In jet engines and gas turbines, the annular shape of the combustion chamber allows the appearance of self-oscillating azimuthal thermoacoustic modes. We report experimental evidence of a new type of modal dynamics characterised by periodic switching of the spinning direction and develop a theoretical model that fully reproduces this phenomenon and explains the underlying mechanisms. It is shown that tiny asymmetries of the geometry, the mean temperature field, the thermoacoustic response of the flames or the acoustic impedance of the walls, present in any real systems, can induce these heteroclinic orbits. The model also explains experimental observations showing a statistically dominant spinning direction despite the absence of swirling flow, or pairs of preferred nodal line directions.

Explicit formulas are given for the hydrodynamic slip lengths associated with longitudinal shear flow over a superhydrophobic grating where the menisci have partially invaded the cavities and are only weakly curved. For flat menisci that have depinned from the top of the grating and have displaced downwards into the cavities, the axial velocity is determined analytically and the slip length extracted from it. This solution is then combined with an integral identity to determine the first-order correction to the slip length when the displaced menisci bow weakly into the cavity. It is argued that the new formulas provide useful upper bounds for quantifying slip in microchannel flows involving partially filled cavities. The new solutions are natural extensions of prior results due to Philip (Z. Angew. Math. Phys., vol. 23, 1972, pp. 353–372) for shear flow over mixed no-slip/no-shear surfaces and due to Bechert & Bartenwerfer (J. Fluid Mech., vol. 206, 1989, pp. 105–129) for shear flow over blade-shaped riblets.

An analysis is presented for the axisymmetric lubrication resistance between permeable spherical particles. Darcy's law is used to describe the flow in the permeable medium and a slip boundary condition is applied at the interface. The pressure in the near-contact region is governed by a non-local integral equation. The asymptotic limit $K=k/a^{2} \ll 1$ is considered, where $k$ is the arithmetic mean permeability, and $a^{-1}=a^{-1}_{1}+a^{-1}_{2}$ is the reduced radius, and $a_1$ and $a_2$ are the particle radii. The formulation allows for particles with distinct particle radii, permeabilities and slip coefficients, including permeable and impermeable particles and spherical drops. Non-zero particle permeability qualitatively affects the axisymmetric near-contact motion, removing the classical lubrication singularity for impermeable particles, resulting in finite contact times under the action of a constant force. The lubrication resistance becomes independent of gap and attains a maximum value at contact $F=6{\rm \pi} \mu a W K^{-2/5}\tilde {f}_c$, where $\mu$ is the fluid viscosity, $W$ is the relative velocity and $\tilde {f}_c$ depends on slip coefficients and weakly on permeabilities; for two permeable particles with no-slip boundary conditions, $\tilde {f}_c=0.7507$; for a permeable particle in near contact with a spherical drop, $\tilde {f}_c$ is reduced by a factor of $2^{-6/5}$.

The structure of the internal boundary layer above long wind farms is investigated experimentally. The transfer of kinetic energy from the region above the farm is dominated by the turbulent flux of momentum together with the displacement of kinetic energy operated by the mean vertical velocity: these two have comparable magnitude along the farm opposite to the infinite-farm case. The integration of the energy equation in the vertical highlighted the key role of the energy flux, and how that is balanced by the growth of the internal boundary layer in terms of energy thickness with a small role of the dissipation. The mean velocity profiles seem to follow a universal structure in terms of velocity deficit, while the Reynolds stress does not follow the same scaling structure. Finally, a spectral analysis along the farm identified the leading dynamics determining the turbulent activity: while behind the first row the signature of the tip vortices is dominant, already after the second row their coherency is lost and a single broadband peak, associated with wake meandering, is present until the end of the farm. The streamwise velocity peak is associated with a nearly constant Strouhal number weakly dependent on the farm layout and free stream turbulence condition. A reasonable agreement of the velocity spectra is observed when the latter are normalised by the velocity variance and integral time scale: nevertheless the spectra show clear anisotropy at the large scales and even the small scales remain anisotropic in the inertial subrange.

When a less viscous miscible fluid displaces a more-viscous one under a pressure-driven channel flow, unstable Kelvin–Helmholtz (K–H)-type billows are formed at the miscible interface. In this paper, we investigate whether such instability can be induced by a simple ($\textbf {A}+ \textbf {B} \rightarrow \textbf {C}$)-type chemical reaction. Here a miscible solution of one reactant $\textbf {A}$ is displacing another isoviscous reactant $\textbf {B}$ and producing a more-viscous product $\textbf {C}$ at the reactive front. It is found that because of a local increase in viscosity gradient due to the formation of more-viscous product $\textbf {C}$, K–H-type billows are formed at the $\textbf {A}$–$\textbf {C}$ interface. The changes in dynamical properties of such billows are examined by varying the governing parameters such as the mobility ratio $R_{c}$, Damköhler number $Da$, Péclet number $Pe$ and Reynolds number $Re$. Interestingly, we have found that even at high reaction rates (sufficiently large $Da$) for $R_{c}=1$, the interface remains stable and for larger values of $R_{c} (=3, 5)$ the K–H billows are observed. It is also noticed that a laminar horseshoe-type vortex develops near the wall at the channel inlet where the less-viscous reactant pushes the more-viscous product. We have computed numerically the onset time ($t_{on}$) of instability to understand the early-stage developments of the K–H billows. For different values of $Da$, we have shown the unstable and stable time zones in the ($t_{on}$–$R_{c}$) space. The bipartite ($t_{on}$–$R_{c}$) space also depicts the critical ($Da$-, $Pe$- and $Re$-dependent) $R_{c}$ value for which instability can be triggered in a finite desirable time. The delay in the onset of instability is observed with increasing $Pe$. Further it is shown that $t_{on}$ can be linearly scaled with $Pe$ to have a modified onset time ($t^{*}_{on}$), which establishes a proportionate dynamics with respect to $Pe$ in the early stages of the instability. Moreover, a reverse dependency of onset on lower $R_{c}$ values for higher Reynolds numbers is observed.

The breakup process of a charged, leaky-dielectric jet subjected to an axial perturbation is computationally analysed from the perspectives of linear and nonlinear dynamics using the arbitrary Lagrangian–Eulerian technique. The linear dynamics of the leaky-dielectric jet is quantitatively predicted by the dispersion relation from the linear stability analysis. Regarding the nonlinear dynamics, it is found that the charge relaxation is responsible for the radial compression of satellite droplets, which is validated by experiments. Two types of charge relaxations, namely, ohmic conduction and surface charge convection, define the pinching process into three breakup modes, i.e. ligament pinching, end pinching and transition pinching. In the ligament-pinching mode, the ohmic conduction dominates the jet breakup since the charge relaxes to the jet ligament instantaneously. In contrast, the surface charge convection takes effect in the end-pinching mode since the surface charge is convected to the jet end via fluid flow. When the ohmic conduction is comparable to the surface charge convection, the breakup occurs simultaneously at the end and the ligament. Finally, the influences of the perturbed wavenumber, the electric field intensity and the viscosity on the breakup mode and the local dynamics at pinch-off are comprehensively discussed.

We use resolvent analysis to develop a physics-based, open-loop, unsteady control strategy to attenuate pressure fluctuations in turbulent flow over a rectangular cavity with a length-to-depth ratio of $6$ at a Mach number of $1.4$ and a Reynolds number based on cavity depth of $10\,000$. Large-eddy simulations (LES) of the baseline uncontrolled flow reveal the dominance of Rossiter modes II and IV that generate high-amplitude unsteadiness via trailing-edge impingement and oblique shock waves that obstruct the free stream. To suppress the oscillations, we introduce three-dimensional unsteady blowing along the cavity leading edge. We leverage resolvent analysis as a linear model with respect to the baseline flow to guide the selections of the optimal spanwise wavenumber and frequency of the unsteady actuation input for a fixed momentum coefficient of 0.02. Instead of choosing the most amplified resolvent forcing modes, we seek a disturbance that yields sustained amplification of the primary response mode-based kinetic energy distribution over the entire cavity length. This necessary but not sufficient guideline for effective mean flow modification is evaluated using LES of the controlled cavity flows. The most effective control case reduces the pressure root mean square level up to $52\,\%$ along cavity walls relative to the baseline and is approximately twice that achievable by comparable steady blowing. Dynamic mode decomposition on the controlled flows confirms that the optimal actuation input indeed suppresses the formation of the large-scale Rossiter modes. It is expected that the present flow control guideline derived from resolvent analysis will also be applicable at higher Reynolds numbers with the aid of physical insights and further validation.

Utilizing the joint singular natures of electric field and hydrodynamic flow around a sharp nanotip, we report new electrohydrodynamic Landau–Squire-type flows under the actions of alternating current (AC) electric fields, markedly different from the classical Landau–Squire flow generated by pump discharge using nanotubes or nanopores. Making use of the locally diverging electric field prevailing near the conical tip, we are able to generate a diversity of AC electrohydrodynamic flows with the signature of a 1/r point-force-like decay at distance r from the tip. Specifically, we find AC electrothermal jet and Faradaic streaming out of the tip at applied frequencies in the MHz and 102 Hz regimes, respectively. Yet at intermediate frequencies of 1–100 kHz, the jet flow can be reversed to an AC electro-osmotic impinging flow. The characteristics of these AC jet flows are very distinct from AC flows over planar electrodes. For the AC electrothermal jet, we observe experimentally that its speed varies with the driving voltage V as V3, in contrast to the common V4 dependence according to the classical theory reported by Ramos et al. (J. Phys. D: Appl. Phys, vol. 31, 1998, pp. 2338–2353). Additionally, the flow speed does not increase with the solution conductivity as commonly thought. These experimental findings can be rationalized by means of local Joule heating and double layer charging mechanisms in such a way that the nanotip actually becomes a local hotspot charged with heated tangential currents. The measured speed of the AC Faradaic streaming is found to vary as V3/2 logV, which can be interpreted by the local Faradaic leakage in balance with tangential conduction. These unusual flow characteristics signify that a conical electrode geometry may fundamentally alter the features of AC electrohydrodynamic flows. Such peculiar electrohydrodynamic flows may also provide new avenues for expediting molecular sensing or sample transport in prevalent electrochemical or microfluidic applications.

The generation of a viscous–inviscid instability through scattering of an acoustic wave by localised and distributed roughness on the upper surface of a NACA 0012 aerofoil is studied with a time-harmonic compressible adjoint linearised Navier–Stokes approach. This extends previous work by the authors dedicated to flat plate geometries. The key advancement lies in the modelling of the inviscid acoustic field external to the aerofoil boundary layer, requiring a numerical solution of the convected Helmholtz equation in a non-uniform inviscid field to determine the unsteady pressure field on the curved aerofoil surface. This externally imposed acoustic pressure field subsequently drives the acoustic boundary layer, which fundamentally determines the amplitudes of acoustic-roughness receptivity. A study of receptivity in the presence of Gaussian-shaped roughness and sinusoidally distributed roughness at Mach number $M_\infty =0.4$ and Strouhal numbers $\mathcal {S} \approx \{46,69,115\}$ shows the effects of various parameters, most notably angle of attack, angle of incidence of the externally imposed plane acoustic wave and geometry of surface roughness; the latter is varied from viewpoint of its placement on the aerofoil surface and its wavelength. The parametric study suggests that non-parallel effects are quite substantial and that considerable differences arise when using parallel flow theory to estimate the optimal width of Gaussian-shaped roughness elements to provoke the greatest response. Furthermore, receptivity amplitudes for distributed roughness are observed to be generally higher for lower angles of attack, i.e. for less adverse pressure gradients. It is also shown that the boundary layer is more receptive to upstream-travelling acoustic waves.

We present experiments and theory relating to transpiration through unrestrained hydrogel beads in contact with a water reservoir below and air above. Experimentally, we find that saturated hydrogel beads shrink until a steady state is reached in which water flows continuously through the beads. The size of the bead in steady state is sensitive to the evaporation rate, which depends on the relative humidity and speed of the surrounding air, and to the pressure head imposed by the fluid reservoir. Specifically, the bead size decreases with increasing pressure head or evaporation rate. Our one-dimensional model proposes that transport in the hydrogel is driven by gradients in osmotic pressure, caused by gradients in polymer concentration in the hydrogel that correspond to gradients in swelling. If the evaporation rate or the pressure head changes, the adjustment of this gradient requires the bead to change shape and size. Smaller beads have larger gradients of osmotic pressure, which drive higher transpiration rates and can draw water against larger pressure heads.

An analytical investigation is performed to study the dynamics of vortex shedding behaviour during two commensurate frequency velocity excitations, with an emphasis on the phenomenon of lock-in. We attempt to theoretically study the dynamical features of lock-in under two-frequency excitations and contrast the behaviour with single-frequency excitation. We employ an existing low-order model to characterise the vortex shedding process behind a step/bluff-body. The continuous-time domain model is transformed to a nonlinear dynamical map that relates time instances of successive vortex shedding. Further, these time instances are converted to phase instances, involving which criteria for a generic p : q phase lock-in is obtained. Four parameters are involved: amplitude and frequency of the two excitation components termed as primary and secondary. Bifurcations occurring are investigated using return maps. The inclusion of secondary excitation leads to the existence of two orders of lock-in within a single lock-in boundary. Furthermore, our results indicate that secondary excitation can be used as a control in order to tailor the 1:1 lock-in region formed by the primary excitation. Finally, analytical expressions are obtained to identify lock-in boundaries and their salient geometrical features. Interesting features such as the occurrence of bistability and change in the order of lock-in are observed, which can be explored further with future experiments.

When a mixture of viscous oil and non-colloidal particles displaces air between two parallel plates, the shear-induced migration of particles leads to the gradual accumulation of particles on the advancing oil–air interface. This particle accumulation results in the fingering of an otherwise stable fluid–fluid interface. While previous works have focused on the resultant instability, one unexplored yet striking feature of the experiments is the self-similarity in the concentration profile of the accumulating particles. In this paper, we rationalise this self-similar behaviour by deriving a depth-averaged particle transport equation based on the suspension balance model, following the theoretical framework of Ramachandran (J. Fluid Mech., vol. 734, 2013, pp. 219–252). The solutions to the particle transport equation are shown to be self-similar with slight deviations, and in excellent agreement with experimental observations. Our results demonstrate that the combination of the shear-induced migration, the advancing fluid–fluid interface and Taylor dispersion yield the self-similar and gradual accumulation of particles.

Single-phase flow inside a Hele-Shaw cell can exhibit inhomogeneous flow fields, for example when actuated by electroosmosis with varying wall mobilities, leading to internal pressure gradients. We derive a two-dimensional dispersion model for a dissolved species in such a non-uniform flow field, utilizing a multiple-scale perturbation approach. The resulting two-dimensional transport equation is an advection–diffusion equation containing an effective dispersion tensor field and additional advection-correction terms. It can be viewed as a generalization of the well-known Taylor–Aris dispersion model. The dispersion model allows for flow fields with both stationary and oscillatory components. For the special case of non-uniform flow induced by both pressure gradients and electroosmosis, we derive expressions for the flow field in the long-wavelength limit. These include arbitrary, time-dependent functions for both the driving field as well as the wall mobilities. We discuss the general characteristics of the model using a sinusoidally varying wall mobility, and derive analytical expressions for the dispersion tensor. Then, in order to validate the model, we compare three-dimensional Lagrangian particle tracing simulations with the dispersion model for several test cases, including stationary and oscillatory shear flow as well as a recirculating flow field. For each test case, a good agreement between the full three-dimensional simulations and the results of the two-dimensional dispersion model is obtained. The presented model has the potential to significantly simplify computations of mass transport in Hele-Shaw flows.

This paper examines the shape of a steady jet with a swirling component, ejected from a circular orifice at an angle to the horizontal. Assuming the Froude number to be large, we derive a set of asymptotic equations for a slender jet. In the inviscid limit, the solutions of the set predict that, if the swirling velocity of the flow exceeds a certain threshold, the jet bends against gravity and rises until the initial supply of the liquid's kinetic energy is used up. This effect is due to the fact that the contributions of the swirl and streamwise velocities to the centrifugal force are of opposite signs, with their sum to be balanced by gravity. As a result, swirl- and streamwise-dominated jets bend in opposite directions. Downward-bending jets also exhibit counter-intuitive behaviour. If the swirling velocity is strong enough (but is still below the above threshold), the streamwise velocity on the jet's axis may decrease with the distance from the orifice, despite the acceleration due to gravity. Eventually, a stagnation point emerges due to this effect, arguably destabilising the jet. Also paradoxically, viscosity-dominated jets can reach higher (if they bend upwards) and farther (in all cases) than their inviscid counterparts, due to the fact that viscosity suppresses formation of stagnation points.

When a deposited layer of granular material fully immersed in a liquid is suddenly inclined above a certain critical angle, it starts to flow down the slope. The initial dynamics of these underwater avalanches strongly depends on the initial volume fraction. If the granular bed is initially loose, i.e. looser than the critical state, the avalanche is triggered almost instantaneously and exhibits a strong acceleration, whereas for an initially dense granular bed, i.e. denser than the critical state, the avalanche's mobility remains low for some time before it starts flowing normally. This behaviour can be explained by a combination of geometrical granular dilatancy and pore pressure feedback on the granular media. In this contribution, a continuum formulation is presented and implemented in a three-dimensional continuum numerical model. The originality of the present model is to incorporate dilatancy as an elasto-plastic normal stress or pressure and not as a modification of the friction coefficient. This allows an explanation of the two different behaviours of initially loose and dense underwater avalanches. It also highlights the contribution from each depth-resolved variable in the strongly coupled transition to a flowing avalanche. The model compares favourably with existing experiments for the initiation of underwater granular avalanches. Results reveal the interplay between shear-induced changes of the granular stress and fluid pressure in the dynamics of avalanches. The characteristic time of the triggering phase is nearly independent of the local rheological parameters, whereas the initial drop in pore pressure and the surface velocity at steady state still strongly depend on them. Finally, the multidimensional capabilities of the model are illustrated for the two-dimensional Hele-Shaw configuration and some of the observed differences between one-dimensional simulations and experiments are clarified.

We present an experimental and numerical investigation of electrokinetic instability (EKI) in microchannel flow with streamwise conductivity gradients, such as those observed during sample stacking in capillary electrophoresis. A plug of a low-conductivity electrolyte solution is initially sandwiched between two high-conductivity zones in a microchannel. This spatial conductivity gradient is subjected to an external electric field applied along the microchannel axis, and for sufficiently strong electric fields an instability sets in. We have explored the physics of this EKI through experiments and numerical simulations, and supplemented the results using scaling analysis. We performed EKI experiments at different electric field values and visualised the flow using a passive fluorescent tracer. The experimental data were analysed using the proper orthogonal decomposition technique to obtain a quantitative measure of the threshold electric field for the onset of instability, along with the corresponding coherent structures. To elucidate the physical mechanism underlying the instability, we performed high-resolution numerical simulations of ion transport coupled with fluid flow driven by the electric body force. Simulations reveal that the non-uniform electroosmotic flow due to axially varying conductivity field causes a recirculating flow within the low-conductivity region, and creates a new configuration wherein the local conductivity gradients are orthogonal to the applied electric field. This configuration leads to EKI above a threshold electric field. The spatial features of the instability predicted by the simulations and the threshold electric field are in good agreement with the experimental observations and provide useful insight into the underlying mechanism of instability.

We study the effect of surfactants on the dynamics of a drop-interface coalescence using full three-dimensional direct numerical simulations. We employ a hybrid interface-tracking/level-set method, which takes into account Marangoni stresses that arise from surface-tension gradients, interfacial and bulk diffusion and sorption kinetic effects. We validate our predictions against the experimental data of Blanchette and Bigioni (Nat. Phys., vol. 2, issue 4, 2006, pp. 254–257) and perform a parametric study that demonstrates the delicate interplay between the flow fields and those associated with the surfactant bulk and interfacial concentrations. The results of this work unravel the crucial role of the Marangoni stresses in the flow physics of coalescence, with particular attention paid to their influence on the neck reopening dynamics in terms of stagnation-point inhibition, and near-neck vorticity generation. We demonstrate that surfactant-laden cases feature a rigidifying effect on the interface compared with the surfactant-free case, a mechanism that underpins the observed surfactant-induced phenomena.

The breaking of internal gravity waves in the abyssal ocean is thought to be responsible for much of the mixing necessary to close oceanic buoyancy budgets. The exact mechanism by which these waves break down into turbulence remains an active area of research and can have significant implications on the mixing efficiency. Recent evidence has suggested that both shear instabilities and convective instabilities play a significant role in the breaking of an internal gravity wave in a high Richardson number mean shear flow. We perform a systematic analysis of the stability of a configuration of an internal gravity wave superimposed on a background shear flow first considered by Howland et al. (J. Fluid Mech., vol. 921, 2021, A24), using direct–adjoint looping to find the perturbation giving maximal energy growth on this evolving flow. We find that three-dimensional, convective mechanisms produce greater energy growth than their two-dimensional counterparts. In particular, we find close agreement with the direct numerical simulations of Howland et al. (J. Fluid Mech., 2021, in press), which demonstrated a clear three-dimensional mechanism causing breakdown to turbulence. The results are shown to hold at realistic Prandtl numbers. At low mean Richardson numbers, two-dimensional, shear-driven mechanisms produce greater energy growth.

The meridional component of the earth's rotation is often neglected in geophysical contexts. This is referred to as the ‘traditional approximation’ and is justified by the typically small vertical velocity and aspect ratio of such problems. Ocean fronts are regions of strong horizontal buoyancy gradient and are associated with strong vertical transport of tracers and nutrients. Given these comparatively large vertical velocities, non-traditional rotation may play a role in governing frontal dynamics. Here the effects of non-traditional rotation on a front in turbulent thermal wind balance are considered using an asymptotic approach. Solutions are presented for a general horizontal buoyancy profile and examined in the simple case of a straight front. Non-traditional effects are found to depend strongly on the direction of the front and may lead to the generation of jets and the modification of the frontal circulation and vertical transport.

We use Navier–Stokes-based linear models for wall-bounded turbulent flows to estimate large-scale fluctuations at different wall-normal locations from their measurements at a single wall-normal location. In these models, we replace the nonlinear term by a combination of a stochastic forcing term and an eddy dissipation term. The stochastic forcing term plays a role in energy production by the large scales, and the eddy dissipation term plays a role in energy dissipation by the small scales. Based on the results in channel flow, we find that the models can estimate large-scale fluctuations with reasonable accuracy only when the stochastic forcing and eddy dissipation terms vary with wall distance and with the length scale of the fluctuations to be estimated. The dependence on the wall distance ensures that energy production and energy dissipation are not concentrated close to the wall but are evenly distributed across the near-wall and logarithmic regions. The dependence on the length scale of the fluctuations ensures that lower wavelength fluctuations are not excessively damped by the eddy dissipation term and hence that the dominant scales shift towards lower wavelengths towards the wall. This highlights that, on the one hand, energy extraction in wall turbulence is predominantly linear and thus physics-based linear models give reasonably accurate results. On the other hand, the absence of linearly unstable modes in wall turbulence means that the nonlinear term still plays an essential role in energy extraction and thus the modelled terms should include the observed wall distance and length scale dependencies of the nonlinear term.