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Overview and exemplify multiphase code GMFIX Hyberbolic-only approach Possible directions "No one believes the results of computational fluid dynamics except the one who performed the calculations, and everyone believes experimental results except the one who performed the experiment." (In the Hollow of a Wave at Kanagawa, Hokusai) Hunting for the Deterministic Template(s) Three General Themes:

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GMFIX (Geophysical Multiphase Flow with Interphase eXchanges) George W. Bergantz, Josef Dufek University of Washington Sebastian Dartevelle, W.I. Rose Michigan Institute of Technology

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KFIX to MFIX to GMFIX 1)Eulerian-Eulerian non-equilibrium multiphase, 3-d, non-steady, enthalpy, reactions 2) SIMPLE algorithm, 2’d order accurate discretization, under-relaxation, variable time-step, iterative linear eq solvers: SOR and conjugate gradient 3) F90, SMP or DMP (MPICH) parallel

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KFIX to MFIX to GMFIX (cont’d.) 4) Convergence criteria- accept only part of solution that does not change with a factor 10 increase in tolerance 5) (V)LES, static Smagorinsky 6) Well validated for fluidized beds at bench scales- but at geological scales to be discussed…

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Our Focus- Improvements in Physics Essential to Validation 1)Reaction-entrainment 2)Numerical improvements, e.g. adaptive gridding 3)Multiphase-turbulence- sedimentation models (Fuji View, Hiroshige)

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Granular Flow Regimes Elastic RegimePlastic RegimeViscous Regime StagnantSlow flowRapid flow Stress is strainStrain rateStrain rate dependentindependentdependent ElasticitySoil mechanicsKinetic theory

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Remarks on multiphase flow features 1)Empirical, complex inter-and– within phase momentum transfer equations allow particle volume fraction to vary significantly 2)But significant challenges for VLES in sedimentation and boundary region (Dragon Escaping on Smoke from Mt. Fuji, Hokusai)

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Remarks on multiphase flow features (cont’d.) "Stokes number is the key dimensionless number for the dynamics of relative particle motions in the global flow parameterization." Kaminski & Jaupart (1997) “In general, fallout of suspended pyroclasts seems reasonably well understood.” (1997) 1)Stokes number can dramatically influence sedimentation (Burgisser & Bergantz, 2002) gives rise to meso-scale structures 2) Turbulence intensity enhanced or attenuated by particles Both challenging to address in a numerical model

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Geometrical setup: Cylindrical Y = 50km height, 100m X = 65km radial, 100m to 1000m Z = 51km arc length, = 1rad Initial Conditions: Vent radius = 400m Particle 50 m, 1500kg/m 3 Dry atmosphere, 298K, 10 5 Pa Tropopause between 11km and 19km Stratospheric T_gradient = -7K/km Boundary Conditions: No-slip at the ground Free-slip at all the other boundaries Mass inflow at the vent: V y = 200m/s T = 900K s = 0.1% 100% of magmatic water at the vent Plinian Column Model

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3 min …

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30 min …

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1 hour … 25 m/s 5 m/s 0 m/s 120 m/s 60 m/s 2 m/s 1.5 m/s -3 m/s

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1 hour …

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Plinian column modeling: Our results are in a good agreement: - with satellite observations of the undercooling at the top of the Plinian cloud (both in magnitude and with time) - with experimental data and previous numerical modeling of buoyant plume (velocity profiles, density profiles, …)

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Plinian column modeling: However, the details of the cloud dynamic reveal unsuspected phenomena: Complex velocity and density distribution within the column Positive buoyancy on the edges of the column (where it is the most turbulent), while the core is collapsing Presence of giant vertical vortices Non-homogenous temperature profiles within the plume (undercooled pockets) The overall altitude is time-dependent and fluctuates with time Complex pressure distribution profiles with time

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Hyperbolic Methods Randy LeVeque, CLAWPACK 1)Advective terms only, excellent for shocks or ‘front tracking.’ 2)Fast, explicit (but semi- implicit coming) 3)Perhaps a terrific tool for field, rapid laptop assessment (Red Fuji, Hokusai)

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Collapsing plume, parabolic initial shape 200 x 200 grid

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Future Directions 1)Invite and enable community with regular workshops, dialog, mutual support 2)Hierarchical modeling tools (Mount Asama, Hiroshige, 1859) Towards a “universal” multi-phase, multi-species flow codes applied to geophysical- volcanological problems It can be used for highly-loaded situations (turbidities, pyroclastic flows) and for dilute ones (pyroclastic surges, plinian column, co-ignimbrites) It does not assume unrealistic physical conditions … it is based on a well accepted physics (Navier-Stokes, continuity, 2 nd law of Mechanics, 1 st law of thermodynamic)

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Development of a water micro-physics model (evaporation-condensation-sublimation) Development of a complete sub-grid multi-phase turbulence model (in collaboration with DOE labs, NETL/ORNL) Development of a multi-grain size model (for unimodal grain-size distribution) Development of better viscous dissipation algorithms for shock waves/fronts Future Directions

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Chapter 3.1 Weakly Nonlinear Wave Theory for Periodic Waves (Stokes Expansion) Introduction The solution for Stokes waves is valid in deep or intermediate.

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