Of the two-phase mixture models used to study deflagration-to-detonation transition in granular explosives, the Baer–Nunziato model is the most highly developed. It allows for unequal phase velocities and phase pressures, and includes source terms for drag and compaction that strive to erase velocity and pressure disequilibria. Since typical time scales associated with the equilibrating processes are small, source terms are stiff. This stiffness motivates the present work where we derive two reduced models in sequence, one with a single velocity and the other with both a single velocity and a single pressure. These reductions constitute outer solutions in the sense of matched asymptotic expansions, with the corresponding inner layers being just the partly dispersed shocks of the full model. The reduced models are hyperbolic and are mechanically as well as thermodynamically consistent with the parent model. However, they cannot be expressed in conservation form and hence require a regularization in order to fully specify the jump conditions across shock waves. Analysis of the inner layers of the full model provides one such regularization [Kapila et al., Phys. Fluids 9, 3885 (1997)], although other choices are also possible. Dissipation associated with degrees of freedom that have been eliminated is restricted to the thin layers and is accounted for by the jump conditions.

1.
M. R.
Baer
and
J. W.
Nunziato
, “
A two-phase mixture theory for the deflagration-to-detonation transition in reactive granular materials
,”
Int. J. Multiphase Flow
12
,
861
(
1986
).
2.
P. Barry
Butler
and
H.
Krier
, “
Analysis of deflagration to detonation transition in high-energy solid propellants
,”
Combust. Flame
63
,
31
(
1986
).
3.
P. Barry Butler, Ph.D. thesis, Technical report, University of Illinois, Urbana, Illinois, 1984.
4.
S. S.
Gokhale
and
H.
Krier
, “
Modeling of unsteady two-phase reactive flow in porous beds of propellant
,”
Prog. Energy Combust. Sci.
8
,
1
(
1982
).
5.
J. M.
Powers
,
D. S.
Stewart
, and
H.
Krier
, “
Theory of two-phase detonation—Part I: Structure
,”
Combust. Flame
80
,
280
(
1990
).
6.
J. M.
Powers
,
D. S.
Stewart
, and
H.
Krier
, “
Theory of two-phase detonation—Part II: Modeling
,”
Combust. Flame
80
,
264
(
1990
).
7.
J. B.
Bdzil
,
R.
Menikoff
,
S. F.
Son
,
A. K.
Kapila
, and
D. S.
Stewart
, “
Two-phase modelling of DDT in granular materials: A critical examination of modeling issues
,”
Phys. Fluids
11
,
378
(
1999
).
8.
J.
Massoni
,
R.
Saurel
,
G.
Baudin
, and
G.
Demol
, “
A mechanistic model for shock initiation of solid explosives
,”
Phys. Fluids
11
,
710
(
1999
).
9.
R.
Saurel
and
R.
Abgrall
, “
A multiphase godunov method for compressible multifluid and multiphase flows
,”
J. Comput. Phys.
150
,
425
(
1999
).
10.
H. B.
Stewart
and
B.
Wendroff
, “
Two-phase flow: Models and methods
,”
J. Comput. Phys.
56
,
363
(
1984
).
11.
D. A. Drew and S. L. Passman, Theory of Multicomponent Fluids (Springer, New York, 1998).
12.
G.-Q.
Chen
,
C. D.
Levermore
, and
T.-P.
Liu
, “
Hyperbolic conservation-laws with stiff relaxation terms and entropy
,”
Commun. Pure Appl. Math.
47
,
787
(
1994
).
13.
A. K.
Kapila
,
S. F.
Son
,
J. B.
Bdzil
,
R.
Menikoff
, and
D. S.
Stewart
, “
Two-phase modeling of DDT: Structure of the velocity-relaxation zone
,”
Phys. Fluids
9
,
3885
(
1997
).
14.
M. R. Baer and J. W. Nunziato, “Compressive combustion of granular materials induced by low-velocity impact,” in Ninth Symposium on Detonation, 1989, pp. 293–305.
15.
J. B. Bdzil and S. F. Son, “Engineering models of DDT,” Technical Report LA-12794-MS, Los Alamos National Lab, 1995.
16.
K. A. Gonthier, R. Menikoff, S. F. Son, and B. W. Asay, “Modeling compaction-induced energy dissipation of granular HMX,” in Eleventh (International) Symposium on Detonation, Snowmass, CO, August 31, 1998.
17.
P.
Embid
and
M.
Baer
, “
Mathematical analysis of a two-phase model for reactive granular materials
,”
Continuum Mech. Thermodyn.
4
,
279
(
1992
).
18.
H. W. Sandusky and R. R. Bernecker, “Compressive reaction in porous beds of energetic materials,” in Eighth International Symposium on Detonation, 1985, pp. 881–891.
19.
J. M. McAfee, B. Asay, W. Campbell, and J. B. Ramsay, “Deflagration to detonation in granular HMX,” in Ninth (International) Symposium on Detonation, Portland, OR, August 28, 1989, pp. 265–279.
20.
J. E. Shepherd and D. R. Begeal, “Transient compressible flow in porous materials,” Technical Report SAND83-1788, Sandia National Lab, 1988.
21.
B. W.
Asay
,
S. F.
Son
, and
J. B.
Bdzil
, “
An examination of the role of gas permeation during convective burning of granular explosives
,”
Int. J. Multiphase Flow
22
,
923
(
1996
).
22.
S. A. Sheffield, R. L. Gustavsen, and R. R. Alcon, “Shock initiation studies of low density HMX using electromagnetic particle velocity and pvdf stress gauges,” in Tenth (International) Symposium on Detonation, Boston, MA, 1993.
23.
E. Kober, J. B. Bdzil, and S. F. Son, “Modeling DDT in granular explosives with a multi-dimensional hydrocode,” in APS Topical Conference of Shock Compression in Condensed Matter-1995, Proceedings of the American Physical Society Topical Conference, Seattle, WA, 1996, pp. 437–441.
24.
D. S.
Stewart
,
B. W.
Asay
, and
K.
Prasad
, “
Simplified modeling of transition to detonation in porous energetic materials
,”
Phys. Fluids
6
,
2515
(
1994
).
25.
S. F. Son, B. W. Asay, and J. B. Bdzil, “Inert plug formation in the DDT of granular energetic materials,” in APS Topical Conference of Shock Compression in Condensed Matter-1995, Proceedings of the American Physical Society Topical Conference, Seattle, WA, 1996, pp. 441–444.
26.
G.-Q.
Chen
and
T.-P.
Liu
, “
Zero relaxation and dissipation limits for hyperbolic conservation laws
,”
Commun. Pure Appl. Math.
46
,
755
(
1993
).
27.
L. F.
Henderson
and
R.
Menikoff
, “
Triple-shock entropy theorem and its consequences
,”
J. Fluid Mech.
366
,
179
(
1998
).
28.
R. Menikoff and E. Kober, “Equation of state and Hugoniot locus for porous materials: P-α model revisited,” in APS Topical Conference of Shock Compression in Condensed Matter-1999, edited by M. D. Furnish, L. C. Chhabildas, and R. S. Hixon (AIP, Melville, NY, 2000), pp. 129–132. Also available on the web at http://t14web.lanl.gov/Staff/rsm/Papers/DDT/Porous.pdf
29.
W.
Herrmann
, “
Constitutive equation for the dynamic compaction of ductile porous material
,”
J. Appl. Phys.
40
,
2490
(
1969
).
30.
M.
Carroll
and
A. C.
Holt
, “
Suggested modification of the P-α model for porous materials
,”
J. Appl. Phys.
43
,
759
(
1972
).
31.
S.
Jin
and
Z.
Xin
, “
The relaxing schemes for systems of conservation laws in arbitrary space dimensions
,”
Commun. Pure Appl. Math.
48
,
235
(
1995
).
32.
R. Menikoff and E. Kober, “Compaction waves in granular HMX,” Technical Report LA-13546-MS, Los Alamos National Lab, 1999. Also available on the web at http://t14web.lanl.gov/Staff/rsm/Papers/CompactWave/CompactWave.pdf
33.
L.
Sainsaulieu
, “
Traveling waves solution of convection-diffusion systems whose convection terms are weakly nonconservative: Application to the modeling of two-phase fluid flows
,”
SIAM (Soc. Ind. Appl. Math.) J. Appl. Math.
55
,
1552
(
1995
).
34.
D. J.
Benson
, “
A mixture theory for contact in multi-material Eulerian formulations
,”
Comput. Methods Appl. Mech. Eng.
140
,
59
(
1997
).
35.
G. B. Wallis, One-Dimensional Two-Phase Flow (McGraw-Hill, New York, 1969).
This content is only available via PDF.
You do not currently have access to this content.