## Abstract

Past decades significantly advanced our understanding of Rayleigh–Taylor (RT) mixing. We briefly review recent theoretical results and numerical modelling approaches and compare them with state-of-the-art experiments focusing the reader's attention on qualitative properties of RT mixing.

## 1. Introduction

Rayleigh–Taylor (RT) mixing controls a variety of processes in fluids, plasmas and materials [1]. These processes can be natural or artificial, their characteristic scales can be astrophysical or atomistic, and energy densities can be low or high. In everyday life, we observe RT mixing while looking at how water flows from an overturned cup [2,3]. Somewhat similar processes influence the formation of ‘hot spot’ in inertial confinement fusion (ICF), limit radial compression of imploding Z-pinches, drive penetration of stellar ejecta into pulsar wind nebula, determine momentum transfer in buoyancy-driven convection, control material transformation under impact and strongly affect reactive and supersonic flows [4–13].

RT mixing develops in a material flow when the gradients of the fields of pressure and density are pointed in opposite directions [2,3,14,15]. The process is driven by acceleration, and may also include strong shocks, blast waves, radiation transport, chemical reactions, diffusion of species and other effects [1]. The flow is characterized, on the one side, by sharp changes of its scalar and vector fields and, on the other side, by relatively small influence of dissipation and diffusion [5,16]. This leads to formation of interfaces separating the flow phases, and to extensive interfacial mixing [17]. RT mixing is inhomogeneous (i.e. the flow fields are essentially non-uniform, even in a statistical sense), anisotropic (i.e. the dynamics in the direction of acceleration differs from that in the normal plane), non-local (i.e. the flow evolution depends on the initial conditions and on the contributions from all the scales) and statistically unsteady (i.e. mean values of the flow quantities vary with time, and there are also time-dependent fluctuations around these means). Its properties are expected to depart from those of canonical turbulence [1,18–28]. Capturing the essentials of RT mixing can improve our knowledge of realistic turbulent processes and can help to elaborate the methods of control of the non-equilibrium dynamics in nature and technology.

With nearly two hundred papers published each year on RT mixing in peer-reviewed mathematical, computational, scientific and engineering journals, our knowledge of this process is still limited [1]. On the side of analysis, evolution of RT flows is an intellectually rich theoretical problem, as it has to balance numerous competing requirements and demands due attention to the multi-scale, nonlinear, non-local and statistically unsteady character of the dynamics (e.g. [4,29,30]). On the side of experiments, one can easily overturn a glass of water, yet it is a challenge to systematically probe and gather precise and accurate data on RT mixing flow owing to its sensitive and transient dynamics (e.g. [17,31–40]). On the side of simulations, numerical modelling of RT instability and mixing is a severe task for both continuous dynamics and particle methods as numerical solution has to track interfaces, accurately account for the dynamics at small scales, capture dissipation and shocks (e.g. [41–55]). Furthermore, on the side of acquiring knowledge from data, a systematic interpretation of RT mixing from experimental and numerical data alone is neither easy nor straightforward, and requires identification of a set of robust parameters that can be precisely diagnosed in the observations [40]. Despite these challenges, significant success was achieved in the past decades on the sides of experiments, technology, theoretical analysis and simulations in our understanding of non-equilibrium turbulent dynamics [1]. This paper serves to synthesize some of our experience on the subject and summarize what is certain and what is not so certain in our understanding of RT mixing. The focus question of this paper—is RT interfacial mixing a disordered process indeed?

## 2. Evolution of Rayleigh–Taylor mixing

Occurring in vastly different physical circumstances, RT flows exhibit a number of similar features in their evolution [1]. RT instability starts to develop when the fluid interface is slightly perturbed near its equilibrium state. The flow transitions from an initial stage, where the perturbation amplitude grows quickly, to a nonlinear stage, where the growth rate slows down and the interface is transformed into a composition of a large-scale coherent structure and shear-driven small-scale vortical irregular structures, and then to the stage of self-similar turbulent mixing [2,3,29,31,33,34,56–59]. Though RT mixing involves many interacting scales, from a ‘far field’ an observer usually identifies two macroscopic length scales associated with the flow—the vertical scale in direction of acceleration (gravity **g**) and the horizontal scale in the normal plane or the amplitude *h* and the spatial period *λ* of the front [60]. The horizontal scale *λ* is induced by the initial perturbation. There is a characteristic wavelength *λ*_{m} associated with the mode of the fastest growth. In incompressible immiscible fluids with kinematic viscosity *ν* and with *g*=|**g**|, this length scale is *λ*_{m} ∼ (*ν*^{2}/*g*)^{1/3} [29,61]. The horizontal scale *λ* may increase if the initial perturbation is broadband and incoherent [56,62,63]. The amplitude *h* is defined by the positions of bubbles (in which the light fluid penetrates the heavy fluid) and spikes (in which the heavy fluid penetrates the light fluid). In the mixing regime, the amplitude is believed to grow self-similarly, *h*∼*gt*^{2}. The pre-factor in this dependence is substantially smaller than that suggested by the free fall law [33,57,59,64].

In RT mixing, the characteristic velocity and integral length scale are *v* and *L*, with *v*∼*gt* and *L*∼*h*∼*gt*^{2}. The flow Reynolds number grows with time as *Re*∼*vL*/*v*∼*g*^{2}*t*^{3}/*ν*, and the rate of dissipation of specific kinetic energy increases as *ε*∼*v*^{2}/(*L*/*v*)∼*v*^{3}/*L*∼*g*^{2}*t*. One may expect the ‘viscous scale’ to decay with time as *l*_{ν}∼(*ν*^{2}/*ε*)^{1/4}∼(*ν*^{2}/*g*^{2})^{1/4}, and the span of scales to grow as *L*/*l*_{ν}∼(*g*^{2}*t*^{3}/*ν*)^{3/4}∼*Re*^{3/4}. The incompressible approximation is bounded by the speed of sound *c* limiting the Reynolds number as *Re*∼*c*^{3}/*gν*. Therefore, RT mixing flow is characterized by the increasing velocity and integral length scale, quickly growing Reynolds number and decaying viscous scale and the span of scales enlarging with time faster than quadratic. These properties are consistent with quantitative criteria at which canonical Kolmogorov turbulence is expected to occur [19,20]. At first glance, one may consider mixing as a process somewhat similar to canonical turbulence [41,42,46,65–68].

Experience with implementation and diagnostics of turbulent flows suggests that the situation may be more complex than it appears. On the one hand, it is known from practice that upon introducing a slight density contrast and pressure gradient the flow may indeed quickly transit to turbulent regime [27,69]. On the other hand, turbulent flows that are sharply accelerated may have a tendency to re-laminarize, as was found by Taylor [70] for flows in curved pipes and by Narasimha & Sreenivasan [71] for accelerated boundary layers. Furthermore, even in canonical turbulence the presence of surfaces (which is essential for RT dynamics) leads to deviations from Kolmogorov scenario as seen in intermittency and multi-fractality [27,72].

An intriguing puzzle of RT mixing dynamics was observed recently in the experiments in strongly accelerated fluids [17,73], in the experiments in high energy density plasmas [74,75], and in the large-scale numerical simulations of the ICF [76,77]. It appeared that RT mixing flow keeps significant degree of order, when it is strongly accelerated and its Reynolds number is high, *Re*>10^{6} [17,73,74,78]. At the same time, in experiments and simulations of RT mixing with relatively low Reynolds numbers *Re*∼10^{3} [50,78,79] the observation of statistically steady turbulence is reported. The theoretical analysis [59,80–82] explained this paradox by considering symmetries and momentum transport in RT flow, and found that RT mixing may indeed exhibit more order and have stronger correlations and weaker fluctuations compared with canonical turbulence. Questions thus appear: Is RT interfacial mixing a fully disordered process? May RT mixing regularize? If so, how does one influence the regularization process? Addressing these questions is necessary to comprehend the fundamentals of a broad variety of natural phenomena (e.g. properties of non-equilibrium dynamics, evolution of the early Universe, explosion of supernovae and convection in stellar and planetary interiors) and to control technological applications (e.g. target design in ICF, mixing enhancement in reactive flows and properties of materials under high strain rates). This review is a physics-based discussion that attempts to synthesize some of our research experience of RT mixing.

## 3. Models, observations and properties of Rayleigh–Taylor mixing

### (a) Turbulent mixing models

Since the time of first hypotheses on the existence of a self-similar regime in RT instability and first endeavours to observe it in experiments, significant efforts were undertaken to understand and model the mechanism underlying RT turbulent mixing and the mixing flow properties. One set of modelling efforts is represented by the interpolation models [56,57,83,84]. These models served to quantitatively describe RT mixing and calibrate experimental and numerical data. The other set of modelling efforts is represented by turbulence models [46,50,65,66,85–91]. Presuming that RT mixing is a process somewhat similar to isotropic turbulence, these models considered RT mixing within the general context of turbulent flows and served for development of the state-of-the-art numerical methods. Momentum model [59,80–82,92] analysed the mechanism and the properties of RT mixing on the basis of symmetry theory and explained a number of experiments and simulations of RT mixing [17,52,73,74,79,93]. It also reproduces the quantitative results of previous empirical models and reconciles the phenomenological approaches for modelling RT mixing with rigorous theories of RT instabilities [6,14,30,94,95].

#### (i) Interpolation and turbulence models

For a detailed consideration of interpolation models of RT mixing, the reader is referred to Refs. [64,69,92]. Briefly, the model of Youngs [57] suggested that RT mixing is induced by a broadband initial perturbation. The model provided excellent data fit with minimal number of free parameters. The interpolation models of Sharp [56] and Glimm & Sharp [62], the statistical model of Alon *et al.* [63,96], and the drag model of Alon *et al.* [84] put forward a hypothesis that an increase in the horizontal scales may trigger the flow transition from the nonlinear to a self-similar regime as with *λ*∼*gt*^{2} leading to *v*∼*gt* and *h*∼*gt*^{2}. The models of earlier studies [56,57,62] considered RT mixing as a multi-scale process dependent on the amplitude and period of the front. The statistical model [63,96] and drag model [84] interpreted RT dynamics as a single-scale process dependent solely on the period of the front.

The importance of the interpolation models is that they identified one of the mechanisms underlying self-similar mixing, quantified the mixing flow evolution and calibrated a broad variety of RT data with nearly the same set of adjustable parameters [69]. The experiments and simulation do not provide a trustworthy guidance whether the scalings *h*/*gt*^{2}, *λ*/*gt*^{2} are indeed universal [64].

Turbulence modelling approaches for RT mixing were initiated by the works of Belenkii & Fradkin [85] and Neuvazhaev [65]. From the equation for energy balance ∂(*ρv*^{2}/2)/∂*t*+*νρv*^{3}/*L*=*ρLvw*^{2} with *w*^{2}=*g*(∂ln*ρ*/∂*z*+*g*/*c*^{2}) being the Brunt–Väisälä frequency, by assuming by analogy with the Prandtl [97] mixing length theory that for an incompressible fluid ∂*ρ*/∂*t*=∂*D*(∂*ρ*/∂*z*)/∂*z* with turbulent diffusion coefficient *D*∼*vL*, they found a self-similar solution for RT mixing with *L*∼*h* and *h*∼*gt*^{2}. Then, the authors of works [46,50,87–89,98] modelled RT mixing in miscible fluids within the Boussinesq approximation. These researchers noted that in accelerated RT mixing the rate of energy dissipation is time dependent, *ε*∼*g*^{2}*t*, and found a similarity solution with *h*=*CAg*(*t*+*t*_{0})^{2}, where *C* is a free parameter and *t*_{0} is the time of the ‘virtual origin’, at which a transition to turbulence is expected to occur.

These results were consistent with Dimotakis [66], who hypothesized that in any high-speed flow, including anisotropic and inhomogeneous RT mixing, at certain values of Reynolds and Taylor numbers a transition should occur to a fully developed isotropic and homogeneous turbulence. Zhou [90] further extended these ideas to the cases of RT mixing under conditions of low and high energy density. For interpreting experimental and numerical data on RT mixing in terms of turbulent power laws, Chertkov [91] attempted to apply Kolmogorov and Bolgiano–Obukhov analyses to RT mixing via a substitution of a time dependence of the integral scale *L*∼*gt*^{2} in the invariants of canonical turbulence. This suggests that with time the viscous scale may decay as approximately *t*^{−1/4} in three dimensions and grow as approximately *t*^{1/8} in two dimensions in RT flow.

Some quantitative agreements were reported between the empirical models of RT mixing and the experiments and simulations. Some qualitative features of the turbulent process remained unclear, such as ordered character of RT mixing at high Reynolds numbers and the presence of coherent structures in high-speed RT flow [34,64,73,74,93,99]. To date, experiments and simulations have not provided a trustworthy guidance on whether the concepts of classical turbulence are fully applicable to RT mixing [59].

The importance of the phenomenological turbulence models is that they considered RT mixing within the general context of turbulent flows and served for development of effective numerical techniques for modelling turbulent flows and for implementation of turbulence in RT instability.

#### (ii) Statistical turbulence models and numerical modelling of turbulence in Rayleigh–Taylor mixing

Numerical modelling of turbulent flow usually employs a so-called ‘fully developed turbulence statistical modelling’ approach or Reynolds averaged Navier–Stokes (RANS) that suggests a methodology for implementation of phenomenological turbulent models of various levels of complexity. These models contain a large number of adjustable constants and have to be developed in close connection with experiments. Several hypotheses have to be fulfilled. Particularly, the flows under study have to be statistically steady (although RANS models are used in unsteady flows), and Reynolds numbers have to be large (although ad hoc extensions were developed for low Reynolds number flows). The flow is then considered as a ‘turbulent material’ and its dynamics is studied. Physical variables are assimilated to random variables and are statistically averaged. For instance, first, by applying Reynolds or Favre decomposition, transport equations are derived for the averaged quantities of the flow and their higher order correlations. Then, the magnitude of various terms is estimated in the transport equations. Closure hypotheses are formulated by assuming quasi-steadiness, weak inhomogeneity and anisotropy. The tensorial nature of correlations is accounted for, realizability conditions are imposed and adjustable constants are introduced. For variable-density flows, a mass weighted average is used, and closure relations are employed similar to the incompressible case. Eventually, the issues that are specific for RT mixing are addressed, such as the choice of the initial conditions, compressibility, stratification, etc.

In the past 50 years, a broad variety of numerical models were developed for implementation and modelling of turbulence in RT mixing [41–43,67,68,85,86,98,100–108]. These models were extensively validated against experiments and served as research tools, for instance, by bringing some insight into the time dependence and magnitude of turbulent kinetic energy [86,98] and the Reynolds stress anisotropy [105,106,108]. The numerical models found the development of homogeneous turbulence in RT mixing layers in the case of miscible fluids with seeded small-scale initial perturbations, as well as in Richtmyer–Meshkov flows (that can be interpreted as RT flow with impulsive acceleration) [109,110]. Depending on the circumstances, homogeneous turbulence in mixing layers can be modulated at large scales [111] and can exhibit structures (the so-called ‘plumes’) with superimposed small-scale mixing, in agreement with experiments [112]. The importance of these numerical models is in development of fast and affordable techniques for numerical implementation of turbulence in a broad variety of flows.

#### (iii) Momentum model and mechanism of transition to mixing

Momentum model was developed to explain the qualitative and quantitative properties of RT mixing that were observed in experiments and to harmonize with one another the rigorous theoretical studies and empirical modelling approaches of RT mixing [59,80–82]. The model employs the fact that RT mixing is driven by the transport of momentum. The specific momentum is gained owing to buoyant force and is lost owing to friction force. The dynamics of a parcel of fluid in the direction of gravity is governed by a balance per unit mass of the rate of momentum gain and the rate of momentum loss *μ* as and . The rate of momentum gain is an effective buoyant force, where is the rate of energy gain induced by the buoyancy. The rate of momentum loss is an effective friction force *μ*=*ε*/*v*, where *ε* is the rate of dissipation of specific kinetic energy *ε*∼*v*^{2}/(*L*/*v*)∼*v*^{3}/*L*, and *L* is the characteristic length scale for energy dissipation, either vertical or horizontal [59,113].

Momentum model showed that self-similar RT mixing with *h*∼*gt*^{2} develops owing to the imbalance of gain and loss of specific momentum, [59]. This imbalance may occur when the horizontal scale grows as *λ*∼*gt*^{2}. It may also occur when the vertical scale *h* is the characteristic scale for energy dissipation *ε*∼*v*^{3}/*L* with *L*∼*h* representing the cumulative contribution of small-scale structures into the flow dynamics [81,82]. Existence of two distinct mechanisms for the mixing development is consistent with the results of theoretical analyses which found that the amplitude *h* and wavelength *λ* both contribute to RT dynamics and that for highly isotropic coherent structures the growth of horizontal scales may not occur [2,6,14,30,31,94,95,114,115]. At the same time, momentum model reproduces in certain limiting cases the results of the interpolation and turbulence models [92], identifies new properties of RT mixing, including the ordered character of accelerated RT mixing and the short span of scales in statistically steady RT flow, and thus explains the experiments [17,73,74,79,93].

### (b) Rayleigh–Taylor mixing properties

#### (i) Experiments on Rayleigh–Taylor mixing in strongly accelerated fluids

The experiments in strongly accelerated fluids in RT mixing flow demonstrate certain features that are associated with the existence of a sharp boundary and a final jump of density across this boundary at the edge of turbulent mixing zone (TMZ). The sharp boundary and the density jump exist in cases of liquid–gas and gas–gas interfaces [73,99,116–118] indicating that mixing dynamics may exhibit order.

Figure 1*a* represents the development of RT mixing at liquid–gas interface. In these experiments, a jelly layer is accelerated with the products of detonation of acetylene–oxygen mixture with high pressure (approx. 1 MPa). Under such pressure, the jelly behaves as a liquid and is strongly accelerated (approx. 7×10^{4} m s^{−2}) leading to RT mixing [17,73,99,118]. The gaseous detonation products (DPs) penetrate the liquid jelly in bubbles, and the liquid jelly penetrates the DP in spikes. The bubble cupola remains smooth keeping the isotropic shape [81,118], and the spikes disperse eventually in little drops. As time progresses, the boundary of the bubble and spike structure remains sharp and has a density jump.

Figure 1*b* represents the interfacial mixing that develops at gas–gas interface between air (heavy gas) and helium (light gas) [17,73,118]. The image is obtained by laser sheet method in diffused light. Figure 1*b* shows that at late stages of flow evolution there exists a sharp boundary separating the heavy gas and the mixing zone. At the same time, no boundary is observed between the light gas and the mixing zone. This suggests that the mixing development in the gas–gas system is associated with the dynamics at the interface between the heavy gas and the zone. Particularly, the heavy gas penetrates the zone in narrow spikes, whose eddies further mix this gas with the light gas. The interface between the heavy gas and the mixing zone beyond the spikes remains sharp and has a density jump. RT mixing appears as a combination of the ordered flow at the interface between the heavy gas and the mixing zone, and disordered mixing of gases in the remaining part of the zone.

In cases of gas–liquid and gas–gas interfaces, the boundary between the media is sharp and has a finite density jump. Existence of this jump is necessary for the development of RT instability. An erosion of this density jump may potentially suppress the mixing development. Inside the mixing zone, in the case of gas–liquid interface, the bubbles float in the fluid and are surrounded by ‘liquid curtains’. In the case of gas–gas interface, a heavy gas spike falls into the mixing zone. It should be emphasized that for continued development of RT mixing the conditions should be preserved that enable the streams of the heavy and light fluids to grow, to propagate in opposite directions and to accelerate. While turbulence may play a role in such a process, its role is unlikely to be the primary one. In all likelihood, the flow dynamics exhibits some order, in agreement with momentum model.

#### (ii) Experiments on Rayleigh–Taylor mixing in high energy density plasmas

This experiment observes blast-wave-driven Rayleigh–Taylor-unstable behaviour in plasmas under conditions of high energy density. It seeks to help understand the behaviour of core-collapse supernovae, in which a spherical blast wave propagates from the centre of a star outwards, through the star's layers of progressively less-dense gases. The interfaces between these gases are RT unstable under the effect of the passing blast wave. It is thought that large-scale mixing owing to this instability could influence the overall evolution of the supernovae. The experiment exploits the concept of Euler similarity to create a system that is hydrodynamically well scaled to the helium–hydrogen interface of the supernova SN1987a, thus demonstrating how perturbation of the star's layered, otherwise spherically symmetric structure can lead to mixing on a much larger scale during a supernova explosion [51,93,119,120].

The experiment was performed at the Omega laser facility [121]. The experimental system is composed of a pair of initially solid materials, such that the interface and its subsequent unstable growth can be carefully controlled. The denser material is a polyimide plastic disc with a machine-seeded perturbation on its surface; the less-dense material is a carbon resorcinol foam cylinder that is pressed against the machined surface of the plastic disc; the plastic and the foam are encased in a polyimide shock tube. The blast wave is driven by 10 beams of the Omega laser producing in total nearly 4500 J of 351 nm laser light that is deposited onto an 800 μm diameter spot on the plastic in a 1 ns temporally uniform pulse. As these beams irradiate the surface of the polyimide disc, they ablate the outer layer of material. The ablation pressure of this newly created plastic plasma drives a shock wave into the disc, and the post-shock pressure is tens of megabars. When the pulse ends, a rarefaction propagating into the plastic occurs. The shock wave has traversed only about half the thickness of the disc in this time, and the rarefaction has overtaken the shock, turning it into a blast wave, by the time it reaches the material interface. Figure 2 represents a typical radiograph of the experiment. The blast wave is propagating in the foam from left to right and is visible to the right of the centre of the image. To the left of the shocked foam, the dark region is the shocked plastic. Figure 2 clearly shows that the interface between the plastic and the foam is RT unstable leading to the development of RT mixing.

In these experiments, the plasma flow is strongly accelerated at the rate of approximately 10^{10} of the Earth's gravity. Its Reynolds number is very high, *Re* > 10^{6} [51,93]. An expectation was for the high-speed plasma flow to transit to a fully turbulent regime [66]. Figure 2 shows that at late stages of the evolution, the strongly accelerated RT mixing is dominated by coherent structures and exhibits significant degree of order, in agreement with momentum model.

#### (iii) Experiments on Rayleigh–Taylor mixing in gases at low and moderate Reynolds numbers

The gas channel experiments [79] study the evolution of turbulence in RT mixing in the case of miscible fluids with similar and contrasting densities for a multi-mode initial perturbation and for flows with the Reynolds number *Re*∼10^{3} varying in different experiments from low to intermediate values as *Re* ∼ 1.5 × 10^{2}–2.5 × 10^{4}. RT mixing is considered as a statistically steady flow similar to canonical turbulence. In the experiments, two gas streams of air (on the top) and helium (at the bottom) initially are separated by a thin splitter plate and move parallel to each other at the same velocity in the gas channel. The streams meet at the end of the splitter plate leading to the downstream formation of RT instability and mixing. The flow is diagnosed by using digital photographs and hot-wire anemometry including both time-averaged and instantaneous statistics. Data are collected on the mean density profile, the growth rate parameters, and the flow statistics, including the spectra of velocity, density and mass flux. The conditional technique enables the analysis of statistical properties of the evolution of the light and the heavy fluids. Figure 3 shows the measured ‘energy density spectra’ [79] for the fluctuations of velocity, density and mass flux at the centreline of the mixing layer in the case of fluids with similar densities (density ratio is approx. 1.06 with corresponding Atwood number *A*=0.03) at two distinct moments of time. The scaling units include the density difference, channel depth *H* and buoyant acceleration *Ag*. The kurtosis *K* of the fluctuations is accounted for. The spectra are compared with canonical turbulence power law *k*^{−5/3}, where *k* is wavevector. Banerjee *et al*. [79] report that in these experiments the distribution of fluctuations follows the behaviour of turbulent fluctuations of the vertical velocity component, and the spectra have power-law dependence for *kH* ∼ 10^{2} with *k* ∼ 10^{2} m^{−1}. Figure 3 shows that in statistically steady RT flow the turbulence-like state has very short dynamic range with the span of scales less than or about one decade, in agreement with the momentum model. The importance of the experiments of Banerjee *et al.* [79] is in gathering detailed statistics on RT mixing for a broad range of experimental parameters and providing data for evaluation of adjustable parameters in numerical turbulence models [67,68,107,108].

#### (iv) Rayleigh–Taylor mixing properties

While complete understanding of interfacial turbulent mixing requires extensive theoretical, experimental and numerical studies [1], recent theoretical results found that accelerated self-similar RT mixing (that is driven by the imbalance of the rate of gain and loss of specific momentum and whose Reynolds number is growing) may indeed exhibit some order [81,82]. A re-laminarization of RT mixing may occur, similar to other hydrodynamic flows, such as flows in curved pipes [70] or accelerated boundary layers [71]. At the same time, in RT flows with balanced gains and losses for the rates of momentum and energy and with relatively low Reynolds number, the properties of statistically steady turbulence can be resembled [81,82].

There are some similarities and distinctions between RT turbulent mixing and canonical Kolmogorov turbulence (see for details [59,80–82]). RT turbulent mixing is anisotropic, inhomogeneous and statistically unsteady. It is driven by the transports of mass and momentum (in real space) and has no external energy sources (other than gravity). In Kolmogorov turbulence, mass and momentum are not transferred (on average) owing to conditions of homogeneity, isotropy and locality. This flow is statistically steady and is driven by the transport of kinetic energy (in wavevector space). It has an external energy source (a constant power ‘motor’). Both RT mixing and Kolmogorov turbulence have a number of symmetries in a statistical sense. In particular, they are invariant under scaling transformation. In the limit of vanishing viscosity in Kolmogorov turbulence, the scaling exponent is *n*=1/3 and the rate of change of specific kinetic energy (e.g. energy dissipation rate) *ε*=*Cv*^{3}/*L* is flow invariant. In RT mixing, the scaling exponent is *n*=1/2 and the invariant corresponding this symmetry is the rate of change of specific momentum (e.g. rate of momentum loss) *μ*=*Cv*^{2}/*L* with *μ*=*ε*/*v* and *μ*∼*g*. Consequently, the *N*th-order velocity structure function scales with length as *N*/3 in canonical turbulence and *N*/2 in RT mixing thus indicating stronger level of correlations in RT mixing flow. In Kolmogorov turbulence, enstrophy is a statistical invariant, whereas in RT mixing it is time dependent, and, furthermore, vortical structures in RT flow are likely to be helices [81].

Here, we mention briefly several factors indicating that accelerated RT mixing can indeed re-laminarize [30,59,80–82]. These include but are not limited to the absence of energy source (other than gravity), relatively large drag, presence and formation of interfaces, time dependence of enstrophy, stronger correlations for the flow quantities, stronger dependence on the initial conditions, finite value of viscous and dissipative time scales that are established by flow acceleration and steeper dimensional-analysis-based spectra. The ordered character of RT mixing flow agrees with the experiments in strongly accelerated fluids and in high energy density plasmas and with the numerical simulations of the ICF [17,59,73,74,76,77,80–82,93].

At the same time, re-laminarization of accelerated RT mixing does not preclude the appearance in RT flow of the state that is like canonical turbulence [59,92]. The properties of canonical turbulence can be resembled in statistically steady RT flow that has balanced rates of momentum and energy. In this state, the characteristic length scale and velocity are *λ* and , the Reynolds number is , the energy dissipation rate is constant at approximately *g*^{3/2}*λ*^{1/2}, the viscous scale is (*ν*^{3}/*ε*)^{1/4}, and the span of scales is approximately (*λ*/(*ν*^{2}/*g*)^{1/3})^{9/8}. Given that 9/8=1.125≈1, this span of scales is well captured by the ratio between the initial perturbation wavelength *λ* and the wavelength corresponding to the mode of the fastest growth as *λ*/(*ν*^{2}/*g*)^{1/3}. In this regime, the flow can appear more coherent or more ‘turbulent’ depending on the Atwood number and the initial conditions in agreement with various studies [30,41,42,46,50,65,66,78,79].

Furthermore, as the evolution of spikes and bubbles may, in principle, have different time dependences (e.g. steadily moving bubbles and accelerated spikes; see [30] and references therein), RT flow can be a combination of (disordered) steadily moving regions and ordered accelerated regions, as was observed experimentally [17,73].

## 4. Conclusion

We considered theoretical, experimental and numerical studies of RT turbulent mixing that have been developed over recent decades. On the one hand, our review confirms that the generic problem of RT turbulent mixing is an exceedingly difficult one. On the other hand, the significant progress that was achieved in our understanding of RT turbulent mixing in terms of analysis, experiment, simulation and data processing suggests new ideas and approaches for grasping essentials of the mixing process and to its better control in technological applications.

The theoretical analysis, experiments and simulations find that the scale coupling in RT mixing flow has a complicated character. On the one hand, the strongly accelerated high Reynolds number mixing flows may indeed keep significant degree of order. On the other hand, at low-to-moderate Reynolds numbers, statistically steady RT flow may resemble properties of canonical turbulence. The ordered character and re-laminarization of strongly accelerated RT mixing is the principal result that may help to comprehend the fundamentals of a variety of natural phenomena (e.g. early Universe evolution, supernova explosions and stellar and planetary convection) and to better control the technological applications (e.g. mixing mitigation in ICF and mixing enhancement in reactive flows). It opens new opportunities for design of experiments and simulations, which may include the realization of either more turbulent or more regular mixing flows in a broad parameter regime. These developments would require further enhancements in the quality and information capacity of experimental and numerical datasets, and would have a potential to bring experiment, numerical modelling and theoretical analysis to a new level of standards. RT mixing thus remains a formidable and multi-faceted problem of modern classical physics and mathematics that is well open for a curious mind.

## Acknowledgements

The authors express their deep gratitude to the National Science Foundation (award no. 1004330), to the US Department of Energy, and to the Ministry of Science and Education of Russia (grant 8836).

## Footnotes

One contribution of 13 to a Theme Issue ‘Turbulent mixing and beyond: non-equilibrium processes from atomistic to astrophysical scales II’.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.