Lei Li ,Xio-xi Lu ,Xio-bin Ren ,Ye-jun Ren ,Shou-tin Zho ,Xio-fng Yn

a State Key Laboratory of NBC Protection for Civilian,Beijing,102205,China

b School of Aerospace Engineering,Tsinghua University,Beijing,100084,China

Keywords: Shell effects.Re flected rarefaction waves Liquid spall fracture Liquid dispersing Shock waves Cavitation layered

ABSTRACT A systematic investigation on the mechanism of dynamic liquid dispersing process via theoretical and experimental approach is presented.The experiments include weak and strong constrained scenarios using the high-speed camera technique and the flash X-ray radiography technique.Based on dynamic analysis,one-dimensional characteristics analysis and some numerical simulations on the propagating processes of blast waves before the container shell rupturing,further and detailed analyses of the experimental results are presented.The effects of the liquid viscosity on the dynamic dispersing flow are also analyzed,and the spall fracture mechanism is explored.Thus,the dominating forces determining the dispersing liquid flow are revealed,that is,the stretching and shearing action due to the interaction of two re flecting rarefaction waves in opposite propagating directions.The in fluence of container shell strength on the dispersing liquid flow is also investigated,and the characters of cavitation layered in liquid before shell rupturing are uncovered.Results revealed that different shell material results in different cavitating layers.Then the different cavitating layers drive the different dynamic liquid dispersing process coming into being.The metastable liquid states caused by pressure drop and cavitation generation are discussed.

1.Introduction

The transient process of liquid atomizing dispersion driven by dynamical shock loading is a complicated phenomenon including not only macroscopic mechanical processes,but also microscopic physicochemical actions as well as mesoscopic effects[1].The multiphase flow dynamic instability has also been cited as the mechanism for the evolution of the turbulent structure of supernova remnants and explosive volcanic rapid release flow with gases and particles[2].In the area of environmental control,the application of the dispersing system includes the fire extinguisher or the suppression of fire in personnel inaccessible area,the disinfection or decontaminating equipment used in toxic region during disaster occurrence.Other applications include the rapid releasing of agricultural chemicals in farmland,as well as the dispersal of liquid fuel to creating a detonable cloud.In these cases,the smaller the sizes of the atomizing droplets are,the more effective such a dispersing system is.To generate an ultra fine aerosol cloud(consisting of submicron and nanosized droplets)rapidly,the typical dispersing system is a cylindrical container filled with liquid substances and a central dynamic shock loading source structure which is showing in Fig.1 as a simpli fied physical model.In this model,the cylinder or canister has a top and a bottom end con fined plate and cylindrical surrounding solid con fined shell.There is an explosive charge located along the central axis of cylinder as dynamic shock loading source,which is detonated by an detonator.

Before 1959,there were only engineering experience summaries and researches about the liquid explosive dispersion.However,in order to control the ef ficiency of vaporization and the generation ef ficiency of aerosol for low volatile liquid,experimental works and theoretic analysis on the mechanism of explosive dissemination of liquid have been carried out since then[3].During the following decade,a series of systematic investigations were conducted,and their research contents involved the equations of state of liquids and calculations of waste heat,the propagation of blast waves,the stability of detonation products-liquid interface,the explosive flash’s in fluence on liquid aerosols,the cavitation on liquid-air interface and spreading into the inside of liquid fill,as well as the mixing dispersing flow of liquid particles and liquid vapor etc.[4-9].These research results indicated that the process of liquid dispersion driven by dynamical shock loading is really complicated.

Fig.1.A simpli fied physical model of liquid dispersion driven by central dynamic shock loading,left is the side view,and right is the overhead view to a cross section of cylinder.

For this reason,in 1969,Zabelka and Smith proposed a hypothesis that during the process of explosive dispersion of liquid,the actions imparted to liquid from shock wave and the mass and heat transfer during dispersion are negligible,while the only work done on the liquid volume in the dispersion process is the expansion work by detonation products(named as “fluid piston” )[10].Based on Zabelka and Smith’s work,Gardner carried out numerical simulations on liquid dispersion process,proposed his dispersal model in which the process of liquid volume dispersal is dominated by expanding power of detonation products,and deduced an analytical solution on the development of the liquid interface which becomes unstable and breaks up into droplets[11].Gardner’s model was called “l(fā)iquid shell instability” model driven by “fluid piston” ,and was supported by Samirant’s experimental results through flash-X radiograph technology in the meantime[12](See Fig.2).

Fig.2.The liquid shell instability model proposed by Gardner and the flash-X radiograph of liquid shell from the experiment by Samirant.

During the next three decades since then,from a macro perspective and some engineering application views,the liquid shell instability model was used in expounding the mechanism of liquid’s primary breakup and in the numerical simulation of the whole liquid’s atomizing dispersion driven by central explosion[13-18]until Li’s research group explored the heterogeneous of dispersing flow composed of a mixture of liquid droplets and vapor according to their experimental observation in a controllable explosion chamber[1,19-22].The results of Li’s research group manifested that the re flected rarefaction wave from liquid outer interface play an important role in fragmentating liquid and dispersing it outward to ambient environment(See Fig.4.).

Associated with above,Stebnovskii studied the liquid’s dispersing flow driven by explosion of a central electrical detonator[23].Based on his experiments,Stebnovskii indicated that only when the magnitude of the speci fic energy released by explosion do not exceeds a threshold,the liquid dispersing flow can be regarded as a hydro-mechanical instability(of the Rayleigh-Taylor type)[24-27].Otherwise,the cavitating flow of liquid will be irreversible to loss the continuity of liquid volume,and develop into block cavitation bubbles or cavitating spalls,so the liquid dispersing flow should be regarded as cavitation disintegration[28].Taking the observations of cavitating flow to be a viscous elastic form character and to have a relaxation time,Stebnovskii proposed a rheological model to approximate the liquid dispersing flow[27-29].Kedrinskii et al.developed a two-phase model-IKW model (Iordanskii-Kogarko-van Wijngaarden, seen in Ref.[30-32])to demonstrate the characteristics of cavitating dispersing flow.Considering the cavitation bubbles behind the rarefaction wave in liquid fill,Vorozhtsov et al.proposed a 1D model of explosive generation of micro atomized aerosols to estimate the size of cavitation bubbles stretched by re flecting rarefaction wave,and to calculate the size of micro atomized aerosol particles by the breakdown of cavitation bubbles[33,34].

Recently,Lu presented their new results on the in fluence of cylindrical shell on to the liquid dispersing flow and some new flash-X radiograph measurement results[35,36].Frost made a systematic review of the problem of multiphase medium dispersion based on many recently investigations[2].In his article,he concluded three classes mechanisms of which the first is shockdriven multiphase instability,such as the Rayleigh-Taylor and Richtmyer-Meshkov instabilities for single-phase flows[37,38];the second is non-uniformities flow induced by the interactions between strong shock and the material(liquid,or particles,or mixture of liquid with particles),such as the dispersal of spallation layer,the fragmentation of compacted material[39-42];the third is explosively dispersion of the light reactive particles embedded within a liquid or solid explosive[43,44].Moreover,some researcher addressed their attention on the interface instability induced by the pressure difference[45]and rarefaction waves[46,47].

Up to the present,the multiscale mechanism of liquid fragmentation and dispersion is still unclear.In Gardner’s model[11,14],the dispersing liquid was considered a continuum which follows the interface instability of “l(fā)iquid shell” breaking into droplets.Later,Ding et al.developed the model,but it is still an approximated model for numerical calculation[15-17].Based on the work of Stebnovskii,Kedrinskii et al.developed the mathematical model of two-phase flow of bubbly liquid[30-32].The modi fied IKWmodel is expressed as follow.

Fig.4.The liquid dispersing flow characters are shown when the explosive equivalents of the central detonating cord are 3 g/m(a),5 g/m(b)and 11 g/m(c),respectively.The recorded times are 2.33,4,16 and 30 ms in(a);2.4,4,18 and 30 ms in(b);1.6,4,5 and 6 ms in(c).They are all quoted from Refs.[1,21,22].

Fig.5.The liquid dispersing flow characters are shown when the central explosives have densities of 54.55 g/m(a)and central explosive charge is 29.3 g(b),respectively.The recorded times are 5 and 6 ms in(a)and(b).They are all quoted from Ref.[1].

Whereρ,p,Tis the average density,pressure and temprature of medium respectively,andvis the average velocity along thezdirection,Ris the radius of bubble,cis the sound speed in the liquid,mgis the mass of bubble gas,n=7.15,Mis molecular weight,μis the viscocity of liquid,psis the pressure in bubble,ρlis the density of liquid,the subscript 0 is the unperturbed situation.These equations are the mass and momentum conservation laws,equation of state in form of Taite taking into account the bubbles,equation of gas state,and Raleigh equation[32].In fact,Lu et al.[35,36]and Xue et al.[42,48]had shown that it is dif ficult to establish a mathematical model by simply dividing the stages of liquid dispersion after the interactions of shock compression and rarefaction tensile.As the same,the IKWmodel cannot be applied to the whole liquid dispersion process.Taking the liquid as the discrete medium,Vorozhtsov et al.presented some approximated models of liquid particles displacement,cavitation development and bubble breaking into microdroplets[33,34].Although the interactions between shock wave and liquid and the tensile roles of rarefaction wave are illuminated in some extent,the further elaborated investigations are needed.As shown by Xi[18],the inertial forces generated by the high-speed motion of the device only affect the motion of so-called “far-field” which means the stage the actions of blast waves vanish.

In this paper,we focus on the multiscale mechanism of shock waves’acting on liquid during the atomizing dispersal driven by central dynamic shock load,and only water and glycerol are considered.Only the stage in which shock waves play a major role in liquid’s motion is considered,so the inertial forces on the liquid are neglected in this paper.In section 2,results of various experiment related to liquid’s dispersing flow are overviewed.The variation characteristics of shock wave pressure in the liquid,the impact compression characteristics of liquids,and the expansion action of detonation products are analyzed in section 3.The onedimensional characteristic analysis,the spallation mechanism for viscous liquid dispersal,the in fluences of cylindrical shell on the formation of cavitation layers,and the nucleation mechanism for cavitation formation are demonstrated in section 4.

2.Overview of the experimental results on liquid dispersing flow

Taking the works in Reference[1,19-22,35,36]as a foundation of this paper,the following sections include our further analysis to the typical experimental results for further revealing the mechanism of dynamic liquid dispersion driven by shock loading.

2.1.Experiment apparatuses

There were four types of constrained containers of different cylindrical shell for high-speed camera record and one container with plastic shell for flash X-ray radiography measurement.All of them are shown in Fig.3.The layout of experiment is sited in an explosion chamber with safety protection.

Fig.3.The containers with thin film shell,Perspex shell,steel shell and PVC shell are shown in(a)-(c)and(e)respectively.The layout of experiment site using high-speed camera is indicated in(d).A container with plastic shell and top lid,the charge shell and RDX columns to be mounted in the shell are shown in(f).And last,the layout of experiment site using flash-X ray radiograph is also indicated in(g).

2.2.Experimental results of thin film case

Acontainer of thin film shell(the diameters are 135-150 mm)as a typical weak constraint to the action of dynamic shock loading,is used to explore some basic characters of liquid dispersal.Here several typical results are exhibited as follows.The dispersed medium is water.

Fig.6.The glycerol(the coef ficient of viscosity is 1.6Pa·s)dispersing flow characters are shown when the central explosives have densities of 5 g/m,the diameter of cylindrical film is 110 mm.The recorded times are 2.5 ms,4 ms and 5.5 ms respectively in(a)and 10 ms,15 ms and 20 ms respectively(b).They are all quoted from Refs.[22].

Fig.7.The dispersing f low characters with Perspex shell((a)-(d),the diameterφis 106 mm with the height h is 92 mm),PVC shell((e)-(h),the diameterφis 109 mm with the height h is 112 mm)and steel shell((i)-(k),the diameterφis 86 mm with the height h is 112 mm)are shown respectively.The central explosive RDX charge in all three cases is the same:9.4g.The recorded times are 0μs,100μs,200μs and 300μs respectively for Perspex shell,100μs,200μs,300μs and 500μs respectively for PVC shell and 100μs,250μs and 500μs respectively for steel shell.They are all quoted from Refs.[35].

Fig.8.The characters of expansion and rupture process of shell are shown.The diameter of container is 47 mm with the height 53.5 mm.The situation of full-filling liquid is shown in(a)and the situation of half-filling liquid is shown in(b).The central explosives charge in two case is the same:6.4g.The recorded times are 10 and 30μs respectively in(a)and 15,20,and 25μs in(b).They are all quoted from Ref.[36].

In Fig.4,the liquid is rushing outward from the cracks of cylindrical shell.The dispersing flows in(a)are jet like,and in(b)and(c) are pin-like spalls along radial direction.In terms of Stebnovskii’s energy thresholds theory,the energy imparting to liquid in jet-like flow is less than 1 J/g[24],in what condition that the cavitation induced by the rarefaction wave is reversible,and the liquid keeps continuous.In this condition the liquid interface instability is the main mechanism for liquid jetting dispersal(It may be a kind of rarefaction driven Rayleigh-Taylor instability[46,47]).In the situation of(b)and(c),the energy imparting to liquid in spalls shell flow is much larger than the energy threshold 1 J/g,and the tensile fracture by rarefaction waves is the main mechanism for liquid pin-like spalls dispersal,and then the spalls gradually disintegrate into droplets or aerosol state.

In Fig.5,firstly,it seems that the liquid has been boiled and vaporized,thus the dispersing flow of mixture of vapor and liquid droplets can be clearly observed.Obviously,the spall flow accompanied by phase transition happens because of the increase in explosion energy.In the same way,the mechanism of tensile fracture by rarefaction waves is reasonable to the mixture flow and the phase change vaporization of liquid.

2.3.Experimental results of viscous liquid dispersal

The in fluence of viscosity on the process of liquid dispersion by central explosion has been described in Ref.[22].Here only the glycerol dispersal experimental results as typical viscous cases are described as follows,in which the constraint shell is also a weak thin film.

In the former three pictures,the pin-like jet front and the annular jetting region are easily identi fied,but there is second annular jetting region appearing which is following the first annular region.From the latter three pictures,the double annular jetting region develops into a cumulative state and makes the front of viscous jetting flow becoming continuous liquid ligaments or sheets which will be undergo interface instability to break into drops.At the same time,the rear part of dual liquid annuli breaks into droplets then.

The tensile fracture by rarefaction waves is also the mechanism of viscous liquid dispersal driven by central explosion.Especially,it is the interaction of two rarefaction waves propagating in opposite directions that causes the spallation of viscous liquid which exhibits the double annular jetting region.

2.4.Experimental results of solid cylindrical shell case

The effects of strength of cylindrical shell constraints on liquid dispersing flow characteristics driven by central explosion have been analyzed in Ref.[35].Here is just another re fined statement.

The solid cylindrical shell is a strong constraint to the action of dynamic shock loading.Fig.7 shows that due to the increase of explosive charge and explosion energy,the moment of fragmented liquid rushed outward from container is much earlier,and moving faster and faster,thus the characters of gas-liquid mixtures become more and more obvious in which the volume fraction of the gas is increasing.In the situation of Perspex shell,the variation process of boundary between liquid inner interface and detonation products can be clearly observed,and the mixing of detonation product and liquid is not obvious and is delayed successively.However,in the situation of PVC and steel shell,the boundary between liquid inner interface and detonation products is hard to identify,and the mixing zone seems to be moving inward.

Fig.9.The manganin Gauge is inserted into an explosive dispersion device with PVC shell which the diameter of container is 110 mm with height 130 mm,and central RDX explosive charge is 50.24 g(a)and(b).The manganin piezo-resistance stress-meter and DPO4034B multichannel oscilloscope is shown in(c).The variation of pressure near explosive column versus time by numerical calculation using AutoDyn software in(d).The measured pressures from channel 1 and channel 2 are presented in(e).

Estimate on the interaction of blast waves shows that after 25-40μs from explosion igniting there appear negative pressure zone in fluid which persists about 5-10μs and whose strength far exceeds the dynamic tensile strength of water.This stretching could undoubtedly tear the liquid,so the tensile fracture by rarefaction waves is likely the decisive mechanism for the dispersion of gasliquid mixtures driven by central explosion.Moreover,it is the interaction of two rarefaction waves in opposite directions that leads to the change of the liquid and detonation products mixing zone.

Fig.10.The shock-Hugoniot curve of water(a),and the shock-Hugoniot curve of glycerol(b).

2.5.Flash-X radiograph results of the explosive experiments

The deformation and fracture of shell material during the early stage of liquid dispersal driven by central explosion can be studied by flash-X radiograph technology which has the advantages of short pulse time,strong penetration ability and the ability to see through the internal information of the object.

As can be seen from the picture,firstly,the internal detonation tube is shaped into an inverted cone to rupture under the action of detonation products,while the outer cylindrical shell expands into a drum under the action of shock wave and then gradually fragments.Secondly,before the cylindrical shell ruptures,the liquid density turns heterogeneous due to the action of the blast wave.Thirdly,the outer cylindrical shell of the upper part which is not filled with liquid first expands and breaks,and the outer cylindrical shell of the bottom part filled with liquid keeps unbroken till 25μs.In the picture at 25μs,the detonation product gas expands to form a gourd shape.These facts indicate that when the shock wave acts on the outer cylindrical shell through different media(air or liquid),the expansion and breaking mechanism of the out shell might be different which should be ascertained in the future study.In addition,no secondary shock waves were captured,but traces of re flected shock waves could be identi fied at the bottom of the cylinder.

3.Dynamic state of liquid under blast waves loading

In order to comprehensively understand the multi-scale mechanism of liquid dispersion driven by explosion,it is necessary to analyze the dynamic state of liquid under the action of blast waves.

Fig.11.The shock wave front and the detonation products boundary in water driven by explosive.The photos were recorded by a shadow photography technique with a high-speed rotating frame camera and are quoted from Ref.[49].

Fig.12.The t-r characteristics diagram of propagation of partial blast waves in detonation products gas,liquid and air.(a)is for weak constraint and(b)is for strong constraint.DJ-detonation wave,TW-Taylor rarefaction wave,RW(1)-re flected rarefaction wave in detonation products,SW-primary shock wave,TRW-transmitted rarefaction wave,RS-re flected shock wave of primary shock wave from the outer shell,RW(2)-re flected rarefaction wave in liquid after the outer shell expanding,TStransmitted shock wave in air,RW(3)-re flected rarefaction wave when re flected shock wave reaches the inner interface of liquid,IS-inner surface of liquid,OS-outer surface of liquid.

Fig.13.The propagations of waves in the experiment of flyer impacting glycerol by light gas-gun technology(a).The free surface velocity due to second shock wave of water and glycerol(b).They are quoted from Ref.[50,51].

3.1.The characteristics of shock wave pressure in liquid

Applying the technology of manganin piezo-resistance gauge,the manganin Gauge is immersed in the liquid contained in the explosive dispersion device in advance,the pressure of the shock wave in the liquid can be measured after initiation(Fig.9(a)-(c)).

Although according to Kamlet formula,the C-J pressure of RDX explosive with density 1.6 g/cm3is about 26.7 GPa,the pressure of the shock wave produced by the detonation product acting on the surrounding medium will decay rapidly.The numerical results show that the pressure of shock wave in water near explosive is less than 8 GPa(explosive without shell)and less than 4 GPa(explosive with shell)(See Fig.9(d)).In Fig.9(e),the measured pressure at the point which is 5 mm from the surface of the explosive column is 4 GPa consistent with the numerical results.In fact,as the primary shock wave propagates outward,the pressure drops gradually,and when it approaches the container outer shell,the pressure has dropped to several hundred MPa,then after it penetrates the outer shell,the pressure has dropped to tens of MPa.This may be the reason Zabelka and Smith think that the effect of shock waves can be neglected[10].

As mentioned above,during the whole process of liquid dispersal driven by central explosion,the liquid is only subjected to the medium and low intensity shock wave.Therefore,in order to understand the mechanism of liquid dispersion by explosion,the dynamic response of the liquid to medium and low shock pressures is very important.

3.2.The shock compression behavior of liquids

In order to understand the dynamic behavior of liquid,obtain the parameters in the equation of state under medium and low pressure shock,and explain how a phase transition occur in the liquid during the dispersion,on the basis of the experimental data by Los Alamos science laboratory[48],the shock-compression experiments for water and glycerol between 0.4 GPa-2 GPa were carried out specially.

Applying the light-gas gun technology,the shock-Hugoniot data were obtained for glycerol at pressure of 1.856 GPa and 1.734 GPa and for water at pressure of 0.951 GPa,0.865 GPa and 0.463 GPa which is shown in Fig.10.In this figure,the red boxes and stars are the measured data in these experiments,while the red circles are the measured data by LASL,and the blue lines are the fitting curve of shock-Hugoniot data for water and glycerol respectively.

It can be seen from the shock-Hugoniot curve in Fig.10 that under medium and low shock pressure,the shock wave cannot directly induce the phase transition of water and glycerol.Furthermore,the results of experiment of underwater explosion with a class of high explosives also revealed that water does not vaporize and the interface between detonation products and water is clear[49].

3.3.The expansion characteristics of detonation products in liquid

It needs to be clari fied that the effects of detonation products on the ambient liquid medium fall into two categories.One is that the rapid expansion of detonation product gas directly produces shock waves in the ambient liquid media,and the other is that the gas of detonation product expands to form bubbles to propel the ambient liquid media.According to the experiment of explosive driving water in Refs.[49],the propagation velocity of shock wave is much higher than the expansion velocity of detonation product itself(See Fig.11).Based on the pulsation theory of expanding gas bubble of underwater explosion,the maximum radiusRmand timetmof expanding bubble can be evaluating.Table 1 shows the evaluating results of the maximum expanding radiusRmand timetmof detonation products bubble for the experiments in section 2.

Table 1 The maximum expanding radius Rm and time tm of detonation products bubble.

Therefore,because the detonation product bubble expands slowly,and the liquid rushes out of the fragmented cylindrical shell quickly,the propelling action of expanding bubble can not become the driving force of liquid dispersion.Whereas,the blast waves formed by shock wave and the re flected rarefaction wave may contribute to liquid dispersion.

Fig.14.The development of cavitation bubbles in water at different moments after initiation which is quoted from Ref.[35].

4.The mechanism for liquid atomized dispersal driven by central explosion

4.1.One-dimension characteristic analysis

Based on the principle of wave propagation and interaction,the mechanism of tensile stress and cavitation fracture of liquid can be qualitatively analyzed.

In Fig.12,FTL means the line of first tension which is determined by the intersection of heads and tails of two rarefaction wave,and the moment when the tensile stress first occurs at different radii in the liquid.In the region above the line,due to the interaction of two rarefaction waves,the liquid is subjected to the tensile stress that increases with time.When the tensile stress is greater than its dynamic tensile strength,cavitation occurs in large quantities and spallation occurs.Under the condition of weak constraint,the stretching region is generated almost immediately after the shock wave reaches the outer surface and re flects,and the cavitation zone expands from the vicinity of the shell to the detonation centre.Under the condition of strong constraint,because of the re flected shock wave of primary shock wave from the outer shell,the propagation of blast waves in liquids is more complicated.FTL is still the location where the tensile stress first occurs in the liquid and is determined by the intersection line of the two wave heads and tails of re flecting the rarefaction waves RW(2)and RW(3).Because of the recompression effect of the re flected shock wave on the liquid,the tension occurrence location in the liquid which induce cavitation and spallation is closer to the region of detonation,and the time is greatly delayed.

4.2.Spallation mechanism of viscous liquid dispersal

From the results of above characteristic analysis,the mechanism of occurrence of two annular jetting regions during the viscous glycerol dispersion process(See Fig.6)can be explained.The interaction of two rarefaction waves propagating in opposite directions causes the spallation of glycerol.The reason why water has no similar two annular jetting regions as glycerol may be that the viscoelasticity of glycerol is higher than that of water and the intercollision strength of the two pieces of glycerol after spallation is lower than that of water.Spall water can be restored by intercollision and come into a coherent medium,while glycerol cannot and be divided into two parts.According to the research results of glycerol fracture mechanism in Ref.[50],the strength of secondary shock wave after glycerol spall is far less than that of water(See Fig.13).

4.3.The in fluence of cylindrical shell’s strength to the liquid dispersing flow

The developing of the internal state of the liquid during its dispersal is dif ficult to observe by experiments.Nevertheless it can be revealed by numerical simulations clearly.Below are the results of numerical simulation to the situations of strong constraint corresponding to the experiments shown in Fig.7.

We focus on the development of cavitation in the cylindrical container.In the case of 1 mm-thick perspex shell,the cavitation occurs almost immediately near the inner surface when the explosive shock wave reaches the interface between liquid and shell.Under the condition of the 3 mm thick PVC shell with,the cavitation occurs with about 5μs delay,at a location away from the inner surface of the shell.Under the condition of the 3 mm thick steel shell,the cavitation occurs with larger delay in the middle of liquid region and close to the detonation product,while cavitation bubbles near the inner surface of shell are rare[35](see Fig.14).

Fig.15.Metastable liquid phase diagram and the realization way of superheated liquid which is quoted from Ref.[52,53](a).Radial distribution of pressure at different times in experiment shown in Fig.7(a)-(d).The variation of rising temperature of liquid with the radius from an example when the radius of explosive and cylindrical container is 25 mm and 250 mm respectively(c).

In addition,the development of the morphology of the interface between detonation products and liquid is also quite different according to different strength of constraint shell.After 100μs of initiation,the interface between detonation products and liquid has developed very irregular for steel shell,while under the conditions for both Perspex and PVC shells,the interface remains smooth till the same time.This fact indicates that the stronger the shell constraint is,the earlier the detonation products gas mixes with the liquid at the interface[35].

These numerical simulation results are consistent with experimental observations on describing the con figuration of the explosive production-liquid interface and expansion of the liquid region.

4.4.Metastable state during liquid dispersion

According to the shock dynamics analysis above,the interaction between the two rarefaction waves in opposite direction is the power of liquid fragmenting and dispersal.Furthermore,the mechanism of liquid phase transition can be further analyzed from the perspective of thermodynamics.

From Fig.15(a),rapid depressurization and rapid temperature rising are two approaches to realize liquid becoming superheat and metastable state,and then leads to a violent phase transition.With some experiments shown above as examples,the results of 2-D numerical calculating show that the absolute value of the negative pressures in the region reached by re flecting rarefaction wave is less than 50 MPa(in Fig.15(b)).As shown in Fig.15(a),their corresponding superheated limit temperatures are all larger than 500 K,but the actual rising temperatures are all less than 443 K(in Fig.15(c)).Therefore,the liquid may not reach the superheated limit and explosive boiling should not occur in liquids during dispersion.However,since the liquid has dynamic tensile strength(according to the literature),there must be a short duration from the liquid’s pressure dropping below the saturated vapor pressure to reach of the dynamic tensile threshold,and the liquid is speculated be shortly at a metastable state before violent phase transitions occur.

Fig.16.The liquid dispersing flow of Perspex shell container(corresponding to experiment shown Fig.7(a)-(d))recorded from side-view which is used for the calculation of cavitation volume fraction(a).Cavitation volume fraction changes with time(b).Cavitation fraction(local maximum and average)changes with time(c).

Fig.16 shows that the cavitation volume fraction from numerical simulation is consistent with the experimental results,and is also consistent with the reports in Ref[33].It is evidently that shortly after explosion ignition violent and developing cavitation(a type of phase transision)occurs in the liquid,which is the possible consequence of a superheated(so is metastable)state.The calculation of volume fraction provides a method for evaluating gasi fication status during liquid dispersal though the generation rate of cavitation bubbles and the phase transition rate need to be achieved through the microscopic nucleation mechanism.

In Ref[31],the classical nucleation theory(CNT)based on homogeneous nucleation theory is applied to explain the mechanism of liquid cavitation induced by rarefaction waves.According to the CNT,a liquid need to be subjected to a high negative pressure to generate considerable nucleated bubbles.Some experiments have demonstrated that for deionized water 60 MPa is needed to induce cavitation[54].But the impure water,such as tap water,can generate cavitation under negative pressure which is not so large,as small as 0.1 MPa[55].The actual liquids are heterogeneous,instead of homogeneous.Therefore,in microscopic view,heterogeneous nucleation theory,instead of CNT,should be used to explain the mechanism of liquid dispersion driven by explosion.

5.Conclusion

In this paper,by reviewing various experiments of liquid dispersion driven by explosion,systematic multi-scale mechanism analyses are presented,such as the shock dynamic state,onedimensional characteristic analysis,spallation,cavitation strati fication,metastable phase transition etc.,and the following conclusions can be summarized.(1)The interaction of two rarefaction waves propagating in opposed direction is the main driving forces for liquid dispersal driven by central explosion.(2)Cavitation strati fication and spallation fragmentation are formed in the liquid,which might resulting from a metastable superheated state induced by the interactions of rarefactions.(3)Different constraint shell strength causes in different location of cavitation strati fication.The stronger the shell is,the more hysteresis the cavitation occures.(4)The microscopic mechanism of liquid cavitation might be explained by heterogeneous nucleation theory.

In view of the above mentioned facts,during the process of explosion-driven liquid dispersion,the liquid is subjected to the shock compression,negative pressure stretching and tensile shear,and it may be under complex mechanical states.However,researches on the multiscale laws of liquid dynamic behaviour are inadequate.The microscopic mechanisms of thermodynamics,nucleation and liquid’s disintegration need more in-depth investigations and explorations in the future.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to in fluence the work reported in this paper.

Acknowledgments

Authors wish to acknowledge the support of National Nature Science Foundation of China,the support numbers are No.10572149 and No.10676120.Meantime,authors also wish to acknowledge the National Key Research and Development program of China(subject no.2017YFC0209901)for its support to the work of this paper.