Experimental measurements and modelling of rapid granular flows

Available online at www.sciencedirect.com

(2008) 182–191www.elsevier.com/locate/powtec

Powder Technology 182

Experimental measurements and modelling of rapid granular flows

R.D. Wildman ⁎, J.M. Huntley

School of Mechanical and Manufacturing Engineering, Loughborough University, Loughborough, Leics. LE11 3TU, United Kingdom

Received 1 May 2007; received in revised form 3 August 2007; accepted 8 August 2007Available online 15 August 2007

Abstract

We present a review of the development of techniques available for the measurement of granular temperature in three dimensional dry granularflows, principally discussing positron emission particle tracking and magnetic resonance imaging. We will show how these techniques have givenunique insight into granular behaviour, demonstrating phenomena such as buoyancy driven flows, non-equipartition and segregation, whilst alsoproviding us with vital benchmark data for validating kinetic theory based models of rapid granular flow.© 2007 Elsevier B.V. All rights reserved.

Keywords: Granular temperature; Rapid granular flows; Experiment; Modelling; Kinetic theory

1. Introduction

When a granular system flows, one generally observes thatthe particles undergo fluctuations in position and velocitysuperposed upon the mean motion of the flow. In order toaccurately model a flow, one has to take into account thestresses and heat transferred due to these fluctuations. One ofthe most successful approaches to tackling the problemtheoretically has been the kinetic theory method, whichconsiders the collision details of particles at the single particlelevel, and is able to predict the constitutive properties of flows[1–7]. The velocity fluctuations can be characterized in anumber of ways, usually in the form of the moments of thevelocity distribution. The 2nd moment of the distribution, or thegranular temperature, is particularly interesting since it providesa measure of the energy contained within these fluctuations.

Over the last 30 years a number of theoretical studies haveattempted to use the kinetic theory to predict granular flowproperties. These have been idealised, generally making strongassumptions about the inelasticity, shape and rotational motion.Event driven (molecular dynamics) simulations have in manycases been in reasonable agreement with the predictions ofkinetic-theory-based models for these rather restrictive cases,but until the past decade three-dimensional experimental datawas not available against which theories and simulation could

⁎ Corresponding author.E-mail address: [email protected] (R.D. Wildman).

0032-5910/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.powtec.2007.08.004

be judged. This paper reviews a series of experiments that havebeen performed by the group at Loughborough on 3D granularflows over the last 10 years, illustrating the progress made inquantifying granular temperature distributions.

2. Experiments on 3D vibro-fluidized beds using positronemission particle tracking

A key development in being able to measure granulartemperature was the availability of high speed cameras thatallowed framing rates beyond 1 kHz and established the basic ideathat – at least in two dimensions – kinetic theory was anappropriate tool for predicting granular behaviour [8–10]. To gaina more complete understanding and move towards more realisticflows, however, analysis of 3D systems was required. Positronemission tomography (PET), and its derivative, positron emissionparticle tracking (PEPT), had been commonly used in theengineering community to look at static and slow flows and ob-tain mean-flow data [11–14]. As we will discuss in the followingsections, improvements in PEPT allowed, for the first time,measurement of granular temperature in 3D flows, and subse-quently, the first observation in polydisperse granular systems.

2.1. Positron emission particle tracking

Positron emission particle tracking was originally developedfrom positron emission tomography as a method for tracking asingle, moving, tracer [15]. The subsequent emergence of

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http://dx.doi.org/10.1016/j.powtec.2007.08.004

Fig. 1. Schematic of a vibro-fluidized granular bed and its location within thePEPT facility.

183R.D. Wildman, J.M. Huntley / Powder Technology 182 (2008) 182–191

cameras that could determine locations to an accuracy of∼1 mm with a time resolution of ∼1 ms, however, meant thatfor the first time PEPT could be used to track single particlemotion between collisions in reasonably dense flows. In a quasi-steady-state system such as a vibrated bed, the technique is ableto generate pseudo-whole-field data, despite only being able tofollow a single particle, due to the long periods for which datacan be logged (up to 6 h). This whole-field coordinate data iscrucial to the calculation of fields such as the packing fractionand the granular temperature. PEPT makes use of radioactivedecay by positron emission, and relies on detecting the pairs ofback-to-back gamma rays produced when positrons annihilatewith electrons. These gamma rays are very penetrating and anaccurate location can be determined from detection of a smallnumber of these back-to-back pairs. Coincident detection of twogamma rays in a pair of position sensitive detectors defines aline passing close to the point of emission without the need forcollimation and allows the tracer location to be determined to anaccuracy of around 1 mm using an iterative algorithm thatdiscards the background of scattered data. Detailed discussionof the PEPT facility and examples of its application to granularflows can be found in the following references [16,14,17,21].

2.2. Granular temperature measurement

Granular temperature is in principle easiest to measurethrough direct calculation of the second moment of the velocitydistribution after calculating the velocity through finite dif-ferencing of the coordinate locations or by fitting an expectedfunctional form to the measured velocity distribution [18,19],where we define the granular temperature, T, as

T ¼ 13mPc2 ð1Þ

and m is the mass of a granular particle and c is the speed.Applying these methods to PEPT data presents some difficulties,however, as the collision events are difficult to determine reliably,leading to the likelihood of introducing errors through theaveraging of the velocity over collision events. A new methodwas developed during experiments on two-dimensional systemsthat allowed both numerical differentiation and collision detectionto be avoided, thereby avoiding several significant error sources.At short times, where t≪ tE, particles behave ballistically

jrðrÞ � rð0Þj2D E

¼Pc2 t2 ð2Þ

where r(t) is the position vector of the particle centre at time t andtE is the Enskog mean free time [20]. At longer times thebehaviour becomes diffusive, but it was shown that, in thetransition period when not all the particles have suffered acollision, the mean squared speed could be extracted from a plotof mean squared displacement versus time plot. This methodis dependent on being able to measure the mean squared dis-placement accurately for times less than or equal to the meancollision time. In two dimensions this procedure was shown to bereliable even at packing fractions near to the jamming transition[10], and in three dimensions, this method has been successfully

used to calculate granular temperature from PEPT coordinatelocations for vibro-fluidised beds with packing fractions up toη∼0.2 [17,21].

2.3. Experimental setup

For many of our experiments a three dimensional granulargas is generated using a Ling Dynamic Systems (LDS) vibrationsystem. A sinusoidal signal is fed through a field power supply[LDS FPS 1] and power amplifier [LDS PA 1000] into a widefrequency band electro-dynamic transducer [LDS V651]. Thissystem has a frequency range of 5–5000 Hz, a maximumacceleration of 100 g and maximum amplitude of 12.5 mm. Acell of dimensions 140 mm internal diameter and 300 mmheight is placed on the upper surface of the vibrating piston,itself placed between the PEPT photon detectors (Fig. 1). Thecell was constructed of polymethyl methacrylate (PMMA)[17,21] or in the case of the annular bed, from aluminium [22].In the case of the PMMA, the walls are coated with conductingcopper tape to reduce electrostatic charging of the grains. Thecell is usually vibrated at a frequency of 50 Hz with an am-plitude of vibration, A0, typically in the range 0.74, to 1.54 mm.Glass ballotini balls of diameter, d=5.0±0.2 mm (with an inter-grain restitution coefficient, e, measured by filming particlecollisions in a 2D vibrated bed using high speed photography, of0.91 and mass, m=1.875×10−4 kg) are normally used as thegranular medium. The coefficients of restitution for particle–wall collisions were found to be e=0.68 and e=0.79 for thePMMA and aluminium container respectively.

Fig. 2a and b show measured x and z granular temperatureprofiles as a function of height, for the case N=1050 grainswithin the PMMA container, whilst Fig. 3 shows the velocityfield and packing fraction distribution for the same system. Toimprove the statistics, the temperatures have been averagedhorizontally through the thickness of the cell. We can see threecharacteristics typical of granular flows in vibrated beds. Firstly,the granular temperature is highest near to the vibrating source

Fig. 2. a. z-component of granular temperature distribution for N=1050, andA0=0.74, 0.94, 1.14 and 1.34 mm. (From [17,21]). b. x-component of granulartemperature distribution for N=1050, and A0=0.74, 0.94, 1.14 and 1.34 mm.(From [17,21]).

184 R.D. Wildman, J.M. Huntley / Powder Technology 182 (2008) 182–191

of energy; as the distance from the base increases, the granulartemperature of the particles decreases since energy is lost duringcollisions. Secondly, inspection of the granular temperature foreach direction shows that there is a directional dependence: thegranular temperature is higher in the z-direction (the direction ofenergy input) than in the x (or y) directions.

There is, however, some evidence that in the z-direction thegranular temperature increases with height at large altitudes.This behaviour appears somewhat unphysical, since the heatflux must go to zero at high altitudes, and Fourier's heat law,

q ¼ �jdTdz

ð3Þ

where q is the heat flux and κ is the thermal conductivity,implies therefore that the granular temperature gradient mustalso go to zero. There are a number of competing theories toexplain this behaviour, some of which we will describe in thefollowing sections.

2.4. Thermally driven convection

The inter-particle dissipation that occurs during collisionsmeans that granular systems are inherently unstable. Anexample demonstrated in simulations is the observation thatdissipation leads to temperature inversion [23]. In an idealisedsystem with elastic walls and thermal bases, this can lead toa shift from a quiescent system to one with convection withcharacteristics resembling a phase transition. In practicalsystems observable in the laboratory, the walls of the containerare imperfectly elastic and this leads to localized dissipation atthe wall. This dissipation means that the isotherms are no longerhorizontal, there is increased density local to the sidewalls andbuoyancy induced convection occurs in which the particlesmove down at the wall and up along the central axis.

A number of experimental, numerical and analytical studieshave shown the existence of this phenomenon. Analytically,Khain and Meerson [24] investigated the stability of thehydrodynamic equations describing the flow. Through a linearstability analysis they found that in an ideal system with elasticwalls, the system can be described in similar way to thermalfluids and developed a criterion for convection, analogous tothe Schwarzschild criterion for classical flows. For practicalgranular flows, however, this idealised situation cannot berealised and as noted above, the system is inherently unstable sothat wall effects can always be expected to result in convectivemotion. Experimentally, the reduction in granular temperatureand increase in packing fraction was demonstrated and asubsequent event-driven Lagrangian simulation showed clearlythat in a system with parameters similar to those in theseexperiments the convection is driven by wall dissipation [17,21],rather than through a transition to Rayleigh–Bernard convection.

In a standard vibro-fluidised geometry, one cannot simplyturn on or turn off the convection. The convection flowdirection is always down at the wall and upwards in the regionclose to the axis of the cylinder, and one can only control themagnitude by varying the dissipation at the wall. By employingan annular geometry, however, with an inner wall as well as theouter wall (Fig. 4), we can gain more control over the directionas well as the amplitude of the convection currents [25]. Bothexperimentally and numerically, such a system is easy toachieve and a series of studies on a vibrated annular bed wereperformed by Wildman et al. [22]. These demonstrated that bychanging the coefficients of restitution of the particle–wallcollisions one can vary the amplitude of the convection andexercise a degree of control. Indeed, in some cases it waspossible to induce double convection rolls [22], a sign perhapsthat other mechanisms beyond those at the wall are becomingimportant. This behaviour was investigated in detail using anevent driven numerical simulation, based on the algorithmdeveloped by Talbot and Viot [25]. These simulations used ahard sphere interaction with a constant coefficient of restitutionand modelling the free motion of the particles using Newton'sLaws. This showed higher densities and lower granulartemperatures near both the inner and outer walls, and weakconvection fields were also observed. A subsequent explorationof the eiw–eow phase space, where the subscripts iw and ow

Fig. 3. Packing fraction contours maps (left) with mean velocity fields (right) for N=1050, A0=0 94 mm and a frequency of vibration of 50 Hz (From [17]).


represent respectively the inner wall and outer wall, showed thatincreasing the number of particles in the system could initiatestrong convection currents, the direction and shape of whichwere dependent on the exact combination of the wall–graincoefficients of restitution. A simple model based on therespective areas and coefficients of restitution of the twowalls was reasonably successful in predicting the direction ofsingle rolls. Double convection rolls were also observed, similarin detail to those observed experimentally (see, for example,Fig. 5). Thus, through suitable balancing of the dissipation ateach wall, the convection could be reduced significantly, but

Fig. 4. Schematic of vertically-vibrated annular cell geometry.

when the inter-particle dissipation rate is increased through theaddition of more layers of particles, the system becomesunstable and susceptible to double convection rolls. The phasemap in Fig. 6 was produced by performing numericalsimulations where we varied the coefficient of restitution forcollisions between particles and the outer and inner walls in asystematic manner. This figure shows the combinations of eiwand eow that result in a clockwise roll, an anti-clockwise roll anddouble convection rolls. There appears to be a boundarybetween clockwise and anti-clockwise behaviour based on thedissipation rate at the wall, but there is also an unstable regionwhere double convection rolls occur and convection cannot betied to dissipation at the wall. At lower numbers of particles, suchdouble-convection-roll behaviour does not occur, suggesting their

Fig. 5. Experimentally determined non-dimensional velocity vector field inannular cell showing evidence for double convection rolls, for the caseA0=1.24 mm. The rolls are observed to go down at the edges and upwards in theregion between the walls.

Fig. 6. Phase diagram showing the number and direction of convection rolls forthe case N=2644, A0=1.5 mm. Crosses indicate a single convection rollmoving in a clockwise direction, circles indicate a single convection roll movingin an anti-clockwise direction and triangles indicate a double convection roll.(From [22]).


formation is linked to interparticle dissipation rather than to wall–particle inelasticity.

2.5. Modelling monodisperse dry granular systems

The behaviour of granular systems such as vibrated granularbeds can be understood through models that incorporate theconcepts of conservation of mass, momentum and energy.Initially, experimental results were compared to simple modelswith strong assumptions such as constant granular temperatureand Boltzmann distributions of density [18]. Experiments andnumerical simulations showed that behaviour of the bed isdependent on the degree of vibration at the base [26,17,21]. Atlow amplitudes, corresponding to the point where fluidisation isabout to occur, the key parameter is the acceleration. At largeamplitudes, however, the behaviour of the bed is related to thevelocity of the base. A simple theoretical analysis suggested thatthe mean granular temperature should scale as the square of thebase velocity. However, both experimentally, and in simula-tions, although a power–law relationship was observed, theexponent was closer to 1.5 than 2. Since the relationship wasbased on a rather simple model of the energy balance, thediscrepancy in the index could be related to, for example, highdensity effects, the equation of state and the isothermal“atmosphere” [27,18,17,21]. Adapting the energy rate balanceequation to include dissipation at the wall, showed that areduced scaling exponent would be expected for grain–wallrestitution coefficients less than unity, in line with experimentalresults, and suggesting that wall effects were at least partlyresponsible for the discrepancy.

A drawback of the balance models described above was theassumption of an isothermal atmosphere of granular particles,since this inherently ignores the dissipative nature of granular

systems. A one-dimensional hydrodynamic model was devel-oped for the prediction of granular temperature and packingfraction profiles within a three-dimensional vibrofluidised bed.This model uses the conservation equations for mass,momentum and energy, combined with constitutive relationsfor three-dimensional granular systems. Two slightly differentapproaches to deriving granular constitutive relations haveappeared in the literature. The first is rooted in a balance ofenergy in shear flows [28], whilst the second is based on aconsideration of freely cooling dissipative gases [1,2,7]. Theresults of these different approaches can be seen most keenly intheir prediction of the heat flux law: the first uses a Fourier typeLaw (Eq. (3)), whereas the second includes a concentrationdependent term

q ¼ �jdTdz

� kdgdz

ð4Þ

where λ is a coefficient that is non-zero when eb1 [29,30,7]. Ina vibro-fluidised bed, the latter constitutive relation combinedwith the zero heat flux upper boundary condition results in aprediction that the granular temperature should increase withheight at sufficiently high altitudes [31]. The model andexperiment were compared by introducing the particle–particle,particle–base, and particle–wall restitution coefficients, togeth-er with the base velocity, cell geometry, and number of particles,into the model allowing direct calculations to be made withoutany adjustable parameters [32]. The results were found to agreereasonably well with profiles determined experimentally usingPEPT. At low excitation levels, the temperatures were found toagree well, while packing fractions agreed over the full range ofexcitation levels studied. Results from the version of the modelthat includes an additional term in the packing fraction gradientwere found in general to be in slightly better agreement thanthose from the one without the term, although effects due toanisotropic temperature distributions, wall friction etc. couldwell be sufficiently large to explain such discrepancies. Onemajor remaining discrepancy, however, was that the experi-ments did not show a steadily increasing temperature at highaltitudes to the extent predicted by the model. This was ex-plained by the observation that in the high altitude region thehydrodynamic description breaks down because the mean freepath is becoming comparable to the size of the container.Criteria for the onset of the Knudsen regime were proposed, andthe resulting modified temperature profiles were found to agreemore closely with the experimental distributions.

The 1D model is able to predict the key fields as a function ofheight, but clearly the presence of the wall induces additionalradial variations. A development of the 1D model has been toinclude both the axial and radial variations in a full 3D axi-symmetric calculation [33]. The energy equation for the system,along with the momentum conservation equation and theboundary conditions, were solved numerically using a commer-cially available finite element (FE) package. The FE results werethen compared to experimental results obtained using PEPT. Twomodels were again compared, and it was found that both modelsbroadly agreed with the experimental granular temperature andpacking fraction fields at low altitudes, with discrepancies, on the

Fig. 7. Comparison of experimental, numerical simulation and analytical mixture granular temperatures (a), solid fraction distributions for 5 mm diameter particles (b)and solid fraction distributions for 4 mm diameter particles (c) for the case N1=350 and N2=540. The solid line represents theory, circles are experiment and squaresshow results from numerical simulation. (For details see Fig. 3 in [28]).


whole, being much smaller than the differences due to theanisotropy in the granular temperatureswhich is not accounted forin these models.

2.6. Polydispersity

The eventual aim of research in this area must be to predictthe behaviour of non-ideal granular systems, since mostgranular systems in an industrial context do not of courseconsist of spherical particles all of identical diameter. There is,therefore, a need to consider more complex particulate systemswhere, for example, there is a distribution of sizes. A first steptowards this goal is to introduce two sizes of particles into avibrated granular bed allowing the testing of theories of granularflows containing two species of particles [28]. Recently, boththeoretical and numerical simulations have shown that oneshould expect non-equipartition of energy between the speciesin granular beds. We reproduced this phenomenon experimen-tally using a vibrated granular bed and analyzed the systemthrough the steady-state energy balance equation with aparticular focus on assessing the scaling relationships betweenthe granular temperature, peak base velocity and the fraction ofeach size phase of grains [34]. Our experiments suggested thatthe differences between the granular temperature distributions isdriven, in this case, not by differences in the restitution co-efficients as might be expected, but as a result of the different

packing fraction distributions, and hence differences in collisionand dissipation rates between species. This is in obvious contrastto thermal systems where the systemwould equilibrate to a singletemperature. One also sees that the complex boundary conditionsin granular systems play a role in determining the granular tem-perature. In a thermal system, the local temperature at theboundarymust be identical for both species, but for a granular gas,the dissipation at the wall is dependent on the collision rate, whichin turn depends on the local temperature and density of eachspecies. Thus, the differences in the dissipation rates, at the walland within the bulk of the granular gas, allow the system to relaxtowards separate granular temperatures. These results haveimportant ramifications for the prediction of behaviour inindustrial systems where distributions of particle sizes are usedand granular temperature would also be distributed amongst thegrain sizes.

The difficulty in deriving models of polydisperse flow is thatthe possible interactions between species introduces consider-able complexity into the problem, both in the derivation ofconstitutive relations and in their solution [2,35,36,28,37]. Anextension of this study was to examine the theoretical model ofa binary granular flow developed by Arnarson and Jenkins [38],implementing the appropriate boundary conditions for oursystem and solving numerically [28]. Dissipation at the wall wasincluded, andwas found to be a necessary ingredient for obtaininga close agreement between theory, numerical simulation and


experimental results. Considering the approximations made inconstructing the theoretical description, and the assumptionsof the theory regarding isotropy and equipartition (sometimesassumed but never observed), the model fits the experimental andnumerical results rather well (Fig. 7, showing the results of acomparison for a vibro fluidized bed containing two species ofparticles, driven with an amplitude A0=1.4 mm and frequency of50 Hz), suggesting that a kinetic theory approach to multi-component granular systems may well be successful.

3. Nuclear Magnetic Resonance (NMR) imaging

The PEPT technique described in the previous section relieson radioactive labeling of a single particle, with all other grainsessentially invisible to the PEPT system. As a result, relativelylong time averages are required and it is necessary to assumesteady-state behaviour over the timescale (typically of order 1 h)of the acquisitions. An alternative technique for visualizing theinternal motion of a rapid granular flow is nuclear magneticresonance (NMR). All the grains in the system contribute signaland measurement of transient behaviour becomes feasibleprovided it is repeatable.

Some of the earlier NMR studies [39–42] focused on thetime-averaged behaviour of rapid granular flows. We haverecently carried out experiments using a novel spin-echovelocity profiling technique to measure the cyclic variationsin velocity distributions, granular temperature and meanvelocity in a vertically vibrated granular bed of mustard grains.

Fig. 8. Signal S(vz,z,t) from NMR measurements of a vibrated granular bed. Each framHorizontal and vertical axes represent vz and z with units of m s−1 and mm, respect

The vertical component of the grain velocity, vz, is measuredwithin the three-dimensional cell as a function of both verticalposition (z) and vibration phase (characterized by time t afterthe trigger pulse). The details of the NMR acquisition tech-nique, including pulse sequence, developed for this study aredescribed in [43]. We chose here the vertical velocity com-ponent (vz) since the granular temperature is normally highest inthis direction, although other components could be measuredequally easily at the expense of additional acquisition time. Thetotal experimental time for a single vibration frequency wasapproximately 38 min.

The results presented in this section come from a singledataset with the following parameters: amplitude of vibrationA0=1.84 mm, frequency f=38.2 Hz, and number of grains inthe cell, Ng=110, which for a grain diameter of d=2.04±0.23 mm, and internal cell diameter of 9 mm, corresponded toapproximately two layers in the condensed phase. Fig. 8 showsthe measured signal, S(vz,z,t), as a series of frames where withineach frame the horizontal and vertical axes represent respec-tively vz and z. The height of the base is given by zb=A0cos(2πft+ϕ), where the time origin (t=0) corresponds to frame 1 inFig. 8, and the phase offset ϕ was estimated as 1.05 rad (see[43] for more details). The interframe time is 2.18 ms. Withineach frame the central ‘blob’ is the signal from the mustardgrains. Some signal from the cell wall and base is also apparent:since the entire cell is moving upwards at the same velocity thisshows up as a vertical line whose horizontal position variesfrom frame to frame.

e is an individual t value corresponding to 1/12 of the frame period of 26.2 ms.ively. The strongest signal appears red, the weakest blue. (From [43]).

Fig. 9. (a) Time-varying velocity probability density function at a fixed height of 6.81 mm, with Gaussian best fits (dashed lines). (b) Experimental nondimensionaltemperature and (c) mean velocity profiles at each of 12 stages of the vibration cycle. Plots are offset vertically with the local zeros indicated by the horizontaldotted lines, and have a tick spacing of unity along the vertical axes. The digit at the right hand end of each plot identifies the frame number defined in Fig. 8. (From[43]).


A horizontal slice through S at fixed time and heightprovides, after subtraction of the side wall signal andnormalization, the probability density function (PDF) for vz,denoted here P(vz). Fig. 9(a) shows P(vz) for each frame at afixed height of 6.81 mm, together with the best-fit Gaussian toeach, Pf (vz), where

Pf ðvzÞ ¼ 1ffiffiffiffiffiffi2p

prexp � vz � vzh ið Þ2

2r2

!; ð5Þ

The granular temperature, Tz=mb (vz−bvzN)2N, and meanvelocity ⟨vz⟩ are estimated, respectively, from the standard de-viation and centre point of the best-fit Gaussian. Despite therapid cyclical changes in these two quantities, the Gaussianfunction provides a remarkably good fit to the measured PDFs.

The time-varying granular temperature and mean velocitydistributions, calculated from the best-fit Gaussians as describedabove, are plotted in Fig. 9(b) and c. All variables have beenconverted into non-dimensional form, denoted by superscript ⁎:in the case of z by dividing by d, in the case of Tz by dividing bymgd, and in the case of ⟨vz⟩by dividing by

ffiffiffiffiffigd

p, where g is the

acceleration due to gravity.The most important general observation is the highly

transient nature of both the granular temperature (Fig. 9(b))and mean velocity profiles (Fig. 9(c)). Fig. 9c shows thepresence of a relatively short intense compressional pulse(region of negative gradient on the bvz⁎N versus z curve)travelling in the positive z direction as a result of the collisionwith the base earlier in the cycle, followed by a longer, weaker

dilatational phase. The compressional sound wave causes localheating as it passes, and hence broadening of the velocitydistribution. For example, frames 11 and 12 in Fig. 9(a) have thebroadest velocity distributions of the 12 and are therefore thehottest phase in the cycle at this particular altitude (z⁎=3.34).Fig. 9(c) shows that this is also the point in the cycle where thecompressional pulse is passing through the same location.Comparison of Fig. 9(b) and (c) shows a local heating wave(a local maximum in the granular temperature) travelling at thesame speed and in phase with the compressional sound wave inFig. 9(c).

Two important points emerged from this study [43]. Firstly,viscous damping of such waves may result in significantvolume heating of the granular gas, particularly in regions, suchas free surfaces, with strong spatial variations in particle density.This contrasts with the usual static modelling procedures[44,32,42], in which heat is assumed to diffuse into the gas fromthe base of the cell, and provides an alternative explanation forthe counter-intuitive upturn in temperature at high altitudewithout the need for the concentration-dependent term in theFourier heat law. Secondly, the ability to acquire time-resolved3-D data demonstrated the significant distortions to the velocitydistributions and the systematic errors in calculated temperaturedistributions that may arise with time-averaged measurementsof a system exhibiting transient behaviour. In the exampleshown in Fig. 8a, the width of the best-fit curve to the cycle-averaged signal implies a non-dimensional granular temperatureof 0.972, which is 23% higher than the average temperaturecalculated from the 12 frames of Fig. 9(a). This effect is more


pronounced at lower altitudes and less pronounced at higheraltitudes.

4. Conclusions

Positron emission particle tracking and magnetic resonanceimaging are powerful tools for probing the behaviour of granularflows, allowing the measurement of granular temperature, as wellas density profiles. They have been instrumental in demonstratingphenomena such as buoyancy driven flows, non-equipartition andsegregation, the internal dynamics of which has previously beenunseen. Additionally, they have demonstrated the power of usingkinetic theory methods to describe granular flows, but have alsoshown the need formore investigation into theories that are able todeal with greater complexity, such as the inclusion of anisotropyand polydispersity.

Acknowledgments

We are pleased to acknowledge continued support fromthe Engineering and Physical Sciences Research Council(EPSRC). One of us (JMH) is also grateful to the Royal Societyand Wolfson Foundation for a Royal Society-Wolfson ResearchMerit Award.

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Experimental measurements and modelling of rapid granular flows

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