# Surface oscillations of a liquid-solid fluidized bed

Loïc Rousseau, Laurence Girolami, Mohammed Boussafir, Frédéric Risso

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Loïc Rousseau, Laurence Girolami, Mohammed Boussafir, Frédéric Risso. Surface oscillations of a liquid-solid fluidized bed. Physical Review Fluids, 2026, 11 (1), pp.014303. 10.1103/jj92-8tlq. hal-05462600

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
Surface oscillations of a liquid-solid fluidized bed

Loïc Rousseau¹, Laurence Girolami¹,², Mohammed Boussafir¹, and Frédéric Risso³

¹ GéHCO, Université de Tours, Campus Grandmont, 37200 Tours, France

² RECOVER, INRAE-Aix-Marseille Université,
13100 Aix-en-Provence, France and

³ Institut de Mécanique des Fluides de Toulouse (IMFT),
Université de Toulouse, CNRS, 31400 Toulouse, France

(Dated: January 6, 2026)
Abstract

We report experimental investigations of an unconfined liquid-solid fluidized bed at low to moderate Reynolds number. The fluidization velocity $U_f$ is measured when an upward flow is imposed, and the sedimentation velocity $U_{sed}$ is measured when the flow is stopped. Only when the level of inlet flow fluctuation is sufficiently low will these two velocities coincide, as expected for an unbound system. A systematic comparison between $U_f$ and $U_{sed}$ should therefore be made when attempting to determine fluidization laws. Comparisons with previous studies show that $U_f$ measurements gather on a single function of the ratio $\Phi/\Phi_{pack}$ between the particle volume fraction $\Phi$ and its packing value $\Phi_{pack}$, except for a factor $K$ that encompasses all effects other than concentration: Stokes number, Reynolds number, confinement and inlet condition. Time-resolved measurements of the fluidized bed surface show two contributions: uncorrelated high-frequency fluctuations and low-frequency oscillations that remain correlated over minutes. The coherent oscillations correspond to the arrival of upward concentration waves at the bed surface. Their wavelength, which characterizes the length scale of large-scale heterogeneities, was known to diverge as the packing state is approached. Here, we found that it also increases with decreasing concentration, suggesting that the loss of global bed stability observed in previous work at high expansions may be related to the development of heterogeneous structures of a size comparable to that of the column.

# I. INTRODUCTION

Dispersed two-phase media, consisting of a population of particles immersed in a fluid, are very common in industrial and natural flows. A key mechanism of their dynamics is the resistance to the motion of the particles through the fluid, which can be seen as an increase of the drag force on the particles due to the presence of their neighbors. When the particles are uniformly distributed in space, the drag increases with the particle concentration because of the enhanced shear in the interstices between the particles. This phenomenon, known as the hindrance effect, can be isolated in two configurations that are particularly well suited to its investigation. In the case of particles that are heavier than the fluid, those are: the fluidization of the particles by a uniform ascending flow or their sedimentation through a fluid at rest. As noted by Batchelor in 1988 [1], assuming there is no significant wall effect, those two situations are symmetric. We will examine both situations here, although even
today their study is still divided in the literature. We refer to the average fluidization
velocity as $U_f$ and the average sedimentation velocity as $U_{sed}$, or we use $U$ for both when
the difference is irrelevant.

The simplest situation is that of a uniform suspension of identical solid particles of density
ρ<sub>p</sub> and equivalent diameter d, in a Newtonian fluid of density ρ<sub>f</sub> and viscosity μ<sub>f</sub>. In that
case, the dynamics can be reasonably characterized by four dimensional parameters: a
Reynolds number, Re = ρ<sub>f</sub>Ud/μ<sub>f</sub>, comparing fluid inertia to viscous forces, a Stokes number,
St = (ρ<sub>p</sub>+1/2ρ<sub>f</sub>)Ud/μ<sub>f</sub>, comparing particle inertia to viscous forces, the current particle volume
fraction Φ, and the particle volume fraction Φ<sub>pack</sub> in the loose packing state reached after
particles have settled. The latter should not be forgotten, as it determines the volume
fraction at which the sedimentation velocity becomes zero. Many attempts have been made
to find a fluidization/sedimentation law, U(Φ). Very often, experimental results are adjusted
by the original Richardson-Zaki empirical correlation [2], U = U<sub>i</sub>(1 − Φ<sup>n</sup>), which can be
improved as U = U<sub>i</sub>(1 − Φ/Φ<sub>pack</sub>)<sup>n</sup> by including Φ<sub>pack</sub>. If such correlations most of the time
provide reasonable agreement with experimental data in an intermediate range of values of
Φ, no general relationship between U<sub>i</sub> and n and the various non-dimensional groups has
been achieved.

In a previous work [3], some of the authors investigated the fluidization and the sedi-
mentation of 8 types of particles in two fluids, water at 20°C and 35°C (to vary the viscos-
ity) and air at 170°C (to cancel inter-particle cohesion due to moisture). The experiments
were conducted in unconfined condition with a ratio between the particle and the con-
tainer sizes between 600 and 3000. The particle Reynolds number was small to moderate
(6 × 10<sup>-3</sup> ≤ Re ≤ 4) while the Stokes number was small to large (1.5 × 10<sup>-2</sup> ≤ St ≤ 70).
The smallest Re and largest St were obtained with volcanic ash particles in air, thanks to
their small size (d ≤ 100 µm) and very large density ratio. The smallest St was reached with
light PMMA particles in water, and the larger Re with 335 µm heavy glass beads in water.
Velocities U<sub>f</sub> and U<sub>sed</sub> were systematically measured in a range of volume fraction between
two limits, Φ<sub>low</sub> and Φ<sub>up</sub>, that depend on the system under consideration. At Φ = Φ<sub>pack</sub>, the
particles cease settling (U<sub>sed</sub> = 0), while the liquid can still go through the bed (U<sub>f</sub> ≠ 0).
Φ<sub>up</sub> was defined as the volume fraction from below which U<sub>sed</sub> = U<sub>f</sub>, its value ranging from
0.80 to 0.95 Φ<sub>pack</sub>. In between Φ<sub>low</sub> and Φ<sub>up</sub>, the surface of the bed is easy to locate and ap-
proximately horizontal, while below Φ<sub>low</sub> the surface becomes diffuse and difficult to define.
$\Phi_{low}$ was thus associated with the limit of stability of a homogeneous suspension. The terms “stable” and “homogeneous” have to be taken with care. In that case, the homogeneity was not only justified by the fact that the interface was approximately flat, but also by the fact that $U_{sed} = U_f$, which implied that the lateral walls did not affect the flow motion. The expression “limit of stability” was justified by the fact that a sudden transition towards a visually inhomogeneous state was observed at $\Phi_{low}$. It is worth mentioning that the value of $\Phi_{low}$ was found to increase with the Stokes number, which characterizes the importance of particle inertia. The larger the Stokes number, the narrower the concentration range [$\Phi_{low}, \Phi_{pack}$] where the homogeneous regime was observed. In this regime, the results of all the systems considered were well described by a single correlation

$$U = \frac{U_0(1 - \Phi)}{F(\frac{\Phi}{\Phi_{pack}})K(St_0)}, \quad (1)$$

where $U_0 = \frac{(\rho_p - \rho_f)gd^2}{18\mu_f}$ is the Stokes velocity for an isolated particle and $St_0 = \frac{(\rho_p + 1/2\rho_f)(\rho_p - \rho_f)gd^3}{18\mu_f^2}$ is the Stokes number based on $U_0$. $F$ is a function that increases from 1 to $\infty$ as of $\Phi/\Phi_{pack}$ increases from 0 to 1. It is exponential at low $\Phi$ and diverges as a power law similar to the Richardson-Zaki law when approaching the packing state. It describes the hindering effect for a system at negligible particle inertia. $K$ is a function that increases from 1 to 3 as $St_0$ increases from 0 to $\infty$. It describes the fact that the hindering effect is enhanced by the inertia of the particle. Most of the experimental results having been obtained at small Reynolds number, the effect of $Re$ is not accounted for in eq. (1), which should be understood as the low $Re$ limit of that sedimentation/fluidization law.

Experiments by [4] have shown that particle inertia also reduces the threshold for the onset of concentration waves, which are considered to be the first step towards complete bed destabilization [5]. The significant effect of particle inertia on both $U$ and the suspension homogeneity means that fluctuations take an important role upon the suspension dynamics, even in the limit of small $Re$. It is thus highly desirable to characterize such fluctuations and to understand how are related average velocity $U$ and concentration fluctuations, and global stability of the homogeneous state. Unfortunately, the presence of numerous solid/fluid interfaces makes the suspension opaque and complicates measurements within it. Some optical techniques can be used, depending on the ratio $w/d$ between the container thickness $w$ and the particle size $d$. Measuring the intensity of a laser beam passing through the suspension, Ham et al. (1990) [4] detected concentration waves for $10 < w/d < 50$. Filming
neon light transmitted through the suspension, Duru et al. (2002) [6] made maps of the concentration average over the gap for $w/d \approx 17$, which enabled the characterization of non-planar waves. Using optical index matching between transparent beads and the suspending fluid, Alméras et al (2021) [7] measured the velocity fluctuations of both the fluid and the particles for $w/d \approx 13$. However, there is a lack of results regarding concentration and velocity fluctuations in unconfined situations. That means that particles are much smaller than the container, let's say $w/d \ge 100$, but the confinement ratio would be better defined from the scale $\Lambda$ of the larger inhomogeneities, as $\Lambda/w \gg 1$.

The present study is an experimental investigation of fluidized/sedimented suspensions, considering the same particles and fluid as Ref. [3], but a different injection system. We have chosen to focus on unconfined systems ($w/d \ge 500$) at the cost of limiting our investigation to measurements of the bed surface motions, which are analyzed using statistical and spectral tools to provide insights into the mean velocity, magnitude and time scales of the fluctuations. The results are compared to previous studies of confined suspensions for which average sedimentation velocity and concentration-wave properties have been measured [4, 8–10]. The paper is organized as follows: experimental setup and procedures are described in section II, sedimentation and fluidization laws are examined in section III, spatio-temporal fluctuations of the suspension surface are investigated in section IV, the spatially-averaged time oscillations of the fluidized bed height are analyzed and discussed with respect to their connection with concentration waves in section V, concluding remarks are given in section VI.

## II. EXPERIMENTAL SETUP AND MEASUREMENTS TECHNIQUES

Three types of particles are investigated (see Table I): two sets of glass beads of diameter 160 µm (GB1) or 335 µm (GB3) and one set of PMMA beads of 210 µm. They are immersed in tap water at ambient temperature, ranging between 18°C and 25°C. The three sets have been chosen to show contrasting inertial effects. With PMMA particles, both the Reynolds and Stokes numbers are low. With GB1, *Re* remains below 1.1 while *St* significantly exceeds unity. With GB3, *Re* is greater than 1 and *St* greater than 2.8.

The experimental setup is depicted in Fig. 1a. It consists of a vertical reservoir made of PMMA of 900 mm height with a rectangular cross section of dimension $w_1=200$ mm in
<table><caption>TABLE I. Particles and fluid physical parameters, as well as ranges of operating conditions. Note that the viscosity of the water varies between the different tests due to slight variations in temperature.</caption><thead><tr><th>Parameters</th><th>GB1</th><th>GB3</th><th>PMMA</th></tr></thead><tbody><tr><td>d [μm]</td><td>160</td><td>335</td><td>210</td></tr><tr><td>ρ<sub>p</sub> [kg m<sup>-3</sup>]</td><td>2500</td><td>2500</td><td>1200</td></tr><tr><td>ρ<sub>f</sub> [kg m<sup>-3</sup>]</td><td>998</td><td>998</td><td>998</td></tr><tr><td>μ<sub>f</sub> [10<sup>-3</sup> Pa.s]</td><td>[0.833, 1.109]</td><td>[0.967, 1.081]</td><td>[1.015, 1.095]</td></tr><tr><td>h<sub>0</sub> [m]</td><td>0.18</td><td>[0.35, 0.48]</td><td>0.09</td></tr><tr><td>h<sub>mid</sub> [m]</td><td>[0.21, 0.44]</td><td>0.54</td><td>[0.12, 0.21]</td></tr><tr><td>φ<sub>pack</sub></td><td>0.59</td><td>0.59</td><td>0.59</td></tr><tr><td>φ<sub>s</sub>/φ<sub>pack</sub> = 1/E</td><td>[0.41, 0.86]</td><td>[0.65, 0.89]</td><td>[0.43, 0.73]</td></tr><tr><td>St<sub>0</sub> = (ρ<sub>p</sub>+1/2ρ<sub>f</sub>)(ρ<sub>p</sub>-ρ<sub>f</sub>)gd<sup>3</sup> / (18μ<sub>f</sub><sup>2</sup>)</td><td>[9.5, 14.5]</td><td>[78, 94]</td><td>[1.4, 1.7]</td></tr><tr><td>U<sub>f</sub> [mm s<sup>-1</sup>]</td><td>[1.1, 6.0]</td><td>[3.1, 8.9]</td><td>[0.37, 1.04]</td></tr><tr><td>Re = ρ<sub>f</sub>U<sub>f</sub>d / μ<sub>f</sub></td><td>[0.19, 1.1]</td><td>[0.95, 2.8]</td><td>[0.05, 0.22]</td></tr><tr><td>St = (ρ<sub>p</sub>+1/2ρ<sub>f</sub>)U<sub>f</sub>d / μ<sub>f</sub></td><td>[0.5, 3.4]</td><td>[2.9, 8.3]</td><td>[0.11, 0.37]</td></tr></tbody></table>

the $x$ direction, and $w_2=300$ mm in the $y$ direction. (Note that the column consists of two connected segments. The connection is smooth and has no effect on the flow within the column. However, it prevents measurements being taken in the range of elevations from 250 mm to 280 mm.)

In fluidization mode, water is supplied by a pump and injected from below through a porous medium, whose role is to homogenize the flow. The four-blade centrifugal pump rotates at speeds between 2,500 and 3,000 rpm, powered by a 50 Hz electrical current. The cyclic nature of the pump may generate low-amplitude fluctuations at frequencies greater than 40 Hz. The porous medium is made of three layers topped by a mesh filter that held them in place: one layer of gravels of roughly 4 mm diameter, one layer of 4 mm iron beads mixed up with 0.336 mm glass beads, covered with a 10-mm thick sponge.

A mass $m_p$ of particles is introduced into the water-filled column. The pump is started and the flow rate $Q_{flow}$ is adjusted. $Q_{flow}$ is measured by means of an electromagnetic flow meter, from which is determined the sedimentation velocity $U_f = Q_{flow}/(w_1w_2)$, with an
<figure><img src="image_2.png" /><figcaption>FIG. 1. Experimental setup. (a) General view, (b) Packed state after sedimentation, (c) steady state during fluidization, (d) Sedimentation process.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>FIG. 2. Snapshots of the bed surface at various concentrations during fluidization.</figcaption></figure>

accuracy of ±2%. After a short transient, the bed of particles reaches steady state (Fig. 1b) and its height $h_{mid}$ is measured by means of a ruler located in the middle of the shorter face of the the column, at $x = x_{mid} = w_1/2$. Then, the water supply is stopped and the particles settle (Fig. 1c) until they form a steady deposit of height $h_0$ and solid volume concentration $\Phi_{pack} = m_p/(\rho_p h_0 w_1 w_2)$, which corresponds to a loose random packing associated with the
<figure><img src="image_4.png" /></figure>

FIG. 3. Time evolution of the bed height $h_{mid}(t)$ at the middle of the column during bed collapse experiments. (The lack of data for the case at $\Phi/\Phi_{pack} = 0.607$ is due to the passage of the interface at a junction of the column.)

<figure><img src="image_5.png" /></figure>

FIG. 4. Bed height time fluctuations $h'(x, t)$ at various distance $x$ from the wall during fluidization experiments.

microstructure of the fluidized bed. The bed expansion $E = h_{mid}/h_0$ of the fluidized bed is measured with an accuracy of ±2% and its solid concentration $\Phi$ is determined through the relation $\Phi/\Phi_{pack} = 1/E$.

The surface of the bed is illuminated by a LED panel located above and inclined at 45 degrees to the vertical, and filmed by means of a Phantom VE-O 340L camera located on the same side of the column. The camera has an horizontal viewing axis and focuses on the
bed surface near to the wall. It images a rectangular window of the (x,z) plane from the left wall ($x_{wall} = 0$) to a point located 20 mm from the column middle ($x_{max} = 120$ mm), with a resolution of 16 pixels per millimeter. Examples of images are provided in Fig. 2. During bed collapse, the camera is operated at 1200 frames per second and the time evolution of the bed height at the middle of the column, $h_{mid}(t)$, is determined. Fig. 3 shows the result for GB1 particles at various concentrations. We observe small oscillations around a linear decay. The fact that the average velocity does not change confirms that the average solid concentration remains unchanged throughout the process. The presence of regular oscillations is reminiscent of the traveling waves observed in the previous works mentioned in the introduction. During fluidization, the acquisition rate is set to 150 frames per second and the spatio-temporal profile of the surface, $h(x,t)$, between $x_{wall}$ and $x_{max}$ is extracted by image processing. In what follows, time averaging is denoted by an overbar, $\bar{h}(x,t)$, and space averaging by brackets, $\langle \cdot \rangle$.

Fig. 4 presents the time fluctuations, $h'(x,t) = h(x,t) - \overline{h(x,t)}$, for GB1 particles at various distances $x$ from the wall for two different solid fractions. The signal shows variation of amplitude less than 2 mm and involves both a regular oscillation and random fluctuations. Its characteristics do not change with time, which means that it is statistically steady, but depends on $\Phi$. In what follows, the analysis of surface fluctuations will be carried out on fluidization experiments, whose signals are better suited to processing because they are not limited to a short duration, unlike collapse tests.

### III. SEDIMENTATION AND FLUIDIZATION VELOCITIES

In this section, we examine the average fluidization and sedimentation velocities, before considering interface fluctuations in the next sections. Fig. 5 presents the measured values of $U_f$ and $U_{sed}$ as a function of the normalized concentration $\Phi/\Phi_{pack}$, for the three sets of particles. Our results (circles) are compared with those of Ref. [3] (squares) for the same particles. As expected, $U_f$ and $U_{sed}$ are decreasing functions of $\Phi/\Phi_{pack}$ and are lower the lighter or smaller the particles.

Comparing $U_f$ to $U_{sed}$, Ref. [3] found that sedimentation and fluidization velocities remained equal over the concentration range studied. Our results show different features depending on the type of particles considered. For PMMA particles, $U_f$ and $U_{sed}$ remain
equal over the concentration range studied. For GB1, they are still close at high concentration but $U_f$ becomes greater than $U_{sed}$ for $\Phi/\Phi_{pack} \le 0.6$. For GB3, $U_f$ is always greater than $U_{sed}$ and the discrepancy increases when decreasing the concentration. Compared to Ref. [3], the present particles and column are the same, but the porous medium through which water is injected is significantly less compact. In our case, it is therefore likely that disturbances in the upstream flow are stronger and less filtered by the porous medium. In Ref. [3], incoming fluctuations are negligible compared to the fluctuations that are naturally present in a suspension due to the interactions between particles. The result is that the fluidization process, where the pump is running, is not different from the sedimentation process, where the pump is switched off. In the present work, additional disturbances during fluidization cause an increase of $U_f$. The effect of the injection condition that has been revealed here highlights the difficulty of quantitative comparisons between fluidization velocities measured in different setups of the literature.

We now compare fluidisation velocities obtained by different authors for different types of particles, which requires making $U_f$ dimensionless. A way to do that is to define the relative sedimentation viscosity of the suspension as

$$ \frac{\mu_{sed}}{\mu_f} = \frac{U_0(1 - \Phi)}{U_f}, \qquad (2) $$

where $\mu_{sed}$ is the viscosity of an equivalent fluid having the same density, $\rho_m = (1 - \Phi)\rho_f + \Phi\rho_p$, of the suspension, in which a single particle settles at a velocity $U_f$ in the limit of vanishing Reynolds number [11]. Using Eq. (2), Eq. (1) can be recast as follows

$$ \frac{\mu_{sed}}{\mu_f} = K F(\frac{\Phi}{\Phi_{pack}}), \qquad (3) $$

where $K$ is a coefficient that takes into account the effect of particle inertia, which can be expressed as a function of only $St_0$ when the Reynolds number is small, the suspension is unconfined and homogeneous, and the inlet condition does not affect the dynamics ($U_{sed} = U_f$). In Fig 6 are plotted the values of $F = \mu_{sed}/(K\mu_f)$ as the function of $\Phi/\Phi_{pack}$ for the present work as well as for experiments of Ref. [3, 4, 8–10], whose physical properties are reported in Table II. A value of $K$ has been adjusted for each series so as to collapse the data on the following empirical curve [12],

$$ F_{fit}(\frac{\Phi}{\Phi_{pack}}) = \exp(1.9\frac{\Phi}{\Phi_{pack}}) + 0.85(\frac{\Phi}{\Phi_{pack}})^2(1 - \frac{\Phi}{\Phi_{pack}})^{-2/3}. \qquad (4) $$
<figure><img src="image_6.png" /><figcaption>FIG. 5. Fluidization and sedimentation velocity measured for all sets of particles as a function of the particle concentration.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>FIG. 6. Experimental values of $F = \mu_{sed}/(K\mu_f)$ determined using values of $K$ given in Fig. 7. The curve corresponds to empirical relation (4).</figcaption></figure>

The corresponding values of $K$ are plotted in Fig. 7 as a function of $St_0$. Expression (4) exhibits a classic power-law divergence, $(1 - \frac{\Phi}{\Phi_{pack}})^{-2/3}$, when approaching $\Phi_{pack}$. However, this asymptotic behavior is not relevant at low concentrations, where an exponential increases, $\exp(1.9\frac{\Phi}{\Phi_{pack}})$, is observed. Adding a quadratic factor, $(\frac{\Phi}{\Phi_{pack}})^2$, to the high-concentration term ensures a smooth transition between these two behaviors.
<figure><img src="image_8.png" /><figcaption>FIG. 7. Values of *K* used to fit Eq. (4) on experimental data plotted in Fig. 6. Dark blue squares correspond volcanic ash or FCC particles fluidized by a hot dry air from [11].</figcaption></figure>

First of all, Fig. 6 shows that most of the data can be collapsed on the master curve (4) found by [3] despite the variety of physical parameters covered by the different studies. As already discussed, the slight discrepancies with our data are due to fluctuations in the incoming flow. Regarding other previous studies, effects of the confinement and Reynolds number are expected according to the values of *w/d* and *Re* reported in Table II. However, only data of Ref. [8] for particles larger than 860 µm and at large concentration are incompatible with the master curve, since they do not follow the expected divergence as *Φ* tends toward *Φ*<sub>pack</sub>. This suggests that the separation, noted by [3], between the effect of the concentration, accounted for by a unique function *F*(*Φ*/*Φ*<sub>pack</sub>), and other parameters, accounted by coefficient *K*, may have a more general validity.

Unlike *F*, the values of *K* shown in Fig. 7 are more dispersed. If the reference results for liquid-solid [3] (light blue squares) and gas-solid [11] (dark blue squares) fluidized beds can be described by a function of only *St*₀, the results of other studies do not collapse on the same curve. The larger *K*, the smaller *U*<sub>f</sub> and the greater the drag force on the particles and the sedimentation viscosity. With the present results (green circles), *K* is also an increasing function of *St*₀, but its values are much less. This enhanced settling may be attributed to the formation of clusters that decreases the drag as it is observed in bubble swarms [13]. It is difficult to draw conclusions for the cases measured under confined conditions,
which may also imply a non-small Reynolds number. The results of Didwania & Homsy [10] (upside-down triangles), performed with beads of 290 to 1100 µm at Re between 1.7 to 170 in a two-dimensional column with a gap of 31.5 mm, also show K smaller than those of Ref. [3, 11] and coincide with the present GB3 at $St_0 = 92$. The results of Anderson & Jackson [8] (triangles), with beads of 640 to 2070 µm in a cylindrical column of 7 mm or 38.1 mm diameter at Re between 5 and 84, show large values of K, from 3.4 to 5. The results of Homsy [9] (crosses), with beads of 590 to 1560 µm in a pipe of 24.5 mm at Re between 6 and 80, show values of K around 3 for $St_0$ less than $10^4$ and of 5.5 beyond. Finally, the results Ham et al. [4] (diamonds), involving beads of 325 µm at Re less than 2 in a cylindrical column of 12.7 mm, are rather close to the results of Ref. [3]. Confinement is expected to increase K since it is known that it causes an increase in drag. All results above the blues squares ([3, 11]) are influenced by confinement. On the other hand, results below the blues squares involve a dominant effect of Reynolds number (inverse triangles from [10]) or inlet condition (circles from this study), which is thereby observed to cause a reduction in drag.

At this stage, we conclude that it is only when $U_f$ is equal to $U_{sed}$ and in the absence of confinement ($w/d \gg 1$) that $U_f$ depends only on the properties of the fluid, those of the particles, concentration and gravity. However, in most of the cases, the main differences turn out to be embedded in the values of K while the evolution of $\mu_{sed}/(K\mu_f)$ can be described by a unique function $F(\frac{\Phi}{\Phi_{pack}})$.
<table><caption>TABLE II. Physical parameters and ranges of operating conditions of previous experimental investigations that are compared with the present work. * Values of $\Phi_{pack}$ are not available in Ref [8] and have been set to 0.6.</caption><thead><tr><th></th><th>d(10<sup>-6</sup>m)</th><th>w/d</th><th>φ<sub>pack</sub></th><th>φ* = φ<sub>s</sub>/φ<sub>pack</sub></th><th>f(Hz)</th><th>U<sub>f</sub>(m.s<sup>-1</sup>)</th><th>Δρ(kg.m<sup>-3</sup>)</th><th>St<sub>0</sub></th><th>Re = ρ<sub>f</sub>U<sub>f</sub>d/μ<sub>f</sub></th></tr></thead><tbody><tr><td colspan="10"><strong>Didwania & Homsy (1981) [10]</strong></td></tr><tr><td>Set c</td><td>290</td><td>29</td><td>0.64</td><td>0.903</td><td>1.6</td><td>0.002</td><td>1903</td><td>92</td><td>1.73</td></tr><tr><td>Set a</td><td>589</td><td>109</td><td>0.63</td><td>0.872</td><td>2.3</td><td>0.003</td><td>2990</td><td>1502</td><td>2.60</td></tr><tr><td>Set b</td><td>1100</td><td>53</td><td>0.62</td><td>0.922</td><td>1.5</td><td>0.0011</td><td>2990</td><td>9738</td><td>25.9</td></tr><tr><td>Set a</td><td>589</td><td>109</td><td>0.63</td><td>0.898</td><td>2.2</td><td>0.016</td><td>2990</td><td>1502</td><td>37.7</td></tr><tr><td>Set b</td><td>1100</td><td>53</td><td>0.62</td><td>0.940</td><td>1.6</td><td>0.026</td><td>2990</td><td>9738</td><td>114</td></tr><tr><td>Set a</td><td>589</td><td>109</td><td>0.63</td><td>0.907</td><td>2.3</td><td>0.038</td><td>2990</td><td>9738</td><td>169</td></tr><tr><td colspan="10"><strong>Anderson & Jackson (1969) [8]</strong></td></tr><tr><td>No.7 ballotini</td><td>640</td><td>19.84</td><td>0.60*</td><td>[0.783, 0.841]</td><td>1.25</td><td>[0.012, 0.008]</td><td>1950</td><td>961</td><td>[5.50, 7.68]</td></tr><tr><td>No.5 ballotini</td><td>860</td><td>14.76, 29.53</td><td>0.60*</td><td>[0.765, 0.885]</td><td>1.1, 1.5</td><td>[0.017, 0.013]</td><td>1860</td><td>2166</td><td>[11.5, 15.2]</td></tr><tr><td>No.3 ballotini</td><td>1270</td><td>10, 20</td><td>0.60*</td><td>[0.786, 0.894]</td><td>0.75</td><td>[0.029, 0.024]</td><td>1860</td><td>6976</td><td>[31.2, 36.4]</td></tr><tr><td>2mm ballotini</td><td>2070</td><td>12.27, 18.40</td><td>0.60*</td><td>[0.905, 0.970]</td><td>1.1</td><td>[0.042, 0.037]</td><td>1700</td><td>26297</td><td>[77.0, 83.6]</td></tr><tr><td colspan="10"><strong>El-Kaissy & Homsy (1976) [9]</strong></td></tr><tr><td>Set A</td><td>590</td><td>38</td><td>0.65</td><td>[0.881, 0.921]</td><td>[0.6, 1.4]</td><td>[0.0112, 0.0095]</td><td>2991</td><td>1892</td><td>[6.27, 7.40]</td></tr><tr><td>Set B</td><td>830</td><td>27</td><td>0.63</td><td>[0.883, 0.932]</td><td>[0.5, 1.6]</td><td>[0.0186, 0.0149]</td><td>2991</td><td>5268</td><td>[13.8, 17.3]</td></tr><tr><td>Set C</td><td>1100</td><td>20</td><td>0.61</td><td>[0.890, 0.909]</td><td>[0.8, 0.9]</td><td>[0.0276, 0.0261]</td><td>2991</td><td>12264</td><td>[32.1, 34.0]</td></tr><tr><td>Set D</td><td>1560</td><td>15</td><td>0.61</td><td>[0.858, 0.910]</td><td>[0.7, 1.5]</td><td>[0.0447, 0.0373]</td><td>2991</td><td>34982</td><td>[65.1, 78.0]</td></tr><tr><td colspan="10"><strong>Ham et al. (1990) [4]</strong></td></tr><tr><td>Set C</td><td>325</td><td>24</td><td>0.59</td><td>[0.708, 0.976]</td><td>[0.05, 0.29]</td><td>[0.0011, 0.0045]</td><td>1490</td><td>75</td><td>[0.358, 1.46]</td></tr><tr><td>Set F</td><td>325</td><td>24</td><td>0.60</td><td>[0.830, 0.957]</td><td>[0.10, 1.50]</td><td>[0.0025, 0.0050]</td><td>3140</td><td>245</td><td>[0.812, 1.63]</td></tr><tr><td colspan="10"><strong>Present study</strong></td></tr><tr><td>GB1</td><td>160</td><td>1250</td><td>0.59</td><td>[0.413, 0.854]</td><td>[0.1, 0.4]</td><td>[0.0011, 0.0060]</td><td>1502</td><td>12</td><td>[0.19, 1.1]</td></tr></tbody></table>

## IV. TIME OSCILLATIONS AND SPATIAL DEFORMATION OF THE SURFACE

In this section, we examine the oscillations of the bed surface, $h'(x, t)$. We are interested to check whether there is a relation between the average sedimentation velocity and the characteristics of these fluctuations. For that reason, we focus on GB1 particles since a transition is observed in the difference between $U_f$ and $U_{sed}$ around $\Phi/\Phi_{pack} = 0.6$ in Fig. 5.

Fig. 8 presents the variance of the bed fluctuations computed by averaging over time at location $x$, $\sigma^2(x) = \overline{h'}^2(x)$. The cases are ordered from left to right and from top to bottom in decreasing order of $\Phi/\Phi_{pack}$, which means in increasing order of bed expansion $E$ and sedimentation velocity $U_f$. Location $x = 0$ corresponds to a wall, while $x = 100$ mm is the middle of the column (marked by a thin dash line). At large concentrations ($\Phi/\Phi_{pack} \gtrsim 0.6$), the variance $\sigma^2$ is almost flat, except for a small increase when approaching the wall, which is localized below $x_{low} = 40$ mm (marked with a thick dashed line) and is more pronounced at smaller expansions. At $\Phi/\Phi_{pack} = 0.541$, $\sigma^2$ starts to increase near the wall, and for $\Phi/\Phi_{pack} \le 0.449 \sigma^2$ is no more constant in the central region ($x \ge x_{low}$), which means
<figure><img src="image_9.png" /><figcaption>FIG. 8. Variance of the bed fluctuations, $\sigma^2(x) = \langle h'^2(x) \rangle$, as a function of location $x$ at various concentrations $\Phi/\Phi_{pack}$.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>FIG. 9. Standard deviation of the bed height normalized by the particle diameter, computed by time and space averaging in the central region of the column.</figcaption></figure>

that the surface is nowhere flat. Another important observation is that neither average nor the fluctuating sedimentation velocity disappears at the wall, whereas the velocity of the suspending fluid does. This proves the existence of a significant slip velocity of the particles at the walls, which seems to be a common feature of concentrated suspensions either at negligible [14] or large [15] Reynolds number.

Fig. 9 shows the standard deviation of the bed height $\sigma$ normalized by the particle
<figure><img src="image_11.png" /><figcaption>FIG. 10. Time cross-correlation of the surface height between two points, $x_i$ and $x_j$.</figcaption></figure>

diameter $d$, computed by averaging over both time and space in the central region ($x_{low} \le x \le x_{max}$). We again see a transition around $\Phi/\Phi_{pack} \approx 0.6$, where the oscillation amplitude is minimum ($\sigma = 0.74d$). Decreasing the concentration, $\sigma$ steeply increases to reach a value of $4.19d$ at $\Phi/\Phi_{pack} = 0.41$. Increasing the concentration above this point, $\sigma$ increases moderately to reach a value of $1.82d$ at $\Phi/\Phi_{pack} = 0.854$.

We now want to know whether the surface fluctuations at different locations are synchronized with each other or not, an information that is not contained in the single-point measurements. In particular, the fact that a variance profile represented in figure 8 is flat, as is the case at intermediate concentrations, does not guarantee that the surface is flat at a given instant. Cross-correlation is a good tool to gain insight into this matter. We define the time correlation between two points $x_i$ and $x_j$ of the surface,

$$C_{x_i, x_j}(\tau) = \frac{\overline{h'(x_i, t)h'(x_j, t + \tau)}}{\sqrt{\overline{h'^2(x_i)}} \sqrt{\overline{h'^2(x_j)}}} \quad (5)$$

Fig. 10 presents $C_{x_i, x_j}(\tau)$ for pairs of points located in the central region ($x_{low} \le x \le x_{max}$). The point $x_i$ is fixed to a location near the end $x_L$ of the measurement window, while $x_j$ is moved to several locations, so that the distance between them is varied from zero to approximately 60 mm. All curves show a dominant frequency, indicating the existence of a deterministic mechanism that will be analyzed in the next section. For now, we note that when the oscillations displayed by the cross-correlations obtained at two different $x_j$ for a given concentration are in phase, we can conclude that the motions of these two points are
<figure><img src="image_12.png" /><figcaption>FIG. 11. Time evolution of the spatially-averaged bed height for the same cases as in Fig. 4.</figcaption></figure>

synchronized. Such a synchronization is observed for $\Phi/\Phi_{pack} \gtrsim 0.6$ up to time delays larger than 20 s. The synchronization degrades at smaller concentrations. At $\Phi/\Phi_{pack} \le 0.449$, even the first oscillations can be in phase opposition, indicating a severe de-synchronization of the fluctuations. Visual inspection of the movies confirms these conclusions. At low expansion, the surface is observed to oscillate in the vertical direction as a whole. At large expansion, sloshing motion develops, breaking the synchronization between adjacent points.

V. VERTICAL SURFACE OSCILLATIONS AND TRAVELING WAVES

From now, we focus on time fluctuations and ignore any spatial heterogeneity by consid-
ering the bed height, $\langle h \rangle(t)$, which is spatially averaged over the central part of the column
($x_{min} \le x \le x_{max}$). Then, we define its time fluctuations, $\langle h' \rangle(t) = \langle h \rangle(t) - \langle \overline{h} \rangle(t)$, by
removing their time average $\langle \overline{h} \rangle(t)$.

Fig. 11 presents two examples of time evolution $\langle h' \rangle(t)$. Compared to the local signals shown in Fig. 4, the spatial average of the height contains less high frequencies and thus appears less noisy. However, it has a similar low frequency content except that the effect of sloshing occurring at small concentration is attenuated, making it suitable for analyzing the main features of the vertical oscillations. The presence of a dominant frequency in $\langle h' \rangle(t)$ is visible at all concentrations. It can be better distinguished from the stochastic part of the signal by considering the autocorrelation function, $C(\tau) = \frac{\langle h' \rangle(t) \times \langle h' \rangle(t+\tau)}{\langle h' \rangle^2(t)}$, which is presented in Fig. 12. For all volume fractions considered, $C(\tau)$ shows a peak at the origin that
ends before $\tau = 1$ s, followed by smooth oscillations with one main and a few secondary
frequencies which do not de-correlate in more than 50 s. This means that the high-frequency
content accounted for by the initial peak corresponds to random fluctuations, whereas the
regular low-frequency oscillations are associated with a deterministic mechanism. The mag-
nitude of the correlated oscillations in $C(\tau)$ is in between 0.11 and 0.13 at $\Phi/\Phi_{pack} \ge 0.808$
and approximately 0.17 at lower concentrations. While the total energy $\sigma^2$ of the fluctua-
tions varies by a factor of 30 between the highest and lowest values (Fig. 9), the relative
contribution of the deterministic oscillations does not change much.

The dominant frequencies are better determined in the spectral domain. Fig. 13 shows
the power spectral density of $\langle h' \rangle(t)$ computed by the periodogram method based on the
Fourier transform of the time signal. Its characteristics vary with concentration. In any
case, the largest frequencies detected are less than 3 Hz, with a dominant frequency of less
than 0.43 Hz. These values are very small compared to the pump’s 40 Hz cyclic frequency,
allowing us to conclude that they are independent of any pump-induced effects. At lower
expansions ($\Phi/\Phi_{pack} \ge 0.797$), the spectrum extends between 0.1 and 1 Hz with a main peak
around 0.4 Hz and a significant secondary peak at a frequency below 0.3 Hz. At intermediate
concentrations ($0.687 \ge \Phi/\Phi_{pack} \ge 0.784$), where the energy of the fluctuations is the lowest,
the spectrum has shifted slightly towards lower frequencies and shows up to 3 secondary
peaks. At larger expansions ($\Phi/\Phi_{pack} \le 0.583$), the highest peak, located in between 0.1
and 0.2, becomes very dominant.

Fig. 14 shows the frequency $f$ of the dominant harmonic as function of $\Phi/\Phi_{pack}$. It
has been normalized by the particle diameter and fluidization velocity in the form $fd/U_f$
to allow comparison with other systems of fluid and particles. The results of previous
studies reported here correspond to frequency measurements of concentration waves traveling
through the suspension. They all concern low expansions close to the minimum fluidization
velocity ($\Phi/\Phi_{pack} \ge 0.8$), with the exception of one measurement from Ham et al. [4] at
$\Phi/\Phi_{pack} = 0.7$. On the other hand, our results cover the range $0.40 \le \Phi/\Phi_{pack} \le 0.85$. We
did not take any measurement for $\Phi/\Phi_{pack} > 0.85$ because satisfactory fluidization could
not be achieved throughout the column at such high concentrations using our system. The
range of concentrations $\Phi/\Phi_{pack}$ studied in common by the literature and the present work
therefore lies between 0.7 and 0.85. In this range, our measurements agree fairly well with
those of previous studies, particularly with those of Ham et al. [4] who considered particles
<figure><img src="image_13.png" /><figcaption>FIG. 12. Fig. Autocorrelations of the bed-height time evolution for various particle concentrations.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>FIG. 13. Fig. Power spectral density of the bed-height time evolution for various particle concentrations.</figcaption></figure>

with similar properties. By taking all the results into account, we can describe a wide concentration range, revealing three stages as $\Phi/\Phi_{pack}$ increases. For $0.4 \le \Phi/\Phi_{pack} \le 0.6$, the normalized frequency slightly increases from approximately $3 \times 10^{-3}$ to $7 \times 10^{-3}$. Above $\Phi/\Phi_{pack} = 0.6$, it jumps to a value greater than $10^{-2}$ and goes on increasing to reach a maximum for a value of $\Phi/\Phi_{pack}$ between 0.8 and 0.9. Then, the frequency strongly decreases while the system is approaching the packing state. Although system parameters, including confinement and inlet conditions, strongly influence fluidization velocity, the evolution of the normalized frequency turns out to be less sensitive to these parameters.
<figure><img src="image_15.png" /><figcaption>FIG. 14. Dimensionless dominant oscillation frequency as a function of the concentration.</figcaption></figure>

The dynamics of fluidized beds is a long-standing subject and the existence of upward concentration waves passing through them is well known [16]. The theoretical equations for plane waves propagating in an unconfined bed have been established by considering perturbations of the mass and momentum balances around the homogeneous state by Anderson and Jackson in 1968 [17] and then by Batchelor in 1988 [1], who followed a more rigorous approach. Linear analysis showed that these waves can give rise to convective or absolute instability [18]. This waves are a first step toward a global destabilization of a homogeneous fluidized bed. They can be followed by a secondary instability that breaks the one-dimensional structure, which has been described theoretically [5, 19–21] and observed experimentally [6].

Following Batchelor [1], the wave dynamics involve several physical parameters, such as the diffusivity, drag and added-mass coefficients, and velocity variance of the particles. How to relate these quantities to the control parameters of the bed is unfortunately not known. From their values, it is possible to determine $Q$, the effective bulk elastic modulus of the suspension divided by the particle mass per unit volume. Two kinds of concentration waves can exist separately according to the operating conditions. Firstly, the kinematic wave results from the transport equation of the particle concentration and has a propagation velocity,

$$c_{kw} = \Phi \frac{dU_f}{d\Phi}. \qquad (6)$$
<figure><img src="image_16.png" /><figcaption>FIG. 15. Ratio of the propagation velocity measured in the experiments and that of the kinematic waves calculated using Eq. (6). (For Ham et al (1990) and the present study, the ratio has been arbitrarily fixed to unity and is only represented to indicate the Reynolds range).</figcaption></figure>

Secondly, the dynamic wave results from the momentum equation, whose propagation velocity is

$$c_{dw} = Q^{1/2}. \qquad (7)$$

In this study, interface fluctuations are interpreted as the arrival of upward concentration waves at the top of the bed, which is confirmed by the good agreement with previous wave propagation studies in terms of frequency. Since coherent oscillations are observed over long periods of time, we can conclude that our bed is unstable with respect to planar waves. The question of the type of waves observed here remains open. Observation of the bed surface does not allow us to measure the wave velocity. We will therefore rely on the results of previous work.

The experimental propagation wave velocity, $c_{exp}$, has been measured in Refs. [8–10]. The kinematic wave velocity $c_{kw}$ is calculated by applying Eq. (6) to the relation between the fluidization velocity and the particle concentration measured in each experiment. Fig. 15 shows the ratio $c_{exp}/c_{kw}$ as a function of the particle Reynolds number. The velocity $c_{exp}$ has not been measured by Ham et al (1990). In their case, as in ours, the ratio $c_{exp}/c_{kw}$ has been fixed at one and only plotted to indicate the range of Re. A clear trend emerges from the available data. The lower Re, the less scatter for a given series and the closer $c_{exp}/c_{kw}$
<figure><img src="image_17.png" /><figcaption>FIG. 16. Wavelength of the propagating waves as a function of the particle concentration.</figcaption></figure>

is to unity. The smallest available value of $Re$ is 5, where $c_{exp}/c_{kw}$ is in between 0.8 and 1. Since the range of $Re$ is between 0.35 and 1.8 in Ham et al (1990) and between 0.19 and 1.1 in this study, we will assume that $c_{exp}/c_{kw} = 1$ in both cases.

We are therefore justified as considering that our fluidized bed and that of Ham et al (1990) are operated in the same regime, where coherent oscillations of the concentration are dominated by kinematic waves. Knowing the wave velocity, we can compute the wavelength as $\lambda = \frac{c_{kw}}{f}$. Fig. 16 shows the normalized wavelength $\frac{\lambda}{d}$ as a function the normalized concentration $\Phi/\Phi_{pack}$. The results of the two studies coincide in the region where their concentration ranges overlap. The wavelength reaches a minimum of a few tens of $d$ around $\Phi/\Phi_{pack} = 0.85$. From this minimum, the measurements of Ham et al (1990) show that $\lambda$ rises sharply to exceed $10^3 d$ as $\Phi$ approaches the onset of coherent oscillations, just before the packing state. This trend is expected from the linear theory, which predicts that long wavelengths are the most unstable. In the other direction, $\lambda$ also increases as the bed expands until it reaches a plateau around $350d$ for $\Phi/\Phi_{pack} < 0.6$, which corresponds to the region where the wave loses its one-dimensional character and the surface begins to slosh.

At this stage, it is important to recall that the range of concentrations studied in common by the literature and the present work is $0.7 \le \Phi/\Phi_{pack} \le 0.85$. Consequently, we cannot determine whether the sharp increase in wavelength when approaching the minimum fluidization velocity observed by Ham et al. (1990) is affected by confinement. Moreover, the
decrease in frequency and the increase in wavelength we observed when decreasing $\Phi_s/\Phi_{Pack}$ below 0.7 is a new phenomenon that cannot be compared with previous results for confined flows.

It is now interesting to discuss the bed homogeneity in the light of these results. Since the time correlation (Fig. 12) and spectrum (Fig. 13) of height fluctuations have shown that the time scales of all coherent structures are of the same order, the wavelength $\lambda$ can be interpreted as the scale of the suspension heterogeneities in the vertical direction. As shown in figure 16, $\lambda$ is much larger than $d$, so the suspension can be considered statistically homogeneous at the particle scale. The picture is different if we look at the scale of the whole bed. The column of Ham et al (1990) is a cylinder a diameter $w$ of 78 mm and the bed height $h$ is between 0.4 and 0.6 m depending on the expansion. Thus, $\lambda$ is always greater than the width of the bed ($1.5 \le \lambda/w \le 60$), but remains less than a tenth of the height of the bed, except for $\Phi/\Phi_{pack} > 0.9$ where it approaches unity. In this study, the sides of the rectangular column ($w_1=200$ mm, $w_2=300$ mm) are comparable to the bed height ($210 \le h_{mid} \le 440$ mm). Therefore, $\lambda$ is lower than a tenth of all the bed dimensions at moderate expansion ($\Phi/\Phi_{pack} > 0.6$). However, at larger expansions, the wavelength reaches $\lambda/h_{mid} \approx 0.15$ and $\lambda/w \approx 0.25$. In addition, we know from the images in Fig. 2 and the analysis of section IV, that the bed surface is no longer flat or moving vertically as a whole during such expansions. It is therefore likely that the overall destabilization of the bed at high expansions occurs when the size of the heterogeneities associated with the rising waves reaches the dimension of the column.

## VI. CONCLUSION

Experimental studies of a liquid-solid fluidized bed were carried out under unconfined conditions. Due to the huge number of interfaces present in the column, it was hardly possible to measure concentration and velocity inside the suspension, and we limited ourselves to measurements of bed surface motion. Both the fluidization velocity, $U_f$, that prevents the suspension from settling and the sedimentation velocity, $U_{sed}$, undergone by the particle when they settle in a fluid at rest were measured. It is possible that $U_f$ and $U_{sed}$ match because the particles are observed to slip along the lateral walls, which therefore allows the average velocity profiles of the particles to remain flat in both cases. However, this matching
requires that the flow injected in the fluidization case does not bring any disturbances that significantly increase the particle random agitation compared to the sedimentation case. When this is the case, $U_f$ becomes greater than $U_{sed}$.

Comparisons of sedimentation velocity were made with previous work dealing with both unconfined and confined systems, at low to moderate Reynolds numbers. It turned out that, when normalized by $U_0(1-\Phi)$ (where $U_0$ is the Stokes falling velocity of an isolated particle), all values of $U_f$ collapse to a single master curve $1/(K \times F(\Phi/\Phi_{pack}))$. In a sense, $K$ plays the same role as the $U_i$ (the value of $U_f$ in the dilute limit) in Richardson-Zaki inspired correlations, while $F$ plays that of exponent $n$. Whereas the function $F$ remains the same, $K$ varies according to Stokes number, Reynolds number, inlet conditions and confinement ratio.

The fluctuations of the fluidized bed surface have been studied in detail for a particle Reynolds number below one. They include uncorrelated high frequencies associated with the suspension microstructure and low frequency coherent oscillations involving a few harmonics. These coherent oscillations are the signature of the arrival at the bed surface of ascending concentration waves, which are known to develop in fluidized beds. The main oscillation frequency $f$ is in fairly good agreement with previous studies. Unlike sedimentation velocity, it seems to be essentially determined by the physical properties of the fluid and particle system, and more robust to confinement and inlet conditions. It increases slowly at first, then steeply with the concentration $\Phi$ to reach a maximum between 0.8 and $0.9\Phi_{pack}$, before decreasing sharply as the system approaches the packing state.

Results from the literature suggest that in the range of parameters concerned, where inertia is moderate, the propagation velocity is that of kinematic waves, $c_{kw}$. Using the theoretical expression (7), $c_{kw}$ is calculated from the experimental sedimentation law $U(\Phi/\Phi_{pack})$. This allows us to determine the wavelength, $\lambda = \frac{c_{kw}}{f}$, in our case and that of Ham et al (1990) [4], which has similar parameters. The wavelength has a general U-shape, with a minimum around $\Phi/\Phi_{pack} = 0.85$, from which it increases as the concentration tends towards both the packing state and the dilute regime. Considering $\lambda$ as the size of bed heterogeneities lead to the following conclusions. On the one hand, the increase of $\lambda$ at small expansions can be related to the divergence of the correlation length between particle locations as the system approach the jamming transition. On the other hand, the increase at large expansions can set the limit for the overall homogeneity of the fluidized bed as $\lambda$ ceases to be
small compared to the column dimensions, in a system where these dimensions are much
larger than the particle size.

ACKNOWLEDGMENTS

This study was supported by the Region Centre-Val de Loire through the Contribution
of “Academic Initiative”: RHEFLEXES/ 2019-00134935.

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