We consider the levitation of cuboidal blocks by means of the viscous stresses that are generated when the block adheres to a vertically moving wall that is coated with oil. We first describe an experimental procedure that reveals the parameter regimes in which long-time levitation can occur. Then a simple model for the relevant lubrication flows is used to explain the theoretical basis for these observations.

The response of asymmetric and planar shock interactions to a continuous excitation of the lower incident shock is investigated numerically. Incident shock waves and centred expansion fans are generated by two wedges asymmetrically deflecting the inviscid free stream flow at Mach 3. The excitations mechanisms considered are (i) pitching of the lower wedge traversing the steady-state dual-solution domain (DSD) of regular interaction (RI) and Mach interaction (MI), (ii) a periodic (sinusoidal) oscillation of the lower wedge deflection with a mean value both within and outside of the steady-state DSD and (iii) a periodic (sinusoidal) streamwise oscillation of the lower wedge location with fixed wedge deflection. A detailed analysis of characteristic unsteady flow features, including the Mach stem growth, pressure evolution across the shock system and corresponding flow deflections and entropy rise, is presented with a focus on the bi-directional RI$\rightleftarrows$MI transition process. For fast pitching conditions, the MI pattern is maintained far inside the steady-state RI domain. The observed $\text{MI}\rightarrow \text{RI}$ transition limit as the rotational velocity decreases does not fully match steady-state theory, however. This is attributed to geometry-related effects. In the opposite case, $\text{RI}\rightarrow \text{MI}$ transition, good agreement with steady-state theoretical predictions is obtained for slow rotations, and a shock polar analysis applied in the (moving) frame of reference of the shock interaction location improves the agreement with fast pitching numerical data significantly. Furthermore, the MI pattern is found to be more robust against periodic perturbations than the corresponding RI configuration for mean flow conditions inside the steady-state DSD, which appears to be a consequence of the dynamics of the Mach stem during a period of excitation. This is not the case for mean flow conditions outside the steady-state DSD in the RI domain for which a periodic $\text{RI}\rightarrow \text{MI}\rightarrow \text{RI}$ alternation occurs instead.

This paper studies the propagation of free, long waves on a potential vorticity front in the presence of a vertical coast, using a $1{\textstyle \frac{1}{2}}$-layer, quasi-geostrophic model with piecewise-constant potential vorticity. The coastal boundary induces flow through image vorticity and a Kelvin wave, either of which can reinforce or oppose the Rossby wave dynamics at the front. The behaviour of the front depends strongly on the relative strengths of these three mechanisms, which are explicit in our model. The richest behaviour, which includes kink solitons (under-compressive shocks) and compound-wave structures, occurs in the regime where vortical effects are dominant. The evolution of the front is described by a fully nonlinear finite-amplitude equation including first-order dispersive effects, which is related to the modified Korteweg–de Vries equation. The different behaviours are classified using the canonical example of the Riemann problem, which we analyse using El’s technique of ‘dispersive shock-fitting’. Contour-dynamic simulations show that the dispersive long-wave theory captures the behaviour of the full quasi-geostrophic system to a high degree of accuracy.

We formulate a model for the dynamic growth of a membrane developing in a flow as the result of a precipitation reaction, a situation inspired by recent microfluidic experiments. The precipitating solid introduces additional forces on the fluid and eventually forms a membrane that is fixed in the flow due to adhesion with a substrate. A key challenge is that, in general, the location of the immobile membrane is unknown a priori. To model this situation, we use a multiphase framework with fluid and membrane phases; the aqueous chemicals exist as scalar fields that react within the fluid to induce phase change. To verify that the model exhibits desired fluid–structure behaviours, we make simplifying assumptions to obtain a reduced form of the equations that is amenable to exact solution. This analysis demonstrates no-slip behaviour on the developing membrane without requiring fluid–membrane interface boundary conditions. The model has applications towards precipitate reactions where the precipitate greatly affects the surrounding flow, a situation appearing in many laboratory and geophysical contexts including the hydrothermal vent theory for the origin of life. More generally, this model can be used to address fluid–structure interaction problems that feature the dynamic generation of structures.

The flow behind streamwise arrays of roughness elements was examined with a hot-wire probe. The roughness elements had heights of approximately 20 % and 40 % of the boundary layer thickness, and different spacings and orientations of these roughness elements were tested. The circular roughness elements were spaced two diameters apart or four diameters apart from centre to centre. Transition moved upstream only when the roughness elements were spaced four diameters apart. The rectangular roughness elements were oriented so that they were at a $45^{\circ }$ angle relative to the leading edge of the plate. Tandem rectangular elements had either the same orientation or opposing orientations. Mean mass-flux and total-temperature profiles of the flow field downstream of the roughness elements were examined for mean-flow distortion. Mass-flux fluctuation profiles showed that a 45 kHz odd-mode disturbance was present downstream of the shorter circular roughness elements. The dominant instability downstream of the taller circular roughness elements was a 65–85 kHz even-mode disturbance. Mass-flux fluctuation profiles showed that the dominant mode downstream of the tandem rectangular roughness elements with the same orientation was similar to that of a single roughness element and centred at a frequency of approximately 55 kHz. The 55 kHz instability appeared to correspond to increased spanwise shear, and thus was determined to be an odd-like mode. The dominant instability downstream of the tandem roughness elements with opposing orientations was centred at a frequency of 65 kHz and did not transition in the measurement region.

Buoyancy-driven exchange flows occur in a variety of natural and industrial situations, including nuclear and hydraulic engineering, oceanography and building ventilation. Balanced exchange flows, whereby there is simultaneously an equal volume flux transferred vertically upwards and downwards through a horizontal opening, have previously been described theoretically. However, until now there has been no theoretical description of unbalanced exchange flows, whereby the volume flux in one direction through an opening exceeds that in the other. The model developed herein examines the growth of perturbations on the density interface at an opening made in a horizontal plane that connects buoyant fluid below with denser fluid above. By considering the interface as it is advected away from the plane of the opening by a bulk flow imposed in the vertical, we quantify the exchange for the unbalanced case. The model successfully predicts the Froude number criterion, which corresponds directly to the minimum dimensionless flow rate of the imposed flow, for the onset of unbalanced exchange across circular openings found experimentally. Additionally, comparisons made between the exchanges predicted and measured show excellent agreement across the entire range of possible flows, from unidirectional flow, through unbalanced exchange to balanced exchange. Consideration is given to applications of the model to ocean outfall design and to the prediction of building ventilation flows. For natural ventilation, the theoretical model we derive for unbalanced exchange bridges the gap in the prediction of air flow rates between displacement flows, where the flow is unidirectional, and balanced exchange flows.

In contrast to common continuous wings, microscopic flying insects such as fairyfly and thrips possess extraordinary wings that comprise several bristles. In the low-Reynolds-number environment inhabited by these insects, a bristled wing benefits from the formation of virtual fluid barriers that prevent flow leakage in the gaps of bristles. In this study, the optimal configuration regarding the aerodynamic performance of simple bristled-wing models is investigated numerically in terms of the Reynolds number and the distance between bristles, which are known to be important parameters for determining the effectiveness of a virtual fluid barrier. Inspired by the inherent characteristics of low-Reynolds-number flow, we provide new insights into the aerodynamics of a bristled wing by using a scaling parameter that combines these two independent parameters: effective gap width. In addition, the force ratio is scaled with the Reynolds number to better characterize the generation of drag and lift by a bristled wing. Despite the Reynolds number and the gap width varying by several orders of magnitude, the scaled drag force converges asymptotically beyond a specific effective gap width, and the lift force ratio has a peak within a very narrow range of the effective gap width. The trend of the lift with the effective gap width is correlated with changes in the asymmetric distributions of pressure and vorticity on the surface of the bristles. In addition, the drag and lift predicted theoretically, by invoking the linear superposition of creeping flows, agree well with numerical results, in particular for large effective gap width.

We study a linear inviscid model of a passively flexible swimmer with distributed flexibility, calculating its propulsive performance and optimal distributions of flexibility. The frequencies of actuation and mean stiffness ratios we consider span a large range, while the mass ratio is fixed to a low value representative of swimmers. We present results showing how the trailing edge deflection, thrust coefficient, power coefficient and efficiency vary with frequency, mean stiffness and stiffness distribution. Swimmers with distributed flexibility have the same qualitative features as those with uniform flexibility. Significant gains in thrust can be made, however, by tuning the stiffness such that a resonant response is triggered, or by concentrating stiffness towards the leading edge if resonance cannot be triggered. To minimize power, the opposite is true. Meaningful gains in efficiency can be made at low frequencies by concentrating stiffness away from the leading edge, since doing so induces efficient travelling wave kinematics. We also speculate on the effects of a finite Reynolds number in the form of streamwise drag. The drag adds an offset to the net thrust produced by the swimmer, causing efficiency-maximizing distributions of flexibility to tend towards thrust-maximizing ones, representative of what is found in nature.

The near-field energetics and directional properties of internal gravity waves (IGWs) radiated from the turbulent wake of a sphere towed through a linearly stratified fluid are investigated using a series numerical experiments. Simulations have been performed for an initial Reynolds number $Re\equiv UD/\unicode[STIX]{x1D708}\in \{5\times 10^{3},10^{5},4\times 10^{5}\}$ and internal Froude number $Fr\equiv 2U/ND\in \{4,16,64\}$, defined using body-based scales – $D$, the sphere diameter; $U$, the tow speed; and $N$, the Brunt–Väisälä frequency. Snapshots of temporally evolving wake flow fields are sampled over the full wake evolution. The energy extracted from the wake through internal wave radiation is quantified by computing the total wave power emitted through a wake-following elliptic cylinder. The total time-integrated wave energy radiated is found to increase with $Re$ and decrease with $Fr$. The peak radiated wave power occurs at early times, near to the onset of buoyancy control, and is found to be approximately unchanged in magnitude as $Re$ is increased. For the two higher $Re$ considered, at late times, IGWs continue to be emitted – accounting for a distinct increase in total radiated wave energy. The most powerful IGWs are radiated out of the wake at a wide range of angles for $Nt<10$, at $20^{\circ }{-}70^{\circ }$ to the horizontal for $10\leqslant Nt\leqslant 25$, and nearly parallel to the horizontal late in the non-equilibrium regime of wake evolution. Internal wave radiation is found to be an important sink for wake kinetic energy after $Nt=10$, suggesting wave radiation cannot be neglected when modelling stratified turbulent wakes in geophysical and ocean engineering applications.

A planar jet issuing from a fully developed two-dimensional turbulent channel flow is studied, with a focus on the transverse flapping of the jet core. The streamwise and transverse velocities were measured with hot-wire anemometry using an X-type probe. The mean velocity field and the velocity covariances were first characterised to assess the undisturbed flow field. Periodic excitations were introduced from a slot mounted at the channel exit and the coherent fluctuating part of the signal was obtained by using a phase-locked averaging technique, where the periodic initial forcing was used as trigger. This enabled the eduction of the coherent structure associated with the introduced perturbation. Its amplitude was found to be directly proportional to the intensity of the initial forcing and, within a certain range of the initial forcing amplitude, the growth curves were identical as well as the spatial distribution of the extracted fluctuations. Parallel and non-parallel linear stability theory captures qualitatively and quantitatively the features of the educed coherent structure. The existence of the linear mode in the turbulent jet implies that the large-scale perturbations observed in natural (unforced) jets can be regarded as an incoherent set of linear modes.

When the cross-section is reduced, a vortex displays spiralling and elasticity, bursting when the external velocity drops. How are these properties affected near a free surface? To answer, a visualization experiment in water is considered where an oscillating obstruction is pitched orthogonally to a rolling oscillation expending minimum energy. We show that short aerated vortices in adverse pressure gradient and shear remain stable during stretching, bursting into bubbles when relaxed after maximum stretching. A vertical aerated double helix (DH) root vortex of contra-rotating vortex tubes is produced near a Rossby number of 0.20. The vortex approaches an inclination of $45^{\circ }$ to the vertical, where stretching intensifies the vorticity to the maximum extent. Vortex inclinations up to $45^{\circ }$ are stable and unstable thereafter. Vortex bursting commences only when the inclination crosses $45^{\circ }$ by the slightest amount. When relaxed, oscillations are produced, breaking the vortex into arrays of bubbles, sometimes precisely at $45^{\circ }$ inclination. The bubble diameter, modelled by equating the effects of the centrifugal force (CF) on the vortex core pressure, to surface tension scales with the inverse of the square of the rotational velocity. When a high CF is withdrawn, the bursting DH aerated vortex cone is crushed due to surface tension. A root vortex and a coiled-up trailing edge vortex together form a compact Hill’s vortex when the highest CF is relieved, bursting into bubbles spectacularly scattered in the unstable sector. The DH vortex breakup is the connection point of vortices in proximity. At high CF, the DH has stacks of Taylor air tubes at $45^{\circ }$ to the external flow bursting when relaxed. Kelvin waves abound when bursting.

We present a non-local ‘microcontinuum’ constitutive equation describing particle suspensions undergoing heterogeneous flows in the low Stokes and particle Reynolds number regime. Most prior models rely on mesoscale averages of structural dynamics that smear out non-local effects or otherwise place strong restrictions on flow type. We relieve this restriction here by ensemble averaging the exact equations that govern the suspension at the microscale, thereby obtaining explicit structure-property connections. The result is a pointwise, heterogeneous ensemble average of the suspension stress valid for suspended particles of arbitrary shape, size, composition and concentration, as well as arbitrary sources of heterogeneity, including non-uniform shear fields, heterogeneous force fields or spatially varying volume fractions. This non-local model accounts for spatial and temporal variations in the flow structure over arbitrary length and time scales. We express the microcontinuum constitutive equation as a superposition of gradient operators that automatically accounts for flow heterogeneity over any length scale larger than the particle size. Batchelor’s result for a homogeneous suspension is recovered in the limit of zero gradients; non-zero gradient terms provide non-local corrections to the average stress and account for statistical heterogeneity in the suspension. We utilize energy methods to compute the influence of these gradient operators on the pointwise-averaged stress tensor, revealing a deep connection to the microhydrodynamics formalism. We apply this general framework to a dilute suspension of spherical particles and show that the resultant constitutive equations are consistent with experimental observations.

We present a detailed analysis of the electroviscous effect in the squeeze out of thin electrolyte films confined between two charged surfaces. The two charged surfaces consist of a curved surface and a flat surface, which closely simulate the tip–substrate configuration in the force measurement of electrolyte solutions with dynamic atomic force microscopy. In the lubrication limit, we find the analytical solution of the electroviscous-effect-modified squeezing flow field in the thin electrolyte film confined between the tip and substrate by solving the Nernst–Planck–Poisson/Navier–Stokes equation under the justified condition of pseudo-steadiness. We also derive the solution of the tip–substrate interaction, which comprises of a conservative electric double layer (EDL) force and an electroviscous-effect-enhanced dissipative hydrodynamic force. The current work focuses on the dissipative hydrodynamic force since the conservative EDL force has been well described by the well-known Derjaguin–Landau–Verwey–Overbeek theory. We introduce a power-law index ($n$) for the tip surface, which enables an unprecedented quantitative characterization of the tip profile effect on the electroviscous effect. We observe a seemingly counter-intuitive effect that, for a given tip–substrate separation, the electroviscous effect is the strongest at one particular value of the zeta ($\unicode[STIX]{x1D701}$) potential and diminishes as the $\unicode[STIX]{x1D701}$ potential departs from this value. We reveal the counterion conductivity of the EDL to be the governing factor for the electroviscous effect under a given $\unicode[STIX]{x1D701}$ potential. In addition to enhancing the dissipative hydrodynamic interaction force, the electroviscous effect modifies the velocity profiles in the thin electrolyte solution films such that they are much sharper.

We investigate the role of resonance in finite-amplitude swimming of a flexible flat plate in a viscous fluid. The role of resonance in performance remains unclear for two reasons: (i) a lack of definition of resonance for the fully coupled fluid–structure interaction system in a viscous flow, and (ii) the presence of nonlinear effects, which makes it difficult to disentangle resonant and non-resonant mechanisms in finite-amplitude swimming. We address point (i) and provide an unambiguous definition for system resonance by computing global linear stability modes of the fully coupled fluid–structure interaction system that account for the viscous fluid, the plate and the coupling between them. We resolve point (ii) by considering high-fidelity nonlinear simulations of systematically increased amplitude. By comparing the results for different amplitudes with one another and with the linear stability modes, we separate linear and/or resonant effects from nonlinear and/or non-resonant effects. Resonant behaviour is observed over a wide range of plate stiffnesses, with peaks in trailing-edge motion and thrust occurring near the resonant frequency defined by the global linear analysis. The peaks broaden and weaken with increasing heave amplitude, consistent with an increased damping effect from the fluid. At the same time, non-resonant mechanisms are present at large heave amplitudes. The input power exhibits qualitatively different dynamics at large heave amplitudes compared to smaller heave amplitudes, where resonance dominates. Moreover, leading-edge separation is present for stiff plates at large heave amplitudes, which can drastically alter the performance characteristics from what one would expect through linear predictions.

Under a steady DC electric field of sufficient strength, a weakly conducting dielectric sphere in a dielectric solvent with higher conductivity can undergo spontaneous spinning (Quincke rotation) through a pitchfork bifurcation. We design an object composed of a dielectric sphere and an elastic filament. By solving an elasto-electro-hydrodynamic (EEH) problem numerically, we uncover an EEH instability exhibiting diverse dynamic responses. Varying the bending stiffness of the filament, the composite object displays three behaviours: a stationary state, undulatory swimming and steady spinning, where the swimming results from a self-oscillatory instability through a Hopf bifurcation. By conducting a linear stability analysis incorporating an elastohydrodynamic model, we theoretically predict the growth rates and critical conditions, which agree well with the numerical counterparts. We also propose a reduced model system consisting of a minimal elastic structure which reproduces the EEH instability. The elasto-viscous response of the composite structure is able to transform the pitchfork bifurcation into a Hopf bifurcation, leading to self-oscillation. Our results imply a new way of harnessing elastic media to engineer self-oscillations, and more generally, to manipulate and diversify the bifurcations and the corresponding instabilities. These ideas will be useful in designing soft, environmentally adaptive machines.

This paper employs an input–output based approach to analyse the convective velocities and transport of fluctuations in turbulent channel flows. The convective velocity for a fluctuating quantity associated with streamwise–spanwise wavelength pairs at each wall-normal location is obtained through the maximization of the power spectral density associated with the linearized Navier–Stokes equations with a turbulent mean profile and delta-correlated Gaussian forcing. We first demonstrate that the mean convective velocities computed in this manner agree well with those reported previously in the literature. We then exploit the analytical framework to probe the underlying mechanisms contributing to the local convective velocity at different wall-normal locations by isolating the contributions of each streamwise–spanwise wavelength pair (flow scale). The resulting analysis suggests that the behaviour of the convective velocity in the near-wall region is influenced by large-scale structures further away from the wall. These structures resemble Townsend’s attached eddies in the cross-plane, yet show incomplete similarity in the streamwise direction. We then investigate the role of each linear term in the momentum equation to isolate the contribution of the pressure, mean shear, and viscous effects to the deviation of the convective velocity from the mean at each flow scale. Our analysis highlights the role of the viscous effects, particularly in regards to large channel spanning structures whose influence extends to the near-wall region. The results of this work suggest the promise of an input–output approach for analysing convective velocity across a range of flow scales using only the mean velocity profile.

In this paper, the unsteady wake of a simplified square-back vehicle, with and without wheels, is investigated using large-scale tomographic particle image velocimetry, at a Reynolds number of $Re_{H}=5.78\times 10^{5}$ (based on the model height). In the no-wheel case, the time-averaged wake features a balanced toroidal shape, with a good level of symmetry in both vertical and lateral directions. However, analysis of the wake dynamics shows this widely accepted result to be a poor model of the wake structure. Application of proper orthogonal decomposition to the unsteady data reveals the existence of the widely reported bi-stable behaviour, consisting of random switches between two lateral symmetry-breaking states. For the first time, the three-dimensional topology of each state is fully characterised and the changes in wake topology during the switches between bi-stable states are also described. Each symmetry-breaking state is shown to feature a characteristic ‘hairpin vortex’ structure that is the result of the merging of two horseshoe vortices, aligned with the vertical edges of the model base. The mutual interactions between these vortices are found to be at the origin of the bi-stable mode. The vertical symmetry is lost when wheels are added to the model, resulting in the formation of an upwash-dominated wake. The bi-stable behaviour is removed but considerable mobility in the near wake remains, in the form of a swinging motion of the rear recirculation.