On the Role of Collisions in the Ejection of Lunar Regolith during Spacecraft Landing

Transcription

1 University of Colorado, Boulder CU Scholar Chemical & Biological Engineering Graduate Theses & Dissertations Chemical & Biological Engineering Spring On the Role of Collisions in the Ejection of Lunar Regolith during Spacecraft Landing Kyle Joseph Berger University of Colorado at Boulder, Follow this and additional works at: Part of the Chemical Engineering Commons Recommended Citation Berger, Kyle Joseph, "On the Role of Collisions in the Ejection of Lunar Regolith during Spacecraft Landing" (2016). 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 On the Role of Collisions in the Ejection of Lunar Regolith during Spacecraft Landing By Kyle Joseph Berger B.S., University of California Berkeley, 2010 A dissertation 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 2016

3 This thesis entitled: On the Role of Collisions in the Ejection of Lunar Regolith during Spacecraft Landing Written by Kyle Joseph Berger Has been approved for the Department of Chemical and Biological Engineering Christine M. Hrenya Alan W. Weimer Charles B. Musgrave Daniel J. Scheeres Philip T. Metzger December 2015 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 Berger, Kyle Joseph (Ph.D., Chemical and Biological Engineering) On the Role of Collisions in the Ejection of Lunar Regolith during Spacecraft Landing Dissertation Directed by Professor Christine M. Hrenya Gas-solid flows are ubiquitous in nature and industry and, despite being widely studied, are still not well understood. One example of such a flow is the spraying of lunar regolith (soil) from a rocket landing on the Moon. Such spray poses a danger to both equipment and personnel for future missions. Previous researchers, using models such as single particle trajectory models and direct simulation Monte Carlo (DSMC), have not directly studied the erosion from the surface or modeled the collisions directly. The goal of this work is to improve upon prior models by developing and validating a new model that can be used to design mitigation systems such that future missions will not be endangered. For this purpose, the discrete element method (DEM) is first used to examine the erosion from the surface and to probe the relevant erosion mechanisms in order to better understand the erosion process. This work is performed using a variety of particle size distributions (PSDs), including monodisperse, binary, and lognormal. The results show that collisions are crucial in correctly modeling both the near-field (surface erosion) and far-field (downstream) effects. However, the DEM model is too computationally expensive to be used for the entire lunar system. Thus, the erosion results from the DEM model are used in a kinetic-theory-based continuum model, similar to the Navier-Stokes model for traditional fluids, using a discretized PSD. Validation of this model is performed against Apollo data and shows discrepancies between the observed and predicted particle velocities. Further work is required to resolve this discrepancy, along with additional validation due to difficulty of obtaining experimental/field data for lunar conditions. iii

5 In addition, the validity of using a single-particle drag law in rarefied conditions is evaluated using the lattice-boltzmann method (LBM) to simulate periodic arrays of spheres. The results suggest that such an assumption may be valid if the Knudsen number (ratio of mean free path of the gas to particle diameter; used to measure rarefication) is sufficiently large, as it may be for the lunar case. However, additional work is needed to fully understand the implications of such an assumption and develop a multi-particle drag law that considers rarefication. iv

6 Dedication To my loving wife, Tina Berger v

7 Acknowledgements I am extremely grateful to those who have helped me as I performed this work. I would like to thank Anshu Anand, who began this work and set up the foundation for everything that I did. The undergraduates, Brendan Brown and Tony Neumayer, who taught me just as much as I taught them. Brendan in particular helped me immensely by showing me how I could do everything so much faster. I would also like to thank my mentors at NASA: Philip Metzger and James Mantovani of Kennedy Space Center and Rudranaryan Mukherjee of Jet Propulsion Laboratories. Phil is an amazing scientist and has taught me a great deal. His willingness to always lend a hand, combined with his immense knowledge, helped me through a lot of tough challenges. James stepped in to replace Phil as our contact for this project when Phil left NASA and helped ensure a smooth transition, along with good insight into the project. Rudra has been instrumental in my success as a researcher. He always knows how to keep me on my toes and pushed me to be my best. His mentorship played a large part in helping me to secure a position after completing my degree. I have also had the pleasure of collaborating with Xiaolong Yin and Lei Wang at the Colorado School of Mines. With their help, I was able to add a valuable chapter to this work and learn a great deal. Xiaolong was always willing to make the time to help me with concepts I knew little about and ensure our project was a success. I m also thankful for all of the researchers within the Hrenya Research Group who have come and gone. Carly Donahue, Aaron Murray, Peter Mitrano, Aaron Morris, Casey LaMarche, vi

8 William Fullmer, Kevin Kellog, Peiyuan Liu, and Aaron Lattanzi have all provided valuable insight into various stages of this project. They have also taught me a great deal about what it means to be a good researcher and I thank you all. I am pleased to have the opportunity to have a lot of great teachers and professors over the years. I would like to thank Kathy Spurrel and Cindy Avol, who helped me on my current path by teaching me science and math in middle school. Tena Bushart, who taught me math in high school and also sponsored the Math Club, was truly amazing. She gave us her time and knowledge and I am thankful for that. I want to give special thanks to Charlotte Shales-Clark, my chemistry teacher in high school. She helped to push me along my path to chemical engineering with her passion and drive. She was always ready to provide her knowledge or advice on anything and everything. Finally, I greatly respect Professor Clayton Radke from Cal. He is an amazing professor and taught me much of what I know about chemical engineering. He always challenged me to be my best and work my hardest. My mother and father always pushed me to be my best in school and put my studies ahead of everything else. They made sure I became the person I am today. I also want to thank my brothers, stepmom, stepdad, and all my other family for supporting me and putting up with me for many years. I know I have not always been the easiest person to be around. A special thanks to my chemical engineering buddy from Cal, JP Cardinal. JP has always been a good friend to me and has taught me a lot both academically and personally. I want to thank my loving wife, Tina, and my children Alaka i, Aolani, and Liam. They have all given me the support and love that got me through each and every day. Even when things vii

9 were difficult, I could always count on Tina to be there for me. She is my partner in crime, my gaming buddy, my food critic, and my shoulder to cry on. I don t think I could have done this without her. Finally, I want to show my thanks to Christine Hrenya for leading me through this tumultuous journey. I remember first talking with her during my recruiting visit and I knew I wanted to work on the NASA project. Even though I wasn t initially selected for a project (and she wasn t even offering the NASA project), she helped make sure that I got the chance to do what I wanted. I want to thank her for always being ready to help however she could and keeping me in check. She gave me the freedom to pursue developing my own software, which led to me further developing my software development skills and ultimately led to an amazing job. Christine s dedication has always inspired me to do better and she has helped me to realize my full potential. I will never regret choosing to work with her. viii

10 Table of Contents 1. Introduction Motivation Modeling Approaches Previous Work System Description Near-field Far-field Gas model Dissertation Objectives Near-field: Impact of inter-particle contacts on monodisperse erosion (Chapter 2; [44, 45]) Near-field: Impact of inter-particle contacts on polydisperse erosion (Chapter 3; [46, 47]) Accuracy of single-particle drag force in rarefied conditions (Chapter 4; [48]) Accuracy of discretizing continuous PSD (Chapter 5; [47]) Far-field: Impact of inter-particle contacts on far-field dynamics (Chapter 6; [46]) Validation of continuum model via DEM (Chapter 6) Validation of continuum model via Apollo data (Chapter 7) References Near-field: Erosion of Monodisperse Solids Introduction Computational Model Base case parameters Results and Discussion Base case: Impact of contacts on erosion Collision-less DEM Effect of Input Parameters Summary References Near-field: Erosion of Polydisperse Solids Introduction Methods Model Inputs Continuous Size Distributions Results and Discussion Binary Case Erosion Mechanisms Lognormal PSD Conclusions References Accuracy of Single-Particle Drag force Introduction ix

11 4.2 Methods Results and discussion Stokes flow over periodic arrays of spheres Relation between surface boundary-slip and drag reduction in periodic cells Small, Finite Re flow over periodic arrays of spheres Conclusions References Accuracy of Discretizing Continuous Particle Size Distribution Introduction Methods Results and Discussion Erosion Flux Comparison Validation of Volumetric Discretization Conclusions References Effect of Inter-particle Contacts on Far-field Dynamics and Validation of Continuum Model via DEM Introduction Methods Far-field DEM model Continuum Model Results and Discussion Far-field DEM Continuum model validation via DEM Conclusions References Validation of Continuum Model via Apollo (Far-field) Data Introduction Methods Continuum Model Simulation Procedure Results and Discussion Conclusions References Summary and future work Summary Recommendations for future work References Appendix A x

12 List of Figures Figure 1.1 Left: Image of lunar surface from Apollo 15 lander prior to plume impingement on surface. Right: Same image 1 second later [8] Figure 1.2 Lognormal (99% of particles), coal, and lunar soil PSDs Figure 1.3 Depictions of modeling approaches used: (left) Lattice Boltzmann method (LBM), (middle) discrete element method (DEM), and (right) continuum model. Red arrows denote gas velocities and blue denote solids velocities Figure 1.4 Schematic of the overall system Figure 1.5 Schematic of the near-field simulations performed via DEM: two-dimensional representation of a three-dimensional system Figure 1.6 Plume velocity profile as calculated from CFD Figure 2.1 Total eroded volume for base case parameters. Dotted line denotes the average erosion during the non-depleted phase. The dashed vertical line demarcates the depleted and nondepleted phases. The solid vertical line demarcates the erosion-only and erosion-andsedimentation phases. These lines are repeated in later plots with the same meaning Figure 2.2 Average fractional collision number for base case vertical lines represent beginning and end of steady erosion Figure 2.3 Cumulative erosion volume for dissipative and non-dissipative collisions, along with the case of no collisions Figure 2.4 Cumulative erosion number for different anchoring plane depths. Penetration depths are 2, 4 (base case), and 6 particle diameters. All other parameters are given in Table Figure 2.5 Cumulative erosion number for different erosion plane heights Figure 2.6 Sensitivity of cumulative erosion volume to gravitational acceleration. Values used: 0, 1.63, 3, 5, 9.8, 15 m/s Figure 2.7 Sensitivity of cumulative erosion volume to seed used for initial particle configuration. Seed 1 refers to base case (Figure 2.1); parameters used for all seeds given in Table Figure 2.8 Cumulative erosion volume per box area versus time for different periodic domain sizes. Note that the number of particles in each simulation is scaled based on the area (linearly compared to base case parameters). All other parameters for all cases are shown in Table Figure 2.9 Cumulative Erosion number for two different initial bed depths (number of particles). All other parameters are shown in Table Figure 2.10 Cumulative erosion number versus time for different spring coefficients. Spring coefficients depicted are 8x10 2, (base case), and kg/s Figure 2.11 Cumulative erosion number versus time for different coefficients of restitution and friction coefficient. (a) enorm = 0.6, (b) enorm = 0.8. Friction coefficients displayed are 0.2 (base case), 0.5, 0.8, and Figure 2.12 Monodisperse: Effect of particle size and particle-particle contacts on erosions flux (90% confidence intervals) Figure 3.1 Lognormal and lunar soil distribution [24] xi

13 Figure 3.2. Binary: Total erosion flux, along with non-interacting mixture approximation (90% CI) Figure 3.3. Binary: Species erosion flux of large (solid lines) and small (dashed lines) particles normalized against non-interacting mixture approximation (90% CI) Figure 3.4. Average fractional collision number for binary and monodisperse cases with same Sauter-mean diameter (size ratio 4; 90% CI) Figure 3.5. Binary: Net rate of vertical momentum transfer from small to large particles during steady erosion (size ratio 4; 90% CI) Figure 3.6. Binary: Net rate of vertical momentum transfer across the erosion plane during steady erosion relative to the monodisperse case (size ratio 4; 90% CI) Figure 3.7. Schematic showing change in lift force (a) before collision, (b) during collision, and (c) after collision due to horizontal momentum transfer. Solid arrow denotes horizontal particle velocity, thick red arrow represents lift force, and dashed arrow represents transfer of momentum Figure 3.8. Binary: Change in erosion flux with respect to the expected non-interacting mixture for the base case (black diamond) and case with fixed anchoring/erosion planes (red square). Size ratio = 4 and 95% CI Figure 3.9. Total erosion flux of lognormal distribution as a function of distribution width. Error bars are 90% confidence intervals (CI) over multiple runs Figure Ratio of family erosion flux to non-interacting erosion flux for σ/µ (90% CI) Figure 4.1 Spatial configuration for a simple cubic lattice with a L= 60 with εs= Figure 4.2. Dimensionless drag vs. Sl for very small Re as a function of solids fraction for TMAC of 0.5, along with the Cunningham correction factor [20] and the Loth correlations [21]. The numbers along the vertical axis (Sl = 0) are in good agreement with Sangani and Acrivos [10] Figure 4.3. The forcing and fluid dissipation terms as a function of Knudsen number and TMAC (σ) for L = 120 and a solids volume fraction of (a) 0.1 and (b) 0.4. The difference between the terms is the boundary-slip term Figure 4.4. Example of dimensionless drag vs Re for varying Sl ( s 0.4, SC array). Points are 2 LBM data and dashed lines are fits to the curve F F0 F1Re Figure 4.5. Fit of equation (4.25) to dimensionless drag vs Re for varying Sl (εs=0.4, SC array) Figure 4.6. Fitting parameters for equation (4.25) Figure 5.1. Ratio of number fraction to particle size for S=10 discretizations for both discretization methods and number-based probability distribution functions for truncated and true lognormal PSD Figure 5.2. Erosion flux of discretized and continuous PSD for method of matching moments (90% CI) Figure 5.3. Erosion flux of discretized and continuous PSD for volumetric discretization (90% CI) Figure 5.4. Dimensionless transport coefficients for the two discretization methods and simulation data xii

14 Figure 6.1 Computational domain used for far-field simulations: top-down view of rectangular domain (solid red and dash-dot red) relative to actual radial domain (dashed black). Dotted red line is periodic wall from near-field DEM that is removed for far-field DEM. See text for more details Figure 6.2. Far-field simulations for binary mixtures: Particle positions for (a) with contacts and (b) without contacts at 0.015s. Initial simulation domain is in the bottom left corner (50 wt% large; size ratio 4) Figure 6.3. Far-field simulations for monodisperse system: Particle positions for (a) with contacts and (b) without contacts at 0.015s. Initial simulation domain is in the bottom left corner (31µm particles) Figure 6.4 Monodisperse: Continuum vs. DEM results for horizontal particle velocity Figure 6.5 Binary: Continuum vs. DEM results for gas volume fraction Figure 6.6 Binary: Continuum vs. DEM results for large particle horizontal velocity Figure 6.7 Discretized Continuous PSD: average horizontal velocity Figure 6.8 Discretized Continuous PSD: average vertical velocity Figure 7.1 Fitting parameters for the horizontal extrapolation of plume data: (a) a, (b) b, and (c) c Figure 7.2 OB-1 distribution and volumetric discretization with 6 species Figure 7.3 Continuum model domain for far-field simulations. See text for details of boundary conditions Figure 7.4 Results of continuum simulation after 10 seconds. (a) total bulk density, (b) total horizontal flux, (c) total vertical flux, (d) horizontal flux of 40.4µm particles, (e) horizontal flux of 119.4µm particles, and (f) horizontal flux of 265.2µm particles. All data is scaled logarithmically for clarity, except vertical flux Figure 7.5 Horizontal particle velocity at 10 seconds. Data is scaled logarithmically for clarity xiii

15 1. INTRODUCTION 1.1 Motivation Gas-solid flows are prevalent in both nature and industry. Examples include fluidized beds, pneumatic transport, cyclone separation, avalanches, landslides, and many more. Another example, which is the focus of this work, is the landing of a rocket on a dusty surface. On the Moon, when an active rocket descends, the exhaust plume causes the soil-like material, known as lunar regolith, to be ejected in all directions. Similar phenomena can occur on other surfaces such as Mars or asteroids. This ejection poses problems for the future of space exploration. The Apollo 12 landed near the deactivated Surveyor III spacecraft (~160km away), which was completely sandblasted by the landing. Based on an analysis of the recovered pieces of the Surveyor III, scouring, pitting, and dust impregnation of nearby hardware during landing is a serious concern for future missions [1-4]. In addition, rough estimates of particle trajectories suggest that particles may travel very large distances and possibly reach escape velocities, which could pose threats to more distant equipment [2, 5]. Finally, the ejected regolith causes an immediate threat to the crew due to reduced visibility and a spoofing of landing sensors caused by particles colliding with the landing sensors of the spacecraft [6, 7]. The extent of the reduction of visibility can be seen in Figure 1.1, which shows two camera views on the Apollo 15 within one second of each other. 1

16 Figure 1.1 Left: Image of lunar surface from Apollo 15 lander prior to plume impingement on surface. Right: Same image 1 second later [8]. Granular flows, such as that of lunar regolith, are further complicated by the presence of multiple sizes, otherwise known as polydisperse granular flows. Such flows exhibit behaviors that have no counterpart in traditional (molecular) fluids such as size segregation [9-13] and momentum transfer between species [14, 15]. Lunar regolith has an extremely wide particle size distribution (PSD), spanning at least 3 orders of magnitude, which leads to interesting physics such as strong cohesive forces for smaller particles and changes in bulk flow properties (e.g., stress) [16-21]. Figure 1.2 shows the PSD of lunar soil, along with a lognormal distribution with a width of 100% (ratio of standard deviation, σ, to mean, µ) and a coal distribution from a gasifier. Note that the lognormal distribution, which is theoretically infinite, is truncated to include 99% of particles for clarity. Clearly, the PSD of lunar soil is extremely wide. In addition to complexities associated with the flow, a wide distribution also presents unique challenges for modeling [22]. Namely, a wider PSD with the same mean results in a larger computational cost compared to a narrower PSD. 2

17 Figure 1.2 Lognormal (99% of particles), coal, and lunar soil PSDs. 1.2 Modeling Approaches Experiments for lunar regolith ejection are difficult and costly [2, 23]. In addition, the physics are nearly impossible to replicate in terrestrial conditions. To overcome these challenges, computational modeling is a good alternative. The modeling approaches used in this work are the Lattice-Boltzmann method (LBM) [24], discrete element method (DEM) [25], and continuum theory [26], as illustrated in Figure

18 Figure 1.3 Depictions of modeling approaches used: (left) Lattice Boltzmann method (LBM), (middle) discrete element method (DEM), and (right) continuum model. Red arrows denote gas velocities and blue denote solids velocities. LBM is a type of direct numerical simulation that is equivalent to solving the gas phase on a grid much smaller than the particle size. As a result, the no slip (or slip) boundary condition is resolved directly at the particle surface. The gas-phase solution in LBM is equivalent to solving the Navier-Stokes equations: ( g g ) ( g g g ) 0 t v (1.1) D 2 gg vg gp g g gg (1.2) Dt where εg is the gas volume fraction, ρg is the gas density, vg is the gas velocity, p is the gas pressure, and τg is the gas fluid viscous stress tensor. Note that LBM can also be used to model a liquid, but this work is focused on gases. The solid particles in LBM are described using Newton s laws and appropriate collision (force) laws. A noteworthy aspect of LBM is that gas-solid drag is an output 4

19 of the model since the detailed nature of the gas flow is resolved at the particle surface (see Figure 1.3). DEM is a Lagrangian method that simulates particles using Newton s laws, just as LBM does [25]. Namely, force balances on each particle are solved simultaneously: ma mg F F (1.3) coll gs where m is the particle mass, a is the particle acceleration, g is the acceleration due to gravity, Fcoll is the force due to inter-particle contacts, and Fgs is the gas-solid interaction force (drag and/or lift). The key difference between DEM and LBM is the treatment of the gas. The gas flow is no longer resolved at the particle surface and is instead solved on a grid that is larger than the particle size (see Figure 1.3) and typically includes several particles. As a result, the drag force can no longer be extracted from the simulation and instead appears as an additional term in the momentum balance. Specifically, the momentum balance now takes the form: D (1.4) N 2 g g vg gp g g gg ns, ifgs Dt i1 where ns,i is the solids number density of solids species i, and N is the number of solids species. This gas-solid interaction force is determined based on a chosen drag and/or lift law, which can be determined via experiments or LBM simulations. The continuum approach uses the same methods to solve for the gas flow as DEM. However, the solid phase is treated as a continuum rather than as discrete particles (see Figure 1.3). Specifically, Navier-Stokes-like balances over mass (via species number density, ni), 5

20 momentum (via species velocity Ui), and granular energy (via the granular temperature Ti, which is a measure of the kinetic energy of fluctuations) of the solid phase take the form: nm i i t ( nmu ) 0 (1.5) i i i N Ui nimi U i U i P i ni F i, ext, cons F Dip (1.6) t p1 N 3 Ti n i U i T i q i P i U i i U i F Dip (1.7) 2 t p1 where the subscript i refers to species i, P is the stress tensor, Fi,ext,cons refers to external forces (weight, drag, lift, etc.), q is the flux of granular energy, γ refers to the collisional dissipation rate of granular energy, and FDip is the collisional source of momentum (between unlike species). The constitutive relations for P, q, γ, Fi,ext,cons, and FDip are obtained via the kinetic-theory analogy; detailed expressions can be found in [26]. These continuum equations can be used to simulate the flow of particles with relatively low computational cost compared to DEM or LBM models, but the drawback is the need for more constitutive relations. In addition, a continuous PSD is impractical in the framework of the continuum model outlined above since the number of balances scales with the number of species. Thus, a method for approximation of such a PSD must be used, such as that developed by Murray et al. [27]. 6

21 1.3 Previous Work A classic example of particle erosion due to gas motion is wind-blown sand and corresponding saltation. Saltation refers to a process in which particles are entrained by the gas via stochastic rolling and hopping and then return to the surface, colliding with other particles and causing them to be ejected into the gas flow. These collisions at the surface are known as saltating (or saltation) splashes. This mechanism is dominant over lift forces as typically lift forces are too small for particles to be lifted under terrestrial conditions. Past work on terrestrial systems involving windblown sand and its saltation have also indicated that mid-air collisions play a large role in the particle dynamics for narrow PSDs [28-32]. Namely, mid-air collisions increase the likelihood of a particle returning to the surface to cause a saltating splash. Before describing previous work related to lunar regolith ejection, it is helpful to define a few key terms. Surface erosion, or just erosion, refers to the process of particles being lifted from the regolith surface by the gas plume. The rate at which this happens is known as the erosion rate and the corresponding flux of particles from the surface is the erosion flux. Ejection dynamics refers to what happens to the regolith particles after they leave the surface. The ejection process on the Moon caused by rocket exhaust is different from analogous terrestrial gas-solid flows in several ways. First, the gas plume from the rocket is characterized by high-knudsen and high-mach number conditions. Also, in the high-velocity/high-shear of a rocket exhaust, aerodynamics (lift and drag) dominate over gravity within the locality of the lander where the erosion is taking place. Thus, particles generally do not fall back to the surface in the immediate vicinity and saltation does not occur at the surface. In addition, particles in the lunar case are easily 7

22 lifted off the surface by the strong plume even in the absence of saltating splashes to mechanically eject them from the surface. This case in which aerodynamic forces dominate over gravity occurs not only with rocket exhaust, but also in cases with sufficiently strong subsonic impinging jets forming scour-holes in a granular material, which is relevant to some terrestrial processes (i.e. scour holes forming under a culvert discharging water, or in a pneumatic excavation using gas jets) [2]. In addition, this case of aerodynamic dominance is different than the most commonly studied case of Aeolian sand transport where gravity and saltation dominate [28-31, 33]. The lack of atmosphere on the Moon also results in an absence of drag and lift once the particles leave the influence of the gas plume. Gravity is also reduced compared to a flow in terrestrial conditions. Finally, the particle size distribution (PSD) is much wider than soil found on Earth (see Figure 1.2). Previous work on predicting the surface erosion and ejection dynamics of lunar regolith relies largely on single-particle trajectory models [34-37], coupled with computational fluid dynamics (CFD) for the gas and a simple erosion rate model for particles at the surface [2]. This combined treatment is known as the Plume Erosion Trajectory (PET) model. First, the gas plume (the P of PET) is simulated using CFD and the resulting shear stress on the surface of the regolith is calculated. The erosion rate is then calculated using dimensionless scaling laws (the E of PET) to scale erosion rate based on the shear stresses calculated (using the CFD model above) and the erosion rates observed and shear stresses estimated for the Apollo landings. The trajectories of different size particles (the T of PET) from each point of erosion are then calculated one at a time. For each particle eroding from the surface, its acceleration and velocity are found by integrating its aerodynamic (drag and lift) and gravitational forces over the (CFD) gas-flow field under the 8

23 assumption that no particle-particle contacts are occurring. The ejection results can then be calculated by scaling the flux at a given point by the erosion rate of a given particle size. Nonetheless, previous investigations into the erosion phenomenon have suggested that particleparticle collisions, which are neglected in single-trajectory models, may play a role in erosion and ejection dynamics. For example, both experiments and examination of Apollo landing videos [23] suggest that collisions are frequent and significantly affect both the erosion flux and ejection dynamics. Furthermore, studies of pieces of the Surveyor III returned from the Moon using scanning electron microscopy and energy dispersive X-ray spectroscopy have concluded that interparticle collisions were responsible for the Surveyor being bombarded by regolith [1]. More specifically, the Surveyor was shielded by the lunar terrain, and thus only particles knocked below the main regolith spray by collisions would have had an opportunity to collide with the Surveyor. In addition, the authors show evidence of clusters of particles colliding with the Surveyor, a wellknown granular phenomenon which can only be predicted by models that include inter-particle collisions [38, 39]. Even in the case of terrestrial erosion dominated by saltation and gravity, interparticle mid-air collisions have been shown to have a significant effect on the erosion [32]. Unlike the PET methodology, in the work of Morris et al. [40], contacts directly above the lunar surface were modeled using a two-phase direct simulation Monte-Carlo method (DSMC) and a polydisperse mixture. They found that contacts between eroded particles do play a significant role in the far-field dynamics. The nature of these contacts was not investigated and thus it is unclear if the effects of particle contacts seen were due to polydispersity or if a monodisperse system would show similar results. Moreover, the DSMC method and techniques is restricted to fine particles (~<10 µm), which represent about 10% of the regolith mass; larger particles were not 9

24 simulated. However, the larger particles may pose a bigger threat to nearby hardware due to danger of pitting and/or puncture of equipment as observed on the Surveyor spacecraft [1]. In addition, without the presence of larger particles, the momentum transfer from the smallest particles to the largest are not considered, which may have a non-negligible effect on the small-particle trajectories. Finally, inherent in DSMC simulations is an assumption of probabilistic collisions. Accordingly, an improved model that includes both large and small particles and does not utilize probabilistic collisions is needed to probe these unanswered questions and unknown effects. Here, we seek to lay the foundation for a model that will simulate a wide size distribution using physically-resolved particle collisions, and one that can be applied to simulate erosion at the lunar surface and beyond. 1.4 System Description Figure 1.4 shows a schematic of the overall system. The impingement point is the point on the lunar surface directly below the landing rocket (along the dotted line in figure). The gas plume first contacts the regolith in the vertical direction at the impingement point as seen in Figure 1.4 and then the gas plume spreads outward in the radial direction. The investigation of this system will be divided into two parts: near-field and far-field. Near-field investigations involve studying the erosion flux at the surface, as well as inter-particle collisions immediately above the surface in a relatively small domain assumed to be located at a single, fixed distance. Far-field investigations involve studying the regolith particles as they travel downstream from the rocket after eroding. 10

25 Figure 1.4 Schematic of the overall system Near-field For the near-field simulations, performed using DEM, the computational domain is a small semi-periodic box at a fixed distance from impingement. Figure 1.5 shows the computational domain, which is periodic in the horizontal x- and z-directions, and semi-infinite in the vertical y- direction. The predominant direction of the gas plume is radially outwards from the impingement point, i.e., in the positive x-direction. The simplification of using a rectangular box, as opposed to a radial wedge, is justified since the horizontal dimensions (order of millimeters) are much smaller than the circumference of a circle with a radius equal to the distance from impingement. The domain contains a collection of spherical particles, which differs from actual lunar soil but is used here as a first approximation due to the significant added complexity of contact detection for nonspherical particles [41]. 11

26 Figure 1.5 Schematic of the near-field simulations performed via DEM: two-dimensional representation of a three-dimensional system. To characterize the erosion process as a function of time, two planes are defined in the computational domain relative to the regolith surface, as displayed in Figure 1.5. The regolith surface is initially defined as the location of the highest particle prior to erosion for simplicity. The anchoring plane, which is a distance hpenetration below the regolith surface, represents the (assumed) depth to which the gas plume penetrates the regolith layer. The assumption of a penetrating depth is a practical consideration in this simplified model to ensure gas forces are adequate to lift particles from the bed in the absence of saltating splashes, which is the case of interest observed in typical experiments [2, 37]. The penetration represents the grain-scale details of the gas flow under and around the individual particles on the surface of the bed as well as the time-varying turbulent interactions with the bed, details that are omitted for simplicity in this model. The anchoring plane divides the regolith into two parts: the perturbed regolith and unperturbed regolith. The particles located above the anchoring plane are under the direct influence of the gas plume and thus form the perturbed regolith layer. The particles located under the anchoring plane do not experience the 12

27 effects of the gas plume. The erosion plane, which is located a distance herosion above the regolith surface, represents the position at which the erosion flux is calculated, and thus is positioned a small distance above the surface of the regolith. The positions of both the anchoring and erosion planes move as the simulation progresses. Specifically, the volumetric erosion flux (i.e., the rate at which particles leave the regolith layer for the eroded layer per unit area) is calculated at every time step. The anchoring plane is then adjusted in the negative y direction until it sweeps through a volume that corresponds to the volumetric loss of particles in the previous time step, including the void space between particles in the regolith layer. In this manner, the height of the erosion plane above the surface of the regolith layer remains fixed throughout the simulation (i.e., the regolith surface and anchoring plane both move down at the same rate). The (adjusted) volumetric erosion flux also feeds into the new position of the anchoring plane. Namely, the anchoring plane is moved downward by the same distance as the erosion plane in each time step, thereby ensuring a constant penetration depth of the anchoring plane into the regolith layer throughout the simulation. Accordingly, new (deeper) layers of regolith are affected by the gas plume throughout the course of the simulation Far-field For the far-field simulations, which will be performed using both DEM and continuum frameworks, the computational domain spans large distances in the direction of the plume flow: on the order of meters for the DEM, and 100 s of meters for the continuum model. The far-field simulations are targeted at understanding the flow after particles have eroded. For far-field DEM, the simulation is initialized with an already eroded near-field simulation. In the case of the 13

28 continuum model, the erosion flux is used as a boundary condition in order to model the introduction of solids into the system. More details will be provided in later chapters Gas model The flow of the rocket exhaust plume was previously modeled by a NASA researcher using the computational fluid dynamics package Fluent. The CFD was performed in two dimensions (axi-symmetric) using the Spalart-Allmaras turbulent model (Re-Normalization Group k-ε model was also used for comparison). The parameters used for the CFD can be found in Table 1.1. The plume profile determined by the CFD simulations is then used as an input to the particle simulations, resulting in a one-way coupling between the particles and gas. This assumption is necessary because minimal work has been done in the regime of particles in rarefied flows. Figure 1.6 gives the variation of the radial plume velocity as a function of distance from impingement and distance from the surface of the regolith as calculated from the CFD analysis. Table 1.1 Parameters for CFD simulations of gas plume. Property Specific Heat (Constant Pressure) Thermal Conductivity Dynamic Viscosity Molecular Weight Ambient Pressure Value 1610 J/K kg W/m K Pa s 20.8 g/mole 0.1 Pa 14

29 Figure 1.6 Plume velocity profile as calculated from CFD Gas-particle interaction model: Drag and lift force For both near-field and far-field simulations, the same model of the gas, described above, will be used. The difference between the near- and far-field is the near-field treats the entire domain as a single distance from impingement and thus the plume will only vary in the vertical, y-direction, while the far-field plume varies in both the horizontal, x-direction, and vertical, y-direction. The force of the plume on regolith, namely Fgs in equation (1.3) and Fi,ext,cons in equation (1.6), is calculated as the sum of a drag and a lift force, Fdrag and Flift respectively. These forces are calculated using the drag and lift coefficients, respectively, developed by Loth [42, 43]. The drag force is calculated in each direction depending on the relative velocity in that direction (i.e., difference between particle and gas plume velocity) and the lift force is assumed to act in the y direction for simplicity (perpendicular to the gas plume velocity). The expressions of Loth were 15

30 obtained using theoretical predictions and empirical corrections for an isolated, spherical particle (i.e., do not take into account the presence of surrounding particles). Before introducing the detailed expressions provided by Loth [42, 43], it is useful to discuss the range of validity of these expressions. The drag coefficient is valid for particle Reynolds number up to 2.0 x The particle Reynolds number, Rep, is defined as Re p vd 0 (8) where ρ is the density of the fluid phase, v0 is the magnitude of the relative velocity between the particle and the gas plume, d is the diameter of the particle, and µ is the viscosity of the fluid phase. Typical particle Reynolds numbers present in our simulations are around 1-10, which is within the range of validity for this correlation. The drag coefficient also depends on the particle Mach number and is valid up to Mach numbers of at least 10. The particle Mach number, Ma, is defined as Ma v 0 (9) RT where γ is the ratio of specific heats of the gas, R is the gas constant, and T is the temperature of the gas. The particle Mach numbers present in our simulations are no more than about 5 and therefore within the valid range. In addition, the drag coefficient depends on the particle Knudsen number, Kn, which is defined as 16

31 Ma Kn (10) 2 Re The typical particle Knudsen number for our simulations is 1-10, which is below the maximum valid Knudsen number of 100. Similar to the drag coefficient, the lift coefficient of Loth [42] also depends on several dimensionless groups, namely the particle Reynolds number and the continuous-phase vorticity,, and is valid for values up to 50 and 0.8, respectively. The continuous-phase vorticity is * shear defined as Dv d * 0 shear (11) v0 where (Dv0) is the gradient of the plume velocity perpendicular to v0 (taken to be in the vertical direction for convenience). Typical values for the continuous-phase vorticity in our simulations are , which is within the bounds for the lift correlations. The drag force and lift force on the particle are expressed as a function of the drag and lift coefficients, CDrag and CLift, and the relative unhindered velocity, v0, as follows F Drag 2 2 CDrag d gv0 xˆ v 0 ˆ 8 x v0 yˆv CDrag d gv0 C 0 Drag d gv0 zˆv 0 ˆ ˆ 8 y 8 z v v 0 0 (12) 17

32 2 2 CLift d gv F 0 ˆ Lift y (13) 8 where g is the plume (gas) density and xˆ, yˆ, and z ˆ are the unit normal vectors pointing in the x, y, and z directions. The relative unhindered velocity of the particle is the velocity of the particle relative to the plume and is given as v v v (14) 0 plume particle where vparticle is the velocity of the particle and vplume is the velocity of the plume. The density of the plume is treated as a constant in the near-field DEM simulations because it does not change significantly within the simulation domain. For the far-field simulations, changes in density are accounted for based on the CFD results. The detailed expressions for the drag and lift coefficients, CD and CL, as functions of the Reynolds, Knudsen, and Mach numbers, can be found in the Appendix A. 1.5 Dissertation Objectives The overall goal of this work is to understand and predict the ejection of lunar regolith during spacecraft landing and develop a tool that can be used to design systems to mitigate the inherent challenges associated with the ejection. To accomplish this goal, the following objectives were addressed, as detailed in following chapters. 18

33 1.5.1 Near-field: Impact of inter-particle contacts on monodisperse erosion (Chapter 2; [44, 45]) DEM is used to examine a system with a monodisperse distribution of particle sizes. The goal of this study is to determine the effect of inter-particle collisions on the erosion flux for the case of a monodisperse distribution. The results show that inter-particle collisions below the surface tend to increase the erosion flux relative to a case without collisions due to the momentum transfer between particles making it easier for particles to erode. However, collisions above the surface are relatively rare and have a minimal impact on the erosion flux. The variation of the erosion flux with a variety of input parameters is also examined Near-field: Impact of inter-particle contacts on polydisperse erosion (Chapter 3; [46, 47]) DEM is used to examine a system with a binary and a continuous PSD. The goal of this study is to determine whether the introduction of unlike particles will have an influence on the erosion flux and/or inter-particle interactions. In addition, the erosion mechanisms relating to a binary PSD are examined in detail Accuracy of single-particle drag force in rarefied conditions (Chapter 4; [48]) In the DEM and continuum model studies, a single-particle drag force is used because no multi-particle drag force laws exist for rarefied conditions. In order to understand the impact of the assumption of a single-particle drag force, LBM is used to probe the drag on spherical arrays of particles in the slip flow, or slightly rarefied, regime. The goal of this study is to determine if the assumption of the single-particle drag force may be reasonable. The case of low and finite 19

34 Reynolds numbers (<10) are both explored, along with an analysis of the low Reynolds number case using a momentum balance Accuracy of discretizing continuous PSD (Chapter 5; [47]) A necessary step to using the desired continuum model is to discretize the continuous lunar soil PSD. In order to determine an accurate discretization method, the DEM model is used to compare the erosion of a discretized PSD to the corresponding continuous PSD. A new method for the discretization of continuous PSDs is developed and related to that of Murray et al. [27]. The comparison is performed for the lunar erosion case. In addition, this new method is validated by comparing the continuum model transport coefficients calculated from the discretized PSD to hard-sphere DEM simulations of the continuous PSD, the same method used by Murray et al. [27] Far-field: Impact of inter-particle contacts on far-field dynamics (Chapter 6; [46]) By initializing a far-field DEM simulation with the data from an eroded near-field simulation, the far-field dynamics are probed in an attempt to confirm inter-particle collisions are important to the far-field dynamics, as has been suggested by previous work [1, 5, 37, 40]. This study is performed for both monodisperse and binary PSDs Validation of continuum model via DEM (Chapter 6) Prior to using the continuum model to study far-field dynamics, the continuum model must be validated. In order to accomplish this validation, the DEM model for the near-field simulations is used. Namely, the DEM is used to generate boundary and initial conditions for the continuum 20

35 model and then the two models are compared after a given amount of simulation time is elapsed in each model. The gas forces are successfully integrated into the continuum model and validated for monodisperse, binary, and continuous PSDs [26] Validation of continuum model via Apollo data (Chapter 7) The previously validated continuum model is used to probe the far-field dynamics of the ejected regolith. The erosion flux calculated from the DEM model serves as a boundary condition to the continuum model. The results are compared with existing data from Apollo videos. Overall, this preliminary validation shows good agreement between the developed model and Apollo data for the particle flux and ejection angle. However, the particle velocities are not consistent with those estimated from Apollo landings and thus more work is needed to resolve the discrepancy. 1.6 References 1. Immer, C., et al., Apollo 12 Lunar Module exhaust plume impingement on Lunar Surveyor III. Icarus, (2): p Metzger, P., et al., Scaling of Erosion Rate in Subsonic Jet Experiments and Apollo Lunar Module Landings, in Earth and Space , American Society of Civil Engineers. p Brien, B.J., S.C. Freden, and J.R. Bates, Degradation of Apollo 11 Deployed Instruments because of Lunar Module Ascent Effects. Journal of Applied Physics, (11): p Jaffe, L.D., Blowing of Lunar Soil by Apollo 12: Surveyor 3 Evidence. Science, (3973): p Immer, C., et al., Apollo video photogrammetry estimation of plume impingement effects. Icarus, (1): p C. Conrad, R.F.G., Jr., A. L. Bean, Apollo 12 Technical Crew Debriefing. 1969: NASA Johnson Space Center, Houston, TX. 21

36 7. D. Scott, J.I., A. Worden, Apollo 15 Technical Debriefing, in Rep. MSC : NASA Manned Space Center, Houston, TX. p T.f. 9. Curtis, J.S. and B. van Wachem, Modeling particle-laden flows: A research outlook. AIChE Journal, (11): p Hrenya, C.M., Kinetic theory for granular materials: Polydispersity, in Computational Gas-Solids Flows and Reacting Systems: Theory, Methods and Practice, M.S. S. Pannala, T. O'Brien, Editor. 2011, IGI Global: Hershey, PA. p Muzzio, F.J., T. Shinbrot, and B.J. Glasser, Powder technology in the pharmaceutical industry: the need to catch up fast. Powder Technology, (1): p Ottino, J.M. and D.V. Khakhar, Mixing and segregation of granular materials. Annual Review of Fluid Mechanics, : p Sundaresan, S., Some outstanding questions in handling of cohesionless particles. Powder Technology, (1): p Chew, J.W., D.M. Parker, and C.M. Hrenya, Elutriation and Species Segregation Characteristics of Polydisperse Mixtures of Group B Particles in a dilute CFB Riser. AIChE Journal, (1): p Peters, I.R., Q. Xu, and H.M. Jaeger, Splashing Onset in Dense Suspension Droplets. Physical Review Letters, (2): p Arslan, H., S. Batiste, and S. Sture, Engineering Properties of Lunar Soil Simulant JSC- 1A. Journal of Aerospace Engineering, (1): p Goodings, D.J. and C.P. Lin, Geotechnical Properties of the Maryland-Sanders Lunar Simulant. Geotechnical Testing Journal, (2): p Hartzell, C.M. and D.J. Scheeres, The role of cohesive forces in particle launching on the Moon and asteroids. Planetary and Space Science, (14): p Li, Y.Q., J.Z. Liu, and Z.Y. Yue, NAO-1: Lunar Highland Soil Simulant Developed in China. Journal of Aerospace Engineering, (1): p Li, Y.Q., et al., Two Lunar Mare Soil Simulants. Acta Geologica Sinica-English Edition, (5): p Rame, E., et al., Flowability of lunar soil simulant JSC-1a. Granular Matter, (2): p

37 22. Berger, K.J. and C.M. Hrenya, Challenges of DEM: II. Wide particle size distributions. Powder Technology, (0): p Metzger, P.T., J. Smith, and J.E. Lane, Phenomenology of soil erosion due to rocket exhaust on the Moon and the Mauna Kea lunar test site. Journal of Geophysical Research: Planets, (E6): p. E Ladd, A.J.C. and R. Verberg, Lattice-Boltzmann Simulations of Particle-Fluid Suspensions. Journal of Statistical Physics, (5-6): p Cundall, P.A. and O.D.L. Strack A discrete numerical model for granular assemblies. Géotechnique, , Iddir, H. and H. Arastoopour, Modeling of multitype particle flow using the kinetic theory approach. AIChE journal, (6): p Murray, J.A., et al., Continuum representation of a continuous size distribution of particles engaged in rapid granular flow. Physics of Fluids, (8): p SØRensen, M. and I.A.N. McEwan, On the effect of mid-air collisions on aeolian saltation. Sedimentology, (1): p Dong, Z., N. Huang, and X. Liu, Simulation of the probability of midair interparticle collisions in an aeolian saltating cloud. Journal of Geophysical Research: Atmospheres, (D24): p. D Huang, N., Y. Zhang, and R. D'Adamo, A model of the trajectories and midair collision probabilities of sand particles in a steady state saltation cloud. Journal of Geophysical Research: Atmospheres, (D8): p. D Kok, J.F. and N.O. Renno, A comprehensive numerical model of steady state saltation (COMSALT). Journal of Geophysical Research: Atmospheres, (D17): p. D Carneiro, M.V., et al., Midair Collisions Enhance Saltation. Physical Review Letters, (5): p Bagnold, R.A., The physics of wind blown sand and desert dunes. Methuen, London, Lane, J., et al., Lagrangian Trajectory Modeling of Lunar Dust Particles, in Earth & Space p Lane, J., P. Metzger, and J. Carlson, Lunar Dust Particles Blown by Lander Engine Exhaust in Rarefied and Compressible Flow, in Earth and Space p

38 36. Lane, J.E. and P.T. Metzger, Ballistics Model for Particles on a Horizontal Plane in a Vacuum Propelled by a Vertically Impinging Gas Jet. Particulate Science and Technology, (2): p Metzger, P.T., J. Smith, and J.E. Lane, Phenomenology of soil erosion due to rocket exhaust on the Moon and the Mauna Kea lunar test site. Journal of Geophysical Research- Planets, Goldhirsch, I., M.L. Tan, and G. Zanetti, A molecular dynamical study of granular fluids I: The unforced granular gas in two dimensions. Journal of Scientific Computing, (1): p Hopkins, M.A., J.T. Jenkins, and M.Y. Louge, On the structure of three-dimensional shear flows. Mechanics of Materials, (1 2): p Morris, A.B., et al., Approach for Modeling Rocket Plume Impingement and Dust Dispersal on the Moon. Journal of Spacecraft and Rockets, 2015: p Guo, Y. and J.S. Curtis, Discrete element method simulations for complex granular flows. Annual Review of Fluid Mechanics, : p Loth, E., Lift of a Spherical Particle Subject to Vorticity and/or Spin. AIAA Journal, (4): p Loth, E., Compressibility and rarefaction effects on drag of a spherical particle. AIAA Journal, (9): p Berger, K.J., et al., Role of collisions in erosion of regolith during a lunar landing. Physical Review E, (2): p Berger, K.J., et al., Erratum: Role of collisions in erosion of regolith during a lunar landing. Physical Review E, (1): p Berger, K.J. and C.M. Hrenya, Impact of a binary size distribution on the particle erosion due to an impinging gas plume. AIChe Journal, In Press. 47. Berger, K.J. and C.M. Hrenya, Predicting regolith erosion due to a lunar landing: Role of a continuous size distribution. Submitted to Icarus. 48. Berger, K.J., et al., Gas-solid drag on multiple particles in the slip flow regime at low and finite Reynolds numbers. Submitted to International Journal of Multiphase Flow. 24

39 2. NEAR-FIELD: EROSION OF MONODISPERSE SOLIDS AB Abstract The supersonic gas plume of a landing rocket entrains lunar regolith, which is the layer of loose solids covering the lunar surface. This ejection is problematic due to scouring and dustimpregnation of surrounding hardware, reduction in visibility for the crew, and spoofing of the landing sensors. To date, model predictions of erosion and ejection dynamics have been based largely on models which ignore collisions or resolve collisions stochastically. In the present work, the parameters affecting the erosion rate of monodisperse solids are investigated using discrete element method (DEM) simulations of the near field. The drag and lift forces exerted by the rocket exhaust are incorporated via one-way coupling. The results demonstrate that inter-particle collisions above the regolith surface are relatively rare with only as many as 0.3% of particles engaged in collisions. Thus, collisions between eroded particles above the surface play a minimal role in the erosion process of monodisperse regolith. In addition, a direct assessment of the influence of particle contacts on the erosion rate is accomplished via a comparison between a collision-less DEM model and the original DEM model. This comparison shows that the erosion flux is dependent upon contacts between particles at or below the surface. A Kyle J. Berger, Anshu Anand, Philip T. Metzger, Christine M. Hrenya, Role of collisions in erosion of regolith during a lunar landing, Physical Review E, 2013, 87, 2 B Kyle J. Berger, Anshu Anand, Philip T. Metzger, Christine M. Hrenya, Erratum: Role of collisions in erosion of regolith during a lunar landing, Physical Review E, 2015, 91, 1 25

40 2.1 Introduction To better understand and begin to quantify the role of particle contacts on the erosion of regolith, the discrete element method (DEM) is used here to simulate the surface erosion of lunar regolith with a monodisperse particle size distribution (PSD). The DEM simulations track individual particle motion according to Newton s laws, including forces arising from one-way particle-gas coupling (gas affects particles but not vice-versa), lunar gravity, and particle contacts (see equation 1.3). The focus of this chapter is on examining the trends associated with interparticle contacts for monodisperse systems. The system under consideration is the near field (Figure 1.5). In addition, a sensitivity study to the various model parameters is performed in order to quantify the effects of these parameters on the results shown. 2.2 Computational Model Base case parameters The simulation proceeds in two steps. First, randomly-placed particles are allowed to settle in the simulation domain (see Figure 1.5) under the action of lunar gravity alone (i.e., no plume effects) in order to obtain an initial settled state. Once the particles have settled, the plume velocity is turned on. In this second step, under the combined action of the plume and lunar gravity, the particles begin to erode from the regolith surface. The periodic box is chosen to be very high in the vertical (y) direction such that the effect of a ceiling is nonexistent (see Table 2.1 for actual lengths). The number of particles is chosen such that the erosion occurs over a sufficient amount of time to investigate the erosion and ejection dynamics near the surface. The particle size is chosen 26

41 to fall within the range of particle sizes found in lunar regolith. The time step used in the simulations is chosen to be the collision time (calculated from other input parameters) divided by 50 to ensure accurate numerical integration. The collision time, tcol, is given by t col 2 k norm norm m eff 2m eff 1/2 (2.1) The initial height of the anchoring plane (see Figure 1.5) is chosen to be 4 particle diameters below the top of the regolith layer to mimic the gas plume penetrating the void spaces in the bed. The initial height of the erosion plane (see Figure 1.5) is chosen to be 1 particle diameter above the top of the regolith layer in order to examine the erosion very near the surface. Thus, the distance between the anchoring and erosion planes is a total of 5 particle diameters. The sensitivity of the erosion dynamics to the location of the anchoring and erosion planes is examined in sections and The spring constant used is typical of spring constants used in other similar DEM simulations and, as shown in section , the qualitative nature of our conclusions do not change with significant changes in the spring constant (several orders of magnitude) [1]. In addition, the percent overlap (relative to the particle radius), which is one possible guide for determining if a spring coefficient is reasonable, remains below 5% (on average) for contacts near the erosion plane [2]. Perhaps more importantly, the conclusions drawn do not change with significant changes in the spring constant (see section ). 27

42 Table 2.1 Baseline parameters Particle Properties Total number, n Diameter, d Density, Time step, Δt Plume Properties Material Density, g Viscosity, Temperature, T µm 2700 kg/m 3 ~ s kg/m Pa*s 500 K Lunar Conditions Acceleration due to gravity on the Moon 1.63 m/s 2 Particle Collision Properties Spring stiffness particle-particle normal, k norm,pp tangential, k tan,pp 8000 kg/s kg/s 2 particle-wall normal, k norm,pw tangential, k tan,pw Friction coefficient particle-particle, pp particle-wall, pw Coefficient of restitution particle-particle, e norm,pp particle-wall, e norm,pw Dashpot Coefficient normal, norm tangential, tan System geometry Distance from impingement point Length of periodic box in x direction Length of periodic box in z direction Length of box in y direction Initial bed height Initial height of anchoring plane Initial height of erosion plane 8000 kg/s kg/s Calculated from coefficient of restitution and spring stiffness (see Appendix A) 0.5 * norm 1 m m m 1 m m m m 28

43 2.3 Results and Discussion Base case: Impact of contacts on erosion An important aim of this work is to assess the role of inter-particle contacts on erosion and ejection dynamics near the surface. The results obtained from the base case simulation are described in detail below in order to gauge the impact of inter-particle contacts in monodisperse systems. Figure 2.1 is a plot of the cumulative erosion volume, which is the volume of particles above the erosion plane, as a function of time after the plume is turned on at t = 0 s. From Figure 2.1, two distinct phases are identified during the course of the simulation: the non-depleted phase and the depleted phase. The non-depleted phase corresponds to the initial section of the plot where the cumulative erosion number is increasing rapidly (t ~ < 0.006s). The depleted phase corresponds to the section of the plot with a nearly constant value of the cumulative erosion number (t ~ > 0.006s). These phases are demarcated by the vertical dashed line in Figure 2.1 as well as in subsequent figures. The depleted phase begins when all of the initially settled particles have eroded above the erosion plane and the continuous supply of particles from the regolith layer has stopped. This phase is not important to the study of the erosion because it is the result of the limited number of particles initially placed in the system. 29

44 Figure 2.1 Total eroded volume for base case parameters. Dotted line denotes the average erosion during the non-depleted phase. The dashed vertical line demarcates the depleted and non-depleted phases. The solid vertical line demarcates the erosion-only and erosion-and-sedimentation phases. These lines are repeated in later plots with the same meaning. The non-depleted phase can be further broken down into an initial transient and pseudo steady-state (demarcated by a dashed vertical line in Figure 2.1). The former refers to the portion of the plot which has a varying slope (t ~ < 0.002s) during erosion and the latter refers to the portion of the plot which has an approximately constant slope (0.002s ~> t ~ >0.006s). The two portions of the plot have different slopes because of the nature of the beginning of the simulation. More specifically, a number of particles are immediately under the influence of the plume (as opposed to the anchoring plane gradually exposing more particles over the course of the simulation), which results in a higher initial erosion flux. However, once all of these particles have been eroded, the erosion flux decreases slightly due to particles being gradually exposed. The erosion flux, G eros, is defined as the mass flux of particles across the erosion plane: 30

45 G v (2.2) eros p, rel, y s s where v p, rel, y is the time-averaged vertical particle velocity relative to the moving erosion plane, s is the solids volume fraction at the erosion plane, and s is the solid particle material density. However, for convenience in calculation, the erosion flux can be rewritten in terms of the average slope of the cumulative erosion volume during the steady erosion phase as G eros V t p s (2.3) LL x z where V p t is the average volume crossing the erosion plane per time and Lx and Lz are the dimensions of the domain in the x and z directions, respectively (see Table 2.1). Note that, in Figure 2.1, the local slope is always positive, suggesting that particles, once eroded, do not return below the erosion plane. In order to understand this trend, it is important to examine the collisions occurring above the erosion plane, since it is those collisions that could result in particles returning to the surface. The ratio of the moving average of the instantaneous number of collisions per time step (for 1000 time steps) to the number of particles above the erosion plane (at the end of the interval) is defined as the average fractional collision number of the system. A collision is said to happen during a given time step if two particles are overlapping at any point during that time step. The average fractional collision number is plotted in Figure

46 Figure 2.2 Average fractional collision number for base case vertical lines represent beginning and end of steady erosion. With the exception of initial transients at the beginning of erosion caused by only a few eroded particles, the average fractional collision number has a value of about Assuming that the contacts between particles are primarily binary in nature (which was verified), only about % of eroded particles are engaged in collisions during the steady erosion phase. This finding, combined with the lack of particles returning to the surface, suggests that particle contacts above the surface play a minimal role in the erosion flux of monodisperse systems Collision-less DEM In order to directly analyze the influence of collisions on the erosion flux, a collision-less version of the DEM was implemented such that particle trajectories were able to cross through one another without contacts being resolved (Eq. (1.3) sans FColl). The cumulative erosion volume plot comparing the collision-less DEM and the original DEM model (for dissipative and non- 32

47 dissipative collisions) is displayed in Figure 2.3. The figure shows that when contacts are removed, there is both a time delay in the erosion, as well as a decrease in the erosion flux. However, this finding is not contrary to the results above because here it is the contacts below the erosion plane that are playing a major role. Namely, contacts below the surface enhance erosion flux by transferring momentum from particles lower in the bed to those near the top, which results in particles near the top gaining momentum more quickly and therefore eroding faster. Figure 2.3 Cumulative erosion volume for dissipative and non-dissipative collisions, along with the case of no collisions. In addition to the change in erosion flux when collisions are removed, it can be seen from Figure 2.3 that the erosion flux increases when there are non-dissipative collisions (i.e., enorm = 1 and µ = 0) compared to dissipative collisions. One way to explain this trend is the effect that collisions have on the particle velocities. As a simple explanation, consider the isolated case of two particles separated vertically, where the top particle is stationary and the bottom is moving 33

48 upwards. If the collision is elastic, then the particles will undergo a momentum swap, leaving the top particle moving up and the bottom particle stationary (ignoring the role of lift, for purposes of illustration). Such a configuration lends itself to a high erosion flux because the top particle erodes while the bottom particle is now more likely to be involved in a collision with a particle below it. However, if the collision is inelastic, then not only will the top particle erode slower, but the bottom particle will also be less likely to be involved in another collision and thus receives less momentum from particles below it (since inelasticity causes a reduction in the relative velocity between particles upon collision). This phenomenon creates a narrower velocity distribution and thus results in fewer collisions, which was confirmed by examining the velocity distribution and collisional data (data not shown for sake of brevity). In particular, these results indicate that the rate at which collisions occur between the anchoring and erosion planes is much higher for the non-dissipative case as opposed to the dissipative case (during the erosion and sedimentation phase). In addition, the delay of the onset of erosion in the no-collisions case is caused by a lack of momentum transfer between particles. Particles can only gain momentum from the plume and cannot transfer that momentum to a more useful particle (i.e., one nearer to the top of the bed) i.e., the top layer of particles will erode first, and in the no collisions case their only source of momentum is drag/lift, whereas in either of the collisions cases, momentum can also be transferred via collisions from particles lower in the bed to these top layers Effect of Input Parameters The parameter space can be broadly divided into three categories: plume properties, system properties and particle properties. Depths of the plume penetration into the regolith, the initial 34

49 height of the erosion plane and particle restitution coefficient denote an example from each of these categories, respectively. In general, by studying the effect of various parameters on cumulative erosion number and instantaneous collision number over time, the role of various parameters on erosion dynamics is assessed Plume properties: Penetration depth As described previously, the anchoring plane denotes the initial depth of penetration of the gas plume into the regolith. Physically, this penetration depth depends on the permeability of the regolith. Specifically, a larger permeability of the regolith will lead to greater penetration depth of the gas plume. Figure 2.4 depicts the effect of varying the initial anchoring plane on the overall erosion flux. As expected, the overall erosion flux increases as the penetration depth increases. In other words, a more permeable regolith will exhibit a significantly higher erosion flux than a less permeable one. 35

50 Figure 2.4 Cumulative erosion number for different anchoring plane depths. Penetration depths are 2, 4 (base case), and 6 particle diameters. All other parameters are given in Table System properties Even though changing the height of the erosion plane above the regolith layer does not affect the particles at all, doing so can yield valuable insight into the erosion dynamics. As described before, the erosion plane in the base case is located at one particle diameter above the top of the regolith layer, which is useful for calculating the erosion flux very close to the surface. For comparison, Figure 2.5 contains the cumulative erosion number plot for two different locations of the erosion plane. The overall erosion flux (average slope; see Figure 2.1) is lower for a higher erosion plane because some particles are unable to reach the higher erosion plane. A smaller change in the erosion plane would yield minimal change in the erosion flux. 36

51 Figure 2.5 Cumulative erosion number for different erosion plane heights. In an effort to be as general as possible, the gravitational acceleration is varied to determine any associated effects. The effects on the cumulative erosion volume are shown in Figure 2.6. The erosion flux decreases marginally for the values investigated (up to 15 m/s 2 ), suggesting that gravity only plays a minimal role in the erosion flux of the particles studied. This result is likely due to the drag and lift forces being much larger than the gravitational force for the particle size studied here and thus it would take a significantly larger gravitational acceleration to modify the erosion flux. In addition, larger particles would likely see a larger influence of gravity due to increased mass. 37

52 Figure 2.6 Sensitivity of cumulative erosion volume to gravitational acceleration. Values used: 0, 1.63, 3, 5, 9.8, 15 m/s Particle properties The other parameters for which a sensitivity analysis was performed include the initial particle arrangement, size of periodic domain, spring constant, coefficient of restitution, particle friction, and particle size. As is apparent in Figure 2.7, starting with a different initial particle configuration (as a result of settling from a different initial state) does not change the overall erosion flux, although small deviations in the plots do exist, as is expected from a different initial state. Similarly, by varying the size of the periodic box in the x and z directions, the approximate flux of particles does not show any change as seen in Figure 2.8. In addition, starting with a different number of particles results in the same qualitative behavior as plotted in Figure 2.9. The initial erosion phase lasts for the same amount of time, as is expected, but the steady erosion phase lasts longer because more particles are available for erosion. 38

53 Figure 2.7 Sensitivity of cumulative erosion volume to seed used for initial particle configuration. Seed 1 refers to base case (Figure 2.1); parameters used for all seeds given in Table

54 Figure 2.8 Cumulative erosion volume per box area versus time for different periodic domain sizes. Note that the number of particles in each simulation is scaled based on the area (linearly compared to base case parameters). All other parameters for all cases are shown in Table 2.1. Figure 2.9 Cumulative Erosion number for two different initial bed depths (number of particles). All other parameters are shown in Table

55 Changes in the spring constant do not result in any qualitative or quantitative changes in the cumulative erosion plot. Figure 2.10 gives the cumulative erosion number versus time for spring constants an order of magnitude smaller and larger than that used in the base case. The figure shows that for spring constants other than the one used in the base case (k = kg/s 2 ), no significant changes are observed in the erosion. Figure 2.10 Cumulative erosion number versus time for different spring coefficients. Spring coefficients depicted are 8x10 2, (base case), and kg/s 2. Changes in both the coefficient of restitution and friction coefficient have minor effects on the erosion, as is evident from Figure The erosion flux increases slightly with increasing friction. This trend is possibly due to the friction coefficient controlling the transfer of momentum in the tangential direction. The erosion flux is also nearly independent of the coefficient of restitution for the values investigated. 41

56 Figure 2.11 Cumulative erosion number versus time for different coefficients of restitution and friction coefficient. (a) enorm = 0.6, (b) enorm = 0.8. Friction coefficients displayed are 0.2 (base case), 0.5, 0.8, and

57 Figure 2.12 shows the erosion flux for monodisperse systems of varying particle sizes with 90% confidence intervals (CI) over multiple runs. Displayed in the figure is a case with particle contacts, without contacts, and for sub-surface contacts only, where the surface is defined as a distance herosion below the erosion plane (see Figure 1.5). The erosion fluxes shown are comparable to those necessary to yield observations of the Apollo landings [3]. This comparison is done by calculating the erosion flux necessary to yield the particle flux estimated to be traveling past a rock during the Apollo 11 landing (~11.6m from impingement). The erosion flux estimate is made by assuming no significant sedimentation prior to the rock, a 2 o ejection angle, and a constant erosion flux from 0 to 3m from impingement. The estimate yields an erosion flux of about 350 kg/m 2 /s, which is about 2 times larger than the values calculated from this study. This result is very close for such a rough estimate, especially given the monodisperse distribution. As expected, the erosion flux decreases with increasing particle size, since particle mass is proportional to 3 d p while the lift force is proportional to d (see Appendix A), which results in an acceleration that is proportional to 2 p 1 d p. The impact of contacts on the erosion flux is also noteworthy. When all contacts are turned off ( Fcoll 0 in Eq. 1.3), the erosion flux decreases significantly relative to the (full) contacts case. However, when only sub-surface contacts are allowed, the observed erosion is larger than the (full) contacts case. This result is consistent with the results shown in Figure 2.3 These trends reaffirm a physical picture in which the sub-surface contacts aid erosion via a transmission of force from particles lower in the bed to particles near the surface, whereas contacts above the surface attenuate erosion due to mid-air collisions that reduce the vertical momentum available for erosion. 43

58 Figure 2.12 Monodisperse: Effect of particle size and particle-particle contacts on erosions flux (90% confidence intervals). 2.4 Summary The discrete element method is used to simulate the erosion of the lunar regolith caused by a landing rocket. A one-way coupling is assumed in which the exhaust plume affects the regolith but not vice versa. DEM simulations of monodisperse particles are used to calculate the erosion flux and collision characteristics of the particles at a distance 1m away from the impingement point. The DEM results establish that inter-particle contacts between uneroded particles (i.e., below the surface) play an important role in determining the erosion flux, but collision between eroded particles (i.e., above the surface) play a minimal role. This result implies that the singleparticle trajectories calculated via the PET model may be adequate for monodisperse systems. 44

59 A direct analysis of the influence of contacts on the erosion flux is achieved via comparison with a collision-less DEM model. The results show that the erosion flux is highest for nondissipative collisions, followed by the dissipative collisions base case, and then the no-collisions case. Non-dissipative collisions result in the largest erosion flux because collisions serve as an effective means of transferring momentum between particles. However, contacts always serve to increase the erosion flux. The no-collisions case has a reduced erosion flux because the lack of momentum transfer between uneroded (sub-surface) particles hinders erosion. An examination of the effect of different parameters on the overall erosion flux suggests that the permeability of the regolith (penetration depth of the plume) and particle size are the most important system parameter that affects the erosion dynamics. The particle properties examined, with the exception of the non-dissipative system and particle size, had a minimal effect on the erosion flux. In addition, periodic box dimensions and initial settling condition have negligible effect on the erosion flux. Finally, gravity is found to play only a marginal role in determining the erosion flux for the particles examined. None of these parameters, however, change the qualitative conclusions drawn above. This work represents a first step in establishing the role of collisions in the erosion and ejection dynamics of regolith spurred by a lunar landing. An important extension of the current effort is to consider the polydisperse nature of the regolith material, since this polydispersity will not only impact collision rates of (unlike) particles, but could also lead to species segregation. These effects are explored in the next chapter. 45

60 2.5 References 1. Garg, R., et al., Documentation of open-source MFIX-DEM software for gas-solids flows from URL 2. Malone, K.F. and B.H. Xu, Determination of contact parameters for discrete element method simulations of granular systems. Particuology, (6): p Immer, C., et al., Apollo 12 Lunar Module exhaust plume impingement on Lunar Surveyor III. Icarus, (2): p

61 3. NEAR-FIELD: EROSION OF POLYDISPERSE SOLIDS CD Abstract Solid particles of non-uniform sizes exhibit behaviors not seen in monodisperse distributions. Following on the work in the previous chapter, the goal of this chapter is to investigate effects of a polydisperse size distribution on erosion of lunar regolith from a rocket landing. The discrete element method is used to examine near-field (near surface) behavior. For these simulations, binary and lognormal distributions are used to examine the erosion flux and erosion mechanisms. The results show that small-particle erosion flux is hindered while large-particle erosion flux is boosted relative to the monodisperse (or non-interacting) case. This result is primarily due to collisions between unlike particles, changes in particle packing, and a change in the penetration rate of the plume due to the presence of multiple particle sizes. Namely, small particles preferentially collide with large particles from below, due to a difference in velocity caused by gas forces, which is analogous to behavior for entrainment in gas-solid fluidized beds. These collisions cause vertical momentum transfer from small to large particles that affects species and total erosion flux. In addition, the total erosion flux in the binary system relative to the monodisperse case is dependent on composition, while the erosion flux of the lognormal distribution is increased relative to the monodisperse case. C Berger, K.J. and C.M. Hrenya, Impact of a binary size distribution on the particle erosion due to an impinging gas plume. AIChe Journal, In Press. D Berger, K.J. and C.M. Hrenya, Predicting regolith erosion due to a lunar landing: Role of a continuous size distribution. Submitted to Icarus. 47

62 3.1 Introduction As discussed in chapter 1, lunar regolith (soil-like material) is characterized by an extremely wide particle size distribution (PSD), spanning at least 3 orders of magnitude [1], which may lead to interesting physics like strong cohesive forces for smaller particles and changes in bulk flow properties (e.g., stress) [2-7]. Terrestrial systems with polydisperse PSDs are also known to exhibit behaviors with no monodisperse counterpart, such as size segregation and momentum transfer between species [8-12]. One example of a system with a bidisperse mixture involving significant momentum transfer between species is an upward, gas-solid flow in a vertical tube, as occurs in a fluidized-bed gasifier. In such a system, fine particles travel faster than coarse particles in the vertical direction due to their smaller terminal velocity. Thus, the fine particles collide preferentially with the larger particles from below, leading to vertical momentum boosts of the coarse particles. This momentum transfer results in the addition of fines causing a counterintuitive increase in the entrainment rate of the coarse particles despite a decrease in composition [12]. Another example of related behavior of a binary PSD is found in the splashing of particle-laden liquid drops [13]. Specifically, in monodisperse systems, large particles tend to escape from the drop at a lower impact velocity than small particles due an increased effect of surface tension on the small particles. However, when mixed, the small particles escape at lower velocities since they receive a large momentum boost from large particles. However, large particles have difficulty obtaining sufficient momentum for escape, since contacts with small particles transfer less momentum and since likelihood of collisions between large particles is reduced with small particles present. 48

63 Past work on terrestrial systems involving wind-blown sand and its saltation has indicated that mid-air collisions play a large role in the particle dynamics for narrow PSDs [14-20]. However, conditions on extraterrestrial surfaces, such as the Moon, are radically different and thus the results are not necessarily transferrable. For example, when considering the erosion of monodisperse particles from the lunar surface, our previous results obtained using discrete-particle simulations suggest that, in the near-field, contacts between eroded particles may play a relatively small role and thus single-particle trajectories may be sufficient to simulate the system [21, 22]. Morris et al. [23] also modeled contacts directly above the surface for the lunar case using a two-phase direct simulation Monte-Carlo method (DSMC) and a polydisperse mixture. They found that contacts between eroded particles do play a significant role in the far-field dynamics. However, the nature of these contacts was not investigated and thus it is unclear if the effects of particle contacts seen were due to the polydisperse nature of the DSMC simulations or if a monodisperse mixture would show similar results. This chapter focuses on understanding the role of a polydisperse particle mixture on the erosion flux using a binary and continuous PSD. The goal of this study is to gauge the effect of multiple particle sizes on the erosion flux, particularly through inter-particle contacts, as well as the effect of such contacts on the far-field dynamics of the erosion. The results indicate that contacts, particularly those between particles of different sizes, play a significant role in the erosion process. Namely, the higher-speed small particles preferentially collide with larger particles in the vertical direction from below. Such contacts increase the erosion flux of large particles, while decreasing the flux of small particles. Overall, these changes to species fluxes result in the total erosion flux for a binary PSD increasing (with many large particles) or decreasing (with few large particles), 49

64 depending on the composition, relative to a system without inter-species interactions. The erosion flux of the lognormal PSD investigated increases relative to the monodisperse case for the nonzero widths examined. In addition, contacts above the surface tend to reduce the erosion flux of both species via collisional momentum transfer that makes it more difficult for particles to erode. 3.2 Methods Model Inputs Table 3.1 shows the parameters that are modified from those used in chapter 2. Namely, the domain sizes are modified to account for larger particle sizes and the packing of particles varies more significantly for systems with polydisperse PSDs. Table 3.1 Model parameters Particle Collision Properties Initial solids volume fraction, εs System Parameters Length of box in x direction, Lx Length of box in z direction, Lz Length of box in y direction, Ly 2.0 x 10-3 m 1.0 x 10-3 m 1 x 10-2 m Initial bed height 7.0 x 10-4 m 8.0 x 10-4 m The binary PSDs used here are chosen by first selecting a Sauter-mean diameter, which is the diameter of the sphere with the same volume-to-surface-area ratio as the mixture. The Sautermean diameter chosen is the same diameter used in the previous monodisperse study, which is 50µm [1, 21, 22]. This diameter is used to calculate the particle sizes for a 50/50 weight% (wt%) mixture of large/small particles at each size ratio. These sizes are then held constant as the 50

65 composition is varied. Table 3.2 shows the particle sizes, dlarge and dsmall, and number of particles, Nlarge and Nsmall, for large and small particles used at each size ratio. Table 3.2. Particle number and size for each size ratio for the 50/50 wt% mixture. Size Ratio Nlarge dlarge(µm) Nsmall dsmall(µm) Continuous Size Distributions The size distributions used in this study are based on the lognormal distribution for which the probability density function is 2 1 (ln d p log ) f( dp) exp 2 d 2 plog 2 log (3.1) where σlog and µlog are distribution parameters. The mean, µ, and standard deviation, σ, of the lognormal distribution can be related to these parameters via log ln 1 2 (3.2) log ln 1 2 (3.3) 51

66 The lognormal distribution is used because it is characteristically similar to the distribution of lunar regolith. Figure 3.1 shows a lognormal distribution with a distribution width, σ/µ, of 2 and the lunar soil distribution from returned samples. The lognormal distribution is truncated such that 99.9% of particle volume is included. The similarity between the distributions is clear from the figure. Figure 3.1 Lognormal and lunar soil distribution [24]. The distribution width determines how varied the distribution is - i.e., a width of 0 refers to a monodisperse distribution and larger values are increasingly polydisperse. The lognormal distribution, due to its theoretical nature, has no defined minimum or maximum size. However, because only a finite number of particles can be simulated, a minimum and maximum size must be defined. In addition, the simulation time is highly dependent on both the smallest particle size 52

67 as well as the size ratio of the smallest and largest particle [1]. Thus, the lognormal distribution must be truncated. The truncation range used is the following: d SM dmin di dsm trunc dmax trunc (3.4) where dmin is the minimum diameter, dsm is the Sauter-mean diameter, ϕtrunc is the truncation ratio (ratio of maximum to minimum diameter), di is the size of particle i, and dmax is the maximum particle size. The Sauter-mean diameter is defined as d SM N i1 N i1 d d 3 i 2 i (3.5) where N is the total number of particles. This diameter is chosen because the external forces in the system, weight and lift/drag forces, are proportional dp 3 and dp 2, respectively. The above truncation range, which for the parameters chosen roughly incorporates particles between 10 and 300 µm, is selected because it captures a large portion of the volume of the distribution (about 90% of the volume of lunar regolith are particles >10 µm). In addition, Immer et al. [25] estimated particles around 100 µm have the largest impulse intensity and therefore are most likely to do the most damage. This peak in impulse intensity is due to a tradeoff between particle speed and particle mass that both factor into how much damage a given particle can do. In order to generate the particle size distribution for use in DEM, a lognormally-distributed random number is generated based on the normally-distributed random number generator in 53

68 MATLAB. The particle size is accepted only if it lies within the range in equation (3.4), otherwise a new number is generated until all particle sizes have been chosen. This is done independently for each run. The range of parameters used to generate these truncated distributions is located in Table 3.3. Table 3.3 Continuous PSD parameters. Property Value dsm 5.0 x 10-5 m ϕtrunc 10, 20, 25 σ/µ 0.5, 1, 1.5, 2 To better identify trends in the data with respect to particle size, particles are arranged into families according to size. The first family has the smallest size particles and the last family has the largest particles. The distribution is divided up by particle number, so if 5 families are used for 1000 particles, the smallest 200 will be in family 1 and the largest 200 in family Results and Discussion Binary Case Figure 3.2 shows the total erosion flux for a binary mixture as a function of composition, along with a non-interacting mixture approximation. The non-interacting mixture approximation is a mass-based average of the monodisperse fluxes associated with each of the two particles sizes, represented as G, - eros non int, which can be defined as G G w G w (3.6) eros, non- int eros, mono, L L eros, mono,s S 54

69 where G eros, mono,i represents the monodisperse erosion flux of species i (see equation 1.3), wi represents the volume fraction associated with species i, and L and S refer to the large and small particles, respectively. Such a flux would result if the two species had no interactions between them, which allows comparison between the monodisperse and binary cases. The total erosion flux appears to increase for the case of 50% and 75% large particles and decrease for the case of 25% large particles relative to the non-interacting mixture, suggesting that the effect of mixing the particle species on the total erosion flux is dependent on composition. Figure 3.2. Binary: Total erosion flux, along with non-interacting mixture approximation (90% CI). Figure 3.3 shows the species erosion flux, normalized with the non-interacting mixture erosion flux, for binary mixtures of two size ratios. Based on Figure 3.3, small-particle erosion flux is reduced compared to non-interacting mixture since the normalized values generally lie below 1. Moreover, the large-particle erosion flux is increased compared to the non-interacting 55

70 mixture. These deviations from the non-interacting case, for both small and large particles, increase with size ratio examined. These trends in species flux indicate that the trend in total flux is due to a combination of the change in the species flux, but the direction of the change in the total flux depends largely on the composition. In section 3.3.2, possible mechanisms for the changes in species flux relative to a non-interactive mixture will be discussed. Figure 3.3. Binary: Species erosion flux of large (solid lines) and small (dashed lines) particles normalized against non-interacting mixture approximation (90% CI). The presence of two particle sizes also leads to an increase in the frequency of contacts relative to the monodisperse case. Figure 3.4 shows the average fractional collision number as a function of time for the monodisperse and binary cases. The average fractional collision number is a moving average of the fraction of eroded particles that are engaged in a (mid-air) collision. 56

71 For the binary case, the contacts are broken down into large-large, small-small, and large-small contacts. From Figure 3.4, it is clear that contacts between large and small particles dominate all contact types. Physically, the small particles travel faster than the large particles in both the horizontal and vertical directions (on average), leading to a larger number of collisions between particles of dissimilar size. The difference in velocity is caused by the difference in the particle mass, which scales as 3 d p, and the lift (vertical) and drag (horizontal) forces, which scale as (see Appendix A). (Note that the lift and drag forces are relatively insensitive to particle velocities since these forces on the particle are based on the particle velocity relative to that of the gas, and this relative velocity is similar for large and small particles.) Thus, the small particles will accelerate to a higher velocity than the large particles. 2 d p Figure 3.4. Average fractional collision number for binary and monodisperse cases with same Sauter-mean diameter (size ratio 4; 90% CI). 57

72 3.3.2 Erosion Mechanisms Here we explore possible mechanisms to explain the trends in species fluxes noted in Figure 3.3 i.e., the enhancement of large-particle flux relative to a non-interacting mixture and vice versa for small particles. Note that these mechanisms are primarily due to contacts between unlike particles and thus play little to no role in the monodisperse case. The mechanisms considered include: (i) vertical momentum transfer arising from large-small contacts beneath the erosion plane, (ii) vertical momentum transfer due to collisions across the erosion plane, (iii) impact of horizontal component of collisional momentum transfer between species on their respective lift forces, (iv) an increase in packing concentration for a binary mixture relative to monodisperse, and (v) a complex interaction between the (moving) anchoring and erosion plane and the particles. Mechanisms (i), (ii), (iii), and (v) are related to particle velocity changes and mechanism (iv) is related to particle concentration changes, both of which impact the observed species flux (see equation 1.3). The first mechanism that may impact the species fluxes relative to a non-interacting mixture involves contacts between unlike particles below the erosion plane. Such contacts result in a net rate of momentum transfer from small to large particles, as shown in Figure 3.5. The figure shows the net rate of vertical momentum transfer from small to large particles during collisions below the erosion plane (as defined in Figure 1.5) for a size ratio of 4. Note that the rate of vertical momentum gained by particles due to the plume forces (drag and lift) is of the same order of magnitude (not shown) as that transferred via contacts, suggesting the latter is not negligible. The most significant momentum transfer occurs in the region just below the erosion plane, which is in 58

73 a fluidized state, thereby allowing significant momentum transfer through mid-air collisions. The net momentum transfer is from small to large particles because small particles tend to travel faster than large particles (both in the vertical and horizontal directions), which allows the small particles to preferentially collide with the large particles from below (see section 3.3.2). These contacts result in a boost in the large particle erosion flux and a reduction in the small particle erosion flux due to an increase in the large-particle velocity at the expense of the small particle, analogous to the fluidized-bed system discussed previously [12]. Overall, based on the comparable magnitudes of the momentum transferred from the gas to particles and the momentum transferred small to large particles, this mechanism has a moderate effect on the erosion flux. Figure 3.5. Binary: Net rate of vertical momentum transfer from small to large particles during steady erosion (size ratio 4; 90% CI). 59

74 In addition to contacts between unlike particles below the erosion plane (mechanism i), contacts of all types across the erosion plane result in a reduction of the erosion flux for both species (mechanism ii), as evidenced by an increase in the erosion flux when contacts do not occur across the erosion plane (not shown for sake of brevity). Contacts across the erosion plane (one particle above and one below) cause a net collisional transfer of momentum from below the erosion plane to above. Figure 3.6 shows the net rate of vertical momentum transfer across the erosion plane for small and large particles, relative to the monodisperse case. With the exception of the large particles in the 25 wt% large particles case, the momentum transfer is increased in the binary case relative to the monodisperse case. This result is due to the increased frequency of collisions, particularly large-small collisions, near the erosion plane (see Figure 3.4). The large number of large-small contacts is a result of the same mechanism described above in which small particles, which travel faster in the vertical direction, catch up with large particles and preferentially collide with them. However, this mechanism reduces both species fluxes and thus does not directly explain the increase of large particle erosion flux. By examining the momentum transferred across the erosion plane and comparing it to that transferred in mechanism (i), the relative importance of the two mechanisms can be compared. The loss of momentum from the small particles is greater via mechanism (ii) than mechanism (i) when there are mostly small particles (25 wt% large), but otherwise mechanism (i) is dominant (50 wt% and 75 wt% large). However, mechanisms (i) and (ii) act in the same direction for small particles. For the large particles, mechanism (ii) is dominant when there are many large particles (75 wt% large), which, independent of other mechanisms, would result in a net decrease in erosion flux. For the case of fewer large particles (25 wt% and 50 wt% large), mechanism (i) is dominant, resulting in a net increase in erosion flux. 60

75 Figure 3.6. Binary: Net rate of vertical momentum transfer across the erosion plane during steady erosion relative to the monodisperse case (size ratio 4; 90% CI). Along with the vertical component of contacts (mechanisms i and ii), the horizontal transfer of momentum between particles in the direction of the plume flow also has an effect on the erosion flux (mechanism iii). Namely, horizontal momentum transfer between large and small particles causes a change in the slip velocity of the particles, where the slip velocity is the difference in velocity between a particle and the gas. This change in the slip velocity causes an increase in the lift force of small particles and a decrease in the lift force on the large particles. Figure 3.7 shows such momentum transfer (dashed arrow) occurring between a large and small particle before collision (subplot a), during collision (subplot b), and after collision (subplot c) and the change in the lift force (thick red arrows). The small particle catches up to and collides with the large particle. Such a collision is typical because, on average the small particles travel faster than the large 61

76 particles (magnitude of velocities denoted by solid arrows). This difference in velocities between different-sized particles is due to the particle mass, which scales as 3 d p, and drag force, which scales as 2 d p, which results in a larger acceleration for small particles (see section III.b). When the small and large particles collide, the momentum transfer decreases the small particle velocity and increases the large particle velocity. This change in particle velocity causes an increase in the slip velocity of the small particle and a decrease in the slip velocity of the large particle. This change in the slip velocity results in an increase of the lift force on the small particle and the opposite for the large particle (see Appendix A). However, this effect is relatively small because the particle velocity is much smaller than the gas velocity and thus the relative velocity between the gas and the particles changes only slightly. Thus, the lift force changes at most 2-3% (results not shown for brevity) and the effect on the erosion flux is minimal, although the effect on the horizontal velocity of the particles is not negligible. Figure 3.7. Schematic showing change in lift force (a) before collision, (b) during collision, and (c) after collision due to horizontal momentum transfer. Solid arrow denotes horizontal particle velocity, thick red arrow represents lift force, and dashed arrow represents transfer of momentum. In addition to the altering of particle velocities caused by the collisional mechanisms (i) (iii), an increase or decrease in the concentration of particles can also result in corresponding increase or decrease in the erosion flux (mechanism iv; see equation 1.3). Particles in a binary mixture pack tighter than a monodisperse system due to small particles filling in gaps between the 62

77 large particles. Table 3.4 shows the corresponding change in the solids volume fraction for binary mixtures relative to their monodisperse counterparts, obtained after the initial settling phase of the simulations. For example, for a 50/50 binary mixture by weight and a size ratio of 4, the relative concentration of the large particles increases by 26.9% relative to monodisperse, which causes a similar increase in the erosion flux. For small particles, the erosion flux is decreased relative to the non-interacting mixture and thus mechanism (iv) must have less of an effect than mechanisms (i) and (ii) combined (since (i) and (ii) decrease the erosion flux, while (iv) increases the erosion flux). For large particles in the case of 75 wt% large particles, mechanism (iv) is more dominant than the net effect due to mechanisms (i) and (ii), given that the large particle flux increases in the binary mixture relative to the non-interacting mixture. However, for the cases 25 and 50 wt% large particles, comparing mechanism (iv) to (i) and (ii) individually is difficult given that both mechanism (iv) and the net effect of (i) and (ii) serve to increase the large particle erosion flux relative to the non-interacting mixture. Table 3.4. Increase in concentration of small and large particles up to two Sauter-mean diameters below the erosion plane during steady erosion relative to the monodisperse case (size ratio of 4, 90% CI). Composition (wt% large) Large Concentration Increase Small Concentration Increase 25% 23.0±4.9% 2.0±2.00% 50% 26.9±3.2% 6.9±2.03% 75% 28.9±3.7% 8.9±0.85% Though more subtle than the change in erosion flux due to changes in velocity and concentration (mechanisms i-iv), the erosion flux is also affected by a change in the rate at which the anchoring and erosion planes move in the binary case compared to the monodisperse case (mechanism v). As described in section II, the anchoring and erosion planes move as particles 63

78 erode. The erosion plane velocity, v eros, which is equal to the anchoring plane velocity, is directly tied to the erosion flux, which is in turn a function of particle velocity relative to the moving erosion plane via v eros G v v eros, l arg e G eros, small p, rel, y, l arg e s,large s, l arg e p, rel, y, small s, small s, small s, l arg e s, l arg e s, small s, small s, l arg e s, l arg e s, small s, small (3.7) Small particles erode faster than large particles and thus large particles will see an increase in the velocity of the erosion plane and small particles will see a decrease compared to the respective monodisperse cases. This change in the erosion and anchoring plane velocity can affect the erosion flux by changing the relative velocity between the particles and erosion plane (see equation 1.3). First, an increase in the magnitude of the erosion/anchoring plane velocity results in a decrease in the time spent under the influence of the plume forces prior to erosion (between the anchoring and erosion planes). This reduction in time leads to a change in the impulse gained from the plume, which can be calculated as t eros m(v v ) F dt F (t t ) (3.8) p,y, f p, y,0 t lift lift eros anch anch where v py is the vertical particle velocity,, v p, y, f is the particle velocity when it crosses the erosion plane, v py is the initial particle velocity when it crosses the anchoring plane (approximately,,0 zero), t eros is the time at which the particle crosses the erosion plane, t anch is the time at which the particle crosses the anchoring plane, F is the lift force, which is assumed constant for simplicity lift 64

79 (no qualitative difference to using a varying lift force). In order to determine teros -t anch, the reference frame in which the erosion plane is fixed and equation (3.7) are used to determine hpenetration herosion Flift (teros t anch ) F lift vp, y v eros (3.9) where < > denotes the time average. Utilizing the assumption that the lift force is constant and thus the average velocity is half of the final velocity, the velocity of the particle as it crosses the erosion plane can be found as F 2 lift vp, y, f veros veros 2( hpenetration herosion ) (3.10) m As the magnitude of the erosion plane velocity, v eros, increases, the velocity of the particle as it crosses the erosion plane, v p, y, f, decreases (note erosion plane velocity is negative). Using equation (3.10), the relative velocity between the particle and erosion plane is F 2 lift vp, y, f veros veros 2( hpenetration herosion ) (3.11) m Thus, as the magnitude of the erosion plane velocity, v eros, increases, the erosion flux, G eros, increases due to an increase in the relative velocity between the particles and erosion plane, v p, y, f v eros, leading to an increase in the erosion flux (decrease in erosion flux as erosion plane velocity decreases). 65

80 To determine whether the change in the anchoring and erosion plane velocities from the monodisperse to the binary case leads to an increase in the large particle erosion and a decrease in the small particle erosion, a system with fixed anchoring and erosion planes is simulated. The anchoring plane was placed at the bottom of the simulation domain and the erosion plane was placed at the same initial location as previous simulations. Figure 3.8 shows the change in the erosion flux for this case and the original simulations relative to the non-interacting mixture. For the large particles, the moving anchoring and erosion planes result in an erosion flux that is equal to or greater than that of the fixed planes. Similarly for the small particles, the erosion flux in the case of the moving planes is equal to or less than that of the fixed planes (when considering the confidence intervals). The change in the small particle erosion flux is smaller than that of the large particles. This difference is due to the large particles having a smaller velocity, which causes changes in the erosion plane velocity to have a larger effect on the flux (see equation (3.11)). Overall, this mechanism appears to be relatively negligible for the small particles and the large particles for the case of 75 wt% large particles, but is significant for the large particles in the cases of 25 wt% and 50 wt% large particles and. Based on the values shown in Table 3.4, this mechanism is of similar magnitude to mechanism (iv) for those two cases. 66

81 Figure 3.8. Binary: Change in erosion flux with respect to the expected non-interacting mixture for the base case (black diamond) and case with fixed anchoring/erosion planes (red square). Size ratio = 4 and 95% CI. Overall, mechanisms (i), (ii), and (iv) play a significant role in changing the species erosion flux in the binary case compared to the monodisperse case. Particularly, mechanisms (ii) and (iii) roughly cancel (due to being in opposite directions), with mechanism (i) leading to the results shown in Figure 3.3. Mechanism (v) also appears to play a role for the large particles (25 wt% and 50 wt% large particles cases only), and is of similar magnitude than the first three mechanisms based on the change in erosion flux shown in Figure 3.8 compared to the concentration changes in Table 3.4. Mechanism (iii) has a negligible effect on the erosion flux. 67

82 3.3.3 Lognormal PSD Figure 3.9 shows the erosion flux for the (truncated) lognormal distribution as a function of distribution width, for different truncation ratios. Also provided is the erosion flux of a monodisperse distribution (width of zero) with the same Sauter-mean diameter. The results indicate that the erosion flux for the continuous PSD increases slightly compared to the monodisperse case. However, for the non-zero widths examined, no statistically significant difference is observed between the erosion fluxes at differing widths. Results from a binary PSD showed similar results for 50% large particles by volume or greater, in that the total erosion flux increased relative to the monodisperse case. The cause of this behavior is considered in greater detail below. Figure 3.9. Total erosion flux of lognormal distribution as a function of distribution width. Error bars are 90% confidence intervals (CI) over multiple runs. The ratio of the erosion flux for each family, Geros,j, to its non-interacting erosion flux, Geros,non-int,j, for the case of 5 families is shown in Figure The non-interacting erosion flux 68

83 refers to the erosion flux of a family that would occur if no interactions between unlike particles occurred. The non-interacting erosion flux of a family j is defined as G N j wg (12) eros, nonint, j i eros, mono, i i1 where Nj is the number of particles in family j, wi is the volume fraction of particle i (fraction of solids volume represented by particle i), and Geros,mono,i is the monodisperse erosion flux of particle i. The monodisperse erosion flux is determined from Figure A function is then fit to the data and used to determine the erosion flux of a given size. The results show the erosion flux of the largest particles (family 5) increases relative to the monodisperse case, while the remaining, smaller-particle families generally decrease relative to monodisperse. This trend results in an overall increase in the total erosion flux because family 5 has a much larger volume than the other families combined. Figure Ratio of family erosion flux to non-interacting erosion flux for σ/µ (90% CI) 69

84 The trends in family erosion flux noted above are similar to our previous findings for a binary case, namely that the large-particle erosion flux increased and the small-particle erosion flux decreased relative to the monodisperse case [26]. Overall, the findings show that the erosion cannot be accurately modeled by a monodisperse distribution. First, the total erosion flux is increased slightly compared to the monodisperse distribution (see Figure 3.9). In addition, it is important to accurately model the species/family erosion flux because changes in the species flux can result in changes to the nature of the eroding PSD. Changes to the eroding PSD can potentially have implications for the particle motion after erosion in the far-field (not modeled here). One example of such an implication is additional erosion of large particles could lead to additional damage that might not be accounted for if the erosion flux of the large particles was determined without considering the increase in large particle erosion flux caused by small particles. Also, the contacts between unlike particles clearly play a large role in the erosion by enhancing the erosion of large particles and reducing the erosion of small particles. 3.4 Conclusions DEM is used to simulate the ejection of regolith from a spacecraft landing on the Moon with a focus on binary and lognormal size distributions. When comparing the erosion flux of the binary case to the monodisperse case, the total erosion flux initially decreases relative to the erosion flux expected based on the monodisperse case as large particles are added. However, the erosion flux is increased above the expected erosion flux as the fraction of large particles is increased further. In addition, the large particle erosion flux increases and that of the small particle decreases relative 70

85 to the expected erosion flux based on the monodisperse case. The dominant mechanisms for this change in the species erosion flux are a collisional transfer of momentum between unlike particles below the erosion plane, contacts across the erosion plane, and a change in concentration due to tighter packing than the monodisperse case. In addition, a notable effect on the erosion flux of large particles is caused by a change in the rate of penetration of the gas caused by the presence of multiple particle sizes. The details of these mechanisms and how they are related to individual particles and particle collisions are discussed in section The size distributions used here (up to size ratio of 4 due to high computational overhead) are quite narrow compared to lunar regolith or other regoliths, which span several orders of magnitude. Given that the change in the small and large particle erosion fluxes increases with increasing size ratio, the trends shown here may be magnified with the larger particle size disparities present in regolith. However, large size ratios are computationally prohibitive due to an increase in particle number, reduction in time step, and increase in cost of parallel computational overhead [1]. A continuum description would help to alleviate these issues and thus is needed to be able to simulate the system. The erosion flux for a truncated, (continuous) lognormal PSD is also examined. The results show that the erosion flux increases slightly from the monodisperse case with the same Sautermean diameter but does not vary significantly for the non-zero distribution widths and truncation ratios examined. Similar to the binary case, the largest particles are boosted and smaller particles hindered. These results show that collisions, particularly those between unlike particles, play an important role in the erosion process. Therefore, the system cannot be modeled without collisions, such as with a single particle trajectory model, or with a monodisperse PSD. 71

86 3.5 References 1. Berger, K.J. and Hrenya, C.M., Challenges of DEM: II. Wide particle size distributions. Powder Technology, (0): p Goodings, D.J. and Lin, C.P., Geotechnical Properties of the Maryland-Sanders Lunar Simulant. Geotechnical Testing Journal, (2): p Li, Y.Q., Liu, J.Z., and Yue, Z.Y., NAO-1: Lunar Highland Soil Simulant Developed in China. Journal of Aerospace Engineering, (1): p Arslan, H., Batiste, S., and Sture, S., Engineering Properties of Lunar Soil Simulant JSC- 1A. Journal of Aerospace Engineering, (1): p Rame, E., Wilkinson, A., Elliot, A., and Young, C., Flowability of lunar soil simulant JSC- 1a. Granular Matter, (2): p Hartzell, C.M. and Scheeres, D.J., The role of cohesive forces in particle launching on the Moon and asteroids. Planetary and Space Science, (14): p Li, Y.Q., Liu, J.Z., Zou, Y.L., Ouyang, Z.Y., Zheng, Y.C., and Yue, Z.Y., Two Lunar Mare Soil Simulants. Acta Geologica Sinica-English Edition, (5): p Ottino, J.M. and Khakhar, D.V., Mixing and segregation of granular materials. Annual Review of Fluid Mechanics, : p Sundaresan, S., Some outstanding questions in handling of cohesionless particles. Powder Technology, (1): p Curtis, J.S. and van Wachem, B., Modeling particle-laden flows: A research outlook. AIChE Journal, (11): p Hrenya, C.M., Kinetic theory for granular materials: Polydispersity, in Computational Gas-Solids Flows and Reacting Systems: Theory, Methods and Practice, M.S. S. Pannala, T. O'Brien, Editor. 2011, IGI Global: Hershey, PA. p Chew, J.W., Parker, D.M., and Hrenya, C.M., Elutriation and Species Segregation Characteristics of Polydisperse Mixtures of Group B Particles in a dilute CFB Riser. AIChE Journal, (1): p Peters, I.R., Xu, Q., and Jaeger, H.M., Splashing Onset in Dense Suspension Droplets. Physical Review Letters, (2): p Bagnold, R.A., The physics of wind blown sand and desert dunes. Methuen, London,

87 15. SØRensen, M. and McEwan, I.A.N., On the effect of mid-air collisions on aeolian saltation. Sedimentology, (1): p Dong, Z., Huang, N., and Liu, X., Simulation of the probability of midair interparticle collisions in an aeolian saltating cloud. Journal of Geophysical Research: Atmospheres, (D24): p. D Huang, N., Zhang, Y., and D'Adamo, R., A model of the trajectories and midair collision probabilities of sand particles in a steady state saltation cloud. Journal of Geophysical Research: Atmospheres, (D8): p. D Kok, J.F. and Renno, N.O., A comprehensive numerical model of steady state saltation (COMSALT). Journal of Geophysical Research: Atmospheres, (D17): p. D Jasper, F.K., Eric, J.R.P., Timothy, I.M., and Diana Bou, K., The physics of wind-blown sand and dust. Reports on Progress in Physics, (10): p Carneiro, M.V., Araújo, N.A.M., Pähtz, T., and Herrmann, H.J., Midair Collisions Enhance Saltation. Physical Review Letters, (5): p Berger, K.J., Anand, A., Metzger, P.T., and Hrenya, C.M., Role of collisions in erosion of regolith during a lunar landing. Physical Review E, (2): p Berger, K.J., Anand, A., Metzger, P.T., and Hrenya, C.M., Erratum: Role of collisions in erosion of regolith during a lunar landing. Physical Review E, (1): p Morris, A.B., Goldstein, D.B., Varghese, P.L., and Trafton, L.M., Approach for Modeling Rocket Plume Impingement and Dust Dispersal on the Moon. Journal of Spacecraft and Rockets, 2015: p Provided by NASA Kennedy Space Center. 25. Immer, C., Metzger, P., Hintze, P.E., Nick, A., and Horan, R., Apollo 12 Lunar Module exhaust plume impingement on Lunar Surveyor III. Icarus, (2): p Berger, K.J. and Hrenya, C.M., Impact of a binary size distribution on the particle erosion due to an impinging gas plume. AIChe Journal, In Press. 73

88 4. ACCURACY OF SINGLE-PARTICLE DRAG FORCE E Abstract The work described in previous and following chapters uses a lift and drag law for a single particle (Loth 2008, Loth 2008). It is well known that drag on a particle is increased due to the presence of nearby particles. However, the influence of rarefication, measured by the Knudsen number (Kn), has only been studied for small Reynolds number (Re). In this chapter, Lattice Boltzmann (LB) simulations of boundary-slip flows through periodic arrays of spheres are used to examine the increase in the drag force on multiple particles compared to a single particle for both low and finite Reynolds numbers (Re). Using a first-order slip boundary condition at the particle surface, the drag force, found via direct solution of the Navier-Stokes equation, is established for three-dimensional simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC) lattices. Starting from the momentum equation, the effect of boundary-slip on the rate of dissipation of the system is determined. The boundary-slip reduces the kinetic energy dissipated through viscous forces, which results in a reduction in the drag force. For the case of low Re with boundary-slip, the change in drag is a function of solids volume fraction, tangential momentum accommodation σ, and the Knudsen number. For finite Re, the change in drag with boundary-slip is a quadratic function of Re as is the case without boundary-slip (Re < 10). The joint influence of boundary-slip and inertia is well approximated by the multiplication of their respective individual contributions in the range Re < 10 and Kn (2 σ) σ < The findings demonstrate as Kn increases for small Kn, the E Berger, K.J., Hrenya, C.M., Wang, L., and Yin, X., Gas-solid drag on multiple particles in the slip flow regime at low and finite Reynolds numbers. Submitted to International Journal of Multiphase Flow,

89 difference between the drag on multiple particles compared to that of a single particle is reduced for both low and finite Reynolds numbers, suggesting that a single particle drag law may be reasonably applied at larger Kn. 4.1 Introduction It is well known that when particles, such as sand grains, are entrained in a fluid with a nonzero relative velocity, the drag force on a given particle changes when other particles are nearby. The simplest documented case of this phenomenon is that of sand grains falling in a stationary fluid. This case is known as hindered settling because grains homogeneously distributed in the fluid will fall slower than a single grain would. This change is due to the increase in the drag force on the grains owning to the presence of nearby grains [1-9]. The effect of nearby particles on the drag is usually addressed by inclusion of the solid volume fraction in drag laws. For particles of a uniform size, drag in the Stokes regime is strictly a function of and particle configuration [10, 11]. Outside the Stokes regime, the Reynolds number is used to describe the effect of fluid inertia. The empirical laws of Ergun [12] and Forchheimer [13] proposed quadratic dependences of drag on fluid-particle relative velocity. The empirical law of Wen and Yu [14], on the other hand, used the power-law hindered settling function developed by Richardson and Zaki [1] and incorporated Re into the exponent. The Gidaspow [15] model widely used in gas-solid flow modeling is a blend of the Wen-Yu and Ergun correlations. The effect of Reynolds number on fluid-particle drag has since then been characterized computationally by a number of authors, leading to several more accurate drag laws [16-19]. Here, we adopt Gidaspow s definition of the Reynolds number 75

90 Re u u d g g s p (4.1) where g is the gas density, u g is the gas velocity, diameter, is the gas dynamic viscosity. u s is the solids velocity, d p is the particle All of the above models apply to flows free of non-continuum effects, i.e., the Knudsen number is less than about The Knudsen number, Kn, is defined as Kn (4.2) r where is the mean free path of the gas and r is the radius of the particle. As the Knudsen number increases, in addition to being a function of the solids volume fraction, the drag force also decreases as the flow begins to become rarefied. The simplest relationship for the drag force on a single particle with Kn > is the Cunningham correction factor, which is used to modify the drag coefficient of a single particle [20]. The Cunningham correction factor, C, is defined as C 1 KnA1 A2e 2 A3 Kn (4.3) where A 1, A 2, and A 3 are experimentally determined coefficients. The Cunnnigham correction factor is generally only valid in the Stokes flow regime. More recent work, performed by Loth [21], combined the Cunningham correction factor, along with other relationships, to form a set of drag relationships that span a wide range of Kn and Re, although they are still only for a single particle. 76

91 Boundary-slip flows that involve multiple particles, to our knowledge, have only been studied in the Stokes regime. Stokes boundary-slip flow through arrays of cylinders or spheres have been studied by several authors [22-27]. The problem of moving particles, which is important for the understanding of the breakdown of hydrodynamic interactions, has been studied by Ying and Peters [28] and Chun and Koch [29]. The former solved the case of two spheres in a slightly rarefied gas in the Stokes regime using a linearized Boltzmann equation, while the latter developed a direct simulation Monte Carlo method and investigated flow around two spheres at moderate Knudsen numbers. The above review indicates that the drag for multi-particle systems that involve not only rarefaction effects but also finite Re (non-stokes flow) has not been studied. While gas-particle boundary-slip is generally observed on small particles that are typically associated with low Reynolds numbers, finite Re and rarefaction can co-exist in, for example, certain aerosol systems under rarefied conditions such as those involved in combustion and propulsion, and extraterrestrial systems such as the soil ejection from a lunar landing or dust transport in the Martian atmosphere [30-37]. The purpose of this study is to investigate the drag force of periodic arrays of particles as it varies with Kn and Re in an effort to determine if a single particle drag law may be applicable in rarefied, or large Kn, flows. In this study, we limit the range of Kn such that the effect of rarefaction is limited to a surface slip. That is, the bulk flow is still governed by the Navier-Stokes equations and the effect of rarefaction is addressed by replacing the no-slip boundary condition on the particle surface with the Maxwell first-order slip boundary condition [38] 77

92 2 us n I nn (4.4) where u s is the slip velocity at the boundary, is the tangential momentum accommodation coefficient (TMAC), is the mean free path, is the dynamic viscosity, is the viscous stress tensor, I is the unit tensor, and n is the unit normal vector pointing from the solid wall to the fluid. for Newtonian fluids is directly proportional to the velocity gradient tensor T u u (4.5) TMAC is the fraction of incident molecules that are reflected diffusely, with a typical value close to 1 [39-43]. We also limit the range of Re such that the Mach number Ma, which is proportional to the product of Kn and Re, is small. As gas compressibility is proportional to Ma 2, flow in the periodic cell is nearly incompressible. The use of an incompressible Navier-Stokes solver is thus justified; it also ensures that the application of equation (4) to a curved surface is the correct reduced form of the higher-order Maxwell-Burnett boundary condition [42]. Results suggest that a single particle drag law may be applicable at larger Kn for both low and finite Re. 4.2 Methods The lattice-boltzmann method, a type of direct numerical simulation method, is used to simulate boundary-slip flow through periodic arrays of spheres. The lattice-boltzmann method is a mesoscopic method that recovers the Navier-Stokes equations though simulation of a lattice model of molecular transport. The method consists of using a lattice of nodes that discretizes the 78

93 computational domain, each of which represents either the fluid or the solid. Each fluid node is occupied by a discretized velocity distribution function, n r, t i, where subscript i refers to the index of discrete velocities, r is the position of the fluid node, and t is the time. In this study, the D3Q19 discrete velocity model is used, meaning 3 dimensions and 19 velocities. The velocity distribution function evolves in time according to a discrete analogue of the Boltzmann equation: n ( r c t, t t) n ( r, t) [ n ( r, t)] (4.6) i i i i i where i is the change in ni due to instantaneous molecular collisions at the lattice nodes and t is the time step [44]. The collision operator, i, used in this study is the two-relaxation-time (TRT) collision operator described in Ladd and Verberg [44]. This model utilizes two relaxation times, s and v, which are related to the shear and bulk viscosities, s and b, via s cst s 2 (4.7) b 3 v 2 2 2cs t 1 1 (4.8) where 36 is the fluid material density, c 13 is the speed of sound, and t is the time s step, all of which are in the lattice units, which are the non-dimensional units of the Lattice- Boltzmann method. All results described below are in terms of non-dimensional numbers and so are not restricted to the values of the parameters chosen. The collision operator and the relaxation 79

94 times describe the decay of the non-equilibrium part of the velocity distribution function due to fluid viscosity. The lattice Boltzmann method utilizes a two-step scheme to simulate the evolution of distributions in each time step. First, the collision step is performed in which the collision operator is applied to each node to simulate collisions. Then, distributions are propagated between neighboring nodes in the propagation step to yield the new velocity distribution function at each node. When distributions are propagated toward a solid surface, several types of boundary conditions can be implemented. The simplest is known as the link-bounce-back rule in which the velocity distribution inbound to a solid node is bounced back to the fluid node with the velocity reversed and modified by a correction term n '( r, t t) n *( r, t) ( r, t) (4.9) i i i b where n ' represents the velocity distribution in the opposite direction of the incident distribution i to the solid node, n i *, and r b is halfway between the fluid node and the solid node. One can choose to model no-slip, free-slip, or partial-slip boundary conditions. Expressions for no-slip i and free-slip boundary conditions can be found in Ladd and Verberg [44] and Yin et al. [45]. The that recovers the Maxwell first-order slip boundary condition, which is used in this study, was i derived in Wang and Yin [27]. 80

95 The inputs to the Lattice-Boltzmann model include the body force, the relaxation times, the tangential momentum accommodation coefficient, the fluid mean free path, and the geometrical parameters. The body force, which is equivalent to the pressure drop, controls the fluid velocity in the simulation. The larger the body force, the larger the fluid velocity and thus the larger the Reynolds number. The relaxation times, s and v, are chosen to yield the desired shear and bulk viscosity as described in equations (4.7) and (4.8). The fluid mean free path is chosen to yield the desired Kn based on the particle size. Finally, the particle systems studied include simple cubic (SC), body-centered-cubic (BCC) and face-centered-cubic (FCC) arrays of spheres. For these arrays, the lattice resolution, L, used to resolve the dimension of a unit cell is first chosen; then, the particle size is selected based on the desired solids volume fraction. Figure 4.1 shows a simple cubic system with L = 60 with s = A single particle is representative of a simple cubic lattice because of the periodic boundaries on each face (two full particles are used for BCC and four for FCC). Table 4.1 shows the range of simulation parameters used in this study and the range of dimensionless parameters. Because it is the combination 2 S Kn that controls the boundary-slip in the Maxwell boundary condition, we define Sl as the dimensionless boundaryslip length and present Sl instead of the individual values of σ and Kn. l 81

96 Figure 4.1 Spatial configuration for a simple cubic lattice with a L= 60 with εs=0.25. Table 4.1 Range of simulation parameters used throughout this study Parameter Simple Cubic Body-centered Cubic Face-centered Cubic Body Force to to to s v Mean free path, Box Size, L Solids fraction, s Fractional Diameter, dp / L Re to to to 5 Kn S Kn l The output from the lattice-boltzmann simulations is the steady-state velocity distribution functions. By taking the moments of the distributions, fluid density ρ, momentum j, and momentum flux can be obtained: 82

97 n (4.10) i i j u c (4.11) ni i i (4.12) nicc i i i In the above equations, tensor is defined as c i are the discrete velocities in the D3Q19 model [44]. The Euler stress E pi uu (4.13) where p is the pressure. The viscous stress tensor is determined using [46] 1 1 E (4.14) 2 s In addition, using the obtained velocity field (via the momentum density), the average velocity and dimensionless drag are found. The dimensionless drag is defined as the ratio of the drag coefficient of each particle in the presence of other particles compared to that for a single particle FGS FGS C ua u dp D, mp 2 8 FGS 1 s F (4.15) C D, sp 3d pu s Re ud p 83

98 where F is the dimensionless drag, C is the drag coefficient for the multi-particle case (i.e. D, mp array), C is the drag coefficient for the single particle case, F D, sp GS is the average per-particle gassolid interactive force, A is the cross-sectional area of the particle perpendicular to the flow, u is the fluid speed averaged over the fluid nodes, is the solids volume fraction and s superficial speed related to u by U u1 s s U is the s. Note that this definition of F has an additional factor of (1- εs) 2 compared to previous definitions by Sangani and Acrivos [10] and Hill et al. [16]. Due to momentum conservation, the gas-solid interaction force on the particle is equivalent to the force on the fluid system due to the body force (force density) f B F GS 1 3 f N f L B fluids B s (4.16) n n where unit cell. N fluids and n are, respectively, the number of fluid nodes and the number of particles in the 4.3 Results and discussion In order to investigate the effects of slip boundary condition on the drag of multiple particles, we begin with examining the drag in the Stokes flow regime. Starting from the momentum equation for the fluid, a simple equation for the dissipation of momentum is established. Finally, the dimensionless drag in the small Re regime (Re < 10) is examined. 84

99 4.3.1 Stokes flow over periodic arrays of spheres Wang and Yin [27] examined boundary-slip flow through periodic arrays of spheres, along with other systems, for the Stokes flow regime. Utilizing the same techniques, the Stokes results are replicated here, with a slight modification. In order to facilitate modeling of gas-solid flows with boundary-slip, the dimensionless drag (equation (4.15)) is used, as opposed to the permeability used in [25]. Figure 4.2 shows the dependence of the dimensionless drag for SC arrays as a function of the solids fraction and the dimensionless boundary-slip length, Sl, in the Stokes flow regime (Re < 0.1). The values shown are based on a linear extrapolation of the dimensionless drag as a function of grid resolution in an attempt to estimate the value of the dimensionless drag at an infinite grid resolution. In order to demonstrate how the drag force changes on a single isolated particle with boundary-slip, the drag calculated via the Cunningham correction factor and the Loth drag correlations, which account for boundary-slip but not multiple particles, are also shown. These correlations are dependent on the Knudsen number and not the dimensionless boundary-slip length so values for Kn used are 0, 0.005, 0.01, 0.025, and The figure demonstrates that the dimensionless drag force decreases with increasing Sl, which is the same as that for a single particle (change with S in single particle case in Figure 4.2 is minimal due to small Sl). However, as the solids fraction increases, the decrease in the drag force with increasing Sl becomes larger. Thus, as Sl increases, the drag becomes more similar to that of a single particle, but admittedly, may or may not asymptotically approach the single particle case. 85

100 Figure 4.2. Dimensionless drag vs. Sl for very small Re as a function of solids fraction for TMAC of 0.5, along with the Cunningham correction factor [20] and the Loth correlations [21]. The numbers along the vertical axis (Sl = 0) are in good agreement with Sangani and Acrivos [10] Relation between surface boundary-slip and drag reduction in periodic cells In an attempt to understand the cause of the reduction in the drag force, we compute the dissipation over the unit cell. Starting from the Eulerian momentum balance in the Navier-Stokes equation, assuming an incompressible fluid U t 2 U U p fb U (4.17) where U is the fluid velocity, t is time, ρ is the fluid density, p is the fluid pressure, and µ is the dynamic viscosity. The dot product of equation (19) with the fluid velocity U gives an equation for the dissipation. Further, by applying volume averaging and the divergence theorem to the 86

101 dissipation equation, for flows through periodic arrays of spheres with no boundary-slip at the particle surface, we have fluid 1 U fb τ τ (4.18) 2 fluid In order to evaluate equation (4.18) using the simulation results, the equation can be discretized on the lattice as fluid nodes 1 U fb τ τ (4.19) 2 flud nodes Each of these terms can be identified in terms of the hydrodynamics of the system. The term on the left represents the body force (pressure drop) applied on the system, which will now be referred to as the forcing term. The term on the right represents the viscous dissipation within the fluid, which will now be referred to as the fluid dissipation term. By applying first-order Maxwell boundary-slip at the particle surface, the dissipation equation becomes U fb τ τ ds 2 n τ I nn τ n (4.20) fluid fluid S nodes nodes The last term on the right in equation (4.20) represents an additional dissipation due to boundaryslip. 87

102 For a given simulation, the difference between the forcing term and the fluid dissipation term therefore gives the boundary-slip term (for non-zero Kn). Figure 4.3 shows the forcing and dissipation terms as a function of Kn for a solids volume fraction of 0.10 and 0.40 for two different values of σ for a SC cell. Note that both terms were non-dimensionalized using d f L p B. The difference between the square symbols and the diamond symbols is indicative of the size of the boundary-slip term. Results obtained for other solids fractions are qualitatively similar. Figure 4.3 shows that, for a given forcing term, when boundary-slip occurs at the surface, the fraction of dissipation due to the fluid dissipation term is reduced. The boundary-slip term grows with increasing Kn and grows with increasing solid fraction. The presence of the boundary-slip term increases the superficial velocity and decreases the drag of the particle. This result may be seen through the following derivations. We can rewrite (4.20) into 1 2 Us τ τ ds 2 f f L n τ I nn τ n (4.21) 3 B B S where U s is the superficial velocity, the volume average of U in the unit cell, and 1 τ τ 2 is the volume average of the viscous dissipation. It is clear that the boundary-slip term now adds to the superficial velocity of the fluid. With equations (4.15) and (4.16), we can derive an expression for F fbl 1 s 1 2 F 3 ds 3dp 2 τ τ L n τ I nn τ n (4.22) S 88 1

103 The boundary-slip term, based on equation (4.22), clearly serves to reduce the drag compared to the no-slip situation. 89

104 Figure 4.3. The forcing and fluid dissipation terms as a function of Knudsen number and TMAC (σ) for L = 120 and a solids volume fraction of (a) 0.1 and (b) 0.4. The difference between the terms is the boundary-slip term. 90

105 4.3.3 Small, Finite Re flow over periodic arrays of spheres When Re is greater than about 0.1, the dimensionless drag is also a function of Re. Hill et al. [16] explored the finite-re drag on multiple particles for the no-slip case. They found that the dimensionless drag can be decomposed into inertial and Stokes contributions as F F Re F (4.23) where the first term is the inertial contribution, with proportionality constant F, and the second 1 term, F, is the Stokes contribution. Note that the scaling with Re 2 is only valid for about Re < 10, 0 and the scaling becomes Re 1 for larger Re. Figure 4.4 shows the dimensionless drag as a function of Re for a single solids fraction (0.2) and varying Sl for a simple cubic lattice. The drag shown is based on values calculated at L = 120, but scaled to represent an infinitely large domain. Such scaling follows the method presented by Wang and Yin [27], who showed that the dimensionless permeability, and correspondingly the inverse of the dimensionless drag, scales linearly with the inverse of the domain size. The fittings in Figure 4 are all quadratic to Re following equation (4.23). It is clear that the inertial scaling still holds for the boundary-slip case. However, the coefficients are now functions of Sl, along with being functions of solids fraction. For the no-slip case, the results match well with those reported by Hill et al. [16]. 91

106 Figure 4.4. Example of dimensionless drag vs Re for varying Sl ( s 0.4, SC array). Points are LBM data and dashed lines are fits to the curve F F F Re In order to generate an equation that can be used to determine the dimensionless drag for any value of Sl and Re, we combined equation (24) with the scaling of F in the absence of Re [27] F0 F= 1S S l 2 l (4.24) leading to the following equation: 2 F0 F1Re F = 1S S l 2 l (4.25) Equation (4.25) combines F1 determined in the absence of boundary-slip, and α and β determined in the absence of fluid inertia; it is built upon the notion that in the limit of small Sl and small-but- 92

107 finite Re the effects of rarefaction and fluid inertia should contribute nearly independently to the drag. We find that this correlation fits the simulation data very well except at the extremes of Sl and Re. Figure 4.5 shows the fits for an SC array with s 0.4. The deviation at Re = 8 for Sl = 0.03 is, in fact, the worst deviation observed in the entire data set. Fit parameters for simple cubic, body-centered cubic, and face-centered cubic for all solids fractions are shown in Figure 4.6. The parameters shown are in good agreement with those presented in Wang and Yin [27], with very small deviation in α and slightly larger deviation in β. However, Wang and Yin presented a fit for a different range of Sl. Figure 4.5. Fit of equation (4.25) to dimensionless drag vs Re for varying Sl (εs=0.4, SC array). 93

108 Figure 4.6. Fitting parameters for equation (4.25). Of note is the nonlinear dependence of the dimensionless drag on the dimensionless boundary-slip, (or alternatively the Knudsen number) as indicated by the parameter β. This result arises despite the use of a first-order boundary condition. Previous numerical studies have shown this nonlinear dependence can arise from the use of a first-order boundary condition [47, 48]. This phenomenon is due to the curved surfaces of the particles. As implied by equation (4.22), 0 holds only when the viscous stress τ is not affected by the surface boundary-slip. While this is true for flows in channels and cylindrical tubes, we do not expect that it holds for flows with curved surfaces. Experimental evidence of this nonlinear behavior has indeed been observed [49]. Figure 4.5 shows that equation (4.25) results in a small deviation, up to 3.8%, from the simulation data for higher Re. In particular, as Sl increases, this deviation is observed at smaller 94

109 Re. This result suggests that additional terms may be necessary, particularly terms that scale with both Sl and Re. However, the form of such a term is not clear. 4.4 Conclusions In the above, we saw that as Knudsen number or dimensionless boundary-slip increases, the dependence of the dimensionless drag force on both solids fraction and Reynolds number is reduced. The influence of Kn or Sl on the drag is nearly independent of that of inertia, and can be correlated by a simple equation. This result signifies that as Kn or Sl increases, the influence of nearby particles on the drag is reduced. This finding is in agreement with the expectation that, as the disturbance felt by a particle due to a neighbor particle is proportional to the neighbor particles drag force in Stokes and weakly inertial flows, reduction in drag should lead to reduced influence of the solid fraction. We expect that as the Kn continues to increase beyond the range explored here, the drag force will continue to become more similar to that of a single particle. If the Knudsen number is sufficiently large, it may be sufficient to model the drag on multiple particles as if the particles were isolated, which simplifies calculations significantly. However, more work is needed to investigate a wider range of Kn using a model like direct simulation Monte Carlo in order to resolve the details at higher Kn and determine the asymptotic behavior toward the limit at higher Kn [50]. 95

110 4.5 References 1. Richardson, J. and W. Zaki, The sedimentation of a suspension of uniform spheres under conditions of viscous flow. Chemical Engineering Science, (2): p Batchelor, G.K., Sedimentation in a dilute dispersion of spheres. Journal of Fluid Mechanics, (02): p R H Davis, a. and A. Acrivos, Sedimentation of Noncolloidal Particles at Low Reynolds Numbers. Annual Review of Fluid Mechanics, (1): p Brady, J.F. and G. Bossis, Stokesian Dynamics. Annual Review of Fluid Mechanics, (1): p Ladd, A.J.C., Hydrodynamic transport coefficients of random dispersions of hard spheres. The Journal of Chemical Physics, (5): p Ladd, A.J., Dynamical simulations of sedimenting spheres. Physics of Fluids A: Fluid Dynamics ( ), (2): p Mo, G. and A.S. Sangani, A method for computing Stokes flow interactions among spherical objects and its application to suspensions of drops and porous particles. Physics of Fluids (1994-present), (5): p Richardson, J.F. and W.N. Zaki, Sedimentation and fluidisation: Part I. Chemical Engineering Research and Design, , Supplement: p. S82-S Yin, X. and D.L. Koch, Hindered settling velocity and microstructure in suspensions of solid spheres with moderate Reynolds numbers. Physics of Fluids (1994-present), (9): p Sangani, A.S. and A. Acrivos, Slow flow through a periodic array of spheres. International Journal of Multiphase Flow, (4): p Koch, D.L. and A.S. Sangani, Particle pressure and marginal stability limits for a homogeneous monodiperse gas-fluidized bed: Kinetic theory and numerical simulations. Journal of Fluid Mechanics, (1): p Ergun, S., Fluid Flow Through Packed Columns. Chemical Engineering Progress, (2): p Forchheimer, P., Wasserbewegung durch boden. Z. Ver. Deutsch. Ing, (1782): p

111 14. Wen, C.Y.a.Y., Y. H., Mechanics of Fluidization. Chem. Eng. Pro. Symp., : p Gidaspow, D., Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions. 1994: Academic Press. 16. Hill, R.J., D.L. Koch, and A.J.C. Ladd, The first effects of fluid inertia on flows in ordered and random arrays of spheres. Journal of Fluid Mechanics, : p Hill, R.J., D.L. Koch, and A.J. Ladd, Moderate-Reynolds-number flows in ordered and random arrays of spheres. Journal of Fluid Mechanics, : p Beetstra, R., M. Van der Hoef, and J. Kuipers, Drag force of intermediate Reynolds number flow past mono and bidisperse arrays of spheres. AIChE Journal, (2): p Tenneti, S., R. Garg, and S. Subramaniam, Drag law for monodisperse gas solid systems using particle-resolved direct numerical simulation of flow past fixed assemblies of spheres. International journal of multiphase flow, (9): p Cunningham, E., On the Velocity of Steady Fall of Spherical Particles through Fluid Medium. Proc R. Soc. Lond. A, (563): p Loth, E., Compressibility and rarefaction effects on drag of a spherical particle. AIAA Journal, (9): p Pich, J., Pressure drop of fibrous filters at small Knudsen numbers. Annals of Occupational Hygiene, (1): p Bruschke, M. and S. Advani, Flow of generalized Newtonian fluids across a periodic array of cylinders. Journal of Rheology (1978-present), (3): p Datta, S. and S. Deo. Stokes flow with slip and Kuwabara boundary conditions. in Proceedings of the Indian Academy of Sciences-Mathematical Sciences Springer. 25. Wang, C.Y., Stokes slip flow through square and triangular arrays of circular cylinders. Fluid Dynamics Research, (5): p Lasseux, D., et al., A macroscopic model for slightly compressible gas slip-flow in homogeneous porous media. Physics of Fluids (1994-present), (5): p Wang, L., Yin, X., Lattice Boltzmann simulation of fluid flow with first-order slip through periodic arrays of spheres. Submitted to Physical Review E,

112 28. Ying, R. and M.H. Peters, Hydrodynamic interaction of two unequal-sized spheres in a slightly rarefied gas: resistance and mobility functions. Journal of Fluid Mechanics, : p Chun, J. and D.L. Koch, A direct simulation Monte Carlo method for rarefied gas flows in the limit of small Mach number. Physics of Fluids, (10): p Sagan, C. and C. Chyba, Triton's streaks as windblown dust. Nature, (6284): p Pratsinis, S.E., W. Zhu, and S. Vemury, The role of gas mixing in flame synthesis of titania powders. Powder Technology, (1): p Schnell, M., C.S. Cheung, and C.W. Leung, Investigation on the coagulation and deposition of combustion particles in an enclosed chamber with and without stirring. Journal of Aerosol Science, (11): p Lobo, P., et al., Physical Characterization or Aerosol Emissions from a Commercial Gas Turbine Engine. Journal of Propulsion and Power, (5): p Petrosyan, A., et al., THE MARTIAN ATMOSPHERIC BOUNDARY LAYER. Reviews of Geophysics, (3): p. RG Jasper, F.K., et al., The physics of wind-blown sand and dust. Reports on Progress in Physics, (10): p Berger, K.J., et al., Role of collisions in erosion of regolith during a lunar landing. Physical Review E, (2): p Berger, K.J., et al., Erratum: Role of collisions in erosion of regolith during a lunar landing. Physical Review E, (1): p Maxwell, J.C., On Stresses in Rarefied Gases Arising from Inequalities of Temperature. Proceedings of the Royal Society of London, ( ): p Davis, M.H., Collisions of Small Cloud Droplets: Gas Kinetic Effects. Journal of the Atmospheric Sciences, (5): p Dahneke, B., Comments on Collisions of Small Cloud Droplets: Gas Kinetic Effects. Journal of the Atmospheric Sciences, (3): p Allen, M.D. and O.G. Raabe, Re-evaluation of millikan's oil drop data for the motion of small particles in air. Journal of Aerosol Science, (6): p Lockerby, D.A., et al., Velocity boundary condition at solid walls in rarefied gas calculations. Physical Review E, (1): p

113 43. Agrawal, A. and S.V. Prabhu, Survey on measurement of tangential momentum accommodation coefficient. Journal of Vacuum Science & Technology A, (4): p Ladd, A.J.C. and R. Verberg, Lattice-Boltzmann Simulations of Particle-Fluid Suspensions. Journal of Statistical Physics, (5-6): p Yin, X., D.L. Koch, and R. Verberg, Lattice-Boltzmann method for simulating spherical bubbles with no tangential stress boundary conditions. Physical Review E, (2): p Krüger, T., F. Varnik, and D. Raabe, Second-order convergence of the deviatoric stress tensor in the standard Bhatnagar-Gross-Krook lattice Boltzmann method. Physical Review E, (2): p Chai, Z., et al., Gas slippage effect on the permeability of circular cylinders in a square array. International Journal of Heat and Mass Transfer, (13 14): p Silin, D., Digital Rock Studies of Tight Porous Media Klinkenberg, L.J., The Permeability Of Porous Media To Liquids And Gases. 1941, American Petroleum Institute. 50. Bird, G.A., Approach to Translational Equilibrium in a Rigid Sphere Gas. Physics of Fluids ( ), (10): p

114 5. ACCURACY OF DISCRETIZING CONTINUOUS PARTICLE SIZE DISTRIBUTION F Abstract The discrete element method (DEM) model used in previous chapters is too computationally expensive to use as a tool to study the full extent of the lunar erosion problem. However, conventional kinetic-theory-based continuum models (i.e., those based on a Chapman-Enskog expansion) cannot directly handle a continuous particle size distribution (PSD) due to the number of differential equations scaling with the number of particle species. In this chapter, two methods of discretizing a continuous PSD are explored in order to facilitate the use of a continuum model: the method of matching moments developed by Morris et al. (Physics of Fluids, 2012) and volumetric discretization, a new method. The former matches the moments of the continuous distribution to a limited number of discrete sizes and the latter divides the particles into bins of equal volume and the sizes are determined by the arithmetic mean of each bin. The volumetric discretization method appears to better match the erosion flux of the truncated lognormal distribution used in this work. In addition, the scalar transport coefficients from the polydisperse continuum theory of Garzό, Hrenya, and Dufty (Physical Review E, 2007) for the two discretizations and the DEM results of Dahl et al. (Powder Technology, 2003) are compared for purposes of validating the discrete approximation. F Berger, K.J. and C.M. Hrenya, Predicting regolith erosion due to a lunar landing: Role of a continuous size distribution. Submitted to Icarus. 100

115 5.1 Introduction Conventional kinetic-theory-based continuum models (i.e. those based on a Chapman-Enskog expansion) require a discrete number of particle species due to the number of differential equations scaling with the number of particle species [1-3]. Thus, in order to predict the flow behavior of a continuous PSD, such as the lunar distribution or the lognormal distribution used in Chapter 3, a method of discretization (i.e., approximating the continuous distribution with a series of discrete particle sizes) must be chosen. Here, two methods are investigated: the method of matching moments from Murray et al. [4] and a new method called volumetric discretization. The method of matching moments under-predicts the erosion flux of the original distribution and underpredicts more severely for wider distributions (ratio of standard deviation to mean). The volumetric discretization method slightly under-predicts the erosion flux for several cases, but the match is overall much better. 5.2 Methods The method of matching moments is used to discretize any continuous distribution to S sizes by matching the first (2S-1) moments of the discrete approximation to that of the continuous distribution under consideration. The sizes and compositions are selected by solving the following equation for each moment S k k xd i p, i i1 (5.1) 101

116 where µk is the k th moment of the distribution, xi is the number fraction of particles of species i, and dp,i is the diameter of species i. The volumetric discretization method instead involves dividing the distribution into S bins of equal volume. The size of the particle representing a given bin is determined by the local arithmetic mean of each bin based on number. The volumetric fraction (fraction of solids volume represented by a particle size) of each particle size is equal and can be simply converted to the number fraction via the particle size. Other schemes were tested, including dividing the bins by number and area and selecting the Sauter-mean diameter, but volumetric discretization method described above performed best and so the others will not be discussed further. As a point of reference, the two discretizatio methods for S=10, along with a truncated lognormal and true lognormal for the case of σ/µ=1 and ϕtrunc=25 are shown in Figure 5.1 The discrete and continuous distributions can not be directly compared quantitatively, however the values in the figure can be compared qualitatively. Note that the volumetric discretization has slightly more of the smallest and largest particles than the moments discretization, which may play a role in the results explored below. 102

117 Figure 5.1. Ratio of number fraction to particle size for S=10 discretizations for both discretization methods and number-based probability distribution functions for truncated and true lognormal PSD. The discretized distributions displayed in Figure 5.1 are generated using the truncated (continuous) lognormal distribution with the parameters in Table 5.1 (see chapter 3 for more details on the truncated lognormal distribution). The other system parameters are the same as those in tables 2.1 and 3.1. Table 5.1 Discretized PSD parameters Property Value dsm 5.0 x 10-5 m ϕtrunc 25 σ/µ 0.25, 0.5, 1 103

118 5.3 Results and Discussion Erosion Flux Comparison The truncated lognormal PSD is first approximated as a discrete-valued distribution via the method of matching moments [4]. The resulting erosion flux is shown in Figure 5.2, along with the erosion flux of the corresponding continuous PSD (see Chapter 3). Note that the moments matched are specific to the truncated PSD, not those of the true lognormal PSD. In addition, in order to keep the simulations practical, those particle sizes which have extremely small number fractions (<0.004%) are not simulated. To simulate all particles in the discretization would require too many particles to be practical (more than particles). For the case of the discretized PSD, the erosion flux decreases with increasing width, which is contrary to the change in erosion flux of the continuous PSD. Note that additional particle species beyond 10 were not used because the moment set becomes unrealizable at about 13 sizes (equations become too difficult to solve numerically). This issue is due to the stiff nature of the equations used to solve for the sizes due to both the large polynomial powers of the sizes, as well as the orders of magnitude differences between higher order moments of the distribution [5, 6]. Note that the results shown here appear to contradict the results of Murray et al. [4], who showed that the discretization matches the kinetictheory-based continuum model transport coefficients very well. However, the transport coefficients do not take the lift or drag force into account and thus likely contribute to the discrepancy shown here. Namely, the gas may be imparting less momentum to the discretized distribution than the original continuous distribution. 104

119 Figure 5.2. Erosion flux of discretized and continuous PSD for method of matching moments (90% CI). For direct comparison to Figure 5.2, the erosion flux resulting from using the volumetric discretization on the truncated lognormal PSD is shown in Figure 5.3, along with the corresponding continuous PSD erosion flux (ϕtrunc=25). The match between the discretized PSD and continuous PSD is much better than for the method based on matching moments. In this case, with as many as 100 particles, the erosion flux matches better than the moments discretization, although the erosion flux does not match perfectly. The better match has two possible sources. First, differences in particle sizes chosen can lead to differences in the gas forces on the particles, which could lead to differences in erosion flux. Second, changes in transfer of momentum between particles of different size could play a role, particularly in the vertical momentum transfer which plays a large role in determining the erosion flux. As mentioned previously, the primary difference between the discretizations is that the volumetric discretization has more of the smallest and largest 105

120 particles, which could cause the difference in gas forces or momentum transfer. However, no clear method exists for determining which of these phenomenon is contributing more to the difference between the discretizations. Figure 5.3. Erosion flux of discretized and continuous PSD for volumetric discretization (90% CI) Validation of Volumetric Discretization To ensure that the volumetric discretization method proposed here is correct, the validation tests used by Murray et al. [4] for the method of matching moments are performed on the new method. Namely, the scalar transport coefficients of the kinetic-theory based model of Garzό, Hrenya, and Dufty [1, 2] for the case of simple shear flow are compared for the volumetric discretization method, the method of matching moments, and simulation data using a discrete particle method collected by Dahl et al. [1, 2, 7]. This analysis is possible on an untruncated 106

121 lognormal distribution since the distribution widths are much smaller than those present in lunar regolith. The results for the dimensionless pressure and shear viscosity are shown in Figure 5.4. The dimensionless pressure is defined via p/(ρsdrmc 2 γ 2 ) where p is the (granular) pressure, drmc is the root-mean-cubed diameter (which changes as a function of width), and γ is the applied shear rate. The dimensionless shear viscosity is defined as η/(ρsdrmc 2 γ 2 ) where η is the (granular) shear viscosity. S=8 is chosen for the method of matching moments because additional species caused convergence issues in the calculation in the transport coefficients. S=100 is chosen for the volumetric discretization because additional species show minimal change and the calculation is very long. The results of the volume discretization asymptotically approach those of the method of matching moments. Note that a much larger number of particles is needed for the volumetric discretization method compared to the method of matching moments to achieve the same result. The other scalar transport coefficients show similar agreement. This result may appear inconsistent with the results shown for the erosion flux in Figure 5.2 and Figure 5.3 since the volume discretization method matched the DEM erosion flux much better than the matching-of moments method; the likely explanation was discussed in section Figure 5.4. Dimensionless transport coefficients for the two discretization methods and simulation data. 107

122 5.4 Conclusions The discretization of continuous PSDs is explored in order to facilitate the use of kinetic-theory based continuum models. For the narrow parameter space of the truncated lognormal distribution used here, the newly developed volumetric discretization method outperformed the method of matching moments developed by Murray et al. [4] when compared via the erosion flux of the original distribution (truncated lognormal). The results show that for sizes, the method of volumetric discretization better predicts the erosion flux of the original PSD. The improved match may be related to changes in momentum transfer between species or changes in gas forces on particles. The volumetric discretization is also validated by comparing the continuum model transport coefficients for a simple shear flow system with the method of matching moments and simulation data. Additional work is needed to determine if and for what other systems the volumetric discretization method should be used over the method of matching moments. 5.5 References 1. Garzó, V., Dufty, J.W., and Hrenya, C.M., Enskog theory for polydisperse granular mixtures. I. Navier-Stokes order transport. Physical Review E, (3): p Garzó, V., Hrenya, C.M., and Dufty, J.W., Enskog theory for polydisperse granular mixtures. II. Sonine polynomial approximation. Physical Review E, (3): p Hrenya, C.M., Kinetic theory for granular materials: Polydispersity, in Computational Gas-Solids Flows and Reacting Systems: Theory, Methods and Practice, M.S. S. Pannala, T. O'Brien, Editor. 2011, IGI Global: Hershey, PA. p Murray, J.A., Benyahia, S., Metzger, P., and Hrenya, C.M., Continuum representation of a continuous size distribution of particles engaged in rapid granular flow. Physics of Fluids, (8): p

123 5. Marchisio, D.L. and Fox, R.O., Computational models for polydisperse particulate and multiphase systems. 2013: Cambridge University Press. 6. Yuan, C., Quadrature-based moment methods for polydisperse multiphase flow modeling. 2013, Iowa State University. 7. Dahl, S., Clelland, R., and Hrenya, C., Three-dimensional, rapid shear flow of particles with continuous size distributions. Powder technology, (1): p

124 6. EFFECT OF INTER-PARTICLE CONTACTS ON FAR-FIELD DYNAMICS AND VALIDATION OF CONTINUUM MODEL VIA DEM G Abstract In this chapter, the importance of mid-air inter-particle collisions on the far-field dynamics is studied using a modified version of the DEM model used in previous chapters. The results indicate that mid-air collisions are important for both monodisperse and polydisperse distributions in the far-field, despite such collisions playing a minimal role in the (surface) erosion flux of monodisperse distributions. Namely, mid-air collisions cause a change in the vertical and horizontal distance traveled by particles relative to a simulation without collisions. Such collisions can significantly alter the particle trajectory, resulting in collisions being important even when relatively rare, such as in the monodisperse case. In addition, the polydisperse continuum model of Iddir and Arastoopour (2005) is validated for use in modeling the particles after they are eroded by comparison to DEM results of chapters 2 and 3. This validation is performed using monodisperse, binary, and discretized continuous PSDs. The continuum model has significantly reduced computational cost and can be later used to study much larger systems. 6.1 Introduction The importance of inter-particle collisions on the erosion and near-field dynamics has been established in previous chapters (see Chapters 2 and 3). For the case of a monodisperse G Berger, K.J. and C.M. Hrenya, Impact of a binary size distribution on the particle erosion due to an impinging gas plume. AIChe Journal, In Press. 110

125 distribution, only sub-surface contacts play a major role in determining the erosion flux. However, collisions both above and below the surface play a major role in systems with a polydisperse PSD. Unfortunately, the discrete element method (DEM) model used for the erosion process cannot be applied to the entire lunar system because the number of particles required would be extremely large, resulting in intractable computational cost. Thus, in order to model the downstream flow of particles, a more computationally efficient model approach must be used. Apollo data and previous studies suggest inter-particle collisions are important to far-field dynamics. The Apollo 12 caused a sandblasting of the deactivated Surveyor 3 lander, about 160m away. Immer et al. [1] showed that the flux at the Surveyor was too small (by orders of magnitude) to have been impacted by the main regolith spray, which likely went over the Surveyor. This result suggests some particles left the main spray and collided with the Surveyor. The most likely cause of this phenomenon is inter-particle contacts that cause some particles to leave the main spray. In addition, Morris et al. [2], in DSMC simulations of the lunar system, showed that inter-particle contacts significantly change the qualitative nature of the particle flow in the far field. These results suggest that a model that includes inter-particle collisions is necessary to fully describe the ejection of lunar regolith. The focus of this chapter is, firstly, to determine whether mid-air inter-particle collisions are important to the far-field dynamics and thus justify the use of the continuum model, which includes collisions. This task will be performed using a modified version of the DEM model used in previous chapters. The results suggest that inter-particle collisions do significantly alter the farfield flow of particles, thus necessitating the use of a model which includes inter-particle collisions 111

126 and is capable of modeling the large system size associated with the far field i.e., a continuum model. In addition, the continuum model is validated in the near-field using the DEM model developed in previous chapters for monodisperse, binary, and discretized continuous PSDs since far-field DEM is computationally prohibitive. However, the continuum model can not be used to simulate the erosion process due to sustained multi-particle contacts below the surface and thus the DEM model, which is able to model such contacts, serves to determine the erosion from the surface, which is used as a boundary condition in the continuum model. 6.2 Methods Far-field DEM model In order to determine whether inter-particle collisions are important to the far-field dynamics, the DEM model used in previous chapters is modified. Namely, the simulation (with all contacts resolved) at 1m from impingement is stopped when nearly all particles have eroded. At this time, the height attained by particles in the periodic domain roughly corresponds to an ejection angle of 3 degrees from the impingement point, consistent with observations from Apollo landing videos [3]. The periodic boundary in the direction of the plume flow is then turned off, and particles are allowed to travel outside the original domain while still under the influence of the gas. The full radial velocity profile of the gas is shown in Figure 1.6. However, because the domain is no longer small in the horizontal direction compared to the distance from impingement, the radial nature of the system must be accounted for. Namely, to account for the lateral spreading of particles as they travel further from the impingement point, the 112

127 concentration of particles in the rectangular domain is decreased accordingly. Figure 6.1 shows a depiction of the extended domain used. Dashed (black) lines indicate a true wedge-like domain, dash-dotted (red) lines indicate domain boundaries shared by the original small domain and the extended domain, the dotted (red) line indicates the edge of the original small domain (which is removed in the extended domain), and the solid (red) lines indicate boundaries of the extended domain. Based on geometrical arguments, a ratio of the volume of the wedge to the domain used can be calculated as 0.5(r + 1m), where r is the radial position relative to 1m from impingement. The probability of removing a particle at a given time step can be found by calculating the fraction of particles that should be removed between the radial positions r and r v r t, where v r is the radial velocity and t is the time step, which yields r 1 Pdeleted ( r, v rt) 1 (6.1) r v t 1 r Within the simulation, a pseudo-random number between zero and one is generated for each particle at each time step. If this number is smaller than the probability calculated in equation (6.1), the particle is removed from the domain. 113

128 Figure 6.1 Computational domain used for far-field simulations: top-down view of rectangular domain (solid red and dash-dot red) relative to actual radial domain (dashed black). Dotted red line is periodic wall from near-field DEM that is removed for far-field DEM. See text for more details Continuum Model The continuum model, rather than modeling particles discretely, solves the solid phase as a series of balances for the hydrodynamic variables, similar to how Navier-Stokes equations treats a gas flow as a continuum. The simulations are performed using the open-source Multiphase Flow for Interphase exchange (MFiX) code [4]. The model of Iddir and Arastoopour [5] is used, in which balances over mass (via species number density, ni), momentum (via species velocity Ui), and granular energy (via the granular temperature Ti, which is a measure of the kinetic energy of velocity fluctuations relative to species mean) of the solid phase take the form: nm i i t ( nmu ) 0 (6.2) i i i 114

129 N Ui nimi U i U i P i ni F i, ext, cons F Dip (6.3) t p1 N 3 Ti n i U i T i q i P i U i i U i F Dip (6.4) 2 t p1 where the subscript i refers to species i, P is the stress tensor, Fi,ext,cons refers to external forces (weight, drag, lift, etc.), q is the flux of granular energy, γ refers to the collisional dissipation rate of granular energy, and FDip is the collisional source of momentum (between unlike species). The constitutive relations for P, q, γ, Fi,ext,cons, and FDip are obtained via the kinetic-theory analogy; detailed expressions can be found in [5]. The lift and drag forces of Loth [6, 7] are incorporated into the continuum model via Fi,ext,cons (see Appendix A). For each computational cell, the appropriate plume properties are calculated using the same method as the DEM model, but assuming the properties at the center of the cell apply to the entire cell. Similarly, the plume force on each particle species is calculated based on the species velocity/density at each computational cell Validation Procedure The continuum model described above is validated against results from the DEM model in the near-field. This validation is done in the near-field, as opposed to far-field, because DEM is not capable of fully simulating the large system size associated with the far field. The far-field DEM model described above is only sufficient for determining whether mid-air collisions are important. The DEM model does not simulate erosion from more than a single location, which would require 115

130 modifications to the erosion simulation (to account for a domain that is large compared to the distance from impingement) and a large number of particles. Thus, the far-field DEM model is too computationally expensive to be used as a physical model of the system. In addition, the continuum model is unable to model the surface erosion process because it has an inherent assumption of binary, instantaneous contacts whereas sustained, multi-particle contacts occur below the surface. Accordingly, the surface erosion results obtained from the DEM model, which does account for sustained and multi-particle contacts, serve as an input to the continuum model. The procedure used to validate the continuum model via DEM of the near field is as follows: (1) DEM simulation is performed, (2) data is collected from DEM run at and above the erosion plane, (3) DEM data collected is used to generate initial (data above erosion plane) and boundary conditions (data at erosion plane) for the continuum model at the onset of steady erosion (see Chapter 2), and (4) continuum simulation is run and results are compared with DEM at the onset of steady erosion. The continuum boundary condition at y=0 (erosion plane; see Figure 1.5 for domain orientation) is a mass inflow condition with the flux equal to the erosion flux obtained from DEM. The boundary condition at y=ly is a mass outflow condition with a zero flux; however, particles do not reach a height of Ly and thus the choice of this boundary does not influence the results. The x and z boundaries are periodic. The parameters used for these simulations are nearly identical to the ones used in tables 2.1 and 3.1, with two primary exceptions. The coefficient of restitution, enorm, is set to 0.99 and the coefficient of friction, µ, is set to 0. These parameters are used because the assumptions of the Iddir-Arastoopour continuum model include near-elastic particles and no friction [5]. While the more recent model of Garzo, Hrenya, and Dufty [8, 9] relaxes the elasticity assumption, the model was found to be extremely computationally slow due 116

131 to slow convergence in the granular energy balance that resulted in intractable computational cost. In addition, no adequate treatment of frictional particles currently exists for continuum models. These assumptions, although limiting, still allow a reasonable first evaluation of the lunar erosion system. 6.3 Results and Discussion Far-field DEM In an effort to determine if mid-air collisions between particles are important to the farfield dynamics and thus whether a model that includes such collisions is necessary, the DEM technique described in section is used to model, in a simplified manner, the far-field dynamics of the lunar erosion. The focus of this study is on mid-air collisions because only the mid-air collisions are modeled in the continuum model. The continuum model is not able to model the erosion process, which involves sustained multi-particle contacts, and so collisions below the surface are not modeled. Figure 6.2 shows the results from two simulations using the DEM technique described above, both with contacts (subplot a) and without contacts (subplot b) for a binary case (50wt% large, size ratio 4). Figure 6.3 shows analogous simulations for the monodisperse case (31µm; size of small particles for binary case). Note that the no-contacts cases have many particles beyond 4m, but those are not shown to ensure the contacts cases were visible. As illustrated in Figure 6.2, the binary case with contacts is slightly more diffuse in the vertical direction compared to the case without contacts, particularly looking close to the point of origin (bottom-left corner). However, the same is not true for the monodisperse case depicted in Figure 6.3. In addition, for both binary and monodisperse systems, the particles in the no-contacts case 117

132 travel further than those in the contacts case. Both of these observations are similar to what was seen by Morris et al. in their DSMC simulations of polydisperse systems [2]. However, they only simulated a single polydisperse distribution and thus did not compare their results with any other distribution. The results above, particularly for the monodisperse case, may seem contrary to the results shown in chapter 2 [10, 11]. Namely, the relatively rare collisions above the surface have a minimal effect on the erosion flux. However, such collisions do occur and have a much larger impact on the far-field dynamics due to each collision significantly altering the path of the particles which collided. The change in the vertical dispersion of particles is due to mid-air collisions, particularly between unlike particles, transferring vertical momentum. Note that this is consistent with the effect of mid-air collisions on the erosion flux in the near-field simulations performed in Chapter 3, which showed a significant vertical transfer of momentum in the case of a polydisperse distribution. This trend is not observed in the monodisperse case because collisions are fewer and transfer less momentum (due to reduced relative velocity between particles). The difference in the horizontal distance traveled is caused by a combination of two collisional mechanisms, one between unlike particles and the other between like particles. First, due to the binary nature of the mixture, small particles are likely to catch up and collide with large particles in the horizontal direction (see Chapter 3), which reduces the total momentum of the small particles. This reduction in the small particle momentum reduces the maximum distance traveled because the small particles travel the furthest. The second mechanism, which occurs 118

133 between like particles, is that contacts between like particles tend to transfer momentum from faster moving particles to slower moving particles, particularly the small-small contacts in the binary case since the large particles do not travel far. The effect of this mechanism is expected to be similar for the monodisperse and binary cases because the small-small collision frequency is similar (see Figure 3.4). Contacts between like particles do not change the mean velocity of the species (due to conservation of momentum), but contacts are most likely to transfer momentum from faster to slower particles because the faster particles catch up with the slower ones. This momentum transfer results in the fastest moving particles traveling slower, which can be confirmed by examining the particle velocities (not shown). Single-particle trajectory models do not account for an exchange of momentum between particles of different sizes. Given that the particles traveled further in the no-contacts case for both binary and monodisperse, the mechanism involving collisions between like particles is likely more dominant for causing the reduction in the horizontal distance traveled. Note that the time simulated is relatively short compared to the time scales of deposition and so the changes between the contacts and no contacts cases noted above are expected to grow with time. It is clear from this analysis that the contacts above the surface are likely to play a significant role in the far-field dynamics of the regolith ejection. 119

134 Figure 6.2. Far-field simulations for binary mixtures: Particle positions for (a) with contacts and (b) without contacts at 0.015s. Initial simulation domain is in the bottom left corner (50 wt% large; size ratio 4). 120

135 Figure 6.3. Far-field simulations for monodisperse system: Particle positions for (a) with contacts and (b) without contacts at 0.015s. Initial simulation domain is in the bottom left corner (31µm particles) Continuum model validation via DEM In order to assess the validity of the continuum model for use in this lunar system, the results of the continuum and DEM models are compared for the following cases: (1) monodisperse PSD, (2) binary PSD, and (3) discretized continuous PSD. Figure 6.4 shows the monodisperse comparison of the horizontal particle velocity from the DEM and continuum model for two different times. Note the time is relative to the beginning of the continuum simulation. The qualitative and quantitative agreement between the models is very good (including for gas fraction and vertical velocity; not shown). Small deviations are seen, especially at larger heights. However, this result is somewhat expected because these regions contain the fewest particles and thus the DEM provides minimal data to compare against. 121

136 Figure 6.4 Monodisperse: Continuum vs. DEM results for horizontal particle velocity. Following the monodisperse PSD comparison above, the continuum model is then validated using a binary PSD. Figure 6.5 and Figure 6.6 show the gas fraction and large particle horizontal velocity for a system with a binary PSD (50 wt% large particles, size ratio 4). Once again, the match between models is very good, with some small deviations at larger heights. The vertical velocity of both phases and the small particle horizontal velocity show similar good agreement (not shown) 122

137 Figure 6.5 Binary: Continuum vs. DEM results for gas volume fraction. Figure 6.6 Binary: Continuum vs. DEM results for large particle horizontal velocity. 123

138 Finally, the continuum model is validated using a discretized continuous PSD. The PSD used is a discretization of a truncated lognormal distribution using the volumetric discretization method (see Chapter 5) with σ/µ=1. The DEM is performed using the continuous distribution and thus a method for relating the particle information from the continuous distribution to its discretization is needed. This transformation is performed by leveraging the volumetric discretization method. Namely, the volumetric discretization divides the distribution into several bins of equal volume. This process is performed on the particles in the DEM simulation and then the average values of each bin are used as initial and boundary conditions for the continuum model. For example, with 2 species, the volumetric discretization splits the distribution into two bins of equal volume, one of the larger particles and one of the smaller particles. For transforming the DEM data into continuum conditions, the average values (velocity, bulk density) of the large half of particles are used to determine the continuum conditions for the large particle of the discretization, and similarly for the small half of particles. Figure 6.7 and Figure 6.8 shows the species averaged horizontal and vertical velocity, respectively, for the DEM and continuum models for 2 species (agreement is similar independent of number of species used up to 10). Overall the qualitative agreement is fairly good and is similarly good independent of the number of discretized species used. However, quantitatively the continuum model over-predicts the horizontal velocity at later times. This result is likely related to the differences between the discrete PSD used in the continuum model and the continuous PSD used in the DEM model. More specifically, a likely cause of this discrepancy is that the particles in the continuum model may gain additional momentum from the gas due to the differences between the distributions. However, the difference is small and thus does not present itself immediately. Nonetheless, the reasonable 124

139 qualitative agreement of the horizontal velocity, combined with the very good quantitative agreement in the vertical velocity and gas solids fraction (not shown) provides confidence in the accuracy of the model. Figure 6.7 Discretized Continuous PSD: average horizontal velocity. 125

140 Figure 6.8 Discretized Continuous PSD: average vertical velocity. 6.4 Conclusions In this chapter, the importance of inter-particle collisions to far-field dynamics is shown using a monodisperse and binary PSD using DEM. Unlike the near-field case, the results indicate that such mid-air collisions are important in the monodisperse case. This phenomenon occurs because the collisions are relatively rare, and thus do not significantly influence the system on short scales, such as those relevant in erosion. However, each collision can significantly alter the trajectory of a particle and thus the relatively rare collisions can still have a major influence on large scales. In addition, such collisions play an even stronger role in the binary case by modifying the vertical dispersion of particles. However, the effects of inter-particle collisions on the horizontal distance traveled by particles is similar in the monodisperse and binary cases is similar due to the similar frequency of small-small contacts. The effects of collisions on the far-field dynamics had not been 126

141 previously investigated. Models that have been used in the past include single-particle trajectory models and DSMC [2, 12]. Single-particle trajectory models inherently do not account for mid-air collisions and therefore lack important physics. The DSMC model includes contacts though they are probabilistic in nature. Similarly, the DEM model used here is not able to simulate a sufficiently large domain nor very wide size disparities due to high computational overhead [13]. As a natural follow-on to the above results, a polydisperse continuum theory for the solids phase of Iddir and Arastoopour [5] is validated against DEM data for near-field simulations. This preliminary validation is successfully performed for monodisperse, binary, and discretized continuous PSDs. The primary advantage of the continuum model is significantly reduced computational cost, relative to DEM, which will allow simulation of systems on a much larger scale, such as that of the lunar surface in the next chapter. 6.5 References 1. Immer, C., Metzger, P., Hintze, P.E., Nick, A., and Horan, R., Apollo 12 Lunar Module exhaust plume impingement on Lunar Surveyor III. Icarus, (2): p Morris, A.B., Goldstein, D.B., Varghese, P.L., and Trafton, L.M., Approach for Modeling Rocket Plume Impingement and Dust Dispersal on the Moon. Journal of Spacecraft and Rockets, 2015: p Immer, C., Lane, J., Metzger, P., and Clements, S., Apollo video photogrammetry estimation of plume impingement effects. Icarus, (1): p Garg, R., Galvin, J., Li, T., and Pannala, S., Documentation of open-source MFIX-DEM software for gas-solids flows from URL 5. Iddir, H. and Arastoopour, H., Modeling of multitype particle flow using the kinetic theory approach. AIChE journal, (6): p

142 6. Loth, E., Lift of a Spherical Particle Subject to Vorticity and/or Spin. AIAA Journal, (4): p Loth, E., Compressibility and rarefaction effects on drag of a spherical particle. AIAA Journal, (9): p Garzó, V., Dufty, J.W., and Hrenya, C.M., Enskog theory for polydisperse granular mixtures. I. Navier-Stokes order transport. Physical Review E, (3): p Garzó, V., Hrenya, C.M., and Dufty, J.W., Enskog theory for polydisperse granular mixtures. II. Sonine polynomial approximation. Physical Review E, (3): p Berger, K.J., Anand, A., Metzger, P.T., and Hrenya, C.M., Role of collisions in erosion of regolith during a lunar landing. Physical Review E, (2): p Berger, K.J., Anand, A., Metzger, P.T., and Hrenya, C.M., Erratum: Role of collisions in erosion of regolith during a lunar landing. Physical Review E, (1): p Lane, J.E. and Metzger, P.T., Ballistics Model for Particles on a Horizontal Plane in a Vacuum Propelled by a Vertically Impinging Gas Jet. Particulate Science and Technology, (2): p Berger, K.J. and Hrenya, C.M., Challenges of DEM: II. Wide particle size distributions. Powder Technology, (0): p

143 7. VALIDATION OF CONTINUUM MODEL VIA APOLLO (FAR-FIELD) DATA Abstract The DEM results from chapter 6 demonstrate the need for a model that includes inter-particle collisions to simulate the far-field dynamics. Additionally in chapter 6, the kinetic-theory-based continuum model of Iddir and Arastoopour (AIChE Journal, 2005) was validated using discrete element method (DEM) simulations of the near-field. The goal of this chapter is to modify the continuum model developed and validated in chapter 6 such that it is applicable to the entire lunar surface (far-field). This model is then compared to data collected from Apollo missions for preliminary validation in the far-field. The results from the model agree well with the data from Apollo missions for the ejection angle and particle flux. However, the velocities predicted by the model and those estimated from the Apollo landings are not consistent and thus more work is needed to resolve this discrepancy. 7.1 Introduction In chapter 6, the continuum model of Iddir and Arastoopour [1] was modified to account for the role of the gas phase at lunar conditions, the predictions of which compared successfully to near-field DEM results for a wide variety of PSDs. The ultimate goal of this work is to validate the newly-developed modeling strategy that treats the erosion/near-field dynamics via DEM and the far-field dynamics via a continuum model for use in modeling the entire system (near field and far field). The focus of this chapter is to extend the continuum model developed in chapter 6 for application to the far field. Then, the resulting predictions are compared to experimental 129

144 measurements from the Apollo landings (far field). The continuum model is not compared to the far-field DEM model from Chapter 6 because the DEM model is unable to simulate the large number of particles required for erosion at more than a single distance from impingement. 7.2 Methods Continuum Model Additional modifications to the continuum model of Iddir and Arastoopour [1] used in chapter 6 are necessary to model the far-field lunar system. First, the gas plume, which previously did not vary in the horizontal direction (parallel to plume flow), must be modified to vary with distance from impingement. The gas plume profile can be seen in Figure 1.6. In addition, the gas plume profile (obtained by previous researchers) only covers up to 10m from impingement and 1.5m above the ground (see Figure 1.6). However, the software used to perform these simulations is not available and thus, in order to allow particles to travel beyond this region, the plume data is extrapolated. To ensure the extrapolation remains reasonable, it is only performed in one direction because a two-dimensional extrapolation would involve too much uncertainty. Particularly, the plume data is extrapolated only in the horizontal (parallel to plume) direction between 10m from impingement and the end of the computational domain in the horizontal direction. No extrapolation is necessary above 1.5m prior to 10m from impingement (vertical direction) because particles do not reach this region in any simulations performed. Extrapolation is not performed in the region greater than 10m from impingement and higher than 1.5m above the ground, which would require a two-dimensional extrapolation. Instead, the 130

145 simulations are stopped when particles reach heights of 1.5m. Further work with another set of plume data is necessary to consider the system further. For the gas density, the dependence in the horizontal direction is found to closely follow a decay with a 1/r 2 dependence, where r is the distance from impingement. Thus, the density at each height near the boundary at 10m is fitted to a curve with the form a/r 2, where a is a fitting parameter, and extrapolated for each computational cell at distances greater than 10m. Similarly, the plume radial velocity is found to decay exponentially in the horizontal direction, except very close to 1.5m. Thus, the radial plume velocity is fit to a function of the form cr be, where b and c are fitting parameters. While this fit is not very realistic near 1.5m, the particles spend very little time in this region before the simulation is halted. All three fitting parameters are shown in Figure 7.1. For the purposes of the simulations, the plume force is considered to be zero above 1.5m. Additionally, note that the plume viscosity is not modified because it is an input to the computational fluid dynamics model used to generate the plume profile (see chapter 1). 131

146 132

147 Figure 7.1 Fitting parameters for the horizontal extrapolation of plume data: (a) a, (b) b, and (c) c. Finally, in order to reduce computational costs, the lunar surface will be modeled as pseudotwo-dimensional (similar to DEM simulation in chapter 6) in which the angular (z-dimension) is very small. This approach is used due to the cylindrical symmetry with a single rocket nozzle. However, similar to the DEM model used in chapter 6, the reduction in the bulk density of particles as they spread in the angular direction must be accounted for. To model this reduction in bulk density, a modification to the continuity equation (equation (6.2)) is made. Namely, the convection in the horizontal direction from cell i to its adjacent cell i+1 is reduced by a factor of xi/xi+1, where xi is the horizontal distance from impingement of cell i. This factor is determined via the same method described in section Simulation Procedure For the purposes of simulating the lunar soil, the OB-1 lunar soil simulant is used for the PSD, which is shown in Figure 7.2. Also shown in the figure is the discretized approximation of the continuous PSD used here, which is the volumetric discretization (see chapter 5) with 6 species. Note that the smallest particle diameter of the discretization is extremely small compared to the next smallest particle (~2 orders of magnitude). To reduce computational time, which increases significantly with large particle size ratios [2], the smallest particle is not included in the simulations. Note that the smallest particles are least likely to do significant damage due to their very small mass, despite their larger velocities, which results in very small impact intensity. In fact, Immer et al. [3] estimated that particles around 100µm do the most damage, with particles of similar size causing large pits and cracks in the Surveyor 3 due to the Apollo 12 landing. 133

148 Figure 7.2 OB-1 distribution and volumetric discretization with 6 species. Similar to the validation performed in chapter 6, DEM data (erosion rate, particle velocities, bulk density) are collected during erosion and used as boundary condition in the continuum model. However, the initial condition used is an empty domain. A schematic of the boundary conditions (BCs) can be seen in Figure 7.3. BC 1, at the regolith surface, is comprised of two sections. Up to about 1.6m, where erosion from the surface is significant, the boundary is a mass inflow condition with data based on DEM simulations. The remaining portion is a wall condition. This wall simulates the regolith at rest beyond the direct influence of the plume and is chosen to be a free slip wall to allow particles to collide and continue to flow. A free slip wall is defined as a boundary at which all velocity gradients vanish. BC 2, far above the regolith surface, is modeled as a mass outflow condition with zero flux. However, because it resides above the limit of the plume at 1.5m, the particles never interact with this condition in the simulations presented here. Future work may necessitate this boundary being higher to ensure particles do not interact with it. BC 3, located at the rocket nozzle, is simulated as a free slip wall, which mimics the symmetrical nature of the system along this axis. However, the choice of this BC is not important 134

149 as particles do not interact in any significant way with this boundary. Finally, BC 4 is treated the same as BC2 because particles are not intended to interact with this boundary and would need to be similarly extended if particles were to travel further downstream. Additionally, the domain is periodic in the z direction. Figure 7.3 Continuum model domain for far-field simulations. See text for details of boundary conditions. The model parameters are shown in Table 7.1 for both the DEM and continuum models. Note that in order to generate the mass inflow condition at BC 1 in Figure 7.3, the DEM model is simulated for distances from impingement from 0.1m to 2.5m in increments of 0.1m. The resulting erosion data is then used as the boundary condition, with each DEM simulation corresponding to one computational boundary cell (computational cell length in x direction is also 0.1m). In addition, erosion is not significant beyond 1.6m, so the region beyond is treated as a free slip wall, as described above. Table 7.1 Model Parameters Particle Properties (DEM and Continuum) 135

150 Density, Coefficient of restitution, e norm Coefficient of friction, µ Particle size, dp Particle Properties (DEM) Spring stiffness particle-particle normal, k norm,pp tangential, k tan,pp particle-wall normal, k norm,pw tangential, k tan,pw Friction coefficient particle-particle, pp particle-wall, pw Coefficient of restitution particle-particle, e norm,pp particle-wall, e norm,pw Dashpot Coefficient normal, norm tangential, tan 2700 kg/m (not used), 40.4, 66.2, 119.4, 187.3, µm 8000 kg/s kg/s kg/s kg/s Calculated from coefficient of restitution and spring stiffness (see Appendix A) 0.5 * norm Number of Particles Lunar Conditions (DEM and Continuum) Acceleration due to gravity on the Moon, g 1.63 m/s 2 Continuum Model System Geometry Length of domain in x direction Length of domain in y direction Length of domain in z direction 100m 2m m Length of computational cell in x direction Length of computational cell in y direction Length of computational cell in z direction 0.1m 0.01m m DEM Model System Geometry Length of domain in x direction Length of domain in y direction Length of domain in z direction m 0.1m m 136

151 7.3 Results and Discussion Figure 7.4 shows the total horizontal flux, total vertical flux, total bulk density, and the species horizontal flux for the 40.4, 119.4, and µm particles after 10s of erosion. The flux is calculated as the appropriate particle velocity times the bulk density of each species. For the case of the figures a-c, the values are summed across all species. Figure 7.4(a) shows how two primary particle streams: one at an angle of about 2.5 o that travels to about 30-35m (when simulation is stopped) and one at an angle of about 1 o that travels only about 15-20m before coming back to the regolith surface. The stream with the higher angle contains only the smallest particles furthest downstream (Figure 7.4(d) and (e)), with larger particles not traveling as far in the same amount of time (Figure 7.4(f)). This difference is due to the difference in the velocities of the particles caused by gas forces. Namely, the drag and lift scale with dp 2, whereas the mass scales with dp 3, resulting in an increase in the particle acceleration, and thus velocity, as the particle size becomes smaller. In addition, the stream with the lower angle has similar amounts of each species (Figure 7.4 (d)-(f)), although the smaller species travel slightly further before hitting the ground (Figure 7.4(d) and (e)). Finally, some particles, after colliding with the wall along with bottom exhibit strange behavior, namely rebounds with large energy (see Figure 7.4(a) from ~25-50m from impingement up to ~0.5m height). This behavior is likely related due to the free-slip wall not dissipating any energy and thus could potentially be alleviated by implementing a boundary condition that dissipates energy, but this was not done in this study. This phenomenon is likely not physical, although some sedimentation-like behavior might be expected, in which particles colliding with the surface knock up other particles, in the real system. However, this behavior is not modeled in this system because the regolith surface is treated as a wall. 137

152 Figure 7.4 Results of continuum simulation after 10 seconds. (a) total bulk density, (b) total horizontal flux, (c) total vertical flux, (d) horizontal flux of 40.4µm particles, (e) horizontal flux of 119.4µm particles, and (f) horizontal flux of 265.2µm particles. All data is scaled logarithmically for clarity, except vertical flux. For purposes of comparison against Apollo data, three pieces of data can be compared between the simulations and Apollo landing data: ejection angle, particle flux, and particle velocity. Data from Apollo videos, presented by Immer et al, [4] shows a typical ejection angle of about 1-3 o for a typical landing on roughly flat ground, with the exception of Apollo 15, which had a higher ejection angle likely due to landing on a crater rim. Clearly the results presented in Figure

153 agree very well with this data. Additionally, one estimate of the flux from Apollo 11, based on optical data and velocities generated by the single-particle trajectory model yielded a flux of ~33 g/cm 3 at ~11.6m from impingement (the location of a large stationary rock provided the reference necessary for this estimate) [3]. The results above show very similar flux in the flow angled at 2.5 o, with flux of g/cm 3 in the main sheet at about 0.5m height (Figure 7.4(b)). Note that while this comparison involves using the single-particle trajectory model results, the velocities obtained from such a model are expected to be reasonably close to the actual particle velocities, especially so close to the rocket. The main issue with using the single-particle trajectory model is that it will not be able to accurately predict the deposition pattern of particles due to inter-particle collisions causing some particles to have vastly different trajectories. Figure 7.5 Species averaged horizontal particle velocity at 10 seconds. Data is scaled logarithmically for clarity. 139