A comparative study of approaches for modeling Rayleigh–Taylor turbulent mixing

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Phys. Scr. T142 (2010) 014012 (13pp) doi:10.1088/0031-8949/2010/T142/014012

A comparative study of approachesfor modeling Rayleigh–Taylorturbulent mixingSnezhana I Abarzhi and Robert Rosner

Department of Astronomy and Astrophysics, Physical Sciences Division, The University of Chicago,Chicago, IL 30337, USA

E-mail: snezha@uchicago.edu and r-rosner@uchicago.edu

Received 20 September 2010Accepted for publication 13 December 2010Published 31 December 2010Online at stacks.iop.org/PhysScr/T142/014012

AbstractThis paper considers similarities and differences in the governing mechanisms and basicproperties of Rayleigh–Taylor turbulent mixing as discussed in recent theoretical and heuristicmodeling studies and briefly discusses how these mechanisms and properties may be exploredin experiments and simulations. We were motivated by a number of stimulating questions,thoughtful comments and sagacious remarks by our colleagues and the anonymous referees,whose contribution to improving this work is warmly appreciated.

PACS numbers: 47.20.−k, 47.20.−i, 47.54.+r, 52.35.Py, 52.57.−2, 52.35.−g, 94.05.Lk

1. Introduction

Rayleigh–Taylor (RT) turbulent mixing is an extensiveinterfacial mixing process that develops when fluids ofdifferent densities are accelerated against their densitygradient [1, 2]. It governs a wide variety of naturalphenomena from astrophysical to atomistic scales andplays an important role in technological applications [3–5].Examples include inertial confinement and magneto-inertialfusion, light–material interaction, supernova explosions,flows in the atmosphere and oceans, and shock–turbulenceinteraction in aerodynamics and free-space opticaltelecommunications [6–13]. RT mixing is a multi-scale,heterogeneous, anisotropic and statistically unsteadyturbulent process with non-local interactions among the manyscales [3, 4]. The development of RT turbulent mixing isusually associated with the conditions of strong gradients ofpressure and density and may also include spatially varyingand time-dependent acceleration, diffusion of species, heatrelease, chemical reactions, etc [1–13]. These conditionsthus depart from the conditions under which one mightexpect canonical Kolmogorov turbulence to occur [4, 14–16].Capturing the mechanisms and properties of RT mixing canlead to a better understanding of realistic turbulent flows andfurther develop our intuition and the methods of controllingnon-equilibrium process in nature and technology.

The RT flows that arise in a variety of diverse applicationsnevertheless exhibit a number of similar features of theirevolution [1–4, 17–27]. The mixing starts to develop whenthe fluid interface is slightly perturbed near its equilibriumstate. The flow transitions from an initial stage, wherethe perturbation amplitude grows relatively quickly (e.g.exponentially in time), to a nonlinear stage where thegrowth rate decreases and the interface is transformed intoa composition consisting of a large-scale coherent structureand small-scale irregular structures driven by shear, and thenfinally to a stage of turbulent mixing, whose dynamics isbelieved to be self-similar [1–4, 17–27].

Over 100 RT-related papers are published per year inpeer-reviewed mathematical, computational, scientific andengineering journals. However, our knowledge of this mixingprocess and progress toward fuller understanding are stilllimited. On the experimental side, a systematic studyof RT flows is a challenging task [17–28]. Due to thesensitivity and transient character of the dynamics, repeatableimplementation of these flows in a laboratory environmentrequire tighter than usual control of the macroscopicparameters and of the initial and boundary conditions.Statistical unsteadiness of turbulent mixing indicates the needto measure spatial and temporal distributions of the flowquantities [28]. Furthermore, these measurements should becarried out with relatively high accuracy and resolution,adequate dynamic range and data acquisition rate [3, 28].

0031-8949/10/014012+13$30.00 1 © 2010 The Royal Swedish Academy of Sciences Printed in the UK

Phys. Scr. T142 (2010) 014012 S I Abarzhi and R Rosner

On the side of numerical simulations, modeling RT mixingis challenging also because of the highly singular nature ofthe instability (namely the rapid initial growth of small-scalestructures) combined with the limited spatial dynamic rangeof exemplary simulations, even those which are based onstate-of-the-art adaptive mesh refinement computations andthe use of ‘leadership’-class massively parallel computers.The numerical solutions are likely to be strongly dependenton the effects of unresolved small-scale structures (especiallyinsofar as they affect local density contrasts) and the possibleanomalous character of energy transport [29–39]. On theside of theoretical analysis, the dynamics of RT flows isan intellectually challenging problem, as it has to balancenumerous competing requirements and demands due attentionto the multi-scale, highly nonlinear and non-local characterof the mixing flows. If possible, we would like to identify‘universal’ asymptotic solutions and to establish whethermemory of the initial conditions exists, to capture a certaindegree of order of the mixing flow and to account for thenoisy character of the turbulent dynamics and so forth. Ouraim is to use physics-based considerations for preventing theanalysis from becoming too mathematical and too empirical,and to help identify a set of robust parameters, whichcould in principle be precisely diagnosed in the observations([3, 4, 28] and references therein).

For a detailed discussion of the state-of-the-art methodsin theoretical, experimental, numerical and computationalstudies of RT flows, see recent reviews [40]. This paperconsiders only one aspect of the multi-faceted RT mixingproblem, specifically the mechanisms and properties of RTmixing suggested by theoretical and heuristic modelingstudies [4, 41–61]. Our discussion will be physics-based andwill attempt to synthesize knowledge of the problem from theperspectives of analysis and modeling, as it was elaborated bythe turbulent mixing community.

2. Approaches for modeling RT turbulent mixing

2.1. Outline of empirical approaches

An overview of experimental, numerical and theoreticalstudies [17–39] suggests that RT flow is characterized bya large-scale structure, small-scale structures and extensivetransport of mass, momentum and energy among the scales.The large-scale structure has two macroscopic length scales:the amplitude h in the direction of gravity g and the spatialperiod λ in the normal plane [3]. The horizontal scale λ

is induced by the initial perturbation and/or by the modeof fastest growth with wavelength ∼(ν2/g)1/3, where ν isthe kinematic viscosity and g = |g|. The horizontal scaleλ may increase if the flow is two-dimensional (2D) andthe perturbation is broad-band and incoherent [47]. Thevertical scale h is believed to grow self-similarly in themixing regime with h ∼ gt2, and it can be regarded asan integral scale representing cumulative contributions ofsmall-scale structures, which are produced by shear at thefluid interface [59–61]. Some other features are inducedin the dynamics by diffusion, compressibility, stratification,finite-size domian, non-uniform acceleration and high energydensity conditions [17–39].

Since the time of first hypotheses on the existence of aself-similar regime in RT flows [41, 62] and first endeavorsto observe it in the experiments and simulations [43, 45,17, 27], tremendous effort has been put by researchers inempirical and theoretical modeling studies of the mixingdynamics, and significant success has been achieved [41–61].In addition to addressing fundamental scientific questions,theories and models [41–61] have a definitive practicalimportance: because of the sensitivity, statistical unsteadinessand transient character of the dynamics, experiments andsimulations on RT mixing require solid benchmarks foranalyzing and calibrating data sets [17–39], even in the casewhen data calibration is performed by means of nonlinearspline interpolation with multiple adjustable parameters.

Virtually all modeling efforts [41–61] have tried toaddress the following questions. (i) What is the mechanismunderlying self-similar mixing? (ii) What are the mixingflow properties? A detailed description of the former set ofefforts can be found in [49]. Representative examples of theseapproaches are the viable works [43, 45] and the modelspresented in [47, 48]. The models [43, 44, 47–49] suggestedthat the growth of the horizontal scale λ varies as λ ∼ h ∼ gt2

and they scaled the coarsest vertical length scale of the flowas h = α f (A)gt2, where α is a constant, f (A) is a functionof the Atwood number A = (ρh − ρl)/(ρh + ρl) and ρh(l) isthe density of the heavy (light) fluid. These interpolationmodels traditionally served for a calibration of experimentaland numerical data sets and for further elaboration of the firstquantitative descriptions of RT flows [34, 49].

The set of modeling efforts related to the turbulencemodels presented in [41, 42, 50–58] were initiated by theseminal works [41, 42, 50, 58]. These models consideredRT mixing within the general content of turbulent flowsand have served to underpin the development of fast andaffordable numerical modeling techniques of the mixingprocess (e.g. [32, 34, 38, 58]). These models regardedRT mixing as an anisotropic turbulent flow with densityfluctuations, which are in turn driven by the velocity fieldas in the case of passive scalar mixing. In order to interpretexperimental and numerical data in terms of turbulent powerlaws [14–16, 63–66], the models [50–57] somewhat modifiedthe classical Kolmogorov theory, including the introduction ofa virtual origin and a time scale for the transition to turbulence,the introduction of time dependence h ∼ gt2 in Kolmogorov’sinvariants, etc.

The principal difference between the turbulencemodels [41, 42, 50–58] and the interpolation models [43–49]lies in how the dynamics of the interface is handled. Themodels [43–49] were developed to capture the dynamicsof the front—a sharp interface between the immisciblefluids. In contrast, the models [41, 42, 50–59] consideredturbulent mixing in a continuous approximation, presumingthat the fluids are miscible and can therefore diffuse into oneanother. For an early time evolution with the perturbationamplitude being significantly smaller than the spatialperiod, h � λ, the sharp-boundary approximation and thecontinuous approximation provide equivalent results [1–3].To capture late-time dynamics, i.e. with h ∼ λ and h � λ, themodels [41–58] applied a number of empirical hypotheses (tobe discussed below) augmented with adjustable parameters,

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and found some quantitative agreement with the observationaldata (which, as an important aside, spanned a relatively shortdynamic range [17–39]).

Concurrent with the empirical modeling efforts, rigoroustheoretical studies were extensively conducted (see [3, 4] andthe review article [40] and references therein). These studiesconfirmed some of the empirical hypotheses [41–57], andfound a number of new qualitative and quantitative propertiesof turbulent mixing. It was shown, for instance, that thenonlinear dynamics of RT flows has an essentially non-localand multi-scale character, characterized by independentcontributions of the horizontal (λ) and vertical (h) scales,and that in highly isotropic coherent structures, growth ofhorizontal scales may not occur [3, 4]. Recently developedmomentum-based considerations accounted for these resultsand identified two distinct mechanisms for the developmentof the mixing process [59–61]. Furthermore, it revealednew invariant, scaling and spectral properties of RT mixingflows and showed the departure of these properties fromthe canonical Kolmogorov scenario [15]. In the following,we compare different approaches [41–61] for modeling RTturbulent mixing and outline some directions for futuredevelopment of the experiments and simulations.

2.2. Conservation laws

As in any physical process, RT mixing is governed by theconservation of mass, momentum and energy. In continuousapproximation, in the fluid bulk the conservation of mass andmomentum has the form

ρ + ∇ · ρv = 0, ρ (v + (v · ∇) v + g) + ∇ p + S = 0, (1)

where ρ, v and p are the density, velocity and pressure ofthe fluid, respectively, S denotes terms induced by viscousstress and other effects, and the dot marks the partialderivative in time t [14]. For compressible fluids, system (1)is augmented with the equation for energy transport and theequation of state [67]; for miscible fluids, the equations forconcentration are also incorporated [14]. For incompressibleimmiscible fluids, the fluid interface is a discontinuity, withρ = ρh H(−θ) + ρl H(θ) and v = vh H(−θ) + vl H(θ), whereH is the Heaviside step function, θ is a scalar function onthe coordinates and time with θ = 0 at the interface and ρh(l)

and vh(l) are the density and velocity of the heavy (light) fluidlocated in the region θ < 0 (θ > 0) [3, 14]. If there is no massflow across the interface, the pressure and normal componentof velocity are continuous at the interface:

ph|θ=0 = pl|θ=0 , vl · n|θ=0 = vh · n|θ=0 = −θ/|∇θ |,

(2.1)

where n = ∇θ/|∇θ | is the interface unit normal vector. Inspatially extended fluid systems the flow has no mass sources

vh|z→ +∞ = vl|z→ −∞ = 0 (2.2)

and can be periodic in the plane (x, y) normal to the directionof gravity, z. The initial conditions at the interface and atthe boundaries of the domain close the set of governingequations (1) and (2). For compressible and/or misciblefluids, the fluid ‘interface’ is a region characterized by

strong gradients of the flow quantities. A self-consistentdescription of such an interface requires establishingnew connections of continuum matter approximation tokinetic processes at microscopic scales as well as abetter understanding of the interplay between Eulerian andLagrangian descriptions in systems that are out of localthermodynamic equilibrium [39, 40].

2.3. Comparative study of theories and models

2.3.1. Governing principles of the models. In order toobtain a rigorous theoretical description of RT mixing,one has to find a solution for a mathematical problemof fundamental complexity. This problem includes athree-dimensional non-stationary system of nonlinear partialdifferential equations augmented with the initial conditionsand with the boundary conditions, which are representedby a sub-set of nonlinear partial differential equations at anonlinear freely evolving interface, and with the conditions atthe boundaries of the outside domain. Asymptotic solutionsfor this mathematical problem are sensitive to the initialconditions and to the influence of secondary instabilities andsingularities, which may develop at the discontinuities, e.g.the fluid interface. While the mixing process is observed tomaintain certain features of coherence and order, associatedprimarily with the dynamics of large scales, it is yet anon-deterministic and statistically unsteady process, whoserandomness results from the interaction of all the scales[3, 4, 14, 60].

The first attempts to capture nonlinear evolution ofthe unstable mixing front were made by Davies andTaylor [2], Fermi and von Neuman [62], Layzer [68],Birkhoff and Carter [69] and Garabedian [70] via applicationof the collocation method, Lagrange dynamics, localspatial expansion and conformal mapping techniques. Nearly30 years later, on the basis of theoretical solutions [68–70]and experimental observations [2, 17], the model presentedby Sharp and coworkers [43] put forward a hypothesis that anincrease of the spatial period may lead to flow accelerationand thus trigger the transition from the nonlinear stage toaccelerated mixing as

v ∼√

g λ, λ ∼ gt2⇒ v ∼ gt, ⇒ h = αAgt2,

(3.1)where v is the characteristic velocity of the front withv ∼ |v|. Good agreement between the renormalizationgroup analysis [43] and 2D front-tracking numericalsimulations [44] was achieved.

To describe the experimental [17] and numerical data [45]on RT mixing, another empirical model was proposed byYoungs [45],

v = Ag − C(A)v2/h, (3.2)

with C(A) being a function of the Atwood number, distinctfor bubbles and spikes [45]. Model (3.2) suggests that mixingdevelopment is induced by a broad-band initial perturbation.This model is known to provide an excellent data fit withminimal use of free parameters [45, 49].

Yet another interpolation model [47, 48] initiated amerger campaign with the use of an empirical equationbalancing the flow inertia, buoyancy and drag as

(v − Ag)Volume = −(C(A)v2)Area, (3.3)

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with C(A) being a function of the Atwood number, distinctfrom C(A) and with Area ∼ λ2, Volume ∼ hλ2 and withthe ‘single-scale’ mixing mechanism λ ∼ h ∼ gt2. Thismodel attempted to provide a universal characterization ofRT mixing with sustained and time-dependent accelerationhistories g = g(t), as well as in the special case of theRichtmyer–Meshkov mixing with g = 0 for both 3D and 2Dflows. The model of [47, 48] relies extensively on adjustableparameters and agrees with the experimental and numericaldata processed by means of nonlinear interpolation [49].

Applying turbulence modeling approaches for RT mixingwas initiated by the works of Belen’ki and Fradkin [41] andNeuvazhaev [42], who derived a self-similar solution for themixing dynamics from the energy balance relation

(1/2)(∂ρv2/∂t) + νρv3/L = ρLvω2, (3.4)

where ω2=g(∂ ln ρ/∂z+g/c2) is the Brunt–Väisälä frequency,

L is the characteristic length scale, ν is kinematicviscosity and c is the speed of sound. Presuming thatthe density of an incompressible fluid obeys the equation∂ρ/∂t = ∂(D∂ρ/∂z)/∂z with the turbulent diffusioncoefficients D ∼ vL and L ∼ h, and integrating overthe mixing zone by analogy with Prandlt’s theory ofshear-induced mixing, the models [41, 42] found self-similarsolutions with h ∼ gt2 in the cases of sustained andtime-dependent accelerations. These results were extensivelyused for the calibration of experimental and numerical datasets [24, 25, 27] and provided good agreement with data in abroad parameter regime.

A more rigorous ‘turbulent approach’ was suggestedby Harlow and co-workers [50, 51] for a two-fluid system.Ristorcelli and Clark [53] applied a similar approach for asingle fluid system and expanded the velocity and densityfields in the vicinity of their mean values within Boussinesqapproximation, presuming the validity of this approximationfor miscible fluids with similar densities A � 1. To the firstorder, the model [53] finds that

∂〈vv〉/∂z=∂p/∂z−Agφ, ∂φ/∂t + ∂〈vφ〉/∂z=D∂2φ/∂z2,

(3.5)

where v is the z-component of velocity, 〈· · ·〉 indicates theFavre averaging and φ is an effective ‘concentration’. Asin [41, 42], equations (3.4) have the similarity solutionh = (C0/4)Ag(t + t0)2 with t0, h0 being the time scale andlength scale of the ‘virtual’ origin, at which the transition to‘turbulence’ is expected to occur and at which h0 = C0 Agt2

0with C0 being a free parameter [53]. The authors of [53] notedthat in accelerated mixing flow, the rate of energy dissipationε is time dependent with ε ∼ g2t .

The turbulent diffusion models [41, 42, 50–53] agreedwith experiments and simulations, similarly to the inter-polation front-tracking models [43–49]. The results presentedin [53] were consistent with the finding of Dimotakis [54],who analyzed a wide variety of turbulent flows and suggestedthat at certain values of Reynolds and Taylor Reynoldsnumbers a transition may occur to a fully developedisotropic turbulence and that such a transition is a universalphenomenon for all turbulent flows including RT mixing.Zhou et al [55, 56] further extended Dimotakis’ ideas to the

cases of RT and Richtmyer–Meshkov mixing in the high andlow energy density regimes.

The importance of the models [50–56] is that thesemodels formulated the problem of RT mixing within thegeneral context of turbulent flows. Furthermore, inspiredby the pioneering numerical approaches of Nikiforov et aland viable works of Gauthier et al on the application ofthe k − ε sub-grid-scale model to numerical simulationsof accelerated and shock-driven flows [58], the turbulentdiffusion models [41, 42, 50–56] helped develop affordablenumerical techniques for large-scale computations of RTmixing and for comparison of simulation results with theexperiments [31, 32, 34, 35, 38].

To interpret experimental and numerical data on RTmixing in terms of turbulent power laws, Chertkov [57]attempted to extend Kolmogorov and Bolgiano–Obukhovanalyses for the case of RT mixing via a substitution of timedependence of the integral scale L ∼ h ∼ gt2 in the invariantsof canonical turbulent flows. This substitution suggested thatthe viscous scale should decay with time as ∼t−1/4 in the 3Dcase and grow as ∼t1/8 in the 2D case [57].

Some quantitative agreements were reported betweenthe models [41–58] and the observations [17–39]. Somequalitative features of the turbulent process remain unclear,such as the relatively ordered character of RT mixing flowat high Reynolds numbers [56]. To date, experiments andsimulations have not provided a trustworthy guidance onwhether the concepts of classical turbulence are applicable toan accelerating RT flow and whether the scalings h/gt2 andλ/gt2 are indeed universal.

In order to explain the qualitative features of RTmixing and to connect empirical models to conservationlaws (1) and (2) and to rigorous theoretical studies [3, 4], amomentum-based consideration was developed [4, 59–61]. Itsuggests that in RT mixing flow, the specific momentum isgained due to buoyancy and is lost due to dissipation. Thedynamics of a parcel of fluid is governed by a balance perunit mass of the rate of momentum gain µ and the rate ofmomentum loss µ as

h = v, v = µ − µ. (3.6)

The rate of momentum gain is the buoyant force, µ = ε/v,where ε is the rate of energy gain induced by buoyancyand µ = g f (A) with f (A) being a function of the Atwoodnumber. The rate of momentum loss is µ = ε/v, where ε

is the rate of dissipation of kinetic energy. Without viscousscale on the basis of dimensional grounds ε = Cv3/L , whereC is a constant and L is the characteristic length scalefor energy dissipation, either in the vertical or horizontaldirection. The model identifies two distinct mechanismsof mixing development, shows that invariant, scaling andspectral properties of RT mixing depart from the canonicalscenario and suggests that turbulent mixing flow has amore regular character compared to isotropic Kolmogorovturbulence [4, 63–66].

Below we perform a comparative study of modelingapproaches [41–61]. First we show that, on the onehand, momentum consideration (3.6) can be reducedin some limiting cases to interpolation and turbulencemodels (3.1)–(3.4), and that, on the other hand, it identifies

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new features of mixing dynamics. We then consider themechanism and properties of turbulent mixing and discusssome macroscopic properties of the flow, such as energybudget, momentum and energy transport, diffusion anddissipation mechanisms, dynamics of the center of mass,and flow drag. We further discuss theoretical aspectsof turbulent mixing dynamics, including symmetries andinvariants, and analyze its scaling, spectra and fluctuations.Similarities and distinctions will be identified betweenthe non-inertial turbulent mixing and inertial Kolmogorovturbulence. Connections of the modeling results to theobservations will also be discussed.

2.3.2. Similarities of the results of the models. As discussedin [59–61], asymptotic solutions for model (3.6) depend onwhether the characteristic length scale of the flow is horizontalor vertical. If the characteristic length scale is horizontal,L ∼ λ, then equations (3.6) have steady solutions withv ∼

√gλ and h ∼ t

√gλ, and the rates of momentum and

energy are balanced: µ = µ = g and ε = ε = (gλ)3/2/λ√

C(hereafter for simplicity we re-scale g f (A) → g). If thecharacteristic scale is vertical, L ∼ h, then asymptoticallyin time, h = agt2/2 and v = agt with a = (1 + 2C)−1. Therates of energy gain and dissipation are time dependent,ε = ag2t and ε = (1 − a)ag2t , and the rates of momentumgain and loss are time invariant and scale invariant, µ = gand µ = Cv2/h [59–61]. Found in many observations, thevalues of a are rather small, a ∼ 0.05–0.15 [17–39]. Thus,in the mixing flow, almost all the energy induced by thebuoyancy dissipates, ε ≈ ε with ε/ε = (1 − a), and there isa slight imbalance between the rates of momentum gainand loss, µ ≈ µ with (µ − µ)/µ = a. Self-similar mixingmay develop when the horizontal scale λ grows with timeas λ ∼ h ∼ gt2 [43, 47] and when the vertical scale h,h ∼ h, is the characteristic scale for energy dissipation thatoccurs in small-scale structures at the fluid interface [59–61].

Results obtained within momentum consideration agreein certain approximations with the results found byinterpolation and turbulent models (3.1)–(3.6). For instance,the momentum model [59–61] reproduces the results ofGlimm and Sharp [43] if one assumes that the characteristiclength scale of the flow is horizontal, L ∼ λ, imposes thegrowth λ ∼ h ∼ gt2 and adjusts the values of dimensionlessparameters as f (A) = A and C = (1 − 2α)/4α to ensurethe dependence h = αAgt2 found in [43, 44]. On the otherhand, if one assumes that the characteristic scale is vertical,L ∼ h, and properly re-scales the values of f (A) and C ,one can reduce the momentum consideration [59–61] to themodel of Youngs [45]. Further, by introducing adjustableparameters to describe the merger process and the volumetricand surface terms, one can reduce momentum model to thedrag models [47–49, 61–63], etc.

With the proper choice of the initial conditions,momentum consideration [59–61] reproduces also resultsof turbulence models [41, 42, 50–53], which consider RTmixing as an effective diffusion process. Furthermore, inagreement with results of Dimotakis [54] and Zhou et al[55, 56], momentum consideration suggests that propertiesof RT turbulent mixing may resemble certain properties ofKolmogorov turbulence [59–61]. This situation may occur,

Table 1. Mechanisms of mixing development.

Model Mechanism

[43, 44, 47, 48] Mixing is induced by growth of horizontalscale λ ∼ gt2

[45, 46] Mixing is induced by a broad-band incoherentinitial perturbation

[59–61] Growth of horizontal scale λ ∼ gt2 is possibleDominance of vertical scale h may lead to flow

acceleration

for instance, when the rates of gain and loss of specificmomentum and energy are balanced, µ = µ and ε = ε, andthe flow is steady, so that µ − µ = 0 and buoyancy forcingsupplies energy at a constant rate ε = ε = (gλ)3/2/λ

√C . This

can account for the observations [23, 26] of statistically steadyturbulence in RT flows with relatively low Reynolds numbersRe ∼ 103. It also suggests that RT instability indeed canbe applied to produce semi-isotropic Kolmogorov type ofturbulent flows—a well-known approach in experimental fluiddynamics [54]. On the other side, in agreement with the modelof Chertkov [57], the momentum consideration indicates thatin accelerated turbulent mixing with h ∼ gt2 the rate of energydissipation grows with time as ε ∼ g2t , and, as it was donein [57], one might choose to substitute these dependenciesin canonical Kolmogorov and Obukhov–Bolgiano scaling andspectra [64–66, 70, 71].

2.3.3. Distinctive properties of the turbulent mixing

2.3.3.1. On the mechanism of turbulent mixing evolution.Agreeing in certain limiting cases with principal results ofthe interpolation and turbulent diffusion models, momentumconsideration identifies some new properties of the mixingflow [59–61]. It suggests that the accelerated turbulentmixing develops due the imbalance of gain and loss ofspecific momentum, µ 6= µ. This imbalance may occur when(i) the horizontal scale grows as λ ∼ gt2 and/or when(ii) the vertical scale h is a characteristic scale for energydissipation, ε = Cv3/L with L ∼ h, and when it representscumulative contributions of small-scale structures into theflow dynamics. Existence of two distinct mechanisms of themixing development reconciles models [43–49] and [50–58].It also agrees with results of theoretical studies [3], whichfound that the amplitude h and period λ provide independentcontributions to the nonlinear RT dynamics and that forhighly isotropic large-scale coherent structure the growth ofhorizontal scales may not occur [80], table 1.

2.3.3.2. On energy budget, transports of energy andmomentum and position of the center of mass. Turbulenceis a property of dissipative systems and it decays if it is notdriven [15, 16, 63–66]. Kolmogorov turbulence is driven byan external energy source, which supplies energy to the flowat a constant rate ε: energy is injected at large scales by anexternal source, and then it is transferred without loss throughthe inertial interval and dissipates at small scales [15, 16,63–66]. Interpolation and turbulent diffusion models [41–56]do not consider the source of energy of turbulent mixing flow.Furthermore, turbulence models [50–56] presume that theenergy and momentum transports in RT mixing are similar to

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Table 2. Energy source, transport of momentum and energy and position of the center of mass.

Model Energy source

[15, 16] Energy is injected at large scales by an external source and then transferred without loss through the inertial interval anddissipated at small scales. Mean velocity of the center of mass of the fluid system is time independent

[50–57] Presume that the transport of energy and of momentum are similar to that in Kolmogorov turbulence[59–61] There is no external energy source other than gravity. Energy and momentum are gained due to buoyancy and lost due to

dissipation. In the steady regime µ − µ = 0 and ε = ε = (gλ)3/2/λ√

C . Accelerated turbulent mixing is driven by imbalancebetween the gain and loss of specific momentum, and at any scale µ 6= µ and ε 6= ε. In accelerated mixing, the mean velocityof the center of mass of the fluid system is time dependent

Table 3. Symmetries of turbulent dynamics.

Model Symmetries

[15, 16] Kolmogorov turbulence is inertial and is invariant with respect to Galilean transformation, translations in time and 3D space andspatial rotations and inversions. It is scale invariant, L → L K , T → T K 1−n , v → vK n with n = 1/3

[50–57] Presume that symmetries of RT turbulent mixing are the same as in Kolmogorov turbulence[59–61] RT turbulent mixing is non-inertial and is invariant with respect to translation, rotations and inversions in the plane normal to

gravity g. It is scale invariant, L → L K , T → T K 1−n , v → vK n with n = 1/2

that in Kolmogorov turbulence. According to the momentumconsideration [59–61], for RT turbulent mixing an externalenergy source (other than gravity) is not required, and thespecific momentum is gained due to buoyancy and is lost dueto dissipation. In accelerated flow at any scale µ 6= µ andε 6= ε, and this imbalance indicates that the mean velocity ofthe center of mass of the fluid entrained in the motion is timedependent, whereas in statistically steady turbulent flow it isinvariable, table 2.

2.3.3.3. On the asymptotic state in space and time.Statistically steady Kolmogorov turbulence is an asymptoticin time state, which is achieved when the memory of theinitial conditions is completely lost, and when the boundariesof the outside domain do not influence the dynamics [15, 65,66]. These conditions can be realized in a spatially extendedsystem or in a finite-size domain, when the span of scalesruns several decades from viscous to integral scale [15].Implementation of these conditions in RT turbulent mixingrequires special attention [28]. In a finite-size domain, anasymptotic in time dynamics corresponds to a stable statewith no motion at all: under the influence of gravity (directedfrom the top to the bottom) the system transits from anunstable configuration to a stable configuration (e.g. froman initial state with heavier fluid located at the top of thedomain and lighter fluid—at the bottom to a reverse state),and the change in the system potential energy dissipatesinto heat. In a spatially extended system (e.g. in a largedomain) the flow may accelerate, however at a certain timecompressibility and stratification start to play a role and resultsin flow stabilization, as discussed in [14, 59, 73, 74]. Toallow for the development of RT turbulent mixing and toenable its diagnostics over substantial span of scales, the sizeof the domain should be large enough yet not so large toprevent mixing stabilization by effects of compressibility andstratification.

2.3.3.4. On effective drag in the mixing flow. Regularizationof accelerated turbulent mixing is at first glance an unusual(and certainly unexpected by turbulent mixing community)concept. However, there is some evidence from previousstudies that is does take place. For instance, re-laminarization

of an accelerated flow is a well-known fluid dynamicsphenomenon discovered in the works of Taylor [75] forflows in curved pipes and Narasimha and Sreenivasan [76]for boundary layers. Another indication of a more regularcharacter of RT mixing follows from the characteristic valueof the flow drag. Coefficient C in the relations ε = Cv3/Land µ = Cv2/L can be viewed as effective drag coefficient,which is related to the growth-rate h = agt2/2 viaa(1 + 2C) = 1 [59–61]. For C → 0 (no drag) the solution isa free-fall with a → 1, whereas for C → ∞ (infinitely largedrag) a → 0 and the flow cannot accelerate. Experiments andsimulations report relatively small values of a ∼ 0.05–0.15(with α ∼ 0.03–0.07 in the relation h = αA gt2 in [49]).These values correspond to drag coefficient of C ∼ 3–8,indicating that flow may tend to be more laminar rather thanturbulent [77]. In canonical Kolmogorov turbulence, the valueof C is calculated from the third-order velocity structurefunction as C = 5/4 and C ∼ 1 [15, 66]. This may lead toa = 2/7 ≈ 0.3 (α ∼ 0.14 in h = αA gt2 in [49]), which issignificantly greater than the values actually observed [49].

2.3.3.5. On symmetries and invariants of RT mixing. Acornerstone of Kolmogorov theory is that the isotropic andhomogeneous turbulent flow has a number of symmetriesin statistical sense [15, 16, 63–66]. It is invariant withrespect to Galilean transformation, to translations in timeand in 3D space, to spatial inversions and rotations and toscaling transformation with L → L K , t → t K 1−n and v →

vK n for any n, where v = |v| [15, 66]. Kolmogorov foundthat n = 1/3, and the measure of the scaling symmetry is therate of change of specific kinetic energy ε ∼ v3/L . Similarlyto Kolmogorov turbulence, RT turbulent mixing has a numberof symmetries [4, 61], however due to the presence of gravity,g 6= 0, and non-inertial character of the dynamics, thesesymmetries are distinct from those of Kolmogorov turbulence.RT mixing flow is invariant with respect to translations,inversions and rotations in the plane normal to g, and toscaling transformation L → L K , t → t K 1−n and v → vK n

with n = 1/2. The measure of this scaling symmetry v2/L hasthe same dimension as g and quantifies the rate of change ofspecific momentum, table 3.

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Table 4. Some invariant properties of turbulent dynamics.

Model Invariants

[15,16] Dynamics is statistically steady. Invariance of energy dissipation rates ε ∼ v3/L is compatible with the existence of the inertialinterval and energy cascade. Enstrophy and helicity are other invariants

[50–57] Presume that the inertial interval and energy cascade exist in accelerated turbulent mixing and substitute the dependenceL ∼ gt2 in energy dissipation rate and other invariants of canonical turbulent flows

[59–61] Dynamics is statistically unsteady. Invariance of rate of momentum loss µ ∼ v2/L leads to non-dissipative specific momentumtransport between the scales. Energy dissipation rate and enstrophy are time dependent. Helicity is invariant for λ ∼ h ∼ gt2

Table 5. Velocity spatial scaling.

Model Velocity scaling Velocity Nth-order structure function

[15, 66] vl/v ∼ (l/L)1/3 based on ε invariance ∼ (l ε)N/3 based on ε invariance[59–61] vl/v ∼ (l/L)1/2 based on µ invariance ∼ (l ε)N/2 based on µ invariance

Table 6. Velocity temporal scaling.

Model Velocity scaling Velocity fluctuations due to turbulent transport

[15, 66] vτ/v ∼ (τ/T )1/3 based on ε invariance vτ ∼ (εvτ)1/3 based on ε invariance and (εvτ)1/3� (ετ )1/2

[59–61] vτ/v ∼ (τ/T ) based on µ invariance vτ ∼ µτ based on µ invariance and µτ ∼ (µ − µ)τ ∼ gτ

In isotropic turbulence, the total momentum is zerobecause of isotropy, and time- and scale-invariance ofthe energy dissipation rate ε ∼ v3/L is compatible withexistence of inertial interval and non-dissipative energytransfer between the scales [15, 16, 63–66]. In acceleratedRT turbulent mixing, the specific momentum is imbalanced,µ 6= µ, and time- and scale invariances of µ ∼ v2/L implythat at any time and length scale the specific momentumis being lost at the same constant rate, and momentumtransfer between the scales is non-dissipative [61]. Enstrophyis another invariant of isotropic turbulence [64–66], whereasin RT mixing this value decays with time. This providesanother indication of a tendency of accelerated mixing flow tore-laminarize [61]. In RT flow, vortical structures form helixesnot vortices. In a flow dominated by the growth of horizontalscales, λ ∼ h ∼ gt2, the helicity is a statistically steady valueand its steadiness may serve as an indicator of achieving amerger-driven self-similarity [61], table 4.

The interpolation models [43–49] and the turbulentdiffusion model [41, 42, 50–57] do not discuss distinctions insymmetries and invariants of Kolmogorov turbulence and RTturbulent mixing. Models [50–57] presume that similarly toKolmogorov turbulence, inertial interval and energy cascadeexist in accelerated turbulent mixing and substitutes thedependence L ∼ gt2 in the quantities of canonical turbulentflows, table 4.

2.3.3.6. On space–time scaling properties and the role offluctuations. For a description of scaling properties, let thelength scale L and time scale T refer to large scales andtimes, with the characteristic velocity v. Let the characteristicvelocity be vl at a small length scales l, and let thecharacteristics velocity be vτ on a short time-scale τ .

In Kolmogorov turbulence [14, 15, 16, 64–66], theinvariance of the energy dissipation rate ε ∼v3/L ∼ v3

l / lyields the velocity scaling vl/v ∼ (l/L)1/3, N th-ordervelocity structure function ∼ (l ε)N/3, and velocity scalingwith time vτ/v ∼ (τ/T )1/3. The relative velocity of two

parcels of fluids involved in the motion is ∼(ετ )1/2 ona time delay τ , and it is substantially smaller than thevelocity fluctuations vτ ∼ (εvτ)1/3 induced by turbulence.This well-known result means that in Kolmogorov turbulence,the main contribution to velocity fluctuations is providedby the turbulence not by the initial conditions [14, 15, 66],tables 5 and 6.

In RT turbulent mixing, the invariance of the rate ofmomentum loss µ ∼ v2/L ∼ v2

l / l yields the velocity scalingvl/v ∼ (l/L)1/2, N -th order velocity structure function∼(l µ)N/2 and the velocity scaling with time vτ/v ∼ (τ/T ).For two parcels of fluids involved in the motion with a timedelay τ , their relative velocity is ∼ (µ − µ)τ ∼ gτ and itis comparable to vτ ∼ µτ induced by the turbulent process,whereas their own velocities grow with time as ∼gt and∼g(t − τ) [14]. We see that in accelerated mixing flow, thevelocity fluctuations are frozen to the level of the initialconditions, and with time the contribution of fluctuations tothe mixing dynamics is reduced, tables 5 and 6.

2.3.3.7. On Reynolds number, viscous scale and integralscale. In Kolmogorov turbulence, Reynolds number is finiteRe = vL/ν = const and local Reynolds number Rel = vll/νscales as Rel ∼ Re(l/L)4/3 leading to the viscous lengthscale lν ∼ (ν3/ε)1/4 and time-scale τν ∼ (ν/ε)1/2 for Rel ∼ 1.In accelerated turbulent mixing the Reynolds number growswith time as Re = vL/ν ∼ g2t3/ν and the local Reynoldsnumber Rel = vll/ν scales as Rel ∼ Re(l/L)3/2. For Rel ∼ 1viscosity plays a dominant role, thus leading to viscouslength scale lν ∼ (ν2/µ)1/3 with the corresponding timescale τν ∼ (ν/µ2)1/3. This viscous length scale is finite.It is set by the flow acceleration and is comparable to thewavelength of mode of fastest growth [1, 2]. Thus despitein accelerated RT mixing the Reynolds number can reachlarge values relatively quickly the flow viscous scale remainsfinite. An upper limit for Reynolds number Re ∼ g2t3/ν

can be estimated at a border of validity of incompressibleapproximation gt ∼ c as Rec ∼ c3/gν, where c is the soundspeed; see table 7.

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Table 7. Reynolds number, viscous scales and integral scale.

Model Reynolds number Viscous and integral scales

[15, 66] Re = vL/ν = const. Invariance of Invariance of ε leads to lν ∼ (ν3/ε)1/4 and τν ∼ (ν/ε)1/2

ε leads to Rel ∼ (vll/ν) ∼ Re(l/L)4/3 An integral scale is the scale at which energy is gained by the flow system[59–61] Re = vL/ν ∼ g2t3/ν. Invariance of µ leads to Invariance of µ leads to lν ∼ (ν2/µ)1/3 and τν ∼ (ν/µ2)1/3

Rel ∼ (vll/ν) ∼ Re(l/L)3/2 An integral scale is the coarsest vertical scale representing cumulativeFor gt ∼ c upper limit is Rec ∼ c3/gν contributions of small scale structures

Table 8. Dimensional-ground-based spectra.

Model Specific kinetic energy Specific momentum

[15, 16] E(k) ∼ ε2/3k−5/3 set by invariance of ε M(k) ≡ 0 due to isotropy[59–61] E(k) ∼ µk−2 set by invariance of µ M(k) ∼ µ1/2k−3/2 set by invariance of µ

Table 9. Pressure fluctuations.

Model Scaling Spectrum

[15, 66] ∼ ε4/3l4/3 set by invariance of ε ∼ ε4/3k−7/3 set by invariance of ε

[59–61] ∼ µ2l2 set by invariance of µ ∼ µ2k−3 set by invariance of µ

In Kolmogorov turbulence the integral scale is the scale,at which energy is gained by the flow system. For turbulentmixing this consideration may not be directly applicable. InRT mixing, momentum and energy are gained and dissipatedat any scale, and imbalance between the rate of momentumgain and loss leads to flow acceleration. The coarsestvertical scale in RT flow can be regarded as an integralcumulative scale, which represents cumulative contributionsof small-scale structures in the flow dynamics, table 7.

2.3.3.8. On dimensional-analysis-based spectral properties.In isotropic turbulence, the invariance of energy dissipationrate leads to kinetic energy spectrum E(k) ∼ ε2/3k−5/3

[14, 15, 66]. In RT mixing accurate determination ofspectra (and corresponding eigenfunctions) is a formidabletheoretical task because the dynamics is statistically unsteady.Dimensional analysis suggests that the spectrum of thespecific kinetic energy has the form E(k) ∼ µk−2, whichis steeper than Kolmogorov; similarly for the spectrumof specific momentum one obtains M(k) ∼ µ1/2k−3/2

(in Kolmogorov turbulence M(k) ≡ 0 due to isotropy),table 8 [61].

2.3.3.9. On pressure fluctuations. In Kolmogorovturbulence, pressure fluctuations are evaluated usingfourth-order velocity structure function so that pressurefluctuates as ∼ε4/3l4/3 with spectrum ∼ε4/3k−7/3 [66].For RT mixing dimensional grounds suggest for pressurefluctuations ∼µ2l2 with spectrum ∼µ2k−3, which is steeperthan in Kolmogorov turbulence, table 9 [61].

2.3.3.10. On scalar transport. An important outcome ofmomentum consideration [59–61] is the distinct roles, whichare played by diffusion and dissipation in RT mixing:diffusion influences the gains of momentum and energy,dissipation affects their losses and the scalar transport departsfrom a standard form of Fickian diffusion.

Based on energy conservation (per unit volume), the heattransport is described by the equation

∂(ρs)/∂t − χ(∇θ/θ)2− [· · ·] = 0, (4.1)

where θ is temperature, s is entropy, χ is thermal conductivityand [· · ·] denote other terms [14, 64, 65]. In the Boussinesqapproximation, one presumes that the change in densityinduced by temperature contrast is substantially greater thanthe change in density induced by change in hydrostaticpressure (e.g. by gravitational stratification), and that both aresmaller than the fluid density itself [14]. In isentropic limitenergy, equation (4.1) is then reduced to diffusion equation ofFickian type:

∂θ/∂t − κ∇2θ + (v∇)θ = 0, (4.2)

where κ is thermal diffusivity. Temperature fluctuationsare described by thermal dissipation function εθ = κ(∇θ)2.For turbulent flow κ ∼ vL and εθ ∼ (v/L)(δθ)2 [14, 65].The passive scalar consideration (4.2) works remarkablywell for canonical problems of isotropic and homogeneousturbulence [65] and turbulent convection [71, 72]. Inthese flows, the thermal dissipation function is time-and scale invariant, similarly to energy dissipation ratein Kolmogorov turbulence. In the case of isotropic andhomogeneous turbulence, the invariance of εθ ∼ (v/L)θ2

∼

(vl/ l)θ2l results in temperature fluctuations with θ2

l ∼

εθε−1/3l2/3. In the case of turbulent convection, accounting for

the constancy of temperature contrast, temperature fluctuatesas θ2

l ∼ ε4/5θ (gβ)−2/5l2/5, where β is a thermal expansion

coefficient [64, 65, 71, 72].Momentum consideration [59–61] suggests that thermal

transport influences the rate of momentum gain µ = gδρ/ρ

and that in the isentropic limit δρ/ρ ∼ δθ/θ the equation (4.1)yields for L ∼ h asymptotically in time

˙µ∼−(v/gL)µ2, ϕ(t)[h/gt2, v/gt, δθ/θ, µ/g, µ/g]∼1,

ϕ2(t)[ε/g2t, ε/g2t, εθ t/θ2] ∼ 1,

(4.3)

where ϕ(t) = ln(gt2/h0) and h0 is an initial length scale.In contrast to problems of isotropic turbulence and turbulentconvection [64, 65, 71, 72], in RT mixing thermal dissipationfunction εθ is not an invariant, and the rates of momentumand energy µ, µ, ε, ε, εθ are time dependent. This limits

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Table 10. Thermal dissipation and temperature fluctuations.

Model Thermal dissipation Temperature fluctuations

[64, 65, 71, 72] In isotropic turbulence and convection thermal In isotropic turbulence θ 2l ∼ εθε

−1/3l2/3;dissipation function εθ = κ(∇θ)2 is an invariant in convection θ 2

l ∼ ε4/5θ (gβ)−2/5l2/5

[59–61] In RT mixing values µ, µ, ε, ε, εθ are time-dependent Constancy of 5 ∼ v2/(δθ/θ)gL can be used in sub-grid-scaleand value 5 = µ/µ = ε/ε ≈ 1 is invariant models to account for scalar transport and contributions of

small scales

Table 11. Dissipative scale.

Model Batchelor scale

[64, 78] lB = (τBκ)1/2 based on invariance of ε and lB = lν Sc−1/2

[59–61] lB∼(νκ2/µ2)1/6 based on invariance of µ and lB = lν Sc−1/2

one’s capabilities to rigorously estimate scaling and spectralproperties of the mixing flow [61]. However, given thatϕ(t) = ln(gt2/h0) is a slowly growing function, at largebut finite times the viscous and dissipative scales of theflow would still remain finite. It is remarkable that inthe mixing flow the ratio 5 = µ/µ = ε/ε is a constantvalue with 5 ≈ 1 at highly nonlinear and turbulent stages,with or without turbulent diffusion accounted for. Parameter5 ∼ v2/(δθ/θ)gh can be thus be used as the flowcharacteristics in the case of sustained and/or time-dependentaccelerations (for instance, for evaluation of the propertiesof Reynolds stress in the sub-grid-scale models), table 10.

We point the reader’s attention to the fact that themomentum consideration [59–61] reproduces the resultsof turbulent diffusion models [41, 42, 50–56] when oneneglects the effect of scalar transports on the gains ofmomentum and energy. This case can be realized inexperiments when the scalar is a neutral buoyancy dyewith zero heat solubility. Then, employing expression forviscous scale lν ∼ (ν2/µ)1/3 and considering the dynamicswithin the context of passive scalar mixing [64, 78], onecan find for RT flows the dissipative (Batchelor) length scalelB ∼ (νκ2/µ2)1/6. Similarly to lν , this scale is finite and setby flow acceleration. For Sc � 1, where Sc = (ν/κ) is theSchmidt number, lB � lν , table 11.

2.3.3.11. On 2D and 3D flows. The 2D and 3D dynamicsoften exhibit drastically different properties, and turbulentflows are no exceptions. For instance, in the case ofisotropic and homogeneous turbulence, 3D flow exhibitsthe direct cascade, whereas 2D flow exhibits the inversecascade [15, 63–66, 79]. Extending these classical resultsto the case of RT dynamics via the use of Boussinesqapproximation and substitution of time dependence L ∼ gt2

in the energy dissipation rate ε ∼ v3/L ∼ g2t and inthe thermal dissipation function εθ ∼ (v/L)θ2

∼ θ2/t ,model [57] claimed that 3D RT mixing is driven byenergy transport, and similarly to isotropic turbulence, it ischaracterized by velocity fluctuations with vl/v ∼ (l/L)1/3

and temperature fluctuations with θl/θ ∼ (l/L)1/3. Thisleads to viscous scale lν ∼ (ν3/ε)1/4

∼ (ν3/g2t)1/4 anddissipative scale lB ∼ (ν3/g2t)1/4Sc−1/2. For the 2D RT flow,model [57] established that ‘the inverse cascade cannot berealized because the “pumping” scale L ∼ gt2 grows toofast for the larger structures to become established’ and

suggested applying the scenario of turbulent convection.Based on the time-dependent thermal dissipation functionεθ ∼ (v/L)θ2

∼ θ2/t , model [57] derives the scalingvl/v ∼ (l/L)3/5 and θl/θ ∼ (l/L)1/5 and identifiesviscous and dissipative scales as lν ∼ (ν/v)5/8L3/8

∼

ν5/8t1/8/g1/4 and lB ∼ Sc−1/2(ν5/8t1/8/g1/4). Momentumconsideration [59–61] finds that RT flow is driven bymomentum transport, and its viscous and dissipative scalesare finite and set by flow acceleration, in either 2D or 3D flow(table 12).

The other turbulent and interpolation models [43, 44,47–49] do not distinguish between the dynamics of 2D and3D flows and suggest the growth of horizontal scale λ ∼ gt2,induced by binary interactions, as the sole mechanismof mixing development. Group theory consideration [80]confirms their results only partially and notes that, in a 3Dflow, the interactions are essentially multi-pole and that thegrowth of horizontal scale may not occur if the large-scalecoherent dynamics is highly isotropic. The latter resultagrees with momentum consideration [59–61], which does notrequire the growth of horizontal scales for mixing evolution(table 12).

2.3.3.12. On statistically steady and statistically unsteadydynamics. To conclude this section, we discuss in moredetail statistically steady and unsteady regimes in RT flows.In a steady regime, the flow can appear more coherentor more ‘turbulent’ depending on the Atwood numberand the initial conditions [3, 4]. For the steady flow, therates of momentum gain and loss as well as energy gainand dissipation are balanced, µ = µ and ε = ε, and thecharacteristic length scale of the flow λ is constant. Thecharacteristic velocity is v ∼

√gλ, the Reynolds number is

Re = vL/ν ∼ λ√

gλ/ν and the energy dissipation rate isconstant ε ∼ (gλ)3/2/λ. This formally corresponds to theviscous scale (ν3/ε)1/4

∼ (ν3λ/(gλ)3/2)1/4, which is smallerthan the mode of fastest growth (ν2/g)1/3 for λ > (ν2/g)1/3.However, as λ/(ν3λ/(gλ)3/2)1/4

∼ (λ/(ν2/g)1/3)9/8 and9/8 = 1.125, the characteristic span of scales in the steadyflow is well captured by the ratio λ/(ν2/g)1/3.

The flow acceleration increases the flow velocity, integrallength scale, Reynolds numbers and energy dissipationrate. At first glance, this may lead to the appearance ofhigh Reynolds number turbulent flow with a significantspan of scales [41–58]. Momentum consideration [59–61]suggests, however, that buoyancy-driven turbulent mixing isaccelerated due to an imbalance between the gain and loss ofmomentum and energy with (µ − µ)/µ = a and (ε − ε)/ε =

a. In this flow the velocity v ∼ gt , the length scale h ∼ gt2,the Reynolds number Re ∼ g2t3/ν and the span of scales∼ gt2/(ν2/g)1/3 indeed increase. Here the viscous scale

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Table 12. 2D and 3D dynamics.

Model 3D flow 2D flow

[57] Substitute L ∼ gt2 in ε ∼ v3/L suggest scaling Substitute L ∼ gt2 in εθ ∼ (v/L)θ2 suggest scalingvl/v ∼ (l/L)1/3 and θl/θ ∼ (l/L)1/3 as in isotropic vl/v ∼ (l/L)3/5 and θl/θ ∼ (l/L)1/5 as in turbulentturbulence and find for viscous and dissipative convection and finds for viscous and dissipative scaleslν ∼ (ν3/g2t)1/4 and lB ∼ (ν3/g2t)1/4 Sc−1/2 lν ∼ ν5/8t1/8/g1/4 and lB ∼ Sc−1/2(ν5/8t1/8/g1/4)

[43, 44] [47–49] Presume that growth λ ∼ gt2 is driven by binary Presume that growth λ ∼ gt2 is driven by binaryinteractions and it is the sole mechanism of the mixing interactions and it is the sole mechanism of the mixingdevelopment development

[80] Find that growth of horizontal scales is driven by Find that growth of horizontal scales ismulti-pole interactions and may not occur driven by binary interactionsfor highly isotropic coherent structures

[59–61] Find that mixing may develop due to the dominance Compared to 3D flows, regularization processof vertical scale. Find that viscous and dissipative scales may be harder to implement in 2D flowsare finite and set by flow acceleration. Suggest thatRT mixing may regularize with proper choice ofthe initial conditions and acceleration history

Table 13. Steady and accelerated RT flows.

[59–61] Flow quantities

Steady flow Balance of momentum and energy µ = µ and ε = ε. Constant length scale λ, velocity v ∼√

gλ,Reynolds number Re ∼ λ

√gλ/ν and energy dissipation rate ε ∼ (gλ)3/2/λ with corresponding viscous scale

(ν3λ/(gλ)3/2)1/4 and span of scales (λ/(ν2/g)1/3)9/8

Accelerated flow Imbalance of momentum and energy (µ − µ)/µ = a and (ε − ε)/ε = a. Time-dependent length scale h ∼ gt2,velocity v ∼ gt , Reynolds number Re ∼ g2t3/ν and energy dissipation rate ε ∼ g2t . Constant rate of momentum lossµ ∼ g with corresponding viscous scale ∼(ν2/g)1/3 and span of scales ∼gt2/(ν2/g)1/3 (upper limit ∼c2/g(ν2/g)1/3)

is ∼ (ν2/g)1/3 and the upper limit for the span of scales is∼ (c/(gν)1/3)2 for c ∼ gt . However, compared to the caseof statistically steady isotropic and homogeneous turbulence,accelerated turbulent mixing exhibits stronger correlations,reduced contribution of fluctuations and steeper spectra, andtends to be more laminar [59–61, 76, 77]; see table 13.

2.4. Connection to observations

As discussed earlier, momentum consideration [59–61] agreesin some limiting cases with the results of interpolation andturbulent diffusion models [41–58]. It can thus be appliedto interpret the existing observations [17–39] to the sameextent as the models [41–58]. Furthermore, the momentumconsideration [59–61] is self-consistent and explains theobservations, which other models do not explicitly address.For instance, a relatively ordered character of RT mixingidentified within momentum consideration agrees with theexperiments [56], which found that at very high Reynoldsnumbers Re > 105 the mixing flow keeps a significantdegree of order. On the other hand, momentum considerationsuggests that at relatively low Reynolds number Re ∼ 103, RTmixing may resemble some of the properties of statisticallysteady turbulence (conditional on whether there is a balancebetween the gains and losses of specific momentum andenergy, as well as on the density ratio and initial perturbation).Departures from the canonical Kolmogorov scenario, whichwere found in recent experiments and simulations [17–39],indicate the importance of momentum transport for theflow dynamics, in agreement with [59–61]. The frozencharacter of fluctuations in RT mixing indicates that thisflow is more sensitive to the initial conditions comparedto the case of canonical turbulent flows, in agreement withobservations [17–39].

Stronger correlations and coupling and reduced level offluctuations are indications of a higher degree of order in RT

mixing compared to the case of isotropic and homogeneousturbulence [59–61]. This principal theoretical result opensup new opportunities for the design of experiments andsimulations, which may include realization of either a moreturbulent and strongly fluctuating mixing flow or a moreregular and better controlled mixing flow at high Reynoldsnumbers [17–58]. Furthermore, as suggested by the relativeindependence of horizontal and vertical scales in the mixingflow, these opportunities can be realized in a broad parameterregime. For instance, horizontal scales can be controlledwith the proper choice of the dispersion relation, initialconditions, isotropy and coherence, whereas vertical scalescan be influenced by the means of time-dependent andspatially varying acceleration and deceleration, diffusion andstratification, etc [3, 4, 59–61]. Details of the potential designof such experiments and simulations will be provided in aforthcoming publication.

The momentum consideration [59–61] indicates thatfor a reliable description of RT mixing it is essentialto account for the interface dynamics (1) and (2),and that the analysis of 4D momentum–energy transportdescribing compressible turbulent mixing may help us tocut through the problem’s complexity. Our comparativestudy of different approaches for modeling RT turbulentmixing [41–61] offers to experiments and simulations anumber of diagnostic parameters and invariant measures,including the position of the center of mass, energybudget, momentum transport, scalar transport, effective drag,scaling and spectra, correlations and fluctuations, etc. Basedon these results, a number of qualitative experimentaltests can be performed (e.g. flow regularization at highReynolds number and distinct mechanisms of mixingdevelopment), and some robust quantitative parameters canbe implemented in numerical simulations to model sub-gridscales and to capture the properties of the Reynolds stress

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(e.g. the value of 5 = µ/µ = ε/ε ≈ 1 at any scale, forsustained and time-dependent acceleration, with and withoutaccounting for turbulent diffusion, in either steady orunsteady regimes [59–61]). Direct testing of the momentumconsideration [59–61] would require further enhancements inflow diagnostics and in the quality and information capacity ofexperimental and numerical data sets, including tight controlof macroscopic parameters, accuracy, precision, acquisitionrates and dynamic range [28]. It therefore has a potential tobring experiments and simulations to a new level of standard.

3. Conclusions

In this work, we have performed a comparative study ofdifferent approaches [41–61] for modeling RT turbulentmixing. Interpolation models, turbulent diffusion modelsand momentum consideration [41–61] agree with oneanother under certain approximations and can be appliedfor interpreting existing experimental and numerical datasets [17–39]. By focusing on momentum transport [59–61],one can identify a number of new properties of RT flows.These include, among others, two distinct mechanisms ofmixing development, non-dissipative transport of specificmomentum between the scales, time- and scale-independence,the ratio between the rates of loss and gain of specificmomentum and energy 5 = µ/µ = ε/ε for sustained andvarying acceleration, and finite values of viscous anddissipative scales. Compared to isotropic, homogeneousand statistically steady turbulence, the anisotropic,inhomogeneous and statistically unsteady RT turbulentmixing is characterized by a different set of symmetriesand invariants. It has stronger correlations and coupling,weaker contribution of fluctuations to the flow dynamics,stronger dependence on the initial conditions, and steeperdimensional-analysis-based spectra. The more orderedcharacter of the mixing dynamics opens up new opportunitiesfor the design of experiments and simulations for therealization of regular or turbulent high Reynolds numbermixing flows.

Acknowledgments

SIA thanks the US National Science Foundation and theOffice of Fusion Energy Sciences of the US Department ofEnergy for support, Dr K R Sreenivasan and S Gauthierfor deep discussions and Drs S I Anisimov, B Alder, ABershadskii, R P Drake, S Fedotov, B Fryxell, Y Fukumoto,G Glatzmaier, G Hazak, L P Kadanoff, A Klimenko, V Lvov,K Nishihara, A Pouquet, B Remington, J Schumacher, AL Velikovich, J Werne and V Yakhot for discussions andcomments.

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