Heavy particle concentration in turbulence at dissipative and inertial scales

0.57 MB·English

by  J. Bec

Heavy particle concentration in turbulence at dissipative and inertial scales
J. Bec,1 L. Biferale,2 M. Cencini,3 A. Lanotte,4 S. Musacchio,5 and F. Toschi6
1CNRS UMR6202, Observatoire de la Cˆote d’Azur, BP4229, 06304 Nice Cedex 4, France.
2Dip. Fisica and INFN, Universit`a “Tor Vergata”, Via Ricerca Scientifica 1, 00133 Roma, Italy.
3INFM-CNR, SMC Dept. of Physics, Universit`a “La Sapienza”, P.zzle A. Moro 2, 00185 Roma, Italy, and
CNR-ISC, Via dei Taurini 19, 00185 Roma, Italy.
4CNR-ISAC and INFN, Sezione di Lecce, Str. Prov. Lecce-Monteroni, 73100 Lecce, Italy.
5Weizmann Institute of Science, Department of Complex Systems, 76100 Rehovot, Israel.
6CNR-IAC, Viale del Policlinico 137, 00161 Roma, Italy, and
INFN, Sezione di Ferrara, Via G. Saragat 1, 44100 Ferrara, Italy.
(Dated: February 8, 2008)
7 Spatialdistributionsofheavyparticlessuspendedinanincompressibleisotropicandhomogeneous
0 turbulent flow are investigated by means of high resolution direct numerical simulations. In the
0
dissipativerange, itis shown thatparticles form fractal clusterswith propertiesindependentof the
2
Reynoldsnumber. Clusteringisthereoptimalwhentheparticleresponsetimeisoftheorderofthe

n Kolmogorovtimescaleτη. Intheinertialrange,theparticledistributionisnolongerscale-invariant.
a It is however shown that deviations from uniformity depend on a rescaled contraction rate, which
J is different from the local Stokes number given by dimensional analysis. Particle distribution is

2 characterizedbyvoidsspanningallscalesoftheturbulentflow;theirsignatureinthecoarse-grained
mass probability distribution is an algebraic behavior at small densities.

]
D
PACSnumbers: 47.27.-i,47.10.-g
C
.
n Understanding the spatial distribution of finite-size cording to the Newton equation [7]
li massiveimpurities,suchasdroplets,dustorbubblessus-
n pended in incompressible flows is a crucial issue in engi- τX¨ =u(X,t)−X˙ . (1)
[
neering [1], planetology [2] and cloud physics [3]. Such
The response time τ is proportional to the square of the
2 particles possess inertia, and generally distribute in a
v strongly inhomogeneous manner. The common under- particlessizeandtotheirdensitycontrastwiththe fluid.
5 Here, we neglect buoyancy and particle Brownian diffu-
standingofthislongknownbutremarkablephenomenon
4 sion. As stressed in [8, 9], diffusion may affect concen-
ofpreferential concentrations reliesontheideathat,ina
0
tration of particles. However we assume that the typical
8 turbulent flow, vortexesactas centrifuges ejecting parti-
lengthscale below which particle diffusion becomes dom-
0 clesheavierthanthefluidandentrappinglighterones[4].
6 This picture was successfully used (see, e.g. [4, 5]) to de- inant is much smaller than the Kolmogorov scale η of
0 the fluid flow,andthan any observationscaleconsidered
scribe the small-scale particle distribution and to show
n/ that it depends only on the Stokes number S = τ/τ , here. In most situations, particles are so massive that
η η
nli which is obtained by non-dimensionalizing the particle sucThhaenfluaipdpvreolxoicmitaytiuonsaitsisffiuellsythjuesitnificeodm.pressibleNavier-
response time τ with the characteristic time τ of the
: η Stokes equation
v small turbulent eddies.
i
X In this Letter, we confirm that this mechanism is rele- ∂ u+u·∇u=−∇p+ν∇2u+F with ∇·u=0, (2)
t
vanttodescribe the particledistributionatlength-scales
r
a whicharesmallerthanthedissipativescaleofturbulence, p being the pressure and ν the viscosity; F is an ex-
η. In particular, maximal clustering is found for Stokes ternal homogeneous and isotropic force that injects en-
numbers of the order of unity. However, we show that ergy at large scales L with a rate ε = hF ·ui. We per-
stationary particle concentration experiences also very formed direct numerical simulations of (2) by means of
strongfluctuationsintheinertialrangeofturbulence. In a pseudo-spectral code on a triply periodic box of size
analogywithsmall-scaleclustering,itisexpectedthatfor L=2π with 1283, 2563 and 5123 collocation points cor-
r ≫η the relevant parameter is the local Stokes number responding to Reynolds numbers (at the Taylor micro-
Sr = τ/τr, where τr is the characteristic time of eddies scale) Rλ ≈65, 105 and 185,respectively. Once the flow
ofsizer [6]. Surprisingly,wepresentevidences thatsuch u is statistically stationary, particles with 15 different
a dimensional argument does not apply to describe how values of the Stokes number in the range S ∈[0.16,3.5]
η
particlesorganizeinthe inertialrangeofturbulence. We (N=7.5millionsofparticlesperS ),areseededhomoge-
η
show that the way they distribute depends on a scale- neously in space with velocity equal to that of the fluid,
dependent rate at which volumes are contracted. and are evolved according to (1) for about two large-
Inverydilutesuspensions,thetrajectoryX(t)ofsmall scale eddy turnovertimes. After this time, particle mass
sphericalparticlesmuchheavierthanthefluidevolveac- distribution reaches a statistical steady state too. Mea-
2
3
2.8
2.6
2
D
2.4 Reλ=65
Reλ=105
2.2 Reλ=185
P
2
0 0.5 1 1.5 2 2.5 3 3.5
S
η
FIG. 1: (Color online) The correlation dimension D2 vs Sη FIG. 2: (Color online) (a) The modulus of the pressure gra-
for the three different R . Also shown the probability P to dient,givingthemaincontributiontofluidacceleration,ona
λ
findparticlesinnon-hyperbolic(rotating)regionsoftheflow, slice 512×512×4. B/W code low and high intensity, respec-
for Re =185 (multiplied by an arbitrary factor for plotting tively. Particle positions in the same slice are shown for (b)
λ
purposes). D2 has been estimated taking into account also Sη=0.16, (c) 0.80 and (d) 3.30. Note the presence of voids
subleading terms, as described in [12]. with sizes muchlarger than the dissipative scale.
surements are then performed. Details on the numerics
values of the fluid acceleration. This phenomenon was
and on the transient are reported in [10].
already evidenced in [10] where a statistical analysis of
Below the Kolmogorovlength-scale η where the veloc-
thefluidaccelerationatparticlepositionswasperformed,
ity field is differentiable, the motion of inertial particles
and also in [13] for 2D turbulent flows.
is governed by the fluid strain and the dissipative dy-
namics leads their trajectories to converge to a dynam- From Fig. 2, it is clear that fluctuations in the par-
ically evolving attractor. For any given response time ticle spatial distribution extend to scales far inside the
of the particles, their mass distribution is singular and inertial range; this confirms the experimental measure-
genericallyscale-invariantwithfractalpropertiesatsmall ments of [14]. Moreover, for Stokes numbers of the or-
scales [8, 11]. In order to characterize particle clusters der of unity (Figs. 2c and 2d), we note that the sizes of
at these scales, we measured the correlation dimension voids span all spatial scales of the fluid turbulent flow,
D , which is estimated through the small-scale algebraic similarly to what observed in 2D inverse cascade turbu-
2
behavior of the probability to find two particles at a dis- lence[13,15]. Togainaquantitativeinsight,weconsider
tance less than a given r: P2(r)∼rD2. The dependence the Probability Density Function (PDF) Pr,τ(ρ) of the
of D on S and Re is shown in Fig. 1. Notice that D particle density coarse-grainedon a scale r inside the in-
2 η λ 2
depends very weakly on Re in the range of Reynolds ertial range, that is the probability distribution of the
λ
numbers here explored. A similar observation was done fraction of particles in a cube of size r, normalized by
in [5],where particleclusteringwasequivalently charac- the volume r3 of the cube. For tracers, which are uni-
terized in terms of the radial distribution function. This formly distributed, and for an infinite number of parti-
indicates that τ , which varies by more than a factor 2 cles N → ∞, this PDF is a delta function on ρ = 1.
η
between the smallest and the largest Reynolds numbers For finite N, the probability to have a number n of par-
consideredhere,istherelevanttime scaletocharacterize ticles in a box of size r is given by the binomial distri-
clustering below the Kolmogorov scale η. For all values bution and leads to a closed form for the PDF of the
ofRe ,amaximumofclustering(minimumofD )isob- coarse-grained density ρ = nL3/(Nr3). In our settings,
λ 2
served for S ≈ 0.6. For values of S larger than those the typicalnumber ofparticles in cells inside the inertial
η η
investigatedhere,D isexpectedtosaturatetothespace range is several thousands. Thus, finite number effects
2
dimension [12]. For small values of S , particle positions are not expected to affect the core of the mass distri-
η
stronglycorrelatewiththelocalstructureofthefluidve- bution ρ = O(1), but they may be particularly severe
locityfield. AquantitativemeasureisalsogiveninFig.1 when ρ≪1 because of the presence of voids. To reduce
by the probability P to find particles in non-hyperbolic this spurious effect, we computed the quasi-Lagrangian
regions of the flow, i.e. at those points where the strain (QL) mass density PDF P(QL)(ρ), which corresponds to
r,τ
matrixhastwocomplexconjugateeigenvalues. Notethat a Lagrangian average with respect to the natural mea-
P attains its minimum for values of S close to the min- sure[16], andis obtainedby weightingeachcellwiththe
η
imum of D . This supports the traditional view relating massitcontains. Forstatisticallyhomogeneousdistribu-
2
particleclusteringtovortexejection. Asqualitativelyev- tion,QLstatisticscanbe relatedtotheEulerianonesby
idenced fromFig.2, particle positions correlatewith low noticing that hρpi =hρp+1i (see, e.g., [16]).
r QL r
3
cordingtoKolmogorov1941theory(K41),velocityincre-
7
mentsbehaveasδ u∼(εr)1/3. K41theorysuggeststhat
101 6 fluctuationsatscarler areassociatedtotimescalesofthe
5
order of the turnover time τ =r/δ u∼ε−1/3r2/3. This
α4 r r
100 3 impliesthatforanyfiniteparticleresponsetimeτ andat
sufficiently large scales r, the local inertia measured by
2
L)ρ ()10−1 10 1S = τ 2/ τ 3 Sasr t=raτc/eτrsr banecdomdiestsrsiboustmeaullntifhoartmplyaritniclsepsascheo[u6l]d. bDeehvaivae-
Qτ η η
(Pr, tions from uniformity for finite Sr are expected not to
10−2 be scale-invariant [8]. In particular, observations from
random δ-correlated in time flows [17] suggest that par-
ticle distribution should depend only on the local Stokes
10−3
number S . However,as explained below, this argument
r
does not seem to apply to realistic flows.
10−4 The presence of inhomogeneities in the spatial distri-
10−2 10−1 ρ 100
butionofparticlesisdue toadynamicalcompetitionbe-
FIG. 3: (Color online) The quasi-Lagrangian PDF of the tween stretching, folding, and contraction due to their
coarse-grained mass density ρ in log-log scale for S =0.27, dissipative dynamics. At small scales where the flow
r η
0.37, 0.58, 0.80, 1.0, 1.33, 2.03, 3.31 (from bottom to top) at is spatially smooth, these mechanisms are described by
scale r =L/16=0.39. The dashed line represents a uniform the Lyapunov exponents and their finite-time fluctua-
distribution. Inset: exponent of the power law left tail α vs
tions, associated to the growth and contraction rates
S . It has been estimated from a best fit of the cumulative
η of infinitesimal volumes. At larger scales, competition
probability which is far less noisy. Data refer to Re =185.
λ
between particle spreading and concentration is also at
equilibrium. The spatial distribution of particle at a
given scale r ≫ η should only depend on the time scale
Figure 3 shows P(QL)(ρ), for various response times
r,τ givenby theinverseofthe contractionrateγ ofasize-
r,τ
τ and at a given scale (r = L/16) within the inertial
r blob B of particles with response time τ. However in
r
range. Deviations from a uniform distribution can be
the inertial range, the flow is not differentiable and an
clearly observed, they become stronger and stronger as
approachbased on Lyapunov exponents cannot be used.
τ increases. This means that concentration fluctuations
In particular, the rate γ should depend on r. To es-
r,τ
are important not only at dissipative scales but also in
timate this contraction rate in the limit of small τ, we
the inertial range. A noticeable observation is that the
make use of Maxey’s approximation [21] of the particle
low-density tail of the PDF (which is related to voids)
dynamics [7]
displays an algebraic behavior P(QL)(ρ) ∼ ρα(r,τ). This
r,τ
meansthattheEulerianPDFofthe coarse-grainedmass X˙ ≈v(X,t) with v =u−τ(∂ u+u·∇u) , (3)
t
density has also a power-law tail for ρ ≪ 1, with expo-
nent α−1. The dependence of this exponent for fixed r meaning that particle trajectories can be approximated
and varying the Stokes number Sη is shown in the inset. by those of tracers in the synthetic compressible veloc-
For low inertia (Sη → 0), it tends to infinity in order ity field v. The inertial correction is proportional to the
to recover the non-algebraic behavior of tracers. At the fluid acceleration that, in turbulent flows, is itself domi-
largest available Stokes numbers, we observe α → 1, in- nated by pressure gradient. The contraction rate of the
dicating a non-zero probability for totally empty areas. blob B with volume r3 is γ = (1/r3)R d3x∇·v(x).
r r,τ Br
Note however that α is expected to go to infinity when As ∇·v ≃τ∇2p, we have γ ∼ (τ/r2)δ p, where δ p
r,τ r r
S →∞ with r fixed, since a uniform distribution is ex-
η denotes the typical pressure increment at scale r. This
pected for infinite inertia. Algebraic tails are relevant to
means that the scale dependence of the contraction rate
variousphysical/chemicalprocessesinvolvingheavypar-
isdirectlyrelatedtothescalingpropertiesofthepressure
ticles. The particle distribution in low density regions is field. Dimensionalargumentssuggestthatδ p∼(εr)2/3,
r
an important effect to be accounted for, e.g., modelling so that the contraction rate scales as γ ∼ τε2/3/r4/3.
r,τ
the growth of liquid droplets by condensation of vapor
However, as stressed in [19], K41 scaling for pressure is
onto aerosolparticles [8] and of aerosol scavenging [3].
observable only when the Reynolds number is tremen-
Fixing the response time τ and increasing the obser- dously large (typically for Reλ >∼ 600). In our simu-
vation scale r reproducesthe same qualitative picture as lations, where Re < 200, pressure scaling is actually
λ
fixing r and decreasing τ. A uniform distribution is re- dominated by random sweeping [20], and we observe
coveredinbothlimitsr →∞orτ →0. Thesetwolimits δ p ∼ U(εr)1/3 (as in other simulations at comparable
r
areactuallyequivalent. Atlength-scalesr ≫ηwithinthe Reynolds, see [19]). This implies that the contraction
inertial range, the fluid velocity field is not smooth: ac- rate is γ ∼τε1/3U/r5/3.
r,τ
4
the spatial structure of turbulent flows.
101
100
We acknowledge useful discussions with G. Boffetta,
− 110−1 ∝ γr9, τ/5 A. Celani and A. Fouxon. This work has been partially
100 ρ〉 QL10−2 supported by the EU under contract HPRN-CT-2002-
〈 00300 and the Galileo programme on Trasport and dis-
ρ) 10−3 persion of impurities suspended in turbulent flows. Sim-
L)( 10−1 10−4 10−3 10−2 ulations were performed at CINECA (Italy) and IDRIS
(QPτr, γr, τ (France) under the HPC-Europa programme (RII3-CT-
2003-506079). Unprocessed data of this study are freely
10−2 available from http://cfd.cineca.it.
10−3
10−2 10−1 ρ 100
[1] C. Crowe, M. Sommerfeld, and Y. Tsuji, Multiphase
FIG. 4: (Color online) Pdf of the coarse-grained mass in the Flows with Particles and Droplets (CRC Press, New
inertialrangeforthreevaluesofthenon-dimensionalcontrac- York, 1998).
tion rate γ τ . From bottom to top: γ = 4.8×10−4/τ [2] I. de Pater and J. Lissauer, Planetary Science (Cam-
r,τ η r,τ η
(different curves refer to S = 0.16, 0.27, 0.37, 0.48), γ = bridge University Press, Cambridge, 2001).
2.1×10−3/τ (for S = 0η.58, 0.69, 0.80, 0.91, 1.0), γr,τ = [3] A. Kostinski and R. Shaw, J. Fluid Mech. 434, 389
η η r,τ
7.9×10−3/τ (for S = 1.60, 2.03, 2.67, 3.31). Inset: devi- (2001).
ation from uηnity of tηhe first-order QL moment for r within [4] J. Eaton and J. Fessler, Int.J. Multiphase Flow 20, 169
the inertial range. For comparison, the behavior ∝ γ9/5 ob- (1994).
r,τ [5] L. Collins and A. Keswani, New J. Phys. 6, 119 (2004).
tainedwhenassuminguniformlydistributedpointlikeclusters
[6] G. Falkovich, A. Fouxon, and M. Stepanov, in Sedimen-
ofparticles isshown asasolid line. Datarefer toRe =185.
λ
tation and Sediment Transport, edited by A. Gyr and
W.Kinzelbach(KluwerAcademicPublishers,Dordrecht,
Figure 4 shows P(QL)(ρ) for three choices of γ ob- 2003), pp.155–158.
r,τ r,τ
tainedfromvarioussetsofvaluesofrandτ. Thecollapse [7] M. Maxey,J. Fluid Mech. 174, 441 (1987).
[8] E. Balkovsky, G. Falkovich, and A. Fouxon, Phys. Rev.
of the different curves strongly supports that the distri-
Lett. 86, 2790 (2001).
bution of the coarse-grained mass density depends only
[9] T. Elperin, N. Kleeorin, V. L’vov, I. Rogachevskii, and
uponthescale-dependentcontractionrateγr,τ. Inpartic- D. Sokoloff, Phys. Rev.E 66, 036302 (2002).
ular, as represented in the inset of Fig. 4, the deviations [10] J. Bec, L. Biferale, G. Boffetta, A. Celani, M. Cencini,
from unity of the first moment of the distribution col- A.Lanotte, S.Musacchio, andF.Toschi, J. Fluid Mech.
lapseforallS investigatedandallscalesinsidetheiner- 550, 349 (2006); M.Cencini, J.Bec,L.Biferale, G.Bof-
η
fetta,A.Celani,A.Lanotte,S.Musacchio,andF.Toschi,
tialrangeofoursimulation. Thisquantityisthesameas
Journ. Turb.7, 1 (2006).
theEulerian2nd-ordermoment,givingtheprobabilityto
[11] T.Elperin,N.Kleeorin,andI.Rogachevskii,,Phys.Rev.
havetwoparticleswithinadistancer. Wenotethatpar- Lett. 77, 5373 (1996).
ticle distribution recoversuniformity at large scales very [12] J. Bec, M. Cencini, and R. Hillerbrand, Physica D, in
slowly, much slower than if particle were distributed as press (2006).
Poissonpoint-like clusters for which hρ2i−1∝r−3∝γ9/5 [13] L. Chen, S. Goto, and J. Vassilicos, J. Fluid Mech. 553,
r,τ
(also shown in the inset for comparison). 143 (2006).
[14] A.Aliseda,A.Cartellier,F.Hainaux,andJ.C.Lasheras,
We presented numerical evidence that at moderate
J. Fluid Mech. 468, 77 (2002).
Reynolds number the distribution of heavy particles at [15] G.Boffetta,F.DeLillo,andA.Gamba,Phys.Fluids16,
lengthscales within the inertial range is fully described
L20 (2004).
in terms of a scale-dependent volume contraction rate [16] J.Bec,K.Gaw¸edzki,andP.Horvai,Phys.Rev.Lett.92,
γ ∝ τ/r5/3. However we expect that γ ∝ τ/r4/3 at 224501 (2004).
r,τ r,τ
sufficiently large Reynolds numbers (Re >600). State- [17] J. Bec, M. Cencini, and R. Hillerbrand (2006), preprint
λ
of-the-artnumericalsimulationscannotcurrentlyattain nlin.CD/0606038.
[18] M. Wilkinson and B. Mehlig, Europhys. Lett. 71, 186
these turbulent regimes, so that only experimental mea-
(2005).
surements of particle distribution can help to confirm [19] T. Gotoh and D. Fukayama, Phys. Rev. Lett. 86, 3775
suchaprediction. Wehaveseenthatparticledynamicsin (2001);Y.TsujiandT.Ishihara,Phys.Rev.E68,026309
the inertialrangecanbe directlyrelatedtothe structure (2003).
of the pressure field (and thus of acceleration). Charac- [20] H. Tennekes,J. Fluid Mech. 67, 561 (1975).
terizing the distribution of accelerationis thus crucial to [21] Fordescribingscalesr>ηtheapproximationisexpected
understandparticleclusters,andconversely,particledis- to be valid whenever Sr ≪ 1, i.e. for essentially all the
explored range of τ’s.
tributioncouldbe usedas anexperimentaltoolto probe