doi10.1017S002211200500844X Printed in the United Kingdom Acceleration statistics of heavy

349 J.Fluid Mech.(2006),vol.550,pp.349–358.c 2006Cambridge University Press

doi:10.1017/S002211200500844X Printed in the United Kingdom

Acceleration statistics of heavy particles

in turbulence

By J.B E C1,L.B I F E R A L E2,G.B O F F E T T A3,A.C E L A N I4, M.C E N C I N I5,A.L A N O T T E6,S.M U S A C C H I O7

A N D F.T O S C H I8

1CNRS Observatoire de la C?ote d’Azur,B.P.4229,06304Nice Cedex4,France

2Department of Physics and INFN,University of Rome“Tor Vergata”,

Via della Ricerca Scienti?ca1,00133Roma,Italy

3Department of Physics and INFN,University of Torino,Via Pietro Giuria1,10125,Torino,Italy 4CNRS,INLN,1361Route des Lucioles,F-06560Valbonne,France 5SMC-INFM c/o Department of Physics,University of Rome“La Sapienza”,Piazz.le A.Moro,2, I-00185Roma,Italy,and CNR-ISC via dei Taurini19,I-00185Roma,Italy 6CNR-ISAC,Sezione di Lecce,Str.Prov.Lecce-Monteroni km1,200,I-73100Lecce,Italy

7Department of Physics,University of Rome“La Sapienza”,Piazz.le A.Moro,2,I-00185Roma,Italy 8CNR-IAC,Viale del Policlinico137,I-00161Roma,Italy and INFN,Sezione di Ferrara,

via G.Saragat1,I-44100,Ferrara,Italy

(Received5August2005and in revised form9December2005)

We present the results of direct numerical simulations of heavy particle transport in homogeneous,isotropic,fully developed turbulence,up to resolution5123(Rλ≈185). Following the trajectories of up to120million particles with Stokes numbers,St, in the range from0.16to3.5we are able to characterize in full detail the statistics of particle acceleration.We show that:(i)the root-mean-squared acceleration a rms sharply falls o?from the?uid tracer value at quite small Stokes numbers;(ii)at a given St the normalized acceleration a rms/( 3/ν)1/4increases with Rλconsistently with the trend observed for?uid tracers;(iii)the tails of the probability density function of the normalized acceleration a/a rms decrease with St.Two concurrent mechanisms lead to the above results:preferential concentration of particles,very e?ective at small St,and?ltering induced by the particle response time,that takes over at larger St.

1.Introduction

Small impurities like dust,droplets or bubbles suspended in an incompressible ?ow are?nite-size particles whose density may di?er from that of the suspending ?uid,and cannot thus be modelled as point-like tracers.The description of their motion must account for inertia whence the name inertial particles.At long times particles concentrate in singular sets evolving with the?uid motion,leading to the appearance of a strong spatial inhomogeneity dubbed preferential concentration.At the experimental level such inhomogeneities have long been known(see Eaton&Fessler 1994for a review)and utilized for?ow visualization(e.g.exploiting bubble clustering inside vortex?laments).The statistical description of particle concentration is at present a largely open question with many industrial and environmental applications, such as spray combustion in diesel engines(Post&Abraham2002)or some rocket propellers(Villedieu&Hylkema2000),the formation of rain droplets in warm clouds

350J.Bec and others

(Pinsky&Khain1997;Falkovich,Fouxon&Stepanov2002;Shaw2003)or the coexistence of plankton species(Rothschild&Osborn1988;Lewis&Pedley2000). Inertial particles are also relevant to spore,pollen,dust or chemicals dispersion in the atmosphere where the di?usion by air turbulence may even be overcome by preferential clustering(Csanady1980;Seinfeld1986).

On the experimental side,the study of particle motion in turbulence has recently undergone rapid progress thanks to the development of e?ective optical and acoustical tracking techniques(La Porta et al.2001,2002;Mordant et al.2001;Warhaft, Gylfason&Ayyalasomayajula2005).In parallel with the experimental e?ort, theoretical analysis(Balkovsky,Falkovich&Fouxon2001;Falkovich&Pumir2004; Bec,Gawedzki&Horvai2004;Zaichik,Simonin&Alipchenkov2003)and numerical simulations(Boivin,Simonin&Squires1998;Reade&Collins2000;Zhou,Wexler& Wang2001;Chun et al.2005)are paving the way to a thorough understanding of iner-tial particle dynamics in turbulent?ows.Recently,the presence of strong inhomogenei-ties characterized by fractal and multifractal properties have been predicted,and found in theoretical and numerical studies of stochastic laminar?ows(Balkovsky et al.2001; Bec2004;Bec2005),in two-dimensional turbulent?ows(Bo?etta,De Lillo&Gamba 2004)and in three-dimensional turbulent?ows at moderate Reynolds numbers in the limit of vanishing inertia(Falkovich&Pumir2004).

Here we present a direct numerical simulations(DNS)study of particles much heavier than the carrier?uid in high-resolution turbulent?ows.In particular,we shall focus on the behaviour of particle acceleration at varying both Stokes and Reynolds numbers.For?uid tracers,it is known that trapping into vortex?laments(La Porta et al.2001;Biferale et al.2005)is the main source of strong acceleration events.On the other hand,little is known about the acceleration statistics of heavy particles in turbu-lent?ows,where preferential concentration may play a crucial role.Moreover,since in most applied cases it is almost impossible to perform DNS of particle transport in realistic settings,it is important to understand acceleration statistics for building stochastic models of particle motion with and without inertia(Sawford&Guest1991).

2.Heavy particle dynamics and numerical simulations

The equations of motion of a small,rigid,spherical particle immersed in an incompressible?ow have been consistently derived from?rst principles by Maxey& Riley(1983).In the limiting case of particles much heavier than the surrounding?uid, these equations take the particularly simple form

d X d t =V(t),

d V

d t

=?

V(t)?u(X(t),t)

τs

.(2.1)

Here,X(t)denotes the particle trajectory,V(t)its velocity,and u(x,t)is the?uid velocity.The Stokes response time isτs=2ρp a2/(9ρfν)where a is the particle radius,ρp andρf are the particle and?uid density,respectively,andνis the?uid kinematical viscosity.The Stokes number is de?ned as St=τs/τηwhereτη=(ν/ )1/2is the Kolmogorov timescale and the average rate of energy injection.Equation(2.1) is valid for very dilute-suspensions,where particle–particle interactions(collisions) and hydrodynamic coupling are not taken into account.

The?uid evolves according to the incompressible Navier–Stokes equations

?u ?t +u·?u=?

?p

ρf

+ν u+f,(2.2)

where p is the pressure?eld and f is the external energy source, f·u = .

Acceleration statistics of heavy particles in turbulence351

Rλu rmsενηL T EτηT tot T tr x N3N t N p N tot 185 1.40.940.002050.010π 2.20.0471440.01251235×1057.5×10612×107 105 1.40.930.005200.020π 2.20.0732040.0242563 2.5×1052×10632×106 65 1.40.850.010.034π 2.20.1102960.0481283 3.1×1042.5×1054×106

Table 1.Parameters of DNS.Microscale Reynolds number Rλ,root-mean-square velocity u rms,energy dissipationε,viscosityν,Kolmogorov lengthscaleη=(ν3/ε)1/4,integral scale L, large-eddy Eulerian turnover time T E=L/u rms,Kolmogorov timescaleτη,total integration time T tot,duration of the transient regime T tr,grid spacing x,resolution N3,number of trajectories of inertial particles for each St N t saved at frequencyτη/10,number of particles N p per St stored at frequency10τη,total number of advected particles N tot.Errors on all statistically?uctuating quantities are of the order of10%.

The Navier–Stokes equations are solved on a cubic grid of size N3for N= 128,256,512with periodic boundary conditions.Energy is injected by keeping constant the spectral content of the two smallest-wavenumber shells(Chen et al. 1993).The viscosity is chosen so to have a Kolmogorov lengthscaleη≈ x where x is the grid spacing:this choice ensures a good resolution of the small-scale velocity dynamics.We use a fully dealiased pseudospectral algorithm with second-order Adam–Bashforth time-stepping.The Reynolds numbers achieved are in the range Rλ∈[65:185].

The equations of?uid motion are integrated until the system reaches a statistically steady state.Then,particles are seeded with homogeneously distributed initial positions and velocities equal to the local?uid velocity.Equations(2.1)and(2.2) are then advanced in parallel.A transient in particle dynamics follows,for about2?3 large-scale eddy turnover times,before reaching Lagrangian stationary statistics.It is only after this relaxation stage has completely elapsed that the measurement starts. We followed15sets of inertial particles with Stokes numbers from0.16to3.5.For each set,we saved the position and the velocity of N t particles every d t=1/10τηwith a maximum number of recorded trajectories of N t=5×105for the highest resolution. Along these trajectories we also stored the velocity of the carrier?uid.At a lower frequency～10τη,we saved the positions and velocities of a larger number N p of particles(up to7.5×106per St at the highest resolution)together with the Eulerian velocity?eld.We have also followed?uid tracers(St=0),that evolve according to the dynamics

d x(t)

=u(x(t),t),(2.3)

d t

in order to systematically assess the importance of the phenomenon of preferential concentration at varying both St and Rλ.

A summary of the various physical parameters is given in table1.

3.Results and discussion

In this paper we focus on the statistics of particle acceleration a(t)=d V/d t. From previous studies on?uid tracers we know that acceleration statistics are very intermittent and strong?uctuations are associated with trapping events within vortex ?laments(La Porta et al.2001,2002;Mordant et al.2001;Biferale et al.2005). How does inertia a?ect acceleration statistics?A good starting point to gain insight into the e?ect of inertia is given by the formal solution of(2.1)in the statistically

352

J.Bec and others

a r m s /( 3/ν)1/4St St Figure 1.(a )The normalized acceleration variance a rms /( 3/ν)1/4as a function of the Stokes number for R λ=185(?);105(?);65(?).The inhomogeneous distribution of particles is quanti?ed for the highest R λin the inset,where we plot the correlation dimension,D 2,as a function of St .The correlation dimension is de?ned as p (r )～r D 2(for r η)where p (r )is the probability of ?nding two particles closer than r (Bec et al.2005).(

b )Comparison between the acceleration variance,a rms (?),as a function of St ,with the acceleration of the ?uid tracer measured at the particle position, (D u /D t )2 1/2(+).The curve (?),approaching the a rms for large St ,is the one obtained from the ?ltered tracer trajectories,a F rms .All data refer to Re λ=185.

stationary state,relating the instantaneous particle velocity to the previous history of ?uid velocity along the particle trajectory.It is expressed as V (t )=1τs t ?∞

e ?(t ?s )/τs u (X (s ),s )d s (3.1)yielding for the acceleration

a (t )=1τ2s t ?∞

e ?(t ?s )/τs [u (X (t ),t )?u (X (s ),s )]d s.(3.2)It is instructive to analyse separately the two limiting cases o

f small and large Stokes numbers.

At small St ,i.e.τs τη,the ?uid velocity along the trajectory evolves smoothly in time and expression (3.2)for the acceleration reduces to a (t ) (d /d t )u (X (t ),t ),i.e.to the derivative of ?uid velocity along the inertial particle trajectory.At su?ciently small St this is indistinguishable from the ?uid acceleration (D u /D t )(X (t ),t )evaluated at the particle positions.The latter,in turn,is essentially dominated by the ??p contribution.Therefore we can draw the following picture for the small-St case:the heavy particle acceleration essentially coincides with the ?uid acceleration;however,inertial particles are not homogeneously distributed in the ?ow and concentrate preferentially in regions with relatively small pressure gradient (low-vorticity regions).As a result,the net e?ect of inertia is a drastic reduction of the root-mean-squared acceleration a rms = a 2 1/2,due essentially to preferential concentration.Indeed,as shown in ?gure 1(a )the acceleration variance has already droped o?very fast at quite small St values.In ?gure 1(b )we give evidence that the value of a rms is very close to (D u /D t )2 1/2for St <0.4when the average is not taken homogeneously in space but conditioned at be on the same spatial positions as the inertial particles.The agreement of the two curves supports the arguments above.Notice that for increasing St the two curves start to deviate from each other,the tracer acceleration conditioned on the particle positions has a minimum for St ≈0.5close to the maximum of clustering

Acceleration statistics of heavy particles in turbulence353 (see inset of?gure1a),eventually recovering the value of a rms of the unconditioned

tracers for larger St.The latter e?ect is a clear indication that inertial particles follow

the small-scale structures of the?ow more and more homogeneously on increasing

St.In this limit a di?erent mechanism is responsible for the reduction of the a rms.

At large St,i.e.τs τη,inspection of(3.2)shows that the main e?ect of inertia on particle acceleration is a low-pass?ltering of?uid velocity di?erences,with

a suppression of fast frequencies aboveτ?1s.In?gure1(b)we also compare the

acceleration variance with that obtained by an arti?cial low-pass?ltering based only

on the?uid tracer trajectories.For each tracer trajectory,x(t),we de?ne a new

velocity,u F,?ltered on a window size of the same order as the Stokes time:

u F(t)=1

τs

t

?∞

e?(t?s)/τs u(x(s),s)d s.(3.3)

The?ltered acceleration is thus given by a F=(d/d t)u F.Of course,in order to extract the e?ect due to?ltering only we need to employ?uid trajectories:(3.3) applied along particle trajectories is the same as(3.1),so that the acceleration would coincide with the particle acceleration by de?nition.The root mean square?uctuation,

a F rms = (d/d t)u F)2 1/2,is thus computed by averaging along the tracer trajectories

without any condition on their spatial positions,i.e.homogeneously distributed in

the whole three-dimensional domain.The curves corresponding to a rms and to a F rms

become closer and closer together as St increases,supporting the conjecture that

preferential concentration for St>1becomes less important.For intermediate St we

expect a non-trivial interplay between the two above mechanisms that makes it very

di?cult to build up a model able to reproduce even the qualitative behaviour.

Another interesting aspect shown in?gure1(a)is the residual dependence of the

normalized particle acceleration on Reynolds number.For the case of?uid tracers it

is known that intermittent corrections to the dimensional estimate a rms=a0( 3/ν)1/4

may explain the Reynolds number dependence(Sawford et al.2003;Hill2002;

Biferale et al.2004).Data suggest that the?uid intermittency may be responsible of

such deviations at St>0as well.This view is supported by the fact that the curves

for the three Reynolds numbers are almost parallel.

A two-parameter formula for the variance of the acceleration as a function

of Stokes number can be derived in the limit of vanishing Stokes numbers:

a2rms(St)=a2rms(0)+C exp[?(D/St)δ](G.Falkovich2005,personal communication).

This expression follows from the acceleration probability distribution function(p.d.f.)

of tracer particles under the assumptions that(i)the main e?ect of inertia is to reduce

the particle concentration in regions where the acceleration is larger thanν1/2/τs3/2;

(ii)the p.d.f.tail is well reproduced by a stretched exponential shape with exponent

β=2/3δ.Although the formula?ts the data well,the limitation of our data-set to

only a few points with St 1does not permit a signi?cant benchmark of the model.

In table2we summarize the values that we have measured for a2 and a4 as a function of all Stokes numbers and for all Reynolds numbers available.Besides

the e?ect of inertia on typical particle acceleration it is also interesting to investigate

the e?ects on the form of the p.d.f.a(t).As shown in?gure2(a),the p.d.f.s become

less and less intermittent as St increases.In the inset of the same?gure we show the

?atness, a4 / a2 2,as a function of St.The abrupt decreasing for St>0is even more evident here(notice that the y scale is logarithmic).

In the limits of small and large St the qualitative trend of the p.d.f.s can be captured

by the same arguments used for a rms.In?gure2(b)we compare the p.d.f.shape for

354

J.Bec and others (a )St 00.160.270.370.480.590.690.800.911.011.121.341.602.032.673.31 ?a 2 3.092.071.801.631.501.391.311.241.171.121.060.970.880.750.610.51 ?a

4 28848.130.522.417.714.512.310.69.208.117.215.774.473.111.941.29(b )St 00.160.270.380.490.600.710.820.931.041.151.371.642.082.743.40 ?a 2 2.631.891.651.451.381.291.211.141.081.030.980.890.800.680.540.4

5 ?a

4 13332.921.616.313.110.99.298.037.016.185.484.373.362.231.390.90(c )St 00.160.260.370.470.580.680.790.891.001.101.311.571.992.623.2

5 ?a 2 2.021.591.401.281.191.111.050.990.940.890.850.770.700.590.470.39 ?a 4 52.819.113.110.18.246.956.015.244.614.113.672.952.321.590.970.63

Table 2.Normalized values of the second and fourth moments of the acceleration ?a

2 = a 2 /[3( 3/ν)1/2], ?a 4 = a 4 /[3( 3/ν)]for (a )R λ=185,(b )R λ=105and (c )R λ=65.The statistical error on all entries are of the order of 5%.

100

10–2

10–410–6

10–80510152025303540

P (a )a r m s (a )

30

1530123

10–2

10010–4

10–6–20–1001020

a r m s P (a )a /a rms (

b )

Figure 2.(a )Acceleration p.d.f.’s for a subset of St values (St =0,0.16,0.37,0.58,1.01,2.03,3.31from top to bottom)at R λ=185.The inset displays the acceleration ?atness, a 4 / a 2 2,at increasing R λfrom bottom to top.(b )The two outer curves correspond to the acceleration p.d.f.for St =0.16(?)and the p.d.f.of the ?uid tracers acceleration measured at the same position of the inertial particles,D u /D t (dashed line).The two inner curves are the acceleration p.d.f.at the highest Stokes,St =3.31(?)and the p.d.f.of the ?ltered ?uid acceleration (solid line).All curves are normalized to have unit variance.

Acceleration statistics of heavy particles in turbulence 355

the smallest Stokes number with the one obtained by using the tracer acceleration measured on the particle position,D u /D t .The two functions overlap perfectly,con?rming that the only di?erence between ?uid particles and inertia particles for small St is due to preferential concentration.In the same ?gure we also compare,for the highest Stokes number,St =3.31,the p.d.f.of the particle acceleration with the one obtained from the ?ltered ?uid trajectories.Now the agreement is less perfect but still fairly good,providing reassurance that this limit can be captured starting from a low-pass ?lter of ?uid tracer velocities.Note that the p.d.f.of tracer acceleration measured on the particle position,D u /D t ,approaches the unconditioned p.d.f.as St increases (not shown).This further con?rms that preferential concentration plays a minor role in the acceleration at these large Stokes numbers.

4.Statistics of acceleration conditioned on the ?ow topology

We now focus on particle acceleration statistics conditioned on the topological properties of the carrier ?ow at the particle positions.In particular,we look at the sign of the discriminant (see e.g.Chong,Perry &Cantwell 1990and Bec 2005):?= det[?σ]2 2? Tr[?σ2]6

3,(4.1)where ?σ

ij =?i u j is the strain matrix evaluated at the particle position X .Note that,in deriving (4.1),we omitted the term proportional to Tr[?σ

]because of incompressibility.For ?60the strain matrix has three real eigenvalues (strain-dominated regions);for ?>0it has a real eigenvalue and two complex conjugate ones (rotational regions).For a similar study,using a di?erent di?erent characterization of the ?ow structures,see (Squires &Eaton 1991).Note that in two dimensions the equivalent of ?is the well-known Okubo–Weiss parameter that di?erentiates elliptic from hyperbolic regions of the ?ow.

In ?gure 3(a –c )we show the acceleration p.d.f.,P (a |?),conditioned on the sign of ?at particle positions,for three di?erent characteristic Stokes number St =0.16,0.48,1.34.In ?gure 3(d )we show the root-mean-squared acceleration, a 2|? /3,as a function of St .A few results are worth noting.The fraction of particles in the two regions (N (??0))varies considerably as a function of the Stokes number (see inset of ?gure 3d ),with a depletion of particles in the regions with some degree of rotation,which becomes less e?ective at large St .This is similar to what is observed in the inset of ?gure 1(a ),where the non-homogeneous particle distribution is characterized in terms of the correlation dimension (Bec et al.2005).Further,although the shape of the p.d.f.for a given Stokes number does not change much as a function of the sign of ?,a noticeable change in the squared acceleration is observed.As shown in ?gure 3(d ),the acceleration is higher in the strain-dominated regions than in the ones with some degree of rotation.Note that the e?ect of inertia is dramatic:for the smallest St the conditional acceleration is larger when ?<0while the opposite behaviour is observed for tracer (St =0).This may be the signature of the expulsion of particles out of intense vortex ?laments (which is more e?ective for St 1)leading to an undersampling of the acceleration in the regions dominated by rotational motion.The same di?erence is also measured for higher moments of the conditioned acceleration (not shown).

These results show that the strong correlation between ?ow structure and particle preferential concentration is more e?ective at low Stokes numbers.At larger St the particle fraction N (??0)approaches the tracer value (the response time is too large to maintain the correlation between particle trajectories and the local ?ow topology)

356

J.Bec and others

10–2

10–410–6

10–8( 3/ν)1/4P (a |?)St 10–210–410–6

10–8( 3/ν)1/4P (a |?)a /( 3/ν)1/4Figure 3.(a –c )Acceleration statistics conditioned on the sign of the discriminant ?de?ned in (4.1).(a )P.d.f.s of acceleration for (a )St =0.16,(b )0.48,(c )1.34,conditioned on strain regions (solid line,?60)and on rotating regions (symbols,?>0)regions,respectively.(d )Normalized root mean square conditional acceleration on ?60(open squares)and ?>0(?lled squares)regions as a function of St .The inset displays the fraction of particles in the rotating regions N (?>0)(N (?60)=1?N (?>0))as a function of St .The conditional acceleration was computed on the data recorded at frequency 10τη(see table 1).For St =0the acceleration a 2|? /3is estimated using the pressure gradient ??p .

and the depletion of acceleration should be ascribed to the e?ect of ?ltering,as discussed in the previous section (cf.?gures 1b and 2b ).

5.Conclusions and perspectives

A systematic study of the acceleration statistics of heavy particles in turbulent ?ows,on changing both Stokes and Reynolds numbers has been presented.The main conclusions are (i)preferential concentration plays an almost singular role at small St .Indeed,even a quite small inertia may su?ce to expel particles from those turbulent regions (vortex cores)where the most intermittent and strong acceleration ?uctuations would be experienced;(ii)for small St ,a good quantitative agreement between the inertial particle acceleration and the conditioned ?uid tracer acceleration is obtained;(iii)at large St ,the main e?ect is ?ltering of the velocity induced by the response Stokes times.For St >1,the statistical properties of ?uid tracers averaged over a time window of the order of τs are in quite good agreement with the inertial particle properties.

Some important questions remain open.It is not clear how to build up a phenomenological model that is able to describe the inertial particle acceleration as a

Acceleration statistics of heavy particles in turbulence357 function of both Stokes and Reynolds numbers.For example,a naive generalization of the multifractal description,successfully used for?uid tracers(Biferale et al.2004), may be insu?cient.It is not straightforward to include in such models the correlation between preferential concentration and the local topological properties of the carrier ?ow.Here such correlations have been studied in terms of the real or complex nature of the eigenvalues of the strain matrix at particle positions.We found that,more e?ectively at small rather than large St values,particles preferentially concentrate in strain-dominated regions and that this preferential concentration has a clear role in determining the acceleration?uctuations.However,this information does not directly lead to a model for the acceleration statistics.

The strong?uctuations of both Kolmogorov time and Kolmogorov dissipative scale are the most interesting aspects which distinguish the statistics of heavy particles in turbulence from those measured in smooth?ows.It would thus be important to also study the statistical properties conditioned on the local Stokes number(de?ned in terms of a‘local’energy dissipation,see e.g.Collins&Keswani2004).

Work in this direction will be reported elsewhere.

We acknowledge useful discussions with G.Falkovich, E.Bodenschatz and Z. Warhaft.This work was partially supported by the EU under the research training network HPRN-CT-2002-00300“Stirring and Mixing”.Numerical simulations were performed with the support of CINECA(Italy)and IDRIS(France)under the HPC-Europa project(R113-CT-2003-506079).We also thank the“Centro Ricerche e Studi Enrico Fermi”and N.Tantalo for support on the numerical computations.

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