A coin always floats in stable equilibrium with its longitudinal axis normal to the air–liquid interface. In contrast, a long thin cylindrical pin floats with its longitudinal axis parallel to the air–liquid interface. In this context, we present a theoretical investigation of the stability of small-scale hydrophobic cylinders floating in vertical and horizontal orientations at various aspect ratios (length/diameter). The study is limited to cylinders denser than water floating at the air–water interface. Our analysis shows that, unlike large-scale vertically floating cylinders, the stability of vertically orientated small-scale cylinders increases with an increase in aspect ratio. A similar trend is observed in the stability of small-scale horizontal cylinders. We also explain the underlying mechanics that leads to a rise in the stability of floating cylinders with an increase in aspect ratio. Unlike large-scale floating cylinders with uniform density, we show that the effect of governing forces (weight, buoyancy and surface tension) in small-scale cylinders changes from a stabilising to a destabilising force with a change in the aspect ratio. For example, in the case of a vertically floating cylinder, the buoyancy force acts as a stabilising force at a small aspect ratio whereas, at large aspect ratios, the buoyancy force has a destabilising influence. Likewise, the body's weight has a destabilising influence at a small aspect ratio and stabilising effect at a large aspect ratio. The reason behind this transformation is that, above a particular aspect ratio, the centre of gravity of small-scale floating cylinders lies below the centre of buoyancy.

The third-order law links energy transfer rates in the inertial range of magneto- hydrodynamic (MHD) turbulence with third-order structure functions. Anisotropy, a typical property in the solar wind, challenges the applicability of the third-order law with the isotropic assumption. To shed light on the energy transfer process in the presence of anisotropy, we conducted direct numerical simulations of forced MHD turbulence with normal and hyper-viscosity under various strengths of the external magnetic field ($B_0$), and calculated three forms of third-order structure function with or without averaging of the azimuthal or polar angles with respect to $B_0$ direction. Correspondingly, three estimated energy transfer rates were obtained. The result shows that the peak of normalized third-order structure function occurs at larger scales closer to the $B_0$ direction, and the maximum of longitudinal transfer rates shifts away from the $B_0$ direction at larger $B_0$. Compared with normal viscous cases, hyper-viscous cases can attain better separated inertial range, thus facilitating the estimation of the energy cascade rates. We find that the widespread use of the isotropic form of the third-order law in estimating the energy transfer rates is questionable in some cases, especially when the anisotropy arising from the mean magnetic field is inevitable. In contrast, the direction-averaged third-order structure function properly accounts for the effect of anisotropy and predicts the energy transfer rates and inertial range accurately, even at very high $B_0$. With limited statistics, the third-order structure function shows a stronger dependence on averaging of azimuthal angles than the time, especially for high $B_0$ cases. These findings provide insights into the anisotropic effect on the estimation of energy transfer rates.

The first long-lived turbulent structures observable in planar shear flows take the form of localized stripes, inclined with respect to the mean flow direction. The dynamics of these stripes is central to transition, and recent studies proposed an analogy to directed percolation where the stripes’ proliferation is ultimately responsible for the turbulence becoming sustained. In the present study we focus on the internal stripe dynamics as well as on the eventual stripe expansion, and we compare the underlying mechanisms in pressure- and shear-driven planar flows, respectively, plane-Poiseuille and plane-Couette flow. Despite the similarities of the overall laminar–turbulence patterns, the stripe proliferation processes in the two cases are fundamentally different. Starting from the growth and sustenance of individual stripes, we find that in plane-Couette flow new streaks are created stochastically throughout the stripe whereas in plane-Poiseuille flow streak creation is deterministic and occurs locally at the downstream tip. Because of the up/downstream symmetry, Couette stripes, in contrast to Poiseuille stripes, have two weak and two strong laminar turbulent interfaces. These differences in symmetry as well as in internal growth give rise to two fundamentally different stripe splitting mechanisms. In plane-Poiseuille flow splitting is connected to the elongational growth of the original stripe, and it results from a break-off/shedding of the stripe's tail. In plane-Couette flow splitting follows from a broadening of the original stripe and a division along the stripe into two slimmer stripes.

A long-wave asymptotic model is developed for the flow of an axisymmetric viscous film lining the interior of a tube for the case where slip occurs at the tube wall. Both the case of a falling film with a passive air core and that of a film driven up the tube by pressure-driven airflow are considered. The impact of slip on the net liquid volume flux is discussed, and linear stability analysis of the evolution equation is conducted to identify the impact of slip on the phase speed and growth rates of disturbances in each case. The presence of slip enhances the growth rates, though its impact on phase speed depends on the film thickness and the strength of the core airflow. For some parameter combinations, slip can modify the phase speed without altering the base flow. The nonlinear evolution of the free surface is then studied numerically. For falling films, increasing the slip length reduces the critical thickness required for plug formation to occur. Families of travelling wave solutions are found via continuation and are used to derive a simple formula for the dependence of this critical thickness on the slip length; this formula is shown to hold for small slip length. For air-driven films, the topology of streamlines in the film can be altered by slip at the wall; if the slip length is large enough, it can prevent regions of recirculation from forming at the wave crest.

When a blunt body impacts an air–water interface, large hydrodynamic forces often arise, a phenomenon many of us have unfortunately experienced in a failed dive or ‘belly flop’. Beyond assessing risk to biological divers, an understanding and methods for remediation of such slamming forces are critical to the design of numerous engineered naval and aerospace structures. Herein we systematically investigate the role of impactor elasticity on the resultant structural loads in perhaps the simplest possible scenario: the water entry of a simple harmonic oscillator. Contrary to conventional intuition, we find that ‘softening’ the impactor does not always reduce the peak impact force, but may also increase the force as compared with a fully rigid counterpart. Through our combined experimental and theoretical investigation, we demonstrate that the transition from force reduction to force amplification is delineated by a critical ‘hydroelastic’ factor that relates the hydrodynamic and elastic time scales of the problem.

Wave turbulence is the study of the long-time statistical behaviour of equations describing a set of weakly nonlinear interacting waves. Such a theory, which has a natural asymptotic closure, allows us to probe the nature of turbulence more deeply than the exact Kolmogorov laws by rigorously proving the direction of the cascade and the existence of an inertial range, predicting stationary spectra for conserved quantities, or evaluating the Kolmogorov constant. An emblematic example is given by fast rotating fluids for which a wave turbulence theory has been derived by Galtier (Phys. Rev. E, vol. 68, issue 1, 2003, p. 015301). This work involves non-trivial analytical developments for a problem that is anisotropic by nature. We propose here a new path for the derivation of the kinetic equation by using the anisotropy at the beginning of the analysis. We show that the helicity basis is not necessary to obtain the wave amplitude equation for the canonical variables that involve a combination of poloidal and toroidal fields. The multiple time scale method adapted to this anisotropic problem is then used to derive the kinetic equation that is the same as the original work when anisotropy is eventually taken into account. This result proves the commutativity between asymptotic closure and anisotropy. In addition, the multiple time scale method informs us that the kinetic equation can be derived without imposing restrictions on the probability distribution of the wave amplitude such as quasi-Gaussianity, or on the phase such as random phase approximation that naturally occurs dynamically.

The general problem of tracer diffusion in non-equilibrium baths is important in a wide range of systems, from the cellular level to geographical length scales. In this paper, we revisit the archetypical example of such a system: a collection of small passive particles immersed in a dilute suspension of non-interacting dipolar microswimmers, representing bacteria or algae. In particular, we consider the interplay between thermal (Brownian) diffusion and hydrodynamic (active) diffusion due to the persistent advection of tracers by microswimmer flow fields. Previously, it has been argued that even a moderate amount of Brownian diffusion is sufficient to significantly reduce the persistence time of tracer advection, leading to a significantly reduced value of the effective active diffusion coefficient $D_A$ compared to the non-Brownian case. Here, we show by large-scale simulations and kinetic theory that this effect is in fact practically relevant only for microswimmers that effectively remain stationary while still stirring up the surrounding fluid – so-called shakers. In contrast, for moderate and high values of the swimming speed, relevant for biological microswimmer suspensions, the effect of Brownian motion on $D_A$ is negligible, leading to the effects of advection by microswimmers and Brownian motion being additive. This conclusion contrasts with previous results from the literature, and encourages a reinterpretation of recent experimental measurements of $D_A$ for tracer particles of varying size in bacterial suspensions.

Two-way momentum-coupled direct numerical simulations of a particle-laden turbulent channel flow are addressed to investigate the effect of the particle Stokes number and of the particle-to-fluid density ratio on the turbulence modification. The exact regularised point-particle method is used to model the interphase momentum exchange in presence of solid boundaries, allowing the exploration of an extensive region of the parameter space. Results show that the particles increase the friction drag in the parameter space region considered, namely the Stokes number $St_+ \in [2,80]$, and the particle-to-fluid density ratio $\rho _p/\rho _f \in [90,5760]$ at a fixed mass loading $\phi =0.4$. It is noteworthy that the highest drag occurs for small Stokes number particles. A measurable drag increase occurs for all particle-to-fluid density ratios, the effect being reduced significantly only at the highest value of $\rho _p/\rho _f$. The modified stress budget and turbulent kinetic energy equation provide the rationale behind the observed behaviour. The particles’ extra stress causes an additional momentum flux towards the wall that modifies the structure of the buffer and of the viscous sublayer where the streamwise and wall-normal velocity fluctuations are increased. Indeed, in the viscous sublayer, additional turbulent kinetic energy is produced by the particles’ back-reaction, resulting in a strong augmentation of the spatial energy flux towards the wall where the energy is ultimately dissipated. This behaviour explains the increase of friction drag in particle-laden wall-bounded flows.

Drag for wall-bounded flows is directly related to the spatial flux of spanwise vorticity outward from the wall. In turbulent flows a key contribution to this wall-normal flux arises from nonlinear advection and stretching of vorticity, interpretable as a cascade. We study this process using numerical simulation data of turbulent channel flow at friction Reynolds number $Re_\tau =1000$. The net transfer from the wall of spanwise vorticity created by downstream pressure drop is due to two large opposing fluxes, one which is ‘down-gradient’ or outward from the wall, where most vorticity concentrates, and the other which is ‘up-gradient’ or toward the wall and acting against strong viscous diffusion in the near-wall region. We present evidence that the up-gradient/down-gradient transport occurs by a mechanism of correlated inflow/outflow and spanwise vortex stretching/contraction that was proposed by Lighthill. This mechanism is essentially Lagrangian, but we explicate its relation to the Eulerian anti-symmetric vorticity flux tensor. As evidence for the mechanism, we study (i) statistical correlations of the wall-normal velocity and of wall-normal flux of spanwise vorticity, (ii) vorticity flux cospectra identifying eddies involved in nonlinear vorticity transport in the two opposing directions and (iii) visualizations of coherent vortex structures which contribute to the transport. The ‘D-type’ vortices contributing to down-gradient transport in the log layer are found to be attached, hairpin-type vortices. However, the ‘U-type’ vortices contributing to up-gradient transport are detached, wall-parallel, pancake-shaped vortices with strong spanwise vorticity, as expected by Lighthill's mechanism. We discuss modifications to the attached eddy model and implications for turbulent drag reduction.

We study the effect of continuum breakdown on the two-dimensional thermoacoustic radiation of a thin plate set in a perfect monatomic gas. The plate is heated harmonically in time and the acoustic field is investigated in the entire range of gas rarefaction rates. Analytical approximations are obtained in the limits of high (free-molecular) and low (continuum limit) gas rarefaction, accompanied by direct simulation Monte Carlo calculations at intermediate flow conditions. While the source acoustic field is of a monopole type in the continuum regime, it turns non-isotropic at non-continuum conditions, exhibiting dipole directivity and exponential decay rate in the collisionless limit. The combined effects of source heating frequency and gas Knudsen number on the far-field acoustic radiation are illustrated and rationalized.

This study investigates the effect of vibration on the flow structure transitions in thermal vibrational convection (TVC) systems, which occur when a fluid layer with a temperature gradient is excited by vibration. Direct numerical simulation (DNS) of TVC in a two-dimensional enclosed square box is performed over a range of dimensionless vibration amplitudes $0.001 \le a \le 0.3$ and angular frequencies $10^{2} \le \omega \le 10^{7}$, with a fixed Prandtl number of 4.38. The flow visualisation shows the transition behaviour of flow structure upon the varying frequency, characterising three distinct regimes, which are the periodic-circulation regime, columnar regime and columnar-broken regime. Different statistical properties are distinguished from the temperature and velocity fluctuations at the boundary layer and mid-height. Upon transition into the columnar regime, columnar thermal coherent structures are formed, in contrast to the periodic oscillating circulation. These columns are contributed by the merging of thermal plumes near the boundary layer, and the resultant thermal updrafts remain at almost fixed lateral position, leading to a decrease in fluctuations. We further find that the critical point of this transition can be described nicely by the vibrational Rayleigh number ${{Ra}}_{vib}$. As the frequency continues to increase, entering the so-called columnar-broken regime, the columnar structures are broken, and eventually the flow state becomes a large-scale circulation (LSC), characterised by a sudden increase in fluctuations. Finally, a phase diagram is constructed to summarise the flow structure transition over a wide range of vibration amplitude and frequency parameters.

We propose a constitutive equation for flows of gases in high-rate regimes where the Navier–Stokes theory breaks down. The model generalizes the Navier–Stokes relation and agrees well with that model in all lower rate flows examined. Our proposed constitutive relation is calibrated with the method of objective molecular dynamics (OMD) using families of compressible and incompressible flows of Lennard-Jones argon. The constitutive relation makes use of the higher-order objective strain rates due to Rivlin and Ericksen (J. Rat. Mech. Anal., vol. 4, 1955, pp. 323–425). The constitutive relation is fully frame-indifferent, and the macroscopic flows corresponding to the OMD simulations are exact solutions for the proposed model. The model is shown to agree with atomistic results much better than the Navier–Stokes equations in the transition regime. The success of our model indicates that it is not higher gradients that become important in the high-rate regime, but rather higher rates of change of the strain rate tensor. While somewhat more complicated to implement than the Navier–Stokes relation, the proposed model is expected to be compatible with existing methods of computational fluid dynamics and may extend those methods to higher rate regimes, while preserving their ability to handle large spatial scales.

Understanding the skin friction in an axisymmetric turbulent boundary layer (ATBL) flow is a key to designing and optimising the flow past axisymmetric bodies, for example, a rocket engine nozzle and a submarine hull. In this study, we propose a universal law of the skin-friction coefficient in an ATBL flow. The flow is steady and fully developed with a zero pressure gradient. The governing equation for the ATBL flow is derived by methodically applying the boundary layer approximation. Subsequently, the scaling law of the Reynolds shear stress, caused by turbulent eddies, at the surface tangential to the wall roughness summits is derived by incorporating the role of transverse curvature. The skin-friction coefficient in a smooth ATBL flow is found to depend on two parameters, namely, the Reynolds number based on the cylinder radius, Rea, and the ratio of boundary layer thickness to cylinder radius, δ/a. The analysis predicts a simple form of the skin-friction coefficient as ${C_f} = 4.56 \times {10^{ - 2}}{[R{e_a}\ln (1 + \delta /a)]^{ - 1/4}}$, which agrees satisfactorily with the available experimental data and the numerical simulations in all the axisymmetric flow regimes. The proposed law, in the limit of infinite radius, is consistent with the classical law of the skin-friction coefficient in a plane turbulent boundary layer flow as ${C_{f0}}\sim Re_\delta ^{ - 1/4}$, where Reδ is the Reynolds number based on the boundary layer thickness. This study reveals that, for δ/a < 1, the relative skin-friction coefficient, $({C_f} - {C_{f0}})/{C_{f0}}$, follows a linear law with δ$/$a.

We investigate steady and oscillatory flow through a hexagonal close-packed arrangement of spheres in the framework of the volume-averaged momentum equation. We quantify the friction and pressure drag based on a direct numerical simulation dataset. Using the pressure decomposition of Graham (J. Fluid Mech., vol. 881, 2019), the pressure drag can be further split up into an accelerative, a viscous and a convective contribution. For the accelerative pressure, a closed-form expression can be given in terms of the potential flow solution. We investigate the contributions of the different drag components to the volume-averaged momentum budget and their Reynolds number scaling. For steady flow, we find that the friction and viscous pressure drag are proportional to $Re$ at low Reynolds numbers and scale with $Re^{1.4}$ for high Reynolds numbers. This is close to the steady laminar boundary layer scaling. For the convective pressure drag, we find a cubic scaling at low and a quadratic scaling at high Reynolds numbers. The Reynolds stresses have a minor contribution to the momentum budget. For oscillatory flow at low and medium Womersley numbers, the amplitudes of the drag components are similar to the steady cases at the same Reynolds number. At high Womersley numbers, the drag components behave quite differently and the friction and viscous pressure drag are relatively insignificant. The drag components are not in phase with the forcing and the superficial velocity; the phase lag increases with the Womersley number. This suggests that new models beyond the current quasisteady approaches need to be developed.

Non-stationarity is the rule in the atmospheric boundary layer (ABL). Under such conditions, the flow may experience departures from equilibrium with the underlying surface stress, misalignment of shear stresses and strain rates, and three-dimensionality in turbulence statistics. Existing ABL flow theories are primarily established for statistically stationary flow conditions and cannot predict such behaviours. Motivated by this knowledge gap, this study analyses the impact of time-varying pressure gradients on mean flow and turbulence over urban-like surfaces. A series of large-eddy simulations of pulsatile flow over cuboid arrays is performed, programmatically varying the oscillation amplitude $\alpha$ and forcing frequency $\omega$. The analysis focuses on both longtime-averaged and phase-dependent flow dynamics. Inspection of longtime-averaged velocity profiles reveals that the aerodynamic roughness length $z_0$ increases with $\alpha$ and $\omega$, whereas the displacement height $d$ appears to be insensitive to these parameters. In terms of oscillatory flow statistics, it is found that $\alpha$ primarily controls the oscillation amplitude of the streamwise velocity and Reynolds stresses, but has a negligible impact on their wall-normal structure. On the other hand, $\omega$ determines the size of the region affected by the unsteady forcing, which identifies the so-called Stokes layer thickness $\delta _s$. Within the Stokes layer, phase-averaged resolved Reynolds stress profiles feature substantial variations during the pulsatile cycle, and the turbulence is out of equilibrium with the mean flow. Two phenomenological models have been proposed that capture the influence of flow unsteadiness on $z_0$ and $\delta _s$, respectively.

Three-dimensional schools of hydrodynamically axisymmetric swimmers self-propelling at a constant velocity are studied. We introduce a low-order model for the induced velocity based on the far-field approximation. We inquire if, by holding suitable relative positions in the three-dimensional space, the swimmers can reduce the overall energy consumption of the school in comparison with the same number of isolated individuals at the same velocity. We find a considerable (several per cent) energy saving achievable by chain formations. The benefit increases asymptotically with the number of individuals, towards a finite limit that is a function of the minimum allowed spacing between each pair of neighbours.

This paper investigates the multilayer Rayleigh–Taylor instability (RTI) using statistically stationary experiments conducted in a gas tunnel. Employing diagnostics such as particle image velocimetry (PIV) and planar laser induced fluorescence (PLIF), we make simultaneous velocity–density measurements to study how dynamics and mixing are linked in this variable density flow. Experiments are conducted in a newly built, blow-down three-layer gas tunnel facility. Mixing between three gas streams is studied, where the top and bottom streams are comprised of air, and the middle stream is an air–helium mixture. Shear is minimized between these streams by matching their inlet velocities. The four experimental conditions investigated here consist of two different density ratios (Atwood numbers 0.3 and 0.6), each investigated at two instability development times (or equivalently, two streamwise locations), and all experiments are with the same middle stream thickness of 3 cm. The growth of the middle layer is measured using laser-based planar Mie scattering visualization. The mixing width is found to grow linearly with time at late times. Various quantitative measures of molecular mixing indicate a very high degree of molecular mixing at late times in the multilayer RTI flow. The vertical turbulent mass flux $a_y$ is calculated. In addition to mostly negative values of $a_y$, typical of buoyancy-dominated flows due to negative correlation between velocity and density fluctuations, positive regions are also observed in profiles of $a_y$ due to entrainment and erosion at the lower edge of the mixing region. Global energy budgets are calculated for the multilayer RTI flow at late times and it is found that the majority of potential energy released has been dissipated due to viscous effects, and a large value of mixing efficiency ($\sim$60 %) is observed.

To elucidate the effect of particle shape on the rheology of a dense, viscous suspension of frictional, non-Brownian particles, experimental measurements are presented for suspensions of polystyrene particles with different shapes in the same solvent. The first suspension is made of spheres whereas the particles which compose the second suspension are globular but with flattened faces. We present results from steady shear and shear-reversal rheological experiments for the two suspensions over a wide range of stresses in the viscous regime. Notably, we show that the rheology of the two suspensions is characterised by a shear-thinning behaviour, which is stronger in the case of the suspension of globular particles. Since the shear-reversal experiments indicate an absence of adhesive particle interactions, we attribute the shear thinning to a sliding friction coefficient which varies with stress as has been observed previously for systems similar to the first suspension. We observe that the viscosity of the two suspensions is similar at high shear stress where small sliding friction facilitates particle relative motion due to sliding. At lower shear stress, however, the sliding friction is expected to increase and the particle relative motion would be associated with rolling. The globular particles attain a higher viscosity at low shear stress than the spherical particles. We attribute this difference to a shape-induced resistance to particle rolling that is enhanced by the flattened faces. Image analysis is employed to identify features of the particle geometry that contribute to the resistance to rolling. It is shown that the apparent rolling friction coefficients inferred from the rheology are intermediate between the apparent dynamic and static rolling friction coefficients predicted on the basis of the image analysis. All three rolling resistance estimates are larger for the globular particles with flat faces than for the spherical particles and we argue that this difference yields the stronger shear thinning of the globular particle suspension.

A solid body in a viscous fluid undergoing oscillatory motion naturally produces a steady secondary flow due to convective inertia. This phenomenon is embodied in the streaming flow generated by a sphere in an unbounded fluid executing rectilinear oscillations. We review the considerable literature on this canonical problem and summarise exact and asymptotic formulas in the small-amplitude limit. These analytical formulas are used to explore the characteristic flow structure of this problem and clarify previously unreported features. A single, toroidal-shaped vortex exists in each hemisphere regardless of the oscillation frequency, which can drive a counter-flow away from the sphere. The vortex centre moves monotonically away from the sphere with decreasing oscillation frequency, and engulfs the entire flow domain for $\beta \equiv \omega R^2/\nu < 16.317$, where $\omega$ is the angular oscillation frequency, $R$ the sphere radius, and $\nu$ the fluid kinematic viscosity. This seemingly abrupt change in flow structure at the critical frequency $\beta _{critical} =16.317$, and its quantification, appear to have not been reported previously. We perform a direct numerical simulation of the Navier–Stokes equations, to (1) confirm existence of this critical frequency at finite amplitude, and (2) examine its variation with amplitude. This reveals a universal relationship between the critical frequency and oscillation amplitude, clarifying previous reports on the structure of this streaming flow. The critical frequency is shown to be identical for the streaming flow and the cycle-averaged particle paths, establishing that the critical frequency is accessible directly using standard measurements.

This study presents a comprehensive analysis on the extreme positive and negative events of wall shear stress and heat flux fluctuations in compressible turbulent boundary layers (TBLs) solved by direct numerical simulations. To examine the compressibility effects, we focus on the extreme events in two representative cases, i.e. a supersonic TBL of Mach number $M=2$ and a hypersonic TBL of $M=8$, by scrutinizing the coherent structures and their correlated dynamics based on conditional analysis. As characterized by the spatial distribution of wall shear stress and heat flux, the extreme events are indicated to be closely related to the structural organization of wall streaks, in addition to the occurrence of the alternating positive and negative structures (APNSs) in the hypersonic TBL. These two types of coherent structures are strikingly different, namely the nature of wall streaks and APNSs are shown to be related to the solenoidal and dilatational fluid motions, respectively. Quantitative analysis using a volumetric conditional average is performed to identify and extract the coherent structures that directly account for the extreme events. It is found that in the supersonic TBL, the essential ingredients of the conditional field are hairpin-like vortices, whose combinations can induce wall streaks, whereas in the hypersonic TBL, the essential ingredients become hairpin-like vortices as well as near-wall APNSs. To quantify the momentum and energy transport mechanisms underlying the extreme events, we proposed a novel decomposition method for extreme skin friction and heat flux, based on the integral identities of conditionally averaged governing equations. Taking advantage of this decomposition method, the dominant transport mechanisms of the hairpin-like vortices and APNSs are revealed. Specifically, the momentum and energy transports undertaken by the hairpin-like vortices are attributed to multiple comparable mechanisms, whereas those by the APNSs are convection dominated. In that, the dominant transport mechanisms in extreme events between the supersonic and hypersonic TBLs are indicated to be totally different.