A closed expression for the average pressure difference (often called the macroscopic dynamic capillary pressure in the literature) is proposed for two-phase, Newtonian, incompressible, isothermal and creeping flow in homogeneous porous media. This upscaled equation complements the average equations for mass and momentum transport derived in a previous article. Consistently with this work, the expression is derived employing a simplified version of the volume-averaging method that makes use of elements of the adjoint method and Green's formula. The resulting equation for the average pressure difference is novel, as it shows that this quantity is controlled by the pressure gradient (and body forces) in each phase, as well as interfacial effects, and is applicable to situations in which the fluid–fluid interface is not necessarily at its steady position. The effective-medium quantities associated with the sources are all obtained from the solution of a single adjoint (or closure) problem to be solved on a (periodic) unit cell representative of the process. The average pressure difference predicted by the derived expression is validated through excellent comparisons with direct numerical simulations performed in a model porous structure.

Recent studies of pressure-driven flows of dilute polymer solutions in straight channels demonstrated the existence of two-dimensional coherent structures that are disconnected from the laminar state and appear through a subcritical bifurcation from infinity. These travelling-wave solutions were suggested to organise the phase-space dynamics of purely elastic and elasto-inertial chaotic channel flows. Here, we consider a wide range of parameters, covering the purely elastic and elasto-inertial cases, and demonstrate that the two-dimensional travelling-wave solutions are unstable when embedded in sufficiently wide three-dimensional domains. Our work demonstrates that studies of purely elastic and elasto-inertial turbulence in straight channels require three-dimensional simulations, and no reliable conclusions can be drawn from studying strictly two-dimensional channel flows.

Analysis of satellite altimetry and Argo float data leads Ni et al. (J. Geophys. Res., 125, 2020, e2020JC016479) to argue that mesoscale dipoles are widespread features of the global ocean having a relatively uniform three-structure that can lead to strong vertical exchanges. Almost all the features of the composite dipole they construct can be derived from a model for multipoles in the surface quasi-geostrophic equations for which we present a straightforward novel solution in terms of an explicit linear algebraic eigenvalue problem, allowing simple evaluation of the higher radial modes that appear to be present in the observations and suggesting that mass conservation may explain the observed frontogenetic velocities.

Fast microjets can emerge out of liquid pools from the rebounding of drop-impact craters, or when a bubble bursts at its surface. The fastest jets are the narrowest and are a source of aerosols both from the ocean and from a glass of champagne, of importance to climate and the olfactory senses. The most singular jets, which we observe experimentally at a maximum velocity of $137\pm 4\ {\rm m}\ {\rm s}^{-1}$ and a diameter of $12\ \mathrm {\mu }{\rm m}$, under reduced ambient pressure, are produced when a small dimple forms at the crater bottom and rebounds without pinching off a small bubble. The radial collapse and rebounding of this dimple is purely inertial, but highly sensitive to initial conditions. High-resolution numerical simulations reveal a new focusing mechanism, which drives the fastest jet within a converging conical channel, where an entrained air sheet provides effective slip at the outer boundary of the conically converging flow into the jet. This configuration bypasses any viscous cutoff of the jetting speed and explains the extreme sensitivity to initial conditions observed in detailed experiments of the phenomenon.

We consider transverse shear flow over a $2L$-periodic array of air-filled grooves, which are separated by solid slats. We address the scenario where the liquid partially invades the grooves. Following Crowdy's solution of the longitudinal problem (J. Fluid Mech. vol. 925, 2021, R2), we focus upon the simplified geometry of infinitely narrow slats. For flat menisci, the ratio $\lambda$ of the slip length to $L$ depends upon a single geometric parameter, namely the ratio $H$ of the invasion depth to $L$. We analyse the singular limit $H\ll 1$ using matched asymptotic expansions, with an outer region on the scale of the period and an inner region on the scale of a single slat. In the outer region, we employ the complex-variable formulation of Stokes flows, obtaining the complex velocity as a divergent series. This formal representation is transformed to an admissible closed-form expression. In the inner region, the problem is equivalent to that of Stokes flow past a finite line segment. Asymptotic matching yields $\lambda = [\ln (2/{\rm \pi} H)-1]/{\rm \pi}$.

A bio-inspired, passively deployable flap attached to an airfoil by a torsional spring of fixed stiffness can provide significant lift improvements at post-stall angles of attack. In this work, we describe a hybrid active–passive variant to this purely passive flow control paradigm, where the stiffness of the hinge is actively varied in time to yield passive fluid–structure interaction of greater aerodynamic benefit than the fixed-stiffness case. This hybrid active–passive flow control strategy could potentially be implemented using variable-stiffness actuators with less expense compared with actively prescribing the flap motion. The hinge stiffness is varied via a reinforcement-learning-trained closed-loop feedback controller. A physics-based penalty and a long–short-term training strategy for enabling fast training of the hybrid controller are introduced. The hybrid controller is shown to provide lift improvements as high as 136 % and 85 % with respect to the flapless airfoil and the best fixed-stiffness case, respectively. These lift improvements are achieved due to large-amplitude flap oscillations as the stiffness varies over four orders of magnitude, whose interplay with the flow is analysed in detail.

In the context of transition analysis, linear input–output analysis determines the worst-case disturbances to a laminar base flow based on a generic right-hand-side volumetric/boundary forcing term. The worst-case forcing is not physically realizable, and, to our knowledge, a generic framework for posing physically realizable worst-case disturbance problems is lacking. In natural receptivity analysis, disturbances are forced by matching (typically local) solutions within the boundary layer to outer solutions consisting of free-stream vortical, entropic and acoustic disturbances. We pose a scattering formalism to restrict the input forcing to a set of realizable disturbances associated with plane-wave solutions of the outer problem. The formulation is validated by comparing with direct numerical simulations of a Mach 4.5 flat-plate boundary layer. We show that the method provides insight into transition mechanisms by identifying those linear combinations of plane-wave disturbances that maximize energy amplification over a range of frequencies. We also discuss how the framework can be extended to accommodate scattering from shocks and in shock layers for supersonic flow.

‘Freeze-out’ of amplitude growth, i.e. the amplitude growth stagnation of a shocked helium–air interface, is realized through a reflected shock, which produces baroclinic vorticity of the opposite sign to that deposited by the first shock. Theoretically, a model is constructed to calculate the relations among the initial parameters for achieving freeze-out. In particular, if the amplitude growth is within the linear regime at the arrival of the reflected shock, the time interval between the impacts of two shock waves is linearly related to the initial perturbation wavelength, and is independent of the initial perturbation amplitude. Experimentally, an air–SF$_6$ (or air–argon) plane interface is adopted to produce a weak reflected shock. Seven experimental runs with specific initial conditions are examined. For all cases, freeze-out is achieved after the reflected shock impact under the designed conditions.

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.