The description of riblets and other drag-reducing devices has long used the concept of longitudinal and transverse protrusion heights, both as a means to predict the drag reduction itself and as equivalent boundary conditions to simplify numerical simulations by transferring the effect of riblets onto a flat virtual boundary. The limitation of this idea is that it stems from a first-order approximation in the riblet-size parameter $s^+$, and as a consequence it cannot predict other than a linear dependence of drag reduction upon $s^+$; in other words, the initial slope of the drag-reduction curve. Here the concept is extended to a full asymptotic expansion using matched asymptotics, which consistently provides higher-order protrusion coefficients and higher-order equivalent boundary conditions on a virtual flat surface. While the majority of this expansion, though nonlinear in $s^+$, remains linear in velocity, and therefore we shall not directly address the shape of the drag-reduction curve, this procedure will also allow us to explore the way nonlinearities of the Navier–Stokes equations first enter the $s^+$ expansion, with somewhat surprising negative results.

The accuracy obtained with computational fluid dynamics and process simulations of flotation critically depends on the quality and robustness of the underlying models for the non-resolved subprocesses. An important issue in flotation is the collision between particles and air bubbles. Many models have been developed, but their accuracy for applications in flotation is limited. In particular, the significant size difference between particles and bubbles and their intricate coupling to the turbulent flow field pose severe challenges. The present paper first reviews presently employed collision models, highlighting their advantages and disadvantages when applied to flotation. On this basis, the `integrated multisize collision model’ (IMSC) is proposed. After a detailed evaluation, it combines existing approaches from various sources and introduces new developments designed to address present shortcomings. The model is validated by own direct numerical simulation data as well as data from the literature. It is shown that, overall, the IMSC provides better predictions for the collision rate in typical flotation conditions than presently employed collision models and covers the entire parameter range of the flotation process very well. Using the available data, some of the underlying modelling assumptions are validated. Finally, a comprehensive overview of the model is provided for further use in Euler–Euler frameworks or process simulations also beyond flotation.

Results of an experimental study investigating issues of Coriolis effects on the fluid dynamics associated with vortex rings propagating through a rotating fluid are presented. The vortex rings are generated at the top of a large, water-filled rotating tank and they propagate downwards along the axis of rotation. The motion and the decay of the rings in a rotating fluid are expected to be accompanied by an inertial-wave field being established in the fluid surrounding the rings. However, the existence of this inertial-wave field had previously never been verified experimentally. Particle image velocimetry measurements were performed with the goal of demonstrating the existence of the inertial-wave field. Datasets were processed to extract individual inertial-wave modes and, for the first time, experimentally construct the dispersion relation for the inertial waves associated with vortex rings. For rotation rates when Coriolis forces dominate the dynamics, the experimental data are found to be in very good agreement with the well-established theoretical dispersion relation for inertial waves. The generation of inertial waves implies that kinetic energy is radiated away from the vortex rings. First results relating to the redistribution process of the kinetic energy are briefly discussed.

Slow viscous flow around a fixed body generates a shape-dependent drag. We explore the drag-minimising shapes of bodies centred between two parallel plates in two-dimensional viscous flow. The channel width introduces a length scale so that the optimal profile is area-dependent. We solve the shape optimisation problem numerically over a wide range of areas. We also compute the optimal elliptical shapes and this identifies how these shapes should be slightly altered to reduce the drag with reductions of up to $3.8\,\%$ attained at high areas. More broadly, we derive two properties of general optimal shapes within the confined flow: the magnitude of the surface vorticity is approximately (but not exactly) constant and the noses have sharp angles that are independent of area. For relatively small bodies, the optimal shape becomes identical to that in an unconfined geometry, but the drag is qualitatively different owing to the influence of confinement; within a channel, it is proportional to the inverse of the logarithm of the body area. At relatively large areas, the optimal body becomes long and its surface is approximately parallel to the channel boundaries, except in the vicinity of the noses. Using a lubrication approximation, we recast the optimisation problem as an Euler–Lagrange equation that is solved to determine the drag-minimising shape, finding that the drag is proportional to the body area in this regime.

Unsteadiness lies at the heart of turbulent fluid dynamics, eddy formation and instabilities in flows, thus making it central to both understanding and controlling fluid systems. In this work, we present an objective measure for the unsteadiness of a time-dependent velocity field, the deformation unsteadiness, derived from a spatio-temporal variational principle, allowing for a frame-independent assessment of the unsteadiness of a given flow field. Additionally, as an application of our main result, we define an objective analogue of the classical $Q$-criterion based on extremisers of unsteadiness minimisation. We apply our results to several examples of analytical flows as well as simulated flow data sets in two and three dimensions. In particular, we apply our newly derived vortex criterion to several explicit, time-dependent solutions of the Navier–Stokes equation and compare the results with existing vortex criteria. We give a physical interpretation of the deformation unsteadiness and discuss future research directions.

Liquid sheets arise in curtain coating, polymer processing and sprays. When a fluid is ejected from a die (nozzle) to form a liquid sheet, its cross-section is rectangular albeit for the two rounded ends. The latter retract due to surface tension. The retraction dynamics is also affected by stresses owing to bulk rheology, which may be viscous and/or viscoelastic in nature, and surface rheology, which may be due to the presence of surface-active agents. We analyse theoretically and numerically the retraction dynamics of highly viscous Newtonian liquid sheets when surface viscous stresses are present. While it has been shown recently that viscoelasticity increases retraction rate, it is demonstrated that surface viscosity operates synergistically with bulk viscosity to decrease retraction rate. As the two surfaces of a retracting sheet remain flat outside of the two tip regions, an exact analytical solution is obtained for the transient sheet thickness in terms of the Lambert W function. An asymptotic solution for sheet thickness, valid for early times, is also obtained and shown to agree well with the analytical solution and simulations. An energy analysis is performed to rationalise that at early times, the rate of energy dissipation due to the action of surface viscous stresses can be dominant in slowing retraction, but it can wane in importance and be overtaken at large times by the rate at which energy is dissipated due to the action of bulk viscous stresses.

The impact of spanwise surface temperature heterogeneity on steady stably stratified Ekman layers is systematically studied using large-eddy simulation (LES). Spanwise varying strips of high and low surface temperature are imposed in idealised LES of stable boundary layers (SBLs), in which a steady state results from a balance between cooling at the ground and heating due to imposed synoptic subsidence. Consistent with previous studies on channel flows with streamwise-aligned surface heterogeneity (e.g. Bon & Meyers, J. Fluid Mech. 2022, pp. 1–38), large-scale secondary circulations develop and extend deep into the stable Ekman layer. Coriolis effects enhance counterclockwise circulations while reducing clockwise ones, thereby tilting the mean secondary flow structures towards the left (in the northern hemisphere). Nevertheless, for the considered surface temperature contrasts of 1.5–12 K and spanwise wavelengths of 100–800 m, the impact on mean SBL structure is substantial. As the surface temperature difference or strip width increases, secondary flows and dispersive fluxes strengthen, eventually reaching the top of the SBL. This augmentation further enhances near-surface gradients, elevates SBL depths and low-level jets, and reduces mean surface heat fluxes. Novel correlations between characteristics of the surface heterogeneity and their impact on the mean SBL structure are proposed. Moreover, the local surface fluxes are shown to significantly deviate from the mean, highlighting that horizontally averaged SBL properties do not capture all important physical processes in a heterogeneous flow. Overall, this work affirms that thermal surface heterogeneity is a crucial factor in governing transport processes within the atmospheric stable boundary layer.

The two-dimensional to three-dimensional wake transition of a circular cylinder in a sinusoidal oscillatory flow arises from the Honji instability at a critical Keulegan–Carpenter number (denoted $\textit{KC}_{cr}$) with a corresponding critical spanwise wavelength (denoted $\lambda _{cr}$) for a given Stokes number (denoted $\beta$) larger than approximately 50. However, significant discrepancies in the $\textit{KC}_{cr}$ and $\lambda _{cr}$ values exist among the theoretical predictions by Hall (J. Fluid Mech., vol. 146, 1984, pp. 347–367), empirical formulae by Sarpkaya (J. Fluid Mech., vol. 457, 2002, pp. 157–180) and other experimental and numerical results in the literature. These long-standing discrepancies are addressed in this study, and new equations for $\textit{KC}_{cr}$ and $\lambda _{cr}$ are proposed for $\beta = 55$–$10^{6}$. The present $\textit{KC}_{cr}$ and $\lambda _{cr}$ values agree well with the Floquet analysis results of Elston et al. (J. Fluid Mech., vol. 550, 2006, pp. 359–389) for $\beta \sim 50$–$100$, and asymptotically converge to theoretical predictions by Hall (1984) as $\beta \to \infty$, but deviate significantly from the empirical formulae by Sarpkaya (2002). The underlying physical mechanisms for these deviations are elucidated. In addition, we reproduce the quasi-coherent structure (QCS) numerically for the first time, and demonstrate that the QCS observed by Sarpkaya (2002), where transient Honji vortices become pronounced near peak flow velocities but diminish during deceleration, is physically induced by ambient disturbances inevitably contained in physical experiments, such that $\textit{KC}_{cr}$ given by Sarpkaya (2002) is specific to the level of disturbance in his experimental setting and is somewhat arbitrary.

In the article, the unsteady flow phenomenon of self-excited and forced oscillations in a rectangular diverging isolator is studied by using large eddy simulation, and the shock train region is analysed particularly. Self-excited oscillations are analysed under four pressure ratios, with pressure statistically processed to reveal shock train oscillation characteristics. The interference factors of the external environment on the unsteady flow in the isolator are investigated, and the function of upstream disturbance and downstream disturbance on the shock train oscillation is studied. Both disturbance types show that pressure amplitude increases oscillation amplitude, while frequency variations have opposing effects. Due to mismatched response speeds, low-frequency disturbances intensify oscillations, whereas high-frequency ones suppress them. The difference is that the pressure frequency excitation upstream is transmitted along the flow direction and directly acts on the shock train in the trough period of each unsteady pressure transformation, which intensifies the negative effect on shock train oscillations. The downstream disturbance arrives at the shock train region after passing through the complex flow coupling in the mixing region. The superposition of the external pressure excitation frequency and the mixing region makes the response of the shock train slower leading to a weakening effect of the shock train oscillation. Moreover, the unsteady flow develops in the mixing district and transmits upstream, and the inhibition effect is stronger than that of upstream pressure frequency excitation.

The stability analysis of multiphase capillary wavetrains on water of infinite depth is performed using two coupled fourth-order nonlinear evolution (NLE) equations. We have investigated analytically the influence of a second wavetrain travelling in a different direction to the first wavetrain. The propagation of multiphase modes is studied for the case when group velocity projections of two wavetrains overlap. Criteria are derived for capillary Stokes wave instabilities and for the existence of a multiphase solitary envelope solution. We have exhibited that the weakly nonlinear multiphase capillary wavetrains in deep water is unstable to oblique disturbances and presented that the dominant modulational instability is two-dimensional in deep water. It is found that the growth rate of modulational instability increases with the increase of the angle of interaction between two wavetrains. The existing fourth-order analysis provides significant deviations on the stability results when compared with the third-order analysis.

We compute particle deposition rates on the back side of a cylinder at Reynolds numbers $\textit{Re}={1685}$, $6600$ and $10\,000$ using direct numerical simulation and Lagrangian particle tracking. We find that the deposition rates for $\textit{Re}={6600}$ and $10\,000$ are highly variable in time, with differences of up to a factor 27 in deposition rates between alternating low- and high-deposition-rate periods. The deposition-rate fluctuations are found at frequencies lower than the vortex-shedding frequency and therefore require long simulation times to be discovered. Additionally, we find that these fluctuations correlate positively with the drag and negatively with the cylinder base pressure. These observations imply that the back-side deposition process is governed by the low-frequency modulation of the cylinder wake. The high-deposition-rate regime is associated with a shorter wake and a more efficient turbulent transport of particles towards the cylinder surface, where the wake length modulation appears to have a more prominent effect. Consequently, the wake modulation controls the deposition rate but does not significantly affect the deposition mechanism. The back-side deposition has a maximum at Stokes number $St = 0.07$, as particles of lower Stokes number have too little inertia to deposit effectively and the deposition rate decorrelates from the wake fluctuations for larger Stokes numbers. These results highlight the strong sensitivity of the back-side deposition process to accurate descriptions of the wake turbulence over long enough times. These observations are critical when constructing accurate datasets for data-assisted methods to predict long-term back-side deposition on bluff bodies.

We derive a depth-averaged equation for the magnetic field induced by long surface gravity waves over variable seabed. The equation is verified using known analytical results and a novel numerical model for magnetic anomalies over variable bathymetry. Unlike amplitude-based theories, our results show that the magnetic response is governed by the forward energy flux associated with the surface gravity wave. This reframes the physics of long-wave magnetics and provides a new basis for interpreting geomagnetic observations.

We investigate the control effects of spanwise heterogeneous roughness on shock-wave/turbulent boundary-layer interactions (STBLIs) using wall-resolved large-eddy simulations. The roughness extends over the entire computational domain and consists of streamwise-aligned sinusoidal ridges alternating with flat valleys. The baseline case is a Mach 2.0 impinging STBLI flow with a 40$^\circ$ impinging-shock angle, for which we consider incoming turbulent boundary layers at two friction Reynolds numbers, $Re_\tau \approx$ 350 and 1200. Multiple roughness configurations are analysed, which maintain consistent geometric characteristics under either inner or outer scaling. The results show that the rough-wall configurations introduce a moderate increase in mean drag, while substantially modifying the dynamics of the interaction. The wall-pressure fluctuations near the separation-shock foot consist of two components: low-frequency fluctuations associated with large-scale shock excursions and high-frequency fluctuations linked to amplified turbulence. We find that both spectral components can be significantly attenuated by the investigated wall roughness. At low Reynolds number, the attenuation of low- and high-frequency components contributes comparably to the overall reduction. At high Reynolds number, an overall stronger reduction of the pressure fluctuation peak is observed and is mainly attributed to the effective suppression of the low-frequency component. Cross-correlation analyses support downstream mechanisms for the low-frequency dynamics in the current strong interaction regime, where large-scale shock excursions are mainly driven by the breathing of the reverse-flow bubble. Large-scale Görtler-like vortices are identified around the reattachment location in all cases. They appear largely unaffected by roughness geometry and contribute to the flow dynamics over a wide range of frequencies.

We investigate two-dimensional vortex merging of three vortices, initially aligned and evenly spaced, with the two outer vortices having the same strength and the middle one having any strength. Based on the vorticity transport equation (VTE) a vortex is identified as an extremum of the vorticity. The vorticity is also investigated through the low-dimensional core-growth model, providing analytical insight into the vorticity patterns and transitions, including explicit formulas of trajectories of the critical points of vorticity. Four distinct vorticity patterns and four types of trajectories of the vorticity are found. For a corotating centre vortex there are two types of trajectories of the vorticity, one where the centre vortex dominates the two outer vortices, and one where the centre vortex is suppressed by the two outer vortices. The two types of trajectories are separated in parameter space by the strength ratio of the inner to outer vortex being $4\exp (-{3}/{2})$. In the case of a counter-rotating vortex centre, the centre vortex is suppressed in the flow transitions for centre vortex strengths less than the sum of the two outer vortices. For a range of vortex strengths of the middle vortex, the three vortex configuration first rotates in one direction and then shifts direction of rotation. In the case of a centre vortex strength exceeding the sum of the two outer vortices, the two outer vortices are pushed away. The core-growth model quantitatively reproduces the VTE flow for low Reynolds number (Re) and topologically provides accurate descriptions up to Re = 1290 where filamentation vortices are created.