## Abstract

This Introduction summarizes and provides a perspective on the papers representing one of the key themes of the ‘Turbulent mixing and beyond’ programme—the hydrodynamic instabilities of the Rayleigh–Taylor (RT) and Richtmyer–Meshkov (RM) type and their applications in nature and technology. The collection is intended to present the reader a balanced overview of the theoretical, experimental and numerical studies of the subject and to assess what is firm in our knowledge of the RT and RM turbulent mixing.

This part II of the Theme Issue consists of the following papers: the paper by Sreenivasan & Abarzhi on acceleration and turbulence in Rayleigh–Taylor (RT) mixing [1]; by Meshkov on experimental studies of unstable interfaces [2]; by Youngs on numerical modelling simulations of self-similar regimes in mixing flows [3]; by Grinstein *et al*. [4] on a pragmatic approach for reproducing complex multiphase flows in simulations; by Glimm *et al*. [5] on the so-called alpha problem; by Nevmerzhitskiy on the implementation and diagnostics of RT/Richtmyer–Meshkov (RM) mixing in experiments [6]; by Livescu on high resolution approaches for numerical modelling of RT instabilities [7]; by Statsenko *et al.* [8] on subgrid scale models applied to RT/RM mixing; by Prestridge *et al*. analysing the RM mixing experiments conducted over the past decade [9]; by Levitas on mixing applications in reactive flows [10]; by Pudritz & Kevlahan on supersonic processes and shock waves in interstellar media [11]; and by Anisimov *et al*. [12] summarizing the status of our understanding of RT mixing.

We observe the development of the Rayleigh–Taylor instability (RTI) when fluids of different densities are accelerated against the density gradient [13,14]. The Richtmyer–Meshkov instability (RMI) develops when the acceleration is driven by a shock wave or when it is impulsive [15,16]. Extensive interfacial material mixing of the fluids ensues with time [17]. Turbulent mixing plays an important role in these instances and in a broad variety of natural phenomena, ranging from astrophysical to microscales and from high to low energy density regimes, as well as in technological applications (for a summary, see [18] and references therein): it influences the formation of the ‘hot spot’ in inertial confinement fusion, limits radial compression of imploding Z-pinches and controls non-equilibrium heat transfer induced by ultrafast, high-power lasers in solids [19,20]. RT and RM mixing governs the transport of mass, momentum and energy in mantle–lithosphere tectonics, dynamics of compressible flows in ocean and atmosphere, evolution of interstellar molecular clouds and flashes in supernovae; it also plays an important role in industrial applications in combustion, aerodynamics and aeronautics. While occurring in vastly different circumstances, RT flows exhibit a number of common features in their evolution [17]. The mixing starts to develop when the fluid interface is slightly perturbed near its equilibrium state. The flow transitions from an initial stage in which the perturbation amplitude grows relatively quickly, to a nonlinear stage where the growth rate slows and the interface is transformed into a composition of a large-scale coherent structure and small-scale irregular structures driven by shear and, finally, to a stage of turbulent mixing whose dynamics is believed to be self-similar. The dynamics of RM flows is more complicated and involves shock–interface interaction, background motion of the compressed materials, appearance of heterogeneous structures in the fluid bulk, etc.

Even with over a hundred papers published each year on RTI and RMI in peer-reviewed mathematical, computational, scientific and engineering journals, our knowledge of these mixing processes is still limited [18]. On the experimental front, a systematic study of RT and RM flows is quite a challenging task. Owing to the sensitivity and transient character of the dynamics, repeatable implementation of these flows in a laboratory environment requires unusually tight control of macroscopic parameters as well as initial and boundary conditions. Statistical unsteadiness of turbulent mixing indicates a need to measure both spatial and temporal distributions of flow quantities. Furthermore, these measurements should be performed with high accuracy and resolution, substantial dynamic range and high data acquisition rate [21]. With respect to numerical simulations, the modelling of RT and RM mixing is challenging because of its singular nature combined with the limited spatial dynamic range of illustrative simulations, even those that use the leadership-class massively parallel computers. These challenges are compounded by the necessities to capture shocks, track interfacial dynamics and accurately account for dissipation processes [17]. For theoretical analysis, the dynamics of RTI and RMI is intellectually challenging, as it has to balance numerous competing requirements and pay attention to the demands of the multi-scale, highly nonlinear and non-local character of the mixing flows. We wish to identify ‘universal’ asymptotic solutions if they exist, establish whether memory of initial conditions persists, capture the order prevalent in the mixing flow despite the noisy character of turbulent dynamics, and so forth. The use of physics-based considerations may help us to keep the analysis from being too mathematical or too empirical, and identify a set of robust parameters that should be precisely diagnosed in observations (see [17] and references therein).

This issue is intended to give the reader a balanced overview of the theoretical, experimental and numerical studies of RT and RM instabilities and attempt a summary of our knowledge of subsequent turbulent mixing. The issue starts with a short paper by Sreenivasan & Abarzhi on acceleration and turbulence in RT mixing. The authors propose that the imbalance in momentum that turbulence acquires because of the continually present acceleration owing to gravity suggests the need for a new paradigm in RT mixing dynamics, wherein the role played by the energy flux in classical turbulence is subsumed by the momentum imbalance owing to gravity.

The first review article in the issue is of the author whose seminal work [16] started the era of experimental studies of shock-driven instabilities. Meshkov's [2] paper analyses experiments on unstable interfaces and turbulent mixing carried out over the last few decades. Starting with a historical overview of the phenomenon, the paper discusses how the instability of the interface can be suppressed by the joint effect of RT and RM instabilities and focuses on a few special cases (‘peculiarities’). The first one is the development of a turbulent mixing zone at the unstable interface of jelly layer accelerated by an acetylene–oxygen mixture. The second experiment is the motion of a large-scale air bubble in a water channel, whereas the third deals with the flow generated by a circular perturbation imprinted on a bubble cupola. These experiments raise the question of the stability of the bubble cupola. Another set of experiments is the turbulent mixing at gas–gas and liquid–gas interfaces. The author notes that there is a density jump between the heavy gas and the turbulent mixing zone and this jump guarantees the development of the instability. The last experiment reviewed is the development of a turbulent mixing zone at an interface accelerated by a non-stationary shock wave where the RMI is superseded by an RTI.

Youngs [3] reviews the history of studies of RTI from considerations of the so-called single mode dynamics to self-similar turbulent mixing and makes an assessment of the applicability of self-similar hypothesis for modelling turbulent mixing layers. To provide data for calibration of engineering models, a sequence of highly resolved three-dimensional numerical simulations has been conducted for fluids with contrast densities and for those with similar densities, in the case of weakly compressible flows, by employing the monotone integrated large eddy simulation (LES) numerical methods. The focus of the studies is the identification of the so-called alpha parameter—the dimensionless pre-factor in the quadratic time dependence of the width of the mixing zone. Youngs distinguishes the case in which the initial perturbation consists of a random combination of short wavelength from that in which large scales are present. He shows that an enhanced self-similar mixing can be obtained if long-wavelength perturbations are added with a specific spectrum. He discusses that the numerical values of the alpha parameter are usually significantly higher in experiments than in simulations and attributes this discrepancy to small-amplitude long-wavelength perturbations present initially. The author also argues that in a finite domain the influence of the initial conditions may persist throughout the duration of the experiment.

The apparent universality of the alpha parameter has indeed been a puzzle for the RT/RM community for nearly two decades now. Its significance in RT mixing is not its universality (which may well not be the case) but that its numerical value is rather small. This small value implies that almost all the energy induced by the buoyant force dissipates and only a small portion of it remains for flow acceleration ([17] and references therein). It is worth noting that in the entire field of physical sciences the laws of nature are represented by scaling dependences; indeed, there exists a single dimensionless constant—the fine structure constant (1/137)—that couples the elementary charge, the reduced Planck constant and the speed of light, and whose universality is considered to be a key for linking quantum mechanics to electrodynamics and relativity [22].

The paper by Glimm *et al.* [5] focuses on the growth of the RT turbulent mixing layer and a comparison of the numerical data against experiments for code validation. A systematic study with increasingly accurate front tracking and LES methods reveals good agreement with experimental data and, indeed, shows no universality of the growth rate. The authors have achieved systematically validated simulations of the RT mixing rate across a wide range of explicitly defined conditions, with a code solving the equations for compressible flows. The effect of long-wavelength perturbation on the mixing growth rate is also studied. The study reveals at most a 5% effect and shows that large effects predicted by some simulations are inconsistent with experimental data. It is worth noting that existing experiments and simulations on RT mixing require substantial increase of dynamic range to convincingly demonstrate that the width of the mixing indeed grows quadratically with time.

Statsenko *et al.* [8] review numerical simulations carried out over the last decades on RT and RM instabilities. The authors note that direct numerical simulation (DNS) of some turbulent flows is not affordable even with modern computers and that LES approaches may offer a promising alternative. One of the LES methods is the implicit large eddy simulations (ILES) approach where no subgrid models are used. The authors present a detailed consideration of the numerical simulation of RTI. The simulations show a self-similar regime after a Reynolds number-dependent transient. The authors also discuss the effect of a large-scale local perturbation on the evolution of turbulent mixing and show that the scaling holds even if the pre-factor varies considerably. A configuration with two immiscible fluids is also studied. The authors note that the degree of mixing homogeneity in turbulent flows is very important for the determination of chemical and nuclear reaction rates. They find considerable spread of the calculated data in terms of homogeneity and discuss this disagreement with experimental data.

Grinstein *et al.* [4] adopt a pragmatic approach for modelling complex flows by recognizing that such flows are characterized by a broad range of length and time scales which are unavoidably underresolved. So the authors take recourse to coarse-grained LES strategies. They emphasize that successful strategies have to adequately capture the ‘physics’ of the process. They briefly outline possible numerical approaches to this end. The results obtained within these approaches in shock-driven turbulent mixing are discussed in greater detail. In particular, the authors demonstrate that the instability evolves into linear or nonlinear regimes depending on the resolution of the initial perturbations of vector and scalar fields. The areas of applicability of the numerical approach include inertial confinement fusion, the collapse of supernovae and supersonic combustion. Ideally, it would be a spectacular achievement if a single control parameter could be identified to describe the mixing flow evolution in such systems. The authors identify that the initial fluctuations of the small-scale material concentration and the gradient of their root-mean-square at the material interface are important parameters influencing the flow the evolution. As a particular example, the authors consider the dynamics of the interface between air and SF_{6} that is shocked and re-shocked, with the latter subjected to a rarefaction wave.

Nevmerzhitskiy [6] reviews experiments on RT and RM mixing and the diagnostics as well as experimental results that can be used as benchmarks for theoretical analysis and numerical code validation. The influence of turbulent mixing on the shock wave stability is studied with experiments carried out at high Mach numbers. A striking result is that the perturbations of the shock wavefront correspond to pressure perturbations at the leading edge of the mixing layer. The influence of the heavy fluid compressibility on a geometrical defect growth such as a step or a groove is also shown. These growths are inhibited at large Mach numbers, and the turbulent mixing width increases with the Mach number shock wave in the heavy fluid. Experiments on RT turbulent mixing at a gas–liquid interface with a large acceleration have also been conducted. The scaling pre-factor for the growth of the mixing region is found to be close to 0.07 for Reynolds numbers ranging from 0.5 to 10×10^{6}, and large scales are clearly visible on the video frames. Self-similar growth of a hemispherical groove perturbation is exhibited at a gas–liquid interface where a low-strength jelly of gelatin–water solution is used as the liquid. Finally, the author considers a configuration in which the initially light fluid is compressed such that its density becomes higher than that of the heavy fluid. In this situation, the Atwood number changes sign and the mixing layer continues to expand by inertia.

The paper by Livescu [7] attempts to outline approaches for modelling RT/RM mixing—a ‘challenge’ problem for numerical methods, as has been stated already. The author provides a review of numerical methods available for the description of RTI with an emphasis on configurations at large density ratios. For that purpose, he refers to quantum molecular dynamics, molecular dynamics particle methods, and then to the statistical approach where the Boltzmann equation is used to describe the particle distribution (see also [18] and references therein). The numerical methods for solving this latter equation are discussed, firstly the direct simulation Monte Carlo approach, and secondly the lattice Boltzmann equation. The multi-component equation of motion for gas mixtures in the continuum limit is derived with two incompressible limit approximations. Molecular level closures of fluxes are specified. The simplifications usually implied by neglecting the pressure gradient are pointed out. The numerical methods for the solution of these continuous equations are reviewed along with their advantages and disadvantages over DNS and some LES approaches. Results obtained by several authors are put in perspective, making useful distinctions between the DNS solutions of the Navier–Stokes equations and the subgrid models. The author concludes that the modelling of turbulent mixing is still an open problem, especially in the case of compressible fluids.

The next paper concludes the discussion of numerical and experimental results on RM mixing. Prestridge *et al.* [9] review and analyse the RM experiments carried out over the last decade and focus on the influence of shock strength, density contrast, initial conditions, and three-dimensional effects on flow evolution. The authors confirm that the width of the mixing zone scales with the Mach number. However, one may have different flows and mixing structures for the same mixing width. Moreover, very little is known about small-scale mixing. As has been already noted, initial conditions play a more important role in turbulent mixing than in canonical turbulent flows in general, and a particular difficulty in RM experiments is the control of these initial conditions. The density ratio has a strong influence as well. The authors emphasize that some available results are firm yet this problem is still open. They discuss the three-dimensional effects in experiments in convergent and annular shock tubes. Increased growth of the mixing zone width is found but measurements are more difficult to conduct in such geometries. The authors highlight that, as modelling and simulations demand more information about the flow quantities (vorticity, enstrophy, mixing fractions) and turbulence quantities (Reynolds stress tensor components, velocity–density cross-correlations, etc.), it becomes more critical that experimental measurements of the fluctuating flow fields be more detailed.

The work of Levitas [10] is devoted to non-equilibrium processes at microscopic scales. Aluminium is an important energy source for industrial systems (e.g. in solid rocket propulsion), and for chemical processes, such as material synthesis and controlled hydrogen generation. For these purposes, aluminium particles are integrated into various energetic formulations, using the reaction heat of aluminium oxidation. However, when the aluminium particle size is reduced from micrometre to nanometre, their reactivity drastically increases. The main issue is to understand this dramatic increase that cannot be explained by the diffusion oxidation mechanism. To explain this behaviour, Levitas [10] suggests a new mechano-chemical mechanism, the so-called melt–dispersion mechanism. This mechanism considers hundreds or thousands of small bare particles instead of a single particle covered by an oxide shell. The consideration successfully describes the dramatic increase in particle reactivity, and the predictive capability of the model is supported by numerous qualitative and quantitative confirmations.

It would be of immense interest to understand how the structure that one observes in galaxies, stars and planets has been created and shaped from the background interstellar medium of gas that is essentially diffuse. It seems self-evident that the mechanisms should involve supersonic mixing, including the propagation and collision of shock waves. This is the subject of the article by Pudritz & Kevlahan [11]. The authors review the observations, simulations, and theories of how turbulent-like processes can account for structures seen in molecular clouds. They then compare traditional ideas of supersonic turbulence with a simpler physical model on the effects of multiple shock waves and their interactions in the interstellar medium. This medium behaves as a Newtonian incompressible fluid although it is far away from it (because of large mean free path, heat conduction, embedded magnetic field and strong compressibility), and the density fluctuations follow a −5/3 spectrum over 11 orders of magnitude. On the other hand, interactions of two plane shock waves of unequal strength may generate a vortex sheet. This vortex sheet is unstable and may lead to three-dimensional turbulence. Two major additional physical effects influencing the structure of star forming gas are gravity and feedback processes from young stars. Both of them can produce power-law tails. Using two basic shock models (intersecting plane shocks and self-focusing curved shocks), the authors arrive at a self-consistent picture of the physics of shock waves in gravitating media. Finally, models involving shock wave interactions provide a more self-consistent picture of the interstellar medium and the conditions leading to star formation than the traditional ideas of supersonic turbulence.

The concluding article in this Theme Issue is by Anisimov *et al.* [12]. In the past few decades, significant advances have been made by theoretical analysis in the understanding of RT dynamics. The work briefly overviews some of these theoretical results and compares them with the state-of-the-art experiments. The comparison is focused on qualitative criteria and revisits some quantitative properties applied traditionally for the quantification of RT mixing. For instance, is RT mixing a deterministic process dependent on initial conditions or a turbulent stochastic process that is independent of initial conditions? Can the RT mixing flow be regularized and in what parameter regime may the flow regularization be achieved? We have currently sufficient information to answer these questions with good confidence, and, based on rigorous analysis, to obtain knowledge from data and to further apply this knowledge for the design of experiments in high energy density plasmas and for simulations of inhomogeneous and anisotropic flows in nature and technology.

We hope that the two parts of the Theme Issue will expose turbulent mixing and beyond research area to a broad scientific community and help to develop new ideas for tackling the fundamental aspects of the problem, assist in application of novel approaches in a wide range of phenomena where the turbulent processes occur, and to have an impact on technology development. We further hope that an inquisitive mind will be captured and fascinated by the opportunity to grasp the universal features of the multi-scale, non-equilibrium dynamics and to elaborate new concepts, whose lucidity and simplicity may cut through the complexity of the problem.

## Acknowledgements

We gratefully acknowledge the support of the National Science Foundation (USA) European Office of Aerospace Research and Development (UK) of the Air Force Office of Scientific Research (USA), Office of Naval Research Global (UK), Department of Energy (USA), Ministry of Science and Education (Russian Federation), Los Alamos National Laboratory (USA), Lawrence Livermore National Laboratory (USA), Argonne National Laboratory (USA), Commissariat l'Energie Atomiqueat aux 'Energies Alternatives (France), The UNESCO-IAEA International Centre for Theoretical Physics (Italy), The University of Chicago (USA), Institute for Laser Engineering (Japan), Joint Institute for High Temperatures of the Academy of Sciences (Russian Federation), and Photron (Europe) Ltd (UK).

## Footnotes

↵† Present address: Carnegie Mellon University, Pittsburgh, PA, USA, and Carnegie Mellon University–Qatar, Doha, Qatar.

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.