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I rewrite the defined thermodynamic properties of the ejected gas and of the particles as follows

Figure 30 I rewrite the defined thermodynamic properties of the ejected gas and of the particles as follows

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Figure 0.0.1: Strombolian eruption at Stromboli Volcano, June 2006, www.photovolcanica.com
Figure 0.0.1: Strombolian eruption at Stromboli Volcano, June 2006, www.photovolcanica.com
Figure 0.0.2: Subplinian eruption, Calbuco Volcano (Chile), April 22, 2015. Carlos F. Gutierrez/AP.
Figure 0.0.2: Subplinian eruption, Calbuco Volcano (Chile), April 22, 2015. Carlos F. Gutierrez/AP.
Figure 0.0.3: Pyroclastic density current generated by the June 1991 Mount Pinatubo Plinian eruption (Philip- pines). Albert Garcia.
Figure 0.0.3: Pyroclastic density current generated by the June 1991 Mount Pinatubo Plinian eruption (Philip- pines). Albert Garcia.
Figure 0.0.4: A relatively small volcanic ash plume and pyroclastic density current generated by early stages of the June 1991 Mount Pinatubo Plinian eruption (Philippines), |volquake . weebly . com/ {mt-pinatubo-1991.html|
Figure 0.0.4: A relatively small volcanic ash plume and pyroclastic density current generated by early stages of the June 1991 Mount Pinatubo Plinian eruption (Philippines), |volquake . weebly . com/ {mt-pinatubo-1991.html|
re 0.0.5: Volcanic ash plume phenomenology, [225]. The three regions of a volcanic plume development are sketched: jet (or gas thrust region), plume (or buoyancy driven region) and umbrella cloud. The turbulent entrainment erodes the dense inner core and mixes the plume mixture with the atmosphere.
re 0.0.5: Volcanic ash plume phenomenology, [225]. The three regions of a volcanic plume development are sketched: jet (or gas thrust region), plume (or buoyancy driven region) and umbrella cloud. The turbulent entrainment erodes the dense inner core and mixes the plume mixture with the atmosphere.
Figure 0.0.6: Subplinian eruption at Mount Etna (Italy), December 5, 2015. Fernando Famiani
Figure 0.0.6: Subplinian eruption at Mount Etna (Italy), December 5, 2015. Fernando Famiani
Figure 0.0.7: Classification of eruption styles, and intensity. In bracket is reported the typical maximum plume height for each eruption style, http://www.vialattea.net/esperti/php/risposta.php?num=7055|
Figure 0.0.7: Classification of eruption styles, and intensity. In bracket is reported the typical maximum plume height for each eruption style, http://www.vialattea.net/esperti/php/risposta.php?num=7055|
Figure 0.0.8: Number of eruptions in the last two centuries. In the upper panel are reported all the registered eruptions, evidencing the first and second world word periods. The lower panel reports only the eruptions with VEI > 3. The bold line is a windowed time-average.
Figure 0.0.8: Number of eruptions in the last two centuries. In the upper panel are reported all the registered eruptions, evidencing the first and second world word periods. The lower panel reports only the eruptions with VEI > 3. The bold line is a windowed time-average.
Figure 0.0.9: Global distribution of volcanoes, http://www.gso.uri.edu/lava/Volcanism/volcanism.html
Figure 0.0.9: Global distribution of volcanoes, http://www.gso.uri.edu/lava/Volcanism/volcanism.html
where p. = p.(x,t) is the bulk density field (which can also be seen as the volumetric fraction times the material density: p. = €.f.), wu. = u.(a,t) the velocity vector field, p = p(a,t) is the fluid pressure field, T = T(a,t) is the stress tensor field (the stress terms are neglected in the solid phase in the approximation of dilute regime), g is the gravitational acceleration, f; = f;(x,t) is the drag force per unity of volume acting on the 7th particle class, e is the internal energy per unity of mass, q = q(a,t) is the fluid heat flux, Q; = Q;(a#,t) is the heat per unity of volume exchanged between the fluid phase and the jth particle class, (wp — u;) - f; is the
where p. = p.(x,t) is the bulk density field (which can also be seen as the volumetric fraction times the material density: p. = €.f.), wu. = u.(a,t) the velocity vector field, p = p(a,t) is the fluid pressure field, T = T(a,t) is the stress tensor field (the stress terms are neglected in the solid phase in the approximation of dilute regime), g is the gravitational acceleration, f; = f;(x,t) is the drag force per unity of volume acting on the 7th particle class, e is the internal energy per unity of mass, q = q(a,t) is the fluid heat flux, Q; = Q;(a#,t) is the heat per unity of volume exchanged between the fluid phase and the jth particle class, (wp — u;) - f; is the
Table 1.1.1: Sutherland law constants obtained by fitting data from where jigth and Ts, are two constants depending on the fluid. Using data from I obtain results presented in Tab. Heat flux As usually done, the heat flux q is defined through the Fourier law In the applications faced in the present work, I will always consider the volume viscosity negligible and the dynamic viscosity of the gaseous component will be considered constant or depending on the temperature by the Sutherland law:
Table 1.1.1: Sutherland law constants obtained by fitting data from where jigth and Ts, are two constants depending on the fluid. Using data from I obtain results presented in Tab. Heat flux As usually done, the heat flux q is defined through the Fourier law In the applications faced in the present work, I will always consider the volume viscosity negligible and the dynamic viscosity of the gaseous component will be considered constant or depending on the temperature by the Sutherland law:
Figure 1.1.1: Function Re;(d;) for a falling particle with: ~; = 2000kg/m? in an ambient fluid with pi = 1.846 * 10—° Pas and pg = 1.17, 0.117, or 0.0117 kg/m?.
Figure 1.1.1: Function Re;(d;) for a falling particle with: ~; = 2000kg/m? in an ambient fluid with pi = 1.846 * 10—° Pas and pg = 1.17, 0.117, or 0.0117 kg/m?.
igure 1.1.2: Selling velocity w(d,) of a falling particle with diameter dj. Properties are: 6; = 2000kg/m° in an ambient fluid with p = 1.846 * 10-5 Pas and pg = 1.17kg/m?. Exact solution of Eq. 1.1.34) (points) and approximation based on Eq. 1.1.36} (solid line). Gas—particle heat transfer
igure 1.1.2: Selling velocity w(d,) of a falling particle with diameter dj. Properties are: 6; = 2000kg/m° in an ambient fluid with p = 1.846 * 10-5 Pas and pg = 1.17kg/m?. Exact solution of Eq. 1.1.34) (points) and approximation based on Eq. 1.1.36} (solid line). Gas—particle heat transfer
Figure 1.1.3: The correction factor 2¢-¢/Nu,; as a function of Re;, fixing the Prandtl number of atmospheric air to Prg = 0.71.
Figure 1.1.3: The correction factor 2¢-¢/Nu,; as a function of Re;, fixing the Prandtl number of atmospheric air to Prg = 0.71.
Lagrangian form:
Lagrangian form:
leading to the classic expressions (recall that both R and Cy are constant along streamlines of w,): where, by defining X,(ao,t) the trajectory starting in a and driven by ug, the adiabatic index of the gas mixture is defined as:
leading to the classic expressions (recall that both R and Cy are constant along streamlines of w,): where, by defining X,(ao,t) the trajectory starting in a and driven by ug, the adiabatic index of the gas mixture is defined as:
There, it is worth noting that the speed of sound of a gas—particle mixture decreases with respect to that of the carrier gas phase because the mixture density pm increases and the specific heat ratio decreases, because of the presence of the solid particles. Since yg and y, are constants along streamlines, Rm/Cmn = Ym — | is also constant along streamlines:
There, it is worth noting that the speed of sound of a gas—particle mixture decreases with respect to that of the carrier gas phase because the mixture density pm increases and the specific heat ratio decreases, because of the presence of the solid particles. Since yg and y, are constants along streamlines, Rm/Cmn = Ym — | is also constant along streamlines:
can be defined when f,, 4 0. This velocity field takes into account the kinematic decoupling v;, correcting the advection term of fin. Continuity equations Advection with respect the mixture velocity
can be defined when f,, 4 0. This velocity field takes into account the kinematic decoupling v;, correcting the advection term of fin. Continuity equations Advection with respect the mixture velocity
Enthalpy equations taking into account of the combined effect due to kinematic decoupling and difference between the enthalpy (v;,) and kinetic energy (vx) of the mixture and of the jth specie. This equation for the momentum balance is the classical compressible Navier-Stokes equation with the substitution ug — Um and the addition of the term V - (pmT;) which takes into account the effects of particle decoupling on momentum (two-way coupling).
Enthalpy equations taking into account of the combined effect due to kinematic decoupling and difference between the enthalpy (v;,) and kinetic energy (vx) of the mixture and of the jth specie. This equation for the momentum balance is the classical compressible Navier-Stokes equation with the substitution ug — Um and the addition of the term V - (pmT;) which takes into account the effects of particle decoupling on momentum (two-way coupling).
Summarizing, the [eulerian] model can be rearranged by using the mixture mass, velocity and enthalpy definitions:
Summarizing, the [eulerian] model can be rearranged by using the mixture mass, velocity and enthalpy definitions:
In order to emulate the behavior of an open boundary, the following conditions 1.5.2 Atmospheric boundary condition — [atmo]
In order to emulate the behavior of an open boundary, the following conditions 1.5.2 Atmospheric boundary condition — [atmo]
If I now consider ¢ > 7;, neglecting the transient phase in which particles reach the equilibrium with the fluid? I obtain and assuming that the series converges uniformly, I get
If I now consider ¢ > 7;, neglecting the transient phase in which particles reach the equilibrium with the fluid? I obtain and assuming that the series converges uniformly, I get
In particular, I will use the Bochner integral to deal with Banach spaces W*»(0, 7; _X) defined as follows: where L?(0,7; X) is the space of strongly measurable functions such that Finally, I define the spaces with weak topologies as follows:
In particular, I will use the Bochner integral to deal with Banach spaces W*»(0, 7; _X) defined as follows: where L?(0,7; X) is the space of strongly measurable functions such that Finally, I define the spaces with weak topologies as follows:
Integration by parts gives us that Veé12 > 0
Integration by parts gives us that Veé12 > 0
The novelty in the results which appeared for the moment as a Technical Report (just in the spring of 2015) [206] is that all regularizing parameter can be taken to zero and the limit will give a weak solution of the model Aueiel, . The details, I am still double-checking are very intricate and they are based on a very specific way of taking limits in a determined order. First one has to pass to the limit in the Galerkin approximation (which I skipped in the system, being the most standard step), then one has to take the following limits
The novelty in the results which appeared for the moment as a Technical Report (just in the spring of 2015) [206] is that all regularizing parameter can be taken to zero and the limit will give a weak solution of the model Aueiel, . The details, I am still double-checking are very intricate and they are based on a very specific way of taking limits in a determined order. First one has to pass to the limit in the Galerkin approximation (which I skipped in the system, being the most standard step), then one has to take the following limits
igure 3.2.1: Sketch of the cylindrical control volume used to find the mean conservation equations. Here B is the cylinder radius, 6z its thickness and z its height with respect the vent elevation.
igure 3.2.1: Sketch of the cylindrical control volume used to find the mean conservation equations. Here B is the cylinder radius, 6z its thickness and z its height with respect the vent elevation.
Table 3.4.1: Independent parameters for a multiphase plume in a stratified atmosphere. It is worth noting that q,, q, > —1 because the specific heats and gas constants are positive (x., w. > 0) and the sum of the initial mass fraction is smaller than 1 (cf. definition of q,, gy in Tab. [3.4.1). Moreover, ¢ > —1 because Cg 91'g.9 > 0. Even if these are the general conditions for such parameters, in Tab. [8.4.1] there are summarized the possible ranges for volcanic eruptions.
Table 3.4.1: Independent parameters for a multiphase plume in a stratified atmosphere. It is worth noting that q,, q, > —1 because the specific heats and gas constants are positive (x., w. > 0) and the sum of the initial mass fraction is smaller than 1 (cf. definition of q,, gy in Tab. [3.4.1). Moreover, ¢ > —1 because Cg 91'g.9 > 0. Even if these are the general conditions for such parameters, in Tab. [8.4.1] there are summarized the possible ranges for volcanic eruptions.
It is worth noting that in the single phase case Cg = C, and Rg = Ry. Thus, the initial enthalpy anomaly reduces to the initial thermal anomaly or equivalently to the density anomaly: If the thermodynamic properties of the ejected fluid are similar to those of the ambient fluid then |qy| ~ |q,| < 1. In this case, model (3.4.1) becomes:
It is worth noting that in the single phase case Cg = C, and Rg = Ry. Thus, the initial enthalpy anomaly reduces to the initial thermal anomaly or equivalently to the density anomaly: If the thermodynamic properties of the ejected fluid are similar to those of the ambient fluid then |qy| ~ |q,| < 1. In this case, model (3.4.1) becomes:
This ordinary differential equation has a first integral of motion] U in both the considered cases for 7,,. I found respectively for the entrainment models of Morton et al. and Ricou and Spalding [160]: 3.7 Non stratified plume regime
This ordinary differential equation has a first integral of motion] U in both the considered cases for 7,,. I found respectively for the entrainment models of Morton et al. and Ricou and Spalding [160]: 3.7 Non stratified plume regime
Table 3.7.1: Column stability parameters for the plumes studied in this thesis.
Table 3.7.1: Column stability parameters for the plumes studied in this thesis.
igure 3.8.1: The height of collapse of a multiphase plume in a non-stratified stable atmosphere as a function of the parameter ag defined in Eq. (3.8.7). Here, the exact formula Eq. (3.8.10) is compared with its asymptotic expansion Eq. (3.8.11), in the case vg = 0.2.
igure 3.8.1: The height of collapse of a multiphase plume in a non-stratified stable atmosphere as a function of the parameter ag defined in Eq. (3.8.7). Here, the exact formula Eq. (3.8.10) is compared with its asymptotic expansion Eq. (3.8.11), in the case vg = 0.2.
Figure 3.8.2: a) The virtual radius by as a function of ag. The virtual radius tends to zero when ag — 0 and increases with a square root law as ag increases (cf. Eq. 3.8.15}). b) Height of the plume radius necking Cneck as predicted by Eq. (3.8.18).
Figure 3.8.2: a) The virtual radius by as a function of ag. The virtual radius tends to zero when ag — 0 and increases with a square root law as ag increases (cf. Eq. 3.8.15}). b) Height of the plume radius necking Cneck as predicted by Eq. (3.8.18).
Figure 3.8.3: Evolution of the plume radius b(z) = q/,/m in all the admissible regimes of model (3.8.2) with Uq = 0.2. Starting from the lower graph, I choose: ag = —1, —10, —50, oo, 50, 10, 1, 0.1, 0.0001.
Figure 3.8.3: Evolution of the plume radius b(z) = q/,/m in all the admissible regimes of model (3.8.2) with Uq = 0.2. Starting from the lower graph, I choose: ag = —1, —10, —50, oo, 50, 10, 1, 0.1, 0.0001.
This model reduces to the same model introduced by Morton in the monophase allowing to write the multiphase plume model in a stratified calm atmosphere:
This model reduces to the same model introduced by Morton in the monophase allowing to write the multiphase plume model in a stratified calm atmosphere:
Table 3.9.1: Plume parameters useful for estimate the plume height of the four plume examples of this thesis.
Table 3.9.1: Plume parameters useful for estimate the plume height of the four plume examples of this thesis.
‘igure 3.9.1: Contour plot of the plume height function §(2o, go) defined in Eq. (3.9.20). This function assumes its maximum in (1,0) = [1 ~ 2.572, and it is a strictly decreasing function of go. When ro > 17 the plume regime is reached. and enough enthalpy to rise until its maximum; 2) when 7 > 1 and 6, < 1, the collapsing plume regime is reached, near the point (Xo, Go) = (—1,0); 3) when 6; « 1 the jet regime is reached, near the line x) = 0. In general, 7. is the parameter controlling the column stability: when 7, < 1 then 0 < x < 1, the column is not collapsing and when 29 — 1 the column behaves as a plume, while x9 > 0*, the column behaves as a jet.
‘igure 3.9.1: Contour plot of the plume height function §(2o, go) defined in Eq. (3.9.20). This function assumes its maximum in (1,0) = [1 ~ 2.572, and it is a strictly decreasing function of go. When ro > 17 the plume regime is reached. and enough enthalpy to rise until its maximum; 2) when 7 > 1 and 6, < 1, the collapsing plume regime is reached, near the point (Xo, Go) = (—1,0); 3) when 6; « 1 the jet regime is reached, near the line x) = 0. In general, 7. is the parameter controlling the column stability: when 7, < 1 then 0 < x < 1, the column is not collapsing and when 29 — 1 the column behaves as a plume, while x9 > 0*, the column behaves as a jet.
with less than 12% of error with respect to the “real” values. ) Analytic solution for a non-Boussinesq plume in a stratified environment
with less than 12% of error with respect to the “real” values. ) Analytic solution for a non-Boussinesq plume in a stratified environment
Table 3.10.1: Comparison between the height of the plume as evaluated from the numerical (ASH1D, Eq. and analytical (ASHOD) one-dimensional plume model.
Table 3.10.1: Comparison between the height of the plume as evaluated from the numerical (ASH1D, Eq. and analytical (ASHOD) one-dimensional plume model.
10.1: [forcedPlume]: Vertical evolution of the non-dimensional fluxes q, m, f (log-log scale), of the plume radius b (log-log scale) and of the dimensional physical parameters U, 8, Tg, Yo (s)) in (linear-log) scale. Solid lines correspond to the numerical solution of model (3.4.1), while dashed lines are evaluated by using the analytic asymptotic solution Eqs. 3.10.1}, (3.10.3), (3.10.5).
10.1: [forcedPlume]: Vertical evolution of the non-dimensional fluxes q, m, f (log-log scale), of the plume radius b (log-log scale) and of the dimensional physical parameters U, 8, Tg, Yo (s)) in (linear-log) scale. Solid lines correspond to the numerical solution of model (3.4.1), while dashed lines are evaluated by using the analytic asymptotic solution Eqs. 3.10.1}, (3.10.3), (3.10.5).
igure 3.10.3: [weakPlume]: Vertical evolution of the non-dimensional fluxes g, m, f (log-linear scale) and of the dimensional physical parameters U, b, 8, Ts, Yo (s): Solid lines correspond to the numerical solution of model (3.4.1), while dashed lines are evaluated by using the analytic asymptotic solution Eqs. {3.10.1}, (3-10-3p, (8.10.5).
igure 3.10.3: [weakPlume]: Vertical evolution of the non-dimensional fluxes g, m, f (log-linear scale) and of the dimensional physical parameters U, b, 8, Ts, Yo (s): Solid lines correspond to the numerical solution of model (3.4.1), while dashed lines are evaluated by using the analytic asymptotic solution Eqs. {3.10.1}, (3-10-3p, (8.10.5).
; Schematic overview of the aims and methodology of the work presented in this chapter. (A.1 Volcanic plume emission is recorded using a TIR camera. Modeling of such phenomenon requires 3D numerical models (A.2), able to reproduce the complex fluid-dynamic behavior at various length and time scales. However because such models require high computational power and time, analytical 1D mean plume models (B.2) may be used to predict the mean behavior of volcanic plumes. Such models are time-averaged stationary models, which describe the mean spatial distribution of flow parameters (e.g., particle concentration, temperature, velocity) given a set of input conditions. By coupling an electromagnetic model to the plume model, the TIR emission of the gas—particle mixture is simulated, and a synthetic thermal infrared image (C.2) computed. The mean plume behavior may also be recovered from the recorded image by constructing a "mean image" (B.1), which is a time-averaged image obtained from averaging a sequence of images in a TIR video sequence. In doing this, the time-dependent dynamic fluctuations of the plume are filtered, leaving an image that reflects the mean plume behavior. Image processing is then applied to obtain an image with a vent-centred metric coordinate system, comparable to that created by the forward model. Recursive minimization of the discrepancy between the observed and modelled TIR images is then performed by application of an inversion model (2D when the entire images are compared, or 1D when only the plume central axis is compared), which searches for the best model input-parameters (e.g., ash mass, particle size distribution, etc.) reproducing the observed data.
; Schematic overview of the aims and methodology of the work presented in this chapter. (A.1 Volcanic plume emission is recorded using a TIR camera. Modeling of such phenomenon requires 3D numerical models (A.2), able to reproduce the complex fluid-dynamic behavior at various length and time scales. However because such models require high computational power and time, analytical 1D mean plume models (B.2) may be used to predict the mean behavior of volcanic plumes. Such models are time-averaged stationary models, which describe the mean spatial distribution of flow parameters (e.g., particle concentration, temperature, velocity) given a set of input conditions. By coupling an electromagnetic model to the plume model, the TIR emission of the gas—particle mixture is simulated, and a synthetic thermal infrared image (C.2) computed. The mean plume behavior may also be recovered from the recorded image by constructing a "mean image" (B.1), which is a time-averaged image obtained from averaging a sequence of images in a TIR video sequence. In doing this, the time-dependent dynamic fluctuations of the plume are filtered, leaving an image that reflects the mean plume behavior. Image processing is then applied to obtain an image with a vent-centred metric coordinate system, comparable to that created by the forward model. Recursive minimization of the discrepancy between the observed and modelled TIR images is then performed by application of an inversion model (2D when the entire images are compared, or 1D when only the plume central axis is compared), which searches for the best model input-parameters (e.g., ash mass, particle size distribution, etc.) reproducing the observed data.
ure 4.2.1: Schematic configuration of the propagation of a TIR ray inside a heterogeneous medium. The absorption coefficient is proportional to particle concentration. 'TIR intensity changes along the path due to particle emission/absorption. I indicate with s the curvilinear coordinate along the path, with Jo being the background intensity and I; being the measured intensity at distance L from the background position. Thus, along a ray, this balance is expressed by:
ure 4.2.1: Schematic configuration of the propagation of a TIR ray inside a heterogeneous medium. The absorption coefficient is proportional to particle concentration. 'TIR intensity changes along the path due to particle emission/absorption. I indicate with s the curvilinear coordinate along the path, with Jo being the background intensity and I; being the measured intensity at distance L from the background position. Thus, along a ray, this balance is expressed by:
sure 4.2.2: Typical spectral response of a FLIR camera (grey-shaded line) in the spectral window 7 + 18 um with the specific absorption spectra of water vapor (solid line), SO2 (dashed line) and COg (dotted line) at standard conditions (296 K, 1 atm). Weighted averages of the absorption coefficients of the three gases over the spectral response A; are reported in the panel. In the small plot, a logarithmic zoom of the water vapor has been drawn. Data are retrieved from the HITRAN 2012 database [164].
sure 4.2.2: Typical spectral response of a FLIR camera (grey-shaded line) in the spectral window 7 + 18 um with the specific absorption spectra of water vapor (solid line), SO2 (dashed line) and COg (dotted line) at standard conditions (296 K, 1 atm). Weighted averages of the absorption coefficients of the three gases over the spectral response A; are reported in the panel. In the small plot, a logarithmic zoom of the water vapor has been drawn. Data are retrieved from the HITRAN 2012 database [164].
ure 4.3.1: Geometric configuration adopted for the calculation of the IR intensity. a) Side view. b) Top view of a cutting plane orthogonal to the plume axis (point 0). The plume radius is represented by the gray-shaded region (top-hat approximation).
ure 4.3.1: Geometric configuration adopted for the calculation of the IR intensity. a) Side view. b) Top view of a cutting plane orthogonal to the plume axis (point 0). The plume radius is represented by the gray-shaded region (top-hat approximation).
ire 4.4.1: a) Averaged image computed from a set of thermal infrared images recorded with a FLIR camera, imaging the stationary emission of a sustained volcanic ash plume at Santiaguito. The red dots represent the extracted plume axis |201|. Temperature values along the plume axis are presented in Fig. |4.4.5] b) Relative fluctuation image computed from the same set of images, evidencing turbulence intensity and turbulent entrainment development.
ire 4.4.1: a) Averaged image computed from a set of thermal infrared images recorded with a FLIR camera, imaging the stationary emission of a sustained volcanic ash plume at Santiaguito. The red dots represent the extracted plume axis |201|. Temperature values along the plume axis are presented in Fig. |4.4.5] b) Relative fluctuation image computed from the same set of images, evidencing turbulence intensity and turbulent entrainment development.
ire 4.4.2: a) Image of the atmosphere above the volcano before the eruption. b) Averaged image of the volcanic eruption. In both images, horizontal and vertical axes represents the position in meters inside the image, the temperature is represented by the color scale in Celsius degrees.
ire 4.4.2: a) Image of the atmosphere above the volcano before the eruption. b) Averaged image of the volcanic eruption. In both images, horizontal and vertical axes represents the position in meters inside the image, the temperature is represented by the color scale in Celsius degrees.
Table 4.4.1: Result of the two-dimensional minimization procedure. Best fit values of the plume parameters. Here I obtained o(p*) = 6.428° C. the correlation matrix is discussed in Sec. That analysis allows to evaluate the sensitivity of the result on the input parameters and the error associated to the solution. For this test case I obtained g = 6.428° C. 4.4.4 Axial inversion
Table 4.4.1: Result of the two-dimensional minimization procedure. Best fit values of the plume parameters. Here I obtained o(p*) = 6.428° C. the correlation matrix is discussed in Sec. That analysis allows to evaluate the sensitivity of the result on the input parameters and the error associated to the solution. For this test case I obtained g = 6.428° C. 4.4.4 Axial inversion
4.3: a,c) Synthetic image of the plume obtained by fitting the experimental data showed in Fig. (a: two-dimensional fitting, c: axial fitting); b,d) Unsigned difference between the synthetic and the experimental images allowing error quantification and localization (b: two-dimensional fitting, d: axial fitting). In both images, horizontal and vertical axes represents the position (in meters) inside the image. The color scale represents the temperature in degrees Celsius.
4.3: a,c) Synthetic image of the plume obtained by fitting the experimental data showed in Fig. (a: two-dimensional fitting, c: axial fitting); b,d) Unsigned difference between the synthetic and the experimental images allowing error quantification and localization (b: two-dimensional fitting, d: axial fitting). In both images, horizontal and vertical axes represents the position (in meters) inside the image. The color scale represents the temperature in degrees Celsius.
4.4.4: Two-dimensional inversion procedure. Variation of o(p) of Eq. around p* as a function of each single parameter, the others being kept fixed (p = (vq, Um, £0, , dx, 4, Ag)). The fit was done using the parameter transformation y = —q,/qy and qg = —qy. Asterisks mark the value of each component of p*; here I obtained o(p*) = 6.428° C.
4.4.4: Two-dimensional inversion procedure. Variation of o(p) of Eq. around p* as a function of each single parameter, the others being kept fixed (p = (vq, Um, £0, , dx, 4, Ag)). The fit was done using the parameter transformation y = —q,/qy and qg = —qy. Asterisks mark the value of each component of p*; here I obtained o(p*) = 6.428° C.
Table 4.4.2: Result of the axial minimization procedure. Best fit values of the plume Teen teenparameters. Here I obtained o(p*) = 0.6596° C. Because the coupled model dependency on input parameters is non-linear, it is difficult, in principle, to ensure that the result of the minimization procedure is unique. This is why the inversion must be done by preliminary constraining the ranges of possible outcomes (third column in tables |4.4.1] and |4.4.2). These can be constrained based on the experience of the operator and on the basis of complementary field observations. In the present application, I have explored also other ranges but only the minima represented in Figures [4.4.6] and [4.4.4] were found. These figures give a view of the multidimensional shape of the minimization function o(p), from which the point of minimum is clearly recognizable. The reason for this good behavior of the optimization procedure is that the brightness image is inverted
Table 4.4.2: Result of the axial minimization procedure. Best fit values of the plume Teen teenparameters. Here I obtained o(p*) = 0.6596° C. Because the coupled model dependency on input parameters is non-linear, it is difficult, in principle, to ensure that the result of the minimization procedure is unique. This is why the inversion must be done by preliminary constraining the ranges of possible outcomes (third column in tables |4.4.1] and |4.4.2). These can be constrained based on the experience of the operator and on the basis of complementary field observations. In the present application, I have explored also other ranges but only the minima represented in Figures [4.4.6] and [4.4.4] were found. These figures give a view of the multidimensional shape of the minimization function o(p), from which the point of minimum is clearly recognizable. The reason for this good behavior of the optimization procedure is that the brightness image is inverted
Figure 4.4.5: Result of the one-dimensional fit (solid line) of the experimental thermal image along the axis (stars)
Figure 4.4.5: Result of the one-dimensional fit (solid line) of the experimental thermal image along the axis (stars)
re 4.4.6: Axial inversion procedure. Variation of o(p) of Eq. 4.4.4) around p* as a function of each single parameter, the others being kept fixed (p = (vq, Um, 40, 6, 4x; Ag))-. The fit was done using the parameter transformation y = —q,/qy and qg = —qy. Asterisks mark the value of each component of p*; here I obtained o(p*) = 0.6596° C.
re 4.4.6: Axial inversion procedure. Variation of o(p) of Eq. 4.4.4) around p* as a function of each single parameter, the others being kept fixed (p = (vq, Um, 40, 6, 4x; Ag))-. The fit was done using the parameter transformation y = —q,/qy and qg = —qy. Asterisks mark the value of each component of p*; here I obtained o(p*) = 0.6596° C.
gure 4.4.7: Contour plot of the optical depth rz, (see below Eq. (4.2.4)) of the synthetic plume obtained from a) axial fitting; b) two-dimensional fitting. The synthetic plume obtained from axial inversion is more opaque than the plume obtained from two-dimensional inversion.
gure 4.4.7: Contour plot of the optical depth rz, (see below Eq. (4.2.4)) of the synthetic plume obtained from a) axial fitting; b) two-dimensional fitting. The synthetic plume obtained from axial inversion is more opaque than the plume obtained from two-dimensional inversion.
Table 4.4.3: Result of the axial and two-dimensional minimization procedure: physical input parameters for the coupled model. jw and ps are the total erupted water vapor and solid masses.
Table 4.4.3: Result of the axial and two-dimensional minimization procedure: physical input parameters for the coupled model. jw and ps are the total erupted water vapor and solid masses.
ure 4.5.1: Function D(X) plotted by fixing dg/dmin in Eq. 4.6.3). The big panel refers to the axial fit Sauter diameter while the small panel has been obtained by using the Sauter diameter from the 2D fit. In each panel, solid black line is associated to the measured Sauter diameter, while solid grey lines corresponds to the error limits. Dashed lines represent the error limits for the selected value of X = dmax/dmin = 3.2 * 10°.
ure 4.5.1: Function D(X) plotted by fixing dg/dmin in Eq. 4.6.3). The big panel refers to the axial fit Sauter diameter while the small panel has been obtained by using the Sauter diameter from the 2D fit. In each panel, solid black line is associated to the measured Sauter diameter, while solid grey lines corresponds to the error limits. Dashed lines represent the error limits for the selected value of X = dmax/dmin = 3.2 * 10°.
Figure 5.1.1: ASHEE parallel efficiency on Fermi and PLX supercomputers at CINECA (www. cine
Figure 5.1.1: ASHEE parallel efficiency on Fermi and PLX supercomputers at CINECA (www. cine
gure 5.2.1: Comparison of a DNS executed with the eight order scheme by Pirozzoli and Grasso [150] and ASHEE code implemented using the C++ libraries of OpenFOAM® at t/Te = 1.093. The L? norm between the two spectra is 4.0 * 10-4. The main parameters are Re, ~ 116, Marms = 0.2. In order to validate the numerical code in the single-phase case, the DNS of com- pressible decaying homogeneous and isotropic turbulence is compared with the well tested numerical solver for Direct Numerical Simulations of compressible turbulence of Sergio Pirozzoli and Matteo Bernerdini [see [7|. For this comparison the ollowing initial parameters are fixed: p= Rm = 1, Ym = 1.4, Pr = 1, Marms = 0.2, Co = 0, u2., = 2Ko = 0.056, ko = 4, Av = 0.5, Te © 3.6596, pp = 5.885846 « 1074, Re, ~ 116, Re,, ~ 100. Thus a grid with N = 256° cells gives kinax &% 127 and kimaxt) & 3.0, big enough to have a DNS. Here I have used the numerical scheme linear]. The simulation has been performed on 1024 cores on the Fermi Blue Gene/Q infrastructure at CINECA |CIN]. It needed about 5 h to be computed completely. In Fig. the comparison is shown between the energy spectrum
gure 5.2.1: Comparison of a DNS executed with the eight order scheme by Pirozzoli and Grasso [150] and ASHEE code implemented using the C++ libraries of OpenFOAM® at t/Te = 1.093. The L? norm between the two spectra is 4.0 * 10-4. The main parameters are Re, ~ 116, Marms = 0.2. In order to validate the numerical code in the single-phase case, the DNS of com- pressible decaying homogeneous and isotropic turbulence is compared with the well tested numerical solver for Direct Numerical Simulations of compressible turbulence of Sergio Pirozzoli and Matteo Bernerdini [see [7|. For this comparison the ollowing initial parameters are fixed: p= Rm = 1, Ym = 1.4, Pr = 1, Marms = 0.2, Co = 0, u2., = 2Ko = 0.056, ko = 4, Av = 0.5, Te © 3.6596, pp = 5.885846 « 1074, Re, ~ 116, Re,, ~ 100. Thus a grid with N = 256° cells gives kinax &% 127 and kimaxt) & 3.0, big enough to have a DNS. Here I have used the numerical scheme linear]. The simulation has been performed on 1024 cores on the Fermi Blue Gene/Q infrastructure at CINECA |CIN]. It needed about 5 h to be computed completely. In Fig. the comparison is shown between the energy spectrum
Figure 5.2.2: Isosurface at T = 1, coloured with the magnitude of u.
Figure 5.2.2: Isosurface at T = 1, coloured with the magnitude of u.
Figure 5.2.3: Evolution of the main integral parameters for DHIT with Re, ~ 116 and Marms = 0.2.
Figure 5.2.3: Evolution of the main integral parameters for DHIT with Re, ~ 116 and Marms = 0.2.
Figure 5.2.4: Isosurface at Qu ~ 19 Hz? and t/t. ~ 2.2, representing zones with coherent vortices Looking through all the figures it is worth noting that the coarse DNS [noM]-32? behaves very well, being the coarse simulation that gives results more similar to the filtered DNS. However, in Fig. tends to accumulate to much energy at the smallest scales. This behavior should cause numerical instability, especially for long duration LES. On the other hand, subgrid models do not show this behavior, tending to dissipate too much in the inertial range while too much little at integral scale. However, dynamical subgrid models (right panels of Fig. and they gives satisfactory results. 5.2.5 5.2.5a| the simulation without a subgrid model behaves much better than non-dynamical ones
Figure 5.2.4: Isosurface at Qu ~ 19 Hz? and t/t. ~ 2.2, representing zones with coherent vortices Looking through all the figures it is worth noting that the coarse DNS [noM]-32? behaves very well, being the coarse simulation that gives results more similar to the filtered DNS. However, in Fig. tends to accumulate to much energy at the smallest scales. This behavior should cause numerical instability, especially for long duration LES. On the other hand, subgrid models do not show this behavior, tending to dissipate too much in the inertial range while too much little at integral scale. However, dynamical subgrid models (right panels of Fig. and they gives satisfactory results. 5.2.5 5.2.5a| the simulation without a subgrid model behaves much better than non-dynamical ones
Table 5.3.1: Stokes time, maximum Stokes number and diameter of the solid particles inserted in the turbulent box.
Table 5.3.1: Stokes time, maximum Stokes number and diameter of the solid particles inserted in the turbulent box.
Figure 5.3.1: Evolution of the Kolmogorov time mi- croscale Ty.
Figure 5.3.1: Evolution of the Kolmogorov time mi- croscale Ty.
Figure 5.3.2: Slice of the turbulent box at t/te ~ 2.2. The two panels represent respectively a logaritmic color map of y3 (Stmax = 0.5) and of |ag|. Moreover, it is worth noting that (P),; vanishes in absence of preferential concentra- tion. By dimensional analysis, preferential concentration is expected to behave as:
Figure 5.3.2: Slice of the turbulent box at t/te ~ 2.2. The two panels represent respectively a logaritmic color map of y3 (Stmax = 0.5) and of |ag|. Moreover, it is worth noting that (P),; vanishes in absence of preferential concentra- tion. By dimensional analysis, preferential concentration is expected to behave as:
igure 5.3.3: Evolution of the degree of preferential concentration (P); as a function of Ste (LES) or St, (DNS) The evolution is shown for the 5 particle classes of Tab. [5.3.1] in the DNS (bold solid lines) and LES (bold dashed lines) cases. The asymptotic trend St? (solid line) is shown with the Lagrangian results by Rani and Balachandar [158] (points) and the fit Eq. (dotted line). A good agreement is obtained between equilibrium—Eulerian LES/DNS and Lagrangian DNS simulations.
igure 5.3.3: Evolution of the degree of preferential concentration (P); as a function of Ste (LES) or St, (DNS) The evolution is shown for the 5 particle classes of Tab. [5.3.1] in the DNS (bold solid lines) and LES (bold dashed lines) cases. The asymptotic trend St? (solid line) is shown with the Lagrangian results by Rani and Balachandar [158] (points) and the fit Eq. (dotted line). A good agreement is obtained between equilibrium—Eulerian LES/DNS and Lagrangian DNS simulations.
Table 5.4.1: Physical parameters for the natural convection test case. D. reports the physical parameters used for this test case.
Table 5.4.1: Physical parameters for the natural convection test case. D. reports the physical parameters used for this test case.
At the beginning of the thermodynamic transformation, Nu, > Nu, because thi hotter wall heats the box interior, while the colder is in equilibrium with it. However when the stationary state is reached, then it must hold Nu, = Nu, = Nu, because o energy conservation. Thus, at the equilibrium, the box acts as a conductor betwee! the two wa ls. If conduction dominates (small Ra), Nu ~ 1; on the contrary i convection dominates (large Ra), Nu increases. In the four simulations the Rayleig] number is eft increase, from Ra = 10° to Ra = 10°, in order to compare th present result with the experimental data found in Costa [40], Deng and Tang [50 Fig. 5.4.1]s hows the 6400 cells mesh used, together with the temperature field o the Ra = 10° case. The final time for each simulation is te. Tab. |5.4.2|/ reports thi resu. ts of t he cited papers, present results and relative errors. Fig. 5.4.2 show: Table 5.4.2: Comparison between the results obtained with ASHEE and that measured by Costa [40] and simulate by Deng and Tang [50]. The relative error in the fifth column is relative to Costa [40].
At the beginning of the thermodynamic transformation, Nu, > Nu, because thi hotter wall heats the box interior, while the colder is in equilibrium with it. However when the stationary state is reached, then it must hold Nu, = Nu, = Nu, because o energy conservation. Thus, at the equilibrium, the box acts as a conductor betwee! the two wa ls. If conduction dominates (small Ra), Nu ~ 1; on the contrary i convection dominates (large Ra), Nu increases. In the four simulations the Rayleig] number is eft increase, from Ra = 10° to Ra = 10°, in order to compare th present result with the experimental data found in Costa [40], Deng and Tang [50 Fig. 5.4.1]s hows the 6400 cells mesh used, together with the temperature field o the Ra = 10° case. The final time for each simulation is te. Tab. |5.4.2|/ reports thi resu. ts of t he cited papers, present results and relative errors. Fig. 5.4.2 show: Table 5.4.2: Comparison between the results obtained with ASHEE and that measured by Costa [40] and simulate by Deng and Tang [50]. The relative error in the fifth column is relative to Costa [40].
Figure 5.4.1: Configuration for the natural convection test case. a) Mesh with 6400 cells; b) stationary tempera- ture configuration for the Ra = 10° case (AT = 10.59825 K).
Figure 5.4.1: Configuration for the natural convection test case. a) Mesh with 6400 cells; b) stationary tempera- ture configuration for the Ra = 10° case (AT = 10.59825 K).
Figure 5.4.2: Evolution of the Nusselt number for all the four simulations presented. For each case, the line above refers to Nuy, while that below is Nuc.
Figure 5.4.2: Evolution of the Nusselt number for all the four simulations presented. For each case, the line above refers to Nuy, while that below is Nuc.
Table 5.5.1: Configuration parameters for the [forcedPlume] experiment.
Table 5.5.1: Configuration parameters for the [forcedPlume] experiment.
5.5.1: 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—°/3 and Str~° are represented with a thick solid and dashed line respectively.
5.5.1: 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—°/3 and Str~° are represented with a thick solid and dashed line respectively.
gure 5.5.3: Two-dimensional slice and streamlines of the velocity field: a) time-averaged velocity field; b) in- stantaneous velocity field at t = 10 s. 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. the evolution of the radial profiles for the mean vertical velocity field. This figure,
gure 5.5.3: Two-dimensional slice and streamlines of the velocity field: a) time-averaged velocity field; b) in- stantaneous velocity field at t = 10 s. 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. the evolution of the radial profiles for the mean vertical velocity field. This figure,
ure 5.5.4: 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 # = 0.141 + 0.001.
ure 5.5.4: 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 # = 0.141 + 0.001.
5/2 Figure 5.5.5: Linear regression between m and q? for the plume simulation with azimuthal forcing
5/2 Figure 5.5.5: Linear regression between m and q? for the plume simulation with azimuthal forcing
re 5.5.6: 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.
re 5.5.6: 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.
Figure 5.5.7: Centerline time-average axial velocity, and temperature profiles with azimuthal forcing. Inset) cen- terline correlations of fluctuating velocity and temperature.
Figure 5.5.7: Centerline time-average axial velocity, and temperature profiles with azimuthal forcing. Inset) cen- terline correlations of fluctuating velocity and temperature.
igure 5.6.1: The Sod’s shock tube density after 0.007 s (here c = 374.348 m/s). Here the analytic solution (solid) is compared with two simulations performed with ASHEE (dashed) and rhoCentralFoam (points) codes. The resolution is 100 cells, while in the inlet it is reported the ASHEE simulation with resolution equal to 1000 cells.
igure 5.6.1: The Sod’s shock tube density after 0.007 s (here c = 374.348 m/s). Here the analytic solution (solid) is compared with two simulations performed with ASHEE (dashed) and rhoCentralFoam (points) codes. The resolution is 100 cells, while in the inlet it is reported the ASHEE simulation with resolution equal to 1000 cells.
Figure 5.7.1: Evolution of MMF and comparison with the DNS presented in Ozgékmen et al. [145]. Point type plots have been obtained by using the LES models described in Sect.|1.4
Figure 5.7.1: Evolution of MMF and comparison with the DNS presented in Ozgékmen et al. [145]. Point type plots have been obtained by using the LES models described in Sect.|1.4
Figure 5.7.2: Evolution of RPE*. Point type plots have been obtained by using the LES models described in Sect. 1.4
Figure 5.7.2: Evolution of RPE*. Point type plots have been obtained by using the LES models described in Sect. 1.4
Figure 5.7.3: Snapshot of the solid phase bulk density in the lock exchange simulations after t/7T ~ 1.114. Dif- ferent LES models are compared with the DNS (a): b) [wale]; c) [dynWale]; d) [dynOneEqEddy]; e) [dynWale] with three PIMPLE iterations.
Figure 5.7.3: Snapshot of the solid phase bulk density in the lock exchange simulations after t/7T ~ 1.114. Dif- ferent LES models are compared with the DNS (a): b) [wale]; c) [dynWale]; d) [dynOneEqEddy]; e) [dynWale] with three PIMPLE iterations.
Table 6.0.1: Vent conditions for the [weakPlume] and [strongPlume] volcanic simulations. e 6.1.1: Atmospheric profiles used for the atmosphere of the two volcanic plume simulations (height z km with respect atmospheric temperature Ty K): left) [weakPlume]; right) [strongPlume]. The domain for the [strongPlume] case extends up to 50 km. The thermal profile gradient from 40 to 50 km is assumed to be the same obtained below 40 km. The parameters obtained in the piecewise fit are reported in Tab.|6.1.1
Table 6.0.1: Vent conditions for the [weakPlume] and [strongPlume] volcanic simulations. e 6.1.1: Atmospheric profiles used for the atmosphere of the two volcanic plume simulations (height z km with respect atmospheric temperature Ty K): left) [weakPlume]; right) [strongPlume]. The domain for the [strongPlume] case extends up to 50 km. The thermal profile gradient from 40 to 50 km is assumed to be the same obtained below 40 km. The parameters obtained in the piecewise fit are reported in Tab.|6.1.1
Table 6.1.1: Configuration parameters for the boundary and initial conditions. The two rows correspond to the [weakPlume] and [strongPlume] numerical experiments, respectively. In both cases g = 9.80665 m/s?. As already done in Sec. |5 5.5} the boundary conditions in the oe A top sides of the domain have atmospheric boundary conditions (see Sect. ean the bottom wall has a thermally insulated no-slip wall condition, ea The inlet velocity has the same profile introduced in (1.5.10 white noise with standard deviation o9 and an asymmetric forcing (cf. Eq. (1.5.13 and smal 1.5.17 with a ea axed ). I choose to introduce anisotropy to the forcing at the inlet to emulate vent instabilities.
Table 6.1.1: Configuration parameters for the boundary and initial conditions. The two rows correspond to the [weakPlume] and [strongPlume] numerical experiments, respectively. In both cases g = 9.80665 m/s?. As already done in Sec. |5 5.5} the boundary conditions in the oe A top sides of the domain have atmospheric boundary conditions (see Sect. ean the bottom wall has a thermally insulated no-slip wall condition, ea The inlet velocity has the same profile introduced in (1.5.10 white noise with standard deviation o9 and an asymmetric forcing (cf. Eq. (1.5.13 and smal 1.5.17 with a ea axed ). I choose to introduce anisotropy to the forcing at the inlet to emulate vent instabilities.
Figure 6.2.1: Two slices of the mesh for the plume domain with ng = 0.16 and tan@g = 0.04. Shaded surfaces represent the area invaded by the inlet extruded mesh.
Figure 6.2.1: Two slices of the mesh for the plume domain with ng = 0.16 and tan@g = 0.04. Shaded surfaces represent the area invaded by the inlet extruded mesh.
Figure 6.2.2: Dependence of the grading factor G on the stretching ¢/N6. I define tan 0g the slope of the spreading of the vent block (from dyn = bo t:
Figure 6.2.2: Dependence of the grading factor G on the stretching ¢/N6. I define tan 0g the slope of the spreading of the vent block (from dyn = bo t:
able 6.2.1: Mesh parameters for the two numerical plume cases studied in this chapter. In the [weakPlume] test case Gg < Goo = 353 (cf. Eq. (6.2.6)) because the stretching is very high and its value is adapted to the mesh resolution. Now, the right radial grading in the outer cylinder Gg must be found, in order to
able 6.2.1: Mesh parameters for the two numerical plume cases studied in this chapter. In the [weakPlume] test case Gg < Goo = 353 (cf. Eq. (6.2.6)) because the stretching is very high and its value is adapted to the mesh resolution. Now, the right radial grading in the outer cylinder Gg must be found, in order to
igure 6.3.1: Three-dimensional numerical simulation of a weak volcanic plume, 400 s after the beginning of the injection (inlet conditions as in Tabs. 6.0.1 and 6.1.1 . Isosurfaces and vertical sections of the fine (light white) and coarse (light sand) ash volume fractions. The isosurfaces are given at es = 10~". The maximum height of the fine class isosurface is approximatively 12.1 km. The two-dimensional plots represent the distribution of the volume concentration of coarse (left) and fine (right) particles across vertical orthogonal slices crossing the plume axis.
igure 6.3.1: Three-dimensional numerical simulation of a weak volcanic plume, 400 s after the beginning of the injection (inlet conditions as in Tabs. 6.0.1 and 6.1.1 . Isosurfaces and vertical sections of the fine (light white) and coarse (light sand) ash volume fractions. The isosurfaces are given at es = 10~". The maximum height of the fine class isosurface is approximatively 12.1 km. The two-dimensional plots represent the distribution of the volume concentration of coarse (left) and fine (right) particles across vertical orthogonal slices crossing the plume axis.
Figure 6.3.2: Vertical section of the instantaneous value of the mixture velocity modulus (in logarithmic scale) at t = 400 s and velocity streamlines.
Figure 6.3.2: Vertical section of the instantaneous value of the mixture velocity modulus (in logarithmic scale) at t = 400 s and velocity streamlines.
e 6.3.4: Time-averaged plume radius and velocity as a function of plume height. The insets display the non-dimensional mass, momentum and buoyancy fluxes (top) and the time-averaged entrainment coefficient. Panels a-c) ASHEE model at different resolutions; panel d) dusty gas model. Weak Plume, dynWaleRes8, averaged between 300--720 s Weak Plume, dynWaleRes16, averaged between 300--720 s
e 6.3.4: Time-averaged plume radius and velocity as a function of plume height. The insets display the non-dimensional mass, momentum and buoyancy fluxes (top) and the time-averaged entrainment coefficient. Panels a-c) ASHEE model at different resolutions; panel d) dusty gas model. Weak Plume, dynWaleRes8, averaged between 300--720 s Weak Plume, dynWaleRes16, averaged between 300--720 s
Weak Plume dusty, dynWaleRes4, averaged between 300--720 s
Weak Plume dusty, dynWaleRes4, averaged between 300--720 s
Figure 6.3.6: Entrainment coefficient of [weakPlume]. This is a zoom of Fig. [6.3.58 The fit graph obtained at high resolution is shown, together with the fit result for the other simulations performed.
Figure 6.3.6: Entrainment coefficient of [weakPlume]. This is a zoom of Fig. [6.3.58 The fit graph obtained at high resolution is shown, together with the fit result for the other simulations performed.
sure 6.4.1: Three-dimensional simulation of [strongPlume], 1000 s after the beginning of the injection (inlet conditions are in Tabs. 6.0.1]and 6.1.1). Isosurface and vertical sections of the fine (light white) and coarse (light sand) ash volume fractions. The isosurfaces are given at ¢; = 1077. The maximum height of the fine class isosurface is approximatively 43 km. The two-dimensional plots represent the distribution of the volume concentration of coarse (left) and fine (right) particles across vertical orthogonal slices crossing the plume axis.
sure 6.4.1: Three-dimensional simulation of [strongPlume], 1000 s after the beginning of the injection (inlet conditions are in Tabs. 6.0.1]and 6.1.1). Isosurface and vertical sections of the fine (light white) and coarse (light sand) ash volume fractions. The isosurfaces are given at ¢; = 1077. The maximum height of the fine class isosurface is approximatively 43 km. The two-dimensional plots represent the distribution of the volume concentration of coarse (left) and fine (right) particles across vertical orthogonal slices crossing the plume axis.
igure 6.4.4: Infrasonic signal generated by the turbulent eddies. Here the infrasonic perturbation is visualized by using the magnitude of the field of acceleration ag, in logarithmic color scale, after 120 s since the eruption started.
igure 6.4.4: Infrasonic signal generated by the turbulent eddies. Here the infrasonic perturbation is visualized by using the magnitude of the field of acceleration ag, in logarithmic color scale, after 120 s since the eruption started.
ure 6.4.5: Infrasonic spectrum of the pressure fluctuations as measured from a probe placed at the ground level 15 km from the vent center, in the time window t € [0,13] min. The frequency is expressed both in its dimensional (f) and non-dimensional (Str = f 2b9/Uo) formulation. Pressure fluctuations Ep are represented both in logarithmic scale and in dB scale relative to 20 wPa. The slope Str7!/4 is shown in bold dashed line. The experimental large-scale similarity spectrum is presented in bold solid line [see [193]. The inset shows the pressure fluctuation signal in [kPa] until ~ 13 min.
ure 6.4.5: Infrasonic spectrum of the pressure fluctuations as measured from a probe placed at the ground level 15 km from the vent center, in the time window t € [0,13] min. The frequency is expressed both in its dimensional (f) and non-dimensional (Str = f 2b9/Uo) formulation. Pressure fluctuations Ep are represented both in logarithmic scale and in dB scale relative to 20 wPa. The slope Str7!/4 is shown in bold dashed line. The experimental large-scale similarity spectrum is presented in bold solid line [see [193]. The inset shows the pressure fluctuation signal in [kPa] until ~ 13 min.
e 6.4.6: (second part) Profiles of all the integral variables of [strongPlume]: influence of the mesh resolution. Here, they are compared by changing the resolution from Ar = D/32 (high res.) to Ar = D/16 (mid res.) and Ar = D/8 (low res.).
e 6.4.6: (second part) Profiles of all the integral variables of [strongPlume]: influence of the mesh resolution. Here, they are compared by changing the resolution from Ar = D/32 (high res.) to Ar = D/16 (mid res.) and Ar = D/8 (low res.).
LES subgrid model
LES subgrid model
Kinematic decoupling
Kinematic decoupling
igure 6.4.9: Entrainment coefficient of [strongPlume]. This is a zoom of Fig. [6.4.6f The fit graph obtained at high resolution with the [dynWale] sub-grid model is shown, together with, in legend, the fit result for the other simulations performed.
igure 6.4.9: Entrainment coefficient of [strongPlume]. This is a zoom of Fig. [6.4.6f The fit graph obtained at high resolution with the [dynWale] sub-grid model is shown, together with, in legend, the fit result for the other simulations performed.
ure 6.4.10: Collapsing dynamics: (a) Isosurface at nil vertical velocity (tm - 2 = 0), colored with the outflow intensity fm(tm-n) in kg/m?s. The streamlines of the velocity field @m are also represented, colored with its magnitude in m/s. (b) Vertical evolution of the mass flux of the ejected mixture (water and ash particles) normalized with its vent value Qo = 1.5* 10° kg/s (solid line). It is also reported in dashed line the vertical evolution of the collapsing flux of the ejected mixture qc(z). It is calculated following Eq. (6.4.1).
ure 6.4.10: Collapsing dynamics: (a) Isosurface at nil vertical velocity (tm - 2 = 0), colored with the outflow intensity fm(tm-n) in kg/m?s. The streamlines of the velocity field @m are also represented, colored with its magnitude in m/s. (b) Vertical evolution of the mass flux of the ejected mixture (water and ash particles) normalized with its vent value Qo = 1.5* 10° kg/s (solid line). It is also reported in dashed line the vertical evolution of the collapsing flux of the ejected mixture qc(z). It is calculated following Eq. (6.4.1).
re 6.4.13: Behavior of conserved quantities of the asymptotic integral model as calculated from three- dimensional simulations. Here, some of the plume parameters reported in Tab.[3.4-2]have been mod- ified to obtain a better behavior of the conserved quantities (see discussion): ag = 0.473 + ag = 2.7, Ye = 0.345 + Yc = 0.445. Moreover, it is used the value v9 = 3.0 * 10-2, obtained averaging the atmospheric profile: wo ~ 2.61 * 10~? Hz.
re 6.4.13: Behavior of conserved quantities of the asymptotic integral model as calculated from three- dimensional simulations. Here, some of the plume parameters reported in Tab.[3.4-2]have been mod- ified to obtain a better behavior of the conserved quantities (see discussion): ag = 0.473 + ag = 2.7, Ye = 0.345 + Yc = 0.445. Moreover, it is used the value v9 = 3.0 * 10-2, obtained averaging the atmospheric profile: wo ~ 2.61 * 10~? Hz.
Related topics:
VolcanologyMultiphase FlowLarge Eddy SimulationMultiphase systemsCompressible Fluid DynamicsTurbulent Jets and PlumesMultiphase flowsPhysical VolcanologySobolev spacesVolcanic ash
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