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Breakup of small aggregates driven by turbulent hydrodynamic stress Matthaus U. Babler,1 Luca Biferale,2 and Alessandra S. Lanotte3 1Dept. of Chemical Engineering and Technology, Royal Institute of Technology, 10044 Stockholm, Sweden 2Dept. of Physics and INFN, University of Rome Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy 3ISAC-CNR, Str. Prov. Lecce-Monteroni, and INFN, Sez. Lecce, 73100 Lecce, Italy (Dated:) Breakupofsmallsolidaggregatesinhomogeneousandisotropicturbulenceisstudiedtheoretically and by using Direct Numerical Simulations at high Reynolds number, Re (cid:39) 400. We show that λ turbulent fluctuations of the hydrodynamic stress along the aggregate trajectory play a key role in 2 determining the aggregate mass distribution function. Differences between turbulent and laminar 1 flowsarediscussed. Anoveldefinitionofthefragmentationrateisproposedintermsofthetypical 0 frequency at which the hydrodynamic stress becomes sufficiently high to cause breakup along each 2 Lagrangianpath. WealsodefineanEulerianproxyoftherealfragmentationrate,basedonthejoint statisticsofthestressanditstimederivative,whichshouldbeeasiertomeasureinanyexperimental n set-up. BothourEulerianandLagrangianformulationsdefineaclearprocedureforthecomputation a J ofthemassdistributionfunctionduetofragmentation. Contrary,previousestimatesbasedonlyon single point statistics of the hydrodynamic stress exhibit some deficiencies. These are discussed by 4 investigating the evolution of an ensemble of aggregates undergoing breakup and aggregation. 2 ] PACSnumbers: 47.27-i,47.27.eb,47.55.df n y d Turbulence has a distinct influence on the aggrega- The main issue we investigate in this paper is how to - tion of colloidal and aerosol particles. It not only en- defineandmeasurethefragmentationratef inaturbu- u ξ l hances the rate of collision among particles, i.e. by in- lentflow. Theoutcomesofouranalysisaremanifold: i)a f . ducing high velocity differences and preferential concen- Lagrangian and an equivalent Eulerian definition of the s tration within the particle field [1, 2], but also it cre- fragmentation rate can be derived, that fall off to zero c i ates hydrodynamic stress that can cause restructuring in the limit of small aggregate mass, while they have a s y and breakup of aggregates [3], a phenomena macroscop- power-law behavior for large masses; ii) the power-law h ically expressed in shear thinning in dense suspensions tail description is crucial to obtain a steady-state aggre- p [4]. Breakup of small aggregates due to hydrodynamic gate mass distribution when considering the full aggre- [ stress in turbulence is of high relevance to various appli- gation/breakup dynamics; iii) turbulent fluctuations al- 1 cations, e.g. processing of industrial colloids, nanoma- low for a broad asymptotic mass distribution, while a v terials, wastewaters, and sedimentation of marine snow much narrower distribution of aggregates is obtained in 1 [3, 5]. In a mean field situation, aggregation-breakup the laminar case. 3 dynamics are described by the Smoluchowski equation. We adopt the simplest possible framework [6], and con- 0 5 Defining nξ(t) = Nξ(t)/N0 where Nξ(t) is the number sider a dilute suspension of aggregates in a stationary . concentrationofaggregatesconsistingofξ primaryparti- homogeneous and isotropic turbulent flow. We con- 1 (cid:82)∞ cles,andN = dξξN (t),theSmoluchowskiequation sider very small aggregates - much smaller then the Kol- 0 0 0 ξ 2 reads as mogorov scale of the flow, in the range of 25 to 100 µm 1 for typical turbulent flows, with negligible inertia. Fur- (cid:34) v: n˙ (t)=−f n (t)+(cid:90) ∞dξ(cid:48) g f n (t)+ 3φ 1(cid:90) ξ ther, we assume that aggregates concentration is such i ξ ξ ξ ξ ξ,ξ(cid:48) ξ(cid:48) ξ(cid:48) 4π 2 0 that they do not modify the flow. Hence, their evolution X (cid:90) ∞ (cid:21) is identical to that of passive point-like particles (unless r dξ(cid:48)k n (t)n (t)−n (t) dξ(cid:48)k n (t) , (1) extreme deviations from a spherical shape are present, a ξ(cid:48),ξ−ξ(cid:48) ξ(cid:48) ξ−ξ(cid:48) ξ ξ,ξ(cid:48) ξ(cid:48) 0 e.g. elongatedfibers). Moreover,theaggregatesarebrit- where φ = 4πa3N is the solid volume fraction, a is tle and breakup occurs instantaneously once being sub- 3 p 0 p jecttoahydrodynamicstressthatexceedsacriticalvalue the radius of the primary particle (assumed monodis- [7]. For small and inertialess aggregates, the hydrody- perse and spherical), and k is the aggregation rate. ξ,ξ(cid:48) namic stress exerted by the flow is ∼µ(ε/ν)1/2, where µ Breakup is accounted for by the fragmentation rate f ξ and ν are the dynamic and kinematic viscosity, respec- g . Determiningthesefunctionsisnoteasyanddespite ξ,ξ(cid:48) tivelyandεisthelocalenergydissipationperunitmass. considerable efforts [6] a basic understanding of breakup Thus,thekeyroleisplayedbytheturbulentvelocitygra- dynamicsisstilllacking. Areasonforthisisthecomplex dients across the aggregate which are known to possess role of turbulence and the way it generates fluctuating strongly non-Gaussian, intermittent statistics [8]. stress to which an aggregate is exposed to. 2 tainaproxyforthefragmentationratewhichiseasierto measure. Given a threshold for the hydrodynamic stress ε , onecanmeasuretheseriesT , T ,...ofdiving-times, cr 1 2 namely the time lags between two consecutive events of istantaneous stress crossing the threshold ε along the cr aggregate motion, Fig. 1. In [11], it was proposed to estimate the fragmentation rate as the inverse of the mean diving-time, fE = 1/(cid:104)T(ε )(cid:105). An important result is εcr cr that(cid:104)T(ε )(cid:105)canbeobtainedusingtheRicetheoremfor cr FIG.1: Pictorialevolutionoftheenergydissipationεalong the mean number of crossing events per unit time of a anaggregatetrajectory. Startingtorecordthestressattime differentiable stochastic process across a threshold [12]. t ,theexit-timeτ andthediving-timesT , T ,...foragiven 0 1 2 Hence, threshold ε are shown. cr (cid:82)∞ 1 dε˙ ε˙p (ε ,ε˙) fE = = 0 2 cr . (3) Let εcr(ξ) be the critical energy dissipation needed to εcr (cid:104)T(εcr)(cid:105) (cid:82)0εcrdε p(ε) break an aggregate of mass ξ. In the simplest case of Here the numerator is the Rice formula giving the mean a laminar flow, ε relates to the critical shear rate for cr number of crossings of ε in terms of the joint proba- cr breakupasG ∼(ε /ν)1/2. Earlierworks[7,9]support cr cr bility of dissipation and its time derivative p (ε,ε˙); the 2 the existence of a constituent power-law relation for ε cr denominator is the measure of the total time spent in implying that larger aggregates break at a lower stress the region with ε < ε . Notice that the integration in cr than smaller ones: ε (ξ) = (cid:104)ε(cid:105)(ξ/ξ )−1/q where the ex- cr s the numerator goes only on positive values in order to ponent q is related to the aggregate structure and ξ is s consider only up-crossing of the threshold ε [12]. An cr the characteristic aggregate mass. An equivalent rela- obvious advantage of Eq. (3) is that it is quasi-Eulerian: tionexistsforsmalldropletsbreakingatcriticalcapillary itdoesnotrequiretofollowtrajectories,sinceitdepends number,ε ∼σ2/(µρξ2/3)(hereξ isthedropletvolume, cr onlythespatialdistributionofdissipationandofitsfirst and σ the interfacial energy). Hence, f can be equally ξ time derivative in the flow. Expressions (2) and (3) are formulated in terms of a critical dissipation f . εcr(ξ) not strictly equivalent. A direct calculation of the mean In this contribution we propose to define the fragmenta- exit-timeintermsofthedistributionofdiving-timesgives tion rate fξ or fεcr in terms of a first exit-time statistics. indeed (cid:104)τ(εcr)(cid:105)ex = (cid:104)T2(εcr)(cid:105)/[2(cid:104)T(εcr)(cid:105)], which relates This amounts to measure the fragmentation rate using the mean exit-time to moments of the diving-time. thedistributionofthetimenecessarytoobservethefirst Once a definition from first principles is set-up, we pro- occurrence of a local hydrodynamic stress strong enough ceed to measure the fragmentation rate for aggregates to break the aggregate. An operational formula of f εcr convected as passive point particles in a statistically ho- readsasfollows: (i)seedhomogeneouslyaturbulent,sta- mogeneous and isotropic turbulent flow, at Reynolds tionary flow with a given number of aggregates of mass number Re (cid:39) 400. Details on the Direct Numeri- λ ξ; (ii) neglect those aggregates in regions where the hy- cal Simulations (DNS) of Navier-Stokes equations with drodynamic stress is too high (ε > εcr); (iii) from an 20483gridpointsandLagrangianparticlesarein[2]. The initial time t0, selected at random, follow the trajectory presentanalysisisobtainedaveragingover6×105 trajec- of each remaining aggregate until it breaks, and count tories,recordedevery0.05τ ,whereτ istheKolmogorov η η the total number of breaking events in a given time in- time of the flow. terval [τ,τ +dτ]. The time τ is the first exit-time: for Figure 2 shows the fragmentation rate measured from an aggregate initally in a region with ε < ε , τ is the cr theDNSdatafollowingtheevolutionofthevelocitygra- time it takes to the hydrodynamic stress seen by the ag- dients along aggregate trajectories. The exit-time mea- gregate along its motion to cross the critical value ε at cr surement (2) and its Eulerian proxy (3) show a remark- first opportunity, Fig. 1. The fragmentation rate is the able, non trivial behavior for small values of the critical inverse of the mean exit-time: threshold, i.e. for large aggregate mass. In this region, (cid:20)(cid:90) ∞ (cid:21)−1 1 thereisacompetingeffectbetweentheeasinessinbreak- fεcr = dτ τPεcr(τ) = (cid:104)τ(ε )(cid:105) , (2) ingalargeaggregateandthedifficultytoobservealarge 0 cr ex aggregate existing in a region of low energy dissipation. where P (τ) is the distribution of first exit-time for a As a result, the estimated fragmentation rate develops a εcr threshold ε , and (cid:104)·(cid:105) is an average over P (τ). The quasi power-law behavior for small thresholds. On the cr ex εcr definition in Eq. (2) is certainly correct but difficult to other hand, for large thresholds the super exponential implementexperimentallyasitisneededtofollowaggre- fall off is expected. It is the realm of very small aggregate trajectories and record the local energy dissipation, gates that are broken only by large energy dissipation something still at the frontier of nowadays experimental bursts. The exit-time (2) and diving time (3) measure- facilities [10]. The question thus arising is if we can ob- ments are very close and we therefore consider the latter 3 FIG. 2: The normalized fragmentation rate fεcrτη versus the FIG.3: TimeevolutionofI0 fordf =2.4andq=0.36with normalized energy dissipation εcr/(cid:104)ε(cid:105). Filled circles are the breakup rates fεEcr (squares); fεIcr (circles); fεIcIr (triangles). definition (2), measured up to thresholds where statistical Runswithdifferentsolidvolumefractionsφareshifted. Inset: convergence of exit-times is obtained; the solid line is fεEcr. I0 atsteadystateasafunctionofφ(samesymbols). Atlarge Squares are fI , while crosses are fII. (Bottom inset) Joint φ, the model with fragmentation (3) relaxes to the predicted εcr εcr distribution p2(ε,ε˙). The continuous line is the dimensional scaling curve I0 ∼ φ1/(1+χ/q−3/df), given by the solid line. estimate ε˙ ∼ ε/τη(ε). (Top inset) Numerator of Eqs. (3) – Here, ξ =104 implying that a/a ∼ξ1/df ≈50 is the value s p s (5); curves colors are the same of the main figure. of the characteristic aggregate size. a very good proxy of the former, the real fragmentation distribution of the energy dissipation only. In Figure 2, rate. The main advantage of the estimate (3) is the very thefragmentationratesobtainedfromtheclosuresIand high statistical confidence that can be obtained since it II are also shown. Both of them reproduce the correct is a quasi-Eulerian quantity. Moreover it gives a reliable behavior for large values of the critical dissipation but estimate also in the region of large thresholds where the deviate for small ones. The reason for such discrepancy convergence of exit-time statistics, requiring very long ismadeclearinthetopinsetofFig.2,wherethenumer- aggregate trajectories, is difficult to obtain. ator of Eqs. (4) and (5) are shown. It appears that, for StartingfromEq.(3),simplemodelscanbeproposedfor small values of the critical stress ε , fI underestimates the statistics of ε˙, whose direct measure requires a very cr εcr the number of breakup events, while the numerator of high sampling frequency along Lagrangian paths. First, fII saturates to a constant value. as it appears from the bottom inset of Fig. 2, the dissi- εcr Since in experiments breakup a fortiori takes place to- pation and its time derivate are significantly correlated. getherwithaggregaterecombination,weexplorehowthe Scaling on dimensional grounds suggests ε˙ ∼ ε/τ (ε) η where τ (ε) ∼ (ν/ε)1/2 is the local Kolmogorov time. actual fragmentation rate Eq. (3) and the two closures, η Eqs. (4)-(5), influence the time evolution of an ensem- It follows that the joint PDF p (ε,ε˙) can be estimated 2 asp (ε,ε˙)= 1p(ε)δ(|ε˙|−ε/τ (ε))wherep(ε)istheprob- ble of aggregates nξ(t). At this purpose, the Smolu- 2 2 η chowski equation (1), subject to the initial condition ability density of energy dissipation. Prefactor 1/2 ap- n (0) = δ(ξ − 1), is evolved in time. To model ag- pears since for a stationary process ε˙ is positive or neg- ξ gregation we use the classical Saffman-Turner expres- ative with equal probability. Plugging this expression in Eq. (3) gives sion kξ,ξ(cid:48) = D0/τη(ξ1/df + ξ(cid:48)1/df)3, where D0 is an O(1) constant and d is the fractal dimension that re- f 1ε p(ε )/τ (ε ) lates the collision radius of an aggregate to its mass, fεIcr = 2 c(cid:82)rεcrcdrε p(ηε)cr . (4) a/ap = ξ1/df. Breakup is assumed to be binary and 0 symmetric, g = 2δ(ξ −ξ(cid:48)/2), which despite its sim- ξ,ξ(cid:48) We refer to it as Closure I. A different approach was pleness represents well the quality of the evolution. To proposedin[6]. Itassumesthatactiveregionsintheflow quantifyourfindingsweconsiderthesecondmomentob- where ε> ε engulf the aggregates at a rate ∼ 1/τ (ε), servablereadilyaccessiblefromstaticlightscattering[7]. cr η which results in Fig. 3 shows the time evolution of I0 for typical values of d and q found in turbulent aggregation of colloids (cid:82)∞ f fII = εcrdε p(ε)/τη(ε). (5) [3]. After an initial growth period, curves obtained with εcr (cid:82)εcrdε p(ε) Eqs. (3) and (5) both relax to a steady state. At small 0 solid volume fraction φ, the two models nearly overlap, We refer to it as Closure II. Both models share the ad- whereas at larger φ Closure II underestimates I . This 0 vantage of being fully Eulerian and based on the spatial behavior is confirmed in the inset of Fig. 3 that shows I 0 4 atsteadystateforbothmodelsandfordifferentvaluesof φ. Clearly, at large φ other phenomena, i. e. modulation of the flow due to the particles, may occur which, however, is beyond the scope of the present work. Closure I shows a very different behavior. At small φ, an evolution similar to previous cases is observed. However, by increasing φ a drastic change appears and I diverges, a 0 direct consequence of the presence of a maximum in the shape of fI , see Fig. 2. εcr The time evolution of the number concentration n (t), ξ governed by Eq. (3), is further examined in Fig. 4. Here, we compare the steady distribution obtained in the tur- bulentflowwiththatofalaminarflowfordifferentvalues of the solid volume fraction. In the turbulent case, n (t) ξ rapidly grows and reaches a stationary state whose peak mode is controlled by the magnitude of aggregation, i.e. FIG.4: Stationarymassdistributionn (t)withd =2.4and ξ f thesolidvolumefractionφ. Increasingφcausesthemode q=0.36, for different solid volume fractions φ. Distributions to broaden and to shift to the right as aggregation gets obtainedintheturbulentflowatReλ (cid:39)400(a)arecompared tothoseofthelaminarcases(b),withtheuniformshearrate morepronounced. Thedistributionthusgraduallymoves εlam =(cid:104)(cid:15)(cid:105). In(a),weplotf (dashedline)asgivenbyEq.(3), into the region where f assumes power-law behavior. cr ξ εcr assumingapower-lawbehaviorinthelimitoflargeaggregate A numerical fit of the left fragmentation tail in Fig. 2 mass. gives f ∼ ξχ/q, with χ = 0.42±0.02 (dashed curve in ξ Fig. 4). Using this latter expression in Eq. (1), one can deriveascalingrelationforintegralquantitiesofn (t)at ξ steady state [14]. Such scaling is reported in the inset of tiative and CINECA (Italy) for technical support. Fig. 3. On the other hand, in the laminar case where a uniformshearrategovernsthebreakup, thesteadystate distribution is much narrower and shows multiple reso- nantmodes. Theseareduetothesharponsetofbreakup [1] G. Falkovich, A. Fouxon, and M. G. Stepanov, Nature once the aggregates grow larger than the characteristic (London) 419, 151 (2002). J. Bec, et al. J. Fluid Mech. aggregatemassξ . Theseresultsclearlydemonstratethe s 646,527(2010).J.Chun,et al.J.FluidMech,536,219 strong influence of turbulent fluctuations on the statisti- (2005). cally stationary mass distribution function. [2] J. Bec, et al. J. Fluid Mech. 645, 497 (2010). We have presented a study of the fragmentation rate of [3] M. Soos et al. J. 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