We use the theory of spectral submanifolds (SSMs) to develop a low-dimensional reduced-order model for plane Couette flow restricted to the shift–reflect invariant subspace in the permanently chaotic regime at ${Re}=187.8$ studied by Kreilos & Eckhardt (2012, Chaos: Interdisciplinary J. Nonlinear Sci., vol. 22, 047505). Our three-dimensional model is obtained by restricting the dynamics to the slowest mixed-mode SSM of the edge state. We show that this results in a nonlinear model that accurately reconstructs individual trajectories, representing the entire chaotic attractor and the laminar dynamics simultaneously. In addition, we derive a two-dimensional Poincaré map that enables the rapid computation of the periodic orbits embedded in the chaotic attractor.

We consider the two-layer quasi-geostrophic model with linear bottom friction and, in certain simulations, a planetary vorticity gradient, $\beta$. We derive energy budgets in wavenumber space for eddy available potential energy (EAPE), baroclinic eddy kinetic energy (EKE) and barotropic EKE, a particular decomposition that has previously been overlooked. The conversion between EAPE and baroclinic EKE, $\widehat {T}^{{W}}$, has a strong dependence on both bottom drag strength and planetary $\beta$. At the deformation scale $\widehat {T}^{{W}}$ is always negative, representing the conversion of EAPE to EKE via baroclinic instability. For strong, linear bottom drag, $\widehat {T}^{{W}}$ is positive at large scales due to frictional energisation of the baroclinic mode, providing a large-scale EAPE source. With weak-to-moderate bottom drag and moderate-to-strong planetary $\beta$, $\widehat {T}^{{W}}$ is the dominant source of EAPE at large scales, converting baroclinic EKE that has experienced a baroclinic inverse cascade back into EAPE, and thus closing a novel and exclusively baroclinic energy loop. With planetary $\beta$, zonal jets form and the dominant large-scale processes in the energy cycle of the system, e.g. barotropic dissipation and the peak of positive $\widehat {T}^{{W}}$, occur at the meridional wavenumber corresponding to the jet spacing, with no zonal wavenumber component, i.e., $k_{x}=0$. Importantly, the traditional source of large-scale EAPE, barotropic stirring of the baroclinic mode, is not a part of this $k_{x} = 0$ energy cycle, and thus plays a secondary role. The results suggest that consideration of horizontally two-dimensional processes is requisite to understand the energetics and physics of baroclinic geophysical jets.

This study investigates the influence of free-stream turbulence (FST) and the thrust coefficient ($C_T$) on wind turbine wakes. Wakes generated at $C_T \in \{0.5, 0.7,0.9\}$ are exposed to turbulent inflows with varying FST intensities ($1\,\% \lesssim {\textit{TI}}_{\infty } \lesssim 11\,\%$) and integral length scales ($0.1 \lesssim {\mathcal L}_x/\!D \lesssim 2$, $D$ is the rotor diameter). For high-${\textit{TI}}_{\infty }$ inflows, a flow region in the wake is observed where a mean momentum deficit persists despite the turbulence intensity having already homogenised with that of the free stream, challenging traditional wake definitions. A ‘turning point’ in the mean wake width evolution is identified, beyond which wakes spread at slower rates. Near-field ($x\!/\!D \lesssim 7$) wake growth rate increases with higher ${\textit{TI}}_{\infty }$ and $C_T$, while far-field ($x\!/\!D \gtrsim 15$) wake growth rate decreases with higher ${\textit{TI}}_{\infty }$ – a finding with profound implications for wind turbine wake modelling that also aligns with the entrainment behaviours observed in bluff- and porous-body wakes exposed to FST. Increasing ${\mathcal L}_x$ delays wake recovery onset and reduces the mean wake width, with minimal effect on the spreading rate. Both $C_T$ and FST influence the high- and low-frequency wake dynamics, with varying contributions in the near and far fields. For low-${\textit{TI}}_{\infty }$ and small-${\mathcal L}_x$ inflows, wake meandering is minimal, sensitive to $C_T$ and appears to be triggered by a shear-layer instability. Wake meandering is enhanced for high-${\textit{TI}}_{\infty }$ and large-${\mathcal L}_x$ inflows, with the integral length scale playing a leading role. This emphasises the complex role of FST integral length scale: while increasing ${\mathcal L}_x$ amplifies meandering, it does not necessarily translate to larger mean wake width due to the concurrent suppression of entrainment rate.

We present an analysis of the coherent structures in Langmuir turbulence, a state of the ocean surface boundary layer driven by the interactions between water waves and wind-induced shear, via a resolvent framework. Langmuir turbulence is characterised by multiscale vortical structures, notably counter-rotating roll pairs known as Langmuir circulations. While classic linear stability analyses of the Craik–Leibovich equations have revealed key instability mechanisms underlying Langmuir circulations, the vortical rolls characteristic of Langmuir turbulence, the present work incorporates the turbulent mean state and varying eddy viscosity using data from large-eddy simulations (LES) to investigate the turbulence dynamics of fully developed Langmuir turbulence. Scale-dependent resolvent analyses reveal a new formation mechanism of two-dimensional circulating rolls and three-dimensional turbulent coherent vortices through linear amplification of sustained harmonic forcing. Moreover, the integrated energy spectra predicted by the principal resolvent modes in response to broadband harmonic forcing capture the dominant spanwise length scales that are consistent with the LES data. These results demonstrate the feasibility of resolvent analyses in capturing key features of multiscale turbulence–wave interactions in the statistical stationary state of Langmuir turbulence.

Applying a sufficiently rapid start–stop to the outer cylinder of the Taylor–Couette system, structures approximately aligned with the rotation axis were recorded in the classic work of Coles (1965 J. Fluid Mech. vol. 21, no. 3, pp. 385–425). These short-lived rolls are oriented perpendicular to the classic Taylor-vortex rolls. In this work we report numerical observation of this instability, guided by a more recent experimental observation. The instability is shown to be related to an inflection in the azimuthal velocity profile, a finding consistent with the experimental observations of its emergence during the deceleration phase. Despite the transient nature of start–stop experiments, we show that the instability can be linked to that of the oscillating boundary layer problem of Stokes. There are several reasons why the instability may have remained elusive, both for experimental observation and for the idealised system. We look in more detail at dependence on the radius ratio for the Taylor–Couette system, $\eta=R_i/R_o$, where $R_i$ and $R_o$ are the inner and outer radii. We find that, in the case where the size of the rolls scales with the gap width, for radius ratios any lower than that used by Coles, $\eta=0.874$, the instability is quickly overrun by axisymmetric rolls of Görtler type.

We discuss flow-induced vibrations of an equilateral triangular prism confined to travel on a circular path when placed in the concave or convex orientations with respect to the flow. In each orientation, we consider three different initial angles for the prism. In Case 1, one side of the prism sees the flow first; in Case 2, one sharp edge sees the flow first; and in Case 3, one side of the prism is parallel to the incoming flow. We show that the response of the structure as well as the observed wake depend heavily on both the orientation and the initial angle of the prism. Case 1 exhibits vortex-induced vibration (VIV) in the concave orientation and galloping in the convex orientation. Case 2 does not oscillate in the concave orientation; however, oscillates about a mean deflection after a critical reduced velocity in the convex orientation. Case 3 exhibits small-amplitude oscillations in the concave orientation about a mean deflection, while in the convex orientation, exhibits VIV at low reduced velocities, followed by an asymmetric response with VIV features in a half-cycle and galloping features in the other half, and divergence at higher reduced velocities. These different types of responses are accompanied by a myriad of vortex patterns in the wake, from two single vortices shed in the wake in each cycle of oscillations to two vortex pairs, two sets of co-rotating vortices, and a combination of single vortices and vortex pairs depending on the prism’s orientation and its initial angle.

To investigate the characteristics of a turbulent boundary layer (TBL) over the curved edge of the bow of submarine technology program office (SUBOFF) model, wall-resolved large-eddy simulation is conducted at a Reynolds number of $\mathop {\textit{Re}}\nolimits _L = 1.1 \times {10^6}$ based on the model length and free-stream velocity. Instead of using a trip wire at the bow surface, turbulent inflow is added to the simulation to induce boundary layer transition. The effects of geometric curvature and inflow turbulence intensity (ITI) are examined. With a low ITI level, natural transition takes place at the rear end of the straight section. With higher ITI levels, turbulence emerges immediately and evolves gradually, following a strong favourable-pressure-gradient (FPG) region near the forehead, which is significantly influenced by the large streamwise curvature. Within the FPG region, the root mean square of the wall pressure fluctuation (WPF) decreases rapidly, with the frequency spectra of WPF exhibiting good scalability with outer variables. Moreover, higher turbulence intensity levels lead to larger skin friction, which is related to the development of the TBL. To elucidate the generation mechanism of skin friction, the dynamic decomposition is derived in the curvilinear coordinate system. The mean convection and streamwise pressure gradient make the largest contributions to the local skin friction. Furthermore, an analysis of the energy transfer process based on the Reynolds stress transport equations in the curvilinear coordinate system is presented, highlighting the significant impact of geometric effects, particularly on the production term.

This study presents an analytical advancement in predicting the growth rate of perturbation amplitude in two-dimensional non-standard Richtmyer–Meshkov instability (RMI), driven by the interaction of a first-phase rippled shock wave at moderate Mach number with a heavy–light interface. We extend the irrotational model to encompass non-standard RMI scenarios, establishing a generalised framework validated through numerical simulations. Distinct from previous models, our model is free of empirical coefficients, and demonstrates superior accuracy across diverse perturbation configurations and Mach numbers. The analyses reveal the fundamental disparity of non-standard RMI from classical RMI: the vorticity deposition mechanism in non-standard RMI arises not only from normal pressure gradients at the shock front but crucially from tangential pressure gradients behind the shock wave. The asymptotic circulations are also well predicted by our model. Moreover, the relationship of the amplitudes between sinusoidal shock and perturbed interface is derived based on the model to realise the freeze-out of interface amplitude. The initial fundamental mode’s amplitude growth is frozen well, and the mixing width is greatly suppressed.

A backward swept shape is one of the common features of the wings and fins in animals, which is argued to contribute to leading-edge vortex (LEV) attachment. Early research on delta wings proved that swept edges could enhance the axial flow inside the vortex. However, adopting this explanation to bio-inspired flapping wings and fins yields controversial conclusions, in that whether and how enhanced spanwise flow intensifies the vorticity convection and vortex stretching is still unclear. Here, the flapping wings and fins are simplified into revolving plates with their outboard 50 $\%$ span swept backward in either linear or nonlinear profiles. The local spanwise flow is found to be enhanced by these swept designs and further leads to stronger vorticity convection and vortex stretching, thus contributing to local LEV attachment and postponing bursting. These results further prove that a spanwise gradient of incident velocity is sufficient to trigger a regulation of LEV intensity, and a concomitant gradient of incident angle is not necessary. Moreover, an attached trailing-edge vortex is generated on a swept wing and induces an additional low-pressure region on the dorsal surface. The lift generation of swept wings is inferior to that of the rectangular wing because the extended stable LEV along the span and the additional suction force near the trailing edge are not comparable to the lift loss due to the reduced LEV intensity. Our findings evidence that a swept wing can enhance the spanwise flow and vorticity transport, as well as limit excessive LEV growth.

Surface tension gradients of air–liquid–air films play a key role in governing the dynamics of systems such as bubble caps, foams, bubble coalescence and soap films. Furthermore, for common fluids such as water, the flow due to surface tension gradients, i.e. Marangoni flow, is often inertial, due to the low viscosity and high velocities. In this paper, we consider the localised deposition of insoluble surfactants onto a thin air–liquid–air film, where the resulting flow is inertial. As observed by Chomaz (2001 J. Fluid Mech. 442, 387–409), the resulting governing equations with only inertia and Marangoni stress are similar to the compressible gas equations. Thus, shocks are expected to form. We derive similarity solutions associated with the development of such shocks, where the mathematical structure is closely related to the Burgers equation. It is shown that the nonlinearity of the surface tension isotherm has an effect on the strength of the shock. When regularisation mechanisms are included, the shock front can propagate and late-time similarity solutions are derived. The late-time similarity solution due to regularisation by capillary pressure alone was found by Eshima et al. (2025 Phys. Rev. Lett. 134, 214002). Here, the regularisation mechanism is generalised to include viscous extensional stress.

The present study deals with the electrophoresis of a non-polarizable droplet with irreversibly adsorbed ionic surfactants suspended in monovalent or multivalent electrolyte solutions. The impact of the non-uniform surface charge density, governed by the interfacial surfactant concentration, along with Marangoni, hydrodynamic and Maxwell stresses on droplet electrophoresis is analysed. At a large ionic concentration, the hydrodynamic steric interactions and correlations among finite-sized ions manifest. In this case the viscosity of the medium rises as the local volume fraction of the finite-sized ions is increased. The governing equations, incorporating these short-range effects, are solved numerically based on the regular linear perturbation analysis under a weak applied electric field consideration. We find that the electrophoretic velocity consistently decreases with an increase in the droplet-to-electrolyte viscosity ratio due to the Marangoni stress caused by inhomogeneous surfactant distribution. This monotonic relationship with droplet viscosity is absent for the case of constant surface charge density, where a low-viscosity droplet may exhibit a lower mobility than a high-viscosity droplet. In the presence of ionic surfactant, a continuous variation of mobility with surfactant concentration is found. For a monovalent electrolyte, mobility decreases significantly at an elevated ionic concentration due to the short-range effects described above. Reversal in mobility is observed in multivalent electrolytes due to the correlations among finite-sized ions, attributed to overscreening of surface charge and formation of a coion-rich layer within the electric double layer. In this case a toroidal vortex develops adjacent to the droplet and the reversed mobility enhances as the Marangoni number is increased. This mobility reversal is delayed for low-viscosity droplets.

Swimming and flying animals demonstrate remarkable adaptations to diverse flow conditions in their environments. In this study, we aim to advance the fundamental understanding of the interaction between flexible bodies and heterogeneous flow conditions. We develop a linear inviscid model of an elastically mounted foil that passively pitches in response to a prescribed heaving motion and an incoming flow that consists of a travelling wave disturbance superposed on a uniform flow. In addition to the well-known resonant response, the wavy flow induces an antiresonant response for non-dimensional phase velocities near unity due to the emergence of non-circulatory forces that oppose circulatory forces. We also find that the wavy flow destructively interferes with itself, effectively rendering the foil a low-pass filter. The net result is that the waviness of the flow always improves thrust and efficiency when the wavy flow is of a different frequency than the prescribed heaving motion. Such a simple statement cannot be made when the wavy flow and heaving motion have the same frequency. Depending on the wavenumber and relative phase, the two may work in concert or in opposition, but they do open the possibility of simultaneous propulsion and net energy extraction from the flow, which, according to our model, is impossible in a uniform flow.

Most turbulent boundary-layer flows in engineering and natural sciences are out of equilibrium. While direct numerical simulation and wall-resolved large-eddy simulation can accurately account for turbulence response under such conditions, lower-cost approaches like wall-modelled large-eddy simulation often assume equilibrium and struggle to reproduce non-equilibrium effects. The recent ‘Lagrangian relaxation-towards-equilibrium’ (LaRTE) wall model (Fowler et al. 2022 J. Fluid Mech. vol. 934, 137), formulated for smooth walls, applies equilibrium modelling only to the slow dynamics that are more likely to conform to the assumed flow state. In this work, we extend the LaRTE model to account for wall roughness (LaRTE-RW) and apply the new model to turbulent flow over heterogeneous roughness and in accelerating and decelerating flows over rough surfaces. We compare predictions from the new LaRTE-RW model with those from the standard log-law equilibrium wall model (EQWM) and with experimental data to elucidate the turbulence response mechanisms to non-equilibrium conditions. The extended model transitions seamlessly across smooth-wall and fully rough regimes and improves prediction of the skin-friction coefficient, especially in recovering trends at roughness transitions and in early stages of pressure-gradient-driven flow acceleration or deceleration. Results show that LaRTE-RW introduces response delays that are beneficial when EQWMs react too quickly to disturbances, but it is less effective in flows requiring rapid response, such as boundary layers subjected to accelerating–decelerating–accelerating free stream conditions. These findings emphasize the need for further model refinements that incorporate fast turbulent dynamics not currently captured by LaRTE-RW.

Fluidic levitation of different types of objects is achieved using laboratory experiments and described using simple mathematical models. Air bubbles, liquid tetrabromoethane droplets and solid spherical polytetrafluoroethylene beads were levitated in flowing water inside vertically oriented cylindrical tubes having diameters of 5, 8 and 10 mm. The centre of mass of all levitated objects was observed to undergo horizontal oscillations once a stable levitation point had been established. A simple model that considers the balance of gravitational, buoyancy and drag forces (as well as wall effects) was used to successfully predict the flow rates that are required to obtain stable levitation of objects with a range of different sizes. Horizontal motion was shown to be driven by vortex shedding of the objects in the tubes, and the dependence of the frequency of oscillation on their size was predicted.