GMDGeoscientific Model DevelopmentGMDGeosci. Model Dev.1991-9603Copernicus PublicationsGöttingen, Germany10.5194/gmd-9-697-2016ASHEE-1.0: a compressible, equilibrium–Eulerian model for volcanic ash plumesCerminaraM.matteo.cerminara@gmail.comhttps://orcid.org/0000-0001-5155-5872Esposti OngaroT.https://orcid.org/0000-0002-6663-5311BerselliL. C.Scuola Normale Superiore, Pisa, ItalyIstituto Nazionale di Geofisica e Vulcanologia, Sezione di Pisa, Pisa, ItalyDipartimento di Matematica, Università degli Studi di Pisa, Pisa, ItalyM. Cerminara (matteo.cerminara@gmail.com)18February2016926977304September201519October201520January201620January2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://gmd.copernicus.org/articles/9/697/2016/gmd-9-697-2016.htmlThe full text article is available as a PDF file from https://gmd.copernicus.org/articles/9/697/2016/gmd-9-697-2016.pdf
A new fluid-dynamic model is developed to numerically simulate the
non-equilibrium dynamics of polydisperse gas–particle mixtures forming
volcanic plumes. Starting from the three-dimensional N-phase Eulerian
transport equations for a mixture of gases and solid dispersed particles, we
adopt an asymptotic expansion strategy to derive a compressible version of
the first-order non-equilibrium model, valid for low-concentration regimes
(particle volume fraction less than 10-3) and particle Stokes number
(St – i.e., the ratio between relaxation time and flow characteristic
time) not exceeding about 0.2. The new model, which is called ASHEE (ASH
Equilibrium Eulerian), is significantly faster than the N-phase Eulerian
model while retaining the capability to describe gas–particle non-equilibrium
effects. Direct Numerical Simulation accurately reproduces the dynamics of
isotropic, compressible turbulence in subsonic regimes. For gas–particle
mixtures, it describes the main features of density fluctuations and the
preferential concentration and clustering of particles by turbulence, thus
verifying the model reliability and suitability for the numerical simulation
of high-Reynolds number and high-temperature regimes in the presence of a
dispersed phase. On the other hand, Large-Eddy Numerical Simulations of
forced plumes are able to reproduce the averaged and instantaneous flow
properties. In particular, the self-similar Gaussian radial profile and the
development of large-scale coherent structures are reproduced, including the
rate of turbulent mixing and entrainment of atmospheric air. Application to
the Large-Eddy Simulation of the injection of the eruptive mixture in a
stratified atmosphere describes some of the important features of turbulent
volcanic plumes, including air entrainment, buoyancy reversal and maximum
plume height. For very fine particles (St→0, when non-equilibrium
effects are negligible) the model reduces to the so-called dusty-gas model.
However, coarse particles partially decouple from the gas phase within eddies
(thus modifying the turbulent structure) and preferentially concentrate at
the eddy periphery, eventually being lost from the plume margins due to the
concurrent effect of gravity. By these mechanisms, gas–particle
non-equilibrium processes are able to influence the large-scale behavior of
volcanic plumes.
Introduction
Explosive volcanic eruptions are characterized by the injection from a vent
into the atmosphere of a mixture of gases, liquid droplets and solid
particles, at high velocity and temperature. In typical magmatic eruptions,
solid particles (also termed tephra) constitute more than 95 % of the
erupted mass and are mostly produced by fragmentation of a highly viscous
magma during its rapid ascent in a narrow conduit
, with particle sizes and densities
spanning over a wide range, depending on the overall character and intensity
of the eruption . In this paper, we
consider eruptive particles mostly in the sub-millimeter range, generally
termed ash. The plume mixture volumetric concentration very rarely exceeds
ϵs∼3×10-3, and the order of magnitude of
the ejected fragments density is ρ^s∼103kgm-3. Thus, the plume mixture can be considered mainly as
a dilute suspension in the sense of .
After injection in the atmosphere, this multiphase eruptive mixture can rise
convectively in the atmosphere, either forming a buoyant volcanic plume or
collapsing catastrophically forming pyroclastic density currents. These two
end-members have different spatial and temporal scales and different impacts
on the surrounding of a volcano. Understanding the dynamics of volcanic
columns is one of the topical aims of volcanology and the main motivation
for this work.
The term volcanic column will be adopted in this paper to
generically indicate the eruptive character (e.g., convective/collapsing
column). Following the fluid-dynamic nomenclature, we will term jet
the inertial regime of the volcanic column and plume the
buoyancy-driven regime. A forced plume is characterized by an
initial momentum-driven jet stage, transitioning into a plume.
In this work, we present a new computational fluid-dynamic model to simulate
turbulent gas–particle forced plumes in the atmosphere. Although the focus of
the paper is on subsonic regimes, the model is also suited to be applied to
transonic and supersonic flows. In many cases, indeed, the eruptive mixture
is injected into the atmosphere at pressure higher than atmospheric, so that
the flow is initially driven by a rapid, transonic decompression stage. This
is suggested by numerical models predicting choked flow conditions at the
volcanic vent , implying
a supersonic transition above the vent or in the crater
and it is
supported by field evidences of the emission of shock waves during the
initial stages of an eruptions . Despite the
importance of the decompression stage for the subsequent development of the
volcanic plume and
for the stability of the eruptive column , our analysis is
limited to the plume region where flow pressure is equilibrated to the
atmospheric pressure. From laboratory experiments, this is expected to occur
within less than 20 inlet diameters above the ground
.
A wide set of numerical tests are presented in this paper
(Sect. ) to assess the adequacy of the model for the intended
volcanological application (Sect. ) and the reliability of the
numerical solution method.
Dusty-gas modeling of volcanic plumes
Starting from the assumption that the large-scale behavior of volcanic
columns is controlled by the bulk properties of the eruptive
mixture, most of the previous models of volcanic plumes have considered the
eruptive mixture as homogeneous (i.e., they assume that particles are
perfectly coupled to the gas phase). Under such a hypothesis, the multiphase
transport equations can be largely simplified and reduce to a set of mass,
momentum and energy balance equations for a single fluid (named
dusty-gas or pseudo-gas) having average thermo-fluid
dynamic properties (mixture density, velocity and temperature) and an
equation of state accounting for the incompressibility of the particulate
phase and gas co-volume .
By adopting the dusty-gas approximation, volcanic plumes have been studied in
the framework of jet and plume theory
. One-dimensional, steady-state dusty-gas models
of volcanic plumes have thus had a formidable role in volcanology to identify
the main processes controlling their dynamics and scaling properties
.
Accordingly, volcanic plume dynamics are schematically subdivided into two
main stages. The lower, jet phase is driven by the initial flow momentum.
Mixture buoyancy is initially negative (the bulk density is larger than
atmospheric) but the mixture progressively expands adiabatically thanks to
atmospheric air entrainment and heating, eventually undergoing a buoyancy
reversal. When buoyancy reversal does not occur, partial or total collapse of
the jet from its maximum thrust height and generation of pyroclastic density
currents are expected.
Above the jet thrust region, the rise of volcanic plumes is driven by
buoyancy and it is controlled by turbulent mixing until, in the stratified
atmosphere, a level of neutral buoyancy is reached. Above that level, the
plume rises up to its maximum height and then starts to spread out as a
gravity current e.g., forming an umbrella ash cloud
dispersing in the atmosphere and slowly falling-out.
In one-dimensional, time-averaged models, entrainment of atmospheric air is
described by one empirical coefficient (the entrainment coefficient) relating
the influx of atmospheric air to the local, vertical plume velocity. The
entrainment coefficient also determines the plume shape
and can be empirically assessed by means of
direct field observations or ad hoc laboratory measurements.
Further development of volcanic plume models have included the influence of
atmospheric stratification and humidity , the effect of crosswind , loss and
re-entrainment of solid particles from plume margins
, wet aggregation and
transient effects . However,
one-dimensional models strongly rely on the self-similarity hypothesis, whose
validity cannot be experimentally ascertained for volcanic eruptions.
To overcome the limitations of one-dimensional models, three-dimensional
dusty-gas models have been developed to simulate volcanic plumes.
have developed a three-dimensional dusty-gas model
(SK-3D) able to accurately resolve the relevant turbulent scales of
a volcanic plume, allowing a first, theoretical determination of the
entrainment coefficient , without the need of an empirical
calibration.
To simulate the three-dimensional large-scale dynamics of volcanic plumes
including particle settling and the complex microphysics of water in volcanic
plumes, the ATHAM (Active Tracer High Resolution Atmospheric Model) code has
been designed . ATHAM describes
the dynamics of gas–particle mixtures by assuming that particles are in
kinetic equilibrium with the gas phase only in the horizontal component,
whereas along the vertical direction they are allowed to have a differential
velocity. Thermal equilibrium is assumed. In this sense, ATHAM relaxes the
dusty-gas approximation (while maintaining its fundamental structure and the
same momentum transport equations) by describing the settling of particles
with respect to the gas.
Multiphase flow models of volcanic plumes
Notwithstanding all the above advantages, dusty-gas models are still limited
by the equilibrium assumption, which can be questionable at least for the
coarsest part of the granulometric spectrum in a plume. Turbulence is indeed
a nonlinear, multiscale process and the time and space scales of
gas–particle interaction may be comparable with some relevant turbulent
scales, thus influencing the large-scale behavior of volcanic plumes.
To model non-equilibrium processes, Eulerian multiphase flow models have been
developed, which solve the full set of mass, momentum and energy transport
equations for a mixture of gas and dispersed particles, treated as
interpenetrating fluids. , and first
analyzed the influence of erupting parameters on the column behavior. By
means of two-dimensional numerical simulations, they individuated a region of
transition from collapsing to convective columns. Lately, two-dimensional
and three-dimensional numerical simulations
have contributed to further modify the view of a sharp
transition between convecting and collapsing columns in favor of that of
a transitional regime, characterized by a progressively increasing fraction
of mass collapsing. However, previous works could not investigate in detail
the non-equilibrium effects in volcanic plumes, mainly because of their
averaged description of turbulence: a detailed resolution of the relevant
turbulent scales in three dimensions would indeed be computationally
prohibitive for N-phase systems.
The main objective of the present work is therefore to develop a new physical
model and a fast three-dimensional numerical code able to resolve the spatial
and temporal scales of the interaction between gas and particles in turbulent
regimes and to describe the kinetic non-equilibrium dynamics and their
influence on the observable features of volcanic plumes. To this
aim, a development of the so-called equilibrium–Eulerian approach
has been adopted. It is a generalization of
the dusty-gas model keeping the kinematic non-equilibrium as a first-order
correction of the model with respect to the Stokes number
of the solid particles in the mixture. Here, we generalize the
model to the compressible two-way case.
The derivation of the fluid dynamic model describing the non-equilibrium
gas–particle mixture is described in detail in Sect. . The
computational solution procedure and the numerical code development are
reported in Sect. . Section focuses on
verification and evaluation issues in the context of applications to
turbulent volcanic plumes. In particular, we discuss three-dimensional
numerical simulations of compressible isotropic turbulence (with and without
particles), experimental-scale forced plumes and the shock tube
problem. Finally, Sect. presents numerical simulations of
volcanic plumes and discusses some aspects related to numerical grid
resolution in practical cases.
The multiphase flow model
To derive an appropriate multiphase flow model to describe gas–particle
volcanic plumes, we introduce the non-dimensional scaling parameters
characterizing gas–particle and particle–particle interactions.
The drag force between gas and particles introduces into the system
a timescale τs, the particle relaxation time, which is the time a particle needs to equilibrate to
a change of gas velocity. Gas–particle drag is a nonlinear function
of the local flow variables and, in particular, it depends strongly on
the relative Reynolds number, defined as
Res=ρ^g|us-ug|dsμ.
Here ds is the particle diameter, ρ^g is the
gas density, μ is the gas dynamic viscosity and ug(s)
is the gas (solid) phase velocity field. With
ρ^g(s) being the gaseous (solid) phase density and
ϵs=Vs/V the volumetric concentration of the
solid phase, it is useful to define the gas bulk density ρg≡(1-ϵs)ρ^g≃ρ^g and the solid bulk density ρs≡ϵsρ^s (even though in our applications
ϵs is order 10-3, ρs is non-negligible
since ρ^s/ρ^g is of order 103).
For an individual point-like particle (i.e., having diameter ds
much smaller than the scale of the problem under analysis), at
Res<1000, the drag force per volume unity can be given
by Stokes' law:
fs=ρsτs(ug-us),
where
τs≡ρ^sρ^gds218νϕc(Res)
is the characteristic time of particle velocity relaxation with respect the
gas, ρ^s is the particle density, ν is the gas
kinematic viscosity and ϕc=1+0.15Res0.687
is a correction factor for particle Reynolds number larger than
1 see and for spherical particles . In
Eq. () we disregard all effects due to the pressure gradient,
the added mass, the Basset history and the Saffman terms, because we are
considering heavy particles:
ρ^s/ρ^g≫1see. Equation () has a linear dependence on the
fluid–particle relative velocity only when Res≪1,
so that ϕc≃ 1 and the classic Stokes drag expression is
recovered. On the other hand, if the relative Reynolds number
Res grows, nonlinear effects become much more important
in Eq. (). The empirical relationship used in
this work has been used and tested in a number of papers
e.g.,, and it is
equivalent to assuming the following gas–particle drag coefficient:
CD(Res)=24Res(1+0.15Res0.687). discussed nonlinear effects due to this correction on the
dynamics of point-like particles falling under gravity in a homogeneous and
isotropic turbulent surrounding. We recall here the terminal velocity that
can be found by setting ug=0 in Eq. () is
ws=4dsρ^s3CDρggg=τsg.
As previously pointed out, the correction used in Eq. () is valid if
Res<103, the regime addressed in this work for ash
particles much denser then the surrounding fluid and smaller than about
1 mm. As shown by , maximum values of
Res are associated with particle gravitational settling (not with
turbulence). Using Eqs. () and (), it is thus
possible to estimate the Res of a falling particle with diameter
ds. We obtain that Res is always smaller than 103
for ash particles finer than 1 mm in air. If regimes with a stronger
decoupling need to be explored, more complex empirical corrections have to be
used for ϕc. It is also worth
noting that ash particles can differ significantly from spheres and terminal
settling velocities of volcanic particles can be up to a factor 2–3 with respect to
spherical assumption. To account for this effect various modifications to
Eq. () have been devised e.g.,.
The same reasoning can be applied to estimate the thermal relaxation time between gas and particles. In terms of the solid phase specific heat
capacity Cs and its thermal conductivity kg, we have
τT=2Nusρ^sCskgds212,
where Nus=Nus(Res,Pr)
is the Nusselt number, usually function of the relative Reynolds number and
of the Prandtl number of the carrier fluid . In terms of
τT, the heat exchange between a particle at temperature Ts
and the surrounding gas at temperature Tg per unit volume is
Qs=ρsCsτT(Ts-Tg).
Comparing the kinetic and thermal relaxation times we get
τTτs=322ϕcNusCsμkg.
In order to estimate this number, firstly we notice that factor
2ϕc/Nus tends to 1 if Res→0, and it remains smaller than ≃2 if Res<103. Then, in the case of ash particles in
air, we have (in SI units) μ≃10-5, Cs≃103,
kg≃10-2. Thus we have that τT/τs≃1, meaning that the thermal equilibrium time is typically of the same
order of magnitude as the kinematic one. This bound is very useful when we
write the equilibrium–Eulerian and the dusty-gas models, because it ensures
that the thermal Stokes number is of the same order as the kinematic one, at
least for volcanic ash finer than about 1 mm.
The non-dimensional Stokes number (St) is defined as the ratio
between the kinetic relaxation time and a characteristic time of the flow
under investigation τL, namely
Sts=τs/τL. The definition of the
flow timescale can be problematic for high-Reynolds number flows (typical of
volcanic plumes), which are characterized by a wide range of interacting
length- and timescales, a distinctive feature of the turbulent regime. For
volcanic plumes, the more energetic timescale would be of the order of τL=L/U, where L and U are the plume diameter and velocity at the vent,
which gives the characteristic turnover time of the largest eddies in
a turbulent plume e.g.,. On the other hand, the smallest
timescale (largest Sts) can be defined by the
Kolmogorov similarity law by τη∼τLReL-1/2, where the
macroscopic Reynolds number is defined, at first instance, by
ReL=UL/ν, with ν being the kinematic
viscosity of the gas phase numerical models. It is also useful to introduce
the Large-Eddy Simulation (LES) timescale τξ, relative to the LES
length scale ξ. This is related to the numerical grid resolution, size of
the explicit filter and discretization accuracy
. At LES
scale ξ, Sts is not as large as at the Kolmogorov
scale, thus the decoupling between particles and the carrier fluid is
mitigated by the LES filtering operation. Dimensional analysis shows (see
Sect. ) that Sts≲0.2 for LES of
volcanic ash finer than about 1 mm.
The model presented here is conceived for resolving dilute
suspensions, namely mixtures of gases and particles with volumetric
concentration VsV≡ϵs≲10-3. We here use the definition of dilute suspension by
and ,
corresponding to regimes in which particle–particle collisions can be
disregarded. This can also be justified by analyzing the timescale of
particle–particle collisions. In the dilute regime, in which we can assume
an equilibrium Maxwell distribution of particle velocities, the mean free
path of solid particles is given by λp-p=162dsϵs.
Consequently, particle–particle collisions are relatively infrequent
(λp-p∼0.1m≫ds), so that we
can neglect, as a first approximation, particle–particle collisions
and consider the particulate fluid as pressure-less, inviscid and
non-conductive.
In volcanic plumes the particle volumetric concentration can exceed by one
order of magnitude the threshold ϵs≃ 10-3 only
near the vent see, e.g.,. However, the
region of the plume where the dilute suspension requirement is not fulfilled
remains small with respect the size of the entire plume, weakly influencing
its global dynamics. Indeed, as we will show in Sect. , air
entrainment and particle fallout induce a rapid decrease of the volumetric
concentration. In contrast, the mass fraction of the solid particles cannot be considered small, because particles are heavy: ϵsρ^s≡ρs≃ρg. Thus,
particle inertia will be considered in the present model: in other words, we
will consider the two-way coupling between dispersed particles and
the carrier gas phase.
Summarizing, our multiphase model focuses and carefully takes advantage of
the hypotheses characterizing the following regimes: heavy particles
(ρ^s/ρ^g≫1) in dilute suspension
(ϵs≲10-3) with dynamical length scales much
larger than the particles diameter (point-particle approach) and relative
Reynolds number smaller than 103.
The Eulerian model in “mixture” formulation
When St≤1 and the number of particles is
very large, it is convenient to use an Eulerian approach, where the carrier
and the dispersed phases are modeled as interpenetrating continua, and their
dynamics is described by the laws of fluid mechanics .
Here we model a polydisperse mixture of i∈[1,2,…,I]≡I gaseous phases and j∈[I+1,I+2,…,I+J]≡J solid phases. From now on, we will use the subscript (⋅)j
instead of (⋅)s for the jth solid phase. Thus, the bulk
density of the mixture reads: ρm=∑Iρi+∑Jρj. The mass fractions will be denoted by the symbol
y – i.e., yj=ρj/ρm. The bulk density of the gas phase
thus is ρg=∑Iρi=∑Iyiρm, while that of the solid phases ρs=∑Jρj=∑Jyjρm. Thus,
ρm=ρg+ρs. The volumetric
concentration of the ith(jth) phase is given by ϵi=ρi/ρ^i. Solid phases represent the discretization of
a virtually continuous grain-size distribution into discrete bins, as usually
done in volcanological studies see. Another
possible approach is the method of moments, in which the evolution of the
moments of the grain size distribution is described. This has
recently been applied in volcanology to integral plume models by
. In the present work we opted for the classical
discretization of the grain size distribution see.
In , we analyze the Eulerian–Eulerian model under
the barotropic regime to show the existence of weak solutions of the
corresponding partial differential equations problem.
In the regime described above, the Eulerian–Eulerian equations for each
phase (either gaseous or solid) are written. In Appendix we
reformulate the Eulerian–Eulerian model in a convenient equivalent
formulation (“mixture” formulation). We use the subscripts
i,j,m to associate a generic field ψ with the gas phases
(ψi), with the solid phases (ψj), and with the mixture
(ψm=∑Iyiψi+∑Jyjψj).
All the fields are defined in a spatial domain x∈Ω and a
temporal interval t∈T. The field variables to be solved are:
the density of the mixture ρm(x,t); the mass fractions
yi(x,t) and yj(x,t); the velocity fields
um(x,t) and uj(x,t); the enthalpy
hm(x,t); the temperature fields Tj(x,t).
The Eulerian–Eulerian model in mixture formulation thus reads
∂tρm+∇⋅(ρmum)=∑j∈JSj;∂t(ρmyi)+∇⋅(ρmugyi)=0,i∈I;∂t(ρmyj)+∇⋅(ρmujyj)=Sj,j∈J;∂t(ρmum)+∇⋅(ρmum⊗um+ρmTr)=-∇p+∇⋅T+ρmg+∑j∈JSjuj;∂t(ρmhm)+∇⋅ρmhm(um+vh)=∂tp-∂t(ρmKm)-∇⋅ρmKm(um+vK)+∇⋅(T⋅ug-q)+ρm(g⋅um)+∑j∈JSj(hj+Kj).
The terms Kg=12|ug|2 and
Kj=12|uj|2 are the kinetic energy per unity of mass of
the gaseous and solid phases, respectively. The acceleration due to gravity
is g. The other terms, needing closure constitutive equations, are:
the pressure of the gas phase p; the stress tensor of the gas phase
T; the heat flux in the gas phase q; the source (or sink)
term for the jth phase Sj.
The first equation is redundant, because it is contained in the second and
third set of continuity equations and ∑Iyi+∑Jyj=1. The system is missing the 4J momentum and enthalpy
equations for the solid phases, needed to find the decoupling velocity
vj=uj-ug and the decoupling
temperature Tj-Tg. They are reported in
Eq. ().
The equilibrium–Eulerian model described in the Sect.
provides a way to solve the decoupling equations in a fast, explicit way,
avoiding the need to solve the complete system of PDEs of the Eulerian–Eulerian
model.
Constitutive equations
To close the system, constitutive equations are needed. They are listed here:
Perfect gas: p=∑Iρ^iRiTg, with Ri the gas constant of the ith gas
phase. This law can be simplified by nothing that
ϵs≪1, thus ϵi≃1 and
ρ^i≃ρisee. Anyway, in
this work we use the complete version of the perfect gas law. It can
be written in convenient form for a poly-disperse mixture as1ρm=∑j∈Jyjρ^j+∑i∈IyiRiTgp.
Newtonian gas stress tensor:T=2μ(Tg)(sym(∇ug)-13∇⋅ugI),where μ(T)=∑Iϰiμi(T) is the gas
dynamic viscosity, μi is that of the ith gas component, and
ϰi is the molar fraction of the ith component
see.
Enthalpy per unit of mass of the gas (solid) phase:hg=∑IρiCiTg/ρg+p/ρg(hj=CjTj), with Ci(Cj) the specific heat at constant
volume of the ith (jth) phase.
Thus: hi=CiTg+p/ρg; ∑Iyihi=yghg; hm=∑IyiCiTg+∑JyjCjTj+p/ρm.
The Fourier law for the heat transfer in the gas: q=-kg∇T, where kg=∑Iϰiki and ki is the conductivity of the ith gas
component.
Sj is the source or
sink term (when needed) of the jth phase. Ki=|ui|2/2 is the kinetic energy per unit of mass of the ith
gas phase (Kj for the jth solid phase).
Equilibrium–Eulerian model
In the limit Stj≪1, the drag terms fj and the
thermal exchange terms Qj can be calculated by knowing
ug and Tg only, and the Eulerian–Eulerian
model can be largely simplified by considering the dusty-gas approximation
. In this approximation, the coupling is so strong that
the decoupling terms vj and Tj-Tg go to zero, making it
unnecessary to solve the 4J equations for the decoupling in
Eq. ().
A refinement of the dusty-gas approximation (valid if Stj≲0.2), has been developed by and is discussed in what follows.
Kinematic decoupling
The Lagrangian particle momentum balance (see Eq. for its
Eulerian counterpart) reads
∂tuj+uj⋅∇uj=1τj(ug-uj)+g.
By using the Stokes law and a perturbation method (see
Appendix ), and by defining a≡Dtug (with Dt=∂t+u⋅∇), we obtain a correction to particle velocity up to first order:
uj=ug+wj-τj(∂tug+uj⋅∇ug)+O(τj2).
It can be restated that
uj=ug+Gj-1⋅wj-τja+O(τj2)Gj≡I+τjα(∇ug)T,
leading to the so-called equilibrium–Eulerian model developed by
, and for
incompressible multiphase flows. Here α is a local correction
coefficient inserted to avoid singularities introduced
by. At the zeroth order we recover uj=ug+wj, where wj is the settling velocity
defined in Eq. ().
It is worth noting from Eq. () that τj depends on
Rej, and it cannot be determined until uj is known. One
solution to this problem has been proposed by , where
τj is evaluated directly by knowing
Rẽj≡ρ^jdj3|a-g|18ν,
by approximating Eq. () in the range 0<Rej<300. We here
improve that approximation in the range 0<Rej<103 by using the
inversion formula:
Rej=Rẽj1+0.315Rẽj0.4072.
Another strategy we use in ASHEE, is to evaluate Rej explicitly
from the previous time step and correct it iteratively (within the PISO loop,
see Sect. ).
Equation () highly simplifies the Eulerian–Eulerian
Eq. () because it gives an explicit approximation of
the decoupling velocity vj, which can be used directly in
Eq. (). There, we keep the term ∇⋅(ρmTr) because of the presence of the
settling velocity wj in vj which is at the leading order.
Thermal decoupling
As pointed out in Eq. () and below, in our physical regime
the thermal Stokes time is of the same order of magnitude as the kinematic
one. However, this regime has been thoroughly analyzed in the incompressible
case by , demonstrating that the error made by
assuming thermal equilibrium is at least one order of magnitude smaller than
that on the momentum equation (at equal Stokes number), thus justifying the
limit Tj→Tg=T as done for the thermal equation in the
dusty-gas model.
This approximation allows to write in a convenient way the constitutive equation for the enthalpy:
hm=CmT+pρmCm=∑i∈IyiCi+∑j∈JyjCj.
Advantages of the equilibrium–Eulerian model
Summarizing, in ASHEE we refer to the compressible equilibrium–Eulerian
model as the PDEs listed in Eq. () with the
constitutive equations described in Sect. , the
decoupling velocity vj=uj-ug written in
Eq. (), and nil thermal decoupling Tj-Tg=0.
It is worth noting that in the Navier–Stokes equations it is critical to
accurately take into account the nonlinear term ∇⋅(ρu⊗u) because it is the origin of the major difficulties
in turbulence modeling. A large advantage of the dusty-gas and
equilibrium–Eulerian models is that in both models the most relevant part of
the drag (∑Jfj) and heat exchange (∑JQj) terms have been absorbed into the conservative derivatives for
the mixture. This fact allows the numerical solver to implicitly and
accurately solve the particles' contribution to mixture momentum and energy
(two-way coupling), using the same numerical techniques developed in
Computational Fluid Dynamics for the Navier–Stokes equations. The dusty-gas
and equilibrium–Eulerian models are best suited for solving multiphase
system in which the particles are strongly coupled with the carrier fluid and
the bulk density of the particles is not negligible with respect to that of
the fluid.
The equilibrium–Eulerian model thus reduces to a set of mass, momentum and
energy balance equations for the gas–particle mixture plus the equations for
the mass transport of the particulate and gaseous phases. In this respect, it
is similar to the dusty-gas equations, to which it reduces for
τs≡0. With respect to the dusty-gas model, the ASHEE
model solves for the mixture velocity um, which is slightly
different from the carrier gas velocity ug. Moreover, it
can compute the kinematic decoupling (i.e., the difference between the fields
ug and uj), responsible for preferential
concentration and settling phenomena (the vector vj includes
a convective and a gravity accelerations terms).
The equilibrium–Eulerian method becomes even more efficient (relative to the
standard Eulerian–Eulerian) for the polydisperse case (J>1). For each
bin of particle tracked, the standard Eulerian method requires four scalar
fields; the fast method requires one. Furthermore, the computation of the
correction to vj needs only to be done for one particle species. The
correction has the form -τja, so once the term a is
computed, velocities for all species of particles may be obtained simply by
scaling the correction factor based on the species' response times
τj. As already stated, the standard Eulerian method needs I+4+5J scalar partial differential equations, while the equilibrium–Eulerian
model needs just I+4+J – i.e., 4J equations less.
LES formulation
The spectrum of the density, velocity and temperature fluctuations of
turbulent flows at high Reynolds number typically spans over many orders of
magnitude. When the smallest turbulent length scales cannot be resolved by
the numerical grid, it is necessary to model the effects of the
high-frequency fluctuations on the resolved flow. This leads to the LES technique, in which a low-pass filter is applied
to the model equations to filter out the small scales of the solution. In the
incompressible case the theory is
well developed see, but
LES for compressible flows is still an open research field. In ASHEE, we apply
a spatial filter, denoted by an overbar (here δ is the filter scale):
ψ‾=∫ΩG(x-x′;δ)ψ(x′)dx′.
Some example of LES filters G(x;δ) used in compressible
turbulence are reviewed in . In compressible
turbulence it is also useful to introduce the so-called Favre filter:
ψ̃=ρmψ‾ρ‾m.
In Appendix , we apply this filter to Eq. (), in
order to obtain the LES filtered version of the equilibrium–Eulerian model,
Eq. ().
By applying the Favre filter to Eq. () (for the
application of the Favre filter to the compressible Navier–Stokes equations
see ,
and ), we obtain
∂tρ‾m+∇⋅(ρ‾mũm)=∑j∈JS̃j;∂t(ρ‾mỹi)+∇⋅(ρ‾mũgỹi)=-∇⋅Yi,i∈I;∂t(ρ‾mỹj)+∇⋅[ρ‾mũjỹj]=S̃j-∇⋅Yj,j∈J;∂t(ρ‾mũm)+∇⋅(ρ‾mũm⊗ũm+ρ‾mT̃r)+∇p¯=∇⋅T̃+∑j∈JS̃jũj+ρ‾mg-∇⋅B∂t(ρ‾mh̃m)+∇⋅[ρ‾m(ũm+ṽh)h̃m]=∂tp¯-∂t(ρ‾mK̃m)-∇⋅[ρ‾m(ũm+ṽK)K̃m]∇⋅(T̃⋅ũg-q̃)+ρ‾m(g⋅ũm)+∑j∈JS̃j(h̃j+K̃j)-∇⋅(Q+QK).
The terms Y,B,Q represent the contribution
of the subgrid turbulent scales (SGS). They must be modeled to close the
system in terms of the resolved fields. In ASHEE, they are
Yi=ρ‾m(yiug̃-ỹiũg)=-μtPrt∇ỹiYj=ρ‾m(yjuj̃-ỹiũj)=-μtPrt∇ỹjB=ρ‾m(um⊗um̃-ũm⊗ũm)=2dρ‾mKtI-2μtS̃mQ=ρ‾m(hmum̃-h̃mũm)=-μtPrt∇h̃mQK=ρ‾m(Kmum̃-K̃mũm)=-μtPrt∇K̃m,
respectively: the subgrid eddy diffusivity vector of the ith phase; of the
jth phase; the subgrid-scale stress tensor; the diffusivity vector of the
enthalpy and kinetic energy. The rate-of-shear tensor is
S̃m=sym(∇ũm)-13∇⋅ũmI. In order to
close the system, the SGS viscosity μt, the SGS kinetic energy
Kt and the SGS Prandtl number Prt must be
expressed in terms of the resolved variables, as detailed in
Appendix .
These coefficients can be computed either statically or dynamically
see. In ASHEE, we implemented
several SGS models . Currently, the code offers the
possibility of choosing between: (1) the compressible Smagorinsky model, both
static and dynamic see, (2) the subgrid-scale K-equation model, both
static and dynamic see, (3) the dynamical Smagorinsky model in
the form used by , (4) the WALE model, both static and dynamic
see.
Throughout this paper, we present results obtained with the dynamic WALE
model (see Fig. and the corresponding section for a study on the
accuracy of this LES model). A detailed analysis of the influence of
subgrid-scale models to simulation results is beyond the scope of this paper
and will be addressed in future works .
Numerical solver
The Eulerian model described in Sect. , is solved numerically to
obtain a time-dependent description of all independent flow fields in
a three-dimensional domain with prescribed initial and boundary conditions.
We have chosen to adopt an open-source approach to the code development in
order to guarantee control on the numerical solution procedure and to share
scientific knowledge. We hope that this will help in building a wider
computational volcanology community. As a platform for developing our solver,
we have chosen the unstructured, finite volume (FV) method, open-source C++
library, OpenFOAM® (version 2.1.1).
OpenFOAM®, released under the Gnu Public License
(GPL), has gained a vast popularity in recent years. The readily
existing solvers and tutorials provide a quick start to using the code and also
to inexperienced users. Thanks to a high level of abstraction in the
programming of the source code, the existing solvers can be freely and easily
modified in order to create new solvers (e.g., to solve a different set of
equations) and/or to implement new numerical schemes.
OpenFOAM® is well integrated with advanced tools
for pre-processing (including meshing) and post-processing (including
visualization). The support of the OpenCFD Ltd, of the
OpenFOAM® foundation and of a wide developers and
users community guarantees ease of implementation, maintenance and extension,
suited for satisfying the needs of both volcanology researchers and of
potential users – e.g., in volcano observatories. Finally, all solvers can be
run in parallel on distributed memory architectures, which makes
OpenFOAM® suited for envisaging large-scale,
three-dimensional volcanological problems.
The new computational model, called ASHEE (the ASH Equilibrium Eulerian model) is
documented in the VMSG (Volcano Modeling and Simulation Gateway) at Istituto
Nazionale di Geofisica e Vulcanologia (http://vmsg.pi.ingv.it) and is
made available through the VHub portal (https://vhub.org).
Finite volume discretization strategy
In the FV method , the governing partial
differential equations are integrated over a computational cell, and the
Gauss theorem is applied to convert the volume integrals into surface
integrals, involving surface fluxes. Reconstruction of scalar and vector
fields (which are defined in the cell centroid) on the cell interface is
a key step in the FV method, controlling both the accuracy and the stability
properties of the numerical method.
OpenFOAM® implements a wide choice of
discretization schemes. In all our test cases, the temporal discretization is
based on the second-order Crank–Nicolson scheme
, with a blending factor of 0.5 (0 meaning
a first-order Euler scheme, 1 a second-order, bounded implicit scheme) and an
adaptive time stepping based on the maximum initial residual of the previous
time step , and on a threshold that depends on the
Courant number (Co<0.2). All advection terms of the model are
treated implicitly to enforce stability. Diffusion terms are also discretized
implicit in time, with the exception of those representing subgrid
turbulence. The pressure and gravity terms in the momentum equations and the
continuity equations are solved explicitly. However, as discussed below, the
PISO (Pressure Implicit with Splitting of Operators;
) solution procedure based on a pressure correction
algorithm makes such a coupling implicit. Similarly, the pressure advection
terms in the enthalpy equation and the relative velocity vj are made
implicit when the PIMPLE (mixed SIMPLE and PISO algorithm,
) procedure is adopted. The same PIMPLE
scheme is applied treating all source terms and the additional terms deriving
from the equilibrium–Eulerian expansion.
In all described test cases, the spatial gradients are discretized by
adopting an unlimited centered linear scheme which is second-order
accurate and has low numerical diffusion
–. Analogously, implicit advective fluxes at
the control volume interfaces are reconstructed by using a centered linear
interpolation scheme (also second-order accurate). The only exception is for
pressure fluxes in the pressure correction equation, for which we adopt a TVD
(Total Variation Diminishing) limited linear scheme (in the subsonic regimes)
to enforce stability and non-oscillatory behavior of the solution. To enforce
stability, the PISO loop in OpenFOAM® usually has
incorporated a term of artificial diffusion for the advection term
∇⋅(ρu⊗u). As studied and suggested
in , we avoid using this extra term which is not
present in the original PISO implementation. We refer to
for a complete description of the discretization strategy adopted in
OpenFOAM®.
Solution procedure
Instead of solving the set of algebraic equations deriving from the
discretization procedure as a whole, most of the existing solvers in
OpenFOAM® are based on a segregated solution
strategy, in which partial differential equations are solved sequentially and
their coupling is resolved by iterating the solution procedure. In
particular, for Eulerian fluid equations, the momentum and continuity equation
(coupled through the pressure gradient term and the gas equation of state)
are solved by adopting the PISO algorithm. The PISO algorithm consists of one
predictor step, where an intermediate velocity field is solved using pressure
from the previous time step, and of a number of PISO corrector steps, where
intermediate and final velocity and pressure fields are obtained iteratively.
The number of corrector steps used affects the solution accuracy and usually
at least two steps are used. Additionally, coupling of the energy (or
enthalpy) equation can be achieved in OpenFOAM®
through additional PIMPLE iterations which derives from the SIMPLE
algorithm by. For each transport equation, the linearized
system deriving from the implicit treatment of the advection–diffusion terms
is solved by using the PbiCG solver (Preconditioned bi-Conjugate Gradient
solver for asymmetric matrices) and the PCG (Preconditioned Conjugate
Gradient solver for symmetric matrices), respectively, preconditioned by
a Diagonal Incomplete Lower Upper decomposition (DILU) and a Diagonal
Incomplete Cholesky (DIC) decomposition. The segregated system is iteratively
solved until a global tolerance threshold ϵPIMPLE is
achieved. In our simulations, we typically use ϵPIMPLE<10-7 for this threshold.
The numerical solution algorithm is designed as follows:
Solve the (explicit) continuity
Eq. () for mixture density
ρm (predictor stage: uses fluxes from previous
iteration).
Solve the (implicit) transport equation for all
gaseous and particulate mass fractions: yi,i=1,…,I
and yj,j=1,…,J.
Solve the (semi-implicit) momentum equation to obtain
um (predictor stage: uses the pressure field from
previous iteration).
Solve the (semi-implicit) enthalpy equation to update the
temperature field T, the compressibility ρm/p
(pressure from previous iteration) and transport coefficients.
Solve the (implicit) pressure equation and the
relative velocities vj to update the fluxes ρu.
Correct density, velocity with the new pressure field
(keeping T and ρm/p fixed).
Iterate from 5 evaluating the continuity error as the
difference between the kinematic and thermodynamic calculation of
the density (PISO loop).
Compute LES subgrid terms to update subgrid transport
coefficients.
Evaluate the numerical error ϵPIMPLE and
iterate from 2 if prescribed (PIMPLE loop).
With respect to the standard solvers implemented in
OpenFOAM® (v2.1.1) for compressible fluid flows
(e.g., sonicFoam or rhoPimpleFoam), the main modifications
required are the following:
The mixture density and velocity replaces the fluid ones.
A new scalar transport equation is introduced for the mass
fraction of each particulate and gas species.
The equations of state are modified as described in
Eq. ().
First-order terms from the
equilibrium–Eulerian model are added in the mass, momentum and enthalpy
equations.
Equations are added to compute flow acceleration
and velocity disequilibrium.
Gravity terms and ambient fluid stratification are added.
New SGS models are implemented.
Concerning point 5, it is worth remarking that, according
to , the first-order term τj in
Eq. () must be limited to avoid the divergence of
preferential concentration in a turbulent flow field (and to keep the
effective Stokes number below 0.2). In other words, we impose at each
time step that |vj-wj|≤0.2|ug+wj|. We test the effect of this limiter
on preferential concentration in Sect. .
ASHEE parallel efficiency on Fermi and PLX
supercomputers at
CINECA (www.cineca.it).
Parallel performances
Figure reports the parallel efficiency of the numerical
tests described in Sect. , on both the Fermi and the PLX (a Linux
cluster based on Intel Xeon ESA- and quad-core processors @2.4 GHz)
machines at CINECA. The ASHEE code efficiency is very good (above 0.9) up to
512 cores (i.e., up to about 30 000 cellscore-1), but
it is overall satisfactory for 1024 cores, with efficiency larger
than 0.8 on PLX and slightly lower (about 0.7) on Fermi probably due to the
limited level of cache optimization and input/output scalability
. The code was run also on 2048 cores on Fermi with
parallel efficiency of 0.45 .
Model verification and evaluation
Evaluation tests are focused on the dynamics of gas (Sect. ) and
multiphase (Sect. ) turbulence and on the mixing properties of
buoyant plumes (Sect. ). Compressibility likely exerts
a controlling role to the near-vent dynamics during explosive eruptions
e.g.,. Although this is not the focus of this work, we
briefly discuss in Sect. the performance of the model on a standard
one-dimensional shock wave numerical test.
Compressible decaying homogeneous and isotropic turbulence
Turbulence is a key process controlling the dynamics of volcanic plumes since
it controls the rate of mixing and air entrainment. To assess the capability
of the developed model to resolve turbulence which requires low
numerical diffusion and controlled numerical errors;,
we have tested the numerical algorithm against different configurations of
decaying homogeneous and isotropic turbulence (DHIT).
In this configuration, the flow is initialized in a domain Ω which is
a box with side L=2π with periodic boundary conditions. As described in
, , ,
and , we chose the initial
velocity field so that its energy spectrum is
E(k)=1632πurmsk0kk04e-2k2k02,
with peak initially in k=k0 and so that the initial kinetic
energy and enstrophy are
K0=∫0∞E(k)dk=12urms2H0=∫0∞k2E(k)dk=58urms2k02.
As reviewed by , the Taylor microscale can be written as
a function of the dissipation ε=2νH:
λT2≡5νurms2ε=5KH,
thus in our configuration, the initial Taylor microscale is
λT,0=5K0H0=2k0.
We have chosen the non-dimensionalization keeping the root mean square of the
magnitude of velocity fluctuations (u′) equal to
urms:
urms≡u′⋅u′Ω≡1(2π)3∫Ωu′⋅u′dx=2∫0∞E(k)dk.
We also chose to make the system dimensionless by fixing ρm,0=1, T0=1, Pr=1, so that the ideal gas law becomes
p=ρmRmT=Rm,
and the initial Mach number of the mixture based on the velocity fluctuations
reads
Marms=urms2cm2=2K0ρmγmp=urms(γmp)-12.
This means that Marms can be modified keeping fixed
urms and modifying p. Following , we define
the eddy turnover time:
τe=3λTurms.
The initial compressibility ratio C0 is defined as the ratio
between the kinetic energy and its compressible component Kc:
C0=Kc,0K0=12(2π)3K0∫Ωuc′⋅uc′dx.
Here, uc′ is the compressible part of the velocity
fluctuations, so that ∇⋅u′=∇⋅uc′ and ∇∧uc′=0.
The last parameter – i.e., the dynamical viscosity – can be given by fixing
the Reynolds number based either on λT,0 or k0:
Reλ=ρmurmsλT,03μRek0=ρmurmsk0μ.
It is useful to define the maximum resolved wavenumber on the selected
N-cells grid and the Kolmogorov length scale based on Rek0.
They Rek0 – respectively,
kmax=N2-12πLNN-1,η=2πk0Rek0-34.
In order to have a Direct
Numerical Simulation (DNS), the smallest spatial scale δ should be
chosen in order to have kmaxη>2.
We compare the DNS of compressible decaying homogeneous and isotropic
turbulence with a reference, well-tested numerical solver for DNSs of compressible turbulence by
and . For this comparison we fix the
following initial parameters: p=Rm=1, γm=1.4, Pr=1, Marms=0.2, C0=0,
urms2=2K0=0.056, k0=4, λT=0.5,
τe≃3.6596, μ=5.885846×10-4,
Reλ≃116, Rek0≃100. Thus a grid with N=2563 cells gives kmax≃127 and kmaxη=2π, big enough to have a DNS. The simulation has been performed on 1024
cores on the Fermi Blue Gene/Q infrastructure at Italian CINECA
super-computing center (http://www.cineca.it), on which about
5h are needed to complete the highest-resolution runs
(2563cells) up to time t/τe=5.465 (about 3500
time steps). The average computing speed on 1024 Fermi cores is about
1–3 Mcells/s, with the variability associated with the number of
solid phases described by the model. This value is confirmed in all benchmark
cases presented in this paper.
Isosurface at Qu≃19Hz2
and
t/τe≃2.2, representing zones with coherent
vortices.
Comparison of a DNS executed with the eight
order scheme by
and our code implemented using the
C++ libraries of OpenFOAM® at
t/τe=1.093. The L2 norm between the two spectra
is 4.0×10-4. The main parameters are Reλ≃116, Marms=0.2.
Evolution of dynamical quantities in DHIT with
Reλ≃116 and Marms=0.2 at t/τe=5.465. (a) Density fluctuations ρrms,
compressibility C and density contrast
ρmax/ρmin; (b) evolution of the
energy spectrum E(k); (c) non-dimensional
kinetic energy K/K0, enstrophy H/H0 and
Taylor microscale λT/λT,0;
(d) Kolmogorov timescale τη.
Figure shows an isosurface of the second invariant of the
velocity gradient, defined as
Qu=12(Tr(∇u))2-Tr(∇u⋅∇u).
The so-called Q-criterion allows the
identification of coherent vortices inside a three dimensional velocity
field.
In Fig. we present a comparison of the energy spectrum
E(k) obtained with the ASHEE model and the model by
after approximatively 1 eddy turnover time; the L2
norm of the difference between the two spectra is 4.0×10-4. This
validates the accuracy of our numerical code in the single-phase and
shock-free case.
Figure shows the evolution of several integral parameters
describing the dynamics of the decaying homogeneous and isotropic turbulence.
Figure a displays the density fluctuations
ρrms=〈(ρ-〈ρ〉Ω)2〉Ω, the density contrast
ρmax/ρmin and the standard measure of
compressibility C=〈|∇⋅u|2〉Ω/〈|∇u|2〉Ω
which takes value between 0 (incompressible flow) and 1 (potential flow)
. All the quantities shown in
Fig. a depend on the initial Mach number and
compressibility. For the case shown, Marms=0.2 and we
obtain very similar results to those reported in Figs. 18 and 19 by
.
Figure b shows the kinetic energy spectrum at
t/τe=0, 1.093,5.465. The energy spectrum
widens from the initial condition until its tail reaches k≃kmax≃127. Then the system dissipates and the maximum
of the energy spectrum decreases. The largest scales tend to lose energy
more slowly than the other scales and the spectrum widens also in the larger-scale
direction.
Figure c presents the evolution of K (total turbulent
kinetic energy), H (enstrophy) and λT (Taylor
microscale). The total kinetic energy decreases monotonically
and at t≃5.5τe just ≃15 % of its initial
value is conserved. On the other hand, enstrophy increases until it reaches
a maximum at 1.5<t/τe<2. It then starts to decrease
monotonically. This behavior is related to the two different stages we have
highlighted in the analysis of the energy spectrum evolution. In the first
stage, viscous effects are negligible and enstrophy increases due to vortex
stretching. During the second stage, viscous diffusion starts to have an
important role and distorted dissipative structures are created
. Also the Taylor microscale reflects this behavior,
reaching a minimum at the end of the first stage and increasing monotonically
during the second stage of the evolution. This is a characteristic of the
magnitude of the velocity gradients in the inertial range: by comparing it
with δ we can have an idea of the broadness of the range of wave
numbers where the flow is dissipative. In this DNS, we have
λT≃10.2δ at t≃5.5τe.
In Fig. d we show the evolution of the Kolmogorov timescale τη during the evolution of the decaying turbulence.
We finally compare in Fig. the DNS described with simulations at
lower resolution with N=323cells and N=643cells. In
this case, it is expected that the spectra diverge from the DNS, unless an
appropriate subgrid model is introduced to simulate the effects of
the unresolved to the resolved scales. Several subgrid models have been
tested , both static and dynamic. Figure
presents the resulting spectrum using the dynamic WALE model
. In this figure, we notice how the dynamic WALE
model works pretty well for both the 323 and 643 LES, avoiding the
smallest scales to accumulate unphysical energy.
Multiphase isotropic turbulence
In this section we test the capability of ASHEE to correctly describe the
decoupling between solid and gaseous phases when Stj<0.2 and to
explore its behavior when the equilibrium–Eulerian hypothesis Stj<0.2 is not fulfilled so that a limiter to the relative velocity
ug-uj is applied. We mainly refer
to for a quantitative assessment of numerical model results.
To this aim, we performed a numerical simulation of homogeneous and isotropic
turbulence with a gas phase initialized with the same initial and geometric
conditions described in Sect. . We added to that configuration five
solid particle classes chosen in such a way that Stj∈[0.03,1], homogeneously distributed and with zero relative velocity:
vj(x,0)=0. From Fig. d, we see that,
during turbulence decay, approximately τη∈[0.6,1.2]. Therefore,
for a given particle class with τj fixed, during the time interval
t/τe∈[0,5.5] we have
Stmax/Stmin≃2. In
Table we report the main properties of the particles
inserted in the turbulent box. To evaluate the Stokes time here we used
τj=ρ^jdj2/(18μ) because in the absence of settling,
Rej<1 when Stj<1. We set
the material density of all the particles to ρ^j=103. In order
to have a small contribution of the particle phases to the fluid dynamics –
one way coupling – here we set the solid particle mass fraction to a small
value, yj=0.002, so that yg=0.99.
Stokes time, maximum Stokes number and
diameter of the solid particles inserted in the turbulent box.
Energy spectrum E(k) at t/τe=5.465 obtained with different spatial resolutions and with/without subgrid-scale LES model.
Slice of the turbulent box at
t/τe≃2.2. The two panels represent respectively a logarithmic color map
of y3(Stmax= 0.5) and of
|ag|.
In Fig. we show a slice of the turbulent box at
t/τe≃ 2.2. Panel (a) displays the solid mass fraction,
highlighting the preferential concentration and clustering of particles in
response to the development of the acceleration field (panel b) associated
with turbulent eddies.
As described in and , a good measure for the degree of
preferential concentration in incompressible flows is the weighted average on
the particle mass fraction of the quantity (|D|2-|S|2), where S is the vorticity tensor – i.e., the skew-symmetric part of the gas velocity gradient – and D is its
symmetrical part. For compressible flows, we choose to consider
〈P〉j≡〈|D|2-|S|2-|Tr(D)|2〉j≡〈yjP-〈P〉Ω〉Ω〈yj〉Ω.
This is a good measure because (use integration by parts, the Gauss
theorem and Eq. ) with wj=0,
〈∇⋅uj〉Ω=-τj∑l,m∂lum∂mul-∂lul∂mumΩ=-τj|D|2-|S|2-|Tr(D)|2Ω.
Moreover, it is worth noting that 〈P〉j
vanishes in the absence of preferential concentration. By dimensional
analysis, preferential concentration is expected to behave as
〈P〉j∝τj/τη3DNSτj/τξ3LES,
because it must be proportional to τj and have a dimension of
[s-2]. As described by Pope (2001), the typical timescale
corresponding to an eddy length scale ξ in the inertial sub-range, can be
evaluated by means of the Kolmogorov's theory as
τξ=τλξλT23,
where the Taylor microscale λT is defined by
Eq. (). Since the time based on the Taylor microscale is
defined as
τλ=3λTurms,
we can evaluate the typical time at the smallest resolved LES scale ξ
knowing the kinetic energy K(t) and λT(t):
τξ(t)=32K(t)ξ23λT(t)13.
Evolution of the degree of
preferential concentration with
Stξ (LES) or Stη (DNS). We obtain a good
agreement between equilibrium–Eulerian LES/DNS and Lagrangian DNS
simulations. The fit for the data by is
found in
Eq. ().
In Fig. we show the time evolution of the
degree of preferential concentration as a function of the Stokes number for
both DNS with 2563cells and the LES with 323cells.
There, we multiply 〈P〉j by τξτj in
order to make it dimensionless and to plot on the same graph all the
different particles at different times together.
At t=0 the preferential concentration is zero for all Stokes numbers. Then,
preferential concentration of each particle class increases up to a maximum
value and then it decreases because of the decaying of the turbulent energy.
The maximum degree of preferential concentration is reached by each particle
class when τη is minimum (at t/τe≃1.7, see
Fig. d). Then, 〈P〉j decreases
and merges with the curve relative to the next particle class at the final
simulation time, when τη is about twice its minimum. Note that the
expected behavior of Eq. () is
reproduced for Stj<0.2 and in particular we find
〈P〉j≃1.52Stjτη-2DNS1.52Stjτξ-2LES.
Moreover, by comparing our results with the Eulerian–Lagrangian simulation
described in , we note that our limiter for the preferential
concentration when St>0.2 is well behaving.
For the sake of completeness, we found that the best fit in the range
St<2.5 for the data found by is
〈P〉j≃1.52×Stj1+3.1×Stj+3.8×Stj2τη-2,
with root mean square of residuals 8.5×10-3.
Regarding the 323 LES simulation,
Fig. shows that the Stokes number of each
particle class in the LES case is much smaller than its DNS counterpart.
In accord with , we have
Stξ=Stηηξ23,
confirming that the equilibrium–Eulerian model widens its applicability
under the LES approximation. Also note that the presented LES is able to
reproduce the expected degree of preferential concentration with
a satisfactory level of accuracy when St<0.2. In particular, the
LES slightly overestimates preferential concentration and the time needed to
reach the equilibrium and to “forget” the particle initial condition.
Turbulent forced plume
As a third benchmark, we discuss high-resolution, three-dimensional numerical
simulation of a forced gas plume, produced by the injection of a gas flow
from a circular inlet into a stable atmospheric environment at lower
temperature (and higher density). Such an experiment allows us to test the
numerical model behavior against some of the fundamental processes
controlling volcanic plumes, namely density variations, non-isotropic
turbulence, mixing, air entrainment and thermal exchange. This study is
mainly aimed at assessing the capability of the numerical model to describe
the time-average behavior of a turbulent plume and to reproduce the magnitude
of large-scale fluctuations and large-eddy structures. We will mainly refer
to laboratory experiments by and and
numerical simulations by for a quantitative assessment of
model results.
Numerical simulations describe a vertical round forced plume with heated air
as the injection fluid. The plume axis is aligned with the gravity vector and
is subjected to a positive buoyancy force. The heat source diameter 2b0 is
6.35cm, the exit vertical velocity on the axis u0 is 0.98
ms-1, the inflow temperature T0 is 568K and the
ambient air temperature Tα is 300 K. The corresponding
Reynolds number is 1273, based on the inflow mean velocity, viscosity and
diameter. Air properties at inlet are Cp=1004.5J(Kkg)-1,
R=287J(Kkg)-1 and μ=3×10-5Pas.
As discussed by the development of the turbulent plume
regime is quite sensitive to the inlet conditions: we therefore tested the
model by adding a periodic perturbation and a non-homogeneous inlet profile
to anticipate the symmetry breaking, and the transition from a laminar to
a turbulent flow. The radial profile of vertical velocity has the form
u(r)=12u01-tanhb04δrrb0-b0r,
where δr is the thickness of the turbulent boundary layer at the
plume inlet, that we have set at δr=0.1b0. A periodical forcing
and a random perturbation of intensity 0.05u0 has been superimposed to
mimic a turbulent inlet.
The resulting average mass, momentum and buoyancy flux are Q0=2.03×10-3kgs-1, M0=1.62×10-3kgms-2 and
F0=1.81×10-3kgs-1.
The computational grid is composed of 360×180×180 uniformly
spaced cells (deformed near the bottom plane to conform to the circular
inlet) in a box of size 12.8×6.4×6.4 diameters. In
particular, the inlet is discretized with 400 cells. The adaptive
time step was set to keep the Co<0.2.
Based on estimates by
, the selected mesh refinement is coarser than the
required grid to fully resolve turbulent scales in a DNS (which would require
about 720×360×360 cells). Nonetheless, this mesh is resolved
enough to avoid the use of a subgrid-scale model. This can be verified by
analyzing the energy spectra of fluctuations on the plume axis and at the
plume outer edges. In Fig. we show the energy spectra of
temperature and pressure as a function of the non-dimensional frequency: the
Strouhal number Str=f× 2b0/u0 (f is the frequency in
[Hz]). We recover a result similar to , where the
inertial–convective regime with the decay -5/3 and the inertial–diffusive
regime with the steeper decay -3 are observable .
Temperature (solid) and pressure (dashed)
fluctuations energy
spectra: (a) at a point along the plume axis (0, 0, 0.5715) m; (b) at a point along the plume outer edge (0, 0.06858,
0.5715) m. The slopes Str-5/3 and
Str-3 are represented with a thick solid and dashed
line respectively.
Three-dimensional numerical simulation of a forced
gas plume at
t=10s. (a) Isosurface of temperature T=305K, colored with the magnitude of velocity, and the temperature
distribution on two orthogonal slices passing across the inlet
center. (b) Isosurface of Qu=100s-2
colored with the value of the velocity magnitude, and its
distribution across two vertical slices passing through the inlet
center.
Model results describe the establishment of the turbulent plume through the
development of fluid-dynamic instabilities near the vent (puffing is clearly
recognized as a toroidal vortex in Fig. a). The breaking of
large-eddies progressively leads to the onset of the developed turbulence
regime – which is responsible for the mixing with the surrounding ambient air,
radial spreading of the plume and decrease of the plume average temperature
and velocity. Figure a displays the spatial distribution of
gas temperature. Mixing becomes effective above a distance of about four
diameters. Figure b displays the distribution of the
vorticity, represented by values of the Qu invariant
(Eq. ). The figure clearly identifies the toroidal vortex
associated with the first instability mode (puffing, dominant at such Reynolds
numbers). We observed the other instability modes helical and
meandering; only by increasing the forcing intensity (not
shown).
Experimental observations by and
reveal that the behavior of forced plumes far enough from the inlet can be
well described by integral one-dimensional plume models
provided that an adequate empirical entrainment
coefficient is used. In the buoyant plume regime at this Reynolds number
obtained an entrainment coefficient of 0.153.
To compare numerical result with experimental observations and
one-dimensional average plume models, we have time-averaged the numerical
results between 4 and 10 s (when the turbulent regime was fully
developed) and computed the vertical mass Q(z), momentum M(z) and
buoyancy F(z) fluxes as a function of the height. To perform this
operation, we define the time-averaging operation (⋅)¯ and the
horizontal domain:
O(z)={(x,y)∈R2|(y‾tracer(x)>0.01×ytracer,0)∧(u‾z(x)>0)},
where (x,y,z)=x are the spatial coordinates, ytracer
is the mass fraction field of a tracer injected from the vent with initial
mass fraction ytracer,0 and uz is the axial component of the
velocity field. We use this definition for O(z) for coherence
with integral plume models, where the mean velocity field is assumed to have
the same direction as the plume axis see. This hypothesis is tested in
Fig. a, where it can be verified that the time-averaged
streamlines inside the plume are parallel to the axis
(Fig. b shows the instantaneous streamlines and velocity
magnitude field).
Two-dimensional slice and streamlines of the
velocity field:
(a) time-averaged velocity field; (b)
instantaneous velocity field at t=10s. The mean
velocity field outside the plume is approximatively horizontal while
in the plume it is approximately vertical. The region where the mean
velocity field change direction is the region where the entrainment
of air by the plume occurs.
Time-averaged plume radius and velocity. The
insets display
the non-dimensional mass, momentum and buoyancy fluxes (top) and the
time-averaged entrainment coefficient. The line in the entrainment
panel is a constant fit, from which results κ=0.141±0.001.
The plume fluxes are evaluated as follows see:
mass flux Q(z)=∫Oρ‾u‾zdxdy,
momentum flux M(z)=∫Oρ‾u‾z2dxdy,
buoyancy flux F(z)=∫Ou‾zρα-ρ‾dxdy,
where ρα=ρα(z) is the atmospheric density. From these
quantities it is possible to retrieve the main plume parameters:
plume radius b(z)=Q(F+Q)πραM,
plume density β(z)=ραQ(F+Q),
plume temperature Tβ(z)=TαF+QQ,
plume velocity U(z)=MQ,
entrainment coefficient κ(z)=Q′2πραUb,
where (⋅)′ is the derivative along the plume axis and
Tα is the atmospheric temperature profile.
Figure displays the average plume radius and velocity. As
previously reported by and , the
plume radius initially shrinks due to the sudden increase of velocity due to
buoyancy (at z=0.1m). Above, turbulent mixing becomes
effective and increases the plume radius while decreasing the average
velocity. The upper inset in Fig. shows the values of
the vertical mass q=Q/Q0, momentum m=M/M0 and buoyancy f=F/F0,
normalized with the inlet values. All variables have the expected trends and,
in particular, the buoyancy flux is constant (as expected for weak ambient
stratification) whereas q and m monotonically increase and attain the
theoretical asymptotic trends shown also in Fig. .
Indeed, have shown that an integral plume model for
non-Boussinesq regimes (i.e., large density contrasts) in the approximation
of weak ambient stratification and adopting the formulation
for the entrainment coefficient, has a first integral such that q2 is
proportional to m5/2 at all elevations. Figure
demonstrates that this relationship is well reproduced by our numerical
simulations, as also observed in DNS by .
The lower inset in Fig. shows the computed entrainment
coefficient, which is very close to the value found in experiments
and numerical simulations of
an analogous forced plume. We found a value around 0.14 in the buoyant plume
region (6.4<z/2b0<16).
The analysis of radial profiles led to a similar conclusions: in
Fig. , we show the evolution of the radial profiles for the
mean vertical velocity field. In this figure, we also report the plume radius
as evaluated from Gaussian fits of these profiles on horizontal slices:
u‾z(x,y)=Ufitexp-x2+y2bfit2.
The slope of the function bfit(z) has been evaluated in the
region 6.4<z/2b0<16, to obtain bfit/z=0.142±0.001 to be
compared with the result of : bfit/z= 0.135 ± 0.010.
Finally, Fig. reports the time-averaged values of the
vertical velocity and temperature along the plume axis. As observed in
laboratory experiments, velocity is slightly increasing and temperature is
almost constant up to above four inlet diameters, before the full development of
the turbulence. When the turbulent regime is established, the decay of the
velocity and temperature follows the trends predicted by the one-dimensional
theory and observed in experiments. The inset displays the average value of
the vertical velocity and temperature fluctuations along the axis. Coherently
with experimental results , velocity fluctuations reach
their maximum value and a stationary trend (corresponding to about
30 % of the mean value) at a lower height (about three inlet diameters) with
respect to temperature fluctuations (which reach a stationary value about
40 % above four inlet diameters).
Linear regression between m5/2 and
q2 for the plume
simulation with azimuthal forcing.
Radial profiles of the time-averaged velocity
field at
various height. The scale for these profiles is indicated by the
up-down arrow on the left in the panel. The thick solid line is the
plume radius evaluated from the mass, momentum and buoyancy fluxes,
while the thick dashed line is the plume radius evaluated from
Gaussian fits of horizontal profiles.
Transonic and supersonic flows
Although not essential in the present application, the ability to solve
transonic and supersonic regimes is also required for the full-scale
simulation of volcanic processes. We here test the behavior of the ASHEE code
in the presence of shocks in the classical shock tube test case,
describing the expansion of a compressible, single-phase gas having adiabatic
index γ=1.4. At t=0 the domain of length 10 m is
subdivided into two symmetric subsets. In the first subset (spatial coordinate
x<0) we set u=0, p=105Pa, T=348.432K, so
that ρ=1kgm-3. In the second subset (x>0), we set u=0, p=104Pa, T=278.746K, so that ρ=0.125kgm-3. We indicate with c=374.348ms-1
the speed of sound of the gas in the x<0 part of the domain. We impose
zero gradient boundary conditions (∂x(⋅)=0) for all the
variables u, p, T. As described in , a reference
analytic solution exists for this problem.
In Fig. we show the density profile obtained with the
ASHEE model after 0.007 s of simulation. We performed two simulations
at different resolution. The first has 100 cells and it is compared
with the OpenFOAM® solver rhoCentralFoam
with a second order semi-discrete, non-staggered central scheme of
for the fluxes, and a total variation diminishing
limiter for the interpolation. We refer to
for a presentation of rhoCentralFoam and of
the shock tube test case. The inset of
Fig. is the simulation with a higher resolution
(1000 cells). In this figure, the code performs
satisfactorily both at low and high resolution. It is able to capture the
shocks pretty well, with a diffusion that is comparable with that obtained
with rhoCentralFoam, a solver conceived for simulating shocks.
3-D simulation of a turbulent volcanic plume
Numerical simulations of volcanic plumes were conducted in the framework of
the IAVCEI (International Association of Volcanology and Geochemistry of the
Earth Interior) plume model intercomparison initiative
, consisting in performing a set of
simulations using a standard set of input parameters so that independent
results could be meaningfully compared and evaluated, and different
approaches discussed. We here study three-dimensional numerical simulation of
a weak volcanic plume in a stratified, calm atmosphere, whose input data were
set assuming parameters and meteorological conditions similar to those of the
26 January 2011 Shinmoe-dake eruption . Initial
conditions and injection parameters are reported in
Table .
The particle size distribution is composed of two individual classes of
pyroclasts in equal weight proportion representing, respectively, fine
(diameter d=0.0625mm; density ρ^=2700kgm-3,
volume fraction ϵ=0.00086821) and coarse ash (diameter
d=1.0000mm; density ρ^= 2200 kgm-3, volume
fraction ϵ= 0.00106553). With respect to the laboratory benchmark
case of Sect. , volcanic plumes are characterized by non-Boussinesq
regimes at the vent and buoyancy reversal (with the initial mixture density
about four times larger than the atmospheric one) and by a stratified atmosphere
(Fig. ). However, the most relevant difference is due to the
significant temperature contrast (900 K) and to the presence of
a high particle content which may strongly affect the mixing properties of
the plume.
Centerline time-averaged axial velocity, and
temperature
profiles with azimuthal forcing. Inset: centerline correlations of fluctuating velocity and
temperature.
The shock tube density after
0.007 s (here c=374.348ms-1). Here we compare the analytic solution
(solid) with two simulations performed with ASHEE model (dashed
line) and the OpenFOAM®rhoCentralFoam solver (circles). The resolution is 100 cells,
while in the inset the solution obtained with the
ASHEE model with a resolution of 1000 cells is reported .
Atmospheric profiles as provided by the Japanese
Meteorological
Agency's Non-Hydrostatic Model for
Shinmoe-dake volcano at 00:00 JST on 27 January 2011.
Close-up of the computational grid used for volcanic
plume simulations.
The Stokes number of the solid particles is, in general, a complex function
of time and space, since the turbulent flow is characterized by a wide
spectrum of relevant length- and timescales. Generally, the Stokes number is
associated with the most energetic turbulent eddy scale which, in analogy
with laboratory plumes, has a typical turnover time of the order of τL∼Str2b0U0≈0.12s, where b0 and
U0 are the plume radius and velocity at the vent, respectively, and
Str is the Strouhal number, of the order Str=0.3. Based on this timescale, and computing the particle
relaxation time from Eq. (), the Stokes number for the two
adopted particle classes is about Stcoarse≈5 and
Stfine≈0.2, so we expect to see non-equilibrium
phenomena for both particles classes, with more evident effects on the
coarsest phase. However, the average value of the Stokes number in the whole
plume is not as high as calculated above. Indeed, by using
Eq. () as reference time for the turbulent dynamics, we obtain
Stcoarse≈0.1 and Stfine≈0.005. This result has been obtained a posteriori for the finer mesh
resolution, having ξ≈40m, K≈218m2s-2 and λT≈231m,
when the plume reaches its maximum height. It is worth recalling here that
the equilibrium–Eulerian approach is accurate and advantageous for particles
having St≤0.2 and that, in our model, we numerically
limit the acceleration field in order to keep the turbulent non-equilibrium
within this limit, as explained in Sect. and tested in
Sect. , Fig. . The averaged value
of this limit – measuring the importance of the decoupling limiter for this
simulation – is approximately 40 %.
Vent conditions for the weak volcanic plume
simulation.
ParameterValueVent elevation1500mVent diameter54mMass eruption rate1.5×106kgs-1Exit velocity135ms-1Exit temperature1273KExit water fraction3 wt.%Mixture density at vent4.85kgm-3
Three-dimensional numerical simulation of a weak
volcanic
plume, 400 s after the beginning of the injection (inlet
conditions as in Table ). The isosurfaces (left)
represent fine (light gray) and coarse (light sand) ash volume fractions
ϵs=10-7. The two-dimensional plots represent the
distribution of the volume concentration of fine (center) and coarse
(right) particles across a vertical orthogonal slice crossing the
plume axis.
Distribution of Ccoarse(a)
and Cfine(b), for the coarsest
particles across a vertical section at t=400s
(see Eq. ).
Time-averaged plume radius and velocity.
The insets display
the non-dimensional mass, momentum and buoyancy fluxes (top) and the
time-averaged entrainment coefficient κ. The results refer
to the simulation at maximum resolution δ=2b0/32.
Time-averaged plume radius and velocity.
The effect of the resolution and of the decoupling model is shown. In
particular, the legend refers to the following simulations: high res. is
δ=2b0/32; mid res. is δ=2b0/16; low res. is δ=2b0/8; [eqEu] refers to the equilibrium–Eulerian model; [dusty] refers
to the dusty-gas model. All simulations have been performed using the same
LES model.
The computational domain is cylindrical and extends to 483b0×765b0 in the radial and vertical directions (b0 being the vent radius). The
numerical grid is non-uniform and non-orthogonal. The discretization of the
vent is represented in Fig. a. For the highest-resolution run,
the cell size increases from a minimum grid size of δ=2b0/32 with no
radial grading factor in the region where the plume is expected to develop
(Fig. b), whose initial radius is equal to 2.5b0. The mesh
size increases linearly in the vertical direction with an angle θ such
that tanθ=0.147, slightly larger than the tanθ=0.096
predicted by the Morton's plume theory with entrainment κ=0.1. Outside this region, a radial grading factor
of 1.0446 is applied. Along z, 2048 cells are utilized. The minimum
vertical cell size is δ=2b0/32, and a grading factor of 1.00187 is
imposed. The azimuthal resolution is constant and equal to 132π
(5.625∘). The resulting total number of cells is N=10 747 904.
This numerical mesh guarantees the accuracy of the results: the solution
procedure utilizes two PISO and two PIMPLE loops to achieve an absolute residual
ϵPIMPLE=10-7 (see Sect. ).
Simulation of 720 s of eruption at the highest resolution required
about 490 000 time steps (imposing a CFL constraint of 0.2, resulting in an
average time step dt≈1.5ms, with a maximum
velocity at the vent of about 150 ms-1) for a total run-time of
about 25 days on 1024 cores on the Fermi architecture at
CINECA (meaning about 2.25 Mcellss-1, consistently with the
estimates of Sect. ). The lowest-resolution test case
(δ=2b0/8), which gave satisfactory results (see
Fig. ), could be run in about 34h on a
quad-core Intel i7@2.8GHz, meaning about
160 000cells/s.
Figure shows the development of the volcanic plume at
t=400s. Because of the atmospheric stratification, the plume
reaches a neutral buoyancy condition at about 10 km above the vent
(i.e., 11.5 km above the sea level, still within the troposphere).
Due to its inertia, the plume reaches its maximum plume height
Hmax≈12km, higher than the neutral buoyancy
level, before spreading radially to form the so-called volcanic umbrella. The
two orthogonal sections highlight the different spatial distribution of the
volumetric fraction of fine (center) and coarse (right) ash particles, due to
the different coupling regime with the gas phase. Coarse particles have indeed
a larger settling velocity ws=τsg which
causes a more intense proximal fallout from the plume margins and a reduced
transport by the umbrella.
Besides settling, the large inertia of the coarse ash is responsible for the
kinematic decoupling, leading to preferential concentration and clustering of
particles at the margins of turbulent eddies. To illustrate this phenomenon,
in the non-homogeneous flow, the instantaneous preferential concentration is
computed as the (normalized) ratio between the jth particle concentration
and the concentration of a tracer (in our case, water vapor), i.e.,
Cj=yjyj,0⋅ytracer,0ytracer,
where the 0 subscript corresponds to the value at the vent.
Figure a shows the distribution of Cj
for the coarsest particles at t=400s. The color scale is
logarithmic and symmetric with respect to 1, which corresponds to the nil
preferential concentration. For Cj<1, the mixture is
relatively depleted of particles (green to blue scale); for Cj>1, particles are clustered (green to red scale), with mass fraction up to five
times larger and 20 times smaller than the value it would have in absence of
preferential concentration. This behavior is expected to affect the mixing
and entrainment process. It is also worth remarking that the more uniform red
area beyond the plume margins corresponds to the region of settling particles
below the umbrella region. On the other hand, the top of the plume is
relatively depleted of coarse particles. The corresponding
Fig. b for fine particles confirms that these are
tightly coupled to the gas phase and almost behave as tracers (value of
Cfine is everywhere around 1). These conclusions are
coherent with the a priori estimate of Stj we gave at the
beginning of this section, based on the Taylor microscale time
(Eq. ).
Finally, we present the results obtained by averaging the volcanic plume flow
field over time (in a time-window [300–720] s where the plume has reached
statistically stationary conditions) and over the azimuthal angle, in order
to allow comparison with one-dimensional integral models
e.g., and discuss the effect of numerical resolution.
The averaging procedure is the generalization to the multiphase case of that
explained in Sect. see.
Figure is analogous to Fig. for the
laboratory plume test case and presents the results of the averaging
procedure for the simulation at the highest resolution δ=2b0/32. The
one-dimensional average clearly highlights the existence of a maximum plume
height, where the averaged plume velocity, the mass flux q and the momentum
flux m go to zero. In the jet stage the velocity decreases to reach a
minimum of about 40ms-1 at z≃2km above the
vent. Above, buoyancy reversal occurs and the plume slightly accelerates,
while the radius increases almost linearly, reflecting the self-similar flow
structure . The computed entrainment coefficient shows
a different behavior with respect to the laboratory case, associated with the
effect of the density contrast. In this case, a maximum value of about
κ∼0.1 is obtained in the buoyant plume region between 2 and
5 km above the vent. Analogously to the laboratory plume case, the
entertainment coefficient is much lower in the jet stage (κ≃0.05÷0.07). Interestingly, we find that in three-dimensional simulations the
entrainment decreases near the NBL and it becomes negative above that level.
This happens because the mass exits from the plume region moving into the
umbrella cloud. We plan to study this behavior more thoroughly in future
studies.
To analyze the effect of the grid resolution, we plot in
Fig. the plume radius b(z) and vertical velocity
U(z) at resolution δ=2b0/16 (mid res.) and δ=2b0/8 (low
res.). In addition, we show in the same plot the results of the dusty-gas
model [dusty] at low res. Results demonstrate that the numerical model is
robust and accurate so that even low-resolution simulations are able to
capture the main features of the volcanic plume development. However, the
maximum plume height systematically decreases from 12 100 m (a), to
11 300 m (b) to 11 000 m (c) when we decrease the
resolution. Analogously, the Neutral Buoyancy Level (NBL) decreases from
7800 m (a) to 7200 m (b) to 7100 m (c). Although the
lowest-resolution run seems to underestimate the maximum plume height and the
plume radius by about 10 %, the average velocity profile is consistent in
the three runs, showing a jet-plume transition at about 2000 m above
the vent, also corresponding to the transition to a super-buoyancy region
.
The dusty-gas model shows a significantly different behavior, with a larger
plume radius, and a more marked jet-plume transition with no further
acceleration (without a super buoyancy transition). The plume height is
slightly lower than the non-equilibrium case at the same resolution having
maximum plume height and neutral buoyancy level of 9900 and 6100 m,
respectively. Numerical simulations thus suggest that the effects of
non-equilibrium gas–particle processes (preferential concentration and
settling) on air entrainment and mixing are non-negligible. These effects are
certainly overlooked in the volcanological literature and will be studied
more thoroughly in future studies, by applying the present model to other
realistic volcanological case studies (see ).
Conclusions
We have developed a new, equilibrium–Eulerian model to numerically simulate
compressible turbulent gas–particle flows. The model is suited to simulate
relatively dilute mixtures (particle volume concentration
ϵs≲10-3) and particles with Stokes number
St≲0.2. It is appropriate to describe the dynamics of volcanic
ash plumes, with kinematic decoupling between the gas and the particles,
assumed in thermal equilibrium.
We have tested the model against controlled experiments to assess the
reliability of the physical and numerical formulation and the adequacy of the
model to simulate the main controlling phenomena in volcanic turbulent
plumes, and in particular: (1) multiphase turbulence (including preferential
concentration and density effects), (2) buoyancy and compressibility effects,
(3) stratification and density non-homogeneity.
The model reproduces the main features of volcanic plumes, namely:
(1) buoyancy reversal and jet-plume transition, (2) plume maximum height and
spreading of the umbrella above the neutral buoyancy level, (3) turbulent
mixing and air entrainment, (4) clustering of particles, (5) proximal
particle fallout. Results demonstrate that the compressible
equilibrium–Eulerian approach adopted in the ASHEE model is suited to
simulate the three-dimensional dynamics of volcanic plumes, being able to
correctly reproduce the non-equilibrium behavior of gas–particle mixtures
with a reduced computational cost with respect to that expected from
Eulerian–Eulerian models.
Finally, the adopted open-source computational infrastructure, based on
OpenFOAM®, will make the model easily portable
and usable and will ease the maintenance and implementation of new modules,
making ASHEE suitable for collaborative research in different volcanological
contexts.
Code availability
The ASHEE code with the input files for some simulation presented in this
work are available for download on the site for collaborative volcano
research and risk mitigation Vhub (https://vhub.org/groups/ashee). Code
documentation is on the Volcano Modelling and Simulation gateway of INGV Pisa
(http://vmsg. pi.ingv.it/). A gallery of movies of numerical
simulations performed with the ASHEE code can be found at
“https://sites.google.com/site/matteocerminara/”.
All the symbols used in this paper are listed here.
Table of symbols.
aacceleration fieldbplume radiusBsubgrid-scale stress tensorcspeed of soundCspecific heat at constant volumeCDdrag coefficientCpspecific heat at constant pressureCjnormalized mass concentration ratio of the jth phasedsolid particles diameterDsymmetrical part of the velocity gradientEturbulent energy spectrumfnon-dimensional buoyancy fluxfjdrag force per volume unity between the gas and the jth solid phaseFbuoyancy fluxggravity acceleration magnitudeggravity accelerationGequilibrium–Eulerian model operatorhenthalpy per unity of massHplume heightHenstrophyiindex running over all the gas phasesInumber of gas phasesIset of all the gas phases indicesIidentity tensorjindex running over all the solid phasesJnumber of solid phasesJset of all the solid phases indiceskwavenumberkgthermal conductivity of the gas phaseKkinetic energy per unity of massLlength scalemnon-dimensional momentum fluxMmomentum fluxNnumber of mesh cellsOhorizontal domain, corresponding to plume sectionsppressurePpreferential concentrationqnon-dimensional mass fluxqheat flux
Continued.
Qmass fluxQusecond invariant of the velocity gradientQsheat exchange between the gas and solid phaseQdiffusivity vector of the enthalpyQKdiffusivity vector of the kinetic energyrradial coordinateRgas constantSsource (sink) termSrate-of-shear tensorSskew-symmetric part of the velocity gradientttimeTtemperatureTstress tensorTtemporal intervaluvelocity magnitudeuvelocity fieldu′velocity fluctuationsurrelative velocity between the gas and the mixtureUvelocity scale; plume axial velocityvjrelative velocity between the jth solid and the gas phasesvψvelocity correcting the mixture advection term of the generic field ψ due the decouplingVvolumewterminal settling velocityxhorizontal coordinatexposition vectorymass fraction; horizontal coordinateYsubgrid eddy diffusivity vectorzaxial coordinateαatmospheric densityβplume densityγspecific heat ratioδsize of the smallest mesh cellδrthickness of the turbulent boundary layer∂tpartial derivative with respect to time∇del operatorϵvolumetric concentrationϵPIMPLEPIMPLE loop residual errorεturbulent dissipationηKolmogorov length scaleκentrainment coefficientϰimolar fraction of the ith gas component
Continued.
λp-pmean free path of the particlesλTTaylor microscaleμdynamic viscosityνkinematic viscosityξLES length scaleρ^densityρbulk densityτtypical timescaleτjStokes time of the jth solid phaseτTStokes time of the thermal decouplingϕccorrection factor to the Stokes timeψgeneric flow fieldΩspatial domainCoCourant numberMaMach numberNuNusselt numberPrPrandtl numberReReynolds numberStStokes numberStrStrouhal number(⋅)0relative to the initial time; relative to the vent level(⋅)ccompressible part(⋅)coarserelative to the coarse particles(⋅)erelative to the eddy turnover scale(⋅)finerelative to the fine particles(⋅)fitrelative to the Gaussian fit(⋅)grelative to the gas phase(⋅)irelative to the ith solid phase(⋅)jrelative to the jth solid phase(⋅)krelative to the wavenumber k(⋅)Lrelative to the length scale L(⋅)mrelative to the mixture(⋅)maxmaximum value(⋅)minminimum value(⋅)rmsroot-mean-square(⋅)srelative to the solid phase(⋅)tracerrelative to the tracer(⋅)zaxial component(⋅)αrelative to the atmosphere(⋅)ηrelative to the Kolmogorov scale η(⋅)λrelative to the Taylor microscale(⋅)ξrelative to the LES scale ξ(⋅)¯filtered quantity; time averaged quantity(⋅)̃Favre-filtered quantity〈⋅〉jspatial averaged quantity, with weight yj〈⋅〉Ωquantity averaged over the domain ΩDerivation of the Eulerian-Eulerian model in “mixture” formulation
In the regime described in Sect. , the Eulerian–Eulerian
equations for a mixture of a gas and a solid dispersed phase are
∂tρi+∇⋅(ρiug)=0,i∈I;∂tρj+∇⋅(ρjuj)=Sj,j∈J;∂t(ρgug)+∇⋅(ρgug⊗ug)+∇p=∇⋅T+ρgg-∑j∈Jfj;∂t(ρjuj)+∇⋅(ρjuj⊗uj)=ρjg+fj+Sjuj,j∈J;∂t(ρghg)+∇⋅(ρghgug)+∇⋅(q-T⋅ug)=∂tp-∂t(ρgKg)-∇⋅(ρgKgug)+ρg(g⋅ug)-∑j∈J(uj⋅fj+Qj);∂t(ρjhj)+∇⋅(ρjhjuj)=Qj+Sjhj,j∈J;
which are the I+4+5J balance laws of mass, momentum and enthalpy of
the gaseous and solid phases, respectively.
In this formulation the equations are solved for each phase singularly.
However, when the Stokes number is small, and consequently the coupling
between the phases is strong, the numerical solution of this formulation can
be demanding, because the coupling terms fj (Eq. )
and Qj (Eq. ) become very important. An
alternative is to reformulate the whole problem in terms of the mixture
fields (ρm,um,hm), using the
explicit form of the coupling terms only to determine the velocity and
temperature difference between the phases.
We show in what follows how Eq. () can be expressed in terms
of the mixture fields.
Advection in mixture formulation
Let the particle velocity field be uj=ug+vj. Recalling the definition for the mass fraction
and the mixture density given at the beginning of Sect. , we
here define the mixture velocity field um and the relative
velocity between the gas and the mixture ur through the
mass weighted average:
um=∑i∈Iyiug+∑j∈Jyjujug=um+urur=-∑j∈Jyjvj.
Using these definitions, the advection of a generic field ψ can be
rewritten; ∑Iρiψi+∑Jρjψj=ρmψm;∑i∈Iρiψiug+∑j∈Jρjψjuj=ρmψmum+ρm∑j∈Jyjvj(ψj-ψm)=ρmψm(um+vψ),
where
vψ=∑Jyjvj(ψj-ψm)ψm
can be defined where ψm≠0. This velocity field takes into
account the kinematic decoupling vj, correcting the advection
term of ψm.
Continuity equations
Summing up over i and j in Eqs. () and (),
we obtain the continuity equation for the mixture:
∂tρm+∇⋅(ρmum)=∑j∈JSj,
while for the phases we have
∂t(ρmyi)+∇⋅(ρmugyi)=0,i∈I;∂t(ρmyj)+∇⋅(ρmujyj)=Sj,j∈J.
It is worth noting that the mixture density follows the classical continuity
equation with velocity field um.
Momentum equation
Summing up over i and j the gas and particle momentum
Eqs. () and (), and using Eq. ()
with ψ=u, we obtain
∂t(ρmum)+∇⋅(ρmum⊗um+ρmTr)+∇p=∇⋅T+ρmg+∑j∈JSjuj,
where Tr=∑J(yjvj⊗vj)ur⊗ur. This
equation is the classical compressible Navier–Stokes equation with the
substitution ug→um and the addition of
the term ∇⋅(ρmTr) which takes into
account the effects of particle decoupling on momentum (two-way coupling).
Enthalpy equation
The same technique can be used for the enthalpy in
Eq. (). By defining
hm=∑Iyihi+∑JyjhjKm=∑IyiKi+∑JyjKj=12|um|2+12∑Jyj|vj|2-12|ur|2,
summing up all the enthalpy Eqs. () and (), and
using Eq. () with ψ=h and ψ=K, we obtain
∂t(ρmhm)+∇⋅ρmhm(um+vh)-∇⋅(T⋅ug-q)==∂tp-∂t(ρmKm)-∇⋅ρmKm(um+vK)+ρm(g⋅um)+∑j∈JSj(hj+Kj),
where
vh=ur+∑Jyjhjvjhm=∑Jyj(hj-hm)vjhmvK=ur+∑JyjKjvjKm=∑Jyj(Kj-Km)vjKm
are the correction velocity fields taking into account the combined effect
due to kinematic decoupling and difference between the enthalpy (vh)
and kinetic energy (vK) of the mixture and of the jth species.
Decoupling
The great advantage of Eqs. () and () is that the
coupling terms fj and Qj cancel out when summing the
equations. The equations for the mixture momentum and enthalpy differ from
the corresponding single-phase equations only because of the new decoupling
terms (those proportional to Tr,vh,vK),
which are small in the strongly coupled regime (they goes to zero in the
dusty-gas and in the one-way coupling approximations). All the I+4+J
equations of mass, momentum and enthalpy conservation for the mixture are
summarized in Eq. ().
However, we started with I+4+5J equations. The remaining 4J equations are
those modeling the decoupling, which depend on fj and
Qj. They are the 3J momentum and J enthalpy equations for the
solid phases. They should be solved together with the mixture equations in
order to find the kinematic decoupling vj and the thermal decoupling
Tj-Tg. Another possibility is to use the equilibrium–Eulerian
model described in Sect. .
Derivation of the equilibrium–Eulerian approximation through
asymptotic expansion
Equation () is nonlinear because of the convective term
uj⋅∇uj but also because of the correction term
ϕc(Rej)(ug-uj) in the Stokes drag
force (see Eq. ). As pointed out and analyzed
in , the latter nonlinear term can be considered as
slowly variable and treated as a constant in the following analysis.
Here we want to solve Eq. () using an asymptotic expansion
technique. Indeed, letting 1/τj→+∞ and considering t≫τj, it is possible to formally solve that equation. In our
volcanological applications there are some zones in the domain where the
gravitational effect (particle fallout) is dominant, thus we must consider
the term wj=τjg at the leading order. In other words,
we must consider g=O(ug/τj) and rewrite
Eq. () in terms of the terminal velocity wj=τjg already defined in Eq. (). Then,
multiplying Eq. () by et/τj and calling V=ujet/τj, we get
∂tV+uj⋅∇V=(1τj(ug+wj))et/τj,
which is a transport equation, with solution
V(X(x0,t),t)=V0(x0)+∫0t1τj(ug(X(x0,s),s)+wj)es/τjds,
with X(x0,t) such that
dtX(x0,t)=uj(X(x0,t),t)X(x0,0)=x0.
Thus we have formally obtained uj:
uj(X(x0,t),t)=uj,0(x0)e-t/τj+(1-e-t/τj)wj+∫0t1τjug(X(x0,t-s),t-s)e-sτjds,
where uj(X(x0,t),t) is the velocity of the
particle “x0” evaluated in its position at time t. In
order to carry out the asymptotic expansion, we perform the Taylor
expansion of ug around s=0:
ug(X(x0,t-s),t-s)=∑n=0+∞(-1)nn!dnugdtn(X(x0,t),t)sn.
Using the relation
∫1τjsne-sτjds=-e-sτj∑k=0nn!k!τjn-ksk
and supposing that the series converges uniformly, we get
∫0t1τjug(X(x0,t-s),t-s)e-sτjds=∫0t1τj∑n=0+∞(-1)nn!dnugdtn(X(x0,t),t)sne-sτjds=-∑n=0+∞(-1)ndnugdtn(X(x0,t),t)e-sτj∑k=0n1k!τjn-ksk|0t=∑n=0+∞(-1)nτjndnugdtn1-∑k=0n1k!tτjke-tτj.
Thus
uj(X(x0,t),t)-ug(X(x0,t),t)-wj=(uj,0(x0)-ug(X(x0,t),t)-wj)e-t/τj-τjdugdt(X(x0,t),t)1-e-t/τj-tτje-t/τj+τj2d2ugdt21-e-t/τj-tτje-t/τj-12tτj2e-t/τj+O(τj3).
If now we consider t≫τj, neglecting the transient phase in
which particles reach the equilibrium with the fluid
For this
reason the model is known as equilibrium–Eulerian model.
, we obtain
uj(X(x,t),t)=ug(X(x,t),t)+wj-τjdugdt(X(x,t),t)+τj2d2ugdt2(X(x,t),t)+O(τj3),
which, using Eq. (), gives us
uj=ug+wj-τj(∂tug+uj⋅∇ug)+O(τj2).
Note that we here obtain the same expansion of reported and
discussed in and .
LES filtering approachFiltering the equilibrium–Eulerian model
In order to obtain the LES filtered version of Eq. (), we apply
the Favre filter Eq. () to the equilibrium–Eulerian model
fundamental Eq. () modified as follows:
uj=ug+wj-τj(∂tum+um⋅∇um+(wr+uj-ug)⋅∇um),
moving the new second-order terms into O(τj2), using
∂tyj+uj⋅∇yj=0, defining
wr=-∑jyjwj,
and recalling that at the leading order ũm≃ũg-w̃r. Multiplying the new
expression for uj by ρm and Favre filtering, at the
first order we obtain
ũj=ũg+G̃-1⋅wj-τjãm+w̃r⋅∇ũm-τjρ‾m∇⋅B,
where we have used ãm=∂tũm+ũm⋅∇ũm,
τ̃j=τj and consequently w̃j=wj
because the Stokes time changes only at the large scale and it can be
considered constant at the filter scale. Moreover, we have used the
definition subgrid-scale Reynolds stress tensor: B=ρ‾m(um⊗um̃-ũm⊗ũm). As
discussed and tested in , the subgrid terms can be
considered O(τj) and neglected when multiplied by first-order terms.
Filtering the ASHEE model
To filter the momentum advection term of the ASHEE model ()
we used the Boussinesq eddy viscosity hypothesis Eq. (), where the
deviatoric part of the subgrid stress tensor can be modeled with an eddy
viscosity μt times the rate-of-shear tensor
S̃m. On the other hand, the first term on the
right-hand side of Eq. () is the isotropic part of the
subgrid-scale tensor, proportional to the subgrid-scale kinetic energy
Kt. While in incompressible turbulence the latter term is
absorbed into the pressure, it must be modeled for compressible flows
(see ; ). showed
another way to treat this term by absorbing it into a new
macro-pressure and macro-temperature (see also
; ).
To filter the advection of a generic field ψ, we used the eddy
diffusivity viscosity model (see ): any scalar ψ
transported by um generates a subgrid-scale vector that can
be modeled with the large eddy variables. We have
ρ‾m(umψ̃-ũmψ̃)=-μtPrt∇ψ̃,
where Prt is the subgrid-scale turbulent Prandtl
number.
Moreover, we used additional approximations to filter all the equations. The
viscous terms in momentum and energy equations, and the pressure-dilatation
and conduction terms in the energy equations are all nonlinear terms and we
here treat them as done by and . The
subgrid terms corresponding to the former nonlinear terms could be neglected
so that, for example, p∇⋅ug‾≃p‾∇⋅ũg. In particular, this
term has been neglected also in presence of shocks
see. We refer to for a priori and
a posteriori analysis of all the neglected terms of the compressible
Navier–Stokes equations. Moreover, in our model the mixture specific heat
Cm and the mixture gas constant Rm vary in the domain
because yi and yj vary. Thus, also the following approximations should
be done, coherently with the other approximations used: h̃m=CmT+p/ρm̃≃C̃mT̃+p‾/ρ‾m and
∑IyiRiT̃≃∑IỹiRiT̃.
Acknowledgements
This work includes some results achieved in the PhD work by the first author
(MC), carried out at Scuola Normale Superiore, Pisa, with a grant by Istituto
Nazionale di Geofisica e Vulcanologia. L. C. Berselli and M. Cerminara are
members of GNAMPA, of the Istituto Nazionale di Alta Matematica “F.
Severi”. We thank A. Neri and M. de' Micheli Vitturi for their suggestions
and help about volcanological models, and M. Bernardini and S. Pirozzoli for
useful discussion on decaying turbulence and for providing DNS data for model
comparison and validation. We thank A. Folch and A. Costa for their thorough
reviews which have greatly improved the paper. We acknowledge the CINECA for
the availability of high-performance computing resources and technical
support on porting OpenFOAM® on HPC architectures
by I. Spisso and M. Culpo. In particular, this work took advantage of the
CINECA infrastructure through the ISCRA projects: IsB06 VolcFOAM, IsC26
VolcAshP and IsC07 GEOFOAM.Edited by: S. Marras
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