Combustion, Explosion, and Shock Waves, Vol. 37, No. 2, pp. 206 213, 2001
Simulation of Convective Detonation Waves
in a Porous Medium by the Lattice Gas Method
A. P. Ershov,1 A. L. Kupershtokh,1 UDC 534.222.2+662.215.1
and D. A. Medvedev1
Translated from Fizika Goreniya i Vzryva, Vol. 37, No. 2, pp. 94 102, March April, 2001.
Original article submitted October 6, 1999; revision submitted January 28, 2000.
 Gas film convective detonation in a rigid porous medium is considered. The mo-
tion of the gas phase is described by a discrete stochastic lattice gas model taking into
account the real laws of friction and heat exchange between the phases. The reaction
kinetics was specified so that the characteristic time of combustion corresponded to
experiment. The model simulates the main characteristics of the phenomenon: a non-
flat (irregular) wave front, smooth increase in the pressure averaged over the charge
cross section, friction-dominated mean flow, slow cooling of combustion products after
completion of the reaction.
Peculiar detonation-like flows in rigid porous media tion products from a separate chamber upon rupture
were observed in [1 6]. The active component of such of a membrane or the action of gas-detonation prod-
media can be a gas mixture that fills pores [1] or a layer ucts on a powder, also generate a wave with a velocity
of a high explosive (HE) on the pore surface [6]. In the of H"1 km/sec and a pressure of H"2 kbar. In charges
present work, we consider a more complex system in of small diameter (3 4 mm) with a very light shell, this
which a gaseous oxidizer in pores reacts with a fuel film wave is rather stable; here a smooth increase in pressure
on the surface of the pore structure [2 5]. and formation of jets were also observed. Although the
The wave regimes of combustion in a rigid porous low strength of HE leads to slight deformation of the
medium are characterized by a complex wave front, porous bed, available data also suggest a jet mechanism
which is a random pulsating relief of hills and valleys,
and by a smooth increase in pressure. The average front
velocity is H"1 km/sec. The front pattern and the pres-
sure profile are shown schematically in Fig. 1.
In opinion of authors of experimental works, the
waves propagate by a convective or jet mechanism. The
wave-propagation conditions are strongly affected by
the porous bed. Because of friction losses, the wave
velocity is not sufficient to initiate a reaction by the
standard shock-wave mechanism. Instead, ignition is
ensured by hot gas jets that burst ahead of  average
front from the combustion zone.
Previously, similar conclusions were made for a dif-
ferent system  a porous explosive [7, 8]. Some initi-
ation regimes, such as an electric discharge or explo-
sion of a conductor inside HE, injection of hot combus-
1
Lavrent ev Institute of Hydrodynamics,
Fig. 1. Diagram of detonation-like flow in rigid
Siberian Division, Russian Academy of Sciences,
porous media and pressure profile.
Novosibirsk 630090.
206 0010-5082/01/3702-0206 $25.00 © 2001 Plenum Publishing Corporation
Simulation of Convective Detonation Waves in a Porous Medium 207
of wave propagation. From the pressure level, the burnt velocity is low compared to the characteristic velocity
fraction of the material is estimated at several percent, of sound and the wave velocity. Thus, the kinetic en-
i.e., the concentration of the reacted HE is close to that ergy and some of the convective terms in the energy
used in the experiments of [6]. equation can be ignored. The so-called  short model,
which employes an approximate integral of the energy
equation, is described in [9, 10].
AVERAGE APPROACH
Traditionally, such flows have been simulated using
LATTICE GAS
a continual approach based on the equations of contin-
uum mechanics. For a multiphase heterogeneous sys-
The averaged approach is of little use for describ-
tem, the equation are written for averaged parameters.
ing a wave propagating in a stochastic manner due to
It is obvious that for flows with a significant random
random ejections of jets. Conventional finite-difference
component, the applicability of averaging is limited.
methods are unsuitable to the same extent. In nu-
Nevertheless, the continual model is a good starting
merical solution of the equations, the ignition front in-
point, from which one can go over to the discrete model
evitably has to be specified artificially, and this is in
studied in the present paper.
conflict with the concept of the convective mechanism.
Physically, an active medium of the  gas film type
These problems can be solved by direct modeling of ran-
is, for example, loose-packed sand or rigid granules
dom processes.
coated with a thin fuel film capable of reacting with
In the present work, we use a discrete lattice gas
an oxidizer in the pores. The system can be initiated
or cellular automata method. In this method, a con-
by  impact of a gas-detonation wave.
tinuous medium is represented as a  gas consisting of
Following [9], we restrict ourselves to the simplest
identical atoms or particles that occupy the nodes of a
case of a fast reaction, where the fuel entering the pore
fixed lattice.
space from the walls burns immediately. In this case,
For the two-dimensional case, the model proposed
the pores can contain an oxidizer or combustion prod-
by Frish, Hasslacher, and Pomeau (FHP model) [11] is
ucts or their mixture. Variation in the porosity of the
used most widely. Here the lattice is composed of equi-
medium during combustion can be ignored with good
lateral triangles, so that six links go out from a node.
accuracy.
Up to six atoms can be present at each node, and all
Under these assumptions, the standard averaged
directions of their velocities must be different. The pres-
equations are written as [9]:
ence of one rest atom is also possible. In a time step, a
Át + (Áu)x = j, rt + (ru)x = j, particle travels a unit distance  from one lattice node
to a neighbor. Collisions at the nodes with conservation
(Áu)t + (Áu2 + p)x = -f, (1)
of the number of particles and momentum lead to estab-
lishment of local equilibrium and information transfer.
(E + Áu2/2)t + (u(E + Áu2/2 + p))x = jQ - q.
As shown in [11, 12], the system described simu-
Here Á is the total density of the gas, r is the part lates two-dimensional viscous gas dynamics in an av-
of Á produced by the fuel entering the pores (assumed eraged sense. The collision rules contain a source of
to be burnt completely), u is the flow velocity, p is the randomness, which is significant for many processes. In
pressure, and Q is the constant volume heat of the com- practice, the lattice model is an extremely simplified
bustion reaction in constant volume (per unit mass of version of the molecular dynamics method with mini-
the fuel). Because the porosity is constant, it is not in- mum computational power requirements. It should be
cluded in the equations; the mass influx j, the friction noted that in some respects, the model is  too simple.
force f, the internal energy of the mixture E, and the The choice of a coordinate system attached to the lat-
heat losses q are defined per unit volume of the pore tice restricts the applicability of the approach to the
space. In the approximation of identical adiabatic ex- slow flows.
ponents Å‚ of the gas components, we have E = p/(Å‚ -1) For flow in a porous medium, the disadvantages of
irrespective of the volume distribution among them and the model are insignificant because the flow velocity is
the degree of mixing. low due to friction. Of course, simulation of fast jets
Further simplification of the continual model is pos- can be only qualitative, but today this is true for de-
sible with allowance for stagnation of the flow. For a terministic finite-difference methods. Some results of
gas flow through a porous medium, friction against the application of the FHP lattice model to the problem of
porous bed is a dominant factor. Therefore, the flow convective waves are reported in [13, 14]. These papers
208 Ershov, Kupershtokh, and Medvedev
deal with the case of  isothermal detonation, where A gas-dynamic block (propagation + collision) was
the active component is an explosive. In this case, the tested in special calculations. Average values of the mo-
temperature of the gas (reaction products) in the com- mentum flux tensor components ik were calculated for
bustion zone is constant. specified equilibrium states. They were close to p´ik 
the main term (´ik is the Kronecker delta). The in-
ertial terms were of the order of Áuiuk, although the
coefficients depended significantly on the distribution
DISCRETE MODEL
of atoms over the levels (i.e., on the temperature). For
FOR A CONVECTIVE WAVE
flow in a porous medium, the error in describing these
components, which are quadratic in velocity, is insignif-
For gas film detonation in the reaction zone, the
icant because the velocity is low due to friction (H"0.1
temperature is obviously variable. It increases during
in natural units).
fuel burnup from the low initial temperature of the ox-
In addition, the velocity of propagation of small
idizer to the temperature of combustion products. It is
perturbations over a homogeneous state was deter-
clear that the isothermal FHP model is inapplicable to
mined. In the range studied, the propagation velocity
this system.
of a  step perturbation was nearly constant (between
Therefore, we implemented a nine-velocity version
0.9 and 1), although the temperature in test calcula-
of the method on a square lattice [15], which is shown
tions varied by at least an order of magnitude. This is
schematically in Fig. 2. Particles move along the sides of
a consequence of the inaccuracy of the model, i.e., the
the square (density n1, velocity" and energy 1/2) or its
1,
limited number of permissible states.
diagonals (density n2, velocity 2, and energy 1). Each
A decrease in temperature (due to prevalence of
of these eight states can be occupied by one or none par-
fixed atoms) did not lead to a noticeable decrease in
ticle. In addition, there are rest particles (density n0),
wave velocity because perturbations were transferred
whose number may in principle be arbitrary (in our cal-
by moving particles, whose velocity along the lat-
culations, it is not more than six). The system simulates
"
a two-dimensional gas with density Á = n0 + n1 + n2 tice axes is equal to unity. An  ideal dependence
c = 2p/Á <" T might be expected for very long waves
and pressure p = n1/2 + n2. The presence of three  en-
when the flow has a chance to attain local temperature
ergy levels makes it possible to introduce a variable
equilibrium. For the problem considered, such waves
temperature T = p/Á. The  diagonal atoms correct to
are of no interest.
some extent the disadvantages of the square lattice by
A complete cycle of calculation ignoring heat losses
producing nondiagonal components of the momentum
consists of four steps. Along with propagation and col-
flux.
lisions, it includes a reaction and friction against the
A standard step in time includes propagation of
porous bed. Combustion was simulated by introducing
atoms to neighbor nodes and collisions at nodes. The
two sorts of gas particles:  blue particles (oxidizer)
result of collision is chosen in a random manner out of
and  red particles (combustion products). Initially,
all possible states (when present) that have the same
the pores contain only the oxidizer. Fuel (which form
number of particles, momentum, and energy and are
a film on the pore walls in the physical system) par-
not identical to the initial states (when present). A
ticipates in the calculation as a source of particles that
table of possible states is computed before calculation.
 evaporate into the gas. In the simplest case, the reac-
Some examples of collisions are shown in Fig. 2.
tion at each node involves formation of two high-energy
 red product particles from one  blue rest particle
and one fuel particle. Red particles cannot turn into
 blue particles (the reaction is irreversible) but  red
and  blue particles can exchange energy during colli-
sions. This simulates the process
A + B - 2C
with the energy effect equal to 2. At a given node,
combustion begins when a certain condition is satisfied
(for example, upon reaching specified temperature and
pressure averaged over the nearest neighborhood of the
given node) with given probability of the reaction w.
Fig. 2. Geometry and examples of collisions for a For each node, once combustion began, the ignition con-
square lattice.
dition was not further verified. This corresponds to the
Simulation of Convective Detonation Waves in a Porous Medium 209
irreversibility of ignition in a given pore. CALCULATION RESULTS
Reaction at the  burning node occurs with the
same probability w. The introduction of this parameter
Although the calculations were performed in di-
reflects to some extent the nonuniformity of the sizes
mensionless form, it is conveniently to assume that the
and geometry of real pores, which should affect ignition
lattice spacing is 1 mm and the time step is 1 µsec. The
and combustion. In most calculations, we used the value
velocity is then expressed in km/sec. For the density,
of w = 0.5. Naturally, for the reaction, it is necessary
any scale can be adopted, and the pressure is then ex-
that unexpended fuel, oxidizer, and two free diagonal
pressed in the units of Áu2. For example, if the unit of
states be present at a given node.
density corresponds to 10-3 g/cm3 = 1 kg/m3, the unit
If three or four free diagonals were available, three
of pressure is 1 MPa. For temperature, the reasonable
diagonal product particles were formed from one fuel
coefficient of conversion can correspond to 3000 K per
particle and two oxidizer rest particles (naturally, if they
unit.
were present). This improves the stoichiometry because
We used a lattice with 1 d" x d" 250 and 1 d"
conventional fuel (for example, of gross-composition
y d" 125. On the top and bottom boundaries, peri-
CH2) is markedly lighter than the oxidizer (1.5O2). For
odic boundary conditions were imposed, and the right
four free diagonals, new particles were randomly di- and left boundary were rigid walls. Initial concentra-
rected.
tions of the fuel f and oxidizer ( blue particles) were
The last step of the cycle simulated friction. In the
specified: in the standard version, f = 1.5, n0 = 3,
range of interest to us, the friction force is proportional
n1 = 0.8, and n2 = 0.32. Moving particles were ar-
to the squared velocity:
ranged according to the probability of occupation. For
the rest particles and the fuel (n0 and f), the integer
Áuu
f = k , part was first distributed uniformly, and the fractional
d
part, when present, was then randomly arranged. Af-
ter several collisions, equilibrium was established in the
where d is the particle size of the porous bed and
gas. The initial concentrations are close to the equilib-
k is the friction coefficient. According to [16], k =
rium values corresponding to the specified density and
1.75(1 - Õ)/Õ2, where Õ is the porosity (about 0.4 for
energy.
loose packing). According to more recent data [17], the
Then, combustion was initiated by specifying a hot
friction coefficient is approximately half the indicated
region with larger values of n1 = 1.2 and n2 = 0.96 for
value. Therefore, we assumed k = 3.5.
x < 7 (which corresponds to an increase in pressure
For each node, we calculated the local flow veloc-
by a factor of 2.17 and an increase in temperature by
ity u (averaged over nine points  a node and eight
a factor of 1.73). At an ignition temperature of 0.4, a
nearest neighbors). Then, the state at the node was re-
threshold pressure of 2.1, and a probability of reaction of
placed with probability w = Äku/(d + Äku) by a new
0.5, this perturbation developed into a quasistationary
one with the same number of particles and the same
wave that  forgot the initial conditions. An example
energy but a random value of the momentum, so that,
of calculation is shown in Fig. 3.
on the average, the velocity in the new state became
The wave is obviously nonuniform, especially at the
zero. This procedure simulates loss of momentum in
beginning. This is a consequence of the randomness in
quadratic friction in time step Ä. At the same time, the
the initial conditions. At t = 50, the hot region looks
stochasticity of flow in a porous medium is simulated.
like two  peninsulas. In fact, because of vertical peri-
The step Ä was always considered unit. In most of the
odicity, this is one hot zone. The front is later smoothed
calculations, d = 1.
but even after attainment of a quasistationary regime,
Thus, lattice gas simulates system (1) (more pre-
it does not become completely flat. At the bottom of
cisely, its two-dimensional analog). But it should not
the figure there are plots of pressure, fuel concentration,
be regarded only as a computational method because
density and velocity averaged over the vertical coordi-
system (1) itself is a rather crude approximation that
125
nate2 [for example, p (x) = p(x, y)/125]. For
lacks randomness, which is important for the problem
y=1
the given kinetics, the increase in the average pressure
considered. Lattice gas would be more properly treated
is smooth and corresponds to the region over which the
as an independent model of a real physical system.
wave front  is spread.
2
Letters Á and f in Figs. 3 5 denote the corresponding
curves, and the scaled quantities Á/2 and f/f0 are laid on
the vertical axis.
210 Ershov, Kupershtokh, and Medvedev
Fig. 3. Position of the wave front (ignition site) and Fig. 4. Slow wave at low initial temperature.
qualitative distributions of local pressure for various
times after the beginning of motion: nodes at which
p > 4 are shown, and for the last time, the averaged
Figure 4 shows the results for a colder initial state
wave structure is given.
with temperature half that used in the previous calcu-
lations. Here the front is also significantly irregular and
the wave velocity is equal to 0.78, which is less than the
The wave velocity was measured from the shift
velocity of sound in the initial state. As a result, there is
of the pressure profile from the time t = 100 and at
a certain increase in pressure and velocity ahead of the
t = 200, it was 0.93 km/sec (in natural units), which is
ignition front. The gas has managed to lead the slow
larger than the perturbation velocity in the initial state
combustion wave. This may be a source of some non-
(0.9) but smaller than that in products (1.0). This cor-
stationarity. The gas ahead of the front favors faster
responds to experiments with the only difference being
ignition and acceleration of the wave front. However,
that in the lattice gas, the range of sound velocities is
because of large friction, the penetration effect is slow,
very narrow. The flow velocity, as noted above, is about
and in the computational domain, acceleration was not
0.1.
observed.
The increase in density corresponds to injection of
fuel. At the wave front there is a small peak due to local
compression.
EFFECT OF HEAT LOSSES
As the probability of the reaction w increases to 1, a
flatter wave front is obtained; accordingly, the pressure
In experiments there is a heat flow from the reac-
rise is sharper. The wave velocity is D = 1.19. Hence,
tion mixture to the porous bed, which leads to cooling
the wave is supersonic with respect to both the cold ini-
of the gas. Elaboration of the lattice gas model allows
tial gas and reaction products. In further calculations,
this effect to be taken into account. The scheme is sup-
we set w = 0.5.
plemented with a fifth step  calculation of heat losses.
Simulation of Convective Detonation Waves in a Porous Medium 211
cause the main dependences are easily scaled (the pres-
sure rise, for example, is nearly proportional to density).
An exception is heat exchange (q <" Á0.7, and the energy
in unit volume is proportional to Á). Because of the de-
creased density, cooling in the calculation is accelerated
by approximately 30%, which can be neglected taking
into account the qualitative character of the calculation
model. Thus, the effect of density is also insignificant
for heat exchange.
The calculated wave velocity (925 m/sec) is close to
the experimental value (940 m/sec). The pressure pro-
file shape is also in qualitative agreement with measure-
ments [3]. The agreement of the pressure rise time sug-
gests a reasonable choice of the kinetics and the agree-
ment of the pressure decrease indicates that the heat
exchange was properly taken into account.
We note that the experimental and calculated pres-
sures agree only in order of magnitude. This difference
is partly related to the lower initial density but even
after multiplication of the calculated pressure by 2.5 
the ratio of the experimental and calculated densities 
a difference of about three times remains. This is of
course a consequence of the inaccuracy of the model.
Because of the discrete nature of the processes and the
stiff bounds for the main constants, it is impossible to
simultaneously obtain agreement for velocity and wave
amplitude.
A better agreement is achieved by correcting the
model and recalculating the results using reasonable
physical considerations. Let us consider the difference
Fig. 5. Combustion wave in the presence of heat
losses.
in the properties between the real and lattice gases. The
real adiabatic exponent of the combustion products is
Å‚ H" 1.3, and the energy release per unit mass of the
The heat flux (per unit volume of the pores) was calcu-
products is Q H" 11 kJ/g. For the lattice model, Å‚ = 2
lated from the formula
and Q = 1 kJ/g (in the adopted units). The reaction
6(1 - Õ) º(T - T0)
occurs in a practically constant volume, and the final
q = Nu,
d Õd pressure is p H" (Å‚ - 1)ÁQ. For a real fuel of stoichio-
metric composition, the final density is Á H" 1.3Á0, and
where d is the particle diameter, º is the thermal con-
in the calculations presented in Fig. 5, Á H" 1.5Á0. With
ductivity of the gas, T0 is the initial temperature, and
equal initial density of the oxidizer Á0, the model should
Nu is the Nusselt number. The standard Denton re-
give a pressure about three times lower than that in the
lation was used [18]: Nu = 2 + 0.6(ÁudÕ/·)0.7, where
real process.
· is the dynamic viscosity of the gas. At each node,
Figure 6 gives curves of pressure versus time.
the heat losses q was calculated in dimensionless units.
Curve 2, showing the average pressure in a certain cross
Then, the energy at a node was decreased by two units
section, is calculated from the data of Fig. 5, and curve 1
with probability q/2, which simulates the heat loss per
is an experimental curve taken from [3]). The initial
unit step in time. The mass and momentum were not
pressure in the calculation is subtracted to simulate a
affected.
piezoelectric gauge record. The calculated pressures are
The calculation results are shown in Fig. 5. The
increased by a factor of 7.5 to compensate for the dif-
initial pressure (1 MPa) and particle size of the porous
ferences in thermodynamics and stoichiometry (coeffi-
medium (2.5 mm) are the same as in [3]. In the calcula-
cient 3) and initial density (coefficient 2.5). After this
tion, the initial density was 5.3 kg/m3 (2.5 times lower
recalculation, quantitative results of modeling practi-
than that in [3]). The difference in density results from
cally coincide with the experiment. We note that the
the inaccuracy of the model. It is not significant be-
212 Ershov, Kupershtokh, and Medvedev
wave structure, is in good agreement with the results of
the simplified continual model of [9]. We note that in
the discrete model, friction is quite real, thus describing
the most important feature of the wave  stagnation
of average flow. At the level of mechanics, the  short
model [9] is supported by direct calculations. The heat
exchange in the model is also real (to an extent to which
it is possible to use the notion of temperature).
The further development of the model should in-
clude allowance for the detachment of the unburnt fuel
from the walls, which can lead to noticeable losses of
momentum.
At the same time, the model is rather crude. Be-
Fig. 6. Curves of p(t) for the experiment of [3] (1) cause of the small number of energy states, the tem-
and the present calculation (2).
perature of the lattice gas is limited (not more than 1).
For a more accurate simulation of the large tempera-
ture and pressure gradients in real mixtures (by several
heat exchange is overestimated by approximately 30%,
tens times), one needs to assign an initial state with a
which explains in part calculated faster decrease in wave
temperature of about 0.01, i.e., a state that practically
pressure. The residual difference has the same order of
consists of particles at rest. The physical meaning of
magnitude as the experimental scatter. This agreement
such formulation is questionable.
is even better than one might expect for the model con-
sidered. Thus, the comparison performed shows that
the lattice model gives a reasonable description of the
CONCLUSIONS
process.
The lattice method is useful for modeling the me-
chanics of the process, because, first of all, it takes into
account fluctuations and randomness at the mesoscale
DISCUSSION OF RESULTS
(pore size). Usually, statistical noise is regarded as a
The calculations show a qualitative fit to the exper- shortcoming of lattice calculations but in the present
imental picture of the phenomenon. The wave generally
problem it is vital. Waves with a reaction in crowded
has an irregular front, whose bulges should be identi- space is an almost ideal object for the lattice approach.
fied with the initiating jets. The protrusion at the front
At the same time, the lattice gas is a qualitative
where the reaction begins and the pressure increases
method with respect to kinetics and thermodynamics.
tends to propagate further. In contrast, friction and
At present, however, due to the inaccuracy of available
lateral expansion of the protrusion stabilize the front.
experimental information there is little point in more
The interaction of the randomness, gas dynamics, and
refined approaches. We believe that there is no ideal
dissipation determines the front shape.
computational methods and it is most reasonable to
The wave velocity is close to the velocity of sound.
combine discrete and continual approaches.
The jet mechanism suggests exactly this order of mag-
This work was supported by the Russian Foun-
nitude for the average velocity of the front [9]. How- dation for Fundamental Research (Grant No. 95-01-
ever, in wave calculations by the continual model, one
00912a).
has to specify the velocity of the front. In the discrete
model with specified kinetics, the motion of the front
is obtained automatically. This, along with simulation
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