4.3 Spatially decreasing porosity
Lastly, predictions are given in Fig. 11 that illustrate the interaction
of an incident compaction wave with initially denser
material; the values φL = φfp, φR = 0.99, and up = 100 m/s
were used for this simulation. The incident compaction wave
propagates through the loose material at speed 334.9m/s.
Because this granular solid is at the volume fraction of the
virgin material, no precursorwave results; thewave is entirely
viscoplastic. The peak grain surface heat flux for the incident
wave is approximately 56.1MW/m2, resulting in a grain scale
temperature rise and explosion length of _T = 81.39K and
lex δ based on the characteristic values  ̄ q = 30.7MW/m2,
D = 334.9m/s,  ̄ u = 67.0m/s, δ = 2.62mm, and R =
100μm. Based on this estimate, it may be concluded that
the incident wave would not result in thermal explosion,
though again it is re-emphasized that this estimate is based
on uniform grain surface heating and is therefore likely an
overprediction. Because of the higher acoustic impedance
of the dense material, the interaction results in both reflected
and transmitted compaction waves that may further influence
combustion initiation. The reflected and transmitted compaction
waves are predicted to have an approximate peak
grain surface heat flux of 57.3 and 13.6MW/m2, respectively.
Importantly, grains within the initially loose material that are
located near the interface will experience the high amplitude
heat pulses of both the incident and transmitted compaction
waves in rapid succession which would enhance initiation.
Similar, and possiblymore pronounced, heating would occur
for the reflection of compaction waves from rigid walls; this
phenomenon is aworthy topic of further study due to its practical
importance to the dynamic loading of confined granular
energetic solids.