Clustering of vertically constrained passive particles in homogeneous, isotropicturbulence

Massimo De Pietro,1 Michel A.T. van Hinsberg,2 Luca Biferale,1

Herman J.H. Clercx,2 Prasad Perlekar,3 and Federico Toschi4

1Dip. di Fisica and INFN, Universita` Tor Vergata,Via della Ricerca Scientifica 1, I-00133 Roma, Italy.

2Department of Applied Physics, J. M. Burgerscentrum,Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands

3TIFR Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research,21 Brundavan Colony, Narsingi, Hyderabad 500075, India

4Department of Applied Physics and Department of Mathematics and Computer Science,Eindhoven University of Technology, 5600 MB Eindhoven,

The Netherlands and IAC, CNR, Via dei Taurini 19, I-00185 Roma, Italy(Dated: May 5, 2015)

We analyze the dynamics of small particles vertically confined, by means of a linear restoring force,to move within a horizontal fluid slab in a three-dimensional (3D) homogeneous isotropic turbulentvelocity field. The model that we introduce and study is possibly the simplest description for thedynamics of small aquatic organisms that, due to swimming, active regulation of their buoyancy, orany other mechanism, maintain themselves in a shallow horizontal layer below the free surface ofoceans or lakes. By varying the strength of the restoring force, we are able to control the thickness ofthe fluid slab in which the particles can move. This allows us to analyze the statistical features of thesystem over a wide range of conditions going from a fully 3D incompressible flow (corresponding tothe case of no confinement) to the extremely confined case corresponding to a two-dimensional slice.The background 3D turbulent velocity field is evolved by means of fully resolved direct numericalsimulations. Whenever some level of vertical confinement is present, the particle trajectories deviatefrom that of fluid tracers and the particles experience an effectively compressible velocity field.Here, we have quantified the compressibility, the preferential concentration of the particles, and thecorrelation dimension by changing the strength of the restoring force. The main result is that thereexists a particular value of the force constant, corresponding to a mean slab depth approximatelyequal to a few times the Kolmogorov length scale, that maximizes the clustering of the particles.

PACS numbers: 47.27.Gs, 47.27.T-, 47.27.ek, 47.63.Gd

I. INTRODUCTION

The problem of the distribution of inertial particlesin a turbulent flow is a crucial topic in many differentfields [19], for instance, modeling the interactions be-tween small particles carried by a turbulent flow for thestudy of cloud formation in the atmosphere, the devel-opment of industrial processes, or the study of the dy-namics of plankton organisms in oceans and lakes. It isknown that while non-inertial particles follow exactly theflow streamlines, and are homogeneously distributed inthe fluid volume, inertial particles lighter than the fluidtend to be trapped inside vortices, as opposed to heav-ier inertial particles, that tend to accumulate in strain-dominated regions of the flow [1, 3, 10, 11]. This prefer-ential concentration has important consequences in thedynamics of particles under the influence of gravity [9],and in all the situations where the clustering of parti-cles may have non-trivial consequences, as in, for exam-ple, cloud formation [1214] or the biology of aquatic

Version accepted for publication (postprint) on Phys. Rev. E 91,053002 Published 4 May 2015

microorganisms [1518].

In this paper we will study an idealized situationthat could be interesting both as a particular case ofparticles falling through a weakly stratified fluid, untilthey reach a buoyant equilibrium, and as a model of thedynamics inside plankton layers. Observations of marineecosystems often report the striking finding of planktonpopulations living confined in horizontally extended andvertically thin layers [19, 20]. Many different physical andbiological mechanisms such as buoyancy regulation, gy-rotaxis [16, 18], or nutrient variability [21] may be at thebasis of the formation of such planktonic layers, and therelevance of the different mechanisms may vary amongstdifferent plankton species [20]. The spatial confinementof living populations has direct consequences on the totalpopulation size (carrying capacity). This is the case alsowhen, independently from the particular physical mech-anisms, an effective compressibility is produced, leadingto preferential accumulation [17, 2228].

In this paper, we propose and analyze a simple modelmeant to describe the effects of preferential concentrationon passive small particles confined to move on a verticalslab inside a chaotic and turbulent flow. We are not in-terested in the biological or structural reason leading tothe confinement; we will limit ourselves to imagine that

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2there exists a bias in the equation of motion that does notallow the particles to move freely in the vertical-directionand analyze the consequences of this fact. To do that,we introduce a linear restoring force, capable of provid-ing a tunable confinement level for particles in a specificdepth interval. These confined particles are advected bya velocity field obtained by a direct numerical simulation(DNS) of homogeneous and isotropic turbulence. Disper-sion and transport processes under real marine conditionsare usually complicated by many more phenomena whichwe have not included in our model. For example, strat-ification due to density differences will be important foroceans and estuarine flows (salinity and temperature gra-dients) and also for lakes (temperature gradients). Den-sity stratified turbulent flows will change the dynamicsof the flow, the particle trajectories, and the dispersionproperties [29, 30]. Moreover, simulation of real planktondynamics should include many biological phenomena likereproduction or nutrient cycles that are not incorporatedin this model.

The paper is structured as follows: in Section II wedescribe in detail the model used for the dynamics of theparticles and the parameters of the simulation. In SectionIII the results of the simulations are presented. Our mainresult shows that there exists a non-trivial correlationbetween the distribution of passive particles as a functionof the degree of vertical confinement and the underlyinghomogeneous and isotropic turbulent flow field.

II. MODELING AND METHODS

A. The flow

The numerical integration of the homogeneous andisotropic turbulent velocity field is performed by meansof DNS of the Navier-Stokes equations employing a stan-dard pseudo-spectral algorithm. The domain size is acube with size L3 = (2pi)3 and a 1283 grid has been usedfor the discretization. Periodic boundary conditions havebeen applied in all three directions. The nonlinear termis dealt with by the 2/3-rule for dealiasing, temporal ad-vancement is based on the Adams-Bashforth second order(AB2) scheme.

Energy is injected at small wave numbers in orderto achieve a stationary state. The external forcing issuch that the energy injected in the system is constant[31]. All values of physical parameters that follow aregiven in units of the numerical simulation. The viscos-ity, in our simulation, was = 0.01 and the energy dis-sipation rate ' 2.39. The Kolmogorov length scaleis calculated using the standard dimensional argument = (3/)1/4 ' 0.025 and the Kolmogorov time scale is ' 0.065. As to the integral scales, the rms velocityis vrms ' 1.32 and the large-scale eddy-turnover time isTL = L/vrms ' 4.7.

The resulting Reynolds number was Re = vrmsL/ '800 (corresponding to a Taylor scale Reynolds number

Re = vrms/ ' 60). Let us notice that the underlyingflow is only moderately turbulent. This is not consid-ered a problem, being in the sequel mainly interested insmall-scales clustering, i.e., at length scales smaller thanthe Kolmogorov scale, where the flow would be smoothanyway (but chaotic in time and with non trivial spatialcorrelations).

In total, Np = 105 particles have been injected at

time t = 0 on a plane of constant height z0. The parti-cles are randomly and homogeneously distributed withinthe chosen plane. The particle equations of motion (seeSection II B) are also advanced in time using the AB2scheme. Both fluid and particle equations of motion werenumerically integrated for about 100 large eddy-turnovertimes TL. To ensure that the dynamics of the systemis statistically stationary (and transient phenomena havedecayed), ensemble averaging starts from time > 4TL.

B. Equations of motion for the particles

The particles are treated as passive (i.e., they produceno feedback on the fluid), point-like tracers with a con-fining force acting along the vertical, z, direction. Theequations of motion are:

dx(t)

dt= u(x, t),

u(x, t) = v(x, t)K(z(t) z0)z , (1)

where u is the velocity of the particle at time t at positionx, v is the velocity of the fluid at the particle position,K is a force constant (determining the strength of theconfinement), and z0 (here z0 = L/2) is the center of thevertical confinement layer.Equation (1) must be understood as the simplest (mini-mal) set of equations that might mimic one of the manycases when small particles are constrained to move ona given layer inside an otherwise three-dimensional (3D)volume. The physical mechanism leading to this con-straint can have a very different origin. Here we limitourselves to notice that, for example, Eq (1) can be for-mally derived from the Maxey-Riley equations [32], inthe case of almost neutrally buoyant particles in a lineardensity profile. If we neglect the Basset history term andthe Faxen corrections, we can write [33]:

du

dt=

Dv

Dt u v

s+ (1 )g , (2)

where =3f

f+2pis the density ratio (with p and f the

particle and fluid density, respectively), s =a2

3 is the

particle relaxation time (or Stokes time), and g = gz isthe acceleration due to gravity. We consider a linear den-sity profile for the flow f (z) ' 0 +(df/dz)(zz0) anduse the definition of the Brunt-Vaisala frequency N , forwriting: |df/dz| = 0N2/g [34] (note that the density

3gradient is negative for stable stratification). We assumethat N 1 and that p = 0, as we release neutrallybuoyant particles at the reference depth z0. With this inmind, the buoyancy term in Eq. (2) can be expressed as:

(1 )g ' 2g(df/dz)30

(z z0) = 23N2(z z0)z .

(3)Multiplying by s and rearranging terms in Eq. (2) weobtain:

v u = s dudt sDv

Dt+K(z z0)z , (4)

where K = 23N2s.

In the limit of small s, Eq. (4) can be further simplified.Following Ref. [32], for small inertia Dv/Dt du/dtand substituting Eq. (3) in Eq. (4) under the furtherhypothesis that (Dv/Dt)/g 1 we obtain our modelequation (1).

Other mechanisms may support or counteract confine-ment. Examples are swimming of algae or buoyancy self-regulation of cells. We assume that it is possible to modelthe combined effect of such mechanisms with a (weak)background density stratification with an effective poten-tial. If this is the case, Eq. (4), and its simplified versionEq. (1) for small Stokes numbers, stay the same, with theonly difference being a modified force constant K. Theresulting equation of motion (1) leads to a Gaussian-likeconcentration profile, as we will show later (Fig. 2). Inthe remaining part of this paper we restrict ourself to anddiscuss the physical aspects of the particle distribution inconfined layers.

III. OBSERVABLES AND RESULTS

First of all we analyze the effects of confinement onthe distribution of particles at large scales, by measuringthe particles probability distribution, NK(z), in the zdirection and its standard deviation, , from the middleplane (also referred to as an effective layer or slab depth).These quantities provide indications on the effective spa-tial extent and the strength of the vertical confinement.In order to characterize the implication of particle con-finement on the particle distribution in the horizontalslab of fluid within a 3D statistically homogeneous andisotropic turbulent flow field, we also analyze both thetwo-dimensional (2D) and 3D compressibility effects aswell as the velocity correlation integrals.

1. Vertical distribution

In Figure 1 we show some representative snapshots ofthe system, for different values of the force constant K.From these pictures it is evident that confinement hasstrong effects both at large and small scales.

FIG. 1: Plot of the spatial distribution of particles for differ-ent values of the force constant K. The magnitude of the slabdepth is also shown for comparison. (top panel) Free tracerscase, corresponding to the choice of parameters K = 0. (mid-dle panel) Intermediate confinement case, with force constantK = 0.125. The associated effective slab depth is ' 21.(bottom panel) Strong confinement case corresponding to pa-rameters K = 6 and ' 0.46. Here, particles are almostperfectly confined in a plane. The presence of the verticalconfinement induces an evident and strong preferential con-centration in the horizontal plane.

40

L/2

L

10-4 10-3 10-2 10-1 1

Z

N

FIG. 2: Plot of the normalized z-distribution of particles,NK(z), for different values of the force constant K (or dif-ferent values of ). The curves corresponds to the values/ = 38, 21, 9.1, 4.7, 3.48, 2.33, 1.41, 0.46; the vertical widthof the distribution monotonically increases. The points showa normalized Gaussian fit for the curve with / = 21. isthe Kolmogorov length scale.

For unconfined particles we observe the classical ho-mogeneous distribution on the whole volume, while inthe extreme case of particles bound to move on a plane,we observe a fractal-like distribution with a dimensionconsiderably smaller than 2. For each different value ofthe force constant we estimated the probability densityof finding a particle with a z coordinate in the range[z; z + z] computing an histogram, NK(z). Figure 2shows the histograms for the z distribution of the parti-cles. The distribution is well approximated by a Gaussianwith center at z = z0 and variance

2.

Each curve in Figure 2 corresponds to a specific valueof . There is a one-to-one correspondence between thevalues of and those of the force constant K. Measuring is thus an intuitive way for quantitatively describingthe strength of the confinement of the particles aroundthe central plane z = z0.

For a linear restoring force, we expect to be inverselyproportional to K. Furthermore, it is possible to calcu-late analytically the value of in two limit cases: un-confined particles or particles restricted to a plane. Inthe limit of particles strictly confined on a plane, i.e.,K , is exactly zero. In the limit of K 0,i.e., no vertically constraining force, particles are freelyadvected over the full domain following the underlyingfluid motion (3D turbulent diffusion). The particle den-sity becomes homogeneous over the cubic domain and evolves to a constant value that is a fraction of L. Figure3 shows the relation between the dispersion and theforce constant K. We see that, for large values of K, isproportional to 1/K, as we expect. This proportionalityshould disappear when K is decreased.

1

4

16

64

10 100 1000

Slab

dep

th

Inverse elastic constant 1/K

FIG. 3: Plot of the slab depth versus the inverse of the forceconstant 1/K. The horizontal dotted line is the asymptoticvalue of for K 0. The slab depth is given in units of ,the Kolmogorov length scale. 1/K is given in units of 1 ,where is the Kolmogorov time scale. The diagonal dashedline is obtained from a linear least squares fit.

2. 2D and 3D compressibility

We measured two different compressibilities for thesystem of particles (the underlying flow is always incom-pressible): the 2D compressibility based only on the hor-izontal velocity components and the full 3D compress-ibility. In both cases, in order to obtain an ensembleaverage, the compressibility has been calculated averag-ing on both particles and time. The error is then givenby the standard deviation of the mean.a. 2D compressibility The 2D compressibility is de-

fined as:

C2D =

(i iui)

2

i,j (iuj)2 , (5)

where i, j = x, y and u is the particle velocity.In the limit K 0 we can calculate analytically thevalue of C2D. This limit corresponds to extracting thevelocity data from a plane in a fully 3D flow field. Fora 3D homogeneous and isotropic turbulent flow (HIT),by simply substituting the relations between the velocitygradients inside Eq. (5), we find CHIT2D =

16 ' 0.167.

Figure 4 shows the relation between the 2D compress-ibility Eq. (5) and particle dispersion (that is inverselyproportional to the force constant K). In the case ofunconfined particles we correctly recover C2D = C

HIT2D .

Increasing the confinement, the 2D compressibility has avalue lower than CHIT2D . This is because the exact valueCHIT2D is calculated in an Eulerian framework, averagingthe velocity field of the flow in the whole plane, while inour case we use the Lagrangian velocity gradients, i.e.,the Eulerian velocity gradients measured at the positionsof the particles (thus not homogeneously distributed in

5 0

0.05

0.1

0.15

0.2

0.25

0.3

0.25 1 4 16 64

2D C

ompr

essib

ility

Slab depth

FIG. 4: Plot of the 2D compressibility c2D versus the slabthickness . Here, is given in units of the Kolmogorovlength scale . Data points are obtained averaging on bothtime and particles. Errors are the standard deviation of themean.

a plane, due to the presence of preferential concentra-tion). Only without preferential concentration do we ex-pect C2D = C

HIT2D .

b. 3D compressibility The 3D compressibility is de-fined as:

C3D =

(i iui)

2

i,j (iuj)2 , (6)

where i, j = x, y, z and u is the particle velocity.Substituting the equations of motion (1) in Eq. (6) weobtain:

C3D =

(xvx + yvy + zvz K)2

[i,j (ivj)

2] 2Kzvz +K2

==

K2i,j (ivj)

2

+K2, (7)

where we explicitly used the incompressibility of the 3Dflow field and the fact that zvz = 0. So we expectC3D = 0 in the K 0 limit, and C3D = 1 in the K limit.

Figure 5 shows the relation between the 3D compress-ibility (6) and the particle dispersion (that is inverselyproportional to the force constant K). As expected, C3Dis zero for tracers, and increases monotonically with theconfinement strength.

3. Accumulation of particles and pair correlations in space

Another important way of characterizing the system isto look at the distribution of particles at small scales, i.e.,

10-5

10-4

10-3

10-2

10-1

0.25 1 4 16 64

3D C

ompr

essib

ility

Slab depth

FIG. 5: Log-log plot of the 3D compressibility C3D versus theslab thickness . Here is given in units of the Kolmogorovlength scale . The dashed line is a fit of Eq. (7).

the tendency of particles to inhomogeneously concentratein space. This behavior is the result of the interplay be-tween the motion induced by the underlying fluid and thepresence of a vertically confining force on the particles.Because of the inhomogeneity on the vertical direction,we decided to characterize the spatial distribution cen-tering the analysis on a small volume around the centralequilibrium plane:

A = {zi [L/2z;L/2 + z]} , (8)with z ' 0.2. In this way, measurements on horizon-tal scales larger (smaller) than z will be mainly two-dimensional (three dimensional).

We defined a pair distribution integral, P2(r) as fol-lows: having defined the set A of all particles falling inthe central volume of vertical width z, one counts howmany pairs with a relative distance r can be formedwith a particle in A and another particle anywhere in thevolume. Formally:

P2(r) =iA

Npj=1

(r |xi xj |) , (9)

where (x) is the Heaviside step function and xi, xj arethe particle coordinates. If one had chosen A equal tothe whole system of particles then one would obtain theclassical correlation dimension [35].

In order to quantify the scale-by-scale clustering prop-erties it is useful to introduce the local scaling-exponent

(r) =d log (P2(r

))d log (r)

r=r

. (10)

In the limit r 0 one expects that the scaling expo-nent recovers the definition of correlation dimension ofthe particle distribution.

6 0.5

1

1.5

2

2.5

3

0.01 0.1 1 10 100

Loca

l sca

ling

expo

nent

Pair distance r

= 0.46 = 1.41 = 2.33 = 3.48 = 4.70 = 9.08 = 21.05 = 38.17 = 72.43

FIG. 6: Log-log plot of the local scaling exponents (r) ver-sus the radial distance of the pairs, for different values of theeffective slab thickness . Continuous lines represent all theavailable data; the purpose of the symbols is to help the readerto distinguish the different curves. Both the slab thickness and the pair distance r are given in units of the Kolmogorovlength scale . Only a few indicative error bars are shown,in order to keep the Figure clear. The error bars have beenestimated by comparing results between two different subsetsof the whole statistics. The curve corresponding to = 4.70is emphasized to stress the fact that the minimum in frac-tal dimension does not correspond to the minimum in slabthickness.

This local scaling exponent expresses how the numberof pairs scales with the distance r, for the particles in theinnermost part of the layer.

Figure 6 shows the local scaling exponent (r) ver-sus the radial distance of the pairs, where the error barshave been estimated by comparing results between twodifferent subsets of the whole statistics. Figure 7 showthe local scaling exponents (r) at different values of thedistance r versus the particle dispersion . This figureconfirms the results of Fig. 6: there exists a minimumin the exponent (corresponding to a maximum in the ac-cumulation of particles) for ' 5, at least in a rangeof scales 0.02 < r/ < 19.6. For & 5 the exponentgrows, corresponding to a decrease in clustering becauseof the reduced confinement effects. On the other hand, if is decreased below the exponent stays constant, sincethe particles are already constrained to be very close tothe central plane.

Let us notice that using the correlation integral asintroduced in [35] to analyze the particle distributionleads to an undesirable effect for our set-up: centeringthe spheres on peripheral particles (far from the centralplane), gives a lower number of pairs inside a given sphereof radius r because of the vertical non-homogeneous dis-tribution. Using our definition to measure the pair distri-bution integral P2(r) corresponds to measuring the orig-inal three-dimensional distribution in such a way thatthe central particles have a larger weight with respect

0.6

1

1.4

1.8

2.2

2.6

3

0.25 1 4 16 64

Local scalin

g e

xponent

Slab depth

r = 0.0020r = 0.020r = 0.20r = 1.96r = 9.80

r = 19.61

FIG. 7: Plot of the value of the power-law exponent (r)versus for different values of r. We can see that the curvesin the range 0.02 < r/ < 19.6 exhibit non-monotonicity, anda minimum around r ' 5. Points are taken intersecting thecurves in Figure 6 with lines r = constant. Both the slabthickness and the pair distance r are given in units of theKolmogorov length scale . The error estimation procedureis detailed at the end of Sec. III 3. The dashed line is merelya guide for the eye.

to the peripheral ones, and the effect of the inhomo-geneity in the z direction is less pronounced. The valuez ' 0.2 has been chosen because it allows us to shiftthe abrupt drop in the scaling exponent ( visible in Fig.6) at large scales, leaving a cleaner power-law behaviorat the scales we are interested in. Increasing z wouldshift the drop in the scaling exponent at smaller lengthscales.

IV. CONCLUSIONS

In this study we have investigated the dynamics of asystem of particles suspended in a turbulent flow andvertically constrained to evolve within an horizontal slabwith a certain thickness depending on an effective linearrestoring force. In particular, we quantified the effectivecompressibility of the particle distribution and particleaccumulation varying the degree of confinement.

Using DNS we have studied a simplified model in whichparticles are suspended in a 3D, isotropic and homoge-neous, turbulent flow. These particles are passive, point-like and confined only in the vertical direction by meansof a restoring force. We studied different situations, vary-ing the thickness of the slab, in order to analyze thecharacteristics of the particle suspension in a range ofconditions from having full accessibility to the 3D in-compressible flow, to a 2D slice of a 3D incompressibleflow. The particle distribution shows a certain degree ofcompressibility, except for K 0.

We have also shown that there exists a particular -optimal- value of the effective slab thickness (or, equiv-

7alently, of the force constant K) that maximizes the accu-mulation of the particles (minimizing the fractal dimen-sion of the system). This happens when the depth of thehorizontal slab is of the order of a few Kolmogorov lengthscales ( 5). Let us point out that viscous effectsare known to be important up to 5 10 in turbulentflows, meaning that the maximum particle accumulationis achieved when their vertical displacement is boundedto be no larger than the size of viscous eddies.

We want to stress how our model, though simple,could be important towards the quantitative under-standing of the phenomenology of passive, point-likeentities in a turbulent marine thin layer. Indeed, ourmodel captures the generic features associated with thepresence of a vertical localization, irrespective of thebiological or physical reason beyond its production andits stability. Thin phytoplankton layers are always muchthicker than the size of individual cells and for thisreason one may question how the discussed confinementmay be relevant at all to plankton population dynamics.Here it must be stressed that what is important is therelation between time scales, and not length scales, in thesystem. Indeed the typical generation time for planktonand bacteria in a marine environment is well within theinertial range and such that over a generation the cellhas been transported to distances much larger than thevertical confinement. Clearly real marine conditions

are complicated by many more phenomena that wehave not included in our model and the real planktondynamics shows phenomena that the simplified modeldiscussed here cannot incorporate. The improvementof our model in order to apply it to more complexsituations and to the modeling of systems more similarto real-life plankton particles in oceans is a challengefor future studies. For example, an obvious follow-up tothis investigation would be to simulate inertial particlesinstead of passive tracers, integrating more physicallyaccurate equations of motion, while keeping a simplelinear restoring force for the vertical confinement.

ACKNOWLEDGMENTS

This work is part of the Research Programs No.11PR2841 and No. FP112 of the Foundation for Funda-mental Research on Matter, which is part of the Nether-lands Organisation for Scientific Research. The workof L.B. and M.D.P was partially funded by EuropeanResearch Council Grant No. 339032. We acknowledgesupport from the European Cooperation in Science andTechnology (COST) Actions MP0806 and MP1305.

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I IntroductionII Modeling and methodsA The flowB Equations of motion for the particles

III Observables and results1 Vertical distribution2 2D and 3D compressibility3 Accumulation of particles and pair correlations in space

IV Conclusions ACKNOWLEDGMENTS References