Experimental affirmation of secondary flows inside granular media

Experimental setup

The experimental flume is split into three sections: a reservoir to produce incoming grains, the circulation area of curiosity, and an outflow area the place grains go away the system and fall right into a bucket. The size of the circulation area is 0.55 m, and it has a width of 0.10 m. The partitions separating these areas are elevated at heights h₁ and h₂ above the belt and the lateral sidewalls. The sidewalls confine the circulation for the entire size of the experimental area, together with the outflow. The conveyor belt is fabricated from rubber with uniformly distributed asperities of dimension 3 mm. The two cross-flow partitions and the flume are all made out of 1cm-thick acrylic sheets, whereas the entrance wall, which accumulates the heap, is fabricated from aluminium. To keep away from vibrations throughout exams, the flume was first pressed in opposition to the conveyor belt beneath its weight after which anchored tightly utilizing beams on the prime of the system, out of the road of x-rays. This anchoring method prevented grains from slipping beneath the partitions and even getting caught through the exams, thus averting any potential belt harm.

The granular materials contains spherical glass beads with a median diameter of d = 3 mm. The take a look at is ready up by first filling up the reservoir with grains, in addition to putting an extra pile of grains with a peak > h₁ subsequent to the reservoir gate, contained in the circulation area of curiosity. As the belt begins transferring, the brand new grains coming into the flowing area be part of the pile and ultimately both influence the outgoing wall or circulation by the downstream opening, so their aggregated bulk ultimately types a gentle heap just a few seconds after influence. The on-line Supplementary Video 1 captures this course of. The regular state concludes when the reservoir empties, which ceases the incoming flux and step by step reduces the heap’s dimension till it ranges and disappears. Through trial and error it was discovered that by taking h₂ = h₁ − d/2 the circulation maintained a sufficiently extended regular state geometry for x-ray imaging that lasted as much as 2 minutes, restricted by the capability of the reservoir.

Multiple exams had been carried out, all with the belt velocity U_b fastened at 2 cm s⁻¹ to eradicate movement blur on the x-ray radiographs. On the opposite hand, we diversified h₁ from 25 to 60 mm, and the peak of the heap h_* from 50 mm to h_* ≈ 100 mm by altering the preliminary quantity of grains on the conveyor belt. The take a look at outcomes present that the heap size L solely is dependent upon the heap peak h_* − h₁, and the heap maintains a continuing slope α ≃ 18^o. Given this pattern, a single take a look at is analysed on this research, with h₁ ≈ 25 mm and h_* ≈ 77 mm. Additional take a look at outcomes confirming the robustness of the mechanism are introduced within the Supplementary Information.

The x-ray sources had been positioned ~2 m from the flume to minimise non-parallel beam results. Similarly, the x-ray detectors had been located round 20 cm from the flume in the wrong way of their respective sources, as depicted in Fig. 1. Steel panels had been positioned to forestall undesired illumination of the detectors from their non-associated sources. During exams, the sources had been set to provide x-ray at a most vitality of 190 keV and 4 mA present for supply 1, and 190 keV and 5 mA for supply 2. Radiographs had been captured at a frequency of 30 fps with a decision of 960 × 768 px at 16-bit with a spatial decision of 0.29 px mm⁻¹ and 0.22 px mm⁻¹ for detectors 1 and a couple of, respectively. In addition to the complete, flowing system, radiographs had been additionally obtained of the empty chute to assist subsequent evaluation. The accuracy of the 2D and 3D velocity measurement strategies introduced on this work has been studied utilizing identified velocity fields^23,40. While the strategies typically introduce quantitative errors, they qualitatively recuperate a variety of underlying fields and subsequently the secondary circulation findings on this paper are believed to be sturdy.

Free floor measurement

Here, we introduce a reconstruction technique for establishing 3D profiles of free surfaces of granular flows utilizing x-ray radiography from two orthogonal instructions. The start line of the strategy assumes that the radiograph depth I_i from the i-th detector follows the Beer-Lambert absorption legislation⁴⁷:

the place I_i,R is the i-th detector’s reference depth of the radiograph of the empty flume, μ ≡ μ(ξ) is the native absorption of the fabric on the ξ location, whereas the integral is over the x-ray beam size.

In the case of detector 2, assuming that there isn’t any important spatial change within the quantity fraction of the dense flowing materials, Eq. (1) simplifies and rearranges to:

the place h_BL(x, y) is the theoretical Beer-Lambert floor peak profile, and μ_c is the efficient materials absorption of the grains on detector 2, which is assumed fixed. The radiograph intensities I_i, used on this course of, are depicted in Supplementary Video 3, together with their values normalised by I_i,R. In order to seize further results comparable to beam hardening and x-ray scattering, which aren’t thought-about by the Beer-Lambert legislation, we take into account the next empirical linear relation for the precise floor peak profile:

with β_c a continuing.

In order to acquire μ_c and β_c, we first calculate a nominal profile of the averaged peak alongside the y path, ({ < h > }_{y}equiv { < h > }_{y}(x)) utilizing detector 1. This is finished from direct thresholding of the time-averaged absorption area ({ < ln ({I}_{1}/{I}_{1,R}) > }_{t}), and is well-defined because of the very completely different absorption of air and grains, and the truth that the fabric peak doesn’t range a lot throughout the y path of the flume, even close to the dip.

To first order, the floor peak on the centre of the flume is estimated by (h(x,0)approx { < h > }_{y}). Then, the efficient materials attenuation coefficient μ_c on detector 2 may be discovered by becoming a linear legislation from the time-averaged absorption area noticed at every pixel alongside the centre line:

the place we receive μ_c = 0.01117 mm⁻¹, in addition to β_c = 0.2304 which was launched to permit for extra results of beam hardening and x-ray scattering. The match may be seen in Supplementary Fig. 2.

As a final step, h(x, y) from Eq. (3) is smoothed with a 2D Gaussian filter of normal deviation 6 px.

DEM simulations

The bodily experiments had been computationally modelled utilizing the DEM by the open supply code YADE⁴⁸. The simulations concerned N ≈ 120,000 spherical particles, which had been modelled utilizing a viscoelastic contact legislation with a linear spring for regular contacts and a Coulomb threshold for tangential contacts⁴⁹. These particles have the identical common diameter 3 ± 0.75 mm because the glass beads utilized in experiments, with some added polydispersity to keep away from crystallisation whereas stopping segregation. Their interparticle friction was set at 0.5, their stiffness at 1 × 10⁷ Pa, their density was set at 2500 kg m⁻³ to imitate the fabric and guarantee most interparticle deformation of 10⁻⁴d. The regular restitution coefficient was set at 0.5, and the Poisson’s ratio at 0.3. To mimic the geometric results of the rubber conveyor belt, its texture of was simulated utilizing a set of spheres with a hard and fast spacing of each two rows alongside the y-axis, at a peak z = 0, with a longitudinal row at a peak of z = − d. These boundary grains had been set in movement at velocity U_b and had an elevated friction coefficient of 1.0. An analogous friction coefficient was set to explain the acrylic partitions, aligning with literature values for the friction coefficient between glass with acrylic and rubber⁵⁰. Fictitious rolling friction was not carried out because the boundary spheres present an precise bodily rolling resistance. However, completely different friction coefficient values and belt preparations had been examined, with the said mixture of parameters above exhibiting the closest similarity to the experimental consequence whereas additionally aligning with the bodily properties of the simulated supplies.

The experimental setup for the simulations mirrored the bodily experiment: a reservoir initially full of grains and a pile of grains located contained in the flume. The pile peak was adjusted to match h_* upon reaching a gentle state. Once a continuing state was achieved, the grains’ areas, velocities, and radii had been recorded to a file at a frequency of 30 fps, mirroring the experimental setup. 2700 frames had been used for the simulations introduced on this work. Next, a coarse-graining technique⁵¹ was employed to acquire the rate and density fields of the recorded flowing spheres, utilizing a Lucy windowing perform with a radius of twod.

Synthetic radiographs had been generated from DEM knowledge as in earlier research^40,42 utilizing the sphere positions and radii to calculate the attenuation alongside the x-ray path. This was primarily based on an idealised system the place the pattern consists of stable particles, every of the identical uniform x-ray attenuation coefficient, and interstitial air of negligible attenuation in comparison with the stable section. Assuming a parallel x-ray beam with no scattering, the Beer-Lambert attenuation legislation then simplifies to

the place I_0,DEM is the preliminary x-ray depth earlier than travelling by any grains, μ_DEM is the fixed attenuation coefficient and D is the overall thickness of grains that the x-ray beam travels by. This thickness will rely upon the pixel place (η, ζ) within the native imaging coordinates and may be calculated by summing every of the N projected particle thicknesses. For detector 1 the depth is then given by

and equally for detector 2 the depth is

the place (x_i, y_i, z_i) and r_i signify the place and radius of particle i. These two thicknesses are calculated for every body and Eq. (5) is then used to generate the corresponding synthetic radiographs with an arbitrary attenuation coefficient of μ_DEM = 10, preliminary depth I_0,DEM = 100,000 and spatial decision 3 px mm⁻¹.

Velocity reconstruction utilizing x-ray rheography

In order to stress the density fluctuations throughout the circulation, the radiographs are first divided by the time-averaged depth of the steady-state frames ({ < I > }_{t}). The absorption fluctuation area (μ_fluct) is then calculated by

as depicted in Supplementary Video 3. The 3D velocity area is reconstructed utilizing the x-ray rheography approach⁴⁰ utilized to the normalised depth fields μ_fluct. We concentrate on the x-axis velocity element, which is widespread to each imaging instructions. X-ray rheography is a correlation-based algorithm that first reconstructs the distribution of in-plane displacements by the out-of-plane path by fixing a deconvolution downside⁴². It then solves an optimisation downside to mix velocity distributions from two perpendicular instructions and reconstruct inner velocity fields. The accuracy and sources of errors of x-ray rheography have beforehand been investigated for each the preliminary distribution reconstruction⁴² and the entire rheography course of⁴⁰ (see Supplementary Information inside).

Here, we apply the preliminary correlation evaluation on successive pairs of photos utilizing an interrogation window dimension of 32 px and a most displacement of  ±16 px. The deconvolution course of was carried out utilising regularisation parameters α = 0.1 and p = 2.0 (see Baker et al.⁴⁰), and this was repeated for a number of pairs of photos to enhance accuracy, as much as a most of 100,000 evaluations. The optimisation downside was solved by averaging the perfect 100 configurations with the smallest errors from a complete of 10,000 to provide the ensuing inner x velocities. Note that this velocity area assumes a uniform free floor throughout the width of the flume, and thus disregards the obtained dip together with some other variation within the free floor. The calculated velocity fields are subsequently trimmed utilizing the free floor measurements so as to eradicate extrapolated velocities out of the granular area.

PIV evaluation of the depth-averaged flows

The experimental depth-averaged velocity fields had been computed from successive x-ray radiographs utilizing the methodology developed by Guillard et al.²³ for granular silos, which was validated in opposition to identified circulation charges from a mass stability. PIV evaluation was carried out on the attenuation fluctuation fields μ_fluct from the experiments and the DEM artificial radiographs utilizing the software program PIVLab⁵². For the DEM radiographs, the evaluation was carried out utilizing auto distinction, with FFT window deformation because the PIV algorithm. The interrogation width space was set at 64 px with a step of 32 px for the primary go, and a width of 32 px with a step of 16 px for a second go. A Gauss 2 × 3 level for the sub-pixel estimator and a normal correlation robustness setting had been utilized. Similar settings had been utilized for the experimental radiographs, with the addition of a CLAHE filter of 64 px throughout picture preprocessing. For the aspect detector evaluation, a masks drawn on the free floor of the aspect view detector (1) was carried out to disregard all displacements above the free floor. Obtained displacements for detector 1 had been calibrated utilizing peak h₂ and for detector 2 utilizing the identified flume width. The remaining outcomes had been averaged in time with no further filtering.