doc fuel air bomb droplet dispersion.pdf

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Defence Sc ~en ceournal, VolS1,No 3, July 2001,pp. 303-314O 2001, DESIDO C

Extended N ear-Field M odelling and Droplet S ize Distribution forFuel-Air Explosive WarheadS.K. Singh and V.P. Singh

Cen tre for A eronautical Systems Studies & Analyses, Bangalore - 560 07 5ABSTRACT

A theo retical model is dev eloped for the prediction of the mean-mass diameter of droplets produced by thefragmentation of liquid fuel sheet and film in a fuel-air explosive (FA E) device after the detonation of the centralburster charge. This model does not contain arbitrarily assumed values for the instabilities as in presentlyavailable mod els. Also, adisqib ution model for the initial distribution of the droplet diameter, which depends onthe design parameters of the FAE d ev ice , is presented.Key wo rds: Fuel-air explosive warhead, mean-mass diameter, dispersal modelling, fuel ejection mode

1. INTRODUCTIONThe fuel-air explosive (FAE) is a relatively

new warhead system which has been recognised aspossessing much h~gher amage potential than theconventional TNT-based warheads for soft targetsspread over large areas. In some quarters(especially in erstwhile USSR) it has beendesignated as a weapon of mass destruction. It maywell be one of the more widely used future weaponsystems.

The FAE mechanism is not new and is oftenencountered in everyday life. The combustion inthe cylinder of an auto-engine is an FAE event andso is the explosion of coal dust in a mine.Historically, its use as a warhead goes back to worldwar 11, when the Germans used a form of FAEagainst the Russians in the siege of Sebastapol.Ini t~ally, hey used arrays of 28 cm and 32 cmbarrage rockets to disperse fuels like gasolene,kerosene and naphtha, followed by 28 cm high

explosive rockets to detonate the fuel-air. TheSoviets noted widespread personnel casualties dueto blast'.

Though the functioning of an FAE device doesnot involve very advanced physics, several phasesof its functioning are still not well understood. Inparticular, the formation of the fuel-air cloud by thebreaking of liquid fuel mas to droplets and thepsubsequent formation of k detonable fuel-airmixture is one such area. Attempts have been madeto model the functioning'.2 of FAE, but a number ofgrey areas still exist. One of the most prominenttrouble spots is a reliable estimation of the dropletsize:

In this paper, a theoretical model for theestimation of mean-mass diameter (MMD) of thedroplets rather than that of a single droplet has beenpresented. Also, a model for an initial distributionof the droplet diameter based on log normaldistribution3 is preselited.

Revised 02 March 2001


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DEF SC1 I, VOL 51, NO 3, JULY 2001

2. FUEL-AIR EXPLOSIVE DEVICEA typical FAE device consists of two coaxial

metallic cylinders. The inner cylinder with muchsmaller radius compared to the outer one contains aburster charge of low velocity of detonation. Theouter cylinder (also called the canister) is providedwith longitudinal serrations to facilitate its openingin a petal-like manner4. A liquid fuel is placed in thespace between the two cylinders. After thedetonation of the central burster charge, a strongshock wave propagates through the liquid fuel. Thecanister is broken when the shock wave impingesupon it after propagating through the fuel. Theliquid fuel is propelled outwards, first as liquidsheets through the cracked serrations and later as acoherent liquid shell when the canister is totallyfragmented. The liquid shell moves outwardsradially, with its thickness decreasing continuouslytill it is reduced to a very thin liquid film whichultimately breaks into a field of droplets. Thesedroplets continue to move outwards, but due totheir size (which is of the order of microns) the airdrag comes into play and they stop after travellingsome distance. The turbulent mixing of thedroplets and the air produces a fuel-air cloud. Whendetonated after a pre-fixed delay, it gives rise to astrong blast wave which though of a relatively lowpeak overpressure has a much longer timeduration (hence higher impulse) and much lessspatial attenuation than TNT warheads. This factmakes it an effective. weapon system against softtargets spread over large areas, such as troopconcentration, mines, parked aircraft, etc.

The formation of a fuel-air cloud following thedetonation of the central burster charge involvesmany complex processes. The dispersal process hasbeen divided into three main regimes: (i) Ejectionregime, in which the explosive forces of the burstercharge predominate and accelerate the fuel mass,(ii) transition regime, in which the explosive forcesare comparable to the fuel-air interaction forceswhich tend to decelerate the fuel mass, and(iii) expansion regime, in which the aerodynamicforces dominate the explosive forces, the fuelconcentration in the cloud becomes low and the

shock waves decay into insignificance. The firstwo regimes together are termed as the near-fieldregime, whereas the third one is termed as far-fieldregifiez' '.3. EXTENDED NEAR-FIELD MODELLING

The initiation of the central burster chargetakes place at one end and the detonation waveproceeds along its length. A strong shock wavepropagates successively through the inner cylinderand the liquid fuel. This shock wave, on reachingthe fuel-canister interface, generates extremelylarge stresses, which cause the canister to openlongitudinally into a number of strips. A detailedmathematical modkl for the propagation of shockwave through various material media in an FAEdevice is available. There are iwo distinctlyrecognisable modes of fuel ejection when thecanister opens up. These are (i) sheet mode, and (ii)shell mode.3.1 Sheet Mode

This mode is applicable when the serrations areopening up but canister has not yet fullydisintegrated. This mode lasts for a very small time.However, a significant mass of the liquid fuel isejected from the longitudinal cracks along theserrations. The liquid rushes out in a nearly sheetform. As the sheet travels into air, instabilities aregenerated on its surface, which grow with time andlead to the n of the sheet into

break into droplets.3.2 Shell Mode

Once the cracks propagating along thelongitudinal serrations reach the other end of thecanister, it is almost entirely fragmented and theleftover liquid is liberated as a radially expandingmass. Due to high initial velocity, this liquid masscontinues to move radially outwards. Its thicknessdecreases continuously as it expands until it isreduced to a thin film. Due to the liquid film-airinteraction, instabilities or perturbations grow onthe film surface, which ultimately lead to itsdisintegration into droplets.


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SINGH & SINGH: DROPLET SIZE DISTRIBUTI( N FOR FUEL-AIR EXPLOSIVE WARHEAD

4. DISPERSION RELATIONS4.1 Sheet Disintegration

The sheet disintegration has been extensivelystudied during the last half century. The firstformulation for the disintegration of a viscous sheetwas given by Dombrowsky and ~ o h n s ~ .s pointedout by Li and an kin', this formulation is valid onlyfor large gas Weber number. Li and Tankin andMitra and ~ i *ave presented a more systematictheory for the growth of instabilities, leading to itsdisintegration into droplets. However, since theyassumed only Rayleigh mode of breakup for theligaments into droplets, their-theory resulted intomuch bigger droplet sizes. Lin and an^^ pointedout that at higher sheet velocities, an atomisationmode of disintegration also exists for the jets andligaments which gives much smaller droplets asobserved in many physical situations. Since in atypical FAE device, the velocity of the liquid fuelsheets issuing from the cracks along the pre-formedserrations is quite high, it is appropriate to use theatomisation mode of ligament disintegration. It isalso w'orth noting that Rayleigh mode is included asa sub-case in the atomisation mode.

Li and Tankin's formulation has been brieflypresented for the growth of instabilities in a planesheet along with atomisation mode of ligamentdisintegration due to Lin and Kang. Though in anFAE device, the width of the crack, and hence thethickness of the liquid sheet changes with time, inthe proposed model, thickness of the liquid sheetwas taken to be approximately constant. Thissimplifying assumption would be valid during theinitial opening of the canister. In any case when thecrack widens substantially, a switchover was madeto the shell mode.

The motion of a 2-D liquid sheet wasconsidered with density p , , viscosity P I , surfacetension 0 nd thickness 2a ,moving with a velocityUo hrough air which was assumed to be an inviscidfluid of density p The sheet was moving in theradial direction, denoted by r. The direction alongthe crack width was denoted by y, with origin at themid-plane of the sheet. Let a small perturbation inthe sheet surface be given by

5= 5 , exp(wt+ ikr) (1)Then the two interfaces are given by

y =+a +5 , where y =2 a is the equilibriumposition of the interfaces, 50is the initial amplitudeof the disturbance ( 5 , a


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DEF SCI J , VOL 51 , NO 3, N L Y 2001suggested separation of the liquid velocity intoinviscid and viscous parts as

where ul, l are the inviscid irrotational solutions inthe liquid, and the leftover part u~,vz ontains theeffect of liquid viscosity. For inviscid irrotationalliquid flow, there exists a velocity potential (@),such that u,= a@ ar , v , = a# lay , and $ satisfies:

The pre;ure is obtained by integrating theinviscid p a r d Eqn (3) wrt r as

The equations for the viscous parts reduce to

Next, Li and Tankin introduced a streamfunction like variable v , defined as u2 &&/.ay,v2= - qlar .

From Eqn (1 I), one gets:

Considering the disturbances given in Eqn (I) ,@ and * are assumed to be of the following form:

# = @(y)exp(ot + ikr)11, = Y(y)exp(@t+ikr) (13)Substituting Eqn (13) into Eqn (8) and Eqn (12)

respectively, one gets:

wheresz= k ' + ( o + i k U o ) l v I

Solving Eqn (14), one gets:Q = (C, e" + C2e-b)'I = (C3e8+ C , e - * ) 1 (From the boundary conditions [Eqns (5) a

( 6 ) ] , he unknown constants are obtained as

Hence, the normal stress in the liquid fueobtained as

= [ { p i ~ + i k ~ ~ ) + 2 , u , k ~ ) ( e "e-" )c,- i , ( e S . ' e-")C,exp(ot+ikr)]The motion of the air, assumed as an invis

and incompressible gas, is given by the followequations:

where the subscript g denotes the quantities forgas medium. The boundary conditions require tacross the liquid-gas interface the normal velocbe continuous, and far away from the liquid shthe effects of the disturbances die out.

For the inviscid gas, the velocity can expressed in terms of a velocity potential, whichaccordance with Eqn ( I ) is assumed to be of following form:

@, = Q,(y)exp(ot+ikr) (


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SlNGH & SMGH: DROPLET SIZE DISTRIBUTION FOR FUEL-AIR EXPLOSIVE WARHEAD

Using the boundary conditions [Eqns (20) and(21)], the velocity potential is given by

o#J s = - - exp[k(a- lyl)] t ,exp(ot + ikr)for y l 2 a (23 )Hence, the normal stress in the gas medium is

given bya#JsZs.rr = -p* = P s at

4.1.1 Pressure Induced by Surface TensionThe pressure induced by the surface tension is

given by

where R is the radius of curvature of the interface.Using Eqn (I ), p , is given by

po = -uk2t,exp(wt + ikr) ( 2 6 )The boundary condition [Eqn (6)] (viz., the

continuity of normal stresses across the interface),on substitution from Eqns (17), (24) and (26) aty = a gives the following relationship between thecomplex growth rate (a) nd the disturbance wavenumber (k) (the dispersion relation) as

By putting p, = 0 in Eqn (27), the invisciddispersion relationship was obtained.4.2 Cylindrical Liquid Film Disintegration

Unlike the liquid film disintegration, not muchliterature is available for the disintegration of acylindrical liquid film. ~ a r d n e r ~roposed ananalytical model for the growth of instabilities inthe liquid film, leading to its breakup into dropletsand also attempted to estimate the diameter of asingle drop. Though the physical phenomenon wasreasonably modelled, the initial values forinstabilities were chosen rather arbitrarily which

introduced uncertainties in the estimation of thedroplet size.

In this paper, an attempt has been made toincorporate the concept of dispersion relation by analternative formulation for the growth ofinstabilities in the cylindrical film which enablesone to get rid of arbitrary assumed initial values forthe instabilities. Further, it also enables one to .establish a relationship with the well-studied sheetcase. Li and an kin' and Mitra and ~ i ' , et a1introduced the viscosity via a specially definedstream function approach. Since stream functionsare not available for a truly 3-D flow, the study wasrestricted presently to inviscid case only. Based ona comparison of results in the plane sheet case, itwas found that the maximum error in the dropletsize does not exceed 10 per cent when the viscousformulation was replaced by an inviscid one.

A cylinder of fluid, denoted fluid 1, of rqdius ri( r ) and of height-z, was considered. This wassurrounded by an annulus of fluid, denoted as fluid2 , of outer radius r,(t) and also of height z,.Surrounding this annulus of fluid 2 was an infinitefluid medium, denoted as fluid 3. The interfacebetween the fluids 1 and 2 was denoted interface 1,and that between the fluids 2 and 3, interface 2.With reference to an FAE device, fluid 1represented the detonation product gases producedby the detonation of the burster charge. Fluid 2revresented the liauid fuel to be dispersed andfluid 3 represented the ambie atmosphere. Thus,the liquid shell consisting f l t 2 expands radiallyafter the detonation of burster charge till this shellwas reduced to a very thin liquid film. Instabilitiesin waveform propagated in the liquid film whoseamplitude grew till it was broken into ligaments,which were subsequently broken into droplets.

Using the usual cylindrical coordinate system(r, 0, z), the basic radial motion of the liquid shellperturbs by the small amount as

r, = 5 + 5 ,exp[ot + i(kz+ no]r, = r, + 6, exp[wt + i(kz + no] (28 )

where k is the axial wave number, n 1s an integerand as in the plane sheet case, o = o, + io, is acomplex variable with its real part a,.s the rate of


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DEF SCI J, VOL 51, NO , JULY 2001growth of the disturbance and the imaginary partm i as 2n times the frequency.

The situation was considered when the filmhad become very thin, i.e. when ro - i was a smallquantity and it was assumed that at this moment51 = 5 2 = 50 (say). The preceding motion wasassumed to be governed by Gardner's model2. The

' radial velocity Uo = r, is the basic velocitycomponent and let v,, ve and v, be the perturbedvelocity components along r, 0 and z-directions,respectively. The linearised equations of motion forthe inviscid fluid are:

3 u avo 1 l a pat O Jr P I r Je (30)

where pl is the density of the liquid and p is theperturbation pressure. Since the flow is inviscid andirrotational, there exists a velocity potential q5, suchthat

J4' J4 84'v = - v =-, v- = -d r ' ae - a~ (32)and 4' satisfies the following Laplace's equation:

In view of the disturbances given by Eqn (28),Q#J may be assumed in the following form:

Using Eqn (34), Eqn (33) reduces to:r 2 @ " + r @ ' - ( n 2 + k 2 r 2 ) @ = (35)

which is modified Bessel's equation with generalsolution cD = ClI,,(kr) + CzK,(kr), where I,, and K,are modified Bessel 's functions of the first and thesecond kind, respectively. For the fluid 1, whichincludes the origin, C2 = 0 (since K,,(r) + m asr + 0). Also, for the unperturbed or the basic radial

motion, the velocity potential is @," = ri4 lThe total velocity potential as the sumunperturbed and perturbed parts is given by

XI,kr) exp[ot+ i(kz+ no)]Similarly for the fluid 3 (i.e. the unboun

air), the velocity potential is given by

The velocity potential for the fluid 2 (i.eliquid fuel) is given byq52 = ioroogr

[K;(bo )In kr) -C(kr, )Kn hk[I;(kri)K:(krO)-I:(~,)

Next, from Eqn (29), the pressure in fluidgiven by

Using Eqn ( 3 8 c ~ r thin film, Eqn reduces to

Since the velocity is continuous at liquid-gas interface, one has at r = ro , Uzns= U=to . he pressure is given by p , = - p, (a$, t

Using Eqn (37), one gets:


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S M G H & SINGH: DROPLET SIZE DISTRIBUTION FOR FUE LA IR EXPLOSIVE WARHEAD

X exp[of+(kz+nO)]1 (41)The radial stresses in the liquid and the gas

medium are: r , - and rZ,,, - p , ,respectively. Since the normal or the radial stressesare continuoushas:

where a is the surface tension and R is the radius ofcurvature which for the thin film is given by

Usmg Eqns (40) and (41) in Eqn (42), one gets:

To the zeroth-order of 5 0 , one obtains thefollowing expression from Eqn (44):

which is satisfied by the unperturbed radial motionof the liquid film. To the first-order of 5 0 , oneobtains the following expression from Eqn (44):

which is the dispersion relation for the cylindricalfilm.5. ESTIMATION OF MEAN-MASS

DIAMETER5.1 Dispersion Relations

The dispersion relations [Eqn (27) or Eqn (46)]play a very important role in the disintegrationprocess of a plane sheet or a thin cylindrical film,respectively as these provide information about themost dominant wave number, which is likely to getamplified, leading to the breakup of the sheet or thefilm.5.1.1 Sheet Case

The wave number k for a typical FAE data is ofthe order of 10' or more, hence tanh (ka)-t 1, tanh(sa)+l; and Eqn (27) reduces to the followingexpression:

where only the real part of Eqn (47) wasconsidered and using w, = (1+ ij)-' (- kU , ? ikm),(see Appendix), where m2 = ijU{ - k(I+ ij)lp,,

P,i j=- ,one gets after some simplifications:PI

uk+-=oP I

where


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DEF SCI J, VOL 51, NO 3, JULY 2001

For the inviscid case (i.e. vl = O), the Eqn (48)assumes a much simpler form in which w, can beexplicitly expressed as

It is obvious from Eq$ (49) that w , = 0 only fork = 0 and k= U~i j l (p; ' ul ( l+ j i ) )= k,,,,,(say). Themaximum value of w , is given by

, which gives

The most unstable wave8 or the wave ofmaximum growth6 occurs at the maximum value ofw, for both viscous and inviscid cases and is givenby the relation [Eqn (SO)], from which thecorresponding wavelength hl is obtained using therelation hl=2nlk,,,. In the case of a plane sheet,waves grow on the sheet until they reach the criticalamplitude. Tears occur in the crests and troughsand fragments of the sheet corresponding toone-half wavelength are broken off6.'. Thefragments contract by surface tension into unstableligaments whose diameter, d~ is given by

5.1.2 Ligament Breaking ModesThe ligaments produced thus further break

down into individual droplets. Mitra and ~ i 'sedthe classical axisymmetric breaking mode due to~ a ~ l e i ~ h ' ~ . ~ ~hich resulted in much bigger dropletdiameters. Lin and Kang on the other hand showedthat Rayleigh breaking mode was valid only atlower speeds, whereas at higher speeds atomisationmode woul'd occur, resulting in much small&droplet diameters.

(a) Rayleigh 's Axisymmetric Breaking ModeThe well-known result for a jet or a ligament

predicts its breaking by amplification of wave typeinstabilities, whose wavelength is given by

where dL is the diameter of the jet or the ligament.(b) Atornisation Mode

Lin and Kang have derived the dispersionrelation in non-dimensional form as

5'-k2 I, k)- we- 'k(1-k2)x-- 05'+k2 Io(k) (54)where c2 = k2 + Re (w- ik), Re being the Reynoldsnumber -Uo dL (2vI), We = 2 c / p l ~ ~ 2 d ~s the Webernumber. It is interesting to note that for the inviscidcas e and for 3-0, Eqn (54) reduces to theRayleigh's result for the breakup of low velocityligament or jet. Further for k+ m, its dimensionalform reduces to Eqn (47), which leads to A, = A , .5.2 Estimation of Mean-Mass DiameterWhen the wavelength hz of the instabilitiesdeveloped on the ligament is known, the mean-massdiameter (MMD), D3a estimated from theconservation of mass as

5.2.1 Cylindrical Film CaseThe dispersion relation for this case is glven by

Eqn (46). Considering only its real part as in thesheet case, the relation between w, and k is obtainedas

where


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SINGH & SINGH,: DROPLET SIZE D ISTRlBUl "ION FOR FUEL-AIR EXPLOSIVE W ARHEAD

Numerical evaluation of the second term inEqn (56) using asymptotic expansion for themodified Bessel's function and its derivative for atypical case showed that it was negligibly smallcompared to the first term. The unstable wave hasbeen given by Eqns (50) and (51). The rest of thecomputation followed the plane sheet case.6. DROPLET DIAMETER DISTRIBUTION

The dispersion relations [Eqns (27) and (46)Jmerely indicate the most dominant wave modes areused to obtain the MMD. In this, it is obvious thatother wave modes are also present and hencedroplets of other diameters would also be present inthe fuel-air cloud. Thus, this part requires astochastic sub-model. ln3, a lognormal distributionfor droplet diameter distribution was presented;however the parameters of the distribution couldnot be directly related to the parameters of othermodules of the mathematical model. It is worthnoting that the estimation of MMD from thedispersion relation enables to establish a directrelationship among the droplet diameterdistribution model, shock model and the extendednear-field model presented here.The MMD is taken as the mean of thelognormal distribution. Its upper and lower limitscome from physical considerations. The upper limit(dl) is taken approximately as the diameter of thedroplet which would travel to the maximum extentof the cloud, taken as 25-30 times of the initialcanister diameter. The lower limit (d,) is assumed tobe the minimum droplet diameter below which theliquid droplets become unstable (i.e. evaporateimmediately). The lognormal distribution is givenby

where, p= In D30, o = (In df - n d,)/6, x = In d, d isthe droplet diameter.

7. RESULTS & DISCUSSIONFollowing Gardner's model, an FAE device is

considered with the following data for the liquidfuel: Density of the liquid, p, = 899 kg/m3, surfacetension, o = 2.42 ~ 1 0 . ~/m, dynamic viscosity,p = 3.1 x kglm-s, air density, p, =1.2 kg/m3.The particle velocity in liquid fuel is taken from theshock model, a typical value for (I0 is 140 mls.Since the sheet thickness 2a is not constant in thepresent case, it varied over a range 1 mm to 10 mm .

Figure 1 shows the disturbance growth factor(w,) against the wave number (k) for both theviscous and inviscid cases. The inviscid solutionfollows from Eqn (53), whereas for the viscoussolution, Eqn (52) was numerically solved. Theinviscid solution gives (k,,,)i.,i, = 6.4707 x lo5,whereas the viscous solution yields (k,,,).i,, =5.70 x loS.

Table 1 shows the inviscid and viscous MMDof the droplets for both Rayleigh and atomisationmode of disintegration for various values of thesheet thickness. It is observed that the MMD fromthe viscous cases result are nearly 8 per cent largerthan the corresponding inviscid cases. Further, it isobserved that the MMDs of the droplets given byRayleigh mode are many times larger than thosegiven by the atomisation mode. It may be noted thatdroplets with diameters larger than 100 pm cannotstay floating in air and will fall down under the

'-1

N U W E R 1k1DUSANOS Ifactor m, versus wave

200 I 00WAVEITHIFigure 1. Amplification growthnumber k.

31 1


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DE F SCI J, VOL 51, NO 3, JULY 2001Table I. Inviscid and viscous mean-mass diameters for bothRayleigh and atomisation modesSheet R R (D.410m)inr (D~r ,) ~ i~thickness(mm) Olm) Olm) lum) lum)

I 148.59071 158.31688 35.57430 38.712432 210.13899 223.89388 44.82081 48.77460

effect of gravity'3. Thus, e v a the formation ofclouds would not be possible for such largedroplets. On the other hand, the MMDs of dropletsgiven by the atomisation mode are more realistic;the droplets with these diameters would result in astable aerosol cloud which may be subsequentlydetonated by a secondary mechanism.

oo'll

0 SO 100 150 2W 250 300 350 bWDROPLET DIAMETER ( ~ m )

Figure 2. Distribution of droplet diameters

In Fig. 2, the lognormal distribution of thedrople t d iameters f or a typical MMD(corresponding to a sheet thickness of 6 mm inatomisation mode) is shown, with the MMD as themean of the distribution.

REFERENCES1. Kennedy, D.R. Warheads: A historicaperspective. In Tactical missile warheads, editeby Joseph Carlene. Progress in Astronautics anAeronautics, Vol. 155,AIAA Inc., USA, 1993.2. Gardner, D. R. Near-field dispersal modelling foliquid fuel-air explosives. Sandia Report NoSAND-90-0686, 1990.3. Singh, V.P. & Singh, S.K. Fuel-air explosives: review of mathematical models. Proceedings othe Symposium on Systems Analysis for DefencCASSA, Bangalore, 25-26 September 199pp. 247- 54.4. Sihot a, B.S. Fuel-air explosive - higperformance force multiplier armament systefor soft targets. Paper presented in Nation

Workshop on Force Multiplier ArmamenSystem, 1996.5. Glass, M.W. Far-field dispersal modelling ffuel-air explosive devices. Sandia Report NSAND-90-0528, 1990.6. Dombrowski, N. &Johns, W.R. The aerodynaminstability and disintegration of viscous liqusheets. Chemical Engg. Sc., 1963, 18,203-147. Li, X. & Tankin, R.S. On the temporal instabiliof a two-dimensional viscous liquid sheet,Fluid Mech., 1991, 226,425- 43.8. Mitra, S.K.& Li, X. Initial distribution of droplsizes in sprays. Proceedings of the 2"International HEMCE, IIT Madras, ChennaDecember 8-10, 1 9 e p p . 535-41.

\9 . Lin, S.P.& Kang, D... Atomisation of a liquid jePhysics of Fluids, 1987,30 (7), 2000-06.10. Drazin, P.G. & Reid, W.H. Hydrodynamstability. Cambridge University Press, 1981.11. Milne-Thomson, L.M. Theoretical hydrodynamics, Ed. 4. Macmillan, London, 1962.12. Strutt, J. W. (Lord Rayleigh) The theory sound, Vol 11. Dover Publications, 1945 .13. Reist, P.C. Aerosol science and technology, Ed.McGraw-Hill Inc., 1993.


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SINGH & SINGH: DROPLET SIZE DISTRIBUTION FOR FUEL-AIR EXPLOSIVE WARHEADAPPENDIX

Capillary Wave at the Interface of Liquid and Gas

The equation for the wave propagation at the m2= pu,2- ok(l+ 3 )liquid-gas interface in the absence of gravity is": Pi@, (U, - c ) ~ oth(ka)+ @sc2coth(ka')= o k y ~ l ) then from Eqn (A3), one gets:where c is the wave propagation velocity, a, a' are c= (1+p)-' (U, + iin)the thicknesses of the liquid and gas layers, (A4)respectively. In the present case, a'+ m (as the air Since c= - wi 1k, one can obtain from Eqn (A4)extends up to infinite extent), hence coth (ka') + 1, ' kU, ikmalso as noted earlier, the wave number k has a large mi=--+- + p - l + p (AS)value in the present case, hence coth(ka) + 1 andEqn (A l) reduces to Since w= w, +iw,, hence the real part of w is:

ok kin( U , - C ) ~ + ~ C ~ = - Rew=w, ?--Pi (A21 I + p (A6)which gives It may be noted that instability occurs (i.e. w,> 0)only whenc = (l+p)- ' U, + u: (1+p) u; --[ { [ ~ ) r ]A3) pu; >( l +p ) - kPi (A71If pU: > ok (l +P )l pi , then the wavevelocity c and r o when pil; a (l+P)uklpi ,becomes imaginary. Let

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