The average wave drag in unsteady motion is studied experimentally with force measurements. Towing hulls of size $L$ at sinusoidal speed, the mean drag is measured for different amplitudes and frequencies of the fluctuating velocity, as well as different Froude numbers $\mathcal {F}_0$ associated with the mean velocity $V_0$ ($\mathcal {F}_0 = V_0/\sqrt {gL}$). The wave drag is reported to be either increased or decreased by velocity fluctuations depending on $\mathcal {F}_0$. For small fluctuation amplitudes, this drag change is proportional to the square of the amplitude. The effect is maximized for a resonance frequency identified as the Wehausen frequency, which scales as $\sqrt {g/L}$ times the inverse of the Froude number. All these results are rationalized by developing an extension to Havelock's theory.

Using laboratory experiments and numerical simulations, we examine the transfer of soluble material from small, spherical particles sinking in homogeneous turbulence at large Péclet number. A theoretical analysis predicts two distinct mechanisms of convective mass transfer: strain due to turbulence and slip due to gravitational settling. Their relative strength is parametrised by the sinking ratio, ${Sr} = w_0 \tau _\eta /a$, where $w_0$ is the quiescent settling velocity, $a$ is the particle radius and $\tau _\eta$ is the Kolmogorov time scale. This analysis predicts that the topology of the concentration wake changes from a symmetric topology at ${Sr} \ll 1$ to an asymmetric topology at ${Sr} \gg 1$ as the dominant mechanism of mass transfer changes. Particle tracking flow visualisations of small spheres releasing dye in turbulence confirm the existence of this change in mechanism at ${Sr} = O(1)$. We complement these experiments with numerical simulations of the mass transfer from sinking particles. The transfer rate predicted by the simulation is found to be in good agreement with literature data for mass transfer to turbulent suspensions of solid particles and is consistent with asymptotic expressions for mass transfer in uniform flow when ${Sr} \gg 1$. A decomposition of the convective fluxes confirms the transition in the transfer mechanism. At ${Sr} = O(1)$, both mechanisms provide comparable contributions to the transfer rate. Cross-correlation analysis reveals that particle-scale knowledge of both the recent strain and velocity history is required to predict the instantaneous transfer rate. Turbulence-induced particle rotation has a modest suppression effect upon convective transfer by sinking.

Horizontal convection at large Rayleigh and Prandtl numbers is studied experimentally in a regime up to seven orders of magnitude larger in terms of Rayleigh numbers than previously achieved. To reach Rayleigh numbers up to $10^{17}$, the horizontal density gradient is generated using differential solutal convection by a differential input of salt and fresh water controlled by diffusion in a novel experiment in which the zero-net mass flux of water is ensured through permeable membranes. This set-up allows us to accurately measure the Nusselt number in solutal convection by carefully controlling the amount of salt water exchanged through the membranes. Combined measurements of density and velocity across more than five orders of magnitude in Rayleigh numbers show that the flow transitions from the Beardsley & Festa (J. Phys. Oceanogr., vol. 2, issue 4, 1972, pp. 444–455), Shishkina & Wagner (Phys. Rev. Lett., vol. 116, issue 2, 2016, 024302) regime to the Chiu-Webster et al. (J. Fluid Mech., vol. 611, 2008, pp. 395–426) regime and frames the present results within the scope of Shishkina et al. (Geophys. Res. Lett., vol. 43, issue 3, 2016, pp. 1219–1225), and the theory of Part 1 (Passaggia & Scotti, vol. 997, 2024, J. Fluid Mech., A5). In particular, we show that, even for large Prandtl numbers, the circulation eventually clusters underneath the forcing horizontal boundary, leaving a stratified core without motion. Finally, the previous regime diagrams (Hughes & Griffiths, Annu. Rev. Fluid Mech., vol. 40, 2008, pp. 185–208; Shishkina et al., Geophys. Res. Lett., vol. 43, issue 3, 2016, pp. 1219–1225) are extended by combining the present results at high Prandtl numbers, the results at low Prandtl numbers of Part 1, together with previous results from the literature. This work sets a new picture of the transition landscape of horizontal convection over six orders of magnitude in Prandtl number and sixteen orders of magnitude in Rayleigh number.

In this paper, a high-level Green–Naghdi (HLGN) model for large-amplitude internal waves in a two-layer fluid system, where the upper-fluid layer is of finite depth and the lower-fluid layer is of infinite depth, is developed under the rigid-lid free-surface approximation. The equations of the present HLGN model follow Euler's equations under the sole assumption that the horizontal and vertical velocity distributions along the vertical column are presented by known shape functions for each layer. The linear dispersion relations of the HLGN model for different levels are presented and compared with those obtained by other strongly nonlinear models for deep water, including the fully nonlinear models that include the dispersion effects $O(\mu )$ (where $\mu$ is the ratio of the upper-fluid layer depth to a typical wavelength) derived by Choi & Camassa (Phys. Rev. Lett., vol. 77, 1996, pp. 1759–1762) and $O(\mu ^2)$ derived by Debsarma et al. (J. Fluid Mech., vol. 654, 2010, pp. 281–303). It is shown that the HLGN model has a wider application range than other models. Solutions of travelling large-amplitude internal solitary waves in the absence and presence of background shear-current are then investigated by using the HLGN model. For the no-current cases, results obtained by the HLGN model show better agreement with Euler's solution on wave profile, velocity profile at the maximum interface displacement and wave speed compared with those obtained by other models. For the background shear-current cases, results obtained by the HLGN model also show good agreement with those obtained by solving the Dubreil-Jacotin–Long equation.

Flexible canopy flows are often encountered in natural scenarios, e.g. when crops sway in the wind or when submerged kelp forests are agitated by marine currents. Here, we provide a detailed characterisation of the turbulent flow developed above and between the flexible filaments of a fully submerged dense canopy and we describe their dynamical response to the turbulent forcing. We investigate a wide range of flexibilities, encompassing the case in which the filaments are completely rigid and standing upright as well as that where they are fully compliant to the flow and deflected in the streamwise direction. We are thus able to isolate the effect of the canopy flexibility on the drag and on the inner–outer flow interactions, as well as the two flapping regimes of the filaments already identified for a single fibre. Furthermore, we offer a detailed description of the Reynolds stresses throughout the wall-normal direction resorting to the Lumley triangle formalism, and we show the multi-layer nature of turbulence inside and above the canopy. The relevance of our investigation is thus twofold: the fundamental physical understanding developed here paves the way towards the investigation of more complex and realistic scenarios, while we also provide a thorough characterisation of the turbulent state that can prove useful in the development of accurate turbulence models for RANS and LES.

In nature and engineering applications, water jet plunging acts as a key process causing interface breaking and generating mixed-phase turbulence. In this paper, high-resolution numerical simulations of the plunging of a water jet into a quiescent pool were performed to investigate the statistical properties of mixed-phase turbulence, with a special focus on the closure problem of the Reynolds-averaged equation. We conducted phase-resolved simulations, with the air–water interface captured using a coupled level-set and volume-of-fluid method. Various cases were performed to analyse the effects of the Froude number and Reynolds number. The simulation results showed that the turbulence statistics are insensitive to the Reynolds number under investigation, while the Froude number influences the flow properties significantly. To investigate the closure problem of the mean momentum equation, the turbulent kinetic energy (TKE) and turbulent mass flux (TMF) and their transport equations were analysed further. It was discovered that the balance relationship of the TKE budget terms remained similar to many single-phase turbulent flows. The TMF is an additional unclosed term in mixed-phase turbulence over the single-phase turbulence. Our simulation results showed that the production term in its transport equation was highly correlated to TKE. Based on this finding, a closure model for the production term of TMF was further proposed.

A novel approach is presented for identifying disturbance sources in wall-bounded shear flows. The underlying approach models the flow state, as measured by sensors embedded in the flow, as a mixture of disturbance sources. The degenerate unmixing estimation technique is adopted as a blind source separation technique to recover the separate sources and their unknown mixing process. The efficiency of this approach stems from its ability to isolate any, a priori unknown, number of sources, using two sensors only. Furthermore, by adding a single additional sensor, the method is expanded to also determine the propagation velocity vector of each of the isolated sources, based on sensor readings from three sensors appropriately located in the flow field. Theoretical guidelines for locating the sensors are provided. The power of the method is demonstrated via computer simulations and wind-tunnel experiments. The numerical study considers disturbances comprising discrete Tollmien–Schlichting waves and wave packets. Linear stability theory is used to model source mixtures acquired by sensors placed in a Blasius boundary layer. The experimental study investigates the flow over a flat plate, with hot wires as sensors, and a loudspeaker and plasma actuators as source generators. Based on numerical and experimental demonstrations, it is believed that the new approach should prove useful in various applications, including active control of boundary layer transition from laminar to turbulent flow.

Just as a solid object would, a liquid jet or a stream of droplets impacting a free surface deforms and perforates it. This generic flow interaction, met in everyday life but also in cutting edge industrial processes, has puzzled scientists for centuries. Lee et al. (J. Fluid Mech., vol. 921, 2021, A8) present an experimental study of a simple droplet train interacting with a liquid bath and identify two stages in the interaction: a first where a cavity elongates and finally bursts, and a second where the interface is steadily punched by the incoming stream. Each of these regimes is explained with elementary but effective models arising from first principles, thereby revealing a full and simple picture of the physics of making holes in liquids.

The transition to dripping in the gravity-driven flow of a liquid film under an inclined plate is investigated at zero Reynolds number. Computations are carried out on a periodic domain assuming either a fixed fluid volume or a fixed flow rate for a hierarchy of models: two lubrication models with either linearised curvature or full curvature (the LCM and FCM, respectively), and the full equations of Stokes flow. Of particular interest is the breakdown of travelling-wave solutions as the plate inclination angle is increased. For any fixed volume, the LCM reaches the horizontal state where it attains a cosine-shaped profile. For sufficiently small volume, the FCM and Stokes solutions attain a weak Young–Laplace equilibrium profile, the approach to which is described by an asymptotic analysis generalising that of Kalliadasis & Chang (J. Fluid Mech., vol. 261, 1994, pp. 135–168) for the LCM. For large volumes, the bifurcation curves for the FCM and Stokes model have a turning point so that the fully inverted state is never reached. For fixed flow rate, the LCM blows up at a critical angle that is well predicted by asymptotic analysis. The bifurcation curve for the FCM either has a turning point or else reaches a point at which the surface profile has an infinite slope singularity, indicating the onset of multi-valuedness. The latter is confirmed by the Stokes model, which can be continued to obtain overturning surface profiles. Overall, the thin-film models either provide an accurate prediction for dripping onset or else supply an upper bound on the critical inclination angle.

The tangential strain rate in premixed flames impacts significantly the flame surface area generation and thus the combustion process. Studies on incompressible isotropic turbulence have revealed that the mean tangential strain rate at material and iso-scalar surfaces is positive and exhibits a universal value when normalized by the Kolmogorov time. This is associated with the preferential alignment of the surface normal with the most compressive principal strain rate. The present study investigates such effects in premixed hydrogen and iso-octane flame kernels using direct numerical simulations. It is shown that the normalized mean tangential strain rate of the investigated flames has a very similar value compared with the incompressible flows. However, in the reaction zone, the flame surface normal aligns preferentially with the most extensive principal strain rate. Furthermore, this alignment depends on the reaction progress variable and the Lewis number, while the tangential strain rate remains independent of these parameters. Such counter-intuitive behaviour is systematically investigated by decomposing the effects of dilatation and residual solenoidal turbulence. It is found that the solenoidal turbulence influences significantly the tangential strain rate. A general effect of turbulence on the tangential strain rate is identified, which is consistent with incompressible flows and independent of the Lewis number and the reaction progress variable. This is a remarkable finding indicating that models of the tangential strain rate developed based on incompressible flows apply also to premixed flames with different Lewis numbers, and, for the modelling, only the solenoidal turbulence should be considered.

Recent studies have shown that superimposing rhythmic perturbations to oscillating tailbeats could simultaneously enhance both the thrust and efficiency (Lehn et al., Phys. Rev. Fluids, vol. 2, 2017, p. 023101; Chao et al., PNAS Nexus, vol. 3, 2024, p. 073). However, these investigations were conducted with a tethered flapping foil, overlooking the self-propulsion intrinsic to real swimming fish. Here, we investigate how the high-frequency, low-amplitude superimposed rhythmic perturbations impact the self-propelled pitching and heaving swimming of a rigid foil. The swimming-speed-based Reynolds number ranges from 1400 to 2700 in our study, depending on superimposed perturbations and swimming modes. Numerical results reveal that perturbations significantly increase swimming speeds in both pitching and heaving motions, while enhancing efficiency exclusively in the heaving motion. Further derived scaling laws elucidate the relationships of perturbations with speeds, power costs and efficiency, respectively. These findings not only hypothesise the potential advantages of perturbations in biological systems, but also inspire designs and controls in biomimetic propulsion and manoeuvring within aquatic environments.

Inclining a fluid layer subjected to a temperature gradient introduces a profusion of fascinating patterns and regimes. Previous experimental and computational studies form the starting point for an extensive numerical bifurcation study by Reetz & Schneider (J. Fluid Mech., vol. 898, 2020, A22) and Reetz et al. (J. Fluid Mech., vol. 898, 2020, A23). Intricate trajectories passing through multiple steady and periodic states organize the dynamics. The consequences for chaotic patterns in large geometries is discussed.

We consider a theoretical model for the settling of rod-shaped particles of a dilute, initially homogeneous, suspension in rapid rotation. The particle Reynolds number and the particle Taylor number of the detailed flow around the particles are assumed small, representing a relevant limit for an industrial centrifugal separation process. By applying a statistical approach using the Fokker–Planck equation, and neglecting particle–particle interactions, we obtain an explicit, analytical solution for the time dependent, spatially uniform particle orientation distribution function. Not only does the volume fraction in the bulk of the suspension decrease with time due to the divergent centrifugal field, as similarly described in the literature for suspensions of spherical particles, the orientation of the rod particles also changes with time from an initially uniform distribution to one where the particles tend to align with a plane perpendicular to the axis of rotation. The corresponding particle trajectories, as also influenced by first-order effects from the Coriolis acceleration and gyroscopic effects, are obtained numerically for different initial particle orientation angles.

Convection from a buoyancy source distributed over a vertical wall has diverse applications, from the natural ventilation of buildings to the melting of marine-terminating glaciers which impacts on future sea level. A key challenge involves determining how the rate and mechanisms of turbulent heat transfer should be extrapolated across a range of scales. Ke et al. (J. Fluid Mech., vol. 964, 2023, A24) explore transitions in the turbulent flow dynamics using direct numerical simulation of a convective boundary layer at a heated vertical wall. A classical regime of heat transfer, consistent with previous laboratory experiments, gives way with increasing accumulation of buoyancy to an ultimate regime with enhanced heat transfer. The key to this transition lies in a near-wall sublayer, with a switch from laminar buoyancy-driven dynamics to a sublayer dominated by turbulence and shear instability from the mean flow.

An important problem in passive scalar transport is to parametrize the effect of a fluctuating component of the flow, in order to overcome a limited resolution. A local effective diffusivity is one such parametrization, and over the years there have been many different suggestions for ‘closures’ that relate the advective flux to gradients of the mean concentration. Souza et al. (J. Fluid Mech., 2023, in press) introduce a stochastic framework where the local effective diffusivity is replaced by an exact effective diffusivity operator. By computing this operator for various examples, they quantify deviations from the local approximation, which can suggest areas of improvement and novel closure models.

The refraction of surface gravity waves by currents leads to spatial modulations in the wave field and, in particular, in the significant wave height. We examine this phenomenon in the case of waves scattered by a localised current feature, assuming (i) the smallness of the ratio between current velocity and wave group speed, and (ii) a swell-like, highly directional wave spectrum. We apply matched asymptotics to the equation governing the conservation of wave action in the four-dimensional position–wavenumber space. The resulting explicit formulas show that the modulations in wave action and significant wave height past the localised current are controlled by the vorticity of the current integrated along the primary direction of the swell. We assess the asymptotic predictions against numerical simulations using WAVEWATCH III for a Gaussian vortex. We also consider vortex dipoles to demonstrate the possibility of ‘vortex cloaking’ whereby certain currents have (asymptotically) no impact on the significant wave height. We discuss the role of the ratio of the two small parameters characterising assumptions (i) and (ii) above, and show that caustics are significant only for unrealistically large values of this ratio, corresponding to unrealistically narrow directional spectra.

Despite the widespread occurrence of pendant drops in nature, there is still a lack of combined studies on their dynamic and static stability. This study focuses on the dynamic and static stability of elongated drops with either a free or pinned contact line on a plane. We first examine static stability for both axisymmetric and non-axisymmetric perturbations subject to volume or pressure constraints. The stability limits for volume and pressure disturbances (axisymmetric) correspond to the maximum volume and pressure of the drops, respectively. Drops with free contact lines are marginally stable to non-axisymmetric perturbations because of their horizontal translational invariance, whereas pinned drops are stable. The linear dynamic stability is then investigated numerically through a boundary element model, restricted to volume disturbances. Results show that when the stability limit is reached, the first zonal mode has a zero frequency, suggesting that the thresholds for static and dynamic stability are essentially equivalent. Furthermore, natural frequencies experience sharp changes as the stability limit is approached. Another zero frequency mode associated with the horizontal motion of the centre of mass is also revealed by the numerical results, reflecting the horizontal translational invariance of drops with free contact lines. Finally, the frequency spectrum modified by gravity is explored, resulting in the identification of five gravity-induced frequency shift patterns. The frequency shifts break the spectral degeneracy for hemispherical drops with free contact lines, leading to various spectral orderings according to polar and azimuthal wavenumbers.

The hydrodynamic influence of surface texture on static surfaces ranges from large drag penalties (roughness) to potential performance benefits (shark-like skin). Although it is of wide-ranging research interest, the impact of roughness on flapping systems has received limited attention. In this work, we explore the effect of roughness on the unsteady performance of a harmonically pitching foil through experiments using foils with different surface roughness, at a fixed Strouhal number and within the Reynolds number ($Re$) range of $17\,000\unicode{x2013}33\,000$. The foils’ surface roughness is altered by changing the distribution of spherical-cap-shaped elements over the propulsor area. We find that the addition of surface roughness does not improve the performance compared with a smooth surface over the $Re$ range considered. The analysis of the flow fields shows near-identical wakes regardless of the foil's surface roughness. The performance reduction mainly occurs due to an increase in profile drag. However, we find that the drag penalty due to roughness is reduced from $76\,\%$ for a static foil to $16\,\%$ for a flapping foil at the same mean angle of attack, with the strongest decrease measured at the highest $Re$. Our findings highlight that the effect of roughness on dynamic systems is very different than that on static systems; thereby, it cannot be estimated by only using information obtained from static cases. This also indicates that the performance of unsteady, flapping systems is more robust to the changes in surface roughness.