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S0032-5910(17)30980-4 doi:10.1016/j.powtec.2017.12.032 PTEC 13021

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Powder Technology

Received date: Revised date: Accepted date:

14 August 2017 31 October 2017 4 December 2017

Please cite this article as: R. Li, H. Yang, G. Zheng, Q.C. Sun, Granular avalanches in slumping regime in a 2D rotating drum, Powder Technology (2017), doi:10.1016/j.powtec.2017.12.032

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ACCEPTED MANUSCRIPT Granular avalanches in slumping regime in a 2D rotating drum R. Li1a, H. Yang1a,b, G Zheng1c, Q. C. Sun2*

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1 a. School of Optical-Electrical and Computer Engineering; b. Shanghai Key Lab of Modern Optical System; c.

Shanghai, 200093, China

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School of Medical Instrument and Food Engineering, University of Shanghai for Science and Technology,

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2 State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, China

* Corresponding author: [email protected]

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Abstract:

In our study of the avalanche dynamics of granular materials in the slumping regime

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of a 2D rotating drum, we employed digital image processing to measure the slope angle, and speckle visibility spectroscopy (SVS) to measure the time-resolved dynamics of the generated avalanches. It was found that steady and unsteady states

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alternately occur in an avalanche. In the steady state, granular temperature is low and

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the flow rate is constant at a given drum rotating speed; whilst in an unsteady state, both the flow rate and granular temperature are significant because there are intense

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relative motions among the grains in the active layer. This leads to the removal of solids in upper active layer and the decrease of the surface slope angle. Based on our previous work on granular avalanche dynamics in 3D rotating drums [Yang et al.,

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Chemical Engineering Science, 146 (2016) 1-9], a model is proposed for the analysis of avalanching characteristics using the duration time, the rest time and the steady flow time. One of the interesting observations of the study was that steady flow durations in granular avalanches exhibit a normal distribution, as well as increasing in value with an increase in the filling degree.

Keywords: avalanche dynamics, granular flows, rotating drum, speckle visibility spectroscopy

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ACCEPTED MANUSCRIPT 1. Introduction Granular flows in rotating drums are of wide interest not only in the study of the mechanics of granular media, but also in investigating the mechanisms of both debris

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flows and snow avalanches in nature [1, 2]. In the case where a drum rotates at very

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low speeds (typically with the Froude number Fr between 10−5 and 10−3 . Fr is

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defined as the ratio of centrifugal force 𝜔2 𝑅 to gravity 𝑔, 𝐹𝑟 =

𝜔2 𝑅 𝑔

. 𝜔 is rotation

speed and 𝑅 is drum radius) , the granular flows are often in the form of intermittent

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bursts of activity characterized by avalanches in the so-called slumping regime. A large number of avalanche models have been proposed for this regime. For example,

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Mellmann developed a mathematical model to predict the avalanche duration time and the transitions between the different forms of transverse motion [1]. Liu et al. put forward an improved model which involved measuring the lower and upper angles of

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repose and suggested these angles are essentially two related parameters [3]. However,

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there still remain some difficulties in improving the measurement accuracy of repose angles. Yang et al took granular temperature as the key indicator of the granular bed

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dynamics and introduced a number of steps to the avalanche model with accurate measurements of duration and rest time [4]. Rajchenbach [2] and Douady [5] found that an avalanche started from some unbalanced particles and propagated in the active

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layer. Based on their work, our group further studied dynamic propagations in the passive layer [6].

Many experimental methods are available for the measurement of particle moments in rotating drums, including particle image velocimetry (PIV), particle tracking velocimetry (PTV) [7], positron emission particle tracking (PEPT) [8], and magnetic resonance imaging [9], some of them are restricted to two dimensions of motion (e.g. PIV and PTV) whilst the other methods can only resolve the granular dynamics to a fine scale with relatively poor temporal resolution, and vice versa. In contrast, speckle visibility spectroscopy (SVS) [10, 11] has the capacity to resolve the average of the three components of motion for grains in dense granular systems with high spatial-temporal resolutions; thus, SVS is an appropriate technique to probe the 2 / 19

ACCEPTED MANUSCRIPT granular dynamics of avalanches. In this work, we report on the SVS and imaging study of granular avalanching in the slumping regime generated in a 2D rotating drum. The dynamical features are

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analyzed with respect to the quantities of granular temperature and the inclined

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surface angles.

2. Experiment setup

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The 2D drum, partly filled with spherical glass beads, is shown in Fig. 1. The filling degree 𝑓 is defined as the portion of the volume occupied by the particle bed in the drum. In this study, we determine the filling degree 𝑓 by using the filling angle

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𝜀 as shown in fig. 2, 𝑓 = 𝜋 (𝜀 − sin 𝜀 cos 𝜀). The inner diameter D and thickness L are 300 mm and 10 mm, respectively. The walls are made of transparent plexiglass to

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permit optical access, and the drum was driven by a DC motor at 0.01~0.10

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revolutions per minute (RPM). The filling degree of glass beads ranged from 15% to

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50%. The diameter of the glass beads d ranged from 0.5mm to 0.6 mm. The drum is made of the same material as the glass beads, and the friction coefficients of the inter-glass beads and glass beads-walls are the same and are measured as 0.45 in this

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work. The density of the glass beads is 2.65 × 103 kg/m3, and the bulk density of the granular system is 1.73 × 103 kg/m3. The profiles of the granular system in the drum were recorded by a CCD camera (1024 × 1024 pixels at a rate of 50 frames/s). The line rate of the CCD is 20 KHz. So the time resolution is 50 μs. The temporal sensitivity of SVS is 2.5 ms after smoothing. The average spatial resolution and sensitive is 266 nm (determined by 1/2 wavelength of the laser). The measurement point is focused on the middle of the slope surface in the active layer as shown in Fig. 1. The measuring field is a round area with a diameter of 0.5 cm. A weight was hung alongside the drum as a plumb line so as to facilitate the measurement of the surface slope angle 𝜃. An example of the snapshots is shown in Fig. 2 (a) and the binarization process to measure 𝜃 is depicted in Fig. 2 (b). The maximum and minimum values of 𝜃 are 𝛼, 𝛽, respectively, and 3 / 19

ACCEPTED MANUSCRIPT the shear wedge angle is defined to be 𝛾 = 𝛼 − 𝛽 (see Fig. 2 (c)), which will be used to describe the dynamics of the granular avalanches. Speckle visibility spectroscopy involves illuminating the granular material with a

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monochromatic diode pumped solid state (DPSS) laser laser (wavelength 𝜆 is 532 nm and power is 500 mW in this experiment) as shown in Fig. 1. The photons that

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emerged from the granular material, after diffusing within it, form a speckle pattern that was detected using an 8-bit line scan CCD camera of 2×1024 pixels and a 50 kHz frame rate, i.e. providing an exposure time of T=18.5 𝜇𝑠. The camera was placed

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with its optical axis normal to the side of the bed and 400 mm from it, such that the ratio of pixel to speckle size was about 0.5. As illustrated in the example shown in Fig.

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3(a), in the absence of any relative motion of these particles, the scattered light is characterised by a constant intensity which is detected in each pixel. However,

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relative motion between the particles leads to the scattered light showing temporal

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fluctuations over the pixels, see Fig. 3 (b). Furthermore, for a given exposure time, the faster the dynamics of the grains, the more the speckle image is blurred and the

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contrast is lowered enabling the capture of rapid changes in the granular material with time such as that which occurs in an avalanche as illustrated in Fig. 3 (c). The intensity 𝐼 is measured by the CCD camera [10], and the variance of 𝐼 can be

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expressed as

𝑉2 (𝑇) ∝ 〈𝐼 2 〉 𝑇 − 〈𝐼〉2

(1)

where 〈𝐼〉 𝑇 denotes the average laser intensity over pixels exposed for a duration T. The proportionality constant of 𝑉2 (𝑇) is set by both the laser intensity and the ratio of speckle to pixel size (i.e. it is set-up dependent). This constant can be eliminated by considering the variance ratio 𝑉2 (2𝑇)⁄𝑉2 (𝑇). For diffusely backscattered light from particles moving with a random ballistic motion, the power spectrum is Lorentzian [12], and the variance ratio can be derived as 𝑉2 (2𝑇) 𝑉2 (𝑇)

𝑒 −4𝛤𝑇 −1+4𝛤𝑇

= 4(𝑒 −2𝛤𝑇 −1+2𝛤𝑇)

(2)

where 𝛤 = 4π𝛿𝑣/λ [10]. The velocity fluctuation is described with the variance of velocity 4 / 19

ACCEPTED MANUSCRIPT 𝛿𝑣 2 = 〈𝑣 2 〉𝑅 − 〈𝑣〉𝑅

2

(3)

where 𝑣 is particle velocity. 〈𝑣〉𝑅 denotes the average velocity over particles in the measurement region 𝑅 . The averaged 𝛿𝑣 2 , i.e.〈𝛿𝑣 2 〉, is the so-called granular

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temperature [13]. 〈𝛿𝑣 2 〉 can be detected with a particular experimental set-up composed of a laser source, lens and a CCD camera. The laser intensity of scattering

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spot on beads is measured by the CCD camera. 〈𝛿𝑣 2 〉 is a function of the wavelength of the light used and the scan speed of the CCD camera which can be derived from Eq. (2). Details can be found in Refs[4, 6]. Fig. 3 (d) is an example of the measured

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〈𝛿𝑣 2 〉. We can, therefore, see that SVS is a much-improved approach for probing

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instantaneous avalanche phenomena in rotating drums.

3. Results and discussions

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Fig. 4 (a) shows the variation of the surface slope angle 𝜃 during two avalanche

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cycles as the drum rotates at a speed of 0.06 RPM. When 𝜃 increases to the maximum repose angle of 28.5° at 6.2s, some particles gradually become unstable

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and start to flow in the active layer (see Fig.1) [6]. Afterwards, the surface slope angle decreases from 28.5° to 25.9° and the first avalanche event occurs. Both the active layer thickness and flow rate vary rapidly during the first avalanche [14]. Similarly,

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𝛿𝑣 2 (i.e. granular temperature) accelerates progressively to a maximum and then decreases, as shown in Fig. 4 (b). This indicates that the relative movement of particles is significant [4]. Wang and Zhang [14] measured 𝛿𝑣 and 𝑣 in experiments, and it was found that the trends of 𝛿𝑣 and 𝑣 were similar and the average temporal correlation is strong. The study by Abate et al. [15] shows that the velocity √〈𝑣 2 〉 is about 10 times of √〈𝛿𝑣 2 〉. In this work, this period is called the unsteady flow. After the unsteady flow, 𝜃 remains constant at 25.9°. This indicates that the granular flow rate in the active layer has a small fluctuation and the particles relative movement, denoted by 𝛿𝑣 2 , is small as well, about less than a half of the maximum value of 𝛿𝑣 2 in the unsteady flow. At the end of the slumping, the flow rate decreases to zero and the slope angle increases 5 / 19

ACCEPTED MANUSCRIPT slightly. This period is called the steady flow. We also note that there is no obvious steady flows can be observed when the drum is rotating at a lower speed of 0.01RPM, as shown in Fig. 4 (c) and (d).

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Based on the above observations, we propose that a granular avalanche at relatively higher speeds can be divided into 1) unsteady flows in which the relative

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movement of particles increases rapidly, and 2) steady flows in which the flow rate maintained by the rotating drum remains constant. In comparison with the 3D drum experiments carried out by Yang et al. [4], there were no steady flows observed in

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their work, which leads to a symmetric avalanche process based on 𝛿𝑣 2 . Because the time resolution of the SVS method is higher and the error in the measurement of the

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slope angle is significant, the half band-width (HBW) of 𝛿𝑣 2 is defined as the width at half of the wave peak and it is used to denote the band-width when the endpoints

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shown in Fig. 4 (b) and (d).

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are influenced by noise. We take double HBW time as the unsteady flow (UF) time, as

To quantitatively analyze the characteristics of unsteady and steady flows, we

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measured the times for both flow states. As shown in Fig. 4 (a), a slumping cycle can be separated into two time phases; firstly, there is the period of elevation of the bed which is denoted by the rest time 𝑡r and, secondly, there is the period of slumping of

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the free surface layer which is denoted by the avalanche duration 𝑡d . The duration time comprises the unsteady flow time 𝑡u , and the steady flow time 𝑡s . Fig. 5 shows the measured values of 𝑡d , 𝑡r , 𝑡u , and 𝑡s as the rotating speed of the drum increases from 0.01 to 0.08 RPM. We can see that the rest time 𝑡r decreases with increasing rotating speed, but the duration time 𝑡d increases and can be represented by the fitted linear function, 𝑡d = 24.70𝜔 + 1.88

(4)

where 𝜔 is the angular velocity of the cylinder in RPM(revolutions per minute) unit. This is in line with the study of avalanches in 3D drums [4]. The unit of 𝑡d is the second. At same time, it can be observed that the unsteady flow time 𝑡u remains constant when the rotating speed varies. The steady flow time increases can be represented by the fitted linear function, 6 / 19

ACCEPTED MANUSCRIPT 𝑡s = 33.39𝜔 + 0.66

(5)

The constant term is a parameter influenced by the 2D drum system, such as the boundary condition (inner diameter, friction and so on). In an ideal quasi-static system,

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there will be no steady flows because grains moving to the active layer are too few to maintain a steady flow. Moreover, it is noticed that the slope of the steady flow time

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curve is slightly larger than the duration time curve by comparing the Eq. (4) with Eq. (5). This indicates that the duration time is mainly affected by the steady flow time. Eqs. (4) and (5) shows the steady flow duration basically proportional with speed 𝜔.

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So steady state duration is too short to be easily measured at low speeds, and in experiments we cannot distinguish it from the unsteady state either using SVS or

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imaging method. At the same time, our rotating motor cannot provide lower rotating speeds than 0.01 rpm without influencing the uniform speed performance. These

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results are in line with the study by Fischer et al. [16].

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As shown in the schematics of the bed profile in Fig. 2 (c), the rest time 𝑡r is the

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rotation time as the body from angle 𝛽 to 𝛼, which can be calculated as 𝛾𝜋

𝑡r = 180𝜔

(6)

where 𝛾 is the shear wedge angle defined as 𝛾 = 𝛼 − 𝛽. Mellmann [1] suggested the

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duration of the avalanche can be evaluated using the relation 2𝑠 ∗

𝑡d = √𝑔(sin 𝜂−𝜇 cos 𝜂) i

(7)

where 𝑔 is the acceleration due to gravity, 𝜂 is the average of the upper and lower angles of repose, 𝜇i is the friction coefficient between the particles, and 𝑠 ∗ is the distance between the center of gravity S1 for sector ABC and S2 for sector CDE, which stands for the average particle path, as shown in Fig. 2 (c). Yang et al. take the distance of particle travel in a single avalanche, which is measured by PEPT [8], into consideration [4]. They studied the number of steps 𝑛 that particles need to travel from the top to the bottom of the bed-free surface in a 3D rotating drum, and provide the following expression for 𝑠 ∗ : 𝑠 ∗ = 𝑛𝑠 = 0.5𝑔(sin 𝜂 − 𝜇𝑖 cos 𝜂)(𝑎𝜔 + 𝑏)2 𝑠

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ACCEPTED MANUSCRIPT where 𝑠 is the distance of the averaged step length. The duration time 𝑡d can be calculated using Eq. 4, and the parameters 𝑎 and 𝑏 are then obtained by using Eqs. 7 and 8. For the experimental setup and the glass bead used in this work, we obtained

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𝑎 = 24.7 and 𝑏 = 1.884.

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This model (Eqs. 7 and 8) is based on a 3D drum of duration time 𝑡d , and only

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takes the unsteady flow into consideration. It can be seen from Fig. 6 that the calculated 𝑡d based on Eqs. 7 and 8 is consistent with the measured 𝑡u in a 2D drum. Thus, there are differences between duration time in 3D and 2D drums. By using Eq.

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5 to calculate the steady flow time, we improve the expression of the duration time in a 2D drum which includes both unsteady and steady flows: 2𝑠∗

𝑙

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𝑡d ′ = √𝑔(sin 𝜂−𝜇 cos 𝜂) + (𝑓(𝑑) × 𝜔 + 𝑐) i

(9)

where 𝑐 is a parameter influenced by the 2D drum system. In an ideal quasi-static

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system 𝑐 = 0 because the grains moving to the active layer is too few to maintain the

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steady flow. 𝑓(𝑙 ⁄𝑑 ) is a parameter with a positive correlation to the dimensionless geometric variable size ratio 𝑙 ⁄𝑑 . And l is the length of the inclined surface and

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equals to the length of AE and BD shown in Fig. 2 (c) 𝑙 = 2𝑅 sin 𝜀

(10)

where 𝜀 is the filling angle, which is positively correlated to the filling degree. In

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this study 𝑓(𝑙 ⁄𝑑) = 33.39 and 𝑐 = 0.66, and were calculated from the linear fit of the steady flow.

Fig. 6 shows 𝑡d based on Eq. 7 and 𝑡d ′ based on Eq. 9 as a function of the rotating speed. The measured unsteady flow time, 𝑡u , is consistent with the calculation based on the model used in Ref [4]. The measurement duration time is consistent with the calculated 𝑡d ′ . The difference between 𝑡d and 𝑡d ′ is caused by errors in the determination of the critical rotation speed in a 2D drum. In this work, the critical rotation speed is judged when there is equilibrium between the filling and emptying of the shear wedge [1, 4]. As reported in Ref. [1], slumping continues in a stable way as long as the shear wedge ABC can empty itself faster than it is filled anew, as shown in fig. 2 (c). This leads to 𝑡𝑑 < 𝑡𝑟 . As rotational speed increases, the 8 / 19

ACCEPTED MANUSCRIPT rest time decreases more than duration time. Therefore, the equilibrium between filling and emptying of the shear wedge can be regarded as a critical state for the transition to continuous rolling. If 𝑡𝑑 ≥ 𝑡𝑟 , continuous rolling occurs.

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Based on the parameters obtained from the experiments, the critical rotation

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speed can be calculated using Eqs. 6 and 7, 𝜔𝑐1 = 0.067 RPM, and the critical

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rotation speed can also be calculated using Eqs. 6 and 9, 𝜔𝑐2 = 0.109 RPM. The difference between 𝜔𝑐1 and 𝜔𝑐2 is related to the transition regime [4], which essentially needs further study.

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Fig. 7 shows the normal distribution of the steady flow time when the filling degree is 15.51%, 20.19% and 24.23% at 0.06 RPM. The average steady flow time for

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each filling degree is 2.16s, 2.67s and 3.06s respectively. This indicates that the steady flow time increases as a function of the filling degree, as detailed in Eqs. 9 and 10. Fig.

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8 shows the steady flow time as a function of rotating speed by using glass beads of

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0.5mm diameter with filling degrees of 15.51%, 20.19% and 24.23%. The size ratio, 𝑙 ⁄𝑑 , is 508, 550 and 578 respectively for each filling degree. 𝑓(𝑙 ⁄𝑑 ) are accordingly

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33.39, 58.84 and 74.90 which is constituent with Eq. 9. The particle-wall friction would affect the time in the equations. We have tried beads and walls made of two different materials. It turns out that the bigger

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particle-wall friction would lead to bigger upper and lower repose angles. So 𝜂 in Eqs. (7) and (9) would be larger if particle-wall friction increased. 𝑡d and 𝑡d ′ would increase as well. But it is difficult to quantitatively analyze the relationship between the particle-wall friction and the repose angles. The related work can be found in Pohlman et al. [17]. Their study shows that bigger particle-wall friction may lead to longer steady flow time and 𝑡d ′ . They also figured out that there would be no significant difference between the particle velocity at the center and the side in a drum if 𝐿⁄𝐷 ≤ 0.52, where 𝐿 is the length and 𝐷 is the diameter of a drum. In our study, 𝐿⁄𝐷 = 1/30, which is far less than 0.52.

3. Conclusions Through the use of speckle-visibility spectroscopy (SVS) and digital image 9 / 19

ACCEPTED MANUSCRIPT processing, the avalanche dynamics of granular beds in the slump regimes of a 2D rotating drum were carefully measured and analyzed using a number of time lengths. We found that, unlike the situation in a 3D drum, an avalanche in a 2D drum can be

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divided into unsteady and steady flows. In the case of unsteady flows, the relative movement of particles increases rapidly; whereas, in steady flows, which are

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maintained by the rotating drum, the flow rate is effectively constant. The duration time is mainly affected by the steady flow time with the drum rotation speed increasing. By introducing the concept of steady flow to avalanche modelling in a 3D

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drum, we propose an expression for the duration time which includes unsteady and steady flows as laid out in Eq. 9. In addition, we suggest that the difference between

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duration time in 3D and 2D drums might be significant in the study of the critical

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rotation speed and the transition regime.

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Acknowledgements:

This work has been supported by the National Natural Science Foundation of China

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(11572201, 11572178, 91634202) and the Innovation Program of Shanghai Municipal Education Commission (15ZZ072).

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Reference: [1] J. Mellmann, The transverse motion of solids in rotating cylinders - forms of motion and transition Behaviour, Powder Technology, 118 (2001) 251-270. [2] J. Rajchenbach, Dynamics of grain avalanches, Physical Review Letters, 88 (2002) 014301. [3] X.Y. Liu, E. Specht, J. Mellmann, Experimental study of the lower and upper angles of repose of granular materials in rotating drums, Powder Technology, 154 (2005) 125-131. [4] H. Yang, R. Li, P. Kong, Q.C. Sun, M.J. Biggs, V. Zivkovic, Avalanche dynamics of granular materials under the slumping regime in a rotating drum as revealed by speckle visibility spectroscopy, Physical Review E, 91 (2015) 042206. [5] S. Douady, B. Andreotti, A. Daerr, From grain to avalanches: on the physics of granular surface flows, Physical Review C, 3 (2002) 177–186. [6] R. Li, H. Yang, G. Zheng, B.F. Zhang, M.L. Fei, Q.C. Sun, Double speckle-visibility spectroscopy for the dynamics of a passive layer in a rotating drum, Powder Technology, 295 (2016) 167-174. [7] N. Jain, J.M. Ottino, R.M. Lueptow, An experimental study of the flowing granular layer in a rotating drum, Physics of Fluids, 14 (2002) 572-582. 10 / 19

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[8] S.Y. Lim, J.F. Davidson, R.N. Forster, D.J. Parker, D.M. Scott, J.P.K. Seville, Avalanching of granular material in a horizontal slowly rotating cylinder: PEPT studies, Powder Technology, 138 (2003) 25-30. [9] K. Yamane, S.A. Altobelli, T. Tanaka, Y. Tsuji, Steady particulate flows in a horizontal rotating cylinder, Physics of Fluids, 10 (1998) 1419-1427. [10] R. Bandyopadhyay, A.S. Gittings, S.S. Suh, P.K. Dixon, D.J. Duran, Speckle-visibility spectroscopy: A tool to study time-varying dynamics, Review of Scientific Instruments, 76 (2005) 093110 - 093110-11. [11] H. Yang, G.L. Jiang, H.Y. Saw, C.E. Davies, M.J. Biggs, V. Zivkovic, Granular dynamics of cohesive powders in a rotating drum as revealed by speckle visibility spectroscopy and synchronous measurement of forces due to avalanching, Chemical Engineering Science, 146 (2016) 1-9. [12] P.A. Lemieux, D.J. Durian, Investigating non-Gaussian scattering processes by using nth-order intensity correlation functions, Journal of the Optical Society of America A, 16 (1999) 1651–1664. [13] I. Goldhirsch, Introduction to granular temperature, Powder Technology, 182 (2008) 130-136. [14] Z. Wang, J. Zhang, Fluctuations of particle motion in granular avalanches-from the microscopic to the macroscopic scales, Soft Matter, 11 (2014) 5408-5416. [15] A.R. Abate, H. Katsuragi, D.J. Durian, Avalanche statistics and time-resolved grain dynamics for a driven heap, Physical Review E, 76(2007) 061301. [16] R. Fischer, P. Gondret, M. Rabaud, Transition by intermittency in granular matter: from discontinuous avalanches to continuous flow, Physical Review Letters, 103(2009) 128002. [17] N.A. Pohlman, J.M. Ottino, R.M. Lueptow, End-wall effects in granular tumblers: From quasi-two-dimensional flow to three-dimensional flow, Physical Review E, 74(2006) 031305.

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ACCEPTED MANUSCRIPT Figure captions

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Fig. 1 Schematics of the experimental setup

Fig. 2 (Color online) Measurement of a granular bed profile during avalanching. (a) A

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photo of the bed. (b) Binarization processing of the photo and the determination of the slope angle. (c) Schematics of the bed profile before and after an avalanche.

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Fig. 3 An example of SVS results of speckles distributed in the CCD pixels vs. time. (a) Stationary drum. (b) Drum rotates at a speed of 0.06 RPM and no avalanche events observed. (c) Drum rotates at a speed of 0.06 RPM, an avalanche event occurs

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in the period of 0.25-1.25 s, and (d) the calculated velocity fluctuation 〈𝛿𝑣 2 〉. Fig. 4 (Color online) Time evolution of two granular avalanches in a 2D drum. (a) 𝜃

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and (b) 𝛿𝑣 2 at 0.06 RPM. (c) 𝜃 and (d) 𝛿𝑣 2 at 0.01 RPM. UF denotes unsteady

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flow. SF denotes steady flow, and HBW denotes the half band-width. A few flow phases are quantified with time-length. The period of bed elevation is denoted by rest

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time 𝑡r , slumping of the free surface layer denoted by avalanche duration 𝑡d . The duration time composes the unsteady flow time 𝑡u , and the steady flow time 𝑡s . Fig. 5 (Color online) Unsteady and steady flow time, rest time, duration time and the

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best fits

Fig. 6 (Color online) Unsteady flow time and duration time. The calculation of 𝑡d is based on Eq. 7 and 𝑡d ′ is based on Eq. 9. Fig. 7 The distribution of the steady flow time 𝑡s at different filling degrees Fig. 8 (Color online) The steady flow time vs. rotating speed at different filling degrees

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Figures

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Fig. 1 Schematics of the experimental setup

Fig. 2 (Color online) Measurement of a granular bed profile during avalanching. (a) A

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photo of the bed. (b) Binarization processing of the photo and the determination of the slope angle. (c) Schematics of the bed profile before and after an avalanche.

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Fig. 3 An example of SVS results of speckles distributed in the CCD pixels vs. time. (a) Stationary drum. (b) Drum rotates at a speed of 0.06 RPM and no avalanche

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events observed. (c) Drum rotates at a speed of 0.06 RPM, an avalanche event occurs

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in the period of 21.6-29.7 s, and (d) the calculated velocity fluctuation 〈𝛿𝑣 2 〉.

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Fig. 4 (Color online) Time evolution of two granular avalanches in a 2D drum. (a) 𝜃 and (b) 𝛿𝑣 2 at 0.06 RPM. (c) 𝜃 and (d) 𝛿𝑣 2 at 0.01 RPM. UF denotes unsteady flow. SF denotes steady flow, and HBW denotes the half band-width. A few flow phases are quantified with time-length. The period of bed elevation is denoted by rest 15 / 19

ACCEPTED MANUSCRIPT time 𝑡r , slumping of the free surface layer denoted by avalanche duration 𝑡d . The

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duration time composes the unsteady flow time 𝑡u , and the steady flow time 𝑡s .

Fig. 5 (Color online) Unsteady and steady flow time, rest time, duration time and the

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best fits

Fig. 6 (Color online) Unsteady flow time and duration time. The calculation of 𝑡d is based on Eq. 7 and 𝑡d ′ is based on Eq. 9.

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Fig. 7 The distribution of the steady flow time 𝑡s at different filling degrees

Fig. 8 (Color online) The steady flow time vs. rotating speed at different filling degrees

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Graphical abstract

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Speckle visibility spectroscopy is used to measure the time-resolved dynamics of

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A steady and a unsteady state alternately occur in an avalanche, which are

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granular avalanches

identified with granular temperature and surface angle.

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Temporal characteristics of state durations and granular temperature are analyzed

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