Computer Science and Methods
Model of Turbulent Combustion of Al Particle Clouds in Explosions
A. L. Kuhl
, J. B. Bell
, V. E. Beckner
and K. Balakrishnan
Lawrence Livermore National Laboratory, Livermore, CA USA
Lawrence Berkeley Natonal Laboratory, Berkeley, CA USA
We consider the problem of combustion in Shock-Dispersed-Fuel (SDF) explosions [1, 2]. The SDF charge consists of a
spherical PETN booster (1/3 the mass), surrounded by flake Aluminum powder (2/3 the mass) with a bulk density of 0.6
g/cc. Detonation of the booster charge creates a blast wave that disperses the Al powder and ignites the ensuing Al-air
mixture—thereby forming a two-phase combustion cloud embedded in the explosion. We model this process with a two-
phase version of our AMR code [3,4].
It is based on the gasdynamic conservation laws for the gas phase and the dilute
continuum conservation laws for the particle phase, as formulated by Nigmatulin . Inter-phase mass, momentum and
energy exchange are prescribed by the phenomenological model of Khasainov. The thermodynamic states are specified by
the locus of states in Le Chatelier diagram of internal energy versus temperature (assuming frozen reactants and
equilibrium products) as calculated by the Cheetah code, along with the perfect gas law. The combustion sub-model is
based on mass conservation laws for fuel, air and products; it includes: (i) an induction-parameter model (to capture
Arrhenius kinetics effects), (ii) an empirical ignition temperature, and (iii) an ignition probability model (to capture multi-
particle effects). The model takes into account both the afterburning of the detonation products of the booster with air, and
the combustion of the Al particles with air. The model equations were integrated by high-order Godunov schemes for both
the gas and particle phases. Since the integrators are based on Riemann solvers, information propagates along
characteristics at the correct wave speeds, and they incorporate nonlinear wave interactions within the cell during the
time-step. They include a limiting step (slope flattening), which automatically reduces the order of the approximation in
the neighborhood of discontinuities, while the scheme remains second order accurate in both time and space in smooth
regions of the flow. These Godunov schemes were incorporated into an adaptive mesh refinement (AMR) algorithm,
which allows one to focus the computational effort in complex regions of the flow (e.g., to resolve reaction zones). AMR
is used to refine turbulent mixing regions; by successive refinements, we are able to capture the energy-bearing scales of
the turbulence on the computational grid—the so called iLES approach first proposed by J. Boris in 1990.
The Model was used to simulate the turbulent combustion of an Al particle cloud in a 1.5-g SDF explosion. A cross-
section of the flow field inside the cloud is shown in Fig. 1. Turbulence effects are quite apparent in the temperature and
density fields. Fireball temperatures reach 4,100 K, corresponding to the adiabatic flame temperature for Al-air
combustion. One can see that the gas-phase vorticity fills the entire cloud, while the particle-phase vorticity is
concentrated in radial shear layers (created by drag effects during the particle-dispersion phase). Figure 2 compares
computed and measured reflected pressure histories from a simulation of a barometric calorimeter experiment. Figure 3
shows computed Al mass histories. In the calorimeter case, virtually all the fuel is consumed (due to continued stirring by
the shock reverberations in the chamber). In the un-confined explosion, about 40 % of the fuel remains; this shows that
there is a limit to how much air a spherical fireball can entrain before the turbulence decays away.
Computer Science and Methods
(c) Gas Vorticity
(d) Particle Vorticity
Figure 1. Cross
section of the flow fields in the Al particle combustion cloud.
Figure 2. Comparison of pressure histories.
Figure 3. Fuel mass remaining.
 A. L. Kuhl, H. Reichenbach, Combustion effects in confined explosions, Proc. Combustion Institute 32 (2) pp. 2291-
 A. L. Kuhl, H. Reichenbach, Barometric calorimeters, Combustion, Explosions and Shock Waves, 4 (2) 2010 pp. 271-278 (also in
Khimicheskaya Fizika 4(2) 2010 pp. 271-278).
 A. L. Kuhl, J. B. Bell, V. E. Beckner, Heterogeneous continuum model of aluminum particle combustion in
explosions, Combustion Explosion and Shock Waves 46 (4), pp. 433-448, 2010 (also in Физика горения и взрыва, 2010,
т. 46, N
 A. L. Kuhl, J. B. Bell, V. E. Beckner, H. Reichenbach, Gasdynamic model of turbulent combustion in TNT
explosions, Proc. Combustion Institute 33 (2) pp. 2177-2185.
 Nigmatulin, R. I., Dynamics of Multi-phase Flows, Vol. 1. Moscow, Nauka, 1987, 464 pp.
This work was performed under the auspices of the U. S. Department of Energy by the University of California, Lawrence Livermore
National Laboratory under Contract No. LLNL-ABS-499158.