This study investigates the transport of particles in turbulent channel flow with friction Reynolds number $Re_\tau = 1000$ by direct numerical simulation. We focus on how large-scale flow structures, namely the $Qs$ structures (Lozano-Durán et al. 2012, J. Fluid Mech., vol. 694, pp. 100–130), affect the wall-normal transport of particles. Despite occupying less than $10\,\%$ of the physical domain, our results highlight the critical role played by $Qs$ structures in the particle transport, namely that the particle number and momentum flux inside the $Qs$ structures are substantially higher than outside. The fraction of particle wall-normal momentum flux inside $Qs$ structures is considerably larger than their volume fraction, suggesting highly efficient transport inside the $Qs$ structures. This prominent role played by $Qs$ structures in the transport of inertial particles is more effective by diminishing the inertia of particles. Notably, the long-distance transport of particles in the wall-normal direction is driven primarily by the continuous effect of $Qs$ structures. In summary, our findings advance the understanding of the effects of $Qs$ structures on particle transport, and demonstrate their significant role in the process.

The electromagnetically driven magnetised spherical Couette flow is studied experimentally, theoretically and numerically in the laminar regime. The working fluid, Galinstan, is contained in the spherical gap between two concentric spheres at rest. The electromagnetic stirring is primarily generated due to the interaction of a direct current, which is injected through two ring-shaped electrodes located at the equatorial zone of each sphere, and a dipolar magnetic field produced by a permanent magnet located inside the inner sphere. The flows were explored experimentally for a Reynolds number ranging from 450 to 2230 and a Hartmann number of 240. Ultrasound Doppler velocimetry and particle image velocimetry were used to characterise the flow. For low Reynolds numbers, given the symmetry of the problem, a one-dimensional analytic solution is obtained in the equatorial plane from the magnetohydrodynamic equations. The analytical solution reproduces the main characteristics of the flow. In addition, a full three-dimensional numerical model is able to reproduce both the analytical solution and the experimental measurements. To the best knowledge of the authors, this is the first time experimental results of the magnetised spherical Couette flow have been reported with electromagnetic forcing using a liquid metal as the working media.

Viscoplastic fluids exhibit yield stress, beyond which they flow viscously, while at lower stress levels they behave as solids. Despite their fundamental biological and medical importance, the hydrodynamics of swimming in viscoplastic environments is still evolving. In this study, we investigate the swimming of an ellipsoidal squirmer and the associated tracer diffusion in a Bingham viscoplastic fluid. The results illustrate that neutral squirmers in viscoplastic fluids experience a reduction in swimming speed and an increase in power dissipation as the Bingham number increases, with swimming efficiency peaking at moderate Bingham numbers. As the aspect ratio of a squirmer increases, ellipsoidal squirmers exhibit significantly higher swimming speeds in viscoplastic fluids. The polar and swirling modes can either enhance or reduce swimming speed, depending on the specific scenarios. These outcomes are closely related to the confinement effects induced by the yield surface surrounding the swimmer, highlighting how both swimmer shape and swimming mode can significantly alter the yield surface and, in turn, modify the swimming hydrodynamics. In addition, this study investigates the influence of viscoplasticity on swimmer-induced diffusion in a dilute suspension. The plasticity enforces the velocity far from the swimmer to be zero, thus breaking the assumptions used in Newtonian fluids. The diffusivity reaches its maximum at intermediate aspect ratios and Bingham numbers, and increases with the magnitude of the squirmer’s dipolarity. These findings are important to understand microscale swimming in viscoplastic environments and the suspension properties.

Hydrodynamic modulation of short ocean surface waves by longer ambient waves significantly influences remote sensing, interpretation of in situ wave measurements and numerical wave forecasting. This paper revisits the wave crest and action conservation laws and derives steady, nonlinear, analytical solutions for the change of short-wave wavenumber, action and gravitational acceleration due to the presence of longer waves. We validate the analytical solutions with numerical solutions of the full crest and action conservation equations. The nonlinear analytical solutions of short-wave wavenumber, amplitude and steepness modulation significantly deviate from the linear analytical solutions of Longuet-Higgins & Stewart (1960 J. Fluid Mech. vol. 8, no. 4, pp. 565–583) and are similar to the nonlinear numerical solutions by Longuet-Higgins (1987 J. Fluid Mech. vol. 177, pp. 293–306) and Zhang & Melville (1990 J. Fluid Mech. vol. 214, pp. 321–346). The short-wave steepness modulation is attributed 5/8 to wavenumber, 1/4 due to wave action and 1/8 due to effective gravity. Examining the homogeneity and stationarity requirements for the conservation of wave action reveals that stationarity is a stronger requirement and is generally not satisfied for very steep long waves. We examine the results of Peureux et al. (2021 J. Geophys. Res.: Oceans vol. 126, no. 1, e2020JC016735) who found through numerical simulations that the short-wave modulation grows unsteadily with each long-wave passage. We show that this unsteady growth only occurs for homogeneous initial conditions as a special case and not generally. The proposed steady solutions are a good approximation of the nonlinear crest-action conservation solutions in long-wave steepness $\lesssim 0.2$. Except for a subset of initial conditions, the solutions to the nonlinearised crest-action conservation equations are mostly steady in the reference frame of the long waves.

Sound entering the ear is known not only to transmit signals to the nerve system, but also to generate vortex-like steady streaming in the cochlea. This streaming has been suggested as the primary vehicle for drug delivery in the inner ear (Sumner, Mestel & Reichenbach, 2021, Sci. Rep., vol. 11, 57). An alternative vehicle by pure diffusion alone has also been suggested by Sadreev et al. (2019, Front. Cell. Neurosci., vol. 13, 161). This paper purports to examine both mechanisms analytically, and compare their relative importance, based on the two-dimensional model of Allen (1977, Acoust. Soc. Am., vol. 61, 110–119). First, we reconstruct the fluid mechanics of the Békséy vortices by an asymptotic theory of multiple scales as a complement to the two-dimensional numerical theory of Edom, Obrist & Kleiser (2014, J. Fluid Mech., vol. 753, 254–278). For discerning the difference between Sumner, Mestel & Reichenbach (2021) and Sadreev et al. (2019), we combine sound-induced streaming and molecular diffusion by modeling the drug as a solute of known diffusivity. It will be shown that due to the high frequency of sound, advection is augmented by the Lagrangian velocity, but molecular diffusion still dominates drug transport in the cochlear duct, unlike Taylor dispersion of pollutant by tides in a shallow river.

Synthetic-aperture radar images and mesoscale models show that wind-farm wakes differ from single-turbine wakes. For instance, wind-farm wakes often narrow and do not disperse over long distances, contrasting the broader and more dissipating wakes of individual turbines. In this work, we aim to better understand the mechanisms that govern wind-farm wake behaviour and recovery. Hence we study the wake properties of a $1.6$ GW wind farm operating in conventionally neutral boundary layers with capping-inversion heights $203$, $319$, $507$ and $1001$ m. In shallow boundary layers, we find strong flow decelerations that reduce the Coriolis force magnitude, leading to an anticlockwise wake deflection in the Northern Hemisphere. In deep boundary layers, the vertical turbulent entrainment of momentum adds clockwise-turning flow from aloft into the wake region, leading to a faster recovery rate and a clockwise wake deflection. To estimate the wake properties, we propose a simple function to fit the velocity magnitude profiles along the spanwise direction. In the vertical direction, the wake spreads up to the capping-inversion height, which significantly limits vertical wake development in shallow-boundary-layer cases. In the horizontal direction and for shallow boundary layers, the wake behaves as two distinct mixing layers located at the lateral wake edges, which expand and turn towards their low-velocity side, causing the wake to narrow along the streamwise direction. A detailed analysis of the momentum budget reveals that in deep boundary layers, the wake is predominantly replenished through turbulent vertical entrainment. Conversely, in shallow boundary layers, wakes are mostly replenished by mean flow advection in the spanwise direction.

Bubble–particle collisions in turbulence are key to the froth flotation process that is widely employed industrially to separate hydrophobic from hydrophilic materials. In our previous study (Chan et al., 2023 J. Fluid Mech. 959, A6), we elucidated the collision mechanisms and critically reviewed the collision models in the no-gravity limit. In reality, gravity may play a role since, ultimately, separation is achieved through buoyancy-induced rising of the bubbles. This effect has been included in several collision models, which have remained without a proper validation thus far due to a scarcity of available data. We therefore conduct direct numerical simulations of bubbles and particles in homogeneous isotropic turbulence with various Stokes, Froude and Reynolds numbers, and particle density ratios using the point-particle approximation. Generally, turbulence enhances the collision rate compared with the pure relative settling case by increasing the collision velocity. Surprisingly, however, for certain parameters the collision rate is lower with turbulence compared with without, independent of the history force. This is due to turbulence-induced bubble–particle spatial segregation, which is most prevalent at weak relative gravity and decreases as gravitational effects become more dominant, and reduced bubble slip velocity in turbulence. The existing bubble–particle collision models only qualitatively capture the trends in our numerical data. To improve on this, we extend the model by Dodin & Elperin (2002 Phys. Fluids 14, 2921–2924) to the bubble–particle case and found excellent quantitative agreement for small Stokes numbers when the history force is negligible and segregation is accounted for.

To elucidate the attenuation mechanism of wall-bounded turbulence due to heavy small particles, we conduct direct numerical simulations (DNS) of turbulent channel flow laden with finite-size solid particles. When particles cannot follow the swirling motions of wall-attached vortices, vortex rings are created around the particles. These particle-induced vortices lead to additional energy dissipation, reducing the turbulent energy production from the mean flow. This mechanism results in the attenuation of turbulent kinetic energy, which is more significant when the Stokes number of particles is larger or particle size is smaller under the condition that the volume fraction of particles is fixed. Moreover, we propose a method to quantitatively predict the degree of turbulence attenuation without using DNS data by estimating the additional energy dissipation rate in terms of particle properties.

Inspired by the need to theoretically understand the naturally occurring interactions between internal waves and mesoscale phenomena in the ocean, we derive a novel model equation from the primitive rotational Euler equations using the multi-scale asymptotic expansion method. By applying the classic balance $\epsilon =\mu ^2$ between nonlinearity (measured by $\epsilon$) and dispersion (measured by $\mu$), along with the assumption that variations in the transverse direction are of order $\mu$, which is smaller than those in the propagation direction, we arrive at terms from the classic Kadomtsev–Petviashvili equation. However, when incorporating background shear currents in two horizontal dimensions and accounting for Earth’s rotation, we introduce three additional terms that, to the best of the authors’ knowledge, have not been addressed in the previous literature. Theoretical analyses and numerical results indicate that these three terms contribute to a tendency for propagation in the transverse direction and an overall variation in wave amplitudes. The specific effects of these terms can be estimated qualitatively based on the signs of the coefficients for each term and the characteristics of the initial waves. Finally, the potential shortcomings of this proposed equation are illuminated.

Direct numerical simulations are performed to explore the impact of surface roughness on inter-scale energy transfer and interaction in a turbulent open-channel flow over differently arranged rough walls. With friction Reynolds number approximately 540, six distinct configurations of roughness arrangements are examined. The results show that the clustered roughness arrangements yield notable changes in large-scale secondary-flow structures, which manifest in the profiles of dispersive stresses, predominantly near the roughness elements. They are marked by the presence of spanwise alternating high-momentum pathways and low-momentum pathways. From the outer peak in the spanwise energy spectra, the size and intensity of turbulent secondary flows are shown to be related to the spanwise spacing of the roughness heterogeneity. The emergence of turbulent secondary flows serves to suppress the original large-scale structures in the outer region of smooth-wall turbulence, paving the way for the development of new turbulent structures at the second harmonic scale. Furthermore, the spanwise triadic interaction analysis reveals the mutual energy exchange between the secondary harmonic scale and the secondary-flow scale. These findings elucidate the underlying mechanisms behind the attenuation of large-scale structures in the outer region influenced by roughness, offering new insights into the dynamic interplay of scale interactions in rough-wall turbulence.