We present a theoretical approach that derives the wavenumber $k^{-1}$ spectral scaling in turbulent velocity spectra using random field theory without assuming specific eddy correlation forms or Kolmogorov’s inertial-range scaling. We argue for the mechanism by Nikora (1999 Phys. Rev. Lett. 83 (4), 734), modelling turbulence as a superposition of eddy clusters with eddy numbers inversely proportional to their characteristic length scale. Statistical mixing of integral scales within these clusters naturally yields the $k^{-1}$ scaling as an intermediate asymptotic regime. Building on the spectrum modelling introduced in Jetti et al. (2025b Z. Angew. Math. Physik. 74 (3), 123), we develop and apply an integral formulation of the general velocity spectrum that reproduces the $k^{-1}$ regime observed in field spectra, thereby bridging theoretical derivation and empirical observations. The model is validated using wind data at a coastal site, and tidal data in a riverine environment where the –1 scaling persists beyond the surface layer logarithmic region. The results confirm the robustness of the model at various flow conditions, offering new insights into the spectral energy distribution in geophysical and engineering flows.

The non-uniform evaporation rate at the liquid–gas interface of binary droplets induces solutal Marangoni flows. In glycerol–water mixtures (positive Marangoni number, where the more volatile fluid has higher surface tension), these flows stabilise into steady patterns. Conversely, in water–ethanol mixtures (negative Marangoni number, where the less volatile fluid has higher surface tension), Marangoni instabilities emerge, producing seemingly chaotic flows. This behaviour arises from the opposing signs of the Marangoni number. Perturbations locally reducing surface tension at the interface drive Marangoni flows away from the perturbed region. Continuity of the fluid enforces a return flow, drawing fluid from the bulk towards the interface. In mixtures with a negative Marangoni number, preferential evaporation of the lower-surface-tension component leads to a higher concentration of the higher-surface-tension component at the interface as compared with the bulk. The return flow therefore creates a positive feedback loop, further reducing surface tension in the perturbed region and enhancing the instability. This study investigates bistable quasi-stationary solutions in evaporating binary droplets with negative Marangoni numbers (e.g. water–ethanol) and examines symmetry breaking across a range of Marangoni numbers and contact angles. Bistable domains exhibit hysteresis. Remarkably, flat droplets (small contact angles) show instabilities at much lower critical Marangoni numbers than droplets with larger contact angles. Our numerical simulations reveal that interactions between droplet height profiles and non-uniform evaporation rates trigger azimuthal Marangoni instabilities in flat droplets. This geometrically confined instability can even destabilise mixtures with positive Marangoni numbers, particularly for concave liquid–gas interfaces, as in wells. Finally, through a Lyapunov exponent analysis, we confirm the chaotic nature of flows in droplets with a negative Marangoni number. We emphasise that the numerical models are intentionally simplified to isolate and clarify the underlying mechanisms, rather than to quantitatively predict specific experimental outcomes; in particular, the model becomes increasingly limited in regimes of rapid evaporation.

This paper theoretically introduces a new architecture for pumping leaky-dielectric fluids. For two such fluids layered in a channel, the mechanism utilises Maxwell stresses on fluid interfaces (referred to as menisci) induced by a periodic array of electrode pairs inserted between the two fluids and separated by the menisci. The electrode pairs are asymmetrically spaced and held at different potentials, generating an electric field with variation along the menisci. To induce surface charge accumulation, an electric field (and thus current flow) is also imposed in the direction normal to the menisci, using flat upper and lower electrodes, one in each fluid. The existence of both normal and tangential electric fields gives rise to Maxwell stresses on each meniscus, driving the flow in opposite directions on adjacent menisci. If the two menisci are the same length, then a vortex array is generated that results in no net flow; however, if the spacing is asymmetric, then the longer meniscus dominates, causing a net pumping in one direction. The pumping direction can be controlled by the (four) potentials of the electrodes, and the electrical properties of the two fluids. In the analysis, an asymptotic approximation is made that the interfacial electrode period is small compared to the fluid layer thicknesses, which reduces the analytical difficulty to an inner region close to the menisci. Closed-form solutions are presented for the potentials, velocity field and resulting pumping speed, for which maximum values are estimated, with reference to the electrical power required and feasibility.

Interactions between shock waves and gas bubbles in a liquid can lead to bubble collapse and high-speed liquid jet formation, relevant to biomedical applications such as shock wave lithotripsy and targeted drug delivery. This study reveals a complex interplay between acceleration-induced instabilities that drive jet formation and radial accelerations causing overall bubble collapse under shock wave pressure. Using high-speed synchrotron X-ray phase contrast imaging, the dynamics of micrometre-sized air bubbles interacting with laser-induced underwater shock waves are visualised. These images offer full optical access to phase discontinuities along the X-ray path, including jet formation, its propagation inside the bubble, and penetration through the distal side. Jet formation from laser-induced shock waves is suggested to be an acceleration-driven process. A model predicting jet speed based on the perturbation growth rate of a single-mode Richtmyer–Meshkov instability shows good agreement with experimental data, despite uncertainties in the jet-driving mechanisms. The jet initially follows a linear growth phase, transitioning into a nonlinear regime as it evolves. To capture this transition, a heuristic model bridging the linear and nonlinear growth phases is introduced, also approximating jet shape as a single-mode instability, again matching experimental observations. Upon piercing the distal bubble surface, jets can entrain gas and form a toroidal secondary bubble. Linear scaling laws are identified for the pinch-off time and volume of the ejected bubble relative to the jet’s Weber number, characterising the balance of inertia and surface tension. At low speeds, jets destabilise due to capillary effects, resulting in ligament pinch-off.

Interactions between hyperelastic bio-membranes and fluid play a crucial role in the flight (or swimming) motion of many creatures, such as bats, flying squirrels and lemurs. Bio-membranes are characterised by high stretchability and micro-bending stiffness, leading to unique fluid–solid coupling properties (Mathai et al., 2023, Phys. Rev. Lett., vol. 131, 114003). This study presents a high-fidelity numerical exploration of the hyperelastic characteristics of a pitching foil inspired by bio-membranes in fluid within a low Reynolds number regime. The focus is on the effect of foil compliance on its self-propulsion performance, mimicking natural propulsion mechanisms, with the foil free to move in the horizontal direction. We find that with certain compliance, the foil may experience a velocity crisis, meaning that its propulsive capability is completely lost. This phenomenon is caused by the loss of beat speed when the foil’s passive deformation is out of phase with the pitching motion. By contrast, the two motions can be in phase at proper compliance, leading to an increased beat speed. This will significantly enhance propulsive velocity up to $33\,\%$ compared with the rigid case. The results demonstrate the feasibility of compliance tuning to circumvent the velocity crisis and improve the propulsive speed, which are helpful in the design of micro aerial robots using biomimetic membranes.

Biologically inspired aero/hydrodynamics attracts considerable interest because of promising efficiency and manoeuvring capabilities. Yet, the influence that external perturbations, typical of realistic environments, can have over the flow physics and aerodynamic performance remains a scarcely investigated issue. In this work, we focus on the impact of free stream turbulence (FST) on the aerodynamics of a flapping wing with a prescribed (heaving and pitching) motion at a chord-based Reynolds number of 1000. The problem is tackled by means of direct numerical simulations using an immersed boundary method and a synthetic turbulence generator. The effect of two key parameters, i.e. the turbulence intensity and integral length scale of FST, is described by characterising the phase- and spanwise-averaged flows and aerodynamic coefficients. In particular, we show how FST effectively enhances the dissipation of the vortices generated by the flapping wing once they are sufficiently downstream of the leading edge. The net (i.e. time-averaged) thrust is found to be marginally sensitive to the presence of FST, whereas the characteristic aerodynamic fluctuations appear to scale linearly with the turbulence intensity and sublinearly with the integral length scale. Moreover, we reveal a simple mechanism where FST triggers the leading-edge vortex breakup, which in turns provides the main source of aerodynamic disturbances experienced by the wing. Finally, we show how the frequency spectra of the aerodynamic fluctuations are governed by the characteristic time scales involved in the problem.

We analyse the energy flux in compressible turbulence by generalizing the exact decomposition recently proposed by Johnson (2020 Phys. Rev. Lett. 124, 104501) to study incompressible turbulent flows. This allows us to characterize the effect of dilatational motion on the interscale energy transfer in three-dimensional compressible turbulence. Our analysis reveals that the contribution of dilatational motion to energy transfer is due to three different physical mechanisms: the interaction between dilatation and strain; between dilatation and vorticity; and the self-interaction of dilatational motion across scales. By analysing numerical simulations of freely decaying and forced turbulence, we validate our theoretical derivations and provide a quantitative description of the role of solenoidal and dilatational motions in energy transfer. In particular, we determine the scaling dependence of the dilatational contributions on the turbulent Mach number. Moreover, our findings provide criteria for tuning the parameters in commonly used Smagorinsky and Yoshizawa models for large-eddy simulations of compressible turbulence.

Gamero-Castaño and colleagues have reported that a large number of calculated shapes for electrified cone jets collapse into a nearly universal geometry when scaled with a characteristic length $R_G$ previously introduced by Gañán-Calvo et al. (J. Aerosol Sci., vol. 25, 1994, pp. 1121–1142). The theoretical reasons for that unexpected success were, however, unclear. Recently, Pérez-Lorenzo & Fernández de la Mora (J. Fluid Mech., vol. 931, 2022, A4) have noted that a slightly different length scale $L_j$ is suggested by the asymptotic jet structure inferred by Gañán-Calvo (Phys. Rev. Lett., vol. 79, 1997, pp. 217–220) from energy conservation and the hypothesis that the asymptotic electric field is that given by Taylor’s static model. This article aims to identify which of these two scales best collapses calculated cone-jet structures, and whether there is an alternative superior one. The characteristic lengths are tested against a large set of numerical solutions of a cone-jet model. The effectiveness of each scaling is determined through analyses based on the standard deviation of the numerical solutions. Despite the slight difference between $R_G$ and $L_j$, this analysis clearly identifies $L_j$ as the most accurate scaling for all cone-jet parameters tested. Differentiating between both scales would not have been possible with experimental measurements, but requires the use of high-fidelity numerical solutions. Surprisingly, the success of $L_j$ is not limited to the jet region, but extends to the cone and the neck. These findings provide a slightly superior scaling enjoying a considerably firmer theoretical basis.

This paper explores the role of barodiffusion in the dynamics of gas bubble growth in highly viscous gas-saturated magma subjected to instant decompression. A mathematical model describing the growth of a single isolated bubble is formulated in terms of the modified Rayleigh–Plesset equation coupled with the mass transfer and material balance equations. The model simultaneously takes into account both dynamic and diffusion mechanisms, including the effect of barodiffusion caused by emergence of a large pressure gradient in the liquid, which, in turn, is associated with formation of a diffusion boundary layer around the bubble. An analytical solution of the problem is found, the construction of which is based on the existence of a quasi-stationary state of the bubble growth process. It is shown that barodiffusion manifests itself at the initial and transient stages and under certain conditions can play a paramount role.

Dynamics of a spherical particle and the suspending low-Reynolds-number fluid confined between two concentric spherical walls were studied numerically. We calculated the particle’s hydrodynamic mobilities at various locations in the confined space. It was observed that the mobility is largest near the middle of confined space along the radial direction, and decays as the particle becomes closer to no-slip walls. At a certain confinement level, the maximal mobility occurs near the inner wall. We also calculated the drift velocity of the particle perpendicular to the external force. The magnitude of the drift velocity normalised by the velocity along the external force was found to depend on particle location and the confinement level; it is observed that the maximal drift velocity occurs near the wall. Fluid vortices in the confined space induced by particle motion were observed and analysed. In addition, we studied particle trajectories in the flow when the walls rotate at constant angular velocities. The externally applied force, rotation-induced flow and centrifugal/centripetal force, and particle–wall interaction lead to various modes of particle motion. This work lays the foundation to understand and manipulate particulate transport in microfluidic applications such as intracellular transport and encapsulation technologies.

Large-scale spanwise motions in shock wave–turbulent boundary-layer interactions over a $ 25^{\circ }$ compression ramp at Mach 2.95 are investigated using large-eddy simulations. Spectral proper orthogonal decomposition (SPOD) identifies coherent structures characterised by low-frequency features and a large-scale spanwise wavelength of $ O(15\delta _{0})$, where $ \delta _{0}$ is the incoming boundary-layer thickness. The dominant frequency is at least one order of magnitude lower than that of the shock motions. These large-scale spanwise structures are excited near the shock foot and are sustained along the separation shock. Global stability analysis (GSA) is then employed to investigate the potential mechanisms driving these structures. The GSA identifies a stationary three-dimensional (3-D) mode at a wavelength of $ 15\delta _{0}$ with a similar perturbation field, particularly near the separation shock. Good agreement is achieved between the leading SPOD mode and the 3-D GSA mode both qualitatively and quantitatively, which indicates that global instability is primarily responsible for the large-scale spanwise structures surrounding the shock. The reconstructed turbulent separation bubble (TSB) using the 3-D global mode manifests as spanwise undulations, which directly induce the spanwise rippling of the separation shock. Furthermore, the coupled TSB motions in the streamwise and spanwise directions are examined. The TSB oscillates in the streamwise direction while simultaneously exhibiting spanwise undulations. The filtered wall-pressure signals indicate the dominant role of the streamwise motions.

Fingering instabilities readily occur if a less viscous fluid displaces a more viscous fluid in a narrow gap due to the action of destabilising viscous forces. If the fluids are miscible, the instability can be suppressed in the limit of large advection as complicated flow structures are formed across the gap. Using a fluid to displace a monolayer of non-colloidal particles suspended in the same fluid, Luo et al. (2025 J. Fluid Mech. vol. 1011, A48) suppress the formation of the cross-gap structures and identify a new fingering mechanism which instead relies on long-range dipolar disturbance flows generated by the particle confinement.

Active colloidal particles create flow around them due to non-equilibrium processes on their surfaces. In this paper, we infer the activity of such colloidal particles from the flow field created by them via deep learning. We first explain our method for one active particle, inferring the $2s$ mode (or the stresslet) and the $3t$ mode (or the source dipole) from the flow field data, along with the position and orientation of the particle. We then apply the method to a system of many active particles. We find excellent agreements between the predictions and the true values of activity. Our method presents a principled way to predict arbitrary activity from the flow field created by active particles.