We investigate by direct numerical simulations the fluid–solid interaction of non-dilute suspensions of spherical particles moving in triperiodic turbulence, at the relatively large Reynolds number of $Re_\lambda \approx 400$. The solid-to-fluid density ratio is varied between $1.3$ and $100$, the particle diameter $D$ is in the range $16 \le D/\eta \le 123$ ($\eta$ is the Kolmogorov scale) and the volume fraction of the suspension is $0.079$. Turbulence is sustained using the Arnold–Beltrami–Childress cellular-flow forcing. The influence of the solid phase on the largest and energetic scales of the flow changes with the size and density of the particles. Light and large particles modulate all scales in an isotropic way, while heavier and smaller particles modulate the largest scales of the flow towards an anisotropic state. Smaller scales are isotropic and homogeneous for all cases. The mechanism driving the energy transfer across scales changes with the size and the density of the particles. For large and light particles the energy transfer is only marginally influenced by the fluid–solid interaction. For small and heavy particles, instead, the classical energy cascade is subdominant at all scales, and the energy transfer is essentially driven by the fluid–solid coupling. The influence of the solid phase on the flow intermittency is also discussed. Besides, the collective motion of the particles and their preferential location in relation to properties of the carrier flow are analysed. The solid phase exhibits moderate clustering; for large particles the level of clustering decreases with their density, while for small particles it is maximum for intermediate values.

Pressure-driven flows of viscoelastic fluids in narrow non-uniform geometries are common in physiological flows and various industrial applications. For such flows, one of the main interests is understanding the relationship between the flow rate $q$ and the pressure drop $\Delta p$, which, to date, is studied primarily using numerical simulations. We analyse the flow of the Oldroyd-B fluid in slowly varying arbitrarily shaped, contracting channels and present a theoretical framework for calculating the $q-\Delta p$ relation. We apply lubrication theory and consider the ultra-dilute limit, in which the velocity profile remains parabolic and Newtonian, resulting in a one-way coupling between the velocity and polymer conformation tensor. This one-way coupling enables us to derive closed-form expressions for the conformation tensor and the flow rate–pressure drop relation for arbitrary values of the Deborah number ($De$). Furthermore, we provide analytical expressions for the conformation tensor and the $q-\Delta p$ relation in the high-Deborah-number limit, complementing our previous low-Deborah-number lubrication analysis. We reveal that the pressure drop in the contraction monotonically decreases with $De$, having linear scaling at high Deborah numbers, and identify the physical mechanisms governing the pressure drop reduction. We further elucidate the spatial relaxation of elastic stresses and pressure gradient in the exit channel following the contraction and show that the downstream distance required for such relaxation scales linearly with $De$.

We consider two-dimensional flows above topography, revisiting the selective decay (or minimum enstrophy) hypothesis of Bretherton and Haidvogel. We derive a ‘condensed branch’ of solutions to the variational problem where a domain-scale condensate coexists with a flow at the (smaller) scale of the topography. The condensate arises through a supercritical bifurcation as the conserved energy of the initial condition exceeds a threshold value, a prediction that we quantitatively validate using direct numerical simulations. We then consider the forced–dissipative case, showing how weak forcing and dissipation select a single dissipative state out of the continuum of solutions to the energy-conserving system predicted by selective decay. As the forcing strength increases, the condensate arises through a supercritical bifurcation for topographic-scale forcing and through a subcritical bifurcation for domain-scale forcing, both predictions being quantitatively validated by direct numerical simulations. This method provides a way of determining the equilibrated state of forced–dissipative flows based on variational approaches to the associated energy-conserving system, such as the statistical mechanics of two-dimensional flows or selective decay.

The Holton–Lindzen–Plumb (HLP) model of the quasi-biennial oscillation (QBO) is investigated in order to assess the impact of introducing intermittency in the wave forcing. Intermittency is introduced to HLP by allowing the amplitude of the waves which force the QBO to evolve according to a stationary random process, driven by a stochastic differential equation (SDE) with an associated time scale $\tau$. Provided that $\tau$ is much shorter than the QBO period, it is shown that the impact on the QBO of the intermittent forcing is captured by a single intermittency parameter $\lambda$, and the value of $\lambda$ is proportional to $\tau$ and otherwise depends upon the details of the SDE. Numerical simulations, using a family of mean-reverting Ornstein–Uhlenbeck processes as the choice of SDE, show that the effect of increasing the intermittency parameter is invariably to decrease the QBO amplitude and increase its period. Changes to the QBO amplitude and period are indeed found to collapse onto a single curve controlled by $\lambda$, as predicted by the theory, provided that $\tau$ is small enough for the approximations used to be valid. The extension to broadband forcing is discussed in the context of stochastic gravity wave parameterisation, with the eventual goal of developing a representation of source intermittency in the most general situation with close fidelity to the physics.

Lubrication theory is adapted to incorporate the large normal stresses that occur for order-one Deborah numbers, $De$, the ratio of the relaxation time to the residence time. Comparing with the pressure drop for a Newtonian viscous fluid with a viscosity equal to that of an Oldroyd-B fluid in steady simple shear, we find numerically a reduced pressure drop through a contraction and an increased pressure drop through an expansion, both changing linearly with $De$ at high $De$. For a constriction, there is a smaller pressure drop that plateaus at high $De$. For a contraction, much of the change in pressure drop occurs in the stress relaxation in a long exit channel. An asymptotic analysis for high $De$, based on the idea that normal stresses are stretched by an accelerating flow in proportion to the square of the velocity, reveals that the large linear changes in pressure drop are due to higher normal stresses pulling the fluid through the narrowest gap. A secondary cause of the reduction is that the elastic shear stresses do not have time to build up to their steady-state equilibrium value while they accelerate through a contraction. We find for a contraction or expansion that the high $De$ analysis works well for $De>0.4$.

Analysing the dynamics of velocity gradients is useful for understanding various nonlinear turbulence processes. This work focuses on how the vibrational non-equilibrium of the constituent molecules in a gaseous medium affects the dynamics of the velocity gradient and the pressure Hessian tensors. We first derive the exact evolution equation of the pressure-Hessian tensor in the presence of the vibrational non-equilibrium process. Subsequently, we perform several direct numerical simulations of compressible isotropic turbulence, including the vibrational relaxation process therein. Using flow fields extracted from these simulations, we conduct several parametric studies over a range of the Damköhler number (ratio of the relevant fluid time scale to that of the mean vibrational relaxation process) and the initial ratio of the vibrational temperature to the mean local temperature. We find that a variation in the initial Damköhler number does influence the evolution of the pressure-Hessian and the velocity gradient tensors. As the vibrational relaxation process becomes more rapid (an increase in the value of the initial Damköhler number), it causes a decrease in the strength of the pressure-Hessian tensor and simultaneous suppression of dilatational fluctuations in the flow field. On the other hand, a variation in the initial value of the ratio of the vibrational temperature to the local temperature does not seem to affect the pressure-Hessian or the velocity gradient tensor. These findings are expected to aid in the development of closure models for the pressure-Hessian tensor in compressible flows under vibrational non-equilibrium conditions.

Microorganisms are ubiquitous in nature and technology. They inhabit diverse environments, ranging from small river tributaries and lakes, to oceans, as well as wastewater treatment plants and food manufacturing. In many of these environments, microorganisms coexist with settling particles. Here, we investigate the effects of microbial activity (swimming E. coli) on the settling dynamics of passive colloidal particles using particle tracking methods. Our results reveal the existence of two distinct regimes in the correlation length scale ($L_u$) and the effective diffusivity of the colloidal particles ($D_{eff}$), with increasing bacterial concentration ($\phi _b$). At low $\phi _b$, the parameters $L_u$ and $D_{eff}$ increase monotonically with increasing $\phi _b$. Beyond critical $\phi _b$, a second regime is found where both $D_{eff}$ and $L_u$ are independent of $\phi _b$. We demonstrate that the transition between these regimes is characterized by the emergence of bioconvection. We use experimentally measured particle-scale quantities $L_u$ and $D_{eff}$ to predict the critical bacterial concentration for the diffusion–bioconvection transition.

A global transient linear stability analysis of the three-dimensional time-dependent flow around an aerofoil undergoing small-amplitude pitching motion is performed using the optimally time-dependent (OTD) framework. The most salient linear instabilities associated with the instantaneous basic state are computed and tracked over time. The resulting OTD modes reflect the variations in the basic state and can be used as predictors of its spatial and temporal evolution, including the formation of a laminar separation bubble and its gradual spanwise modulation via primary global instability, leading to secondary instability and finally rapid breakdown to turbulence. The study confirms and expands upon earlier stability analyses of the same case based on the local properties of spanwise averaged velocity profiles in the bubble that predicted the onset of absolute instability soon followed by rapid breakdown of the separation bubble. The three-dimensional structure of the most unstable OTD mode is extracted, which compares well with both the locally absolutely unstable mode and the evolution of the basic state itself.

When a sphere rotates near a rigid boundary coated with a thin layer of viscous liquid, ‘tracks’ are generated both behind and over the sphere. This paper describes a theory for the simplest one-track configuration which can occur under particular experimental conditions. The theoretical predictions are in good agreement with experimental observations.

We study the stability of a zero-pressure gradient boundary layer subjected to free-stream disturbances by means of local stability analysis. The dataset under study corresponds to a direct numerical simulation (DNS) of a flat plate with a sharp leading edge in realistic wind tunnel conditions, with a turbulence level of 3.45 % at the leading edge. We present a method to track the convective evolution of the secondary instabilities of streaks by performing sequential stability calculations following the wave packet, connecting successive unstable eigenfunctions. A scattered nature, in time and space, of secondary instabilities is seen in the stability calculations. These instabilities can be detected before they reach finite amplitude in the DNS, preceding the nucleation of turbulent spots, and whose appearance is well correlated to the transition onset. This represents further evidence regarding the relevance of secondary instabilities of streaks in the bypass transition in realistic flow conditions. Consistent with the spatio-temporal nature of this problem, our approach allows us to integrate directly the local growth rates to obtain the spatial amplification ratio of the individual instabilities, where it is shown that instabilities reaching an $N$-factor in the range [2.5,4] can be directly correlated to more than 65 % of the nucleation events. Interestingly, it is found that high amplification is not only attained by modes with high growth rates, but also by instabilities with sustained low growth rates for a long time.

In 1945–1949, Lars Onsager made an exact analysis of the high-Reynolds-number limit for individual turbulent flow realisations modelled by incompressible Navier–Stokes equations, motivated by experimental observations that dissipation of kinetic energy does not vanish. I review here developments spurred by his key idea that such flows are well described by distributional or ‘weak’ solutions of ideal Euler equations. 1/3 Hölder singularities of the velocity field were predicted by Onsager and since observed. His theory describes turbulent energy cascade without probabilistic assumptions and yields a local, deterministic version of the Kolmogorov $4/5$th law. The approach is closely related to renormalisation group methods in physics and envisages ‘conservation-law anomalies’, as discovered later in quantum field theory. There are also deep connections with large-eddy simulation modelling. More recently, dissipative Euler solutions of the type conjectured by Onsager have been constructed and his $1/3$ Hölder singularity proved to be the sharp threshold for anomalous dissipation. This progress has been achieved by an unexpected connection with work of John Nash on isometric embeddings of low regularity or ‘convex integration’ techniques. The dissipative Euler solutions yielded by this method are wildly non-unique for fixed initial data, suggesting ‘spontaneously stochastic’ behaviour of high-Reynolds-number solutions. I focus in particular on applications to wall-bounded turbulence, leading to novel concepts of spatial cascades of momentum, energy and vorticity to or from the wall as deterministic, space–time local phenomena. This theory thus makes testable predictions and offers new perspectives on large-eddy simulation in the presence of solid walls.

We investigate the global instability mechanism of the flow past a three-dimensional stepped cylinder. A comprehensive study is performed for different diameter ratios of the two joined cylinders ($r=D/d$) ranging from $r=1.1$ to $r=4$. Independently of $r$, the spectrum of the linearised Navier–Stokes operator reveals a pair of complex conjugate eigenvalues, with Strouhal number $St \approx 0.11$. The initial transition is triggered by a two-dimensional mechanism of the larger cylinder only, not affected by the presence of the junction and the smaller cylinder ($Re_{D,cr}\approx 47$). The structural sensitivity analysis is used to identify where the instability mechanism acts. The onset of transition is solely localised in the large cylinder wake (L cell), where the wavemaker has two symmetric lobes across the separation bubble. When the Reynolds number increases, a second and a third unstable pair of complex conjugate eigenvalues appears. They are localised in the small cylinder (S) wake and modulation (N) region. For any $r$, the appearance of unstable eigenmodes resembling the three cells S–N–L in the wake is observed. The nonlinear simulation results support this finding, in contrast with the previous classification of the laminar vortex shedding in direct (L–S) and indirect (L–N–S) modes interaction Lewis & Gharib (Phys. Fluids, vol. 4, 1992, pp. 104–117). This result indicates that each cell undergoes a supercritical Hopf bifurcation for any $r$. As $r$ approaches $1$, the modal linear stability results also show an unstable eigenmode in the wake of the small cylinder resembling a new modulation cell, named N2, similar to the N cell but mirrored with respect to the junction plane.

Spray formed by a myriad of secondary droplets generated by the impact of raindrops on a deep-water pool is studied with a laboratory rain facility. Experiments are performed with two rain rates and raindrops fall on the water surface at a nearly constant velocity. The secondary droplets at various heights above the pool's water surface are recorded with a cinematic digital in-line holographic technique that consists of a high-speed camera, a pulsed Nd:YLF laser and associated optics. The experimental results show that in the heat-map scatter plots of radius versus velocity near the water surface of the pool, the droplets are distributed into three regions, corresponding to distinct physical mechanisms of droplet generation. It is found that the diameter distribution of the droplets in the rain field changes with height above the pool's water surfaces. Both numerical simulation and experimental data reveal that the liquid water content, due to the presence of secondary droplets, in the atmospheric surface layer decreases exponentially with increasing height.

We present data from direct numerical simulation (DNS) of the fully turbulent flow through nominally two-dimensional channels containing longitudinal, surface-mounted, rectangular ribs whose widths ($W$) are either one third of or equal to the gap ($S-W$) between consecutive ribs across the domain, where $S$ is the span (centre-to-centre spacing) of the ribs. A range of the ratio of channel half-height ($H$) to span ($S$) is considered, covering $0.25\le H/S\le 2.5$. In each case, a fixed rib height ($h$) of $0.1H$ was used, but a number of cases with much smaller heights, $h/H=0.025$ or 0.05, were also studied. The secondary flows resulting from the presence of the ribs are examined, along with their sources in terms of the axial vorticity transport equation, which highlights the effects of spanwise inhomogeneity in the Reynolds stresses. We show that the strength of the secondary flows depends strongly on $H/S$ (and, correspondingly, on $W/S$) and that the major sources of axial vorticity arise near the top corners of the ribs, with convection of that vorticity dominating its spread. We also show that for smaller ribs, the secondary flow strengths are similar to those predicted by Zampino et al. (J. Fluid Mech., vol. 944, 2022, A4) using a linearised model of the Reynolds-averaged equations, which does not include the vorticity convection process; the behaviour of secondary flow topology and strength with varying $W/H$ is thus noticeably different.

A theory is presented for wave-driven propulsion of floating bodies driven into oscillation at the fluid interface. By coupling the equations of motion of the body to a quasipotential flow model of the fluid, we derive expressions for the drift speed and propulsive thrust of the body which in turn are shown to be consistent with global momentum conservation. We explore the efficacy of our model in describing the motion of SurferBot (Rhee et al., Bioinspir. Biomim., vol. 17, issue 5, 2022), demonstrating close agreement with the experimentally determined drift speed and oscillatory dynamics. The efficiency of wave-driven propulsion is then computed as a function of driving oscillation frequency and the forcing location, revealing optimal values for both of these parameters which await confirmation in experiments. A comparison with other modes of locomotion and applications of our model with competitive water sports is discussed in conclusion.

We study self-similar viscous fingering for the case of divergent flow within a wedge-shaped Hele-Shaw cell. Previous authors have conjectured the existence of a countably infinite number of selected solutions, each distinguished by a different value of the relative finger angle. Interestingly, the associated solution branches have been posited to merge and disappear in pairs as the surface tension decreases. For the first time, we demonstrate how the selection mechanism can be derived based on exponential asymptotics. Asymptotic predictions of the finger-to-wedge angle are additionally given for different sized wedges and surface-tension values. The merging of solution branches is explained; this feature is qualitatively different to the case of classic Saffman–Taylor viscous fingering in a parallel channel configuration. Moreover, because the asymptotic framework does not highly depend on specifics of the wedge geometry, the proposed theory for branch merging in our self-similar problem likely relates much more widely to tip-splitting instabilities in time-dependent flows in circular and other geometries, where the viscous fingers destabilise and divide in two.

The present study investigates the modal stability of the steady incompressible flow inside a toroidal pipe for values of the curvature $\delta$ (ratio between pipe and torus radii) approaching zero, i.e. the limit of a straight pipe. The global neutral stability curve for $10^{-7} \leq \delta \leq ~10^{-2}$ is traced using a continuation algorithm. Two different families of unstable eigenmodes are identified. For curvatures below $1.5 \times 10^{-6}$, the critical Reynolds number ${{Re}}_{cr}$ is proportional to $\delta ^{-1/2}$. Hence, the critical Dean number is constant, ${{De}}_{cr} = 2\,{{Re}}_{cr}\,\sqrt {\delta } \approx 113$. This behaviour confirms that the Hagen–Poiseuille flow is stable to infinitesimal perturbations for any Reynolds number and suggests that a continuous transition from the curved to the straight pipe takes place as far as it regards the stability properties. For low values of the curvature, an approximate self-similar solution for the steady base flow can be obtained at a fixed Dean number. Exploiting the proposed semi-analytic scaling in the stability analysis provides satisfactory results.

Rain drops form in clouds by collision of submillimetric droplets falling under gravity: larger drops fall faster than smaller ones and collect them on their path. The puzzling stability of fogs and non-precipitating warm clouds with respect to this avalanche mechanism has been a longstanding problem. How can droplets of diameter around 10 $\mathrm {\mu }$m have a low collision probability, inhibiting the cascade towards larger and larger drops? Here we review the dynamical mechanisms that have been proposed in the literature and quantitatively investigate the frequency of drop collisions induced by Brownian diffusion, electrostatics and gravity, using an open-source Monte Carlo code taking all of them into account. Inertia dominates over aerodynamic forces for large drops, when the Stokes number is larger than $1$. Thermal diffusion dominates over aerodynamic forces for small drops, when the Péclet number is smaller than $1$. We show that there exists a range of size (typically 3–30 $\mathrm {\mu }$m for water drops in air) where neither inertia nor Brownian diffusion are significant, leading to a gap in the collision rate. The effect is particularly important, due to the lubrication film forming between the drops immediately before collision, and secondarily to the long-range aerodynamic interaction. Two different mechanisms regularise the divergence of the lubrication force at vanishing separation: the transition to a non-continuum regime in the lubrication film, when the separation is comparable to the mean free path of air, and the induction of a flow inside the drops due to shear at their surfaces. In the gap between inertia-dominated and diffusion-dominated regimes, dipole–dipole electrostatic interactions becomes the major effect controlling the efficiency of drop collisions.

Turbulence in gravity-driven film flow is usually discussed in terms of three-dimensional solitary-wave pulses as they are frequently observed in flows along smooth walls. Here, we show that free-surface turbulence arises in films along rippled substrates, as they are commonly employed in process engineering applications, already at rather low Reynolds numbers from the irregular break-up of the solitary-wave fronts. Short waves in the capillary regime replace the broken solitary waves beyond a certain Reynolds number. The crossover coincides with the occurrence of steady three-dimensional surface patterns upstream, which suppress travelling waves. The waves show power spectral densities with power-law exponents typical for weak capillary-wave turbulence. With increasing Reynolds number, the steepness of the power law declines to lower levels.

Two simulations of turbulent Couette flows were performed at friction Reynolds numbers of 1000 and 2000 in a large box of dimensions $L_x=16{\rm \pi} h$, $L_y=2h$ and $L_z=6{\rm \pi} h$, where h is the semi-height of the channel. The study focuses on the differences in the intensity and scaling of turbulence at these two Reynolds numbers. The 2000 case showed a lack of a clear log layer with a higher value of the Von Kármán constant $\kappa$ than Poiseuille channels. The intensities were well-scaled in the buffer layer and below, with a second maximum of the streamwise intensity at approximately 350 wall units. Contrary to Poiseuille channels, the dissipation scales close to the wall in wall units. This fact can be attributed to the constant value of the derivative of the streamwise intensity in wall units. The intensities of the 2000 case showed remarkable differences compared with those at Reynolds number 1000 at the channel centre, likely due to the organization of large scales of the streamwise fluctuactions, $u$. These large scales were thought to be considered ‘infinite’. However, for the 2000 case, while all the structures have a width of $\ell _z \approx 6/8{\rm \pi} h$, their length varies from $\ell _x \approx 6{\rm \pi} h$ to $\ell _x \approx 16{\rm \pi} h$, which clearly contradicts the trends obtained in the past. This is a new effect that has not been reported for turbulent Couette flow and points to the uncertainty and sensitivity that is observed for certain statistical quantities.