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Collisional Granular Flows with and Without Gas Interactions in Microgravity

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Collisional Granular Flows with and Without Gas Interactions in Microgravity COLLISIONAL GRANULAR FLOWS WITH AND WITHOUT GAS INTERACTIONS IN MICROGRAVITY A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Ful¯llment of the Requirements for the Degree of Doctor of Philosophy by Haitao Xu August 2003 c 2003 Haitao Xu ° ALL RIGHTS RESERVED COLLISIONAL GRANULAR FLOWS WITH AND WITHOUT GAS INTERACTIONS IN MICROGRAVITY Haitao Xu, Ph.D. Cornell University 2003 We studied flows of agitated spherical grains in a gas. When the grains have large enough inertia and when collisions constitute the dominant mechanism for momentum transfer among them, the particle velocity distribution is determined by collisional rather than by hydrodynamic interactions, and the granular flow is governed by equations derived from the kinetic theory. To solve these equations, we determined boundary conditions for agitated grains at solid walls of practical interest by considering the change of momentum and fluctuation energy of the grains in collisions with the wall. Using these condi- tions, solutions of the governing equations captured granular flows in microgravity experiments and in event-driven simulations. We extended the theory to gas-particle flows with moderately large grain in- ertia, in which the viscous gas introduces an additional dissipation mechanism of particle fluctuation energy. When there is also a mean relative velocity between the gas and the particles, the gas gives rise to particle fluctuation energy and to mean drag. Solutions of the resulting equations compared well with Lattice- Boltzmann simulations at large to moderate Stokes numbers. However, theoretical predictions deviated from simulations at small Stokes number or at large particle Reynolds number. Using a method analogous to the integral treatment of laminar boundary lay- ers, we derived averaged equations to study the development of granular and gas- particle flows in rectangular channels. The corresponding predictions of the stream- wise evolution of averaged flow variables such as the mean particle velocity, the granular temperature, and the mean gas velocity agreed well with event-driven simulations. Finally, we used our analyses to prescribe microgravity experiments in which to test theories for gas-particle interactions with large to moderate particle inertia and small gas inertia. We also predicted uncertainties in measuring the granular mean and fluctuation velocities from a computer vision analysis of video images of the flow. BIOGRAPHICAL SKETCH The author was born in April, 1974, in Suizhou, China. He received his B.E. in 1993 from Huazhong University of Science and Technology, China. In August, 1998, he joined the Sibley School of Mechanical and Aerospace Engineering of Cornell University. He was enchanted by \gorgeous" Ithaca and decided to stay at Cornell as a postdoc after graduation. iii To my wife, For her consistent love, understanding and support iv ACKNOWLEDGEMENTS I am grateful to Professor Michel Louge, the chairman of my special committee, for his advice, patience and constant help. His ideas initiated this work and his accessibility encouraged numerous discussions. I am also grateful to him for his help and valuable advice on my professional and personal development and for teaching me writing skills. I am indebted to Professor James Jenkins and Professor Donald Koch, both served as members of my special committee, for their inspiring ideas and sugges- tions. I learned a lot from them. I would also like to thank Professor Anthony Reeves for providing us the image analysis algorithm, and Steve Keast for building the shear cells that we used in the experiments. Special thanks to Dr. Rolf Verberg, who generously provided us the result of his Lattice-Boltzmann simulations. Finally, I would like to thank all of my friends at Cornell. Their friendship made my stay here a pleasant and memorable experience. v TABLE OF CONTENTS 1 Introduction 1 2 Boundary Conditions for Collisional Granular Flows 11 2.1 Smooth Bumpy Boundaries . 13 2.2 Granular Flows of Disks interacting with Flat Frictional Walls . 23 2.3 Granular Flows of Disks interacting with Bumpy, Frictional Walls . 39 2.4 Three-Dimensional Flows of Spheres interacting with Bumpy, Fric- tional Walls . 51 3 Solutions of the Kinetic Theory for Bounded Collisional Granular Flows 55 3.1 Flow of Identical Spheres . 58 3.1.1 Conservation laws and constitutive relations . 58 3.1.2 Comparison with simulations and experiments . 61 3.2 Flow of Binary Mixtures . 76 3.2.1 Exact theory . 76 3.2.2 Simpli¯ed theory . 82 3.2.3 Comparison of theory, simulations and experiments . 85 3.3 Conclusion . 94 4 Flow of Collisional Grains in a Viscous Gas 96 4.1 Governing Equations . 97 4.2 Bounded Shear Flows . 106 4.2.1 One-Dimensional Rectilinear Flow . 106 4.2.2 Comparison with Lattice-Boltzmann Simulation . 108 4.2.3 E®ects of Stokes number . 111 4.2.4 Two-Dimensional Rectilinear Flow . 126 5 Flow Development of Bounded Collisional Granular Flows 136 5.1 Granular Flow in a Rectilinear Channel . 137 5.1.1 E®ects of flat side walls . 145 5.1.2 E®ects of section length . 149 5.2 Granular Flow in an Axisymmetric Shear Cell . 151 5.3 Granular Flow in a Shear Cell Shaped as a Race Track . 158 5.4 Gas-particle Flow in an Axisymmetric Shear Cell . 161 6 Measurement Errors in the Mean Velocity and Granular Temper- ature 170 6.1 Vision Algorithm . 171 6.2 Measurement Errors . 175 6.2.1 Imperfect Tracking . 175 6.2.2 Finite Pixel Size . 181 vi 6.2.3 Collisions . 185 6.2.4 Strip Statistics . 188 6.3 Tradeo®s . 191 7 Design of Experiments to Study Gas-particle Interaction in Mi- crogravity 195 7.1 Experiment Objectives . 196 7.1.1 Viscous dissipation experiments . 196 7.1.2 Viscous drag experiments . 200 7.2 Constraints on the design of experiments . 201 7.2.1 Particle inertia . 201 7.2.2 Gas inertia . 202 7.2.3 Duration of microgravity . 203 7.2.4 Quality of microgravity . 208 7.2.5 Sphere properties . 215 7.2.6 Continuum flow . 217 7.2.7 Accuracy of gas flow rate . 218 8 Conclusions and Recommendations 221 A Collision Integrals 225 B Approximations of Stresses and Heat Flux at Flat, Frictional Walls 227 C Determination of Granular Pressure 231 C.1 Iteration Scheme . 233 C.2 One Dimension . 233 C.3 Two Dimensions . 239 D Expressions in the Exact Mixture Theory 245 E Tentative Test Matrices 247 E.1 Viscous Dissipation Experiments . 247 E.2 Viscous Drag Experiments . 253 Bibliography 275 vii LIST OF FIGURES 1.1 Categories of fluid-particle flow. 2 2.1 Geometry of the bumpy boundary. 14 2.2 Stress ratio and heat flux at smooth, bumpy boundaries: compar- ison of the present study (solid lines) and the results of Richman & Chou [105] (dashed lines). Top plot: stress ratio. Bottom plot: dimensionless heat fluxes. The circles are simulation data with σ = d = 2; s = 0; the squares are σ = 2; d = 3:2; s = 0; ew = 0:96 in both cases. 21 2.3 Geometry of the flat boundary. 24 2.4 Distributions of the y-component of velocity. 31 2.5 E®ects of the particle velocity distribution on the stress ratio and flux of fluctuation energy at flat, frictional walls. Dotted lines: twin-± distribution; solid lines: Maxwellian distribution; dashed lines: Weibull distribution. Figures in the top row show stress ratios and heat fluxes with ¹ = 0:1 and 0:3, and ¯xed e = 0:9, and ¯0 = 0. Figures in the middle row correspond to e = 1 and 0:7, ¯xed ¹ = 0:3 and ¯0 = 0. Figures in the bottom row are for ¯0 = 0 and 0:4, ¯xed e = 0:9 and ¹ = 0:4. 34 2.6 Comparison of stress ratios and fluxes of fluctuation energy calcu- lated in the two limits of \all-sticking" and \all-sliding" with the exact calculation assuming a twin-± distribution. Solid lines: ex- act calculation; dashed lines: two-limit approximation. Figures in the top row show the stress ratio and heat flux corresponding to ¹ = 0:1 and 0:3 with ¯xed e = 0:9, and ¯0 = 0. Figures in the middle row correspond to e = 1 and 0:7, ¯xed ¹ = 0:3 and ¯0 = 0. Figures in the bottom row are for ¯0 = 0 and 0:4, ¯xed e = 0:9 and ¹ = 0:4. 38 2.7 Geometry of a bumpy, frictional boundary of half-cylinders in \pla- nar" granular flows. 40 2.8 Stress ratio S=N (top) and dimensionless flux of fluctuation energy Q=NpT (bottom) for planar flows of spheres at a bumpy, frictional boundary. The dashed and dotted lines represent the respective contributions of bumps and friction. The solid lines show the total S=N or Q=NpT . The boundary parameters are σ=d = 1, s=d = 1=2, e = 0:9, ¹ = 0:1, and ¯0 = 0. 49 2.9 Stress ratio S=N for 3-D flows of spheres at a bumpy, frictional boundary. Top plot: σ = 2, d = 3:2, s = 0; bottom plot: σ = d = 2, s = 0. Lines are the superposition Eq. (2.86), symbols are simulation results. Solid lines and squares: ¹w = 0:1; dashed lines and circles: ¹w = 0:0. 52 3.1 Fully developed, steady flows in a rectilinear cell. 62 viii 3.2 Mean and fluctuation velocities in the cross-section of a rectilinear cell. Top: mean velocity in x-direction made dimensionless with the velocity U of the top boundary. Bottom: fluctuation veloc- ity made dimensionless with U. The symbols and lines represent, respectively, the results of the simulations and the predictions of the theory. Solid lines and squares are y=H = 0; dashed lines and upward triangles, y=H = 1=3; dash-dotted lines and circles, y=H = 2=3; dotted lines and downward triangles, y=H = 1.
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