Vol48/04/2005def 843 ANNALS OF GEOPHYSICS, VOL. 48, N. 4/5, August/October 2005 Key words pyroclastic flow – fluidization – gravi- ty current 1. Introduction Considerable progress has been made over the last decade in mechanistic understanding of the establishment and flow of pyroclastic gravity currents (Druitt, 1998; Burgisser and Bergantz, 2002; Neri et al., 2002b). In particular, the role of gases – and of their interaction with pyroclastic solids – on the establishment of dense versus di- lute suspension currents as a result of explosive volcanism has been highlighted. On one side, dense pyroclastic gravity currents result from limited amount of gas in the eruptive mixture leaving the crater, insufficient to produce a fully convective buoyant column. On the other, the amount of gas entrained by the current in the at- mosphere is largely responsible for its mobility and hazard. The role of fluidization in the emplacement of dense pyroclastic flows has long been recog- nized (Spark, 1976; Wilson, 1980, 1984; Mar- zocchella et al., 1998; Roche et al., 2002; Druitt et al., 2004; Gravina et al., 2004). Notwith- standing, quantitative assessment of fluidiza- tion in pyroclastic flows is still extremely poor. Broad uncertainties still characterize funda- mental aspects of pyroclastic flows, like: a) the prevailing nature and source of the fluidizing gas, either endogenous or associated with en- trainment phenomena; b) the rheology of aerat- ed/fluidized gas-solid two-phase dense flows under high-velocity strongly turbulent condi- tions; c) the perturbation to the classical phe- nomenology of fluidization determined by the highly sheared flow conditions; d) the influence of solids polydispersity on fluidization, segre- gation and flow rheology; e) the expected com- plexity of bifurcation/dynamical patterns of rapid unsteady granular flows on account of the strong nonlinearity of the governing equations. The current lack of fundamental understanding, despite the extensive published literature on Assessment of motion-induced fluidization of dense pyroclastic gravity currents Piero Salatino Dipartimento di Ingegneria Chimica, Università degli Studi di Napoli «Federico II», Napoli, Italy Abstract The paper addresses some fundamental aspects of the dynamics of dense granular flows down inclines relevant to pyroclastic density currents. A simple mechanistic framework is presented to analyze the dynamics of the frontal zone, with a focus on the establishment of conditions that promote air entrainment at the head of the cur- rent and motion-induced self-fluidization of the flow. The one-dimensional momentum balance on the current along the incline is considered under the hypothesis of strongly turbulent flow and pseudo-homogeneous behav- iour of the two-phase gas-solid flow. Departures from one-dimensional flow in the frontal region are also ana- lyzed and provide the key to the assessment of air cross-flow and fluidization of the solids in the head of the cur- rent. The conditions for the establishment of steady motion of pyroclastic flows down an incline, in either the fluidized or «dry» granular states, are examined. Mailing address: Prof. Piero Salatino, Dipartimento di Ingegneria Chimica, Università degli Studi di Napoli «Fe- derico II», P.le V. Tecchio 80, 80125 Napoli, Italy; e-mail: salatino@unina.it 844 Piero Salatino rapid granular flows (Savage, 1979; Campbell, 1990), is largely due to the fact that investiga- tions in this field mostly addressed steady flows of granular materials down inclines at velocities far smaller than those typical of pyroclastic flows. Accordingly, these studies completely missed the complex dynamics of the frontal zone, which is bound to be one key to the onset of fluidization phenomena. Moreover, the rela- tive extent of frictional, collisional, streaming and turbulent stresses, which dictates the rheol- ogy of the flow, does not even approach condi- tions relevant to pyroclastic flows. Due to the intrinsic phenomenological com- plexity and the lack of experimental results and of adequate theoretical frameworks, modelling of dense pyroclastic currents is still at an embry- onic stage and lags far behind the more success- ful and comprehensive modeling of the dynam- ics of particle-lean gas-solid suspensions of py- roclastic nature (ash cloud) (Neri et al., 2002a). The present paper addresses some of the open issues related to quantitative assessment of dense pyroclastic flows from a fundamental per- spective. The dynamics of the frontal zone is considered, with an emphasis on the establish- ment of conditions that promote air entrainment and motion-induced self-fluidization of the flow. An approximate one-dimensional momentum balance on the current steadily flowing along the incline is proposed. It is based on the hypothesis of strongly turbulent flow, of pseudo-homoge- neous behaviour of the two-phase gas-solid flow, embodying some of the features that bring about departures from one-dimensional flow. The con- ditions for the establishemnt of the steady mo- tion-induced fluidization of the pyroclastic flow down an incline are examined. The proposed mechanistic framework pro- vides the starting point for a broader discussion of research needs and priorities essential to fill the gap of quantitative understanding of un- steady rapid granular flows. 2. Basic hydrodynamical and rheological features of a dense pyroclastic flow A dense pyroclastic flow is a two-phase gravity current moving down an incline (whose slope is θ) at a velocity U (fig. 1). The gravity current generates, as it moves, two macroscop- ic shear layers: a free (i.e. unconfined) shear layer at the top of the gravity current; a con- fined shear layer at the bottom of the gravity current. Momentum exchange between the current and the environment takes place via entrain- ment of the surrounding medium at the upper free shear layer. Momentum exchange between the current and the ground takes place via wall friction at the basal shear layer. The rheological behaviour of the two-phase gas-solid flow arises from a combination of the following contributions: yield strength due to interparticle friction; stresses due to collision- al/kinetic and streaming contributions of the particle phase; viscous/turbulent stresses within Fig. 1. Outline of a pyroclastic density current. 845 Assessment of motion-induced fluidization of dense pyroclastic gravity currents the interstitial gas phase; turbulent «Reynolds» stresses at the macroscopic scale. The first contribution is associated with par- ticle-to-particle stresses of frictional nature. The yield strength is generally expressed as a function of the local normal stress within the particle phase according to the Coulomb Law: (2.1) where µ is an internal friction coefficient and θo is the angle of internal friction of the granular solids. The second contribution is due to momen- tum transfer via particle-to-particle impact (col- lisional/kinetic component) or to momentum transfer of convective nature associated with particle velocity fluctuations (streaming com- ponent). The shear stress due to this contribu- tion is usually expressed as (Bagnold, 1954; Patton et al., 1987) (2.2) where (2.3) is the effective viscosity of the gas-solid sus- pension. The third contribution accounts for defor- mation of the interstitial gas phase induced by the flow. This is generally neglected on ac- count of the large value of the Bagnold number for flow condi- tions of practical interest. The fourth contribution arises from the on- set of turbulence over length scales much larg- er than the particle scale. It is a momentum transfer of convective nature but, differently from the streaming component, is related to the motion of particle pockets rather than of indi- vidual particles. Altogether, the intrinsic rheological behav- iour of the two-phase gas-solid system deter- mined by the combined contribution of the above components is that of a Bingham fluid, non-newtonian, shear-thickening. If the flow is in the fluidized state, then µ = θ°= 0 . Accordingly, the yield strength van- /( ( / ) > )Ba d dv dy 1p z g2= t h > C d dy dv dy dv Kf p z z2$ $= =h t c m C d dy dv dy dv ,zy c f p z z2 2 $ $= =x t hc cm m o( )tan,zy y y$ $= =x v n v if ishes. For typical parameters of dense pyroclas- tic flows (h ≅ 1 m; dp ≅ 500 µm; ρm ≅ 100 kg/m3; ρp ≅ 1000 kg/m3; U ≅ 30 m/s; Cf = 0.01 according to Patton et al., 1987) the relevant dimension- less numbers governing the flow of the gravity current take the following values: (2.4) (2.5) where the expression of the Reynolds number holds for a power-law shear-thickening fluid whose viscosity is given by eq. (2.3) (Levenspiel, 1998). The large values of Richardson number suggest that momentum exchange via air entrain- ment at the upper free shear layer is negligible (Britter and Linden, 1979). On the other hand, the very large values of Reynolds number indicate that the flow is dominated by inertia. 3. Momentum balance in the one- dimensional flow approximation The hypothesis is made that the flow of the current can be regarded as steady one-dimen- sional. Furthermore, it is assumed that the cur- rent is in the fluidized state and, according to conclusions drawn in the previous section, mo- mentum exchange is dominated by turbulence. The momentum balance equation reads (3.1) where z is a spatial coordinate along the incline. Equation (3.1) equates the component along the incline of the gravitational force (LHS) to the force exerted on the current by friction at the ground (RHS). Equation (3.1) yields, upon inte- gration (3.2) Both laboratory data on two-phase fully devel- oped turbulent flows (Wallis, 1969) and field data relative to large scale simulated dry dense snow avalanches (Hopfinger, 1983) suggest that .1000 m/s2.p ( ) ( ) .sin sinU f g h h 2 = =i p i ( )sing hdz f U dz 2m m 2 =t i t 10-1Ri ,=:ichardson ( ) R cosg U h a 2t t i∆ :Re Reynolds K h 2 3 10 10 m 2 7 8,= - t 846 Piero Salatino For small slopes and relevant internal fric- tion the literature on dry dense snow avalanch- es (Hopfinger, 1983) suggests an alternative form of eq. (3.2) (3.3) 4. Departure from the one-dimensional flow approximation The one-dimensional framework within which the momentum balance on the gravity current was developed falls short when the de- tailed flow structure of the frontal region is an- alyzed. Phenomenological observation and nu- merical simulation of dense pyroclastic gravity currents highlight the following features of the flow in the frontal region (fig. 2): – Flow elements at increasing distance from the ground move faster, possibly overtak- ing the front. Accordingly 2D and 3D macro- scopic flow patterns are established in the vicinity of the front (Valentine and Wohletz, 1989; Todesco et al., 2002). These patterns in- clude fall-out of coarse solids to the ground and vigorous recirculation of the suspension in the head of the current. ( ) ( ) .sin cosU h= -p i n i6 @ – The fall-out of coarse solids compresses the underlying air, which can be either «squeezed out» from the basal region of the front or forced to percolate through it. – Coupled with the gas flow, finer solids are elutriated through the front contributing to the formation of the upper ash cloud. Fines can also contribute to the momentum exchange be- tween the uprising gas and the coarse solids. In particular, the second feature deserves at- tention in the context of the present theoretical development, on account of the possibility that extensive cross-flow of air is driven by over- pressures which establish at the base of the cur- rent. In turn, cross-flow of air might be large enough to promote fluidization of solids in the head of the current affecting the rheology of the current itself. 5. An approximate criterion for self- fluidization of a dense granular flow moving down an incline It is speculated that, as a consequence of the «average» motion of the front and of the fall- out of the solids to the ground (fig. 3), the basal region of the front experiences a value of pres- Fig. 2. Solids and gas flow patterns in the head a pyroclastic density current. 847 Assessment of motion-induced fluidization of dense pyroclastic gravity currents sure which is approximately given by (5.1) This value is obtained by application of the Bernoulli equation as the stagnation pressure of a flow element initially moving at the front ve- locity U as it is brought to rest at the ground. There is general consensus that dynamic pres- sures in the frontal region of pyroclastic densi- ty currents can assume values ranging from a few kPa up to several tens of kPa (Valentine, 1998; Esposti Ongaro et al., 2002; Nunziante et al., 2003). When the two-phase nature of the flow is considered, it can be assumed that the stagnation pressure pstagnation is shared between the interstitial gas phase pg = αpstagnation and the solid phase ps = (1−α) p stagnation according to a pressure partition coefficient α. In the case α departs significantly from 1, the assumption of an isotropic particle phase pressure is clearly a gross simplification: it is likely that particle pres- sure be largely anisotropic under the strongly sheared flow conditions that characterize the motion of dense pyroclastic gravity currents. The simple idea underlying the self-flu- idization criterion is that the overpressure pg in the gas phase associated with the stagnation condition provides the driving force for the flow of air from the basal region of the front across the current. This cross-flow might be large enough to promote fluidization of the cur- rent. To this end, the usual condition for flu- idization of stationary beds of granular solids is considered: fluidization is established as fluid is supplied at the base of the bed at a relative pres- sure exceeding the buoyant weight of the bed divided by its cross-sectional area. Of course, one should additionally require that fluid be .p U 2stagnation m 2 . t supplied at the base of the front at a flow rate in excess of that corresponding to incipient flu- idization, in order to establish steady cross-flow under self-fluidized conditions: this condition is easily fulfilled if one considers that incipient fluidization velocities of pyroclastic solids typ- ically range between a few mm/s and cm/s (Gravina et al., 2004), whereas the front of the density current progresses at velocities in the order of tens of m/s. By combining these con- cepts, the approximated criterion for the mo- tion-induced self-fluidization of the front is for- mulated as follows: «Self-fluidization of the flow occurs if the basal gas pressure pg in the frontal region of the current exceeds the hydrostatic head imposed by the gas-solids suspension». Or, equivalently (5.2) Equation (5.2) yields, upon rearrangement and simplification of terms (5.3) Equation (5.3) gives the lower limit of the slope θc of the incline beyond which self-fluidized flow of the gravity current may onset. The lim- it value of the slope turns out to be independent of the depth of the current h and of its speed U, which cancel out with each other in eq. (5.2). Remarkably, for given flow conditions θc de- pends on the value of the pressure partition co- efficient α, whose meaning and significance will be addressed later. ( ) ( ) .sin cos g2 1$ ap i i ( ) ( ) .sin cos p U h gh 2 2 1 g m m m 2 $= =a t a t p i i t Fig. 3. Assessment of the stagnation pressure in the basal layer of the head of a pyroclastic density current. Sol- id line: trajectory of a flow element brought to rest as the front is shifted; dashed line: cross-flow of air driven by the stagnation pressure. 848 Piero Salatino It is worth noting that, according to the pres- ent analysis, fulfilment of the criterion given by eq. (5.3) is a necessary but not sufficient condi- tion for the establishment of steady self-flu- idized flow of the density current, which is also dependent on the initial conditions of the flow. More specifically, self-fluidization establishes provided that the initial conditions of the flow correspond to a fluidized state. This require- ment is met, for instance, if the current results from the fall-out of solids (as in the collapse of an eruptive column). 6. Implementation of the criterion and discussion of results Figure 4 demonstrates the application of the self-fluidization criterion expressed by eq. (5.3). The LHS of eq. (5.3) is plotted versus the slope of the incline θ, the partition coefficient α being treated as a parameter. Three values of the pressure partition coefficient α have been considered in the computations: 1, 0.1, 0.01. According to the criterion, regions in the plot where exceeds 1 corre- spond to slopes of inclines on which self-flu- idized motion of the density current may estab- lish. ( / ) ( ) ( )sin cosg2ap i i Analysis of fig. 4 suggests that for the smallest value of the pressure partition coeffi- cient, α = 0.01, self-fluidization is never estab- lished, regardless of the value of θ. When the partition coefficient α is set at 0.1, self-fluidization is established on inclines of slopes exceeding about θc = 13°. Self-fluidization is established at incline slopes of θc = 2° and larger when the upper lim- iting value of α, namely 1, is assumed in the computations. The significance and implications of the self- fluidization criterion are further clarified by fig. 5. The speed U of the density current is plotted in this figure as a function of the slope θ. The speed U is computed according to eqs. (3.2) or (3.3) depending on whether the flow is assumed to be in the self-fluidized state or not, respective- ly. Computations refer to arbitrary (though real- istic) values of the depth of the flow (h = 5 m) and of the angle of internal friction of the gran- ular material (θ 0 = 27°). The latter value corre- sponds to a coefficient of internal friction of the solids µ = 0.5. Again, plots corresponding to three values of the partition coefficient α (namely 1, 0.1, 0.01) are reported in the figure. For α = 0.01, the flow is never self-flu- idized, whatever θ. Steady flow cannot estab- lish on inclines whose slope is smaller than the Fig. 4. Implementation of the self-fluidization criterion. 849 Assessment of motion-induced fluidization of dense pyroclastic gravity currents angle of internal friction θ 0 (U=0). Steady gran- ular flow establishes for θ > θ 0, at a speed U computed according to eq. (3.3). For α = 0.1 and α = 1, steady fluidized flow can be established at slopes larger than θc (where θc = 13° for α = 0.1 and θc = 2° for α = 1). The following flow regimes can be recognized, depending on the slope: θ < θc – Self-fluidization and steady flow of the current cannot be established. Regardless of the initial conditions, the asymptotic condition of the flow is the rest: U = 0. θc <θ < θ0 – Self-fluidization and steady flow of the density current can be established, provided that the initial conditions of the flow correspond to a fluidized state. The speed U is represented, for any given slope, by the upper curve, corresponding to the self-fluidized state of the front. If the initial conditions of the flow correspond to a granular (non-fluidized) state, self-fluidization and steady motion of the cur- rent cannot establish and the flow comes to rest. θ > θ0 – Steady motion of the current is estab- lished anyway. If the initial conditions of the flow correspond to a fluidized state, this state will be self-preserved by the motion of the current. The speed U is given in this case by the upper curve in fig. 5 corresponding to the self-fluidized state. If the initial conditions of the flow correspond to granular (non-fluidized) flow, the flow will keep being granular. The speed U is given by the low- er curve corresponding to the granular state. It is interesting to note that the critical slope θc plays the role of a bifurcation abscissa. Multi- ple steady states may establish for θc < θ depend- ing on the initial conditions of the flow. In partic- ular it is noteworthy that steady flow of the den- sity current can be established even at slopes smaller than the angle of internal friction θ0, pro- vided that the initial conditions of the flow corre- spond to a fluidized state. This finding puts some emphasis on the proper choice of the initial con- ditions in experiments aiming at the characteriza- tion of unsteady granular flows along inclines. Fig. 5. Implementation of the criterion: computation of the front speed. Depth of the current: h = 5 m; angle of internal friction θ 0 = 27°. 850 Piero Salatino Similarly, these arguments shed some doubt on results obtained in experiments where the initial state of the flow was not properly documented (Takahashi and Tsujimoto, 2000). The significance of the pressure partition co- efficient α is better appreciated as one considers its physical meaning in the light of the rheologi- cal behaviour of the two-phase gas-solid system. The limiting case α = 1 corresponds to pseudo- homogeneous two-phase flow (Wallis, 1969). This limiting case is characterized by the impor- tant feature that momentum transfer between the particulate phase and the interstitial gas phase is effective to the point that the slip velocity be- tween the phases is negligible. On the other hand, α = 0 corresponds to the case where the stagnation pressure is entirely withstood by the granular solids. This corresponds to purely gran- ular flow, with an expected strong influence of frictional stresses of Coulombian nature. According to the criterion set by eq. (5.3), on- set of self-fluidization is favoured by large values of α, that is, as pseudo-homogeneous flow condi- tions are approached. On the other hand, estab- lishment of pseudo-homogeneous flow is greatly enhanced by: a) fineness and lightness of the bed material; b) polydispersion of the bed material as regards particle size distribution, as momentum transfer from gas to coarser particles may occur, as in «dusty» gas, via the finer particles; c) sheared flow of the suspension, on account of the property of shear to stabilize homogeneous, i.e. bubble-free, fluidization of solids (Apicella et al., 1997). Features a) and b) are remarkable proper- ties of pyroclastic granular materials. Maybe the recognition of the role of mutual momentum ex- change between the particulate phase and the in- terstitial gas phase in the strongly accelerated flows at hand is the main outcome of the present study. It is recommended that parameters of the two-phase system which may affect interphase momentum exchange be carefully controlled in any experimental study aiming at the characteri- zation of unsteady granular flows along inclines. 7. Conclusions A simple criterion for the establishment of motion-induced self-fluidized flow of a pyroclas- tic density current is presented. It is based on a simplified formulation of the momentum balance equation on the frontal region of the dense flow, on the assessment of overpressures establishing in the basal shear layer of the flow as a conse- quence of solids fall-out, on the application of the criterion for fluidization of beds of granular solids. The criterion highlights the importance of the momentum exchange between the interstitial gas phase and the discrete particulate phase, ex- pressed through a pressure partition coefficient α. This parameter significantly affects the results of the application of the criterion. Its value depends on properties of the two-phase gas-solid system like particle size and density as well as polydis- persity of the population of bed solids. The application of the criterion is demonstrat- ed. It is shown that steady self-fluidized motion of the density current can be established even on slopes smaller than the angle of internal friction of the granular solids – but larger than a critical slope θc which depends on the pressure partition coefficient – provided that the initial conditions of the flow correspond to the fluidized state. The critical slope θc corresponds to a bifurcation point, marking the limit beyond which multiple steady flow conditions, keeping memory of the initial conditions of the flow, may establish. The proposed criterion is admittedly simple and its predictive capabilities have yet to be as- sessed. It might be helpful to shed light on some dynamical features of dense pyroclastic flows and to highlight aspects that need to be taken under control in experiments aiming at the characterization of dense granular flows. Acknowledgements The paper is part of an ongoing research pro- gram on «Dynamics of pyroclastic density cur- rents and their impact on buildings», carried out under the umbrella of AMRA, Centro Regionale di Competenza sul Rischio Ambientale, Regione Campania. The author is grateful to Prof. Lucio Lirer, Prof. Antonio Marzocchella, Prof. Luciano Nunziante, Dr. Paola Petrosino, Dr. Teresita Gra- vina for useful discussion. The author is grateful to Augusto Neri and Olivier Roche for careful and constructive review of the paper. 851 Assessment of motion-induced fluidization of dense pyroclastic gravity currents REFERENCES APICELLA, E., M. D’AMORE, G. TARDOS and R. MAURI (1997): Onset of instability in sheared gas fluidized beds, AIChE J., 43, 1362-1365. BAGNOLD, R.A. 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