Instability and Mixing of Gas Interfaces Driven by Cylindrically Converging Shock Wave
IInstability and Mixing of Gas Interfaces Driven by CylindricallyConverging Shock Wave
Wei-Gang Zeng, Yu-Xin Ren , and Jianhua Pan
1, 2, a) Department of Engineering Mechanics, School of Aerospace, Tsinghua University, Beijing 100084,China. William. G. Lowrie Department of Chemical and Biomolecular Engineering, Koffolt Labs, The Ohio State University,US. (Dated: 9 February 2021)
In the present paper, an efficient method to generate "pure" cylindrically converging shock wave without a followingcontact surface is proposed firstly. Then, the Richtmyer-Meshkov instabilities of two interfaces driven by the generatedcylindrically converging shock wave and the associated fluids’ mixing behaviors are numerically studied. The resultsshow that the instability of the interface is characterized by the growth of perturbation amplitude before re-shock.However, the mixing of fluids is enhanced dramatically after re-shock, which is manifested not only by the evolutionsof flow structures but also by the temporal behaviors of mixing parameters. Further investigation shows that, althoughthese two cases are of different initial perturbations, their evolutions of mixing width and other mixing parameters suchas molecular mixing fraction, local anisotropy and density-specific volume correlation could achieve the same lawsof temporal behavior, especially during the later stage after re-shock. These results to some extent demonstrate thatthere also exist scaling law and temporal asymptotic behaviors in the mixing zone for cylindrically converging shockwave driven interface. Moreover, the analyses of turbulent kinetic energy spectrums in the azimuthal direction at latestage also witness the k − / decaying law of turbulent kinetic energy for the present inhomogeneity flows driven bycylindrically converging shock wave, which further manifests that the fluids’ mixing is indeed enhanced at later timeafter re-shock. I. INTRODUCTION
The interaction between shock and gases interface with ini-tial perturbations is complicated, which involves interfacialinstabilities of various kinds and the possible turbulent mixingat late stage. At the beginning of interaction, the Richtmyer-Meshkov instability (RMI) dominates the flow, resulting in thedeposition of baroclinic vorticities and the growth of the per-turbation. Then, the accumulated vortices cause the formationof the primary Kelvin–Helmholtz (KH) billows, which wouldlead to the dramatic mixing of fluids when the more complexinstabilities occur in these billows.In the last three decades, the instabilities of shock drivengases interface and the associated fluids’ mixing behaviorshave been widely studied due to their importance in Iner-tial Confinement Fusion (ICF) , supernova explosions , andsupersonic combustion . Many theoretical models are pro-posed for the vortex generation and growth rate of pertur-bation amplitude of RMI flows, most of which are man-ifested not only by numerical simulations but also byexperiments . Recently, the essential development of flu-ids’ mixing at late stage attracts great attention . Therefore,the morphological behaviors of turbulent mixing and the cri-terion of turbulent mixing transition are extensively studiedduring past ten years . However, most of these investi-gations are mainly focused on interface instabilities and theassociated fluids’ mixing induced by planar shock wave. Bycontrast, the studies on interface instabilities and the associ-ated fluids’ mixing induced by converging shock wave, which a) Electronic mail: [email protected] would be more relevant to engineering applications such asICF, are much less abundant. Obviously, these research top-ics are gaining increasing attention recently. For example, re-cent studies have manifested that, due to the Bell-Plesset ef-fect and Rayleigh-Taylor effect, the perturbation amplitude ofinterface driven by converging shock wave would grow in adifferent way both at early and later stages . However,the fluids’ mixing behaviors of interface driven by convergingshock wave at late stage remain to be open issues.The motivation of the present paper is to investigate the in-stability and the associated fluids’ mixing of gases interfacesdriven by cylindrically converging shock wave (CCSW) us-ing high resolution finite volume (FV) method. To this end,we adopt an efficient way to generate “pure” CCSW whichcan be used as the incident shock wave for further numeri-cal study of interaction between CCSW and gases interface.Then, the RMI and the associated fluids’ mixing behaviors oftwo interfaces driven by such CCSW are numerically inves-tigated in detail. The evolutions of flow structure and fluids’mixing behaviors highlight that the instability of the interfaceis characterized by the growth of perturbation amplitude be-fore re-shock, while the fluids’ mixing is enhanced dramati-cally after the interface being re-shocked. The enhanced flu-ids’ mixing is also manifested by the exponential scaling lawsof mixing width as well as the temporal asymptotic behaviorsof mixing parameters such as molecular mixing fraction, lo-cal anisotropy and density-specific volume correlation at laterstage after re-shock. Moreover, the turbulent kinetic (TKE) inthe azimuthal direction also decays with a slop of k − / in arelatively broader range of low wave numbers at later stage af-ter re-shock, which further confirms the enhanced fluids’ mix-ing since the inertial range is extended during the developingprocess. a r X i v : . [ phy s i c s . f l u - dyn ] F e b The remainder of this paper is organized as follows. Thenumerical framework based on high resolution FV method forcompressible two fluids is presented in Section II. The methodto generate “pure” CCSW and the verification for its usage inCCSW/interface interaction are introduced in Section III. Thenumerical simulations for the CCSW induced RMI flows andthe corresponding results which include the wave patterns, thefluids’ mixing behaviors and the decaying law of TKE spec-trums are well discussed in Section IV. And finally, the con-clusions remarks are given in Section V.
II. NUMERICAL FRAMEWORKA. Governing equations
Following our previous work , the integral form of gov-erning equations for compressible two fluids with consistenttreatment of the convective terms at a material interface canbe written as ∂∂ t (cid:90) Q d Ω = (cid:73) ∂ Ω ( F c − F v ) · n dS = (cid:90) W d Ω . (1)In the above formula, Q = (cid:2) ρ ρ Y i ρ u ρ E θ (cid:3) T is the vec-tor of quasi-conservative variables in the control volume Ω ,where ρ is the density of mixture, Y i is the mass fractionof specie i , u = [ u v w ] T is the vector flow velocity, E isthe total energy of mixture and θ = γ − γ . It should be notedthat, due to the quasi-conservative form of θ , a source term W = (cid:2) θ ∇ · u (cid:3) T should be added to the right-hand-side of Eq. (1). In fact, the equation for θ is introduced toachieve a consistent treatment of the material interfaces and toremove non-physical oscillations in the vicinity of the mate-rial interfaces, which was initially proposed by Abgrall andlately improved by Johnsen for FV method using high orderreconstructions. Additionally, F c and F v are, respectively, theinviscid flux and viscous flux on the control surface S with theunit outward normal vector n . Their definitions are given asfollows F c = ρ u ρ Y i u ρ uu + p I ρ H u θ u , (2) F v = ρ D i ∇ Y i ττ · u − q c − q d . (3)In the above two equations, I is the unit tensor, H = E + p / ρ isthe total enthalpy of the mixture, τ = µ S − µ ( ∇ · u ) I is the viscous stress tensor, and S = ( ∇ u + ( ∇ u ) T ) is the strainrate tensor. The heat conduction and diffusion flux are givenas follows q c = − κ ∇ T , (4) q d = − ∑ l = ρ h l ( D l ∇ Y l − Y l ∑ m = D m ∇ Y m ) , (5)where, T is the static temperature of the fluids’ mixture. Forspecies i = l or m , h i and D i are the individual enthalpy andthe effective binary diffusion coefficient , respectively. Addi-tionally, the models for dynamic viscosity coefficient µ , ther-mal conductivity κ and the effective binary diffusion coeffi-cient for the mixture are well documented by Tritschler andShanka , and one can also refer to our previous paper fordetails.To close the governing equations, the equation of state(EOS) for the mixture of ideal gases is adopted in the presentpaper. Its formula is given by p = ρ RT , (6)where R = R u M is the gas constant of the mixture with R u beingthe universal gas constant and M = / ∑ i = Y i M i being the meanmolecular mass of the mixture. As proposed by Johnsen , theinternal energy is related to pressure in the following form: E = p ( γ − ) ρ + | u | , (7)where the specific heat ratio of mixture, γ , is given by1 γ − = ∑ i = Y i M ( γ i − ) M i . (8)In the above formulas, M i and γ i are, respectively, the molec-ular mass and specific heat ratio of species i . B. Numerical Method
In the framework of FV method, the semi-discretized formof Eq. (1) is used to update the cell-averaged physical statesat the cell center, which is in the following form dQdt = − Ω ∑ n f = ( F ∗ c , n f − F ∗ v , n f ) (cid:52) S n f + W . (9)In the above formula, Q is the average state of Q at each cellcenter, F ∗ c , n f = F c · n n f and F ∗ v , n f = F v · n n f are, respectively,the numerical convective flux and viscous flux at the cell inte-face n f with area of (cid:52) S n f and unit outward normal vector of n n f , and W = (cid:20) ∑ n f = θ ( u n f · n n f (cid:52) S n f ) (cid:21) T is the av-erage source term of the cell with θ being the correspondingaverage of the function of specific heat ratio for the mixtureand u n f being the fluids’ velocity at cell interface.Combined with the fourth-order MDCD reconstruction pro-posed by Wang , the Harten-Lax-van Leer-Contac (HLLC)Riemann solver for quasi-conservative form of governingequations of multi-fluids, which is initially proposed byAbgrall , is used to calculate the numerical flux of convec-tion, F ∗ c , n f . Additionally, Green’s theorem is used to integratethe numerical viscous flux, F ∗ v , n f . Moreover, to evaluate theaverage source term, the formula proposed by Johnse for thefluids’ velocity at cell interface, u n f , is used, which is highlyconsistent with the HLLC Riemann solver for the numericalflux of convection. Once all the terms in Eq.(9) are evaluated,we update the flow states temporally using the third-order totalvariation diminishing (TVD) Runge-Kutta method proposedby Shu . In terms of the accuracy of the present simulationcode, it has already been well demonstrated in our publishedwork . III. GENERATING CCSW AND VERIFICATION OFCCSW/INTERFACE INTERACTIONA. Generating CCSW
Generating cylindrically converging shock wave or spher-ically converging shock wave is somewhat complicated, nomatter in experiment or in numerical simulation .Based on Guderley’s theory of converging shock wave , Lombardini and Pullin successfully set up the initial condi-tions for numerical study on the turbulent mixing driven byspherical implosions . However, there are some defects forGuderley’s method in generating converging shock wave sincethe condition for the validity of Guderley’s theory is that theshock wave must be strong enough. Another efficient methodof generating converging shock wave is based on the theory ofshock tube, which is initially used by Bhagatwala and Lele to generate the spherically converging shock wave. In thismethod, one just needs to increase the pressure and densityratios at a specific radius to generate the "pure" convergingshock wave without a following contact surface. In this paper,we follow the approach of Bhagatwala and Lele to generateCCSW.According to the theory of shock tube, the region of lowpressure gas (initially in static state with thermal states ofpressure p , density ρ , and the ratio of specific heats γ ) andthe region of high pressure gas (initially in static state withthermal states of pressure p , density ρ , and the same ratioof specific heats γ ) are initially separated by the diaphragm.As soon as the diaphragm is broken, a CCSW will propagateinward into the region of low pressure gas, while an expansionfan will propagate outward into the region of high pressuregas. To generate a "pure" CCSW with desired Mach num-ber M s and no following contact surface, the pressure p anddensity ρ of the gas in the intermediate region (the region be-tween the CCSW and the expansion fan) at initial time shouldmeet the following set of implicit formulations ρ ρ = ( γ + ) M s ( γ − ) M s + , (10) p p = + γγ + ( M s − ) , (11) ρ ρ = ρ ρ − a a ( γ − )( ρ ρ − ) (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) ρ ρ (cid:20) γ + − ( γ − ) ρ ρ (cid:21) − γ − , (12) p p = p p − a a ( γ − )( p p − ) (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) γ (cid:20) γ + ( γ + )( p p − ) (cid:21) − γγ − . (13)In the above equations, φ / φ and φ / φ , where φ ∈ { ρ , p } ,are jump ratios across the initial diaphragm and the desired CCSW, respectively, and a i = (cid:112) γ p i / ρ i (with i = , . The initial radius of circular shock (correspondingto the diaphragm position) for this simulation is R = m . Togenerate a converging shock wave with initial Mach numberof M s = .
5, the thermal states of initially static gas (air withspecific heat ratio γ = .
4) inside the circular shock wave is setto be: ρ = . kg / m , p = pa , a = . m / s .The corresponding thermal states of initially static air outsidethe circular shock wave, ρ , p and a , can be given by iter-atively solving Eq.(10)-Eq.(13). Once these initial conditionsare given, a converging shock wave and a expansion fan willpropagate inward and outward respectively from the initial po-sition. The thermal states of the air in the intermediate regionat the very beginning time, ρ , p and a , can also be givenby Eq.(10)-Eq.(13).Obviously, the strength of the converging shock wave willbe enhanced during its propagation since the area of shocksurface is decreased. The Mach number ( M s ) of the converg-ing shock wave at any radius r during its inward propagationin our numerical simulation can be derived from the followingformulation p b p = + γγ + ( M s − ) , (14)where, p b is the pressure behind the inward propagatingCCSW at the corresponding radius.Additionally, based on the theory of Guderley , during theinward propagation for a pure converging shock wave initiallyplaced at the radius of R , its radius at given time t , r ( t ) , canbe addressed as r ( t ) = R ( − tt ) α . (15)In the above equation, t is the total propagation time from theinitial radius to the center for converging shock wave, whichis about 0.865 ms in the present simulation. Additionally, forpure CCSW propagating in gas with γ = .
4, the Guderley ex-ponent α ≈ . M s = a α R t (cid:18) rR (cid:19) ( α − ) / α , (16)where, a is the sound speed of gas inside the CCSW.Based on the above formulations, both r ( t ) and M s wouldbe sensitive to the value of α . Particularly, when the shock ispropagating near to the center, a marginal change of α wouldresult in a dramatical variation of r ( t ) and M s . In order totake the effect of α into account, we additionally chose an-other two values of Guderley exponent α ± = α ( ± ) , with α = . ln [ r ( t ) / R ] versus nondimensionalized time ln ( − t / t ) and the evolu-tion of Mach number M s versus r of CCSW in our numericalsimulation with the corresponding results obtained from the FIG. 1. Evolution of nondimensionalized radius ln [ r ( t ) / R ] versusnondimensionalized time ln ( − t / t ) .FIG. 2. Evolution of Mach number M s ( r ) versus r . theory of Guderley , respectively. As shown in Fig.1 andFig.2, the evolutions of nondimensionalized radius and Machnumber in our simulation agree well with those of Guderley’stheory when the Guderley exponent is in marginal range of [ α − , α + ] , which could demonstrate that the method used byBhagatwala and Lele to generate the spherically convergingshock wave is also efficient in generating CCSW. B. Verification of CCSW/interface interaction: Studies onthe amplitude growth of CCSW driven interfaces
One of most important features for RMI flow is the growthof perturbation amplitude. Due to the impulsive accelera-tion of incident shock wave, the initial perturbation will bestretched, resulting in the possible linear growth of perturba-tion amplitude at early stage. Initially studied by Richtmyer and later confirmed experimentally by Meshkov , the growthrate of perturbation amplitude for interface driven by pla-nar shock wave is extensively studied during past threedecades . However, the perturbation growth of interfacedriven by converging shock wave remains to be an attractivetopic with many open issues. Based on the CCSW generatedby the method mentioned above, using 2D numerical simula-tions, we try to explore some features of perturbation growthfor CCSW driven interface before re-shock in this subsection.In addition to numerical simulations, the theoretical modelproposed by Mikaelian is also used to predict the amplitudegrowth of CCSW driven interface for further mutual confir-mation.In the present numerical studies of amplitude growth, theinitial perturbation of gases interface is in the following form η = r − a [ . − cos ( n ϕ )] , (17)where, r is the initial radius of the outer interface (radius ofcrest at initial time), a is the initial amplitude, n is the az-imuthal mode number, and ϕ is the azimuthal angle. To makeour results be more general, we take the effects of initial modeand initial amplitude into account. Therefore, three cases withdifferent initial azimuthal mode number and/or initial ampli-tude are studied. The parameters of initial perturbations forthe three cases are well listed in Table I. TABLE I. Parameters of initial perturbations for study of amplitudegrowthCase Index r ( m ) a ( cm ) nI 0.38 1 . . . Moreover, the CCSW with desired Mach number of M s = . R = . m . The gas inthe region between gases interface and the CCSW is air withinitial thermal states of ρ = . kg / m , p = pa , a = . m / s and γ = .
4. Once more, the initial ther-mal states of gas (air) outside the CCSW is iteratively solvedbased on Eq.(10)-Eq.(13). Additionally, the gas inside the ini-tial interface is a mixture of air (20% in mass fraction) andsulphur hexafluoride (SF , with specific heat ratio γ = . . ms for the CCSW to ini-tially strike the gases interface when its Mach number approx-imately reaches 1.51. For boundary conditions, a viscous cir-cular wall with radius of 1 cm is placed around the center. Ad-ditionally, the circular boundary at the outside of the computa-tional domain is large enough for the propagations of all possi-ble waves during the durations of simulation. Consequently, azero-gradient boundary condition is used for the outer bound-ary. A body-fitted mesh is used for all three simulations, with4096 cells in azimuthal direction and 2440 cells in radial di-rection. The amplitude of interface in numerical simulations FIG. 3. Crest radius and trough radius in simulations. is then given by a ( t ) _CFD ≈ (cid:2) r crest ( t ) − r trough ( t ) (cid:3) /
2, where r crest ( t ) = max { r Y SF = . } and r trough ( t ) = min { r Y SF = . } are,respectively, the radius of the outer interface (crest) and innerinterface (trough) of simulation results as shown in Fig.3.As an alternative to numerical simulations, linear modelsare also widely used for predicting the perturbation devel-opment of CCSW driven gases interface . Based on thepioneered work of Bell and Plesset , Mikaelian modelledthe amplitude growth rate of interface (with small ratio of ini-tial amplitude to the initial wave length) driven by CCSW asfollows d a ( t ) dt = − rr da ( t ) dt + ( nA − ) ¨ rr a ( t ) . (18)In the above formula, n is the mode number in azimuthaldirection, A = ( ρ in − ρ out ) / ( ρ in + ρ out ) is the Atwood num-ber with ρ in and ρ out respectively being the density insideand outside the already shocked interface. Additionally, a ( t ) and r are, respectively, the amplitude and average radius ofthe gases interface at time t . There is no analytical formulafor r ( t ) , consequently, it is approximately given by r ( t ) = (cid:2) r crest ( t ) + r trough ( t ) (cid:3) / t > t + (where t + is the time when the incident CCSW passes throughthe trough of interface), Eq.(18) can be integrated in the fol-lowing form a ( t ) − a + = ˙ a + r (cid:90) tt + r dt + ( nA − ) (cid:90) tt + (cid:82) tt + a ( τ ) r ¨ rd τ r dt , (19)where a + ≈ a ( − V si / V is ) and ˙ a + = a + ( nA − ) V si / r are, re-spectively, the initial amplitude and growth rate of amplitudeat time t + , and r ≡ r − a ≈ r for small a . Additionally,in the above formulations, V is is the velocity of CCSW whenit initially strikes the gases interface and V si is the velocityjump of the shocked gases interface. There are two terms onthe right hand side of Eq.(19), of which the first term denotesthe Bell-Plesset (BP) effect on the growth of amplitude andthe second term denotes Rayleigh-Taylor (RT) effect on thegrowth of amplitude . These two terms are correspondingto the effects of the first and second term on the right handside of Eq.(18), respectively.For the given a + and ˙ a based on initial conditions, the per-turbation amplitude of Mikaelian’s model can be obtained bydirectly integrating Eq.(18) using standard 4 th order Runge-Kutta method. In order to identify the BP effect and RT ef-fect on the growth of perturbation more clearly, we integrateEq.(18) in two ways. In what follows, a ( t ) _BP stands forthe theoretical amplitude calculated by only integrating thefirst term of Eq.(18) on the right hand side, while a ( t ) _BPRTstands for the theoretical amplitude calculated by integratingboth terms of Eq.(18) on the right hand side.Obviously, the BP effect is fully resulted from the geo-metric convergence effects on flows, since this term will bezero for the planar shock driven interface . Additionally, aswe will discuss below, Rayleigh-Taylor effect will not be en-hanced by the geometric convergence at early stage. However,it, indeed, plays an important role in the growth of pertur-bation amplitude at later stage. Fig.4 shows the evolutionsof perturbation amplitude obtained from numerical simula-tions as well as theoretical model of Mikaelian for all threecases. As shown in Fig.4, the results of Mikaelian’s theoreti-cal model agree well with our numerical results at early stagefor all three cases (see Region I for each case). Additionally,there are slight differences between a ( t ) _BP and a ( t ) _BPRTduring this stage, which means that BP effect is the domi-nant factor for the growth of perturbation amplitude at earlystage. However, at later stage (see Region II for each case),the evolutions of a ( t ) _BP will increasingly diverge from thenumerical results, while the evolutions of a ( t ) _BPRT can stillmimic the numerical results although there are some differ-ences between them. These results indicate that the RT effectplays an important role in in the development of perturbationat later stage before re-shock (even overwhelms the BP effectat the very later stage before re-shock since the amplitudes forall three cases are decreasing at the end of Region II).Moreover, according to the theoretical model of Mikaelian[see Eq.(19)], at the early stage when the BP effect domi-nates the amplitude growth, the nondimensionalized ampli-tude a ( t ) / a can be approximately given by a ( t ) a ≈ ( − V si V is )[ + C ( nA − ) V si r ] . (20)In the above formula, r and C = (cid:82) tt + r dt would become ap-proximately constant since the initial amplitude is small. Forall three cases in the present study, V si and V is are the samesince the Mach number of the incident CCSW and the initialdensities inside and outside the interface are the same. Con-sequently, during this stage, a ( t ) / a should be approximatelyproportional to the wave number of initial interface. Fig.5compares the evolutions of nondimensionalized amplitude forthree numerical simulations (versus nondimensionalized time t / t wall , where t wall = . ms is the approximate time when theCCSW strikes the inner wall boundary). As shown in Fig.5, due to the same wave numbers of initial perturbations, thenondimensionalized amplitudes of Case I and Case II growalmost in the same way at the early stage. On the other hand,due to a larger wave number of initial perturbation, the nondi-mensionalized amplitude of Case III is larger than those ofCase I and Case II during the corresponding stage.According to the above analyses, we can see that, at earlystage, the BP effect is the dominant factor of the perturbationgrowth for CCSW driven interfaces, and the amplitude growthobtained from the theoretical prediction of Mikaelian’s modelagrees well with our numerical simulation results. However,at later stage, the RT effect becomes important and wouldeven overwhelm the BP effect. Moreover, during the laterstage, although there are some differences between the resultsof theoretical prediction (including RT effect) and our nu-merical simulations, the evolutions of a ( t ) _BPRT can mimicthe trend of our numerical results on the whole. In fact,such differences of amplitude growth have also been ob-served between the results of Mikaelian’s theoretical modeland shock tube experiments . There are several factorswhich can account for the above differences between the evo-lutions of a ( t ) _BPRT and the results of numerical simula-tions/experiments. The main one is that the amplitude of per-turbation becomes larger at later stage as the flows evolve.Consequently, the assumption of Mikaelian’s model, whichrequires that the amplitude of perturbation should be small, isnot quite valid anymore. Another one is that, in practice, wecan not have the analytical formula for r ( t ) . Consequently, theapproximate formulation of r ( t ) could introduce errors, espe-cially for the term of RT effect since it involves the second-order derivative of r ( t ) . In summary, all the results mentionedabove, to some extent, not only could demonstrate the featuresof amplitude growth of perturbation for CCSW driven inter-faces, but also can manifest that the CCSW derived from theaforementioned theory of shock tube indeed can be used forthe studies of CCSW induced RMI flows. IV. FLUIDS’ MIXING OF CCSW DRIVEN INTERFACES
The fluids’ mixing in the mixing zone of gases interface isanother crucial topic for shock-driven inhomogeneous flows.Better understandings of the mixing behaviors of shock-drivenflows can shed light on the mechanisms of turbulent mixing as well as the turbulence modeling for such kind of flows .Recently, the mixing behaviors of planar shock driven inter-face are widely studied . However, the corresponding be-haviors of gases interface driven by converging shock wave,which are more important for some scientific disciplines suchas ICF and supernova explosions, remain to be further inves-tigated. In this section, we follow the approach of implicitlarge eddy simulations (ILES) to have a primary study on themixing behaviors of three-dimensional (3D) gases interfacesdriven by CCSW. Moreover, to take the effects of mode ofinitial perturbation on the mixing behaviors into account andto make our results be more general, two cases with differentamplitudes and mode numbers are studied. (a) Case I (b) Case II(c) Case III FIG. 4. Comparisons of amplitude growth between numerical results and theoretical results
A. Problems setup
As remarked by Mikaelian , the fluids’ mixing of gasesinterface driven by converging shock wave would not happendramatically if the wave length of initial perturbation is muchlarger than the amplitude of initial perturbation. In order toenhance the fluids’ mixing at later stage after the gases inter-faces are re-shocked, the initial perturbation for both cases inthe present study are set to be a linear combination of "egg-carton" , which, in the cylindrical coordinate system, can be addressed as η ( ϕ , z ) = r − a × (cid:12)(cid:12) cos ( n ϕ ϕ ) cos ( n z z ) (cid:12)(cid:12) + δ − r δ . (21)In the above formula, r and a are, respectively, the crestradius and the amplitude of the initial interface; n ϕ and n z are, respectively, the azimuthal mode number and axial modenumber; r = (cid:112) x + y is the radius away from the center and ϕ is the azimuthal angle. Combined with an initial diffusionlayer (with thickness δ = . cm ) of the form proposed byLatini , the mass fraction of SF inside the CCSW can be FIG. 5. Evolutions of nondimensionalized amplitude for all threecases in numerical simulations. addressed as Y SF = Y SF , i f η (cid:62) . Y SF , × ( . − e | η | ln β ) i f (cid:54) η < . i f η < , (22)where β is the machine zero. According to the above formu-lation, at initial time, the mass fraction of SF for the mix-ture of SF and air inside the interface (the diffusion layer)is Y SF , ( Y SF , = .
75 for both cases), while the gas is pureair outside the interface. The parameters of the initial per-turbations for both cases are listed in Table II. Furthermore,we initially place the CCSW with desired Mach number of M s = . R = . m for both cases. The thermalstates of air in the region between the initial gases interface(diffusion layer) and the CCSW are the same as that we set inthe second part of Section III, which will result in the samethermal states of air outside the initial CCSW. Moreover, forthe present 3D simulations, we extend the axial width, L z ,to be 0.128 m . The configurations of initial flow field areshown in Fig.6. As for boundary conditions, a viscous cylin-drical wall with radius of 1 cm is placed around the center,and the periodic boundary condition is used along axial di-rection. Additionally, in order to avoid the effects of wavesreflected by the outer boundary, the cylindrical outer bound-ary is far away from the flow structure evolving region (theregion with fine grid). To reduce computational costs, a hy-perbolic mesh stretching is applied between the fine-grid do-main and the outer boundary along the radial direction. Thewhole computational domain is discretized by a body-fittedmesh with total cells of 1024 × ×
128 (azimuthal cells × radial cells × axial cells).Due to the moderate grid resolution, we hereby remark thatthe numerical dissipation would overweigh the physical onefor the present simulations. Consequently, the present studies TABLE II. Parameters of initial perturbations for CCSW inducedinterfacial fluids’ mixingCase Index r ( m ) a ( mm ) n ϕ n z . . (a) Case 1(b) Case 2 FIG. 6. Configurations of initial flow field for both cases. can be categorized as a class of ILES, in which the equationsare implicitly filtered by the discretization and the numericaldissipation is treated as a surrogate for an explicit subgrid-scale model . As remarked by Grinstein and Attal , al-though ILES could only resolve the length scales of turbu-lent mixing driven by advection and convective stirring, manystudies show that they are indeed suited to (moderate) high-Reynolds-number flows in which shocks and interfaces arepresent . Moreover, our previous study on reshockedheavy gas curtain shows that the fluids’ mixing behaviors ofcoarse grid (corresponding to the results of ILES) agree wellwith those of fine grid (corresponding to the results of directnumerical simulation) statistically , which, to some extent,manifests that our numerical method is appropriate for ILES. B. Results and Discussions
1. Wave patterns and flow structures evolutions
The wave patterns of RMI induced by converging shock aremore complicated than those of RMI induced by planar shock,which would affect the resulting evolutions of flow field tosome extent. Fig.7 shows the typical wave patterns of CCSWinduced RMI (results of Case 1 in the axial view). At earlystage ( t = . ms ) shown in Fig.7(a), the reflected expansionfan (REF), which is associated with the incident CCSW andgenerated at the initial time, would propagate outward all thetime. Additionally, as the results of interaction between theincident CCSW and initial interface, the reflected shock wave(RSW) will always propagate outward while the first trans-mitted shock wave (FTSW) will propagate inward initially.Obviously, the REF and the RSW would not impose much ef-fects on the later evolutions of the flows since they propagateoutward all the time and would not be reflected by the outerboundary which is large enough. However, the FTSW will bereflected by the inner wall boundary and re-impact the gasesinterface, which will result in the second transmitted shockwave (STSW) and the reflected rarefaction wave (RRW) atlater time ( t = . ms ) as shown in Fig.7(b).To identify the propagations of shock waves and their ef-fects on the evolutions of flow structures more clearly, Fig.8and Fig.9 show the details of flow evolutions and the propaga-tions of shock waves for Case 1 and Case 2, respectively. Asshown in Fig.8(a) and Fig.9(a), at early stage ( t = . ms ),the gases interfaces will move inward with growth of perturba-tion amplitudes since the FTSWs will induce inward radial ve-locities. However, the inward propagating FTSWs will be re-flected by the inner wall boundary. Then, the reflected FTSWswill propagate outward and begin to re-impact the gases inter-faces, which are well shown in Fig.8(b) and Fig.9(b). Afterthe reflected FTSWs re-impact the gases interfaces, the gasesinterfaces will move outward since the resulting STSWs willpropagate outward and induce outward main radial velocities[see Fig.8(c)-(d) and Fig.9(c)-(d)]. Moreover, based on theevolutions shown in Fig.8 and Fig.9, we can see that the flowstructures are characterized by the growth of perturbation am-plitudes and the fluids’ mixing is not intensive at early stage(Therefore, we just show one quarter of the flow fields). How-ever, the fluids’ mixing is dramatically enhanced by the sec-ond RMI after the gases interfaces are re-shocked [see the SF mass fraction iso-surface shown in Fig.8(c)-(d) and Fig.9(c)-(d)].The morphological patterns of waves and the motions offlow structures (the evolutions of positions of inner and outerinterfaces) mentioned above for both cases are depicted quan-titatively in Fig.10. As shown in Fig.10, there are someunique features for the CCSW induced RMI flows. One isthat the FTSW will move faster as it propagates inward dueto the deformation (decreasing area) of shock surface. This isquite different from the RMI flows induced by planar shockwave, since, for planar shock driven RMI flows, the transmit-ted shock wave will propagate forward with a nearly constantvelocity . Another feature is that the movements of innerand outer gases interfaces are nonlinear versus time before re-shock, while, for planar shock driven RMI flows, the shockedinterface will move forward with an approximately constantvelocity as well . Obviously, the nonlinear movementsof inner and outer gases interfaces before re-shock will resultin the nonlinear growth of perturbation amplitudes or mixingzone width at the very beginning (see Fig.11 shown below). (a) t = ms (b) t = ms FIG. 7. Typical wave patterns of the CCSW induced RMI.
Moreover, we hereby remark that the model of Mikaelianmentioned in the second part of Sections III would not be ap-plicable to the nonlinear growth of perturbation amplitudesfor these two cases, since, for both cases, their ratios of initialamplitude to initial wave length are not small enough.
2. Scaling law of mixing width
The self-similar scaling laws of mixing width for theRMI flow induced by planar shock wave are extensivelyinvestigated . For the planar shock induced RMIflows without re-shock, Dimonte initially demonstrated thatthe mixing width is an exponential function versus time δ ( t ) ∼ t σ . For the planar shock induced RMI flow with re-shock, later studies of Thornber and Young show that themixing width after re-shock scales as δ ( t ) ∼ ( t − t (cid:48)(cid:48) ) σ r , where t (cid:48)(cid:48) is a virtual time and always set to be the time of reshock-ing instant. Moreover, the exponents σ and σ r for the abovescaling laws would be varied depending on some factors such0 (a) t = ms (b) t = ms (c) t = ms (d) t = ms FIG. 8. Evolutions of flow structures and shock waves propagations for Case 1. as Mach number of incident shock wave . Recently, the evo-lutions of mixing width for the RMI flows induced by con-verging shock wave are paid more attention , while thecorresponding scaling laws are not fully reported and remainto be open issues. Consequently, in this part, we try to gainfurther insights into the scaling laws of mixing width for RMIflows induced by CCSW.In the cylindrical coordinate system, the mixing width forRMI flows induced by CCSW can be defined as δ ( t ) = (cid:90) r IMZmax r IMZmin < Y SF > ϕ z ( − < Y SF > ϕ z ) dr . (23)In the above formulation, r IMZmin and r IMZmax are, respectively, theminimum radius and maximum radius of the inner mixingzone (IMZ) in which the average mass fraction of sulphur hex-afluoride < Y SF > ϕ z ∈ [ . , . ] . Additionally, for arbitraryscalar φ , < φ > ϕ z is the ensemble average of φ on the cylin- drical shell (in ϕ z plane), which is defined as < φ > ϕ z ( r , t ) = π L z (cid:90)(cid:90) φ ( r , ϕ , z , t ) d ϕ dz . (24)The evolutions of mixing width versus time for both casesand their corresponding evolutions on Log-Log scale areshown in Fig.11(a) and Fig.11(b), respectively. As depictedin Fig.(9), the evolutions of mixing width for both RMI flowsinduced by CCSW almost follow the same scaling laws. Atearly stage, the scaling law of mixing width is δ ( t ) ∼ t . ,which is quite similar to the scaling law of mixing width forplanar shock driven RMI flow with 3D broadband perturba-tions on an initial interface . Additionally, at the earlierstage after re-shock, the mixing width for both cases scalesas δ ( t ) ∼ ( t − t (cid:48)(cid:48) ) . (with re-shocking instant t (cid:48)(cid:48) ≈ . ms ).Actually, this scaling law of mixing width for CCSW inducedRMI flows is also widely reported for RMI flows driven byplanar shock during the earlier stage after re-shock .However, at the later stage after re-shock for present two1 (a) t = ms (b) t = ms (c) t = ms (d) t = ms FIG. 9. Evolutions of flow structures and shock waves propagations for Case 2. (a) Case 1 (b) Case 2
FIG. 10. Morphological wave patterns and quantitative evolutions of flow structures. (a) Normal evolutions versus time(b) Evolutions versus time on a Log-Log scale FIG. 11. Temporal evolutions of mixing width.
CCSW driven RMI flows, the mixing width seems to scaleas δ ( t ) ∼ t σ again, while the scaling exponent becomes σ ≈ .
75. This recovering scaling law of δ ( t ) ∼ t σ at later stage af-ter re-shock seems to be unique for CCSW driven RMI flowssince no other similar results have been reported for planarshock driven RMI flows. It should be noted that, although wewitness the same scaling laws of mixing width for the presenttwo CCSW induced RMI flows, the scaling exponents at cor-responding stages would be varied for other cases since theywould largely depend on the mode of initial perturbations, im-pulsive Mach numbers and other factors .
3. Temporal asymptotic behaviors of mixing parameters
As mentioned above, although the flow structures are char-acterized by the growth of perturbation amplitude before re-shock, the fluids’ mixing is dramatically enhanced after re-shocked. To figure out the level of fluids’ mixing as well asthe isotropy/homogeneity properties of the mixing zone, wequantitatively investigate the temporal asymptotic behaviorsof molecular mixing fraction Θ , local anisotropy a i , vol anddensity-specific volume correlation b vol in this subsection.The molecular mixing fraction can characterize the relativeamount of molecularly mixed fluid within the mixing layer. Itcan be interpreted as the ratio of molecular mixing to large-scale entrainment by convection motion. Following the defi-nition of Youngs , in the cylindrical coordinate system, theformulation of molecular mixing fraction can be expressed as Θ ( t ) = (cid:82) r IMZmax r IMZmin < Y SF ( − Y SF ) > ϕ z dr (cid:82) r IMZmax r IMZmin < Y SF > ϕ z ( − < Y SF > ϕ z ) dr . (25)The temporal asymptotic behaviors of the molecular mix-ing fraction for present two CCSW induced RMI flows areshown in Fig.12. As shown in Fig.12, the ratios of molec-ular mixing to large-scale entrainment by convection motionare relatively small for both cases at the early stage beforere-shock, while they increase as the instabilities evolve. Ad-ditionally, the molecular mixing between fluids for both casesis sharply increased, to some extent, after the gases interfacesare re-shocked by the reflected FTSW (after t ≈ . × − s ).Moreover, at the later stage after re-shock, the evolutions ofmolecular mixing fraction for both cases become asymptotic,with a final value being 0.93 approximately. Actually, theabove asymptotic behavior of molecular mixing fraction forthe present CCSW induced RMI flows highly resembles thatfor some planar shock driven RMI flows which are well re-ported numerically and experimentally For mixing flows, the anisotropy and inhomogeneity are ofsignificance since both of them are important to large-eddyand Reynolds-averaged Navier-Stokes modeling . To fig-ure out the properties of local anisotropy and inhomogene-ity of fluids’ mixing for the present two CCSW driven RMIflows, we investigate the temporal asymptotic behaviors of thevolume-averaged anisotropy a i , vol and the volume-averageddensity-specific volume correlation b vol , respectively. Theirformulations are given as follows a i , vol = ∆ r (cid:90) r IMZmax r IMZmin < (cid:12)(cid:12)(cid:12) u (cid:48)(cid:48) i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u (cid:48)(cid:48) r (cid:12)(cid:12) + (cid:12)(cid:12) u (cid:48)(cid:48) ϕ (cid:12)(cid:12) + (cid:12)(cid:12) u (cid:48)(cid:48) z (cid:12)(cid:12) − > ϕ z dr , (26) b vol = ∆ r (cid:90) r IMZmax r IMZmin ( < ρ > ϕ z < ρ > ϕ z − ) dr . (27)In the above two equations, ∆ r = r IMZmax − r IMZmin is the length ofthe inner mixing zone in the radial direction. Additionally, for3
FIG. 12. Temporal evolutions of the molecular mixing fraction. arbitrary scalar φ , its fluctuating part, φ (cid:48)(cid:48) , is given by φ (cid:48)(cid:48) = φ − φ , (28)where φ = < ρφ > ϕ z / < ρ > ϕ z is the ensemble Favre av-erage of φ on the cylindrical shell. Moreover, for i = r , φ and z , u (cid:48)(cid:48) i respectively indicates the fluctuating part of radial,azimuthal and axial velocity, and, a i , vol respectively denotesthe corresponding volume-averaged anisotropy in radial, az-imuthal and axial direction.Based on Eq.(26), the volume-averaged anisotropy a i , vol would range from − to , manifesting the ratio of the TKEin a specific direction to the total TKE. Larger value of a i , vol implies larger fluctuation or TKE in the corresponding direc-tion. Moreover, the positive value of a i , vol indicates that theTKE in the corresponding direction is dominant, while thenegative value of a i , vol implies that the TKE in the correspond-ing direction is less important. For isotropically mixing flow, a i , vol should be almost zero in all directions. Fig.13 showsthe temporal evolutions of the volume-averaged anisotropy inall three directions for both cases. As shown in Fig.13, thevolume-averaged anisotropy in each direction would achievea final asymptotic value for both case. However, the temporalasymptotic behavior of a i , vol for the present CCSW inducedRMI flows is different from that for planar shock driven RMIflows. For the planar shock driven RMI flows, the magnitudeof a i , vol in all directions would achieve a very small asymp-totic value at later stage , implying that the flows would be-come much less anisotropic. However, for the present CCSWdriven RMI flows during the duration of simulations, the tem-poral asymptotic values in three directions follow the law of a r , vol > > a ϕ , vol > a z , vol and the magnitudes of a r , vol and a z , vol are much larger than zero to some extent. These re-sults indicate that, during the duration of simulations, the flu-ids mixing would always be anisotropic for the present CCSWdriven RMI flows. Actually, the above results are consistent FIG. 13. Temporal evolutions of the anisotropy. with the corresponding evolutions of TKE. For CCSW drivenRMI flows, the TKE in the radial direction and the TKE inthe azimuthal direction are always more important than theTKE in the axial direction, simply because the converging andexpanding effects will continuously perturb the flow field inthese two directions.The volume-averaged density-specific volume correlationis critical in second-moment turbulence modeling for variabledensity flows . According to Eq.(27), b vol is a non-negativeparameter. For nearly homogeneous flow, b vol would becomevery small. However, if the fluids’ mixing is spatially inhomo-geneous, the value of b vol would be large. Fig.14 shows thetemporal evolutions of the volume-averaged density-specificvolume correlation for the present two CCSW induced RMIflows. As shown in Fig.14, the volume-averaged density-specific volume correlation for both cases is decreasing onthe whole. Moreover, at the later stage after re-shock, thevolume-averaged density-specific volume correlation for bothcases would asymptotically achieve the same relatively smallvalue of 0.04. These results imply that the fluids’ mixing forthe present CCSW driven flows would become much less in-homogeneous at later stage after re-shock.
4. Turbulent kinetic energy spectrums
As mentioned by Tritschler , a fully isotropic mixing zoneis never obtained for shock induced fluids’ mixing flows, al-though the fluid’s mixing would become less anisotropy andless inhomogeneous at later stage. However, the theory ofTKE spectrum for homogeneous isotropic turbulence is oftenused as the theoretical framework for the numerical analysesof shock induced RMI flows . According to this theory, therewould be a broadened inertial range in TKE spectrums oncethe fluids’ mixing is enhanced. Therefore, alternative to the4 FIG. 14. Temporal evolutions of the density-specific volume corre-lation. aforementioned analyses in physical space, the enhanced flu-ids’ mixing after re-shock is analyzed in Fourier representa-tion for both cases in this subsection. Additionally, due to themoderately small cell number in the axial direction, the 2DTKE spectrums analyses in ϕ z plane would conceal the char-acteristics of TKE spectrums at relatively high wave numbersin the azimuthal direction (which would be more importantfor the present CCSW induced RMI flows since, as mentionedabove, the TKE in the azimuthal direction would be more im-portant to some extent). Consequently, only the TKE spec-trums in the azimuthal direction after re-shock are analyzedfor both cases in this subsection.In the cylindrical coordinate system, the average TKE spec-trum for the inner mixing zone in the azimuthal direction isgiven by E a ( k ϕ , t ) = ∆ r (cid:90) r IMZmax r IMZmin < E ( r , k ϕ , z , t ) > ϕ z dr , (29)where E ( r , k ϕ , z , t ) is the TKE spectrum of wave number k ϕ in azimuthal direction at time t on specific radial r and axialposition z . The formulation of TKE spectrum is given by E ( r , k ϕ , z , t ) = (cid:98) u (cid:48)(cid:48) r (cid:98) u (cid:48)(cid:48) r ∗ + (cid:99) u (cid:48)(cid:48) ϕ (cid:99) u (cid:48)(cid:48) ϕ ∗ + (cid:98) u (cid:48)(cid:48) z (cid:98) u (cid:48)(cid:48) z ∗ . (30)In the above equation, for arbitrary scalar φ , (cid:98) φ denotes itsFourier transform in the azimuthal direction, and (cid:98) φ ∗ indicatesthe corresponding complex conjugate of (cid:98) φ . Fig.15 shows theaverage TKE spectrums at three instants of the later stage af-ter re-shock for the present two CCSW driven RMI flows. Asdepicted in Fig.15, at the relatively early stage after re-shock ( t = . ms ) , the decaying law of k − / for the TKE spec-trums of these two cases only locates in a narrowband of wavenumbers. However, as the flows evolve, the TKE spectrums (a) Case 1(b) Case 2 FIG. 15. Average TKE spectrums in the azimuthal direction. for both cases would decay with a slope of k − / in a broaderrange of wave numbers. These results imply that the inertialrange is extended during the developing process and manifest,to some extent, that the fluids’ mixing is enhanced at later timeafter re-shock. V. CONCLUSIONS
Based on the theory of shock tube, we successfully gener-ate "pure" CCSW without a following contact surface. Addi-tionally, studies on the amplitude growth of gases interfacesmanifest that the generated CCSW is efficient for studying on5CCSW induced RMI flows. Then, the instabilities and fluids’mixing behaviors of two gases interfaces driven by CCSWare numerically investigated using high resolution FV method.The morphological wave patterns and the evolutions of flowstructures imply that the instabilities of the interfaces are char-acterized by the growth of perturbation amplitude before re-shock, while the fluids’ mixing is dramatically enhanced afterre-shock. Detailed analyses of the fluids’ mixing parametersshow that the evolutions of mixing width and other mixingparameters could achieve the same laws of temporal behaviorfor the present two CCSW induced RMI flows, which indi-cates the existences of scaling law and temporal asymptoticbehaviors for fluids’ mixing parameters in the mixing zone.Additionally, due to the converging/expanding effects on theflow fields, the motions of shock wave and inner/outer gasesinterfaces for the present CCSW driven RMI flows are non-linear versus time, which is quite different from the resultsof planar shock wave induced RMI flows. These nonlineardevelopments of flow fields would lead to some unique fea-tures for the fluids’ mixing of CCSW driven RMI flows, oneof which is that, at later stage after re-shock, the fluids’ mixingwould be less isotropic than that of planar shock wave inducedRMI flows although both of them would reach final temporalasymptotic behaviors in each direction. Further analyses ofTKE spectrums in the azimuthal direction at later stage afterre-shock also witness the k − / decaying law of TKE spec-trums for the present CCSW driven RMI flows. Both the tem-poral behaviors of mixing parameters and the decaying lawof TKE spectrums manifest that the fluids’ mixing is indeedenhanced at later time after re-shock. AVAILABILITY OF DATA
Raw data were generated at the TH-2 Supercomputer. De-rived data supporting the findings of this study are availablefrom the corresponding author upon reasonable request.
ACKNOWLEDGMENTS
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