Through triglobal resolvent analysis, we reveal the effects of wing tip and sweep angle on laminar separated wakes over swept wings. For the present study, we consider wings with semi-aspect ratios from $1$ to $4$, sweep angles from $0^\circ$ to $45^\circ$ and angles of attack of $20^\circ$ and $30^\circ$ at a chord-based Reynolds number of $400$ and a Mach number of $0.1$. Using direct numerical simulations, we observe that unswept wings develop vortex shedding near the wing root with a quasi-steady tip vortex. For swept wings, vortex shedding is seen near the wing tip for low sweep angles, while the wakes are steady for wings with high sweep angles. To gain further insights into the mechanisms of flow unsteadiness, triglobal resolvent analysis is used to identify the optimal spatial input–output mode pairs and the associated gains over a range of frequencies. The three-dimensional forcing and response modes reveal that harmonic fluctuations are directed towards the root for unswept wings and towards the wing tip for swept wings. The overlapping region of the forcing–response mode pairs uncovers triglobal resolvent wavemakers associated with self-sustained unsteady wakes of swept wings. Furthermore, we show that for low-aspect-ratio wings optimal perturbations develop globally over the entire wingspan. The present study uncovers physical insights on the effects of tip and sweep on the growth of optimal harmonic perturbations and the wake dynamics of separated flows over swept wings.

Electrophoretic motion of a particle carrying a weak but arbitrary non-uniform surface charge density in an Oldroyd-B fluid is analysed here in the thin electrical double layer limit. A semi-analytical generic framework, based on regular perturbation, the Lamb's general solutions and the generalized reciprocal theorem, assuming the viscoelastic effects to remain subdominant, is developed for tracing the particle's trajectory using its instantaneous translational velocity and accounting for the temporal evolution of its surface charge driven by rotation. Our results reveal that in a viscoelastic medium, non-uniformly charged particles may generally follow distinct trajectories depending on their sizes, which is in stark contrast to Newtonian fluids. By considering the multipole moments of the surface charge, we show that the particle may initially rotate until its dipole moment becomes collinear with the imposed electric field, and the nature of the surrounding medium does not alter this fundamental behaviour. However, during the course of rotation, the excess polymeric stresses within the electrical double layer and the bulk may cause the particle to migrate perpendicular to the applied field, by forcing the multipole moments of the surface charge to interact with each other. The final steady-state trajectory of the particle and its possible migration normal to the applied electric field are also largely governed by these interactions and more specifically, presence of non-zero quadrupole moments. The present framework may be helpful towards designing tools for particle separation and sorting, relevant in many biological applications.

We present experimental results of irregular long-crested waves propagating over a submerged trapezoidal bar with the presence of a background current in a wave flume. We investigate the non-equilibrium phenomenon (NEP) induced by significant changes of water depth and mean horizontal flow velocity as wave trains pass over the bar. Using skewness and kurtosis as proxies, we show evidence that an accelerating following current could increase the sea-state non-Gaussianity and enhance both the magnitude and spatial extent of the NEP. We also find that below a ‘saturation relative water depth’ $k_p h_2 \approx 0.5$ ($k_p$ being the peak wavenumber in the shallow area of depth $h_2$), although the NEP manifests, the decrease of the relative water depth does not further enhance the maximum skewness and kurtosis over the bar crest. This work highlights the nonlinear physics according to which a following current could provoke higher freak wave risk in coastal areas where modulation instability plays an insignificant role.

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

Instantaneous features of three-dimensional velocity fields are most directly visualized via streamsurfaces. It is generally unclear, however, which streamsurfaces one should pick for this purpose, given that infinitely many such surfaces pass through each point of the flow domain. Exceptions to this rule are vector fields with a non-degenerate first integral whose level surfaces globally define a continuous, one-parameter family of streamsurfaces. While generic vector fields have no first integrals, their vortical regions may admit local first integrals over a discrete set of streamtubes, as Hamiltonian systems are known to do over Cantor sets of invariant tori. Here we introduce a method to construct such first integrals approximately from velocity data, and show that their level sets indeed frame vortical features of the velocity field in examples in which those features are known from Lagrangian analysis. Moreover, we test our method in numerical datasets, including a flow inside a V-junction and a turbulent channel flow. For the latter, we propound an algorithm to pin down the most salient barriers to momentum transport up to a given scale providing a way out of the occlusion conundrum that typically accompanies other vortex visualization methods.

A direct numerical simulation of an oblique shock wave impinging on a turbulent boundary layer at Mach number 2.28 is carried out at moderate Reynolds number, simulating flow conditions similar to those of the experiment by Dupont et al. (J. Fluid Mech., vol. 559, 2006, pp. 255–277). The low-frequency shock unsteadiness, whose characteristics have been the focus of considerable research efforts, is here investigated via the Morlet wavelet transform. Owing to its compact support in both physical and Fourier spaces, the wavelet transformation makes it possible to track the time evolution of the various scales of the wall-pressure fluctuations. This property also makes it possible to define a local intermittency measure, representing a frequency-dependent flatness factor, to pinpoint the bursts of energy that characterise the shock intermittency scale by scale. As a major result, wavelet decomposition shows that the broadband shock movement is actually the result of a collection of sparse events in time, each characterised by its own temporal scale. This feature is hidden by the classical Fourier analysis, which can only show the time-averaged behaviour. Then, we propose a procedure to process any relevant time series, such as the time history of the wall pressure or that of the separation bubble extent, in which we use a condition based on the local intermittency measure to filter out the turbulent content in the proximity of the shock foot and to isolate only the intermittent component of the signal. In addition, wavelet analysis reveals the intermittent behaviour also of the breathing motion of the recirculation bubble behind the reflected shock, and allows us to detect a direct, partial correspondence between the most significant intermittent events of the separation region and those of the wall pressure at the foot of the shock.

We present a family of exact inviscid three-dimensional Beltrami flows in a horizontally periodic domain lying between two parallel free-slip boundaries. Significantly, these flows are not stress free: the horizontal vorticity varies on each boundary. Using direct numerical simulations (employing horizontal hyperdiffusion only for numerical stability), we find that the largest-scale member of the family is unstable and breaks down into anisotropic turbulence, with relatively large horizontal vorticity at and near each boundary, and associated surface frontal features. We conjecture that all members of the family are similarly unstable. The free-slip boundaries play an important role in the late stages of the instability by constraining the deformation of vortex lines near the boundaries. This study appears to be the first to consider the role of boundary horizontal vorticity in an inviscid context.

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.

This paper gives, in the limit of large Froude number, a closed-form, analytical solution for steady, two-dimensional, inviscid, free-surface attached flow over a submerged planar hydrofoil for arbitrary angles of attack and depths of submergence. The doubly connected flow domain is conformally mapped to a concentric annulus in an auxiliary plane. The complex flow potential and its derivative, the complex velocity, are obtained in the auxiliary plane by considering their form at known special points in the flow, and the required conformal mapping is determined by explicit integration. The four real solution parameters are determined as the simultaneous roots of four real nonlinear algebraic equations arising from the flow normalisation. The explicit form allows accurate evaluation of various flow quantities, including the lift on the foil, and these are related to the large-Froude-number results in recent numerical solutions.

Boundary conditions at a liquid–solid interface are crucial to dynamics of a liquid film coated on a fibre. Here, a theoretical framework based on axisymmetric Stokes equations is developed to explore the influence of liquid–solid slip on the Rayleigh–Plateau instability of a cylindrical film on a fibre. The new model not only shows that the slip-enhanced growth rate of perturbations is overestimated by the classical lubrication model, but also indicates a slip-dependent dominant wavelength, instead of a constant value obtained by the lubrication method, which leads to larger drops formed on a more slippery fibre. The theoretical findings are validated by direct numerical simulations of Navier–Stokes equations via a volume-of-fluid method. Additionally, the slip-dependent dominant wavelengths predicted by our model agree with the experimental results provided by Haefner et al. (Nat. Commun., vol. 6, issue 1, 2015, 7409).

The turbidity current (TC), a ubiquitous fluid–particle coupled phenomenon in the natural environment and engineering, can transport over long distances on an inclined terrain due to the suspension mechanism. A large-eddy simulation and discrete element method coupled model is employed to simulate the particle-laden gravity currents over the inclined slope in order to investigate the auto-suspension mechanism from a Lagrangian perspective. The particle Reynolds number in our TC simulation is $0.01\sim 0.1$ and the slope angle is $1/20 \sim 1/5$. The influences of initial particle concentration and terrain slope on the particle flow regimes, particle movement patterns, fluid–particle interactions, energy budget and auto-suspension index are explored. The results indicate that the auto-suspension particles predominantly appear near the current head and their number increases and then decreases during the current evolution, which is positively correlated with the coherent structures around the head. When the turbidity current propagates downstream, the average particle Reynolds number of the auto-suspension particles remains basically unchanged, and is higher than that of other transported particles. The average particle Reynolds number of the transported particles exhibits a negative correlation with the Reynolds number of the current. Furthermore, the increase in particle concentration will enhance the particle velocity, which allows the turbidity current to advance faster and improves the perpendicular support, thereby increasing the turbidity current auto-suspension capacity. Increasing slope angle will result in a slightly larger front velocity, while the effect of that on the total force is insignificant.

This paper presents a novel theoretical framework, based on the concept of the Schwarz function of a wave, for understanding water waves with vorticity in the absence of gravity and capillarity. The framework leads naturally to a taxonomy of three subcases, herein referred to as cases 1, 2 and 3, into which fall three existing studies of water waves incorporating uniform vorticity and submerged point vortices. This provides a theoretical unification of several seemingly unrelated results in the literature. It also provides a route to finding new exact solutions with this paper focussing on new solutions falling within the case 2 category. Among several presented here are a submerged point vortex pair cotravelling with a solitary deep-water wave, von Kármán point vortex streets cotravelling with a periodic deep-water wave and a point vortex row cotravelling with a wave in water of finite depth. Some other more exotic waveforms are also constructed. All these new solutions generalize those of Crowdy & Roenby (Fluid Dyn. Res., vol. 46, 2014) who found steady waves in deep water cotravelling with a submerged point vortex row for which the free surface shapes turn out to coincide with those of pure capillary waves on deep water found by Crapper (J. Fluid Mech., vol. 2, 1957). The new exact solutions are likely to provide a useful basis for asymptotic or numerical studies when additional effects such as gravity and capillarity are incorporated.

Piezoelectric macro-fibre composite (MFC) actuators are employed onto the flexible leeward surface of an airfoil for active control. Time-resolved aerodynamic forces, membrane deformations and flow fields are synchronously measured at low Reynolds number (Re = 6 × 104). Mean aerodynamics show that the actively controlled airfoil can achieve lift-enhancement and drag-reduction simultaneously in the angle of attack range of 10° ≤ α ≤ 14°, where the rigid airfoil encounters stall. The maximum increments of lift and lift-to-drag ratio are 27.1 % and 126 % at the reduced actuation frequency of ${f^ + } = 3.52$. The unsteady coupling features are further analysed at α = 12°, where the maximum lift-enhancement occurs. It is newly discovered that the membrane vibrations and flow fields are locked into half of the actuation frequency when ${f^ + } > 3$. The shift of the dominant vibration mode from bending to inclining is the reason for the novel ‘half-frequency lock-in’ phenomenon. To the fluid–structure interaction, there are three characteristic frequencies for the actively controlled airfoil: $S{t_1} = 0.5{f^ + }$, $S{t_2} = {f^ + }$, and $S{t_3} = 1.5{f^ + }$. Here, St1 and its harmonics (St2, St3) are coupled with the natural frequencies of the leading-edge shear layer, resulting in the generation of multi-scale flow structures and suppression of flow separation. The lift presents comparable dominant frequencies between St1 and St3, which means the instantaneous lift is determined by the flow structures of St1 and St3. The local membrane bulge and dent affect the instantaneous swirl strength of flow structures near the maximum vibration amplitude location, which is the main reason for the variation of instantaneous lift.

Fundamental properties of the multicomponent diffuse-interface model (DIM), such as the maximum entropy principle and conservation laws, are used to explore the basic interfacial dynamics and phase transitions in fluids. Flat interfaces with monotonically changing densities of the components are proved to be stable. A liquid layer in contact with oversaturated but stable vapour is shown to either fully evaporate or eternally expand (depending on the initial perturbation), whereas a liquid in contact with saturated vapour always evaporates. If vapour is bounded by a solid wall with a sufficiently large contact angle, spontaneous condensation occurs in the vapour. The external parameters of the multicomponent DIM – e.g. the Korteweg matrix describing the long-range intermolecular forces – are determined for the water/air combination. The Soret and Dufour effects are shown to be negligible in this case, and the interfacial flow, close to isothermal.

We calculate the net energy per unit time exchanged between two sets of modes in a generic system governed by a three-wave kinetic equation. Our calculation is based on the property of detailed energy conservation of the triadic resonant interactions. In a first application to isotropic systems, we re-derive the previously used formula for the energy flux as a particular case for adjacent sets. We then exploit the new formalism to quantify the level of locality of the energy transfers in the example of surface capillary waves. A second application to anisotropic wave systems expands the currently available set of tools to investigate magnitude and direction of the energy fluxes in these systems. We illustrate the use of the formalism by characterizing the energy pathways in the oceanic internal wavefield. Our proposed approach, unlike traditional approaches, is not limited to stationarity, scale invariance and strict locality. In addition, we define a number $w$ that quantifies the scale separation necessary for two sets of modes to having negligible mutual energy exchange, with potential consequences in the interpretation of wave turbulence experiments. The methodology presented here provides a general, simple and systematic approach to energy fluxes in wave turbulence.

The unsteady wake dynamics of two finite wall-mounted cylinders of unequal height and arranged in tandem are investigated using time-resolved particle image velocimetry (TR-PIV). The cylinders were fully submerged in a turbulent boundary layer with a Reynolds number based on the cylinder diameter (d) of 5540 and boundary layer thickness of $\delta /d = 8.7$. The centre-to-centre spacing between the cylinders was fixed at 4d. The degree of sheltering was studied by varying the height of the upstream cylinder (UC) $(h/d \in [0.7\unicode{x2013} 7.0])$ while keeping the height of the downstream cylinder (DC) constant at $H/d\; = \; 7.0$. The resulting height ratios were $h/H = 0.10,0.25,0.50,0.75$ and $1.00$ denoted as HR10, HR25, HR50, HR75 and HR100, respectively. The wake dynamics of the DC were also compared to those of an isolated single cylinder (SC) with similar Reynolds number, aspect ratio and submergence ratio. TR-PIV measurements were performed in the symmetry plane for all test cases and five spanwise planes along the height of HR75. The results showed that as the height ratio increases, the downwash from the free end of the UC impinges directly on the frontal surface of the DC and induces a strong upwash on the opposite rear side of the DC. The induced upwash impedes the downwash from the free end of the DC much earlier than observed behind the SC, resulting in a reduced reverse flow area and high velocity deficit in the sheltered portion of the DC. Nonetheless, the reverse flow area behind the UC and DC undergoes a quasi-periodic pumping motion with frequencies that are synchronized for HR25 to HR100. Spectral analysis and proper orthogonal decomposition also reveal that the attachment of the shear layers of the UC on the DC also promotes a lock-in anti-symmetric vortex shedding behaviour.