Direct simulation of ––solid flow with a free surface lattice Boltzmann method

Citation for published version (APA): Bogner, S. P. M., Harting, J. D. P., & Rüde, U. (2017). Direct simulation of liquid–gas–solid flow with a free surface lattice Boltzmann method. International Journal of Computational Dynamics, 31(10), 463-475. https://doi.org/10.1080/10618562.2018.1424836

DOI: 10.1080/10618562.2018.1424836

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Direct simulation of liquid–gas–solid flow with a free surface lattice Boltzmann method

Simon Bogner, Jens Harting & Ulrich Rüde

To cite this article: Simon Bogner, Jens Harting & Ulrich Rüde (2017) Direct simulation of liquid–gas–solid flow with a free surface lattice Boltzmann method, International Journal of Computational Fluid Dynamics, 31:10, 463-475, DOI: 10.1080/10618562.2018.1424836 To link to this article: https://doi.org/10.1080/10618562.2018.1424836

Published online: 18 Jan 2018.

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Direct simulation of liquid–gas–solid flow with a free surface lattice Boltzmann method

Simon Bognera,JensHartinga,b and Ulrich Rüdec aForschungszentrum Jülich, Helmholtz-Institut Erlangen-Nürnberg für Erneuerbare Energien, Nürnberg, Germany; bFaculteit Technische Natuurkunde, Technische Universiteit Eindhoven, Eindhoven, The Netherlands; cLehrstuhl für Systemsimulation, Universität Erlangen-Nürnberg, Erlangen, Germany

ABSTRACT ARTICLE HISTORY Direct numerical simulation of liquid–gas–solid flows is uncommon due to the considerable compu- Received  July  tational cost. As the grid spacing is determined by the smallest involved length scale, large grid sizes Accepted  December  become necessary – in particular, if the bubble–particle aspect ratio is on the order of 10 or larger. KEYWORDS Hence, it arises the question of both feasibility and reasonability. In this paper, we present a fully par- Lattice Boltzmann method; allel, scalable method for direct numerical simulation of bubble–particle interaction at a size ratio free surface flow; particle of 1–2 orders of magnitude that makes simulations feasible on currently available super-computing suspension simulation; resources. With the presented approach, simulations of bubbles in suspension columns consisting of liquid–gas–solid flow; more than 100,000 fully resolved particles become possible. Furthermore, we demonstrate the signif- bubble simulation icance of particle-resolved simulations by comparison to previous unresolved solutions. The results indicate that fully resolved direct numerical simulation is indeed necessary to predict the flow struc- ture of bubble–particle interaction problems correctly.

1. Introduction structures in every detail, and provides the most accurate solutions. Due to the computational complexity of fluid–solid and Todate,onlyfewDNSmodelsforLGSflowscanbe liquid–gas–solid (LGS) flow problems, numerical solu- found in the literature. A numerical method for this tionsareusuallybasedonhomogenisedmodels(Panetal. class of flows must combine a two-phase flow solver 2016; Panneerselvam, Savithri, and Surender 2009;Liand (Scardovelli and Zaleski 1999;Tryggvason,Scardovelli, Zhong 2015).Homogenisedmodelsdonotresolveall and Zaleski 2011) with a structural solver for the sus- involved scales and model the phase interaction based pended solid phase. Most LGS simulation approaches on closure relations (drag correlations) instead. The clo- (Li, Zhang, and Fan 1999;ChenandFan2004;vanSint sure relations, in turn, are obtained from experiments, Annaland, Deen, and Kuipers 2005a; Xu, Liu, and Tang or – with the advent of high-speed computers – by direct 2013; Sun and Sakai 2015;LiandZhong2015)donot numerical simulation (DNS) of systems of smaller size. fullyresolvetheparticlegeometrywithintheflow.This DNS techniques allow the most accurate predictions by means that hydrodynamic interaction between particles resolving even the smallest relevant length scales. For cannot be captured fully in these models which thus do fluid–solid particulate flows, the smallest typical length not count as DNS models according to the narrower defi- scale is the particle diameter. For the case of fluid–solid nition applied here. Nevertheless, these approaches make flows, new drag correlations have been derived from use of discrete particle methods (Bicani´ c´ 2004;Deenetal. numerical data (Beetstra, van der Hoef, and Kuipers 2007; 2007) to resolve particle–particle collisions. The first Tenneti, Garg, and Subramaniam 2011;Bogner,Mohanty, models to resolve both bubble and particle geometries and Rüde 2014;Tangetal.2015). Also, DNS has helped have been presented by Deen, van Sint Annaland, and to investigate the behaviour of particle suspensions and Kuipers (2009)andBaltussenetal.(2013). These DNS to study hydrodynamic interaction in particulate flows models combine a front-tracking liquid–gas method with (Aidun and Clausen 2010; Tenneti and Subramaniam an immersed boundary approach (Mittal and Iaccarino 2014) in full detail. Due to computational costs, the sys- 2005) to couple the flow with the particle simulation. tem sizes that can be realised by DNS are limited com- Recently, the same group applied their methodology to paredtounresolvedandhomogenisedmodels.Neverthe- study the effective drag on bubbles and particles in less, DNS is an important tool that allows the study of flow liquid flow (Baltussen, Kuipers, and Deen 2017). CONTACT Simon Bogner [email protected] ©  Informa UK Limited, trading as Taylor & Francis Group 464 S. BOGNER ET AL.

Presumably due to computational limitations, the sim- model of Ladd (1994) and Ladd and Verberg (2001). The ulated systems contain bubbles and particles of similar computational domain is subdivided into three disjoint size only. In many situations of practical relevance, how- regions, corresponding each to the space occupied by liq- ever, the particle size is much smaller than the bubble uid, gas, or solid phase, respectively. size.Alternatively,thereareeffortstocombinediffusive multiphase models with particle models (Stratford et al. 2.1. Hydrodynamic lattice Boltzmann model 2005;JansenandHarting2011;JoshiandSun2009). These DNS models work with fully resolved particles, To solve the hydrodynamic equations for the liquid but additional limitations arise from the necessity to region, we use a D3Q19 lattice Boltzmann model (Wolf- resolve also the liquid–gas interface – especially at a Gladrow 2005; Qian, d’Humieres, and Lallemand 1992) high-density difference. The density ratio is on the order on a Cartesian grid. The lattice velocities are denoted by O( ) of 10 in these models, which is much smaller than cq with q = 0, … , 18 and have units of grid spacing δx per the density ratio of most liquid–gas two-phase flows. time step δt.Thedatafq with q = 0, … , 18 of the scheme is Only recently, Connington, Lee, and Morris (2015)have also called particle distribution function (PDF). The lattice reached high-density ratios for special cases. Boltzmann equation of the model can be written as Inthefollowing,wepresentaDNSmodelforLGSflow that allows the simulation of bubble–particle interaction (x + c δ , + δ ) = ∗(x, ), in containing liquid. The model is based on the free sur- fq q t t t fq t (1a) face lattice Boltzmann method (FSLBM) of Körner et al. ∗(x, ) = (x, ) + λ neq,− + λ neq,+, (2005) for high liquid–gas density ratios combined with fq t fq t − fq + fq (1b) the particulate flow model of Ladd1994 ( )andLaddand ∗(x, ) Verberg (2001). Based on a previous effort (Bogner and where f t has been substituted, and is referred to as + − Rüde 2013), we have developed a model that is inherently the post-collision state. The upper indices ‘ / ’ denote parallel and allows bubble sizes one order of magnitude the even/odd parts of the respective function. The right- larger than the particle size while still fully resolving the hand side of Equation (1b) corresponds to the two relax- single-particle geometries. Since the grid spacing must ation time collision operator of Ginzburg, Verhaeghe, be smaller than the particle diameter, the total number and d’Humieres (2008), with the odd and even eigen- λ λ ࢠ − of lattice sites is necessarily large and the computational values −, + ( 2, 0). These eigenvalues thus con- cost is considerable. Therefore, the model is implemented trol the relaxation of the even and odd parts of the non- based on a parallel software framework (Feichtinger et al. equilibrium, 2011), that has already been used to realise massively par- neq(x, ) = (x, ) − eq(x, ), allel simulations of suspensions (Götz et al. 2010)and fq t fq t fq t (2) bubbly flows (Donath et al. 2009). The new LGS model enables detailed studies of particle transport in the defined as the deviation from the equilibrium function eq = (ρ(x, ), u(x, )) of rising bubbles. We demonstrate that our model is capa- fq eq t t , given as the polynomial ble of predicting important suspension properties, such ,α α cq u uαuβ 2 as increased effective with solid volume frac- (ρ, u) = ρw + + ( ,α ,β − δαβ ) , eq q 1 2 4 cq cq cs tion correctly. The terminal rise velocity of a gas bub- cs 2cs (3) ble decreases accordingly in simulations. Furthermore, where the w , q = 0, … , 18, are a set of lattice weights, we validate the free surface model for different bubble q √ and the constant c = δ /( 3 δ ) is called the lattice speed regimes according to the classification of Grace (spheri- s x t of sound.Themacroscopic flow variables of pressure and cal, ellipsoidal, skirted, dimpled), and present examples of velocityaremomentsofthePDF,thatis, particle transport and mixing in the wake of a single ris- ing bubble for the different regimes. The cost of the DNS 18 is considerable. However, a comparison of the results to (x, ) = 2ρ(x, ) = 2 (x, ), p t cs t cs fq t (4a) previous unresolved simulations indicates the necessity of q=0 DNS to predict the full characteristics of the flow and the 18 induced particle transport. (x, ) = 1 (x, ). uα t ρ cq,α fq t (4b) q=0 2. Method It can be shown that the velocity field is a second-order Inthefollowing,weuseahybridmethodbasedonthe accurate solution to the incompressible Navier–Stokes FSLBM of Körner et al. (2005)andtheparticulateflow equations (Frisch et al. 1987; Holdych et al. 2004;Junk, INTERNATIONAL JOURNAL OF COMPUTATIONAL FLUID DYNAMICS 465

I(t) gas cells (inactive)

interface cells

liquid cells

Figure . A fictitious interface I(t) and its discrete FSLBM representation consisting of gas, interface, and liquid nodes.

Klar, and Luo 2005) with kinematic viscosity ThesetofinterfacenodesCi is exactly the set of boundary nodes that possess a neighbour in the gas sub- domain. If an interface node x ∈ C has a gas neighbour ν =− 1 + 1 2δ . b i cs t (5) x + δ c ∈ λ+ 2 t q Cg, then the boundary condition of Körner et al. (2005), λ While the first relaxation parameter + is chosen accord- ∗ + f ¯(x , t + 1) =−f (x , t) + 2e (ρw, uw ), (7) ingtothedesiredflowviscosity,thesecondparameterλ− q b q b q is fixed to satisfy the equation ¯ isappliedfortheoppositedirectionq with −cq = cq¯. Here, pw = c2ρw defines the boundary value for pressure, 1 1 1 1 3 s + + = . and uw represents the flow velocity at the boundary. It can λ λ (6) + 2 − 2 16 beshownthatEquation(7) yields a first-order approx- imation of a free boundary (Bogner, Ammer, and Rüde This ‘magic’ parameterisation is optimal for straight-axis- 2015). aligned wall boundaries (Ginzbourg and Adler 1994), and The interface capturing scheme is updated according yields viscosity-independent solutions in general geome- to the flow simulation in every time step. The indica- tries (Ginzburg and d’Humieres 2009). tor function ϕ(x, t) is updated directly from the lattice Boltzmann data. ⎛ ⎞ Q−1 2.2. Free surface lattice Boltzmann method (FSLBM) 1 ϕ(x , t + 1) = ϕ(x , t) + ⎝ m (x , t)⎠ , i i ρ(x , t + 1) q i The FSLBM is an interface capturing scheme that is based i q=1 on the volume of fluid (Hirt and Nichols 1981;Tryggva- (8a) son, Scardovelli, and Zaleski 2011)approach.Thefill level with the direction-dependent exchange mass or volume fraction ϕ(x) serves as indicator function. For x ϕ(x) (x , ) each node ,thefilllevel is defined as the volume mq⎧i t fraction of liquid within the cubic cell volume around x. ⎪ x + c ∈/ ( ∪ ), ⎪0ifi q Ci Cl Figure 1 shows that three different types of cells can be ⎨⎪ 1 (ϕ(x + c ) + ϕ(x )) if x + c ∈ C , distinguished: = 2 i q i i q i ⎪ ( ¯(x + c ) − (x )) ⎪ fq i q fq i r ⎩ (x + c ) − (x ) x + c ∈ , C (t): the set of gas nodes,whereϕ = 0. fq¯ i q fq i if i q Cl r g C (t): the set of liquid nodes,whereϕ = 1. r l (8b) Ci(t): the set of interface nodes,where0<ϕࣘ 1. x An interface node always has a liquid neighbour The sets Cl, Ci,andCg are updated according to x + δ c ∈ x + δ c ∈ t q Cl and a gas neighbour t p Cg for the rules illustrated in Figure 2, where each arrow cor- some p, q = 1, … , 18. responds to a possible state transition: The transitions between gas, liquid, and interface state are triggered The nodes in ClCi are active lattice Boltzmann according to the fill levels ϕ of the interface cells. When- nodes. Introducing further the set of obstacle nodes Cs(t) ever the fill level of an interface cell becomes equal to that are not part of the fluid domain (e.g. walls or nodes 0(equalto1),thenaconversionintoagascell(liquid that are blocked out by particles), CgCs forms the set of cell) is triggered. If needed, inverse transitions from liquid inactive nodes within the simulation domain. (gas) into interface state are performed in order to close 466 S. BOGNER ET AL.

ϕ(t + δt) ≥ 1.0 ϕ(t + δt) ≤ 0.0

liquid interface gas

neighbor con- neighbor converts verts into gas into liquid

Figure . Possible cell state conversions in FSLBM simulations. Conversions of interface cells are triggered by the fill level ϕ and, for gas and liquid cells, by conversions of neighbouring interface cells into liquid or gas, respectively (Bogner ).

the layer of interface cells. If a gas node changes to inter- ωP(t). Any grid node inside of a particle is called obsta- face state, then its LBM data is initialised based on the cle node. Whenever a liquid node xb is next to an obsta- equilibrium, Equation (3). Details can be found in Körner cle node covered by particle P, then the bounce-back rule et al. (2005)andinBognerandRüde(2013)formoving with velocity term, particles. Since the flow of the gas phase is not simulated in the (x , + ) = ∗(x , ) − −(ρ , u ), fq¯ b t 1 fq b t 2eq w w (11) free surface model, a special treatment of the individual bubbles, i.e. connected regions of gas, is necessary (cf. is used to impose the particle surface velocity, Anderl et al. 2014;Caboussat2005;Körneretal.2005). Such a bubble model conserves the mass in the gas phase uw = uP + ωP × (xw − xP ), (12) and provides the local gas pressure pg needed to define the boundary condition, Equation (7). ρ For the simulation of capillary flows, the Laplace at the boundary. In Equation (11), w is substituted with pressure jump across the interface can be included in thedensityvalueoftheboundarypointfromtheprevious Equation (7). The boundary value for the pressure is time step. then For the time integration of the particle data, the hydro- dynamic force FP and torque TP are computed from the lattice Boltzmann data. Based on the pw = p (x, t) + 2σκ(x, t), (9) g exchange principle,onecomputes κ where pg is the pressure in the gas bubble and is the local δ3 curvature of the interface. Following Brackbill, Kothe, F = j (x) x , P q δ (13) x∈ t andZemach(1992), the curvature can be computed from BP q∈IP (x) the fill levels, based on the equations δ3 T = (x − x ) × j (x) x , P P q δ (14) x∈ ∈ (x) t n =∇ϕ, (10a) BP q IP κ(x) =−(∇·nˆ ). (10b) where

Toevaluatetheseexpressions,weuseanoptimised ∗ jq(x) := cq f (x, t) − cq¯ f (x, t + 1) (15) finite difference scheme according to Parker and Youngs (1992). At the solid particles, perfect wettability is is used to approximate the momentum transferred along assumed. Further details and alternative curvature com- x putation schemes can be found in Cummins, Francois, a single boundary-intersecting link at a boundary node (Ladd 1994; Ladd and Verberg 2001). In Equations (13) and Kothe (2005), Popinet (2009) and Bogner, Rüde, and and (14), the set B Harting (2016). P is the set of all nodes surrounding the particle P with a nonempty set IP(xb) of particle sur- face intersecting links. If BP contains lattice nodes on the 2.3. Particulate flow simulation inside of another particle, the equilibrium distribution is assumed in Equation (15). Each (spherical) particle is defined by its radius RP,spe- The hydrodynamic lubrication forces obtained by ρ cific density P, and a Lagrangian description consist- Equation (13) are valid only if the gap between two par- ing of position xP(t),velocityuP(t), and angular velocity ticles P1, P2 is sufficiently resolved. Hence, if the gap size INTERNATIONAL JOURNAL OF COMPUTATIONAL FLUID DYNAMICS 467 becomes smaller than c = 2/3 δx,thenalubrication cor- Kromkamp et al. 2006). Here, we reproduce as validation rection, experiment the relative shear viscosity of a spherical par- ticlesuspensioninashearflowbetweenplates.Similar 6πμ(R R )2 1 1 Flub =− P1 P2 − xˆ to Kromkamp et al. (2006), a domain is initialised with P ,P 1,2 1 2 (R + R )2 |x , |−R − R P1 P2 1 2 P1 P2 c a random particle bed of N spherical particles of radius · (u − u )xˆ , , P1 P2 1 2 (16) RP = 8 δx and specific density ρs = 8.Thedomainsizeis fixed to V = 114 δx × 116 δx × 180 δx, altering N to realise F x , = x − x is added to the net force P1 ,where 1 2 P2 P1 is the different solid volume fractions relative position of the particles (Ladd and Verberg 2001). μ = ρν NV Here, is the dynamic viscosity. This improves  = P , (17) the simulation of hydrodynamic interaction between par- V ticles (Aidun and Clausen 2010). Since Equation (16) where VP istheparticlevolume.Theflowisinitiallyat diverges for |x , |→0, the gap size is limited from below 1 2 rest, and driven by imposing a constant velocity u = to be at least 0.2 . Furthermore, to increase stability at x c ±0.01δ /δ on the boundary planes at z = 0andz = 180 higher solid volume fractions, the time integration of the x t in opposed directions, while applying periodicity along x particles proceeds in up to 10 time steps per LBM step. and y directions. The effective viscosity μ of the numer- The inclusion of wetting boundaries is described in s ical suspension model is evaluated by measuring the net Brackbill, Kothe, and Zemach (1992)andBogner,Rüde, force F¯ on the boundary walls, and Harting (2016). However, we only study fully wetting x particles in the following. F¯ μ = x , (18) s Aγ˙ 3. Validation of the numerical model where γ˙ is the shear rate. The resulting force values oscil- The simulations of bubbles in moderately dense sus- late due to non-trivial interaction between particles, and pensions and bubble–particle interaction are found in must be averaged over a number of time steps T (typically Section 4. Here, we first validate the correct behaviour γ˙ T ≥ 100). The (particle) Reynolds number is defined of solid–liquid suspension simulations (Section 3.1)and as gas–liquid simulations (Section 3.2)withourmodel. ρ γ(˙ )2 = f 2RP , ReP μ (19) 3.1. Validation of particle suspension model

It has been demonstrated in the past that the LBM is valid where ρf isthefluiddensity.InBogner(2017), the same in the simulation of particle suspensions (e.g. Aidun and setup was repeated with various solid volume fractions Clausen 2010; Ladd and Verberg 2001; Harting et al. 2014; and Reynolds numbers. As shown in Figure 3,themodel

ReP =0.011 Φ=0.1 8 ReP =0.1 Φ=0.2 8 /μ . /μ s s ReP =1 Φ=03 μ μ ReP =2 Φ=0.4 6 Equation (20) 6

4 4 relative viscosity 2 2 relative viscosity

0.10.20.30.4 10−2 10−1 100

solid volume fraction Φ Reynolds number ReP (a) (b)

Figure . Simulation of a thickening particle suspension in shear flow. The relative viscosity of the simulated suspension increases with solid volume fraction and Reynolds number (Bogner ). (a) Relative viscosity at various solid volume fractions. (b) Relative viscosity at various Reynolds numbers. 468 S. BOGNER ET AL.

open (pressure) as

gμ4 ρ Mo = , (21) ρ2σ 3

μ ρ free-slip where g is the gravitational constant, and are the viscosity and density of the surrounding liquid, and ρ

Ωl isthedensitydifferenceofgasandliquid.TheEötvös free-slip number Ωg 2R ρ 2 = gd , Eo σ (22)

no-slip where d is the diameter of the bubble, characterises the ratio of and forces. Finally, the Figure . Boundary conditions around liquid column for rising =  bubble Reynolds number is defined as bubble simulations. At t , a spherical bubble g of diameter R is initialised, surrounded by quiescent liquid  . l ρ = u∞d , Reb μ (23) correctly predicts the expected increase of effective vis- cosity with increased solid volume fraction and Reynolds where u is the terminal rise velocity of the bubble. number. Figure 3(a) also displays the empirical correla- The bubble diameter d is understood as the diameter of tion of Eilers (Stickel and Powell 2005), a volume-equivalent spherical bubble, unless otherwise  noted. Based on Grace (1973), the behaviour of gas bub- μ () .  2 bles rising in a liquid column can be classified and allows s = + 1 25 , 1 (20) a prediction of the bubble shape, e.g. spherical, ellipsoidal, μ 1 − /max sphericalcap,skirted,dimpled,etc.,orcanbeusedtoesti- matetheterminalrisevelocityu of the bubble, if Eo and where max = 0.63 is assumed as the maximal packing Mo are given. Alternatively, one can work with the corre- fraction for random sphere packings. lation of Fan and Tsuchiya (1990),

⎡   ⎤ / − √ −n/2 1 n 3.2. Validation of free surface model −1/4 n ⎣ Mo Eo 2c Eo ⎦ u˜∞ = + √ + , The FSLBM described in Section 2.2 has been vali- Kb Eo 2 dated for the case of single rising bubbles in liquid (24) columns in Bogner (2017), from which the following where u˜∞ isthenondimensionalrisevelocity.Velocity results are adopted. The behaviour of single rising bubbles and diameter are made nondimensional using in a quiescent (infinite) liquid is characterised by three- dimensionless numbers (Clift, Grace, and Weber 1978; 1/4 ρ ˜ ρg 1/2 Fan and Tsuchiya 1990). The Morton number is defined u˜ = u , and d = d . (25) σ g σ

Table . Test cases for different bubble regimes. Reb is the expected Reynolds number for infinite domains according to Equation () with n = ., c = . and = ∗ Kb  in Equation (). Reb is the value obtained in FSLBM simulations in a finite domain with free-slip walls. Simulations were parameterised according to the given viscosity μ, liquid–gas surface tension σ , and gravitational constant g, assuming a liquid mass density of ρ =  kg/m.

 ∗ Case μ (Pa s) σ (N/m) g (m/s ) Mo Eo Reb Reb

A (spherical) . . . . × − . . . B (ellipsoidal) . . . . . . . C (skirted) . . . . . . . D (dimpled) . . .  . . . INTERNATIONAL JOURNAL OF COMPUTATIONAL FLUID DYNAMICS 469

= δ Figure . Bubble shapes obtained from simulations of the bubble regimes from Table . (a) Case A (spherical), t  t;(b)CaseB = δ = δ = δ (ellipsoidal), t  t; (c) Case C (skirted), t  t; (d) Case D (dimpled), t  t.

= δ Figure . [Colour online] Velocity field in the slice y  x of the simulations of the bubbles regimes from Table . Red colour indi- = δ cates high flow velocity, blue indicates low-velocity magnitude. Black lines indicate the free surface. (a) Case A (spherical), t  t; = δ = δ = δ (b) Case B (ellipsoidal), t  t; (c) Case C (skirted), t  t; (d) Case D (dimpled), t  t. 470 S. BOGNER ET AL.

liquid). The value of Kb is adapted as 0.54 = ( , −0.038),

∞ Kb max 12 Kb0 Mo (26) u . 0 52 where Kb0 depends on the liquid (e.g. Kb0 = 14.7 for water). Like the Grace diagram, correlation equation (24) is obtained from experimental data, and can predict u 0.5 with an error of about ±10%. van Sint Annaland, Deen, and Kuipers (2005b)suggest terminal velocity ¯ four different bubble regimes for the validation of a two- 0.48 phasevolumeoffluidsolver,thatareusedasareference in the following. The Mo and Eo numbers used in the fol- 0% 2% 4% 6% 8% 10% lowing simulations (Table 1)havebeenchosentorepre- solid volume fraction Φ sent the test cases suggested by van Sint Annaland, Deen, and Kuipers (2005b). In this reference, the surface ten- Figure . Dimensionless terminal rise velocity at solid volume frac- sion modelling is based on Brackbill, Kothe, and Zemach tions from % to %. (1992), similar to our finite difference model. The authors In Equation (24), the parameters Kb, c,andn are cho- suggest free-slip boundary conditions for all lateral direc- sen to account for special material properties not covered tions, no-slip at the bottom, and a pressure boundary at by Re, Mo, and Eo. The parameter n ranges from 0.8 to thetopofthedomain.Asketchofthedomainandini- 1.6 depending on the liquid purity, while c is chosen as tial conditions is shown in Figure 4.Thefluidparame- 1.2 (single-component liquid) or 1.4 (multi-component ters for the simulation collected in Table 1 are given with

Figure . Bubble rise without particles, at selected time steps (ordered from left to right). The spherical cap shape agrees well with the prediction according to Grace (). (a) Three-dimensional view of the bubble shape within liquid column. (b) Slice through the centre = δ of the domain (y  x). INTERNATIONAL JOURNAL OF COMPUTATIONAL FLUID DYNAMICS 471

Figure . [Colour online] Bubble rise from a bed consisting of , particles ( = .%) at selected time steps (times chosen identical to Figure ). (a) Three-dimensional view of bubbles and particles. Particle colour indicates the initial z coordinate of particle position at = = δ t . (b) Slice through the centre of the domain (y  x). respect to a liquid of density ρ = 1000 kg/m3.Thegrid 4. Results −3 −4 spacing is δx = 10 mandthetimestepisδt = 10 s. As a compromise between computational cost and influ- 4.1. Bubble particle interaction ence of the finite domain size on the bubble dynamics, × × Wenow study cases of bubble–particle interaction. Again, adomainsizeof40 40 100nodesisrecommended thebasicsetupconsistsofaliquidcolumncontaininga by the authors. Here, we directly adopt the resolution single bubble of radius R = 0.01 m rising within a bed of and the domain size of the original. The initial condition = × −4 = δ spherical particles of radius RP 8 10 m. Using a grid consists of a spherical bubble of R 6 x centred around δ = × −5 δ spacing of x 8 10 m,thesizeofthecomputational the position (20, 20, 10) x in a column of quiescent domain is 500 × 500 × 1300 lattice cells surrounding the liquid. δ ࣙ initially spherical bubble at (250, 250, 250) x.Thismeans For t 0, the initially spherical bubble starts to accel- that each particle is resolved by 5 lattice cells per diame- erate due to the pressure gradient until it reaches a ter- ter. The surface tension σ is 0.145 N/m and the gravity is minal velocity and the bubble shape does not change any assumed to be 0.981 m/s2, such that the spherical bubble more. Figure 5 shows the simulated bubble shapes using regime is expected, in a fluid of density ρ = 1000 kg/m3 a triangulation of the smoothed indicator function con- μ = ϕ = and viscosity 0.25 kg/(m s). Within the liquid tour surface 0.5. For each case, the simulated bubble column,ahomogeneousparticlebedisinitialisedby shape agrees well with the predictions according to the choosing random positions. The bed density  varies Grace diagram. The velocity field around the bubble is withtheparticlenumberN, shown in Figure 6. As reported also by van Sint Anna- land, Deen, and Kuipers (2005b), the terminal veloci-  = NVP , ties obtained from simulations agree reasonably well with (27) V − Vb the predictions of experimental relations. Table 1 lists the ∗ simulated terminal Reynolds number Re in comparison = / π 3 = b where VP 4 3 RP istheparticlevolumeandVb to the prediction according to Equation (24). 4/3πR3 is the bubble volume. The solid mass density is 472 S. BOGNER ET AL.

Figure . [Colour online] Bubble rise from a bed consisting of , particles ( = %) at selected time steps (times chosen identical to Figure ). (a) Three-dimensional view of bubbles and particles. Particle colour indicates the initial z coordinate of particle position at = = δ t . (b) Slice through the centre of the domain (y  x).

3 ρs = 3000 kg/m . At the given bubble–particle size ratio, in this work, solid volume fractions of 20% often devel- the liquid–solid system surrounding the bubble can be oped instabilities and nonphysical behaviour. The reason viewed as a homogeneous medium of increased density seemstobethedistributionofparticlesnexttotheliquid– and viscosity. Notice that the effective time scale of par- gas interface that can make the free-surface algorithm ticle sedimentation is low compared to the expected rise ineffective by covering the cells containing the liquid– velocity of the bubble. gas interface. This is currently a limitation of the model, In a series of simulations, the bubble is released within which might be improved by altering the surface tension theparticlebed,andtheterminalrisevelocitydepending modeltosatisfytheperfectwettabilityofparticlesmore on the bed solid volume fraction is evaluated. Figure 7 accurately. shows the decrease of bubble velocity with increased bed 4.2. Simulation of bubble-induced particle mixing density. Due to the presence of the particles, the aver- agemassdensityoftheparticlesuspensionaroundthe Theparticlebedisnowlimitedtotherangez = [0, bubble increases, and the buoyancy force on the bubble 500] δx that includes the initially spherical bubble. We is increased. This explains the small increase in veloc- assume a liquid density of ρ = 1000 kg/m3,viscosityμ = ityfrom0%to1%solidvolumefraction.Moresignif- 10−1 kg/(m s), liquid–gas surface tension σ = 0.1 N/m, icantly, with higher solid volume fraction, the higher and a gravity g =−9.81 m/s2 along the z-axis. Choosing −5 effective viscosity of the suspension reduces the terminal thetimestepasδt = 2.5 × 10 s, the lattice relaxation velocity reached. This is in agreement with experiments time becomes τ ࣈ 1.672. The dimensionless numbers from the literature, that reports a decreasing rise veloc- for the bubble are Mo = 9.81 × 10−4 and Eo = 39.24, ity with increased suspension thickness (Tsuchiya et al. with an expected terminal Reynolds number Reb = 61.73 1997). according to Equation (24)(withn = 1, c = 1.2). This We remark that the model is currently limited to low setup has been chosen similar to a test case of Deen, van solid volume fractions. For the spatial resolution applied Sint Annaland, and Kuipers (2007). INTERNATIONAL JOURNAL OF COMPUTATIONAL FLUID DYNAMICS 473

Figure. [Colour online] Particles mixing in the wake of bubbles at different bubble regimes (initial bed density  = %). The simulations allow a detailed investigation of the wake structure and the accompanying mixing process of the particles. In the spherical case (a), only a thin filament of particles follows the upward motion of the bubble. Hardly any particles from lower layers (green or blue) of the bed are carried upwards. In the ellipsoidal case (b), the effect is stronger and one observes a significant portion of green particles following the bubble wake. Cases (c) and (d) feature a circulating flow in the wake of the bubble. This recirculating region in the wake of the bubble can carry a larger number of particles. The effect is strongest in case (d), where particles from lower layers of the bed (green) change relative position with particles from the top (red). (a) Spherical, Eo = 2.71, Mo = 1.26 × 10−3; (b) ellipsoidal, Eo = 27.1, Mo = 9.09 × 10−2; (c) dimpled, Eo = 271, Mo = 1.02 × 103;(d)skirted,Eo = 39.2, Mo = 9.81 × 10−4.

Without any particles ( = 0), the terminal rise Scenarios of this complexity require a considerable ∗ = Reynolds number obtained from simulations is Reb 48. amount of computational cost. The run time in each of Again,thelowervelocitycanbeattributedtowalleffects. the above cases is approximately 6.66 hours when utilis- Figures 8–10 show the process without particles and at ing 2000 cpu cores of a distributed system in parallel. The bed densities  = 2.5% and  = 10%. The terminal rise considerable complexity stems from the high resolutions velocity is hardly affected by the presence of the particles required by the DNS. Here, we have used 3.25 × 108 grid in this case, after the top of the bubble is uncovered from points, whereas the unresolved simulations of Deen, van particles.Intheseimages,theparticlecolourindicatesthe Sint Annaland, and Kuipers (2007)requireonly1.6× 105 initial z-position of the particle, to show the mixing of grid points. different bed layers in the wake of the bubble. A circulat- ing motion is observable in the wake of the bubble. This effect seems to be absent in the unresolved simulations of 5. Conclusion Deen, van Sint Annaland, and Kuipers (2007), where the DNS of LGS flows offers the possibility of detailed recirculation region seems to have no influence on the studies of bubble–particle interaction in . The particles.Also,alargenumberofparticlesfromthemid- parallel model proposed in this paper allows, to the best of dleandlowerlayerofthebedarecarriedinthewakeof ourknowledge,forthefirsttimeparticle-resolvedsimula- thebubble.Thatis,wecanobservesubstantialchanges tions of gas bubbles within slurry columns. This is possi- in the relative positions of particles during the mixing ble, thanks to the parallel design of the model that allows process. the exploitation of the parallelism of modern supercom- Figure 11 shows the same setup with altered fluid prop- puters. It has been demonstrated that the model can sim- erties corresponding to different bubble regimes. The ulate particle mixing in the wake of rising gas bubbles. wake and particle structure differs significantly in the Thestructuresformedbyparticlesinthewakeofthebub- different regimes. The bubble wake is strongest for the bles can be studied in great detail. The effect of differ- skirted regime that generates the largest recirculation ent bubble regimes, i.e. bubble size and surface tension, region that also displays the highest solid mass transport. ontheparticletransportcanbeanalysed.Acompari- At the other extreme, we find that the spherical case has son with previous, unresolved simulations indicates that the least mixing effect and generates only a thin cone of particle-resolved DNS is indeed necessary to predict this lifted particles. flow structure correctly. 474 S. BOGNER ET AL.

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