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Particle-resolved simulations of inertial suspensions of spheres and polyhedrons : analysis and modeling Seyed-Ahmadi, Arman 2020

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Particle-resolved simulations of inertial suspensions ofspheres and polyhedrons: Analysis and modelingbyArman Seyed-AhmadiM.Sc., Mechanical Engineering, University of Tabriz, 2015B.Sc., Mechanical Engineering, Urmia University of Technology, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Chemical and Biological Engineering)The University of British Columbia(Vancouver)December 2020© Arman Seyed-Ahmadi, 2020The following individuals certify that they have read, and recommend to theFaculty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Particle-resolved simulations of inertial suspensions of spheres andpolyhedrons: Analysis and modelingsubmitted by Arman Seyed-Ahmadi in partial fulfillment of the requirements forthe degree of Doctor of Philosophy in Chemical and Biological Engineering.Examining Committee:Anthony Wachs, Mathematics and Chemical and Biological EngineeringSupervisorGwynn Elfring, Mechanical EngineeringUniversity ExaminerRajeev Jaiman, Mechanical EngineeringUniversity ExaminerAndrea Prosperetti, Mechanical Engineering, University of HoustonExternal ExaminerAdditional Supervisory Committee Members:James Feng, Mathematics and Chemical and Biological EngineeringSupervisory Committee MemberIan Frigaard, Mathematics and Mechanical EngineeringSupervisory Committee MemberiiAbstractParticle-laden flows where a dispersion of a solid phase is carried by a fluidphase are at the core of numerous industrial and natural processes, such as flu-vial sediment transport and fluidized-bed reactors. The dynamics of each phaseis intimately coupled with that of the other phase, leading to the emergence ofcomplex, nontrivial interactions that can span wide ranges of spatial and temporalscales. The focus of this thesis is two-fold; namely, analysis of particle shape ef-fects, and modeling hydrodynamic forces and torques in particle-laden flows. Tothis end, direct numerical simulations are performed for the generation of high-fidelity data, based on which all analyses of this thesis are carried out. In the firstpart, we scrutinize the dynamics of an isolated polyhedron, i.e. a cube, in highlyinertial regimes and various density ratios. Robust helical motions and wake pat-terns are found for Reynolds numbers at which a sphere moves rectilinearly. Anisolated cube exhibits remarkably larger rotational and lateral motions comparedto a sphere, by which the effective drag on the particle is greatly affected. We thenextend the analysis to inertial suspensions of cubes, where detailed comparisonsare made with their counterpart sphere suspensions for various solid volume frac-tions. While strong clustering occurs in sphere suspensions, cube suspensions arefound to be remarkably more homogeneous, as evident from their microstructureand momentum transfer properties. As demonstrated by their intensive transversevelocity fluctuations, cubes are more likely to break up and escape clusters, thusresisting local accumulation and making suspensions better mixed. In the secondpart, we develop a novel probability-driven point-particle model for the predic-tion of hydrodynamic forces and torques based on local microstructure in denseiiiarrays of spheres. Following probabilistic arguments, necessary statistical infor-mation is extracted from particle-resolved simulations to construct force/torque-conditioned probability distribution maps, which are in turn used as basis func-tions for a regressive-type model. We subsequently show that our model is capableof predicting a substantial part of the observed force and torque variations, and isthus conceived to be highly promising for accurate interphase coupling in Euler-Lagrange simulations.ivLay summaryFrom rain formation and scattering of volcanic ash, to particle and powder mo-tion in chemical processing devices, fluid flows carrying solid materials are foundeverywhere. Depending on their properties, these particle-fluid suspensions showa variety of different behaviors: particles can aggregate or disperse, settle fast orslow, follow the fluid or resist. To control and exploit these features, we need tounderstand and quantify the physics underlying such phenomena. In this thesis, wefirst study how particle shape can influence the behavior of suspensions. Particu-larly, we show that cube suspensions are more agitated and better mixed comparedto sphere suspensions. The second part of the present work concerns the develop-ment of a physically-inspired data-driven model that is capable of predicting forcesand torques in particulate flows. While presently impossible, this model serves asa step towards bringing numerical simulation of industrial-scale systems withinreach.vPrefaceThe content of this thesis represents the results of my PhD research project thatI undertook over the course of four years at the University of British Columbia,with the guidance and mentorship of my supervisor, Professor Anthony Wachs.The contributions of the main chapters of this thesis have resulted in the publica-tion/submission of the following papers:• Arman Seyed-Ahmadi and Anthony Wachs, “Dynamics and wakes offreely settling and rising cubes”, Physical Review Fluids 4 (2019).doi: 10.1103/PhysRevFluids.4.074304Simulations, data analysis and manuscript preparation of this paper was doneby me, while Anthony Wachs supervised the research and contributed tothe reviews and edits. The research published in this paper is based on thecontent of chapter 2.• Arman Seyed-Ahmadi and Anthony Wachs, “Microstructure-informedprobability-driven point-particle model for hydrodynamic forces and torquesin particle-laden flows”, Journal of Fluid Mechanics 900 (2020).doi: 10.1017/jfm.2020.453Theory development, simulations, and manuscript preparation of this paperwas done by me, while Anthony Wachs supervised the research and con-tributed to the reviews and edits. The research published in this paper isbased on the content of chapter• Arman Seyed-Ahmadi and Anthony Wachs, “Sedimentation of inertialmonodisperse suspensions of cubes and spheres”, Submitted for review inPhysical Review Fluids (2020).Simulations, data analysis and manuscript preparation of this paper was doneby me, while Anthony Wachs supervised the research and contributed to thereviews and edits. This submitted paper is based on the content of chapter 3.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . 31.1.1 Particle shape effects in liquid-solid suspensions . . . . . 31.1.2 Microstructure-informed point-particle model . . . . . . . 52 Dynamics of a freely settling & rising cube . . . . . . . . . . . . . . 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Single particle systems: The state of the art . . . . . . . . . . . . 92.2.1 Spheres immersed in a fluid . . . . . . . . . . . . . . . . 92.2.2 Cubes immersed in a fluid . . . . . . . . . . . . . . . . . 122.3 Mathematical formulation and numerical algorithm . . . . . . . . 152.3.1 Governing equations . . . . . . . . . . . . . . . . . . . . 152.3.2 Numerical method . . . . . . . . . . . . . . . . . . . . . 16viii2.3.3 Problem setup . . . . . . . . . . . . . . . . . . . . . . . 192.4 Regimes of motion . . . . . . . . . . . . . . . . . . . . . . . . . 212.4.1 Oblique regime . . . . . . . . . . . . . . . . . . . . . . . 222.4.2 Unsteady vertical regime . . . . . . . . . . . . . . . . . . 242.4.3 Helical regime . . . . . . . . . . . . . . . . . . . . . . . 272.4.4 Chaotic regime . . . . . . . . . . . . . . . . . . . . . . . 302.5 Force balances . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.5.1 Horizontal forces . . . . . . . . . . . . . . . . . . . . . . 332.5.2 Normal forces . . . . . . . . . . . . . . . . . . . . . . . . 362.5.3 Drag force . . . . . . . . . . . . . . . . . . . . . . . . . 402.6 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . 483 Inertial settling of cube and sphere suspensions . . . . . . . . . . . . 523.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.2 Computational methodology . . . . . . . . . . . . . . . . . . . . 593.2.1 Governing equations . . . . . . . . . . . . . . . . . . . . 593.2.2 Numerical Method . . . . . . . . . . . . . . . . . . . . . 603.2.3 Simulations setup . . . . . . . . . . . . . . . . . . . . . . 643.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.3.1 Settling velocity . . . . . . . . . . . . . . . . . . . . . . 673.3.2 Velocity fluctuations . . . . . . . . . . . . . . . . . . . . 693.3.3 Angular velocities . . . . . . . . . . . . . . . . . . . . . 733.3.4 Microstructure . . . . . . . . . . . . . . . . . . . . . . . 773.3.5 Drag force . . . . . . . . . . . . . . . . . . . . . . . . . 843.4 Discussion & final remarks . . . . . . . . . . . . . . . . . . . . . 894 Microstructure-informed probability-driven point-particle model . 934.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.2 PR-DNS of fixed beds of spheres . . . . . . . . . . . . . . . . . . 1034.2.1 Governing equations . . . . . . . . . . . . . . . . . . . . 1034.2.2 Numerical method . . . . . . . . . . . . . . . . . . . . . 1044.2.3 Simulation setup . . . . . . . . . . . . . . . . . . . . . . 1064.2.4 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . 108ix4.2.5 Dataset construction . . . . . . . . . . . . . . . . . . . . 1104.3 Probability-driven model . . . . . . . . . . . . . . . . . . . . . . 1114.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 1114.3.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.3.3 Probability distribution maps . . . . . . . . . . . . . . . . 1204.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . 1274.4.1 Practical aspects of implementation . . . . . . . . . . . . 1274.4.2 Model assessment . . . . . . . . . . . . . . . . . . . . . 1314.4.3 Performance of the MPP model . . . . . . . . . . . . . . 1324.5 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . 1385 Conclusions & prospects . . . . . . . . . . . . . . . . . . . . . . . . 1435.1 Particle shape effects in liquid-solid suspensions . . . . . . . . . . 1435.2 Microstructure-informed point-particle model . . . . . . . . . . . 147Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151A Supporting material . . . . . . . . . . . . . . . . . . . . . . . . . . . 172A.1 Wake transitions of a fixed cube . . . . . . . . . . . . . . . . . . 172A.2 Force data filtering process . . . . . . . . . . . . . . . . . . . . . 174xList of TablesTable 3.1 Summary of the parameters used for PR-DNS of suspensions ofcubes and spheres . . . . . . . . . . . . . . . . . . . . . . . . 64Table 3.2 Terminal settling Reynolds number for all simulations based onaverage suspension settling velocity. . . . . . . . . . . . . . . 68Table 4.1 Summary of the parameters used for PR-DNS of fixed beds ofspheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Table 4.2 Statistics of the drag and lift data for cases presented in table 4.1 110Table 4.3 Performance of the MPP model represented by the coefficientof determination R2 for cases considered in this work . . . . . 133xiList of FiguresFigure 1.1 Industrial examples of the cost associated with experimentalversus numerical design of fluidized beds [2] . . . . . . . . . 2Figure 2.1 Schematics of the computational domain for a sample settlingcube case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Figure 2.2 The evolution of the vertical velocity of the cube for Ga = 160and various density ratios . . . . . . . . . . . . . . . . . . . . 20Figure 2.3 Flow-map of the rising and settling cube in the Ga−m param-eter space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Figure 2.4 (a) & (b) Side-view of the trajectory of the cube in the obliqueand unsteady vertical regimes (c) Left: (Ga,m) = (70, 0.8) -symmetric vortex structure; Right: (Ga,m) = (70, 4) - plane-symmetric vortex structure. Top row: Contours of −0.005 6ωz 6 0.005 (ωz ≈ 0 in green regions) on a z-normal plane at adistance of d = 2 downstream of the cube center; Bottom row:Iso-surfaces of stream-wise vorticity for ωz = ±0.01 . . . . . 23Figure 2.5 Side-view of the trajectory of a cube for various density ratiosin the unsteady vertical and helical regimes. The color code isthe same as in figure 2.2 . . . . . . . . . . . . . . . . . . . . 25Figure 2.6 Vortex structure for various cases in the unsteady vertical regime.Visualization is done using iso-surfaces of stream-wise vortic-ity for ωz = ±0.02 (the first and the second image) and Q-criterion for Q = 0.003 (the third and the fourth image). . . . 26xiiFigure 2.7 Various views of wake structures and trajectories of the cubein the helical regime. The vortices are visualized using iso-surfaces of Q-criterion for Q = 0.003. . . . . . . . . . . . . . 28Figure 2.8 Different views of the trajectories of a cube at various valuesof Ga and m. The color code is the same as in figure 2.2 . . . 29Figure 2.9 Transient zigzag paths in the initial stages of the helical regime.The vortices are visualized using iso-surfaces of Q-criterionfor Q = 0.003. . . . . . . . . . . . . . . . . . . . . . . . . . 29Figure 2.10 Different views of the trajectories of a cube in the chaoticregime. Please note the difference in the ranges of x and yvalues which is necessary to be able to show the entirety of thetrajectories for Ga = 250. The color code is the same as infigure 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Figure 2.11 Schematics of the velocity and force vectors of the cube, alongwith the horizontal and normal planes . . . . . . . . . . . . . 32Figure 2.12 Force balance in the horizontal direction for the lightest anddensest cube in the helical and chaotic regimes . . . . . . . . 34Figure 2.13 Time average of the horizontal component of the angular ve-locity (open-circle markers) after initial transient period. Theerror bars represent the magnitude of fluctuations, i.e., the stan-dard deviation of |Ω · eˆh| over the averaging time window. . . 35Figure 2.14 Time average of the horizontal force magnitude (open-circlemarkers) after initial transient period. The error bars representthe magnitude of fluctuations, i.e., the standard deviation of|Fv · eˆh|/FB over the averaging time window. . . . . . . . . 36Figure 2.15 Evolution of the normal force in the helical and chaotic regimes 37Figure 2.16 Time average of the normal force (open-circle markers) afterinitial transient period. The error bars represent the magnitudeof fluctuations, i.e., the standard deviation of |Fv · eˆn| over theaveraging time window. . . . . . . . . . . . . . . . . . . . . 38xiiiFigure 2.17 Visualizations of FM,h and F(v,n),h, the horizontal compo-nents of the Magnus force vector and the normal vortex forcevector; respectively, after the transient period. The light purplelines indicate the trajectory of the cube during this window oftime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Figure 2.18 Variations of the drag coefficient with Galileo number for dif-ferent density ratios. The marker styles on the contour plotrepresent different regimes of motion and are the same as infigure 2.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Figure 2.19 Variations of the drag coefficient with Reynolds number fordifferent density ratios. For the fixed sphere, the standard dragcorrelation proposed by Turton and Levenspiel [65] is used. . 42Figure 2.20 The wake structures, together with velocity and force vectorsof the cube with (Ga,m) = (140, 0.2) at two different stagesof its rising. The thick purple line shows the trajectory of thecube, and Fid denotes the induced drag vector. The wakes arevisualized using iso-surfaces of Q-criterion for Q = 0.003. . . 45Figure 2.21 Contour plot of the relative error  between Ft and Ft,pr asa function of the vortex and incidence angle for (Ga,m) =(140, 0.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Figure 2.22 Contour plot of horizontal velocity fluctuations for various val-ues of Ga and m. The marker styles, which represent differentregimes of motion, are the same as in figure 2.3 . . . . . . . . 48Figure 3.1 Computational domains used for PR-DNS of cube and spheresuspensions with different volume fractions . . . . . . . . . . 65xivFigure 3.2 Suspension settling velocities normalized by the terminal ve-locity of an isolated particle as a function of solid volume frac-tion for the two Galileo numbers of Ga = 70 and Ga = 160.The dashed lines in (a) and (b) show the Richardson & Zaki[69] formula presented in equation (3.2), with the prefactork = 0.85, (c) the vertical velocity of an isolated cube as afunction of time at Ga = 160 normalized by the gravitationalvelocity scale. . . . . . . . . . . . . . . . . . . . . . . . . . . 67Figure 3.3 Velocity fluctuations (a) in the gravity direction and (b) normalto the gravity direction. The particle and fluid data are shownby solid and open symbols, respectively. Also, red lines andsymbols demonstrate the results for cube suspensions, whereasblack lines and symbols represent results for sphere suspen-sions. The inset of each plot shows the same data on a linearscale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Figure 3.4 Time evolution of ensemble-averaged particle velocity fluctu-ations, with the left and right panels showing vertical and hor-izontal components, respectively. In these plots, solid linesrepresent simulations with Ga = 160, whereas dotted linesshow data for simulations with Ga = 70. Also, red and blacklines demonstrate the results for cube and sphere suspensions,respectively. In order to enhance visual representation of theplots, darker lines show a filtered version of the data, while theoriginal data are plotted with a lighter color. . . . . . . . . . . 70Figure 3.5 Time evolution of ensemble-averaged local fluid velocity fluc-tuations, with the left and right panels showing vertical andhorizontal components, respectively. In these plots, solid linesrepresent simulations with Ga = 160, whereas dotted linesshow data for simulations with Ga = 70. Also, red and blacklines demonstrate the results for cube and sphere suspensions,respectively. In order to enhance visual representation of theplots, darker lines show a filtered version of the data, while theoriginal data are plotted with a lighter color. . . . . . . . . . . 72xvFigure 3.6 Anisotropy of (a) particle and (b) fluid velocity fluctuationsindicated by the ratio of the vertical to horizontal velocity vari-ance. The insets extended ranges of values on the vertical axis. 73Figure 3.7 Distribution of particle angular velocities for different solidvolume fractions. The inset of each plot shows the same datawith y-axis scaled logarithmically. Solid lines represent sim-ulations with Ga = 160, whereas dotted lines show data forsimulations with Ga = 70. Also, red and black lines demon-strate the results for cube and sphere suspensions, respectively.The mean value of each distribution is shown using short ver-tical lines on each plot. . . . . . . . . . . . . . . . . . . . . . 74Figure 3.8 Joint probability distribution functions of (a) cube and (b) spheresuspensions both obtained for (Ga, φ) = (160, 0.05), (c) Vari-ation of Pearson’s correlation coefficient as a function of thesolid volume fraction for all cases. The dotted and solid linesrepresent data for Ga = 70 and Ga = 160, respectively, whereasred and black lines demonstrate the results for cube and spheresuspensions, respectively. . . . . . . . . . . . . . . . . . . . . 75Figure 3.9 Pair and radial distribution functions for Ga = 160. The left-most and center columns pertain to cube and sphere suspen-sions, respectively. . . . . . . . . . . . . . . . . . . . . . . . 78Figure 3.10 Pair and radial distribution functions for Ga = 70. The left-most and center columns pertain to cube and sphere suspen-sions, respectively. . . . . . . . . . . . . . . . . . . . . . . . 79Figure 3.11 Time evolution of ensemble-averaged local solid volume frac-tion φloc. Solid lines represent simulations with Ga = 160,whereas dotted lines show the data for simulations with Ga =70. Also, red and black lines demonstrate the results for cubeand sphere suspensions, respectively. . . . . . . . . . . . . . 82xviFigure 3.12 Drag force of the cube and sphere suspensions as a function ofsolid volume fraction for two Galileo numbers of Ga = 70 andGa = 160. Open symbols show data for dynamic suspensions,while solid symbols represent data obtained from fixed beds.Along with data obtained from our PR-DNS, we also showdrag correlations of Wen & Yu [107] and Tang et al. [116] forstatic arrays of spheres, and that proposed by Chen & Müller[119] for fixed beds of cubes. . . . . . . . . . . . . . . . . . . 86Figure 3.13 Pair distribution functions of cube and sphere suspensions fortwo different solid volume fractions. In each of the four panels,the left and right plots pertain to cube and sphere suspensions,respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Figure 4.1 Depiction of the concept of multiscale strategy in modelingparticle-laden flows . . . . . . . . . . . . . . . . . . . . . . . 95Figure 4.2 Fixed beds of spherical particles at Re = 40 and solid volumefractions of φ = 0.1 (left) and φ = 0.4 (right). The streamlinesshown are colored with respect to the fluid velocity magnitude. 106Figure 4.3 Validation of the drag data obtained from PR-DNS of the presentwork with correlations of [115], [118], [116] and [117] . . . . 109Figure 4.4 (a) Unconditioned PDF of the first closest neighbor positionand PDF of the first closest neighbor position when the ref-erence particle experiences a (b) higher than average, or (c)lower than average drag force. The PDFs are obtained for thecase of Re = 40 and φ = 0.1. . . . . . . . . . . . . . . . . . 122Figure 4.5 PDF of the first closest neighbor position when the referenceparticle experiences a (a) higher than σ or (b) lower than σ lift,with σ being the standard deviation of the data. (c) PDF of thefirst closest neighbor position when the lift is either higher orlower than σ. The PDFs are obtained for the case of Re = 40and φ = 0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . 123xviiFigure 4.6 PDF of the first closest neighbor position when the referenceparticle experiences a (a) higher than σ and (b) lower than σlateral torque, with σ being the standard deviation of the data.(c) PDF of the first closest neighbor position when the lateraltorque is either higher or lower than σ. The PDFs are obtainedfor the case of Re = 40 and φ = 0.1. . . . . . . . . . . . . . . 123Figure 4.7 PDFs of various neighbor positions for (a) ∆Fx < −σ, and(b) ∆Fx > σ, with σ being the standard deviation of the data.Note that the numbering represented by j is based on proxim-ity to the reference particle, where j = 1 shows the closestneighbor. The PDFs are obtained for Re = 40 and φ = 0.1. . . 125Figure 4.8 PDFs of positions of 15 closest neighbors for two differentsolid volume fractions. . . . . . . . . . . . . . . . . . . . . . 127Figure 4.9 Regression plots for the drag, lift and lateral torque for variousReynolds numbers and solid volume fractions. The oblique redline in each plot shows an ideal fit. . . . . . . . . . . . . . . . 134Figure 4.10 PDF of the first closest neighbor position (a) having a secondneighbor (drawn as a dashed white circle) deliberately fixedon x − y plane at r2 = A = (−1, 1, z), (b) PDF of the firstclosest neighbor position, same as in figure 4.4c. The PDFsare obtained for Re = 40, φ = 0.1 . . . . . . . . . . . . . . . 136Figure 4.11 Coefficient of determination R2 as a function of the number ofincluded neighbors M used for construction of the model . . . 141Figure A.1 Evolution of the wake of a fixed cube with increasing the Reynoldsnumber. Left: Contours of stream-wise vorticity for −0.05 6ωz 6 0.05 (ωz ≈ 0 in gray regions) at a distance of d = 1.5downstream of the cube center, Right: Vortex structures visu-alized using iso-surfaces of Q-criterion with Q = 0.003 forRe = 215 and Q = 0.02 for Re = 225. Please note thatReynolds number values are all based on the edge length ofthe cube. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173xviiiFigure A.2 Drag coefficient data from the simulations performed in thepresent study for a cube (open-circle markers), together withcorrelations proposed for a fixed cube in the literature (solidlines). The standard drag law of a fixed sphere [65] is alsoshown (dotted line). . . . . . . . . . . . . . . . . . . . . . . . 174Figure A.3 (a) A demonstration of the original vertical force data and thefiltered data for (Ga,m) = (160, 0.2), (b) A magnified viewof the same data over a short time interval. . . . . . . . . . . . 175xixChapter 1Introduction“The arrival time of a space probe traveling to Saturn can bepredicted more accurately than the behavior of a fluidized-bedchemical reactor!” – Derek GeldartAs the quote signifies [1], the behavior of particle-laden flows where a fluidphase carries solid or deformable particles is highly complex by nature. Each phasedynamically interacts with the other phase in many intricate ways that are far frombeing readily understandable. The resulting phenomena typically involve wideranges of spatial and temporal scales, and a clear separation of scales is usually noteasily established. A fluidized-bed reactor is a prominent example of a multiphaseflow system commonly used in chemical, pharmaceutical and food processing in-dustries, where the knowledge of phenomena occurring at the level of a laboratoryscale device is rarely sufficient to give a complete picture of the fluid-particle inter-actions in a full-size industrial scale device. Furthermore, there are several naturalprocesses involving particulate flows that have a significant impact on the every-day life of people and on the ecosystem. Typical examples include the transportand dispersion of air pollutants by the atmospheric boundary layer, eruption of vol-canic ash and soot, formation of rain in the clouds and sedimentation of sand andgravel in rivers, as well as turbidity currents near continental shelves. Whether it bethe economical and environment-friendly design of a fluidized-bed reactor, or pre-1Project Pilot-scale coal gasifierLocationPower Systems Development Facility, Alabama, USAGoalPrediction of the effect of the design changes to prevent oxygen breakthrough, increase mixing and residence times in the riser section of the gasifierRequired Time and CostNumerical Simulation2 weeks$10,000Actual Modification and/or Construction 3-4 months$6 millionProject HydrogasifierLocationArizona PublicService, Arizona, USAGoalEvaluation of 17 design variations in of the hydrogasifierRequired TimeNumerical Simulation 2-3 weeksActual Modification and/or Construction 51 monthsProjectFluidized-bed for coatingnuclear fuel particlesLocation BWXT, Virginia, USAGoalDevelopment of an optimal design for producing high-quality particlesRequired TimeNumerical Simulation 1 yearActual Modification and/or Construction 20-30 yearsFigure 1.1: Industrial examples of the cost associated with experimental versus nu-merical design of fluidized beds [2]dicting and harnessing the behavior of natural systems and processes, these effortsare heavily reliant on our understanding of the dynamics involved in particle-ladenflow systems.Experimental techniques such as radiation absorption and digital image analysisprovide invaluable information in suspensions; however, these methods are limitedto pseudo-two-dimensional setups [3, 4]. Moreover, the extent to which the detailsof particulate flows are available to experimental measurements is quite limited[5], as either extreme or inaccessible conditions are commonly encountered. Dueto the lack of a comprehensive knowledge of these systems, meeting particularoutput parameters in the effective design a fluidized-bed reactor, for example, oftenrequires a trial-and-error design process. This incurs huge expenses for buildingand testing designs, which are usually pilot prototypes of small sizes compared toindustrial-scale devices (see figure 1.1). A viable alternative is to take advantage ofnumerical tools which provide a robust and efficient solution for modeling physicsof particulate flows. In the present work, we make use of computational fluiddynamics tools capable of accommodating the motion of rigid particles within thefluid domain to directly simulate and study inertial suspensions of particles. Inwhat follows, we give a brief overview of the essence of the problems that areconsidered in this thesis, while complete descriptions and details of each will bepresented in their dedicated chapters of this document.21.1 Outline of this thesisThe present thesis generally follows two main themes; namely, understandingthe effects of particle shape in the dynamics of inertial suspension flows, and mod-eling hydrodynamic forces and torques in dense particle-laden flows based on thelocal microstructure of individual particles. The material in chapters 2 and 3 per-tains to the former theme, whereas chapter 4 is devoted to the latter theme. Finally,the thesis ends with a discussion of the main conclusions and contributions in chap-ter 5, and potential prospective research directions are delineated. In the followingsections, motivations and general outlines of each corresponding sub-project ispresented.1.1.1 Particle shape effects in liquid-solid suspensionsIn most real-life scenarios involving particle-laden flows, particles are not spher-ical. It is estimated that more than 70% of the raw granular materials in modernindustries are non-spherical [6]. Nevertheless, the majority of research studieson particle suspensions are limited to spherical particles, owing to a number ofreasons. First, the numerical implementation of a spherical geometry is ratherstraightforward, especially in terms of contact detection and collision handling.In addition, the geometry of a sphere is fully determined by a single parameter,i.e. its diameter. In contrast, most non-spherical shapes (e.g. cylinders, oblateand prolate spheroids) additionally require some sort of aspect ratio to be uniquelydefined, which significantly expands the parameter space to be swept. Further-more, orientation of a non-spherical object with respect to the mean flow adds yetanother level of complexity, posing major difficulties in the development of hydro-dynamic closure laws for point-particle models. In this thesis, we have chosen toinvestigate the hydrodynamics of suspensions of cubes. While a few studies haverecently reported results for suspensions of spheroidal objects, the realm of non-smooth shapes in suspensions such as polyhedrons has remained almost entirelyunexplored. The shape of a cube is chosen as being representative of a polyhedralshape having sharp edges, while its isometry implies that similar to a sphere, nonotion of aspect ratio is involved.Prior to analyzing the collective behavior of cubes, we begin by characterizing3the inertial motion of an isolated cube in chapter 2. In this chapter, we presentnumerical simulations of freely settling and rising cubes in a quiescent Newto-nian fluid in various inertial regimes and solid-to-fluid density ratios. Ultimately,we obtain a comprehensive two-parameter flow-map for a freely moving cube andcharacterize prominent features of each regime of motion such as trajectories andwake structure. We find that regardless of the density ratio, the cube trajectorytransitions from oblique to unsteady vertical, helical and finally random/chaotic.Unlike the case of a sphere, helical motion is observed for all density ratios, mark-ing it as a characteristic type of motion for a cube. Furthermore, we present anin-depth force analysis relevant to the induced lateral motions, and we show thatthere is a significant jump in the drag coefficient coincident with the onset of thehelical regime where large-amplitude lateral displacements appear. We attempt toexplain these significant variations of the drag coefficient in connection with thepath and vortex structures of the moving cube. Consequently, the enhancement ofthe drag coefficient is explained to be a combined effect of the vortex-induced dragand the orientation angle of the cube.Subsequently, we proceed to chapter 3 where particle-resolved simulations ofinertial settling of monodisperse suspensions of cubes and spheres are considered.While strong columnar clustering is observed for the most inertial and most dilutecase in a sphere suspension, such vertical structures are less prominently present ina cube suspension. We find that in all cases, cube suspensions tend to be more ho-mogeneous compared to sphere suspensions, as indicated both by their microstruc-ture and momentum transfer properties. This is found to be mainly due to pro-nounced rotational motions of cubes and the resulting Magnus forces, which pro-mote transverse motions and the likelihood of escaping from clusters. Higher an-gular velocities of cubes thus play a major role in the transfer of momentum fromthe gravity to the transverse direction, as demonstrated by the lower anisotropyof particle velocity fluctuations in cube suspensions. In more dilute cases, cubesinduce significantly stronger pseudo-turbulence in the flow, especially in the trans-verse direction. The drag of dynamic cube suspensions is found to be similar tostatic beds of cubes in most of the cases, the reason for which is speculated to berelevant to their motion freedom and the more homogeneous microstructure.41.1.2 Microstructure-informed point-particle modelIn a particle-resolved simulation method, also referred to as a microscale de-scription, computational cells are usually an order of magnitude smaller than theparticle diameter. In such a formulation, fluid stresses can be directly integrated toobtain hydrodynamic loads. This approach is very desirable, yet computationallyprohibitive when a large number of particles are concerned. In a mesoscale, orEuler-Lagrange simulation, the computational cells are typically an order of mag-nitude larger than the particle diameter, and particles are treated as point sourcesor sinks of momentum. In this method, the computational cost is greatly reduced,which is why a much larger number of particles can be simulated. Nevertheless, in-teraction forces and torques cannot be computed directly and thus a closure modelis needed to supply this crucial piece of information. The physical fidelity of anEuler-Lagrange simulation is therefore mainly dependent on how accurately theexchange of momentum between the fluid and particles is accounted for.Point-particle closure models have proven to be fairly adequate to reliably pre-dict hydrodynamic forces given that the particle diameter is smaller than the rele-vant flow scales, and the Reynolds number based on the slip velocity is close to theStokes limit. These conditions imply low particle concentration and small particleresponse times (i.e. small Stokes numbers). However, for inertial suspensions athigher solid volume fractions, particles are closer to each other and the perturbationof each particle easily influences the flow field around a neighboring one. Sincethe flow field now varies over the scale of a particle diameter, particles have a finitesize with respect to the flow, and hence can no longer be treated as point particles.These flow perturbations, also known as pseudo-turbulence, substantially modifyhydrodynamic loads experienced by the adjacent particles. For example, the fluc-tuation of the drag experience by individual spheres in a dense stationary array canbe as large as 50% mean drag itself [7]. Notably, the existing closure laws in theliterature are only able to give a mean value for the drag regardless of the neigh-borhood of each particle. Furthermore, the lift and torque induced by the presenceof surrounding particles are completely neglected. Consequently, the ignorance ofmicrostructure causes significant underestimation of particle velocity fluctuations,and leads to physically incorrect evolution of the suspension in an Euler-Lagrange5simulation [8, 9, 10, 11].Motivated by the foregoing challenges, in chapter 4 we present a novel deter-ministic model that is capable of predicting particle-to-particle force and torquefluctuations in a fixed bed of randomly distributed monodisperse spheres. First,we generate our dataset by performing particle-resolved simulations of arrays ofstationary spheres in moderately inertial regimes and a solid volume fraction rangecorresponding to densely populated beds. The key idea exploited by our modelis that while the arrangement of neighbors around each particle is uniform andrandom, conditioning forces or torques exerted on a reference sphere to specificranges of values results in the emergence of significantly non-uniform distribu-tions of neighboring particles. Based on probabilistic arguments, we take advan-tage of the statistical information extracted from resolved simulations to constructforce/torque-conditioned probability distribution maps, which are ultimately usedas basis functions for regression. Given the locations of surrounding particles as in-put to the model, our results demonstrate that the present probability-driven frame-work is capable of predicting up to 85% of the actual observed force and torquevariation in best cases. Since the precise location of each particle is known in anEulerian-Lagrangian simulation, our model would be able to estimate the unre-solved subgrid force and torque fluctuations reasonably well, and thereby consid-erably enhance the fidelity of EL simulations via improved interphase coupling.6Chapter 2Dynamics of a freely settling andrising cube12.1 IntroductionMultiphase flows are encountered in a wide range of industrial, environmentaland biological situations where rigid particles, bubbles or drops of various shapesand densities are carried by a continuous phase. The dynamics of one phase in-fluences the other phase in a variety of ways leading to complex interactions andoverall behaviors. Examples of such situations include sediment transport in rivers,cloud and rain formation in the atmosphere [5], fluidized beds and reactors [13],bubbly flows [14] and red blood cells in a vessel [6].The motion of a single rising or settling particle is of particular interest in manyengineering fields. Various patterns of path oscillation are known to occur as aresult of coupling between the particle motion and the surrounding flow. Such mo-tion patterns are often accompanied by significant modifications of the settling rate,interaction forces and particle fluctuations, as well as heat and mass transfer char-acteristics of the moving particle [15]. Furthermore, small scale mechanisms are1A version of this chapter has been published in Physical Review Fluids [12].7known to influence the dynamics and collective behavior of sizable group of parti-cles at a larger scale. As an example, significant side motions of spheres movingobliquely could promote wake attraction and clustering even in a dilute suspension[16]. There are many open questions to be addressed with regards to the free mo-tion of single particles, including mechanisms by which geometry and frequencyof path oscillations are chosen, effects of background flow, vicinity of walls [17],distinction between motion characteristics of spherical and non-spherical particles[18], and interactions with a nearby particle such as the Drafting-Kissing-Tumblingphenomenon [19]. Significant conclusions may be drawn for multi-particle systemsfrom a clearer view of the physics governing the motion of a freely moving singleparticle.The sphere has been selected as the preferred ideal geometric shape in numer-ous studies due to its omni-directional symmetry; expectedly though, it is estimatedthat at least 70% of raw material particles in industry are non-spherical [6]. Chemi-cal catalysts of arbitrary shapes, deformed bubbles in liquids and frazil ice in riversand oceans [20] are a few examples of buoyancy-driven motion of non-sphericalobjects. The geometric description of such shapes often requires additional param-eters (e.g., aspect ratio for axisymmetric shapes), and the already complex problemof fluid-particle interactions hence becomes even further challenging as more de-grees of freedom are introduced. Furthermore, the orientation of a non-sphericalparticle with respect to the local fluid velocity becomes another factor in explain-ing the induced forces and torques [21, 22, 23]. Remarkably, angular particles thathave sharp edges exhibit much larger rotational velocities, that in turn gives rise toadditional side-forces such as the Magnus force [18]. In the case of multi-particlesystems, collisions that occur between non-spherical particles require different andmore involving modeling strategies. The foregoing considerations highlight the in-herent difficulties of handling particle-flow configurations with non-spherical par-ticles.The vast majority of the previous published literature on dynamics of freelymoving particles is devoted to spheres. Inspired by a previous study from ourgroup on the free motion of angular particles [18], we intend to extend beyond thelimited number of cases presented in that study. The current chapter is dedicated to8give a dense coverage of the parameter space and an improved picture of transitionscenarios for a freely settling and rising cube in a numerical framework. Beinginterested in settling and rising of angular particles, we have selected the shape ofa cube for the two following reasons: Firstly, it is an angular particle which can bedefined with only a single parameter similar to a sphere; thus simplifying a com-prehensive exploration of the governing parameters. Secondly, since the cube hasthree orthogonal planes of symmetry and possesses similar geometrical propertiesin all three directions, added-mass effects only act as additional resistance to bodyacceleration without cross-coupling force and torque balances [17]. This meansthat wake instability would be the only contributing factor to any occurrence ofpath instability.2.2 Single particle systems: The state of the art2.2.1 Spheres immersed in a fluid2.2.1.1 Fixed spheresAn idealized example of viscous flow past a three-dimensional bluff body isa sphere held fixed in a stream of incoming fluid. The wake instability behind asphere has been extensively studied both experimentally and numerically in thepast [24, 25, 26, 27, 28, 29], and the transition scenario is now well established.The flow behind a sphere becomes separated at Re ≈ 24 and an axisymmetrictoroidal vortex region appears downstream of the body [29]. This closed circula-tion region expands up to Re ≈ 212, where the wake undergoes a primary regularbifurcation, or an “exchange of stability” [25] to another steady vortex structurethat is no longer axisymmetric. This regime is characterized by the planar sym-metry of the wake and the existence of a pair of counter-rotating filaments of vor-ticity (also termed the “bifid” wake in [30]) extending downstream of the sphere.Small-amplitude sustained oscillations have been observed experimentally withoutvortex shedding slightly below Re = 270 [25]. These oscillations precede the nexttransition which occurs at 270 < Re < 280 through a secondary Hopf bifurca-tion [25, 26, 28]. The fully time-dependent wake now exhibits periodic shedding9of hairpin vortices while the planar symmetry observed after the first transition isstill maintained. The sequence of transitions is different from that of the flow pasta two-dimensional cylinder, where the wake directly becomes time-dependent atRe ≈ 47 through a Hopf bifurcation with no intermediate steady regime. Whileexperimental flow visualizations consistently suggest that hairpin vortices are al-ways shed on the same side, numerical simulations reveal two-sided vortex shed-ding with alternating sign and different magnitudes. This disparity is associatedwith a relative weakness of the hairpin vortices on one side in dye visualizationexperiments [17]. Nevertheless, the two sides of shed vortices seem to originatefrom different sources: one side resulting directly from the shedding of the ringvortex attached to the rear of the sphere [31], while the oppositely oriented side issuggested to appear as a result of the interaction of the near-wake region with theouter flow [26]. Freely moving spheresThe fluid-solid system of a freely moving body in an otherwise motionless fluidis expected to display interesting behaviors, since the object has now six degreesof freedom influenced by the hydrodynamic forces and torques arising from thefluid-solid interaction. Moreover, instead of just one (i.e., the Reynolds numberin the case of a fixed sphere), two non-dimensional numbers govern the prob-lem: The density ratio m = ρs/ρf , and the Galileo number defined as Ga =√|1−m|gD3/ν, with ρs and ρf denoting the solid and the fluid density respec-tively, g the gravitational acceleration, D the diameter of the sphere and ν thekinematic viscosity of the fluid. We note that the Galileo number is simply theReynolds number defined based on the gravitational velocity ug =√|1−m|gD.Several recent studies have focused on settling and rising of a single sphere bothnumerically [32, 30, 33] and experimentally [15, 34]. The study of Jenny et al.[33] is among the first to investigate the instabilities and transitions numericallyfor a freely moving sphere. They showed that the threshold of the regular bifur-cation is slightly below that of a fixed sphere based on the asymptotic Reynoldsnumber. In their extensive follow-up study [30], they numerically explored a widerange of Galileo numbers between 150 6 Ga 6 350 and various density ratios10between 0 6 m 6 10. Along with a few experiments, they provided a flow-mapfor different regimes of motion of a freely settling or rising sphere. Jenny et al.[30] also found that the trajectory of the sphere is planar in early stages, and thata steady oblique motion arises after the first regular transition. The critical Recritfor the onset of unsteadiness was reported to increase from Recrit ≈ 224 for mass-less spheres (m = 0) to Recrit ≈ 273 for the case of fixed spheres (m → ∞).A similar observation was made in the dye visualization experiments carried outwith relatively dense, freely settling liquid drops [35]. The thresholds of instabil-ity were reported to be in good agreement with those obtained for the flow past afixed sphere. In the oblique-oscillating regime, periodic oscillations of the sphereabout a mean oblique direction are observed in various works [36, 32, 30]. Theknowledge of freely moving spheres was further advanced by a number of impor-tant investigations in the literature. In their experimental work, Veldhuis et al. [37]examined the free motion of settling and rising spheres (0.5 6 m 6 2.63) at highReynolds numbers (450 6 Re 6 4623). This was followed by another experimen-tal study by Veldhuis et al. [34] on the rising of very light spheres (m = 0.02) atrelatively high Galileo numbers (612 6 Ga 6 1712), in which the authors sug-gested a mechanism for the unusually increased drag coefficient of light spheresreported in the literature [38, 39] and also in an earlier work by the same group[40]. Horowitz & Williamson [15] also conducted a large number of experimentsin their notable work on freely moving spheres. According to their results, allspheres move vertically when Re < 210, as expected from the axisymmetry of thewake. The two trailing vortices that appear after the first wake transition exert asteady lift force which causes the spheres to move in oblique paths. Two criticaldensity ratios were identified; namely, mcrit ≈ 0.4 for Re = 260 − 1550 andmcrit ≈ 0.6 for Re = 1550 − 15000. For m < mcrit, highly periodic zigzagmotions were reported to occur. They concluded that this regime corresponds to anew vortex-shedding mode with four distinct vortex rings per each shedding cycle(and hence the name “4R”). The 4R regime was also recognized to be unique inthat spheres rising in this regime manifest drag coefficients close to CD = 0.75,compared to CD ≈ 0.5 obtained for fixed spheres and for spheres that move instraight paths. Furthermore, the chaotic states of a freely moving sphere were stud-ied thoroughly by Zhou & Dušek [32] and the existence of a highly regular helical11regime beyond Ga > 375 for mass-less spheres in their numerical simulations wasevidenced. Quite recently, Auguste & Magnaudet [36] have focused solely on thepath instabilities as well as drag enhancement of rising spheres for a wide range ofGalileo numbers (125 6 Ga 6 700). They identified that only one type of pathgeometry, namely, the helical path causes the significantly increased drag. A moredetailed discussion about drag coefficient enhancement of moving objects will bepresented later in section Cubes immersed in a fluid2.2.2.1 Fixed cubesAs noted before, several theoretical, numerical and experimental studies aredevoted to the problem of a stationary sphere in flow or a freely moving spherein an otherwise quiescent fluid. However, a striking lack of similar investigationsfor angular objects such as cubes is evident in the literature. As one of the fewworks on this subject, Raul et al. [41] performed numerical simulations of flowpast a stationary cube in the range of 10 6 Reedge 6 100, with Reedge denoting theReynolds number defined based on the edge length of the cube. The first systematicinvestigation of the wake transitions behind a cube was done by Saha [42]. Itwas then followed by another work by the same author [43] that included a studyon the heat transfer from the cube and the corresponding effects associated withtransitions. In the foregoing study, it was shown that the wake conserves symmetryabout two pairs of orthogonal planes (four planes of symmetry) up to Reedge =216 − 218. Four pairs of stream-wise oppositely signed vortices are found in thisregime. Saha [42] also observed two leg-like counter-rotating threads, similar tothe same structure found downstream a sphere after the first regular transition.According to that study, the flow loses temporal stability for Reedge = 270 andshedding of hairpin vortices is seen. Recently, Koltz et al. [31] conducted the firstfully experimental investigation of the wake behind a cube for 100 6 Reedge 6400 using Laser-Induced Fluorescence (LIF) and Particle Image Velocimetry (PIV)visualization methods. They utilized Landau’s instability model to determine theonset of the two bifurcations in this range of Reynolds numbers. The threshold12of Reedge ≈ 184 that they obtained for the first bifurcation is markedly differentfrom the value of Reedge ≈ 216 − 218 reported earlier by Saha [42], which wasattributed to the use of less sensitivity for visualizing the pair of counter-rotatingvortices.Shapes other than a sphere would experience different forces depending ontheir orientation with respect to the flow direction. The effects of the incidenceangle on drag, lift and momentum coefficients for various non-spherical objects,including cubes, were first studied numerically by Hölzer & Sommerfeld [23, 44]using a Lattice-Boltzmann method at four different Reynolds numbers in the range0.3 6 Re 6 240. Motivated by the scarcity of studies concerning the effect ofparticle shape on the Nusselt number Nu, Richter & Nikrityuk [45] performed nu-merical simulations of non-spherical particles, including cubes and ellipsoids, fixedin cross-flow. Focusing on the steady regime in 10 6 Re 6 250, they providedcorrelations for drag coefficient and Nusselt number as functions of Re and geo-metrical parameters of the objects. Moreover, they extended this study to includeeffects of particle orientation on momentum and heat transfer and proposed corre-lations for CD and Nu as functions of the incidence angle of the object [21]. Thedrag coefficient was found to be substantially affected by the particle orientation incase of an ellipsoid, but less so for a cube. The Nusselt number, on the other hand,was found to be only slightly influenced by the incidence angle of the object. Freely moving cubesAmong non-spherical objects, axisymmetric bodies such as oblate and prolatespheroids [19, 46], disks [47], cylinders [48] and cones [49] are the ones whosefree sedimentation (and rising in case of oblate bubbles [50, 46]) have receivedattention. Surprisingly though, investigations of the buoyancy-driven motion ofangular particles (cubes and other polyhedral shapes) are rare. The shape of acube is mentioned in a few drag correlations in [51, 52, 53]. One of the first dragcorrelations for a wide range of Reynolds numbers (Re 6 25000) and various non-spherical particles was proposed by Haider & Levenspiel [53]. Their experimentalstudy was based on data obtained from free settling of particles. They used ameasure of the sphericity of a particle (which is defined as the ratio of the surface13area of the equivalent-diameter-sphere to the actual surface area of the particle) inorder to include the effects of non-sphericity in their correlation. More recently,Yow et al. [54] suggested a correlation for a Reynolds number range of Re =10−2 − 105 and particle sphericity of 0.006 − 1. They demonstrated that as theparticle sphericity decreases, the drag coefficient increases compared to the case ofa sphere. In another work, Tran-Cong et al. [52] utilized groups of packed spheresin order to approximate the geometry of non-spherical shapes, including cubes, insteady free fall experiments. They quantified the drag coefficient in terms of theReynolds number (0.15 6 Re 6 1500) and geometrical parameters such as particlecircularity and the ratio of the surface-equivalent-sphere diameter to the nominaldiameters.Apparently, all the drag correlations given by the experimental studies above arebased on data for settling cubes, and to the authors’ best knowledge, there has beenno account of rising cubes in the literature. Furthermore, none of the aforemen-tioned investigations have attempted to recognize different regimes of free motionof a single cube, as a representative angular shape. The subject of path instabilityof angular shapes was partially covered in a recent work by Rahmani & Wachs[18], where they turned their attention to non-spherical angular shapes, i.e., cubesand tetrahedrons, as opposed to non-spherical axisymmetric shapes such as oblateor prolate spheroids [19, 50, 55] or disks [56]. They reported that objects withhigher angularity, i.e., tetrahedrons, show chaotic behavior at considerably lowerGa compared to less angular shapes such as cubes. Moreover, the rotation rates ofangular particles and the resulting forces are much more accentuated for cubes andtetrahedrons compared to spheres. As a first study on the path oscillations of an-gular particles, the aforementioned investigation presented results only for settlingcubes at a few selected Galileo numbers. In the present work, we attempt to fillthe identified gap in our knowledge on free motion of a cubic particle by extendingthe parameter space to a broader range, along with detailed regime characterizationand relevant force analysis.In remainder of this chapter, we first present the mathematical description ofthe problem, along with introducing our numerical method in section 2.3. In sec-tion 2.4, we provide a flow-map for the free motion of a cube, followed by detailed14characterization of each regime of motion. The force balance analysis is coveredin section 2.5, where we discuss the hydrodynamic forces exerted on the cube indynamically interesting regimes. We will demonstrate that these forces play a cru-cial role in inducing lateral motions and modifying the drag force as well. Also,a complementary discussion is provided in appendix A.1, where we present wakestructures, transition thresholds and drag coefficient variations of a stationary cubefor various values of the Reynolds number. Consequently, we compare our resultswith published works in the literature.2.3 Mathematical formulation and numerical algorithm2.3.1 Governing equationsThe incompressible Navier-Stokes and continuity equations for a Newtonianfluid are given in non-dimensional form as∂u∂t+ u · ∇u = −∇p+ 1Rec∇2u (2.1)∇·u = 0 (2.2)where u and p are the fluid velocity vector and the pressure, respectively. Thereference values used for non-dimensionalizing equation (2.1) are the fluid densityρf , the volume-equivalent diameter of the cubeD, the reference velocity uc and thereference pressure ρfuc2. The reference velocity and the characteristic Reynoldsnumber appearing in equation (2.1) are given asuc =√43|1−m|gD (2.3)Rec =ρfucDµ(2.4)with µ the dynamic viscosity of the fluid. In the above equations, m denotes thesolid-to-fluid density ratio, which together with the Galileo number are the two15governing parameters of the problem defined asm =ρsρf(2.5)Ga =√|1−m|gD3ν(2.6)Furthermore, the rigid body motion of the cube is described by the Newton-Eulerequations which in the body frame of reference aremvpdVdt= F + (m− 1)vpg (2.7)mIpdΩdt+ Ω× (Ip ·Ω) = T (2.8)where V, Ω, vp, Ip and g stand for the object’s translational and angular veloc-ity vector, particle volume, inertia tensor and gravitational acceleration vector,respectively. Note that with Ip being a scalar matrix for a cube, it follows thatΩ × (Ip · Ω) = 0. Also, F and T denote the hydrodynamic force and torqueexerted on the particle, and are given asF =∫S(−pI + 1Rec(∇u +∇ut)).n dS (2.9)T =∫Sr×(−pI + 1Rec(∇u +∇ut)).n dS (2.10)with I being the identity matrix, n the unit vector normal to the boundary of thesolid object, and S the surface enclosing the solid body.2.3.2 Numerical methodOur numerical tool, PeliGRIFF (Parallel Efficient Library for GRains in FluidFlow), is based on the Distributed Lagrange Multiplier/Fictitious Domain (DLM/FD)formulation proposed by Glowinski [57]. Here, we employ a Finite-Volume andStaggered Grid (FV/SG) variant of PeliGRIFF (see [58] for more details). Insteadof fitting a computational grid to the boundary of solid objects, the main idea ofDLM/FD method is to extend the fluid domain to contain solid bodies as well,while enforcing rigid-body motion constraint on the fictitious fluid inside the solid16body region. This way of treating solid boundaries obviates the need for successiveremeshings required in conventional body-fitted methods to accommodate the mo-tion of an immersed object. For the specific case of a single cube in a nonvariationalframework we have1. Combined momentum equations∂u∂t+ u · ∇u = −∇p+ 1Rec∇2u− λ in D (2.11)(m− 1)vpdVdt−∫Pλ dx = F + (m− 1)vpg in P (2.12)(m− 1)IpdΩdt−∫Pr× λ dx = 0 in P (2.13)u− (V + Ω× r) = 0 in P (2.14)2. Continuity equation∇ · u = 0 in D (2.15)where the solid domain and whole fluid/particle domain are denoted by P and D,respectively and r represents the position vector relative to the particle center ofmass. Also, λ shows the distributed Lagrange multiplier vector which is usedto enforce the rigid-body motion constraint equation (2.14). With the DLM/FDmethod, the Lagrange multiplier λ can be used directly to obtain the hydrodynamicforce and torque exerted on the particle P:F =∫Pλ dx + vpdVdt, (2.16)T =∫Pr× λ dx + IpdΩdt. (2.17)It should be noted that the general form of the set of DLM/FD governing equationsshown above is for a single cube (i.e., Ω × IpΩ = 0), without any contact force17since there are no particle-particle or particle-wall collisions in the present study.As for the numerical time-marching algorithm, we employ a two-step classicaloperator-splitting scheme. For each time tn+1, we solve:1. A classical projection scheme for the solution of the Navier-Stokes problem:find un+1/2 and pn+1 such thatu∗ − un∆t− 12Rec∇un+1/2 = −∇pn + 12Rec∇un,− 12(3un · ∇un − un−1 · ∇un−1)− λn, (2.18)∇2ψ = 1∆t∇ · u∗ , ∂ψ∂n= 0 on ∂D, (2.19)un+1/2 = u∗ −∆t∇ψ, (2.20)pn+1 = pn + ψ − ∆t2Rec∇2ψ. (2.21)2. A fictitious domain problem: update the particle translational position Xn+1and angular position θn+1 using Vn and Ωn, and find un+1, Vn+1, Ωn+1and λn+1 such thatun+1 − un+1/2∆t+ λn+1 = λn, (2.22)(m− 1)vpVn+1 −Vn∆t−∫Pλn+1dx = (m− 1)vpg, (2.23)(m− 1)IpΩn+1 −Ωn∆t−∫Pr× λn+1dx = 0 (2.24)un+1 − (Vn+1 + Ωn+1 × r) = 0. (2.25)where ∆t denotes the time step, ψ the pseudo-pressure and ∂D the domain bound-ary. In the Navier-Stokes sub-problem (step 1), the viscous and advective terms arediscretized with second-order time accurate Crank–Nicolson and Adams-Bashforthschemes, respectively. Considering the high order correction of the pressure (thirdterm on the right-hand side of equation (2.21)), the projection scheme is alsosecond-order accurate in time. However, the first-order time discretization of the18xyz816458gFigure 2.1: Schematics of the computational domain for a sample settling cube casefictitious domain sub-problem (step 2) and the first-order operator splitting methodreduce the global time-accuracy of our algorithm to first-order only.2.3.3 Problem setupThe computational domain for this problem is defined to have the lengths ofLx = Ly = 8 in the horizontal direction and a vertical length of Lz = 25. Asfor the boundary conditions, the domain is set to be bi-periodic in the lateral direc-tions, whereas zero velocity and outflow conditions are imposed on the upstreamand downstream boundaries, respectively. A grid resolution of 24 mesh points perequivalent diameter (i.e., D/∆x = 24 with ∆x indicating the grid spacing) is usedinside a computational box around the particle with dimensions of 8 × 8 × 4, de-picted in figure 2.1 as a gray volume surrounding the cube. This resolution wasshown to be sufficient for capturing essential dynamics of an angular particle [18].To keep the computational cost reasonable, a linearly stretched mesh is used only inthe vertical direction. The resolution of the stretched mesh starts fromD/∆x = 24on either side of the gray volume and decreases to D/∆x = 12 on the inlet and190 100 200 300 400 500 60000. 2.2: The evolution of the vertical velocity of the cube for Ga = 160 andvarious density ratiosoutlet boundaries, resulting in 192 × 192 × 448 ≈ 17 × 106 grid points per eachsimulation. In lateral directions, the mesh resolution is kept fixed at D/∆x = 24.The computational domain along with different meshing zones are shown schemat-ically in figure 2.1. As the cube moves, it would exit a domain with such a limitedsize fairly quickly. In order to circumvent this issue, we use a domain translationtechnique already discussed in [18, 59, 60], instead of using an extended domainin the vertical direction or a moving frame of reference. With this method, if thedistance between the particle and the inlet boundary becomes smaller than a spec-ified threshold (lcr = 6 in our simulations), the computational domain is movedin the vertical direction so as to keep the particle at least a distance of lcr awayfrom the inlet boundary. The translation is achieved by removing a certain numberof grid layers from the outlet and adding the same number of layers to the inlet,while the magnitude of the domain translation is chosen to be a multiple of thegrid size (2∆x for all considered cases). The latter ensures that errors associatedwith the projection of the computed solution onto the translated grid are avoided atleast in the gray volume shown in figure 2.1, where the grid size is kept constant.The time step ∆t is always kept smaller than 1 × 10−3 for light cubes (m < 1)and smaller than 2 × 10−3 for dense cubes (m > 1). Most of our simulations are2070 85 100 120 140 160 180 200 2500. Vertical Helical ChaoticFigure 2.3: Flow-map of the rising and settling cube in the Ga−m parameter spacerun up to at least t = 200, while in many cases they are resumed up to t = 400to observe possible long-time behavior of the particle, as indicated in figure 2.2.This is particularly important when it comes to computing the terminal rising orsettling velocity and consequently the drag coefficient. As an example, prematuretermination of the simulations of (Ga,m) = {(160, 1.2), (160, 2)} shown in fig-ure 2.2 could result in measuring a misleading terminal velocity. In these cases, theapparent terminal velocity would be approximately 21% off compared to the valueit converges to after passing the initial transient period, which in turn, would yielda totally different value for the drag coefficient.2.4 Regimes of motionIn this section, we explore the Ga − m parameter space for the range 70 6Ga 6 250 and 0.2 6 m 6 7, obtained with 76 different combinations of these twonumbers. The state-diagram for these ranges of parameters is shown in figure 2.3.For the smallest Ga simulated here, we observe an oblique trajectory in most ofthe cases except for m = 0.2 and m = 0.8. The path chosen by the cube in thesetwo simulations is seen to be vertical, and they seem to belong to the neighboring21vertical regime in figure 2.3. As Ga is increased, the cube with any density ratiom passes through the vertical and helical regimes. For sufficiently large Ga, thetrajectory of all cubes becomes chaotic; meaning that the motions evolve towardnon-predictability even if the appearance of their trajectory does not indicate anentirely random path. We also find that in general, lighter particles make theirtransition to neighboring regimes at relatively smaller values of Ga.2.4.1 Oblique regimeThe leftmost side of the state diagram (figure 2.3) with Ga = {70, 85} belongsto the oblique regime, where the cube moves with a constant angle relative to thevertical axis with no recognizable fluctuations in any directions. It is documentedin the literature that sufficiently heavy spheres start to settle obliquely at Re ≈ 210[15]; here however, we find that the cube shows a similar trajectory at a muchsmaller value of Re = 65. This is equivalent to the onset of an oblique regimefor a sphere at Ga ≈ 160 − 190, which is almost twice the threshold of Ga = 70found here for a cube. This early transition appears to be triggered by the vorticityfluctuations generated at the sharp edges of the cube. Comparing to the reportedinclination angles of 3.5◦ − 4.3◦ for spheres [18, 36], the cube drifts with a muchsmaller angle between 0.6◦ and 0.7◦ throughout the oblique regime region.The horizontal force that constantly pushes the particle to one side has its originin the structure of the wake, as visualized in figure 2.4c as contours ofω·eˆz = ωz orthe stream-wise vorticity, withω indicating the vorticity vector and eˆz a unit vectorin the z direction. In the top panel of figure 2.4c, contours of ωz are shown on aplane normal to the vertical axis, downstream of the object. In the bottom panel ofthe same figure, three-dimensional contours of ωz are visualized. The wake revealsits plane-symmetric structure, which seems identical to the wake of a fixed cubeafter the first wake instability (see appendix A.1). After the first transition, the fourpairs of counter-rotating vortices are replaced by the plane-symmetric wake, whichis characterized by two large bean-shaped vortices in the middle and four smallervortices on the sides [31]. The asymmetric shape in the x direction is also evidentfrom the three-dimensional contours in the right-hand side panel of figure 2.4c, asstronger vortices take up a larger volume on one side than the other side. After22-4 -2 0 2 4-400-300-200-1000100200300(a)-2 -1 0 1 2-250-200-150-100-50050100150200(b)(c)Figure 2.4: (a) & (b) Side-view of the trajectory of the cube in the oblique and un-steady vertical regimes (c) Left: (Ga,m) = (70, 0.8) - symmetric vortex struc-ture; Right: (Ga,m) = (70, 4) - plane-symmetric vortex structure. Top row:Contours of −0.005 6 ωz 6 0.005 (ωz ≈ 0 in green regions) on a z-normalplane at a distance of d = 2 downstream of the cube center; Bottom row: Iso-surfaces of stream-wise vorticity for ωz = ±0.0123reaching steady state, the wake conserves its plane-symmetric form throughout thesimulation time and keeps applying a steady side force in the y direction. Trajec-tories and wake structures of other oblique cases in the state-diagram (figure 2.3)are almost identical to that shown in figure 2.4, with the plane of symmetry beingchosen randomly in x or y directions. Unlike a fixed cube after the first transition,no bifid wake or leg-like structure is observed in the Q-criterion [61] visualizationof the vortices, no matter what contour value is used. It should be noted that forcases with small Reynolds numbers, no vortex structure in the wake region (apartfrom that attached to the body) is identifiable with the Q-criterion method; there-fore, we have visualized only the contours of stream-wise vorticity downstreamof the cube. At higher Reynolds numbers, however, the Q-criterion method is su-perior in visualizing vortical structures and in revealing useful information withregards to the shape and relative orientation of the wake. In this regime, the an-gular velocities in the horizontal direction, denoted as Ωh =√Ω2x + Ω2y are ef-fectively zero and the cube moves with a steady orientation, making a constantincidence angle of φ ≈ 2◦. This can be contrasted with the case of a sphere mov-ing obliquely with a steady horizontal angular velocity of Ωh = 0.015 reported in[30] and 0.0149 6 Ωh 6 0.0082 for 0 6 m 6 10 in [32]. No rotation is observedabout the vertical axis for any of the cases.2.4.2 Unsteady vertical regimeWith further increasing the Galileo number Ga, the oblique regime is replacedby the neighboring unsteady vertical regime (figure 2.3) where the cube translatesvertically while manifesting very small lateral fluctuations. As evidenced in fig-ure 2.5, the amplitude of side movements remains smaller than 0.1, with a nethorizontal displacement of 6 0.5 over a vertical distance of 200− 400 traveled bythe object. The tendency of drifting to one side by a small amount seems to be aleftover trace from the oblique regime; this is probably due to the random devel-opment of a plane-symmetric wake that persists long enough to induce noticeableside motion. In fact, analysis of the wake structure for (Ga,m) = (85, 0.8) revealsthat despite being unstable, the wake appears to take on the same plane-symmetricform sporadically with a preference for a specific plane. What is observed here24-2 -1 0 1 2-600-500-400-300-200-1000100200300(a)-2 -1 0 1 2-500-400-300-200-1000100200300400(b)-2 -1 0 1 2-600-500-400-300-200-1000100200300400(c)Figure 2.5: Side-view of the trajectory of a cube for various density ratios in theunsteady vertical and helical regimes. The color code is the same as in figure 2.2in the vertical regime significantly resembles the “Regime A” described in [17],which is characterized by small, irregular transverse motions of the moving ob-ject. The common feature of all cases in this regime is that the wake becomestime-dependent, and as vorticity builds up in the wake, small-scale vortices areshed randomly. This pattern of occasional, directionally random vortex sheddingis what causes the small lateral irregular fluctuations. The trajectory then becomesmore regular for (Ga,m) = (140, 3) on the boundary between two regimes, thoughit still seems to be intermittent spiraling with small amplitudes (e.g., O(0.1)). Thewake in this case progresses from a shedding state towards two relatively persistentinter-twined threads of vorticity, as seen in figure 2.6.It is worthwhile to note that the unsteady vertical regime described here is com-pletely different from that observed by [32, 15, 30] in the flow-map of a sphere. Thetrajectory of a sphere in that regime is reported to be perfectly vertical without anyoscillations, and the wake is steady and more importantly, axis-symmetric. This iswhy the term “unsteady vertical” is used here to describe the observed motion, aswe clearly find time-dependent wake and intermittent vortex shedding. The onlyexception in our simulations is the case (Ga,m) = (70, 0.8) which exhibits abso-lutely no fluctuations, while the wake structure remains steady with four pairs of25(Ga,m)=(100,0.2) (Ga,m)=(100,4) (Ga,m)=(140,2) (Ga,m)=(140,3)Figure 2.6: Vortex structure for various cases in the unsteady vertical regime. Vi-sualization is done using iso-surfaces of stream-wise vorticity for ωz = ±0.02(the first and the second image) and Q-criterion for Q = 0.003 (the third andthe fourth image).counter-rotating vortices of the same size and four planes of symmetry identical tothat of a fixed cube before the first transition. The cube in this case behaves similarto a sphere in the vertical regime; hence this case probably belongs to a steady ver-tical regime that occurs for smaller Galileo numbers (e.g., Ga < 70) not simulatedhere. It is remarkable that for a sphere, the onset of the oblique regime coincideswith the wake transition at the same Reynolds number, but these transitions occurmuch sooner for a cube. The specificity of the cube geometry, which producesa different vortex structure and the enhanced sensitivity to small body vibrationsmight trigger these early transitions. The flow-map in figure 2.3 also shows an “or-phan” point at (Ga,m) = (70, 7) that is expected to be located within the obliqueregime. The same applies to (Ga,m) = (70, 0.2) and (Ga,m) = (70, 0.8) to someextent. These cases seem to lie at the transition between the oblique regime and theunsteady vertical regime, thus making Ga = 70 a sensitive threshold. We concludethat this regime is characterized mainly by vertical motion with insignificant pathoscillations, and a time-dependent wake. For smaller Ga, the wake switches back26and forth randomly between having four or one plane of symmetry. As Ga is in-creased, the wake starts shedding small vortices. For the highest Ga in this regime,the inter-twined vortices appear behind the cube coinciding with small-amplitudespiralling path; a precursor of the helical regime motion that follows by increasingGa. The unsteady vertical regime appears to be a transitional region in the flow-map shown in figure 2.3, where traces of the oblique and helical regime are bothevident in the wake structure as well as in side-drifts and small-amplitude spirallingmotions. In the unsteady vertical regime, angular velocities in the horizontal direc-tion are higher with an average of magnitude in the range 0.005 6 〈Ωh〉 6 0.01 forGa = 100 and 0.01 6 〈Ωh〉 6 0.02 for Ga = 120 (Note that 〈Ωx〉 = 〈Ωy〉 = 0 forall cases). The oscillations of angular velocities are random at Ga = 100, whereasat Ga = 120 a more regular pattern of oscillations is observed. This is a sign ofthe evolution towards the ordered helical motion in the neighboring regime, whichis mature enough for the smallest m at Ga = 120 to be confidently placed in thehelical regime region of the flow-map.2.4.3 Helical regimeHelical motion of freely moving objects has been reported for various shapes inthe past, such as spheres [36, 32, 34], spheroidal [46] and oblate ellipsoidal bubbles[50] and short cylinders and disks [17]. For rising cubes, the helical motion isfirst seen at Ga = 120 with a relatively small amplitude shown in figure 2.5band for settling cubes at Ga = 130 for m = 7. The helix at Ga = 120 has anellipsoidal shape with a relatively high eccentricity. At Ga = 140, the path of therising cubes become distinctively circular when viewed from above (figure 2.7).The cube with m = 4 still shows only small scale helical motion whereas form = 7, a large-amplitude helical motion is clearly established. This circularity ofthe paths is preserved up to Ga = 170, which is the highest value of the Galileonumber at which ordered helical motion is still observed. Notably, the two caseswith (Ga,m) = (158, 0.8) and (Ga,m) = (170, 1.2) exhibit intermittency in theirtrajectory, also seen in figure 2.8. In other words, although still following helicalpaths most of the time, the cube momentarily drifts off and starts another episodeof helical motion some distance away from the initial starting point.27-202-202050100150200250(Ga,m)=(140,0.2) (Ga,m)=(158,0.8) (Ga,m)=(170,4)-202-202050100150200250300350-600-500-400-300-200-1002020 0-2-2xyxyxyxzxzyxzyxzyxzxzFigure 2.7: Various views of wake structures and trajectories of the cube in the he-lical regime. The vortices are visualized using iso-surfaces of Q-criterion forQ = 0.003.Analysis and visualization of the wake reveals mainly two types of vortexstructure in the helical regime; namely, the twisted double-threaded wake and anew vortex shedding mode that consists of four hairpin vortices per each cycle.From this point onwards, we choose to call this type of wake the “Four Hair-pins” structure or 4H, for convenience. The 4H (or so-called “horseshoe”) vorticeshappen to be arranged at approximately 90◦ angle with respect to each other for(Ga,m) = (140, 0.2). In figure 2.7, three such vortices are clearly seen, while thelast one is on the verge of being detached from the cube. By the time this vortexis shed, the last one has been convected out of the domain, considering its limitedvertical length. That is why not all four vortices are captured in one snapshot. It isremarkable that the wake of spheres and ellipsoidal bubbles along helical pathshave always been reported to contain the twisted, wrapped-around-one-anotherthreads of vorticity that never change their sign [36, 32, 50, 46] similar to whatwe observe in certain cases belonging to the lower Galileo numbers such as in(Ga,m) = (120, 0.2). Horowitz & Williamson [15] argued that since shedding ofvortex loops cannot provide a steady centripetal force, helical motions might notappear beyond Re = 270. Such helical paths have been later found to exist, at leastnumerically, for Ga > 400 and m < 1 [36, 32] equivalent to Re u 600 − 800 in[36]. None of these studies showed any evidence on shedding of vortex rings forspheres.We find that the amplitude of the lateral motion is relatively small (6 1) when-ever an inter-twined wake is seen downstream of the cube. Examples of such casesinclude (Ga,m) = {(120, 0.2), (120, 0.5), (130, 0.5), (140, 4)}. The cubes with28-4 -2 0 2 4-4-2024 Rising-4 -2 0 2 4-4-2024 Settling-4 -2 0 2 4-500-400-300-200-1000100200300400(a)-2 0 2-202Rising-2 0 2-202Settling-4 -2 0 2 4-600-500-400-300-200-1000100200300400(b)Figure 2.8: Different views of the trajectories of a cube at various values of Ga andm. The color code is the same as in figure 2.2(Ga,m)=(140,0.2) (Ga,m)=(170,1.2)-101 -10105010015020025010-250-200-1501-1000.5-5000-1-0.5-1xyyzxyyzyxzyxzFigure 2.9: Transient zigzag paths in the initial stages of the helical regime. Thevortices are visualized using iso-surfaces of Q-criterion for Q = 0.003.the 4H wake often begin moving along a straight path initially and as the cubereaches its maximum vertical velocity (which is achievable only before the onsetof vortex shedding), hairpin vortices very similar to those of a fixed cube start toappear. These vortices are indeed quite strong, and exert a significant side force onthe cube which causes the zigzag motion shown in figure 2.9. This zigzag motionis never stable; in fact, an oscillation normal to the plane of motion soon devel-29ops and helical path ensues shortly after. We never observe such large-amplitudezigzag motions for the inter-twined wake; however, not all 4H cases begin with azigzag motion. In a number of cases, the cube starts with the twisted wake andevolves into the 4H vortex, while some other cases switch from zigzag to helicalmotion. The mean value of the angle between the velocity vector and the verti-cal axis in this regime ranges from 2◦ ± 1◦ for small-amplitude helices such as in(Ga,m) = (140, 4) and 6◦ ± 2◦ in (Ga,m) = (120, 0.2), to almost steady 14◦ in(Ga,m) = (140, 0.2) or (Ga,m) = (140, 0.5). At higher Galileo numbers, this an-gle decreases with increasing the density ratio; 8◦± 1◦ for (Ga,m) = (160, 4) and(Ga,m) = (170, 4) and 12◦ ± 1◦ for (Ga,m) = (160, 2) and (Ga,m) = (170, 2).In a chaotic case such as (Ga,m) = (200, 0.2), the mean angle is 13◦, almost thesame as large-amplitude helices in lower Ga, but the magnitude of the fluctuationsis also around 10◦. As for the rotation rates, the cube in the helical regime rotateswith an almost constant Ωh (i.e., with notably small fluctuations about the aver-age value) in the horizontal direction (see section 2.5.1, figure 2.13). Although theangular velocity component in the vertical direction is still small, it rises to about25% of the horizontal component in the case of the lightest cube with m = Chaotic regimeFirst observed at Ga = 170, the chaotic regime prevails for Ga > 180. In thehelical regime, the path geometry is preserved and the motion seems to be pre-dictable once established. The main characteristic of the chaotic regime, however,is that while for some cases (e.g., (Ga,m) = (170, 0.2)) the path appears to behelical for the most part, the cube suddenly changes direction and starts followinganother helical/spiralling path and exhibits a lateral drift. Carrying “finger-prints”of the neighboring ordered regimes was also noted by Zhou & Dušek [32] forspheres moving in chaotic paths in their simulations. In these cases, the sense ofthe helix also changes; meaning that a right-handed helix turns into a left-handedone and vice-versa. This intermittency and randomness of the path is also reflectedin the wake of the cube where we are unable to identify any persistent structuresuch as those in the helical regime. In cases that are more ordered, hairpin vorticessimilar to the ones seen in the helical regime are observed for brief periods of time.30-2 0 2-202Rising-2 0 2-202Settling-4 -2 0 2 4-500-400-300-200-1000100200(a)-4 -2 0 2 4-4-2024 Rising-4 -2 0 2 4-4-2024 Settling-10 0 10-350-300-250-200-150-100-50050100150(b)Figure 2.10: Different views of the trajectories of a cube in the chaotic regime.Please note the difference in the ranges of x and y values which is necessaryto be able to show the entirety of the trajectories for Ga = 250. The color codeis the same as in figure 2.2The trajectories of the cubes at Ga = 180 and Ga = 250 are shown in figure 2.10.In both cases, the lateral displacements of the rising cubes barely exceed ±2 in xand y directions. The side drifts are more amplified for the settling cubes and muchmore so for Ga = 250. As seen from figure 2.10b, the densest cube reaches a max-imum displacement of close to 12. As will be discussed later, one of the featuresof chaotic cubes is that fluctuations in the magnitude of quantities such as angularvelocity are significantly higher (see section 2.5.1).2.5 Force balancesThe total hydrodynamic force acting on a moving object with constant velocityoriginates entirely from vortex forces. When the object is also subject to acceler-ation, the forces acting on the object now contain a contribution arising from theinertia of the surrounding fluid, referred to as the “added-mass” of the object. Assuggested by Govardhan & Williamson [62], we split the total hydrodynamic force31en etVFFnFtzxyFigure 2.11: Schematics of the velocity and force vectors of the cube, along with thehorizontal and normal planesF into two parts; namely, the vortex force Fv and the added-mass force FAMF = Fv + FAM = Fv − pi6CAMdVdt(2.26)where CAM is the added-mass coefficient. We set CAM = 0.7 for the presentanalysis according to the experimentally reported values of CAM ≈ 0.67, 0.7 [63].Making the distinction between the vortex force and the added-mass force allows arealistic estimation of the forces that exist due to the presence of vortical structures.This is especially important in analyzing helical motions of a constantly accelerat-ing object, where the added-mass force is always present due to the unsteadinessof the motion. The presence of relatively high angular velocity for Ga > 140 (seesection 2.5.1) provides the motivation to consider the role of the Magnus force aswell, which contributes to the vortex force Fv:Fv = FM + F(Ω→0) = CM (Ω×V) + F(Ω→0) (2.27)where CM denotes the Magnus coefficient and F(Ω→0) the non-rotational partof vortex forces that would have existed even if Ω = 0. Clanet [64] gives theMagnus coefficient to be CΩ ≈ 1.7 for spherical objects, which is equivalent toCM = (1/8)CΩ ≈ 0.2 with our non-dimensionalization of the governing equa-tions. This selected value for CM is the only available estimate, considering thatthe Magnus coefficient of a rotating cube has never been studied before to the au-thors’ best knowledge. As noted in [18], however, the generality of the discussion32is not affected by using an estimate value forCM . For the purpose of force analysis,the notations eˆx, eˆy, eˆz are utilized to show the unit vectors in x, y, z directions.We also define the following unit vectorseˆt =V|V| (2.28)eˆn =eˆF − (eˆF · eˆt)eˆt|eˆF − (eˆF · eˆt)eˆt| (2.29)eˆh =eˆF − (eˆF · eˆz)eˆz|eˆF − (eˆF · eˆz)eˆz| (2.30)where eˆt is the unit vector tangential to motion path and eˆF the direction of anyvector of interest. eˆn and eˆh denote the projection of eˆF onto a plane normal tothe direction of motion and a horizontal plane, respectively (see figure 2.11 fora schematic view of these vectors). In order to better demonstrate the relativemagnitude of the forces, we plot the forces with respect to the magnitude of the netbuoyancy force, denoted by FB . Moreover, since the numerical output of forcesacting on the cube is usually quite noisy and difficult to visualize properly, thepresented data in the following analyses are filtered. More detail on this matter andthe filtering procedure is included in appendix A. Horizontal forcesThe force balance for a cube in the vertical regime mainly demonstrates thatthe drag force balances the buoyancy force in the vertical direction, with smalland mostly random oscillations of the horizontal forces about a zero mean in thevertical regime (see section 2.4.2). In the oblique regime, the plane-symmetricwake is responsible for exerting a lift force on the cube with a non-zero meanwhich causes the side drift. When the cube starts to follow a helical trajectory,the forces oscillate accordingly in x and y directions. figure 2.12 shows variousforces acting on the cube along helical and chaotic paths for two extreme densityratios of m = {0.2, 7}. For all values of Ga, the most notable difference betweenm = 0.2 and m = 7 is the magnitude of the total hydrodynamic force on the cubein the horizontal direction (denoted by ΣF in figure 2.12), which is very small form = 0.2 and relatively large for m = 7. Such a variation in magnitude is expected33-0.200.20 50 100 150 200 250-0.200.2(a)-0.200.20 100 200 300 400 500-0.200.2(b)-0.200.20 20 40 60 80 100 120 140 160-0.200.2(c)-0.200.20 100 200 300 400 500-0.200.2(d)-0.200.20 50 100 150 200-0.200.2(e)-0.200.20 50 100 150 200 250 300 350 400 450-0.200.2(f)Figure 2.12: Force balance in the horizontal direction for the lightest and densestcube in the helical and chaotic regimes340 2 4 6 800.10.20.30 2 4 6 800.10.20.30 2 4 6 800.10.20.30 2 4 6 800.10.20.30 2 4 6 800.10.20.3Figure 2.13: Time average of the horizontal component of the angular velocity(open-circle markers) after initial transient period. The error bars representthe magnitude of fluctuations, i.e., the standard deviation of |Ω · eˆh| over theaveraging time window.because of the huge difference in inertia between these two density ratios. Thatis to say, when the cube is in helical motion, the side force needed to change thedirection of motion is proportional to the density of the cube. When m = 0.2(figures 2.12a, 2.12c and 2.12e), the cube inertia is very small (and so is the netforce) and the force balance reduces to the vortex force balancing the added-massforce. The cube with m = 7, being 35 times denser/more inertial than a cube withm = 0.2, requires a much greater centripetal force to sustain the helical motionwhich is also seen evident in figures 2.12b, 2.12d and 2.12f.Since the Magnus force appears due to the simultaneous translation and rotationof an object, we also examine the angular velocity of the cube in various regimes,shown in figure 2.13. In all cases, the z component of Ω is small compared to thenoticeably greater magnitude of the horizontal component. That is why we haveonly shown |Ω · eˆh| in figure 2.13; namely, the projection of the angular velocityvector onto a horizontal plane. Figure 2.13 indicates that as soon as the cube isin a helical (or helical/chaotic for Ga > 160) regime, the mean value of |Ω · eˆh|does not depend on Ga anymore, while it generally decreases as the cube becomesdenser. Interestingly, the mean value of |Ω · eˆh| seems to be the almost samein a helical and a chaotic regime, but the level of fluctuations is much higher inthe chaotic regime. This was previously discussed as one of the features of thechaotically moving cubes in section 2.4.4. The fact that V points mainly in thevertical direction and Ω in the horizontal direction means that a significant portionof FM lies in the horizontal plane. Obviously, the Magnus force follows the sametrend as Ω with changing Ga and m, meaning that its magnitude decreases with m350 2 4 6 800. 2 4 6 800. 2 4 6 800. 2 4 6 800. 2 4 6 800. 2.14: Time average of the horizontal force magnitude (open-circle markers)after initial transient period. The error bars represent the magnitude of fluc-tuations, i.e., the standard deviation of |Fv · eˆh|/FB over the averaging timewindow.but only slightly with Ga. This is also evident from the Magnus force curve (shownin red) between the left column and the right column of figure 2.12 showing theforces for the lightest and the densest cube. A notable observation in figure 2.12 isthat the Magnus force is almost equal in magnitude and phase to the vortex forcefor light cubes, meaning that the centripetal force for light cubes is provided by theMagnus force through the rotation of the cube. For the case of the densest cubewith m = 7, the Magnus force still contributes considerably to the total vortexforce Fv · eˆh. As will be further discussed in section 2.5.2, however, a majorportion of Fv · eˆh is provided by the normal forces pointing inwards (with respectto the circular path in a horizontal plane).2.5.2 Normal forcesInspection of figures 2.12 and 2.14 reveals that the horizontal vortex force in-creases with m, reaching approximately twice as high for m = 7 compared tom = 0.2. Given that in each plot the value of Ga is the same for all density ratios(and that the Reynolds number differs only by 10 – 15 between the lightest anddensest cube), this huge increase in vortex force cannot be solely attributed to thegrowth of side forces with Re. Wake structure visualizations on the other hand, forinstance in figures 2.7 and 2.9, show that the shedding of vortex loops occurs at aconsiderable angle with respect to the vertical axis, especially with lighter cubes.This presents a clue as to what component of Fv should be expected not to vary thatgreatly between cubes with the same Ga. Hence, we look at the projection of thetotal vortex force Fv on a plane normal to the direction of motion which is shown360 100 200 300 400 500 60000. 100 200 300 400 500 60000. 100 200 300 400 50000. 2.15: Evolution of the normal force in the helical and chaotic regimesschematically in figure 2.11. Remarkably, the magnitude of this normal force isseen to be much less sensitive to changes in m, with a variance of ≈ −20% (theside force is greater for m = 0.2 this time) compared to ≈ 100% difference on ahorizontal plane form = 0.2 andm = 7. The evolution of this normal force, alongwith its mean and standard deviation for various values of Ga and m are given infigures 2.15 and 2.16. Whenever a helical path is present, the normal force is sig-nificant and its magnitude decreases slowly with increasing of m. We know fromour analysis that the inclination angle of the cube with (Ga,m) = (140, 0.2) inthe helical regime is almost three times higher than for (Ga,m) = (140, 7) whilethe relative angle between F and V does not exceed 5◦ between the two cases. Asmuch as the velocity vector deviates from the vertical axis, the normal force vectoralso deviates from the horizontal plane, thus explaining the trends in figures 2.14and 2.16. We believe that the remaining variance in |Fv · eˆn| for the lightest and theheaviest cube stems from the difference in coupling and dynamics of cubes eachwith a different density ratio. One other interesting feature of figure 2.16 is thatthe magnitude of the |Fv · eˆn| fluctuations is vanishingly small for perfect heli-cal paths whereas although equal in the mean value, the cubes moving in randomchaotic/helical paths show noticeably large fluctuations.Now that we have identified the normal force as the one that vortex shedding isdirectly responsible for, it is also beneficial to look at visualizations of the Magnusforce FM and the normal force (Fv · eˆn)eˆn vectors. In figure 2.17, the vectorsshown are FM,h = FM · eˆh and F(v,n),h = [(Fv · eˆn)eˆn] · eˆh, which representthe horizontal projections of the corresponding forces (i.e., the top-views of thevectors). Hence, the following comments all apply only to the horizontal com-370 2 4 6 800. 2 4 6 800. 2 4 6 800. 2 4 6 800. 2 4 6 800. 2.16: Time average of the normal force (open-circle markers) after initialtransient period. The error bars represent the magnitude of fluctuations, i.e.,the standard deviation of |Fv · eˆn| over the averaging time window.ponents of these forces. Once more, we select the lightest and densest cubes tohighlight the differences between settling and rising cases. For both light cubesin figures 2.17a and 2.17c, the normal vortex force deviates slightly with respectto the path, while in figures 2.17b and 2.17d, the angle between the path and thenormal force is remarkably higher. It should also be noted that the magnitude ofthe normal force vectors appear to be visually almost the same for both m = 0.2and m = 7. Consequently, these observations suggest that the lighter cube tendsto shed vortices typically in the direction of the velocity vector (black arrows infigures 2.17a and 2.17c) whereas for the denser cubes, the normal force vector isclearly seen to have a significant inward orientation (black arrows in figures 2.17band 2.17d). This major centripetal component is required dynamically to supportthe circular motion in the horizontal plane for a denser particle. Having this inmind, we also find that neither the direction nor the magnitude of the Magnusforce changes much, if visually discernible at all. Since the Magnus force in fig-ures 2.12a, 2.12c and 2.12e is seen to be in phase with the total vortex force andsimilar in magnitude, it is identified as the main force supporting the curvilinearmotion for a light particle. However, as m increases, the normal vortex force be-comes more and more important in sustaining the helical motion.In their comprehensive experimental analysis of settling and rising spheres,Horowitz & Williamson [15] pointed out that “...One might suggest that such heli-cal motions (for Re > 270) would not be compatible with the induced fluid forcescoming from unsteady vortex loops and rings shed from the sphere, since such awake may not be able to provide the steady centripetal force required for a spi-38-0.5 0 0.5 1-0.500.51F(v,n),hFM,h(a)-0.2 0 0.2 0.4 0.6- 0 0.5 1-0.500.51F(v,n),hFM,h(c)-0.5 0 0.5-0.8-0.6-0.4- 2.17: Visualizations of FM,h and F(v,n),h, the horizontal components of theMagnus force vector and the normal vortex force vector; respectively, afterthe transient period. The light purple lines indicate the trajectory of the cubeduring this window of time.ralling path.” Nevertheless, in our case of a moving cube, the object travels he-lically while clearly shedding hairpin-like vortices (see figure 2.7), which seemsto disagree with the foregoing statement. This apparent incompatibility might beexplained in light of the current analysis with two following comments. First,the vortices shown in figure 2.7 for (Ga,m) = (140, 0.2) present the wake struc-ture with a relatively smoother and more “continuous” appearance, in contrast to(Ga,m) = (170, 4) in the same figure with more distinctively separate hairpinvortices. The smoother vortex structure is likely to exert a less fluctuating force,also reflected in figure 2.16. Second, oscillations in the vortex shedding of a light39cube barely affects lateral acceleration because the resulting force mainly acts inthe direction of velocity (when viewed from above, as in figure 2.17), while theMagnus force provides a steady centripetal force. Interestingly, denser particlesmight indeed be affected by the unsteady vortex shedding: the rightmost trajectoryin figure 2.7 which pertains to (Ga,m) = (170, 4) is a helical trajectory, but itappears to be a helical motion on which another slow lateral oscillation is super-posed. This is seen for nearly all settling helical trajectories with distinctive vortexshedding, indicating that despite the prominent presence of helical trajectories, thepaths of the settling cubes are not as “perfect” as those of the rising cubes.2.5.3 Drag forceThe correlation describing variations of the drag coefficient with the Reynoldsnumber for a fixed sphere is known as the standard drag curve [65], which is asteadily decreasing function for Re < 1000. The drag coefficient of a freely mov-ing sphere, on the other hand, turns out to be not so “standard” and therefore, it hasbeen a topic of debate in recent years. Experimental investigations of Karamenev& Nikolov [39] and Karamanev et al. [38] showed that beyond a certain thresholdof Re, spheres with sufficiently small m appear to have a constant drag coeffi-cient of CD ≈ 0.95, which is more than twice as high as the drag coefficient of afixed sphere at the same Reynolds number. The numerical study of Jenny et al. [30]showed that mass-less spheres (m = 0) experience a drag coefficient ofCD = 0.65for Re > 300, which is not as dramatically high and remains almost constant withincreasing the Reynolds number. As mentioned before in section, the com-prehensive experimental study of Horowitz and Williamson [15] paints a qualita-tively similar picture, where spheres with density ratios smaller than a critical value“vibrate” with large amplitudes while they rise. Whenever such a zigzag motionoccurs, it is accompanied by a drag coefficient of CD = 0.75 which no longer de-pends on the Reynolds number. Recently, Auguste & Magnaudet [36] numericallyinvestigated 250 combinations of Ga and m in order to obtain a comprehensivemap of motion regimes of a rising sphere for Re 6 103. In line with the find-ings of Zhou & Dušek [32], they found that a new helical regime characterized bylarge-amplitude oscillations emerges for very light spheres when Ga > 400 and4070 85 100 120 140 160 180 200 2500.80.911. 100 120 140 160 180 200 250012345671. 2.18: Variations of the drag coefficient with Galileo number for different den-sity ratios. The marker styles on the contour plot represent different regimesof motion and are the same as in figure 2.3.m < 0.2; the range of parameters for which chaotic motion is expected. The mostdrastic change in the drag behavior happens when the sphere enters a spirallingregime. With the onset of spiralling motion, CD starts to increase with Reynoldsnumber, reaching a maximum of CD = 0.75 for the lightest sphere at Re = 103.All spheres with m > 0.25 were seen to follow the normal drag behavior through-out the entire range of Re.As previously pointed out in section, the drag coefficient variation of afreely moving cube has rarely been studied before. Here, we present the drag forcecoefficient as a function of the Galileo number Ga as well as the terminal Reynoldsnumber in figures 2.18 and 2.19. The average drag coefficient and the terminalReynolds number are obtained asCD =1Vm2 (2.31)Ret =√43GaVm (2.32)where Vm = 〈|V · eˆz|〉 indicates the average vertical velocity in the steady state.The dependence of CD on the density ratio is clear from figures 2.18 and 2.19:lighter cubes consistently show greater CD for sufficiently high values of Ga. Thefirst deviation from the drag curve of a fixed cube is observed as a jump at Ret ≈128. This occurs for the lightest cube with m = 0.2, coincident with the first4150 100 150 200 250 300 350 4000.80.911. 2.19: Variations of the drag coefficient with Reynolds number for differentdensity ratios. For the fixed sphere, the standard drag correlation proposed byTurton and Levenspiel [65] is used.appearance of a helical path. Every time a cube with a specified m enters thehelical regime, the drag coefficient increases significantly which corresponds to thesudden rise of CD seen in figure 2.19. In some cases, increasing Ga even results ina decrease of the terminal Reynolds number, as a more mature helical motion setsin. Evidenced numerically for settling cubes [18], this remarkable increase hadbeen regarded as a “scatter” of data due to wake instabilities in the experimentalwork of Tran-Cong et al. [52]. In most cases, the local maximum of the dragcoefficient pertains to largest Ga for which an ordered helical regime is observed.Beyond that value of Ga, the CD curves seem to reach a plateau. Also shown infigure 2.19, the correlation of Haider & Levenspiel [53] for a cube does not captureany of these observations, it nevertheless yields more acceptable results for heaviercubes than for lighter cubes. Comparing figure 2.19 and figure 2.18 also suggeststhat using the CD − Ga graph instead of the CD − Ret graph is beneficial in thecontext of settling/rising particles, as Ga is one of the two governing parametersbeing specified a priori, while Ret is an outcome of the simulations. Moreover, aCD −Ga graph prevents ambiguities such as ending up with two values of CD forthe same value of Ret.The wake structure in each regime indicates that even though vortex sheddingor the inter-twined wake is seen in the vertical regime, the drag coefficient of the42moving cube is almost identical to that of a fixed cube. The CD curves start to de-part from the fixed cube drag curve as the helical regime and the ordered sheddingof hairpin vortices appear. The drag coefficient of the lightest cube can be up to50% higher than a heavy cube (e.g., (Ga,m) = (140, 4)), as the rising cube hasentered the helical regime while the settling one is still in the vertical regime. Sinceall cubes pass through the same sequence of regimes, we find that the drag curves ofall cubes with a specifiedm follows a more or less similar trend: a normal decreaseat the beginning (vertical regime), followed by a steep rise and a local maximum(appearance and maturing of the helical regime), leading to a final plateau (chaoticregime). Origin of drag enhancementAs with any shape other than a sphere, the drag force would depend on theorientation of the object with respect to the flow. For the cube as well, it seemstempting to attribute the increase in the drag coefficient to the changes in orienta-tion. As discussed previously in section, Richter & Nikrityuk [21] providea regression formula of the form f(Re, φ) =(0.0582Re0.295)sin2(2φ)1 valid upto Re = 200 for a cube which gives the additional contribution of the orientationangle φ to the drag coefficient. The orientation angle φ is defined as the anglebetween the cube’s velocity vector and the normal vector of the front face of thecube. Given that the correlation provided by Richter & Nikrityuk [21] only con-siders rotation about one axis, our computed drag enhancement due to the cubeorientation will be an estimate at best. Taking the case of (Ga,m) = (140, 0.2)with Ret ≈ 135 as an example, we find the drag coefficient to be ≈ 34%2 higherthan CD of a fixed cube at the same Reynolds number. For an orientation angleof φ = 25◦ (which is the observed average value for this case), the correlation ofRichter & Nikrityuk [21] gives a relative increase of ≈ 13.5%. This means thatabout 40% of the overall enhancement in CD could have come from the variationof the incidence angle of the cube, while the remaining 60% should have originated1In the cited work, this correlation is reported as f(Re, φ) =(0.059/Re0.292)sin2(2φ), whichturns out to contain typographical errors. Following our request, the authors of [21] have kindlyprovided us with the correct form of the equation which is used in the present section.2Please note that here, we have computed CD with the magnitude of the tangential force |F · eˆt|and the total velocity |V|, instead of FB and Vm used in equation (2.31).43from another source.In their computational work on rising oblate bubbles, Mougin & Magnaudet[50] compared the actual drag force on a bubble to the one obtained using a dragcorrelation of steadily moving bubbles and measured about 50% difference be-tween the two values for a helical motion. They associated this increase with thesuction of the oblate bubble into the low-pressure cores of the counter-rotating vor-tices. In an experimental study on rising spheres, Veldhuis et al. [34] invoked theconcept of vortex-induced drag to explain the discrepancy between the drag forceexperienced by a moving sphere and a fixed sphere. This concept basically statesthat if the two major threads of vorticity are not aligned with the instantaneousdirection of motion or eˆt, the force generated by them will not be exactly perpen-dicular to eˆt. This would result in a resistive component in the direction eˆt, denotedfrom here onwards by Fid. Figure 2.20 shows the wake structure of the cube with(Ga,m) = (140, 0.2), both at its initial zigzagging and final helical stage. In thisfigure, Ψ is the apparent angle between the plane that contains the two major vor-ticity threads and the velocity vector. From figure 2.20, it is evident that the rootof the vortex threads immediately downstream of the cube makes a notable anglewith the velocity vector in both case: Ψ = 27◦ for the zigzag and Ψ = 22◦ for thehelical path, are visual estimates based on the wake seen in figure 2.20 at an instantof time. As a result, the total vortex force shown as Fvortex can be obtained as|Fvortex| = |Fn|cos(Ψ)(2.33)and the induced-drag force would therefore be|Fid| = |Fn|cos(Ψ)sin(Ψ) = |Fn| tan(Ψ) (2.34)where we use the notation Fn = Fv · eˆn for the normal force. Since the drag cor-relation proposed by Richter & Nikrityuk [21] is indistinguishable from our datapoints for a fixed cube (see appendix A.1), we use their formula here. Moreover,we also utilize their angle-dependent correction function for our drag force calcu-lations. Therefore, Ft,pr, the predicted drag force exerted on the cube traveling44zxzyVHelical Zig-zagΨ ΨFnVFnFvortex FvortexFid FidFigure 2.20: The wake structures, together with velocity and force vectors of thecube with (Ga,m) = (140, 0.2) at two different stages of its rising. The thickpurple line shows the trajectory of the cube, and Fid denotes the induced dragvector. The wakes are visualized using iso-surfaces of Q-criterion for Q =0.003.along a curved path, is written asFt,pr = Ffixed + Fid (2.35)Ft,pr =pi8|V|2CD,fixed + Fid (2.36)In the above relations, CD,fixed contains both the steady drag force of a cube ori-ented face-forward with respect to the flow and the additional drag due to its inci-4520° 25° 30° 35° 40° 45°10°15°20°25°30°35°111112.<1%Figure 2.21: Contour plot of the relative error  between Ft and Ft,pr as a functionof the vortex and incidence angle for (Ga,m) = (140, 0.2)dence angle:CD,fixed = CD + CD,φ =(20.4Re+8.19√Re+ 0.216)+(0.0582Re0.295)sin2(2φ)(2.37)A knowledge of vortex and incidence angles (Ψ, φ) and the magnitude of the ve-locity vector would enable us to compare the predicted drag force (equation (2.35))and the numerically computed value of the instantaneous drag force on the cube,defined as Ft = |Ft| = |Fv · eˆt|. While Mougin & Magnaudet [50] arrived at adifference of 0.3FB between the measured drag force and the one computed froma drag correlation, for our case we obtain a difference of approximately 0.22FB .Since our values for φ and Ψ could only represent visual estimates for a few snap-shots of the vortices at best, we take another route to analyze the results. In fig-ure 2.21, we present contour plots of the relative percent error between the numer-ically computed drag and the predicted drag, i.e.,  = |Ft−Ft,pr|/Ft× 100 for thecase (Ga,m) = (140, 0.2). Note that all force values here represent averages overa sufficiently long time after the magnitude of the forces has become steady withsmall oscillations. The green band in figure 2.21 shows the range of values of φ andΨ for which the predicted tangential force Ft,pr falls within 1% of Ft, i.e., the dragforce directly computed from the simulation output. Notably, the visual estimation46of (Ψ, φ) = (22◦, 25◦) is indeed in range to put Ft,pr within 1% of the actual valueof Ft. Considering the uncertainty in the visual estimation of (Ψ, φ) = (22◦, 25◦)due to three-dimensionality and its time-variation, we conclude that this analysisgives a reasonably accurate prediction of the real value of Ft. The range of  infigure 2.21 with an orientation angle of φ = 25◦ demonstrates that even an errorof ±5◦ in the estimation of Ψ still predicts the actual drag force with an error ofno greater than 3%. The results of this case can be contrasted with those obtainedfor (Ga,m) = (140, 7) based on a similar analysis. By examining snapshots of thevortex structure and trajectory of the cube, we have found the vortex and orientationangles to be (on average) approximately equal to (Ψ, φ) = (5◦, 28◦) for a settlingcube with (Ga,m) = (140, 7). Using equations (2.35) and (2.37), the drag forcecomputed with this set of (Ψ, φ) also agrees quite well with its true value. In thiscase, the drag force is ≈ 19% higher than that of a fixed cube, about 85% of whichis due to the incidence angle of the cube and the remaining 15% is coming fromthe vortex-induced drag. The foregoing analysis shows that the drag enhancementin figures 2.18 and 2.19 occurs because of two reasons; namely, the orientation ofthe cube, and the misalignment of the two strong vorticity threads with the veloc-ity vector of the cube. The relative contribution of these two factors depends onthe density ratio, which is clearly seen in figures 2.18 and 2.19. The orientationangle of the cube in both cases of (Ga,m) = (140, 0.2) and (Ga,m) = (140, 7) ismeasured to be φ ≈ 25◦− 30◦. However, the amplitude of the lateral excursions issignificant for the smallest density ratio and Ψ can be as large as 22◦ − 27◦, whileΨ ≈ 4◦ − 6◦ for the densest cube. Horizontal velocity fluctuationsThe mean value of Vh, the horizontal velocity vector, and root-mean-square ofV′h, the horizontal velocity fluctuations, are given as〈Vh〉 = (〈V · eˆx〉, 〈V · eˆy〉) = 1T∫ t0+Tt0Vhdt (2.38)V′h = Vh − 〈Vh〉 (2.39)(〈V′h2〉)1/2=[1T∫ t0+Tt0(V′h)2dt]1/2(2.40)4770 100 120 140 160 180 200 250012345670. 2.22: Contour plot of horizontal velocity fluctuations for various values of Gaandm. The marker styles, which represent different regimes of motion, are thesame as in figure 2.3Inspection of figures 2.18b and 2.22 that show contour plots of(〈V′h2〉)1/2andCD respectively, better indicates that the decreasing trend of the drag coefficientis preserved up to the onset of large amplitude helical motions, where a steep gra-dient exists at the boundary between the unsteady vertical and the helical regime.Notably, the largest increase in the drag force in figure 2.18b occurs for the lightestcubes, which also happen to show the highest fluctuations. Similar to observationsof Zhou & Dušek [32] for a sphere, we find high horizontal velocity fluctuationsas soon as the helical regime emerges; contrary to the case of a sphere however,the magnitude of these fluctuations remain approximately the same in the chaoticregime for a cube. This could be justified by considering that chaotic motions atlower values of Ga resemble the curved spiralling paths of the helical motions atrelatively smaller time-scales, except being mostly random.2.6 Summary and conclusionDespite being abundant in many industrial and natural situations, free settlingor rising of non-spherical particles have received less attention due to the inherent48challenges. Among non-spherical shapes, investigations involving angular objects,such as cubes or other polyhedrons, are almost absent with very few exceptions.Motivated by the interesting dynamics of angular particles reported in a previousstudy published by our group [18], we have provided a comprehensive picture ofmotion regimes of a cube for the first time. We have also presented detailed analysisof vortex structures and resulting force balances. This work can be considered as astep towards understanding the dynamics of settling and rising regular polyhedralshapes.In the present study, we have found that cubes pass through the same sequenceof regime types at any density ratio; namely, the oblique, unsteady vertical, he-lical and chaotic trajectories. This is in contrast to the case of spiralling sphereswhich have only been observed for m < 1. Also, no persistent zigzag motion isobserved except for early transient periods preceding the onset of helical motionin some cases. The inclination angle of a cube in the oblique and helical regimesis generally smaller than, but comparable with the inclination angle of a spheremoving with the same path style. One of the remarkable features of cubes mov-ing in helical trajectories is the appearance of a new vortex shedding mode, where“four Hairpin” (4H) vortices are shed per motion cycle. This also presents a dif-ference with helically moving spheres which exhibit inter-twined wakes withoutobservable shedding [36, 32]. The nearly perfect helical trajectories become inter-mittent and unpredictable as the cube enters the chaotic regime. Any trace of theneighboring helical regime has faded away completely at Ga = 250.We have also examined the forces acting on the cubes in selected regimes. Dueto significant angular velocity of the cube, the Magnus force is found to provide thecentripetal force for lighter cubes with negligible inertia. Conversely, heavier par-ticles having higher inertia are supported mainly by the normal (lift) forces so as tobe capable of sustaining helical motions. We have revealed that along the lines ofconstant Ga on the flow-map, the magnitude of the normal force is similar for thelight and heavy cubes, indicating a common nature of the force generation in thehelical regime. Moreover, we find that the drag coefficient increases abruptly whenthe cube starts to move helically, manifested by a jump in curves of CD for all val-ues of m, although more pronounced for lighter cubes. This jump shown here is49consistent with previous experimental investigations which have reported a scatterof data around a similar range of parameters. This is reminiscent particularly ofthe drag coefficient of rising spheres which have been known to be substantiallyhigher than that of settling spheres. We have sought to explain the origin of thisincreased drag force by invoking two contributing factors: (i) the concept of in-duced-drag, resulting from the misalignment of the main vorticity threads with theparticle velocity vector and (ii) the orientation of the cube with respect to directionof motion which alters the drag force. With approximate visual estimates of theinvolved angles, our analysis showed reasonable agreement between the predicteddrag force and its directly computed value if the above two factors are taken intoaccount. We explain the higher drag manifested by lighter cubes to be mainly dueto the fact that the vorticity threads in the wake of the lightest cube make a muchlarger angle with the velocity direction compared to the case of the densest cube,resulting in a considerably higher vortex-induced drag.The present study has highlighted the differences between the dynamics of anangular object, i.e., a cube and a sphere. These differences include the dominantpresence of helical motions for both light and dense cubes, absence of sustainedzigzag motions, lower transition thresholds of Ga and enhancement of the drag co-efficient regardless of density ratio. An interesting extension of this work could beto examine polyhedrons with increasing numbers of faces to see if there is a smoothor sharp transition of the dynamics towards that of a sphere. As for a sphere, thestudy of a single-particle system is deemed crucial towards understanding large-scale many-particle systems. In our case of a cube, the significant lateral motionsthat have been observed for Ga > 120 in both the helical and chaotic regimes arelikely to play a central role in close-range inter-particle interactions in a suspension[16]. The higher velocity fluctuations in these regimes could also induce consider-able pseudo-turbulence in the fluid, which might in turn alter the dynamics of thesurrounding particles. We have also shown the importance of side forces in deter-mining the tangential resistance, and thus the settling or rising velocity of the cube.Existence of strong vortices downstream of the cube was shown to cause a greaterdrag when not aligned with the direction of velocity. Considering all that has beendiscussed, we are greatly interested in finding out the connections between the50present observations for a single cube and a multi-particle system of cubes in thesame ranges of Ga and m and various concentrations. Other inherent particulari-ties between a group of cubes compared to spheres, including the different natureof contact-induced forces and torques or the ability of cubes to inter-lock may alsoinfluence their collective behavior when suspended in a fluid.51Chapter 3Inertial settling of cube andsphere suspensions13.1 IntroductionThe gravity-driven motion of sedimenting particles is one of the most com-monly occurring phenomena in numerous natural and industrial processes. Rainformation and precipitation, sediment transport in rivers and particle settling influidized beds and waste-water treatment tanks are examples of situations wherecollective sedimentation of suspended particles plays a central role. Settling sus-pensions of particles are complex systems rife with rich dynamics that stem fromvarious contributing factors. While a large number of parameters are typicallyinvolved (e.g. mass/volume loading, flow regime, particle shape/rigidity), thereis also a wide range of spatial and temporal scales associated with the occurringphysical phenomena. The time scale of particle collisions, for instance, is gener-ally an order of magnitude smaller than that of the fluid evolution. Moreover, fluiddisturbances on the scale of particle dimensions are known to influence the overallhydrodynamics of suspensions and collective behavior of particles. For instance,clusters which form as a result of wake interactions may span several particle di-1A version of this chapter has been submitted for review in Physical Review Fluids.52ameters and persist over relatively long periods of time [16, 66].The most idealized picture of particle settling has emerged from the pioneeringwork of Stokes, giving the drag force acting on a single rigid sphere settling in anunbounded quiescent fluid to be 3piµDVz,s, where µ and D are the dynamic vis-cosity of the fluid and the particle diameter, respectively, whereas Vz,s denotes theparticle settling velocity. This relation is valid under the assumption of vanishinginertia or Re→ 0, with Re being the Reynolds number defined as Re = ρfVz,sD/µand ρf the density of the fluid. For a particle cluster in an infinite domain, the set-tling velocity is always greater than that of a single settling particle as a result ofhydrodynamic cooperation between the particles [67]. When particles settle in acontainer with a fixed bottom wall, however, the motion is hindered since the down-ward flux of the solid particles and the adjoining fluid needs to be compensated forby an upward fluid flux (also called “back-flow”), thus making the mixture averagevelocity vanish over the entire suspension. A general theoretical treatment of set-tling suspensions has been beyond reach due to the complexity of Navier-Stokesequations with the presence of nonlinear inertial terms. Even when inertia is ig-nored (i.e. Stokes flow) and the governing equations consequently become linear,superposition of velocity disturbances of individual spheres leads to a divergingintegral due to the slow decay of perturbations in Stokes flow. Using probabilityarguments, Batchelor [68] circumvented the divergence problem and showed thatthe hindered settling velocity of a dilute suspension of rigid spheres for Re 1 isgiven byVzVz,s= 1− 6.55φ (3.1)where Vz denotes the ensemble average of the suspension settling velocity, Vz,s theterminal settling velocity of an isolated sphere and φ the solid volume fraction. Forhigher solid volume fractions and Reynolds numbers, empiricism is inevitable toobtain the settling velocity as a function of the solid volume fraction. The mostwidely known formula is the power-law relation of Richardson & Zaki [69]VzVz,s= (1− φ)n (3.2)53which was obtained from experimental data of sedimentation and fluidization ofdense suspensions (i.e. φ > 0.1), and is believed to be accurate for Re 6 25[70] and concentrations up to φ = 0.25 [71]. Based on extensive experimentaldata [69], values of the exponent n were shown to depend on the flow regime,having a constant value for both very low (Stokes regime) and very high (Newtonregime) Reynolds numbers, and a transition region for intermediate values of Re.A compact form of the dependence of n on Re was given by Garside & Al-Dibouni[72] as the following logistic curve5.1− nn− 2.7 = 0.1Re0.9 (3.3)which improves the accuracy of equation (3.2) for higher Reynolds numbers. Later,Di Felice [73] showed through settling experiments that the exponent n in dilutesuspensions with φ 6 0.05 is approximately 1.5 times larger than that for densersuspensions, although having a similar functional dependence on Re. Di Felice[73] also noted that with n being higher for dilute suspensions, the hindrance func-tion in equation (3.2) needs a correction pre-factor k for concentrated suspensions,values of which were suggested to be between 0.8 and 0.9. The rapid decay of thesettling velocity for Re > 1 especially at low concentrations was later confirmedby Yin & Koch [74] and Hamid et al. [75] through numerical simulations. Thedeparture from the power-law relation of equation (3.2) was observed to be morepronounced for higher Re. Consequently, this was attributed to the formation ofanisotropic structures in the suspensions as a result of enhanced wake interactionbetween spheres in more inertial flows [75, 74], hinting that a power-law formfor the hindrance function is possibly associated with a hard-sphere (i.e. random)distribution of the particles.In Stokes regime, the flow field produced around a settling sphere has fore-aftsymmetry. Additionally, two nearby settling particles will maintain their relativeorientation and separation, and will always fall faster than an isolated particle ac-cording to symmetry and reversibility properties of Stokes flow. As the flow be-comes inertial upon increasing the Reynolds number, the symmetry is lost and alow-pressure wake region develops downstream of the sphere. A particle that hap-54pens to be in the wake of another experiences less drag compared to an isolatedparticle, which consequently causes it to accelerate (“drafting” phase) towards theleading particle. The trailing particle is also subject to a lift force due to the shearflow in the wake region and is hence pushed outwards at the same time. If themagnitude of the induced lift force is sufficiently high, the trailing particle touches(“kissing” phase) the leading one. Since the vertical arrangement of the pair is un-stable, the particle pair rotates into a horizontal orientation (“tumbling” phase) andthen they repel each other due to the source flow pushing fluid away from the parti-cles. This robust sequence of Drafting, Kissing and Tumbling (DKT) [76] has beenidentified in several investigations in the past [77, 78, 79, 16, 66, 75, 74, 80, 81] asthe principal mechanism underlying the emergence and evolution of anisotropic,inhomogeneous particle configurations in sedimenting suspensions. The afore-mentioned deviation of the settling velocity from equation (3.2) in inertial sus-pensions is in fact explained by the frequent occurrence of the DKT phenomenon[75, 74]. Wake attraction and the ensuing horizontal arrangement, also known as“rafts” [82], effectively reduce the likelihood of particles being in the wake of oth-ers. Well-separated particles in cross-stream configuration are more exposed to theback-flow of the fluid, thus experience higher drag force and reduced settling ve-locity. The prevalence of horizontal structure of particle clusters has been reportedby Yin & Koch [74] at low Reynolds numbers, and recently observed by Willen andProsperetti [77] in moderately dense suspensions with φ = 0.087, and Reynoldsnumbers as high as Re ' 111.The forgoing scenario regarding settling rate reduction resulting from DKT-induced dispersion has turned out to be entirely different for dilute suspensionswhen the Reynolds number is increased to Re = O(100). On the one hand, thecombination of stronger wakes in higher Re and low particle concentration allowsparticles to interact over longer distances without being interrupted by other parti-cles. On the other hand, the occurrence of wake instabilities and resulting lateralmotions in highly inertial regimes increases the probability of particles crossingpaths. The wake of an isolated sphere first undergoes a transition from an ax-isymmetric to a double-threaded plane-symmetric wake at Re ' 210 [28, 35],inducing a steady side force. The subsequent transition at Re ' 275 initiates55periodic vortex shedding and oscillating side force. These wake instabilities giverise to a variety of different path geometries for a freely settling particle. In caseof a freely moving particle, the regimes of motion are characterized by two di-mensionless numbers; namely, the density ratio m = ρs/ρf , and the Galileonumber Ga =√|1−m|gD3/ν, with ρs showing the solid density, g the grav-itational acceleration and ν the kinematic viscosity of the fluid. As a result ofthe wake transitions, a vertically settling sphere switches to a steady oblique pathfor Re > 210 [15] or equivalently for Ga > 150 [30], while various patternsof oblique-oscillating paths for 185 6 Ga 6 215 emerge depending on the den-sity ratio m [32]. Chaotic paths are usually observed for settling spheres beyondGa ' 250 [32, 18, 30]. The collective behavior of particles in such highly in-ertial regimes was first investigated numerically by Kajishima & Takiguchi [81].Through their PR-DNS of dilute (i.e. φ = 0.002) sphere suspensions, they evi-denced strong particle accumulation and increased settling velocities for Re > 300,for which wake attraction was identified as the responsible mechanism. More re-cently, PR-DNS of Uhlmann & Doychev [16] showed that in very dilute suspen-sions (φ = 0.005), settling velocity is enhanced by up to 12% at Ga = 178, whileno significant change was seen for a similar suspension at Ga = 121. This wasexplained by the fact that at Ga = 178, spheres settle in oblique paths which in-creases the probability of getting trapped in the wake of another sphere and conse-quently undergoing DKT-type interaction. Frequent DKT between spheres resultsin the formation of sizable clusters which settle considerably faster than an iso-lated sphere, or compared to the other simulated case at a lower Galileo number ofGa = 121. This increase of settling is clearly in stark contrast with the aforemen-tioned reduction of settling rate in dilute suspensions with Re = O(10). Similarwake-induced clustering and enhanced settling rates for dilute and highly inertialsuspensions were also consistently reported by Zaidi et al. [66], and also experi-mentally confirmed later by Huisman et al. [79]. Interestingly, the authors of thelatter study reported less enhanced settling velocities for Ga = 310 compared toGa = 170. They suspected the observed behavior to be due to the chaotic wakes ofthe particles in the Ga = 310 case, that might increase the likelihood of particlesto break free from clusters.56Another level of complexity in studying suspension flows originates from theintroduction of non-spherical particles. Freely moving non-spherical objects ex-hibit various path geometries owing to both the anisotropy of their added-masstensor and wake instabilities [17]. Moreover, their nature of hydrodynamic in-teractions in multi-particle systems could be greatly different from spheres. Theorientation of a non-spherical particle also plays an important role in its overalldynamics, not only by affecting hydrodynamic forces but also through inducinghydrodynamic torques that would otherwise be either negligible or absent [83, 84,23]. The introduced torques promote particle rotation, which in turn, give rise toadditional hydrodynamic loads such as the Magnus force [18]. While the dynamicsof a free sphere is reasonably well fleshed out for various regimes [36, 32, 15, 30],fewer studies have been devoted to the characterization of motion regimes of iso-lated non-spherical particles in the existing body of literature. Thus far, free motionof objects such as disks [56, 85, 86], short cylinders [87], long cylinders [88, 48],oblate and prolate spheroids [19], and angular objects such as cubes and tetrahe-drons [18, 12] have been investigated. Expectedly, the motion and flow character-istics associated with non-spherical particles bear significant implications for theircorresponding settling suspensions. This is exemplified, for instance, through therecent work of Ardekani et al. [19] on free settling of oblate and prolate spheroids.They showed that as oblate particles have much wider wake regions compared tospheres, they become drawn into the wake of one another over considerably largerhorizontal separations. The higher susceptibility for wake attraction was shownto be translated into the remarkable formation of columnar clusters in dilute sus-pensions of oblate spheroids at Ga = 60 [70]. Consequently, the settling speedwas reported to be ' 30% larger than that of a single oblate particle. As for othershapes, there are a limited number of studies that consider suspensions of non-spherical particles such as the settling of particles shaped as red blood cells withRe ' 1 and high volume fractions [89], cylindrical particles with 1 6 Re 6 10and 0.1 6 φ 6 0.48 [90], rod-like objects with Re = 0.07 and 0.01 6 φ 6 0.1[91], long fibers with Re 6 10−4 [92] and flexible long fibers at Ga = 160, both indilute and semi-dilute regimes [93]. Evidently, most of the available works on non-spherical particle suspensions are carried out for systems at low Reynolds numberswith the exception of a few.57Overall, a general understanding of suspensions of spheres has been progres-sively accumulating over the past years, especially concerning low to moderateReynolds numbers (i.e. Re 6 50). Numerous studies have been dedicated to,and concur on matters such as volume fraction dependence of settling velocity,higher velocity fluctuations in the gravity direction, dependence of spatial distri-bution of particles on solid volume fraction and Reynolds number and dominanceof wake-induced structures in inertial suspensions of spheres [77]. However, nosuch knowledge is available for suspensions of non-spherical particles. In a re-cently published work, we provided a comprehensive flow-map of the free mo-tion of a cube [12], which complemented a previous study by our group on an-gular particles [18]. Along with identifying a robust regime of helical motion for140 6 Ga 6 170, we found that while a settling sphere is still in the oblique regimeat Ga = 180 [94, 15], the cube has already started moving chaotically with largevelocity fluctuations at the same Ga. Moreover, a cube acquires angular velocitiesthat are at least an order of magnitude larger than that of a sphere for Ga > 160,which induce significant Magnus forces normal to the direction of motion. Addi-tionally, the magnitude of lateral forces caused by intense vortex shedding rises toabout 25% of the net buoyancy force driving the motion. These remarkable fea-tures have motivated the present work, where we study suspensions of cubes fortwo different Galileo numbers; namely, Ga = 70 and Ga = 160, and solid volumefractions in the range of 0.01 6 Ga 6 0.2. In order to establish grounds for com-parison, we have also simulated suspensions of spheres for the same set of valuesof Ga and φ.In what follows, first the numerical methodology implemented in our PR-DNStool and the setup of the simulations are described in section 3.2. We then pro-ceed to the simulation results in section 3.3, beginning with particle dynamics insections 3.3.1 to 3.3.3 in terms of settling rates, velocity fluctuations and rotationrates of particles. Subsequently, detailed description of suspension microstructureis presented in section 3.3.4, followed by drag analysis in section 3.3.5. Finally, weconclude the article in section 3.4 with a summary of our findings in the presentstudy, along with a discussion of possible mechanisms for the observed differencesbetween suspensions of cubes and spheres.583.2 Computational methodology3.2.1 Governing equationsThe conservation of momentum and mass for an incompressible Newtonianfluid is described through Navier-Stokes equations, which are given as∂u∂t+ u · ∇u = −∇p+ 1Ga∇2u (3.4)∇·u = 0 (3.5)where u and p are the fluid velocity vector and pressure, respectively. In equa-tion (3.4), Ga may be thought of as a Reynolds number based on the gravitationalvelocity scale ug, the definitions of which areGa =ρfugDµ(3.6)ug =√|1−m|gD (3.7)with D now being the volume-equivalent diameter of a particle. Based on ug, agravitational time scale tg = D/ug may also be defined. In equation (3.7), mshows the solid-to-fluid density ratio, i.e. m = ρs/ρf , which together with theGalileo number Ga and solid volume fraction φ are the three governing parame-ters for the problem of gravity-driven evolution of a suspension of monodisperseparticles. Furthermore, the rigid-body motion of each particle is described by theNewton-Euler equations, which in the body frame of reference may be written asmvpdVdt= Fh + Fc + (m− 1)vpg (3.8)mIpdΩdt+ Ω× (Ip ·Ω) = Th + Tc (3.9)where V, Ω, vp, Ip and stand for translational and angular velocity vectors, vol-ume and inertia tensor of the particle, respectively, and g shows the gravitationalacceleration vector. Note that for spheres and cubes Ip is a scalar matrix, henceΩ × (Ip · Ω) = 0. In equation (3.8), Fc and Tc show forces and torques arising59from collision between the particles, the computation of which is handled by con-tact mechanics approaches and will be presented in the next section. Furthermore,Fh and Th denote the hydrodynamic force and torque exerted on a particle, andare given asFh =∫S(−pI + 1Ga(∇u +∇uT ))· n dS (3.10)Th =∫Sr×(−pI + 1Ga(∇u +∇uT ))· n dS (3.11)with I being the identity matrix, n the unit vector normal to the boundary of thesolid, S the surface enclosing the solid, ( · )T the matrix transpose, and r the posi-tion vector relative to the solid center of mass.3.2.2 Numerical MethodOur PR-DNS tool, PeliGRIFF (Parallel Efficient Library for GRains in FluidFlow), incorporates a finite volume variant of the Distributed Lagrange Multiplier-Fictitious Domain (DLM-FD) formulation proposed by Glowinski et al. [57] forfluid-solid coupling, while the inter-particle collisions are handled by an efficientDiscrete Element Method (DEM) granular solver. In the framework of the DLM-FD method, fluid domain is extended to the solid region where rigid-body motionsare enforced in the fictitious fluid inside the particles through a set of Lagrangemultipliers collocated in the particle domain. Here, we present a brief descriptionof the method, and refer the interested reader to [58, 95, 96, 59] in which thedetails of our implementation have been elaborated. In a nonvariational form, thecombined momentum and continuity equations are given as∂u∂t+ u · ∇u = −∇p+ 1Ga∇2u− λ in D (3.12)(m− 1)vpdVdt−∫Pλdx = Fh +∑jFcj + (m− 1)vpg in P (3.13)(m− 1)IpdΩdt−∫Pr× λdx =∑jrj × Fcj in P (3.14)u− (V + Ω× r) = 0 in P (3.15)60∇ · u = 0 in D (3.16)where the solid domain and combined fluid/particle domain are denoted by P andD, respectively. Also, λ shows the distributed Lagrange multiplier vector which isused to enforce the rigid-body motion constraint shown in equation (3.15). Withthe DLM-FD method, the Lagrange multiplier λ can be directly used to obtain thehydrodynamic force and torque exerted on a particle P as follows:Fh =∫Pλdx + vpdVdt, (3.17)Th =∫Pr× λdx + IpdΩdt. (3.18)As for the numerical time-marching algorithm, we employ a four-step classicaloperator-splitting scheme. For each time tn+1, we solve1. A classical L2-projection scheme for the solution of the Navier-Stokes prob-lem: Find un+1/2 and pn+1 such thatu˜n+1/2 − un∆t− 12Ga∇2u˜n+1/2 =−∇pn + 12Ga∇2un − 12(3un · ∇un − un−1 · ∇un−1)− λn(3.19a)∇2ψn+1 = 1∆t∇ · u˜n+1/2∂ψn+1∂n= 0 on ∂D(3.19b)un+1/2 = u˜n+1/2 −∆t∇ψn+1 (3.19c)pn+1 = pn + ψn+1 − ∆t2Ga∇2ψn+1 (3.19d)2. A purely granular problem, predictor step: Find Vn+1/3 and Ωn+1/3 suchthat(m− 1)vpVn+1/3 −Vn∆t= (m− 1)vpg +∑jFcj (3.20)61(m− 1)IpΩn+1/3 −Ωn∆t=∑jrj × Fcj − (m− 1)Ωn × Ip ·Ωn (3.21)and update particles position Xn+1/3.3. A fictitious domain problem: Find un+1, Vn+2/3, Ωn+2/3 and λn+1 suchthatun+1 − un+1/2∆t+ λn+1 = λn (3.22)(m− 1)vpVn+2/3 −Vn+1/3∆t−∫Pλn+1dx = 0 (3.23)(m− 1)IpΩn+2/3 −Ωn+1/3∆t−∫Pr× λn+1dx = 0 (3.24)un+1 − (Vn+2/3 + Ωn+2/3 × r) = 0 in P (3.25)4. A purely granular problem, corrector step: Set Xn+2/3 = Xn, and findVn+1 and Ωn+1 such that(m− 1)vpVn+1 −Vn+2/3∆t=∑jFcj (3.26)(m− 1)IpΩn+1 −Ωn+2/3∆t=∑jrj × Fcj (3.27)and update particles position Xn+1.In the above equations, ∆t denotes the time step, ψ the pseudo-pressure and ∂Dthe domain boundary. In equation (3.19a), we use Crank-Nicolson and Adams-Bashforth schemes which are second-order accurate in time to discretize the vis-cous and advective terms, respectively, while the saddle-point problem in step 3is handled by an Uzawa algorithm [58]. Considering the high-order correction ofthe pressure, the projection scheme in step 1 is also second-order accurate in time.However, the first-order time discretization of the fictitious domain sub-problemin step 3, and the overall first-order Marchuk-Yanenko operator-splitting methodreduce the global time accuracy of our algorithm to first-order only. Equations62presented in step 1 are spatially discretized with a second-order central schemefor the diffusion term, whereas the advective term is treated with a Total VariationDiminishing (TVD) scheme combined with a Superbee flux limiter. Despite thesecond-order discretization of the flow equations, the accuracy of our method isbetween first and second order due to the presence of rigid bodies immersed withinthe domain.In order to handle inter-particle collisions, we consider the total contact forceto be the sum of three contributions; namely, a normal Hookean elastic restoringforce, a normal dissipative force, and a tangential friction force. The elastic restor-ing force is given asFe = knδnij (3.28)where kn represents the equivalent spring stiffness, δ the overlap between particlesi and j, and nij the normal unit vector along the line connecting the centers of thetwo particles. Furthermore, the viscous dissipative force is defined asFdn = −2γnMijVn (3.29)where γn denotes the normal damping coefficient, and Mij = (mimj)/(mi +mj)the reduced mass of particles i and j having masses mi and mj , respectively. Inequation (3.29), Vn shows the normal component of the relative velocity betweenparticles i and j. Finally, the tangential friction force is given asFt = −min{µc|Fn|, |Fdt|}tij (3.30)with Fn being the total normal force and Fdt the tangential dissipative force de-fined asFdt = −2γtMijVt. (3.31)In equations (3.30) and (3.31), µc is the Coulomb friction coefficient, Vt the tan-gential component of the relative velocity between particles i and j, tij the unitvector along Vt, and γt the tangential damping coefficient. The sum of the three63φ Ga D/∆x L/D AR Np0.01 70 20 15 2 1280.05 70 24 15 2 6440.10 70 32 10 2 3810.20 70 32 10 1 3810.01 160 24 15 2 1280.05 160 24 15 2 6440.10 160 32 10 2 3810.20 160 32 10 1 381Table 3.1: Summary of the parameters used for PR-DNS of suspensions of cubes andspherescontributions introduced above yields the contact force experienced by particle ifrom collision with a single particle j:Fcj = Fe + Fdn + Ft (3.32)When multiple particles make contact with particle i, the total contact force issimply the sum of forces exerted on particle i by each neighbor j:Fc =∑jFcj (3.33)For more details regarding our DEM granular solver, i.e. Grains3D, the interestedreader is referred to [95].3.2.3 Simulations setupFor our particle-resolved simulations of cubes and spheres, we have selectedtwo values of the Galileo number; namely, Ga = 70 and Ga = 160, and a sin-gle value of the density ratio m = 2. These chosen values correspond to steadyoblique and steady vertical settling of a single cube and a single sphere at Ga = 70,respectively. At Ga = 160, on the other hand, a single cube settles in a helical path[12], while a single sphere settles obliquely [32, 33]. A summary of the simulatedcases along with their parameters are presented in table 3.1.64ϕ = 0.01 ϕ = 0.05 ϕ = 0.20x yzgFigure 3.1: Computational domains used for PR-DNS of cube and sphere suspen-sions with different volume fractionsThe simulations of the present work are performed in tri-periodic rectangularcuboid computational domains with horizontal edge lengths Lx = Ly = L andvertical extent of Lz = {L, 2L}, a depiction of which is shown in figure 3.1. Themean volumetric flux of the suspension in the z direction, i.e. the direction ofgravity, is kept zero via a dynamically adjusted pressure drop at each time step.With Np particles each occupying a volume of vp = pi/6, the corresponding solidvolume fraction is given as φ = Npvp/(L3AR), where AR = Lz/Lx denotes theaspect ratio of the domain. The length L and aspect ratio AR are chosen such thata balance is achieved between the sufficiency of the number of particles Np forstatistical analysis and the computational cost of the simulations. Another impor-tant consideration is the divergence of velocity fluctuations with the domain size65[97]. At low Re, the fluctuations grow linearly due to long-range hydrodynamicinteractions. In contrast, it was shown that for a randomly-distributed dilute sus-pension at finite Re, velocity fluctuations follow a much weaker logarithmic trend[98], and actually saturate to finite levels for Re = O(10) [80]. This saturationis attributed to the inertial screening effect due to the formation of a non-randommicrostructure in the suspension [98, 80]. For a dilute concentration of φ = 0.01at Re = 50, particle velocity fluctuations are reported to be insensitive to the do-main size beyond ten particle diameters [99], thanks to the perturbation of particlewakes by their neighbors. Considering the preceding discussion, for the cases withφ = {0.01, 0.05} the domain edge lengths are chosen to be 15D in the horizontaldirection, and elongated twice as much in the vertical direction in order to suf-ficiently account for the particle wakes. We hence ensure that especially for therelatively high Re in our simulations, periodicity does not introduce any significantspurious effects.Initial particle positions are obtained using a random number generation algo-rithm with the condition that particles cannot overlap. In all simulations, the stiff-ness kn is adjusted so that the maximum dimensionless overlap between the parti-cles, δmax/D, is always less than 0.1%. Moreover, the granular time step is chosensuch that δt = (1/20)Tc, with Tc being the contact duration [95], whereas the fluidtime step is set to ∆t = (1/500)tg. The chosen ∆t also satisfies the Courant-Friedrichs-Lewy (CFL) condition of CFL < 0.2 for the grid resolutions pertainingto cases listed in table 3.1. While in cases with φ = {0.10, 0.20}, a pseudo-stationary state is attained and sustained long enough by running the simulationsup to 1000tg, the less concentrated cases with φ = {0.01, 0.05} and Ga = 160required to be run up to ≈ 2000tg to reach and maintain a pseudo-steady state fora sufficient length of time such that reliable statistics could be collected.3.3 ResultsIn the following sections, all particle-related quantities are presented as ensemble-averaged values unless stated otherwise. Therefore, no particular notation is usedfor averaging operation over particles in the system. Time averaging, on the otherhand, is denoted by angular brackets shown as 〈 · 〉. Additionally, the colors red660 0.1 0.2φ0.50.751〈 V z〉/〈 V z,s〉Ga=70CubesSpheresR & Z (1954)(a)0 0.1 0.2φ0.50.751〈 V z〉/〈 V z,s〉Ga=160Cubes (initial)Cubes (final)SpheresR & Z (1954)(b)FinalTransientInitial(c)Figure 3.2: Suspension settling velocities normalized by the terminal velocity of anisolated particle as a function of solid volume fraction for the two Galileo num-bers of Ga = 70 and Ga = 160. The dashed lines in (a) and (b) show theRichardson & Zaki [69] formula presented in equation (3.2), with the prefactork = 0.85, (c) the vertical velocity of an isolated cube as a function of time atGa = 160 normalized by the gravitational velocity scale.and black on all plots are reserved for the data pertaining to cube and sphere sus-pensions, respectively.3.3.1 Settling velocityAs discussed earlier in section 3.1, the settling velocity of a suspension is hin-dered as a result of the upward flux of the fluid compensating for the downwardmotion of the particles. The average hindered settling rates of cube and sphere sus-pensions, 〈Vz〉, normalized by the terminal velocity of the corresponding isolatedparticle, 〈Vz,s〉, are shown in figures 3.2a and 3.2b. Moreover, the Reynolds num-ber based on the pseudo-steady settling velocity for each suspension is reported intable 3.2. For an isolated sphere, 〈Vz,s〉 is computed using the empirical relationavailable for Ga and Ret given in [74], and for a cube, we use the results from ourprevious study on the motion of a single cube [12]. As can be seen, the dependenceof settling rate on the solid volume fraction for sphere suspensions is well describedby equations (3.2) and (3.3) when φ > 0.05 in both figures 3.2a and 3.2b. Notethat in using equation (3.2), we have also incorporated the pre-factor proposed byDi Felice [73] in equation (3.2) with a value of k = 0.85.For Ga = 70, both suspensions of cubes and spheres settle slower than an iso-67Re = ρf 〈Vz〉D/µ φ = 0.01 φ = 0.05 φ = 0.1 φ = 0.2Ga= 70Spheres 64.7 53.7 43.3 30.0Cubes 56.8 44.4 35.6 23.8Ga= 160Spheres 228.1 160.9 127.9 89.7Cubes 172.1 123.3 98.9 68.1Table 3.2: Terminal settling Reynolds number for all simulations based on averagesuspension settling velocity.lated particle at the same Ga as shown in figure 3.2a, while cube suspensions arealways somewhat slower than sphere suspensions. The situation is, however, differ-ent for Ga = 160. In case of spheres, we find that the suspension with the lowestφ settles faster than an isolated sphere. Such a behavior has also been observedby Uhlmann & Doychev [16], where the settling velocity of the suspension with(Ga, φ) = (178, 0.005) was reported to be 12% higher than a single sphere. In ourcase with (Ga, φ) = (160, 0.01), the relative enhancement of settling velocity turnsout to be approximately 7%. For cube suspensions, the choice of velocity scale fornormalization of the settling rate affects the interpretation of results, owing to thedifferent dynamics of the motion of a single cube compared to a single sphere.Figure 3.2c shows three stages in the time evolution of the settling of a single cubein an unbounded fluid at Ga = 160 and m = 2. The cube initially settles at a rel-atively high velocity in a vertical path (denoted as “initial” in figure 3.2c), whichis followed by the appearance of vortex shedding and rotational motion leadingto a significant deceleration (denoted as “transient” in figure 3.2c). Eventually, thecube establishes a sustained helical motion. If 〈Vz〉 for the cube suspensions is nor-malized using 〈Vz,s〉 obtained from the initial stage of figure 3.2c, we essentiallysee a trend similar to figure 3.2a at Ga = 160 (i.e. red squares in figure 3.2b). Onthe other hand, using the final stage 〈Vz,s〉 of figure 3.2c results in higher valuesof 〈Vz〉/〈Vz,s〉 for cubes compared to spheres (i.e. orange squares in figure 3.2b).Moreover, the cube suspension with φ = 0.01 is now seen to settle faster than asingle cube at its final stage of motion. The reason for such a trend for φ = 0.01 istwofold. First, the helical motions of a single cube, which requires a delicate bal-ance of forces and torques acting on the cube, cannot be established and sustained680 0.1 0.2φ10-310-210-1100〈 σ2 z〉/〈 V z〉2Ga=700 0.1 0.2φ10-310-210-1100Ga=1600 0.1 0.200.510 0.1 0.200.51(a)0 0.1 0.2φ10-310-210-1100〈 σ2 x〉/〈 V z〉2Ga=700 0.1 0.2φ10-310-210-1100Ga=1600 0.1 0.200.510 0.1 0.200.51(b)Figure 3.3: Velocity fluctuations (a) in the gravity direction and (b) normal to thegravity direction. The particle and fluid data are shown by solid and open sym-bols, respectively. Also, red lines and symbols demonstrate the results for cubesuspensions, whereas black lines and symbols represent results for sphere sus-pensions. The inset of each plot shows the same data on a linear the presence of perturbations of other nearby cubes. Consequently, such effec-tive deceleration of the cube as in figure 3.2c does not conceivably occur. Secondly,even though cubes are significantly more likely to break free from clusters com-pared to spheres as will be discussed in section 3.3.4, there is still a considerableprobability of finding a cube in the wake of another, which acts towards reducingthe overall drag force experienced by the cube suspension.3.3.2 Velocity fluctuationsTime averaged values of particle and fluid velocity fluctuations of the cube andsphere suspensions, represented by velocity variance σ2, are given in figure 3.3.Fluid fluctuations are obtained by computing the fluid velocity variance in a cubiccontrol volume centered around each particle with a diameter of 4D, excludinggrid points lying inside the solids. Figures 3.3a and 3.3b show fluctuations inthe gravity and normal directions, respectively. Moreover, time evolution of thesame quantities are presented in figures 3.4 and 3.5. The variances in all casesare normalized by the square of the corresponding settling rate of each suspension,which is the reason that the resulting quantity is occasionally referred to as “relativefluctuation” in the literature [80, 100].Overall, we find that the velocity of cubes fluctuates with higher magnitudecompared to spheres particularly in the transverse direction, and that the differ-690 500 1000 1500 200000.1σ2 Vz/〈 V z〉2φ=0.010 250 500 750 100000.1σ2 Vz/〈 V z〉 2 φ=0.050 250 500 750 100000.1σ2 Vz/〈 V z〉 2 φ=0.100 250 500 750 1000 1250t/tg00.2σ2 Vz/〈 V z〉 2 φ=0.200 500 1000 1500 200000.02σ2 Vx/〈 V z〉 2 φ=0.010 250 500 750 100000.025σ2 Vx/〈 V z〉 2 φ=0.050 250 500 750 100000.05σ2 Vx/〈 V z〉 2 φ=0.100 250 500 750 1000 1250t/tg00.1σ2 Vx/〈 V z〉 2 φ=0.20Figure 3.4: Time evolution of ensemble-averaged particle velocity fluctuations, withthe left and right panels showing vertical and horizontal components, respec-tively. In these plots, solid lines represent simulations with Ga = 160, whereasdotted lines show data for simulations with Ga = 70. Also, red and black linesdemonstrate the results for cube and sphere suspensions, respectively. In orderto enhance visual representation of the plots, darker lines show a filtered versionof the data, while the original data are plotted with a lighter color.ences between cube and sphere suspensions in terms of particle fluctuation levelsdiminish as φ increases. In figure 3.3a, particle fluctuations pertaining to the regimewhere (Ga, φ) = (160, 0.01) appear to be higher than might be expected. As willbe highlighted in section 3.3.4, strong columnar clustering occurs for both cubeand sphere suspensions at this regime, which underlies the high values of verticalparticle fluctuations as a result of prevalent DKT-type interactions. In figure 3.4,we observe more clearly that fluctuations of spheres in both directions stay almostthe same upon increasing Ga, except for the most dilute of φ = 0.01. For cubes,however, enhancement of fluctuations with Ga are seen up to φ = 0.05, particularly70in the horizontal direction. Moreover, cube suspensions at (Ga, φ) = (160, 0.01)and (Ga, φ) = (160, 0.05) exhibit long periods of high vertical velocity fluctua-tion, reminiscent of those observed by Willen & Prosperetti [77] for suspensions ofspheres. As particles interact more frequently at higher solid volume fractions, themagnitude of fluctuations increases with concentration for both types of particles,as seen in figure 3.3. While hydrodynamic interactions are mainly responsible forthis increase at lower solid volume fractions, contact and collision between parti-cles become more pronounced at higher φ, thus contributing greatly to the increaseof particle velocity fluctuations.For both types of particles, fluid velocity variances (shown with open sym-bols in figure 3.3) are significantly larger than that of particles for φ > 0.05. Forφ = 0.01, the difference is smaller since particles are expected to better followlarge-scale fluid motions in dilute cases. In more concentrated cases, particles lessfaithfully adapt to smaller fluid structures with length scales approximately equalto D [98]. In general, the fluid pseudo-turbulence induced by cubes is seen to beconsiderably higher than that of spheres. That is, on average, σ2uz/〈Vz〉2 is 41%higher for cubes for both Galileo numbers, whereas σ2ux/〈Vz〉2 is 52% and 72%higher for Ga = 70 and Ga = 160, respectively. For φ > 0.05, vertical fluctua-tions are quite similar in their average values when Ga increases from 70 to 160 forboth cubes and spheres. In contrast, cubes manifest large differences in the fluidvelocity variance between Ga = 70 and Ga = 160 in the horizontal direction forall solid volume fractions, except φ = 0.2. Sphere suspensions, on the other hand,do not experience such a dramatic amplification of fluctuations upon increasingGa, except for the most dilute case.Figure 3.6 shows the anisotropy of particle and fluid velocity fluctuations ob-tained from the ratio of vertical to horizontal components of velocity variance. Theamplification of velocity fluctuations with solid volume fraction shown in figure 3.3is also accompanied by a reduction in the level of anisotropy in all suspensions,consistent with results obtained for sphere suspensions in the literature [77, 101,98]. More importantly, it is remarkable that the anisotropy of fluctuations in cubesuspensions is always smaller than in sphere suspensions, whether it be for parti-cle (figure 3.6a) or fluid (figure 3.6b) velocity fluctuations. This indicates that in710 500 1000 1500 200000.1σ2 uz/〈 V z〉 2 φ=0.010 250 500 750 100000.2σ2 uz/〈 V z〉 2 φ=0.050 250 500 750 100000.5σ2 uz/〈 V z〉 2 φ=0.100 250 500 750 1000 1250t/tg01σ2 uz/〈 V z〉 2 φ=0.200 500 1000 1500 200000.02σ2 ux/〈 V z〉 2 φ=0.010 250 500 750 100000.05σ2 ux/〈 V z〉 2 φ=0.050 250 500 750 100000.1σ2 ux/〈 V z〉 2 φ=0.100 250 500 750 1000 1250t/tg00.2σ2 ux/〈 V z〉 2 φ=0.20Figure 3.5: Time evolution of ensemble-averaged local fluid velocity fluctuations,with the left and right panels showing vertical and horizontal components, re-spectively. In these plots, solid lines represent simulations with Ga = 160,whereas dotted lines show data for simulations with Ga = 70. Also, red andblack lines demonstrate the results for cube and sphere suspensions, respec-tively. In order to enhance visual representation of the plots, darker lines showa filtered version of the data, while the original data are plotted with a lightercolor.cube suspensions, particle momentum and energy is more effectively transferredfrom the gravity to the transverse direction, thus making these suspensions moreisotropic. For the most dilute case with Ga = 70 in figure 3.6a, anisotropy of parti-cle velocity fluctuations is quite high for spheres due to the limited lateral motionsat this regime, whereas the cube suspension is≈ 50% more isotropic. The value ofanisotropy in our sphere suspension is ≈ 12, which is close to the value of 12.96obtained by Yin & Koch [98], though at a significantly smaller Reynolds number ofRe = 10. We also observe that increasing Ga renders almost all suspensions moreisotropic due to the emergence and enhancement of transverse fluctuations. Fur-720 0.1 0.2φ2345〈 σ2 V z〉/〈 σ2 Vx〉Ga=700 0.1 0.2φ2345Ga=160SpheresCubes0 0.1 0.20612(a)0 0.1 0.2φ246810〈 σ2 u z〉/〈 σ2 ux〉Ga=700 0.1 0.2φ246810Ga=160SpheresCubes0 0.1 0.201020(b)Figure 3.6: Anisotropy of (a) particle and (b) fluid velocity fluctuations indicatedby the ratio of the vertical to horizontal velocity variance. The insets extendedranges of values on the vertical axis.thermore, the differences between cube and sphere suspensions vanish for φ = 0.2,meaning that both types of particles begin to behave similarly in terms of velocityfluctuations. Finally, studies that report values pertaining to the anisotropy of par-ticle velocity fluctuations for sphere suspensions in regimes relevant to the presentwork are scarce. While Zaidi [101] obtained values of 〈σ2Vz〉/〈σ2Vx〉 in the range of≈ 5.7 – 6.76 for 0.05 6 φ 6 0.2, Willen & Prosperetti [77] report ≈ 1.2 – 1.7 for0.087 6 φ 6 0.262. As it turns out, there is a large discrepancy between these twosets of values. Nevertheless, our results in figure 3.6a lie between those reportedby others, albeit much closer to the results of Willen & Prosperetti [77].3.3.3 Angular velocitiesIn our previous study of the motion of an isolated cube [12], we found outthat a cube rotates much more vigorously than a sphere at high Galileo num-bers. This is due to the existence of sharp edges that induce strong hydrodynamictorques as the cube changes its orientation relative to the surrounding flow. In fig-ure 3.7, the distributions of horizontal angular velocities of particles, defined asΩh =√Ω2x + Ω2y, are plotted for all simulated cases. We choose to demonstratethe horizontal components, since it is this component that together with the verti-cal velocity of the particles generates the major contribution to the Magnus forcein the horizontal direction. It is immediately obvious that in all cases, cubes notonly have higher angular velocities on average (e.g. up to ≈ 3 times higher for(Ga, φ) = (160, 0.01)), but are also more likely to experience more extreme val-ues. Similar to velocity fluctuations, the differences become less pronounced with730 0.25 0.5ΩhD/〈Vz〉05101520PDFφ=0.010 0.510-21000 0.25 0.5ΩhD/〈Vz〉0246PDFφ=0.050 0.5 110-21000 0.25 0.5 0.75 1ΩhD/〈Vz〉0123PDFφ=0.100 0.5 110-21000 0.5 1 1.5ΩhD/〈Vz〉00.511.52PDFφ=0.200 0.75 1.510-2100Figure 3.7: Distribution of particle angular velocities for different solid volume frac-tions. The inset of each plot shows the same data with y-axis scaled logarith-mically. Solid lines represent simulations with Ga = 160, whereas dotted linesshow data for simulations with Ga = 70. Also, red and black lines demonstratethe results for cube and sphere suspensions, respectively. The mean value ofeach distribution is shown using short vertical lines on each plot.the increase in solid volume fraction, as inter-particle contact and collisions be-gin to dominate particle kinematics. We can see that for φ = 0.01 in figure 3.7,the distribution of angular velocities of spheres remains almost identical betweenGa = 70 and Ga = 160, whereas for cubes there is a substantial shift towardshigher Ωh. This is similar to the case of an isolated cube, with the difference thatan isolated cube at Ga = 70 only shows negligible oscillations, while in a suspen-sion of cubes the collective effect of the wake of the surrounding cubes inducesconsiderable rotational motions. It can also be seen in figure 3.7 that the average74-0.5 0 0.5ΩxD/〈Vz〉-0.500.5Vy/〈 V z〉ρ=0.43Cubes(a)-0.5 0 0.5ΩxD/〈Vz〉-0.500.5ρ=0.63Spheres(b)0 0.1 0.2φ0.20.40.6ρ(c)Figure 3.8: Joint probability distribution functions of (a) cube and (b) sphere sus-pensions both obtained for (Ga, φ) = (160, 0.05), (c) Variation of Pearson’scorrelation coefficient as a function of the solid volume fraction for all cases.The dotted and solid lines represent data for Ga = 70 and Ga = 160, respec-tively, whereas red and black lines demonstrate the results for cube and spheresuspensions, respectively.magnitude of horizontal angular velocities for spheres turn out to be somewhathigher at Ga = 70 compared to Ga = 160, which might seem counter-intuitive.However, the suspensions simulated here are by no means homogeneous, and as wewill see in section 3.3.4, the level of inhomogeneity increases with Ga. Therefore,we believe that the reduction of angular velocities from Ga = 70 to Ga = 160 isa consequence of the stronger clustering of spheres at the higher Ga. One possiblereason could be that since spheres at Ga = 160 settle in larger groups, it is moreprobable for a sphere to be surrounded horizontally by neighboring spheres. It isthus more likely that the net microstructure-induced hydrodynamic torque attainssmaller values at Ga = 160 compared to Ga = 70.To further elucidate the influence of rotation on transverse forces and sub-sequent lateral motions, we present the Joint Probability Distribution Functions(JPDF) of horizontal rotational and translational particle velocities in figure 3.8.Since the Magnus force is by definition directed along Vslip×Ω = (u−V)×Ω,the rotational velocity Ωx is expected to give rise to a Magnus force and conse-quently a translational velocity in the positive y-direction. Therefore, we considerJPDFs of Ωx and Vy in figure 3.8, but similar information can also be equivalently75obtained with Ωy and Vx, with the only difference of the appearance of a nega-tive correlation. In figure 3.8, ρ denotes the Pearson’s correlation coefficient [102]defined as ρX,Y = cov(X, Y)/σXσY , which shows the covariance between twovariables X and Y normalized by their standard deviations. The variation of ρ forboth cube and sphere suspensions is given as a function of the solid volume frac-tion for Ga = 70 and Ga = 160 in figure 3.8c. In all cases, ρ is expectedly found tobe ≈ 0 for rotational and translational velocities in the same direction, e.g. for Ωxand Vx. We immediately notice in figure 3.8c that the normalized covariance is al-ways stronger at Ga = 160 than in Ga = 70 for both types of suspensions, thoughmore strongly so for spheres. Moreover, ρ is found to be consistently greater forspheres compared to cubes. We attribute this finding to the fact that unlike the caseof a sphere, the rotation of a cube creates both a Magnus (i.e. rotation-induced)force and an orientation-induced lateral force [21]. Since the rotation of a cube isalso accompanied by a change in its orientation, a constant rotation would create alateral force that periodically changes sign. Therefore, even in the absence of theMagnus force, the rotation of a cube causes a sign-changing lateral force resultingin zero covariance. We speculate that this underlies the weaker correlation betweenrotational and translational velocity of cubes compared to spheres in figure 3.8c.Additionally, the wider distribution of velocities in figure 3.8a compared to fig-ure 3.8b is notable, implying that larger horizontal velocities are more frequentlyencountered in cube suspensions despite their weaker correlation with their rota-tional velocities. As signified by the less than perfect covariance of variables shownin figure 3.8c, it should be emphasized that the rotational velocity of a particle andthe corresponding forces are only partly responsible for its lateral motions. Othercontributions include shear and vortex-induced as well as microstructure-inducedlateral forces, which we investigated thoroughly in a previous work [103]. Whilerotation, shear and vortex-induced forces are more influential in dilute regimes,microstructure-induced loads become increasingly more important in denser sus-pensions. An indication of this relative importance is reflected by the decreasingcovariance of rotational and translational velocities with solid volume fraction asshown in figure 3.8c.763.3.4 MicrostructureIn order to characterize the microstructure of the suspensions, we make use ofthe pair distribution function P (r). This quantity shows the ratio of ρn(r), thenumber density of neighboring particles in a volume ∆V located at r, to ρn, theaverage number density in the bulk of the suspension. In spherical coordinates,P (r) is given as [104]P (r) = P (r, θ, ψ) =〈ρn(r, θ, ψ)〉ρn=〈H(r, θ, ψ)〉ρn∆v(3.34)where r is the pair separation vector pointing to the center of a neighboring particlegiven a reference particle at the origin r = 0, θ the azimuthal angle measured frompositive z axis, ψ the polar angle measured from positive x axis, ρn the averagenumber density of particles in the computational domain, ∆v = r2 sin θ∆r∆θ∆ψthe volume element and H(r, θ, ψ) the histogram of particle pairs. Since in oursimulations, P (r, θ, ψ) is axisymmetric about the gravity direction, we only com-pute P (r, θ), which is P (r) averaged over all values of ψ. To obtain P (r, θ, ψ), thespace is first discretized with ∆r = 0.1D and ∆θ = pi/60. At each sampling time,we create periodic and multi-periodic images of the particle positions, followedby a loop for each particle and over the surrounding space to count the number ofneighbors happening to be in the subvolume ∆v at any given r and θ. The his-togram H(r, θ) is thus progressively constructed, and finally averaged over at least3000 time snapshots for each case. Another informative quantity can be derivedfrom equation (3.34) by averaging P (r, θ) over θ, yielding the radial distributionfunction:g(r) =12∫ pi0P (r, θ) sin θdθ (3.35)which shows the ratio of the number density in a spherical shell with thickness ∆ra distance r away from the reference particle to the bulk number density, regardlessof the relative orientation.The pair and radial distribution functions are plotted for Ga = 160 and Ga = 70in figures 3.9 and 3.10, respectively. The color bars have been set to represent the77pi/6pi/3pi/20246810r/D00.511.5P(r,θ)(Ga, φ) = (160, 0.01)pi/6pi/3pi/20246810r/D00.511.5P(r,θ)(Ga, φ) = (160, 0.01)1 2 3 4 5r/D00.511.52g(r)(Ga, φ) = (160, 0.01)CubesSpheresHard spherespi/6pi/3pi/2012345r/D00.511.5P(r,θ)(Ga, φ) = (160, 0.05)pi/6pi/3pi/2012345r/D00.511.5P(r,θ)(Ga, φ) = (160, 0.05)0.5 1 1.5 2 2.5r/D00.511.52g(r)(Ga, φ) = (160, 0.05)CubesSpheresHard spherespi/6pi/3pi/2012345r/D00.511.5P(r,θ)(Ga, φ) = (160, 0.10)pi/6pi/3pi/2012345r/D00.511.5P(r,θ)(Ga, φ) = (160, 0.10)0.5 1 1.5 2 2.5r/D00.511.52g(r)(Ga, φ) = (160, 0.10)CubesSpheresHard spherespi/6pi/3pi/2012345r/D00.511.5P(r,θ)(Ga, φ) = (160, 0.20)pi/6pi/3pi/2012345r/D00.511.5P(r,θ)(Ga, φ) = (160, 0.20)0.5 1 1.5 2 2.5r/D00.511.52g(r)(Ga, φ) = (160, 0.20)CubesSpheresHard spheresFigure 3.9: Pair and radial distribution functions for Ga = 160. The leftmost andcenter columns pertain to cube and sphere suspensions, respectively.same range of values for all cases so that direct comparisons can be made. Thestrong clustering of particles in the vertical direction for the sphere suspension at(Ga, φ) = (160, 0.01) in the form of a columnar structure is the most remarkablefeature of the pair distribution functions in figure 3.9. Such a prominent verti-cal clustering of spherical particles was reported by Uhlmann & Doychev [16] at(Ga, φ) = (178, 0.005), and later confirmed experimentally by Huisman et al. [79].In figure 3.9, we also note an increased concentration of spheres in the horizontaldirection relative to the reference particle. Interestingly, this was also observedby Uhlmann & Doychev [16], despite the fact that their simulated suspension wasmore dilute and the Galileo number that they used was slightly higher. The forma-78pi/6pi/3pi/20246810r/D00.511.5P(r,θ)(Ga, φ) = (70, 0.01)pi/6pi/3pi/20246810r/D00.511.5P(r,θ)(Ga, φ) = (70, 0.01)1 2 3 4 5r/D00.511.52g(r)(Ga, φ) = (70, 0.01)CubesSpheresHard spherespi/6pi/3pi/2012345r/D00.511.5P(r,θ)(Ga, φ) = (70, 0.05)pi/6pi/3pi/2012345r/D00.511.5P(r,θ)(Ga, φ) = (70, 0.05)0.5 1 1.5 2 2.5r/D00.511.52g(r)(Ga, φ) = (70, 0.05)CubesSpheresHard spherespi/6pi/3pi/2012345r/D00.511.5P(r,θ)(Ga, φ) = (70, 0.10)pi/6pi/3pi/2012345r/D00.511.5P(r,θ)(Ga, φ) = (70, 0.10)0.5 1 1.5 2 2.5r/D00.511.52g(r)(Ga, φ) = (70, 0.10)CubesSpheresHard spherespi/6pi/3pi/2012345r/D00.511.5P(r,θ)(Ga, φ) = (70, 0.20)pi/6pi/3pi/2012345r/D00.511.5P(r,θ)(Ga, φ) = (70, 0.20)0.5 1 1.5 2 2.5r/D00.511.52g(r)(Ga, φ) = (70, 0.20)CubesSpheresHard spheresFigure 3.10: Pair and radial distribution functions for Ga = 70. The leftmost andcenter columns pertain to cube and sphere suspensions, respectively.tion of these horizontal configurations was attributed to the tumbling phase of thewell-known DKT phenomenon. Conversely, while the cube suspension in the sameregime, i.e. (Ga, φ) = (160, 0.01) also exhibits increased concentration in the ver-tical direction, it is obvious that the intensity of clustering is significantly smallerthan what is seen for the sphere suspension. The pair distribution function for thecube suspension at (Ga, φ) = (160, 0.01) is also seen to be more uniformly dis-tributed, and shares some similar qualities as well. For both suspensions, we findthat the radial distribution function attains significantly larger values compared toa hard-sphere distribution [105]. Furthermore, for all solid volume fractions shownin figure 3.9, a sharp peak is persistently observed for sphere suspensions. Inspec-79tion of pair distribution functions reveals that there exists a “contact rim”, i.e. acircular high density band, at r/D = 1 for all sphere suspensions (weaker in caseof φ = 0.05). This region exists as spheres spend a relatively long time in con-tact with each other during the tumbling phase of DKT. Obviously, this point ofcontact is always located at r/D = 1 for spheres which causes the sharp peakat r/D = 1, whereas for cubes, contact can occur at any center-to-center dis-tance in the range√pi/6 6 r/D 6√pi/2 due to the geometry of a cube. Thisis evident from radial distribution functions of cubes appearing as approximatelyoblique lines about r/D = 1. At higher solid volume fractions, the wakes of par-ticles become much more frequently disrupted by their neighbors. Additionally,the settling Reynolds number also decreases with increasing φ due to the strongerhindrance effect for the same Ga (as noted in table 3.2), leading to weaker particlewakes. This is why the prominent vertical configuration of particles at φ = 0.01quickly fades away with increasing of the suspension concentration, even thoughthere still remains a trace of vertical accumulation for both cube and sphere sus-pensions at (Ga, φ) = (160, 0.05). Since the vertical arrangement of particles isunstable, both spheres and cubes quickly tumble into a stable raft, or horizontalconfiguration which causes a deficit in the vertical direction of the pair distribu-tion functions. This is reflected quite clearly for sphere suspensions in the formof horizontal high density regions. In case of cubes, while the same horizontalstructures can be still detected in almost all cases, not only the pair distributionfunctions are much fainter, but also the horizontal extension of these regions aresignificantly more limited. This observation is also confirmed by the reduced mag-nitude of the radial distribution functions for cube suspensions compared to theirsphere counterparts. Notably, the pair distribution function of the cube suspensionfor (Ga, φ) = (160, 0.20) in figure 3.9 is almost uniform except for r/D 6 2,whereas the extended light blue and red regions and their margin is still clearlydiscernible in the sphere suspension. We will demonstrate in section 3.3.5 that thisparticular difference in microstructure is, at least partly, responsible for the dif-ferences observed in the drag force experienced by cube and sphere suspensions.While at relatively low solid volume fractions the suspensions are markedly in-homogeneous, the microstructure shows resemblance to random distributions (i.e.hard-sphere distributions in case of sphere suspensions, and their equivalent for80cube suspensions) at φ = 0.2, as many-body interactions begins to dominate thedynamics of the suspensions.At a lower Galileo number of Ga = 70, we conversely find a strong deficit ofparticles in the vertical direction near the reference particle for both sphere andcube suspensions, especially in the most dilute case. Comparison of radial distri-bution functions reveals that except for φ = 0.20, this deficit is remarkably morepronounced compared to suspensions at Ga = 160. We also find that the pair dis-tribution functions for both sphere and cube suspensions at (Ga, φ) = (70, 0.01)are starkly different from those obtained for (Ga, φ) = (160, 0.01). In figure 3.10,a slender vertical region depleted of particles is visible for the sphere suspensionat (Ga, φ) = (70, 0.01), which is the complete opposite of our observation for(Ga, φ) = (160, 0.01). From figure 3.10, it can be seen that this low concentrationregion is immediately adjacent to a higher concentration region. More importantly,such a well-defined depletion zone cannot be detected in the case of the cube sus-pension at (Ga, φ) = (70, 0.01), although weak signs of vertical voids are visible.Another notable observation is that in contrast to sphere suspensions at Ga = 160,the contact rim does not appear as strongly at Ga = 70. This means that whileparticles undergo DKT, they either do not touch as frequently, or spend as muchtime in contact even if they do. Correspondingly, the peaks of the radial distribu-tion functions of sphere suspensions are higher and wider at Ga = 160 comparedto Ga = 70. For the highest solid volume fraction of φ = 0.20, while cube sus-pensions have an almost uniform pair distribution function for separation distancesgreater than r/D = 2, the horizontal sector of higher particle concentration is stilldistinguishable in the sphere suspension at Ga = 160 and φ = 0.20.In order to demonstrate the microstructure evolution, we also present in fig-ure 3.11 the ensemble-averaged local solid volume fraction φloc. This quantity isobtained by considering a cubical control volume of edge length 4D centered onthe location of each particle, and then computing the ratio of the number of gridpoints lying in the solid objects to the total number of points in that control volume,and then performing ensemble averaging at each time step. It is noted that althoughthis method inevitably results in a pixelated (i.e. step-wise) representation of theparticles, it still yields a reliable estimation of the local solid volume fraction for810 500 1000 1500 2000 2500t/tg0.010.0150.02φlocφ=0.010 250 500 750 1000t/tg0.050.0550.06φlocφ=0.050 250 500 750 1000t/tg0.090.10.12φlocφ=0.100 250 500 750 1000 1250t/tg0.180.20.23φlocφ=0.20Figure 3.11: Time evolution of ensemble-averaged local solid volume fraction φloc.Solid lines represent simulations with Ga = 160, whereas dotted lines showthe data for simulations with Ga = 70. Also, red and black lines demonstratethe results for cube and sphere suspensions, respectively.comparison purposes. Despite the overall similarity of pair distribution functionsof cubes and spheres for φ > 0.05 presented in figures 3.9 and 3.10, the localsolid volume fraction in sphere suspensions undergoes several high peaks that lastfor long periods of time, particularly at Ga = 160. In contrast, such noticeablepeaks are not seen in the values of φloc for cube suspensions. Moreover, the plotsin figure 3.11 unequivocally show that spheres are always locally closer to eachother than cubes. For φ > 0.05, there is a distinctive change in values of φloc insphere suspensions when increasing Ga from 70 to 160, while for cubes the onlynoticeable change appears to be in the most dilute case with φ = 0.01. We believethat the high peaks seen for sphere suspensions correspond to the persistent con-tact rim seen in figures 3.9 and 3.10 (and equivalently the sharp peaks of the radialdistribution functions), and hence exist due to the fact that at Ga = 160 spheresspend a substantial amount of time in the kissing phase of DKT. Since the contactrim also exists, although not as prominently, for Ga = 70 as well, we conclude thatat Ga = 160 a greater number of particles are in direct contact thus causing theappearance of high values of local solid volume fraction noted in figure 3.11.The preceding descriptions of the microstructure of simulated systems confirm82that inertial liquid-solid suspensions of spheres and cubes are non-random, partic-ularly for lower solid volume fractions. It also turns out that the hydrodynamicsof dilute suspensions can be very different at Ga = 70 and Ga = 160, an ex-ample of which is the narrow vertical depletion region of sphere suspensions at(Ga, φ) = (160, 0.01) (figure 3.10) as opposed to the high concentration columnof spheres at (Ga, φ) = (70, 0.01) (figure 3.9). At the higher Galileo numberof Ga = 160, the wake is strong enough to attract particles from relatively longdistances. As noted in [16], an isolated sphere moves obliquely at this Galileonumber, which might be a contributing factor in leading particles into the wakeregion of another particle. The trapped spheres spend a sufficiently long time inthe formed clusters such that a substantial imprint is left on the pair distributionfunction of their corresponding suspension. These particles then collide with theleading particle, and tumble into a horizontal position. At the lower Galileo num-ber of Ga = 70 and for φ = 0.01, however, spheres get repelled from the wakeregion. We believe that due to the existence of shear in the wake region, particlesexperience a lift force before getting drawn into the wake of another sphere, whicheffectively pushes them away from the wake region. Furthermore, in contrast toGa = 160, an isolated sphere moves vertically at Ga = 70 instead of obliquely. Inboth of these scenarios, cube suspensions are considerably more homogeneous. Inour previous study of the free motion of an isolated cube [12], we have shown thata cube experiences much larger rotation rates and transverse forces (e.g. Magnusforce) due to its geometry compared to a sphere. A slight change in the orientationof a cube can induce large hydrodynamic torques and subsequent rotations. Eventhough the helical motion of an isolated cube at Ga = 160 is unlikely to be sus-tained in the presence of other cubes, the inherent susceptibility of cubes to rotationis still very likely to cause significant side motions. This is why we suggest thatthe less pronounced clustering of cube suspensions originates from high rotationalvelocities of cubes, which consequently enhances their chances of effectively es-caping clusters. This conclusion is also in line with the results of Kajishima [106]in PR-DNS of dilute sphere suspensions at high Reynolds numbers. Kajishimafound irrotational spheres to maintain cluster structures, as opposed to rotationalspheres which tended to cause periodic break-up of clusters.833.3.5 Drag forcePrior to presenting the numerical results, it is necessary to stress that in theliterature, two different quantities have been referred to as drag force; namely,the total drag and the flow-induced drag denoted by F and Fd, respectively. In asuspension system where a static macroscale pressure gradient exists, the total dragforce acting on a particle is the sum of a buoyancy-type force −vp∇P (which isexperienced even in the absence of a flow) and the flow-induced interaction forceFd. The net drag force exerted on all particles is balanced by the pressure dropover the whole system such that−V∇P = NpF = Np(−vp∇p+ Fd) (3.36)where V is the total volume of the system. From the above equation, we note that−vp(1− φ)∇P = φFd. Therefore, it follows thatF =11− φFd (3.37)In the present study, we have chosen to report the magnitude of the total drag forceF for our results, which is the quantity that is directly computed by integratinghydrodynamic stresses in the simulations as in equation (3.10).Several drag correlations have been proposed in the past for suspensions ofspheres for varying ranges of Re and φ. However, the systems from which thesecorrelations have been extracted have not always been similarly set up. The earliestdrag correlation is the one proposed by Wen & Yu [107] based on fluidizationexperiments with the following formFFs= (1− φ)−β−1 (3.38)where Fs is the drag force of a single stationary sphere, for which reliable correla-tions exist [108, 109]. Equation (3.38) is in fact the same relation that representsthe dilute part of the drag correlation proposed by Gidaspow [110]. Wen & Yu[107] suggested that the exponent β is constant with a value of β = 3.7 over theentire range of Reynolds numbers, based on the observation that β turned out to84be almost the same for the extremities of the Reynolds number range. Based onthe available experimental data, Di Felice [111] later found out that for the inter-mediate range of Re, β is a weak function of the Reynolds number obeying thefollowing relationβ = 3.7− 0.65 exp(−(1.5− log10 Re)22)(3.39)where β goes through a minimum of β ' 3 for 20 6 Re 6 80. Due to obvi-ous practical limitations, fixed-bed data such as those proposed by Ergun [112]can only be obtained for highly concentrated system close to packing solid volumefractions. In recent years, the availability of massive computational power has pro-vided the opportunity to model fluid flow in fixed arrays of spheres over extendedranges of solid volume fractions and Reynolds numbers. The PR-DNS of Hill etal. [113, 114], Beetstra et al. [115], Tang et al. [116] and Bogner et al. [117]based on lattice-Boltzmann method and that of Tenneti et al. [118] based on theimmersed boundary method have provided correlations for drag force over station-ary arrangement of spheres for dense systems (i.e. φ > 0.1) and Reynolds numbersmostly up to Re = 300. Of particular relevance to the present work is the PR-DNSof Chen & Müller [119] on the drag force of assemblies of cubic particles. In theirstudy, super-quadric cubes were used (having relatively round corners and edges)to obtain a drag correlation for 0 6 Re 6 200 and 0.1 6 φ 6 0.45.The drag force data collected from the present simulations are shown in fig-ure 3.12, where the values are normalized by the Stokes drag relation FSt =3piµD〈Vz〉. On the same graphs in figure 3.12, we also show values of drag com-puted using correlations reported in the literature for spheres, and also using theaforementioned correlation proposed for cubes [119] with parameters correspond-ing to our simulations. Since these correlations require the Reynolds number (in-stead of the Galileo number) as an input, we use the Reynolds number based onthe pseudo-steady settling rate of the suspension in each simulation presented intable 3.2. In doing so, it should be noted that for correlations derived from fixed-bed systems, the Reynolds number based on superficial velocity is equivalent tothe Reynolds number computed using the average settling velocity of the particles850.01 0.10 0.20φ57.51012〈 F/FSt〉Ga=70SpheresCubesFixed cubes Wen & Yu (1966)Bogner et al. (2015)Di Felice (1994)Chen & Müller (2018)(a)0.01 0.10 0.20φ5101520〈 F/FSt〉Ga=160SpheresCubesFixed cubesFixed spheres Wen & Yu (1966)Bogner et al. (2015)Di Felice (1994)Chen & Müller (2018)(b)Figure 3.12: Drag force of the cube and sphere suspensions as a function of solidvolume fraction for two Galileo numbers of Ga = 70 and Ga = 160. Opensymbols show data for dynamic suspensions, while solid symbols representdata obtained from fixed beds. Along with data obtained from our PR-DNS,we also show drag correlations of Wen & Yu [107] and Tang et al. [116] forstatic arrays of spheres, and that proposed by Chen & Müller [119] for fixedbeds of cubes.(relative to a fixed container) in sedimenting suspensions [74].From figure 3.12, we can immediately recognize a number of distinct features.The most notable is the atypically lower drag experienced by both spheres andcubes at (Ga, φ) = (160, 0.01) compared to their fixed-bed counterparts. As il-lustrated in section 3.3.4, this is due to the formation of fast-settling clusters inthis particular regime. In addition to the lower drag compared to fixed beds ow-ing to the inhomogeneous microstructure, it turns out that both suspensions at(Ga, φ) = (160, 0.01) on average settle faster even than an isolated particle ac-cording to the data in table 3.2. At this Galileo number and density ratio, a singlesphere settles rectilinearly with a terminal Reynolds number of Re ' 213, whilea single cube decelerates significantly after an initial vertical motion and eventu-ally settles in a helical path at Re ' 155 [12] (see also figure 3.2c). While thehigher-than-normal settling rate or equivalently the lower drag in dilute sphere sus-pensions have been lately reported by other researchers [79, 16], we find that tobe also true for a cube suspension at (Ga, φ) = (160, 0.01), which appears to ex-perience less drag compared to a fixed bed of cubes. Remarkably, no such dragreduction is observed for the similar solid volume fraction at Ga = 70. Anotherfeature of figure 3.12 is that while the drag of sphere suspensions is higher than that86obtained from fixed beds except for (Ga, φ) = (160, 0.01), the drag curve of cubesuspensions closely follows the drag data of fixed beds of cubes in all cases, withthe only exception of the case with (Ga, φ) = (160, 0.01). Note that figure 3.12shows two sets of drag data for fixed beds of cubes. One set, shown by solid graysquares, pertains to the simulations we performed with random cube arrangementsat the same Reynolds numbers as the dynamic cases. The other set, represented bysolid red squares, shows the drag of fixed beds of super-quadric cubes with a round-ing factor of 5 obtained by Chen & Müller [119]. As can be seen, our drag valueslie well above those reported in [119]. This discrepancy might have appeared atleast partly owing to the fact that the cubes in our simulations are represented withsharp edges and corners, while the super-quadric cubes in [119] are rounded bydefinition. Although for a different setup, Jaiman et al. [120] also showed that asingle square cylinder with sharp corners experiences maximum drag compared toits counterparts with varying degrees of corner roundness.The similarity between drag values of stationary and moving cubes is rather sur-prising, as one might expect moving cubes to experience a larger overall drag dueto stronger velocity fluctuations, as is the case for an isolated moving cube com-pared to a fixed one. However, we believe that this trend may be justified basedon that the relative magnitudes of factors contributing to the alteration of drag aredifferent for cubes and spheres. In general, the drag acting on mobile particlesin a dynamic suspension would be different compared to the drag obtained froma stationary array (i.e. a fixed bed) of particles. The effect of particle motion ondrag in a suspension has been a long-standing challenge for developing accuratedrag correlations, as touched upon in the review article by Di Felice [121], andinvestigated by Tang et al. [122] and Tavanashad et al. [82] more recently. Thedifference between the drag of dynamic versus static systems stems from three fac-tors; namely, particle mobility, velocity fluctuations and suspension microstructure.First, the adaptation of moving particles to the flow field changes as opposed to theresistance of fixed particles is known to reduce the drag of sphere suspensions asdemonstrated by Rubinstein et al. [123] in Stokes regime, and by Tavanashad etal. [82] for high Reynolds numbers. Second, particle velocity fluctuations cre-ate additional shear in the flow and also introduce unsteady effects such as history87(a) φ = 0.1(b) φ = 0.2Figure 3.13: Pair distribution functions of cube and sphere suspensions for two dif-ferent solid volume fractions. In each of the four panels, the left and right plotspertain to cube and sphere suspensions, respectively.and added-mass forces, which would otherwise be absent in a suspension withfrozen particles. Third, when particles are free to move, they show a preferencefor particular arrangements that are markedly different from a random (i.e. hard-sphere) distribution, as discussed previously in section 3.3.4. These non-randomstructures develop as a result of inter-particle hydrodynamic interactions such asDKT, leading either to clustering or dispersion [77, 16, 74], whereas dynamic gas-solid systems retain their random particle distributions [122, 124]. Two of theaforementioned factors are readily obtained and analyzed in this work; namely, themicrostructure and velocity fluctuations. First of all, as cubes are less likely to re-main in clusters, the extended horizontal accumulation that enhances drag in caseof spheres is extremely weak in cube suspensions, if detectable at all (e.g. see fig-ure 3.9). In order to demonstrate this point more clearly, we present broader views(i.e. up to r/D = 8) of the pair distribution functions in figure 3.13 for two solidvolume fractions of φ = 0.1 and φ = 0.2, respectively. As evident in figures 3.13aand 3.13b, high density sectorial regions between pi/3 6 θ 6 pi/2 can be found insphere suspensions at Ga = 160 even in the densest suspension, extending all theway up to r/D = Lx. In cube suspensions, though, any horizontal accumulation of88neighbors is hardly seen beyond r/D = 2. These observations indicate that in cubesuspensions, there is a considerably lower probability of finding neighbors orientedhorizontally compared to sphere suspensions. The sharp peaks in radial distribu-tion functions, together with high density regions in the horizontal direction forsphere suspensions evident for φ > 0.05 and both Galileo numbers in figures 3.9and 3.10, and consequently the time evolution of local solid volume fraction infigure 3.11 overall point to the fact that spheres cling to one another horizontallyfor long periods of time, a behavior that amplifies their drag force significantly, butis not found as prominently in cube suspensions. In section 3.3.2, we presentedparticle velocity fluctuations for the present simulations. Even though the veloc-ity variances are generally higher for cubes compared to spheres, the differencesare most striking in the transverse direction, and for more dilute cases. Moreover,Tavanashad et al. [82] noted that while in homogeneous suspensions higher fluctu-ations enhance the drag, this is not necessarily true for clustered suspensions. Thefinal factor, i.e. the mobility of cubes in terms of their adaptation to the flow fieldhas not been studied in this work. We speculate that freedom of cubes to translateand rotate in response to the flow changes should have a greater influence towardsthe reduction of drag compared to similar mobility effects in sphere suspensions[82], thus explaining the lower-than-expected drag of dynamic cube suspensions.This might also be linked to the orientation of each individual cube, which can onaverage be such that a nearly face-normal orientation occurs more frequently in adynamic suspension than a completely random orientation in a fixed bed of cubes.Nevertheless, mobility and its ensuing effects in cube suspensions certainly needfurther specifically targeted investigations in order to firmly establish the mecha-nisms underlying the drag data trend seen in the results of this section.3.4 Discussion & final remarksA cube is an interesting shape for studying the effects of non-sphericity in sus-pension hydrodynamics. While manifesting rich dynamics [12], the cube repre-sents a class of geometric shapes, i.e. regular polyhedrons, for which specifyingonly one parameter suffices to fully define its geometry. Similar to the case ofthe sphere, this feature makes the exploration of the parameter space for a cube, or89any other regular polyhedron for that matter, considerably easier compared to othernon-spherical shapes such as cylinders or spheroids. The existence of sharp edgesand rapid variation of hydrodynamic loads upon changes in its orientation makethe cube prone to strong translational and rotational displacements and velocityfluctuations. In this investigation, we have simulated monodisperse liquid-solidsuspensions of cubes and spheres with a solid-to-fluid density ratio of m = 2,for two Galileo numbers of Ga = 70 and Ga = 160 and a range of solid vol-ume fractions of 0.01 6 φ 6 0.2. Through detailed microstructure analysis, wehave found that while suspensions of cubes bear qualitative structural similarity tosphere suspensions, cubes appear to be less likely to stay in clusters. In line withthis observation, we have shown that horizontal particle velocity fluctuations arein all cases significantly higher in cube suspensions, due mainly to their higherrotation rates and the ensuing Magnus forces. At the highest Galileo number ofGa = 160 and most dilute case of φ = 0.01, we found long vertically alignedparticle clusters that stand out remarkably in sphere suspensions, but less conspic-uously in cube suspensions. These columnar structures as well as the observedhorizontal clusters arise due to the occurrence of the DKT mechanism. While thecolumnar arrangement quickly fades away with the increase of solid volume frac-tion owing to frequent wake disruptions by nearby neighbors, horizontal clusterspersist for all concentrations. In contrast, there is a remarkable depletion of spheresin the wake of a reference sphere for Ga = 70 and φ = 0.01, which we attributeto a repelling lift force induced by the wake shear region. In all cases, preferen-tial concentration is unanimously less intense in cube suspensions, whether it bean accumulation or a deficit of particles in a certain region. In a sense, one mayconclude that cube suspensions remain more homogeneous or better mixed. Sucha conclusion bears important implications in real-life applications of particle-ladenflows. As an example, in situations where particle agglomeration is to be avoidedin order to enhance mixing or maximize surface transfer properties, angular shapesfor the dispersed phase would be a better choice.The effect of particle rotation on characteristics of sphere suspensions was firstinvestigated by Kajishima [106], where it was shown that irrotational spheres de-velop and maintain clusters, whereas rotational spheres tend to break up clusters90frequently as a result of the Magnus lift force and horizontal fluctuations. Our ob-servations with cube suspensions appear to be well aligned with the conclusionsdrawn by Kajishima [106]. We have demonstrated that compared to spheres, cubeson average rotate with considerably higher angular velocities, specifically at lowersolid volume fractions and higher Galileo numbers (e.g. Ωh up to three timeshigher in cube suspensions versus sphere suspensions for (Ga, φ) = (160, 0.01)).Such high rotation rates are accompanied by similarly high values of horizontalparticle velocity fluctuations, conceivably resulting from the induced Magnus liftforces. The existence of stronger transverse forces promotes the chances of escap-ing clusters for cubes, which we believe is the underlying reason for the remarkablymore homogeneous structure of cube suspensions in comparison with sphere sus-pensions. Furthermore, we also find that cubes are more effective in transferringmomentum from the gravity to the horizontal direction, as shown by the ratio ofvertical to horizontal particle velocity fluctuations. The strong transverse motionsof cubes, which are caused both by rotation-induced and orientation-induced liftforces, not only homogenize the suspension structure but also make particle mo-mentum properties more isotropic compared to sphere suspensions.It is well established in the literature that microstructure undoubtedly affectsthe average drag force experienced by particles in a suspension. We confirmedin the present study that drag on sphere suspensions can be even lower than theirfixed-bed counterparts, (e.g. for (Ga, φ) = (160, 0.01)) due to the shielding effectcaused by the strong vertical clustering, whereas it can also be higher than fixedbeds when particles aggregate horizontally. Except for (Ga, φ) = (160, 0.01),however, drag of cube suspensions appears to closely follow that obtained fromfixed-bed simulations with the same parameters and random particle distributions.The greater homogeneity of cube suspensions is certainly an important factor re-sponsible for this interesting and rather unexpected behavior. We have depicted infigures 3.13a and 3.13b the marked presence of wide horizontal accumulation ofspheres especially at Ga = 160, while such regions can be hardly detected in cubesuspensions. Nonetheless, one might still anticipate higher drag for dynamic cubesuspensions compared to fixed beds of cubes, given the higher velocity fluctuationsand that the microstructure is still non-random. However, it was recently shown by91[82] that velocity fluctuations do not necessarily increase drag in clustered suspen-sions. Another factor that needs to be taken into account is the motion freedomof cubes in response to the flow in dynamic cube suspensions. Similar to the find-ings of Rubinstein et al. [123] in Stokes conditions, Tavanashad et al. [82] havereported that mobility of spheres always decreases drag also at high Reynolds num-bers. For cubes, mobility might also result in more frequent occurrence of a certainorientation that might cause less resistance to the flow around the cubes. Conse-quently, since the effect of horizontal clustering and higher velocity variance is toenhance the drag, we suggest that the effect of motion freedom for cubes shouldhave a stronger influence in drag reduction in dynamic cube suspensions comparedto sphere suspensions. This matter, along with several others that have not beenaddressed in the present study need to be considered in future works concerningsuspensions of cubes.92Chapter 4Microstructure-informedprobability-driven point-particlemodel14.1 IntroductionThe ubiquity of flows where a dispersion of particles is carried by a fluid phasehas motivated many theoretical, experimental and numerical studies. Such sys-tems, also referred to as particle-laden flows, are widely encountered in naturaland industrial settings. Sediment transport, rain and drop formation, fluidized bedsand slurry flows are only a few examples among many where particle-laden flowsoccur. Analytical treatment of these systems is generally limited to asymptoticcases with very low Reynolds numbers and solid volume fractions, whereas thepractical interest usually lies in the opposite end of this spectrum. Conversely, theunprecedented availability of massive computational power in the past two decadeshas substantially promoted the utility of numerical methods for investigation ofparticle-laden flows.1A version of this chapter has been published in Journal of Fluid Mechanics [103].93In a dispersed multiphase system, particles interact with each other locally onthe length scale of the particle diameter d, which could lead to the occurrence ofclose-range phenomena such as Drafting-Kissing-Tumbling (DKT) due to wakeattraction in inertial regimes [76, 125]. Interactions of this nature are known tocontribute to the formation of particle clusters that extend several diameters [70,16, 81]. These structures, in turn, potentially interact with each other and thusbring about an integral length scale that could be one or two orders of magnitudelarger than d. Clustering of water droplets in clouds is an interesting natural ex-ample, which is known to greatly enhance coalescence and thereby explaining thegrowth rate of droplets [126]. On the other hand, clusters have a remarkable ef-fect on particle residence time, and heat and mass transfer in circulating fluidizedbeds [127, 128]. A fluidized bed reactor in a Fluid Catalytic Cracking (FCC) unitis typically 14 m high and 6 m in diameter. Within such a device, fluid-solid in-teractions and particle collisions in the sub-millimeter scale can directly influenceflow structures such as clusters in the order of meters [13]. The enhancement oflocal concentration of particles in clusters also leads to increased collision rates.Clearly, the time scale of particle collisions is much shorter than convective or dif-fusive time scale of the flow, creating yet another level of scale separation. Thecascade of such multi-scale interactions modifies the character of particle suspen-sions and significantly affects quantities of interest, such as settling rate and particlevelocity fluctuations [77, 101, 66]. Due to the presence of a wide range of lengthscales, a single description of the physics would fail to provide balance betweenthe required complexity or resolution, and the associated computational cost forall involved scales. This is the motivation for development of various numericalmethodologies aiming at resolving the flow at three major length scales; namely,the micro, meso and macroscale. Such a multi-scale methodology is schematicallyshown in figure 4.1.A micro-scale flow description assumes that particles are at least an order ofmagnitude larger than fluid grid cells, meaning that the fluid-solid interface is wellresolved. Since the hydrodynamic forces and torques can be directly computedfrom the integration of pressure and viscous stresses available as field variables,transfer of momentum between phases involves no approximation or modeling, and94MacroscaleMicroscale MesoscaleFigure 4.1: Depiction of the concept of multiscale strategy in modeling particle-laden flowsis dictated by satisfying the no-slip condition on the solid boundary. Micro-scalemethods (i.e. PR-DNS) were first introduced in the framework of body-conformingmoving-mesh methods [129, 130, 131] where Navier-Stokes equations are solvedon the fluid mesh, and particles are treated as boundaries of the flow. As the systemevolves, frequent re-meshings become inevitable so as to adapt the computationalgrid to the new configuration of the fluid-solid system, rendering such methodsinefficient for simulating particle-laden flows. Fixed-grid methods, also collec-tively called Fictitious Domain Methods (FDM) [3], were proposed to relieve thisburden by extending the fluid domain to include the particles and solving Navier-Stokes equations over the entire fluid-particle domain, thus completely eliminat-ing the need for re-meshing. Immersed Boundary Methods (IBM) [132, 133] andDistributed Lagrange Multiplier/Fictitious Domain (DLM/FD) methods [96, 57]are two such techniques which differ in the enforcement of rigid body motion inthe particle domain and in the computation of interaction forces. Another popu-lar class of PR-DNS tools are the Lattice Boltzmann Methods (LBM) which arebased on kinetic theory of gases [134]. A significant number of fixed-bed stud-ies of particulate flows in the literature are carried out using LBM [117, 115, 135,113, 114]. Assuming particles of typical size in the range of [200 µm, 1000 µm],the micro-scale approach is suitable for simulating laboratory scale devices witha size of O(0.01 m) [13], or O(103 − 104) particles in the context of gas-solidfluidization. Although PR-DNS provides a complete and model-free description ofparticle-laden flows, handling systems at the intermediate (i.e. meso) scale with aphysical size of O(0.1 m) [13] containing O(105− 106) well-resolved particles [6,95136] has only recently been feasible with PR-DNS, and requires massively parallelcomputing resources [137, 138, 139, 140]. The EL technique, on the other hand,attempts to reduce computational costs by taking the volume of each fluid cell tobe generally an order of magnitude larger than that of an individual particle or atleast to be of the order of magnitude of that of an individual particle. Becauseflow in the immediate vicinity of the particles is not resolved, direct computationof hydrodynamic interaction force and torque is not possible. Consequently, in-terphase coupling needs to be established via a suitable force closure model. Thefluid phase sees particles only as point sources and sinks of momentum, with thevolume of particles appearing solely through the local porosity in volume-averagedmass and momentum conservation equations and the employed drag force corre-lation [141]. In such an approach, volume-averaged Navier-Stokes equations aresolved on an Eulerian grid for the fluid phase, whereas positions of the particlesare tracked using Newton’s equations of motion in a Lagrangian manner. For thecase of dense suspensions, this is usually handled by the Discrete Element Method(DEM) with a proper contact model accounting for particle collisions (and hencethe name “DEM-CFD”). Since particles are treated as points suspended in fluidcells, the EL method is also referred to as the Point-Particle (PP) method. This ter-minology, however, may not be descriptive enough for the newer EL methods thatare capable of simulating finite-size particles with a diameter of the order of fluidcells [142]. Even with EL methods, modeling engineering scale pilot devices with asize of O(1 m) containing O(109) particles is impractical at present, since trackingbillions of individual particles in a Lagrangian manner poses a serious computa-tional challenge. With that being the case, one could alleviate this issue by account-ing for the presence of particles indirectly. The Two-Fluid Model (TFM) [110] isa numerical model employing an Eulerian-Eulerian (EE) approach, where the fluidand the solid are both assumed to behave as interpenetrating continua. Since thesolid phase is also modeled as a fluid continuum, details of particle-particle inter-actions are embedded in the effective solid pressure and shear and bulk viscosityclosure terms. The two phases, in turn, interact through an appropriate drag model[13, 141].The up-scaling of simulations can be a viable alternative for PR-DNS only if96the closure models to be used are sufficiently accurate and faithful to the actual un-derlying physics. In an EL simulation, the governing equations have to be closedwith appropriate terms accounting for the fluid-solid momentum transfer. Multi-phase flow modeling enters the stage to bridge the gap between PR-DNS and theEL approach by supplying this missing piece of crucial information. The classicalStokes drag F = 3piµdu is only valid at Re = 0 for a steady uniform flow arounda fixed sphere. When the macroscale undisturbed flow1 u is no longer steady, onecan use the Basset-Boussinesq-Oseen relation given asF = Fun + FD + Fam + Fh (4.1)where the terms on the right hand side account for the undisturbed flow force, thequasi-steady drag, added-mass and Basset history forces, respectively. The ele-gant Faxén’s law [145] makes it possible to extend the validity of equation (4.1)to spatially non-uniform flows by replacing the undisturbed flow u by us and uv,i.e., the average value over the particle’s surface or volume. This formulation wasderived by Maxey and Riley [146] and Gatignol [147] and is rigorously valid in theStokes limit. When particles are much smaller than the macroscopic length scaleof the flow, the particle Reynolds number based on relative slip velocity typicallybecomes very small. This is the case when a dilute dispersion of particles are sus-pended in a turbulent flow such that d/η  1, where η shows the Kolmogorovscale. In this situation, the Maxey-Riley-Gatignol (MRG) equation accurately pre-dicts hydrodynamic forces experienced by the particle. For inertial regimes withfinite Reynolds numbers, force contributions in MRG equation should be modifiedand empiricism is inevitable due to a lack of theoretical analysis. The standarddrag curve of an isolated sphere given as CD = (24/Re)(1 + 0.15Re0.687) [109]is such a modification that characterizes the Reynolds number dependence of thequasi-steady drag term FD in the MRG equation. For Re > 0, ambient shear orvorticity in the background flow gives rise to the Saffman lift force [148], which isabsent in the MRG equation since there is no such lift force in Stokes flow [146].Furthermore, rotation of a particle induces an excess lift even in uniform flow,1The undisturbed flow is defined as the flow that would have existed in the absence of a particle[143, 144].97which is attributed to the Magnus effect [149]. The Magnus force should as wellbe included as an additional term alongside the Saffman lift in the MRG equationwhen inertial regimes are considered. The interested reader is referred to [11] fora detailed overview of various approaches of modeling dispersed multiphase flow.The PP approach faces a serious challenge when the size of particles becomescomparable to the scales of the macroscopic flow. When the carrier flow is alreadyturbulent, this occurs when d/η ≈ 1. The other scenario is when the particle con-centration increases and the suspension can no longer be regarded as dilute. Evenat a solid volume fraction of 1%, the average distance to the closest neighbor isonly 3.7d [7]. The likelihood of hydrodynamic interaction hence increases sub-stantially. The disturbances created by the particles results in the appearance ofpseudo-turbulence, i.e., a non-uniform flow that not only varies spatially on thescale of the particle diameter, but is also potentially prone to temporal fluctuationsfor high enough Re. Extending the PP model to finite-size particles is hamperedwith significant complications associated with the application of analytical or em-pirical force relations, the expression of which are all in terms of undisturbed flow.In the context of finite-size particles, the undisturbed flow is difficult to obtainsince it requires evaluating the same system but with a specific particle removed.Moreover, now that the particle is relatively large, the undisturbed flow varies onthe particle scale. How such a spatially non-homogeneous flow would affect theresulting forces at finite Re remains an open question [143].Parameterizing the drag law in terms of solid volume fraction, in addition to theReynolds number, has been the first step towards the prediction of particle-ladenflow behavior where collective effect of particles cannot be neglected. Theoreti-cal studies are limited to the Stokes flow conditions and very dilute suspensions;namely, Re → 0 and φ → 0 [150, 68, 151]. Proposed correlations by [107] fordilute suspensions and Ergun’s equation [112] for denser systems are the earliestexperimental efforts in this regard. Exponential growth of computing power in thelast two decades has made PR-DNS a preferable alternative for developing moreaccurate drag correlations over wider ranges of Re and φ. Contrary to the exper-imental approach, arbitrary ideal flow conditions can be imposed, and forces arecomputed directly in PR-DNS [116] instead of indirect measurement based on set-98tling velocity [69] or pressure drop [112]. Among PR-DNS techniques, LBM hasbeen the method of choice in numerous studies on drag correlation. [114] simu-lated a fixed bed of spheres with ordered and random arrangements up to the closepacking limit at low Re, and later for moderate Re [113]. Bi-dispersity in randomarrays of fixed spheres was also addressed at very low Re using LBM by [135], andsubsequently for Re up to 1000 by [115]. Recently, other drag correlations wereproposed by [117] using LBM, and by [116] and [118] using IBM.Even though the idea of simulating fixed beds of spheres instead of realisticmoving particles is justified by drawing analogy with high Stokes number gas-solidflows, this simplification was challenged by [116]. They showed that at φ = 0.5,deviation between the actual drag experienced by mobile particles and the dragcomputed from static bed correlations is significant. A similar observation hadbeen previously made by [10], that not only the gas-solid forces are underestimatedby conventional drag laws in an EL simulation, but there is also a large scatter ofthe drag data in a PR-DNS of fluidized beds which the EL approach fails to cap-ture. They noted that the higher drag force seems to correlate with local granulartemperature or particle agitation, which in turn is an outcome of subgrid force fluc-tuations. In a recent work, [8] attempted to alleviate the suppression of granulartemperatures by introducing a stochastic component in the drag closure, the pa-rameters of which were extracted from PR-DNS. Comparisons of their stochasticmodel with conventional DEM-CFD indicated better prediction of granular tem-peratures in liquid-solid regimes. Ultimately, the key missing component seems topertain much more conspicuously to the physical fidelity of the drag model ratherthan the accuracy of fixed-bed drag correlations.The preceding discussion signifies that unless the physics is properly accountedfor, any effort towards further improving EL simulations would be futile regardlessof the accuracy of conventional drag correlations. From all the proposed drag mod-els, we know that the mean drag force experienced by the particles demonstrates astrong correlation with solid volume fraction. In other words, the functional depen-dence of the drag model on φ is able to account for the presence of other particles,but only and strictly in an average sense. It does not matter whether a particle isshielded by an upstream neighbor or exposed entirely to the oncoming flow, the99drag law predicts the same force in both situations. Therefore, it would be highlydesirable to construct a force model that is capable of accounting for the specificarrangement of surrounding particles; namely,Fi = f(Re, φ, {rj=1, . . . , rj=M}) (4.2)where Fi is the force experienced by particle i and rj the position vector of neigh-bor j relative to particle i, while M denotes the number of influential neighbors,and f the functional dependence. [7] showed by analysis of PR-DNS data for20 6 Re 6 180 and 0.11 6 φ 6 0.44 that there is a substantial scatter in thehydrodynamic force experienced by individual particles due to the particular ar-rangement of surrounding spheres, and that the local volume fraction had almost nocorrelation with the force fluctuations. By utilizing a simple anisotropic measure ofeach particle’s neighborhood, they were able to capture some of the drag variation,whereas results for the lift force were less accurate. The notable work of [152]and their Pairwise Interaction Extended Point-particle (PIEP) model is the first tosystematically account for the effect of neighboring particles on drag and lift in adeterministic manner. Their model involved linear superposition of perturbationscreated by each neighboring particle in a pairwise manner in order to obtain theundisturbed flow, then using Faxén’s law to compute various contributions to thetotal hydrodynamic force from the non-uniform undisturbed flow. The PIEP modelwas shown to be capable of predicting up to 75% of the drag force variations forthe (Re, φ) = (0.1, 38), whereas for a denser case of (Re, φ) = (0.21, 87), about56% of the variations were captured. For the lift, however, the results were not aspromising. Subsequently, they extended their work to include modeling of hydro-dynamic torques as well, and also tested their model for sedimentation of 2, 5 and80 spheres [153]. Remarkably, they were able to reproduce DKT of two sphereswith the PIEP model in contrast to the inability of standard PP approach to do so.Quite recently, the same group [154] attempted to improve the shortcomings ofthe PIEP model particularly at high volume fractions by combining a data-drivenapproach based on non-linear regression with their original physics-driven model[152, 153]. The resulting hybrid model was shown to considerably enhance theaccuracy of the PIEP model particularly at higher solid volume fractions.100The fast-growing trend of Machine Learning (ML) and data-driven algorithmshas brought new prospects to fluid flow modeling. Data-based methods have al-ready been present in the context of dimensionality reduction techniques and arehence not alien in the fluids community [155]. Neural Networks are a popular sub-set of ML techniques that have been applied with ground-breaking success to image[156] and speech [157] recognition tasks. Neural networks are shown to be univer-sal function approximators [158], and are thus capable of mapping input features tooutput variables in complex multidimensional problems rife with strong inherentnonlinearities. The tempting power of ML has motivated a major effort towardsits applications in fluid flow simulations, especially turbulent flow modeling. Sev-eral works have focused on improving closure terms in the widely-used Reynolds-Averaged Navier-Stokes (RANS) models using ML algorithms [159, 160, 161,162], and fewer on Large Eddy Simulations modeling (LES) [163, 164, 165, 166,167]. There has also been attempts in the context of multiphase flows to developclosure models for TFM simulations [168, 169]. Another direction pursued bysome researchers is to take advantage of Convolutional Neural Networks (CNN)[170], commonly used in image recognition tasks, for direct approximation of flowfield variables [171, 172]. Despite its seeming success, a crucial fact about ML isthat its high accuracy and flexibility is achieved at the expense of supplying largeamounts of data for the purpose of training. ML algorithms such as neural networksrequire large volumes of data to find fitting functions through adjustment of theirparameters (i.e., weights and biases). This process, also known as the “learning” or“training” phase, is an iterative optimization procedure aimed at minimizing errorsbetween the real data and those predicted by the algorithm. This approach workssuccessfully for computer vision tasks, for example, due to an abundance of labeleddata and the interpolatory nature of the problem [173]. However, the applicationof off-the-shelf ML techniques to flow dynamics problems (or any other physicalphenomena for that matter) inevitably suffers from non-interpretability in terms ofgoverning physics due to ML’s “black-box” nature. Another issue that practicallyimpedes ML application in physical systems such as particle-laden flows is thatwe can generate, at best, no more than a few thousand or a few tens of thousandsof samples (e.g. force and torque on each particle) with PR-DNS. Consider thateven for the most idealized case of mono-dispersed spheres, we would still have to101sweep the parameter space of Re and φ. In such a “small data” regime [174], thefull power of neural networks may not be exploited unless the physics equationsare incorporated, or “hard-coded” in the structure of the algorithm. This can beachieved by directly minimizing residuals of the governing equations through lossfunctions and hence ensuring the physical fidelity of the predictions, giving rise to“physics-informed” deep learning algorithms [174, 175, 176]. Even if successful inprediction (within the range of training dataset at best), the initial spark of a theory-blind conventional ML model quickly fades away since even at its peak, it still isa “glorified curve-fitting” procedure [177] and without guiding theory, such pureempiricism fails to provide knowledge [178]. A physics-informed ML approach[174, 175, 176] offers advantage in that respect due to having physical fidelity en-graved in its core. Even if an accurate ML model is at hand, there exists anotherissue in the framework of EL simulations. In the DEM-CFD approach, the hydro-dynamic force on each particle has to be evaluated at each time step. A deep neuralnetwork typically runs the input through hundreds or even thousands of pre-tunedparameters to output a single prediction, incurring a significant computational coston the EL simulation.In the present work, we propose a novel data-driven model that relies onforce/torque-conditioned probabilities of particle arrangements extracted from PR-DNS in order to correlate hydrodynamic forces and torques to the unique neighbor-hood of each particle. In the remainder of this work, we refer to our model as themicrostructure-informed probability-driven point-particle (MPP) model. We willprovide probabilistic arguments for the prediction of force/torque fluctuations, andapply our method to the data from PR-DNS of fixed beds of randomly distributedspherical particles at various Reynolds numbers and solid volume fractions. Aftervalidating our results with the existing literature, we then evaluate the accuracyof our model’s predictions and demonstrate its performance by providing compar-isons with PR-DNS force and torque data.1024.2 PR-DNS of fixed beds of spheres4.2.1 Governing equationsConservation of momentum and mass for the fluid phase is expressed in termsof incompressible Navier-Stokes and continuity equations for a Newtonian fluid asfollows:∂u∂t+ u · ∇u = −∇p+ 1Re∇2u (4.3)∇·u = 0 (4.4)where u, p and Re respectively denote the fluid velocity vector, pressure and parti-cle Reynolds number, defined asRe =ρUdµ=ρ(1− φ)usdµ(4.5)where ρ, µ are the fluid phase density and dynamic viscosity. The particle Reynoldsnumber is defined based on the superficial velocity U = (1 − φ)us, and us repre-sents the average interstitial fluid velocity. In equations (4.3) and (4.4) and whatfollows, all variables are non-dimensionalized with respect to the particle diameterd as the length scale, U as the velocity scale, ρU2 as the pressure scale and ρU2d2as the force scale. The hydrodynamic force and torque exerted on each particledenoted by F and T are given asF =∫S[−pI + 1Re(∇u +∇uT )]· ndS (4.6a)T =∫Sr×[−pI + 1Re(∇u +∇uT )]· ndS (4.6b)with S denoting the surface enclosing the solid body, I the identity matrix, ( . )Tthe matrix transpose, n the unit vector normal to the boundary of the solid bodyand r the position vector relative to the particle center of mass.1034.2.2 Numerical methodAs our PR-DNS tool, we use PeliGRIFF (Parallel Efficient Library for GRainsin Fluid Flow) which is a multiphase flow solver based on the Distributed LagrangeMultiplier-Fictitious Domain (DLM-FD) formulation proposed by [57]. In ourimplementation, we employ a finite-volume staggered-grid scheme for the fluidconservation equations [58]. Similar to IBM, particles are immersed in the fluiddomain in the DLM-FD method and rigid body constraints on the fictitious fluidinside the solid region are enforced through a set of Lagrange multipliers collocatedin the particle domain. For a fixed array of particles, the combined momentum andcontinuity equations in a nonvariational form are given as∂u∂t+ u · ∇u = −∇p+ 1Re∇2u− λ in D, (4.7a)∇ · u = 0 in D, (4.7b)u = 0 in P, (4.7c)where the solid domain is denoted by P and the fluid-particle domain byD. Further-more, λ shows the distributed Lagrange multiplier vector which is used to enforcethe rigid-body motion constraint in equation (4.7c). For the temporal discretiza-tion, we employ a first-order Marchuk-Yanenko operator-splitting algorithm. Ateach time tn+1, we solve:1. A classical L2-projection scheme for the solution of the Navier-Stokes prob-lem: find un+1/2 and pn+1 such thatu˜n+1/2 − un∆t− 12Re∇2u˜n+1/2 =−∇pn + 12Re∇2un − 12(3un · ∇un − un−1 · ∇un−1)− λn,(4.8a)∇2ψn+1 = 1∆t∇ · u˜n+1/2,∂ψn+1∂n= 0 on ∂D,(4.8b)un+1/2 = u˜n+1/2 −∆t∇ψn+1, (4.8c)104pn+1 = pn + ψn+1 − ∆t2Re∇2ψn+1. (4.8d)2. A fictitious domain problem: Find un+1and λn+1 such thatun+1 − un+1/2∆t+ λn+1 = λn, (4.9a)un+1 = 0 in P. (4.9b)where ∆t denotes the time step, ψ the pseudo-pressure and ∂D the domainboundary. In equation (4.8a), second-order in time Crank-Nicolson and Adams-Bashforth schemes are used to discretize the viscous and advective terms, respec-tively, and the saddle-point problem in step 2 is handled by an Uzawa algorithm[58]. Considering the high-order correction of the pressure, the projection schemein step 1 is also second-order accurate in time. Nonetheless, the first-order timediscretization of the fictitious domain sub-problem in step 2 and the first-orderMarchuk-Yanenko method reduce the global time accuracy of our algorithm tofirst-order only. Equations presented in step 1 are spatially discretized with asecond-order central scheme for the diffusion term, whereas the advective termis treated with a Total Variation Diminishing (TVD) scheme combined with a Su-perbee flux limiter. Despite the second-order discretization of the flow equations,the accuracy of our method is between first and second order due to the presence ofrigid bodies immersed within the domain. It can be shown that with the DLM-FDmethod, the Lagrange multiplier λ can be directly integrated over the volume ofeach particle to obtain the hydrodynamic force and torque acting on the particle P:F =∫Pλ dx, (4.10a)T =∫Pr× λ dx. (4.10b)105x yzFigure 4.2: Fixed beds of spherical particles at Re = 40 and solid volume fractionsof φ = 0.1 (left) and φ = 0.4 (right). The streamlines shown are colored withrespect to the fluid velocity magnitude.4.2.3 Simulation setupA summary of all considered cases is given in table 4.1. For our PR-DNSsimulations, we consider triply-periodic cubic domains of edge lengthL containingNp spherical particles each taking up a volume of vp = pi/6, corresponding toa solid volume fraction of φ = Nvp/L3. A constant flow rate so as to attainthe desired Reynolds number is imposed in the x direction using a dynamicallyadjusted pressure drop. The x direction hence corresponds here to the stream-wisedirection. As noted by [7], [118] demonstrated that using a domain size of only2.4d guarantees the decorrelation of fluid velocities for (Re, φ) = (20, 0.2) and(Re, φ) = (300, 0.2). Our computational domains given in table 4.1 all extend farbeyond 2.4d containing ≈ 2500− 3000 particles each, ensuring both the decay offluid correlations and statistical reliability. Initialization of particle locations forcases where φ ∈ {0.1, 0.2} is performed by distributing spheres randomly in thedomain according to a random number generation algorithm without allowing anyoverlap. For the highly dense cases with φ = 0.4, we start with a structured array ofparticles where each particle is given a random translational and rotational velocity,and we let the system reach an asymptotic motionless state through dissipativecollisions. Visualizations of two sample cases in the present study are shown infigure 4.2. For all solid volume fractions, pair correlation functions have beenobtained and verified to be satisfactorily matching theoretical radial distribution106φ Re d/∆x L Np0.1 2 24 25 29840.1 10 24 25 29840.1 40 24 25 29840.1 150 32 25 29840.2 2 24 20 30550.2 40 32 20 30550.2 150 40 20 30550.4 2 32 15 25780.4 40 40 15 25780.4 150 48 15 2578Table 4.1: Summary of the parameters used for PR-DNS of fixed beds of spheresfunctions of hard spheres [179, 105]. The time-step is taken to be smaller than∆t = 2 × 10−3 in all cases, ensuring time accuracy of the simulations along withsatisfying the Courant-Friedrichs-Lewy (CFL) condition; namely, CFL < 0.4 forthe spatial resolutions presented in table 4.1. The simulations have been run untilsteady-state is achieved, and force and torque data are all collected from the steadypart. For Re = 150 at all volume fractions, the flow becomes time-dependent andoscillating. This occurs because particles in close proximity act as a single obstacleand hence increase the effective length scale and the Reynolds number. Data fromthese cases is collected after a statistically pseudo-stationary state is establishedby averaging over sufficiently extended time intervals. In terms of computationaldemand, the mesh resolution and the domain size in each case along with the loadper CPU core dictate resource requirements. With each processor core handling512×103 grid cells, the smallest simulations with≈ 110×106 grid cells were runon 192 cores, whereas the largest ones with 512×106 grid cells required 960 cores.All computations were carried out on groups of 48-core nodes each equipped with≈187 GB of memory, provided by the Cedar supercomputer as a part of ComputeCanada’s advanced research computing infrastructure1.1http://www.computecanada.ca1074.2.4 ValidationThe aim in this section is to provide a comparison of the results of our PR-DNS simulations with the reported drag correlations in the literature [117, 116,118, 115]. In terms of spatial resolution, our PR-DNS is comparable with thework of [117] and [118] for the range of (Re, φ) considered here, but generallybetter resolved than simulations done by [116, 115]. Nevertheless, validations arepresented merely to establish the accuracy of our code rather than intending tooffer benchmarking data. The ensemble average drag is obtained by summing thedrag force acting on each particles in the bed and dividing by the total number ofparticles Np:〈Fd〉 = 1NpNp∑i=1Fi · eˆx (4.11)Obviously, the normal components of the mean force 〈Fy〉 = 〈F · eˆy〉 and 〈Fz〉 =〈F · eˆz〉 are both expected to be vanishingly small. In order to compare our datawith the available drag correlations, normalization of forces is done with respect tothe Stokes drag given asFst = 3piµdU (4.12)As shown in figure 4.3, the drag computed for our cases listed in table 4.1 all liewithin the range of existing correlations and generally indicate good accordance.In particular, our data seems to agree well with correlations proposed by [117] and[180] at Re = 150 where significant discrepancies are observed between differentcorrelations. In addition, we have also examined force distributions and the extentof data scatter for each case in table 4.2. The magnitude of the ensemble averagelift force given as 〈FL〉 = 〈Fy〉 or 〈Fz〉 is practically zero for all cases, as expected.However, the ratio of the standard deviation of drag and lift shown by σFd/〈Fd〉and σFL/〈Fd〉 are both significant and σFd/〈Fd〉 turns out to be always greaterthan σFL/〈Fd〉. In cases where a rough comparison is possible, our results forσFd/〈Fd〉 and σFL/〈Fd〉 in table 4.2 show a difference of ≈ 2% − 4% with thoseobtained by [7]. A contributing feature that might explain the difference is that our108      Re〈 F d〉/Fstφ=0.1%HHWVWUDHWDO7HQQHWLHWDO7DQJHWDO%RJQHUHWDO3UHVHQWZRUN      Re〈 F d〉/Fstφ=0.2      Re〈 F d〉/Fstφ=0.4Figure 4.3: Validation of the drag data obtained from PR-DNS of the present workwith correlations of [115], [118], [116] and [117]sample sizes for each case are significantly larger by a factor of 5 − 6, renderingthe results statistically more converged. This is obvious from the 〈FL〉 values intable 4.2 compared to those reported by [7], as their mean lift data (while still quiteclose to zero) are orders of magnitude greater than present results. According to[7], realization dependence alone could cause variance in σFd/〈Fd〉 and σFL/〈Fd〉of up to 2.7% and 1.7%, respectively.109φ Re 〈Fd〉/Fst σFd/〈Fd〉 〈FL〉/Fst σFL/〈Fd〉0.1 2 2.89 21.32% 2.7× 10−10 15.14%0.1 10 3.57 24.86% 1.6× 10−9 14.78%0.1 40 5.56 26.71% 3.0× 10−5 15.19%0.1 150 10.49 26.09% 7.1× 10−4 16.48%0.2 0.2 5.48 17.66% 7.5× 10−5 15.21%0.2 2 5.53 17.56% −9.3× 10−9 14.97%0.2 40 9.59 23.77% 2.3× 10−10 16.33%0.2 150 17.76 25.97% −4.2× 10−6 18.86%0.4 2 18.71 22.96% 3.7× 10−8 18.22%0.4 40 30.99 26.07% 5.3× 10−7 18.84%0.4 150 59.73 27.68% 2.3× 10−5 19.62%Table 4.2: Statistics of the drag and lift data for cases presented in table Dataset constructionFor the purpose of analysis and development of our model, we first need to con-struct datasets corresponding to each simulation. Each dataset contains as manyrows as the number of particles Np (which we call samples), while columns repre-sent the input and output variables. For each particle i, the set of position vectorsof the neighboring spheres denoted by {rj=1, . . . , rj=M} along with the averagefluid velocity ui around particle i constitute the inputs, whereas the hydrodynamicforces and torques experienced by particle i are the outputs that we aim to model.The position vectors of neighbors are expressed relative to the location of parti-cle i. The first 30 nearest neighbors are identified for each particle i by loopingover all other particles in the simulation. These neighbors are then numbered fromj = 1 to j = 30 depending on their relative distance, with j = 1 being the clos-est. It is worthwhile to note that since our simulations are performed in tri-periodicdomains, forces and torques on particles near boundaries could be affected by pe-riodic images of particles on the opposite side of the domain whose positions donot exist in the simulation data. For this reason, prior to performing the neighboridentification loop, we create the periodic image positions manually and explicitlyinsert them in the data so that the periodic effects are correctly accounted for.1104.3 Probability-driven model4.3.1 MotivationAs pointed out earlier, we seek to predict the hydrodynamic forces and torqueson each particle as a function of the flow conditions, and more importantly, ofthe unique neighborhood of each particle as highlighted by equation (4.2). Thehydrodynamic force and torque exerted on each sphere may be decomposed andexpressed as followsFi = 〈Fi〉(Re, φ) + ∆Fi(Re, φ, {rj=1, . . . , rj=M}) (4.13a)Ti = ∆Ti(Re, φ, {rj=1, . . . , rj=M}) (4.13b)where ∆Fi and ∆Ti show the fluctuating contribution to the force and torque dueto specific arrangement of neighbors surrounding particle i. Statistically, the lateralcomponents of 〈Fi〉 and all components of 〈Ti〉 are close to zero for a sufficientlylarge number of particles in each simulation. In the stream-wise direction, theensemble average force 〈Fi〉 is identical for all particles by definition and there-fore only depends on macroscopic variables, i.e. the Reynolds number and solidvolume fraction. Many correlations exist that describe such a dependence as dis-cussed in section 4.1. For this reason, we henceforth focus on ∆Fi and ∆Ti inequation (4.13); namely, the deviations from the mean arising due to the uniquelocal neighborhood of each particle.The key question posed here may be expressed as follows: How does chang-ing the location of each neighbor affect the force and torque experienced by aparticle? The most obvious and naive strategy would be attempting multiple sim-ulations where the location of a single neighbor particle is changed systematicallyand the resulting effect on forces and torques are recorded. The outcome wouldbe similar to, for example, the two-dimensional interaction maps given by [181]for two neighboring cylinders characterizing forces for various locations of a sec-ond cylinder. Assuming a three-dimensional system, adding only one or two moreneighbors to the scenario makes the number of simulations needed for populatingthe modeling dataset an almost intractable computational task. For their physics-111based PIEP model, [152] needed to account for 15−40 closest neighbors to obtaingood accuracy, which in three dimensions translates into 45− 120 input variables,respectively. It is well known that in a data-driven framework, such systems areafflicted with the “curse of dimensionality” [182]. It means that constructing areliable dataset for a regression-type model requires millions of samples for eachcase, while we are only able to model a few thousand particles with PR-DNS atbest. The pairwise interactions and order-invariance approximations utilized by[154] for their data-driven model is a good demonstration of a strategy for reduc-ing the number of input variables to a manageable level.4.3.2 TheoryConsider a monodisperse random array of spherical particles with a volumefraction φ subject to fluid flow with a Reynolds number Re. Let X and Y betwo continuous random variable vectors in a d-dimensional space (X,Y ∈ Rd)with marginal probability density functions (PDFs) fX(x) and fY(y) wherefX(x), fY(Y) : Rd → R. By definition, PDFs in general are required to sat-isfy the following conditions:fX(x) > 0∫fX(x) dx = 1(4.14)where the integration is performed on all possible values of x. The value fX(x)dxrepresents the probability of X falling in the infinitesimal volume dx about x.Therefore, the probability that the random variable X will happen to be in A isgiven byP [X ∈ A] =∫AfX(x) dx (4.15)The same properties similarly hold for fY(y). The expected value of the randomvariable X is the weighted average of all possible values of X, each value beingweighted according to its probability of occurrence [183]. The expected value of112X is thus governed by its probability density distribution fX(x), and is given asE [X] = 〈X〉 =∫x fX(x) dx. (4.16)The probability associated with a random variable may also be conditioned on an-other random variable. For this case, fX|Y(x | y) may be defined as the conditionalPDF of X given that Y assumes a particular fixed value y.E [X |Y = y] = 〈X | Y = y〉 =∫x fx|y (x | y) dx. (4.17)Referring to equation (4.13), the three components of the force/torque fluctua-tions pertaining to particle i are shown by ∆Fi = (∆Fi,x,∆Fi,y,∆Fi,z)T or∆Ti = (∆Ti,x,∆Ti,y,∆Ti,z)T . A particular configuration of the neighborhoodsurrounding particle i is denoted by Ri = {rj=1, . . . , rj=M}. Here, we presentthe analysis only for force fluctuations; namely, ∆F, since the derivation of similarequations is similarly done for torque fluctuations ∆T. From this point onwards,the subscript i is dropped from all variables in order to minimize cluttering in thenotations. Upon setting X = ∆F and Y = R, the following PDFp∆F |R (∆F |R) (4.18)represents the probability distribution of the force/torque fluctuations given the lo-cations of M neighbors. In the above equations, R shows the set of position vec-tors {r1, r2, r3, . . . , rM} belonging to the neighbors numbered according to theirdistance from a reference particle, starting with j = 1 being the closest. Eachdistribution p (∆F |R) is to be obtained from collecting ∆F values experiencedby particles that happen to have the same neighbor configuration R, while no con-straints are imposed on neighbors {rj : j > M} located further away. The ex-pected value of the PDF in equation (4.18) gives a prediction of each fluctuatingforce/torque component as follows:E [∆F |R] = 〈∆F | R〉 =∫ξ p∆F |R (∆F |R) dξ. (4.19)Two extreme cases ofM , the number of included neighbors, are worth elaborating:113• M = 0: This means that no conditions are imposed on p, hence E[∆F ] = 0and the predicted force/torque is the same for all particles in the array. Sucha situation corresponds to, for instance, the conventional microstructure-ignorant correlations of the form F · eˆx = 〈Fx〉(Re, φ) in case of the dragforce.• M → Np−1: The positions of all neighbors in the particle array are imposedas a condition on p, hence the variance of p tends to zero and the value of∆F is uniquely determined.Since the problem is fully constrained in the latter case, the predicted value of ∆Fwould be free of uncertainty. Put otherwise, p would no longer be a distributionper se, but rather resembling a Dirac delta function instead:limM→Np−1E [∆F |R] =∫ +∞−∞ξ δ∆F |R (ξ −∆FDNS |R) dξ = ∆FDNS (4.20)which is equivalent of having a dataset populated with an infinite number of sam-ples. In other words, this “ideal” dataset would contain samples for each and ev-ery combination of neighbor locations. Such a dataset would conceivably providea full and exact description of the problem, thus representing a solution to theNavier-Stokes equations. Consequently, equation (4.18) can be viewed as a data-driven description of forces/torques exerted on each particle within an array derivedfrom probabilistic arguments, which is potentially capable of predicting fluctua-tions with varying degrees of accuracy, depending of how constrained we force pto be. It is crucial to realize that unless M = Np − 1, our best estimate of thefluctuating component E [∆F |R] would be an average (i.e., the central tendency)of the distribution, implying that for M < Np − 1, influence of neighbors whosepositions are not constrained is only accounted for in a statistical sense. Havingestablished the notion and properties of the conditioned force/torque PDFs, theprobability-driven prediction of a fluctuating component based on its correspond-ing distribution is given as∆FMPP = E [∆F |R] =∫ +∞−∞ξ p∆F |R (ξ |R) dξ. (4.21)114The integral above may be split to cover the positive and negative force/torquecontributions separately; yielding∆FMPP =∫ 0−∞ξ p∆F |R (ξ |R) dξ +∫ +∞0ξ p∆F |R (ξ |R) dξ. (4.22)The impeding hurdle with equation (4.22) is that computing ∆FMPP requires apriori knowledge of the force/torque distributions for each and every possible con-figuration R. Such an approach is clearly not feasible and certainly would notfulfill the goal of constructing a model based on a limited dataset. An elegant alter-native is to take advantage of Bayes’ theorem for probabilities, which serves to con-vert a conditional probability problem to its reverse case. According to the Bayes’formula for distributions of random variables [184], fX|Y(x | y) = fY|X(y |x)fX(x)/fY(y). Therefore, it follows thatp∆F |R (∆F |R) =pR|∆F (R |∆F ) p∆F (∆F )pR (R)(4.23)Substituting for p∆F |R (ξ |R) in first integral in equation (4.22) yields1pR (R)∫ 0−∞ξ pR|∆F (R | ξ) p∆F (ξ) dξ (4.24)Assuming the continuity of both pR|∆F and p∆F and realizing that pR|∆F , p∆F >0, the generalized mean value theorem for integrals can be applied to the aboveequation to giveα p∆F (α)pR (R)∫ 0−∞pR|∆F (R | ξ) dξ =[α p∆F (α)pR (R)]pR|∆F (R |∆F < 0) (4.25)where ξ = α is a constant value that belongs to the interval (−∞, 0]. Following thesame operations for the second integral of equation (4.22), there exists a value ξ =β in [0,+∞) for which ξ p∆F (ξ) can be brought out of the integral. Substituting115results back in equation (4.22), we arrive at the following:∆FMPP =[α p∆F (α)pR (R)]pR|∆F (R |∆F < 0) +[β p∆F (β)pR (R)]pR|∆F (R |∆F > 0)(4.26)Note that according to equation (4.17), pR|∆F (R |∆F < 0) gives the PDF ofparticle positions provided that the force/torque fluctuations obey ∆F < 0 (sameapplies to pR|∆F (R |∆F > 0) when ∆F > 0). Since particles are assumed to berandomly, yet uniformly distributed, pR (R) is expected to have an approximatelyconstant distribution. This can be verified by inspection of unconditioned parti-cle location PDFs for various neighbors. Therefore, both terms inside the squarebrackets in equation (4.26) are constant values and can be expressed as two coeffi-cients cα and cβ . The final form of the force/torque model reads∆FMPP (R,Re, φ) = cα p˜ (R |∆F < 0) + cβ p˜ (R |∆F > 0) (4.27)where p˜ represents the conditional PDF pR|∆F . It should be noted that the p˜ prob-ability maps depend on the Reynolds number and solid volume fraction, which iswhy ∆FMPP is shown as ∆FMPP (R,Re, φ). Thus far, no approximation wasinvolved in the derivation of ∆FMPP , which means that in case a full knowledgeof the aforementioned distributions exists, equation (4.27) would be theoreticallyexact. Practically, however, for a dataset with a few thousand samples the probabil-ity distributions p˜ (R |∆F < 0) or p˜ (R |∆F > 0) would be extremely sparse, asalluded to in section 4.3.1. Taking N and D to denote sample size and number ofinput dimensions, the data density would be proportional toN1/D. AsD increases,it is straight-forward to see that maintaining a constant sample density requires ex-ponentially more data points. In our case, each neighbor adds three inputs to theproblem, each representing one component of the position vector rj = (xj , yj , zj).If, for instance, N1 = 1000 is deemed a sufficiently dense dataset to form reliablelocal averages when D1 = M × 3 = 1 × 3 (i.e. a single neighbor considered),in case of 15 neighbors D2 = 15 × 3 = 45, thus N2 = N (D2/D1)1 = 100015.Ordinary PR-DNS-generated simulations of particle arrays are far from being suf-ficiently dense for creating meaningful, reliable functional forms for p˜. As a result,116the curse of dimensionality precludes the construction of PDFs in equation (4.27) intheir current form. Any such effort is in fact bound to produce a severely over-fittedmodel with almost no generalizable prediction capability. Consequently, instead ofattempting to obtain p˜ (R |∆F < 0) and p˜ (R |∆F > 0) directly, we suggest thatp˜ (R |∆F < 0) ≈cα,1 p˜1 (r1 |∆F < 0) + cα,2 p˜2 (r2 |∆F < 0) + . . .=M∑j=1cα,j p˜j (rj |∆F < 0)p˜ (R |∆F > 0) ≈cβ,1 p˜1 (r1 |∆F > 0) + cβ,2 p˜2 (r2 |∆F > 0) + . . .=M∑j=1cβ,j p˜j (rj |∆F > 0)(4.28)In the above equation, each function p˜j (rj | · ) is equivalent to a marginal proba-bility distribution which is obtained by integrating out the positions of neighborsrks in p˜ (R = {rk=1, . . . , rk=M} | · ) where k 6= j. In other words, this approxi-mation only considers the effect of a single neighbor on the functional form of p˜,while accounting for the presence of all other surrounding particles in an averagemanner. This marginal distribution can be expressed asp˜j (rj | · ) =∫p˜ (R | · ) dr1 . . . drj−1drj+1 . . . drM (4.29)Equation (4.28) is reminiscent of the pairwise approximation employed by [154]for the purpose of reducing the number of independent variables by accounting forthe influence of only one neighbor at a time. In their data-driven model, the authorsalso invoked the “order invariance” approximation, which removes the dependenceof the model functions on the neighbor number. This was done by weighting thefunctionals according to the probability of a particular neighbor being the j-th clos-est neighbor. In the MPP model, the neighbors first need to be sorted based on theirdistance to the reference particle. Each neighbor’s position can be then passed tothe appropriate PDF.117Substitution of equation (4.28) in equation (4.27) leads to∆FMPP (R,Re, φ) =M∑j=1cα,j p˜j (rj |∆F < 0) +M∑j=1cβ,j p˜j (rj |∆F > 0)(4.30)In equation (4.30), cα,j and cβ,j are constant unknown coefficients. With the fi-nal form of ∆FMPP , we have assumed an additive nature for the effect that eachneighbor might have on the force/torque fluctuation experienced by a reference par-ticle. Notably, both distributions p˜j (rj | · ) (this notation is used to refer to a PDFwith any arbitrary force/torque conditioning) are now functions of three variablesonly; that is, the three components of the position vector rj = (rj,x, rj,y, rj,z). Thismeans that the functional form of the PDFs in equation (4.30) can be convenientlyinferred with a data-driven approach, as the high-dimensionality of the input spacehas been evaded. To this end, a dataset can be constructed by running PR-DNS offixed beds for desired sets of parameters. The discrete estimation of the distribu-tion p˜j can be extracted by filtering neighbor positions according to the conditions{∆F < 0,∆F > 0}. The discrete form of the distribution is then fitted with amultivariate kernel density estimation (KDE) function [185] of the formp˜ (r | · ) = 1mm∑q=1KH(r− rq) (4.31)whereKH(r) = |H|−1/2K(H−1/2 r) (4.32)In the above equations,m shows the total number of samples after filtering with theaforementioned conditions, H the bandwidth matrix and K is a symmetric densityfunction of choice. We select a Gaussian distribution as the kernel function definedasK(r) =1(2pi)D/2exp{−12rT r}(4.33)118We will demonstrate and discuss particular examples of these functions in sec-tion 4.3.3. Now that the appropriate functional forms are identified and known,it remains to decide the values of the constant coefficients cα,j and cβ,j in equa-tion (4.30). The values are obtained using the ordinary least-squares method forlinear regression:{cα,j , cβ,j}Mj=1 = argminNp∑i=1(∆FMPP,i −∆FDNS,i)2 (4.34)which minimizes the residual sum of squares between values of force/torque fluc-tuations predicted by the MPP model and the true values from PR-DNS. Afteroptimal values for cα,j and cβ,j are found, equation (4.30) can be used to makepredictions for each particle based on the particular microstructure of the surround-ing particles. It is of great importance to realize that in the presented approach,the model relies solely on the unique configuration of neighbors surrounding eachparticle, and the PR-DNS-generated force/torque data (as opposed to velocity andpressure fields) in order to make predictions. Consequently, the MPP model ac-counts for the combined effect of various force contributions such as the undis-turbed flow, quasi-steady and added-mass forces. This can be contrasted with thePIEP physics-driven model [152] which uses perturbed velocity and pressure fieldstogether with the Faxén’s theorem to obtain hydrodynamic forces experienced byeach particle.The above analysis applies identically to all force and torque fluctuation compo-nents; namely, (∆Fx,∆Fy,∆Fz) and (∆Tx,∆Ty,∆Tz). Let Cα denote a 1×Mvector that contains the cα,j coefficients in equation (4.30):Cα =(cα,1 · · · cα,j · · · cα,M)(4.35)and let P∆F<0 denote an M × 1 vector that contains the PDFs p˜j (rj |∆F < 0)119in equation (4.30):P∆F<0 =p˜1 (r1 |∆F < 0)...p˜j (rj |∆F < 0)...p˜M (rM |∆F < 0)(4.36)Together with their counter-parts, i.e., Cβ and P∆F>0, the above vectors can beultimately defined for all force components. The complete model equations forforce fluctuations can thus be expressed in the following form:∆FMPP (R,Re, φ) =∆Fx∆Fy∆Fz =Cxα P∆Fx<0 + Cxβ P∆Fx>0Cyα P∆Fy<0 + Cyβ P∆Fy>0Czα P∆Fz<0 + Czβ P∆Fz>0 (4.37)where the super-scripts of the coefficients indicate the corresponding component.Following the above definitions, torque fluctuations are given as∆TMPP (R,Re, φ) =∆Tx∆Ty∆Tz =Dxα P∆Tx<0 + Dxβ P∆Tx>0Dyα P∆Ty<0 + Dyβ P∆Ty>0Dzα P∆Tz<0 + Dzβ P∆Tz>0 (4.38)where Dα, Dβ , P∆T>0 and P∆T<0 for each torque component are equivalent toCα, Cβ , P∆F>0 and P∆F<0 in equation (4.37). Note that the vectors Cα, Cβ ,Dα,Dβ , P∆F<0, P∆F>0, P∆T<0, P∆T>0 each contain as many coefficients andPDFs as the number of included neighbors M , the values of which depend on Reand φ of each case represented in table Probability distribution mapsIn this section, we scrutinize a few of the probability distribution maps obtainedfor different force/torque components and various neighbors. Before proceedingfurther, we first describe the necessary steps taken for generating the foregoingdistributions. In order to construct the maps, those particles that satisfy the desired120condition (e.g. ∆Fx < 0 or |∆Fy| > σ, with σ being the standard deviation ofthe corresponding variable) are identified and filtered by looping over the entirearray of spheres. In this filtered subset of the original data, the positions of thechosen neighbors are recorded. These positions constitute data points to whichKDE functions in equations (4.31) to (4.33) are then fitted, thus giving p˜j (rj | · ).The generated PDFs are functions of the three spatial coordinates, which makesit impossible to visualize these functions directly. Inevitably, we resort to contourplots of the PDFs; however, three-dimensional contour surfaces are also neitherefficient nor convenient to visualize. Consequently, we have opted for depictingtheir projections on two-dimensional planes. In doing so, the contour surfaces areentirely compressed onto an x − y plane. This is why the reference particle willbe shown as a blank circle in contrast to a solid disk, since the neighboring spherescan happen to be located even at (x, y) = (0, 0) when |z| > 0.5. In the case ofstream-wise force, all maps are in fact axisymmetric about the x axis; meaning thatthe choice of the plane on which the maps are shown does not make any difference,as long as it is parallel to the x axis. As for the lateral forces, the maps are plane-symmetric with respect to the same plane whose normal has the same directionas the force component (e.g., p˜ (rj | |∆Fy| > σ) would be symmetric about the yaxis). For the hydrodynamic torques, the maps are also plane-symmetric but withrespect to the plane that contains the torque component. Owing to the symmetryabout the stream-wise direction, the probability distribution map for |∆Fy| > σshown on x − y plane is essentially almost identical to that for |∆Fz| > σ ony − z plane. The same is also true for |∆Tz| > σ on x − y and |∆Ty| > σon y − z. Noting the existence of this similarity, only projections on the x − yplane will be shown in this section. With an ideal densely populated dataset, thePDFs are expected to exhibit perfect symmetry. The limited number of samples inour dataset, however, causes some PDFs to show minor yet noticeable deviationsfrom symmetry. This would be particularly evident in PDFs generated for fartherneighbors due to the lower data density compared to closer neighbors. In whatfollows, statistical symmetry is hence enforced in all PDF depictions.Examination of the probability maps in all cases remarkably reveals quite dis-tinct, non-uniform and physically meaningful functional forms. Figure 4.4 shows121(a) (b) (c)Figure 4.4: (a) Unconditioned PDF of the first closest neighbor position and PDFof the first closest neighbor position when the reference particle experiences a(b) higher than average, or (c) lower than average drag force. The PDFs areobtained for the case of Re = 40 and φ = 0.1.the distribution of the first (i.e. closest) neighbor positions, conditioned on two dif-ferent ranges of the stream-wise force, while the blank circle in the middle of rep-resents the reference particle which experiences the hydrodynamic force or torque.According to these probability maps, it is considerably more likely for a referenceparticle to experience higher than the mean drag force when the closest neighboris located laterally, and slightly downstream of the particle. The right plot in fig-ure 4.4 appears more intuitive: drag force diminishes if the reference particle isshielded by the closest neighbor directly in front of it, as expected. A somewhatless intuitive observation is that even if the neighbor happens to be immediatelydownstream of the reference particle, drag force would be lower than average, dueto the suppression of the low-pressure region behind the reference particle. It iscrucial to stress this point once more that these maps demonstrate the higher likeli-hood of coming across neighbors at particular regions depending on the conditionimposed on force/torque fluctuation. This means that, for instance, even if theclosest neighbor occurs to be located laterally, the drag force might be lower thanaverage because the second neighbor happens to directly shield the reference par-ticle. It signifies the fact that while accounting for the influence of one neighborprovides a great deal of valuable information about the fluctuations, effects of otherneighbors are indispensable in explaining the force/torque fluctuation accurately.Figures 4.5 and 4.6 show PDFs for the distribution of closest neighbors whenconditions are imposed on the lateral force in the y direction and on the lateral122(a) (b) (c)Figure 4.5: PDF of the first closest neighbor position when the reference particleexperiences a (a) higher than σ or (b) lower than σ lift, with σ being the standarddeviation of the data. (c) PDF of the first closest neighbor position when the liftis either higher or lower than σ. The PDFs are obtained for the case of Re = 40and φ = 0.1.2 1 0 1 2x21012yp˜1(r1 |∆Tz >σ) 1 0 1 2x21012yp˜1(r1 |∆Tz <−σ) 1 0 1 2x21012yp˜1(r1 | |∆Tz|>σ) 4.6: PDF of the first closest neighbor position when the reference particleexperiences a (a) higher than σ and (b) lower than σ lateral torque, with σ beingthe standard deviation of the data. (c) PDF of the first closest neighbor positionwhen the lateral torque is either higher or lower than σ. The PDFs are obtainedfor the case of Re = 40 and φ = 0.1.torque in the z direction, respectively. Note that for demonstration purposes, wehave chosen more extreme conditions in figures 4.5 and 4.6, e.g. ∆Fy > σ and∆Fy < −σ, and ∆Tz > σ and ∆Tz < −σ, respectively, in order to make thePDFs stand out more prominently. However, the MPP model (equations (4.37)and (4.38)) is always constructed with the original positive and negative condition-ing (e.g. ∆Fy > 0 and ∆Fy < 0 for the lateral force in the y direction). On theleftmost plot of figure 4.5, the reference particle is seen to experience a positive lat-eral force in the y direction when the first neighbor is located within the region thatlies mostly in the bottom right quadrant where x > 0 and y < 0, whereas a nega-tive lateral force results when the first neighbor happens to be on the opposite side123where x, y > 0 depicted in the middle plot. As pointed out by [186] for a staggeredarrangement of only two spheres, a higher than normal pressure field develops inthe gap between the two particles, which causes a repulsive force acting mainly onthe upstream particle. Upon increasing the distance between the particles, the fluidcan penetrate the gap more easily. The nozzle effect hence creates a low pressureregion in the gap which gives rise to an attraction force. The latter does not appearin figure 4.5 as the first neighbor is quite close to the reference particle, but regionscorresponding to such a condition show up remarkably on the PDFs for the lateralforce in case we consider more distant neighbors. The right most plot shows thePDFs conditioned with |∆F | > σ, which is a combination of the foregoing plotswith {∆F < −σ,∆F > σ}. These are regions for which the reference particleexperiences a non-zero lateral force in the y direction. In figure 4.6, we can seeessentially the same kind of information as in figure 4.5. The most critical regionsfor the modification of the torque are located where the highest amount of vorticityis generated in the flow over a single sphere. As with the lateral force, the higherpressure in the gap decreases the fluid velocity and the resulting vorticity. The vor-ticity imbalance between the top and bottom regions of the reference particle bringsabout a net torque exerted on the particle. The interpretation of the PDFs emergingfrom our analysis is consistent with the observations made with a binary systemof two spheres exhibiting the same dependency of the forces on relative positionsof the spheres [187, 186]. Such modification of the forces and torques were alsoelucidated by [7]. Our results imply that even when the system being modelled isa dense multi-particle system, the forces and torques acting on each particle stillvary in the same manner on average, in response to the positioning of the closestneighbor. For instance, when a particle happens to experience less drag force, itis most probably shielded by an upstream neighbor (or supported by a neighborimmediately downstream with a comparatively less probability) as shown on theright plot in figure 4.4. Although this may not be always true, it is the most likelyreason. Similar arguments apply in regards to all other conditional PDFs involvinglateral forces and torques as well.Interestingly, similar patterns are found for neighbors located further away. Anexample of the PDFs for farther neighbors is shown in figure 4.7. While the dis-12442024yj=1 j=3 j=54 2 0 2 4x42024yj=74 2 0 2 4xj=124 2 0 2 4xj= j=3 j=54 2 0 2 4x42024yj=74 2 0 2 4xj=124 2 0 2 4xj= 4.7: PDFs of various neighbor positions for (a) ∆Fx < −σ, and (b) ∆Fx >σ, with σ being the standard deviation of the data. Note that the numberingrepresented by j is based on proximity to the reference particle, where j = 1shows the closest neighbor. The PDFs are obtained for Re = 40 and φ = 0.1.125tributions are spread out and have become fainter for surrounding particles locatedfarther away from the reference particle, regions with higher probability are dis-cernible and turn out to be generally in accordance with those obtained for j = 1,i.e., the closest neighbor. The fact that the densities are smaller for j > 1 istwo-fold. Firstly, farther particles are distributed over a larger volume around thereference particle. The second and more important reason is that the more distantparticles are comparatively less likely to alter the force/torque fluctuations. Thefunctional forms of p˜ in equation (4.28) has been kept distinct for each neighbor,meaning that p˜1 is a different function compared to p˜2, and they both are differentfrom p˜3 and so on. This is also reflected in the probability maps shown in fig-ure 4.7. A possible simplification can be made by combining all functions p˜j intoa single function p˜ that still contains the contribution of all included neighbors,except for not making distinction based on the ordering of the neighbors. In thismanner, instead of filtering the locations of the j-th neighbor, we filter the loca-tion of all the included neighbors (up to j = M ) based on the desired condition.It is to be noted that we still retain the contribution of each neighbor to the netforce/torque fluctuation by allowing separate terms for every included neighbor inequation (4.28). That is to say, we construct only one PDF, namely, p˜ using theentire neighborhood, while each term p˜ (rj | · ) is assigned a separate unknowncoefficient and influences the prediction of fluctuations differently. An exampleof such a combined PDF is shown for the stream-wise force in figure 4.8a forφ = 0.1. Comparing with figure 4.4, it is immediately obvious that the significantregion around the reference particle has expanded due to the influence of more dis-tant neighbors. Nevertheless, the general form of the maps that were obtained forj = 1 in figure 4.4 still prevails even when all the neighbors up to j = M are in-cluded all at once. Another remarkable observation can be made from figure 4.8b,where PDFs similar to those in figure 4.8a have been shown for the highest solidvolume fraction of φ = 0.4. The PDFs in figure 4.8b have gained fore-aft sym-metry and are shrunk towards the reference particle, reminiscent of the force mapsobtained with the data-driven model of [154] at φ = 0.45.126(a) Re = 40 and φ = 0.13 2 1 0 1 2 3x3210123yp˜(∪15j=1rj |∆Fx >σ) 2 1 0 1 2 3x3210123yp˜(∪15j=1rj |∆Fx <−σ) Re = 40 and φ = 0.4Figure 4.8: PDFs of positions of 15 closest neighbors for two different solid volumefractions.4.4 Results and discussion4.4.1 Practical aspects of implementation4.4.1.1 Computational efficiency of KDE evaluationsSince the MPP model is to be used in EL simulations, at each time step it mightbe necessary to compute the forces and torques for several thousand, or perhapsmillions of particles at once. The computational effort required for making pre-dictions with the model is thus an important matter. The MPP model summarizedin equation (4.30) consists of a handful of constant coefficients, multiplied by thesame number of KDE-estimated PDFs. For m evaluations given n sample points,evaluation of a KDE-estimated PDF by naive kernel summation in equation (4.31)(which we use for model computations in this work) requires a quadratic O(mn)operations, which may be computationally prohibitive for the practical implemen-127tation of the MPP model. However, this issue can be circumvented to a great extentby using efficient approaches that have been proposed over the past years, includ-ing data binning with Fast Fourier Transform (FFT), fast sum updating, fast Gausstransform and dual-tree method [188]. The computational cost can be reducedfrom quadratic O(mn) operations to linear O(m + n) or O(m logm + n log n),resulting in a vast improvement of computational efficiency by orders of magnitudecompared to naive kernel summation.Another possibility for speeding up probability density estimations is to approx-imate the density distributions with well-known functional forms such as polyno-mials or exponential functions. As KDE is a non-parametric estimation method,the dataset based on which predictions are made need to be stored in order to ob-tain KDE-estimated probability densities. Using simpler functional forms obviatesthe need for storing the dataset, albeit at the expense of potential loss of accuracy.Nevertheless, one needs to study the model degradation due to such approximationsversus the gained speed-up in computations, and decide how much of a trade-off isacceptable in a particular situation. Model hyper-parametersIn our model, there are a few parameters that need to be given values beforemaking predictions, and finely tuned to deliver optimal performance. These pa-rameters are to be distinguished from the set of unknown constants of regressionin equation (4.34), which can be deterministically found from the minimizationof sums of errors. Two important hyper-parameters involved in our model are thekernel bandwidth indicated by H in equation (4.32) and the number of includedneighbors shown by M .In its most general form the bandwidth is given by a symmetric, positive definitematrix when multidimensional data is concerned. The bandwidth H significantlyinfluences the obtained KDEs through controlling the orientation and smoothingintensity of a kernel, equivalent to the standard deviation of a Gaussian distribu-tion. A very narrow bandwidth results in an KDE with high variance, whereasan over-smoothed estimation is generated if the bandwidth is too wide. The for-128mer tends to capture specificities of a particular dataset resulting in an over-fittedmodel performing poorly on other datasets, while the latter fails to acquire the cru-cial patterns in the data, giving an overall weak prediction capability. Since in thepresent work, the neighbor positions are uniformly distributed in all three dimen-sions, choosing the same bandwidth for all directions is sensible. In this case, thebandwidth matrix is determined by a scalar h such that H = hJ with J being anall-ones matrix. A rule of thumb for estimating an optimal value for the bandwidthis given by [189] as h = σN−1/(D+4), with N as the number of samples and D asthe number of dimensions. In most of the cases in this work, we have used Scott’srule multiplied by a factor of 1− 1.8. In most cases, a factor of > 1 was necessaryto avoid over-fitted results.The other critical parameter to determine before attempting to make predictionsis the number of neighbors to include in the modeling process. In low Reynoldsnumber regimes, the force/torque fluctuations for each particle depend on a largenumber of surrounding particles (e.g. theoretically depending on each and everyparticle in the system for Stokes flow) due to the dominance of the elliptic natureof the governing equations at such regimes. In relatively inertial regimes, though,only a limited number of neighboring particles influence the deviation of the hy-drodynamic forces and torques acting on a particle from the average values. [152]concluded that ≈ 15 − 40 and ≈ 10 closest neighbors were required for optimalmodeling of drag and lift, respectively, and that inclusion of more surroundingparticles did not improve the quality of their physics-driven PIEP model in termsof the coefficient of determination R2. For their data-driven PIEP model [154],≈ 13 neighbors for drag and ≈ 15 neighbors for lift resulted in the best coefficientof determination. We have overall observed a similar trend for our MPP modelconcerning the number of neighbors to be included. A large number of neighborshave to be accounted for in lower Re for achieving maximum R2, while this num-ber decreases with the Reynolds number. Furthermore, we typically require fewerneighbors to include in order to model the lateral forces and torques compared tothe stream-wise force. Full details of the variation of the model quality with thenumber of neighbors will be given in the following sections.Finally, another choice to be made is the type of kernel in equation (4.32).129Different types of kernels such as parabolic (Epanechnikov), cosine, exponentialand linear kernels can be alternatively used to give the weighted-average value ateach point. As pointed out by [190], bandwidth selection is much more of a con-cern compared to the choice of the kernel, which can be legitimately based on thesmoothness requirements or computational effort involved in making evaluations.In line with [190], our experimentation with different types of kernels indicatesthat the choice of the kernel type does not remarkably affect the performance ofthe model, given that optimal values of bandwidth are used. Inclusion of average velocityThe channeling of the flow through the pores of a bed causes the fluid velocityexperienced by the particles to fluctuate from particle to particle. This varianceof undisturbed fluid velocity is partly dictated by the immediate neighborhood ofeach particle, but not entirely. A bundle of a few particles might be collectivelyfully exposed to the fluid flow, while another group are mostly blocked by upstreamneighbor groups. In the former case, the drag on a particle inside the bundle is sig-nificantly affected by its neighbors in the group, while in the latter case the particlewould have experienced a relatively small or perhaps no drag, with little influencefrom its neighbors as the drag was not great to begin with. If we were able toinclude the effects of several neighbors around each particle instead of the approx-imation employed in equation (4.28), the channeling effect would theoretically becaptured solely from the configuration of the neighbors. In the present work, thisinformation can be directly supplied to the model by including a measure of thevelocity seen by each particle. Since equation (4.30) is an approximation, linearlyadding the average velocity seen by the particle may enhance the model. Therefore,equation (4.30) can be modified as follows:∆FMPP (R,Re, φ) =M∑j=1cα,j p˜j (rj |∆F < 0) +M∑j=1cβ,j p˜j (rj |∆F > 0) +3∑d=1cγ,duvd (4.39)130where each cγ,d is an additional unknown coefficient, and uvd denotes an averagevelocity component of uv in x, y or z direction. The estimation of fluid velocityseen by a finite-size particle (i.e., undisturbed fluid velocity) within a multiparticlesystem is not straight-forward, resulting in several definitions in the literature [191,192]. [191] suggest the average of fluid velocity taken on a spherical shell centeredat the particle location as the characteristic fluid velocity; a definition also utilizedby [16] in their analysis of settling suspensions of spheres. Here, we perform thephase-averaging in a spherical volume about the location of each particle to obtainthe fluid velocity seen by the particle:uv =Nl∑l=1Φlul/ Nl∑l=1Φl (4.40)where Φ is the phase indicator function, which is utilized in order to avoid samplingvelocity data inside the solid region, andNl the number of grid points falling withinthe spherical volume. The diameter of the averaging volume is chosen to be 4d forφ = {0.1, 0.2} and 3d for φ = 0.4. The inclusion of the average velocity uvimproves the predictions in all cases, reflected in the increasing of R2 by up to 5%.This improvement is more pronounced for φ = 0.4, resulting in an increase of upto 15% in the coefficient of determination.4.4.2 Model assessmentAs a measure of performance, we will use the coefficient of determination de-fined asR2 = 1−N∑i=1(∆FMPP,i −∆FDNS,i)2N∑i=1(∆FDNS,i − 〈∆FDNS,i〉)2(4.41)where N is the number of samples used for the computation of R2. Also, we notethat 〈∆FDNS,i〉 = 0 by definition. The coefficient of determination can be in-terpreted as the fraction of variations that are explained by the model. As such,131R2 = 0 implies that the particle-to-particle force/torque fluctuations are not cap-tured at all and the model does not perform any better that giving a single averagevalue for all particles. On the other hand, R2 = 1 would indicate a perfect fit; i.e.fluctuations predicted by the model exactly match those obtained from PR-DNS.With every data-driven model, it is necessary to ensure that the model has notonly fitted the data in a satisfactory manner, but is also able to generalize its pre-dictions to unseen data. Complex data-driven models in the realm of ML suchas neural networks often have numerous unknown parameters which makes themprone to overfitting, particularly when dealing with small datasets. Put anotherway, an over-fitted model simply memories the dataset thanks to its huge set of pa-rameters, while it performs poorly when presented with unseen data. One strategyto prevent over-fitting and to assess the generalization capability of a model is tosplit the dataset into a training set and a smaller validation set, so as to fit the modelusing the former and test its performance on the latter. This technique, commonlyknown as the hold-out method, works well if used with sufficiently large datasets.For datasets limited in size, such as data obtained from PR-DNS here, not onlythe validation set might not be sufficiently representative, the dataset is only par-tially utilized to train the model [193]. K-fold cross-validation is an assessmenttechnique that randomly partitions the dataset into K subsets or “folds”, so thatone of these folds is held out for testing while the model is trained on the rest ofK − 1 folds. This process is repeated with every partitioning such that the entiredataset has been used both for training and for validation. Any measure of erroror accuracy and also the model’s parameters may be averaged to give a more reli-able estimation of the performance. To ensure the prediction reliability of the MPPmodel, we employ the K-fold cross validation with K = 8 where multiple roundsof cross-validation are performed using K subsets of the data. All reported resultsare thus the averaged values over multiple rounds so that a reliable estimate of themodel’s predictive performance is achieved.4.4.3 Performance of the MPP modelThe results of the MPP model’s predictions and comparison with our PR-DNSdata are presented in terms of the coefficient of determinationR2 in table 4.3. Note132φ Re ∆Fx ∆Fy ∆Tz0.1 2 0.84 0.78 0.850.1 10 0.80 0.76 0.800.1 40 0.70 0.69 0.640.1 150 0.57 0.53 0.380.2 0.2 0.71 0.71 0.850.2 2 0.76 0.74 0.860.2 40 0.71 0.66 0.700.2 150 0.62 0.55 0.480.4 2 0.54 0.53 0.690.4 40 0.67 0.58 0.640.4 150 0.61 0.47 0.52Table 4.3: Performance of the MPP model represented by the coefficient of determi-nation R2 for cases considered in this workthat we have deliberately avoided inclusion of data for ∆Fz and ∆Ty, as the resultsobtained for these variables are equivalent to those for ∆Fy and ∆Tz , respectively.The green and light green cells in table 4.3 refer to cases for which R2 > 0.7 andR2 > 0.6, respectively, and cells highlighted in red represents cases for whichR2 6 0.5. Overall, the MPP model exhibits a remarkable ability to correlate theobserved force/torque deviations from the average values to the specific neighbor-hood of each particle. Our proposed model is able to explain up to 84%, 78%and 85% of the drag, lift and torque variations, respectively, in the best cases witha mean coefficient of determination of R2 = 0.68, 0.63 and 0.67 averaged overall cases. Except for three cases (shown in light red in table 4.3), at least 50%of the particle-to-particle force/torque variations are captured by the model, whilein most cases the coefficient of determination is seen to be greater than 60%. Infigure 4.9 correlation scatter plots are shown for the stream-wise force or drag, thelateral force or lift and the lateral component of the torque. For each hydrodynamicload, we have selected three cases representing a range of Reynolds numbers andsolid volume fractions in increasing order, in order to demonstrate cases wherethe model exhibits varying degrees of performance. The horizontal coordinate of133Figure 4.9: Regression plots for the drag, lift and lateral torque for various Reynoldsnumbers and solid volume fractions. The oblique red line in each plot shows anideal fit.134each point on the plots shows the value of the force/torque deviation obtained fromPR-DNS, whereas the vertical coordinate shows the value predicted by the presentMPP model for the same sample in the dataset. The red bisector in figure 4.9 in-dicates a perfect fit, for which ∆FMPP = ∆FDNS and R2 = 1. Notably, theclassical microstructure-ignorant drag correlations of the form F = 〈F 〉(Re, φ) insection 4.2.4 would all lie on a horizontal line given by ∆F = 0 in figure 4.9,since these correlations only give an average value for an entire ensemble of par-ticles and are thus unable to explain any particle-to-particle variation of the drag.Clearly, the MPP model is not perfect in any of the cases, but the improvementsover conventional correlations are substantial, reaching up to R2 = 0.84, 0.77 and0.85 for ∆Fx, ∆Fy and ∆Tz in best cases where (Re, φ) = (2, 0.1). As both Reand φ are increased from left to right in figure 4.9, the performance of the modelsuffers to some extent. Remarkably, however, the drag variation is still captured upto 61% at the extreme case of (Re, φ) = (150, 0.4), while R2 remains in the rangeof 0.66− 0.71 for all variables at (Re, φ) = (40, 0.2).The approximation in equation (4.28) that p˜ (R | · ) (i.e., the PDF dependingon locations of all included neighbors) may be estimated as a linear combinationof p˜j (rj | · ) functions (i.e. marginal PDFs each depending on the location of onlya single neighbor) imposes an important limit on how accurate the model can be.For instance, in case the joint distribution p˜1,2 (r1, r2 | · ) is practical to obtain, thefunctional form of p˜ for the first neighbor would be different depending on wherethe second neighbor is located. In such a situation, the PDF for the drag in fig-ure 4.4 would no longer be axisymmetric about the x axis; it would take on differ-ent forms depending on where the second closest neighbor is located. An exampleof such a case is shown on the left-hand side of the PDF in figure 4.10. On the leftplot, we have fixed the location of the second neighbor (shown as the gray circle)to stay within a small radius of r2 = A = (−1, 1, z) on x− y plane, while the thez coordinate is left free. On the right, we have reiterated the right-hand side plot offigure 4.4, where the PDF is obtained by conditioning only the drag and observingthe locations of the first neighbor. Obviously, the presence of the second neigh-bor has affected the probability distribution of the first neighbor’s position which isindicated by pushing the upstream probability density peak downwards and shrink-135(a) (b)Figure 4.10: PDF of the first closest neighbor position (a) having a second neighbor(drawn as a dashed white circle) deliberately fixed on x−y plane at r2 = A =(−1, 1, z), (b) PDF of the first closest neighbor position, same as in figure 4.4c.The PDFs are obtained for Re = 40, φ = 0.1ing it simultaneously. This alteration of the PDF is partially, but not merely, dueto an exclusion of the space occupied by second neighbor, but also owing to themodification of the flow field induced by the particular positioning of the secondneighbor. Notably, the first neighbor can still happen to be located immediatelyupstream of the reference particle. In short, the second neighbor influences howthe first neighbor affects the reference particle. This type of interaction dubbed as“ternary effects”, and also higher-order interactions of this type are ingredients ofthe flow physics sacrificed in equation (4.28) in exchange for a functional form thatis practical to estimate.In Stokes flow and dilute conditions, binary interactions prevail and high-orderinteractions are negligible, and the net effect arising from different neighbors canbe given as a linear superposition of the influence of each individual neighbor. Asthe Reynolds number increases, linear addition of effects can no longer accountfor ternary and higher-order interactions, thus diminishing the validity of the linearcombination of effects employed in equation (4.28). Consider the relatively lowsolid volume fraction cases with φ = 0.1 in table 4.3. As the Reynolds numberincreases, the performance of the MPP model deteriorates as expected as a result ofthe superposition of each neighbor’s influence in equation (4.28). The coefficientof determination R2, however, remains almost constant for φ = 0.2 and counter-136intuitively improves for φ = 0.4 when the Reynolds number increases. Whenφ = 0.4, the average interparticle distance is considerably smaller than for φ =0.1 making the PDFs more flattened and uniform, thus reducing the variability ofthe extracted probability map. The increase of the Reynolds number renders thedistribution more anisotropic to some extent (also reflected in the higher standarddeviation of higher Re cases for φ = 0.4 in table 4.2) which leads to an improvedR2.In order to be able to practically implement the MPP model, we separated theinfluence of each neighbor on the force/torque fluctuations of a reference parti-cle by marginalization of the probability distribution in equation (4.28), and ex-pressed the net deviation as the linear combination of all effects. Although this as-sumption resembles the pairwise interaction assumption of [152] in their physics-driven PIEP model, it is nevertheless inherently different. The pairwise interac-tion approximation employed by [152] is based on binary interaction maps wherethere is no incorporated notion of solid volume fraction. In equation (4.28), how-ever, each functional p˜j (rj | · ) is obtained for neighbor j in the presence of allother neighboring spheres. In other words, p˜j (rj | · ) appears as a binary inter-action in which the effects of all other neighbors are averaged, but still statisti-cally present. This implies that the functional form of each p˜j is in fact depen-dent on the solid volume fraction, which may explain the superior performanceof the present MPP model (and the data-driven PIEP model, for that matter) in ahigher particle concentrations compared to physics-driven PIEP model. For thepurpose of comparison, with the MPP model R2 = 0.67 and 0.61 for drag in(Re, φ) = (40, 0.4) and (Re, φ) = (150, 0.4), respectively; whereas R2 = 0.12and 0.24 in (Re, φ) = (21, 0.45) and (Re, φ) = (115, 0.45), respectively, withthe physics-driven PIEP model. Although Re and φ are not exactly the same forthe cases here and for those considered by [154], a general comparison of resultsshows that the performance of the MPP model is on par with the hybrid PIEP modelwhich combines the physics-driven and data-driven approaches.Lastly, it must be pointed out that the present MPP model only considers staticarrays of spheres. In a general particle-laden flow situation, however, particles willbe in motion and the dynamic evolution of the system will depend on the positions137as well as the motion of the particles. With the current formulation, inclusion of thetranslational and rotational velocity and acceleration of each neighbor in the PDFsof equation (4.30) will increase their dimensionality, thus rendering them imprac-tical to use in their present form. Future investigations (similar to what has beendone by [8, 9] for conventional EL techniques) are needed to evaluate the MPPmodel’s performance when implemented in EL simulations through comparisonswith corresponding PR-DNS cases. Such an assessment would be greatly benefi-cial in quantifying the significance of the inclusion of translational and rotationalvelocity and acceleration of the neighbors, and also to identify situations wheresuch contributions might be more or less important.4.5 Summary and conclusionNumerical simulations are indispensable tools in analyzing particle-ladenflows, as experimental investigation of this type of flow is both very costly if notentirely impractical, while the extent to which details of these complex flows areavailable to experimental measurements is also quite limited. Particle transport inmicrofluidic separation devices or in highly dense particulate flows and heat trans-fer properties of combustion fluidized beds are a few examples for which resortingto numerical tools is inevitable in order to obtain the details of the physical pro-cesses. Most particle-laden flows of practical interest host billions of particles andspan a space orders of magnitude larger than the size of individual particles. Evenwith the exponential growth of computing power, resolving all relevant scales isnot within reach in the foreseeable future. In a higher intermediate (i.e. meso)scale, the fluid governing equations are averaged and solved in sub-volumes eachcontaining a few particles, treating the fluid in an Eulerian way while still trackingeach individual particle in a Lagrangian manner. Consequently, the computationalcost is substantially lowered at the expense of the need for supplying closure mod-els for fluid-solid momentum exchange. As field variables are not available atthe particle level, hydrodynamic interaction forces and torques cannot be directlycomputed and should hence be appropriately modeled. In relatively dense particle-laden systems, the flow varies on the scale of the particle dimension due to thepseudo-turbulence created by the neighbors. Conventional point-particle models138that are parameterized only in terms of the Reynolds number and average solidvolume fraction fail to account for the effects of the complex undisturbed flow onthe drag. Microstructure-induced lateral forces and torques, on the other hand,have been neglected entirely in such models by definition. Given the fact that thephysical fidelity of EL simulations directly relies on the accuracy of the interphasecoupling scheme, developing force/torque models capable of incorporating the in-fluence of local neighborhood of particles is crucial.In the present work, we have attempted to develop a deterministic model basedon probabilistic arguments for hydrodynamic forces and torques exerted on eachindividual particle within a random array of fixed spheres. Owing to the uniqueneighborhood of each particle, the flow in its vicinity is modified in a distinct way,giving rise to significant force and torque variations. The principal idea exploitedby our MPP model is that conditioning force or torque deviations to positive ornegative ranges results in the emergence of particularly interesting, non-uniformdistributions of neighbor locations. That is to say, if particles (i.e. samples) thatexperience particular ranges of force/torque fluctuations are filtered, and in turnthe positions of their neighbors are examined, we will find that their surroundingparticles are always spatially distributed in remarkably non-uniform manners con-sistent with our physical understanding of binary interactions between two spheres.For instance, a neighbor located immediately upstream shields a particle from theoncoming flow and eliminates the frontal high pressure region in a binary system.Analysis of the neighbors distribution when the drag force is lower than averageclearly demonstrates that this is still true even in dense arrays of sphere, whichsignifies that invaluable information can be extracted from PR-DNS for the depen-dence of forces/torques on the local neighborhood. If a particle experience a lowdrag force, it is highly likely that the particle is shielded by an upstream neigh-bor. The same argument holds for other force/torque-conditioned cases as well. Itis critically important that while in a binary system these observations are certainevents, for an array of particles the occurrence of such events is probabilistic innature. Upon fixing the location of one neighbor (i.e. M = 1), the configurationof all other neighbors is still free to randomly change. Therefore, for a given lo-cation of a neighbor, the force/torque fluctuation on the reference particle would139be represented by a distribution, not a fixed value unlike in a binary system. Thisis why the notion of the expected value is invoked in order to provide an averagevalue of the experienced force/torques for a given location of one neighbor. Twoconceptual extremes are worth reiterating:• All neighbors are free (M = 0). In this case, the expected value for force/-torque deviation is zero, as the distribution is centered on ∆F = 0.• All neighbors are fixed (M = Np − 1). The distribution of force/torquedeviation becomes extremely narrow (Dirac’s delta function) as all sourcesof fluctuation are held fixed. The expected value for force/torque deviationapproaches ∆F = ∆FDNS .We note that in a binary system,M = Np−1 = 1, therefore ∆F = ∆FDNS . The-oretically then, this framework can have varying degrees of accuracy approachingthat of PR-DNS; practically however, obtaining probability distribution maps forM > 1 requires exponentially more samples to sufficiently cover the input spaceof the problem. We recognize the case M = 1 as representing first-order effects,while higher-order effects may be captured by fixing the positions of two or moreneighbors simultaneously (i.e. M > 2, which translates into, for instance, obtain-ing p˜ (r1, r2 | · ) instead of p˜ (r1 | · ) only). In the present work, we have consid-ered M = 1 as a first step of improvement over microstructure-ignorant classicalcorrelations. In order to estimate forces and torques, we establish a frameworkto obtain the expected value of the force/torque deviations from the mean by tak-ing advantage of the force/torque-conditioned probability distribution functions ofneighbor locations. These distributions are approximated using KDEs which serveas basis functions for regression. The effect of each neighbor on the deviations isthen linearly combined and the value of unknown coefficients is found through anordinary least-squares regression method. With M = 1, adding more neighborsresults in the linear addition of more terms, meaning that the functional form of thePDFs do not change. For M = 2, the PDFs themselves would change dependingon where the second neighbor is located in addition to the firs neighbor, as alludedto in figure 4.10.We have generated a dataset consisting of several cases at various Reynolds140      MR2Re=10, φ=0.1∆Fx∆Fy∆Tz      MR2Re=40, φ=0.2∆Fx∆Fy∆TzFigure 4.11: Coefficient of determinationR2 as a function of the number of includedneighbors M used for construction of the modelnumbers and solid volume fractions relevant to dense particle-laden systems ofinterest. Following the aforementioned discussion, the model is not expected tobe perfect when M = 1. Nevertheless, the MPP model demonstrates remarkableperformance by explaining up to 60% − 70% of particle-to-particle force/torquevariation in most cases, while for a few cases in the low Re and φ the percent-age of explained variance in the data rises to up to 85%. We experimented withthe model to examine the dependence of the model performance on the number ofneighbors included in the superposition of effects1. As shown in figure 4.11, wefound out that inclusion of ≈ 20 neighbors is adequate for gaining optimal perfor-mance for the stream-wise force. The jump in R2 is quite steep for lateral forcesand torques, as the inclusion of ≈ 5 − 10 neighbors are sufficient to achieve themaximum performance for most cases. Consequently, the MPP model requires twoprobability distribution functions for negative and positive contributions, and a fewconstant coefficients (≈ 10 − 40) to predict a significant portion of variations inforce/torque components for given Re and φ. Computing the probability densitiesrequires evaluation of the KDEs in equation (4.31), which in turn needs the datasetsamples based on which the model is constructed. The samples needed for KDEestimation can be easily stored and utilized along with the constant coefficients tomake predictions in an EL simulation.1Note that in the present work, the probability distribution functions are all generated by con-sidering only one neighbor, i.e. M = 1. The number of included neighbors in figure 4.11 onlydetermines how many of these PDFs are superposed to make a final prediction.141According to the results obtained by our MPP model and also by the PIEPmodel [154, 152, 153], the performance experiences a significant enhancement byonly considering the first-order effects, hinting an asymptotic behavior with inclu-sion of higher-order effects. Even though M = Np − 1 is theoretically needed(in construction of PDFs) to reach the PR-DNS accuracy, we anticipate that evenconsidering the second-order effects properly renders the performance levels sat-isfactorily high to obviate the need for making the model more complex, as thegain would probably not be significant for M > 3. On the one hand, accountingfor the first-order effects of local microstructure on forces and torques can effec-tively predict the occurrence of phenomena such as wake attraction or DKT, whichis deemed the dominant mechanism in preferential concentration and clustering ofsuspensions [16, 66, 98, 74]. On the other hand, we have previously shown thattransverse particle velocity fluctuations and granular temperature are considerablyunderestimated by using conventional drag correlations in EL simulations [9, 8]due to the unavailability of microstructure-induced lateral forces. The MPP modelcan play a promising role in alleviating both challenges in current meso-scale sim-ulation tools.142Chapter 5Conclusions & prospectsIn this thesis, we have made an effort in two different directions towards con-tributing to the current understanding and multiscale modeling of particle-ladenflows. The first part revolved around physical analysis, where we have attemptedto shed light on the dynamics of suspensions laden with non-spherical particlesshaped as a specific polyhedron, namely, a cube. The second part focused on mo-mentum closure modeling, where we have developed a novel point-particle ap-proach capable of accounting for the effect of local microstructure on hydrody-namic forces and torques experienced by individual spheres in dense suspensions.The subsequent sections summarize the findings and conclusions of the foregoingresearch projects, each followed by a discussion of potential future directions.5.1 Particle shape effects in liquid-solid suspensionsThe sphere has served as a canonical shape in studying the behavior of particlesin suspensions. However, this is a limiting idealization, as the majority of real-lifesuspension flows contain non-spherical particles. Particle-resolved simulations ofsettling/rising of non-spherical objects have thus far been mostly limited to iso-lated disks, cylinders and spheroids, while inertial suspensions of non-sphericalparticles are rarely examined. Motivated by the rich dynamics of the motion ofnon-spherical particles, we chose to investigate the behavior of a regular polyhe-143dron, i.e. a cube, both as an isolated particle and in dilute and dense suspensions.In chapter 2, we began with a detailed analysis of motion regimes of a singlecube along wake structure characterization as well as in-depth inspection of forcebalance. Consequently, we learned about a number of important features of the in-ertial motion of a cube. The critical Galileo numbers for regime transition of a cubeare found to be generally lower than a sphere. A slight change in the orientationof a cube causes force and torque imbalance, which makes the cube more prone tooscillating translational and angular displacements. This triggers wake transitionsat lower thresholds of the Galileo number than expected. While a fixed cube un-dergoes the first transition at Re ≈ 190, the wake of a moving cube already showsevidence of transition at Re ≈ 65 and settles obliquely. We also find distinct helicalmotion regimes, accompanied by a specific vortex shedding pattern. It is interest-ing that while vortex shedding is believed to be incompatible with helical motion,we find it to be a dominant motion pattern for a cube at high enough Galileo num-bers. Based on force balance analysis, it is evident that a steady centripetal forceis supplied predominantly by the Magnus force. This highlights the remarkablerole of high rotational velocities of a cube in inertial regimes. Moreover, as soonas large lateral motions appear, the drag experienced by the cube increases by upto ≈ 50% compared to a fixed cube, about half of which may be explained by thechange in orientation. We suggest that the other contributing factor to drag en-hancement is the vortex-induced drag, which results due to the fact that the lateralforce generated by vortex shedding always makes an angle with the normal direc-tion. More precisely, the vortex-generated force has a component in the directionof velocity, which acts as increased resistance and hence decelerates the cube.Subsequently, we moved on to simulating inertial suspensions of cubes at vari-ous solid volume fractions in chapter 3. Companion simulations of spheres with thesame set of governing parameters were also performed to serve as reference caseswith which cube suspension simulations could be compared. In both the cubeand sphere suspensions, the wake attraction phenomenon initiates the sequenceof drafting, kissing and tumbling between pairs of particles, which creates non-homogeneity in the suspension in the form of long-lived clusters of particles. Inthe most dilute and highly inertial regime, the particle wakes are strong and less144affected by neighbors, leading to columnar clusters spanning the entire computa-tional domain. In denser suspensions, close-range wake attraction and the tumblingof particles causes the formation of horizontal arrangements of particles. The hall-mark of inertial cube suspensions is their more homogeneous microstructure com-pared to sphere suspensions. In other words, cubes are more likely to escape clus-ters. While remarkable columnar clustering of spheres at (Ga, φ) = (160, 0.01) isevidenced, vertical clustering is appreciably weaker for cubes in the same regime.Similarly, horizontal accumulation of cubes in denser regimes is also less promi-nent compared to spheres. These observations reveal that cube suspensions areinherently more resistant to local aggregation and exhibit enhanced mixing behav-ior, as also indicated by their higher average particle and fluid velocity fluctuations.Consequently, we associated these findings with the fact that the average rotationalvelocity of cubes is always greater, by a factor of 2–3 in the more dilute cases, com-pared to spheres, as anticipated from the initial study on the behavior of an isolatedcube. On the one hand, the greater rotational velocities significantly promote thegeneration of the Magnus force that acts in the transverse direction. On the otherhand, rotation of the cubes is also accompanied by a fluctuating orientation-inducedlift force. While flow perturbations created by surrounding particles contribute tothe lateral motions of both cubes and spheres, we attribute the substantially largerhorizontal particle velocity fluctuations of cubes to the transverse forces result-ing from rotation and orientation of cubes. Furthermore, cube suspensions arenot only more homogeneous, but also significantly less anisotropic compared tosphere suspensions, in terms of the ratio of vertical to horizontal velocity variance.This quantity, being always smaller in case of cube suspensions, is found to be upto ≈ 50% less compared to sphere suspensions in the most dilute regime, wheretransverse motions of cubes are most influential.While the preceding summary highlights the salient features of suspensionsof cubes, the results may also be viewed in the context of suspensions ladenwith polyhedral particles in general. A cube experiences significantly larger hy-drodynamic torques and hence remarkably greater rotational velocities, the effectof which translates into the generation of larger transverse forces, i.e. rotation-induced (i.e. Magnus) and orientation-induced forces. Collectively, these factors145result in higher horizontal velocity fluctuations, and consequently lead to bettermixing and a reduced clustering when compared to sphere suspensions. Similarmechanisms are expected to govern the behavior of suspensions of other regularpolyhedrons. We anticipate that for polyhedrons with more faces (e.g. icosahe-dron), the total transverse forces are more affected by the rotation-induced forcesrather than orientation-induced forces. Nevertheless, regardless of the exact inter-play between these contributions and their relative importance, a central questionis whether or not the suspension homogeneity increases with angularity. That isto say, for instance, are suspensions laden with tetrahedrons better mixed and lessclustered than those laden with cubes? Addressing questions of this sort in futureworks will eventually serve to yield an adequate understanding of how angularityinfluences the behavior of suspensions.In our study, we learned that the differences between cube and sphere suspen-sions gradually fades away as the solid volume fraction increases and many-bodyinteractions prevail. We hence suggest that the more dilute regimes, i.e. φ < 0.05,be explored more meticulously. For example, while helical motions were robustlyseen for an isolated cube, no evidence of similar motions was observed in our sim-ulated cube suspensions. We are curious to know if such a path geometry can beestablished in lower solid volume fractions, and whether or not the drag is similarlyaffected. Furthermore, it is hypothesized for spheres that their oblique motion playsa role in the emergence of large-scale vertical clusters by increasing the likelihoodof particles getting caught in the wake of one another [16]. It is open to questionwhether in more dilute regimes (i.e. φ < 0.01), the large transverse motions ofcubes could have a similar role in promoting clustering, or if on the contrary, theycontribute to the break-up of clusters. This is also worth exploring in conjunc-tion with the dynamics of suspensions at higher Galileo numbers, where vortexshedding itself is a major source of transverse forces. In addition, a worthwhilequestion could be whether sphere suspensions eventually exhibit less clusteringupon increasing the Galileo number due to wake-induced transverse forces, andsimilarly, whether enhanced clustering could be achieved for cubes in a regimewhere rotations are less vigorous.1465.2 Microstructure-informed point-particle modelParticle-resolved simulation methods are very desirable, since all scales rele-vant to particle-fluid interactions are resolved and directly accounted for. That is,no closure law or modeling effort is required to deliver a physically accurate repre-sentation of a particulate flow. However, this faithfulness to the underlying physicscomes at the expense of the need for enormous computational resources. Resolvedsimulations of real-life particle-laden flow scenarios where the system size spanslength scales several orders of magnitude larger than the particle diameter is thusout of reach for the foreseeable future. Therefore, resorting to multiscale simula-tions is inevitable, for which reliable force/torque closure modeling is a decisiveprerequisite. Though entirely ignored by conventional drag correlations, varianceof the drag among individual particles in dense suspensions can be of the sameorder of magnitude as the mean values. In addition, microstructure-induced liftand hydrodynamic torques are also completely disregarded in the current Euler-Lagrange simulation tools. These considerations provided us with a strong moti-vation to develop a point-particle model that is capable of accounting for the effectthe local microstructure on the hydrodynamic loads experienced by individual par-ticles.In chapter 4, we presented the principal idea and the development of ourMicrostructure-informed Probability-driven Point-Particle (MPP) model. TheMPP model is based on the observation that while neighboring particles surround-ing a reference particle are uniformly distributed in a random stationary suspen-sion, the positions of these neighbors show strong tendency for particular regionsif the force/torque exerted on a reference particle is conditioned to certain rangesof values. We found that these inhomogeneous distribution maps are in accordancewith our understanding of the hydrodynamic interactions of particles. Therefore,we formulated a statistical framework based on the probability distribution maps ofparticle positions to predict the forces and torques experienced by particles basedon their local neighborhood. In doing so, the required information for construct-ing probability maps is extracted from particle-resolved simulations for a set ofReynolds numbers and solid volume fractions. These maps are then approximatedusing kernel density estimation functions, each of which acts as a basis function147for regression. The unknown coefficients of regression are then found using least-squares method based on the dataset acquired from particle-resolved simulations.One of the most important features of the present approach is that in contrast to con-ventional machine-learning based solutions to such complex problems, our modelis physically inspired and completely explainable. For example, each basis func-tion represents the positive or negative contribution of a particular neighboringparticle to the total deviation of the force or torque.Our analysis for the development of the MPP model clearly demonstrates thateven though the collective effect of several neighbors on the force/torque exertedon a reference particle is a highly nonlinear problem, the way each neighbor con-tributes to these deviations in a statistical sense closely matches the binary hydro-dynamic interactions of two particles. This to say, for example, the shielding effectof an upstream neighbor is statistically still a drag-reducing effect on average, eventhough other neighbors are present. This is a very important conclusion, in thatit shows that the first-order effects can still be fairly accurately approximated byaccounting for binary interactions. We have in fact shown that by only account-ing for the first-order effects, our model is able to predict 65%–80% of the actualforce/torque variations experienced by each particle. It should be emphasized thatour microstructure-informed probability driven approach is not only able to makereasonably accurate predictions based on first-order interactions, but also preciselydelineates what is needed to be supplied as higher-order information to make themodel more accurate, if larger datasets become available. This means that themodel is not merely an ad-hoc solution to the problem, but a statistically completedescription of hydrodynamic interactions between particles in a dense array.Several potential directions are conceivable for future work. The model devel-opment process in this thesis was intended mainly to demonstrate its derivation andperformance. For the purpose of practical implementation, the probability distri-bution maps and the regression coefficients obtained for particular sets of Re andφ need to be appropriately interpolated, such that they are continuously applicablefor other encountered values in the tested range. Regarding the accuracy of themodel, it could also be of interest to statistically inspect the cases where the errorof the model is significantly high. This may be done to identify possible patterns148that might be responsible for such large errors, and to provide potential correc-tions to the model in order to enhance its prediction accuracy. Considering thepossibility of extending the MPP model to suspensions with mobile particles, theapproach presented in this thesis is readily applicable to gas-solid flows with solid-to-fluid density ratios of the order of O(100). This is the case since the responsetime of the particles would be much larger than that of the carrier fluid in suchflows, rendering particles effectively stationary compared to the time scale of thefluid evolution. For liquid-solid suspensions, however, the full description of thehydrodynamic force and torque on each particle involves relative translational androtational velocities and accelerations of the neighbors surrounding a test particle,in addition to their relative positions. Although the probability distribution mapsin the MPP approach can theoretically include any number of additional variables,the increased dimensionality of the functions would necessitate the availability ofsubstantially larger PR-DNS datasets. That being said, a critical next step is the im-plementation of the MPP model in an Euler-Lagrange simulation tool, followed bydetailed and carefully designed comparative tests with PR-DNS cases with identi-cal setups. Studies of this kind would be greatly beneficial to quantify the signif-icance of the inclusion of translational and rotational velocities and accelerationsof the neighbors, and to identify the relative importance of such contributions forconfigurations of interest.In conclusion, it is important to realize that the proposed MPP model, and otherrecent approaches in this field [194, 154, 152] have all taken advantage of super-position of effects in one way or another, to avoid complexities associated withnonlinear interactions. With this assumption incorporated in these models, it isassumed that the effect of each neighbor on the force or torque exerted on a testparticle is independent of the positioning of other neighbors, or is accounted for inan average manner at best. In this way, the total hydrodynamic effect of multipleneighboring particles may be obtained by linearly summing up the contributionsof each neighbor. The superposition assumption thus accounts for first-order ef-fects only and disregards higher-order interactions, enabling the MPP model topredict on average 70% of the total variation of the hydrodynamic loads. It isstriking, however, that a more or less similar level of performance has been shown149to be achieved by entirely different recent approaches [194, 154, 152], as well asin an unpublished work by the author concerning a superposition-based custom-architecture neural network model. The apparent similarity in the performance ofthese quite different models hints that we may have reached the upper limit withapproaches that employ superposition of effects and first-order interactions only.Therefore, the prediction of the remaining unexplained variations inevitably de-mands accounting for higher-order nonlinear effects. 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As depicted in figure A.1, the four pairs of counter-rotating vorticesand the corresponding four planes of symmetry can be identified at Reedge = 150.At higher Reynolds number up to Reedge = 180, the regions of vorticity with thesame signs show a tendency to join together, but the net lift force remains insignif-icant. Klotz et al. [31] also noted this gradual evolution from the basic flow to thesubsequent plane-symmetric flow. The double-threaded vortex structure, which isa characteristic of the wake after the first transition, is not identifiable in the Q-criterion visualizations of the wake. Between Reedge = 180 and Reedge = 200,the lift coefficient steadily increases. The double-threaded wake appears weakly atReedge = 190, and more strongly at Reedge = 200. The lift force also becomes172Re=150Re=200 Re=210 Re=215Re=215Re=170 Re=180Figure A.1: Evolution of the wake of a fixed cube with increasing the Reynolds num-ber. Left: Contours of stream-wise vorticity for −0.05 6 ωz 6 0.05 (ωz ≈ 0in gray regions) at a distance of d = 1.5 downstream of the cube center, Right:Vortex structures visualized using iso-surfaces of Q-criterion with Q = 0.003for Re = 215 and Q = 0.02 for Re = 225. Please note that Reynolds numbervalues are all based on the edge length of the cube.quite significant in direction of the symmetry plane due to the two major counter-rotating vortices at Reedge = 210. An example of this structure can be seen infigure A.1 for Reedge = 215. With further increasing the Reynolds number, theflow becomes time-dependent and shedding of “hair-pin” vortices can be observedat Reedge = 225. Our results indicate that the first transition must occur between180 < Reedge < 190, which agrees with Reedge = 184 reported in [31]. However,the second transition in our case is already present at Reedge = 225. In anotherstudy, Richter & Nikrityuk [45] found steady wake for Reedge = 200 and vortexshedding for Reedge = 250 which seems to confirm, roughly at least, our findinghere for the threshold of the second instability. Both of [42] and [31], however,report a higher value for the onset of unsteady regime. In our case, refining thecomputational mesh up to D/∆x = 40 or decreasing the time-step did not changethe outcome. The variations of the drag coefficient as a function of the Reynoldsnumber (based on the volume-equivalent diameter here) is shown in figure A.2.The correlations proposed in [21, 42] are used for plotting instead of the discreetdata points. Our results overlap with the correlation of Richter & Nikrityuk [21] al-most perfectly. The relative difference between our values of CD and those of [42]increases with the Reynolds number. The curve plateaus after Re = 250, while nosuch trend is observed for the correlation of [42], and the relative difference for the17350 100 150 200 250 300 350 4000.60.811.21.41.6 Fixed sphereFixed cube (Present study)Saha (2004)Richter & Nikrityuk (2013)Haider & Levenspiel (1989)Figure A.2: Drag coefficient data from the simulations performed in the presentstudy for a cube (open-circle markers), together with correlations proposed fora fixed cube in the literature (solid lines). The standard drag law of a fixedsphere [65] is also shown (dotted line).highest Re reaches 26%. In contrast, the correlation of Haider & Levenspiel [53]overestimates the drag coefficient almost throughout the entire range of Re, exceptfor Re < 90. This might be explained by the fact that the data used in [53] wasobtained from sedimentation experiments with mobile particles, which are shownto have higher drag coefficients compared to their fixed counterparts at sufficientlyhigh Reynolds numbers.A.2 Force data filtering processThe force data output with the DLM/FD method employed here is prone tospurious noise due to the particle traversing the computational grid, which is awell-known issue when dealing with moving boundaries in Immersed-Boundarymethods [195, 196, 197]. This is why smoothing/filtering on the force data is nec-essary as a post-processing step in order to reveal the actual trend of the data, whichmight otherwise remain masked under the numerical noise. A smoothed versionof the force data is therefore achieved by means of a Gaussian-weighted movingaverage filter with a window width of w = 1000∆t, where the standard deviationof the Gaussian function is set to σ = w/5. The value of w = 1000∆t is carefully1740 20 40 60 80 100-1.2-1-0.8-0.6-0.4-0.20Raw dataFiltered data(a)95.95 95.96 95.97 95.98 95.99 96-1.1-1.05-1-0.95-0.9Raw dataFiltered data(b)Figure A.3: (a) A demonstration of the original vertical force data and the filtereddata for (Ga,m) = (160, 0.2), (b) A magnified view of the same data over ashort time interval.chosen to ensure that the magnitude of the hydrodynamic forces is not artificiallydampened by the smoothing technique. Such an averaging window amounts tow ≈ 1 for rising and w ≈ 2 for settling cubes in terms of the physical time-scale ofthe system. An example of the original force output and the filtered data is shownin figure A.3 for the case (Ga,m) = (160, 0.2). As evident from figure A.3b, theperiod of these oscillations is O(∆t) = O(0.001), i.e., the numerical time-step,whereas the magnitude of the oscillations is controlled by the grid spacing ∆x.175


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