Shaking conditions required for flame structure formation in a water-immersed granular medium

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 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.

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 ${\ddot{z}}_{\text{peak}}$ 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 $\delta z^{\prime }\propto {({\ddot{z}}_{\text{peak}})}^{1.04}$, 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 ${\ddot{z}}_{\text{peak}}=20.7$ 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 ${\ddot{z}}_{\text{peak}}=39.6$ 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 ${\ddot{z}}_{\text{peak}}=5.0$ 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 ${\ddot{z}}_{\text{peak}}=40.0\pm 0.8$ 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.

The origin of the flame structure

We consider that the flame structure in our experiments forms as a consequence of Rayleigh-Taylor type instability and not from surface instability, resonance, or granular convection. There seem to be at least three requirements for Rayleigh-Taylor instability to occur: the presence of a permeability barrier, a large fluid viscosity, and granular matter being dense. We explain each of these in detail below.

First, whenever a flame structure developed, it always formed at the permeability barrier. This indicates that the accumulation of water at the permeability barrier is necessary for the flame structure to form, from which we infer that the instability is of Rayleigh-Taylor type.

Second, granular convection did not occur in the water-immersed experiments, whereas it did occur in the dry experiments. On the other hand, in the dry experiments, the small particles comprising the upper layer settled downwards and the flame structure did not form. Here, we evaluate the importance of viscosity by comparing the terminal velocity given by the Stokes velocity V _s (Equation 1) and the free fall velocity $V_{\mathrm{f}}=\sqrt{2(\mathrm{\Delta }\rho /\rho )\mathit{\text{gd}}}$ where the particle diameter scale d is taken as the height scale. The ratio of these two velocity scales is the Stokes number St = V _s/V _f (e.g., Andreotti et al. 2013). If S t ≪ 1, viscous drag is large such that the terminal velocity is attained within a particle diameter scale (viscous regime), and vice versa for S t ≫ 1 (free-fall regime). If we use the particle size d of the upper layer (d = 0.05 mm), for the water-immersed case, we obtain S t ∼ 0.08 < 1. In contrast for the dry case, we obtain S t ∼ 6 > 1. This suggests that the suppression of particle motions by viscosity played a role for the accumulation of water at the permeability barrier.

Third, in our experiments, the particles were initially closely packed, and even when they were shaken, we did not observe any standing surface waves indicating that the particles remained close to a close packed state. Even for the dry experiments, granular matter did not detach from the bottom of the case or bounce (e.g., Eshuis et al. 2007). The thickness of the whole layer scaled by the particle size is N ∼ 270. A large value of N results in a larger frictional dissipation (Eshuis et al. 2007), which causes the granular matter to remain dense even when shaken.

An interpretation of a regime diagram

Here, we consider the origin of the critical acceleration and its frequency dependence. Our experiments indicate that the instability occurs when Γ > Γ_c = 0.58, which we indicate by vertical lines in Figure 7a,b. The critical value Γ_c = 0.58 can be understood in terms of the force balance. The inertial stress of a granular layer is $\sim \rho h{\ddot{z}}_{\text{peak}}=\rho \mathit{\text{Ah}}{(2\pi f)}^2$ where h is the thickness scale. Frictional stress is ∼μ Δρ gh, where μ is the friction coefficient. μ estimated from the angle repose θ _r of a pile of glass beads is μ = tanθ _r ∼ 0.4 to 0.5 (Samadani and Kudrolli 2001; Higashi and Sumita 2009). It follows that the ratio of the inertial stress to the frictional stress is ∼ Γ /μ, where we used the definition for Γ (Equation 3). Inertial stress must exceed the frictional stress to displace a particle from which we expect Γ > μ is required for instability to occur. Our experimental result of Γ > 0.58 for instability to occur is consistent with Γ > μ ∼ 0.4 to 0.5.

We use Equation 6 and draw a critical line in Figure 7a,b for S _c = 0.01, which separates regimes Ib and II fairly well. We note however that the high frequency limit may alternatively be interpreted in terms of critical shaking velocity v _c (Hejmandy et al. 2012), since the critical velocity line also scales as $f\propto {\ddot{z}}_{\text{peak}}/v_{\mathrm{c}}$.

In Figure 7a,b, we draw a line for J _c = 290, which separates regimes Ib and II fairly well. The conditions under which these three dimensionless parameters are supercritical, i.e., Γ > Γ_c, S > S _c, and J > J _c, correspond to the colored domain bounded by the three lines in Figure 7a,b.

Finally, we derive the frequency of approximately 100 Hz at which the critical acceleration is minimum. From combining the conditions S _c ∼ 0.01, J _c ∼ 290, and Γ ∼ 0.22, which is the value of Γ at the intersection of the critical S, J lines (see Figure 7b), we obtain f^′ = f · d/V _s ∼ 3, which corresponds to approximately 100 Hz. We also obtain A ∼ 0.05d which is the amplitude of shaking relative to the particle size comprising the upper layer under this condition.

Comparison with the sedimentary rocks

Our experimental parameters (particle size pair, densities, layer thickness ratios) were not chosen to closely match or scale those of the sedimentary rocks in Figure 1. Therefore, our regime diagram (Figure 7) cannot be directly applied to these outcrops. If one wishes to constrain the exact shaking condition required for the flame structure in these outcrops to form, a similar set of experiments using the same particles from these outcrops need to be conducted. Nevertheless, we consider that the basic features of the regime diagram will remain unchanged.

Here, we point out several features which are common with the flame structures in Figure 1 and in Figure 3. First is that they both have upward pointing cusps. Similar cusps have been previously observed in Rayleigh-Taylor instability of a liquid-immersed granular medium (e.g., Michioka and Sumita 2005; Shibano et al. 2012). In addition, for both cases, the preserved flame structures do not fully penetrate through the upper layer. Second is that in the experiments, even when liquefaction occurs, only the uppermost part of the lower layer seems to be fluidized. We may approximate the situation in which water accumulates beneath the permeability barrier as a thin low viscosity layer with a thickness δ underlying a thick high viscosity layer. Here, we use the results of linear stability analysis (Whitehead and Luther 1975), from which we can constrain δ from the wavelength of instability λ as δ < λ/π. It follows that for λ ∼ 21 mm, δ < 7 mm, implying that only the uppermost part of the coarse lower layer is fluidized. Similarly, for the outcrops, using the wavelength of λ ∼ 40 mm, we obtain δ < 13 mm, suggesting that only the uppermost one fifth or less of the tuff layer (layer 2 in Figure 1) became fluidized. Field observations indicate that the thickness of the tuff layer varies in the range of 20 to 80 mm, whereas the wavelength is found to be around λ ∼ 40 mm. The constancy of the wavelength suggests that the thickness of the fluidized layer is also approximately constant, regardless of the thickness of the tuff layer.