The flow past a cylinder in proximity to a plane wall is investigated numerically for small gap ratios. Three vortex dynamic processes associated with different hairpin vortex generation mechanisms are identified for the first time, and the wake-induced turbulent transition is analysed. The vortex shedding is suppressed at $G/D = 0.1$, while the spanwise vortex is generated via a Kelvin–Helmholtz instability and evolves into hairpin vortices. For $G/D= 0.3$, the upper and lower rollers alternatively shedding from the cylinder, interact with the secondary vortex. The split secondary vortex merges with the upper roller and results in a new vortex downstream, which develops into hairpin vortices. When $G/D = 0.9$, the secondary vortex interacts with the lower roller and then evolves into hairpin vortices. A tertiary vortex induced by the secondary vortex is observed, rotating in the opposite direction to the secondary vortex the wake-induced transitions share the same route. The velocity fluctuations deviate from the optimal growth theory in the pre-transitional region. In the transitional region low-frequency disturbances penetrate the sheltering edge to generate streaks where the disturbance energy declines. In the turbulent region the logarithmic layer is formed, indicating that the turbulent equilibrium is established.

We revisit the construction of discrete kinetic models for single-component isothermal two-phase flows. Starting from a kinetic model for a non-ideal fluid, we show that, under conventional scaling, the Navier–Stokes equations with a non-ideal equation of state are recovered in the hydrodynamic limit. A scaling based on the smallness of velocity increments is then introduced, which recovers the full Navier–Stokes–Korteweg equations. The proposed model is realized on a standard lattice and validated on a variety of benchmarks. Through a detailed study of thermodynamic properties including co-existence densities, surface tension, Tolman length and sound speed, we show thermodynamic consistency, well-posedness and convergence of the proposed model. Furthermore, hydrodynamic consistency is demonstrated by verification of Galilean invariance of the dissipation rate of shear and normal modes and the study of visco-capillary coupling effects. Finally, the model is validated on dynamic test cases in three dimensions with complex geometries and large density ratios such as drop impact on textured surfaces and mercury drops coalescence.

The dispersed phase in turbulence can vary from almost inviscid fluid to highly viscous fluid. By changing the viscosity of the dispersed droplet phase, we experimentally investigate how the deformability of dispersed droplets affects the global transport quantity of the turbulent emulsion. Different kinds of silicone oil are employed to result in the viscosity ratio, $\zeta$, ranging from $0.53$ to $8.02$. The droplet volume fraction, $\phi$, is varied from 0 % to 10 % with a spacing of 2 %. The global transport quantity, quantified by the normalized friction coefficient $c_{f,\phi }/c_{f,\phi =0}$, shows a weak dependence on the turbulent intensity due to the vanishing finite-size effect of the droplets. The interesting fact is that, with increasing $\zeta$, $c_{f,\phi }/c_{f,\phi =0}$ first increases and then saturates to a plateau value which is similar to that of the rigid particle suspension. By performing image analysis, this drag modification is interpreted from the point of view of droplet deformability, which is responsible for the breakup and coalescence effect of the droplets. The statistics of the droplet size distribution show that, with increasing $\zeta$, the stabilizing effect induced by interfacial tension becomes substantial and the pure inertial breakup process becomes dominant. The measurement of the droplet distribution along the radial direction of the system shows a bulk-clustering effect, which can be attributed to the non-negligible coalescence effect of the droplet. It is found that the droplet coalescence effect could be suppressed as $\zeta$ increases, thereby affecting the contribution of interfacial tension to the total stress, and accounting for the observed emulsion rheology.

We present a numerical analysis of the lateral movement and equilibrium radial positions of red blood cells (RBCs) with major diameter 8 $\mathrm {\mu }$m under a Newtonian fluid in a circular channel with 50 $\mathrm {\mu }$m diameter. Each RBC, modelled as a biconcave capsule whose membrane satisfies strain-hardening characteristics, is simulated for different Reynolds numbers $Re$ and capillary numbers $Ca$, the latter of which indicates the ratio of the fluid viscous force to the membrane elastic force. The effects of initial orientation angles and positions on the equilibrium radial position of an RBC centroid are also investigated. The numerical results show that depending on their initial orientations, RBCs have bistable flow modes, so-called rolling and tumbling motions. Most RBCs have a rolling motion. These stable modes are accompanied by different equilibrium radial positions, where tumbling RBCs are further away from the channel axis than rolling ones. The inertial migration of RBCs is achieved by alternating orientation angles, which are affected primarily by the initial orientation angles. Then the RBCs assume the aforementioned bistable modes during the migration, followed by further migration to the equilibrium radial position at much longer time periods. The power (or energy dissipation) associated with membrane deformations is introduced to quantify the state of membrane loads. The energy expenditures rely on stable flow modes, the equilibrium radial positions of RBC centroids, and the viscosity ratio between the internal and external fluids.

This paper provides a comprehensive explanation for the lift force acting on a freely deformable bubble rising in a linear shear flow and examines how the lift force scales with the undisturbed shear rate in cases governed by different lift force mechanisms. Four distinct flow mechanisms are identified from previous studies, and the associated bubble-induced vorticity dynamics are outlined. We provide a theoretical framework to qualitatively explain the lift force acting on a bubble in terms of moments of the bubble-induced vorticity. We support our theoretical framework with three-dimensional multiphase direct numerical simulations to illustrate how the vorticity dynamics associated with the four mechanisms generate the lift force. These findings provide a comprehensive explanation for the behaviour of the lift force in a wide range of relevant governing parameters. Additionally, our simulation results show how differently the lift force scales with the shear rate, depending on the dominating lift force mechanism. These results indicate that the shear rate should, in general, be accounted for in highly viscous flows (low Galilei numbers) or at significant bubble deformations (moderate-to-high Eötvös numbers) when modelling the lift force coefficient.

The classical Gill's stability problem for stationary and parallel buoyant flow in a vertical porous slab with impermeable and isothermal boundaries kept at different temperatures is reconsidered from a different perspective. A three-layer slab is studied instead of a homogeneous slab as in Gill's problem. The three layers have a symmetric configuration where the two external layers have a high thermal conductivity, while the core layer has a much lower conductivity. A simplified model is set up where the thermal conductivity ratio between the external layers and the internal core is assumed as infinite. It is shown that a flow instability in the sandwiched porous slab may arise with a sufficiently large Rayleigh number. It is also demonstrated that this instability coincides with that predicted in a previous analysis for a homogeneous porous layer with permeable boundaries, by considering the limiting case where the permeability of the external layers is much larger than that of the core layer.

We study a gravity-driven viscous flow coating a vertical cylindrical fibre. The destabilisation of a draining liquid column into a downward moving train of beads has been linked to the conjunction of the Rayleigh–Plateau and Kapitza instabilities in the limit of small Bond numbers $Bo$. Here, we focus on quasi-inertialess flows (large Ohnesorge number $Oh$) and conduct a linear stability analysis on a unidirectional flow along a rigid eccentric fibre for intermediate to large $Bo$. We show the existence of two unstable modes, namely pearl and whirl modes. The pearl mode depicts asymmetric beads, similar to that of the Rayleigh–Plateau instability, whereas a single helix forms along the axis in the whirl mode instability. The geometric and hydrodynamic thresholds of the whirl mode instability are investigated, and phase diagrams showing the transition thresholds between different regimes are presented. Additionally, an energy analysis is carried out to elucidate the whirl formation mechanism. This analysis reveals that despite the unfavourable capillary energy cost, the asymmetric interface shear distribution, caused by the fibre eccentricity, has the potential to sustain a whirling interface. In general, small fibre radius and large eccentricity tend to foster the whirl mode instability, while reducing $Bo$ tends to favour the dominance of the pearl mode instability. Finally, we compare the predictions of our model with the results of some illustrative experiments, using highly viscous silicone oils flowing down fibres. Whirling structures are observed for the first time, and the measured wavenumbers match our stability analysis prediction.

We study the flow above non-optimal riblets, specifically large drag-increasing and two-scale trapezoidal riblets. In order to reach large Reynolds numbers and large scale separation while retaining access to flow details, we employ a combination of boundary-layer hot-wire measurements and direct numerical simulation (DNS) in minimal-span channels. Although the outer Reynolds numbers differ, we observe fair agreement between experiments and DNS at matched viscous–friction-scaled riblet spacings $s^+$ in the overlapping physical and spectral regions, providing confidence that both data sets are valid. We find that hot-wire velocity spectra above very large riblets with $s^+ \gtrsim 60$ are depleted of near-wall energy at scales that are (much) greater than $s$. Large-scale energy likely bypasses the turbulence cascade and is transferred directly to secondary flows of size $s$, which we observe to grow in strength with increasing riblet size. Furthermore, the present very large riblets reduce the von Kármán constant $\kappa$ of the spanwise uniform mean velocity in a logarithmic layer and, thus, reduce the accuracy of the roughness-function concept, which we link to the near-wall damping of large flow structures. Half-height riblets in the groove, which we use as a model of imperfectly repeated (spanwise-varying) riblets, impede in-groove turbulence. We show how to scale the drag optimum of imperfectly repeated riblets based on representative measurements of the true geometry by solving inexpensive Poisson equations.

Since its development, Stokesian dynamics has been a leading approach for the dynamic simulation of suspensions of particles at arbitrary concentrations with full hydrodynamic interactions. Although developed originally for the simulation of passive particle suspensions, the Stokesian dynamics framework is equally well suited to the analysis and dynamic simulation of suspensions of active particles, as we elucidate here. We show how the reciprocal theorem can be used to formulate the exact dynamics for a suspension of arbitrary active particles, and then show how the Stokesian dynamics method provides a rigorous way to approximate and compute the dynamics of dense active suspensions where many-body hydrodynamic interactions are important.

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.

In this study, the effects of rotor–rotor interaction on wake characteristics were investigated experimentally for a twin-rotor configuration in axial descent. The wake velocities were measured at descent rates (descent speed/induced velocity at the rotor disk during hover) from 0.87 to 1.52, and the rotor–rotor interaction strength was controlled by adjusting the distance between the rotor tips. As the descent rate increased, the wake of the isolated rotor gradually entered the vortex ring state (VRS), where the flow established an extensive recirculation zone. Correlation analysis was performed to distinguish the rotor wake between tubular and VRS topologies. The flow states for the isolated rotor were classified into pre-VRS, incipient VRS, and fully developed VRS, depending on the probability of vortex ring formation. The results reveal that the effects of rotor–rotor interaction on the wake characteristics of twin rotors differ depending on the descent rate, distance between rotor tips, and wake region. In the outer region, the flow state of the rotor wake remains consistent with that of the isolated rotor, irrespective of the distance between rotor tips. Conversely, the strong rotor–rotor interaction changes the flow state in the inner region by disrupting the vortex ring structure, intensifying the wake asymmetry about the rotational axis. The thrust measurements show that under the VRS, as the two rotors get closer, the thrust coefficient increases until vortex ring disruption occurs, and then decreases after the vortex ring is disrupted.

The hydrodynamic cavitation in semidilute solution flows of a flexible polymer additive in water was experimentally explored in a mesoscale converging–diverging nozzle to elucidate the cavitation reduction effects of polymer additives. Rheological measurements demonstrated that polymer solutions were shear-thinning, with infinite viscosities larger than pure water. The polymer additives significantly mitigated the intensity of cloud cavitation and the growth of violent cavity structures in the tested solution concentrations. Under conditions of supercavitation, the tested polymer solutions could not suppress the growth of large structures but showed a reduction in the population of cavitation bubbles. The temporal evolution and spatial variation of cavitation structures in different concentrations were captured using high-speed imaging. Statistical analysis of the images showed that polymers reduce cavitation via three main mechanisms. (1) The longitudinal expansion of cavities downstream is attenuated relative to the pure water. The streamwise distribution of vapour-ratio fluctuations was flattened, and its peak was shifted upstream in the solutions. (2) Mean collapse and growth rate of cavitating bubble pockets and their fluctuations were noticeably relaxed by polymer additives. For a 400 p.p.m. solution (parts per million (p.p.m.)), a reduction of 65 % was measured relative to pure water flow at the highest tested flow rate. (3) Spectral analysis of the downstream pressure indicated that the shedding frequency at the cavitation inception was reduced as the solution's concentration increased. This reduction was as high as 70 % for a 400 p.p.m. solution. These results highlight the strong interplay between polymer additives and the generation of cavitation-related structures.

The radiation force exerted on an object by an acoustic wave is a widely studied phenomenon since the early work of Rayleigh, Langevin and Brillouin, and has led in the last decade to tremendous developments for acoustic micromanipulation. Despite extensive work on this phenomenon, the expressions of the acoustic radiation force applied on a particle have so far been derived only for a steady particle, hence neglecting the effect of its displacement on the radiated wave. In this work, we study the acoustic radiation force exerted on a monopolar source translating at a constant velocity that is small compared to the sound speed. We demonstrate that the asymmetry of the emitted field resulting from the Doppler effect induces a radiation force on the source opposite to its motion.

This paper presents a detailed analysis of the flows induced in a long two-dimensional cavity heated from below in the presence of streaming due to ultrasound acoustic waves emitted by a source. The problem is tackled by using performing spectral element codes, allowing continuation of steady solutions, bifurcation points and periodic cycles. For a given dimensionless source size, the governing parameters are the acoustic streaming parameter $A$ which modulates the acoustic force generating the Eckart streaming and the Rayleigh number ${\textit {Ra}}$ which quantifies the buoyant force responsible for the convection. The streaming flow, which goes to the right along the horizontal axis and returns along the lower and upper boundaries, influences the instability thresholds, which are first strongly stabilized above the pure Rayleigh–Bénard threshold ${\textit {Ra}}_0$ when $A$ is increased, before a destabilization to reach the pure streaming threshold $A_c$ at ${\textit {Ra}}=0$. The steady multi-roll convective flow generated without streaming is replaced by periodic waves when $A$ is increased, forward waves for moderate $A$ and backward waves for large $A$. The transition between these waves induces a specific dynamics involving steady flows, which has been elucidated. The waves also eventually disappear for a sufficient increase of the Rayleigh number, replaced by steady multi-roll flows hardly influenced by the streaming flow. A very rich dynamics is thus observed with the competition between the waves and the steady flows.

We revisit and extend the turbulent Froude number ($Fr_k$) scaling for the mixing coefficient ($\varGamma$) introduced by Garanaik & Venayagamoorthy (GV) (J. Fluid Mech., vol. 867, 2019, pp. 323–333) by directly incorporating the effects of mean shear through the non-dimensional shear parameter $S_{\ast } = S k/\epsilon _k$. For flows where the effects of mean shear are stronger than the background vertical stratification, we find $\varGamma \sim Fr_k^{-2} S_\ast ^{-1}$ for weakly stratified sheared turbulence and $\varGamma \sim Fr_k^{-1}S_\ast ^{-1}$ for moderately stratified sheared turbulence. The scaling procedure is inconclusive for strongly stratified sheared turbulence, but using two independent datasets of homogeneous, sheared, stably stratified turbulence, we empirically observe $\varGamma \sim Fr_k^{-0.5} S_\ast ^{-1}$. Our revised scaling better collapses both datasets compared with the original GV scaling, and we note that the moderately stratified sheared regime is extremely narrow (or maybe even non-existent). We also apply our scaling to the time-varying open channel simulations of Issaev et al. (J. Fluid Mech., vol. 935, 2022) and observe $\varGamma \sim Fr_k^{-2}S_\ast ^{-1}$ for weakly stratified sheared turbulence, but we observe deviations from our revised scaling for moderate and strong stratifications due to time-varying mean shear and vertical transport. Finally, we apply our revised scaling to field measurements of Conry, Kit & Fernando (Environ. Fluid Mech., vol. 20, 2020, pp. 1177–1197) and observe $\varGamma \sim Fr_k^{-2} S_\ast ^{-1}$. We emphasize that our revised scaling is applicable only for stably stratified, vertically sheared turbulence with weak spatio-temporal variations of the mean shear and stratification, and we expect different scaling to apply when additional effects such as depth-varying radiative heating/cooling are present or when the orientation of the mean shear relative to the gravity vector is modified (e.g. horizontal shear).

A recent experiment by Wang et al. (Soft Matt., vol. 17, 2021, pp. 2985–2993) shows that a self-propelled compound drop in a surfactant-laden solution can autonomously change its motion from a straight line to a spiraling trajectory, enhancing its capability for chemical detection, catalytic reaction and pollutant removal in a large fluid region. To understand the underlying physics of this peculiar motion, we develop a two-dimensional minimal model to study the swimming dynamics of a compound droplet driven by a self-generated Marangoni stress. We find that, depending on the Péclet number ($Pe$) and the viscosity and volume ratios of the two compound phases, the drop can swim in a variety of trajectories, including straight lines, circles, zigzag curves and chaotic trajectories. The drop moves in circles when its two components have comparable volumes. Otherwise, it shows other types of motions depending on $Pe$. Our simulation results for the circular motion at small $Pe$ are qualitatively comparable to the experiment. The transition between zigzag and circular trajectories is mainly determined by the orientation of high-order modes with respect to the drop's swimming direction. For most compound drops, the speed decays as $Pe^{-1/3}$ at high Péclet numbers as it does for a single-phase drop. A drop with two equal components undergoes a run-and-reorient motion due to the competition between the even and odd modes.

Understanding the fluid dynamics associated with a circular cylinder oscillating normal to a plane wall is important for safe design of offshore infrastructure, such as power cables and pipeline risers. This paper investigates the fluid dynamics of an oscillating cylinder with no imposed incident current experimentally using flow visualisation and force measurements where the ratio of the cylinder Reynolds number ($Re$) to Keulegan–Carpenter number ($KC$) is $\beta =500$ and $KC$ varies between 2 and 12. The minimum distance between the cylinder and wall was between 12.5 % and $50\,\%$ of the diameter. Across this parameter space three primary vortex flow regimes were observed: (i) for $KC\leq 5$, the flow field is approximately symmetric about the cylinder centreline and the velocity field between the cylinder and the wall resembled a pumping flow in phase with cylinder motion, which is well predicted by potential theory for most of the cycle; (ii) for $5< KC<8$, the flow field is increasingly asymmetric but with frequent switching of the side associated with vortex shedding; and (iii) for $KC\geq 8$, the flow field is consistently asymmetric due to vortex shedding. The in-line force increases when the cylinder is near the wall due to dynamic pressures associated with pumping. This increase can be estimated using potential theory superimposed onto the force time history for an isolated cylinder at the same $KC$ and $Re$. This study complements recent numerical modelling focused on low Reynolds number conditions and provides important insights into the fluid mechanics associated with trenching beneath cable and pipeline risers.

The propagation of the gravity current generated from a moving source of buoyancy is of interest in deep-sea mining and related technologies. The study by Ouillon et al. (J. Fluid Mech., vol. 924, 2021, A43) elucidated some salient patterns of the flow concerning a source close to the bottom on the basis of direct numerical simulation on a supercomputer. Here, we present a simple box model that provides further insights and useful analytical approximations for this gravity-current flow system. We show that this flow is very different from that produced by a moving source at the top, studied by Hogg et al. (J. Fluid Mech., vol. 539, 2005, pp. 349–385). The model confirms that the main governing parameter is the ratio $a$ of speed of source to that of buoyancy propagation. The model points out dependency also on the front-jump Froude number (which implies dependency on the height of the ambient fluid). For a sufficiently large $a >a_{crit}$, a supercritical regime appears in which the gravity current forms a wedge behind the moving source; in the subcritical regime, the upstream propagation attains a maximum $x_m$ at time $t_m$. The model predicts the value $a_{crit}$, the distance and time $x_m$ and $t_m$ in the subcritical case, and the shape of the wedge in the supercritical case, without any adjustable constant. Comparisons with the numerical data show fair agreement.

We characterise the incompressible turbulence cascade in terms of the concurrent inter-scale and inter-space exchanges of the scale-by-scale energy, helicity and enstrophy. The governing equations for the scale-by-scale helicity and enstrophy are derived in a similar fashion to that of the second order structure function following Hill (J. Fluid Mech., vol. 468, 2002, pp. 317–326). We examine the instantaneous dynamics, applying these equations to forced periodic turbulence and a von Kármán flow focusing on scales in the dissipative range $r=2.5\eta$, the near-dissipative range $r=0.5\lambda$ and the onset inertial range $r=\lambda$ (where $\eta$ and $\lambda$ are the Kolmogorov and Taylor length scales, respectively). The signature of the random sweeping effect is observed in all three individual budgets and between the energy and enstrophy transfers. As in the energy cascade, the anti correlation of the pressure transport and non-linear transfer is identified also in the helicity cascade. Owing to its lack of positive definiteness, the helicity transfers are found to be decorrelated from the others. However a connection between the energy cascade and helicity is identified kinematically. This connection reveals the large-scale sweeping motions are a key element in the overall energy cascade and underpins previous observations of large-scale intermittency. Taken together, this work extends a classic framework to gain novel insight on turbulence dynamics that underlay the statistically steady state, and demonstrates how transfers are interconnected.

We show that in free surface flows, a uniform, streamwise current over small-amplitude wavy bottom topography generates cross-stream drift velocity. This drift mechanism, referred to as the current–bathymetry interaction-induced drift (CBIID), is specifically understood in the context of a simplified nearshore environment consisting of a uniform alongshore current, onshore-propagating surface waves and monochromatic wavy bottom making an oblique angle with the shoreline. The CBIID is found to originate from the steady, non-homogeneous solution of the governing system of equations. Similar to Stokes drift induced by surface waves, CBIID also generates a compensating Eulerian return flow to satisfy the no-flux lateral boundaries, e.g. the shoreline. The CBIID increases with an increase in particle's initial depth, bottom undulation amplitude and the strength of the alongshore current. Additionally, CBIID near the free (bottom) surface increases (decreases) with an increase in bottom undulation's wavelength. Maximum CBIID is obtained for long-wavelength bottom topography that makes an angle of approximately ${\rm \pi} /4$ with the shoreline. Unlike Stokes drift, particle excursions due to current–bathymetry interactions might not be small, and hence analytical expressions based on the small-excursion approximation could be inaccurate. We provide an alternative $z$-bounded approximation, which leads to highly accurate expressions for drift velocity and time period of particles especially located near the free surface. Realistic parametric analysis reveals that in some nearshore environments, CBIID's contribution to the net Lagrangian drift can be as important as Stokes drift, implying that CBIID can have major implications in cross-shelf tracer transport.