Multiparticle simulation of collapsing volcanic columns and ... - Wiley

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. B4, 2202, doi:10.1029/2001JB000508, 2003

Multiparticle simulation of collapsing volcanic columns and pyroclastic flow Augusto Neri,1 Tomaso Esposti Ongaro,2 Giovanni Macedonio,3 and Dimitri Gidaspow4 Received 27 December 2000; revised 2 August 2002; accepted 22 January 2003; published 17 April 2003.

[1] A multiparticle thermofluid dynamic model was developed to assess the effect of a

range of particle size on the transient two-dimensional behavior of collapsing columns and associated pyroclastic flows. The model accounts for full mechanical and thermal nonequilibrium conditions between a continuous gas phase and N solid particulate phases, each characterized by specific physical parameters and properties. The dynamics of the process were simulated by adopting a large eddy simulation approach able to resolve the large-scale features of the flow and by parametrizing the subgrid gas turbulence. Viscous and interphase effects were expressed in terms of Newtonian stress tensors and gasparticle and particle-particle coefficients, respectively. Numerical simulations were carried out by using different grain-size distributions of the mixture at the vent, constitutive equations, and numerical resolutions. Dispersal dynamics describe the formation of the vertical jet, the column collapse and the building of the pyroclastic fountain, the generation of radially spreading pyroclastic flows, and the development of thermal convective instabilities from the fountain and the flow. The results highlight the importance of the multiparticle formulation of the model and describe several mechanical and thermal nonequilibrium effects. Finer particles tend to follow the hot ascending gas, mainly in the phoenix column and, secondarily, in the convective plume above the fountain. Coarser particles tend to segregate mainly along the ground both in the proximal area close to the crater rim because of the recycling of material from the fountain and in the distal area, because of the loss of radial momentum. As a result, pyroclastic flows were described as formed by a dilute fine-rich suspension current overlying a dense underflow rich in coarse particles from the proximal region of the flow. Nonequilibrium effects between particles of different sizes appear to be controlled by particle-particle collisions in the basal layer of the flow, whereas particle dispersal in the suspension current and ascending plumes is determined by the gas-particle drag. Simulations performed with a different grain-size distribution at the vent indicate that a fine-grained mixture produces a thicker and more mobile current, a larger runout distance, and a greater elutriated mass INDEX TERMS: 3220 Mathematical Geophysics: Nonlinear than the coarse-grained mixture. dynamics; 3230 Mathematical Geophysics: Numerical solutions; 8404 Volcanology: Ash deposits; 8409 Volcanology: Atmospheric effects (0370); 8414 Volcanology: Eruption mechanisms; KEYWORDS: eruption dynamics, collapsing columns, pyroclastic flows, numerical simulation, multiphase model Citation: Neri, A., T. Esposti Ongaro, G. Macedonio, and D. Gidaspow, Multiparticle simulation of collapsing volcanic columns and pyroclastic flow, J. Geophys. Res., 108(B4), 2202, doi:10.1029/2001JB000508, 2003.

1. Introduction [2] Collapsing volcanic columns and associated fountainfed pyroclastic flows are among the most complex pro1 Consiglio Nazionale delle Ricerche, Istituto di Geoscienze e Georisorse, Dipartimento di Scienze della Terra, Universita` degli Studi di Pisa, Pisa, Italy. 2 Dipartimento di Scienze della Terra, Universita` degli Studi di Pisa, Pisa, Italy. 3 Istituto Nazionale di Geofisica e Vulcanologia, Osservatorio Vesuviano, Naples, Italy. 4 Department of Chemical and Environmental Engineering, Illinois Institute of Technology, Chicago, Illinois, USA.

Copyright 2003 by the American Geophysical Union. 0148-0227/03/2001JB000508$09.00

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cesses that can occur during explosive eruptions. (Here we will use the term pyroclastic flow in the more general sense to define a hot and denser-than-air mixture of pyroclasts and gas produced by the collapse of the eruptive column.) This complexity is due to two main characteristics of these phenomena: (1) the multiphase and multicomponent nature of the erupted fluid and (2) the transient and multidimensional features of the process. [3] As far as the first point is concerned, the eruptive mixture typically consists of gas components, liquid droplets of magma and condensates, and solid particles of juvenile and lithic material [Wright et al., 1980; Fisher and Schmincke, 1984; Cas and Wright, 1987]. This mixture of liquid-solid particles dispersed in a gas continuum phase is the result of the magma fragmentation process which

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NERI ET AL.: MULTIPARTICLE SIMULATION OF EXPLOSIVE ERUPTIONS

occurs in the volcanic conduit during magma ascent. Pyroclasts can exhibit a substantial range of sizes from several decimeters in diameter (blocks) down to a few microns (ash) [Walker, 1971; Sparks, 1976; Kaminsky and Jaupart, 1998]. When such a mixture breaks out into the atmosphere, it begins to mix with the surrounding air and its density decreases. If the mixture density at the top of the jet falls below the atmospheric density at the same height, the volcanic column forms a buoyant convective column that continues to ascend into the atmosphere. If, on the other hand, the density of the jet remains above the atmospheric density, the column collapses and forms a fountain and radially spreading pyroclastic flows. In both cases, pyroclasts of different sizes and properties exhibit different dispersal dynamics and produce specific distributions in the deposit. [4] With regard to the second point, collapsing columns are characterized by significant unsteady interactions between the jet leaving the vent and the collapsing stream. These interactions can produce recycling of eruptive material into the jet as well as fountain height oscillations and intermittent feeding of pyroclastic flows [Valentine et al., 1991; Neri and Dobran, 1994]. Similarly, during their propagation away from the vent, pyroclastic flows can produce coignimbritic or phoenix clouds and particle sedimentation and sorting along the flow [Walker, 1971; Sparks, 1976; Rosi, 1996; Druitt, 1998; Freundt and Bursik, 1998]. Most of these processes are indeed characterized by transient and multidimensional dynamics. [5] Since the mid-1970s, theoretical works have been combined with geological studies in an attempt to describe, on a more physical basis, the dynamics of volcanic processes. The development of one-dimensional, steady state, and homogeneous flow models, as well as the carrying out of analogue experiments in the laboratory, were able to describe quantitatively the principal processes occurring during the collapse of the column and the emplacement of pyroclastic flows [Sparks and Wilson, 1976; Sparks et al., 1978; Wilson, 1980; Huppert et al., 1986; Denlinger, 1987; Carey et al., 1988; Anilkumar et al., 1993; Woods and Bursik, 1994; Bursik and Woods, 1996; Dade and Huppert, 1996]. Important mechanisms such as fluidization and sedimentation of particles in the flow as well as entrainment of air were described and recognized as crucial processes in pyroclastic flow dynamics. [6] Parallely, the development of numerical multiphase codes allowed the analysis of new features of explosive volcanism. A two-dimensional, transient, and two-phase flow model was first used to numerically simulate a caldera-forming eruption [Wohletz et al., 1984]. Similar models were developed by Valentine and Wohletz [1989] and Horn [1989] to model Plinian columns and pyroclastic flows. Dobran et al. [1993] extended these models by implementing a two-component description of the gas phase and a kinetic theory description for the dense gas-particle regime [Ding and Gidaspow, 1990]. More recently, Neri and Macedonio [1996] considered nonequilibrium effects between particles of two different sizes, whereas Oberhuber et al. [1998] and Herzog et al. [1998] developed a transient three-dimensional flow model of pyroclastic dispersal on a large scale considering two-sized particles in dynamic equilibrium with the gas phase.

[7] Multiphase flow models are particularly suitable for describing strongly transient and multidimensional nonequilibrium processes involving gas-solid-liquid mixtures such as those commonly produced by explosive eruptions. The model approach is based on the extension of the fundamental Navier-Stokes continuum mechanics equations to a multiphase mixture. According to this approach, gas and solid particles are treated as interpenetrating continua with specific constitutive equations and exchange mass, momentum, and energy with each other. As a consequence, particle segregation and air entrainment into the flow are no longer directly expressed by time- and space-averaged coefficients, but result indirectly from the prescribed transport and constitutive equations. Similarly, elutriation (i.e., the separation of particles according to their size and density when they are subject to the concurrent action of a convective fluid and gravity) is obtained through the solution of the conservation and balance equations described below and is shown by the different spatial and temporal distributions of the particulate phases over the physical domain. Like the above mentioned steady state, one-dimensional descriptions, this type of model has been extensively tested by laboratory experiments and test cases in the last 30 years and there is a wide literature regarding their applications in engineering, physics, and meteorology [Gidaspow, 1994; Crowe et al., 1998; Jacobson, 1999]. [8] Even though multiphase models have clarified some of the multidimensional, unsteady, and nonequilibrium features of collapsing columns and pyroclastic flows, some major simplifications still need to be removed [Dobran, 1993; Sparks et al., 1997; Valentine, 1998; Gilbert and Sparks, 1998; Macedonio and Neri, 2000; Burgisser and Bergantz, 2002]. The aim of this paper is to present an extension of the above mentioned Dobran et al. [1993] and Neri and Macedonio [1996] models so as to be able to describe the thermofluid dynamics of a mixture of generic N different solid particulate phases dispersed in a continuous gas phase. This extension actually tries to remove one of the main limits of previous models, i.e., the consideration of only one or two particle sizes, and to assess the effect of a range of particle size on the dispersal process. [9] Several numerical simulations have been performed in order to assess both the capability of the model to discriminate between different particulate phases and the sensitivity of results to the way constitutive equations have been parameterized. Model results allow the quantification of significant mechanical and thermal nonequilibrium processes occurring in collapsing columns and pyroclastic flows. Some of the most significant include the elutriation of particles from the fountain and the flow, the generation of a dense underflow at the bottom of a diluted suspended current during pyroclastic flow emplacement and the thermal nonequilibrium between gas and particles in regions of the flow characterized by significant entrainment of air.

2. Model Equations and Solution Procedure [10] The model developed, PDAC-2D (Pyroclastic Dispersion Analysis Code, two dimensions), is based on works by Syamlal [1985] and Gidaspow [1994], who developed multiphase flow models for the simulation of industrial fluidization systems, as well as on works by Dobran et al.


[1993] and Neri and Macedonio [1996], who extended that work to the simulation of collapsing volcanic columns. Most of these models derive from Rivard and Torrey [1977], who developed the original K-FIX code used in the analysis of vapor-liquid flow in nuclear reactors. [11] The model describes the injection and dispersal of a hot and high-velocity gas-pyroclast mixture in a steady standard atmosphere. The gas phase can be composed of several chemical components leaving the crater (such as water vapor, carbon dioxide, etc.) and atmospheric air (considered as a single chemical component). The pyroclasts are described by N phases of solid particles, each one characterized by a diameter, density, specific heat, thermal conductivity, and viscosity, and considered representative of a granulometric class commonly present in the eruptive mixture. Particles are supposed to maintain their original size thus neglecting the effect of any secondary fragmentation or aggregation process on the large-scale dispersal dynamics [Kaminsky and Jaupart, 1998; Freundt, 1998]. [12] Model equations are essentially the generalization to N phases of the conservation and constitutive equations solved by Dobran et al. [1993] and Neri and Macedonio [1996]. Additional differences from previous models are the adoption of a more accurate description of the particleparticle interaction and gas-particle heat transfer coefficient and the implementation of a more realistic ground boundary condition. The model formulation is presented in a form consistent with ‘‘model A’’ of Gidaspow [1994], even though ‘‘model B’’ was also adopted in some simulations. The major difference between the two formulations is that model A assumes that the gas pressure acts on both the gas and solid phases, whereas in model B the total pressure drop is only in the gas phase. However, for the simulations performed in this study, the two formulations give very similar results. 2.1. Conservation Equations [13] The mass conservation equations for the gas and the kth phase, k = 1, 2, . . ., N, and for the ith, i = 1, 2, . . ., M, gas chemical component are @ g rg þ r g rg vg ¼ 0; @t

@ ðk rk Þ þ r ðk rk vk Þ ¼ 0; @t

k ¼ 1; 2; . . . ; N ;

@ g rg yi þ r g rg yi vg ¼ 0; @t

i ¼ 1; 2; . . . ; M ;

ð1Þ

ð2Þ

ð3Þ

with g þ

N X k¼1

k ¼ 1;

M X

yi ¼ 1;

ð4Þ

i¼1

where is the volumetric fraction, r is the microscopic density, v is the velocity vector, y is the mass fraction of the ith gaseous component, and t is time. No phase change process or chemical reaction has been considered in the model.

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[14] Similarly, the momentum balance equations for the gas phase and the kth solid phase can be written as @ g rg vg þ r g rg vg vg ¼ g rPg þ rTg þ g rg g @t N X þ Dg;k vk vg

ð5Þ

k¼1

@ ðk rk vk Þ þ r ðk rk vk vk Þ ¼ k rPg þ rTk þ k rk g Dg;k @t N X vk vg þ Dk; j vj vk ; k; j ¼ 1; 2; . . . ; N ; j 6¼ k; j¼1

ð6Þ

where the four terms on the right-hand side of the two equations account for the pressure gradient, viscous effects, gravitational force, and drag forces between the phases (please refer to the notation section for symbol explanation). As regards the kth solid phase, interphase forces are considered with both the gas phase and the remaining solid phases. [15] The energy balance equations for the gas phase and the N solid phases, in terms of enthalpy, can be written as @Pg @ þ vg rPg g rg hg þ r g rg hg vg ¼ g @t @t N X Qk Tk Tg þ r kge g rTg þ

ð7Þ

k¼1

@ ðk rk hk Þ þ r ðk rk hk vk Þ ¼ r ðkke k rTk Þ Qk Tk Tg ; @t k ¼ 1; 2; . . . ; N ;

ð8Þ

where h is the enthalpy, T the temperature, and Qk (k = 1, 2, . . ., N) are the heat transfer coefficients between the gas and the kth particulate phase. It should be noted that heat transfer between different solid phases, as well as viscous dissipation effects, are now neglected due to their secondorder effect in comparison with convection, conduction, and gas-particle heat exchange [Harlow and Amsden, 1975; Valentine and Wohletz, 1989]. This is also justified in the light of the minor influence of the conductive term with respect to the interphase gas-particle term in the right-hand side of equations (8), as proved by simulations performed without the conductive term. The gas and solid thermal conductivities, kge and kke, are representative of effective values related to the subgrid scale turbulence and to the particle-to-particle heat transfer by conductance and radiation, respectively. 2.2. Constitutive Equations 2.2.1. Equations of State and Transport Properties [16] The gas phase was supposed to obey the ideal gas law, whereas particulate solid phases are considered incompressible, i.e., rg ¼

Pg ; ~ g RT

rk ¼ rsk ;

k ¼ 1; 2; . . . ; N

ð9Þ

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~ is the gas constant of the mixture of gaseous where R components and rsk is the constant density of the kth solid phase. [17] The temperature of each phase was derived from its enthalpy as hg Tg ¼ ; Cpg

hk Tk ¼ ; Cpk

k ¼ 1; 2; . . . ; N ;

ð10Þ

where Cpg and Cpk, (k = 1, 2, . . ., N), i.e., the specific heats of the gas and particles, were assumed constant and corresponding to average values due to their minor sensitivity on temperature. The specific heat of the gas phase was computed as a function of the specific heats of the M chemical components which, in turn, were assumed as average values [Reid et al., 1986]: Cpg ¼

M X

yi Cpg;i :

ð11Þ

i¼1

On the other hand, molecular viscosity and conductivity of the gas phase were computed as a function of gas composition and temperature by using the Wilke and the Mason and Saxema methods [Kadoya et al., 1985; Reid et al., 1986]. 2.2.2. Gas-Phase Stress Tensor [18] The gas phase stress tensor is modeled by adopting a turbulent subgrid scale model. This choice is strictly related to the modeling approach we have followed, commonly known as the large eddy simulation (LES) approach [Ferziger, 1977; Mason, 1994]. Typically, turbulence fluctuations in a volcanic column occur on very different length scales, ranging from millimeters up to hundreds of meters. These are often referred to as eddies or vortices. Unlike direct numerical simulation (DNS), which tries to resolve any structure of the flow by the use of a numerical grid spacing finer than the smallest scale of the motion, the LES approach tries to solve only the large structures of the flow leaving small-scale turbulence to be accounted for by a subgrid scale model. The LES is based on the assumption that the Kolmogorov similarity hypothesis applies to subgrid scale fluctuations that are assumed to be statistically homogeneous and isotropic [Ferziger, 1977; Mason, 1994]. The separation between large and small eddies is obtained by applying a filter function, of a given characteristic length, to the transport equations. The filter used in this work is the ‘‘box filter,’’ based on the original work by Deardorff [1970, 1971]. This operation gives rise to an extra turbulent term, which needs to be parameterized. Similarly to the molecular dissipation, the subgrid turbulent dissipation is described by the introduction of an eddy turbulent viscosity, first proposed by Smagorinsky [1963] in weather prediction studies. [19] According to Smagorinsky’s model, the turbulent viscosity is represented in the form 1=2 ~g T ~g mgt ¼ l 2 rg 2tr T ;

ð12Þ

where l represents the subgrid length scale of the turbulent ~ g is the deformation rate tensor. Away from motions and T

the solid boundaries, the subgrid length scale can be related to the grid size by l ¼ lS ¼ cS ;

ð13Þ

where cS is the so-called Smagorinsky’s constant and = (r z)0.5 with r and z representing the radial and vertical sizes of the computational cell. Therefore the complete gas stress tensor is h T i 2 Tg ¼ g mge rvg þ rvg g mg r vg I 3

ð14Þ

mge ¼ mg þ mgt :

ð15Þ

with

The effective gas conductivity appearing in equation (7) is accordingly determined through the turbulent Prandtl number, Prt, (here assumed to be 0.5), as kgt ¼

Cpg mgt ; Prt

kge ¼ kg þ kgt :

ð16Þ

An additional comment is required regarding the Smagorinsky’s constant, cS, which appears in the expression of the subgrid length scale. Its value, based on theoretical and experimental studies, has been found to vary between 0.1 and 0.2 according to the specific system investigated [Deardorff, 1971; Mason and Callen, 1986]. Lower values have been found more appropriate for systems where turbulence is produced by the mean shear, whereas higher values perform better where turbulence is produced by thermal instabilities. In the present study we have adopted a values of 0.1 in all simulations even though a value of 0.15 has also been used in one case to test the sensitivity of results to this parameter. The relevance of the subgrid scale model has also been investigated performing a simulation without turbulent dissipation (see section 3.1). [20] More sophisticated models, based on the original Smagorinsky model adopted here, have also been developed in recent years. For instance, Germano et al. [1991] developed a dynamic subgrid scale eddy viscosity model where Smagorinsky’s constant is dynamically computed according to the local properties of the flow pattern, whereas Mason and Thomson [1992] investigated the backscattering of energy from small to large scales. However, despite its relatively simple turbulence description, Smagorinsky’s model proved able to describe the principal effects of turbulence at high Reynolds numbers, as shown by the successful simulation of the turbulent atmospheric boundary layer [Mason, 1994; Hobson et al., 1999] and of several engineering flow systems [Deardorff, 1970; Mason and Callen, 1986; Ding and Gidaspow, 1990]. 2.2.3. Solid Stress Tensor [21] A rigorous representation of the viscous effect in the solid phases can be obtained by the application of kinetic theory for dense and dilute flows [Lun et al., 1984; Gidaspow, 1994]. However, the application of this theory to a mixture of N different particulate phases has not yet been sufficiently developed and validated. For this reason, in the present study, the stress tensor of the kth particulate

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phase was simply described in terms of a Newtonian viscous component and a Coulombic repulsive component. The tensor can be expressed by Tk ¼ Tv;k tc;k I;

k ¼ 1; 2; . . . ; N ;

ð17Þ

4-5

whereas for g < 0.8, the famous Ergun [1952] equation was implemented:

Dg;k

2k mg k rg vg vk ¼ 150 þ 1:75 ; dk g dk2

k ¼ 1; 2; . . . ; N ð23Þ

where the viscous tensor Tv,k is Tv;k

where the particle Reynolds number is defined as

1 1 T ¼ 2mk rvk þ ðrvk Þ ðr vk ÞI ; 2 3

k ¼ 1; 2; . . . ; N

ð18Þ

and the Coulombic component gradient is defined by rtc;k ¼ G g rk ; G g ¼ 10ag þb

g rg dk vg vk Rek ¼ ; mg

k ¼ 1; 2; . . . ; N

N m2

ð19Þ

where G(g) is a solid elastic modulus able to account for repulsive forces when low void fraction values are reached in the mixture and to make the numerical problem wellposed [Gidaspow and Ettehadieh, 1983; Bouillard et al., 1991]. In the present model we assumed a = 8.76 and b = 5.43 [Gidaspow, 1994]. [22] As regards the solid viscosity value to be used in the viscous tensor, semiempirical correlations based on experimental work and on the above mentioned kinetic theory studies were adopted [Miller and Gidaspow, 1992; Gidaspow and Huilin, 1996]. In detail, particulate viscosities were expressed as mk ¼ ck k ;

ðPa sÞ;

k ¼ 1; 2; . . . ; N ;

ð20Þ

g k rg vg vk 2:7 3 Dg;k ¼ Cd;k g ; dk 4

Cd;k ¼

24 ; 1 þ 0:15 Re0:687 k Rek

Cd;k ¼ 0:44;

Rek 1000;

k ¼ 1; 2; . . . ; N

ð21Þ

ð24Þ

The accuracy of both correlations is more than satisfactory as proved by their many successful applications and as clearly reported in the original papers and discussed by Bird et al. [1960] and Crowe et al. [1998]. 2.2.5. Particle-Particle Drag Coefficient [25] As outlined above, interactions between particles of different sizes results in the introduction of a particleparticle drag coefficient between each pair of solid phases. Consideration of this term was proved to be important in several laboratory experiments [Soo, 1967; Arastoopour et al., 1980, 1982; Arastoopour and Cutchin, 1985; Aldis and Gidaspow, 1989; Anilkumar et al., 1993]. Such a contribution was already included by Neri and Macedonio [1996] by adopting the expression of Nakamura and Capes [1976]. In this study, we implemented an extension of the correlation of Syamlal [1985]. Such a correlation has the following form: Dk; j

2 dk þ dj vk vj ; ¼ Fkj að1 þ eÞrk k rj j 3 3 rk dk þ rj dj

k; j ¼ 1; 2; . . . ; N ;

where the constant ck, relating the particle viscosity with the volume fraction, ranges between 0.5 and 2 (larger values apply to the coarser particles). [23] It is worth noting that as emerges from the above reported descriptions of turbulence in the gas and solid phases, the model does not consider any direct interaction between the two formulations, even though it has been proved that in specific flow regimes and particle concentrations, particles can significantly affect gas turbulence and viceversa [Eaton and Fessler, 1994; Crowe et al., 1998]. 2.2.4. Gas-Particle Drag Coefficient [24] The expression of the drag coefficient between the gas and the kth (k = 1, 2, . . ., N) particulate phase, Dg,k, consists of two well-established semiempirical correlations for dilute and dense regimes. For g 0.8, the drag expression by Wen and Yu [1966] was adopted, i.e.,

k ¼ 1; 2; . . . ; N :

j 6¼ k;

ð25Þ

where a is an empirical coefficient accounting for nonhead-on collisions, e is the restitution coefficient for a collision, and Fkj is a complex function of the volume fraction of the two phases and of the maximum volume fraction of a random closely packed mixture, kj, i.e., 1=3 1=3 3kj þ k þ j Fkj ¼ 1=3 ; 1=3 2 kj k þ j

ð26Þ

where, for Xk k/[k + (1 k)j], k þ 1 j k kj ¼ k j þð1 aÞð1 k Þj Xk þ j ; k ð27Þ

and, for Xk > k/[k + (1 k)j], kj ¼ ð1 aÞ k þ ð1 k Þj ð1 Xk Þ þ k

ð28Þ

with Rek < 1000 ð22Þ

sffiffiffiffiffi dj ; a¼ dk

dk dj ;

Xk ¼

k k þ j

ð29Þ

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Figure 1. Schematic representation of the computational domain and boundary conditions employed in the simulations. The enlarged computational cell shows the presence of a gas continuous phase and N different particulate phases each characterized by its own concentration, velocity, and temperature (modified from Dobran et al. [1993]). and k, j, representing the solid volume fraction at maximum packing in a single particle system for phase k and j, respectively (here assumed to be 0.63 for all phases). [26] Despite its apparent complexity (we refer to the original work for its complete derivation), the correlation employed is based on a simple momentum balance between particles and has been shown to describe correctly particleparticle interactions occurring in fluidized and dense particle systems [Gidaspow et al., 1986; Syamlal and O’Brien, 1988; Aldis and Gidaspow, 1989]. It should also be noted that as dilute conditions are approached, i.e., k and j 1, the coefficient Fkj tends to 3/2, i.e., the Syamlal correlation reduces to the Nakamura and Capes [1976] expression, making equation (25) applicable in both dilute and dense conditions. Sensitivity tests performed on the effect of the restitution coefficient (varying between 0.5 and 1) and a (between 1 and 0.8) have shown a limited influence of these parameters on the large-scale dispersal dynamics. 2.2.6. Gas-Particle Heat Transfer Coefficient [27] The heat transfer between the gas and the solid phases in equations (7) and (8) is written in standard form as the product of a transfer coefficient, Qk, and a driving force, (Tk Tg). Qk represents the volumetric interphase heat transfer coefficient which equals the product of the specific exchange area and the fluid-particle heat transfer coefficient expressed in terms of the Nusselt number Nuk. As for the drag, the heat coefficient was generalized to the N particulate phases as Qk ¼ Nuk ¼

6kg k ; dk2

k ¼ 1; 2; . . . ; N

ð30Þ

where the Nusselt number depends on particle concentration and Reynolds and Prandtl numbers. In this study, such a dependence has been described by the following empirical correlation of Gunn [1978]: 1=3 1=3 þ 0:13 þ 1:22k Re0:7 Nuk ¼ 2 þ 52k 1 þ 0:7Re0:2 ; k Pr k Pr Rek 105

ð31Þ

with rg dk vg vk Rek ¼ ; mg

Pr ¼

Cpg mg ; kg

k ¼ 1; 2; :::; N :

ð32Þ

With respect to previously employed correlations [e.g., Dobran et al., 1993; Neri and Macedonio, 1996] based on a set of different expressions related to different experimental works [see Zabrodsky, 1966], Gunn’s correlation can be applied to both dilute and dense particulate mixtures and is able to correlate experimental data up to the Reynolds number of 105. 2.3. Initial and Boundary Conditions [28] At time t = 0, a standard atmosphere, vertically stratified in pressure and temperature, and therefore in density, was considered throughout the domain. The atmosphere was composed of dry air and no particle of any size was considered present on the computational domain. At time t > 0, appropriate boundary conditions has to be defined (see Figure 1). First, the volumetric fraction, veloc-


ity, and temperature of each phase considered to enter the domain have to be imposed at the vent. Gas pressure and composition need to be specified as well. One half of the volcanic vent, of diameter Dv, is located in the lower lefthand corner of the physical domain. As will be further discussed, vent conditions were assumed to be constant in time and mechanical and thermal equilibrium between the phases was assumed. Such assumptions can easily be removed once vent conditions are specified as a function of the magma ascent dynamics along the conduit [Wilson et al., 1980; Dobran et al., 1994; Papale et al., 1998; Neri et al., 1998; Clarke et al., 2002]. [29] At the left-hand side boundary a symmetry axis is present. Such a condition imposes null radial gradients of all dependent variables at R = 0 and, of course, no mass flux through it. Such a condition is certainly an idealization, since collapsing columns are characterized by a strong interaction between the jet leaving the vent and the collapsing stream thus leading to strongly transient and threedimensional dynamics (an example of the effect of the symmetry axis on the modeling of a gas-particle coreannulus flow along the riser of a fluidized bed is given by Neri and Gidaspow [2000]). However, the two-dimensional axisymmetric assumption represents, at this time, a good trade-off between the required computational resources and a realistic description of the process. [30] At the ground boundary, no mass and heat transfer are allowed and no-slip conditions are assumed for the radial velocity of each phase. However, in order to account properly for the presence of ground and due to the significant effect of turbulence parameterization on the large-scale dynamics of the process, the gas turbulence model has been improved, with respect to previous models. In fact, at a distance z from the ground rigid boundary, the length scale of the subgrid motions is limited by the distance z itself. A common relationship usually applied for high Reynolds flows is the so-called ‘‘law of the wall,’’ according to which the characteristic Prandtl mixing length of the turbulent motion, l, becomes l ¼ lB ¼ kð z þ z0 Þ

ð33Þ

where k is the von Ka´rmań constant (here assumed to be 0.4), z the distance from the surface, and z0 the roughness length. Equations (13) and (33) were then matched by the commonly adopted equation [Mason, 1994; Hobson et al., 1999] 1 1 1 ¼ þ : l 2 lS2 lB2

ð34Þ

Wieringa [1993] presents a complete set of values of roughness length coming from field data and valid for many terrain types. In this study we assumed, in all simulations, a roughness value of 1.0 m, which is representative of a homogeneous pine forest or a regularly built town. For such terrain types, the top of the roughness elements acts as the effective plane, usually called the ‘‘aerodynamic ground plane.’’ Such a plane is actually displaced from the actual ground plane by a distance d, called displacement height, of the order of the average height of the roughness elements (5 – 10 m in our case). Therefore the bottom boundary

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represented in our simulations has to be interpreted as the aerodynamic ground plane of the terrain type considered, whereas the flow dynamics within the interfacial layer, i.e., the layer between the actual ground plane and the aerodynamic ground plane, can not be explicitly resolved on the scale of our model and using a two-dimensional representation. Such an approach has been successfully adopted for the description of the basal layer of atmospheric boundary layers [Mason, 1994; Hobson et al., 1999], as well as of turbulent channel flow [Deardorff, 1970; Mason and Callen, 1986]. A more complete discussion of the soil boundary condition in the case of flow emplacement over a urban area is reported by Todesco et al. [2002] and Esposti Ongaro et al. [2002]. [31] At the upper and right-hand side boundaries, continuous free outflow/inflow conditions were assumed. In the case of outflow, mass and momentum equations of the mixture were solved for pressure, assuming a null velocity gradient along the boundary frame. As regards inflow streams, they were presumed to consist of standard air at conditions typical of the inflow altitude. The influence of these boundary conditions on the global dynamics of the process proved to be very minor. 2.4. Numerical Resolution and Computational Parameters [32] The model equations described above were numerically solved by a first-order finite difference algorithm according to the implicit multifield (IMF) scheme developed by Harlow and Amsden [1975] for the solution of multiphase flow systems. The numerical scheme adopted treats continuity equations and pressure and drag terms in the momentum equations implicitly, whereas stresses and convective terms are treated explicitly. These features of the scheme allow for calculations in all flow speed regimes as well as for all degrees of coupling between the phases. The energy balance equations for the enthalpies of the (N + 1) phases are subsequently solved fully explicitly. The implicit nonlinear coupling between the momentum and continuity equations is solved by using an iterative Gauss-Seidel algorithm with overrelaxation. Pressure is then corrected at each step by using a combination of nonlinear solvers until the prescribed mass residual is achieved in all cells [Rivard and Torrey, 1977; Syamlal, 1985]. [33] In the performed simulations, different nonuniform grid sizes, computational domains, and time steps were employed. As shown in section 4, parametric studies were carried out on these computational variables in order to illustrate the dependence of results on them and to guarantee the robustness of the main conclusions. Simulations were typically performed on a nonuniform axisymmetric grid extending up to 20 km in both radial and vertical directions. In the radial direction, the cell size starts at 20 m and thereafter slowly increases until it reaches a cell width of 150 m, whereas vertically the typical cell size distribution starts at 4 m and increases up to 100 m. A minimum time step of 0.02 s was employed.

3. Simulations Performed and Results [34] Simulations were performed in order to investigate the effects of the multiparticle formulation of the model on

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Figure 2. Grain-size analyses of flow and surge deposits of the Upper Bandelier Tuff (modified from Cas and Wright [1987]). the dispersal process and to assess the robustness of results with respect to some physical and numerical parameters. Special emphasis is placed here on the analysis of two simulations, named simulation PF-1 and PF-2, characterized by a different particle size distribution at the vent, but with constant vent diameter, mass flow rate, and water content. 3.1. Definition of Input Data [35] As outlined above, vent conditions can be deduced from the modeling of the magma ascent along the volcanic conduit. However, in the present study, in order to limit the number of variables involved and to highlight the net influence of grain-size distribution, vent conditions were fixed following a simplified approach, where total mass flow rate leaving the vent, gas content and composition, and vent diameter are chosen as independent variables. Further independent variables are the weight fractions and densities of the N particulate phases considered. As a consequence, by using the mass conservation equation for each phase and assuming a pressure-balanced jet with all phases in mechanical and thermal equilibrium at the vent, the exit velocity and the volume fraction of each phase can be computed. These assumptions on vent conditions do not limit the generality of the model [see Neri et al., 2002]. [36] The choice of the number and size of the particulate phases considered was also the result of a compromise between a realistic description of the eruptive mixture and some physical modeling limitations. Stratigraphic studies on pyroclastic flow deposits indicate that pyroclast sizes can range from a few microns up to several decimeters [Walker, 1971; Sparks, 1976; Kaminsky and Jaupart, 1998]. As an example, Figure 2 shows two typical grain-size analyses of gravity currents from samples of the Upper Bandelier Tuff [Cas and Wright, 1987]. Figures 2a and 2b show the granulometric and component distribution of a pyroclastic flow in a strict sense and a pyroclastic surge of that sequence, respectively. As a general qualitative rule, pyroclastic flows are usually poorly sorted and polymodal, while pyroclastic surges are usually fine-rich and quite well-sorted [Walker, 1971; Sparks, 1976; Cas and Wright, 1987; Druitt, 1998]. In the case reported, almost all pyroclasts are of

submillimeter size, even though a significant coarse tail up to about 2 cm is present in the flow distribution. Many similar distributions are reported in the volcanological literature showing similar grain-size and component histograms for flow and surge deposits [Walker, 1971; Sparks, 1976; Sigurdsson and Carey, 1989; Rosi, 1996]. [37] From the modeling point of view, the continuous grain-size distribution observed in nature needs to be discretized and described by using a manageable number of solid phases. A large number of phases directly affects the number of transport equations to be solved and the computational time required for their solution. The particle sizes employed in the simulations also need to satisfy some modeling constraints. As an example, some of the above described constitutive equations, i.e., equations (20) and (25), have been obtained and calibrated in the laboratory by using maximum particle size of a few millimeters. In addition, we will see that particle sizes greater than a few hundreds of microns rapidly tend to concentrate in a basal layer of the flow thus requiring, in principle, the use of finer grids and therefore, again, larger computational resources. [38] In the light of all these considerations, the two granulometric compositions, named GC1 and GC2, used in simulations PF-1 and PF-2, respectively, were described by using six different solid phases, each characterized by given diameter, density, and viscosity. Table 1 summarizes the properties employed in each particulate phase. In detail, a diameter of 32 mm (Ø = 5 where d = 2Ø) was assumed for the finest particle phase, phase 1, whereas the coarser phase, phase 6, was assumed to be 1500 mm particles (mean diameter between Ø = 1 and Ø = 0). Phases 2, 3, 4, and 5 have intermediate diameters representative of the most common granulometric classes found in pyroclastic flows and surges (Ø = 0– 4). Density values corresponding to each phase were supposed as average values between the components and estimated from common pumice densities data [Wilson and Huang, 1979] and by assuming a 10 wt % of lithics plus crystals in each phase. As outlined in section 2.2, solid viscosities were estimated on the basis of the experimental and theoretical work by Miller and Gidaspow [1992] and Gidaspow and Huilin [1996]. For simplicity, in


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Table 1. Employed Properties of the Six Solid Particulate Phasesa Phase 1 2 3 4 5 6

dk, mm

rk, kg m3

ck (Equation (20)), Pa s

2100 2100 1800 1100 1100 1100

0.5 0.5 1.0 1.0 2.0 2.0

32 95 190 375 750 1500

tT, s

tV, s 6.5 5.7 2.0 4.7

103/2.6 103 102/2.3 102 101/7.7 102 101/1.8 101 1.9/7.4 101 7.5/2.9

8.6 7.6 2.6 6.2

103/2.9 103 102/2.6 102 101/8.7 101 101/2.1 101 2.5/8.3 101 9.9/3.3

a Specific heat equal to 1255 (J kg1 K1) and thermal conductivity equal to 2.2 W (m K)1 were assumed for all six phases. The two estimates of the mechanical and thermal response times, tV and tT, refer to 300 and 1200 K, respectively, and are valid for low Reynolds numbers.

this work, only three values of the constant ck of equation (20) were assumed. Table 1 also reports the mechanical and thermal response times of the six particulate phases at 300 and 1200 K [Crowe et al., 1998]. These two variables indicate the time required for a particle to react to a change in velocity and temperature, respectively, and give a first approximation of the expected coupling with the gas phase. It is clear from Table 1 that 32 mm particles are indeed mechanically and thermally coupled with the gas, whereas 1500 mm particles require several seconds to equilibrate with the continuous gas phase. It is also interesting to note that response times are greater at low temperature due to the lower values of gas viscosity and conductivity. [39] The weight fraction of each solid phase of the two granulometric distributions is shown in Table 2. The first distribution, GC1, assumes the same weight fraction for all the six phases, whereas the second one, GC2, assumes that the mixture is richer in fine particles (with 50 wt % of particles finer than 100 mm). From the grain-size point of view, the two distributions could be considered as representative of flow-like and surge-like grain-size distributions if we associate to the former term a coarser and more uniform distribution and to the latter a good sorting and a higher content of fine pyroclasts (as a consequence, in this contest, the two terms are not related to the generation mechanisms of the density current). [40] As far as the other input variables are concerned, such as vent diameter, gas content, mass flow rate, and temperature, they were chosen as representative of mediumsize Plinian eruptions and will not be discussed further here [Sparks and Wilson, 1976; Sparks et al., 1997]. A summary of vent conditions is shown in Table 3.

the continuum transport equations in such a wide concentration range is still correct since the Knudsen number, defined as the ratio of the mean free path between particles to the macroscopic length scale of the process, is always much less than one [Gidaspow, 1994; Rosner and Papadopoulos, 1996]. [42] After 50 s from the beginning of the eruption, the volcanic jet has lost its vertical thrust and, at a height of about 1.2 km, starts to collapse. At this point of time, the collapsed portion of the column forms a radially spreading pyroclastic flow and a recirculating stream able to recycle some material into the main flow leaving the vent. At 200 s (not shown in Figure 3), the pyroclastic flow has reached a distance of about 8 km from the vent and a buoyant plume, with particle volume fraction above 103, forms above the fountain. At about 400 s, a continuous elutriation of fine particles from the upper part of the pyroclastic flow is observed. At 500 s (not shown), a main phoenix column starts to form at about 8.5 km from the vent. At this point the pyroclastic flow has reached its maximum distance from the vent of about 12.5 km. After 600 s, the phoenix column has reached a height of about 5 km and slowly moves toward the ascending plume above the fountain. The velocity field at this time clearly illustrates the complex largescale circulation pattern of the dispersal process. At about 650 s, the ascending phoenix column and the central plume are almost joined, whereas the pyroclastic flow head has stopped at about 12.5 km from the vent.

3.2. Simulation PF-1: Coarse-Grained Case [41] Simulation PF-1 is characterized by the granulometric composition GC1 and other vent variables as shown in Table 3. Figure 3 illustrates the distribution of the total particle volumetric fraction and the associated gas flow field in the atmosphere at different times from the beginning of the eruption. Values range from a minimum of 108 (with 107 to 108 approximately corresponding to the visible boundary of the cloud [Horn, 1989]) to a maximum exceeding 101 in the denser part of the flow. The application of

Independent Variables 3 108 500 0.018 GC1 1.0 1100

3 108 500 0.018 GC2 1.0 1100

Dependent Variables 141 10.8 0.992478 0.000846 0.000846 0.000987 0.001614 0.001614 0.001615

141 10.8 0.993366 0.001270 0.001270 0.001185 0.001454 0.000970 0.000485

Table 2. Weight Fractions of the Six Solid Phases for the Two Granulometric Compositions Used in the Simulations dp, mm GC1 GC2

Phase 1

Phase 2

Phase 3

Phase 4

Phase 5

Phase 6

32 0.167 0.25

95 0.166 0.25

190 0.166 0.20

375 0.166 0.15

750 0.166 0.10

1500 0.166 0.05

Table 3. Flow Conditions at the Volcanic Vent for the Performed Simulationsa Variable

m_ v , kg/s Dv , m Yg,v Granular composition yH2 O;v Tv , K vv , m s1 rm,v , kg m3 g,v 1,v 2,v 3,v 4,v 5,v 6,v

PF-1

PF-2

a At the vent, the volcanic jet was considered to be pressure-balanced (Pg = 0.1 MPa) and the gas and solid phases are assumed to be in thermal and mechanical equilibrium (vg,v = vk,v = vv , Tg,v = Tk,v = Tv , k = 1,2,. . .,N ).

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Figure 3. Distribution of the total particulate volumetric fraction and associated gas velocity field in the atmosphere at 50, 400, and 600 s of simulation PF-1. The color contour levels shown are the exponents to the base 10 and correspond to 8, 7, 6, 5, 4, 3, 2, and 1, as reported in the legend. [43] A clear description of the multiphase nature of the dispersal process is given in Figure 4, which shows the distribution of the solid volumetric fraction and the associated velocity field of each particulate phase at 600 s. From Figures 4a –4f it is evident how particle distribution dramatically changes as a function of particle diameter. At 600 s, particles of 32 mm (Figure 4a) are present, with volume concentration above 105, both in the ascending plume above the fountain and in the phoenix column. Their tendency to follow the hot buoyant gas is clearly due to the

strong mechanical and thermal coupling with the gas phase. Similar behavior is observed also for the 95 mm particles (Figure 4b), but with a lower particle concentration in the ascending plumes. With particles of 190 mm (Figure 4c) this trend starts to change. In fact, these particles are almost absent in the phoenix column, but they still show a significant concentration in the ascending plume above the fountain. This tendency is even more evident for 375, 750, and 1500 mm particles (Figures 4d, 4e, and 4f, respectively), which are basically absent in the phoenix column, but have


a volume concentration of about 105 in the central plume. As further discussed in section 4, these coarse particles mostly concentrate in a basal ground layer of the flow forming a highly concentrated granular mixture. [44] A comparison between the gas velocity and the velocity of each particulate phase allows also the quantification of the slip velocity, i.e., the velocity vector difference between the gas and particle velocities. Typical modulus values computed within the pyroclastic flow range between a few centimeters per second, for 32 mm particles, up to 2 – 3 m/s, for 1500 mm particles. Similarly, at the top of the fountain, slip velocities range in the interval 0.1 –6 m/s with the larger values applying to coarser particles. It is worth noting that these slip velocity values are significantly smaller than the settling velocities of particles that can be roughly estimated as the mechanical response time, reported in Table 1, times the acceleration of gravity. This is due both to the influence of particle concentration in the gas-particle and particle-particle drag terms (equations (21) – (23) and (25), respectively) and to the consideration of viscous and pressure terms in the momentum equations (equation (5)). [45] In addition to mechanical nonequilibrium effects that generate the above described particle distributions, thermal nonequilibrium effects also occur. Figure 5 shows the distribution of the temperature difference between particles of different sizes and the gas phase at 100 s from the injection of the mixture. Specifically, Figures 5a, 5b, and 5c refer to 32, 190, and 1500 mm particles, respectively. Figure 5 clearly illustrates that 32 mm particles are practically in thermal equilibrium with the gas phase, whereas 1500 mm particles exhibit a significant thermal disequilibrium (up to 80C in some regions). Such a temperature difference between coarse particles and gas appears to be concentrated in specific portions of the system, specifically the outer and inner edges of the collapsing stream forming the fountain, the pyroclastic flow head, and the upper layer of the density current. These regions are characterized by a very effective air entrainment that produces a temperature drop in the gas phase and therefore interphase thermal disequilibrium. The difference between fine and coarse particles can be clearly explained in terms of thermal response time, which is in the order of a few ms for 32 mm and several seconds for 1500 mm (see Table 1). The behavior of 190 mm particles is intermediate and shows some minor disequilibrium only along the collapsing stream of the fountain and at the head of the flow. It is worth noting that thermal equilibrium between phases exists in most of the inner part of the fountain and pyroclastic flow. [46] Several sensitivity tests have also been performed in order to assess the dependence of results on the constitutive equations and associated semiempirical parameters. As discussed above, one of the most uncertain and critical parameters employed in the model is Smagorinsky’s constant used in the subgrid turbulence model. Figure 6 shows the total solid volumetric fraction and gas velocity at 250 s of simulation PF-1, as obtained by adopting a value of cS = 0.1 (Figure 6a), cS = 0.15 (Figure 6b), and cS = 0 (Figure 6c), keeping constant any other physical/numerical parameter. The plots focus on the emplacement of the pyroclastic flow and formation of buoyant plumes above it, leaving out the fountain region. The comparison between the three distributions shows a quite similar large-scale behavior of the process, but also some major differences. In detail, since

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the run using a larger value of the Smagorinsky’s constant (Figure 6b) is characterized by a more than double turbulent viscosity (note the power two in equation (12)) with respect to the reference case (Figure 6a), the mixing of the erupted mixture with the atmospheric air is, in that case, much more effective. This in turn favors an early and closer to the vent formation of buoyant plumes from the flow (and, at later time, of the phoenix column), a slightly shorter runout of the flow, and a more effective elutriation of fines from the flow. This is better explained in section 4. However, major differences occur when the subgrid scale model is neglected (Figure 6c). In this case the amount of fines elutriated from the flow drastically decreases together with the flow thickness due to the minor turbulent mixing of the flow with the surrounding air. On the contrary, the flow runout greatly increases, reaching 13 km after 250 s compared with about 9 km for cases in Figures 6a and 6b. This clearly indicates the need to take into account subgrid turbulence effects whenever high Reynolds flows are modeled. [47] Finally, Figure 7 shows the effect of different computational grids on the results. Figure 7 illustrates the distribution of the total volumetric particle fraction at 600 s by using, close to the ground, a minimum vertical grid size of 10 m (Figure 7a), 4 m (Figure 7b), and 2 m (Figure 7c) and keeping the same horizontal spacing. As regards the large-scale flow pattern, some notable differences can be observed between the 10 m distribution and the other two distributions which, by contrast, are quite similar to each other. In detail, the 10 m grid produces a larger runout of the flow (about 14 km versus 12.5 km of the other two grids) and an early generation of the phoenix column. On the other hand, the distance at which the phoenix column forms is approximately the same for all three simulations. [48] A significant difference between the three simulations is represented by the values of particle concentrations computed in the lower layer of the pyroclastic flow (and particularly the concentration of the coarser particles). This effect was actually known [see, e.g., Dobran et al., 1993, Figure 12] and is due to the tendency of coarse particles to sediment and form a dense flow unit at the base of the flow. Therefore volumetric concentration of these particles is directly affected by the vertical discretization of the computational domain close to the ground. As a consequence, predicted figures of coarse particle concentrations in the basal layer of the resolved flow need to be handled with caution. In addition, as described in section 2.3, the existence, for the assumed roughness values, of an interfacial layer between the actual ground plane and the resolved aerodynamic plane prevents, a priori, the description of the flow dynamics within this layer. However, it is important to stress here that, despite the impossibility of describing the dynamics of this lower portion of the flow, the large-scale pattern of the dispersal process, as well as the amount of particle elutriated, do not prove to be significantly affected by such a limit. In detail, a variation of the minimum vertical size of the cells from 2 to 10 m produces a variation of about 7% of the fraction of particles elutriated. 3.3. Simulation PF-2: Fine-Grained Case [49] Simulation PF-2 differs from simulation PF-1 only in terms of the granulometric distribution of solid particles, which, in this case, is richer in fines (see Table 3). Figure 8

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Figure 4. Distribution of solid volumetric fraction and velocity field of particulate phases (a) 1, (b) 2, (c) 3, (d) 4, (e) 5, and (f) 6 at 600 s of simulation PF-1. The color contour levels shown are the exponents to the base 10 and correspond to 8, 7, 6, 5, 4, 3, 2, and 1, as reported in the legend. illustrates the total particle volumetric distribution and associated gas velocity field at three different times. The global dynamics of this collapsing column are quite similar to that of simulation PF-1 during the first 200 s, but significant differences emerge at later times. After 400 s, the pyroclastic flow has reached a distance of 13 km from the vent, about 1.5 km more than simulation PF-1 (see Figure 4). Above the fountain a hot particle-rich plume rises in a similar way to simulation PF-1. After 600 s, the pyroclastic flow head has reached a distance of more than

14 km, almost 2 km more than simulation PF-1. At this time, the phoenix column starts to form at about 11 km from the vent, whereas the same coignimbrite column is already formed and well developed at 8.5 km in simulation PF-1. At 800 s, the coignimbrite column is clearly evident (it is about 7 km high) also in simulation PF-2 and the pyroclastic flow head has stopped at about 14.5 km. Finally, at 850 s (not shown), the phoenix column is well formed and has reached a height of about 10 km. At this time the plume of hot gas and pyroclasts above the


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Figure 4. (continued)

fountain and the phoenix column form a single volcanic plume of about 12 km in radius. [50] As described for simulation PF-1, particle dispersal strongly changes as a function of particle diameter (see Figure 4) and is not reported here for the sake of brevity. However, due to the fine-rich granulometric distribution used, such changes are now even more evident. In detail, 32 mm particles have now larger concentrations both in the plume above the fountain and in the phoenix cloud, whereas 190, 375, 750, and 1500 mm particles are now completely absent in the phoenix column. The concentration of the

coarser particles above the fountain is still significant, but smaller than that computed for simulation PF-1.

4. Result Analysis and Discussion [51] Previous work carried out by using two-/three-phase flow models showed the important effect of particle size on the large-scale evolution of pyroclastic flows and atmospheric dispersal processes [Valentine and Wohletz, 1989; Dobran et al., 1993; Neri and Dobran, 1994; Neri and Macedonio, 1996; Oberhuber et al., 1998; Herzog et al.,

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Figure 5. Distribution of gas-particle thermal nonequilibrium, Tk Tg, and associated particle velocity field of simulation PF-1 at 100 s for three different particle sizes: (a) dp1 = 32 mm, (b) dp3 = 190 mm, and (c) dp6 = 1500 mm. The color contour levels indicate temperature differences of 1, 2, 5, 10, 30, 50 and 70C, as reported in the legend.

1998]. The model presented here extends the above models by considering a magmatic mixture of generic N different particulate phases and a gas phase potentially formed by M nonreacting species. [52] Computational constraints and the limits of the constitutive equations employed restrict the present study to the

simulation of the first 10– 15 min of the phenomenon, as well as to the description of a practical number of particulate phases with diameter smaller than a few millimeters (i.e., six different solid phases). Each phase was considered representative of a specific granulometric class of pyroclasts commonly found in deposits and its physical properties were


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Figure 6. Distribution of total solid volumetric fraction and gas velocity field of simulation PF-1 at 250 s obtained by using three different values of the Smagorinsky constant cS: (a) cS = 0.1, (b) cS = 0.15, and (c) cS = 0. The color contour levels are the exponent to the base 10 and correspond to 8, 7, 6, 5, 4, 3, 2, and 1, as reported in the legend.

estimated as average values of the flow components. Such a simplified way of applying the model can easily be improved in the future by describing each particulate component of the pyroclastic mixture (i.e., pumice, crystals, and lithics) through a different phase with its own specific properties. [53] Despite these limits, simulation results already highlight the capabilities of the multiparticle formulation and allow us to preliminarily compare such results to field-based measurements and interpretations of the real event. In

sections 4.1 –4.3, some specific aspects deriving from the multiphase formulation of the model will be analyzed and discussed more fully. 4.1. Particle Segregation in Pyroclastic Flows and Coignimbritic Plumes [54] Analysis of Figure 4 pertaining to simulation PF-1 illustrates clearly the different dispersal of each employed particulate phase in the pyroclastic flow and the atmosphere.

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Figure 7. Distribution of total solid volumetric fraction of simulation PF-1 at 600 s obtained by using a minimum vertical grid size at the ground of (a) 10 m, (b) 4 m, and (c) 2 m. The contour levels are the exponents to the base 10 and correspond to 8, 7, 6, 5, 4, 3, 2, and 1.


A more quantitative illustration of the relative segregation and deposition of particles in the basal layer of the flow can be obtained by plotting radial and vertical distributions of their concentrations at given sections. Figures 9a and 9b show the distribution at 300 and 600 s, respectively, and at the ground level (i.e., in the first cell above the aerodynamic ground plane), of the volumetric fraction of the six particulate phases. Figure 9 shows the different distribution of the various phases, as well as their concentration increase in time in the more distal area of the flow. Two main sedimentation regions are highlighted. The first one is located very close to the vent rim and extends just a few hundreds of meters from the vent. This region is related to the recycling of material from the collapsed stream of the fountain back into the jet. The second region is located several kilometers away from the vent with a maximum density of the flow at about 10.5 km. This second region is clearly associated with the loss of momentum of the flow. In both regions, 375, 750, and 1500 mm particles have the higher volumetric content, whereas 32 mm particles have the lower one, even though they are still present in a significant amount. It is also interesting to note that particle concentrations do not significantly change in time in the more proximal area of the flow, thus indicating a sort of equilibrium between sedimentation and resuspension processes. As discussed above, the reported absolute values of the particle volume concentration need to be handled with caution due to their significant sensitivity to the grid size close to the ground. [55] Similarly, Figures 10a and 10b illustrate the vertical distribution of particles for the same simulation PF-1, at 600 s, and at a distance of 2 and 10 km, respectively, from the vent. Concentrations are again presented as exponents to the base 10 of the volume fraction so as to describe both dense and dilute regions. Simulation results show the existence, from the very first minutes and in both proximal and distal regions, of a dense underflow, concentrated in the first few cells above the aerodynamic ground plane, and a dilute suspension current overlying the former and extending vertically for a few hundreds of meters (see also Figure 4). As already shown in Figure 9, the dense underflow is richer in coarse particles and becomes denser as it moves away from the vent. On the other hand, the overlying dilute flow is richer in fine particles, which are distributed on a wider flow front with respect to the coarser particles and tend to elutriate from the body of the flow. However, in the proximal area (i.e., 2 km from the symmetry axis), the concentration of the six phases is quite similar, thus indicating a very effective turbulent mixing of the flow. Viceversa, in the distal region (10 km in our figure), the low inertia of the flow favors an effective decoupling between fine and coarse particles with the latter present only in the dense underflow. [56] A different representation of the dispersal of each particulate phase in a given atmospheric region can be obtained by spatial integration of the solid distribution reported in Figure 4 [Neri et al., 2002]. Such a technique is used here for the analysis of pyroclast distribution in the plumes developing from the collapsing column, i.e., the phoenix column and the plume above the fountain (Figure 11). Figure 11a refers to simulation PF-1 and shows the timewise behavior of the ratio between the total mass of

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a given solid phase present in the phoenix column and the total amount of mass of that phase injected into the atmosphere. If Fph,k(t) is such a ratio at time t, mph,k is the mass of phase k contained in the phoenix, and m_ v,k is the constant mass flow rate of that phase at the vent, we have Fph;k ðt Þ ¼

mph;k ðt Þ : m_ v;k t

ð35Þ

Figure 11a shows that at about 600 s, only 32 and 95 mm particles are present in the phoenix column, whereas the amount of coarser particles is negligible. From Figure 11a it is evident how the above defined ratios increase from 0, at about 300 s, i.e., the time of generation of the phoenix column, to quasi-constant values at about 650 s. The maximum value is reached by the 32 mm particles with about 30% of the erupted mass entering the phoenix column, whereas only about 5% of 95 mm particles is dragged up; 190 – 1500 mm particles are absent in the phoenix column. At 650 s, the curves in the diagram are interrupted since at this time the phoenix column is merging with the convective plume above the fountain and therefore the spatial integration would no longer be representative of the phoenix cloud. However, the asymptotic trend of the defined ratios have been clearly observed in other similar simulations performed with two particle sizes and run for a longer time [Neri et al., 2002]. [57] Similarly, Figure 11b shows the trend of the above defined ratios for the convective plume above the fountain. In this case, all six particulate phases are significantly present in the plume. Observed fractions range between 5% for 32 mm and about 1% for 1500 mm. In other words, the model predicts a lower and more uniform concentration of particles in the plume above the fountain with respect to the phoenix column. [58] The above distributions can be explained in terms of the modeled nonequilibrium effects between gas and solid phases. For instance, from the inspection of simulation results, as well as from the analysis of the gas-particle and particle-particle drag coefficients reported in section 2.2, turns out that the particle-particle term is largely prevailing on the gas-particle term in the lower dense portion of the flow, whereas the condition is reversed as regards the particle dispersal into the overlying suspension current, the phoenix column, and the plume above the fountain where the drag term basically controls the relative dispersal of the different phases. This result appears to be consistent with previous modeling works regarding the dynamics of concentrated granular flows [Lun et al., 1984; Gidaspow, 1994; Straub, 2000] and diluted ash flows [Denlinger, 1987; Bursik and Woods, 1996; Dade and Huppert, 1996; Burgisser and Bergantz, 2002] as well as with experience of particle segregation for military applications [Jayaswal et al., 1990; Sun et al., 1994]. 4.2. Effects of Grain-Size Distribution [59] The effect of pyroclast fragmentation on large-scale dispersal was investigated by defining a second granulometric distribution of the solid particles. This second distribution is richer in fines and presents only a short coarse tail of millimeter size particles. [60] The behavior of simulation PF-2 is similar to that of simulation PF-1, but with some significant differences.

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Figure 8. Distribution of the total particulate volumetric fraction and associated gas velocity field in the atmosphere at 400, 600, and 800 s of simulation PF-2. The color contour levels shown are the exponents to the base 10 and correspond to 8, 7, 6, 5, 4, 3, 2, and 1, as reported in the legend.

First, the radial dispersal of the flow of simulation PF-2 is greater (about 2 km) than that of simulation PF-1. Second, the flow thickness and velocity are also greater. Such a different velocity propagation also affects the time of formation of the phoenix column, which is delayed by about 3 min for simulation PF-2. [61] Again in this simulation, as in simulation PF-1, ratios defined by equation (35) for the phoenix column and the convective plume above the fountain were com-

puted. Results are of the type shown in Figure 11. Also in this case, only 32 and 95 mm particles are transported by the phoenix. The percentage of 32 mm particles collected by the phoenix tends again to about 30%, whereas the percentage of 95 mm particles is now reduced to about 3%. No major differences are reported with respect to simulation PF-1 also about the convective plume above the fountain. Particle concentration of the six phases are quite uniform and constant in time with maximum percentages


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Figure 9. Radial distributions of the solid volumetric fraction of each particulate phase of simulation PF-1 at the ground level and at (a) 300 and (b) 600 s from the beginning of the simulation. Distributions refer to the simulation performed by using a minimum vertical grid size of 2 m (see text for explanation).

of about 4% for 32 mm particles and 1% for 1500 mm particles. [62] A useful comparison between different simulations can be made by using Figure 12, which shows the timewise behavior of the weight fraction of total pyroclasts dragged up by the phoenix column. The three curves refer to simulations PF-1, PF-2, and again simulation PF-1 but performed by using a Smagorinsky coefficient of 0.15 in the turbulence model (see Figure 6). Figure 12 indicates fractions ranging between 5 and 8 wt % for all three simulations. The asymptotic-type behavior of these fractions is only outlined in the figure, since curves were interrupted at the point when the phoenix column and plume above the fountain were indistinguishable. However, Figure 12 clearly shows that simulation PF-2 is able to elutriate a larger amount of mass than simulation PF-1, consistently with its higher content in fines. Similarly, the use of a greater Smagorinsky coefficient induces an early and larger elutriation of particles as also shown in Figure 6. The same technique, applied to the analysis of the plume above the fountain, indicates a total fraction of pyroclasts elutriated of about 2 – 3 wt % for the collapsing columns simulated. 4.3. Comparison With Field-Based Studies [63] Although the simulations performed are not representative of any specific eruption and describe just a limited

time period of an idealized representation of the process, their results appear to be qualitatively, and in some cases quantitatively, consistent with volcanological data obtained from stratigraphic studies. As regards the pyroclast distribution in the phoenix column, Figures 10b and 11a pertaining to simulation PF-1, as well as results from simulation PF-2, show that only pyroclasts with diameters smaller than about 100– 200 mm actually form the cloud. In addition, the amount in weight of 32 mm particles is about five times greater than that of 95 mm particles. These results are consistent with several stratigraphic works. Sparks and Walker [1977] found that typically 70 wt % of fine-grained ash fall from coignimbrite columns is finer than 1/8 mm, i.e., 125 mm. Self and Rampino [1981] found that the coignimbrite ash of the famous 1883 eruption of Krakatau consists of vitric fragments commonly from a few to 50 mm in diameter. The particle size distribution predicted in the phoenix column also appears to be reasonably consistent with theoretical estimates by Denlinger [1987] based on a comparison between the fall velocity of pyroclasts and the vertical air velocity of the flow, even though flow and particle properties are somehow different from those assumed here. [64] As regards the grain-size analysis of the plume above the fountain, a comparison between model output and real system is harder to achieve due to the difficulty in obtaining

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Figure 10. Vertical distributions of the solid volumetric fraction of each particulate phase of simulation PF-1, at 600 s, and at (a) R = 2 km and (b) R = 10 km from the vent. Distributions refer to the simulation performed by using a minimum vertical grid size of 2 m (see text for explanation).

these data. Qualitatively, this plume appears to be the outcome of two out of three sources of fine ash already proposed by Sparks and Walker [1977]: (1) the escape of fine pyroclasts and hot gas above the fountain and (2) the escape of fine ash from the proximal area close to the vent where the flow starts to segregate (the third source being the above discussed elutriation as a coignimbrite cloud). Simulations are not able to distinguish between the two sources, due to the complex flow pattern, but they can actually quantify their combined contribution. Results indicate a quite uniform concentration of the six particulate phases (see Figure 11b) with about 2 –3 wt % of the total erupted mass forming this plume for the collapsing columns simulated. It is worth mentioning that such a percentage can be much greater as we move from the collapsing to the buoyant regime. Specifically, in the transitional regime, the pyroclast mass can be equally split between the collapsing and convective systems [Neri et al., 2002]. [65] As regards the total mass of pyroclast elutriated from the plume, in the simulations performed, about 30– 40% of the erupted fines of 32 mm is elutriated forming the phoenix column and the plume above the fountain. Because of the presence of the other particles, only about 10 wt % of the total material erupted is found in the convective columns. In the light of this, we expect that 40% might approximately

represent a realistic value of the total percentage of pyroclasts elutriated from a collapsing column when the eruptive mixture is very rich in fines. These predictions compare also reasonably well with estimates found in the literature which indicate that typically 20– 40 wt % of erupted pyroclasts are dispersed in the atmosphere during the formation of large ignimbrites [Sparks and Walker, 1977; Woods and Wohletz, 1991]. Some examples are several flow units of the Vulsini Ignimbrite (Italy) and Minoan Ignimbrite of Santorini (Greece) [Sparks and Walker, 1977], the 1886 eruption of Krakatau [Self and Rampino, 1981], and the 75 ka eruption of Toba [Rose and Chesner, 1987]. Results are also consistent with data from Mount St. Helens indicating that the elutriation of fine ash from the flow is the main source of ash fall during the flow phases [Hoblitt, 1986; Sparks et al., 1986]. [66] Model predictions can also be interpreted as qualitatively consistent with the most common volcanological models of pyroclastic flows, which describe the gravity current as a basal avalanche of a concentrated dispersion of coarse material, underlying a more dilute ash cloud of fine particles [Sparks and Wilson, 1976; Hoblitt, 1986; Denlinger, 1987; Bursik and Woods, 1996; Sparks at al., 1997; Druitt, 1998; Freundt and Bursik, 1998; Burgisser and Bergantz, 2002]. As shown by Figure 10, such a two-layer


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5. Conclusions

Figure 11. Timewise distribution of the (a) phoenix mass fraction and (b) plume above the fountain mass fraction for each particulate phase of simulation PF-1.

structure of the flow is already observable in the more proximal area of the flow close to the impact zone of the collapsing stream with the ground. However, in such a proximal area, the high turbulence of the flow is able to produce a quite uniform distribution of particles with a significant suspension of millimeter-size particles. As regards the effect of the grain-size distribution at the vent, simulation results also appear to be qualitatively consistent with several field evidences, indicating pyroclastic surges as more mobile, faster, and expanded than pyroclastic flows [Cas and Wright, 1987; Rosi, 1996; Sparks et al., 1997; Druitt, 1998], and with one-dimensional, homogeneous, steady state flow models, indicating that finer-grained ash flows tend to have longer run-out distances than coarsegrained flows [Bursik and Woods, 1996]. [67] Further comparisons between model predictions and observations are given by Clarke et al. [2002] regarding the dynamics of Vulcanian explosions and column collapse. The model has also been applied to the assessment of pyroclastic flow hazard at Vesuvius (Italy) [Todesco et al., 2002; Esposti Ongaro et al., 2002]. We refer to these papers for the discussion of further model applications.

[68] A multiparticle, transient, and axisymmetric flow model of pyroclastic dispersal has been developed and applied to the study of the dynamics of collapsing columns producing high-speed and high-temperature pyroclastic flows. The novel feature of the model consists in the full mechanical and thermal nonequilibrium between a continuous gas phase and generic N particulate phases representative of pyroclasts with different sizes and properties. The results allowed the spatial and temporal description of the dispersal of the six particulate phases, considered representative of the finer portion of the mixture, within the pyroclastic flow and the convective plumes. [69] The model predicts that pyroclastic flows are formed by a dense underflow rich in coarse particles, but with still a significant fraction of fines, underlying a dilute fine-rich suspension, where coarse particles are practically absent. This structure of the flow appears to be present from the very proximal region of the flow, although this region appears to be characterized by a quite well-mixed distribution of particles of different sizes. In the more distal area, the decoupling between particles is much more evident, as shown by the total absence of the coarser particles in the buoyant plumes. [70] As regards particle elutriation, very fine particles tend to follow the ascending hot gas, mainly in the phoenix cloud and, secondarily, in the plume above the fountain. A percentage of about 30% of the amount injected into the atmosphere was predicted for 32 mm particles in the phoenix column, whereas only a small percentage of them escape from the fountain and proximal region. Particles greater than about 200 mm in diameter are absent in the phoenix column, whereas their amount is comparable to that of the finer particles in the plume above the fountain. About 10– 15 wt % of the total mass leaving the vent is predicted to be elutriated into the atmosphere from the fountain plus flow systems, but a higher percentage, up to about 30– 40 wt %,

Figure 12. Timewise distribution of the total solid mass fraction carried on by the phoenix cloud of simulation PF-1 and PF-2. The distribution obtained for simulation PF-1 by using a Smagorinsky coefficient of 0.15 is also shown.

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appears to be possible, if a greater fine content is assumed at the vent. [71] A significant influence of the granulometric composition was observed both on pyroclastic flow mobility and the elutriation process. The fine-grained particle distribution at the vent produces a runout significantly greater than the coarser distribution and a considerable delayed formation of the phoenix column. The fine-grained flow is also significantly thicker and faster with respect to the coarse-grained flow. The total amount of mass elutriated from the flow and the fountain also increases by using a mixture richer in fine particles. [72] From the analysis of model results, it is evident that particle dynamics in the basal layer of the flow are strongly affected by particle-particle interactions, whereas particle dispersal in the overlying dilute suspension current and particle elutriation from the flow and the fountain are controlled by the action of the gas-particle drag. Significant thermal nonequilibrium effects are also predicted between gas and coarser particles in the regions of effective air entrainment. [73] Finally, simulation results appear to be in qualitative, and in some respects quantitative, agreement with field data and observations. Further understanding of the processes investigated appears to be tied to the development of more accurate physical models as well as to the carrying out of model applications to well-documented test cases. As regards the modeling work, in the light of the present results, further improvement is necessary in the description of the lower portion of the flow. The modeling approach adopted here is not able to describe accurately the threedimensional dynamics of such dense underflow. This is essentially due to the presence of the interfacial layer, for rough surfaces, and to computational constraints that prevent, at this time, the adoption of grids of submeter size. The physics of such multiparticle flows also needs to be improved, possibly by the adoption of a more accurate description of turbulence for gas-particle mixtures and the consideration of a wider particle size range. Further work, in terms of more effective modeling approaches and computational techniques, also appears necessary to achieve a simulation time in the order of some hours and the threedimensional description of the phenomenon.

Notation a, b, c constants. cS Smagorinsky’s constant. Cd,k drag coefficient between the gas and the kth solid phase. Cp specific heat at constant pressure. d particle diameter. Dg,k interfacial drag between the gas and the kth solid phase. Dk,j interfacial drag between the kth and jth solid phase. Dv vent diameter. e restitution coefficient. F mass fraction. Fkj function used in the particle-particle drag coefficient. g gravitational acceleration.

G h I k l lB lS m_ m M N Nu P Pr Qk r ~ R Re t T T v y Y z z0 a kj

solid elasticity modulus. enthalpy. unit tensor. thermal conductivity. length scale of turbulence. length scale of turbulence close to a solid boundary. length scale of turbulence according to Smagorinsky model. mass flow rate. mass. total number of gas components. total number of solid particulate phases. Nusselt number. pressure. Prandtl number. volumetric heat transfer rate between the gas and the kth solid phase. radial coordinate. gas constant of the gas mixture. Reynolds number. time. temperature. stress tensor. velocity vector. mass fraction of a gas component. mass fraction of gas phase in the mixture. vertical coordinate. roughness. coefficient for nonhead particle collisions. volumetric fraction. maximum volume fraction of a two-population mixture. von Ka`rma`n constant. viscosity. granulometric unit. maximum volume fraction at packing of phase k. density. mechanical and thermal response times. viscous tensor. deformation rate tensor.

k m Ø k r tV, tT T ~ T Subscripts a air. c coulombic. g gas phase. ge gas effective. gt gas turbulent. k kth solid phase. ke kth solid effective. i ith gas component. j jth solid phase or gas component. m mixture. ph phoenix column. pl plume above the fountain. s solid or pyroclastic phase. t turbulent. tot total. v vent, viscous. w water vapor.

[74] Acknowledgments. A.N. carried out most of his work at the Illinois Institute of Technology, Chicago, with the support of two NATO-

NERI ET AL.: MULTIPARTICLE SIMULATION OF EXPLOSIVE ERUPTIONS CNR fellowships (Advanced Fellowship Program 1994 and Senior Fellowship Program 1997). The work was also partially supported by the Gruppo Nazionale per la Vulcanologia, Italy, Project 2000-2/09 and the European Community project EVRI-CT-2002-40026 (Exploris). The authors wish to thank Claude Jaupart, Takehiro Koyaguchi, Francis Albarede, and one anonymous referee for their thorough reviews and Franco Barberi for his continuous support.

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D. Gidaspow, Department of Chemical and Environmental Engineering, Illinois Institute of Technology, 10 W 33rd Street, Chicago, IL 60616, USA. ([email protected]) G. Macedonio, Osservatorio Vesuviano, Istituto Nazionale di Geofisica e Vulcanologia, Via Diocleziano 328, I-80124 Napoli, Italy. (macedon@ ov.ingv.it) A. Neri, Istituto di Geoscienze e Georisorse, Consiglio Nazionale delle Ricerche, Dipartimento di Scienze della Terra, Via S.Maria 53, I-56126 Pisa, Italy. ([email protected]) T. Esposti Ongaro, Universita’ di Pisa, Dipartimento di Scienze della Terra, Via S.Maria 53, I-56126 Pisa, Italy. ([email protected])

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Finite Difference Simulation of Implosive Collapsing for Aluminum

Scaling for faster macroparticle simulation in longitudinal multiparticle

Seasonality of volcanic eruptions - Wiley Online Library

Collapsing Glomerulopathy

Multiparticle Production and Percolation of Strings

Simulation on Cavitation Bubble Collapsing with Lattice Boltzmann ...

Vanishing polyhedron and collapsing map

Hard Unknots and Collapsing Tangles

Separability and distillability of multiparticle quantum systems

Columns

columns

Patterns of late Cenozoic volcanic and tectonic ... - Wiley Online Library

Correlation femtoscopy of multiparticle processes

Multiparticle Dynamics 2004