development of an explosion from an initiation source in a liquid high explosive

ISSN 1990�7931, Russian Journal of Physical Chemistry B, 2010, Vol. 4, No. 6, pp. 909–915. © Pleiades Publishing, Ltd., 2010.Original Russian Text © A.V. Dubovik, 2010, published in Khimicheskaya Fizika, 2010, Vol. 29, No. 11, pp. 34–41.

909

INTRODUCTION

In [1, 2], a detailed analysis of the excitation ofexplosion under different conditions of mechanicalinfluences on the reactive system was performed.No less important for analysis of the mechanicalsensitivity of high explosives (HEs) is the issue ofdevelopment of an explosion from a local source ofinitiation. With a sufficient degree of certainty, onecan assert that the resulting explosion processbegins as layer�by�layer burning at a rate deter�mined by the pressure and temperature in the reac�tion zone of the HE material. In this case, in a rela�tively large volume of HE, the process of burningmanifests itself as an expanding cavity filled with thereaction products.

The dynamics of development and stability of theburning of a spherical cavity in a liquid HE was the�oretically analyzed in the [3–5]. The authors ofthese works derived the equation of the process andexamined a number of special cases. For example,based on the general properties of HEs, Margolin[4] showed that, depending on the value of the pres�sure exponent ν in the burning rate law for HEs andthe relationship between the characteristic times ofburning and hydrodynamic expansion of the cavity,there are two possibilities. In one case, the solutiondiverges, and, therefore, a transient increase ofpressure in the cavity is observed. In the other case,it belongs to the focus, and then an oscillatory modeof burning sets in, the rate of which is asymptoti�cally approaches a mean value.

Clearly, these possibilities do not exhaust theentire variety of explosion types in liquid HEs. In

addition, not all of the properties of the HE chargeand its environment that significantly affect theprocess of the explosion were taken into account in[3–5]. Therefore, we will consider a detailed pictureof an explosion developing from a reaction hotspot,first in the bulk and then in a thin layer of HE mate�rial.

1. DEVELOPMENT OF A REACTION HOTSPOT IN THE BULK OF A LIQUID HE

Formulating the problem, we made the followingassumptions:

(1) the HE under study is a weakly compressibleliquid with consistency m and flow index n;

(2) the movement of fluid around the burningbubble of radius a and (а(0) = а0) is spherically sym�metric;

(3) the pressure Р and temperature Т of the com�bustion products (CP) are uniformly distributed overthe volume of the bubble, but change with time;

(4) the CP obey the equation of state of a real gaswith covolume �;

(5) the surface tension is small compared with theforce of pressure in the bubble;

(6) the burning rate is a preset function of only thepressure:

W = BР ν. (1.1)

To describe the parameters of the CP in the bub�ble, we used the equation of state and the laws of

COMBUSTION, EXPLOSION, AND SHOCK WAVES

Development of an Explosion from an Initiation Source in a Liquid High Explosive

A. V. DubovikSemenov Institute of Chemical Physics, Russian Academy of Sciences, Moscow, 119991 Russia

e�mail: a�[email protected] January 27, 2009; in final form, April 7, 2010

Abstract—A numerical analysis of the development of combustion and explosion from a reaction hotspot inthe bulk and in a thin layer of a liquid explosive placed between two flat solid surfaces is performed. A com�parison of the theoretical results with experimental data shows that a satisfactory agreement between the mea�sured and calculated flame speeds is possible only under the assumption of a multifold increase in the burningsurface area due to its instability. An estimate of the parameters of the shock wave generated by an acceleratingflame shows that the mechanisms of shock�wave and cavitation initiation of detonation cannot be ignored inanalyzing the regimes of unsteady combustion of liquid explosives.

Keywords: liquid high explosives, explosion development.

DOI: 10.1134/S1990793110060060

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RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY B Vol. 4 No. 6 2010

DUBOVIK

conservation of mass and energy:

Р(1/ρ – �) = RT/Mg, (1.2)

dM/dt = ρ0WSω, M = Ωρ, (1.3)

(1.4)

where ρ0 is the HE density, М is the mass of the gas inthe bubble, Ω = (4/3)πa3 and S = 4πa2 are the volumeand surface area of the bubble, ρ and Mg are the den�sity and average molecular weight of the CP, R is theuniversal gas constant, ω ≥ 1 is the relative increase inthe burning surface area due to its instability (themeaning of this parameter will be clarified below),Сv

and Cp are heat capacities of CP at constant volumeand pressure, α is the coefficient of heat transferbetween the CP and HE, Тb is the combustion temper�ature of the HE at constant pressure, and Ti is the ini�tial temperature.

In the first approximation, we assume that the liq�uid is incompressible. Then, from the continuity equa�tion, we obtain the velocity distribution around theburning bubble:

v = Vа2/r2, (1.5)

where V = (dа/dt – W) is the hydrodynamic (mass)velocity at the boundary of the expanding bubble.

Further, we assume that the transient stress fieldwith components σr and σϕ in the liquid is describedby the Euler equation:

ρ0dv/dt = ∂σr/∂r + (2/r)(σr – σϕ). (1.6)

Let the relationship between the stress tensor com�ponents σij and strain rates ξij be given by the powerlaw [6]

σij = –р + 2m|H|(n – 1)/2ξij, (1.7)

where Н = 2ξij ξji is the shear rate intensity.In our case, ξr = ∂v/∂r = –2Vа2/r3 and ξϕ = v/r =

Vа2/r3 and, consequently, =

d MCvT( )/dt

= CpTbρ0WSω PdΩ/dt– αS T Ti–( ),–

( )ϕ= ξ + ξ

2 2 2 2rН

12V2а4/r6. Substituting these expressions into (1.7) givesthe stresses σr and σϕ. Their substitution into (1.6) leadsto an ordinary differential equation for the pressure р;integrating this equation over r from а to ∞ and over рfrom Рl to Р∞ yields the following equation for V(t):

(1.8)

In deriving (1.8), we took into account the rela�tionship between the pressures in the liquid Рl and gasPg, which results from the balance of mass andmomentum at the bubble boundary:

Рl = Pg – 4m(2 � 31/2V)n – 1V/an + ρ0W2(ρ0/ρ – 1). (1.9)

The compressibility of liquid outside the bubblewas described as Herring and Trilling [7] had done.According to their views, on the right�hand side ofboundary condition (1.9), the term (a/c0)dРl/dt (c0 isthat speed of sound in the liquid) should be intro�duced, which is obtained from a joint analysis of thegeneral equation of continuity and the Euler equationin the limit V � c0. In view of (1.1), (1.2), (1.9), equa�tion (1.8) becomes cumbersome, and, therefore, it isnot presented here. Further, we use the Herring–Trill�ing acoustic approximation and calculate the parame�ters of a burning cavity in a typical liquid HE, such asnitroglycerin (NG), which is a normally viscous fluidwith n = 1 and m = μ = 0.03 Pa s. Let ρ0 = 1.6 kg/dm3,с0 = 1.73 km/s, Ср = 1.1 kJ/(kg K), B = 1 cm/(s MPa),ν = 1, Mg = 0.03 kg/mol, γ = Ср/Сv = 1.25, and α =5 kW/(m2 K). The initial parameters of the hotspot areas follows: the size, a0 = 0.1 mm; pressure, Р0 = 10 MPa;and temperature. Т0 = Тb = 3000 K. The characteristicexpansion velocity of the burning bubble and the Rey�nolds number of the flow it generates are V0 =(P0/ρ0)

1/2 = 79 m/s and Re = ρ0a0V0/4μ = 105. Thehigh value of Re is indicative of a strong influence ofthe inertia effects of the motion of the liquid on thecharacteristics of expansion of the burning cavity. Thecalculated parameters as functions of the time of com�bustion of NG are displayed in Fig. 1.

As can be seen, because of the inertia effects asso�ciated with the movement of the bubble, NG combus�tion occurs in the oscillatory mode with a nonmono�tonic increase in the volume of burning. At momentsof bubble compression, CP pressure and temperaturepeaks with gradually decreasing amplitude are observed.Therefore, at long times, t > 1 s, the combustion param�eters tend to constant values: Рs = 0.108 MPa, Ts =424 K, and Js = (V + W)s = 1.89 m/s, so that, at t =1.265 s, the bubble radius is аs = 2.38 m and then onlyincreases. The phase diagram corresponding to thiscalculation is shown in Fig. 2 in the (P/Р0, J/V0) coor�dinates. The arrows in the spiral curve show the direc�tion of motion of the phase point, which starts at coor�dinates (1, 0) and reaches a focus at (0.011, 0.034),

dV/dt P P∞–( )/ρ0 W2ρ0/ρ 1–( )+[=

– 4m 2 31/2V⋅( )n 1–

V/ nρ0an( ) 3V2

/2– 2VW ]/a.–

0.8

0.6

0.4

0.2

0

0.60.40.2 t, µs

1

2

3

4

Fig. 1. Time evolution of the parameters of a burningcavity in NG: (1) pressure Р/Р0, (2) temperature Т/Т0,(3) CP–NG interface velocity J/V0, and (4) interfaceposition а/a0.


DEVELOPMENT OF AN EXPLOSION FROM AN INITIATION SOURCE 911

coordinates that correspond to the indicated pressureand velocity.

The steady combustion parameters are practicallynot influenced by an increase or decrease in the vis�cosity, surface tension, heat transfer coefficient, com�pressibility of NG, initial size of the hotspot, and pres�sure in it. Halving the value of Т0 produces only amoderate reduction of these parameters, while dou�bling it lead to a twofold rise in Рs and Ts and anincrease in Js by an order of magnitude.

Strong influence on the steady combustion param�eters has the value В in the burning rate law of HEs. Itsincrease by an order of magnitude leads to the follow�ing steady combustion parameters: Рs = 43.9 MPa,Ts = 2955 K, Js = 138 m/s, with the burning rate Jsbeing higher that the rate of hydrodynamic expansionof the cavity Vs by ~5 m/s. Even a stronger effect on thesteady combustion parameters of NG has the expo�nent ν. When it is increased from 1 to 1.2, at 3.43 μsafter the start of combustion, the pressure in the CPand the cavity expansion rate reach 4.29 GPa and2.14 km/s, respectively, which actually means anexplosion in the NG medium. Note that the high val�ues of pressure and velocity obtained here and in somefurther calculations go far beyond our assumptions onthe model properties of the studied liquid HE and CPand should be considered only as estimates of thecombustion characteristics of liquid explosives.

A similar effect was observed upon a relativeincrease in the burning surface area ω by a factor of 25or more. This increase may be a consequence of thecurvature of bubble surface or of the dispersion of liq�uid into tiny droplets due to the wave effects associatedwith the unsteady motion of the bubble and due to theturbulization of pre�front flow. Estimates show that, atnot too large ω (if, for example, the size of emergingroughness on the inner surface of the bubble is muchsmaller than its radius), the volume of the bubble Ωchanges insignificantly compared to its smooth(spherically symmetric) walls. This remark is made toemphasize that conservation equation (1.3) and (1.4)need no adjustment if ω increases only moderately.The calculations performed at ω = 25 show that theexplosion occurs within 6.24 μs after ignition. Accord�ing to the same estimates, a higher value of ω can beobtained if ~103 submicron droplets are formed in theburning bubble.

As noted above, a low viscosity has practically noinfluence on the mode of HE combustion. However,when it is high enough, the oscillatory mode becomesaperiodic, with a well�pronounced pressure peak atthe start of combustion. Indeed, at μ = 10 kPa s, themaximum pressure in the bubble at 10.15 μs after ini�tiation reaches 7.44 GPa and then decreases mono�tonically. But already at μ = 20 kPa s, the unlimitedgrowth of the parameters is observed at 9.51 μs afterthe beginning of combustion. In this case, it is possible

to determine the critical value of Re, below which thedisruption of NG combustion occurs: Recr = 1.6 × 10 –4.

The anomaly of the viscosity does not make sub�stantial adjustments to the laws of combustion of HEs.For example, at n = 0.5 and m < 10 kPa s0.5, an oscilla�tory mode of combustion and its evolution to steadycombustion is observed, as in the case of combustionof low�viscosity liquid explosives, whereas at a highervalue of the consistency, aperiodic combustion with astrong peak at the beginning of the process occurs.

A further increase in the pseudoviscosity, to m =33 MPa s0.5, gives rise to explosion at 9.11 μs aftercombustion begins. It is noteworthy that the criticalvalue of Re, Recr = (ρ0a0V0/4m)(2 � 31/2V0/а0)

1 – n =1.6 × 10–4, is the same as for combustion of a normallyviscous liquid explosive.

The physical interpretation of the results is basedon the fact that the viscous fluid around the bubble hasno time to move aside ahead of combustion front, and,therefore, the pressure in the bubble rises sharply,which, in turn, leads to an explosion.

Deflagration to explosion transition (DET) is sub�stantially facilitated by the dispersion of liquid in thevolume of the burning bubble. The possibility of exist�ence of such an effect was discussed by Andreev [8].

Using velocity distribution (1.5), we roughly esti�mated the mass flux at the front of a spherical shockwave (SW) formed by an expanding gas bubble: v(A) =Ja2/А2 where А is the radius of the diverging shockwave front. Based on the known Hugoniot curve ofNG D = c0 + bv(A), (c0 the speed of sound and b = 2),we obtain an equation for determining A(t) by usingthe kinematic condition dA/dt = D. Then, knowing D,we define the pressure at the wave front: П = ρ0Dv(A).Estimates of the parameters of the shock wave per�formed by this method showed that the shock wavegenerated by the steady combustion of NG is close toan acoustic one, propagating at sound speed (с0 =1.73 km/s) and having a pressure at the front of a few

1.0

0.5

−0.15 0.23 0.6J/V0

F

P/P0

Fig. 2. Phase diagram of NG burning, F is the point offocusing.

912


DUBOVIK

or tens of MPa. Moreover, as expected, the pressure atthe wave front is lower than the pressure in the CP dueto the divergence of the flow of liquid.

However, upon disruption of combustion and itstransition into an explosion, the velocity and pressurein the shock increase sharply (hundreds�fold), but stilldo not exceed the parameters of CP in the burningbubble (Fig. 3). Thus, in the above example of DET atω = 25, the flow at t = 5.94 μs is characterized by thefollowing parameters: at the flame front, а = 6.0 mm,Р = 18.3 GPa, and J = 6.25 km/s; at the shock front,A = 12.2 mm, v(A) = 1.52 km/s, D = 4.76 km/s, andП = 11.6 GPa. Note that the adiabatic inductionperiod of explosion at the shock front with a pressureof П = 11.6 GPa is shorter than 1 μs. As noted above,these estimates do not claim being unquestionable, letalone a substantiation of the proposed DET theory;nevertheless, they are indicate of the possibility ofshock�wave initiation of detonation ahead of a rapidlyaccelerating spherical flame.

Thus, at a certain combination of initial condi�tions, the steady combustion of liquid explosivesbecomes impossible: it transforms into an explosionwith a high probability of transformation into normaldetonation.

2. THE DEVELOPMENTOF THE REACTION CHAMBER

IN A THIN LAYER OF LIQUID EXPLOSIVE

In contrast to the problem of bulk combustion, letus consider a somewhat more complicated problem ofthe motion of an axisymmetric flame front in a HElayer of thickness h between solid surfaces of radiusR � h. This type of burning apparently takes place inthe gap between the rollers of devices for testing thesensitivity of HEs to mechanical impacts [1].

We assumed that, at t = 0, the radial motion of fluidcaused by the ignition of a near�axis layer gives rise toa cylindrical cavity of radius а0 filled with combustionproducts at a pressure Р0 and temperature Т0. Outside

the layer of liquid HE, the ambient pressure is constantand equal to Ра ≤ Р0.

As before (Section 1), the liquid is assumed to beincompressible, but with an appropriate allowance forpressure, introduced according to Herring and Trilling[7]. Let the origin of cylindrical coordinates (r, z) be atone of the solid surfaces. Then, for the radial velocityof the flow of an anomalously viscous fluid, we have

∂u/∂t + u∂u/∂r

= –(1/ρ0)∂p/∂r + (m/ρ0)∂(|∂u/∂z|n – 1∂u/∂z)/∂z, (2.1)

(1/r)∂(ur)/∂r = 0, ∂p/∂z = 0. Given the layer being thin, the pressure in the fluid

is a function of only the radius r. Taking into accountthe slip condition at the contact surfaces, we assumethe velocity distribution in the HE layer being depen�dent on z according to a power law. Using the continu�ity equation gives

(2.2)

where U is the velocity of the fluid at the boundary withthe cavity and f(t) is an unknown function of time. Letus satisfy the condition at the boundary on the average

(V = dη, integration from η to 2), thereby avoiding

a cumulative spike typical of power profiles of velocity.As a result, expression (2.2) yields f(t) and, hence,

u = [(2n + 1)/(n + 1)](aV/r)(1 – |1 – η|(n + 1)/n), (2.3)

where V = (dа/dt – W), is the mass velocity at the cav�ity boundary and W is the linear burning rate of HE,which depends on the pressure as specified by for�mula (1.1). Substituting u from (2.3) into Eq. (2.1) andaveraging the results over η from 0 to 2 gives an equa�tion for р(r). Then, integrating over r from а to R withconsideration given to the HE compressibility in theacoustic approximation, we obtain the pressure Pl atthe boundary with the gas cavity:

(2.4)

Since the pressure in the liquid Pl and in the CP Рare related as Pl = Р + ρ0W

2(ρ0/ρ – 1), (2.4) is theequation for V(t). To determine its constituent param�eters, Р and ρ (CP density), we used expressions (1.3)and (1.4) with Ω = πa2h and S = 2πah. Simple trans�formations lead to a system of equations for determin�ing the combustion characteristics, Р, Т, V and а, as

u aU/r( )=

= a/r( )f t( ) 1 1 η– n 1+( )/n–( ), η 2z/h,=

u∫

( )

( )

( )

( ) ( )

2 2 2

1

/ /

/ /

2 1 / 3 2 1 /

2 / 1 2 1 /

2 / 1 /

0

0 [( ( ))

(( ) ( )]

[ ( )][( ) ]

) .

l a l

n n

P P a c dP dt

adV dt V V W ln R a

n n V a R

mR h n n n

aV Rh a R −

− +

⎧=ρ + +⎨

⎩

− + + −

+ − +

⎫⎡ ⎤× − ⎬⎣ ⎦

⎭

43210 t, µs

400

200 1

2

3

4

Fig. 3. Propagation of the shock wave ahead of an acceler�ating flame: (1) pressure in the burning bubble, (2) pres�sure at the shock wave front П/P0, (3) flame speed 3J/V0,(4) mass velocity behind the shock wave front v/V0.



functions of the time of burning. One of the governingparameters of this system is the Reynolds number,

Re = (ρ0h /2mR)((n/(2n + 1)Rh/2а0V0)n, where

V0 = (P0/ρ0)1/2 is the characteristic flow velocity. An

analysis of Eq. (2.4) suggests that the cavity velocityV(t) has a logarithmic singularity at the finite point ofpropagation r = R. Physically, this means that, as theflame front approaches the periphery of the HE layer,hydrodynamic resistance to its movement falls rapidly(mainly due to the action of the inertia and viscousforces), thereby causing its progressive acceleration.This phenomenon is often observed in the experiment.Fig. 4 shows a streak photograph of the combustion ofa weakly gelatinized (with 1% Plexiglas) bis(2�fluoro�2,2�dinitroethyl) formal (FEFO) placed as a 0.5�mm�thick layer between the end faces of a transparentPlexiglas cylinders, through one of which photographswere taken. FEFO was ignited at the center of the layerwith a point electric discharge with energy of 15 J. Theacceleration of the flame at the beginning and endof combustion is clearly seen, with the velocity inthe intermediate region being approximately con�stant, ~180 m/s. The time of burning of the layerwas ~140 μs. From the foregoing, it is obvious that,unlike the combustion of HEs in a bulk medium, thesteady�state parameters of combustion in a thin layerare not reached.

To understand the role of various factors influenc�ing the character of the combustion of thin layers ofliquid HEs, we examined the results of numerical cal�culations of the system equations derived by varyingthe initial parameters of the original version with thefollowing data: h = a0 = 0.1 mm, R = 10 mm, Р0 =10 MPa, Т0 = 3000 К, ρ0 = 1.6 g/cm3, μ = 0.03 Pa s,В = 1 cm/(s MPa), ν = 1, с0 = 1.73 km/s, and α =5 kW/(m2 K). These data approximately characterizethe conditions of combustion of a normally viscousliquid (n = 1), similar to NG. The ignition of such aliquid occurs from an adiabatic�compression�heatedgas bubble with a size equal to the thickness of thelayer, as in experiments on the impact sensitivity of liq�uid explosives on a vertical impact�testing machine [1,2]. For this example, V0 = 79 m/s and Re =ρ0h

2V0/12μR = 35.1.

The calculation results are displayed in Fig. 5,which shows the evolution of the velocity (curve 2),pressures (curve 1), and temperature (curve 3) of aburning cavity with the distance traversed by it s alongthe layer of liquid explosive. At the beginning, theflame speed increases rapidly to JM = 27.6 m/s. Then,because of flow divergence, it reduces to a minimum,Jm = 0.36 m/s, at a distance of s = 1.15 mm (t =0.21 ms) from the initiation site, after which, as aresult of ongoing gas inflow and attenuation of dissipa�tive effects, it increased slowly to Jf = 27.8 m/s at theend of the layer (sf = 9.95 mm). The time of NG burn�out was estimated as tf = 2.46 ms. The pressure andtemperature do not follow the changes in the flame

V02

speed. Since the beginning of burning, Р diminishes toa minimum, Рm = 0.10 MPa (0.11 ms), then increasesto a maximum, РМ = 0.62 MPa (t = 1.50 ms), and fur�ther reduces to Pf = 0.32 MPa. Temperature correlateswith changes in pressure, featuring extrema at Тm =416 К, ТМ = 875 K, and Тf = 698 K.

With other input data, the regularities of changes inthe combustion characteristics were generally similarto those considered above.

The table lists some calculation results obtainedwhen one of the parameters of the original version waschanged. An analysis of the table shows that, as forcombustion in a bulk liquid explosive, an increase inthe initial pressure and the size of the reaction, as wellas a decrease in its initial temperature, do not affectthe basic characteristics of the combustion: the maxi�mum pressure and time of burnout of the liquid. Muchstronger influence is produced by the parameters ofthe burning rate law, В and ν. At ν = 1.2, an unlimitedincrease in the pressure and velocity of the flame wasconditionally accepted as DET. The same effect is pro�duced by a significant increase in the burning surfaceor in the viscosity of the explosive. In the latter case,we can determine the critical value of the Reynoldsnumber, Re = 0.024, below which stable combustionof a liquid explosive layer is impossible. It is notewor�thy that this same value of Recr characterizes the dis�ruption of combustion in an anomalously viscous(pseudoplastic) fluid.

At small μ or elevated h, pulsating combustion isobserved, during which the flame speed periodicallybecomes negative: the rapid expansion of the cavityleads to a decrease in the pressure below atmospheric,resulting in a contraction of the cavity, although theliquid continues to burn. Then, due to gas inflow, thepressure increases again and the cavity expands. Fig�ure 6 shows the time evolution the characteristics ofthe combustion in a 1�mm�thick layer of NG. Notethat this pattern of burning is possible only if cavitycontraction is accompanied by a reduction in fluidinflow from the outside. In other words, the HE layermust go beyond the solid surfaces into the area r > R.

10050

10

20

s, mm

t, µs

Fig. 4. Streak photograph of the combustion of a thin layerof gelatinized FEFO.

914


DUBOVIK

Variant no. 17 of the calculations (table) is pre�sented to quantitatively illustrate the experimentshown in Fig. 4. The calculations were carried out inaccordance with the experimental data on the rheo�logical properties of weakly gelatinized NG [2]. It isshown that the flame speed first increases rapidly to204 m/s and then decreases to 154 m/s. Furthermore,it increases very weakly and only at the end of the com�bustion rises sharply to 426 m/s. At the beginning ofthe combustion, the pressure also increases rapidly to342 MPa, but then slowly decreases to 41 MPa at the

end of the process. Following Р, the temperature ofthe CP first rises to 3550 K and then decreases slowlyand monotonically to 2740 K.

Importantly, the agreement between the calculatedand experimental data is achieved by assuming a ten�fold (ω = 10) increase in the burning surface of the liq�uid, without which the combustion FEFO proceeds ata rate of several tens of meters per second. This asser�tion makes sense, since even at a pressure of 100 MPa,the rate of the layer�by�layer burning of FDEF is only1 m/s.

To estimate the parameters of the shock wave gen�erated by the accelerated combustion of a thin layer ofliquid explosive, we used the mathematical proceduredescribed at the end of Section 1. We assume here thatthe mass velocity of the flow behind the shock front isrelated to the flame speed as u(A) = (V + W)(a/A).However, when the finite size of the explosive layer andthe shock moving ahead of the flame are taken intoaccount, the pattern of burning can change signifi�cantly, as shown in Fig. 4, especially for low�viscosityliquids. Examining variant no. 17 in the table, one cansee that, only at 3.13 μs after the start of combustion(а = 2.3 mm, J = 167 m/s, and Р = 103 MPa), a shockwave with D = 1.83 km/s and П = 150 MPa reaches theperiphery of the HE layer (А = 25 mm), where it trans�forms into a rarefaction wave, which moves opposite to

0.60.40.2 s, cm

0.8

0.4

1

2

3

0

Fig. 5. Combustion parameters as functions of the distancetraversed along a NG layer: (1) pressure 10Р/Р0, (2) frontvelocity 5J/V0, and (3) temperature 3T/T0.

Parameters of burning of thin layers of liquid explosives

No. Calculation version РМ,MPa Jf, m/s tf, μs Re Notes

0 Initial 0.62 27.8 2460 35.1 Burning to the end of the layer

1 P0 = 100 MPa 0.60 33.7 2340 111 Burning to the end of the layer

2 a0 = 1 mm 10.5 36.8 1190 3.51 Burning to the end of the layer

3 T0 = 1500 K 0.37 19.0 4180 35.1 Burning to the end of the layer

4 h = 0.01 mm 172 161 606 0.35 Burning to the end of the layer

5 h = 1 mm 2.0 13 3060 3513 Oscillatory burning to the end of the layer

6 R = 100 mm 14.9 154 8475 35.1 Burning to the end of the layer

7 B = 10 cm/(s MPa) 200 363 52.5 35.1 Burning to the end of the layer

8 ν = 1.2 6100 8260 0.40 35.1 DET at s = 0.38 mm

9 μ = 0.003 Pa s 0.46 3.8 4690 351 Oscillatory burning to the end of the layer

10 μ = 0.3 Pa s 14.9 40 780 3.51 Burning to the end of the layer

11 μ = 45 Pa s 5700 86 96 0.024 DET at s = 3.6 mm

12 ω = 10 253 375 52 35.1 Burning to the end of the layer

13 ω = 15 1800 2100 1.6 35.1 DET at s = 0.51 mm

14 n = 0.5; m = 3.6 Pa s0.5 0.39 15 2860 35.1 Oscillatory burning to the end of the layer

15 n = 0.5; m = 26.7 Pa s0.5

(μe = 0.03 Pa s)11.0 65.6 1111 7.45 Burning to the end of the layer

16 n = 0.5; m = 7000 Pa s0.5 774 44 361 0.028 DET at s = 86 mm

17 n = 0,6; m = 20 Pa s0.6;h = 0.5 mm R = 25 mm; ω = 10

340 425 141 31.5 Burning to the end of the layer

Note: µe is the effective viscosity of the plastic determined as m(V0/h)n – 1.



the flame with an initial velocity of D – u(A) =1.78 km/s. Negative pressure in the unloading wave<–1 kPa) may prove sufficient for the appearance ofcavitation in the liquid ahead of the flame front. Attime tm = 27.3 μs, unloading wave encounters theflame front at point r = а = 4.6 mm. The subsequentcollapse of the cavitation bubbles in the zone of highpressure (85 MPa) of the combustion wave is the realmechanism of low�velocity detonation in heteroge�neous liquid explosives such as NG [2]. Note, how�ever, that, if by the time tm, the pressure of the CP inthe expanding cavity reduces significantly (below acertain critical value), no detonation of the layer ofburning HE will occur, as is seen in Fig. 4.

CONCLUSIONS

The results of numerical analysis of the develop�ment of explosion from an initiation source in the bulkand in thin layer of liquid explosive between solid sur�faces led us to the following conclusions:

(1) If thermorheological parameters and coeffi�cients of the burning rate law for a HE do not gobeyond values typical of the majority of HEs, its com�bustion occurs in a fairly quiet mode with damped

oscillations of the velocity, pressure and, other charac�teristics around some stationary points;

(2) Deflagration�to�explosion transition occursonly upon strong changes in the combustion charac�teristics: the viscosity and coefficients В and ν in theburning rate law for HEs and also a multifold increasein the burning surface area due to various destabilizingfactors, such as fluid flow dispersion and pre�front tur�bulence. The sharp pressure rise is due to the inabilityof the liquid quickly diverges before expanding gasbubbles;

(3) For the bulk combustion of the liquid undercertain conditions, the pressure rise in the shock wavemay turn out to be sufficient for the shock initiation ofnormal detonation in liquid explosives. Burning in athin layer leads to the propagation of wave distur�bances that could lead to the appearance of cavitationin low�viscosity liquid explosives. The interaction ofhigh pressure in the combustion zone with a cavitatingfluid is the major mechanism of initiation of low�speed detonation in a layer of a burning explosive.

REFERENCES

1. F. R. Bowden and A. D. Yoffe, Initiation and Growth ofExplosion in Liquids and Solids (Cambridge Univ.,Cambridge, 1952; Inostrannaya Literatura, Moscow,1955).

2. A. V. Dubovik and V. K. Bobolev, Sensitivity of LiquidExplosive Systems to Shock (Nauka, Moscow, 1978) [inRussian].

3. L. I. Sedov, Mechanics of Continuous Media (Nauka,Moscow, 1994), Vol. 2 [in Russian].

4. A. D. Margolin, Fiz. Goreniya Vzryva 15 (3), 72 (1979).5. A. N. Kiryushkin and Yu. A. Gostintsev, in Combustion

of Condensed Systems, Proc. 8th All�Union Symp. onCombustion and Explosion (OIKhF AN SSSR, Cher�nogolovka, 1986), p. 49 [in Russian].

6. A. A. Il’yushin, Mechanics of Continuous Media (Mosk.Gos. Univ., Moscow, 1978) [in Russian].

7. R. T. Knapp, J. W. Daily, and F. G. Hammit, Cavitation(McGraw�Hill, New york, 1970; Mir, Moscow, 1974).

8. K. K. Andreev, Thermal Decomposition and Combustionof Explosives (Oborongiz, Moscow, 1956) [in Russian].

21 t, ms

0.8

0.4

01

2

3

4

Fig. 6. Oscillations of a burning cavity in NG: (1) pressureР/Р0, (2) temperature T/T0, (3) flame speed J/V0, and(4) cavity boundary a/a0.

development of an explosion from an initiation source in a liquid high explosive

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