Vibration-induced dynamical weakening of pyroclastic flows: Insights from rotating drum experiments

Vibration induced dynamical weakening of pyroclastic flows:1 Insights from rotating drum experiments2 Jose Manuel Valverde∗, Carlos Soria-Hoyo3 Faculty of Physics. University of Seville. Avenida Reina Mercedes s/n, 41012 Seville, Spain4 (Dated: September 27, 2016)5 Abstract Pyroclastic flows are characterized by their high mobility, which is often attributed to gas-fluidization of the usually fine and/or low density particles. However, the physical mechanism that might drive sustained fluidization of pyroclastic flows over extraordinarily long runout distances is elusive. In this letter it is proposed that a powerful mechanism to weaken the frictional resistance of pyroclastic flows would arise from the prolonged and intense mechanical vibrations that commonly accompany these dense gravitational fluid-particle flows. The behavior of fine powders in a slowly rotating drum subjected to vibrations suggests that fluid-particle relative oscillations in granular beds can effectively promote the pore gas pressure at reduced shear rates. Dynamical weakening, as caused by the enhancement of pore fluid pressure, may be a powerful mechanism in any geophysical process that involves vibrations of granular beds in a viscous fluid. This is particularly relevant for granular flows involving large amounts of fine and/or light particles such as pyroclastic density currents. ∗Corresponding author at: Faculty of Physics, Avda. Reina Mercedes s/n, 41012 Seville (Spain). Phone no. +34 954550960. Fax no. +34 954239434. Email: [email protected] 1 I. INTRODUCTION6 Understanding the dynamics of pyroclastic flows (PF) triggered by the collapse of lava domes7 and explosive eruptions is crucial for hazard mitigation in populated areas around volcanoes. A8 main characteristic of PF is their high mobility leading to astonishingly long runout distances even9 on subhorizontal slopes [3, 8, 12, 33, 34, 36]. PF commonly contain large amounts of fine ash10 (particle size dp∼1-100 µm and density ρp∼2500 kg/m3) [28] and light pumice rock fragments11 (ρp=500 kg/m3,dp∼10 mm) [6] whose dynamics is essentially influenced by the gas-solid hy-12 drodynamic interaction [9, 28, 32–34, 36]. Thus, an excess of pore gas pressure above atmospheric13 pressure may easily lead to a fluidization state in which the drag force exerted by the interstitial14 gas on the particles counterbalances their weight and frictional forces become negligible. In the15 fluidization regime, the granular flow acquires the behavior of a low viscosity fluid lacking any16 resistance to shear. Fluidization has long been considered a key process to explain the enhanced17 mobility of PF [33, 36]. Lab-scale observations on fluidized granular flows in horizontal flumes,18 where a high pore gas pressure is artificially imposed by continuously injecting gas through the19 substrate, demonstrate that a state of sustained fluidization leads to essentially infinite runouts20 [34]. Fluidized PF would propagate as low viscosity fluids over most of their emplacement [33].21 Accordingly, ignimbrite deposits from prehistoric PF show the signatures of negligible friction22 and suppressed turbulence at the depositional boundary layer, which is indicative of a fluidization23 governed dynamics [8].24 Several mechanisms have been proposed as responsible for the enhancement of pore gas pres-25 sure leading to fluidization such as the hindered settling of fine particles from the initially flu-26 idized bed, exsolution of gas from juvenile clasts, engulfing of air at the avalanche front, released27 gas from rough substrates and high shear rates [3, 12, 34, 36]. Yet, it is uncertain whether the28 mechanisms considered so far would be intense enough to sustain fluidization of PF over runout29 distances up to 100 km and across topographic obstacles as inferred from some PF deposits [8].30 Once the pore gas pressure within a finely grained mixture (with typically low hydraulic perme-31 ability and high porosity compressibility) is increased by any mechanism it could be maintained32 for long durations due to retarded pore pressure diffusion, which could explain the considerable33 runout distances of PF [28, 32]. The key question is then what mechanism plays the major role on34 rising the pore gas pressure.35 A further important characteristic of PF is the generation of intense mechanical vibrations usu-36 2 ally by collisions of pyroclasts onto the mountain slope or other obstacles [15, 40]. Seismic signals37 associated with PF are generally distinguished by significantly long durations and large amplitudes38 proportional to the volume of PF. Remarkably, seismic signals of similar features are generated39 by snow avalanches, which also exhibit long runout distances [37]. In this letter we explore the40 possibility that mechanical vibrations excited by the propagation of PF promote their mobility41 by enhancing the pore gas pressure. To this end, we have observed the behavior of two types of42 powders in a rotating drum subjected to mechanical vibrations. The rotating drum setup has been43 already used in previous studies within the context of PF to assess the role of particle size and44 density on fluidization [4] and the generation of ash by abrasion of volcanic rock fragments [21].45 Here, we use the rotating drum setup to shed light on the effect of mechanical vibrations on the46 dynamical behavior of PF.47 II. MATERIALS AND METHODS48 Two diverse types of granular materials have been tested in our work. On one hand, we have49 observed the behavior of cornstarch (particle size dp≃15 µm and density ρp≃1550 kg/m3)50 as representative of an easily fluidizable fine powder. Cohesiveness of the cornstarch powder is51 reduced by mixing it with 0.5% by weight of flow control additive (Aerosilrfrom Evonik) [13].52 When subjected to a gas flow, a bed of this fine powder reaches a highly expanded fluidized state53 characterized by a nonbubbling and liquid-like frictionless behavior [41]. On the other hand, we54 have tested the behavior of glass beads (particle size dp≃100 µm and density ρp≃2500 kg/m3).55 Fluidization of these glass beads is characterized by the development of large gas bubbles just56 beyond the onset of fluidization, which hampers bed expansion. This is the characteristic behavior57 of Geldart B granular materials [18, 41].58 Figure 1 illustrates a schematic representation of the experimental setup. In our work we have59 used a cylindrical Plexiglas drum (4.5 cm internal radius and 2cm depth), which is driven by60 a motor that allows a maximum angular velocity of 100 revolutions per minute (rpm) around61 its horizontal axis. The drum is rotated by a shaft supported on the base of an electromagnetic62 vibration exciter through a pair of bearings. The vibrator is driven by a signal generator that63 provides sinusoidal, vertical vibrations of controlled amplitude ξ1, and frequency f(in the range64 25-200 Hz). The shaft is fitted at the other end to the motor axis by means of an elastic cardan.65 Peak vibration velocity u1=ξ12πfis monitored using a piezoelectric accelerometer. A CCD66 camera interfaced to a computer for image processing records the profile of the powder as affected67 3 by rotation and vibration.68 When a bed of a cohesive powder is tilted the avalanche angle depends generally on the length69 of the slope since cohesion leads to a coherence length of size not negligible as compared to the70 system size for small scale systems [30, 45]. The smaller the length of the slope is, the larger is71 the angle it can sustain. In the case of the glass beads used in our study cohesion is negligible72 and therefore effects on the avalanche angle related to the finite size of the drum are not expected.73 On the other hand, cohesion of the fine powder used in our work has been artificially reduced,74 which serves to minimize the dependence of the maximum slope angle on its length. As reported75 in previous studies [45] for a powder similar to the one used in the present work, the avalanche76 angle of a tilted bed becomes roughly independent of the length of the slope for lengths above77 ≃8 cm. Since the diameter of the drum used here is 9 cm, it may be expected that effects on the78 avalanching behavior due to the finite size of the drum are neither relevant for this material.79 III. RESULTS AND DISCUSSION80 As reported in a previous study [9], the two types of materials used in the present work display81 also contrasting behaviors with increasing drum rotating velocity (see Fig. 2). The fine powder82 is progressively fluidized by the air that becomes engulfed in the powder by each avalanche. The83 extent of fluidization increases with the rotation velocity and the whole bed becomes fluidized at84 a rotation velocity Ω≃90 rpm (Fig. 2(a2)). At this point the granular bed looses any frictional85 resistance to shear. As may be seen in Fig. 2(a2), the gas-solid mixture acquires an expanded86 state with a nearly horizontal slope. Particle volume fraction is decreased from φ≃0.4 in the87 settled (solid) state to φ≃0.2 in the fully expanded (fluidized) state. Using L∼1 cm for the88 characteristic size of avalanches, the shear rate is estimated to vary between ˙ γ∼ΩR/L∼1 s−1 89 at the lowest rotation velocity (Ω≃4.8 rpm) and ˙ γ∼40 s−1in the full fluidization state (Ω≃9090 rpm). According to field observations, the thickness of natural PF is usually on the order of tens91 to several hundred meters whereas PF velocities are in the range ∼10 ms−1-300 ms−1[8, 20].92 Thus, the maximum expected values of shear rate in gas-fluidized PF would be ˙ γ∼10s−1as in the93 recent Merapi eruption where fine ash PF around 10 m thick propagated at velocities of about 12594 m/s [20]. Bearing in mind that comparing the observations of small-scale experimental flows and95 large-scale natural phenomena presents both temporal and spatial scaling problems [8], our results96 suggest that shear rates in PF are not high enough to cause full fluidization of fine powders.97 The behavior of fluidizable fine powders in a rotating drum has been analyzed in detail in98 4 [10]. Experimental results for several materials and using drums of different diameters show that99 the interstitial velocity of the air that continuously escapes from the fluidized powder while the100 drum is rotating scales on average proportionally to the tangential velocity of the drum ui∼αΩR,101 where α∼0.01. At the smallest rotation velocity used in the experiments described in the present102 paper (Ω=4.8 rpm), ui∼0.02 cm/s, which is not large enough as to fluidize the powder. Thus,103 the material displays a plastic behavior. Fluidization starts to occur when the rotation velocity is104 further increased leading to interstitial gas velocities similar to the minimum fluidization velocity,105 which is um f ∼0.1 cm/s for the powder analyzed in this work [13]. Figure 2(a3) shows that106 the addition of vibration to the slowly rotating drum (Ω=4.8 rpm) has a relevant effect on the107 dynamical behavior of the fine powder. As seen when full fluidization occurs at large rotation108 velocities (Fig. 2(a2)), the slowly rotated bed looses progressively its frictional strength as the109 intensity of vibrations is increased. For sufficiently strong vibrations, the angle of the slope drops110 to zero (Fig. 2(a3)) even at the relatively small shear rates ( ˙ γ∼1 s−1) corresponding to the111 lowest rotation velocity. Note however that powder expansion is not observed as in the case of full112 fluidization in the rapidly rotating drum (Fig. 2(a2)).113 In contrast to the fine powder behavior in the rotating drum, the avalanching dynamics of the114 glass beads is determined by inertial stresses [9]. In this case, the average angle of the slope is115 increased with the rotation velocity as the centrifugal acceleration builds up (Fig. 2(b2)). However,116 vibration is seen to weaken the frictional resistance of the glass beads in the slowly rotating drum117 too (see Fig. 2(b3)). The minimum vibration intensity to reduce the angle of the slope to zero118 depends on the vibration frequency and the material. Thus, it is u=4.2 mm/s at f=50 Hz for119 the fine powder (Fig. 2(a3)) whereas, by extrapolating the experimental results (shown below), it120 may be estimated that a vibration velocity of about u≃10 mm/s would be needed to completely121 nullify the frictional resistance of the avalanching glass beads.122 If a powder bed is sheared it yields plastically when the shear stress reaches a critical value123 τc(yield strength) usually related to the normal stress σby means of the Coulomb friction law124 τc=µsσ, where µs=tanθsis the static friction coefficient and cohesion is neglected [2]. After125 failure, the granular bed acquires a state of higher porosity and continues to deform at a slightly126 decreased shear stress τd<τc. The coefficient of dynamic friction µd=τd σis thus somewhat127 smaller than the static friction coefficient. Assuming that µdis a constant which depends only128 on the material, its value may be estimated from the average angle of the slope. In the slowly129 rotated drum and in the absence of vibrations we obtain µd≃tanθd≃0.48 for the fine powder and130 5 µd≃0.54 for the glass beads . Data on the relative variation of the dynamic friction coefficient ∆µd µd 131 (estimated from the average angle of the slope) are plotted in Fig. 3a as a function of the relative132 increase of velocity ∆u u0, being ueither the tangential rotation velocity (in the absence of vibrations)133 or the peak vibration velocity in the slowly rotating drum (Ω=4.8 rpm fixed). Remarkably, the134 data from both type of experiments for the fine powder conform to a common linear trend, which135 suggests that the effect of vibration obeys also to the enhancement of the gas-solid hydrodynamic136 interaction. On the other hand, vibration and rapid rotation yield contrasting effects for the glass137 beads. Inertial stresses prevail in the rapidly rotating drum for the glass beads, which leads to138 dynamical strengthening as the centrifugal acceleration increases. Contrarily, the vibrated rotating139 bed looses progressively frictional resistance as the intensity of vibrations is increased. Thus,140 one might wonder whether gas-solid hydrodynamic interactions might play also a role on the141 dynamical weakening of vibrated granular beds of relatively large inertia particles, which would142 otherwise reach a bubbling fluidization regime only when subjected to large gas-solid relative143 velocities (u&0.5 m/s) [9].144 The question on the role of gas effects on vibrated granular beds of large inertia grains is145 longstanding. The interested reader may find a recent review on this subject in [42]. It dates146 back to Faraday [17] who already observed the onset of convective currents within the bulk of147 thick layers of large inertia sand grains subjected to vertical vibrations. A strong indication of the148 relevant role of gas effects was that convection disappeared when air was pumped out and appeared149 again as the air was readmitted as more recently confirmed by other works [24, 29, 42]. Convective150 currents in a vibrated granular bed give rise to the formation of a surface heap along which particles151 avalanches down to be subducted into the bed at its lowest point. In close analogy with our152 observations, the angle of this slope is smaller than the dynamic friction angle of the material153 (in the absence of vibrations) and decreases further as the intensity of vibrations is increased.154 Moreover, chemical engineering studies have long reported that the gas-solid drag can be notably155 promoted by oscillations even with no net fluid flow [16, 22, 39]. For example, the settling of156 large inertia beads is substantially slowed down by vertical oscillations of the surrounding fluid157 and eventually stopped [16, 39]. The oscillating surrounding fluid generates an additional drag158 that retards settling of the beads and the observed retardation was much greater than that expected159 from the fluid-solid drag under steady conditions. Thus, it seems likely that an enhancement of160 the fluid-particle hydrodynamic interaction due to oscillations in dense gravitational flows could161 lead to dynamical weakening.162 6 Our observations demonstrate that dynamical weakening occurs as the rotating velocity is in-163 creased due to fluidization in the case of the fine powder [10]. Fluidization is caused by the164 increase of pore gas pressure over atmospheric pressure ∆p, which leads to a a reduction of the165 effective friction coefficient µe f with the rotation velocity. The rise of the pore gas pressure acts by166 decreasing the effective normal stress σe f =σ−∆p, where σ=ρpφgL is the powder weight per167 unit area, φis the particle volume fraction and Lis the typical thickness of the bed. Thus, the bed168 would exhibit an effective dynamic friction coefficient given by µe f σ=µd(σ−∆p). The relative169 decrease of µdcan be thus obtained from the pressure drop per unit length: ∆µd µd=−∆p σ=1 ρpφg ∆p L.170 When a fluid flows steadily across a granular bed at low Reynolds numbers, the Carman-Kozeny171 equation [7, 31] applies:172 ∆p L=Eφ2 (1−φ)3 η d2 p us=Λn0FS(1) where usis the superficial gas velocity defined as the gas flow rate per unit area. Eis an empirical173 constant (E≃180 for spheres), dp=2Ris particle size, n0=3φ 4πR3is the number of particles174 per unit volume, FS=6πηRusis the Stokes drag force, ηis the fluid’s dynamic viscosity and175 Λ≃10φ (1−φ)3is the factor that takes into account the hydrodynamic interactions within the bed176 [42]. In the case of a vibrated bed, gas-particle relative oscillations (oscillatory flow) would be177 established instead of a steady flow. The drag force on a sphere of radius Rundergoing oscillations178 in a viscous fluid can be calculated in the limit of either small oscillations amplitude (ξ1 R<1) or179 small Reynolds number (Re1=u1ρR η=ξ1 RR δ2<1) as180 F1(t) = 6πηR1+R δu1(t)+3πR2r2ηρ ω1+2R 9δdu1 dt (2) where u1(t)is the instantaneous oscillation velocity, u1=ξ12πfis the peak oscillation velocity,181 ρis the fluid density, and δ=qη ρω is the thickness of the Stokes boundary layer surrounding the182 sphere across which the fluid flow becomes irrotational [23]. If the interaction between the Stokes183 boundary layers developed around neighbor spheres is neglected, the Carman-Kozeny equation184 can be adapted to oscillatory flows through granular beds [44], which leads to a root mean square185 (rms) pressure drop per unit length186 ∆p0 1 L=Λ ϒ n0F0 S(3) where F0 S=6πηRu0 1,u0 1is the rms oscillation velocity (u0 1=u1 √2)and187 7 ϒ="1+R δ2 +R δ21+2R 9δ2#1/2 (4) Thus, we can write188 ∆µd µd =−βϒu0 1(5) where β=6πηRΛn0 ρpφg.189 For a randomly packed bed (φ≃0.6) of the noncohesive glass beads used in our tests β≃190 120. On the other hand, the analysis of the fine powder is more subtle. Fine cohesive particles191 are usually agglomerated due to the prevalence of the attractive force between the particles as192 compared to their weight [11]. By considering agglomerates as effective spheres, the volume193 fraction filled by the agglomerates φ∗is simply related to the particle volume fraction φby φ∗=194 φk3 N, where Nis the average number of particles per agglomerate and k=d∗ dpis the ratio of the195 agglomerates size to the size of the individual particles. Fluidized agglomerates screen the external196 flow field and can be treated as effective particles of density ρ∗=ρpN k3[46]. Settling tests reported197 elsewhere [43] yield N≃9.1, k≃2.44 for the powder used in the present work. Thus we may198 use d∗≃37 µm and ρ∗≃990 kg/m3for the size and density of the agglomerates as effective199 particles, respectively. The packing density measured for this powder under flow conditions [43]200 is in the range φ≃0.2−0.3, which yields φ∗≃0.3−0.5. Using these numbers, we would predict201 β≃200 −800.202 Experimental data of ∆µd µdversus ϒu0 1for the slowly rotating drum with added vibrations of203 increasing intensity are plotted in Fig. 3b. The best linear fits yield β≃116 for the glass beads204 and β≃285 for the fine powder, which are close to the theoretically expected values as caused205 by an increase of the rms pore air pressure. Data obtained for the fluidizable fine powder in the206 non-vibrated drum with increasing rotation velocity are plotted also in Fig. 3b. In this case ϒ=1207 and u0 1is replaced by the superficial gas velocity us, which is related to the interstitial gas velocity208 uiby us= (1−φ∗)ui. Remarkably, the data fits to the same trend of the vibrated drum data209 for us=0.0065ΩR, which is consistent with the scaling of the interstitial gas velocity with the210 tangential rotation velocity of the drum reported elsewhere (ui∼0.01ΩR) [10].211 As a general comment, it must be noticed that the diameter of the drum used in our experiments212 is small. Ideally, only drums of very large size would allow dismissing any effect of boundary213 curvature on powder flow [4]. However, the application of vibrations to heavy drums would be214 8 technically difficult using our setup. Nevertheless, the experiments carried out in our work serve215 to capture the essential effect of vibrations on dense granular flows, which can be explained from216 a simple physical model where boundary effects are neglected. Additional experimental work217 should be pursued in future studies to address the possible role of surface flow curvature. A key218 issue in the interpretation of our results is that high pore gas pressure was generated within the219 granular media by vibrations, which leads to a reduction of the dynamic friction angle. It should220 be remarked however that pore gas pressure was not measured in our experiments and this would221 be a further interesting subject for future work. On the other hand, recent experimental results do222 show that the pore gas pressure in a granular bed subjected to a gas flow is enhanced when high223 intensity sound vibrations are applied [47]. As a result, the minimum gas fluidization velocity224 is decreased in a similar way that fluidization is observed in our rotating drum experiment at re-225 duced rotation velocities when mechanical vibration is applied. Application of either mechanical226 or sound vibrations would equivalently promote the pore gas pressure due to the enhancement of227 drag by gas-solid relative oscillations. Finally, it must be stressed that vibration induced dynamical228 weakening could be promoted by other physical mechanisms that might prevail in some particular229 situations as proposed elsewhere [1, 24, 26, 27]. For example, according to the acoustic flu-230 idization mechanism proposed by Melosh [26, 27], high-amplitude and high-frequency vibrations231 would be also capable of generating transient mechanical stresses intense enough as to facilitate232 failure under a reduced shear stress in geophysical granular flows (regardless of the presence of233 interstitial fluids) although the consistency of this theory was later questioned [35].234 IV. CONCLUSIONS235 Our observations suggest that vibrations may promote fluidization of dense granular flows by236 enhancing the pore fluid pressure. Thus, the intense and prolonged mechanical vibrations that can237 arise in nature as dense granular flows propagate on terrains with irregular surfaces [48] could238 contribute to the sustained fluidization of these flows over long runout distances. Vibration en-239 hanced fluidization would be a main mechanism specially in the case of pyroclastic flows mostly240 consisting of a matrix of easily fluidizable fine ash and light pumice particles. In particular, if the241 ratio R δis very large, vibrations would lead to a notable enhancement of the rms pore gas pressure242 at small shear rates (by a factor ϒ≃2 9R δ21). This can be the case of large and light pumice243 rock fragments found in ignimbrite deposits of pyroclastic flows with very large runout distances,244 which are easily fluidizable due to their very low density [8] (using R=2 cm, it is ϒ∼100 for245 9