the rayleigh-taylor instability driven by an accel-decel-accel profile

The Rayleigh-Taylor Instability driven by an accel-decel-accel profileP. Ramaprabhu, V. Karkhanis, and A. G. W. Lawrie Citation: Physics of Fluids (1994-present) 25, 115104 (2013); doi: 10.1063/1.4829765 View online: http://dx.doi.org/10.1063/1.4829765 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/25/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Simulation of Rayleigh-Taylor instability by Smoothed Particle Hydrodynamics: Advantages and limitations AIP Conf. Proc. 1479, 90 (2012); 10.1063/1.4756070 The late-time dynamics of the single-mode Rayleigh-Taylor instability Phys. Fluids 24, 074107 (2012); 10.1063/1.4733396 An experimental study of small Atwood number Rayleigh-Taylor instability using the magnetic levitation ofparamagnetic fluids Phys. Fluids 24, 052106 (2012); 10.1063/1.4721898 Study of ultrahigh Atwood-number Rayleigh–Taylor mixing dynamics using the nonlinear large-eddy simulationmethod Phys. Fluids 23, 045106 (2011); 10.1063/1.3549931 Nonlinear evolution of the magnetohydrodynamic Rayleigh-Taylor instability Phys. Fluids 19, 094104 (2007); 10.1063/1.2767666

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PHYSICS OF FLUIDS 25, 115104 (2013)

The Rayleigh-Taylor Instability driven by anaccel-decel-accel profile

P. Ramaprabhu,1 V. Karkhanis,1 and A. G. W. Lawrie2,a)

1Department of Mechanical Engineering & Engineering Sciences, University of NorthCarolina, Charlotte, North Carolina 28223, USA2Laboratoire de Mecanique des Fluides et d’Acoustique, Ecole Centrale de Lyon, 36,avenue Guy de Collongue, 69134 Ecully cedex, France

(Received 21 May 2013; accepted 28 October 2013; published online 14 November 2013)

We describe numerical simulations of the miscible Rayleigh-Taylor (RT) instabilitydriven by a complex acceleration history, g(t), with initially destabilizing accelera-tion, g > 0, an intermediate stage of stabilizing deceleration, g < 0, and subsequentdestabilizing acceleration, g > 0. Initial perturbations with both single wavenumberand a spectrum of wavenumbers (leading to a turbulent front) have been consideredwith these acceleration histories. We find in the single-mode case that the instabil-ity undergoes a so-called phase inversion during the first acceleration reversal fromg > 0 to g < 0. If the zero-crossing of g(t) occurs once the instability growth hasreached a state of nonlinear saturation, then hitherto rising bubbles and falling spikesreverse direction and collide, causing small-scale structures to emerge and enhancingmolecular mixing in the interfacial region. Beyond the second stationary point of g(t)where once again g > 0, the horizontal mean density profile becomes RT-unstableand the interfacial region continues to enlarge. Secondary Kelvin-Helmholtz-unstablestructures on the near-vertical sheared edges of the primary bubble have an Atwood-number-dependent influence on the primary RT growth rate. This Atwood numberdependence appears to occur because secondary instabilities strongly promote mix-ing, but the formation of these secondary structures is suppressed at large densitydifferences. For multi-mode initial perturbations, we have selected an initial inter-facial amplitude distribution h0 (λ) that rapidly achieves a self-similar state duringthe initial g > 0 acceleration. The transition from g > 0 to g < 0 induces signifi-cant changes in the flow structure. As with the single-mode case, bubbles and spikescollide during phase inversion, though in this case the interfacial region is turbulent,and the region as a whole undergoes a period of enhanced structural breakdown. Thisis accompanied by a rapid increase in the rate of molecular mixing, and increasingisotropy within the region. During the final stage of g > 0 acceleration, self-similarRT mixing re-emerges, together with a return to anisotropy. We track several turbulentstatistical quantities through this complex evolution, which we present as a resourcefor the validation and refinement of turbulent mix models. C© 2013 AIP PublishingLLC. [http://dx.doi.org/10.1063/1.4829765]

I. INTRODUCTION

An initially sharp interface separating two fluids of different densities is susceptible to theRayleigh-Taylor (RT) instability1–3 when an acceleration field g(t) is applied such that the lighter(ρ l) and heavier (ρh) fluids interpenetrate. The turbulent mixing arising from the development of themiscible RT instability is of key importance in the design of Inertial Confinement Fusion capsules,4

and to the understanding of astrophysical events, such as Type Ia supernovae.5 From Taylor’s seminal

a)Current address: Queen’s School of Engineering, University of Bristol, Queen’s Building, University Walk, Clifton,Bristol BS8 1TR, United Kingdom.

1070-6631/2013/25(11)/115104/33/$30.00 C©2013 AIP Publishing LLC25, 115104-1


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115104-2 Ramaprabhu, Karkhanis, and Lawrie Phys. Fluids 25, 115104 (2013)

work2 that considered the case g(t) = g0, it follows that a sinusoidal perturbation with amplitude h0

and horizontal wavenumber k = 2π /λ will grow as

h(t) = h0 cosh(�t). (1)

During this initial linear stage, the growth rate of the perturbation is given by � = √Agk, where

A is the Atwood number A = ρh − ρl

ρh + ρl. As the amplitudes grow larger and nonlinearities become

increasingly significant, the sinusoidal form, and exponential growth, of the original interface is lost.For kh ∼ 1, lower density structures moving parallel to g0 but in the opposite direction, known asbubbles for the rounded and relatively low-aspect-ratio appearance, differentiate themselves fromhigher density fluid structures moving with g0, known as spikes, that are characterized by a higheraspect ratio and faster relative growth rate. During the nonlinear stage, the saturated velocitiesof bubble and spike structures may be obtained from a potential flow analysis6–10 of the flowsurrounding a cylindrical structure. At late-time, a terminal velocity for bubbles, Vb, and spikes, Vs,of the following form is obtained

Vb/s =√

2Ag

(1 ± A) k. (2)

Interestingly, Eq. (2) does not predict a spike velocity as A → 1, i.e., as ρ l → 0. This drawback is

overcome by the potential flow model of Ref. 10 that accurately predicts spike free-fall (hs ∼ 1

2gt2)

in this limit. The nonlinear stage of single-mode RT growth may also be conveniently characterizedby defining a Froude number,

Frb/s = Vb/s√Agλ

/1 ± A

. (3)

Combining Eqs. (2) and (3) gives Frb/s ∼ 1√π

for low Atwood number flows.

Equation (2) for terminal velocity may also be obtained by making the modeling assumptionthat penetrating structures can be well-represented as elongating cylinders, and assuming that adynamic balance of inertia, buoyancy, and drag forces11–13 has reached a steady state at late time.This approach predicts that for large density differences, spikes should accelerate in agreementwith Zhang6 and consistent with recent numerical simulations14, 15 and experiments.16 Furthermore,as recent numerical simulations15, 17–19 and experiments16, 20 have demonstrated, the appearance ofshear instability between the counter-flowing spike and bubble streams can profoundly alter thedynamics in the nonlinear stage. Surfaces of uz = 0 lie approximately parallel to g0 and are Kelvin-Helmholtz (KH) unstable. At low to moderate Atwood numbers, the velocity field induced aroundthe surface organizes into the familiar array of “roll-up” vortices and these have been observed

to yield significantly higher Froude numbers (Frb/s ∼ 1) than the Frb/s ∼ 1√π

prediction made by

the potential flow models.7 Note that the secondary KH instability is inertially suppressed at largedensity ratios (A → 1), as the bubble velocity reverts to the prediction from Eq. (2).

When the initial interface between the heavy and light fluids is perturbed with a spectraldistribution of wavenumbers rather than one single spatial scale, one can obtain a first estimate of theensemble behavior by assuming a linear superposition of individual, independently growing modesaccording to Eqs. (2) and (3). However, such models ignore nonlinear interactions21, 22 betweenflow structures of different characteristic wavenumbers, and these interactions profoundly alterthe ensemble behavior. There is substantial evidence23, 24 that multi-mode perturbations develop amixing layer that asymptotes to a self-similar form, where the height h(t) of the mixing layer scales

with λ(t) = 2π

k(t), representing the dominant horizontal wavelength in the flow at any given time.

Invoking this assumption of self-similarity for nonlinearly saturated modes and using Eq. (2) forthe bubble or spike terminal velocity, it can be shown that the mixing layer height should evolveaccording to

hb/s = αb/sAgt2, (4)


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where αb/s is a growth factor. While this relationship is trivial to obtain from dimensional arguments,a more detailed picture emerges by following the evolution of saturated, self-similar modes, and onemay develop an expression for α in terms of the Froude number of the dominant structures,

αb = Fr2b

8

ρh + ρl

ρh

Db

hb, (5)

where Db represents the diameter of a typical rising bubble.The extent to which the flow conforms to the assumption of saturated self-similarity can be

captured by the ratio β ≡ λ

h= 2π

h(t)k(t), and together, α and β emerge as key parameters that describe

the growth of the turbulent mixing region. Previous studies25, 26 have reported that in the multimodecase, leading bubbles (i.e., those that collectively form the outer edge of the mixing region) have Fr∼ 1, substantially higher than the classical result from Eq. (3), though consistent with values fromsingle-mode isolated bubbles. It remains unclear if the mechanism for these elevated Fr values isdue to the vortex-induced acceleration from secondary instabilities, as described above, or due toleading bubbles behaving as isolated plumes in a low-drag environment as speculated in Ref. 25.

In this paper, we report on the behavior of miscible RT instability when the imposed accelerationfield swaps direction twice. Consequently the vector field representing baroclinic torque swaps sign,so over a very short time scale previously de-stabilizing torques become stabilizing. The initialdirection acts to destabilize the interface g > 0, then after some time the fluid is subjected to a periodof stabilizing deceleration g < 0, and thereafter destabilizing acceleration, g > 0, resumes. Theacceleration profile we have chosen to examine is inspired by the Linear Electric Motor experimentalresults described in Ref. 27 that investigated RT dynamics subject to acceleration reversals, and isapproximated here as

g = g0 {1 − tanh(η(t − T1)) − tanh(η(t − T2))} , (6)

where the reversals in g(t) occur at t = T1 (unstable → stable) and t = T2 (stable → unstable), andη sets the time scale over which the reversals occur. Since the time scales of RT flows vary inverselywith g0, we use g0 as a proxy to control the stage of evolution of the flow at which accelerationreversals are applied. We denote these accel-decel-accel profiles in this work as “ADAn” to indicate aconfiguration with g0 = n cm/s2, and we consider both single-mode and multi-mode initial interfaceperturbations. As RT flows evolve, nonlinearities become increasingly important and their responseto acceleration reversal varies accordingly. The variation of g0 studied here clarifies this behavior.

The importance to turbulent mix model validation of the particular acceleration profiles weexamine here has been underscored by several earlier studies.27–30 However, they have remainedchallenging to study numerically due to demands on grid resolution placed by the sudden proliferationof small scale structures during the deceleration phase. Additionally, transient periods of stabilizingdeceleration are thought also to occur in experiments, particularly those that involve a moving testcell,31, 32 and a fuller understanding of the influence such periods have on subsequent evolution isnecessary to correctly interpret experimental data.

The numerical techniques employed in our study are described in Sec. II, along with detailsof the problem configuration. In Sec. III, we discuss results from our study of single-mode initialperturbations, which is sub-divided in to Sec. III A (Constant g results), Sec. III B (Low Atwoodnumber results), and Sec. III C (High Atwood number results). Results from multi-mode simulationsare presented in Sec. IV, which is in turn organized in to Sec. IV A (constant g), and Sec. IV B(results from ADA profiles). Some conclusions and further discussion are drawn together in Sec. V.

II. NUMERICAL METHOD AND PROBLEM SETUP

A. MOBILE

MOBILE is a parallelized, three-dimensional, variable-density, finite-volume, incompressibleNavier-Stokes solver, and has been described in detail in Ref. 33–35. Mass and momentum areconserved subject to an incompressibility constraint, and a fractional step approach is employed,


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decoupling hyperbolic (advective transport), parabolic (viscous dissipation and scalar diffusion),and elliptic (pressure/velocity correction) components. The advection algorithm is monotonicity-preserving, using a nonlinear velocity-biased flux calculation that converges at between 2nd and3rd order. The parabolic operators implemented in MOBILE can use either an explicit or semi-implicit update, depending on the relative computational efficiency. Our choice of projection ofan intermediate velocity field on to the “nearest” divergence free vector field exactly conservesdiscrete angular momentum, though not discrete linear momentum, using the well-known Hodgedecomposition. Full multi-grid acceleration is used to efficiently solve the pressure Poisson equationthat arises from this projection. Parallelization is implemented with the MPI protocol, while post-processing of flow fields, calculating derived quantities and reducing them to concise statistics andvisualization is performed using a macro-language interpreter. The interpreter syntax is transparent tothe parallelization, save for the initial serial/parallel distribution of memory allocated for interpretedvariables.

B. Advection algorithm

The hyperbolic operator is decomposed into sequential X-Y-Z-Z-Y-X updates, following theapproach of Strang36 to improve temporal orders of convergence. Each 1D advection sub-problemis Total Variation Bounded (TVB), both eliminating unphysical oscillations and maintaining nu-merical stability for the full 3D problem, even when gradients are not properly resolved. The localRiemann problem across cell faces is solved with Godunov’s exact solution, which for this contact-discontinuity system is trivial, and high spatial order is achieved by modifying the left and rightstates of the Riemann problem using piecewise polynomial reconstruction of the spatial field.37 Themost obvious linear gradient to choose for a linear reconstruction in the cell at xi-1 has the form

mi−1 = ϕi − ϕi−2

2x. (7)

A higher order estimate of the gradient38 can be constructed by using the fluxed volume per unitarea ui-1/2t as a weighting that biases the mi-1 gradient as far towards a central difference over thecell face as possible. This more sophisticated gradient has the form

mi−1 = (1 + ui−1/2t

x)ϕi−1 − ϕi−2

3+

(2 − ui−1/2t

x

)ϕi − ϕi−1

3, (8)

which is the default choice in MOBILE. Empirical tests have shown that the error scales approx-imately with O(xn), 2 < n < 3, even though the stencil retains the spatial compactness of thestandard stencil for second order. The method applied in the flow is spatially dependent accordingto local velocity gradients, and a van-Leer-type limiter interpolates the flux between low and highorder. Mutually staggered grids are used, so that scalar fluxes are computed accurately with onlyone elliptic pressure solve – scalar quantities are stored at cell centers, while face-normal velocitiesare stored on their respective faces, but for each momentum component the fluxes are calculatedconservatively on each of the displaced grids.

The numerical method used in this study belongs to the class of Large Eddy Simulation schemessometimes referred to as Implicit Large Eddy Simulations, which have been extensively validatedover a wide range of flow conditions.39–44 Small-scale dissipation is modeled numerically in thesemethods, thus eliminating the need for an explicit sub-grid filter with tunable coefficients. Aspdenet al.39 developed a scaling analysis for the grid-dependent, numerical dissipation observed in suchmethods, from which an effective Kolmogorov scale is inferred. A similar approach was adoptedin the code comparison study by Dimonte et al.,25 who deduced an effective numerical dissipationεnumerical from the implied Kolmogorov scales in the energy spectra. MOBILE, as well as theILES (Implicit Large Eddy Simulation) codes evaluated in Refs. 25 and 39, produce dissipativespectra consistent with the expected E(k) ∼ k−3 scaling observed in experiments45, 46 and DNS47, 48

calculations. In such numerical methods, the energy dissipation (and associated small-scales) arenot grid-independent, but we made a conscious choice to use MOBILE in ILES mode because itmakes better use of available resolution to represent aspects of the physical problem that we regardas important. One attractive feature of ILES methods relevant to RT simulations is their ability to


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preserve monotonicity of the solution, even in the presence of discontinuous sharp flow featuressuch as shocks or material interfaces.

C. Single- and multi-mode configuration

The single-mode simulations described in Sec. III A and III B (A = 0.15) were performed in a

rectangular box with x-y-z dimensions of λ × λ × 4λ, where g acts along the z-axis, and λ = 2π

kis the wavelength associated with the initial perturbation:

h(x, y, t = 0) = h0

{cos

(2πx

λ

)+ cos

(2πy

λ

)}. (9)

The computational domain was extended to an aspect ratio of 8 for the large Atwood number(A = 0.9) simulations described in Sec. III C, while the interface was positioned off-center atz = 2λ to allow sufficient room for the spikes to grow. All the single-mode simulations described inSec. III A and III B were performed with 128 zones/λ to ensure excellent resolution of the small scalesthat emerge during stabilizing deceleration. The single-mode calculations were performed with a

scaled physical kinematic viscosity νSC L ≡ ν/√

Agλ3 = 1e − 3. Periodic boundary conditions were

enforced in the x-y directions, while the vertical (z - axis) boundaries were treated as outflow surfaces,though flow structures are seldom in proximity with the vertical boundaries on the time scales ofour calculations.

The multimode simulations were initialized with the perturbation function first presented in theso-called “alpha-group” study,25 and specified according to

h(x, y, t = 0) =∑kx ,ky

⎡⎢⎢⎢⎢⎢⎣

ak cos(kxx) cos(kyy) +bk sin(kxx) cos(kyy) +ck cos(kxx) sin(kyy) +dk sin(kxx) sin(kyy)

⎤⎥⎥⎥⎥⎥⎦, (10)

where the amplitudes ak, bk, ck, dk are generated pseudo-randomly to give h0,rms ∼ 3 e-4 L. Thesummation in the above equation is restricted to modes 32–64, thus corresponding to an annulus onthe 2D spectral plane of energetic modes consisting solely of high-k perturbations. This perturbationis expected to generate self-similar turbulence through the mode-coupling of saturated modes,22 andis conjectured49 to constitute a universal, initial-condition-independent solution. For the multi-modesimulations, we have selected our numerical approach to maximize the range of resolved scales.We achieve this by confining energy dissipation to length-scales of the order of the grid spacing,and using our nonlinear numerical scheme to perform this dissipation rather than making directcomputation of dissipation due to shear stress.

Our validation and grid convergence tests that justify this approach are documented in theAppendix. All the multimode simulations were performed with a mesh resolution of 512 × 512× 2048 zones in the (x, y, z) directions, respectively. For all the single mode simulations reported inthis work, the box dimensions were chosen to be one wavelength across, and we arbitrarily assign thesimulated horizontal plane an area of 1 cm2, with the vertical dimensions of the domain in proportionto the aspect ratio chosen in each case. We used the same relative box size for the correspondingmulti-mode simulations.

To establish a baseline for comparison (and for validation purposes), in both the single-mode and multi-mode studies, a constant acceleration simulation was performed with g(t) = gbase

= 2 cm/s2. For the set of simulations with acceleration reversal, g0 = {gbase, 2gbase, 4gbase} cm/s2

were used in Eq. (6) to study the effect of reversals in g(t) applied at different time scales relativeto the nonlinear development of the flow. We refer to these cases as ADA2, ADA4, and ADA8,respectively, according to the naming convention outlined earlier. We chose to study flows subjectto equal durations of acceleration, deceleration, and acceleration before a suitable final simulationtime Tend, with acceleration reversals at T1 = 1/3 Tend and T2 = 2/3 Tend. For the single-mode


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TABLE I. Summary of numerical simulations.

Atwood number Description Aspect ratio T1 (s) T2 (s) Tend (s)

Single-mode A = 0.15 Constantg 4ADA2 4ADA4 4ADA8 4 3.33 6.66 10

A = 0.9 Constantg 8ADA2 8ADA4 8 2 4 6ADA8 8

Multi-mode A = 0.5 Constantg 4ADA2 4ADA4 4 2 4 VariableADA8 4

calculations, we use the linear time scale1

�= 1√

Ag0kderived from Eq. (1), that describes the

time taken for the interfacial amplitude to grow by a factor e. We can thus quantify the degree ofnonlinear development of the RT instability at any dimensional time t by the number of e-foldings.As g0 increases, this measure of nonlinearity, τ ≡ t

√Agk, will be larger for an identical dimensional

time, and in particular at the first acceleration reversal t = T1. The values of T1, T2, and Tend usedfor all the simulations described here are listed in Table I.

Following Dimonte et al.27 and Mikaelian31 we employ a velocity-scale that characterizes themotion of the initial interface throughout the sequence of acceleration reversals

U(t) =t∫

0

g(t′)dt′. (11)

By integrating once more, a corresponding length-scale can be constructed

Z(t) =t∫

0

U(t′′)dt′′ =t∫

0

t′′∫0

g(t′)dt′dt′′, (12)

where the intermediate variables t′ and t′′ are essentially arbitrary. Thus, in a multimode RT, thebubble and spike structures scale with Z(t), under the assumption that these structures arise frommodes that are saturated and self-similar. We can view this definition of Z(t) as a direct generalizationof the hb/s ∼ gt2 length-scale obtained in Eq. (4) that might account for non-constant g. Equivalently,one can relate Z(t) to the required displacement of a moving test cell in a zero-gravity environmentto generate a desired acceleration profile g(t). In the special case of constant acceleration g(t) = gbase

(e.g., representing planetary gravity), the test-cell position Z(t) must displace quadratically in time.Thus, the mixing zone height hb/s(t) of the RT instability which grows quadratically in time(Eq. (4)), must grow linearly with Z(t) according to hb/s(t) = 2αb/sAZ(t). Table I lists details ofthe all simulations described in this work.

III. SINGLE-MODE SIMULATIONS

While most experiments and applications are likely initialized with “multimode” perturbations,the properties of the single-mode RT flow are important in developing a fundamental understandingof the corresponding multimode problem. This has motivated several studies of the single-modeproblem including experiments,20, 50 simulations,14, 15, 17–19 and theory.6–10, 12, 31, 51, 52 Miscible RTexperiments have measured a turbulent growth rate αb of 0.05–0.08, where αb is defined fromEq. (4). As argued in several works (Ref. 49, and references therein), Eq. (4) may be obtained


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by requiring that (a) the RT turbulent front is dominated by leading bubbles with behavior akinto single-mode structures that have reached nonlinear saturation with a constant Froude number,and (b) that these dominant bubbles evolve with Db ∼ hb (self-similarity constraint). Combiningthese two constraints, we can arrive at Eq. (4). Note that there are two schools of thought that relatesingle-mode evolution to turbulent RT growth described by Eq. (5). In the so-called “bubble-merger”models,12, 53 Db ∼ hb is achieved through the merger of saturated bubbles (and spikes) where theseinteractions at a prescribed merger rate cause the bubbles to grow in size. In contrast, with ‘bubblecompetition’ models21, 49, 54], single modes are taken to grow independently, to yield the gt2 scalingfor both the amplitude and wavelength of the dominant bubbles. Thus, our results of the single-modeRT inform the calibration of several widely used theoretical frameworks that are central to simplifiedengineering models of the corresponding multimode problems.

A. Constant g

To establish a benchmark against which the RT instability undergoing an ADA profile canbe compared, we present in this section results from our g(t) = gbase simulation with A = 0.15.Vertical planes of fluid volume fraction, f1 = 1 - f2, taken on the diagonal (x, x, z) with 0 ≤ x≤ λ, during the initial acceleration are realized at the set of non-dimensional times t

√Agk = {2,

6, 8.9} in Figs. 1(a)–1(c), respectively, and show the evolution of the fundamental mode throughthe linear, nonlinear and late-time stages. While the interface retains its sinusoidal shape (Fig. 1(a)),it exhibits exponential growth of its amplitude, according to Eq. (1). A short time later however,as shown in Fig. 1(b), the interfacial growth rate has reached nonlinear saturation, where distinctbubble and spike structures emerge. In these low Atwood number calculations, bubbles and spikes areapproximately symmetric. The late-time evolution shown in Fig. 1(c) is dominated by the maturationof short wavelength (∼λ/2) KH-unstable vortex rollups that have formed on the sides of the originalfundamental wavelength bubble and spike.

In Fig. 2(a), we plot the time evolution of bubble (〈hb〉) and spike (〈hs〉) amplitudes, measuredas the z-locations of the 1% and 99% iso-surfaces of the x-y planar-averaged light fluid volumefraction. We note this is a standard choice widely used in experimental45, 46, 55, 56 and numericalinvestigations15, 23, 25–27, 47, 48 of single-mode and multimode RT. The corresponding bubble and spikeFroude numbers are plotted individually in Fig. 2(b). The evolution in Fig. 2 is consistent with thethree stages of RT growth identified in Fig. 1(a)–1(c). During the linear growth stage, hb/s ∼ e

√Agkt

(a) (b) (c)

FIG. 1. Diagonal slices of volume fraction contours showing bubble and spike structures at (a) t√

Agk = 2, (b) t√

Agk = 6,and (c) t

√Agk = 8.9. The simulations were performed with A = 0.15 and a constant acceleration (g0 = 2 cm/s2).


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(a) (b)

FIG. 2. Evolution of bubble and spike (a) amplitudes and (b) Froude numbers from simulations at A = 0.15 with constantacceleration (g0 = 2 cm/s2), plotted against the scaled time t

√Agk. Dashed line indicates potential flow result from Ref. 7.

and so Frb/s ∼ et. The onset of the “nonlinear growth” phase is clearly identified by the saturation

of the Froude numbers to Frb/s ∼ ± 1√π

, in good agreement with the potential flow models of

Goncharov,7 (shown as dashed horizontal lines in Fig. 2(b)). At later time, there is an increase inFrb/s, due to increasing bubble and spike growth rates. The mechanism for these increases is attributedto the action of KH vortices, which appear as illustrated in Fig. 1(c), on the elongating side-surfacesthat lie between rising and falling structures. The KH-induced velocity field superimposes on thebaroclinically driven base flow, and supplies a pressure gradient that accelerates the bubble andspike structures. This mechanism of bubble/spike reacceleration has been described in detail inrecent numerical studies.15, 17–19

B. Accel-decel-accel (A = 0.15)

The acceleration profiles g(t) we consider are shown in Fig. 3(a), where the three lines correspondto Eq. (6) for the chosen set g0 = {gbase, 2gbase, 4gbase}. By construction, the three profiles g(t) aresymmetric about zero, and the acceleration reversals occur at common time scales, but at differentvalues of the displacement Z(t). The variation in g0 linearly scales the time-rate of development ofthe flow, and allows us to examine how a RT flow responds to a period of deceleration occurring atdifferent stages during its nonlinear development.

In Fig. 3(b), we plot the time-integral of the acceleration profile as given by Eq. (11), which weinterpret as a representative velocity scale for the RT development under arbitrary g(t). The acceler-ation is linear in velocity, and neglecting the period of transition from acceleration to decelerationset by the coefficient η in Eq. (6), the deceleration between T1 = 1/3 Tend and T2 = 2/3 Tend is linearand symmetric to U(T2) = 0. The subsequent re-acceleration duplicates the initial phase 0 ≤ t ≤ T1

so that U(Tend) = U(T1).In Fig. 3(c), we plot Eq. (12), the displacement Z(t) that relates characteristic time- and length-

scales of the flow. Since initial acceleration and deceleration periods have been chosen to be equalin duration, Z(t) is monotonic, with an inflexion point at T2 = 2/3Tend. Z(t) grows quadraticallyinitially, and again neglecting the period of transition, also decelerates quadratically. The subsequentacceleration is once again quadratic, commencing from T2 = 2/3Tend with zero gradient. The finalinterface position is dependent on g0.


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-10

-5

0

5

10

0 2 4 6 8 10

g

(cm/s2)

t (s)

g0 = 2

g0 = 4

g0 = 8

0

5

10

15

20

25

30

0 2 4 6 8 10

U(cm/s)

t (s)

g0 = 2

g0 = 4

g0 = 8

-20

0

20

40

60

80

100

120

140

0 2 4 6 8 10

Z (cm)

t (s)

g0 = 2

g0 = 4

g0 = 8

(a)

(c)

(b)

(d)

0

10

20

30

40

50

0 2 4 6 8 10

S (cm)

t (s)

g0 = 2

g0 = 4

g0 = 8

FIG. 3. Time histories of (a) the acceleration g, (b) the interface velocity U, (c) the interface displacement Z, and (d) thescaling width S, used for accel-decel-accel simulations with g ∼ g0 {1-tanh(η(t-T1))-tanh(η(t-T2))}, T1 = 3.33 s, T2 = 6.66s, and Tend = 10 s.

An alternative length scale is the mix width S proposed elsewhere in the literature,31, 32, 53 whereit has been defined as

S ≡⎧⎨⎩

t∫0

√g(t)dt

⎫⎬⎭

2

. (13)

The scalings S and Z are two different linearizations of the well-known inertia-buoyancy-drag equation, and Eq. (13) emphasizes the balance between drag and buoyancy forces, while thedefinition of Z (Eq. (12)) emphasizes the balance between inertia and buoyancy. Note that both Z andS integrate to gt2 for constant acceleration RT. There is some debate over the appropriate selection ofa scaling for time-dependent g(t), and it may not be independent of the Atwood number or the formof the function g(t). Of course, Eq. (13) breaks down for g(t) < 0, during the deceleration phase ofour simulations, and thus cannot be used directly. We have instead explored a minor variation:

S ≡⎧⎨⎩

t∫0

sgn(g(t))√

|g(t)|dt

⎫⎬⎭

2

. (14)

This definition of S is symmetric about g(t) and thus returns the mix width to zero at the end ofthe deceleration stage. In Figure 3(d), we plot the time evolution of S against time for the three g(t)functions used in our ADA simulations. Clearly, while Z is monotonic (Fig. 3(c)) with an inflexionpoint at g(T2.), where it changes sign, S is piecewise parabolic and non-monotonic. In Sec. IV B, we


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(a) (b) (c)

FIG. 4. Diagonal slices of volume fraction contours from simulations with accel-decel-accel histories, showing bubble andspike structures at (a) t = 3.33 s (T1), (b) t = 6.66 s (T2), and (c) t = 10s (Tend). The simulations were performed withA = 0.15 and g0 = 2 cm/s2.

further explore the appropriateness of the scaling lengths defined here in explaining our multimodesimulation results.

Figures 4 (g0 = gbase) and 5 (g0 = 2gbase) illustrate the complexity of the secondary structuresthat arise when employing the acceleration profiles examined herein. In each case, the images arerealized at t = T1 (end of the first acceleration phase), t = T2 (end of the deceleration phase), andt = Tend at the end of the reacceleration phase (which marks the end of the simulation). Fig. 4(a)corresponds to t

√Agk ∼ 4.5 e-foldings of the original perturbation, whereas Fig. 5(a) is realized

at t√

Agk = 6.46, and thus its nonlinear development is more advanced when the decelerationcommences. The qualitative contrast in flow structure is shown in the series of images in Figs. 4and 5.

For the g0 = gbase ADA case shown in Fig. 4, at t = T1 the interfacial displacement h(x,y,T1)(as defined by the 50% iso-surface of volume fraction) is everywhere single-valued, and by t = T2

in Fig. 4(b), the primary mode has fully inverted. We use Trev to denote the phase reversal time, T1

< Trev < T2, and when points that satisfy the condition h(x, y, 0) − 〈h〉 > 0 exclusively satisfy h(x,y, Trev + ε) − 〈h〉 < 0, where 〈h〉 is the planar-averaged z-location of the 50% iso-surface and ε

is a small parameter, we denote this a “clean” phase reversal. Since t = T2 is an inflexion point ofZ(t), the final re-acceleration can be viewed as a nonlinear finite-amplitude initialization of the RTinstability. As a result of the phase reversal, coherent flow structures previously called bubbles risingbuoyantly will after t = T2 become falling spikes, and vice versa. Thus, the position of bubblesand spikes for t > T2 are displaced horizontally by a half wavelength from their initial positions,given by Eq. (9). The image in Fig. 4(c) of the final time, t = Tend, illustrates the onset of KH-driven secondary structures as the vertical surfaces separating the bubbles and spikes grow longer. InFig. 5(c), where the nonlinear development is further advanced, similar KH-unstable vortical struc-tures appear, though at smaller length-scales. In both cases, these secondary instabilities may beexpected to influence the bubble and spike Froude numbers, and hence the growth rate of the primaryRT mode through the processes described in Sec. III A.

For g0 = 2gbase (Figs. 5(a)–5(c)), the deceleration time T1 arrives when the RT interface is alreadymulti-valued with substantial vertical surfaces and KH-unstable vortical structures developing onthem, hence a significantly broader spectrum of length-scales is present in the flow at late times.


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(a) (b) (c)

FIG. 5. Diagonal slices of volume fraction contours from simulations with accel-decel-accel histories, showing bubble andspike structures at (a) t = 3.33 s (T1), (b) t = 6.66 s (T2), and (c) t = 9 s. The simulations were performed with A = 0.15 andg0 = 4 cm/s2.

At the end of the deceleration (Fig. 5(b)), the primary mode has inverted, but meanwhile secondaryfeatures of the bubble and spike collide, more rapidly creating small-scale structure and consequentlyincreasing the rate of mixing (visible as large regions of gray in Fig. 5(b)). During the reacceleration(Fig. 5(c)), the late-time bubble and spike structures grow as one would expect for well-developedclassical RT, with increasingly complex small-scale structure continuing to emerge.

The time evolution of the bubble and spike amplitudes for ADA2 is shown in Fig. 6, wheredownward (upward) triangles indicate t = T1 (T2), the times at which the reversals in g occur. Inthis figure, the amplitudes are traced by following the extrema of the volume fraction gradient signalon the symmetry axes. Measuring amplitude in this manner is meaningful, so long as the volumefraction gradients ∇f1 = −∇f2 remain relatively steep within the period of interest. For t > 7.5 s, thebubble and spike Froude numbers were estimated to be 0.56 and 0.62, respectively, in agreement withpotential flow7–10 and drag-buoyancy11–13 model estimates for RT flow at A = 0.15. This suggeststhat following the second reversal of g to g(t) > 0, the flow behaves as a classical nonlinear RT witha terminal velocity.

Unfortunately, as the number of e-foldings grows, there can be large regions of mixed fluidand shallow volume fraction gradients, so amplitude identification by the above approach becomesambiguous. Instead, we choose to infer amplitudes from the profile of the planar-averaged volumefraction, and trace the displacement along the vertical coordinate (z) of the 1% and 99% thresholds.This is robust to more substantial interfacial mixing, generalizes to multi-mode initial conditions,and is in line with common practice.25 These results are shown in Fig. 7(a) for the ADA cases g0

= {gbase, 2gbase, 4gbase} discussed above. For comparison, results from the benchmark g(t) = gbase

simulation are also shown as the solid line. As expected, the ADA case is indistinguishable from thebenchmark simulation until T1, the instant of the first phase reversal.

In common with Fig. 6, amplitudes shown in Fig. 7(a) exhibit a decrease during the decelerationperiod T1 > t > T2, and this is consistent with phase reversal. However, the planar-averaged measureof amplitudes is not able to capture phase reversals of the primary mode. Once the interface is multi-valued it must be viewed as occupying a finite region spread by chaotic transport, and the associatedmixing processes reduce the gradients ∇f1 and ∇f2. Thus, during a phase reversal, the amplitudes


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-1.5

-1

-0.5

0

0.5

1

1.5

0 2 4 6 8 10

h (cm)

t (s)

Bubbles

Spikes

FIG. 6. Evolution of the bubble and spike amplitudes for the accel-decel-accel simulations with A = 0.15 and g0 = 2cm/s2. The amplitudes were deduced by tracking the bubble and spike tips, which show the phase reversal seen in Figure 4.Downward triangles indicate onset of deceleration at t = T1, while upward triangles indicate return to acceleration at t = T2.

cannot pass h = 0, but instead have a stationary point that reflects the size of the mixing region at itsmost vertically compact. The tendency for the mixing region to decrease in size under deceleration issomewhat misleadingly referred to as “demixing,” but for clarity we avoid this term, and emphasizethat variable density mixing is an energetically irreversible fluid process, and compression of thevolume of the mixing region due to body forces is distinct from re-arranging the distribution ofdensities in the fluid. The asymptotic recovery to RT growth is robust as all the cases approach thesolid line in Fig. 7(b), which is plotted against the generalized length-scale Z(t).

To quantify the mixing due to collision of coherent structures during deceleration, we computethe global atomic mix parameter,23

=hb∫

−hs

〈flfh〉〈fl〉〈fh〉dz, (15)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 2 4 6 8 10

G2ADA2ADA4ADA8

<h> (cm)

t (s)

Bubbles

Spikes

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 20 40 60 80 100

G2ADA2ADA4ADA8

<h> (cm)

Z (cm)

Bubbles

Spikes

(a) (b)

FIG. 7. Evolution of planar-averaged bubble and spike amplitudes for simulations with g ∼ g0 {1-tanh(η(t-T1))-tanh(η(t-T2))} and A = 0.15. The amplitudes are plotted against (a) time, and (b) the interface displacement Z. Downward trianglesindicate onset of deceleration at t = T1, while upward triangles indicate return to acceleration at t = T2.


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FIG. 8. Evolution of atomic mix fraction for the accel-decel-accel simulations with A = 0.15, plotted against the interfacedisplacement Z. Downward triangles indicate onset of deceleration at t = T1, while upward triangles indicate return toacceleration at t = T2.

where 〈 · 〉 refers to planar averaging over the horizontal x-y plane. Defined thus, = 0 for the caseof pure fluids fl = 1, fh = 0 and fl = 0, fh = 1, and = 1 when fl = fh = 0.5, correspondingto completely mixed fluid. In all of our simulations, starts at a high level caused by the discreterepresentation of the initial interface. This initial value varies according to the position and amplitudeof the initial interface relative to cell faces. The only contributions to the integral in Eq. (15) comefrom cells with impure or partially mixed fluids, and initially these are exclusively located aroundthe interface so the measure is not well posed at t = 0. However, after a rapid decrease during thevery earliest stages of growth, has a stationary point, after which it becomes a meaningful measureof atomic mixing, i.e., as though the initial interface had been represented as perfectly sharp. Forthe g(t) = gbase case, after passing its stationary point, increases gradually with Z(t), as shown inFig. 8, as the emergence of small-scale structures contribute to greater atomic mixing, though therate of growth decays somewhat after Z(t) = 40 cm.

For the ADA case g0 = gbase, which is single-valued at T1, peaks at a time corresponding tothe instant of phase reversal. Due to cell-averaged mixing on the interface in our numerical model,and molecular diffusion in the equivalent physical system, the amplitude will not be precisely zero,hence is not singular at the instant of phase reversal. Following the phase reversal, the gradientd /dt is not dissimilar to that for the benchmark g(t) = gbase case, so it appears from Fig. 8 thatfor single-valued h(x, y, T1) the deceleration phase has a rather limited effect on the growth rate ofthe atomic mix parameter. This is consistent with the benchmark g(t) = gbase, and g0 = gbase ADAprofiles for h(t) in Fig. 7(a) having similar gradients at late time.

The ADA cases g0 = 2gbase and g0 = 4gbase have multi-valued h(x,y,T1) because KH-unstablesecondary vortices form due to the velocity difference across the density surface separating bubbleand spike. During deceleration, the direction of baroclinically applied torque is reversed, now oppos-ing growth of the primary mode, and hence the velocity difference that generates the KH-unstablevortical structures decreases. The further development of secondary vorticity by this mechanism isnow suppressed. However, the atomic mix parameter grows astonishingly rapidly during deceler-ation compared with the benchmark g(t) = gbase case. This occurs because these secondary vorticesundergo quasi-turbulent breakdown as the primary mode reverses direction, and much more rapidmixing ensues. A comparison between Figs. 4(b) and 5(b) illustrates clearly the increase in structuralcomplexity at t = T2 due to multi-valued h(x, y, T1).

The final reacceleration phase is marked by a return to RT-like growth of the atomic mixparameter, consistent with finite-amplitude initialization. As shown in Fig. 8 the benchmarkg(t) = gbase case forms a lower bound for , while the multivalued cases emerge from the


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(a) (b) (c)

FIG. 9. Diagonal slices of volume fraction contours from simulations with accel-decel-accel histories, showing bubble andspike structures at (a) t = 2 s (T1), (b) t = 4 s (T2), and (c) t = 6 s. The simulations were performed with A = 0.9 andg0 = 2 cm/s2.

reacceleration with substantially increased atomic mixing over the benchmark case and the single-valued case (ADA2).

C. Accel-decel-accel (A = 0.9)

When the density difference between the fluids is large (ρ2/ρ1 = 19), the secondary KHinstability in the vertical interfacial region between rising bubbles and falling spikes is inertiallysuppressed,15 resulting in bubble and spike structures with no susceptibility to KH “rollups.” Thisis consistent with the results of linear theory3 for Kelvin-Helmholtz instability, that predicts forlarge density ratios the KH growth rate → 0. Under such conditions, when the acceleration field isreversed, the interface is monotonic leading to a “clean” phase inversion and relatively little mixing.This can be seen in the series of images from the simulation with g0 = gbase (ADA2) presented inFigs. 9(a)–9(c), which show the nearly sinusoidal interface at t = T1 undergoing a phase reversal byt = T2. However, unlike the flowfield shown in Figs. 4 and 5, this process is not accompanied bysmall-scale structure formation or increased mixing over any significant portion of the domain. Evenat late times (Fig. 9(c)), the flow is almost entirely occupied by pure fluid without any secondarystructures. When the simulation is repeated with g0 = 2gbase (ADA4 shown in Figs. 10(a)–10(c)),the advanced nonlinearity of the primary waveform creates more convoluted nonlinear structure andmixing than the corresponding g0 = gbase case, but still less than the A = 0.15 simulations in Figs. 4and 5. In particular, slender jets of the heavy fluid form and penetrate the light fluid shortly after thefirst acceleration reversal (Figs. 10(b) and 10(c)). These narrow spikes flank the primary structure inthe flow, and are formed ahead of it, as shown in Fig. 10(b). The primary spike has a higher effectiveλ and a higher Fr (Eq. (3)), thus at later time (Fig. 10(c)), it moves more quickly than the slendersecondary spike structures and by the end of the simulation (Fig. 10(c)), will dominate them.

Planar averaged bubble and spike amplitudes are shown in Figs. 11(a) and 11(b), plotted,respectively, against time and displacement Z, respectively. As discussed in Sec. III B, resultsfrom the constant acceleration simulation are also shown here as a benchmark for comparison


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(a) (b) (c)

FIG. 10. Diagonal slices of volume fraction contours from simulations with accel-decel-accel histories, showing bubbleand spike structures at (a) t = 1.6 s, (b) t = 3.25 s, and (c) t = 4.9 s. The simulations were performed with A = 0.9 andg0 = 4 cm/s2.

with the ADA flows. When driven by constant g, high Atwood RT develops asymmetrically, asbubble amplitudes follow Eq. (2), while spikes are in free-fall (hs ∼ gt2) in Fig. 11(a) (thus, hs ∼ Z inFig. 11(b) since Z = 1/2 gt2 for constant g). Furthermore, the three stages of ADA growth described inSec. III B are observed here, but with asymmetric amplitudes. Once again, the constant g amplitudesform an outer envelope to which the ADA amplitudes asymptotically recovers. Finally, the evolution

-5

-4

-3

-2

-1

0

1

2

0 1 2 3 4 5 6

G2ADA2ADA4ADA8

<h> cm

t (s)

Bubbles

Spikes

-5

-4

-3

-2

-1

0

1

2

0 10 20 30 40 50

G2ADA2ADA4ADA8

<h> cm

Z (cm)

Bubbles

Spikes

(a) (b)

FIG. 11. Evolution of the bubble and spike amplitudes for simulations with g ∼ g0 {1-tanh(η(t- T1))-tanh(η(t-T2))} and A= 0.9. The amplitudes are plotted against (a) time, and (b) the interface displacement Z. Downward triangles indicate onsetof deceleration at t = T1, while upward triangles indicate return to acceleration at t = T2.


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FIG. 12. Evolution of atomic mix fraction for the accel-decel-accel simulations with A = 0.9. Downward triangles indicateonset of deceleration at t = T1, while upward triangles indicate return to acceleration at t = T2.

of the atomic mix fraction with respect to Z is shown in Fig. 12, for the constant g and the ADAcases. Due to the absence of secondary structures described above, the peaks achieved immediatelyfollowing the g-reversal are lower than the A = 0.15 counterparts. Furthermore, the peaks inFig. 12 are short-lived compared to Fig. 8, suggesting they may be more an artifact of the temporaryflattening of the interface during phase reversal. In each of the ADA cases, the mix fraction showsa tendency to return to the trend line established by the constant g flow, although for ADA4 andADA8, the simulations were not run long enough (due to bubbles encountering the top wall) toascertain this unambiguously.

IV. MULTI-MODE SIMULATIONS

A. Constant g

When initialized with Eq. (10), our simulations yield self-similar, turbulent RT growth con-sistent with Eq. (4). The images in Figs. 13(a)–13(e) are x-slices (at the midplane) of volumefraction contours from calculations with g(t) = gbase, and A = 0.5. The sequence of images areobtained at (a) t = 0.4 s (Z = 0.16 cm), (b) t = 1.0 s (Z = 1.0 cm), (c) t = 2.0 s (Z = 4.0 cm),(d) t = 2.8 s (Z = 8 cm) and (e) t = 6.0 s (Z = 36 cm). Following the growth and saturation of theearly dominant modes49 in Figs. 13(a) and 13(b), we observe continued growth that is apparentlymodified by mode-coupling of larger scales in the flow (Figs. 13(c)). The scale distribution of thedominant modes in Figs. 13(d) and 13(e)appears to validate the self-similarity hypothesis (λ ∼ h), asonly a small number of large structures remain by the end of the simulation. Due to the high degreeof nonlinearity and interaction between scales, there is significant mixing between the fluids byZ = 36 cm, as has been well documented in earlier studies.24, 25, 45

B. ADA results (A = 0.5)

x-slices of the volume fraction field from our ADA2 multimode simulation are shown inFigs. 14(a)–14(d), while the corresponding horizontal slices (x-y) at the interface location areshown in Figs. 15(a)–15(d). The sequence of images are obtained at (a) t = 0.4 s (Z = 0.16 cm),(b) t = 2.0 s (Z = 4 cm), (c) t = 3.9 s (Z = 8 cm), and (d) t = 8.2 s (Z = 25 cm). In eachcase, the first two images show the development of the flow field driven by a positive, constant g(t < T1), with features similar to the images in Fig. 13. The reversal in g induces a rapid changein the distribution of length-scales in the flow, as shown in Figs. 14(c) and 15(c). Previously well-organized, large-scale structures evident at earlier times disintegrate due to collisions in the reversed


160.36.178.25 On: Sun, 21 Dec 2014 15:23:22


(a)

(b)

(c)

(d)

(e)

FIG. 13. X-slices of volume fraction contours from multimode simulations with constant g at (a) t = 0.4 s, (b) t = 1.0 s, (c)t = 2.0 s, (d) t = 2.8 s and (e) t = 6.0 s. The corresponding values of Z are (a) 0.16 cm, (b) 1.0 cm, (c) 4.0 cm, (d) 8 cm, and(e) 36 cm. The simulations were performed with A = 0. 5 and g0 = 2 cm/s2.

(a)

(b)

(d)

(c)

FIG. 14. X-slices of volume fraction contours from multimode simulations with accel-decel-accel histories at (a) t = 0.4 s,(b) t = 2.0 s, (c) t = 3.9 s, and (d) t = 8.2 s. The corresponding values of Z are (a) 0.16 cm, (b) 4 cm, (c) 8 cm, and (d) 25cm. The simulations were performed with A = 0. 5 and g0 = 2 cm/s2.


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(a)

(c)

(b)

(d)

FIG. 15. Horizontal slices of volume fraction contours from multimode simulations with acceldecel-accel histories at(a) t = 0.4 s, (b) t = 2.0 s, (c) t = 3.9 s, and (d) t = 8.2 s. The corresponding values of Z are (a) 0.16 cm, (b) 4 cm, (c) 8 cm,and (d) 25 cm. The simulations were performed with A = 0. 5 and g0 = 2 cm/s2.

gravity field, and are replaced by well-mixed, small-scale structures. The inertia of coherent movingstructures increases with length-scale, so at the moment of acceleration reversal, the response-timeof modes to the new acceleration direction is scale-dependent. During this transition period, localshear gradients are greatly enhanced in regions with a broad spectral distribution of modes, andgrowth of secondary instabilities is dominated by response to shear rather than baroclinic torque.These secondary instabilities are short-lived and small-scale, leading rapidly to mixing.

Another consequence of the enhanced reversal-induced mixing and associated shredding ofcoherent directional structures is that velocities in the interfacial region become less anisotropic,as can be seen qualitatively by comparing Fig. 14(b) with Fig. 14(c). Upon re-acceleration, largescale structures once more dominate the growth; this can be viewed as a re-initialization with afinite coupled velocity and density perturbation of a spectral distribution prescribed by the precedingflow. From comparison of Figs. 13(d) and 13(e)) with Figs. 14(c) and 14(d)), this re-initializationappears, at least qualitatively, to have negligible influence on the late-time evolution. We describethese features quantitatively below.

In Fig. 16(a), we plot vertical profiles of the planar-averaged volume fractions of the lightfluid from our ADA calculation with g0 = gbase (ADA2). The vertical coordinate z is scaled withhb, the instantaneous height of the bubble ensemble, while the profiles are realized at Z-valuescorresponding to the contour images in Figs. 14 and 15 (Z = 0.16 cm, 4 cm, 8 cm, and 25 cm,respectively). Consistent with the vertical slices displayed in Figs. 14, the cross-stream profilesof planar-averaged volume fractions show asymmetry between the bubble and spike fronts. ByZ = 4 cm, self-similarity has been established as the profiles collapse when the vertical coordinateis scaled with hb. In Fig. 16(b), vertical profiles of 〈flfh〉 (z) are plotted at the same stages of RTdevelopment. At Z = 4 cm, 〈flfh〉 peaks at the centerline with a value of ∼ 0.2, where a maximumvalue of 0.25 would indicate perfectly mixed fluids (i.e. 〈fl〉 = 〈fh〉 = 0.5). From Figs. 16(a) and

16(b), this corresponds to a local value of the atomic mix coefficient (z = 0) = 〈flfh〉〈fl〉〈fh〉 ∼ 0.8.


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(a) (b)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-1.5 -1 -0.5 0 0.5 1 1.5

Z = 0.16 cmZ = 4 cmZ = 8 cmZ = 25 cm

< flfh >

z/hb

0

0.2

0.4

0.6

0.8

1

-1.5 -1 -0.5 0 0.5 1 1.5

Z = 0.16 cmZ = 4 cmZ = 8 cmZ = 25 cm

< fl >

z/hb

FIG. 16. Cross-stream profiles of the planar-averaged (a) volume fraction of light fluid and (b) 〈flfh〉 at Z = 0.16 cm, 4 cm,8 cm, and 25 cm. The simulations were performed with A = 0. 5 and g0 = 2 cm/s2.

Following the reversal in g (Z = 8 cm), nearly perfect mixing is achieved at the centerline dueto the aforementioned breakup of coherent structures, resulting in 〈flfh〉 (z = 0) → 0.25 (i.e.

(z = 0) → 1). Upon restoration of the destabilizing acceleration (Z = 25 cm), the mixing becomesless than ‘perfect’, with 〈flfh〉 reverting back to a value of 0.2 ( (z = 0) ∼ 0.8), comparable with theself-similar statistics prior to the g-reversal (Z = 8 cm). Note that at late time (Z = 25 cm), the flowis dominated by a small number of leading bubbles and spikes across any single horizontal plane(see Figs. 14(d) and 15(d)), resulting in fluctuations in 〈fl〉 and 〈flfh〉 due to paucity of structuresincluded in the planar averages.

The planar-averaged amplitudes of bubble and spike fronts from all of our multimode casesare plotted against time, the interface displacement Z, and the scaling width S in Figs. 17(a)–17(c),respectively. In Fig. 17(a), the bubble- and spike-side displacements scale with g0 in the ADAprofiles during the initial and final stages of acceleration. For A = 0.5 (ρ2/ρ1 = 3), the spikefrontslightly outpaces the bubblefront with hs/hb ∼ 1.3 by the end of the simulation for the cases reportedhere. Similar to our results for the single-mode case, the deceleration phase between T1 and T2

induces a phase reversal of individual structures as described in Sec. III B. However, since in this

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 2 4 6 8 10

G2ADA2ADA4ADA8

<h> (cm)

t (s)

Bubbles

Spikes

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 10 20 30 40 50 60 70 80

G2ADA2ADA4ADA8

<h> (cm)

Z (cm)

Bubbles

Spikes

(a) (b)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 10 20 30 40 50 60

G2ADA2ADA4ADA8

<h>cm

S (cm)

(c)

FIG. 17. Evolution of the bubble and spike amplitudes for multimode simulations with g ∼ g0 {1-tanh(η(t-T1))-tanh(η(t-T2))} and A = 0.5. The amplitudes are plotted against (a) time, (b) the interface displacement Z and (c) the scaling width S.Downward triangles indicate onset of deceleration at t = T1, while upward triangles indicate return to acceleration at t = T2.


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multi-mode case these structures are distributed over a broad range of length-scales, the time scalesof their inertial response are also distributed. Thus the deceleration phase merely promotes morerapid breakup of the structures. For Z > 20 cm, the constant g simulation achieves asymptoticbubble- and spike-front growth rates of αb = 0.02 ± 0.004 and αs = 0.03 ± 0.01, respectively, whenthe data from Fig. 17 are compared with the equation for self-similar growth hb/s = 2αb/sAZ, and

using the definition αb/s ≡ 1

2A

dhb/s

d Z. In contrast, the ADA2 simulations asymptote to growth rates

of αb = 0.02 ± 0.01 and αs = 0.03 ± 0.007, respectively, when the data corresponding to the secondacceleration phase are obtained by averaging for Z ≥ 24 cm. Similarly for ADA4, the asymptotic(Z > 30 cm) values of growth rates were determined to be αb = 0.022 ± 0.01 and αs = 0.04 ± 0.01,respectively, while the ADA8 does not appear to have reached full self-similarity in Fig. 17.

The bubble and spike amplitudes from the ADAn and g(t) = g0 simulations are plotted againstthe width S in Figure 17(c). As expected, during the deceleration stage, the amplitudes exhibithysteresis, as S → 0. However, Eq. (14) suggests the mix width must return to its initial value ofzero at the end of deceleration, whereas the simulation data indicate a finite mix width at t = T2.The reacceleration stage t > T2 is therefore initialized with a large amplitude velocity and densityperturbation. We draw the conclusion that inertia—the forcing ignored in the S linearization–isresponsible for the discrepancy between S and the simulation data. The question following thisobservation is whether S or Z would be a better predictor of RT behavior. One feature we noticeis that the late-time trajectory of h(t) from our ADA simulations asymptote towards that of theg(t) = const. case, only when compared against Z (Figure 17(b)), and not when compared againstS. Scaled against S, the amplitudes have a permanent late-time offset.

To further illustrate this point, we plot the evolution of time scales√

h/g0 against√

Z/g0

and√

S/g0 in Figures 18(a) and 18(b), respectively. The ADA simulation data collapse well;the remaining discrepancies between individual simulation trajectories arise because of a weakdependence on the Reynolds number. More relevant to the present discussion, we interpret deviationof the ensemble of simulated amplitudes from a straight diagonal line as a measure of the significanceof the forcing excluded by each linearization, S and Z, respectively. The very earliest stages of theg(t) = g0 simulation have steeper gradients dh/dZ and dh/dS than in subsequent stages, and thisindicates a transitory regime during which artificially perturbed initial conditions settle. Thereafterthe gradient dh/dZ is fairly constant. We treat this as a baseline case, with a small virtual origincorrection. In Figure 18(a) there is a pronounced hysteresis during deceleration and a corresponding

(a) (b)

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

0 1 2 3 4 5 6 7 8

G2ADA2ADA4ADA8

(h/g0)1/2

(s)

(Z/g0)1/2 (s)

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

0 1 2 3 4 5 6 7 8

G2ADA2ADA4ADA8

(h/g0)1/2

(s)

(S/g0)1/2 (s)

FIG. 18. Evolution of the bubble and spike time scales (h/g0)1/2 plotted against (a) (Z/g0)1/2 and (b) (S/g0)1/2 for multimodesimulations with g ∼ g0 {1-tanh(η(t-T1))-tanh(η(t-T2))} and A = 0.5. Downward triangles indicate onset of deceleration att = T1, while upward triangles indicate return to acceleration at t = T2.


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FIG. 19. Evolution of the self-similarity parameter β for multimode simulations with g ∼ g0 {1-tanh(η(t-T1))-tanh(η(t-T2))}and A = 0.5, plotted against the interface displacement Z. Downward triangles indicate onset of deceleration at t = T1, whileupward triangles indicate return to acceleration at t = T2.

long-term offset from the baseline diagonal, indicating that inertia plays an important role duringdeceleration, and memory of such inertial forcing is retained. In contrast, Figure 18(b) shows muchsmaller deviation from the baseline diagonal during deceleration and no significant long-term offset,indicating that the Z scaling correctly captures the inertially dominated response to abrupt changesin acceleration g(t).

In Fig. 19 we plot the evolution of the self-similarity ratio βb/s ≡ λb/s

hb/s, where λb/s may be

interpreted as the dominant length scale at the bubble/spike front, while hb/s are the correspondingamplitudes. Thus for self-similarity, βb/s is a constant and has been reported in earlier studies26, 55 tofall within the range 0.41 (−0.19) − 0.70 (−0.58) for bubbles (spikes). The higher (lower) valuesfor bubbles (spikes) were found to occur at large Atwood numbers. Following Refs. 26 and 55,we compute λb/s from an autocorrelation-based analysis of the leading bubble- and spike-fronts,identified as the 1% and 99% isosurfaces of fl(x, y), respectively. For our calculation withg(t) = g0, βb asymptotes to a value of 0.42 ± 0.03 for Z > 20 cm, while βs reaches a value of−0.22 ± 0.04 at late times. Both the bubble and spike values are in good agreement withexperimental55 and numerical26, 27 data of turbulent RT at A = 0.5, although some variation inresponse to the particular choice of initial conditions is to be expected. Note that the number ofbubble (spike) structures used in computing β decreases as the mixing layer grows, but even atlate times our bubble/spike finding algorithm detects over ∼10–20 structures. At earlier times, thenumber of dominant bubbles/spikes included in the ensemble averages is much higher.

Fig. 19 also traces the evolution of the self-similarity parameter β from our ADA simulations,and shows behavior that is different from the constant g case. Following the g-reversal, the amplitudesof the bubble/spike front decrease dramatically (Fig. 19), resulting in a sudden increase (decrease)in β to a maximum (minimum) value for bubbles (spikes). However, the structures at the dominantlength-scale in the horizontal direction λ have substantially greater inertia, and react to reversals ofg over longer time scales than the smaller structures, and inevitably self-similarity is temporarilylost and accounts for the variation in the parameter β. As reversed bubbles and spikes break updue to collisions, smaller scales dominate the flow, marked by a gradual decrease in β. Finally,for late-times, destabilizing acceleration is once again applied, and self-similarity in the flow isrestored as β approaches a nearly constant value. For ADA2 (ADA4), the saturation of the self-similarity parameter β to an asymptotic value occurs for Z ≥ 24 (30) cm, and we report severalturbulent quantities of interest averaged over this window. In contrast for ADA8, self-similarity wasnot achieved by the end of the simulation in Fig. 19, as the flow is still clearly recovering from


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FIG. 20. Evolution of the global molecular mix fraction for multimode simulations with g ∼ g0 {1-tanh(η(t-T1))-tanh(η(t-T2))} and A = 0.5, plotted against the interface displacement Z. Downward triangles indicate onset of deceleration at t = T1,while upward triangles indicate return to acceleration at t = T2.

the g-reversal. The ADA2 and ADA4 calculations report saturated values of β of 0.39 ± 0.04 and0.38 ± 0.03 for bubbles and −0.23 ± 0.03 and −0.14 ± 0.01 for spikes, respectively, close to theasymptotic values observed in the constant g simulation.

For a constant acceleration, the global atomic mix parameter defined in Eq. (15) reaches anasymptotic (Z > 20 cm) value of ∼0.80 ± 0.004 for the multimode simulations (Fig. 20). Notethat for the multimode simulations we present here, we model atomic mix through mesh-scale cell-averaging of advected volume fractions, exploiting properties of our nonlinear numerical schemeto ensure a good proxy for molecular processes. The corresponding plot for the ADA cases show asudden increase in during deceleration due to the shredding of bubble/spike structures, describedabove, reaching a maximum value close to 1. Reacceleration appears to return the value of the mixparameter to the value observed at constant g (e.g., = 0.78 ± 0.02 for Z > 24 cm for ADA2 and0.84 ± 0.01 for Z > 30 cm for ADA4), as the driven flow reorganizes itself in to anisotropic bubbleand spike fronts.

The distribution of energy among scales may be inferred from the power spectrum of volumefraction fluctuations at the midplane. Following Refs. 25 and 26, we compute the power spectrumof fluctuations in the volume fraction of the light fluid as

P(n) ≡ 2πn⟨fl(n)2

⟩θ, (16)

where fl(n) is the Fourier transform of light fluid volume fraction, n is the mode number, and 〈 · 〉θrepresents averaging over the azimuth in mode number space. The results are shown in Fig. 21(a)for g(t) = g0, and in Fig. 21(b) for the ADA simulation with g0 = gbase. To emphasize the inertialrange, we plot the so-called compensated power spectra defined as P(n) n5/3. In Fig. 21(a), the powerspectra at early times (Z = 0.16 cm) show a narrow distribution of energy approximately centeredaround the mode numbers 32–64. At this early stage of evolution, instability growth follows Eq. (1),with the presence of two distinct and unmixed fluids (see, for example, the planar contour images inFig. 13). This is reflected in the spectral peak associated with the early-time volume fractionfluctuations in Figs. 21(a) and 21(b). The onset of self-similarity in Fig. 21(a) is observed as thebroadening of the power spectra through nonlinear mode interactions, and the appearance of asubstantial inertial range, evident in the figure as the horizontal portion of the spectra. The attendantmixing reduces the energy associated with fluctuations, while an inverse cascade towards low modenumbers driven by the self-similar growth is also visible in Fig. 21(a). A distinct dissipative tail isalso evident, with an apparent scaling of P(n) ∼ n−3.


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0.0001

0.001

0.01

0.1

1

10

1 10 100 1000

Z = 0.16 cmZ = 4 cmZ = 8 cmZ = 25 cm

P(N) N5/3

N

0.0001

0.001

0.01

0.1

1

10

1 10 100 1000

Z = 0.16 cmZ = 4 cmZ = 8 cmZ = 36 cm

P(N) N5/3

N

(a) (b)

FIG. 21. Compensated power spectra of volume fraction fluctuations at the mid plane from multimode simulations with (a)constant g and (b) g ∼ g0 {1-tanh(η(t-T1))-tanh(η(t-T2))}, g0 = 2 cm/s−2, and A = 0.5. The spectra are computed for timesshown in Figures 13 and 14.

Compensated power spectra from the ADA simulation with g0 = gbase are shown in Fig. 21(b)at Z-locations before and after the g-reversal. As expected, the spectra realized at early- (Z = 0.16cm) and intermediate times (Z = 4 cm) are identical to the corresponding data from the constant gsimulation. At Z = 8 cm, following the introduction of deceleration, the energy distribution acrossall scales is significantly reduced due to enhanced mixing. When the destabilizing acceleration isagain applied for t > T2, the fluctuation spectra recovers to energy levels comparable to self-similarRT (Z = 25 cm). Thus, the power spectra at Z = 4 cm and Z = 25 cm are structurally similar, withslightly more energy concentrated at low mode numbers for the late-time spectra.

The evolution of components of the large scale anisotropy tensor for the constant g and ADAcases with g0 = {2, 4, 8} gbase are shown in Figs. 22(a)–22(d), respectively. From Ref. 57, theanisotropy tensor Bij may be defined as

Bij =⟨uiuj

⟩2Ek

− 1

3δij, (17)

where Ek is the turbulent kinetic energy, while 〈uiuj〉 represents elements of the Reynolds stresstensor. Thus, for isotropic turbulence

⟨u2⟩ = ⟨

v2⟩ = ⟨

w2⟩ = 2Ek/3, and Bii = 0. In general, Bij is

bound between −1/3 ≤ Bij ≤ 2/3, where the lower and upper limits represent 2D and 1D turbulence,respectively. Thus for directed RT turbulence with g(t) = g0, where previous studies45 and current

simulations have reported⟨u2⟩ ∼ ⟨

v2⟩ ∼

⟨w2

⟩4

, we expect Buu ∼ Bvv ∼ −1/6, while Bww ∼ 1/3 across

the mixing layer. This is indeed observed in Fig. 22(a) for the benchmark simulation, which showsthe evolution of Bii evaluated at the centerline of the mixing layer, and plotted against interfacedisplacement Z. When averaged for Z > 20 cm, we obtain Buu = −0.17 ± 0.015, Bvv = −0.18± 0.019, and Bww = 0.36 ± 0.026 in good agreement with our estimates above for energy partitionin RT turbulence. When g-reversals are involved, the results are dramatically different as the datafrom our simulations with g0 = {gbase, 2gbase, 4gbase} show in Figs. 22(b)–22(d). For each of thesecases, following the first g-reversal, Bii evaluated at the center plane of the mix region drops tozero (after some transients), indicating a high degree of isotropy during the deceleration phase.When the destabilizing acceleration is once again applied, anisotropy is restored, as Bii assumeslimits consistent with classical RT. During this stage, we report values of Buu = −0.17 ± 0.02,Bvv = −0.16 ± 0.03, and Bww = 0.33 ± 0.05 from ADA2 when averaged over Z ≥ 24 cm. Thecorresponding values from the ADA4 simulation are Buu = −0.15 ± 0.04, Bvv = −0.14 ± 0.03,and Bww = 0.29 ± 0.06, respectively, with the averages reported for Z > 30 cm in this case. This


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-0.4

-0.2

0

0.2

0.4

0.6

0 10 20 30 40 50 60 70 80

Buu

Bvv

Bww

Z (cm)

-0.4

-0.2

0

0.2

0.4

0.6

0 10 20 30 40 50 60

Buu

Bvv

Bww

Z (cm)

-0.4

-0.2

0

0.2

0.4

0.6

0 10 20 30 40 50 60 70 80

Buu

Bvv

Bww

Z (cm)

-0.4

-0.2

0

0.2

0.4

0.6

0 10 20 30 40 50 60

Buu

Bvv

Bww

Z (cm)

(a) (b)

(c) (d)

FIG. 22. Evolution of diagonal components of the anisotropy tensor Bij from multimode simulations with (a) constant g (b)ADA2, (c) ADA4 and (d) ADA8 and A = 0.5. Downward triangles indicate onset of deceleration at t = T1, while upwardtriangles indicate return to acceleration at t = T2.

behavior is consistent with the return to self-similarity observed earlier in Figs. 14–21, followingthe application of the second stage of acceleration.

C. RT energetics

The unstable density initial condition in RT may be expressed as a store of potential energy, partof which is converted to kinetic energy associated with the eddy structures in the turbulent flow. Thepotential energy that is released between the initial condition and some end time may be written as

P E = P Einitial − P E f inal =0∫

−h

(ρl − 〈ρ〉)gzdz +h∫

0

(ρh − 〈ρ〉)gzdz. (18)

Similarly, the total instantaneous kinetic energy currently in the flow at time t is computed from

K E = 1

2L2

h∫−h

⟨ρ(V 2

x + V 2y + V 2

z

)⟩dz. (19)


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FIG. 23. Evolution of KE/PE from multimode simulations with g(t) = g0 and g ∼ g0 {1-tanh(η(t-T1))-tanh(η(t-T2))} andA = 0.5 plotted against the interface displacement Z. Downward triangles indicate onset of deceleration at t = T1, whileupward triangles indicate return to acceleration at t = T2.

KE/PE, at any time t represents the fraction of the potential energy already released and manifestingas kinetic energy in the turbulent flow at that instant. The remainder of the released potential energycan only have been dissipated before time t, thus (1-KE/PE) measures a cumulative time-averageof dissipation. In Figure 23, we plot KE/PE for our simulations with g(t) = g0 and the ADAncases. At late times (Z > 20 cm), we obtain KE/PE ∼ 0.46 ± 0.03 from our simulation witha constant acceleration. These values are in good agreement with several numerical23, 25, 26 andexperimental24, 45 studies. For the ADAn simulations, the onset of deceleration is marked by asudden decrease in KE/PE as the density profile is now reversed leading to a partial “restoration”of the potential energy. Interestingly, following reacceleration the ADAn simulations recover theasymptotic values from the constant g case.

Note that as demonstrated by Winters et al.58 and Lawrie and Dalziel,59 in a variable density flowsubject to acceleration, only a portion of the potential energy released becomes kinetic energy. Theremainder is consumed in reorganizing the density field through scalar diffusion. This transformationis thermodynamically irreversible, analogous to raising the system entropy. Following Winterset al.,58 we partition the potential energy into two components according to its present ability to douseful work in the future. The component that is thermodynamically available to do work is called“available potential energy” (APE), and the remainder, from which no work is recoverable, is referredto as “background potential energy” (BPE), such that APE+BPE = PE. In an incompressible system,there are thus two pathways by which energy can be irreversibly transformed, namely by viscousdissipation to internal energy (IE), or through scalar diffusion from APE to BPE. In analogy with apendulum, KE and APE can be reversibly exchanged. We reproduce the equations constituting thisdecomposition of energy from59

d I E

dt= ε, (20)

d K E

dt= −φ − ε, (21)

d AP E

dt= φ − ζ, (22)

d B P E

dt= ζ, (23)


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0

0.1

0.2

0.3

0.4

0.5

0.6

0 10 20 30 40 50 60

G2ADA2ADA4ADA8

Z (cm)

χ

FIG. 24. Time evolution of the cumulative mixing efficiency η from multimode simulations with g(t) = g0 and g ∼ g0

{1-tanh(η(t-T1))-tanh(η(t-T2))} and A = 0.5, plotted against the interface displacement Z. Downward triangles indicate onsetof deceleration at t = T1, while upward triangles indicate return to acceleration at t = T2.

where ε, ζ , and φ represent the instantaneous energy exchange fluxes due to viscous dissipation,scalar diffusion and buoyancy, respectively. Since PE, BPE, and KE are measurable from simulationdata, the system of Eqs. (20)–(23) may be integrated in time to determine each of the above energyfluxes. Then, an instantaneous mixing efficiency χ may be defined by combining ζ and ε accordingto,

χ = ζ

ζ + ε. (24)

An integral measure less prone to noise, known as the ‘cumulative mixing efficiency’, is plottedin Figure 24 for both the g(t) = g0 and the ADAn simulations. Here, both the numerator anddenominator of (24) are integrated in time, making this quantity analogous to the KE/PE measurethat is commonly25 used for self-similar Rayleigh-Taylor instability. Mixing efficiency enablesbook-keeping of historical energy conversions due to scalar diffusion as well as those due to viscousdissipation, so the pool of energy that remains thermodynamically available for future work to bedone is less than one might interpret from examining the KE/PE alone.

In this work, MOBILE has been used in ILES mode, where the numerical scheme is tailoredto ensure that for turbulent flows, simulated energy dissipation is in certain senses consistent witha real fluid. This removes the need to simulate additional viscous diffusive terms that would reducethe range of dynamically significant scales that can be captured. We infer from the simulations animplicit numerical viscosity, enabling us to plot a representative Reynolds number as a function oftime.

For high Reynolds number flows, the energy spectrum E(k) may be modeled as57

E(k) = Cε23 k

−53 fL (kL) fη(kη) (25)

where

fL (kL) =(

kL√(kL)2 + cL

) 53 +p0

, (26)

fη(kη) = exp(−β

{(kη)4 + C4

η

} 14 − Cη

), (27)

and C, β, cL, cη, and p0 are constants with typical values of 1.5, 2.1, 6.78, 0.4, and 2, re-spectively. In Eq. (25), η refers to the Kolmogorov scale, L is the integral length scale so that


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(a) (b)

FIG. 25. The fractional cumulative energy dissipation for (a) a model spectrum and (b) simulation data from MOBILE.Vertical line in figure (b) indicates the location of kη ∼ 1.

L ∼ h(t) for RT turbulence, while ε is the energy dissipation rate per unit mass. Thus ε =∫

ρεdV

∼ 〈ρ〉h∫

−h

εdz, and for our simulations we use the nominal box dimensions Lx = Ly = 1 cm. The

corresponding dissipation spectrum is given by D(k) = 2vk2 E(k), while the fractional cumulativeenergy dissipation for the flow is independent of ν(t) and obtained from

ε(0, k)

ε(0,∞)=

k∫0

D(k ′)dk ′

∞∫0

D(k ′)dk ′. (28)

In Figure 25(a), we plot the fraction from Eq. (28) for the model energy spectrum functionspecified by (25), and note that approximately 95% of the energy is dissipated at scales withkη ≤ 1. Applying the same standard to the fractional cumulative dissipation functions computedfrom kinetic energy spectra generated by our numerical simulations for Z > 20 cm (see, for instance,Figure 25(b)), our ILES equivalent of a Kolmogorov length scale was found to be nearly constant andoccurs at mode number N ∼ 178 or η ∼ (2.88 ± 0.08), where is the zone width employed in thecalculations. This is consistent with the findings of the alpha group study25 and in good agreementwith estimates from a survey of ILES simulations and methods by Aspden et al.39 who reported η

∼ 2.8 for a variety of grid resolutions.The corresponding physical viscosity that would be required to produce the observed energy

dissipation in these simulations can be computed from57

ν ≡ (η4ε

) 13 , (29)

so a representative Reynolds number for this RT flow would be

Re ≡ have|•have|ν

. (30)

This definition of Reynolds number requires as input an estimate of the dissipative flux. InRT flows, for which there is no steady state, Re(t) is determined by the instantaneous flux ε(t) for


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0

1000

2000

3000

4000

5000

0 1 2 3 4 5 6 7

G2ADA2ADA4ADA8

Re

(Z/g0)1/2

FIG. 26. Evolution of the Reynolds number for multimode simulations with g(t) = g0 and g ∼ g0 {1-tanh(η(t-T1))-tanh(η(t-T2))} and A = 0.5, plotted against the time scale (Z/g0)1/2. Downward triangles indicate onset of deceleration at t = T1,while upward triangles indicate return to acceleration at t = T2.

corresponding have(t) and•

have(t). A reasonable approximation for self-similar flows (such as theconstant g case) as suggested by Dimonte et al.25 is to take the cumulative time-averaged dissipativeflux inferred from KE/PE shown in Figure 23. The cumulative dissipated energy Ed ≡ PE – KEimplies

ε ∼ 1

ρh + ρl

d

dt

(Ed

have

), (31)

and computing with ν(t) and Re(t) computed from Eqs. (29) and (30).However this method is inappropriate for the ADA simulations, which are not self-similar. As

shown in Figure 23, the cumulative time-averaged dissipation does not tend to a constant valueover the lifetime of the simulations. Instead, we track all the energy fluxes explicitly by integratingEqs. (20)–(23) in time. Doing so ensures we correctly account for that portion of KE at time t, that maybe converted to APE at some later time through the reversible flux associated with buoyancy. Oncetransformed to APE, it could either irreversibly become BPE immediately, or reversibly exchangewith KE several more times before irreversibly being dissipated as heat. Integrating (20)-(23) directlyremoves any ambiguity in the ultimate destination of both KE and APE forms of available energyin the system, and allows us to determine the instantaneous dissipative flux ε(t).

The Reynolds numbers Re(t) calculated using the instantaneous dissipative fluxes59 plotted inFigure 26 for the constant g and the ADAn simulations. When plotted against the Z scaling toillustrate the consistency of Re(t) growth rate, the results are consistent with earlier experimental45

and simulation47 work. During the deceleration phase Re(t) decreases, since KE in the flow issuddenly performing work against (rather than receiving energy from) the density stratification.Thereafter (t > T2), Re(t) recovers a growth rate consistent with the constant g case until top andbottom boundaries begin to play a role.

V. SUMMARY

We have reported on detailed numerical simulations that explore the Rayleigh-Taylor instabilitywhen subjected to changes in the direction of the applied acceleration. The test problem is termedaccel-decel-accel (ADA) to indicate alternating stages of destabilizing acceleration sandwiching astage of stabilizing deceleration, and defined by Dimonte et al.27 The investigation of such flows


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is relevant to turbulent mix model validation, as well as understanding experiments where the testcell may be subjected to temporary deceleration. We have investigated the single-mode as well asthe multimode cases in detail, and attempt to interpret the results in the turbulent case through thebehavior of individual bubbles and spikes in the single-waveform problem.

When a single-mode RT is subjected to the ADA profile, the resulting behavior is complex,and dependent on the Atwood number. At low Atwood numbers (A = 0.15), the phase reversal ofthe interface is accompanied by mixing as bubble and spike structures collide resulting in breakupin to small-scale structures. The degree of mixing achieved during phase reversal is dependent onthe nonlinearity of the bubble/spike structures immediately prior to the application of the reversedg, and the presence of secondary structures such as the Kelvin-Helmholtz vortices that form on thesides of the original primary bubbles and spikes. Thus, the reversal of a corrugated structure leadsto greater mixing, while a streamlined, non-KH-unstable bubble/spike undergoes phase reversal inthe absence of such a shredding process. This is indeed the case when the simulations were repeatedat larger density ratios (A = 0.9). When the density difference between the fluids is large, theKelvin-Helmholtz instability between bubble/spike streams is inertially suppressed, and as a resultthe coherent structures are everywhere single-valued in x and y, leading to a clean phase reversal.The planar-averaged amplitudes in both cases scale with the displacement Z, with the amplitudesfrom the g(t) = g0 case forming an outer envelope to which the ADA simulations return to, onceg(t) > 0 is restored.

When the simulations were initialized with interfacial perturbations over an annular spectrumof wavenumbers, self-similarity would be achieved in finite time through the coupling of saturatedmodes. When such a flow is subjected to g-reversal, the self-similarity ceases during the phasereversal, when the aforementioned collision leads to a nearly isotropic mixing layer. We quantifythe degree of isotropy during this stage by computing the isotropy tensor before and after thedeceleration. For t > T2, the second reversal of g(t) to a destabilizing acceleration returns the RTflow to its self-similar, directed evolution with a nearly constant αb and αs, typical for such flows.The amplitude plots show a complex evolution through these stages, and we have reported turbulentstatistics that we hope will be useful for turbulent model validation.

ACKNOWLEDGMENTS

This work was supported in part by the (U.S.) Department of Energy (DOE) under ContractNo. DE-AC52-06NA2-5396. A.G.W.L. would like to acknowledge the support of Ecole Centrale deLyon during the preparation of this paper. The simulations reported here were performed on Krakenat the National Institute for Computational Sciences, an advanced computing resource supported bythe National Science Foundation.

APPENDIX: NUMERICAL VALIDATION AND CONVERGENCE STUDY

In this appendix, we catalog the numerical validation test cases we performed as part of thisstudy. In Fig. 27(a), we plot the linear growth rates from 3D single-mode simulations with h0 =0.01λ. The simulations were performed at a range of Atwood numbers, where the linear growth rateswere determined from fitting numerical data in the range 0 ≤ kh(t) ≤ 0.1, to Eq. (1). The simulationswere performed with no physical viscosity, and a grid resolution of 128 zones/λ. We also performednumerical simulations with an applied physical viscosity, which was varied systematically. Theresults for different values of the scaled viscosity νSCL are shown in Fig. 27(b). The simulationresults are compared with both the inviscid theory (Eq. (1)) and the following dispersion relation forgrowth rate in the presence of viscosity:51, 52

�viscous = −νk2 +√

Agk + ν2k4, (A1)

where ν = (μ1 + μ2) / (ρ1 + ρ2), and k is the perturbation wavenumber defined earlier. As expected,our linear growth rates from numerical simulations computed in the manner described above are inexcellent agreement with Eq. (A1).


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FIG. 27. Plot of RT linear growth rate from MOBILE against (a) the Atwood number, and (b) the scaled viscosity

ϑSC L = ϑ√Agλ3

.

For the single-mode cases, a detailed convergence study was performed with mesh resolu-tions of 8, 16, 32, 64, and 128 zones/λ, respectively. To characterize the behavior of numericalviscosity in MOBILE, these calculations were performed without any explicit physical viscosity.The bubble and spike amplitude evolution vs. Z from all the simulations are shown in Figs. 28(a)and 28(b), and show convergence in the amplitude behavior for a zoning parameter greater than32 zones/λ. The numerical convergence with respect to the linear growth rate is shown in Fig. 29,plotted against the non-dimensional zone width k, with = λ/(Nx-1). From Fig. 29, for a meshresolution ≥64 zones/λ, we find the simulations deviate from the inviscid linear growth rate by lessthan 8%.

(a) (b)

0

0.5

1

1.5

2

0 20 40 60 80 100

8 zones16 zones32 zones64 zones128 zones

hb

(cm)

Z (cm)

0

0.5

1

1.5

2

0 10 20 30 40 50 60 70 80

8 zones16 zones32 zones64 zones128 zones

hs

(cm)

Z (cm)

FIG. 28. Convergence study: (a) Bubble and (b) spike amplitudes from 3D single-mode RT simulations at different zoning.A = 0.15.


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FIG. 29. Convergence study: RT linear growth rates from MOBILE simulations for different values of the zoning parameterk.A = 0.15.

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