Combustion, Explosion, and Shock Waves, Vol. 36, No. 6, 2000
Calculation of the Cellular Structure of Detonation
of Sprays in an H2 O2 System
S. A. Zhdan1 and E. S. Prokhorov1 UDC 534.222.2
Translated from Fizika Goreniya i Vzryva, Vol. 36, No. 6, pp. 111 118, November December, 2000.
Original article submitted August 7, 2000.
On the basis of the mathematical model of a two-phase two-velocity medium, deto-
nation of a cryogenic mixture (gaseous hydrogen drops of liquid oxygen) was studied
numerically. The dynamics of formation and the special features of the structure of
the two-dimensional reaction zone of the detonation wave are discussed. The cellular
structure of detonation is modeled for the first time for a cryogenic hydrogen oxygen
spray.
Cellular structures of detonation in gases are FORMULATION OF THE PROBLEM
well known in experiments [1]. Cells in gas droplet
systems were found only in sprays of small (d0 H"
Let a plane channel of width y0 contain a mix-
5 �m) droplets of decane [2]. The stability of det-
ture of liquid oxygen droplets (with a diameter d0,
onation waves (DW) in homogeneous and hetero-
true density �0, and volume concentration ą20) in
2
geneous explosives is studied theoretically in many
gaseous hydrogen with an initial pressure p0 and tem-
papers [3 9]. These waves may be described using
perature T10. A DW propagates along the channel.
the model of mechanics of a one-velocity medium.
We have to find its dynamics and structure depend-
The stability of DW for multivelocity heterogeneous
ing on the channel width, droplet diameter, and ini-
reactive media, for example, sprays, has not yet
tial composition of the mixture.
been studied analytically. This is primarily caused
Within the framework of the model of mechan-
by mathematical difficulties arising even in the lin-
ics of a two-phase multivelocity medium [12], we con-
ear analysis of detonation stability in these me-
sider a two-dimensional unsteady motion of monodis-
dia. Therefore, the only constructive tool for solv-
persed oxidizer droplets in a gaseous fuel. A com-
ing the problem of stability of a DW in a heteroge-
bustion region is formed in the gas droplet mixture
neous multivelocity medium is direct numerical simu-
behind the shock wave (SW) after a chemical delay
lation. Thus, the one-dimensional instability of det-
of ignition. The rates of chemical reactions in this
onation in a cryogenic hydrogen oxygen spray was
region are much greater than the mass-transfer rate
studied in [10], where, in particular, self-sustained
between the phases.
one-dimensional autooscillatory regimes of heteroge-
We use the following assumptions in the math-
neous detonation were obtained. Based on the analy-
ematical model: 1) there is no energy release in the
sis of these regimes, Zhdan [11] proposed a hypothesis
zone of chemical induction; 2) chemical processes at
on the possibility of existence of cellular structures in
each point behind the ignition front proceed instan-
this spray in a wide range of droplet sizes.
taneously, and the composition of the gas phase is
Results of a numerical study of the two-
in chemical equilibrium; 3) the energy-release rate
dimensional instability of detonation in a cryo-
in the gas droplet mixture is limited by the phase-
genic hydrogen oxygen spray are described in the
transition velocity, and the thermal effect of chemical
present paper. The dynamics of formation of a two-
reactions per unit mass of evaporated droplets and
dimensional reaction zone is discussed.
the molar weight of the gas are variable quantities.
1
Lavrent ev Institute of Hydrodynamics,
Siberian Division, Russian Academy of Sciences,
Novosibirsk 630090.
0010-5082/00/3606-0777 $25.00 � 2000 Plenum Publishing Corporation 777
778 Zhdan and Prokhorov
0 0
The equations of a two-dimensional unsteady U2,ch = -E2/�O2, and c2, l2, and E2 are the heat ca-
two-phase flow are written in the following form: pacity, the latent heat of vaporization, and the energy
of dissociation of molecules of the condensed phase,
(�i)t + (�iui)x + (�ivi)y = (-1)i+1j,
respectively.
If the internal energy of the gas phase U1 is reck-
nt + (nu2)x + (nv2)y = 0,
oned from the maximally dissociated composition at
(�iui)t + (�iu2)x + (�iuivi)y + ąipx
zero temperature, the following relations are valid in
i
the induction zone:
= (-1)i(fx - ju2),
p
U1,th = ,
2 (ł1 - 1)�0
1
(�ivi)t + (�iuivi)x + (�ivi )y + ąipy (1)
(4)
0 0
E2z E1(1 - z)
= (-1)i(fy - jv2),
U1,ch = - - .
�O2 �H2
(�iEi)t + [�iui(Ei + p/�0)]x + [�ivi(Ei + p/�0)]y
i i
Here ł1 is the ratio of specific heats in the induction
0
= (-1)i(fxu2 + fyv2 - jE2), zone and E1 and �H2 are the dissociation energy and
the molar weight of hydrogen, respectively.
Yt + u1Yx + v1Yy = -1/tind,
According to [14, 15], the thermodynamic and
chemical components of internal energy of the gas in
zt + u1zx + v1zy = (1 - z)j/�1,
the region of chemical transformations have the form
2
Ei = Ui + (u2 + vi )/2, �i = ąi�0, ą1 + ą2 = 1. � 1 - �
i i
U1,th = +
�min 2
Here ąi and �i and �0 are the volume concentration
i
and the mean and true densities of the phases, respec-
� �/T1 RT1
tively, ui and vi are the components of the velocity
+ + � - 1 ,
�min exp(�/T1) - 1 �
vector ui, Ui is the total internal energy of the ith
phase (i = 1 and 2), which is reckoned from the max-
1 1
imally dissociated composition at zero temperature,
U1,ch = Ed - , (5)
� �min
p is the gas pressure, n is the number of droplets per
unit volume, f and j are the intensities of the force
where � = �max(� - �min)/(�max - �min), �min and
and mass interactions between the phases, z is the
�max are the molar weights of the gas in the max-
mass fraction of the evaporated substance of the sec-
imally dissociated (in this case, atomic) and max-
ond phase in the gas regardless of the component it
imally recombined states, respectively, �max is the
enters, and Y is the fraction of the induction period
molar fraction of triatomic molecules in the maxi-
(Y = 1 at the SW front and Y = 0 at the ignition
mally recombined state, � is the effective tempera-
front).
ture of excitation of vibrational degrees of freedom
Based on the experimental data of [13], the
of molecules, and Ed is the mean energy of disso-
chemical delay of ignition for a hydrogen oxygen
ciation of the reaction products. All the parame-
mixture behind the SW front, in the induction zone
ters are uniquely determined by the atomic composi-
(Y > 0), has the form
tion of the mixture (by the mass fraction z of evap-
orated oxygen). For the hydrogen oxygen mixture,
Ka�O2 �a
tind = exp , (2)
we have �-1 = (1-z)/�H2 and �max = 2z�max/�O2
�0z RT1 max
1
for z < 8/9, �-1 = z/�O2 + 0.5(1 - z)/�H2 and
max
where �a is the activation energy, R is the universal
�max = (1 - z)�max/�H2 for z 8/9, �-1 =
min
gas constant, Ka is the preexponent, and �O2 is the
2(z/�O2 + (1 - z)/�H2), Ed H" E10 H" E20, and
molar weight of oxygen.
� = 3000 - 500�max.
We supplement system (1) by the equations of
Since the rates of chemical reactions behind
state of the phases
the ignition front are much greater than the mass-
p = �0RT1/�, �0 = const,
transfer rate between the phases, the composition of
1 2
(3)
the gas phase may be considered to be in chemical
Ui = Ui,th + Ui,ch,
equilibrium. Then behind the ignition front (in the
where Ti and Ui,th and Ui,ch are the temperature and reaction zone), we may use the approximate equa-
thermodynamic and chemical components of inter- tion for the shift in chemical equilibrium, which is
nal energy of the ith phase, � is the current mo- in agreement with the second principle of thermody-
lar weight of the gas phase, U2,th = c2T20 - l2, namics [15]:
Calculation of the Cellular Structure of Detonation of Sprays in an H2 O2 System 779
�1 (1 - �/�max)2 Ed
�i = �i0, ui = vi = 0,
exp
� �/�min - 1 RT1
Ui = Ui0 (i = 1, 2), p = p0.
�
The boundary conditions are the no-slip condi-
T1 �/2 �
= K1 1 - exp - . (6)
tion of the gas phase (v1 = 0) at the lower (y = 0) and
T10 T1
upper (y = y0) boundaries, transposition of param-
Here � = 1 + �max�min/(�max - �min) and K1 is the
eters from the solution domain at the left boundary
equilibrium constant.
moving with the mean SW velocity (this is correct
The closing relations for the intensities of the
since the gas flow is supersonic relative to the moving
mass (J = j/�2) and force (f) interactions between
boundary), and relations on a strong discontinuity at
the phases have the following form [12, 16, 17]:
the right moving boundary [19].
ńł
�ł 6(1.5Ą)1/2�2(�0/�0)1/3(�1/�2)2/3Re1/2/d2
Problem (1) (7) was solved numerically. To in-
�ł 1 2
�ł
�ł
�ł
tegrate the system of equations that describe the
�ł
�ł for We > 10,
gas flow, we used the second-order Godunov Kolgan
J =
�ł
scheme in moving grids [19, 20] with capturing of the
1/3
�ł
�ł 3Kvap(1 + 0.27Pr Re1/2)/d2
�ł
�ł
leading shock front and ignition front, where the ther-
�ł
ół
for We < 10,
mal decay of the discontinuity was calculated using
the procedure of [21]. The equations that describe
f = nĄd2�0CD|u1 - u2|(u1 - u2)/8, (7)
1 the motion of droplets were solved by the method of
ńł
coarse particles [22]. The accuracy of solving the dif-
27Re-0.84 for Re < 80,
�ł
ferential equations (1) was controlled by disbalance
CD = 0.27Re0.21 for 80 Re < 104,
in integral laws of conservation of mass and energy,
ół
2 for Re 104.
which was less than 1% in all calculations.
Here d is the current diameter of droplets, Re =
�0dw/�1 is the Reynolds number, We = d�0w2/Ł
1 1
is the Weber number, Pr is the Prandtl number,
CALCULATION RESULTS
Kvap is the vaporization constant, w = |u1 - u2|,
The numerical study was performed for a
�i = �i/�0 and �i are the kinematic and dynamic
i
cryogenic hydrogen oxygen mixture, which was a
viscosities of the ith phase, respectively, the dynamic
monodispersed spray of oxygen droplets in gaseous
viscosity of the gas has a power-law dependence on
hydrogen (2H2 + O2) for the initial temperature of
temperature �1 = �0(T1/300)0.7, and �0 is calculated
the phases T10 = T20 = 80 K with the following
using Herning s method [18]: �0 = (1 - a)�H2 + a�O2
thermodynamic parameters [23]: (a) for gas phase,
and a = z"�O2/((1 - z)"�H2 + z"�O2).
�H2 = 2 kg/kmole, �H2 = 0.9 � 10-5 kg/(m � sec),
Note that the expressions for the mass-transfer
0 0
�O2 = 2 � 10-5 kg/(m � sec), E1 H" E2 H" Ed =
intensity J correspond to the model of droplet-mass
110 kcal/mole, K1 = 1.81 � 102 kmole/m3, ł1 = 1.4,
entrainment by the mechanism of boundary-layer
and Pr = 0.75; (b) for droplets of O2, �O2 =
stripping, which transforms to the mechanism of
32 kg/kmole, �0 = 1135 kg/m3, c2 = 1680 J/(kg � K),
droplet vaporization for We 10, and are valid [16]
2
l2 = 0.213 MJ/kg, �2 = 2.5 � 10-4 kg/(m � sec),
for small droplets (d0 100 �m). Exactly this range
Ł = 0.0131 N/m, and Kvap = 2 � 10-6 m2/sec. The
of oxygen droplet diameters will be considered in the
constants in Eq. (2) for the chemical delay of igni-
present paper.
System (1) (7) is closed and completely de- tion were Ka = 5.38 � 10-11 (mole � sec)/liter and
�a = 17.15 kcal/mole [13].
termines the two-dimensional unsteady two-velocity
For these thermodynamic parameters of the
motion of a reactive gas droplet mixture with vari-
phases, the solution of system (1) (7) with initial
able heat release in the DW reaction zone.
and boundary conditions depends on three governing
parameters: initial pressure of gaseous hydrogen p0
and two scale factors  oxygen droplet diameter d0
INITIAL AND BOUNDARY CONDITIONS
and channel width y0. For a fixed value of p0, the
The initial conditions in the solution domain initial mean densities of gaseous hydrogen �10 and
(0 < y < y0, x > 0) are as follows: a one-dimensional oxygen droplets �20 = ą20�0 are determined unam-
2
steady solution in the reaction zone of ideal heteroge- biguously. In particular, for p0 = 1 atm, we have
neous Chapman Jouguet detonation for a monodis- �10 = 0.305 kg/m3 and �20 = 2.44 kg/m3; the ve-
persed spray of oxygen droplets in gaseous hydrogen locity of the ideal Chapman Jouguet detonation is
for 0 < x < x" and constant initial values for x > x": DCJ = 2.97 km/sec.
780 Zhdan and Prokhorov
Fig. 1. Velocity of the front of the one-
dimensional wave versus time for an autooscil-
latory detonation mode (p0 = 1 atm and d0 =
100 �m).
One-Dimensional Instability of the DW.
It is known [10] that detonation of a cryogenic
hydrogen oxygen spray in the one-dimensional for-
Fig. 2. Longitudinal SW velocity versus time on the
mulation (in contrast to heterogeneous detonation upper (solid curves) and lower (dashed curves) walls of
the channel for y0 = 26.6 (a), 22.15 (b), and 17.7 mm
of hydrocarbon oxygen sprays [11]) is unstable and
(c).
propagates in a pulsating (autooscillatory) mode.
The unstable element in the wave structure, which
leads to periodic oscillations of parameters in the
Two-Dimensional Instability of the DW.
reaction zone, is the ignition front. A typical de-
The numerical algorithm and the code developed for
pendence of the wave-front velocity D on the time t
solving the two-dimensional unsteady problem (1)
for p0 = 1 atm and d0 = 100 �m is shown in
(7) allow one to study the dynamics of propagation
Fig. 1. It also follows from one-dimensional calcu-
of a heterogeneous DW in a plane channel with var-
lations that the time ("t) and space (L0) periods of
ied channel width and oxygen droplet diameter. First
longitudinal oscillations of the wave velocity, its am-
we consider the influence of the channel width on the
plitude, and detonation velocity averaged over the
dynamics of the detonation process for fixed values
period (D0 = L0/"t) depend on the initial diameter
p0 = 1 atm and d0 = 100 �m. Figure 2 shows the
of oxygen droplets. The qualitative information on
calculated longitudinal velocity of the shock front D
these quantities for a number of values of d0 is given
versus the time t on the upper and lower walls for
in Table 1.
several values of the channel width: y0 = 26.6, 22.15,
It is seen that the values of Dmax, Dmin, and
and 17.7 mm. It is seen that the dependences D(t)
D0 depend weakly on d0, whereas the period of self-
differ not only from the one-dimensional solution (see
induced oscillations is an almost linear function of
Fig. 1) but also from one another. As the DW (ini-
the oxygen droplet diameter: L0 <" d1.1. It should
0
tially one-dimensional) propagates, a crossflow insta-
be noted that, because of the nonmonotonic ther-
bility of the ignition front develops in the reaction
mal effect of chemical transformations in the reac-
zone, which leads to transverse oscillations of gas-
tion zone [10], the mean velocity of one-dimensional
dynamic parameters of the flow. The flow pattern
oscillating detonation D0 is almost 8% greater than
becomes two-dimensional, and the shock-front veloc-
the velocity of the steady Chapman Jouguet detona-
ity experiences irregular oscillations (see Fig. 2c). By
tion (DCJ).
varying y0, we show that oscillations of the SW ve-
locity can appear for t > 0.5 msec, which are regu-
lar with respect to the x axis and have different (see
TABLE 1
TABLE 2
d0, "t, Dmax, Dmin, D0, L0,
�m �sec km/sec km/sec km/sec mm
d0, Dmax, Dmin, D ,
a a/b
25 2.581 4.71 2.43 3.29 8.523 �m km/sec km/sec km/sec
50 5.809 4.66 2.40 3.22 18.68 50 7.34 1.94 3.09 21.1 0.585
100 12.08 4.47 2.42 3.19 38.50 100 5.58 2.05 3.04 44.3 0.50
Calculation of the Cellular Structure of Detonation of Sprays in an H2 O2 System 781
Fig. 3. Two-dimensional cellular structure of the DW in a heterogeneous mixture 2H2 + O2 during
one period (d0 = 100 �m and a = 44.3 mm): the solid curve shows the calculated trajectory of the
triple point; (a) isobars of the gas phase; (b) isochores of the gas phase; (c) isochores of oxygen
droplets; .
Fig. 2a) and identical (see Fig. 2b) amplitudes on the width y0 from y" by ą5%, the velocity profiles on the
opposite walls of the channel. There exists a mini- opposite walls of the channel do not coincide, whereas
mum value of the channel width y0 = y" = 22.15 mm the shape of the leading front is still unchanged. The
for which the SW velocity along the x axis at a dis- fact that the SW velocity in a heterogeneous DW
tance x/y0 > 70 in each longitudinal section changes has two maxima during one period (see Fig. 2b) is
periodically (see Fig. 2b) with a time period "t = worth noting. This fact will be explained below by
29.15 �sec and a spatial period b = 88.6 mm. A analyzing the structure of the reaction zone of the
regular transverse perturbation is formed in the DW heterogeneous DW.
structure, which is alternatively reflected from the The calculated periodic two-dimensional DW
channel walls so that the SW velocity and other gas- structure (y0 = y") in a heterogeneous mixture
dynamic parameters on the opposite walls change in 2H2 + O2 during one period is shown in Fig. 3. The
the same manner but in the opposite phase (shifted isobars (P = p/p0) and isochores (R1 = �1/�20) of
by half a period). The calculated values of Dmax and the gas phase are shown in Fig. 3a and b, and the iso-
Dmin, and the wave velocity averaged over the pe- chores of oxygen droplets (R2 = �2/�20) are plotted
riod D = b/"t are listed in Table 2. The quantity in Fig. 3c. To control the reliability of the periodic
y", which will be called the eigenvalue of problem solution found, we performed additional calculations,
(1) (7), orders the two-dimensional instability of a where the channel width was doubled (y0 = 2y" = a)
heterogeneous DW in the form of a regular periodic for a fixed moment of time, and a mirror reflection
solution (going a few steps forward, we note that y" of the distribution of gas-dynamic parameters from
corresponds to half of the transverse size of the deto- the solution domain y" < y < 2y" was transferred
nation cell a/2). The periodic solutions found in nu- to the region y" < y < 2y". The initial data ob-
merical simulation are of special interest, and we will tained in this way were used to calculate an unsteady
dwell on them in more detail. We note that the longi- problem. It is shown that a self-sustained heteroge-
tudinal velocity of the shock wave near the wall D(t) neous DW propagates over the channel. The reaction
is rather sensitive to variation of the channel width; zone of this DW has a two-dimensional periodic so-
therefore, it is a convenient parameter for control- lution with two opposing transverse waves, which is
ling the accuracy of determining y". It follows from symmetric about the channel centerline 0 < y < y".
the calculations that, for a deviation of the channel The internal structure of this DW coincides identi-
782 Zhdan and Prokhorov
Fig. 4. Cellular DW structure in the heterogeneous mixture 2H2 + O2 for one period (d0 = 50 �m
and a = 21.1 mm).
cally with that shown in Fig. 3. Thus, a classical is formed by rapid release of energy due to the con-
detonation cell was obtained for the first time in nu- striction of the chemical induction zone. In contrast
merical simulation of heterogeneous detonation in a to the gas-phase parameters, the distribution of pa-
cryogenic hydrogen oxygen gas droplet mixture. rameters of the condensed phase in the reaction zone
Analysis of the Cellular Structure of a is more smooth (see Fig. 3c). For the heterogeneous
Heterogeneous DW. In a heterogeneous DW, as mixture 2H2 + O2, only 10 15% of the mass of liquid
in gas detonation [1], collisions of transverse waves oxygen droplets are gasified by the end of the induc-
moving over the leading front in opposite directions tion period of chemical reactions. The calculations
lead to reproduction of the front structure in time. show that the length of the zone of complete energy
The main element of this structure is a triple config- release in the heterogeneous DW is determined by
uration consisting of a Mach stem, an incident wave, the intensity of mass exchange (rate of mass addition
and a reflected (transverse) wave adjacent to the first from oxygen droplets to the gas phase). For oxy-
two elements at the triple point. The calculated tra- gen droplets 100 �m in diameter, this length is an
jectory of the triple point is shown in Fig. 3 by the order of magnitude greater than lind and amounts to
solid curve. The special feature of the cellular struc- 2 2.5 a. It is of interest that the gas-phase velocity at
ture for a heterogeneous DW in a cryogenic mixture the end of the energy-release zone at all times is su-
2H2 + O2 is that the induction zone of chemical reac- personic relative to the mean wave velocity D , and
tions lind has a finite length (0.04 0.2 a) at all times the Mach number M = ( D - u1)/c1 reaches 1.4.
because of the low temperature of the gas phase be- This means that the transition from subsonic to su-
hind the SW front. Therefore, the triple configura- personic flow occurs in the reaction zone. An analy-
tion at the leading front is always a shock-wave con- sis of the numerical solution shows a curve LM(x, y):
figuration. The location of the end of the induction u1 + c1 = D exists at all times inside the reaction
period is clearly seen in Fig. 3b in regions of high zone; to the left of this curve, we have u1 + c1 < D .
gradients of the gas density, and the maximum in The mean position of the curve LM(x, y) is at a dis-
lind is always observed at the moment of arrival of tance of 0.75 a from the front, and its curvature does
the triple point at the cell axis. The transverse wave not exceed ą0.25 a. As a result, a significant por-
consists of the shock wave, which emanates from the tion of liquid oxygen (up to 50% of the mass flux
triple point and moves over the induction zone, and for droplets with d0 = 100 �m) enters the chemical
the adjacent compression wave with energy release reaction in the supersonic region of the DW.
in the reaction zone. It is seen from Fig. 3 that the The parameter characterizing the cell shape is
first maximum of the SW velocity (see Fig. 2b) is ob- usually the ratio of the transverse to the longitudinal
served at the moment when the triple point arrives size of the cell (a/b). In the case considered, we have
at the channel wall (which is equivalent to the plane a/b = 0.5 for p0 = 1 atm and d0 = 100 �m, i.e.,
of symmetry of the cell for y0 = a) and the second the cell in a heterogeneous mixture may be more ex-
maximum is observed at the moment of arrival of tended in the longitudinal direction than in reactive
the compression wave from the reaction zone, which gases [1].
Calculation of the Cellular Structure of Detonation of Sprays in an H2 O2 System 783
Variation of the Droplet Diameter. The cell REFERENCES
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