Turbulent mixing is the generic name for processes by which two initially distinct fluids tend towards a homogeneous substance under the action of a vigorous stirring agent. The fluids may be miscible or immiscible, their molecular diffusivities may be comparable or disparate, they may be in single phase or multiphase, and may be contained in simple or complex geometries. Moreover, the thermodynamic and fluid dynamic conditions may be different: incompressible or compressible, low or high speeds, with the dominant stirring mechanism arising from buoyancy, shear or inertial effects. Each of these circumstances has its physical characteristics and requires specific mathematical tools of investigation, but there are also some generic features. Turbulent mixing is an intellectually challenging problem in terms of the underlying physics as well as the tools needed to describe, simulate and predict it. The understanding of turbulent mixing is important since it occurs in numerous and diverse circumstances, involving elementary and idealized flows, on the one hand, and a variety of complex flows in technological applications, on the other. Mixing occurs in many contexts such as inertial confinement, magnetic fusion and non-equilibrium heat transport, material transformation under the action of high strain rates, strong shocks, explosions, blast waves, supernovae and accretion disks, stellar non-Boussinesq and magneto-convection, planetary interiors in astrophysics, mantle-lithosphere tectonics, volcanic eruptions, atmospheric and oceanographic flows in geophysics, premixed and non-premixed combustion, unsteady boundary layers, pollution transport in urban areas, as well as hypersonic and supersonic flows in aerodynamics. A deep understanding of turbulent mixing requires one to go above and beyond studies of canonical turbulence, and include walls, non-equilibrium situations, interfaces, strong and isolated vortices, combustion, and so forth. In this article we briefly review various aspects of Turbulent Mixing that were discussed at the Second International Conference and Advanced School 'Turbulent Mixing and Beyond', TMB-2009, held in summer 2009 at the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy. The papers are arranged by TMB themes and within each theme they are ordered alphabetically by the last name of the first author, with tutorials following research contributions. Canonical turbulence and turbulent mixing. The theme of canonical turbulence and turbulent mixing is considered by several authors. Casciola et al investigate the dynamics of inertial particles dispersed in a turbulent jet and compare their numerical modeling results with the classical similarity theory of the jet far-field. Remarkable agreement is found between the theory and the direct numerical simulations (DNS), including decay of Stokes numbers with the distance from the origin, self-similarity of the mean axial particle velocity profile, etc. Nagata considers complex turbulent flows, which are known to exhibit no linear critical point for the laminar states, and which are linearly stable at finite Reynolds numbers. Square duct flow and sliding Couette flow in an annulus are considered and nonlinear traveling-wave states are found for the flows with the use of the homotopy approach developed by the author. These states may constitute a skeleton around which a time-dependent trajectory in the phase space is organized. Teitelbaum and Mininni study a decaying 3D incompressible turbulence, which mimicks turbulent mixing in geophysical flows, with rotation rendering the flow anisotropic at large scales. The authors analyze three DNS results (without and with rotation, and with helicity), observe a decoupling of the modes normal to the rotation axis, and show that the helicity decreases the decay rate of turbulence. Wang and Peters investigate the structure of turbulence by studying strain rates of various scalars, including a passive scalar, a velocity component, turbulent kinetic energy and dissipation rate. The analyses of the DNS data for homogeneous shear flows show that statistically the gradient vectors with large magnitudes align with each other, while gradients with small magnitudes tend to be randomly organized. Zybin et al study turbulence structure through a model of vortex filament. In this way, they show that contraction and stretching out of a filament provide an energy flux from larger to smaller scales. The authors obtain the scaling exponents for both Lagrangian and transverse Eulerian structure functions and report good agreement with the existing data. Wall-bounded flows. Six papers are focused on the theme of wall-bounded flows. Cassel and Obabko perform numerical simulations of the two-dimensional flow induced by a thick-core vortex. This problem is important for studies of unsteady separation in the vortex-induced flows. Their accurate investigations convicingly justify that the Rayleigh instability does exist at large Reynolds numbers. Cvitanović and Gibson study the effects of geometry on transitional turbulent flow and focus on wall-bounded shear flows at moderate Reynolds numbers. The authors determine a set of unstable periodic orbits from close recurrences of the turbulent flow, identify a few equilibria that resemble frequently observed but unstable coherent structures, and construct a low-dimensional state-space projection from the extremely high-dimensional data sets. The approach developed by the authors can be a useful tool for understanding massive data sets. Seidel et al focus on developing feedback flow control strategies, i.e., they attempt to achieve a desired flow state for the turbulent shear layer behind a backward facing step. The authors show that the Proper Orthogonal Decomposition (POD) of the density field is a better marker than that for the velocity field, as in the former case the contribution of small scale structures is effectively eliminated. Tuğluk and Tarman use solenoidal bases to numerically solve incompressible fluid flow problems. Within this approach the solution remains strictly solenoidal throughout the solution domain. The approach effectively eliminates possible errors which can be induced by the continuity equation. Voropayev and Zagumennyi investigate the receptivity of a laminar boundary layer over an actively deforming surface by means of the stability analysis and the DNS. The study is focused on tracking the energy transport between turbulent fluctuations of the velocity components as well as the energy transfer from the mean flow to fluctuations, and vice versa. Ziaei-Rad presents a parallel finite-volume/finite-element method for compressible turbulent flows with a modified k − turbulent model. Some test cases (open flow and a transient flow generated after an accidental rupture in a pipeline) show the efficiency of the method. Non-equilibrium processes. A number of papers considers non-equilibrium turbulent processes. Abarzhi and Rosner perform a comparative study of modeling approaches of Rayleigh–Taylor turbulent mixing. The authors consider similarities and differences in governing mechanisms and basic properties of turbulent mixing, as discussed in recent theoretical and heuristic modeling studies, and briefly outline how these mechanisms and properties may be explored in experiments and simulations. Grinstein presents numerical simulations of turbulent velocity fields based on subgrid modeling implicitly provided by a class of high-resolution finite-volume algorithms. The approach is successfully applied to the problem of turbulent mixing. Lim et al present a study of the verification and validation of front tracking numerical simulations on two experiments on turbulent mixing. The experiments include the quasi-immiscible case at a very large Schmidt number and the case of miscible fluids. The simulations successfully reproduce results of both experiments and find that the dominant short wavelength of the initial perturbations may significantly influence the mixing process. Livescu et al provide an overview of the variable density effects in buoyancy-driven turbulence at low to moderate Atwood numbers within a single-fluid approximation. The two cases considered are the classical Rayleigh–Taylor (RT) case and an idealized triply periodic RT-flow, between which several important differences are found. Among them, there is a mixing asymmetry in variable density flows and an anomalous small-scale anisotropy. Nevmerzhitsky et al report new experimental results obtained at the experimental facilities at the VNIIEF (Sarov, Russia). In particular, the authors investigate turbulent mixing induced by the Richtmyer–Meshkov (RM) instability in gases with weak and strong shock waves. They obtain accurate quantitative results which can be used as benchmarks for theoretical analysis and the validation of numerical codes. Scagliarini et al present results from numerical simulations of RT-type turbulence, performed with the use of the lattice Boltzmann method, which is able to describe consistently a thermal compressible flow subject to an external forcing. The authors show that the presence of the adiabatic gradient terminates the mixing process, in agreement with results of theoretical analysis and other computational approaches. Interfacial dynamics. Four papers are focused on the dynamics of interfaces and hydrodynamic instabilities. Bazarov et al report an experimental study of the dynamics of a gas bubble rising in a water channel. The experiments are designed to accurately analyze the effects induced by the joint action and development of gravitational and shear instability in a two-dimensional flow. It is shown that initial short-wavelength perturbations on the dome of a rising bubble quickly decay. This decay is not attributed to dissipative mechanisms, such as viscosity or surface tension, but is rather a manifestation of a hydrodynamic consequence of the so-called sub-harmonic instability. Igonin et al study the perturbation growth at a free surface of condensed matter with deterministic initial perturbations under the effect of a shock wave, which induces the RM instability. Two- and three-dimensional initial perturbations at the surface are imposed, and pulsed radiography and a two-piston shock tube technique are applied for experimental diagnostics. The authors quantify the dependency of the linear growth-rate on shock strength and geometry of the perturbation, show that the growth of the perturbations strongly depends on material compression in the shock tube, and find that 3D perturbations grow faster than 2D perturbations in the nonlinear regime of RMI. Matsuoka uses analytical and numerical methods to investigate the RT and RM instabilities in the incompressible limit. The author considers the interfacial dynamics in planar geometry and accounts for the effect of surface tension. Under certain conditions in the parameter regime, a mode–mode interaction leads to a 'resonance' type of behavior in the interfacial dynamics, and this resonant motion is studied in detail. Nevmerzhitsky et al report new experimental results on turbulent mixing induced by the RT instability at the gas-liquid interface. The width of the mixing zone spans a substantial dynamic range, thus allowing for accurate quantification of the mixing growth-rate. The authors find that the prefactor in the gt2 scaling law varies with time and depends on the Reynolds number in the range Re~104–106. Some interesting features of the front dynamics are observed, including front pulsation and formation of secondary structures. High energy density physics. The theme of high energy density physics is of special interest to the TMB community. Huete Ruiz de Lira investigates the classical problem of turbulence generation by a shock wave interacting with a random density inhomogeneity field, and proposes an exact small-amplitude linear theory to describe such interaction. The analysis is applied to study time-space evolution of the perturbed quantities behind a corrugated shock front, and yields the closed-form exact analytical expressions for the turbulent kinetic energy, degree of anisotropy of velocity and vorticity fields in the compressed fluid, shock amplification of the density non-uniformity, and the sonic energy flux radiated downstream. Ktitorov obtains self-similar solution of isentropic compression of a gas in convergent geometry. The stability of shell motion is considered by means of two-dimensional numerical simulations. The perturbation amplitude growth is given for both plane and cylindrical geometries. Lebo and Lebo apply a model of energy transport in turbulent sub-critical laser plasmas of porous targets. This model is proposed for studying powerful laser pulse interaction with a low-density porous target. The interaction is strongly inhomogeneous and turbulent. Material science. Three papers consider the materials aspect of turbulent mixing. Aprelkov et al investigate the RM instability development on the free surface of a metal (lead). The disturbances are regular grooves of a triangular cross-section and a pulsed radiography method is used to visualize the interface after the passage of a strong shock. Numerical computations are in good agreement with experiments. Demianov et al make an attempt to describe RT instability in solid state by the volume-of-fluid numerical method, and their hydrodynamic simulation are based on the Bingham rheological model to include plastic effects. Desai et al report the possibility of laser generated craters to investigate planetary events such as meteorite impact craters. Experiments are performed using a laser beam on aluminum foil targets and results are well explained by two-dimensional hydrodynamic numerical simulations including radiation. Astrophysics. Astrophysical problems are considered in the following papers. Brandenburg et al study the transport in hydromagnetic turbulence and dynamos and examine the predictive power of a mean-field theory by comparing its outcome with simulations under controlled conditions. A recently developed test-field method is used to extract turbulent transport coefficients in kinematic and quasi-kinematic cases. The latter is illustrated by magnetic buoyancy-driven flows. Chernyshov et al provide a survey of various subgrid models for strong compressible magneto-hydrodynamic (MHD) turbulence and perform the large eddy simulation (LES) of weakly compressible turbulence in a local interstellar medium. They observe that density fluctuations exhibit a Kolmogorov-like spectrum over a range of scales with a spectral index close to −5/3, presumably because the density fluctuations behave like a passive scalar. Gibson deals with turbulence and turbulent mixing in natural fluids, i.e., fluids in the Universe. He claims that many recent observations show that the standard cosmological model must be strongly modified to take basic fluid mechanics into account. Ustyugov carries out three-dimensional numerical simulations of solar magneto-convection using a realistic physical model (fully compressible radiation MHD equations, dynamical viscosity, equation of state and opacities of stellar matter) and provides a detailed discussion of the results. Magneto-hydrodynamics. The following four contributions consider the problems of magnetic field line reconnection and turbulence in magnetized plasmas. Gekelman et al study experimentally the reconnection of magnetic field lines in plasma current systems. The authors present experimental results on undriven reconnection, which occurs when two magnetic flux ropes are generated from initially adjacent pulsed current channels in a background magnetoplasma. They also present 3D magnetic fields and currents associated with the colliding laser produced plasmas. The reconnection regions (which are three-dimensional) are directly observed in the experiments. The authors argue that reconnection is not an independent topic but is part of a variety of phenomena associated with the much broader subject of 3D current systems in plasmas. Malyshkin presents a two-fluid magneto-hydrodynamic (MHD) model of quasi-stationary, two-dimensional, magnetic reconnection in an incompressible plasma composed of electrons and ions. The author finds two distinct regimes of slow and fast reconnection, which may serve to explain the initial slow build up and subsequent rapid release of magnetic energy frequently observed in cosmic and laboratory plasmas. Malyshkin and Kulsrud present two theoretical approaches for the calculation of the rate of quasi-stationary, two-dimensional magnetic reconnection with nonuniform anomalous resistivity within the framework of incompressible MHD. These MHD equations are solved for the entire reconnection layer and across the reconnection layer. The approaches give the same approximate answer for the reconnection rate and agree with some recent simulations. Krommes reviews the state-of-the-art in the field of nonlinear gyrokinetics, which is a description of low-frequency dynamics in magnetized plasmas, providing the fundamental basis for numerical simulations of micro-turbulence in magnetic-confinement fusion and astrophysical applications. The author sketches the derivation of the novel dynamical system comprising the nonlinear gyrokinetic equation and the coupled electrostatic gyrokinetic Poisson equation with the use of modern perturbative approaches. This tutorial is accessible to a broad audience and allows the reader to fully appreciate the gyro-kinetic equation as a powerful theoretical tool for studies of turbulence in magnetized plasmas. Canonical plasmas. Several contributions are devoted to experiments and observations on canonical plasmas, including hydrodynamics of discharges and ionosphere. Bagautdinova et al study the influence of the RT instability and turbulent plasma-electrolyte mixing on multichannel discharges in various experimental conditions of particular interest. Their experiments were carried out at atmospheric pressure. They conclude that the process of development of turbulent mixing instability is significantly influenced by discharge current value, the composition and concentration of the electrolyte, a well as the immersion depth of the electrode. Baryshnikov et al investigate the shock wave instabilities in glow discharge, which play an important role in applications as they may lead to a significant reduction of aerodynamic drag. The authors conduct experiments in which a shock wave enters the region of positive glow-discharge column with a lowered density of gas, and study the RM instability developing at the interface between the two media. The authors consider the effects of the shock strength, plasma decay, humidity and the dustiness of air, and show that humidity and small degree of the dustiness have little influence on the evolution of the instability. Kayumov et al have developed an experimental device to investigate the stability of the discharge plasma between a droplet cathode and electrolytes and the induced turbulent mixing of the electrolytes. Physical properties of the plasma discharges and their characterization in the atmospheric and lower pressure regimes is an "unexplored" research area. The authors perform a systematic experimental study spanning a broad parameter regime, of the influence of multi-channel discharge plasma on the creation of plasma swirls and on turbulent mixing, which develops on the boundary between the droplet cathode and electrolyte anode. Son and Tereshonok present a theoretical study of the thermal and plasma effect from discharge influence on air flow. The electron energy distribution function has been found numerically, using the parallel package Gas Dynamics Tool and solving the Boltzmann equation in the two-term approximation. These theoretical studies provide a better understanding of the experimental results described earlier. Cohen et al report an investigation of ionospheric ducts having the shape of large plasma sheets, generated by the vertically transmitted High Frequency Active Auroral Research Program (HFAARP) heater waves in several experiments conducted in Gakona (Alaska). Depending on the polarizations of the heater waves, these large-scale ionospheric plasma structures have different configurations. The authors study in detail the effect of the plasma sheets on the ionosonde signals in the presence of distant plasma blobs, and report good agreement between their theoretical studies and field experiments. Physics of atmosphere. Three manuscripts are devoted to turbulence and turbulent mixing in the atmosphere. Mukund et al deal with the interaction between turbulence and radiative processes within the nocturnal atmospheric boundary layer. They propose a flux-emissivity formulation that eliminates the near-ground anomalous cooling ('Ramdas paradox'). O'Kane and Frederiksen apply a statistical dynamical closure theory of turbulence to the problem of data assimilation in strongly nonlinear settings. Data assimilation aims to obtain a near-optimal estimate of the state of the atmosphere, based on observations and short term forecasts, and to provide the so-called background states with information in the data-void areas. Sofieva et al recall that the main source of turbulence in the stratosphere is the breaking of gravity waves which leads to effective turbulent mixing of the atmosphere. The authors propose the new methodology for reconstructing gravity waves and turbulence spectra parameters from scintillation satellite measurements, and discuss the results of their unique state-of-the-art observations. Geophysics and Earth science. One paper details geophysics and Earth science. To study geophysical phenomena in the laboratory, Cotel performs mixing experiments by impinging a turbulent jet across a stratified interface. The author distinguishes between entrainment and mixing, and suggests that entrainment can be characterized by a new parameter called 'persistence' to allow a proper interpretation of the experiments. Combustion. The following papers are devoted to combustion and mixing in the turbulent regime. Chorny et al verify the potentialities of the Reynolds averaged Navier–Stokes approach to model incompressible turbulent mixing at large Schmidt numbers in a co-axial mixer. Two different mixing regime modes can be observed, with and without a recirculation zone developing just behind the tube. Hicks and Rosner compare the evolution of a burning interface between denser fuel and less dense ashes to the evolution of a non-burning interface, in the presence of gravity. The developments of the subsequent 2D turbulent flows are compared. Meshram investigates the mixing of chemical elements of the type A + B → C by using the two-point closure method. The equations describing the turbulence under study are written in terms of two-point correlation functions and two-point triple correlation functions. Various length scales involved can be evaluated by integrating these equations. Zhang et al study the importance of scalar dissipation rate on the quenching of the steady laminar flamelet model at different stoichiometric ratios in a one-step reversible reaction with Arrhenius rate. The difference between the mixing and quenched states is investigated. Mathematical aspects of non-equilibrium flows. Dynamics of turbulent flows is an intellectually rich problem, and four contributions analyze the mathematical aspects. Fukumoto et al study the stability of a vortex tube embedded in a strain flow (the Moore–Saffman–Tsai–Widnall instability). The Lagrangian approach of the weakly nonlinear analysis is developed. It is shown that this approach facilitates the calculation of the wave-induced mean flow and allows one to study the evolution of three-dimensional disturbances. Goldobin studies the transport of a pollutant in a fluid layer by spatially localized two-dimensional thermo-convective currents appearing under frozen parametric disorder in the presence of an imposed longitudinal advection. The author employs the eddy diffusivity approach and shows that the effective diffusivity can be several orders of magnitude larger than that in the absence of advection. Troshkin exhibits a new exact solution of the Navier–Stokes equations for a rotating gas tube. This solution improves the well-known rigid-body rotation with a constant temperature, and applies to the centrifugation of a gas mixture. Zakharov presents theoretical results about the self-consistent analytical theory for wind-driven sea. He offers answers to some outstanding questions of great importance for the development of self-consistent analytical theory for wind-driven sea, without which experiments and theory cannot make useful contact. Stochastic processes and probabilistic description. The following papers are devoted to stochastic processes, probabilistic description and data analysis aspects of turbulent mixing and beyond. Kim et al present a statistical theory of self-organization of shear flows, modeled by a nonlinear diffusion equation with a stochastic forcing. A non-perturbative method based on a coherent structure is utilized for the prediction of the probability distribution functions. The results are confirmed by numerical simulations. Klimenko introduces non-conservative and competitive mixing within the framework of stochastic simulations where particles move with a fluid flow and are engaged in a random-walk. Traditional mixing is conservative (i.e., the total amount of scalar is preserved during mixing) while, in non-conservative mixing, the post-mixing average of the particles becomes biased towards those participating in mixing. Vesper and Khokhlov remark that many physical objects have a dynamical scale and cannot be numerically simulated with a fixed computational grid. They suggest the use of a proportional-integral-derivative to automatically control the expansion or contraction of the computational grid. The example of the rarefaction wave is discussed. Advanced numerical simulations. Several works are devoted to numerical methods and their applications in TMB-related problems. Belotserkovskaya and Konyukhov carry out three-dimensional numerical simulations of branching patterns that occur when a less viscous fluid filtrates through a porous medium saturated by a more viscous one. They use a finite-volume weighted essentially non-oscillatory scheme (WENO). Belotserkovskii presents a review of numerical modeling studies performed under his leadership at the Institute for Computer Aided Design of the Russian Academy of Sciences. This work describes effective parallel algorithms for the solution of complex problems governed by nonlinear partial differential equations. The algorithms allow a dramatic reduction of the computational time and effective use of the multiprocessing computing resources. Examples from fluid dynamics (RT, RM and Kelvin–Helmholtz instabilities, as well as transitional and turbulent flows) and from medicine (modeling of circulatory and respiratory systems of human organism, and of cranial trauma) are displayed. Fortova studies the initial stage of the onset of turbulence in three-dimensional free shear flows of an ideal compressible gas. It turns out that the birth of turbulence is connected with large vortex structures. Griffond et al build a statistical numerical model of fully developed turbulence in compressible flows. The authors develop a Reynolds stress model that matches shock-turbulence interactions to the predictions of the linear interaction analysis, which, 'à la Ribner', relies on Kovasznay's decomposition and allows for the computation of the waves transmitted or produced at the shock front. The authors demonstrate close agreement between the linear interaction analysis and the Reynolds stress model for any shock strength. Jayakumar et al developed a hybrid (structured/unstructured) finite-volume method capable of handling turbulent flows and conjugate heat transfer. A two-equation turbulence model is implemented and the backward-facing step flow is simulated and studied, since this configuration plays an important role in the design of heating or cooling equipment. Jin et al make an assessment of the LES capability in capturing preferential concentration of heavy particles in isotropic turbulent flows. In such flows, heavy particles tend to accumulate preferentially in regions of high strain rate and low vorticity due to the inertial bias. The authors call for new subgrid-scale models including particle-flow interactions. Lim et al combine a front tracking method with a dynamic subgrid-scale model to compensate the unresolved scales in LES methods. As a result, the authors observe a converging trend for the micro observables for the reshocked RM turbulent mixing flow. They compare their results to a simple model based on 1D diffusion that takes place in the geometry that is defined statistically by the interface between the two fluids. Liu carries out a numerical study of the turbulent mixing in a convergent shock tube, induced by the RM instability. The author characterizes the turbulent mixing flow by a number of quantities and mimics in the simulations the presence of macroscopic perturbations by imposing at the interface some artificial perturbations. He observes that these disturbances influence the mixing flow characteristics significantly. Reckinger et al present an extension of the adaptive wavelet collocation method to simulations of the RT instability. Such a method seems to be promising due to the localized nature of this instability. Numerical tests show that the method successfully captures the characteristics of a weakly compressible single-mode perturbation. Suzuki et al investigate turbulent mixing in regular as well as fractal grid turbulence by means of DNS, and relate to experiments performed in a water channel (see below). Turbulent mixing is enhanced in the fractal grid, especially at large times. They point out the usefulness of employing fractal grids in high-performance mixers. Grinstein et al present a tutorial on implicit large-eddy-simulations methods, i.e., numerical simulations of turbulent velocity fields based on subgrid modeling implicitly provided by a class of high-resolution, finite-volume algorithms. Experimental diagnostics. A substantial part of modern discoveries is provided by state-of-the-art experimental capabilities. The four contributions below are devoted to the methods of experimental design and diagnostics. Haehn et al present experiments performed at the Wisconsin Shock Tube Laboratory, which study the behavior of a twice-shocked spherical density inhomogeneity. High speed cameras are used to observe the development of the vortex ring after reshock. Nevmerzhitsky et al report experiments performed at the Federal Nuclear Center (VNIIEF) in Sarov, Russia, on a dispersion of liquid drop under the effect of an air shock wave. This shock wave is created through the explosion of C2H2+2.5O2 mixture in a shock tube. The drop liquid is tributyl phosphate, and the flow is recorded by high-speed filming. The experimental results have remarkable accuracy and precision, and their high quality allows for the direct comparison with the results of theoretical and numerical models. Suzuki et al perform experiments on high Schmidt number scalar transfer in regular and fractal grid turbulence. The time-series particle image velocimetry and the planar laser induced fluorescence technique are used to measure the velocity and concentration fields. The authors show that turbulent mixing for the fractal grid turbulence is strongly enhanced compared to that for a regular grid turbulence. Niemela considers turbulent flows produced with the use of gaseous 4He as a working fluid. His tutorial highlights some of the motivations, advantages and disadvantages of this experimental approach. It discusses how the use of cryogenic helium enables advances and explorations in classical (especially in high Rayleigh number convection) and quantum turbulence, and how it helps to gain a better understanding of the fundamental aspects of both. Some practical examples are also outlined. Conclusion. In conclusion, the authors of the introductory article hope that this Topical Issue will expose Turbulent Mixing and Beyond phenomena to a broad scientific community and will serve to integrate our knowledge of the field and further enrich its development.