We investigate the dynamics of an oscillatory boundary layer developing over a bed of collisional and freely evolving sediment grains. We perform Euler–Lagrange simulations at Reynolds numbers ${\textit{Re}}_\delta = 200$, 400 and 800, density ratio $\rho _{\!p}/\rho _{\!f} = 2.65$, Galileo number ${\textit{Ga}} = 51.9$, maximum Shields numbers from $5.60 \times 10^{-2}$ to $2.43 \times 10^{-1}$, based on smooth wall configuration, and Keulegan–Carpenter number from $134.5$ to $538.0$. We show that the dynamics of the oscillatory boundary layer and sediment bed are strongly coupled due to two mechanisms: (i) bed permeability, which leads to flow penetration deep inside the sediment layer, a slip velocity at the bed–fluid interface, and the expansion of the boundary layer, and (ii) particle motion, which leads to rolling-grain ripples at ${\textit{Re}}_\delta = 400$ and ${\textit{Re}}_\delta = 800$. While at ${\textit{Re}}_\delta = 200$ the sediment bed remains static during the entire cycle, the permeability of the bed–fluid interface causes a thickening of the boundary layer. With increasing ${\textit{Re}}_\delta$, the particles become mobile, which leads to rolling-grain ripples at ${\textit{Re}}_\delta = 400$ and suspended sediment at ${\textit{Re}}_\delta = 800$. Due to their feedback force on the fluid, the mobile sediment particles cause greater velocity fluctuations in the fluid. Flow penetration causes a progressive alteration of the fluid velocity gradient near the bed interface, which reduces the Shields number based upon bed shear stress.

The inertial migration of neutrally buoyant spherical particles in viscoelastic fluids flowing through square channels is experimentally and numerically studied. In the experiments, using dilute aqueous solutions of polymers with various concentrations that have nearly constant viscosities, we measured the distribution of suspended particles in downstream cross-sections for the Reynolds number ($\textit{Re}$) up to 100 and the elasticity number ($El$) up to 0.07. There are several focusing patterns of the particles, such as four-point focusing near the centre of the channel faces on the midlines for low $\textit{El}$ and/or high $\textit{Re}$, four-point focusing on the diagonals for medium $\textit{El}$, single-point focusing at the channel centre for relatively high $\textit{El}$ and low $\textit{Re}$, and five-point focusing near the four corners and the channel centre for high $\textit{El}$ and very low $\textit{Re}$. Among these focusing patterns, various types of particle distributions suggesting the presence of a new equilibrium position located between the midline and the diagonal, and multistable states of different equilibrium positions were observed. In general, as $\textit{El}$ increases from 0 at a constant $\textit{Re}$, the particle focusing positions shift from the midline to the diagonal in the azimuthal direction first, and then inward in the radial direction to the channel centre. These focusing patterns and their transitions were numerically well reproduced based on a FENE-P model with measured values of viscosity and relaxation time. Using the numerical results, the experimentally observed focusing patterns of particles are elucidated in terms of the fluid elasticity-induced lift and the wall-induced elastic lift.

This study is devoted to the analysis of capillary oscillations of a gas bubble in a liquid with an insoluble surfactant adsorbed on the surface. The influence of the Gibbs elasticity, the viscosities of the liquid and gas, as well as the shear and dilatational surface viscosities, on the damping of free oscillations is examined. Dependences of the frequency shift and the damping rate on the parameters of the problem are determined. In the limit of small viscosities and neglecting the surfactant surface diffusion, a simplified dispersion relation is obtained, which includes finite parameters of surface viscosities and Gibbs elasticity. From this relation, conditions are identified under which the damping of capillary oscillations can occur with a small frequency. Numerical solutions of the full dispersion relation demonstrate that a non-oscillatory regime is impossible for the considered configuration. An additional mode associated with Gibbs elasticity is discovered, characterized as a rule by low natural frequency and damping rate. Approximate relations for the complex natural frequency of bubble oscillations in a low-viscosity liquid in the presence of a surfactant are derived, including an estimate of the contribution of the gas inside the bubble to viscous dissipation. An original Lagrangian–Eulerian method is proposed and used to perform direct numerical simulations based on the full nonlinear Navier–Stokes equations and natural boundary conditions at the interface, accounting for shear and dilatational viscosities. The numerical data on the damping process confirm the results of the linear theory.

We conduct three-dimensional numerical simulations on centrifugal convection (CC) in a closed annular container, incorporating gravity and no-slip top and bottom boundaries, to systematically investigate rotation-induced secondary flow. The Stewartson layer, identified by an elongated circulation in mean vertical velocity plots, emerges near the inner and outer cylinders only beyond a critical gravitational forcing. Quantitative analyses confirm that the layer thickness scales as $\delta _{\,\!\textit{st}}\sim {\textit{Ek}}^{1/3}$ due to rotational effects, consistent with results from rotating Rayleigh–Bénard convection, where $Ek$ represents the Ekman number. The internal circulation strength, however, is determined by both gravitational buoyancy and rotational effects. We propose that gravitational buoyancy drives the internal flow, which balances against viscous forces to establish a terminal velocity. Through theoretical analysis, the vertical velocity amplitude follows $W_{\,\!\textit{st}}\sim {\textit{Ek}}^{5/3}\,Ro^{-1}\,{\textit{Ra}}_g\,Pr^{-1}$, showing good agreement with simulation results across a wide parameter range. Here, $Ro^{-1}$ represents the inverse Rossby number, ${\textit{Ra}}_g$ is the gravitational Rayleigh number, and ${\textit{Pr}}$ is the Prandtl number. The theoretical predictions match simulations well, demonstrating that the Stewartson layer is gravity-induced and rotationally constrained through geostrophic balance in the CC system. These findings yield fundamental insights into turbulent flow structures and heat transfer mechanisms in the CC system, offering both theoretical advances and practical engineering applications.

Vertical thermal convection exhibits weak turbulence and spatio-temporally chaotic behaviour. For this configuration, we report seven new equilibria and 26 new periodic orbits. These orbits, together with four previously studied in Zheng et al. (J. Fluid Mech., 2024b, vol. 1000, p. A29) bring the number of periodic-orbit branches computed so far to 30, all solutions to the fully nonlinear three-dimensional Navier–Stokes equations. These new and unstable invariant solutions capture intricate spatio-temporal flow patterns including straight, oblique, wavy, skewed and distorted convection rolls, as well as bursts and defects. These interesting and important fluid mechanical processes in a small flow unit are shown to also appear locally and instantaneously in a chaotic simulation in a large domain. Most of the solution branches show rich spatial and/or spatio-temporal symmetries. The bifurcation-theoretic organisation of these solutions is discussed; the bifurcation scenarios include Hopf, pitchfork, saddle-node, period-doubling, period-halving, global homoclinic and heteroclinic bifurcations, as well as isolas. Furthermore, these orbits are shown to be able to reconstruct statistically the core part of the attractor, so that these results may contribute to a quantitative description of transitional fluid turbulence using periodic orbit theory.

In this paper, we showcase how flow obstruction by a deformable object can lead to symmetry breaking in curved domains subject to angular acceleration. Our analysis is motivated by the deflection of the cupula, a soft tissue located in the inner ear that is used to perceive rotational motion as part of the vestibular system. The cupula is understood to block the rotation-induced flow in a toroidal region with the flow-induced deformation of the cupula used by the brain to infer motion. By asymptotically solving the governing equations for this flow, we characterise regimes for which the sensory system is sensitive to either angular velocity or angular acceleration. Moreover, we show the fluid flow is not symmetric in the latter case. Finally, we extend our analysis of symmetry breaking to understand the formation of vortical flow in cavernous regions within channels. We discuss the implications of our results for the sensing of rotation by mammals.

Turbulent Rayleigh–Bénard convection in an extended layer of square cross-section with moderate aspect ratio $L/H=8.6$ ($L$ is the length of the cell, $H$ is its height) is studied numerically for Rayleigh numbers in the range ${\textit{Ra}}= 10^6{-}10^8$. We focus on the influence of different types of boundary conditions, including asymmetrical ones, on the characteristics of Rayleigh–Bénard convection with and without an immersed freely floating body. Convection without a floating body is characterised by the formation of stable thermal superstructures with preferred location. The crucial role of the symmetry of the boundary conditions is revealed. In the case of thermal boundary conditions of different types at the upper and lower boundaries, the flow pattern in Rayleigh–Bénard convection has a regular shape. The immersed body makes random wanderings and actively mixes the fluid, preventing the formation of superstructures. The mean flow structure with an immersed body is similar for all combinations of boundary conditions except for the case of a fixed heat flux at both boundaries. The floating disk does not change the tendency of turbulent convection to form a circulation of the maximal available scale under symmetric Neumann-type conditions. The type of boundary conditions has a weak influence on the Nusselt and Reynolds number values, significantly changing the ratio of the mean and fluctuating components of the heat flux. As the Rayleigh number increases, the motions of the body become more intensive and intermittent. The increase of $Ra$ also changes the structure of the mean flow without the body but the additional mixing provided by the floating body preserves the flow structure.

We performed three-dimensional simulations to study the motion and interaction of microswimmers (pulling- and pushing-type squirmers) and spheres for Reynolds numbers ranging from 0.01 to 1 under conditions in which all particles were axially aligned with each other. We show that pullers are attractive and pushers are repulsive, in terms of the pressure at the front and rear of the squirmers. Correspondingly, the pullers always come close to each other and form a string that swims slightly faster than does a single puller. A possible reason for this finding is discussed. In contrast, whether a leading puller touches a trailing pusher depends primarily on its strength. When the two have similar strengths, they come into contact and form a stable doublet with finite inertia. The speed of the doublet is substantially higher than that of a single pusher owing to the additional force stemming from the fore and aft pressure differences of the doublet. We also demonstrate how a leading pusher interacts with a trailing puller, which is quite different. In contrast, a sphere can be directly or hydrodynamically ‘pushed’ to run by a puller or a pusher. In particular, we reveal that the sphere exhibits the highest speed when ‘pulled’ by a leading puller and ‘pushed’ by a trailing pusher simultaneously. Grouping behaviours reflect the interacting nature of the microswimmers and spheres from different aspects. A bunch of pushers/pullers eventually appears in pairs or forms a string depending on the Reynolds number, similar to groups of pushers/spheres and pullers/spheres.

In the standard picture of fully developed turbulence, highly intermittent hydrodynamic fields are nonlinearly coupled across scales, where local energy cascades from large scales into dissipative vortices and large density gradients. Microscopically, however, constituent fluid molecules are in constant thermal (Brownian) motion, but the role of molecular fluctuations in large-scale turbulence is largely unknown, and with rare exceptions, it has historically been considered irrelevant at scales larger than the molecular mean free path. Recent theoretical and computational investigations have shown that molecular fluctuations can impact energy cascade at Kolmogorov length scales. Here, we show that molecular fluctuations not only modify energy spectrum at wavelengths larger than the Kolmogorov length in compressible turbulence, but also significantly inhibit spatio-temporal intermittency across the entire dissipation range. Using large-scale direct numerical simulations of computational fluctuating hydrodynamics, we demonstrate that the extreme intermittency characteristic of turbulence models is replaced by nearly Gaussian statistics in the dissipation range. These results demonstrate that the compressible Navier–Stokes equations should be augmented with molecular fluctuations to accurately predict turbulence statistics across the dissipation range. Our findings have significant consequences for turbulence modelling in applications such as astrophysics, reactive flows and hypersonic aerodynamics, where dissipation-range turbulence is approximated by closure models.

Deformable microchannels emulate a key characteristic of soft biological systems and flexible engineering devices: the flow-induced deformation of the conduit due to slow viscous flow within. Elucidating the two-way coupling between oscillatory flow and deformation of a three-dimensional (3-D) rectangular channel is crucial for designing lab-on-a-chip and organ-on-a-chip microsystems and eventually understanding flow–structure instabilities that can enhance mixing and transport. To this end, we determine the axial variations of the primary flow, pressure and deformation for Newtonian fluids in the canonical geometry of a slender (long) and shallow (wide) 3-D rectangular channel with a deformable top wall under the assumption of weak compliance and without restriction on the oscillation frequency (i.e. on the Womersley number). Unlike rigid conduits, the pressure distribution is not linear with the axial coordinate. To validate this prediction, we design a polydimethylsiloxane-based experimental platform with a speaker-based flow-generation apparatus and a pressure acquisition system with multiple ports along the axial length of the channel. The experimental measurements show good agreement with the predicted pressure profiles across a wide range of the key dimensionless quantities: the Womersley number, the compliance number and the elastoviscous number. Finally, we explore how the nonlinear flow–deformation coupling leads to self-induced streaming (rectification of the oscillatory flow). Following Zhang and Rallabandi (J. Fluid Mech., vol. 996, 2024, p. A16), we develop a theory for the cycle-averaged pressure based on the primary problem’s solution, and we validate the predictions for the axial distribution of the streaming pressure against the experimental measurements.

We explore the fundamental flow structure of temporally evolving inclined gravity currents with direct numerical simulations. A velocity maximum naturally divides the current into inner and outer shear layers, which are weakly coupled by momentum and buoyancy exchanges on time scales that are much longer than the typical time scale characterising either layer. The outer layer evolves to a self-similar state and can be described by theory developed for a current on a free-slip slope (Van Reeuwijk et al. 2019, J. Fluid Mech., vol. 873, pp. 786–815) when expressed in terms of outer-layer properties. The inner layer evolves to a quasi-steady state and is essentially unstratified for shallow slopes, with flow statistics that are virtually indistinguishable from fully developed open channel flow. We present the classic buoyancy–drag force balance proposed by Ellison & Turner (1959, J. Fluid Mech., vol. 6, pp. 423–448) for each layer, and find that buoyancy forces in the outer layer balance entrainment drag, while buoyancy forces in the inner layer balance wall friction drag. Using scaling laws within each layer and a matching condition at the velocity maximum, the entire flow system can be solved as a function of the slope angle, in good agreement with the simulation data. We further derive an entrainment law from the solution, which exhibits relatively high accuracy across a wide range of Richardson numbers, and provides new insights into the long runout of oceanographic gravity currents on mild slopes.

Surface roughness of fairly small (micron-sized) height is known to influence significantly three-dimensional boundary-layer transition. In this paper, we investigate this sensitive effect from the viewpoint that roughness alters the base flow thereby inducing new instabilities. We consider distributed roughness in the form of a wavy wall with its height being taken to be of $\mathit{O} (R^{-1/3 } \delta ^{\ast })$, where the Reynolds number $R$ is defined using the local boundary-layer thickness $\delta ^{\ast }$. Despite having a height much smaller than $\delta ^{\ast }$, the roughness is high enough to induce nonlinear responses. The roughness-distorted boundary-layer flow is characterised by a wall layer (WL) – a thin layer adjacent to the surface – the main layer and a critical layer (CL) – the vicinity of a special position at which a singularity of the Rayleigh equation occurs. The widths of both the WL and CL are of $\mathit{O} (R^{-1/3} \delta ^{\ast })$. Surface roughness alters the base flow significantly, leading to $\mathit{O} (1)$ vorticity distortions in these layers. We show for the first time that the nonlinearly distorted flows in these layers support small-scale local instabilities due to the roughness-induced $\mathit{O} (1)$ vorticities. Two types of modes, CL and WL modes, are identified. The CL modes have short wavelengths and high frequencies, with the spatial and temporal instabilities being governed by essentially the same equation. Thus, we focus on the former, which can be formulated as a linear generalised eigenvalue problem. The WL modes have short wavelengths but $\mathit{O} (1)$ frequencies. The temporal WL mode is governed by a linear eigenvalue problem similar to that for the CL modes, while the spatial WL mode is described by a nonlinear eigenvalue problem. The onset of these small-scale fluctuations could form a crucial step in the transition to turbulence.

Transonic buffet is a complex and strongly nonlinear unstable flow sensitive to variations in the incoming flow state. This poses great challenges for establishing accurate-enough reduced-order models, limiting the application of model-based control strategies in transonic buffet control problems. To address these challenges, this paper presents a time-variant modelling approach that incorporates rolling sampling, recursive parameter updating and inner iteration strategies under dynamic incoming flow conditions. The results demonstrate that this method successfully overcomes the difficulty in designing appropriate training signals and obtaining unstable steady base flow. Additionally, it improves the global predictive capability and identification efficiency of linear models for nonlinear flow-system responses by more than one order of magnitude. Furthermore, two adaptive control strategies – minimum variance control and generalised predictive control – are validated as effective based on the time-variant reduced-order model through numerical simulations of the transonic buffet flow over the NACA 0012 aerofoil. The adaptive controllers effectively regulate the unstable eigenvalues of the flow system, achieving the desired control outcomes. They ensure that the shock wave buffet phenomenon does not recur after control is applied, and that the actuator deflection, specifically the trailing-edge flap, returns to zero. Moreover, the control results further confirm the global instability essence of transonic buffet flow from a control perspective, thereby deepening the cognition of this nonlinear unstable flow.

In this paper, we study experimentally the dispersion of colloids in a two-dimensional, time-independent, Rayleigh–Bénard flow in the presence of salt gradients. Due to the additional scalar, the colloids do not follow exactly the Eulerian flow field, but have a (small) extra velocity $\boldsymbol{v}_{{dp}} = D_{{dp}}\, \boldsymbol{\nabla }\log C_s$, where $D_{{dp}}$ is the phoretic constant, and $C_s$ is the salt concentration. Such a configuration is motivated by the theoretical work by Volk et al. (2022, J. Fluid Mech., vol. 948, A42), which predicted enhanced transport or blockage in a stationary cellular flow depending on the value of a blockage coefficient. By means of high dynamical range light-induced fluorescence, we study the evolution of the colloids concentration field at large Péclet number. We find good agreement with the theoretical work, although a number of hypotheses are not satisfied, as the experiment is non-homogeneous in space, and intrinsically transient. In particular, we observe enhanced transport when salt and colloids are injected at both ends of the Rayleigh–Bénard chamber, and blockage when colloids and salt are injected together and phoretic effects are strong enough.

A heaving and pitching wing encountering effective angle-of-attack perturbations at the Reynolds numbers of 2000 and 20 000 is numerically studied by using an immersed boundary–lattice Boltzmann method. The perturbations are introduced as an abrupt heaving or pitching motion superposed on the baseline motion. It is found that the lift increment scales with the increase in the perturbation effective angle of attack, especially during the heaving perturbation. The pitching perturbation is more likely to disrupt this scaling due to the transition of the leading-edge vortex (LEV) detachment mechanism, where the detachment mechanism of the LEV transitions from bluff-body shedding dominant to vorticity layer eruption dominant. Despite the same variation in the effective angle of attack for the heaving and pitching perturbations, vorticity layer eruption is more likely to occur under the fast pitching perturbation. When the Reynolds number is increased to 20 000, the time histories of aerodynamic force are similar to those at the Reynolds number of 2000. Moreover, the boundary layer under the LEV is more resistant to the adverse pressure gradient, leading to greater variability in vorticity layer eruption.

Understanding the flow behaviour of wet granular materials is essential for comprehending the dynamics of numerous geological and physical phenomena, but remains a significant challenge, especially the transition of these flow regimes. In this study, we perform a series of rotating drum experiments to systematically investigate the dynamic observables and flow regimes of wet mono-dispersed particles. Two typical continuous flows including rolling and cascading regimes are identified and analysed, concentrating on the impact of fluid density and rotation speed. The probability density functions of surface angles, $\theta _{\textit{top}}$ and $\theta _{\textit{lo}w\textit{er}}$, reveal distinct patterns for these two flow regimes. A morphological parameter thus proposed, termed angle divergence, is used to characterise the rolling–cascading regime transition quantitatively. By integrating quantitative observables, we construct the flow phase diagram and flow curve to delineate the transition rules governing these regimes. Notably, the resulting nonlinear phase boundary demonstrates that higher fluid densities significantly enhance the likelihood of the system transitioning into the cascading regime. This finding is further supported by corresponding variations in flow fluctuations. Our results provide new insights into the fundamental dynamics of wet granular matter, offering valuable implications for understanding the complex rheology of underwater landslides and related phenomena.

Linear-stability modelling suggests that all sufficiently large riblets promote maximally growing spanwise rollers (García-Mayoral & Jiménez 2011 J. Fluid Mech. vol. 678, 317–347), yet direct numerical simulations (DNS) have shown that this is not the case (Endrikat et al. 2021 J. Fluid Mech. vol. 913, A37) some riblet shapes do not form spanwise rollers at all. Thus, the drag-reduction breakdown across all riblet shapes cannot be solely attributed to maximally growing spanwise rollers, prompting a reappraisal of the modelling. In this paper, comparing DNS data with riblet-resolving linear-stability predictions shows that the spanwise rollers are actually marginal modes, not maximally growing instabilities. This riblet-resolved linear analysis also predicts that not all riblet shapes promote spanwise rollers, in agreement with DNS, and unlike earlier linear-stability modelling, which relied on a one-dimensional (1-D) mean flow and on an over-simplified effective wall-admittance boundary condition. These riblet-resolved calculations further inform how to capture the effect of the riblet shape in a 1D model. Once captured, predictions with an effective boundary condition match riblet-resolved results, but still do not indicate what features of the riblet geometry promote the roller instability. Thus, the wall admittance is measured near the riblet crests, in both the riblet-resolved linear analysis and DNS, to show that the in-groove dynamics is dominated by a balance between the overlying pressure and unsteady inertia, and not viscous diffusion, as previously assumed. This pressure–unsteady-inertia balance sets the linear scaling of the wall admittance with riblet size, as observed in DNS, and is a key factor in setting the streamwise wavelength of the spanwise rollers. Furthermore, modelling this pressure–unsteady-inertia balance in the wall admittance reveals the role of riblet slenderness in promoting spanwise rollers, which provides the missing link in previous correlations between the riblet geometry and the presence or lack of rollers.

The Richtmyer–Meshkov instability (RMI) develops when a planar shock front hits a rippled contact surface separating two different fluids. After the incident shock refraction, a transmitted shock is always formed and another shock or a rarefaction is reflected back. The pressure/entropy/vorticity fields generated by the rippled wavefronts are responsible of the generation of hydrodynamic perturbations in both fluids. In linear theory, the contact surface ripple reaches an asymptotic normal velocity which is dependent on the incident shock Mach number, fluid density ratio and compressibilities. In this work we only deal with the situations in which a shock is reflected. Our main goal is to show an explicit, closed form expression of the asymptotic linear velocity of the corrugation at the contact surface, valid for arbitrary Mach number, fluid compressibilities and pre-shock density ratio. An explicit analytical formula (closed form expression) is presented that works quite well in both limits: weak and strong incident shocks. The new formula is obtained by approximating the contact surface by a rigid piston. This work is a natural continuation of J. G. Wouchuk (2001 Phys. Rev. E vol. 63, p. 056303) and J. G. Wouchuk (2025 Phys. Rev. E vol. 111, p. 035102). It is shown here that a rigid piston approximation (RPA) works quite well in the general case, giving reasonable agreement with existing simulations, previous analytical models and experiments. An estimate of the relative error incurred because of the RPA is shown as a function of the incident shock Mach number $M_i$ and ratio of $\gamma $ values at the contact surface. The limits of validity of this approximation are also discussed. The calculations shown here have been done with the scientific software Mathematica. The files used to do these calculations can be retrieved as Supplemental Files to this article.

The Boltzmann kinetic equation is considered to compute the transport coefficients associated with the mass flux of intruders in a granular gas. Intruders and granular gas are immersed in a gas of elastic hard spheres (molecular gas). We assume that the granular particles are sufficiently rarefied so that the state of the molecular gas is not affected by the presence of the granular gas. Thus, the gas of elastic hard spheres can be considered as a thermostat (or bath) at a fixed temperature $T_g$. In the absence of spatial gradients, the system achieves a steady state where the temperature of the granular gas $T$ differs from that of the intruders $T_0$ (energy non-equipartition). Approximate theoretical predictions for the temperature ratio $T_0/T_g$ and the kurtosis $c_0$ associated with the intruders compare very well with Monte Carlo simulations for conditions of practical interest. For states close to the steady homogeneous state, the Boltzmann equation for the intruders is solved by means of the Chapman–Enskog method to first order in the spatial gradients. As expected, the diffusion transport coefficients are given in terms of the solutions of a set of coupled linear integral equations which are approximately solved by considering the first Sonine approximation. In dimensionless form, the transport coefficients are nonlinear functions of the mass and diameter ratios, the coefficients of restitution and the (reduced) bath temperature. Interestingly, previous results derived from a suspension model based on an effective fluid–solid interaction force are recovered when $m/m_g\to \infty$ and $m_0/m_g\to \infty$, where $m$, $m_0$ and $m_g$ are the masses of the granular particles, intruders and molecular gas particles, respectively. Finally, as an application of our results, thermal diffusion segregation is exhaustively analysed.

We investigate Lighthill’s proposed turbulent mechanism for near-wall concentration of spanwise vorticity by calculating mean flows conditioned on motion away from or toward the wall in an (friction Reynolds number) ${\textit{Re}}_\tau =1000$ database of plane-parallel channel flow. Our results corroborate Lighthill’s proposal throughout the entire logarithmic layer, but extended by counter-flows that help explain anti-correlation of vorticity transport by advection and by stretching/tilting. We present evidence also for Lighthill’s hypothesis that the vorticity transport in the log layer is a ‘cascade process’ through a scale hierarchy of eddies, with intense competition between transport outward from and inward to the wall. Townsend’s model of attached eddies of hairpin-vortex type accounts for half of the vorticity cascade, whereas we identify necklace type or ’shawl vortices’ that envelop turbulent sweeps as supplying the other half.