The large-eddy simulation technique was used to investigate the dynamics of unsteady flow separation on a flat-plate turbulent boundary layer. The unsteadiness was generated by imposing an oscillating, wall-normal velocity profile at the top of the computational domain, and a range of reduced frequencies ($k$), from a very rapid flutter-like motion to a slow quasi-steady oscillation, was studied. Ambrogi et al. (J. Fluid Mech., vol. 945, 2022, A10) showed that the reduced frequency greatly affects the transient separation process, and at a frequency $k=1$, the separation region became unstable and was advected periodically out of the domain. In this paper, we discuss the causes of the observed advection process and the effects of the unsteadiness on the second moments. The time evolution of turbulent kinetic energy, for instance, reveals that an advection-like phenomenon is also present at a very low reduced frequency, but its dynamic behaviour is completely different from that of the intermediate frequency ($k=1$). At the intermediate frequency the entire recirculation region is advected downstream, keeping its shape. The advected structure is rotational in nature, and moves at constant speed. In contrast, in the low-frequency case the advected fluid originates at the reattachment point, and the structure is shear-dominated. Particle pathlines reflect the fact that the flow at the low frequency is quasi-steady-state, but show peculiar differences at the intermediate frequency, in which the flow response to the freestream forcing depends on the particle positions in the wall-normal direction.

We present an analytical study of two-dimensional flow in a wedge driven by a time-dependent surface heat flux as a model problem to understand buoyancy-induced cross-shore flow. Besides the turbulent Prandtl number and the relevant Rayleigh number, both assumed to be of order unity, the solution is seen to depend on the geometry through a parameter $\beta$, representing the bottom slope. An analytic solution is sought in the asymptotic limit $\beta \ll 1$ for a water layer bounded by an adiabatic bottom surface subject to a harmonic heat flux on the upper surface. The analysis reveals that the motion at leading order can be expressed as the sum of a harmonic component and a steady component, the latter driven by nonlinear advection. This steady-streaming motion includes a near-shore vortex with associated recirculating motion that can affect cross-shore transport and dispersion in coastal environments. The analytical solution is compared with numerical solutions of the complete conservation equations for small values of $\beta$. Excellent quantitative agreement is found for values of the Rayleigh number below a critical value at which the periodic solution undergoes a period-doubling bifurcation, leading to the establishment of thermal-instability cells that dominate the offshore flow dynamics, while the near-shore dynamics remains well described by the analytical solution. The analysis illustrates that a periodic heat input that leads to a vertically inhomogeneous temperature distribution can result in residual motion, net heat fluxes and persistent temperature structure in the cross-shore direction.

We investigate how inhomogeneity influences the $k^{-5/3}$ inertial range scaling of turbulent kinetic energy spectra (with $k$ the wavenumber). For weak statistical inhomogeneity, the energy spectrum can be described as an equilibrium spectrum plus a perturbation. Theoretical arguments suggest that this latter contribution scales as $k^{-7/3}$. This prediction is assessed using direct numerical simulations of three-dimensional Kolmogorov flow.

In this paper a fully coupled particle-size segregation model for granular flows (Barker et al., J. Fluid Mech., vol. 909, 2021, p. A22) is used to simulate the development of the patterns in a triangular rotating drum. The results are compared with the experimental patterns formed with bidisperse and tridisperse granular mixtures, and with varying compositions and fill heights. In all cases the agreement between the simulations and experiments is remarkably good. The experimental patterns are generated in a narrow gap between transparent front and back sidewalls. These prevent three-dimensional motion, but also impose friction on the flow, making it thinner and faster than it would otherwise be. This promotes segregation, as it simultaneously increases the shear rate and reduces the local pressure. To obtain the correct flow dynamics and segregation, width-averaged sidewall friction is incorporated into the two-dimensional simulations, which are performed in OpenFOAM$^{\circledR}$. The free-surface avalanche forms a boundary layer within which all the segregation occurs. Material in the lower reach of the avalanche is continuously deposited into an underlying solid body of grains, which rotates with the drum, and is eventually re-entrained into the avalanche along its upper reach. The changing geometry of the granular region (as the drum rotates) implies that the avalanche is constantly adjusting its length, position and depth. This generates a complex quasi-periodic flow, which when combined with particle-size segregation generates amazing patterns in the solid rotating granular body after only two drum rotations.

With the severity and frequency of significant weather events increasing, methods for alleviating unsteady wind loading for high-rise buildings are gaining interest. This study numerically investigates the three-dimensional flow structures around a canonical high-rise building immersed in an atmospheric boundary layer at different oncoming wind angles, using wall-resolved large eddy simulations. A synthetic jet located on the top surface is used as open-loop active actuation with the aim of suppressing the building's side-force fluctuations when exposed to oncoming wind variations. Three different frequencies of jet forcing are considered, all half an order of magnitude larger than the vortex shedding frequency. The behaviour of the synthetic jet and its effect on the building's unsteady side force, time-averaged flow fields and unsteady flow structures are investigated numerically. The synthetic jet actuation is found to reduce the side-force fluctuation of the building, enhance the downwash flow and successfully attenuate the antisymmetric vortex shedding. This was achieved to different extents across the range of oncoming wind angles considered and may motivate future attempts to explore experimental active control strategies for attenuation of unsteady wind loading.

We study the motion and deformation of a single bubble rising inside a cylindrical container filled with a viscoelastic material. We solve numerically the mass and momentum balances along with the constitutive equation for the viscoelastic stresses, without assuming axial symmetry to allow the growth of three-dimensional disturbances. Hence, we may predict the emergence of the notorious knife-edge shape of the bubble, which is a result of a purely elastic instability triggered in the locality of the trailing edge. Our results compare well with existing experiments. We visualize, for the first time to the best of our knowledge, the flow kinematics and dynamics that arise downstream of the bubble. We propose two quantities, one kinematical and one geometrical, for the determination of the onset of the instability. We demonstrate that extension-rate thinning in the constitutive law is necessary for the emergence of the instability. Moreover, our results indicate that increasing (a) the deformability of the bubble and (b) the initial extension rate hardening of the viscoelastic material, prior to thinning, triggers the instability earlier. These novel findings help us formulate and propose a mechanism that controls the onset of the instability and explain why the knife-edge shape is not encountered as frequently.

Three-dimensional simulations of the ordering of elastic capsule suspensions within planar Poiseuille flow channels are reported. The simulations utilize the immersed boundary method coupled with the lattice Boltzmann method to capture the complex flow-induced capsule deformations and hydrodynamic interactions within the suspensions. A parametric study is presented whereby the confinement ratio and the particle deformability are varied independently within a two-dimensional range relevant to this ordering phenomenon. The initial distribution of capsules is random, and the simulations evolve the system from a disordered state to an ordered one, while an order parameter that quantifies the fraction of capsules belonging to one-dimensional train assemblies is computed throughout time. A monotonic increase in ordering is observed with increasing deformability. However, an optimal confinement ratio is identified corresponding to a peak in the order parameter. This peak is attributed to the competition between increasing long-range capsule attractions and decreasing in-plane capsule density (with fixed volume fraction) as the confinement ratio increases. Simulations are also performed to understand how dispersity in capsule size and deformability impact the degree of ordering. It is shown that ordering is quite sensitive to dispersity in capsule size, and much less sensitive to dispersity in deformability. Overall, the results provide important insights for the design of microfluidic devices.

We re-examine the model proposed by Alexander et al. (Phys. Fluids, vol. 18, 2006, 062103) for the closing of a circular hole in a molecularly thin incompressible Langmuir film situated on a Stokesian subfluid. For simplicity their model assumes that the surface phase is inviscid which leads to the result that the cavity area decreases at a constant rate determined by the ratio of edge tension to subfluid viscosity. We reformulate the problem, allowing for a regularising monolayer viscosity. The viscosity-dependent corrections to the hole dynamics are analysed and found to be non-trivial, even when the monolayer viscosity is small; these corrections may explain the departure of experimental data from the theoretical prediction when the hole radius becomes comparable to the Saffman–Delbrück length. Through fitting, under these relaxed assumptions, we find the edge tension could be as much as six times larger ($\sim$4.0 pN) than reported previously.

The three-dimensional flow over a low-aspect-ratio (low-$A\!R$) trapezoidal plate is investigated experimentally with a focus on how the tip effects impact the structure and dynamics of the separation bubble. The chord-based Reynolds number is $5800$, and the angle of attack varies from $4^\circ$ to $10^\circ$. Once the flow separates, the separation bubble emerges and features a swallow-tailed structure that shrinks near the midspan, which is first found for the flows over low-$A\!R$ plates. This structure develops into the conventional single-tailed structure as the angle of attack increases. Moreover, the vortex shedding within the swallow-tailed separation bubble is restored from multiple asynchronously measured local velocity fields. It is revealed that the leading-edge vortex undergoes the novel transformation from a C-shape vortex into an M-shape vortex. This vortex transformation stems from the mass transport of the near-wall spanwise flow, which affects the fluid motion on the windward side of the C-shape vortex head, strengthening and accelerating the vortex head. The strengthened vortex head facilitates the entrainment of high-momentum fluid from the outer flow. This is responsible for the formation of the swallow-tailed structure. These findings help to fill the gaps left by the downwash at low angles of attack for low-$A\!R$ wings, and are of value in improving the cruising and gliding performance of micro-air vehicles.

When a fluid-filled cube rotating rapidly about an axis passing through two opposite vertices is subjected to harmonic modulations of its rotation rate (librations) at a modulation frequency that is $2/\sqrt {3}$ times the mean rotation frequency, all walls of the cube have critical reflection slopes. As such, all inertial wave beams emitted from edges and vertices of the cube in response to the librations are trapped in thin oscillatory boundary layers for forcing amplitudes (Rossby numbers) below a critical value which depends on the Ekman number (ratio of rotation to viscous time scales). How the resulting oscillatory boundary layer flow, referred to as a boundary-confined wave, depends on Ekman and Rossby numbers is examined in detail over several decades. Of particular interest is how the mean flow grows with increasing forcing amplitude, leading to instability resulting from nonlinear interactions between the mean flow and waves in the oscillatory boundary layers, injecting intense small-scale structures throughout the cube.

Data from an energetic tidal flow are used to investigate the appropriateness of the classical scaling of wall turbulence for very high Reynolds numbers and fully rough conditions. The boundary layer occupied the entire channel depth and the Reynolds numbers were several orders of magnitude larger than typically attained in the laboratory. Profiles of the mean velocity, turbulence quantities and spectra exhibit consistencies with the wall-similarity hypothesis and the attached-eddy model, as well as the universality of the near-wall turbulence structure for geophysical flows. The wall-normal velocity spectra and intensities near the free surface are also strongly attenuated at low wavenumbers, consistent with the constraints imposed by any boundary which may be approximated as a plane slip surface. The results can be used to inform the development of turbulence inflow models for assessing the dynamic loading on tidal turbines and the layout of tidal energy farms.

A model for the third-order mixed velocity–scalar structure function $-\overline {(\delta u)( \delta \phi )^{2}}$ ($\delta a$ represents the spatial increment of the quantity $a$; $u$ is the velocity and $\phi$ is a scalar) is proposed for closing the transport equation of the second-order moment of the scalar increment $\overline {( \delta \phi )^{2}}$. The closure model is based on a gradient-type hypothesis with an eddy-viscosity model which exploits the analogy between the turbulent kinetic energy and a passive scalar when the Prandtl number, $Pr$, is equal to 1. The solutions of the closed transport equation agree well with both measurements and numerical simulations, when $Pr$ is not too different from 1. However, the agreement deteriorates when $Pr$ differs significantly from 1. Nevertheless, the calculations capture the effect expected when $Pr$ varies. For example, as $Pr$ increases, a range of scales emerges where $\overline {( \delta \phi )^{2}}$ remains constant. This emerging scaling range should correspond to the $k^{-1}$ spectral range in the three-dimensional scalar spectrum commonly denoted the viscous–convective range. Also when $Pr$ decreases below 1, the calculations reproduce the emergence of an expected inertial–diffusive range for scales larger than the Kolmogorov length scale.

Electrohydrodynamic sedimentation of simple drops has been a topic well-studied by researchers. However, electrohydrodynamic sedimentation of a compound drop would be critically influenced by the density of the involved phases and this has hitherto remained unaddressed. Herein, we develop a semi-analytical model for an eccentric compound drop settling under the action of gravity and an electric field using bispherical coordinates. The sedimentation velocity of the two drops (shell and core) is determined, and the same is applied to capture the influence of concomitant physical, hydrodynamic and electric properties on compound droplet sedimentation. The findings indicate that the compound drop may either sediment or de-sediment depending on the amount of eccentricity and its interplay with electrohydrodynamic parameters. Thereafter, the critical limit of eccentricity and time within which similar results are furnished by concentric and eccentric configurations is determined. It is found that, based on the property ratios, the eccentricity remains lower than 0.1 up to a non-dimensional time range of the order of 102–103, within which both the configurations can furnish a similar solution.

This work studies the detachment of a micron-sized spherical particle from a surface with concave roughness in a linear shear flow. The concave roughness is described as regularly spaced hollow hemispheres below a flat surface and is characterised by two dimensionless parameters, i.e. dimensionless asperity distance and asperity size ratio. The hydrodynamic force and torque on the particle are calculated by performing lattice Boltzmann simulations for particle Reynolds numbers ranging from 0.02 to 40. Empirical correlations of the drag, lift and torque coefficients of the particle as functions of the particle Reynolds number and the asperity size ratio are proposed. For detachment by lifting, sliding and rolling, a numerical approach to calculate the critical particle Reynolds number (i.e. above which the particle can detach from the surface) is proposed. It is found that the dimensionless asperity distance and the distribution of asperities on the rough surface have a minor influence on the hydrodynamic force and torque on the particle, and the detachment of the particle becomes more difficult as the particle sits deeper in a larger hole. Both the empirical correlations and the numerical approach can be implemented into Lagrangian particle tracking and can accurately predict the detachment of particles from the surface with concave roughness or the detachment of particles embedded in a flat surface.

Owing to the Taylor–Proudman theorem, it is generally believed that rotating flows should have two-dimensional dynamics for rapid background rotation. Yet, we show that two infinitely long counter-rotating vertical vortices always remain unstable to three-dimensional perturbations even for large rotation rate about the vertical axis. The dominant instability consists of quasi-antisymmetric displacements of the two vortices. Its growth rate is independent of the rotation rate when it is sufficiently large, while the most amplified vertical wavelength scales like the rotation rate. Direct numerical simulations show that the instability leads ultimately to a full three-dimensional breakdown of the vortex pair. In other words, a two-dimensional vortex pair will spontaneously develop three-dimensional variations even in the limit of infinite rotation rate but the wavelength will tend to infinity. This implies that the Taylor–Proudman theorem can be strictly valid only in vertically bounded flows. The scaling for the wavelength is next generalized by showing that the typical vertical scale in rapidly rotating unbounded flows is $L_v \sim L_h/Ro$, where $L_h$ is the horizontal scale and $Ro$ the Rossby number. This scaling law is shown to derive from a self-similarity of the Navier–Stokes equations for $Ro \ll 1$. The resulting reduced equations are identical to those obtained first by Julien et al. (Theor. Comput. Fluid Dyn., vol. 11, issue 3–4, 1998, pp. 251–261) and Nazarenko & Schekochihin (J. Fluid Mech., vol. 677, 2011, pp. 134–153) by means of multiscale analyses in the cases of rotating convection or turbulence. The self-similarity demonstrated herein suggests that these reduced equations are valid for any rapidly rotating unbounded flow.

The problem of a plane unsteady potential flow of an ideal incompressible fluid bounded by free boundary segments with a constant pressure and by solid walls moving in accordance with a known law is considered. External forces are absent, and capillary forces are neglected. An approach to constructing exact solutions for this type of problem is proposed. The corresponding solutions can be treated as nonlinear perturbations of a certain base flow. As an example of the application of this approach, nonlinear perturbations in a known problem of a fluid flow with a linear velocity field in the region bounded by a straight-line free boundary and parallel approaching or receding solid walls are considered. It is demonstrated that perturbations grow, which leads to variants of the formation of singularities on the free surface of the fluid within a finite time: formation of droplets, bubbles or cusps. A solution describing the collapse of a bubble in a fluid layer bounded by two approaching solid walls has also been found and studied. Thus, a new method of studying nonlinear stability of complicated unsteady fluid flows with combined boundary conditions is proposed and tested.

We present a detailed derivation of the Reynolds equation and its corresponding energy equation for three-dimensional, steady, laminar, compressible flows of single-phase Navier–Stokes fluids in thrust bearings. These equations are shown to be valid over most of the dense and supercritical gas regime except for the vicinity of the thermodynamic critical point. It is shown that the primary thermodynamic function governing the lubrication flow of high-pressure gases is the effective bulk modulus defined as the ratio of the bulk modulus to the shear viscosity. Numerical solutions to our Reynolds equation are obtained using a finite difference scheme for both moderate and high-speed flows. Approximate solutions to our Reynolds equation for high-speed flows are also derived through a perturbation analysis. It is found that boundary layers form on three out of four edges of the thrust pad. At the inner and outer radii of the pad, the flow is governed by a nonlinear heat equation. As the main flow leaves the pad, the flow is governed by a nonlinear relaxation equation. These three boundary layer solutions are rendered consistent by the construction of boundary layer solutions in the corner regions. A composite solution is developed which provides a single approximation and has the same accuracy as the individual approximations in their respective regions of validity.

A gravity-driven water film falling down an ice sheet is considered within the framework of a long-wave approximation. The integral-boundary-layer method, modified with the account of the phase transition, is adopted to describe the evolution of both the free surface of a water film and the interface between the ice and water. A set of governing equations consisting of five coupled nonlinear partial differential equations is established. The linear instability analysis of the uniform base flow is performed, and the result is in good agreement with the Orr–Sommerfeld analysis of the linearized Navier–Stokes equations. The phase transition at the interface between the ice and water plays a role in stabilizing the system linearly with long-wavelength perturbations. The nonlinear solutions of the steady travelling waves are constructed numerically. The phase transition tends to suppress the dispersion of the interfacial wave. Comparisons to direct numerical simulation of the Navier–Stokes equations, which are performed with an extended marker and cell method, show a remarkable agreement. The integral-boundary-layer method captures the water film thickness and the topography of the ice sheet satisfactorily. The phase transition is observed to enhance the backflow phenomenon in the capillary region of the solitary-like interfacial wave.

The stability of sliding spreading films of Herschel–Bulkley fluid is investigated theoretically, motivated by a dramatic fingering pattern observed experimentally and proposed theoretically to originate from an extensional flow instability of shear-thinning fluids. Considering the thin-film limit, we construct axisymmetric base states and then test their stability towards non-axisymmetric perturbations by numerically solving the initial-value problem. We complement the numerics with analytical solutions for early and late times. The stability analysis demonstrates that spreading thinning films are unstable. At late times, where the spreading of the base state becomes self-similar, non-axisymmetric patterns can develop strongly if the fluid has a yield stress or is sufficiently shear thinning.