particle dynamics in a vibrated submerged granular bed as revealed by diffusing wave spectroscopy

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Page 1: Particle dynamics in a vibrated submerged granular bed as revealed by diffusing wave spectroscopy

This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 142.58.129.109

This content was downloaded on 11/11/2014 at 22:40

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Particle dynamics in a vibrated submerged granular bed as revealed by diffusing wave

spectroscopy

View the table of contents for this issue, or go to the journal homepage for more

2009 J. Phys. D: Appl. Phys. 42 245404

(http://iopscience.iop.org/0022-3727/42/24/245404)

Home Search Collections Journals About Contact us My IOPscience

Page 2: Particle dynamics in a vibrated submerged granular bed as revealed by diffusing wave spectroscopy

IOP PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 42 (2009) 245404 (8pp) doi:10.1088/0022-3727/42/24/245404

Particle dynamics in a vibratedsubmerged granular bed as revealedby diffusing wave spectroscopyV Zivkovic1, M J Biggs2,3 and D H Glass1

1 Institute for Materials and Processes, The University of Edinburgh, Sanderson Building,King’s Buildings, Mayfield Road, Edinburgh EH9 3JL, UK2 School of Chemical Engineering, The University of Adelaide, Adelaide, SA, 5005, Australia

E-mail: [email protected]

Received 15 June 2009, in final form 21 October 2009Published 26 November 2009Online at stacks.iop.org/JPhysD/42/245404

AbstractThe behaviour of agitated submerged granular media is of wide relevance technologically andin nature. Recent modelling of such systems has invoked a kinetic theory description. Suchdescriptions involve the so-called granular temperature, which is directly related to themean-square of the velocity fluctuations about the mean flow velocity. The better formulationof these models and their subsequent validation demand the experimental elucidation of thedynamics and granular temperature of submerged granular media undergoing excitation. Suchelucidation, based on the non-invasive optical technique of diffusing wave spectroscopy(DWS), is reported here. The particle dynamics and granular temperature have been studiedfor a periodically forced submerged granular bed as a function of the forcing conditions.Rather unexpectedly, it was found that the granular temperature scaled with the square ofthe acceleration of the forcing rather than the square of the peak forcing velocity as in dryvibro-fluidized beds. It was also observed that the granular temperature increased withdistance above the base of the bed where the forcing was applied.

1. Introduction

The dynamic behaviour of particles submerged in a liquidis of relevance across industry and nature. Processing ofconcrete, ceramics, pulp fibre suspensions and particle settlingin water treatment and minerals processing are just a fewexamples of technological importance. Some geophysical-related examples include mudslides, submarine landslides andthe response of the seafloor and artificial earth and rock-filldams to seismic activity, in all of which velocity fluctuationscan have profound effects (Hampton and Locat 1996, Ozkan1998).

An important phenomenon affecting the flow behaviourof granular systems is the random motion of particles resultingfrom interactive collisions between them. This random motionis quantified by the so-called ‘granular temperature’, which isdirectly related to the mean-square of the velocity fluctuationsabout the mean flow velocity, 〈δv2〉. This quantity, which

3 Author to whom any correspondence should be addressed.

was first introduced by Ogawa (1978) in the earliest ofkinetic theories of granular flow, now underpins many ofthe current theories of granular media (e.g. Lun et al 1984,Ding and Gidaspow 1990, Goldhirsch 2003). Althoughthese theories are predominately for dry granular systems,granular temperature is of great relevance also for submergedgranular systems undergoing agitation or flow (Garcia-Aragon1995, Iverson 1997, Jenkins and Hanes 1998, Armanini et al2008, Berzi and Jenkins 2008). The validation of thesetheories demands the experimental elucidation of the particledynamics in general and determination of granular temperaturein particular for submerged granular systems.

Whilst the behaviour of dry granular systems subjectto vibration has received much attention (see, for example,Zivkovic et al 2008 and references therein), the same cannotbe said for their submerged counterparts, despite their widerelevance as illustrated above. What work has been done hasbeen focused on either the investigation of interactions betweena small number of spheres (Voth et al 2003, Klotsa et al 2007) orbulk dynamic phenomena such as heaping dynamics (Kozlov

0022-3727/09/245404+08$30.00 1 © 2009 IOP Publishing Ltd Printed in the UK

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J. Phys. D: Appl. Phys. 42 (2009) 245404 V Zivkovic et al

Figure 1. Schematic of the experimental setup.

et al 1998, Ivanova et al 2000, Schleier-Smith and Stone2001), travelling waves (Smith et al 2005) and Faraday tilting(Milburn et al 2005). Our study, on the other hand, presentsfor the first time, as far as we are aware, experimental granulartemperature data in agitated submerged granular media.

The structure of this paper is as follows. We will firstoutline the experimental details, including an overview ofdiffusing wave spectroscopy (DWS) and details pertaining tothe apparatus, the particulate material and the experimentalprocedures used. This is followed by a presentation of theresults obtained and their discussion.

2. Experimental apparatus and procedures

2.1. Experimental apparatus

The experimental apparatus is illustrated in figure 1. Thesubmerged granular material was held in a thin rectangularcolumn with smooth plexiglass walls, fixed so that only verticalmotion was possible. The column inner cross-section was15 × 200 mm and its height was 500 mm. The column wasfilled with semi-transparent spherical glass particles of densityρs = 2650 kg m−3 and diameter d = 475 ± 25 µm to a depthh and tap water to a depth hw. The column was shaken torelease any air bubbles and then carefully refilled with waterto a desired water level so as to ensure no further air bubbleswere trapped in the bed. The mean granular bed height wash = 95 ± 1 mm, while the water level was normally set up tobe hw = 125 ± 1 mm unless otherwise specified.

The degree of expansion of the submerged granular systemwas minute (less than 1 mm in all experiments performed)and practically impossible to measure. However, even thisinvisible dilation provided enough free volume for the particlesto flow. For example, if we assume the expansion of the systemto be 1 mm, we can estimate that the free volume is only around2% of the particles’ actual volume but is still enough to allowflow, in line with previous practical experience (Petekidis et al2002). Thus, the solid volume fraction of the system, φ,

is slightly less than the random close packing limit of 0.64.Furthermore, there was an indication of a concentration profileas a function of bed height—we shall return to this point later.

The submerged bed, including the container, wassubject to vertical vibrational forcing provided by anair-cooled electromagnetically driven shaker (V721, LDSLtd., Hertfordshire, UK). The vibrational forcing wascontrolled by a Dactron COMET USB controller (LDSLtd) with feedback from two integrated-circuit piezoelectricaccelerometers (model 353B03, PCB Piezotronics Inc., NY,US) attached to the base of the column. The accelerationand frequency were controlled to a resolution of 0.005 g and0.01 Hz, respectively. The shaker was capable of deliveringa range of different vibrational modes at frequencies in therange 5–4000 Hz, accelerations up to 70g and amplitudes aslarge as 12.7 mm.

The dynamics of the particles in the vibro-FB were studiedusing DWS in transmission mode (Maret and Wolf 1987, Weitzand Pine 1993). This method involves illuminating one sideof the bed at the point of interest with an ∼2 mm diameterlaser beam and collecting the scattered light from the oppositeside of the bed over a time, t , with a single mode optical fibre(OZ Optics Ltd., Ottawa, Canada). A 400 mW diode pumpedsolid state linearly polarized laser (Torus 532, Laser QuantumLtd., Cheshire, UK), operating at a wavelength of 532 nmin single longitudinal mode, was used. The collected lightsignal was bifurcated and fed into two matched photomultipliertubes (PMTs) to reduce spurious correlations due to possibleafter-pulsing effects of the detector. The outputs from thePMTs were amplified and fed to a multi-tau digital correlator(Flex 05, Correlator.com, US), which performed a pseudocross-correlation analysis in real time to give the intensityautocorrelation function (IACF), g2(t), that was stored on aPC for further offline analysis as detailed below. Both the laserand the fibre optic cable were mounted on linear stages so thatdifferent points above the base of the bed could be investigatedwith ease.

2

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J. Phys. D: Appl. Phys. 42 (2009) 245404 V Zivkovic et al

2.2. Experimental procedure

The submerged vibrated bed was subject to sinusoidalvibrations in all cases reported here. The vertical position ofthe bed, at time t , for such vibrational motion is expressed by

yp = A sin(ωt), (1)

where A and f = ω/2π are the amplitude and frequency,respectively. The associated peak vibrational velocity andacceleration of the bed are vp = ωA and ap = ω2A,respectively; the latter is presented here in the non-dimensionalform, � = ap/g, where g is the acceleration due to gravity.

The IACF was determined along the centreline of thebed, x = 0 mm, at four different vertical positions, y = 40,50, 60 and 70 mm, above the base of the bed. Before eachmeasurement the system was vibrated for at least 10 min,allowing the bed to reach a stationary state. The IACFs wereobtained by collecting and correlating ten blocks of data of30 s long each.

The normalized electric-field autocorrelation function(FACF), g1(t), was obtained from the IACF, g2(t), using theSiegert relationship (Berne and Pecora 1976, Weitz and Pine1993)

g2(t) ≡ 〈I (0)〉〈I (t)〉〈I 〉2

= 1 + β|g1(t)|2, (2)

where β is a phenomenological parameter determined from theintercept of the IACF; this phenomenological parameter wasalways found to be β ≈ 0.5, as expected.

The mean-square displacement (MSD) of the particles,〈�r2(t)〉, was determined by inverting the FACF using theformula (Weitz and Pine 1993)

g1(t)

=L/l∗ + 4/3

z0/l∗ + 2/3

[sin h

(z0

l∗√

X)

+2

3

√X cos h

(z0

l∗√

X)]

(1 +

4

9X

)sin h

(L

l∗√

X

)+

4

3

√X cos h

(L

l∗√

X

) ,

(3)

where X = 〈�r2(t)〉k2 + 3l∗/la, L is the sample thickness(15 mm here), l∗ is the transport mean free path, la is theabsorption path length, zo = γ l∗, the distance over whichthe incident light is randomized and k = 2π/λ the light wavevector. The scaling factor, γ , was set to unity in line withcommon practice (Weitz and Pine 1993, Xie et al 2006).

The particle velocity fluctuations about the mean flowvelocity (i.e. the Ogawa granular temperature to within aconstant) can be derived straightforwardly from the ballisticregion of the MSD (Menon and Durian 1997), provided it isresolved, using the expression

〈�r2〉 = 〈δv2〉t2. (4)

Equation (3) requires knowledge of the transport mean freepath and absorption path length at the positions and conditionsconsidered. In order to determine these parameters themethod of static transmission (Weitz and Pine 1993, Leutzand Ricka 1996) was employed. In principle, measuring the

total transmitted light, T (L), as a function of the samplethickness L determines the parameters l∗ and la by fitting anappropriate theoretical expression to the experimental data.Measuring of the total transmitted light is not trivial, so usuallya small portion of the light transmitted in the forward directionis detected, I (L), and compared with the light transmittedthrough a reference sample of the same thickness, IR(L),measured with the identical optical arrangement (Weitz andPine 1993). The first step is experimental measurement of thetransport mean free path of the reference sample, l∗R which canbe independently determined by DWS measurement using theStokes–Einstein relation to calculate the diffusion coefficientof the scatterers and henceforth the characteristic diffusiontime of the scatterers (Leutz and Ricka 1996, Zivkovic et al2008). The absorption path lengths of both the experimentaland the reference sample, la and laR, respectively, were reliablydetermined from the slope of the experimental graphs of I (L)

and IR(L) versus L, which represents the second step in thisDWS calibration procedure. This is possible as in a case ofnon-negligible light absorption, which is the case here, thefunctional form of T (L) is an exponential exp(−L/la) withdecay parameter 1/la (Leutz and Ricka 1996). Finally in thethird step, the transport mean free path of the experimentalsample l∗ can be determined using (Leutz and Ricka 1996)

la

l∗= laR

l∗R

IR(L)

I (L)+ 2βR

(IR(L)

I (L)− 1

), (5)

where the parameter βR accounts for the reflection of thediffusive light at the surface of the sample (Leutz and Ricka1996), while the reference sample data are identified with laR,l∗R and IR(L) and experimental sample data with la, l∗ and I (L).

In order to have better accuracy both quantities l∗R and laR

should be close to the unknown l∗ and la so a reference sampleof 0.1 vol% aqueous suspension of latex spheres (0.6 µmpolystyrene, G Kisker GbR., Germany), whose mean freepath and absorption path length are l∗R = 2.30 mm andlaR = 11.2 ± 0.1 mm, respectively (Zivkovic et al 2008),was used with packed beds of thickness L = 15, 20, 25and 30 mm. Our previous work on air and water fluidizedbeds (Xie et al 2006, Biggs et al 2008, Zivkovic et al 2009)suggests that the absorption path length in a granular systemis not particularly sensitive to spatial position or solids densityeven for large degrees of expansion. On this basis and takinginto account that the degree of expansion experienced in thewater-immersed vibrated bed considered here was negligible,we assumed here that the value obtained for packed beds wasalso valid for all points and conditions in the granular bed;i.e. la = 9.8 ± 0.3 mm. Further, transmitted light intensityI (L) was found to vary little with any of the parameters of theapplied vibrational agitation, indicating that l∗ is a functionof height above the bed base only. The determined transportmean free path for four different measuring points, y = 40 mm,50 mm, 60 mm and 70 mm above the base of the box, wasl∗ = 1.36 ± 0.05 mm, 1.37 ± 0.06 mm, 1.42 ± 0.07 mm and1.48 ± 0.06 mm, respectively. This indicates a non-uniformsolid fraction distribution along the bed height as our resultsfor the liquid fluidized bed (Zivkovic et al 2009) show thatthe transport mean free path is in inverse proportion to the

3

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J. Phys. D: Appl. Phys. 42 (2009) 245404 V Zivkovic et al

solid volume fraction. Therefore, local solid volume fractionis decreasing with the height, i.e. the system is denser closerto the base. A possible explanation for this is that the weightof the layers of particles above is suppressing dilation of theparticle layers below.

3. Results and discussion

Figure 2 shows an IACF, g2(t), typical of mid-rangeacceleration along with the electric-FACF, g1(t), MSD,〈�r2(t)〉, obtained from the analysis outlined in section 2.2.The example IACF, shown in figure 2(a), first decays fromg2 ≈ 1.5 over the timescale of 10−6–10−4 s to an intermediateplateau where it remains before once again decaying over thetimescale of 100–101 s, this time towards unity. The interceptsof g2 for all considered vibrational conditions were close to1.5, the expected value for depolarized light (Weitz and Pine1993). This value indicates that we are imaging one coherencearea and that enough decorrelation cycles have been taken toensure good statistics.

Figure 2(b) shows that the double decay and timescalesseen in the IACF are reflected and enhanced in the FACF, asone would expect given the Siegert relationship, equation (2).Tests (Biggs et al 2008) were undertaken to check that thedouble decay and intervening plateau observed here werenot an experimental artefact (Lemieux and Durian 1999) but,rather, a true reflection of the physics involved. Rigorousinterpretation of the long time behaviour is difficult due tonon-ergodic sampling, but previous work by us (Zivkovic et al2008) and others (Abate and Durian 2006) suggests that itis reasonable to attribute the intermediate plateau and seconddecay in the ACFs of figure 2 to caged motion of particles atintermediate times and, at longer times, particles breaking freeof their cages only to become trapped once again in new cagesnearby. A better, more quantitative, picture of this long timebehaviour would be obtained using alternative light scatteringapproaches more suited to these longer timescales such asmultispeckle DWS (Viasnoff et al 2002) and speckle visibilityspectroscopy (Dixon and Durian 2003).

The MSD obtained from inversion of equation (3) usingthe FACF is shown in figure 2(c) up to 2 ms where itis quantitatively meaningful. Quantitative analysis of theballistic region of this MSD using equation (4) gives a particlevelocity fluctuation 〈δv2〉1/2 = 0.713 mm s−1. We also fittedthe experimental data with the empirical relation (Menon andDurian 1997, Cowan et al 2000)

〈�r2〉 = 〈δv2〉t2

1 + (t/τdec)2(6)

to obtain the same value of the velocity fluctuation and adecorrelation time of τdec = 32.6 µs. Although it is tempting toreport this decorrelation time as a mean time between collisionsand the product of granular temperature and decorrelation time(�s = 23.25 nm obtained from figure 2(c)) as a mean free pathof the particles (Menon and Durian 1997, You and Pak 2001,Xie et al 2006, Zivkovic et al 2008), the following argumentshows that experimenters should be very careful in DWS datainterpretation. The characteristic length scale probed by DWS

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101

1.0

1.1

1.2

1.3

1.4

1.5

g 2

(a)

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101

0.0

0.2

0.4

0.6

0.8

1.0

g 1

(b)

10-7 10-6 10-5 10-4 10-3

10-19

10-18

10-17

10-16

10-15

t [s]

<r2 (t

)> [

m2 ]

(c)

Figure 2. (a) The IACF, g2(t), for the point x = 0 mm andy = 50 mm above the base at � = 2 and f = 100 Hz. (b) Thenormalized electric-FACF, g1(t), obtained from g2(t) using theSiegert relationship, equation (2). (c) The MSD obtained from g1(t)by inverting equation (3). The dashed line is a fit to the empiricalformula, equation (6).

can be estimated using the formula ldws ≈ λl∗/L (Weitzand Pine 1993) to be around 50 nm. This is indeed onlya small fraction (0.01%) of the mean particle diameter. Asthe solid fraction for our system is very close to the randomclose packing limit (solid volume fraction around 0.64), itis difficult to estimate inter-particle distance for comparison

4

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J. Phys. D: Appl. Phys. 42 (2009) 245404 V Zivkovic et al

10-5 10-4 10-3 10-2 10-1 100 101 102 103

0.0

0.2

0.4

0.6

0.8

1.0

1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

g 1

tf

Figure 3. The normalized electric-FACF, g1(t), for the pointx = 0 mm and y = 50 mm above the base at � = 0.9 andf = 100 Hz. The insert shows a fraction of the oscillatorydata on a linear time axis to emphasize that the period of theoscillations is the same as the forcing. Note that the timeaxes have been non-dimensionalized by the forcing frequency.

with the length scale ldws. Nevertheless, the surface roughnessof particles can be estimated to be 0.1–1% of the particlediameter (Smart and Leighton 1989), indicating that ldws isalmost certainly smaller than the mean free path of the particles.Thus, local in-cage motions of the particles at a length scaledefined by the technique, ldws, lead to complete decay ofthe correlation function (Petekidis et al 2002).Therefore, theintermediate plateau cannot be used for quantitative analysisdue to the limitations of the DWS technique. Finally, we notethat a resolution of our measurements, �s, is half the expectedone, ldws. This lost of resolution is probably due to the seconddecay.

The nonlinearity of the MSD beyond t ≈ 7 µs may bean indication of multi-particle collisions (Campbell 2006);increase in the typical inter-particle contact lifetime from shortbinary to longer multi-body histories will lead to a decrease inthe slope of MSD on a log–log plot from the value of 2 (i.e.〈�r2〉 ∼ t2) typical of the ballistic regime. An alternativepossible explanation for this discrepancy is that the ballistictrajectories of the particles are affected by the fluid flow(Benavides and van Wachem 2008)—the computed Stokesnumber of St = 2/9ρsd〈δv2〉1/2/µ ≈ 0.3 where µ is thefluid viscosity, suggests eddies may be a possible source ofthis nonlinearity. However, in our previous study in a liquidfluidized bed (Zivkovic et al 2009) with similar values ofStokes number, we did not observe this deviation from ballisticbehaviour.

Figure 3 shows an FACF typical of the lower accelerationsconsidered here. The FACF exhibits an initial decay at shorttimes followed by a finite number of decaying oscillationsor echoes (Hebraud et al 1997, Hohler et al 1997, Petekidiset al 2002) whose period is identical to that of the forcing(see insert of this figure). For a regular periodic motion ofthe scatterers, the oscillations in the FACF will not decay.However, if only a fraction of the scattering particles undergoes

0.0 0.2 0.4 0.6 0

Γ.8 1.0 1.2 1.4

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

g 1(tf =

1)

Figure 4. Variation of the height of the first echo in the FACF,g1(tf = 1), with dimensionless acceleration, �, at points x = 0 mmand y = 40 mm (squares), y = 50 mm (circles), y = 60 mm(triangles) and y = 70 mm (diamonds) for a fixed frequency off = 100 Hz.

periodic reversible motion, with the rest undergoing aperiodicmotion, the height of the echoes will decay with time (Hebraudet al 1997) as seen here. This can be used to characterize thedegree of fluidization and the jamming transition in a highlydense vibrated granular bed (Kim et al 2005, Zivkovic et al2008).

The height of the first echo, g1(tf = 1), shown in figure 4,decreases monotonically with dimensionless acceleration � orall vertical positions, y, investigated. The fraction of particlesthat are undergoing non-periodic irreversible motion, which isinversely related to the height of the echoes and indicates adegree of fluidization, increases with vibrational excitation �

as in dry granular systems (Kim et al 2005). Kim et al (2005)observed solid-like behaviour of a dry vibrated bed for � < 1,a perfectly reversible periodic motion which is characterizedby an echo height that does not change with time and it is equalto 1. However, even for very small vibrational excitations, e.g.� = 0.1 we did not detect perfect solid-like behaviour of ourgranular system. This is probably due to higher viscous forcesin our slurry system compared with the dry granular systemand a complex flow pattern of the water (Klotsa et al 2007)which may fluidize a small fraction of particles even at minutevibrational excitations.

Moreover, at a fixed acceleration, the height of the echoesdecreases with the distance above the base of the bed, at leastfor points above y = 40 mm, which indicates that the degree offluidization in effect increases with height in the bed probablyas a consequence of the observed concentration profile, asdiscussed above. These results are in line with experimentalfindings in a dry vibrated bed (Kim et al 2005) as well as withthe visual observation of Kozlov et al (1998) and Ivanova et al(2000) that only layers near the free surface are completelyfluidized. At a certain critical dimensionless acceleration �m,the height of the first echo completely disappears, indicatingthat the scattering volume becomes completely fluidized. Thecritical acceleration is, as expected, a function of the vertical

5

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J. Phys. D: Appl. Phys. 42 (2009) 245404 V Zivkovic et al

50 75 100 125 150 175 2000.0

0.5

1.0

1.5

2.0

<δν

2 >1/

2 [m

m/s

]

f [Hz]

(a)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

<δν

2 >1/

2 [m

m/s

]

Γ

(b)

Figure 5. (a) Variation of the mean velocity fluctuations about themean, 〈δv2〉1/2, with the frequency, f , at the point x = 0 mm andy = 50 mm for various accelerations: � = 0.75 (filled squares), 1(squares), 1.25 (filled circles), 1.5 (circles), 2 (filled triangles), 2.5(triangles), 3 (filled diamonds), 3.5 (diamonds) and 4 (filledsquares). Lines are a guide for the eye only. (b) The same data as in(a) averaged over all frequencies for the different accelerations andplotted versus dimensionless acceleration, � he line is a linear fit tothe data.

position as in a dry vibrated bed (Kim et al 2005) but there aretwo differences. The dependence is not linear as there is nosignificant difference between y = 40 mm and y = 50 mm.In addition, for the points near the free surface �m is smallerthan the gravitational acceleration which is considered as athreshold for fluidization in a dry system (Kim et al 2005).

Figure 5(a) shows that the fluctuation velocity, 〈δv2〉1/2, isalmost constant for a given vibrational acceleration whilst thereis no correlation with the frequency and, consequently, withthe two associated vibrational parameters of peak vibrationalvelocity, vp, and amplitude, A. The peak in data at f = 175 Hzis due to the resonant frequency of the experimental apparatusas determined by means of a transmissibility measurement(Blake 1996). The weak dependence of the fluctuation velocityon frequency for � > 3.5 seen in figure 5(a) coincides withvisual observations of heaping. This visual observation is inline with previous observations in a similar experimental setup(Schleier-Smith and Stone 2001).

0 1 2 3 40.0

0.5

1.0

1.5

2.0

<δv2 >

1/2 [

mm

/s]

Γ

Figure 6. Variation of the mean velocity fluctuations about themean, 〈δv2〉1/2, with the dimensionless acceleration � at pointsx = 0 mm and y = 40 mm (squares), y = 50 mm (circles),y = 60 mm (triangles) and y = 70 mm (diamonds) for the fixedfrequency of f = 100 Hz. The lines are a linear fit to the data.

The same frequency averaged data are plotted againstthe dimensionless acceleration � in figure 5(b). There aretwo distinct regions with different dependences of granulartemperature on acceleration. The data for � > 1.5 showlinear dependence and is described very well by 〈δv2〉1/2 =0.289� + 0.14 (coefficient of determination R2 = 0.999 68).This shows that the granular temperature scales with the squareof acceleration, in contrast to dry systems where the granulartemperature scales with the square of the peak vibrationalvelocity as theoretically predicted and experimentally verified(see Zivkovic et al (2008) and references therein). Thevariation below this acceleration is non-linear. This crossoverpoint, �c, at which the relationship ceases to be linear is closeto the critical acceleration �m at which echoes in the ACFscompletely disappear (see the data for y = 50 mm in figure 4),suggesting that it is near the jamming point (Kim et al 2005,Abate and Durian 2006).

Further experiments were performed for a fixed frequencyof f = 100 Hz as a mid-range frequency far away fromthe resonant frequency of the experimental setup. Variationof the velocity fluctuation with the acceleration at this fixedfrequency, for various vertical positions, is shown in figure 6.Firstly, the granular temperature is clearly a function ofposition, which is again in contrast to our previous DWS studyof dry granular beds (Zivkovic et al 2008). Table 1 indicatesthat the crossover point and critical acceleration both decreasewith height in a non-linear manner, whilst the slope of thefluidized region increases with height, indicating once againthat the system near the free-surface is more fluidized.

Figure 7, which shows variation of the mean velocityfluctuations about the mean with the dimensionless acceler-ation � for three different water levels hw, suggests thatthe granular temperature is a strong function of the waterlevel. The data are qualitatively the same, the crossoverpoint is identical at �c = 1.5 with non-linear dependencein the jammed region below and linear dependence in the

6

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J. Phys. D: Appl. Phys. 42 (2009) 245404 V Zivkovic et al

Table 1. The critical acceleration �m, crossover point �c andslope of the 〈δv2〉1/2–� line for the fully fluidized region, withcorresponding coefficient of determination of the fit, as a functionof vertical position y.

Vertical positiony (mm) �m �c Slope R2

40 1.3 1.5 0.281 0.998 5650 1.2 1.5 0.299 0.999 1260 0.9 1.1 0.368 0.999 2970 0.6 0.7 0.452 0.999 12

0 1 2

0.0

0.5

1.0

1.5

2.0

2.5

<δv

2 >1/

2 [m

m/s

]

Γ

Figure 7. Variation of the mean velocity fluctuations about themean, 〈δv2〉1/2, with the dimensionless acceleration � for differentwater levels hw = 120 mm (squares), hw = 150 mm (circles) andhw = 200 mm. Experiments performed for fixed frequencyf = 100 Hz at point x = 0 mm and y = 50 mm. Lines are linearfit to the data.

fully fluidized region above the crossover point. The slopesof the linear part of the data for water levels of hw = 120 mm,150 mm and 200 mm are, respectively, 0.300 (coefficient ofdetermination R2 = 0.999 12), 0.490 (R2 = 0.999 52) and0.589 (R2 = 0.998 71). Thus, the slope of the linear partincreases monotonically with water level, indicating a higherdegree of fluidization. Further work is necessary to understandthe origin of this behaviour.

4. Conclusion

Using DWS, we have determined and discussed the dynamicsof particles in a vibrated submerged dense granular bedand have determined the granular temperature as a functionof the vibrational conditions and height above the base ofthe bed. Our results revealed interesting differences andsimilarities between the particle dynamics in vibrated dryand submerged granular beds. Similar to vibrated dry beds,glassy dynamics and jamming transitions were observed inthe vibrated submerged bed here. However, the granulartemperature in the submerged bed was found to scale withthe square of the vibrational acceleration in contrast to drybeds where it scales with the square of the vibrational velocity.

The granular temperature was also observed to vary withheight above the base of the submerged bed, whilst ourprevious DWS study of dry beds showed little variation withheight. Moreover, we found strong dependence of the granulartemperature on the water level in the box. Finally, ourresults indicate that the near-surface region of the vibratedsubmerged bed is fluidized even for accelerations smaller thanthe gravitational acceleration, which is contrary to experiencewith vibrated dry beds. This suggests that the interstitial fluideffect should be taken into account in future theoretical andcomputational considerations of vibrated submerged granularmedia.

Acknowledgment

The authors are grateful to EPSRC (EP/C546849) for thesupport of this research.

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