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):
mp
dv
dt = FD+ FVM+ FB+ FP+ FG. (1.1) The first member is the variation of momentum of a particle having a mass mp, 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:
FD =
1
2ρfCDA|u − v|(u − v), (1.2)
where ρf is the density of the fluid, CD 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 CDis expressed as a function of the Reynolds number and
the sphericity [11]. The advantages of this choice lie in the relatively simple way to obtain the dependency of the CD from the shape of the particles.
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:
FStokes= 3πµD(u − v), (1.3)
where D is the particle diameter, while the drag coefficient is CD =
24 Re. When the Reynolds number becomes greater than one, the drag coefficient must be corrected as follows:
CD =
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 (CD ' 0.44) until a critical Reynolds
number is reached, typically of the order of 105 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(6Vp) 2 3 Ap , (1.5)
where Vp and Ap 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 CD = 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 (K1) was defined by the equation:
CD = 4dvg(ρp− ρf) 3v2ρ f = 24 ReK1 . (1.6)
The middle expression in Eq.(1.6) is the working formula for the drag coefficient when a particle with density ρp is settling at velocity v in a fluid
of density ρf with acceleration due to gravity given by g. The right-side
expression in Eq.(1.6) is the usual Stokes law for a sphere (K1 = 1) modified
K1was modeled as a function of the sphericity and, for isometric particles,
it can be expressed as follows:
K1 = ( 1 3+ 2 3ψ −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 (K2) 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 105 (Eq.(1.8)) [20] .
K2 =
CD
CDS
, (1.8)
where CDS 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 K2. Using experimental results K2 was estimated as a function of the
sphericity as follows:
K2 = 101.8148(−log(ψ)
0.5743)
. (1.9)
Taking into account the previous results, a simplified version of the CD
was proposed for both spherical and non spherical particles for a wide range of Reynolds numbers, specifically a generalized Reynolds number (ReK1K2).
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 dP, fluid viscosity µ, fluid density ρf), it is possible to extract K1 and K2
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 ReK1K2 ≤ 105 applicable to all particles shapes is:
CD K2 = 24 ReK1K2 (1 + 0.1118(ReK1K2)0.6567) + 0.4305 1 + ReK3305 1K2 , (1.10)
where K1 and K2 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: FF axen = µπ D3 8 ∇ 2u. (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:
FP = −
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: FP = Z V ∇pdV = −∇pVp. (1.13) Body Force
The gravitational force acting on a body of mass mp is expressed by the
following relation:
FB = mpg = ρpVpg, (1.14)
where g is the gravitational acceleration and ρp 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 Vp 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 2ρf Z V u2dV, (1.15) dKE dt = FV MU. (1.16)
Omitting the intermediate steps and under the hypothesis of incompress-ible and irrotational fluid, it is possincompress-ible to write the force that the body exerts on the fluid as:
FV M =
Mf
2 dU
dt , (1.17)
where Mf 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, dUdt. Finally we can write the added mass force acting on a particle as:
FV M =
ρfVp
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: ˜ FV M = − ρfVpD 80 d dt∇ 2u. (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:
FBasset = 3 2D 2√πρµ Z t 0 d dt√(u − v) t − t0 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:
mp dv dt = 1 2ρfCDA|u − v|(u − v) + µcπ D3 8 ∇ 2 u+ +ρfVp 2 d dt u − v − D 2 40 d dt∇ 2 u + 3 2D 2√ πρµ Z t 0 d dt√(u − v) t − t0 dt 0 + −∇pVp+ ρpVpg. (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 equacorrec-tion 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 = vx dz dt = vz 1 + 1 2γ dvx dt = −CD 3 4 |u − v| γD (ux− vx) − 1 ρp dp dr + 1 2γ dur dt 1 + 1 2γ dvz dt = −CD 3 4 |u − v| γD (uz− vz) − g − 1 ρp dp dz + 1 2γ duz dt (1.22)
where vx and vz are the horizontal and the vertical components of the
velocity of the particles, while ux and uz 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 = ρp
ρf, while A and Vp have been replaced by
A = π(D2)2 and V
p = 43π(D2)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 :
mp
U(t + ∆t) − U(t)
∆t = FD+ FVM+ FB+ FP+ FG, (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 1.5: 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 vorticveloc-ity 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.
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 (ux, uz)
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 104 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 104 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 104 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 104 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 104 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 104 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 104 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 104 1000 2000 3000 4000 5000 x(m) z(m)
Vertical Wind Component (t = 10600 s)
m/s −4 −2 0 2 4
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 104 0 1000 2000 3000 4000 x (m) z (m)
