Turbulence exhibits a striking duality: it drives concentrated substances apart, enhancing mixing and transport, while simultaneously drawing particles and bubbles into collisions. Little experimental data exist to clarify the latter process due to challenges in techniques for resolving bubble pairs from afar to coalescence via turbulent entrainment, film drainage and rupture. In this work, we tracked pairs of bubbles across nearly four orders of magnitude in spatial resolution, capturing the entire dynamics of collision and coalescence. The resulting statistics show that critical variables exhibit scalings with bubble size in ways that are different from some classical models, which were developed based on assumptions that bubble collision and coalescence only mirror the key scales of the surrounding turbulence. Furthermore, contrary to classical models which suggest that coalescence favours slow collision velocity, we find a ‘Goldilocks zone’ of relative velocities for bubble coalescence, where there is an optimal coalescence velocity that is neither too high nor too low. This zone arises from the competition between bubble–bubble and bubble–eddy interactions. Incorporating this zone into the new model yields excellent agreement with experimental results, laying a foundation for better predictions for many multiphase flow systems.

Interface-resolved direct numerical simulations are performed to investigate bubble-induced transition from a laminar to elasto-inertial turbulent (EIT) state in a pressure-driven viscoelastic square channel flow. The Giesekus model is used to account for the viscoelasticity of the continuous phase, while the dispersed phase is Newtonian. Simulations are performed for both single- and two-phase flows for a wide range of Reynolds (${Re}$) and Weissenberg (${\textit{Wi}}$) numbers. In the absence of any discrete external perturbations, single-phase viscoelastic flow is transitioned to an EIT regime at a critical Weissenberg number ($Wi_{cr})$ that decreases with increasing ${Re}$. It is demonstrated that injection of bubbles into a laminar viscoelastic flow introduces streamline curvature that is sufficient to trigger an elastic instability leading to a transition to an EIT regime. The temporal turbulent kinetic energy spectrum shows a scaling of $-2$ for this multiphase EIT regime, and this scaling is found to be independent of size and number of bubbles injected into the flow. It is also observed that bubbles move towards the channel centreline and form a string-shaped alignment pattern in the core region at the lower values of ${Re}=10$ and ${\textit{Wi}}=1$. In this regime, there are disturbances in the core region in the vicinity of bubbles while flow remains essentially laminar. Unlike the solid particles, it is found that increasing shear-thinning effect breaks up the alignment of bubbles.

Turbulent convection under strong rotation can develop an inverse cascade of kinetic energy from smaller to larger scales. In the absence of an effective dissipation mechanism at the large scales, this leads to the pile up of kinetic energy at the largest available scale, yielding a system-wide large-scale vortex (LSV). Earlier works have shown that the transition into this state is abrupt and discontinuous. Here, we study the transition to the inverse cascade at Ekman number ${Ek}=10^{-4}$ and using stress-free boundary conditions, in the case where the inverse energy flux is dissipated before it reaches the system scale, suppressing the LSV formation. We demonstrate how this can be achieved in direct numerical simulations by using an adapted form of hypoviscosity on the horizontal manifold. We find that, in the absence of the LSV, the transition to the inverse cascade becomes continuous. This shows that it is the interaction between the LSV and the background turbulence that is responsible for the earlier observed discontinuity. We furthermore show that the inverse cascade in absence of the LSV has a more local signature compared with the case with LSV.

Particle motions under nonlinear gravity waves at the free surface of a two-dimensional incompressible and inviscid fluid are considered. The Euler equations are solved numerically using a high-order spectral method based on a Hamiltonian formulation of the water-wave problem. Extending this approach, a numerical procedure is devised to estimate the fluid velocity at any point in the fluid domain given surface data. The reconstructed velocity field is integrated to obtain particle trajectories for which an analysis is provided, focusing on two questions. The first question is the influence of a wave setup or setdown as is typical in coastal conditions. It is shown that such local changes in the mean water level can lead to qualitatively different pictures of the internal flow dynamics. These changes are also associated with rather strong background currents which dominate the particle transport and, in particular, can be an order of magnitude larger than the well-known Stokes drift. The second question is whether these particle dynamics can be described with a simplified wave model. The Korteweg–de Vries equation is found to provide a good approximation for small- to moderate-amplitude waves on shallow and intermediate water depth. Despite discrepancies in severe cases, it is able to reproduce characteristic features of particle paths for a wave setup or setdown.

In this study we focus on the collision rate and contact time of finite-sized droplets in homogeneous, isotropic turbulence. Additionally, we concentrate on sub-Hinze–Kolmogorov droplet sizes to prevent fragmentation events. After reviewing previous studies, we theoretically establish the equivalence of spherical and cylindrical formulations of the collision rate. We also obtained a closed-form expression for the collision rate of inertial droplets under the assumption of inviscid interactions. We then perform droplet-resolved simulations using the Basilisk solver with a multi-field volume-of-fluid method to prevent numerical droplet coalescence, ensuring a constant number of droplets of the same size within the domain, thereby allowing for the accumulation of collision statistics. The collision statistics are studied from numerical simulations, varying parameters such as droplet volume fraction, droplet size relative to the dissipative scale, density ratio and viscosity ratio. Our results show that the contact time is finite, leading to non-binary droplet interactions at high volume fractions. Additionally, the contact duration is well predicted by the eddy turnover time. We also find that the radial distribution at contact is significantly smaller than that predicted by the hard-sphere model due to droplet deformation in close proximity. Furthermore, we show that for neutrally buoyant droplets, the mean relative velocity is similar to the mean relative velocity of the continuous phase, except when the droplets are close. Finally, we demonstrate that the collision rate obeys the appropriate theoretical law, although a numerical prefactor weakly varies as a function of the dimensionless parameters, which differs from the constant prefactor from theory.

We develop an asymptotic theory of a compressible turbulent boundary layer on a flat plate, in which the mean velocity and temperature profiles can be obtained as exact asymptotic solutions of the boundary-layer equations, which are closed using functional relations of a general form connecting the turbulent shear stress and turbulent enthalpy flux to the mean velocity and enthalpy gradients. The outer region of the boundary layer is considered at moderate supersonic free-stream Mach numbers, when the relative temperature difference across the layer is of order one. A special change of variables allows us to construct the solution in the outer region in the form of asymptotic expansions at large values of the logarithm of the Reynolds number based on the boundary-layer thickness. As a result of asymptotic matching of the solutions for the outer region and logarithmic sublayer, the velocity and temperature defect laws are obtained, which allow us to describe the profiles of these quantities in the outer and logarithmic regions by universal curves known for the boundary layer of an incompressible fluid. Similarity rules for the Reynolds-tensor components and root-mean-square enthalpy fluctuation are given. The recovery and Reynolds-analogy factors are calculated. A friction law is established that is valid under arbitrary wall-heat-transfer conditions.

Hierarchical parcel swapping (HiPS) is a multiscale stochastic model of turbulent mixing based on a binary tree. Length scales decrease geometrically with increasing tree level, and corresponding time scales follow inertial range scaling. Turbulent eddies are represented by swapping subtrees. Lowest-level swaps change fluid parcel pairings, with new pairings instantly mixed. This formulation suitable for unity Schmidt number $Sc$ is extended to non-unity $Sc$. For high $Sc$, the tree is extended to the Batchelor level, assigning the same time scale (governing the rate of swap occurrences) to the added levels as the time scale at the base of the $Sc=3$ tree. For low $Sc$, a swap at the Obukhov–Corrsin level mixes all parcels within corresponding subtrees. Well-defined model analogues of turbulent diffusivity, and mean scalar-variance production and dissipation rates are identified. Simulations idealising stationary homogeneous turbulence with an imposed scalar gradient reproduce various statistical properties of viscous-range and inertial-range pair dispersion, and of the scalar power spectrum in the inertial-advective, inertial-diffusive and viscous-advective regimes. The viscous-range probability density functions of pair separation and scalar dissipation agree with applicable theory, including the stretched-exponential tail shape associated with viscous-range scalar intermittency. Previous observation of that tail shape for $Sc=1$, heretofore not modelled or explained, is reproduced. Comparisons to direct numerical simulation allow evaluation of empirical coefficients, facilitating quantitative applications. Parcel-pair mixing is a common mixing treatment, e.g. in subgrid closures for coarse-grained flow simulation, so HiPS can improve model physics simply by smarter (yet nearly cost-free) selection of pairs to be mixed.

Turbulence amplification is crucial in shock-wave/turbulent boundary layer interaction (SWTBLI). To examine the impact of interaction intensity on turbulence amplification and inter-component energy transfer, direct numerical simulations of impinging oblique shock reflections at strong ($37^\circ$) and weak ($33.2^\circ$) incident angles are conducted. The results indicate that strong interaction generates a larger permanent separation zone, featuring the unique ‘oblique platform’ in Reynolds stress peaks and ‘secondary turbulence amplification’ downstream. Reynolds stress budget and spanwise spectral analyses reveal that $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ and $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ amplify primarily by production terms. $u''$, $v''$ and $w''$ represent the streamwise, wall-normal and spanwise velocity fluctuations. At the investigated Reynolds number, deceleration effect dominates the initial amplification of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$, influencing multi-scale wall-bounded turbulence structures, while shear effect remains active along the shear layer and may primarily affects streaky structures. The initial amplification of $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ is driven by the adverse pressure gradient, which reshapes the velocity profile and affects the wall-normal velocity. The primary energy for $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ and $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ amplification originates from $\widetilde{ u^{\prime \prime}u^{\prime \prime}}$ via the pressure-strain term. The delayed amplification of $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ is influenced by its production term and energy redistribution, with $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ exhibiting higher spectral consistency with $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ and receiving more energy. In strong interaction, the ‘oblique platform’ serves as a stable dissipation region, formed by increased separation–incident shock distance, characterised by progressively concentrated stress spectra and the transition to large-scale streaks. The downstream ‘secondary amplification’ process resembles the initial amplification near the separation shock foot, driven by intermittent compression waves that strengthen shear instabilities and the deceleration effect. These findings detail the streamwise stress evolution, providing a more comprehensive turbulence amplification mechanism in SWTBLI.

Motivated by the need for a better understanding of marine plastic transport, we experimentally investigate finite-size particles floating in free-surface turbulence. Using particle tracking velocimetry, we study the motion of spheres and discs along the quasi-flat free-surface above homogeneous isotropic grid turbulence in open channel flows. The focus is on the effect of the particle diameter, which varies from the Kolmogorov scale to the integral scale of the turbulence. We find that particles of size up to approximately one-tenth of the integral scale display motion statistics indistinguishable from surface flow tracers. For larger sizes, the particle fluctuating energy and acceleration variance decrease, the correlation times of their velocity and acceleration increase, and the particle diffusivity is weakly dependent on their diameter. Unlike in three-dimensional turbulence, the acceleration of finite-size floating particles becomes less intermittent with increasing size, recovering a Gaussian distribution for diameters in the inertial subrange. These results are used to assess the applicability of two distinct frameworks: temporal filtering and spatial filtering. Neglecting preferential sampling and assuming an empirical linear relation between the particle size and its response time, the temporal filtering approach is found to correctly predict the main trends, though with quantitative discrepancies. However, the spatial filtering approach, based on the spatial autocorrelation of the free-surface turbulence, accurately reproduces the decay of the fluctuating energy with increasing diameter. Although the scale separation is limited, power-law scaling relations for the particle acceleration variance based on spatial filtering are compatible with the observations.

The recent discovery of polymer diffusive instability (PDI) by Beneitez et al. (2023 Phys. Rev. Fluids 8, L101901), poses challenges in implementing artificial conformation diffusion (ACD) in transition simulations of viscoelastic wall-shear flows. In this paper, we demonstrate that the unstable PDI is primarily induced by the conformation boundary conditions additionally introduced in the ACD equation system, which could be eliminated if a new set of conformation conditions is adopted. To address this issue, we begin with an asymptotic analysis of the PDI within the near-wall thin diffusive layer, which simplifies the complexity of the instability system by reducing the number of the controlling parameters from five to zero. Then, based on this simplified model, we construct a stable asymptotic solution that minimises the perturbations in the wall sublayer. From the near-wall behaviour of this solution, we derive a new set of conformation boundary conditions, prescribing a Neumann-type condition for its streamwise stretching component, $c_{11}$, and Dirichlet-type conditions for all the other conformation components. These boundary conditions are subsequently validated within the original ACD instability system, incorporating both the Oldroyd-B and the finitely extensible nonlinear elastic Peterlin constitutive models. Finally, we perform direct numerical simulations based on the traditional and the new conformation conditions, demonstrating the effectiveness of the latter in eliminating the unstable PDI. Importantly, this improvement does not affect the calculations of other types of instabilities. Therefore, this work offers a promising approach for achieving reliable polymer-flow simulations with ACD, ensuring both numerical stability and accuracy.

Thermal forcing in natural environments, such as Earth’s surface, exhibits complex spatiotemporal variations due to daily and seasonal cycles. This motivates our study of Rayleigh–Bénard convection with hybrid spatiotemporal modulation at the thermal boundary, achieved by applying a travelling thermal wave to a bottom plate with modulated wavenumber $k$ and frequency $f$. At low frequencies, spatial modulation dominates, organising coherent thermal plumes. At high frequencies, the rapid propagation of the thermal wave smooths out the plumes, thereby reducing convective efficiency. We find that the emergence of the ‘smoothing’ effect is governed by the ratio between the wave speed ($c = f/k$) and the pseudo-speed of thermal diffusion, $c_{\textit{diff}} = 4\pi k/\sqrt {\textit{RaPr}}$, a scale-dependent measure of thermal damping. By comparing these speeds, we identify distinct regimes: (i) a spatially modulated-dominated regime ($c\lt c_{\textit{diff}}$), in which the slow movement of the boundary thermal wave allows coherent thermal plumes to follow the wave, maintaining coherence in both time and space; and (ii) a travelling-wave-dominated regime ($c\gt c_{\textit{diff}}$), where the fast-moving thermal wave disrupts the spatial coherence of thermal structures near the boundary layer. These findings establish a new framework for understanding the interplay of spatial and temporal modulation, advancing our knowledge of heat transfer in systems with complex boundary conditions.

Shear-thinning fluids flowing through pipes are crucial in many practical applications, yet many unresolved problems remain regarding their turbulent transition. Using highly robust numerical tools for the Carreau–Yasuda model, we discovered that linear instability can arise when the power-law index falls below 0.35. This inelastic non-axisymmetric instability can universally arise in generalised Newtonian fluids that extend the power-law model. The viscosity ratio from infinite to zero shear rate can significantly impact instability, even if it is small. Two branches of finite-amplitude travelling-wave solutions bifurcate subcritically from the linear critical point. The solutions exhibit sublaminar drag reduction, a phenomenon not possible in the Newtonian case.

We provide a rigorous analysis of the self-similar solution of the temporal turbulent boundary layer, recently proposed by Biau (2023 Comput. Fluids 254, 105795), in which a body force is used to maintain a statistically steady turbulent boundary layer with periodic boundary conditions in the streamwise direction. We derive explicit expressions for the forcing amplitudes which can maintain such flows, and identify those which can hold either the displacement thickness or the momentum thickness equal to unity. This opens the door to the first main result of the paper, which is to prove upper bounds on skin friction for the temporal turbulent boundary layer. We use the Constantin–Doering–Hopf bounding method to show, rigorously, that the skin-friction coefficient for periodic turbulent boundary layer flows is bounded above by a uniform constant which decreases asymptotically with Reynolds number. This asymptotic behaviour is within a logarithmic correction of well-known empirical scaling laws for skin friction. This gives the first evidence, applicable at asymptotically high Reynolds numbers, to suggest that Biau’s self-similar solution of the temporal turbulent boundary layer exhibits statistical similarities with canonical, spatially evolving, boundary layers. Furthermore, we show how the identified forcing formula implies an alternative, and simpler, numerical implementation of periodic boundary layer flows. We give a detailed numerical study of this scheme presenting direct numerical simulations up to a momentum Reynolds number of $\textit{Re}_\theta = 2000$ and implicit large-eddy simulations up to $\textit{Re}_\theta = 8300$, and show that these results compare well with data from canonical spatially evolving boundary layers at equivalent Reynolds numbers.

We examine the dispersion of prolate spheroidal microswimmers in pressure-driven channel flow, with the emphasis on a novel anomalous scaling regime. When time scales corresponding to swimmer orientation relaxation, and diffusion in the gradient and flow directions, are all well separated, a multiple scales analysis leads to a closed form expression for the shear-enhanced diffusivity, $D_{\it{eff}}$, governing the long-time spread of the swimmer population along the flow (longitudinal) direction. This allows one to organize the different $D_{\it{eff}}$-scaling regimes as a function of the rotary Péclet number (${\it{{\it{Pe}}}}_r)$, where the latter parameter measures the relative importance of shear-induced rotation and relaxation of the swimmer orientation due to rotary diffusion. For large ${\it{{\it{Pe}}}}_r$, $D_{\it{eff}}$ scales as $O({\it{{\it{Pe}}}}_r^4D_t)$ for $1 \leqslant \kappa \lesssim 2$, and as $O({\it{{\it{Pe}}}}_r^{ {10}/{3}}D_t)$ for $\kappa = \infty$, with $D_t$ being the intrinsic translational diffusivity of the swimmer arising from a combination of swimming and rotary diffusion, and $\kappa$ being the swimmer aspect ratio; $\kappa = 1$ for spherical swimmers. For $2 \lesssim \kappa \lt \infty$, the swimmers collapse onto the centreline with increasing ${\it{{\it{Pe}}}}_r$, leading to an anomalously reduced longitudinal diffusivity of $O({\it{{\it{Pe}}}}_r^{5-C(\kappa )}D_t)$. Here, $C(\kappa )\!\gt \!1$ characterizes the algebraic decay of swimmer concentration outside an $O({\it{{\it{Pe}}}}_r^{-1})$ central core, with the anomalous exponent $(5-C)$ governed by large velocity variations occasionally sampled by swimmers outside this core. Here, $C(\kappa )\gt 5$ for $\kappa \gtrsim 10$, leading to $D_{\it{eff}}$ eventually decreasing with increasing ${\it{{\it{Pe}}}}_r$, in turn implying a flow-independent maximum, at a finite ${\it{{\it{Pe}}}}_r$, for the rate of slender swimmer dispersion.

Inertial sedimentation of a cloud of cylinders released within a confined fluid-filled cell is experimentally investigated. Various cylinder numbers, $N_c$, aspect ratios, $\xi$, solid-to-fluid density ratios, $\rho _c / \rho _{\!f}$, and settling velocities corresponding to moderate Reynolds numbers are examined. The parameters correspond to two distinct path regimes for isolated cylinders: oscillatory trajectories for higher-density cylinders and rectilinear sedimentation for lower-density cylinders. In both cases, we observe the formation of subgroups (termed objects of class $N$) composed of $N$ cylinders in contact, as well as their recombination due to splitting or merging. Depending on the parameters, specific distributions of class-$N$ objects are found. In addition, beyond the formation of individual objects, large-scale vertical columnar structures emerge, made of densely packed objects and alternating regions of ascending and descending fluid. These structures, driven by complex interactions between local clustering and global flow organisation, which persist throughout the sedimentation process, are highly sensitive to $\xi$. Despite its inner complex dynamics, the group is observed to sediment as a collective entity, with a constant velocity exceeding that of an isolated cylinder. This velocity may be predicted from multi-scale information. Fluctuating velocities of the objects are further analysed. Different mechanisms for horizontal and vertical components are identified. Horizontal fluctuations are related to intrinsic particle mobility, while vertical fluctuations are attributed to strong wakes and vertical streams. Both fluctuations are mainly influenced by the cylinders’ aspect ratio, which also affects the structural and spatial distribution of the objects.