RT-RM-Menon

A multiphase buoyancy-drag model for the study ofRayleigh-Taylor and Richtmyer-Meshkov instabilitiesin dusty gases

KAUSHIK BALAKRISHNAN1AND SURESH MENON2

1Computing Sciences, Lawrence Berkeley National Laboratory, Berkeley, California2School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, Georgia

(RECEIVED 16 October 2010; ACCEPTED 4 January 2011)

Abstract

A new multiphase buoyancy-drag model is developed for the study of Rayleigh-Taylor and Richtmyer-Meshkovinstabilities in dusty gases, extending on a counterpart single-phase model developed in the past by Srebro et al.(2003). This model is applied to single- and multi-mode perturbations in dusty gases and both Rayleigh-Taylor andRichtmyer-Meshkov instabilities are investigated. The amplitude for Rayleigh-Taylor growth is observed to becontained within a band, which lies within limits identified by a multiphase Atwood number that is a function of thefluid densities, particle size, and a Stokes number. The amplitude growth is subdued with (1) an increase in particlesize for a fixed particle number density and with (2) an increase in particle number density for a fixed particle size.The power law index for Richtmyer-Meshkov growth under multi-mode conditions also shows dependence to themultiphase Atwood number, with the index for the bubble front linearly decreasing and then remaining constant, andincreasing non-linearly for the spike front. Four new classes of problems are identified and are investigated forRayleigh-Taylor growth under multi-mode conditions for a hybrid version of the model: (1) bubbles in a pure gasrising into a region of particles; (2) spikes in a pure gas falling into a region of particles; (3) bubbles in a region ofparticles rising into a pure gas; and (4) spikes in a region of particles falling into a pure gas. Whereas the bubblesaccelerate for class (1) and the spikes for class (4), for classes (2) and (3), the spikes and bubbles, respectively,oscillate in a gravity wave-like phenomenon due to the buoyancy term changing sign alternatively. The spike or bubblefront, as the case may be, penetrates different amounts into the dusty or pure gas for every subsequent penetration, dueto drag effects. Finally, some extensions to the presently developed multiphase buoyancy-drag model are proposed forfuture research.

Keywords: Buoyancy-Drag model; Dusty gas; Hydrodynamic instability; Rayleigh-Taylor; Richtmyer-Meshkov

1. INTRODUCTION

The Rayleigh-Taylor (RT) instability occurs when an inter-face between two fluids with different densities is acceleratedin a direction normal to the interface from the heavy to thelight fluid. This instability was first investigated by LordRayleigh (1883) and later by Taylor (1950). The Richtmyer-Meshkov (RM) instability occurs when the interface is im-pulsively accelerated, such as for instance by a shock wave.This instability was first theoretically shown by Richtmyer(1960) and later experimentally verified by Meshkov(1969). Both single-mode as well as multi-mode RT and

RM have been investigated in the past, where single-moderefers to the presence of only one wavelength, λ, in the initialspectrum of perturbation length scales, and multimode refersto the presence of multiple wavelengths. In the case of multi-mode perturbations, smaller structures compete and merge,resulting in the formation of larger structures—an inversecascade process. Here, competition and coalescence ofcoherent structures dictates the overall evolution of themixing layer, because of the reduced drag per unit volumeof the larger structures (Alon et al., 1995). In both RT aswell as RM, for multi-mode perturbations, small hydrodyn-amic structures grow into larger scales, and later result in aturbulent mixing layer.

Across an interface separating two fluids with densities ρ1 andρ2, the Atwood number, defined as A= (ρ2− ρ1)/(ρ2+ ρ1),

201

Address correspondence and reprint requests to: Kaushik Balakrishnan,Computing Sciences Department, Lawrence Berkeley National Laboratory,1 Cyclotron Road, Berkeley, CA 94720. E-mail: [email protected]

Laser and Particle Beams (2011), 29, 201–217.©Cambridge University Press, 2011 0263-0346/11 $20.00doi:10.1017/S0263034611000176

has been identified in past studies to play a significant rolein the overall growth trends of the mixing layer. Both RTand RM mixing fronts grow as “bubbles” of lighter fluid“rising” into the heavier fluid, and “spikes” of heavier fluid“falling” into the lighter fluid. Both bubbles and spikesgrow with time, thereby resulting in a mixing layer. Thebubble and spike fronts in the classical single-mode RTgrow exponentially initially, and later transition to a lineargrowth regime (Oron et al., 2001; Goncharov, 2002). Pastnumerical studies of multimode RT have also reportedmemory loss of the perturbations with time (Youngs, 1984,1989, 1991, 1994), where a spectrum of short initial wave-length grow into fewer perturbations corresponding tolarger wavelengths at later times due to the competitionand coalescence of the coherent structures. Furthermore,the mixing zone amplitude, h, was reported to grow as ∼t2

for RT, where t denotes the time; this is now a well estab-lished result and serves as a useful model/simulation vali-dation for multi-mode RT. These studies also confirm thatthe growth rates and the overall mixing phenomena aredifferent for 2D and 3D simulations. See Sharp (1984) fora detailed review on RT. Layzer (1955) investigated therise of a single bubble in a heavy fluid using a potentialflow model, but limited to A= 1. It was shown for abubble rising between parallel plates that the vertex heightincreases exponentially initially, and later at constant rates;analytical expressions based on Bessel function solutionswere derived. Detailed theoretical analysis of perturbationgrowth can also be found in the classical work of Chandrasekhar(1981). Experiments on RT have also been carried out byDalziel (1993) and Dimonte & Schneider (2000), confirmingthe h∼ t2 growth.Concurrent to the above studies, hydrodynamic instabil-

ities have also been investigated theoretically, numerically,as well as by experiments. Extending on Layzer’s work,Alon et al. (1994, 1995) used theoretical models applicablefor all A to study RT and RM, and showed that whereasthe RT mixing front grows as h∼ t2, the RM mixing frontgrows as h∼ tθ, with θ∼ 0.4 in 2D and∼ 0.25 in 3D.Later, the same research group employed numerical simu-lations as well as two different theoretical models: (1) a stat-istical mechanics bubble-merger model; (2) buoyancy-dragmodel to investigate RT and RM (Oron et al., 2001). Differ-ent mixing layer growth trends were reported for 2D and 3Dscenarios. Later, they extended the buoyancy-drag model(Srebro et al., 2003) to more generic RT and RM cases,and investigated the linear and non-linear stages of the evol-ution. Self-similar growth was reported at late times, i.e.,when the bubble amplitude (hB) grows in proportion to thewavelength (hB/λ= b(A)), with the proportionality constant,b, being only a function of A. Mikaelian (2003) investigatedRT and RM bubble growths using analytical expressions,including the linear as well as the non-linear regimes for arange of Atwood numbers, A. Experiments on RM havealso been carried out by Erez et al. (2000), Chapman andJacobs (2006), Wilkinson and Chapman (2007) and Leinov

et al. (2009), with the first and the last study also focusingon re-shocked RM, where the shock wave reflects from anend wall and re-shocks the mixing zone. Latini et al.(2007) and Schilling et al. (2007) have also studied the re-shocked RM using robust numerical simulations, and pre-sented late time turbulent kinetic energy spectra amongother results. A detailed review of the theory of RMgrowth can be found in Brouillette (2002).RT and RM have also been investigated theoretically and

numerically under blast wave driven conditions. Miles (2004,2009) studied the growth of RT and RM in supernovaexplosions using a bubble merger and a buoyancy-dragmodel, accounting for the spherical nature of the problem.Self-similar growth was reported at late times, includingpartial retention of memory of the initial conditions. Very re-cently, these studies were extended to the investigation of RTensuing from multiphase chemical explosions by the currentauthors (Balakrishnan & Menon, 2010; Balakrishnan et al.,2011; Balakrishnan, 2010), and partial memory retentionof the initial perturbations was reported, thereby drawing asimilarity between the observations reported earlier for RTin supernovae, and chemical explosions. In another recentstudy, Ukai et al. (2010) investigated RM in dilute gas-particle mixtures by extending studies applied earlier forKelvin-Helmoltz (KH) instabilities in dusty gases. A theor-etical model was derived and the predicted growth rateswere in accordance with 2D simulations, thereby openingup a new class of problems involving RT and RM in dustygases.To complement experiments and simulations, theoretical

models that can predict growth rates can be very useful, in-cluding also for dusty gases; for the remainder of thispaper, we will interchangeably use the words “dust” or “par-ticles” with the understanding that both refer to the same.Theoretical models must account for the different physicalphenomena that are of relevance to the dusty gas RT andRM, such as flow acceleration effects on bubbles andspikes, drag effects as bubbles rise and spikes fall, inter-phase interaction effects, particle/dust loading effects, etc.Of particular interest in this study is to extend the Buoyancy-drag model developed by Srebro et al. (2003) to dusty gases.Doing so enables the evaluation of RT and RM evolutionfronts in dusty gases, for future comparisons with numericalsimulations. Such theoretical models can be easily used topredict growth rates and offer valuable insights into thedust induced mixing of fluids.Saffman (1962) developed a theoretical two-phase model

to study perturbation growth in dusty gases. Then, Michael(1964) applied this theory to plane Poisuelle flow of dustygases. Recently, Ukai et al. (2010) extended this model tothe investigation of RM instabilities in dusty gases. The pri-mary motivation of this study is to extend ideas from thesethree studies and develop a simple, analytical, two-phasemodel that can predict perturbation growth rates in both RTand RM instabilities in dusty gases. Doing so enables theinvestigation of RT and RM in many two-phase natural

K. Balakrishnan & S. Menon202

and engineering applications such as internal confinementfusion, chemical explosions, nuclear explosions, spray com-bustion engines, etc.This paper is organized as follows. In Section 2, the gov-

erning equations and the numerics are presented, including abrief overview of the baseline buoyancy-drag (BD) modeldeveloped for single phase flows by Srebro et al. (2003).In Section 3, the multiphase buoyancy-drag model is derived,mutatis mutandis to the formulation presented by Srebroet al. (2003). In Section 4, the results obtained by the multi-phase BD model are presented and the intricacies of the two-phase mixing phenomena are elucidated. Parametric studiesare also conducted, with the identification of new classesof dusty gas problems. Finally, in Section 5, the conclusionsdrawn from this study are elaborated.

2. METHOD OF STUDY

In this study, we focus on extending the classical BD modelof Srebro et al. (2003) to the study of RT and RM in multi-phase mixtures; i.e., to derive and apply a multiphaseBuoyancy-Drag (MBD) model. For constant and continuousacceleration flows, we can directly apply the BD/MBDmodels with a chosen acceleration. However, for impulsiveor time varying acceleration flows, it is essential to computethe exact acceleration profile as it changes with time; thisrequires the solution of the 1D multiphase/dusty gas Eulerequations in order to compute the “unperturbed” interfacemotion, from which the interface acceleration profiles canbe obtained to serve as an input to the BD/MBD models.In this section, the classical BD model as proposed bySrebro et al. (2003) is first discussed, followed by the 1Dmultiphase Euler equations, and the numerical methodologythat is employed to solve these governing equations.

2.1. Buoyancy-Drag Model of Srebro et al. (2003)

Extending on Layzer’s (1955) work, Srebro et al. (2003)generalized the BD model to obtain the following equationsfor the bubbles and spikes:

ρ1 + Caρ2( ) duB

dt= ρ2 − ρ1

( )g(t)− Cdρ2

u2Bλ, (1)

ρ2 + Caρ1( ) duS

dt= ρ2 − ρ1

( )g(t)− Cdρ1

u2Sλ, (2)

where uB and uS denote, respectively, the velocities of thebubbles and spikes, g(t) is the time varying interface accel-eration, and λ is the perturbation wavelength. Ca and Cd

denote, respectively, the added mass and bubble or spikedrag coefficients; essentially, these equations state that thenet acceleration or deceleration of a bubble or spike is thedifference between the buoyancy and drag forces acting.The left-hand side represents the total inertia of the bubbleor spike and the inertia of the added mass (i.e., the mass of

the fluid that is pushed by the rising bubble or fallingspike). The coefficients Ca and Cd take the followingvalues depending on 2D or 3D (Srebro et al., 2003):

Ca = 2(2D); Ca = 1(3D), (3)

Cd = 6π(2D); Cd = 2π(3D).

The bubble amplitude, hB and the spike amplitude, hS arethen obtained by integrating the expressions:

uB = dhBdt

; uS = dhSdt

. (4)

Srebro et al. (2003) then extend the BD model by includingthe amplitude dependence through the parameter E(t)=e−CekhB, where k= 2π/λ is the wavenumber, thereby obtain-ing the generalized BD model equations:

CaE(t)+ 1( )ρ1 + Ca + E(t)( )ρ2[ ] duB

dt

= 1− E(t)( ) ρ2 − ρ1( )

g(t)− Cdρ2u2Bλ,

(5)

CaE(t)+ 1( )ρ2 + Ca + E(t)( )ρ1[ ] duS

dt

= 1− E(t)( ) ρ2 − ρ1( )

g(t)− Cdρ1u2Sλ,

(6)

with the coefficient, Ce= 3 (2D), 2 (3D).The BD model described hitherto is valid for single mode

perturbations only. For multimode perturbations, Srebroet al. (2003) replace λ in the above equations with a charac-teristic wavelength, λ, with the assumption that the BDmodelgoverning equations, Eqs. (5)–(6), are applicable with thismodified λ. During the early linear growth regime, λ remainsconstant, but during the late time asymptotic regime, λ growsin a self-similar fashion, i.e., in proportion to the bubble am-plitude, hB. Thus, hB/λ = b(A), a function of the Atwoodnumber only. The primary assumption behind the use ofthe same λ for bubbles and spikes is that they have thesame periodicity, which results from the fact that the domi-nant bubbles inevitably generate the dominant spikes; seeAlon et al. (1995) and Srebro et al. (2003) for more discus-sions. Based on the experiments of Dimonte and Schneider(2000) and theoretical observations of Oron et al. (2001),Srebro et al. (2003) use the following values for b(A):

b(A) = 0.51+ A

(2D); b(A) = 1.61+ A

(3D). (7)

The characteristic wavelength for multimode perturbationsgrow as (Srebro et al., 2003):

dλ

dt=

0, hB < λb(A);uBb(A)

, hB ≥ λb(A).

⎧⎨⎩ (8)

A Multiphase buoyancy-drag model 203

Under this assumption, λ starts to increase through bubblecompetition, i.e., coalescence/merging of contiguousbubbles, only after the bubble amplitude hB has reached thevalue λb(A). This equation is solved in addition to Eqs. (5)and (6) for the multimode perturbation cases.In summary, the generalized BD model described here is

suitable for the study of both RT and RM, by using anappropriate acceleration g(t) profile. For RT, g(t) can remainconstant such as for instance in a gravitational field, or canchange in time if the interface is driven by a blast wave(Balakrishnan & Menon, 2010; Miles, 2009). For RM, onthe other hand, g(t) is impulsive and therefore starts from anon-zero value, but rapidly decays to zero. In order to obtainprofiles for g(t) that will be inputs to the BD model, it is re-quired to solve the 1D Euler equations; since this study focuseson RT and RM in dusty gases, the multiphase Euler equationsare considered and are now elaborated.

2.2. Numerical methodology

To solve the 1D multiphase Euler equations, the formulationpresented by Miura and Glass (1982) is used in this study. Inthis formulation, the gas is assumed to behave perfect, andviscosity and heat conductivity are neglected except for thecomputation of the interaction terms. The particles areassumed to be perfectly spherical of uniform size, and areassumed to obey continuum laws; the volume fraction ofthe particles are neglected. Stated in these terms, the 1Dmultiphase Euler equations for the continuity, momentumand energy equations for both phases are:

∂ρ∂t

+ ∂ ρu( )∂x

= 0, (9)

∂σ∂t

+ ∂ σv( )∂x

= 0, (10)

∂ ρu( )∂t

+ ∂ ρu2( )∂x

+ ∂p∂x

= − σ

mF, (11)

∂ σv( )∂t

+ ∂ σv2( )∂x

= σ

mF, (12)

∂ ρ CvT + (1/2)u2( )[ ]∂t

+ ∂ ρu CvT + (1/2)u2( )[ ]∂x

+ ∂ pu( )∂x

= − σ

mvF + Q( ),

(13)

∂ σ CmΘ+ (1/2)v2( )[ ]∂t

+ ∂ σv CmΘ+ (1/2)v2( )[ ]∂x

= σ

mvF + Q( ),

(14)

where (ρ, u, T, Cv) and (σ, v,Θ, Cm) are the (density, velocity,temperature, specific heat) of the gas and dust, respectively,and p is the gas pressure. The terms that appear on the right-hand side of the above equations are the inter-phase inter-action terms; here, m is the dust/particle mass, obtained asm = (π/6)d3pσp, where dp is the particle diameter, and σp

material density of the dust particles. F is the drag forceacting on a particle and Q is the heat transfer rate betweenthe two phases, and these terms are computed as follows:

F = π

8d2p ρ u− v( )|u− v|CD, (15)

Q = πdpNuκ T − Θ( ), (16)

where κ is the thermal conductivity of the gas; CD and Nudenote, respectively, the drag coefficient and Nusseltnumber, and are computed as:

CD = 24Re

1+ 0.15Re0.687( )

, (17)

Nu = 2+ 0.6Re0.5Pr0.333, (18)

where Re is the Reynolds number computed as Re= ρ|u−v|dp/μ, Pr is the Prandtl number which is assumed to be0.7, and μ is the viscosity of the gas assumed to be 1.5 ×10−5 Kg/ms. κ is computed as κ= μ Cp/Pr, where Cp isthe specific heat of the gas at constant pressure, obtained asCp= Cv+ R, where R is the gas constant. Thermodynamicclosure is obtained from the perfect gas equation of state:

p = ρRT . (19)

To solve the above governing equations, we use the MUSCLscheme (Monotone Upstream-centered Schemes for Conser-vation Laws) (van Leer, 1979) with a flattening procedure toreduce post-shock oscillations (Colella & Woodward, 1984).The gas fluxes are computed using the HLLC Riemannsolver (Toro, 1999) and the dust/particle fluxes are evaluatedusing the Rusanov flux scheme (Rusanov, 1961). Further-more, a predictor-corrector scheme is used for time inte-gration. Overall, the scheme is second order accurate intime and space. To validate the 1D two-phase methodology,many canonical tests have been performed, and one suchstudy is presented in the Appendix.The 1D numerical methodology presented here is used to

determine the initial conditions for the BD and MBD modelsused in the study of RM. The RM studies considered in thispaper correspond to a shock-tube configuration, and a gridresolution of 1000 is considered to resolve a 1 m longdomain. Analysis shows that with 5000 grid points used toresolve a 1 m long domain, the parameters of interest are in-sensitive with their corresponding values predicted with the1000 grid. Thus, 1000 grid points are used for resolving a1 m long domain. One can also employ high order accuratemethods like the spectral volume (Kannan & Wang, 2009,2010) and the spectral difference (Liang et al., 2009)methods, since they (1) utilize a spatially high order rep-resentation to resolve the physics in a better fashion, (2)can deliver very accurate solutions with smaller degrees offreedom. It is worth mentioning that Kannan was able toobtain extremely accurate solutions with substantially lesser


degrees of freedom for a variety of problems (Kannan &Wang, 2009, 2010). However, since the crux of the currentarticle is not on the above, we will limit our approach tousing classical second order solvers and will consider theabove mentioned high order methods in the future. Thechoice of boundary conditions are not of relevance in thisanalysis, since the 1D code is used only to determine theimmediate post-shock parameters that are inputs to the BDand MBD models for RM analysis, after which the modelpredicts subsequent growth trends. Having summarized theBD model and the 1D code methodology, we now focuson developing the MBD model.

3. THE MULTIPHASE BUOYANCY-DRAG MODEL

The BD model of Srebro et al. (2003) is now extended to ac-count for multiphase effects, and is appropriately referred toas the MBD model. The basic formulation stems from thedusty gas investigations of Saffman (1962), Michael(1964), and Ukai et al. (2010). Saffman applied the formu-lation to laminar flow by deriving the multiphase Orr-Sommerfeld equation; Michael extended this work to thestudy of plane Poisuelle flow of dusty gases; and Ukaiet al. applied the formulation to two kinds of dusty gasRichtmyer-Meshkov instabilities. Here, the effect of dust isdescribed by two parameters—the dust concentration (ornumber density, N ) and a relaxation time (essentiallyStokes number, St). The basic assumptions involved in thecurrent formulation are summarized as follows:

1. the dust particles are spherical in shape and are of a uni-form size;

2. the bulk concentration of the dust particles, i.e., the dustvolume fraction, is negligible;

3. sedimentation effects are assumed to not occur and sothe gas-particle mixture stays as a mixture at all times;

4. the gas-particle mixture is assumed to be incompressi-ble for the analysis, as also done so by Saffman (1962)and Ukai et al. (2010);

5. the vortex rings around the RT and RM hydrodynamicstructures are assumed to not cluster the particles (seeBalakrishnan & Menon, 2010; Balakrishnan et al.,2011 for more discussions on clustering);

6. the dust particles move along the gas streamlines, andso the mean velocities of the dust and the local gasare identical;

7. the number density of the particles is constant every-where before the disturbance starts to evolve;

8. the dust particles do not cause additional perturbationsin the gas, but rather only modify the waves whichalready exist in the gas (see Saffman, 1962 for morediscussions on this).

Stated in these terms, we extend on Srebro et al.’s (2003)work to formulate the MBD model, also deriving ingredientsfrom Ukai et al. (2010). Recalling the formulation of

Saffman (1962) and Ukai et al. (2010), small perturnationsare considered for the flow variables and the governingequations are linearized; these equations are not summarizedhere for brevity. The above references then apply boundaryconditions at the far-field and at the species interface, follow-ing which the first order general expression applicable for RTas well as RM involving dusty gases is obtained as (see Ukaiet al., 2010 for a detailed derivation):

ρ1 1+ f11− ikτ1c

[ ]g− kc2( ) = ρ2 1+ f2

1− ikτ2c

[ ]

× g+ kc2( )

, (20)

where f1 and f2 denote, respectively, the particle mass loadingin the light and heavy gases and are evaluated as f1=mNo/ρ1and f2=mNo/ρ2, where m is the dust particle mass, and No isthe initial dust concentration in number per unit volume. Theother terms in Eq. (20) represent the acceleration g(t), wave-number k (= 2π/λ), wave speed c, particle relaxation timescale τ (subscripts 1 and 2 correspond to fluids 1 and 2,respectively), and i is the complex number√(−1). The relax-ation times are obtained as τ1= τ2=m/(6π rpμ); note thatwe have assumed τ1= τ2, which need not necessarily bealways true; the other assumption made is that Stokes dragis valid, as also done so in Saffman (1962) and Ukai et al.(2010). Following this, we generalize the formulation ofUkai et al. (2010) in which the small |kτc| limit was assumed.This formulation can be generalized with the use of the par-ticle Stokes number, St, obtained as St=−ikτc (see thederivation in Ukai et al., 2010 for RM). Thus, Eq. (20) canalso be written as:

ρ1 1+ f11+ St1

[ ]g− kc2( ) = ρ2 1+ f2

1+ St2

[ ]g+ kc2( )

, (21)

where St1= St2 is assumed in this study. Furthermore, Ukaiet al. (2010) define a multiphase Atwood number, Am, underthe small St (≪1) limit as:

Am = ρ2 1+ f2( )− ρ1 1+ f1

( )ρ2 1+ f2( )+ ρ1 1+ f1

( ) , (22)

which, for a generic St, can be written as:

Am = ρ2 1+ ( f2/(1+ St2))( )− ρ1 1+ ( f1/(1+ St1))

( )ρ2 1+ ( f2/(1+ St2))( )+ ρ1 1+ ( f1/(1+ St1))

( ) . (23)

Thus, the multiphase effect in the formulation leads to theextension of the classical Atwood number,A, to the multiphaseAtwood number, Am, as defined above in Eq. (23). We notethat essentially ρ is replaced by ρ(1+ ( f /1+ St)) in the mul-tiphase formulation. When τ1 and τ2, or equivalently, St1 andSt2 ≪ 1, the multiphase replacement for ρ simply becomesρ(1+ f ); on the other hand, when the particle number den-sity, N, is small, f→ 0, resulting in Am→ A. Furthermore,


when St is very large, the multiphase correction for ρ will besimplify to ρ(1+ f/St). These limiting cases conform to Mi-chael’s (1964) results. We use the definition of Am as pre-sented in Eq. (23) to formulate the MBD model, notingthat ρ can be replaced by ρ[1+ f/(1+St)], with subscripts 1and 2 used as appropriate.The total mass per unit volume of a bubble enhanced

by the particles with number density No present in it isρ1+mNo= ρ1(1+ f1); similarly, the spike mass per unitvolume augmented by particles with number density No pre-sent within it can be expressed as ρ2(1+ f2). However, sincethe particles have a delay in response due to finite relaxationtime scales (i.e., St≠ 0), only a fraction of the total particlemass can be used to drive the bubble or spike. We choosethis factor as (1+ St) in order to be consistent with Eq.(23). Thus, the effective multiphase bubble and spikemasses per unit volume are ρm1 = ρ1(1+ ( f1/(1+ St1)))and ρm2 = ρ2(1+ ( f2/(1+ St2))), respectively. FollowingSrebro et al. (2003) by considering the added mass term,buoyancy, and drag effects, we can analogously and intui-tively obtain the following two equations for the bubbleand spike motion in dusty gases:

ρm1 + Caρm2

( ) duBdt

= ρm2 − ρm1( )

g(t)− Cdρm2

u2Bλ, (24)

ρm2 + Caρm1

( ) duSdt

= ρm2 − ρm1( )

g(t)− Cdρm1u2Sλ, (25)

where, as before, uB and uS denote, respectively, the vel-ocities of the bubbles and spikes, g(t) is the time varying in-terface acceleration, and λ is the wavelength. We assume thatCa and Cd remain unaffected by the presence of the particlesand use the same values as before. As done in Srebro et al.(2003), we then extend the formulation to include amplitudedependence through the parameter E(t) = e−CekhB :

CaE(t)+ 1( )ρm1 + Ca + E(t)( )ρm2[ ] duB

dt

= 1− E(t)( ) ρm2 − ρm1( )

g(t)− Cdρm2u2Bλ,

(26)

CaE(t)+ 1( )ρm2 + Ca + E(t)( )ρm1[ ] duS

dt

= 1− E(t)( ) ρm2 − ρm1( )

g(t)− Cdρm1u2Sλ,

(27)

with the coefficient Ce remaining the same as before.For multimode perturbations, we extend Eq. (8) by also ac-

counting for multiphase effects through the b(Am) parameter,defined as:

b(Am) = 0.51+ Am

(2D); b(Am) = 1.61+ Am

(3D), (28)

and also appropriately modify the characteristic wavelengthfor multimode perturbations to account for multiphase

effects:

dλ

dt=

0, hB < λb(Am);uB

b(Am), hB ≥ λb(Am).

⎧⎨⎩ (29)

Here, we are using the same definitions for λ, with the onlyreplacement for A by Am. Thus, for the MBDmodel, we solveEqs. (26) and (27) and, in addition, we solve Eq. (29) formultimode cases. Furthermore, the definition b(Am) is usedin place of b(A).During the early linear perturbation growth, khB is small,

but increases to larger values thereafter as the perturbationswitches to the asymptotic stage. Even though the equationsderived above for both BD as well as MBD models are non-linear, the early-time linear regime can also be captured. Thisis because in the linear stage, expanding Eqs. (26–27) to firstorder in khB, the perturbation evolution equations for bubblesand spikes take the form:

duBdt

= AmkhBg(t);

duSdt

= AmkhSg(t), (30)

which is consistent with theoretical behavior since theequations are equivalent to h(t)= Amkg(t)h(t). Duringthe early linear growth, λ is not changed with time, but thegrowth is self-similar in the asymptotic stage. A similarreasoning was used by Srebro et al. (2003) to demonstratethe validity of the baseline BD model for both linear aswell as the asymptotic stages of evolution. Thus, the non-linear Buoyancy-Drag theory is also applicable for theearly linear stages with the small khB and constant λapproximations.In summary, the generalized MBDmodel described here is

suitable for the study of both RT and RM in dusty gases. ForRT, the MBD model can be directly applied to evaluatebubble and spike growth. For RM, on the other hand, knowl-edge of g(t) as well as the post-shock ρ1, ρ2, N1, and N2 arealso required, which are evaluated from the 1D simulationsdescribed in Section 2.2. Here, N1, and N2 denote, respect-ively, the post-shock particle number densities in the fluids1 and 2. In the following sections, the baseline BD modelof Srebro et al. (2003) is verified for both RT and RM invol-ving both single and multi-mode perturbations, followingwhich the MBD model is tested.

4. RESULTS AND DISCUSSION

4.1. Verification of the Srebro et al. (2003) BD model

First, we verify the efficacy of the baseline BD model asdeveloped by Srebro et al. (2003) for RT and RM appli-cations. To this end, we focus on single-mode RT(SMRT), multi-mode RT (MMRT), single-mode RM


(SMRM), and multi-mode RM (MMRM). It is customary toensure that the choice of the time step, Δt, is sufficientlysmall so that the BD model simulation results are meaning-ful. For RT, we ensure that Δt ≪ τRT, and for RM, Δt ≪τRM, where τRT and τRM denote, respectively, the timescales for RT and RM, given by τRT = ��

λ/2πAg√

(Ramshaw,1998) and τRM = λ/2πAΔv (Ukai et al., 2010), where Δv isthe velocity of the interface after the shock passage.For SMRT, we consider the experimental results of

Wilkinson and Jacobs (2007), which correspond to A∼0.15. In their experiment, the initial amplitude, ao was inthe range 0.248–2.718 mm; we assume ao= 0.25 mm forthe BDmodel analysis as the exact value from the experimentis not known. Figure 1 shows the non-dimensional amplitude(ka, k= 2π/λ) versus non-dimensional time for a constantacceleration, g. As evident, the BD model reasonably pre-dicts the overall growth trends although slight deviationsare observed at later times, (Akg)1/2t∼ 3.5. These discrepan-cies may result from differences in the choice of the initialamplitude used in the experiments and the current BDmodel, or presumably from acceleration on the fluid notbeing exactly constant (in the experiment of Wilkinson andJacobs (2007), an average measured acceleration was used).The growth trend of ka conforms to an exponential variationwith t until ka∼ 1, after which transitions to a linear variationwith t, in agreement with theory (Goncharov, 2002). Thus,the overall growth trends of the amplitude are in reasonableagreement both with experiments as well as theory, therebyexemplifying the efficacy of the BD model to predict SMRT.Next, we apply the baseline BD model to investigate

MMRT, focusing on the simulation results of Youngs(1991), where three different density ratios, ρ2/ρ1= 1.5, 3and 20 were considered. For the BD model, we use λ =1cm and hb= hs= 0.1 mm as initialization, so that the initialka= 0.06 ≪ 1, resulting in linear growth rates. Figure 2shows the bubble height (hb), spike height (hs), andamplitude (a) versus Agt2; as evident, the heights and the

amplitude vary linearly with Agt2, a well established classicalresult for MMRT. The slope, α, of the bubble height, hb, iscomputed for the three chosen ρ2/ρ1 values, and are summar-ized in Table 1, along with the corresponding slopes obtainedin the simulations carried out by Youngs (1991). As evident,α≈ 0.05, a classical RT result; even the BD model α predic-tions are in reasonable agreement with Youngs (1991), withthe deviations widening for higher ρ2/ρ1 ratios. From thisresult, we believe that the BD model is better suited forlower density ratios (and lower A); as A→ 1, some modifi-cations may be required, and will have to be addressed inthe future. This study verfiies the baseline BD model forMMRT.

We next focus on SMRM; in particular, we are interestedin the experiments of Erez et al. (2000) and the analysis doneby the same research group in Shvarts et al. (2000). Here,RM experiments involving both 2D and 3D perturbationsin air/SF6 combinations are considered, with an incidentshock Mach number of 1.25 and an initial λ =26 mm. The1D code described in Section 2.2 is used to estimate theinitial interface velocity, ΔV, that is used as an input tothe BD model, i.e., the initial velocity of the interface afterthe passage of the incident shock. Figure 3 presents the BDmodel results along with experimental results of Erez et al.(2000) and Shvarts et al. (2000); we apply the 2D and 3Dversions of the BD model as appropriate. As evident, theBD model predictions are in reasonable agreement with theexperiments, thereby verifying the application of the baseline

Fig. 1. Single-mode RT. Experimental results are from Wilkinson andJacobs (2007).

Fig. 2. Multi-mode RT: hb, hs and a versus Agt2 for ρ2/ρ1= 20.

Table I MMRT slopes, α, predicted by the current BD model andas obtained by Youngs (1991)

Case ρ2/ρ1 BD Model α α from Youngs (1991)

1 1.5 0.0519 0.0522 3 0.0487 0.0503 20 0.0473 0.054


BD model for SMRM. hb for the 3D initial perturbation isabout 25% greater vis-à-vis the 2D initial perturbation, dueto the smaller drag coefficient for 3D (Alon et al., 1995).Finally, we verify the baseline BD model for the study of

MMRM, focusing on the air/SF6 experiments of Erez et al.(2000), as also done in Srebro et al. (2003). Due to the lack ofexact information on the nature of the initial perturbation, theinitial choice of λ and h= hb+ hs are varied to match withthe experiments along with the assumption hb= hs initially(a similar approach was undertaken by Srebro et al., 2003).Figure 4 presents the total mixing zone width, h, versustime using both the 2D and 3D versions of the baselineBD model; two sets of experiments from Erez et al. (2000)and Srebro et al. (2003) are presented. Analysis shows that

λ = 0.05 cm and h= 0.07 cm results in reasonable agree-ment with the experimental data. Although the origianlexperiments of Erez et al. (2000) involved both incidentand re-shocks, we, however, focus only on the incidentshock in this study, and therefore terminate the BD model

simulation just before the arrival of the reshock. In thefuture, the BD model for re-shocked systems needs to berevisited so as to ascertain the model’s prediction of the ex-pected phase reversal growth trends. We curve-fit the bubbleheight hb, spike height hs, and amplitude a = 1/2 hb + hs( ) aspower laws with time, i.e., hb∼ tθb, hs∼ tθs, a∼ tθa. With the3D version, we obtain θb= 0.164, θs= 0.297, and θa=0.262; the corresponding values with the 2D version areθb= 0.222, θs= 0.327, and θa= 0.294. These values com-pare reasonably well with Srebro et al. (2003), who statethat the experimental θa= 0.24 and their numerical θa=0.29 and 0.32, respectively, with the 3D and 2D models,for the same initial conditions. This study exemplifies theapplicability of the baseline BD model for MMRM.In summary, we have verified the ability of the baseline

BD model to predict the growth trends encountered inSMRT, MMRT, SMRM, and MMRM. The focus now is totest the MBD model for similar cases and to understandthe underlying physics. Of particular interest is to investigatethe effect of particle loading, N, and particle radius, rp, on theamplitude growth trends. To this end, the next sectionfocuses on applying the MBD for the study of similar pro-blems in dusty gases. It is emphasized that to the best ofthe authors’ knowledge, no experimental results existfor RT and RM growth in dusty gases; thus, we are unableto compare MBD model predictions with any dusty gasexperiments.

4.2. Dusty Gas RT Using the MBD Model

The role played by solid particles in the mixing layer growthin RT is investigated, first for SMRT, and then for MMRT. Itis of interest to study the effects of particle loading, N, andparticle size, rp. We apply the MBD model for cases corre-sponding to A= 0.5, ρ1 =1 Kg/m3, g= 1 m/s2, initialwavelength, λ0 =1 cm (λ for SMRT; λ for MMRT) andinitial amplitude, a0= 0.1 mm. For MMRT, the ratio,

b Am( ) = hb/λ as defined in Eq. (28) is used.

4.2.1 SMRT

We apply the MBD model for the above chosen par-ameters; first, we investigate the effect of N on the amplitudegrowth. rp= 40 μm is considered, and this corresponds toSt∼ 1. We consider a baseline particle-free (N= 0 m−3)case in addition to N in the range 108—1013 m−3. Figure 5shows the amplitude, a, versus time. As expected, a growsslower for higher N, since particles serve as an obstructionto the bubble and spike motion. For N= 108 m−3, the ampli-tude growth tends to the particle-free growth as too few par-ticles are present to offer resistance to the growth of themixing layer. This trend is expected, since only f dependson N and rp is independent; therefore, the term 1+ f/(1+St)→ 1 as N → 0. On the other hand, as N increases,1+ f/(1+ St)→ f/(1+ St). We also compute the latetime slopes of the bubble and spike amplitudes to obtain

Fig. 3. Single-mode RM: hb for 2D and 3D initial perturbations. Experimen-tal results are from Erez et al. (2000) and Shvarts et al. (2000).

Fig. 4. Multi-mode RM: hb +hs versus time. Experimental results are fromErez et al. (2000); Srebro et al. (2003).


the respective late time constant bubble and spike terminalvelocities (URT(B/S )). The MBD model predictions forURT(B/S) are in accordance with the value obtained by equat-ing the buoyancy and drag terms:

URT(B/S) =��2Am/(1± Am)gλ/Cd

√, (31)

consistent with the results obtained byOron et al. (2001), albeitwith Am in place of A (the+ sign corresponds to the bubblesand the− sign for the spikes). It is also of interest to considerthe trends in the growth of the non-dimensional amplitude(ka) versus non-dimensional time (t(Amkg)

1/2), and is plottedin Figure 6. It is interesting to note that as N increases, the non-dimensional profiles also tend to converge, thereby creatinga band of solutions between Am→ A and Am→ 0. Thestudy is repeated for a different choice of rp and similar katrends are observed, albeit for a different range of N; this is

because in the term f/(1+ St), f∼ rp3 and St∼ rp

2, and so fora different rp, f/(1+ St) will follow a different trend for agivenN. However, the results are observed to still be containedwithin the same band, i.e., Am→ A (upper limit) and Am→ 0(lower limit); these results for different rp are not presented herefor brevity. Thus, a band of solutions is identified for the growthof the non-dimensional amplitude versus non-dimensionaltime, for dusty gas SMRT.

Next, we investigate the effect of rp for a fixed N=1010 m−3. We consider particle sizes in the range rp= 4–400μm, corresponding to different St. Here, rp= 4 μmcorrespondsto St∼ 0.01, rp= 40 μm to St∼ 1, and rp= 400 μm to St∼100. Figure 7 shows the non-dimensional amplitude versusnon-dimnesional time. Again, as evident, the results are con-tained within the bands corresponding to Am→ A and Am

→ 0. For very small rp, too little particle mass is present toinfluence bubble and spike motion, and so the results conformto the particle-free case (Am= A), the upper limit. Note that forvery small rp, 1+ f/(1+ St)→ 1. On the other hand, whenvery large particle sizes are considered, 1+ f/(1+ St)→6πrpμN/ρ. Thus, for too large an rp, ρ[1+ f/(1+ St)]→ 6πrpμ N, i.e., independent of ρ. Hence, the initial density ratioloses significance and the mixing layer evolves tending tothe Am→ 0 limit (lower limit) for large particle sizes. Eventhe bubble and spike steady-state velocities at late times areobserved to conform to Eq. (31). This study demonstratesthat a band of solutions is observed also for the MMRT industy gases, and that the choice of Am for dusty gases can bephysically analogous to A for pure (dust-free) gases.

4.2.2. MMRT

To investigate multiphase MMRT, as noted in Eq. (28), weuse b(Am). The same initial conditions are used for this study,with the difference being the use of λ instead of λ, and thecorresponding equation for the wavelength growth rate(Eq. (29)). Figure 8 displays the growth of amplitude, a,

Fig. 6. Multiphase single-mode RT: non-dimensional amplitude versusnon-dimensional time for different particle loading with rp= 40 μm. Thelegend denotes the value of N in number per m3. The result correspondingto N= 108 m−3 is coincident with N= 0 m−3 and so is not clearly visible.

Fig. 5. Multiphase single-mode RT: amplitude versus time for different par-ticle loading with rp= 40 μm. The legend denotes the value of N in numberper m3.

Fig. 7. Multiphase single-mode RT: non-dimensional amplitude versusnon-dimensional time for different particle sizes with N= 1010 m−3. Thelegend denotes the value of rp in μm.


versus Amgt2, for a fixed rp= 40 μm (this corresponds to

Stokes number, St∼ 1), for a range of N. As evident,higher N results in subdued mixing layer amplitude growthas more particles obstruct the rise of bubbles and the fall ofspikes. The linear trend between a and Amgt

2 suggests thatthe well established a∼ t2 result also holds for multiphasesystems with the use of Am in place of A. Again, the resultsare contained within a band, with the upper limit correspond-ing to Am→ A (for N → 0) and the lower limit to Am→ 0(for N →∞). The slopes of these curves, α, are evaluated tobe α= 0.0606 for the upper limit and α= 0.0491 for thelower limit. Interestingly, these values are similar to the clas-sical result of α∼ 0.05 reported by Youngs (1984, 1989,1991, 1994). Thus, it is possible that some of the establishedtheories on hydrodynamic instability growth can be extendedto multiphase systems as well by replacing A with Am.Next, we fix the particle loading at N= 1011 m−3 and vary

particle sizes in the range rp= 4–400 μm, for the same ao,initial λ, A, ρ1, and g. Here, rp= 4 μm corresponds to St∼0.01, rp= 40 μm to St∼ 1, and rp= 400 μm to St∼ 100.Figure 9 shows the amplitude variation versus Amgt

2 and, asevident, linear trends (in t2) are observed at later times, withthe amplitudes contained within a band. For very small particlesizes, the total particle mass is negligible to have an effect onthe bubble and spike growth and so the result converges tothe particle-free case; for very large particle sizes, on theother hand, the results again converge to the Am→ 0 solutionas before. The slopes of the band boundaries are determined tobe α= 0.0606 for the upper limit (Am→ A) and as α= 0.0469for the lower limit (Am→ 0). Such banded solutions indicatethat the slope, α, increases with Am for a given A.It is also of interest to consider the trends in the slope of the

amplitude versus Amgt2 curves for the different 3D cases con-

sidered for the multiphase MMRT analysis, for this directlyindicates dependence of the t2 law for different particlesizes and loadings. Figure 10 shows α= a/(Amgt

2) versusAm for these different MMRT cases considered; it is observed

that for Am> 0.2, a linear trend is observed. Note that theseresults correspond to a fixed A= 0.5. In the future, we willbe investigating the α trend versus Am for different choicesof A. In Figure 10, for very small Am, however, a scatter isobserved in the α predictions, due to which we believe thatthe MBD model may not be well suited for Am→ 0. Thisis because when the total particle mass is large (as is forAm→ 0 cases), some of the assumptions that were stated ear-lier in the formulation (Section 3) can fail. For instance, theassumption that the particle-gas mixture remains uniform atall times is of concern under high particle mass cases.Furthermore, when Am→ 0, since the particle mass out-weighs the gas mass, the treatment of the dusty gas mixtureas a fluid presumably fails. Due to these effects, the MBDmodel needs further improvement for Am→ 0 cases.

4.3. Dusty gas RM using the MBD model

4.3.1. SMRM

Here, we extend on the 3D cases considered earlier forthe single phase RM, and investigate the bubble and spike

Fig. 10. Variation of α with Am for different cases considered in MMRT.A= 0.5 for all these cases.

Fig. 8. Multiphase multi-mode RT: amplitude versus Amgt2 for different

particle loading with rp= 40 μm. The legend denotes the value of N innumber per m3.

Fig. 9. Multiphase multi-mode RT: amplitude versus Amgt2 for different

particle sizes with N= 1011 m−3. The legend denotes the value of rp in μm.


growth when particles are present. In both the BD and MBDmodels, the acceleration profile can be crucial in decidingwhether RM can be assumed to be an impulsive case ofRT. For instance, it has been demonstrated by Wouchuk(2001) that RM involving weak shocks behave as an impul-sive case of RT, with impulsive predictions showing devi-ations even for Mach 1.5 shocks. For very strong shocks,models that treat RM as an impulsive RT case need to berevisited in the future, with perhaps the use of a g(t) profilevarying over a finite albeit small time interval. Such profilesmay be essential for strong shock RM in order to more accu-rately predict the perturbation growth rates; studies to this endwill be considered in the future. In the present investigation,only relatively weak shock (Mach number< 1.5) RM casesare considered.The choice of rp or N used in these cases will be different

from those considered for RT due to different time scales in-volved for RT and RM. Thus, comparing the results betweenRM with RT for the same rp or N does not correspond to anyphysical significance; the range of rp and N of interest to theanalysis is thus different between RT and RM for the samereason. Figure 11 shows the (a) hb and (b) hs growth withtime for a fixed rp= 1.5 μm for different values of N. As evi-dent, both hb and hs show subdued growth with time as thenumber of particles increases. Whereas the bubble height isonly affected by∼ 20% for the particle number density

range considered, the spike heights show more pronouncedvariations, with hs being only one-fourth for the densercases considered vis-à-vis the particle-free case. This studyillustrates that the spike growth shows higher sensitivity tothe presence of particles than the bubbles, presumably dueto the higher inertia involved for the spikes. Next, we inves-tigate the effect of rp for a fixed N= 1013 m−3; the bubbleand spike heights are presented in Figure 12. Even forthese cases, both hb and hs show subdued growth with an in-crease in rp, with the spikes showing a higher dependence.It is also interesting to note from Figure 12 that whereasthe particle-free and small particle cases show a rapid in-crease in hb and hs at early times followed by slower expo-nential growth at later times—typical for RM instabilities,for larger particles, on the other hand, the growth trendsare more gradual even at early times—similar to RTgrowth. Essentially, this means that RM with large particlesizes (therefore, slow response times) behave in some phys-ical sense similar to RT. This observation was also made inUkai et al. (2010) based on 2D simulations, and is owingto the slow response of the larger particles, due to whichthe bubble and spike growth does not stay impulsive, butrather switches gradually to a more continuous growth. It isnoteworthy of mention that the MBD model is able to predictthis particle-size sensitivity on the bubble and spike growthtrends at early times.

Fig. 11. Multiphase single-mode RM: (a) bubble amplitude and (b) spike amplitude growth with time for a fixed rp= 1.5 μm. The legenddenotes different values for N in m−3. In (a), the N= 1012 case is coincident with the “No particles” case and is thus not clearly visible.

Fig. 12. Multiphase single-mode RM: (a) bubble amplitude and (b) spike amplitude growth with time for a fixed N= 1013 m−3. Thelegend denotes different values for rp in μm.


4.3.2. MMRMFinally, we shift our focus to the investigation of multiphaseMMRM. The effect of rp on the growth of hb and hs are inves-tigated for a fixed N= 1016 m−3 in Figure 13, for a fixed A=0.7. As before, with the increase in particle size—and there-fore particle mass—subdued hb and hs are observed, withthe spikes showing a higher sensitivity to the presence of par-ticles. Similar results are observed also for higherN for a fixedrp, not shown here for brevity. Power law growth trends withtime are observed for hb, hs, and a for the different 3D multi-phase MMRM cases considered, and these are plotted versusAm in Figure 14; here, hb∼ tθ b; hs∼ tθs; a∼ tθa. As observed,θb linearly decreases for low Am from∼ 0.17 to 0.165, andstays nearly constant with Am for Am> 0.5. θs, and θa, onthe other hand, are observed to non-linearly increase withAm in the Am= 0.1 to 0.7 range. Similar trends were reportedfor the single-phaseMMRM considered in Oron et al. (2001).Thus, theMBDmodel can be used to obtain power law depen-dence for multiphase MMRM. Although only A= 0.7 is con-sidered in this study, in the future such power law coefficientsfor a wider range of A can be investigated.

4.4. Four New Classes of Problems

We have hitherto analysed RT and RM in pure gases using theBD model, and in dusty gas mixtures using the MBD model.This leads to a hybrid scenario wherein a combination of thetwo studies is possible, i.e., when RT or RM involves puregases in certain regions, and dusty gases otherwise. Thus,we develop a hybrid BD/MBD model that uses BD in pure(dust-free) regions and MBD in dusty gas regions. Based onthe current analysis of multiphase RT and RM, 4 new classesof problems are identified that are tractable to analysis withsuch a hybrid BD/MBD model. These are summarized as:

1. Bubbles in a pure gas RT or RM rising into a region ofparticles;

2. Spikes in a pure gas RT or RM falling into a region ofparticles;

3. Bubbles in a multiphase RT or RM rising into a regionof pure gas;

4. Spikes in a multiphase RT or RM falling into a regionof pure gas.

Here, bubbles or spikes, corresponding to single- or multi-mode, RT or RM, in a pure gas (or dusty gas) can rise orfall, respectively, into a dusty (pure) gas, as the case maybe. Under such scenarios, the BD model must be used forpure gas regions, and the MBDmodel in dusty regions, there-by leading to a hybrid model. In this sub-section, only theMMRT will be studied for these identified possibilities, asthe goal here is to demonstrate the efficacy of the hybridsolver for the study of such kinds of RT and RM, whichare, to the best of the author’s knowledge, new in the litera-ture. In addition, the physics of these parametric studies arenot elaborated, as the goal of the current analysis is to demon-strate the application of the BD/MBD hybrid model andidentify four new classes of problems related to dusty gasMMRT. In the future, such possibilities for SMRT,SMRM, and MMRM needs to be revisited.We consider the same set of initial conditions for λ, ao, g,

ρ1, ρ2, and A considered in the MMRT analysis from Section2, with differences in the particle distribution in space. rp=40 μm is used as it corresponds to St∼ 1, which is of interest

Fig. 13. Multiphase multi-mode RM: (a) bubble amplitude and (b) spike amplitude growth with time for a fixed N= 1016m−3. Thelegend denotes different values for rp in μm.

Fig. 14. Variation of θ with Am for different 3D cases considered inMMRM. A= 0.7 is fixed.


here; the particle number density is taken to be N=1010 m−3. For this analysis, we assume, for bubbles risingor spikes falling into a region of particles, the interface be-tween the pure and dusty gases are located at a height of1.5 m from the initial pure gas interface that separates thetwo fluids corresponding to ρ1 and ρ2. For ease of discussion,we refer to the initial interface between the two fluids corre-sponding to ρ1 and ρ2 (when amplitude, a= ao) simply as“interface,” and the interface between the pure fluids andthe dusty gases as “multiphase interface (MI).” Note thatfor the analysis of bubbles rising into a particle region, theMI lies on the side of the bubbles (side of ρ2). Likewise,for spikes falling into a region of particles, the MI is locatedon the side of the spikes (side of ρ1). Similar distinction isemphasized for the analysis of bubbles rising or spikes fall-ing from a dusty gas into a region of pure gas—here, thepure gas is identified by the gas that is present on the sameside as the bubble rise or spike fall.

4.4.1. Bubbles in a Pure Gas MMRT Rising into a Region ofParticles

First, we analyze the rise of pure gas bubbles into a regionof dusty gas located 1.5 m away from the initial interface. Forthis analysis, we use the BD model for both bubbles andspikes until hb reaches 1.5 m, after which the MBD modelis employed for the bubbles, but the BD model is continuedfor the spikes, since only the bubbles encounter the dusty gasregion. The parameter b(Am) is used as this case involvesbubbles encountering both pure fluids as well as dustygases. Figure 15 shows (a) hb, hs, a and (b) ub, us, and as evi-dent, the bubble amplitudes do not show noticable differ-ences as they enter into the dusty gas region. FromFigure 15(b), ub only shows a minor “kink” around 7.5 s,and quickly adjusts to a linear velocity profile thereafter, re-sulting in continued hb∼ t2 growth. The buoyancy term inthe MBD model increases in magnitude as ρ2

m– ρ1

m increaseswhen the bubbles rise into the particle region. This inevitablyaccelerates the bubble front, albeit only to a small extent inthe present case, and causes the kink in the velocity profile.Thus, the dusty gas region has an accelerating influence,albeit small, on the rising bubbles for the case considered.

4.4.2. Spikes in a Pure Gas MMRT Falling Into aRegion of Particles

Next, we focus our attention on the fall of spikes into aregion of particles, using the same initial conditions noted.MI is now switched to the side of the spike fall; b(A) isused instead of b(Am), since the parameter b is related tobubbles, which rise only into pure gases for the presentcase. Figure 16 displays hb, hs, and a. It is observed thateven though hs> hb at early times as expected, once thespikes enter the dusty gas region (hs> 1.5 m) they slowdown and oscillate about the MI (located at 1.5 m), therebyallowing for hb> hs at later times. This oscillation of thespike front is owing to Am becoming negative for thespikes inside the dusty gas, thereby reversing the buoyancyforce. For the BD/MBD analysis, the sign of the drag termfor the spikes is reversed based on the direction of motion,so that the drag term always opposes its motion.Subsequently, the spikes again enter the pure gas region,after which the buoyancy term once again changes sign,causing the spikes to fall again into the dusty region. Then,the sign of the buoyancy term again changes, and such oscil-lations are repeated, leading to a gravity wave-like phenom-enon. The amplitude of the oscillation for every subsequentpenetration decreases in time as the spikes lose momentumto the surrounding dusty gas due to drag. We zoom hs inthe vicinity of the MI and present in Figure 17 a closerview of the oscillatory nature of hs. As evident, the localmaxima and minima in the spike front decays for every sub-sequent penetration into the dusty gas due to drag effects.Even the oscillation frequency decreases for every sub-sequent penetration, as the buoyancy effects decrease forevery subsequent penetration, due to which the spikes requirelesser time to slow down to rest and thereafter reverse direc-tion. In the future, the dependence of this oscillation fre-quency on rp and N needs to be revisited.

4.4.3. Bubbles in a Multiphase MMRT Rising intoa Region of Pure Gas

We now investigate the rise of bubbles in a multiphaseMMRT into a region of pure gas. Again, the MI is located1.5 m above the initial interface that separates the two

Fig. 15. Bubbles in a pure gas MMRT rising into a region of particles: (a) hb, hs and a; (b) ub and us.


fluids, ρ1 and ρ2. The parameter b(Am) is used since this studyinvolves bubbles encountering dusty gases. Figure 18 showshb, hs, and a versus time. As before, the bubble height, hb,shows a gravity wave-like phenomenon as the term ρ2

m−ρ1m changes sign alternatively, thereby changing the sign ofthe buoyancy term. For the BD/MBD analysis, the sign ofthe drag term is changed to ensure that it always opposesthe bubble motion. Even for the bubbles considered here,both the oscillation amplitude and frequency decrease withtime.

4.4.4. Spikes in a Multiphase MMRT Falling intoa Region of Pure Gas

Finally, we investigate spikes in a multiphase MMRT fall-ing into a region of pure gas. The parameter b(Am) is used inthe analysis since the bubbles always remain in a dusty gasfor this case. Figure 19 shows hb, hs and a versus time.The spike front accelerates as it enters into the pure gasbeyond hs> 1.5 m, as ρ2

m− ρ1m in the buoyancy term in-

creases in magnitude, hence accelerating the spike front.

Overall, these parametric studies have identified four newclasses of problems that can be investigated using the cur-rently developed MBD model. Since the goal of this paperis to demonstrate the ability of the MBD model for problemsof this kind, the four parametric studies are not studied inelaborate detail. Furthermore, studies along these lines forSMRT, SMRM and MMRM also needs to be investigatedin the future. The MBD model offers leverage for suchproblems.

4.5. Extensions to the MBD Model

Through the course of this paper, we have formulated and ap-plied the MBD model for a wide variety of RT and RM pro-blems and the model is able to accurately predict the bubbleand spike growths under different multiphase conditions.However, experiments on multiphase RT and RM are limitedin the literature and so the model predictions could not bedirectly verified with experiments. As shown earlier, theMBD model needs to be revisited for Am→ 0, i.e., whenthe particle mass is much higher than the gas; under this

Fig. 17. Oscillatory hs behavior as spikes in a pure gas MMRT fall into aregion of particles.

Fig. 18. Bubbles in a multiphase MMRT rising into a region of pure gas.

Fig. 19. Spikes in a multiphase MMRT falling into a region of pure gas.

Fig. 16. Spikes in a pure gas MMRT falling into a region of particles.


scenario, the treatment of the gas-particle mixture as apseudo-gas mixture is stretched, and so the model needs re-visions for this limiting end. Furthermore, for RM, we haveapplied the MBD model only for a single shock system,and so the MBD model needs further testing for re-shockedRM which causes phase-reversal (Srebro et al., 2003; Leinovet al., 2009). For strong shock RM in dusty gases, the use of anon-impulsive g(t) profile may be warranted in order to moreaccurately predict growth rates. Such modifications can be in-vestigated in future studies with the MBD model.Other applications include the study of explosions in mul-

tiphase environments (Balakrishnan & Menon, 2010;Balakrishnan et al., 2011; Balakrishnan, 2010), for whichthe MBD model needs to be extended to account for geo-metrical divergence effects. For this, the decompressionterm introduced in Miles (2009) for the single-phase BDmodel can be used in the MBD model. Other physical pro-blems of interest include the application of the MBDmodel to reactive systems, where the Stokes number canchange with time for burning particles, which can alsoresult in interesting results; in addition, heat release effectsassociated with burning particles can also drive the bubbleand spike growth due to volumetric expansion, and these pro-blems can also be investigated with the MBD model withsome minor corrections incorporated. Furthermore, for blastwave driven systems, a time-varying acceleration profilecan be considered and the MBD model can be appliedunder such conditions as well to investigate blast drivenRT and RM in dusty gases. Studies along these identifiedlines will be conducted in the near future.

5. CONCLUSIONS

In this paper, a newMBDmodel is developed and is applied toinvestigate the growth trends in Rayleigh-Taylor andRichtmyer-Meshkov instabilities in dusty gases. Bothsingle- and multi-mode perturbations in dusty gases arestudied for Rayleigh-Taylor as well as Richtmyer-Meshkov

growth. A multiphase Atwood number, Am, is identified, asa function of the fluid densities, particle size and Stokesnumber, which is a critical variable for the dusty gas analysis.For Rayleigh-Taylor growth in dusty gases, a band of ampli-tude growth are observed, and the upper and lower limits ofthis band lie within limits identified by Am. The amplitudegrowth with time is subdued when larger particle sizes and/or larger particle number densities are used, which is directlyrelated to Am. For dusty gas Richtmyer-Meshkov growthunder multi-mode conditions, the power law index, θ, for am-plitude shows dependence to Am as well. Whereas θ linearlydecreases and then asymptotes to a constant value for higherAm for bubbles, for spikes, θ increases non-linearly with Am.

Four new classes of problems are identified and investi-gated for Rayleigh-Taylor growth under multi-mode con-ditions, using a hybrid version of the model, with theclassical BD model for pure (dust-free) gas, and the currentlydeveloped MBD model in dusty gas regions. These newclasses of problems are summarized as: (1) bubbles in apure gas rising into a region of particles; (2) spikes in apure gas falling into a region of particles; (3) bubbles ina region of particles rising into a pure gas; and (4) spikesin a region of particles falling into a pure gas. For bubblesin a pure gas rising into a region of particles, as well as forspikes in a region of particles falling into a pure gasregion, the bubble or spike front, respectively, acceleratesonce it crosses the multiphase interface. This is owing toan increase in the buoyancy term, due to which the bubbleor spike front, as the case is, rapidly grows after passingthe multiphase interface. On the other hand, for spikes in apure gas falling into a particle region, and bubbles in a par-ticle region rising into a pure gas, gravity wave-like oscil-lations are observed. Such oscillations are caused due tothe sign change in the buoyancy term as the bubble orspike front, as the case may be, penetrates from the regionof pure or dusty gas to the other. The amplitude of theseoscillations decays with time due to drag effects, indicatingthat the net acceleration of the bubble or spike front forevery subsequent penetration is lesser. Finally, potential ex-tensions to the presently developed multiphase buoyancy-drag model are proposed for future research that can beapplied for a range of problems, inter alia, blast wavedriven instabilities, re-shock systems, etc.

ACKNOWLEDGMENTS

The first author acknowledges the private communications withDr. Guy Malamud of the Negev Nuclear Research Center, Israeland Dr. Oren Sadot of the Ben-Gurion University of the Negev,Israel.

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APPENDIX

The 1D two-phase code is validated using the experimentalresults of Sommerfeld (1985). Here, a 7.81 m long shocktube is considered and is divided into a sequence of three re-gions: a 2 m long high pressure driver section, followed by a1.05 m long region filled with ambient air, and last, a 4.76 mlong section filled with a mixture of ambient air and dust par-ticles. The Mach number of the incident shock is 1.49, andtwo different particle mass loading ratio are considered,

n= 0.63 and 1.25. The particles are assumed to be 27 μmin diameter (dp), with a material density, σp= 2500 Kg/m3

and a specific heat, Cm of 766 J/Kg-K. Based on trial anderror, a resolution of ΔX= 0.01 m is found to suffice forthe analysis. After the high pressure section is released, ashock wave propagates into the low pressure region, and at-tenuates as it propagates through the gas-particle mixture be-cause of momentum and energy loss to the particles. Theshock wave Mach number as it attenuates with distance isof interest, and the numerical predictions are presented inFigure 20 along with the experimental data from Sommerfeld(1985). As evident, the numerical predictions are in goodaccordance with the experiments. In the far downstream re-gions (X> 3.5 m) the shock wave tends to attain an equili-brium Mach number as it propagates through the gas-particle mixture, in accordance with results of Sommerfeld(1985). Furthermore, as expected, for a higher particlemass loading ratio (n), the equilibrium shock Mach numberis lower due to the greater momentum and energy lossfrom the shock wave to the particles. These results validatethe 1D two-phase code for applications of the like.


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