Lagrangian Volcanic Particle Dispersion in the Atmosphere

Dipartimento di Ingegneria Civile e Industriale Sezione Ingegneria Aerospaziale

Tesi di Laurea Magistrale

Uncertainty Quantification in

Lagrangian Volcanic Particle Dispersion in the Atmosphere

Relatori Allievo

Prof. Maria Vittoria Salvetti Federica Pardini Dott. Mattia de’ Michieli Vitturi

Anno Accademico 2013/2014

The aim of this thesis is to deal with the Uncertainty Quantification in vol-

canic ash dispersion and deposition. The Lagrangian code, LPAC, used to

calculate the particles trajectories has been coupled with DAKOTA toolkit,

a code which is able to perform the uncertainty analysis. The uncertainties

we have studied are those related to the initial conditions of the particles

released by the volcanic plume. In particular, we have considered as uncer-

tain the mean diameter of the inlet ash distribution, its standard deviation

and the fragment sphericity. The mathematical technique employed to study

the propagation of the uncertainties is the so called Generalized Polynomial

Chaos Expansion method (PCEg). The main outputs of our analysis are

the the statistical parameters that permits to reconstruct the distribution

of parcels in air and of deposited parcels in 4 vertical stripes at different

distances from the ash source and for different time instants after parcels

emission. We have observed that the changes in the outputs are mainly due

to input mean diameter and sphericity variations. We have also concluded

that the input ash characteristics of the volcanic plume should be extrapo-

lated/reconstructed by considering the whole ash deposit, which means also

the proximal and the distant areas from the vain, areas which in general are

not simple to be sampled. Even if this work represents a first step to be

developed by future researches, we have tried to show a methodology useful

to support not only the study of ash dispersion, but also the evaluation of

the grain size distribution of tephra fall deposits.

Introduction 10

1 Description of the Lagrangian Code 13

1.1 LPAC (Lagrangian Particles Advection Code) . . . . 13

1.1.1 Equation of motion: Basset-Boussinesq-Oseen Equa- tion . . . . 14

1.1.2 Numerics . . . . 22

1.2 Eulerian flow field . . . . 25

1.2.1 Kelvin-Helmholtz Instability . . . . 25

1.2.2 Characteristics of the wind field . . . . 28

2 Uncertainty Quantification and Sensitivity Analysis 34 2.1 Introduction . . . . 34

2.2 Uncertainties and Errors . . . . 35

2.3 Uncertainty Quantification . . . . 37

2.3.1 Generalized Polynomial Chaos Expansions (PCEg) . . 37

2.4 Sensitivity Analysis . . . . 44

2.4.1 Global Sensitivity Analysis: Variance-based decompo- sition (VBD) . . . . 44

3 Simulation Set Up 46 3.1 Grain Size Analysis . . . . 46

3.1.1 Particles Size Distribution . . . . 50

3.1.2 Statistical Parameters . . . . 53

3.3 Definition of the Uncertain Input Parameters . . . . 59

3.3.1 Creation of the input parcels size distribution . . . . . 60

3.3.2 Choice of the range of variation . . . . 62

3.4 Output quantities of interest . . . . 66

4 PCEg Setting Procedure 67 4.1 Method . . . . 67

4.2 Choice of Quadrature Order . . . . 68

5 Analysis of the Results 74 5.1 Some Deterministic Solutions . . . . 77

5.1.1 Effect of the mean size . . . . 78

5.1.2 Effect of the standard deviation . . . . 80

5.1.3 Effect of the sphericity . . . . 83

5.2 Polynomial Response Functions . . . . 86

5.3 Statistical Analysis of the Response Functions . . . . 89

5.3.1 Cumulative and Probability Distribution Functions . . 89

5.4 Sensitivity Analysis of the Output Values . . . 101

5.4.1 Main Sobol Indices . . . 101

5.4.2 Total Sobol Indices . . . 103

5.5 Analysis of the Drag Coefficient . . . 108

Conclusions 112

1.1 Drag coefficient of a spherical particle as a function of Reynolds number [8]. . . . . 15 1.2 Drag coefficient for spherical and nonspherical particles trans-

formed in generalized drag coefficients and generalized Reynolds number [11]. . . . . 18 1.3 (a) Staggered grid adopted by the PDAC code used to describe

the carrier flow field. Three different interpolations of the carrier flow field are required to determine the local radial velocity (b), the vertical velocity (c), and the pressure gradient (d) of a particle with position (x; z) [9]. . . . 23 1.4 Sketches of a strong and a weak plume [7]. . . . . 25 1.5 Plume of Mt.Etna eruption of 24 November 2006. . . . 26 1.6 Cloud ash of 24 November 2006 eruption captured by MOSIS

satellite, source NASA . . . . 26 1.7 KHI present in natural phenomena. . . . 27 1.8 Mechanism of formation of KHI. . . . 27 1.9 Setting table of WRF for the generation of the carrier flow

field [18]. . . . . 28 1.10 Wind field at the beginning of the Lagrangian simulation. The

Kelvin-Helmholtz instability are found to originate in corre- spondence of the volcanic plume, centered at ∼ 4000m a.s.l. . 29 1.11 Wind field after little more than an hour of simulation. The

KHI are evident and they will remain stable also in the next

times. . . . . 30

1.12 Wind field at time t = 9100s. Not significant differences can

be found with respect to the previous instant of time. . . . . 31

1.13 Wind field at the end of the simulation, KHI is still present. . 32

1.14 Trajectories of two groups of particles of different diameters. . 33

2.1 Clenshaw-Curtis grid used in the present work [1]. . . . . 43

2.2 Plot of f (x ₁ , x ₂ , x ₃ ) = x ^1/2 ₁ x ₃ + 3x ³ ₃ + 10x ₂ evaluated for 1000 random samples. . . . . 45

3.1 Common instruments used to evaluate the grain size of a deposit. 48 3.2 Udden-Wentworth grain-size classification of terrigenous sedi- ments [22]. . . . 49

3.3 Total grain-size distribution of tephra-fallout deposits esti- mated for the Plinian eruption of Mount St.Helens1980 [7]. The scale reported in abscissas is the Krumbein scale. . . . 50

3.4 Discrete number frequency distribution. . . . 51

3.5 Number of parcels distribution of a sediment composed by 10000 parcels with a mean diameter of 0φ and a Standard deviation of 1φ. . . . 51

3.6 Density trend as a function of the particle diameter [7]. . . . . 52

3.7 Gaussian distribution characterized by three different value of skewness: equal to zero, positive and negative skewness. . . . 55

3.8 Variation of the Gaussian PDF according to the kurtosis value. 56 3.9 Representation of the computational domain. . . . . 57

3.10 Positions from which the parcels are released. Each red point represents an initial position of the parcels. . . . . 58

3.11 Different sketches of parcels distribution. . . . 61

3.12 Grain size distributions referred to Mount Etna eruptions. . . 62

3.13 Particle sphericity versus roundness [14]. . . . 64

3.14 Input distribution variables . . . . 65

4.1 CDFs for 5 different quadrature orders. . . . 69

4.2 Sketch of the quadrature grid. . . . . 72

4.3 Image of a polynomial response function present in the output file of DAKOTA. For each coefficient, the P i are the unidimen- sional Legendre polynomials which are mixed together in order to generate a multivariate basis. . . . . 73 5.1 Plot of the parcels positions at time t=7600s. The input

parameters of this simulation are the mean ones: µ = 1φ, σ = 1.5φ, ψ = 0.7 . . . . 75 5.2 Parcels positions at time 9100 s. . . . . 75 5.3 Parcels positions at time 10600 s. . . . 75 5.4 Parcels size distribution plotted at three time instants. The

simulation taken into account is still the one with the mean input values. . . . 76 5.5 Parcels size distribution in each region of the domain associ-

ated with the input value of σ and ψ equal to 1.5φ and 0.7 respectively, while µ varies from the minimum value of 0φ to the maximum one of 2φ. Also the mean solution associated to µ = 1φ is reported. . . . 78 5.6 Parcels size distribution associated to µ = 1φ and ψ = 0.7

(mean input value), while σ varies from its minimum to its maximum value, which are 1.3ψ and 1.7ψ respectively. . . . . 81 5.7 Parcels size distribution in each region of the domain associ-

ated with the input value of µ and σ equal to 1φ and 1.5φ respectively, while ψ varies from the minimum value of 0.5 to the maximum one of 0.9. . . . 83 5.8 Sketch of the polynomial response function associated to the

mean value of the parcels diameter in Stripe 1-Air. The do- main reported on x-y axis refers to the range of existence of Legendre polynomials[-1,+1]. . . . 87 5.9 Mean parcels diameter associated to Stripe 1-Ground. The

domain reported on x-y axis refers to the range of existence of

Legendre polynomials[-1,+1]. . . . 88

5.10 CDFs (a) and PDFs (b) related to the mean parcels diameter. 90

5.11 CDFs (a) and PDFs (b) of the standard deviation of the parcels distribution. . . . 93 5.12 Scketchs of the CDFs (a) and PDFs (b) associated with the

skewness value in each cell. . . . 95 5.13 CDFs (a) and PDFs (b) which show the trend of the kurtosis

value on the domain. . . . 97 5.14 CDFs (a) and PDFs (b) of the number of parcels distribution

over the domain. . . . . 99 5.15 Main Sobol Indices of the output values concerning the air

distributions.The variability of the output values (i.e. mean diameter of parcels distribution µ, standard deviation σ, skew- ness (Sk), kurtosis (Ku) and number of parcels (P)) is due to the variation of the input values of µ and ψ. . . 102 5.16 Main Sobol Indices of the output values concerning the ash

deposit on the ground. In this case the contribution of ψ variation is leader also with respect to µ variation. . . . 102 5.17 Total Sobol Indices related to the air parcels distributions. . . 103 5.18 Total Sobol Indices related to the air parcels distributions. . . 103 5.19 Main Sobol Indices and Total Sobol Indices computed for the

output values of Stripe 1-Air. The three graphs represent the contribution to the output variability due to µ, σ and ψ re- spectively . . . 104 5.20 Comparison between Main Sobol Indices and Total Sobol In-

dices computed for the output values of Stripe 2-Air. . . 105 5.21 Comparison between Main Sobol Indices and Total Sobol In-

dices computed for the output values of Stripe 3-Air. . . 105 5.22 Comparison between Main Sobol Indices and Total Sobol In-

dices computed for the output values of Stripe 4-Air. . . 106 5.23 Comparison between Main Sobol Indices and Total Sobol In-

dices computed for the output values of Stripe 1-Ground. . . . 106 5.24 Comparison between Main Sobol Indices and Total Sobol In-

dices computed for the output values of Stripe 2-Ground. . . . 107

5.25 Comparison between Main Sobol Indices and Total Sobol In- dices computed for the output values of Stripe 3-Ground. . . . 107 5.26 Comparison between Main Sobol Indices and Total Sobol In-

dices computed for the output values of Stripe 4-Ground. . . . 108 5.27 Trend of C _D as a function of the particles diameter for the

three different values of sphericity. . . 109 5.28 Trend of C _D as a function of the particles diameter varying

the mean input diameter. . . . 109

5.29 Drag coefficient as a function of Reynolds number. . . 110

2.1 Askey Scheme [1]. . . . 38 3.1 Classification of grain size distribution considering the stan-

dard deviation [16]. . . . 54 3.2 Classification of grain size distribution as a function of the

skewness value [16]. . . . 55 3.3 Summary of the input parameters together with their range

of variation. . . . 65 3.4 Summary of the output quantities of interest computed in each

cell. . . . . 66 4.1 Total number of quadrature points associated to each quadra-

ture order. . . . . 68 4.2 Variation of the maximum polynomial order and of the re-

sponse coefficients number as a function of the quadrature order. . . . . 69 4.3 Variation of the mean value of parcels diameter as a function

of the quadrature order. . . . 70 4.4 Simulation time for the different quadrature order. . . . 70 4.5 Weights values associated to the 7 quadrature points. . . . 71 5.1 Maximum and minimum values of the initial distributions. . . 77 5.2 Statistical parameters of parcels distribution referred to the

lower input value of µ, (coarse particles). . . . . 79

5.3 Statistical parameters of parcels distribution referred to the mean input value of µ (1φ). . . . 79 5.4 Statistical parameters of parcels distribution referring to the

upper input value of µ (2φ). . . . 80 5.5 Output results referred to the input triplet: µ = 1φ, σ = 1.3φ

and ψ = 0.7. . . . 81 5.6 Output results referred to the input triplet: µ = 1φ, σ = 1.5φ

and ψ = 0.7. . . . 82 5.7 Output results referred to the input triplet: µ = 1φ, σ = 1.7φ

and ψ = 0.7. . . . 82 5.8 Output values referred to the input triplet: µ = 1φ, σ = 1.5φ

and ψ = 0.5. . . . . 84 5.9 Output values referred to the input triplet: µ = 1φ, σ = 1.5φ

and ψ = 0.7. . . . 84 5.10 Output values referred to the input triplet: µ = 1φ, σ = 1.5φ

and ψ = 0.9. . . . 85 5.11 Mean value and standard deviation referred to the PDFs as-

sociated to the mean parcels diameter in every cell. . . . 91 5.12 Mean value and standard deviation referred to the PDFs as-

sociated to the standard deviation of the parcels distribution in every cell. . . . 94 5.13 Mean value and standard deviation referred to the PDFs as-

sociated to the skewness value of the parcels distribution in every cell. . . . 96 5.14 Mean value and standard deviation referred to the PDFs asso-

ciated to the kurtosis value of the parcels distribution in every cell. . . . . 98 5.15 Mean value and standard deviation referred to the PDFs as-

sociated to the number of parcels present in every cell. . . . . 100 5.16 Values of the relative mass fractions computed in every cell. It

is evident that the mass is mostly concentrated on the ground. 100

In order to investigate the problem of Uncertainty Quantification (UQ) in volcanic ash dispersion and deposition, Chapter 1 of this thesis presents the Lagrangian code, LPAC, used to perform the 2D simulations. The trajec- tories of the ash particles have been estimated by numerically solving the Basset-Boussinesq-Ossen Equation of motion [8], in which we have taken into account the drag force, the pressure gradient, the gravity and the Von- Mises forces. In particular the drag coefficient of ash fragments has been shaped as a function of the Reynolds number and of the particles sphericity [11].

The wind field used to perform the simulation has been generated by WRF code and it tries to reproduce the atmospheric conditions related to the Mount Etna eruption of 24 November 2006. In this occasion Kelvin- Helmholtz instabilities have been observed at the height of about 4000m from the sea level [18].

Chapter 2 presents the mathematical tool used in the Uncertainty Quan-

tification process, the Generalized Polynomial Chaos Expansion method (PCEg)

[1]. Having characterized the uncertainty parameters by an appropriate prob-

ability density function, this stochastic expansion technique is able to build

the output quantities of interest as polynomial functions which reflect the

propagation of the input uncertainty in the model. The UQ has been carried

out by DAKOTA toolkit, a code developed by Sandia Lab. that permits to

conduct also the Sensitivity Analysis (SA) of the response functions.

Having to treat the phenomenon of ash dispersion and deposition, in Chapter 3 the common tools used by sedimentologists to study the charac- teristic of a tephra fall deposit are presented. Since the particles dimension is the most important factor that influences the ash movement, the Total Grain Size Distribution curves are briefly described together with the Fre- quency Distribution ones and the Number of Particles Distributions [10].

These graphs are used in order to organize the fragments distribution ac- cording to their size expressed using the Krumbein scale. The statistical parameters that characterized the particles distributions (i.e. the mean di- ameter, its standard deviation, skewness and kurtosis) are then introduced and their physical interpretation is shown.

It is fundamental to specify that, in the numerical simulation, we do not consider the single particles, but we study the dispersion of ash packages (parcels). Each parcel is composed by an indeterminate number of particles so that all the parcels have the same mass. This has been done in order to do not treat with a huge number of ash particles. Chapter 3 also presents the description of the domain used for the simulation. The total dimensions of the two-dimensional grid over which the particles dispersion take place are: 40Km horizontally and 6Km vertically. This space has been equally divided into 4 vertical equal stripes and the features and the distributions of the particles in air and of those deposted on the ground have been analyzed separately.

The parcels are released every 10 seconds in groups of 41 units equally spread from a height of 3300m to an height of 4300m. The simulation lasts 2 hours and the output quantities of interest are computed in every cell at three different time instants; however the results have been studied only in corre- spondence of the last time (once the Kelvin-Helmholtz instability has been generated, the output statistics are approximately steady).

The uncertain input parameters we have chosen to study are those related

to the geometrical characteristics of the parcels input distribution: the mean

diameter (µ), its standard deviation (σ) and the sphericity of the particles

(ψ). From literature studies of tephra fall deposit of Mount Etna 2006 erup- tion [7, 6, 4, 3], the range of uncertainty for each parameter has been set to be the following way: µ = [0φ, 2φ], σ = [1.3φ, 1.7φ] and ψ = [0.5, 0.9]. Be- cause of lack of available information, the probability density functions used to describe the input parameters were chosen to be uniform in the ranges previously specified. In order to study the ash dispersion, the output quan- tities are the statistical parameters that permits to reconstruct the parcels distribution in every cell (i.e. the mean diameter, its standard deviation, skewness and kurtosis) and the number of parcels present in each cell.

Chapter 4 presents the setting procedure of DAKOTA. Since the input variables have been described by uniform PDF, the polynomials used by DAKOTA in order to build the response functions are Legendre polynomi- als. The low number of input variables allowed us to evaluate the polynomial coefficients using a Gaussian quadrature technique with 7 quadrature points.

The results of the simulation are illustrated in Chapter 5. First of all

some deterministic solutions obtained by keeping constant an input value

and varying the other two are presented. Then the results of the UQ analy-

sis are considered. In particular the statistical parameters associated to the

output values are plotted, together with their Cumulative Distribution Func-

tions and their Probability Distribution Functions. Analyzing these data it is

possible to observe that the initial variability of the input parameters tends

to disappear with increasing distance from the inlet and the particles tend

to have more uniform geometrical characteristics. Studying the results of

SA (Main and Total Sobol Indices) the main parameters which affect ash

dispersion and deposition are µ and ψ, while the initial sorting of the distri-

bution σ does not play a fundamental role. While ψ uncertainty dominates

the variation of the drag coefficient, µ characterizes the way in which the

particles are coupled with the carrier flow. So, while larger sediments show

a behavior which is almost independent of the wind field, the finer ones are

more sensitive to the flow and thus they are able to reach the largest distance

Description of the Lagrangian Code

1.1 LPAC (Lagrangian Particles Advection Code)

In the present work the numerical simulations have been made using a code developed by INGV-Pisa called LPAC (Lagrangian Particles Advection Code) which, using a simplified version of Basset-Boussinesq-Oseen equa- tion, allows to compute the trajectory of each particle. A characteristic of this code lies in his capability of a one-way coupling with the wind field in which the particles move.

This means that while the properties of the background field influence

the motion of the particles, the particles can not have an active effect on the

characteristics of the wind field which is known during all the simulation.

1.1.1 Equation of motion: Basset-Boussinesq-Oseen Equa- tion

The motion equation of a particle in a steady flow at low Reynolds numbers is represented by Basset-Boussinesq-Oseen equation (BBO):

m _p dv

dt = F _D + F _VM + F _B + F _P + F _G . (1.1) The first member is the variation of momentum of a particle having a mass m _p , while the second member contains all the forces which, acting on the particle, are the cause of the variation (i.e. steady-state drag, virtual mass force, Basset force, pressure gradient force and body force).

Let us examine these forces in more detail.

Drag force

The drag force acting on a particle can be written using the well-known relation:

F _D = 1

2 ρ _f C _D A|u − v|(u − v), (1.2) where ρ _f is the density of the fluid, C _D is the drag coefficient, A is the characteristic area of the particle, u and v are the fluid velocity and the par- ticle velocity, respectively.

One of the most important parameters which influence the residence time of a particle in the air is the drag coefficient. In the past the determination of the drag coefficient for both spherical and non spherical particles was made using general formulas which involved a dependence on the Reynolds number (Re) and on one or more shape descriptors matched together in complicated functions of at least two variables.

In this work the C _D is expressed as a function of the Reynolds number and

the sphericity [11]. The advantages of this choice lie in the relatively simple

Spherical particles

The dynamics of small spherical particles is strongly affected by their Reynolds number. If the Reynolds number is much lower than one (Stokes flow ) the drag force acting on a spherical particle is expressed by Stokes as:

F _Stokes = 3πµD(u − v), (1.3)

where D is the particle diameter, while the drag coefficient is C _D = 24 Re . When the Reynolds number becomes greater than one, the drag coefficient must be corrected as follows:

C _D = 24

Re 1 + 0.15Re ^0.687
. (1.4) Eq.(1.4) accurately describes the drag force on a sphere up to a Reynolds number of about 1000 (Transition region). The first term 24

Re reproduces the Stokes formula. For Reynolds number greater than 1000 (Newton flow ) the drag coefficient becomes constant (C _D ' 0.44) until a critical Reynolds number is reached, typically of the order of 10 ⁵ for spherical particles. After this limit the boundary layer around the particle becomes turbulent and the drag coefficient is reduced.

The dependence of the drag coefficient on the Reynolds number is shown in Fig.(1.1).

Figure 1.1: Drag coefficient of a spherical particle as a function of Reynolds

number [8].

Even if the previous formulas allow a good estimation of the drag coeffi- cient for spherical particles, it is necessary to consider also the case in which the shape of the particle is not a sphere.

There are a lot of researches dealing with the estimation of the drag coef- ficient of small particles, but the geometrical characteristics of the particles are taken into account by complex coefficients and only for a particular range of Reynolds numbers.

Having compiled data from literature, the drag coefficient was expressed as a function of Reynolds number for particles of different shapes [12]. The differences in particle shapes were measured in terms of sphericity (ψ), de- fined as the ratio of the surface area of a sphere with equivalent volume to the actual surface area of the particle:

ψ = π ^{1 3} (6V _p ) ^{2 3}

A _p , (1.5)

where V _p and A _p are the volume and the surface area of the particle respectively; ψ = 1 means that the particle is a sphere.

In [12], a universal function of the two variables, Reynolds number and sphericity C _D = f (Re, ψ), was constructed for the prediction of drag on any particle.

Another approach focused on the drag of non spherical particles in Stokes regime [15]. With the aim of taking into account the shape of the particles, a Stokes shape factor (K 1 ) was defined by the equation:

C _D = 4d _v g(ρ _p − ρ _f ) 3v ² ρ f

= 24 ReK 1

. (1.6)

The middle expression in Eq.(1.6) is the working formula for the drag

coefficient when a particle with density ρ p is settling at velocity v in a fluid

of density ρ _f with acceleration due to gravity given by g. The right-side

expression in Eq.(1.6) is the usual Stokes law for a sphere (K ₁ = 1) modified

for non spherical shapes.

K ₁ was modeled as a function of the sphericity and, for isometric particles, it can be expressed as follows:

K ₁ = ( 1 3 + 2

3 ψ ^{− 1 2} ) ⁻¹ . (1.7)

The limit of Eq.(1.6) lies in the narrow range of Reynolds numbers of its applicability (only Stokes regime).

At a later stage, it was observed that all particles experience a Newton regime and the so-called Newton shape factor (K ₂ ) was introduced, defined as the ratio between the drag coefficient of particles of the same shape and the drag coefficient of a sphere, both at a Reynolds number of 10 ⁵ (Eq.(1.8)) [20] .

K ₂ = C _D C DS

, (1.8)

where C _DS is the value of the drag coefficient for a sphere in Newton’s regime.

The drag coefficient is thus determined as a function of Reynolds number and K ₂ . Using experimental results K ₂ was estimated as a function of the sphericity as follows:

K 2 = 10 1.8148(−log(ψ) ^0.5743 )

. (1.9)

Taking into account the previous results, a simplified version of the C D

was proposed for both spherical and non spherical particles for a wide range of Reynolds numbers, specifically a generalized Reynolds number (ReK ₁ K ₂ ).

The basic assumption of this formulation is that every isolated particle expe- riences a Stokes regime where drag is proportional to relative velocity and a Newton regime where drag is proportional to the square of relative velocity.

In addition, the way a particle behaves in these two regimes can be used

to predict the drag for a large range of Reynolds numbers. Once the main

physical parameters of the problem have been identified (i.e. particle size

d P , fluid viscosity µ, fluid density ρ f ), it is possible to extract K 1 and K 2

from the behavior of the particle in the Stokes and Newton regimes with a dimensional analysis.

Without showing the mathematical steps, the generalized drag coefficient valid for ReK ₁ K ₂ ≤ 10 ⁵ applicable to all particles shapes is:

C _D

K ₂ = 24

ReK ₁ K ₂ (1 + 0.1118(ReK ₁ K ₂ ) ^0.6567 ) + 0.4305 1 + _ReK ³³⁰⁵

1 K 2

, (1.10)

where K ₁ and K ₂ are reported in Eq.(1.7) and in Eq.(1.9) respectively.

The power of Eq.(1.10) lies in his capability to give the drag coefficient only as a function of sphericity and for a wide range of Reynolds numbers.

The plot of the drag coefficient as a function of the generalized Reynolds number is reported in Fig.(1.2).

Figure 1.2: Drag coefficient for spherical and nonspherical particles trans-

formed in generalized drag coefficients and generalized Reynolds number [11].

Faxen correction

The spatial variation of the flow field across the particle can be taken into account by a correction term, namely the Faxen force correction[8], as follows:

F _{F axen} = µπ D ³

8 ∇ ² u. (1.11)

Pressure Force

If a pressure gradient is present, a force acting on the particle in the direction of this gradient is generated [8]. This force can be written as:

F P = − Z

S

pndS. (1.12)

By applying first the divergence theorem and then the assumption that the pressure gradient can be considered constant in the proximity of a single particle, we obtain:

F _P = Z

V

∇pdV = −∇pV _p . (1.13)

Body Force

The gravitational force acting on a body of mass m _p is expressed by the following relation:

F _B = m _p g = ρ _p V _p g, (1.14) where g is the gravitational acceleration and ρ _p is the particle density.

Added Mass Force

In BBO equation the contribution of the added mass force, a non steady force acting on a body immersed in a non steady flow, is also present. This force is generated by the disturbance that is made on the fluid by the body because of its presence [8].

A body of volume V _p moving with a velocity v (not necessary constant)

in a initially quiet fluid, causes the origin of a fluid displacement since the

fluid particles have to move to permit the passage of the body.

If the body moves with a non steady velocity, then the kinetic energy transferred to the fluid by the body, Eq.(1.15), changes over time, Eq.(1.16).

Thus a work is done by the body on the fluid.

KE = 1 2 ρ _f

Z

V

u ² dV, (1.15)

dKE

dt = F _{V M} U. (1.16)

Omitting the intermediate steps and under the hypothesis of incompress- ible and irrotational fluid, it is possible to write the force that the body exerts on the fluid as:

F V M = M _f 2

dU

dt , (1.17)

where M f is the mass of the fluid moved by the body. At the same time the fluid exerts on the body a force of equal intensity but with opposite orientation. Usually the relative acceleration of the fluid with respect to the body is ˙ u − ˙v, where ˙ u is the material derivative of the velocity, ^dU _dt . Finally we can write the added mass force acting on a particle as:

F _{V M} = ρ _f V _p

2 ( ˙ u − ˙ v). (1.18)

Even in this case the curvature effect of the carrier fluid can be taken into account by a correction term which can be included in the virtual mass force:

F ˜ _{V M} = − ρ _f V _p D 80

d

dt ∇ ² u. (1.19)

Basset Force

This term takes into account the delay in the growth of the boundary layer due to the relative acceleration between the body and the fluid. It is

’history’ term and it can be written as follows:

F = 3

D ² √ πρµ

Z t d

dt (u − v)

√ dt ⁰ . (1.20)

By inserting Eqs.((1.2), (1.11), (1.18), (1.19), (1.20), (1.13), (1.14)) in Eq.(1.1), we obtain the following expression for the BBO equation:

m p

dv dt = 1

2 ρ f C D A|u − v|(u − v) + µ c π D ³ 8 ∇ ² u+

+ ρ f V p

2 d dt



u − v − D ² 40

d dt ∇ ² u

 + 3

2 D ² √ πρµ

Z t 0

d

dt (u − v)

√ t − t ⁰ dt ⁰ +

−∇pV _p + ρ _p V _p g.

(1.21) Since Eq.(1.21) is quite complex, it is necessary to try to simplify it in order to obtain lower computational costs for its resolution.

First of all we notice that, since the dimensions of the particles are small

compared to the characteristic dimension of the carrier flow, the Faxen correc-

tion and the term related to the curvature in the Added Mass Force equation

can be neglected. Also the history term can be omitted; in fact, even if at

low Reynolds number it increases the drag force, it has been demonstrated

that its effects are less important than originally thought.

With the above simplifications, in a 2D Cartesian coordinate system (x, z), Eq.(1.21) can be rewritten as:



 

 

 

 

 

 

 

 



 

 

 

 

 

 

 

 

 dx

dt = v _x dz dt = v z

 1 + 1

2γ

 dv _x

dt = −C _D 3 4

|u − v|

γD (u _x − v _x ) − 1 ρ _p

dp dr + 1

2γ du r

dt

 1 + 1

2γ

 dv _z

dt = −C _D 3 4

|u − v|

γD (u _z − v _z ) − g − 1 ρ _p

dp dz + 1

2γ du _z

dt

(1.22)

where v _x and v _z are the horizontal and the vertical components of the velocity of the particles, while u _x and u _z are the components of the wind velocity field. The ratio between the density of the particle and the density of the fluid is expressed by γ = ^ρ _ρ ^p

f , while A and V _p have been replaced by A = π( ^D ₂ ) ² and V _p = ⁴ ₃ π( ^D ₂ ) ³ respectively.

1.1.2 Numerics

The simplified set of BBO equations, Eq.(1.22), is numerically integrated using a fourth order method, the Runge-Kutta method. Runge-Kutta inte- gration methods are simple, stable and self-starting, and their accuracy is sufficient for this kind of application.

While the characteristics of the particles and their initial conditions are set in LPAC by the user, the characteristics of the background flow field (i.e.

wind velocity, temperature, pressure etc) are given to LPAC by the code

PDAC. In this code the carrier field is constructed over a bidimensional grid

composed by a certain number of cells. The grid is staggered, this means

that while the velocities are computed at the cell sides, the pressures are

computed at the center of the cells, see Fig.(1.3).

Figure 1.3: (a) Staggered grid adopted by the PDAC code used to describe

the carrier flow field. Three different interpolations of the carrier flow field

are required to determine the local radial velocity (b), the vertical velocity

(c), and the pressure gradient (d) of a particle with position (x; z) [9].

To compute the trajectory of each particle, it is necessary that the prop- erties of the carrier flow are known at each Runge-Kutta step. Since the grid is staggered, if the particle is located at a generic position (x; z), different kinds of interpolations are required to evaluate the characteristics of the wind field at the point where the particle is. In particular a bi-cubic interpolation is adopted in the inner portion of the computational domain, whereas a bi- linear interpolation is used in the cells near the topography and along the boundary of the domain.

The BBO equation solved by LPAC is :

m _p U(t + ∆t) − U(t)

∆t = F _D + F _VM + F _B + F _P + F _G , (1.23)

where ∆t is the time step of integration, U(t + ∆t) and U(t) are the

velocities of the particle computed at the end and at the beginning of the

time step respectively, while the second member contains the forces acting

on the particle. At each Runge-Kutta step the new velocity of the particle

[U(t + ∆t)] is computed and thus the change of position of the particle is

evaluated step by step. In this way it is possible to reconstruct the trajectory

of each particle. Concerning the pressure gradient force and the Von Mises

force, LPAC solves the equation of motion with an explicit integration, while

the drag force is treated with an explicit procedure. This means that, while in

the first case the forces used to evaluate the new velocity are those computed

at the time t, in the second case the new velocity is evaluated taking into

account the drag force computed at the new time (t + ∆t). The time step of

integration is not fixed, but it is set by an adaptive stepsize technique: if the

particle does not move to an adjacent cell, but it has a greater relocation,

the time step is reduced, while the it is increased if the particle remains in

the same cell during the Runge-Kutta steps.

1.2 Eulerian flow field

The section describes the atmospheric conditions and the Eulerian flow field used in the Lagrangian simulation.

1.2.1 Kelvin-Helmholtz Instability

The wind field used in this work tries to reproduce what happened during the 24 November Mt. Etna eruption of 2006. The kind of plume observed during Mt. Etna eruption and treated in our work is a weak plume (see Fig.(1.4)) bent by the wind with a volcanic cloud dispersed in the atmosphere at an altitude centered at ∼ 4000m above the sea level. In this occasion the volcanic clouds showed horizontal stripes oriented perpendicularly to the prevailing wind direction (see Fig.(1.5)). Phenomenons like this have been observed during many volcanic events (e.g. Klyuchevskaya in Kamchatka and Eyjafjallajäkull in Iceland), but in the the Mt. Etna one, observed by the NASA satellite MODIS (see Fig.(1.6)), it was particularly evident.

(a) Sketch of a strong plume, strong plumes are characterized by suver- tical convective region that spreads laterally around the level of neural buoyancy [7].

(b) Sketch of a weak plume, weak plumes are characterized by vertical velocities lower than the wind veloc- ity [7].

Figure 1.4: Sketches of a strong and a weak plume [7].

Figure 1.5: Plume of Mt.Etna eruption of 24 November 2006.

Figure 1.6: Cloud ash of 24 November 2006 eruption captured by MOSIS

satellite, source NASA

Referring to Mt. Etna 2006 eruption, it has been shown that the in- stabilities can be produced by the volcanic cloud itself which modifies the surrounding atmosphere and generates the favorable conditions to Kelvin- Helmholtz instabilities (KHI) [19].

Kelvin-Helmholtz Instability (Helmoltz 1868, Kelvin 1871) is a hydro- dynamic instability observed in a wide range of natural phenomena (see Fig.(1.7)). It is due to the shear present at the interface between two fluids moving at different velocities. This discontinuity in the (tangential) veloc- ity induces vorticity at the interface; as a result, the interface becomes an unstable vortex sheet that rolls up into a spiral (Fig.(1.8)).

(a) Kelvin Helmholtz Clouds, photography copyright Brooks Martner, NOAA Environ- mental Technology Laboratory.

(b) KHI observed on Jupiter’s atmosphere, photo credit: NASA.

Figure 1.7: KHI present in natural phenomena.

Figure 1.8: Mechanism of formation of KHI.

1.2.2 Characteristics of the wind field

The driving wind field has been generated by WRF (The Weather Research and Forecasting Model), an Eulerian fully-compressible non-hydrostatic at- mospheric model [18]. Because the horizontal structures observed for the 24 November 2006 eruption at Mt. Etna were oriented approximately perpendic- ular to the mean wind direction, and the wind direction was nearly constant after 12 : 00P M , we can investigate the phenomenon using two-dimensional simulations. In particular, the Eulerian wind field used in the simulation covers a domain of about 40 Km horizontally and almost 6 Km vertically. It covers a time interval of about 4 hours and the Kelvin-Helmholtz instability becomes evident one hour after the beginning of the Eulerian simulation (see Fig.(1.9)).

Figure 1.9: Setting table of WRF for the generation of the carrier flow field

[18].

The time interval covered by the Lagrangian simulation is of 2 hours. It starts 1 hour after the beginning of the Eulerian simulation and it ends 2 hours later. In this range of time the KHI is present and it is located in the vertical range [3300m; 4300m]. The plot of the velocity components (u _x , u _z ) at four instants of time is represented figures below (Fig.(1.10), Fig.(1.11), Fig.(1.12), Fig.(1.13)).

0.5 1 1.5 2 2.5 3 3.5

x 10 ⁴ 1000

2000 3000 4000 5000

x(m)

z(m)

Horizontal Wind Component (t = 3600 s)

m/s

5 10 15 20

0.5 1 1.5 2 2.5 3 3.5

x 10 ⁴ 1000

2000 3000 4000 5000

x(m)

z(m)

Vertical Wind Component (t = 3600 s)

m/s

−0.3

−0.2

−0.1 0 0.1 0.2 0.3

Figure 1.10: Wind field at the beginning of the Lagrangian simulation. The

Kelvin-Helmholtz instability are found to originate in correspondence of the

volcanic plume, centered at ∼ 4000m a.s.l.

0.5 1 1.5 2 2.5 3 3.5 x 10 ⁴ 1000

2000 3000 4000 5000

x(m)

z(m)

Horizontal Wind Component (t = 7600 s)

m/s

5 10 15 20

0.5 1 1.5 2 2.5 3 3.5

x 10 ⁴ 1000

2000 3000 4000 5000

x(m)

z(m)

Vertical Wind Component (t = 7600 s)

m/s

−4

−2 0 2 4

Figure 1.11: Wind field after little more than an hour of simulation. The

KHI are evident and they will remain stable also in the next times.

0.5 1 1.5 2 2.5 3 3.5 x 10 ⁴ 1000

2000 3000 4000 5000

x(m)

z(m)

Horizontal Wind Component (t = 9100 s)

m/s

5 10 15 20

0.5 1 1.5 2 2.5 3 3.5

x 10 ⁴ 1000

2000 3000 4000 5000

x(m)

z(m)

Vertical Wind Component (t = 9100 s)

m/s

−4

−2 0 2 4

Figure 1.12: Wind field at time t = 9100s. Not significant differences can be

found with respect to the previous instant of time.

0.5 1 1.5 2 2.5 3 3.5 x 10 ⁴ 1000

2000 3000 4000 5000

x(m)

z(m)

Horizontal Wind Component (t = 10600 s)

m/s

5 10 15 20

0.5 1 1.5 2 2.5 3 3.5

x 10 ⁴ 1000

2000 3000 4000 5000

x(m)

z(m)

Vertical Wind Component (t = 10600 s)

m/s

−4

−2 0 2 4

Figure 1.13: Wind field at the end of the simulation, KHI is still present.

Fig.1.14 shows an example of particles trajectories computed by the La- grangian code LPAC. The simulation concerns 2 groups of particles of differ- ent diameters: D = 2mm (red spots on the sketch) and D = 0.06mm (blue spots on the sketch). Each group is composed by 10 particles which, at the beginning of the simulation, are equally spread in the range [3300m, 4300m].

The initial velocity as been supposed equal to zero and all the particles have the same density equal to 1500Kg/m. As we can see the finer particles are able to reach the greatest distances from the inlet, while the coarser ones show an independent behavior from the carrier fluid and they fall down close to the inlet.

0 0.5 1 1.5 2 2.5 3 3.5

x 10 ⁴ 0

1000 2000 3000 4000

x (m)

z (m)

Figure 1.14: Trajectories of two groups of particles of different diameters.

Uncertainty Quantification and Sensitivity Analysis

2.1 Introduction

Although the numerical analysis proves to be an extremely powerful tool in the common engineering practice, it is important to get the results that derive from it with proper precautions.

The limitations of this technique lie in the inevitable discrepancies be- tween the physical model and the numerical one and in the frequent lacking of whole knowledge of the problem itself.

Also in a simplified context, many sources of uncertainty exist: those related to the creation of the model, those related to the definition of the boundary conditions and those related to the choice of the initial conditions.

This issue can be partially overcome with two different approaches which are actually complementary: Uncertainty Quantification (UQ) and Sensitiv- ity Analysis (SA) .

The aim of both these techniques is to understand how changes in input

parameters can influence the output of a particular engineering problem. In

the UQ approach all or some of the input variables are considered uncertain

and they are represented by appropriate probability distribution functions.

Also the output of interest can be described in a statistical way. SA does not require a physical characterization of the input uncertainties and this analysis can be done on a purely mathematical way. In particular, the aim of SA is to understand how the variability of an output is due to a particular input and which input dominates the system response.

2.2 Uncertainties and Errors

First of all it is important to distinguish uncertainties from errors. According to the American Institute of Aeronautics and Astronautics ( AIAA ) " Guide for the Verification and Validation of CFD Simulations " errors are obvious deficiencies in the model and / or in the construction of algorithms, while the uncertainties are potential failures due to an incomplete knowledge of the problem. This definition does not distinguish between mathematics and physics. A better approach is to associate the birth of the errors with the implementation of the mathematical model into a computational algorithm (or code), such as round-off errors, problems of convergence, and all the bugs in the simulation code. Uncertainties arise from the choices that we make when we try to describe the physical problem and when we have to choose the input parameters. An example would be the definition of the boundary conditions of a given problem, conditions that are not always clearly and unambiguously defined by observations and experiments.

It is clear that when we speak about uncertainties we refer to an extremely broad and multifaceted concept which, however, can be organized in a more rigorous way by the distinction between epistemic and aleatory uncertainties.

The aleatory uncertainties (also called stochastic or irreducible uncertain- ties) are associated with the physical variability of the system being analyzed.

They are not due to a lack of knowledge and for this reason they can not be eliminated. The random uncertainties occur when we make choices with the aim of characterizing the properties of the system or its operating conditions.

Generally these variations are described by probabilistic approaches.

due to a lack of knowledge. They may arise from the assumptions introduced during the development of the mathematical model, from the simplifications made and so on. These uncertainties can be reduced by trying to refine the physical model, by more accurate observations, by enhancing the experiments and by more complete understanding of the problem. Their description can be made by non-probabilistic techniques.

In the present work we have considered the characteristics of the particles emitted from the volcano as random uncertainties.

The code used to make the UQ and SA analysis is the so-called DAKOTA (Design Analysis Kit for Optimization and Terascale Applications) toolkit [1].

This open source code, developed by the Sandia Lab, provides mathematical algorithms for:

• Uncertainty Quantification with sampling, reliability, stochastic expan- sion and epistemic methods;

• Optimization with gradient- and non gradient-based methods;

• Parameter estimation with nonlinear least squares methods;

• Sensitivity/variance analysis with design of experiments and parameter

study methods.

2.3 Uncertainty Quantification

DAKOTA gives the opportunity to make UQ analysis with a lot of different techniques which briefly are:

• Sampling techniques (Monte Carlo, Latin Hypercube Samplig);

• Stochastic expansion methods: Polynomial Chaos Expansion (PCE), Stochastic Collocation (SC).

2.3.1 Generalized Polynomial Chaos Expansions (PCEg)

In the present work we have chosen to adopt as UQ technique the so-called Generalized Polynomial Chaos Expansions (PCEg) which is included within the Stochastic Expansion Methods [1]. PCE was developed by Norbert Wiener in 1938 [23] and it soon had a big spread because it was more efficient and accurate than the Monte Carlo method. The term "Chaos" simply refers to the uncertainties in input, while the word "Polynomial" is used because the propagation of uncertainties is described by polynomials. The first step is to model the input variables through appropriate probability distribution functions, then the key point of the procedure is to write the output in a polynomial form. The uncertainties which can be taken into account can be the boundary conditions, initial conditions, forces and parameters. PCE is able to take into account only a limited number of uncertain variables, but it can give a detailed study of their behavior.

Once the uncertain variables have been chosen it is necessary to model each of theme through a proper PDF (Probability Distribution Function) that tries to reproduce the physics of the analyzed problem. If not enough information are available it is advisable to use a uniform distribution.

Polynomial Basis (Askey Scheme)

The choice of the polynomial basis for the output functions is a function of the type of PDF which represents the input variables.

At the beginning Wiener modeled the input PDF as Gaussian distribu-

Polynomial Chaos Expansions (PCE), is quite different from the General- ized Polynomial Chaos Expansions (PCEg) which uses the Wiener-Askey approach. According to this approach, after a PDF has been chosen, an appropriate polynomial basis is selected as reported Tab.(2.1).

Distribution Density function Polynomial Weight function Support range

Normal 1

√ 2π e

−x ²

2 Hermite e

−x ²

2 [−∞, ∞]

Uniform 1

2 Legendre 1 [−1, +1]

Beta (1 − x) ^α (1 + x) ^β

2 ^α+β+1 B(α + 1)(β + 1) Jacobi (1 − x) ^α (1 + x) ^β [−1, +1]

Exponential e ^−x Laguerre e ^−x [0, ∞]

Gamma x ^α e ^−x

Γ(α + 1) Generalized Laguerre x ^α e ^−x [0, ∞]

Table 2.1: Askey Scheme [1].

We can see that for each input distribution (normal, uniform, beta and so on) there is a corresponding polynomial basis. The coupling between the input variable and the type of polynomial is due to the correspondence between the weight function of the polynomial basis and the probability distribution that characterizing the input. The PDF and the weight functions differ by a multiplying factor since it is necessary that the integral of the first one on its support range be equal to one .

This table is derived from the family of orthogonal hypergeometric poly- nomials also known as Askey scheme; the Hermite polynomials used originally by Wiener are a subset of the Askey family.

If the input variables can not be characterized by known probability dis- tributions, additional techniques are required with the aim of transforming the variables in such a way to apply the scheme of Askey to this transformed space.

If ξ = (ξ _{i 1} , ξ _{i 2} , . . . , ξ _{i n} ) is the vector of input variables (each term is a

particular input variable which is associated with an appropriate PDF), the

dexing):

R = a ₀ B ₀ +

∞

X

i 1 =1

a _{i 1} B ₁ (ξ _{i 1} )+

∞

X

i 1 =1 i 1

X

i 2 =1

a _{i 1 i 2} B ₂ (ξ _{i 1} , ξ _{i 2} )+

∞

X

i 1 =1 i 1

X

i 2 =1 i 2

X

i 3 =1

a _{i 1 i 2 i 3} B ₃ (ξ _{i 1} , ξ _{i 2} , ξ _{i 3} )+. . . . (2.1)

The polynomial is unbounded and each set of nested summations is an additional order of the expansion. The above expression can be simplified as:

R =

∞

X

j=0

α _j Ψ _j (ξ), (2.2)

where there is a direct correspondence between α _{i 1 i 2 i 3 ,...,in} and α _j and between B _n (ξ _{i 1} , ξ _{i 2} , . . . , ξ _{i n} ) and Ψ _j (ξ).

Each Ψ j (ξ) is a multivariate polynomial which in composed by the prod- uct between the one-dimensional polynomials (ψ i ) associated to the input PDFs (i.e. Legendre, Hermite, Laguerre, etc).

The polynomial expression of Eq.2.2 must be truncated to a finite order expansion P :

R '

P

X

j=0

α _j Ψ _j (ξ). (2.3)

Generally, PCEg contains a complete basis of polynomials up to a spec- ified total order p, this approach is the so-called total order approach. This means that, for an expansion of total order p involving n input variables, the multi-index term which defines the set Ψ _j (ξ) is limited in the following way:

n

X

i=1

t ^j _i ≤ p, (2.4)

where t ^j _i is the order of the one-dimensional polynomials ψ i involved in

the Ψ j (ξ) term.

For example, the Hermite polynomial basis for an expansion of the second order which involves two random variables is :

Ψ ₀ (ξ) = ψ ₀ (ξ ₁ )ψ ₀ (ξ ₂ ) = 1 Ψ ₁ (ξ) = ψ ₁ (ξ ₁ )ψ ₀ (ξ ₂ ) = ξ ₁ Ψ ₂ (ξ) = ψ ₀ (ξ ₁ )ψ ₁ (ξ ₂ ) = ξ ₂ Ψ ₃ (ξ) = ψ ₂ (ξ ₁ )ψ ₀ (ξ ₂ ) = ξ ₁ ² − 1

Ψ ₄ (ξ) = ψ ₁ (ξ ₁ )ψ ₁ (ξ ₂ ) = ξ ₁ ξ ₂ Ψ ₅ (ξ) = ψ ₀ (ξ ₁ )ψ ₂ (ξ ₂ ) = ξ ₂ ² − 1

The total number of terms (N _t ) presents in an expansion of total order p involving n input variables is:

N _t = 1 + P = 1 +

p

X

1

1 s!

s−1

Y

r=0

(n + r) = (n + p)!

n!p! . (2.5)

In the tensor product approach, each term Ψ j (ξ) is composed by all the combinations of the one-dimensional polynomials. This means that the multi- index term of expansion which defines the set Ψ _j (ξ) is limited in the following way:

t ^j _i ≤ p, (2.6)

where p _i is the maximum order of the polynomial of the n-th dimension.

In this case, for a maximum order of expansion p = 2 and for a number of input variables n = 2, the Hermite base of polynomials is the following one:

Ψ 0 (ξ) = ψ 0 (ξ 1 )ψ 0 (ξ 2 ) = 1 Ψ 1 (ξ) = ψ 1 (ξ 1 )ψ 0 (ξ 2 ) = ξ 1

Ψ ₂ (ξ) = ψ ₂ (ξ ₁ )ψ ₀ (ξ ₂ ) = ξ ₁ ² − 1

Ψ ₄ (ξ) = ψ ₁ (ξ ₁ )ψ ₁ (ξ ₂ ) = ξ ₁ ξ ₂ Ψ 5 (ξ) = ψ 2 (ξ 1 )ψ 1 (ξ 2 ) = (ξ ₁ ² − 1)ξ 2

Ψ 6 (ξ) = ψ 0 (ξ 1 )ψ 2 (ξ 2 ) = ξ ₂ ² − 1 Ψ 7 (ξ) = ψ 1 (ξ 1 )ψ 2 (ξ 2 ) = ξ 1 (ξ ₂ ² − 1) Ψ ₈ (ξ) = ψ ₂ (ξ ₁ )ψ ₂ (ξ ₂ ) = (ξ ₁ ² − 1)(ξ ₂ ² − 1) The total number of terms N _t is:

N _t = 1 + P =

n

Y

i=1

(1 + p), (2.7)

where n is the number of the input variables and p is the maximum order of the one-dimensional polynomials. The second approach is able to handle explicitly the possible anisotropy of the problem since the order of the polynomial basis can be specified for each direction. However, it is possible to consider the anisotropy of the problem with an expansion of total order by cutting the polynomials that satisfy the highest order but violate the requirement for that particular direction.

Estimation of PCEg Coefficients using Spectral Projection

One of the most crucial point of PCEg is the calculation of the coefficients of the expansion. In particular there are two main approaches that are spectral projection and linear regression.

In the first case the response function is projected along each basis func- tions using internal products and each coefficient is extracted using the prop- erties of orthogonal polynomials. Linear regression, also known as point col- location or stochastic response surface, computes the coefficients that best permit to reproduce a set of values of the response evaluated numerically.

In this work we have decided to use the spectral projection as the linear

regression is a suitable technique when the number of input variables is high

(a number of inputs greater than 5) [1].

Spectral Projection

The generic coefficient α j of the PCEg can be evaluated as follows:

α _j = hR, Ψ _j i

Ψ ² _j  = 1 Ψ ² _j 

Z

RΨ _j ρ(ξ)dξ, (2.8)

where each inner product involves a multidimensional integral on the domain of support of the weight functionρ(ξ).

The denominator of Eq.(2.8) is the square norm of the basis of multi- variate polynomials and can be computed analytically using the product of univariate norms squared :

Ψ ² _j  =

n

Y

i=1

ψ ² _{t i}  . (2.9)

The first true computational effort lies in the evaluation of the numerator;

solutions can be found numerically using sampling, quadrature, cubature, or sparse grid approaches. As will be explained in the next Chapter, we have considered three random variables as input, thus the technique we have cho- sen is the Gaussian quadrature. The sparse grid approach is suitable for a higher number of input variables, while sampling is not enough accurate.

Tensor Product Quadrature

The simplest technique for approximating multidimensional integrals is a tensor product of one-dimensional quadrature rules, in particular we used a Gaussian Quadrature.

If n is the number of the input parameters, for each of them a number m _i of quadrature points can be defined , the subscript i refers to the i-th input.

So, for each input, let (ξ ₁ ⁱ , . . . , ξ _m ⁱ _i ) ⊂ Ω _i be a sequence

of quadrature points coordinates abscissas (Ω i is the i-th dimension of the domain Ω = Ω 1 N · · · N Ω n ).

Considering the i-th input, the integral of a function f ∈ C ⁰ (Ω _i ) can be

evaluated by a sequence of one-dimensional quadrature operators:

U ⁱ (f )(ξ) =

m i

X

j=1

f (ξ _j ⁱ )w ⁱ _j , (2.10)

where U ⁱ is the quadrature operator referred to the i-th input, (ξ ⁱ _j ) is the i- th random variable evaluated at the quadrature point j, w _j ⁱ is the weight asso- ciated to the j-th quadrature point and m _i ∈ N is total number of quadrature abscissas referred to the i-th input variable. Having specified an expansion order p, a minimal Gaussian quadrature order of p + 1 is required to obtain good accuracy in the PCEg coefficients estimation. Now, in the multivariate case n  1, for each f ∈ C ₀ (Ω) and for each multi-index i = (i ₁ , . . . , i _n ) ∈ N ⁿ it is possible to define the full tensor product quadrature formulas

Q ⁿ _i f (ξ) = (U ^{i 1} O

· · · O

U ^{i n} ) = f (ξ)

m i1

X

j 1 =1

· · ·

m in

X

j n =1

f (ξ _j ^{i 1}

1 , . . . , ξ _j ^{i n} _n )(w ⁱ _j ¹

i

O · · · O w _j ^{i n} _n ).

(2.11) The above product requires the evaluation of Q n

j=1 m _{i j} functions. When the number of input random variables is small, full tensor product quadrature is a very effective numerical tool.

In order to compute the quadrature points, the grid used in our work is the Clenshaw-Curtis grid (Fig.(2.1)) [1], which, dealing with a Gaussian quadrature and with a small number of variables, is a good solution.

Figure 2.1: Clenshaw-Curtis grid used in the present work [1].

2.4 Sensitivity Analysis

More information about the dependence of the output from the inputs can be obtained by the Sensitivity Analysis (SA). SA consists of a lot of different techniques; in this work we consider the Global SA, which is supported by DAKOTA [1]. SA is a powerful tool to investigate which of the input param- eters are more decisive for the output results. However this technique gives good informations if a very large number of the output values are known (a good number can be 10000 samples) and, most of times, this requires very high computational costs. In this sense a link between UQ and SA proves to be extremely useful; in fact, once the PCEg functions have been built, it is possible to use the polynomials as emulators in order to obtain a large num- ber of output values, without doing the simulations, but simply evaluating the polynomials for the input values.

2.4.1 Global Sensitivity Analysis: Variance-based de- composition (VBD)

The Variance-based decomposition (VBD) is a global sensitivity technique that allows to understand how the variability of output values is due to spe- cific input parameters or to a combination of them. There are two key mea- sures that are S _i (Main Effect Sensitivity Index ) and T _i (Total Effect Index ).

If Y is an output function and x _i is the generic i-th input, S _i (Eq.(2.12)) indi- cates how much the output variability is due only to x _i , while T _i (Eq.(2.13)) expresses the percentage of uncertainty of Y that is due to x _i and to its interaction with other variables. Having indicated with V ar _{x i} [E(Y |x _i )] the variance of the conditional expectation and with V ar(Y ) the variance of the output function, the formulas are:

S _i = V ar _{x i} [E(Y |x _i )]

V ar(Y ) , (2.12)

T _i = E(V ar(Y |x _i ))

V ar(Y ) = V ar(Y ) − V ar[E(Y |x _i )]

V ar(Y ) , (2.13)

where Y = f (x) and x − i = x ₁ , . . . , x − i, x _i , . . . , x _m .

For example, Fig.(2.2) shows the plot of the function f (x 1 , x ₂ , x ₃ ) = x ^1/2 ₁ x ₃ + 3x ³ ₃ + 10x ₂ and the influence that each input variable has on the output is taken into account. From the figure it is evident that x ₂ is the variable with the highest Sobol Index since, for each value of x ₂ , the range of variation of the function value (blue spots on the plot) is very close to the variance of the conditional expectation referred to x ₂ (red line on the plot).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 5 10 15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 5 10 15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 5 10 15

Y

Y

Y Y

X 1

X ₂

X ₃

Figure 2.2: Plot of f (x ₁ , x ₂ , x ₃ ) = x ^1/2 ₁ x ₃ + 3x ³ ₃ + 10x ₂ evaluated for 1000

random samples.

Simulation Set Up

In this chapter we present the choices we have made in order to develop a methodology suitable for the study of ash dispersion. First of all, we introduce the tool used to describe the mass partitioning of the ash particles, the so-called Grain Size Distribution. Then we present the way in which the computational domain has been defined and treated with the aim to have a clear view of what happened to the particles. Finally, we show the input and the output parameters we have chosen as object of our analysis.

3.1 Grain Size Analysis

When we deal with the texture of a volcanic ash deposit, a variety of char- acteristics are involved. However the grain size is the most fundamental property of sediment particles, affecting their entrainment, transport and deposition. Grain Size Analysis gathers all the techniques used in the study of an ash deposit.

The importance of the calculation of the grain size distribution for a tephra fall deposit lies in the following motivations:

• possibility to infer the eruption style by studying the connection be-

tween the particle size and the initial gas content and water magma