At high surface Péclet numbers, it is common to associate the presence of surfactants with surface immobilization, where a free surface becomes indistinguishable from a no-slip surface. A different mechanism has recently been proposed for longitudinal shear flow along a unidirectional trench (Baier & Hardt, J. Fluid Mech., vol. 949, 2022, A34) wherein, at high Marangoni numbers, the meniscus spanning the finite-length trench becomes a constant-shear-stress surface due to contamination by incompressible surfactant. That model predicts recirculating interfacial flows on the meniscus, a phenomenon that has been observed experimentally (Song et al., Phys. Rev. Fluids, vol. 3, issue 3, 2018, 033303). By finding an explicit solution to the constant-shear-stress model at all protrusion angles and calculating the effective slip length for a dilute mattress of such surfactant-laden trenches, we show that those effective slip lengths are almost indistinguishable from those for a surface whose menisci have the same deflection but have been completely immobilized (i.e. they are no-slip surfaces). This means that, despite the presence of non-trivial recirculating vortical flows on the menisci, the aggregate slip characteristics of such surfaces are that they have been effectively immobilized. This surprising result underscores the need for caution in comparing theory with experiments based on effective slip properties alone.

We derive a scaling law for the characteristic frequencies of wall pressure fluctuations in swept shock wave/turbulent boundary layer interactions in the presence of cylindrical symmetry, based on analysis of a direct numerical simulations database. Direct numerical simulations in large domains show evidence of spanwise rippling of the separation line, with typical wavelength proportional to separation bubble size. Pressure disturbances around the separation line are shown to be convected at a phase speed proportional to the cross-flow velocity. This information is leveraged to derive a simple model for low-frequency unsteadiness, which extends previous two-dimensional models (Piponniau et al., J. Fluid Mech., vol. 629, 2009, pp. 87–108), and which correctly predicts growth of the typical frequency with the sweep angle. Inferences regarding the typical frequencies in more general swept shock wave/turbulent boundary layer interactions are also discussed.

When a cylinder is mounted on an elastic support within a current, vortex-induced vibrations (VIV) may occur down to a Reynolds number (Re) close to $20$, based on the body diameter ($D$) and inflow velocity ($U$), i.e. below the critical value of $47$ reported for the onset of flow unsteadiness when the body is fixed. The impact of a forced rotation of the elastically mounted cylinder on the system behaviour is explored numerically for $Re \leqslant 30$, over wide ranges of values of the rotation rate (ratio between body surface velocity and $U$, $\alpha \in [0,5]$) and reduced velocity (inverse of the oscillator natural frequency non-dimensionalized by $D$ and $U$, $U^\star \in [2,30]$). The influence of the rotation is not monotonic, but the most prominent effect uncovered in this work is a substantial enhancement of the subcritical-Re, flow-induced vibrations beyond $\alpha =2$. This enhancement is twofold. First, the rotation results in a considerable expansion of the vibration/flow unsteadiness region in the $({Re},U^\star )$ domain, down to $Re=4$. Second, the elliptical orbits described by the rotating body are subjected to a major amplification, with a transition from VIV to responses whose magnitude tends to increase unboundedly with $U^\star$, even though still synchronized with flow unsteadiness. The emergence of such galloping-like oscillations close to the onset of vibrations disrupts the scenario of gradual vibration growth with Re, as amplitudes larger than $10$ body diameters may be observed at $Re=10$.

Using the theory of wave turbulence for rapidly rotating incompressible fluids derived by Galtier (Phys. Rev. E, vol. 68, 2003, 015301), we find the locality conditions that the solutions of the kinetic equation must satisfy. We show that the exact anisotropic Kolmogorov–Zakharov spectrum satisfies these conditions, which justifies the existence of this constant (positive) energy flux solution. Although a direct cascade is predicted in the transverse ($\perp$) and parallel ($\parallel$) directions to the rotation axis, we show numerically that in the latter case some triadic interactions can have a negative contribution to the energy flux, while in the former case all interactions contribute to a positive flux. Neglecting the parallel energy flux, we estimate the Kolmogorov constant at $C_K \simeq 0.749$. These results provide theoretical support for recent numerical and experimental studies.

Energy-containing eddies (energy-eddies) are the elementary structures of wall turbulence that carry most of the kinetic energy and momentum. Despite the consensus that energy-eddies can self-sustain at each relevant length scale, their precise origin and spatial evolution are currently not well understood. In this study, we examine the spatial evolution of energy-eddies by quenching them at the inflow of a turbulent channel flow. Our study shows that the eddies involved in the energy cascade cannot be sustained without the energy-eddies. The streamwise velocity spectra of the evolving flow start to recover at a spanwise wavelength of $\lambda _z^+ \simeq 100$, equal to the near-wall spacing of streaks in the buffer layer located at $y^+ \simeq 15$, whereas there are no active vortical motions in the streamwise vorticity spectra until the energy at the streak location is re-established. Hence, the present study demonstrates that in a spatially evolving flow, the formation of near-wall streaks is the primary process necessary in the recovery of energy-eddies.

Active elements in active nematics can impose forces on immersed bodies and move them accordingly. We numerically investigate the vibrational motion of a cantilever beam placed in active nematics. The continuous energy transfer from vortices to the beam results in beam oscillation, whose direction and amplitude depend on the vortex strength, size and position. Referring to the kinetic-energy spectrum, we indicate that both the large- and small-scale vortices are the primary mechanism for the energy transfer between the fluid and beam, leading to the beam oscillatory motion, with the contribution from the large-scale vortices being higher. We investigate the effect of fluid properties such as activity, viscosity and elastic constant on the oscillation frequency. We show that the intensification of the activity increases peak frequency, and there is a linear correlation between the peak frequency and activity. We further demonstrate the reciprocal relationship between viscosity and peak frequency. Subsequently, we relate the increase and decrease in the peak frequency to the energy injection/dissipation by activity/viscosity. Moreover, we reveal the negligibly small dependency of beam peak frequency on the elastic constant and discuss free energy's role in accounting for this behaviour. The findings clearly demonstrate that active fluids can impose an oscillatory motion on flexible bodies, which might be used as a novel method for measuring the critical properties of active nematics.

The role of the Lagrangian mean flow, or drift, in modulating the geometry, kinematics and dynamics of rotational and irrotational deep-water surface gravity waves is examined. A general theory for permanent progressive waves on an arbitrary vertically sheared steady Lagrangian mean flow is derived in the Lagrangian reference frame and mapped to the Eulerian frame. A Lagrangian viewpoint offers tremendous flexibility due to the particle labelling freedom and allows us to reveal how key physical wave behaviour arises from a kinematic constraint on the vorticity of the fluid, inter alia the nonlinear correction to the phase speed of irrotational finite amplitude waves, the free surface geometry and velocity in the Eulerian frame, and the connection between the Lagrangian drift and the Benjamin–Feir instability. To complement and illustrate our theory, a small laboratory experiment demonstrates how a specially tailored sheared mean flow can almost completely attenuate the Benjamin–Feir instability, in qualitative agreement with the theory. The application of these results to problems in remote sensing and ocean wave modelling is discussed. We provide an answer to a long-standing question: remote sensing techniques based on observing current-induced shifts in the wave dispersion will measure the Lagrangian, not the Eulerian, mean current.

We derive a limit on energy savings in controlled channel flow. For flow in a channel driven by pressure, shear or any combination of the two, and controlled via wall transpiration or spanwise wall motion, the uncontrolled laminar state requires the least net energy (accounting for the energetic cost of control). Thus, the optimal control solution is to laminarize the flow. Additionally, we raise the possibility of beating this limit. By simultaneously applying wall transpiration and spanwise wall motion, we show that it may be possible to attain sustained sub-laminar energy expenditure in a controlled flow. We provide a necessary design criterion for net energy savings.

Convection occurs ubiquitously on and in rotating geophysical and astrophysical bodies. Prior spherical shell studies have shown that the convection dynamics in polar regions can differ significantly from the lower latitude, equatorial dynamics. Yet most spherical shell convective scaling laws use globally-averaged quantities that erase latitudinal differences in the physics. Here we quantify those latitudinal differences by analysing spherical shell simulations in terms of their regionalized convective heat-transfer properties. This is done by measuring local Nusselt numbers in two specific, latitudinally separate, portions of the shell, the polar and the equatorial regions, $Nu_p$ and $Nu_e$, respectively. In rotating spherical shells, convection first sets in outside the tangent cylinder such that equatorial heat transfer dominates at small and moderate supercriticalities. We show that the buoyancy forcing, parameterized by the Rayleigh number $Ra$, must exceed the critical equatorial forcing by a factor of ${\approx }20$ to trigger polar convection within the tangent cylinder. Once triggered, $Nu_p$ increases with $Ra$ much faster than does $Nu_e$. The equatorial and polar heat fluxes then tend to become comparable at sufficiently high $Ra$. Comparisons between the polar convection data and Cartesian numerical simulations reveal quantitative agreement between the two geometries in terms of heat transfer and averaged bulk temperature gradient. This agreement indicates that rotating spherical shell convection dynamics is accessible both through spherical simulations and via reduced investigatory pathways, be they theoretical, numerical or experimental.

In Rayleigh–Bénard convection, it has been found that the amount of heat passing through the fluid has a power-law dependence on the imposed temperature difference. Modifying this dependence, either enhancing or reducing the heat transfer capability of fluids, is important in many scientific and practical applications. Here, we present a simple means to control the vertical heat transfer in Rayleigh–Bénard convection by injecting heat through one lateral side of the fluid domain and extracting the same amount of heat from the opposite side. This horizontal heat flux regulates the large-scale circulation, and increases the heat transfer rate in the vertical direction. Our numerical and theoretical studies demonstrate how a classical Rayleigh–Bénard convection responds to such a perturbation when the system is near or well above the onset of convection.

Direct numerical simulations of the flow past an impulsively started cylinder at high Reynolds numbers (25k–100k) reveal an intriguing portrait of unsteady separation. Vorticity generation and vortex shedding entails a cascade of separation events on the cylinder surface that are reminiscent of Kelvin–Helmholtz instabilities. Primary vortices roll up along the cylinder surface as a result of instabilities of the initially attached vortex sheets, followed by vortex eruptions, creation of secondary vorticity and formation of dipole structures that are subsequently ejected from the surface of the cylinder. We analyse the vortical structures and their relationship to the forces experienced by the cylinder. This striking cascade of vortex instabilities may serve as reference for reduced-order models of flow separation and as guide for flow control of separated flows at high Reynolds numbers.

A small spheroid settling in a quiescent fluid experiences an inertial torque that aligns it so that it settles with its broad side first. Here we show that an active particle experiences such a torque too, as it settles in a fluid at rest. For a spherical squirmer, the torque is $\boldsymbol {T}^\prime = -{\frac {9}{8}} m_f (\boldsymbol {v}_s^{(0)} \wedge \boldsymbol {v}_g^{(0)})$ where $\boldsymbol {v}_s^{(0)}$ is the swimming velocity, $\boldsymbol {v}_g^{(0)}$ is the settling velocity in the Stokes approximation and $m_f$ is the equivalent fluid mass. This torque aligns the swimming direction against gravity: swimming up is stable, swimming down is unstable.

Instabilities of fluid flows often generate turbulence. Using extensive direct numerical simulations, we study two-dimensional turbulence driven by a wavenumber-localised instability superposed on stochastic forcing, in contrast to previous studies of state-independent forcing. As the contribution of the instability forcing, measured by a parameter $\gamma$, increases, the system undergoes two transitions. For $\gamma$ below a first threshold, a regular large-scale vortex condensate forms. Above this threshold, shielded vortices (SVs) emerge within the condensate. At a second, larger value of $\gamma$, the condensate breaks down, and a gas of weakly interacting vortices with broken symmetry spontaneously emerges, characterised by preponderance of vortices of one sign only and suppressed inverse energy cascade. The latter transition is shown to depend on the damping mechanism. The number density of SVs in the broken symmetry state slowly increases via a random nucleation process. Bistability is observed between the condensate and mixed SV-condensate states. Our findings provide new evidence for a strong dependence of two-dimensional turbulence phenomenology on the forcing.

The well-known self-propulsion, or swimming, of a deformable body in Stokes flow (i.e. at low Reynolds number) can be understood and modelled from the variation in the configuration-dependent hydrodynamic resistance tensor throughout the period of deformation. Remarkably, at the other extreme of high Reynolds number, a deformable body may also self-propel without doing any net work on the fluid in potential flow. As a body deforms, the mass of fluid displaced – the so-called added mass – depends on the instantaneous body configuration, and a net displacement is possible over a period of deformation. This potential swimming takes a form identical to that for Stokes swimmers with the configuration-dependent added mass replacing the hydrodynamic resistance tensor. Analytical insight into the swimming of a deformable body is obtained through an expansion of the nonlinear spatial dependence of the hydrodynamic interactions and connections between previous studies of swimming in Stokes flow to those in potential flow are made.

A new two-equation model for gravity-driven liquid film flow based on the long-wave expansion has been derived. The novelty of the model consists in using a base velocity profile combining parabolic (Ruyer-Quil & Manneville, Eur. Phys. J. B, vol. 15, issue 2, 2000, pp. 357–369) and ellipse (Usha et al., Phys. Fluids, vol. 32, issue 1, 2020, 013603) profile functions in the wall-normal coordinate. The dependence on a free parameter $A$ related to the eccentricity of an ellipse serves as an adjustable parameter. The resulting models are consistent at $O(\varepsilon )$ for inertia terms and at $O(\varepsilon ^2)$ for viscous diffusion effects, and predict accurately the primary instability. Appropriate tuning of the adjustable parameter helps to recover accurate predictions for the asymptotic wave celerity of nonlinear solitary waves. Further, the model is shown to capture the closed separation vortices that can form underneath the troughs of precursory capillary ripples.

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.

Turbulent motions enhance the diffusion of large-scale flows and temperature gradients. Such diffusion is often parameterized by coefficients of turbulent viscosity ($\nu _{t}$) and turbulent thermal diffusivity ($\chi _{t}$) that are analogous to their microscopic counterparts. We compute the turbulent diffusion coefficients by imposing sinusoidal large-scale velocity and temperature gradients on a turbulent flow and measuring the response of the system. We also confirm our results using experiments where the imposed gradients are allowed to decay. To achieve this, we use weakly compressible three-dimensional hydrodynamic simulations of isotropically forced homogeneous turbulence. We find that the turbulent viscosity and thermal diffusion, as well as their ratio the turbulent Prandtl number, $\textit {Pr}_{t} = \nu _{t}/\chi _{t}$, approach asymptotic values at sufficiently high Reynolds and Péclet numbers. We also do not find a significant dependence of $\textit {Pr}_{t}$ on the microscopic Prandtl number $\textit {Pr} = \nu /\chi$. These findings are in stark contrast to results from the $k{-}\epsilon$ model, which suggests that $\textit {Pr}_{t}$ increases monotonically with decreasing $\textit {Pr}$. The current results are relevant for the ongoing debate on, for example, the nature of the turbulent flows in the very-low-$\textit {Pr}$ regimes of stellar convection zones.

We consider nonlinear dynamical systems driven by stochastic forcing. It has been largely evidenced in the literature that the linear response of non-normal systems (e.g. fluid flows) may exhibit a large variance amplification, even in a linearly stable regime. This linear response, however, is relevant only in the limit of vanishing forcing intensity. As the intensity increases, the frequency distribution and the variance of the response may be strongly modified due to nonlinear effects. To go beyond this limitation, we propose a theoretical approach to derive an amplitude equation governing the weakly nonlinear evolution of these systems. This approach does not rely on the presence of an eigenvalue close to the neutral axis, applying instead to any sufficiently non-normal operator, and the Fourier components of the response are allowed to be arbitrarily different from any eigenmode. The methodology is outlined for a generic nonlinear dynamical system, and the application case highlights a common non-normal mechanism in hydrodynamics: convective non-normal amplification in the flow past a backward-facing step.

An isolated charge-neutral droplet in a uniform electric field experiences no net force. However, a droplet pair can move in response to field-induced dipolar and hydrodynamic interactions. If the droplets are identical, the centre of mass of the pair remains fixed. Here, we show that if the droplets have different properties, the pair experiences a net motion due to non-reciprocal electrohydrodynamic interactions. We analyse the three-dimensional droplet trajectories using asymptotic theory, assuming spherical droplets and large separations, and numerical simulations based on a boundary integral method. The dynamics can be quite intricate depending on the initial orientation of the droplets line-of-centres relative to the applied field direction. Drops tend to migrate towards a configuration with line-of-centres either parallel or perpendicular to the applied field direction, while either coming into contact or indefinitely separating. We elucidate the conditions under which these different interaction scenarios take place. Intriguingly, we find that in some cases droplets can form a pair (tandem) that translates either parallel or perpendicular to the applied field direction.

We present an experimental study of Rayleigh–Bénard convection using liquid metal alloy gallium-indium-tin as the working fluid with a Prandtl number of $Pr=0.029$. The flow state and the heat transport were measured in a Rayleigh number range of $1.2\times 10^{4} \le Ra \le 1.3\times 10^{7}$. The temperature fluctuation at the cell centre is used as a proxy for the flow state. It is found that, as $Ra$ increases from the lower end of the parameter range, the flow evolves from a convection state to an oscillation state, a chaotic state and finally a turbulent state for $Ra>10^5$. The study suggests that the large-scale circulation in the turbulent state is a residual of the cell structure near the onset of convection, which is in contrast with the case of $Pr\sim 1$, where the cell structure is transiently replaced by high order flow modes before the emergence of the large-scale circulation in the turbulent state. The evolution of the flow state is also reflected by the heat transport characterised by the Nusselt number $Nu$ and the probability density function (p.d.f.) of the temperature fluctuation at the cell centre. It is found that the effective local heat transport scaling exponent $\gamma$, i.e. $Nu\sim Ra^{\gamma }$, changes continuously from $\gamma =0.49$ at $Ra\sim 10^4$ to $\gamma =0.25$ for $Ra>10^6$. Meanwhile, the p.d.f. at the cell centre gradually evolves from a Gaussian-like shape before the transition to turbulence to an exponential-like shape in the turbulent state. For $Ra>10^6$, the flow shows self-similar behaviour, which is revealed by the universal shape of the p.d.f. of the temperature fluctuation at the cell centre and a $Nu=0.19Ra^{0.25}$ scaling for the heat transport.