Dimensionality and morphology of particle and bubble clusters in turbulent flow
Enrico Calzavarini, Martin Kerscher, Detlef Lohse, Federico Toschi
aa r X i v : . [ n li n . C D ] F e b Dimensionality and morphology of particleand bubble clusters in turbulent flow
By Enrico Calzavarini , , Martin Kerscher , Detlef Lohse , , andFederico Toschi , Physics of Fluids Group, Department of Science and Technology, J.M. Burgers Center forFluid Dynamics, and Impact-Institute,University of Twente, P.O Box 217, 7500 AE Enschede, The Netherlands, Mathematisches Institut, Ludwig–Maximilians–Universit¨at, Theresienstrasse 39, D–80333M¨unchen, Germany, IAC-CNR, Istituto per le Applicazioni del Calcolo, Viale del Policlinico 137, I-00161 Roma,Italy and INFN, via Saragat 1, I-44100 Ferrara, Italy International Collaboration for Turbulence Research(Received 4 November 2018)
We conduct numerical experiments to investigate the spatial clustering of particles andbubbles in simulations of homogeneous and isotropic turbulence. Varying the Stokes pa-rameter and the densities, striking differences in the clustering of the particles can beobserved. To quantify these visual findings we use the Kaplan–Yorke dimension. This localscaling analysis shows a dimension of approximately 1.4 for the light bubble distribution,whereas the distribution of very heavy particles shows a dimension of approximately 2.4.However, clearly separate parameter combinations yield the same dimensions. To over-come this degeneracy and to further develop the understanding of clustering, we performa morphological (geometrical and topological) analysis of the particle distribution. Forsuch an analysis, Minkowski functionals have been successfully employed in cosmology,in order to quantify the global geometry and topology of the large-scale distribution ofgalaxies. In the context of dispersed multiphase flow, these Minkowski functionals – be-ing morphological order parameters – allow us to discern the filamentary structure of thelight particle distribution from the wall-like distribution of heavy particles around emptyinterconnected tunnels.
1. Introduction
Even in homogeneous and isotropic turbulence particles, drops, and bubbles (all fromnow on called “particles”) do not distribute homogeneously, but cluster , see Crowe et al. (1996) for a classical review article. The clustering has strong bearing on so diverse issuessuch as aerosols and cloud formation (Falkovich et al. (2002); Celani et al. (2005); Vail-lancourt et al. (2002)), plankton distribution in the deep ocean (Malkiel et al. (2006)),sedimentation and CO deposition in water (Upstill-Goddard (2006)). Considerable ad-vances in particle tracking velocimetry (Porta et al. (2001); Hoyer et al. (2005); Bourgoin et al. (2006); Bewley et al. (2006); Ayyalasomayajula et al. (2006)) and in numerics (El-ghobashi & Truesdell (1992, 1993); Wang & Maxey (1993); Boivin et al. (1998); Druzhinin& Elghobashi (2001); Marchioli & Soldati (2002); Mazzitelli et al. (2003 a , b ); Biferale et al. (2005); Chun et al. (2005); van den Berg et al. (2006); Bec et al. (2006 b , a )) now allowfor the acquisition of huge data sets of particle positions and velocities in turbulence. Calzavarini, Kerscher, Lohse, and Toschi (a) (b) (c)
Figure 1. (color) Snapshots of the particle distribution in the turbulent flow field for St = 0 . β = 3 (bubbles), (b) β = 1 (tracers), and (c) β = 0 (heavy particles), all for Re λ = 75.Corresponding videos are shown in the accompanying material (or can be downloaded fromhttp://cfd.cineca.it/cfd/gallery/movies/particles-and-bubbles). In our numerical experiments the fluid flow is simulated by the incompressible Navier-Stokes equation on a 128 , a 512 , and a 2048 grid with periodic boundary conditionsand a large scale forcing, achieving Taylor-Reynolds numbers of Re λ = 75, 180 and400, respectively. One-way coupled point-particles are included which experience inertiaforces, added mass forces, and drag, i.e., the particle acceleration is given by (Maxey &Riley (1983); Gatignol (1983)) d v dt = β DDt u ( x ( t ) , t ) − τ p ( v − u ( x ( t ) , t )) , (1.1)where v = d x /dt is the particle velocity and u ( x ( t ) , t ) the velocity field. Equation (1.1)holds in the limit of small ( ≪
1) particle Reynolds number and for particles whose size issmall as compared to the Kolmogorov length scale η – for finite size particles the resultswill naturally differ. Lift, buoyancy, two-way-coupling and particle-particle interactionsare ignored. Next to the Reynolds number the control parameters are the particle radius a , the density of the fluid ρ f and of the particle ρ p . The dimensionless numbers used tocharacterize the particle in the turbulent flow are the density ratio β = 3 ρ f / ( ρ f + 2 ρ p )and the Stokes number St = τ p /τ η = a / (3 βη ), where τ p = a / (3 βν ) is the particle timescale, ν the viscosity, and τ η the Kolmogorov time scale. In our numerical study we treat β and St as independent parameters. The case β = 0 (and finite St ) corresponds to veryheavy particles and β = 3 to very light particles, e.g. (coated) bubbles (i.e., with no-slipboundary conditions at the interface). β = 1 means neutral tracers and if in addition St = 0 we have fluid particles.In figure 1 snapshots of the resulting particle distributions from a simulation with Re λ = 75, St = 0 . N = 10 are shown. For β = 0(heavy particles) and β = 3 (light bubbles) we observe clustering – but of different type.No clustering is observed for the neutral tracers with β = 1.
2. Dimensionalities of particle and bubble clusters
How to quantify and characterize the clustering? One way borrowed from dynamicalsystem theory is to calculate the Kaplan-Yorke dimension D KY in the six-dimensionalspace spanned by the particle positions and their velocities, see Bec (2003, 2005). TheKaplan-Yorke dimension follows from the Lyapunov exponents (Eckmann & Ruelle(1985)), λ i ( i = 1 ,. . . , D KY = K + P Ki =1 λ i / | λ K +1 | , K being the largest integer such that P Ki =1 λ i ≥ β = 0 was studied. imensionality and morphology of particle and bubble clusters in turbulent flow KY β StD KY β St Figure 2. (color) The Kaplan-Yorke dimension D KY and the correlation dimension D (bottom)of the particle distribution as function of St and β . The black line in D KY marks the value D KY ( β, St ) = 3, the red line the value 2. For the D KY plot 10 particles were integrated alongthe tangent space and for the D plot 5 · particles were followed. In both cases the averagingtime was tens of large eddy turnovers. The particles were grouped in about 500 different typescharacterized by their ( β, St )-values. Re λ = 75. In fig. 2, upper, the full landscape of D KY as function of β and St is shown for Re λ = 75; in fig. 3 cuts through this landscape for fixed β = 0, 1, and 3 are shown forboth Re λ = 75 and Re λ = 180, revealing at most a minute Reynolds number dependenceof D KY .We now discuss the dependence D KY ( β, St ): Point-particles ( St = 0, β = 1) do notexperience any drag ( v = u ) and accordingly D KY = 3. For fixed β the contractionis strongest for a Stokes parameter St in the range between 0.5 and 1.5, i.e., when the Calzavarini, Kerscher, Lohse, and Toschi D KY St β =1 β =0 β =3 Figure 3.
The Kaplan-Yorke dimension D KY vs. St for three β values: β = 0 ( ◦ ) , △ ) , (cid:3) ).Results at two different Re λ are reported: Re λ = 75 (filled symbols), Re λ = 180 (open symbols).(a) (b) (c) Figure 4. (color) Projection of the particles in a slice of 19 η thickness and side 640 η × η with Re λ = 180, for (a) St = 0 .
1, (b) St = 0 .
6, and (c) St = 4. Heavy particles ( β = 0) are shown inred, bubbles ( β = 3) in blue. To illustrate the different clustering, particles and bubbles in thesame simulation are plotted on top of each other – note that they do not mutually interact, i.e.,collisions and feedback on the fluid are neglected. characteristic time scale τ p of the particle roughly agrees with the Kolmogorov timescale τ η . For smaller St the particles follow the small scale turbulent fluctuations andthe clustering decreases and D KY increases. For large St the particles are so big thatthey average out the small scale turbulent fluctuations (see figure 4); correspondingly, theclustering decreases and D KY increases. The dynamical evolution of the heavy particles( β = 0) shows the strongest contraction for St ≈ . D KY = 2 .
6. Also that of thelight particles ( β = 3) has the strongest contraction for St ≈ .
5. Now even D KY = 1 . D KY which is worth beingmentioned: D KY ( St, β = 1) slightly increase with increasing St , reflecting the fact thatlarge particle, even if neutrally buoyant, do not exactly follow the flow, see the discussionin Barbiano et al. (2000).An analysis of the spatial distribution performed with the correlations dimension D imensionality and morphology of particle and bubble clusters in turbulent flow D KY , see fig.2, lower (see also Bec et al. (2007) for further details on this observable). The D iscomputed by fitting the separation probability P ( r ), which is the probability that thedistance between two particles is less than r . It is assumed that P ( r ) ∼ r D as r → D , the curve D ( β, St ) isslightly less smooth as compared to D KY ( β, St ). The finiteness of the particle samples,here roughly 20 snapshots of 10 particles, leads to noisy measures especially for thehighly clustered cases ( β ∼ P ( r ).Due to its local character a metric measurement with the Kaplan–Yorke or correlationdimension cannot supply us with global morphological information. Indeed, as seen fromfigure 2, clusters of heavy and light particles can have the very same D KY , though theylook very different. Indeed, the striking morphological differences of the particle andbubble distribution within the same turbulent velocity field are illustrated in figure 4.
3. Morphology of particle and bubble clusters
With global geometrical and topological order parameters we are able to distinguishthe clustering on spatially extended, eventually interconnected, sheets from clouds orfilamentary clustering. Consider the union set A r = S Ni =0 B r ( x i ) of balls of radius r around the N particles at positions x i , i = 1 , , . . . N , thereby creating connections be-tween neighboring balls. The global morphology of the union set of these balls changeswith radius r , which is employed as a diagnostic parameter. It seems sensible to re-quest that global geometrical and topological measures of e.g. A r are additive, invariantunder rotations and translations, and satisfy a certain continuity requirement. Withthese prerequisites Hadwiger (1957) proved that in three dimensions the four Minkowskifunctionals V µ ( r ), µ = 0 , , ,
3, give a complete morphological characterization of thebody A r . The Minkowski functional V ( r ) simply is the volume of A r , V ( r ) is a sixthof its surface area, V ( r ) is its mean curvature divided by 3 π , and V ( r ) is its Eulercharacteristics. Volume and surface area are well known quantities. The integral meancurvature and the Euler characteristic are defined as surface integrals over the mean andthe Gaussian curvature respectively. This definition is only applicable for bodies withsmooth boundaries. In our case we have additional contribution from the intersectionlines and intersection points of the spheres. Mecke et al. (1994) discuss the definitionsof the integral mean curvature and the Euler characteristic for unions of convex bodies.The Euler characteristic as a topological invariant allows for several other definitions,like χ = − et al. (1994) and have successfully been used in cos-mology by Kerscher et al. (1997); Kerscher (2000); Kerscher et al. (2001), porous anddisordered media by Arns et al. (2002), dewetting phenomena by Herminghaus et al. (1998), statistical physics in general (Mecke (2000)) and have been recently employed tostudy magnetic structure in small scale dynamos (Wilkin et al. (2007)).There is an interesting relation to fractal dimensions. The scaling behaviour of thevolume density for r → D and D KY , as usedin our case, are much more reliable measures for the dimension. Since the Minkowski Calzavarini, Kerscher, Lohse, and Toschi
Poisson(cid:3)Particles(cid:3) (cid:3)
Bubbles
Figure 5. (color) Volume densities of the Minkowski functionals v µ ( r ), µ = 0 , , ,
3, forpassive tracers ( β = 1, black dotted, corresponding to those of a Poisson distribution Mecke &Wagner (1991)), heavy particles ( β = 0, red solid), and bubbles ( β = 3, blue dashed). In allcases St = 0 . Re λ = 78. Distances are in Kolmogorov units ( η ). To estimated the errorsin the Minkowski functionals we looked at several realizations of a Poisson process. The onesigma error bars are smaller than the linethickness of the shown curves which illustrates therobustness of the Minkowski functionals. – The code used for the calculations of the Minkowskifunctionals is an updated version of the code developed by Kerscher et al. (1997), based on themethods outlined in Mecke et al. ∼ kerscher/software/ . functionals are dimensional quantities, one is able to define other scaling dimensionsbeyond the volume. This has been formalized and investigated for special models byMecke (2000). It remains an open question how to apply this to points decorated withspheres.In figure 5 we show the volume densities of the four Minkowski functionals v µ ( r ) = V µ ( r ) /L , µ = 0 , , ,
3, determined from the particle distributions with St = 0 . β = 3 (“bubbles”), β = 0 (heavy particles), and β = 1 (neutraltracers). For the neutral tracers ( β = 1) the functionals coincide with the analyticallyknown values (Mecke & Wagner (1991)) for randomly distributed objects. As the radiusincreases, the volume is filled until reaching complete coverage where the volume density v ( r ), i.e. the filling factor, reaches unity. This increase is considerably delayed for heavyparticles and even more for bubbles, which is a characteristic feature of a clusteringdistribution produced from the empty space in between the clusters.The density of the surface area, measured by v ( r ), increases with the radius r . Asmore and more balls overlap the growth of v ( r ) slows down and the surface area reachesa maximum. For large radii the balls fill up the volume and no free surface area is left.For both the bubbles and the heavy particles the maximum of v ( r ) is smaller comparedto the Poisson case. For intermediate and large radii we observe the skewed shape of imensionality and morphology of particle and bubble clusters in turbulent flow v ( r ) with a significant excess of surface area on large scales compared to Poisson. Theparticles cluster on clumpy, filamentary and sheet like structures and the surface area ofthe balls is growing into the empty space in between. Especially for the bubbles ( β = 3)the maximum of v ( r ) is attained for considerably larger r compared to the distributionswith β = 0 ,
1, suggesting mainly separated filamentary shaped clusters.The density of the integral mean curvature v ( r ) allows us to differentiate convex fromconcave situations. For small radii the balls are growing outward. The main contributionsto the integral mean curvature is positive, coming from the convex parts. Increasing theradius further we observe a maximum for all three cases, but as with the surface area, theamplitude of the maximum is reduced for heavy particles and especially for the bubbles.For tracer particles ( β = 1) the empty holes start to fill up and the structure is growinginto the cavities. Now the main contribution to the integral mean curvature is negative,stemming from the holes and tunnels through the structure. This concaveness is lesspronounced for the heavy particle case ( β = 0). Typically interconnected networks oftunnels show such a reduced negative contribution. For bubbles ( β = 3) A r is hardlyconcave, i.e., no holes and no tunnels develop, just as expected from isolated (filamentary)clusters. For large radii the balls fill up the volume, no free surface and hence no curvatureis left.The topology undergoes a number of changes which we measure with the Euler char-acteristic. For small radii r ≈ v ( r ≈
0) equals the number density of the particles. As the ra-dius increases, balls join and the Euler characteristic decreases. Both for heavy particles( β = 0) and especially for bubbles ( β = 3) the decrease of v ( r ) with increasing r is moredramatic, due to the clustering. When further increasing the radius r , more and moretunnels start to form resulting in a negative v ( r ). This is observed for neutral particles( β = 1) and less pronounced for the heavy particles ( β = 0). No tunnels seem to formin the bubble distribution ( β = 3). For neutral particles this behavior reaches a turn-ing point when these tunnels are blocked to form closed cavities and a second positivemaximum of v ( r ) can be seen. Neither bubbles nor heavy particles show a significantpositive v ( r ).To develop a physical picture we first look at the heavy particles. One expects that theparticles are dragged out of the turbulent vortices gathering on structures around thevortices. A dimension D KY ≈ . v ( r ) for large r we conclude that these 2d-structures are mainlyconcave, i.e. growing inward. Hence these 2d-structures are not small 2d-patches, butenclose the vortices. The particles around the vortices form the enclosure of tunnels butthey do not fully enclose the vortices (they do not form the “casing of a sausage”).Completely anclosed cavities would leed to a v ( r ) larger zero for large r , which we donot observe. There have to be regions connecting the vortices devoid of any particles.In other words there exists an empty interconnected network of tunnels surrounded byparticles.For the bubble distribution one expects that the bubbles are sucked into the vortices.The dimensional analysis with D KY ≈ . v ( r ) is almost everywhere positive, hence the1d-structures remain convex for all radii. Similar v ( r ) is positiv on all scales. Hence,neither tunnels devoid of bubbles nor empty cavities enclosed by bubbles form. The 1d-structures built from the bubbles are separated filamentary clusters. This also shows thatthe bubbles do not form a percolating structure throughout the simulation.Minkowski functionals calculated from points decorated with spheres do depend on thenumber density of the point distribution. One can derive explicit expression in terms of Calzavarini, Kerscher, Lohse, and Toschi
Figure 6. (color) Volume densities of the Minkowski functionals v µ ( r ), µ = 0 , , , β = 0 and St = 0 .
6. We compare the Minkowski functionals for samples with a differentnumber of points, obtained by randomly thinning the original data set with 100k points. 100kpoints (black solid), 90k points (red solid), 80k points (blue solid), 50k points (black dotted),10k points (red dotted), 1k points (blue dotted). high order correlation functions quantifying the non-trivial dependence on the numberdensity (see e.g. Mecke (2000)). In our analysis we always compare data sets with the samenumber of points. However one may ask the question how well our Minkowski functionalsare converged. In Fig. 6 the Minkowski functionals for a randomly subsampled data setare shown. Considering only 80% of the points we obtain nearly identical result as for thefull sample. This can be understood as follows: We decorate the particles with spheres.In our case the particles follow well defined structures. For a radius equal to zero, v equals the number density. That is why the v curves fan out for small radii. Already for r ≈ . η the curves for 100%, 90% and 80% overlap again. In this sense our results arewell converged.Obviously, the convergence must get lost if we further and further reduce the number ofpoints. Randomly subsampling (thinning) will inevitably lead to a convergence towardsPoisson (see the blue dotted line in fig. 6) – the discreteness effects become more andmore important.Up to now we investigated three extreme cases with St = 0 . β = 0 , ,
3. Thesethree particle distributions also show different D KY , hence their morphological differencesdo not come as a surprise. Now we investigate the morphology of the particle distributionswith similar D KY but quite different physical parameters β and St . In figure 7 we compareheavy particles ( β = 0, St = 0 .
6) with D KY = 2 .
59 with light bubbles ( β = 3, St = 0 . D KY = 2 .
55. Clearly, their morphological properties differ. Although both have imensionality and morphology of particle and bubble clusters in turbulent flow Figure 7. (color) Volume densities of Minkowski functionals v µ ( r ), µ = 0 , , , β = 0 and St = 0 . β = 3 and St = 0 .
103 (redshort dashed): Whereas for these two particle types D KY basically is the same, the Minkowskifunctionals clearly differ. As a second pair with nearly identical D KY but different Minkowskifunctionals we compare two light particle distribution with differing Stokes parameter: β = 3, St = 0 . β = 3, St = 1 .
75 (green dashed-dotted). Poisson distribu-tion behavior is reported for reference (black dotted). similar D KY , the distribution of the heavy particle shows a stronger deviation from thePoisson distributed points than the light bubbles. As another example, in figure 7 wealso compare the distribution of two sets of light bubbles, both with β = 3 and identical D KY = 1 .
65, but with different Stokes number. The data set with St = 1 .
75 shows adistribution dominated by isolated clusters whereas the data set with St = 0 . Re λ = 75 withsimulations at higher Reynolds numbers Re λ = 180 and Re λ = 400, keeping the param-eters β = 0 and St = 0 . β = 3, St = 0 .
6. This result agreeswith the very weak Reynolds dependence of D KY already revealed in Fig. 3 and in Bec et al. (2006 a ).
4. Conclusions and outlook
Often the small-scale behavior in a turbulent dispersed multiphase flow is of interest.Then scaling or contraction indices like the Kaplan–Yorke Dimension D KY are the main0 Calzavarini, Kerscher, Lohse, and Toschi
Figure 8. (color) Volume densities of the Minkowski functionals v µ ( r ), µ = 0 , , , β = 0 and St = 0 . Re λ = 75 (red solid), Re λ = 180 (blue short dashed), Re λ = 400 (blue long dashed) . Poisson distribution behavior isreported for reference (black dotted). All point sets have been subsampled to the same numberdensity. tools. With D KY as a function of β and St a simple picture emerges: For heavy particles( β <
1) we observe in the extreme cases a scaling reminiscent of local planar structures,for light bubbles ( β >
1) we find indications for local linear structures. The strength ofthe deviation from D KY = 3 is further modulated by the Stokes parameter St . If one isinterested in the connectivity and other (global) morphological features of the particledistribution, the picture becomes significantly more complex. The distribution of heavyparticles seems to allow interconnected empty tunnels, whereas the distribution of bub-bles typically shows isolated filamentary structures. But, depending on both β and St ,also interconnected structures appear in the bubble distribution. Using Minkowski func-tionals as morphological order parameters we are able to quantify these geometrical andtopological features in a unique way. Especially, we are able to overcome the degeneracyseen in D KY as a function of β and St ∼ kerscher/software/. Raw data can be down-loaded from the International CFD database, iCFDdatabase (http://cfd.cineca.it). imensionality and morphology of particle and bubble clusters in turbulent flow Acknowledgements:
We thank Herbert Wagner and Massimo Cencini for interesting andhelpful discussions and the computer centers CASPUR (Rome), SARA (Amsterdam),and CINECA (Bologna) for CPU time.
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