In this study, we tackle the challenge of inferring the initial conditions of a Rayleigh–Taylor mixing zone for modelling purposes by analysing zero-dimensional (0-D) turbulent quantities measured at an unspecified time. This approach assesses the extent to which 0-D observations retain the memory of the flow, evaluating their effectiveness in determining initial conditions and, consequently, in predicting the flow’s evolution. To this end, we generated a comprehensive dataset of direct numerical simulations, focusing on miscible fluids with low density contrasts. The initial interface deformations in these simulations are characterised by an annular spectrum parametrised by four non-dimensional numbers. To study the sensitivity of 0-D turbulent quantities to initial perturbation distributions, we developed a surrogate model using a physics-informed neural network (PINN). This model enables computation of the Sobol indices for the turbulent quantities, disentangling the effects of the initial parameters on the growth of the mixing layer. Within a Bayesian framework, we employ a Markov chain Monte Carlo (MCMC) method to determine the posterior distributions of initial conditions and time, given various state variables. This analysis sheds light on inertial and diffusive trajectories, as well as the progressive loss of initial conditions memory during the transition to turbulence. Furthermore, it identifies which turbulent quantities serve as better predictors of Rayleigh–Taylor mixing zone dynamics by more effectively retaining the memory of the flow. By inferring initial conditions and forward propagating the maximum a posteriori (MAP) estimate, we propose a strategy for modelling the Rayleigh–Taylor transition to turbulence.

Turbulent mixing in a supercritical CO$_2$ shear layer is examined using both experimental and numerical methods. Boundary conditions are selected to focus on the rarely studied near-critical regime, where thermophysical properties vary nonlinearly with respect to temperature and pressure. Experimental results are obtained via Raman spectroscopy and shadowgraphy, while numerical results are obtained via direct numerical simulation. The shear layer growth rate is found to be 0.2. Additionally, density profiles indicate a relaxation of density gradients between the mixed fluid and heavy fluid as the flow evolves downstream, which runs counter to existing supercritical shear layer data in the literature. The computational results identify significant anisotropy in the turbulence in the shear layer, which is discussed in terms of the development of regions of high density gradient magnitude. The Reynolds-averaged enstrophy budget at various streamwise locations indicates no significant dilatational or baroclinic contribution within the shear layer.

Local shearing motions in turbulence form small-scale shear layers, which are unstable to perturbations at approximately 30 times the Kolmogorov scale. This study conducts direct numerical simulations of passive-scalar mixing layers in a shear-free turbulent front to investigate mixing enhancements induced by such perturbations. The initial turbulent Reynolds number based on the Taylor microscale is $ Re_\lambda = 72$ or 202. The turbulent front develops by entraining outer fluid. Weak sinusoidal velocity perturbations are introduced locally, either inside or outside the turbulent front, or globally throughout the flow. Perturbations at this critical wavelength promote small-scale shear instability, complicating the boundary geometry of the scalar mixing layer at small scales. This increases the fractal dimension and enhances scalar diffusion outward from the scalar mixing layer. Additionally, the promoted instability increases the scalar dissipation rate and turbulent scalar flux at small scales, facilitating faster scalar mixing. The effects manifest locally; external perturbations intensify mixing near the boundary, while internal perturbations affect the entire turbulent region. The impact of perturbations is consistent across different Reynolds numbers when the amplitudes normalised by the Kolmogorov velocity are the same, indicating that even weaker perturbations can enhance scalar mixing at higher Reynolds numbers. The mean scalar dissipation rate increases by up to 50 %, even when the perturbation energy is only 2.5 % of the turbulent kinetic energy. These results underscore the potential to leverage small-scale shear instability for efficient mixing enhancement in applications such as chemically reacting flows.

We report the first measurement of turbulent mixing developing from the convergent Richtmyer–Meshkov (RM) instability using time-resolved planar laser-induced fluorescence in a semi-annular convergent shock tube. A membraneless yet sharp interface with random short-wavelength perturbations, but controllable long-wavelength perturbations, is created by an automatically retractable plate, enhancing the reproducibility and reliability of RM turbulence experiments. The cylindrical air/SF$_6$ interface formed is first subjected to a convergent shock, then to its reflected shock and subsequently transits to turbulent mixing. It is found that the mixing width after reshock has a linear growth rate more than twice the rate in planar geometry. Also, the mixing width does not present power-law growth at late stages as in a planar geometry. However, the scalar spectrum and transition criterion obtained are similar to their planar counterparts. These findings indicate that the geometric constraint greatly affects the large scales of the flow, while having a weaker effect on the small scales. It is also found that the reflected shock significantly increases both scale separation and Reynolds number, explaining the rapid transition to turbulence following reshock.

In this work, we investigate the mixing of active scalars in two dimensions by the stirring action of stochastically generated weak shock waves. We use Fourier pseudospectral direct numerical simulations of the interaction of shock waves with two non-reacting species to analyse the mixing dynamics for different Atwood numbers (At). Unlike passive scalars, the presence of density gradients in active scalars alters the molecular diffusion term and makes the species diffusion nonlinear, introducing a concentration gradient-driven term and a density gradient-driven nonlinear dissipation term in the concentration evolution equation. We show that the direction of concentration gradient causes the interface across which molecular diffusion occurs to expand outwards or inwards, even without any stirring action. Shock waves enhance the mixing process by increasing the perimeter of the interface and by sustaining concentration gradients. Negative Atwood number mixtures sustain concentration gradients for a longer time than positive Atwood number mixtures due to the so-called nonlinear dissipation terms. We estimate the time until that when the action of stirring is dominant over molecular mixing. We also highlight the role of baroclinicity in increasing the interface perimeter in the stirring dominant regime. We compare the stirring effect of shock waves on mixing of passive scalars with active scalars and show that the vorticity generated by baroclinicity is responsible for the folding and stretching of the interface in the case of active scalars. We conclude by showing that lighter mixtures with denser inhomogeneities ($At\lt 0$) take a longer time to homogenise than the denser mixtures with lighter inhomogeneities ($At\gt 0$).

We study the transition from the momentum- to buoyancy-dominated regime in temporal jets forced by gravity. From the conservation of the thermal content and of the volume flux, we develop a simple model which is able to describe accurately the transition between the two regimes in terms of a single parameter representing the entrainment coefficient. Our analytical results are validated against a set of numerical simulations of temporal planar forced plumes at different initial values of Reynolds and Froude numbers. We find that, although the the pure jet-scaling law is not clearly observed in simulations at finite Froude number, the model correctly describes the transition to the buoyancy-dominated regime which emerges at long times.

We have investigated the dynamics of floating tracer in an idealised turbulent quasi-geostrophic ocean by advecting Lagrangian particles in a high-resolution velocity field enhanced by the potential flow associated with vortex stretching. At first order in the Rossby number expansion, this component of the ageostrophic circulation can be derived through a diagnostic equation in terms of the geostrophic velocities. Borrowing methods from the theory of Lagrangian coherent structures, we identify coherent material loops around strong vortex cores using the Lagrangian averaged vorticity deviation (LAVD). Building on studies of clustering in kinematic, stochastic velocity fields, we utilise methods from statistical topography to show that the coherent vortices dominate the distribution of extreme values of the concentration field. We find that the presence of clusters and voids in a coherent vortex depends on more than just the sense of rotation, but also on the full evolution of the vorticity over its lifecycle. We identify the mechanism behind the cluster formation that respects the symmetries of the quasi-geostrophic equations but can be expected to hold robustly in more complicated regimes, due to the simple physical description. The association of cluster formation with vortex stretching implies that LAVD is a particularly relevant metric for floating tracer dynamics. The detection of intense clustering also has implications for reaction rates between ocean-borne flotsam, meaning that our results are relevant to understanding the general risk of floating microplastics and marine biological populations.

In freely decaying stably stratified turbulent flows, numerical evidence shows that the horizontal displacement of Lagrangian tracers is diffusive while the vertical displacement converges towards a stationary distribution, as shown numerically by Kimura & Herring (J. Fluid Mech., vol. 328, 1996, pp. 253–269). Here, we develop a stochastic model for the vertical dispersion of Lagrangian tracers in stably stratified turbulent flows that aims to replicate and explain the emergence of a stationary probability distribution for the vertical displacement of such tracers. More precisely, our model is based on the assumption that the dynamical evolution of the tracers results from the competing effects of buoyancy forces that tend to bring a vertically perturbed fluid parcel (carrying tracers) to its equilibrium position and turbulent fluctuations that tend to disperse tracers. When the density of a fluid parcel is allowed to change due to molecular diffusion, a third effect needs to be taken into account: irreversible mixing. Indeed, ‘mixing’ dynamically and irreversibly changes the equilibrium position of the parcel and affects the buoyancy force that ‘stirs’ it on larger scales. These intricate couplings are modelled using a stochastic resetting process (Evans & Majumdar, Phys. Rev. Lett., vol. 106, issue 16, 2011, 160601) with memory. More precisely, Lagrangian tracers in stratified turbulent flows are assumed to follow random trajectories that obey a Brownian process. In addition, their stochastic paths can be reset to a given position (corresponding to the dynamically changing equilibrium position of a density structure containing the tracers) at a given rate. Scalings for the model parameters as functions of the molecular properties of the fluid and the turbulent characteristics of the flow are obtained by analysing the dynamics of an idealised density structure. Even though highly idealised, the model has the advantage of being analytically solvable. In particular, we show the emergence of a stationary distribution for the vertical displacement of Lagrangian tracers. We compare the predictions of this model with direct numerical simulation data at various Prandtl numbers $Pr$, the ratio of kinematic viscosity to molecular diffusion.

We propose a computational framework for simulating the self-similar regime of turbulent Rayleigh–Taylor (RT) mixing layers in a statistically stationary manner. By leveraging the anticipated self-similar behaviour of RT mixing layers, a transformation of the vertical coordinate and velocities is applied to the Navier–Stokes equations (NSE), yielding modified equations that resemble the original NSE but include two sets of additional terms. Solving these equations, a statistically stationary RT (SRT) flow is achieved. Unlike temporally growing Rayleigh–Taylor (TRT) flow, SRT flow is independent of initial conditions and can be simulated over infinite simulation time without escalating resolution requirements, hence guaranteeing statistical convergence. Direct numerical simulations (DNS) are performed at an Atwood number of 0.5 and unity Schmidt number. By varying the ratio of the mixing layer height to the domain width, a minimal flow unit of aspect ratio 1.5 is found to approximate TRT turbulence in the self-similar mode-coupling regime. The SRT minimal flow unit has one-sixteenth the number of grid points required by the equivalent TRT simulation of the same Reynolds number and grid resolution. The resultant flow corresponds to a theoretical limit where self-similarity is observed in all fields and across the entire spatial domain – a late-time state that existing experiments and DNS of TRT flow have difficulties attaining. Simulations of the SRT minimal flow unit span TRT-equivalent Reynolds numbers (based on mixing layer height) ranging from 500 to 10 800. The SRT results are validated against TRT data from this study as well as from Cabot & Cook (Nat. Phys., vol. 2, 2006, pp. 562–568).

The evolutionary process of mixing induced by Rayleigh–Taylor (RT) and Richtmyer–Meshkov (RM) instabilities typically progresses through three stages: initial instability growth, subsequent mixing transition and ultimate turbulent mixing. Accurate prediction of this entire process is crucial for both scientific research and engineering applications. For engineering applications, Reynolds-averaged Navier–Stokes (RANS) simulation stands as the most viable method currently. However, it is noteworthy that existing RANS mixing models are primarily tailored for the fully developed turbulent mixing stage, rendering them ineffective in predicting the crucial mixing transition. To address that, the present study proposes a RANS mixing transition model. Specifically, we extend the idea of the intermittent factor, which has been widely employed to integrate with turbulence models for predicting boundary layer transition, to mixing problems. Based on a high-fidelity simulation of a RT case, the intermittent factor defined based on enstrophy is extracted and then applied to RANS calculations, showing that it is possible to accurately predict mixing transition by introducing the intermittent factor to the turbulence production from the baseline K-L turbulence mixing model. Furthermore, to facilitate practical predictions, a transport equation has been established to model the spatio-temporal evolution of the intermittent factor. Coupled with the K-L model, the intermittent factor provided by the transport equation is applied to modify the Reynolds stress in RANS calculations. Thereafter, the present transition model has been validated in a series of tests, demonstrating its accuracy and robustness in the capturing mixing process in different types and stages of interfacial mixing flows.

Turbulent mixing is a pivotal phenomenon in fusion research with profound implications for energy gain. A Reynolds-averaged Navier–Stokes model capable of predicting realistic mixing transition processes is of significant importance for fusion applications, yet such a model remains elusive. This work addresses the limitations of prevalent global transition criteria, proposing a new idea to quantify local transition characteristics based on the mixing state, recognizing its direct relevance to fusion reaction rates. We delve into the description and analysis of the spatiotemporal evolution of the mixing state and its interplay with the transition process. Then, a local transition indicator is developed and compared with conventional global criteria using the large-eddy simulation (LES) of Rayleigh–Taylor turbulent mixing. Building upon this foundation, we introduce a novel eddy viscosity model based on the local transition indicator. A posterior assessment using LES data validates that it significantly outperforms standard gradient transport models during the transition stage. Consequently, we integrate this new eddy viscosity model with the Besnard–Harlow–Rauenzahn model to construct a comprehensive transition model, which demonstrates reasonably good performance in comparison with LES results. This work paves the way for future research in developing advanced modelling strategies that can effectively address the complexities of transitional flows in fusion engineering applications.

In the astrophysics community it is common practice to model collisionless dust, entrained in a gas flow, as a pressureless fluid. However, a pressureless fluid is fundamentally different from a collisionless fluid – the latter of which generically possess a non-zero anisotropic pressure or stress tensor. In this paper we derive a fluid model for collisionless dust, entrained in a turbulent gas, starting from the equations describing the motion of individual dust grains. We adopt a covariant formulation of our model to allow for the geometry and coordinate systems prevalent in astrophysics, and provide a closure valid for the accretion disc context. We show that the continuum mechanics properties of a dust fluid corresponds to a higher-dimensional anisotropic Maxwell fluid, after the extra dimensions are averaged out, with a dynamically important rheological stress tensor. This higher-dimensional treatment has the advantage of keeping the dust velocity and velocity of the fluid seen, and their respective moments, on the same footing. This results in a simplification of the constitutive relation describing the evolution of the dust rheological stress.

We investigate the concentration fluctuations of passive scalar plumes emitted from small, localised (point-like) steady sources in a neutrally stratified turbulent boundary layer over a rough wall. The study utilises high-resolution large-eddy simulations for sources of varying sizes and heights. The numerical results, which show good agreement with wind-tunnel studies, are used to estimate statistical indicators of the concentration field, including spectra and moments up to the fourth order. These allow us to elucidate the mechanisms responsible for the production, transport and dissipation of concentration fluctuations, with a focus on the very near field, where the skewness is found to have negative values – an aspect not previously highlighted. The gamma probability density function is confirmed to be a robust model for the one-point concentration at sufficiently large distances from the source. However, for ground-level releases in a well-defined area around the plume centreline, the Gaussian distribution is found to be a better statistical model. As recently demonstrated by laboratory results, for elevated releases, the peak and shape of the pre-multiplied scalar spectra are confirmed to be independent of the crosswind location for a given downwind distance. Using a stochastic model and theoretical arguments, we demonstrate that this is due to the concentration spectra being directly shaped by the transverse and vertical velocity components governing the meandering of the plume. Finally, we investigate the intermittency factor, i.e. the probability of non-zero concentration, and analyse its variability depending on the thresholds adopted for its definition.

This work reports an experimental study of the turbulent entrainment into the planar wake of a circular cylinder, exposed to various turbulent backgrounds, from the near- to the far-field. The background turbulence features independently varying turbulence intensity and integral length scale, thereby rendering different turbulent/turbulent interfaces (TTIs) between the background and the primary flow (wake). Combined, simultaneous particle image velocimetry and planar laser induced fluorescence measurements were conducted to quantify the entrainment characteristics across these various TTIs at an inlet Reynolds number of 3800. The primary focus was on understanding how turbulent entrainment evolves spatially in conjunction with the rapid development of the large-scale coherent vortices in the planar wake, and how such evolution is affected by the background turbulence. It is found that TTIs can establish two layers when the background turbulence is sufficiently intense, which distinguishes TTIs from the turbulent/non-turbulent interface (TNTI). The two layers are underpinned by different physical mechanisms but have the same thickness and appear to scale with the local Kolmogorov length scale after the wake spreading transition position (Chen & Buxton, J. Fluid Mech., vol. 969, 2023, A4). It is also found that the probability density functions of the entrainment velocity for both TTIs and a TNTI display power law tails, which are associated with extremely large entrainment velocities occurring more frequently than for a Gaussian process. These intermittent, extreme entrainment velocities make a remarkable contribution to the mean entrainment velocity, particularly in the near wake, which leads to a much higher mean entrainment velocity than farther downstream, for both a TNTI and the TTIs. Conditionally averaged analysis reveals that these extreme events of the entrainment velocity are directly associated with intense enstrophy structures close to the interface.

The entrainment hypothesis states that the mean inflow velocity across the boundary of a turbulent flow is proportional to a characteristic velocity of the flow. Proposed by G. I. Taylor approximately 80 years ago, it is still a common model of turbulence closure widely used in environmental engineering and geophysical fluid mechanics. Although it is a very simple concept and mathematical model, it has proven to be able to predict the entrainment in a variety of geophysical flows, e.g. convective clouds and plumes from erupting volcanoes in the atmosphere; dense water overflows and turbidity currents in the ocean; magma injection in a magma chamber in the interior of the Earth, to name just a few. In a seminal paper, Turner (J. Fluid Mech., vol. 173, 1986, pp. 431–471) presents a variety of laboratory and geophysical flows to illustrate the success of the entrainment hypothesis and discusses why such a simple hypothesis works so well even when the original assumptions are no longer valid.

We present direct numerical simulations of a three-layer Rayleigh–Taylor instability (RTI) problem with a configuration based on the experiments of Suchandra & Ranjan (J. Fluid Mech., vol. 974, 2023, A35) and Jacobs & Dalziel (J. Fluid Mech., vol. 542, 2005, pp. 251–279). The problem consists of a layer of light fluid between two layers of heavy fluid with an Atwood number of 0.3. These simulations are first validated through comparison with available experimental data. The validated simulations are then utilized to analyse statistics in this three-component flow. First, length scales are examined utilizing spectra and two-point spatial correlations of velocity and species concentration fluctuations. Next, joint probability density functions (p.d.f.s) of species concentration are compared against several model p.d.f.s representing generalizations of the bivariate beta distribution. Notably, the joint p.d.f.s do not appear to be accurately described by a Dirichlet distribution, indicating the marginal distributions do not conform to a beta distribution. Finally, similarity of the present configuration to three-component mixing found in inertial confinement fusion (ICF) applications is exploited to develop and validate an improved model for the impact of multicomponent mixing on thermonuclear (TN) reaction rates. A single time instant from the present simulations is chosen for a TN burn calculation under the hypothetical assumption of ICF materials and temperatures. Total TN output from this second calculation is then compared against the prediction of the improved model. The new model is found to accurately predict TN reaction rates in both premixed and non-premixed configurations.

The present study offers a twofold contribution on counter-gradient transport (CGT) of turbulent scalar flux. First, by examining turbulent scalar mixing through synchronized particle image velocimetry and planar laser-induced fluorescence on an inclined jet in cross-flow, we clarify the previously unexplained phenomenon of CGT, revealing key flow structures, their spatial distribution and modelling implications. Statistical analysis identifies two distinct CGT regions: local cross-gradient transport in the windward shear layer and non-local effects near the wall after injection. These behaviours are driven by specific flow structures, namely Kelvin–Helmholtz vortices (local) and wake vortices (non-local), suggesting that scalar flux can be decomposed into a gradient-type term for gradient diffusion and a term for large-eddy stirring. Second, we propose a new approach for reconstruction of turbulent mean flow and scalar fields using continuous adjoint data assimilation (DA). By rectifying model-form errors through anisotropic correction under observational constraints, our DA model minimizes discrepancies between experimental measurements and numerical predictions. As expected, the introduced forcing term effectively identifies regions where traditional models fall short, particularly in the jet centreline and near-wall regions, thereby enhancing the accuracy of the mean scalar field. These enhancements occur not only within the observation region but also in unseen regions, underscoring present DA approach's reliability and practicality for reproducing mean flow behaviours from limited data. These findings lay a solid foundation for adjoint-based model-consistent data-driven methods, offering promising potential for accurately predicting complex flow scenarios like film cooling.

Experiments on the Richtmyer–Meshkov instability (RMI) in a dual driver vertical shock tube (DDVST) are described. An initially planar, stably stratified membraneless interface is formed by flowing air from above and sulfur hexafluoride from below the interface location using the method of Jones & Jacobs (Phys. Fluids, vol. 9, issue 1997, 1997, pp. 3078–3085). A random three-dimensional, multi-modal initial perturbation is imposed by vertically oscillating the gas column to produce Faraday waves. The DDVST design generates two shock waves, one originating above and one below the interface, with these shocks having independently controllable strengths and interface arrival times. The shock waves have nominal strengths of $M_L=1.17$ and $M_H=1.18$ for the shock wave originating in the light and heavy gas, respectively, with these strengths chosen to result in arrested bulk interface motion following reshock. The influence of the length of the shock-to-reshock time, as well as the order of shock arrival, on the post-reshock RMI is examined. The mixing layer width grows according to $h\propto t^\theta$, where $\theta _H=0.36\pm 0.018$ (95 %) and $\theta _L=0.38\pm 0.02$ (95 %) for heavy and light shock first experiments, respectively, indicating no strong dependence on the order of shock wave arrival. Volume integrated specific turbulent kinetic energy (TKE) in the mixing layer versus time is found to decay according to $E_{tot}/\bar {\rho }\propto t^p$ with $p_H=-0.823\pm 0.06$ (95 %) and $p_L=-1.061\pm 0.032$ (95 %) for heavy and light shock first experiments, respectively. Notably, the 95 % confidence intervals do not overlap. Analysis on the influence of the shock-to-reshock time on turbulent length scales, transition criteria, spectra and mixing layer anisotropy are also presented.

Turbulent entrainment at the turbulent/non-turbulent interface (TNTI) plays an important role in understanding the turbulent diffusion. While entrainment in fully developed canonical turbulent flows has been extensively studied, the evolution of entrainment in spatially developing flows remains poorly understood. In this work, characteristics of entrainment and the effect of vortices on entrainment of the shear layer separated from a wall-mounted fence are studied by the experiment in a water channel. The shedding vortex experiences a series of stages, including generation, growth, deformation and breakdown into smaller vortices. With the development of the flow, entrainment varies correspondingly. The prograde vortex near the TNTI is found to suppress entrainment but have little effect on the detrainment process, while the retrograde vortex promotes entrainment and suppresses detrainment as well. Consequently, the local entrainment velocity is decreased by the prograde vortex and increased more significantly by the retrograde vortex. Along the streamwise direction, the time-mean entrainment velocity is smallest where the prograde vortex is strongest in the vortex deformation stage. However, the largest time-mean entrainment velocity is located where the enstrophy gradient near the TNTI is greatest after reattachment, rather than where the retrograde vortex is strongest shortly after the breakdown of the shedding vortex, because the scarcity of retrograde vortices in the vicinity of the TNTI makes their long-time cumulative contribution not as significant as their local enhancement. The present study reveals how entrainment evolves in the separated and reattaching flow, and improves our understanding of the effect of vortices on entrainment.

In convergent geometry, the effect of convergence and compression on the Rayleigh–Taylor instability (RTI) and Richtmyer–Meshkov instability (RMI) modifies the growth rate and behaviour of the instabilities. In order to better understand how compression/expansion caused by axial strain rates (i.e. strain rates normal to the interface) change the instability dynamics, axial strain rates are applied to RMI in planar geometry, isolated from the effects of convergence. Potential flow theory for the linear regime shows the growth rate of the instability is modified to include the background velocity difference of the instability's width. Resolved two-dimensional simulations of single-mode RMI showed the potential flow model is accurate whilst the amplitude is small compared with the wavelength. The application of strain rate to an RMI-induced mixing layer was investigated using three-dimensional implicit large eddy simulations (ILES) of the quarter-scale $\theta$-group case by Thornber et al. (Phys. Fluids, vol. 29, 2017, 105107). Whilst the background strain rate contributed to the mixing layer's growth, it was to a smaller extent than expected. The shear production of axial turbulent kinetic energy from the strain rate modified the rate of bulk entrainment, affecting the mixing layer's growth and mixedness, such that the strained simulations no longer attained the same self-similar state. The capability of the buoyancy-drag model by Youngs & Thornber (Physica D, vol. 410, 2020, 132517) to predict the mixing layer width was investigated, using a model calibrated to the unstrained case. New terms were introduced into the buoyancy-drag model, which correspond to the shear production of turbulent kinetic energy.