Growth of an instability
Figure
3 and Additional file
1 show a typical example of an experiment in which the instability develops at the two-layer interface and a flame structure forms. We start shaking the case at t = 0 s and find that the heights of the lower and upper layers decrease with time. Initially the two-layer interface remains flat. This indicates that the interstitial water is percolating upwards in a laterally uniform manner, and the granular medium is compacting. After t ∼ 0.5 s, undulations start to develop. The photo at t = 1.6 s in Figure
3a shows the occurrence of sand boils at the surface of the granular layer, indicating an upward channelized flow. Figure
3b is a close-up image at t = 1.6 s, indicating the formation of an instability with a wavelength of λ∼2.7 mm. Since this instability forms at the permeability barrier, we interpret this as the onset of a Rayleigh-Taylor type instability, which occurred as a consequence of the formation of a thin buoyant liquefied layer at the two-layer interface. The amplitude of the instability increases with time (Figure 3d), and undulations merge to form plumes (Figure
3c). However, the amplitude growth is suppressed and the plumes do not fully penetrate through the upper layer, in contrast to the Rayleigh-Taylor type instability of superimposed viscous fluids (Whitehead and Luther
1975). Here, we note that the shape of the flame structure deviates from the sinusoidal shape and is characterized by upward pointing cusps.
Additional file 1:
Movie 1 in QuickTime. An example of an experiment in which a flame structure forms at an acceleration of 40.5 m/s2, a frequency of 40 Hz, and an amplitude of 0.64 mm, replayed at ×0.2 speed. This is the same experiment which was shown in Figures
3 and
4. (MOV 10 MB)
Figure
4a shows how the heights of the granular layer surface and the two-layer interface evolve with time for the same experiment shown in Figure
3. Here, thin vertical sections of the images, which are separated by a time interval of 0.033 s, are aligned horizontally as a function of time. The figure shows that in the initial stage, both the heights of the granular layer surface and the two-layer interface decrease with time as a result of compaction. The instability starts to grow at t ∼ 0.5 s, but the height of the two-layer interface starts to increase at a later time, t ∼ 1.8 s. This is because the instability growth and compaction occur simultaneously, and the height increase of the two-layer interface does not commence until the former exceeds the latter. Figure 4b shows the growth of the interfacial amplitude (δ z) with time which was calculated as follows. We define the height at t = 0 as the reference and subtract it from each height data. We then subtract the linear fit to the data and define the amplitude by its standard deviation. The figure shows that the initial amplitude is δ z ≃ 0.068 mm, which is intermediate between the particle sizes which comprise the upper and lower layers. The amplitude initially increases exponentially with time, but its growth stops at t ∼ 4.3 s. The flame structure remains after we stop shaking at 5 s. Here, we define the relative amplitude δ z′ = δ z-δ z
0, where δ z
0 is the amplitude at t = 0.1 s. Using δ z′, we define the following stages I to III: stage I for δ z′ < 0.1 mm, stage II for 0.1 ≤ δ z′ < 0.6 mm, and stage III for δ z′ ≥ 0.6 mm. The threshold of δ z′ = 0.1 mm is close to the spatial resolution limit of the images (≃0.12 mm). For δ z′ ≥ 0.6 mm, a fully developed flame structure forms. We consider these flame structures fully developed because a peak amplitude can be identified in the amplitude vs time plot during shaking, for most (14 out of 15) of the experiments which transit to stage III. Examples of such peaks are indicated by arrows in Figures
4b,
5b, and
6.
Acceleration and frequency dependence
We next show how the resulting instability changes when the peak acceleration and frequency of shaking are varied. Figure
5a and Additional file
2 show the acceleration dependence at a fixed frequency of 50 Hz. The figure shows that for a small acceleration of 2.1 m/s2, the interstitial water percolates upwards, the granular layer compacts, and the instability ends in stage I. However, for the acceleration of 19.3 m/s2, the instability evolves from stage I to stage II. Finally, for 40.5 m/s2, the instability evolves through stages I and II, and ends in stage III. The growth of amplitudes for the three experiments shown in Figure
5a and two additional experiments at the same frequency is shown in Figure
5b. The figure shows that as peak acceleration increases, both the growth rate (initial slope of the curves in Figure
5b) and the amplitude of the instability increase.
Additional file 2:
Movie 2 in QuickTime. Acceleration dependence at a fixed frequency of 50 Hz replayed at ×0.4 speed. These are four selected examples from those shown in Figure
5b. Accelerations in m/s2 (and the resulting regime) are 2.1 (Ib: percolation), 7.9 (II: transition), 19.3 (II: transition), and 40.5 (III: flame), respectively. (MOV 7 MB)
Figure
6 shows the frequency dependence at a fixed acceleration of 5.1 ± 0.3 m/s2, where the error is given by the standard deviation. Additional file
3 shows four selected examples from the experiments shown in Figure
6. Here, we note that at a constant acceleration, the amplitude (A) of shaking decreases with frequency (f) as A ∝ f-2. Figure
6 shows that at a frequency of 150 Hz, the instability growth is fastest and attains the largest amplitude.
Additional file 3:
Movie 3 in QuickTime. Frequency dependence at a fixed acceleration of 5.0 ± 0.4 m/s2 replayed at ×0.4 speed. These are four selected examples from those shown in Figure
6b. Frequencies in Hz (and resulting regime) are 10 (Ib: percolation), 100 (II: transition), 150 (III: flame), and 3,000 (Ib: percolation), respectively. Note that the flame structure forms at the intermediate frequency of 150 Hz. (MOV 7 MB)
We conducted a total of 73 experiments with different combinations of shaking acceleration and frequency, and classified the results into the following four regimes using the relative amplitude δ z′ and the total compaction of the whole granular layer δ h at t = 9.9 s (i.e., 4.9 s after the shaking stops): regime Ia (no change), when δ z′ < 0.1 mm and δ h < 0.1 mm; regime Ib (percolation), when δ z′ < 0.1 mm and δ h ≥ 0.1 mm; regime II (transition), when 0.1 ≤ δ z′ < 0.6 mm; and regime III (flame), when δ z′ ≥ 0.6 mm. Here, regimes I to III are defined using the same threshold values of δ z′ used to define stages I to III, but at t = 9.9 s. δ h is used to subdivide regime I into regimes Ia and Ib. For all experiments in regimes II to III, δ h ≥ 0.1 mm, and therefore, we do not use δ h to classify these regimes. For the amplitude data shown in Figures
5b and
6, the different resulting regimes are indicated by the line thicknesses.
In Figure
7a, we plot the classified regimes using different marker shapes and δ z′ ranges using different marker sizes (colors), in the parameter space of acceleration and frequency. The figure shows that there is a critical acceleration above which the instability grows and regime Ib (percolation, in square markers) transits to regime II (transition, in triangle markers). In addition, we find that the critical acceleration is frequency dependent and is minimum (3.4 m/s2) at a frequency of approximately 100 Hz.
In Figure
7b, we show the same regime diagram using the dimensionless parameters. Here, we non-dimensionalize acceleration by the reduced gravity (Δρ g/ρ) to obtain a dimensionless acceleration
(3)
where
is the peak acceleration. Similarly, we non-dimensionalize frequency by V
s/d
s(sm) to obtain a dimensionless frequency
Similar non-dimensionalizations have been used previously (e.g., Pak and Behringer
1993; Melo et al.
1995; Duran
2000; Schleier-Smith and Stone
2001; Burtally et al.
2002; Voth et al.
2002; Leaper et al.
2005; Eshuis et al.
2005,
2007). Here, we define the onset of instability when the relative amplitude is δ z′ ≥ 0.1 mm (i.e., onset of the transitional regime). Figure
7b indicates that the minimum acceleration for the instability to occur is Γ = 0.58 at a frequency of f′ = 2.5.
In Figure
8, we plot the relative amplitudes δ z′ at t = 9.9 s, which were used to classify the regimes, as a function of acceleration
for five selected frequencies in the range of 40 to 200 Hz. The figure shows that the transition from percolation to flame regime is continuous. In particular, for the 100 Hz data, we obtain
, with an exponent close to a linear dependence.
Effects of other parameters
In addition to the shaking conditions, there are many other changeable parameters in our experiments. Here, we briefly describe the effects of other parameters which help to better understand the dynamics occurring in our experiments. Here, we define the experiments for the parameters shown in Figure
2 as the reference case.
First, we describe the results of two dry experiments (i.e., without water) and otherwise the same as the reference case. We find that the results are strikingly different. Most importantly, flame structure does not form in the dry experiments. At
m/s2 and f = 50 Hz, the small particles comprising the upper layer settle into the pore space of the large particles comprising the lower layer and formed a mixture layer whose thickness increased with time. Furthermore, the granular layer as a whole gradually tilts. At
m/s2 and f = 50 Hz, in addition to tilting, a large-scale whole layer granular convection with a length scale comparable to the case width occurs.
Second, we describe the results of experiments using a wider case (width 194 mm, height 104 mm, thickness 26 mm) and otherwise the same as the reference case. From a series of experiments at fixed accelerations of
and 8.8 m/s2 and different frequencies, we find that the final amplitude reached a maximum at a frequency of approximately 100 Hz, the same as the reference case. We also find a similar frequency dependence of the critical acceleration and a similar wavelength of the instability of λ ∼ 10 mm, indicating that the effects of width dimension are not evident.
Third, we describe the results of different layer thicknesses. When there is no permeability barrier (i.e., one layer consisting of a particle size d = 0.22 mm with a thickness 21.9 mm), neither sand boil nor flame structure formation occurs. We also compare three experiments with the same thickness 33.8 ± 2.0 mm of the granular layer but with different thickness fractions of the upper layer: 0.19, 0.28, and 0.78. For
m/s2 and f = 50 Hz, for all three cases, a similar flame structure forms. However, we find that the relative amplitude at t = 9.9 s decreases by 35% as the upper layer becomes thicker.