Numerical study on shock-accelerated heavy gas cylinders with diffusive interfaces

Dongdong Li·Ben Guan·Ge Wang

Abstract Interactions of shock waves and heavy gas cylinders with different diffusive inter faces are numerically investigated.Comparisons among these interfaces are made in terms of cylinder morphology,wave system evolution,fluid mixing, and circulation generation.Navier–Stokes equations are solved in the present work to simulate the complex multi-fluid flow.A fifth-order weighted essentially non-oscillatory scheme is used to compute the numerical flux.The influence of interface diffusion is revealed by numerical results.Cylinders with similar geometric scale but different diffusion interface have significant similarities in hydrodynamic characteristics,including the inter face morphology,shock focusing, and molecular mixing,as well as circulation deposition.For cases with more severe interface diffusion,the cylinder develops into more regular vortex pairs.The diffusive inter face significantly mitigates the strength of the reflected shock wave and weakens the shock focusing capability.Some inter face evolution features are also recorded and analyzed.The diffusive interface brings about slower molecular mixing and less circulation generation.The circulation deposition on different interfaces is quantitatively investigated and compared with the theoretical models.The theoretical models are found to be applicable to the scenarios of diffusive interfaces.

Keywords Shock–bubble interaction ·Interfacial instability ·Diffusive interface·Circulation model

1 Introduction

Whenever a planar incident shock wave impulsively accelerates a spherical or cylinder density inhomogeneity,the baroclinic vorticity deposits on the interface between the two fluids,enhances the fluid mixing, and finally results in a complex,turbulent flow field.This is known as the classical shock–bubble interaction(SBI),which has been a hot issue since the 1950s[1]due to its academic value.Over the years,the SBI studies have provided insights into the vortex dynamics,compressible turbulence,astronomic phenomena[2],inertial confinement fusion[3], and supersonic combustion ram jets[4].In SBI,the bubble is initially compressed and accelerated, and the n stretched by developing vertices,which leads to a final turbulent flow.The Richtmyer–Meshkov(RM)instability[5,6]plays a crucial role in the bubble morphology evolution because of the baroclinic mechanism.

SBI studies aid not only in understanding each of the specific physical processes independently but also in understanding how the elements interact and couple with each other to influence the post-shock–bubble morphology.Extensive pioneering studies on SBI have been conducted to promote an understanding of fundamental fluid flow problems and stimulate the booming of the experimental and numerical technologies.The first SBI experiment was performed by Rudinger and Somers[7]in the 1960s.In their study,a complex turbulent flow field with long-living vortex rings was observed.Subsequently,Haas and Sturtevant[8]carried out their seminal experiments on SBI.In their experiments,cylinder and spherical bubbles of both light and heavy gas inhomogeneities were sharply confined in space using thin plastic membrane or soap film technique.Images of the wave systems and interface deformations after it was impacted by the planar incident shock wave were obtained via a shadowgraph technique.Up to now,their experimental images have been widely used as benchmarks for various numerical methods[9–12].Using the soap film technique,Layes et al.[13]investigated the shock-induced velocity and volume changes of spherical bubbles filled with helium,nitrogen, and krypton.Following Layes’work,Giordano and Burtschell[14]presented their numerical simulations in which the vorticity generation,advection, and dilatation mechanisms were analyzed.The n Zhu et al.[15]studied the influence of the incident shock intensity on the interaction processes between the shock and SF6bubble.A numerical study on the vortex dynamics and the turbulent mixing of planar shock-accelerated triangular heavy gas inhomogeneity was conducted by Zengetal.[16].In recent years,a series of experimental and numerical studies on the interaction of shock(both planar and converging)with spherical bubbles have been performed[17–19].

In the studies mentioned above,soap film or plastic membranes were used to sharply separate the test gas and the ambient gas for SBI experiments.In practice,a membraneless technique has also been commonly used in SBI experiments.This method was adopted by Jacobs[20]to study the gas cylinder evolution influenced by planar incident shock waves.In his experimental setup,a round laminar helium jet was used to produce the light gas cylinder, and a planar laser-induced fluorescence(PLIF)technique was employed to visualize the flow in a two-dimensional manner.High-quality flow field images were obtained, and based on the se,quantitative parameters including cylinder area change, and mixing rate were presented and studied.The work of Tomkins et al.[21]adopted the PLIF technique to study the interaction of shock wave and heavy gas cylinder,where SF6was adopted as the testing gas.The instantaneous scalar dissipation rate and total mixing rate defined by molecular diffusivity and concentration were introduced to identify the most intensive mixing region.Three distinct mixing mechanisms were identified in the process of cylinder deformation derived from the vortex core,the bridge section connecting the two vortices, and the secondary instability occurring at the outer interface,respectively.The work of Santhosh et al.[22]numerically reproduced Tomkins’experiment to investigate the influence of tracer particles on hydrodynamic instability and mixing,which indicated that the influence of tracer particles was non-negligible.An experimental study on elliptic SF6gas cylinders using vertical gas jets by Zou et al.[23]revealed the influence of the elliptic aspect ratio on long-term cylinder evolution.In the se studies,the interfaces are diffusive,however,the effect of interfacial diffusion on SBI has rarely been studied systematically.It is hoped that the study on this issue will provide insight into the SBI.

In the published literature,the oretical models have emerged for characterizing various aspects of the SBI.Among the m,the circulation model which estimates the circulation generation at the material interface is a challenging task.Based on a series of assumptions,several circulation models have been proposed,generally known as the PB model by Picone and Boris[10],the YKZ model by Yang,Kubota,and Zukoski[24], and the SZ model by Samtaney and Zabusky[25].In addition,an analytical approach for calculating the final volume of the bubble was proposed by Giordano and Burtschell[14].

Differences in the initial interface will lead to differences in the wave pattern,vorticity behavior, and interface deformation.The membrane technique introduces the influence of fragments into the inter face evolution once the membrane is broken,while the membraneless technique introduces an inevitable interfacial transition layer(ITL)[26]between the testing gas and its ambient gas.The existence of the IT Lalters many properties of the typical SBI in sharp interface scenarios.In order to further understand the influence of initial interfacial diffusion on the interfacial evolution,interfaces with different degrees of diffusion are numerically investigated in the present work.The evolution of interface,wave pattern and deposited circulation,as well as mixing,is discussed in detail.

2 Numerical model

2.1 Governing equations

Compressible multicomponent Navier–Stokes(NS)equations are solved in a conservative form to reproduce the present SBI problem.Viscosity is considered to make the simulations more practical.A system of conservation laws in two dimensions is

with

where ρ denotes the mixture density,u and v are the velocity components in x- and y-directions,E is the total energy,p is the pressure, and Yiis the mass fraction of species i?1,2,...,Ng,with Ngthe total number of species.qxand qyare the components of the thermal diffusion vector q in x- and y-directions.σxx,σxy,σyx, and σyyare the components in the viscous stress tensor τ.Jx,iand Jy,iare the components of species diffusion flux Jiin x- and y-directions,respectively.The ideal gas equation of state for gas mixture is used to close the equations

Specific gas constant of the mixture is defined by?R,with R the universal gas constant, andthe molar mass of the mixture calculated by

where Xiand Miare the mole fraction and molar mass of species i.Equation(4)is used to determine the total energy from the primitive variables

with CPthe specific heat at constant pressure

The viscous stress tensor τ for a Newtonian fluid is given by

where the first term on the right-h and side is heat conduction derived from the Fourier law, and the second term is the interspecies diffusional heat flux.hiis the enthalpy of species i and is the thermal conductivity of the gas mixture.represents diffusion velocity of species i

with

where Di,mixis the effective binary diffusion coefficient of species i.The species diffusion Jiis given by

2.2 Molecular transport properties and multicomponent mixing rules

For individual fluid,the viscosity μiof species i is assumed to obey the Chapman–Enskog law

where σiis the collision diameter of species i and Ωμ,iis the collision integral

in which the coefficients are A?1.16145,B??0.14874,C ?0.52487,D ??0.7732,E ?2.16178,F ??2.43787, and?T/(ing/k)i.(ing/k)iis the Lennard-Jones energy parameter for species i.Thermal conductivity of species i is calculated by

The mass diffusion coefficient of a binary mixture is computed from constitutive empirical law

with

where the collision integral for diffusion ΩD,ijis given by in which the parameters are A?1.06036,B??0.1561,C?0.19300,D??0.47635,E?1.03587,F??1.52996,G?1.76474,H??3.89411, and?Tingijis obtained from the Lennard-Jones energy parameters for species i and j

The mixture-averaged transport properties,,are calculated using a simplified mixing rule

where the parameter ξirepresents a transport coefficient(i.e.shear viscosity μ or thermal conductivity λ) and n is equal to 6 and 4 for the case of μ and λ,respectively.For species i in a multicomponent gas mixture,mixture-averaged diffusion coefficient Di,mixis given by

2.3 Numerical methods

The governing Eq.(1)is decoupled,using second-order Strang time-splitting scheme[27],into the hyperbolic and parabolic steps(Eqs.(23) and (24))

The hyperbolic step is split on a dimension-by-dimension basis

A finite difference scheme is used to discretize Eq.(1).For the hyperbolic step,the numerical fluxes are reconstructed via the fifth-order weighted essentially non-oscillatory scheme[28].The time integration is performed using the third-order total variation diminishing Runge–Kutta scheme[29].For the parabolic step,the fourth-order central difference scheme is adopted to handle the viscosity term and the second-order explicit Runge–Kutta–Chebyshev[30]scheme is used to integrate the unsteady time term.

Fig.1 Schematic of the computational domain and boundary conditions

3 Computational setups

3.1 Computational domain and boundary conditions

The computational domain and boundary conditions of the numerical simulation are presented schematically in Fig.1.The Cartesian mesh is used in the present study on which the Navier–Stocks equations are discretized.Only the upper half of the cylinder is simulated owing to the physically span wise symmetry.The computational domain covers a two dimension space in the Cartesian coordinate system with X×Y?[?9.0,45.0]mm×[0.0,12.0]mm.The center of the gas cylinder is initially located at(x,y)?(0.0,0.0).The initial gas cylinder is located at the right(downstream)side of the incident shock wave and is surrounded by quiescent air with temperature 298 K and pressure 0.8 atm(1 atm?1.01×105Pa).The incident shock Mach number is 1.20 which launches at x??4.5mm.The post-shock thermodynamic states are given by Rankine–Hugoniot conditions.The left,right, and upper boundaries are set to be no-reflection boundaries,where gradients of physical parameters are zero.The lower boundary(x-axis)is treated as a symmetric boundary.

where Rdis the radius of the initial cylinder(Rd?3mm),r is the radius vector measured from the circle center,is the initial maximum concentration of SF6which is 0.83.Index I,which controls the diffusivity of the cylinder,equals to 1.54 for ITL1(follow Ref.[22]to reproduce Tomkins’experiment[21]),2.5 for ITL2,3.5 for ITL3, and 100.0 for ITL4.It is worth noting that for the interface type ITL4,the

The gas mixture inside the cylinder is composed of SF6,N2, and O2.The mole ratio of N2and O2is maintained at 3.76:1,while the distribution of SF6mole fraction is different.As shown in Fig.2,ITL1,ITL2,ITL3, and ITL4 are four types of diffusive interfaces.Their initial SF6mole fraction profiles follow the definition presented by Tomkins et al.[21]large index number results in very little diffusivity,so that it can be considered a sharp interface.Time is initialized to zero(t?0μs)when the incident shock moves to x??3mm.

Fig.2 Distributions of SF6 mass fraction for different diffusive interfaces

Fig.3 Grid convergence testing results.Plots a–d indicate that the re are 60,120,240, and 480 grids within the cylinder radius.Schlieren images,80μs after the shock impact

3.2 Grid resolution test

A comparison of the simulation on interface ITL1 deformation using four different grid sizes is shown in Fig.3.Grids a–d correspond to the cases in which 60,120,240, and 480 grids are arranged within the cylinder radius(within 3 mm).Generally,the large-scale structures,i.e.,the large crescent shape and the middle jet,remain the same in the four cases.However,small-scale structures,including the jet hat and the upstream surface of the cylinder,become sharper as the grids become finer.The density along the symmetric lines(y?0.0mm) of the four testing cases at this moment is illustrated in Fig.4.It is seen that the density curves derived from the cases of grid c and grid d nearly overlap.Considering the computational cost and resolution requirement comprehensively,the same grid size as grid c is used in the following simulations.

Fig.4 Plots of density along the symmetric lines of the gas cylinders with different grid sizes at 80μs

4 Results and discussion

4.1 Interface morphology

The experimental images obtained by Tomkins et al.[21] and numerical results with the four interfaces(ITL1–ITL4)are shown in Fig.5.The comparison of the experimental images presented in the first row and the numerical results presented in the second row(ITL1)shows that although some differences exist,the present numerical methods can reproduce the experimental results to a good extent.At 130μs,a very pointed jet is formed, and the two “arms” of the crescent shaped cylinder are about to roll inward.Interestingly,the jet flattens at 220μs and smears out after 310μs.The jet evolution is determined by two factors.First,counter-rotating vortices induce air flow towards the middle leeward surface of the cylinder,which goes against the jet growth, and second,the vortices stretch the middle part of the heavy gas bulk(usually called the “bridge”)which connects the two vortices,making the middle part thinner,so that the jet volume is reduced.The formation of multi-layer vortices can be observed after 200μs, and their spiral structures exp and as time elapses.Kelvin–Helmholtz instability(KHI)can be clearly observed on the spiral belts at 400μs in the numerical results,which will further increase the fluid mixing.It is worth noting that the rolling of counter-rotating vortices shown in the numerical results is obviously stronger than that shown in the experimental images, and this discrepancy derives mainly from the influence of tracer particles used in the experiment,as stated in Ref.[22].

The second to fifth row shown in Fig.5 present the numerical results of ITL1–ITL4,respectively.As clearly seen in Fig.5,the detailed evolution of interface morphology is strongly depended on the diffusivity of the initial interface.From the four numerical results,similar evolution can be observed in which typical flow characteristics including crescent-shaped cylinder,pointed jet,counter-rotating vortices, and KHI coexist,although differences in details cannot be ignored.With the increase in diffusivity,the cylinder evolution slows,the deformation becomes smooth, and the KHI occurs late.For the ITL4 interface,which is fairly sharp,the primary vortex pair becomes chaos at t?220μs and the tail of crescent-shaped portion wrinkles at t?310μs because of the generation of KHI.

Fig.5 Temporal evolution of SF6 mass fraction of experimental result[21](in the first row) and numerical results in present study(ITL1–ITL4 in the second–fifth rows)

To record the interface deformation,the contour line of YSF6?0.1 is used to mark the cylinder interface,as shown in Fig.6 by the red solid line.Figure6 also provides the definitions of several parameters,i.e.,the leftmost point,the rightmost point,the interface w idth W, and the interface length L.

Fig.6 Definitions of the characteristics scales

Figure 7a presents the displacements of the leftmost point,Xleft.It reflects the streamwise distance from the in-time leftmost point of the deformed interface to the initial leftmost point of the intact interface.It is seen that the displacements grow with nearly constant velocities.The minor difference in Fig.7a shows that the interface moves slightly faster for the case with more serious diffusion.This is caused by the lower acoustic impedance and smaller density gradient at the upstream interface.The displacements of the rightmost point,Xright(which reflects the distance from the in-time rightmost point of the deformed interface to the initial rightmost point of the intact interface)hold still before the shock passes and move with nearly constant velocities.It can be found that the difference of these two scales caused by interfacial diffusion is fairly small.

Fig.7 Evolutions of a X left,b X right,c L/L0, and d W/W0 of four different interfaces

Fig.8 Variation in bubble volume of the four different interfaces

The interface length L is defined as the distance between the leftmost and rightmost points of the deformed interface.The evolutions of the length are normalized by the initial diameter L0and shown in Fig.7c.The length decreases initially due to the shock compression.This decrease ceases when the downstream interface gains a comparable velocity via the transmitted shock impact.Notably,a slight and short termed increase is seen which is caused by the sudden occurrence of the jet.The n,L decreases a little due to the gradual disappearance of the jet until the downstream edges of the primary vortices exceed the jet tip.After this,the interface length L increases because of the velocity of downstream is larger than that of the upstream interface.During these increasing processes,several“kinks”exist because of the development of KHI(as shown in Fig.5),especially for the cases ITL3 and ITL4.For the ITL1 and ITL2 interfaces,the outer boundaries are smooth,so that the changes of interface length L are also smooth.The evolutions of the interface width W normalized by the initial cylinder width W0are plotted in Fig.7d.At the beginning,W/W0maintains the initial width before the incident shock wave reaches the uppermost of the interface.Later,the width decreases due to the compression of the shock wave.After that,owing to the continuous rotation of primary vortex pairs,the W increases for all four cases.It is obvious that with the increase of the interfacial diffusion degree,both the width W?W0and length L?L0decrease in their growth rates.

Fig.9 Shock wave system evolution of interface ITL1.a t?3.6μs,b t?7.6μs,c t?13.6μs,d t?19.6μs,e t?23.6μs

Fig.10 Shock wave system evolution of interface ITL2.a t?3.6μs,b t?7.6μs,c t?13.6μs,d t?19.6μs,e t?23.6μs

Fig.11 Shock wave system evolution of interface ITL3.a t?3.6μs,b t?7.6μs,c t?13.6μs,d t?19.6μs,e t?23.6μs

Fig.12 Shock wave system evolution of interface ITL4.a t?3.6μs,b t?7.6μs,c t?13.6μs,d t?15.6μs,e t?19.6μs,f t?23.6μs

Volume compressibility γcis calculated and shown in Fig.8.Generally,similar compression histories and final stable values are observed.At the early stage,cylinder volume sharply decreases due to the compression of the incident shock wave.The n short-period oscillation is observed because of the early-stage shock reverberation through the cylinder.After the reverberation stage,the γcvalues tend to stabilize and reach a final value of about 72%.The final volume agrees with the one-dimensional gas dynamics the ory derived in Ref.[14](which is 71.32%).This result shows that the volumetric compression is greatly restricted by the flow states rather than the interfacial diffusivity.

4.2 Wave pattern

Figures9,10,11 and 12 show the evolutions of wave system for ITL1,ITL2,ITL3, and ITL4 at the very early stage.In each subfigure,cylinder interface is denoted by SF6mass fraction contour YSF6?0.1(the solid line), and the dashed line represents the initial interface.

Substantial similarities in wave systems can be observed for the four diffusive interfaces according to the numerical Schlieren photographs presented in Figs.9,10,11 and 12,especially for ITL1,ITL2,ITL3 and ITL4.After the incident shock wave touches the cylinder upstream pole,only a transmitted shock wave is seen propagating into the cylinder.The reflected shock wave is too weak to be detected at t?3.6μs shown in Figs.9a,10a,11a, and 12a.As time progresses,the nonlinear acoustic effects result in deformation of both the transmitted and incident shock waves through refraction and diffraction,respectively,at t?7.6μs and 13.6μs,as shown in Figs.9b,c,10b,c,11b,c, and 12b,c.The incident shock wave is diffracted around the downstream portion of the interface with a near-normality condition.At the same time,the transmitted shock in the volume intersects the diffracted shock wave at the cylinder interface during this period.At 19.6μs of all cases,the shock focusing inside the cylinder has completed, and the diffracted shock converges at the cylinder downstream area.The internally reflected shock wave the n exp and s and forms the secondary transmitted shock wave that exp and s outside the cylinder,as shown at t?23.6μs.Interestingly,after the internally reflected shock wave exp and s out of the diffusive cylinder,the re is no shock wave detected in the cylinder.In addition,the jet forms in the diffusive ITL region,where no distinct interface exists.This indicates that the jet formation mechanism can be attributed more reasonably to a result of shock–shock interaction rather than shock-interface interaction,as inferred in previous studies.Differences are observed for the interface of ITL4 compared with the other three interfaces owing to its sharp interface.An internal triple point connecting the transmitted shock wave,shock 1 and shock 2 forms because of the increase in the shock front concavity and intensity in Fig.12d.

To determine the shock focusing intensity of these cases,the pressure of the transmitted shock front along the bubble axis is recorded and presented in Fig.13.The figure shows that the intensity of the transmitted shock increases slowly at the very early stage.Later,a sudden increase is generated owing to the converging of the transmitted shock wave near the downstream pole.The maximum pressure is228,694 Pa,243,586 Pa,269,023 Pa, and 304,384 Pa for ITL1,ITL2,ITL3, and ITL4,respectively,which indicates that the more diffusive the interface,the weaker the energy-gathering ability the cylinder has.

Fig.13 Time histories of the maximum pressure along the symmetric lines of cylinders with different interfaces

4.3 Molecular mixing

Molecular mixing fraction(MMF)Θ(t)is used to estimate the influence of the interface diffusion on mixing

where XSF6and XO2are molecular fractions of SF6and O$2.Th?e$ bar denotes the averaging manipulation?

Time histories of MMF for cylinders with different interfaces are plotted in Fig.14.It is not surprising that a significant difference in MMF values is seen among the four cases.Within the time computed,the MMF value for the interface ITL1 maintains a high level at the early stage due to the existence of a wide ITL.The MMF values increase with the respective growth rate,where the more serious the interface diffusion,the slower the growth rate.Until 450μs,the MMF values increase 3%,11%,16%, and 31% for cases from ITL1 to ITL4.The MMF value increase can be attributed to two factors,i.e.,the local concentration gradient and the total length of the interface.For the weak diffusive interface,the concentration gradient is large and the interface is severely deformed,which mutually results in a high MMF value,especially for the ITL4 interface.

4.4 Circulation calculation

The integral of vorticity generated by the interaction of a shock wave and the interface can be characterized by the circulation that is widely used in SBI to demonstrate the fluid mixing intensity.Several theoretical models have been proposed for predicting circulation generation in SBI.By assuming that there was little change happening in either the cylinder shape or density in the process of incident shock sweeping,a PB model[10] and YKZ model[24]were proposed in which the vorticity production was decoupled from shock refraction and diffraction.Another approach was taken in the SZ model[25],in which the gas interface was assumed to be composed of many straight segments.The study by Niederhaus et al.[31]proposed their theoretical model(R model)by integrating velocities along a fixed path,where the velocities after shock passage were reconstructed through one-dimensional gas dynamics analog.

Fig.14 Time histories of molecular mixing fractions for cylinders with different interfaces

Table1 Comparison of numerical and theoretical circulation values for cylinder with ITL4 interface.E1 denotes the prediction error

These models are directly used to estimate the circulation values of ITL4 interface.Numerically,the circulation value of ITL4 interface is?0.467m2/s.The model predicted values and their prediction errors are listed in Table1,which reveals that the prediction errors obtained by the PB and YKZ models are larger than that obtained by the SZ model, and the R model presents the best-predicted value,with prediction error E1? ?0.4%.In addition,the numerical circulation Γnumof the other three cases are?0.4287 m2/s,?0.4597 m2/s, and ?0.465 m2/s for ITL1,ITL2 and ITL3,respectively,which is close to the circulation value of ITL4.

Fig.15 Numerical circulation histories of the cases with different interfaces.a Positive circulation Γ+,b negative circulation Γ?, and c total circulation Γ

Figure15 shows the circulation histories of positive(Γ+),negative(Γ?), and total(Γ )circulations calculated from the numerical results for the cases with different interfaces.In the present computation domain,negative circulation dominates the flow field.For all cases,the negative circulations dramatically increase at the early stage because of incident shock wave sweeping.Later,the negative circulation grows more slowly because of the weaker reflected shock waves and the self-induced vortex development.The positive circulations,on the other hand ,increase much more slowly than the negative circulation, and the total circulation is obtained by summing the negative and positive circulation values.

By taking the vorticity production equation into account

,where w denotes the vorticity,?ρ and ?p are the gradients of the density and pressure),the density gradient at the diffusive interface is greatly mitigated,which directly results in greatly reduced circulation production at the early and interim stage.At the latest age,the growth rates of both positive and negative circulations are suppressed due to the well-developed KHI which makes the density gradient moderate,especially for the ITL3 and ITL4 interface.

5 Conclusions

Numerical simulations of two-dimensional SBI are conducted in the present work.The main emphasis is placed on investigating the influence of the interfacial diffusion on long-term flow features.Four diffused mass fraction distributions are adopted.For all cases,the incident shock wave Mach number is 1.20, and a multicomponent mixture(SF6,N2, and O2)is used in the gas cylinder.The evolution of gas cylinder morphologies and development of shock wave patterns,especially shock focusing,are qualitatively analyzed and compared.The molecular mixing and circulation generation histories are recorded.

Obvious differences among the cases with different diffusive interfaces are illustrated by the present numerical results.For the serious diffusive interface,the initial reflected shock wave for the serious diffusive interface can hardly be detected,the shock focusing is greatly mitigated, and the appearance of KHI is delayed to a very late stage.This results in a very smooth interface evolution for a long period.The interface structural parameters Xleft,Xright,interface length L and width W are recorded.These parameters are found to be closely related to the compression and vorticity generation.The molecular mixing fractions between the SF6and O2are found to strongly depend on the density gradient of the initial interface.The diffusive interface not only slows the development of cylinder morphology but also prevents the cylinder from further mixing.

The intensities of circulation accumulated on the different initial interfaces are quantitatively estimated.Good agreement is found between the numerical circulation and that obtained from the theoretical model.The circulations are found to increase rapidly due to the impact of the incident shock wave at the early stage.A long-term and approximately linear increase can be observed later until the KHI is fully developed.The circulation generation is found to be weaker for the diffusive interfaces than for the sharp interface.

In future work,more extensive investigations will be conducted using the present numerical method.The influence of the initial mole fraction of the testing gas,the influence of strong incident shock wave by which high pressure ratios are imposed, and their mutual influence combined with the interfacial diffusivity will be comprehensively studied.