Lagrangian tracer dynamics in a closed cylindrical turbulent convection cell

Mohammad S. Emran and Jörg Schumacher*Institut für Thermo- und Fluiddynamik, Technische Universität Ilmenau, Postfach 100565, D-98684 Ilmenau, Germany

�Received 16 February 2010; published 7 July 2010�

Turbulent Rayleigh-Bénard convection in a closed cylindrical cell is studied in the Lagrangian frame ofreference with the help of three-dimensional direct numerical simulations. The aspect ratio of the cell � isvaried between 1 and 12, and the Rayleigh number Ra between 107 and 109. The Prandtl number Pr is fixed at0.7. It is found that both the pair dispersion of the Lagrangian tracer particles and the statistics of theacceleration components measured along the particle trajectories depend on the aspect ratio for a fixed Ray-leigh number for the parameter range covered in our studies. This suggests that large-scale circulations presentin the convection cell affect the Lagrangian dynamics. Our findings are in qualitative agreement with existingLagrangian laboratory experiments on turbulent convection.

DOI: 10.1103/PhysRevE.82.016303 PACS number�s�: 47.55.pb, 47.27.te, 47.27.ek

I. INTRODUCTION

The motion of a fluid can be described either in the Eu-lerian or Lagrangian frame of reference. In case of fluid tur-bulence, the Lagrangian point of view brought interestingand new insights on the small-scale structure and statistics�see e.g., Yeung �1�, Toschi and Bodenschatz �2� for compre-hensive reviews�. These studies included intermittency oftemporal velocity increments and accelerations �3–5� or thegeometry of particle tracks �6�. Furthermore, two-particleand multiparticle dispersion have been analyzed and com-pared with classical predictions by Batchelor �7� and Rich-ardson �8�.

Almost all of the experimental and numerical investiga-tions in turbulent thermal convection have been conducted inthe Eulerian frame. This includes studies on the turbulentheat transfer and large-scale circulations �9,10� as well as onsmall-scale structures and dynamics of thermal plumes�11–13�. Recently, first Lagrangian laboratory experimentson convection were conducted in a closed cylindrical vessel�14� and numerical simulations were carried out in Cartesianslabs with periodic side walls and free-slip boundary condi-tions at the top and bottom �15,16�. The focus of these stud-ies was on the local variations of the heat transfer which canbe measured along the trajectories of the Lagrangian tracers.In the numerical studies, these results could be connected tothe local tracer accelerations. In the laterally infinitely ex-tended layer �i.e., the configuration with periodic side walls�,slightly less intermittent vertical accelerations az were de-tected in comparison to the lateral ones, ax and ay. This wasmanifested in the sparser tails of the probability density func-tion �PDF� of az.

In the present work, we want to take these numerical La-grangian studies to the next level. We report investigations ina closed cylindrical cell with no-slip boundary conditions atall walls. This is the characteristic setup for almost all labo-ratory experiments. It will allow for a closer comparisonwith the work by Gasteuil et al. �14�. In addition, we want toexplore systematically how the finite size of the cell affects

the Lagrangian dynamics, such as the long-time behavior ofthe two-particle dispersion. The latter point is also motivatedby recent Eulerian studies, in which a systematic dependenceof the turbulent heat transfer on the aspect ratio � is foundfor small and moderate values of � �17�. The dependence isin line with morphological changes in the large-scale circu-lation �LSC�, e.g., from one-roll to multiroll patterns, in theconvection cell. Our boundary conditions will differ fromthose in Refs. �15,16�. This difference might also affect thestatistics of the acceleration components. It is thus anotheropen question if a different velocity boundary layer structurehas an impact on the intermittency of lateral and verticalacceleration components.

The outline of the paper is as follows. In Sec. II wepresent the equations of motion, the numerical model and thetracer advection scheme for the nonuniform cylindricalmesh. Section III is focused on the particle dynamics andlocal heat transfer followed by studies of the pair dispersionin Sec. IV and the aspect ratio dependence of the accelera-tion statistics in Sec. V Finally, we summarize our findingsand give a brief outlook.

II. EQUATIONS OF MOTION AND NUMERICAL MODEL

A. Boussinesq equations

We solve the Boussinesq equations for Rayleigh-Bénardconvection numerically with a second-order finite differencescheme in cylindrical coordinates �� ,r ,z� �19,20�. The equa-tions are given by

�u

�t+ �u · ��u = − �p + ��2u + �gTez, �1�

� · u = 0, �2�

�T

�t+ �u · ��T = ��2T , �3�

where p�x , t� is the �kinematic� pressure, u�x , t� the velocityfield, � the thermal expansion coefficient and g the gravityacceleration. The velocity field has a no-slip boundary con-*Corresponding author; joerg.schumacher@tu-ilmenau.de

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dition at all walls. The temperature field is isothermal at thetop and bottom plates and adiabatic at the side wall. Follow-ing Refs. �13,17�, we have chosen the direct numerical simu-lation �DNS� grid such that the global maximum of the geo-metric mean of the grid spacings i.e.,

� = max�,r,z

��3 ���r�z� , �4�

satisfies the resolution threshold by Grötzbach �21�, namely,�� /�1 where � is the Kolmogorov dissipation lengthwhich is calculated via the mean energy dissipation rate ��= �3

H4 �Nu−1�RaPr−2. The details are provided in Table I.The resolutions of the runs are exactly the same as in �17�.

Note that the turbulence is strongly inhomogeneous with re-spect to the vertical direction z. This results in amplitudes ofthe energy dissipation rate which are by orders of magnitudelarger close to the plates than in the bulk. A local Kolmog-orov scale is therefore much larger in the bulk than in theboundary layers. It motivated us in �17� to relate the gridspacing to ��z�=�3/4 / ��z��A,t

1/4 rather than �.The dimensionless parameters and their range of values in

our numerical study are: the Rayleigh number Ra=�g�Tbottom−Ttop�H3 / ���� between 107 and 109, the Prandtlnumber Pr=� /�=0.7 and the aspect ratios �=D /H between1 and 12. D is the cell diameter and H is the height of thecell. The Nusselt nu7mber is denoted by Nu.

B. Numerical integration scheme

We conduct direct numerical simulations �DNS� of theBoussinesq equations. The equations are solved in a cylin-drical coordinate frame. The spatial discretization is per-formed on a staggered grid and solved by a second-orderfinite difference scheme �19,20�. The pressure field p is de-termined by a two-dimensional Poisson solver �22� after ap-plying one-dimensional fast Fourier transformations in theazimuthal direction. The time advancement is done by athird-order Runge-Kutta scheme. The grid spacings are non-equidistant in the radial and axial directions. In the verticaldirection, they correspond to Tschebycheff collocationpoints. Some parameters and the corresponding grid resolu-tions are given in Table I.

C. Tracer advection

The equations of Lagrangian tracer motion are given by

dxp�t�dt

= u�xp,t� . �5�

Here u= �u���p� ,ur�rp� ,uz�zp��T is the velocity of the particleat position xp= ��p�t� ,rp�t� ,zp�t��T with the symbols �, r,and z corresponding to the azimuthal, radial and axial direc-tions, respectively. As discussed before, the Eulerian fieldsare well resolved on a staggered and nonuniform grid in thesimulations. The velocity components, as shown in Fig. 1,are then given on particular faces of the grid cells only, whilethe Eulerian temperature field is always centered in the gridcell. All these fields are required at the Lagrangian particleposition to advance �Eq. �5�� and study local heat transfer.We therefore applied an interpolation scheme for the velocitycomponents which makes direct use of the staggered setup.The interpolation is then of second order and the maximalerror can be estimated to be of the order of O��2� �see Eq.�4��. Given the Eulerian velocities at the mid points of theedges of the mesh cell �see Fig. 2�, we calculate the corre-sponding Lagrangian components at an arbitrary position in-side a cell as

ur�rp� =

wj−1/2r ur�i, j +

1

2,k + wj+1/2

r ur�i, j −1

2,k

�r�i, j,k�,

TABLE I. Simulation parameters for various Ra and � with a fixed Pr=0.7. The grid resolution, theRayleigh number Ra, the aspect ratio �, the ratio �� /�, and NBL, the number of horizontal grid planes insidethe thermal boundary layer thickness, are given. NBL satisfies the criteria discussed in �18� very well.

Run N��Nr�Nz Ra � �� /� NBL

1 193�97�128 107 1 2.26 14

2 257�165�128 107 3 1.43 14

3 361�257�128 107 5 1.32 14

4 401�311�128 107 8 1.06 14

5 601�401�128 107 12 1.01 14

6 271�151�256 108 1 1.73 20

7 361�181�310 109 1 1.03 17

FIG. 1. Sketch of a 3D grid cell in the cylindrical coordinates, inwhich the positions of the velocity components, the temperature andthe pressure are shown on the staggered grid.

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u���p� =

wi−1/2� u��i +

1

2, j,k + wi+1/2

� u��i −1

2, j,k

���i, j,k�,

uz�zp� =

wk−1/2z uz�i, j,k +

1

2 + wk+1/2

z uz�i, j,k −1

2

�z�i, j,k�. �6�

Indices i , j ,k denote the grid positions with respect to � ,r ,z,which run from 1 to N�, Nr, and Nz, respectively.

In order to progress further, we rewrite the three ordinarydifferential equations �we provide here details of the equa-tion in r-direction only� �23� as

drp

dt= ur�rp� =

rp − rL

rR − rLur,R + �1 −

rp − rL

rR − rLur,L, �7�

where ur,R and ur,L are the Eulerian velocities at face posi-tions rR and rL, respectively and rp is the particle’s positionin between. The subscripts L and R refer to the left and rightfaces of a three-dimensional �3D� cell �see Fig. 1�, respec-tively. Rearranging Eq. �7� yields �23�

drp

dt− rp

�ur

�r= ur,L − rL

�ur

�r, �8�

with �ur=ur,R−ur,L and �r=rR−rL. Multiplying both sidesof Eq. �8� with the integrating factor exp�−

�ur

�r t� gives

d

dt�rpe−��ur/�r�t� = �ur,L − rL

�ur

�re−��ur/�r�t. �9�

Now integrating both sides of Eq. �9� with respect to t givesthe r-position of the particle as

rp�t� = rL − ur,L�r

�ur+ Ce��ur/�r�t. �10�

At t= t0

rp�t0� = rL − ur,L�r

�ur+ Ce��ur/�r�t0, �11�

and thus

�r = rp�t� − rp�t0� = C�e��ur/�r�t − e��ur/�r�t0� . �12�

From Eq. �11�, with rp�t0�=ur,0, and Eq. �8� at t= t0 the con-stant C follows to

C = ur,0�r

�ure−��ur/�r�t0. �13�

and thus the radial increment for �t= t− t0 is

�r = ur,0�r

�ur�e��ur/�r��t − 1� �14�

Expanding the exponential term, this scheme is first order intime for the leading contribution. Similarly, for the azimuthaland axial directions, the distances are given by

�� = u�,0��

�u�

�e��u�/r0����t − 1� , �15�

�z = uz,0�z

�uz�e��uz/�z��t − 1� . �16�

The equations of the particle trajectories in the Lagrangianframe are

rp�t� = rp�t0� + �r , �17�

�p�t� = �p�t0� + �� , �18�

zp�t� = zp�t0� + �z . �19�

The temperature field T is stored at the cell center of thestaggered grid as shown in Fig. 1. At first, we interpolate T atthe vertices of the Eulerian grid and then apply trilinear in-terpolation to calculate the Lagrangian temperature at aknown particle position.

III. TRACER DYNAMICS IN THE CELL

A. Tracer trajectories and large-scale circulation

For the Lagrangian simulations, we have selected cylin-drical cells which are given in Table I keeping the Prandtlnumber constant �Pr=0.7�. We advect Np105 tracer par-ticles in the cell �see Table II for more details�. They areinitially seeded in the whole cell and integrated in time formore than 150 time units t / tf where tf =H /Uf is the free falltime. Here, Uf =�g��Tbottom−Ttop�H. The statistical analysisis thus conducted over 3000 independent Lagrangiansamples, which are separated by �t=0.05tf which corre-sponds to �t=0.25 �, 0.33 �, and 0.43 � for Ra=107, 108,and 109, respectively.

We visualize the time-evolution of the trajectory of a La-grangian particle in Fig. 3 for a simulation at Ra=107 in theRayleigh-Bénard cell with aspect ratio �=1. It seems thatthis particle travels preferentially along the large-scale circu-

FIG. 2. Position of the velocity components on a 2D staggeredgrid in r−z plane, which is the midsection of a typical 3D cell asshown in the top panel of the figure. The indices i, j, and k corre-spond to the azimuthal, radial and axial directions, respectively. Theparticle velocity ur�rp� at position rp is obtained by the linear inter-polation of the Eulerian velocities ur�i , j− 1

2 ,k� and ur�i , j+ 12 ,k�

with the widths wj−1/2r and wj+1/2

r as given in Eq. �6�.

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lation �LSC�. Figure 4 plots the time series of the verticalposition and various quantities along the same particle pathas shown in Fig. 3. The results are qualitatively in agreementwith the Eulerian measurements �24� and the Lagrangianmeasurements �14�. Sharp spikes of the temperature signalappear due to rising and falling plumes close to the hot andcold plates of the cell. They are correlated with the maximaand minima in the vertical position. The vertical velocityuz�t� has zero mean. The convective heat fluxes are shown inFigs. 4�d�–4�f�. The signals are highly irregular with sharpintermittent peaks. The factor �RaPr is multiplied to all fluxterms uxT, uyT and uzT in order to get the values of thedimensionless local convective heat flux. Similar to Gasteuilet al. �14� we observe strong fluctuations about the meanvalue which are 4.8, −1.7 and 23.4 for uxT, uyT, and uzT,respectively. It demonstrates that the vertical flux uzT isdominantly responsible for the net heat transfer in the cell.

The time series of the vertical tracer position z�t� shows arather regular oscillatory form which is caused by the meanwind in the cell. One can therefore extract a typical loop time

from all the time series. Motivated by the analysis ofSreenivasan et al. �25� for mean wind reversals, we havetherefore performed a statistics of the zero crossings for zi�t�with i=1. .Np. The mean time interval between two subse-quent zero crossings for each tracer track has been deter-mined, and a subsequent Lagrangian ensemble mean, � · �Lhas been taken. The latter, which we denote as �Tzc�L, gives acharacteristic loop time of the tracers, Tl=2�Tzc�L. For a fixedRayleigh number Ra=107, Tl=19.4tf, 17.8tf, and 19.0tf for�=1, 3, and 8, respectively.

Translating into a convective time unit tc=H / �u2�V,t1/2 �see

Ref. �17� for a discussion� gives Tl�3.7tc, 3.4tc, and 3.6tcfor �=1, 3 and 8, respectively. For a fixed aspect ratio �=1, we find Tl=19.6tf at Ra=108 and Tl=20.6tf at Ra=109.We see that the characteristic loop time of the tracer is

TABLE II. The total integration time t / tf, the Eulerian and La-grangian values of the Nusselt number plus errorbars NuE and NuL,and the total number of tracers Np are listed.

Run t / tf NuE�� NuL�� Np

1 150 16.73�0.08 16.77�2.95 104160

2 150 16.06�0.05 15.86�1.37 81024

3 150 16.36�0.03 15.70�0.57 202860

4 150 17.44�0.02 16.00�0.54 208425

5 150 17.49�0.03 16.06�0.54 272811

6 150 32.21�0.32 31.75�4.29 112320

7 139 64.31�0.64 65.12�19.53 179850

FIG. 3. �Color online� Three-dimensional trajectory of a La-grangian tracer particle for the simulation at Ra=107 in the cell withaspect ratio �=1. The tracer track is displayed for 100 free-fall timeunits tf.

FIG. 4. �Color online� Time series of various quantities alongthe trajectory of a single Lagrangian tracer particle: �a� the verticalposition z�t� normalized by the cell height H, �b� the vertical veloc-ity uz�t�, �c� the temperature T�t�, products of velocity componentsand temperature �d� uzT, �e� uyT and �f� uxT. The dotted lines insome of the panels show the corresponding value of the time aver-age which enters the Lagrangian definition of the Nusselt number asgiven in Eq. �21�. Data are taken along the tracer track which isshown in Fig. 3

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slightly varying but no clear trend when � is varied and Ra isfixed. It seems, therefore, to be independent of the presenceof a single-roll or multiroll LSC. In case of the fixed aspectratio, a slight increase of Tl with increasing Ra is detected.We recall that the velocity fluctuations amplitude decreaseswith increasing Ra as reported for example by Verzicco andCamussi �20�. Our observation of a slightly longer loop timeseems to be consistent with a reduced amplitude of velocityfluctuations with increasing Ra. On the basis of the presentdata, it is however difficult to draw a firm conclusion aboutthe robustness of this trend.

B. Heat transfer in the Lagrangian frame

Similar to the Eulerian case, where the Nusselt number isgiven by

NuE = 1 +H

��T�uzT�V,t, �20�

with � · �V,t being a volume and time average, the Nusseltnumber in the Lagrangian frame, NuL, is given by �16�

NuL = 1 +H

��T�uzT�L,t = �NuL�t��t, �21�

where the symbol � · �L,t denotes an averaging over all trajec-tories and time. We compute the time series of the heat trans-fer in the Lagrangian frame and compare with its value in theEulerian frame in Fig. 5 for a simulation at Ra=107 in thecell with aspect ratio �=1. The time averaging of NuL�t�=1+H / ���T��uz�t�T�t��L curve from t=40 to 200 yieldsNuL=16.77, which almost exactly matches the correspond-ing value of the Eulerian simulation, which was NuE=16.73 as provided in Table II. Similar result is achieved forthe simulation conducted at Ra=108 in the same cell, forwhich NuE=32.21 and NuL=31.75 with a deviation of 1.5%from NuE. The results also validate our interpolationschemes and the time step sizes for the Lagrangian simula-tions, which are accurate enough to reproduce flow quantities

of the corresponding Eulerian simulations in turbulent con-vection. In contrast, recent experiments by �14� obtained avalue of NuL almost twice as large as the Eulerian case. Thisdiscrepancy is due to the fact that their �14� mobile sensortraversed preferentially along the mean flow circulation pathand hence the contribution from the rest of the volume wasmissing in their Nusselt number measurement.

At the beginning of the NuL�t� curve in Fig. 5, there is alarge overshoot from mean, which generally appears whenthe majority of the particles follow the LSC path or the par-ticles are not seeded uniformly in the flow. We also observestrong oscillations in NuL�t� for all the simulations listed inTable II. Strong fluctuations around the mean result in largestandard deviation �. The value of � decreases with increas-ing �. When the number of particles was increased �case:Np�2.5�105, Ra=107 and �=1; the results are not shownhere�, the convergence of NuL�t� was achieved within ashorter period of time. It can be concluded that the numberof tracer particles should be sufficiently large and theyshould be uniformly seeded in the domain for faster statisti-cal convergence of NuL�t�.

IV. LAGRANGIAN PAIR DISPERSION

The pair dispersion measures the relative separation of apair of tracer particles traversing along their trajectories andis defined as

R2�t� = ��xp�t� − xp�t0��2�L. �22�

Since convective turbulence is inhomogeneous, we decom-pose the distance vector into two parts—the lateral partRxy�t� and the vertical part Rz�t�. Their contributions to thetotal dispersion R2�t� are given by

Rxy2 �t� = ��xp�t� − xp�t0��2 + �yp�t� − yp�t0��2�L, �23�

Rz2�t� = ��zp�t� − zp�t0��2�L. �24�

This decomposition is chosen in order to relate our results tothe findings in �15,16�. We also compute the autocorrelationfunctions of the velocity components as

Cxy� � =�uxy�t + � · uxy�t��L,t

��uxy�2�L,t, �25�

Cz� � =�uz�t + �uz�t��L,t

�uz2�L,t

. �26�

Here Cxy� � is the autocorrelation coefficient for the lateraltracer velocities uxy =uxex+uyey, with ex and ey are the unitvectors in x and y directions, respectively, Cz� � is for thevertical velocity uz, and is the time lag. The vertical andlateral Lagrangian times, TL

z and TLxy are obtained by integrat-

ing Cz� � and Cxy� � as

TLz =

0

�

Cz� �d �27�

FIG. 5. �Color online� Graph of NuL�t� which is averaged over105 tracer paths. The temporal mean value of NuL= �NuL�t��t �seeEq. �21�� between time t=40–200 is NuL=16.77, which almostexactly matches with the corresponding Eulerian value NuE

=16.73 as listed in Table II.

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TLxy =

0

�

Cxy� �d , �28�

and the Kolmogorov time � is given by

� =� �

��. �29�

Figures 6�a� and 6�b� plot the total dispersion, R2, as well asthe lateral, Rxy

2 , and vertical, Rz2, contributions. Two aspect

ratios are selected, namely, �=1 and 3 for simulations atRa=107. Initially, the quantities grow with R2 t2, whichcorresponds to the ballistic dispersion. After the ballisticgrowth, a transition to a regime with a growth law of R2

t3 occurs. Similar observations were made for fluid turbu-lence �26� and for convective turbulence in an extended layer�15,16�. The range is however too short to conclude if this aRichardson-like regime �8� or not. The horizontal dashedlines mark the square of the cell diameter, the limit whichcannot be exceeded by pair separation. The finite size of thecell suppresses also an eventual Taylor diffusion regime withR2 t and as a result the dispersion levels off, which is dif-ferent from the Cartesian cells with periodic sidewalls�15,16�. The Kolmogorov time � and the lateral Lagrangianintegral time TL

xy as a function of the aspect ratio are listed inTable III. While the Kolmogorov time remains nearly un-changed with an increase of �, the lateral Lagrangian inte-gral time increases by almost 100% for an increase of � from1 to 5. This would be in line with a stronger correlation of

the lateral velocity components which could arise in the mul-tiroll LSC case.

Recent numerical Lagrangian studies in homogeneousisotropic turbulence by Sawford et al. �27� yielded a sensi-tive dependence of the pair dispersion on the initial pairseparation. This was confirmed in the case of convectiveturbulence in an extended layer �16�. In addition it was foundthere that the intermediate evolution of the dispersion in ther-mal convection depends on the seeding height of the tracerpairs since convective turbulence is inhomogeneous in thevertical direction. Here, we repeat these studies. In Fig. 6�c�,the pair dispersion normalized by ��t3 is plotted for threedifferent initial separations, namely, �, 2�, and 4�, with �=�3/4 / ��1/4 the Kolmogorov length. The figure shows that aplateau, i.e., R2 t3, is reached for the smallest initial sepa-ration only.

The autocorrelation functions of the lateral and verticalvelocities are shown in Fig. 6�d� for one run. The results arein agreement with �15�. The vertical velocity correlation isstrongly anticorrelated. This strong anticorrelation is in linewith the rapid upward and downward motions as indicatedby the signals of velocity components along a tracer path inFig. 4, in which fluctuations in uz�t� are stronger than thosein ux�t� and uy�t� �time signals of ux and uy are not shownhere�. In order to demonstrate this more quantitatively, weplot in Fig. 7 the vertical displacement of the Lagrangianparticles and the corresponding integrand of definition TL

�z�

�see Eq. �27�� for six tracers that are initially seeded close tobottom boundary layer. The pronounced minimum ofuz�0�uz� � coincides with the reversal point of z�t� close tothe top boundary layer.

Figure 8 shows the dependence of the long-time behaviorof the lateral dispersion on the aspect ratio. Only for thelargest aspect ratio ��=12� a diffusion limit with Rxy

2 �t� t isobtained. It underlines an important difference in contrast toconfigurations with periodic boundary conditions in the lat-eral directions. In our case, the dispersion is additionallyconstrained by the sidewalls for small-aspect-ratio systems.

V. ACCELERATION STATISTICS

Finally, we report the probability density function �PDF�of acceleration components along three-dimensional trajecto-ries. In Fig. 9 we compare data for three different Rayleighnumbers with fixed aspect ratio �=1. Both lateral and verti-cal acceleration components show no systematic trend withRa. The stretched exponential form is similar to those re-ported in laboratory experiments �3–5� and DNS simulationsof turbulent convection �15,16�.

A critical issue of numerical and experimental Lagrangianstudies—not only in turbulent convection—is the statistical

FIG. 6. Pair dispersion as in Eqs. �22�–�24�: �a� aspect ratio �=1, �b� �=3, �c� pair dispersion normalized by ��t3 for three initialseparations, �, 2� and 4�, with � the Kolmogorov length and �d�the autocorrelation functions Cr� � of the lateral �r=xy� and vertical�r=z� velocities for �=1 as given in Eqs. �25� and �26�, respec-tively. The Kolmogorov time � �see Eq. �29�� and the lateral La-grangian time TL

�L��=TLxy� �see Eq. �28�� are marked on the time axis

in �a� and �b�. The horizontal dashed lines in �a� and �b� denoteD2=�2H2. The vertical dashed line in �d� represents the character-istic time required by a particle to travel the cell of height H. Theanalysis is conducted at Ra=107 with 4.16�108 and 3.24�108

statistical samples for �=1 and �=3, respectively.

TABLE III. Kolmogorov time scale and lateral Lagrangian in-tegral times. Data are for Ra=107 and different aspect ratios.

� 1 3 5 8

� 0.211 0.215 0.213 0.206

TLxy 0.68 0.98 1.21 1.16

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convergence, which is required to determine, for example, afourth-order moment, such as the flatness F�ak�. In Lagrang-ian turbulence, the situation becomes particularly problem-atic since the tails of the PDFs are found to be stronglyscattered �28,29�. The statistical uncertainty is also mani-fested in the scattered tails of the PDFs in Fig. 9 althoughmore than 2�108 data points have been included. In Fig. 10,we ran the simulation for significantly longer period suchthat the number of samples became one order of magnitudelarger. The tails become less scattered and more extendedwith increasing number of samples. The data indicate that theconvergence in the vertical direction is slower than in thelateral direction. Nevertheless, the scatter for the fourth-ordermoments remains so strong that the flatness is obtained witha big error bar only. The flatness is defined as

F�ak� =�ak

4�L,t

�ak2�L,t

2 �30�

with k=x, y and z. We found that F�az�=89�18 for 4�109 samples.

The dependence on the aspect ratio � is demonstrated inFig. 11. It can be seen that the tails of the PDF of ax growwith increasing aspect ratio, which implies that less con-strained lateral motion causes larger acceleration magni-tudes. Meandering of the tracers among different LSC rolls�17� will probably contribute to larger amplitudes in the ac-celeration as well. Interestingly, a similar but weaker trendholds for the vertical acceleration component as well.

VI. SUMMARY AND DISCUSSION

We have presented DNS studies on the Lagrangian tracerparticle dynamics in turbulent Rayleigh-Bénard convection.The main objective of this work was to investigate how thefiniteness of the cell size in convection impacts the motion,dispersion and acceleration of the tracer particles. We devel-oped an interpolation scheme for the staggered computa-tional mesh, which was used in the Eulerian simulation. Ourinterpolation scheme reproduces accurately the global heattransfer in the Lagrangian frame NuL, which matches thecorresponding value in the Eulerian frame NuE. We detectedhowever large fluctuations of NuL�t� about the mean value

FIG. 7. �Color online� Relation between the vertical coordinateand the anticorrelation of vertical velocity. In the top panel, weshow the z�t�-component of six particle trajectories, which wereseeded close to the bottom plate. The corresponding productuz�0�uz� � along each trajectory which enters into the definition ofthe Lagrangian integral time is shown in the bottom panel.

FIG. 8. �Color online� Time evolution of the lateral pair disper-sion as a function of the aspect ratio �. Data are for Ra=107. Thelowermost curve is for �=1 and the uppermost for �=12.

FIG. 9. �Color online� The probability density functions of theacceleration components for different Rayleigh numbers at �=1.Amplitudes are normalized by the corresponding root-mean-squarevalue. �a� Component ax. �b� Component az.

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since the majority of the tracers is moving along the meancirculation path in the cell. Our studies suggest that either alarge number of particles or a very long time integrationperiod are necessary for faster convergence of NuL�t�. Theconvergence is found to be slower for a nonuniform initialtracer seeding.

In the pair dispersion analysis, we confirmed the initialballistic regime �R2 t2� and a very short-range regime withR2 t3 which have also been found in other studies on con-vection �15,16� and simulations of homogeneous isotropicturbulence in a periodic cube �26,27�. The finite size of theconvection cell suppresses the dispersion limit; hence aTaylor-like regime �R2 t� is absent for ��8 in our studiesas opposed to �15�. The autocorrelations of the vertical ve-locity component are strongly anticorrelated. This is in linewith �15� and seems to be associated with the characteristicstructures—the thermal plumes—present in convection.These structures have a strong influence on the vertical tracermotion. In particular they cause the reversals in the tracertracks, namely, when a particle hits the top or bottom plates.While we did not observe a Rayleigh-number-dependence ofthe acceleration statistics, a systematic dependence of thePDFs of the acceleration components on � was obtained. To

conclude, the basic Lagrangian properties we studied in thepresent work agree qualitatively with those in previous stud-ies, e.g., in �15,16�. The additional variation of the aspectratio is in line with changes in the large-scale circulation inthe convection cell, which affects the Lagrangian tracer dis-persion and acceleration.

Our studies complemented the only existing Lagrangianlaboratory experiment on convection by Gasteuil et al. �14�.We could confirm qualitatively some results obtained in theirexperiments. Further experimental and numerical studies inthe large aspect ratio regime are desirable and necessary.Similar to the Eulerian case, we can expect the aspect ratiodependence to disappear for sufficiently large values of �.

ACKNOWLEDGMENTS

We wish to thank Oleg Zikanov for discussions and theJülich Supercomputing Centre �Germany� for support withcomputing resources on the JUROPA cluster under grantHIL03. This work is also supported by the DeutscheForschungsgemeinschaft �DFG� under grant SCHU1410/2-1and by the Heisenberg Program of the DFG under Grant No.SCHU 1410/5-1. JS acknowledges partial travel support bythe European COST Action MP0806 “Particles in turbu-lence.”

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FIG. 10. �Color online� The convergence of the probability den-sity functions of one lateral and the vertical acceleration compo-nents for different Ra=107 and �=1. Data are for 2�108, 4�108,109, and 2�109 sample events.

FIG. 11. �Color online� The probability density functions of theacceleration components. �a� Component ax. �b� Component az. Theuppermost graph is for the largest aspect ratio in both panels.

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