predictions given by Eq. (6.3). (Blue) circles/solid line, (red) triangles/dashed line, and (black) squares/dot-dashed line correspond to e = 0.9, e = 0.8, and e = 0.7, respectively. Error ranges are the size of the data points and are omitted Figure 5.3 Critical length scale for velocity vortices as a function of (a) the ratio of diameters d 1 /d 2 with m 1 /m 2 = 2, x 1 = 0.5, and = 0.2 and (b) the mole fraction x 1 with m 1 /m 2 = 6, d 1 /d 2 = 1, and = 0.2. The meaning of symbols and lines is the same as that of Fig Figure 6.1 Visualizations of three-dimensional, stable (L/d=18) flows and clustering (L/d=40) flows in a homogeneous cooling system at long times with restitution coefficient of 0.7 and solids volume fraction of 0.1: (a) stable and (b) unstable DEM simulations showing slice of thickness L/10, (c) stable and (d) unstable kinetictheory-based simulations showing one layer of numerical cells (of thickness d) Figure 6.2 Maximum difference in cell concentration normalized by the average concentration plotted as a function of time for e=0.7 and φ =0.1. Dashed (red) and solid (blue) lines represent L/d=20 and L/d=18, respectively, with a (black) vertical dashed line indicating the onset time of the clustering instability, t*= Figure 6.3 Critical dimensionless length scale for clustering instabilities as a function of solids fraction for (a) e=0.9 and (b) e=0.7. The (black) hollow squares, (blue) squares, and (blue) line represent predictions from the continuum simulations, DEM simulations, and linear stability analyses, respectively. Error ranges are the size of data points and omitted Figure 6.4 Visualizations of the onset of clustering (showing a slice of the threedimensional domain): (a) plot of (coarse-grained) velocity field in two-dimensional slice of thickness d, (b) granular-temperature-field T( x, y ) color map, and (c) xv

17 corresponding concentration-field φ ( xy, ) grey-scale plot with e=0.7, φ =0.1, L/d=20, and t*= Figure 6.5 Spatial color maps of concentration and granular temperature Knudsen numbers at the onset of clustering (corresponding to Fig. 5.4): (a) in the x- and (b) y-spatial direction, (c) in the in the x- and (d) y-spatial direction with e=0.7, φ =0.1, L/d=20, and t*= Figure 6.6 Spatial color maps of the velocity-field Knudsen numbers at the onset of clustering (corresponding to Fig. 5.4) corresponding the gradients: (a)! x U x, (b)! y U x, (c)! x U y, and (d)! y U y with e=0.7, φ =0.1, L/d=20, and t*= Figure 6.7 Cumulative distribution of the x-, y-, and z-velocity components of throughout the three-dimensional domain at the onset of clustering with e=0.7, =0.1, L/d=20, t*= Figure 7.1 Particle kinetic energy levels versus dimensionless time: The thin solid line corresponds to a DNS simulation (Re T = 3, φ = 0.2, e = 0.9, ρ p /ρ g = 1000); the thin dashed line corresponds to a granular DEM simulation (φ = 0.2, e = 1.0). Thick solid and dashed lines are the respective analytical solutions. Diamonds mark the onset of the vortex instability Figure 7.2 Left snapshots of particle positions at three different times in DNS illustrates evolution of particle cluster. Right evolution of the coarse-grained particle velocity field in DNS at three different times. Re T = 30, e = 0.8, φ = 0.2, ρ p /ρ g = Figure 7.3 Collisional vs. viscous dissipation for six DNS simulations with varying Re T and e. The phenomenon that vortex instability (diamonds) precedes cluster formation (circles) is observed in both viscous dominated and collision dominated regimes. Particle-fluid density ratio ρ p /ρ g = 1000 and φ = Figure 7.4 Kinetic energy evolution prior to onsets of instability. Left: Gas-solid systems (DNS) with inelastic collisions (e=0.9) at Re T = 3 ( ) and 30 (+); gas-solid systems (DNS) with elastic collisions (e=1.0) at Re T = 3 ( ) and 30 ( ); a granular system (DEM) with inelastic collisions (e=0.9, solid line). For all systems, φ = 0.4 and ρ p /ρ g = Right: Gas-solid systems (DNS) ρ p /ρ g = 800( ) and 1500 (Δ). For both systems, Re T = 30, e = 0.9, and φ = 0.4. The granular (DEM) counterpart with the same e and φ (solid line) is also given Figure 7.5 Average dimensionless onset times in systems with different dissipation mechanisms. The 10 solid lines correspond to the 10 gas-solid DNS simulations with both dissipative collisions and gas-phase viscosity (Re T = 3, φ = 0.2, e = 0.9, ρ p /ρ g = 1000). The average dimensionless onset time for the gas-solid DNS is marked by a vertical line at 31.7 (±8.5). The 9 dashed line shows 9 sample granular DEM simulations (φ = 0.2, e = 0.9), and the vertical line at 44.9 (±2.8) marks the average dimensionless onset time of 150 granular simulations. The ( ) symbols represent a gas-solid DNS without dissipative collisions (Re T = 3, φ = 0.2, e = 1.0, ρ p /ρ g = 1000), where the vertical line marks the onset time of The diamonds show the distribution of vortex onset times and the circles represent cluster formations xvi

18 Figure 7.6 Cases of kinetic energy level crossovers observed during evolution of the instabilities. (a) crossovers between gas-solid DNS with dissipative collisions and gas-solid DNS with elastic collisions and granular DEM simulations; (b) crossovers among DNS with different ρ p /ρ g ; (c) crossovers between DNS with different Re T ; (d)-(f) crossovers between granular DEM and gas-solid DNS with different e Figure 7.7!! max = (! max "! min ) /! as a function of dimensionless time for TFM simulations with e=0.9,! s /! f = 1000, Re T =5, and! =0.2 of various system sizes. The cutoff value of 0.01 is shown as a dashed line Figure 7.8 Visualization of velocity field for L/d=14 at t*=10 from a TFM simulation with e=0.9,! s /! f = 1000, Re T =5, and! = Figure 7.9 Critical dimensionless length scale for clustering instabilities as a function of solids concentration for Re T =5 and! s /! g =1000 with e=0.9 (red) and e=0.8 (blue). The hollow squares, circles, and lines represent predictions from the TFM continuum simulations, DNS, and linear stability analyses (LSA), respectively. Error ranges are the size of data points and omitted Figure 7.10 Critical dimensionless length scale for clustering instabilities as a function of thermal Reynolds number for! =0.3 and! s /! g =1000 with e=0.9 (red) and e=0.8 (blue). The hollow squares and circles represent predictions from the TFM continuum simulations and DNS, respectively Figure 7.11 Critical length scale for vortices as a function of solids fraction in granular and gas-solid (! s /! g =1000, Re T = 5) flows with e=0.8 (red) and e=0.9 (blue). Lines represent the gas-solid (GS) linear stability analysis (LSA). Diamonds represent the granular LSA. Open squares are (granular) DEM simulations. Squares are DNS. 138 Figure 7.12 Granular temperature as a function of t* = T t m for the gas-solid flow d corresponding to Fig. 7.5 (! s /! g =1000, Re T = 5, and φ = 0.2) for e=0.1 (blue) and e=0.9 (red) Figure 8.1 Critical dimensionless length scale for clustering instabilities (predicted by MFIX two-fluid simulations) as a function of mean flow Reynolds number with! s /! g =1000,! =0.2, Re T = 3. Error bars indicate the range for the critical length scale xvii

19 1. Introduction 1.1 Motivation Since Maxwell first asserted, via a series of stability analyses, that the rings of Saturn consist of discrete particles rather than a collection of thin, solid ringlets, the stability of particulate flows has been studied [1]. Flow instabilities stemming from dissipative particle interactions have implications ranging from density waves in the (granular) rings of Saturn [2], early-stage planetary formation [3], and structure formation in ice floes [4,5] to pharmaceutical mixing, food processing, and state-of-the-art energy industries. One such instability, particle clustering (see Fig. 1.1), refers to transient inhomogeneities in particle concentration [6,7] and is commonly observed in gas-solid circulating fluidized beds (CFBs) [8-11]. CFBs have a range of applications from metallurgical processes, and polymerization of olefins to the gasification of biomass and coal for energy production [12]. Because the presence of clustering instabilities in CFBs influences process performance via gas-solid contact area, mixing dynamics, and heat/mass transfer rates, it is important to accurately describe flow instabilities. Such a description requires a fundamental understanding of the mechanisms giving rise to clusters. 1

20 (a) (b) Figure 1.1 Clustering in a circulating fluidized bed (CFB): (a) depiction of the freeboard section of a riser with a cluster located at top right (circled) and (b) camera image of particle clustering instability taken in the cross section of the freeboard with cluster at the middle left (circled). Dark circles are particles. The mark on the bottom right is 150 microns in width. While discrete-particle simulations offer a means to gain a physical understanding of such mechanisms, the computational limitations prevent any feasible predictive studies of the large-scale systems ubiquitous to industrial CFBs. A more efficient framework is necessary. Hydrodynamic (continuum) models for particle flows offer such a framework. Simulations using such models have been shown repeatedly to predict particles clusters qualitatively similar to what has been observed in experiments, though quantitative assessments of their accuracy are noticeably lacking. Moreover, some concerns exist over the applicability of hydrodynamic models to CFBs due to the assumptions inherent in such models. One potentially problematic assumption is that of small-gradient flow (i.e., the small-knudsen-number assumption) because significant gradients in velocity and concentration are observed in CFBs (e.g., core-annulus flow, bubbling, particle clustering). The study of instabilities (such as velocity vortices and particle clustering) 2

21 offers an excellent opportunity to quantitatively assess the ability of hydrodynamic (continuum) models to predict the dynamics of high-gradient flows because developed velocity vortices and particle clusters give rise to gradients in the hydrodynamic fields. 1.2 Velocity Vortices and Particle Clusters Two types of dissipation-driven instabilities, shown in Fig. 1.2, that have no counterpart in (non-dissipative) molecular systems have been observed in the homogeneous cooling system (HCS) [7,13]. The HCS is a periodic flow with zero mean motion, and is described further in Section 1.4. Fig. 1.2 shows snapshots of the velocity field and particle locations in an initially homogeneous flow (Fig. 1.2a-b) that becomes unstable later in time with respect to velocity vortices (Fig. 1.2c) and particle clusters (Fig. 1.2d). Figure 1.2 Visualizations from a granular, discrete-element method simulation of a binary mixture: (a) stable, coarse-grained velocity field at an early time, (b) stable 3

22 particle positions at an early time, (c) unstable, coarse-grained velocity field at a later time, and (d) cluster systems at a later time. A cell size of one-fifth of the domain size is used for local velocity averaging. The velocity vortex instability represents local organization of particle momentum due to dissipation of particle kinetic energy. Consider two frictionless, inelastic particles with u 1,z = u 2,z = 0, u 1,x = u 2,x = 1, and u 1,y =!u 2,y = 1 (where u i,j indicates the j- component velocity of particle i) such that a collision occurs, as depicted in Fig While the tangential components of the particle velocities (for example, u 1,x and u 2,x ) will remain unchanged after the collision due to the frictionless nature of particles, the normal components will decrease in magnitude, such that the alignment of the particles will be greater after the collision compared to before. Over time, an arbitrarily preferred direction of flow may manifest. As stray particles interact with those flowing in the preferred direction, the velocities of stray particles becomes closer to those in the pack. From a continuum perspective, vortices form because the transversal components of a velocity perturbation decay more slowly than the longitudinal component [14]. From a qualitative perspective, the varying decay rates may be understood by considering the more head-on, dissipative nature of a longitudinal wave compared to the tangential nature of a transversal one. Specifically, based on a linear stability analysis [15,16], The growth rate of a transversal wave increases with the cooling rate and wavelength of perturbation and decreases with shear viscosity. From these dependencies, one can see that a long wavelength perturbation combined with a small shear viscosity will give to growth of the 4

23 Figure 1.3 Visualization of before (left), during (middle), and after (right) a particleparticle collision with a restitution coefficient e=0.5. transversal velocity (relative to the longitudinal mode) for a sufficiently large cooling rate. Consider that transversal growth will be dampened for large shear viscosity while a large cooling rate exacerbates the difference in decay rates between the longitudinal and transversal modes. Additionally, a long wavelength transversal growth will give rise to a relatively smaller shear gradient compared to a short one. The resulting growth of the two (perpendicular) transversal components of velocity in either a clockwise or counterclockwise formation will act to create circular motion. This circular motion forms a vortex. In addition to vortices, particle-clustering instabilities (local regions of relatively high solids concentration) may be observed in particulate flows. Picture a flow containing homogenous fields of velocity (Fig. 1.2a) and concentration (Fig 1.2b). Consider a local 5

24 region of relatively high solids concentration. This higher concentration of particles will cause an increased collision frequency, which is proportional to the square of concentration, in the local region being considered. Because dissipative collisions reduce the granular temperature (proportional to the square of the particle velocity fluctuation and thus analogous to a molecular temperature), the local cooling is expedited. As is the case for traditional fluids, the granular pressure (analogous to the traditional pressure for molecular systems) is proportional to the granular temperature. The reduced granular temperature will give to a relatively small pressure in this local region of high particle concentration. For a sufficiently large system, the cooling rate is faster than the rate of diffusion [7], and particles will be driven into this region of low pressure from the surrounding high-pressure environment, leading to a further increase in particle concentration, local cooling, etc. In this way, a cluster of particles forms. From a particle perspective, clustering can be understood in a similar manner. In the perturbed region of high concentration, the collision frequency increases. Because the collisions are inelastic, the magnitude of the relative, normal velocity is decreased during the collision. This decrease in relative motion means that particles in the highconcentration region are less likely to escape. As a result, a further increase in the local concentration and collision frequency can lead to cluster formation. Instabilities can also form due to gas-solid interactions. As is the case for particleparticle collisions, models of the lubrication force between particles indicate that head-on interactions are more dissipative than glancing ones [17]. Thus, velocity vortices form in gas-solid flows in the same manner as for granular ones. The mechanism for particle clustering is also in line with the granular case because gas-solid drag causes a decrease 6

25 in the magnitude of normal relative velocity after collision as compared to before [18]. Mean drag can also lead to clustering. Particles strung together in the face of mean flow provides a path of reduced resistance for the particles to accelerate due to gravity, while in the relatively dilute regions, the fluid can rise relative to the solids [19]. 1.3 Mechanisms Relevant to the Formation of Instabilities Of interest here are the instabilities in particulate flows known as velocity vortices and particle clusters [20-24]. Previous and current work has identified several mechanisms for the particle clustering instability. Specifically, inelastic (dissipation with respect to normal contacts) [6,13] and frictional (dissipation with respect to tangential contacts) (Chapter 4; Ref. [25]) particle collisions give rise to instabilities in granular flows. For flows in which an interstitial gas is also relevant, previous work has indicated that mean drag [26] and thermal (referring here to the granular temperature) drag [27] also lead to instabilities in gas-solid flows. These gas-solid drag mechanisms are discussed in depth in Section ; simply put, thermal drag (proportional to the granular temperature T = m ( u! u) 2 / 3, where m is the particle mass, u is the particle velocity, and both the over bar and brackets indicate averaging) refers to drag associated with random fluctuations (about the mean) of the particle velocities, while mean drag refers to drag associated with (and proportional to) the mean relative velocity between with solid and fluid phases. An understanding of how velocity vortices and particle clusters form is provided in Section 1.2. The role of energy dissipation in the formation of instabilities in rapid granular flows has been examined in some detail [6,7,9,13,15,28-30] for frictionless and inelastic 7

26 particles. For such systems, the level of energy dissipation is dictated by the normal restitution coefficient, 0! e = u 1,normal,post " u 2,normal,post u 1,normal " u 2,normal! 1, which relates the normal, relative, post-collision velocity of two colliding grains to the pre-collisional value. For a given level of dissipation (dictated by e) and concentration! of particles in the domain, a critical dimensionless length scale exists for both the onset of velocity vortices (L Vortex / d) and particle clusters (L Cluster / d) where L is the domain length and d is the particle diameter. Above this critical length scale, the instability of interest will be observed after long times. A crude analogy can be made here with a critical Reynolds number (Re c ) for molecular fluids, where the Reynolds number is directly proportional to a length scale such as tube diameter; namely, laminar (stable) flow is exhibited below Re c and turbulent (unstable) flow is exhibited above Re c. Adding a fluid phase to this flow provides two additional contributions to clustering: mean drag [19,26] and thermal drag [27], which are governed by the mean flow Reynolds number Re M = (1!!)" g d "U µ g and thermal Reynolds number Re T =! g d µ g T 0 m, respectively. These additional gas-phase effects increase the propensity of the system to develop instabilities. The thermal Reynolds number, mean flow Reynolds number, and density ratio! s /! g characterizes the gas effects where! g is the gas density, d is the particle diameter, T = m ( u! u) 2 / 3 is the granular temperature, which is a measure of the fluctuating velocity of particles just as traditional temperature 8

27 is of molecules, µ g is the gas viscosity, m is the mass,! s is the solid density, u is the solids velocity, and!u is the mean relative velocity between phases. 1.4 System Description For this work, particulate flow systems are chosen specifically to isolate the mechanisms for instability and elucidate the relative importance of each mechanism. While a system with dissipative walls would inherently give rise to spatially inhomogeneous flow (due to slower particle velocities after colliding the the wall), a fully periodic, granular system without external forces will exhibit homogeneous flow (with respect to both concentration and momentum) in the absence of instabilities. Such a flow with zero net momentum and inelastic grains is referred to as the homogeneous cooling system (HCS) because the kinetic energy KE = m(u! u) 2 / 2 associated with random particle motion dissipates (and, thus, the granular temperature cools ) over time. 1.5 Modeling Approaches A range of modeling techniques, as illustrated in Figure 1.4, is incorporated in this thesis. One focus of this work is quantitatively testing the validity hydrodynamic (continuum) models rather than using the qualitative visual comparison often used in previous work. Furthermore, an objective is to better understand the interplay of various mechanisms to drive the development of future hydrodynamic models. Chapters 1-6 contain the study of granular (solids-only) hydrodynamics and test against discrete element method (DEM) simulations. In Chapters 7 and 8, the hydrodynamic (continuum) equations are coupled to the Navier-Stokes equations of fluids to study gas-solid flows and test against direct numerical simulation (DNS) and DEM combined with 9

28 computational fluid dynamics (DEM-CFD). In both DNS and DEM-CFD, particles are tracked individually so the most marked difference between DEM-CFD and DNS is in the treatment of the fluid phase. In DNS, the fluid flow is fully resolved by implementing a no-slip condition along the particle surface, whereas in DEM-CFD a grid-based CFD technique utilizes the exact particle velocity and local-average fluid velocity. The DNS (Fig. 1.4a), granular DEM, DEM-CFD (Fig. 1.4b), and hydrodynamic (continuum) modeling (Fig. 1.4c) approaches to modeling particulate flows are discussed below. Figure 1.4 Depictions of various approaches to modeling gas-solid flows: (a) direct numerical simulation (DNS) in which the detailed motion of the fluid is resolved around each particle, (b) discrete element method-computational fluid dynamics (DEM-CFD) in which individual particles are tracked while averaged fluid motion is solved on a grid larger than inter-particle spacing, and (c) hydrodynamic (continuum) modeling of both gas and solid phases. Blue and black arrows indicate the fluid and particle velocity, respectively. Dashed lines in (b) and (c) represent the numerical grid used to solve the continuum balances Granular Flow Discrete Element Method (DEM) The discrete element method (DEM), which is applied to granular flows (no fluid), tracks the positions and velocities of individual particles over time while resolving 10

29 particle-particle interactions via Newton s laws of motion. This method is a wellestablished and highly detailed model of particle interactions. In systems free of external forces like the HCS, particles follow linear trajectories and the (event-driven) simulation skips from collision to collision. This hard-sphere treatment repeatedly performs 3 steps: (i) the time until the next contact between two particles is calculated, (ii) all particles are advanced in space according to this elapsed time, and (iii) the collision is resolved by calculating post-collisional velocities for each particle. This hard-sphere algorithm is outlined in the following paragraph. The reader interested in further details on eventdriven DEM simulations (e.g., accelerated search techniques used for large N systems) is referred to the appropriate texts [31,32]. The time until collision is calculated for every combination of particles in the system from the initial positions, radii, and velocities of each particle [31]. The minimum time calculated represents the time at which the next two particles will be in contact. All particles are advanced along a linear trajectory by this period of time. The resulting particle collision is resolved through momentum and kinetic energy balances using the hard-sphere collision rule, in which collisions are treated as instantaneous and binary. Specifically, for a frictionless collision, the post-collisional velocity for each colliding particle, namely u 1,post and u 2,post is determined by u 1, post = u 1! u 2,post = u 2 + (1+ e) 2 (1+ e) 2 11 [ k "(u 1! u 2 )]k, (1.1) [ k!(u 1 " u 2 )]k, (1.2) where k is the unit vector pointing from the center of particle 1 to the center of particle 2, u 1 and u 2 are the pre-collisional velocities for particles 1 and 2, and e is the restitution

30 coefficient. The restitution coefficient e = u 1,normal, post! u 2,normal, post u 1,normal! u 2,normal (where u normal is the speed of the particle in the normal direction and the subscript post indicates the speed after the collision of particle 1 and 2) ranges from zero (i.e., perfectly dissipative collisions resulting in identical post-collisional normal velocities) to one (i.e., perfectly elastic collisions). The next time until contact is then calculated with these postcollisional velocities, and this process is repeated until the desired simulation duration is reached. The addition of frictional interactions is detailed in Section Kinetic-Theory-Based Hydrodynamic (Continuum) Modeling Hydrodynamic (continuum) models derived from kinetic theory (KT) offer an efficient framework for predicting large-scale behavior of particulate flows. The transient, kinetic-theory-based continuum equations developed by Garzó & Dufty [33] for monodisperse, granular flows are solved numerically. The balance equations for this continuum model [33] are given by!n!t + u"#n + n#"u = 0, (1.3) %! "u & ' "t + u#$u ( ) * = +$P, (1.4)!T!t + u"#t = $ 2 3n # " q + P :#u ( ) $%T, (1.5) where n is number density, t is time, u is velocity, ρ = nm is the bulk solids density, P is the pressure tensor, T is the granular temperature, q is the heat flux, and ζ is the cooling rate. Detailed constitutive relations for the terms P, q, and ζ can be found in 12

31 [33]. The theory assumes hard spheres that interact via instantaneous, binary collisions with a constant restitution coefficient e. Previous studies have shown the ability of such models to qualitatively predict clustering in granular [29] and gas-solid flows [8,9,11,12,34] via visual snapshots from a transient simulation. However, concerns over the quantitative accuracy of hydrodynamic (continuum) theories exist due to the assumptions involved in the starting kinetic equation (e.g., Enskog equation) and assumptions used during the derivation process. Practical application of hydrodynamic models based on such theories involves a truncation of the Chapman-Enskog (CE) perturbation expansion [35] used to obtain the constitutive quantities. This truncation is typically performed at Navier-Stokes order (i.e., first order in gradients) such that fluxes are proportional to first-order hydrodynamic gradients (e.g. the heat flux depends on!t ), although higher-order expressions exist. This truncation represents a small-gradient assumption (i.e., higher-order components of the flux definition are small with respect to first-order ones) and brings into question the applicability of Navier-Stokes-order hydrodynamics to flows containing large gradients. The study of instabilities represents an ideal avenue to quantitatively assess the ability of hydrodynamic (continuum) models to predict the behavior of high-gradient flows because developed velocity vortices and particle clusters give rise to relatively large gradients in the hydrodynamic fields (i.e., velocity, concentration, and granular temperature). For example, the interface of a particle cluster is characterized by a relatively large gradient in concentration between the clusters and surrounding dilute regions. 13

32 1.5.2 Gas-Solid Flow Direct Numerical Simulation (DNS) As illustrated above in Figure 1.4a, direct numerical simulation (DNS) is the most detailed modeling approach considered in this work for the study of gas-solid flows (this approach is not relevant to granular flows). Particle-particle interactions are resolved through Newton s laws of conservation of momentum. The discretized Boltzmann equation applied over a lattice mesh describes the fluid phase, and the dynamics of this fluid phase are fully resolved by implementing a no-slip boundary condition along the surface of each particle. Professor Xiaolong Yin of the Colorado School of Mines and coworkers have performed the DNS data presented in this thesis. They used the Susp3D lattice- Boltzmann-based technique developed by Ladd and coworkers [36,37] for the DNS Discrete Element Method- Computational Fluid Dynamics (DEM-CFD) Compared to DNS, DEM-CFD simulations contain less detail on the fluid phase. As illustrated in Figure 1.4b, the fluid phase quantities represent averages over regions containing many particles rather than having the fluid motion resolved around each particle. As a result, DEM-CFD requires a model for the drag force, whereas the drag forces is an output of DNS simulations. Also, in contrast to the previous hard-sphere simulations performed in granular DEM, a soft-sphere collision model [38] is used in DEM-CFD because the fluid calculations are more suited to a constant time step. In softsphere simulations, a force balance is invoked at each time step Namely, the governing equations for particle motion are given by, 14

33 dx i dt = u i, m i du i dt = m i g + F D,i + F C,i, (1.6) where X is the particle position, i denotes the i th particle in the system, t is time, u is the instantaneous particle velocity, m is the particle mass, g is gravity, F D is the total fluid force, and F C is the force from particle collisions. The fluid drag force is of the form! ( u g! u), and can be obtained empirically or from DNS simulations. The collisional force of particle i (with diameter D i ) is given by the spring-and-dashpot model, F C,i = F S + F with F =!k dx DA S n" n n and F DA =! d where k n is the normal spring dt coefficient,! n = 0.5( D i + D j )! X i! X j is the normal, linear overlap, n = X j! X i X j! X i is the normal vector pointing from particle i to particle j, and! d is the damping coefficient.! d is related to the normal restitution coefficient, collision time, a function of, k n, and m [38].! d " e = exp!! d # $ m t col % & ' where t col is the Two-Phase Hydrodynamic (Continuum) Modeling Hydrodynamic (continuum) models derived from kinetic theory (KT) offer an efficient framework for predicting large-scale behavior of gas-solid flows. The transient, kinetic-theory-based gas-solid continuum equations developed by Garzó, Tenneti, Subramaniam, and Hrenya [39] for monodisperse gas-solid flows are solved numerically. The balance equations for the solids and gas phases of this monodisperse, two-fluid model (TFM) [39] are given, respectively, by 15

34 !n!t + u"#n + n#"u = 0, (1.7) $!!u % &!t + u"#u ' ( ) = *#P * n!+u + "g, (1.8)!T!t + u"#t = $ 2 (#"q + P : #u) $!T $ 2T " + m#, (1.9) 3n m!" f!t + " f # $u f + u f $#" f = 0, (1.10) % "u! f f ' & "t ( + u f # $u f * ) = +#P f + n,-u +! f g, (1.11) where n is number density of solids, t is time, u is solids velocity, ρ = nm is the bulk ( ) " solids density, is the solids pressure tensor,! = 3"µd 10! 1!! + ( 1!! % P $ ) ! ' # & is the mean-flow drag coefficient,!u = u - u g is the relative velocity between the solid and fluid phase, g is gravity, T is the granular temperature, q is the granular heat flux, ζ is the cooling rate,! is the thermal drag coefficient,! is the coefficient associated with neighbor effects,! f is the bulk fluid density, u f is the fluid velocity, P f is the pressure tensor, C p, f is the heat capacity, T f is the temperature, and q f is the fluid heat flux. Detailed constitutive relations for the terms P, q, ζ,!, and! can be found in [39]. The theory assumes hard spheres that interact via instantaneous, binary collisions with a constant restitution coefficient e. In the absence of the fluid phase (i.e.,! = " = # = 0 ), the hydrodynamic model [39] reduces to that of monodisperse, granular flows developed by Garzó & Dufty [33]. 16

35 Within this context of macroscopic balance equations for the solid phase, fluid effects can be decomposed into constituent parts: (i) mean drag, (ii) thermal drag, and (iii) the influence of neighboring particles on the fluctuating gas-phase velocity. Mean drag manifests in the solid-phase hydrodynamic equations via a term in the momentum balance that is proportional to mean relative velocity between phases (second term on right-hand side of Eq. 1.8). Thermal drag manifests as a sink term, proportional to T, in the granular energy balance (penultimate term of Eq. 1.9). Perhaps least obviously, the influence of neighboring particles appears as a source term in the energy balance (final term in Eq. 1.9) because the gas-phase fluctuations in pressure and velocity that are caused by neighboring particles will affect the momentum imparted on particles. While the gas-solid theory described above presents one of the most advanced attempts to incorporate fluid effects into particle flow hydrodynamics, many previous attempts have been made. Early attempts, via settling experiments [40,41], focused on adding an empirical mean drag to the momentum balance of the form where β is the drag coefficient, u g is the gas phase velocity, u is the solid-phase velocity, and an overbar represents a mean value. This method inherently fails to incorporate the fluid influences on the granular energy balance or constitutive relations. A more robust procedure is to include the fluid force in the starting (Enskog) kinetic equation. In this way, the fluid effects arise fundamentally from the kinetic theory derivation. Furthermore, the mean phase velocities can be replaced by instantaneous ones, giving to the form! ( u g! u)! ( ug " u) [42,43]. However, a recent investigation [44] indicates that this 17

36 presumed drag form is not appropriate for instantaneous quantities since the influences of neighboring particles are not properly considered. An improved drag form is possible in the limit of Stokes flow, where thermal drag and the influence of neighboring particles are included analytically in the starting kinetic equation [45,46]. However, this methodology is not practical beyond the Stokes limit. Authors Garzó, Tenneti, Subramaniam, and Hrenya ( GTSH ) [39] of a recent gas-solid hydrodynamic model have proposed casting the instantaneous fluid force in the form of a Langevin equation, based on direct numeric simulations, whereby the influence of neighboring particles can be considered via a stochastic term. For the case of Stokes flow, this model has been validated via comparison [39] with results from the aforementioned Stokes description. Most importantly, this DNS-based GTSH approach [39] is not restricted to low-reynolds-number flow since DNS can be carried out over a wide parameter space. The GTSH starting equation incorporates an instantaneous fluid force of the form, F fluid dt = m du =! ( ug " u) dt " # $ ( u fluct ) dt + mb$dw, (1.12) where t is time, m is the particle mass, u fluct is the fluctuating velocity ( u! u ),! is the viscous loss coefficient, W is a (stochastic) Wiener process increment, and B is the stochastic coefficient [39]. The first term on the RHS is the conventional drag force (proportional to the mean relative velocity), the second term is the so-called thermal drag (proportional to the granular temperature), whereby the fluctuating particle velocity decays even in the absence of mean drag, and the third term is a stochastic term that accounts for the influences of neighboring particles via gas-phase fluctuations. 18

37 1.6 Dissertation Objectives This thesis begins with the typical monodisperse, frictionless, granular HCS (Chapters 2, 3, and 6). Chapter 2 begins with a test of the ability of hydrodynamic (continuum) models to describe flows of moderate dissipation and particle concentration. The ability of such models to describe extreme levels of dissipation is tested in Chapter 3. The effects of frictional particle collisions are studied in Chapter 4 by adding frictional interactions via a tangential restitution coefficient!. The influences of a polydisperse particle distribution are examined in Chapter 5 by allowing non-uniform particles of number fraction x 1 and x 2. While Chapters 2-5 focused on the onset of the velocity vortex where flow gradients are small, in Chapter 6 the focus is on testing the ability hydrodynamics to describe high-gradients flows where vortices have evolved. Large gradients present a challenge to the small-gradient (i.e., small-knudsen-number) assumption invoked during the derivation of the hydrodynamic (continuum) equations. In the following chapters, the fluid phase is included in this standard HCS to examine different mechanisms for instability. The effects of thermal drag are studied in Chapter 7 by including the fluid phase (in a zero-net-momentum flow) with thermal Reynolds number Re T, and the effects of mean drag are examined in Chapter 8 by including gravity in a gas-solid flow such that mean flow occurs with a mean Reynolds number Re M. 19

38 1.6.1 Assessing Ability of Hydrodynamics to Describe Particulate Flows Granular: Moderate Concentration and Dissipation (Chapter 2; [47]) Two important differences between the well-established Boltzmann and Enskog equations, upon which kinetic-theory-based hydrodynamics are based, are the inclusion of finite-size effects and dissipative collisions. The ability of hydrodynamics to predict the critical length scale for the onset of velocity vortices in flows of moderate concentration and dissipation are apt tests of these new complexities associated with the Enskog equation and represents the first quantitative test of hydrodynamics via instabilities. Predictions of granular DEM in the HCS. L Vortex / d from linear stability analysis [16] are compared to Granular: Extreme Dissipation (Chapter 3; [48]) Since deviations from the elastic-particle velocity distribution, upon which the distribution of the starting kinetic equation is based, increase as inelasticity decreases, a more accurate representation of the distribution may be required for highly dissipative flows. Such a modified Sonine polynomial approximation is tested against the standard Sonine approximation in Chapter 3. Predictions of L Vortex / d and L Cluster / d from linear stability analysis [16] are compared to granular DEM in the HCS Granular: Frictional Particle Collisions (Chapter 4; [25]) Since real grains dissipate energy in both the normal and tangential component of velocity, a theory for inelastic, frictional particles is studied in Chapter 4. Predictions of L Vortex / d and L Cluster / d from a linear stability analysis [16,49] are compared to granular 20

39 DEM in the HCS in an effort to understand the frictional (tangential) mechanism relative to its inelastic (normal) counterpart in an effort to understand the frictional (tangential) mechanism relative to its inelastic (normal) counterpart Granular: Polydisperse Particles (Chapter 5; [50]) The introduction of additional particle species invokes uncertainty in the choice of hydrodynamic variables. While traditional fluid hydrodynamics use a species concentration, mixture velocity, and mixture temperature, the relevant energy balance is unclear due to the dissipative nature of particulate flows. Predictions of L Vortex / d from a linear stability analysis of a hydrodynamic theory [51-53] using a mixture temperature in the energy balance are compared to granular DEM in the HCS to assess the appropriateness of using a mixture temperature Granular: High Gradients (Chapter 6; [54]) As is the case for the typical closures of the Navier-Stokes equations, those of the hydrodynamic model are based on a truncation at first order in hydrodynamic gradients. The validity of this Navier-Stokes-order truncation is assessed by comparing predictions from hydrodynamics with discrete-particle simulations in flows with developed velocity vortices such that large gradients are present. Note that reference to Navier-Stokes hydrodynamics in this dissertation generally refers to the first-order-in-gradients (Navier- Stokes-order) truncation of the solids-phase constitutive quantities rather than the Navier- Stokes equations for fluids. Theoretical predictions of L Cluster / d from a linear stability analysis [16] and transient numerical simulation are compared to granular DEM in the HCS. 21

40 Gas-solid: Thermal Drag (Chapter 7) The inclusion of a fluid phase (in the absence of a mean relative velocity between the fluid and solid) introduces a thermal drag that is proportional to the granular temperature. This dissipation of energy via thermal drag has been shown to cause clustering in flows of elastic particles [27]. Predictions of L Cluster / d from transient numerical simulations of hydrodynamics are compared to direct numerical simulation in the gas-solid HCS to assess the accuracy of the proposed influence of drag Identification of Dominant Mechanisms for Instability Granular: Frictional Particle Collisions (Chapter 4; [25]) The introduction of friction leads to a new source for collisional energy dissipation. The importance of frictional interactions is the formation of instabilities relative to normal inelasticity is studied via DEM and a linear stability analysis [16,49] of hydrodynamics in the granular HCS Gas-solid: Thermal Drag (Chapter 7; [55]) The dissipation of energy caused by thermal drag, which is proportional to the granular temperature, is a mechanism for instability [27]. The importance of thermal drag in the formation of particle clustering compared to collisional dissipation is studied via transient numerical simulations of hydrodynamics and direct numerical simulation in the gas-solid HCS. 22

41 Gas-solid: Mean Drag (Chapter 8) Mean drag, which is proportional to the mean relative velocity between phases, is a mechanism for instability [19,26]. The importance of mean drag relative to collisional dissipation is studied via transient numerical simulations of hydrodynamics in a gas-solid settling flow of monodisperse, frictionless spheres. 1.7 References 1. Goldreich, P. & Tremaine, S The dynamics of planetary rings. Annual Review of Astronomy and Astrophysics 20, Schmidt, J., Salo, H., Spahn, F. & Petzschmann, O Viscous overstability in Saturn's B-ring: II. Hydrodynamic theory and comparison to simulations. Icarus 153, Lissauer, J. J Planet formation. Annual Review of Astronomy and Astrophysics 31, Herman, A Molecular-dynamics simulation of clustering processes in sea-ice floes. Phys. Rev. E 84, Bartels-Rausch, T., Bergeron, V., Cartwright, J. H. E., Escribano, R., Finney, J. L., Grothe, H., Gutiérrez, P. J., Haapala, J., Kuhs, W. F., Pettersson, J. B. C., Price, S. D., Sainz-Díaz, C. I., Stokes, D. J., Strazzulla, G., Thomson, E. S., Trinks, H. & Uras-Aytemiz, N Ice structures, patterns, and processes: A view across the icefields. Reviews of Modern Physics 84, Hopkins, M. A. & Louge, M. Y Inelastic microstructure in rapid granular flows of smooth disks. Phys. Fluids A 3, Goldhirsch, I., Tan, M. L. & Zanetti, G A molecular dynamical study of granular fluids I: The unforced granular gas in two dimensions. J. Sci. Comput. 8, Gidaspow, D. Multiphase flow and fluidization : continuum and kinetic theory descriptions. (Academic Press, 1994). 9. Jackson, R. The dynamics of fluidized particles. (Cambridge Univ. Press, 2000). 10. Sundaresan, S Modeling the hydrodynamics of multiphase flow reactors: current status and challenges. AIChE Journal 46, Sundaresan, S Instabilities in fluidized beds. Annu. Rev. Fluid Mech. 35, Kunii, D. & Levenspiel, O. Fluidization engineering. Vol. 2 (Butterworth- Heinemann Boston, 1991). 13. Goldhirsch, I. & Zanetti, G Clustering instability in dissipative gases. Phys. Rev. Lett. 70, Brey, J. J., Dufty, J. W., Kim, C. S. & Santos, A Hydrodynamics for granular flow at low density. Phys. Rev. E 58,

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43 37. Nguyen, N.-Q. & Ladd, A. J. C Lubrication corrections for lattice-boltzmann simulations of particle suspensions. Phys. Rev. E 66, Garg, R., Galvin, J. E., Li, T. & Pannala, S Documentation of open-source MFIX-DEM software for gas-solids flows. From URL Garzó, V., Tenneti, S., Subramaniam, S. & Hrenya, C. M Enskog kinetic theory for monodisperse gas-solid flows. J. Fluid Mech. 712, Richardson, J. F. & Zaki, W. N The sedimentation of a suspension of uniform spheres under conditions of viscous flow. Chemical engineering science 3, Wen, C. Y. & Yu, Y. H Mechanics of Fluidization. Chem. Eng. Prog. Symp. Ser. 62, Ma, D. & Ahmadi, G A kinetic model for rapid granular flows of nearly elastic particles including interstitial fluid effects. Powder Technol. 56, Lun, C. K. K. & Savage, S. B. in Granular gas dynamics (Springer, 2003). 44. Tenneti, S., Garg, R., Hrenya, C. M., Fox, R. O. & Subramaniam, S Direct numerical simulation of gas-solid suspensions at moderate Reynolds number: Quantifying the coupling between hydrodynamic forces and particle velocity fluctuations. Powder Technol. 203, Koch, D. L Kinetic theory for a monodisperse gas-solid suspension. Phys. Fluids A 2, Koch, D. L. & Sangani, A. S Particle pressure and marginal stability limits for a homogeneous monodisperse gas-fluidized bed: kinetic theory and numerical simulations. J. Fluid Mech. 400, Mitrano, P. P., Dahl, S. R., Cromer, D. J., Pacella, M. S. & Hrenya, C. M Instabilities in the homogeneous cooling of a granular gas: A quantitative assessment of kinetic-theory predictions. Phys. Fluids 23, Mitrano, P. P., Garzó, V., Hilger, A. M., Ewasko, C. J. & Hrenya, C. M Assessing a hydrodynamic description for instabilities in highly dissipative, freely cooling granular gases. Phys. Rev. E 85, Goldshtein, A. & Shapiro, M Mechanics of collisional motion of granular materials. Part 1. General hydrodynamic equations. J. Fluid Mech. 282, Mitrano, P. P., Garzó, V. & Hrenya, C. M Instabilities in granular binary mixtures at moderate densities. Phys. Rev. E 89, Garzó, V., Dufty, J. W. & Hrenya, C. M Enskog theory for polydisperse granular mixtures. I. Navier-Stokes order transport. Phys. Rev. E 76, Garzó, V., Hrenya, C. M. & Dufty, J. W Enskog theory for polydisperse granular mixtures. II. Sonine polynomial approximation. Phys. Rev. E 76, Murray, J. A., Garzó, V. & Hrenya, C. M Enskog Theory for Polydisperse Granular Mixtures. III. Comparison of dense and dilute transport coefficients and equations of state for a binary mixture. Powder Technol. 220, Mitrano, P. P., Zenk, J. R., Benyahia, S., Galvin, J. E., Dahl, S. R. & Hrenya, C. M Kinetic-theory predictions of clustering instabilities in granular flows: beyond the small-knudsen-number regime. J. Fluid Mech. 738, R2. 25

44 55. Yin, X., Zenk, J. R., Mitrano, P. P. & Hrenya, C. M Impact of collisional versus viscous dissipation on flow instabilities in gas-solid systems. J. Fluid Mech. 727, R2. 26

45 2. Granular Flows: Effect of Moderate Concentration and Dissipation a Abstract Previous work has indicated that inelastic grains undergoing homogeneous cooling may be unstable, giving rise to the formation of velocity vortices, which may also lead to particle clustering. In this effort, the discrete element method (DEM) is performed over a wide parameter space to determine the critical system size demarcating the stable and unstable regions. Specifically, a system of monodisperse, frictionless, inelastic hard spheres is simulated for restitution coefficients e 0.6 and solids fractions! 0.4. Simulations for each e,! pairing are then carried out over a range of system sizes to determine the critical dimensionless length scale for velocity vortices L Vortex / d (L is the system length and d is the particle diameter), above which velocity vortices appear (unstable system) and below which they are suppressed (stable system). The results show excellent agreement with the theoretical predictions obtained by Garzó [Physical Review E, 72, (2005)] using a linear stability analysis of kinetic-theory-based (continuum) equations that were derived from the Enskog equation. Finally, the time required for onset of the unstable behavior is also explored via DEM and found to be a universal function of the ratio of L/d to L Vortex / d. a Mitrano, P. P., S. R. Dahl, D. J. Cromer *, M. S. Pacella *, and C. M. Hrenya, Instabilities in the homogeneous cooling of a granular gas: A quantitative assessment of kinetic-theory predictions, Physics of Fluids 23, (2011). 27

46 2.1 Introduction Particulate flows play a critical role in industry (energy production, pharmaceuticals, food processing, agriculture), space (planetary formation, ring dynamics, regolith ejection) and nature (avalanches, landslides). Despite this pervasiveness, particulate systems are not well understood relative to their fluid counterparts, often due to unique phenomena that arise as a consequence of dissipative collisions. Of particular interest here are the instabilities in rapid granular flows that stem from inelastic collisions [1-5]. A telltale characteristic of instabilities in such flows is the transient non-uniformities in the solids concentration as illustrated in Figure 2.1 for a homogeneous cooling system (HCS; Figure 2.1a) and the simple shear flow (SSF; Figure 2.1b) of a granular material. These instabilities were first reported via the discrete element method (DEM) by Goldhirsch and co-workers [6,7] for the case of a homogeneous cooling system and by Hopkins and Louge [8] for the case of simple shear. Since then, the existence of such clustering instabilities has been widely documented in both experiments [9-11] and discrete element method simulations (e.g., [12-17]). Furthermore, the important impact of the instabilities on the overall flow behavior is evident. For example, it has been shown that the presence of such instabilities leads to a significant change in the solid-phase pressure and viscosity for granular flows [18]. These results are not surprising given the large impact of instabilities in more traditional flows, such as the role of turbulence (Reynolds) stresses in single-phase fluid flows. 28

47 Figure 2.1 Snapshot of particles extracted from DEM simulations of the clustering instability in (a) the HCS with e=0.6,! =0.05, and N=40,000 and (b) a SSF with e=0.6,! =0.2, and N=2064, where N refers to the total number of particles. A natural follow-on question to the aforementioned findings is the following: are existing continuum (hydrodynamic) descriptions of these flows able to predict the instabilities observed in experiments and DEM simulations? A quick answer to the question is yes, though previous comparisons are largely qualitative in nature. For example, consider the system examined here, the homogeneous cooling system, in which inelastic grains with finite random velocities are placed in a force-free, fully periodic system with zero momentum (and thus the system cools with time due to the dissipative collisions). DEM simulations have indicated that a critical system size exists for the HCS such that smaller systems remain stable with time whereas larger systems exhibit instabilities [6]. Furthermore, DEM simulations of the granular HCS, an initially homogeneous system as depicted in Fig. 2.2a, have indicated that a velocity vortex instability, as is re-produced in Fig. 2.2b, precedes the formation of the particle clustering 29

48 instability. The former (Fig. 2.2b) is characterized by a velocity field with vortices but a relatively uniform distribution of particles, whereas the latter (Fig. 2.1a) is also characterized by strong concentration inhomogeneities. Now considering previous work on the continuum description of such flows, numerous kinetic-theory-based predictions have shown features consistent with the DEM simulations. In one such study, the transient, two-dimensional form of the kinetic-theory-based conservation equations has been applied to the HCS [16], giving rise to predictions of transient particle clusters as exemplified in Fig. 2.1a. A similar effort was later undertaken using a dilute-limit form of the governing equations; the solution exhibited a singularity in the concentration field that was presumed indicative of particle clustering [19]. In another related set of studies, stability analyses of kinetic-theory-based equations applied to the HCS have been carried out, several of which result in expressions for the critical length scale (or system size) that demarcate stable and unstable behavior (e.g., [12,20]). Moreover, the works of [16,21-23] provide separate critical length scales for the velocity-vortex and particleclustering instabilities that indicate the former precedes the latter (under all but extremely dissipative conditions), which is qualitatively consistent with the DEM findings. Nonetheless, a quantitative comparison between the theoretical predictions for the critical length scale and DEM data has not been performed. As part of an overall effort to assess the quantitative ability of a continuum description to predict instabilities in granular flows, the objective of this chapter is to assess the kinetic-theory-based predictions for the critical length scale associated with instabilities in HCS. Accordingly, a comprehensive suite of DEM simulations is performed over a wide range of solids concentrations, restitution coefficients, and system 30

49 sizes. The resulting values of the critical length scale (system size) are compared to the kinetic-theory-based predictions obtained from the linear stability analysis of Garzó [23]. Excellent agreement is found, indicating the ability of kinetic-theory-based predictions at the onset of the instabilities in granular flows and the corresponding appropriateness of a linear stability analysis. Figure 2.2 (Color Online) Snapshots of particle positions, shown as dots (grey online), and spatially-averaged local-velocity vectors, shown as arrows (red online), extracted from DEM simulations of (a) homogeneous concentration and velocity fields at 10 collisions per particle (cpp) and (b) homogenous concentration and inhomogeneous velocity fields at 210 cpp with e=0.98,! =0.05, and N=40, Computational Methods System of Interest The typical monodisperse, inelastic, frictionless HCS is studied in three dimensions as described in Section 1.3 and depicted in Fig While collisions between elastic particles (i.e., molecules) would continue to be characterized by homogeneous velocity 31

50 and concentration fields, dissipative grains may instead exhibit instabilities under certain conditions, such as the particle clustering instability illustrated in Fig. 2.3b. Figure 2.3 DEM snapshot of particle positions in a three-dimensional HCS showing a two-dimensional slice of thickness= L/10 for (a) 1 cpp and (b) 50 cpp with e=0.6,! =0.05, and N=40, Discrete Element Method Initially, N particles with diameter d are placed on a cubic grid within a periodic domain of volume L 3. Particle positions are then slightly and randomly altered in a manner that ensures no overlap. Regarding velocity, individual particle x-, y-, and z- velocities are initially drawn randomly from a uniform distribution. The initial kinetic energy KE 0 of the system is also required as input as a means of setting the starting magnitude of the random velocity, though its value does not impact the nature of the dimensionless results. This expected independence of the dimensionless results to KE 0 has been confirmed through the DEM simulations. Because the initial velocity 32

51 distribution does have a random nature, a small non-zero momentum exists. To instead set the momentum to zero prior to the simulation start, the i-component of each particle velocity is reduced by p i /N, where p i is the total momentum in the i-direction at initialization. For a description of the hard-sphere collision model and simulation algorithm, please see Section The final cpp is chosen to ensure that results are independent of simulation duration, as further described in Section Because DEM simulations involve the tracking of discrete particles, the position and velocity of each particle (and thus the kinetic energy of the system) is known at any given time. For purposes of this chapter, and as detailed in Section below, the DEM code outputs the ratio of the system kinetic energy KE to the initial kinetic energy KE 0 as a function of the number of collisions per particles cpp (i.e., a measure of time) that have occurred in the system. Due to the long duration of these simulations, the amount of net momentum initially present, which is negligible relative to KE 0 (and zero to machine precision), can become significant relative to particle momentums due to continually decreasing particle velocities if the simulation duration is long enough. In order to avoid a loss of relative precision, the net momentum is regularly measured and reset to zero in a manner similar to that done at the simulation start. This correction is similar to an alternative method [20] used to avoid numerical precision issues after long times. It is worthwhile to note that the hard-sphere algorithm presented here provides a straightforward comparison with the kinetic-theory (continuum) model, since both involve inelastic, smooth spheres involved in instantaneous, binary collisions. Moreover, the hard-sphere model also allows for event-driven simulations, which are 33

52 computationally more efficient for rapid granular flows than the soft-sphere alternative Parameter Space Considered The stability of the HCS will depend on the dimensionless inputs characterizing the system, namely the dimensionless system length scale L/d, the restitution coefficient e, and the solids fraction!. A simulation domain of volume L 3 that contains N particles with diameter d has a solids fraction! as defined by! = N"d3 6L 3. (2.1) DEM simulations are performed for a range of dissipation levels, namely e= , and a range of solids fractions, namely! = For each pair of e and!, simulations are run with a range of L/d to identify the critical length scale L Vortex / d, which demarcates stable and unstable systems. More dissipative systems (i.e., e < 0.6) are not as practically relevant and may suffer from inelastic collapse, which is an unphysical artifact of a diverging collision frequency [24-26]. Accordingly, more dissipative systems are not examined here. Furthermore, the range of dilute to moderately dense! considered for this chapter is consistent with the validity of the kinetic-theory (continuum) predictions, which are based on the Enskog equation. Finally, the examined parameter space was also restricted to sufficiently large L/d (or equivalently N) to avoid poor statistics. More specifically, systems with fewer than approximately 200 particles were not studied Detection of Instability As a granular gas in the HCS evolves, its stability can be determined via a comparison of its rate of energy dissipation with Haff s law [12,16,27-29]. According to 34

53 Haff s law, the natural logarithm of the kinetic energy of a granular gas with homogeneous concentration and velocity fields decays linearly with the number of collisions [30]. Accordingly, in the initial stages of DEM simulation, while particle positions and velocities are still homogeneous, the rate of energy decay is measured. This decay rate, which represents the Haff s-law prediction (i.e., homogeneous cooling), is extrapolated throughout the entire simulation for comparison with the actual kinetic energy obtained from the simulation, as shown in Fig In particular, Fig. 2.4a shows good agreement between Haff s law and DEM simulations, which is indicative of a stable system. For simulations performed with the same e and! but for a larger system, the discrepancy displayed in Fig. 2.4b indicates unstable behavior. It is interesting to note that for this unstable system, the kinetic energy in the DEM simulations decays at a slower rate than the Haff s-law prediction. For the vortex instability, this behavior is expected since the alignment of velocities results in more glancing collisions, which are less dissipative than their more head-on counterparts (since energy loss is proportional to the square of the magnitude of normal component of velocity). The same behavior is also true for the clustering instability, since particle clusters are also characterized by aligned velocities, which slow the energy decay [6]. Accordingly, the type of instability present cannot be determined from these plots since both instabilities will decrease the rate of energy dissipation relative to Haff s law (i.e., both will result in deviation above Haff s law). Nonetheless, a significant deviation is indicative of an instability, and in this chapter a 5% deviation between the DEM simulations and Haff s-law values of ln(ke/ke 0 ) is used as the criterion for the onset of an instability. Because the initial Haff s-law measurement is extrapolated throughout the simulation, slight noise in such 35

54 measurements will produce a DEM-Haff s deviation that grows with time even if no instability is present. While a smaller deviation criterion (< 5%) would detect the onset of instability earlier, the 5% criterion was chosen as a conservative estimate to account for the presence of noise in the initial Haff s-law measurements. Figure 2.4 Illustration of the use of Haff s law to detect instability in the HCS for e=0.8,! =0.3: (a) stable behavior for L/d=7.04 and (b) unstable behavior for L/d=9.09. DEM data is shown as a solid line (blue online) and Haff s law is shown as a dashed line Hydrodynamic (Continuum) Description A linear stability analysis of the HCS was carried out by Garzó [23] based on the kinetic-theory-based hydrodynamic (continuum) description developed by Garzó and Dufty [31]. The Garzó and Dufty [31] theory uses the Enskog equation as its starting point for the Chapman-Enskog expansion, and the resulting description is applicable to (moderately) dense flows. Furthermore, because the derivation does not involve a nearly elastic assumption, it is applicable to a wide range of restitution coefficients. Finally, the Garzó and Dufty [31] theory is targeted at inelastic, frictionless spheres involved in 36

55 binary and instantaneous collisions. All of these treatments mimic those of the DEM simulations, thereby providing an apples-to-apples comparison between the two. Previous stability analyses have been either restricted to the dilute limit [32,33] or elastic limit [12,14,20,34]. In his stability analysis of the HCS, Garzó [23] assumes small perturbations in the hydrodynamic fields such that terms quadratic in the perturbation variables and/or their derivatives are negligible. The resulting linear stability analysis is valid during the onset of unstable behavior since products of these fluctuations are still small [6,35]. From this analysis, two length scales emerge, namely L vortex and L cluster, which refer to the critical length scale associated with the shear (velocity vortex) mode and the heat (particle clustering) mode, respectively. For e (as is relevant for the parameter space considered here), L vortex is less than L cluster and three regions of stability are identified: (i) for L/d < L vortex /d, the system is linearly stable to perturbations, (ii) for L vortex /d < L/d < L cluster /d, the HCS is unstable to vortices but linearly stable to particle clusters so that concentration inhomogeneities can form only through a nonlinear coupling [6,12,32], and (iii) for L/d > L cluster /d, vortices precede clusters, with the HCS unstable to both. Accordingly, for the parameter space examined here (e 0.375), the linear stability analysis indicates that L vortex serves as the critical length scale demarcating the stable and unstable regions and that the type of instability forming at onset is a velocity vortex (shear mode). The theoretical prediction for this critical length scale is given by: L vortex = 5 4!! " #* $ %, (2.2) *

56 where! is the pair correlation function at contact,! 0 * is a dimensionless cooling rate,! * is a dimensionless shear viscosity, and! 0 is the mean free path,! = 2!" 2 1!" ( ) 3, (2.3)! * 0 = 5 " 1! e2 12 " ( ) 1+ 3 # $ 32 c % & ' (2.4)!! * =! * k 1+ 4 $ #!" ( 1+ e) " 5 & % + 3 5! * (2.5)! 0 = ( 2"nd 2! )!1 (2.6) where c is only a function of e,! k * is the kinetic contribution to! *, and! * is a dimensionless bulk viscosity, c = 32(1! e)(1! 2e 2 ) 81!17e + 30e 2 (1! e) (2.7) "! * k =! * "! 1 2 # *% 0 # $ & '!1 ( 1! 2 + * (1+ e)(1! 3e)!" ) 5 -, (2.8)! * = 128 5" # " 2 $(1+ e) 1! c % # $ 32& ' (2.9) where! " * is a dimensionless collision frequency, "! * " = # 1! 1 %( $ (1! e)2 # 4 ' & 1! c + ) * 64, - (2.10) 2.4 Results and Discussion The results presented below cover the determination of the critical length scale 38

57 L Vortex / d from DEM simulations (Section 2.4.1), a comparison of this DEM-based L Vortex / d with that obtained from the linear stability analysis (Section 2.4.2), and DEM results for the time needed for instability formation (Section 2.4.3). As discussed in Section 2.3, these results are expected to correspond to the onset of the vortex instability. Although not unexpected, it is worthwhile to point out that Figures 2.3 and 2.7 provide the first evidence of particle clusters and velocity vortices, respectively, in threedimensional DEM simulations of the HCS (as previous work [6,12,13,16] was restricted to two dimensions, namely disks traveling in a plane) Determination of the Critical Length Scale in DEM The detailed evolution of granular flows depends on the initial particle positions and velocities (i.e., initial system conditions). A small change to only the initial conditions will alter the course of collisions and may change the prediction of stability for systems near the critical length scale. Because relatively small systems (i.e., small L/d or equivalently N) are used in this chapter in order to study the behavior near L Vortex / d, multiple realizations of each simulation are required to obtain good statistics. Accordingly, the simulation of a given set of system parameters (i.e., L/d, e,! ) is repeated 10 times with different initial conditions. The differences in initial conditions are achieved by feeding different seeds into a random number generator, which dictates the initial particle positions and velocities. Accordingly, a probability of instability can be determined for a given set of system parameters by evaluating the stability of simulations that differ only in their initial conditions. This probability of instability is simply the number of simulations that exhibit unstable behavior divided by the total number of 39

58 simulations. The dependency of this probability with system size is illustrated in Fig More specifically, Fig. 2.5 shows that systems with L/d! 8 are always stable (probability of instability = 0) while systems with L/d 9 are always unstable (probability of instability = 1). In between these values lies a transition region from stability to instability, which corresponds to a small range of transition length scales. More specifically, based on a linear fit of all intermediate probabilities (values not equal to zero or one, as represented in Fig. 2.5 by the squares containing points inside), the lower and upper bounds of the transition length scale range are determined from the L/d values of the fitted line at probabilities of 0 and 1, respectively. Because the theoretical predictions from a linear stability analysis consider a given L/d as unstable if any possible set of initial conditions leads to instability, the lower bound of the transition length scale range corresponds to the critical length scale ( L Vortex / d ). Consequently, when comparing DEM with theory, the lower bound of the transition range ( L Vortex / d ) is shown as a point. The upper bound is also included as a vertical error bar (Figures 2.8 and 2.9). An advantage of this fitting procedure is that it allows a single value of L Vortex / d to be obtained (as opposed to a range of values) without performing an infinite number of simulations. More importantly, increasing the resolution in the region 7 < L/d < 8 does not significantly change the prediction for L Vortex / d or impact the comparison with theory in Figures 2.8 and 2.9. For reference, the theoretical prediction for the onset of instability (see Section 2.3) corresponding to Fig. 2.5 is L Vortex / d =

59 Figure 2.5 DEM determination of L Vortex / d at 800 cpp for e=0.8 and! =0.3. The length scales associated with the 0 and 1 probabilities (as predicted by the linear fit) represent the lower ( L Vortex / d ) and upper bound, respectively, of the transition range. This range is represented as vertical error bars in Figures 2.8 and 2.9. One interesting characteristic of Fig. 2.5 is the two values of L/d (near 8.5) that give rise to the same probability of instability (0.2). Fig. 2.6a shows that a five-fold increase in the number of DEM replicate simulations (i.e., 50 seeds instead of 10) remedies this issue and results in a more linear prediction of the transition from stability to instability, yet the quantitative prediction of each bound of the transition range is relatively unaffected (i.e., upper bound changes by only 1.1% while the lower bound, L Vortex / d, changes negligibly). Of course, an insufficient number of seeds will result in poor averaging for the probability of instability, though Fig. 2.6a and similar tests (not shown) provide evidence that 10 replicates are sufficient to obtain an accurate prediction of L Vortex / d. Similarly, the DEM prediction of L Vortex / d may depend on the duration (i.e., total 41

60 number of collisions) of a given simulation. In particular, because unstable behavior evolves over time, a short simulation may not exhibit instability, whereas a longer simulation with the same initial conditions may reveal instability. An example of this dependence is illustrated in Fig. 2.6b, where a doubling of the simulation duration results in a less than a 2% change in either bound of the transition range. A rule-of-thumb, namely that the total number of simulated collisions (or equivalently cpp) is doubled until the prediction of each bound of the transition range changes by less than 2%, is applied to all simulations reported here to ensure that the simulation duration is not significantly impacting the prediction of L Vortex / d. The DEM determination of L Vortex / d obtained from Fig. 2.5 was also verified via Fourier analysis. This spectral analysis of density and momentum fields allows for the differentiation of unstable heat (particle clustering) and shear (velocity vortex) modes, respectively, from stable modes (see Section 4 of [6]). Specifically, Goldhirsch et al. studied systems far beyond the onset of instability (L/d >> L Vortex / d ) and showed that a peak in the momentum mode (P) develops at wave vector k = 2! in the presence of vortex instabilities. Fourier analysis of systems with L/d < L Vortex / d shows that the value of P increases monotonically with k (in contrast to the plots shown for L/d >> L Vortex / d in reference [6]. Accordingly, an obvious choice for a quantitative criterion to distinguish a stable momentum mode from one in which vortices are present is to require that the value of P at k = 2! is greater than the next point (k = 4! ). With this Fourier-based criterion, no instabilities are detected for L/d! 7.75 but a vortex instability is detected 42

61 for L/d = 8, suggesting 7.75 < L Vortex / d < 8. This result is in striking agreement with the lower bound of the transition range ( L Vortex / d ) and provides validation for the procedure to determine L Vortex / d (the onset length scale for vortices) presented here. It is also interesting to note that no particle-clustering instabilities are detected in this analysis. Figure 2.6 Sensitivity of instability probability to the (a) number of replicates (seeds) and (b) duration of simulation (total number of collisions) for a system with e=0.8 and! = Critical Length Scale and Type of Instability Numerous investigators have found that DEM simulations of grains in the HCS exhibit cluster and/or vortex formation [6,7,16]. As mentioned above (Section 2.3), the linear stability analysis predicts that for L Vortex / d < L/d < L Cluster / d, the HCS is unstable to velocity vortices and linearly stable to particle clusters. Recall that for the parameter range examined in this chapter, L Vortex / d < L Cluster / d and simulations are intentionally performed near L Vortex / d, so the L Vortex / d < L/d < L Cluster / d region is the most relevant unstable region of the two mentioned in Section 2.3 (the other being one in which the system is linearly unstable to both velocity vortices and particle clusters, with the former 43

62 preceding the latter). Fig. 2.7, which is a representative DEM snapshot of the velocity profile at the time instability was detected, shows the expected vortex instability. It is interesting to note that obvious particle clusters, such as displayed in Fig. 2.3b, were not visually observed in any stage of the DEM simulations performed, nor were any unstable heat (particle clustering) modes observed in the limited Fourier analysis in this chapter, which is consistent with the linear stability analysis. Figure 2.7 A spatially averaged velocity field at the time of instability detection (80 cpp) for e=0.8,! =0.3, and L/d=9.56 with the domain split into square cells with sides of length L/5 for purposes of averaging. Although this qualitative consistency regarding the type of initial instability is encouraging, the primary contribution of this chapter is the quantitative comparison between DEM and theory over a wide range of dissipation levels and solids fractions, as detailed below. 44

63 First, Fig. 2.8 shows the length scale demarcating the stable and unstable regions, namely L Vortex / d, as a function of! for restitution coefficients of e=0.8 (Fig. 2.8a) and e=0.7 (Fig. 2.8b). Error bars associated with DEM predictions correspond to the transition range of L/d as described in Section and depicted in Fig The theoretical predictions of Garzó are obtained from Ref. [23]. Excellent agreement is obtained for the entire range of volume fractions examined (! = ). One notable trend in Fig. 2.8 is that more dilute systems form instabilities at larger L/d. This trend can be physically understood as follows. Particles in more dense systems have shorter mean free paths, while the relevant system time scale (proportional to KE 1/2 ) is unaffected. The increased collision frequency, which results from a shorter average distance between collisions, increases the organization of momentum via an increased decay of the normal component of particle velocities. This trend can also be understood from an analysis of the theoretical prediction; the shear viscosity and mean free path decrease with increasing!, while the cooling rate increases. Thus, the! dependency of each term contributes to decreasing L Vortex / d with increasing!. 45

64 Figure 2.8 Comparison of the critical length scale ( L Vortex / d ) for (a) e=0.8 and (b) e=0.7. The error bars correspond to the transition length scale range obtained from DEM simulations (see Section and Figure 2.5). Theoretical predictions are obtained from the linear stability analysis of Garzó [23]. To complement the information in Fig. 2.8, Fig. 2.9 gives L Vortex / d as a function of e for constant! =0.1 (Fig. 2.9a) and! =0.4 (Fig. 2.9b). Agreement is again excellent and notably extends to high dissipation (i.e., e=0.6). An obvious trend in Fig. 2.9 is that more dissipative systems form vortex instabilities at smaller length scales. This behavior can be physically understood as follows. Vortices form because collisions systematically remove energy in the normal direction while the tangential component of particle velocities is unaffected. As e decreases, more energy is dissipated with each collision so that the redirection of particle momentum to the tangential direction is augmented, such that instability forms more readily. This trend can also be understood in the context of the theoretical prediction; the magnitude of the cooling rate decreases with increasing e and thus, causes an increase in L Vortex / d. 46

65 Figure 2.9 Comparison of the critical length scale ( L Vortex / d ) for (a)! =0.1 and (b)! =0.4. The error bars correspond to the transition length scale range obtained from DEM simulations (see Section and Fig. 2.5). Theoretical predictions are obtained from a linear stability analysis of Garzó [23]. The agreement presented in Figures 2.8 and 2.9 provides evidence that two important continuum assumptions are not severely violated during the onset of vortex instability. The Enskog equation assumes the velocities of colliding particles are not correlated (i.e., molecular chaos), which is known to breakdown with increasing!, and the Chapman-Enskog expansion assumes small Knudsen numbers, defined as the ratio of the mean free path to the length scale that characterizes spatial variations in hydrodynamic variables [36]. The velocity gradients present in unstable flows, as illustrated in Fig. 2.7, along with agreement at moderate densities, as shown in Fig. 2.8, demonstrate the utility of the Enskog equation for such systems. This finding supports previous comparisons with continuum theory that suggest Enskog-based predictions are applicable to systems of moderate densities and also to systems where Knudsen effects appear important [27,37-39]. 47

66 It is worthwhile to note that such good agreement between DEM and kinetic theory over a range of! and e (Figures 2.8 and 2.9, respectively) is not entirely unexpected because the kinetic- theory-based description [31] used for the linear stability analysis is not restricted to the dilute or elastic limits in contrast to much previous work Time to Instability in DEM Although unavailable from stability analyses, the characteristic time needed for instability to manifest can be obtained from DEM simulations, and is plotted in Fig This time has been non-dimensionalized using the initial system kinetic energy, particle mass and diameter: t* = t d KE 0 m, (2.11) where t is the time elapsed prior to the onset of instability (i.e., 5% deviation from Haff s law as outlined in Section 2.4.1). Analysis of t* when DEM simulations reach 5% deviation from Haff s law shows that t* is dominated by the ratio of L/d to L Vortex / d. More specifically, the ratio of L/d to the midpoint of the transition range (L M /d) is examined to ensure that the simulations are beyond the transition length scale range. Fig. 2.10a shows that the dependence of t* on L/d exhibits variations of approximately 12 orders of magnitude, while varying e and! with a constant ratio of L/d to L M /d (i.e., a constant propensity toward instability) alters t* by about an order of magnitude (Figures 2.10b and 2.10c, respectively). This strong dependence on the ratio of L/d to L M /d is not entirely unexpected since varying L/d from the stable regime to the unstable will essentially change t* from infinity (stable behavior) to a finite value (unstable). This 48

67 dimensionless time to instability (t*) is expected to provide important benchmark data with which to compare transient solutions to the three-dimensional form of the governing equations (fully nonlinear solution). Figure 2.10 Dimensionless time until the onset of the instability (t*) extracted from DEM simulations with (a) e=0.8 and! =0.3, (b) e=0.8 and L/d=1.2*(L M /d), and (c)! =0.3 and L/d=1.2*(L M /d). 2.5 Concluding Remarks In this effort, a comprehensive suite of DEM simulations was carried out to explore the onset of flow instabilities in the homogeneous cooling of a granular material. A 5% violation of Haff s law was used to detect the onset of the instability. Simulations were carried out over a range of solids volume fractions, restitution coefficients, and system sizes in order to identify, for the first time, the DEM-based critical length scale ( L Vortex / d ) demarcating the stable and unstable regions. The resulting dependency of L Vortex / d with! and e showed excellent quantitative agreement with predictions obtained by Garzó 49

68 [23] using a linear stability analysis of kinetic-theory-based (continuum) equations. Moreover, at the onset of the instability, the DEM simulations revealed the presence of velocity vortices, which is also consistent with predictions of the linear stability analysis [23] for the type of instability expected at onset. Although encouraging, this chapter represents a first step in assessing the quantitative ability of kinetic-theory-based predictions to accurately predict instabilities in granular flows. In particular, the current effort has focused on the onset of the instabilities, when a linear treatment of the perturbation variables is appropriate and when the low-knudsen (small-gradient) assumption used in the derivation of the continuum description is expected to be upheld. (For example, see [40] for impact of Knudsen effects on kinetic-theory-based constitutive relations). Several of the subsequent chapters will continue to focus on the onset of instabilities, but for more complex systems (extremely dissipative particles in Chapter 3, frictional particles in Chapter 4, and polydisperse particles in Chapter 5), whereas the focus of later chapters (Chapters 6-8) is on evolved instabilities where nonlinear and higher-kn effects may play a role. 2.6 References 1. Campbell, C. S Rapid granular flows. Annu. Rev. Fluid Mech. 22, Jaeger, H., Nagel, S. & Behringer, R The physics of granular materials. Phys. Today 49, Ottino, J. & Khakhar, D Mixing and segregation of granular materials. Annu. Rev. Fluid Mech. 32, Goldhirsch, I Rapid Granular Flows. Annu. Rev. Fluid Mech. 35, Goldhirsch, I., Noskowicz, S. H. & Bar-Lev, O Theory of granular gases: some recent results and some open problems. J. Phys. Cond. Matter 17, S Goldhirsch, I., Tan, M. L. & Zanetti, G A molecular dynamical study of granular fluids I: The unforced granular gas in two dimensions. J. Sci. Comput. 8,

69 7. Goldhirsch, I. & Zanetti, G Clustering instability in dissipative gases. Phys. Rev. Lett. 70, Hopkins, M. A. & Louge, M. Y Inelastic microstructure in rapid granular flows of smooth disks. Phys. Fluids A 3, Kudrolli, A., Wolpert, M. & Gollub, J. P Cluster formation due to collisions in granular material. Phys. Rev. Lett. 78, Olafsen, J. & Urbach, J Clustering, order, and collapse in a driven granular monolayer. Phys. Rev. Lett. 81, Falcon, E., Wunenburger, R., Evesque, P., Fauve, S., Chabot, C., Garrabos, Y. & Beysens, D Cluster formation in a granular medium fluidized by vibrations in low gravity. Phys. Rev. Lett. 83, Brito, R. & Ernst, M Extension of Haff's cooling law in granular flows. Europhys. Lett. 43, Luding, S. & Herrmann, H. J Cluster-growth in freely cooling granular media. Chaos 9, van Noije, T. & Ernst, M Cahn-Hilliard theory for unstable granular fluids. Phys. Rev. E 61, Conway, S. L. & Glasser, B. J Density waves and coherent structures in granular Couette flows. Phys. Fluids 16, Brilliantov, N., Saluena, C., Schwager, T. & Pöschel, T Transient structures in a granular gas. Phys. Rev. Lett. 93, Rice, R. B. & Hrenya, C. M Characterization of clusters in rapid granular flows. Phys. Rev. E 79, Liss, E. & Glasser, B The influence of clusters on the stress in a sheared granular material. Powder Technol. 116, Fouxon, I., Meerson, B., Assaf, M. & Livne, E Formation and evolution of density singularities in hydrodynamics of inelastic gases. Phys. Rev. E Soto, R., Mareschal, M. & Mansour, M. M Nonlinear analysis of the shearing instability in granular gases. Phys. Rev. E 62, Brey, J. J., Dufty, J. W., Kim, C. S. & Santos, A Hydrodynamics for granular flow at low density. Phys. Rev. E 58, Wakou, J., Brito, R. & Ernst, M. H Towards a Landau-Ginzburg-Type Theory for Granular Fluids. J. Stat. Phys. 107, Garzó, V Instabilities in a free granular fluid described by the Enskog equation. Phys. Rev. E 72, McNamara, S. & Young, W. R Inelastic collapse and clumping in a onedimensional granular medium. Phys. Fluids 4, McNamara, S. & Young, W. R Inelastic collapse in two dimensions. Phys. Rev. E 50, McNamara, S. & Young, W. R Dynamics of a freely evolving, twodimensional granular medium. Phys. Rev. E 53, Dahl, S. R., Hrenya, C. M., Garzó, V. & Dufty, J. W Kinetic temperatures for a granular mixture. Phys. Rev. E 66,

70 28. Huthmann, M., Orza, J. & Brito, R Dynamics of deviations from the Gaussian state in a freely cooling homogeneous system of smooth inelastic particles. Granul. Matter 2, Orza, J. A. G., Brito, R., VanNoije, T. P. C. & Ernst, M. H Patterns and long range correlations in idealized granular flows. Internat. J. Modern Phys. C 8, Haff, P Grain flow as a fluid-mechanical phenomenon. J. Fluid Mech. 134, Garzó, V. & Dufty, J. W Dense fluid transport for inelastic hard spheres. Phys. Rev. E 59, Brey, J. J., Ruiz-Montero, M. J. & Cubero, D Origin of density clustering in a freely evolving granular gas. Phys. Rev. E 60, Petzschmann, O., Schwarz, U., Spahn, F., Grebogi, C. & Kurths, J Length scales of clustering in granular gases. Phys. Rev. Lett. 82, Brey, J. J., DomÌnguez, A., GarcÌa de Soria, M. I. & Maynar, P Mesoscopic theory of critical fluctuations in isolated granular gases. Phys. Rev. Lett Kundu, P. & Cohen, I. (Elsevier Academic Press, New York, 2008). 36. Chapman, S. & Cowling, T The Mathematical Theory of Non-Uniform Gases.. Cambridge: Cambridge University. 37. Rericha, E., Bizon, C., Shattuck, M. & Swinney, H Shocks in supersonic sand. Phys. Rev. Lett. 88, Xu, H., Louge, M. & Reeves, A Solutions of the kinetic theory for bounded collisional granular flows. Continuum Mechanics and Thermodynamics 15, Wildman, R., Martin, T., Huntley, J., Jenkins, J., Viswanathan, H., Fen, X. & Parker, D Experimental investigation and kinetic-theory-based model of a rapid granular shear flow. J. Fluid Mech. 602, Hrenya, C. M., Galvin, J. E. & Wildman, R. D Evidence of higher-order effects in thermally driven rapid granular flows. J. Fluid Mech. 598,

71 3. Granular Flows: Effect of Extreme Dissipation b Abstract Whereas the previous chapter assessed the ability of the hydrodynamic model to quantitatively predict flow for moderate dissipation levels, the focus of this chapter is on extremely dissipative flows. Specifically, in this chapter, an assessment of a modified- Sonine approximation recently proposed [Garzó et al., Physica A 376, 94 (2007)] is performed for a granular gas via an examination of system stability. In particular, the critical length scale associated with the onset of two types of instabilities vortices and clusters is determined via stability analyses of the Navier-Stokes-order hydrodynamic (continuum) equations by using the expressions of the transport coefficients obtained from both the standard and the modified-sonine approximations. The impact of both Sonine approximations is examined over a range of solids fraction! <0.2 for small restitution coefficients e= , where the standard and modified theories exhibit discrepancies. The theoretical predictions for the critical length scales are compared to discrete element method (DEM) simulations. Results show excellent quantitative agreement between DEM and the modified-sonine theory, while the standard theory loses accuracy for this highly dissipative parameter space. The modified theory also remedies a (high-dissipation) qualitative mismatch between the standard theory and DEM for the instability that forms more readily. Furthermore, the evolution of cluster size is b Mitrano, P.P., V. Garzó, A. M. Hilger *, C. J. Ewasko *, and C. M. Hrenya, Assessing a dynamic description for instabilities in highly dissipative, freely cooling granular gases, Physical Review E 85, (2012). 53

72 briefly examined via DEM, indicating that domain-size clusters may remain stable or halve in size, depending on system parameters. 3.1 Introduction Instabilities, such as dynamic particle clusters, occur in both granular and gassolid rapid flows [1-4]. In granular flows, such non-uniformities in concentration (bulk density) can be traced to the dissipative nature of collisions [1,2,5,6], whereas in gassolid flows, both mean drag and viscous damping can also lead to clustering instabilities [7-9]. Regardless of the source, clustering instabilities impact the performance of industrial operations, such as gas-solid fluidized beds and pneumatic conveyers [10]. Hydrodynamic (continuum) descriptions derived from kinetic theory of rapid particulate flows predict inhomogeneities that are qualitatively similar to previous experiments and discrete-particle simulations [6,11]. Here, the quantitative ability of a modified kinetic theory applied to a simple granular system is considered. To date, much qualitative work has been done on instabilities in the homogeneous cooling of granular flows. The absence of external forces in the HCS gives rise to a homogeneous system when stable, though homogeneity may be lost due to the formation of velocity vortices (i.e., transversal- or shear-mode instabilities, as illustrated in Fig. 3.1b) or particle clusters (i.e., longitudinal- or heat-mode instabilities, as illustrated in Fig. 3.2b and 3.2c) [11-17]. For a given e,! pairing, a critical dimensionless system scale L Vortex /d demarcates (stable) homogeneous flow (Fig. 3.1a) from one with velocityvortex instabilities (Fig. 3.1b), while a separate critical value L Cluster /d demarcates a 54

73 (stable) homogeneous particle distribution (Fig. 3.2a) from one exhibiting the clustering instability (Fig. 3.2b & 3.2c) [11-19]. Figure 3.1 Spatially averaged velocity fields from DEM simulations with e=0.4,! =0.05, and number of particles N=2096 at (a) 5 collisions per particle (cpp) and (b) 90 cpp, and (c) corresponding Fourier momentum spectra (P) normalized to the value at k=4!. Square cells with length L/5 were averaged for visualization in (a) and (b). Figure 3.2 Snapshots of particle positions (showing only a slice of thickness L/10) extracted from three-dimensional DEM simulations with e=0.6,! =0.2, N=2000 at (a) 2 cpp, (b) 40 cpp, (c) 800 cpp, and (d) corresponding Fourier mass spectra (R). In Chapter 1, the first quantitative assessment [19] of predictions of the critical system size for velocity-vortex instabilities (L Vortex /d) was made. The results show excellent agreement between discrete element method (DEM) simulations and a linear stability analysis [17] of the hydrodynamic equations derived from the Enskog kinetic 55

74 theory [20] over moderate ranges of dissipation (0.6 e 0.9) and solids fraction (0.05! 0.4). It is worth noting that the difference between the previous treatment and the modified version lies in the modified kinetic theory used here, and how the integral equations defining the transport coefficients were evaluated. In particular, in order to obtain an analytical solution for integrals defining the transport coefficients, Ref. [20] approximated the first-order velocity distribution as the product of the Maxwell- Boltzmann distribution and a truncated Sonine polynomial; this procedure will henceforth be referred to as the standard-sonine approximation. In spite of this simple approximation, such predictions compare well with Direct Simulation Monte Carlo (DSMC) results [21,22], except for heat flux transport coefficients in highly dissipative systems (e<0.6) at the dilute limit. Motivated by this disagreement, a slight modification to the standard Sonine approximation has been recently proposed for monocomponent [23] and multicomponent [24] granular gases. The idea behind the modified-sonine approximation is to assume that the isotropic part of the first-order distribution function is mainly governed by the HCS distribution rather than by the Maxwellian distribution. This modified-sonine approach significantly improves the e-dependence of the heat flux transport coefficients and corrects the disagreement observed in dilute systems between theory and DSMC results. It is worth stressing that DSMC, a numerical solution of the Boltzmann or Enskog equation (from which the hydrodynamic equations and transport coefficients are derived analytically), is ideal for testing mathematical assumptions used in the derivation process, whereas DEM, based on Newton s equations of motion, is entirely independent of the starting kinetic equation. DEM thus serves as an ideal test bed 56

75 for the kinetic equation itself along with assumptions used to derive the hydrodynamic equations. In this chapter, L Vortex /d and L Cluster /d are determined via DEM simulations to quantitatively assess the ability of the standard [20] and modified-sonine [23] approximations of the Enskog equation to predict vortex and cluster instabilities. Building on the earlier comparison [19], which was limited to velocity vortices, a previously unexplored range of high dissipation (e= ) and dilute-to-moderate volume fractions (! = ) is considered, where the new modification [23] strongly impacts the theoretical transport coefficients. It is found that the modified theory performs considerably better than the standard one in predicting both the quantitative and qualitative nature of the instabilities in this region of high dissipation and moderate concentrations. 3.2 Computational Methods Simulation Method Monodisperse, frictionless, spherical particles with a constant, normal restitution coefficient e are simulated via event-driven DEM in a three-dimensional HCS. The assumption of instantaneous, binary collisions, which is shared by kinetic theory and DEM, allows for straightforward comparison between theory and simulation. A more detailed description of hard-sphere DEM is available in Refs. [25,26] and Section Here, the range of e considered is ensured to be low enough to explore the effects of the modifications [23] to the standard-sonine approach [20], yet high enough that inelastic collapse (an unphysical artifact of the hard-sphere model [27-29]) does not 57

76 interfere with the reported results. Specifically, system parameters that allow at least 85% of simulation replicates to advance a minimum of 400 collisions per particle, or cpp, without collapse are chosen; the results of collapsed systems are ignored Stability Detection: Fourier Analysis Goldhirsch et al. [2] introduced a technique to characterize the intensity of instabilities in DEM simulations of HCS based on a Fourier analysis. In particular, the Fourier transform of the momentum density ˆp and the mass density (concentration) ˆ! are given by [2] N ( ) ˆp(k) = m # u j exp ik " x j, (3.1) 2! j=1 N ( ) ˆ!(k) = m $ exp ik # x j, (3.2) 2" j=1 where x j and u j are the position and velocity of the j th particle, and m is the particle mass. Periodic boundaries allow for the wavenumbers, k = ( 2a! / L x, 2b! / L y, 2c! / L z ), (3.3) where a, b, and c are integers. Stability in the momentum and concentration fields can be deduced from the k-dependence of the momentum spectrum P and mass spectrum R, respectively, P(k) = R(k) = 2! " 0 2! " 0! " " ˆp 2 r 2 sin# dr d# d$, (3.4) 0! 0 k+dk 0 k+dk " " ˆ# 2 r 2 sin$ dr d$ d%, (3.5) 0 58

77 where P and R are the norm squared of ˆp and ˆ!, respectively, integrated in spherical coordinates from a sphere with radius k to k + dk where k = k. Here, a computation is performed of the corresponding Riemann sums for spherical shells (between each concentric, k-space sphere) with an outer radius that is a factor of 2! greater than the inner radius, which is a 3-dimensional extension of the previous work [2,30]. For a homogeneous system, P and R increase monotonically with k as the volume associated with each spherical shell increases. However, as noted by Goldhirsch et al. [2], unstable HCS displays a non-monotonic variation. This observation will be used as a basis for the stability criterion, as detailed below. Figure 3.1 and 3.2 show snapshots of velocity and concentration fields, respectively, that are stable (Fig. 3.1a and 3.2a) and unstable (Fig. 3.1b, 3.2b, and 3.2c), along with Fourier spectra of momentum (Fig. 3.1c) and mass (Fig. 3.2d). Considering first the velocity-vortex instability (Fig. 3.1), note that a momentum mode (P) excitation appears at k=2! (Fig. 3.1c) for the system exhibiting velocity vortices (Fig. 3.1b) but not for its stable counterpart (Fig. 3.1a). It follows that the presence of this excitation demarcates an unstable, vortex flow (Fig. 3.1b) from a stable, homogenous flow (Fig. 3.1a). This criterion agrees with a previous method [19] based on Haff s law. The Fourier spectra are measured throughout the simulation at intervals of 10 cpp, a value to which the results are insensitive. Averaging is begun of the percent difference (with respect to the k=4! mode) between the excited-wavenumber (k=2! ) mode and the following (k=4! ) mode value once 3 consecutive measurements show k=2! excitations in the 59

78 momentum spectrum. If the averaged percent difference is positive, the system is deemed unstable to vortices. For each pairing of e,! and L/d, 50 simulations are run with varied initial configurations. The error bars shown in Fig. 3.3 and 3.4 and throughout this thesis represent a bracketing of the critical value where the lower bound is the largest L/d without any vortices present in the 50 simulations and the upper bound is the smallest L/d with at least 1/50 simulations exhibiting vortices. The data points simply represent the average of the two bracketing L/d values. Detection of clustering instabilities (Fig. 3.2) follows a similar methodology as the velocity vortices (Fig. 3.1), though some interesting differences exist. While vortex structures were always observed to grow to the system size (i.e., no repeating vortex patterns), the same is not true for clusters. For relatively large L/d, vortices are sufficiently excited to shear the system-size concentration wave (i.e., a cluster and void region) into a wave with twice the frequency (i.e., two cluster and two void regions). This structural change gives to an excitation of the Fourier mass spectrum at k=4! [2], a DEM manifestation of the (non-linear) clustering theory derived by Soto et al. [14]. An example of this behavior is displayed in the particle position snapshot of Fig. 3.2c and the corresponding Fourier mass spectrum of Fig. 3.2d (i.e., 800 cpp). Note that this concentration wave with period=l/2 (Fig. 3.2d at 800 cpp) is preceded in time by the larger wave of period L (Fig. 3.2c at 40 cpp). However, sustained clusters are observed of period L (as predicted by the linear theory [17] and analogous to Fig. 3.2b, but not evolving to higher-frequency clusters as in Fig. 3.2c) in simulations with domains slightly larger than L Cluster /d. To the best of our knowledge, such clusters have not been displayed 60

79 previously through DEM simulations; previous work [2,31] showed only higherfrequency, or smaller, clusters (see Fig. 2 of Ref. [2]) because large systems (i.e., L/d>>L Cluster /d) were studied. In this chapter, care is taken to detect either type of sustained cluster (i.e., k=2! and k=4! excitations); with this exception, the criteria for L Cluster /d follow from the description of L Vortex /d Hydrodynamic Description As mentioned previously, Garzó [17] used the Enskog kinetic equation to perform a linear stability analysis of the Navier-Stokes hydrodynamic equations. This stability analysis gave rise to critical system lengths associated with vortices and clusters, L Vortex = 5! 4! "# 0 $ * % 0 *, (3.6) L Cluster = 5! 4 5! 2 "# 0 $ * C p % µ * ( 2g % C p ), (3.7) & 0 * where! is the pair correlation function at contact,! 0 is the mean free path,! * is the dimensionless shear viscosity,! 0 * is the dimensionless zeroth-order cooling rate,! * is the dimensionless thermal conductivity, C p and g are functions of! equal to unity in the dilute limit, and µ * is the dimensionless heat-flux transport coefficient associated with gradients in concentration. Each of these quantities is defined in terms of e and! ; see Ref. [17] for detailed forms. Equations (3.6) and (3.7) show that the critical lengths depend on the transport coefficients. Explicit forms for these coefficients have been obtained by means of the standard [20] and modified-sonine [23] approximations. Both approaches differ 61

80 significantly for small values of the restitution coefficient in the case of the heat flux transport coefficients. In this chapter, the standard [20] and modified [23] forms of the transport coefficients in Eq. (3.6) and (3.7) are used for the critical lengths and comparison of the predictions of L Vortex /d and L Cluster /d with DEM data is performed. 3.3 Results Unlike the previous evaluation of L Vortex /d using the standard theory at moderate dissipation and dilute-to-moderate concentrations [19], highly-dissipative systems are examined here. Focusing on this parameter space allows us to compare the standard and modified-sonine approaches in the region where the discrepancy is expected to be most significant. Moreover, these results represent the first time that L Cluster /d has been obtained from DEM simulations. This knowledge is important since predictions from the standard theory indicate that systems of moderate dissipation and volume fraction are more prone to the vortex instability, whereas highly dissipative, dilute systems are more prone to the clustering instability [17]. As detailed below, the DEM and modified theory results indicate otherwise. In Fig. 3.3, the critical system length scale for particle clustering (L Cluster /d) is plotted for e=0.25 (Fig. 3.3a) and e=0.4 (Fig. 3.3b) over a range of! = Fig. 3.3a shows that the modified-sonine approximation [23] improves the theoretical prediction for L Cluster /d relative to the standard prediction [17] in moderately dilute systems. This agreement provides evidence that the modified-sonine approximation successfully accounts for non-gaussian corrections to the particle velocity distribution, and that such corrections are important at high dissipation levels. 62

81 Figure 3.3 Critical clustering length scale (L Cluster /d) as a function of solids fraction for (a) e=0.25 and (b) e=0.4. The solid and dashed lines correspond to the modified and standard theories, respectively. The data points correspond to DEM. It is intriguing to note that Fig. 3.3b appears to suggest a negligible difference between the standard and modified theories for e=0.4 although theoretical transport coefficients (e.g.,! * or µ * ) have been shown to differ for such dissipation [23]. The apparent contradiction can be explained by a closer examination of how the transport coefficients, namely! * and µ *, impact L Cluster /d. In particular, Garzó et al. [23] shows that both transport coefficients are overestimated by the standard theory [20], such that the deviation caused by the modified theory appears to cancel out in the computation of L Cluster /d via Eq. (3.7). Figure 3.4 shows the L Vortex /d and L Cluster /d associated with the standard and modified theories, and DEM for e=0.25. Note that the standard theory predicts L Cluster /d<l Vortex /d, while DEM shows that vortices manifest at smaller L/d than clusters. The modified-sonine approach remedies this qualitative disparity in highly dissipative 63

82 systems between DEM and the standard theory; both DEM and the modified theory suggest L Vortex /d<l Cluster /d. Finally, the results presented here are robust. In particular, the quantitative agreement between DEM and the modified theory shown in Fig. 3.3 and 3.4 is also representative of results obtained at e=0.3 and! = , (not shown for the sake of brevity). Figure 3.4 Critical length scale for vortex and cluster instabilities (i.e., L Vortex /d and L Cluster /d, respectively) plotted as a function of solids fraction for e=0.25. The solid and dashed lines correspond to the modified and standard theories, respectively. 3.4 Concluding Remarks In summary, the current chapter reveals that a recent modification [23] to the standard-sonine method [20] for obtaining the Navier-Stokes transport coefficients improves the quantitative predictive ability for highly dissipative (e<0.4) systems of dilute-to-moderate concentrations (0.05! 0.15). Comparison with DEM shows that the modified theory improves the quantitative prediction of L Cluster /d relative to the standard 64

83 theory [20] and also correctly identifies the leading instability in a region where the standard theory does not. The excellent quantitative agreement between theory and DEM for the onset of vortices and clusters exemplifies the wide range of applicability of granular hydrodynamics derived from a modified kinetic theory and supports the claim that hydrodynamics (which is based on the separation between microscopic and macroscopic length and time scales) is not only limited to nearly elastic particles. Furthermore, the agreement with the Navier-Stokes theory (linear relations between fluxes and spatial gradients) is not entirely unexpected since the systems studied here are near the critical scale for instability such that large gradients have not yet developed. The present chapter, which is restricted to rigid-particle systems, also adds to a growing body of evidence [19,32,33] that the Enskog equation (which neglects velocity correlations between the particles that are about to collide) accurately describes moderately dilute and moderately dense systems. Finally, the results reported here are of practical interest since the high dissipation levels (e= ) examined in this chapter are also exhibited by wetted particles [34], which are critical in processes such as coagulation, spray coating, filtration, and pneumatic conveying. The focus of the previous and current chapter was on the onset of instability in flows where the dissipation was caused by inelastic yet frictionless particles. The next chapter continues to examine the onset of instability, but for particles that are also frictional. 65

84 3.5 References 1. Hopkins, M. A. & Louge, M. Y Inelastic microstructure in rapid granular flows of smooth disks. Phys. Fluids A 3, Goldhirsch, I., Tan, M. L. & Zanetti, G A molecular dynamical study of granular fluids I: The unforced granular gas in two dimensions. J. Sci. Comput. 8, Gidaspow, D. Multiphase flow and fluidization : continuum and kinetic theory descriptions. (Academic Press, 1994). 4. Jackson, R. The dynamics of fluidized particles. (Cambridge Univ. Press, 2000). 5. Kudrolli, A., Wolpert, M. & Gollub, J. P Cluster formation due to collisions in granular material. Phys. Rev. Lett. 78, Conway, S. L. & Glasser, B. J Density waves and coherent structures in granular Couette flows. Phys. Fluids 16, Glasser, B. J., Sundaresan, S. & Kevrekidis, I. G From bubbles to clusters in fluidized beds. Phys. Rev. Lett. 81, Wylie, J. J. & Koch, D. L Particle clustering due to hydrodynamic interactions. Phys. Fluids 12, Wylie, J. J., Koch, D. L. & Ladd, J. C Rheology of suspensions with high particle inertia and moderate fluid inertia. J. Fluid Mech. 480, Agrawal, K., Loezos, P. N., Syamlal, M. & Sundaresan, S The role of mesoscale structures in rapid gas-solid flows. J. Fluid Mech. 445, Brilliantov, N., Saluena, C., Schwager, T. & Pöschel, T Transient structures in a granular gas. Phys. Rev. Lett. 93, Goldhirsch, I. & Zanetti, G Clustering instability in dissipative gases. Phys. Rev. Lett. 70, Brey, J. J., Dufty, J. W., Kim, C. S. & Santos, A Hydrodynamics for granular flow at low density. Phys. Rev. E 58, Soto, R., Mareschal, M. & Mansour, M. M Nonlinear analysis of the shearing instability in granular gases. Phys. Rev. E 62, Wakou, J., Brito, R. & Ernst, M. H Towards a Landau-Ginzburg-Type Theory for Granular Fluids. J. Stat. Phys. 107, Brilliantov, N. & Pöschel, T. Kinetic theory of granular gases. (Oxford University Press, 2004). 17. Garzó, V Instabilities in a free granular fluid described by the Enskog equation. Phys. Rev. E 72, Brey, J. J., Ruiz-Montero, M. J. & Cubero, D Origin of density clustering in a freely evolving granular gas. Phys. Rev. E 60, Mitrano, P. P., Dahl, S. R., Cromer, D. J., Pacella, M. S. & Hrenya, C. M Instabilities in the homogeneous cooling of a granular gas: A quantitative assessment of kinetic-theory predictions. Phys. Fluids 23, Garzó, V. & Dufty, J. W Dense fluid transport for inelastic hard spheres. Phys. Rev. E 59, Brey, J. J. & Ruiz-Montero, M. J Simulation study of the Green-Kubo relations for dilute granular gases. Phys. Rev. E 70,

85 22. Brey, J. J., Ruiz-Montero, M. J., Maynar, P. & Garcia de Soria, M. I Hydrodynamic modes, Green-Kubo relations, and velocity correlations in dilute granular gases. J. Phys. Cond. Matter 17, S Garzó, V., Santos, A. & Montanero, J. M Modified Sonine approximation for the Navier-Stokes transport coefficients of a granular gas. Physica A 376, Garzó, V., Reyes, F. V. & Montanero, J. M Modified Sonine approximation for granular binary mixtures. J. Fluid Mech. 623, Allen, M. P. & Tildesley, D. J. Computer simulation of liquids. (Clarendon Press, 1989). 26. Pöschel, T. & Schwager, T. Computational granular dynamics: models and algorithms. (Springer, 2005). 27. McNamara, S. & Young, W. R Inelastic collapse and clumping in a onedimensional granular medium. Phys. Fluids 4, McNamara, S. & Young, W. R Inelastic collapse in two dimensions. Phys. Rev. E 50, McNamara, S. & Young, W. R Dynamics of a freely evolving, twodimensional granular medium. Phys. Rev. E 53, Tan, M. L. in doctoral dissertation (Princeton University, 1995). 31. Luding, S. & Herrmann, H. J Cluster-growth in freely cooling granular media. Chaos 9, Dahl, S. R., Hrenya, C. M., Garzó, V. & Dufty, J. W Kinetic temperatures for a granular mixture. Phys. Rev. E 66, Murray, J. A., Garzó, V. & Hrenya, C. M Enskog Theory for Polydisperse Granular Mixtures. III. Comparison of dense and dilute transport coefficients and equations of state for a binary mixture. Powder Technol. 220, Donahue, C. M., Hrenya, C. M., Davis, R. H., Nakagawa, K. J., Zelinskaya, A. P. & Joseph, G. G Stokes' cradle: normal three-body collisions between wetted particles. J. Fluid Mech. 650,

86 4. Granular Flows: Effect of Frictional Particle Collisions c Abstract Flow instabilities driven by the dissipative nature of particle-particle interactions have been well documented in granular flows. The bulk of previous studies on such instabilities, as well as the previous two chapters, has considered the impact of inelastic dissipation only and has shown that instabilities are enhanced with increased dissipation. The impact of frictional dissipation on the stability of grains in a homogeneous cooling system is studied in this chapter using discrete element method (DEM) simulations and kinetic-theory-based predictions. Surprisingly, both DEM simulations and theory indicate that high levels of friction actually attenuate instabilities relative to the frictionless case, whereas moderate levels enhance instabilities compared to frictionless systems, as expected. Additionally, at moderate friction levels, increasing friction can attenuate instabilities. The mechanism responsible for both behaviors is identified as the coupling between rotational and translational motion. These results have implications not only for granular materials, but more generally to flows with dissipative interactions between constituent particles cohesive systems with agglomeration, multiphase flows with viscous dissipation, etc. c Mitrano, P. P., S. R. Dahl, A. M. Hilger *, C. J. Ewasko *, and C. M. Hrenya, Dual role of friction in granular flows: attenuation versus enhancement of instabilities, Journal of Fluid Mechanics 729, 484 (2013). 68

87 4.1 Introduction Instabilities in molecular fluids (e.g., turbulence) have been studied extensively, and their impact on numerous applications is without question. Granular materials also display flow instabilities [1-11], some of which have similarities to those found in their molecular counterparts [12-14] and others that do not. The latter arises due to the dissipative nature of collisions between grains [1,15]. Previous work has shown that dissipation-driven instabilities can take the form of velocity vortices and particle clusters [4-6], as illustrated in Fig.1 for a homogeneous cooling system (HCS). Moreover, previous experiments [16,17], simulations [2-5,11,15], and theories [1,7-11] have shown a monotonic dependence of instabilities on dissipation. Put simply, instabilities are well known to intensify with increasing dissipation. Figure 4.1 Visualizations from a discrete element method simulation of a HCS with normal restitution coefficient e=0.7, solids fraction φ =0.3, tangential restitution coefficient β =-0.7, and L/d=18, where L is the domain length and d is the particle diameter, after 100 collisions per particle: (a) coarse-grained velocity field with cell size L/5 (vortex instability) and (b) particle positions within a L/10 domain slice (clustering instability). Dissipation in non-cohesive systems occurs via inelastic or frictional particleparticle contacts. Previous simulations and theoretical analysis [1-11,15] of instabilities 69

88 focused on inelastic dissipation, characterized by a normal restitution coefficient, which governs the ratio of the normal, relative velocity on rebound to that on approach. However, these works ignored the influence of friction, which impacts the tangential component of relative velocity. Surprisingly, recent experiments indicate that clusters, forming in streams of particles falling from a nozzle, were less pronounced for rougher copper particles compared to smoother glass particles [18]. At first glance, this attenuation of clustering for increased roughness appears contradictory to the many previous works in which increased inelasticity (dissipation) leads to enhanced clustering. This chapter confirms and expands on the counterintuitive effect of friction on instabilities and identifies the underlying mechanism. First, the discrete element method (DEM) simulations of the HCS indicate that highly frictional particles actually serve to attenuate instabilities (i.e., increase the critical dimensionless length scales LVortex / d and LCluster / d) relative to the frictionless case, while particles with moderate friction enhance instabilities compared to frictionless ones as expected. Second, it is found that increasing friction from moderate to higher levels also attenuates instabilities, consistent with recent experiments [18]. Both behaviors are explained here in terms of the coupling of rotational and translational dynamics. Additionally, a simple granular hydrodynamic (continuum) predictions is obtained by inserting a cooling rate [19] (also available for a polydisperse systems [20]) that includes friction and inelasticity into a linear stability analysis [7] obtained for frictionless particles. These theoretical predictions, which use the same collision rules as DEM, support such findings. 70

89 4.2 Computational Methods The event-driven, hard-sphere DEM simulations of the HCS employed here are composed of N monodisperse spheres in a three-dimensional, cubic volume L 3 with solids 3 3 fraction φ = Nπd / (6 L ). Each collision is resolved via a constant normal 0 e 1 and tangential 1 β 1restitution coefficient: nu ' 12 e =, (4.1) nu 12 where u12 u1 u2 ( Rω1 Rω2 ) n u' 12 β =, (4.2) n u 12 = + is the relative velocity at the point of contact, u1 and u2 are the pre-collisional translational velocities of particle 1 and 2, R is the particle radius, ω 1 and ω 2 are the pre-collisional angular velocities of particle 1 and 2, a prime indicates post-collisional velocities, and n is the unit vector pointing from the center of particle 2 to that of particle 1. β governs the relative tangential velocity at the point of contact (i.e., relative surface velocity), where β = 1 represents frictionless particles (zero tangential impulse, J t ), β = 0 gives to maximum energy dissipation (zero relative tangential velocity), and β = 1 represents elastically rough particles (maximum J t and fully reversed, relative tangential velocity). Accordingly, the post-collisional velocities [21,22] are given by: ( 1+ e) ( 1+β ) u1' = u1 un ut, (4.3)

90 where n ( 1 2) u = u u n n ( 1 2 ( ω1 ω2) ) ( 1+ e) ( 1+β ) u2' = u2 + un + ut, (4.4) 2 7 is the normal component of relative velocity, ut = u u R + R n t t is the tangential component of relative velocity, and t is the tangential unit vector. Collisions resolved in this manner are known as sticking (even if the relative tangential velocity is nonzero). After confirming that the DEM predictions for the ratio of particle rotational to translational energy coincide with previous theory [21], this expression for initialization was used. To determine if the DEM simulations of a given domain size become unstable, a Fourier analysis (see Section 3.2.2) of the momentum and concentration fields is used to detect vortices and clusters, respectively. For each integer value of L/d studied, 10 simulation replicates with varied initial conditions are run to determine a one-unit range for the critical length scale (i.e., LVortex / d or LCluster / d). At the bottom of this range, the instability of interest has not been detected in any of the 10 replicates, while the instability has been detected at least once at the upper bound. Simulations end when the total number of collisions reaches 800N. 4.3 Results From this Fourier analysis, LVortex / d and LCluster / d were determined for 0.7 e 0.9, 1 β 1, and 0.1 φ 0.3. This DEM data is plotted in Fig. 4.2 as a function of β for e=0.9 and φ =0.3. Surprisingly, the inclusion of friction does not always enhance instabilities relative to the frictionless case. First, nearly frictionless (!! "1) and highly 72

91 rough (! ~1) systems attenuate instabilities compared to their frictionless counterparts (! =-1); in other words, larger domains are needed at these extremes for the instabilities to occur. Second, increasing the level of friction does not always enhance instabilities. Specifically, for β >0, instabilities are attenuated as roughness (β) increases. This latter result is perhaps not as surprising as the former since as β increases from zero, frictional dissipation decreases (i.e., more energy is put toward reversing tangential relative velocity as opposed to being dissipated). Regarding the first observation, the attenuation of instabilities relative to the frictionless case is physically meaningful in highly rough ( β 1) systems, whereas its nearly frictionless counterpart (!! "1) is an artifact of the friction model, which only allows for sticking collisions. This β 1 attenuation would be observed in practice only for β 1 combined with a quite high Coulomb-friction coefficient, which is unrealistic for practical materials. As further described below, the resulting accumulation of rotation that occurs in the HCS for β 1 is also responsible for the discontinuity in the critical length scale observed at! =!1. Figure 4.2 DEM data for critical dimensionless length scale for vortex, LVortex / d (blue) and cluster instabilities, LCluster / d (red) as a function of tangential restitution coefficient 73

92 for e=0.9 and φ =0.3. Lines represent critical dimensionless length scales in the frictionless limit ( β =-1). The intriguing attenuation of instabilities noted at the high-friction limit ( β 1) in Fig. 4.2 can be understood by considering the particle motion associated with velocity vortices and the collision outcomes that help to induce (or hinder) such instabilities. In particular, vortices (Fig. 4.1a) are characterized by the alignment of tangential particle translation. Such motion is not directly dependent on particle rotation. Instead, collisions that increase the alignment of tangential translation will help induce vortices. Such alignment can be quantified via another restitution coefficient, e t ( ' 1 ' 2) ( 1 2) n u u = n u u, (4.5) which relates this pre-collisional, tangential component of the relative translational velocity to the post-collisional value. In contrast to β (Eq. 4.2), e t does not directly depend on particle rotation since only translational motion is considered. Furthermore, e t is a measurement characterizing collisional outcomes rather than dictating outcomes, as do the material-property inputs e and β. Collisions that align particle motion (and induce vortices) give rise to e t < 1, while collisions that hinder vortex formation lead to e t > 1. Frictionless interactions lead to e t = 1. Hence, averaging e t over a simulation provides insight into the critical length scales of Fig A log 10 average is more appropriate than traditional averaging, which would inappropriately weight high values of e t. In Fig. 4.3, log 10 e 10 t et is plotted as a function of β for systems with L/d << LVortex / d. As expected, 74

93 the shape of this curve mimics that of L / d (Fig. 4.2) with values of e t >1 occurring at the β 1 and β 1 limits. Vortex Figure 4.3 Translational tangential restitution coefficient extracted from DEM simulations as a function of β for e=0.9 and φ =0.3 in a domain with L/d=4 (red) and L/d=10 (blue). e t An explanation of the β -dependence of e t (Fig. 4.3) stems from a consideration of the ratio of the rotational to translational kinetic energy (RE/KE), plotted in Fig. 4.4 as a function of β for e=0.9 and φ =0.3. A theoretical prediction [19] of this ratio in HCS, based on identical collision rules, is also shown. Excellent agreement was found between DEM simulations and theory [19], supporting a previous assessment [21]. This energy ratio plays an important role in the development of instabilities via the frictional coupling of rotation and translation. First consider the behavior when RE>KE, which occurs for highly rough ( β >0.75 for e=0.9) particles (Fig. 4.4). (Note that β >0.75 is also the region where L / d & L / d (Fig. 4.2) and e t (Fig. 4.3) become greater than their Vortex Cluster corresponding frictionless values.) For such systems of fast-spinning, highly frictional 75

94 particles, collisions actually serve to increase the tangential component of relative translational velocity (Fig. 4.3), similar to a tennis ball with topspin converting rotation into tangential translation. Specifically, the relative tangential translation between the ball and ground is increased via collision compared to a non-rotating or frictionless ball. As β increases toward unity, particles transfer greater portions of tangential momentum ( J (1 + β) u ; see Eq ) to collision partners, resulting in increased relative t t tangential translation. This behavior causes decreased alignment of particle motion ( e t >1) and explains the attenuation of instabilities (Fig. 4.2) as particles approach perfect roughness, β =1. Figure 4.4 The ratio of rotational kinetic energy (RE) to translational kinetic energy (KE) as a function of β for e=0.9 and φ =0.3. Data points are averages from DEM simulations run for 1600 collision per particle with 124 particles (blue) and 990 particles (red). The line is the theoretical prediction [19]. Now consider the tangential momentum exchange J (1 + β) u as β à -1. The extremely high rates of rotation in this region (Fig. 4.4), despite small (1 + β ) values, give rise to tangential impulses that, again, increase relative tangential translation (at the t t 76

95 cost of rotation), as shown in Fig With such high rotation, particles with aligned tangential translation will tend to be less aligned after colliding. In this way, high rotational velocities, which give rise to high tangential impulses, hinder vortex formation. This result corroborates a previous finding in which rotational driving hindered instabilities [23]. This consideration of e t in conjunction with RE/KE provides an explanation of the attenuation of vortices (increased LVortex / d ) observed in the low- and high-friction limits (Fig. 4.2), and a similar argument applies to cluster attenuation. Namely, the increased tangential component of relative translation ( e t >1) for β 1 and β 1 leads to increased tangential separation after particles collide. This increased separation hinders cluster formation relative to the frictionless case (where e t =1). Another mechanism for cluster attenuation, since vortices precede clusters, is the attenuation of vortices themselves since viscous heating, which leads to cluster formation [1,2,10,24], is smaller for a more homogeneous velocity field. Recall the previously described attenuation of instabilities (Fig. 4.2) at the high friction limit ( β 1) is physically meaningful, while the analogous behavior at low friction levels ( β 1) is an artifact of the friction model used. It is hypothesized that this artifact stems from the sticking collisions (Eq. 4.2) allowed by the model and would not be present in more robust friction models that also allow for (sliding) Coulombfriction contacts. Specifically, the extremely high RE/KE at β 1 (Fig. 4.4) would have a severely diminished effect in a more robust model since such high levels of rotation would result in sliding collisions. Sliding collisions would lead to smaller tangential 77

96 impulses (than sticking collisions) near β=-1 since such impulses are proportional to the relative normal velocity at contact rather than its tangential counterpart. To test this hypothesis, additional simulations were performed with a more robust friction model [22], which allows for both sticking and sliding. The remainder of this paragraph will focus on the results from this more robust model, in which two J t values are calculated for each collision, the smaller of which is used to resolve the collision: (i) J t based on sticking interactions (with constant β =0-1), and (ii) J t based on sliding interactions (with constant Coulomb friction coefficient µ =0- ). It is found that many similarities exist between this more robust model and the simpler β-only model. Most importantly, instabilities are attenuated with highly frictional particles ( β and µ on the order of 1), while moderate friction levels (moderate values of µ for all values of β ) enhance instabilities compared to the frictionless case. However, the more robust friction model indicates that small levels of friction (small values of µ for all values of β ) give to critical length scales that coincide with those of the frictionless case, which is contrary to the β-only model and consistent with this interpretation that the behavior of the β-only model near the frictionless limit is a model artifact. One final aspect worth discussing is that the increase in relative tangential translation imposed by the extremely high temperature ratio at β à -1 does not affect vortex and cluster formation equally. As outlined above, vortices depend on the relative tangential translation between colliding particles, while cluster formation depends on particle separation. Thus, collisions that selectively increase relative tangential translation (rather than normal translation) hinder vortices to a greater extent than clusters. This 78

97 selective attenuation of vortices is exemplified in Fig. 4.2 where, for approximately β -0.95, clusters manifest more readily (at smaller L/d) than vortices. Overall, these DEM findings of the attenuation of instabilities in highly rough systems compared to their smooth counterparts are surprising given the conventional wisdom that instabilities are enhanced with increased dissipation. As a first attempt to corroborate these findings theoretically, a simple theoretical construct is used to estimate the critical length scales for instability. Specifically, Garzó performed a linear stability analysis [7] of the granular hydrodynamic theory [25] for inelastic, frictionless particles. In this chapter, a cooling rate ( ζ ) is inserted that incorporates both inelasticity and friction [19] into the stability-analysis expressions, * 0 L Cluster = 5! 4 5! 2 "# 0 $ * C % & µ * ( ), (4.6) ' 0 * 2g & C % L Vortex * 5π η = πχλ, (4.7) 4 0 ζ * 0 All other quantities are for frictionless particles [7]: * * η is the shear viscosity, κ is the * thermal conductivity, µ is the Dufour coefficient, which relates concentration gradients to the heat flux, χ is the pair correlation function at contact, λ 0 is the mean free path, g = 1+! " "! ln #, and C = 1+ g " g. Asterisks indicate non-dimensionality.! 1+ 2#(1+ e)$ The results from this simplified theory are provided in Fig. 4.5, along with DEM Vortex Cluster predictions of L / d and L / d, all normalized to their respective frictionlessparticle DEM predictions. Excellent agreement occurs for the case of velocity vortices. 79

98 This (somewhat surprising) result suggests that the influence of friction on the cooling rate is critical to accurately describing granular flows via hydrodynamics. Such agreement is encouraging since it implies that complexities associated with incorporating friction into other transport coefficients could potentially be avoided without a significant loss of accuracy. The discrepancy between the linear stability analysis of theory and DEM simulations for the case of clusters is also telling. In particular, previous studies [1,2,10,24] suggest nonlinear mechanisms (e.g., viscous heating) are important in cluster formation. The results contained in Fig. 4.5 are consistent with this notion, since the linear theory over predicts LCluster / d compared to DEM (due to the importance of nonlinear mechanisms). Also note the striking similarity between the dependence of instabilities (Fig. 4.5) and e t (Fig. 4.3) on β. In Figure 4.6, the non-dimensional cooling rate normalized to the smooth-particle value is plotted against the tangential restitution coefficient. The figure shows that adding a new mechanism for dissipation can actually lead to a reduction in the cooling rate. Figure 4.5 LVortex / d (blue) and LCluster / d (red), normalized to the respective frictionlessparticle DEM prediction (see Fig. 4.2), as a function of the tangential restitution coefficient, as predicted by DEM (data points) and theory (lines) for e=0.9 and φ =0.3. Values greater than 1 refer to attenuation of instabilities relative to the frictionless case. 80

99 Figure 4.6 Theoretical non-dimensional cooling rate normalized to the smooth-particle value as a function of tangential restitution coefficient for φ = Concluding Remarks The work presented in this chapter shows, via DEM simulations and theory, that high levels of friction can actually attenuate instabilities relative to the frictionless case. This result is surprising since previous work shows increased (inelastic) dissipation leads to enhanced instabilities, whereas here it is seen that adding another source of dissipation (friction) on top of inelastic dissipation can result in the opposite behavior. It is also found that when sticking collisions dominate, increasing roughness (β) from moderate levels ( β > 0) attenuates instabilities. The physical origin of both findings is traced to the coupling of rotational and translational motion. The results presented here are representative of findings for lower restitution (e=0.7) and solids fraction (φ=0.1). The typical crossovers between enhancing and attenuating instabilities (for this parameter 81

100 space) occur when β / e 0.8 1(i.e., when normal and tangential dissipation are of the same order) and are insensitive to φ. Follow-on work is needed to confirm the robustness of these findings beyond the HCS, though some previous work appears encouraging in this regard. Specifically, Ref. [26] showed through a linear stability analysis of an ad-hoc frictional, hydrodynamic model that specific levels of friction can actually stabilize an inelastic granular shear flow, though they also reported that particles that only dissipate energy through friction are stable regardless of the level of friction, which is in conflict with the current findings for e=1 (not shown here for the sake of brevity, but they have a qualitatively similar form to Fig. 4.2 for -1< β <1). The observation that friction may attenuate or enhance instabilities is expected to impact not only granular flows that display such instabilities (e.g., planetary rings, asteroid belts, and ejection of lunar soil upon spacecraft landing [27]), but also their gas-solid counterparts. Examples of the latter include the welldocumented clustering phenomenon in circulating fluidized bed reactors [28-30] and roping observed in pneumatic transport lines [31-33]. It is also worth noting that the highroughness levels investigated here occur in a range of materials, including biomass feedstock for energy production [34] and materials made with customized roughness [35]. Finally, the results may have implications to other forms of energy dissipation experienced by the particles, including cohesion and fluid drag. Chapters 2-4 have been limited to monodisperse flows, while the following chapter studies the ability of hydrodynamic models to incorporate polydispersity. 82

101 4.5 References 1. Goldhirsch, I. & Zanetti, G Clustering instability in dissipative gases. Phys. Rev. Lett. 70, Goldhirsch, I., Tan, M. L. & Zanetti, G A molecular dynamical study of granular fluids I: The unforced granular gas in two dimensions. J. Sci. Comput. 8, Petzschmann, O., Schwarz, U., Spahn, F., Grebogi, C. & Kurths, J Length scales of clustering in granular gases. Phys. Rev. Lett. 82, Luding, S. & Herrmann, H. J Cluster-growth in freely cooling granular media. Chaos 9, Mitrano, P. P., Dahl, S. R., Cromer, D. J., Pacella, M. S. & Hrenya, C. M Instabilities in the homogeneous cooling of a granular gas: A quantitative assessment of kinetic-theory predictions. Phys. Fluids 23, Mitrano, P. P., Garzó, V., Hilger, A. M., Ewasko, C. J. & Hrenya, C. M Assessing a hydrodynamic description for instabilities in highly dissipative, freely cooling granular gases. Phys. Rev. E 85, Garzó, V Instabilities in a free granular fluid described by the Enskog equation. Phys. Rev. E 72, Brito, R. & Ernst, M Extension of Haff's cooling law in granular flows. Europhys. Lett. 43, Brey, J. J., Ruiz-Montero, M. J. & Moreno, F Instability and spatial correlations in a dilute granular gas. Phys. Fluids 10, Soto, R., Mareschal, M. & Mansour, M. M Nonlinear analysis of the shearing instability in granular gases. Phys. Rev. E 62, Brilliantov, N., Saluena, C., Schwager, T. & Pöschel, T Transient structures in a granular gas. Phys. Rev. Lett. 93, Shinbrot, T., Alexander, A. & Muzzio, F. J Spontaneous chaotic granular mixing. Nature 397, Goldfarb, D. J., Glasser, B. J. & Shinbrot, T Shear instabilities in granular flows. Nature 415, Ciamarra, M. P., Coniglio, A. & Nicodemi, M Shear Instabilities in Granular Mixtures. Phys. Rev. Lett. 94, Hopkins, M. A. & Louge, M. Y Inelastic microstructure in rapid granular flows of smooth disks. Phys. Fluids A 3, Kudrolli, A., Wolpert, M. & Gollub, J. P Cluster formation due to collisions in granular material. Phys. Rev. Lett. 78, Conway, S. L., Shinbrot, T. & Glasser, B. J A Taylor vortex analogy in granular flows. Nature 431, Royer, J. R., Evans, D. J., Oyarte, L., Guo, Q., Kapit, E., M bius, M. E., Waitukaitis, S. R. & Jaeger, H. M High-speed tracking of rupture and clustering in freely falling granular streams. Nature 459, Goldshtein, A. & Shapiro, M Mechanics of collisional motion of granular materials. Part 1. General hydrodynamic equations. J. Fluid Mech. 282,

102 20. Santos, A., Kremer, G. M. & Garzó, V Energy production rates in fluid mixtures of inelastic rough hard spheres. Prog. Theor. Phys. No McNamara, S. & Luding, S Energy nonequipartition in systems of inelastic, rough spheres. Phys. Rev. E 58, Hoomans, B. P. B., Kuipers, J. A. M., Mohd Salleh, M. A., Stein, M. & Seville, J. P. K Experimental validation of granular dynamics simulations of gasfluidised beds with homogenous in-flow conditions using Positron Emission Particle Tracking. Powder Technol. 116, Cafiero, R., Luding, S. & Herrmann, H. J Rotationally driven gas of inelastic rough spheres. Europhys. Lett. 60, Brey, J. J., Ruiz-Montero, M. J. & Cubero, D Origin of density clustering in a freely evolving granular gas. Phys. Rev. E 60, Garzó, V. & Dufty, J. W Dense fluid transport for inelastic hard spheres. Phys. Rev. E 59, Alam, M. & Nott, P. R The influence of friction on the stability of unbounded granular shear flow. J. Fluid Mech. 343, Immer, C., Metzger, P., Hintze, P. E., Nick, A. & Horan, R Apollo 12 Lunar Module exhaust plume impingement on Lunar Surveyor III. Icarus 211, Gidaspow, D. Multiphase flow and fluidization : continuum and kinetic theory descriptions. (Academic Press, 1994). 29. Jackson, R. The dynamics of fluidized particles. (Cambridge Univ. Press, 2000). 30. Fan, L.-S. & Zhu, C. Principles of gas-solid flows. (Cambridge University Press, 2005). 31. Yilmaz, A. Roping Phenomena in Lean Phase Pneumatic Conveying. (Lehigh University, 1997). 32. Schallert, R. & Levy, E Effect of a combination of two elbows on particle roping in pneumatic conveying. Powder Technol. 107, Yilmaz, A. & Levy, E. K Formation and dispersion of ropes in pneumatic conveying. Powder Technol. 114, Phani, A., Lope, T. & Schoenau, G Physical and frictional properties of nontreated and steam exploded barley, canola, oat and wheat straw grinds. Powder Technol. 201, Majidi, C., Groff, R. E., Maeno, Y., Schubert, B., Baek, S., Bush, B., Maboudian, R., Gravish, N., Wilkinson, M., Autumn, K. & Fearing, R. S High Friction from a Stiff Polymer Using Microfiber Arrays. Phys. Rev. Lett. 97,

103 5. Granular Flows: Effect of Polydispersity d Abstract In this chapter, the complexity associated with polydispersity is included in the study of the onset of velocity vortices in the HCS. A linear stability analysis [1] of the Navier- Stokes-order (NS) granular hydrodynamic (continuum) equations is used to determine the critical length scale for the onset of vortices and clusters instabilities in granular dense binary mixtures. The theoretical predictions for the critical length scales are compared to discrete element method (DEM) simulations in flows of moderate dissipation (e ij! 0.7) and solid volume fractions (!! 0.2). Excellent agreement is found between DEM and kinetic theory for the onset of velocity vortices, indicating the applicability of NS hydrodynamics to polydisperse flows even for strong inelasticity, finite density, and particle dissimilarity. d Mitrano P. P., V. Garzó, C. M. Hrenya. Instabilities in moderately dense granular binary mixtures. Physical Review E 89(2): , Rapid Communication (2014). 85

104 5.1 Introduction Although hydrodynamic (continuum) models are frequently used to describe rapid granular flows, there are still some open questions about the domain of validity of this description [2]. As for ordinary fluids at moderate densities, the constitutive equations for the fluxes and the forms of the transport coefficients can be derived from the revised Enskog kinetic theory (RET) [3] conveniently adapted to account for the dissipative dynamics. The derivation of such fluxes from the corresponding kinetic equation assumes the existence of a hydrodynamic regime where all space and time dependence of the distribution function only occurs through the hydrodynamic fields (normal solution). A first-order Chapman-Enskog expansion [4] provides the Navier-Stokes-order (NS) hydrodynamic equations. However, there are still some concerns regarding the transition from kinetic theory to hydrodynamics beyond the quasielastic limit [2]. The reason for this concern resides in the fact that the inverse of the cooling rate, which measures the rate of energy loss due to collisional dissipation, introduces a new time scale not present for elastic collisions. The variation of the granular temperature over this new time scale is faster than over the usual hydrodynamic time scale. However, as the inelasticity increases, it is possible that the system could lack a separation of time scales between the hydrodynamic and the pure kinetic excitations such that there is no aging to hydrodynamics or, in the language of kinetic theory, there is no normal solution at finite dissipation. An approach to assess the hydrodynamic theory is to compare it to DEM simulations. The latter does not utilize the RET and thus provides a strong test for the validity of the theory. The determination of the critical length scale for the onset of 86

105 instabilities (which is generally driven by the transversal shear mode) in freely cooling flows offers one of the best opportunities to assess NS hydrodynamics. This kind of instability, which can be traced to the dissipative nature of collisions, is perhaps the most striking phenomenon that makes dissipative flows so distinct from ordinary (elastic) gases [5-8]. Moreover, the comparison between kinetic theory and computer simulations for the critical size can be considered as a clean gauge of the former since both approaches (theory and simulation) are restricted to the linear regime where the deviations of the hydrodynamic fields from their values in the HCS are small. The accuracy of the prediction of the critical length scale given by kinetic theory has been verified for a low-density monodisperse granular gas by DSMC [9] and more recently by DEM simulations for a granular fluid at moderate density [10,11]. In both cases, the theoretical predictions for the critical size compare well with computer simulations even for strong dissipation. On the other hand, the results for polydisperse granular systems (namely, when the system is constituted by grains of different masses, sizes, and coefficients of restitution) are more scarce. Polydispersity introduces phenomena that have no counterpart in monodisperse flow but may dramatically influence system behavior, such as species segregation [12-14]. Polydispersity also introduces complexities in the treatment of the velocity distribution and species energy balance to the hydrodynamic description [15]. Specifically, many authors have assumed either a Maxwellian velocity distribution [16,17] (at least between unlike particles [18-20]) and/or an equipartition of energy between particle species [21-24], which implies that the species masses are similar and (collisional) dissipation is small. As for the energy balance, the theory of Zamankhan [22] involves a balance of species granular energy, 87

106 while Jenkins & Mancini [21] and Arnarson & Willits [23] utilize a mixture energy balance, reducing the number of governing equations by a factor s, the total number of particle species. A recent polydisperse theory (see Ref. [25] and [26], with corrections in [27]), which is derived from the revised Enskog theory and does not contain the assumptions of Maxwellian velocity distribution or equipartition, will be the focus of polydisperse work. Furthermore, this theory features a Navier-Stokes-order CE expansion, a first-order Sonine polynomial expansion, and a mixture energy balance. Hydrodynamic stability analyses (e.g., [28]) offer a convenient means of assessing this model. To the best of our knowledge, the only comparison for shearing instability for binary systems has been recently carried out in Ref. [29] in the dilute limit case where the collisional contributions (due to density effects) to the NS transport coefficients are neglected. As for Ref. [9], theory compares well with the DSMC simulations of the Boltzmann equation. However, given that most solids flows present in nature are dense and polydisperse, a proper theoretical framework for these systems is critical to obtain an accurate description of practical particle flows. The aim of this chapter is to assess the ability of hydrodynamics to predict the critical length scale via a comparison with DEM simulations in a binary granular mixture at moderate density. The theoretical results are based on a recent solution to the RET [25-27] that takes into account the nonlinear dependence of the transport coefficients on dissipation. This Chapman Enskog solution [25-27] differs from some previous theoretical attempts for dense granular flows [21-23] that were obtained for quasielastic particles and so, none of the transport coefficients depend on the coefficients of 88

107 restitution. Thus, the present theory subsumes all previous analyses [9-11,21-23,29], which are recovered in the appropriate limits. Given that DEM simulations avoid any assumptions inherent in the kinetic theory (e.g., molecular chaos) or approximations made in solving the RET by means of the DSMC method, the comparison between kinetic theory and DEM simulations carried out here can be considered as the most stringent quantitative assessment of kinetic theory to date for conditions of practical interest. In this context, the results reported in this chapter provide strong evidence of the reliability of hydrodynamics for a wide range of densities and inelasticities in a quite complex (polydisperse) system. 5.2 Computational Methods A binary mixture of inelastic, smooth, hard spheres of masses m 1 and m 2, and diameters d 1 and d 2 is considered. The inelasticity of collisions among all pairs is characterized by three independent constant coefficients of normal restitution e ij. At a kinetic level, the relevant information on the state of the mixture is given through the one-particle distribution functions which obey the RET. From it, one can derive the NS hydrodynamic equations for the granular binary mixture with explicit expressions for the hydrostatic pressure, the cooling rate, and the complete set of transport coefficients. The detailed form of the above quantities can be found in Ref. [25-27]. As for ordinary fluids, all these quantities (which have been approximately obtained by considering the leading terms in a Sonine polynomial expansion) are given in terms of the mole fraction x 1 = n 1 / (n 1 + n 2 ) (n i being the number density of species i), the mass ratio m 1 / m 2, the size ratio d 1 / d 2, the solid volume fraction!, and the coefficients of restitution e ij. 89

108 The hydrodynamic equations admit a nontrivial solution, which corresponds to the so-called HCS. It describes a uniform state (!x 1H =!n H =!T H = 0) with vanishing flow field (U = 0) and a granular temperature T H decreasing monotonically in time, namely, (! t +! H (0) )T H = 0. Here, the subscript H denotes the homogeneous state, n H = n 1H + n 2H, and! (0) H! T H is the zeroth-order contribution to the cooling rate. However, it is well known [5] that the HCS is unstable with respect to long enough wavelength perturbations. To obtain quantitative estimates on the first stages of this instability, a (linear) stability analysis of the NS hydrodynamic equations with respect to the HCS can be performed. As usual, it is assumed that the deviations! y " (r,t) = y " (r,t)! y H" (r,t) are small, where! y " (r,t) denotes the deviation of {x 1,n,U,T} from their values in the HCS. The resulting equations are then written in dimensionless form by using the (dimensionless) time t '! = 1 " H (t ') 2 # dt ' 0 and the (dimensionless) length l =! H (t). Here,,, and 2! H (t) r! (t) = T (t) / m m = (m H H 1 + m 2 ) / 2! H = [8" (d#1)/2 / 5$(3 / 2)]n H d 2 12 % H where d 12 = (d 1 + d 2 ) / 2. A set of Fourier transformed dimensionless variables are introduced by! 1,k =! x 1k / x 1H,! k =!n k / n H, w k =!u k /! H, and! k =!T k / T H, where! y k!! {" 1,k,! k,w k,! k } is defined as! y k" (# ) = # dle!ik"l! y " (l,# ), (5.1) As DEM simulations (see Section ) carried out in this chapter clearly show, the origin of the instability is associated with the transversal components of the velocity 90

109 field w k! = w k " (w k #k)k. As such, the critical length scale is, the critical length L Vortex scale for velocity vortex instabilities. As expected [9-11,29], the two shear (transversal) modes w k decouple from the other four longitudinal modes, greatly simplifying the theoretical analysis of the onset of instability. The evolution equation of w k is!w k"!! + $ 1 2 "* k 2 *' #! % & 0 ( ) w k" = 0, (5.2) where! * = (" H! H ) / (#H$ 2 H ) is the dimensionless shear viscosity of the mixture and! * 0 =! (0) H /" H [25-27]. Here,! H = m 1 n 1H + m 2 n 2H is the total mass density. The solution to Eq. 5.2 is w k! (k,! ) = w k! (0)exp[s! (k)! ], where s! (k) = " * 0 # 1. This identifies a 2 $* k 2 critical wave number k! c = 2! 0 * /" * such that a linear excitation of the (scaled) transversal velocity with k < k! c grows in time. Since the simulations made here consider periodic boundary conditions, the smallest allowed wave number is 2π/L, where L is the system length. Hence, for given values of the parameters of the mixture (masses, diameters, composition, coefficients of restitution, and volume fraction), a critical length L Vortex can be identified such that the system becomes unstable when L > L Vortex. The value of L Vortex is ( ) "1 L Vortex = 5 2 2!(3 / 2)! * 2 n * H d 12 " 0, (5.3) 5.3 Results In order to assess the accuracy of the theoretical predictions, DEM simulations of the homogeneous cooling system were performed. A cubic, periodic domain of length L 91

110 that consists of a total number of N spheres is simulated via hard-sphere DEM [30]. The parameter space over which Eq. (5.3) has been verified is the mole fraction x 1, mass ratio m 1 /m 2, ratio of diameters d 1 /d 2, solid volume fraction! = (" / 6)(n 1 d n 2 d 3 2 ), and the common coefficient of restitution e = e 11 = e 22 = e 12. Two different values of the solid volume fraction! have been considered here,! = 0.1 and! = 0.2, both representing a granular fluid with moderate concentration. In addition, three different values of e have been studied: e = 0.9 (weak dissipation), e = 0.8 (moderate dissipation), and e = 0.7 (strong dissipation). To determine the stability of a given simulation with respect to velocity vortices, a Fourier analysis [31] is used. Specifically, an integrated Fourier transform of the momentum field is used to assess the magnitude of contributions to given wavelengths. A monotonic increase in the magnitude of contribution with respect to wavelength corresponds to a homogeneous flow. Relatively large contributions (or excitations) at small wavelengths (2π/L or 4π/L in the HCS) correspond to velocity vortices, organized structures in the momentum field. For each set of particle parameters simulated, a critical dimensionless length scale exists that distinguishes stable systems from ones unstable to velocity vortices. To determine this critical scale, 24 replicate simulations (that only differ in initial conditions) are considered for a range of domain length scales. If any one of the 24 replicates is unstable, the corresponding L Vortex / d 12 is considered unstable. Thus, a range for L Vortex / d 12 can be determined where the higher L / d 12 (i.e., upper bound) of this range is unstable and the smaller L / d 12 (i.e., lower bound) is stable. As an illustration, Fig. 5.1 shows snapshots of velocity and concentration fields. Small systems will remain stable 92

111 [see Figures. 5.1(a) and 1(b) for homogeneous velocity field and particle positions, respectively], while instabilities [see Figures. 5.1(c) and 1(d) for vortices and clusters, respectively] will manifest in large systems after long times. Figure 5.1 Visualizations from a DEM simulation of an equimolar mixture (x 1 = 0.5) with m 1 /m 2 = 2, d 1 /d 2 = 3,! = 0.2, and e = 0.7 of (a) stable, coarse-grained velocity field at five collisions per particle (or cpp ), (b) stable particle positions at five cpp, (c) unstable, coarse-grained velocity field at 400 cpp, and (d) cluster systems at 400 cpp. A cell size of L/5 is used for local velocity averaging. Next, the (linear) hydrodynamic predictions of L Vortex given by Eq. (5.3) are compared to results from DEM simulations. Figure 5.2 shows L Vortex as a function of the 93

112 mass ratio and coefficient of restitution with d 1 /d 2 = 1, and x 1 = 0.1. It is quite apparent that Fig. 5.2(a) (! = 0.1) shows excellent agreement between hydrodynamics and DEM simulations throughout the parameter space studied, even for significant dissipation in combination with large mass ratios. For moderate densities (! = 0.2) and dissipation (e = 0.9) [see Fig. 5.2(b)], excellent agreement is observed up to quite extreme mass ratios (m 1 /m 2 = 10), while for higher dissipation (e = 0.7), strong agreement is observed to significant mass ratios (m 1 /m 2 = 4). An interesting qualitative agreement is also displayed in Fig For both DEM simulations and hydrodynamics, the L Vortex / d 12 predictions for α = 0.7 and α = 0.8 begin to converge for large mass ratios and eventually crossover. Figure 5.2 Critical length scale for velocity vortices as a function of the mass ratio m 1 /m 2 with x1 = 0.1, d 1 /d 2 = 1 for (a)! = 0.1 and (b)! = 0.2. The data points correspond to DEM simulations, while the lines are the theoretical predictions given by Eq. (6.3). (Blue) circles/solid line, (red) triangles/dashed line, and (black) squares/dot-dashed line correspond to e = 0.9, e = 0.8, and e = 0.7, respectively. Error ranges are the size of the data points and are omitted. Figure 5.3(a) shows L Vortex / d 12 as a function of the ratio of diameters with x 1 =0.5, m 1 /m 2 = 2, and! = 0.2. Excellent agreement is observed throughout the conditions 94

113 studied. Figure 5.3(b) shows the critical size as a function of species composition x 1 for a relatively large mass ratio (m 1 /m 2 = 6) with d 1 /d 2 = 1, and! = 0.2. Even in the extreme case of small composition (x 1 = 0.1) and large dissipation (e = 0.7), hydrodynamics and DEM deviate by less than roughly 10%. For moderate dissipation (e=0.9), the theory compares very well with DEM for all species compositions. Figure 5.3 Critical length scale for velocity vortices as a function of (a) the ratio of diameters d 1 /d 2 with m 1 /m 2 = 2, x 1 = 0.5, and! = 0.2 and (b) the mole fraction x 1 with m 1 /m 2 = 6, d 1 /d 2 = 1, and! = 0.2. The meaning of symbols and lines is the same as that of Fig The comparison carried out in Figures. 5.2 and 5.3 for L Vortex shows in general an excellent agreement between theory and simulation when physical properties of particles are similar, while only good (at worst 20% error) for the most extreme conditions studied [e.g., Fig. 5.2(b); m 1 /m 2 = 10, x 1 = 0.1,! = 0.2, and e = 0.7]. On the other hand, based on the results derived from the RET for ordinary fluid mixtures [32,33], one would expect that the accuracy of the first Sonine approximation to the shear viscosity η [which is a 95

114 transport coefficient involved in L Vortex ; see Eq. (5.3)] would decrease as the mass ratio becomes more disparate. In this context, the leading order truncation of the Sonine polynomial expansion may be responsible for the discrepancies found between theory and DEM rather than assumptions inherent to the RET, such as the absence of velocity correlations (molecular chaos hypothesis). In fact, recent results [34,35] for the tracer diffusion coefficient D have shown that in general the second Sonine approximation to D improves significantly the prediction of the first Sonine solution, especially for high dissipation and/or extreme mass or diameter ratios. The inclusion of the second-order Sonine corrections to η could mitigate part of the discrepancies observed here, especially in the case of quite extreme values of the mass ratio. 5.4 Concluding Remarks In summary, the comparison addressed here between the predictions of linear hydrodynamics (derived from the RET) and discrete-particle computer simulations provides the most stringent test to date for hydrodynamic description of multicomponent granular fluids. This hydrodynamic description continues to be a source of controversy, from the appropriateness of the RET (which is derived under the molecular chaos assumption) for flows with moderate densities to the appropriateness of the NS equations. Here, DEM simulations (which do not rely on any of the above assumptions) are the ideal data set. The system is complex; hydrodynamic instabilities in a transient, polydisperse system at moderate density with significant dissipation levels. The good agreement found in this paper between the predictions of linear hydrodynamics (with the NS transport coefficients derived from the RET) and DEM simulations must be considered as a 96

115 nontrivial example of the reliability of hydrodynamics as a quantitative predictive tool for moderate dense and highly dissipative granular binary mixtures. Therefore, although the theoretical method used here (Chapman-Enskog) is formal and does not strictly establish the existence of hydrodynamics, the results (theory and DEM simulations) clearly indicate that the granular temperature can still be considered as a slow hydrodynamic variable such that the normal solution to the RET is still applicable. Hydrodynamic descriptions derived from kinetic theory are critical tools in the description of numerous industrial processes involving solid particles. These descriptions are now standard features of commercial, multiphase computational fluid dynamics codes (such as FLUENT) and open-source, research codes (such as MFIX). Given that such codes rely on accurate expressions for the transport coefficients, it is evident that the results displayed in the present chapter are of great value not only to the granular physics community working on kinetic theory but also to more applied scientists interested in more practical problems (e.g., biomass gasification, mixing of pharmaceutical powders, heat exchange in concentration solar power plants, synthesis of fine chemicals, pollution control, and ejection of lunar soil from rocket landings). Because of this pervasiveness, the applicability of hydrodynamics to complex polydisperse systems should resonate throughout the fluid dynamics community. While the focus of the Chapters 2-5 has been on the onset of instability, the following chapter studies the ability of hydrodynamic models to predict the behavior of high-gradient flow where instability has evolved such that the assumption of small Knudsen number is assessed. 97

116 5.5 References 1. Mitrano, P. P., Garzó, V. & Hrenya, C. M Instabilities in granular binary mixtures at moderate densities. Phys. Rev. E 89, Goldhirsch, I Rapid Granular Flows. Annu. Rev. Fluid Mech. 35, Brey, J. J., Dufty, J. W. & Santos, A Dissipative dynamics for hard spheres. J. Stat. Phys. 87, Chapman, S. & Cowling, T The Mathematical Theory of Non-Uniform Gases.. Cambridge: Cambridge University. 5. Goldhirsch, I. & Zanetti, G Clustering instability in dissipative gases. Phys. Rev. Lett. 70, Brilliantov, N., Saluena, C., Schwager, T. & Pöschel, T Transient structures in a granular gas. Phys. Rev. Lett. 93, Burton, J. C., Lu, P. Y. & Nagel, S. R Energy Loss at Propagating Jamming Fronts in Granular Gas Clusters. Phys. Rev. Lett. 111, Pathak, S. N., Jabeen, Z., Das, D. & Rajesh, R Energy decay in threedimensional freely cooling granular gas. Phys. Rev. Lett. 112, Brey, J. J., Ruiz-Montero, M. J. & Moreno, F Instability and spatial correlations in a dilute granular gas. Phys. Fluids 10, Mitrano, P. P., Dahl, S. R., Cromer, D. J., Pacella, M. S. & Hrenya, C. M Instabilities in the homogeneous cooling of a granular gas: A quantitative assessment of kinetic-theory predictions. Phys. Fluids 23, Mitrano, P. P., Garzó, V., Hilger, A. M., Ewasko, C. J. & Hrenya, C. M Assessing a hydrodynamic description for instabilities in highly dissipative, freely cooling granular gases. Phys. Rev. E 85, Ottino, J. & Khakhar, D Mixing and segregation of granular materials. Annu. Rev. Fluid Mech. 32, Kudrolli, A Size separation in vibrated granular matter. Reports on progress in Physics 67, Ottino, J. M. & Lueptow, R. M On mixing and demixing. Science 319, Hrenya, C. M Kinetic theory for granular materials: polydispersity. Computational Gas-Solids Flows and Reacting Systems: Theory, Methods and Practice, Jenkins, J. T. & Mancini, F Balance laws and constitutive relations for plane flows of a dense, binary mixture of smooth, nearly elastic, circular discs. J. Appl. Mech. 54, Huilin, L., Gidaspow, D. & Manger, E Kinetic theory of fluidized binary granular mixtures. Phys. Rev. E 64, Mathiesen, V., Solberg, T., Arastoopour, H. & Hjertager, B. H Experimental and computational study of multiphase gas/particle flow in a CFB riser. AIChE Journal 45, Rahaman, M. F., Naser, J. & Witt, P. J An unequal granular temperature kinetic theory: description of granular flow with multiple particle classes. Powder Technol. 138,

117 20. Iddir, H. & Arastoopour, H Modeling of multitype particle flow using the kinetic theory approach. AIChE Journal 51, Jenkins, J. T. & Mancini, F Kinetic theory for binary mixtures of smooth, nearly elastic spheres. Phys. Fluids A 1, Zamankhan, P Kinetic theory of multicomponent dense mixtures of slightly inelastic spherical particles. Phys. Rev. E 52, Arnarson, B. O. & Willits, J. T Thermal diffusion in binary mixtures of smooth, nearly elastic spheres with and without gravity. Phys. Fluids 10, Serero, D., Goldhirsch, I., Noskowicz, S. H. & Tan, M. L Hydrodynamics of granular gases and granular gas mixtures. J. Fluid Mech. 554, Garzó, V., Dufty, J. W. & Hrenya, C. M Enskog theory for polydisperse granular mixtures. I. Navier-Stokes order transport. Phys. Rev. E 76, Garzó, V., Hrenya, C. M. & Dufty, J. W Enskog theory for polydisperse granular mixtures. II. Sonine polynomial approximation. Phys. Rev. E 76, Murray, J. A., Garzó, V. & Hrenya, C. M Enskog Theory for Polydisperse Granular Mixtures. III. Comparison of dense and dilute transport coefficients and equations of state for a binary mixture. Powder Technol. 220, Garzó, V Instabilities in a free granular fluid described by the Enskog equation. Phys. Rev. E 72, Brey, J. J. & Ruiz-Montero, M. J Shearing instability of a dilute granular mixture. Phys. Rev. E 87, Allen, M. P. & Tildesley, D. J. Computer simulation of liquids. (Clarendon Press, 1989). 31. Mitrano, P. P., Dahl, S. R., Hilger, A. M., Ewasko, C. J. & Hrenya, C. M Dual role of friction in granular flows: attenuation versus enhancement of instabilities. J. Fluid Mech. 729, López de Haro, M. & Cohen, E. G. D The Enskog theory for multicomponent mixtures. III. Transport properties of dense binary mixtures with one tracer component. The Journal of Chemical Physics 80, Kincaid, J. M., Cohen, E. G. D. & López de Haro, M The Enskog theory for multicomponent mixtures. IV. Thermal diffusion. The Journal of Chemical Physics 86, Garzó, V. & Vega Reyes, F Segregation of an intruder in a heated granular dense gas. Phys. Rev. E 85, Garzó, V., Murray, J. A. & Reyes, F. V Diffusion transport coefficients for granular binary mixtures at low density: Thermal diffusion segregation. Phys. Fluids 25,

118 6. Granular Flows: Effect of High-Gradients e Abstract This chapter assesses, via instabilities, a Navier-Stokes-order (small-knudsen-number) continuum model based in a flow where high gradients exist due to the development of velocity vortices. Predictions for the critical length scales required for particle clustering obtained from transient simulations of the continuum model are compared with discrete element method (DEM) simulations. The agreement between continuum simulations and DEM simulations is excellent, particularly given the presence of well-developed velocity vortices at the onset of clustering. More specifically, spatial mapping of the local velocity-field Knudsen numbers ( Kn u ) at the time of cluster detection reveals Kn u >> 1 due to the presence of large velocity gradients associated with vortices. Although kinetictheory-based continuum models are based on a small-knudsen-number (i.e., smallgradient) assumption, the findings suggest that, similar to molecular gases, Navier- Stokes-order continuum theories provide surprisingly accurate predictions outside of their expected range of validity. e Mitrano, P. P., J. R. Zenk *, S. Benyahia, J. E. Galvin, S. R. Dahl, C. M. Hrenya. Kinetic-theory-based predictions of clustering instabilities in granular flows: beyond the small-knudsen regime. Journal of Fluid Mechanics (Rapids) 738, R2 (2014). 100

119 6.1 Introduction Continuum models based on kinetic theory (KT) offer a practical framework for predicting instabilities in practically sized systems. Previous studies have shown the ability of such models to qualitatively predict clustering in granular [1] and gas-solid flows [2-6] via visual snapshots from a transient simulation. However, the ability of continuum models to quantitatively describe flows containing high gradients is lacking in the literature and is uncertain due to the assumptions inherent in their derivation. Of specific interest here is the small-knudsen-number, or small-gradient, assumption invoked by kinetic-theory-based models of Navier-Stokes (NS) order, where Knudsen number (Kn) is defined as the ratio of the (local) mean free path (! 0 ) to the characteristic length ( L grad ) associated with the spatial gradient of a given hydrodynamic variable (velocity, concentration, etc.). This assumption of small Kn allows for truncation of the constitutive fluxes at first order in gradients (i.e., Navier-Stokes order) [7]. Whereas previously chapters have focused on the onset of inhomogeneities before gradients have developed, large Knudsen numbers may be the norm in clustering flows. Specifically, a large gradient in concentration exists at the interface of the clustered and dilute regions. This large-kn feature of the instability would seem at odds with the small- Kn assumption of NS-order continuum models. Therefore, one may expect kinetictheory-based predictions to deteriorate in the presence of clustering, yet a Navier-Stokes description of molecular gases is known to provide reliable predictions well outside of the expected range of validity. The focus of the current effort is to assess the quantitative 101

120 accuracy of NS-order predictions for clustering instabilities that give rise to Kn of O(1) or greater. Previous researchers have pointed to an inherent lack of scale separation, which gives rise to moderate and large Kn, as an inherent feature of even stable granular flows (see review by [8]). Indeed, many earlier works on granular flows have documented telltale signs of higher-order-kn (free molecular) effects such as Knudsen layers, shocks, etc. [9-16]. Moreover, previous studies of systems with seemingly large gradients in concentration have found that NS-order, KT-based models agree well with simulation [10,17] and experiment [15,17]. These works bode well for the accuracy of NS-order models outside of the expected range of applicability (akin to molecular gases), but have not included a quantitative assessment of the Kn fields. The current chapter aims to explicitly address the applicability of NS-order, KTbased models [18] to large-kn flows via quantitatively studying the particle clustering instability at a time when Knudsen numbers are large. As a first step towards assessing KT-based predictions in gas-solid fluidized systems, the focus is on granular flow. Particle clustering (see Fig. 1), occurs in dissipative granular flows where the domain length L is sufficiently large compared to the particle diameter d [19,20]. In this chapter, the critical dimensionless length scale Lcluster / d that demarcates spatially homogeneous (stable) and clustering flows (unstable) is studied. Specifically, predictions from transient, NS-order continuum simulations in the granular homogeneous cooling system (HCS) are compared to DEM simulations, while quantifying the Knudsen numbers throughout the domain at the time of cluster detection. 102

121 Figure 6.1 Visualizations of three-dimensional, stable (L/d=18) flows and clustering (L/d=40) flows in a homogeneous cooling system at long times with restitution coefficient of 0.7 and solids volume fraction of 0.1: (a) stable and (b) unstable DEM simulations showing slice of thickness L/10, (c) stable and (d) unstable kinetic-theorybased simulations showing one layer of numerical cells (of thickness d). 6.2 Computational Methods Transient Simulation of Hydrodynamic Model The transient, kinetic-theory-based hydrodynamic (continuum) equations developed by Garzó & Dufty [18] are numerically solved via a first-order accurate (and second-order accurate for convection and diffusion terms), finite-volume method for discretization [21,22]. Specifically, the computational fluid dynamics code MFiX, developed at the National Energy Technology Laboratory (mfix.netl.doe.gov) is used. The continuum model [18] is derived from the Enskog kinetic equation, which assumes molecular chaos (i.e., that the velocities of colliding particles are uncorrelated) and is applicable to flows up to moderate concentrations. Constitutive equations are derived via the Chapman-Enskog perturbation expansion. By invoking an assumption of small Knudsen number (i.e., small spatial gradients in hydrodynamic variables), the constitutive quantities are truncated at first order in spatial gradients. The balance equations of this continuum model [18] are given by 103

122 n + u n+ n u= 0 t (6.1) $! &!u %!t + u "#u ' ) = *#P (6.2) (!T!t + u "#T = $ 2 # "q + P :#u 3n where n is number density, t is time, u is velocity, ( ) $%T (6.3) ρ = nm is the bulk solids density, P is the pressure tensor, T is the granular temperature, q is the heat flux, and ζ is the cooling rate. Detailed constitutive relations for the terms P, q, ζ can be found in [18] and are not duplicated here for the sake of brevity. The theory assumes hard spheres that interact via instantaneous, binary collisions with a constant restitution coefficient e. The system simulated is the standard HCS of cubic volume L 3 with domain solids fraction φ that consists of a number of particles N. Since beginning a continuum simulation with (unperturbed) homogeneous initial conditions would result in a perpetually spatially homogeneous solution, a perturbed initial state for the continuum simulation is obtained by extracting field variables from the initial state of a DEM simulation, also performed via MFiX [23], where particles are randomly placed in the domain. Care is taken to ensure no net momentum develops due to numerical round off (by adjusting each cell velocity by an amount p / m, where p i is the total momentum in the i th direction and m j i j is the mass of the j th cell). Continuum simulations use a numerical grid size of 2d, and the results are insensitive to increased resolution. 104

123 Particle clustering instabilities are assessed via the evolution of!! max = (! max!! min ) /!, the maximum difference in solids concentration between any two cells in the domain normalized by the domain concentration, as shown in Fig In stable systems (L/d = 18),!! max decays to zero (i.e., homogeneity) from the initially perturbed state, while in unstable systems (L/d = 20),!! max levels to a non-zero value after some time. Long-time non-zero asymptotic behavior in!! max indicates clustering. t T0 Continuum simulations are performed up to a non-dimensional time t* = d m =5x10 7, and the results are insensitive to increased simulation duration and the magnitude of initial perturbation. Figure 6.2 Maximum difference in cell concentration normalized by the average concentration plotted as a function of time for e=0.7 and φ =0.1. Dashed (red) and solid (blue) lines represent L/d=20 and L/d=18, respectively, with a (black) vertical dashed line indicating the onset time of the clustering instability, t*=

124 6.2.2 Linear Stability Analysis of Hydrodynamic Model To complement the continuum simulations, a linear stability analysis [24] of the HCS of the same NS-order theory [18] outlined above is included. This stability analysis gives the critical length scale required for particle clustering, L cluster, and is outlined in Section It is worth emphasizing that since stability analysis incorporates a linear assumption, or small perturbations of the hydrodynamic variables, the stability analysis does not consider nonlinear contributions to the formation of the particle clustering instability. In contrast, the numerical simulations carried out using the continuum model (see previous section) provide a transient, three-dimensional solution that includes linear and nonlinear effects alike Discrete Element Method Simulations The theoretical predictions obtained from the previously outlined techniques will be assessed via comparison with event-driven, hard-sphere DEM simulations, detailed in Section Stability with respect to clustering is determined via analysis of the concentration Fourier spectra [25] for varying domain sizes, as is further described in Section (Ref. [26]). Each given domain size is tested for stability with 10 replicate simulations that differ only by initial conditions. A range is developed where at the lower bound all ten simulation replicates are stable, while at the upper bound at least one of the replicates is unstable. 106

125 6.3 Results Lcluster Fig. 6.3 shows the dimensionless critical length scales for particle clustering, / d, obtained from continuum simulations, DEM simulations, and linear stability analysis as a function solids fraction for e= 0.9 (Fig. 3a) and e=0.7 (Fig. 6.3b). Excellent quantitative agreement between NS-order continuum simulations and DEM simulations is observed, whereas predictions from the linear stability analysis show significant overprediction relative to continuum and DEM simulations. Each of these observations will be discussed in turn below. Figure 6.3 Critical dimensionless length scale for clustering instabilities as a function of solids fraction for (a) e=0.9 and (b) e=0.7. The (black) hollow squares, (blue) squares, and (blue) line represent predictions from the continuum simulations, DEM simulations, and linear stability analyses, respectively. Error ranges are the size of data points and omitted. The discrepancy between continuum simulations and the linear stability analysis, which uses the same set of balance equations [18] as their starting point, can be traced to the lack of nonlinear effects in the stability analysis. One such term, viscous heating ( P :!u in Eq. 6.3), has been shown to be an important contributor to the clustering 107

126 instability via continuum theory [27,28] and discrete-particle simulations [28,29]. The negligence of nonlinear contributions to clustering, namely viscous heating, causes the over prediction of Lcluster / d (i.e., less prone to cluster) relative to continuum simulations. This nonlinear contribution, which is proportional to the square of the velocity gradient, plays a role due to the presence of velocity vortices, which are known to precede particle clustering in the HCS [25]. The mechanism by which viscous heating contributes to the onset of clustering via vortex motion can be understood in the following way. First, consider vortex flow in the HCS; since momentum is conserved, multiple shear bands (regions of aligned, convective flow, which can be seen in Fig. 6.4a) develop in opposing directions. The regions between shear bands of opposing direction must contain a relatively large velocity gradient such that (!u) 2 is significant relative to!u, and a linear analysis, which neglects the former, may not be appropriate. The large velocity gradients in these regions give to an increased rate of granular energy production (via viscous heating). Conversely, the regions within shear bands exhibit strong velocity alignment. Since the gradient in velocity is small within these regions, viscous heating is also small. Therefore, the granular temperature within shear bands is relatively small such that local gradients in temperature develops across the domain, which at this time is still homogenous with respect to concentration. A corresponding granular pressure gradient develops, with high-pressure (high T) regions outside of shear bands driving particle flow into the low-pressure (low T) regions within shear bands. Thus, the interiors of shear bands become relatively dense. Fig. 6.4 shows a mapping between the locally aligned velocity field and small granular temperature (Fig. 6.4a & 6.4c) as well as the inverse 108

127 relationship between the granular temperature and concentration (Fig. 6.4b & 6.4c), all at the onset of clustering. Specifically, Fig. 6.4 shows visualizations of the L/d = 20 simulation depicted in Fig. 6.2 at the time of cluster onset, t*=107. Note that dense regions of Fig. 6.4c generally correspond to regions of local velocity alignment in Fig. 6.4a. Figure 6.4 Visualizations of the onset of clustering (showing a slice of the threedimensional domain): (a) plot of (coarse-grained) velocity field in two-dimensional slice of thickness d, (b) granular-temperature-field T( x, y ) color map, and (c) corresponding concentration-field φ ( xy, ) grey-scale plot with e=0.7, φ =0.1, L/d=20, and t*=107. To evaluate the validity of NS-order theories, maps of the local Knudsen numbers associated with gradients in concentration ( Kn! ), granular temperature ( Kn T ), and velocity ( Kn u ) are computed at the onset of clustering. Recall that Kn is the ratio of the mean free path to the characteristic length scale associated with the gradient (Kn =! 0 / L grad ). Here L grad is defined as the length over which the hydrodynamic field of interest (concentration, x-, y-, z-component velocity, or temperature) changes by 20%. This length is determined by extrapolating a local, linear, (x-, y-, or z-direction) spatial gradient from the field of interest. Fig. 6 shows the maps of Kn! and Kn T corresponding 109

128 to Fig As expected at the onset of clustering, the concentration and granular temperature are nearly homogeneous, as shown by the relatively small scale in Fig. 6.4b & 6.4c. Correspondingly, Kn! and Kn T are less than O(1), adhering to the small-kn assumption in a Navier-Stokes-order Chapman-Enskog truncation. Figure 6.5 Spatial color maps of concentration and granular temperature Knudsen numbers at the onset of clustering (corresponding to Fig. 5.4): (a) Kn! in the x- and (b) y- spatial direction, (c) Kn T in the in the x- and (d) y-spatial direction with e=0.7, φ =0.1, L/d=20, and t*=107. Regarding the agreement displayed in Fig. 6.3 between the continuum simulations and DEM for Lcluster / d, recall the constitutive quantities of NS-order continuum theories rely on a small-kn (small-gradient) assumption [7]. Although Fig. 6.5 shows that Kn! and Kn T are indeed small with respect to one, the previous assessment of viscous heating suggests that higher-order velocity gradients are significant such that Kn u may be of order one or greater (i.e., contributions beyond NS order may be important). Therefore, the continuum-dem agreement suggests either that this small-gradient assumption is not 110

129 being challenged or that the NS-order theory is accurate beyond the expected range of validity. To assess the magnitude and distribution of velocity gradients at the time of cluster detection, Kn u has been evaluated throughout the domain. Fig. 6.6 shows maps of Kn u corresponding to the domain slice and conditions of Fig Figure 6.6 Spatial color maps of the velocity-field Knudsen numbers Kn u at the onset of clustering (corresponding to Fig. 5.4) corresponding the gradients: (a)! x U x, (b)! y U x, (c)! x U y, and (d)! y U y with e=0.7, φ =0.1, L/d=20, and t*=107. Significant gradients exist in the velocity field at the time clustering is first detected, as demonstrated in Fig Fig. 6.7 shows this distribution of Kn u more explicitly. Specifically, Fig. 6.7 shows the cumulative distribution of each component of Kn u for the entire three-dimensional domain for the conditions and time of Fig In contrast to the relatively small values of Kn! and Kn T throughout the domain, 76% of local measurements have Kn u! O(1) and 20% have Kn u! O(10). DEM simulations exhibit a similar distribution of Kn u at the time of clustering but are not shown for 111

130 brevity. The presence of such large gradients is beyond the assumption of the small Knudsen number used to derive the NS-order theory. Furthermore, Fig. 6.6a clearly shows that velocity vortices, and thus correlations in the velocities of colliding particles, exist in the flow. These correlations are at odds with the assumption of molecular chaos, inherent to the starting (Enskog) kinetic equation. Figure 6.7 Cumulative distribution of the x-, y-, and z-velocity components of Kn u throughout the three-dimensional domain at the onset of clustering with e=0.7, φ =0.1, L/d=20, t*= Concluding Remarks In this chapter, the ability of a Navier-Stokes-order (small-kn) continuum description [18] to predict the onset of the particle clustering instability is assessed via transient, three-dimensional simulations. Specifically, the continuum predictions of Lcluster / d in the granular HCS are compared to results obtained from DEM simulations. DEM and continuum simulations each explicitly demonstrate the presence of Kn u of order 10 or greater. Moreover, continuum simulations show that 76%, 20%, and 1.9% of 112

131 the domain contain Kn u of order 1, 10, and 100 (or greater), respectively, at the onset of particle clustering. Such significant velocity gradients can be traced to the presence of velocity vortex instabilities, which manifest at earlier times than clustering instabilities. Despite such large Kn u at the onset of clustering, DEM and continuum simulations are in excellent agreement for Lcluster / d. It is worth noting that previous works [26,29,30] showing good agreement between DEM simulations and KT-based predictions for the onset of instabilities were limited to systems in which gradients in all hydrodynamic variables are relatively small. The excellent continuum-dem agreement observed here is somewhat surprising within the context of the Navier-Stokes-order theory since its derivation is based on a perturbative expansion about small Kn. Furthermore, the starting kinetic equation presumes that the velocities of colliding particles are not correlated (i.e., molecular chaos), whereas the coarse grained velocity field of Fig. 6.4a suggests local velocity correlation due to the presence of vortices. The excellent agreement observed in this chapter between the continuum model and DEM simulations is very encouraging for the applicability of the kinetic-theory-based, Navier-Stokes-order model [18] beyond what a direct interpretation of the assumptions may imply, as is well accepted for NS-order theories of molecular gases. While analogous theories can, in principle, be derived beyond NS order (Burnett, super Burnett, etc.), the complexity of the derivation (see, for example, [31]) and the application of such higher-order theories remains a significant challenge. Specifically, the computation of higher-order transport coefficients in such theories typically requires an inherently unstable, numerical solution [32] and boundary 113

132 conditions that are difficult to physically recognize (e.g., higher-order slip/temperature jump boundary conditions [33]). It remains to be seen if Navier-Stokes-order, kinetictheory-based predictions continue to perform well once particle clusters further develop (when L / d >> L cluster / d ) since Kn! and Kn T will be large in addition to the large Kn u observed here in the velocity field. While the focus of previous chapters was on the onset of instability, the onset of clustering in flows with developed velocity vortices was studied here in order to target a system with large gradients. Chapters 7 and 8 continue with the study of clustering instabilities, in which the vortex instability has already evolved, but with the addition of a gas phase. As discussed in the introductory chapter, the presence of the gas phase lends to two additional mechanisms giving rise to instabilities, thermal and mean drag. These will be considered in succession in the following two chapters. 6.5 References 1. Brilliantov, N., Saluena, C., Schwager, T. & Pöschel, T Transient structures in a granular gas. Phys. Rev. Lett. 93, Gidaspow, D. Multiphase flow and fluidization : continuum and kinetic theory descriptions. (Academic Press, 1994). 3. Kunii, D. & Levenspiel, O. Fluidization engineering. Vol. 2 (Butterworth-Heinemann Boston, 1991). 4. Jackson, R. The dynamics of fluidized particles. (Cambridge Univ. Press, 2000). 5. Sundaresan, S Instabilities in fluidized beds. Annu. Rev. Fluid Mech. 35, Gidaspow, D. & Jiradilok, V. Computational techniques: The multiphase CFD approach to fluidization and green energy technologies. (Nova Science Publishers, 2009). 7. Chapman, S. & Cowling, T The Mathematical Theory of Non-Uniform Gases.. Cambridge: Cambridge University. 8. Goldhirsch, I Rapid Granular Flows. Annu. Rev. Fluid Mech. 35, Brey, J. J., Ruiz-Montero, M. J. & Moreno, F Hydrodynamics of an open vibrated granular system. Phys. Rev. E 63, Rericha, E., Bizon, C., Shattuck, M. & Swinney, H Shocks in supersonic sand. Phys. Rev. Lett. 88,

133 11. Wassgren, C. R., Cordova, J. A., Zenit, R. & Karion, A Dilute granular flow around an immersed cylinder. Phys. Fluids 15, Martin, T. W., Huntley, J. M. & Wildman, R. D Hydrodynamic model for a vibrofluidized granular bed. J. Fluid Mech. 535, Galvin, J. E., Hrenya, C. M. & Wildman, R. D On the role of the Knudsen layer in rapid granular flows. J. Fluid Mech. 585, Hrenya, C. M., Galvin, J. E. & Wildman, R. D Evidence of higher-order effects in thermally driven rapid granular flows. J. Fluid Mech. 598, Wildman, R., Martin, T., Huntley, J., Jenkins, J., Viswanathan, H., Fen, X. & Parker, D Experimental investigation and kinetic-theory-based model of a rapid granular shear flow. J. Fluid Mech. 602, Almazán, L., Carrillo, J. A., Salueña, C., Garzó, V. & Pöschel, T A numerical study of the Navier-Stokes transport coefficients for two-dimensional granular hydrodynamics. New Journal of Physics 15, Xu, H., Louge, M. & Reeves, A Solutions of the kinetic theory for bounded collisional granular flows. Continuum Mechanics and Thermodynamics 15, Garzó, V. & Dufty, J. W Dense fluid transport for inelastic hard spheres. Phys. Rev. E 59, Hopkins, M. A. & Louge, M. Y Inelastic microstructure in rapid granular flows of smooth disks. Phys. Fluids A 3, Goldhirsch, I. & Zanetti, G Clustering instability in dissipative gases. Phys. Rev. Lett. 70, Syamlal, M MFIX documentation: Numerical technique. US Dept. of Energy, Patankar, S. V. Numerical heat transfer and fluid flow. (Hemisphere Pub, 1980). 23. Garg, R., Galvin, J. E., Li, T. & Pannala, S Documentation of open-source MFIX-DEM software for gas-solids flows. From URL Garzó, V Instabilities in a free granular fluid described by the Enskog equation. Phys. Rev. E 72, Goldhirsch, I., Tan, M. L. & Zanetti, G A molecular dynamical study of granular fluids I: The unforced granular gas in two dimensions. J. Sci. Comput. 8, Mitrano, P. P., Garzó, V., Hilger, A. M., Ewasko, C. J. & Hrenya, C. M Assessing a hydrodynamic description for instabilities in highly dissipative, freely cooling granular gases. Phys. Rev. E 85, Brey, J. J., Ruiz-Montero, M. J. & Cubero, D Origin of density clustering in a freely evolving granular gas. Phys. Rev. E 60, Soto, R., Mareschal, M. & Mansour, M. M Nonlinear analysis of the shearing instability in granular gases. Phys. Rev. E 62, Mitrano, P. P., Dahl, S. R., Hilger, A. M., Ewasko, C. J. & Hrenya, C. M Dual role of friction in granular flows: attenuation versus enhancement of instabilities. J. Fluid Mech. 729,

134 30. Mitrano, P. P., Dahl, S. R., Cromer, D. J., Pacella, M. S. & Hrenya, C. M Instabilities in the homogeneous cooling of a granular gas: A quantitative assessment of kinetic-theory predictions. Phys. Fluids 23, Sela, N. & Goldhirsch, I Hydrodynamic equations for rapid flows of smooth inelastic spheres, to Burnett order. J. Fluid Mech. 361, Hrenya, C. M Kinetic theory for granular materials: polydispersity. Computational Gas-Solids Flows and Reacting Systems: Theory, Methods and Practice, Agarwal, R. K., Yun, K. Y. & Balakrishnan, R Beyond Navier-Stokes: Burnett equations for flows in the continuum-transition regime. Phys. Fluids 13,

135 7. Gas-Solid Flows: Effect of Thermal Drag f Abstract Unlike previous chapters in which the focus was on instabilities in granular systems, the systems studied here include a gas phase. In particular, flow instabilities encountered in the homogeneous cooling of a gas solid system are considered via direct numerical simulations. Unlike previous efforts, the relative contribution of the two mechanisms leading to instabilities is explored: thermal drag (fluid-phase effects) and collisional dissipation (particle-phase effects). The results indicate that the instabilities encountered in the gas solid system mimic those previously observed in their granular (no fluid) counterparts, namely a velocity vortex instability that precedes in time a clustering instability. The onset of the instabilities is quicker in more dissipative systems, regardless of the source of the dissipation. Somewhat surprisingly however, a crossover of the kinetic energy levels is observed during the evolution of the instability. Specifically, the kinetic energy of the gas solid system is seen to become greater than that of its granular counterpart (i.e. same restitution coefficient) after the vortex instability sets in. Additionally, the prediction of the onset length scale for particle clustering from direct numerical simulation is compared to the (hydrodynamic) two-fluid model. The strong agreement observed indicates the ability of the two-fluid model to incorporate fluid effects. f Yin, X., J. R. Zenk *, P. P. Mitrano, C. M. Hrenya, Impact of collisional versus viscous dissipation on flow instabilities in gas-solid systems, Journal of Fluid Mechanics (Rapids) 727, R2 (2013). 117

136 7.1 Introduction Flow instabilities such as particle clusters are widely documented in both granular (no interstitial fluid) and gas-solid systems. For granular flows, early observations of clustering include dissipative discrete element method (DEM) simulations of simple shear flow [1] and homogenous cooling [2], experiments on a vibrated plate [3], and theoretical predictions of homogeneous cooling using kinetic-theory-based hydrodynamic (continuum) models [4]. For the case of gas-solid flows, observations of particle clustering go back even further than their granular counterparts. As early as the 1970 s, transient particle clusters known as streamers were recorded in vertical pipe flows [5]. From a practical standpoint, the presence of such clusters can dramatically change the performance of the associated unit operation. For example, in gas-solid circulating fluidized beds (CFBs), clusters have a significant effect on gas-solid contacting, which in turn impacts both heat transfer and reaction rates. The origin of instabilities in granular flows traces to the dissipative nature of particle-particle collisions [6]; similar instabilities are not observed in molecular (nondissipative) systems. Dissipation can arise from inelastic and/or frictional contacts, as was the focus of previous chapters. For gas-solid flows, the presence of the gas phase gives rise to two additional sources of instabilities [7]: the relative motion between the gas and solid phases (mean drag) [8], and the dampening of fluctuating particle motion by fluid (viscous damping or thermal drag) [9]. To date, however, studies on the relative importance of these instability-enabling mechanisms in gas-solid flows, namely dissipative collisions vs. gas-phase effects, are lacking. 118

137 In this chapter, instabilities of a gas-solid, homogeneous cooling system (HCS) are studied via lattice Boltzmann simulations. Special attention is given in this chapter to gas-phase effects arising from thermal drag (viscous damping), whereas the focus of the next chapter is on gas-phase effects arising from mean drag. Unlike previous HCS studies in which only the role of dissipative collisions (granular flows) [2,4,10-12] or viscous damping (gas-solid flows with non-dissipative collisions) [9] on instabilities was considered, here the interplay between the two mechanisms are considered. 7.2 Computational Methods System Description The HCS examined here consists of solid particles suspended in a gaseous medium. The system is a three-dimensional rectangular domain with periodic boundaries. Instabilities in the HCS can be readily distinguished since no inherent inhomogeneities are associated with the system. Adding a fluid phase to this flow provides an important contribution to clustering, thermal drag [9], which is governed by thermal Reynolds number Re T =! g d µ g T 0 m. The thermal Reynolds number and density ratio! s /! g characterizes the gas effects where! g is the gas density, d is the particle diameter, T = m ( u! u) 2 / 3 is the granular temperature, µ g is the gas viscosity, m is the mass,! s is the solid density, and u is the solids velocity. 119

138 7.2.2 Granular DEM Simulations Though the bulk of the simulations were performed for gas-solid systems as detailed below, a number of granular (no fluid) simulations were also carried out for purposes of comparison. Specifically, event-driven, hard-sphere simulations [13,14] were performed using inelastic, frictionless spheres (see Section ). A range of dimensionless parameters, namely normal restitution coefficients e = and solids volume fractions φ = , were studied. Domain sizes were also varied to match the gas-solid simulations, as described below. The occurrence of instabilities during the simulations was detected via a Fourier analysis of the particle concentration and velocity fields (see Section 3.2.2) Gas-solid Direct Numerical Simulations As illustrated in Figure 1.4a, direct numerical simulation (DNS) is the most detailed modeling approach considered in this work for the study of gas-solid flows. Particle-particle interactions are resolved through Newton s laws of conservation of momentum. The discretized Boltzmann equation applied over a lattice mesh describes the fluid phase, and the dynamics of this fluid phase are fully resolved by implementing a noslip boundary condition along the surface of each particle. Professor Xiaolong Yin and Xiaoqi Li of the Colorado School of Mines have performed the DNS data presented in Chapters 7-8. They used the Susp3D lattice- Boltzmann-based technique developed by Ladd and coworkers [15,16] for the DNS. Simulations are performed in a domain with L x /d = 30, L y /d = 30, and L z /d = 4. See Ref. [17] for a more detailed description of this simulation method. 120

139 7.2.4 Two-Fluid Model (TFM) simulations In stark contrast to the discrete-particle simulations of the previous section, the two-fluid model views the two phases as interpenetrating continua. The transient, kinetictheory-based hydrodynamic (continuum) TFM equations (Eq ) developed by Garzó, Tenneti, Subramaniam, and Hrenya [18] for monodisperse gas-solid flows are solved numerically via a first-order accurate (and second-order accurate for convection and diffusion terms), finite-volume method for discretization [19,20]. Specifically, the computational fluid dynamics code MFiX, developed at the National Energy Technology Laboratory (mfix.netl.doe.gov) is used. The continuum model is derived from the Enskog kinetic equation, which assumes molecular chaos (i.e., that the velocities of colliding particles are uncorrelated) and is applicable to flows up to moderate concentrations. Constitutive equations are derived via the Chapman-Enskog perturbation expansion. By invoking an assumption of small Knudsen number (i.e., small spatial gradients in hydrodynamic variables), the constitutive quantities are truncated at first order in spatial gradients. The balance equations of this continuum model are outlined in Section Since beginning a continuum simulation with (unperturbed) homogeneous initial conditions would result in a perpetually spatially homogeneous (stable) solution, a perturbed initial state for the continuum simulation is obtained by extracting field variables from the initial state of a DEM simulation, also performed via MFiX [21], where particles are randomly placed in the domain. Care is taken to ensure no net momentum develops due to numerical round off (by adjusting each cell velocity by an amount p / m, where p i is the total momentum in the i th direction and m j is the mass of i j 121

140 the j th cell). Continuum simulations use a numerical grid size of 2d, and the results are insensitive to increased resolution. Particle clustering instabilities are assessed via the evolution of!! max = (! max!! min ) /!, the maximum difference in solids concentration between any two computational cells in the domain normalized by the domain concentration, as described in Section Verification of DNS Computations and Instability Detection The accuracy of the granular and gas-solid simulation codes, as well as the Fourier-based method used to detect the onset of instabilities, was verified via a comparison with an analytical solution for a stable HCS (i.e., HCS with no spatial variation of hydrodynamic variables). A recent kinetic-theory-based description of gassolid flows by Garzó and co-workers [18] is utilized to obtain the analytical solution. Specifically, when applied to a stable HCS, the energy balance takes the form dt 2T = ζt γ dt m, (7.2) where m is the particle mass and the granular temperature T is defined by T = 1/3 m<v 2 >, where <V 2 > is the locally averaged square of the particle velocity. The first term on the right-hand side represents a sink of granular energy arising from inelastic (dissipative) particle-particle collisions, and the second term represents the sink due to fluid viscous forces, or thermal drag. Note that the more general form of this balance also contains a source term due to fluid-particle interactions. This additional source term is proportional to the mean relative velocity between the gas and solid phases, which is zero for the HCS. 122

141 For HCS, the constitutive relation for the cooling rate [18] is given by 8φ 2 3 T ζ = ( 1 e ) χ 1+ a2 π d 16 m (0) (0) 1 φ /2 µ 4 5µ 2 where χ =, a2 =, 1 φ (1) 95 (0) µ 4 µ 2 16, µ (0) 9 2 (0) 4 2, and. 2 e = + µ (1) 2 (0) µ 3 4 ( ) 2 = + e + µ e Similar to the granular energy balance, the term a 2 (and thus the cooling rate ζ) contains an additional term in its most general form that accounts for the effects of the fluid phase. This additional term is proportional to the mean relative velocity between the phases, which is zero for the HCS, and thus not included in the expression for ζ shown above. Finally, for gas-solid HCS, which has zero mean relative velocity between phases, γ is expressed as γ = 3 πµ dr [18,22,23] where [24] and [25] ( ) µ 2 = 2πχ 1 e (0) 2 ( ) 3 g diss Rdiss = Rdiss0 + ReT K φ, (7.3) = 1+ 3 φ + φlnφ φ( 1 5.1φ φ 21.77φ ) φχlnε, (7.4) 2 64 R diss 0 m ( ) K ( φ) φ = ( 1 φ) (7.5) With these expressions, the granular energy balance (Eq. 7.2) takes the general form dt AT dt = 3/2 BT, (7.6) 123

142 where A and B, which are independent of T, can be deduced from the expressions given above. The analytical solution to Eq. (7.6) is given by T T 0 2 Bt B e = Bt 2 B+ A T0 1 e 2. (7.7) This analytical solution to the stable HCS obtained using the Garzó et al. theory [18] is compared with the dissipative DEM simulations of granular flows and the DNS simulations of gas-solid flows in Fig In particular, KE/KE 0 is shown as a function of dimensionless time, where KE refers to the average kinetic energy of all particles in the domain. Note that T/T 0 (as predicted by theory) and KE/KE 0 (as obtained in simulations) reduce to the same quantities for the stable HCS, since the local mean flow is zero throughout the domain. In other words, the granular temperature is defined relative to a local mean velocity while the kinetic energy is not, but this point is moot if the local mean velocity is zero (as it is for a stable HCS). Also indicated on the figure is the time at which the first instability was detected via the Fourier analysis referred to earlier (diamonds). Two important conclusions can be drawn from this plot, which is representative of the results obtained for other parts of the parameter space. First, both the granular and DNS simulations display excellent agreement with the Garzó et al. theory [18] prior to the detection of the first instability, which provides verification of the code (simulation) accuracy as well as validation for the recent theory of Garzó et al. [18], which is the first to rigorously incorporate the gas phase into the starting kinetic (Enskog) equation for the derivation of the conservation laws and constitutive relations alike. Second, at the times when the simulations begin to deviate from the theoretical 124

143 predictions, onsets of the vortex instability are also detected via the Fourier-based method (diamonds), thereby providing a verification for this detection method KE/KE (KE 0 /m) 1/2 t/d Figure 7.1 Particle kinetic energy levels versus dimensionless time: The thin solid line corresponds to a DNS simulation (Re T = 3, φ = 0.2, e = 0.9, ρ p /ρ g = 1000); the thin dashed line corresponds to a granular DEM simulation (φ = 0.2, e = 1.0). Thick solid and dashed lines are the respective analytical solutions. Diamonds mark the onset of the vortex instability. 7.3 Results While the latter portion of the results will provide a quantitative comparison between DNS and TFM simulations for the onset length scale for clustering, the first focus of the results is on assessing the behavior of instabilities in the gas-solid HCS via DNS. The types and progression of instabilities observed in the DNS were consistent throughout the parameter space explored. Specifically, in the gas-solid DNS, the instability always begins with the appearance of velocity vortices, as illustrated in Fig. 125

144 7.2. Snapshots of the particle x-y positions (left) and coarse-grained velocity field (averaged over the z direction, right) are shown at three non-dimensionalized times via DNS. At short times (top), both particle concentration and velocity fields appear randomized. At intermediate times (middle), the distribution of particles remains homogenous, while velocity vortices are now evident. At longer times (bottom), particle clusters are now visible in addition to vortices. This instability progression in DNS mimics that previously documented for a granular HCS [2,12] throughout the parameter space. The findings also represent the first report of velocity vortex instabilities in gassolid systems. 126

145 Figure 7.2 Left snapshots of particle positions at three different times in DNS illustrates evolution of particle cluster. Right evolution of the coarse-grained particle velocity field in DNS at three different times. Re T = 30, e = 0.8, φ = 0.2, ρ p /ρ g = Figure 7.3 shows the ratio of collisional to viscous dissipation in DNS as function of dimensionless time in several systems. Because the kinetic energy decays with time in all HCS systems (continual cooling ; see also Fig. 7.1), viscous dissipation plays a more 127

146 dominant role with time relative to its collisional counterpart. More specifically, based on existing closures established for homogeneous systems, viscous dissipation is O(T) and collisional dissipation is O(T 3/2 ) (as described above in Computational Methods section). Moreover, for systems with higher Re T, collisions are seen to play a more dominant role due to the higher particle inertia in DNS. The explanation for the relative behavior observed in systems with different e can be traced to the reduced level of collisional dissipation (and thus larger role of viscous dissipation) for systems with higher e. Similar to the role of e but not shown for the sake of brevity is the effect of particle density ratio in DNS: systems with higher ρ p /ρ g are dominated by collisional dissipations for longer period of time than those with lower ρ p /ρ g due to the increased role of particle inertia. Collision/Viscous Dissipation Re T =3,e=0.8 Re T =3,e=0.9 Re T =10,e=0.8 Re T =10,e=0.9 Re T =30,e=0.8 Re T =30,e= (KE 0 /m) 1/2 t/d Figure 7.3 Collisional vs. viscous dissipation for six DNS simulations with varying Re T and e. The phenomenon that vortex instability (diamonds) precedes cluster formation (circles) is observed in both viscous dominated and collision dominated regimes. Particlefluid density ratio ρ p /ρ g = 1000 and φ =

147 Prior to the detection of instability, gas-solid systems including both dissipative mechanisms are always observed via DNS and DEM to lose energy at quicker rates than those with only one source of dissipation (with all other system conditions fixed), as illustrated in Fig Specifically, in the left panel of Fig. 7.4, several cases are considered: gas-solid systems with e=0.9 at Re T = 3 and 30 (inelastic and viscous dissipation), gas-solid systems with e=1.0 at Re T = 3 and 30 (viscous dissipation only), and a granular system with e=0.9 (inelastic dissipation only). While all of the cases show cooling with time with both sources of dissipation exhibit lower kinetic energy levels than those with only one source of dissipation. Furthermore, for those when both sources of dissipation are present, systems with lower values of Re T (and all other quantities fixed) dissipate energy quicker than their higher-re T counterparts due to the increased level of viscous dissipation. Similarly, the effect of ρ p /ρ g is exhibited in the right panel of Fig When ρ p /ρ g is decreased from 1500 to 800, the gas-solid system (DNS) becomes more dissipative and further from its granular counterpart (DEM). The effect of ρ p /ρ g, however, is small and not as significant as that of Re T in the range studied ( ). 129

148 KE/KE KE/KE (KE 0 /m) 1/2 t/d (KE 0 /m) 1/2 t/d Figure 7.4 Kinetic energy evolution prior to onsets of instability. Left: Gas-solid systems (DNS) with inelastic collisions (e=0.9) at Re T = 3 ( ) and 30 (+); gas-solid systems (DNS) with elastic collisions (e=1.0) at Re T = 3 ( ) and 30 ( ); a granular system (DEM) with inelastic collisions (e=0.9, solid line). For all systems, φ = 0.4 and ρ p /ρ g = Right: Gas-solid systems (DNS) ρ p /ρ g = 800( ) and 1500 (Δ). For both systems, Re T = 30, e = 0.9, and φ = 0.4. The granular (DEM) counterpart with the same e and φ (solid line) is also given. The onset of instabilities marked in Fig. 7.5, where diamonds and circles represent the onset of vortices and clusters, respectively, is a typical illustration of the sequence of instabilities in DEM and DNS: vortices are seen in all systems, followed by clusters (if observed). Furthermore, by obtaining the confidence intervals of the vortex onset points, it is shown that systems with dissipation arising from two sources (DNS), namely dissipative collisions and gas-phase viscosity, develop instabilities quicker than (DNS or DEM) simulations with only one of these mechanisms (with all other system parameters fixed). The onset points (diamonds) shown in Fig. 7.5 for systems with both dissipative collisions and gas phase viscosity were based on 10 DNS simulations. The dimensionless average onset time is 31.7 with a 90% confidence interval of ±8.5. For those systems with a single source of dissipation, the elastic gas-solid system shows a 130

149 dimensionless onset time of 102.3, clearly higher than 31.7; the granular systems (DEM) show a dimensionless average onset time of 44.9 with a 90% confidence interval of ± 2.8 (based on 150 granular DEM simulations, onset points not shown in Fig. 7.5). This observation again indicates that the onset of instability is directly related to levels of energy dissipation KE/KE (KE 0 /m) 1/2 t/d Figure 7.5 Average dimensionless onset times in systems with different dissipation mechanisms. The 10 solid lines correspond to the 10 gas-solid DNS simulations with both dissipative collisions and gas-phase viscosity (Re T = 3, φ = 0.2, e = 0.9, ρ p /ρ g = 1000). The average dimensionless onset time for the gas-solid DNS is marked by a vertical line at 31.7 (±8.5). The 9 dashed line shows 9 sample granular DEM simulations (φ = 0.2, e = 0.9), and the vertical line at 44.9 (±2.8) marks the average dimensionless onset time of 150 granular simulations. The ( ) symbols represent a gas-solid DNS without dissipative collisions (Re T = 3, φ = 0.2, e = 1.0, ρ p /ρ g = 1000), where the vertical line marks the onset time of The diamonds show the distribution of vortex onset times and the circles represent cluster formations. Perhaps the most interesting results are revealed when considering the evolution of the instabilities in DEM and DNS. As exemplified in Fig. 7.1, for all conditions, a 131

150 comparison between the simulations and the analytical solution for its stable counterpart indicates that the rate of energy dissipation is significantly reduced after the onset of instabilities (i.e., during evolution). Fig. 7.6 shows via DEM and DNS that, for many cases, a system with a relatively low kinetic energy exhibited at early times due to higher levels of energy dissipation can exhibit, at later times, kinetic energy levels equal to and then even higher than the kinetic energy levels of its lesser dissipative counterparts. The cases presented in Fig. 7.6 show that the resulting crossovers of kinetic energy levels in DEM and DNS are quite robust with respect to the various parameters of the simulations (Re T, e, and ρ p /ρ g ). These crossovers, though counterintuitive, are consistent with the following physical picture: the presence of instabilities significantly reduces the rate of energy dissipation, and thus energy dissipation after the onset of instability may become lower for systems with higher initial dissipations. More specifically, at the onset of the instability, velocity vortices develop. These vortices are associated with more aligned motions of particles i.e., collisions are more glancing in nature. For dissipation due to collisions, more glancing collisions results in lower energy dissipation since inelasticity is associated with normal component of relative velocity (i.e., head-on collisions are more dissipative than glancing collisions). Similarly, for viscous dissipation, although there is no general formula for the viscous dissipation for all particle-particle relative motions at all conditions, the particle-particle lubrication force models also predict that head-on motions are more dissipative than glancing ones at short range [26]. Thus, regardless of the source of dissipation, the onset of vortex motion leads to a reduction in the rate of kinetic energy dissipation. 132

151 KE/KE Re T =30,φ=0.4,e=0.9,ρ p /ρ f =1000 Re T =30,φ=0.4,e=1.0,ρ p /ρ f =1000 Granular: φ=0.4,e= (KE 0 /m) 1/2 t/d KE/KE Re Re T p /ρ f T =30,φ=0.4,e=0.9,ρ p /ρ f =1000 Re Re T p /ρ f T =30,φ=0.4,e=0.9,ρ p /ρ f = Re T =30,φ=0.4,e=0.9,ρ p /ρ f =1500 Re T =30,φ=0.4,e=0.9,ρ p /ρ f =1500 Granular: Granular: φ=0.4,e= (KE 0 /m) 1/2 t/d 10-1 Re T =30,φ=0.2,e=0.8,ρ p /ρ f =1000 Re T =10,φ=0.2,e=0.8,ρ p /ρ f = Granular: φ=0.2,e=0.8 Granular: φ=0.2,e=0.9 KE/KE KE/KE (KE 0 /m) 1/2 t/d (KE 0 /m) 1/2 t/d KE/KE KE/KE Re T =10,φ=0.2,e=0.9,ρ p /ρ f =1000 Re T =10,φ=0.2,e=0.8,ρ p /ρ f = (KE 0 /m) 1/2 t/d Re 10-4 T =30,φ=0.2,e=0.9,ρ p /ρ f =1000 Re T =30,φ=0.2,e=0.8,ρ p /ρ f = (KE 0 /m) 1/2 t/d Figure 7.6 Cases of kinetic energy level crossovers observed during evolution of the instabilities. (a) crossovers between gas-solid DNS with dissipative collisions and gassolid DNS with elastic collisions and granular DEM simulations; (b) crossovers among DNS with different ρ p /ρ g ; (c) crossovers between DNS with different Re T ; (d)-(f) crossovers between granular DEM and gas-solid DNS with different e. 133

152 In addition to the above study, where DEM and DNS methods are used as ideal computational data to provide insight on the behavior of instabilities in gas-solid flows, the predictions for the critical length scale for clustering from (hydrodynamic) TFM simulations are compared with DNS to quantitatively assess the ability of the TFM to capture the influence of the fluid phase. As outlined in Section 7.2.4, stability in TFM simulations is determined via the evolution of!" max = (" max #" min ) /". Specifically, a cutoff value of 1% is used. If!! max is greater than this cutoff value at the end of the simulation, the system is deemed unstable. TFM simulation duration was chosen to match corresponding DNS simulations. As shown in Figure 7.7, the behavior of!! max groups together in simulations with!! max greater than the cutoff. Additionally!! max increases Figure 7.7!! max = (! max "! min ) /! as a function of dimensionless time for TFM simulations with e=0.9,! s /! f = 1000, Re T =5, and! =0.2 of various system sizes. The cutoff value of 0.01 is shown as a dashed line. 134

153 at the end of these simulations. Figure 7.8 shows the velocity field for the L/d=14 TFM simulation of Figure 7.7 at t*=t 1/2 t/d=10 and indicates that velocity vortices and large gradients in velocity are present in the flow. Figure 7.8 Visualization of velocity field for L/d=14 at t*=10 from a TFM simulation with e=0.9,! s /! f = 1000, Re T =5, and! =0.2. Figure 7.9 shows the critical length scale for particle clustering in such a flow as predicted by DNS, TFM simulations (transient three-dimensional solutions of GTSH theory), and a linear stability analysis (LSA) [27] of the GTSH theory for the gas-solid HCS [18]. There is strong agreement between DNS and TFM simulations, while a significant discrepancy is observed for the linear stability analysis. Figure 7.10 shows the critical length scale for clustering as a function of thermal Reynolds number. As expected, the contribution of collisional dissipation is more significant for higher thermal Reynolds numbers where the fluid plays a smaller role. 135